Increasing Returns and Inframarginai Economics - Vol. 3 <*"*
Inframarginai Contributions to
Development Economics
edited by
Christis Tombazos Xiaokai Yang
Inframarginal Contributions to
Development Economics
Increasing Returns and Inframarginal Economics Series Editors: James Buchanan, Yew-Kwang Ng, (Xiaokai Yang) Associate Editor: Guang-Zhen Sun
Published Vol. 1 An Inframarginal Approach to Trade Theory Edited by Xiaokai Yang, Wenli Cheng, Heling Shi & Christis G. Tombazos Vol. 2
Readings in the Economics of the Division of Labor: The Classical Tradition Edited by Guang-Zhen Sun
Vol. 3
Inframarginal Contributions to Development Economics Edited by Christis Tombazos & Xiaokai Yang
Increasing Returns and Inframarginal Economics - Vol. 3 f ^ f a j
Inframarginal Contributions to
Development Economics
edited by
Christis Tombazos Xiaokai Yang Monash University, Australia
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INFRAMARGINAL CONTRIBUTIONS TO DEVELOPMENT ECONOMICS Increasing Returns and Inframarginal Economics — Vol. 3 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ToXiaofpi
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Preface
As in the case of any anthology of reprinted contributions, primary credit for this volume goes to the authors of the various chapters. We are also grateful to the original publishers for their permission to reprint articles that appear in this volume. Unlike the remaining contributions, Chapter 1 is written specifically for this volume. It discusses the origins of inframarginal analysis, and provides an introduction to the implementation of the inframarginal methodology in the context of a simple example. The production of the camera-ready copy of this manuscript was a difficult and labourious task that was managed, almost single-handedly, by our in-house copy editor Aiko Yoshibayashi. We thank Aiko for her extraordinary attention to detail, and her hard work. We also gratefully acknowledge the generous financial support of the Department of Economics and the Faculty of Business and Economics of Monash University. Finally, we thank the Research Centre for Increasing Returns and Economic Organisation of Monash University for providing the hardware and office space without which this project would not have been possible.
Christis G. Tombazos Managing Editor
vn
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Contents
Preface
vii
Part 1. Introduction 1. Returning to the Classical Tradition: The Relevance and Application of Inframarginal Analysis to Development Economics Christis G. Tombazos and Stephen Matteo Miller
3
Part 2. The Origins of Inframarginal Applications to the Study of Economic Development 2. The Marginal Cost Controversy Ronald H. Coase 3. Economics and Biology: Specialization and Speciation Hendrik S. Houthakker 4. Externality James M. Buchanan and Wm. Craig Stubblebine
29
47 55
Part 3. Development Strategies, Income Distribution, and Dual Structures 5. Economic Development, International Trade, and Income Distribution Xiaokai Yang and Dingsheng Zhang IX
77
Contents
X
6. Pursuit of Relative Utility and Division of Labor Jianguo Wang andXiaokai Yang
111
Part 4. Urbanization 7. Development, Structural Changes and Urbanization 137 Xiaokai Yang 8. An Equilibrium Model Endogenizing the Emergence of a Dual Structure between the Urban and Rural Sectors 167 Xiaokai Yang and Robert Rice 9. Agglomeration Economies, Division of Labour and the Urban Land-Rent Escalation: A General Equilibrium Analysis of Urbanisation 195 Guang-Zhen Sun andXiaokai Yang Part 5. Entrepreneurship and the Firm 10. Theory of the Firm and Structure of Residual Rights Xiaokai Yang and Yew-Kwang Ng 11. The Theory of Irrelevance of the Size of the Firm Pak-Wai Liu andXiaokai Yang 12. Specialization, Product Development, Evolution of the Institution of the Firm, and Economic Growth Jeff Borland and Xiaokai Yang
231 259
291
Part 6. Endogenous Transaction Costs and Property Rights 13. Endogenous Specialisation and Endogenous Principal-Agent Relationship Xiaokai Yang and Yeong-Nan Yeh 14. A Model Formalizing the Theory of Property Rights Xiaokai Yang and Ian Wills
329
363
Contents
15. Economy of Specialization and Diseconomy of Externalities C. Y. Cyrus Chu and C. Wang
XI
391
Part 7. Investment, Endogenous Growth, and Social Experiments 16. The Division of Labor, Investment and Capital Xiaokai Yang 17. A New Theory of Industrialization Heling Shi and Xiaokai Yang
409 437
Part 8. Infrastructure, Labor Surplus, Insurance, and the Trade-off Between Leisure and Income 18. Population Density and Infrastructure Development 463 C. Y. Cyrus Chu 19. Uncertainty, Insurance, and Division of Labor 479 Monchi Lio 20. Push or Pull? The Relationship between Development, Trade and Primary Resource Endowment 497 Mei Wen and Stephen P. King Index
531
Parti
Introduction
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CHAPTER 1 RETURNING TO THE CLASSICAL TRADITION: THE RELEVANCE AND APPLICATION OF INFRAMARGINAL ANALYSIS TO DEVELOPMENT ECONOMICS
Christis G. Tombazos and Stephen Matteo Miller* Monash University 1. Why Consider the Inframarginal Approach to Economic Development? Economic development is an observable phenomenon, improvement in the well-being of people, while development economics is the study of that phenomenon. A relatively new sub-discipline of economics known alternatively as "inframarginal economics", "new classical economics" or "the economics of the division of labour," may offer its deepest insights in the study of economic development. This reason alone justifies compiling a collection of essays relating to economic development from the perspective of inframarginal economics, which differs in important ways from more traditional frameworks. We do not expect that inframarginal economics will be readily embraced by all people interested in the field of development. Arndt1 provides a fascinating account of the history of the idea of economic development, though it concentrates especially on the 20th century. It shows just how much recent debate there has been about how to define development. For the most part, until the collapse of the Soviet Union, economic development as practiced has been part of a political response from the West to counter the spread of communism to developing countries. Yet, as convincing and detailed as Arndt's story sounds, it does not seem likely that the idea of development only emerged within the last one hundred years. For instance, Arndt makes little mention of Adam Smith. * We thank Xiaokai Yang for useful discussions. 3
4
C. G. Tombazos, S. M. Miller
As Yang and Ng2 note in the introductory chapter of their well known book, neoclassical economics has focused primarily on the problem of resource allocation, while inframarginal economics, or the economics of the division of labour, gives primary attention, just like Adam Smith, to issues of economic organisation. Economic development is much less about a simple allocation of resources, and much more about explaining first the type of economic organisation that emerges, and only then analysing the resource allocation problem. Put another way, economic organisation puts boundaries on how resources are allocated, and therefore, the underlying reason for variations in economic development are ultimately related to organisational issues about economic activity. This line of reasoning is also present in the new institutional economics (NIE) view of economic development, including Bates3 and North4, among other efforts prior to and since then (see http://coase.org for more references). The motivations are different though, as work in the NIE tradition frequently begins with observations on economic history, while the new classical economics begins from the history of economic thought. Both approaches share common criticisms of standard neoclassical economics. Yet, work among practitioners of the former often make minor adjustments of neoclassical models to fit the historical fact that transaction costs matter for economic performance. Practitioners of the latter typically begin by attempting to mathematically formalise certain aspects of Smith's framework, and then search for evidence to verify inframarginal, and hence, Smithean views of economic performance. So while the issues and explanations of economic performance are often the same, the approaches are different, but it is important to bear in mind that neither approach fully embodies what Smith actually said about economic development. 2. Smith's View of Economics and Economic Development Many economists today would certainly point to Adam Smith as an early contributor to, if not the founder of, economics. Smith was also preoccupied with where improvements in economic performance, or growth, come from. Hence, he might also be viewed as the first development economist. Economic ideas had, of course, been circulating
Returning to the Classical Tradition
5
before him. Levy5 even claims that the anti-economic content of Plato's Republic was aimed at the economic ideas found in Homer's Illiad. There, the Gods are perfect rational agents, and understand the benefits of positive sum trade, as they make deals with each other, while humans are primitive, and understand only negative sum trade, or war. What Plato's Republic did was to establish an elitist view, that the division of labour for all occupations must be planned, and the agents needed to do so were Philosopher Kings. This view was dominant for roughly the next two thousand years. In The Wealth of Nations6, Smith challenged this view by appealing to our egalitarian sensibilities. In short, he suggested that the division of labour, which was constrained by market size and population, could arise if each person were left to choose their own occupation. If transaction costs are not prohibitive, a larger market and population makes it possible for people to reap the returns to specialisation. However, he was a great synthesizer of economic ideas, and to appreciate his contribution, it is necessary to consider his three works, The Theory of Moral Sentiments (TMS)7, Lectures on Jurisprudence (LJ)8, and An Enquiry into the Nature and Consequences of the Wealth of Nations (TWN)6. Smith's interest in development has its origins in his work on moral philosophy. As Levy9'10 puts it: It seems to me that Smith is relying on a belief that modestly informed spectators would reach a common judgment about the level of wellbeing of the median member of different societies to defend economic growth. Smith has told us three things. First, he tells us that the as-if well-being of children is easy for everyone to impute. Second, he has told us that the children of the median member of society will do best in a rapidly growing society. Third, he has told us that one and two suffice for us to conclude that rapid growth is good (see page 313). So, herein lies the justification for growth. In order to understand where growth comes from, it is first necessary to point out that for Smith, it is man's inherent desire to trade that drives the division of labour, and volume of trade that determines the extent of the division of labour. Furthermore, trade is possible because people share language; whether it's Chinese, French, Jamaican, English, or Swahili. As long as people
6
C. G. Tombazos, S. M. Miller
can communicate, it is language (see Smith TWN I. ii par. 26, Levy1' for an extensive interpretation of Smith's views). Thus, a larger market size with lots of commercial activity means a greater division of labour (see Smith TWN I. iii par. I6). The division of labour increases worker productivity because of improvements to human capital ("dexterity") and physical capital ("the invention of a great number of machines"), as each factor becomes more specialised, and because specialisation itself implies time is no longer lost in switching from one task to another (see Smith TWN I. i par. 56). Rising productivity is of course an indicator of a growing economy. Lectures on Jurisprudence* includes an early draft of the Wealth of Nations that discusses the role of specialisation in economic growth. It also includes a noteworthy discussion of the impediments to growth. Specifically, any factor that drives a wedge between the natural (distortion-free) price, and the observed market price, slows down commercial activity, and hence, the potential for economic growth. These wedges might be caused by monopoly, by taxes or tariffs, or by prices or risk due to various forms of uncertainty including property rights (see Juris-Prudence Part II. Of Police in Smith8 where he discusses among other things, the relationship between the division of labour and growth, as well as retardants to growth.). 3. Where has the Division of Labour Gone? Stigler12 observes that specialisation is the key facet of Smith's explanation of economic growth and development that is surprisingly missing from all of neoclassical economics. For instance, it is notably absent from the neoclassical framework of Marshall and Jevons. Reasons for this anomaly have been offered by a number of authors. Buchanan and Yoon13 speculate that the neoclassical thinkers sidestepped the issue of the division of labour because they considered it irreconcilable with their theory of distribution. This theory depends critically on constant returns. Yet, division of labour, and the benefits of specialisation, suggest increasing returns to scale. Others, including Houthakker (p.62)14 and Yang (p.8)15, add technical hurdles to what may have discouraged the founders of marginalist economics from addressing issues relevant to the division of labour.
Returning to the Classical Tradition
7
These hurdles originate from the fact that division of labour ultimately boils down to a binary decision of whether or not to produce a particular good. Mathematically, such outcomes are represented by corner solutions and require models that can accommodate such discontinuities. Yet, the marginal revolution relied on calculus that, in turn, requires smooth and continuous functions. Such functions can investigate interior outcomes, but are not suitable to study the discontinuities that characterise the division of labour. Clearly, nineteenth century mathematical formalism represented a genuine hurdle in reconciling the theory of distribution with classical questions of economic organisation. However, Buchanan and Yoon13 have recently argued that Marshall's decision to banish division of labour from the core preoccupations of the marginal revolution on the basis that it would undermine the requirement of constant returns was unnecessary. As these authors explain, improvements in the division of labour are in fact consistent with constant returns to scale. This is so because enhancements in the division of labour are generally realised when agents switch from self-production to market production. Hence, as the division of labour intensifies, the economy-wide value of the final output relative to the total value of all inputs increases. This leads to what Buchanan and Yoon refer to as "generalised increasing returns". Still, given any particular pattern of the division of labour, as the quantity of inputs employed by any one firm expands, output increases proportionally. It is the reorganisation of the economy that redefines what is produced and exchanged in the market that generates generalised increasing (economy-wide) returns, not the increase in inputs used by any given firm in the absence of any change in the economy-wide division of labour. In other words, generalised increasing returns and constant returns to scale can happily coexist. Applications of corner solutions and discontinuities to economic models can be traced to Coase16 and Koopmans17. Following Stigler12, a number of papers from faculty and students at the University of Chicago, including Rosen18' 19, Becker and Murphy20, and Tamura21' 22, among others, began to explore the economics of specialisation. Just as this research was beginning at the University of Chicago, Xiaokai Yang, imprisoned in China as a dissenter to the ideas that defined the Cultural
8
C. G. Tombazos, S. M. Miller
Revolution, began a remarkable journey to recreate economic theory using a framework that establishes a bridge between modern economics and Smith (see Yang15 for an excellent summary of relevant contributions). Unlike Smith, inframarginal economics makes no claims about the relationship between language and trade, nor does it ask what the theoretical justifications for growth are. Instead, it begins with the idea that there are gains from specialisation, that are possible in market economic activity, but there are also transaction costs from using the market. People are consumer-producers. They choose how much of their waking hours to allocate to a specific activity. As the division of labour evolves, individuals move away from self-producing all of their consumption towards producing only one good, which earns income that is used to purchase a wide variety of consumption goods. If transaction costs are high, people are unlikely to find it worthwhile to engage in market activity, and will instead self-produce their consumption in autarky. If, on the other hand, transaction efficiency is high enough, people who otherwise start off identical, will become increasingly specialised in particular occupations, thereby capturing the benefits of specialisation. Table 1 below compares some aspects of inframarginal and neoclassical models with Smith's theory of economic growth. Clearly, the inframarginal literature is closer than neoclassical theory in capturing Smith's thoughts on development, notwithstanding the absence of any moral content. The aim of this volume therefore is to bring together a number of recent articles on this topic in an effort to summarise the inframarginal explanation of the origins of economic development. As illustrated by the articles included in this anthology, inframarginal analysis has profound implications on the study of economic development. To assist the non-specialist reader with these contributions, the remaining of this article serves as an introduction to the practical implementation of inframarginal analysis. In particular, the next section outlines the key "steps" of inframarginal analysis, and section 5 implements these steps in the context of a simple model.
Returning to the Classical Tradition
9
Table 1: Classical, Inframarginal, and Neoclassical Perspectives Classical
Inframarginal
(Adam Smith)
(New Classical)
Neoclassical
Not explicitly
Yes
Yes
Moral justifications for growth
Yes
Not Yet
Not Yet
Firms are collections of people
Yes
Yes
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
No?
Yes
Yes
No
No
No
Yes?
Use of models is necessary
Assume people start off identical and choose to be different in their occupation Growth occurs by realising gains from specialisation Transaction costs determine the level of specialisation and hence growth Trade, and hence growth can occur even without exogenous differences Population
growth
adversely
affects economic performance
4. The "Steps" of Inframarginal Analysis Commonly, inframarginal analysis of economic models involves the following steps: i.
Design the model [important features of any model include items such as: types of agents, dimensions, production and consumption characteristics, and institutional structure (in particular: rules of exchange — e.g. presence of transaction costs, tariffs, etc.)]
ii.
Identify the prime configurations and market structures {note: prime configurations are patterns of production and consumption that may characterise individual agents within countries, and which may be consistent with meaningful frameworks of
C. G. Tombazos, S. M. Miller
exchange — or market structures). Autarky is viewed as meaningful "pattern of exchange" — i.e. a pattern of exchange with zero flows For each configuration, within each structure, employ marginal analysis to optimise for each agent's objective function Using demand and supply from previous step, solve for the market clearing conditions, which determine the price that would prevail if the corner equilibrium that is represented in each structure is in fact the general equilibrium. Of course, depending on the production function, prices may be determined prior to evaluation of market clearing conditions (as per the nonsubstitution theorem) Still, step (iv) may facilitate the determination of necessary conditions for the structure at hand to prevail Undertake a demarcation of parameter space in parameter value subsets within each of which the local corner equilibrium is the general equilibrium. The general equilibrium requires that under the relative price that prevails in any given corner or interior equilibrium, all individuals in all countries prefer the configurations consistent with the structure at hand, than any other possible shadow configurations. (Note that shadow configurations include all possible configurations other than the configuration prevalent in the structure and for the agent under examination) The final step entails examination of the resulting inframarginal comparative statics
Returning to the Classical Tradition
11
5. Even a Simple Inframarginal Model has Much to Say About Development The two good inframarginal model has been summarised in a number of contributions (Yang and Ng2, Yang15), but typically the purpose is to provide an intuition for more complicated models. Here, the two good inframarginal model is presented to summarise its implications for economics development. The context in which it is used here is to highlight the differences between an economy based on subsistence agriculture, and one that is more specialised, and hence developed. Just as Adam Smith pointed out that the difference between the philosopher and the common porter is really much smaller than the philosopher would lead us to believe, the model begins with Smith's inherent assumption that each person starts of the same. Differences arise from individual choices about how to specialise in productive activities. What follows represents a simple implementation of the steps outlined in section 4. Step i — Model design: o o
There is a continuum of identical consumers-producers that has massM Agents ex ante identical, and have the following utility function for food (denoted/) and non-food («) commodities: max ui = (f + kfd)(n
o
+ knd)
f, n: goods that are self provided fd, nd: goods bought (demanded) from the market 0 < k < 1 is the trading efficiency coefficient (the lower k is, the higher the transaction cost) Each consumer-producer has an allocation of labour-time (/) and land (/) that is exhausted across the two goods:
tf+t„=l
12
C. G. Tombazos, S. M. Miller
o
The production functions for food and non-food for a consumerproducer are variants of the Cobb-Douglas production function, in terms of labour (/,-) and land (?,): f + fs =max 0,(lf-ajtbf
\(ln-*)'t]
n + ns = max 0.
o
where the term a is a fixed learning cost. Finally there is a budget constraint such that in general:
Pffs+Pnns=[pffd+Pnnd] The idea behind inframarginal models is typically that the efficiency of transactions, often denoted by k, determines the extent of the division of labour, which in turn, determines income levels. The lower the efficiency of transactions (the higher the costs of transacting), the more expensive it is to rely on the market, and hence, people find selfproduction makes them better off. This is depicted below: k is low
k is medium range
i
i
Autarky
Partial specialisation
k is high
i Complete specialisation
The inframarginal approach typically calls for backing out the transaction efficiency parameter, to find the point at which the economy switches between configurations. These range from autarky, when each individual self-produces its consumption, to extreme specialisation, when each individual produces one-good, from which the proceeds are used to finance consumption of all other goods. To back out those transaction efficiency thresholds (the points at which the model switches behaviour), the first required step is to find the optimal supply and demand for each, initially identical, consumer-producer under autarky, under partial
Returning to the Classical
specialisation (for specialisation.
more
complicated
13
Tradition
models),
and
complete
Autarky
Partial specialisation
Complete specialisation
i
LIL
i Optimal
Optimal
Optimal S&D
S&D
S&D
Hence, there is an optimal supply and demand for each person in each configuration, which corresponds to a different extent of the division of labour. These optimal supply and demand conditions are then used to get an indirect or real utility/welfare function for each individual in each configuration: Autarky
Partial specialisation
Complete specialisation
i
1 Welfare
Welfare
Welfare
Once the real utility is known for each individual, it is then possible to determine the thresholds of transaction efficiency at which people in the economy find themselves in autarky, partial specialisation and complete specialisation. With this overview in mind, the model can be analysed in different configurations (in this case two). Steps ii — Configurations and market structures
(a) Structure A, autarky
(b) Structure D, division of labour
Figure 1: Configurations and Structures
C. G. Tombazos, S. M. Miller
14
Step Hi — Marginal solutions From the general problem, it is easy to see that autarky is a much simpler world because there are no markets, and hence, there are no prices and there is no budget constraint. Hence, human behaviour is not governed by relative price changes. People don't respond to what is not observable. Still, they have preferences so that opportunity cost is still very much in their minds. Structure A (Autarky): gen<;ral problem simplifies to: max u, = f -n if1-
f = (lf-ajtbf n = {ln-a)°tbn lf+ln=\ tf + tn=\ This constrained maximisation problem can be simplified to an unconstrained optimisation problem, in this case solved for the non-food crop by substituting in all the constraints: a maxu tbn x uAA=({l-l = 111n)-a)" — <„| — a |(l-tj i i ~ / „ {ln-a)
The four first order necessary Kuhn-Tucker conditions are: | ^ = -fl.((l-/i,)-a)"-1 (!-/„)*•(/„-«)"<+
a.({l-ln)-a)a{l-tn)b.{ln-artbn<* dl
Returning to the Classical Tradition
15
^L = ^ . ( ( l - / . ) - a H l - / . r •(/.-«)•< + b.((l-ln)-a)a(l-tn)b.(ln-a)aCl<0
"K In autarky, because of the Cobb-Douglas utility specification, both goods must be consumed, and hence, both goods must be produced, as there is no market for goods that the household decides not to produce. Looked at another way, if either /„ or tn is either zero or one, then the first order condition is not satisfied. Hence, an interior solution must exist in autarky, such that the household produces all goods that it consumes. Rearranging and simplifying the first order necessary condition for labour yields:
|^((iH)-«r'-a-«r=((i-'.)-«H'»-<*r' From the labour constraint, this implies that:
Since the consumption shares are equal in this specification, the model implies that labour is therefore allocated equally to both food and non-food goods. Hence, half of each person's labour is devoted to each good, but less than half of that labour is actually used in producing the goods, because there are learning costs, which are incurred twice, once for each good. Turning now to the land first order necessary condition:
£ ^ L : ( l - , ) 6 - V =(!_,)>.,*-i
From the land constraint, this implies that:
16
C. G. Tombazos, S. M. Miller
uA=({l-ln)-a)\x-tn)b.{ln-a)atbn={*.5-af What is interesting to note about this real utility function is that, in addition to the absence of prices (since markets are not used), well-being is, marginally, independent of transaction efficiencies. This condition will be recalled later so that the threshold at which the economy switches from autarky to the complete division of labour can be determined. Structure D (division of labour): Food producers Under division of labour, everything changes. Now households face a budget constraint, and they face transaction costs in purchasing consumption goods. Now households specialise in producing one crop (food or non-food), such that all labour is allocated to a single activity, the proceeds from which are used to purchase all other goods (in multiple good models), in this case one. Now, labour shares are no longer the choice variable, instead the choice is how much of the good produced to supply to the market, and how much to keep for domestic consumption. The model in this configuration also makes room for Adam Smith's brilliant insight, that you can still have trade (within a country) between people who start off identical. The driving force behind this is that people now choose to be different in their occupations, because there are gains from specialising, a feature that is not inherent to the typical neoclassical model. Under specialisation there are two cases, the food crop and the nonfood crop producers. The mathematics for multiple good models can quickly become cumbersome (for the model with m goods there are 2m possible zero/non-zero combinations to consider). Fortunately, there is a so-called Wen theorem, proven by Wen (1998), which holds that no individual buys and self-provides the same good. This simplifies the mathematics of the model significantly. Applying this theorem to the
Returning to the Classical Tradition
17
food crop specialist, since a food producer would not buy that same food crop, but would instead purchase all other goods, the general problem simplifies to: maxwD = ( / + (0))((0) + knd) = f -knd (lf-a)atbf-f
f =
/ / + (0) = 1
f/+(0) = l
Pfr+{0) = {0) +
d
Pnn
The next step is to substitute the second and third constraints for labour and land, into the production function. So now / = (l — or) - fs, since all labour and land is allocated to food production. This can be substituted into the utility function. Additionally, the budget constraint can be solved for non-food purchased from the market n = —— so that it too can be substituted into the utility function. After making these substitutions, the problem is now to maximise the following utility function. max uDJ nf =
,((!-„)"-/•).*.
f
V Pn
This problem calls for maximising utility in terms of food supplied. Hence, du
D,f
= -k-
df
r
Pff Pn
du
DJ
df
^.((l-a)a-r).k<0
+
Pn
=o
In autarky, the marginal cost of producing more of one good entails consuming less of the other. Here the marginal cost of supplying more to the market requires having less of that good to consume at home (since the Wen theorem implies that no person buys and self-provides the same good). Applying the Kuhn-Tucker theorem to these two first order
18
C. G. Tombazos, S. M. Miller
necessary conditions reveals that food supplied to market must be positive. If it were zero,' then — • ((1 - a)")- k<0, which is infeasible if prices, transaction efficiency, and the learning costs are all positive. Hence, the individual's optimal food supply to the market is:
duD,f dfs 'J
.fs=il-aT 2
The greater the returns to specialisation (a), or the lower the learning cost of producing food, the greater amount of food is supplied. When combined with the budget constraint, this implies that the amount of food demanded is: n'
=
l a a =Pf( ~ )
Pff' Pn
Pn
2
As in neoclassical economics, the first law of demand, that generally requires the quantity of the good demanded to be negatively related to its relative price, is satisfied. Also, as under autarky, the point here is to find the real utility function, in terms of prices, for food crop producers. Inserting the optimal supply and demand into the utility function yields, uDf=f-knd
=k-^-± '— 4 Pn Points to observe about this measure of well-being is that, ceteris paribus, food producers are better off as the relative price of food rises, as the transaction efficiency rises, as the returns to specialisation {a) rises, or as the learning cost for producing food crops falls. Structure D (division of labour): Non-food producers Following the logic from above, after simplifying the general problem, the problem for non-food producers is symmetric:
maxuD,n=((0) + fd)(n + (0)) = kfd-n n = {ln-a)"tbn-ns
(0) + / „ = l
Returning to the Classical Tradition
19
(<>) + *„ = 1 {0)
s
=pffd+(0)
Pnn
+
which, after substituting in the constraints, reduces to: max ur
Pf
By similar reasoning, the symmetry of the model implies that for nonfood crop producers, their supply function is characterised by: duD,n_^s dns
=
(l-a)a
'
2
and their demand for the food crop is: fd
=
P„ns Pf
=
Pn C 1 -")" 2
Pf
Substituting in the optimal supply and demand functions into the utility function yields: U
D,n
-
*
Pn {l-afa .
4 Pf As with the real utility function for food producers, ceteris paribus, non-food crop specialists are better off as the relative price of the crop they produce rises, as the transaction efficiency rises, as the returns to specialisation (a) rises, or as the learning cost for producing non-food crops falls.
Step iv — Market demand and supply Compared to the neoclassical view, market demand and supply are defined in a slightly different, but consistent way. First, note that markets in this model only exist under complete specialisation. In the two good case, the market demand for one good actually equals the individual demands for that good multiplied by the mass of suppliers of the other
20
C. G. Tombazos, S. M. Miller
good, since no person buys and self-produces the same good. Hence for food, market demand is given by:
Pn (l~aT
Fd=Mn-fd=Mn
l
Pf
where again, the first law of demand holds, and also, the quantity of food demanded is increasing in the population of consumers, the number of non-food producers. Market supply is defined by:
From this, it is evident that market supply of food is increasing in the number of food producers, a reduction in fixed learning costs of producing food, and an increase in the returns to specialisation. While the first is due to the fact that more producers means more supply. The second is because a reduction in learning costs means more time per person is actually devoted to producing the crop. Finally, an increase in the returns to specialisation results from the fact that for each unit of time allocated, a higher amount of output is achieved. Equating demand and supply results in:
= Mf Pf
(1-a)-
= FS
2
which implies that:
K M
f
_Pf Pn
Since Walras's Law holds, in the two-good case presented here, it is sufficient to consider only the market clearing conditions for food to determine equilibrium. However, just to be sure, with properties that are symmetric to food crops, market demand for non-food crops is given by: Nd =Mf-nd
=Mf-^-^ Pn
'— 2
Returning to the Classical Tradition
21
where again the first law of demand for non-food crops holds, and demand is once more increasing in the population of consumers, in this case food producers. Non-food crop supplies are defined by:
(l-a)a Ns =M
-ns =M-±
'— 2 which is likewise increasing in the number of producers, declining in the costs of learning to produce the crop, and increasing in the returns to specialisation. Market equilibrium for non-food crops is defined by:
pf (i-a)a
, f
Pn
2
(i-«r " 2
which yields an identical condition: MIL=PL M
f
Pn
From this, it is possible to determine the relative price between food and non-food crops. Consider rearranging that condition as follows: M
nPn =
M
fPf
Since everyone is identical, and no two people are charged different prices, then it must be the case that this condition holds between any pair of food and non-food specialists, hence, the two prices are equal: Pn=Pf With market equilibrium defined, it is now possible to address the final step, which is to identify the point in transaction efficiency space at which the world shifts from autarky to the division of labour. Step v-vi — Autarky vs specialisation Recall that it is the indirect utility function that helps to determine in which configuration the individuals find themselves. The indirect utility functions for each subsistence farmers, food crop specialists, and nonfood crop specialists are, respectively:
22
C. G. Tombazos, S. M. Miller
= (0.5-a)2 (1 -a)
uDf
4
Pn
\2a U
D,n
( I " -a) 4
=
Pf The logic concerning how to interpret these results is simple. If the individual is "happier", that is, has a higher real income, in one configuration than in any other, she will choose that configuration. Also, within a particular configuration (the division of labour in this case), it must be the case that the utility of a particular occupation equals that of all others. Hence, for two goods and two configurations, there are only two conditions to check. Within the division of labour configuration it must be the case that utility across occupations is identical:
. (I-af
, (I-af
4
4
The more important case to consider entails a comparison between real utility under autarky with that under the division of labour. If the real income under autarky is greater than real income under specialisation, then autarky will be chosen: v
(0 S
IT
1
^ /r.
V
'
~
U
D,f
~
U
D,n
Alternatively, division of labour chosen is: ^-(0.5
a)
'
= UDj
=
UDn
Isolating both cases in terms oik, suggests that autarky is chosen over the division of labour if the transaction efficiency parameter k falls below:
Returning to the Classical Tradition
23
, fas-«r (l-af and division of labour is chosen if:
{l-af Hence, the model implies that a fundamental problem in generating higher real income is to reduce transaction inefficiencies, which illustrates the similarity between the inframarginal and transaction cost approaches to economic development. So again, even this simple model can provide a more or less formal presentation of some of Smith's ideas about development: 1. Transaction inefficiencies are a key impediment to realising higher levels of well-being. 2. Higher levels of well-being are attainable if people make choices about their professional specialisation, as opposed to having that decision made for them by someone else. 3. Trade occurs even between ex ante identical individuals. 6. Concluding Remarks As illustrated in the context of the simple model discussed in the previous sections, the inframarginal perspective facilitates a reconciliation between division of labour considerations and neoclassical frameworks of distribution. The resulting analysis sheds light on a host of topics relevant to development economics that have been largely neglected by marginalist orthodoxy. Examples include the benefits from specialisation when trading partners are ex ante identical, the interaction of technological comparative advantage, factor endowments, and economies of specialisation, the role of transaction costs in the organisation of the economy, and so forth. Naturally, this field of research has generated considerable interest in recent years. However, little has been done by way of organising the
24
C. G. Tombazos, S. M. Miller
accumulated knowledge in a single volume. This anthology fills this gap by collecting key inframarginal contributions to development economics. As such, this volume serves both as an introduction of the field to the new researcher as well as a useful source of reference to those actively involved in this area.
References 1.
H. W. Arndt, Economic Development: The History of an Idea (University of Chicago Press, Chicago, 1987).
2.
X. Yang and Y. K. Ng, Specialization and Economic Organization: A New Classical Economic Framework (North-Holland, Amsterdam, 1993).
3.
R. Bates, States and Markets in Africa (University of California Press, Berkeley, 1981).
4.
D. C. North, Institutions, Institutional Change and Economic Performance (Cambridge University Press, Cambridge, 1990).
5.
D. M. Levy, Economic Ideas of Ordinary People: From Preferences to Trade (Routledge, London, 1992).
6.
A. Smith, An Inquiry into the Nature and Causes of the Wealth of Nations (Liberty Fund, Indianapolis, 1981).
7.
A. Smith, The Theory of Moral Sentiments (Liberty Fund, Indianapolis, 1982).
8.
A. Smith, Lectures on Jurisprudence (Liberty Fund, Indianapolis, IN, 1982).
9.
D. M. Levy, The European Journal of the History of Economic Thought, 2 (1995).
10.
D. M. Levy, How the Dismal Science Got Its Name: Classical Economics and the Ur-Text of Racial Politics (The University of Michigan Press, Ann Arbor, 2001).
11.
D. M. Levy, Economic Inquiry, 35 (1997).
Returning to the Classical Tradition
25
12.
G. Stigler, Journal of Political Economy, 84 (1976).
13.
J. M. Buchanan and Y. J. Yoon, History of Political Economy, 31 (1999).
14.
H. S. Houthakker, Kyklos, 9 (1956).
15.
X. Yang, Economics: New Classical Versus Neoclassical Frameworks (Blackwell Publishers, Maiden and Oxford, 2001).
16.
R. Coase, Economica, 13 (1946).
17.
T. C. Koopmans, Three Essays on the State of Economic Science (McGraw-Hill, New York, 1957).
18.
S. Rosen, Economica, 45(1978).
19.
S. Rosen, Journal of Labor Economics, 1 (1983).
20.
G. S. Becker and K. M. Murphy, The Quarterly Journal of Economics, 107 (1992).
21.
R. Tamura, Journal of Political Economy\ 99 (1991).
22.
R. Tamura, Journal of Economic Theory, 58 (1992).
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Part 2
The Origins of Inframarginal Applications to the Study of Economic Development
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CHAPTER 2 THE MARGINAL COST CONTROVERSY*
Ronald H. Coase University of Chicago Law School 1.
The State of the Debate
I wish to discuss in this article the question of how prices ought to be determined in conditions of decreasing average costs. In particular, I wish to discuss one answer to this question which is by now familiar to most economists and which may be summarised as follows: (a) The amount paid for each unit of the product (the price) should be made equal to marginal cost. (b) Since, when average costs are decreasing, marginal costs are less than average costs, the total amount paid for the product will fall short of total costs. (c) The amount by which total costs exceed total receipts (the loss, as it is sometimes termed) should be a charge on the Government and should be borne out of taxation. This view has been supported by Professor H. Hotelling,1 Professor A. P. Lerner,2 Mr. J. E. Meade and Mr. J. M. Fleming.3 It has aroused * Reprinted from Economica, 13, Ronald H. Coase, "The Marginal Cost Controversy," 169-182, 1946, with permission from Blackwell. ' "The General Welfare in Relation to Problems of Taxation and of Railway and Utility Rates," Econometrica, July, 1938. 2 In his book, The Economics of Control (1944). Professor Lerner had earlier set out this view in articles in the Review of Economic Studies and in the Economic Journal. 3 "Price and Output Policy of State Enterprise," Economic Journal, December, 1944. See also J. E. Meade, Economic Analysis and Policy, 1936, pp. 182-186, American edition (1938), pp. 195-199.
29
30
R. H. Coase
considerable interest and has already found its way into some textbooks on public utility economics.4 But despite the importance of its practical implications, its paradoxical character, and the fact that there are many economists who consider it fallacious, it has so far received little written criticism.5 It may have been the sheer quantity of literature in favour of this solution, and the relatively small amount of written adverse criticism which led Mr. J. M. Fleming to claim that it "is not, I think, open to serious criticism" and to lament the fact that it was not more widely understood and accepted "outside the narrow ranks of the economists." But a different solution, which I believe in essentials to be the correct one, had already been suggested by Mr. C. L. Paine in 19376 and by Professor E. W. Clemens in 1941. 7 I wrote in 1945 a short note
See C. Woody Thompson and Wendell R. Smith, Public Utility Economics, pp. 271-273, and Irston R. Barnes, The Economics of Public Utility Regulation, pp. 586-588. See also Professor Emery Troxel, "I Incremental Cost Determination of Utility Prices," "II Limitations of the Incremental Cost Patterns of Pricing," "III Incremental Cost Control under Public Ownership," Journal of Land and Public Utility Economics, November, 1942, February, 1943, and August, 1943; and Professor James C. Bonbright, "Major Controversies as to the Criteria of Reasonable Public Utility Rates," Papers and Proceedings, American Economic Association, December, 1940. Professor Bonbright points out that the "extreme social conservatism of most public utility and railroad specialists had prevented" this solution "from gaining wide acceptance, or even from receiving any considerable notice, in the literature of rate theory." However, he thought that it might become a live issue in the next few years (after 1940) as a result of Professor Hotelling's article, which Professor Bonbright considered to be "one of the most distinguished contributions to rate-making theory in the entire literature of economics." 5 It is true that Professor Ragnar Frisch criticized Professor Hotelling's article shortly after it appeared. But, though much of interest emerged in Professor Frisch's note and the subsequent discussion with Professor Hotelling, it appears, at least to the non-mathematical reader, that Professor Frisch's attack was not directed at the foundations of Professor Hotelling's argument but rather to what seemed to him to be defects in its formulation. See Ragnar Frisch, "The Dupuit Taxation Theorem" and "A Further Note on the Dupuit Taxation Theorem" and H. Hotelling, "The Relation of Prices to Marginal Costs in an Optimum System" and "A Final Note", Econometrica, April, 1939. 6 See C. L. Paine, "Some Aspects of Discrimination by Public Utilities," Economica, November, 1937. 7 See E. W. Clemens, "Price Discrimination in Decreasing Cost Industries," American Economic Review, December, 1941.
The Marginal Cost
Controversy
31
criticising the solution as set out by Mr. Meade and Mr. Fleming, and a further note by Dr. T. Wilson9 underlined the fact that agreement among economists had not yet been reached. I now propose to examine the Hotelling-Lerner solution, as I shall call it, in greater detail and to point out the fundamental defects which I believe it contains. 2.
Isolation of the Problem
Any actual economic situation is complex and a single economic problem does not exist in isolation. Consequently, confusion is liable to result because economists dealing with an actual situation are attempting to solve several problems at once. I believe this is true of the question I am discussing in this article. The central problem relates to a divergence between average and marginal costs. But, in any actual case, two other problems usually arise. First, some of the costs are common to numbers of consumers and any consideration of the view that total costs ought to be borne by consumers raises the question of whether there is any rational method by which these common costs can be allocated between consumers. Secondly, many of the so-called fixed costs are in fact outlays which were made in the past for factors, the return to which in the present is a quasi-rent, and a consideration of what the return to such factors ought to be (in order to discover what total costs are) raises additional problems of great intricacy.10 These are, I think, the other two problems which usually exist simultaneously with a divergence between average and marginal costs. They are, however, separate or at least separable questions. Thus, the example used by Professor Hotelling, the problem of pricing in the case of a bridge,11 is in fact an extremely complex case rather than the simple one it appears to be on the surface.
"Price and Output Policy of State Enterprise: A Comment," Economic Journal, April, 1945. 9 "Price and Output Policy of State Enterprise," Economic Journal, December, 1945. 10 See F. A. Hayek, Collectivist Economic Planning, the section entitled "The Criterion of Marginal Costs," pp. 226-231. 11 This example was originally used by Dupuit in an article in the Annates des Ponts et Chaussees (1844) which was reprinted in De Vutilite et de sa mesure (1933).
32
R. H. Coase
I propose to isolate the question at issue by examining an example in which, although there is a divergence between marginal and average costs, all costs are attributable to individual consumers; in which all costs are currently incurred; and in which, to avoid a further complication which might trouble some readers concerning the meaning of marginal cost, all factors are in perfectly elastic supply. Assume that consumers are situated around a central market in which a certain product is available at constant prices. Assume that roads run out from the central market but that each road passes only one consumer of the product. Assume also that a carrier can carry on each journey additional units of the product at no additional cost (at least to a point beyond the limit of consumption of any individual consumer).12 Assume further that the product is sold at the point of consumption. It is clear that the cost of supplying each individual consumer would be the cost of the carrier plus the cost at the central market of the number of units consumed by that particular consumer of the product. The marginal cost would be equal to the cost of a unit of the product at the central market. The average cost would be higher than the marginal cost and would decline as the cost of the carrier was spread over an increasing number of units.13 The Hotelling-Lerner solution would presumably be that the amount which consumers should pay for each unit of the product should be equal only to marginal cost. The effect would be for consumers to pay for the cost of the product at the central market and for the Government, or rather the taxpayer, to bear the costs of carriage. It is the validity of this solution that I wish to examine. But first it is necessary to turn to a consideration of fundamentals.
12
An indivisibility must be present in all cases of decreasing average costs. Although I assume that it is not possible to employ less than a carrier, his services may be assumed to be in perfectly elastic supply in that payment will vary proportionately with the time he is employed and that the additional employment of carriers will not raise their price. 13 The assumption that the total costs consist of two distinct kinds, one of which enters into marginal cost while the other does not, is not essential. We could have assumed that the costs of carriage increased as additional units were carried but that the marginal costs of carriage were below the average. It will, however, aid in exposition if we keep to the original assumption.
The Marginal Cost
3.
Controversy
33
What is Optimum Pricing?
I take a pricing system to be one in which individual consumers have command over various sums of money which they use to obtain goods and services by spending this money in accordance with a system of prices. It is, of course, not the only method of allocating goods and services, or more properly, the use of factors of production between consumers. It would be possible for the Government to decide what to produce and to allocate goods and services directly to consumers. But this would have disadvantages as compared with the use of a pricing system. No Government could distinguish in any detail between the varying tastes of individual consumers (which is, of course, why a "points" system of rationing in wartime is adopted for many items);14 without a pricing system, a most useful guide to what consumers' preferences really are would be lacking; furthermore, although a pricing system puts additional marketing costs on to consumers and firms, these may in fact be less than the organising costs which would otherwise have to be incurred by the Government.15 These are the reasons which would lead an enlightened Government to adopt a pricing system — and we shall see later that they are very relevant to the problem we are considering. If it is decided to use a pricing system, there are two main problems that have to be solved. The first is how much money each individual consumer shall have — the problem of the optimum distribution of income and wealth. The second is, what is to be the system of prices in accordance with which goods and services are to be made available to consumers — the problem of the optimum system of prices. It is with the second of these problems that I am concerned in this article. The first is partly, though not entirely, a question of ethics. But it is important to realise that there are these two problems and that both have to be solved if a pricing system is to produce satisfactory results. As I am in this section dealing with the second only of these problems, I shall assume that the distribution of income and wealth can be taken to be the optimum. For an individual consumer, the system of prices represents the terms on which he can obtain various goods and services. According to what 15
Compare Lerner, op. cit., p. 53. See R. H. Coase, "The Nature of the Firm," Economica, November, 1937.
34
R. H. Coase
principles should prices be determined? The first would appear to be that for each individual consumer the same factor should have the same price in whatever use it is employed, since otherwise consumers would not be able to choose rationally, on the basis of price, the use in which they prefer a factor to be employed. The second would appear to be that the price of a factor should be the same for all consumers since otherwise one consumer would be obtaining more for the same amount of money than another consumer. If the optimum distribution of income and wealth had been obtained, the effect of charging different prices for the same factor to different people would be to upset that distribution. It is a more subtle application of this second rule that the price fixed should be such as to allow factors to go to the highest bidders. That is, the price should be one which equates supply and demand and it should be the same for all consumers and in all uses.16 This implies that the amount paid for a product should be equal to the value of the factors used in its production in another use or to another user. But the value of the factors used in the production of a product in another use or to another user is the cost of the product. We thus arrive at the familiar but important conclusion that the amount paid for a product should be equal to its cost. It will be this principle which will enable us to discuss the problems of individual pricing without tracing throughout the economic system all the changes consequent upon the alteration of a single price. 4.
The Argument for Multi-Part Pricing
How does this general argument for basing prices on costs apply to the case we are considering — the case of decreasing average costs? The writers whose views I am considering seem to assume that the alternatives with which one is faced are to charge a price equal to marginal cost (in which case a loss is made) or to charge a price equal to average cost (in which case no loss is made). There is, however, a third possibility — multi-part pricing. In this section, I set out the argument for multi-part pricing when there are conditions of decreasing average costs.
Compare also Lerner, op. cit., pp. 45-50.
The Marginal Cost
Controversy
35
It is clear that if the consumer is not allowed to obtain at the marginal cost additional units of products, produced under conditions of decreasing average costs, he is not being allowed to choose in a rational manner between spending his money on consuming additional units of the product and spending his money in some other way, since the amount which he would be called upon to spend to obtain additional units of the product would not reflect the value of the factors in another use or to another user. But for the same reason it can be argued that the consumer should pay the total cost of the product. A consumer does not only have to decide whether to consume additional units of a product; he has also to decide whether it is worth his while to consume the product at all rather than spend his money in some other direction. This can be discovered if the consumer is asked to pay an amount equal to the total costs of supplying him, that is, an amount equal to the total value of the factors used in providing him with the product. To apply this argument to our example, the consumer should not only pay the costs of obtaining additional units of the product at the central market, he should also pay the cost of carriage. How can this be brought about? The obvious answer is that the consumer should be charged one sum to cover the cost of carriage while for additional units he should be charged the cost of the goods at the central market. We thus arrive at the conclusion that the form of pricing which is appropriate is a multi-part pricing system (in the particular case considered, a two-part pricing system), a type of pricing well known to students of public utilities and which has often been advocated for just the reasons which I have set out in this article.17
See H. F. Havlik, Service Charges in Gas and Electric Rates, and references therein. See also Barnes, op. cit., p. 588. Havlik himself appears to support the view that costs which are attributable to individual consumers should be charged to those consumers. He does, however, use a variant of the Hotelling-Lerner solution when dealing with the case in which what he terms marginal customer costs, "the additional costs of taking on a customer and maintaining the connection, without actually supplying him with electricity," are less than average customer costs. In this case, "revenues from a customer charge would be less than total customer costs" and it would be "justifiable" for the Government "to give a subsidy" (pp. 92-93). Havlik does not discuss how the subsidy ought to be raised. In this article I am, however, concerned simply with the case in which all costs are attributable to individual consumers and to this case Havlik's variant of the Hotelling-Lerner solution, which is concerned with common costs, does not apply.
36
R. H. Coase
Now it is, I think, extremely significant that none of the advocates of the Hotelling-Lerner solution should have examined the possibilities of multi-part pricing as a solution of the problem they are considering. They write as though the only possible method of pricing is to charge a single price per unit and the problem they have to solve is what that price should be. It may be that their reason for not examining multi-part systems of pricing was that they were sure they had in fact found the optimum system of pricing. We must therefore compare the results of adopting the Hotelling-Lerner solution with those of using multi-part pricing. 5.
Multi-Part Pricing Compared with the Hotelling-Lerner Solution
The Hotelling-Lerner solution, if adopted in the case of my example, would mean that the cost of the goods at the central market would be paid for by consumers but that the cost of carriage would be borne out of taxation. My objections to this solution as compared with adopting a two-part system of pricing fall under three heads: first, that it leads to a maldistribution of the factors of production between different uses; second, that it leads to a redistribution of income; and third, that the additional taxation imposed will tend to produce other harmful effects. First, the Hotelling-Lerner solution would appear to remove the means whereby consumers make a rational choice between the use as carriers and the use for some other purpose of the factors which enter into the cost of carriage. In this use, the factor would be free; in another use (provided that it entered into marginal cost) it would have to be paid for. Similarly, this solution would mean that consumers would choose between different locations without taking into account that the costs of carriage vary as between one location and another. The answer which the supporters of the Hotelling-Lerner solution would make to this objection would appear to be that the Government should estimate for each individual consumer in my example whether he would buy the product and also what location he would prefer, if he had
The Marginal Cost
Controversy
37
to pay the total cost.18 Only if the consumer would thus have been prepared to pay the total cost of supplying the product to a given location will provision for supplying it to that location be made under the Hotelling-Lerner scheme. Professor Hotelling points out that to decide whether the demand was sufficient to warrant the costs of building a bridge "would be a matter of estimation of vehicular and pedestrian traffic originating and terminating in particular zones, with a comparison of distances by alternative routes in each case, and an evaluation of the saving in each class of movement".19 If it were possible to make such estimates, at low cost and with considerable accuracy and without knowledge of what had happened in the past when consumers had been required to pay the total cost, this would be likely to lead, in my opinion, not to a modification of the pricing system but rather to its abolition. The pricing system, as I pointed out earlier, is a particular method of allocating the use of factors of production between consumers and the arguments for its adoption derive their main force from the view that such estimates of individual demand by a Government would be very inaccurate. It should be noted here that neither Professor Lerner nor Mr. Meade in fact make any considerable claim for the accuracy of these estimates. Indeed, Professor Lerner in an earlier section of his book argues for a pricing system on precisely the grounds that it is impossible for a Government to make such estimates.20 Neither Professor Hotelling, Professor Lerner nor Mr. Meade give in my view sufficient weight to the stimulus to correct forecasting which comes from having a subsequent market test of whether consumers are willing to pay the total cost of the product. Nor do they recognize the importance of the aid which the results of this market test give in enabling more accurate forecasts to be made in the future. Professor Hotelling says: "Defenders of the current theory that the overhead costs of an industry must be met out of the sale of its products or services hold that this is necessary in order to find out whether the creation of the See Lerner, op. cit., pp. 186-199 and Meade, loc. cit., pp. 324-325. And it would seem that Professor Hotelling's mathematical formulation comes to much the same thing, see loc. cit., p. 262 and p. 268. 19 Loc. cit., pp. 247-248. 20 Op. cit., pp. 61-64.
38
R. H. Coase
industry was a wise social policy. Nothing could be more absurd." This, he says, "is an interesting historical question".21 And he adds later: "When the question arises of building new railroads or new major industries of any kind or of scrapping the old, we shall face, not a historical, but a mathematical and economic problem".22 Nowhere in Professor Hotelling's article does one find recognition of the fact that it will be more difficult to discover whether to build new railroads or new industries if one does not know whether the creation of past railroads or industries was wise social policy. And it is certainly not absurd to take into account the fact that decisions are likely to be better made if afterwards there is some test of whether such decisions were wise social policy than if such an enquiry is never made. I do not myself believe that a Government could make accurate estimates of individual demand in a regime in which all prices were based on marginal costs. But it may be well to consider what would be likely to be done if a Government attempted to carry out the Hotelling-Lerner policy. Consider the example I have been discussing. Certain consumers would have to be designated as able to buy the product. The Government would then undertake to pay whatever costs for carriage were incurred on behalf of these consumers. A Government would have a difficult task in deciding where to draw the line. If it adopted a narrow view of the qualifications required of those allowed to consume this product, consumers who really preferred to use the factor employed in the carriage of the product in this way, would be prevented from doing so. If on the other hand it was liberal in its view, many would find that they were no longer deterred from consuming the product or living at a greater distance from the central market by the cost of the factor used in carriage, that is, by its value in alternative uses or to an alternative user. It would, of course, be possible for the Government to follow a liberal policy to one class of consumers and a narrow policy to others at the same time. It is not easy to guess what policy a Government would be likely to follow. But in Great Britain I suspect that it would
21 22
hoc. cit., p. 268. hoc. cit., p. 269.
The Marginal Cost
Controversy
39
tend to err on the liberal side and that there would consequently be too great an employment of the factor used in the carriage of the product. But even if the Government were able to estimate individual demands accurately, the Hotelling-Lerner solution would be subject to another objection. The Government is supposed to estimate which consumers would be willing to pay the cost of carriage (and we shall assume for the moment that it estimates correctly). But it does not in fact ask these consumers to pay this sum. This money is then available for these consumers to spend on some other commodity. Consumers who buy products which are produced under conditions of decreasing average costs will therefore obtain products for any given expenditure embodying a greater value of factors than those who do not. There is a redistribution of income in favour of consumers of goods produced under conditions of decreasing average costs.24 There would not, I think, be any dispute that what is equivalent to a redistribution of income does in these circumstances take place. Professor Hotelling is, however, the only one of the writers whose views I am examining who deals explicitly with this point. I shall therefore examine his reasons for thinking that this objection is of little substance. First of all, I believe that Professor Hotelling considers this objection to be largely irrelevant because the initial distribution of income, at least in the United States, is not in fact the optimum. He does not directly say this but it is evident from his whole approach to the question.25 When he argues that the loss resulting from an application of the marginal cost rule should be borne out of income taxes, inheritance taxes and taxes on the site value of land, he is, I think, partly doing so because he believes that the wealthy and the landlords already have too large a share of the total wealth and income. But why should consumers of goods produced under conditions of decreasing average costs be the only ones to benefit from this All the essentials of this argument have been set out in another connection by Cannan, in his History of Local Rates in England (Second Edition). See Chapter VIII, "The Economy of Local Rates" and especially his remarks on p. 187. 24 This assumes that the taxes from which the loss is made good do not fall entirely on consumers of goods produced under conditions of decreasing average costs. This is, of course, so because it is proposed that the taxes to be used should be income and similar taxes. 25 See, for example, his remarks, loc. cit., p. 259.
40
R. H. Coase
redistribution? The reason why Professor Hotelling sees little harm in using pricing policy partly as a means of redistributing income is, I think, that he does not consider the distinction between consumers of products produced under conditions of decreasing average costs and consumers of products produced under conditions of constant or increasing average costs to be of great importance. He argues that a Government carrying out his policy would undertake a great variety of public works. "A rough randomness in distribution would be ample to ensure such a distribution of benefits that most persons in every part of the country would be better off by reason of the programme as a whole".26 This comes to saying that in a regime of marginal cost pricing, all consumers will buy goods produced under conditions of decreasing average costs; that what is lost by any particular consumer in the redistribution involved in one scheme will be offset as a result of the redistribution following on another scheme; and that as a consequence, the significant redistribution would be from the wealthy and landlords to all others. It would be indeed pedantic to object to the achievement of a desirable aim merely because it is done in an unusual way. But this argument stands or falls by the assumption that there will be no significant redistribution as between consumers of different kinds of products. There is no reason to assume that this will be so. The gain which individual consumers would derive from the Hotelling-Lerner policy would depend on the extent to which they were willing to pay the total cost for products produced under conditions of decreasing average costs (given their initial income); and on the absolute divergence between marginal and average costs in the case of these goods; and on the extent to which the additional income derived as a result of the Hotelling-Lerner policy was spent on goods produced under conditions of decreasing average costs; and on the absolute divergence between marginal and average costs in these cases. It would be possible to appraise the character of the redistribution only after a detailed factual enquiry. There seems, however, no reason to suppose that it would be a negligible redistribution. The public utility industries provide some of the most striking instances of products supplied under conditions of decreasing average costs. Let us assume that they are the only industries in which these Loc. cit., p. 259.
The Marginal Cost
Controversy
41
conditions are found. Consumers who live in regions of low density of population would probably not be willing to pay the total costs of supply of public utility services which in their case would be very high, and would consequently gain nothing as a result of the Hotelling-Lerner policy because they would not be given the services. Consumers who live in cities would find their gains limited because, equipment there being relatively intensively used, the divergence between marginal and average cost would probably be much less than elsewhere; while since they probably already use all the public utility services, the additional income would be likely to be spent on other than public utility services. It would be those living in small towns, which have some but not all the public utility services and where the divergence between marginal and average cost was great, who would, I think, tend to gain most from the Hotelling-Lerner policy. I see no reason to suppose that there would not be some redistribution, possibly very considerable, as a result of this policy if it were generally applied. Professor Hotelling admits this possibility but claims that by a subsequent redistribution a situation could be produced in which everyone was better off than before.27 He does not describe how this redistribution would be effected. But it would obviously be an inferior arrangement to adopting a multi-part system of pricing which makes it unnecessary to have subsequent redistributions of income at all. I am, however, at a loss to understand how ordinary taxation procedures could be used to redistribute income from consumers of goods produced under conditions of decreasing average costs to all other consumers. An attempt to do this might be made by means of a tax on the consumption of goods produced under conditions of decreasing average costs. But this would either be equivalent to introducing multi-part pricing (if a lump sum tax was levied on consumers) or, if a tax per unit of consumption is imposed, it would bring about a divergence between the amount paid for additional units and marginal cost, a result which it is the object of the Hotelling-Lerner solution to avoid. I now turn to the third objection to the Hotelling-Lerner solution. The loss incurred is, it is said, to be made good by increased taxation. The taxes which Professor Hotelling and the others who support this solution have in Loc. cit., pp. 257-258.
42
R. H. Coase
mind are income taxes, inheritance taxes and taxes on the site value of land. Let us assume for the time being that the form of tax used to make good the loss is an income tax. But income taxes are usually so framed that marginal units of income are taxed, and therefore an income tax will have the same unfortunate effect on consumers' choice as a tax on goods and will produce results similar in character to those which follow from charging an amount for additional units of output greater than marginal cost. After the appearance of Professor Hotelling's first article, he seems to have had his attention drawn to this point by Professor Lerner. Professor Hotelling says in the discussion with Professor Frisch which followed his original article that "an income tax of the usual kind is a sort of excise tax on effort and on waiting, as well as on other less defensible ways of getting an income. An income tax is to some extent objectionable because it affects the choice between effort and leisure and the choice between immediate and postponed consumption. Thus some of the same kind of loss attaches to an income tax as to excise taxes proper. How serious this effect may be is a question for factual research; but there is some reason to suppose an income tax superior to excise taxes on individual commodities in this respect....".28 Professor Hotelling does not give any reasons why he thinks income taxes will tend to be less harmful in this respect than excise taxes. It may be so but it is obviously desirable to know what the circumstances are in which income taxes are less harmful and when they are likely to be found before applying the Hotelling-Lerner solution — if, that is, this policy would lead to increases in income taxes.29 Professor Hotelling attempts to avoid this difficulty by suggesting that "the public revenues, including those required to operate industries with sales at marginal cost, should be derived primarily from rents of land and other scarce goods, inheritance and windfall taxes, and taxes designed to reduce Econometrica, April, 1939, pp. 154-155.1 would add that income taxes also affect the choice between doing a job for oneself and employing some one to do it for one and in consequence an income tax dissipates some of the advantages of specialisation. See F. W. Paish, "Economic Incentive in Wartime," Economica, August, 1941, p. 244. 29 This problem seems to have been overlooked in the theory of public finance. The usual discussion of the burden of indirect taxation assumes that the alternative is a lump sum payment. See for example, M. F. W. Joseph, "The Excess Burden of Indirect Taxation," Review of Economic Studies, June, 1939. Compare also J. R. Hicks, Value and Capital, p. 41.
The Marginal Cost
Controversy
43
socially harmful consumption." This is not a very satisfactory solution. First of all, it assumes that such taxes will be sufficient to raise the sum required. Secondly, it assumes that the disturbance to the distribution of income and wealth due to the additional taxation on those who derive their incomes in these ways is better than the loss which would occur if the additional taxation was spread more evenly over people in the country. Alternatively, Professor Hotelling's suggestion involves the assumption that the optimum distribution of income and wealth has not already been achieved and that those who derive their incomes in these ways have not been taxed enough in the past. But, of course, if this is so, this further taxation is desirable quite apart from questions of pricing policy and there is little need to link it to the problem of pricing under conditions of decreasing average costs. Furthermore, the question would still remain of how the pricing problem should be solved when the optimum distribution of income and wealth was achieved. Professor Hotelling's suggestion for avoiding the loss which would result from increased income taxes is one of limited validity. In this section, I have compared the results of using a multi-part pricing system with those which would follow from the Hotelling-Lerner policy. I have shown that the Hotelling-Lerner solution would bring about a maldistribution of the factors of production, a maldistribution of income and probably a loss similar to that which the scheme was designed to avoid, but arising out of the effects of increased income taxes. These results would be avoided by the use of a multi-part system of pricing. 6.
Average Cost Pricing Compared with the Hotelling-Lerner Solution
Professor Hotelling, Professor Lerner, Mr. Meade and Mr. Fleming do not seem to have realised that many of the problems which they were trying to solve could have been dealt with by means of multi-part pricing and that this system of pricing would in fact have produced results not open to the objections which could be brought against the Hotelling-Lerner solution. But in fairness to them it must be pointed out that their attack was directed Econometrica, April, 1939, p. 155.
44
R. H. Coase
against charging a single price which was based on average cost and not against multi-part pricing. Is the argument valid in this case? If multi-part pricing is not possible, is it not preferable to adopt the Hotelling-Lerner solution rather than to adopt pricing based on average costs? In this case, the argument for the Hotelling-Lerner solution is considerably strengthened — and this in two respects. First of all, it is clear that if consumers are not allowed to buy additional units at marginal cost, there is a maldistribution of the factors of production. The nature of the gain which would accrue in this respect through the adoption of the Hotelling-Lerner solution has already been discussed in earlier sections.31 The second respect in which the argument for the Hotelling-Lerner solution is strengthened concerns the effectiveness of average cost pricing in providing a market test of the willingness of consumers to pay the total costs. In the previous section, I pointed out that multi-part pricing furnished such a test. How does this apply to the case of average cost pricing? The fact that consumers are willing to buy at a price which covers average costs certainly shows that they prefer to obtain that value of factors in that form rather than in any other which is open to them.32 The difficulty is, as Professor Hotelling points out, that the reverse is not true. It has long been known to economists that in cases in which the demand curve lies at all points below the average cost curve, it may be possible, by means of price discrimination, to raise the average revenue sufficiently to bring it up to average cost. If therefore pricing is on an average cost basis, there will be certain cases in which consumers would have been willing to pay the total cost but in which, owing to the limitations of this particular method of pricing, this would not be possible. Production could be undertaken in such cases if the Hotelling-Lerner policy was followed. These are the advantages of the Hotelling-Lerner solution as compared with average cost pricing. But the disadvantages which were examined in the previous section still remain. These have to be balanced 31
It might be thought that if all goods were priced on an average cost basis, since all prices would be raised above the marginal cost level, the choice of the consumers would be unaffected. But this would be true only if the rise in price were proportionate to marginal cost and this is most unlikely to be true. See the discussion between Professor Frisch and Professor Hotelling in Econometrica, April, 1939. 32 Compare Wicksteed, The Common Sense of Political Economy, pp. 675-676.
The Marginal Cost Controversy
45
one against the other. The first advantage which the Hotelling-Lerner solution possesses as compared with average cost pricing is that it allows a better choice at the margin in consumption. But this advantage would be reduced and might be offset by the loss which would result if the Hotelling-Lerner solution involved increased income taxes. The second advantage is that a Government could undertake production in cases in which consumers would be willing to pay the total cost but which could not be undertaken with average cost pricing. But it has to be remembered that this policy is one in which the Government estimates individual demands and is therefore subject to the limitations which we discussed in the previous section. Not all cases in which production would not be undertaken with average cost pricing ought to be undertaken. A Government which made many errors in its estimates of individual demands could easily offset any good such a policy might produce. Average cost pricing may prevent some things from being done which perhaps ought to be done but it is also a means of avoiding certain errors in production, some of which would inevitably be made if the Hotelling-Lerner policy were followed. As I indicated earlier, I do not myself believe that it is reasonable to assume that the Government could make accurate estimates of individual demands if all prices were based on marginal cost. Finally, there is the redistribution of income and wealth which the Hotelling-Lerner solution would involve and which, as I pointed out in the previous section, would appear to be difficult to rectify in the absence of multi-part pricing, without reintroducing the kind of tax which would prevent that rational choice at the margin which the Hotelling-Lerner solution aims to achieve. It will be seen from the discussion in this section that the question of average cost pricing as against the Hotelling-Lerner solution does not present any clear-cut case. The claim which is made for the Hotelling-Lerner solution as inevitably superior to average cost pricing must therefore be rejected. 7.
The Problems that Remain
In this article, I have been examining the problem of pricing under conditions of decreasing average costs. I have, however, confined myself
46
R. H. Coase
to one particular case, that in which all costs are attributable to individual consumers and in which all costs are currently incurred. Given these assumptions, I showed that the Hotelling-Lerner solution was inferior to a multi-part system of prices and that as compared with average cost pricing the balance of advantage was not clear. The next steps would appear to be to examine the problem of pricing when there are common costs. If there are costs which cannot be attributed to individual consumers, does the Hotelling-Lerner solution then come into its own, as Mr. H. F. Havlik has suggested?33 Should such common costs be borne out of taxation? Or is the right approach to discover some basis in accordance with which these costs should be allocated among consumers? Finally, there is the question of expenditures which have already been incurred for factors. Are these costs to be borne out of taxation? Or should they be borne by consumers? If the analysis in this article is accepted, these would seem to be the next questions to be examined.
See footnote 17.
CHAPTER 3 ECONOMICS AND BIOLOGY: SPECIALIZATION AND SPECIATION*
Hendrik S. Houthakker* Stanford
University
It is well known that Charles Darwin's work on evolution, according to his own statement, was partly inspired by Malthus' theory of population. To this extent economics may therefore count itself among the sources of modern biology. Apart from this initial link, however, economics has had less contact with biology than with almost any other major science. With the physical sciences, particularly classical mechanics and thermodynamics, economics at least has some conscious affinity of method, and with the social sciences it shares the subject matter (though little else), but with biology it appears to have nothing in common. It would be presumptuous for an economist to argue that closer relations between economics and biology would benefit the other field. All I want to point out here is that economists may derive some useful insight from observation of the non-human living world. This is particularly true for that much-neglected but centrally important chapter of economics: the division of labor, or specialization as it may be more appropriately called in the present context. Specialization, as we shall see, is closely, connected with what biologists call speciation, or the formation of species.
Reprinted from Kyklos, 9 (2), Hendrik S. Houthakker, "Economics and Biology: Specialization and Speciation," 181-189, 1956, with permission from Blackwell. * I am indebted to my colleagues, Melvin Reder and Tibor Scitovsky for useful comments.
47
48
H. S. Houthakker
Adam Smith, in his unsurpassed discussion, declared the division of labor to be peculiar to human societies. He viewed it as the result of a mysterious "propensity to truck," from which animals are somehow immune. We do not, he observed, see two dogs make a fair and reasonable exchange of one bone for another. Related arguments to this effect are also supported by canine examples. It seems that Smith's interest in the animal world did not go beyond dogs; otherwise he might have thought of ants and bees. Even in his day something like division of labor was known to exist in insect societies, and Linnaeus had already described aphids as ants' cows. Smith was carefully ambiguous about the question whether the propensity to truck can itself be reduced to more immediately rational considerations, but he did point out the advantages of following that propensity. These advantages are the increase in skill, the saving of time otherwise necessary to switch from one job to another, and the enhanced possibilities of using tools and machinery. Without much explanation he also indicated how far specialization can go in the famous statement that "the division of labor is limited by the extent of the market." It is not to the credit of economists that in the 180 years following the publication of the Wealth of Nations so little should have been done to clarify this statement, the simplicity of which is quite deceptive. Most economists have probably regarded the division of labor, in Schumpeter's words, as an "eternal commonplace," yet there is hardly any part of economics that would not be advanced by a further analysis of specialization and related phenomena. It should be added, however, that such an analysis involves the use of methods that are rather unlike those by which the classical questions of economics are discussed. These classical questions are treated with the aid of traditional calculus methods (often disguised in literary form) but the latter are not suited to deal with indivisibilities. It is in fact from indivisibilities that the division of labor takes its start, and the basic indivisibility is that of the individual, whether human or animal. This may seem like a play on words, or what is almost the same thing, bad metaphysics, but it is more serious than that. For our purpose we may regard an individual as a coordinated complex of activities. The indivisibility of the individual consists in the fact that, although it may be capable of a great many different activities, it can perform only few activities simultaneously because most activities utilize the same resources and more particularly that coordinating resource which is
Economics and Biology: Specialization and Speciation
49
known as the brain. The larger the number of simultaneous activities, the greater the difficulty of coordinating them and of carrying out each one properly, and the smaller therefore the output from each activity. This applies not only to simultaneous activities, but also to activities that are spread out over time. In the first place some shorter or longer interval is usually needed to switch from one activity to another; in the second place it is usually easier to perform activities that are known from previous experience than to perform them for the first time. All this, the economist will note at once, can be put in terms of increasing returns. We have increasing returns to the extent that, if several activities are replaced by a single one, there is less need for coordination and switching time and more scope for acquiring experience. The output of the single activity may thus be raised above the combined outputs of the several activities. If we have two activities, for instance, the simplest shape of the production possibility curve exhibiting these features will be as in Figure 1. If the individual produces only the first commodity he can obtain AP units, and if only the second AQ units, but if he produces both simultaneously he is bounded by the straight line P'Q' The segments PP or QQ' therefore represent the loss of production due to the need for coordination, etc. If the individual cannot trade with others he will normally be unable to specialize, at least if both commodities are necessary to him. He will then be at a point such as R, where an indifference curve touches the production possibility curve.
Figure 1
50
H. S. Houthakker
As soon as another individual appears on the scene specialization may arise (Figure 2). The individual may then get to a point such as V, whose exact location depends on the respective offer curves (cases of incomplete specialization are also possible). The post-trade point V is better than either of the pretrade points R and U, and both parties therefore gain, but it is possible that only one party gains and the other remains where he was. The case here described differs from the Ricardian case of constant cost in that there will be a tendency to specialization even if there are no differences in relative efficiency (that is, even if P'Q' and S'T are parallel). Specialization may even go against the comparative advantage (that is,P'g'may have a flatter slope than S T ' ) . Under the Ricardian assumptions the benefits of specialization consist wholly in the utilization of comparative advantages, but in the present case, which is essentially that of Adam Smith, the benefit comes mainly from the avoidance of coordination costs (in the widest sense) by the two individuals. It may be noted that Smith went out of his way to deny the existence of innate differences between individuals; in his theory such differences were unnecessary. Like the Declaration of Independence, which dates from the same year 1776, the Wealth of Nations is based on the premise that all men are born equal.
Figure 2
Although specialization will lead to the avoidance of individual coordination costs, it may in turn call for coordination between the two individuals. The simplest example is transportation cost. These external
Economics and Biology: Specialization and Speciation
51
coordination costs may be heavy enough to outweigh the saving in internal coordination costs. In a box diagram such as Figure 2, transportation costs may be represented by shifting all points referring to the second individual to the left, thus narrowing the area between P'Q' and S'T' within which specialization is advantageous. The division of labor is thus limited by the extent of the market. Let us consider a slightly different example of specialization, which has attracted the attention of students of spatial competition such as Hotelling1 and Loesch.2 Imagine a population which is spread out, not necessarily evenly, along a road and which consumes a single commodity. This good is sold by firms who deliver it at the customer's door. All firms have the same cost function, consisting of a part proportional to their sales, a part proportional to the aggregate distance over which they have to deliver, and a part that is constant. There is free entry, so that no firm can make a profit over its total cost. Consumers buy in the cheapest market, but otherwise their demand is inelastic. A complete solution of this problem would have to specify the location of each firm and the prices it charges to its various customers. For our purpose we need only mention some general characteristics of the solution. The area served by each firm has to be just large enough to provide a margin of revenue over variable costs equal to the fixed costs. At the boundary between the territories of two firms both must charge the same price, since otherwise customers near the boundary would switch to the cheaper firm. This price at the boundary has to be equal to variable cost (the cost of the commodity plus transportation cost from the firm's location), without any allowance for fixed costs. The latter have to be recouped from the customers inside the territory, and the prices charged to them will be the highest that will prevent new firms from establishing themselves at the boundaries between existing firms. In the case of a uniform distribution of demand, for instance, the price charged will be the same everywhere. What matters to us at the moment is not the level of prices, but rather the fact that the above conditions determine the number of firms, though no explicit general formula for that number has as yet been derived. It is clear, however, that the number of firms depends critically on the ratio 1
H. Hotelling, "Stability in Competition," Economic Journal, 1929, p. 41. A. Loesch, Die raumliche Ordnung der Wirtschaft, Jena 1940 (English transl. The Economics of Location, New Haven, Conn., 1954). 2
52
H. S. Houthakker
between the fixed costs and the transportation costs, or more generally, between the external and the internal coordination costs. Another example may serve to show that the above case is of wider importance than may appear at first sight. People's feet are of different sizes, and shoes are the more comfortable the more their size corresponds to the size of the foot. On the other hand, the unit cost of a pair of shoes of a given size decreases with the number of pairs of that size that is produced, because it is expensive to switch from one size to another. Hence people may find it preferable to wear cheap shoes that do not fit their feet exactly rather than expensive shoes made to their measure. Under free entry (or even under monopoly) there will again be a definite number of sizes that is actually produced. In this case the transportation cost from the previous example has its parallel in the reduction of price consumers are willing to pay for badly fitting shoes. By now the listener may well wonder what all this has to do with biology. The connection is not really very remote. One of the most striking phenomena of the living world is its organization into species, of which there are hundreds of thousands, or even millions, each of them with a greatly varying number of members. Though vastly more complicated, this pattern has some analogy to that of the firms along a road, or of the discretely varying shoe sizes. Each species may be compared to one size of shoe, and its members to the shoes produced of that size. The advantages and disadvantages of the biological pattern can therefore not be too dissimilar from those of the economic pattern. We may, for instance, consider each species as adapted to one particular range of foods (though this is by no means the whole story). The wider the range of foods, the more complicated the anatomical and physiological arrangements necessary to obtain and digest each type of food efficiently; or in the above terminology, the higher the internal coordination cost. On the other hand, a wider range of foods makes possible a larger number of members of the species, because the danger of starvation is smaller. For each species there will consequently be an optimum range and an optimum number of individuals, as there was in the case of shoes. In reality this balance between range and number is only one element, though probably a basic one, in the structure of advantages and disadvantages. It may of course be asked: advantages and disadvantages to whom? When it comes to the explanation of the characteristics of separate species biologists tend to use the word "advantage" quite freely, and
Economics and Biology: Specialization and Speciation
53
usually in the sense of something which promotes survival. This may easily lead to paradoxical results; thus R. A. Fisher in his Genetical Theory of Natural Selection3 maintains that evolution leads to an increase in fitness, which he defines as the chance of survival, from which it follows, oddly enough, that a species has the best chance of survival just before it becomes extinct. If we consider not separate species, but the whole animal world, the notion of advantage becomes even more tenuous. Since the purpose of nature is unknown, and it may indeed be meaningless to ask if there is a purpose, teleological explanations may appear to be ruled out. This, it is often held, contributes the dividing line between the social and the biological sciences. It appears, however, that this view overestimates the purposefulness of human behavior. It is true that the rational man has long been a favorite with economists, but recently his standing has somewhat declined. The development of the theory of choice, for instance, has cast doubt on many of the qualities with which the rational man was traditionally endowed. The realization of the importance of uncertainty has further undermined the concept of rationality, for which no satisfactory definition appropriate to uncertainty has yet been found. Moreover, it is sometimes argued that economic activities are primarily distinguishable by the manner in, rather than by the motive with which they are carried out. Although nature may have no discernible motives, it may nevertheless operate in the same way as if it had a motive. Since there are therefore no major objections, what is the advantage of extending economic analysis to biological phenomena? There is, I think, one important advantage. In economics we also observe species, such as different commodities, occupations, etc., but the boundaries between them are often vague for lack of a precise criterion of classification. Consequently any attempt to count the number of "individuals" in a "species" would meet with considerable conceptual difficulties. In biology, however, there is a more definite criterion, even though it is not absolutely precise. Species are defined by the criterion of interfertility, the ability to produce fertile offspring. The species of biology are therefore much more clearly defined than those of economics, and this is their principal advantage as an object of research.
3
Oxford, 1930.
54
H. S. Houthakker
In this lecture I have indulged in rather wild speculation, but I hope I need not apologize for it. Provided it is administered in small doses and counterbalanced by large quantities of more solid inquiry, speculation is healthy for any science even though most of its results turn out to be wrong.
CHAPTER 4 EXTERNALITY*
James M. Buchanan 3 and Wm. Craig Stubblebine b "George Mason University
Claremont McKenna
College
Externality has been, and is, central to the neo-classical critique of market organisation. In its various forms — external economies and diseconomies, divergencies between marginal social and marginal private cost or product, spillover and neighbourhood effects, collective or public goods — externality dominates theoretical welfare economics, and, in one sense, the theory of economic policy generally. Despite this importance and emphasis, rigorous definitions of the concept itself are not readily available in the literature. As Scitovosky has noted, "definitions of external economies are few and unsatisfactory".1 The following seems typical: External effects exist in consumption whenever the shape or position of a man's indifference curve depends on the consumption of other men. [External effects] are present whenever a firm's production function depends in some way on the amounts of the inputs or outputs of another firm.2 It seems clear that operational and usable definitions are required.
Reprinted from Economica, 29, James M. Buchanan and Wm. Craig Stubblebine, "Externality," 371-384, 1962, with permission from Blackwell. 1 Tibor Scitovosky, "Two Concepts of External Economies", Journal of Political Economy, vol. LXII (1954), p. 143. 2 J. de V. Graaf, Theoretical Welfare Economics, Cambridge, 1957, p. 43 and p. 18.
55
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J. M. Buchanan, W. C. Stubblebine
In this paper, we propose to clarify the notion of externality by defining it rigorously and precisely. When this is done, several important, and often overlooked, conceptual distinctions follow more or less automatically. Specifically, we shall distinguish marginal and inframarginal externalities, potentially relevant and irrelevant externalities, and Pareto-relevant and Pareto-irrelevant externalities. These distinctions are formally developed in section 1. As we shall demonstrate, the term, "externality", as generally used by economists, corresponds only to our definition of Pareto-relevant externality. There follows, in section 2, an illustration of the basic points described in terms of a simple descriptive example. In section 3, some of the implications of our approach are discussed. It is useful to limit the scope of the analysis at the outset. Much of the discussion in the literature has been concerned with the distinction between technological and pecuniary external effects. We do not propose to enter this discussion since it is not relevant for our purposes. We note only that, if desired, the whole analysis can be taken to apply only to technological externalities. Secondly, we shall find no cause for discussing production and consumption externalities separately. Essentially the same analysis applies in either case. In what follows, "firms" may be substituted for "individuals" and "production functions" for "utility functions" without modifying the central conclusions. For expositional simplicity only, we limit the explicit discussion to consumption externalities. 1. We define an external effect, an externality, to be present when, u*=u*(XxtX2,...,Xm,YJ.
(1)
This states that the utility of an individual, A, is dependent upon the "activities", (Xl,X2,....,Xm), that are exclusively under his own control or authority, but also upon another single activity, Yx , which is, by definition, under the control of a second individual, B, who is presumed to be a member of the same social group. We define an activity here as any distinguishable human action that may be measured, such as eating bread,
Externality
57
drinking milk, spewing smoke into the air, dumping litter on the highways, giving to the poor, etc. Note that A's utility may, and will in the normal case, depend on other activities of B in addition to Y\, and also upon the activities of other parties. That is, A's utility function may, in more general terms, include such variables as (Y2,Y3,....,Ym; Z,,Z 2 ,....,Z m ). For analytical simplicity, however, we shall confine our attention to the effects of one particular activity, Yu as it affects the utility ofA. We assume that A will behave so as to maximise utility in the ordinary way, subject to the externally determined values for Yh and that he will modify the values for the Xs, as Yx changes, so as to maintain a state of "equilibrium". A marginal externality exists when, «jj*0.
(2)
Here, small it's are employed to represent the "partial derivatives" of the utility function of the individual designated by the super-script with respect to the variables designated by the subscript. Hence, u§ = duA/8Y] , assuming that the variation in Y\ is evaluated with respect to a set of "equilibrium" values for the Xs, adjusted to the given value for Yx. An infra-marginal externality holds at those points where, «$=0,
(3)
and(l) holds. These classifications can be broken down into economies and diseconomies: a marginal external economy existing when, u* > 0 ,
(2A)
that is, a small change in the activity undertaken by B will change the utility of A in the same direction; a marginal external diseconomy existing when, u*<0.
(2B)
An infra-marginal external economy exists when for any given set of values for (Xl,X2,....,XJ,say,(Cl,C2,...,Cm),
58
J. M. Buchanan, W. C. Stubblebine
u*=0, and \u§dY^ >0. (3A) o This condition states that, while incremental changes in the extent of 5's activity, Yh have no affect on A's utility, the total effect of 5's action has increased A's utility. An infra-marginal diseconomy exists when (1) holds, and, for any given set of values for (Xx,X2,...,Xm), say, (C,,C2,...,Cm), then, u* = 0 , a n d \u*dYi < 0 .
(3B)
o
Thus, small changes inS's activity do not changer's level of satisfaction, but the total effect of 5's undertaking the activity in question is harmful to A. We are able to classify the effects of B's action, or potential action, on A's utility by evaluating the "partial derivative" ofA's utility function with respect to Y\ over all possible values for Y\. In order to introduce the further distinctions between relevant and irrelevant externalities, however, it is necessary to go beyond consideration of A's utility function. Whether or not a relevant externality exists depends upon the extent to which the activity involving the externality is carried out by the person empowered to take action, to make decisions. Since we wish to consider a single externality in isolation, we shall assume that B's utility function includes only variables (activities) that are within his control, including Y\. Hence, B's utility function takes the form, u»=u*(Xx,Y2,...,Ym).
(4)
Necessary conditions for utility maximisation by B are,
KK=f*lfrBt>
&
where 1} is used to designate the activity of B in consuming or utilising some numeraire commodity or service which is, by hypothesis, available on equal terms to A. The right-hand term represents the marginal rate of substitution in "production" or "exchange" confronted by B, the party taking action on Yu his production function being defined as,
/*=/*(^,y2,...,yj,
(6)
Externality
59
where inputs are included as activities along with outputs. In other words, the right-hand term represents the marginal cost of the activity, Yu to B. The equilibrium values for the Yi 's will be designated as Yt 's. An externality is defined as potentially relevant when the activity, to the extent that it is actually performed, generates any desire on the part of the externally benefited (damaged) party (A) to modify the behaviour of the party empowered to take action (B) through trade, persuasion, compromise, agreement, convention, collective action, etc. An externality which, to the extent that it is performed, exerts no such influence is defined as irrelevant. Note that, so long as (1) holds, an externality remains; utility functions remain interdependent. A potentially relevant marginal externality exists when,
*0.
uA
(7)
This is a potentially relevant marginal external economy when (7) is greater than zero, a diseconomy when (7) is less than zero. In either case, A is motivated, by 5's performance of the activity, to make some effort to modify this performance, to increase the resources devoted to the activity when (7) is positive, to decrease the quantity of resources devoted to the activity when (7) is negative. Infra-marginal externalities are, by definition, irrelevant for small changes in the scope of B's activity, Y\. However, when large or discrete changes are considered, A is motivated to change S's behaviour with respect to Y\ in all cases except that for which, u*\ _ = 0 , a n d
(8) K
M\y-V
7
u*(C1,C2,...,Cmjl)Zu*(Cl,C2,...,Cm,Yl)JoT&\l Y^%. When (8) holds, A has achieved an absolute maximum of utility with respect to changes over Y\, given any set of values for the X's. In more prosaic terms, A is satiated with respect to Y\3. In all other cases, where infra-marginal external economies or diseconomies exist, A will have 3
Note that, ui\
_ = 0 , is a necessary, but not a sufficient, condition for irrelevance.
60
J. M. Buchanan, W. C. Stubblebine
some desire to modify 5's performance; the externality is potentially relevant. Whether or not this motivation will lead A to seek an expansion or contraction in the extent of 5's performance of the activity will depend on the location of the infra-marginal region relative to the absolute maximum for any given values of the JTs.4 Pareto relevance and irrelevance may now be introduced. The existence of a simple desire to modify the behaviour of another, defined as potential relevance, need not imply the ability to implement this desire. An externality is defined to be Pareto-relevant when the extent of the activity may be modified in such a way that the externally affected party, A, can be made better off without the acting party, B, being made worse off. That is to say, "gains from trade" characterise the Pareto-relevant externality, trade that takes the form of some change in the activity of B as his part of the bargain. A marginal externality is Pareto-relevant when5 (-)«£ /uXj > [ « * / « * - f*m
KK
>(r)[KK
]
~frJfrB\=?^
_ and when u{ /uXj < 0, and
(9)
u*/uXj >0.
In (9), Xj and Yj are used to designate, respectively, the activities of A and B in consuming or in utilising some numeraire commodity or service that, by hypothesis, is available on identical terms to each of them. As is indicated by the transposition of signs in (9), the conditions for Pareto relevance differ as between external diseconomies and economies. This is because the "direction" of change desired by A on the part of B is different in the two cases. In stating the conditions for Pareto relevance under ordinary two-person trade, this point is of no significance since trade in one good flows only in one direction. Hence, absolute values can be used.
4
In this analysis of the relevance of externalities, we have assumed that B will act in such a manner as to maximise his own utility subject to the constraints within which he must operate. If, for any reason, B does not attain the equilibrium position defined in (5) above, the classification of his activity for A may, of course, be modified. A potentially relevant externality may become irrelevant and vice versa. 5 We are indebted to Mr. M. McManus of the University of Birmingham for pointing out to us an error in an earlier formulation of this and the following similar conditions.
61
Externality
The condition, (9), states that^'s marginal rate of substitution between the activity, Yu and the numeraire activity must be greater than the "net" marginal rate of substitution between the activity and the numeraire activity for B. Otherwise, "gains from trade" would not exist between A and B. Note, however, that when B has achieved utility-maximising equilibrium, U»/!*%=/*/f°.
(10)
That is to say, the marginal rate of substitution in consumption or utilisation is equated to the marginal rate of substitution in production or exchange, i.e., to marginal cost. When (10) holds, the terms in the brackets in (9) mutually cancel. Thus, potentially relevant marginal externalities are also Pareto-relevant when B is in utility-maximising equilibrium. Some trade is possible. Pareto equilibrium is defined to be present when, (-)u$Juj
= W*/H£ - fyj fr. \> and when u^luj, < 0 , and (11)
KK =(-)[%1/M" -frjfy^\
when U
ZK
>0
-
Condition (11) demonstrates that marginal externalities may continue to exist, even in Pareto equilibrium, as here defined. This point may be shown by reference to the special case in which the activity in question may be undertaken at zero costs. Here Pareto equilibrium is attained when the marginal rates of substitution in consumption or utilisation for the two persons are precisely offsetting, that is, where their interests are strictly opposed, and not where the left-hand term vanishes. What vanishes in Pareto equilibrium are the Pareto-relevant externalities. It seems clear that, normally, economists have been referring only to what we have here called Pareto-relevant externalities when they have, implicitly or explicitly, stated that external effects are not present when a position on the Pareto optimality surface is attained.6
6
This applies to the authors of this paper. For recent discussion of external effects when we have clearly intended only what we here designate as Pareto-relevant, see James M. Buchanan, "Politics, Policy, and the Pigovian Margins", Economica, vol. XXVIX (1962),
62
J. M. Buchanan, W. C. Stubblebine
For completeness, we must also consider those potentially relevant infra-marginal externalities. Refer to the discussion of these as summarised in (8) above. The question is now to determine whether or not, A, the externally affected party, can reach some mutually satisfactory agreement with B, the acting party, that will involve some discrete (non-marginal) change in the scope of the activity, Y\. If, over some range, any range, of the activity, which we shall designate by AYX, the rate of substitution between Y\ and Xj for A exceeds the "net" rate of substitution for B, the externality is Pareto-relevant. The associated changes in the utilisation of the numeraire commodity must be equal for the two parties. Thus, for external economies, we have AuA
I AuA /
AY, J AXj
, N AuB lAuB > (-) AY / AYj X
AfB lAfB AY, / AYj
(12)
and the same with the sign in parenthesis transposed for external diseconomies. The difference to be noted between (12) and (9) is that, with infra-marginal externalities, potential relevance need not imply Pareto relevance. The bracketed terms in (12) need not sum to zero when B is in his private utility-maximising equilibrium. We have remained in a two-person world, with one person affected by the single activity of a second. However, the analysis can readily be modified to incorporate the effects of this activity on a multi-person group. That is to say, 5's activity, Yu may be allowed to affect several parties simultaneously, several's, so to speak. In each case, the activity can then be evaluated in terms of its effects on the utility of each person. Nothing in the construction need be changed. The only stage in the analysis requiring modification explicitly to take account of the possibilities of multi-person groups being externally affected is that which involves the condition for Pareto relevance and Pareto equilibrium. For a multi-person group i
pp. 17-28, and, also, James M. Buchanan and Gordon Tullock, The Calculus of Consent, Ann Arbor, 1962.
Externality
( - ) Z < / " £ >[ M */ W * - / y f / / ^ l 1=1
63
- when w | / w £ < 0, and, (9A) 1_
Z < K >(-)[«?/"? -fi/tf]r=y
]
when
0 -
That is, the summed marginal rates of substitution over the members of the externally affected group exceed the offsetting "net" marginal evaluation of the activity by B. Again, in private equilibrium for B, marginal externalities are Pareto-relevant, provided that we neglect the important element involved in the costs of organising group decisions. In the real world, these costs of organising group decisions (together with uncertainty and ignorance) will prevent realisation of some "gains from trade" — just as they do in organised markets. This is as true for two-person groups as it is for larger groups. But this does not invalidate the point that potential "gains from trade" are available. The condition for Pareto equilibrium and for the infra-marginal case summarised in (11) and (12) for the two-person model can readily be modified to allow for the externally affected multi-person group.
The distinctions developed formally in section 1 may be illustrated diagrammatically and discussed in terms of a simple descriptive example. Consider two persons, A and B, who own adjoining units of residential property. Within limits to be noted, each person values privacy, which may be measured quantitatively in terms of a single criterion, the height of a fence that can be constructed along the common boundary line. We shall assume that 5's desire for privacy holds over rather wide limits. His utility increases with the height of the fence up to a reasonably high level. Up to a certain minimum height, A's utility also is increased as the fence is made higher. Once this minimum height is attained, however, A's desire for privacy is assumed to be fully satiated. Thus, over a second range, ^4's total utility does not change with a change in the height of the fence. However, beyond a certain limit, A's view of a mountain behind B's property is progressively obscured as the fence goes higher. Over this third range, therefore, A's utility is reduced as the fence is constructed to
64
J. M. Buchanan, W. C. Stubblebine
higher levels. Finally, A will once again become wholly indifferent to marginal changes in the fence's height when his view is totally blocked out. We specify that B possesses the sole authority, the only legal right, to construct the fence between the two properties. The preference patterns for .4 and for B are shown in Figure 1, which is drawn in the form of an Edgeworth-like box diagram. Note, however, that the origin for B is shown at the upper left rather than the upper right corner of the diagram as in the more normal usage. This modification is necessary here because only the numeraire good, measured along the ordinate, is strictly divisible between A and B. Both must adjust to the same height of fence, that is, to the same level of the activity creating the externality. As described above, the indifference contours for A take the general shape shown by the curves aa, a'a', while those for B assume the shapes, bb, b'b'. Note that these contours reflect the relative evaluations, for A and B, between money and the activity,Yx. Since the costs of undertaking the activity, for B, are not incorporated in the diagram, the "contract locus" that might be derived from tangency points will have little relevance except in the special case where the activity can be undertaken at zero costs. Figure 2 depicts the marginal evaluation curves for A and B, as derived from the preference fields shown in Figure 1, along with some incorporation of costs. These curves are derived as follows: Assume an initial distribution of "money" between A and B, say, that shown at Mon Figure 1. The marginal evaluation of the activity for A is then derived by plotting the negatives (i.e., the mirror image) of the slopes of successive indifference curves attained by A as B is assumed to increase the height of the fence from zero. These values remain positive for a range, become zero over a second range, become negative for a third, and, finally, return to zero again.7
7
For an early use of marginal evaluation curves, see J. R. Hicks, "The Four Consumer's Surpluses", Review of Economic Studies, vol. XI (1943), pp. 31-41.
65
Externality
V M*
H,
H,
H,
H<
Height of Fence (B' s Activity)
Figure 1
5's curves of marginal evaluation are measured downward from the upper horizontal axis or base line, for reasons that will become apparent. The derivation of 2?'s marginal evaluation curve is somewhat more complex than that for A. This is because B, who is the person authorised to undertake the action, in this case the building of the fence, must also bear the full costs. Thus, as B increases the scope of the activity, his real income, measured in terms of his remaining goods and services, is reduced. This change in the amount of remaining goods and services will, of course, affect his marginal evaluation of the activity in question. Thus, the marginal cost of building the fence will determine, to some degree, the marginal evaluation of the fence. This necessary interdependence between marginal evaluation and marginal cost complicates the use of simple diagrammatic models in finding or locating a solution. It need not,
66
J. M. Buchanan, W. C. Stubblebine
ME,
\
ME, NMEA=MSCB NMEB= MSC„ MC Figure 2
however, deter us from presenting the solution diagrammatically, if we postulate that the marginal evaluation curve, as drawn, is based on a single presumed cost relationship. This done, we may plot 5's marginal evaluation of the activity from the negatives of the slopes of his indifference contours attained as he constructs the fence to higher and higher levels. 5's marginal evaluation, shown in Figure 2, remains positive throughout the range to the point H5, where it becomes zero. The distinctions noted in section 1 are easily related to the construction in Figure 2. To A, the party externally affected, B's potential activity in constructing the fence can be assessed independently of any prediction of 5's actual behaviour. Thus, the activity of B would, (1) exert marginal external economies which are potentially relevant over the range OHi,
Externality
67
(2) exert infra-marginal external economies over the range HiH2, which are clearly irrelevant since no change in 5's behaviour with respect to the extent of the activity would increase A's utility; (3) exert marginal external diseconomies over the range H2Ha, which are potentially relevant to A; and, (4) exert infra-marginal external economies or diseconomies beyond H4, the direction of the effect being dependent on the ratio between the total utility derived from privacy and the total reduction in utility derived from the obstructed view. In any case, the externality is potentially relevant. To determine Pareto relevance, the extent of S's predicted performance must be determined. The necessary condition for 5's attainment of "private" utility-maximising equilibrium is that marginal costs, which he must incur, be equal to his own marginal evaluation. For simplicity in Figure 2, we assume that marginal costs are constant, as shown by the curve, MC. Thus, S's position of equilibrium is shown at// B , within the range of marginal external diseconomies for A. Here the externality imposed by B's behaviour is clearly Pareto-relevant: A can surely work out some means of compensating B in exchange for B's agreement to reduce the scope of the activity—in this example, to reduce the height of the fence between the two properties. Diagrammatically, the position of Pareto equilibrium is shown at H^ where the marginal evaluation of A is equal in absolute value, but negatively, to the "net" marginal evaluation of B, drawn as the curve NMEB. Only in this position are the conditions specified in (11), above, satisfied.8
This diagrammatic analysis is necessarily oversimplified in the sense that the Pareto equilibrium position is represented as a unique point. Over the range between the "private" equilibrium for B and the point of Pareto equilibrium, the sort of bargains struck between A and B will affect the marginal evaluation curves of both individuals within this range. Thus, the more accurate analysis would suggest a "contract locus" of equilibrium points. At Pareto equilibrium, however, the condition shown in the diagrammatic presentation holds, and the demonstration of this fact rather than the location of the solution is the aim of this diagrammatics.
68
J. M. Buchanan, W. C. Stubblebine
3. Aside from the general classification of externalities that is developed, the approach here allows certain implications to be drawn, implications that have not, perhaps, been sufficiently recognised by some welfare economists. The analysis makes it quite clear that externalities, external effects, may remain even in full Pareto equilibrium. That is to say, a position may be classified as Pareto-optimal or efficient despite the fact that, at the marginal, the activity of one individual externally affects the utility of another individual. Figure 2 demonstrates this point clearly. Pareto equilibrium is attained at H3, yet B is imposing marginal external diseconomies on A. This point has significant policy implications for it suggests that the observation of external effects, taken alone, cannot provide a basis for judgment concerning the desirability of some modification in an existing state of affairs. There is not a prima facie case for intervention in all cases where an externality is observed to exist.9 The internal benefits from carrying out the activity, net of costs, may be greater than the external damage that is imposed on other parties. In full Pareto equilibrium, of course, these internal benefits, measured in terms of some numeraire good, net of costs, must be just equal, at the margin, to the external damage that is imposed on other parties. This equilibrium will always be characterised by the strict opposition of interests of the two parties, one of which may be a multi-person group. In the general case, we may say that, at full Pareto equilibrium, the presence of a marginal external diseconomy implies an offsetting marginal internal economy, whereas the presence of a marginal external economy implies an offsetting marginal internal diseconomy. In "private" equilibrium, as opposed to Pareto equilibrium, these net internal economies and diseconomies would, of course, be eliminated by the utility-maximising acting party. In Pareto equilibrium, these remain because the acting party is being compensated for "suffering" internal
9 Cf. Paul A. Samuelson, Foundations of Economic Analysis, Cambridge, Mass., 1948, p. 208, for a discussion of the views of various writers.
Externality
69
economies and diseconomies, that is, divergencies between "private" marginal costs and benefits, measured in the absence of compensation. As a second point, it is useful to relate the whole analysis here to the more familiar Pigovian discussion concerning the divergence between marginal social cost (product) and marginal private cost (product). By saying that such a divergence exists, we are, in the terms of this paper, saying that a marginal externality exists. The Pigovian terminology tends to be misleading, however, in that it deals with the acting party to the exclusion of the externally affected party. It fails to take into account the fact that there are always two parties involved in a single externality relationship. 10 As we have suggested, a marginal externality is Pareto-relevant except in the position of Pareto equilibrium; gains from trade can arise. But there must be two parties to any trading arrangement. The externally affected party must compensate the acting party for modifying his behaviour. The Pigovian terminology, through its concentration on the decision-making of the acting party alone, tends to obscure the two-sidedness of the bargain that must be made. To illustrate this point, assume that A, the externally affected party in our model, successfully secures, through the auspices of the "state", the levy of a marginal tax on S's performance of the activity, Yx. Assume further that A is able to secure this change without cost to himself. The tax will increase the marginal cost of performing the activity for B, and, hence, will reduce the extent of the activity attained in B's "private" equilibrium. Let us now presume that this marginal tax is levied "correctly" on the basis of a Pigovian calculus; the rate of tax at the margin is made equal to the negative marginal evaluation of the activity to A. Under these modified conditions, the effective marginal cost, as confronted by B, may be shown by the curve designated as MSCB in Figure 2. A new "private" equilibrium for B is shown at the quantity, H3, the same level designated as Pareto equilibrium in our earlier discussion, if we neglect the disturbing interdependence between marginal evaluation and marginal costs. Attention solely to the decision calculus of B here would suggest, perhaps,
This criticism of the Pigovian analysis has recently been developed by R. H. Coase; see his "The Problem of Social Cost", Journal of Law and Economics, vol. Ill (1960), pp. 1-44.
70
J. M. Buchanan, W. C. Stubblebine
that this position remains Pareto-optimal under these revised circumstances, and that it continues to qualify as a position of Pareto equilibrium. There is no divergence between marginal private cost and marginal social cost in the usual sense. However, the position, if attained in this manner, is clearly neither one of Pareto optimality, nor one that may be classified as Pareto equilibrium. In this new "private" equilibrium for B,
u
?Jui=ffJfrBru$K>
<13)
where uYi ux represents the marginal tax imposed on B as he performs the activity, Y\. Recall the necessary condition for Pareto relevance defined in (9) above, which can now be modified to read, (-)u$ juXj > [uBYJuBYj - f»/f*
+ u$JuXj ]
and uY> juAXj > ( - ) [ « * / « * - f*j'f^
+ u^luXj
, when u{ juXj < 0, (9B) _ , when u* lux > 0.
In (9B), Y1 represents the "private" equilibrium value for Yu determined by B, after the ideal Pigovian tax is imposed. As before, the bracketed terms represent the "net" marginal evaluation of the activity for the acting party, B, and these sum to zero when equilibrium is reached. So long as the left-hand term in the inequality remains non-zero, a Pareto-relevant marginal externality remains, despite the fact that the full "Pigovian solution" is attained. The apparent paradox here is not difficult to explain. Since, as postulated, A is not incurring any cost in securing the change in B's behaviour, and, since there remains, by hypothesis, a marginal diseconomy, further "trade" can be worked out between the two parties. Specifically, Pareto equilibrium is reached when,
(r)u$lux
= u§lul - ffl'fl,+uYlJux~\
< / < =(-)[
whenw^/wf <0, and (HA) -f'/f'+ufJu^whenu'/u^X).
Diagrammatically, this point may be made with reference to Figure 2. If a unilaterally imposed tax, corresponding to the marginal evaluation of A, is placed on 5's performance of the activity, the new position of Pareto
Externality
71
equilibrium may be shown by first subtracting the new marginal cost curve, drawn as MSCB, from 5's marginal evaluation curve. Where this new "net" marginal evaluation curve, shown as the dotted curve between points Hi and K, cuts the marginal evaluation curve for A, a new position of Pareto equilibrium falling between H2 and H3 is located, neglecting the qualifying point discussed in Footnote 8. The important implication to be drawn is that full Pareto equilibrium can never be attained via the imposition of unilaterally imposed taxes and subsidies until all marginal externalities are eliminated. If a taxsubsidy method, rather than "trade", is to be introduced, it should involve bi-lateral taxes (subsidies). Not only must B\ behaviour be modified so as to insure that he will take the costs externally imposed on A into account, but A's behaviour must be modified so as to insure that he will take the costs "internally" imposed on B into account. In such a double tax-subsidy scheme, the necessary Pareto conditions would be readily satisfied.11 In summary, Pareto equilibrium in the case of marginal externalities cannot be attained so long as marginal externalities remain, until and unless those benefiting from changes are required to pay some "price" for securing the benefits. A third point worthy of brief note is that our analysis allows the whole treatment of externalities to encompass the consideration of purely collective goods. As students of public finance theory will have recognised, the Pareto equilibrium solution discussed in this paper is similar, indeed is identical, with that which was presented by Paul Samuelson in his theory of public expenditures.12 The summed marginal rates of substitution (marginal evaluation) must be equal to marginal costs. Note, however, that marginal costs may include the negative marginal evaluation of other parties, if viewed in one way. Note, also, that there is nothing in the analysis which suggests its limitations to purely collective goods or even to goods that are characterised by significant externalities in their use. 11
Although developed in rather different terminology, this seems to be closely in accord with Coase's analysis. Cf. R. H. Coase, loc. cit. 12 Paul A. Samuelson, "The Pure Theory of Public Expenditure", Review of Economics and Statistics, vol. XXXVI (1954), pp. 386-9.
72
J. M. Buchanan, W. C. Stubblebine
Our analysis also lends itself to the more explicit point developed in Coase's recent paper.13 He argues that the same "solution" will tend to emerge out of any externality relationship, regardless of the structure of property rights, provided only that the market process works smoothly. Strictly speaking, Coase's analysis is applicable only to inter-firm externality relationships, and the identical solution emerges only because firms adjust to prices that are competitively determined. In our terms of reference, this identity of solution cannot apply because of the incomparability of utility functions. It remains true, however, that the basic characteristics of the Pareto equilibrium position remain unchanged regardless of the authority undertaking the action. This point can be readily demonstrated, again with reference to Figure 2. Let us assume that Figure 2 is now redrawn on the basis of a different legal relationship in which A now possesses full authority to construct the fence, whereas B can no longer take any action in this respect. A will, under these conditions, "privately" construct a fence only to the height H0, where the activity clearly exerts a Pareto-relevant marginal external economy on B. Pareto equilibrium will be reached, as before, at H3 determined, in this case, by the intersection of the "net" marginal evaluation curve for A (which is identical to the previously defined marginal social cost curve, MSC, when B is the acting party) and the marginal evaluation curve for B.u Note that, in this model, A will allow himself to suffer an internal marginal diseconomy, at equilibrium, provided that he is compensated by B, who continues, in Pareto equilibrium, to enjoy a marginal external economy. Throughout this paper, we have deliberately chosen to introduce and to discuss only a single externality. Much of the confusion in the literature seems to have arisen because two or more externalities have been handled simultaneously. The standard example is that in which the output of one firm affects the production function of the second firm while, at the same time, the output of the second firm affects the production function of the first. Clearly, there are two externalities to be analysed in such cases. In R. H. Coase, loc. cit. The H3 position, in this presumably redrawn figure, should not be precisely compared with the same position in the other model. We are using here the same diagram for two models, and, especially over wide ranges, the dependence of the marginal evaluation curves on income effects cannot be left out of account. 14
Externality
73
many instances, these can be treated as separate and handled independently. In other situations, this step cannot be taken and additional tools become necessary.15
15 For a treatment of the dual externality problem that clearly shows the important difference between the separable and the non-separable cases, see Otto Davis and Andrew Whinston, "Externalities, Welfare, and the Theory of Games", Journal of Political Economy, vol. LXX (1962), pp. 241-62. As the title suggests, Davis and Whinston utilise the tools of game theory for the inseparable case.
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Part 3
Development Strategies, Income Distribution, and Dual Structures
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CHAPTER 5 ECONOMIC DEVELOPMENT, INTERNATIONAL TRADE, AND INCOME DISTRIBUTION*
Xiaokai Yang" and Dingsheng Zhang b * "Monash University
b
Monash and Wuhan
University
1. Introduction The purpose of this paper is twofold. First, we develop a general equilibrium model with endogenous specialization to show that the relationship between economic development and inequality of income distribution is neither monotonic nor an inverted U-curve. Second, we use the general equilibrium model to investigate the interdependence between the extent of the market, the level of division of labor, productivity and the degree of inequality of income distribution. Let us motivate the two tasks. Krugman (1995a, b; 1996) and Krugman and Venables (1995) vividly document the fact that the relationship between economic development and income distribution came to the focus of public concern in the 1970s and 1990s. However, this was always a controversial issue in economic literature. Some models and theories were developed, with empirical evidence sometimes, to show that there is a positive correlation between economic
* Reprinted from Journal of Economics, 78 (2), Xiaokai Yang and Dingsheng Zhang, "Economic Development, International Trade, and Income Distribution," 163-190, 2003, with permission from Springer-Verlag. * We are grateful to Guangzhen Sun, Lin Zhou and Meng-Chun Liu for helpful discussion. We also gratefully acknowledge the Australian Research Council for financial support (Large Grant A79602713). We are solely responsible for the remaining errors.
77
78
X. Yang, D. Zhang
development and inequality of income distribution (Banerjee and Newman, 1993; Lewis, 1955; Palma, 1978; Li and Zou, 1998; Chang and Ram, 2000, for instance). Other models were developed, with empirical evidence sometimes, to show that there is a negative correlation between inequality and international trade, which relates to economic development (Aghion, Caroli and Garcia-Pena Losa, 1999; Alesina and Rodrik, 1994; Galor and Zeira, 1993; Thompson, 1995, and Fei, Ranis and Kuo, 1979; Frank, 1977; Balassa, 1986, for instance). Not only data from developing and newly industrialized countries are contradictory, but data from developed countries also generate controversy. Kuznetz (1955) proposed the hypothesis of an inverted Ucurve of the relationship between inequality and per capita income and provided some support for it from US data. Krugman and Venables (1995) use a general equilibrium model with global economies of scale to predict such an inverted U-curve of inequality. Greenwood and Jovanovic (1990) use a dynamic equilibrium model to predict the inverted U-curve. Some theories and data show that there is a negative or insignificant correlation between trade or development and inequality in the developed country (see, for instance, Krugman and Lawrence, 1994, and Katz and Murphy, 1992). Others show a positive correlation between trade and inequality in the developed country (see, for instance, Grossman, 1998; Murphy and Welch, 1991; Borjas, Freeman and Katz, 1992; Karoly and Klerman, 1994; Sachs and Shatz, 1995).1 However, Ram (1997) provides new empirical evidence for an uninverted-U pattern in the developed countries, taking into account data gathered since the Second World War. Jones (1998, p. 65) also shows that the ratio of GDP per worker in the 5th-richest country to GDP per worker in the 5th-poorest country fluctuated from 1960 to 1990. The controversy and new evidence call for new models and theories that can resolve it. In this paper, we develop a general equilibrium model to show that the relationship between inequality and trade that relates to economic development is neither monotonic nor an inverted U-curve.
1
Recent surveys on the relationship between trade and income distribution in developed countries can be found in Cline (1997), Williamson (1998), and Burtless (1995).
Economic Development, International Trade, and Income Distribution
79
In our general equilibrium model of endogenous specialization, economic development is described as an evolutionary process of division of labor that is driven by improvements in trading efficiency. Because of differences in trading efficiencies between countries and between different groups of residents in the same country, those individuals with better trading efficiencies will be involved in the division of labor and related trade before others are. Inequality increases because of the dual structure. As latecomers catch up, the dual structure disappears, so that inequality declines. As the leading group goes to an even higher level of specialization, leaving others behind, dual structure emerges and inequality increases again. As latecomers catch up, the dual structure disappears and inequality decreases again. This ratcheting process of inequality and equality generates fluctuation of the degree of inequality of income distribution. This implies that the relationship between inequality and economic development is neither monotonically positive nor monotonically negative. It may not be a simple inverted Ucurve. The intuition behind the model is as follows. If the relationship between inequality and economic development is monotonic, then in the very developed country, income distribution must be either extremely equal or extremely unequal. This is true too for an inverted U-curve between inequality and per capital income. However, we do not see either of these two extreme cases. Hence, inequality of income distribution must fluctuate as the equilibrium level of division of labor evolves. This evolution puts a group of individuals in a higher level of division of labor earlier than others. Hence, a dual structure in which some individuals are more specialized (or more commercialized) than others keeps emerging and disappearing as the equilibrium level of division of labor increases. A new data set in Deininger and Squire (1996) supports our hypothesis. It indicates that there is no systematical link between growth and changes in aggregate inequality. Recent regressions of Barro (1999), and Banerjee and Duflo (1999) also suggest that no monotonic correlation between inequality and growth performance can be supported by data.
80
X. Yang, D. Zhang
The second purpose of the current paper is to explore the intrinsic relationship between the development of division of labor and changes in inequality of income distribution. Kuznetz (1955) explained inequality of income distribution by per capita income. This is certainly not a general equilibrium view since per capita income is endogenous in a general equilibrium model, which itself should be explained by parameters. Murphy, Shleifer, and Vishny (1989) show that unequal income distribution restricts the extent of the market, so that economies of scale cannot be fully exploited and economic development is retarded. We put this idea together with Allyn Young's conjecture that "not only division of labor is dependent on the extent of the market, but also, the extent of the market is determined by the level of division of labor", to develop a general equilibrium mechanism that simultaneously determines the extent of the market, the level of division of labor, productivity and the degree of inequality of income distribution. We will examine effects of the coexistence of exogenous and endogenous comparative advantages on inequality of income distribution. Yang (1994), and Yang and Borland (1991) have drawn the distinction between David Ricardo's exogenous comparative advantage (Ricardo, 1817) and Adam Smith's endogenous comparative advantage (Smith, 1776).2 There is an extensive literature on exogenous comparative advantage in trade theory (see, for instance, Dixit and Norman, 1980). Separately, there are many models of endogenous comparative advantage in the growing literature on endogenous specialization (see Yang and Ng, 1998 for a recent survey on this literature and references there). This paper develops a general equilibrium model with the trade off between transaction costs and endogenous and exogenous comparative advantages. We shall show that as trading efficiencies are improved, the scope for efficiently balancing this trade off is enlarged. Hence, the equilibrium level of division of labor increases. The existence of exogenous comparative advantage implies that different
2
Endogenous comparative advantage is associated with economies of specialization and referred to by Grossman and Helpman (1991) as acquired comparative advantage, whereas exogenous comparative advantage is associated with constant returns to scale in production, referred to by them as natural comparative advantage.
Economic Development, International
Trade, and Income Distribution
81
types of individuals are sequentially involved in the division of labor. Part of the population is first involved in a low level of division of labor, the rest being left behind. The emergence of this dual structure increases inequality of income distribution. Then the rest of the population is involved in this low level of division of labor, catching up to rich fellows. This catch up process reduces duality and related inequality. Then, part of the population goes to an even higher level of division of labor, duality and related inequality increases. Then the rest who are left behind catch up; duality and inequality reduces again. This ratcheting process goes on, generating fluctuation of the degree of duality of economic structure and of the degree of inequality of income distribution. In this development process, the most important determinant of equilibrium aggregate productivity and welfare is trading efficiency rather than inequality of income distribution. This is called irrelevance of inequality (or equality) of income distribution to economic development. Early studies of structural changes and dual structure relied on the assumption of disequilibrium in some markets to predict dual structure and structural changes. For instance, Lewis (1955) tried to explain the dual structure between commercialized and self-sufficient sectors by the evolution of the division of labor and the related productivity progress (see also Ranis, 1988). Due to lack of appropriate analytical tools, he ended up with a model based on disequilibrium in the labor market caused by institutional wage. Chenery (1979) used market disequilibrium to explain structural changes. Recently, general equilibrium models have been used to study dual structure. In some of these models, such as in Khandker and Rashid's equilibrium model (1995), dual structure is exogenously assumed.3 They cannot predict the emergence and evolution of dual structure. In a recent literature on formal equilibrium models of high development economics, evolution of dual structure between the manufacturing sector with economies of scale in production and the agricultural sector with constant returns to scale can be predicted (see Krugman and Vanables, 1995, 1996, and Fujita 3
Also, in Dixit's dynamic planning model (1968), the existence of labor surplus is exogenously assumed.
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and Krugman, 1995). The equilibrium models with endogenous geographical location of economic activities of Krugman and Venables (1995), and Baldwin and Venables (1995) attribute the emergence of dual structure to the geographical concentration of economic activities in economic development that marginalizes peripheral areas. Kelly (1997), based on Murphy, Shleifer, and Vishny (1989), develops a dynamic general equilibrium model that predicts spontaneous evolution of a dual structure between the modern sector with economies of scale and the traditional sector with constant returns technology. As trading efficiencies are sufficiently improved, the level of division of labor increases and dual structure disappears. Our model in this paper is complementary to these general equilibrium models that predict the emergence and evolution of dual structure. We pay more attention to the effects that the evolution of individuals' levels of specialization and the coexistence of exogenous and endogenous comparative advantages exert on the emergence and evolution of dual structure.4 The rest of this paper is organized as follows, section 2 presents the 2 x 2 Ricardian model with transaction costs and endogenous and exogenous comparative advantages. Section 3 solves for general equilibrium and its inframarginal comparative statics. Section 4 extends the analysis to consider not only income distribution in the developing country but also that in the developed country. The concluding section summarizes the findings of the paper and suggests possible extensions.
4
Mokyr (1993, pp. 65-66) documents the evolution of individuals' level of specialization during the Industrial Revolution in Britain. This evolution is sometimes referred to as the "industrious revolution" which implies that self-provided home production is replaced with commercialized production. Yang, Wang and Wills (1992) find empirical evidence for this evolution from Chinese data. A difference in empirical implications generated by the Krugman and Venables model and our model is that the former yields scale effects (productivity of the manufacturing firm increases if and only if its size increases), but the latter does not.
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2. A Ricardian Model with Endogenous and Exogenous Comparative Advantages Consider a world consisting of country 1 and country 2, each with Mt (i= 1, 2) consumer-producers. The set of individuals is a continuum and the mass of population is 1. There are two types of individuals in country 1. The mass of type a persons is Ma and that of type b persons is Mb, where Ma + Mb + M2 = 1. All individuals in country 2 are assumed to be identical, but different from individuals in country 1. The utility function for individuals of type i is Ut = (*, + ktfYiy,
+ klyttP,
»= a, b, 2,
(1)
where xh yt are quantities of goods x and y produced for selfconsumption, xt , yt are quantities of the two goods bought from the market and kt is the trading efficiency coefficient for a type i individual. The transaction cost is assumed to take the iceberg form: for each unit of good bought, a fraction 1 - kt is lost in transit, the remaining fraction kt is received by the buyer. Type a persons and type b persons are distinguished by their trading efficiencies. We assume that ka = k,kb = t, and k > t. In other words, each type a persons has higher trading efficiency than each type b person. Also, we assume that k2 = k. The assumptions can be interpreted as follows. Country 2 is a developed country that has greater trading efficiencies. Country 1 is a less developed country where some residents (who might be urban or coastal residents) have the same trading efficiency as in the developed country, but the rest of the population has a lower trading efficiency. The production conditions are the same for all individuals in the same country, but different between the countries. Hence, the production functions for a type i consumer-producer are X +X
J J=L%'
yj+ysj=Ljy'
x2+xs2=If2x,
y2+ys2=rL2y,
j=*>b'
(2a)
(2b)
where x., yst are respective quantities of the two goods sold by a type i person; Lis is the amount of labor allocated to the production of good s (= x, y) by a type / person, and Lix + Liy = B > 1. For simplicity, we assume that B = 2. It is assumed that r, c > 1. This system of production
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functions and endowment constraint displays economies of specialization in producing good x and constant returns to specialization in producing good y. But an individual in country 2 has a higher productivity in producing good y than an individual in country 1. Economies of specialization are individual specific and activity specific, that is, they are localized increasing returns, which are compatible with the Walrasian regime.5 If all individuals allocate the same amount of labor to the production of each good, then an individual in country 1 has the same average labor productivity of good x as an individual in country 2. But the average and marginal labor productivity of good y for an individual in country 2 is higher. This is similar to the situation in a Ricardian model with exogenous comparative advantage. Country l's productivities are not higher than country 2 in producing all goods, but may have exogenous comparative advantage in producing good x. But if an individual in country 2 allocates much more labor to the production of x than an individual in country 1, her productivity is higher than that of the latter. Similarly, if an individual in country 1 allocates more labor to the production of good x than an individual in country 2, her productivity of good x will be higher. This is referred to as endogenous comparative advantage, since individuals' decisions on labor allocation determine 5
Marshall was aware of the incompatibility between his marvellous marginal analysis of demand and supply and classical economic thinking about the implications of specialization and division of labor. As a result he drew the distinction between internal economies of scale and external economies of scale. The latter is supposed to relate to economies of division of labor in the market place. However, as Allyn Young (1928) argued, "the view of the nature of the processes of industrial progress which is implied in the distinction between internal and external economies is necessarily a partial view. Certain aspects of those processes are illuminated, while, for that very reason, certain other aspects, important in relation to other problems, are obscured." As Young and Yang (2001) have pointed out, economies of specialization are diseconomies of a person's scope of production activities and economies of division of labor consist of economies of specialization of all individuals and economies of diversity of different occupations. They are different from (internal or external) economies of scale, though related to local economies of scale of an individual labor in producing a good. Hence, it seemed to Young that the concept of external economies of scale is a misrepresentation of the classical concept of economies of specialization and division of labor.
Economic Development, International Trade, and Income Distribution 85 difference in productivity between them. But an individual in country 1 has no endogenous comparative advantage in producing good y since her marginal and average productivity of y is always 1, lower than r, independent of her labor allocation. However, country 1 has exogenous comparative advantage in producing x and country 2 has exogenous comparative advantage in producing y. The decision problem for a type i individual involves deciding on what and how much to produce for self-consumption, to sell and to buy from the market. In other words, the individual chooses six variables xnx° ,xf ,yt,y° ,yf > 0 . Hence, there are 26 = 64 possible corner and interior solutions. As shown by Wen (1998), for such a model, an individual never simultaneously sells and buys the same good, never simultaneously produces and buys the same good, and never sells more than one good. We refer to each individual's choice of what to produce, buy and sell that is consistent with the Wen theorem as a configuration. There are three configurations from which the individuals can choose: (1) self sufficiency. Configuration A, where an individual produces both goods for self-consumption. This configuration is defined by x,,yt >0,
x' = x," = yt' = yf =0,i
= a,b,2.
(2) specialization in producing good x. Configuration (x/y), where an individual produces only x, sells x in exchange for y, is defined by Xi,x;,yf>o,
x?=yi=y;=o.
(3) specialization in producing good y. Configuration (y/x), where an individual produces only y, sells y in exchange for x, is defined by
yny:,x?>Q,
yf=Xt=x,'=Q.
The combination of all individual's configurations constitutes a market structure, or structure for short. Given the configurations listed above, fourteen structures may occur in equilibrium. Structure AAA, as shown in panel (1) of Figure 1, is an autarky structure where individuals in both countries choose self-sufficiency (configuration A). Here, the first two letters denote the configurations chosen by type a and type b persons, respectively in country 1 and the
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X. Yang, D. Zhang
00- g)fc^ fcw0' ^ 'g) g) g) country 1 country 2 (1) Structure AAA
country 1 country 2 (7) Structure XXP
country 1 country 2 (2) Structure PAY
country 1 country 2 (8) Structure XXD
Figure 1: Configurations and Structures
country 1 country 2 (3) Structure XAP
country 1 country 2 (9) Structure XXY
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third letter denotes the configuration chosen by individuals in country 2. In each panel upper-left circles represent configurations chosen by type a individuals, lower-left circles represent configurations chosen by type b individuals, and circles on the right hand side represent configurations chosen by individuals in country 2. Structure PAY, shown in panel (2) of Figure 1, means that some type a persons choose configuration (x/y) and others choose autarky, all type b persons choose autarky, and individuals in country 2 choose configuration (y/x). Letter P stands for a type of person partially involved in the division of labor: some of them completely specialize and others choose autarky. This structure involves three types of dual structure. In a type I dual structure ex ante identical individuals are divided between specialization and autarky. In a type II dual structure, different types of individuals in the same country are divided between specialization and autarky. In a type III dual structure, one country is completely involved in the division of labor, while some residents in the other country are self-sufficient. Structure XAP, shown in panel (3) of Figure 1, means that all type a persons choose configuration (x/y), all type b persons choose autarky, some individuals in country 2 choose (y/x) and others choose autarky. Structure DAY, shown in panel (4) of Figure 1, means that some type a persons choose (x/y) and others choose (y/x), all type b persons choose autarky, and individuals in country 2 choose (y/x). Letter D stands for division of individuals of a certain type between (x/y) and (y/x). Structure XAY, panel (5) of Figure 1, means that all type a persons choose (x/y), all type b persons choose autarky, and individuals in country 2 choose configuration (y/x). Structure XAD implies that all type a persons choose (x/y), all type b persons choose autarky, some individuals in country 2 choose (y/x) and others choose (x/y). Structure XPY, panel (6), means that all type a persons choose (x/y), some type b persons choose autarky and others choose (x/y), and individuals in country 2 choose configuration (y/x).
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X. Yang, D. Zhang
Structure XXP, panel (7), means that all type a and type b persons choose (x/y), some individuals in country 2 choose configuration (y/x), and others choose autarky. Structure XXD, panel (8), means that all type a and type b persons choose (x/y), some individuals in country 2 choose configuration (y/x), and others choose (x/y). Structure XXY, panel (9), implies all type a and type b persons choose (x/y) and all individuals in country 2 choose configuration (y/x). Structure XDY means that all type a persons choose (x/y), some type b individuals choose (x/y) and others choose (y/x), all individuals in country 2 choose configuration (y/x). Structure YDY means that all type a persons choose (y/x), some type b individuals choose (x/y) and others choose (y/x), all individuals in country 2 choose configuration (y/x). Structure DXY means that some type a persons and all type b persons choose (x/y), other type a persons choose (y/x), all individuals in country 2 choose configuration (y/x). Structure DYY means that some type a persons and all type b persons choose (y/x), other type a persons choose (x/y), all individuals in country 2 choose configuration (y/x). It can be shown that other structures cannot occur in equilibrium. 3. General Equilibrium and its Inframarginal Comparative Statics According to Sun, Yang and Zhou (1998), a general equilibrium exists and is Pareto optimal for the kind of models in this paper under the assumptions that the set of individuals is a continuum and preferences are strictly increasing and rational; both local increasing returns and constant returns are allowed in production and transactions. Also, the set of equilibrium allocations is equivalent to the set of core allocations. An equilibrium in this model is defined as a relative price of the two goods and all individuals' labor allocations and trade plans, such that (i) Each individual maximizes her utility, that is, the consumption bundle generated by her labor allocation and trade plan maximizes utility function (1) for given p.
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(ii) All markets clear. For simplicity, we assume that J3= 0.5. Let the number (measure) of type i individuals choosing configuration (x/y) be Mix, choosing (y/x) be Miy, and choosing A be MA. Since the interior solution is never optimal in this model of endogenous specialization and there are many structures based on corner solutions, we cannot use standard marginal analysis to solve for a general equilibrium. We adopt a two step approach in solving for a general equilibrium. In the first step, we consider a structure. Each individual's utility maximizing decision is solved for the given structure. Utility equalization condition between individuals choosing different configurations in the same country and market clearing condition are used to solve for the relative price of traded goods and numbers (measure) of individuals choosing different configurations. The relative price and numbers and associated resource allocation are referred to as a corner equilibrium for this structure. According to the definition, a general equilibrium is a corner equilibrium in which all individuals have no incentive to deviate, under the corner equilibrium relative price, from their chosen configurations. Hence, in the second step, we can plug the corner equilibrium relative price into the indirect utility function for each configuration, then compare corner equilibrium values of utility across all configurations. The comparisons are called a total cost-benefit analysis which yields the conditions under which the corner equilibrium utility in each constituent configuration of this structure is not smaller than any alternative configuration. This system of inequalities can thus be used to identify a subspace of parameter space within which this corner equilibrium is a general equilibrium. With the existence theorem of general equilibrium proved by Zhou, Sun and Yang (1998), we can completely partition the parameter space into subspaces, within each of which the corner equilibrium in a structure is a general equilibrium. As parameter values shift between the subspaces, the general equilibrium will discontinuously jump between structures. The discontinuous jumps of structure and all endogenous
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variables are called inframarginal comparative statics of general equilibrium. We now take the first step of the inframarginal analysis. As an example, we consider structure XAP. Assume that in this structure, M2y individuals choose configuration (y/x) and M2A individuals choose autarky in country 2, where M2y + M2A = M2. Ma individuals choose configuration (x/y) and Mb individuals choose configuration A in country 1. Since all individuals in country 2 are ex ante identical in all aspects, the maximum utilities in configurations A and (y/x) must be the same in country 2 in equilibrium. Marginal analysis of the decision problem for an individual in country 2 choosing autarky yields her maximum utility in configuration A: U2A = (r2c+1y)0'5, where y- ccl(c + l) e+1 . Marginal analysis of the decision problem for an individual in country 2 choosing configuration (y/x) yields the demand function xd2 = rip, the supply function^ = r, and indirect utility function: U2y = r(k/p)05. The utility equalization condition U2y = U2A yields p = pjpy = kr/2°ny. Similarly, the marginal analysis of the decision problem of a type a individual choosing configuration (x/y) yields the demand function, yd _ jc-\ ip ^ m e SUppiy function xsa = 2C_1 , and indirect utility function: Uax =2c~l(kp)05 . The marginal analysis of the decision problem for a type b individual choosing configuration A in country 1 yields her maximum utility in autarky UbA = (2c+lyf5. Inserting the corner equilibrium relative price into the market clearing condition for good x, Msx x" - M2y %d2 , yields the number of individuals selling good y, M2y = kMJAy, Mm = Ma, and MbA = Mh. Indirect utility functions for individuals choosing various configurations in the two countries are listed in Table 1.
Configurations Type a person Type b person Country 2
Table 1: Indirect Utility Functions Indirect utility functions (x/y) (y/x) t/„ = 2c-\kpf5 U!Ly = (k/pf5 $ UbK = ¥-\tpf Uby = (t/pf5 c 5 U2x = 2 ~\kpf U2y = r(k/pf5
A U^ = (2c+lrf5 UbA = (T+lrf5 U2A = (l^ryf5
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Following this procedure, we can solve for corner equilibrium in each structure. The solutions of all corner equilibria are summarized in Table 2. We can then take the second step to carry out total cost-benefit analysis for each corner equilibrium and to identify the parameter subspace within which the corner equilibrium is a general equilibrium. Consider the corner equilibrium in structure XAP as an example again. In this structure Ma individuals choose configuration (x/y) and Mh individuals choose autarky in country 1, and M2y individuals choose configuration (y/x) and M2A individuals choose autarky in country 2. For a type a person in country 1, equilibrium requires that her utility in configuration (x/y) is not smaller than in configurations (y/x) and A under the corner equilibrium relative price in structure XAY. Also equilibrium requires that all type b persons in country 1 prefer autarky to configurations (x/y) and (y/x) and that all individuals in country 2 are indifferent between configurations (y/x) and A and receive a utility level that is not lower than in configuration (x/y). In addition, this structure occurs in equilibrium only if M2ye (0, M2). All the conditions imply Um > Uay, Uax > UaA, UhA > UbM UbA > Uby, U2A=U2y>U2x,
M2ye(0,M2),
where indirect utility functions in different configurations and corner equilibrium relative price are given in Tables 1 and 2. The conditions define a parameter subspace: kt < 16/Vr, k e (4yVr, Min{4f, 4M2y/Ma}), where y = cc/(c + l) c+1 . Within this parameter subspace, the corner equilibrium in structure XAP is the general equilibrium. Following this procedure, we can carry out total cost-benefit analysis for each structure. The total cost-benefit analysis in the second step and marginal analysis of each corner equilibrium in the first step yields inframarginal comparative statics of general equilibrium, summarized in Table 3. From this Table, we can see that the parameter subspaces for structure AAD, XAA, and all structures in which country 1 exports y and country 2 exports x to occur in general equilibrium are empty.
92
Structure AAA PAY XAP DAY XAY XAD XPY XXP XXD XXY XDY YDY DXY DYY
X. Yang, D. Zhang Table 2: Comer Equilibria Numbers of individuals choosing various configurations A4A = Ma, MbA = Mb, M2A = M2 r23~c/k Mix = M2kr/4y. M^ = M a - M2kr/4y, MbA = Afb, M2A = M2 krlyl^1 MK = Ma, MbA = Mb, M2y = MJdAy, M2A= M2- (MJdAy) 21M ax = (Ma+ rM2)/2, May= (M a - rM2)/2, MbA = Mb,M2v = M2 •Mx = Ma, MbA = Mb, M2v = M2 M2r2'^/Afa ^ = M , MbA = Mb, M2y = (Ma + M2)/2 r2'" c Mx = Ma, Mbx = (foM2/4x) - Ma, M2v = M2 yl^lk krly2M M« = Ma, M,x = Mb, M2y = *(Ma + Mb)/4r, M 2 A =M 2 -WM a +M b )/4^ Max = M,, Mbx = Mb, M2y = (Ma + Mb + M2)I2, a1-* M2x = (M 2 - M - M)/2 Mx = Ma, Mbx = Mb, M2v = M2 M2r2i-c/(M!i+ Mb) 21Mx = Ma, Mbx = (Mb - Ma + rM2)/2, Mbv = (Ma + Mb - rM2)/2, M2v = M2 May = Ma, Mbx = (Ma + Mb + rM2)/2, 21Mbv = (Mb - Ma - rM2)/2, M2v = M2 2l-c Mbx = Mb, Max = ( M - Mb + rM2)/2, Mav = (Ma + Mh - rM2)/2, M2v = M2 2l-c Mbx = Mb, M« = ( M + Mb + rM2)/2, Mav = (Ma - Mb - rM2)/2, M2v = M2
Relative price of xto y
When a dual structure occurs in equilibrium, individuals in autarky appear as underemployed or underdeveloped in the sense that they receive none of the gains from trade and income differential between them and other individuals who are involved in the division of labor increases as a result of a shift of equilibrium from AAA to a dual structure. These self-sufficient individuals cannot find a job to work for the market. Figure 1 illustrates the equilibrium structures.6 We say the 6
The coexistence of two equilibria distinguishes the model of endogenous specialization from a standard Arrow-Debreu equilibrium model in which the number of multiple equilibria is always odd.
w
i« -^ £
A
x •*
s.
ft
13
^
^
11
T
^J->
I*
V
us
-
S3
Economic Development, International Trade, and Income Distribution
V V
*
5tv
•
^
V
fs
Si
X
"V.
5?*!'
v 1
x
V -?
ST
93
Table 3: {Continued) Mi/M2 >r
MJ M2< 1
XAP
MJM2 < Vr
AAA
MJ M2 e
ihr)
MJM2 > Vr
k< 4yM2IMa XAP
; < AyMJrMi
AyMJrMi
k> 4rM2/Ma XAY
XAY
XPY
4yMJrM2 PAY
4j-Ma/f-M2 XAY
MJ Mi >r
X
XPY
PAY
X
Economic Development, International Trade, and Income Distribution
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level of division of labor increases if occurrence of letter A or P decreases and/or the occurrence of letter D, X, and/or Y increases in the equilibrium structure. We can identify five levels of division of labor. The lowest one is associated with structure AAA where all three types of individuals are self-sufficient. The second level is associated with structures PAY and XAP in each of which two types of individuals are self-sufficient. The third level is associated with structures DAY, XAY, XAD in each of which all of individuals of a certain type are selfsufficient. The fourth level is associated with structures XPY and XXP in each of which only part of the population of a certain type are selfsufficient. The fifth level is associated with structures XXD, XXY, XDY, YDY, DXY, DYY, in each of which all individuals are specialized. In this table y= ccl(c + l) c+1 . X stands for configuration (x/y) chosen by individuals of a certain type, Y stands for (y/x) chosen by individuals of a certain type, A stands for autarky, P stands for the partial division of labor where individuals of a certain type are divided between autarky and specialization, D stands for the division of individuals of a certain type between (x/y) and (y/x). Structure XAP involves type I and type II dual structures, structure XXP involves type I and type III dual structures. Structures DAY, XAY, XAD involve type II and type III dual structures. Structures PAY and XPY involve all three types of dual structure. With the definition of division of labor in mind, we can now closely examine Table 3. As trading efficiencies are improved, the equilibrium level of division of labor increases. If the equilibrium level of division of labor increases from level 1 to level 2, then dual structure emerges, or duality of economic structure increases. If the equilibrium level of division of labor increases from 2 to 3, duality decreases as those individuals who are left behind catch up. If the equilibrium level of division of labor increases from 3 to 4, duality increases again as some of individuals in a group that was in autarky before choose specialization. As the equilibrium level of division of labor increases from 4 to 5, duality decreases again as those who were left behind catch up. But it is also possible for improvements in trading efficiencies to generate a jump of equilibrium from level 2 to level 4 in the division of labor. Such a
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X. Yang, D. Zhang
jump will reduce duality. A jump from level 3 in the division of labor to level 5 will reduce duality as well. This analysis implies that as different groups of the same type or different types of individuals gradually choose a higher level of specialization one by one, the degree of duality fluctuates. If we use the difference between the highest and the lowest per capita real income i.e. the difference in per capital real income between type a and type b individuals in country 1 as a measure of income inequality, its increase will raise inequality of income distribution and its decrease will reduce inequality. Hence, as different individuals gradually become involved in the network of the division of labor as a result of improved trading efficiencies, the equilibrium degree of inequality fluctuates. It does not monotonically increase or decrease. The relationship between inequality and per capita real income may not be an inverted U-curve. Inserting the corner equilibrium relative price in Table 2 into indirect utility functions in Table 1, we can compare the difference of per capita real incomes (equilibrium levels of utility) of two types of individuals in country 1, considering the parameter subspaces that demarcate the structures. The comparisons confirm our analysis. We first consider the path of evolution in the division of labor: AAA-»XAP->XAY-»XPY —»XXY. The difference in per capita real income between type a and type b individuals in country 1 is 0 in structure AAA. It is positive in structure XAP. It decreases as the equilibrium jumps from XAP to XAY. It then increases as the equilibrium jumps from XAY to XPY. It decreases again as the equilibrium jumps from XPY to XXY. We can find other evolutionary paths of the division of labor with a fluctuating degree of inequality of income distribution. Also, comparisons of ratios of the highest to the lowest real income between corner equilibria generate similar results. All of the results on the evolution of the division of labor, dual structure and trade pattern are summarized in the following proposition, illustrated in Figure 1. Proposition 1: As trading efficiency increases from a very low to a very high level, the equilibrium level of the domestic and international
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division of labor increases from complete autarky in both countries to complete division of labor in both countries. As different individuals are gradually involved in the division of labor, the duality of economic structure fluctuates, generating fluctuation of the equilibrium degree of inequality of income distribution in the less developed country. This degree is neither monotonically increasing nor monotonically decreasing. The relationship between the inequality and per capita real income might not be of an inverted U-curve. The inframarginal comparative statics of general equilibrium in Table 3 can be used to establish two corollaries. The first is that evolution in the division of labor generated by improvements in trading efficiencies will raise equilibrium aggregate productivity. In order to establish the above statement, we consider the aggregate PPF for individual 1 (from country 1) and individual 2 (from country 2). As shown in Figure 2 where c = 2, the PPF for individual 1 is curve BC, that for individual 2 is curve AC. In autarky, the two persons' optimum decisions for taste parameter j6 G (0, 1) are functions of /?. Let f3 change from 0 to 1; we can calculate equilibrium values of Y = yx + y2 and X=X\ + x2 as functions of J3. The values of X and Y for different values of /? constitute curve EGD in Figure 2. The equilibrium aggregate production schedule in structure AAA is a point on the curve, dependent on the value of j5. But the aggregate PPF for the two individuals is the curve EFD. Since in a structure with complete division of labor the equilibrium production schedule is point F which is on the aggregate PPF, the aggregate productivity in a structure with the complete division of labor is higher than in structure AAA. The difference between EFD and EGD can be considered as economies of the division of labor. Following the same reasoning, we can prove that the equilibrium aggregate productivity in a dual structure is lower than the PPF. Hence, Proposition 1 implies that as trading efficiencies are improved, the equilibrium level of division of labor and equilibrium aggregate productivity increase side by side.
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X. Yang, D. Zhang y/
2+2a
2&A
C 2
0
B 4
8
^x
Figure 2: Economies of Division of Labor Based on Endogenous and Exogenous Comparative Advantage
The second corollary is that deterioration of a country's terms of trade and increase of gains received by this country from trade may concur. Suppose that the initial values of parameters satisfy (Ma + Mh)/M2 e (Vr, r), MJM2 e (1, r), kt > I6f/r. Within this parameter subspace, suppose that the initial value of/ satisfies /'< 4){M!i+ Mb)/M2r. Table 3 indicates that structure XPY occurs in equilibrium where country l's terms of trade are p' = y23~c/t'. Now assume that / increases to the value /", so that t"> 4XM a + Mb)/M2. According to Table 3, the equilibrium jumps to structure XXY where country l's terms of trade arep" = M2r21_7(Ma+ Mb). It can then be shown that country l's terms of trade deteriorate as a result of the change in / within the parameter subspace. But this shift of the equilibrium from XPY to XXY increases utility of each individual in country 1 if k'lt' < k"M2r/4f{Ma+ Mh). This has established the claim that there exists some parameter subspace within which the deterioration of a country's terms of trade may concur with an increase of gains that this country receives from trade. There are other parameter subspaces within which changes in parameters may
Economic Development, International
Trade, and Income Distribution
99
generate concurrence of the deterioration of one country's terms of trade and an increase in its gains from trade.7 This corollary generates the following policy implications. In the transitional stage of economic development and globalization, the terms of trade are against the less developed country that has relatively low trading efficiency: some residents in the less developed country receive autarky utility and most gains from trade go to the developed country. There are two policies to change this inferior position. One is to impose a tariff to improve the terms of trade and the other is to improve trading efficiency to expand the network of trade. The former is to increase the share of gains received by the less developed country from a shrunk pie because of the deadweight caused by the tariff. The latter is to provide a greater share of gains from trade by enlarging the pie. The expanded network of division of labor can generate productivity gains. As long as productivity improvements outpace the deterioration of the terms of trade, the less developed country can receive more gains from trade not only because of productivity gains, but also because of more equal division of gains from trade between the countries as all individuals are involved in the international and domestic division of labor. 4. Effects of Trade on Income Distribution in the Developed Country Since we assume that there is only one type of individuals in the developed country that has greater trading efficiency, the model in the previous section cannot explore effects of economic development and international trade on income distribution in the developed country. In this section we extend the model in the previous section by specifying two types of individuals in the developed country. The new model is exactly the same as the previous one except that there are two types of
7
Empirical and theoretical research on economic development and terms of trade can be found, for instance in Morgan (1970), Kohli and Werner (1998), Sen (1998).
Table 4: Equilibrium Structure
0
kt < 16 fir
AAAA
XAPA
0
ks> 16/Vr kt> 16 f/r
16AXPYA
16^/r XXYP
st < 16 fir
AADA
f< r
^ XAYA
s> 4y1r st> 16f
?>4
^ XPYA
XX
Economic Development, International Trade, and Income Distribution 101
individuals in country 2 as well. Types la and lb of individuals are in country 1 and types 2a and 2b of individuals are in country 2. For type 2a individuals, trading efficiency is k, for type lb individuals, trading efficiency is t, and for type 2b and la individuals, trading efficiency is s, where 1 > k > s > t > 0. For simplicity, we assume that M]a = Mu, = M2a = M2\, = 1/4. Following the method in the previous section, the inframarginal comparative statics of this new model can be solved as in Table 4, where y = ccl (1 + c)c+1, X denotes configuration (x/y), Y denotes (y/x), P denotes a division of individuals of a certain type between specialization and autarky, D denotes a division of individuals of a certain type between (x/y) and (y/x). The first of four letters denoting a structure represents the configuration chosen by type la persons, the second represents that chosen by type lb persons, the third represents that chosen by type 2a persons and the fourth represents that chosen by type 2b persons. Figure 3 gives an intuitive illumination of the evolution of the division of labor and economic development as a result of improvements in trading efficiencies. In this figure, an arrow -» denotes the direction of the evolution of the division of labor, the numbers 1 and 2 next to the arrows denote the two countries and the signs + and - denote increase and decrease, respectively, of the inequality of income distribution in a country. A question mark implies that either an increase or decrease of inequality is possible. From the results we can see that although evolutionary paths of division of labor AAAA—»AADA—»XAYA-» XAYD—»XXYY generates an inverted U-curve of inequality in the less developed country, path AAAA->XAPA->XPYA->XAYP->XXYY generates fluctuation of inequality. Other paths with ? may generate such fluctuation, too. Again, inframarginal comparative statics of general equilibrium confirm Proposition 1 even if income distribution in the developed country is considered. This yields Proposition 2.
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X. Yang, D. Zhang
1:0 2: AAAA 1 2:0
AADA
-XAYD
1:.2:XXYY
XAPA
1:2-A
-XPYA
1:+ 2:1
-XXYP
1:2:-
Figure 3: Evolution in Division of Labor
Proposition 2: As trading efficiency is improved, the equilibrium level of division of labor and aggregate productivity increase. The degree of inequality of income distribution in the less developed country and the developed country fluctuates as duality of economic structure fluctuates. The degree of inequality is nonmonotonic and it might not be an inverted U-curve. An interesting feature of our inframarginal analysis of general equilibrium is that economic development and structural changes can take place in the absence of changes of tastes and production functions. All development and trade phenomena can be generated by changes in trading efficiencies that are neither on the demand side nor on the supply side. All recent general equilibrium models of "high development economics" share this feature (see Fujita and Krugman 1995; Krugman and Venables, 1995; Kelly, 1997; Shi and Yang, 1995; Yang, 1991, and Yang and Rice, 1994, for instance). Hence, they explore the limitation of the analysis that explains trade and development phenomena by changes in demand and/or supply sides. 5. Conclusion and Policy Implication In this paper, we have proposed a theory of ratcheting inequality and equality of income distribution using inframarginal analysis of a model of endogenous specialization. According to our view, there are two resources concerning the source of the income inequality. One is the
Economic Development, International Trade, and Income Distribution 103
difference in transaction condition between groups; the other is the relative number of people of different types. This theory accommodates the empirical evidence for the coexistence of positive and negative correlation between inequality and economic development which relates to trade. Because of the trade off between economies of the division of labor and transaction costs, the equilibrium level of the division of labor increases as trading efficiencies are improved. If trading efficiencies are different between countries or between different groups of individuals in the same country, different individuals will be involved in the division of labor sequentially. If some individuals have a higher level of specialization (or commercialization) than other individuals, a dual structure will increase inequality of income distribution. As those poor individuals increase their levels of specialization, the dual structure disappears and inequality decreases. But as the level of division of labor increases further, while some individuals' levels of specialization increase before others' do, a new dual structure will occur, which will disappear as latecomers catch up. This ratcheting process generates fluctuation of the equilibrium degree of inequality as the division of labor evolves. Therefore, there is neither a monotonic relationship between inequality of income distribution and economic development nor an inverted U-curve. In other words, inequality (equality) of income distribution is irrelevant to economic development. We should pay more attention to improvements in trading efficiencies that will enhance aggregate productivity and all individuals' welfare than to the relationship between inequality and economic development (or trade). Our model has policy implications. The government can reduce inequality of income distribution by implementing policies that affect trading efficient parameters. The government may increase trading effciency of a poor population by removing institutional obstacles of free migration from inland or rural areas to coast or urban areas, or by constructing a transportation infrastructure in poor areas. A logical extension of this paper is to add more goods and/or more countries and introduce a tariff into the Ricardian model. More structures may occur in equilibrium and the ratcheting process will be more pronounced. If political economics of income distribution and rent
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seeking is considered, negative effects of unfair income distribution on economic development may be analyzed.
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Krugman, P. (1995a): "Growing World Trade: Causes and Consequences." Brookings Paper on Economic Activity 1: 327-362. Krugman, P. (1995b): "Technology, Trade, and Factor Prices." NBER Working Paper No. 5355, National Bureau of Economic Research. Krugman, P., and Lawrence, R. (1994): "Trade, Jobs and Wages." Scientific American 270: 44-49. Krugman, P., and Venables, A. J. (1995): "Globalization and the Inequality of Nations." Quarterly Journal of Economics 110: 857-880. Kuznets, S. (1955): "Economic Growth and Income Inequality." American Economic Review 45: 1-28. Lewis, W. (1955): The Theory of Economic Growth. London: Allen and Unwin. Li, H., and Zou, H. (1998): "Income Inequality is not Harmful for Growth: Theory and Evidence." Review of Development Economics 2: 318-334. Mokyr, J. (1993): "The New Economic History and the Industrial Revolution." In The British Industrial Revolution: An Economic Perspective, edited by J. Mokyr. Boulder and Oxford: Westview Press. Murphy, K., Schleifer, A., and Vishny, R. (1989): "Income Distribution, Market Size, and Industrialization." The Quarterly Journal of Economics 104: 537-564. Murphy, K., and Welch, F. (1991): "The Role of International Trade in Wage Differentials." In Workers and Their Wages: Changing Patterns in the United States, edited by M. Kosters. Washington, American Enterprise Institute. Palma, B. (1978): "Dependency: A Formal Theory of Underdevelopment or a Methodology for the Analysis of Concrete Situations of Underdevelopment?" World Development 6: 899-902. Puga, D., and Venables, A. (1998): "Agglomeration and Economic Development: Import Substitution vs. Trade Liberalisation." Centre for Economic Performance, Discussion Paper No. 377. Ram, R. (1997): "Level of Economic Development and Income Inequality: Evidence from the Postwar Developed World." Southern Economic Journal 64: 576-583. Ranis, G. (1988): "Analytics of Development: Dualism." In Handbook of Development Economics, edited H. Chenery and T. Srinivasan, vol. 1, 4.
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CHAPTER 6 PURSUIT OF RELATIVE UTILITY AND DIVISION OF LABOR*
Jianguo Wang a and Xiaokai Yang b * ^National University of Singapore
Monash
University
1. Introduction Adam Smith (1776) argued that the development of the division of labor is due to humankind's inclination for exchange. By contrast, Emile Durkheim (1933) argued that the division of labor may be the result of individuals' pursuit of relative economic standing. According to Durkheim, the division of labor is a result of the struggle for existence; different occupations can coexist without being obliged mutually to destroy one another because they pursue different objects. Frank (1984, 1985a,b) also argues that people's pursuit of relative economic standing generates a positive effect on specialization because individuals intend to open new fields to avoid competition and to rank themselves at higher positions. Henri Saint-Simon, Joseph Fourier, Robert Owen, and Karl Marx (e.g., 1844) and many modern management scientists believe that individuals normally dislike specialization but prefer diverse activities. However, the existing theories in the literature of this field have not explored the relationship between changes in preference for relative utility and the division of labor even in a descriptive stage, not to mention in the * Reprinted from Journal of Comparative Economics, 23 (1), Jianguo Wang and Xiaokai Yang, "Pursuit of Relative Utility and Division of Labor," 20-37, 1996, with permission from Elsevier. * We are grateful to two anonymous referees and to Yew-Kwang Ng, Robert Frank, Jeff Borland, Ben Heijdra, and Murray Kemp for helpful comments. Any remaining errors are ours. Ill
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context of general equilibrium. In this paper, we shall develop two general equilibrium models to explore the relationship between the degree of individuals' desire for relative utility and the development of the division of labor. Unfortunately, the neoclassical microeconomic framework cannot be used to achieve this for the following reason. In the neoclassical microeconomic framework in which a production function or technology, either with constant returns to scale or with increasing returns to scale, is employed to describe the relationship between output and inputs, total factor productivity is independent of the level of specialization and of the internal organization of a firm. This paper shall combine Yang's (1990) framework with consumerproducers, economies of specialization, and transaction costs with Wang's (1993) approach to modeling individuals' behavior pursuing relative utility to formalize classical thoughts about the relationship between pursuit of relative position and the division of labor. Yang's framework can be used to investigate the relationship between total factor productivity, the level of specialization, and trade dependence. Wang's (1993) approach, which is built on the literature of pursuit of relative economic standing (see, for example, Basmann (1988), Frank (1984, 1985a,b, 1989), Mezias (1988), Miner (1990), Ng and Wang (1993), Stadt et al. (1985), Schoeck(1969), Veblen (1899), and Veenhoven (1991)), can be used to explore equilibrium implications of pursuit of relative utility. The blend of the two analytical vehicles will enable us to inquire into the relationship between individuals' pursuit of relative utility and the development of the division of labor. Two key concepts in this paper, division of labor and relative utility, should be given here first. Roughly speaking, an economic organizational pattern is said to involve the division of labor if it allocates the labor of different individuals to different activities. Hence, the levels of specialization of individuals and the number of professional activities are two sides of the level of division of labor. Relative utility is the utility ratio between individuals, e.g., the ratio of person i's utility to person/s utility. Note that relativity of utility and relative utility are two distinct concepts. Relativity of utility means that formation of preferences is relative. We have the relativity of utility so long as individuals pursue relative
Pursuit of Relative Utility
113
economic standing, which includes relative utility but may include something else such as relative income. The main finding of our analysis is that, given the existing structure of individuals' preferences and the degree of increasing returns to specialization, (i) a stronger desire for relative economic standing, e.g., envy, will promote the level of division of labor and productivity if it reduces the differential between preferences for goods or if it weakens an individual's preference for variety of her productive activities; and (ii) the consumption of relative economic standing may or may not generate positive effects on the division of labor, depending on specific conditions. In the first model, with two goods and M consumer-producers (section 2), each person as a consumer prefers diverse consumption and as a producer prefers specialized production. An increase in the division of labor increases, of course, the number of transactions. Since there is a cost occurring from each transaction, i.e., transaction cost, a higher level of the division of labor must generate a higher level of transaction costs. The trade-off between economies of specialization and transaction costs implies that the level of division of labor is determined by transaction efficiency. If transaction efficiency is low, then individuals choose autarky because transaction costs outweigh economies of specialization generated by the division of labor. If transaction efficiency is sufficiently improved, then individuals choose the division of labor because transaction costs are outweighed by economies of specialization. Each individual's utility is a function of the quantities of goods consumed as well as a ratio of her utility level to others' utility levels. The behavior pursuing relative utility is equivalent to behavior with envy. This implies that a person's utility can be expressed as a function of the quantities of all goods consumed and others' utility levels, which have negative effects on her utility level if she pursues relative utility. Each individual maximizes her utility function given others' utility levels under a well-specified and enforced private property system so that stealing is impossible. The equilibrium relative prices of traded goods, relative numbers of different specialists, and equilibrium level of division of labor are then functions of the parameters that characterize pursuit of relative utility. The comparative statics of general equilibrium yield the
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equilibrium implications of individuals' pursuit of relative utility. Since, in the context of preference interdependence, an increase in the desire for relative utility implies a relative decrease in the desire for direct consumption, the increase in the desire for relative position will affect the structure of preferences for goods. The first equilibrium model with two goods is developed in section 2 to show that if an increase in the desire for relative position increases the differential between preferences for various goods, then the equilibrium is more plausibly to be associated with a lower level of division of labor. On the other hand, if an increase in the desire for relative position reduces the differential between preferences for various goods, then a higher level of division of labor is more plausibly the equilibrium. In the second model, section 3, the number of goods is assumed to be N instead of 2, and each individual is assumed to prefer a variety of her productive activities in addition to direct consumption and relative position. An increase in the desire for relative position will change the structure of preferences. If this increase weakens the preference for a variety of activities, then the equilibrium level of division of labor will be increased by the increased desire for relative position. 2. A Model with Two Consumer Goods and M Consumer-Producers 2.1. Specification of the model Consider an economy with M ex ante identical consumer-producers and two consumer goods. The self-provided amounts of the two goods are x and y, respectively. The amounts of the goods sold at the market are xs and ys, respectively. The amounts of the goods purchased in the market are xd and yd, respectively. The fraction 1 - k of a unit of goods purchased disappears in transit because of transaction costs. Hence, a person receives k when she buys one unit of goods. Each person is equipped with a system of production functions and an endowment constraint. xt + xf = Laxi,
yt + yf - Layi
(production function)
Pursuit of Relative
Lxj + Lyi = 1
Utility
115
(endowment constraint), (1)
where Lri is person i's labor share in producing good r. We define the share as person z's level of specialization in producing good r. The parameter a represents the degree of economies of specialization. This system of production functions displays economies of specialization if a > 1. This implies that, for a > 1, labor productivity of a good increases as a person's level of specialization in producing this good increases. Note that economies of specialization are individual specific or increasing returns are localized. rt + r/ is the quantity of good r(r = x, y) that is produced by person /. Due to transaction costs, krf is the quantity that is received by person i from the purchase of rf, and rt + ty is the quantity of good r consumed by person i. Hence, each consumer-producer's utility function is specified as1 f Y U^ix^kxfYiy^kyfY n^T >
(2a)
where J is a set of all individuals except person i, a and/? are respective preference parameters for the two goods, and y is the parameter that represents the preference for relative utility. We assume a + f3 + y -1 for simplicity or for normalization. However, this constant-returns- to-scale assumption is not essential. What is essential is that these preference parameters are interdependent. Even if we assume a + jB + y = g where g may be larger or smaller than 1, our main results will remain the same. 1
The strategic interactions are not taken into account in this model mainly because they do not make much difference since no uncertainty is considered and with a large population the strategic effect is negligible in the models. It may be argued that the utility function in (2a) involves circularity and information problems since, in a model of relative utility, Uj itself depends on Ut. Moreover, it might also be argued that individual;' cannot observe Uj. However, as shown in Wang (1993), the existence of envy is equivalent to the existence of pursuit of relative utility. If we start from a model allowing for real incomes as the source of envy, it can be converted into a model with utilities as the source of envy. Moreover, ex ante identical individuals who can only take others' utility levels as given when they make their optimum decision have no information problems for their decision making. Thus, individual / may know Uj via individual j's real income. Also, (2a) is not an acceptable formation if Mis a variable. However, since Mis a constant here, (2a) is acceptable. See Wang (1993) for the case of a variable M. In fact, relative utility may be more observable than relative income. For example, if we use one's smile frequency as one of her happiness indicators, obviously, her smile frequency is easier to be seen than her income.
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Obviously, our utility function is an application of the Cobb-Douglas log-linear hypothesis, and its functional form is very similar to what Khinchin (1957) discusses. However, we do not use the logarithm of the objective function mainly because its algebra is nearly the same, and the results will be the same. The utility function characterizing the pursuit of relative utility can be rearranged as N -r-|V(l-(M-l)/)
U,=
a
{xi+kx?) {yi+ky?y
(2b)
where M is the population size. This utility function characterizes an individual's envy, which can be derived from behavior pursuing relative position, characterized by the utility function in (2a). Let/? be the price of good y in terms of x, then person i's decision problem is - i/(i-(w-i)r)
a
{xi+kxt) {yi+kyfY\YlU U, = lx,+kxfYlv,+kyfY \\U..j xt+xf -If.,xi' L„.+L = 1
ys=La.
v. + J i
(utilityfunction)
Si
xf + pyf = xf + pyf
(production function)
yi
(endowment constraint) (budget constraint). (3)
2.2. Individuals' decision problems and corner solutions The model is similar to Yang's except that the pursuit of relative utility is allowed. Therefore, our analysis focuses mainly on the relationship between the pursuit of relative utility and the development of division of labor and skips those relationships analyzed by Yang. Due to the great number of ex ante identical individuals and individual-specific economies of specialization or localized increasing returns, no individual has monopoly power to manipulate market supply and demand and, therefore, to influence prices of traded goods. Thus, the regime here is a special Walrasian one: the numbers of individuals selling different goods play the same role as prices do in the conventional Walrasian regime.
Pursuit ofRelative Utility
117
According to Yang (1990), an individual's optimum decision is always a corner solution in the models of this kind. Facing many corner solutions, she must decide which one is optimal. Since the utility levels are discontinuous from one corner solution to another, the conventional marginal analysis works only within a corner solution but not for choosing the optimum decision out of many corner solutions. Yang (1990) has solved this problem by applying the Kuhn-Tucker theorem. He first identifies a set of candidates for the optimum solution, rules out the interior solution and many corner solutions from this set, and then identifies an optimum decision by comparing utility levels across many corner solutions. This total benefit-cost analysis across corner solutions in addition to the marginal analysis of each corner solution is referred to by Buchanan and Stubblebine (1962) as inframarginal analysis. To follow Yang's approach, it is necessary to adopt some important concepts from him. A pattern of economic organization has a higher level of division of labor than another pattern if all individuals' levels of specialization in the former pattern are higher than in the latter. A pattern of economic organization is said to be autarky if every individual produces her own consumption. A combination of zero and nonzero variables that is compatible with the Kuhn-Tucker conditions and the budget constraint is called a configuration. A combination of configurations that is compatible with the market-clearing condition is called a market structure or simply a structure. If n goods are traded, n - 1 relative prices and n - 1 relative numbers of individuals choosing different configurations are determined by n - 1 independent market clearing conditions and n - 1 utility equalization conditions across n configurations. The n - 1 relative number and the n - 1 relative prices define a corner equilibrium in a given structure. All the corner equilibria constitute a set of candidates for the general equilibrium. General equilibrium is defined as a fixed point that satisfies the following conditions, (i) Each individual uses inframarginal analysis, which means total benefit-cost across corner solutions in addition to marginal analysis for each corner, to maximize her utility with respect to configurations and quantities of each good produced, consumed, and traded for a given set of relative prices of traded goods and a given set of the number of individuals selling different goods; and (ii) the
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set of relative prices of traded goods and the set of number of individuals selling different goods clear the markets for traded goods and equalize utility for all individuals. According to this definition, the general equilibrium is the corner equilibrium that yields the highest per-capita real income because it satisfies the two conditions for general equilibrium and other corner equilibria do not satisfy condition (i). We will see in the following sections that computation of the equilibria is rather complicated and computational cost is quite high. In reality, equilibria are approached through some iterative improvements. However, this dynamic process must be described by a dynamic model (see Yang and Ng (forthcoming)) who show in a dynamic model that individuals can search for the efficient pattern of division of labor through the price system over time if pricing efficiency is not too low). In our static model, this dynamic iterative process is not described, and we can see only the outcome of the process. For a person's optimum decision, Yang (1990, p. 206) has proven the following lemma. Lemma: According to the necessary condition for an optimum decision, an individual does not produce and purchase a good at the same time, and sells at most one good. According to this lemma, there are only three configurations in the present model. Following an approach for solving for the equilibrium based on corner solutions developed by Yang (1990), we can solve for the equilibrium in two steps. First, we can solve for a corner solution for each configuration and a corner equilibrium for each structure. Then the general equilibrium is identified from the set of corner equilibria. A. Autarkic configuration Configuration autarky or simply A is a profile of zero and nonzero variables defined by xs=ys - xd=yd = 0, x,y, Lx, Ly > 0. In configuration A, each consumer-producer self-provides all goods she needs. Let xs = ys - xd = yd = 0 in (3), and person /'s decision problem for configuration A becomes
Pursuit ofRelative Utility
P maxC/, = L"*{\-L.T XI
XI
V
)
(
m
V JeJ
Yr'
119
i/(i-(Af-i)r)
(4a) J
The first-order condition dUldLxi = 0 yields the corner solution for structure A as
P
a a+p
n
' x =
L* =1-L* =x a + p' v
a
a + ft
(
y
v
P
(4b)
« + /?
where subscript / is dropped when no confusion is caused. Inserting (4b) into (4a) and using the factt/. = [/. for all i andy yields the per-capita real income in configuration A, UA, given by a a +p
U,
P
\"P
(4c)
a+p
B. Configuration(x\y) Configuration(x|y\ is defined by x, xs,yd> 0, Lx -1, xd - y = ys - Ly = 0 . A person choosing this configuration sells good x and buys good y. Inserting x, xs, yd > 0, Lx = 1 , xd = y = ys = Ly = 0 into (3), person f s decision problem for(x| jA is ,
maxC/,= (\-x!)a(kx;/pY
v y
-|i/(HM~1)r)
YlUj eJ
vJ
(5a)
• )
The first-order condition dUxi jdx\ = 0 yields the optimum decisions xs
- •
P
a + j3
y
d
—.
P
P{cc + PY
r
u
a ^af kp p(a + p) a+p
where Ux(p) is the indirect utility function for(x|_y).
,
(5b)
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J. Wang, X. Yang
C. Configuration (y\x) Configuration (y\ x) is defined by y, ys, xd > 0, Ly = 1, yd = x = xs = Lx = 0. A person choosing (y\x) sells good y and buys good x. Inserting y, ys, xd >0 , Ly = 1, yd -x = xs = LX = 0 into (3), person z's decision problem for (y\x\ is max £/,
(1-j/fteO 0 \Uuj)
•
( 6a )
The first-order conditionrfC/^./<s^f = 0 yields the optimum decisions a a + /3
r=
,
pa pa ' a + (3
„ ( U =
fi Y kpa a+p {a + P)
(6b)
whereUy(p)is the indirect utility function for(j|x). 2.3. Corner equilibrium and general equilibrium and its comparative statics A. Market structures and corner equilibria There are two structures in the model: autarky denoted by A and the division of labor denoted by D. Due to the absence of trade, the comer solution for autarky is a comer equilibrium. The comer equilibrium for structure D is determined by two conditions: the market clearing condition, which is established by equilibrating market demand and supply, and the utility equalization condition, which comes from the assumption of free entry and utility-maximization behavior. We can solve for the relative price p and the relative number of individuals choosing (x| y) and yy\ x) from the market-clearing and utility-equalization conditions. The utility-equalization condition Ux(p) = Uy(p) yields p* = £(^-«)/(1->') . The market-clearing condition Mxxs - Myxd yields M^ - ap/p ,2 where Mx and My are the respective numbers of 2
Mmust be an integer. Yang and Ng (1993) have shown that if the integer condition is not satisfied, the equilibrium may not exist.
121
Pursuit of Relative Utility
individuals selling good x and good y and M^ = Mx/My . The market-clearing condition for goody is not independent of that forx due to Walras' law. Inserting the corner equilibrium level ofp into indirect utility function Ux(p) or U (p) yields the corner equilibrium utility level for structure D as UD =
\a
P a+P) \a+P a
V
fc2aft/(a+fi)
(V)
B. General equilibrium and its comparative statics Since the general equilibrium is the corner equilibrium that generates maximum per-capita real income, which can be identified by comparing the utility levels across structures, hence, having compared UD with UA, we have
uD>uA
iff
r
\
(a-\)(a+p)l7P
a k>k° = a + (3
P
\
a +p
(a-\)(a+p)l2a
(8)
Expression (8) yields the following proposition. Proposition 1: Equilibrium is the division of labor if the coefficient of transaction efficiency k> k° and is autarky ifk < k°. Figure 1 provides an intuitive illustration of Proposition 1. Two individuals in Figure la are used to illustrate autarky. In fact there are more individuals than shown in Figure 1. Circles represent configurations. The lines represent flows of goods self-provided. In autarky there is no market demand and supply. Productivity is low because of a low level of specialization. However, transaction costs do not exist because people do not do business with each other. Figure lb represents the division of labor, structure D. The lines represent the flows of goods. For structure D, there is a local business community composed of two persons. There are two markets, one for x and the other for j / .
122
J. Wang, X. Yang
a
b
A yC 3 33f Xs
>V )
f market V. for x .
(yx
1 market
v
for
y >
Figure 1: (a) Autarky and (b) the Division of Labor
There are two distinct professional sectors. The per-capita output of each good is higher than in autarky because of a higher level of specialization for each person. But transaction costs are higher than in autarky too, because each person must undertake two transactions to obtain the necessary consumption. Moreover, the degree of interdependence between individuals, the degree of production concentration of each good, and the degree of integration of society are higher for structure D than in autarky. The most important implication of the inframarginal analysis and framework with consumer-producers, economies specialization, and transaction costs is to predict many macroeconomic phenomena on the basis of a sound microfoundation; see Yang and Ng (1993, Chaps. 17 and 18). For the static simple model in this paper, it is predicted that aggregate demand and supply, the extent of the market will shrink as transaction efficiency k is reduced due to, say, oil crisis. According to our theory, demand and supply and the division of labor, and the division of labor and the extent of market are two sides of the same coin. Hence, macroeconomic phenomena that relate to aggregate demand can be explained by a change in the parameter of pursuing relative position through its effect on the equilibrium level of division of labor. Unfortunately, due to the limitation of space, we need another paper to investigate the macroeconomic implications of the model.
Pursuit of Relative
123
Utility
The effects of the degree of individuals' desire for relative utility, y, on the development of the division of labor can be seen from the effects of a change in y on k° via its effect on a and fl . If the effect of an increase in y on k° is negative (positive), the requirement in transaction efficiency for the division of labor to be the equilibrium is lower (higher). A total differentiation of k° with respect to y yields dink0 da
d\nk°
d\nk° d/3 —, (9) dy da dy dfi dy wheredfijdy = -(dajdy)-1 due toa + fi + y-l. The division of labor is more plausibly the equilibrium for a stronger desire for relative utility3 if d In k°/dy < 0 and less plausibly to be equilibrium if J In k°jdy > 0. A differentiation of k° in (8) yields 51nA:° a-\ '.ta-S^-Zb. » \ >0 2 da fS a + j3 a a + fi =
+
dink0
and^-^-O
dfi
if£a>0,y
V(10)
'
which implies that (i) dlnk°jdy <0 , if a < fi , da/dy>0 , and/or d0/dy / ? , da/dy<0 , and/or dfijdy >0 ; (ii) dlnk° Idy>0 , if a0 , or if a> fi , da/dy>0, and/or dfijdy < 0 ; (iii) dlnk°jdy-0 ifa = fl, no matter what sign dajdy and dfijdy are; and (iv) for the case where both dajdy<0 and dfijdy <0 , d\nk°jdy may be positive or negative, depending on the degree of dajdy < 0 anddfijdy < 0 . It is also easy to see that91nA:°/5a<0. Hence, a greater degree of increasing returns to specialization promotes the division of labor. This deduction is summarized in the following proposition. Proposition 2: A stronger desire for relative utility promotes the development of division of labor if it reduces the differential between preferences for different goods. It has no effects on the development of division of labor if this differential is zero. It has negative effects on the 3
A stronger desire for relative utility means that the parameter for relative utility ( y ) is increased relative to other preference parameters (e.g., a and /J).
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J. Wang, X. Yang
development of division of labor if it increases the differential between preferences for different goods. If both da/dy < 0 and d/5/dy < 0, d In k^/dy may be positive or negative; i.e., a stronger desire for relative utility may or may not promote the development of division of labor, depending on the relative magnitude of da/dy < 0 to dfi/dy < 0. A greater degree of increasing returns to specialization promotes the development of division of labor. As we have seen, whether individuals choose autarky or the division of labor depends on the utility differenceU D -U A . UD-UA = 0 determines a critical value of transaction efficiency &° to generate the division of labor in a general equilibrium. If a stronger desire for relative utility lowers k°, it must promote the development of division of labor. But why does an increase in y lower k° only if it reduces the differential between preferences for goods? The reason is as follows. Because there exists a trade-off between a productivity gain from a higher level of division of labor due to increasing returns to specialization and a cost from more transactions, a differential between preferences for different goods, through its effects on the trade-off, will generate a tension, i.e., a changeable scope or room for the trade-off between allocational efficiency and organizational efficiency4 that restrains the development of division of labor. Because the smaller the differential between the preferences for different goods, the larger the scope for trading off allocational efficiency against organizational efficiency, a higher level of division of labor can be achieved by a change in y that reduces the differential. A change in y that increases this differential will slow down the development of the division of labor. When this differential vanishes, the division of labor can no longer be promoted by an increase in y, but it can still be made more favorable by a larger degree of increasing returns to specialization. However, if we allow a trade-off between work and leisure
4
In Yang (1990), a competitive equilibrium is defined as organization-efficient if the equilibrium level of the division of labor is the same as the Pareto-efficient level, while it is allocation-efficient if the equilibrium relative prices are the same as the price in the Pareto-optimum corner equilibrium.
Pursuit of Relative Utility
125
in the model, the development of division of labor can be generated by a change in y even if this differential equals zero. According to Proposition 2, some of the intuitions and conclusions of Marx, Durkheim, Veblen, and other economists are theoretically defensible, and others are indefensible. For example, Veblen (1899) and Frank (1985b) argue that the pursuit of relative economic standing, in general, generates social welfare losses, and Durkheim (1933) and Frank (1984, 1985a,b) argue that the pursuit of relative position, in general, promotes the level of specialization. Our model shows that these conclusions are not correct without qualification. Therefore, it becomes a matter for empirical economists to decide on the applicability of various conjectures to the historical record. 3. A Model with N Consumer Goods and M Consumer-Producers An increase in the desire for relative utility may generate implications for the level of division of labor via its effect on preferences for a variety of each person's activities. This section is devoted to the investigation of the effects in the context of an equilibrium model with N goods. Let us consider an economy with M consumer-producers and N consumer goods. The self-provided amount of good r by person i is xir. The amount sold in the market of good r by person i is x"ir .The amount purchased in the market of good r by person i is xdir. There are n traded goods and N- n nontraded goods. The transaction cost coefficient is 1 - k. Thus, kxir is the amount person i obtains when she purchases xfr . According to Lemma 1, each individual buys « - 1 goods, self-provides N - (w - 1 ) goods, and produces and sells 1 good. The amount consumed of good r by person i is xir +kxfr. The utility function is identical for all individuals (11) where a, a, and y e (0, 1) and aN + <J + y = 1; a and y are her preference parameters for consumer goods and for relative utility, respectively, and a is her preference parameter for the variety of her productive activities
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J. Wang, X. Yang
because N- (n - 1) is the number of self-produced goods, where n is the number of traded goods and n - 1 is the number of goods purchased by an individual. Here, there are two assumptions different from those for the previous model: (i) a consumer's preferences for all consumer goods are the same, and (ii) an individual has a preference for the variety of productive activities. Thus, the effects of the changes in preference differentials between different consumer goods from a stronger desire for relative utility on the development of division of labor cannot be investigated in this model. The reason for this assumption is that, in this general model, our analysis is to focus on the effects of the pursuit of relative utility on the relationship between preferences for consumer goods as a whole and preference for diversely productive activities, and in turn, on the development of division of labor. According to Karl Marx (1844) and other economists, individuals, indeed, have preferences for the variety of productive activities. Then an individual as a consumer-producer must find a balance between preference for diverse activities and preferences for consumer goods as a whole. Moreover, if an individual's tastes differ across goods, this general model cannot be solved. The production functions are similar to that in the preceding section Xir+K=L%> r=l,...,N,a>l.
(12)
The transaction cost coefficient 1 - k can be seen as a variable transaction cost since 1 - k is proportional to the quantity purchased. For the model with N goods, a fixed transaction cost is needed to ensure that the second-order condition for an interior value of the number of traded goods is satisfied in equilibrium. If the second-order condition is not satisfied, then the equilibrium is either autarky or complete specialization and gradual evolution of the division of labor is impossible. A fixed labor transaction cost coefficient for each purchase is assumed to be c, which can be interpreted as, for example, a fixed communication cost for each transaction between the Walrasian auctioneer and a person. Assuming that an individual spends a fixed fraction c of her available hours on each purchase, the labor share an individual spends in n - 1 purchases isLc = c(n - 1 ) . The endowment constraint of labor is given by
127
Pursuit of Relative Utility
L c +JTz, f r =l,
Z.fr,Z.c e[0,l].
(13)
The budget constraint for a consumer-producer selling good r is
where g e G , G is a set of all goods purchased, and pr and pg are the respective prices of good r and good g. In our symmetric model, we have assumed that goods 1,2,..., n are traded by individuals selling good r, and goods n + 1, n + 2,..., N are not traded (« e [1, AT)). Let / / be a set of all nontraded goods. Applying Lemma 1 cited in section 2.2 to the model, for a person selling good r, we have xir>0,
xl >0,
x
ig=xig=Lig=°>
x?r=0, 4
Lir>0,
> 0
%,A*>0, x>Ol=x$l=0
V
£
G G
'
V/zei/.
(15)
Hence, the decision problem for an individual selling good r becomes
m a X :c/ / =x-nK)°n^( gcG heH
jv w+i ( uY
- r riTT
\jeJUjJ
s.t. xir + xsir - Lair, xih =Laih, \/h&H Lc = c(n -1)
(production function)
(total fixed cost in trade)
Lir + ~Y^Lih+Lc=\->
(endowment constraint)
heH
prx\r = ]T Pgx?g
(budget constraint),
(16)
where pr is the price of good r, and the decision variables are xir,xl,xfg,Ljr,Lih,xih,and n. Having inserted all constraints into [/and differentiating[/, with respect toLir,xsir,xfg, individual optimal decisions are given by
Lir=n{\-Lc)/N, X* =Kl(n-\)
= Llln,
x°r=(n-l)Li/n xih
=[(l-Lc)/Nj
128
J. Wang, X. Yang 4 =(Pr/Pg)xl/(n-l)
= {Pr/Pg)Ll/",
VgeG,
(17)
where Lc = c(n -1) . Inserting these optimal solutions into Uj , and noticing pr = pg in a symmetric model, Ui can be expressed as a function of n \an
U,=
ka[n-\)
'l-c(n-l)'
N
aa(N-n)
N
x(N-n + iy.
(18)
Maximizing Ui with respect to n yields the optimal number of traded goods, n , for an individual selling good r dhxU,L = -a(]nn*+\) dn
aaNc — -+ • l-c(«*-l)
N~n'+\ +a\nk = F^0.
(19)
It is not difficult to identify the sign of dn*/dy from (19). Since the larger optimal number of traded goods, n , means the higher level of division of labor, we can use the sign of dn*jdy to investigate the effect of a stronger desire for relative utility on the level of division of labor. Because n cannot be solved explicitly, we must apply the implicit function theorem to (19) so that dn* __ dF/dy dy dF/dn'
(20)
where y = \- aN - a . The second-order condition for interior optimal solution n is dF/dn* = d2 In Ul /dn2 < 0 . Since we assume a, a, and^ are interdependent and the sum of them is a constant, we have F = F[a(y), o(y)]. Umce,dF/dy = (dF/da)(da/dy) + (dF/da) (da/dy). Differentiating F and using F = 0 in (19) yields dF/da = a/a (N - n +1) > 0. Also dF/da < 0 can be derived from (19). Therefore, dF — <0, dy dF n — >0, dy
da if
da <0<
(21a)
dy dy ..da da n if >0> . dy dy
(21b)
Pursuit of Relative
129
Utility
Expressions (21a) and (21b) together with the second-order condition for the interior n , implies that dn*
n
.. da
< 0 if
dy This leads us to
n
da
<0<
dy
dn* ;
dy
da > 0 if —
dy
dy
_. da >0>
.
(22)
dy
Proposition 3:5 A stronger desire for relative utility has positive effects on the level of division of labor n {or the number of traded goods) if it weakens the preference for variety of each person's productive activities and increases the preference for direct consumption of goods. Proposition 3 is intuitive. If a stronger desire for relative utility weakens the preference for the variety of activities, a decrease in a, individuals will prefer more specialization so that the equilibrium level of division of labor is promoted. Figure 2, where it is assumed that m = N=A, provides an intuitive illustration of the evolution of division of labor when an increase in the degree of the desire for relative utility reduces the degree of the desire for variety of a person's productive activities and increases the desire for direct consumption of goods. The lines signify goods flows. The small arrows indicate direction of goods flows. The numbers beside the lines signify goods involved. A circle with number / signifies a person selling good i. Figure 2a denotes autarky, where each person self-provides four goods, i.e., n - 1 = 0, because of a low degree of the desire for relative utility that leads to a high degree of the desire 5 Compared to Proposition 2, no difference in preferences across consumer goods is assumed for the general model from which Proposition 3 is drawn. That is to say, an individual has the same preference for all consumer goods. Hence, the effects of the size of preference differentials between different consumer goods on the development of division of labor cannot be investigated in this general model. Otherwise, the conclusions from this more general model must be more ambiguous. This assumption is made since our main purpose is to focus on the effects of the size of preference differential between all consumer goods as a whole and the variety of her productive activities on the development of division of labor (in fact, here our suggestion is based on, as many economists have argued, that "individuals could also have preferences for variety in their activities"). Also if we assume preferences are different across N goods, the model is not tractable. Of course, in reality, the preferences are not symmetrical and solving for the equilibria is a dynamic process of iterative improvement.
130
J. Wang, X. Yang
'$>,'$>.
'$'$
Figure 2: The evolution of division of labor, (a) autarky, n = 1, four communities; (b) partial specialization, n = 2, two communities; (c) complete specialization, n = 4, the integrated market.
for variety of productive activities. Figure 2b denotes partial specialization, where each person sells one good, buys one good, trades two goods, and self-provides three goods, i.e., n = 2, because of a higher degree of the desire for relative utility that leads to a lower degree of the desire for variety of activities. Figure 2c denotes extreme specialization, where each person sells and self-provides one good, buys three goods, and trades four goods, i.e., n = 4, because of a very high degree of the desire for relative utility that leads to a very low degree of the desire for variety of activities. Note that here there are multiple equilibria but all of them yield the same utility and symmetric comparative statics. Since our model is symmetrical, only the number of traded goods is relevant, and patterns of
Pursuit of Relative
Utility
131
trade do not make any differences. For instance, for an equilibrium structure in Figure 2b, it is indeterminate if goods 1 and 2 or goods 2 and 3 are traded. Also, it is indeterminate who is specialized in which profession. An infinitesimal differential in parameters of tastes, production, and transactions across goods and individuals will rule out the multiplicity of equilibria. Since there exists a positive relationship between the level of division of labor and productivity and trade dependence as shown in Yang (1990), Yang and Borland (1991), and Borland and Yang (1992), the following corollary can be derived from Proposition 3. Corollary: A stronger desire for relative utility has positive effects on labor productivity and trade dependence if it decreases the desire for a variety of each person's productive activities and increases the desire for quantities of consumption goods. It can be shown that as long as economies of specialization are individual specific, that is, increasing returns to specialization are localized, and individuals pursue their relative utility under perfect competition, the equilibrium is Pareto optimal. For detailed proof of this proposition, see Yang and Ng (1993) and Wang (1993).6 4. Concluding Remarks We have explored the relationship between the pursuit of relative utility and the development of division of labor in two equilibrium models in this paper. It is shown that the equilibrium level of division of labor depends on some critical level of transaction efficiency, which is in turn determined by preference parameters for consumption, for a variety of activities, and for relative utility. An increase in the degree of the desire for relative utility will change the structure of preferences for consumption 6
According to Debreu's theory of value, the existence of global increasing returns and/or externalities distorts market. However, Yang and Ng (1993) and Wang (1993) show that (i) if increasing returns to specialization instead of to scale and are localized to each of many individuals who have no monopoly power (this is just the case in this paper), a Walrasian regime prevails; and (ii) relative utility effects do not distort market under perfect competition but relative consumption effects and relative income effects do.
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and for a variety of activities. This increase will generate positive effects on the development of division of labor if it reduces the differential between preferences for different goods or if it reduces the desire for a variety of each person's productive activities. Our finding also means that the consumption of relative position may or may not generate positive or negative effects, depending on specific conditions. Thus, some of the intuitions and conclusions of Marx, Durkheim, Veblen, and other economists on the relationship between pursuit of relative position and division of labor are not correct without qualification.
References Basmann, Robert L., Molina, David J., and Slottje, Daniel J., "A Note on Measuring Veblen's Theory of Conspicuous Consumption." Rev. Econom. Statist. 70, 3:531-535, Aug. 1988. Borland, Jeff, and Yang Xiaokai, "A New Approach to Economic Organization and Growth." Amer. Econom. Rev. 82, 2:460-482, May 1992. Buchanan, James, and Stubblebine, Wm G., "Externality." Economica 29, 116:371-384, Nov. 1962. Durkheim, Emile, The Division of Labour in Society. New York: The Free Press, 1933. Frank, Robert H., "Are Workers Paid Their Marginal Products?" Amer. Econom. Rev. 74, 4:549-571, Sept. 1984. Frank, Robert H., "The Demand for Unobservable and Other Non-Positional Goods." Amer. Econom. Rev. 75, 1:101-116, Mar. 1985a. Frank, Robert H., Choosing The Right Pond: Human Behaviour and The Quest for Status. New York: Oxford Univ. Press, 1985b. Frank, Robert H., "Frames of Reference and The Quality of'Life." Amer. Econom. Rev. 79, 2:80-85, May 1989. Khinchin, Aleksandr I., Mathematical Foundations of Information Theory. New York: Dover, 1957.
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Marx, Karl, In Dirk J. Struik, Ed.; Martin Milligan, Trans., Economic and Philosophic Manuscripts of 1844. New York: International Publishers, 1971. Mezias, Stephen J., "Aspiration Level Effects: An Empirical Investigation." J. Econom. Behav. Organiz. 10, 4:389^100, Dec. 1988. Miner, Frederick C , Jr., "Jealousy on the Job." Personnel. J. 69, 4:88-95, Apr. 1990. Ng, Yew-Kwang, and Wang, Jianguo, "Relative Income, Aspiration, Environment Quality, Individual and Political Myopia: Why May the Rat-Race for Material Growth Be Welfare-Reducing." Math. SocialSci. 26, 1:3-23, July 1993. Schoeck, Helmut, Envy: A Theory of Social Behaviour. New York: Harcourt, Brace & World, 1969. Smith, Adam, Reprint, In E. Cannan, Ed. An Inquiry into The Nature and Causes of The Wealth of Nations. Chicago: Univ. of Chicago Press, 1976. Stadt, Huib van de, Kapteyn, Arie, and Geer, Sara van de, "The Relativity of Utility: Evidence from Panel Data." Rev. Econom. Statist. 67, 2:179-187, May 1985. Veblen, Thorstein, The Theory of The Leisure Class. New York: Macmillan Co., 1899. Veenhoven, Ruut, "Is Happiness Relative?" Social Indicators Res. 24, 1:1-34, February 1991. Wang, Jianguo, "Pursuit of Relative Economic Standing," Ph.D. Dissertation, Department of Economics, Monash University, 1993. Yang, Xiaokai, "Development, Structure Change, and Urbanisation." J. Develop. Econom. 34, 1-2:199-222, Nov. 1990. Yang, Xiaokai, and Borland, Jeff, "A Microeconomic Mechanism for Economic Growth," J. Polit. Econom. 99, 3:460^182, July 1991. Yang, Xiaokai, and Ng, Yew-Kwang, Specialization and Economic Organisation: A New Classical Microeconomic Framework, Contributions to Economic Analysis Series. Amsterdam: North Holland, 1993. Yang, Xiaokai, and Ng, Yew-Kwang, "Specialization, Information, and Growth: A Sequential Equilibrium Analysis."/. Econom. Behav. Organiz., forthcoming.
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Part 4
Urbanization
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CHAPTER 7 DEVELOPMENT, STRUCTURAL CHANGES AND URBANIZATION*
Xiaokai Yang* Monash
1.
University
Introduction
As far as the problem of production was concerned, Adam Smith (1776) and Allyn Young (1928) emphasized the productivity implications of economic organization (the division of labor). Neoclassical microeconomics cannot explore such implications for the following reason. For production functions with constant returns to scale, an agent's productivity of a good is not greater when he produces only this good than when he produces many goods. On the other hand, production functions with increasing returns to scale cannot be used to characterize the level of specialization within a firm. The concept of economies of scale presupposes a complete separation of pure consumers from pure producers. 'Scale' relates to a firm which is a pure producer, but is irrelevant to a pure consumer. The separation of pure producers from pure consumers is a basis of Debreu's theoretical framework and neoclassical microeconomics. This artificial separation has perhaps misled economic theory. In autarky, there is neither a pure consumer nor a pure producer; each individual is a
Reprinted from Journal of Development Economics, 34 (1-2), Xiaokai Yang, "Development, Structural Changes and Urbanization," 199-222, 1990, with permission from Elsevier. * The author is grateful to Gene Grossman, Barry Nelebuff, Edwin Mills. Yew-Kwang Ng, Roslyn Anstie, and two referees for their comments. Thanks also go to the Ford Foundation and the Open Society Fund for financial support. The author is responsible for the remaining errors. 137
138
X. Yang
producer/consumer. The division of labor will increase the portion of a person's production that is not consumed by himself (i.e. the portion sold to other people) and increase the portion of a person's consumption that is not produced by himself (i.e. the portion purchased from other people). We can view this change as an increase in the degree of separation between production and consumption though each person is a producer/consumer even in the division of labor. The degree of such separation depends on the level of division of labor (or inversely on the degree of self-sufficiency). As to endogenizing the level of division of labor and thereby the degree of such separation, Debreu's framework is irrelevant since in his framework, pure consumers are completely separated from pure producers and the degree of the separation of consumption from production cannot be defined. In order to capture the ideas of Smith and Young, this paper specifies production functions for each producer/consumer such that an individual's productivity increases with the level of specialization and the aggregate transformation curve for the whole economy depends positively on the level of division of labor. There are several implications of this method of specifying production functions for the theory of equilibrium. First, each individual is a producer/consumer. He must decide how many goods are self-provided, i.e. what is the level of specialization. Hence, the model in this paper can be used to endogenize the level of division of labor. According to conventional microeconomics, pure consumers cannot choose the level of specialization since they must buy all goods from firms. Second, a Cobb-Douglas utility function is specified for each producer/consumer. Consequently each individual as a consumer prefers diverse consumption and as a producer prefers specialized production. This implies that the division of labor will incur great transaction costs. Therefore, there is a trade-off between economies of specialization and transaction costs. In other words, our method of specifying production functions makes the level of division of labor crucial for productivity; while transaction efficiency is critical for the determination of the level of division of labor. Because of increasing returns to specialization, the production possibility frontier (PPF) is associated with extreme
Development,
Structural Changes and Urbanization
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specialization. Extreme specialization will, however, incur prohibitively great transaction costs since people prefer diverse consumption. Hence, the welfare frontier may differ from the PPF.1 A natural conjecture is that a competitive equilibrium will balance the trade-off between economies of specialization and transaction costs, and that improvement of transaction efficiency will move the equilibrium closer to the PPF, resulting in an increase in the division of labor. The major purpose of this paper is to prove this conjecture. In other words, Smith's conjecture of the 'invisible hand' and his insights into increasing returns to specialization will be reconciled in this paper. A crucial assumption leading to this result is that labor is specific for each person who is able to produce all goods. This assumption combined with the assumption of free entry ensures the first welfare theorem even if there exist increasing returns to specialization. Intuitively, we can see that if increasing returns to specialization are specific for each individual, economies of scale are limited. Since production functions are specified for each producer/consumer, prices are determined by the numbers of individuals selling different goods. This number cannot be manipulated by any individual because of the assumption of free entry and the assumption that each person is able to produce all goods. Hence, nobody is able to manipulate prices. Nevertheless, if labor can be divided in fine detail and the population size is very large, economies of division of labor may be very large. Hence, a competitive market may be compatible with the substantial economies of the division of labor. Therefore, we can use this method to develop the concepts of equilibrium and Pareto optimum in relation not only to the resource allocation for a given level of division of labor, but also to the determination of the level of division of labor and productivity. The implications of this method for other fields of economic analysis are important. The level of division of labor based on increasing returns to specialization is intimately related to the extent of market, trade dependence, trade pattern, market structure, and economic structure, so these can be made endogenous in our model. Productivity is related to 1 According to conventional microeconomics, the welfare frontier coincides with the PPF.
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the level of division of labor which depends on transaction efficiency, which is in turn affected by urbanization, government policies, and institutional arrangements. Hence, our model can be used to investigate the impacts of urbanization, government policies, and institutional arrangements on the equilibrium level of division of labor (related to the market structure, trade dependence, and so on) and productivity. This paper shows that the equilibrium trade volume depends positively on the absolute degree of increasing returns to specialization in production and transaction, and negatively on the average distance between a pair of neighbors. The trade pattern is determined by the relative degree of increasing returns to specialization in producing different goods and relative preference for different goods. In addition, it is shown that increases in diversification of the economic structure, concentration of production, integration of the economy, specialization, and the output share of roundabout productive activities are different versions of the evolution of division of labor resulting from improvements of transaction efficiency which are in turn caused by urbanization, liberalization policies, or changes in institutional arrangements. Many economists have proposed similar ideas. Nevertheless few among them have been successful in formalizing them. On the other hand, the formal models proposed by mainstream economists are often inconsistent with these ideas. This may be due to the difficulty of formalizing the ideas of Smith and Young. For example, many economists [see, e.g. Helpman and Krugman (1985) and Herberg and Tawada (1982)] point to problems which are considered to make an equilibrium model with increasing returns to specialization and transaction costs unmanageable. Such problems include the issue of corner solutions based on increasing returns to specialization, the problem of infinite combinations of individual corner solutions in solving for equilibrium, notorious complications in dealing with indexes of variables in models with transaction costs, and the problem of existence of equilibrium. Formalizing the notion of increasing returns to scale is much easier than formalizing the notion of increasing returns to specialization. This might explain why it is hard to find microeconomic equilibrium models that formalize the theory of production proposed by Smith and Young.
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In this paper, we try to formalize the essence of the ideas of Smith and Young as well as to keep an equilibrium model tractable. Our techniques for achieving these two goals are to specify a specific transaction technology and to devise a multiple-step approach to handling the issue of combinations of corner solutions. In order to get around the problem of the existence of equilibrium, we propose a specific model. Using this model, we can prove the existence of equilibrium although it is impossible to reach a general conclusion on this issue for a model with increasing returns to specialization. Fortunately, these measures are not only effective in keeping the model tractable, but also useful in working out the meaningful comparative statics of equilibrium. Many interesting economic phenomena which are not addressed in the conventional theory can be explained by the kind of model in this paper. This paper is organized as follows. Section 2 sets out a model with three goods. Sections 3-7 develop a multiple-step approach to handling the model with increasing returns to specialization. Section 3 solves for the corner solutions for the individual decision problem. Section 4 solves for all candidates for equilibrium in various market structures. Section 5 solves for the restricted Pareto optimum in each market structure and the full Pareto optimum. Section 6 investigates the relationship between equilibria and the Pareto optima. Section 7 solves for an equilibrium and investigates its comparative statics. Some simple conclusions are summarized in the final section. 2.
A Model with Three Goods
Let us first consider an economy with M consumers/producers and three consumer goods. The self-provided amounts of these goods are x, y, and z, respectively. By self-provided we shall mean that quantity of a good produced by an individual for his own consumption. The amounts of these goods sold at the market are xs, /, and z\ respectively. The amounts of these goods purchased in the market are xd, y\ and z , respectively. An 'iceberg' type of transaction technology is characterized by the coefficient k. Fraction i of a shipment disappears in
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transportation. Thus, (1 - k)xd, (1 - k)yd , and (1 - k)zd are the amounts a person receives from the purchases of three goods, respectively. Furthermore, we assume that (1 - k) depends on the quantity of labor used in transaction. (1 - k) can be viewed as transaction service. Such services are categorized into self-provided ones and traded ones. Let 1 - k = T + Td where T is the self-provided quantity of transaction service and T1 is the quantity purchased of transaction service. The more transaction service T + Td, the greater portion of a purchase is received by its buyer. Here, T + Td < 1. Signifying the quantity sold of transaction service by Ts , transaction technology and production functions are thus given by x + xs=Lax,
y + ys=Lby,
z + z"=Lcz,
T + TS=L'T,
(2.1a)
where x + x5, y + ys, z + zs, and T + TS are the output levels of four goods and service, respectively. Ls is the amount of labor used in producing good (or service) s where s - x,y,z,T . (2.1) is assumed to be identical for all individuals. In such iso-elasticity production functions parameters a, b, c, and t characterize the returns to specialization. If a, b, c,t>\, then there are increasing returns to specialization. Adopting the concept of localized technology proposed by Sah and Stiglitz (1986), we assume that the total quantity of labor available for an individual is specific for him. Let this quantity be one; there is an endowment constraint of the specific labor for an individual: Lx+Ly+Lz+LT=l,
0
i = x,y,z,T.
(2.1b)
This method of specifying production functions is substantially different from the conventional one. This system of production functions differs from the production functions associated with the U-shaped average cost curve and those with global increasing returns to scale. The production possibility frontier (PPF) of this system of production functions is associated with extreme specialization. Therefore, if there is no transaction cost, equilibrium is the extreme division of labor (each individual has extreme specialization and different individuals specialize in producing different goods). In our model the reason that people prefer some division of labor is that internal economies of scale are very limited because labor is specific
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for each individual. This point distinguishes increasing returns to specialization from increasing returns to scale. Assume, further, that the transaction service T (or 7t/) is related only to the quantity traded and there are transaction costs related to the distance between a pair of trade partners; thus there is a location problem. Suppose that all people are evenly located and the geographic distance between a pair of neighbors is a constant. The distance between a pair of trade partners may differ from the distance between a pair of neighbors. If all trade partners of an individual are located in a circle with radius R and with his location as the origin, it can be shown that the average distance between this individual and his trade partners is proportional to R and the number of these trade partners N is proportional to R2. Hence, the average distance between this individual and his trade partners is proportional to yJN. If all the trade partners supply different goods to this individual, the number of traded goods for him is n = N + 1. For simplicity, we assume that the number of trade partners of an individual, N=n-\, where n is the number of traded goods for him. Assume that the transaction cost coefficient K characterizes the transaction cost related to the average distance between a pair of trade partners; then the K fraction of (T + Td)xd or (T + Td)yd , or (T + Td)zd disappears on the way from a seller to a buyer. The relation between K and the number of trade partners of an individual, N, is given by K = s4N,
(2.2)
where s is a constant depending on the distance between a pair of neighbors and n. Taking (2.2) into account, the amounts consumed of the three goods are x + (l-K) (T + Td)xd , y + (\-K) (T + Td)yd , and z + (l-K) (T + Td)zd , respectively. The utility function is identical for all individuals: U = [x + (l-K)(T x[z + (l-K)(T
+ Td)xdJ[y + Td)zdJ,
+ (l-K)(T
+
Td)ydJ (2.3)
where0
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We assume free entry for all individuals into any sector and that M is large. These assumptions imply that individuals treat prices parametrically. 3.
The Individual Optimal Decision
This section and the four sections to follow are devoted to devising a multiple-step approach to handling the model with increasing returns to specialization. In our model, the individual decision problem implicitly includes three problems: to choose the optimum level of specialization (in other words, the optimal number of traded goods is a decision variable), to choose the optimal composition of traded goods, and to choose the optimal quantities of consumption, production, and trade. This problem is unmanageable in a step. If there are increasing returns, some variables will take zero values in the individual optimal decision and in equilibrium. Hence, an individual needs to enumerate all possible combinations of zero and non-zero variables before identifying his optimal decision. Also we need to enumerate all possible combinations of these individual combinations of zero and non-zero variables before identifying the Pareto optimum and equilibrium although we can exclude some of these combinations from the list of candidates for equilibrium by application of the Kuhn-Tucker theorem (as shown in Appendix A). There are five steps in solving for equilibrium. This section will solve for individually rational decisions for given prices and for each combination of zero and non-zero variables. Section 4 solves for the candidate for equilibrium (or corner equilibrium) for each combination of the above individual combinations. Section 5 solves for the Pareto optimum candidate for equilibrium. Section 6 investigates the relationship between equilibrium and the Pareto optimum. Finally, equilibrium and its comparative statics are solved. The individual decision problem is maxU = [x + (\-K)(T
+ Td)xdJ[y
x[z + (\-K)(T + Td)zdJ,
+ (l-K)(T
+ Td)y<
(3.1)
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s.t. x + xs = Lax, y + ys =Lby, z + zs =LCZ S
T + T =L'T
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(production function), (transaction technology),
Lx + L +Lz+LT=\
(endowment constraint), d
pTT> + Pxx° + pyy + pzz° = pTT + pxx" + pyyd + pzzd (trade balance), where/,/*,/'', and Lt (i = x,y,z,T) are decision variables, which may take on zero or positive values. pt is the price of good (or service) /. If a, b, c, t > 1, the optimal decision is certainly a corner solution. By combination of zero and non-zero values of the variables, there are several possible corner solutions. We shall call such a combination a 'structure'. An individual needs to enumerate and compare utilities in all structures before choosing a structure. Therefore, an individual must solve for the corner solution for each structure and his decision making process consists of two stages. In the first stage all structures are enumerated. An individual solves for the efficient allocation (how much should be produced, consumed, and traded of each good) for given prices and for each structure. In the second stage, he decides what should be produced, and what should be sold and purchased, i.e. which structure should be chosen. Section 4 will discuss this problem.2 There are 29 structures of four general types: (1) Autarky (x,y,z), i.e. an individual self-provides three goods. For this structure xs = y* = zs = xd = yd = zd = T = Td = Ts = LT = 0. In other words, the amounts sold and purchased of the three goods are zero, as are the amounts sold, purchased, and self-provided of transaction services. (2) Structure (i/j), i.e. an individual sells good (or service) / and purchases good (or service)/, i,j = x,y,z,T. For such structures ks =kd =id =j = j s =Lj=0
2
for
k*i,j,
This two-stage set up is just for convenience of exposition. In fact all these decisions are determined simultaneously.
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where index k denotes the goods other than goods / and j . By two permutations of four factors, we obtain 12 structures of this type: (xly), (y/x), (x/z), (z/x), (y/z), (z/y), (T/x), (x/T), (T/y), (y/T), (T/z), and (zIT). (3) Structure (iljk), i.e. an individual sells good (or service) i and purchases goods (or services) j and k. i, j , k = x, y, z, T. For such structures rs =rd =id =j = k = j s =ks =Lj=Lk=0
for
r*i,j,k,
where index r denotes the goods other than goods i,j, and k. There are 12 structures of this type: (xlyz), (x/yT), (x/zT), iylxz), iylxT), iylzT), (zlxy), (zIxT), (zlyT), (Tlxy), (Tlxz), and (T/yz). (4) Structure (i/jkr), i.e. an individual sells good (or service) i and purchases goods (or service) j , k, and r, i, j , k, r = x, y, z, T. For such structures ^ = j = k = r = j s = ks = rs = Lj =Lk=Lr=0
for j,k, r ± i.
There are four structures of this type: (xlyzT), (y/xzT), (z/xyT), (T/xyz). Appendix A has proven Proposition 1: According to the necessary condition for an optimum decision, an individual does not produce and purchase a good at the same time, and sells only one good {if any). This proposition implies that the remaining structures, e.g., (ijk/ijk), (ij/jk), (ij/k), etc. do not satisfy the necessary condition for maximizing individual utility. Hence, these structures will not be concerned. Letting the relevant variables in problem (3.1) take on a zero value, an individual can solve for his optimal decisions for each structure. These individual decisions include individual supply, which is a constant depending on a, b, c, a, P, y, t, and s, and individual demand, depending positively on the relative prices of the goods he sells to the goods he purchases, and his supply. Inserting the individual optimal decision for each structure and for given prices into the utility function gives an indirect utility function for each structure. The indirect utility functions differ from structure to structure although the original utility function is identical for all individuals. In an indirect utility function, the number of relative prices is
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one less than the number of traded goods. Here there are 29 alternative structures for each individual that might be rational. 4. The Markets and Candidates for Equilibrium (Corner Equilibria) As in solving for the individual decisions, if there are increasing returns, we need to enumerate all combinations of structures before identifying the equilibrium and the Pareto optimum. This section first investigates how structures are combined to constitute markets. We will then enumerate all combinations of structures and solve for the 'candidates' for equilibrium (corner equilibrium). The corner equilibrium is an analogue to the corner solution in the optimization problem for an individual. It is a concept used to solve for the equilibrium in a calculation of several steps. Of course, a corner equilibrium would never come into being if it was not a full equilibrium3. Defining a combination of several mutually consistent structures as a market, there are many market configurations, such as those shown in Figure 1. Figure 1(a) is a market that combines structure (x/y) (an individual sells x and buys y, and self-provides z and T) and structure (y/x) (an individual sells y and buys x, and self-provides z and 7). We refer to this market as P. Figure 1(b) is a market that combines structure (x/yz) (an individual sells x and buys y and z, and self-provides 7), structure (y/x) (an individual sells y and buys x, and self-provides z and 7) and structure (z/x) (an individual sells z and buys x, and self-provides y and T). We refer to this market as B. Note that some structures are not mutually consistent, e.g., structure (x/y) and (x/z) are not mutually consistent, i.e. there is no demand for x and no supply of y and z although there are supply of x and demand for y and z for a combination of these two structures. Therefore, a combination of (x/y) and (x/z) cannot constitute a market.
3
Many papers on international trade, e.g. Ethier (1986), show that in models with increasing returns to scale, there are multiple equilibria and some of them are unstable. In our model all corner equilibria except the Pareto optimum one are not full equilibria. This point distinguishes our model from the models with increasing returns to scale.
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(a) Market P
(b) Market B Figure 1
In the analysis of competitive equilibrium, we will use the concept of a basic market. A basic market is defined as a market for which one cannot obtain another market by dropping any of its structure. In Figure 1, the two markets are basic markets. However, a combination of structures (x/yz), iy/xz), (y/x) and (z/x) is not a basic market because we can obtain market B in Figure 1(b) by dropping structure (y/xz). Many structures can be combined with a basic market to obtain a new, non-basic market. For example, by combining market B with any of the other six structures which do not involve trade in T and are not in market B we can obtain ^
C{ - 26 markets, where C( isj combination of
six factors. For a basic market, the number of traded goods equals the number of structures. For a non-basic market, the number of traded goods is smaller than the number of structures. We will enumerate all possible markets, and solve for all candidate equilibria (corner equilibria), then find the Pareto optimum corner equilibrium and the full equilibrium.
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From the individual optimal decisions, the individual supply and demand functions for each structure can be derived. Let Mt be the number of individuals selling good i; market clearing conditions can be specified for each basic market. For example, if the market is P, a combination of structure (x/y) and structure (y/x), shown in Figure 1(a), there are the market clearing conditions in market P: Mxxs = Myxd,
Mxyd =Myys,
(4.1 a)
where Mx and My are the numbers of the individuals selling x and y, respectively. The individual budget constraints for the two component structures are Pvx'=y,
Pvx*=y,
(4.1b)
where pxy=pxi'py. One of these two equations implies the other due to Walras' law. From (4.1a) and (4.1b), it can be derived that pxy=ydlx°=ylrxs,
(4.2)
where r = Mx I My is the ratio of the number of individuals choosing structure (x/y) to the number of individuals choosing structure (y/x). We need solve only for the relative numbers of individuals choosing different structures because we can derive Mx = rM/(l + r) and M - Mj (l + r) from Mx+ My = M. Hence, Mx and My can be found if r is known. According to the individual optimal decisions, xs and ys are constants depending upon a, b, c, a, /?, y, s, and /. Hence, (4.1b) implies p^ depends inversely on r. Inserting (4.2) into indirect utility functions in structures (x/y) and (y/x), the utilities may be expressed as functions of r. Setting U(x/y) = U(y/x), we have U(x/y) = r-f>G(xly) = r"G(ylx) = U(y/x),
(4.3)
where G's depend on a, b, c, a, ft, y, t, and s. The intersection of U(x/y) and U(y/x) determines a corner equilibrium value of r. Inserting the value of r back into the utility function, (4.3) gives a corner equilibrium utility U*, which is real income as well as the real returns to labor in this market because of the assumption that each person has one unit of labor.
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For the category of non-basic markets, there is the following proposition: Proposition 2: There does not exist a corner equilibrium for any non-basic market. For a non-basic market, there are m structures and n traded goods, and m > n. According to Appendix A, in any optimal structure an individual sells at most one type of good. This implies that the types of structures selling the same type of good exceed one if m > n. The necessary conditions for a corner equilibrium lead to m - 1 conditions of utility equalization. These m - \ equations contain n - 1 relative prices. Noting the fact that these m - 1 equations are log-linear, nonhomogeneous, and independent of one another, this system has no consistent solution since m > n. Therefore, Proposition 2 has been established. Noting the following two points, we can prove the existence of a corner equilibrium for a given basic market. (A) For a given basic market with n traded goods and n component structures, the individual supplies of all traded goods are constants and the individual demands for all traded goods are functions of relative prices. The indirect utility functions are determined only by relative prices of the traded goods. Letting the n indirect utility functions equal one another, we obtain (n - 1) log-linear equations containing (n - 1) relative prices. Again, noting that these equations are non-homogeneous and independent of one another, we can solve for a vector of log-relative prices. The relative prices are positive. (B) Given the relative prices solved in (A), (n - 1) independent market clearing conditions can be transformed into a system of equations containing (n - 1) relative numbers of individuals selling the different goods. This system looks like a system of linear equations associated with an input-output system. For example, in the market with three traded goods, such a system is
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-filia + -y(a + r)
1
b^M^y s
V
151
(4.4a)
Mzxpzxz
If the number of goods traded by individuals selling good z is less than that traded by the whole market, e.g. structure (z/x) trades two goods and structures (x/y, z) and (y/x, z) trade three goods, then (4.4a) becomes MyxPyxys
1
fx^
(4.4b)
where My= MJMj is the relative number of individuals selling good i to those selling good j . Myx and Mzx are unknown. It is easy to see that the solution vector (Myx, Mzx)' for this kind of system of linear equations is positive. Considering the positiveness of equilibrium relative prices shown in (A), the positiveness of the solution of (4.4) guarantees the existence of corner equilibrium in a basic market. This leads us to Proposition 3:
There exists a corner equilibrium for any basic market.
There are multiple corner equilibria. All these corner equilibria comprise a set of the candidates for equilibrium. These candidates satisfy the following conditions (i) All excess demands for goods are zero for uniform positive relative prices of goods and a uniform positive price of labor (the real returns to labor). (ii) Individuals maximize their utilities for given prices and for a given basic market. Note that we have not yet imposed full maximization by individuals at the moment since this can be done only when all candidates for equilibrium are enumerated. By enumerating all corner equilibria, we will find the Pareto optimum corner equilibrium and prove that it is a full equilibrium in the next two sections.
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The Pareto Optimum Corner Equilibrium
This section first proves that a corner equilibrium is Pareto efficient for a given basic market. Then, we prove that all non-basic markets are neither equilibria nor Pareto optimal. Finally the relationship between the real return to labor in a corner equilibrium and the preference and technology parameters will be investigated. If the market is P as shown in Figure 1(a), i.e., some individuals choose structure (x/y) and other individuals choose structure (y/x), then we can derive the necessary conditions for a restricted Pareto efficient allocation from the problem in Appendix B. By 'restricted', we shall mean that the market is given. From the necessary conditions of that problem, it can be shown that the corner equilibrium found in sections 3 and 4 is the restricted Pareto optimum for the market. Following this procedure, we can show that each corner equilibrium is the restricted Pareto optimum for a given basic market. Moreover, Appendix B has proven Proposition 4: Each corner equilibrium is Pareto efficient for a given basic market and all non-basic markets cannot satisfy the necessary conditions for the Pareto optimum. Combining this with Proposition 2, we conclude that neither equilibrium, nor the Pareto optimum is associated with a non-basic market. This leads us to Proposition 5: Both candidates for an equilibrium and for the Pareto optimum market are associated with some basic market. By comparing real returns to labor in different basic markets, we can find the Pareto optimum corner equilibrium: it is the corner equilibrium with the maximum real return to labor. This Pareto optimum corner equilibrium depends upon t, s and the differences among (a,a), (b, /?), (c, y). Given the number of traded goods, it can be shown that (i) the corner equilibrium including as traded goods those with larger preference
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parameters is Pareto superior to that including as traded goods those with smaller preference parameters; and (ii) the corner equilibrium including as traded goods those with a high return to specialization is Pareto superior to that including as traded goods those with a small return to specialization, (i) and (ii) lead us to Proposition 6: The corner equilibrium including as traded goods those with a large preference parameter has a greater real return to labor, and the corner equilibrium including as traded goods those with higher returns to specialization has a greater real return to labor. Proof: See Appendix C. Assume that a = b = c = t and a = /? = y, then differences in the real returns to labor in various markets depends only upon the number of traded goods and service. The composition of traded goods has no effect on the real returns to labor. With this assumption, there are five possible market configurations: (1) Autarky. We refer to it as market A. This market consists of structure (x, y, z). (2) Partial division of labor in production. We refer to it as market P. This market consists of structure (x/y) and (y/x). (3) Complete division of labor in production. We refer to it as market C. This market consists of structure (x/yz), (y/xz) and (z/xy). (4) Partial division of labor in production and transaction. We refer to it as market PT. This market consists of structure (x/yT), (y/xT), and (Tlxy). (5) Complete division of labor in production and transaction. We refer to it as market CT. This market consists of structure (xlyzT), (ylxzT), (z/xyT), and (T/xyz). Having solved the corner equilibria in these five markets and compared the real returns to labor, we obtain
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Proposition 7: (1) For s (the average distance between a pair of neighbors) < 0.54, the real returns to labor in markets P and PT cannot be the maximum. (i) Market A has the maximum real return to labor if a < 0.41 0.251og(l-1.414s). (ii) Market C has the maximum real return to labor z/0.41 - 0.251og (1 - 1.4145) < a < 2.77 - 1.091og (1 - 1.732*). (Hi) Market CT has the maximum real return to labor if 2.11 - 1.091og (1 - 1.7325) < a. (2) For 0.54 < s < 0.58, the real return to labor in market P cannot be the maximum. (i) Market A has the maximum real return to labor if a < 0.41 0.25 log (1-1.4145). (ii) Market C has the maximum real return to labor ifOAl - 0.251og (1 - 1.414s) 0.71, market A has the maximum real return to labor for any value of a.
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Note, the Pareto optimum corner equilibrium has the maximum real return to labor. The next section will prove that all non-Pareto optimum corner equilibria are not full equilibria and that the Pareto optimum corner equilibrium is a full equilibrium. 6.
The Relationship between the Equilibrium and the Pareto Optimum
This section will establish Proposition 8: For an increasing returns economy, the Pareto optimum corner equilibrium is an equilibrium and equilibrium is Pareto optimal. Moreover, the Pareto optimum allocation with equal utilities for all individuals is an equilibrium. Because all non-basic markets are incompatible with equilibrium and the Pareto optimum, only basic markets are concerned in proving this proposition. To prove this proposition, it suffices to establish that (1)
if an equilibrium exists, it is an element of a set that consists of the corner equilibria in all basic markets; (2) the Pareto optimum corner equilibrium exists and is an equilibrium; (3) all non-Pareto optimum corner equilibria are not equilibria. On the basis of these three statements, it is trivial to show the final part of Proposition 8 because the only difference between the Pareto optimum and equilibrium is that the former may have the utilities different from individual to individual, but for the latter all people have equal utilities. It is easy to show (1) because the set of all markets includes all possible combinations of corner solutions for individual optimal decisions following the definition of a market. The equilibrium is certainly associated with an element of the market set since the definition of equilibrium requires optimization of individual decisions. This,
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combined with Proposition 5, yields that the equilibrium, if it exists, is an element of a set that consists of the comer equilibria in all basic markets. To justify (2), we need to prove the existence of a Pareto optimum comer equilibrium and prove that this comer equilibrium satisfies the full maximization of individual utilities. Such full maximization will ensure that this comer equilibrium is a full equilibrium because a comer equilibrium satisfies all conditions for equilibrium except ensuring optimal choice of structure by an individual. Propositions 3 and 5 can be used to justify the existence of a Pareto optimum comer equilibrium. The definition of a Pareto optimum comer equilibrium combined with Proposition 4 implies that this comer equilibrium ensures the full maximization of individual utilities. Therefore, (2) can be proven. That is, there exists a Pareto optimum comer equilibrium and this comer equilibrium is an equilibrium. (3) is easy to prove. Since the notion of equilibrium in this paper is extremely neoclassical, equilibrium has to ensure that each person chooses the comer solution that maximizes his utility. Non-Pareto optimal comer equilibria are not equilibria because in each of these equilibria every person's utility is not maximized with respect to his comer solutions. Proposition 8 can thus be established. This section has developed an approach to analyzing a model with increasing returns to specialization. Such an approach has been used to show that the equilibrium achieves a Pareto optimum. By solving for all comer equilibria, we can analyze under what conditions the equilibrium will shift from one comer equilibrium to another. Therefore, this approach is useful for analyzing the development of market structure and equilibrium. The next section applies this approach to investigate the comparative statics of this model. 7.
Equilibrium and its Comparative Statics
Propositions 5, 7, and 8 lead us to Corollary 1: For sufficiently small s, increasing economies of specialization will cause the equilibrium to evolve from autarky first to
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complete division of labor in production {market Q , then jump to complete division in production and transaction {market CT). For sufficiently large s, equilibrium is autarky for any degree of increasing returns to specialization. For the values of s in between, increasing economies of specialization will make equilibrium gradually evolve from autarky first to market C, then to market PT, finally to market CT. In this evolution, the goods with relatively large parameters of preference and degree of increasing returns to specialization will be traded before other goods are involved in the market. Here, s is the average distance between a pair of neighbors. Moreover, Propositions 4, 5, and 6 lead us to Corollary 2: For sufficiently large economies of specialization, decreasing s will involve more goods in the division of labor and the market. The two corollaries tell us that the number of traded goods is determined by the absolute degree of increasing returns to specialization and the average distance between a pair of neighbors, while the composition of traded goods is determined by the relative degree of increasing returns to specialization and the relative preference for different goods. Also, the corollaries say that a sufficiently large degree of increasing returns to specialization in production as well as in transaction is a necessary but not sufficient condition for the division of labor, while a sufficiently small s is a necessary but not sufficient condition for the division of labor. Sufficient increases in a as well as in Ms will produce a 'take-off of the division of labor and productivity. Moreover, these two corollaries mean that there is some substitution between a and lis. For example, if the degree of increasing returns to specialization is not large, then urbanization can promote the division of labor and productivity by increasing lis. If a is large, then we may still 4
Yang (1988) shows that introducing intermediate goods and the market for labor into the model in this paper, we can justify Coase's theory of the firm; while all results in this paper still hold.
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have the developed division of labor even if population is dispersed (large s). If the degree of increasing returns to specialization in production as well as in transaction is a sufficiently large constant, then urbanization can decrease s, thereby producing a 'take-off of the equilibrium level of division of labor. This take-off will enhance trade dependence (the ratio of trade volume to income), the extent of market (demand for traded goods), and per capita real income (the real returns to labor, or real productivity). We have worked out the formula for these three variables in different markets. They are increasing when the market evolves to the complete division of labor from autarky as s decreases. Moreover, our model can be used to show that the following variables also change with the evolution. (1) Self-sufficiency decreases as specialization develops. (2) Diversification of the economy increases as the number of traded goods and professional sectors increase. (3) Degree of concentration of production increases as the total output share of a traded good produced by one producer rises. (4) Degree of integration of the economy develops as the number of trade partners of each individual increases. (5) Transaction efficiency increases and the ratio of the value of transaction service (or roundabout production, or intermediate production) to the value of consumer goods increases as the division of labor evolves. All these phenomena, some of them apparently contradictory, are different versions of the evolution of division of labor. This evolution is caused by the increases in lis and a. In order to explore the difference between the theory of structural change proposed here from the conventional one, two examples are discussed. First, we compare our theory of structural change with the theory of Lewis and Fei and Ranis. According to Lewis (1955) and Fei and Ranis (1964), development is a process in which surplus labor in the agricultural sector is transferred to the industrial sector. Sufficient surplus of agricultural output is a necessary condition for starting this
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process. In order to compare their models with our theory, let us assume that in our model x is food, y is clothing, and z is house. If a is very small and s is very great, then our equilibrium is autarky. All individuals self-provide three goods and live scattered throughout the rural area. We can call this a traditional agricultural sector. In this economy each individual produces all goods he needs. A decrease in s generated by urbanization or an increase in a generated by a technical innovation will shift our equilibrium to the division of labor, e.g. to market CT. In this market the previous 'natural agents' are changed to professional farmers, workers specializing in the production of clothing or housing, and professional traders producing T, respectively. In this transition, we see that the professional farmers are less than the previous 'natural agents' (producing all goods) and population is concentrated in urban areas. This looks like a shift of labor from the agricultural sector to the industrial sector. But this actually is a process of the evolution of division of labor. Each individual switches to professional activity from self-sufficient activity. In autarky, low productivity is not because of 'labor surplus', but because of the low level of division of labor. Hence, 'labor surplus' as well as 'surplus of agricultural products' are not necessary conditions for this shift of economic structure. From our theory proposed in this paper, the necessary conditions for this transition of economic structure are the sufficiently high degree of increasing returns to specialization (a, b, c, t) and sufficiently small s. Therefore, the key issue for economic development and transition of economic structure is the initiation and speeding up of the evolution of the division of labor rather than the existence of a labor surplus. If s is interpreted as the transaction cost related to government interference with the market system, e.g., a trade tax and restrictions on market exchange, our model will tell us that a liberalization policy will substantially decrease s and raise the equilibrium level of division of labor thereby generating economic development. Indeed, such transaction costs imposed by a government in a less developed country are large and the success of liberalization policies in the development of 'four small tigers' (Taiwan, Hong Kong, South Korea, and Singapore) is the best support for this theory.
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The second example is the theories of Kuznets (1966) and Chenery (1979). According to their theory of structural changes, the transition of economic structure is based on an increase in per capita income. According to our theory, however, the increase in per capita income and all other phenomena listed above are different versions of the evolution of the division of labor. It does not make sense to explain one version of this evolution by another. This paper explains the evolution by the improvements of transaction efficiency resulting from changes in the level of urbanization, policy, or institutional arrangements and explains structural changes by this evolution. Yang (1988) explores a mechanism behind the evolution of the division of labor in the context of a dynamic equilibrium model without exogenous changes in a and s. 8. Conclusions The major accomplishment in this paper is to distinguish increasing returns to specialization from increasing returns to scale and to develop a multiple-step approach to handling the model with increasing returns to specialization. Also, this paper shows that a competitive market can efficiently integrate economies of specialization (which is endogenous to individuals) into economies of division of labor in the whole society (which is exogenous to individuals). A free market endogenously determines an efficient level of division of labor by balancing a trade-off between increasing returns to specialization and transaction costs. In addition, the comparative statics of our model shed new light on urban economics and the issue of structural change.
Appendix A: The Choice of Structures This appendix proves Proposition 1 in section 3. First, the first part of this proposition is established. Assume xd> 0; xd can be solved from the budget constraint. Inserting this expression of xd and the rearranged production function of x into the utility function yields
Development, Structural Changes and Urbanization U = {LX -x> +(l-K)(l-k)[x° + pz(zs-zd)/px]}a[y
+pT(T> -T")/px
+ (l-K){l-k)y"Y[z
+ py{y>
+ (l-K)(l-k)^y.
161
-yd)/px (A.1)
s
Differentiating (A. 1) with respect to x yields dU/dxs = - A [1 - (1 - K)(\ - k)] < 0,
(A.2a)
where 0 < ( l - ^ ) ( l - & ) < 1, and A is a positive magnitude independent ofXs. Canceling yd or zd, orTdby using the budget constraint, it can be shown that dU/di' < 0 if i" > 0, i = x, y,z,T.
(A.2b)
Eq. (A.2) implies that the optimum amount sold of a good is zero if an individual buys this good. In other words, an individual will not buy and sell a good at the same time. Assume xd > 0 [this implies Xs = 0 due to (A.1-A.2)]; then the optimum quantity sold of at least another good has to be positive because of the budget constraint. Without loss of generality, we supposey"> 0.8U/dys = Ogives the necessary condition for the optimum ys . Inserting this condition into 8U/dLx and differentiating the resulting first-order derivative with respect to Lx again yields d2U/8L2x>0
if 8U/dLx=0.
(A.3a)
This implies that the optimum value of Lxis either zero or one if xd > 0. Lx = 1 conflicts the assumptions^ 0, implied by the assumption xd > 0, and requiring Ly > 0. Hence, (A. 3 a) means the optimum value of Lx is zero if xd> 0. (A.3b) In other words, an individual will not produce and purchase a good at the same time. This is just the first part of the above proposition. Next, the second part of the proposition is proven. Without loss of generality, we assume ys, zs > 0 . Because / cannot be negative for i = x, y, z, T and because i" = 0 if id > 0 due to (A.2), it can be shown that yd = zd = 0 if ys , zs > 0 . dU/dys = dll/dzs = 0 gives the necessary conditions for the optimum ys and zs. Inserting these conditions into dU/dLy and differentiating the resulting first-order derivatives with respect to Ly again yields d2U/dL2y>0
if dU/dLy=0.
(A.4a)
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This implies that the optimum value of Ly is either zero or one. Ly = 0 conflicts the assumption ys > 0, and Ly = 1 conflicts the assumption z > 0. Hence, (A.4a) means ys and zs cannot be positive at the same time.
(A.4b)
In other words, an individual will not sell two goods at the same time. This is just the second part of the above proposition. Appendix B: Corner Equilibrium and the Restricted Pareto Optimum If market is P in Figure 1(a), i.e., some individuals choose structure (x/y), signified by subscript 1 and other individuals choose structure (y/x), signified by subscript 2, then we can obtain the necessary conditions for the Pareto efficient allocation from the problem below: d
max. z
i
Ux=x?\T(\-K)y?Yzl, i
(B.l)
Z
JV*;? > 2>>'2. >.. --• T S.t.
A | ~T J l | — Ltx,
Z | — ^1
Ljj
l^x J
y2+y°2=L»y,z2=(l-LT-Lyy,
,
T = LT,
MjXf = M2x( or rx{ = x(, M2y\=Mxydx
or ys2=ryf,
U2 = yp2 U(l - K)xd2 \a z\>u. where u is a constant. The first-order conditions are MRSI=MRSIJ\ MRSlx = MRTz\, MRSl=MRT$, MRS\y=MRFfy, MRSyx=MRTyx, 2 MRTL=MRTIT , yx yx
(B.2) MRSl=MRT>, MRS^MRT^,
(B.3) (B.4)
where MRS* =(dUk/dxJ)/(dUk/dxi) is the marginal rate of substitution between good i and j for individuals choosing structure k; MRT* = (dxJdL)/(dXj/dL) is the marginal rate of transformation between good i and j for individuals choosing structure k.
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These are just the necessary conditions for corner equilibrium. From the corner equilibrium for market P, solved in subsection LB andI.C., we can derive that MRS^/T = Py/px=
MRS%T or MRS\X =
MRS^.
This is just (B.2). From this equilibrium we can obtain (B.3) and (B.4) too. Actually, the relative number of individuals, r, is also a decision variable in the problem of Pareto efficient allocation. However, from the problem (B.l), we can solve for the unique xf, x{ , ys2, and yf; they determine a unique r through the market clearing condition since the number of traded goods equals the number of structures in the basic market. For a non-basic market, e.g., that consists of structures (z/xy), (x/yz), (z/xy), and (x/y), we have a similar problem of Pareto optimal allocation as (B.l) which maximizes U(z/xy) subject to all individual production functions, transaction technologies, and balance between consumption and production giving that other utilities are not smaller than some constants. However, the relative numbers of individuals in structures (x/yz), and (x/y) are flexible. Name these two numbers as M\ and M2, respectively. For the relevant Lagrange function associated with this restricted optimization problem, LA, we can show dLA/dMt \M _o < 0 if AdU (z/xy)/dx + BdU (z/xy)/by < 0, 8LA/8M2 \M=0<0
(B.5a)
if AdU (z/xy)/dx + BdU (z/xy)/By > 0, (B.5b)
where we use the facts thatM = ^TM(. , M is the total number of individuals, and Mt is the number of individuals in structure /. LA is the Lagrange function. (B.5) implies that for any value ofAdU(z/xy)/dx + B8U(z/xy)/dy, the Pareto optimum requires either Mi = 0 or M2 = 0. In other words, this non-basic market cannot satisfy the necessary conditions for the Pareto optimum. Applying the above procedure to each market, we can show Proposition 4 in section 5.
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Appendix C: A Proof to Proposition 6 Assume, for example, that the market is P, as shown in Figure 1(a); it is not difficult to show (i) the comer equilibrium including as traded goods those with larger preference parameters is Pareto superior to that including as traded goods those with smaller preference parameters; and (ii) the comer equilibrium including as traded goods those with great return to specialization is Pareto superior to that including as traded goods those with small return to specialization. (i) Assuming a = (5 and a = b = c = /, we have utilities for structure (x/y) and (y/x): log[/(x/j) = a(21ogx + l o g r - l o g r ) + ^logz + «log(l-icr), (C.la) logU(y/x) = a(2logx + logT + logr) + ylogz + a l o g ( l - K ) , (C.lb) where x = [2a/( 3 a + j ' ) ] / 2 , z = y/(3a + y)
andT = [a/(3a + y)J.
The condition of utility equalization becomes E = logU(x/y) - logU(y/x)
= 2alogr = 0.
(C.2)
(i) will be established, if it can be shown that d\ogU*/da\a=y
=dlogU*/da\a=r+(d\ogU*/dr*)(dr*/da)\a__r>0,(C3)
where U* is real income and r* = M*xJM*y is relative number of the individuals choosing different structures in the comer equilibrium. From (C.2) it can be derived that dr'lda\a__f = -(dE/da)/(8E/dr%__r where 8E/da\
=
dlogU'/da\
= 0,
(C.4a)
= 2 log r = 0 because r = 1 if a = y, and >0.
(C.4b)
(C.4) ensures that (C.3) holds. (C.3) implies that if individuals prefer x and y to z, the comer equilibrium with x and y as traded goods will have a greater real return to labor than that with x and z or with y and z as traded goods.
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(ii) Assuming a = p = y and a = b = t, (C.l) holds if x = [2a/(3a + c)]72, z = [cl{3a + c)]c, and 71 = [2a/(3a + c)]°. Also, there is the condition of utility equalization (C.2). (ii) will be established, if it can be shown that d]ogU*/da\aae=dlogU'ma=c+(d\ogU'/drtXdr'/da^g=c>0.
(C.5)
From (C.2) with x = [2a/(3a + c)]72, z = [c/(3a + c)]c, and T = \2aLI (3a + c)]a, it can be derived that. dlogU*/da\a=c >0, and dr*jda\a=c =-(dE/da)/(dE/dr*)\a=c where 8E/da\ '
=0, (C.6)
=0.
\a=c
(C.6) ensures (C.5) to hold. (C.5) implies that if the returns to specialization in producing x and y are greater than that in producing z, then the corner equilibrium with x and y as traded goods will have a greater real return to labor than that with x and z or with y and z as traded goods. Proposition 6 can be established by using (i) and (ii).
References Chenery, M., 1979, Structural change and development policy (Oxford University Press, Oxford). Ethier, W.J., 1986, The theory of international trade, Discussion paper no. 1 (International Economics Research Center, University of Pennsylvania, Philadelphia, PA). Fei, J. and G. Ranis, 1964, Development of the labor surplus economy (Irwin, Homewood, IL). Helpman, E. and P. Krugman, 1985, Market structure and foreign trade (The MIT Press, Cambridge, MA). Herberg, H., M.C. Kemp and M. Tawada, 1982, Further implications of variable returns to scale in general equilibrium theory, International Economic Review 9, 261-272.
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Kuznets, S., 1966, Modern economic growth (Yale University Press, New Haven, CT). Lewis, W., 1955, The theory of economic growth (Allen and Unwin, London). Sah, R. and J. Stiglitz, 1986, On leaning to learn, localized learning, and technological progress, Mimeo. (Department of Economics, Princeton University, Princeton, NJ). Smith, A., 1976, An inquiry into the nature and causes of the wealth of nations, edited by E. Cannan (The University of Chicago Press, Chicago, IL). Yang, X., 1988, A microeconomic approach to modeling the division of labor based on increasing returns to specialization, Ph.D. dissertation (Department of Economics, Princeton University, Princeton, NJ). Young, A.A., 1928, Increasing returns and economic progress, The Economic Journal 1, 527-542.
CHAPTER 8 AN EQUILIBRIUM MODEL ENDOGENIZING THE EMERGENCE OF A DUAL STRUCTURE BETWEEN THE URBAN AND RURAL SECTORS*
Xiaokai Yang and Robert Rice* Monash University
1. Introduction The intimate relationship between the emergence of cities and the trade-off between economies of specialization and transaction costs is noted by Mills and Hamilton [9] and others. Some formal models have been developed to capture different aspects of the relationship in isolation. Kendrick [6] examines the trade-off between economies of scale and transaction costs in a planning model of plant site location, but the relationship between the trade-off and urbanization is not explored. A general transaction technology is introduced into Debreu's general equilibrium framework with constant returns to scale by Hahn [4], Karman [5], Kurz [8], and others. These models cannot formalize the trade-off between economies of specialization and transaction costs because of the assumption of constant returns to scale. Schweizer [11] develops some equilibrium models to address the trade-off between
* Reprinted from Journal of Urban Economics, 35 (3), Xiaokai Yang and Robert Rice, "An Equilibrium Model Endogenizing the Emergence of a Dual Structure between the Urban and Rural Sectors," 346-368, 1994, with permission from Elsevier. * The authors are grateful to two anonymous referees, Jan Brueckner, Jeff Borland, Pak Wai Liu, Yew-Kwang Ng, and Victor Bottini for helpful comments. Thanks also go to Edwin Mills and Gene Grossman who encouraged the first author's research that relates to the paper. We are responsible for any remaining errors.
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economies of scale and transaction costs. Krugman [7] explores the implications of the trade-off between economies of scale and transaction costs for geographical concentration of economic activities.1 However, economies of scale differ from economies of specialization, a sort of diseconomies of scope, despite the connection between the two concepts.2 The paper develops a general equilibrium model which derives the emergence of a dual structure between the urban and rural sectors within Yang's framework of consumer-producers, economies of specialization, and transaction costs. In Yang [12], this framework is used to formalize a trade-off between economies of specialization and transaction costs. The labor productivity of goods and transaction services depends on the distance between neighbors. If this distance is large, then transaction costs generated by the division of labor outweigh economies of specialization, so that individuals will choose a low level of specialization. A decrease in this distance, which may be associated with urbanization, can increase the equilibrium level of specialization and labor productivity by reducing transaction costs.3 In that model, as urbanization develops, the distance between all neighbors decreases by the same amount. Hence, Yang's model cannot endogenize the emergence of a dual structure between the urban and rural sectors. In this paper, we extend the model, introducing a differential in transaction cost parameters between the agricultural and manufacturing sectors. A dual structure between the urban and rural sectors may emerge with the level of urbanization, depending on the transaction cost parameters. In the model to be considered there are four consumer goods and many ex ante identical consumer-producers. Each good is essential in 1
In Krugman [7], economies of scale are realized within a firm; in this paper, economies of specialization result from the division of labor between individuals. 2 The distinctions between economies of scale and specialization and between economies of specialization and diseconomies of scope are identified by Yang and Ng [15]. Scale can be specified as the output level of a firm, while level of specialization is specified as a person's labor share in producing a certain good, which decreases with the scope of his activities. Scope can be specified for a firm, but level of specialization must be specified for each person as well as for each activity. This implies that the scope of a firm and the levels of specialization of many different specialists in the firm may be great at the same time although each person's level of specialization decreases with his scope of activities. 3 Specialization in production at the level of the individual differentiates Yang's model [12] from the general equilibrium models in location theory presented in Schweizer [11].
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consumption. Each individual, as a consumer, prefers diverse consumption and, as a producer, prefers specialized production due to economies of specialization in the production of each good. Assume that goods 1, 2, and 3 are manufactured goods and their production is not land intensive; the producers of these goods can either reside in a single city or be dispersed in the countryside. We refer to them as C-type producers. Good 4 is food and its production is land intensive, so that its producers have to reside dispersedly in the countryside. We refer to them as R-type producers. The assumption that good 4 is land intensive and goods 1, 2, and 3 are not implies that the transportation distance and thereby the transaction cost coefficient of a unit of good is much smaller among C-type producers than it is among R-type producers and between R-type and C-type producers. We draw the distinction between the quantities of the same good that are self-provided, sold, and purchased. This distinction leads to corner solutions of consumer-producers. Using the Kuhn-Tucker approach to corner solutions, we can show that each individual sells at most one good and does not simultaneously buy and sell nor simultaneously buy and self-provide the same goods. The combinations of individuals' corner solutions that are consistent with this condition generate many candidates for the general equilibrium which satisfy the market-clearing conditions but not all conditions for full utility maximization of individuals. Following the approach developed in Yang [12], we can identify four types of market structures as candidates for the general equilibrium. The first is autarky where all people reside dispersedly and no city exists. The second is the partial division of labor where each R-type producer trades good 4 for good 1, each C-type producer trades good 1 for good 4, and the level of specialization is the same for all individuals. All people reside dispersedly in the countryside and no city exists. Each and every individual self-provides three goods and trades two goods. The third is associated with a higher level of division of labor where each C-type producer resides in the city and the level of specialization is lower for all R-type producers than for all C-type producers. The city emerges from the division of labor. The fourth is a complete division of labor where everybody trades all four goods in a dual structure between the urban and rural sectors.
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Applying the approach developed in Yang [12], we prove that the equilibrium is autarky if a variable transaction cost coefficient for a unit of good and/or a fixed cost for each transaction are sufficiently large, since in this case transaction costs outweigh economies of specialization generated by the division of labor. As the transaction cost coefficients are slightly lowered, the general equilibrium shifts to the partial division of labor without cities where no division of labor exists between C-type persons. If the transaction cost coefficients are further lowered, the general equilibrium entails a city where C-type specialists have the division of labor. For this general equilibrium, a dual structure between the urban and rural sectors emerges from the partial division of labor. If the transaction cost coefficients are sufficiently small, the general equilibrium entails the complete division of labor with a city since economies of specialization outweigh the transaction costs generated by the complete division of labor. As the transaction cost coefficients are reduced, the division of labor evolves, resulting in an increase in the ratio of urban to rural population and per capita real income. Section 2 of this paper describes the model which has consumerproducers, economies of specialization, and transaction costs. Section 3 investigates individuals' decision problems and equilibrium. In section 4 the comparative statics of the equilibrium are used to explore the implications of the model for the emergence of a dual structure between the urban and rural sectors from the division of labor. 2. A Model with Consumer-Producers, Economies of Specialization, and Transaction Costs Consider an economy with M consumer-producers and four consumer goods where Mis large. Each consumer-producer's self-provided amount of good i is Xj. The amounts of good i sold and purchased in the market are xf and xf , respectively. Assume that goods 1, 2, and 3 are manufactured goods and that not much land is needed in producing them; then the producers of these goods can be either concentrated in a city or dispersed in many localities. However, good 4 is a land-intensive agricultural good. Its producers can only reside dispersedly in the rural
Dual Structure between Urban and Rural Sectors
171
area. The production functions for the four goods, which are the same for each consumer-producer, are given by xt+x^=L%
Lt+L2+L3+L4=l, a>\,
L,e[0,l],
1 = 1,2,3,4,
(1)
where x, + xf is the output level of good i and Li is a person's labor share in producing good i, which we term his level of specialization in producing good i. Each consumer-producer is endowed with one unit of labor. Parameter a represents the degree of economies of specialization. There are several distinctive features to this system of production functions and endowment constraint. First, this system is specified for an individual rather than for a firm. Consequently, each individual is a consumer-producer. Demand (or supply) depends not only on the utility (or production) function, but also on the production (or utility) function. Specifying production functions as identical for all individuals makes the distinction between our production function and the traditional production function clearer. The model in this paper does not involve comparative advantage in the conventional sense. According to neoclassiccal microeconomics, such a model with identical technology and preferences for all individuals is too trivial to be interesting. However, this paper will show that such a model is capable of not only generating gains from trade, but also endogenizing the level of division of labor and trade dependence.4 Second, it is assumed that the available time input is specific for each individual and for each activity. Therefore, the time available to an individual is fixed and there is an endowment constraint in this system. This point ensures that the economies of specialization are limited and are individual specific as well as good specific. Third, the production of any two goods is separable. This implies that an individual can choose extreme specialization. A comparison between this functional form and alternative specifications will make this point clearer. Suppose the production function is specified as x°x* = L , where 4
The models developed by Dixit and Stiglitz [1], Ethier [2], Krugman [7], and Grossman and Helpman [3] can also generate gains to trade between ex ante identical agents. However, our model here is based on the concept of economies of specialization, which is a sort of diseconomies of scope and differs from the concept of economies of scale specified in those models.
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a + b < 1 for increasing returns to scale and L is the total amount of labor allocated to the production of goods /' andy. This production function is not well defined forx,. = Oorxy = 0, implying that individuals cannot choose extreme specialization. Suppose alternatively that the production function is specified as xf + x* = L , where a, b < 1 for increasing returns to scale and L is the total amount of labor allocated to the production of goods i and/. With this functional form, the labor allocated to produce good i cannot be distinguishable from that allocated to produce goodj. Thus, this production function will not be able to capture the stylized fact that learning by doing which generates economies of specialization is not only specific for an individual, but also specific for a good. There are two types of transaction costs. The first is a variable transaction cost, characterized by the coefficient K e (0, 1). The second is a fixed transaction cost, characterized by the coefficient c e (0, 1). We first examine the variable transaction cost. Fraction 1 - K of a shipment disappears in transit because of transaction costs. Thus, Kxf is the amount a person receives from the purchase of good i. The individuals producing good 4 reside in the rural area. We refer to them as R-type persons. Here, a city is defined as the area where many individuals have the division of labor and reside in close proximity to one another. If some individuals specialize in producing good 1,2, or 3, they can reside either in a city or in the rural area. We refer to these individuals as C-type persons. If they reside within a city, then the distance between each pair of C-type neighbors will be short. This short distance between a pair of C-type trade partners saves on transaction costs if C-type persons choose to have the division of labor among themselves.5 If there is no division of labor among producers of goods 1, 2, and 3, nobody will choose to reside in a city since living closely together has no benefit. A C-type person cannot reside closely to a R-type person because each R-type person has to occupy a sufficiently large area for agricultural activities. Thus, if C-type persons choose to reside in a city, then the distance between a pair of R-type and C-type neighbors is much greater than that between a pair of
5
The distance between a person and his trade partner may be greater than the distance between neighbors. The former may increase with the number trade partners, which increases in turn with the level of division of labor.
Dual Structure between Urban and Rural Sectors
173
C-type neighbors. Hence, the distance between a pair of R-type and C-type trade partners is much greater than that between a pair of C-type trade partners who live in the same city. Let the transaction efficiency coefficient between C-type trade partners be k and that between a pair of R-type and C-type partners be s. Then, s is much smaller than k if C-type persons reside in a city. Here, the transaction efficiency parameter is related to the technical conditions of transactions and has nothing to do with the concept of efficiency in the sense of welfare analysis. To summarize, we have K = k between C-type trade partners, and K = s between R-type and C-type partners,
(2)
with k being much larger than s if all C-type persons reside in a city. Later we will see that ex ante identical individuals can choose between becoming R-type persons and becoming C-type persons as they choose different levels of specialization and different professions. An individual's consumption of good i is xi + Kxf . The utility function is identical for all individuals:
V = Y\(xi+Kxf).
(3a)
Here, 1 - K is the coefficient of the first type of transaction cost. The second type of transaction cost is a fixed cost in terms of utility loss, incurred in transactions.6 The fixed transaction cost is proportional to the number of goods purchased by each individual and is independent of the quantities of traded goods. For each good purchased, a fraction c of utility V disappears, so that an individual enjoys only the fraction (1 - nc) of utility V when he buys n goods. This fixed cost coefficient can be interpreted as the cost of creating a new market or of finding the price of a particular good; for example, the investment cost of facilities and instruments that are necessary to implement trade of a particular good or utility loss in discovering the price of the good. Hence, the utility level that is enjoyed by an individual is 6
Introducing only the variable transaction cost without the fixed transaction cost means that the condition for the market structures with an intermediate level of division of labor to be the equilibrium will not be satisfied, so that the equilibrium will be at a corner, either autarky or complete division of labor. This point will be shown in Proposition 1.
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U = (l-nc)V,
(3b)
where n is the number of goods purchased. We assume that there is free entry for all individuals into any sector and the population size Mis large. These assumptions imply that ex ante identical individuals treat prices parametrically. 3. Configurations and Market Structures In this section we investigate the relationship between individuals' decisions of their levels and patterns of specialization and the level and pattern of division of labor and urbanization for society as a whole. It is assumed that each consumer-producer maximizes his utility with respect to the quantities of goods produced, traded, and consumed by him for a given set of relative prices of traded goods subject to the budget constraint, production functions, and endowment constraint. Since the quantities self-provided, produced, and traded of the same good are distinguished from one another, an individual's decision is always a corner solution which involves zero values of some decision variables. Combinations of zero and nonzero values of 12 decision variables xnxf,xf (i = 1, 2, 3, 4) generate 212 = 4096 profiles of variables and thereby one interior solution and 4095 corner solutions. We call a combination of zero and nonzero variables a configuration. Yang [12] has proven that for the models of the kind in this paper, a consumer-producer does not simultaneously buy and sell nor simultaneously buy and self-provide the same goods; also, he sells one good at most (see Proposition 1 and its proof in Yang [12]). Using this proposition, the interior solution and many corner solutions can be ruled out from the set of candidates for the optimal decision. Since preference and production parameters are identical for all goods and for all individuals and the transaction cost coefficient is the same for all C-type persons, the symmetry between C-type persons who sell different goods can be used to rule out more corner solutions from consideration. Having taken this and the budget constraint into account, the candidates for the optimal decision are reduced to only 29 corner solutions. A combination of configurations in which demand for any traded good is matched by supply of that good is called a market structure or simply a
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structure. Free entry and competition among identical utility maximizers will establish utility equalization and market-clearing conditions for each structure. The utility equalization and market-clearing conditions determine a set of relative prices for the traded goods and the relative numbers of individuals selling different goods. The set defines a corner equilibrium for each structure. Yang [12] has proven that the corner equilibrium that generates the highest utility level is the general equilibrium, whereas corner equilibria that do not generate the highest utility level cannot be the general equilibrium because individuals have incentives to deviate from them.7 In this section, individuals' corner solutions and the corner equilibrium are solved for each structure, and the general equilibrium is identified by comparing all corner equilibrium levels of utility. Having taken Proposition 1 in Yang [12] into account, there are four categories of configurations. (i) Configuration autarky or A. An individual who chooses this configuration produces four goods for his own consumption. (ii) Configuration (ilj). An individual who chooses this configuration sells good i, buys goody, and self-provides other goods, where i,j = 1,2, 3, 4; i ^j; i before the slash denotes a good sold, and j after the slash denotes a good purchased. There are 12 configurations of this category: (1/2), (1/3), (1/4), (2/1), (2/3), (2/4), (3/1), (3/2), (3/4), (4/1), (4/2), (4/3). (iii) Configuration (i/jr). An individual who chooses this configuration sells good i, buys goods j and r, and self-provides the other goods, where i,j, r = 1,2,3,4, and i £j ^ r. There are 12 configurations of this category: (1/23), (1/24), (1/34), (2/13), (2/14), (2/34), (3/12), (3/14), (3/24), (4/12), (4/13), (4/23). 7
Yang and Ng [15, pp. 161-165] have shown how a decentralized market sorts out the efficient pattern and level of division of labor from many corner equilibria. Since the concept of corner equilibrium is a vehicle for solving for general equilibrium so long as the concept of corner solution is a vehicle for a person to solve for the optimum decision, multiple corner equilibria do not imply multiple general equilibria as long as each person's multiple corner solutions do not imply multiple optimum decisions. Yang and Ng [15, Chap. 6, Proposition 6.5] have shown that the general equilibrium is Pareto optimal. The assumption that economies of specialization are limited to each consumer-producer is crucial for this result.
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(iv) Configuration (i/jrt). An individual who chooses this configuration sells and self-provides good i and buys goods j , r, and t, where i,j, r, t = 1, 2, 3, 4, and i ^ j ± r ^ t. There are 4 configurations of this category: (1/234), (2/134), (3/124), (4/123). Combinations of the configurations that are compatible with the market-clearing conditions (i.e., the demand for any traded good is matched by the supply of that good) generate a myriad of structures. Since all preference, production, and transaction parameters are identical for all individuals of the same type, many structures with different trade compositions generate the same per capita real income (utility). Hence, there is indeterminacy of trade composition, and only the number of traded goods matters. Thus, the distinct structures can be characterized by the numbers of traded goods for the two types of persons. Let (nm) denote a structure in which a C-type person trades n goods and a R-type person trades m goods; there are 10 distinct market structures that generate different utility levels. 1. Autarky, depicted in Figure la, which consists of configuration A and is referred to as structure A. 2. Structure (22), depicted in Figure lb, in which each C-type person and each R-type person trade two goods, respectively. This structure consists of configurations (1/4) and (4/1). In this structure, R-type persons exchange good 4 for good 1 with C-type persons. The structures consisting of (z'/4) and (4//), where z = 2 or 3, are symmetric with this structure and generate the same corner equilibrium per capita real income. Hence, we do not draw the distinction between the three structures. This structure is referred to as balanced partial division of labor without cities since the two types of persons have the same low level of specialization. 3. Structure (23), in which each C-type person trades two goods and each R-type person trades three goods. This structure consists of configurations (z'/4) and (4/if), where i,j = 1, 2, 3, and i £j. 4. Structure (24), in which each C-type person trades two goods and each R-type person trades four goods. This structure consists of configurations (z'/4) and (4/123), where z = 1, 2, 3.
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177
4
Figure 1: Emergence of a dual structure between urban and rural sectors, (a) Autarky, (b) Structure (22), without city and with balanced partial division of labor, (c) Structure (32), with a city and unbalanced division of labor, (d) Structure (33), with a city and balanced partial division of labor, (e) Structure (43), where urban residents are completely specialized and rural residents are partially specialized, (f) Structure (44), with complete division of labor.
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5.
Structure (32), depicted in Figure lc, in which each C-type person trades three goods and each R-type person trades two goods. This structure consists of configurations (i/4j) and (4//), where i,j = 1, 2, 3, and i ± j . This structure is referred to as a low level of unbalanced partial division of labor with a city since all C-type persons have a higher level of specialization than R-type persons. 6. Structure (33), depicted in Figure Id, in which each C-type person and each R-type person trade three goods. This structure consists of configurations (z'/4/) and (4/y), where i,j = 1, 2, 3, and i ^j. This structure is referred to as balanced partial division of labor with a city since two types of persons have the same level of specialization in a dual structure between the urban and rural sectors. 7. Structure (34), in which each C-type person trades three goods and each R-type person trades four goods. This structure consists of configurations (i/4j) and (4/123), where i,j = 1, 2, 3, and i i^j. 8. Structure (42), in which each C-type person trades four goods and each R-type person trades two goods. This structure consists of configurations (i/4jr) and (4/z), where i,j, r= 1,2,3, and / £j ^ r. 9. Structure (43), depicted in Figure 1 e, in which each C-type person trades four goods and each R-type person trades three goods. This structure consists of configurations (i/4jr) and (4/z)'), where i,j, r = 1, 2, 3, and i £j ^ r. This structure is referred to as a high level of unbalanced division of labor with a city since all urban residents have a higher level of specialization than rural residents and all individuals have a higher level of specialization than in structure (32). 10. Structure (44), depicted in Figure If, in which each C-type person and each R-type person trades four goods. This structure consists of configurations (z'/4/>) and (4/123), where i,j, r= 1,2,3, and z ^ j ^ r. This structure is referred to as complete division of labor. There are some structures that are symmetric to and generate the same corner equilibrium per capita real income as each of the structures listed above. We do not distinguish between these structures and the corresponding ones that we have listed.
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Letting the number of individuals selling good i be M., a product of M, and individual supply (demand), given by the individual optimal decisions, yields the market supply of (or demand for) each traded good. Equilibrating the market demand to the market supply yields the market-clearing conditions for each market structure. Furthermore, free entry will ensure utility equalization through individuals' behavior of maximizing utilities. The market-clearing and utility equalization conditions determine the relative prices of traded goods and the relative numbers of individuals choosing different configurations. The set of relative prices and relative numbers defines a comer equilibrium in each structure. The concept of comer equilibrium for a structure is an analogue of comer solution for an individual configuration. A comer equilibrium is not a general equilibrium unless each individual's utility is maximized with respect to comer solutions because a comer solution is not an optimal decision unless it maximizes an individual's utility. Let us investigate these 10 structures one by one. (1) In structure A (autarky) there is only one configuration, A. The individual decision problem for this configuration is max U - xxx2x3xA = L°La2L° (1 - L, - L2 - L3 )a s.t. x(. =Lj, i = \, 2,3,4 i , + L2 + L3 + L4 = 1 The optimum decision is /,={,
1 = 1,2,3,4,
(4a)
(production function) (endowment constraint). U(A) = 2-*°,
(4b)
where U(A) is the per capita real income in autarky. (2) In structure (22), there are two different configurations, (1/4) and (4/1). The decision problem for (1/4) is max U = (1 - cn)xxx2x3sxf = (1 - c){Lax - x[ )La2La3 sx{ /p4l s.t. x, + xf = L°, xi -Lat, i = 2,3 Lx + L2 + L3 = 1 x{ = p^xf
(5a)
(production function) (endowment constraint) (budget constraint),
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where p„ is the price of good 4 in terms of good 1. Note that all transactions here are between R-type persons and C-type persons. Hence, K = s. Moreover, the number of goods purchased, n, is one for this configuration. The optimum decision is A = i . A = j , i = 2,3, xf = x, = L>/2, xf = xf/pn
(5b)
Ul(p4l) = s(l-c)2-*"-ip-4}, where£/,Cp41) is the indirect utility function for configuration (1/4). The decision problem for configuration (4/1) is symmetric to (5). The optimum decision is
U4(pj =
s(l-c)2-*^ptl,
where f/4(/>41) is the indirect utility function for configuration (4/1). The market-clearing and utility equalization conditions for structure (22) are MlX;=MAx?(pJ
(7a)
t/ 1 (p 41 ) = C/4(/>41).
(7b)
Due to Walras' law, the market-clearing condition for good 4 is not independent of (7a). Equation (7a) ensures a special Walras regime. Prices are determined by market demand and supply, which are determined by the numbers of persons selling any specific good. Each person is able to produce all goods, and thereby the numbers of persons selling different goods cannot be manipulated by any individual. Combined with the fact that economies of specialization are limited to each individual and to each good, this then implies that prices are parameters to individuals. Here the relative numbers of persons selling different goods play the same role as the relative prices in the conventional Walras regime. Inserting the individual optimum decisions into the relevant variables in (7), we find the corner equilibrium in structure (22), p41=M4/M1=l,
C/(22)= 5 (l-c)2-^-2,
(8)
where U(22) is the per capita real income in structure (22). Repeating this procedure, we have solved for the corner equilibria in other structures. The per capita real income, relative prices, relative numbers of different type
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persons, and level of specialization for individuals choosing different configurations are listed in Table 1. In the table, ^ . Mi /M4 , i = 1,2,3, is the ratio of urban population size to rural population size for those structures involving a city, which is defined as the degree of urbanization, andL; is the level of specialization for a person selling good /. We have multiple corner equilibria. All of these corner equilibria are candidates for the general equilibrium. These candidates satisfy the following conditions: (i) Excess demands for all traded goods are zero for the uniform positive relative prices of the traded goods, the relative numbers of individuals choosing different configurations, and the uniform positive price of labor (real return to labor), and (ii) individuals maximize their utility for a given set of relative prices and for a given structure. Note that we have not yet imposed full maximization of utility by individuals across structures. Now we allow individuals to choose configurations across structures. Then each individual will choose a configuration that generates the highest real income, taking a set of relative prices of traded goods as given. By enumerating all candidates for the general equilibrium, we will find the corner equilibrium that generates the highest per capita real income and prove that it is the general equilibrium. The details of this proof are in Yang [12] or in Yang and Ng [15, Chap. 6, Proposition 6.5]. From the individual decision problems in this section, we can see the effects of our method of specifying production functions for individual demand and supply, indirect utility functions, and the theory of equilibrium. First, individual demand depends on individual supply since each individual is a consumer-producer. Demand and supply functions and indirect utility functions are not continuous across structures. Second, an individual not only solves for the efficient resource allocation for a given level of specialization, but also chooses a configuration in order to find the efficient level of specialization. The indirect utility function differs from configuration to configuration despite ex ante identical utility functions for all individuals. A corner equilibrium in this paper is equivalent to a general equilibrium in neoclassical microeconomics. A corner equilibrium determines an efficient allocation of resources for a given level of division of labor. However, for our model, the general equilibrium will determine the efficient level of division of labor. Many economic phenomena can be explained by changes in organizational
Table 1: Corner Equilibria in 10 Market St Structure type
Per capita real income
Corner equilibrium relative prices
A (Fig. la) 2-8° Structure (22) (Fig-lb) s(l-c)2- 6 "- 2
p4i =
1
Structure (23) , 4 / 3 ( 1 _ c f 3 ( l _ 2 c f
^
^
x 3„-l2-(20o+4)/3
S
™ic)(32)
5
[Ml-^)(l-2c)f
X3
(
g
}
=
x 31-«, 2
[(l_c)A(1_2c)]>/3 2
("-')/3
A, = P« = [k(l-2c)/(l-c)f2(^ p4l = pa= (k/sf
x33(«-l)2-8«
Structure (24) ^3/2 (j _ C)V« ^ _ 3cy/4 x2-(9«+7)/2
{
X31*-')
IJ(-I)2-7-I
Structure(33) ^ / 3 ( l _ 2 c )
=
M
Al
= Pa = PG = .y-I/2 ["(! _ c y ( ! _ 3 c ) j ' / 4 21.5(l-«)
=
(
Table 1: (Continued) Structure type Structure (42)
Per capita real income
^[(l- C )(l-3c)]'
/2
x2-3a-5
Structure (34)
c)f
x2 (»-')A:
s^k^(l-2cfA
P*l = PA2 = Pn
(skf(l-2cf 2/3
= [(l-3 C )/(l-2 C )*f x 2 l/4 x l/2 2 2(»-l)33(l~»)/4
x39(„-l)/4
x(l-3c)
A i = P 4 2=P43 = [ ( 1 - 3 c ) / ( 1 3
x(l-3c) 1 / 4 2-o«-' Structure (43) (Fig.le)
Corner equilibrium relative prices
(M
P« = Pt2 = Pil, 2 8
3°-'2- - °/3
3
= H [(l-3c)/5(l-2c)]
V3
=
x 31-a 2 -l+8«/3
Structure (44) ( ^ ) 3 / 2 ( l - 3 c ) 2 - " (Fig. If)
P« = Pa = Pn = {klsT =
Note. ^ ; MJMA, / = 1,2,3, is the ratio of urban population size to rural population siz defined as the degree of urbanization, and L-L is the level of specialization for a person s
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patterns of division of labor even if utility and production functions are ex ante identical for all individuals. Therefore, we refer to this approach as organization oriented while neoclassical microeconomics is resource allocation oriented. 4. The General Equilibrium and its Comparative Statics By comparing per capita real income in all structures, we can identify the corner equilibrium with the maximum per capita real income. We refer to this corner equilibrium as the Pareto-optimum corner equilibrium. The Pareto-optimum corner equilibrium is the general equilibrium. A comparison among the per capita real incomes in the 10 structures yields Proposition 1: (1) For c > 1, the general equilibrium is structure A; (2) For c e ( \ , 1), the general equilibrium is structure A if a < a\ and is structure (22) if a > a\\ (3) For c G ( y , | ) , where c is sufficiently close to 1/3, the general equilibrium is structure A if a < a2, structure (22) if a e (a 2 , a3), structure (32) if a e (a 3 , a4), structure (33) if a > a4; (4) For c G (0, j ), where c is sufficiently close to 0, the general equilibrium is structure Aif[a< a5 andsk > b] or [a < a6 andsk< b], structure (43) if a G (a5, a7) andsk> b, structure (44) if [a > aq andsk > b] or [a > a6 andsk < b]. Here a is the degree of economies of specialization, s is the transaction efficiency parameter between C-type and R-type persons and between R-type persons, k is the transaction efficiency parameter between C-type persons when they reside in the same city, and c is the fixed transaction
Dual Structure between Urban and Rural Sectors
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cost coefficient.8 The critical values of a that make the general equilibrium shift between structures are listed below. or, sl-0.721ns,
a2 = l - 0 . 3 5 1 n [ s ( l - c ) ] ,
a3 s0.05 + 0.091n[(l-c)/ifc(l-2c)], aA = 1 + 0.52 In [(l - c)/(l - 2c)] - 0.35 In s, a5 =0.52 -0.28 In (sk), a6=l-0.27h\(sk),
an = 4.08 - 0.12 In (sk),
6 = 1/4603119211, a2 < a3 < a4,
if c e (|,^) and c is sufficiently close to ~,
a5
if s£ > b, c e (0,1J, and c is sufficiently close to 0,
a5>a1,
if sk
Proposition 1 is illustrated in Figure 2. As shown in Figure 2, as a and/or sk increase and/or as c decreases, the equilibrium shifts from autarky to structures with increasingly higher levels of division of labor and urbanization. Holding a constant, a sufficient decrease in c causes the equilibrium to shift to a higher level of division of labor. Holding c constant, a sufficient increase in a causes the equilibrium to shift to a higher level of division of labor. Since all critical values at decrease withs and/or k, a sufficient increase in s and/or k causes the equilibrium to shift to a higher level of division of labor, holding a and c constant. Proposition 1 implies that as transaction efficiency is improved, division of labor is first developed between a farmer and a manufacturer with a low level of specialization, but not among manufacturers. Hence, no city emerges from the low level of division of labor. As transaction efficiency is further improved, a city emerges from the division of labor between specialist manufacturers. A dual structure between the urban and
Proposition 1 has not exhausted all possibilities. The Appendix has examined the case where c e (-j,-j)and c is not close toy and the case where c e (0,-j) and c is not close toO.
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X. Yang, R. Rice C i
A
=1
to
A
(22)
c=l/2 1/3 c=0
ax (22)
A
(32) a3
a2
(43)
A
(33) a4
-a
(44)
ac
c=m c= 0
1441
^
-•a
Figure 2: Illustration of Proposition 1
rural sectors in the sense of unequal population density emerges from a higher level of partial division of labor. In the transitional period from a low level of balanced division of labor to a high level of balanced division of labor, unbalanced partial division of labor will generate a dual structure in the sense of differentials in the level of specialization and labor productivity between the urban and rural sectors. In this evolutionary process, the level of urbanization will increase. Figure 1 provides an intuitive illustration of how a city emerges from the evolution of the division of labor generated by improvements in transaction efficiencies. The numbers in circles represent the goods sold by persons and the numbers with lines represent flows of goods. Dashed triangles, boxes, or circles represent cities. In Figure la, each individual self-provides all goods he needs, all individuals reside in the countryside, and no city exists. For structure (22), depicted in Figure lb, a circle enclosing number one represents a person selling manufactured good 1, buying agricultural good 4, and self-providing other goods. A circle enclosing number four represents a person selling good 4, buying good 1, and self-providing other goods. He resides in the countryside and exchanges good 4 for good 1.
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For structure (32), depicted in Figure lc, there is a division of labor between C-type persons in producing manufactured goods as well as a division of labor between C-type and R-type persons. Hence, a city where all C-type persons reside can be used to reduce transaction costs generated by the division of labor between C-type persons. For this structure, all C-type persons have a higher level of specialization than R-type persons. Hence, there is a dual structure not only in terms of unequal population density between the urban and rural sectors but also in terms of an unequal level of specialization between the two sectors. Economies of specialization imply that the labor productivity of a good increases with the level of specialization. Hence, this unequal level of specialization implies that the labor productivity of urban residents in producing manufactured goods is higher than that of rural residents in producing food. For structure (33), depicted in Figure Id, all individuals have the same level of specialization and are partially specialized. The difference between the balanced partial division of labor in (22) and that in (33) is that the latter involves a city and has a higher level of division of labor than the former. The level of specialization for any individual in (33) is higher than that for any individual in structure (22). No dual structure in any sense exists in structure (22). But a dual structure in the sense of an unequal population density between the urban and rural sectors exists in structure (33), although the dual structure in the sense of an unequal level of specialization and productivity between the two sectors does not occur in structure (33). In structure (43), depicted in Figure le, all producers of industrial goods are completely specialized and reside in a city. Rural residents are not completely specialized. A dual structure in both the senses of a differential in the level of specialization and productivity and unequal population density between the urban and rural sectors emerges from the unbalanced partial division of labor. In structure (44), depicted in Figure If, all individuals are completely specialized and the differential in the level of specialization and labor productivity disappears, but a dual structure between the rural and urban sectors in terms of unequal population density still remains.
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The proof of Proposition 1 is included as an appendix. Let us examine Proposition 1 in more detail. While holding the fixed transaction cost coefficient c constant, increasing the degree of economies of specialization and/or increasing the transaction efficiency characterized by k and s will cause the equilibrium to evolve from structure A to an increasingly higher level of division of labor. This is shown in Figure 1. If the fixed transaction cost coefficient is sufficiently large, then the evolution in division of labor will end up with a balanced partial division of labor without a city, which is structure (22), where all individuals have the same level of specialization and no transactions occur between C-type persons, so that no city exists. In other words, if the fixed transaction cost coefficient is large, the complete division of labor and other organizational patterns with a high level of division of labor and a city never emerge from the evolutionary process. If the fixed transaction cost coefficient is sufficiently small, the equilibrium will jump from autarky to structure (43) in which urban residents are completely specialized and rural residents are partially specialized, followed by the complete division of labor, which is structure (44), where all individuals are completely specialized in a dual structure between the rural and urban sectors. If the fixed transaction cost coefficient is at an intermediate level, then as the degree of economies of specialization and/or transaction efficiencies increase, the equilibrium will evolve from autarky, first to a balanced partial division of labor without cities, which is structure (22), followed by unbalanced partial division of labor, which is structure (32), where all C-type persons reside in a city and have a higher level of specialization than rural residents, and finally ending up with the balanced partial division of labor with a city, which is structure (33), where all C-type persons reside in a city and the C- and R-type persons have the same level of specialization. If the degrees of economies of specialization and/or transaction efficiencies are sufficiently large, then a decrease in the fixed transaction cost coefficient will make the equilibrium jump from a low level of division of labor to a higher one. Since Proposition 1 has not exhausted all possible combinations of values of a, c, sk, the holes are qualitatively filled in the Appendix. It is shown that if the fixed transaction cost coefficient c e (} ,-j) and c tends to \ from the left-hand side, then the equilibrium will jump from structure A over structure (22) and/or (32) to structure (33) as transaction
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efficiencies k and s are improved. If the fixed transaction cost coefficient c G (o, j ) and c tends to -j from the left-hand side, then some structures with an intermediate level of division of labor between structures A and (43) may emerge from the evolution of the division of labor generated by a sufficiently large increase in transaction efficiencies k and/or s. As c tends to zero, the evolutionary process will involve a jump from autarky, over the intermediate structures, to complete division of labor. This is why we specify the fixed transaction cost coefficient. An intermediate value of c will make the structures with an intermediate level of division of labor emerge from the evolution of the division of labor. The intuition behind this feature is straightforward. With the specification of the fixed transaction cost coefficient, total transaction cost may increase more than proportionally as more economies of specialization are exploited, so that the second-order condition for an interior value of the level of division of labor is fulfilled. This is an analogue to the second-order condition for the interior optimum decision for a firm: marginal cost increases faster than marginal revenue as the output level increases within the neighborhood of the interior maximum point. It is shown in the Appendix that those structures with a higher level of specialization for rural residents than for urban residents (such as structures (23) and (24)) cannot be equilibria since transaction efficiency for the urban sector, k, is much higher than that for the rural sector, s. The assumption that k is much greater than s is necessary for proving Proposition 1. The evolution in the division of labor resulting from the increases in a, k, and/or s for a given value of c will raise the equilibrium relative number of urban residents to rural residents. For c e ( J , - J ) , this relative number increases from 0 in structures A and (22) to a positive value in structure (32),(M 1 +M 2 )/M 4 =3-< a+1 V3[A:(l-2c)/(l-c)f 5 , followed by a larger value of the relative number in structure (33), (M, + M2)/M4 = 2(k/sy, as transaction efficiencies k and s are improved (see Table 1). For c G (o,y) , the relative number is increased from 0 in structures A and (22) to a positive value in structure (43), ( M , + M 2 + M 3 ) / M 4 = 2<2a^'2 k2'3 [s(l - 3c)/(l - 2c)]1/3, followed by a larger value of the relative number in structure (44),(M, +M2+ M3)/M4 = 3(k/s)1/2, as transaction efficiencies k and s are improved (see Table 1).
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Because of multiple corner equilibria in our model with consumer-producers, economies of specialization, and transaction costs, the comparative statics summarized in Proposition 1 involve shifts (evolution) of market structure from a corner equilibrium to another as transaction efficiencies are improved. This feature of our analytical framework significantly enhances the predictive power of our model.9 The intuition behind Proposition 1 is straightforward. A continuous improvement in transaction efficiencies (k, s, and lie) generates evolution in the division of labor. In this process a low level of division of labor is developed first between a non-specialized producer of manufactured goods and non-specialized farmer and then between different specialist manufacturers as transaction efficiency is further improved. Urbanization can be used to promote the division of labor between different professional manufacturers by reducing transaction costs. Finally, a higher level of urbanization emerges from a higher level of division of labor between the urban and rural sectors. It is interesting to see that the division of labor is necessary but not sufficient for the emergence of a city. For a low level of balanced division of labor, which is structure (22), the division of labor is between a C-type person and a R-type person who reside dispersedly in the rural area and is not between different C-type specialists. Hence, cities are not needed to reduce transaction costs caused by the division of labor. A sufficient development of division of labor between C-type persons is sufficient for the emergence of a city. In other words, the development of division of labor within the manufacturing sector is sufficient for the emergence of a city.
9
The emergence of many cities and a hierarchical structure of cities from the evolution of division of labor is investigated in Yang and Hogbin [14]. The evolution of division of labor in this paper is exogenous because it never occurs in the absence of the exogenous evolution of transaction efficiency. Endogenous evolution of division of labor is investigated in Yang and Borland [13]. Here, endogenous evolution is a spontaneous evolutionary process that is not driven by any exogenous changes in parameters and exogenous evolution is driven by such exogenous changes.
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5. Conclusion A framework with consumer-producers, economies of specialization, and transaction costs has been used to formalize a trade-off between economies of specialization and transaction costs in this paper. The comparative statics of the general equilibrium based on corner solutions involve shifts of the general equilibrium across several corner equilibria. Having introduced a differential in transaction efficiency between the agricultural and manufacturing sectors, we have endogenized the emergence of a city from the evolution in division of labor which is driven by improvements in transaction efficiency. As transaction efficiency is improved, division of labor is first developed between a farmer and a manufacturer who have a low level of specialization, but not among manufacturers. Hence, no city emerges from the low level of division of labor. As transaction efficiency is further improved, a city emerges from the division of labor between specialist manufacturers. This results in the first type of dual structure — unequal population density between the urban and rural sectors. In the transitional period from a low level of balanced division of labor to a high level of balanced division of labor, unbalanced partial division of labor generates a second type of dual structure with differentials in the level of specialization and labor productivity between the urban and rural sectors. In this evolutionary process, the level of urbanization will increase. To our knowledge, the model in this paper is the first general equilibrium model that endogenizes the emergence of a city from the evolution of the division of labor. If a data set of transaction efficiency can be collected, then a regression of the degree of urbanization on transaction efficiency may test the hypothesis generated by the theory in this paper against empirical observations. Yang and Shi [16] have developed an approach to modeling concurrent evolution in product diversity, in the level of specialization, and in productivity. If that approach is applied, an extended model may generate simultaneous evolution of productivity, urbanization, and product diversity as different aspects of the evolution of the division of labor. Yang and Borland [13] have developed an approach to modeling endogenous evolution of the division of labor. Following that approach, a dynamic version of the model in this paper may be developed to predict
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concurrent and endogenous evolution in productivity, in the extent of the market, in trade dependence, and in the degree of urbanization as different aspects of the endogenous evolution of the division of labor.
Appendix: A Proof of Proposition 1 From Table 1, it can be seen that per capita real incomes in all structures except in autarky are non-positive if c > 1. Hence, structure A is the general equilibrium if c > 1. Also, per capita real incomes in all structures except in structures A and (22) are non-positive if c>\ . Hence, the set of candidates for the general equilibrium is composed of only structures A and (22). A comparison of per capita real income between the two structures indicates that (7(A) > 0(22) iff a < ax. If c e (y ,y), then the per capita real incomes in structures (24), (42), (43), (34), and (44) are non-positive. A comparison of the per capita real incomes in structures (22), (23), and (32) indicates that 0(23) < either 0(22) or 0(32). Thus, the set of candidates for the general equilibrium consists of structures A, (22), (32), and (33). A comparison between [/(A), 0(22), 0(32), and 0(33) establishes the statements that A generates the highest per capita real income if a < a2 structure (22) generates the highest per capita real income if a e (az, a3), structure (32) generates the highest per capita real income if a e (a3, a4), and structure (33) generates the highest per capita real income if a > a4, where the values of a, are given in Proposition 1. A comparison between a2, a3, and a4 shows that a2 < a3 < a4 if c is sufficiently close to y from the right-hand side. Either or both of the two inequalities may not hold if c G (y, j) and c is sufficiently close to \ . For a3 > a4, we have 0(33) > 0(32) > 0(22) if a > a3, and 0(32) < 0(22) if a < a3. This implies that structure (32) cannot be the equilibrium. The equilibrium will jump from (22) over (32) to (33) as transaction efficiency improves. If c is sufficiently close to zero, then it can be shown that structures (22), (32), (23), (33), (24), and (42) cannot generate the highest per capita real income. This statement can be established in several steps. Suppose c = 0. First, it can be shown that 0(24) < 0(42), so that structure (24) cannot
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generate the highest per capita real income. Then, it can be established that (7(23) < (7(32) if a > 1 and k> s, which is assumed in this paper. Therefore, structure (23) cannot be the equilibrium. Furthermore, comparisons among U(32), (7(42), and U(22) indicate that (7(42) > U(32) if a > 1 - 0.48 In k and (7(22) > (7(32) if a < 1 - 0.48 In k. This implies that structure (32) cannot be the equilibrium. A comparison establishes that U(33) < (7(43) if a > 1. Also, (7(34) < t/(43) if a > 1 and k > s. This means that structures (33) and (34) cannot be the equilibrium. Now, the set of candidates for equilibrium is narrowed down to structures A, (22), (42), (43), and (44). Comparisons among (7(A), (7(22), and (7(43) yield that (7(43) > (7(22) if a > ag = 0.88 - 0.1 In s - 0.39 In k and (7(22) > (/(A) if a >a9 = 1 - 0.72 In s. It is straightforward that ag ag, £7(22) < C/(A) and U(43) if a e (a 8 , a9), and (7(22) < t/(A) if a < as. Therefore, structure (22) cannot be the equilibrium. Similarly, it can be shown that (7(42) < C/(43) if a >an = 1 - 0.29 \n(sk); U(42) < (7(43) and (/(A) if a G (a10, ^ l), where a10 = -1.31 - 0.25 \n(sk); and (7(42) < (/(A) if a b, then A is the equilibrium when a a7. Ifsk < b, then (7(43) < (/(A) or (7(44), so that A is the equilibrium if a < a6 and (44) is the equilibrium if a > a6. If c e (o,j) and c is sufficiently close to \ from the left-hand side, then more structures with an intermediate level of division of labor may occur in equilibrium. Here, the values of b and all at for i < 8 are given in Proposition 1.
References 1.
Dixit and J. Stiglitz, Monopolistic competition and optimum product diversity, American Economic Review, 67, 297-308 (1977).
2.
W. Ethier, Internationally decreasing costs and world trade, Journal of International Economics, 9, 1-24 (1979).
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G. Grossman and E. Helpman, Product development and international trade, Journal of Political Economy, 97, 1261-1283 (1989).
4.
F. Hahn, Equilibrium with transaction costs, Econometrica, 39, 417-39 (1971).
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A. Karman, "Competitive Equilibria in Spatial Economics," Oelgeshloger, Cambridge (1981).
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D. A. Kendrick, "The Planning of Industrial Investment Programs," Johns Hopkins Univ. Press, Baltimore (1978).
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P. Krugman, Scale economies, product differentiation, and the pattern of trade, American Economic Review, 70, 950-959 (1980).
8.
M. Kurz, Arrow-Debreu equilibrium of an exchange economy with transaction costs, Econometrica, 42, 45-54 (1974).
9.
E. Mills and B. Hamilton, "Urban Economics," Scott, Foresman, Glenview, IL (1984).
10. Adam Smith, "An Inquiry into the Nature and Causes of the Wealth of Nations" (E. Cannan, Ed.), University of Chicago Press, Chicago (1976). 11. U. Schweizer, General equilibrium in space, in "Location Theory." (J. J. Gabszewicz et al., Eds.), Harwood, London (1986). 12. X. Yang, Development, structural changes, and urbanization, Journal of Development Economics, 34, 199-222 (1990). 13. X. Yang and J. Borland, A microeconomic mechanism for economic growth, Journal of Political Economy, 99, 460-482 (1991). 14. X. Yang and G. Hogbin, The optimum hierarchy, China Economic Review, 2, 125-40(1990). 15. X. Yang and Y. Ng, "Specialization and Economic Organization," Elsevier Science, Amsterdam (1993). 16. X. Yang and H. Shi, Specialization and product diversity, American Economic Review, 82, 392-398 (1992).
CHAPTER 9 AGGLOMERATION ECONOMIES, DIVISION OF LABOUR AND THE URBAN LAND-RENT ESCALATION: A GENERAL EQUILIBRIUM ANALYSIS OF URBANISATION*
Guang-Zhen Sun a and Xiaokai Yang b * "Max Planck Institute and University of Macau
h
Monash
University
1. Introduction The purpose of the paper is to explain several important phenomena of urbanization, including the urban land-rent escalation, decreases in the relative per capita consumption of land in the urban and rural area, increases in the population size of the urban area compared to the rural area, and the absolute increase of diversity of occupations in the urban area as well as relative to that in the rural area, as different aspects of evolution in the division of labour. The intimate relationship between cities and the division of labour has indeed long been recognized by Xenopnon (Gordon 1975), William Petty (1682), Alfred Marshall (1890, Ch. 9-10), Mills (1972), Scott (1988) and others. However, as noted by
Reprinted from Australian Economic Papers, 41 (2), Guang-Zhen Sun and Xiaokai Yang, "Agglomeration Economies, Division of Labour and the Urban Land-Rent Escalation: A General Equilibrium Analysis of Urbanisation," 164-184, 2002, with permission from Blackwell. * We are grateful to Laurent Calvet, Don Snodgrass, Guoqiang Tian, our colleagues at Max Planck Institute and Monash University, seminar participants at Duke University and Harvard University and the anonymous referee for helpful comments on previous versions. Sun gratefully acknowledges the financial support from the Research Committee of the University of Macau. The usual disclaimer applies.
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Stigler (1976, pp. 1209-1210), there was no formal theory of division of labour and specialisation in the mainstream economics: "The last of Smith's regrettable failures is one for which he is overwhelmingly famous — the division of labour (A)lmost no one used or now uses the theory of division of labour, for the excellent reason that there is scarcely such a theory.... Smith gave the division of labour an immensely convincing presentation — it seems to me as persuasive a case for the power of specialization today as it appeared to Smith. Yet there is no evidence, so far as I know, of any serious advance in the theory of the subject since his time, and specialization is not an integral part of the modern theory of production, which may well be an explanation for the fact that the modern theory of economies of scale is little more than a set of alternative possibilities." In the recent two decades, there emerged a growing literature on specialisation and the division of labour, including Becker (1981), Rosen (1978, 1983), Baumgardner (1988), Kim (1989), Locay (1990), Yang (1990), Becker and Murphy (1992) and Yang and Ng (1993) among others, with accentuation of the endogenisation of individuals' levels of specialisation. In this paper, we address the progressive division of labour and urbanisation process using the framework developed by Yang (1990) and Yang and Ng (1993) which is featured by the concept of consumer-producers (rather than consumers and firms) and increasing returns to labour specialisation. Although considerable progress has been made in modelling urbanisation and agglomeration, particularly within the New Economic Geography, there have been few analyses that address the emergence and growth of cities resulting from the evolution in division of labour in a general equilibrium setting. Remarkable exceptions are Yang and Rice (1994) and Fujita and Krugman (1995).1 Taking evolution of division of labour as evolution in the number of traded goods and in individuals' levels of specialisation, Yang and Rice's story runs as follows. Due to the trade off between economies of specialisation and transaction costs, 1
Other related works in one way or another are Fujita (1985), Fujita and Mori (1997), Hochman (1997), Krugman and Venables (1996) and Yang (1990).
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as a unit transaction cost coefficient falls, each individual reduces her number of self-provided goods and increases her level of specialisation, so that the equilibrium level of division of labour evolves. In addition, the production of agricultural goods is land intensive and hence farmers must have their residence dispersed, while producers of manufactured goods can freely choose between dispersed residences and concentralised residences. If the partial division of labour between the production of the agricultural good and the production of one manufactured good occurs in equilibrium, then each manufacturer resides nearby a farmer, no city exists. If a high level of division of labour between manufacturers as well as between farmers and manufactures emerges from sufficiently improved transaction conditions, all manufactures will reside together in cities to reduce transaction costs between them. Their model can predict the following phenomena as different aspects of the evolution in division of labour and urbanisation. Productivity of all goods and per capital real income increase, the trade dependence, the extent of the market, individuals' levels of specialisation, and the degree of diversity of economic structure increase. However, this model cannot predict some important phenomena of urbanisation such as increases in the land rent differential between the urban and rural areas and decreases in relative per capita consumption of land of urban and rural residents, as the consumption of land does not enter the utility functions of the agents in their model. These two phenomena have been confirmed by many empirical works, of which Colwell and Munneke (1997) is the latest one among others.2 Similarly, in Fujita and Krugman (1995), there are also tradeoffs between global economies of scale, utility benefit of consumption variety of manufactured goods, and transport costs. An increase in population size or in transaction efficiency will enlarge the scope for trading off one against others among the conflicting forces, thereby increasing productivity, per capita real income, and consumption variety. In addition, there is a trade off between transaction costs that can be 2
von Thunen (1826), Beckmann (1969), Ben-Akiva et al. (1989) and others have studied the land price differentials without taking account of the equilibrium structure of division of labour.
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saved by concentrated residences of manufacturers in a city and transaction costs between rural farmers and urban manufacturers that is increased by the concentrated urban residences. The increase in the number of manufactured goods, an aspect of division of labour, will move the efficient balance of the latter trade off toward a more concentrated residential pattern of manufacturers, making a city more likely to emerge. The benefit of concentrated residences of manufactures caused by an increase in the number of manufactured goods is called economies of agglomeration. In fact, their model is a modern version of von Thunen's (1966) isolated state, in which the driving force of city formation is the circular linkage between economies of agglomeration and concentration of production. They conducted a very interesting comparative static analysis demonstrating that a decrease in the unit transaction cost, a larger population size, and a higher productivity of the agricultural sector will make a city more likely to emerge from a greater number of manufactured goods and will increase the population share of urban residents. But the city in Fujita and Krugman (1995) is a dimensionless single point and hence the residential land market could not be addressed. As in Yang and Rice (1994), the consumption of residential land does not enter the utility functions of agents. Furthermore, the degree of market integration is not endogenised because all agents are always connected by an integrated market. Increases of individuals' specialisation could not be addressed in Fujita and Krugman either. Both the Yang and Rice model and Fujita and Krugman model cannot predict the increases in relative land rent and population density in the urban and rural areas since economies of agglomeration in their models come from the concentration of manufacturers' residences (refereed to as type I economies of agglomeration hereafter). But in reality, we can observe economies of agglomeration that are generated by geographical concentration of transactions (type II economies of agglomeration hereafter). The geographical concentration of transactions will generate transaction cost advantage to residing in a city since urban residents can considerably benefit from informational spillovers, and save on travelling faraway as most transactions are executed at the city.
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The competition for the urban land that is a central market place where transactions are executed will then bid up urban land rent. Hence, type II economies of agglomeration may be the main driving force behind land rent differential between the urban and rural areas.3 In the current paper, a general equilibrium model with endogenous urban-rural occupation structure and the division of labour is developed to endogenise the following phenomena. Concomitantly with the urbanisation resulting from the expansion of the network of division of labour that is in turn driven by improvements in transaction efficiency, the land rent in the urban area increases absolutely as well as relative to that in the rural area, the number of occupations in the urban area increases absolutely as well as relative to that in the rural area, per capita consumption of land in the city decreases and the per capita consumption of rural residents increases, the number of traded goods for each individual as well as for the society as a whole increases, and the extent of endogenous comparative advantages between individuals of different occupations increases. Our story runs as follows. Each agent is a consumer-producer, and consumes both consumption goods and land for residence. The trade off between economies of specialisation and transaction costs implies that the equilibrium level of division of labour increases as a unit transaction cost coefficient of goods falls. A larger network size of division of labour will generate a larger number of transactions per person, so that the concentrated pattern can save on travelling cost per person by shrinking a large transaction network into a concentrated area (the city). Those residents at the central market place can trade all goods without travelling faraway and benefit from intra-city informational spillovers (Quigley 1998). Hence, competition for residing in the city will bid up the land rent of the urban area. Free migration between the urban and rural areas and between different professions will equalise per capita real income of all individuals such that transaction advantage of urban residents are offset by a higher land-rent for residence and smaller per 3
Our distinction between the two types of economies of agglomeration echoes Lindsey et al. (1995), McCann (1995) and Nakamura (1985) who argue that the concepts of economies of agglomeration and externalities of urbanisation need to be refined.
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capita consumption of land. As transaction conditions are improved, a larger network size of division of labour will be chosen and type II economies of agglomeration increase, so that all concurrent phenomena mentioned above take place as different aspects of the evolution in division of labour. In short, stripping the process of urbanisation and concentration of residence down to its bare essentials, it is the complicated interplay among the division of labour, geographical patterns of transactions and residence, and the trading efficiency that gives rise to urban land-rent escalation and the structural rural-urban shift in terms of residents, occupations, and concentration of economic activities.4 In an interesting study by Brueckner and Zenou (1999) of the augmented Harris-Todaro model with a land market, the urban land-rent escalation also provides a force against the rural-urban migration. The Brueckner and Zenou model is nevertheless a partial equilibrium model in which the product prices are exogenously given. Moreover, in their model, the evolution of the diversity of occupations, a remarkable feature concomitant with the urbanisation, is not addressed. Neither is the division of labour and specialisation. By combining Henderson's (1974) modern classic piece and Krugman (1991), one of founding models of the new economic geography, Tabuchi (1998) recently contributes a general equilibrium analysis of urban agglomeration economies due to product variety and agglomeration diseconomies due to intra-city congestion in a two-city system framework. It demonstrates dispersion may take place when the interregional transportation cost is sufficiently low. The economies of agglomeration in Tabuchi's model come from the concentration of manufacturing, whereas in our model what matters is the type II economies of agglomeration coming from the concentration of transactions (more discussion on this point will be given in the end of section 5, below). More importantly, the urban land rent escalation and the evolution of the diversity of urban occupations, most remarkable features among others concomitant with the 4
As far as we know, Scott (1988) is the first one who spells out, in a descriptive way, the interdependence among the structure of division of labour, location patterns of economic activities and transaction conditions.
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urbanisation, are not addressed in Tabuchi (1998). Neither is the diversity of the traded goods for consumption and individuals' specialisation. The paper proceeds as follows. Section 2 is devoted to an analysis of the interdependence among the geographical pattern of transactions, trading efficiency and the network size of the division of labour. The model is specified in section 3. Section 4 solves for the general equilibrium urban-rural occupation structure and the network of division of labour. A comparative statics is conducted in the section 5. The final section concludes. 2. Economies of Transaction Agglomeration and Division of Labour In the preceding section, we highlight that type II economies of agglomeration, which are associated with the network effects of the division of labour and the concentrated location pattern of transactions, may be the most important driving force of the land-rent differentials between the urban and rural areas. This section is motivated to further spell out the mechanism of the intrinsic relation among the network size of the division of labour, the trading efficiency and the economics of geographical concentration of transactions. Presumably, if the geographical pattern of individuals' residences is fixed and each pair of trade partners trade in the geographical mid-point of their residences, total travel distance and related cost will increase more than proportionally as the network of transactions required by a particular level of division of labour is enlarged. But if all individuals conduct their transactions at a central place, the large network of transactions can be geographically shrunk and be concentrated in that central place to significantly reduce the total travel distance of all individuals. To be sure, the economies of transaction agglomeration differ from economies of scale. Some economists call them positive externalities of cities. Indeed, they are generated by interactions between the positive network effects of division of labour and the effects of the geographical concentration of transactions. On the one hand, the extent of the division of labour is determined by trading efficiency, which itself depends on the geographical pattern of transactions; but, the effect of the
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geographical pattern of transactions on trading efficiency is in turn determined by the level of division of labour, on the other. Hence, trading efficiency, geographical pattern of transactions, and level of division of labour are interdependent and should be simultaneously determined in a general equilibrium model, as is shown in the following sections. One important insight among others that could be drawn from such a perspective is that the crucial determinant of the land rent of a city is the size of network of division of labour that is associated with the city as its center of transactions. The above analysis regarding how geographical concentration of transactions can improve the trading efficiency through the network effects of the division of labour could be illustrated by considering a simple numerical example. An Example. Consider a local community with n goods (n = 2, 3,..., 7) traded among agents, as shown in Figure 1. The distance between each pair of neighbours is assumed to be 1. For simplicity, we restrict ourselves to the symmetric case in which each agent is assumed to sell only one good to exchange for the other (« - 1) traded goods. The number of types of specialists producing some particular goods, n, is a proxy for the degree of the division of labour. Abstracting from irrelevant complication, suppose that exogenous transaction costs are proportional to the travel distance of individuals in conducting the transactions required by the division of labour and that one unit of travel distance costs $1. In panel (a), each pair of individuals trades at the geographical middle point of their residences, which is represented by a small circle. In panel (b), all individuals go to the centre of the local community, represented by the small circle, which is the residence of an individual, to trade with each other. Now, consider the case n = 7. In panel (a), each of the six individuals residing at the periphery of the community has a farthest trade partner. She travels to the centre, which is the middle point between her and the partner, to trade with the partner. She can trade with the person at the centre by the way of the trip. The travel distance for the return trip cost $2. She has other two neighbouring trade partners. It costs her $1 to trade with each of them. The distance between her and each of the
A General Equilibrium Analysis of Urbanisation »=2
-e
•
n=3
«=4
n=5
n=6
n=l
(a) Dispersed location of transactions
(b) Concentrated location of transactions
Figure 1: Dispersed vs. Concentrated Location Patterns of Transactions
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other two trade partners is V3. A return trip to the middle point between her and each of them is thus V3. Hence, it cost her $2V3 to trade with the two trade partners. Her transaction costs with six trade partners then total $(2 + 2 + 2A/3) = $7.46. For the person at the centre, transaction costs are zero, since all other individuals will trade with her at the centre as they stop by there to trade with their farthest partners. In the geographical pattern of panel (b), all individuals not residing at the centre bring their goods there to trade. Total transaction cost for each of them is $2. A comparison of transaction costs between panels (a) and (b) indicates that geographical concentration of transactions can save on transaction costs if the size of the network of division of labour is sufficiently large. Williamson refers to the pattern of transactions in panel (b) as the pattern of the wheel, and that in panel (a) as the pattern of all channels. If the purpose of travelling is to obtain information about products, prices, and partners, then increasing returns to the geographical concentration of transactions will be more significant. We may define a central market place as a geographical location where many trade partners conduct transactions. This definition implies a corresponding geographical concentration of transactions. Geographically dispersed bilateral transactions are not associated with the market according to this definition. In fact, it can be shown that the market is not needed if the level of division of labour is low. Consider, for instance, the case n = 2. It can be shown that if the geographical pattern of transactions is such that each pair of trade partners trade at the mid-point between their residences, the transaction cost to each of them is only $ 1. But if all individuals go to the central market place to trade, as shown in Figure 14.2, panel (b), for the case with n = 7, then each individual's transaction cost is $2. This illustrates that for a low level of division of labour, geographically concentrated transactions will generate unnecessary transaction costs. Thus, the capacity of a geographically concentrated pattern of transactions to save on transaction costs depends on the level of division of labour. In other words, trading efficiency is dependent not only on the geographical pattern of transactions, but also on the level of division of labour. But the level of division of labour is itself determined by trading
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efficiency. This interdependence among the level of division of labour, the geographical pattern of transactions, and trading efficiency implies that the three variables are simultaneously determined in a general equilibrium environment. It is analogous to the interdependence between the prices and quantities of goods that are consumed and produced in the standard general equilibrium model, where the optimum quantities of goods consumed and produced are dependent on prices, while the equilibrium prices are themselves determined by individual agents' decisions on the optimum quantities. 3. The Model Consider an economy with M ex ante identical agents in terms of all characteristics.5 Each agent is a consumer-producer, being endowed with a unit of labour. The sizes of residential land in the urban and rural areas are assumed to be A and B, respectively.6 Each individual can freely choose any occupation configuration and residence location between rural and urban areas. There are m consumption goods. The number of goods that are actually traded, however, is to be endogenised by the complicated interplay among the location pattern of transactions, the division of labour and the trading efficiency. Traded goods produced in the urban and rural areas are referred to as A-type and B-type goods, respectively. For each individual, the job choice is associated with the choice between working in the urban area producing A-type goods and working in the rural area producing B-type goods. The utility function of each individual is given by u(z, R), where z represents the amount of the composite consumption good and R the 5
The ex ante identity assumption allows us to formalise Smith's notion of endogenous comparative advantages, which, different from Ricardo's exogenous comparative advantages, means that differences in productivity among individuals are the outcome rather than cause of the division of labour (1776/1976, Chapter 2, p.28). Indeed, as argued forcefully by Smith (1776, Ch.1-3), the labour heterogeneity is not a necessity for the division of labour to emerge. Unfortunately, this point seems to have been simply neglected by many contemporary economists. 6 The assumption of exogenously given lot sizes simplifies the algebra significantly. We leave the endogenisation of the sizes to future research.
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consumption of residential land, i.e., the lot size of the house. Specifically, we assume thatw(z,i?) = zi?andzis composted in such a way from different consumption goods that, z = Y\ ™ 1 xf, where xf is the amount of good/the individual consumes. Thus, the utility function is f m
\
Y\xf R
(1)
V i'=i
where xct - xt + ktxf is the amount of good i consumed, xt is the amount of good i self-provided, xf is the amount of good i purchased from the market, /c, is an iceberg transaction efficiency coefficient (fraction 1 - kt of a unit of good purchased disappears in transit) of good i, and R the amount of land consumed. The production function is identical for all goods, no matter whether they are produced in the city or in the rural area, with a constant marginal return and a fixed set-up cost, Xf + xf -I-a
(2)
where /. is the amount of labour employed in producing good /, which is referred to as the individuals level of specialisation in producing good /, xf is the amount of good i sold to the market, and a is the fixed learning cost, which generates economies of specialisation in production. As shown by Houthakker (1956), Becker (1981), Rosen (1983), the fixed learning cost implies that the division of labour can save on total learning cost for society as a whole by avoiding duplicated learning. The aggregation economies of specialisation for all individuals can generate economies of division of labour which are sometimes referred to as positive network effects. A city, in the urban economics literature, is usually defined as an agglomeration of transactions and manufacturing.7 As Glaeser (1998, p. 140) puts it, 'Conceptually, a city is just a dense agglomeration of people and firms.' Yang and Rice (1994, p.350) hold a similar view, 'a city is defined as the area where many individuals have the division of 7
Nakamura (1985, p. 108) typifies the idea that the benefit from agglomeration of economic activities constitutes the basic driving force of the city formation and urbanisation, 'Agglomeration economies are the most important in explaining modern cities.'
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labour and reside in close proximity to one another'. For residents in the urban area, they benefit not only from the low transport cost as they exchange their produce for other goods that they need in the concentrated urban area but also from the informational spillovers that is much more pronounced in the urban area than in rural areas.8 Those who reside in the rural area pay a higher commuting cost, as they have to travel from the dispersed rural area to the city for their transaction activities, and they suffer from the less spillover of information compared to urban residents. In short, the transaction efficiency for the urban residents is higher than that for rural residents. For simplicity, the ratio of the transaction efficiency coefficient of urban residents, being assumed the same for all traded goods, denoted as kA, and that of rural residents for all traded goods, denoted as kB, is assumed to be a constant greater than one, kAlkB=X>\
(3)
Although individuals might have incentives to reside in the urban area for the facility in transaction, competition for residence in the urban area bids up the residential land rent in the city which in turn means less space for residence in the urban area. The tradeoff between economies of agglomeration and the disutility from less consumption of land due to the land rent escalation in the urban area leads to an equilibrium residence structure and an equilibrium location pattern of transactions, while the efficient trade off between economies of specialisation and transaction costs determines the equilibrium network size of division of labour. Here, the location pattern of residences, the geographical pattern of transactions, the consumption pattern of land, relative land rent between urban and rural areas, and the network size of division of labour are interdependent. Therefore, the concept of general equilibrium is a Glaeser (1998, p. 140) argues that, 'All of the benefits of cities come ultimately from reduced transport costs for goods, people and ideas' and further points out that even now 'the positive impact of agglomeration that comes from reducing the costs of moving people and ideas appear to be as important as ever.' Quigley (1998) emphasises that agglomeration economies are associated with the benefits from the diversity of economic activities in cities. Fujita and Thisse (1996) provide a rather comprehensive overview of economies and diseconomies of agglomeration of economic activities.
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powerful vehicle for figuring out a mechanism that simultaneously determine all of the interdependent variables, as to be shown below in section 4. Lastly, a public ownership of the residential land in both urban and rural areas is assumed.9 Total rent of residential land within the urban (rural) area is equally shared among urban (rural) residents. Hence, each individuals income consists of a component from selling her professional produce and a component from the equally-shared land rent. 4. Equilibrium Residence Structure and Division of Labour What we consider in this paper is a framework with consumerproducers, economies of specialisation in production, and transaction costs in which each individual is allowed to decide upon her specialisation pattern, i.e., to choose the numbers of goods self-provided and purchased from the market. That is, each individual may choose the zero value of some decision variables. The invisible hand coordinates the decentralised decision of individuals on their specialisation pattern and resource allocation and leads to the general equilibrium in which the residence pattern and the network size of division of labour for the society are endogenised. This generates the high degree of complexity of modelling as well as high explanatory power of the model. When the zero value of each decision variable is allowed, there are 23m possible corner and interior solutions for the problem with m decision variables. Fortunately, as proved by Wen (1998), in this kind of model, each individual sells at most one good and does not self-provide and purchase the same good. Although Wen's (1998) proof is somehow complicated mathematically, the economic intuition is rather straightforward: the agent would otherwise bear some unnecessary transaction costs. 9
This assumption, which was also made in Sun and Yang (1998) for simplifying the algebra, indeed can be to some extent justified. 'By institutional facts, half of urban land is in the public domain and the other half is controlled by planning and zoning' (Henderson, 1996, p.32). Note the assumption of a public land ownership is also adopted in Fujita and Thisse's (1986, section 5), in which a Stackelberg strategy location model is developed with firms as leaders and households as followers, as well as in Helsley and Strange (1990).
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209
As is in section 3, traded goods produced in the urban (rural) area are denoted as A-type goods (B-type goods). Then, producers of A-type goods (B-type goods) have incentives to reside in the urban (rural) area to save on commuting cost between the residence and working location. Therefore, for each consumer-producer, the decisions on residence location and job choice become independent: one chooses to reside in the urban (rural) area if and only if she chooses to produce A-type (Btype) goods. To make the algebra tractable, in the remainder of this paper, we assume that no goods are of both type A and type B. Let nA (nB) be the number of A-type (B-type) traded goods, and n = nA+nB be the number of all traded goods. Clearly, the values of nA, nB, and n are not independent. Given a pair of (nA, nB), each agent can optimise on her resource allocation for the given structure of division of labour with nA occupation configurations in the urban area and nB occupations in the rural area. Each pair of (nA, nB) thus could be used to describe an occupation structure between urban and rural areas. But in a general equilibrium setting, the equilibrium structure of division of labour, which is fundamentally important for understanding the urbanisation process and residence patterns, is endogenously determined. In the following, we will use a two-step approach to solve for the equilibrium pattern of residence and social division of labour. First, we define a value profile of nA, nB, and n as a structure. All individuals' utility maximising decisions are solved for given values of nA, nB, and n. Then market clearing conditions and utility equalisation conditions are used to solve for the equilibrium for a given urban-rural occupation structure (nA, nB). Secondly, taking account of possible value profiles of nA, nB, and n, of which each profile corresponds to a rural-urban occupation structure, we solve for the general equilibrium, which, as is shown below is actually the structure associated with the highest per capita real income.
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a) An equilibrium analysis for a given rural-urban occupation structure The symmetry of the model implies that we need to consider only two types of decision problems. For a urban resident producing a A-type good, the decision problem is Max: uA ^{xA)(kAxdA)"^(kAxdB)^xJA" s.t. xA + xA = lA — a xjA = ljA - a
R
A
(4)
(production function for the good sold)
(production function for a non-traded good)
lA+(m —ri)ljA< 1 (endowment constraint for working time) PAXA (HA
~ 1) + PB4nB
+ r R
A A= PAXA + EA
(budget constraint)
where nA(nB) is the number of A-type (B-type) goods, n = nA+nB is the number of all traded goods, xA is the quantity of the A-type good the individual self-provides, xA is her amount of the A-type good sold to the market, and lA is her quantity of labour allocated to the production of this good, defined as her level of specialisation in producing this good, xA is her amount of a A-type good purchased from the market, xB is her amount of a B-type good purchased from the market, xjA is her amount of a non-traded good produced and consumed, ljA is her quantity of labour allocated to the production of a non-traded good. The symmetry implies that xdA is the same for the (nA - 1) A-type goods purchased, xB is the same for nB B-type goods purchased, and ljA is the same for m - n of non-traded goods. pA and pB are the respective prices of A-type and B-type goods. rA (rB ) is the land rent in the urban (rural) area. RA is the lot size of each urban resident's house. EA= rAAIMA is the land rent distributed to each urban resident, where MA (MB) is the number of urban (rural) residents. The decision variables are xA, xA, lA, an XJA, UA, XA 5 XB d RA10
Given the rather complicated interplay among the geographic pattern of transactions, the network size of the division of labour and the trading efficiency, it is natural for variables characterising specialisation patterns, time allocation, land rents, prices of
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211
The solution of the decision problem for given values of nA, nB, and n in (4) is called an individual's resource allocation for a given rural-urban occupation structure. Standard marginal analysis is applicable to such resource allocation problems. It yields the demand and supply functions and indirect utility function as follows npA [l - (m - n + \)a\ -{m-n
+ \)EA
{m + \)pA pA(l + na) + EA (m + \)pA
(5b)
pA\-(m-n + X)a\ + E,A xdA = ^—-. —-± (m + l)pA PA[l-(m-n
+l a
)]
+
EA
{m + \)pB pA[l-(m-n + Y)a] + EA {m + \)rA f
K
A
r
APl
P.A
^" B
EA+ pA[\-(m-n
+ \)a}}
m+\
\PBJ
(5a)
(5c)
(5d) ^
(5f)
The optimum decision for a rural resident is symmetric to (5), that is, it can be obtained by exchanging subscripts A and B and letting kB=k. Due to the free rural-urban migration, the equilibrium for structure (nA, nB) can be obtained from the utility equalisation between urban and rural residents and the market clearing conditions for residential lands and goods. The land market clearing conditions are MARA=A
(6a)
MBRB=B
(6b)
The market clearing condition for A-type goods yields goods and so on to be incorporated. Appendix A contains a glossary of all the variables, categorised into parameters and endogenous variables, for the readers' reference.
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npA[l-(m-n
+ l)a]-(m-n
+ l)EA MA
{m + \)pA
nA
PA[\-(m-n -
+ \)a] + EA MA. -(nA-l)-\ (m + \)pA
pB[l-(m-n + l)a] + EB ;{m + \)p ; MB A
(7) The market clearing condition for B-type goods is not independent of (6) and (7) duo to Walras' law. Free rural-urban migration implies the equalisation of utilities for residents in urban and rural areas, from which one obtains,
^ + ^ [ 1 - ( " » - w + l)g]| EB+pB[l-(m-n + \)a]\
\"-B J
r
\
(8)
\PBJ
i.e.
X
„_, \EA+pA[\-(m-n
+ \)a\
E
B +pB[i-(m~n
+ l)a],
r
\
(8)'
PB
vny (6)-(8), the population equation MA+ MB = M, and the definitions EA = rAAIMA and EB = rBBIMB yield an equilibrium for a given structure defined by nA, nB, and n (let /? s = 1 , i.e., the price of B-type goods is used as the numeraire), PA =
h=lB=—
B 0 A 1-0
l-(m-n
_[l-(m-n
m
X«
+ \)a
—+ «
+ l)a]0M mA
[\-{m-n + \)a](\- 8)M mB M^^-^OM
(9a) (9b) (9c)
(9d) (9e)
A General Equilibrium Analysis of Urbanisation
MB=^-
(9f)
= (\-0)M
RA^AMB^A(l~0) RB BMA BO u* = k/~l
l-(m-n m
where / =
-r
6 j v 1-0
+ Y)a
,(\-0)„
213
(9g) A_ \ + f M f (l-0)n+l
(9h)
6= nAln is the urban share of traded goods.
A general equilibrium of residence pattern and division of labour in this paper is comprised of two components. The first component consists of a set of relative prices of traded goods and land, and a set of numbers of individuals choosing different configurations of occupation and residence, and a location pattern of transactions that satisfy the market clearing and utility equalisation conditions. The second is an equilibrium rural-urban occupation structure defined by (nA, nB), ox{n,6), which is endogenised by the interaction among economies of the division of labour, transaction costs and the residence pattern. In the next subsection we will take account of the second component and solve for the general equilibrium, based on which a comparative statics analysis is to be conducted in section 5. b) The general equilibrium residence structure and social division of labour In the preceding subsection, both the number of all traded goods and that of A-type traded goods are fixed. However, both the emergence and evolution of the urban system result from the expansion of the network of division of labour, which is in turn associated with the structural shift of transaction location patterns and the increase of the diversity of occupations (see, e.g., Quigley 1998). In this subsection a general equilibrium analysis is done to solve for the urban-rural structure and the total number of occupations.
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Note that in our model occupation structures are Pareto rankable in terms of utility due to the utility equalisation within any structure, which in turn results from the ex ante identity of agents. In fact, the general equilibrium is in the structure that yields the maximum per capita real income, as shown by the following Lemma.11 Lemma: The occupation structure with the highest per capita real income is a general equilibrium. Any Pareto inefficient structure is not a general equilibrium. This Lemma indeed establishes the claim that a decentralised market can fully utilise economies of agglomeration (which look like externality) and the network effect of division of labour by choosing the Pareto efficient pattern of individuals' residences, the efficient location pattern of transactions, and the efficient network size of division of labour. It also rules out multiple equilibria with different per capita real incomes and shows that there is no coordination failure in a static general equilibrium model with network effects of division of labour and location pattern.12 The very function of the market is to coordinate individuals' decisions in choosing their specialisation patterns and location patterns of residence and transactions in order to fully utilise network effects of the division of labour and economies of agglomeration.13 11
Sun, Yang and Zhou (2001) have shown that both the existence of general equilibrium and the First Welfare Theorem still hold for a broad class of models in the framework with consumer-producers, transaction costs and increasing returns to specialisation in production. One can modify their argument regarding the First Welfare Theorem by considering the augmented model (with a land market incorporated) presented in this paper, and prove the following Lemma. For an alternative proof of the Lemma, see Sun and Yang (1998). 12 But because of indeterminacy about who specialises in which activity, which n out of the m goods are traded, and which nA out n traded goods are produced in the urban area in equilibrium, we have multiple equilibria that generate the same per capita real income. 13 The two-step approach adopted in this section allows us to sort out the general equilibrium, in which nobody has incentives to deviate from her current occupation (location) and resource allocation. It could be shown that if the representative A-type agent is allowed to maximise her utility with respect to numbers of A-type goods and Btype goods she purchases from the market at the first step of our algorithm, and so is the
A General Equilibrium Analysis of Urbanisation
215
With the above Lemma, we can solve for the general equilibrium by maximising the per capita real income given in (9) with respect to nA and nB. Since nA + nB = n, this is equivalent to maximising per capita real income in (9h) with respect to n and 9 = nAln, for which the first order condition leads to (see Appendix B for details), A-i = * • - * A 1-0
(10)
and In* +
— = -#ln/l \-{m-n + Y)a
(11)
Note that (10) implies \h\X\dn =
1 0(1-0)
d0
(12)
that is, 0 = nAln increases with n. In other words, the number of traded goods produced in the urban area increases more than proportionally as the network size of division of labour n increases. (10) and (11) uniquely determine the general equilibrium values of n and 0, which characterise the size and pattern of the network of division of labour and the location pattern of individuals' residences and transactions.14 Consequently, the general equilibrium values of the number of A-type goods and B-type goods (nA and nB), the urbanisation extent (MA or MA/M), the prices of traded goods (pA and pB, note the latter is used as the numeraire), the residential land rents in the urban B-type representative, then their optimal decisions would not match with each other, and hence such an one-step approach does not lead to an occupation equilibrium. Individuals may have coordination failure in deciding their occupation and trade patterns in such an asymmetric dual structure. But that does not mean the coordination failure of the 'invisible hand' in sorting out the equilibrium structure of occupations and division of labour, as is further discussed below. 14 A careful reader may be concerned with the possibility of multiple equilibrium structures characterised by («*,#*) satisfying (10) and (11). As to be shown in section 5, for any («*,#*) satisfying (10) and (11), bothw* and 0* are increasing with k. But from (12), n* and 6* are increasing with each other. Thus the equilibrium occupation structure is unique.
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G.-Z. Sun, X. Yang
and rural areas (rA and rB), the per capita lot size of residence for urban and rural areas (RA and RB), and the specialisation level of individuals (IA and lB) can be obtained by using (10)—(11) and the equilibrium solution for a given structure derived in the preceding subsection, as below (the subscript" " is suppressed). nA=0n
(13a)
nB = {l-0)n
(13b)
K t=0M
(13c)
PB=1
PAZ =
(13d)
_[\-(m i-n + \)a]i9M 'A
mA
[l-(m- n + \)a](\-0)M 'B
mB B A
0 1-0
(13h)
K
RB=
.
(13i)
(l-0)M
\-(m-n
lA=lB=—i
(13f) (13g)
RA=— A 0M B
(13e)
+ \)a L- + a
,,_.. (13j)
m Note that not only the number of traded goods produced in the urban area increases more than proportionally as the network size of division of labour n increases, but more interestingly, both the absolute urban land-rent (rA) and the relative urban-rural land-rent {rA/rB) increase even faster than the urbanisation speed (0 = nA/n-MAIM) . That is, concomitant with the urbanisation process and the expansion of the network of social division of labour, is the even faster escalation of the land rent in urban areas.
A General Equilibrium Analysis of Urbanisation
217
5. Comparative Statics In this section we conduct a comparative statics with respect to the transaction efficiency to investigate the implication of the institutional arrangements and infrastructure for the urbanisation and the network size of the social division of labour. Intuitively, as the transaction efficiency is improved, the unit transaction cost involved in each transaction is decreased, and therefore individuals are allowed a larger scope for trading off economies of specialisation against transaction costs. That means, more goods would be traded through the market. One may naturally expect more traded goods to be produced in the urban area because of the better facility for transaction in the urban area compared with that in the rural area. But the more trade dependent, the more attractive the urban area as residence location to individuals. That means, as the transaction condition is improved, more and more people would migrate from the rural area to the city, thereby further bidding up the urban land-rent. Consequently, the lot size of residence in the urban area decreases. The balance of the tradeoff between economies of the transaction agglomeration and the disutilities from the smaller per capita consumption of land in the urban area leads to a new equilibrium for the improved transaction condition. The following comparative static analysis confirms this conjecture and provides some important insights into other phenomena associated with urbanisation, among which is the increase of specialisation levels of individuals. Our comparative statics starts with an analysis of changes of the occupation diversity and the urban-rural structure of traded goods, i.e., n (=nA+nB) and 0{= nAln) , in response to the improvement of transaction condition, i.e, the increase of the transaction efficiency coefficient k. As shown by (10) and (12) in section 4, the A-type share of traded goods, 0, increases (more than proportionally) with the number of all traded goods, n. Differentiating both sides of (10) with respect to k yields,
f-rai-tfUn*]^ ak
dk
(.4)
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G.-Z. Sun, X, Yang
Inserting (12) into the total differentiation of (11), together with the second order condition for maximising the per capita real income (9h) with respect to 0 and n that requires the negative definiteness of the Hessian matrix (D2u*(n,6) < 0), leads to (see Appendix C for proof) ^=ldk
k
-2
^ [l-(m-n
l
-
>0
(15)
-0(l-0)(ln^ + l)a]
Combining (14) and (15), we have, dk (15) and (16) summarise the comparative statics of general equilibrium network size of division of labour and the equilibrium location pattern of residence and transactions. That is, not only that the number of traded goods for each individual as well as for society as a whole, n, increases with transaction efficiency, but the number of traded goods produced in the urban area increases more than proportionally as improvements in transaction conditions enlarge the network size of division of labour, n. Using (9), the comparative statics of general equilibrium values of endogenous variables can be easily worked out, as shown below. ^ = —*- = — increases with transaction efficiency k nB
MB
1-6
nA = nO increases with transaction efficiency k nB = n(\ - 0) ambiguous — = — = — increases with transaction efficiency k r,
1-0
A
_ [ \m *—"J— increases with transaction efficiency k A
mA \\-{m-n+\)a\a-e)M
rB = ^-^
mB
..
ambiguous
-^- = ——- decreases with transaction efficiencyJ k R„
BO
A General Equilibrium Analysis of Urbanisation
219
RA = — decreases with transaction efficiency k RB =
increases with transaction efficiency k
h=h=
~n+ 'a + a increases with transaction efficiency k
u (per capita real income) increases with transaction efficiency k Note that the envelope theorem is used to derive the positive effect of A: on u from (9h) and (10). Besides the implication of the improvement in transaction facility for the urban land-rent escalation, the structural rural-urban shift of production of traded goods and the urbanisation, it can be further shown that the following concurrent phenomena are also different aspects of the evolution in division of labour driven by improvements in transaction conditions. The diversity of occupations, the differentials between different occupations, and the number of markets for different goods, increase. The number of transactions for each individual, the trade dependence, which is defined by trade volume to real income, interdependence among individuals of different occupations, the extent of the market, which is defined by per capita aggregate demand from the market, and the degree of commercialisation, which is defined by the ratio of income from the market to total real income, increase. The extent of endogenous comparative advantage, which is defined as the difference in productivity between sellers and buyers of traded goods, increases. Production concentration, which is defined as the reciprocal of the number of producers of each traded goods, the per capita real income and productivity of each good increase also. In summary, we have Proposition: Urban land rent increases absolutely as well as relative to the land price in the rural area in the urbanisation process which in turn is a result of the expansion of the network of division of labour driven by improvement in transaction efficiency. At the same time the per capita land consumption decreases for urban residents and increases for rural residents, and the number of occupation in urban areas increases
220
G.-Z. Sun, X. Yang
absolutely as well as relative to that in the rural area. Improvement in transaction conditions also generates the following concurrent phenomena. The number of traded goods for each individual as well as for the society as a whole increases. The number of traded goods produced in the urban area increases absolutely as well as relative to that in the rural area. The population ratio of urban and rural residents increases. Each individual's level of specialisation and the extent of endogenous comparative advantage between individuals of different occupations increase. The number of markets, the number of transactions for each individual, the trade dependence and the interdependence among individuals of different occupations increase. The extent of the market, the geographical concentration of both production and transaction increase. Per capita real income and productivity of each good increase. It seems worthwhile to point out that a distinguishing feature of type II economies of agglomeration in this model is that even if it is indeterminate in equilibrium which «B out of n traded goods are produced in the rural area, a high level of division of labour will make the concentration of transactions and residences in the urban area to emerge from ex ante identical individuals with identical production technologies. The assumptions made in Yang and Rice (1994) and Fujita and Krugman (1995) that land-intensive agricultural production must be dispersed in the rural area is not necessary for the existence of type II economies of agglomeration that mainly result from the agglomeration of transactions and residences. Furthermore, a variety of manufactured goods that is not land intensive may possibly be produced in the rural area, like what have happened in most developed countries after 1970s that is sometimes termed as suburbanisation (see, e.g., Glaeser 1998, p. 145 and Tabuchi 1998, p.334), because of the trade off between economies of agglomeration and high land rent in the city. This has a flavour of the theory of complexity: some phenomena that do not exist for each individual element of a system emerge from a complex structure of a collection of numerous ex ante identical individual elements. This is just like different DNA structure of the same molecules generating many species of animals. This suggests that a hierarchical system of cities
A General Equilibrium Analysis of Urbanisation
221
might be developed based upon type II economies of agglomeration alone. 6. Concluding Remarks A general equilibrium model with consumer-producers, economies of specialisation, and transaction costs is developed in this paper to address the urban land rent, trade concentration in urban area and other issues associated with the urbanisation process. Urbanisation results from the expansion of the network of the social division of labour driven by improvements in transaction conditions. It is shown that as urbanisation develops, the urban land rent increases absolutely as well as relative to the land rent in the rural area, the land consumption decreases for urban residents and increases for rural residents, and the number of occupations in urban areas increases absolutely as well as relative to that in the rural areas. Improvement in transaction conditions also generates the following concurrent phenomena as different aspects of the evolution in division of labour. The number of traded goods for each individual as well as for society as a whole increases. The number of traded goods produced in the urban area increases absolutely as well as relative to that in the rural area. The population ratio of urban and rural residents increases. Each individual's level of specialisation and the extent of endogenous comparative advantage between individuals of different occupations increase. The number of markets, the number of transactions for each individual, the trade dependence and the interdependence among individuals of different occupations increase. The extent of the market, the degrees of commercialisation, the geographical concentration of both production and transactions increase. Per capita real income and productivity of each good increase. We have shown that the invisible hand can fully exploit the network effect of division of labour and economies of agglomeration (which look like externality of urbanisation) by efficiently coordinating individuals' decisions in choosing their individual networks of transactions and their location patterns of residence and transactions. Interpreting agglomeration economies as increasing returns, Ades and Glaeser's (1999) empirical investigation of the (positive) relationship
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G.-Z. Sun, X. Yang
among extent-of-market, the division of labour and economic growth effectively provides a support to our insight that the interplay between geographical patterns of transactions and trading efficiency leads to urbanisation and increase of productivity. Furthermore, the theory developed in this paper has some important policy implications for urbanisation and economic development. Concomitant with the urbanisation is usually the diversity of economic activities in the city that is conducive to the economic development and hence enhances the welfare of the economy. As is demonstrated in China in the post-Mao era when the government interventions with economic activities attenuates and thus the transaction efficiency is improved, the population share of urban residents increases from 17.9 percent in 1978 to 30.4 percent in 1998 (PRCY 1999, p. 783). Potentially, the theory developed in this paper also has important implications for the real estate business practice. Once elabourated, it may hopefully provide a vehicle to predict the potential for price increase of the urban land. The potential is critically dependent on the equilibrium network size of the division of labour that is in turn determined by transaction conditions. It should be pointed out that our model is simple and limited in some aspects. One possible way to expand the model is to relax the assumption of fixed land sizes in urban and rural areas, and then the city size in terms of the size of residential land would be endogenised. That will significantly enrich the urbanisation story and allow for more insights into the geographic structure of the dual economy. As such, the presumption taken in this paper that the land rent within the city (and the rural area) is the same could also be relaxed. Indeed, as far as transaction costs are concerned, only the distinction between rural and urban areas is made, but within the urban (rural) area the distance from the central market place is assumed irrelevant to the transaction costs in our model. Thus, the distance-related transaction costs could be addressed in the extended model. Another extension in to modify the model to allow for the endogenisation of the diversity of consumptions by introducing a CES utility function. By doing so, increase in the diversity of consumption product, a notable feature with the urbanisation process, would be endogenous rather than exogenously given as is in this paper.
A General Equilibrium Analysis of Urbanisation
223
That is actually an appealing attempt but the mathematics may become quite complicated. Thirdly, no intermediate goods are allowed in the current model, and therefore a very important aspect of the division of labour, i.e. the roundaboutness of production and its connection to urbanisation process is ignored. One possible way to undertake this task is to integrate the present model with Sun and Lio's (1998) model of industrialisation that centres around the co-evolution of the division of labour and roundaboutness in production to address the interplay between urbanisation and industrialisation as an important aspect of the progressive division of labour. We attempt to conduct further analyses in sequels to this paper.
Appendix A Parameters: kA transaction efficiency coefficient for urban residents kB = k transaction efficiency coefficient for rural residents X = kA/kB urban-rural transaction efficiency ratio a set-up cost in all production functions M population size of the economy A{B) size of residential land in the urban (rural) area Endogenous variables: z composite consumption goods consumption of residential land of the urban (rural) resident /, labour employed to produce good i lA(lB) specialisation level of the urban (rural) resident xA(xB) amount of the traded good self-provided by the urban (rural) resident xsA (xsB) amount of a traded goods sold by the urban (rural) resident RA{RB)
224
G. -Z. Sun, X. Yang
nA{nB) number of traded goods produced in the urban (rural) area n = nA + nB number of all the traded goods 9= njn urban share of traded goods m number of all necessity consumption goods MA(MB) population of urban (rural) residents PA{PB) prices of traded goods produced in the urban (rural) area rA(rB) land rent in the urban (rural) area Appendix B Maximisation of the utility (9h) with respect to the A-type share of traded goods, # yields In
X"~ —
e
1
1
+—: 9
e1
(Al)
\-e
Note that the left hand side of (A 1) is an increasing function while the right hand side is a decreasing function of — ^ — - , so that (Al) hold if < r-'A(i-8). and only if =1, i.e. BO A--*.-' A 1-9
(A2)
K
Inserting (A2) into the first order condition for maximising (9h) with respect to the number of the traded goods, n ,yields ma ln& + = -9\nA 1 - (m - n + \)a
(A3)
Appendix C Insertion of (A2) and (A3) into the second order derivatives of the utility (9h) with respect to the number of traded goods (n) and the A-type share of traded goods (9) yields
A General Equilibrium Analysis of Urbanisation
d2u dO1
n(n + 2) (n + l)2
=
d2u* dOdn
225
1 0(1-0)
n(n + 2) In X (n + l)2
(A5)
ma2
6(l-6)(lnX)2 = — (A6) T +— dn2 [\-(m-n + \)a\ (n + l)2 The negative definiteness of the Hessian matrix D2u* (n, 6) requires that (A4) and (A6) be negative and15
d2u* d2u* de2
dn2
f d2u* ^ dOdn
>0
(A7)
which holds if and only if ma.
[l-(m-n
2 + l)a] >6{\-9)^X)
(A8)
Thus, we establish (15).
References Ades, A.F. and Glaeser, E. 1999, 'Evidence on Growth, Increasing Returns, and the Extent of the Market', Quarterly Journal of Economics, vol. 114, pp. 1025-1045. Baumgardner, J.R. 1988, 'The Division of Labor, Local Markets, and Worker Organization', Journal of Political Economy, vol. 96, pp. 509-527. Becker, G. 1981 A Treatise on the Family, Harvard University Press, Cambridge. Becker, G. and Murphy, K. 1992, 'The Division of Labour, Coordination Costs, and Knowledge', Quarterly Journal oj'Economics, vol. 107, pp. 1137-1160.
Indeed we can choose the parameter X sufficiently close to (yet greater than) one to guarantee that (A6) is negative and that (A8) holds.
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Beckmann, M. 1969, 'On the Distribution of Urban Rent and Residential Density', Journal of Economic Theory, vol. 1, pp. 60-67. Ben-Akiva, M., de Palma, A. and Thisse, J.-F. 1989, 'Spatial Competition and Differentiated Products', Regional Science and Urban Economics, vol. 19, pp. 5-19. Brueckner, J.K. and Zenou, Y. 1999, 'Harris-Todaro Models with a Land Market', Regional Science and Urban Economics, vol. 29, pp. 317-339. Colwell, P.F. and Munneke, H.J. 1997, "The Structure of Urban Land Prices', Journal of Urban Economics, vol. 14, 3: pp. 321-336. Fujita, M. 1985, 'Towards General Equilibrium Models of Urban Land Use', Revue Econmique, vol. 36, pp. 135-167. Fujita, M. and Krugman, P. 1995, 'When is the Economy Monocentric: von Thunen and Chamberlin Unified', Regional Science and Urban Economics, vol. 25, pp. 505-528. Fujita, M. and Thisse, J. -F. 1986, 'Spatial Competition with a Land Market: Hotelling and von Thunen Unified', Review of Economic Studies, vol. 53, pp. 819-841. Fujita, M. and Thisse, J. -F. 1996, 'Economies of Agglomeration', Journal of Japanese and International Economies, vol. 10, pp. 339-378. Glaeser, E.L. 1998, 'Are Cities Dying?' Journal of Economic Perspectives, vol. 12, 2: pp. 139-160. Gordon, B. 1975, Economic Analysis before Adam Smith. Macmillan, London. Henderson, J.V. 1974, "The Size and Types of Cities', American Economic Review, vol. 64, pp. 640-656. Henderson, J.V. 1996, 'Ways to Think About Urban Concentration: Neoclassical Urban Systems versus the New Economic Geography', International Regional Science Review, vol. 19, 1-2: pp. 31-36. Helsley, R.W. and Strange, W.C. 1990, 'Matching and Agglomeration Economies in a System of Cities', Regional Science and Urban Economics, vol. 20, pp. 189-212. Hochman, O. 1997, 'More on Scale Economies and Cities', Regional Science and Urban Economics, vol. 27, 4-5, pp. 373-398. Houthakker, M. 1956, 'Economics and Biology: Specialization and Speciation', Kyklos, vol. 9, pp. 181-189.
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Kim, S. 1989, 'Labor specialization and the Extent of the Market', Journal of Political Economy, vol. 97, pp. 692-705. Krugman, P. 1991, 'Increasing Returns and Economic Geography', Journal of Political Economy, vol. 99, 3: pp. 483-499. Krugman, P. and Venables, AJ. 1996, 'Integration, Specialization, and Adjustment', European Economic Review, vol. 40, pp. 959-967. Lindsey, J.H., Pratt, J.M. and Zeckhauser, R.J. 1995, 'Equilibrium with Agglomeration Economies', Regional Science and Urban Economics, vol. 25, pp. 249-260. Locay, L. 1990, 'Economic Development and the Division of Production between Households and Markets', Regional Science and Urban Economics, vol. 98, pp. 965-982. Marshall, A. 1890, Principles of Economics. McCann, P. 1995, 'Rethinking the Economics of Location and Agglomeration', Urban Studies, vol. 32, 3: pp. 563-577. Mills, E. 1972, Urban Economics, Scott, Foresman and Company, Glenview. Nakamura, R. 1985, 'Agglomeration Economies in Urban Manufacturing Industries: A Case of Japanese Cities', Journal of Urban Economics, vol. 17, pp. 108-124. Petty,W. 1682, Another Essay in Political Arithmetick, Concerning the Growth of the City of London. PRCY, 1999, People's Republic of China Yearbook 1999, Yearbook Publisher, Beijing. Quigley, J.M. 1998, 'Urban Diversity and Economic Growth', Journal of Economic Perspectives, vol. 12, 2: pp. 127-138. Rosen, S. 1978, 'Substitution and the Division of Labor', Economica, vol. 45, pp. 235-250. Rosen, S. 1983, 'Specialization and Human Capital', Journal of Labor Economics, vol. 1, pp. 43-49. Scott, A.J. 1988, Metropolis: From the Division of Labor to Urban Form, University of California Press. Smith, A. 1776/1976, An Inquiry into the Nature and Causes of the Wealth of Nations. Reprint, edited by E. Cannan, University of Chicago Press, Chicago.
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Stigler, G. 1976, 'The Successes and Failures of Professor Smith', Journal of Political Economy, vol. 84, pp. 1199-1213. Sun, G.-Z. and Lio, M. 1998, 'The Division of Labor and Roundabout Production: Allyn Young Revisited', Working Paper, Department of Economics, Monash University. Sun, G.-Z. and Yang, X. 1998, 'Evolution in Division of Labor, Urbanization and Land Price Differentials between the Urban and Rural Areas', Development Discussion Paper #639, Harvard Institute for International Development. Sun, G.-Z., Yang, X. and Zhou, L. 2001, 'General Equilibria in Large Economies with Endogenous Structure of the Division of Labor', Working Paper, Department of Economics and Finance, City University of Hong Kong. Tabuchi, T. 1998, 'Urban Agglomeration and Dispersion: A Synthesis of Alonso and Krugman', Journal of Urban Economics, vol. 44, pp. 333-351. von Thunen 1826/1966, The Isolated State, Translated by Carla M.Wartenberg, Pergamon, NewYork. Wen, W. 1998, 'An Analytical Framework with Consumer-Producers, Economies of Specialization, and Transaction Costs', In K. Arrow et al, Eds., Increasing Returns and Economic Analysis, Macmillan, London. Yang, X. 1990, 'Development, Structural Changes and Urbanization', Journal of Development Economics, vol. 34, pp. 199-222. Yang, X. and Ng, Y.-K. 1993, Specialization and Economic Organization, A New Classical Microeconomic Framework, North-Holland, Amsterdam. Yang, X. and Rice, R. 1994, 'An Equilibrium Model Endogenizing the Emergence of a Dual Structure between the Urban and Rural Sectors', Journal of Urban Economics, vol. 35, 3: pp. 346-368.
Part 5
Entrepreneurship and the Firm
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CHAPTER 10 THEORY OF THE FIRM AND STRUCTURE OF RESIDUAL RIGHTS*
Xiaokai Yang and Yew-Kwang Ng* Monash
University
1. Introduction This paper uses a formal version of the Coase-Cheung theory of the firm to investigate the productivity implications of the structure of residual rights. According to Coase (1937), the rationale for the existence of the firm is the differences in transaction costs between the market and the institution of the firm. If the 'market' involves greater transaction costs than the firm, then the division of labor will be organized within a firm rather than via the market. Hence, the boundary of the firm is determined by the equalization condition of marginal benefits (saving on transaction costs relative to the market) and marginal bureaucracy costs of the firm. For a further development, Cheung (1983) argued that the firm does not replace the market with a non-market institution. Rather, it replaces the market for intermediate goods with the market for labor hired to produce the intermediate goods. Hence, the firm emerges if transaction costs are higher in trading intermediate goods than in trading the labor hired to produce the intermediate goods. Thus, the argument that the firm internalizes externalities does not make sense since externalities in the
* Reprinted from Journal ofEconomic Behavior & Organization, 26 (1), Xiaokai Yang and Yew-Kwang Ng, "Theory of the Firm and Structure of Residual Rights," 107-128, 1995, with permission from Elsevier. * The authors are grateful to Jeff Borland, Richard Day, Gene Grossman, Edwin Mills, Paul Milgrom, Yingyi Qian, M. Aoki, and an anonymous referee for their comments. Thanks also go to the Ford Foundation and the Open Society Fund for financial support. The second author wishes to acknowledge the predominant contribution of the first author. 231
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market for goods may be replaced by externalities in the market for factors.1 Given the existence of the firm, it may have many different structures of ownership of a firm. Does the structure of ownership make a difference? According to Coase (1960), the structure of ownership matters if transaction costs are not negligible. We show here that the existence of transaction costs is necessary but not sufficient for the structure of ownership to have productivity implications.2 Our story runs as follows. There are many ex ante identical consumer-producers in an economy. Each individual as a consumer must consume a final good, called cloth, the production of which requires an intermediate good, called management of production, as an input. Although each individual as a producer can produce either or both cloth and management service, he prefers specialized production due to economies of specialization. Each individual's optimal decision is a corner solution. He may choose autarky which implies that he selfprovides cloth and self-manages the production. This pattern of organization generates a low productivity level because economies of specialization cannot be exploited. Individuals can choose the division of labor which implies that some individuals become professional producers of cloth and others become professional producers of management service. 1
The principal-agent theory and related contract theory formalize a tradeoff between efficient risk sharing and effective incentive mechanism to show that the second best partial equilibrium contract generates transaction costs (see Hart and Holmstrom, 1987). The procurement model (see, for example, Laffont and Tirole, 1986 and Lewis and Sappington, 1991) addresses the decision problem in choosing between outside procurement and self-provision within a firm by formalizing a tradeoff between exploitation of exogenous comparative advantage and distortions resulting from information asymmetry. Zhang (1993) identifies the condition under which capitalists act as principals. All the models do not explain why and how firms emerge from the division of labor. 2 Grossman and Hart (1986) and Hart and Moore (1990), built on the work of Williamson (1975) and Klein, Crawford, and Alchian (1978), have established the statement that the structure of ownership makes a difference, although their theory is a theory of optimum ownership structure rather than a theory of the firm. However, their models established the statement on the basis of asset specificity argument. Our model will establish a similar statement on the basis of indirect pricing of intangible intellectual property rights. In the Holmstrom and Milgrom model (1994) and our model here the labor contract is emphasized as an essential feature of the institution of the firm.
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This pattern of organization generates a high productivity of both goods, on the one hand, and incur transaction costs, on the other. Hence, there is a tradeoff between economies of specialization and transaction costs. If transaction efficiency is high, then the division of labor will occur in equilibrium because economies of specialization outweigh the transaction costs generated by the division of labor. Otherwise autarky will be chosen as the equilibrium. Suppose that transaction efficiency is so high that individuals prefer the division of labor to autarky. Then there are three different structures of residual rights which can be used to organize transactions required by the division of labor. The first (structure 1) is comprised of markets for cloth and management services. Specialist producers of cloth exchange cloth for management consultant service with specialist producers of management service. For this market structure, residual rights to any contracts and authority are symmetrically distributed between trade partners and no firms and labor market exist. The second structure of residual rights (structure 2) is comprised of the market for cloth and the market for labor hired to produce management service within a firm. The producer of cloth is the owner of the firm and specialist producers of management services are employees. Residual rights and authority are asymmetrically distributed between the employer and his employees. The employer claims the residual of the firm which is the difference between revenue and wage bill. The third structure of residual rights (structure 3) is comprised of the market for cloth and the market for labor hired to produce cloth within a firm. The professional manger is the owner of the firm and specialist producers of cloth are employees. For the final two structures of residual rights, the firm emerges from the division of labor. Compared with structure 1, the two structures involve a labor market but not a market for management services. As Cheung argued, the firm replaces the market for intermediate goods with the market for labor hired to produce the intermediate goods. Although both structures involve the firm and asymmetric structure of residual rights, they have different structures of ownership of a firm. Assuming that transaction efficiency is much lower for management service than for labor, then the institution of the firm can be used to organize the division of labor more efficiently because it avoids trade in
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management services. Suppose further that transaction efficiency for labor hired to produce management services is much lower than for labor hired to produce cloth because it is prohibitively expensive to measure efforts exerted producing intangible management (a sort of intellectual property) and to measure output level (quality and quantity) of management services. Then the division of labor can be more efficiently organized in structure 3 than in structure 2 because structure 3 involves trade in cloth and in labor hired to produce cloth but not trade in management services and in labor hired to produce management services, while structure 2 involves trade in cloth and in labor hired to produce management services. Hence, structure 3 will occur in equilibrium if the transaction efficiencies for labor hired to produce cloth and for cloth are sufficiently high. The claim to the residual of the firm by the manager is the indirect price of management services. Therefore, the function of the asymmetric structure of residual rights is to get the activity with the lowest transaction efficiency involved in the division of labor while avoiding direct pricing and marketing of the activity, such that the division of labor and productivity are promoted. In a sense, the function of the asymmetric structure of residual rights is similar to that of a patent law which enforces rights to intangible intellectual property thereby promoting the division of labor in research and development. However, the asymmetric structure of residual rights can indirectly price those intangible intellectual properties which are prohibitively expensive to protect even through a patent law (can we enforce a patent law of intangible management knowledge?). Although transaction costs play an important role in the analysis, we do not explicitly model the source of the transaction costs. Rather, 'iceberg'-type transaction costs are assumed. These transaction costs may be interpreted as anticipated costs of ex post strategic or opportunistic behavior associated with trade in a particular market. The exogeneity of transaction costs allows us to capture in a simple way the main ideas which seem to have emerged from the transaction cost literature, namely that transaction costs exist and that they may differ across goods and factors and across institutional structures for production and exchange.3 3
For example, Williamson (1985) argued that since relationships between economic agents are governed by incomplete contracts, there is scope for opportunistic behavior by
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235
After presenting our analysis, we use it for the concluding section to explain specific institutional settings, appertaining to Chinese history and to more recent Marxian issues in modern socialist economies. 2. A Model with Intermediate Goods Let us consider an economy with M identical consumer-producers. There is one consumer good and one intermediate good (or service) in this economy. The self-provided amounts of the consumer and intermediate goods are y and x respectively. The quantities of the two goods sold in the market are ys and xs respectively. The quantities of the two goods purchased in the market are yd and xd respectively. In order to produce the final good, the intermediate good is a necessary input. An individual's production function for the final good is y + y°={[x + (l-t)xd]Ly}a,
a<\,4
(la)
where xd is the amount of the intermediate good purchased from the market and t is its transaction cost coefficient. The fraction / of xd disappears in transaction.5 Hence, (l-t)xd is the amount an individual receives from the purchase of this intermediate good. The amount self-provided of this good is x. Ly is the labor share in producing the consumer good and y + ys is the output level of this consumer good. A person's labor share in producing a good is defined as his level of specialization in producing this good. A production function for a good is said to exhibit economies of specialization if total factor productivity of the good is an increasing function of the level of specialization in producing this good. Total factor productivity of good y, (y + ys y(XaD~a J = L2"'1 increases with the level the parties to such contracts, and that this possibility of opportunistic behavior constitutes a source of transaction costs. The scope for opportunistic behavior, and hence the size of transaction costs, will differ across contracts and across institutions which specify different rules for governing a relationship when unforeseen circumstances arise, and according to the frequency of trade and the degree of asset specificity and uncertainty involved in the particular relationship. 4 Equilibrium may not exist if a > 1. 5 This specification of 'iceberg' transaction technology is necessary for avoiding notorious indexes of origins and destinations of deliveries.
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of specialization!^ if a > 1/2, w h e r e X s x + (l-^)x r f is the input level of the intermediate good. The parameter a represents the degree of economies of specialization in producing the final good.6 The production function of the intermediate good is x + x s + Lbx,
(lb)
where x + Xs is the output level of the intermediate good and Lx is a person's level of specialization in producing the intermediate good. This production function displays economies of specialization if b > 1. The parameter b represents the degree of economies of specialization in producing the intermediate good. It is assumed that labor is specific for an individual and there is a constraint on the total labor share, given by Lx+Ly=l,
0
i = x,y.
(lc)
By choosing a proper unit of labor, we can assume that each person is endowed with a unit of labor. (1) specifies a system of production functions for an individual. According to these production functions, each individual may self-provide all goods. Transaction technology of the final good is the same as that of the intermediate good. The fraction A: of a shipment disappears in transit. Thus, (l - k)yd is the amount an individual receives from the purchase of the consumer good. The amount consumed of the final good is y +(l - k)yd. The utility function is identical for all individuals and given by U = y + (l-k)ydl.
(2)
Free entry for all individuals into any sector and a large Mare assumed.
6
Parameter a may be divided into two components a\ and a2, which represent output elasticities of the two inputs. However, this separation complicates the algebra and contributes little to the results. 7 In Yang (1990), the case of multiple final goods is tackled by adopting a Cobb-Douglas utility function, giving rise to a trade-off between preference for diverse consumption and economies of specialization.
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3. Configuration and Corner Solution versus Market Structure and Corner Equilibrium The set of feasible market structures can now be characterized. Each individual makes a decision about which goods to produce and on his demand for and supply of any traded good to maximize. A given structure of production and trade activities for any individual is defined as a configuration. There are 26 = 64 combinations of zero and non-zero values of x, xs, xd, y, /, yd and therefore 64 possible configurations. The combination of configurations of the M individuals in the economy is defined as a market structure, or simply a structure. A feasible market structure consists of a set of choices of configurations by individuals such that for any traded good, demand for the good is matched by supply of the good. There exists a corner equilibrium for each structure. Corner equilibrium is defined as a set of relative prices and relative numbers of individuals choosing different configurations such that (i) for any traded good market supply equals market demand (i.e. market clearing); and (ii) each individual maximizes utility at the given prices and for a given structure. With the assumption of free entry, (ii) implies that the utility of all individuals is equalized in any corner equilibrium. Using the Kuhn-Tucker theorem it is possible to rule out the interior solution and many corner solutions from the list of candidates for the optimum decision. Proposition 1: Ifb > 1 and a e (0.5, I), an individual sells at most one good and does not buy and self-provide the same good; he self-provides the consumer good when he sells it; and he does not self-provide the intermediate good unless he self-provides the final good. The proof is given in the Appendix. In this paper, it is assumed that b > 1 and a e (0.5, 1). Taking into account this proposition and the possibility for setting up firms, there are four structures. Here, institution of the firm is defined as a structure of transactions that satisfies the following conditions. (a) One party (employer) has control rights of other party's (employee's) labor; (b) In relevant contracts, only payment to employees is specified. An employer claims residuals of returns; and (c) An employer sells goods
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or services, produced using employees' labor, in good market. This section first solves for corner equilibria in these four structures, then identifies the general equilibrium from these corner equilibria.8 Structure A (autarky) Each individual self-provides all goods, so xs = ys = xd = yd = 0. Assuming this in the decision problem (l)-(2) yields MaxU = y = x°L°y =Lf (\-Lx)°,
(3a)
Lx = 6/(1 + b), Ly = 1/(1 + b), x = [b/{\ + b)]b,
(3b)
which implies
UA=y = [b>/(l + b)M'_ Here the individual maximum utility UA is percapita real income as well as the maximum percapita output level of the final good in autarky. Structure D Individuals specialize in either the intermediate or final good. Denoting a configuration in which an individual sells the intermediate good and buys the final good by (x/y) and a configuration in which an individual sells the final good and buys the intermediate good by (y/x), structure D consists of configurations (x/y) and (y/x). Authority and residual rights are equally distributed between the two parties to a contract. There are two steps in solving for the corner equilibrium. The individual optimum decisions for these two configurations are first solved. Then individual demand and 8
Note that the specific model in this paper can be used to solve for the explicit equilibrium, so that a general proof on the existence of equilibrium is not necessary. On the other hand, the major concern of this paper is the evolution of the division of labor and institutional arrangements. A specific model is necessary for comparative statics that describes such an evolution because the comparative statics generate shifts of general equilibrium across market structures and it can be characterized only by comparing corner equilibrium utilities in different structures. This is another restriction imposed by the complexity of non-linear programming technique.
Theory of the Firm and Structure of Residual Rights
239
supply, generated by the optimum decisions, and the market clearing conditions, and utility equalization condition are used to solve for the corner equilibrium. Letxs,yd > 0, Lx = 1, x = xd = y = ys = Ly = 0; (lb,c) and (2) give the individual decision for configuration (x/y) as pyd _ xs ^ (budget constraint) xs = Lbx, Lx = 1, (production function and endowment constraint) d
> (4)
b
Ux = y (\- k) = L x(\- k)/ p = (l- k) p, (indirectuntilityfunction) where/? is the price of the final good in terms of the intermediate good and Ux is the indirect utility function for configuration (x/y). Let ys,y,xd >0, Ly=\, yd =x = xs = Lx = 0 in (la,c) and (2); the individual decision problem for configuration (y lx) is MaxUy=y = [(\-t)xdJ-xd/p, s.t. y + ys -\{\-t)xdLy\
, L -\,
(5a) (productionfunction)
pyS _ xd ^ (budget constraint) where / is the transaction cost coefficient for the intermediate good. The optimum decisions are Ly =1, x" = [ a / ? ( l - / ) f l ] 1 / ( M , y =xd/p,
(5b)
Uy=y = (l-a)[ap{\-t)YX-"\ where Uy is the indirect utility function for configuration (ylx). Let Mx and My be the numbers of individuals selling the intermediate and final goods respectively. Multiplying Mt (i = x, y) with individual demand or supply gives market demand or supply. Equalizing the market demand to the market supply establishes the market clearing conditions. The assumption of free entry combined with the individuals' behaviour of maximizing utility ensures the utility equalization condition. From the utility equalization condition and market clearing condition, we can thus solve for the corner equilibrium relative number of individuals choosing two configurations, M^ = MxJMy , and relative price/?. The values of Mx and My are determined by Mxy and the population size M -Mx+M .
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The utility equalization condition is Uy= (1 - *)[(l - t)apfX-a)
= (1 - k)/p = Ux.
(6)
(6) gives the corner equilibrium p and the percapita real income in structure D, UD: p=
[(l-k)/(l-a)fa/[a(l-t)J,
UD=
a"(\-afa(\-t)a(\-k)a.
The market clearing condition for the intermediate good is Mxxs = Myxd .
(8)
Note that the market clearing condition for the final good is not independent of (8) due to Walras' law. Inserting xd and Xs, given by the individual optimum decisions (4) and (5), andp, given by (7), into (8), we have Mv = xd/xs = xd = a(\ - jfc)/(l - a).
(9)
This Mxy is the corner equilibrium relative number of individuals selling different goods in structure D. Structure FY Let (y/Lx) denote a configuration in which an individual sells the final good, buys labor, produces the final good, and employs workers to produce the intermediate good. Similarly, (LJy) denotes a configuration in which an individual sells his labor, buys the final good, and becomes a worker producing the intermediate good. Structure FY consists of configuration (y/Lx) and configuration (LJy). In this structure, the producer of y claims the residual rights in the relationship. The individual decision problem for configuration (ylLJ) is MaxUy=y
= [(\-v)Nj
s.t. y + r=[{\-v)DxNLy^,
-N/q,
(10a) Lx=l,
Ly=\,
(production function and endowment constraint)
Theory of the Firm and Structure of Residual Rights
qys = NLx = N,
241
(budget constraint or trade balance)
where v is the transaction cost coefficient of labor hired to produce the intermediate good, N is the number of workers hired by an employer, q is the price of the final good in terms of labor. Ly is the level of specialization of an individual choosing configuration (y/Lx) in producing the final good and Lx is the level of specialization of an individual choosing configuration (LJy) in producing the intermediate good. Lx and Ly are the employer's decision variables because of asymmetric distribution of control rights. The optimum decisions are Lx=Ly=\, y>=N/q, Uy=y = [(l-v)Nj-N/q, r -WO-") N= (1-v) aq ,
(10b) (10c)
where Uy is the indirect utility of an employer. For an individual choosing configuration (LJy), all variables are fixed, given by U^={\-k)y=(l-k)/q, d
qy =LX=\,
(11a)
(budget and endowment constraints)
where UL is the indirect utility of an individual choosing configuration (LJy). Here, an individual has one unit of labor and labor is the numeraire.9 Manipulating the market clearing condition and utility equalization condition yields the corner equilibrium in structure FY, given by N = Miy={l-k)a/(l-a), q = [(l-k)/{l-a)f/[a(l-v)J
, (lib)
UFY=[(\-v)(l-k)aJ(l-a)l-\ whereUFYis the percapita real income in structure FY.M^ = MxlMy is the number of individuals choosing configuration (LJy) relative to those choosing (ylLx), Mx is the total number of employees, and M is the total number of employers. It should be noted that an employer cannot Note that for convenience in manipulating the algebra, the intermediate good is assumed to be the numeraire in market D, and labor the numeraire in structures FY and FX.
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X. Yang, Y.-K. Ng
manipulate M^ although he can choose Af (the number of workers hired by an employer) and that the corner equilibrium value of N is the same as the corner equilibrium value of Mxy.l° While more than one worker may be employed, economies of specialization do not extend beyond a specific individual. Structure FX Let (y/Ly) denote a configuration in which an individual sells the final good, buys labor, produces the intermediate good, and hires workers to produce the final good. Similarly, (Ly/y) denotes a configuration in which an individual sells his labor, buys the final good, and becomes a worker employed producing the final good. Structure FX consists of configurations (y/Ly) and (Ly/y). The difference between structures FY and FX is that in structure FY a producer of the final good is the owner of the firm, whereas in structure FX a producer of the intermediate good is the owner of the firm. Repeating the procedure of solving for the corner equilibrium in structure FY, the corner equilibrium in structure FX is given by Myx={l-k){\-a)/a, q = [{l-k)/a{l-r)j/{l-a)X-a,
(12)
UFX=[(l-r)aJ[(l-k)(l-a)f\ where r is the transaction cost coefficient of labor hired to produce the final good in structure FX, UFX is the percapita real income in structure FX, and q is the price of the final good in terms of labor. Myx is the number of individuals producing the final good and choosing configuration (Ly/y) relative to those producing the intermediate good and choosing configuration (y/Ly). A comparison between the percapita real income in structures A, D, FY, and FX leads us to
10
To mistakenly believe that a single employer can chose Mv since he can choose N is to commit the fallacy of attribution discussed by Ng (1982).
Theory of the Firm and Structure of Residual Rights
243
Proposition 2: (1) Structure A generates the maximum percapita real income if transaction efficiency and economies of specialization are sufficiently small. (2) The structure with the division of labor and without firms generates the maximum percapita real income if transaction efficiency and economies of specialization are sufficiently large and the transaction efficiency for the intermediate good is higher than for labor. (3) The structure with the division of labor and with producers of the final good as the bosses of firms generates the maximum percapita real income if transaction efficiency and economies of specialization are sufficiently large and the transaction efficiency for the labor used to produce the intermediate good is great compared to thatfor the intermediate good and for the labor used to produce the final good. (4) The structure with the division of labor and with producers of the intermediate good as the bosses of firms generates the maximum percapita real income if transaction efficiency and economies of specialization are sufficiently large and the transaction efficiency for the labor used to produce the final good is great compared to that for the intermediate good and for the labor used to produce the intermediate good.u The corner equilibrium with the maximum percapita real income is the general equilibrium because all corner equilibria satisfy all conditions for equilibrium, except that individuals' utilities are not maximized with
1
' More technically, Proposition 2 states: (1) Structure A generates the maximum percapita real income if j{k, t, a) < g(b), where f{k, U a)= (l -k)(\ -1) a(\ - a)V"~', df/dk < 0, df/dt< 0, 8f/da> 0, g{b) = b»l(\ + bf , dg/8b< 0. (2) Structure D generates the maximum percapita real income if fik, t, a) > g(b), t < v and h(t, r, k, a) > 1, where h(t,r,k,a) = [(1 - /)/(l - r)J (1 - kf"^ , dh/dk < 0 , dh/dt < 0 , dh/dr > 0 . (3) Structure FY generates the maximum percapita real income if J(k, v, a) > g(b), t>v and h(v, r, k, a) > 1, where df/dv < 0 , dh/dv < 0 . (4) Structure FX generates the maximum percapita real income if J{k, r, a) > g(b), h(t, r, k, a) < 1, and My, r, k, a) < 1, where df/dr < 0.
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X. Yang, Y.-K. Ng
respect to the choice of configurations across structures. Proposition 2 implies.
Thus,
Corollary 1: (1) The general equilibrium is the corner equilibrium in autarky if transaction efficiency and economies of specialization are sufficiently small. (2) The general equilibrium is the corner equilibrium in the structure with the division of labor and without firms if transaction efficiency and economies of specialization are sufficiently large and the transaction efficiency for the intermediate good is higher than for labor. (3) The general equilibrium is the corner equilibrium in the structure with the division of labor and with producers of the final good as the bosses of firms if transaction efficiency and economies of specialization are sufficiently large and the transaction efficiency for the labor used to produce the intermediate good is great compared to that for the intermediate good, andfor the labor used to produce the final good. (4) The general equilibrium is the corner equilibrium in the structure with the division of labor and with producers of the intermediate good as the bosses of firms if transaction efficiency and economies of specialization are sufficiently large and the transaction efficiency for the labor used to produce the final good is great compared to thatfor the intermediate good andfor the labor used to produce the intermediate good. Corollary 1 implies that an economy will evolve from autarky to the division of labor as transaction efficiency is improved. The equilibrium institution based on the division of labor is comprised of the markets for the final and intermediate goods if transaction efficiency of the intermediate good is higher than that of labor. It is comprised of the markets for the final good and for the labor hired by firms to produce the final good if transaction efficiency of labor hired to produce the final good is higher than that hired to produce the intermediate good. It is comprised of the markets for the final good and for the labor hired by firms to 12
Indeed, Yang (1988a, 1990) has proved that all corner equilibria that do not generate the maximum percapita real income are not general equilibrium.
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produce the intermediate good if transaction efficiency of labor hired to produce the intermediate good is higher than that hired to produce the final good. Firms will emerge from the latter two institutional arrangements. Figure 1 illustrates market structures A (autarky), D (division of labor without firms), FY (division of labor with firms owned by producers of the final good), and FX (division of labor with firms owned by producers of the intermediate good). Firm
(a) autarky
(b) structure D
(c) structure FY
(d) structure FX
Figure 1: Market Structures and the Institution of the Firm
4. Structure of Residual Rights, Economies of Specialization, Economies of Division of Labor, and Economies of the Firm This section investigates first the relationship between economies of specialization, economies of division of labor, and economies of the firm, and then the implications of a structure of residual rights for the theory of the firm. As shown in (la), the production function for good y displays economies of specialization if a > 1/2 since the total factor productivity of goody increases with a person's level of specialization in producing good y. Also, (lb) implies that the production function for good x exhibits economies of specialization in producing the intermediate good if b > 1 since labor productivity of good x increases with a person's level of specialization in producing this good.
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Economy of division of labor is a concept different from economy of specialization and economy of scale. There is a need for defining the concept of economy of division of labor before examining the difference between these concepts. The system of production function (1) is said to exhibit economies of division of labor if the maximum output level of the final good with zero transaction cost is greater in structure D than in structure A. In order to ascertain if production function (1) exhibits economies of division of labor, it is necessary to solve for the maximum output level of fmal good per capita in structure D. Since there is only one final good (and therefore utility is determined by percapita output level of the final good), the Pareto-efficient allocation in structure D will generate the maximum output level of the final good per capita. Maximizing utility for a configuration subject to the balance constraint between aggregate consumption and production and to the constraint that utility in another configuration is not smaller than a constant, it can be shown that a corner equilibrium in a structure is Pareto efficient for this given structure. As there is only one final good in our model, this implies that the corner equilibrium in structure D maximizes the percapita output level of the final good in this structure. This, combined with a fixed population size, implies that the maximum output level of the final good in structure D is given by the corner equilibrium in structure D. Assume that there is no transaction cost, i.e. k = t = 0 in (3b) and (7); a comparison of the maximum percapita output level of final good (which equals the percapita real income) between structures D and A yields the result that the system of production functions (1) exhibits economies of division of labor if F = a{\-ajla-\\ + bfb/b»>\,
(13)
where dF/da > 0 and dF/db > 0. This leads us to Proposition 3: The system of production functions (1) exhibits economies of division of labor if either (i) economies of specialization in producing the final good are sufficiently great, provided that b is not too small, or (ii) economies of specialization in producing the intermediate good are sufficiently great, provided that a is not too small.
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The implications of expression (13) are that economies of specialization in producing a single good (either the final or intermediate good) are neither necessary nor sufficient for the existence of economies of division of labor. Economies of specialization in producing a single good are not sufficient for the existence of economies of division of labor since we can find some a > 1/2 (i.e. economies of specialization exist in producing the final good) and a sufficiently small b such that (13) does not hold (i.e. no economies of division of labor). However economies of specialization in producing both goods are sufficient for the existence of economies of division of labor because (13) must hold if a > l/2and&> 1. Economies of specialization in producing the final good are also not necessary for the existence of economies of division of labor because we can find some a < 1/2 (i.e. economies of specialization do not exist in producing the final good) and a sufficiently large b such that (13) holds (i.e. economies of division of labor exist). In addition we can find some b < 1 (economies of specialization do not exist in producing the intermediate good) and a value of a that is sufficiently close to one (i.e. economies of specialization in producing the final good exist) such that a < 1 and (13) holds. This implies that the input-output relationship between goods provides the possibility that a sector with a significant degree of economies of specialization can promote productivity in other sectors which have no economies of specialization. In structure D, if b < 1, i.e. economies of specialization do not exist for (x/y) (or a < 111, i.e. economies of specialization do not exist for (y/x)), the economies of division of labor are external to configuration (x/y) (or (y/x)). However, if economies of specialization do not exist in any configurations, economies of division of labor cannot arise because (13) cannot hold if a < 1/2 and b < 1. In other words, economies of division of labor cannot be external to all individuals in our model. This provides a formal underpinning for Knight's argument (1925) that the concept of increasing returns that are external to all firms is an 'empty economic box'. On the other hand, even though economies of division of labor may be external to some individuals, our model will show that a decentralized market can trade off economies of specialization against ex ante or anticipated ex post transaction costs in order to exploit such externalities and diffuse the economies of
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specialization which exist in one configuration to other configurations which do not have economies of specialization. This is an important function of a free market system. In order to see this function, we have to show that a general equilibrium is Pareto optimal. Since a corner equilibrium is a restricted Pareto optimum for a given structure, and a general equilibrium is the corner equilibrium with the maximum percapita real income, a general equilibrium is certainly Pareto optimal. This, together with Corollary 1, implies that a decentralized market will fully exploit economies of division of labor if these economies outweigh the transaction costs. If there are economies of division of labor and there are no transaction costs, the division of labor can be organized through markets for final and intermediate goods. The institution of the firm is not required. However, if there are transaction costs and there exist economies of division of labor, the free market will search for the most efficient organizational structure in transactions among structures D, FY, and FX. In order to highlight the conditions for the emergence of the institution of the firm, we need to define economies of the firm. The system of production (1) and transaction technology is said to display economies of the firm if the percapita real income is greater in structure FY or FX than in structures A and D. This definition, combined with Propositions 1 and 3, and the comparison of percapita real income across the various structures yields. Corollary 2: No economies of the firm exist if there are no transaction costs. Economies of division of labor between production of the final good and production of the intermediate good are necessary but not sufficient for the existence of economies of the firm. There exist economies of the firm if economies of division of labor outweigh the transaction cost in structure FY (or FX) and the transaction efficiency is lower in trading the intermediate good than in trading the labor hired to produce the intermediate good. Coase (1937) surmises that economies of division of labor are not sufficient for the existence of economies of the firm. It is also necessary, he argues, that transaction costs are positive. Corollary 2 has formalized
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and refined these ideas. Cheung (1983) has developed Coase's theory of the firm by pointing out that the firm is replacing a goods market with a labor market rather than replacing market with non-market institutions. Our model has formalized the idea of Cheung. In addition, our model can be used to verify the Coase theorem (1960) if we interpret the Coase theorem in a special way that relates to our model. In the model presented in this paper, the structures of production and the division of labor are the same, but the structures of residual rights differ between market structures D, FX, and FY. In structure D, no employer-employee relationship exists, or in Grossman and Hart's words (1986), authority and residual rights of control and returns are equally distributed between all parties to a contract. In market structure FX or FY, authority is asymmetrically distributed between an employer and his employees. The employer claims the residual rights to returns and control of labor. If we identify the ownership of a firm with the claim to residual rights of a firm, then there are two structures of ownership of a firm in our model. In structure FX a producer of x claims residuals and owns the firm. Alternatively in structure FY a producer of y claims residuals and owns the firm. Corollary 2 implies that an asymmetric distribution of residual rights to returns and control is more efficient than a symmetric distribution of residual rights to returns and control if economies of the firm exist. Proposition 1 and Corollary 1 state that the equilibrium must be one of structures FX and FY if economies of the firm exist. Together with our discussion of the Pareto optimality of any equilibrium, this implies that the structure of ownership of a firm will affect the equilibrium and the Pareto optimum. This conclusion is the same as in Grossman and Hart (1986) which shows that the structure of ownership matters. However, our focus differs from that of Grossman and Hart (1986) and Hart and Moore (1990) who emphasize the role of specificity of tangible assets in understanding the implications of a structure of ownership for the theory of the firm. Consistent with the Coase theorem, our model shows that the structure of ownership makes no difference if transaction cost is absent, but the structure of ownership may matter if transaction costs are not zero. However, our model shows that the existence of transaction costs is necessary but not sufficient for the equilibrium implications of the
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structure of ownership. For instance, structures D, FX, and FY generate the same percapita real income, so that the structure of ownership makes no difference if t = v and h(t, r, k, a) = 1, where h(t, r, k, a) is given in footnote 11. The implication of the analysis in this paper is that claims to residual rights can avoid the pricing of efforts of the claimant of residual rights, thereby reducing transaction costs if such pricing is prohibitively expensive. Let us use an example to illustrate this point. Suppose y is a tangible consumer good, such as cloth and x is an intangible management service in producing cloth. It is prohibitively expensive to measure and price x or the labor used to produce x, or to measure the relationship between efforts exerted producing x and the quantity and quality of x. For instance, it is impossible to tell if a manager is thinking about management strategy or about his girl friend when he is sitting in office. This assumption is equivalent to a large value of t, the transaction cost coefficient of the intermediate good, and a large value of v, the transaction cost coefficient of labor hired to produce x. Assume, on the other hand, that it is very cheap to price labor hired to produce y, because it is easy to measure the relationship between efforts exerted producing y and the quantity and quality of the tangible final goods. For instance, the quality and quantity of cloth produced by a worker is easy to measure and it is easy to tell that the worker is shirking if he does not move his hands. This assumption is equivalent to small values of k, the transaction cost coefficient for trading the final good, and r, the transaction cost coefficient for trading labor hired to produce y. For the structure of residual rights in market structure FX, the final good y is exchanged for labor hired to produce y. Hence, goody and efforts exerted producingy have to be priced, but the producer good x and efforts exerted producing x need not be priced in this structure. Therefore, this structure of residual rights is a special way of replacing the pricing of intangible x and the efforts exerted in producing x with the pricing of the tangible good y and the effort exerted in producing y. The claim to residuals of a firm owned by the producer of x is an indirect price of the employer's effort. This indirect pricing of management efforts via claims to residual rights avoids the prohibitively
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high costs in pricing x and in pricing the labor used to produce x. Under our assumptions of the particular value ranges of parameters, Proposition 1 and Corollary 1 imply that structure FX is the equilibrium and is Pareto efficient. That is, under our assumption of the value range of transaction cost coefficients, a producer of management service owns a firm and claims its residuals in the equilibrium where there are markets for the consumer good and for the labor used to produce consumer good y and there is no market for the management service and for the labor used to produce the management service x. Therefore, this market structure avoids the problem of pricing management services and the labor used to produce management services. Under our assumptions, structure FY is not efficient and cannot be an equilibrium because it involves a market for the labor used to produce the management service which is associated with a prohibitively high transaction cost. Structure D is not efficient and cannot be an equilibrium because structure D involves a market for management services which is associated with a prohibitively high transaction cost. Intuitively, the division of labor in producing producer and final goods not only generates transaction costs, but also creates more scope for economic players to choose a method of organizing transactions in order to save on transaction costs. The method of organizing transactions can be a two combination of four factors. The four factors are: trading the final good, trading the producer good, trading the labor used to produce the final good, and trading the labor used to produce the producer good. The first feasible combination of the four factors is trading the final and producer goods (structure D); the second is trading the final good and the labor used to produce the producer good (structure FY); the final is trading the final good and the labor used to produce the final good (structure FX). An equilibrium structure of residual rights will avoid pricing output (good) and input (labor) of the activity that has the lowest transaction efficiency. The implications of this example for productivity progress and for the evolution in the division of labor are straightforward. Suppose that the transaction cost coefficient of the management service, t and the transaction cost coefficient of labor used to produce management service, v, are extremely large, then when structure FX is not allowed the division
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of labor will generate a lower percapita real income (utility) than autarky. The emergence of structure FX will allow the division of labor to generate a higher percapita real income than in autarky. This implies that a particular structure of residual rights may promote the division of labor and productivity via improving transaction efficiency. The structure of residual rights achieves this by allowing individuals to specialize in the production of x while avoiding the direct pricing and marketing of x and related efforts. This analysis yields the following corollary: Corollary 3: The structure of residual rights and the structure of ownership of a firm are critical for the determination of equilibrium and the Pareto optimum if transaction cost differs across goods and factors. Provided economies of the firm exist, structure FX is the equilibrium if transaction efficiency of labor hired to produce the final good is higher than that hired to produce the intermediate good; otherwise structure FY is the equilibrium. Although a structure of residual rights associated with the firm can avoid the direct pricing and marketing of the activity that has the lowest pricing efficiency or transaction efficiency, only one activity can be excluded from direct pricing and marketing. If there are many goods and services and increasingly more goods are traded as transaction efficiency is improved, then the activity which is indirectly priced via claims to residual rights of a firm may be changed. For instance, the producer of management service is the claimant of residual rights before the division of labor between production management and portfolio management emerges. The portfolio managers, or shareholders may become the claimants of residual rights as the division of labor between production management and asset portfolio management emerges. This can be explained by a higher transaction cost of pricing efforts of portfolio management than transaction costs of pricing efforts of production management and by the fact that only one activity with the lowest transaction efficiency can be excluded from direct pricing and marketing by a structure of residual rights.
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This story cannot be generated by simply assuming different values of the transaction cost coefficient in structure D since even if the transaction cost coefficient is the same for structures D, and FX, structure D will generate a higher percapita real income due to a differential in the structure of transactions between D, where the specialist producers of the final good self-provide the final good, and FX, where the specialist producers of the final good have to buy the final good from the market. The theory of the firm developed in this paper will be referred to as 'theory of indirect pricing' and the theory of the firm developed by Grossman, Hart, and Moore will be referred to as 'theory of asset specificity'. A complete story of the firm that occurs in reality may be then predicted by a blend of the theory of indirect pricing and the theory of asset specificity. It is a useful exercise to extend this model to the case with several final goods; this is undertaken in Yang (1988b). 5. Implications and Conclusions In this paper, a framework with consumer-producers, economies of specialization, and transaction costs has been used to explore the role of a structure of residual rights that is associated with the institution of the firm. An asymmetric structure of residual rights can be used to improve transaction efficiency and to promote the division of labor by excluding the activity with the lowest transaction efficiency from direct pricing and trading. In other words, a structure of residual rights that is associated with a firm may get an activity whose outputs and effort inputs are intangible involved in the division of labor but avoid direct pricing and marketing of this activity. In this way, the institution of the firm plays a role similar to that of a patent law through improving pricing efficiency of intellectual properties. According to the theory of indirect pricing, legislation of a law that protects free association and the residual rights of owners of firms is a driving force in economic development. The intuition behind the theory is straightforward. If an entrepreneur has a good idea to develop a new business to make money but it is extremely expensive to sell this idea in the market because it is extremely expensive to enforce his rights to the intangible intellectual property, then the best way for him to develop the business meanwhile enforcing his intellectual property rights is to set up a
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firm and to hire workers to do whatever he wants them to do. By doing so, he indirectly sells his intangible intellectual property to the market while avoiding its direct pricing and marketing. The theory developed in this paper is motivated by two observations. First, it can be used to resolve a puzzle in Chinese economic history. Elvin (1973) has documented that Song China (960-1270 A.D.) possessed both the scientific knowledge and the mechanical ability to have experienced a full-fledged industrial revolution some four centuries before it occurred in Europe. Chinese developed very elaborate contracts and sophisticated commercial organizations at that time. However, all Emperors were extremely sensitive to unofficial free associations because dissidents tended to use such associations to develop underground anti-government movements due to the characteristics of the dynasty cycle in Chinese history. Hence, there was no legal system which protected the residual rights of entrepreneurs to any manufacturing firms. Instead, the government tended to infringe arbitrarily upon such residual rights. This discouraged investments especially in manufacturing industries. According to the theory of the firm developed in this paper, it is the absence of a legal system which protects residual rights to firms, compounded by the absence of patent laws, that prevented China from expanding its technical inventions to large scale commercialized production, which might have brought about an industrial revolution. The second observation is related to the development implications of the absence of a legal system that protects residual rights to firms in a Soviet style socialist economy. Due to Marx's ideology, such residual rights are considered as the source of exploitation in a Soviet style socialist country. Also, a Soviet style government prohibits unofficial free association (including free enterprises) for political reasons. According to our theory of indirect pricing, it is this institutional arrangement rather than distorted relative prices of tangible goods that is responsible for the shortage of innovative entrepreneurial activities in such an economy.
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Appendix: Proof of Proposition 1 There are 26 = 64 combinations of zero and non-zero values of x, xs, xd, y, ys, yd. Applying the Kuhn-Tucker theorem, we may rule out many of the combinations from the list of candidates for an individual's optimum decision. We need not count these combinations in solving for equilibrium. Using (1), (2), and the budget constraint xs + pys = xd + pyd ;
(A-l)
an individual's utility can be written as
U = ^(\-Ly)b -x°+{l-t)x*yyJ - (xd - xs + pyd )/p, if ys > 0,
U = { [\~Ly)b -xd -pyd + (l-k)yd,
if ys=0.
+(l-k)yi
(A-2a)
+(\~t)xd (A-2b)
Differentiating (A-2), it follows dU/dyd <0 if ys>0 (A-3a) dU/dxd < 0 if dU/dxs = 0 and ys > 0 or if xs > 0 and ys = 0 (A-3b) According to the Kuhn-Tucker condition (dU/dyd)yd = 0 at the optimum, (A-3a) implies that the optimum yd is zero if y* > 0. Similarly, (A-3b) implies that the optimum xd is zero if the optimum xs > 0. Assume xs and yare positive at the same time, then (A-3) implies that xd = yd = 0 which contradicts the budget constraint because a person will not sell if he does not buy. In summary, (A-3) and the budget constraint implies that an individual sells only one good and does not buy and sell a good at the same time and thereby the optimum decision is either xd = ys =0, xs ,yd >0, (A-4a) d s s d or x ,y >0, x =y =0, (A-4b) or xd = xs = ys = yd = 0, (A-4c) where (A-4c) is autarky; (A-4b) is configuration (y/x) except we are not sure y > 0; (A-4a) is configuration (x/y) except we are not sure x = L = 0. Therefore we can prove that the optimum decision is one among
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configurations autarky, (x/y), and (y/x) if we have shown that y > 0 if (A-4b) is true and that x = Ly = 0 if (A-4a) is true. Assume (A-4b) holds true; the quantity consumed of the final good^ + (1 - k)yd is zero ify = 0. Therefore a positive utility level implies y must be positive if (A-4b) holds true. (A-5) In other words, we rule out combination of xd,ys > 0 and >> = 0 from the list of candidates for the optimum. Assume that (A-4a) holds true and note (1), (2) and (A-l); (A-2) can be written as
U=
[xLyJ+(\-k)\(l-Ly)b
/p.
Assume x > 0; the optimum value of x is given by dU(x,Ly)/dx
= 0.
(A-6a)
Inserting this optimum value of x into (A-5) and differentiating U with respect to Ly, we have d{du[Ly,x(Ly)]/dLy}/dLy>0
if
dU(x,Ly)/dLy=0,
0.5 < a < 1 and b > 1
(A-6b)
where x(Ly) is given by (A-6a). It can be shown that d{dU[Ly, x(Lyj\jdLy\jdLy > 0 if the Hessian of U(x,Ly)wiih respect to x and Ly is negative. This, combined with (A-6b), implies that the interior extreme of Ly is not the maximum and the optimum Ly may be at a corner, i.e. either Ly= 0 orL y = 1 ifx > 0, 0.5 < a < 1 and b > l.Ly= 1, i.e. Lx = 0 contradicts the assumption of xs > 0. Thus, the optimum Ly = 0 if x > 0. (A-6c) Since dU/dx < 0 for any x if Ly=0,
(A-6d)
the optimum x is zero ifLy= 0. This contradicts (A-6c). Hence, a positive optimum value of x is impossible if (A-4a) is true. Assume that x = 0; it is easy to see that the optimum Ly = 0 because dU/8Ly < 0 for any Ly. Hence, (A-6) means Ly = x = 0 if (A-4a) is true, b>\, and 0 . 5 < a < l . (A-7)
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(A-4,5,7) lead us to Proposition 1 which implies that the list of candidates for the optimum consists of configurations autarky (x/y), and (y/x). Similarly, we can prove that the list consists of the seven configurations in section 3 if we take into account the possibility for establishing firms.
References Alchian, Armen and Harold Demsetz, 1972, Production, Information Costs, and Economic Organization, American Economic Review 62, 777-795. Cheung, Steven, 1983, The Contractual Nature of the Firm, The Journal of Law and Economics 1, 1-21. Coase, Ronald, 1937, The Nature of the Firm, Economica 4, 386-405. Coase, Ronald, 1960, Social Costs, Journal of Law and Economics 3, 1-44. Coase, Ronald, 1991, The Nature of the Firm: Origin, Meaning, Influence, in: Oliver Williamson and Siney Winter (eds.), The Nature of the Firm (Oxford University Press, New York). Demsetz, Harold, 1967, Toward A Theory of Property Rights, American Economic Review 57, 347-359. Elvin, Mark. 1973, The Pattern of the Chinese Past (Eyre Methuen, London). Grossman, Sanford and Oliver Hart, 1986, The Costs and Benefits of Ownership: A Theory of Vertical and Lateral Integration, Journal of Political Economy 94,691-719. Hart, Oliver and B. Holmstrom, 1987, The Theory of Contracts, in: T. Bewley (ed.), Advances in Economic Theory (Cambridge University Press, Cambridge). Hart, Oliver and John Moore, 1990, Property Rights and the Nature of the Firm, Journal of Political Economy 98, 1119-1158. Holmstrom, B. and P. Milgrom, 1994, Authority, Compensation and Ownership: The firm as an Incentive System, American Economic Review, forthcoming.
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Klein, Benjamin, Robert Crawford and Armen Alchian, 1978, Vertical Integration, Appropriable Rents and the Competition Contracting Process, Journal of Law and Economics 21, 297-326. Knight, Frank, 1925, Decreasing Cost and Comparative Cost: A Rejoinder, Quarterly Journal of Economics 39, 332-333. Laffont, Jean-Jacques and Jean Tirole, 1986, Using Cost Observation to Regulate Firms, Journal of Political Economy 94, 614-641. Lewis, Tracy and Devid Sappington, 1991, Technological Change and the Boundaries of the Firm, American Economic Review 81, 887-900. Ng, Yew-Kwang, 1982, A Micro-Macroeconomic Analysis Based on a Representative Firm, Economica 49,111-128. Smith, Adam, 1776, Edwin Carman (ed.), An Inquiry into the Nature and Causes of the Wealth of Nations, (University of Chicago Press, 1976). Williamson, Oliver, 1975, Markets and Hierarchies (Free Press, New York). Williamson, Oliver, 1985, Economic Institutions of Capitalism (Free Press, New York). Yang, Xiaokai, 1988a, A Microeconomic Approach to Modeling the Division of Labor Based on Increasing Returns to Specialization, Ph.D. Dissertation, Dept. of Economics, Princeton University. Yang, Xiaokai, 1988b, An Approach to Modeling Institutional Development, Discussion Paper of the Yale Economic Growth Center. Yang, Xiaokai, 1990, Development, Structural Changes, and Urbanization, Journal of Development Economics 34, 199-222. Yang, Xiaokai and Jeff Borland, 1991, A Microeconomic Mechanism for Economic Growth, Journal of Political Economy 99, 460-482. Zhang, Wei Ying, 1993, Why Are Capitalists The Principals?, mimeo., Nuffield College, Oxford University.
C H A P T E R 11 THE THEORY OF IRRELEVANCE O F T H E SIZE O F T H E FIRM*
a
Pak-Wai Liu and b Xiaokai Yang
"Chinese University of Hong Kong
Harvard and Monash
University
1. Introduction Many scholars have observed that average employment of labor by firms (average firm size) in many industries in both advanced countries and newly industrialized countries has declined. Data clearly show that in numerous developed countries, such as the US, UK, France, Austria and Belgium, employment share of small and medium enterprises in their manufacturing industries exhibits a U-shape pattern. It declined initially in the 1960s and early 1970s but started to increase in the late 1970s and early 1980s.1 Among the newly industrialized countries, South Korea's manufacturing industries exhibit the same pattern as the developed countries, with a turning point in the early 1980s.2 In Taiwan, average firm size in manufacturing measured in terms of average employment increased from 7.3 persons per firm in 1954 to a peak of 28.2 in 1971
* Reprinted from Journal of Economic Behavior & Organization, 42 (2), Pak-Wai Liu and Xiaokai Yang, "The Theory of Irrelevance of the Size of the Firm," 145-165, 2000, with permission from Elsevier. 1 For data on the US, UK and France, see Loveman and Sengenberger (1991). For data on Austria, see Aiginger and Tichy (1991) and for Belgium, see Storey and Johnson (1987). 2 See Kim and Nugent (1993).
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after which it followed a decline trend, reaching 24.2 in 1984. 3 Similarly in the 1980s, manufacturing firms in Singapore were getting smaller in size on the average. This inverted U-shaped trend is not confined to manufacturing industries. Liu (1992) shows that in Hong Kong, besides manufacturing, industries such as business services, import/export and restaurant exhibit a trend of declining average firm size. Table 1 shows the change in firm size of these four industries measured in two ways, the average number of employees per firm and the percent of employees engaged in firms with employees larger than 50. The trend since the late 1970s when data became available is generally downward. The decline in the average size of firms is associated with an increase in per capita real income and total factor productivity. Table 1: Declining Firm Size in Hong Konga Year Manufacturing Import/export Business services Average number of employees per firm 1978 23.72 6.35 10.67 6.74 1981 20.61 11.30 1984 19.10 6.07 10.42 6.21 9.78 1987 18.60 5.45 8.50 1990 14.47 1993 13.13 5.10 7.38
Restaurants 25.27 27.42 28.31 23.49 21.30 21.47
Percent ofemployees in industry engaged,infirms with employees larger• than 50 59.03 1980 58.72 23.45 52.85 57.72 1984 59.07 20.20 51.20 56.25 1987 57.77 23.14 50.47 51.62 1990 53.36 21.15 52.31 42.74 18.98 45.45 1993 52.73 a Source: Census and Statistics Department, Annual Digest of Statistics, Hong Kong Government, various years.
This trend of declining average firm size is contrary to the common perception that due to technological change and economies of scale, firm size should get larger over time. For instance, Kim (1989) shows from his model that because there are economies of scale arising from 3
See Hu (1991).
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increasing returns due to specialization, firm size gets larger with specialization and the extent of the market. Our study provides an explanation of why firm size may get smaller over time on the basis of an analysis of the division of labor and transaction cost. We will show that the institution of the firm will emerge if the transaction efficiency for labor is higher than that for intermediate goods. By transaction efficiency we measure the fraction of the purchased good which disappears in transit due to transaction cost. Transaction competes for time and management. For instance, a producer of an intermediate good can either pay the search cost to find a supplier of the primary good used in its production or the managerial monitoring cost of in-house production of the primary good. We will show that given the emergence of firms, their size (measured by employment) decreases if the transaction efficiency for intermediate goods becomes higher than that for labor. The decrease in the size of firms is driven by two forces. First, each firm becomes more specialized if division of labor develops among firms, so that each firm's scope of activities is reduced. However, if division of labor and specialization are developed within each firm, the level of specialization of each worker in a firm and the scope of the firm may increase hand in hand. But if the transaction efficiency for intermediate goods is higher than that for labor, then organizing division of labor between more specialized firms will be more efficient than organizing division of labor within a firm since the former involves more transactions of intermediate goods while the latter involves more trade in labor, thus generating a decline in the size of firms.4 Several recent studies including Grossman and Hart (1986); Hart and Moore (1990); Milgrom and Roberts (1990); Kreps (1990), and Lewis and Sappington (1991) explain the size of the firm in various ways. None of these models, however, can predict the negative relationship between an increase in per capita real income and the average size of
4
A firm's decision to increase its level of specialization which leads to the equilibrium phenomenon of disintegration and increasing division of labor between firms is called 'outsourcing', 'downsizing', 'returning to core competencies', and 'diverting unrelated businesses' in the literature of management. See for instance, Milgrom and Roberts (1992).
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firms, nor can they explore the interaction between that negative relationship and the level of specialization. Our model extends the Camacho's (Camacho, 1991, 1996) analysis of the trade-off between benefits of division of labor and co-ordination costs as an explanation of the optimal size of the firm. Echoing Jones' empirical works (Jones, 1995 a, b) which conclusively reject a scale effect, our theory explains economic development on the basis of the division of labor instead of economies of scale. It in effect formalizes AUyn Young's (Young, 1928) criticism of the notion of economies of scale. It is also in keeping with the arguments of Young, Coase (1937); Stigler (1951); Cheung (1983), and Langlois (1988) which imply that the size of the firm may increase or decrease, depending on whether division of labor is organized through the labor market or through the market for intermediate goods. In section 2, we use an example to illustrate the story behind the formal model. In section 3 a model of consumer-producers with economies of specialization in production and transaction costs in trading is presented. Section 4 characterizes eight market structures that may emerge from the development in division of labor and identifies the general equilibrium by comparing per capita real incomes in different structures. The main theoretical results of the paper on the emergence of the institution of the firm and average employment are contained in section 5. The paper concludes in section 6. 2. The Theories of the Firm and the Story behind the Formal Model There are three strands of the literature on the institution of the firm and transaction costs that closely relate to our paper. Recent literature of general equilibrium models with economies of scale and endogenous number of goods, represented by Dixit and Stiglitz (1977); Krugman (1980, 1981); Ethier (1982); Judd (1985); Romer (1990); Grossman and Helpman (1989, 1990); Fujita and Krugman (1995), and Krugman and Venables (1995), generates some important empirical implication, which is referred to as scale effects. There are several types of scale effects. Type I scale effect exists if productivity or growth performance is positively correlated with the
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average size of firms. Type II scale effect exists if the growth rate of per capita income or per capita consumption goes up as the size or growth rate of population increases. Type III scale effect exists if the growth rate of per capita income goes up as the size of the research and development sector increases. Type IV scale effect exists if the growth rate of per capita income goes up as the investment rate increases. All four types of scale effects are wildly at odds with empirical evidence. The AK model and R&D based model have generated scale effects of types II, III, and IV. The empirical work of Jones (Jones, 1995a, b), and National Research Council (1986) has rejected the three types of scale effect. Jones (Jones, 1995a, b), Alwyn Young (Young, 1998), and Segerstrom (1998) suggest several ways to avoid Type III scale effect in the R&D based model. But the modified models still have a Type II scale effect. As Jones has shown, endogenous growth cannot be preserved in the neo-classical endogenous growth models in the absence of scale effect. On the ground of the empirical evidence, it is concluded that the neo-classical endogenous growth models have not provided a convincing explanation of the driving mechanism of economic growth (Jones, 1995a, pp. 508-509). This class of models with economies of scale generates other types of scale effects too. For instance, the Fujita and Krugman model (1995) predicts a positive correlation between the degree of urbanization and the average size of firms. The Krugman and Venables model (1995) predicts a positive correlation between the degree of industrialization and the average size of firms. A preliminary examination of data does not support all of the scale effects. Coase, Stigler, Young, and Cheung have proposed a theory of irrelevance of the size of the firm. According to them, if division of labor develops within the firm, then the average size of firms goes up as productivity increases. But if division of labor develops between firms, then as productivity rises the average size of firms falls. This paper will use a general equilibrium model to formalize this idea and provide a theoretical foundation for empirical work that rejects various scale effects. The second strand of the literature relates to the notion of endogenous transaction costs. Many economists of information
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economics claim that the principal-agent model with moral hazard, the model of adverse selection, and other models with information asymmetry provide microeconomic theories of the firm. However, as Hart (1991) suggests, the principal-agent model has nothing to do with the theory of the firm. The contingent contract in the principal-agent model is not labor contract because of the assumption that labor effort is not observable or is non-verifiable in the court when dispute arises. Also, both parties of the contingent contract claim part of residual returns. In addition, decision authority is symmetrically distributed, so that asymmetric distribution of residual returns and residual control rights, which are two essential components of the firm, does not occur in the model. All models of information asymmetry do not explain why and how firms emerge, though they study endogenous transaction costs associated with information problems. In this paper, we consider endogenous transaction costs as a departure of the equilibrium from the Pareto optimum. But recent dynamic game models with information asymmetry show that information asymmetry may reduce endogenous transaction costs. For instance, the Milgrom and Roberts model (1982) shows that information asymmetry can reduce the distortions caused by monopoly power. From the game models we can learn that information asymmetry may not be a source of endogenous transaction cost. At least the endogenous transaction costs caused by information asymmetry are not as serious as the naive model of adverse selection, which ignores complicated interactions between information and strategies, predicts. As Hurwicz (1973) points out, the distortion in the Walrasian equilibrium with extreme information asymmetry (each player does not know others' utility and production functions and endowment and has no information about the distribution functions of their characteristics) is smaller than any other possible mechanisms and also involves lower information cost. Hence, information asymmetry and related endogenous transaction costs are not essential for the theory of the firm. This claim has been supported by two research lines. First, Cheung (1983) argues that the institution of the firm is to replace trade in goods with trade in labor. According to this argument, if trading of goods involves endogenous transaction costs caused by moral
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hazard or other types of information asymmetry, trading in labor may involve endogenous transaction costs as well. Great measurement costs of quantity and quality of goods may cause information problem, so that endogenous transaction costs result. But quantity and quality of labor, in particular, labor allocated to research and entrepreneurial activities, involve prohibitively high measurement costs. This may cause information problem too. Hence, endogenous transaction cost per se is not enough for explaining the emergence of the firm. The key point is to explain why trading of goods is replaced with trading in labor since this replacement is essential for explaining the emergence of the firm. Second, the recent literature of incomplete contract was believed to point to a hopeful research direction that may provide a theory of the firm. However, Maskin and Tirole (1997) poses a convincing challenge against the theory of incomplete contract. According to them, this theory does not have sound theoretical foundation. It involves logical inconsistency, so that it cannot provide a theoretical foundation for the theory of the firm. Also, Holmstrom and Milgrom (1995) and Holmstrom and Roberts (1998) point out that labor trade, which is essential for the existence of the firm, is not essential in the model of incomplete contract. Hence, the theory of incomplete contract is a theory of optimum ownership structure rather than a theory of the firm. Yang and Ng (1995) have developed a general equilibrium model to formalize Coase and Cheung's theory of the firm. Since many readers are not familiar with the theory and its technical substance, we briefly outline the story behind their model in this section. According to Yang and Ng, the institution of the firm is a structure of transactions based on the division of labor that satisfy the following three conditions. (i) There are two types of trade partners who are associated with a firm: the employer (she) and the employee (he). There is an asymmetric distribution of control rights or authority. The employer has control rights to use the employee's labor. He must do what he is told to do, if she so demands. That is, the ultimate rights to use the employee's
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labor are owned by the employer, subject to the employee's freedom to quit the job and to other terms of the labor contract. (ii) The contract that is associated with the asymmetric relationship between the employer and the employee never specifies how much the employer receives from the relationship. It only specifies how much the employee receives from the relationship. The employer claims the residual returns, defined as the revenue of the firm net of the wage bill and other expenses. The employer is referred to as the owner of the firm. One of the most important components of the ownership of a modern firm is the entitlement to the business name of the firm. The exclusive rights to the business name are enforced through a business name search process when the firm is registered with the government and through recognition of the name in legal cases in the judicial process. The exclusive rights are also enforced through the laws of brand. (iii)A firm must involve a process that transforms labor of the employee into something that is sold in the market by the owner of the firm. In the process, what is produced by the employee is owned by the employer (residual returns). The relationship between a professor and his housekeeper does not involve the institution of the firm since it does not involve such a resale process of services provided by the housekeeper. The professor directly consumes what the housekeeper produces and never resells it to the market. Yang and Ng's story of the firm runs as follows. There are many ex ante identical consumer-producers. Each individual as a consumer must consume a final good, called cloth, the production of which requires an intermediate good, called management service, as an input. There is a trade-off between economies of specialization and transaction costs. If a transaction cost coefficient for a unit of goods or labor traded is small, then division of labor will occur at equilibrium. Otherwise autarky will be chosen as the equilibrium. There are two ways to interpret the transaction cost. First, the transaction cost can be considered as resource allocated to measure quantity and quality of goods and labor. Following
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common practice in the literature of equilibrium models of geography and monopolistic competition, we can use an iceberg transaction cost coefficient to represent such transaction cost. Second, the distortions caused by moral hazard and information asymmetry can be considered as endogenous transaction cost. Since simultaneous endogenization of transaction costs, specialization and the emergence of the firm involves formidable algebra, we use the iceberg transaction cost represents the anticipated endogenous transaction cost. For trade in goods, the anticipated endogenous transaction cost may be associated with the distortion caused by information asymmetry about the quality of goods, as in the model of lemons (Akerlof, 1970). For trade in labor, the anticipated endogenous transaction cost may be associated with moral hazard and shirking in the model of efficiency wage (Shapiro and Stiglitz, 1984). There are three different structures of residual rights which can be used to organize transactions required by the division of labor. Structure D, shown in Figure lb, comprises markets for cloth and management services. Specialist producers of cloth exchange that product for the specialist services of management. For this market structure, residual rights to returns and authority are symmetrically distributed between the trade partners, and no firms or labor market exist. Structure E, shown in Figure 1 c, comprises the market for cloth and the market for labor hired to produce the management service within a firm. The producer of cloth is the owner of the firm and specialist producers of management services are employees. Control rights over employees' labor and rights to the firm's residual returns are asymmetrically distributed between the employer and her employees. The employer claims the difference between revenue and the wage bill, has control rights over her employees' labor, and sells goods that are produced from employees' labor. Structure F, shown in Figure Id comprises the market for cloth and the market for labor hired to produce cloth within a firm. The professional manager is the owner of the firm and specialist producers of cloth are employees. For the final two structures of residual rights, the firm emerges from the division of labor. Compared with structure D, these involve a labor market but not a market for management services.
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As Cheung (1983) argues, the firm replaces the market for intermediate goods with the market for labor. Although both structures E and F involve a firm and an asymmetric structure of residual rights, they entail different firm ownership structures. Firms
•/\i
(a) Autaky
(c) Structure E Figure 1: Emergence of the Firm from the Division of Labor
Suppose that the transaction cost coefficient is much larger for management service than for labor. This is very likely to be the case in the real world, since the quality and quantity of the intangible entrepreneurial ideas are prohibitively expensive to measure. Potential buyers of the intellectual property in entrepreneurial ideas may refuse to pay by claiming that these are worthless as soon as they are acquired from their specialist producer. Under this circumstance, the institution of the firm can be used to organize the division of labor more efficiently because it avoids trade in intangible intellectual property. Suppose further that the transaction cost coefficient for labor hired to produce management services is much larger than for labor hired to produce cloth because it is prohibitively expensive to measure the efforts exerted in producing intangible management services (can you tell if a manager sitting in the office is pondering business management or his girlfriend?). Then the division of labor can be more efficiently organized in structure F than in structure E. This is because structure F involves trade in cloth and in labor hired to produce cloth but not trade in management services nor in labor hired to produce management services,
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while structure E involves trade in cloth and in labor hired to produce management services. Hence, structure F will occur at equilibrium if the transaction cost coefficients for labor hired to produce cloth and for cloth are sufficiently small. The claim to the residual return of the firm by the manager is the indirect price of management services. Therefore, the function of the asymmetric structure of residual rights is to get the activity with the lowest trading efficiency involved in the division of labor while avoiding direct pricing and marketing of the output and input of that activity, thus promoting the division of labor and productivity. In a sense, the function of the asymmetric structure of residual rights is similar to that of a patent law which enforces rights to intangible intellectual property, thereby promoting the division of labor and productivity in producing the intangible. However, the asymmetric structure of residual rights to returns and control can indirectly price those intangible intellectual properties which are prohibitively expensive to price even through a patent law. Intuitively, there are two ways to do business if an individual has an idea for making money. The first is to sell the entrepreneurial idea in the market. This is very likely to create a situation in which everybody can steal the idea and refuse to pay for its use. The second way of proceeding is for the entrepreneur to hire workers to realize the idea, while keeping the idea to herself as a business secret. Then she can claim as her reward the residual returns to the firm, which represent the difference between revenue and the wage bill. If the idea is a good one, then the entrepreneur will make a fortune. If the idea is a bad one, she will become bankrupt. The residual return is the precise price of the idea and the entrepreneur gets what she deserves, so that stealing and over or under pricing of intellectual property is avoided. To understand this, you may image that Bill Gates did not set up a company to hire others to realize his entrepreneurial ideas about how to make money from computer software. Instead he sold his ideas as consultant services. Would anybody pay billion dollars (which is the real market value of the ideas indicated by Bill Gates residual returns) to buy the ideas? This theory is referred to as the theory of indirect pricing. The Yang-Ng model (1995) formalizes the Cheung-Coase theory of the firm
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(Coase, 1937; Cheung, 1983) that the institution of the firm can save on transaction costs by replacing trade in intermediate goods with trade in labor if the former involves higher transaction costs than the latter. The model does not endogenize transaction costs if these are defined as a departure from the Pareto optimum. But as discussed before, endogenous transaction cost is not essential for telling story of the firm though we can use some transaction cost coefficient to represent anticipated endogenous transaction cost. The essence of the story is that why endogenous or exogenous transaction costs of goods are replaced with endogenous or exogenous transaction costs of labor as the institution of the firm emerges. The essence of the theory of indirect pricing can be summarized as follows. If individuals engage in division of labor in producing two goods x and y, trade in two of the four elements comprising outputs x, y, and inputs 4, ly can be used to organize the division of labor. Hence, there are six possible structures of transactions (two combinations of four elements) that can be used to organize the division of labor: x and y (structure D), y and lx (structure E), y and ly (structure F), x and /x (infeasible since a specialist of x who must consume y cannot buy y), x and ly (infeasible too), ly and lx (a structure that may be seen in some collective organization with direct exchange of labor but is rarely seen in the real world). Hence, individuals can at least consider three of the structures and choose one of them to avoid trade that involves the lowest trading efficiency. Yang and Ng (1993, chapter 13) have shown that as the number of traded goods and the level of division of labor increase, the number of possible structures of transactions increases more than proportionally. This implies that choice of transaction structure is increasingly more important than choice of production structure and resource allocation for a given structure of production for economic development as division of labor develops. This is why in a highly commercialized society (high level of division of labor) there are more opportunities for entrepreneurs to make fortunes from playing around with structures of transactions. In general, the number of possible pricing structures of goods and factors increases more than proportionately as the network size of division of labor is enlarged. For instance, one of
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recent five patented internet trading practices pay reviewers of advertisements for clicking an ad or provide free email services. The generous pricing policies can make a lot of money using positive network effects that pay the companies via other channels. Hence, conventional marginal analysis of optimum price of a single good is outof-date. In a large and highly interconnected network of division of labor and communication, players can price a subset of the set of all goods and services. It is a very complicated optimization decision problem to identify which subset of all goods and services should be priced and which can be provided free of charge. The function of the market in sorting out the efficient pricing structure is even much more complicated than a player's problem of optimum pricing structure. The Yang-Ng model cannot predict a negative correlation between productivity and the average size of firms since it involves only two goods. In the next section we introduce three goods in to the model to formalize the theory of irrelevance of the size of the firm. 3. A Model with Consumer-Producers, Economies of Specialization and Transaction Costs We consider an economy with M ex ante identical consumer-producers (where M is assumed large). There is a single consumer good z, the amount of this good that an individual self-provides is denoted by z and the amounts sold and purchased are zs and zd, respectively. A fraction 1 k of zd disappears in transit because of transaction costs. Hence, the quantity that a person receives from the purchase of zd is kzd. Total consumption is thus z + kzd. 1 - k is referred to as the transaction cost coefficient for food and k is referred to as the transaction efficiency coefficient or simply transaction efficiency for the consumer good. As discussed in section 2, the transaction cost can be considered as the measurement cost of quantity and quality of goods or anticipated distortion caused by moral hazard or other opportunistic behavior. An alternative formulation is to introduce an explicit time constraint on the consumer-producers in such a way that producing the good takes a
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Yang
certain amount of time but purchasing it takes more or less time.5 The individual maximizes the objective function u (which can be interpreted as utility) given by total consumption: u = z + kzd (1) The consumer good is produced using labor Lz and an intermediate goodx, i.e. z + zs = (x + rxd)aLz(1 - a)a,
a e (0, 1) and a > 1
(2)
d
where x and x are the respective quantities of the intermediate good self-provided and purchased, r is the transaction efficiency of the intermediate good, 1 - r is the transaction cost coefficient for a unit of the intermediate good purchased and, therefore rxd is the quantity received by a person who buys xd. The parameter a is the elasticity of output with respect to input of the intermediate good and can be interpreted as the relative importance of the roundabout productive sector compared to labor. The degree of economies of specialization is represented by a. Production of the intermediate good requires labor Lx and a primary good y. x + xs = (y + tyd)aLx(l ~ a)a, s
a e (0, 1) and a > 1
(3) d
where x is the quantity of the intermediate good sold and y and y are, respectively, the quantities of the primary good self-provided and purchased. 1 - /is the transaction cost coefficient for the primary good (t is the transaction efficiency) and tyd is the amount that a person receives from the purchase of yd. It is also assumed that fraction 1 - s of goods produced by an employee within a firm disappears when it is delivered to the employer because of transaction cost of labor. The production function for the primary good is y + ys=Lay,
a>\
(4)
where ys is the quantity of the primary good sold. The labor endowment constraint for each individual is 5
The alternative formulation was suggested by the Editor which we gratefully acknowledge. Our specification of iceberg transaction technology avoids formidable indices of origins and destinations of deliveries and other technical difficulty.
The Theory of Irrelevance of the Size of the Firm Lz + Lx + Ly=l,
L ; e [ 0 , 1],
i = z,x,y.
273 (5)
where Lt is a person's level of specialization in producing good i. This system of production exhibits economies of specialization. The total factor productivity of the consumer good and the labor productivity of the primary good increase with the level of specialization in producing a good concerned. But economies of specialization are individual specific and do not extend beyond the scale of an individual's working time. Also, they are related to diseconomies of an individual's scope of activities. There are 2 9 = 512 combinations of zero and non-zero values of z, zs, d z , x, xs, xd, y, ys and yd. Following the approach of Borland and Yang (1995), we can exclude many combinations of zero and nonzero variables from the list of candidates for an individual's optimal decision. 4. Configurations, Market Structures and General Equilibrium A combination of zero and nonzero variables that is compatible with the Kuhn-Tucker condition and other conditions for the optimum decision is called a configuration. For each configuration, an individual can solve for a corner solution for a given set of relative prices of traded goods. A combination of configurations that satisfies the market clearing conditions is called a market structure, or simply structure. There are eight market structures if the institution of the firm is allowed. We define the firm as contractual arrangements which involve asymmetric relationship between an employer and his employees and the production of intermediate goods using labor. The employer has control rights of his employees' labor and claims the difference between the firm's revenue and wage bill. The eight market structures include autarky, two structures involving partial division of labor with no firm and two with firm, one structure involving complete division of labor with no firm and two with firms. The eight market structures and the combination of configurations that make up these structures are categorized and characterized as follows:
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Autarky 1. Structure A Structure A consists of a configuration with zs = xs = ys = zd = xd = yd = 0. In this structure each individual self-provides all goods he needs. Partial division of labor with no firm 2. Structure PI Configuration 1: an individual sells the consumer good, selfprovides the intermediate good, and buys the primary good Configuration 2: an individual sells the primary good, and buys the consumer good 3. Structure P2 Configuration 1: an individual sells the intermediate good, selfprovides the primary good, and buys the consumer good Configuration 2: an individual sells the consumer good, and buys the intermediate good Partial division of labor with firm 4. Structure PF1 Configuration 1: an individual sells the consumer good, selfprovides the intermediate good, buys labor, and directs workers to produce the primary good in his firm Configuration 2: an individual sells labor which is hired to produce the primary good by a firm, and buys the consumer good 5. Structure PF2 Configuration 1: an individual sells labor which is hired by a firm to produce the intermediate good and the primary good in the firm, and buys the consumer good Configuration 2: an individual sells the consumer good, buys labor, and directs each worker to produce both the primary and the intermediate goods.
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Complete division of labor with no firm 6. Structure C Configuration 1: an individual sells the consumer good, and buys the intermediate good Configuration 2: an individual sells the primary good, and buys the consumer good Configuration 3: an individual sells the intermediate good, and buys the primary good and the consumer good Complete division of labor with firm 7. Structure CF1 Configuration 1: an individual sells the consumer good, buys the primary good, hires workers and directs them to produce the intermediate good using the primary good in his firm Configuration 2: an individual sells the primary good, and buys the consumer good Configuration 3: an individual sells labor which is hired by a firm to produce the intermediate good, and buys the consumer good 8. Structure CF2 Configuration 1: an individual sells the consumer good, buys labor, and directs some of the workers to produce the primary good and others to produce the intermediate good using the primary good Configuration 2: an individual sells labor which is hired by a firm to produce the intermediate good, and buys the consumer good Configuration 3: an individual sells labor which is hired by a firm to produce the primary good, and buys the consumer good A corner equilibrium for a certain structure is defined by a set of relative prices of traded goods and a set of relative numbers of individuals choosing different configurations that equilibrate total corner-demand to total corner-supply of each traded goods and equalize all individuals' consumption levels. Here, competition among ex ante identical consumption-maximizing individuals, combined with free entry into each configuration, implies that all configurations in a structure will
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be chosen only if consumption is equalized across the configurations. In this section, the corner equilibria for the eight structures are first solved. A two-step approach developed in Yang and Ng (1993) is used to solve for the corner equilibrium for each of the eight market structures. As an illustration, the algebra for solving for the corner equilibrium for structure PF1 is shown as follows. Solving for the corner equilibrium for structure PF\ Structure PF1 consists of configurations 1 and 2 as summarised earlier. The individual decision problem for configuration 1 is as follows Max: u2 = z
(objective function)
s.t. z + zs = x% a ( 1 " a ) ,
x = (syd)aLxa(l-a)
yd = nys
(production function)
(material balance in a firm)
Lx + Lz = 1
(endowment constraint for the employer's labor)
s
y = Ly
(production function for an employee)
Ly = 1
(endowment constraint for an employee)
pzzs = wn (budget constraint for the firm) where Lx and Lz are the employer's labor allocation to the production of x and z, respectively, n is the number of workers hired by an employer, Ly employee's labor which is subject to the employer's control, ys each employee's output of y within the firm, yd the total output level of y in the firm or input requirement for the production of x in the firm, w is the wage rate, and s is the transaction efficiency for labor or 1 - s is the transaction cost coefficient of labor in terms of the loss of goods produced out of labor in transit. If 1 - s is assumed to be in terms of the loss of labor in transit, the essence of the results will not change, but the algebra will be more complicated. The solution of the problem is a L
L
'"(1 + a)'
z
l "(l + a)'
L=i y
'
The Theory of Irrelevance of the Size of the Firm (
n =
2
A
\
A
a P, W
zs = — , Pz
277
l/(l-a)(l+a)
, A = [sadl-a)T{\
+ a)• (1 - a)(\
+ a)a
u,=(l-a2)A (0
where n is demand for labor by the firm. zs is supply of z by the firm, and uz is the indirect objective (utility) function for configuration 1. The decision for configuration 2 is fixed as follows. uy = kzd = — Pz
(obj ective function)
why = pzzd, Ly=\ (budget and endowment constraints) where uy is the indirect objective function (utility) of an employee. The consumption equalization condition uz = uy yields the corner equilibrium relative price wlpz and the market clearing condition for good z, Mzzs = MyZd, yields the relative number of individuals choosing the two configurations. Here, MyIMz= n is the corner equilibrium relative number of individuals choosing to be employees to those choosing to be employers, which happens to be the same as the corner equilibrium value of n, a decision variable for an employer. The solution for the corner equilibrium is summarized in Table 2. Following this procedure, the corner equilibria in all market structures can be solved. Table 2 summarises these solutions. A general equilibrium is defined as a fixed point that satisfies the following two conditions: (i) each individual uses inframarginal analysis to maximize his consumption respect to configurations and quantities of each good produced, consumed, and traded for a given set of relative prices of traded goods and a given set of the numbers of individuals selling different goods; and (ii) the set of relative prices of traded goods and the set of individuals selling different goods clear the markets for traded goods and equalize consumption for all individuals selling different goods. (Yang and Ng, 1993, Chapter 6) have proved that in this kind of models, the general equilibrium is the corner equilibrium that generates
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Table 2: Solutions to Corner Equilibria of Market Structures3 Relative price Labor allocation Real per capita income Lx = d\ -a)
«(A) = /?'(1 - a)a
J _ ~2
PI
price of primary good in terms of consumer good t a [ ( l - a ) ( l + a) 1 -1 <1 ~ aX1 + a) price of intermediate good in terms of consumer good pa=k2a-lpl-{l-a)a wage rate in terms of consumer
Lz = 1 - a relative number of sellers of «(P1) = (ki)aaa'A2aHl ~a)a] primary good to sellers of [(1 - a)(l + a) 1 -"]* 1 ""* 1 " 0 ' consumer good = ko?l{\ - a2)
Pyz~-
P2
PF1
,
PF2
a1
a[a+(l-a)a]
relative number of sellers of w(P2) = (krfp'+ aa intermediate good to sellers of consumer good = kcd{\ - d) number of workers hired by a w(PFl) = (sk)aaa'42a+(l ' a)a] firm = kdl{\ -J) [(!-«)(! + a)1 -"f + aX1" a)
good= s a [ ( l - a ) ( l + a) 1 " a //t] (1 "* 1 + ^ = transaction efficiency for labor number of workers hired by a wage rate in terms of consumer firm = kal(\ - a) good= [sapa\a[{\ - a)/*]1" a price of primary good in terms of intermediate goodp^ = fit"; price of intermediate good in terms of consumer good/t = {kctf lrapa
relative number of sellers of «(C) = (Jcrfffi'+ intermediate good to sellers of consumer good = ok relative number of sellers of primary good to sellers of intermediate good= a/(\ - a)
number of workers hired by a price of consumer good in firm = &a terms of primary good Pzy = k{- a(st°) - 7T 1 " a wage rate in terms of primary good = 1 number of workers hired by a price of intermediate good in CF2 firm = £a(l - a) terms of consumer good Pxz = (ksa)a~ \rpf; wage rate in terms of consumer a l+a good =--k -\rSyp ^oteP=ef(l-ay
CF1
w(PF2) = (sk)"P('+ m)
a(CFl) = (kstyp[
a
+
"
w(CF2) = sa{' + a)kap'+ "
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the highest per capital real income.6 The other corner equilibria are not general equilibria. A revealed preference argument can be used to show that each corner equilibrium is locally Pareto optimal for a given structure (See Yang and Ng, 1993). Hence, the general equilibrium is globally Pareto optimal. This implies that the competitive market can efficiently co-ordinate division of labor to fully utilize network division of labor. This result critically depends on the assumption that economies of specialization are individual specific and different from economies of scale. 5. Development in Division of Labor and Average Employment A comparison between per capita real incomes generated by the eight market structures yields Proposition 1: (1) As transaction efficiency is improved, the general equilibrium shifts from structure A first to structure PI or PF1, followed by P2 or PF2, and finally to structure C, CFl, or CF2. This process increases per capita real income, productivity, trade dependence, individuals' level of specialization, and the number of specialized activities. (2) The institution of the firm will emerge from the development in division of labor if transaction efficiency is higher for labor than for intermediate goods. (3) Given the emergence of the institution of the firm from the development in division of labor, the average employment increases if the transaction efficiency for labor is higher than that for intermediate goods. It decreases if transaction efficiency is improved in such a way that the transaction efficiency for intermediate goods becomes higher than that for labor. Proposition 1 is proved in the Appendix. The intuition of Proposition 1 is as follows. There is a trade-off between economies of specialization and transaction costs. Individuals will choose a low level of 6
The proof of this statement is available from the authors.
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specialization and a large scope of production activities if transaction efficiency is low because economies of specialization are outweighed by transaction costs generated by a high level of division of labor. As transaction efficiency is improved, the equilibrium level of specialization increases and the equilibrium scope of each individual's production activities becomes narrower. As the level of division of labor increases with improvements in transaction efficiency, two types of market structure can be used to organize a higher level of division of labor. The first involves markets for final and intermediate goods and the second involves markets for final goods and for labor. If the transaction efficiency for labor hired to produce an intermediate good is higher than that for the intermediate good, the institution of the firm will emerge from division of labor. If the institution of the firm emerges, but the transaction efficiency for intermediate goods is improved more quickly than that for labor, then the equilibrium size and scope of firms will decrease as each individual increases his level of specialization and narrows down his scope of production activities. The decrease in the average employment is driven by two forces. First, each firm becomes more specialized if division of labor between firms and individuals develops, so that each firm's scope of activities is reduced. Second, if the transaction efficiency is improved in such a way that the transaction efficiency for intermediate goods becomes higher than that for labor, then organizing division of labor between more specialized firms and individuals will be more efficient than organizing division of labor within a firm since the former involves more transactions of intermediate goods while the latter involves more trade in labor. Hence, an increase in the level of specialization of firms and an increase in division of labor between small scale firms and individuals will occur, generating a decline in average employment of firms. To illustrate the shifting of market structures. Suppose the initial values of parameters are such that the transaction efficiencies for final and intermediate goods are very low but the transaction efficiency for the intermediate good is higher than for labor. Then the general equilibrium will be autarky, as shown in the Appendix. If the transaction efficiencies for final and intermediate goods are improved while the
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transaction efficiency for labor is still lower than for intermediate goods, then the general equilibrium will be structure PI where firms and labor market do not exist and completely specialized producers of primary good exchange it for final good with non-specialized producers of intermediate and final goods in the market. Suppose the transaction efficiencies for final goods and labor are improved relative to the transaction efficiency of the intermediate good, then the general equilibrium will shift to structure PF2 where a specialist producer of final good hires kal{\ - a) workers and directs each of them to produce primary and intermediate goods within a firm. In PF2, each firm produces three goods and the level of specialization of workers is not high although it is higher than in autarky. If the transaction efficiency for primary good is sufficiently improved such that it is higher than the transaction efficiency for labor which in turn is higher than that for the intermediate good, then the general equilibrium will shift from PF2 to structure CF1 where a specialist producer of final good hires ak workers, buys the primary good, and directs the workers to specialize in producing the intermediate good using the primary good within his firm. For this particular exogenous changing pattern of values of parameters, the general equilibrium will shift along the path represented by the structure sequence A=>P1=>PF2=> CF1 in Figure 2. Structure PI
\ . Structure P2
x Structure C
Figure 2: Development in Division of Labor and Structure of Firms
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The development of specialization in output associated with a decrease in the scope of a firm's activities, an increase in specialization of labor assignment within firms and replacement of self-provision of intermediate goods with market procurement, are common in the economy. For instance, law firms may give up providing a full range of legal services to specialize in only one or two aspects of the legal practice such as conveyancing or family law. Specialization in labor assignment is also common in manufacturing assembling and processing. A manufacturing firm may purchase intermediate services such as advertising, storage and distribution in the market instead of advertising its own products and operating its own warehouses and fleet of trucks. To summarise, all patterns of changes in division of labor that involve a decrease in average employment of firms are listed in (6a), and those patterns that involve a monotonic increase in average employment are listed in (6b). The values of S are the average employment of firms in the relevant structures. A=>P1^
A=>
PF2 =>\^
PF1
S=a2k/(l~a2)
(6a)
=>PF2 P2:
A => PI => S =
A=>
PF1
S=a2k/(\-a2)
PF2 =>CF2 **/('-«)
fCFl, S=ak < [CF2 S = ak/(l-a)
(6b)
=>PF2=> CF2 S=ak/(\-a)
where the average employment of firms is akl{\ - a) in structure PF2, ok in CF1, dkl{\ - d) in PF1, and ak/(l - a) in CF2, and akl{\ - a) > ctkl{\ - a 2 ), ak. (6) is proved in the Appendix.
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6. Conclusions In this paper we use inframarginal comparative static equilibrium analysis to show, after the manner of Coase and North, a decline in the transaction cost coefficient leads to organizational changes from selfsufficient production to specialization in production.7 If we assume that each individual maximizes total discounted utility and transaction efficiency changes over time, our model in this paper will become a dynamic general equilibrium model which can predict evolution of firm size over time. Another way to extend the static model is to assume bounded rationality and interactions between dynamic decisions and information of prices. Zhao (1998) has developed such a Walrasian sequential equilibrium model that can predict spontaneous evolution of the institution of the firm and of society's knowledge of institution. Our model predicts that the average employment of firms will decrease as division of labor develop if transaction efficiency is improved in such a way that the transaction efficiency for intermediate goods becomes higher than that for labor. To test the model one needs to have good measures of the relative transaction conditions of goods to labor. However, these data are difficult to obtain. Nevertheless, empirical work of Murakami et al. (1996), and Yang et al. (1992) have provided indirect evidence for this theory. Murakami, Liu, and Otsuka show that in China's machine tool industry the average size of firms declined while division of labor and productivity rose in the 1980s. Yang, Wang, and Wills show that an institution and policy index (a weighted average of 12 subindices of transaction conditions for four types of properties and three components of property rights) that affects transaction efficiency in China increased in the 1980s. The subindex for transaction efficiency of goods was higher than that of labor during that period. These two pieces of empirical work indirectly support the hypothesis based on our theory. More empirical work needs be done to test our theory. 7
Development in division of labor in this paper is based on comparative statics. It is exogenous and cannot take place in the absence of exogenous change of transaction efficiency parameters. A dynamic version of the model is developed by Borland and Yang (1995) to generate endogenous evolution of the firm.
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Appendix A: Proof of Proposition 1 Since the corner equilibrium that generates the highest per capita real income is the general equilibrium, the general equilibrium can be identified by comparisons among per capita real incomes in all structures. A comparison between w(Pl) and «(PF1), or between w(P2) and w(PF2) yields w(PFl) > w(Pl) if s > t and w(PF2) > w(P2) if s > r. (7) This implies that for a given pattern of partial division of labor, the institution of the firm will be used in the general equilibrium to organize the division of labor if the transaction efficiency for labor (s) is higher than the transaction efficiency for intermediate goods (r or t). Comparisons among u(C), w(CFl) and w(CF2) yield the following results. w(C) > w(CFl) and w(CF2) if s < Min{r, (rt°)m
+ a)
}
w(CFl) > w(C) and w(CF2) if s e (r, /) & / > r. m
(8a) (8b)
+ a)
w(CF2) > M(C) and w(CFl) if s > Max{/, (rf) } (8c) (8) together with Table 1 imply that the larger the transaction efficiency for labor compared to that for intermediate goods, the more likely the general equilibrium is a structure involving the institution of the firm, and that the larger the transaction efficiency for labor compared with that for intermediate goods, the larger is the employment of a firm. Since the general equilibrium, which is the corner equilibrium with the highest per capita real income, will shift between corner equilibria as transaction efficiency parameters reach some critical values, there are many possible paths of such development in the division of labor, indicated by the arrows with numbers in Figure 2. The critical values can be identified by comparing per capita real incomes in all structures. Such comparisons yield the following results. Shift 1, A=>P1, will take place if kt increases to the critical value kt>/30 Shift 2, A=>PF1, will take place if ks increases to the critical value ks> J30
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Shift 3, P1=>P2, will take place if k and r increase to the critical value
f>Px Shift 4, P1=>PF2, will take place if k and s increase to the critical value
f>Px Shift 5, PF1=>P2, will take place if k and r increase to the critical k{l~a)r value Shift 6, PF1=>PF2, will take place if ks increases to the critical value (ks)l~a > fa Shift 7, P2=>C, will take place if k increases to the critical value t {X a) r - >j32'2 Shift 8, P2=>CF1, will take place if & and t increase to the critical sta value Shift 9, P2=>CF2, will take place if s and r increase to the critical s0+a)
value
r>P2 Shift 10, PF2=?>C, will take place if r and / increase to the critical 1 value
r f
s>P2 Shift 11, PF2=>CF1, will take place if t increases to the critical value f > J32 Shift 12, PF2=>CF2, will take place if s increases to the critical value sa>j32 (9)
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where ^ f f V ( l - a ) " ^ w , # ^ [of1 " 2a)(l - a)a " 'f " ' x (1 + a)a-^x^D/a; a n d ^ s [ a « ( 1 _ a y - Y " ' . It can be shown that the size and scope of firms will decrease if the development in division of labor follows the structure sequences: A=>P1=>PF2=>C (Paths 1, 4, 10 in Figure 2), A=>P1=>PF2=>CF1 (Paths 1, 4, 11 in Figure 2), A=^>PF1=>PF2=>C (Paths 2, 6, 10 in Figure 2), and A=>PF1=>PF2=>CF1 (Paths 2, 6, 11 in Figure 2). According to (7-9), this will take place if transaction efficiencies for goods and labor increase but the transaction efficiency for labor does not increase too fast. On the other hand, the size of firms will increase if the development in division of labor follows the structure sequences A=>P1=>P2=>CF1 (Paths 1, 3, 8 in Figure 2), A=>P1=>P2=>CF2 (paths 1, 3, 9, in Figure 2), and A=>PF1^PF2^>CF2 (paths 2, 6, 12 in Figure 2). According to (7-9) in the Appendix, this will take place if transaction efficiencies for goods and labor increase or the transaction efficiency for labor increases more quickly than the transaction efficiency for intermediate goods. The results are summarized in (6). Figure 2 or (9) in the Appendix, combined with Table 1 imply that as transaction efficiencies for labor, and intermediate and final goods are improved, the level of division of labor increases. Individuals' level of specialization, trade dependence, per capita real income, and the number of specialized activities increase. This change will not involve the institution of the firm if the transaction efficiency for labor is lower than that for intermediate goods. The institution of the firm will emerge from the development in division of labor if the transaction efficiency is higher for labor than for intermediate goods. (6), (7), (8), and (9) are sufficient to establish Proposition 1.
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References Aiginger, K., Tichy, G., 1991. Small firms and the merger mania. Small Business Economics 3 (2), 83-102. Akerlof, G., 1970. The market for lemons: quality uncertainty and the market mechanism. Quarterly Journal of Economics 89, 488-500. Borland, J., Yang, X., 1995. Specialization, product development, evolution of the institution of the firm, and economic growth. Journal of Evolutionary Economics 5, 19-42. Camacho, A., 1991. Adaptation costs, co-ordination costs and optimal firm size. Journal of Economic Behavior and Organization 15, 137-149. Camacho, A., 1996. Division of Labor, Variability, Co-ordination, and the Theory of Firms and Markets, Boston, Kluwer Academic Publishers, Dordrecht, MA. Cheung, S.N.S., 1983. The contractual nature of the firm. Journal of Law and Economics 26, 1-21. Coase, R., 1937. The nature of the firm. Economica 4, 386-405. Dixit, A., Stiglitz, J., 1977. Monopolistic competition and optimum product diversity. American Economic Review 67, 297-308. Ethier, W., 1982. National and international returns to scale in the modern theory of international trade. American Economic Review 72, 389-405. Fujita, M., Krugman, P., 1995. When is the economy monocentric: von Thiinen and Chamberlin unified. Regional Science & Urban Economics 25, 505-528. Grossman, G., Helpman, E., 1989. Product development and international trade. Journal of Political Economy 97, 1261-1283. Grossman, G., Helpman, E., 1990. Comparative advantage and long-run growth. American Economic Review 80, 796-815. Grossman, S., Hart, O., 1986. The costs and benefit of ownership. Journal of Political Economy 94 (4), 691-719. Hart, O., 1991. Incomplete contract and the theory of the firm. In: Williamson, O., Winter, S. (Eds.), The Nature of the Firm. Oxford University Press, New York.
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Hart, O., Moore J., 1990. Property rights and the nature of the firm. Journal of Political Economy 98 (6), 1119-1158. Holmstrom, B., Roberts, J., 1998. The boundaries of the firm revisited. Journal of Economic Perspectives 12, 73-94. Holmstrom, B., Milgrom, P., 1995. The firm as an incentive system. American Economic Review 84, 972-991. Hu, M.M., 1991. A study of small and medium enterprises in manufacturing of Taiwan, Ph. D. Dissertation. National Taiwan University, Taipei, (in press). Hurwicz, L., 1973. The design of mechanisms for resource allocation. American Economic Review 63, 1-30. Jones, C.I., 1995a. Time series tests of endogenous growth models. Quarterly Journal of Economics 110, 525-695. Jones, C.I., 1995b. R & D-based models of economic growth. Journal of Political Economy 103, 759-784. Judd, K., 1985. On the performance of patents. Econometrica 53, 579-585. Kim, L., Nugent, J.B., 1993. Clues to the success of Korean SMEs in manufacturing exports. Paper Presented at the Far Eastern Meeting of Econometric Society, Taipei. Kim, S., 1989. Labor specialization and the extent of the market. Journal of Political Economy 92 (3), 692-705. Kreps, D., 1990. Corporate culture and economic theory. In: Alt, J., Shepsle, K. (Eds.), Perspectives on Positive Political Economy, Cambridge University Press, Cambridge, pp. 90-143. Krugman, P., 1980. Scale economies, product differentiation, and the pattern of trade. American Economic Review 70, 950-959. Krugman, P., 1981. Infra-industry specialization and the gains from trade. Journal of Political Economy 89, 959-973. Krugman, P., Venables, A. J., 1995. Globalization and the inequality of nations. Quarterly Journal of Economics 110, 857-880. Langlois, R., 1988. Economic change and the boundaries of the firm. Journal of Institutional and Theoretical Economics 144, 635-657.
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Lewis, T., Sappington, D., 1991. Technological change and the boundaries of the firm. American Economic Review 81, 965-982. Liu, P.W., 1992. Extent of the market, specialization, transaction efficiency and firm size in Hong Kong. Working Paper No. 20, Department of Economics, Chinese University of Hong Kong. Loveman, G., Sengenberger, W., 1991. The re-emergence of small-scale production: an international comparison. Small Business Economics 3 (1), 1-38. Maskin, E., Tirole, J., 1997. Unforeseen contingencies, property rights, and incomplete contracts. Harvard Institute of Economic Research Discussion Paper No. 1796, Harvard University, USA. Milgrom, P., Roberts, J., 1990. Bargaining, influence costs, and the organization of economic activity. In: Alt, J., Shepsle, K. (Eds.), Perspectives on Positive Political Economy. Cambridge University Press, Cambridge, pp. 57-89. Milgrom, P., Roberts, J., 1992. Economics, Organization, and Management, Englewood Cliffs, Prentice-Hall, Englewood Cliffs, New Jersey, USA. Murakami, N., Liu, D., Otsuka, K., 1996. Market reform, division of labor, and increasing advantage of small-scale enterprises: the case of the machine tool industry in China. Journal of Comparative Economics 23, 256-277. Romer, P., 1990. Endogenous technological change. Journal of Political Economy 98, S71-S102. Segerstrom, P., 1998. Endogenous growth without scale effects. American Economic Review 88, 1290-1310. Stigler, G.J., 1951. The division of labor is limited by the extent of the market. Journal of Political Economy L1X (3), 185-193. Storey, D.J., Johnson, S., 1987. Job Generation and Labour Market Change, Macmillan, London. Yang, X., Ng, Y.K., 1993. Specialization and Economic Organization: A New Classical Microeconomic Framework, Amsterdam, North-Holland. Yang, X., Ng, Y.K., 1995. Theory of the firm and structure of residual rights. Journal of Economic Behavior and Organization 26, 107-128. Yang, X., Wang, J., Wills, I., 1987. Economic growth, commercialization, and institutional changes in rural china, 1979-1987. China Economic Review 3, 1-37.
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Young, A.A., 1928. Increasing returns and economic progress. Economic Journal 38, 527-542. Young, A., 1998. Growth without scale effects. Journal of Political Economy 106, 41-63. Zhao, Y., 1998. Concurrent evolution of division of labor and information of organization. Review of Development Economics, in press.
CHAPTER 12 SPECIALIZATION, PRODUCT DEVELOPMENT, EVOLUTION OF THE INSTITUTION OF THE FIRM, AND ECONOMIC GROWTH*
Jeff Borland" and Xiaokai Yang b * "University of Melbourne
b
Monash
University
1. Introduction An important aspect of the development of endogenous growth models has been the emphasis placed on the nexus between the division of labor and economic growth. While one class of endogenous growth models has considered the relation between economic growth and the division of labor that occurs through product development (for example, Romer 1986a; Grossman and Helpman 1989a, 1989b, 1990, 1991; Segerstrom et al. 1990; Young 1991a; Rivera-Batiz and Romer 1991) an alternative approach has associated the division of labor with increases in the level of specialization of individual agents (Yang and Borland 1991a). The former approach is able to generate evolution in the variety of goods in the economy, but does not endogenize the range of goods produced by each agent; on the other hand, the latter approach which allows the range of goods produced by each agent to vary, cannot generate evolution in the variety of goods. To achieve a more comprehensive coverage of the sources of growth than is available in existing studies, this paper provides a dynamic analysis * Reprinted from Journal of Evolutionary Economics, 5 (1), Jeff Borland and Xiaokai Yang, "Specialization, Product Development, Evolution of the Institution of the Firm, and Economic Growth," 19-42, 1995, with permission from Springer-Verlag. * We are grateful to Yew-Kwang Ng, Sherwin Rosen, James Schmitz, Jr., and two anonymous referees for comments. Any remaining errors are our responsibility.
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of growth and welfare in an environment where both aspects of the division of labor, individual specialization and product development, are endogenous. It extends previous work (Yang and Borland 1991b) which considered evolution of the division of labor in an economy with only consumption goods to the case of an economy with consumption and intermediate goods. In the model to be considered, each individual is assumed to be a consumer/producer and to derive utility from a single consumption good which uses as inputs labor and either one or two intermediate goods. Each individual is endowed with a single unit of labor in each period, and has a system of individual-specific production functions for producing the intermediate and consumption goods which display increasing returns to specialization and learning-by-doing. In addition to being self-produced, consumption and intermediate goods can be purchased, but in this case two types of transactions costs may be incurred. The first is a variable transaction cost which is proportional to the quantity of the good traded. The second is a fixed transaction cost which is incurred in the first period in that an individual engages in trade. In each period an individual makes a decision on which goods to produce, and on demands for and supplies of any traded good. The aggregate outcome of the decisions of all individuals in the economy is then an endogenously determined division of labor and variety of intermediate goods. In this framework, growth in the division of labor can be interpreted as either a shift toward greater specialization in production by an individual, or as an increase in the variety of producer goods employed by a producer of final goods. In each period there are tradeoffs involved in an individual's decisions on production and trade activities. By increasing his degree of specialization, an individual is able to increase productivity by speeding up accumulation of experience via specialized learning-by-doing; however, greater specialization will necessitate trade (in either intermediate or consumption goods) and thereby incur transaction costs which is undesirable due to an individual's preference for current consumption. At the same time, an individual who produces the consumption good must decide upon the range of intermediate goods with which to produce the consumption good. If transaction efficiency is low,
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then transactions costs outweigh economies of specialization and therefore individuals will choose autarky. The benefits from expanding the range of intermediate goods derives from economies of complementarity between the intermediate goods; however increasing the variety of intermediate goods may limit an individual's capacity to exploit economies of specialization in autarky. As transaction efficiency is improved, the division of labor between different specialists can be used to avoid the tradeoff between economies of specialization and economies of complementarity, so that the variety of goods and specialization can be increased simultaneously. The interaction of these tradeoffs determines the nature of the dynamic equilibrium for the economy. If transaction efficiency is sufficiently high, in the dynamic equilibrium the economy is in a market structure in which individuals specialize in the production of either an intermediate or the final good and trade occurs in each of these goods in all periods; if transaction efficiency is sufficiently low, the economy is in autarky in all periods - depending upon the degree of economies of specialization and complementarity this may involve individuals producing the consumption good with a single or both intermediate goods in all periods, or the division of labor with an expanded range of intermediate goods over time. If transaction costs are at an intermediate level, the dynamic equilibrium involves evolution of the economy from autarky to a state in which individuals specialize in production activities and all goods are traded. Depending upon the relative magnitudes of economies of specialization and complementarity, this evolution may involve an increase in the number of intermediate goods employed by a producer of the consumption good. In any dynamic equilibrium with evolution in the division of labor, either through changes in individual specialization, or in the range of intermediate goods, per capita real income will increase between periods; hence, the evolution of division of labor is associated with economic growth. There are a number of important determinants of the results obtained in this paper. First, it is assumed that individuals' production functions for the intermediate and final goods exhibit increasing returns to an individual's own labor; usually such an assumption would be incompatible with price-taking behaviour since it implies that individuals
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would want to purchase an infinite amount of that input. However, since learning by doing is individual-specific and thereby labor is individual-specific and each individual has a finite labor endowment, this problem does not arise. Second, in a dynamic equilibrium which involves the division of labor and trade, learning-by-doing and increasing returns will mean that monopoly power accrues to individual producers. However, this monopoly power will be nullified by the assumption that all trade is mediated through a futures market which operates in period 1 and which establishes contracts that cannot be renegotiated in a later period. Since all decisions are therefore made in period 1, and all individuals are ex-ante identical, at the time at which equilibrium prices of traded goods are determined there is no monopoly power and hence a Walrasian regime prevails.1 In the dynamic equilibrium just described, as the division of labor evolves, there are two ways to organize transactions which are required by the division of labor. The first is to exchange consumer goods for intermediate goods in the goods market. The second is to exchange consumer goods for labor, where that labor is then used to produce intermediate goods. This latter pattern of transactions may be associated with the institution of the firm whereby the market for intermediate goods is replaced with the market for labor.2 For instance, a producer of a consumer good may choose the division of labor by exchanging his product for an intermediate good. Alternatively he may hire workers and command them to specialize producing intermediate goods within a firm. In the latter case, the institution of the firm emerges from the division of 1
Earlier studies have dealt with the issue of increasing returns either by assuming that such returns are 'external' so that returns at the level of the firm are non-increasing and a price-taking equilibrium is supported, or by departing from price-taking behaviour and assuming that markets are monopolistically competitive. For surveys of this literature, see Romer (1989) and Sala-i-Martin (1990a, 1990b). 2 Cheung (1983) argues that the institution of the firm can be interpreted as a replacement of a goods market with a labor market, rather than a replacement of the market with non-market institutions, as argued by Coase (1937). According to Cheung's theory of the firm, the argument that the firm internalizes externalities arising in the market place is problematic as the firm may replace the externalities in a goods market with the externalities in a labor market.
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labor between production of consumer goods and production of intermediate goods. This paper shows that if transaction efficiency in the market for an intermediate good is lower than in the market for labor that is used to produce the intermediate good, then the evolution of the division of labor may be associated with the emergence of the institution of the firm. Previous studies of economic growth generated by increasing returns to scale have ignored the role of specialization and have concentrated instead on operation scale or the variety of goods. The level of specialization is much more difficult to be endogenized than the operation scale or the variety of goods as it involves corner solutions and the equilibrium based on a combination of many corner solutions. Yang and Borland (1991a) propose an approach to handling the dynamic equilibrium with corner solutions. As calculus of variation is not appropriate for this kind of model, it is necessary to adopt control theory to solve for the dynamic equilibrium. However, it is extremely difficult to apply control theory in a model with CES production functions and intermediate goods. If either the CES function or asymmetry is introduced into the model, it is impossible to obtain analytical results, yet an asymmetry between the decision problem for specialists of final goods and the decision problem for specialists of intermediate goods is unavoidable in this model. In this paper, the technical problem just described is avoided by applying dynamic programming instead of the control theory. The issue of specialization, which is at the center of this paper, has previously been examined in a range of contexts. In a partial equilibrium framework, Rosen (1978) has shown how specialization can occur because of comparative advantage and the different endowments of economic agents, and Stigler (1951), Rosen (1983), Barzel and Yu (1984), and Edwards and Starr (1987) have examined the incentives for specialization that arise due to increasing returns. Other studies, most notably Baumgardner (1988), Kim (1989), and Yang (1990) endogenize the level of specialization in a static general equilibrium economy. Becker and Murphy (1992) consider the effect of a tradeoff between the benefits of division of labor and coordination costs on the optimal degree of division of labor.
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The role of learning-by-doing in generating economic growth has also been previously considered by Arrow (1962), Young (1991a, 1991b), Murphy et. al.(1991), and Stokey (1991). In these studies knowledge is assumed to be a by-product from the production of capital so that increasing returns are external to the individual firm. By contrast, in the present paper learning-by-doing gives rise to an internal economy. Important early contributions to establishing the determinants of the structure of production and exchange were made by Young (1928) and Coase (1937). The conditions under which the production process will be coordinated through the institution of the firm rather than through a series of exchange transactions in the market have been investigated by Klein et. al. (1978), Williamson (1979, 1985) and Grossman and Hart (1986). A range of studies — Grossman and Hart (1986), Milgrom and Roberts (1987) and Yang and Ng (1995) - have also examined why the institution of the firm can save on transactions costs. The rest of the paper is organised as follows. Section 2 presents the model. Section 3 investigates an individual's decision problem within a given period and derives conditions which characterize the static equilibrium of various market structures. In section 4 the nature of the dynamic equilibrium is investigated. The conditions under which the institution of the firm emerges from evolution in the division of labor is analysed in section 5. In the final section the paper's conclusions are summarized. 2. Model We consider a finite horizon (two-period) economy with M producer/ consumers. There is a single consumer good which has as inputs labor and either one or two intermediate goods.3 Individuals can self-provide any goods or alternatively, can purchase them on the market. The self-provided amounts of the consumption good and of the two
3
This model can be extended to incorporate many consumer and producer goods and many periods in order to predict incessant evolution in the division of labor and product development. This type of extension is restricted by the availability of computer software that can solve analytically for a dynamic programming problem.
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intermediate goods in period t are denoted respectively zt, xt, and yt. The respective amounts sold and purchased of good z in period / are z\ and zd , those of good x are xst and xdt , and those of good y are y] and ydt . It is assumed that a fraction (1 - k) of any shipment of the consumption good disappears in transit due to transactions costs so that kzd is the amount available for consumption after purchasing zd. The total amount of a good consumed is therefore zt + kzd. Similarly a fraction (1 - s) of any shipmen of an intermediate good disappears in transit so that sy? mdsxd are the amounts available of the intermediate goods after purchasing respectively yd and xd. The total amounts of intermediate goods available are therefore xt + sxd and yt + syd. In addition, it is assumed that a fixed cost, c, is incurred in the period in that an individual first engages in trade.4 This fixed cost coefficient can be interpreted as the cost of creating a new market; for example, the investment cost of facilities and instruments that are necessary to implement trade. The utility function in period / is therefore assumed to be equal to the amount of good z consumed net of the fixed transaction cost: ut=zt+kzd-c,5
(1)
where c = 0 if an individual has either engaged in trade prior to period t or has never engaged in trade up to and including period /, and c > 0 if an individual engages in trade for the first time in period t. It is assumed that all trade in this economy is mediated through contracts signed in futures markets which operate in period 1. These contracts cannot be renegotiated in some later period. Assume that the futures market horizon and any individual's decision horizon are for two
4
Introducing only the variable transaction cost without the fixed transaction cost will mean that the condition for a dynamic equilibrium with the gradual evolution of specialization will not be satisfied and the dynamic equilibrium will be at a corner, either autarky or specialization in production, forever. Together with the discounting of future consumption, and fixed transaction cost introduces an efficient timing problem of when to shift from autarky to market exchange so that the gradual evolution of specialization may occur. 5 The results in the paper do not depend on the specific linear form of utility function. As shown in Yang and Borland (1991a, b), a non-linear CES or Cobb-Douglas utility function can be used to generate similar results.
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periods. The objective function for an individual's decision problem is therefore total discounted utility, given by: U-ux+ru2, (2) where C/is total discounted utility, andr e (0,1) is the discount factor. The production functions for an individual are assumed to exhibit learning-by-doing and increasing returns to specialization: z, +z; = [(xt+sxff xt+x;=(Ljxly,
+
(yt+sydt)efle{LJztr
y,+y;=(Lyliyly,
/„+/,+/*=l, 0
aa>\,
«>i
(3a)
Lu=Lu_x+lu,
where lit is the labor allocated to production of good / (/ = x, y, z) in period t, and Lu is the labor accumulated in producing good / prior to period t. Each person is assumed to be endowed with one unit of labor in each period, so that/,., is a person's labor share in producing good i at t as well. We define this share as a person's level of specialization in producing good i at /. Production of a good in period t is impossible in the absence of some labor input to the production of the good in that period.6 It is assumed that learning-by-doing and increasing returns to specialization are specific to each individual and each activity. The production function for z is a Cobb-Douglas function of a composite intermediate input and a composite labor input. The elasticity of output with respect to the composite intermediate input is ae(0, 1), and that with respect to current and accumulated labor is aa. Since it is assumed that aa > 1, economies of specialization, which are defined to exist when there is a positive relationship between the productivity of labor in producing a particular good and an individual's level of specialization in that production activity, are present in the production function for each good. The composite intermediate input is produced with a CES production function with a variety of goods; the elasticity of substitution between intermediate goods is 1/(1 - 9). Since the elasticity of substitution and degree of economies of complementarity between the intermediate goods are inversely related, 1/6
6
This represents an extension of the analysis in Yang and Borland (1991a) where current labor input is not necessary to produce a good. We are grateful to Gene Grossman and James Schmitz for suggesting this modification.
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can therefore be interpreted as the degree of economies of complementarity. In order to solve for an equilibrium in this dynamic model, it is necessary to specify initial values for all variables. It is assumed that: 4i=4.=£,i=i.
(3b)
so that an individual is able to produce a positive amount of each good in period 1, and begins with an equivalent endowment of human capital for undertaking each activity. 3. Individual Decisions and Static Equilibrium This section considers an individual's production and trade decision problem within a given period. We begin by arguing that the assumption that trade is determined in a futures market which operates in period 1 is sufficient to ensure price-taking behaviour by individuals. Hence any dynamic equilibrium will be characterized in each period by a set of market-clearing conditions and a set of utility-equalization conditions. A structure of production and trade activities for any individual in a given period is defined as a configuration. Lemma 1 establishes that an optimal configuration for any individual must be a corner solution with the properties that the individual sells only one good and does not buy and sell or self-provide a good at the same time. The combination of configurations of the M individuals in the economy is defined as a market structure. A feasible market structure consists of a set of choices of configurations by individuals such that the supply of a traded good is matched by the demand for that good. The remainder of this section characterizes the static equilibrium trade and production outcomes for each feasible market structure. In period 1, all individuals are ex-ante identical and hence there are many potential producers of each good. That is, no individual has experience in any production activity in period 1 so that competition for the rights to produce a good in the future is between identical peers rather than between 'experts' and 'novices'. Since all trade is entirely determined in a futures market which operates in period 1, although over time producers will gain monopoly power due to learning-by-doing, at the
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time at which contracts are signed, no such monopoly power exists. This ensures a Walrasian regime with price-taking behaviour in period 1. Since the model is complicated by corner solutions and interactions between learning by doing and specialization, we solve for dynamic equilibrium in two steps. First we solve for the dynamic equilibrium under the following assumption. Assumption Al: In period 1, where an individual's production decision involves allocating labor between a number of production activities, an individual will choose the configuration optimally with reference to his future behaviour, but will allocate his time between different activities for a given configuration without reference to future behaviour. Then Proposition 2 in section 4 characterizes the full dynamic equilibrium when assumption Al is relaxed. Since we consider a two-period model, the assumption of partial myopia will affect the characterization of an individual's decision problem only in the initial period; optimal decisions in the terminal period are consistent with the full dynamic optimization. In the model described in the previous section, the number of producers of each good and the extent of trade are endogenously determined. Since the quantity of a good that is self-provided is distinguished from the quantity traded of the same good, individuals' optimal decisions in any period are always corner solutions. Although there are 29 = 512 possible combinations of zero and non-zero values ofz,, z], zdt , xt, x"t, xdt, yt, y"t , andyf, and thereby 511 possible corner solutions and one interior solution in each period, with the Kuhn-Tucker theorem it is possible to rule out the interior solution and many corner solutions from the list of candidates. In Appendix A it is shown that the optimal pattern of consumption, production, and trade for an individual is such that: Lemma 1: In any period, an individual does not buy and produce the same good. He self-provides the consumer good if he sells it, and does not self-provide any intermediate good if he does not produce the consumer good or if he sells the consumer good. He sells at most one good although he may produce several goods.
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Given Lemma 1 there are six feasible market structures; these are depicted in Figure 1.
© (a) Autarky
(b) Structure C
(c) Structure D
Figure 1: Market Structures without Firms
In the first type of market structure, autarky, there is no trade and each individual self-provides any good that is required. That is, autarky implies a configuration with x* = xf = z] = zdt = y] = ydt = 0. Three configurations, A, A', and B, meet the criterion for autarky. For configuration A the quantities traded of all goods are zero, yt = 0, andz,, xt > 0 so that each individual self-provides a single intermediate good which is an input along with labor in production of the consumption good. Configuration A' is symmetric to configuration A, withx, = 0, andz t , y, > 0. For configuration B, the quantities traded of all goods are zero, butz,, xt, y, > 0; in this case each individual self-provides both intermediate goods and uses these along with labor in production of the consumption good. Each of these autarkical configurations constitutes a market structure. In the second type of market structure, there is trade in the intermediate goods and the consumption goods. Denote a configuration in which an individual sells an intermediate good and buys the final good by (x/z) or {ylz), a configuration in which an individual sells the final good and buys a single intermediate good by (z/x) or (z/y), and a configuration in which an individual sells the final good and buys both intermediate goods by (zlxy).
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Given these possible configurations, there are three feasible market structures with trade: (i) the structure denoted D consists of the division of the M individuals between configurations {xlz), (ylz) and {zlxy) - that is, there are individuals who specialize in the production of each of the intermediate goods and in production of the consumption good producers of the intermediate goods sell their output to producers of the consumption good, and vice-versa with producers of the consumption good retaining an amount for their own consumption; (ii) the structure denoted C consists of a division of the M individuals between configurations {xlz) and (z/x); and (iii) the structure denoted C consists of a division of the M individuals between configurations {ylz) and {zly). A dynamic equilibrium is defined as the sequence of configurations and associated market structures over the two periods which maximize an individual's total discounted utility. Given assumption (Al), it is possible to solve for the dynamic equilibrium in two steps: first, to solve for an individual's per capita real income in each configuration in the static equilibrium of a given market structure in any period; and second, to solve for the optimal sequence of configurations that determines the equilibrium sequence of market structures. Within each feasible market structure, a static equilibrium is defined as a set of relative prices and an allocation of resources such that: (i) for any traded good aggregate demand equals aggregate supply (i.e., market clearing); and (ii) each individual maximizes utility at the given prices and for a given market structure. With the assumption of free entry, condition (ii) implies that the utility of all individuals is equalized in equilibrium. In the remainder of this section, the static equilibria of each of the feasible market structures are solved for, and an expression is derived for an individual's utility in each static equilibrium. 1. Market Structures A and A': An equilibrium in a period when autarky prevails is defined by the solution to an individual's utility-maximizing labor decision for that configuration. In configuration A, the quantities traded of all goods are zero, yt - 0, and zt, xt > 0 ; inserting these values into Eqs. (1) and (3), and substituting from (3) into (1) gives the decision problem for configuration A in period t as:
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Max.«, = (x,Y(LJztr
= {LJJL^l-lJT
•
(4a)
'x!
The solution of (4a) islxt = lzt = 1/2 so that: u1(A) = (LxlLzt/4r, (4b) where ut (A) is the per capita real income when an individual chooses configuration A in period t. The per capita real income for configuration A' is symmetric to that for A, and is given by: ut(A') = (LylL2t/4r. (4c) Note that due to the absence of trade no transaction costs are incurred in these configurations. 2. Market Structure B: In configuration B, the quantities traded of all goods are zero, andz r , xt, yt > 0. From Eq. (3) the decision problem for configuration B in period / is: Mfx:ul=[(xl)e+(yi)er>e(LJzir 'xryt
= [(LJxir
+ (Lyllytrfs[Lzt(l-!xt
-lyt)T b
•
(5a)
The solution to (5a) is /„ = l/2 , lxt = .5L yt/(L xt +L yt), lyt=.5Lbxt j[lbxt + Lbyl), where b = ad/(a0 -1). Hence: u,(B) = (LxtLylLzl/4r(Lbyl
+Lbyf"e-),
b
h
(5b)
where ut (B) is the per capita real income in configuration B for period /. 3. Market Structures C and C : The structure C consists of configurations (x/z) and (z/x). There are two steps in solving for the static equilibrium of a market structure with trade: first, for each type of configuration in the structure the utility-maximizing labor allocation decision and demands and supplies for each good (and hence the indirect utility of an individual who chooses that configuration) are derived; and second, given the demands and supplies of an individual in each configuration, the market clearing conditions and utility equalization conditions are used to solve for the set of static equilibrium relative prices and numbers or individuals choosing each configuration.
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3.1 Configuration (z/x): In this configuration z,, z], xdt> 0,lzt = 1 , and zf =xt= xst - lxt -yt- ydt - y] — lyt - 0 . Hence the decision problem for this configuration is: Max:«, =z,—c subject to:
z, + z] = [(sxf f f/e (LJzl )aa
(6a)
Ptx1 = <»
where pt is the price or x in terms of z in period /. The parameter c is positive if a person engages in trade for the first time in this period, and is zero otherwise. The solution to (6a) is ut [(z/x)] = [a{Lzt f s/p, J'0'"' (\-a)-c, z;=a^[s(LJ°/Plf«-a\
x?=z;/pt,
(6b)
where ut [(z/x)] is the indirect utility function for configuration (z/x) . 3.2 Configuration (x/z): By a similar process, for configuration (x/z) w,[(x/z)]
= £p,(4,y- c ,
x:=(Lxty, zf=Pl(Lx,y,
(?)
where ut [(x/z)] is the indirect utility function for configuration (x/z). Utility maximization by individuals and the assumption of free entry have the implication that the utility of individuals is equalized across configurations. That is,
«,[(*/*)]=«,|>/*)].
(8)
Solving the set of utility equalization conditions gives a set of equilibrium relative prices. Substituting these equilibrium relative prices into the set of market clearing conditions yields an equilibrium set of relative numbers of individuals choosing each configuration. Let M{ represent the number of individuals selling good i. Multiplying M(. by individual demands and supplies gives total demands and supplies. The market clearing condition for the intermediate good is:
Emergence and Evolution of the Firm
Mxx]=Mzxdt,
305 (9)
where Mxx] is total supply of that intermediate good and Mzxd is total demand for the same good. Note that due to Walras' law the market clearing condition for the consumption good is not independent of (9). The equilibrium relative number of individuals choosing each configuration is represented by Mxz = Mx/Mz . Substituting for Eqs. (6) and (7) into the utility equalization condition determines the equilibrium relative price: Pt=(asLzty[k{Lztyl(\-a)T-\
(10)
and hence: u, {C)=aa(l-aya(sknLxlL2tr
~c •
(11)
The equilibrium for structure C is symmetric to that for structure C so that: qt=(asLyty[k(Lyty/(l-a)r\ ut(C') = aa(\-ay-a(sky(LytLziy°-c
(12) (13)
where ql is the equilibrium price of good y in terms of good z in period t, andw, (C) is per capita real income in market structure C . 4. Market Structure D: By an analogous two-step procedure to that used to solve for the equilibrium of market structure C, it is possible to solve for the equilibrium of market structure D which has: Pr[as(LziyJ
[{Lxtye+{LytyeTle [{\-a)lktal2a{Lj
q=[(Lxl/Lyiy]pt
u(D)=a^l-ar[sk(Lj/2j[(LxrHLyirfe-c,
(14a) (14b)
(15)
where ut (D) is the per capita real income in market structure D. Equations (4), (5), (11), (13), and (15) characterize the optimal production and trade decisions and corresponding utility levels for each configuration and associated market structure as a function of relative prices and the levels of human capital. From this information each individual solves a dynamic programming problem to derive his optimal
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sequence of configurations which maximizes total discounted utility over the two periods. 4. Dynamic Equilibrium The dynamic programming problem for an individual is specified as follows: the objective function is specified in Eq. (3), where the value of ut is given by (4b) or (4c) respectively if configuration A or A' is chosen, given by (5b) if configuration B is chosen, and given by (11), (13) or (15) respectively if a configuration within market structures C, C or D is chosen. As all optimal values of quantities consumed, produced, and traded are characterized in (4), (5), (11), (13) and (15), an individual can solve a dynamic programming problem by maximizing total discounted utility with respect to the sequence of configurations over the two periods. There are 82 = 64 patterns of sequences of configurations over the two periods for an individual's dynamic decision. However, since it has been shown that all configurations in the same market structure receive the same real income, we may ignore the distinction between configurations within the same market structure, such as between (z/xy), (x/z) and (y/z) in market structure D, as well as between symmetric configurations and market structures such as A and A'. Recognizing the existence of this symmetry there are only 42 = 16 distinct sequences of configurations. Signifying the order of configurations chosen over the two periods by the order of the letters, the sixteen patterns are AA, AB, AC, AD, BB, BA, BC, BD, CC, CA, CB, CD, DD, DA, DB and DC, where, for example, AA denotes that an individual chooses autarky producing the consumption good with a single intermediate good in each period, and AD denotes that an individual chooses configuration A in period 1 and a constituent configuration of market structure D, (x/z), (y/z) or (z/xy), in period 2. The dynamic equilibrium is the sequence which maximizes total discounted utility over the two periods. As already described in assumption Al, individuals' labor allocation decisions within any configuration are made without reference to future behaviour; however, it should be emphasized that individuals' choices of configurations are assumed to be made taking into account future behaviour. For example, in sequence DD if an individual will choose configuration (x/z) in period 2,
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307
that individual will choose the same configuration in period 1 as this will allow the individual to exploit economies of specialized learning-by-doing. It is shown in Appendix B that since a market structure where individuals specialize in a single production activity allows the tradeoff between economies of complementarity and specialization which is present in autarky to be avoided, both intermediate goods are always produced in that market structure with specialization; hence a sequence of configurations with market structure C cannot constitute a dynamic equilibrium. It is also the case that the economy will not evolve from specialization to autarky. In the absence of transaction costs the economy would move immediately to specialization since this allows the benefits from specialized learning-by-doing to be exploited; hence if the economies of specialization are sufficient to outweigh the transaction costs incurred through trade in period 1, specialization in production must also be optimal in period 2. From a comparison of the remaining sequences of market structures, Appendix B has established: Proposition 1: If transaction efficiency is sufficiently high the economy will move immediately to the market structure with trade and specialization in production (DD). If transaction efficiency is sufficiently low, the economy remains in autarky forever with self-production of a single intermediate good if economies of specialization outweigh economies of complementarity (AA), of two intermediate goods if economies of complementarity outweigh economies of specialization (BB), or with evolution in the number of intermediate goods if economies of specialization are balanced by economies of complementarity (AB). If transaction efficiency is at an intermediate level, the dynamic equilibrium involves evolution from autarky to a state in which individuals specialize in production activities, all goods are traded, and the consumption good is produced with both intermediate goods. This evolution involves an increase in the number of intermediate goods used by a producer of the consumption good if economies of complementarity are not too large relative to economies of specialization (AD); otherwise this evolution involves no expansion in the range of intermediate goods used by a producer (BD).
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Proposition 1 provides a characterization of the conditions under which the various sequences of market structures will constitute the dynamic equilibrium. For example, the dynamic equilibrium will involve the gradual evolution of trade and specialization if in period 1 the discounted value of the benefits from specialization is not significant compared to the associated utility losses which arise due to transaction costs incurred so that each individual chooses to self-produce any good that is required. Later, the effects of learning-by-doing on productivity will increase an individual's ability to absorb the costs of foregone consumption associated with trade, and hence markets for the intermediate and consumption goods will evolve. In the final state with specialization and trade, both intermediate goods are used to produce the consumption good. The evolution of division of labor can also arise in a dynamic equilibrium through an expansion in the range of intermediate goods: the economy may evolve from a state where individuals use a single intermediate input to self-produce the consumption good to a state where both intermediate inputs are used for self-production of the consumption good. This will occur if economies of specialization and complementarity are sufficiently close. In this case, the benefits of specializing in production of a single intermediate good, which are greatest in period 1 when there is a future period in which the benefits of learning-by-doing can be captured, will exceed the benefits due to economies of complementarity from using both intermediate goods to produce the consumption good in period 1, but are outweighed by those economies of complementarity in period 2. In any dynamic equilibrium with evolution in the division of labor, either through changes in individual specialization (AD or BD) or in the range of intermediate goods (AB), per capita real income will increase between periods to a greater extent than if there was no increase in the division of labor. For example, from Proposition 1 it is evident that in the absence of any transaction costs, the dynamic equilibrium will be for individuals to immediately engage in trade and to adopt the highest degree of specialization as this allows the benefits of specialized learning-by-doing and a wider range of intermediate goods to be exploited from the earliest stage. This has the implication that growth in real per
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309
capita income is highest in the dynamic equilibrium where the economy shifts most quickly from autarky to a state with markets of the goods. Hence, the evolution of specialization is associated with a higher rate of economic growth than if the economy remains in autarky forever. An important question is the impact of assumption Al on the characterization of the dynamic equilibrium in Proposition 1. Proposition 2, which is proved in Appendix C, addresses this issue. Proposition 2: If assumption Al was to be relaxed so that individuals take into account future behaviour not only in deciding upon the choice of configuration in period 1, but also in their discretionary labor allocation decisions within any configuration, the relative likelihood that a configuration sequence which involves the evolution of specialization or an increase in the variety of intermediate goods is the dynamic equilibrium, is increased. That is, in any sequence of configurations which involves a choice of labor allocation between activities in period 1 and a shift of configuration over periods (AB, AD, BD) the calculations of total discounted utility upon which Proposition 1 are based understate the level of utility which would be achieved if the assumption of partial myopia was to be removed (since this can be interpreted as a reduction in the set of constraints on an individual's decision problem, its effect on utility is unambiguosly non-negative). It should be noted however that removing assumption Al would not enlarge the set of sequences of market structures which constitute a dynamic equilibrium (for example, any sequence of market structures involving structure C would still be dominated by a sequence replacing that market structure with structure D). Although Proposition 2 can be used to identify exactly the direction of bias in the characterization of the full dynamic equilibrium caused by assumption Al, this proposition also indicates that the essence of Proposition 1 will not be changed by removing assumption Al. An important determinant of economic growth in the dynamic model that has been examined is transaction efficiency. Therefore, if transaction efficiency differs across countries due, for example, to differences in institutional arrangements, the evolution of trade will commence earliest
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in those countries with highest transaction efficiency. Hence a disparity in growth rates across countries with different transaction efficiencies can arise due to a divergence in the speed of evolution of market transactions. Since government policies, institutional arrangements, and urbanization all affect transaction efficiency in important ways, their effects on the evolution of trade and economic growth are critical. For example, if a decrease in k is interpreted as an increase in a sales tax, then having noted that the probability of a dynamic equilibrium involving specialization and trade is an increasing function of the transaction efficiency parameter, k, therefore an increase in sales tax will raise the likelihood that AA is the dynamic equilibrium relative to DD. In other words, a sales tax will retard the evolution of trade and the division of labor.7 5. Emergence of Firms from the Evolution in Division of Labor If the division of labor emerges in a dynamic equilibrium, then there are two institutional mechanisms through which that division of labor could be organized. In the preceding section, one mechanism that organizes the division of labor via the markets for consumer and intermediate goods has been investigated. Suppose alternatively, the producer of the consumer good does not buy intermediate goods from the goods market, but instead hires labor and commands workers to specialize in producing intermediate goods in a firm and claims residual that is the difference between sales revenue of the consumer good and wage bill. Then the institution of the firm will replace the market for intermediate goods with the market for labor. If such a method of organizing the division of labor is allowed, then there are more feasible market structures than have been examined thus far. Each of the extra market structures simply replaces a market for intermediate goods with a market for labor. Suppose that trade in labor is allowed, so that there are five more feasible market structures which are denoted FC, FC, FD(z), FD(xz), 7
Barro (1991) finds evidence that the rate of growth is inversely related to the share of government consumption in GDP. This may provide some support for the argument advanced that through the mechanism of transaction efficiency, higher taxes will lower the rate of growth and the division of labor can speed up human capital accumulation via specialized learning-by-doing.
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FD(yz). Let (z / lx) denote a configuration in which an individual sells the final good and buys labor; he produces the final good and employs workers to produce x. Similarly, (/2 / z) denotes a configuration in which an individual sells his labor and buys the final good; he becomes a worker producing intermediate good x. Structure FC consists of configuration (z/lx) and configuration (/^/z) . In this structure, the producer of z claims residual rights and becomes an employer. Authority is asymmetrically distributed between the employer and the employees. It is easy to see that market structure FC is exactly the same as structure C as far as the level and pattern of the division of labor and the variety of producer goods are concerned. The only difference between structures FC and C is in the organizational structure of transactions. Structure FC involves the institution of the firm, the market for labor, and the market for z, but does not involve the market for x, while structure C involves the markets for x and z, but does not involve the institution of the firm and the market for labor. In other words, the institution of the firm replaces the market for x with the market for labor used to produce x. Market structures involving firms are depicted in Figure 2. The dashed circle represents a firm and the dot in the dashed circle represents a process that transforms labor into intermediate goods. Panel (a) depicts structure FC. Structure F C is symmetric to structure FC. It consists of configurations (z/ly) and {ly/z), where (z/ly) denotes a configuration in which an individual sells the final good and buys labor, and produces the final good and employs workers to produce y; {ly/z) denotes a configuration in which an individual sells his labor and buys the final good, and becomes a worker producing intermediate good y. Replacing letter x with letter y, panel (a) in Figure 2 depicts structure FC. Structure FD(z) consists of configurations {z/ljy), {IJz), and {ly/z), where (z/lxly) denotes a configuration in which an individual sells the final good and buys labor, and produces the final good and employs workers to produce x and y. Configurations (IJz) and {ly/z) are the same as in structures FC and FC. Structure FD(z) is the same as structure D except that it replaces the markets for x and y with the market for labor. Structure FD(z) is depicted in panel (b) of Figure 2. Structure FD(xz) consists of configurations {z/xly), (x/z), and (//z), where {z/xly) denotes a configuration in which an individual sells the final
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J. Borland, X. Yang
Firm
Firm
t? <2j Market for labor
.
Market ^ forz .
(a) Structure FC
Market for labor
,
Market forz ,
(b) Structure FD(z)
(c) Structure FD(xz)
Figure 2: Market Structures with Firms
good and buys labor and x; he produces the final good and employs workers to produce y. Configuration (x/z) is the same as in configuration C or D and (//z) is the same as in structure F C or FD. Structure FD(xz) is the same as structure D except that it replaces the markets for y with the market for labor. Structure FD(xz) is depicted in panel (c) of Figure 2. Structure FD(yz) consists of configurations (z/ylx),(y/z)and(lx/z), where (z/ylx) denotes a configuration in which an individual sells the final good and buys labor and y; he produces the final good and employs workers to produce x. Configuration (y/z) is the same as in configuration C or D and(/ ;c /z) is the same as in structure FC or D. Structure FD(yz) is the same as structure D except that it replaces the market for x with the market for labor. Exchanging letters x and y in panel (c) of Figure 2, the graph for structure FD(yz) can be obtained. It can be shown that if transaction efficiency coefficient were the same for each method of organizing the division of labor, then each method would generate the same per capita real income. The method of organizing the division of labor makes a difference only if transaction efficiency differs across various market structures. To draw the difference between transaction cost coefficient for different market structures, let
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313
1 - s0 and c0 denote variable and fixed transaction cost coefficients, respectively, in structure C, 1 - s, and c, denote those in structure C , 1 -s2 and c2 denote those in structure D. Let us consider structure FC first. The individual decision problem in period t for configuration(z/lx) can be figured out from (1), (2), (3) by noting that configurations (z/x) and (z/lx) have the same structure of human capital, consumption, and trade, the only difference is in the method of organizing transactions. The decision problem for configuration(z/lx) is Max: u,
=z,-c,
N,zt,z, ,x
s.t. z,+z;=(s3x?y(Ljztr = (s3NLaxl )a La2t =
hi
=
1 Kt 1
(production function) (endowment constraint)
s
pzz = wNlxt = wN (budget constraint or trade balance) (16) where 1 -s3 is the transaction cost coefficient of labor hired to produce the intermediate good. This transaction cost 1 -s3 encompasses all costs that relate to measurement of the relationship between workers' effort and quantity and quality of the intermediate good. c3 represents a fixed transaction cost coefficient for structure FD(z), N is the number of workers hired by an employer, pz is the price of the final good, w is the wage rate, and utis the utility of an individual choosing (z// x ). Here, l2t is the labor share of an individual choosing configuration (z/lx) in producing the final good andlxt is the labor share of an individual choosing configuration (lx/z) in producing the intermediate good. The optimum decisions are z' = wN/p,, u, {z/lx) = z = (s3NL"xtyL::-(wN/pz)-c3
NHapM^ls^LlJ1^
(17)
where Lxl is an employee's human capital accumulated in producing the intermediate good, L z/ is the employer's human capital accumulated in producing the final good, andw, (z/lx)is the employer's indirect utility function. For an individual choosing configuration (lx jz), all variables are fixed, given by
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J. Borland, X. Yang ut(lx/z)=kzd
-c3 = kw/pz-c3
pzzd = wlxt = w
(18) (budget and endowment constraints)
where ut (lx/z)is an employee's indirect utility. Here, an individual has one unit of labor. Manipulating the market clearing condition and utility equalization condition yields the equilibrium in structure FC in period t, given by N=ka/(l-a),
PJw
ut(FQ =aa(\-a)Xa
= {kl{\-a)ta\as,LxtLX,
(19)
[s3kLaxlLazlJ - c3
where ut (FC) is the per capita real income in structure FC. N is the number of workers hired by an employer as well as the relative number of individuals choosing configuration(lx/z)to those choosing(z/lx) . The total number of employees and the total number of employers can be solved from N and the population size M.8 A comparison between w, (FC) from (19) and ut(C) from (10) in section 3 indicates that u,(FC) >u,(Q iff (s 3 > sQand/orc3
(20)
Following the two step approach to solving an equilibrium based on corner solutions, it is also possible to find the equilibria in structures FC, FD(z), FD(xz), and FD(yz). For structure FD(z), equilibrium in period / is given by w/Pz =aa[(\-a)/kta
[(s5/2)Ll(I%+I%ye]a
u,[FD(z)]=aa(l-a)l-a
[(ksj2)Ll(Lax° +LayyeT -c5
(21)
where 1 - ^5 is the transaction cost coefficient of labor, c5 is the fixed transaction cost coefficient in structure FD(z), and U[FE(z)] is the per capita real income in this structure. Here, we have used the symmetry and the fact thatZ^ = Lyt in equilibrium. 8
It should be noted that an employer cannot manipulate the relative number of workers to employers for society as a whole although he can choose N and the equilibrium value of N is the same as the equilibrium value of the relative number just by coincidence. While more than one worker may be employed, economies of specialization do not extend beyond a specific individual.
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For structure FD(xz), equilibrium in period t is given by w/Pl = aa [(1 - d ) j k t a [{sj2)L°zl(L"° + La6JeX,
( 22 )
PJP^K, u.iFDixz^a^l-a)1-"
Kksj2)LULl0, +L;ety/BY -c6
where 1 - s6 and c6 are the variable and the fixed transaction cost coefficients, respectively, in structure FD(xz). U[FD(xz)] is the per capita real income in this structure. Here, we have used the symmetry and the fact that Lxt = Lyt in equilibrium. Replacing px with py (22) gives equilibrium for structure FD(yz) which is symmetric to FD(xz). A comparison between equilibria in structures C , D, FD(z), FD(xz), and FE(yz) yields u, (FC) > u, (C) iff (5 4 > sx and/or c4 < cx) u, [FD(z)] > u, (P) iff ( s5 > s2 and/or c5 ut(£)) iff (s6 > s2 and/orc 6 ut(D) iff (s7 > s2 and/orc 7
(23)
where 1 - s4 and c4 are the variable and fixed transaction efficiency coefficients in structure F C and 1 -s7 andc7 are those in structure FD(yz). Equations (20) and (23), together with Proposition 1, imply that any market structure that involves the division of labor in Proposition 1 will be replaced by the corresponding structure with firms if transaction efficiency for an intermediate good is lower than transaction efficiency of labor that is used to produce the intermediate goods. This result is summarized in the following proposition. Proposition 3: The division of labor is necessary, but not sufficient for the emergence of the institution of the firm. If transaction efficiency is lower for an intermediate good than for the labor employed to produce that intermediate good, then the emergence of the division of labor is associated with the emergence of the institution of the firm. Otherwise, the institution of the firm will not emerge from the evolution of the division of labor.
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In addition, Yang (1990) has shown that the institution of the firm is not needed in the absence of intermediate goods. This result, together with Proposition 3, implies that the division of labor between production of consumer goods and production of intermediate goods is necessary but not sufficient for the emergence of the institution of the firm. 6. Conclusion This paper has investigated the dynamic effects of trade and the division of labor on growth and welfare in an environment where three aspects of the division of labor, individual specialization, product development, and the emergence of firms, can occur concurrently. The nature of the dynamic equilibrium of this economy was shown to depend on the interaction of transaction efficiency, economies of specialization and economies of complementarity. For example, provided transaction efficiency and economies of specialization are sufficiently high, a dynamic equilibrium will involve the evolution of trade and the division of labor through specialization. The emergence of the division of labor will be associated with a higher rate of economic growth. If transaction efficiency for an intermediate good is lower than that for labor employed to produce this good, the emergence of firms will be one aspect of the evolution of division of labor and related economic growth. The concurrent evolution of specialization, trade, and product innovation does appear to have been an important aspect of the historical process of economic development. For example, Chandler (1977, pp 19-28) in describing the growth of the cotton industry in the United State between 1790-1850 emphasizes that this involved the rise of specialization, increased trade, and finished product development. This paper also suggests that specialization will increase economic growth. Romer (1986a) provides empirical evidence that across those countries estimated to have had the highest level of productivity growth in each of four different epochs since 1700, growth rates have consistently increased along with the degree of specialization in economy.
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Appendix A: A Proof of Lemma 1 By assumption Al an individual's allocation of labor within a given period is time-independent. Hence the time subscript is omitted throughout this proof. Claim 1: z andzd cannot be positive at the same time. Suppose z, zd > 0. We may then establish Claim 1 by showing a contradiction. Replacing zd in the utility function with its equivalent in the budget constraint: px(xs - xd) + py(ys - yd) = pz(zd - zs), therefore: u = z + kz«=z + kzs+k{[px(xs-xd)
+ py(ys-yd)]/pz}.
(Al)
Substituting for zs from the production function for z+zs, (Al) becomes: u = (l-k)z
+ k{[px(xs-xd)
s
Py(y
+
-yd)]/p2}
+k{[(x + sxd)e +(y + syd)0f0(LJ2r}.
(A2)
Hence d u/d z > 0 so that the optimum value of z is as large as possible. From the endowment constraint, this impliesx s ,y s , a n d z i = 0. But zero values for these variables contradicts zd > 0 due to the budget constraint so that z and zd cannot be positive at the same time. Claim 2: xs and ys are zero if z and zs are positive. A positive value of z implies zd = 0 due to claim 1. Hence substituting from the production function for z and the budget constraint: u = {(x + sxd)0 + [y + s((px(xs-xd)
s
+ Pzz
)/py)
+
ys)frle(LJzr-z\
(A3) where y =(Lylyfa -y5. From (A3) sgn{du/dys}= sgn{ -1 + s} < 0 V / > 0. From the Kuhn-Tucker theorem this implies y" = 0 if z, zs > 0. By using the budget constraint to eliminate xd rather than yd from u, it is also possible to show thatx* = 0 if z, zs > 0. Claim 3: If zd > 0, therefore x = y = zs = xd = yd = 0.
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If zd > 0, by claim 1, z = 0; hence u = kzd. Since zd and zs are quantities of the same good, due to the transaction cost incurred, if zd > 0, zs = 0. Substituting from the budget constraint and production functions for x andj>:
«=[*//>,] [/'x((Vx) fl -'-^)+M^ra-'xr-j'-/)]- (A4) Hence dw/cbc < OVx > 0 ; du/dy < OVy > 0 ; du/dxd < QNxd > 0 ; and, du/8yd < 0 V / > 0 . Therefore x = y = xd = yd = 0. Claim 4: xs and ys cannot be positive at the same time if zd > 0. From (A4): 8u/dlx\lo<0
(A5a)
9"/ 9/ x| /i=1 >°.
(A5b)
and fmmdu/dlx=0 it is possible to derive the unique solution of lx. (A5) implies that the interior maximum point of lx exists only if there are at least three interior extremes. However, there is a unique interior extreme point of lx. Hence the optimum lx is at either zero or one. Since lx = 1 implies / = 0, therefore xs and ys cannot be positive at the same time if zd>0. Claims 1-4 are sufficient to establish Lemma 1. Appendix B: Proof of Proposition 1 Let ut (i) denote a person's utility in period t when she chooses configuration /', and v, (0 be the maximum discounted value of ut (i) for given levels of human capital. According to Bellman's optimality principle and assumption (Al), v, (/) can be calculated using the solution to the decision problem taking as given human capital in each period. Using the solutions to individuals' decision problems given in (4), (5), (11), (13) and (15), and the relative prices in (10), (12) and (14), an individual's total discounted utility level for configuration sequence U(y) can be summarized as follows (where the definition of the parameters G, E, Q, ci and/?; are provided at the end of the Appendix):
Emergence and Evolution of the Firm
319
U(Aj)=Vl(A)+v2(f) v, (A) = 2-2aa and v2 (A) = r (3/4) 2m v2(5)-v2(^)Ga v2(0 = r[e^(3/2)2aa-c] v2 (£>) s r[ 2 a(1 ^ )/fl g/? a (3/2) 2aa - c],
(i)
so that A maximizes v2 if 0 > 1/a and either c > c, and p > /?, or /? < /?,, B maximizes v2 if # < 1/a and either c > c4 and /? > /? 5 , or p 1, and D maximizes v2 if either c > max{c 15 c 4 } andy# > max{ fix, fi5} or if p < min {/?,, £ }. U(Bj)=v,(B)+v2(j-) Vj ( O s g / T a - c, and v2 (^) = r 2~2aa v2(B) = r2aW (2/3) 4ao v2(0-r[g^(4/3)2--c] v2 (£>) = r[ 2a{X-0)le Qpa (4/3) 2aa - c],
(ii)
a(1 e)/
so thatv2 (A) 1, and D maximizes v2 if either c > max{ q , c4 } and /? > max{ /?,, P5} or if p < min ( Q ) = v , ( C ) + v 2 0) v, ( Q = g ^ - a - c, and v2 (^) = r 2" aa v 2 (5) = r2 a(1 - 2ae)/e
v2(Q = r e ^ 2 2 a a v2 (£>) = r 2 a( '- e)/e QP" 22aa , where v2 (C) < v2 (D) since 2 a ( 1 ^ ) / e > 1. f/(I)/-)=v1(Z))+v2(/') v,(D)s Qpa 2aie-
c, and v 2 (4)s r 2~aa v2{B)=rS 2~aa v2(Q= r Qpa 2laa
(iii)
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J. Borland, X. Yang
v2 (D) = r 2a°-$)/e Qf5a 22aa, where v2 (A)
(iv)
v2 (C) < v2 (D) and D
maximizes v2 if /? > S. An examination of the payoff sequences above establishes that the payoff from choosing market structure C in period 1 is always below that from choosing structure D. Thus, a dynamic equilibrium will not involve structure C being chosen in any period. Hence the remaining sequences of market structures to be considered are AA, AB, AD, BB, BD and DD. A comparison of the discounted utilities generated by these sequences market structures yields the following results from which the statement of Proposition 1 is derived: l.For 9a>\: (a) AA is the dynamic equilibrium if c > max{ cx,c2 } and /? > max{ /?,,/?2} or if < min{ j0 p> 0 2 }. (b) AD is the dynamic equilibrium if (i) cx< c < c3 and J3 > max{/?!,/?3}or if ct> c and /?,<(5< J33, and (ii) if r < r{ and /? r > rx. (c) DD is the dynamic equilibrium if c < min {c2,c3} and ft > max
2. For 1 >6 a > log(2)/2[log(3) - log(2)]: (a) AB is the dynamic equilibrium if c > max{c 4 ,c 5 } and/5 > max {&,/? 6 }, °r if/? < min { & , & } . (b) AD is the dynamic equilibrium if (i) c4 > c > c3 and /? > max {p 3 , y?6}, or c r > (c) DD is the dynamic equilibrium if c < min {c3,c6}and f5 > max {/?3>/U3. For log(2)/2[log(3) -log(2)] >9a: (a) BB is the dynamic equilibrium if c > max{c 6 ,c 7 } and/? > max {/? 8 ,/? 9 },orif /?<min{/? 8 ,/? 9 }-
321
Emergence and Evolution of the Firm
(b) BD is the dynamic equilibrium if (i) c7 > c > c8 and /? > max {/?8,y#10},or c 7 >cand $,/3n,orifr >r 5 and/? max where: c, s QP" (3/2)2aa 2a(i/e~l)-(3/4)2aa; c,=2ale[Qpa
c2 = Qj3a 2al9[\ + r2a{2a~{) ] -K;
2a/e(l+rG)-3-2aa]/(l
c5 = Qj3a 2aieJ-2-2aa-r(3/4)2aa c6 = 2aie [JQ/3a-
- r ) ; c4 s c,£ a ; Ea;
3-2aa-r(2/3)*aa];
c7 = QP" (4/3)2oa 2(Xle~X)-2aie (2/3) 4aa ; c8=
[e^(l+^)-3-2aa]2a/e/(l-r);
jffsjjfc; /? 1 ^[2 1 - 2 a " 1 / e ]/e 1 / a ; fl2 = [(1 - r)(3/4) 2aa -2" 2 a a ] x,a IQxla 2Va [(3/2) 2aa 2~a- 1 - r2~ a F| 1 / a ; /?3 = 21/9+2a [1 + r2aF\ ~x,a Q~x,a; P4 = [K/QJ]x/a 2~xle; fis = 2x~2a-(Xla) Q-X/0E; J36 = [1 +r(3/2) 2 a a Ea]x/a (Qjyi/a
2-(xm~2a;
p1 = {[(1 -r)(3/2) 2 a o Ea- l]/(3 2aa 2- a -r2 4 a a - a - 1 )} 1 / a ; PB =
{2/Q)x/a3-2a;
P9 s [3" 2aa + r(2/3) 4 a a ] 1 / a (gJ)- 1 / a ; fi0 = 3~2a [Q{\ + rG)] ~x'a ; A, - {[(1 - r)(2/3) 4 "" ]/g(/?(l - r) - 1 - rG}x/a ; r, s 1 -(2/3) 2 a a ; r2 = (3/4) 2ao -2 a ( 1 " 2 a ) ; r3 = 1 -(2/3) 2 a a £ a ; r4 = (3/4) 2aa -2a{X~2a); r5=l -(3/4) 2 a a ; r6 = (7? - l)/(i? + G); g = a" (1 - a ) 1 - * ; K= 2~2aa + r(3/4)2aa ; F= 22aa -(3/2) 2 a a ;
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J. Borland, X. Yang
Appendix C: A Proof of Proposition 2 This appendix establishes Proposition 2 which concerns the implication of assumption Al for dynamic decisions. In this model only accumulated value of lit turns up. Hence, choice of values of all decision variables except ljt are time independent. Moreover, the value of lu is variable only for configurations A, A', and B. Also, the specific specification of functional functions make decisions on lit for configurations A, A', and B time independent if a configuration sequence does not involve shifts between configurations over time. Hence, we are concerned with only configuration sequences that involve shifts between A, A', B, and other configurations. Due to Bellman's backward decision rule, if A, A', or B is chosen in the terminal period, then the decision on lu is still time independent in this period, so that full optimization solutions in the absence of assumption Al for all configuration sequences that do not involve A, A', or B in period 1 are indifferent from one solved in section 3. That is, we are concerned with only configuration sequences AB, AC, AD, BA, BC, and BD, when we examine the implication of assumption Al for dynamic decisions. Suppose that assumption Al is not imposed. Assume further that a person chooses A in period 1 and (z/x) in period 2. According to the Bellman backward decision rule, the decision and equilibrium in period 2 is given by (10), dependent on human capital which is determined by time allocation between production of x and z in period 1. Total discounted utility for sequence AC is U(AC) = w, (A) + ru2 (C) = z, + r[aa (1 - a) 1 "" (sky (Lz2L'x2 )aa - c]
= [ 4 ^ 4 , ( 1 - W +r{a«(l-ay-« W K 4 , +lzl)L'x2)"° -c]} = Uzld-U]aa + r{a"(l-ay-«(skn(l + lzi)L'x2y° -c]} (A5) where we have used (1), (2), (3), the initial condition Ln = 1, the definition of Lj2 = Ln + ln , the definition of (z/x) that requires lzt = 1 and lxt - 0, and the endowment constraint lxt +lzt=\. L'x2 is human capital of another person who chooses (x/z) in period 2. Hence L'x2is independent of the decision of the person choosing (z/x) in period 2. It is easy to see that the optimum decision based on assumption Al is lxl - lzl - .5 and the optimum decision in the absence of assumption A 1 is / , >.5 because
Emergence and Evolution of the Firm
8U(AC)/dl2l
> 0 if /zl = .5
323
(A6)
(A6) implies that the optimum value of U(AC) is greater when assumption Al is not imposed than when it is. Similarly, we can prove that the maximum values of U(AB), U(AD), U(BA), U(BC), and U(BD) are greater, relative to U for other sequences, when assumption Al is not imposed than when it is. Hence all sequences that involve the evolution of division of labor and/or an increase in the variety of producer goods are more likely to be the dynamic equilibrium if assumption Al is not imposed. This leads us to Proposition 2.
References Arrow KJ (1962) The Economic Implications of Learning-by-Doing. Rev Econ Studies 29: 155-173. Barro RJ (1991) Economic Growth in a Cross-Section of Countries. Quarterly J Econ 106: 407-^44. Barzel Y, Yu BT (1984) The Effect of the Utilization Rate on the Division of Labor. Econ Inquiry 22:18-27. Baumgardner JR (1989) The Division of Labor, Local Markets, and Worker Organization. J Political Econ 96: 509-527. Becker G, Murphy K (1992) The Division of Labor, Coordination Costs, and Knowledge. Quarterly J Econ 107: 1137-60. Chandler A (1977) The Visible Hand: The Managerial Revolution in American Business. Harvard University Press, Cambridge. Cheung S (1983) The Contractual Nature of the Firm. J Law Econ 26: 1-21. Coase R (1937) The Nature of the Firm. Economica 4: 386^105. Edwards BK, Starr RM (1987) A Note on Indivisibilities, Specialization and Economies of Scale. Am Econ Rev 77: 192-194.
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Grossman S, Hart O (1986) The Costs and Benefits of Ownership. A Theory of Vertical and Lateral Integration. J Political Econ 94: 691-719. Grossman GM, Helpman E (1989a) Product Development and International Trade. J Political Econ 97: 1261-1283. Grossman GM, Helpman E (1989b) Growth and Welfare in a Small Open Economy. NBER Working Paper #2970. Grossman GM, Helpman E (1990) Comparative Advantage and Long-Run Growth. Am Econ Rev 80: 796-815. Grossman GM, Helpman E (1991) Quality Ladders and Product Cycles. Quarterly J Econ 106: 557-586. Kim S (1989) Labor Specialization and the Extent of the Market. J Political Econ 97: 692-705. Klein B, Crawford R, Alchian A (1978) Vertical Integration, Appropriable Rents, and the Competitive Contracting Process. J Law Econ 78: 293-326. Milgrom P, Roberts J (1987) Bargaining, Influence Costs, and the Organization of Economic Activity. Working Paper #8731. Department of Economics, University of California, Berkeley. Murphy KM, Schleifer A, Vishny RW (1991) The Allocation of Talent: Implications for Growth. Quarterly J Econ 106: 503-530. Rivera-Batiz, LA, Romer PM (1991) Economic Integration and Endogenous Growth. Quarterly J Econ 106: 531-556. Romer PM (1986) Increasing Returns, Specialization, and External Economies: Growth as Described by Allyn Young. Working Paper #64, Center for Economic Research, University of Rochester. Romer PM (1989) Increasing Returns and New Developments in the Theory of Growth. NBER Working Paper #3098. Rosen S (1978) Substitution and the Division of Labor. Economica 45: 235-250. Rosen S (1983) Specialization and Human Capital. J Labor Econ 1: 43^49. Sala-i-Martin X (1990a) Lecture Notes on Economic Growth: Introduction to the Literature and Neoclassical Models. Discussion Paper #621, Economic Growth Center, Yale University.
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Sala-i-Martin X (1990b) Lecture Notes on Economic Growth: Five Prototype Models of Endogenous Growth. Discussion Paper #622, Economic Growth Center, Yale University. Segerstrom PS, Anant TCA, Dinopoulos E (1990) A Schumpeterian Model of the Product Life Cycle. Am Econ Rev 80: 1077-1091. Stigler GJ (1951) The Division of Labor is Limited by the Extent of the Market. J Political Econ 59: 185-193. Stokey NL (1991) Human Capital, Product Quality, and Growth. Quarterly J Econ 106: 587-616. Williamson O (1979) Transactions Cost Economics: The Governance of Contractual Relations. J Law Econ 22: 233-261. Williamson O (1985) The Economic Institution of Capitalism. The Free Press, New York. Yang X (1990) Development, Structural Changes, and Urbanization. J Development Econ 34: 199-222. Yang X, Borland J (1991a) A Microeconomic Mechanism for Economic Growth. J Political Econ 99: 460-482. Yang X, Borland J (1991b) The Evolution of Trade and Economic Growth. Mimeo, University of Melbourne. Yang X, Ng Y-K (1995) Theory of the Firm and Structure of Residual Rights. J Econ Behavior & Organization 26: 107-128. Young Allyn (1928) Increasing Returns and Economic Progress. Econ J 38: 527-542. Young Alwyn (1991a) Learnins-by-Doing and the effects of International Trade. Quarterly J Econ 106: 369-406. Young Alwyn (1991b) Invention and Bounded Learning-by-Doing. NBER Working Paper #3712.
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Part 6
Endogenous Transaction Costs and Property Rights
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CHAPTER 13 ENDOGENOUS SPECIALISATION AND ENDOGENOUS PRINCIPAL-AGENT RELATIONSHIP*
Xiaokai Yang a and Yeong-Nan Yeh b * "Monash and Harvard University
Academia
Sinica
1. Introduction The purpose of the paper is to endogenise the principal-agent relationship and transaction costs in a general equilibrium model with consumerproducers, economies of specialisation, and transaction costs. Two phenomena that cannot be explained by existing models of principal-agent motivate this research. From our daily experience, we can see an interesting phenomenon that when an individual works for himself in the absence of moral hazard, he may be not as diligent as working for the other in the market with moral hazard. Why, under a certain condition, does an individual choose a higher effort level when moral hazard is present than when moral hazard is absent? If specialisation and transaction costs are simultaneously endogenised in a general equilibrium model, we can then predict this phenomenon as follows. In the present paper we call a departure of equilibrium from the Pareto optimum endogenous transaction cost and refer to exogenous transaction costs as a kind of
Reprinted from Australian Economic Papers, 41 (1), Xiaokai Yang and Yeong-Nan Yeh, "Endogenous Specialisation and Endogenous Principal-Agent Relationship," 15-36, 2002, with permission from Blackwell. * Thanks to Ken Arrow, Paul Milgrom, Yingyi Qian, the anonymous referee, and the participants of the seminars at Stanford University, Hong Kong University of Sciences and Technology, and Monash University for comments and criticisms. We are solely responsible for the remaining errors.
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X. Yang, Y.-N. Yeh
transaction costs that can be identified before individuals have made decisions. Endogenous transaction cost cannot be identified before individuals have made decisions and the economy has settled down in equilibrium. The trade offs between positive network effect of division of labour, endogenous transaction costs caused by moral hazard, and exogenous transaction costs imply that interactions between two types of transaction costs are crucial determinants of the equilibrium level of division of labour. If an exogenous transaction cost coefficient for each unit of goods traded is very large, the transaction cost outweighs productivity gains from the division of labour. Hence, individuals choose autarky where there is no transaction and related exogenous and endogenous transaction costs. Because of each person's limited time, each person has a narrow scope for trading off positive contribution of effort in reducing low productivity risk against leisure in autarky. Thus, the efficient trade off between the positive contribution and the disutility of the effort may end up with a low effort level in autarky. As the transaction cost coefficient falls, individuals will choose a higher level of division of labour, which implies a higher aggregate productivity, so that the scope for trading off the positive contribution against disutility of risk avoiding effort is enlarged by the higher level of specialisation. This is because each individual has more time for both leisure and working in the activity that he chooses when he increases his level of specialisation (reducing the number of production activities that he undertakes). Also, the decrease in the exogenous transaction cost coefficient enlarges the scope for trading off economies of specialisation against moral hazard. Hence, he may choose a higher level of effort in reducing risk when he chooses to work for others in a larger network of division of labour where endogenous transaction costs are present due to moral hazard. In other words, individuals can afford a higher endogenous transaction cost when aggregate productivity increases as a result of a larger network of division of labour. We may refer to the phenomenon as that a man works harder for others in the presence of moral hazard than working for himself in the absence of moral hazard. The second phenomenon that motivates the paper relates to the first phenomenon. We can see that in a large and very commercialized city (or with a very sophisticated network of division of labour), a greater
Endogenous Specialisation and Endogenous Principal-Agent
331
interdependence between individuals and a larger number of transactions create more scope for endogenous transaction costs caused by moral hazard and adverse selection than in a remote countryside village in a poor country where the number of transactions is small and no much scope for moral hazard to take place. As a general equilibrium phenomenon, we may observe that residents in that large city are more opportunistic than village folks. But yet people in the city are Pareto better off than those in the poor country. In other words, an equilibrium with moral hazard, which is not Pareto optimal, may be Pareto superior to an equilibrium with no moral hazard under different exogenous transaction conditions. This phenomenon may be referred to as Pareto improvement associated with an increase in moral hazard. This interesting phenomenon can be predicted by our general equilibrium model with endogenous level of division of labour and the trade offs between endogenous comparative advantage and endogenous and exogenous transaction costs. If the exogenous transaction cost coefficient for each unit of good traded is large, then exogenous transaction cost outweighs endogenous comparative advantage, so that the Pareto optimum as well as the equilibrium is autarky where there is no transactions and moral hazard. As the exogenous transaction cost coefficient falls the scope for trading off endogenous comparative advantage against transaction costs is enlarged, so that the Pareto optimum shifts to a higher level of division of labour which involves more interactions and transactions between individuals. The transactions must involve endogenous transaction cost caused by moral hazard, so that the equilibrium is not Pareto optimal. But as long as endogenous comparative advantage that can be exploited is sufficiently large compared to endogenous and exogenous transaction cost such that individuals can afford the moral hazard caused by the division of labour, the division of labour with more moral hazard occurs at general equilibrium and is Pareto superior to autarky. Hence, productivity, welfare, and moral hazard are simultaneously increased by the fall of the exogenous transaction cost coefficient.
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X. Yang, Y.-N. Yeh
hi the existing literature of moral hazard, the principal-agent relationship is exogenously given.1 The principal cannot take care of his own business and he has to ask an agent to do the job, while the agent cannot have his own business and he has to work for others. Since individuals' levels of specialisation and related principal-agent relationships are not endogenised in this literature, the implications of the endogenous transaction costs caused by moral hazard for the equilibrium level of division of labour are not explored.2 Hence, the above two phenomena cannot be predicted by the existing models of principal-agent. The literature of endogenous specialisation has endogenised individuals' levels of specialisation and the level of division of labour for society as a whole on the basis of the trade off between economies of specialisation and exogenous transaction costs. But it has not endogenized transaction costs.3 Hence, this literature cannot explore the implications of endogenous transaction costs for the level of division of labour and related number of principal-agent relationships either. The current paper introduces moral hazard into the general equilibrium model of endogenous specialisation and develops a general equilibrium principal-agent model with endogenous specialisation to explore the
1 The surveys of this literature can be found from Hart and Holmstrom (1987), Arrow (1985), Holmstrom and Roberts (1998), Bolton and Scharfstein (1998), and Gibbons (1998). The tradeoff between exogenous monitoring cost and endogenous transaction costs caused by moral hazard is considered by Holmstrom and Milgrom (1991) and Cowen and Glazer (1996). Implications of residual rights in the model with two sided moral hazard is considered by Gupta and Romano (1998) and Hart (1995), the tradeoff between insurance and exogenous and endogenous transaction costs is considered by Milgrom and Roberts (1992) and others. Unemployment insurance in the principal-agent relationship is considered by Hopenhayn and Nicolini (1997). Laffont and Tirole (1986) and Lewis and Sappington (1991) develop models with hidden information (rather than hidden actions) to endogenise principal-agent relationship by formulating the trade off between exogenous comparative advantage and endogenous transaction costs caused by information asymmetry. Moral hazard in general equilibrium model is considered in Helpman and Laffont (1975), Kihlstrom and Laffont (1979), and Legros and Newman (1996). 2 One exception is Legros and Newman (1996) which allows players to choose between agent and principal, though it does not endogenise emergence of the principal-agent relationships. 3 Surveys of this literature can be found from Yang and Ng (1998) and Yang (2001).
Endogenous Specialisation and Endogenous Principal-Agent
333
implications of endogenous transaction costs caused by moral hazard for the equilibrium level of division of labour. The concept of division of labour is in essence a general equilibrium concept. Hence, an individual's level of specialisation and level of division of labour can be endogenised only in a general equilibrium framework. The equilibrium level of division of labour is dependent on endogenous transaction costs (for instance an individual will not choose specialisation if it involves too high endogenous transaction cost), while the endogenous transaction costs are determined by the level of division of labour (for instance endogenous transaction cost does not occur if all individuals choose autarky where no transaction takes place). Hence, a general equilibrium mechanism that simultaneously determines the interdependent variables should be used to explore the implication of endogenous transaction costs for the equilibrium network size of division of labour. But in the existing literature, endogenous transaction costs are investigated for a given pattern of division of labour between the principal and the agent. As shown in the literature of endogenous specialisation, the extent of the market, trade dependence, the degree of market integration, production concentration, the degree of diversification of economic structure, the number of markets, the degree of endogenous comparative advantage, the number of traded goods, the degree of interpersonal dependencies, productivity, and individuals' levels of specialisation are associated with the level of division of labour which is determined by the efficient trade off between economies of specialisation and transaction costs. Also, it is shown that emergence of money, business cycles, and unemployment can be explained by evolution in the level of division of labour. In a general equilibrium model with endogenous comparative advantage and exogenous and endogenous transaction costs, complicated trade offs among the three counteracting forces may generate much richer stories than told by the existing literature of principal-agent and by the existing literature of endogenous specialisation. Yang (1994) has drawn the distinction between endogenous and exogenous comparative advantage. The endogenous comparative advantage is positive network effect of division of labour that can be exploited only if different individuals, who might be ex ante identical in
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all aspects, choose different levels of specialisation in producing different goods. Endogenous comparative advantage is much more important than exogenous comparative advantage since it is determined by individuals' decisions and level of division of labour. A general equilibrium model that can predict endogenous emergence of principalagent relationship from evolution in division of labour can then be used to explore the implications of endogenous transaction costs for exploitation of endogenous comparative advantage. In most existing principal-agent models, exogenous comparative advantage which is based on ex ante differences between individuals is the driving force of the principal-agent relationship. The principal is ex ante different from the agent before decisions have been made. Hence, the existing literature has not explored the implications of endogenous transaction costs for exploitation of endogenous comparative advantage. Endogenous transaction costs will affect in important ways the equilibrium pattern and level of division of labour which generate some concurrent phenomena (evolution in endogenous comparative advantage, in the degree of market integration, in the degree of diversity of economic structure, and so on) which cannot be predicted by standard marginal analysis of principal-agent models. Hence, the implications of endogenous transaction costs and contingent pricing for productivity progress and for all of the above economic phenomena can be explored using our general equilibrium model of principal-agent. In our general equilibrium model, players' choices between contingent pricing and pure pricing is endogenised. Within a certain parameter subspace, individuals may prefer pure pricing to contingent pricing in general equilibrium. This feature of our general equilibrium model may be motivated by casual observations. We can see contingent pricing as well as pure pricing in real world under different conditions. The condition might not be as simple as the presence or absence of moral hazard. Our model will be used to identify the dividing line between contingent and pure pricing when moral hazard is present. Section 2 specifies the model, section 3 solves for corner equilibria in six market structures, section 4 solves for the general equilibrium and its comparative statics. In section 5, the implications of endogenous
Endogenous Specialisation and Endogenous Principal-Agent
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transaction costs for the equilibrium level of division of labour are explored. The final section concludes the paper. 2. A General Equilibrium Model of Principal-Agent Relationship In our model there are two ex ante identical consumer-producers. Each of them can produce a final good y and an intermediate good x. Each person's utility function is specified as follows. U=ln[(y + kyd)s] (1) where y is self-provided quantity of the final good, yd is the quantity of the final good purchased from the market. A fraction 1 - k of a unit of goods purchased disappears in transit because of exogenous transaction costs. Hence, kyd is the quantity of the final good that is received from the purchase ofyd. (y + kyd) is thus the quantity of the final good that is consumed. U is his utility level and s is his level of leisure. This strictly concave utility function represents a preference with risk aversion. Each person is equipped with the same production functions for the final and intermediate goods. f = y+ys={x
+ kxd)LyA
•fr a *u J» j . » K with probability/^ if Lx = pH, then x = x + x =\ [6L with probability 1 — pH
(2) (3)
\&H with probability p. F 3 HL \fLx = /3L, then ^ = x + x*= \ H \9L with probability 1 - p L where y° and xf are respective output levels of the final and intermediate goods, ys and Xs are respective quantities of the two goods sold, x is selfprovided quantity of the intermediate good, xd is the quantity of the intermediate good purchased, Ly is the amount of labour allocated to produce good y. Lx is effort level in reducing risk for a low productivity If functional forms of the model are not explicitly specified, inframarginal comparative statics of general equilibrium cannot be solved because of complexity generated by many possible corner solutions. The result is not sensitive to the specification of functional forms as long as the trade-off between economies of specialisation and transaction costs exists. However, if Grossman's first order condition approach is adopted, result may be altered (see Yang, 2001, Ch. 9).
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of x, which can be either a higher level f3H or a lower of J5L. We call Lt an individual's level of specialisation in producing good i.5 fiH, PL, OH, 6L, pH, and pL are parameters. The production function (2) indicates that total factor productivity of y, yp /(x + kxd)V2Lxly2 , increases with an individual's level of specialisation in producing good y, Ly. This nature of production function is referred to as economies of specialisation. Here, the exponentially weighted average of two types of inputs, (x + kx ) Ly , is total factor employed to produce x. Since economies of specialisation are individual specific and will not extend beyond the size of each person's limited amount of working time, they are localised increasing returns. A person's level of specialisation in producing a particular good increases as his range of activities is narrowed down. Hence, it is different from scale of his labour, despite the connection between the scale of labour and level of specialisation. The distinction between economies of specialisation and economies of scale is discussed in more details in Yang (1994). Each individual is endowed with L units of time that can be allocated between working and leisure, so that the endowment constraint for each person is Lx + Ly + s=L,
(4)
where s is time allocated for leisure. This individual specific endowment constraint of time highlights the distinction between economies of specialisation and economies of scale. Finally, we assume Ly e [0, L], Lx = fiH or fiL, s > 0, x, xs, xd, y, y\ yd > 0, L>{5H>f3L > 0 , 0H>0L>O, \>PH >PL>0. (5) Each individual's self-interested behaviour is represented by a non-linear programming problem that maximises his expected utility with respect to Lh s, x, xs, xd, y, y\ yd, subject to constraints (2)-(5).6 When the decision variables x, Xs, xd, y, ys, yd take on 0 and positive values, there are 26 = 64 combinations of 0 and positive values of the six variables. If two values of Lx are considered too, there are 128 possible corner and interior solutions for each person's non-linear programming problem. 5
Lx represents a person's level of specialisation as well as his effort level in avoiding risk in producing x. 6 Note that due to the budget constraint and market clearing condition, one of the decision variables is not independent of the others.
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There are three periods in the model. Two players sign a contract on terms of trade in period 1. In period 2 they choose risk avoiding effort levels to maximise expected utility for given contractual terms. In period 3, nature chooses a realised state and contractual terms are implemented. We assume that terms of trade are determined by a Nash bargaining game in period 1 when two players have free entry into the production of each good. Since two players are ex ante identical, it is easy to show that two players' expected utilities will be equalised in the Nash bargaining equilibrium. However, two players cannot change occupation in periods 2 and 3 after the contract is signed. The combination of moral hazard and endogenous specialisation generates a formidably large number of corner solutions. We have to reduce the number of corner solutions that we must consider to keep the model tractable. Wen (1998) has shown that in this kind of models, the interior solution is never optimal and that a person never simultaneously buys and sells the same good, never simultaneously buys and self-provides the same goods, never sells more than one goods, and never self-provides the intermediate good if he sells that good. Following this theorem, we can rule out the interior solution and most corner solutions from consideration. A profile of 0 and positive values of decision variables that is consistent with the theorem is referred to as a configuration. Having considered possibilities for pure and contingent pricing for goods, we can identify 10 configurations that need to be considered. There is a corner solution for each configuration. A profile of matching configurations that is compatible with market clearing conditions is referred to as a market structure or simply a structure. There are 6 structures that need to be considered. The market clearing conditions, the incentive compatibility constraint, and utility equalisation conditions determine a corner equilibrium for each structure. The general equilibrium in the Nash bargaining game is a corner equilibrium in which nobody has incentive to deviate from his chosen configuration. We will adopt a two step approach to solving for the general equilibrium.
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First, all corner equilibria in the six structures are solved. Then the general equilibrium and its comparative statics will be identified.7 3. Corner Equilibria in Six Structures In this section, we specify, for each structure, a Nash bargaining game where two consumer-producers sort out terms of trade, sign contract, and then decide on their quantities of goods to consume, to produce, and to trade in period 1. They implement the contract after a state of xf is realised in period 2. In this regime, each person can observe outputs of the other person, but cannot see his effort level Lt. The trade off between leisure and income from working is assumed, which together with nonverifiability of effort generates moral hazard. In the Nash bargaining game, each player, taking the relative price as given, maximises expected utility with respect to quantities of goods produced, traded, and consumed. The optimum decisions generate indirect utility functions for different occupation configurations. Then the Nash bargaining game will maximise the Nash product of the difference between two players' indirect utility functions and their utilities in autarky. Since two players are ex ante identical with the same autarky utility, the maximisation of the Nash product establishes utility equalisation between the two occupation configurations. Hence, the market clearing condition, utility equalisation, and incentive compatibility conditions determines a corner equilibrium for each structure. The Nash bargaining game will choose the structure with the highest expected utility. We first identify all structures that need to be considered. Then all corner equilibria in the structures will be solved. (A) There are two autarchic structures AH and AL, where there is no market and principal-agent relationship, and each individual selfprovides all goods he consumes. Structure AH is composed of two individuals choosing configuration AH, which implies a profile of decision variables with Lx = /?» Ly; x, y, s > 0; Xs = xd = ys = yd = 0. Structure AL is composed of two individuals choosing configuration AL, 7
This inframarginal approach is used by Dixit (1987, 1989a, b), Grossman and Hart (1986), and Hart and Moore (1990) too.
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which implies a profile of decision variables with Lx = f5L\ Ly, x,y,s> 0; xs = xd = ys = yd = 0. The decision problem for a person choosing configuration AH is Max: EU= lay + Ins (utility function) s.t. y = xLy
(production function of y)
\6„ with probability pH Lx = f3H, x = \ (production function of x) (6) [9L with probability 1 - p H Lx + Ly + s = L
(endowment constraint)
where E denotes expectation, Lh x, y, s are decision variables. The optimum solution for this problem and maximum expected utility for this configuration are listed in Table 1. The decision problem for configuration AL can be obtained by replacing Lx = j3H in problem (6) with Lx = ftL and by replacing pH with pL. Its optimum solution and expected maximum utility are listed in Table 1. (B) There are two market structures with the division of labour and unique relative price of the two goods, DL and DH Structure DL is comprised of a division of two individuals between configurations (x/ y)uL and (y I x)"L . Configuration (x/ y)"L implies that Lx = j3L;, s, x, xs, yd > 0; Ly = xd = y = ys = 0 and an individual choosing this configuration sells x, buys y, and accepts only a single relative price of the two goods. Configuration (y I x)"L implies that Ly, s, y, ys, xd > 0, Lx = yd = x = Xs = 0 and an individual choosing this configuration sells y, buys x, and accepts only a single relative price of the two goods.8 Structure DH consists of a division of two individuals between configurations (x/ y)"H and (y/x)uH . Configuration {xl y)"H is the same as (x I y)"L except L = J5H. Configuration (y I x)"H is the same as (y I' x)"L in structure DL except that value of xd will equal value of xs that is given by Lx = J3H instead of by Lx = f5L?
If uncertainties are specified for the production of good y and related configuration (y/x), then two sided moral hazard can be used to extend the Grossman-Hart-Moore model, as show in Yang (2000). 9 As Hart (1995) indicated in a seminar, ownership of property does not make difference in the principal-agent model. The assumption that the specialist of x or the specialist of;y
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X. Yang, Y.-N. Yeh
Let us consider structure DH first. In this structure the decision problem for a person choosing configuration (x I y)"H is Max:Et/,= ln(ik)^) + ln5 s.t. yd = pxs
(budget constraint)
\0H with probability pH Lx = PH, x =< (production function of x) (7) \0L with probability 1 - p H Lx + s = L
(endowment constraint)
s
where Lx, X , s are decision variables and p = pjpy is the price of good x in terms of good y which is determined by Nash bargaining. Because of the market clearing condition yd = y\ he takes ys as given. The optimum decision for this configuration and expected indirect utility function are listed in Table 1. The decision problem for configuration (y/x)u is Max : Et/„ = In y + In s y
s
s
s.t. y = px
d
(budget constraint)
s
y + y = (kx )Ly (production function of y) L +s = L (endowment constraint) where Ly and s are decision variables and p = pjpy is the price of good x in terms of good y. Because of the market clearing conditions xd = xs and the budget constraints for two configurations: pxd = / and yd = pxs, xd can be replaced with xs, andys is not independent of xs. Hence, the person choosing this configuration takes Xs, which is chosen by a person in (x/y), and ys = pxd = pxs as given, where p is determined by Nash bargaining which equalises utility between the two configurations. The optimum decision for this configuration and expected indirect utility function are listed in Table 1. The utility equalisation condition gives the corner equilibrium solution of relative pricey. Plugging the corner equilibrium relative price
owns x and/or y will not alter our results. For the implication of ownership when contracts are incomplete, see Grossman and Hart (1986) and Hart and Moore (1990).
Endogenous Specialisation
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and Endogenous Principal-Agent
341
Table 1: Comer Solutions in 10 Configurations Demand Expected indirect utility Self-provided quantities and supply x = 0, y = 0.5(L --PL)X pL\n6H+(\-pL)\n9L + 2[ln(L-/?i)-ln2] s = 0.5(L-/3L)
AH
x = 0, y = 0.5(L --PH)X, s = 0.5(L-pH)
(x/y)Lu
J=£-A
x°=0 yd=pxs
\np + pL\n0H+{\ - pL)\n0L + ln(L - pL) + \nk
iylx)Lu
s = {kL-p)l2k y = (kL-p)xd/2
xd=xs ys=pxs
21n(A£ -p) + pL\nOH + (l-p i )ln<9 i -21n2-lnfc
(x/yh"
s = L-pH
x°=0 yd=pxs
\np + pHln0H + (1 - pH)\n0L + ln(L-pH) + \nk
iyix)H"
s = (kL+p)/2k y = (kL+p)xd/2
xd=xs ys=px°
+ (l-p. ff )ln<9 I -21n2-lnA:
s = L-pL
xs=0 yd=0p
p,\n(pHdH) + (\-pL)\n(pLeL) + ln(L-pL) + lnk
xd=xs
pL\r\[(kly-pH)dH] + (l-pL)ln[(kly-pL)0L]
(*/y)L
(y/x)L'
s = (kL-p)/2k, y = (kL-p)/2
(x/yh'
s = L-pH
(ylx)„l
s = {kL-p)l2k, y = (kL-p)/2
pH\n&H+ (1 - pH)ln0L + 2[ln(I-^)-ln2]
f=p0 xf=e yd=6p xd=xs
f=p
2\n(kL -p) + PH^QH
pH\n{pH6H)+ (1 -pH)\n(pL9L) + \n(L-pH) + \nk pHln[(kly-pH)0H] + (\-pH)\n[(kly-pL)0L]
into utility function yields the comer equilibrium expected real income in stmcture DH. All of the information on this comer equilibrium is in Table 2. Replacing Lx = fiH in stmcture DH with Lx = /3L and following the procedure for solving for the comer equilibrium in DH, the comer equilibrium in stmcture DL can be solved. The comer equilibrium relative price and expected real income in DL are listed in Table 2.
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Structure AL
X. Yang, Y.-N. Yeh Table 2: Corner Equilibria in 6 Structures Corner equilibrium relative price Expected real income pLln&H+(1 - pL)ln0L + 2[ln(L- y tf i )-ln2]
AH
pH\ndH+{\ - pH)\ndL + 2[\n(L-pH)-\n2]
DL
p L is given by (10)
\apL + pLln0H+ (1 + \n(L - /3L) + \nk
DH
pHis given by (11)
\npH + pH\n 0H + (1 - p ^ l n 0L + ln(L-/3H) + \nk
-pL)\ndL
CL
p " and p"H are given by (9) and (12)
Piln( pLH 0H) + (1 - pL)ln( p[ 0L) + ln(L - pL) + Ink
CH
p " and p " are given by (9) and (13)
pH\n( p " 9H) + (1 - pH)\n{ p " 6L) + \n(L-pH) + \nk
(C) There are two market structures with the division of labour and with two relative contingent prices, CL and CH. Structure CL consists of a division of two individuals between configurations (x I y)'L and (y I x)'L . Configuration (x I y)'L is the same as (x I y)"L except that an individual choosing this configuration accepts relative price pn when output level of x is high, or is 6H, and accepts relative p r i c e y if output level of x is 9L. Configuration (y/x)'L is the same as {ylxfL except that an individual choosing this configuration accepts relative price pu when output level of x is 6H, and accepts relative price pi if output level of x is 0L. Structure CH is comprised of a division of two individuals between configurations (x/y)'H and (y I x)'H . Configuration (x/y)'H is the same as (x/y)'L except that Lx = fiL is replaced with Lx = f3H. Configuration (y I x)'H is the same as in structure CL except that value of xd will equal value of x" that is given by Lx = fiH instead of by Lx = J3L. The trade off between sharing risk and providing incentive is standard. Although the two players are ex ante identical, their occupations involve different ex post risk since configuration (x/y) involves uncertainty in production, while (y/x) does not. Hence, the specialist producer of y has a trade off between providing
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the specialist producer of x with strong incentive and sharing risk in producing x with him via contingent terms of trade. The procedure for solving for the corner equilibria in the two structures is the same as that for structures DL and DH except that the incentive compatibility condition is used, together with utility equalisation condition, to determine two corner equilibrium relative prices. Assume that the relative price isp'H when output level of x is 6H and that it is p'L when the output level is 6L, in structure C„ where i = L, H. Let expected utility of the specialist choosing (x/ y)'H , given in Table 1, equal his expected utility for (xl y)'L , the incentive compatibility condition for Ct can be derived as follows. l n l ^ - = \ntj. P'„
(9)
where 77 is defined by (pH - pL) In rj = (pH - pL )ln ^- - In ^±-, and P'„ >Pi i f f ? < l . All information on the six corner equilibria is summarised in Table 2. ^ = ^ { [ L + 2*(L-^)]-2>/*Z-(i-/?i) + ^ ( i - A ) 2 }
(10)
p« =k{[L
(11)
A In
2k{L-PH)]-24kL(L-(iH)
+
P « L kl y-pLH
kpL
|
k(l-pL)
klLy-pLH pH\n
Wy-pl
P H " kl«-p«
kpH U?-Pg
+(1-A)ln =
kl»-p»
V(L-PHy}
P
PL) " +\nKL =0, U$-pZ L-lj;
1
(12)
L-lj;
+ ( l - pyH> g)ln
+k(l-pH)_
+
P " kl»-p»
1 L-l"
+\nk(L'^H)=0, L-l»
(13)
where p'H is the price of good x in terms of good y when output level of x is 6H and p'L is the relative price when output level of x is 9L in structure Ct. pH is the price of good x in terms of good y in structure DH and pL is that in structure DL. In (12) or (13), ll =0 + y[y ,where and y are
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X. Yang, Y.-N. Yeh
dependent onp' H ,p' L ,p,L,k . We take ly = (/> + y[y to be the solution of V since kly - p'j > 0 can be used to rule out the other solution. The corner equilibrium in structure DH does not exist if the following condition holds. 7]<\. (14) where rj is defined in (9). This condition implies that for a unique relative price of the two goods a specialist of x will choose Lx = f3L, which is incompatible with the definition of structure DH, if his expected utility is higher for Lx = fiL than that for Lx = J3H. This is referred to as the problem of moral hazard. Similarly, the corner equilibrium in structure DL does not exist if TJ > 1. Hence, (14) gives ranges of parameter values that define the dividing line between the two corner equilibria. (14) implies that if benefit of effort in reducing risk of low productivity is outweighed by disutility of such effort, then the specialist of x will choose the low effort level under unique relative price of the two goods. 4. General Equilibrium In this section we first define general equilibrium, then analyse comparative statics and welfare implications of the general equilibrium. General equilibrium is defined as a set of contingent or pure prices of goods and two players' labour allocations and trade plans that satisfy the following conditions, (i) Each individual's consumption plan generated by his labour allocation and trade plan maximises his expected utility for a given set of relative prices of traded goods; (ii) The set of relative prices of traded goods maximizes the Nash product of the two players' differences of utilities between a chosen structure and autarky and clear the markets for traded goods subject to the incentive compatibility constraint. It is trivial to show that the Nash bargaining equilibrium is the corner equilibrium with the highest expected utility provided the incentive compatibility constraint is met. Therefore, solving for the general equilibrium becomes identifying the corner equilibrium with the highest expected real income subject to the incentive compatibility
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and Endogenous Principal-Agent
345
condition.10 Comparisons of expected real incomes in 6 corner equilibria, given in Table 2, and expressions (9)-(14) yield the following theorem. Theorem 1: (l)For TJ< ?70<1 (la) The general equilibrium is AL if /3L < j , or if/?x >j and IX
***. / C i j
y
(lb) The general equilibrium is DL if J3L > j and k>kXL. where kXL is given by 4k2 Vjbf + 2k- 2Ak-^
+ k2 J = 1 (see
Lemma 7 in Appendix 1 for proof) and TJ0 is given by EU(CH) EU(DL) = 0. (2) For TJ&(T]0, 1)
(2a) The general equilibrium is AL if k < k2 and rj > TJ{ , or if 77 < 7^; (2b) The general equilibrium is C# if k > k2 and rj>?]l; where &2 is given by (9), (13), mdEU(CH) = EU(AL) , rjx is given by EU{CH) - EU(AL) = 0 and k = 1. (3) For 7 e ( l , TJ2), where ^ is defined by In ?72 ^ ^ ^ l n ^ . (3a) The general equilibrium is ALifk< k3; (3b) The general equilibrium is DH if k > k3; where h is given by ln[4#(*, ^, /?«)] - In ( 7 ^ - ) ji!^)"^
=0
(see Lemma 8 in Appendix 1 for proof). (4) For t] > T]2
(4a) The general equilibrium is AH if fiH < j , or if k < klH and (4b) The general equilibrium is DH if k > klH and fiH > j . where kiH is given 4k2(j~
+ 2k-2Jkj~
+ k2 I = 1 (see (23) in
Lemma 7 in Appendix 1).
The existence of equilibrium is not a problem in the explicitly specified model with nonlinear pricing so long as (10)—(13) have solutions.
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X. Yang, Y.-N. Yeh
There are two types of comparative statics of general equilibrium of this model. Theorem 1 summarises the first type of them, referred to as inframarginal comparative statics. Inframarginal comparative statics imply that the general equilibrium will discontinuously jump across five of the six corner equilibria. The second type, referred to as marginal comparative statics, are conventional: endogenous variables continuously change in response to those changes in parameters that are within a parameter subspace demarked by inframarginal comparative statics. Theorem 1 has partitioned a 7 dimension parameter space into 8 subspaces and identifies which of 5 corner equilibria is general equilibrium within which parameter subspace. Solving for inframarginal comparative statics of general equilibrium is to solve for a system of simultaneous inequalities if the corner equilibrium expected per capita real income can be explicitly solved as functions of 7 parameters. But this is not easy for our model for three reasons. First, effects of 7 parametersL,/3L,f3H,pL,pH,6L,0H on many endogenous variables must be considered. Second, when we analyse discontinuous jumps of the general equilibrium, 5 corner equilibria need to be considered, each of them is equivalent to a conventional general equilibrium. Finally, the corner equilibria in structure C, (i = L, H) cannot be solved analytically, but it is extremely difficult to identify the parameter subspace within which a particular structure generates the highest per capita real income in the absence of analytical expressions of the per capita real incomes in 6 corner equilibria. Hence, we need to prove 9 lemmas before proving Theorem 1. Since the proof of the 9 lemmas and Theorem 1 is quite cumbersome, we put it in the Appendix. Table 3 summarises inframarginal comparative statics of general equilibrium, given by Theorem 1. The bottom line in this table gives the equilibrium structure for each particular parameter subspace. In Table 3, AL (or AH) represents autarky with a low (or a high) effort level in avoiding low productivity risk, DL (or DH) represents the structure with the division of labor, pure price, and a low (or a high) risk reducing effort level, CH represents the structure with the division of labour, contingent prices, and a high risk reducing effort level. Recalling the definition of r/ in expression (9), we can see that 77 represents benefit of risk reducing effort compared to its disutility. Bearing these definitions in mind, we
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347
Table 3: General Equilibrium and its Comparative Statics 7?<
'
PL
or J3L>L/5
and k
•
ne (1 - %)
rje (%, 1)
%
J3L>L/5 and k>klL
DL
TJ
or n> r\x and k
V > and k>k2
CH
m
k
AL
k>k3
DH
n> % PH L/5 and k
PH>LI5
and k>km
DH
can see from Table 3 that the dividing line between autarky and division of labour is the exogenous transaction efficiency coefficient k. If the transaction efficiency coefficient is small, the general equilibrium is autarky where no market and principal-agent relationship exists and productivity is low. If it is large, a structure with the division of labour and related reciprocal principal-agent relationships occurs in general equilibrium. If 77 is very small (77 < rj0), or if the benefit of risk reducing effort is insignificant compared to its cost, this effort level is always low in equilibrium, irrespective of the level of division of labour. If 77 is very large (77 > TJ2), or if the benefit of risk reducing effort is great compared to its cost, this effort level is always high in equilibrium, irrespective of the level of division of labour. If 77 is neither very large nor very small (77 e (/70, 772), see columns 3-6 in Table 3), then it is possible that as the exogenous transaction efficiency coefficient (/c) increases from a low level to a high level, the general equilibrium jumps from autarky (column 3 or 5) where risk reducing effort level is low (structure AL) to the division of labour where this effort level is high (structure CH in column 4 or DH in column 6 of Table 3). Having compared cases (l)-(4) in Theorem 1 (or columns 1-8 in Table 3), it can be seen that the emergence of the principal-agent relationship will involve two contingent relative prices if benefits of effort in reducing risk are neither too great nor too small compared to its disutility, that is, if 77 is not too large nor too small. Otherwise, the emergence of the principal-agent relationship will involve a unique relative price of the two goods (D). An examination of (9) and Table 3 indicates that if structure CH is the general equilibrium, then 77 < 1 which
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X.Yang, Y.-N. Yeh
implies pH > pL . Since CL is never a general equilibrium structure, this result implies thatpH> pL when two contingent relative prices take place in general equilibrium. As we promised in the introductory section, our model can predict simultaneous increases in moral hazard, productivity, and equilibrium per capita real income within the parameter subspace 77 e (770, 1) which implies that productivity benefit of a high effort level in reducing risk is neither too great nor too small compared to its disutility. Within this parameter subspace, as transaction efficiency increases from a value smaller than k2 to a value larger than k2, the general equilibrium jumps from autarky with low effort level AL to the division of labour with high effort level and contingent pricing (structure CH). This story is quite intuitive. If transaction efficiency is low and benefit of working hard is not that significant compared to its disutility, then individuals will choose autarky with low effort level. As transaction efficiency is sufficiently improved, individuals will choose the division of labour. The higher level of division of labour enlarges the scope for trading off productivity benefit of high effort level against its disutility since each individual now produces only one good and has more time to afford leisure. Also, the increase in transaction efficiency enlarges the scope for trading off positive network effect of division of labour against moral hazard, so that moral hazard, per capita real income, and productivity increase side by side as division of labour evolves. This predicts the phenomenon that a man works harder for others than for himself as the degree of commercialisation and related moral hazard increase. The analysis of corner and general equilibrium can be summarised as follows. Each corner equilibrium sorts out resource allocation and contractual terms for a given level of division of labour and a given contractual regime. The general equilibrium sorts out the level of division of labour and contractual regime (pure price, Dh or contingent prices, C„ low or high level of effort in producing x) by efficiently trading off economies of specialisation against endogenous and exogenous transaction costs and by trading off benefits of a high level of effort against its disutility. The equilibrium level of division of labour is mainly determined by transaction efficiency compared to other parameters. The main determinant of contractual regime is parameter rj
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which represents benefits of a high level of effort compared to its disutility. Figure 1 gives an intuitive illustration of exogenous evolution of division of labour based on the inframarginal comparative statics of the general equilibrium and related emergence of the reciprocal principalagent relationships. Two individuals in panel (a) are used to illustrate autarky. Circles represent configurations. The lines represent flows of goods self-provided. In autarky there is no market demand and supply. Productivity is low because of a low level of specialisation. However, transaction costs do not exist because people do not do business with each other. Panel (b) represents the division of labour (structure D or C). The lines represent the flows of goods. For structure D or C, there are two markets, one for x and the other for y. There are two distinctive professional sectors. The per capita output of each good is higher than in autarky because of a higher level of specialisation for each person. But transaction costs are higher than in autarky too because each person has to undertake two transactions to obtain the necessary consumption. The transactions incur exogenous as well as endogenous transaction costs. Moreover, the degree of interdependence between individuals, the degree of production concentration of each good, the degree of diversity of economic structure, the degree of endogenous comparative advantage, and the degree of integration of society are higher in structure D or C than in autarky.
f
^-^—N
(x/y J
y/x ) x* (a) Structure A, autarky
Xs
(b) Structure D, division of labour
Figure 1: Evolution in Division of Labour and Emergence of Principal-Agent Relationships
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X. Yang, Y.-N. Yeh
The implications of moral hazard for the equilibrium level of division of labour will be more significant if the number of goods is more than 2. The numbers of configurations and corner equilibria will increase more than proportionally as the numbers of goods increases in the model. 5. Welfare Analysis and Endogenous Transaction Costs We define a departure of equilibrium from the Pareto optimum as endogenous transaction cost and define exogenous transaction costs as a kind of transaction costs that can be identified before individuals have made decisions. Endogenous transaction cost cannot be identified before individuals have made decisions and the economy has settled down in equilibrium. In our model the transaction cost for a unit of goods purchased, 1 - k, is exogenous transaction cost. Two features of the endogenous transaction costs in this model deserve special attention. A decrease in exogenous transaction cost 1 - k (or an increase in k) may increase endogenous transaction costs and promote productivity as well as division of labour at the same time. For instance, in case (2) of Theorem 1 (columns 3 and 4 of Table 3), if k< k3, then Lemma 8 in the Appendix indicates that structure AL is the Pareto optimum. As k increases, so that k e {k^, k2), the equilibrium is still AL, but Lemmas 2 and 8 in Appendix indicate that DH is the Pareto optimum. Hence, the endogenous transaction costs keep the equilibrium structure and productivity from the efficient ones. This implies that an increase in k creates endogenous transaction costs. As k increases further, so that k > k2. Then the equilibrium is CH and the Pareto optimum is DH. This increase in k generates endogenous transaction costs when it promotes division of labour and productivity, and increases each individual's utility. This predicts the phenomenon of simultaneous increases in moral hazard, productivity, and welfare. A member of a developed society with a greater degree of commercialisation (a higher level of division of labour) and more moral hazard is better off than in an autarchic less developed society with no much commercialisation and moral hazard. However, in case (2b), structure CH may save on endogenous transaction costs if it is compared to AL and DH. For this circumstance, the Pareto optimum that is associated with structure DH cannot occur in
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351
equilibrium due to moral hazard. If structure CH is not allowed, then the general equilibrium will be autarky with an even lower real income than in structure DH. Hence, in this circumstance, structure CH maximises economies of division of labour net of endogenous and exogenous transaction costs despite the existence of endogenous transaction costs in structure CH. This result substantiates the second part of the Coase theorem (Coase, 1960) that the contractual structure in the market will maximise benefit of trade net of endogenous and exogenous transaction costs. It implies that it is not efficient to reduce endogenous transaction costs, which are considered as distortions, as much as possible because of the trade offs among three conflicting forces: economies of division of labour, endogenous transaction costs, and exogenous transaction costs. When benefits of effort in reducing risk are great, compared to its disutility, the conflict between a decrease in endogenous transaction costs and a decrease in exogenous transaction costs no longer exists, so that there is no trade off between economies of division of labour and endogenous transaction costs despite the existence of the trade off between economies of division of labour and exogenous transaction costs. In this sense, the model in this paper endogenises moral hazard and contingent prices. For cases in columns 5-8, the general equilibrium is Pareto optimal and endogenous transaction costs are not incurred. The second of the features is that for the story in columns 3-6 of Table 3, a sufficient improvement in transaction efficiency generates two kinds of gains from trade. It creates a greater scope for efficiently trading off economies of division of labour against endogenous and exogenous transaction costs, on the one hand, and creates greater scope for efficiently trading off benefits of effort in reducing risk against disutility of such effort. Hence, not only the equilibrium level of division of labour increases, but also the equilibrium level of effort in reducing risk increases. Both of such increases improve productivity. As we claimed in the introductory section, the reciprocal principalagent relationships may endogenously emerge from evolution in division
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X. Yang, Y.-N. Yeh
of labour. In contrast, in the existing literature, the principal-agent relationship is exogenously given.11 6. Concluding Remarks This paper has shown that the most important function of the price system is to efficiently trade off economies of division of labour against exogenous and endogenous transaction costs to sort out the equilibrium level of division of labour, the equilibrium extent of the market, and the equilibrium contractual arrangements that maximise positive network effect of the division of labour net of transaction costs. It is not efficient to reduce distortions (endogenous transaction costs) as much as possible because of the trade offs among economies of division of labour, endogenous transaction costs, and exogenous transaction costs. An inframarginal analysis is developed in this paper to sort out the inframarginal comparative statics of the general equilibrium that involve discontinuous jumps of general equilibrium between market structures as parameters shift between parameter subspaces. Each individual uses total benefit-cost analysis to choose his level and pattern of specialisation and contractual regime in addition to marginal analysis for sorting out resource allocation and contractual terms for a given pattern of specialisation and a given contractual regime. This model not only can predict emergence of reciprocal principal-agent relationships from evolution in division of labour, but also can endogenise comparative advantage and individuals' choice between a pure relative price and contingent relative prices. We use complicated trade offs between endogenous comparative advantage and endogenous and exogenous transaction costs to predict two interesting phenomena: (i) a man works harder for others in the presence of moral hazard than working for 11 As Hart (1991) indicates, the principal-agent model in the present paper should not be considered as part of the formal theory of the firm since they have not endogenised asymmetric distribution of residual control rights and labour contract is not essential in the principal-agent model. The agent may claim part of residual returns from the contingent contract. The reciprocal principal-agent relationships in the model may exist in the absence of the institution of the firm, which is associated with asymmetric residual control and residual return and labour contract. Grossman and Hart (1986), Hart and Moore (1990), and Yang and Ng (1995) have developed formal models of institution of the firm.
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353
himself in the absence of moral hazard; (ii) simultaneous increases in welfare, productivity, and moral hazard. The two phenomena may occur if positive contribution of high effort level in reducing risk for low productivity is neither too great nor too small compared to its disutility and if transaction efficiency is improved. A promising extension of our model is to follow Hart (1995) and Gupta and Romano (1998) to introduce two sided moral hazard and incomplete contracts into the model. The extended model may be able to explain the emergence of the institution of the firm from evolution in division of labour and to explore the implication of ownership structure for the equilibrium level of division of labour and related endogenous transaction costs.
Appendix: Proof of Theorem 1 We first prove 9 lemmas that are essential for proving Theorem 1. Lemma 1: The corner equilibrium in DH cannot be a general equilibrium if In 77 > 0; the corner equilibrium in structure DL cannot be a general equilibrium if In 7 > 0. Proof: The moral hazard indicated in (14) implies that for In 77 > 0, specialists of x have an incentive to choose Lx = /3L, so that the corner equilibrium in DH cannot be a general equilibrium. Similarly, for In 77 > 0, structure DL cannot be a general equilibrium. Lemma 2: In the absence of moral hazard, the division of labour with a unique relative price of the two goods (structure DH or DL) generates a greater expected real income than the corresponding structure with two contingent relative prices (structure CH or CL). Proof: From Table 2, it can be shown that EC/(Q < EU(D,) iff
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X. Yang, Y.-N. Yeh
Ri=pH]npiH+(\-pH)]npiL<]npi
(14)
where i = L, H, p'H is the relative price when xs = 6H in structure C„ p[ is the relative price when x" = 9L in structure C„ p' is the relative price in structure Dt. pfj and/?f are given as functions of all parameters by (9) and (12), and pL is given as a function of parameters by (10). p% and/>f are given as functions of parameters by (9) and (13), and pH is given as a function of parameters by (11). Rt is a function of 7 which relates to parameters /?,, Gt, pt. (9) and (13) gives that p'„=PiL=Pi,vrheaTj
=l
(15)
where p' is the relative price in structure Dh given by (10) or (11). Hence, p' can be considered as a special solution of (9) and (13) for 7 = 1. (15) and the definition of Rt in (14) together implies that (16)
R(TJ = 1) = P<
where the subscript of R is removed since it does not make any difference when 7 = 1 . Since i?, in (14) is valued at any value of 7 ^ 1 the left hand side of the inequality in (14) can be redefined as a function of 7, or ^.(7>l)ori?,.(7
(17)
Replacing the left hand side of the inequality in (14) with (17) and the right hand side with (26) yields i?,(7 > 1) < R{TI = 1) or R^
< 1) < R(rj = 1)
(18)
It is not difficult to see that (18) holds and therefore (14) holds if dRt Idrj < 0 for 7 > 1 anddRt/dTj>0for TJ< 1. Using (9), the definition of 7 in (9), and the first equation in (12) or in (13), we obtain an equation of 7, V and p'H . Total differentiating this equation with respect to 7 and using the second equation in (12) or (13) yields ^ < drj
0for7> l a n d ^ > 0for7< 1 i f ^ < 0 drj dt]
(19)
Using (15) and differentiating the first equation in (12) or in (13) with respect to 7, then using the second equation in (12) or in (13) yields
Endogenous Specialisation and Endogenous Principal-Agent
^ < 0 drj (14), (18)-(20) are sufficient for establishing Lemma 2.
355
(20)
Remark: Lemmas 1 and 2 imply that the corner equilibrium in CL cannot be a general equilibrium for In 77 < 0 and that CH is ruled out for In 77 < 0. (14) is analogous to the Jesen inequality, but it is much more difficult to prove than the Jesen inequality since p' is not necessarily a weighted average of p'H and p'L . Lemma 3: EU(CL) < EU(D„) for 77 > 1. Proof: From Table 2, it can be shown EU(CL)<EU(DH)ffiRL(Tj)-hip»
-{PH-PJIMJKO
(21)
where i?i(?7) and 77 are defined in (14) and (9), respectively. From Lemma 2 RL(t]) < ]npL . From (15), RL(rj) —> \xipL as 77 —»1. According to (11) and (12), the corner equilibrium value of pl is a function of /?,. Denoting the function as p'ifid a n d differentiating (11) and (12) with respect to /?, yields
dpWdfi
> 0, so thatpVff) > / ( A ) .
(22)
ff
Therefore, i?L (77) - In p - (pH - pL) In 77 < 0 for 77 > 1 . This establishes Lemma 3. Lemma 4: The corner equilibrium in AH cannot be a general equilibrium if ln77<0. Proof: From Table 2, it can be shown EU(AH) > EU{AL) iff In77 > PH-PL
InL~PL L
~PH
where P^_PL \njzjlw > 0 • This implies that the corner equilibrium in AH cannot be a general equilibrium if In 77 < 0.
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X. Yang, Y.-N. Yeh
Lemma 5: EU(DH) > EU(DL) ifEU(AH) > EU(AL). Proof: From Table 2, it can be shown EU(DH)-EU(DL)-[EU(DH)-EU(DL)]>Oiff L [L-J3H
\
L
+k+l—^— + k-2<Jk>0 \L-fiL V \L-fiH The last inequality certainly holds because of assumption/^ > fiL > 0. L-ftL J
>fk\ J
Lemma 6: EU(CH) - EU{DL) > 0 as 77 -> 1 and EU(CH) - EU(DL) < 0 as 77-^0. There exists 770 e (0,1), such that EU{CH) - EU{DL) > 0 iff 77 > 770. Proof: From Table 2, it can be shown EU(CH)>EU(DL)
iffRHto)-\npL
+(pH
~pL)\nTj>0
where RH iv) and 77 are defined in (14) and (9), respectively. From Lemma 2 RH(77)\npH asrj^l. From H L (28), p >p . Therefore, ^(?7)-ln/?i+(Ptf-A)ln/7>Oas?7-»l. Also, it can be seen from the definitions of/?// and 77 that PH RH (?) ~ In PL + (PH ~ PL ) l n 7 =to—r + 0 - A ) In 7 where (1-/9^)^177 —»-00 as 77 -> 0 . From (11) and (12), it can be shown that pHlpL tends to infinity only if pH - pL tends to infinity. However, the assumption pH, /3L < L implies fiH - f3L cannot be infinite. This means pHlpL cannot be infinite. Therefore, Et/(Q)-Et/(Z>i) = l n ^ - + ( l - p i ) l n ^ < 0 a s i 7 - > 0 . pL Hence, there exists 770 e (0, 1), such that EU(CH) - EU(DL) > 0 iff 77 > 770, where 770 is given by EU(CH) - EU(DL) = 0. This establishes Lemma 6. Lemma 7: ' ^ w ^ w l > 0. EU(D,) > EU(A,) iff k > ku, where ku is given by
Endogenous Specialisation and Endogenous Principal-Agent
4kf(k,L,ft)
357
=h
(23a)
I
^
1
2 f(k,L,fi) = k -^—+ 2&-2 \k-^— + k
(23b)
L-A
L-Pi
ku e (0, 1) if fit > f. EU(D,) < EU(A,) ifk is sufficiently close to 0. Here i = L,H. Proof: From Table 2, it can be shown that d[EU(Di)-EU(Ai)]>l
mdf(k,L,fit)>0
dk
.ff
>0.(24)
L-fi,
dk
where i = L, H. It is straightforward that the last inequality in (24) always holds because of the assumption L > /?.. Also, it can be shown that E£/(Z).)-E(/(4) = \n[4kf(k,L,/?,)] -»-co, as k-> 0, I
(25)
for jfc = 1, C/(Z).)-C/(4) > 0 iff fit > (24) and (25) are sufficient for establishing Lemma 7. Lemma 8: given by
4EC/( ff
° j; E t / w ] > 0 . U(DH) > U(AL) if k > k3, where h is L-PH^
\n[4kf(k,L,/3H)]-\n
L-PL
and k3 e (0, l ) . Here f(k,L,P()
2
/y„
\PH-PL
1 = 0. \9LJ
is given in (23)
Proof: From Table 2 and expressions (11), it can be shown that dk
>0.
(26)
Also, it can be shown that EU{DH)<EU(AL) if jfc->0
(27)
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X. Yang, Y.-N. Yeh
and when k= \,EU{DH)>EU{AL) iff 4 ( ^ + 2 - 2 ^ + 1)
( ^ f
> (jijfr) • It is not difficult to show that the last inequality always holds. This implies that when k = 1, we always have U(D„) > U(AL).
(28)
(26)-(28) are sufficient for establishing Lemma 8. Lemma 9: rflEC/< W ^ W l > 0 . EC^Cff) > EU(A,) if k>k2 and 77 > 77^ EU(CH) < EU(AL) if kT]x, or if 7 < 77,, where &2 is given by (9), (13) and EU(CH) = EU(AL) and 71 is given by EU(CH) - EU(AL) = Oandfc=l. Proof: Differentiating the first equation in (12) or in (13) with respect to k yields dp) 21'-L -^->0iff—y>0. (29a) dk k(L-Vy) where i = L,H and j = L, H. It can be verified from the second equation in (12) or in (13) that /;>|.
(29b)
(29) implies that dk Differentiating EU(CH) - EU(A,) with respect to k yields dvH d[EU(CH) - EU(A,)]/dk > 0 iff - ^ - > 0
(31)
dk
(30) and (31) implies d[EU{CH) - EU(A,)]/dk > 0 Also, it can be shown that For k = 1, EU(CH) -> EU(DN) as 77 -> 1 For £ = 1, Ef/(Cff) < EE/(4) as 77 -> 0
(32) (33) (34a)
Endogenous Specialisation and Endogenous Principal-Agent
(33) and Lemma 2 imply EU(CH)<EU(Al)ifk^>0, (28) and (33) imply EU(CH) > EU(A,) for k = 1 and rj -> 1.
359
(34b) (34c)
(34) implies that There exists 77, e (0, 1) such that when k = 1 EU(CH) - EU(AL) > 0 iff T1>T1U
(35)
where 71 is given by EU(CH) - EU(AL) = 0 and k = 1. (32) and (35) are sufficient for establishing Lemma 9. Now, we are ready to prove Theorem 1. Proof of Theorem 1: Lemma 1 implies that for 77 < 1, the set of candidates for general equilibrium consists of AL, AH, DL, CL, CH. Lemmas 2 and 3 can be used to rule out CL from the set. Lemma 4 rules out AH from this set. Then Lemmas 6, 7, and 9 are sufficient for establishing claims (1) and (2) in Theorem 1. Lemma 1 implies that for 77 > 1, the set of candidates for general equilibrium consists of AL, AH, DL, CL, CH. Lemmas 2 rules out CM from the set. Now we consider case EU(AH) > EU(AL) which holds iff ?]> rj2 and case EU(AH) < EU(AL) which holds iff t] > rj2, separately. For case 1 < t] < r/2, Lemma 3 can be used to rule out CL from the set of candidates for equilibrium. Then the set consists of only AL and DH. Lemma 8 is thus sufficient for establishing claim (3) in Theorem 1. For case rj > tj2> 1, Lemma 5 can be used to rule out DL from the set of candidates for equilibrium. Lemma 3 can be used to rule out CL from the set. Then the set consists of only AH and DH. Lemma 7 is thus sufficient for establishing claim (4) in Theorem 1.
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References Bolton, Patrick, and David Scharfstein 1998, 'Corporate Finance, the Theory of the Firm, and Organization,' Journal of Economic Perspectives, vol. 12, pp. 95-114. Coase, R. 1960, 'The Problem of Social Cost,' Journal of Law and Economics, vol. 3, pp. 1-44. Dixit, A. 1987, 'Trade and Insurance with Moral Hazard,' Journal of International Economics, vol. 23, pp. 201-20. Dixit, A. 1989a, 'Trade and Insurance with Adverse Selection,' Review of Economic Studies, vol. 56, pp. 235-48. Dixit, A. 1989b, 'Trade and Insurance with Imperfectly Observed Outcomes,' Quarterly Journal of Economics, vol. 104, pp. 195-203. Grossman, S. and Hart, O. 1986, 'The Costs and Benefits of Ownership: A Theory of Vertical and Lateral Integration,' Journal of Political Economy, vol. 94, pp. 691-719. Gibbons, Robert 1998, 'Incentives in Organization,' Journal of Economic Perspectives, vol. 12, pp. 115-32. Gupta, Srabana and Romano, Richard 1998, 'Monitoring the Principal with Multiple Agents,' Rand Journal of Economics, vol. 29, pp. 427-42. Hart, O. 1991, 'Incomplete Contract and the Theory of the Firm,' in O. Williamson and S. Winter (eds.) The Nature of the Firm, Oxford University Press, New York. Hart, O. 1995, Firms, Contracts, andFinancial Structure, Clarendon Press, Oxford. Hart, O. and Holmstrom, B. 1987, 'The Theory of Contracts,' in T. Bewley (ed.) Advances in Economic Theory, Cambridge University Press, Cambridge. Hart, O. and Moore, B. 1990, 'Property Rights and the Nature of the Firm,' Journal of Political Economy, vol. 98, pp. 1119-1158. Helpman, Elhanan and Laffont, Jean-Jacques 1975, 'On Moral Hazard in General Equilibrium Theory,' Journal of Economic Theory, vol. 10, pp. 8-23. Holmstrom, Bengt, and Milgrom Paul, 1991, 'Multitask Principal-Agent Analysis: Incentive Contracts, Asset Ownership, and Job Design,' Journal of Law, Economics, and Organization, vol. 7, pp. 24-51. Holmstrom, Bengt and Roberts, John 1998, 'The Boundaries of the Firm Revisited,' Journal of Economic Perspectives, vol. 12, pp. 73-94.
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and Endogenous Principal-Agent
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Hopenhayn, Hugo and Nicolini, Juan 1997, 'Optimal Unemployment Insurance,' Journal of Political Economy, vol. 105, pp. 412-38. Kihlstrom, Richard and Laffont, Jean-Jacques 1979, 'A General Equilibrium Entrepreneurial Theory of Firm Formation Based on Risk Aversion,' Journal of Political Economy, vol. 87, p. 719. Laffont, J. and Tirole, J. 1986, 'Using Cost Observation to Regulate Firms,' Journal of Political Economy, vol. 94, pp. 614-41. Lewis, T. and Sappington, D. 1991, 'Technological Change and the Boundaries of the Firm,' American Economic Review, vol. 81, pp. 887-900. Legros, Patrick and Newman, Andrew 1996, 'Wealth Effects, Distribution, and the Theory of Organization,' Journal of Economic Theory, vol. 70, pp. 312-41. Milgrom, P. and Roberts, J. 1992, Economics, Organization and Management, PrenticeHall, Englewood Cliffs. Wen, M. 1998, 'An Analytical Framework of Consumer-Producers, Economies of Specialization and Transaction Costs,' in K. Arrow, Y-K. Ng, X. Yang (eds.) Increasing Returns and Economic Analysis, Macmillan, London. Yang, X. 1994, 'Endogenous vs. Exogenous Comparative Advantages and Economies of Specialization vs. Economies of Scale,' Journal of Economics, vol. 60, pp. 29-54. 2000, 'Incomplete Contingent Labour Contract, Asymmetric Residual Rights and Authority, and the Theory of the Firm.' Seminar Paper, Department of Economics, Monash University. 2001, Economics: New Classical versus Neoclassical Frameworks, Blackwell, Cambridge, MA. Yang, X. and Ng, S. 1997, 'Specialization and Division of Labour: A Survey,' in K. Arrow, et al, (ed.) Increasing Returns and Economic Analysis, Macmillan, London. Yang, X. and Ng, Y-K. 1995, "Theory of the Firm and Structure of Residual Rights,' Journal of Economic Behavior and Organization, 26, 107-28.
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CHAPTER 14 A MODEL FORMALIZING THE THEORY OF PROPERTY RIGHTS*
Xiaokai Yang and Ian Wills* Monash University
1. Introduction The purpose of this paper is to formalize the economics of property rights. The theory of property rights is presented in papers collected in Furubotn and Pejovich, Eds. (1974), Manne, Ed. (1975), North (1987), Cheung (1983), Demsetz and Lehn (1985). North (1987) argues that economic growth based on economies of specialization depends on efficiency in specifying and enforcing property rights. Furubotn and Pejovich (1974, Chapt. 1) suggest the use of the neoclassical equilibrium model to describe how vagueness in specifying and enforcing property rights affects the utility frontier and the production possibility frontier. Cheung (1970, 1983) and Furubotn and Pejovich (1974, Chapt. 1) surmise that a competitive equilibrium will balance a tradeoff between the transaction costs involved in specifying and enforcing property rights, including the costs of defining, negotiating, measuring, and policing the claim, and the welfare loss resulting from vagueness in specifying and
* Reprinted from Journal of Comparative Economics, 14 (2), Xiaokai Yang and Ian Wills, "A Model Formalizing the Theory of Property Rights," 177-198, 1990, with permission from Elsevier. * The authors are grateful to Yoram Barzel, Steven Cheung, Yew-Kwang Ng, and two anonymous referees for comments and are responsible for the remaining errors.
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enforcing property rights.1 For example, property rights are more vaguely specified for a monthly train ticket or an hourly-wage contract than for a one-ride ticket or a piece-rate wage contract. However, taking the costs of specifying and enforcing property rights into account, a monthly ticket may sometimes be more efficient than a one-ride ticket and an hourly-wage contract than a piece-rate wage contract. Hence it may be efficient to have property rights vaguely specified and enforced in competitive equilibrium. Moreover, a new technology used to improve transaction efficiency, for example, magnetic cards used in monitoring payments, or changes in the legal system and institutional arrangements, for example, the passage of a patent law, may alter the efficient degree of vagueness in specifying and enforcing property rights. Conventional microeconomics does not address such problems since vagueness in specifying and enforcing property rights is assumed to be absent, i.e., the budget constraint is assumed to be binding. In Yang (1988), an equilibrium model is used to formalize a tradeoff between increasing returns to specialization and transaction costs. Yang shows that an improvement in transaction efficiency will raise the equilibrium level of division of labor and of productivity. If we introduce vagueness in specifying and enforcing property rights and the costs of reducing such vagueness into a model with increasing returns to specialization, we can formalize Cheung's (1983) theory of efficient contractual arrangements. According to Cheung, contractual arrangements are efficient when the degree of vagueness in specifying and enforcing property rights balances the tradeoff between transaction costs and the welfare loss resulting from vagueness in specifying and enforcing property rights (Cheung, 1983, pp. 9-10). 2 We surmise that an improvement of efficiency in specifying and enforcing property rights will increase the equilibrium level of the division of labor and extend the welfare frontier because of the tradeoff between the costs in specifying 1
Cheung points out (1970) that distortions caused by vagueness in specifying and enforcing property rights look like the externalities extensively discussed in the literature on market failure. 2 Such welfare loss seems to be the same as that resulting from externalities. However, Cheung holds that the word "externality" is inappropriate for addressing this very general economic problem.
Formalizing Property Rights Theory
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and enforcing property rights and increasing returns to specialization. The purpose of this paper is to prove this conjecture. No formal model of the economics of property rights has been developed since economists proposed the theory of property rights in the 1960s, for two reasons. The first is the restricted technological assumptions of the standard neoclassical framework. The second is the mathematical complexities involved in a realistic formal model. We deal with these in turn. In a neoclassical theoretical framework, gains from specialization and associated transactions are based either on comparative advantage with constant returns to scale, or on increasing returns to scale. Comparative advantage with constant returns to scale is based on exogenously-given differences of technology and endowments between agents. This notion, proposed by David Ricardo (1817), substantially differs from the ideas of Adam Smith (1776) and Allyn Young (1928) about economies of the division of labor based on increasing returns to specialization. According to Smith and Young, people specializing in producing different goods may acquire a comparative advantage even if they initially possess the same technology and endowment. According to Ricardo, no comparative advantage exists if all people have the same technology and endowment. We refer to the theory of Smith and Young as the theory of endogenous comparative advantage and that of Ricardo as the theory of exogenous comparative advantage.3 For a model with exogenous comparative advantage, the number of transactions cannot be endogenized and therefore the impacts of transaction efficiency and the corresponding property rights structure on the number of transactions and related gains to specialization cannot be formalized. The concept of economies of scale presupposes an artificial separation of pure producers from pure consumers. Scale relates to a firm that is a pure producer, but is irrelevant to pure consumers. This artificial separation has misled economic theory. In autarky, there are neither pure consumers nor pure producers; each individual is a consumer and producer. The division of labor will increase the portion of a person's 3
The difference between exogenous and endogenous, or natural and acquired, comparative advantages is discussed in Yang (1988) and Grossman and Helpman (1988).
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production sold to other people and the portion of a person's consumption purchased from other people. We can view this change as an increase in the degree of separation of production from consumption though each person is a consumer/producer even when there is a division of labor. The degree of such separation depends on the level of division of labor, or, inversely, on the degree of self-sufficiency. An increase in the degree of separation increases the number of exchanges between agents and therefore requires a more elaborate property rights structure. In a model with economies of scale, however, the artificial separation of pure producers from pure consumers implies that each consumer has to buy all goods. Hence, the number of transactions is determined by the number of consumers of final and intermediate goods and the number of goods, which are exogenously given. The number of transactions cannot be endogenized. Therefore, the neoclassical concept of economies of scale cannot be used to explore the impact of the number of transactions on productivity and the impact of efficiency in specifying and enforcing property rights on the number of transactions.4 In this paper we specify production functions with increasing returns to specialization and a preference for diverse consumption for each consumer/producer. Hence a tradeoff between economies of specialization and individuals' preferences for diverse consumption will lead to a positive relationship between economies of specialization and the number of transactions. The assumption that each transaction involves a risk of losing property rights allows us to establish a tradeoff between economies of specialization and the transaction costs of specifying and enforcing property rights. Thus, we can formalize the economics of property rights. This idea motivates this paper. The second reason why economists have failed to formalize the economics of property rights may be the technical difficulties in formalizing the tradeoff between economies of specialization and 4
Baumgardner (1988), Rosen (1983), Barzel and Yu (1984), Krugman (1981), and Romer (1986) represent new efforts to distinguish economies of specialization from economies of scale. However, their models cannot be used to formalize the economies of property rights, either because the tradeoff between economies of specialization and costs related to the number of transactions is not specified or because their models artificially separate pure consumers from pure producers.
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transaction costs in an equilibrium model. With increasing returns to specialization, corner solutions raise problems of the existence of equilibrium and difficulty in characterizing equilibria. In section 2 of the paper, we set out a model with increasing returns to specialization and without the artificial separation of pure consumers from pure producers. By specifying a symmetric and specific model, all the technical difficulties mentioned above are solved or avoided.5 The comparative statics of this model are reported in section 3. We show that productivity depends on the level of division of labor, and the equilibrium level of division of labor depends on efficiency in specifying and enforcing property rights. 2. A Model with Increasing Returns to Specialization The model is set out in subsection 2A. In subsection 2B we investigate utility maximization for a single consumer/producer. In the final subsection we solve for equilibrium. 2A. A model with m goods Consider an economy with M consumers/producers, m consumer goods, and M= m.6 The self-provided amount of good i is x.. The amount of good i sold in the market is xst . The amount of good / purchased in the market is xf . The transportation cost coefficient is k. A fraction k of a shipment disappears in transportation. Thus,(l-k)xf is the amount an individual obtains when he buys xf . The amount of good i consumed is xi + (1 - k)xf. The utility function is identical for all individuals;
By symmetry, we mean that the parameters which characterize production, transaction, and transport conditions and preferences are the same across all goods and across all individuals. In this paper a specific model is used to solve for an explicit equilibrium, so that we can get around the issue of existence of equilibrium. This is at the cost of generality of results. Hence, the model is confined to a formal demonstration of some interesting and important ideas. 6 A number of goods that equals the population size means that the division or activity is limited by the population size. This assumption simplifies the algebra.
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X. Yang, I. Wills m
U = Y\[xi+(l-k)xf].
(2.1)
1=1
This utility function represents a preference for diverse consumption. The quantity of each and every good consumed must be positive for utility to be positive. This preference combined with economies of specialization and transaction costs leads to a tradeoff between economies of specialization and costs that increase with the number of transactions. We assume that all consumers/producers have an identical system of production functions given by xt+x* =(EL.)a,
i = l,...,m,
a>\,
(2.2)
where xt + xf is the output level of good i. An individual allocates a fraction Lt of his total available hours to the production of good i. The share of labor used in an activity equals the labor cost of the activity if each individual has one unit of labor. Because of uncertainty about the amount of labor spent specifying and enforcing property rights and a fixed total labor endowment, Lt is a contingent variable. We will specify the uncertainty and the endowment constraint later. Therefore the output level is a function of expected labor share ELi .7 This system of production functions exhibits increasing returns to specialization. We assume that, for each purchase, there is a risk to the buyer of losing property rights to the good that he expects to receive. We assume that this is not due to opportunistic behavior as described by Williamson (1975), that is, consumers/producers do not take advantage of imperfect information to cheat their exchange partners.8 However, vagueness in specifying and enforcing property rights in purchase contracts may lead to
7
Here, xt + xf is not contingent. If we specify x, + xf as a random variable, the resulting mathematical complexities will make the model intractable. 8 We could allow for opportunism by specifying a probability that a seller of a good fails to deliver the good to the buyer because of the seller's opportunist behavior (cheating), and/or a probability that a buyer of a good fails to pay the seller for the good (equivalent to stealing). Then, we would have to specify the payoff to a seller or a buyer from such opportunist behavior. This will involve a binomial distribution of probability for a seller to gain some benefits or to lose property rights in the transactions with many buyers and thereby make the algebra in solving for equilibrium intractable, see (Sah and Stiglitz, 1985).
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369
misunderstandings about the category of goods to be delivered, about delivery times, and so on. We represent such delivery failures by specifying a probability qr that a seller fails to deliver good r to this buyer due to vagueness in specifying and enforcing property rights in the purchase contract. In other words, a buyer has a risk of losing his rights to the good in a single transaction, qr, due to vagueness in specifying and enforcing property rights. He will not receive the good with probability qr although he has paid for it. A seller of the good receives payment for it whether the buyer receives it or not. The risk of losing property rights means that a buyer's expenditure and demand are uncertain. These uncertainties are specified in subsections 2B and 2C. A large qr is associated with great vagueness of a contract. A small qr is associated with little vagueness of a contract. Suppose that there are Nr individuals selling traded good r and a buyer can immediately shift to other producers if a producer fails to deliver the goods ordered by the buyer. Hence, for a single buyer, there areNr parallel producers who supply good r. The probability that Nr independent sellers fail to deliver the good to the buyer is q^ . This is the risk to the buyer of losing property rights to the good which he expects to receive from Nr sellers. The probability that a buyer actually receives the good he buys is9 1-q*.
(2.3)
The probability that a consumer receives all traded goods bought by him is a product of (2.3) over r. Let the probability that an individual receives all goods bought by him be P; we have
^IK1-^')'
(2-4a)
where R is a set of all goods bought by an individual. P is the probability that an individual's property rights are perfectly specified and enforced in all transactions. An individual will enjoy U given by (2.1) if he receives all goods bought by him. Because of the property of the Cobb-Douglas utility function, he will enjoy zero utility if he fails to receive one or more goods 9
In effect traded good r disappears with probability qr on the way to a single buyer from a single seller though the seller always receives payment for the good sold.
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he buys. Suppose that his preference is represented by a von Neumann-Morgenstern utility function V=EU, or V = PU + (l-P)xO = UxY[{l-q?'),
(2.4b)
reR
where 1 - P is the probability that an individual has not received at least one of the goods he buys. 1 - P can be viewed as a risk of coordination failure in all transactions of an individual. The portion 1 - P of U seems to disappear in transactions. Hence, we define (1 - P) as the distortion rate. Later, we will show that a distortion of this sort substantially differs from a distortion based on opportunistic behavior because the former may be compatible with the Pareto optimum but the latter causes equilibrium to deviate from the Pareto optimum. This expected utility function is risk neutral. We assume that the more labor is used in stipulating and enforcing a contract, the less vaguely the property rights are specified and enforced, and the lower the probability that the buyer loses his property rights in a single transaction. Specifically, we assume that the share of an individual's labor endowment required to ensure that a buyer receives a good from a single seller with probability \-qr is given by c{l~qr),
(2.5)
where c is a constant coefficient and lie is a measure of efficiency in specifying and enforcing property rights. Assume that qr is the same across all sellers of a traded good. Since a single purchase involves uncertainties of delivery, the expected labor cost in purchasing a good involves several contingent states. When an individual buys good r, he has to spend the labor cost c(\-qr) in stipulating and enforcing the contract with the seller he contacts first. Hence, no uncertainty is associated with the cost c(l - qr). However, the seller will fail to deliver at probability qr. The conditional probability that the buyer has to ask for a delivery from the second seller when the first fails to deliver isqr. The labor cost in this state is c(l-qr) also. In general, the conditional probability that the buyer has to ask for a delivery from the next seller and incur a cost c[l-qr) when the first / sellers fail to deliver is q'r. Therefore, the expected labor cost in a transaction that involves Nr incumbent and potential sellers is
Formalizing Property Rights Theory
c(l-qr){l
371
+ q,+q?+... + q?r) = c(l-q?>).
The expected labor cost for all purchases of a consumer/producer is 1
'EC "**)-
Suppose that the sum of this expected labor cost and labor costs in production equals an individual's labor endowment; the endowment constraint for a consumer/producer endowed with one unit of labor is m
cZO-^O + ZA-l,
0
(2.6)
where Lt = ELt because contingent labor costs in specifying and enforcing property rights are replaced with their expected level in this endowment constraint. This model formalizes several important tradeoffs. First, there is a tradeoff between economies of specialization and the labor costs involved in specifying and enforcing property rights for increasing numbers of purchase contracts, given individual preferences for diverse consumption. Second, there is a similar tradeoff between economies of specialization and transportation costs for increasing numbers of purchase contracts. Third, given a certain level of vagueness in specifying and enforcing property rights, there is a tradeoff between economies of specialization and the reliability of exchange coordination across all transactions which indirectly affects individuals' expected utilities. Finally, there is a tradeoff between economies of specialization and the use of labor to reduce vagueness in specifying and enforcing property rights. To see the third of these tradeoffs, let the number of traded goods for an individual be n. We will show that the equilibrium n is the same for all individuals, that the equilibrium number of sellers, N, is the same for all goods, and the equilibrium q is the same for all goods. This combined with Proposition 1 in section 2, each individual sells only one good (if any), implies that M = nN andP-{\-qN)" . An increase in the number of traded goods, n, will generate more gains to specialization. However, this will increase the probability of coordination failure in exchange, for two reasons. On the one hand, an increase in n means a decrease in N= Mln for a fixed population size M. Hence, the risk of delivery failure in purchasing
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a traded good sold by N individuals, qN, will increase and thereby the reliability that property rights are perfectly protected in n - 1 transactions, P, will decrease. On the other hand, an increase in n implies a decrease in P = (l-qN)" via the direct effect of n on P. This economic model is an analogue of electronic equipment that consists of many components connected in series. Such a series connection will reduce the reliability of this equipment if each component has an independent probability of breaking down. According to the theory of reliability, some duplicate components in parallel connection with each working component will substantially reduce the probability of equipment breakdown even if the probability of individual component breakdowns remains constant. In our economic model, multiple sellers of a single traded good are analogues of such components in a parallel connection, 1 - q is an analogue of the reliability of a single component, and P is an analogue of the reliability of the equipment. The number of traded goods is an analogue of the number of working components connected in series. What distinguishes our economic theory from the theory of reliability in electronic engineering is that the reliability of a component, 1 - q, in our model is endogenously determined by the labor share devoted to specifying and enforcing property rights. In the theory of reliability in electronics, \-q is a given constant. Also the theory of reliability in electronics has no analogue to economies of specialization in production. Before investigating individuals' decision problems and equilibrium, we outline the nature of the market regime in our model. The number of producers of each traded good and thereby the market structure are endogenously determined. Later, we will show that the equilibrium number of producers of each traded good is a decreasing function of transaction efficiency. Thus, for sufficiently great transaction efficiency, only one individual produces each good in equilibrium, while the number of consumers of each good always equals the population size because of preference for diverse consumption. The market regime in this situation seems similar to monopolistic competition. However, monopoly powers are related to a perceived downward sloping aggregate demand (or supply) curve which is based on asymmetry of position between pure producers and pure consumers of a good. The asymmetry is in turn based on an artificial separation of pure producers from pure consumers. With such
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separation, pure consumers of a good cannot choose to produce the good and pure producers of a good cannot choose to consume the good. If there are globally increasing returns to scale, free entry for all pure producers into any sector will drive profits to zero although a producer reaps monopoly rents due to the lack of free entry for pure consumers into the production of goods. Such asymmetry of position between pure producers and pure consumers leads to the downward sloping demand curve perceived by a monopolist producer. The specification of production functions in our model allows free entry for all consumers/producers into the production of each good. Nobody perceives downward sloping demand curves because of the lack of asymmetry between pure consumers' position and pure producers' position. Nobody has monopoly powers even if only one individual produces a good in equilibrium, due to the absence of the perceived downward sloping aggregate demand curve. Prices of traded goods in our model are determined by aggregate supply and aggregate demand, which are in turn determined by the numbers of individuals who sell and buy different goods. These numbers cannot be manipulated by any individual because each individual is able to produce all goods and increasing returns to specialization are specific to an individual as well as to a good. Thus, prices of traded goods cannot be manipulated by any individual. The regime here is a special Walrasian regime where the numbers of individuals selling and buying different goods play the same role as prices in the conventional Walrasian regime. The difference here is that we do not need the Walrasian auctioneer to adjust prices. Individuals make decisions about which goods they sell and buy. Consequently, the impersonal market determines the numbers of individuals selling and buying different goods, which, in turn, determines the prices of traded goods. These numbers are parameters to each individual because his decision about which goods he trades is trivial in comparison to the great number of individuals who are able to produce and purchase each and every good. The regime here substantially differs from that in the models with the Ethier production function, see (Ethier, 1979; Krugman, 1981), where globally increasing returns to scale, free entry, and separation of pure consumers from pure producers of final and intermediate goods lead to monopolistic competition.
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2B. Individual decisions In our model the quantity of a good self-provided is distinguished from the quantity of the good sold or bought. In the appendix we prove that utility maximizers' decisions are corner solutions in such a model. There are 23m possible corner solutions for an individual consuming m goods. Applying the Kuhn-Tucker theorem, the appendix proves Proposition 1: Each individual sells only one good (if any) and does not buy and produce the same good. By employing the proposition, we can rule out many corner solutions from the list of candidates for an individual's optimum decision. Because of complete symmetry of the model, an individual's optimum trade composition is indeterminate. Without loss of generality, we assume that goods 1, 2,..., /,..., n are traded and goods n+ \,n + 2,...,m(0
prxt{\ + qr+q}+... + q^) = prx?{\-q?>)l{l-qr). An individual's total expected expenditure on goods 1, 2,..., n except good i is
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375
Because of risk neutrality which means EU(-) = £/[£(•)], we may use the expected expenditure to specify the budget constraint of an individual selling good i as follows
Taking the budget constraint into account and letting Lt = ELt (i - \,...,m), the decision problem for an individual selling good / is stated as Max:F,=C/1.i>=xin[(l-^] s.t. xi+xf=L%
Xj=L°.9 for j = n + l,..,,m
m
n (production function)
n
L(+ ^T Lj +c V (l - q^ ) = 1 j=n+l
(endowment constraint)
r=\,r*i n
Pixi = E Prx"iI1 -
(budgetconstraint), (2.7a)
n
where Pt- \\{}-qfr)
is the probability that the individual enjoys C/„
Xj is the self-provided amount of traded good i, x/ is the amount of traded good i sold, xdH is the amount of traded good r purchased by an individual who sells good /, Lt is the fraction of labor used in producing traded good i, Xj is the self-provided amount of nontraded goody, qr is the probability of losing property rights in purchasing a good delivered by a single seller, and pr is the price of good r. The decision variables are *,., x- , xdri, Lt, Xj, Lj, qr, and n. There is a subscript i for U and P because the form of the utility function and P may differ from individual to individual if they sell different goods although all people are initially identical, n and qr may differ across individuals who sell different goods.
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Later, we will show that optimum n and qr are the same for all people. For simplicity of notation, we omit subscript i of n and qr. Solutions to the problem in (2.7a) give individual demand for and supply of the n traded goods. Because of the symmetry, the optimal trade composition is indeterminate, i.e., the selection of which goods are traded has no effect on the utility level. Only the number of traded goods n matters. The individual demand and supply and n are functions of n - 1 relative prices of traded goods. Their forms are symmetric for all individuals selling different goods. Using the first order conditions for the problem (2.7a), we can write (2.7a) as Max: Vt = rf-*» [(1 - *)(l - qr)]"'' {p* lf[p
am
r){A/m)
P,,
(2.7b)
n
where A = \-c
^
(l-q^^.P.
n = n(pl/p,,Nl/N,,t
is given in (2.7a). Equation (2.7b) gives
= l,...,n)
qr=Ni/Nt>t
= [
' >->n)> r = \,...,n except/.
(2.7c)
Inserting (2.7c) into (2.7b), we have n indirect utility functions Vt=Vt{ptlp„NtlN„t
= \,...,n),
i = \,...,n.
(2.7d)
Moreover, (2.7a)-(2.7c) give an individual's demand and supply functions: xd n = Pi (nA/mJ jpr, I = \,...,n, r -\,...,n except / x? =(n-\)(nA/m)ajn,
i = l,...,n.
(2.7e)
Equations (2.7c)-(2.7e) are symmetric over i and r. 2C. Equilibrium With free entry, individual utility maximizers will equalize n indirect utility functions across individuals selling different goods because all people are initially identical, and the production, preference, and transaction parameters are the same for all goods. Hence, we have utility equalization conditions
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311
Vl{pJpt,NjNt,t = l,...,n) = V2{pJp,,N2/Nl,t = \,...,n) = ... = Vn{pJPl,NjNt,t
= \,...,n)
(2.8a)
Here, there are n - 1 equations. Because of the uncertainty of single purchases, total demand involves uncertainty. Assume that the quantity of good i demanded by a person selling good r is xfr . Initially he will demand xfr from the first seller. He will demand another xfr with probability qr when the first seller fails to deliver, a third xdir with probability q2r when the first two sellers fail to deliver, and so on. Hence the expected demand for good i by a person selling good r is
<{\+qr+ql+-+q?'-l)=<{\-/= £ Nrxi{\-q^)l(\-qr),
i = 2,...,n.
(2.8b)
Note, the market clearing condition for good 1 is not independent of (2.8b) due to Walras' law. Here, we assume that n and N are continuous. Equation (2.8b) is exactly symmetric for index i. Since the trade composition is indeterminate, we assume that all individuals trade the same bundle of goods l,...,n. The symmetry of the utility equalization conditions and the market clearing conditions implies that the prices of all traded goods are equal, JV,. is the same for / = 1,..., n, n is the same for all individuals, and qr is the same for all traded goods and all individuals. In other words, (n - 1) equations in (2.8a) and (n - 1) equations in (2.8b) ensure that the {n - 1) relative prices of traded goods are one and (« - 1) relative numbers of individuals selling different goods are one. Also, (2.7c) gives equal n and q for individuals selling different goods if N and p are identical for all goods. Thus, we have
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X. Yang, I. Wills n is the same for all individuals Ni=N = m/n
for
i-l,...,n,
pt =p
for i = l,...,n,
qr=q
for all individuals and for r - \,...,n ,
Pi=P = {\-qN)nX
for i = \,...,n
(for all individuals).
(2.9a)
Equations (2.9a) and (2.1 e) lead us to xfr=\n\\-c(n-\)(\-qNy\lm\
n,
xsi={n-\)\n\_\-c{n-\){\-qN)\lm}a
forall/andr jn,
for alii,
(2.9b)
where m = Mis the population size as well as the number of goods and A^= m/n. Using (2.9a) and (2.9b), (2.7b) can be written as Max : ^ F s n<"^ f(l - k)(l L
n,q
q)T' J
x{[l-c(«-l)(l-^^)]/w}am,
for i = \,...,n,
(2.9c)
where N= m/n. The first order condition for the problem (2.9c) is (dVIdn)/V = {a-\)(\ogn
+ \) + \og(\-k){\-q)
-amc[\-qN
+N(n-l)qN\ogq/n]/[l-c(n-l)(l-qN)] (dV/dq)/V =
=0
(2.10a)
{amc(n-l)NqN-l/[\-c(n-l)(l-qN)] - ( « - l ) / ( l - 9 ) } = 0.
(2.10b)
Equation (2.10) combined with JV= m/n gives the equilibrium n and q as functions of a, c, k, and m. Note that V converges to zero as k and c tend to one, and V is smaller than the utility level in autarky for any n> 0 if a is sufficiently small. The optimum V is m~am in autarky or if n = 0. Moreover, dV/dq<0 for any q if c = 0. This means equilibrium g = 0 if c = 0. For q = 0, dV/dn is positive for any n < m if k and/or c are sufficiently close to zero and/or a is sufficiently great. This implies equilibrium n is zero (its minimum) if a, \/k, and/or 1/ c are sufficiently small
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379
equilibrium n is m (its maximum) if a, l/k, and lie are sufficiently great. (2.11a) Differentiating (2.10) again, we find that the second order conditions for the interior n and q are satisfied if a, l/k, and lie are neither too great nor too small. Therefore, equilibrium n is between 0 and m if a, l/k, and 1/c are neither too great nor too small. (2.1 lb) Letting i = 1,..., n in (2.7a), we can maximizeVlwith respect to all quantities of individual's consumption, production, and trade subject to the individuals' production functions and endowment constraints, balance between the total consumption and production of each traded good, and the constraint that the Vt {i = 2,...,n) are not smaller than some constants. The necessary conditions for the maximization are the necessary conditions for the Pareto optimum which equalizes the marginal rate of substitution of traded goods among all individuals. The equilibrium solved in this section satisfies these necessary conditions for the Pareto optimum. This result depends crucially on the assumption of continuity of n and 7Y. This Pareto optimum involves a special category of distortions (1 - P is positive) caused by vagueness in specifying and enforcing property rights which results in human errors. However, if the number of producers of a traded good, N, can be manipulated by any individual agent, e.g., a government planner who can manipulate TV in a communist country, equilibrium will not be Pareto optimal because such an agent will manipulate prices via his control over N, thereby resulting in conventional distortions which cause equilibrium to deviate from the Pareto optimum. The equilibrium solved in this section determines the level of division of labor, n, the degree of vagueness in specifying and enforcing property rights in a contract, q, each individual's cost of specifying and enforcing property rights, c(n-\)(\-qN), and the number of producers of each traded good, N = m/n. We now explore the effects of changes in values of parameters on the equilibrium.
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X. Yang, I. Wills
3. Comparative Statics In this section, we examine the effects of varying the parameters c, the inverse of efficiency in specifying and enforcing property rights, k, the inverse of transportation efficiency, and a, the degree of increasing returns to specialization, on the equilibrium level of the division of labor, «*, the equilibrium number of producers of each traded good, N*, which is related to the conventional notion of degree of competition, the equilibrium degree of vagueness of property rights in a single transaction, q , and the equilibrium probability that property rights are perfectly specified and enforced for all transactions of an individual./?* (q and 1 -P are distortion rates in a single transaction and in all transactions, respectively). Differentiating (2.10) and noting (2.11), we have that if a, c, and k are neither too small nor too great, then Vir = dW/didr,
i, r = n, q, a, k, c
K„
V =V >0
^<°>
K„>o, ^ < o Kq>o, ^vnnvqq ,-^4> a
^o,
nq
qn
^=°
(3.1)
where we have used the first order conditions (2.10). Using (3.1) and (2.10), we can derive comparative statics of the equilibrium: dn'/dc =
(VcqVng-VgqVcn)/A<0
dn*/dk =
(VkqVnil-VggVh,)/A<0
dnlda = (VaqVnq-VqqVan)/A>0
(3.2a)
dq*/dc = (VcnVnq - VmVcq )/A is ambiguous dq
Idk^V^-VJ^jA<0
dq'/da = (VanVnq - VmVaq )/A > 0.
(3.2b)
We have rearranged the concrete form of (VcqVnq-VqgVcnj and have manipulated and employed the concrete forms of the first order conditions
Formalizing Property Rights Theory
381
and of V < 0, the second order condition, when we derive dn'/dc < 0. The same technique has been employed to show Vnq > 0. Since N = m/n and P = (l - qml")" , we can use (3.2) to get dtT/dc = (dN/dn)(dn*/dc) > 0 dN'/dk = (dN/dn)(dn/dk)
>0
dN'/da = (dN/dn)(dn/da)<0
(3.3a)
dP'/dc = (dP/dn)(dn*/dc) + (dPy'dq){dq* /'dc) < 0 if(dP/dn)(dn/dc) dP'/dk = (dP/dn)(dn'/dk)
< \(dP/dq)(dq*/dc)\
+ (dP/dq)(dq*/dk) < 0 if (dP/dn)(dn/dk)
dP'/da = (dP/dn)(dn*/da) + (dP/dq)(dq'/da)
< \(dP/dq)(dq*/dk)\ < 0,
(3.3b)
where dP/dn<0 and 3P/dq<0. Applying the envelope theorem to (2.9c) yields dV*/dc = 8V/8c<0 dV'/dk = dV/dk<0,
(3.4)
where V* is the equilibrium expected real productivity in terms of utility (per capita real income), which is a function of parameters a, c, and k, and n and q* which are also functions of a, c, and k. Equations (3.2)-(3.4) lead us to Proposition 2: Increases in transportation efficiency {Ilk) and/or efficiency in specifying and enforcing property rights (1/c) will raise the equilibrium level of division of labor («), thereby improving productivity and decreasing the number of producers of each traded good. Their effects on the equilibrium probability that property rights are perfectly specified and enforced in all transactions (P) are ambiguous. Increased transportation efficiency will increase the equilibrium degree of vagueness of property rights in a transaction (q); the effect of increased efficiency in specifying and enforcing property rights on the equilibrium degree of vagueness is ambiguous. An increase in the degree of economies
382
X. Yang, I. Wills
of specialization will raise the equilibrium level of division of labor and the equilibrium degree of vagueness of property rights in a transaction and lower the equilibrium number of producers of each traded good and the equilibrium probability that property rights are perfectly specified and enforced in all transactions. Kornai (1980) surmises that economic development, which involves an increase in productivity V and an increase in the level of division of labor, n, will in general make budget constraints increasingly softer. Since vagueness in specifying and enforcing property rights affects the real budget constraint, the concept of softness of the budget constraint is equivalent to the concept of vagueness in specifying and enforcing property rights. However, it is not clear whether Kornai's notion of softness of the budget constraint relates to the vagueness of property rights in a single transaction, q, or to the probability that property rights are not perfectly protected in all transactions, 1 - P. According to Proposition 2, improved efficiency in specifying and enforcing property rights, lie, has an ambiguous effect on the vagueness of property rights in a transaction, q, and a positive effect on real productivity, V. This means that Kornai's conjecture is unproven as far as the relationship between changes in q, V, and lie is concerned. On the other hand, Kornai's conjecture is correct if it concerns the relationship between changes in q, n, V, 1 - P, \lk, and a because an increase in a will raise n, q, and 1 - P and an increase in Ilk will raise n, q, and V. Also, the correctness of his conjecture depends on the particular values of parameters in the case of the relationship between changes in 1 -P, V, 1/c, and l/k, because of the ambiguous effects of 1/c andl/Aronl-P. If 1/c differs from good to good but a and Ilk are identical for all goods, then both the number of traded goods and the composition of trade are important for determining equilibrium. Assume that the number of traded goods is given; (3.4a) implies that per capita real income increases with 1/c. With the assumption of fixed a and Ilk, this means that the per capita real income in a market that trades goods with higher 1/c is greater than that in a market that trades the same number of goods with lower 1/c. Using the approach proposed in Yang (1988), we can show that in equilibrium the goods with greater 1/c will be traded if not all goods are
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383
traded.10 In other words, the development of exchange for a certain good depends crucially on efficiency in specifying and enforcing property rights to this good. This leads us to Proposition 3: The level of efficiency in specifying and enforcing property rights affects the number of markets in existence. There is no market for a particular good if the level of efficiency in specifying and enforcing property rights to that good is sufficiently low. 4. Remarks Proposition 3 differs from the conventional wisdom of market failure which does not explain why the markets for some commodities, for example, clean air and intangible information, do not exist. Our model formalizes Cheung's point (1983) that the determination of contractual forms is a matter of the degree of vagueness in specifying and enforcing property rights, or, in less illuminating words, the degree of externality. The difference between the externalities in buying a pound of oranges and buying clean air is a difference of degree rather than a difference of substance. When people buy oranges, there are externalities resulting from vagueness in weighing oranges and in estimating their quality (Barzel, 1982). However, the equilibrium degree of vagueness in property rights to oranges is much lower than in property rights to clean air because of much greater efficiency in specifying and enforcing property rights to oranges than to clean air. Our model also formalizes Cheung's (1970) idea that a decentralized market under a private property system can discover an efficient extent of externality. According to Cheung, for some economic activities, the costs of specifying and enforcing property rights are extremely high, so that the
In Yang (1988), the author defines a combination of zero and nonzero variables for an individual's decision problem as a structure and a combination of such structures as a market. For each market, there exists a corner equilibrium which is an analog of corner solution for an individual structure. Hence, the market structures trading the same number and different composition of goods are related to different corner equilibria. Yang shows that full equilibrium is the corner equilibrium with maximum per capita real income and other corner equilibria are not equilibria.
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rights to contracting cannot be delimited. Hence, no markets for such activities exist. The benefits and costs of such activities, not registered in markets, are commonly termed externalities. However, eliminating all such externalities is not efficient because of the costs of specifying and enforcing exclusive rights to contracting. A decentralized market will find an efficient extent of externality. This efficient extent of externality will balance the tradeoff between the welfare loss caused by absence of markets and the costs of specifying and enforcing the property rights required for markets. Alternatively, the government may determine the appropriate tradeoff between welfare losses resulting from externality and costs of reducing it by a tax/subsidy scheme or by direct regulation. A positive economic analysis may compare the efficiencies of alternative economic systems in balancing this tradeoff. This property rights approach to the problem of externalities is much more insightful than the theory of market failure, which attributes externalities to the inefficiency of the market. A legal system affects the equilibrium level of division of labor, and thereby the equilibrium level of productivity, via its impacts on efficiency in specifying and enforcing property rights 1/c. A good legal system will increase real productivity V, the utility frontier, by increasing 1/c. The major functions of a legal system in a market economy are to secure great efficiency in specifying and enforcing property rights and to guarantee the freedom of people to search for the efficient level of vagueness in specifying and enforcing property rights. In other words, the legal system in a market economy gives individuals the freedom to choose contractual arrangements, so that property rights are efficiently vague, rather than requiring that the degree of vagueness (q and 1 - P) is as low as possible. Efficiency in specifying and enforcing property rights is determined by both the legal system and technical conditions. For example, low efficiency in specifying and enforcing property rights in China is attributed to a legal system that restrains free trade in labor, land, and capital, while low efficiency in specifying and enforcing property rights to clean air is due to the high cost of technology used to measure pollution. If a legal system does not permit the achievement of efficiency in specifying and enforcing property rights, 1/c, then the division of labor and productivity cannot develop. This is consistent with the idea that the
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modern private property system was shaped under the pressure from entrepreneurs to develop the division of labor in the manufacturing sector on the eve of the Industrial Revolution, as argued by institutional economists, see (Gorden, 1980). This legal system raises efficiency in specifying and enforcing property rights by providing protection against arbitrary loss at the whim of people who have political or military power. In this sense, the modern private property right system is a powerful driving force for economic growth.11 The propositions in the preceding section formalize the tradeoffs mentioned in section 2: the tradeoff between economies of specialization and the labor and transportation costs involved in additional transactions, the tradeoff between economies of specialization and the use of labor to decrease vagueness in specifying and enforcing property rights in single transactions, and the tradeoff between economies of specialization and the number of producers of each traded good, which contributes to the reliability of coordination in exchange. There are two kinds of costs of single transactions. The first is the cost borne by the buyer when the seller fails to deliver. The second is the labor cost involved in less vague specification and enforcement of property rights. The first of these costs can be reduced either by increasing the number of sellers of each traded good, thereby sacrificing economies of specialization, or by less vague property rights, requiring an increase in the second type of costs. A decentralized market will search for the efficient market structure defined by N*, n \ q', and p* in order to balance the tradeoffs described above. Therefore, a decentralized market does not operate to eliminate the distortions which relate to q and 1 - P completely and to maximize the number of sellers of each traded good, but rather determines the efficient numbers of sellers, so as to balance the tradeoff between distortions and economies of specialization. Neoclassical microeconomics interprets Smith's invisible hand in terms of a decentralized market that is capable of avoiding all distortions by so-called pure competition (N = oo) and a private property system without vagueness in specifying and enforcing 11 Demsetz (1967) provides empirical evidence from American history about a positive correlation between trade dependence, which depends positively on the level of division of labor, and the importance of a private property right system.
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property rights (q = 1 - P = 0). We hold that this is a misinterpretation of Smith. The model in this paper can be used to provide a more appropriate interpretation of Smith. Conventional microeconomics formalizes the functions of a decentralized market in balancing two simple tradeoffs: a tradeoff between quantities of different goods in raising utility and a tradeoff between quantities of different factors in raising output or revenue. In this paper, the invisible hand balances tradeoffs among increasing returns to specialization, distortions in trade, and transaction costs. It determines the efficient level of division of labor, the efficient distortion rate, the efficient degree of vagueness in specifying and enforcing property rights, and the efficient number of producers of each traded good. Here, the conjecture of the invisible hand is consistent with Smith's insight into the implications of the division of labor based on increasing returns to specialization for general welfare. Interestingly, the equilibrium real productivity in this model may differ from the production possibility frontier. Because of increasing returns to specialization, the production possibility frontier is associated with extreme specialization. Because of transaction costs, the equilibrium, and Pareto optimum, may not be associated with extreme specialization. An improvement in transaction efficiency will move the utility frontier closer to the production possibility frontier. Therefore, a legal system that determines transaction efficiency has an impact on the equilibrium level of productivity. In terms of the comparative statics, there is a positive relationship between economic growth and efficiency in specifying and enforcing property rights. The critical assumptions that lead to this result are: (i) each individual is a consumer/producer; (ii) there are increasing returns to specialization in production; (iii) each individual prefers diverse consumption; (iv) there are transportation and transaction costs. These assumptions generate a tradeoff between increasing returns to specialization and transaction costs of two kinds. Thus, the level of division of labor affects productivity, and transaction efficiency affects the equilibrium level of division of labor. Therefore, transaction efficiency affects productivity growth.
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Appendix: A Proof of Proposition 1 First, we establish the second part of Proposition 1. Assume xf > 0; we can solve for xf from the budget constraint. Inserting this expression for xf and the rearranged production function of xi into the expected utility function, we can show dV/dxf < 0.
(A.l)
According to the Kuhn-Tucker conditionxf (dV/dxf) = 0, (A.l) implies The optimum value of xf is zero if xf > 0.
(A.2)
Equation (A.2) implies that an individual will not buy and sell a good at the same time. Assume xf > 0 (this implies xf = 0 due to (A.2)); then the optimum quantity sold of at least one of the other goods has to be positive because of the budget constraint. Without loss of generality, we suppose Xj >0 (j ^ i). dV/dxf - 0 gives the necessary condition for the optimum xsj. Letting L, = ELt and inserting the necessary condition into dVjQLi and differentiating the resulting first order derivatives with respect to Li again, we have d2V/dL2>0
if dV/dLt=0.
(A.3a)
This implies that the optimum value of Li is either zero or one if xf > 0. Lt = 1 conflicts with Lj > 0 which is required by the assumption xj > 0, implied by the assumption xf > 0. Hence, (A.3a) means The optimum value of L. is zero if xf > 0.
(A.3b)
In other words, an individual will not produce and purchase a good at the same time. This is just the second part of Proposition 1. Next, we prove the first part of Proposition 1. Without loss of generality, we assume xj,x;> 0. Because x j , x; cannot be negative and because xj (or xf) = 0 if xj (orxf) > 0 due to (A.2), it can be shown that xj = xf = 0 if xj, xf > 0. dV/dxf = 0 and dV/dx) = 0 combined with xj - xf = 0 give the necessary conditions for the optimum xj and xf . Inserting these conditions into dV/dLj and differentiating the resulting first order derivative with respect to L. again, we have d2V/dL2j>0
if dV/dLj=0.
(A.4a)
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This implies that the optimum value of Lj is either zero or one. Lj = 0 conflicts with the assumption x'j > 0 and Lj = 1 conflicts with the assumption xsr > 0. Hence, (A.4a) means xsj and*; cannot be positive at the same time.
(A.4b)
In other words, an individual will not sell two goods at the same time. This is just the first part of Proposition 1. The intuition behind this proposition is plausible. Selling one good in exchange for other goods is more efficient than selling many goods because selling many goods means dispersing limited labor into many activities, or reducing the economies of specialization. If the transaction costs are too great for a person to buy many goods and thereby he self-provides several goods, then buying these goods is not efficient. If transaction efficiency is sufficiently great for a person to buy a good, then self-providing (producing) this good is not efficient because of economies of specialization.
References Alchian, A., and Demsetz, H., "Production, Information Costs, and Economic Organization." Amer. Econ. Rev. 62, 5:777-795, Dec. 1972. Barzel, Y., "Measurement Cost and the Organization of Markets." J. Law Econ. 25, 1:27-48, April 1982. Barzel, Y., and Yu, B. T., "The Effect of the Utilization Rate on the Division of Labor." Econ. Inquiry 22, 1:18-27, Jan. 1984. Baumgardner, J., "The Division of Labor, Local Markets, and Worker Organization." J. Polit. Econ. 96, 3:509-527, June 1988. Cheung, S., "The Structure of A Contract and the Theory of a Non-Exclusive Resource." J. Law Econ. 13, 1:49-70, April 1970. Cheung, S., "The Contractual Nature of the Firm." J. Law Econ. 26, 1:1-21, April 1983. Demsetz, H., "Toward A Theory of Property Rights." Amer. Econ. Rev. 57, 2:347-359, May 1967.
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Demsetz, H., and Lehn, K., "The Structure of Corporate Ownership: Causes and Consequences." J. Polit. Econ. 93, 6:1155-1177, Dec. 1985. Furubotn, E., and Pejovich, S., eds., The Economics of Property Rights. Cambridge, MA: Ballinger, 1974. Gorden, W., Institutional Economics. Austin: University of Texas Press, 1980. Grossman, G., and Helpman, E., "Comparative Advantages and Long-Run Growth." Department of Economics, Princeton University, 1988. Kornai, J., Economics of Shortage. Amsterdam: North-Holland, 1980. Krugman, P., "Intraindustry Specialization and the Gains from Trade." J. Polit. Econ. 89, 5: 959-973, Oct. 1981. Marine. H., ed., The Economics of Legal Relationships. St. Paul: West, 1975. North, D., "Institutions, Transaction Costs and Economic Growth," Econ. Inquiry 25, 3:419-428, July 1987. Ricardo, D., The Principles of Political Economy and Taxation, 1817. London: Gaernsey, 1973. Romer, P., "Increasing Returns, Specialization, and External Economies: Growth as Described by Allyn Young." University of Rochester, Rochester Center for Economic Research, Working Paper No. 64, 1986. Rosen, S., "Specialization and Human Capital."/. Labor Econ. 1, l:43-49,Jan. 1983. Sah, R., and Stiglitz, J., "Human Fallibility and Economic Organization." Amer. Econ. Rev. 75, 2:292-297, May 1985. Smith, A., In E. Cannan, Ed., An Inquiry into the Nature and Causes of the Wealth of Nations, The University of Chicago Press, 1776 (original ed.), 1976 (this ed.). Williamson, O., Markets and Hierarchies. New York: Free, 1975. Yang, X., "A Microeconomic Approach to Modeling the Division of Labor Based on Increasing Returns to Specialization." Ph.D. Dissertation, Dept. of Economies, Princeton University. University Microfilms International, Order #8816042, Ann Arbor, 1988. Young, A., "Increasing Returns and Economic Progress." Econ. J. 38, 1:527-542, Dec. 1928.
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CHAPTER 15 ECONOMY OF SPECIALIZATION AND DISECONOMY OF EXTERNALITIES*
C. Y. Cyrus Chu a and C. Wang b "National Taiwan University and Academia National Taiwan University
Sinica
1. Introduction There are three well-known solutions to the problem of externalities:1 (i) making the parties involved 'internalize' the external effect, (ii) creating an externality market, and (iii) imposing a Pigovian corrective tax. Take the classical cattle trespassing case as an example, in which the cattle causes some negative externalities to the neighborhood vegetable field. The first solution suggests that the externality problem will be solved if the cattle and the vegetable field are owned by the same agent. Suppose such ownership refers to an internal command relationship under a farm corporation director. To internalize the externality, the director can order the 'cattle branch' to pay the 'vegetable branch' some internal transfers, which can induce the former to take into account the negative externality imposed upon the latter. An alternative ownership scenario is that a peasant owns both the cattle and the vegetable field. This peasant, taking into account the cattle-vegetable negative externality, will determine an optimal proportion of the land, say a, for herding cattle, and (1 - a) proportion of the land for growing vegetables. In this case, the peasant
* Reprinted from Journal of Public Economics, 69 (2), C. Y. Cyrus Chu and C. Wang, "Economy of Specialization and Diseconomy of Externalities," 249-261, 1998, with permission from Elsevier. 1 See for instance Varian (1994a, chap. 24). See also Varian (1994b) for some alternative ideas.
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also takes into account the implicit shadow cost of keeping cattle on the vegetable field. The negative externality caused by the cattle should be revealed by a reduction in the optimal a. In either of the above cases, the externality problem is internalized and solved. If we try to solve the externality problem by imposing a Pigovian subsidy or setting up an externality market, the subsidy rate or the market price must also be related to the above-mentioned internal transfer rate; the only distinction is that the agent and process of making such transfers are different. Thus, 'internalizing the externality' is the key idea of the traditional solutions to the externality problem. From the above discussion, we see that there is one point missed by the traditional approaches: since any peasant can choose to both herd cattle and grow vegetables to internalize the externality in question, the fact that he did not do so reveals that there may exist an economy of specialization, and the peasant realizes that it is not efficient to be involved in two production activities.2 The above-mentioned traditional solutions to the externalities problem more or less take the existing institutional practices (such as the pattern of the division of labor, specialization, ownership, etc.) as given, whereas they are in reality endogenously determined. Suppose i's product causes an externality to the production process of/, and suppose / and/ are separate firms. A corrective Pigovian tax may cause a change in the institutional structure, which is a factor neglected in most previous analysis.3 The fact that the cattle herdsman and the vegetable grower are both specialized reveal some kind of non-convexity in the theory of externalities. Suppose the production of x by peasant i causes an externality to/'s production of y. Presumably, i could choose to produce both x mdy, and the production possibility frontier (PPF) of i should be a two-dimensional diagram on the x-y plane. This is also true for peasant/. 2
Similarly, the fact that the cattle herdsman does not merge with the vegetable grower means that there may be some principal-agent cost involved if they do merge. 3 Our concern here is different from that in Schulze and d'Arge (1974) and Buchanan and Faith (1981), which consider the entry/exit decision of firms caused by externality-correction policies. Their contribution was to extend the focus of externality research from the original firm level to the broader industry level, and consider possible changes in the size of the industry. This is different from the institutional changes we are interested in here.
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The fact that both i and/ decide to produce only one product implies that there is an economy of specialization which, according to Yang and Ng (1993), implies a possible non-convexity in the production possibility set. Thus, a Pigovian tax based on the comparison of marginal cost and benefit does not apply in such a non-convex situation. Furthermore, if an externality-correction policy causes a change in the originally efficient specialization, it is certainly another kind of distortion. The purpose of this short paper is to demonstrate and elaborate the above idea. We set up a model with economies of specialization, and study the impact of an externality-corrective policy on the endogenouslydetermined institutions. To simplify our analysis, through most of this paper we concentrate upon the institutional changes in the division of labor. Other more complicated institutional arrangements (such as integration, procurement, or other contracts of coalitions) will be briefly discussed in the last section. The organization of this paper is as follows. In the next section, we set up a model in the spirit of Yang and Shi (1992) to characterize the trade-off between various divisions of labor. Then in section 3 we analyze the three possible equilibrium scenarios. In section 4 we discuss the impact of a Pigovian tax or a liability rule on the possible changes in the division of labor and institutional structure. The final section contains discussion and extensions. 2. A Two-Person Two-Sector Model 2.1. Specification Suppose there are two commodities in the economy, x and y. One can think of x as vegetables and y as meat. There are two representative peasants in the economy, i andy, both have one unit of labor endowment. If peasant i wants to produce x(y), he has to spend A(B) unit of time to learn the production technique. If peasant j wants to produce x(y), his learning cost is B(A). Without loss of generality, we assume that A < B. This implies that i(j) has the relative (also absolute) advantage in producing x(y).
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Suppose that the production technology which transforms labor input to commodity output is linear with a unity transformation rate.4 Then peasant i faces the following production constraints: x = lx-A-ty y = ly-B (la) 1 = /,+/,, where lx(ly) is the labor time devoted to the x(y) production, and ty characterizes the negative externality from the production of meat to the production of vegetables (e.g. the unavoidable trespassing by the cattle). As we mentioned, peasant/ faces exactly the same technology as i except that j is more efficient in learning how to produce y. Soy has the following constraints: x = lx-B-ty y = ly-A
(lb)
1 = /,+/,, Both / and/ are assumed to have the same utility function:
U = {x
2.2. Specialization and transaction Since these people all have a Cobb-Douglas utility function, they always prefer to consume two commodities instead of one. As one can see, the existence of fixed learning costs {A and B) for the production of x mdy 4
This production specification with specialization is a slight modification of the one adopted in Yang and Ng (1993).
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makes it inefficient to produce both commodities, hence a division of labor between i and/ is more efficient. This is a typical case of economy of specialization. A specialized individual has to trade with others, and the transaction loss (1 - k) constitutes a negative factor for the division of labor. If the transaction efficiency is low (k is small), most people would prefer not to trade, which results in autarky. Since i(f) has an absolute advantage in the production of x(y), if /(/') specializes, he will definitely specialize in the production of x(y). In the autarky situation, there is no need to trade and peasants can live far apart, so that neither one's production ofy will affect the production of x of the other peasant. But when people are specialized and trade with each other, they move closer to each other to save on transportation costs, which makes the production of the ^-specialist cause an external damage to the production of the x-specialist. A peasant may choose to produce x and consume only part of it, and the amount not consumed will be sold, denoted xs. When a peasant is involved in market transactions, other than the production constraint specified in (1), he has to face another budget constraint. Let the price of y be 1 and that of x be p. Each peasant's budget constraint can be written as pxs + ys - pxd + yd. In our later discussion, the parameter k, interpreted by Yang and Ng (1993) as the transaction efficiency coefficient, plays an important role in the determination of the equilibrium scenario. As one can imagine, in ancient times when transaction efficiency was low, people would choose to self-provide most of their consumption goods, which is essentially an autarky. Individuals would be willing to specialize in some production activities only when the transaction efficiency was sufficiently high, which in turn hinged upon the development of transportation technology and the density of the population (see Chu and Tsai, 1997). But externalities would never be a problem in an autarky economy with isolated self-sufficient individuals, since each rational individual must have endogenized the externalities involved. There are externality problems only when the two parties (individuals or firms) involved are not far apart and are separate agents. But the fact they choose to be separate
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agents implies that separate production is an efficient outcome. Most discussion of externalities and their remedies, however, takes the externality creator and the externality receiver as separate entities and imposes remedies on one or two sides, without considering the fact that 'separation' itself is an endogenous decision. 2.3. The non-convexity of the PPF Given our specification of production technology in (1), the economy of specialization can be characterized in Figure 1. In this figure, an individual's PPF in autarky is the triangle shown by the vertical lines excepting the two boundary points. If i is specialized in producing x{y), he can get to point a(b). If i etadj are both in autarky, the joint PPF is the shaded area again with the exception of the two boundary points.
\ l-A-B 2-2A-B Figure 1: The Non-Convexity of the PPF l-A-B \-A
2-2A-2B
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and Diseconomy of Externalities
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If z is specialized andy produces both x andy, the aggregate PPF is the triangle with dots. If/ is specialized and i produces both x andy, the PPF is the triangle with horizontal lines. The isolated point d is the output combination that can be reached when both i and/ are specialized. Point d is isolated because a complete specialization saves the learning cost of the other production techniques, and there is no continuity around d. Finally, point c(e) can be reached if both i andj are specialized in producing x(y). Notice that c and e are also isolated points.5 It is easy to see from Figure 1 that the aggregate PPF of / and j is typically non-convex. For instance, nowhere around an isolated point can be reached by linear combinations of two other feasible points. Such a non-convexity comes from the economy of specialization. If the original equilibrium is at d, then a marginal change in prices or government taxes cannot cause any change in the equilibrium. Alternatively, only policies that can cause decision changes in the division of labor may have any real impact on resource allocation. This will become clear as we proceed. 3. Possible Equilibrium Scenarios We shall separate our discussion of equilibrium into three exhaustive scenarios: (1) an autarky equilibrium in which both i andy self-provide x and y; (2) an equilibrium with division-of-labor but without legal imposition of externality remedy; and (3) an equilibrium with both division-of-labor and the imposition of some externality remedies. The remedy in question may either be a liability rule protection or the imposition of a Pigovian tax. 3.1. Scenario 1: Autarky In this case peasant i will spend A + B unit of time learning how to produce both x and y. Suppose peasant i uses lx of his time in the x production, then the time left for y production is \-lx. Deducting the learning time of each sector, the net production time for both sectors will be lx-A and
5
Other minor technical details of the figure are omitted.
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\-lx-B . Thus, peasant-z's optimization problem is to choose lx to maximize Ut=(xy)V2
= {\lx-A-t{\-lx-B)J\-lx-B]}
>'/2
.
It is easy to see that the above expression is concave in lx, and a simple differentiation generates the optimal /*:
AH}-m+2o 2(1 + 0 Substituting /* back into the formula of Ui yields the corresponding indirect utility:
u:
l-A-B
2VT
• t
Similarly, one can show that peasant/ will choose his optimal /* the same as the expression on the right hand side of (2), and his indirect utility level is also the same as U*:
^•-^W'
(3)
where the subscript a indicates autarky. We see from Eq. (3) that the highest utility under autarky U is decreasing in /, the degree of negative externality. This is intuitively clear: the more the cattle invade the vegetable field, the worse off an autarkic individual. We also note that U is decreasing in production learning costs A and B, which is again intuitively appealing. 3.2. Scenario 2: Division of labor without externality remedy Now we consider the situation when i and j are each specialized in the production of a good. Because of our specification in (1), it must be the case that / is specialized in the production of x, and/ is specialized in the production of y. In this subsection we consider the case in which/ is not required to compensate / for the externality caused by his production of y.
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Since/ is specialized in the production of y and no remedy is imposed, j will certainly exert all available time 1 - A on the production of y. Therefore the externality caused is /(l -A). Suppose j sells ys on the market in exchange for k(ys/p) units of x for consumption. Then, y's utility is kys U
J=-
[l-A-y>]\
Maximizing U. with respect to ys yields ys* = (l-A)/2 , which transforms into (1 - A)/2 p units of market demand for x, denoted xd*: xd =
. 2P Substituting ys = ys* = (l - A)/2 back into f/. yields the maximum utility:
_y/k(l-A) i=
24p- '
For peasant i, his output of x is \ - A -1{\ - A), of which he sells xs in exchange for kpxs units of y. So fs utility is ,1/2
t/,. = {[(i-^)(i-0-^](^)} . Maximizing f7; with respect to xs yields
._(i-^)(i-0
x> --
2
s
Substituting x * back into thet/. expression above yields /'s maximum utility: •(i-^)(i+0 In equilibrium, xJ* must equal xd' , which solves the market equilibrium price level p*:p* = 1/(1 - /). Substituting this back intof/,. and Uj yields the corresponding indirect utility of/ and/:
u*d=u; = u* =
l
-^jk{\-t),
(4)
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where the subscript Vindicates division of labor. Notice that the maximum utility is a decreasing function of t and also a decreasing function of A.6 3.3. Scenario 3: Division of labor with externality remedy Given the existing division of labor, peasant fs marginal benefit of producing y is p = 1. But j does not have any private marginal cost because, without spending the learning cost B, he cannot produce any x anyway. However, the social marginal cost of producing y is tpx = tp, the value of the externality. Since there is a gap between the social marginal cost and the private marginal cost, without paying attention to the endogeneity of the division of labor, one may be tempted to impose a liability rule on/, or to give i the property right, which requires/ to pay i the externality cost caused by/. If the Pigovian tax collected is paid back to peasant i, then in our scenario there is no difference between a Pigovian tax and a liability rule. To simplify our analysis, we assume that this is the case in our later discussion. Given the liability rule specified above, peasant i's objective function becomes
U, =
{(l-A-ty-x°)[kp(xs+ty)]}12,
in which pty is the externality compensation i receives from/. Taking the first order condition yields fs optimal xs*. Substituting this value back into Ui yields fs maximum utility: 2 For peasant/, the objective function is 6
In view of (4) and (6), we see that the size of k does not affect the comparison between scenarios 2 and 3. But in view of (3) and (4), we see that U*d > (<) U'a if and only if
(\-A-B)/(\-A) <(>) jk(\-t2) . Thus, the smaller k, the more likely autarky (scenario 1) will become the equilibrium. An extreme case is when the externality is very minor (t -> 0), and the above expression reduces to (1 - A - B)/{\ -A)<{>)4k. In this case, since (1 - A - 5)/(l - A) is less than 1, it is necessary to assume k < 1 in order to make scenario 1 a possible equilibrium.
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where / is the labor timey spends on the production ofy. ly may be less than 1 because j , facing the liability rule, may not want to devote all his time to the production of y. Maximizing U- with respect to ys, and substituting the optimal solution back into the Z7. expression, we get fs maximum utility: yfk(l-pt)(l-A)
MP 4. The Impact of a Pigovian Tax So far we have not yet solved the equilibrium price p for different values of / under scenario 3. But for our purpose, this is not necessary. Notice that 1 - pt must be positive for scenario 3 to be an equilibrium, otherwise peasant j can choose to return to autarky, which guarantees him a utility level U'a> 0 (see Eq. (3)). Given that 1 - pt > 0, U]> 0 in (5) is an increasing function of / . Thus, j will certainly choose ly = 1. But we already see from the discussion in scenario 2 that when / = 1 the equilibrium price isp = 1 / (1 - t). Therefore, as long as t l-pt
=l
\-2t n , „ n = > 0 , o r 1 - 2r > 0
\-t
\-t
we must have an equilibrium with full division of labor. In this case, the indirect utilities of i and j become
4k{\-A) _yfk(l-2t)(l-A)
ULJ = 2VT \ fA
".
(6b)
where the subscript L indicates a liability rule. Comparing (4) with (6), one can easily see that, if 1 > 2/ so that the regime with a division of labor remains under the externality remedy, then ULj < U* = U* < ULi, and ULi + ULj = £/* + U) . Thus, the liability rule
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makes / better off and/ worse off, but the utility frontier has not changed because the specialization decision has not changed. If, however, 1 - 2t < 0, peasant/ would no longer be willing to remain specialized in the production of y. He will return to the autarky situation and get utility Summarizing the above analysis, we have the following observations. First, when the degree of externality is not serious (1 > 2f), a liability rule will not distort the equilibrium of full division of labor. The impact of a liability rule or a Pigovian tax is not to improve efficiency, but to transfer some income from/ to i. When the degree of externality is serious (1 < 2t), a liability rule will make peasant/ move back to the autarky, start producing both x andy, and destroy the efficient division of labor. Even if peasant/ is not efficient in producing x, he is willing to do the difficult job just to avoid the liability associated with the production ofy. Without a type-/ peasant to trade with, a type-/ peasant will also be forced to produce y, and the economy becomes an autarky equilibrium. If the government wants to impose a liability rule while retaining the efficient division of labor, it has to impose some additional lump-sum transfers from i to /, or make / only partially liable. The above discussion tells us that whatever the value of t is, a liability rule or a Pigovian tax equal to the difference between social and private marginal cost cannot improve upon efficiency. This is a natural consequence of the non-convexity of the production possibility frontier: when the PPF is non-convex due to economies of specialization, a Pigovian tax is either ineffective or causes a jump in the equilibrium.
7
t > 1/2 (or / < 1/2) corresponds to the case when the externality in question is severe (or minor). In the case of our specified Cobb-Douglas utility function, 1/2 characterizes the budget share spent on each commodity. If our specification of the utility function changes, then in general the critical value will not be t = 1/2.
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5. Extensions and Discussion 5.1. Other related research The well-known fundamental non-convexity in the theory of externalities was first analyzed by Starrett (1972) and Baumol and Bradford (1972). In their analysis, the non-convexity of the PPF of two goods may arise even if the input-output production technology of each good is convex. Specifically, since the presence of negative externalities means that increases in the output of one good raise the other good's cost of production, as the extent of externality increases, the PPF of the two goods would be pushed toward the origin. With a sufficiently strong negative externality, the PPF must become non-convex. The cause of non-convexities discussed in Starrett (1972) and Baumol and Bradford (1972) is certainly different from ours, but the policy implications are rather similar: (i) a Pigovian tax based on the comparison of marginal cost and benefit is not necessarily efficient, and (ii) a marginal Pigovian tax may cause a discrete jump in equilibrium. Richter (1995) recently applied an analytical framework similar to ours to show that taxing basic needs will make autarkic form of provision of these needs attractive, which leads to an inefficient division of labor. The assumption driving such an inefficiency result is that basic needs are something which have to be consumed, and a tax imposed on these goods will certainly discriminate against the market provision of such goods, thereby inducing some people to self-provide such basic needs. Besides the existence of an autarky alternative, the focus as well as the analytical framework of Richter's are distinct from ours. Richter investigated whether the tax base should include basic needs, whereas we study whether a production externality should be remedied. 5.2. An extreme example: The prohibition of alcohol in the US According to Blum et al. (1973), during the Prohibition period (1920-1933) in the US, because the production, transportation and sales of alcohol were all prohibited, most of the existing large alcohol producers had to cease their business. But since the consumption had not been
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reduced significantly (Miron and Zwiebel, 1991), most of the alcohol demand was fulfilled either by smuggling from Canada, or by small domestic bootleggers. Many consumers even made their own brews at home (Blum et al., 1973, p. 595). The Prohibition amendment was enacted because of the assumed negative externality associated with alcohol. In fact, the externality involved was believed to be so severe and the corresponding optimal corrective Pigovian tax so large that the production and consumption of alcohol should be prohibited altogether. This is an extreme example of externality-remedy.8 As noted above, however, such a prohibitive Pigovian tax not only reduced alcohol consumption, but also changed the pattern of alcohol production, from professional and presumably more efficient alcohol businesses to household side-jobs and bootleggers, who often produced low-quality or even poisonous liquor (Blum et al., 1973, p. 595). 5.3. More general institutional changes In this paper we only discuss the institutional changes that are related to the endogenous division of labor. In reality, a remedy for an externality may cause other kinds of institutional changes such as coalition, procurement, and vertical or horizontal integration. All these changes may alter the boundary which differentiates what is internal and what is external. For instance, suppose the production of y b y j causes some negative externality on the production of a downstream i. An imposition of a Pigovian tax on j may increase / s intention to integrate with the downstream i. The integration in question unavoidably involves some principal-agent costs, and the fact that / andy did not integrate presumably suggests a high agency cost. When firm i is integrated under firm j , according to Yang and Ng (1993), essentially the owner of/ becomes an employee of/. Person j , who is originally specialized in producing y, now has to split his time between the production of y and the monitoring of i. Again, this is a change in the 8
Of course, the Prohibition law should not be understood from efficiency ground alone; there may be paternalistic concerns involved. Here we only focused upon its efficiency implications.
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and Diseconomy of Externalities
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division of labor, and similar analysis can be carried out. It is impossible for us to exhaust the discussion of all institutional changes here. Our general message is: a Pigovian tax or other kinds of externality-correction interventions may facilitate a change in the boundary which differentiates what is internal and what is external. As a result of the intervention, the externality disappears, but the total social costs may not have been reduced; they were merely shifted from external to internal costs.
References Baumol, W.J., Bradford, D.F., 1972. Detrimental externalities and non-convexity of the production set. Economica 39, 160-176. Blum, J.M., Morgan, E.S., Rose, W.L. et al., 1973. The National Experience: A History of the United States, 3rd ed. McMillan, New York. Buchanan, J.M., Faith, R.L., 1981. Entrepreneurship and the internalization of externality. Journal of Law and Economics 24, 95-111. Chu, C.Y.C., Tsai, Y., 1997. Productivity, investment in infrastructure, and population size: Formalizing the theory of Ester Boserup. In: Arrow, K. et al. (Eds.), Increasing Returns and Economic Analysis. McMillan, New York, 1998. Miron, J.A., Zwiebel, J., 1991. Alcohol consumption during prohibition. American Economic Review 81, 242-247. Richter, W., 1995. Expenditures for basic needs: an efficiency case for tax exemption. Universitat Dortmund, working paper. Schulze, W., d'Arge, R., 1974. The Coase proposition, information constraint, and long-run equilibrium. American Economic Review 64, 763-772. Starrett, D.A., 1972. Fundamental non-convexities in the theory of externalities. Journal of Economic Theory 4, 180-199. Varian, H.R., 1994a. Microeconomic Analysis. Norton, New York.
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Varian, H.R., 1994b. A solution to the problem of externalities when agents are well-informed. American Economic Review 84, 1278-1292. Yang, X., Ng, Y.-K., 1993. Specialization and Economic Organization. North Holland, Amsterdam. Yang, X., Shi, H., 1992. Specialization and product diversity. American Economic Review 82, 392-398.
Part 7
Investment, Endogenous Growth, and Social Experiments
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CHAPTER 16 THE DIVISION OF LABOR, INVESTMENT AND CAPITAL*
Xiaokai Yang* Harvard and Monash
University
1. Introduction The purpose of the current paper is to use a dynamic general equilibrium model to assess 'saving and investment fundamentalism' which claims an unconditional positive relationship between current saving and future productivity. This investment fundamentalism is taken as granted in the growth models of Ramsey (1928), Solow (1956, 1964), Lucas (1988), Romer (1986, 1987, 1990) and Grossman and Helpman (1989, 1990, 1991). The specification of production functions in all the models implies that saving and investment will increase productivity in the future by increasing capital per person. We may ask, however, why productivity in the future can be increased by saving today. This positive relationship did not exist two thousand years ago. For instance, two thousand years ago, peasants invested corn seeds each year. But that investment could only maintain simple reproduction without much increase in productivity. Also,
Reprinted from Metroeconomica, 50 (3), Xiaokai Yang, "The Division of Labor, Investment and Capital," 301-324, 1999, with permission from Blackwell. * The author is grateful to the participants of the International Conference on Dynamic Modeling and of the seminars at University of London, Monash University and Australian National University, and the two referees for Metroeconomica for their comments and criticisms. Special thanks are due to Don Snodgrass, Yew-Kwang Ng and Jeff Borland for helpful discussion. I am responsible for the remaining errors.
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Chinese peasants invested in houses which were completely selfprovided in the 1970s. Productivity based on such investment in durable houses was extremely low (Yang et al. (1992)). To the question, Lucas and Romer will respond by pointing to human capital generated by saving and investment. However, we will again use the Chinese case to argue that investment in human capital and education does not necessarily lead to an increase in productivity. Chinese people have a special preference for saving and for investment in education. However, this had not generated significant productivity increases until the modern school and university system was introduced into China at the end of the nineteenth century. In traditional Chinese schools, there was no division of labor between teachers. Each teacher taught students a broad range of knowledge, from literature to philosophy. But in a modern university, there is a very high level of division of labor between different specialist teachers and between different specialized colleges. Also educated individuals are very specialized in their professions after their graduation from universities. It is the high level of division of labor that ensures high productivity in providing education, so that investment in education can contribute significantly to productivity progress. Recent empirical evidence supports our observation. Pritchett (1997) shows that empirical evidence from macro data rejects the unconditional positive relationship between educational capital and the rate of growth of output per worker, despite the positive effects of education on earnings demonstrated with micro data. To our question above, Grossman and Helpman might respond by pointing to investment in research and development (R&D). However, Marshall attributed the invention of the steam engine by Boulton and Watt to a deep division of labor in the inventing activities (1890, p. 256). Edison's experience is more evidence for the implication of the division of labor for successful inventions. Not only did Edison himself specialize in inventing electrical machines for most of his life, but he also organized a professional research institution with more than a hundred employees who specialized in different inventing activities (Josephson(1959)). These observations imply that investment in physical capital goods, in education or in research would not automatically increase productivity
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in the future if the investment were not used to develop the right level and pattern of division of labor. Hence, the essential question around the notion of capital is not so much how much we invest and save, but rather what level and pattern of division of labor are used to invest in machines, education and research. The recent empirical evidence that rejects so-called scale effects associated with the positive relationship between current saving and future productivity supports our observation. A type I scale effect exists if there is a positive relationship between growth rates in per capita GDP and investment rates. The AK model generates the type I scale effect which is conclusively rejected by empirical evidence (see Jones (1995a)) . This suggests that 'the AK models do not provide a good description of the driving forces behind growth' (Jones (1995a, pp. 508-509)). The R&D-based model generates a positive relationship between the growth rates in per capita GDP and the level of resources devoted to R&D, referred to as a type II scale effect. The type II scale effect is also rejected by the empirical observations (Jones (1995b)). Jones (1995b) and Alwyn Young (1998) have developed two models to salvage the R&D-based model, but the modified models still have a type III scale effect, a positive relationship between the growth rate in per capita GDP and the growth rate of population, which is also wildly at odds with the empirical evidence surveyed by Dasgupta (1995). As Jones (1995b) indicates, endogenous growth cannot be preserved if the scale effect in the R&D-based model is eliminated. Now endogenous growth economists are busy with developing new models that can avoid scale effects. The current paper will show there is a simple way to avoid scale effects: formalize classical economic thinking on investment, division of labor, and growth. The growth mechanisms described by most classical economists do not have an unconditional positive relationship between saving and 1
According to Jones (1995a, 1995b) and Barro and Sala-i-Martin (1995), the Romer model (1987), the Rebelo model (1991), the Barro model (1991) and the Benhabib and Jovanovic model (1991) can be considered as AK models since their reduced forms are the same as the AK model. 2 Judd (1985), Romer (1990), Grossman and Helpman (1990, 1991) and Aghion and Howitt (1992), among others, are along this line.
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productivity. Instead they emphasize the connection between the division of labor and investment. Smith (1776) and Allyn Young (1928) explicitly spelled out the relationship between the division of labor, investment and capital. According to them, capital and investment are a matter of the development of the division of labor in roundabout productive activities. If there is a division of labor between the production of final consumption goods (say food) and the production of producer goods (say tractors) and if the production of tractors takes time to complete due to, for instance, a significant fixed learning cost, then the specialist producers of tractors cannot survive in the absence of investment which is used to provide the specialists with food before they can sell tractors. Hence, capital is a vehicle for society to increase the level of division of labor in roundabout productive activities. A high level of division of labor can speed up the accumulation of knowledge through specialized learning by doing, thereby generating progress in productivity. The current paper will formalize the story of investment and capital. A dynamic general equilibrium model will be used to address the following questions. What is the relationship between capital, which relates to saving and investment, and the division of labor, which determines the extent of the market, trade dependence and productivity? What is the mechanism that simultaneously determines the investment level and the level of division of labor and what are determinants of the equilibrium investment (saving) rate, interest rate, growth rate, and the equilibrium level of division of labor? Our story of investment runs as follows. There are many ex ante identical consumer-producers in an economy where food can be produced out of labor alone or out of labor and tractors. In producing 3
Smith stated (1776, p. 371): 'When the division of labor has once been thoroughly introduced, the produce of a man's own labor can supply but a very small part of his occasional wants. The far greater part of them are supplied by the produce of other men's labor, which he purchases with the produce,... of his own. But this purchase cannot be made till such time as the produce of his own labor has not only been completed, but sold. A stock of goods of different kinds, therefore, must be stored up somewhere sufficient to maintain him, and to supply him with the materials and tools of his work, till such time, at least, as both these events can be brought about'.
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each good, there are economies of specialized learning by doing. A fixed cost is incurred in the period when an individual engages in a job for the first time or when job shifting takes place. Each individual can choose between specialization and self-sufficiency. The advantage of specialization is to exploit economies of specialized learning by doing and to avoid job shifting costs. However, it increases productivity in the future at the expense of current consumption because of an increase in transaction cost caused by specialization. Moreover, in producing a tractor, there is a significant fixed learning cost. The production of a tractor cannot be completed until the learning cost has reached a threshold level. Hence, there are trade-offs between economies of specialized learning by doing, economies of roundaboutness, transaction costs, and fixed learning costs. Each consumer-producer maximizes total discounted utility over the two periods with respect to the level and pattern of specialization and quantities of goods consumed, produced and traded in order to efficiently trade off one against others among the four conflicting forces. The interactions of these trade-offs determine the nature of the dynamic equilibrium for the economy. If the transaction cost coefficient is sufficiently great, the economy is in autarky in all periods — depending upon the level of fixed learning cost and the degree of economies of roundaboutness this may involve each individual selfproviding food, or each individual self-providing both food and tractors, or an evolution in the number of goods. If the transaction cost coefficient is sufficiently small and economies of specialized learning by doing and of roundaboutness are significant, in the dynamic equilibrium the economy is in a market structure in which individuals specialize in the production of either tractors or food and trade occurs. For the division of labor there are two patterns of investment and saving. If the fixed learning cost in producing tractor is not large, each individual will sacrifice consumption in period 1 to pay transaction costs in order to increase the level of division of labor so that productivity in period 2 can be increased. This is a self-saving mechanism which does not involve the transfer of a saving fund from one individual to another. Also, an evolution in the level of specialization and/or in the number of goods
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may take place in the dynamic equilibrium if the transaction cost coefficient and the degree of economies of specialization and of roundaboutness are neither too large nor too small. If the fixed learning cost in producing tractor is so large that the production of a tractor can only be completed when the time for specialized learning by producing tractor is longer than one period, then an explicit saving arrangement which involves a loan from a specialist producer of food to a specialist producer of tractors in period 1 is necessary for specialization in producing roundabout productive tractors. Under the assumptions of a great fixed learning cost in producing tractors, a small transaction cost coefficient, and significant economies of specialized learning by doing and roundaboutness, dynamic general equilibrium yields the following picture. A specialist producer of food produces food using his labor only and makes a loan in terms of food to a specialist producer of tractors in period 1 when the production of tractors is not completed. In period 2, a specialist producer of tractors sells tractors to a specialist farmer in excess of the value of his purchase of food in period 2. The difference is his repayment of the loan received in period 1. Per capita consumption of food in period 1 is lower than in an alternative autarkic pattern of organization. But in period 2, tractors are employed to improve productivity of food. The discounted gains will be more than offset the lower level of per capita consumption in period 1 if the transaction efficiency coefficient and economies of specialized learning by doing and roundaboutness are great. Economic growth takes place not only in the sense of an increase in per capita real income between periods, but also in the sense that total discounted real income is higher than in alternative autarkic patterns of organization.4 The social organization of 4
In our model, not only does division of labor depend on the extent of the market, but also the extent of the market is determined by the level of division of labor (Allyn Young, 1928). In addition, not only does the level of division of labor in roundabout production depend on the saving rate, but also returns to saving are dependent on the level of division of labor. These interdependences, together with interplay between observable prices and production functions in the market and individuals' dynamic decisions in choosing occupation configurations, make our dynamic equilibrium much more complicated than interactions between information and dynamic strategies in sequential
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division of labor that is created by interpersonal loans and that speeds up specialized learning by doing and increases productivity can be considered as capital in our model. The model generates the following empirical implications. Returns to investment are higher when division of labor evolves than when the potential for further evolution of division of labor has been exhausted. Hence, the returns to capital in a developing economy experiencing evolution of division of labor are higher than in a developed economy where the potential for the evolution has been nearly exhausted. Since transaction efficiency determines whether there is a more lucrative opportunity for the evolution of division of labor, returns to investment used to develop division of labor are higher in a developing economy with higher transaction efficiency than in a developing economy with low transaction efficiency even if the latter is short of capital compared with the former. In addition our model avoid all kinds of scale effects that are the common features of the AK models and R&D-based models and that are rejected by empirical evidence. The rest of the paper is organized as follows. Section 2 presents the model. Section 3 solves for dynamic equilibrium. In section 4, the model is extended to endogenize the decision horizon. A final section concludes. 2. Model We consider a finite horizon (two-period) economy with M ex ante identical consumer-producers. There is a single consumer good (called food) produced by labor alone or by labor and an intermediate good (called tractor) together. Individuals can self-provide any goods or alternatively can purchase them on the market. The self-provided amounts of food and of tractor in period / are denoted respectively y, and equilibrium models. Hence, it is extremely difficult to obtain analytical solutions of comparative dynamics if the decision horizon is long and the number of goods is more than two, although theoretically our analysis can be extended to the case with many goods and a very long horizon. One technical difficulty is that no general mathematical method is available for working out the analytical solutions of individuals' dynamic programming problems.
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x,. The respective amounts sold and purchased of food in period t are yst and ydt , and those of tractor are xst and xf . It is assumed that a fraction 1 - k of any shipment of a good disappears in transit due to transaction costs so that&y, is the amount available for consumption after purchasing yf . The total amount of food consumed by an individual is therefore yr + kyt . Similarly kxdt is the amount available of tractor after purchasing xt . The total amount of tractor available is therefore x, + kxt . The utility function in period / is therefore assumed to be equal to the natural log of the amount of food consumed: u, = \sx{y,+ ky1)
(1)
where u, will be negative infinity ifyt= yt = 0. It is assumed that all trade in this economy is mediated through contracts signed in futures markets which operate in period 1. These contracts cannot be renegotiated in some later period. Assume that the futures market horizon and any individual's decision horizon are of two periods. The objective function for an individual's decision problem is therefore total discounted utility, given by U=ui + u2/(l+r)
(2)
where Uis total discounted utility, and r is a subjective discount rate. It is assumed that a fixed learning cost in terms of labor, A, is incurred in producing tractor. The fixed learning cost in producing food is B. The production functions for an individual are assumed to exhibit economies of specialized learning by doing:5 y?=y, +yst=max[(xt+kx?y(Lyt
-
{Lyt -
ae(0, xf =xt+ Xs, = max[(4, - oA)b, 0]
5
b>\,
l),Be(0,
l)
Ae(0, 1]
(3)
It can be assumed that output is zero if current labor input is zero even if accumulated past labor input is great (see Borland and Yang (1995)). This will avoid the unreasonable case that output is positive even if current labor input is zero.
Division of Labor, Investment and Capital
All
where y, + yst and x, + xst are respective total output levels of food and tractor in period /, /,-, is the amount of labor allocated to the production of good i (i = x, y) in period t, and Lit is the amount of labor accumulated in producing good i up to period /. We define lit as a person's level of specialization in producing good i at /. In producing a good in period /, a= 0 if an individual has engaged in producing the good in period / - 1 and does not shift between different activities in periods t and t - 1; a = 1 if an individual changes jobs in period t or t - 1 or engages in producing the good for the first time in period t. Each person is assumed to be endowed with / units of labor in each period. The assumption A e (0, /] implies that it is possible that the production process of tractor cannot be completed in period t = 1 if A = /, so that a story of investment, saving and capital may be told. It is assumed that economies of specialized learning by doing and the fixed learning cost are specific to each individual and to each activity. The elasticity of output of food with respect to input of tractor, a, can be interpreted as the degree of type I economies of roundaboutness. Type II of economies of roundabout production are said to exist if labor productivity of food is higher when tractor is employed than when it is not employed. It can be shown that there are type II of economies of roundabout production if the amount of tractor employed by each farmer is sufficiently large and a is large. The amount of tractor per farmer is determined by the productivity of tractor, which in turn is dependent on the level of specialization of producers of tractor. But each person's level of specialization is determined by the efficient trade-off between economies of specialized learning by doing and transaction costs. Hence, specification of the production functions sets up interdependence between division of labor, production roundaboutness, and transaction costs. We assume that A+ B> I - 1, which implies that self-provision of the two goods by each individual in the two periods is not optimal. A Walras regime prevails because economies of specialized learning by doing are individual specific (increasing returns are localized) and the population size is assumed to be very large, so that competition among
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many peer specialists in each professional sector nullifies the monopoly power which might occur from specialized learning by doing. 3. Dynamic Equilibrium and Comparative Dynamics In this section we consider an individual's production and trade decision problem and dynamic equilibrium. We follow Borland and Yang (1995) and call a profile of zero and positive values of decision variables in each individual's decision problem in a period a configuration. As shown by Borland and Yang (1995), in this kind of model, each individual's dynamic optimum decision is associated with a sequence of configurations over periods. There are six configurations, shown in Figure 1, which constitute four feasible market structures in a period. The circles represent configurations and lines represent flows of goods.6 There are two autarky structures E and F, depicted in Figure 1(a). In structure E each individual self-provides food without using tractor. In structure F, each individual self-provides both tractor and food. Two structures involve the division of labor. Structure D consists of configuration (xly), shown in Figure 1(c), selling tractor and buying food in a period, and configuration (y/x) selling food and buying tractor. Structure C consists of configuration (y/0), which denotes a farmer providing a loan in terms of food, and configuration (0/y), which denotes a tractor producer receiving food as a loan when he learns how to produce tractor. Feasible structure sequences over two periods are: CD, which means structure C is chosen in period 1 and structure D is chosen in period 2; DD, involving trade and division of labor between specialist producers of tractors and professional farmers over the two periods; EE, in which all individuals self-provide food over the two periods; FF, where all individuals self-provide tractors and food over the two periods; FE, which means each individual self-provides food in period 1 and selfprovides tractor and food in period 2. Sequence CD involves explicit 6
Morishima (1996) used the approach to equilibrium based on comer solutions to prove the existence of Walras's general equilibrium of capital and credit. A recent survey on the literature of infra-marginal analysis of endogenous specialization can be found in Yang and S. Ng (1998). A model of endogenous evolution of division of labor with an infinite horizon can be found in Yang and Borland (1991).
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ylx
(
Market
\
^ fay J
./
I Market V forx
(b)
(c)
Figure 1: Configurations and Market Structures (a) autarky; (b) structure C; (c) structure D
saving since a specialist producer of tractor buys food and sells nothing in period 1. The one-way trade can be considered as a loan made by a specialist farmer to a specialist producer of tractor. Other structure sequences are either unfeasible or cannot occur in equilibrium (see Yang (1996) for proof of this statement). Structure sequences are shown in Figure 2. A dynamic equilibrium is defined as a fixed point that satisfies the following conditions: (i) for a given profile of the sequences of configurations chosen by individuals, a relative number of individuals choosing different sequences of configurations and a sequence of relative prices of traded goods at different points in time clear the market for goods and equalize total discounted utility of all individuals; (ii) for a given relative number of individuals choosing different sequences of configurations and for a given set of sequences of relative prices of traded goods, individuals maximize total discounted utility with respect to the sequences of configurations and quantities of goods produced, consumed and traded. It is possible to solve for the dynamic equilibrium in two steps. First, we can solve for a local dynamic equilibrium for each structure sequence.
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Sequence EF period 1
period 2
Sequence ED period 1
Sequence CD period 1 period 2
period 2
Sequence FD period 1
period 2
Sequence DD period 1 period 2
Figure 2: Dynamic General Equilibrium and its Comparative Dynamics
This is given by the market clearing conditions for all traded goods in each period and equalization condition of total discounted utility across individuals choosing different configuration sequences. Then, we can compare each individual's total discounted utility under the local equilibrium prices in a given structure sequence between all possible configuration sequences. By the definition of dynamic equilibrium, a local equilibrium in a structure sequence is a general equilibrium if nobody has an incentive to deviate from his or her configuration sequence in this structure sequence. Hence, the comparisons can be used to identify parameter subspaces within which the local equilibrium in a structure sequence is a general equilibrium. The solution generates comparative dynamics of the general equilibrium which describe how the dynamic equilibrium jumps between structure sequences as parameter values shift between the parameter
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subspaces that demarcate structure sequences. The algebra to substantiate this is available from the author upon request. The comparative dynamics are summarized in Table 1. EE denotes that all individuals self-provide food in the absence of tractors in two periods. EF denotes that all individuals choose autarky in two periods, but they self-provide only food in period 1 and self-provide both food and tractors in period 2. In other words, tractors emerge in period 2 and there is evolution in the number of goods over time. ED denotes that individuals self-provide food in period 1 and some of them specialize in producing food and others specialize in producing tractors in period 2. Evolution of division of labor takes place through increases in both individual specialization and the number of goods. FD denotes that individuals self-provide both food and tractors in period 1 and choose specialization and trade of the two goods in period 2. In other words, evolution of division of labor takes place through an increase in individual specialization in the absence of changes in the number of goods. DD denote the division of labor and trade of the two goods in the two periods without its evolution but with an implicit self-saving in period 1. CD denotes the division of labor in the two periods with an explicit saving and a loan in period 1. All of the evolutionary patterns are shown in Figure 2. y = ab\n{l - A)l{\ + r) increases with the degree of economies of specialization in producing tractor, which is b, and with the degree of economies of roundaboutness, which is a, and decreases with the discount rate r. Here ab > po means that the degree of economies of specialization and roundaboutness is greater than a critical value. Y> 7t means that the degree of economies of specialization and roundaboutness is greater and the discount rate is smaller than some critical values. k> k0 means that the transaction efficiency parameter is larger than a critical value. Note that Y\ and Yi decrease as k increases, so that DD is more likely to be the equilibrium than ED or FD if transaction efficiency is higher. Also, k0 decreases with ab, so that CD is more likely to be equilibrium than EE if the degree of economies of specialization and roundaboutness is greater.
Table 1: Equilibrium and its Comparative Dy A<1 k is large, equilibrium involves division of la
k is small
the absence of explicit saving A+B is close to / ab<po
ab> po
r
r>n
A, B are r
EE
EF
ED
DD
FD
Autarky with
Autarky
Evolution in
Division of
Evolution in
only food
with tractor
specialization,
labor without
specialization
produced in 2
emerging in
tractor emerges
evolution
with 2 goods
periods
t=2
int = 2
produced in 2 periods
Notes: po= [ln(2/) - ln(2/-^) + ln(ab + l)]/[ln(2/-^) -\n{ab + 1) + \n(ab)] n = -[fi + abln(2l)/(l + r)]/r, /}= a{2\nk+ lna) + (1 - a)ln(l - a) Yi = [To-P+ ab\n(2l)l(\ + r)](l + rf1 Yo = {abln[ab(2l -A-B) + l\ + \n[(ab + 2)1-A- B)] + (ab + l)[ln(/ -A-B)-(2 + r)\n(ab + 1)] -ln(2/)}/(l + r) + [abln(ab) -ln( \nk0 = [(2 + r)ln(2 + r) - (2 - a + r)ln(2 -a + r) - alna - abia(2l)]/2a
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Table 1 provides a characterization of the conditions under which the various sequences of market structures will constitute the dynamic equilibrium. There are five parameters that determine the comparative dynamics of the equilibrium: A (fixed learning cost in producing tractor), B (fixed learning cost in producing food), k (transaction efficiency), ab (degree of economies of specialization and roundaboutness) and r (discount rate). Transaction efficiency determines whether specialization is more likely to take place in the equilibrium compared to autarky. The fixed learning cost A determines whether an explicit saving is essential for the division of labor. If A is small, then explicit saving and a loan between individuals are not necessary for the division of labor. When individual specialization and related trade takes place in DD, there is an implicit investment for increasing specialization, which is in terms of a decrease in consumption of food in period 1 (compared to consumption of food in an alternative autarkic structure), caused by a larger transaction cost. However, such investment does not involve the transfer of investment funds between individuals. Hence, investment comes from self-savings rather than from commercial savings. If A = /, then explicit saving and a loan between individuals are necessary for the division of labor. If A < I but A + B is close to /, then FD is infeasible and ED will be the equilibrium provided k is large and ab is not great compared with r. The degree of economies of specialization and roundaboutness ab determines whether a larger number of goods and/or a higher level of specialization is more likely to take place in the equilibrium. Gradual evolution in individual specialization and/or in the number of goods will take place in the equilibrium if k is large but ab is small compared with r, or if k is small but ab is large. A larger discount rate r will make a small number of goods and a low level of specialization more likely to take place in the equilibrium. It is interesting to see that increases in the extent of the market, in trade dependence, in the extent of endogenous comparative advantage, in productivity, in the degree of production roundaboutness, in the variety of goods and different professions, in the degree of production concentration, and in the degree of market integration are different aspects of the evolution in division of labor. Detailed discussion of the
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concurrent phenomena can be found in Borland and Yang (1995) and Yang and Y.-K.Ng (1993). If transaction efficiency k and the fixed learning cost A are sufficiently large, then the equilibrium sequence is CD where a loan in terms of food is made from a professional farmer to a professional producer of tractor in period 1 and repaid in terms of tractor in period 2. It can be shown that in this structure sequence a farmer's saving level and saving rate in period 1 increase with the degree of economies of roundaboutness, and decrease with the subjective discount rate. But more importantly, transaction efficiency k determines whether CD or EE is the equilibrium. If transaction efficiency k is very small due to a deficient tax system or a deficient legal system, then EE instead of CD will be the equilibrium, which implies a zero saving rate. Also, the magnitude of the fixed learning cost determines whether implicit or explicit saving is essential for a high growth rate of per capita real income. The real saving level in CD can be calculated as the difference in utility between EE and CD in period 1. The real return on saving can be calculated from the difference in discounted utility between CD and EE in period 2. Again, there are two types of comparative dynamics of the equilibrium. If values of the parameters of transaction efficiency and economies of specialization and roundaboutness decline and reach a threshold level, then the dynamic equilibrium shifts from CD to EE (autarky) and real returns to saving drop down to zero. The second type of comparative dynamics is that, as the parameters change within the range defined by the threshold values such that equilibrium stays in CD, then it can be shown that the real interest rate increases with transaction efficiency and with the degree of economies of specialization. This result can be used to theorize the successful practice of the Hongkong and Taiwan governments in carrying out a liberalization and internationalization policy which stimulates investment and trade by reducing tax. The theory of capital and investment can be used to explain a sudden decline in interest rates. Suppose that the transaction efficiency coefficient and the fixed learning cost are sufficiently large, so that CD
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is the equilibrium and the saving rate and real interest rate on saving are positive. However, if the decision horizon is longer than two periods, then the opportunity for investment will suddenly disappear in period t > 2 as structure D, which is the highest level of division of labor in the model, has been reached. If the number of goods is more than two in the model, then opportunity for lucrative investment may last longer than two periods. However, there is a limit for the evolution in division of labor. For instance, the number of professional sectors cannot be larger than the population size, so that the highest level of division of labor will eventually be reached and no more opportunity for investment for increasing the division of labor can be pursued if the number of professional sectors is close to the population size. This is an interesting explanation of a decline in interest rates, which is consistent with the classical theory of capital based on the intimate connection between capital and division of labor, but is substantially different from Keynes's (1936) view of interest rates which focuses on the implication of pure consumers' preference for liquidity. 4. Endogenous Horizon If the decision horizon is a decision variable and uncertainty of realizing the gains from investment for increasing division of labor is introduced, there is another interesting trade-off between gains from a longer decision horizon which can amortize investment cost over a longer period of time and a decreasing probability for realizing the gains. Suppose the fixed learning cost A = I in an extended model which is the same as the original one except that the decision horizon n is each individual's decision variable. The discount rate is assumed to be zero and / is assumed to be 1 for simplicity. The probability that gains of total discounted utility from a sequence with the division of labor and savings against autarky are realized is/(«) = \ln and the probability that the gain is zero is 1 - / ( « ) . Each individual maximizes expected gains in total discounted utility from choosing the division of labor. Then, it can be shown that the necessary condition for the equilibrium decision horizon is equivalent to the first-order condition for maximizing
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V=f(n)F(n,k,a,b) F (n, k, a, b) = [n - a {n - l)]ln[« - a(n - 1)] + a{n - l)(ln2 + lna + 2 M ) + ^[0.5(1 + n)n - 1] where F is the difference in total utility over n periods between division of labor and autarky. Application of the envelope theorem to V=f(n)F (n, k, a, b) and the implicit function theorem to F yields the comparative dynamics -d2Vldkdn dV(n,k) n A dV(n\k) =— — >0 -= - ^->0 dk d2Vldn2 dk dk where 8 Vldkdn> 0, d2V I' dn2 < 0. n is the equilibrium decision horizon, which efficiently trades off the gains from a longer horizon against a decreasing probability for realizing them. The policy implication of the result is straightforward. A liberalization and internationalization policy which reduces tariffs or a better legal system which improves transaction conditions will increase individuals' efficient decision horizon and increase opportunities for lucrative investment for increasing the division of labor. dn
5. Conclusion A dynamic general equilibrium model based on corner solutions is used to show that investment does not necessarily increase future productivity. Productivity in the future can be increased by an investment that is used to create a higher level of division of labor which can speed up accumulation of professional experience (human capital) through specialized learning by producing roundabout productive equipment or services. Self-saving is enough for investment in raising the division of labor if the fixed learning cost in roundabout productive activities is not large. A loan from a producer of consumption goods to specialist producers of producer goods is essential for such investment if the fixed learning cost is large. If the transaction cost coefficient is large due to a deficient legal system or to a protectionist tariff, such opportunities for lucrative investment for increasing the division of labor do not exist, so that investment may not increase real income. A
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decrease in the degree of economies of specialization and roundaboutness, an increase in the transaction cost coefficient, and/or exhaustion of the potential for further evolution of the division of labor will reduce real return rates on investment and reduce opportunities for lucrative investment. Sachs and Warner (1995) provide preliminary empirical evidence for our model. They use cross-country data to show that growth performance (and therefore related returns to capital) is positively affected by an index of institutional quality that affects transaction conditions. Gallup and Sachs (1998) show that the population share in coastal regions and other geographical conditions that affect transaction conditions positively contribute to growth performance too.
Appendix: Proof of Comparative Dynamics of Equilibrium We use structure sequence CD, in which structure C is chosen in period 1 and D is chosen in period 2, to show how to solve for a local dynamic equilibrium in a given structure sequence. The structure C consists of configurations (0/y) and (y/0) and structure D consists of configurations (x/y) and (y/x). Hence, this structure sequence consists of the sequence of configurations (0/y) and (x/y) and the sequence of configurations (y/0) and (y/x). There are two steps in solving for the local equilibrium in the structure sequence: first, for each sequence of configurations the utility-maximizing labor allocation decision and demands and supplies for each good in each period (and hence the indirect total discounted utility of an individual who chooses that configuration sequence) are derived; and second, given the demands, supplies and indirect utility function of an individual in each configuration sequence, the market clearing conditions and utility equalization conditions are used to solve for the set of corner equilibrium relative prices and number of individuals choosing different configuration sequences.
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Sequence of configurations (0/y) and (x/y) In this sequence x2, ydt > 0, lxt = /, and x[ = x, = lyt = yt = xf = yst = 0. This sequence will be chosen by an individual only ifA = l which means that the production process of a tractor cannot be completed in period 1 due to the large fixed learning cost. If A < /, a sequence of (x/y) over two periods will be chosen since it does not make sense to choose (0/y) which delays sale of tractor when a tractor can be produced in period 1. Hence, we assume A = I in this configuration the sequence, so that the decision problem for sequence of (0/y) and (x/y) is max U = ln( kyf ) + ln( ky2 )/(1 + r) yf
subject to xj* = 0 and xs2 = (4i + lx2)b
(production function)
h\ = hi= I
(endowment constraint)
Px2xi= yf
+
Pyiyi.
(budget constraint)
where pit is the price of good i in period / in terms of food in period 1 which is assumed to be the numeraire and yx is a loan in terms of food made by a specialist farmer to the specialist producer of tractor. The specialist tractor producer's sale in period 2,px2xs2, is greater than his purchase in period 2, py2y2. The difference is repayment of the loan. The solution to (6a) is x*2=(2l)b (l + r)(2l)bpx2 yi
~
2+ r
(2l)px2 y2
~(2
+
r)py2
(2 + r ) [in k + b In (2/) + In px2 - In (2 + r )] - In py2 _ _
Ux=
+ ln(l + r )
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where xst and yf are the supply of tractor and the demand for food in period t, respectively, and Ux is the indirect total discounted utility function for the sequence. Sequence of configurations (y/0) and (y/x) By a similar process, the optimum decision for this sequence can be found as follows. xd2=(2akalpy2/px2) l-B-(l
»1/(1-*)
2py2l(ak/px2)a
+ r)(l-a)
K =-
i/(i- a )
2+ r
(l + a + r) 2l(py2ak/px2)a y\=-
i/(i- a )
~{l-B)jpy2
2+ r (2 + r ) I n \ l - B + (l-a)
2py2l(ak/px2)a
i/d-fl)
-ln(2 + r)
U=1+ r ln(l + r ) -
1+ r
where Uy is the indirect total discounted utility function for the sequence. Utility maximization by individuals and the assumption of free entry (when individuals make decisions at t = 1) have the implication that the total discounted utility of individuals is equalized across the two sequences of configurations. That is, Ux=Uy Let M( represent the number of individuals selling good /'. Multiplying Mt by individual demands and supplies gives market demands and supplies. The market clearing conditions for the two goods over two periods are Mxxs2 = Myxd2
Mxy? = Myy:
t=l,2
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where Mxxs2 and Myxd2 are market supply and demand, respectively, for tractor in period 2, and Mxyd and My] are market demand and supply, respectively, for food in period /. Note that in view of Walras's law one of the three market clearing conditions is not independent of the others. There are three independent equations generated by the market clearing and utility equalization conditions, which determine three unknown variables: MJMy, px2, pyi, where pit is the price of good i in period t in terms of food in period 1 and Mt can be solved using the population size equation M = Mx + My as soon as MJMy is determined. The corner equilibrium values of the three variables are thus given as follows. = Px2
~
(2-a + r)(l-B) b
(\ + r)k(2l)
Mv
ak
My
2-a+r
=
[(2-a + r)/ak2(2l)bJ
Py2
(l-B)
+ab
~
(l + r)(2iy
t/(CD)=ln(/-5) (2-a + r)ln(2-a
+ r)-(2 + r)\n(2 + r) + (ab + l)
ln(2/) + a(lna + 21nfc)] \ +r where U(CD) is the maximum total discounted utility for structure sequence CD, which is derived by inserting the corner equilibrium relative prices into the expression for indirect total discounted utility. Following the procedure for solving for the local equilibrium in market sequence CD, total discounted utility levels in all market sequences can be derived as follows. £ / ( E E ) = ^ ^ + In(/-5) 1+r U(FF)=^^-\ab\n(ab)-(ab \+r
+ l)\a.(ab + l)\
+^ ^ - r ( l +r)ln(/-^-5) + ln(2/-^-5)l 1+r
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ab\n(ab) + (ab + l)\ln(2l-A)-]n(ab + \)] U(EF) = -^^ 11 + hi(/ - B) 1+ r a(21n^ + lng) + ( l - q ) l n ( l - a ) + aZ>ln(l-^) + ln(2/) 1+ r + MI-B) a(2\nk + \na) + (\-a)\n(\-a) C/(FD) =
+ab In l\ab(2l -A-B) -—
+ l] + \n Uab + 2)1 i ^ 1+ r
+abln(ab) + (ab + \)
Hl-A-B)-(2
^DD) = (2 + r ) [ a < 2 t a * + ^
+(
A-B] l
+ r)Hab
(Al)
+ l)
1+ r
'- a ) l n ( 1 ~ a ) UMn(;-^
1+ r U(CD)=\n(l-B) (2-a + r) ln(2 - a + r) - (2 + r) ln(2 + r)
+-
+ (aZ) +1) ln(2/) + a(ln a + 2 In k) 1+ r
Following the method used by Sun et al. (1998), we can prove that in this model the general equilibrium is the local equilibrium that generates the highest total discounted utility. Hence, comparisons between utilities in different structure sequences can partition the parameter space into subspaces within each of which a particular structure sequence occurs in equilibrium. Here we exclude unfeasible sequences, such as EC, FC, CE, CF and CC, which are incompatible with the budget constraint. Also, those sequences that are obviously incompatible with equilibrium are excluded. For instance, all sequences that involve devolution in specialization or in the number of goods cannot generate the highest total discounted utility because they cannot exploit economies of specialized learning by doing and economies of roundaboutness nor save on transaction costs in period 1 compared with alternative sequences. (Al) shows that total discounted
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utility levels in all structure sequences with the division of labor are increasing functions of In k. Hence, they tend to negative infinity as k converges to zero, and In k tends to zero as k converges to one. But all structure sequences with autarky are independent of k. This implies that all sequences with division of labor (ED, FD, DD, CD and others with D or C) cannot be equilibria and only autarkic sequences (EE, FF and EF) can be candidates for general equilibrium if k is sufficiently close to zero. We consider first the case with a value of k that is sufficiently close to zero, and then the case with a value of k that is sufficiently close to one. Parameter k is sufficiently close to zero, so that only EE, FF, and EF need to be considered (i) A comparison between [/(EF) and [/(FF) indicates that [/(EF) > [/(FF) holds if A + B > I - 1 which is assumed in the model. Therefore, FF cannot be a general equilibrium. (ii) A comparison between [/(EF) and [/(EE) yields [/(EF) > [/(EE) .„ , iff ab>pn
l n ( 2 / ) - l n ( 2 / - ^ ) + ln(aZ> + l) =— ^ (A2) 0 ln(2l-A)-\n(ab + \) + \n(ab) where po decreases with ab, which implies that EF is more likely to be the equilibrium than to EE if ab is larger, po becomes negative and ab > po certainly holds if b is sufficiently large, po is a positive number smaller than one if ab is sufficiently small. For the latter case, J7(EE) > [/(EF) becomes possible. Parameter k is sufficiently close to one (i) A is not close to /. (ia) Assume that B < I, A + B is sufficiently close to /, and k is not very large; then [/(FD) tends to be negative and FD cannot be an equilibrium. A comparison between [/(ED) and [/(DD) yields [/(DD) > [/(ED) iff y> n
(A3)
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wherey = abln(/ - A)/(l + r) and yx=-[P + ab ln(2/)/(l + r)]Ir, J3 = a(2\nk + ]na) + (l-a)\n(l-a). (ib) Assume that A and B are not too large and k is close to 1; then
(A4)
ro > n where j ; 0 s { a Z > l n [ a Z > ( 2 / - ^ - 5 ) + / ] + ln[(a6 + 2 ) / - ^ - J 8 ] +(ab +
l)[ln(l-A-B)
- ( 2 + r) ln(a£ +1)] - ln(2/)} /(l + r) + [a6ln(afe)-ln(/-5)] Comparisons between U(ED), C/(FD) and f/(DD) yield [/(ED) > t/(FD) iff ^> y0 C/(DD) > C/(ED) iff r > ri
(A5)
E/(DD) > C/(FD) iff x> ^2 where _y2 =[_y0 - / ? + aMn(2/)/(l + r)](l + r ) ^ . (A5) implies that ED can be an equilibrium only if C/(ED) > E/(FD), £/(DD), or only if Y\ > /o, which contradicts (A4). Hence, under the assumption (ib), ED cannot be an equilibrium. Thus, under assumption (ib) DD is the general equilibrium if y > y~ and FD is the equilibrium if y < y2
(A6)
(ii) Assume that A = I; then only EE and CD are feasible. A comparison between [/(CD) and £/(EE) yields [/(CD) > C/(EE) iff k > k0 (A7) where \nk0 = [(2 + r)ln(2 + r) - (2 - a + r)ln(2 - a + r) - alna ab\n(2l)]/2a. (A1)-(A7) are sufficient for establishing the comparative dynamics of equilibrium in Table 1.
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References Aghion P., Howitt P. (1992): 'A model of growth through creative destruction', Econometrica, 60, pp. 323-351. Barro R. (1991): 'Economic growth in a cross section of countries', Quarterly Journal of Economics, 106:2, pp. 407-443. Barro R., Sala-i-Martin X. (1995): Economic Growth, McGraw-Hill, New York. Benhabib J., Jovranovic B. (1991): 'Externalities and growth accounting', American Economic Review, 81, pp. 82-113. Borland J., Yang X. (1995): 'Specialization, product development, evolution of the institution of the firm, and economic growth', Journal of Evolutionary Economics, 4, pp. 19-42. Dasgupta P. (1995): 'The population problem: theory and evidence'. Journal of Economic Literature, 33, pp. 1879-1902. Gallup J., Sachs J. (1998): 'Geography and economic development', Working Paper, Harvard Institute for International Development. Grossman G. M., Helpman E. (1989): 'Product development and international trade', Journal of Political Economy, 97, pp. 1261-1283. Grossman G. M., Helpman E. (1990): 'Comparative advantage and long-run growth', American Economic Review, 80, pp. 796-815. Grossman G., Helpman E. (1991): 'Quality ladders and product cycles', Quarterly Journal of Economics, 106, pp. 557-586. Jones C. I. (1995a): 'Time series tests of endogenous growth models', Quarterly Journal of Economics, 110, pp. 695-725. Jones C. I. (1995b): 'R&D-based models of economic growth', Journal of Political Economy, 103, pp. 759-784. Josephson M. (1959): Edison; a Biography, McGraw-Hill, New York. Judd K. (1985): 'On the performance of patents', Econometrica, 53, pp. 579-585. Keynes J. (1936): The General Theory of Employment, Interest and Money, Macmillan, London.
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Lucas R. E. Jr (1988): 'On the mechanics of economic development', Journal of Monetary Economics, 22, pp. 3-42. Marshall A. (1890): Principles of Economics, 8th edn, Macmillan, New York, 1948. Morishima M. (1996): Dynamic Economic Theory, Translation, Cambridge University Press, Cambridge. Pritchett L. (1997): 'Where has all the education gone?', World Bank Policy Research Working Paper. Ramsey F. (1928): 'A mathematical theory of saving', Economic Journal, 38, pp. 543-559. Rebelo S. (1991): 'Long-run policy analysis and long-run growth', Journal of Political Economy, 99, pp. 500-521. Romer P. M. (1986): 'Increasing returns and long-run growth', Journal of Political Economy, 94, pp. 1002-1037. Romer P. (1987): 'Increasing returns, specialization, and external economies: growth as described by Allyn Young', American Economic Review, Papers and Proceedings. Romer P. M. (1990): 'Endogenous technical change', Journal of Political Economy, 98, pp. S71-S102. Sachs J., Warner A. (1995): 'Economic reform and the process of global integration', Brookings Papers on Economic Activity, 1. Sachs J., Warner A. (1997): 'Fundamental sources of long-run growth', American Economic Review, Papers and Proceedings, 87, pp. 184-188. Smith A. (1776): An Inquiry into the Nature and Causes of the Wealth of Nations. Reprint, edited by E. Cannan, University of Chicago Press, Chicago, IL, 1976. Solow R. (1956): 'A contribution to the theory of economic growth', Quarterly Journal of Economics, 70, pp. 65-94. Solow R. (1964): Capital Theory and the Rate of Return, Tand McNally. Sun Yang and Yao (1998): 'General equilibrium network of division of labor', Seminar Paper, Department of Economics, Monash University. Yang X. (1996): 'The division of labor and the nature of capital', Seminar Paper, Department of Economics, Monash University.
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Yang X., Borland J. (1991): 'A microeconomic mechanism for economic growth', Journal of Political Economy, 99, pp. 460-482. Yang X., Ng S. (1998): 'Specialization and division of labor: a survey,' in Arrow K., Ng Y.-K., Yang X. (eds), Increasing Returns and Economic Analysis, Macmillan, London. Yang X., Ng Y.-K. (1993): Specialization and Economic Organization, North-Holland, Amsterdam. Yang X., Wang J., Wills I. (1992): 'Economic growth, commercialization, and institutional changes in rural China, 1979-1987', China Economic Review, 3, pp. 1-37. Young Allyn (1928): 'Increasing returns and economic progress', Economic Journal, 38, pp. 527-542. Young Allyn (1998): 'Growth without scale effects', Journal of Political Economy, 106, pp. 41-63.
CHAPTER 17 A NEW THEORY OF INDUSTRIALIZATION*
Heling Shi and Xiaokai Yang* Monash University 1.
Introduction
Casual observation indicates a positive correlation between the level of specialization, the degree of the roundaboutness of production, and the variety of available goods in an economy. The variety of available goods is small and the degree of production roundaboutness is low in a less-developed and autarkic economy where the level of division of labor and specialization is extremely low and everybody self-provides the goods they need. In contrast, the variety of available goods is great and the degree of production roundaboutness is high in a developed economy where the division of labor and specialization are extremely high. This positive correlation may reflect some important mechanism that is essential for the theory of industrialization and that has not been well understood by economists. Two recent lines of research treat two aspects of the mechanism in isolation. The first line, developed by Dixit and Stiglitz (1977), Ethier (1982), Krugman (1981), Romer (1986, 1990), and Grossman and Helpman
Reprinted from Journal of Comparative Economics, 20 (2), Heling Shi and Xiaokai Yang, "A New Theory of Industrialization," 171-189, 1995, with permission from Elsevier. * The authors are grateful to Sherwin Rosen, Wilfred Ethier, Yew-Kwang Ng, T.N. Srinivasan, Ian Ward, Jeff Borland, and two anonymous referees for helpful comments. Thanks also go to the participants of seminars at the University of Chicago, Yale University, University of Pennsylvania, the University of Melbourne, and the University of Hong Kong. The remaining errors are our responsibility.
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(1989) endogenizes the variety of goods by formalizing a tradeoff between the distortions arising from economies of scale and preferences for diverse consumption or economies of complementarity in production. The crucial element of the models is the CES utility or production function in which all goods are not necessities individually, so that the number of goods can be endogenized. However, these models cannot endogenize the level of specialization for each individual. To endogenize the level of specialization, the degree of self-sufficiency of consumers and the range of production of each individual have to be endogenized. Rosen (1978), Baumgardner (1988), Kim (1989), Locay (1990), Yang (1991), Yang and Wills (1990), and Yang and Borland (1991) have endogenized the level of specialization, leaving the number of goods exogenously fixed. Yang's (1991), Yang and Wills' (1990), and Yang and Borland's (1991) static and dynamic equilibrium models have endogenized the level of specialization by formalizing a tradeoff between economies of specialization and transaction costs. However, the variety of available goods is not endogenized in their models because the Cobb-Douglas utility function assumed in their models entails an exogenously fixed number of goods.1 Yang and Shi (1992) have endogenized the variety of consumer goods, the number of traded goods, and the level of specialization by specifying a CES utility function in an equilibrium model with transaction costs, economies of specialization, and consumer-producers. The number of producer goods and the length of the roundabout production chain is not endogenized in their model because producer goods and the CES production function are absent. A natural conjecture is that a synthesis of the thinking along these two lines may enable us to decipher the mechanism behind the concurrent increases in specialization, production roundaboutness, product diversity, and productivity. This idea motivates the paper. According to Smith (1776) and Young (1928), there are three aspects of the division of labor: the level of specialization of individuals, the Their framework involves corner solutions for consumer-producers' decision problems and a myriad of combinations of individual corner solutions as candidates for equilibrium. This makes the tractability of comparative statics rely on completely symmetric models and the existence of equilibrium rely on the explicit specification of models at the cost of generality of model.
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degree of diversity of professions, and what was referred to by Young as the degree of roundaboutness.2 Roundaboutness may be related to the vertical division of labor between upstream and downstream sectors in a long production chain and the second aspect, diversity of professions, may be related to the horizontal division of labor between professional sectors at the same level in this chain. Suppose there is a hierarchical structure of goods and factors, a production chain. At a downstream layer of the hierarchy there are many consumer goods and, at an upstream layer, there are many producer goods. The endogenization of the level of specialization in Baumgardner (1988), Locay (1990), Yang (1991) and Yang and Borland (1991) captures the first aspect of the division of labor. The endogenization of the number of consumer goods by employing the CES utility function in Dixit and Stiglitz (1977) and Yang and Shi (1992) captures the second aspect of the division of labor at the downstream layer of the hierarchy. However, the third aspect of the division of labor and the second aspect of the division of labor at the upstream layer of the hierarchy do not seem to have been captured in previous studies. In order to endogenize roundaboutness we have to endogenize the number of layers of the hierarchy of goods or the length of the production chain in a general equilibrium model. The number of producer goods as well as the level of specialization can be endogenized by introducing the CES production function and producer goods into the Yang model (1991). This is pursued in this paper. In the model to be considered, each individual is assumed to be a consumer-producer and to derive utility from a single consumption good, called food, which uses as inputs labor and either one or two producer goods. One of the two producer goods, called a hoe, can be produced out of labor and the other, called a tractor, needs as inputs labor and another producer good, called a machine tool. Finally a machine tool can be produced out of labor. Each individual is endowed with a fixed amount of labor and has a system of individual-specific Cobb-Douglas-CES production functions for producing the producer and consumption goods. The relationship between labor and producer goods is specified by a 2
As Smith (1776, Ch. I, p. 14) argued, the division of labor leads to the invention and use of a greater variety of roundabout productive machines.
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Cobb-Douglas function. The relationship between hoes and tractors is specified by a CES function. In producing a tractor, machine tools are needed. In producing a machine tool, only labor is needed. Owing to the property of the CES function, hoes and tractors are not individually necessities for the production of food. Hence, if tractors are not employed, then machine tools will not be produced. Where there are economies of specialization, economies of complementarity between producer goods in producing the final good, and transaction costs, a tradeoff exists. A greater degree of horizontal division of labor between the production of hoes and the production of tractors may generate more opportunities for the vertical division of labor between the production of tractors and the production of machine tools, which is associated with a larger number of layers of the hierarchy. This implies higher productivity, generated by a greater variety of sophisticated professional equipment and machines, but, at the same time, greater transaction costs. If transaction efficiency is extremely low, then the gains to introducing more layers of the hierarchy and further horizontal and vertical division of labor are outweighed by transaction costs. In this case, each individual will choose autarky, that is, he will self-provide all producer goods and consumer goods. A tradeoff still exists in autarky between economies of specialization and increasing returns to a variety of producer goods. If a large number of producer goods are produced in autarky, a person's level of specialization in producing each good must be low. Thus, in autarky the foregone economies of specialization due to the production of many producer goods at many layers of the hierarchy of goods outweigh the gains to a variety of producer goods. Therefore, in autarky, each individual will choose a hierarchy of goods with a small number of layers and a small number of producer goods at each layer, so that he can capture more economies of specialization by concentrating his limited labor in a few activities. This implies that some sophisticated producer goods based on a large number of layers of the hierarchy are unlikely to exist in autarky. For example, it is likely that only hoes will be produced but no tractors or machine tools will be produced in autarky. If transaction efficiency is extremely high, then people may choose a greater degree of horizontal as well as vertical division of labor and in
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the meantime maintain each individual's level of specialization at a high level through the division of labor between many different specialists. Therefore, a high transaction efficiency may bring out some new layers in the hierarchy of goods and new producer goods at each layer in the hierarchy. For example, tractors and hoes are produced if a higher transaction efficiency generates a higher level of horizontal division of labor in producing tractors and hoes. The higher level of horizontal division of labor will create scope for the vertical division of labor in producing both downstream goods, tractors, and upstream goods, machine tools. The emergence of the new layers and new producer goods implies new technology and new industries that are associated with an industrialization process. Hence, a general equilibrium model may be used to predict concurrent increases in the number of producer goods at each layer, in the level of specialization, and in the number of layers of the hierarchy of goods. We will show that, for a sufficiently large elasticity of substitution between producer goods, the number of producer goods at each layer and the number of layers of the hierarchy of goods increase as transaction efficiency is improved. Further, trade dependence, productivity, and the level of specialization for individuals increase with improvements in transaction efficiency. In section 2, a model with consumer-producers, increasing returns to specialization, and transaction costs is specified. Section 3 investigates individuals' decisions and equilibrium. The final section concludes the paper. 2.
A Model with a Variable Hierarchical Structure of Goods
Consider an economy with M ex ante identical consumer-producers. There is a hierarchical structure of consumer and producer goods and factors. There are at least three and at most four layers of the hierarchy. The first layer consists of a consumer good z and a utility function. Good z may be considered as food. The second layer consists of at least one and at most two producer goods: x, called a tractor, and y, called a hoe, and labor spent producing food at the first layer. In producing tractors, producer good w, called a machine tool, at the third layer and labor are needed as inputs. In producing hoes, only labor is needed. In
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producing machine tools w, labor at the fourth layer is needed. Hence, the third layer consists only of labor and the fourth layer does not exist if tractors x are not produced. The fourth layer will exist only if tractors are produced. The hierarchy for autarky where each person self-provides all goods is depicted in panel (b) of Figure 1. Let us specify the hierarchy layer by layer. The first layer consists of food and a utility function. The respective amounts of food self-provided, sold, and bought are z, zs, and zd. The fraction 1 - k of a unit of good purchased disappears in transit because of transaction costs. Thus, kzd is the amount an individual receives from the purchase of food. The amount consumed of food is z + kzd. The utility function is u = z + kz - en,
(1)
where n is the number of goods purchased by an individual and c is a fixed cost for a transaction in terms of utility loss. The fixed transaction cost coefficient can be taken to be a cost for finding the price of a good, which is proportional to the number of transactions and independent of quantities traded. Later, it will be shown that the equilibrium will jump between autarky and the complete division of labor as transaction efficiency changes and will not involve a gradual evolution of division of labor if c = 0. The second layer consists of at least one and at most two producer goods. A, Cobb-Douglas-CES production function establishes the relationship between the top layer and the second layer z~z
(x + kxdj+(y
+ kydj
lf~S),
p,Se
(0,1),
a>\,
(2) where x and y are the respective amounts of tractor and hoe selfprovided and xd and yd are the respective amounts of the two producer goods purchased from the market. The transaction cost coefficient for
(c) Structure C
(a) Structure/4
(e) Structure
z
y
K
t
(xlz) X
^ x X
y
I
l
> ly
(x/wz) J
z
X
y i w
l
t'
t* I (zlx)
(b) Structures
(d) Structure/)
(w/z) (f) Structure
Figure 1: Different Patterns of Industrial Structure
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H. Shi, X. Yang
those goods is 1 - k. Hence, kxd or kyd is the amount an individual receives from the purchase of a producer good. The superscript s stands for sold. 4 is the labor spent producing food and z + / is the output level of food. zs is the amount of food sold. 4 is the quantity of labor allocated to the production of good z. As in Yang (1991), /, is defined as a person's level of specialization in producing good i. Taking [(x + kxd)p + (y + kyd)p]Vp to be a composite producer good with a variety of inputs, (2) is a Cobb-Douglas production function with the composite producer good and labor 4 as inputs. The degree of economies of specialization is represented by a. The composite producer good is produced with a CES production function with a variety of producer goods. The elasticity of substitution between producer goods is 1/(1 - p). This elasticity increases with p, whose reciprocal XIp can be taken to be a measure of the degree of economies of input variety or complementarity between producer goods. Note that food z is a consumption input into the utility function at the top layer of the hierarchy and is an output of the production function in (2). The third layer of the hierarchy consists of, at most, machine tools w which are used to produce tractors x, and labor spent producing tractors and hoes. The relationship between this layer and the second layer is specified by the Cobb-Douglas production functions for tractors and hoes. Labor and machine tools are necessities in producing tractors. For the production of hoes, only labor is needed. The production functions are x + xs =(w+kwd)slf-S)
and y + f = 1° ,
(3)
where x + xs and y + y5 are the respective output levels of the two producer goods, and w and wd are the respective amounts of machine tools self-provided and bought, and 4 and ly are the respective quantities of labor spent producing the two producer goods. The transaction technology for good w is the same as for goods x and y. The degree of economies of specialization is represented by a > 1. Note that x and y are inputs into the production function in (2) and are outputs of the production functions in (3). The fourth or bottom layer of the hierarchy consists only of labor spent producing machine tools. The relationship between this layer and the third layer is specified by the production function for machine tools
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w +^= C
(4)
s
where w + w is the output level of machine tools, lw is the labor spent producing machine tools, and a > 1 is the degree of economies of specialization in producing machine tools. Note that machine tools are an input into the production function of good x in (3) and an output of the production function in (4). The labor endowment constraint is l2+lx+ly+lw=
I, h e [0, /],
0 < i, f, f,
V/ = z,x,y, w,
(5)
where / > 1 is a constant. According to (2)-(5), the total factor productivity of z and labor productivity of x and w increase with a person's level of specialization in producing these goods. As in Yang (1991) and Yang and Borland (1991), free entry for all individuals into any sector and a large M are assumed. Combined with the assumptions that each consumer-producer is able to produce any and every good and all consumer-producers are ex ante identical in all aspects, these assumptions ensure a Walrasian regime. 3.
Individual Decision Problems and Equilibrium
This section charaterizes the set of feasible market structures for the economy described in the previous section. Each individual makes a decision on which goods to produce and on his demand for, and supply of, any traded good. A given pattern of production and trade activities for any individual is defined as a configuration. There are 212 = 4096 combinations of zero and nonzero values of x, Xs, xd, y, y*, yd, z, zs ,zd, w, w5, wd and therefore 4096 possible configurations. Individuals make decisions on production and trade in order to maximize utility. Since the quantity of a good that is self-provided is distinguished from the quantity traded of the same good, individuals' optimal decisions are always corner solutions. It is assumed that an individual sells at most one good, does not buy and sell or self-provide the same good, and does not self-provide intermediate goods if he does not produce the final good or if he sells the final good. This assumption rules out the interior solution and myriad corner solutions from the list of candidates for the optimum decision. The combination of configurations of the M individuals in the economy
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is defined as a market structure or simply structure. A feasible market structure consists of a set of choices of configurations by individuals such that for any traded good, demand for the good is matched by supply of the good. There are 13 possible configurations. The combinations of the configurations generate 10 structures. As shown in Yang (1991), for each structure, there is a corner equilibrium that equilibrates demand and supply and yields the same utility for all individuals. In this section we first solve for a corner equilibrium in each of the 10 structures. Then, we identify the equilibrium from amongst these corner equilibria. Seven of the structures are depicted in Figure 1, where the rectangles represent configurations and the curves that connect rectangles represent the flow of goods. The other three structures are variants of particular structures depicted in Figure 1. In order to save space, we do not depict them in Figure 1. 3.1. Autarkical structures A andB Autarky implies xs = y = zs = w" = xd = yd = zd = wd = 0. There are two configurations that meet the criterion for autarky: (i) a configuration, denoted A, with zero quantities traded of all goods, x = w = 0, and y,z> 0, and (ii) a configuration, denoted B, with zero quantities traded of all goods, x, y, w, z > 0. Panels (a) and (b) in Figure 1 depict the two structures. It is easy to see that there are only three layers in structure A and there are four layers in structure B. The first layer in structure A consists of food z. Its second layer consists of hoes y and labor lz. Its third layer consists of labor ly. The first layer in structure B consists of food. Its second layer consists of tractors x, hoes y, and labor lz. Its third layer consists of machine tools w and labor lx and ly. Its fourth layer consists of labor lw. Let the quantities traded of all goods be zero, x = w = 0, and y, z > 0 in (l)-(5) and insert (2)-(5) into (1); the decision problem for configuration A can be identified. This problem yields the optimum individual decisions for configuration^, given by 4 = (1 - a)l,
ly = al, (1
z = [(l-a)/r -°y, y = (al)\
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Industrialization
u(A)=z = [(l-af-a)aa[]a,
(6a)
where u(A) is the per capita real income in configuration, or structure, A. Similarly, an individual can solve for the corner solution for configuration B, given by lx = STp-il/(g ly=flx,
lw =
u(B) = [I'/^lf
+
Sfp^h). Slx/(\-S),
+ la/ J'P(l-lx-ly-
lw)a(1 ~ \
(6b)
where/= S,Sp!(ap'l\l - sfx-a5p)l{ap-x\ g = (1 - S){[S/(l - S)]aSp +f(S, a, p)ap), h = [5I(\ - S)] + 1 + / , and u(B) is the per capita real income in structure B. 3.2. Structure C Denoting a configuration in which an individual sells hoes and buys food (y/z) and a configuration in which an individual sells food and buys hoes (z/y), structure C consists of configurations (y/z) and (zly). Panel (c) in Figure 1 depicts this structure. There are two steps in solving for the corner equilibrium in structure C. Individuals' optimum decisions for each of the two configurations are first solved. Then individual demand and supply, the market clearing conditions, and utility equalization condition are used to solve for the corner equilibrium. Configuration (y/z) Letys, zd> 0, ly = L = I,y = yd = x = xs = xd = lx = z = zs = lz= w = ws = w = lw = 0; the individual decision for configuration (y/z) is therefore given by z
u
Py/
(budget constraint or trade balance)
ys = l",
ly = I (production function and labor constraint)
y(Py) = kzd - c = kpyf - c (indirect utility function), (7) where/*, is the price of the producer good i in terms of food and uy(py) is the indirect utility function for configuration (y/z). Note that a person
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H. Shi, X. Yang
choosing (y/z) buys only one good, so that the coefficient of c in the indirect utility function is 1. Configuration (z/y) Let z\ z, yd>0,lz=l,zd = y=ys=ly = x=Xs = xd=lx=W!= wd = Iw= 0; the individual decision problem for configuration (z/y) is maxu z = z-c = lf~S)(kyd)s-zs-c = lf~S)(kyd)5 - p yd -c yd
y
s.t. z + zs = lf~6) (kyd ) s , s
d
z =pyy
lz = l
(production function)
(budget constraint or trade balance). (8)
The optimum decisions are 4 = /, / = p / , yd =
(6k?/py)m-»r
uz(py) = kXsk'ipyf^ r -Py(SksiPyf(x - V - c,
(9)
where uz(py) is the indirect utility function for configuration (z/y). The corner equilibrium in structure C The utility equalization and market clearing conditions uy(py) = ujjyy) and Myys = M^ yield the comer equilibrium py = <7*(1 - S)x ~ slk2S'\ u(Q = S?(l-S)iSk25la-c,
(10)
Myz = (1 - d)ISk (11)
where u(C) is the percapita real income in structure C, py is the comer equilibrium price of a hoe in terms of food, and Myz is the comer equilibrium relative number of individuals selling hoes to those selling food. 3.3. Structure D This structure consists of configuration (xlz) and configuration (zlx). A person choosing configuration (xlz) sells tractors, self-provides machine tools, and buys food. A person choosing configuration (zlx) sells food, self-provides food and hoes, and buys tractors. Panel (d) in Figure 1
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depicts structure D. Following the two-step approach developed in Subsection 3.2, we can solve for the corner equilibrium of this structure, given by the system of simultaneous equations 5lj?p-1 = (i _ S)Vxd + t?\xdfly =
map
~ °,
d (p m p 1
p\x ) - " - \ d
Px = ae(x
Mj' =
r- \i - iyf ~ ^[(kxOy + i;p r~\
Mxfs(\-8ff-l-e>r,
Mz + Mx = M,
u(D) = [(kxdr+ r/fv-iyf'-^-p^-c,
(12)
where fi= [kp+ ' ^ ( 1 -
-kspx]r = c,
Py=Pxs,
H. Shi, X. Yang
450
MJMX = &\\u(E) = kp/a(1
where s = ff\\ - S)
-\q
5)a('"s>lalxd,
MyIMz = ///",
c,
(13) (X
p)W
= {\- S)(Skf ~ \l + s^~ X'-
~ \ xd
= \dks [l + s"ir-» ]*"'}Uil~S) l°pW-*, and / = xV / ( "' 1 } . xd a n d / are the respective quantities of tractors and hoes bought by a person choosing configuration (z/xy), and u(E) is the per capita real income in structure E. 3.5. Structure F This structure consists of configuration (x/zw), configuration (w/z), and configuration (z/x). Configuration (z/x) is the same as in structure D. A person choosing configuration (x/zw) sells tractors and buys machine tools and food. A person choosing configuration (w/z) sells machine tools and buys food. Panel (f) in Figure 1 depicts structure F. Following the two-step approach in subsection 3.2., the corner equilibrium for this structure is given by the system of simultaneous equations px = Sa/pklay/p-^a°-p\l
-
lyf'p-ws*
x[S(l- ly)l(l - S)ly- 1]<"-l)lp(\ px = (pJS)\(klapw + c)l(\ -
8?-alp,
S)lat%
pw=[si;p-x /(i - *)]'" (/ - /„ fp+(l-S)a i Ma -Px[sryp-l(i-iy)/(\-s)-i;pJp xd = [sr/~l (i-iy) i(\ wd =
/k2i°,
s)~i;pJp/k,
[5kYx-%lPwf^\
MJMX = wdlla, MJMX = (kwd)sf(1" V , u(F) = kpJa-c,
Mx + Mw+Mz = M, (14)
where /_,, is the labor spent producing hoes by a person choosing configuration (z/x), xd is his demand for tractors, wd is the demand for machine tools of a person choosing configuration (x/zw),pt is the price of
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good i in terms of food, and u(F) is the per capita real income in structure F. The system of nine equations yields the corner equilibrium values of pw,px, xd, wd, ly, Mx, Mw, Mz, and u{F) as functions of a, S, p, k, I, and M. Structure E involves complete specialization in the production of food and hoes and incomplete specialization in the production of machine tools and tractors, while structure F involves complete specialization in the production of tractors and machine tools and incomplete specialization in the production of food and hoes. Incomplete specialization for a person in producing good i is defined as /, < / for this person. The number of layers of the hierarchy of goods and factors and the number of goods are the same for the two structures. 3.6. Structure G This structure consists of configurations (z/xy), (y/z), (xlzw), and (wlz). Configurations (x/zw) and (wlz) are the same as in structure E. Configurations (x/zw) and (wlz) are the same as in structure F. Panel (g) in Figure 1 depicts structure G. The corner equilibrium for this structure is given by wd =
[Sksla(l-%lpw]m^
/=
{8ks[i+(pjPy^p%x^S)r,
yd = (pjpw)m-p)xd, P, = (PylS)X(c + klapy)lf(\ - S)]l~Sk-\ Py = [(k>-%f*l-d-lf-^Px, pw = py, My/Mz = ylMx = yd(kwd)slxdf\ MJMX = wd/la, MJM, = (kwd)sla(l ~ S)lxd, u(G) = kpja - c, d
d
(15)
where x and y are the respective demands for tractors and hoes by a person choosing configuration (zlxy), pt is the price of good / in terms of food, and u(G) is the per capita real income in structure G. The 10 equations in (15) yield the corner equilibrium values of pw, px, py, MJMX, My/Mx, MJMX, wd, xd, yd, and u(G).
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H. Shi, X. Yang
Structure G involves complete specialization in the production of all goods. There are four layers and four goods are produced in this structure. We refer to this structure as the complete division of labor. 3.7. Structures
B',E',G'
Configuration or structure B' is exactly the same as structure B except y is not produced. Structure E is exactly the same as structure E except y is not produced. Structure G' is exactly the same as structure G except y is not produced. As shown in Yang (1991), the corner equilibrium with the maximum per capita real income is the general equilibrium. A comparison between per capita real income in structures A,B,C, D, E, F, and G will provide Table 1: Simulation of Structural Transformation a: *
p.
0.3
1.2 0.5
1.5 0.7
0.3
1.8
0.5
0.7
0.3
0.5
0.7
8=03 0.1-0.2
B
B
B
B
B
A
B
B
A
0.3
B
B
B
B
B
A
F
F
A
0.4
B
B
B
B
B
A
G
F
A
0.41 ~ 0.44
B
B
B
B
B
A
G
G
F
0.45-0.5
B
B
B
B
B
A
G
G
G
0.6
B
B
B
B
B
C
G
G
G
0.61-0.62
B
B
B
B
F
C
G
G
G
0.63-1.0
B
B
B
G
G
C
G
G
G
0.1-0.29
B
B
B
B
B
A
B
B
A
0.3-0.33
B
B
B
B
B
A
F
D
A
B
A
F
F
A
D
A
G
G
G
G
G
G
G
G
G
£=0.5
0.34 - 0.5
B
B
B
B
0.6 - 0.62
B
B
B
F
0.7
B
B
B
G
F
A
0.8
B
B
B
G
G
C
0.9
B
B
B
G
G
G
G
G
G
1.0
G
B
B
G
G
G
G
G
G
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Table 1: {Continued) a: k
p.
1.2
1.5
0.3
0.5
0.7
0.1-0.3
B
B
B
0.4
B
B
0.5
B
B
0.6 ~ 0.65
B
0.66
B
0.67 ~ 0.7 0.8
0.3
1.8
0.5
0.7
0.3
0.5
0.7
B
B
A
B
B
A
B
B
B
A
B
D
A
B
B
B
A
D
D
A
B
B
D
D
A
F
F
A
B
B
F
D
A
G
G
A
B
B
B
F
F
A
G
G
A
B
B
B
G
G
A
G
G
G
0.9
B
B
B
G
G
G
G
G
G
0.91 ~ 0.92
F
B
B
G
G
G
G
G
G
0.93-1.0
G
B
B
G
G
G
G
G
G
<5=0.7
Note. Structure A: Autarky with one consumer good, one producer good, three layers; Structure B: Autarky with one consumer good, three producer goods, four layers; Structure C: Three layers and complete specialization in producing z, y; Structure D: Four layers of the hierarchy and no complete specialization; Structure E: Four layers of the hierarchy and complete specialization in producing z, y, but incomplete specialization in producing x, w; Structure F: Four layers of the hierarchy, complete specialization in producing x, w, but incomplete specialization in producing z, y; Structure G: Complete division of labor.
the comparative statics results, namely the relationship between equilibrium structure and values of parameters k, a, S, and p. However, since the corner equilibria in structures D, E, F, and G can only be solved numerically, comparisons can be made only via simulations of the corner equilibria on the computer. The results of such comparisons are summarized in Table 1. The first three rows give values of parameters S, output elasticity of the composite producer good; a, degree of economies of specialization; and p, positively related to the elasticity of substitution between producer goods. The first column of Table 1 gives values of parameter k, transaction efficiency. The other entries indicate equilibrium structures of industrial structure and are denoted A, B, C, D, E, F, and G. Here, it is assumed that / = 100 and c = 30. The simulation demonstrates that the per capita real income in structures E, B', E', and G' is always smaller than in some other
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H. Shi, X. Yang
structures if transaction efficiency is the same for all goods. Therefore, the corner equilibria in these four structures cannot be the general equilibrium. Additional simulations show that if c = 0 or if c is too large, then structures with an intermediate level of division of labor, such as structures C, D, and F, will not be the general equilibrium, while the general equilibrium is either autarky, if k is sufficiently small and c = 0 or if c is sufficiently large, or the complete division of labor, if k is sufficiently large and c = 0. The level of division of labor for the economy as a whole can be characterized by three components: the diversity of professions (the number of goods at a layer), the roundaboutness (the number of layers of the hierarchy of goods and factors), and the level of specialization for individuals. According to this definition, structure C, which involves complete specialization in producing z and y but produces only two goods and generates three layers of the hierarchy of industrial structure, is roughly equivalent, in terms of the level of division of labor, to structure D, which generates four layers of the industrial structure, produces four goods, and involves incomplete specialization in the production of all goods. With this definition, Table 1 yields the following result. Proposition 1: (i) If the fixed transaction cost is too high, then the equilibrium is autarky. For a high elasticity of substitution and significant economies of specialization, tractors and machine tools are not produced in autarky. (ii) If the fixed transaction cost is sufficiently low, then the equilibrium jumps from autarky to the complete division of labor as transaction efficiency is improved. (iii) For an intermediate level of fixed transaction cost, the level of division of labor evolves as transaction efficiency is improved. The path of such an evolution depends on the elasticity of substitution parameter p and the degree of economies of specialization a. The evolution takes the path A=> C (F) => G if p and a are large and takes the path B => Z> => F => G if p and a are small.
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(iv) The higher the degree of economies of specialization, a, the faster the evolution in the division of labor. This proposition implies that if the elasticity of substitution between producer goods and the degree of economies of specialization are high, then the evolution of division of labor involves the evolution of specialization as well as the evolution of industrial hierarchy. New layers of the hierarchy and new producer goods at each layer emerge as the division of labor evolves with an improvement in transaction efficiency. If the elasticity of substitution and the degree of economies of specialization are low, then the evolution of the division of labor involves only the evolution of specialization, keeping the number of goods at each layer and the number of layers unchanged. Part (ii) of the proposition implies that if the fixed transaction cost is small, economies of division of labor increase faster than transaction cost does as division of labor increases, so that the second-order condition for an intermediate level of division of labor, which says that transaction cost increases faster than benefit of division of labor does, is not satisfied. Hence, the level of division of labor jumps between the two corners: autarky and complete division of labor. Part (iii) of the proposition implies that if a hoe is not a good substitute for a tractor, then both tractor and hoe have to be produced in autarky, which is the equilibrium when transaction efficiency is low, and therefore the evolution in the number of goods is impossible as transaction efficiency is improved. If a hoe is a good substitute for tractor, indivduals can produce only hoes to exploit economies of specialization in autarky, so that the evolution in the number of goods is possible when transaction efficiency is improved. Where the elasticity of substitution between producer goods and the degree of economies of specialization are quite high, Proposition 1 predicts an industrialization process in which an economy evolves from autarky, where each individual self-provides few goods and does not use sophisticated producer goods, into an industrialized stage, where each individual specializes producing one good and is heavily dependent on the market and trade, and the economy is highly commercialized. In the industrialized stage, there are many layers of the industrial hierarchy, and many sophisticated producer goods are produced by specialist producers.
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Furthermore, computer simulations show that the income share of producer goods, or the capital-labor ratio, the income share of transaction volume, the extent of the market or trade dependence, and the per capita real income increase as the economy evolves from autarky to the complete division of labor with an improvement in transaction efficiency.3'4 As transaction efficiency is improved, the number of professional sectors that are distinct from each other increases. That is, the degree of diversity of economic structure increases. Meanwhile, the level of specialization and the degree of integration of the economy increase. In addition, this model shows that a dualistic economic structure emerges from the transitional stage from autarky to the complete division of labor. For instance, individuals choosing configuration (zlx) have a lower level of specialization than those choosing configurations (x/wz) and (w/z) in structure F. Hence the labor productivity of x and w are higher than j> and z and people selling x and w have a higher per capita commecial income than those selling x despite the same real income for all individuals. Such a dualistic structure does not occur in structures A, B, C, D, or G. Therefore, many economic phenomena associated with the process of industrialization can be predicted by the model presented in this paper. Looking at a particular example from Table 1 where S= 0.5, a = 1.5, and p= 0.7, it is possible to see how equilibrium evolves as k increases from 0 to 1. If k < 0.7, then the equilibrium is structure A and no tractors or machine tools are produced. Each person self-provides food using hoes as an input. As k increases to 0.8, the equilibrium changes to structure C where it is still the case that no tractors or machine tools are produced. However, professional farmers who produce only food and workers who produce only hoes emerge. People have higher trade dependence and more diverse professions. As k tends to 0.9, the equilibrium structure changes to G and professional producers of tractors and machine tools emerge. At this stage, new producer goods both at the second and third layers emerge. Also, a new layer of the industrial 3
As Young (1928) argued, the increase in the capital-labor ratio is in essence an aspect of the evolution of the division of labor in producing producer goods and the increase in the roundaboutness of production or in the length of the production chain. 4 The tables which report the simulations are available from the authors on request.
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hierarchy emerges, so that the level of division of labor is higher than in structure C. Trade dependence, the degree of integration, the degree of diversity of economic structure, and productivity increase. Moreover, the income share of producer goods increases. Note that if the output elasticity of intermediate goods, 8, is low, the elasticity of substitution is high, and the degree of economies of specialization is at an intermediate level, the evolution of the division of labor may end up with structure C rather than G. For instance, this happens for 5= 0.3, a = 1.5, and p = 0.7. Also, a jump in the level of division of labor may happen as transaction efficiency is improved. For instance, the equilibrium jumps from structure B to G as k increases from 0.9 to 1.0 when S= 0.5, a = 1.2, and p = 0.3, and it jumps from structure A to G as k increases from 0.5 to 0.6 when S- 0.5, a = 1.8, and p= 0.7. Not surprisingly, the general equilibrium is Pareto optimal. This result, together with Proposition 1, implies that a decentralized market will fully exploit economies of division of labor, consisting of economies of specialization, economies of complementarity, and economies of roundaboutness, if the economies of division of labor outweigh the transaction costs. 4.
Concluding Remarks
A distinctive feature of our approach is that labor is taken to be specific for an individual person and an individual activity, and each individual is a consumer-producer. This, combined with increasing returns to specialization and transaction costs, makes the concrete forms of utility and production functions differ from configuration to configuration despite ex ante identical endowments, preferences, and production functions for all individuals. Our framework, without the dichotomy between pure consumers and pure producers and with a hierarchical structure of the Cobb-Douglas-CES production function, has important implications for the concepts of equilibrium and Pareto optimum. For our framework, the concepts of equilibrium and Pareto optimum are not only related to efficient allocation within a certain market structure, but are also related to the determination of the efficient market structure, the optimum
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diversity of producer goods at a layer and the optimum length of the production chain. Hence, our model demonstrates that an important function of the free market is to exploit economies of the division of labor, which consist of economies of specialization, economies of a variety of producer goods at a layer, and economies of roundaboutness. Our model is powerful in predicting many of the phenomena associated with the industrialization process. It can generate increases in productivity, specialization, the number of goods, the length of production chain, the diversity of economic structure, trade dependence, the capital-labor ratio, the degree of economic integration, and the degree of commercialization, and the emergence of new goods and related new technology. In addition, our model predicts a dual economic structure over the transition stage from autarky to the complete division of labor.5 The cost of such predictive power is enormous. The concurrence of so many interesting phenomena can be predicted only by comparative statics of our model, while no analytical solution of the comparative statics can be obtained even for an extremely specific model. Many works on rethinking trade and growth theory (Dixit and Stiglitz, 1977; Krugman, 1981; Ethier, 1982; Grossman and Helpman, 1989) show that the analytical results may be obtained from a specific symmetric model that endogenizes the number of goods. Yang (1991) has shown that analytical results may be obtained from a specific asymmetric model involving only final goods that can endogenize the level of specialization. Yang and Shi (1992) have shown that analytical results may be obtained from a specific symmetric model involving only final goods that can endogenize the level of specialization as well as the number of goods. Our experience indicates that it is impossible to obtain analytical results if we attempt to simultaneously endogenize the number of producer goods, the number of layers of a hierarchy of goods, and the level of specialization in a general equilibrium model. There are big tradeoffs between the degree of endogenization, the degree of generality, and the degree of tractability. However, it seems acceptable to reduce generality and tractability in order to achieve an increase in the degree of 5
The models of Murphy et al. (1989a,b) can also predict a dual economic structure and increases in productivity and the income share of the modern sector.
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endogenization that substantially contributes to the predictive power of our model.
References Baumgardner, James, "The Division of Labor, Local Market, and Worker Organization." Journal of Political Economy 96, (3): 509-527, June 1988. Borland, Jeff and Yang, Xiaokai, "The Organization of Production and Exchange." working paper, Department of Economics, Monash University, 1990. Dixit, Avinash and Stiglitz, Joseph, "Monopolistic Competition and Optimum Product Diversity." American Economic Review 67, (3): 297-308, June 1977. Ethier, Wilfred, "National and International Returns to Scale in the Modern Theory of International Trade." American Economic Review 72, (3): 950-959, June 1982. Grossman, Gene, and Helpman, Elhanan, "Comparative Advantage and Long-Run Growth." Journal of Political Economy 97, (6): 1261-1283, Dec. 1989. Kim, Sunwoong, "Labor Specialization and the Extent of the Market." Journal of Political Economy 97, (3): 692-709, June 1989. Krugman, Paul, "Intra-industry Specialization and the Gains from Trade." Journal of Political Economy 89, (5): 959-973, Oct. 1981. Locay, Luis, "Economic Development and the Division of Production between Households and Markets." Journal of Political Economy 98, (5): 965-982, Oct. 1990. Murphy, Kevin, Shleifer, Andrei, and Vishny, Robert, "Industrialization and the Big Push." Journal of Political Economy 97, (5): 1003-1026, Oct. 1989a. Murphy, Kevin, Shleifer, Andrei, and Vishny, Robert, "Income Distribution, Market Size, and Industrialization." Quarterly Journal of Economics 104, (3): 537-564, Aug. 1989b. Romer, Paul, "Increasing Returns and Long-Run Growth." Journal of Political Economy 94, (5): 1002-1037, Oct. 1986. Romer, Paul, "Endogenous Technological Changs." Journal of Political Economy 98, (5): S71-S102, Oct. 1990.
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Rosen, Sherwin, "Substitution and the Division of Labor." Economica 45, (179): 235-250, Aug. 1978. Rosen, Sherwin, "Specialization and Human Capital." Journal of Labor Economics 1, (1): 43-49, Jan. 1983. Smith, Adam, In E. Carman, Ed., "An Inquiry into the Nature and Causes of the Wealth of Nations." Chicago: The Univ. of Chicago Press, 1976,1776. Yang, Xiaokai, "Development, Structural Changes, and Urbanization." Journal of Development Economics 34: 199-222, 1991. Yang, Xiaokai, and Borland, Jeff, "A Microeconomic Mechanism for Economic Growth." Journal of Political Economy 99, (3): 460-482, June 1991. Yang, Xiaokai, and Ng, Yew-Kwang, "Theory of the Firm and Structure of Residual Rights." Journal of Economic Behaviour and Organization 1994 (forthcoming). Yang, Xiaokai, and Shi, Heling, "Specialization and Product Diversity." American Economic Review 82, (2): 392-398, May 1992. Yang, Xiaokai, and Wills, Ian, "A Model Formalizing the Theory of Property Rights." Journal of Comparative Economics 14, 177-198, 1990. Young, Allan, "Increasing Returns and Economic Progress." Economic Journal 38, (152): 527-542, Dec. 1928.
Part 8
Infrastructure, Labor Surplus, Insurance, and the Trade-off Between Leisure and Income
CHAPTER 18 POPULATION DENSITY AND INFRASTRUCTURE DEVELOPMENT*
C. Y. Cyrus Chu* National Taiwan University and Academia
Sinica
1. Introduction Many economists have observed the close connection between population density and early economic development. Kuznets (1960), Simon (1977, 1981) and Ng (1986) all argued that a large population size spurs technological changes. The key idea is that the technologies invented have the nondepletable property: when the population density in an economy is large, the per capita costs are low both for the ex ante R&D activities which facilitate an invention, and for the ex post maintenance of a new technology. Although the above idea is intuitively appealing, so far the existing formal models cannot generate the features of economic development described in the work of Ester Boserup (1981). Boserup's book on population and technological change might have become a "classic," which by an anecdotal definition refers to something "everyone cites but few read carefully." Indeed in the recent literature concerning the relationship between population and growth, Boserup's book is in the reference list of each paper; but we believe that the existing literature has not embodied the propositions in her book, and hence the implications obtained are not always compatible with her observations.
* Reprinted from Review of Development Economics, 1 (3), C. Y. Cyrus Chu, "Population Density and Infrastructure Development," 294-304, 1997, with permission from Blackwell. * I thank Xiaokai Yang for his helpful comments and suggestions.
463
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C. Y. C. Chu
Boserup reviewed technical changes in America, in Arab and African countries, in East and South Asia, in Europe, and in Oceania, from ancient hunter-gather societies to the mid-twentieth century contemporary world. The assortment of technologies she studied was also broad, including agriculture and general industrial sectors, and technologies related to urbanization and transportation. She repeatedly argued that only with a large population can an economy support various types of infrastructure, which are usually very labor-intensive. Boserup also noticed the nondepletable property of technology, and she put more emphasis on the nondepletability of the infrastructure (particularly transport) technology. She asserted that before the Industrial Revolution, the main advantage of a dense population was "the better possibilities to create infrastructure" (p. 129). In fact, Boserup argued that the irrigation technology for agriculture (p. 66), the building and maintenance of roads (p. 67), the canalization of a river (pp. 68, 97), and the laying of the railroad system (p. 132) were all possible only with the support of a large population.1 The same point has also been made by Lee (1986, p. 102), who wrote "the larger the population engaged in non-food-producing activities, the greater the possible division of labor, and the greater the possibilities for technological advance." The growth of infrastructure improves overall transaction efficiency, which in turn facilitates the division of labor and economic development. Such an important connection between population density and the division of labor was noted by Goodfriend and McDermott (1995), but the analysis therein did not have a microeconomic foundation compatible with what was described in Boserup. As Boserup (p. 120) noted, before the Industrial Revolution agriculture was only a part-time occupation; members of peasant families must have spent a considerable share of their time producing tools, clothing, and household equipment, and repairing dwellings. After the Industrial Revolution, most of the abovementioned commodities and services could be provided by specialized individuals or factories. Since individual 1
For instance, Boserup (1981, p. 68) mentioned that more than one million workers were mobilized for the construction of the imperial canal in China in the sixth century AD. This construction made long-distance bulk transport of food and other products possible. A sparse population can never construct and maintain such a labor-intensive infrastructure.
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specialization is an endogenous decision in a free market, the synchronized pattern of infrastructure construction and individual specialization should be modeled with care. The purpose of this paper is to set up a model which captures the following features: (1) there is a publicly accessible transport technology; (2) such a technology is more advanced when there is more specialized labor in it; (3) a large population size (or density) facilitates the development of the transport technology; (4) an improved transport technology can improve the transaction efficiency of other goods which in turn facilitates a finer division of labor; and (5) because there are economies of specialization, a finer division of labor can increase per capita income. It is important to note that the amount of labor specialized in the infrastructure sector is a result of each individual's micro job-decision. When an individual is "specialized," his or her job decision is in fact a corner solution. The scenario Boserup described is a process of an ever-finer division of labor, beginning before the existence of the infrastructure sector and continuing as this sector advances. Since such a process involves switches from one (job) corner decision to another, it is natural to resort to the inframarginal analysis of Yang and Ng (1993) to proceed with our analysis. Later I will explain why a higher population density encourages a finer division of labor. It is also important to note that the positive impact of population density on economic development proposed here is different from the size effect mentioned in Romer (1990) or Backus et al. (1992). The size effect in modern growth theory emphasizes the property of increasing return in the production process, which does not have much to do with the size of the population. In the present model, the tradeoff between economies of specialization and transaction costs implies that the extent of the market and the level of division of labor are determined by transaction efficiency. But transaction efficiency is itself endogenously determined by the extent to which a specialized infrastructure sector can survive and develop, which is in turn determined by the extent of the market. The notion of general equilibrium is powerful to describe the mechanism that simultaneously determines the abovementioned variables: the extent of the market, the level of division of labor, the development of infrastructure,
C. Y. C. Chu
466
the transaction efficiency, and the progress of productivity. The population size or density plays a very sophisticated role in the mechanism that simultaneously determines all these variables. 2. The Model Following the notation in Yang and Ng (1993), let us consider an economy with M ex ante identical individuals. There are two goods in each individual's utility function, denotedXand Y, and an infrastructure sector which helps improve transaction efficiency may emerge in equilibrium. Each individual has the following Cobb-Douglas utility function: u = (x + kxd)(y + kyd), where x and y are the amounts of self-provided goods X and Y, and xd and j''are the amounts of goods Xand Thought from the market. Fraction 1 - k of xd{oryd) disappears in transit because of transaction costs. Hence k e (0, 1) is a parameter of transaction efficiency. Thus x + kxd {ory + kyd ) is the quantity of good X (or Y) that is consumed. Each person has one unit of time available for the production. lx and / are respective amounts of labor allocated to the production of goods X and Y. Thusl = lx+ly. In order to produce good X or good Y, an individual has to spend a units of time to learn the production process. The total output will be either self-consumed or sold to the market. Let xs and ys denote the amounts sold; then the production condition of an individual can be specified as follows: x + xs =lx—a, y + ys =ly-a, /x+/y=l. There is also an infrastructure sector which determines the value of k prevailing in the economy. Let Mk be the number of people devoted to the infrastructure sector. Following Chu and Tsai (1996), we assume that the level of k is determined by f l - l / M , k if M,k >1 k =\ '
(1)
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467
It is clear from the above equation that k is an increasing function of Mk provided Mk > 1. As shown in Yang and Ng (1993), a fixed learning cost a generates economies of specialization. But the existence of the transaction cost 1 - k makes people less willing to transact in the market. The tradeoff between economies of specialization and transaction cost can be used to endogenize individuals' levels of specialization. The endogenization of specialization creates a room for endogenous emergence of the professional infrastructure. If k and a are small, then economies of specialization are outweighed by transaction costs. Individuals will choose autarky where transactions and the infrastructure sector are absent. If k and a are sufficiently large, economies of specialization outweigh transaction costs, so that individuals will choose specialization and division of labor. The final equilibrium will be determined partly by the relative sizes of a and k. In what follows, I shall denote the number of people specialized in the production of X and Y, respectively, as Mx and My. Given the above specification of preferences and technology, each person specialized in the production of X or Y has the following budget constraint: PS' + Pyys = P*xd + Pyyd + Pk, where pk is the per-person fee for using the infrastructure. For each individual specialized in the infrastructure sector, his or her budget constraint is pxxd + pyyd
= pk (Mx + My )JMk.
(2)
In the above expression, pkyMx+ My\ is the total revenue collected from all producers of goods, which is divided evenly among all A:-sector workers. This yields the per-person revenue on the right-hand side of (2). In a competitive economy, all individuals maximize their utilities to determine what to produce and how much to sell and buy. Possible scenarios are discussed in the next section.
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C. Y. C. Chu
3. Corner Equilibria in Three Structures As far as each individual is concerned, there are six possible work/ transaction choices: (i) producing both X and 7, and does not trade, denoted A (autarky); (ii) producingXand buying Fin a world with primitive infrastructurekQ, denoted x/y; (iii) producing 7 and buying X in a world with primitive infrastructure £ 0 , denoted j/x; (iv) producing X and buying 7 in a world with improved infrastructure k, denoted xlyk; (v) producing 7 and buying X in a world with improved infrastructure k, denoted y/xk; (vi) specialized in the production of k and buying both X and 7, denoted k/xy. Corresponding to the above six work/transaction choices, there are three possible equilibrium structures. The first is an autarky structure, in which there are only type-/ individuals. The second structure is specialization with no professional infrastructure sector, where individuals choose either x/y ory/x. This structure is denoted/). The third case is specialization with a professional infrastructure sector, in which individuals choose either xlyk, or y/xk, or k/xy. There exists a corner equilibrium for each structure, and all structures are illustrated in Figure 1. In this figure, the arrows indicate flow of goods. For instance, structure D says that the x/y individuals self-provide good x, and sell (buy) xs (y d ) to (from) the ylx individuals.
(a) Autarky
(b) Structure D Figure 1: Three Structures
(c) Structure C
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469
Individuals' demand, supply and indirect utility functions for the six cases are listed in Table 1, and the corner equilibrium relative prices, number of specialists and per capita real incomes for the three structures are listed in Table 2. I will provide algebra for just one typical structure; calculation details for other structures are similar and are available from the author on request. Let us consider structure C in which the population is divided into three types: x/yk, y/xk, and k/xy. For those who choose x/yk, their objective is to maximize xky*=(\-a-x°)k{p*XS~Pk),
u=
Py s
s
d
where x + x = 1 - a and pxx = pyy + pk are used to obtain the second equality. Let pt/Pj = Py • Simple differentiation of the above equation with respect to xs yields x° = [\-a +
Pkx]/2.
This, together with the budget constraint and the production function of X, yields vd
_ (l-a)pv-pfy
_
2
\-a-Phc 2
Substituting the optimal decisions back into the objective function, we have the indirect utility (IU) function for X-specialists below: _(l-«-Pfa)2foy 4
I(J
Following the same procedure, it is easy to see that individuals specialized in the production of Y have y'=[l-a
+
pfy]/2,
2
y
A
Table 1: Six Possible Work/Transaction Sce Demandfunction Supply
Type
Self-supply
A
x = y = {\- 2a)/2
xly
x = (l-a)/2
y'=[(l-a)p„]/2
x°=(
ylx
y = (l-a)/2
x*=[(l-a)Pyx]/2
/=(
xlyk
x=
(l-a-Pkx)/2
ylkx
y=
{^-a-pty)/2
klxy
k = {l-l/Mt)
0
y=[{l-a)pv-Pi,]/2
*=(l-
xd=[(l-a)Pyx-Pkx]/2
y*=(l-
x<=(Mxk +
Myt)Pkt/2
y={Mxk+Myk)Pkyl2
Population and Infrastructure
All
As to individuals specialized in the production of k, their per-person income is on the right-hand side of (2). Let My =MtjMj ; then the income of ^-specialists can be rewritten as pk(Mxk + Myk). The CobbDouglas utility specification will make the ^-specialists split their budget equally betweenXand Y. Thus, their demands forXand Fare respectively _(Mxk+Myk)pkjMxk+Myk)Pkx
xd
2/>, yd=-
2
(Mxk+Myk)pk_(Mxk
+ Myk)Pky
2py
2
The ^-specialists' indirect utility is
_[k(Mxk+Myk)]2PkxPky 4
Since people are ex ante identical, and since they have the freedom of choosing jobs, the indirect utilities of ^-specialists, X-specialists and 7-specialists must be equal: IUx=IUy = IUk.
(3)
Furthermore, we have the population size identity Mx + My+Mk=M,
(4)
and the market clearing condition Mxxs = Myxd + Mkxd , Myys = Mxyd + Mkyd .
(5)
Equations (3)-(5) together solve the corner equilibrium relative prices between X and Y and the relative numbers of specialists: px = py and Mx=My ={M-Mk)/2 . H e n c e ^ =\,Pla = Pky,andMxk =Myk. Using the above information, we can rewrite (3) as (\ — a — p^) -k
~
f^—^Mlpl.
The corner equilibrium relative price p^ therefore is
472
C. Y. C. Chu
Pk*
\-a M-Mk
\ + 4k
as shown in Table 2. Substituting this comer equilibrium price back into the formula fovIU, we see that the comer equilibrium utility in structure C, denoted V(Q, can be expressed as a function of Mk: V(C) =
\-a
Mt k(M~Mk)
1
+ yfk -
where A: is a function ofMk, as given in (1). SubstitutingA: = l-(l/Af i t ) into V(Q and differentiating V(Q with respect to Mk yields the following first-order condition: M {M-Mk)(Mk-\)
+
2Mk[Mk{Mk-\)f
= 0. (6)
(M-Mk)
From the above equation the comer equilibrium number of workers in the infrastructure sector (M*k) can be determined (see Table 2). Table 2: Three Equilibria Relative prices
Number of specialists
Utility
A
—
—
[(l-2«)/2f
D
/v = 1
Mx = My= M/2
[(l-a)/2fk0
C
/v = 1
Structure
Mx=My=(M-Mk)/2
ffi
Pto=
Mk \-a l+
>[k(M/Mt-\)
k(M-Mk)
1"
Jk
-2
Population and Infrastructure
473
The general equilibrium is defined as a vector of relative prices and numbers of various specialists that satisfies the following conditions: (a) Each individual maximizes utility with respect to quantities of goods that are produced and consumed, given the relative prices of traded goods and the numbers of specialists. (b) The relative prices of goods and the numbers of specialists clear the markets of all traded goods and equalize the utility of all individuals. Following Yang and Ng (1993, ch. 6), it can be shown that the corner equilibrium with the maximum per capita utility is the general equilibrium, and individuals have incentives to deviate from other corner equilibria. 4. The General Equilibrium and its Comparative Statics Let the utility in structures D and A be respectively V{D) and V(A). From Table 2 we see that both V(D) and V(A) are independent of the size of total population (M). From the envelop theorem, however,
dV(C)JV(C)^ dM dM Thus, as the size of the population gets larger, it is more likely that V(C) will be larger than V(D) and V(A), and hence the structure with division of labor and a professional infrastructure sector is more likely to appear. This is consistent with the observations in Boserup (1981). More specifically, if we equalize V(C) and V(A), we obtain a critical population M0, which is the minimum population size needed to establish the inequality V(C) > V(A). Similarly, we can equalize V(C) with V(D) and obtain a critical number Mx, which is the minimum population size needed to establish the inequality V(Q > V(D). Furthermore, we know that V(D) >F(^)ifandonlyif
,
fl-2aV
Combining the above information, we can identify the interval of parameters within which a certain corner equilibrium is the general
474
C. Y. C. Chu
equilibrium, as shown in Table 3. It is interesting to see that the general equilibrium discontinuously jumps across corner equilibria as parameters reach some critical values. From Table 3 it is also clear that a finer division of labor and a professional infrastructure sector will appear only when the size of the population is sufficiently large (M> max {M0, Mx}). Therefore, population density forms a necessary support for economic development. This has essentially formalized Boserup's theory. Table 3: General Equilibrium and Comparative Statics
° \ \ - a ) M<M0 Equilibrium structure
A
0>
[l-a
J
M > m a x {M0,M,}
M<M, n
r
Another interesting result is the Pareto-optimality of the infrastructure investment in the present model. Note that the corner equilibrium utility in structure C is derived by maximizing V(C) with respect to Mk, the number of workers in the infrastructure sector. Given the freedom of choosing jobs, when Mk is less than the optimal M*k for instance, a worker moving from the X or Y sector to the k sector will increase his or her utility. As long as Mk =£ M*k continues to hold, the abovementioned job-transfer process will continue, until V(C) is maximized with respect to Mk. It is trivial to show that each corner equilibrium is locally Pareto-optimal for the given structure (Yang and Ng, 1993, ch. 6). This result, together with the fact that the general equilibrium is the corner equilibrium with the greatest per capita income, implies that the general equilibrium in the present model is also globally Pareto-optimal. The traditional wisdom concerning public goods as stated by Samuelson (1954) is that a competitive economy cannot generate optimal public good, because the marginal conditions forced by individuals do not match those for the society. But in the inframarginal analysis here, the marginal analysis of each corner solution is not enough for decision-making. Total cost-benefit analysis across corner solutions is essentially needed. Provided the infrastructure sector can charge the
Population and Infrastructure
475
users,2 and as long as people have the freedom of choosing jobs, they will change jobs so as to achieve a Pareto-optimum. There is no free-rider problem, because the infrastructure sector can charge fees. Neither is there a marginal incentive problem of insufficient provision of the infrastructure investment, because the infrastructure is "produced" by the ^-sector workers, who compare total benefit and costs across various structures in addition to the marginal analysis within each structure. 5. Conclusions Pryor and Maurer (1982), Lee (1986, 1988), and Kremer (1993) have attempted to provide a rigorous analysis for Boserup's theory. Although the focus of their research is slightly different, a common feature of their papers is that they put population size as a variable in the right-hand side of the technology change equation. This setting inevitably makes the relationship between population size and technological change an if-and-only-if one. Pryor and Maurer and Lee tried to synthesize the theories of Malthus and Boserup, which is not within the scope of this paper. A model of the Malthusian theory has to contain an equation for population growth, whereas the main focus of Boserup's theory is the impact of population size or density on technological change. If, as Lee (1986, p. 102) pointed out, the division of labor involved in Boserup's analysis was essentially a static one, then it is not necessary to discuss the Boserup and the Malthus theory in a single model. Both Lee (1986, p. 102) and Boserup (1981, ch. 6) noted the necessity of a sufficiently large number of non-food-producing workers, which is referred to as "excess labor" in the development literature, for the advancement of technology. In this paper I point out that whether or not people will be specialized in food production or agriculture is endogenously determined, and I explicitly characterize the relationship between population size and the possible advancement of technology. The infrastructure in the present model is a public good because it is not depletable; but it is not a pure public good because it is excludable. The subtle distinction between depletability and excludability can be found in Mas-Colell et al. (1995). In the present framework, excludability is an institutional property of a good, and rivalry is a technical property of a good. Nondepletability is not the source of distortion, but nonexcludability is.
476
C. Y. C. Chu
Yang and Ng (1993) has a detailed discussion of the economies of specialization, but their model does not leave much room for the role of population. The very limited role of population in their model is simply that "the total number of specialized sectors cannot be more than the total population size." Yang and Ng do not treat the publicly-supported infrastructure sector as a variable affecting transaction efficiency which, however, is a typical phenomenon described in Boserup (1981). Population size plays an important role in the present model because it helps generate more aggregate demand for a professional infrastructure sector, which in turn improves transaction efficiency and facilitates the division of labor. The analysis here also sheds light on the private provision of public goods. Conventional models of public goods assume a difference between individual and societal (marginal) optimality conditions in a competitive environment. I argue in this paper that the number of individuals devoted to the construction of infrastructure will be Pareto-optimal in a competitive economy, if nondepletable services can be provided exclusively to those who pay fees.
References Backus, David K., Patrick Kehoe, and Timothy J. Kehoe, "In Search of Scale Effect in Trade and Growth," Journal of Economic Theory 58 (1992):377-409. Boserup, Ester, Population and Technological Change, Chicago, IL: University of Chicago Press, 1981. Chu, C. Y. Cyrus and Yao-Chou Tsai, "Productivity, Investment in Infrastructure, and Population Size," in X. Yang and Y. Ng (eds), Increasing Returns and Economic Analysis: Essays in Honor of Kenneth Arrow (1996, forthcoming). Goodfriend, Marvin and John McDermott, "Early Development," American Economic Review 85 (1995):116-33. Kremer, Michael, "Population Growth and Technological Change: One Million B.C. to 1990," Quarterly Journal of Economics 108 (1993):681-716.
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Infrastructure
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Kuznets, Simon, "Population Change and Aggregate Output," in Universities-National Bureau of Economic Research, Demographic and Economic Change in Developed Countries, Princeton, NJ: Princeton University Press, 1960. Lee, Ronald D., "Malthus and Boserup: A Dynamic Synthesis," in David Coleman and David Schofield (eds), The State of Population Theory: Forward from Malthus, Oxford: Basil Blackwell, 1986. , "Induced Population Growth and Induced Technological Progress: Their Interaction in the Acceleration Stage," Mathematical Population Studies 1 (1988):265-88. Mas-Colell, Andreu, Michael D. Whinston and Jerry R. Green, Microeconomic Theory, New York: Oxford University Press, 1995. Ng, Yew-Kwang, "On the Welfare Economics of Population Control," Population and Development Review 12 (1986):247-66. Pryor, Frederic L. and Stephen B. Maurer, "On Induced Economic Change in Precapitalist Society," Journal of Development Economics 10 (1982):325-53. Romer, Paul M., "Endogenous Technological Change," Journal of Political Economy 98 (1990):S71-102. Samuelson, Paul A., "The Pure Theory of Public Expenditure," Review of Economics and Statistics 36 (1954):387-9. Simon, Julian, The Economics of Population Growth, Princeton, NJ: Princeton University Press, 1977. , The Ultimate Resource, Princeton, NJ: Princeton University Press, 1981. Yang, Xiaokai and Yew-Kwang Ng, Specialization and Economic Organization, Amsterdam: North-Holland, 1993.
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CHAPTER 19 UNCERTAINTY, INSURANCE, AND DIVISION OF LABOR*
Monchi Lio* National Taiwan
University
1. Introduction The purpose of this paper is twofold. First, an equilibrium model is developed to investigate the relationships between transaction uncertainties, risk aversion, specialization, and per capita real income. The effects of insurance on specialization and division of labor are then explored. Arrow (1965) suggests that the existence of insurance is one of the basic needs in developing the theory of uncertainty. Economic historians (John, 1958; North, 1968; Price and Clemens, 1987) have found that insurance against transaction risks increases the extent of the market. For example, insurance for ocean shipping has been an important factor for the extension of world trade. According to Smith (1776), the division of labor is limited by the extent of the market, and productivity is determined by the level of division of labor. When observing the impact of insurance on increasing productivity and the extent of the market in economic history, one should expect that the theory of uncertainty must be able to formalize the relationship between uncertainty, insurance, and the level of division of labor.
* Reprinted from Review of Development Economics, 2 (1), Monchi Lio, "Uncertainty, Insurance, and Division of Labor," 76-86, 1998, with permission from Blackwell. * I would like to thank C. Y. Cyrus Chu and Yew-Kwang Ng for helpful comments. I am responsible for any remaining errors. 479
480
M. Lio
The theory of uncertainty in most economic textbooks, however, does not provide a satisfactory explanation of how the extent of the market and the level of division of labor interact with uncertainties in transactions. The traditional analytical framework in most textbooks is devised to address the question of how the market sorts out the relative quantities of goods consumed and produced for a given degree of scarcity and a given pattern of economic organization. Therefore, it is impossible for economists to investigate within this framework the relationship between uncertainty, insurance, the degree of scarcity, and the level and pattern of division of labor. Since the traditional theory neglects the effects of uncertainties on the extent of the market and the division of labor, the risks considered in this framework, in Laffont's words (1989, p. 121), "have no effect on society's aggregate resources but can seriously affect the welfare of the unfortunate individuals." As Young (1928) pointed out, the securing of increasing returns depends on the progressive division of labor. If we recognize that the division of labor is a function of transaction risks, we will find that uncertainties affect a society's aggregate resources and the welfare of all individuals. In this paper we provide a model in which the evolution of specialization and division of labor is restricted by transaction uncertainties, and, because of economies of specialization, the productivity and the level of real income are determined by individuals' levels of specialization and society's level of division of labor. Our model takes the size of the market network as the outcome of interactions of individuals' optimal decisions, and different network structures are represented by different levels of specialization and division of labor. That is, although productivity is increased by higher levels of specialization and division of labor, also it means more unwelcome risks for coordination failure of the network of division of labor. An economic organization such as insurance that can be used against transaction uncertainties enlarges the scope for individuals to trade off economies of specialization for reliability of coordination of the network of division of labor. Hence, the equilibrium level of division of labor, the size of market network, and economic development will be promoted by insurance. Recent years have seen an important economic analytical framework developed by Borland, Ng, Shi, and Yang that can endogenize
Uncertainty, Insurance, and Division of Labor
481
individuals' levels of specialization and society's level of division of labor as a whole on the basis of the tradeoff between economies of specialization and exogenous transaction costs. Their works have demonstrated how the economic structure in equilibrium is determined by interactions of individuals' optimal decisions. Yang andNg (1993, chs 10 and 11) have discussed some aspects of transaction uncertainties, but in their models the implications of insurance are not explored. The models in the literature of uncertainty are concerned with insurance's influence on the optimal decisions of risk-averse individuals while neglecting its impact on the network size of division of labor, because these models have not endogenized levels of specialization and division of labor. This paper synthesizes these two literatures and develops a model to explain how insurance increases productivity and the level of specialization by encouraging risk-averse individuals to undertake risky transactions. The paper is organized as follows. An equilibrium model in which the level of division of labor is determined by the tradeoff between economies of specialization, transportation costs, and transaction risks is specified and the equilibrium solution and its comparative statics are reported in section 2. Then the relationship between risk aversion and the level of division of labor is investigated. The impacts of the emergence of insurance on the size of the network of division of labor are analyzed in section 3. Section 4 concludes the paper. 2. Transaction Uncertainties, Risk Aversion, and Division of Labor
The model Consider an economy with Mex ante identical consumer-producers and m consumer goods. Each consumer good can be either purchased in the market or self-provided. For consumer good i, the self-provided amount is xi, and the amounts sold and purchased in the market are x' and xf , respectively. In purchasing good i, a fraction 1 - kt of any quantity purchased disappears in transportation, and the amount an individual obtains from the purchase is ktxf. The total amount consumed of good i is x] = x{ + k{xf. Assume that, in trading good i, there are two possible
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states for the transportation efficiency coefficient /c,. : a higher level denoted as kH and a lower level denoted as kL . Assume also that 1 > kH > kL > 0 and that the probabilities for kH and kL are 0 and 1 - 0, respectively. It is assumed t h a t # e ( 0 , 1). Therefore, there exists uncertainty in the trading activities. We further assume that individuals' preferences satisfy the expected utility hypothesis. Each individual is assumed to have an identical von Neumann-Morgenstern utility function, given by u
=Yl{xi) _jk„ '~[kL
'xi
-xi+k>x?>P>1>
with 0 with 1-0,
v
'
where parameter p represents the degree of risk aversion. Lemma 2 will assert that individuals are more risk averse when p is larger. The system of production and endowment of time for each consumer-producer is specified as x. +x* =max{/ ; -a, 0}, a e ( 0 , l), i = l,2,...,m (2)
E',=u,e[o,i], where x. + x* is the output level of good z; term /. , representing the individual's level of specialization in producing good /, is the amount of labor spent at producing good /; and parameter a is a fixed learning or training cost and represents the degree of economies of specialization. This system of production functions and endowment constraint displays economies of specialization since labor productivity increases with an individual's level of specialization. The budget constraint for each person is given by m
m
Z(A^) = Z ( ^ ) . i=l
(3)
i=\
where pt is the price of good i. As in Yang and Shi (1992), in this model a Walrasian regime is assumed since economies of specialization are individual-specific or increasing returns are localized.
Uncertainty, Insurance, and Division of Labor
483
The following two subsections identify the equilibrium and its comparative statics. The influence of transaction risks on the division of labor is explored in the first subsection; the relationship between risk aversion and the division of labor is investigated in the second subsection. Uncertainty and division of labor The following lemma has been established by Wen (1997). Lemma 1: According to Kuhn-Tucker conditions, for an individual's optimum decision, an individual sells at most one good and does not buy and sell or self-provide the same good. Taking Lemma 1 into account and signifying the utility of a person selling good i by «., we can obtain the decision problem for an individual selling good i, given by -\VP~
maxEut=E< x, ]^[ (krxdr j]~[(x } ) (4) s.t. xt + x. = max{/;. - a, 0], Xj = maxU. - a, 0| V/ eJ, (production function) /. +^lj
= 1>
ptxst ='^J(Prxr)>
(endowment constraint) (budget constraint)
where R, consisting of n - 1 elements, is the set of goods the individual buys in the market; J, consisting of m-n elements, is the set of nontraded goods the individual self-provides; and n is the number of goods traded by the individual. We can rewrite, using symmetry, the expected utility function for a person selling good i as follows: Eu, = Wi/p
e-x (kH )(""1)/p + cr'0"-2 (i - e){kH f~2)lp (kL fp
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+...+c::l(i-dy-\kLr-»/»] = ^ v "[^f+(l-<9)(* i ff 1 ,
(5)
where W = x. (xdr ) "_1 ( xs)m~". Inserting (5) into (4), we can use the first-order conditions for this decision problem to express the optimum values of lt, /., xi, x\, Xj, xdr , and n as functions of relative prices of all traded goods. The optimum x\ and xdr represent individual's supply and demand functions, respectively. Inserting the optimum values of decision variables into the expected utility function yields an indirect expected utility function. In equilibrium all markets must clear and the expected real incomes of all individuals selling different goods must be equal. The « - 1 expected utility equalization conditions for the n - 1 types of individuals selling different goods determine the n - 1 relative prices of n traded goods. The n - 1 market clearing conditions determine the n - 1 relative numbers of individuals selling n traded goods: pt/p,=l,
MJM,=\,
V/> = 1,...,«,
(6)
where M. is the number of individuals selling good /, and Ms is the number of individuals selling good s which can be solved by using (6) and the population equation £,•Ml.=M . The other market clearing condition is not independent of (6) owing to Walras' Law. Inserting the equilibrium relative prices into the first-order conditions for the maximization problem yields the equilibrium values for all decision variables, given by /,. = n+a(n2 d
JC. = x r
-mn + m-n) \/m, lj = [ ! + « ( « - l ) ] / m ,
- Xj = [l - a (l + m - n )J im, n
«=(l-l/«) + m{l-l/{/>ln[^+(l-6')^]}j, Eu = {[1 -a(l + m- n)]/m}mlP [Okf + (l - 0)k^j~l,
(7)
where n , representing the level of division of labor, is the equilibrium number of traded goods; Eu is the equilibrium expected real income. The comparative statics of this equilibrium are given by dn/d0>O, dn/dks>0,
dl,/d0>O,
dljdks>0,
Uncertainty, Insurance, and Division of Labor
d[M(n-\)xdr]/d0>O,
485
d[M(n-\)xdr]/dk5>0,
dEu/de > 0, dEu/dks > 0, s = K,H,
(8)
where M{n - \)xdr , representing the extent of the market, is the aggregate demand for all goods by all individuals. From (8) the following proposition can be derived. Proposition 1: The level of division of labor, the level of specialization, the extent of the market, productivity, and per capita expected real income in equilibrium increase as transportation efficiency is improved or as the probability for high transportation losses declines. Figure 1 provides an intuitive illustration of how the evolution of division of labor may proceed. The lines in each panel signify flows of goods. The small arrows indicate direction of flows. The numbers beside the lines signify goods involved. A circle with the number i signifies a person selling good z. Panel (a) denotes autarky where each individual self-provides three goods because of low transaction efficiency or because of high probability for high transportation losses. As transaction efficiency or the probability for low transportation losses is slightly increased, the economy evolves to the state depicted in panel (b), where each individual sells one good, buys one good, trades two goods, and engages in two production activities. From panel (a) to (b), the level of specialization and productivity increase since the number of production activities for each person is reduced from three to two. When transaction efficiency improves or the probability for high transportation losses falls further, the economy evolves to panel (c), where each individual sells one good, buys two goods, trades three goods, and engages in only one production activity. The level of specialization and productivity in panel (c) are higher than in panel (b) since each person's number of production activities is reduced from two to one.
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(a) Autarky, n = 1, m - 3
(b) Partial Division of Labor, n = 2,m = 3
(c) Complete Division of Labor, n = 3,m = 3 Figure 1: The Evolution of Division of Labor
Risk aversion and division of labor In investigating the relationship between risk aversion and division of labor, first we establish the following lemma. Lemma 2: Individuals are more risk averse when p is larger. Proof: Let u' be the von Neumann-Morgenstern utility function corresponding to p = pt, i = 1,2. Suppose that p2< px, and it is easy to see that u1 can be obtained from ux by a concave and monotone increasing transformation. Since ul andw2 both represent the same ordinal preferences over the set of bundles of goods consumed, according to Kihlstrom and Mirman (1974), u2 is more risk-averse thanw1. Therefore, parameter p represents the degree of risk aversion. Differentiating n with respect to p, we can obtain dn/dp < 0 (a detailed proof is given in the Appendix). The inequality dnjdp < 0 implies that the
Uncertainty, Insurance, and Division of Labor
487
equilibrium level of division of labor is lower when people are more risk-averse. The following proposition can then be established. Proposition 2: The level of specialization, the level of division of labor, the extent of the market, and productivity decrease with the degree of risk aversion. Now we relax the assumption of/? > 1, and allow p to be equal to or less than unity. Following Kihlstrom and Mirman (1974), we define a risk premium r , analogous to the Arrow-Pratt risk premium, by the equation VP
Eu, = E
*n(M0nw &
kT=Ek-r,
i/p
•ui(r)
=
^nM)n(^)
Ek = OkH+(l-0)kL,
jeJ
(9)
where Ek is the expected value of the transaction efficiency coefficient. Inserting ul. (r) into the decision problem and solving for the equilibrium value of ui. ( r ) , by comparing the equilibrium value of ui (r) with£z7 in (7), we can prove that r = 0 if /? = 1 , T > 0 if p > 1, and T < 0 if p < 1. The equilibrium number of traded goods when the transaction efficiency coefficient equals Ek is given by n(Ek) = (l-\/a)
+ m(l-\/lnEk).
(10)
Comparing n(Ek) withn in (7) yields n(Ek)>n
iff p>\, n(Ek) = n iff p = \.
(11)
We say that the transaction uncertainties decrease the level of division of labor — that is, the level of division of labor is lower when transaction uncertainties exist, if n(Ek)>n ; the transaction uncertainties do not affect the level of division of labor if n[Ek) = n ; and the transaction uncertainties increase the level of division of labor if n(Ek)
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3. Insurance and Division of Labor As in Helpman and Laffont (1975), this equilibrium model is provided with the necessary institutions that enable individuals to purchase insurance against transaction uncertainties. For transporting good r, an individual can purchase an insurance contract that will pay him crxdr in the event of low transportation efficiency; i.e., if the amount kLxdr is received. The premium he has to pay for crxdr of insurance coverage is 7trxdr . By the assumption of symmetry in this model, we can obtain nr = it and cr = c for all r e R, and the expected utility for a person selling good i in (4) can be rewritten as follows: l -wW /p Eu, =
en-\kH-7t)
+cnl-le"-2(\-e)(kH-7t) („-!)/„
n
+...+c:l{\-e)
= wxlp
-\kL-7t+c)
9(kH-7t)
.<"
.
{kL-7t+c)lP
-w.
+(\-e){kL-7t
Wp-"-1
+ c)'
(12)
where W = ^(x?)" -1 (*,)"'". Following Kreps (1990, ch. 3), we specify the insurance contract as follows:
/? =
flr/[(l-6>)(c-^)]>
(13)
where p, characterizing the insurance contract, is not smaller than unity. The expected payout (l-d)c equals the premium n if J3 = l, and the expected payout is less than the premium if /? > l. Therefore, the insurance contract can be said to be complete if /? = l and be incomplete if,fl>l. Given two insurance contracts /3a and j5b, we say that contract Pa is more incomplete than/?6 if Pa> Pb • Given an insurance contract/?, inserting (13) into (12) and solving for the optimization problem with respect to n, we can obtain the optimum premium and outcomes, given by
7t={\-e)(kH
-
pPl(p-x)kL)/{\-e+pxl{p-l)e),
kH-7t = pPl^\kL+C~7t)
= pll{p-X]
Uncertainty, Insurance, and Division of Labor
489
[ekH +p{\-e)kL\l(\-e+pxl(p-x)9).
(H)
Since n > 0 for a viable insurance and dnjdfi < 0, a feasible insurance contract must satisfy (kH lkLfp~X)lp > j3 > 1. People purchase no insurance if p < 1, and only full insurance can survive if p -1. Under the assumption (p-\)lp
p<\, people purchase no insurance as n = 0,/3>(k H lk L ) , partial insurance as 7r>0 , (kH /kL)(p~l)/p > / ? > l , and full insurance as 7r = (\-0)(kH-kL) , P = \ . Assuming (kHIkJ"^'" >(3>\, then the decision problem for an individual selling good / and purchasing insurance becomes max Eu,
^(^flxJ-fWMWf^
ci{p,e,kH,kL)=/3-l[\-e+pyl(p-l)e)p~[ekH+p(\-e)kL] s.t. xt+xst = max{/i -a,0),
x} = max{//. -a,o\,
V/e J,
By applying the method used in section 2, we can solve for the equilibrium values of endogenous variables, given by /, = rip +a\rip2 -mnp + m-np} x(. = xdr =Xj = \l-a{\ np={l-\/a) Eufi={[l-a(l
+
\/m, l} = \ + a{rip - l ) Urn,
+ m-rip} Urn, m{l-l/]n[Q(j3,0,kH,kL)]},
+ m-nfi)ym}m/P[a(p,0,kH,kL)f^)/p,
(16)
where np, representing the level of division of labor when the insurance contract is (3 , is the equilibrium number of traded goods; and Eufi is the equilibrium expected real income. The comparative statics of this equilibrium are given by drip/dO > 0, drip/dks > 0, dlJdO > 0, dlt I dks > 0,
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d[M(nfi-\)xdryd0>O, dEup /d$>0,
dEup /dks>0,
s = K,H
(17)
dlJd/3
dnp/d/3<0, dEup/dp<
d[_M(np -l)x r r f ]/dt s >0,
0 iff (kH lkL)(p-l)lp
> p.
(18)
There will be no insurance contract if/? > (kH/kL)
(j3 = (kH/kLf-^)
= [0k^+(\-0)k^J
. Note that Q.
and np(0 =
(kH/kL)W)
= n . That is, if we insert f3 = {kH lkL) into (16) we can obtain (7), and the equilibrium given in (7) is a special case of the equilibrium given in (16). It is easy to see that the comparative statics given by (17) are consistent with Proposition 1. From (18), we can find that, if there exists an insurance contract in equilibrium, the level of division of labor, the level of specialization, the extent of the market, productivity, and per capita expected real income will be lower if the insurance contract is more incomplete. (P-\)IP
We say that an insurance contract is feasible if(kHlkL) the following proposition can thus be established.
> /3, and
Proposition 3: In an equilibrium where a feasible insurance contract exists, either complete or incomplete, the level of division of labor, the level of specialization, the extent of the market, productivity, and per capita expected real income will be increased by the insurance contract. Proof: If the insurance contract is complete (i.e.,/?= 1), people will purchase full insurance, and the equilibrium number of traded goods in full insurance, denoted as np (/? = l ) , is given by n0(p = l) = (l-l/a)
+ m{\-l/\n[0kH+(l-0)kL]}.
(19)
It is straightforward to see that np (ft = 1) > n . If the insurance contract is feasible but incomplete (i.e., (kH/kL)(p~X)lp >/?> 1), people will choose partial insurance. We can prove that the equilibrium number of traded goods in partial insurance is greater than n by noticing dnp jdj3 < 0, (kH/kL)("1,/P > J3> 1, andnp{(3 = l)>n p (J3 = (kH IkL)(^1)/p) = n . Since
Uncertainty, Insurance, and Division of Labor
491
the level of specialization, the extent of the market, and productivity increase with the level of division of labor, the remaining part of the proposition can thus be established. If, in equilibrium, people purchase insurance against transaction risks, the effect of a greater degree of risk aversion on the level of division of labor is not clear: the level of division of labor will not be affected by the degree of risk-aversion if people can purchase full insurance. However, it is easy to prove that people can accept a more incomplete insurance contract if they are more risk-averse by noticing that the upper bound of jB, (kH I kL) " , increases with p . Therefore, people can accept a more incomplete insurance contract if they are more risk-averse. The equilibrium insurance contract This subsection considers how the insurance contract is determined in the market. We assume that the insurance industry is in perfect competition, and there is a fixed management cost, denoted as 5, for the insurance company to pay per contract. In equilibrium, perfect competition will force the expected profit of the insurance industry to zero. Let (\-0)c-7i = 8 , and from (13) it can be shown that the insurance contract is determined by S=
{\-\lP)6n
= 6{\-9){\-\lp)(kH
-/3pl{p-x)kL)/[\-e
+ Pxl{p-l)e),
(20)
where f3 e \\,(kH lkLfp~l)lp). We can establish the following proposition, which is proved in the Appendix. Proposition 4: There exists no insurance contract in equilibrium if the management cost for the contract is too large. For a given sufficiently small positive management cost, there exists at most one partial insurance contract in equilibrium if the insurance industry is in perfect competition. If there exists a partial insurance contract for a given positive management cost, the level of division of labor and specialization, the
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extent of the market, productivity and per capita real income will increase as the management cost decreases. 4. Conclusions This paper has investigated the interactions between transaction uncertainties, risk-aversion, insurance, the level of specialization, and per capita real income. It has been found that, in equilibrium, if the individual is willing to pay an amount of premium to avoid the uncertainties in transactions, he will choose a lower level of specialization and division of labor when there are transaction uncertainties. The emergence of insurance that enables individuals to pool the uncertainty will promote the evolution of division of labor. In this process the level of specialization and division of labor, the extent of the market, productivity, and per capita expected real income are increased. The existence of insurance, therefore, plays an important role in determining the size of the market network and how far the economic development can reach. Insurance can increase the welfare not only by shifting risks away from risk-averse individuals, but also by increasing the level of specialization and division of labor, the extent of the market, and productivity. Also, the degree of risk-aversion can affect the level of division of labor; if there is no insurance, the level of division of labor will decrease as people become more risk-averse. The level of division of labor will decrease with a more incomplete partial insurance contract that is caused by a greater cost for implementing each insurance contract in equilibrium.
Appendix
Proofof
dn/dp<0
From (7) we can derive that dn/dp < 0 if and only if
Uncertainty, Insurance, and Division ofLabor
<[9klp\nklp +{\-e)kx[p\nkx[p\
493
(Al)
Let kL = akH, a e (0, l ) , and (Al) is satisfied if and only if [O + (l-e)avp]hi[0
+ (l-e)al/p]-(l-0)a1/plna1/p
Denoting the left-hand side of (A2) a^(6,p,a),
(A2)
we can obtainlirn1? = 0
and l i m ^ = 6 In 00
x
'
*F < 0 for a e (0, l), dn/dp > 0 is thus established. Proof of Proposition 4 Let f(J3) = «kH-ppl'p-l)kL)l(\-e + Pll(p-l)e) , g{f?) = l-l/0 /?(/?) = f(fi)g(fi). Equation (20) can be rewritten as h{P) = f{fi)g{P)
= Sl[e{\-0)~\,
, and (A3)
wheredf(fi)/d/3<0 and dg(p)/dp>0. To prove Proposition 4, we need to establish the following three claims. Claim 1: No feasible insurance contract can yield nonnegative expected profit if the management cost is sufficiently large. Claim 2: Given a positive management cost, only partial insurance contracts can yield non-negative expected profit; only the insurance contract that favors the insured most will exist in equilibrium. Claim 3: If there exists an equilibrium insurance contract for a given level of management cost, there is an equilibrium insurance contract for any lower level of management cost; the equilibrium insurance contract with the lower level of management cost will be more complete than the contract with the higher level of management cost. To establish Claim 1, we only have to prove that /?(/?) has an upper bound. Since(kH/kLfp'l)/p >p>\, it can be shown that 0
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- kL and 0< g(0) < 1 -{kJkH){p^)lp, and from (A3) it is straightforward (p 1)/p toseethatO 8/[(0(\ - 0)], h(j3 = 1) = 0, and h(JJ) is continuous. Since there exists some contract/? e[l,/?) that satisfies h(JJ) = 8/[0{l - 0)], we know that any contract with (5 >/? that also satisfies h(J3) = 8/[0(l - 0)] will not be the equilibrium insurance contract because of competition. Finally, from Claim 2, we know that only the contract that favors the insured most will be the equilibrium insurance contract. Claim 3 is thus established. From Claim 3, we know that the insurance contract will be more complete if the management cost decreases. From (18), we know that the level of division of labor, the level of specialization, the extent of the market, productivity, and per capita expected real income will increase if the insurance contract is more complete. The last statement of Proposition 4 is thus established.
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References Arrow, Kenneth J., Aspects of the Theory of Risk-Bearing, Helsinki: Yrjo Johnsson Lectures (1965). Helpman, Elhanan and Jean-Jacques Laffont, "On Moral Hazard in General Equilibrium Theory'," Journal of Economic Theory 10 (1975):8-23. John, A. H., "The London Assurance Company and the Marine Insurance Market of the Eighteenth Century," Economica, 25 (1958): 126-41. Kihlstrom, Richard E. and Leonard J. Mirman, "Risk Aversion with Many Commodities," Journal of Economic Theory 8 (1974):361—88. Kreps, David M., A Course in Microeconomic Theory, Hemel Hempstead: Harvester Wheatsheaf, 1990. Laffont, Jean-Jacques, The Economics of Uncertainty and Information, Cambridge, MA: MIT Press, 1989. North, Douglass C , "Sources of Productivity Change in Ocean Shipping, 1600-1850," Journal of Political Economy 76 (1968):953—70. Price, Jacob M. and Paul G. E. Clemens, "A Revolution of Scale in Overseas Trade: British Firms in the Chessapeake Trade, 1675-1775," Journal of Economic History, March (1987):l-43. Smith, Adam, An Inquiry into the Nature and Causes of the Wealth of Nations, 1776 (reprint edited by E. Cannan), Chicago: University of Chicago Press, 1976. Wen, Mei, "An Analytical Framework with Consumer-Producers, Economies of Specialization, and Transaction Costs," in K. Arrow, Y-K Ng and X. Yang (eds.), Increasing Returns and Economic Analysis, London: Macmillan, forthcoming. Yang, Xiaokai and Jeff Borland, "A Microeconomic Mechanism for Economic Growth," Journal of Political Economy 99 (1991):460-82. Yang, Xiaokai and Yew-Kwang Ng, Specialization and Economic Organization: A New Classical Microeconomic Framework, Amsterdam: North-Holland, 1993. Yang, Xiaokai and Heling Shi, "Specialization and Product Diversity," American Economic Review 82 (1992):392-8. Young, Allyn, "Increasing Returns and Economic Progress," Economic Journal 152 (1928):527-42.
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CHAPTER 20 PUSH OR PULL? THE RELATIONSHIP BETWEEN DEVELOPMENT, TRADE AND PRIMARY RESOURCE ENDOWMENT*
Mei Wen a and Stephen P. King b * "Australian National University
b
University of Melbourne
1. Introduction "... where uncultivated land is still to be had upon easy terms, no manufactures for distant sale have ever yet been established in any of their towns....From the artificer he becomes planter... In countries, on the contrary, where there is either no uncultivated land, or none that can be had upon easy terms, every artificer who has acquired more stock than he can employ in the occasional jobs of the neighborhood endeavours to prepare work for more distant sale." (Smith, 1776, p. 339). The relationship between resource endowment, transaction efficiency, specialization and development is at the heart of economics. As Smith noted more than 200 years ago, increased per capita resources may act as a disincentive to specialization and trade. Lewis (1955) and Fei and Ranis (1964) also noted the relationship between a labor surplus in agriculture and economic growth through the transfer of labor to the
* Reprinted from Journal of Economic Behavior and Organization, 53 (4), Mei Wen and Stephen P. King, "Push or Pull? The Relationship between Development, Trade and Primary Resource Endowment," 569-591, 2004, with permission from Elsevier. * We would like to thank two anonymous referees, Franklin M. Fisher, Murray Kemp, Yew-Kwang Ng, Jonathan Pincus, Heling Shi, Guangzhen Sun, Guofu Tan, and Xiaokai Yang, for their helpful comments.
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M Wen, S. P. King
industrial sector. Ashton (1948) argued that the industrial revolution first occurred in Britain due to a severe shortage of wood in the 17th century. Sachs and Warner (1999) present evidence from Latin America suggesting that natural resource booms may be accompanied by declining per capita GDP.1 The above observations suggest that inadequate levels of primary resources can push a country into specialization and trade. Indeed, countries and areas like Japan and Hong Kong have developed with a scarcity of primary resource and become some of the most affluent places in the world.2 However, natural resource abundance is not always a curse. History shows that other countries with abundant primary resources like America and Sweden are also prosperous. As Mikesell (1997, p. 191) notes, "there is nothing inherent in resource abundance that condemns countries to either low growth or nonsustainability". Further, a lack of primary resource is not sufficient for development. Eighteenth century Ireland, 19th century Egypt, India, China and some modern African countries all failed to achieve rapid development despite high population density and low per capita resource endowments. Hence, it seems to us that a lack of primary resources and a surplus of labor may be neither necessary nor sufficient for economic development. Murphy et al. (1989) (MSV hereafter) and Yang (1990) provide alternative explanations for development. MSV formalize RosensteinRodan's (1943) idea of big-push industrialization. When labor can be drawn from unemployment or a low-paying traditional sector to a modern industrial sector, industrialization can 'spontaneously' occur. In the MSV framework, infrastructure investment, such as investment in the railroads, and government coordination, encouraging the simultaneous investment in modern technology are crucial for big-push industrialization. See also Auty (1995). A number of factors behind the inverse relationship between primary resource endowment and development have been suggested in the empirical literature. For example, Gylfason (2001) argues that the relationship operates through education with 'natural capital' crowding out 'human capital', see also Gylfason and Zoega (2001). On the relationship between human capital and resource boom, see Asea and Lahiri (1999). 2 For example, Ranis (1997) considers Taiwan's economic performance and argues that the relative paucity of natural resources has benefited the growth of Taiwan's economy.
Development,
Trade and
Endowments
499
Yang (1990) follows Adam Smith's idea that the individual level of specialization determines a nation's wealth. People can be stuck in autarky when the costs of transactions outweigh the efficiency gain from specialization. When transaction efficiency improves through infrastructure investment, the division of labor evolves. It is the development of institutions that facilitate trade that drives specialization and growth.3 In both the MSV and Yang models, transaction conditions play a key role in a nation's development. If transaction costs are high, then there is little gain to specialization. However, if trade operates smoothly and transportation is relatively cheap, specialization can raise individual and social welfare.4 Clearly, improvements in institutions that facilitate trade do help specialization and development. The United States might not be an economic giant today if it had not virtually completed its national railway system by 1910. It is unlikely that Hong Kong would have reached its present per capita income if it were not a tariff free area.5 Empirical studies reinforce the relationship between institutional improvement, transaction efficiency and development. For example, see Lall et al. (2000) and Fischer et al. (2001). However, institutions that improve transaction efficiency do not appear to be sufficient for development. British colonies in India, Africa and Australia had similar capital and institutional frameworks to govern trade but developed at substantially different rates in the nineteenth centuries. This suggests that the interaction between resource endowment and development that are absent from the MSV and Yang models cannot be ignored.6 3
See also Yang and Ng (1993). Yang and Borland (1991) use a similar framework where labor specialization evolves spontaneously over time. 4 This institutional approach can partially explain the 'curse' of abundant natural resources and Sachs and Warner (1997, 1999) suggest an inverse relationship between natural resource abundance and growth as well as a positive relationship between institution quality and growth. 5 Baxter (1984) also attributes the under-development of Bangladesh to its old institutions and infrastructure. Pryor (1994) argues that the high transactions costs associated with the former communist countries in eastern Europe led to their relatively slow growth. 6 Sachs and Warner (1999) develop a model with resource endowments based on the MSV model. They show how the resource endowment of a country can change the labor allocation between traded and non-traded sectors through the increase of the national
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In this paper, we explicitly model the interaction between transaction efficiency and primary resources in economic development. We delineate four patterns of economic development based on the combination of primary resources and the development of efficient trading institutions. Consider first countries with low levels of natural resources. In such countries, relatively small improvements in institutions can result in the country being 'pushed' into specialization and development. But development is not guaranteed. If the necessary transactions efficiencies do not arise, the country can remain under-developed. In contrast, consider countries with relatively abundant natural resources. Even with reasonably efficient trading institutions these countries may fail to develop. A high level of transactions efficiency and well-evolved institutional structures are needed before such a country is 'pulled' into development. In particular, institutions that might enable development in resource poor countries may be inadequate in resource-rich countries. Thus resource rich countries, like some oil producing nations today, may fail to specialize and industrialize despite access to highly efficient transaction technology. The resource-push pressure is simply absent in these countries. At the same time, a resource poor country like Japan may develop a powerful economy even with an infrastructure which is not the most advanced for trade. To model the interplay between transactions efficiency and resource endowment, we adopt Yang's (1990) approach, but introduce a primary resource factor into production. Individuals have economies of specialization in production (we call it economies of scale for simplicity hereafter7) that provide an incentive to specialize. However, if transaction costs 'eat up' the potential gains from trade, specialization will not occur. One source of individual economies of scale in our model is a minimum input requirement for production. An individual needs to invest a fixed amount of labor and primary resource into each activity wealth created by exporting the resource. They do not address the role of inadequate primary resource in pushing specialization, which is the key feature of the model developed here. 7 Yang (1994) provides a detailed distinction between these two concepts.
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before he starts to produce. For example, a farmer needs to provide at least a minimum amount of land and labor to allow any production of livestock. A blacksmith needs to invest a minimum amount of labor in training and capital in equipment before he can produce any output. The fixed investment of the primary factor in each activity an individual undertakes provides the relationship between resource endowment and specialization.8 A reduction in the per capita resource endowment makes an individual's investment of this resource into a range of productive activities relatively more expensive. As pressure builds on the scarce primary factor, it pays all individuals to economize on their use of the factor by specialization. Those individuals with an excess of the primary resource after specialization can then trade this surplus with individuals who specialize in relatively resource intensive activities. At the same time, specialized producers trade final product with each other. The efficiency of this trade will be a crucial factor to the specialization decision. 2. The Model Consider an economy with a continuum of ex ante identical individuals who are both consumers and producers.9 There are two consumption goods X and Y that are produced by using labor (/) and a primary resource (s). Each individual is endowed with a given quantity of labor / and a certain amount of the primary resource J . Production technology for each person is given by x + X°=a(lx-AxT{sx-Bf
and y + y =b(ly - Ayf
(sy - 5 f ,
where x + xs and y + ys are respective output levels of the two goods and /; and 5. are the amount of labor and the primary resource employed to produce good i. As noted in section 1, production of each good requires a fixed investment of both labor and primary resource. For good Rosen (1983) indicates that the division of labor can be used to save on fixed learning cost by avoiding duplicated learning and training. 9 For example, the consumers might be represented by the line interval between zero and unity. Assuming a continuum of consumers avoids issues of the existence of equilibrium, see Zhou etal. (1999).
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i, the investment of labor is given by At. It simplifies the analysis if we assume that the investment in primary resource is identical between goods and is given by B. With trade, xs and ys are the quantities of each good produced for sale while x and y are output used for own consumption. Clearly, under autarky, xs and ys are both equal to zero. Restrictions on production parameters are needed to guarantee both increasing returns from individual specialization and the feasibility of unspecialized production. For increasing returns to scale, assume that Aj > 0 (/• = x, y), B > 0, at > 0 , 0 < /?, < 1, and at + /?, > 1, i = 1, 2. For feasibility of production of both goods in autarky, assume that / >AX+Ay and J > 2B . If individuals specialize and trade in final goods, they can also trade in the primary resource. Thus, with specialization, the amount of primary resource used in production is not limited to the individual endowment.10 Individual utility is given by U - XY = (x + kxd)(y + kyd), where X and Y are the quantities consumed of the two goods and x and y are selfprovided quantities and xd and yd the quantities of the two goods purchased if there is trade. Transactions efficiency is represented by the parameter k that shows the fraction of goods received from purchasing one unit of final goods. The transaction costs per unit of final good purchased are given by (1 - k).u There are two possible economic structures in equilibrium for this economy: autarky and the structure with specialization and division of labor. In autarky, each individual both produces goods and consumes the goods he produces. There is no specialization and trade. With division of labor, each individual produces only one good, sells part of the good he produces in exchange for the other good.12 In autarky, each individual chooses a production and consumption bundle to maximize utility. Specialization, however, involves the individual choosing a 'corner'
For simplicity we assume that there are no transaction costs in trading the primary resource. This contrasts with the transaction costs for final goods presented below. 11 This is sometimes referred to as 'ice-berg' transaction costs — a fraction of each unit purchased 'melts' before consumption. 12 It follows from increasing returns at the individual level that no individuals will both produce and buy a good. Once an individual chooses to specialize, his utility is maximized by complete specialization.
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solution, producing a particular good and trading that good at market prices.13 As noted above, the primary resource is also traded with division of labor. It is useful to define an individual's level of specialization in good i by the labor share devoted to producing that good: /,.// , i - x,y. Clearly if an individual is specialized in producing one good then the level of specialization is unity for that good and zero for the other good. We can measure an individual's total factor productivity (TFP) in good / by the index TFP =
/ + is —
In Appendix A we show that the TFP in good i is increasing in the level of specialization. Thus the division of labor leads to an increase in TFP in our model14. We refer to this as endogenous productivity progress. Of course, TFP may also change if one of the parameters in the individual production function changes. This would be an exogenous change in productivity. 3. Comparing Autarky and the Division of Labor Our focus in this paper is the interaction between the per capita endowment of the primary resource, s, the level of transaction efficiency, k, and the degree of equilibrium specialization. In order to compare how exogenous levels of s and k affect the endogenous level of specialization, we need to determine when the economy will be in autarky and when equilibrium will involve division of labor.
13 The comparison between utility levels at different corner solutions that underlies the individual's decision is sometimes called inframarginal analysis; Buchanan and Stubblebine (1962) first used this term. 14 There are other works implying that a more extensive division of labor raises productivity, e.g. Becker and Murphy (1992).
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3.1. Autarky In autarky each individual solves the following problem: maxs j U = xy, subject to x = a(lx - Ax)"' {sx - B f , y = b{ly-Ay)ai(sy-B)P\
lx+ly=l,
sx+sy=J,
x>0,
y>0.
The optimization problem yields an equilibrium level of utility:15 C;
= a 6
_£f£_
(a1+«2f
+a2
^^l(T-Ax-Ayr^s-2Br\{l)
(A+/?2)
5.2. Specialization If an individual specializes then he can either produce and sell good x and buy good y (denoted x/y) or produce and sell good y and buy good x (denoted y/x). Individuals may also exchange resources. In each case the individual chooses levels of production and trade to maximize his utility: r \ x maxsydU = x{kyd), s.t. x + xs = a(T - Ax Y1 (J + sx - B ) 0 ' , Pssi+Pyyd^Pxxs, y
x>0,
yd>0,
maxs x, U = (kxd)y,
yXj
&X. y + y =b{l -A^ (s+s2-Bf
Pss2+pxxd^pyys,
,
x d > o , y>o.
These two optimization problems yield demand and supply functions for goods x and y as well as for factor s.16 Equilibrium is determined by the The detailed algebra in solving for the comer equilibrium is in Appendix B. For details see Appendix B.
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market clearing. Also, for equilibrium in the choice of activity, utility must be equalized regardless of the direction of specialization. If Mt is the population size of individuals who choose to specialize in the production of good i, then the relative population size of individuals specializing in each good in equilibrium is given by
Mx_\_ •A My 1-fr
(2)
The population size of individuals choosing configuration (x/y) will be smaller than that choosing configuration (y/x) if the primary resource is used more efficiently in the production of x than in the production of y. In equilibrium, relative prices are
• = ka{l-A^
2-px-P2){__B) (i-A)(A+A)
-|A-i
&M
^ - = kb(T-Ayf fi2"AN-A) {s-B)
(3) h-i
(l-AX/UA)
(4)
Individual equilibrium trade in the primary resource is given by
(J-B),
(i-A)(A+A) A-A (s-B). (i-A)(A+A)
(5) (6)
From (5) and (6), an individual who specializes in the production of good x will buy the primary resource while an individual who specializes in the production of good y will sell the resource if /?, > jB2. In other words, specialists in the production of the more resource intensive good buy the primary resource in equilibrium. The amount of each good sold in equilibrium is given by ••M.
|A-A (s-B), (i-A)(fl+/?2)
(7)
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A(2-A-A)/_ (s-B) .(l-AKA+A)
(8)
(J-B)
(9)
(I-AKA+A) Each individual's utility with specialization is given by 1 +V
i - Ai J
(10)
Itlt
NAH
i+
A
A+A '
(A+A)'
i - A2 7
The general equilibrium of the economy is either autarky or division of labor. Given the equilibrium relative prices (3) and (4), individuals will choose to specialize if this maximizes their utility (i.e. if U^ >U*A).17 From (1) and (10), this will occur if and only if B
k>k = E\l
^"
(11)
J-B
where a,
E= v
«,+«2y V
1-3-
17
'i f
1 +i - A
a, a
\
+ a
2 J V
i-A
i-A
1 +i-A
\i-A
i - A2 7
(12)
1-=^-
In Murphy et al. (1989), government coordination for simultaneous investment in many modern industries is one of the sufficient conditions for the big-push industrialization. In Sachs and Warner (1999), social expectations about future coordinate the investments. Transaction efficiency in an economy can be improved endogenously through government taxation and investment in infrastructure in development process as described in Wen (1997). Once the unit transaction efficiency makes U'B > U[ and some people in the society start to try to specialize in two different professions, a decentralized market adjustment process like that described in Appendix B can coordinate number of different specialists towards the equilibrium.
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and k is the threshold value of transaction efficiency for the division of labor to be the general equilibrium and E e (0, 1). 4. Resource Endowment, Transaction Efficiency and Specialization Equation (11) summarizes the relationship between resource endowment, transaction efficiency and specialization.... Formally, we the following lemma. Lemma 1: General equilibrium has division of labor and specialization if the transaction efficiency k is higher than the threshold value k and has autarky if the transaction efficiency is lower than the threshold value k . To interpret Lemma 1, suppose an economy has a given level of primary resource. If transaction efficiency is initially below the threshold level, k , then rises to exceed the threshold, equilibrium discontinuously jumps from autarky, where no market and trade exist and productivity is low, to the specialization with increased productivity and trade dependence. 19 The increase in productivity involves endogenous productive progress, albeit that specialization is driven by the exogenous (to our model) change in transaction efficiency. Ex ante identical individuals acquire their ex post comparative advantage in producing different goods through choosing different professions. In our model, every person in the economy is identical in autarky: each of them has exactly the same tastes, production technology, and endowments. They make the same optimal decision about production and consumption and have the same average and marginal productivity in the production of each good. With the division of labor, a person who specializes in the production of x has higher productivity in the production of this good than in autarky. This reflects the economies of specialization in the production technology. Similarly, a person who specializes in good y has
See Appendix C for the proof. The increase of trade dependence is obvious. The increase of total factor productivity is shown in Appendix D.
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a higher productivity in this good than in autarky. For society as a whole specialization increases the factor productivity of both x andy. We can rewrite the threshold level of transaction efficiency as k=E
fl-2BT+p I-B
This expression captures the trade-off between transaction efficiency and the resource savings due to specialization. In Autarky, a minimum of 25 of the primary resource must be devoted to production before any net output for consumption can bring the individual a positive utility. In contrast, with specialisation, each individual only incurs a fixed cost B and can trade some of the produce for the other consumption good. However, the transactions costs of trade must be incurred. As J increases, the relative primary resource cost of autarky falls. The significance of specialised use of primary resources decreases. Consequently, transaction efficiency must be higher to induce specialization. The relationship between per capita primary resource endowment, J, and the threshold level of transaction efficiency for specialization, k , is given by "*• = (&+P2)E\l ffl v Hl)
"
—
V I-B)
— ^ > 0 .
(13)
(I-B)2
As the per capita resource endowment increases, the threshold value of transaction efficiency rises. In other words, a relatively resource poor country needs a lower level of transaction efficiency to achieve specialization than a resource rich country. In this sense, for a given set of trading institutions, a resource poor country is more likely to have division of labor in general equilibrium than a resource rich country. The institutions that enable specialization in a resource poor country might be inadequate for division of labor in a resource rich country. This relationship is illustrated in Figure 1. Holding /?,, j32 and B fixed, Figure 1 plots the threshold level of transaction efficiency against per capita resource endowment. Clearly by our assumption that autarky is feasible, the relevant range of per capita resource is I> 25. From (13) k is increasing in I over this range of per capita endowment. Figure la
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illustrates the situation when J5X + /?2 > 1 . k approaches zero as J approaches 25. The slope of A: ( J ) also approaches zero as J approaches 2B but is increasing so that k (J) is convex for J close to IB \iPi + p2 > 1. k is bounded above by E < 1 and asymptotes to E as J approaches infinity. There is also a point of inflexion at 1= 0.55(3 + /?/ + /? 2 )when# + ,02>l. Figure lb illustrates the case of /?/ + f52 < 1. k(I) is concave and the slope of A: (J) approaches infinity as J approaches 2B. The locus of & divides Figure 1 into two subspaces. Above this locus (region I) specialization is achieved in general equilibrium. Below (region II) general equilibrium is autarky. The dotted vertical line further divided the parameter space. Left of this line (denoted by a) involves a relatively low resource endowment while to the right (denoted by b) involves relative natural resource abundance. The four regions in Figure 1 represent four potential paths of development. First, consider la. This involves countries with a relatively low level of primary resource but whose transaction efficiency is sufficient to gain specialization. Such countries and areas, which might include say Japan and Hong Kong, have sufficient transaction efficiency to develop despite their relative lack of natural resources. In contrast, region lb reflects countries with both abundant natural resources and high transactions efficiency, such as the USA. Such countries face the 'curse' of abundant natural resources but have highly efficient trading institutions that more than offset their resource endowment. Transaction efficiency is so high that it offsets the negative effects of high per capita resources on the division of labor. Region lib involves countries with high per capita resource endowment whose transaction efficiency is inadequate to create specialization and the division of labor. For example, some South American countries fit into this situation. These countries have low productivity compared to countries in region lb due to the failure of development and specialization. Even so, they may have developed trading institutions and transaction efficiency that would have led to
When /?! + p^ = 1, the slope of k (s) approaches to (fix + PifE/B as S approaches 25.
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05B(3+^+p2)
(b)
Figure 1: Partition of (k,s) space when (a) J3X + /?2 > 1 ; (b) # + /?2 < 1
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511
specialization and increased productivity if they were in region la — in other words, if they were resource poor. In this sense, their trading institutions are inadequate given their relative resource abundance. Finally, region Ha reflects resource poor countries with low transaction efficiency. This description describes some African nations. These nations are resource poor but their trading institutions are so inadequate that they have not attained even the moderate transaction efficiency needed to propel specialization. The following proposition summarizes the relationships captured in Figure 1. Proposition 2: General equilibrium with specialization will occur in countries with either (la) a low level of per capita resource endowment and institutions that involve moderate to high transaction efficiency or (lb) a high level of per capita resource endowment and a high level of transaction efficiency. General equilibrium without specialization and development will occur in countries with either (Ha) a low level of per capita resource endowment and institutions that involve a low level of transaction efficiency or (lib) a high level of per capita resource endowment and a low or moderate level of transaction efficiency. Transaction efficiency will often depend on the social infrastructure and political institutions of a country. Laws that protect private property and contracting are likely to raise transaction efficiency. The development of banks and professional middlemen, the creation of communication and transport networks, and the liberalization of domestic and international trade will tend to raise transaction efficiency and promote specialization and the division of labor. However, as shown in Figure 1 and Proposition 2, the ability of such institutional development to 'pull' an economy towards specialization and development interacts with the tendency for a lack of natural resources to 'push' development. These two factors relating to development cannot be considered in isolation. The interaction between resource endowment and transaction efficiency has implications for the welfare of countries on different development paths. For example, suppose a country in region II of
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Figure 1 develops institutions that raise transaction efficiency and promote specialization and trade. Given its per capita resource endowment, these institutional improvements might allow the country to jump into region I with increased productivity and specialization. This change is clearly welfare improving. People switch from autarky as trade becomes more efficient. Specialization is in their individual interest. People could remain in autarky without a change in welfare, but they find it welfare improving to specialize. In contrast, suppose a country in region II suffers a decline in the level of per capita resource holding fixed its level of transaction efficiency. This may occur, say, due to a rapid increase in population. The decline in per capita resource may allow the country to move into region I (i.e. push it into development) but it also lowers welfare. Overall, the decline in the per capita resource level makes people worse off. Specialization in such a situation 'slows' the decline in welfare but does not prevent it. The switch from autarky to specialization involves declining standards of living. Proposition 3: (I) If specialization occurs because the level of transaction efficiency rises for a given level of per capita resource, then this change will lead to a rise in individual welfare. (II) If specialization occurs because the level of per capita resource falls for a given level of transaction efficiency, then this change will lead to a fall in individual welfare. Proof: For (I), suppose the initial level of transaction efficiency is k\ and the final level is k2. Initially, UD (kx }UA (k2) as specialization is the equilibrium after the change in k. But, from (1), UA(k{)-UA(k2) so UD (k2 )>UA (kx) and welfare increases after the increase in transaction efficiency. For (II), from (10), dU*D/ds=[(ft + J32)/(J-B)]U*D >0 . Suppose the per capita endowment of the primary resource is initially J} and it then falls to J2. As autarky is the original equilibrium, UA (J,) > U*D {Jx). As 5J>J2 it follows that U^ (J{)>U^(J2). Hence, U*A (Sj) > U^, (J2), the
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utility level having decreased after the fall in the level of per capita primary resource. 5. Technology and Specialization The incentives for a country to specialize will alter over time as technology changes. However, technological progress need not increase the gains to specialization. Given the level of a country's per capita primary resource, technological change may either increase or decrease the threshold level of transaction efficiency that is needed for specialization. To see this, Proposition 4 considers two alternate forms of technical progress. First, technology may improve by reducing the establishment costs involved in producing a good. This efficiency improvement is reflected by a decrease in B in our framework. Second, production becomes more efficient if there is an increase in the 'inputoutput' elasticity of production. In our model this is equivalent to an increase in a, or /?,. As shown by the proposition, improvements in the efficiency of production due to a change in an input-output coefficient lead to a fall in the threshold level of transactions efficiency required for specialization. This type of technical progress makes specialization more likely. However, a reduction in the 'set-up' cost of producing each good raises the threshold level of A:. Proposition 4: For a given level of per capita primary resource J, the threshold level of transactions efficiency for specialization k is: (I) decreasing in the fixed resource cost B and (II) decreasing in each of the production coefficients at and fy. Proof: • For (I):
* = _ £(/S , +A) _Lr 1+ ^_i[ 1 _^_r* ,
H]
H2
s-Bl
J-BJl
J-Bj
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For (II):
a,
+ ln 1-=<0. V I-A,j
So k is decreasing in ax. Similarly, da2 d\nk~ dpx d\nk
= ln 1 — V
r
«,
+ In
A
A
5
\
<0,
a
\ + a2 J
-In 1 +
+ ln 1
= -ln i+
+ ln i j
<0.
J-B B J-B
<0.
There has been significant debate in the development literature about the relationship between population and innovation, particularly in agricultural societies. 21 Our model can shed some light on the relationship between increasing population, as represented by a decreased per capita resource endowment, and the incentive to take up innovations that improve transaction efficiency. Suppose there are two societies that both have an identical level of transaction efficiency K and are in autarky (region II in Figure 1). One society, however, has a lower level of per capita resource endowment than does the other society. Either society can adopt an innovation that will increase transaction efficiency by Ak but adoption is costly, so each society will only choose to adopt the innovation if the benefits outweigh the cost. From Figure 1, a society will adopt the innovation only if K + Ak exceeds the critical level of transaction efficiency for that society.22 In this sense the society with the lower level of per capita resource is more likely to adopt the 21
For example, see Boserup (1965, 1981), Simon (1992), Darity (1980) and Pryor and Maurer(1982). 22 This is a necessary but not sufficient condition for adoption. The society will also consider whether the increased utility gained by adoption and specialization exceeds the adoption cost.
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innovation, so that our model is consistent with the idea that population pressure may 'push' innovation. 6. Conclusions We have developed a model that combines transaction efficiency and the institutional features of development with the level of resource endowment. By bringing these two factors together we have modeled patterns of development that have been noted in the empirical literature. Our model can explain the well-documented examples of resource rich countries that have failed to develop and resource poor countries that have developed. At the same time the model can also show why a resource rich country might develop; how a resource poor country may be mired in poverty; and why institutions that accompany development in one country might be inadequate to spur development in another country.23 Our model captures both the idea that a lack of natural resources might 'push' an economy to develop and the idea that improved institutions and increased transaction efficiency can 'pull' the development process. However, it shows that both these ideas alone are inadequate to explain the pattern of development. Rather, development depends on the interaction of transaction efficiency and resource endowment. A country that is relatively resource poor may only need a relatively moderate improvement in transaction efficiency to spur development. In contrast, a resource rich country may need greater transaction efficiency and more advanced institutions before development takes place. Development is not guaranteed for a resource poor country. It can fail to develop due to inferior institutions preventing transaction efficiency from appropriate improvement. Similarly, a resource rich country might overcome the 'curse' of its resources by developing sufficiently advanced trading institutions. Easterly and Levine (2001) argues that many long-run growth facts do not support models with diminishing returns, constant returns to scale and "something else" rather than physical and human capital accumulation explains a lot for divergence of countries. Our model based on endogenous productivity progress due to economies of specialization belongs to what they call "something else".
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Our framework is highly simplified. While we believe that it captures many 'stylized facts' from historic studies of development, it is clear that many issues cannot be fully addressed in our framework. This said, it both overcomes empirical 'anomalies' that have faced alternative theoretical models and provides a useful starting point to consider the complex interactions between different forces for development.
Appendix A: Proof of the Economies of Specialization in the Production of Each Good When the TFP in good i is increasing in the level of labor share input in the production, we say the production technology exhibits economies of specialization. From the definition ofTFP„ ^(TF^)
=
a
qj
i
(/i-(V(",+A)) _ A /-'/(«,+A)f1"1 (V-(i/(«>+A))
_ & -I/(« 1 + A) y* [( f f i +p x
_ I)/;I/(«,+A) + ^/- o
when / >J_ and s„>B , X
d[TYYy)
b
_
X
q2
/
(l/(g2+A))
a(iyuy a2+p2^
_
/_1/(a2+A)\
^
a
» - 1 / 1.(1/(0,+/%))
> ys>
_&-V(«2+A))A [(« 2 +p1 _i)/-V(«2+A) + ^ / - ( I / ( « 2 + A ) H ] > o when / > Ay and sy> By. By definition, there are economies of specialization in these productions. Appendix B: Solving for Corner Equilibria This appendix is to solve for the equilibria under autarky and division of labor. For autarky we solve a standard utility maximization problem for each person. With trade, we solve for a corner equilibrium. This is defined as a group of relative prices of traded goods and relative number
Development, Trade and Endowments
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of specialists such that every individual maximizes his utility through the freedom of choosing between different configurations or 'professions', market demand for each traded good equals market supply, and the relative prices are determined by free choice among different professions and market clearing conditions. In autarky each agent self-provides both goods. Every agent maximizes his utility through allocating his labor and resource in the two productions: maxs , U = xy, s.t. x = a{lx-Axy(sx-B)\
(B.l)
y = b[ly-Ay)a2(sy-B)Pl,
lx+ly=T,
sx+sy=I,
x>0,
y>0.
The first-order conditions yield the following solution: a,
-{I-A,-A,), «, +a2
(B.2)
>, = * , + - *
(B.3)
l=A+-
ax + a2 s=B S
A A+A
(J-2B),
(B.4)
A
nr (s-2B),
(B.5)
+-
B
V=
A+A "1
a, x=a (f-4-4) a, + a2 y=b
a,
A
A+A 1
p-A-^)
«2
A A+A
nr (J
-IB)
(J-2B)
(B.6)
(B.7)
Substituting x andy into the objective function, we get the agent's utility level in autarky as Ul=ab-
a
(al+a2) ^(Bl
P?P£ ^(T-A-Aj^is-lBY^. +
B2y
(B.8)
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In the structure with division of labor there are two configurations. In one configuration xly, the agents self-provide and sell good x, but buy good y. We call these agents x sellers. In the other configuration xly, the agents self-provide and sell good y, but buy good x. We call these agents y sellers. B.l. Solving for the optimal decisions ofx sellers The problem that x sellers need to solve is
x + xs =a(l -AxY1(l
s.t.
Ps^i+Pyyd^Pxxs,
-B)A,
+ s1
(B.9)
yd>0.
x>0,
From the first-order conditions, we can find (B.10) Px p
Px l
y
+ sx-B)A
—y =-z a{l-Axf(l P* 2 dd
Ps_
(B.ll)
Px
and f
s
y/(A-i) Ps
+s,-B I - A\
(B.12)
.
Rearranging (B.l 1) yields r
y/(A-i) Ps
\PxcAj
-J + B .
(B.13)
Applying (B.10)-(B.12) and rearranging it, we find the demand for good y is
y
Px 2Py
(i-A)c,
l/(A-0
r
(B.14) PxCA
Px
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519
Substituting (B.13) and (B.14) into (B.10) yields the sale's plan: X , = I( 1 + A ) C ]
Ps
Ps
PAA
(^ (s-B)
(B.15)
2P>
From production function of x and (B.15), the amount of self-provided x can be derived as (/(A-')
*=i(i-A)c,
+ A. •(s-B).
PAA
(B.16)
Substituting (B.14) and (B.16) into the objective function yields the indirect utility of x sellers as Ux,y=x(kyO) = kp*
\MA-i)
(i-A)c,
4/\
+
KPAAJ
. (B.17)
^-(S-B)
Px
B.2. Solving for the optimal decisions ofy sellers Similarly, by solving y sellers' optimal problem max
s2,x*Uy/x={kxd)y> y=b(T-Ay)'h(s+S2-B)fil,
s.t. y + Pss2+Pxxd
^Pyys> xd>0,
(B.18)
y>0.
We can derive that the optimal trade plan of the primary resource, the optimal purchase of good x, the self-provided amount as well as the sale's plan ofy seller are as follows: r
y/(A-i)
Ps PyC K 2P2; x
a
=•
P
2PX
-I + B.
(B.19) b'(A-i)
(i-A)c 2
Ps
\PyC2P2j
(•=• (s-B)
,
(B.20)
520
M. Wen, S. P. King A/(A-i)
y=±{\
+ /32)C2
(s-B), yPyCiPij Ps
y=l-(\-fr)C2
Ps
yPyC2P2j
(B.21)
2A (;
2P/
(B.22) '
where C 2 =ft(/ -AyJ2 . The corresponding indirect utility is MA-') (1-/? 2 )C 2
Uylx=kx*y =
(B.23) \PyC2P7J
4P;
Py
B.5. Solving for the corner equilibrium of the structure with division of labor Due to the freedom of entry, there is utility equalization between the two configurations so that y/(A-0
1-fl
A
1/(^-1)
1-&
A vftQA
PA&
(B.24)
Denoting the number of x sellers as Mx, and the number of y sellers as My and letting m = MxlMy , market clearing conditions are msl + s2 = 0 ,
(B.25)
wxs = xd .
(B.26)
Substituting (B.13) and (B.19) into (B.25) and applying (B.24) yields
A(i-A)" r ft V'(A-0 = (i + i«X5'-5). m+ A(i-A). ftQA1 /
(B.27)
Substituting (B.15) and (B.20) into (B.27) and applying (B.24) yield m
i+fl A
i-A A
ftQA
:(l + »fX5"-5).
(B.28)
Development, Trade and Endowments
521
Solving (B.27) and (B.28) simultaneously, we can find the relative number of x specialists to y specialists and the relative price of resource to final good x: M, —'- = M2
1-A m-—— \-fi2
(B.29)
A(2-A-A)
• = CA (l-AXA + A) (s-B)
(B.30)
Substituting (B.30) into (B.24) and rearranging it yields the relative price of resource to final goody as -|A-i Ps
• = cj2
(i-A)(A+A) v
(B.31)
.
Applying (B.30) and (B.31) to (B.13) and (B.19), respectively, the trades of the resource of these two configurations are
A~A (J-B) (i-A)(A+A)
(B.32)
A-A (s-B), (i-/?2)(A+A)
(B.33)
and
respectively. If jix > fi2, sx > 0 and s2 < 0 ; if /?, < J32, s, < 0 ; and s2 < 0 ; /?, = /?2 implies s, = s2 = 0. (B.30) and (B.31) imply Py_
A
C.AM1-A)1 C 2 y9 2 A(l-A) 1_
A-A 2
(B.34)
-A-*(T-B)
A+A
.
By substituting (B.30) and (B.34) into (B.14)-(B.16), we can derive the corner equilibrium trade plan and the amount of self-provision for x sellers as
(i-A)c, A ( 2 - A - A ) 2 - A - A (i-A)(A+A)
(s-B)
(B.35)
522
M. Wen, S. P. King
y* =
(i-A)c 2 v 2 - A - A (i-A)(A + A)
(B.36)
;
and
(i-A)c, W-*-*\l-B) 2 - A - A (i-A)(A+A)
(B.37)
Similarly, the trade plan and the amount of self-provision forj sellers are y -
(i-A)c v 2 - A - A (i-A)(A+A)
.
(i-A)c, v 2 - A - A _(i-A)(A+A)
.
(B.38)
(B.39)
and y-
(i-A)Q A ( 2 - A - A ) (5--B) 2 - A - A (l-AKA + A)
(B.40)
Finally, substituting the comer equilibrium prices into the indirect utility function yields the comer equilibrium utility in the structure of division of labor: A+A
Uv=kClC2(s-B)
i-A 1+ i-A i J
A-i
\A-i
{l-Bf+t
kab(7-Axf(T-Ayf
1+\
1 +i-A
\A-i
i-A * J
AA# (A+A) A + A '
AA# A+A (A+A)
1 + i-A i-A 2 7
2
?2
i-A i
/
(B.41)
It is easy to see that the solution solved for the stmcture of division of labor is also a comer equilibrium because the relative prices
523
Development, Trade and Endowments
• = CA A
= c2/?2
(l-AXA + A)
and the relative number of specialists MjM2 = m = (l-/3l)/(l-J32) clear the markets and maximize the utilities of both x sellers andy sellers through individual choice between the professions. In fact, in the structure of division of labor, freedom of choosing profession and individual utility maximization will establish market clearing conditions. Since the excess demand function of resource s can be derived from (B.13) and (B.19) as y/(A-i)
r
S^ = M,sx + M2s2 = M2 m
-J + B
\PAxP\j y/(A-i) -J + B
(B.42)
yPyC2/32
and the excess demand function of x can be derived from (B.15) and (B.20) as X^ = M2x - M.x* = ^^~ 1-A d
A
2Px
-m
+ (s-B) \PyC2fi2j \I/(A-I)
i+A p\
y/(&-i)
/
PAA 1 /
-(T-B)
, (B.43)
the market clearing conditions S*0 - 0 and X^ = 0 yield i/(A-i)
PAA and
MJ~B) ft+fi:2
'l + V
^ m
(B.44)
524
M. Wen, S. P. King i/(A-i)
MJ~B)
(m + \).
(B.45)
PyC2& \fy^lH2)
As fi] < 1, /32 < 1, (B.44) and (B.45) indicate dm
(B.46)
•<0
and *(P,/P.) > 0 , (B.47) dm (B.46) and (B.47) mean that individual utility maximization and market competition make the relative prices change with the relative number of specialists and form the following negative feedback adjustment mechanism of the price system: -> T m =M ^ i 2
Ps
Ps
E±l
Ry.^
Uy
and l
M
\ +
The arrow on the right-hand side of M\IM2 can be a starting point of a change and the arrow on the left-hand side is then the end point of the change. If market relative prices and the actual relative number of specialists are different from the corner equilibrium relative prices and the relative number of specialists, the negative feedback adjustment mechanism functions to adjust the over demand of resource and goods toward market clearing situations and adjust the relative prices and the relative number of specialists until all markets are cleared and no person wants to change his professions. Then the utility equalization condition makes the relative number of specialists equal to the corner equilibrium number and relative prices equal to the equilibrium prices. Thus, the
525
Development, Trade and Endowments
solution solved for the structure of division of labor is the corner equilibrium in the structure of division of labor. Appendix C: Proof of Ee (0,1) a~
E=
a,
t / i "T" KA"J
V
a
\
+ a
2 J
i + L-E*. V i - Ai J
i + IzA V
\i-A
2y
\«2
A, / -A,y J
/ -A.x J
It is obvious that E > 0 . We only need to show E < 1. Denote #>(«,, a 2 ) =
a, va,+a2y
a, Ka,+a2j
Total differentiate In cp yields J(ln #>) = [in ax - ln(«, + a2)] Jaj + [in a2 - ln(«t + a2)] d a 2 . Hence, In (p as well as #? decrease in a, and a2. From assumption, we have at>\- f3t. Therefore, «2
<*2
X
ax+a2
V
1j
V
a,
a,1 +a ' "7
/
/
i-A /
i-A
2-A-Ay
\i-A
i-A
2-A-A 2 7
That is 1-
a.,
a, v
As
«i+«2y
1 +i - A
i-A
(vt'-^rnvt^))
\i-A
i-A 1-
i-A
<1.
l-Ay < 1, we have E < 1.
Appendix D: Proof of the TFP Increase after the Division of Labor This appendix shows that the TFP increases after the division of labor. Here we adopt the usual measurement of multi-product productivity. So in autarky,
526
M. Wen, S. P. King TFPA
Mx -TFP, + - J ^ L _ T F P y Mx + My Mx + My
(D.l)
and in the structure of division of labor Mx(x + xs) TFPD = -TFP„ + Mx (x + Xs) + M (y + ys) Mx(x + xs) + My(y + ys)
y
'
(D.2) Since the general case is technically too complicated, we use the simple symmetric production case to show the productivity improvement (i.e. a = b , Ax=Ay=A, at=a2=a and /?, = J32= J3). In this simplified case, we have TFP = TFP^. = TFP^ . Therefore, it can be derived that
a(F-2A)a(I-2By
TFPA =-2(«+/9-D/
(D.3)
al(a+P)Jpi(a+P)
and TFPn
a(l
-A)a(J-BY
(D.4)
/ al(a+PYgPI(a+P)
Obviously, TFPD > TFPA if both T and J are same in (D.3) and (D.4) (i.e. the division of labor is promoted by the improvement in transaction efficiency). If a population growth promotes the division of labor, say at population M, and per capita resource s~x =S/M1 , the equilibrium structure of the economy is autarky; but at population M2 (M2 > M,) and per capita resource I2= S/M2, the equilibrium economic structure is the one with division of labor, then as long as a+fi S-2BM, y c M v-w( fi S-BM, 2 J v ^ . y
< 2(«+/?-D V
1+, / -2A
(D.5)
we have TFPD > TFPA . It is easy to verify that if M2 2M,, we still have productivity improvement if the population increase can promote the division of labor.
Development,
Trade and
Endowments
527
References Asea, P., Lahiri, A., 1999. The precious bane. Journal of Economic Dynamics and Control 23, 823-849. Ashton, T., 1948. The Industrial Revolution 1760-1830. Oxford University Press, Oxford. Auty, R., 1995. Industrial policy, sectoral maturation, and postwar economic growth in Brazil: the resource curse thesis. Economic Geography 71, 257-272. Baxter, C , 1984. Bangladesh: A New Nation in an Old Setting. Westview Press, London. Becker, G., Murphy, K., 1992. The division of labor, coordination costs, and knowledge. Quarterly Journal of Economics 107, 1137-1160. Boserup, E., 1965. The Conditions of Agricultural Growth. Allen & Unwin, London. Boserup, E., 1981. Population and Technological Change: A Study of Long-term Trends. The University of Chicago Press, Chicago, IL. Buchanan, J., Stubblebine, W., 1962. Externality. Economica 29, 371-384. Darity, W., 1980. The Boserup theory of agricultural growth. Journal of Development Economics 7, 137-157. Easterly, W., Levine, R., 2001. Its not factor accumulation: stylized facts and growth models. World Bank Working Paper. World Bank, Washington, DC. Fei, J., Ranis, G., 1964. Development of the Labor Surplus Economy. Richard D. Irwin, New York. Fischer, S., Alonso-Gamo, P., Erickson von Allmen, U., 2001. Economic developments in the West Bank and Gaza since Oslo. Economic Journal 111, 254-275. Gylfason, T., 2001. Natural resources, education and economic development. European Economic Review 45, 847-859. Gylfason, T., Zoega, G., 2001. Natural resources and economic growth: the role of investment.Working Paper No. 142. Central Bank of Chile. Lall, P., Featherstone, A., Norman, D., 2000. Productive efficiency and growth policies for the Caribbean. Applied Economics 32, 1483-1493. Lewis, W., 1955. The Theory of Economic Growth. George Allen & Unwin, London.
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Mikesell, R., 1997. Explaining the resource curse with special emphasis to mineralexporting countries. Resources Policy 23, 191-199. Murphy, K., Shleifer, A., Vishny, R., 1989. Industrialization and the Big Push. Journal of Political Economy 97, 1003-1026. Pryor, F., 1994. Growth deceleration and transaction costs: a note. Journal of Economic Behavior and Organization 25, 121-133. Pryor, F., Maurer, S., 1982. On induced economic change in precapitalist societies. Journal of Development Economics 10, 325-353. Ranis, G., 1997. Reflections. Discussion Paper No. 786. Yale Economic Growth Center, New Haven. Rosen, S., 1983. Specialization and human capital. Journal of Labor Economics 1, 4 3 ^ 9 . Rosenstein-Rodan, P., 1943. Problems of industrialization of eastern and southeastern Europe. Economic Journal 53, 202-211. Sachs, J.,Warner, A., 1999. The big push, natural resource booms and growth. Journal of Development Economics 59, 43-76. Sachs, J., Warner, A., 1997. Fundamental sources of long-run growth. American Economic Review 87, 184-188. Simon, J., 1992. An integration of the invention-pull and population-push theories of economic demographic history. In: Population and Development in Poor Countries: Selected Essays. Princeton University Press, Princeton, NJ. Smith, A., 1776. An Inquiry into the Nature and Causes of the Wealth of Nations. Reprinted by J.M. Dent and Sons Ltd., London. Wen, M., 1997. Infrastructure and evolution in division of labor. Review of Development Economics 1, 191-206. Yang, X., 1990. Development, structural changes, and urbanization. Journal of Development Economics 34, 199-222. Yang, X., 1994. Endogenous vs. exogenous comparative advantage and economies of specialization vs. economies of scale. Journal of Economics (Zeitschrift-furNationalokonomie) 60, 29-54. Yang, X., Borland, J., 1991. A microeconomic mechanism for economic growth. Journal of Political Economy 99, 460^82.
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Yang, X., Ng, Y.-K., 1993. Specialization and Economic Organization: A New Classical Microeconomic Framework. North-Holland, Amsterdam. Zhou, L., Sun, G., Yang, X., 1999. General equilibrium in large economies with transaction costs and endogenous specialization. Working Paper. Department of Economics, Monash University, Clayton.
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Index
Arrow-Debreu (AD) model, 306-309, 320-323, 464 Asea P., 498, 527 Ashton T., 498, 527 asset specificity, 232, 235, 253 asymmetric distribution endogenous, 352 asymmetric residual control, 352 autarky equilibrium, 397, 402 Auty R., 498, 527 average cost, 29, 31-34, 39-41, 44-46, 142 constant, 40 decreasing, 29, 32-35, 39-40, 43, 45 increasing, 40 average cost pricing, 43-46 average firm size, 259-263, 271, 283
absolute advantage, 395 AdesA. F., 221, 225 adverse selection, 264, 331 agglomeration, 196-200,206-207, 214, 220-221 economies of, 198-200, 207, 214, 220-221 economies of transaction, 201 transactions and residences, 220 agglomeration diseconomies, 200 agglomeration economies, 195, 207, 221, 226-227 urban, 200 agglomeration of transactions, 206, 220 aggregate demand, 122, 219, 302, 372-373, 377, 476, 485 aggregate inequality, 79 aggregate productivity, 97, 102-103, 330 equilibrium, 81, 97 aggregate supply, 122, 302, 373, 377 AghionP.,78, 104,411,434 AK model, 263, 411,415 Alchian Armen, 232, 257-258, 324, 388 allocational efficiency, 124 Arndt H. W., 3, 24 Arrow Kenneth J., 92, 108-109,194, 228, 296, 323, 329, 332, 361, 405, 436, 476, 479, 487, 495
Backus David K., 465, 476 Baldwin R. E., 82, 104 Banerjee A., 78-79, 104 Barro model, 411 Barro R., 79, 104, 310, 323, 411, 434 basic market, 148-152, 155-156, 163 non-basic market, 148,150, 152, 155, 163 Bates R., 24 Baumgardner James, 196, 225, 295, 323, 366, 388, 438-439, 459
531
532 Baumol W. J., 403, 405 Baxter C , 499, 527 Becker G. S., 7, 25, 104, 196, 206, 225, 295, 323, 503, 527 Bellman backward decision rule, 322 Bellman optimality principle, 318 BenhabibJ.,411,434 Benhabib-Jovanovic model, 411 big push industrialization, 498, 506, 528 Blum J. M., 403-405 Bonbright James C , 30 Borland J., 80, 109, 111, 131-133, 167, 190-194, 231, 258, 273, 283, 287, 291-292, 295, 297-298, 325, 409, 416, 418, 424, 434, 436-439, 445, 459, 460, 480, 495, 499, 528 Boserup Ester, 405, 463-465, 473-477, 514,527 Boserup theory, 474-475 bounded rationality, 283 Bradford D. F., 403, 405 Brueckner J. K., 167, 200, 226 Brueckner-Zenou model, 200 Buchanan James M., 6-7, 25, 55, 61, 104, 117, 132, 392, 405, 503, 527 budget constraints, 12-18, 117, 127, 160-161, 174, 179, 210, 239, 255, 276, 313, 317-318, 336, 340, 364, 374-375, 382, 387, 395, 428, 431, 447-448, 467, 469, 482-483 individual, 149 real, 382 Camacho A., 262, 287 capital-labor ratio, 456, 458 Chandler A., 316, 323 Chenery H., 81, 105, 107, 160, 165 Cheng W. L., 8, 25
Index Cheung Steven, 231, 233, 249, 257, 262-264, 268-269, 287, 294, 323, 363-364, 383, 388 Cheung theory, 364 Chu C. Y. Cyrus, 391, 395, 405, 463, 466, 476, 479 Clemens E. W., 30, 479, 495 Coase Ronald H., 25, 29, 33, 69, 71-72, 105,157,231-232, 248-249, 257, 262-263, 265, 269, 283, 287, 294, 296, 323, 351, 360, 405 Coase theorem, 249, 351 Coase-Cheung theory, 265 Cobb-Douglas utility specification, 15 Cobb-Douglas production function, 444 comparative advantage acquired, 80, 365 endogenous, 80-84, 98, 199, 205, 219-221,331-334,349,352, 365, 423 exogenous, 80-84, 98, 205, 232, 332-333, 365, 528 natural, 80, 365 comparative dynamics, 415, 418, 420-427, 433 comparative statics, 113, 120-121, 130, 141, 144, 156, 160, 170, 184, 190, 191, 198, 201, 213, 217-218, 238, 283, 334, 338, 344-347, 367, 380, 386, 438, 453, 458, 473-474, 481, 483-484, 489-490 analysis, 198,213,217 inframarginal, 10, 82, 88-94, 97, 101,283,335,346,349,352 inframarginal comparative static equilibrium analysis, 283 marginal, 346
Index comparative technological advantage, 23 competitive economy, 467, 474, 476 complete specialization {cf. incomplete specialization), 126, 130, 397, 449, 451^154,502 constant elasticity of substitution (CES), 222, 295, 298, 438-439, 442, 444, 457 non-linear, 297 production functions, 295, 439 utility functions, 222, 295, 438-440 constant returns to scale (CRS), 7, 80-82, 88, 112, 137, 167, 365, 515 constant returns to specialization, 84 consumption bundle, 88, 502 consumption effects relative, 131 consumption externality, 56 consumption variety, 197 contingent contracts, 264, 352 contingent prices, 346, 348, 351 relative, 342, 347, 352-353 contingent pricing, 334, 337, 346, 348, 351 contract locus, 64, 67 contract theory, 232 contractual arrangements, 273, 352, 364, 384 efficient, 364 control theory, 295 conventional microeconomics, 138-139,364,386 convex functions, 393, 397, 402-403, 509 coordination costs, 50, 52, 295, 527 external, 51-52 internal, 51-52 corner decision, 465
533 corner equilibrium relative price, 89-91, 96, 182-183, 277, 340-343, 427, 430,469,471,524 solution, 340 corner solutions, 7, 85, 89, 116-120, 140-141, 145, 147, 155-156, 169, 174-175, 179, 191, 208, 232, 237, 273, 295, 299-300, 314, 335-337, 341, 367, 374, 383, 418, 426, 438, 445, 447, 465, 474, 503 corner equilibrium solution, 340 multiple, 175 correlation negative, 78, 271 positive, 77-78, 103, 263, 385, 437 Crawford Robert, 232, 258, 324 Dasgupta P., 411,434 Davis Otto, 73 Debreu G., 92, 131, 137, 167, 194 decision horizon, 297, 415-416, 425-426 equilibrium, 425-426 decision problem, 85, 90, 116, 118-120, 127, 179-180,210-211, 232, 238, 295, 299, 302-304, 313, 318, 339-340, 375, 377, 418, 428, 438, 446, 483-484, 487, 489 individual, 84, 141, 144, 170, 174, 179, 181,214,221,239-240, 276,296,298,300,309,313, 318,334,372,383,416,418, 441, 445, 448 optimization, 271 decision variable, 127, 144-145, 163, 174, 208, 210, 241, 277, 322, 336-340,375,418,425,484 degree of economies of specialization, 272
534 Deininger K., 79, 105 demand functions, 19, 90, 211, 376, 470, 504, 523 individual, 149, 484 Demsetz H., 257, 363, 385, 388-389 derivative first-order, 161 partial, 57-58 second-order, 224 development economics, 3, 23-24 direct pricing, 234, 252-253, 269 discount factor, 298, 416, 421, 423-425 disequilibrium, 81 diverse consumption, 113, 138, 169, 236, 366, 368, 371-372, 386, 438 division of labor balanced, 186,190-191 balanced partial, 176-178, 187-188 complete, 16, 97, 153, 157-158, 169-170, 173, 177-178, 188-189,273,275,442, 452-456, 458 degree of, 202, 295 developed, 158 development of, 80, 111-112, 116, 123-126, 129,131,190, 412 domestic, 97 economics of, 3-4 economies of, 84, 97-98, 103, 139, 160, 206, 213, 245-248, 351-352,365,455,457-458 economy wide, 7 efficient level of, 160,181,386 endogenous, 404 endogenous evolution of, 190-191 endogenous level of, 331
equilibrium level of, 79-80, 95, 97,102-103, 113-114, 122, 124, 129, 131,140, 158-159, 197, 199, 330-335, 348-353, 364, 367, 380-384, 386, 412, 480, 487 equilibrium network size of, 207, 222, 333 equilibrium structure of, 197, 209 evolution in the level of, 333 evolution of, 81, 96-97, 101-103, 126, 129-130, 140, 158-160, 170, 186, 188-191, 195-196, 200,219-223,228,238,251, 290-296,308,310,315-316, 323, 334, 349, 352-353, 415, 418, 421-427, 442, 455-457, 485-486, 492, 499, 528 exogenous evolution of, 190, 349 extreme, 142 general equilibrium network size of, 218 high level of, 79, 81,113-114, 117, 124, 128, 169, 185-190, 197,220,270,280,330-331, 348, 350, 410, 412, 426 highest level of, 425 horizontal, 439-441 increasing, 261, 425 inefficient, 403 intermediate level of, 173, 189, 193, 454-455 international, 99 larger network of, 330 larger network size of, 199 level of, 77-82,103, 112-113, 125-131, 138-140, 171-175, 181, 189,202,204,270,280, 286, 332-334, 347-348, 366-367, 379, 382-386,
535
Index 412-414, 437, 454-457, 465, 479-484, 487-494 low level of, 81,114, 159, 185, 188-191,204,492 network effects of, 201-202, 214, 221 network of, 96,99,199,201,204, 213,215,219,271,330,435, 480-481 network size of, 200-202, 207-210,214-218 organizing, 261, 280 partial, 95, 153, 169-170, 177, 186, 188, 197,273-274,284 pattern of, 118,175,311,333, 392,411,480 positive network effects of, 201, 330, 333, 348, 352 social, 209, 213 society's level of, 480^481 unbalanced, 177-178 unbalanced partial, 178, 186-191 vertical, 439-440 Dixit A., 80-81,105,171, 193, 262, 287, 338, 360, 437, 439, 458-459 dual economy, 222, 458 dual externality problem, 73 dual structures, 75, 79, 81, 87, 92, 95-97, 103, 168-170, 177-178, 185-191,215 evolution of, 81 Duflo E., 79, 104 Dupuit Jules, 30-31 Dupuit taxation theorem, 30 DurkheimEmile, 111, 125, 132 dynamic decisions, 283, 306, 322, 414 dynamic model, 118, 299, 309 dynamic optimization full, 300 dynamic planning model, 81
dynamic programming, 295-296, 305-306 individual problem, 415 problem, 296, 305-306 Easterly W., 515,527 economies of agglomeration, 198-200, 207, 214, 220-221 Type 1,198 Type II, 198,200-201,220 economies of complementarity, 293, 299, 307-308, 316, 438, 440, 444, 457 degree of, 298 economies of division of labor, 84, 97-98, 103, 139, 160, 206, 213, 245-248,351-352,365,455, 457-458 economies of roundaboutness (cf. roundabout production), 223, 413-414, 417, 421-424, 427, 431, 437-439, 454, 456- 458 Type I, 417 Type II, 417 economies of scale, 80-81, 84, 137, 139, 167-168, 171, 194, 196, 201, 246, 260, 262-263, 279, 288, 336, 365-366, 438, 500, 528 external, 84 global, 78, 197 internal, 84, 142 local, 84 economies of scope diseconomies of scope, 168, 171 economies of specialization (ES), 7, 23, 80,84,108,112-115,131,138, 157, 160, 167-175, 180, 187-191, 196, 199, 206-208, 217, 221, 228, 232-236, 242-247, 253, 262, 266, 271-273, 279, 293, 298, 307-308, 314, 316, 329, 332-336, 348, 361,
536 363, 366-372, 385-397, 402, 421-424, 438, 440, 454-458, 465-467, 476, 480-482, 495, 500, 507,515-516,528 aggregation, 206 degree of, 115, 171, 184, 188, 236, 247,272,293,382,414,421, 423-424, 427, 444-445, 453-457, 482 economies of specialized learning by doing, 413-417, 431 increasing, 156 individual-specific, 116 economies of the firm, 245, 248-249, 252 economies of transaction agglomeration, 201, 217 economy wide division of labor, 7 Edgeworth box diagram, 64 efficient contractual arrangements, 364 elastic supply, 32 elasticity of output, 272, 298, 417 elasticity of production, 513 elasticity of substitution, 298, 441, 444, 453-457 Elvin Mark, 254, 257 endogenous comparative advantage, 80-84, 98, 199, 205, 219-221, 331-334,349,352,365,423 endogenous evolution of the firm, 283 endogenous growth models, 263, 288, 291,411,434 development of, 291 neoclassical, 263 endogenous productivity progress, 503, 515 endogenous specialization, 77-80, 89, 92, 102, 329, 332-333, 337, 467, 529 inframarginal analysis, 418
Index endogenous transaction costs, 263-264, 267,270,329,331-335,348, 350-353 endowment constraint, 84, 114-115, 126-127, 142,171, 174, 179,210,241, 276-277,313-314,317,322, 336, 339-340, 368, 371, 374-375, 379, 428, 482-483 fixed total labor, 368 individual, 502 individual's labor, 370-371 labor, 272, 294, 393, 445 per capita resource, 498, 501, 508-514 primary resource, 498 resource, 497, 499-501, 507, 509, 511,514-515 envelope theorem, 219, 381, 426 equilibrium aggregate productivity, 81, 97 analysis, 210 autarky, 397, 402 competitive, 124, 139, 148, 363 corner, 10, 89-92, 96, 117-121, 144, 147-156, 162-165, 175-184,190-191,237-248, 275-279, 284, 334, 337-350, 353, 355, 383, 427, 430, 446-454, 468-469, 471-474, 504, 516, 520-524 corner equilibrium solution, 340 decision horizon, 425-426 dynamic, 78,160,293-310, 316, 320-323,413-415,418-424, 427 dynamic general, 82, 283, 409, 412,414,426 dynamic general equilibrium model, 82, 283, 409, 412, 426
Index full, 147-156,383 general, 10,77-97, 101-102, 112-113,117-121,124, 165-170, 175, 179, 181, 184-185, 191-196, 199-209, 213-218,221,226,238, 243-244, 248, 262-263, 273, 277, 279-284, 329, 331-337, 344-355, 359-361, 418, 420, 431-435,439,441,452,454, 457-458, 465, 473-474, 495, 506-509,511-512,529 general equilibrium principalagent model, 332 general equilibrium residence structure, 213 general equilibrium theory, 165 individual, 505 inframarginal comparative static equilibrium analysis, 283 insurance contract, 491, 493-494 interior, 10 Khandker-Rashid, 81 level of division of labor, 79, 80, 95,97,102-103,113-114,122, 124,129,131, 140,158-159, 197, 199, 330-335, 348-353, 364,367,380-384,386,412, 480, 487 level of specialization, 168,280 level of utility, 504 local, 420, 427, 430-431 market, 21, 399 multiple, 92, 130, 147, 214-215 Nash bargaining, 337, 344 neoclassical, 363 network size of division of labor, 207, 222, 333 non-Pareto optimum corner, 155 occupation, 215 occupation structure, 215
537 Pareto, 61-63, 67-72 Pareto optimum corner, 148, 151-152, 155-156 Pareto-optimum corner, 124,184 partial, 200, 232, 295 partial analysis, 200,232, 295 partial equilibrium contract, 232 price-taking, 294 private, 63 relative price, 89, 113, 124, 151, 304-305, 484, 506 residence structure, 208 sequential, 415 static, 302, 438 static equilibrium relative price, 303 static general, 214, 295 structure of division of labor, 197, 209 utility maximizing, 61-62, 67 Walrasian, 264 Walrasian sequential, 283 equilibrium residence structure, 208 equilibrium specialization, 503 Ethier W., 147, 165, 171, 193, 262, 287, 373, 437, 458-459 existence theorem, 89 exogenous comparative advantage, 80-84, 98, 205, 232, 332-333, 365, 528 exogenous monitoring cost, 332 exogenous transaction cost, 202, 270, 329,331-335,349-352,481 coefficient, 330-331 exogenous transaction efficiency coefficient, 347 extent of market, 122, 139, 158 external cost, 405 external diseconomy, 55, 60, 62 marginal, 57, 67-68
538 external economy, 55, 60, 62, 66, 84, 435 marginal, 57, 68, 72 Pareto-relevant marginal, 72 external effects, 55, 61, 68 technological, 56 externality, 55-60, 62, 64, 67-73, 104, 131-132, 199, 214, 221, 231, 247, 294, 364, 383, 391^105, 434, 527 consumption, 56 inter-firm externality relationships, 72 marginal, 56-57, 60-61, 63, 69, 71 market, 391-392 negative, 391, 394, 398, 403-404 Pareto-relevant, 56, 60-62, 69 Pareto-relevant marginal, 70 potentially relevant, 59-60, 67 potentially relevant marginal, 59 problem, 391-392, 395 problem of, 384, 391,406 production, 56, 403 relevant, 56, 58, 60-61 single, 58, 69, 72 technological, 56 externality remedy, 397-401, 404 externality-correction intervention, 405 policy, 392-393 extreme specialization, 130, 139, 142, 171,386 factor endowments, 23 factors of production, 33, 36-37,43-44 Fei J., 78, 105, 158, 165, 497, 527 finite horizon (two-period) economy, 296,300,415 First Welfare Theorem, 214
Index first-order conditions, 15, 119-120, 162, 215, 224, 376, 378, 380, 400, 425,472,484,517-518 Grossman, 335 first-order derivative, 161 Fisher R. A., 53 fixed costs, 31, 51-52, 127, 170, 173, 297, 413, 442, 508 coefficient, 173,297 fixed learning costs, 12,20,206, 394, 412-417, 423-428, 467, 482, 501 fixed management costs, 491 fixed total labor endowment, 368 fixed training costs, 482 fixed transaction costs, 126, 172-173, 185, 188,292,297,313-315,442, 454-455 coefficient, 185, 188, 313-315, 442 fixed transaction efficiency, 315 Fleming J. M., 29, 43 Fourier Joseph, 111 Frank Robert H., 78, 105, 111-112, 125, 132,258 free-rider problem, 475 Frisch Ragnar, 30, 42, 44 FujitaM., 81, 102, 106,196-198, 207-208, 220, 226, 262-263, 287 Fujita-Krugman model, 198, 263 fundamentalism investment, 409 saving, 409 Furubotn E., 363, 389 Gallup J., 106, 427, 434 game theory, 73, 264 general equilibrium residence structure, 213 generalized increasing returns (GIR), 7
Index geographical concentration of transactions, 198, 201-202, 204 increasing returns, 204 Glaeser E. L., 206-207,220, 225-226 Goodfriend Marvin, 464, 476 Greenwood J., 78 Grossman Gene, 78, 80, 104, 106, 137, 167, 171, 194, 231-232, 249, 253, 257, 261-262, 287, 291, 296, 298, 324, 338-340, 352, 360, 365, 389, 409-411,434,437,458-459 Grossman Sanford, 232, 249, 261, 296, 338, 340, 352 Grossman-Helpman model, 80, 171, 262, 291, 365, 409-411, 437, 458 growth models endogenous, 263, 288, 291, 411, 434 neoclassical endogenous, 263 Gupta Srabana, 332, 353, 360 Gylfason T„ 498, 527 Hart Oliver, 106, 232, 249, 253, 257, 261, 264, 287-288, 296, 324, 332, 338-339, 352-353, 360 Helpman Elhanan, 106, 140, 165, 194, 287, 324, 332, 360, 389, 434, 459, 488, 495 Henderson J. V., 200, 208, 226 Hessian determinant, 218, 225, 256 high development economics, 81, 102 Holmstrom B., 232, 257, 265, 288, 332, 360 Holmstrom-Milgrom model, 232 Hotelling H., 29-32, 35-39, 41, 43-46, 51,226 Houthakker H. S., 6, 25, 47, 206, 226 HowittP.,411,434 Hurwicz L., 264, 288
539 iceberg transaction costs, 83, 206, 267, 272 identical individuals, 23, 87, 115-116, 173-174,220,466,501,507 identical production technology, 220 implicit function theorem, 128, 426 incentive compatibility, 337-338, 343-344 conditions, 338, 343, 345 constraint, 337, 344 income distribution, 33, 39, 43, 75, 77-82, 99, 101, 103, 105-107, 459 equality, 102 inequality, 77-80, 96-97, 101-103 irrelevance of inequality, 81 income effects relative, 131 incomplete contract, 234, 265, 289, 353 incomplete specialization (cf. complete specialization), 50, 451, 453-454 increasing economies of specialization, 156 increasing returns, 6-7,49, 84, 88,112, 115-116, 124, 131, 137-144,147, 155-157, 160, 166, 172, 196,204, 214, 221, 247, 261, 292-296, 298, 336,364-368,373,386,417, 440-441,457,480,482,502 generalized increasing returns (GIR), 7 to labor specialization, 196 increasing returns to scale model (IRS), 6, 88, 112, 137, 140-143, 147, 160, 172,295,365,373,502 increasing returns to specialization, 20, 124, 131, 138-144, 156-157, 160, 166, 214, 261, 292, 298, 364-368, 373, 386, 441, 457
540 absolute degree of, 140, 157 degree of, 113,123-124, 157-158,380 relative degree of, 140, 157 sufficiently high degree of, 159 sufficiently large degree of, 157 incremental cost, 30 indirect pricing, 232, 250, 253-254, 269-270 indirect taxation, 42 indirect utility functions, 21, 89-91, 96, 119-121, 146, 149-150, 180-181, 211, 239, 241, 303-304, 313-314, 338, 340-341, 376, 398-401, 427, 447-448, 469-471, 519-520, 522 indirect utility level, 398 individual decision problem, 84, 141, 144,170,174,179,181,214,221, 239-240, 276, 296, 298, 300, 309, 313, 318, 334, 372, 383, 416, 418, 441,445,448 individual endowment, 502 individual pricing, 34 individual specialization, 292-293, 308,316,421,423,465,502 information asymmetry, 232, 264-265, 267, 332 extreme, 264 inframarginal analysis, 8-9, 12, 90, 102, 117, 122, 277, 338, 352, 418, 465, 474, 503 endogenous specialization, 418 inframarginal comparative statics, 10, 82, 88-94, 97, 101, 283, 335, 346, 349, 352 equilibrium analysis, 283 inframarginal economics, 3, 4, 8 inframarginal external diseconomy, 58-59, 67 inframarginal external economy, 57, 59,67
Index inframarginal externality, 57, 59, 62 input-output production technology, 403 institution of the firm, 231-233, 237, 245, 248, 253, 261-262, 264-270, 273, 279-280, 284, 286, 291, 294, 296, 310-311,315-316, 352-353 evolution of, 283, 287, 291, 434 insurance incomplete, 491 insurance contract, 488-494 equilibrium, 491-494 partial, 491-494 integer condition, 120 inter-firm externality relationships, 72 interior equilibrium, 10 interior solutions, 15, 89, 117, 174, 237, 300, 337, 445 intermediate production, 158 internal diseconomy, 68 marginal, 68, 72 internal economy, 68, 84, 142, 296 marginal, 68 investment fundamentalism, 409 invisible hand, 139, 208, 215, 221, 385 Jesen inequality, 355 Jevons W. S., 6 Jones C. I., 262-263 Jovanovic B., 78, 106 JuddK., 262, 288,411,434 Kelly M„ 82, 102, 106 Kendrick D. A., 167, 194 Keynes J., 425 Khandker-Rashid equilibrium model, 81 Kihlstrom Richard E., 332, 361, 486^87, 495 Kim Sunwoong, 196, 260, 295, 438 Klein Benjamin, 232, 258, 296, 324
Index Knight Frank, 247, 258 Koopmans T. C , 25 Kornai J., 382, 389 Kreps David M., 261, 288, 488, 495 Krugman P., 77-78, 81-82, 102, 106-107, 140, 165, 168, 171, 194, 196-197, 200, 220, 226-227, 262-263, 287-288, 366, 373, 389, 437, 458-459 Krugman-Venables (KV) model, 82, 263 Kuznetz S., 78, 80 labor allocation utility maximizing, 303, 427 labor contract, 232, 264, 266, 352 labor endowment, 272, 294, 393, 445 individual's, 370-371 labor surplus, 81, 159, 165, 497 Laffont Jean-Jacques, 232, 258, 332, 360-361,480,488,495 Lagrange multiplier, 163 Lahiri A., 498, 527 learning by doing, 172, 292, 294, 296, 298-300, 307-308, 310, 412-413, 415,417-418 economies of specialized, 413-414,416-417,431 learning costs, 15, 18-20, 206, 393, 397-400, 413, 416, 423-426 fixed, 12, 20, 206, 394, 412-417, 423^128, 467, 482, 501 total, 206 Lee Ronald D., 464, 475, 477 lemons model, 267, 287 Lerner A. P., 29, 32-46 level of production, 504 level of specialization, 9, 103, 112, 125, 137-138, 168-169, 173, 181, 183, 186-187,191,235,241,
541 261-262, 273,280-281,291, 295, 330, 334, 336, 417, 437-438, 441, 454-458, 480-481, 485, 490-492, 494, 503 endogenous, 439, 503 equilibrium, 168, 280 evolution in, 413 high, 79, 96, 103, 122, 178, 187-189,330,349,423,480 individual, 82, 103, 117,467, 480-481 individuals', 82, 103, 117, 181, 196,220-221,279,286,298, 332-333,336,438,441,467, 480-482, 503 low, 121,168,176, 185,191,280, 423, 456, 492 optimal, 144 person, 115, 168, 236, 245, 273, 298, 417, 440, 444-445 person's, 115, 168, 236, 245, 273, 298,336,417,440,444-445 unequal, 187 level of utility equilibrium, 504 LevineR., 515,527 Lewis W. Arthur, 78, 81, 107, 158, 166, 232, 258, 261, 289, 332, 361, 497, 527 liberalization policy, 140, 159 Lio M., 228, 479 List of candidates for an optimal individual's decision, 273 Liu D., 283 Liu Pak-Wai, 77, 167, 259-260, 289 localized technology, 142 location theory, 168 Locay Luis, 196, 227, 438-439, 459 Loesch A., 51 Lucas R. E. Jr, 409-410, 435
542 Malthus Thomas Robert, 475, 477 Malthusian theory, 475 management costs, 491, 493-494 fixed, 491 higher level, 493 lower level, 493 positive, 491,493,494 marginal analysis, 10, 84, 89-91, 117, 211,334,352,474 conventional, 117, 271 marginal cost, 17, 29-45, 59, 61, 65-69,71, 189,393,400,403 curve, 71 private, 55, 69-70, 400, 402 social, 55, 69-70, 72, 400 marginal diseconomy, 70 marginal evaluation, 63-72 curve, 64-67, 71-72 negative, 69, 71 marginal external diseconomy, 57, 67, 68 potentially relevant, 59 marginal external economy, 57, 68, 72 Pareto-relevant, 72 potentially relevant, 59 marginal externality, 56-57, 60-63, 69, 71 Pareto-relevant, 70 potentially relevant, 59, 61 marginal internal diseconomy, 68, 72 marginal internal economy, 68 marginal private cost, 55, 69-70 marginal rate, 58, 61, 63, 71, 162, 379 marginal rate of substitution, 58, 61, 63,71, 162,379 marginal revolution, 7 marginal social cost, 55, 69-70, 72 curve, 72 marginal tax, 69-70 marginalist economics, 6, 23
Index market clearing conditions, 10, 20, 89-90,117,120,149-150,163, 169, 175-176, 179-180,209-213, 237-241, 273, 277, 299, 302-305, 314, 336-340, 377, 420, 427-430, 447-448, 471, 484, 505, 517, 520, 523-524 market configurations, 147, 153 market failure, 364, 383-384 market integration, 198, 333-334, 423 market place, 204 market structures, 9, 13, 85, 117, 120, 139, 141, 156, 165, 169, 173-176, 179, 182-183, 190, 233, 237-238, 245,249-251,262,267,273, 276-280,293,296,299, 301-312, 315, 320, 334, 337, 339, 342, 352, 372, 383, 385, 413, 418-419, 423, 445, 457 Marshall A., 6, 84, 195, 227, 410, 435 Marshallian framework, 7 Marx Karl, 111, 125-126, 132-133, 254 Marxian economics, 235 Maskin Eric, 265, 289 mathematical formalism, 7 maximization full, 151, 156, 181 maximization problem, 484 maximum utility, 90, 238, 339, 399, 400-401 absolute, 59 McDermott John, 464, 476 McManus M., 60 Meade J. E., 29-30, 37, 43 Mikesell R., 498, 528 Milgrom P., 231, 257, 261, 264-265, 288-289, 296, 324, 329, 332, 360-361 Milgrom-Roberts model, 264
Index Mirman Leonard J., 486-487, 495 MokyrJoel, 82, 107 monitoring cost exogenous, 332 monopolistic competition, 193, 267, 287, 372-373 monopoly power, 116, 131, 264, 294, 299,372-373,418 Moore John, 232, 249, 253, 257, 261, 288, 338-340, 352, 360 moral hazard, 264-267, 271, 329, 331-332, 334, 337-338, 344, 348, 350-353 endogenous, 351 two sided, 332, 339, 353 moral philosophy, 5 multi-part pricing, 34-36, 41-45 Murakami N., 283, 289 Murphy K. M., 25, 78, 80, 82, 106-107, 196,225,295-296, 323-324, 458-459, 498, 503, 506, 527-528 Murphy-Shleifer-Vishny (MSV) model, 498-499 myopia partial, 300, 309 NakamuraR., 199,206,227 Nash bargaining, 337-340, 344 bargaining equilibrium, 337, 344 bargaining game, 337-338 product, 338, 344 natural comparative advantage, 80, 365 neighbourhood effects, 55 neoclassical economics, 4-9, 16, 18, 23,112,137,181, 184,365,385 neoclassical equilibrium, 363 neoclassical trade theory, 8 network effects, 201-202, 214
543 new classical economics, 3, 4, 9, 24-25,109,133,228,289,361, 495, 529 New Economic Geography, 196, 226 new institutional economics (NIE), 4 newly industrialized countries, 78, 259 Ng Yew-Kwang, 24, 80, 108-112, 118, 120, 122, 131,133,137, 167-168, 175,181,194,196,228,231,242, 258, 265, 269-271, 277, 279, 289, 291, 296, 325, 332, 352, 361, 363, 393-395, 404, 406, 409, 418, 424, 436-437, 460, 463, 465-467, 473-480, 495-499, 529 non-convexity, 392, 396-397, 402-403, 405 PPF (cf. production possibility frontier), 396, 402-403 non-Pareto optimum corner equilibrium, 155 nonsubstitution theorem, 10 North D. C , 24, 104, 106, 133, 228, 283, 289, 363, 389, 406, 436, 477, 479, 495, 529 objective function, 10, 116, 272, 276-277, 298, 306, 400, 416, 469, 517,519 occupation configurations, 209, 338, 414 operation scale, 295 opportunism, 368 opportunistic behavior, 234, 271, 368, 370 optimization decision problem, 271 optimization problem, 147, 163, 398, 488, 504 optimization solutions, 322 optimum decisions, 97, 115-120, 144-146, 174-175, 179-180,211,
544 215, 232, 237, 239, 241, 255, 273, 300, 313, 322, 338, 340, 429, 445, 448,469,481,507,518-519 dynamic, 418 individual, 117, 127, 144, 146, 149, 155, 179-180,238,240, 255, 300, 374, 445, 447, 480-483 interior, 189 multiple, 175 optimum distribution, 33-34, 43 income, 33-34, 43 optimum pricing, 33, 36 optimum solution, 16, 117, 128, 339, 401 interior, 128 optimum system, 30 optimum value, 161-162, 256, 317, 323, 387-388, 484 organizational efficiency, 124 original utility function, 146 Otsuka K., 289 Owen Robert, 111 Paine C. L., 30 parameter space, 10, 89, 346, 509 subspace, 89, 91, 96, 98, 334, 346, 348,352,420-421,431 parameters preference, 115, 123, 125, 131, 153, 157, 164, 174 production, 502 technology, 152 transaction, 176, 376 transaction cost, 168 transaction efficiency, 12, 22, 173, 184,283-284,310,421 Pareto efficient, 68, 152, 162-163, 214, 246,251 allocation, 152, 162-163, 246 level, 124
Index Pareto equilibrium, 61-63, 67-72 Pareto inefficient, 214 Pareto irrelevance, 60 Pareto optimality, 61, 70, 249 Pareto optimum, 61, 68, 70, 88, 124, 131, 139, 141, 144, 147-148, 151-152, 155-156, 163, 175, 184, 248-249, 252, 264, 270, 279, 329, 331,350,370,379,386,457, 474-476 corner equilibrium, 124, 148, 151-152, 155-156, 184 non-Pareto optimum corner equilibrium, 155 restricted, 162 Pareto relevance, 60 Pareto relevant, 56, 60-63, 67, 70, 72 externality, 56, 60-62, 69 marginal external economy, 72 marginal externality, 70 Pareto superior, 153, 164, 331 partial derivatives, 57-58 partial equilibrium analysis, 200, 232, 295 partial equilibrium contract, 232 patent law, 234, 253-254, 269, 364 Pejovich S., 363, 389 per capita consumption, 199, 263,414 relative, 195, 197 smaller, 200, 217 per capita primary resource, 508, 513 per capita real income, 96-97, 118-119, 121, 158, 170, 176-184, 192, 197, 199, 209, 214-215, 219-221, 238-243, 246-253, 260-262, 279, 284, 286, 293, 302-305, 308, 312-315, 346, 348, 381, 382, 414, 424, 447^153, 456, 469, 479, 492 equilibrium utility level, 89, 121, 149, 472, 522
Index maximum, 184, 214-215, 218, 243-244, 248, 383, 452 per capita resource endowment, 498, 501, 508-514 Pigovian analysis, 69 calculus, 69 corrective tax, 391 discussion, 69 internalize, 391-392 margins, 61 solution, 70 subsidy, 392 tax, 70, 392-393, 397, 400-405 terminology, 69 population density, 186-187, 191, 198, 463-465, 474, 498 positive externalities of cities (cf. economies of transaction agglomeration), 201 positive management costs, 491-494 positive network effects, 201, 206, 271 potential relevance, 60, 62 potentially relevant externality, 59-60, 67 marginal external diseconomy, 59 marginal external economy, 59 marginal externality, 59, 61 price-taking behavior, 293-294, 299-300 equilibrium, 294 pricing efficiency, 118, 252-253 pricing system, 33, 37, 118, 352, 524 average cost pricing, 43—46 individual pricing, 34 multi-part pricing, 34-36, 41, 43-45 two-part pricing, 35-36 prime configurations, 9 principal-agent cost, 392, 404
545 principal-agent model, 232, 264, 329, 332-334, 338-339, 347, 352, 392, 404 emergence of endogenous principal-agent relationship, 334 endogenous principal-agent relationship, 329, 332 general equilibrium, 332 principal-agent cost, 392, 404 principal-agent relationship, 329, 332-335, 338, 347, 349, 352 reciprocal principal-agent relationship, 347, 349, 351-352 PritchettL.,410,435 private cost marginal, 55, 69, 70, 400, 402 private equilibrium, 63 private marginal cost, 400, 402 private property system, 113, 383, 385 procurement model, 232 producer goods, 251, 292, 296, 311, 323, 412, 426, 438-444, 453, 455-458 product development, 287, 291-292, 296,316,434 production concentration, 122, 219, 333, 349, 423 production conditions, 83, 466 production constraint, 394-395 production externality, 56, 403 production functions, 10, 12, 17, 55-58, 72, 83-84, 102, 112-115, 126-127, 137-139, 142, 160, 163, 171, 174, 179, 181, 184,206,210, 223, 235-236, 239, 245-246, 264, 272,276,298,313,317-318, 335-336, 339-340, 366, 368, 373-375, 387, 409, 414-417, 428, 438, 444-448, 457, 469, 482-483, 503,519
546 CES (cf. constant elasticity of substitution), 295, 298, 438-439, 442, 444, 457 Cobb-Douglas, 444 individual, 293, 379 individual-specific, 292 production parameter, 502 production possibility frontier (PPF), 97, 138-139, 142, 363, 386, 392, 396-397, 402-403 individual, 396 non-convexity, 396, 402-403 production technology, 394, 396, 501, 507,516 identical, 220 input-output, 403 productivity progress endogenous, 503, 515 programming dynamic, 295-296, 305-306 linear, 336 non-linear, 238, 336 prohibition law, 404 property rights, 6, 72, 232, 253, 283, 289, 363-375, 379-386, 400 economics of, 363-366 enforcing, 363-372, 379-386 theory of, 363, 365 protectionist, 426 Pryor F., 475, 477, 499, 514, 528 public expenditure theory, 71 public goods, 474, 476 pure competition, 385 pure consumer, 137-138, 365-367, 372-373, 425, 457 pure producer, 137, 365-367, 372-373, 457 Quigley J. M., 199, 207, 213, 227
Index R&D based model, 263, 410-411, 415, 434, 463 Ram Rati, 78, 105, 107, 409, 435 Ranis G., 78, 81, 105, 107, 158, 165, 497-498, 527-528 real productivity, 158, 381-386 real return to labor, 149,151-154, 158, 164-165, 181 maximum, 152-155 real returns to labor, 149,151-154, 158 Rebelo model, 411 RebeloS.,411,435 relative consumption effects, 131 relative contingent prices, 342 relative economic standing, 111, 113 pursuit of, 111-112, 125 relative importance of the roundabout productive sector, 272 relative income effects, 131 relative price, 10, 14, 18-21, 88-89, 117, 120,146,150,174-175, 179-181,213,237,239,254, 273-277, 302, 304-305, 318, 338-344, 347, 353-354, 376-377, 419, 473, 484, 505, 516, 521-524 contingent, 347, 352-353 corner equilibrium, 89-91, 96, 182-183, 277, 340-343, 427, 430,469,471,524 equilibrium, 89, 113, 124, 151, 304-305, 484, 506 pure, 352 static equilibrium, 303 uniform, 151, 181 relative prices, 472 relative utility, 111-116, 123-125, 128-131 pursuit of, 112-113, 115-116, 126,131
Index relativity of utility, 112 relevant externality, 56-61 residual control asymmetric, 352 rights, 264, 352 residual returns, 264-269, 352 residual rights, 231-234, 238, 240, 245, 249-254, 267, 269, 289, 311, 332 asymmetric structure of, 233-234, 253, 268-269 resource endowment, 497, 499-501, 507-511,514-515 per capita, 498, 501, 508-514 primary, 498 restricted Pareto optimum, 162 returns to labor specialization increasing, 196 returns to scale constant (CRS), 7, 80-81, 112, 137, 167,365,515 increasing (IRS), 6, 112, 137, 140-143,147,160, 172,295, 365, 373, 502 returns to specialization, 139-142, 153, 160, 164-165, 364, 386 constant, 84 increasing, 20, 124, 131, 138-144, 156-157, 160,166,214,261, 292, 298, 364-368, 373, 386, 441,457 Ricardian model, 8, 50, 80-84, 103, 205 Ricardo David, 80, 106, 108, 205, 365, 389 Rice R., 102, 167, 196, 198, 206, 220, 228 Richter W., 403, 405 risk aversion, 335, 479, 481-483, 486-487,491-492 risk premium, 487
547 risk reducing effort level, 330, 335, 346-348,351,353 Roberts J., 261, 265, 288-289, 296, 324, 332, 360-361 Romano Richard, 332, 353, 360 Romer model, 411 Romer Paul M., 262, 289, 291, 294, 316, 324, 366, 389, 409^111, 435, 437, 459, 465, 477 Rosen S., 25, 108,196,206,227,291, 295, 324, 366, 389, 437-438, 460, 501,528 Rosenstein-Rodan P., 498, 528 roundabout production (cf. economies of roundaboutness), 140, 158,223, 272, 412-414, 417, 421-426, 431, 437-439, 454-458 machines, 439 specialized learning by producing roundabout productive, 426 Sachs J., 25, 78, 106, 108,427, 434-435, 498-499, 506, 528 SahR., 142,166,368,389 Saint-Simon Henri, 111 Sala-i-Martin X., 294, 324-325, 411, 434 Samuelson Paul A., 68, 71, 474, 477 saving fundamentalism, 409 scale effects, 82, 262-263, 289-290, 411,415,436 Type I, 262-263, 411 Type II, 263, 411 Type III, 263,411 Type IV, 263 scale of labor, 336 Schmitz James, 291, 298 Schweizer U., 167-168, 194 Scitovosky Tibor, 55 Scott A. J., 194-195, 200, 227
548 second-order conditions, 126-129, 189, 218,379,381,455 second-order derivatives, 224 Segerstrom P., 263, 289, 291, 325 self sufficiency, 85, 138, 158, 366, 413, 438 sequential equilibrium Kreps and Wilson, 415 shadow configurations, 10 Shi He-Ling, 102, 191, 194, 393, 406, 437-439, 458, 460, 480, 482, 495, 497 Shleifer A., 80, 82, 105, 459, 528 single externality, 58, 69, 72 size effect, 465 size of the firm, 259, 261-262 optimal, 262 Smith Adam, 3-11, 16, 23-24, 30, 48, 50,80, 108,111,133,137-141, 166, 194, 196, 205, 226-227, 258, 365, 385, 389, 412, 435, 438-439, 460, 479, 495-499, 528 Smithian framework, 4-8, 11, 23, 48, 106, 139, 196, 205, 385 social cost, 69, 105, 257, 360 marginal, 55, 69, 70, 72, 400 total, 405 social division of labor, 209, 213 network of, 216, 221 network size of, 217 specialized learning by producing roundabout productive, 426 specialized production, 113, 138, 169, 232 spillover effects, 55, 207 Squire L., 79, 105 Stackelberg strategy location model, 208 Starrett D. A., 403, 405
Index static equilibrium, 296, 299, 302-303, 438 static equilibrium relative price, 303 static model, 118,283 Stigler G. J., 25, 196, 228, 262-263, 289, 295, 325 Stiglitz J., 142, 166, 171, 193, 262, 267, 287, 368, 389, 437, 439, 458-459 Stubblebine Wm G., 55, 104, 117, 132, 503, 527 substitution marginal rate, 58, 61, 63, 71, 162, 379 suburbanization, 220 Sun Guangzhen, 77, 88-89, 108, 195, 208, 214, 223, 228, 431, 435, 497, 529 supply functions, 19, 90, 181, 470 individual, 149, 484 Tabuchi model, 200 Tabuchi T., 200, 220, 228 take-off, 157-158 Tamura R., 25 taxsubsidy method, 71 double taxsubsidy, 71 technological external effects, 56 technological externality, 56 terms of trade, 98-99, 337-338, 343 theory of irrelevance, 259, 263, 271 of the size of the firm, 263, 271 theory of value, 131 Thisse J.-F., 207-208, 226 Tirole Jean, 232, 258, 265,289, 332, 361 total corner-demand, 275 total corner-supply, 275 total cost-benefit analysis, 89, 91, 117, 352, 474
Index total discounted utility, 283, 298, 302, 306, 309, 318, 413, 416,419-420, 425,429-431 indirect, 427-430 total factor productivity, 112, 235, 245, 260, 273, 336, 445, 503, 507 total labor endowment fixed, 368 total social cost, 405 total utility, 63, 67, 426 discounted, 283, 298, 302, 306, 309,318,413,416,419-420, 425,429-431 trade balance, 145, 313, 447-448 trade dependence, 112, 131, 139, 158, 171, 192, 197, 219-221, 279, 286, 333, 385, 412, 423, 441, 456, 458, 507 trading efficiency, 11, 79-83, 95-103, 200-205, 210, 222, 269-270 coefficient, 11, 83 training costs fixed, 482 transaction agglomeration economies of, 217 transaction conditions, 103, 197, 200, 217-222, 283, 331, 426-427, 499 transaction cost parameters, 168 transaction costs, 4-5, 8-11, 16, 23, 80-83, 103,108,112-115, 121-122, 125-126, 138-143, 159-160, 167-174, 187-191, 194-199, 204, 207-208, 213-214, 217, 221-222, 228, 231-235, 239-242, 246-253, 261-272, 276, 279, 283, 292-293, 296-297, 303, 307-308, 312-314, 318, 327-335, 349-352, 361-368, 386-389, 413-417, 423, 426, 431, 438-442, 455, 457, 465-467, 495, 499-502, 508, 528-529
549 advantage, 198 coefficient, 125-126, 143, 169-170, 174, 188-189, 197, 199, 235, 239-242, 250-253, 266-272,276,283,312-314, 330-331,413-414,426,442 endogenous, 263-267, 270, 329-335, 348-353 exogenous, 202, 270, 329-335, 350-352,481 exogenous transaction cost coefficient, 330-331 fixed, 126, 172-173, 185, 188, 292,297,313-315,442, 454-455 fixed transaction cost coefficient, 185, 188,313-315,442 iceberg, 83, 206, 267, 272 variable, 126, 170-173, 292, 297 transaction efficiency, 8, 12-22, 113, 121-124, 131, 138, 140, 158, 160, 173, 184-192, 197, 199, 206-207, 217-223, 233-234, 243-244, 248, 251-253, 261, 271-272, 276-286, 289, 292-293, 295, 307-312, 315-316, 347-353, 364-365, 372, 386, 388, 395, 414-415, 421-424, 440-442, 453-457, 464-466, 476, 485, 487, 497-500, 503, 506-515, 526 coefficient, 173, 206-207, 217, 223,271,312,347,395,414, 424, 487 exogenous evolution of, 190 exogenous transaction efficiency coefficient, 347 fixed, 315 parameter, 12, 22, 173, 184, 283-284,310,421 transaction service, 142-145, 158, 168
550 transaction technology, 141-142, 235-236, 248, 444, 500 general, 167 iceberg, 272 transaction uncertainty, 479-481, 487^188, 492 transportation efficiency, 380-482, 485, 488 Tsai Yao-Chou, 395, 405, 466, 476 two-part pricing, 35-36 two-period (finite horizon) economy, 296,300,415 two-person model, 60-63, 393 two-sector model, 393 uncertainty, 6, 53, 63, 115, 235, 287, 339, 342, 368-370, 374, 377, 425, 479^182, 487, 492 transaction, 479-481, 487-488, 492 unity transformation rate, 394 urban land-rent escalation, 195, 200, 219 urban-rural occupation structure endogenous, 199 general equilibrium, 201 utility equalization condition, 89-90, 117,120, 150, 164-165,175, 179-180, 209-214, 239-241, 299, 303-305, 314, 337-340, 343, 376-377, 427, 430, 447-448, 484, 520, 524 utility functions, 11, 16-21, 56-59, 72, 83,90, 113-116, 125,138,143, 146, 149-150, 160, 173, 181, 197-198, 205, 236, 297, 317, 335, 338-341, 367-370, 375, 387, 394, 402, 416, 438, 441-444, 448, 466, 482-486 CES (cf. constant elasticity of substitution), 222, 438-439
Index indirect, 21, 89-91, 96, 119-121, 146, 149-150,180-181,211, 239,304,313,338,340,376, 427, 447-448, 469, 522 original, 146 utility level, 91, 113-117, 121, 173-176, 305, 335, 376, 378, 401, 432,503,513,517 indirect, 398 positive, 256 utility maximization, 58, 61-62, 67-68, 88-89, 120, 169, 209, 237, 246, 302-304, 367, 427, 429, 473-376, 516 decision, 89 equilibrium, 61-62, 67 full, 181 identical, 175 individual, 523-524 labor allocation, 303, 427 labor decision, 302 Varian H. R., 391, 405-406 Veblen Thorstein, 112, 125, 132-133 Venables Anthony J., 77-78, 82, 102, 104, 107, 196, 227, 262, 288 Vishny R„ 80, 82, 107, 324, 459, 528 von Neumann-Morgenstern utility function, 482 von Thiinen J. H., 106, 197-198, 226, 228, 287 Walrasian auctioneer, 126, 373 equilibrium, 264 law, 20, 180,417-418,430 regime, 84, 116, 131,294,300, 373, 445, 482 sequential equilibrium, 283 Wang Jianguo, 82, 109-112, 115, 131, 133, 283, 289, 391, 436
Index Warner A., 108, 427, 435, 498-499, 506, 528 welfare analysis, 350 welfare economics, 55 welfare theorem, 139 Wen M., 16-17, 85, 108, 208, 228, 337, 361, 483, 495, 497, 506, 528 Wen theorem, 16-17, 85, 208, 337 Whinston Andrew, 73 Williamson Oliver, 78, 108, 204, 232, 234, 257-258, 287, 296, 325, 360, 368, 389 Wills Ian, 82, 109, 283, 289, 363, 436, 438, 460 Wilson T., 31 Yang model, 439, 499 Yang Xiaokai, 3-7, 11, 24-25, 77, 80, 82,84,88-89,102,108-112, 116-124, 131-133, 137, 157, 160, 166-170, 174-175, 181, 190-191, 194-198, 206, 208, 214, 220, 228, 231,236,244,253,258-259, 265-266, 269-273, 277, 279, 283, 287-292, 295-298, 316, 325, 329,
551 332-336, 339, 352, 361-365, 382-383, 389, 393-395,404,406, 409-410, 416-419, 424, 434-439, 444-446, 452, 458-460, 463-467, 473-477, 480, 482, 495-500, 528-529 Yang-Ng model general equilibrium model, 265 story of the firm, 266 two-step approach, 276 Yang-Rice model, 198 Yoon Y. J., 7, 25 Young Allyn, 80, 84, 109, 137-141, 166, 228, 262-263,290-291,296, 324-325,365,389,412,414, 435-438, 456, 460, 480, 495 Young Alwyn, 263,411 Zenou Y., 200, 226 Zhang Wei Ying, 232 Zhao Y., 283,290 Zhou Lin, 77, 88, 89,108, 214, 228, 501,529 Zoega G., 498, 527
Inframarginal Contributions to Development Economics The core of classical economic analysis represented by William Petty and Adam Smith concentrated on the field of development economics. This classical footing of the study of development is different from the neoclassical perspective in two important respects: (a) it focuses on division of labor as the driving force of development, and (b) it emphasizes the role of the market (the "invisible hand") in exploiting productivity gains that are derived from division of labor. However these aspects have received little attention in the body of literature that represents the modern field of development economics — which largely represents the neoclassical application of marginalism. A notable exception is research that utilizes inframarginal analysis of individuals' networking decisions in an attempt to formalize the classical mechanisms that drive division of labor. This book is a first attempt to collect relevant key contributions and is intended for active researchers in the field of development economics.
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