Nonlinear Models for Economic Decision Processes

Non inear Mode s or E onomi Decis on Processes

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Nonlinear Models for Economic Decision Processes Ionut Pu ca a a

ICP

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Imperial College Press

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

NONLINEAR MODELS FOR ECONOMIC DECISION PROCESSES Copyright © 2010 by Imperial College Press 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.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-1-84816-427-7 ISBN-10 1-84816-427-0

Printed in Singapore.

T o: o my father who provided the will for nonlinear research, Professor Giuseppe Furlan who provided the opportunity, Professor Herman Haken who provided some of the means, my family who provided the time

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Contents

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5.

Introduction: Reasons for Writing this Book, a Decision Theory Approach ................................................................................................. 1.1. Introduction ................................................................................. 1.2. Why the Structure of This Book – a Decision Theory Approach ...................................................................................... Nonlinear Models for the Labour Market ................................................ 2.1. Introduction ................................................................................. 2.2. Nonlinear Models and Examples for the Labour Market ............. 2.3. Conclusion ................................................................................... Second Order Effects in Population Migration ........................................ 3.1. Nonlinear Migration Behaviour ................................................... 3.2. Cases of Reverse Migration ......................................................... 3.3. A (Not So) Simple Model ............................................................ 3.4. Results ......................................................................................... 3.5. Conclusion ................................................................................... Cities: Reactors for Economic Transactions ............................................ 4.1. Transaction Environment ............................................................. 4.2. Diffusion Equation ....................................................................... 4.3. The Reflector (Albedo) ................................................................ 4.4. Decrease in Income ...................................................................... 4.5. Dynamic Evolution Equation ....................................................... 4.6. Conclusion ................................................................................... Annex 4.1 . .............................................................................................. A.4.1. The Coefficient K ........................................................................ Considerations on the Reform in the Power Sector (Avoiding Chaos in the Path to an Optimal Market Structure) ............... 5.1. Introduction ................................................................................. 5.2. From Power Sector to Power Market ........................................... 5.3. Non-linear Effects in Market Penetration ..................................... vii

3 3 13 23 23 24 30 33 33 35 35 39 41 45 45 47 56 58 60 62 62 62 71 71 73 81

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Contents

5.4. Conclusion ................................................................................... Appendix 5.1 . ......................................................................................... 6. A Model of Non-linear Dynamics in the Implementation of Decisions for the Evolution of Energy Technologies .............................................. 6.1. Introduction ................................................................................. 6.2. Description of the Model ............................................................. 6.3. Criteria for Energy Development Strategies ................................. 6.4. An Energy Planner’s Perception of Risks and Benefits ................ 6.5. Numerical Examples .................................................................... 6.6. Energy Policy and Technological Profile ..................................... 6.7. Perception of Alternatives and Strategic Conduct ........................ 7. Non-linear Effects in Knowledge Production .......................................... 7.1. Implementation of New Technologies ......................................... 7.2. Essentials of Chaotic Behaviour .................................................. 7.3. Complex Cyclical Patterns ........................................................... 7.4. The Industrial Production and the Production of Technologies .... 7.5. Measuring Technological Information and Entropy ..................... 7.6. Conclusion ................................................................................... 8. Institutional Structures as Benard–Taylor Processes ............................... 8.1. Epistemic Sense and Ontological Sense ....................................... 8.2. Social Reality and Collective Behaviour ...................................... 8.3. Dynamics of Memes .................................................................... 8.4. Conclusion ................................................................................... 9. Oscillatory Processes in Economic Systems............................................ 9.1. Cycles in Dynamics of Economic Systems .................................. 9.2. Optimality Conditions and Associated Equations ........................ 9.3. Production Potential and Quantization ......................................... 9.4. Oscillatory Behaviour – Some Numerical Results ....................... 9.5. Conclusion ................................................................................... Appendix 9.1. Second-order Systems ..................................................... 10. Final Thoughts on a Different Way of Looking at Economic Processes ................................................................................................

86 87 93 93 94 100 102 104 106 107 111 112 115 116 116 121 124 129 129 129 130 135 141 141 143 145 148 150 150 155

General References ......................................................................................... 159 Index ............................................................................................................... 167

Dubito ergo cogito... Descartes

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Chapter 1

Introduction: Reasons for Writing this Book, a Decision Theory Approach

1.1. Introduction The development of human infrastructure (energy, transport, communications, etc.) at a planetary level and the capability to work with large amounts of data has revealed the existence of limits with regard to both socio-technological and environmental developments. If the limits of technological development have proven to be of the saturation type, i.e., new technologies penetrate to replace the old, saturated ones, environmental evolution shows very clearly that this planet is all we’ve got. In our striving to master energies at the planetary level, which is not expected to happen in the foreseeable future, we are left with the hope that the environment will be resilient enough to absorb our errors due to our lack of knowledge and inability to accept our limits. Several decision reactions are possible:

‘Whatever you do will change nothing’ ‘Anything you do will change everything’

‘Certain things you do will push the system beyond stable equilibrium, others will not’ The perception of change in complex systems and, accordingly, the reaction, show bifurcation-like behaviour especially when one acquires an awareness of 3

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Nonlinear Models for Economic Decision Processes

the limits (environmental, technological, social, etc.), Linear mentalities like ‘whatever you do will change nothing’ or ‘anything you do will change everything’ must be changed in more subtle ways of acting that take into account second-order effects which characterize the behaviour of complex systems. For instance, when building dams to protect against sea-level increase, we should consider the fact that the production of cement for dams represents a source of CO2, which contributes to the rise in sea level. During the Middle Ages, the indicator of welfare was the quantity of gold one possessed. Accordingly, the ‘research programmes’ of those days were aimed at changing everything to gold. Since the emergence of energy-availability limitations, the indicators have changed. Also, the increased complexity of interactions among the various systems (energy, population, economy, environment, etc.) has lead to the introduction of aggregated indicators. The planetary view we have today requires the consideration of the meteogeographical conditions of each region and the normalization of the specific indicator values in order to make better comparisons. This suggests a personalization of new energy-supply technologies being implemented in various regions, taking into account not only the geographical conditions, but also the social ones, in order to achieve maximum efficiency. Taking decisions for development has always been based on some type of representation of the process. Various models have served as tools to devise or justify decisions. The mathematics behind these models is usually linear. Since the behaviour of the processes involved is highly non-linear, the approximations made were valid for restricted areas and time intervals. These models were not able to predict the limits beyond which a discontinuous behaviour would occur in systems evolution. Decisions of the type ‘quit financing a technology and enhance others’ are common in the economy. Only in recent years, non-linear models based on non-linear mathematical tools have made possible the prediction of discontinuous decisions which occur when certain system parameters cross some limit. Although the mathematics involved is more complicated with respect to the linear one, the representation of systems evolution among limits is more straightforward. Even if the limits are not accepted, they may sometimes be avoided or, in rare cases, crossed with the associated shocks. The capability to absorb shocks and still perform normally (resilience) measures the impact of our decisions for development on the environment, the economy, etc. Alternatively, accepting the limits opens the way to understanding the mutual interactions among the various

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systems, thus making it possible to change those limits in a sustainable symbiotic evolution. Negotiating between energy and environmental concerns in development involves information which is not always available, and time constants that may be longer than what we have dealt with. The costs and financial measures implied may lead, for example, to capital accumulations which we are not prepared to control yet, lacking appropriate administrative structures, or may lead to unusually long payback times and the prospect of irreversibly damaging the environment. The present changes in energy generation, transmission, distribution and end-use systems, leading to more players in the market, have raised questions about the role of a regulator which would prevent chaos in the process and thus prevent shocks to the economy. Correlating global change with energy is one of the first projects to consider the interactions among various systems at a planetary scale, opening the way for closer international cooperation. Geographical research on fuels along with scientific research on conversion technologies were one of the main reasons that led to the development of infrastructure, such as transport, telecommunications, etc., which in turn gave us the consciousness of, and the possibility to monitor, the influence of our activities on the environment. The ability to perceive changes in the complex systems we interact with has influenced our ways of understanding and consequently modelling more complex behaviours. One should not forget that one of the first classes of models which show ‘chaotic’ behaviour was aimed at describing meteorological behaviour. (Lorenz, 1963) 1.1.1. Development in a limited environment One of the basic concepts in non-linear behaviour is that the limit (separatrix, discontinuity) is seen not in the asymptotic way, as in linear theories, but as a drastic change in the behaviour of the system. The limits separate various basins of coherent (possibly linear) behaviour. Crossing the limits is the normal way of evolution for the system. As an example, our decisions for development are, frequently enough, to abandon a certain energy programme and intensify another, or are asymmetric, intensifying a programme with respect to others, as in the case of the nuclear energy in Italy and France. Such decisions may be seen as crossing various types of limits in the ‘phase space’ of the characteristic parameters of that system.

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Nonlinear Models for Economic Decision Processes

1.1.1.1. Social limits When talking about social limits we have a spectrum of quantitative and qualitative parameters to consider. Population is the main quantitative one and in fact refers to two things: first, the purpose of the production in the economic system; second, one of the elements of the production. We must note that the projections we make assume a certain dynamic for the parameters influencing population evolution. The impact of fluctuation events like a Third World War or an epidemic of an unknown virulent disease without adequate medical technology (e.g. the plague in Europe in the Middle Ages) are not considered when making such projections. On a different line, the economic interactions may be described with models that take into account the geometry of the process. Cities are seen as reactors of economic interactions where the agents are described with neutron physics models. It is shown that the dimension of a city depends on the intensity of the interactions, that the residents’ saving profile occurs as a natural saturation process, and that specific indicators such as reaction cross-sections explain in a wider framework Zipf’s ‘one over income’ power-law. Another parameter which, although measurable, has a more qualitative aspect in the sense of describing socio-economic structuring and order is energy. Lately, the structuring of energy markets has been ongoing in different economies. This process is shown to have optimality as well as the potential to generate chaotic (deterministic chaos) behaviour in the penetration of privates into a contestable market, thus leading to a better explanation of the role of regulatory agencies. 1.1.1.2. Technological limits The creation and penetration of technologies have been regarded from both economic and anthropologic viewpoints. Here, too, we encounter the type of nonlinear behaviour in the dynamics of technology penetration in a given economy. This penetration of new technologies to replace the old, saturated ones is typical non-linear behaviour. The same applies to the shift of choice towards one technology although there existed two or more similar technologies in that field at the beginning. A good example is the video recording systems which started with both Beta and VHS; after a relatively short time the choice shifted to VHS although the other system was perfectly comparable from a technical point of view.

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This non-linear behaviour in the dynamics of technology evolution shall be considered later in a more formalized way through a qualitative model. New technologies have been frequently issued in response to a perceived limit in development; they refocus the economy on different resources, some of which were totally unused up to that moment, having even a restructuring effect. Since we now have the perception of having reached the limits of the environment, a new wave of technologies is supposed to arrive, with a restructuring effect on the economies involved. 1.1.1.3. Environmental limits Although certain limits in local ecosystems have been known for a long time, the planetary environment system is still too complex for us to assess (based on the relatively scarce amount of data we have), This is especially regarding the real importance, in the short and long term, of the limits we have presently identified. It is certain that there is a strong correlation between the emissions of CO2 (together with other gases) and the air temperature of the planet. Also it has been ascertained that the integrity of the ozone layer is perturbed by the presence of CFCs in the atmosphere. Butthe extent to which the earth may absorb these perturbations while maintaining conditions innocuous to humans is a thing we would not dare to find out, given our present capability to control the planet. 1.1.2. The perception of changes in complex systems 1.1.2.1. Perception of a complex process The greenhouse (GH) effect is characterized by a large spectrum of time constants of the involved processes. Certain economic actions may produce an effect on temperature over decades; others may act continuously and screen the long-term tendency of the evolution. The perception of the interaction such as the one between human society and the planetary system is polarized by its high level of complexity. At one end the interaction may be perceived as ‘do whatever you please; nothing will happen’ or, in other words, a totally absorbent (completely resilient) anthropo-planetary system. However big the shock induced to the system by a certain decision, it will not produce a loss of equilibrium. A physical representation would show the state point of the system inside an infinitely high potential well.

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Nonlinear Models for Economic Decision Processes

At the other end there is a perception of low resilience: ‘anything you do will ruin everything’. The physical representation of such a case shows the system’s state point on top of a potential hill: any shock will move it out of the equilibrium position. Between these two extremes the interaction may be summed up as ‘certain things you do will push the system beyond stable equilibrium, others will not’. This involves a finite depth of the characteristic potential well; i.e., certain shocks may push the state point of the system out, either to non-equilibrium positions or to other equilibriums. 1.1.2.2. From simple to aggregate The simplest method used for getting an image of the dynamics is to show the evolution of each indicator separately. Although this offers a good way to predict the future values of each indicator, it does not allow foreseeing the global system behaviour. To characterize the interaction we have to start using combined/aggregated indicators. If, for example, in the hydro-dynamical systems we are very much accustomed to criterial numbers (Reynolds, Prandl, etc.) resulting as a combination of the system parameters (indicators), in the economical systems, although highly dynamic, the normally used characterization of the interactions is done by aggregated indicators resulting from the simple division of only pairs of the simple indicators (e.g. energy/capita, energy/GNP, CO2 emissions/capita, CO2 emissions/unit of energy, GNP/capita, etc.), We must note that the simple division indicators’ evolution is easy to verify intuitively; thus, even if there is no model that sustains the interpretation of the indicator related to data, one can use intuition to draw conclusions about the future behaviour of the economic system. 1.1.2.3. Rich and poor trigger second-order effects in labour or population migration Nowadays, the problem under study is the difference between the specific behaviours of rich and poor economies. It includes the use of child labour, superseding of moral values, as well as migration of population (in and out of various areas) in relation to economic characteristics (e.g. infrastructure, GNP, etc.) of the respective areas.

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Regarding the use of child labour, it has been shown that this is merely one of two equilibrium points in a bifurcation model on the use of adult labour versus the use of adult and child labour, depending on the family subsistence level. Regarding migration, what is the potential gap that produces a variation of concentration (a flux) of persons from one area to another? With the political transitions we are witnessing, we shall consider that people are simply going from poor to rich areas. So the difference that is intensively perceived between the West and East or between the North and South is one of welfare/poverty. Taking into account the welfare/poverty barrier between the West and East or the North and South, and the consequent migration of population from poor to rich areas, we may identify some non-linear behaviour like the one described below. The infrastructure measures the efficiency with which an economy makes labour (active population) produce GNP, expressed as GNP/capita. Increasing the population by immigration leads to an increase in the active population (labour), Over a certain saturation value of the infrastructure’s efficiency the increase in population shall be greater than its capacity to produce GNP. So the GNP/capita will diminish, this being perceived as poverty. Thus, along with migration from poor to rich areas there is also an import of poverty into the rich economy. In parallel, the investments of rich economies to create (or develop) infrastructures in either the East or South contributes to the increase in efficiency of those economies. Thus, the increase in efficiency will produce a greater GNP/capita, perceived as an import of welfare from the rich economy into the poor one. If this perception is strong enough the outflow of population might reverse. These reversals may create cycles of immigration and emigration in initially poor regions where investments are being made to develop the infrastructure. An example is Italy where the emigration of the fifties was reversed in the midseventies, signalling that the infrastructures were set up and operational. Another typical example is the south of Italy where, 20 years ago, emigration was the rule for workers of that area. Investments done by the government to create infrastructure has led to a slowdown, if not a stop, of emigration. Based on the ideas above we might expect an immigration to the south of Italy after a certain number of years. This may seem unbelievable to a 40-something-year-old Italian, but similarly the disintegration of communist-like structures would have seemed unbelievable to an Eastern European 20 years ago.

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Nonlinear Models for Economic Decision Processes

1.1.2.4. Implementation of new technologies We also analyse the implementation of new technologies into an economy. This will lead to more complex aggregated parameters, but will lead to a simpler layout of the model which allows a better understanding of the occurrence of discontinuities. This process has a dynamic characterized by the successive saturation of old technologies and emergence of new ones (Kondratiev cycles; see Grubler and Nowotny, 1990), This may be seen as a succession of hysteresis-type cycles chained in three-dimensional space (π,u,v), The decision of allocating funds between two technologies in competition (or one and the other available technologies) is described with a Fokker–Planck equation whose stationary solution leads to a cusp catastrophe. The evolution of the system’s trajectory may have a sudden discontinuity associated with the decision to abandon a project. If one considers the production of technologies along with the production of other goods, it can be demonstrated that a similar relation to the Cobb–Douglas function may model the generation of technologies where instead of labour, one considers intelligence, and instead of production means, one takes research means. (Purica, 1988) Moreover, the interplay between the production of GDP and the one of technologies is shown to be ruled by a Hénon ‘strange attractor’. Also, the gain of information by the experimenter of technologies is measured in a Minkowski space by an associated Lorentz transformation. This is a measure of information resulting from applying multivalent modal logic and not the typical bivalent one. 1.1.3. Decisions for development 1.1.3.1. Models for development – linear versus non-linear Looking at the papers of 20 years ago which tried to forecast development, one is struck by the linear behaviour suggested by those results, i.e., uphill with varying slopes. The first type of forecasting that involved some non-linearities was resource utilization. In this case after an initial slow increase, due to technology implementation followed a period of fast increase which ended in a saturation due to the depletion of the resource reservoir. Another area where saturation occurs is the penetration of technologies into the economy. This process was extensively described by Marchetti and Nakicenovic. The results gave the

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possibility to predict limits, although only saturation ones, which was a great difference from the former linear models. Generalizations of these models (Gheorghe and Purica, 1979, UN-GPID Project; Ursu et al., 1985) have shown that limits are due not merely to saturation but also to strong non-linear correlations among the system’s control parameters, as seen in the example of the economic model described above. The interplay between aggregated parameters creates limits in the evolution which include the saturation ones but are not limited to them. 1.1.3.2. Avoid or cross the limits – system resilience When limits may be predicted one is tempted, in the first instance, to avoid them. On second thought, after having estimated the shock to the economy, crossing certain limits may be a better decision. Thus, if the system’s capability to absorb shocks is good enough, the decision will have to specify not only where to invest, but also when, in order to avoid certain limits or/and cross preselected ones. The possibility of non-linear approaches to include the moment of time as an element of the decision is more similar to what we are faced with in day-to-day life, giving a higher predictability level for these models. 1.1.3.3. Sustainability – accepting the limits Once the consciousness of the system’s dynamics is created in a systematic way, showing how to control the evolution trajectory, another possibility occurs, representing an interaction of a superior level. It refers to the fact that by designing the parameters of a system, one may control the position and amplitude of the limits. Thus, the alternative of accepting the limits and trying to control them by influencing the parameters proves to be the best long-run decision. The one word which now, after the issue of the 1988 Brundtland Report (Our Common Future), describes this mentality is ‘sustainability’. This approach might lead to better coping with the reality of climate and energy conversion pattern changes. 1.1.3.4. Indicators of sustainability At this point, along with Pearce et al. we may say that sustainable development enters as a fundamentally different approach, shifting the focus from economic growth, as narrowly construed in traditional attitudes, to economic policy. It speaks

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Nonlinear Models for Economic Decision Processes

of development rather than growth, of the quality of life rather than real incomes alone. That is, sustainable development makes it clear that the very antithesis of growth and the environment is not the issue. Sustainable development accepts that what we have been calling ‘economic growth’ in the past has been measured by some very misleading indicators. The tendency has been to use a measure of gross national product (GNP) as the basis for economic growth calculations. If GNP increases that is economic growth. But GNP is constructed in a way that tends to divorce it from one of its underlying purposes: to indicate, broadly at least, the living standard of the population. If pollution damages health and, health care expenditures rise, that represents an increase in GNP – interpreted as a rise in the ‘standard of living’ – not as a decrease. If we use up natural resources then, that is capital depreciation, just as if we have machines we count their depreciation as a cost to the nation. Yet depreciation on man-made capital is a cost while depreciation of environmental capital is not recorded at all. (Pearce, Markandya, Barbier, 1990) The message above is that from now on, along with man-made capital, the material and ‘know-how’, we will have to seriously take into account the environmental capital. Finding and implementing indicators to show the depreciation of the latter capital will not be an easy task. Implementing environmental standards will certainly mean important costs for all nations involved. These new concepts need to penetrate and spread into the various economies, so that they become a state of mind. Thus, there is a need to institutionalize them in the broader sense of the word. The next chapter concentrates on describing the occurrence of institutional structures using the Brusselator model, where the reaction–diffusion is done with the memes of Richard Dawkins (1976), and showing that institutions occur as Benard–Taylor dynamic stabilities far from thermodynamic equilibrium, in the memespace. The evolution of institutional structures viewed from this perspective recalls Heraclitus’s ‘panta rhei’ (‘all things are in flux’) model of the world dynamic. We have been talking about cycles in economy. Considering the product cycles and their superposition onto the full economy cycles, one may introduce an interpretation based on the fact that the so-called ‘velocity of rotation of money’ is actually a frequency correlated to the oscillatory behaviour of the economy. Dividing general costs into production and transaction costs one may find a conservation law of the monetary mass, similar to that of energy, over the duration of an economic cycle. The associated equation for this process recalls

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the well-known Schrödinger equation in quantum physics. It is also shown that along with production functions there may be transition functions introduced for products, these being similar to potential and kinetic energy, respectively. Various other findings occur from such considerations related to the discrete (meaning not continuous) nature of economic activity in a finite resource environment. Let’s start now with an example of applying decision theory. 1.2. Why the Structure of This Book – a Decision Theory Approach As decision theory has its own vocabulary that is not sufficiently popular among economists and physicists, we will try to introduce such models with an example below. The example used is the one on the manner I used to decide on the structure of this book when I decided to put together ideas developed from various papers I had published in the last 25 years. I was reasoning in the following way: If I write this book it may be either: a normal standard book (N) or, a ‘crazy’ book (C) I shall call these strategies (Si) After reading the book the reaction of the readers may be: take an interest (I) or, show politeness (P) I shall call these outcomes (Q), The consequences may be the following: (Q1) - I write a normal book and the readers take an interest, it means: “They consider me having a comparable expertise with other authors”, or, shortly, experienced. (Q2) - I write a normal book and the readers show politeness, which means: “They consider me under the level”.

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Nonlinear Models for Economic Decision Processes

(Q3) - I write a ‘crazy’ book and the readers take an interest, it means: “They consider me inspired” (Q4) - I write a crazy book and the readers show politeness: “They consider me crazy. I had represented these alternatives in a decision tree: Strategies \ Outcomes

I(A1)

P(A2)

S1(N)

Q1

Q2

S2(C)

Q3

Q4

N

C

I

Q1

P

Q2

I

Q3

P

Q4

Please note that the outcomes and the consequences are exclusive and complete and we shall call the assembly of such consequences a variable. The second step was to try to order the consequences using my performance to be able to decide what strategy I will adopt. As we like to use functions I had associated with my performance a function V, called value function that associates real numbers to consequences: V(Qj ) → number I postulated that when I prefer Qj to Qk, then V(Qj) > V(Qk) and that when the occurrence of Qj or Qk is indifferent to me, then V(Qj) = V(Qk),

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I also considered that the relative value of the consequences is unique up to a linear increasing transformation V(Qj) = aV(Qj) + b. I may establish the following order of performance: I prefer first to be considered with experience (Q1), if not, to be inspired, if not, to be considered myself crazy and lastly to be considered under the level. V(Q1)>V(Q3)>V(Q4),V(Q2) In the following step I shall try to associate the same kind of numbers to assess the possibility (P) that a given consequence occurs when I choose a given strategy. P(Qj/Sk) = Pkj. I construct the matrix: Q1

Q2

Q3

Q4

S1

P11

P12

P13

P14

S2

P21

P22

P23

P24

Because when I choose the strategy S1 only one of the consequences may occur, the numbers can satisfy: Sj Pj1 = 1 And, similarly when S2 is chosen: Sj Pj2 = 1 No conditions are imposed on the column, but: Sk (Sj Pjk ) = 2 Up to now I have no possibility to associate numbers to Pjk. Before trying to do this, please remark that we can simplify, a little, my problem if we notice that in this case each strategy consists in one action to be done: write a normal book - A1 = S1 write a ‘crazy’ book - A2 = S2 Each consequence is associated to one couple (Ai,Ok) where Ok is an outcome: Q1 = (A1,O1); Q2 = (A2,O2) Q3 = (A2,O1); Q4 = (A2,O2)

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Nonlinear Models for Economic Decision Processes

We easily see that when S1 is adopted or the identical action A1, it is not possible that Q3 = (A2,O1) or that Q4 = (A2,O2), Thus it is reasonable to put: P(O3/S1) = P(O4/S1) = 0 Equally, if S2=A2 is adopted: P(O1/S1) = P(O2/S2) = 0 Now the previous matrix reads: Q1 = (A!,O1)

Q2 = (A1,O2)

Q3 = (A2,O1)

Q4 = (A2,O2)

S1 = A1

P(Q1/S1)

P(Q2/S2)

0

0

S2 = A2

0

0

P(Q3/S2)

P(Q4/S2)

Collapsing the matrix above we have the matrix of action and outcomes:

A1 A2

O1

O2

P(O1/A1) P(O1/A2)

P(O2/A1) P(O2/A2)

The numbers P(Qj/Sk) or P(Oj/Ak) are like the probabilities because they are considered 0 when the consequence is impossible and 1 when we are sure that only the Qj consequences will occur after action Ak is done. But, I had no reason to put any P(Oj/Ak) = 1, which means I was not sure that if I did an action e.g. A2, it will be surely followed by outcome e.g. O2. On the other hand I am sure that after action A1, one and only one of the O1 or O2 will follow and this was the reason I considered P(O1/A1) + P(O2/A1) = 1. Any name may be used for these numbers but, they have the properties of the Kolmogorov’s axiomatic definition of the probability. I shall use the same name and I will consider P(Oj/Ak) as a conditional probability of the occurrence of Oj, when I knew that Ak was done. “Probability in and by itself is neither a desirable nor an undesirable thing. However, when related to relative values of consequences, we see it in a different perspective. For example, under a specific strategy, one would like to have a high probability (near 1) for attaining the most desirable consequences and, respectively, a low probability (near 0) for

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attaining the least desirable consequences. Since there is a fixed amount of probability, namely 1, that, is distributed over the consequences for each strategy” (P.C.Fishburn, Decision and Value Theory, 1964), This way, our probability concept is nearer to the de Bayes probability definition than to the Laplace probability definition. I will remind the two below: - Laplace, Théorie analytique des probabilités, Paris 1st ed.1812 “La théorie des probabilités consiste a réduire tous les événements qui peuvent avoir lieu dans une circonstance donnée, a un certain nombre des cas également possibles, c’est a dire tels que nous soyons également indécis sur leur existence et a déterminer parmi ces cas, le nombre de ceux qui sont favorables a l’événement dont on cherche la probabilité. Le rapport de ce nombre a celui de touts les cas possibles, est la mesure de cette probabilité, qui n’est donc qu’une fraction dont le numérateur est le nombre de cas favorables et dont le dénominateur est celui des touts le cas possibles.” - T. de Bayes (1763), An essay towards solving a problem in the doctrine of chance. Phyl Trans.Royal Soc.Vol.53, p.370. “5. A probability of any event is the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon its happening. 6. By chance I mean the same as probability.” Coming back to my problem, it consisted, in fact, of choosing one of the two strategies of action; supposing I knew the conditional probabilities and the relative value function. It is reasonable to use the mathematical expectation to compare the strategies: E(Sk) = E(Ak) = Σk P(Qj/Sk),V(Qj) I prefer the strategy having a higher expectation value. So, when: E(S1)>E(S2) or E(S1)-E(S2)>0 I will prefer S1. My capacity to use this criterion is limited by my knowledge about the conditional probabilities and/or relative value function.

18

Nonlinear Models for Economic Decision Processes

Fortunately, such criterion needs not the knowledge of figures for all the conditional probabilities and/or for all values of the relative value function. We have: E(S1) - E(S2) = Σj=1,t P1jVj - Σj = 1,t P2jVj = Σj = 1,t (P1j - P2j)Vj If we remember the Abel identity:

Σj = 1,n ajbj = Σk = 1, k - 1(Σj = 1, k(Σj = 1, k aj)(bk-bk+1) + (Σj = 1,n aj)bn And use it to transform the previous equation: E(S1) - E(S2) = Σk = 1, n - 1 Σj = 1, k(P1j - P2j) (Vk - Vk + 1) we see that any information in (Pij) imposes conditions to (Vk) to have a dominance of one of these strategies. In the case of my problem an order of preference was established among different outcomes. We rewrite the matrix taking into account the outcomes order of preference: Q1

Q3

Q4

Q2

S1 = A1

P11

0

0

P12

S2 = A2

0

P21

P22

0

When we consider, in the Abel identity, the Vk in the ordered chain, all VkVk+1 are positive. It is easy to see that the coefficient (Vk-Vk+1) is obtained by forming into the matrix the partial sum by adding in rows the left term, and then taking the column differences. Q1

Q3

Q4

Q2

S1 = A1

P11

P11

P11

P12 + P11 = 1

S2 = A2

0

P21

P21 + P22 = 1

P21 + P22 = 1

P11

P11 - P21

P11 - 1

0

The conditions to have a positive sign for the expectations’ values, meaning to prefer S1 strategy, are: P11 > 0 P11-P21 > 0

P11 > P21

P11-1 > 0

P11 > 1

Introduction – Reasons for Writing this Book, a Decision Theory Approach

19

Since every P must be less or equal to 1: P11 = 1 and P21 < P11 The conditions for having a negative sign, which means preferring strategy S2, are: P11 < 0 P11-P21 < 0

P11 < P21

P11-1 < 0

P11 < 1

From the first and last relations we have: P11 = 0 while the second relation gives: P21 > 0 Now, to be able to make a decision I must have the information on P11. Looking at the literature in the field of econophysics I assume that all authors are more experienced in the topic than I am. With this information I put: P11 = 0 and: E(S1)-E(S2) < 0 E(S1) < E(S2) This means I had to choose the second strategy or the second action: A2 = ‘Write a ‘crazy’ book’ And what you will read follows this conclusion. References Brundtland H., (1988), Our Common Future, UN Report. Dawkins R. ( 1976), The Selfish Gene, Oxford University Press. Fishburn P.C., (1964), Decision and Value Theory, John Wiley & Sons, New York. Gheorghe A., Purica I., (1979), Decisions for Development in Energy Systems, UN-GPID Project papers. Grubler A., Nowotny H. (1990), Towards the Fifth Kondratiev Upswing: Elements of an Emerging New Growth Phase and Possible Development Trajectories, International Journal of Technological Management 5(4):431– 471.

20

Nonlinear Models for Economic Decision Processes

Laplace, (1812), Théorie analytique des probabilités, Paris. Lorenz, E. (1963), Deterministic Nonperiodic Flow, Journal of Atmospheric Sciences, 357, 130–141. Pearce D., Markandya A., Barbier E., (1990), Blue Print for a Green Economy, Earthscan Publications Ltd., London. Purica I. (1988), Creativity, Intelligence and Synergetic Processes in the Development of Science. Scientometrics 13(1–2): 11–24. de Bayes, T. (1763), An essay towards solving a problem in the doctrine of chance. Phyl Trans.Royal Soc., 53, 370. Ursu I., Purica I., et.al., (1985), Risk Analysis, vol. 5, nr.4.

If you want truly to understand something, try to change it. Kurt Lewin

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Chapter 2

Nonlinear Models for the Labour Market

In the last years several mathematical models describing nonlinear behaviour (deterministic chaos, catastrophe theory, bifurcation theory) have found applications in economics. This was generated also by the increase in the complexity of the economic systems’ evolution. 2.1. Introduction The concept of a fundamentally continuous world is, conceptually, in contrast to the one where the world is discontinuous. A world that is calm represents the Newtonian–Victorian dream of a gradual and regular ascendant movement of reality through a smooth process. In economic science this concept was the essence of Alfred Marshall’s writings. The discontinuous world is defined by strong contrasts and sudden changes. Generally speaking the principal current in economic thinking had emphasized continuity as fundamental to economic processes. The standard model of general equilibrium assumes continuity as the basic feature of behaviour and technology that, combined with convexity, generates the continuous curves of supply and demand and the existence of equilibrium. The associated linear models have generated continuous dynamics, or as Marshall puts it, ‘nature does not make jumps’. Progressively, though, this status of economic science has been subject to contestation. The main pressures came from disciplines outside the corpus of economic science, such as mathematics and physics. Generally, two major approaches of dynamic disciplines have developed within mathematics. They are based on the idea of bifurcation or breaking of equilibrium at critical points, on one hand, and on the idea that many of the functional relations prove to be non-linear rather than linear, on the other hand. Economists have traditionally shown preference for using linear models, or at least linearizable models, in the vicinity of a solution. Obviously, the use of 23

24

Nonlinear Models for Economic Decision Processes

models having only one solution or equilibrium position, that can be solved explicitly, rather than by iterative algorithms, and having relatively simple statistical properties, is explained by the incipient stages of economic science. Such models may be analysed using a limited arsenal of techniques and unambiguous results may be frequently obtained. Early non-linear models having multiple solutions, like the ones proposed by Marshall and Walras, were completely ignored for a long time. In the analysis of dynamic processes, emphasis was placed on linear differential equations that produce regulated cycles. Moreover, a distinction was made between deterministic systems producing regular, hence predictable, behaviour, and the statistical series that reflect a stochastic, hence unpredictable, behaviour. The chaotic behaviour of these dynamical series was simply interpreted as stochastic and in estimating the linear models the inconvenient observations were classified as accidental and, consequently, ignored. 2.2. Nonlinear Models and Examples for the Labour Market The beginning of economists’ interest in complex dynamics stems from papers on economic cycles that try to define those basic conditions which may generate oscillatory behaviour in the studied economic systems. Starting in 1930, the papers by Frish (1933), Lundberg (1937) and Samuelson (1939) began to use differential equations in models generating deterministic time trajectories of interest parameters. A non-trivial example is the model of Samuelson that is based on three standard relations: Yt = Ct + It Ct = cYt + k It = b(Yt-1 - Yt-2) Where Y is national income, C is consumption, c is the rate of consumption (propensity) and I is investment. The equation of C is a linear function of production with a delay of one period. The investment function is linear in the investment assumed to be proportional to the rate of increase in income in the previous period. Substitution of the first two equations into the last one gives: Yt = (c + b)Yt-1 - bYt-2 + k which is the linear equation in second order differences of Samuelson. Out of this type of models, four types of qualitatively different behaviours may occur: 1. Oscillating and stable (convergent through decreasing amplitude oscillations to an equilibrium value)

Nonlinear Models for the Labour Market

25

2. Oscillating and divergent (cycles having divergent amplitudes) 3. Non-oscillating and stable 4. Non-oscillating and unstable It was soon understood, when trying to model the behaviour of real economic systems, that the four possible evolutions above did not represent a sufficiently rich set of behaviours to model complex situations encountered in reality. Models that bring a larger variety of behaviour descriptions are the nonlinear ones. In the 1970s René Thom introduced catastrophe theory as a way to describe the occurrence of discontinuities in the evolution of various systems. This was based on Hopf’s bifurcation theory. Later on, the theory of deterministic chaos was crystallized based on work done by Hénon, Lorentz, Ruelle and Takens, followed by Mandelbrot’s fractals, as well as other nonlinear approaches in the systems of physics, mathematics, etc. (e.g. Haken, Prigogine, Kauffman), In economics a series of authors studied the occurrence of complex situations and their modelling by deterministic chaos. (Baumol and Benhabib, 1989; Benhabib, 1980; Grandmont, 1986) Catastrophe theory and bifurcation theory were successfully applied by authors like Zeeman (1977), Gandolfo (1997), Albu (2002) and Gheorghe and Purica (1985), Also, complexity and the occurrence of self-organization in economic systems was described by Krugman (1996), The list above is not exhaustive. We will show a simple model for the occurrence of a bifurcation from an equilibrium point, in the labour supply and demand space, to two equilibrium points. Let us consider the situation of a family having available labour supplied by adults and children. This family lives in a labour market where the labour demand is typical, i.e., decreasing with the rise in the wage rate. The family offers its labour and its subsistence is based only on salary gains. The interesting thing in the model we are building is the introduction of a subsistence level of the family income, beyond which the family will offer child labour along with the adult one. For the linear case, noting L, the income from labour force, and w’ (()’ describes time variation) the variation of the wage (wage rate), we consider that the demand is described by a linear equation: L=-k·w’; where k is a constant that describes the will of employers to take more labour if the wage is lower. In a linear market we may consider that the supply equation is similar: L=l·w’; where l measures the increase in the labour supply with the increase of the wage rate. For the case considered, the point of intersection of the two equations will be found in the origin of the coordinate system.

26

Nonlinear Models for Economic Decision Processes

We move on now to define the nonlinear aspects of the model. For a description of the market dynamic, as mentioned above, we introduce a limit level of the family income from using its labour resource. This is the subsistence level; it is considered proportional to the wage w and denoted by s·w; where s is a constant coefficient. Thus, the dynamic equation becomes: -kw’ = L-sw or kw’ = sw-L. We see that the dependence of L on w determines the linearity of the behaviour of the supply in the labour market. In a linear market L is proportional to the wage rate (we note L = l·w’), The final equation becomes: w’ = (s/(k + l))·w. This is a firstorder differential equation which, for w’ = 0, shows only one equilibrium point in the origin of the coordinate system – thus verifying the behaviour in the linear case. We move on to the case where the existence of a subsistence limit influences the expression of L; and consider that a factor of intense perception of family survival occurs, that increases the supply of labour with child labour, proportional to a term in wage at third power; with the notations above, k1·w3; where k1 is constant. The equation for this case becomes: w’ = s·w - k1·w3. Determining the equilibrium points is done letting w’ = 0. From the algebraic analysis of the third-order equation in w (considering k1 is always positive), it is seen that there are two different situations that depend on the sign of s; i.e.,

Fig. 2.1. Bifurcation in the labour market

Nonlinear Models for the Labour Market

27

If s < 0 (situation eliminated by the fact that the subsistence level cannot be negative) or s = 0 (no subsistence level exists), only one stable solution is present in w = 0, hence a single equilibrium point:

Fig. 2.2. One-equilibrium case

If s > 0, three solutions show up: w = 0 (unstable equilibrium) and w1,2 = √(|s|/k1) (two points of stable equilibrium),

Fig. 2.3. Two-equilibrium case

The occurrence of two stable equilibrium points, in the case where a subsistence level of the family income exists, makes the family use all its labour resources to overcome it. This leads in a non-linear way, in contrast to the classical labour market, to situations when an equilibrium point occurs in poor economies at low wage values that involves intensifying child labour.

28

Nonlinear Models for Economic Decision Processes

As the wage increases, the subsistence level is overcome, and the equilibrium in the labour market shifts toward the other stability point where children are not obliged to work. 2.2.1. Case examples The first cases of children being officially integrated into the labour market date from the beginning of the Industrial Revolution. The co-inventor of the Roller Spinning machine initially used in textile factories, English mechanic John Wyatt, declared that this innovative device was so easy to use that even 5 or 6year-old children could operate it. At the end of the 19th century the use of children in the labour market was already declining in the majority of industrialized nations, but at the global level, the problem was not fully solved. According to the International Labour Organization (ILO) in 2000, there were 186 million children aged between 5 and 14 years – on average, 1 in 6 children – working in the black market in developing countries. Of these, 111 million worked in high-risk sectors like mining, construction and agriculture, sectors that may have negative consequences on the children’s health for the rest of their life. Moreover, about 8 million were obliged to work or were used as kid-soldiers, or were forced to prostitute. These figures are the result of long-time analysis because the ILO tried not to include in the above numbers children fulfilling household-related work. According to various viewpoints the estimations are either above or below the effective numbers of working children. Although the sectors defining a child as working are reduced in number, one should not neglect the cases of household work that sometime go beyond the normal attributions thus exploiting numerous children. The question to ask is what may be done to eradicate this phenomenon. The answer depends firstly on the identification of the factors that maintain and even amplify this situation. Finally, we review the ideas described mathematically above. The curves of supply and demand are basic concepts in economics. The ‘curves’ are actually (in a linear interpretation) two straight lines crossing, normally at one point representing the equilibrium price in a free market economy. But in markets where children are seen as a potential labour force, the supply and demand may intersect at more than one point. Children are sent to work by the families whose

29

Nonlinear Models for the Labour Market

income, coming only from adult labour, is very small thus assessing that child labour is but a fraction of adult one. Based on this we may draw a graph where the curve of the supply is no longer a straight line but a non-linear third-order curve.

Percentage of children working

700

19%

600 Millions of children between 5 and 14 years

500 400 655

300 200 100 0 Percentage Total

29% 2%

16%

4% 119

108

62

15% 167

88

Developed Countries

Excommunist bloc

Asia and Pacific

Latin America

Africa subSaharan

2%

4%

19%

16%

29%

15%

119

62

655

108

167

88

Middle East

Regions

Fig. 2.4. Percentage of children working

In this labour market where child labour may show up, two equilibriums may occur. The first represents the big salaries – above the subsistence level – where only adults work, while the second, at small salaries – below the subsistence level – shows both adults and children working. In the second case the family as a whole acts toward maximizing its total income, and issues of morals and child protection are superseded by the need for survival, in contrast to the situation when the family income goes beyond the subsistence level.

30

Nonlinear Models for Economic Decision Processes

2.3. Conclusion The nonlinear behaviour of economic processes has started, in the last few years, to be described with nonlinear models that, with the variety of situations encompassed, are more adapted to today’s economic reality. Development of non-linear models in mathematics, physics and chemistry have recently lead to applications in economics with good prediction results. This first chapter has presented a simple application of a bifurcation model to an important process in the labour market, i.e., child labour. Even the simple model presented gives more than just the linear approach in terms of decisions on how to effectively tackle the issue. We will proceed in the next chapter to something slightly more complex, a model where cyclic behaviour occurs, and apply it to migration cycles. References Albu L.-L. (1998), Tranzitia economiei sau tranzitia stiintei economice. Editura IRLI (Institutul Roman pt Libera Intreprindere), Bucharest. Albu L.-L. (2002), Macroeconomie non-liniara si prognoza – teorie si aplicatii. Academia Romana – Centrul roman de economie comparata si consens, Editura Expert, Bucharest. Basu K. (2003), The Economics of Child Labor, Scientific American, 289(4):66– 73. Baumol W.J., Benhabib J. (1989), Chaos: Significance, Mechanism and Economic Applications, Journal of Economic Perspectives, 3(1):77-100. Benhabib J. (1980), Adaptive monetary policy and rational expectations. Journal of Economic Theory, 23:261–266. Grandmont J.M. (1986), Periodic and Aperiodic Behaviour in Discrete, Onedimensional, Dynamical Systems, North Holland, New York, 1:3-12. Klein L.R., et al. (2003), Principiile modelarii macroeconomice. Editura Economica, Bucharest. Takens F. (1980), Dynamical Systems and Turbulence, Mathematics 898: 366– 382.

I prefer to be first in a village than second in Rome Julius Caesar

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Chapter 3

Second Order Effects in Population Migration

Migration becomes an increasingly significant process that triggers various types of complex behaviour. After analysing the process, especially with regard to the occurrence of nonlinear behaviour, a model is built to include the features that may lead to the occurrence of cycles of migration reverse. The results of a simulation show patterns of behaviour similar to the Italian case where large exmigrations in the fifties were reversed in the mid-seventies. A set of potential applications to migration from new EU entrants to the EU-15 is possible, especially with regard to actions that may speed up the moment of migration cycle reverse. 3.1. Nonlinear Migration Behaviour Nowadays population migration is an increasingly important process with wideranging impact. Migration is described in or between various areas in relation to economic characteristics (such as GDP, infrastructure, etc.) that are likely to generate nonlinear behaviour, at least regarding the occurrence of cycles where the flow of ex-in migrants from and into a country may be reversed. Let us build a model for migration which contains some of the features that give it a nonlinear character. The most frequent type of migration, occurring in peacetime, is the one driving people from poor countries to ex-migrate to rich ones. The difference in GDP/capita between the North and South or the East and West leads to population movement toward higher GDP/capita. Taking into account intensive internal migration in areas like the EU, or, even at a lower level within countries, we may identify a typical behaviour such as the one described in the diagram below:

33

34

Nonlinear Models for Economic Decision Processes Poor

Rich Ex-migration of poor labour

- GDP up faster - capita down → GDP/capita up faster

- GDP up slower - capita up Perception by poor labour

→ GDP/capita up slower

In-migration of poor labour

Investment in poor infrastructure by rich

Saturation of rich infrastructure efficiency leads to import of poverty, while investment in poor infrastructure increases perception of well-being in poor country to the level of reversing migration.

Diagram 3.1. Migration behaviour process

The quality of an economic infrastructure is determined by the efficiency with which an economy is able to make labour (active population) produce GDP, expressed in GDP/capita. The population increase because of migration represents an increase in labour. Beyond a certain saturation limit of the infrastructure the rate of population increase will exceed that of the GDP. Thus, GDP/capita will diminish, this being perceived as poverty. We may say that a massive increase of migrants into rich economies may bring, over a certain limit, an influx of poverty. Simultaneously, the investments made by rich economies to create/develop infrastructures in poor economic areas contribute to the increase in efficiency in those areas. Consequently the GDP/capita will increase, being perceived as an import of well-being into the poor economies.

Second Order Effects in Population Migration

35

3.2. Cases of Reverse Migration If this perception is strong enough, the migration flux from the poor economy to the rich may reverse. Several cycles of this sort may show up in ex-in migrations to and from initially poor countries. A conclusive example is the one of Italy, where migration waves of the fifties were reversed in the mid-seventies (Fig. 3.1.), this being a sign that more efficient infrastructures were set up and operational (partially an effect of the Marshall Plan during that period).

3.E+05 2.E+05 2.E+05 1.E+05 5.E+04 0.E+00 -5.E+04

Fig. 3.1. Italy Ex-In patriated (Source: ISTAT)

Another typical example is the south of Italy where, 20 to 30 years ago, emigration was the rule for workers in the area. Investments made by the government have led today to a significant slowdown or even a reverse of migration (which may have seemed impossible to an Italian 40 years ago), Probably it is as unbelievable as the fall of communism would have seemed 20 years ago. Obviously, the enlarged EU is witnessing a similar process from the poorer Eastern Europe to the richer Western Europe. 3.3. A (Not So) Simple Model Behaviour such as the one described above may be included in a scheme like the one below, done based on the iThink software.

36

Nonlinear Models for Economic Decision Processes

Diagram 3.2. Migration model

The relations among various model components were defined such as to reflect the comments made in the previous paragraphs, and are listed in the equations below (we are giving the notation in iThink program for those who wish to use the model): Equations for Ex-In migration INIT Poor_labour = 5E6 INIT Rich_GDP = 40E9 INIT Poor_GDP = 2.5E9 INIT Perception_by_poor = (Rich_GDP-Poor_GDP)/Rich_GDP INIT Rich_labour = 2E6 exmigration = Poor_labour*Perception_by_poor inmigration = (Rich_labour-Poor_labour)*Perception_by_poor INIT Investment_from_rich = 0.001*Rich_GDP

37

Second Order Effects in Population Migration

The following parameters are given in the form of a graph reflecting the saturation trend mentioned above. These graphs are described numerically in the three equations that follow. Invest eff poor Restore

Rate of investment = 0.000

0.005 0.010

Labour eff poor

Graph 2

Labour efficiencÉ

Diagram 3.3. Control and saturation parameters

Invest_eff_poor = GRAPH(Investment_from_rich) (0.00, 900), (5e+006, 1350), (1e+007, 2700), (1.5e+007, 4200), (2e+007, 11100), (2.5e+007, 19650), (3e+007, 23850), (3.5e+007, 25350), (4e+007, 26700), (4.5e+007, 27900), (5e+007, 28350) Labour_eff_poor = GRAPH(Poor_labour) (0.00, 0.00), (700000, 750), (1.4e+006, 4650), (2.1e+006, 14100), (2.8e+006, 21300), (3.5e+006, 24750), (4.2e+006, 26700), (4.9e+006, 26700), (5.6e+006, 26850), (6.3e+006, 27150), (7e+006, 27000)

38

Nonlinear Models for Economic Decision Processes

P_GDP_change = (Invest_eff_poor+Labour_eff_poor)*Poor_labour-Poor_GDP Labour_efficiency_rich = GRAPH(Rich_labour) (0.00, 3000), (700000, 4050), (1.4e+006, 7950), (2.1e+006, 18600), (2.8e+006, 23250), (3.5e+006, 26250), (4.2e+006, 28200), (4.9e+006, 28500), (5.6e+006, 28500), (6.3e+006, 28500), (7e+006, 28200) R_GDP_change = (Labour_efficiency_rich-Rich_GDP/ Rich_labour)*Rich_labour chg_GDP_per_cap_poor = P_GDP_change/Poor_labour chg_GDP_per_cap_rich = R_GDP_change/Rich_labour Change_in_perception = (chg_GDP_per_cap_poor-chg_GDP_per_cap_rich)/ Perception_by_poor Rate_of_investment = 1E-3 Change_in_investment = Rate_of_investment*R_GDP_changeInvestment_from_rich Ex_In_migration = exmigration-inmigration Poor_labour(t) = Poor_labour(t - dt) + (inmigration - exmigration) * dt Rich_GDP(t) = Rich_GDP(t - dt) + (R_GDP_change) * dt Poor_GDP(t) = Poor_GDP(t - dt) + (P_GDP_change) * dt Perception_by_poor(t) = Perception_by_poor(t - dt) + (Change_in_perception) * dt Rich_labour(t) = Rich_labour(t - dt) + (exmigration - inmigration) * dt Investment_from_rich(t) = Investment_from_rich(t - dt) + (Change_in_investment) * dt exmigration = Poor_labour*Perception_by_poor inmigration = (Rich_labour-Poor_labour)*Perception_by_poor

Second Order Effects in Population Migration

39

Invest_eff_poor = GRAPH(Investment_from_rich) (0.00, 900), (5e+006, 1350), (1e+007, 2700), (1.5e+007, 4200), (2e+007, 11100), (2.5e+007, 19650), (3e+007, 23850), (3.5e+007, 25350), (4e+007, 26700), (4.5e+007, 27900), (5e+007, 28350) Labour_eff_poor = GRAPH(Poor_labour) (0.00, 0.00), (700000, 750), (1.4e+006, 4650), (2.1e+006, 14100), (2.8e+006, 21300), (3.5e+006, 24750), (4.2e+006, 26700), (4.9e+006, 26700), (5.6e+006, 26850), (6.3e+006, 27150), (7e+006, 27000) P_GDP_change = (Invest_eff_poor+Labour_eff_poor)*Poor_labour-Poor_GDP Labour_efficiency_rich = GRAPH(Rich_labour) (0.00, 3000), (700000, 4050), (1.4e+006, 7950), (2.1e+006, 18600), (2.8e+006, 23250), (3.5e+006, 26250), (4.2e+006, 28200), (4.9e+006, 28500), (5.6e+006, 28500), (6.3e+006, 28500), (7e+006, 28200) R_GDP_change = (Labour_efficiency_rich-Rich_GDP/Rich_labour)*Rich_labour chg_GDP_per_cap_poor = P_GDP_change/Poor_labour chg_GDP_per_cap_rich = R_GDP_change/Rich_labour Change_in_perception = (chg_GDP_per_cap_poor-chg_GDP_per_cap_rich)/ Perception_by_poor Change_in_investment = Rate_of_investment*R_GDP_change-Investment_from_ rich Ex_In_migration = exmigration-inmigration The values chosen for the parameters above are arbitrary but they obey the behaviour trends described e.g. saturation of GDP generated by increasing labour. 3.4. Results The dynamic regimes that result from solving the equations in the model (using a 4th -order Runge–Kutta method) are given in Fig.3.2 below. One may see that a large out migration from the poor country is followed by a ‘return home’, and then by more cycles driven by the investment from the rich country and by attaining the saturation of efficiency in the rich economy.

40

Nonlinear Models for Economic Decision Processes 1: Investment from rich 1: 2: 3: 4:

2: Poor GDP

1.41e+008 2.46e+011 1.20e+011 5000000.00

3: Rich GDP 2

4: Poor labour 2

4

4

4

3

2

3

4

3

3 2 1: 2: 3: 4:

9.05e+007 1.24e+011 8.00e+010 3667208.15

1

1 1

1

1: 2: 3: 4:

4.00e+007 2.50e+009 4.00e+010 2334416.29 0.00

5.00

10.00

Graph 2: Page 4

15.00

Years

20.00

8:06 PM 12/16/2007

Fig. 3.2. Migration cycles

Looking at the evolution of poor labour and Poor GDP depicted below (Fig. 3.3), one may see the occurrence of an attractor for the evolution trajectory of the system.

1: Poor labour v. Poor GDP 2.46e+011

1.24e+011

2.50e+009

2334416.29

Graph 1: Page 4

3667208.15

5000000.00

Poor labour

Fig. 3.3. Dynamic stability

8:06 PM 12/16/2007

Second Order Effects in Population Migration

41

The fact that dynamic stability may be shown to exist in such systems leads to the following conclusions. 3.5. Conclusion After having described the migration behaviour of population between poor and rich economic areas, with a view to better understand it, we were driven to a model having nonlinear behaviour generating features. Solving the resulting system of first-order differential equations led us to regimes of dynamic stability and cycle occurrence of the ex-in migrations from the poor country that replicate the behaviour described and analysed in the case of Italy (as a living example where the cycle has closed in the last 50 years), Finally, this chapter only shows that by analysing economic processes with nonlinear, complex behaviour, one may generate models having a higher degree of predictability, that encompass natural behaviour, observed as real cases. These models may be used for new entrant countries into the EU, whose development may bring hope that, with appropriate action, we may witness reverse migration cycles. The population lives in cities and countries where they interact and make transactions within an economic space. In the next chapter, we present a more elaborate model for this basic economic process. References ISTAT (1994), Anuario Statistico Italiano, Rome. IThink User Manual (1994), High Performance Systems Inc., USA. Purica I. (1992), Environmental Change and the Perception of Energy System Dynamics. Proceedings of ICTP Trieste Conference on Global Change and Environmental Considerations for Energy System Development, ICTP Trieste. Purica I. (2002), Environmental Change and the Perception of Energy System Dynamics. Millennium 2000 6:23-38. Purica I. (2005), Schimbarea mediului inconjurator si perceptia dinamicii sistemelor de energie, Dezvoltarea durabila in Romania, modele si scenarii pe termen mediu si lung, Emilian Dobrescu, Lucian-Liviu Albu, coordonatori, Academia Romana, INCE, IPE, chapter 11, Editura Expert, Bucharest.

When in Rome, do as the Romans do Proverb

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Chapter 4

Cities: Reactors for Economic Transactions

The distribution of persons in each economy on their income is similar to the Maxwell–Boltzmann equilibrium distribution encountered in various physical systems. The transactions the persons make to buy things that enable them to survive (in all sorts of ways) are making them ‘poorer’, diminishing the amount of money they have. The work-and-get-paid type of transactions rebuild the financial capacity of the persons. Describing this process as a diffusion equation in a cylindrical geometry results in a Bessel function J0(r) solution which matches the density distribution of persons in Paris (as a typical circular pattern city), The analysis of the decrease in income shows a saving/spending behaviour function which saturates around 15%–20% of the total income. This is a possible explanation for the frequent savings value of 15%–22% of GDP found in various economies. Moreover, a simple equation for the dynamic behaviour of a city, on which a 365-day period is imposed, results in one week as the period after which persons have to be paid to restart transactions. The transaction cross-section σ is shown to have a ‘1/income’ behaviour, being a measure of the capability to make transactions, proportional to the probability of enterprises to have an income greater than a given value – this behaviour has recently been shown to happen in various economies and is known as Zipf’s law. Using neutron physics methods in describing the economic transactions environment opens up an alternative view on the models forecasting behaviour of economic systems, and shows that the geographical dimension of a city is determined by the economic transaction behaviour/environment in that city. 4.1. Transaction Environment In a World Bank publication (Milanovich, 1997) it was shown that the population of various countries has a given distribution per income values. The closer the income, corresponding to the maximum of this distribution, is to a given common 45

46

Nonlinear Models for Economic Decision Processes

poverty value, the greater the impact of income reduction will be on the population, i.e., more persons will be affected each year. When faced with this type of problem the first thing I do is to try representing the process in the ‘phase space’ and in time. At this moment something strikes me: the time constant. We are not getting poorer every year, but every time we spend money for all the things which help to keep us alive and well, this is happening with totally different time constants than a year. All this spending represents interactions with various companies. After each such interaction we remain with less money than before. In this way, we will, sooner or later, spend all our money. Is there a type of interaction in which we gain money? Of course; some of us gain money every fortnight, others more or less frequently, depending on the type of work we do and on the frequency and amount of the payment. We are living in a space (e.g. city) where there are various companies with which we interact by doing various transactions. Let’s consider, in order to make things simple, first, that the distribution of the companies is uniform. In a volume (we may consider a surface since the height of a city may be considered constant) of thickness dx, there are C transactions taking place. Out of a number I of persons who are passing through the space, dI are having less money than before. So, there is a negative increase (a decrease) -dI in the number of persons I having the initial amount of money, which is equal to the number of transactions C: -dI = C or

dI = -C

A sensible assumption would be that the number of transactions is proportional to the distance dx and to the number of persons I. But it does not mean a transaction will be done every time a person passes a company. We thus define a mean free pass between two transactions which we call λ. The shorter the mean free pass is, the more transactions take place in a given space. The equation above becomes: dI = -I dx/λ Solving it and letting I0 be the initial number of persons entering the space, each with a certain amount of money, we get the law of variation of persons making transactions (considering also that any person’s amount of money diminishes after the transaction) when passing through a transaction space: I = I0 exp(- x/ λ ) Going ahead with this reasoning we may say that the greater the number N of companies, the greater is the probability Σ of having more transactions, i.e.,

Cities: Reactors for Economic Transactions

47

a smaller λ. The probability of making transactions will also depend on the capability of a company to attract transactions, which we will call σ (see Annex 1) and which may be considered specific to every type of transaction, including, this time, the ones in which the person receives money. So, we write: 1/ λ = σ N = Σ. The process described here is showing that the persons who pass with money through the transactions space will come directly from work or from some other place in the transaction space (diffusion) and will ‘disappear’ (absorption) – going back to work and earn more money – after having spent their initial amount of money. Given the above, let us consider how the number of persons with income used for transactions vary in a given space. 4.2. Diffusion Equation First, let us calculate the number of persons diffused through an element of surface ∆S placed around the origin O in the plane XOY (Fig. 4.1.) coming from the upper half space (z>0),

Fig. 4.1. We decompose this space into elements of a cone of thickness r·dt and height dr. All points of these elements play the same role versus ∆S. Such a crown has a volume dV = 2πr sin θ. r dθ. dr

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Nonlinear Models for Economic Decision Processes

The number of transactions in this volume element is: I/λ dV Each point in the crown sees the surface ∆S with an apparent surface ∆S cos θ, such that the probability that a person, after isotropic diffusion in the crown, is sent toward ∆S is:

∆S cos θ / 4πr2 But we saw above that the number of persons who arrive at the surface ∆S is multiplied by a factor exp(-r/λ) due to the diffusion in that transaction space. Thus, the number of persons that pass ∆S is: dn = I/λ. 2πr sin θ. r dθ. dr ∆S cos θ /(4πr2), exp(-r/λ) Let us call the current density the number of persons that pass through the unit of surface in one direction. Here, the direction is toward the negative z and will be denoted by J-. We have: J- = ∫0

θ=π/2

∫0

r= ∞

I/λ/2. sin θ. cos θ. dθ. exp(-r/λ) dr

This integration requires the knowledge of the distribution of I(θ,r), But the exponential term makes negligible the contribution of faraway regions and, if we assume that the function I has no singularity in O and varies smoothly around this point (condition which is met at some distance from borders and from sources), we may put: I = I0 + x(∂I∂x)0 + y(∂I∂y)0 + z(∂I∂z)0 But: x = r sin θ. cos φ ; y = r sin θ. sin φ ; z = r cos θ . The integration of terms in x and y containing φ should give the same variable. But the integration of cos φ or sin φ from 0 to 2π leads to zero, thus eliminating the two terms. Wherefrom: J- = (I/(2λ)) ∫0

θ=π/2

∫0

r= ∞

sin θ. cos θ. dθ. exp(-r/λ) dr

+ (1/(2λ)) (∂I∂z)0 ∫0

θ=π/2

∫0

r=∞

sin θ. cos θ. dθ. exp(-r/λ) dr

Integrating by parts we have: J- = I0/4 + (λ/6) (∂I∂z)0

Cities: Reactors for Economic Transactions

49

The calculation toward positive z values is the same, having only to change the sign of the gradient ((∂I∂z)0), Thus: J+ = I0/4 - (λ/6) (∂I∂z)0 Thus the current in any given direction is: J = J+ - J- = - (λ/3) (∂I∂z)0 The persons diffusing from an elementary cube (see Fig. 4.2) in the direction x are passing through the surface dydz. Z

J(x)

J(x + dx)

dz

X dy Y

Fig. 4.2.

The persons entering at x = 0 are given by: J(x) dydz The persons exiting at x + dx are: J(x + dx) dydz Thus the number of persons exiting along the x-axis (letting dv be the volume element): (J(x + dx) - J(x)) dydz = (∂J∂x) dxdydz = - (λ/3) (∂2I∂x2)0 dv The same reasoning applies to directions y and z so the total number of persons coming out of dv is: - (λ/3) (∂2I∂x2 + ∂2I∂y2 + ∂2I∂z2)0 dv = - (λ/3) ∇2 I0 dv Having I, the density of persons with money, the variation of their number in a volume dv and a time dt is given by the balance equation:

∂n/∂t dv = (production + diffusion losses + absorption) dv

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Nonlinear Models for Economic Decision Processes

We consider that the persons with money are gaining their money in the given environment at a rate of S per unit of volume. So: Production = S dv From above: Absorption = IΣa dv The index ( )a comes from ‘absorption’, the name used in physics for the ‘disappearance’ process described above, which is actually a representation of the fact that a transaction changes the person who does it. The balance equation becomes:

∂ n/ ∂ t = S + (λ/3) ∇ 2 I - Σa I Let us consider a person coming from a unique source S (e.g. place of work) with a certain amount of money. He starts zigzagging (diffuse) (see Fig. 4.3.) in the transactions space until it spends all the money and, thus, ‘disappears’ from the transactions making capability point of view. It is interesting to calculate the expression of the source and the radius (straight line) of the circle which the person needs to diffuse as we have described above. We have that the stationary diffusion equation without sources in the environment is given by: d2u/dr2 – (1/L2)u = 0 where: (we call L the diffusion length) and we noted I = u/r to simplify the equation which should be written in spherical coordinates to account for the unique source taken into consideration. The solution of this type of equation is: u = A1 exp(-r/L) + A2 exp(+r/L) from which: I = A1 exp(-r/L)/r + A2 exp(+r/L)/r The condition that I is finite when r → ∞ gives A2 = 0, and the condition that the current density of persons from an infinitely small sphere around the source (which ‘emits’ n persons) is n, gives: lim r → 0 (4 π r2 J) = n from which: J = - (λ/3) ( ∂ I ∂ r) = A1 (λ/3) exp(-r/L) (r/L +1)/r2 giving: A1 = (3n)/(4πλ)

Cities: Reactors for Economic Transactions

51

Finally: I = (3n)/(4πλ) exp(-r/L)/r Now, using the result above we calculate the mean square r2 of the radius (straight line) of the circle which the person needs to diffuse. The number of persons who ‘disappear’ – from doing transactions point of view – in the spherical crown at the distance r is: dn = 4 π r2 dr IΣa where, IΣa is the number of ‘disappearances’ per unit of volume.

M r dr

S

Fig. 4.3.

By definition we have: ∞

r2 = (1/n) ∫0 r2 dn where, with the value of the number of persons given by a point source, i.e., having L2 = λaλ/3 I = (3n)/(4πλ) * (1/r) exp(-r/L) the expression above becomes: ∞

r2 = (1/n) ∫0 r2 4 π r2 dr Σa (3n)/(4πλ) * (1/r) exp(-r/L) ∞

r2 = L2 ∫0 r3 exp(-r/L) dr which after integration by parts gives:

r2 = 6 L2 We see that L is one sixth of the medium square straight line trajectory of the person in the transactions space. Let us make things more complex and consider that the place of work, the source, is not unique but that there are a uniform number of working places in the

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Nonlinear Models for Economic Decision Processes

transaction space. It is interesting to note here that the enterprises are both absorbers of money through the sale of their products, and sources of income for their employees, shareholders, etc. We start by considering a person who has spent his money and goes to work to earn more money. He gets paid and raises by a factor K (which may be greater or smaller than one) his financial capacity. Also, from his ‘birth’, for transactions, to his ‘disappearance’, he travels a length M, defined for this fictive person as similar to L above. If the number of persons is constant in time, ∂n/∂t = 0, then the persons who pass with money through the transaction space will come directly from work or from diffusion and will ‘disappear’ – going back to work to gain more money – after having spent their initial amount. Writing this balance, which is constant in time, as we said above, we have: S + λ/3 ∇ 2 I – (1/λa) I = 0 where S are the persons who ‘disappear’ – go to work – and come back with new money. As per the consideration above: S = K I/λa wherefrom

∇ 2 I + 3(K-1)/(λλa) I = 0 or, with M2 = (λλa)/3 ∇ 2 I + (K-1)/M2 I = 0 We see that the coefficient of I depends only on the structure of the work. If we put B2m = (K-1)/M2 the equation becomes:

∇ 2 I + B2m I = 0 This type of diffusion equation has to satisfy zero limit conditions on the border of the transactions space, i.e., nobody goes outside the city to make transactions (this condition implies that the city limits act like a reflector for the transaction space but we will keep things simple for the moment), Under these conditions the geometry of the city gives increasing eigenvalues of B. We will call Bg the minimum of these geometry-determined values. The condition for the existence of solutions is then: Bm = Bg Based on the above relations let us calculate the distribution of persons in a transaction space having a cylindrical geometry. If one looks at a map of large cities, several of them have a circular base (remember we said the height is

Cities: Reactors for Economic Transactions

53

taken as constant), We denote with R and H the dimensions of the cylinder (see Fig. 4.4.):

z

H

r R

Fig. 4.4.

and the diffusion equation becomes, in cylindrical coordinates (r,z):

∂2I/∂r2 + (1/r)∂I/∂r + ∂2I/∂z2 + B2g I = 0 We let I = Z(z) * T(r) and have: (1/T)d2T/dr2 + (1/r) dT/dr + (1/Z) d2Z/dz2 + B2g = 0 Each of the two terms in Z and T must be independently constant and so we write: –a2 –b2 + B2g = 0 The term in T becomes: d2T/dr2 + (1/r) dT/dr + a2 T = 0 or: d2T/d(ar)2 + (1/(ar)) dT/d(ar) + T = 0 while the one in Z is: d2Z/dr2 + b2 Z = 0 We have, for T, the Bessel differential equation of order zero. The solution is given by: T = A J0(ar) + C N0(ar) The two Bessel functions of order zero, i.e., J0 and N0, are represented in Fig. 4.5. We see that N0 is infinite for r = 0 which is physically unacceptable for describing the distribution of persons in the city, hence C = 0. The equation in Z is easily solved to Z = A’ cos(bz) , so we have: I = A’’ J0(ar) . cos(bz)

54

Nonlinear Models for Economic Decision Processes J0(r) and N0(r) 1 0.5 0 -0.5 0 -1 -1.5 -2 -2.5 -3

2

4

6

8

10

12

Fig. 4.5.

The limit conditions are: 1. cos(bH/2) = 0 , wherefrom b = π/H; 2. J0(aR) = 0; the first zero of the Bessel function J0 is at aR = 2.405 hence a = 2.405/R. Thus the Laplacian is determined by: B2g = (2.405/R)2 + (π/H)2 From the above calculations we obtained that the radial distribution of persons in a city (a transactions space) is given by I = J0(2.405 r/R), What would be the critical dimension R of a city which is characterized by a given B2m? From the equality B2g = B2m we can determine the value of R: First we calculate the minimum critical volume. From the above relation we may write: R2 = ((2.405)2 H2) / (B2m H2 – π2) The volume V results as: V = π ((2.405)2 H3) / (B2m H2 – π2) Deriving with respect to H we find the minimum volume V for: B2m H2 = 3 π2 With these values of H and R we have the minimum critical volume given by: Vc = 148/ B3m

55

Cities: Reactors for Economic Transactions

The volume may also be written as a function of R (considering the relation between R and H resulting from the above relations, i.e., R = (2.405/π/√2) H = 0.54 H): π ((2.405)2 (R/0.54)3) / (3 π2 – π2) = 148/ B3m i.e. 5.85 R3 = 148 / (K-1) 1.5 * M3 Considering that we may measure M2 from the relation: M2 = λ2/3 where λ = - 2 / ln(I/I0) = 2 / ln(I0/I) we have that: R3 = 38.97 / ((K-1) 1.5 (ln(I0/I))3) We consider for the moment that H has no significance for the transaction space and consider only the radial behaviour (distribution) of persons. To find the value of R we need measured values of I0 and I which are specific to various cities. If we take Paris, for example, and consider that the distribution of the number of the persons making transactions is proportional to the density distribution of the population of the city shown in Fig. 4.6 , and that K is 1.02 (see Appendix 1), then we may determine the critical dimension of the city, from the formula above (where we take I0=290pers/ha and I=50pers/ha)(World Bank, 1999i), as: R = 13.64 km We are now in the position to draw the radial distribution of persons in Paris resulting from the calculations above and to compare it with the real data of distribution of persons (World Bank, 1999i), The Bessel function is scaled

pers / ha

Paris distribution of persons

350 300 250 200 150 100 50 0 0

5

10

15

R : km from city center

Fig. 4.6. (Source: World Bank, 1999i)

20

56

Nonlinear Models for Economic Decision Processes

by I0 = 290 persons/ha. A striking match between the real and the calculated distributions is obtained, as seen in Fig. 4.7:

persons / ha

Paris population distribution 350 300 250 200 150 100 50 0 0

2

4

6

8

10

r : km from city center

Paris [cap/ha]

Bessel J(2.405/R*r,0)

Fig. 4.7.

Actually the figure above only gives the distribution for the first 11 km from the city centre (the critical radius is actually giving the distance at which the theoretical distribution becomes zero), The size of Paris is greater than the critical radius, as may be seen in Fig. 4.4. We should note, though, that after approximately 11 km, the distribution drops very smoothly. It is now the time to talk about the city limit as having a reflector effect on the distribution. Let us see how this can be taken into account, to look at what is the distribution beyond the 11 km limit. 4.3. The Reflector (Albedo) The city is finite and the distribution of persons is not zero on the border. This situation can be represented by considering that there is a reflector at the border of the city which makes persons diffuse back into the city. In this environment there are no enterprises, i.e., no sources of money, so the diffusion equation is:

∂2I’/∂r2 + (1/r)∂I’/∂r + ∂2I’/∂z2 - 1/L’2 I’ = 0 where 1/L’2 = 3/(λ’λa’) and where the (’) marks the values in the reflector.

Cities: Reactors for Economic Transactions

57

If we make the change of variable T = I’/r and take only the radial part of the equation we have:

∂2T/∂r2 - 1/L’2 T = 0 We will not solve the equation here but will calculate the positive J+ (outward from the city) and the negative J- (inward from the border) currents of persons. The ratio of these currents is the albedo, i.e., the ratio of inward-coming persons; it is written:

β = J-/J+ Considering the expressions for the current densities determined above we can define the albedo as

β = [I0/4 + (λ/6) (∂I∂z)0]/[ I0/4 - (λ/6) (∂I∂z)0] If we assume the reflector as an infinite environment, then I = a1 exp(-x/λr). So, we have

β = [1 - 4(λ/6) (1/λr)]/[ 1 + 4(λ/6) (1/λr)] which may also be written as

β = [3 - 2(λ/λr)]/[ 3 + 2(λ/λr)] If now we set the condition that there is an inward current equal to J-, i.e., all the persons who reach the limits of the city are reflected back, then J+ - J- = Jand we have: J+ = 2 J- which gives β = 1/2. Determining (λ/λr) for the above value of β, we find that (λ/λr) = 1/2 or that λ = 2λr The value of λ is

λ = (R) / ln(I0/I) which, with R = 13.64 km, I0 = 290 and I = 50, gives λ = 7.8 km. We have that λr = λ/2 = 3.9 km and, from here, letting Lr2 = λr2/3, we have that Lr = 2.25 km The thickness (t) of the reflector which results in an effect equivalent to a reflector of infinite thickness is t > 3 Lr (Soutif, 1962), Hence the thickness of the reflector, in the case of Paris, would be a minimum of 6.75 km beyond the

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critical radius. This gives a total city radius of 13.64 + 6.75 = 20.4 km. From the conditions above, in the reflector there would be a smoothly falling distribution I = a1 exp(-r/(R+t)), where a1 is determined from the measured data, to ensure border continuity (at 11 km radius), at a value of 156. The final distribution (theoretical and experimental) is shown in Fig. 4.8 below:

persons / ha

Paris population distribution 350 300 250 200 150 100 50 0 0

5

10

15

20

r : km from city center

Paris [cap/ha]

Bessel J(2.405/R*r,0) and exp(-r/(R+t))

Fig. 4.8.

4.4. Decrease in Income Up till now we have described the behaviour in the transactions space from one transaction to another, which led to good predictions of the population density distribution in a city (viewed as a transaction environment), Let us see now what happens with the income, which we said decreases after each transaction, that is not of the ‘work and get paid’ type. We start with a given income E0 and assume that after a transaction, the income will pass from an initial value E0 to an income interval dE around a value E which lies in a band [E0, αE0]. The parameter α measures the loss of income in a transaction and will be discussed later. The basic consideration here is that the probability to have a given decrease in income is proportional to the size of the band in which the remaining income will be found. This gives: P(E) = (E+dE – E)/(E0-αE0) = dE/(E0(1-α))

Cities: Reactors for Economic Transactions

59

The relative decrease of income, from E0 to E, is given by: E

∫E0 dE/E = ln(E/E0) We define, thus, the logarithmic income decrease ξ = - ln(E/E0) and will calculate below the average logarithmic decrease in income for the interval [E0, αE0], i.e., for the α transaction. The average of ξ is calculated for the probability determined above, giving

ξ = - ∫E0

αE0

dE/(E0(1-α)) ln(E/E0) = 1 + (α/(1-α)) lnα

The average logarithmic decrease in income does not depend on the initial income but only on α which characterizes the transaction. If n transactions are made in succession with the income decreasing from E0 to E1, E2, .., En-1, En, then we may write: ln(E0/En) = ln(E0/E1) + ln(E1/E2) + ….+ ln(En-1/En) = nξ Consider now that the part of the income remaining after spending αE is saved (i.e., ξ = ln(E/(1-α)E)), In this context 1-α represents the proportion of saved income. Below (Fig. 4.9), we give the variation of n(α), resulting from the formulae above, to be: n(α) = ln(1/(1-α))/(1+α/(1-α) lnα) One interesting comment should be made here i.e. we see from Fig. 4.9 that after circa 80% of the initial income is spent, it takes to spend the rest of 20% a very fast increasing number of transactions (a saturation below approx. 20% to 15% of savings), Is this a description of the spending behaviour of persons n(a) 50.00 40.00 30.00 20.00 10.00 0.00 0

0.5 α

Fig. 4.9.

1

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Nonlinear Models for Economic Decision Processes

in different economies? Is 15%–20% a saturation limit for savings in a well operating economy? (See Annex 4.1), We hope all these questions may result in more research on the physics of human economic transactions. 4.5. Dynamic Evolution Equation The calculations made above for the solution of the diffusion equation assumed that it was a stationary state situation, i.e., ∂I/∂t = 0. Let us see now a simple dynamic behaviour situation. We start with the diffusion equation

∂I/∂t = S + (λ/3) ∇ 2 I - Σa I and assume all transaction environment evolves in phase – no local perturbations of I will be considered. Under these conditions we may put: I(x,y,z,t) = I(t) ι(x,y,z) The function ι(x,y,z) changes very little for a slow variation of the steady state distribution; thus, we have at any time

∇ 2 ι(x,y,z) = - B2 ι(x,y,z) We may thus replace the value of ∇2I in the dynamic diffusion equation with the one above. The remaining equation will be in I(t) only, since the source S is also proportional to ι(x,y,z), After dividing by ι(x,y,z) we get: dI/dt = S - (λ/3) B2 I - Σa I We will make now another consideration which is important when we consider the dynamic case: not all the persons are paid on the same day (prompt pay), We made this assumption for the steady state case, but in the dynamic situation the fact that a proportion β of persons is paid later (delayed pay) is not negligible, as we shall see below. The source of promptly paid persons is: S = (1-β ) K Σa I The source of delayed pay persons is given considering that the concentration of the persons having delayed pay is C(I) (we take only one group of delayed pay persons for simplicity), The decrease of this concentration over time is done with a constant λC. The variation of this concentration is: dC/dt = β KΣa I - λCC

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61

Considering a solution of the type C = C0exp(ωt) and I = I0exp(ωt), where ω is a real or imaginary constant, we have: C = KΣa I (βλC/(ω+λC)) that is the source of delayed pay persons. With the above expressions for the sources, and after some algebra, the diffusion equation becomes

θω = K(1-β) – 1 + K(βλ/(ω+λC)) or with δK = K-1:

δK/K = θω/K + (βω/(ω+λC)) We denote with θ the time used by a person, having spent his money making transactions, till the moment he goes to work to rebuild his financial capacity. This is the disappearance (death) of the person (although it does not mean physical death in the real world), We may say that every person is entitled to be paid only once, but after he gets paid he becomes a ‘different person’ who then repeats the process. The way we defined I(t) shows there is a period of time variation of the transaction environment which is given by T = 1/ω From the expression above we obtain a second-order equation for ω, with two real roots, one positive and small, and the other negative and relatively big. The evolution of I is determined only by the positive exponential, i.e., after a short time the negative ω exponential component becomes negligible. We come back to the case of Paris, for which we have K=1.02, δK=0.02, and consider the limit value of β = 0 (all persons paid on the same date), We have from the expression above that:

θ = δK/ω = δK * T Let us assume that the period T of the city is 365 days (one year for which the values of K are actually calculated), then the value of θ is:

θ = 0.02*365 = 7.3 days. Is 7 days the time after which the persons in Paris have to come back to work and replenish their source of money? This result serves once more to emphasize the need for an in-depth program to develop the details of the physics of economic transactions systems.

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Nonlinear Models for Economic Decision Processes

4.6. Conclusion If a city with persons making transactions, in which they lose or gain money, is regarded as an environment where transactions (reactions) are taking place between enterprises and persons, then the critical dimensions of the city may be calculated as well as the distribution of persons in the city. The ‘experimental’ data from Paris – a city which observes the condition of a uniform distribution of transaction supply (enterprises) – result in a very good match of the population distribution data with the calculated theoretical distribution in a cylindrical geometry, i.e., a J0 Bessel function. Describing the decrease in income resulting from making transactions, it was shown that a saturation level of saved income occurs at 15–20% of the initial income, which matches closely the value in the USA, and most of the EU countries. The dynamic diffusion equation provides also a value for the average time the persons in Paris take to return to work to replenish their financial resources, which is very close to one week. This time value results if a period of 365 days is considered for the mentioned city. The variables and parameters introduced in this approach are related to economic data from which they can be determined. One recommendation to be made here is to start some program to measure parameters like σ, λ, N, K, etc., which may lead to the possibility of understanding better the dynamics of the ‘transactions space’, be it a city, an economy or a number of interacting economies. This may show, as a result, the possibility to predict crises and ways of avoiding them or absorbing the shocks in a more resilient manner. Some inferred suggestions for possible correlation between these parameters and some economic behaviour are given in Annex 4.1. Annex 4.1. A.4.1. The Coefficient K We have considered above that a person, who is in the situation of having spent his money, goes to work to get more money. He gets paid and raises by a factor K (which may be greater or smaller than one) his financial capacity. The simplest way to assess K, based on the available data, is to consider that the increase in K is equal to the increase in GDP for France over a long enough period to eliminate short-term fluctuations. We state it again here: the time

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Cities: Reactors for Economic Transactions

constants for the process are very important, but in the example we considered in the paper, the values taken into account for the given economy are average ones. It is very important to have the values of K measured specifically for Paris in order to increase the accuracy of the data, but for the time being we assume that Paris behaves like the whole French economy. So, from the World Development Indicators (World Bank, 1999ii), we have that the increase in GDP in France, between 1980 and 1990, was 2.3%/year, while for the following period, 1990–1998, it was 1.3%/year. A time interval weighted average of these values gives an increase of GDP, in France, of 1.9%/year for the whole period. Another correction to make to the above data would be given by taking into consideration the private consumption increase per year – this value would probably reflect better the evolution of the financial capacity of the persons making transactions. The values, also given in World Bank (1999ii), are 2.6%/year for the first period above and 1.2%/year for the second period. The weighted average is 2.06%/year. From the two values above it is reasonable to take K = 1.02, the value which is used to calculate the distribution of persons in Paris. A.4.1.1. Gross domestic savings Table A1. (World Bank, 1999ii) below gives the gross domestic savings (in % GDP) for the USA and various countries taken at random from Europe: Table A1 Country

Gross domestic savings [% GDP] 1997

Country

Gross domestic savings [% GDP] 1997

France

20

Poland

18

Germany

22

Italy

22

Spain

21

Hungary

27

United Kingdom

15

Romania

14

USA

16

Russian Federation

25

14

27

The interval 15%–22% is containing most of the values presented above.

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Nonlinear Models for Economic Decision Processes

A.4.1.2. The behaviour of σ We introduced σ as a characteristic of an enterprise. It measures the probability of that enterprise to make transactions. How can we relate the value of σ with the ‘propensity’ of an enterprise for transactions? Let us assume that the more transactions an enterprise makes, the higher its revenue is. For the moment we consider that the majority of these transactions are profitable, i.e., the enterprises with positive income (i.e., we will select the viable enterprises in a given economy or sector of activity), The probability of an enterprise having the income greater than a given value is determined by various factors like the advertising budget, the costs of its activity, and is also proportional to the number of transactions it makes. In previous years, various papers (Mandelbrot, 1960; Montroll and Schlesinger, 1983; Okuyama, Takayasu and Takayasu, 1999) have explored the probability distribution of having an income greater than a certain value, over the values of the incomes for enterprises in different economies. These data are the basis for understanding the behaviour of σ as a function of the income of enterprises. We mentioned that σ is dependent on the type of transaction, but since the measured values are scarce we will limit the discussion to general behaviour for the whole economy. A typical representation of the above mentioned relation is given below: P(>x)

Romania 1995 : enterprises

1.E+00 -1.07

y~x

1.E-01 1.E-02 1.E-03 1.E-04 1.E+00

1.E+02

1.E+04

1.E+06

Income [M.ROL]

Fig. A1

The dependence of probability on income may be described by a power law which applies across various economies. The function is of the form P(i>x) ~ xb

65

Cities: Reactors for Economic Transactions

Below are given the values of b for various economies: Economy

Sector

b

Source

USA

Overall

-1.4

Montroll and Schlessinger (1983)

Japan

Overall Construction Electrical products

-1 -1.13 -0.72

Okuyama et al. (1999) Okuyama et al. (1999) Okuyama et al. (1999)

Romania

Overall

-1.07

Purica (Fig. A1)

Italy

Overall

-1

Okuyama et al. (1999)

The power-law dependence of the probability of having an income greater than a given value of income could be called the ‘1/income’ dependence. If we assume (see above) that σ is measured by this probability, then the variation law of σ is of the type ‘1/income’. From the data available we do not know if this law is valid for the whole domain of income and for each economic sector (e.g., a resonance variation may be encountered as a different type of behaviour), If we consider the annual individual GDP/capita in France of US$22210 (World Bank, 1999ii) and that there are N = 20000 entities in the city with which any individual may interact, we have very roughly that λ = 1/Σ = 1/σN. With σ ~ 1/[GDP/cap] we have that λ = 1.1. If we express λ depending on I (density of persons in Paris) to give the same value, we have:

λ = - 2 / ln(I/I0) = 2 / ln(I0/I), where I = 50 and I0 = 290 This value will be used in the calculation of the city radius, made later in the book. A.4.1.3. Distribution of persons on their income values From Milanovic (1997) an extensive database of distributions may be constructed for the countries in Eastern Europe and the Newly Independent States. Some of these distributions are given in Fig. A2 below. If we consider that in the transaction space, the persons are at equilibrium with the enterprises, in the frame of the economic processes taking place there, then we may import from physical systems the typical equilibrium distribution, i.e., the Maxwell–Boltzmann one. This distribution, unscaled, is represented in Fig. A2 and alone in Fig. A3.

66

Nonlinear Models for Economic Decision Processes Income distributions 1988 % population 45% 40%

Theory

35%

Estonia

30%

Lithuania

25%

Latvia

20%

Russia

15%

Ukraine

10%

Moldova

5%

Kazakhstan

0% 0

200

400

600

US$'88/cap/mo

Fig. A2

The expression for the distribution is given below (using the notation E for income): dn(E)/n = (2π √E)/(πkT)3/2exp(-E/kT) dE Again, we are faced with a striking similarity between the theoretical distribution and the ones of the countries in Fig. A2: Theoretical distribution sqrt(E) exp( E)

0.5 0.4 0.3 0.2 0.1 0 0

1

2

Fig. A3

3

4 E

Cities: Reactors for Economic Transactions

67

This exercise should be taken into consideration only from the potential similarity it shows between economies and other physical systems whose behaviour we have already analysed and know how to predict. The full correlation of the distributions determined above and the usual economic indicators should be the focus of a separate analysis. References Mandelbrot B.B. (1960), Int. Econom. Rev. 1:79. Milanovic B. (1998), .Income, Inequality and Poverty during the Transition from Planned to Market Economy, World Bank Publications, Washington, DC. Montroll E.W., Schlesinger M.F. (1983), J. Stat. Phys. 32:209. Okuyama K., Takayasu M., Takayasu H. (1999), Zipf’s law in income distribution of companies. Physica A 269. Soutif M. (1962), Physique Neutronique. Presses Universitaires de France, Paris. World Bank (1999), World Development Indicators, World Bank, Washington, DC. World Bank (2000), Entering the 21st Century, World Development Report (1999/2000), Oxford University Press, Oxford.

Useless laws weaken the necessary laws Montesquieu

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Chapter 5

Considerations on the Reform in the Power Sector (Avoiding Chaos in the Path to an Optimal Market Structure)

The reform of a one-player power sector (i.e., a natural monopoly) into a multiple-player power market brings to the clients not only the benefits of competition but also the costs of complexity. In between the two, an optimal number of players is found, corresponding to the minimum price of power for the clients. Considering time as the third dimension, the optimum curve becomes a potential surface on which the evolution of the market entities is seen as oscillations (mergers and unbundling) along the valley of minimum price. Every oscillation triggers a price burst which is detrimental to the clients. To avoid this, the role of the regulator is better defined in the sense of smoothing the transition from monopoly to market. The example of the US and EU power sector evolution is relevant here. In the above approach, long-range competition resulting from the future opening of power markets in Europe, or from the penetration, 70 years ago, of interconnection technology in the USA, is compared with short-range (local) competition. Finally the price limits are determined to ensure that (i) the new entrants in the market are not eliminated and (ii) that the market avoids oscillations (chaotic behaviour) which may drastically shock a non-resilient economy. A case study calculation is done for a transition economy (Romania). 5.1. Introduction A lot is happening these days in the power industries both in Europe (East and West) and in the United States, Australia, etc. The main trend is toward the change of the monopoly dominated national power sectors into power markets. The benefits of the competition, implemented through this change, are measured 71

72

Nonlinear Models for Economic Decision Processes

by the decrease, in the long run, of the price of energy to the clients. Alas, there’s no such thing as a free lunch! That is why we try to assess here the price to pay for the benefits of competition, resulting from the costs of the increased complexity of the market. Can this cost be minimized? Is there an optimal structure of the market which results from the interplay between the benefits of competition and the costs of complexity? We try to answer these questions below, first by defining the behaviour of the process and, second, by building the conceptual tools that may allow the determination of the best strategies to face the new power market. The role of the regulator is presented in the light of these strategies. 5.1.1. General comments on (market-monopoly-market) cycles From the point of view of the information, the cycle of passing from a market economy to a monopoly dominated one (the outmost extreme is a centrally planned one) and back to a market economy shows the hysteresis effect. The pass from market to planned economy is done by nationalization which triggers a process of information flow from the enterprise level in the market to the central planning entity. In time, no enterprise will know any longer who the manufacturers of raw materials are and who the clients for its products are; they will only know that raw materials are taken from a certain storehouse and that products are to be delivered to another specified storehouse. It is only the central planner who will have full, real knowledge about the market. Free Market

Market Freedom

Liberalization

Nationalization

Centrally Planned Market Market Information

Fig. 5.1. Economic structure hysteresis

Considerations on the Reform in the Power Sector

73

To reverse this process, i.e., go from planned to market economy, one cannot simply reverse nationalization into liberalization. If the liberalization is done before having re-introduced all the market information back to the level of the enterprises, one will not get a market economy. The only thing obtained is a conglomerate of disconnected enterprises and a number of market information holders, which will use the information to get rich fast. This fast enrichment comes from a high transaction cost resulting precisely from the lack of information. Situations may be encountered where there is almost a monopoly on the transaction costs established by the market information holder. 5.2. From Power Sector to Power Market At present the power sector, in some economies, is acting as an 'economic sector' where the state is managing through the help of a natural monopoly instrument. Any natural monopoly in power has historically developed by concentrating on the benefits resulted from economies of scale. This concept has been reflected both on the supply and the demand sides. On the supply side the nominal power of power plants has continuously increased, and the fuel, whether imported or bought from the country, was taken in bulk, which allowed negotiation of lower prices. On the demand side, the interconnection technology was extensively used to create the grid for transport and distribution, while more and more customers were connected, increasing the scale of the market. Also, regarding the safety of supply, blackouts experienced in operation have triggered measures resulting in a significant improvement of the grid resilience. Of course, passing to a real market needs competition. Outsiders may come in, both on the supply and on the demand sides. This requires a favourable legal environment carefully thought out so as to lower risk perception. The institutionalization that follows will definitely have to create a regulator for the power market as well as a system operator to manage the power pool. 5.2.1. Costs of complexity An important observation has to be made here: in a completely unbounded power generation, part of the benefit of scale, regarding the bulk supply of fuels, is lost. The sum of the costs of numerous smaller quantities of fuel (at higher prices) will be greater at the macroeconomic level than the cost of larger quantities bought at lower prices. Of course, this will happen unless competition is not implemented in

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Nonlinear Models for Economic Decision Processes

the fuel supply too, by, for example, diversifying the local and the foreign sources of supply. This situation raises an important point regarding the costs of transition and the costs of complexity versus the benefits of scale and of competition. Creating a market for power leads to a large number of players in competition. This brings the advantage of competition to the newly defined ‘market clients’, coming from the former ‘sector customers’. At the same time, there is a loss in the benefits of scale, which is felt at the macroeconomic level (e.g. in the impact of the total cost of fuel on the country’s balance of payments), This loss comes from the increased complexity of the market and can thus be associated to a measure of the cost of complexity. As shown below, other mechanisms contribute to this cost. Going to the extremes we may see that the greater the number of market players, the greater the competition will be, lowering the price to the clients. But the greater the market complexity, the higher the complexity cost (e.g. loss of benefits of scale, transaction costs and/or other mechanisms) which tends to increase the price to the clients (the increase could be direct or indirect through macroeconomic influences), Since two opposite trends have been identified for the price to the client, we may define an optimum price corresponding to a market complexity where the benefits of competition are balanced by the cost of complexity. A qualitative behaviour of the process is shown in Fig. 5.2 below: Price to client

Elimination of state subsidies

Minimal price of market

Costs of complexity

Benefits of titi Monopoly - 1 player

Optimal market structure

Market - N players

Fig. 5.2. Minimal price to the client optimization of the market structure

Considerations on the Reform in the Power Sector

75

As more players are penetrating the market, the cost of complexity accumulates. The processes that generate the cost of complexity are, in a nonexhaustive list, the following (the estimated figures are given for the cases of various countries, based on mentioned international sources): The loss of scale benefits for the fuel supply - we think that due to the size of the coalmines, or of the oil tankers, or oil fields, the quantity of fuel delivered is covering the fuel storage capacity of one power plant. So the loss of scale benefits for the fuel supply may not be substantial in the price of fuel if the size of the generation entities is suitably chosen. Diversification of fuel sources may also contribute positively. The estimations made for UK show a loss of benefits compared to pro CEGB situation of US$2.1 billion for 15 years (1995–2010), (Newbery and Pollitt, 1997) The need for an information network, which has to be set in place at the level of each player in the market as well as for the whole market, is high. The cost of this information system is substantial and the lack of such a system in the market is liable to produce losses which may add to the cost. Immediately after the unbundling in the UK there were significant investments in IT development and implementation (e.g. companies like National Power report figures close to £450 million. (Source: National Power and UK Electricity Association) Along with the information there is a need for more metering. Setting up manufacturing facilities and installation and maintenance capabilities for that equipment adds other costs. This situation, though, creates jobs which may help to absorb the redundant personnel from the power entities, reducing the social conversion costs. Just for your information, in Norway the charge for metering is US$360/year. (Livik et al., 1998) Creating and maintaining a market mechanism, e.g. a power pool, as well as a regulatory agency, represents costs which have also got to be sustained. We will discuss later, in more detail, the issue of regulation. As for the costs of founding and operating a regulatory agency, there are also some effects on the price of power which are reported (Navarro, 1996) to have led to firms in some EU countries paying over 50% more for their electricity than their American counterparts. It is important to note here that passing from a monopoly to a deregulated market cannot be done without a regulatory body, which bears costs, and improves stability during the transition. The approach developed here is aimed at reducing these costs by defining in a better context the role of the regulator.

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Nonlinear Models for Economic Decision Processes

The professional relocation of the personnel laid off from the previous monopoly as well as the training of the remaining ones in order to increase their competitiveness bring more costs into the picture. The order of magnitude could be roughly £2.8 billion for a 15-year period – as estimated for the UK (Newbery and Pollitt, 1997), Finally, we mention here the transaction cost. It usually results from the increasing number of intermediaries doing the retail wheeling in the market. For example, the electricity bill of a New Hampshire residential customer includes the following charges (Conger, 1996): Description of Charge

Quantity

Meter Charge

Rate

Amount

9.16

9.16

Transmission Charge

786 kWh

0.00389

3.06

Distribution Charge

786 kWh

0.01900

14.93

Acquisition Premium

786 kWh

0.02970

23.34

Stranded Cost

250 kWh

0.02069

5.17

536 kWh

0.06252

33.61

Pilot Participation Credit

786 kWh

-0.01480

-11.63

PSNH Energy Energy Charge

786 kWh

0.03300

25.94

Total Current Charges

$103.58

Also, transaction costs are brought in by the new power projects; being estimated at 5–10% of total project costs (Klein, So and Shin, 1997), i.e., US$2 billion to US$3 billion a year assuming that investments worldwide exceed US$35 billion a year. Most of these costs ultimately will be borne by consumers and taxpayers, although investors may have to swallow the consequences of serious miscalculations. Where are the benefits which may compensate for these costs, i.e., the benefits of competition? They come from the following trends: 1. The increase of efficiency stemming from better organization and management of the market players; we estimate it to generate a price decrease of cca.10%. (Source: UK Electricity Association, 1994; Chisari et al., 1997) 2. The increase of technological efficiency resulting from the implementation of modem technologies; the assessed value based on the

Considerations on the Reform in the Power Sector

77

British and Argentinean data is cca.30% decrease in the toe/GWh. (See sources for point 1 above) 3. The competitiveness effect of the market on the supply side; is estimated to reduce prices of electricity by cca.10%. (Same sources as above) 4. The use of the benefits of scale by users who are joining in order to increase the scale of the demand, thus lowering the price, may bring another cca.10% reduction. (Same sources as above, plus Conger, 1996) Two other points should be mentioned here: the security of supply and the need for guarantees by the state which will foster the flow of capital into the power sector. A good example of the influence that the government’s energy investments guarantee policy has, on the speed of the reform, is the Temelín Nuclear Power Plant in the Czech Republic. This case evolved as follows: CEZ, the power company of the country was in a process of spinning off its generation. At the same time the Czech government decided to finalize the NPP Temelín project. A foreign company came in with a good offer but required a state guarantee. The Czech government did not want to give that guarantee (in order not to increase its already big foreign debt) and asked CEZ to provide it. At this moment the downsizing of the company stopped because CEZ had to have enough revenue to be able to cover the required guarantee. So, the generation which was taken away amounted to merely 20% of the total. The process may continue after the Temelín plant is finished. (Source Pro-Democratia Foundation, 1997) A conclusion of the story above is that the creation of a power market with several smaller players is only possible if the government, or any designated entity, assumes the cost of guarantees for investments in power projects. If this doesn’t happen, then either the power companies must be big enough to sustain guarantees, or the new power projects (e.g. power plants) built into that economy will be downsized. Guarantee responsibility represents an increased cost to the government, but it buys out the future of the power system’s operational capability. The other important point to mention is the safety of supply. From fuel abundance in the market, coming from diversified sources, to the availability of power at any time, through to the existence of a continuous distribution service and appropriate maintenance, this involves various economic layers working in intercorrelation.

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Nonlinear Models for Economic Decision Processes

We may identify the:
Physical layer of the technologies used to convert energy
Information layer of the data related to the system operation, finance, etc.
Commercial layer of the actions ensuring the interrelations among the parties involved (generators, operators, clients, etc.)
Financial layer of the fluxes of money serving, among other things, to maintain the working capability of the physical layer, etc. It is important to note here the fact that the nominal operation of the power system is influenced by the processes within each layer, while the safety of supply depends strongly on the intercorrelation among the layers. Considering the four layers one may define the safety of supply in relation to each of them. Thus, at the physical layer the main parameter to consider is the reliability of the technologies used – it is the only remedy against low frequency, high consequence events like a total blackout; at the information layer it is the timelines of the data allowing fast quality decisions. The interplay of information with technology may better help to avoid the low frequency, high consequence events. Moving toward the commercial and the financial layers, we enter the field of high frequency, low consequence events. The protection against these can be achieved through: − −

contract design to minimize risk (this also depends on the market structure) at the commercial level, and the setup of a sound insurance policy for covering and distributing risk. A captive insurance company may generate some more financial resources which could be used for direct investments and/or guarantees.

5.2.2. Time evolution of market entities Let us look at Fig. 5.3, to which we add a third dimension, i.e., time. The optimum curve becomes a surface having a valley of optimum. In this representation the time evolution of the number of entities in the market, which could increase with new entrants or decrease (e.g. by merging, buy-out, etc.), is seen as a curve oscillating on the two sides of the optimum valley, eventually converging to the valley. If we project this curve on the two planes respectively describing the time evolution of the number of enterprises and the time evolution of the price, we see that each change in the number of entities leads only to the increase in the price over the optimal value. These price shocks can only be detrimental to the clients.

Considerations on the Reform in the Power Sector

79

To mitigate them, a new special entity is needed in the market, i.e., the regulator. Considering this approach, the role of the regulator becomes better defined in relation to the market evolution. The regulator must speed up the convergence time to the market optimum by diminishing the number of oscillations of the number of market entities. It should also ensure a smoother penetration of newcomers into the market. By doing this, the number of price shocks to the market is diminished, with a beneficial effect on the economy as a whole. Price

Time

No. of entities Fig. 5.3. Time evolution of the market

One other behaviour resulting from the interpretation above concerns the elimination of borders in the European Union. This is creating more competition which leads to a reduction of the local market power costs to the clients, without actually increasing the cost of local complexity too much. This leads to a displacement of the minimum in the price–entities surface towards the left, i.e., towards fewer entities in the market. This shows that the expected effect of the overall opening of the European grid is a merging of power entities in the local markets. The same effect in the USA is given by the penetration of interconnection technology, which leads to private power entities merging, being exposed to more fierce, long-range competition, while the local entities did not merge to the same level, being confined to servicing mainly their local areas, without being interested in long-range competition.

80

Nonlinear Models for Economic Decision Processes 4,500

Publicly Owned Investor Owned Cooperative

4,000 3,500 3,000 2,500 2,000 1,500 1,000 500

1995

1985

1975

1965

1955

1945

1935

1925

1915

1905

1895

1885

-

Fig. 5.4. Evolution of the power market in the USA (Source: APPA, 1996)

To substantiate the above statement, we give below the evolution of the number of entities in the US power sector over the years. The occurrence of regulation (FERC) and of interconnection technology has had a strong damping effect on the oscillation of public utilities and has led to the smooth penetration of the latecomers, rural power cooperatives. One observation to make here relates to the fact that by contrast to the USA at the end of the 19th century, the power sectors of today’s Europe are not starting from nothing. The companies are not forming themselves as new technology penetrates, but the existing natural monopolies in power are segregating and the whole market is restructuring. What are the limits of the speed of penetration of private power companies into a monopoly dominated market? Based on the argument of economic resilience (capability of an economy to absorb shocks and still operate) there are, presently, two approaches: −

−

The small steps approach, which tries to minimize the shock by distributing it in time. A possible criticism of this approach is that by taking small steps, one may never reach the new, better structure in a finite amount of time. The risk of the extinction of privates is very big in this case. The sudden change approach, which tries to minimize the shock by reducing it to only one – even if relatively big – shock instead of having

Considerations on the Reform in the Power Sector

81

to suffer several smaller, successive ones. The critics here are related to the capability of a weak economy to resist this first shock without being severely damaged. One shock could be beneficial, if it would take the market directly to the optimal structure. If this is not the case, the market will tend to its optimal structure inducing, thus, subsequent shocks which add up to the first. The following is an attempt to identify a third way out, where the penetration may be smooth enough, through the limited involvement of a regulator. This may be done through the identification and use of non-linearity in the market behaviour. 5.3. Non-linear Effects in Market Penetration Recent years have revealed the possibility of applying a number of newly found mathematical concepts, like ‘deterministic chaos’, ‘catastrophe theory’, ‘bifurcation theory’ etc., to the description of economic systems’ behaviour. These instruments have been proved to be more accurate descriptors of several types of non-linear dynamic systems, thus arriving at the identification of certain specific mechanisms by which the outcome of complex, ‘chaotic’ behaviour may be predicted. One such mechanism is shown to occur in the process of privatization of monopolies or, equivalently, it may be applied to the transition from centrally planned economies to market ones. Some comments on the rate of privatization and its role in producing chaotic evolution conclude the chapter. Appendix 5.1 gives some comments on the earlier dynamic models in economics and the essentials of ‘deterministic chaos’ behaviour generated by quadratic phase diagrams. This behaviour occurs also in various economic situations. 5.3.1. Quadratic phase diagrams in economics Let us analyse the way second-order dynamic relationships can arise in economics starting with a simple example showing this pattern: Consider the relationship between a firm’s profits and its advertising budget decision. Suppose that without any expenditure on advertising, the firm cannot sell anything. As advertising outlay rises, total net profit first increases, then

82

Nonlinear Models for Economic Decision Processes

gradually levels off and finally begins to decline, yielding the traditional hillshaped profit curve. If Pt represents total profit in period t and yt is total advertising outlay, Pt can, for illustration, be expressed as Pt = ayt(l-yt), If the firm devotes a fixed proportion, b, of its current profit to advertising outlays in the following period so that yt+l = bPt, the first equation is transformed into our basic chaos one with w = ab. The reason the slope of the phase graph turns from positive to negative in this case is clear and widely recognized. Even if an increase in advertising outlay always raises total revenue, after a point its marginal net profit yield becomes negative and hence the phase diagram exhibits a hill-shaped curve. Giving it some thought, one may see why the time path of yt can be expected to be oscillatory. Suppose the initial level of advertising, yo, is an intermediate one that yields a high profit figure Po. That will lead to a large (excessive) advertising outlay yl in the next period, thereby bringing down the value of profit figure P1. That, in turn, will reduce advertising again and raise profit and so on ad infinitum. The thing to be noted about this process is that it gives good reason to expect the time paths of profit and advertising expenditures to be oscillatory. But it does not give us any reason to expect that these time paths need either to be convergent or perfectly replicatory. This is an example of how chaotic behaviour patterns can arise. Another example has been provided in the theory of productivity growth. (Baumol and Wolff, 1983) It involves the relationship between the rate of productivity growth (Pt) and the level of R&D expenditures by the private industry (r), Obviously a rise in r can be expected to increase Pt . However, since research can be interpreted as a service activity with a more or less fixed labour component, its costs will be raised by productivity growth in the remainder of the economy and the resulting stimulus to real wages. This, in turn, will cut back the quantity of R&D demanded. The result, as a formal model easily confirms, is a cycle with high productivity rates leading to high R&D prices which restrict the next period’s productivity growth, and so reduce R&D prices and so on. If R&D costs ultimately increase disproportionately with increases in productivity growth, it is clear that the relation Pt+1 = f(Pt) can generate the sort of hill-shaped phase graph that is consistent with a chaotic regime. Another model that can generate cyclic or chaotic dynamics is a standard growth model of Solow type in which the propensity to save out of wages is lower than that for profits. Suppose that at low levels of capital stock K one obtains increasing marginal returns to increased capital, and the elasticity of

Considerations on the Reform in the Power Sector

83

substitution of labour for capital is initially low; but diminishing returns eventually set in and the elasticity of substitution moves the other way. Then total profits can rise, at first, relative to total wages, but later profits may fall both relative to wages, and even absolutely. This can generate a hill-shaped relationship between Kt+l and Kt as rising Kt at first elicits rising savings and then eventually depresses them as profits fall. Other similar models may be constructed exhibiting chaotic behaviour. The following model relates to the privatization of a monopoly. (Purica, 1994) 5.3.2. Privatization of a monopoly The characteristic Total Revenue (Rm) of a monopolist is another typical case of a hill-shaped process: at zero quantity demanded (Q) total revenue is zero, while at a price of zero, Rm is zero although the quantity demanded is positive. The slope of any line from the origin to the Rm(Q) curve is the marginal price of the item at the corresponding output level. If we consider now the perfect competition market, it is obvious that the total revenue Rc will have a linear dependence on the quantity demanded. The process of passing from the monopolistic market to the perfect competition one is characterized by the shift of the total quantity demanded from the monopolistic production to the private one. So if at year yt the monopolistic production dominates the market providing all the quantity demanded, it obtains a total revenue given by the parabolic equation Rm(Q) = m.Q(l-Q) where Q is the ratio between the actual demand and the total potential demand of the economy. Let us suppose that the next year yt+l, part of the demand Q (i.e. rQ) is provided from private sources which gives a total perfect competition revenue Rc = p.(r.Q), with p being the price of the private production. Further on we may consider that the perfect competition revenue Rc(t+l) is proportional to the previous year’s monopolistic revenue q.Rm(t), where q is an indication of the rate of privatization. Resuming the suppositions above, one obtains: Rm(t) = m.Q(l-Q)

(1)

Rc(t) = p.(r.Q)

(2)

Rc(t+l) = q.Rm(t)

(3)

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Nonlinear Models for Economic Decision Processes

From Equations (1) and (2) we have Rm(t) = m/(p.r), Rc(t) (1-Rc(t)/(p.r))

(4)

which, together with (3), gives: (5) Rc(t+l) = q.m/(p.r), Rc(t) (1-Rc(t)/(p.r)) Relation (5) represents a typical process where ‘chaotic’ behaviour may occur. In order to bring it to the canonical form we have, as a first alternative, to make the assumption that p.r = l. This actually says that the share of the demand r that the private competitors may hope to cover is inversely proportional to the price p that they use on the market, which represents a sensible conclusion. We have that q.m = w is the parameter we have discussed above regarding the outcome of chaos. Considering the limits emerging in the dynamics and focusing on the relation between the monopolistic parameter m and the privatization parameter q, we have: q.m3 where the slope will be less than -1; the private revenue will start oscillating, first experiencing, various doublings of the period then passing to a chaotic regime. Analysing how this translates into considerations on the privatization rate q, it follows easily from the above inequalities that: q<(l/m) gives an extinction of the privatization process, this representing a lower limit of privatization, below which it is inefficient to pursue it. (1/m)< q <(3/m) gives an interval where the private revenue will converge to a stable value. Since this value will increase with the privatization parameter value going up, there is a strong tendency toward a fast privatization. (3/m)< q which means that beyond this limit the private revenue starts oscillating, tending to a very complex (‘chaotic’) behaviour, and is thus seemingly unpredictable.

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Considerations on the Reform in the Power Sector

Extinction

Stable

1

Chaotic

2

3

w

Fig. 5.5. Privates penetrating the monopoly dominated market

Figure 5.5 synthesizes an image of the behaviour described. If now we consider Equation (5) and, instead of putting the condition p.r = l, we re-scale Rc as R = Rc/(pr), we obtain R = (q.m/(p,r)),R(l-R)

(6)

Next we will present the numerical data obtained from analysing the historical data from the power sector of Romania, where RENEL, the local monopoly, was corporatized (into CONEL) and is now unbundled and privatized. 5.3.2.1. Numerical data – the Romanian case In Romania the Electrical Energy Authority (CONEL) was generating cca.96% of the electricity and 40% of the heat required by the economy. Considering the 1989–1996 data on the sales of electrical energy for low, medium and high voltage, and the total revenue of the public utility presented in Appendix 5.1, we may determine the values of m and of w = (q.m/p.r),

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These values result from a parabolic regression of, respectively, the data R[M$] = R(Q[MWh]) and R(t + l) = f(R(t)), The resulting values are m = 0.047 [$/kWh] and w = 1.61. The following conclusions were drawn: 1. Since w = 1.61 > 1 we have the confirmation that it is now a good time to have private entities penetrate the power generation market. 2. Considering that the fraction of the private supply of electrical energy (r), is the same with the fraction of private income (q), hence that (q/r) = l, we have the following limits for the price p of electricity from private suppliers in a hypothetical privatization of the energy domain: 2.1. q/r*m/p < 1 => p > m; p > 0.047 [$/kWh] means the extinction of privates; from a technological prospective the present prices of energy from combined cycle plants are significantly lower; 2.2. 1 < (q/r*m/p) < 3 => 0.016 < p < 0.0468 [$/kWh] means the penetration of the private suppliers into the energy market; 2.3. q/r*m/p > 3 => p < 0.016 [$/kWh] where the penetration of private suppliers may oscillate in a chaotic manner, hardly controllable by energy policy makers (and by the regulator), This price limit is given by cca.25% of hydro generation existing in Romania (in Brazil the average power prices went to even lower values between 1992 and 1998), 5.4. Conclusion The important conclusions resulting from the approach developed above stress: −

−

−

the fact that the benefits of competition are balanced by the cost of increased complexity of the market. There exists an optimal number of players in the market, giving a minimum price to the clients. In various markets where, for example, there is too much unbundling, the trend of the market would be towards merging to reach the optimal structure (Poland may be an example of this these days.); the process of privatizing monopolies, especially in economies whose structures are rapidly changing, may lead to complex dynamic regimes (‘chaotic’) that cannot be controlled by the policy makers; the privatization rate is bounded both below and above: privatization that is too slow leads to extinction of the privates while one that is too fast leads to chaotic regimes liable to produce shocks on an economy with low

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−

−

87

resilience. The penetration of too many privates may reach a point where the shock would not be sustainable anymore and ‘chaotic’ regimes may show up mainly at the commercial and financial levels (the past situation of the IPPs in Pakistan may be a relevant example); the safe price range for the consistent penetration of the privates is determined for the case of electricity production in Romania. By keeping the power prices within these limits, the regulator may achieve a smooth change in the power sector, avoiding unbearable shocks to the economy. As the market evolves these limits may change; the existence of an optimal market structure (number of entities for a minimum price to the clients) and of an optimal time path (inflicting a minimum shock to the economy) may create a basis for the design of a power market and of its regulatory framework before a natural monopoly is broken. This possibility shows that the one-large-step approach is the best, provided the path trajectory from monopoly to market, and the target structure of the market, are the optimal ones. Thus subsequent shocks are eliminated and the path is smooth.

Appendix 5.1. Comments on Regulation The regulatory aspect is very important to the market. Regulation, as this chapter shows, has got to enhance competition, balancing between the short-range and the long-range players in the market. From the point of view of the market player’s action range, regulation is usually associated with short-range action, while deregulation is associated with letting longer-range players compete in the local markets. There are significant opportunities for gains in deregulating power markets. Table A.5.1 below shows electricity prices in Europe and the United States. (Guasch and Hahn, 1997) To the extent that these prices reflect incremental costs, there are likely to be significant gains from reducing entry barriers into different markets. For example, strict regulations in Germany require domestic companies to purchase electricity from regional producers, even though power is often available at lower cost nearby. The extent of the potential gains for consumers is difficult to estimate, but in the United Kingdom, energy deregulation resulted in a 70% increase in productivity and an 18–21% reduction in franchise contract prices. (Organization for Economic Cooperation and Development, 1996) The absence of similar deregulation in other European

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Union countries has led to firms paying over 50% more for their power than their American counterparts. Moreover, the impact of higher energy prices on the overall economy can be quite significant. For example, a 30% increase in electricity prices tends to raise the price of goods such as paper and pulp, metals, chemicals and glass by roughly 2.5%. (Navarro, 1996). Table A.5.1. Effects of too much (protective) energy regulation in the European Community (cost rounded to the nearest cent per kWh) Country

Cost

Country

Cost

Germany

12

France

7

Italy

10

Netherlands

7

Portugal

10

United States

7

Belgium

9

Greece

7

Spain

9

Denmark

6

Britain

8

Finland

6

Luxembourg

8

Norway

5

Ireland

7

Sweden

4

Source: Electricity Association Services Ltd., 1996. Source: Guasch and Hahn, 1997.

References Chisari O., Estache A., Romero C. (1997), Winners and Losers from Utility Privatization in Argentina. Policy Research Working Paper No. 1824, The World Bank, Washington, DC. Conger J.K. (APPA) (1996), Competitive Benefits of Coexisting Private and Publicly Owned Electric Utilities. Romania National Energy Conference. Electricity Association Services Ltd. (1996), International Electricity Prices, Issue 23, London. Electricity Association (1995), U.K. Electricity ’94, London. Guash J.L., Hahn W.R. (1997), The Costs and Benefits of Regulation. Policy Research Working Paper No. 1773, The World Bank, Washington, DC. Klein M., So J., Shin B. (1997), The Cost of Privatization Transactions – Are They Worth It? The Private Sector in Infrastructure Strategy Regulation and Risk. The World Bank Group, Washington, DC.

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Livik K. et al. (1998), Consequences for tariffs and end-use after deregulation – Experience from the Norwegian utility industry, Deregulation of the Nordic power market, Implementation and experience 1991–1997, World Bank Seminar. Navarro P. (1996), Electric Utilities: The Argument for Radical Deregulation. Harvard Business Review 74(1):112–125. Newbery D.M., Pollitt M.G. (1997), The Restructuring and Privatization of the U.K. Electricity Supply – Was It Worth It?, The Private Sector in Infrastructure Strategy Regulation and Risk. The World Bank Group, Washington, DC. ProDemocratia Foundation (1997), Electric Industries development in Eastern Europe, Seminar, Bucharest, 1997.

The Tao gives birth to the One, the One gives birth to the Dyad, the Dyad gives birth to the Triad, and the Triad to all things. Lao Tzu

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Chapter 6

A Model of Non-linear Dynamics in the Implementation of Decisions for the Evolution of Energy Technologies

6.1. Introduction Today, a great deal of attention is being given to predicting scenarios for energy systems development which will be able to cover future energy needs for both basic and non-basic human needs. We are witnessing a ‘transition phase’ of the energy systems both quantitatively from GW to the TW scale of magnitude, and qualitatively in a new interplay between hard and soft ways of generating energy. While electricity, gas and oil systems are interconnecting geographically, from a structural and organizational point of view they are disaggregating into a much more flexible and dynamic meta-system. Various old and new energy technologies and resources are fighting to survive or penetrate the energy market. This process was intensively studied by Marchetti and Nakicenovich (1978), They conclude that ‘the penetration of the market by new technologies is a very complex interplay between producers and consumers’. On a different line of research, Haefele (1977) uses the concept of resilience, which ‘qualitatively...means a system’s capability to absorb impacts from outside without ceasing to exist as a system. Absorbing impacts from outside certainly changes the system’s state, possibly dramatically, but not the fact that the system exists’. As Haefele concludes, ‘by going beyond the concept of resilience one may be able to arrive at policy judgements that would be based on mathematical models 93

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and not on numbers’. Such models should be envisaged as ‘topological comparisons of strategies (energy policies)’. But in the world of today, planning the development of energy systems means having to take into account a large and intricate pattern of various indicators not only connected with technological and economical aspects, but also with the political, sociological, environmental, etc. The good fitting of the statistical data by the logistic function is only providing the energy planner with a method to predict the evolution of energy technology penetration in the future. It does not show how to change and control such an evolution. So, it represents just an experimental assertion, a very important and necessary one, but not a consistent theory which would provide the criteria and the means for deciding and influencing the evolution of energy systems. We are trying to provide a means to fill this gap by analysing the case of funds allocation, measured in monetary units, for the intensification of one or the other of two energy technology projects in competition, with the imposed or wished variations of the external parameters represented by indicators of benefits and costs of risk control. This way, development decisions may be taken to avoid sudden, unprepared and large discontinuities resulting in impacts that may affect the evolution of the energy programmes. 6.2. Description of the Model We shall now review the main ideas which have led to the construction of the model; for a more detailed presentation the reader is referred to Gheorghe and Purica (1979). 6.2.1. Experimental grounds In a 1970 paper, Fisher and Pry tried to fit the statistical data with two-parameter logistic functions of the type: F/(1-F) = exp(α t + β)

(1)

where t is the independent variable measuring the time, α and β are the coefficients of the logistic curves, F is the portion of the technological market occupied by the new technologies, and (1-F) is the portion occupied by the old technology.

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This paper was the base for Marchetti and Peterka, who extended the logistic penetration model to more than two technologies. The basic supposition states that if the substitution of an old technology with a new one takes place, then the process continues following a logistic curve. Extending this to a decision level we analyse the decision to intensify one or the other of two technologies in competition by transferring monetary funds between their development programmes. The possibility to predict the result of abandoning the funding of one of the programmes is thus opened up, among others, to the decision maker. The competition between technologies in each moment of time is described by a distribution function of the probability to have, at that moment, a certain partition function of the total fund between the two technology development programmes. The distribution function is the stationary solution of a Fokker–Planck equation whose form depends on the expression of the transition probabilities associated with the decision of transferring funds between the two technologies. If we consider that the replacement of a technology by another is the result of a fund transfer decision from one technology to the other one, we notice that F/(1-F) represents the frequency of occurrence of the event of partial replacement of a technology by the other. At the limit we may associate with the decision a transition probability expressed as an exponential function. So, our model is in close resemblance to the Ising models in physics, also used for the description of the polarization of opinion processes in sociology. Within this approach the coefficients α , β used in the expression of the transition probabilities have precise meanings, their variation (given or imposed) determining the system’s behaviour. 6.2.2. The master equation of the process We start by admitting that the structure of an economy’s energy system may be described as a collection of conversion and transport technologies weighted by a given distribution. In terms relevant to decision-making, the weight of a given technology or group of technologies may be measured by the amount of money put at each moment t into the research, development, demonstration and deployment of that particular technology.

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For the sake of simplicity we assume that – –

the total amount of money M allocated to the energy sector is practically constant within the observation time lapse; we consider only two types of energy technology, e.g. ‘hard’ versus ‘soft’ in the sense of Lovins, or nuclear versus coal, or fast breeder nuclear reactors versus thermal breeder ones, etc.

So, we may write (2)

M = M1 + M2

with Mi , i = 1,2 being the allocated funds for the two types of technology. Intensifying, for example, the type one technology is represented by the transition (M1, M2) → (M1 + 1, M2 - 1), The decision process leading to an allocation is stochastic and cooperative since a given decision is the result of a complex interaction of opinions among individuals and/or groups, sensitive to their respective monetary power of influence of each of them, in terms of sound arguments, number, effectiveness, etc., and to external circumstances like market, policy, etc. That is why one is interested in describing the probability distribution function f to have, at a moment t, decided on an allocation (M1, M2), In order to find it we write the master equation of this process: f(M1,M2,t)/t = w21(M1 + 1, M2 - 1),f(M1 + 1, M2 - 1,t) + w12(M1 - 1, M2 + 1), f(M1 - 1, M2 + 1, t) - (w21(M1, M2) + w12(M1, M2)), f(M1, M2, t) (3) which represents the balance between entering the state (M1, M2) from and leaving it for the neighbouring states (M1-1, M2 + 1) and (M1 + 1, M2-1); see Fig. 6.1.

(M1 -1,M2 +1)

(M1,M2 )

(M1 +1,M2 -1)

Fig. 6.1. Funds allocation

w21 and w12 are dependent on the probabilities per unit time of the transitions 1 → 2 and 2 → 1, respectively. They are obviously proportional to the effective

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amount of money in the initial state and must also account for the cooperative character of the decision process underlining transitions. As we have seen from par.6.2.1 above, the transition probabilities are of exponential form. By similarity to the Ising model in physics we may say that the chance of a 1 → 2 transition is enhanced by options already made in favour of the 2nd kind of technology while discouraged by options already made in favour of the 1st kind of technology. The opposite goes for a 2 → 1 transition. With these assumptions w12 and w21 read: w12(M1, M2) = M1/M exp[-(1/θ),(I.(M1/M - M2/M)/2 + B]

(4)

w21(M1, M2) = M2/M exp[+(1/θ),(I.(M1/M - M2/M)/2 + B] 6.2.3. Identification of the system variable and control parameters The parameter I is a measure of the intensity of the energy system, B is a ‘preference parameter’ accounting for external influences on the decision process, and θ is a ‘decision climate’ parameter: the higher θ is, the higher the temperature of the debate, owing to a higher perception of the crisis. Choosing the specific indicators for I and B, in the case of energy technologyies, must take into account both the integrative character of these indicators and the scale of magnitude of the energy technology domain under investigation. For example, when describing the interplay of hard and soft energy technologies one should choose the indicators from the set of economics indicators related to a national energy system (such as energy per capita, import– export balance, etc.), In the case of analysing thermal breeding versus fast breeding, the indicators should refer to investment level, efficiency, etc. In order to solve the master equation (3) we define for large M the continuous variable: ξ = M1/M - M2/M)/2 (5) From (5) and (2) we have M1/M = 1/2 + ξ (6) M2/M = 1/2 - ξ Expanding all functions of ξ up to the second order in 1/M, one gets a Fokker–Planck-type equation:

δf/δt = - δJ/δξ

(7)

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Nonlinear Models for Economic Decision Processes

where the current J is given by J(ξ) = 1/M.(w21 - w12)f - 1/(2M)2.δ/δξ.((w21 + w12)f)

(8)

The stationary solution (J = 0) is found to be f(ξ) = const.[1/M.(w21(ξ) + w12(ξ))] -1/2

- 1exp[2M ∫ξ

dx[(w21(x) - 12(x))/((w21(x) - w12(x))]

(9)

where: w12(ξ) = (1/2 + ξ) exp(- 1/θ.(I.ξ + B)) w21(ξ) = (1/2 + ξ) exp(1/θ.(I.ξ + B)) 6.2.4. Change of stationary solution with variations of the control parameters – the ‘cusp’ type catastrophe A numerical analysis of F given by Haken (1975) for B = 0 and various I and θ shows some remarkable features (see Fig. 6.2), fst (ξ)

I/θ = 0

a)

ξ

I/ θ = 2

b)

ξ fst (ξ) I/ θ = 2.5

c)

ξ Fig. 6.2. Stationary solution behaviour

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99

1) It says essentially that the higher the energy intensity of a given economy, the higher the chances that strong stands can be taken for more than one energy option; 2) and, subsequently, that clear decisions are to be expected (sharp probability peaks) when the decision climate is favourable (θ is small, i.e., the ‘temperature of the debates’ is kept low), Letting u = I/θ and v = B/θ, the function f becomes f(ξ,u,v), Watching the bifurcation in Fig. 6.2, we may, along with Thom (1972), admit that this shape of the function f implies the existence in the space (ξ,u,v) of a topological surface of the points corresponding to the extremum states of the function f. This surface corresponds to the ‘cusp’ type catastrophe. Because its points represent the extremes of the function f, which are the equilibrium states of the system, the evolution of the system may be represented by trajectories on this topological surface (see Fig. 6.3),

ξ v u

Fig. 6.3.The cusp fold

As it was shown by Thom, this topological surface is determined by a potential whose derivative has the form

δV(ξ,u,v)/dξ = ξ3 + u.ξ + v

(10)

The projection of the topological surface in Fig. 6.3 on the plane (u,v) is a characteristic bifurcation whose branches represent the limit when a small variation of the parameters implies a sudden change of the system’s state, i.e., a sudden large reallocation of funds between the two energy technologies.

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The evolution of the energy development programme is represented by the trajectories on the manifold P of the topological surface and their projections on the parameter plane. 6.3. Criteria for Energy Development Strategies 6.3.1. Choice of control parameters As it was shown in Section 6.2.3., above, the choice of control parameters for our model is dependent on the magnitude of the domain which contains the two energy technologies in competition. If one is interested in analysing the evolution of fast breeder nuclear reactors versus thermal breeder reactors, the control parameters would have to be chosen from indicators such as investment level (i) and efficiency (e) of the technologies. An analysis of the French Phénix–Superphénix fast breeder programme versus the American Clinch River fast breeder–modified Shippingport thermal breeder shows that a sudden reallocation of funds occurred in the USA due to an increase in the Clinch River investment with a non-corresponding rise in efficiency. This did not happen with the French programme (see Purica, 1978), Of course, when wishing to describe the interplay of ‘soft’ and ‘hard’ energy technologies at a national level, in the sense of Lovins, the model will have as control parameters indicators related to this scale of magnitude, i.e., energy per capita (E/c), energy import–export balance (E/I), We see that for an economy with a large E/c, such as the USA, a growth in E/I due to a decreasing I (energy imports) leads to a moment when a large investment is necessary for ‘soft’ energy technology development programmes. For countries with a small E/c but with presently or potentially large E/I due to large exports, mainly in oil, the behaviour trajectory on the manifold is quite different. If one takes the case of Saudi Arabia with a large E/I large and relatively small E/c, then one can see that they are trying to avoid the discontinuities of a large investment in large unitary power stations by developing an intensive solar energy programme. Developing countries which try to raise their E/c, by industrialization policies, for example, reach a moment of high investment in ‘hard’ energy technologies (coal, hydro, nuclear), (Gheorghe and Purica, 1979) If one chooses v = (E - I)/c (energy exports minus energy imports per capita) as did Ursu and Vamanu (1979), then some very interesting conclusions can be

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drawn about attaining energy independence (E – I = 0) and maintaining it as a stable state. For a better understanding, we give a numerical example based on finitehorizon (five years) data on an economy with a relatively large E/c, and an E/I with a tendency of decreasing (Gheorghe and Purica, 1979), The values are given relative to ubase = (E/c)base = 2 kWh/capita; vbase = (E/I)base = 50 GWhexp/GWhimp. Through controlled actions, one can move the system away from the cusp branches via investments reshaping the energy programmes, and/or an intensive conservation policy, etc. 6.3.2. Beyond resilience – decisions for safety Going through the cases described in Sec. 6.3.1, we have seen that each sudden jump from one fold to the other means a large reallocation of funds between the two technologies. Every such reallocation represents an effort which must be made by that specific nation changing the evolution strategy of its energy systems. The question arises of what are the amplitude and frequencies of such shock like efforts which the nation's economy can still absorb and sustain without being completely perturbed. We have here the very definition of resilience as given by Haefele (1977), But our model goes beyond that by first being able to discern between the amplitudes of shocks and their frequencies of occurrence, thus giving a limit on amplitude – transferred funds – a limit on frequency, and a limit on the total number of shocks. When any of these are reached, the economy is drastically perturbed. By extending the mechanical analogy we may define a fatigue limit, measured by the number of cyclic shocks (funds reallocation) an economy can sustain before becoming completely exhausted and being forced to change its whole development in order to recover. Combined amplitude and frequency effects of shocks may be accommodated within our model. Another feature of this approach is the possibility to predict the arrival of shocks and, based on their predicted amplitude, to decide the most appropriate variations of the control parameters in order to avoid the shock or to mitigate its consequences if it is accepted. Being able to make such decisions gives the energy system’s planner the possibility to optimize the social effort for energy development, thus contributing to an increase in the nation’s economic, social and ecological safety.

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Nonlinear Models for Economic Decision Processes

6.4. An Energy Planner’s Perception of Risks and Benefits As Cullingford, Niehaus and Vuori (1982) write, ‘One important consideration in decisions concerning proposed new technologies is whether the risks are out of proportion to those from existing technologies which society presently accepts,’ while ‘when seeking to increase the safety (reduce the risk) of existing technologies an important factor is deciding where to invest available resources to achieve the greatest reduction in risk’. Of course in real processes there is a strong interdependence among the total funds invested to enhance an energy conversion technology, the benefits of that technology and the costs of controlling its risks. Within this complex dynamical pattern of fast variables, like the funds reallocated at each moment between energy technologies, and slow parameters, such as costs of risk control and benefits, moments may arrive when the planner questions not only where to invest but also when to invest, in order to achieve greater benefits and/or lower costs of risk control. Situations may occur when the enhancement of technology temporarily implies greater costs of risk control than those finally reached when the process of technology development attains a stability stage [between the risks control involved for certain benefit. Let us see how these situations can be accommodated by our model. 6.4.1. Dynamic interdependence – development trajectories First we note that we can no longer consider the control parameters u,v as independent. Finding a certain dynamic dependence between the parameters also implies the fast variable ξ imposes a flow of trajectories on the smooth surface of the cusp. We said in Sec. 6.2.3. that u is a parameter for the ‘intensity of the energy technology’ and in what follows we take it as measured by the benefits of that technology, while v is a ‘preference parameter’ which we measure by the costs of risk control. We have thus established the control parameters. The dynamic equations resulting from the interdependence of ξ, u and v are given below for a qualitative example based on the interplay between ξ, u, and v. We first consider that the time variation (ξ') of the fast variable is proportional to δV/δξ; V being the potential corresponding to the cusp catastrophe. Then we take the variation of the benefits (u') as proportional to the value of the benefit and to the value of the funds allocated to the technology.

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103

equilibrium

slow smooth return

fast action

threshold

Fig. 6.7. A flow of trajectories on the cusp fold

The variation of the costs of risk control (v') depends on the benefits obtained, from which is subtracted a constant value v0, defined as the limit value of the costs of risk control that makes the technology acceptable to society. The dynamical system we obtain reads: ξ' = - (ξ 3 + ξ.u + v) u' = a.u + bξ v' = c.u - v0

(11)

Figure 6.7 shows the flow of trajectories on the topological surface. We see that the equilibrium is given, putting x' = u' = v' = 0, and is the point of the coordinates: ξe = (av0)/(bc) ue = v0/c 2 ve = - (av0 )/(bc2),((a/b)2(v0/c) - 1) The branches of the cusp are given by 4.u3 + 27.v2 = 0

(12)

From Fig. 6.7 we see that by raising v (costs of risk control), we reach a critical value when a large reallocation of funds to a different type of technology is necessary. The critical border is not crossed if the benefits of the first technology are increased by, for example, finding new applications for it that are of interest to the society.

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Nonlinear Models for Economic Decision Processes

If we consider a, b, and c constants, we see that the equilibrium point (ξe,ue,ve) depends on v0 which is the measure of a society’s acceptance of a technology from the point of view of its cost of risk control. The example above assumes a linear dependence on u and v. Even in this rather simple case, it results in a strong dependence of the planner’s decision on the society’s perception of level of risk (v0), A possible method of assessing the acceptance level of a technology may be based on an evaluation of occupational and public man-days lost per GWa for different energy systems. 6.5. Numerical Examples Let us, for the sake of illustrating the process, give a numerical example based on a consistent set of data for Sweden. We consider an eight-year horizon (1973– 1980) for the following data: funds invested in nuclear power (M1) versus total invested funds in energy development (M) per year; the percentage of nuclear power in the total energy generated each year, assumed to be proportional with the benefits u; the average exposure per reactor unit per year, assumed to be inversely proportional with the costs of risk control v. ξ is given as defined above (M1-(M-M1))/2M = (M1/M)-0.5 while the time derivatives are calculated for each year as the difference from the previous one. The values are given in the table below: Table 6.1. Data for Sweden t (year)

ξ

u

u'

v

v'

1973 1974 1975 1976 1977 1978 1979 1980

-.52 -.48 -.44 -.40 -.48 -.56 -.43 -.31

.03 .09 .16 .19 .23 .24 .25 .29

.06 .07 .03 .04 .01 .01 .04

3.33 .77 2 .67 .5 .83 .67 .91

2.44 1.23 -1.33 -.17 .33 -.16 .24

The coefficients a and b and c and v0 are obtained by multiple linear regression for the expression of u and by simple linear regression for v. The results are the following:

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a = - 0.4976 b = - 0.2194 (we neglect the value of the resulting constant - 0.00314) c = 11.341 v0 = 2.718 With these values the coordinates of the equilibrium point are

ξe = 0.5435 ue = - 0.2397 ve = - 0.0303 while the corresponding vcr and ucr for ue and ve, respectively, are given by vcr = (- 4.ue3/27)1/2 = 0.1884 ucr = (- 27.ve2/4)1/3 = - 0.1837 It is important to mention that in the conditions above, the equilibrium point is a stable one that all evolution trajectories converge towards. The critical points give an indication of the path to equilibrium, which may be either smooth or discontinuous if the trajectory is crossing from one fold to another. One other relevant result is the value of v0 = 2.718 which marks the public’s perception limit of the risks concerning nuclear energy technology. It is easy to explain the strong reaction against nuclear power plants in Sweden by comparing the limit value of v0 with the ones achieved after 1973. Let us see another example, also on nuclear power, this time considering an economy which is supposed to take the decision for implementing the technology, i.e., start a nuclear programme. We will consider Romania in the period immediately preceding the 1977 decision to start the construction of nuclear power plants. This time, since the whole economy is considered, we take u to be the energy per capita and v to be the ratio of export to import of energy. The data are given in the table below; this time we are only interested in the trajectory evolution in the (u,v) control parameter plane: Data for Romania (Source: RENEL) t (year)

u = E/c [u base = 2 kWh/cap]

v = E/I [v base = 50 GWh exp/GWh imp]

1971

1.002

1.00594

1972

1.0045

0.1818

1973

1.1795

0.2979

1974

1.2

0.08

1975

1.3545

0.1198

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The evolution trajectory in the control parameter plane is clearly heading for the critical line, suggesting a new energy conversion technology should be implemented, i.e., nuclear, the only one available with an established technology at that time. Nuclear power plants projects started to be developed in 1977 and construction of actual units took place in 1979. 6.6. Energy Policy and Technological Profile The critical lines in the field (u,v) of energy policy should be naturally interpreted as the resilience limits of the different phases of the system. In our terms, it means that a system with a dominant type of technological structure accepts continuous reallocation of money from the this first type of technologies to the second type without essentially changing its type 1 profile until the resilience border is closed, due, for example, to a continuous pursuance of its traditional energy trade course (import–export), Then the structure of the system has to undergo a sudden transition to a dominant type 2 profile, thus facing the undesirable consequences of sudden change. The same applies for an evolution from the second type to the first type of profile in connection with the other resilience border. The ‘topology of behaviour’ suggested here is merely a way to keep the planner’s thoughts concentrated on the various risks which the model allows him to see, thus giving him the possibility to know not only where to invest, but also when, in order to avoid or mitigate the effects of an evolution with too many and/or too big discontinuities. An interesting thing to remark on is the natural occurrence of the notion of resilience (as the capability to absorb shocks), and the capability of this approach to identify the point (0,0) in the (u,v) plane as the one where bifurcation has not taken place yet and the energy planner is faced with a large spectrum of alternatives to choose from. This point corresponds to a large plateau of the probability distribution f(ξ,u,v), meaning that a whole mix of technological decisions is compatible with the energy status of the economy. It seems that achieving such a state is more at hand for economies with a low ‘energy income’; indeed they can choose an evolutionary path with no or little discontinuity in their striving to increase the energy income. The options are different in the case of more energy affluent economies which, instead of reducing their standards of living (unwanted decrease in energy per capita), may devise policies based on increasing the security of energy consumption by better

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balancing the energy import–export trade. Also, more efficient use of energy may be considered to avoid reducing the standards of living. 6.7. Perception of Alternatives and Strategic Conduct To summarize: through a variety of paths, energy economies will probably strive toward a status of maximum energy security, or even ‘over security’ (net importers turned into net exporters on the account of lower energy intensity in the economy, new generation technologies, etc.), taking advantage of their increased margins of resilience as their energy consumption increases. All paths would, in principle, require timely adjustments in the technological infrastructure of the energy economy. The amplitude of the adjustment varies with the energy condition of an economy and with the kind of path chosen. If the need for adjustment is ignored, some paths, when followed persistently and blindly, can drive the energy system and the economy into critical states of disruption. Faced with these findings, it is clear that a prompt and correct perception of the need and the size of structural adjustments in the energy system is a prerequisite for a sound and smooth energy policy. In terms of this approach, long-term planning of the energy systems means, among other things: (i) perceiving where one’s path is to enter the critical area of the cusp, since, at that moment, the development of potential alternatives to the established energy technology infrastructure becomes both possible and increasingly advisable; (ii) perceiving where the respective resilience limit is in order to avoid crossing it through current swings in policy, which may prove costly and painful for society. Such a generic strategy, advocating R&D programmes in principle with not much of an apparent short-term market justification, might have a number of unappealing features. While ignoring the market reality and indulging in wishful thinking is certainly risky, a non-anticipative, non-creative attitude towards the market looks equally unadvisable, for there is also a risk with being too late in giving the green light to latent, sound alternatives that take some unforgiving lead time to develop. On the other hand, it is only obvious that no need for crash programmes exists when the energy economy path is chosen appropriately, thus avoiding any accidental and unprepared crossing of the resilience borders of the energy system.

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While there is a cost and risk to any change, the fact that there is also a cost and risk to ignoring the need for changes seems inescapable. An attitude that treats alternatives not as subversions of the existing establishment, but as safeguards of its stability and also generators of future, ever more appropriate patterns, is of great importance for both the planner and the public. References Cullingford M. Niehaus F. Vuori S., (1982), Use of Risk Analysis in Safety Decisions, Presented at the Annual Congress of the French Society of Radioprotection, 18–22 October, Avignon, France. Fisher J.C., Pry R. H. (1971), A Simple Substitution Model of Technological Change, Report 70-C-215, General Electric Company, Research and Development Center, Schenectadv, New York. Gheorghe A., Purica I. (1979), Decision Behaviour in Energy Development Program, Preprint, University of Bucharest, Division of System Study, The University Press, Bucharest. Haefele W. (1977), On Energy Demand. IAEA General Conference, Vienna. Haken H. (1975), Co-operative Phenomena in Systems far from Thermal Equilibrium and in Non-Physical Systems. Rev. 01 Mod. Phys. 47(1), Huang K. (1963), Statistical Mechanics. John Wiley and Sons, Inc., New York: London. Marchetti C., Nakicenovic N. (1978), The Dynamics of Energy Systems and the Logistic Substitution Model. IIASA AR-78-18. Peterka V. (1977), Macro-dynamics of Technological Change: Market Penetration by New Technologies. IIASA RR-11-22. Thom R. (1972), Stabilite Structurelle et Morphogenese. W.A. Benjamin, Inc. Advanced Book Program, Reading, Massachusetts. Ursu I., Vamanu D. (1979), Motivations and Attitudes in the Long-Term Planning of Alternative Energy Systems, Proc UNITAR Conf on Long-Term Energy Resources, 11, CF7/XXI/2. Ursu I. (1975), Energy Today – New Goals and Challenges for Physics. Proc 3rd General Conf European Phys Soc. Ursu I. (1982), A Chairman’s Statement, in Toward an Efficient Energy Future. Proc Int Energy Symposium III May: 24–27. (Ballinger Publishing Company, Cambridge, Massachusetts, 1983).

To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true; so that he who says of anything that it is, or that it is not, will say neither what is true or what is false. Aristotle

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Chapter 7

Non-linear Effects in Knowledge Production

The generation of technological knowledge is paramount to our present development. Economic science concentrates on representing the functions of production applied to all sectors, e.g. the well-known Cobb–Douglas model, associated with parameters such as capital and labour. Based on the paradigm, demonstrated in a previous paper Purica 1989, that the production of technological knowledge is governed by the same Cobb–Douglas-type model, with the means of research and the intelligence level replacing capital, respectively labour, we explore the basic behaviour of present-day economies that produce technological knowledge along with the ‘usual’ industrial production. Considering the intercorrelations of technology and industrial production, we determine a basic behaviour that turns out to be a ‘Hénon attractor’, well-known as one of the first analysed systems that present chaotic behaviour confined to strange attractors. The behaviour inside the basin of the attractor’s dynamic shows some interesting features, such as the fact that too little effort in technological knowledge production is associated to low industrial production, while too much resource allocation to technological production also leads to low industrial production. This effect clearly shows that insufficient allocation of resources to research is equivalent to an excessive allocation of resources to research, in that both hamper industrial production. Moreover, there is an area of large industrial production that corresponds to a certain rate of technology production which, in some way, optimizes development. Two measures are introduced, one for the gain of technological knowledge and the other for information of technological sequences. These are based on the underlying multi-valued logic of the technological research and on non-linear thermodynamic considerations. We have witnessed in the last decades several cases of economies, e.g., Ireland and Finland in Europe, the Asian tigers and China in Asia, which had a moment in their history when research (both means

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and intelligence) was the main priority. Luckily, globalization acted as a stabilizer that kept them close to the optimum of ‘as high as reasonably acceptable’ technological production. In contrast to the ALARA (as low as reasonably acceptable) principle that applies in risk analysis, we may introduce here the AHARA principle resulting from the non-linear behaviour of technological production vs. industrial production. 7.1. Implementation of New Technologies We continue by first analysing the implementation of new technologies in an economy. This will lead to more complex aggregated parameters, but to a simpler layout of the model which allows for a better understanding of the occurrence of discontinuities. a) The production function: we consider that for an active population L the national product is P = pL; the production means is M = mL where p and m are, respectively, production per capita and production means per capita. If we denote po as the primitive production when m = 0, we have the well-known Cobb– Douglas relation: P = kpM a Lb + poL with a + b = 1 (1) The coefficient kp includes the technical progress and will be discussed later. b) The variation of the production means: denoting the annual investment with I(t), and the function of the utility value at the moment t of an investment made at the moment t' with Km(t-t'), we have for the production means m(t): t

M(t) = ∫ -∞ dt' I(t') Km(t-t')

(2)

(it is the sum of the investments made at different time intervals dt', taking into account their depreciation), Changing the variable t-t' = τ we have: ∞

M(t) = ∫ o dτ Km(τ) I(t-τ)

(2')

where τ is the age of the investment. c) The accounting relation is a conservation relation which exists between the national product P(t), the investments I(t), the consumption C(t) and the reserves R(t): P(t) = I(t) + C(t)+R(t) (3) In what follows we consider R(t) as part of C(t).

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d) The consumption function which we take as proportional to L, with a proportionality coefficient which depends on (p-poc), where poc is the specific production for which the consumption is proportional only to L. The consumption may be expressed as: C(t) = CML/(1 + m exp(- a(p-poc)) (4) In a first approximation we may say: C(t) = CoL + C1(P-Poc)

(4')

where Poc = Lpoc. e) The demographic function: the active population varies with a time constant which is a function of the specific production. If this is under a certain value, poL, then there will be a decrease in the population. dL/dt = L/τL ; τL = τoL/g(p - poL) (5) or dL/dt = γ(P - PoL)/τoL

(5')

f) The variation of the national product is done with a time constant τp which we may consider as a control parameter: dP/dt = P/τp (6) We notice that in the case of socio-economic systems the solutions that are of interest are the ones characterized by a dynamic equilibrium (where the velocities are fixed) and not those of the stationary regimes. We shall use an approximation of the variation of production means for the case when the age of the investment is relatively short. We shall develop the function I(t) in Taylor series under the integral, keeping the first two terms: ∞

M(t) = ∫ o dτ Km(τ) (I(t) - τdI/dt + ....) = KmI(t) - τmKmdI/dt

(2")

where we noted ∞

∞

Km = ∫ o dτ Km(τ) ; τm = (1/Km) ∫ o dτ Km(τ)τ τm is a specific time constant for the investment depreciation. If we substitute in (2") the value of M(t) from (1), eliminate I and C using Equations (3) and (4), and substitute dP/dt and dL/dt with (5') and (6'), then we obtain the equation which the national production per capita (of active population), p = P/L, must satisfy: (p-po)1/α = u (p-po) + v (7)

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where u = kp1/a Km((1-C1)(1-τm/τp) + γCoτm/τoL) (8) v = kp1/αKm(((1-C1)(1-τm/τp) + γCoτm/τoL)po-Co(1 + γpoLτm/τoL) + C1poC) Since 0.25 < = α < = 0.4, we may take the mid-interval value: α = 0.33; hence 1/α = 3. Thus we obtain a third-order equation in π = p-po:

π3-uπ-v = 0

(9)

π is the behaviour parameter of the system and u and v are control parameters. They depend on the magnitudes with which we may guide the behaviour of the economic system. Equation (7) or (9) represent a surface in the space (π, u, v) and have different characteristics as a function of the variation domains of the control parameters u, v which dictate the type of the equation above (9), So noting (u/2)2 - (v/3)3 = D we have: D > 0 one real root D = 0, u ≠ 0, v ≠ 0 one simple root and two double roots D < 0 three real roots The state of the system is represented by a point on the behaviour surface. For u > 0 we have, for certain values of v, two possible states, and on the curves that limit this domain, sudden jumps may be witnessed from one state to another, either in the sense of decreasing π or increasing it, which corresponds to states of crisis or of revolutionary jumps in the technological progress. If τm < τp, then u is positive and hence the system is in the domain where discontinuities may occur. In order to have a technical revolution, v must be positive and large enough. But v is proportional to kp, which reflects the technical progress in the Cobb-Douglas formula. The parameter v may become negative only if the factor Co, which describes the consumption, is sufficiently large, as it results from Equation (8), A negative v may lead to an economic crisis, with the introduction of new technologies as a way out (Fig. 7.1). This process has a dynamic characterized by the successive saturation of the old technologies with the emergence of new ones (Kondratiev cycles; see Grubler and Nowotny, 1990), This may be seen as a succession of hysteresis-type cycles chained in the three-dimensional space (π, u, v). If one considers the production of technologies along with the production of other goods, it can be demonstrated (Purica, 1988) that a similar relation to the Cobb–Douglas one may model the generation of technologies where one

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Fig. 7.1.

considers intelligence instead of labour, and one takes research means means instead of production means (Fig. 7.2). In the above we have postulated the environment as an infinite sink for our society’s pollutants and as an infinite source of resources. Anthropogenic climate change indicates that we are now going to reach some limit in the environment. 7.2. Essentials of Chaotic Behaviour In essence, chaos theory shows that a simple relationship that is deterministic but non-linear, such as a first-order non-linear difference equation, can yield an extremely complex time path. Intertemporal behaviour can acquire an appearance of disturbance by random shocks and can undergo violent, abrupt qualitative changes, either with the passage of time or with small changes in the values of the parameters. Chaotic time paths can have the following attributes, among others: a) a trajectory (time path) can sometimes display sharp qualitative changes in behaviour, like those we associate with large random disturbances (e.g. very sudden changes from small-amplitude to large-amplitude cycles and vice versa), so at least some tests of randomness cannot distinguish such chaotic patterns of change from ‘truly random’ behaviour; b) a time path is sometimes extremely sensitive to microscopic changes in the values of the parameters which can completely completely transform the qualitative character of the path; c) they may display in a bounded region an oscillatory pattern which is very ‘disorderly’.

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Chaos theory provides both for the economic analyst and for the policy designer, warnings like, that apparently random behaviour may not be random at all, demonstrating the dangers of extrapolation and showing the difficulties that can beset economic forecasting generally. 7.3. Complex Cyclical Patterns The simplest and most common chaos model involves a non-linear one-variable difference equation of first order, that is, one of the form: yt+1 = f(yt) whose graph (the phase diagram), showing f(yt) as a function of yt, is hill-shaped and tunable; in other words, the height, steepness and location of the hill can be adjusted as desired by a suitable modification in the values of the parameters of f(yt), This phase diagram is the geometric instrument used to analyse the chaotic behaviour time path generated by a difference equation model. The function most commonly used to illustrate the chaos phenomenon is the quadratic equation with a single parameter, w: yt+l = f(yt) = wyt (1-yt) ; where dyt+l /dyt = w(1-2yt) As may be seen from the equation, if w < l, the phase curve will lie below the 45° line in the first quadrant; if w > l, there will be a positive value equilibrium point at the intersection of the 45° line and the parabolic curve; if 1 <w < 2, the phase curve slope at the intersection point will be positive; if 2 <w < 3, the slope will be negative but less than unity in absolute value; if w > 3, the slope will be less than -1. The last case is the one where chaotic behaviour may set in after a number of frequency bifurcations of the emerging limit cycles (from two frequencies to four frequencies, etc.), Grandmond (1986) and Baumol and Benhabib (1989) are giving extensive descriptions of the phase space of this equation for the economic dynamic case. 7.4. The Industrial Production and the Production of Technologies We will not insist here on the demonstration that the production of technological knowledge behaves similarly to the industrial production modelled by the Cobb–

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Douglas formula, with different meanings of the parameters and variables. Purica (1988) gives an in-depth analysis and his results will be taken here and used to develop a simple model, that is beyond a separate analysis, of the two types of production in order to assess their mutual correlations. First, to visualize the correlation between the two types of production, the figure below (Purica, 1988) gives the correlation diagram of the industrial production and the production of technologies. The symmetric relation between the two is straightforward.

Fig. 7.2. Production of knowledge and of GDP

Having presented the similar behaviours of industrial production and the production of technological knowledge, we will consider how the two are correlated. A simple consideration would state that the production of technologies is proportional to the allocated part, of the previous year’s industrial production to technological research. Denoting t(n) as the production of technologies of the current year and p(n-1) as the industrial production of the previous year, we may write: t(n) = b*p(n-1)

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On the other side, we assess that this year’s industrial production will depend on the result of the previous year’s production of technologies, t(n-1), and also on a more complex term reflecting the combined influence of the previous year’s production and this year’s technologies. Thus we may write, using the above notation: p(n) = 1 + t(n-1) + a’*p(n-1)*t(n) and, considering the relation for t(n), the equation for p(n) becomes: p(n) = 1 + t(n-1) + a*p(n-1)*p(n-1) In the equation above we introduced a constant, scaled to 1, that brings the expression of p closer to the usual Cobb–Douglas expressed as a sum of logarithms. For example, we may consider that the disappearance of unqualified labour (periodic training of labour to cope with higher productivity brought by new technologies) shows that a measure of labour is given by the amount of technological knowledge produced, while the capital becomes more efficient, i.e., the square of previous production, due to the implementation of new technologies. These last statements are just conjectures that need to be proved by experimental analysis of existing data for various economies. Just for reference, the data on industrial production and patents for the USA, from 1860 till 1976, given by Haustein and Neurath in 1982, show, as analysed here, a dependence that is non-linear (see Fig. 7.3). Patents vs. industrial production in USA 100000

t(n)

80000 60000 40000

y = -0.0055x 2 + 34.402x + 17501

20000 0 0

1000

2000

3000

p(n) t(n)

Poly. (t(n))

Fig. 7.3. Patents vs. Industrial production in USA

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Finally, grouping the two equations we have: p(n) = 1 + t(n-1) t(n + a*p(n-1)^2 t(n) = b*p(n-1) By contrast to the behaviour described by the quadratic equation with one parameter, the above system is called a Hénon attractor and has a behaviour presented in Fig. 7.4 .4 below, for given values of the coefficients.

Fig. 7.4. Hénon Attractor: A typical typi representation

This type of basins of behaviour has drawn the name of ‘strange attractors’ to the multitude of such complex mathematical entities found later. The interesting thing about the occurrence of this behaviour in the economies that, along with th the industrial production, are allocating resources to enhance the production of technologies, is that one may describe various patterns of behaviour that are not allowed by the usual linear approach. The behaviour inside the basin of the attractor’s dynamic shows some interesting features, such as the fact that too little effort in technological knowledge production is associated with low industrial production, while too much technological production also also leads to an area of low industrial production. This effect clearly shows that too little allocation of resources to research, that does not allow a large industrial production due to lack of knowledge, is equivalent to a disproportionate allocation of resources to research that hampers industrial production because of lack of resources for production. A badly managed economy may fall into the trap of producing large oscillations in

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allocating its resources, thus creating dynamic regimes whose consequences are hard to absorb, leading to an erratic evolution. By contrast, there is an area of large industrial production that corresponds to a given rate of production of technologies which, in some way, optimizes development. We have witnessed in the last decades several cases of economies, e.g. Ireland and Finland in Europe, the Asian tigers and China in Asia, that had a moment in their history when research (both means and intelligence) was the main priority. Luckily, globalization acted as a stabilizer that kept them close to the optimum of ‘as high as reasonably acceptable’ production of technologies. In contrast to the ALARA (as low as reasonably acceptable) principle that applies in risk analysis, here we may introduce the AHARA principle resulting from the non-linear interplay between technological production and industrial production. Although remaining parabolic in essence, the real behaviour of the system may show different patterns for various coefficients determined from the experimental data. Data from the reference above may show a behaviour described in Fig. 7.5 below.

Fig. 7.5. Production of technological knowledge and industrial production in the USA

One may see that a strange attractor behaviour is seen in this representation too, having, in excess of the features explained above, the limitation of a certain value of industrial production over which the production of technological knowledge increases rapidly while still remaining within the attractor basin.

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7.5. Measuring Technological Information and Entropy The comments above on resource allocation lead in a wider context to the problem of the creation of technological information and further on to entropy, as a measure of economic transitions, taken in the sense of Georgescu–Roegen. Since, in the case of the correlation between technological information and GDP productions described above, we have encountered the possibility of the occurrence of a chaotic regime with several trajectories of evolution confined to a ‘strange attractor’ (of the Hénon type), it is useful to analyse the specific parameters for such nonlinear dynamics. The approach taken here will involve escort distributions and Rényi information measures but we will first introduce a measure of information gain that is more appropriate, in our view, for economic processes that are strongly anchored in the experimental reality and have an underlying modal logic. There is in economic processes, like the one described above, an underlying logic that is not always bivalent. By recreating the conditions for technological knowledge generation, information is gained on the state of truth or falseness of the technological sequences under research and implementation. Let us consider that there are not only two states, i.e., true (functions, generates more GDP, is more efficient, etc.) and false (does not work, does not generate more GDP, is less efficient, etc.) of the technological sequence of events under consideration; but any number of possible values between true and false. Each of the intermediate values can be expressed as a combined measure of the measure of true and the one of the false technological sequences. From the point of view of research and implementation, we may define the specific measures of the two states as the probabilities that technological sequences will be true (in the sense above) defining, for true, the normalized probabilities for the technological sequence i as Pit = (pi)β/Σ(pj)β where pi are not zero. Here the distributions p are given by the observed relative frequencies of the technological experimental tests. In the same manner we define the measure of false Pif for the sequence where instead of p we take 1-p. The two measures are actually resulting from the research and development activity that separates the working technological events from the ones that do not work in the process to create useful technologies. In fact the research activity is creating conditions for testing the technological sequences and is gaining technological information from the outcome of the tests. The gain of information will be described below.

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P Pif

Pit

True

Fig. 7.6. Knowledge vector for technological sequence

We may define a vector of the state of knowledge for each experimental stage in technological development and associate with it, a measure that results from typical vector calculus as: P2 = Pit2 + Pif2. Further on, we consider that research and development is repeating the conditions for testing technologies and this is adding to the two dimensions (true and false) a third one, which marks the passage of time. We associate with this dimension a measure of time that results from the frequency of repeating the tests, denoted by iP0 (with i = √(-1) marking a rotation by 90 degrees in the complex space), Since after each development test, the vector of knowledge changes by gaining technological information, we may describe the change in information by the change of the knowledge vector in a space that has been defined as a Minkowski space in physics. As shown in Purica, (1990) the gain in information is described by a Lorentz transform. The resulting vector has a magnitude P2 = Pit2 + Pif2 - P02. Those familiar with relativity theory will recognize the specific three-dimensional cones that separate the space into three regions. These have a specific meaning in our interpretation: Inside the cone, we have technological information that is generated from tests and still keeps a certain incertitude described by P02 > Pit2 + Pif2. On the cone, the technological information is coherent and leads to implementable technological sequences, while outside the cone, P02 < Pit2 + Pif2 (this would be an unreal situation when the frequency of technological sequence tests is lower than the number of tests on technological knowledge).

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Fig. 7.7. Evolution of the state of knowledge

Coming back to the definition of the probabilities P, we may take 1/β to represent the equivalent of the temperature in the thermodynamic space. (Beck and Schlogl, 1993) Thus the escort distribution considered above may be represented as dependent on a function that varies as a power of β. Moreover, for sequences of conditional technological events it may be factorized into two or more conditional probabilities, leading to the equivalent of the grand canonical ensemble with chemical potentials. Measuring information of the process may be done by using the Rényi information of order β of the original distribution given as Iβ(p) = 1/(β-1), ln(Σ(pj)β). (Beck and Schlogl, 1993) As a property of this measure, when β = 1 the expression reduces to one of Shannon information for the distribution p. The value of β is associated with the inverse of temperature, and serves to scan various behaviour patterns of p and its partition function that may determine the Helmholtz free energy of the escort distribution as F(β) = -1/β. ln(Σ(pj)β).

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7.6. Conclusion Nowadays, the production of technological knowledge, through integrated research in economic activities, is a must for the countries striving for coherent and consistent development, aimed at bringing them in the forefront of civilization. In the interplay between the ‘usual’ industrial production and the production of technological knowledge, the rule is non-linear behaviour and not the linear one. The simplistic thinking, that more and more resources allocated to technological research will result in more industrial production, may jeopardize development. By introducing a measure of the generation of technological knowledge through research and development, we showed that gaining knowledge is associated with a Lorentz transformation in Minkowski space of the frequencies of tested sequences of technological events in time. The information measure for the associated escort distributions to these technological sequences is the Rényi one is better suited to the nonlinear dynamics of the process. To better specify the difference between information measure for the escort distributions and the knowledge gain, one must stress that ‘knowledge’ refers to the experimental process which describes the interaction of the researching structures of the society with the technological sequences that aim to produce implementable technologies; while the information measure just describes each technology sequence in itself, allowing comparison among them. Obviously, a technological jump (on the logistic curve) may boost industrial production, but to enter the logistic trend it needs accumulated research. The decision to reallocate resources to a new technology is in itself a non-linear discontinuity in evolution. The ‘Aha!’ moment of creation acts as a butterfly effect that generates a technological typhoon in the industry, but it may also bring costs that impinge on industrial production and trade a decent present for a better future. We balance between ALARA, in relation to accepting the risks, and AHARA, in relation to accepting the costs of development. But in any case, knowledge remains the basic element for setting both the lower and upper limits of our evolution.

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References Baumol W.J., Wolff E.N., (1983), Feedback from Productivity Growth to R&D, Scandinavian Journal of Economics 85(2):147–57. Baumol W.J. (1986), Unpredictability, Pseudorandomness and Military Civilian Budget Interactions. Revista Internationale di Science Economiche et Commerciali April XXXIII(4):297–318. Baumol W.J., Benhabib J. (1989), Chaos: Significance, Mechanism, and Economic Applications. Economic Perspectives – A Journal of the American Economic Association 3: l. Beck C., Schlogl F. (1993), Thermodynamics of Chaotic Systems. Cambridge University Press, Cambridge. Grandmond J.M. (1986), ‘Periodic and Aperiodic Behaviour in Discrete Onedimensional Dynamical Systems’ in Hildenbrand W., Mas-Colell A. (eds), Contributions to Mathematical Economics, North Holland, New York. Haustein H.-D., Neurath E. (1982), Long Waves in World Industrial Production, Energy Consumption, Innovations, Inventions and Patents and Their Identification by Spectral Analysis, Technological forecasting and social change 22:53–89. Purica I. (1977), Legile gândirii modale (The Laws of Modal Thought), Editura Academiei, Bucharest. Purica I. (1988), Creativity, Intelligence and Synergetic Processes in the Development of Science. Scientometrics 13(1–2): 11–24. Purica I. (1989), Creativity and the socio-cultural niche. 15(3–4):181–187.

Sic transit gloria mundi

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Chapter 8

Institutional Structures as Benard–Taylor Processes

To consider the epistemic and ontological sense of principles and functions and the way in which they contribute to the creation and evolution of institutional structures, a model was developed in which the reaction–diffusion of memes (Dawkins, 1976) in a human niche (Popper, 1973) is described as a Brusselator model that presents far-from-equilibrium stabilities of a Benard–Taylor type. These dynamical stabilities are associated with the formation and evolution of institutional structures leading to a new interpretation of Heraclitus’s ‘panta rhei’ principle in visualizing human institutional history. 8.1. Epistemic Sense and Ontological Sense The environment contains things that exist without needing to be experienced continuously by a conscious subject. This could be called objective ontology. A statement may be established as true or false by objective fact. Other statements may have a truth value based on subjective opinions. The ontology of perceptions is objective if statements on the existence of things are made about things that exist, regardless of our feelings about them; it is subjective if statements refer to truth values about the opinion of one or several persons. In both cases one may have objective knowledge coming out of the associated sciences (e.g. social sciences as well as natural sciences), 8.2. Social Reality and Collective Behaviour In a collectivity, individual beliefs, desires and decisions triggering actions are correlated into collective ones.

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General sets of social facts may have subsets of institutional facts that result from a collective assignment of functions which may be performed not only in virtue of their physical nature but also in virtue of their social acceptance by the collectivity as having a status assigned to them. As an example, the noise made by a judge knocking at the end of a trial, without a definite status of behaviour assigned to it, would only represent a natural noise. Conversely, taken inside the set of functions associated with the legal institution, the mentioned noise acquires a special meaning in terms of behaviour. Thus, in order for functions to have action value that ensures the forming of an institutional structure, a number of reactions need to happen with ideas and functions (described below as memes) that spread among individuals in a collectivity, creating deontic powers such as defined: rights, duties, obligations, permissions, requirements. Not all of these are only institutional but without them, institutions would not exist as stable dynamic structures. (Searle, 2005) By creating institutional reality, human power is increased by its extended capacity for action. But the possibility to fulfil desires within the institutional structures (like getting rich in an economical structure or becoming president in a political structure) is based only on the recognition, acknowledgement and acceptance of the deontic relationships. The above statements require a basic intellectual cohesion that connects the members of a collectivity. This involves language as well as the media. 8.3. Dynamics of Memes The movement of ideas and principles among the members of a collectivity creates a dynamic where socio-cultural niches are formed as described by Popper (1973), To look deeper into the dynamic we will consider the memes introduced by Dawkins (the ‘virus-like sentences’, in Douglas Hofstadter’s terminology) that describe the basic conceptual framework for such an analysis. (Purica, 1988) There exists a certain intercorrelation of memes in a society that reacts and diffuses among the individuals. Let us consider that the objective ontology memes, A, contribute to the generation of X, new memes with social subjective ontology. A→X

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131

The new memes, X, react with existing subjective ontology memes to produce new function memes, Y, along with existing function memes, D. B+X→Y+D Further on, we consider that the existence of new memes of subjective ontology, X, in sufficient number, say 2X, combined with the function memes, Y, creates more subjective ontology memes. Actually this process says that once people start believing in new ideas, the associated functions reinforce this belief (e.g. one needs both a bible and a church to have a religion), 2X + Y → 3X We must mention that by new ideas we mean those subjective ontology principles that lay the basis for institutional structures (like new religion principles, new economic principles, new political principles, etc.). Finally, the existing new memes become traditional memes, E, in the created institutions. X→E Grouping the above reactions we have (also known as the Brusselator model): A→X B+X→Y+D 2X + Y → 3X X→E We consider the concentration of the memes as a for A; b for B; n1 for X; n2 for Y. (Haken, 1977) Concentrations a and b will be fixed as representing the existing institutions’ memes, while n1 and n2 will be variables. Thus the sources and sinks for X and Y are given below for each equation (based on typical chemical considerations(Nicolis and Prigojyne, 1977)): Reactions A→X B+X→Y+D 2X + Y → 3X X→E dn1/dt dn2/dt

Source X a

Source Y

Sink X

b.n1

b.n1

2

Sink Y

n12.n2

n1 .n2 n1 -b. n1 - n1

a + n12.n2 b.n1

- n12.n2

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Nonlinear Models for Economic Decision Processes

We will also consider that there is a diffusion of memes in a society among its members x. The diffusion coefficients are, respectively, D1 and D2 for the memes X and Y. The reaction diffusion process presented above is described by the equations given below (Haken, 1977): dn1/dt = a - (b + 1)n1 + n12n2 + D1d2n1/dx2 dn2/dt = bn1 - n12n2 + D2d2n2/dx2 The boundary conditions considered for concentrations n1(x,t) and n2(x,t) may be of two kinds: n1(0,t) = n1(1,t) = a n2(0,t) = n2(1,t) = b/a or nj; j = 1,2 remain finite for x → +/-infinity. The first set of conditions stems from the stationary-state (dnj/dt = 0 and Dj = 0) solutions of the equations which are: n10 = a and n20 = b/a They say simply that the dynamic of new principles and functions starts from the existing ones; while the second set limits, to a finite value, the concentration level of new principles and functions over the diffusion space of the niche of persons exposed to these new principles and functions In order to check if new spatial or temporal structures show up, we do a stability analysis on the differential equations. To this purpose we let n1 = n10 + q1 ; n2 = n20 + q2 and linearize the equations with respect to q1 and q2. Thus, we have: dq1/dt = (b-1)q1 + a2q2 + D1d2q1/dx2 dq2/dt = - bq1 - a2q2 + D2d2q2/dx2 The boundary conditions become: q1(0,t) = q1(1,t) = q2(0,t) = q2(1,t) = 0 or qj finite for x → +/-∞ Letting q = (q1 q2), the equations become: q = Lq

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133

where the matrix L is defined as: L=

|| D1d2/dx2+b-1 a2|| || || || -b D2d2/dx2-a2 ||

In order to satisfy the boundary conditions we let q(x,t) = q0exp(λlt)sin(lπx) with l = 1, 2,... Inserting the above expression for q(x, t) into the equation for q' yields a set of homogeneous linear algebraic equations for q0. Non-vanishing solutions are only possible if the determinant is zero: |-D1 + b-1-λ a2 | | |=0 |-b - D2-a2-λ | where λ = λl and D'j = Djl2π2, j = 1, 2. To have a null determinant λ must obey the characteristic equation λ2-αλ + β = 0 where

α = (-D'1 + b-1-D'2-a2) β = (-D'1 + b-1)(-D'2-a2) + ba2 Instability occurs if Re(λ) > 0. A dynamic of interest for our analysis is the one with a constant, a, (the concentration of objective ontology memes rarely changes nowadays; although moments like ‘e pur si muove’ may be interesting to look at), The following will consider the change of b, subjective ontology memes, to reach a critical level bc when the solution becomes unstable. So λ = α/2 + /-1/2.√(α2-4β) We first consider λ is real, which requires α2-4β > 0, and λ > 0 requires α + √(α2-4β) > 0. If λ is complex, then α2-4β < 0, and we need for instability α > 0. After transforming the inequalities above back to expressions in a, b, D'1, D'2, we obtain: 1) soft-mode instability: λ is real; λ > = 0, (D'1 + 1)(D'2 + a2)/D'2 < b (which results from α2-4β < 0 above)

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2) hard-mode instability: λ is complex; Re(λ) > = 0, D'1+D'2+1+a2 < b < D'1-D'2+1+a2+2a √(1+D'1-D'2) The left inequality comes from α<0 above while the right one comes from α2-4β < 0. Considering the inequalities for b, the instability occurs for such a wave number for which the smallest b fulfils the inequalities for the first time. As results from the analysis, a complex λ is associated with a hard-mode excitation while a real λ is associated with a soft-mode one. Since soft-mode instability occurs for k ≠ 0 and a real λ, a static partially inhomogeneous pattern arises. We follow Haken (1977) in applying an adiabatic elimination of the stable modes to evidence the soft-mode instability resulting in a bifurcation, after a critical value. So, we put: q(x,t) = ζuq0,u√(2)sin(lcπx)+Σ’jlζsjlq0sjl√(2)sin(lπx) where the index u refers to ‘unstable’, and the sum over j contains the stable modes which are eliminated adiabatically leaving us in the soft-mode case, for an even l, with ζ’u = c1(b-bc)ζu-c3ζ u 3 describing the behaviour of the order parameter ζu. The coefficients c1 and c3 are functions of a, bc, D’1, D’2 and lc, where lc is the critical value of l for which instability occurs first. A plot of ζu as a function of b is given in Fig. 8.1.

ξn

bc

Fig. 8.1. Occurrence of instability

b

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135

1.5 1

n1

0.5 0 0

5

10

15

20

25

-0.5 -1 -1.5 x

Fig. 8.2.

At b = bc a point of bifurcation occurs and a spatially periodic structure is established (Fig. 8.2). If l is odd then the equation for ζu reads:

ζ’u = c1(b-bc)ζu + c2ζ u 2 - c3ζ u 3 where c1 and c3 are the same as above, while c2 depends also on a, bc, D1, D2 and lc. Hard and soft regimes are coming out that we will not continue to describe here, but refer the reader to the abundant mathematical literature in the field. 8.4. Conclusion The point we wanted to make is largely explained by the analysis above. We found out, by applying a reaction–diffusion model to describe the dynamics of memes in a society, that institutions occur as space-time structures with a Benard–Taylor type of stability far from equilibrium . (Smirnov, 1997; Purica, 1991) The process of institutional structuring comes as a bifurcation, occurring over a critical value of the existing concentration of subjective ontology memes that allows the setting up of a spatial-temporal structure of the new memes concentration describing the acceptance of principles and functions of an institution.

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The approach we take opens a way to describe not only the space structures (in the sense of socio-cultural institutions), but also their time dynamics. The conditions of change of political structures, of administrative ones, or even the onset of new religious sects, that are generally referred to as revolutions and changes, may be accommodated within this approach. It is interesting to note that the onset of dynamically stable structures of memes concentration in societies occurs when the critical values are crossed, leading to bifurcation. The dynamic stabilities far from thermal equilibrium, that associate with the behaviour described by these processes, may lead to a better understanding of institution dynamics in the framework of socio-economic evolution within the socio-cultural niches of societies. See Fig. 8.3 that represents a simulation of this type of behaviour for a typical physical system.

Fig. 8.3. Illustrative Benard–Taylor dynamic – 2 cells

Finally, we state that the present-day physical models, if applied to socioeconomic evolution, show that Heraclitus’s ‘panta rhei’ may represent patterns of dynamic institutional structures that change from one into another with the time constants of human history. References Haken H. (1977), Synergetics, Springer-Verlag, Berlin. Nicolis G., Prigojyne I. (1977), Self-Organization in Nonequilibrium Systems, Wiley, London. Popper K. (1973), La logique de la decouverte scientifique, Payot, France. Purica I. (1988), Creativity, intelligence and synergetic processes in the development of science, Scientometrics 13(1–2):11–24.

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Purica I. (1991), ‘Synergetic application of complex ordered processes, Proceedings of ENEA Workshops on Non-linear Systems, vol. 3.’ in Maino G. (ed) Simulation of Non-linear Systems in Physics, World Scientific Publishing, Singapore. Searle J.R. (2005), What is an institution? Journal of Institutional Economics 1(1):1–22. Smirnov B.M. (1977), Introduction to Plasma Physics, MIR Publishers, Moscow.

Anyone can hold the helm when the sea is calm

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Chapter 9

Oscillatory Processes in Economic Systems

We have considered the behaviour of a system of products, being produced and transitioned to consumption supported by a monetary mass used at a given frequency. The introduction of transaction functions along with the production ones (e.g. Cobb–Douglas) has been suggested. The products were considered from the viewpoint of their cycles as having a specific frequency that is single for some simple products and the frequency of a wave packet for more complex products. Imposing optimum requirements for the transaction to consumption to be minimal, and for the positive interference of the environment to give stationary number of waves, we have obtained an equation for the transport of products and a general equation for the probability to find the product in the given space. The solution of this last equation depends both on the frequency and on the monetary mass. To analyse the behaviour one may consider cases having a constant M over a given cycle, and consider the way the frequency spectrum of products changes in space and time (from one cycle to the next), Similarities to the Schrödinger equation approach in quantum mechanics are noted. 9.1. Cycles in Dynamics of Economic Systems All through the development of economic thinking, the notion of the cycle was present, stemming from the dynamics of the surrounding world and the way human society coped with it. Nature provides us with cycles, whether circadian, seasonal, annual, etc. The basic activities that we develop, such as production of goods and information, have the notion of the cycle embedded in our consideration of them. We frequently talk about production cycles, life cycles, as well as transaction cycles, and include the notion in our description of economic activity, even starting to consider it already understood without needing to be

141

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mentioned, e.g., when we talk about depreciation or the time value of money, there is cyclical behaviour implied although not explicitly mentioned. Looking at the definition of the speed of rotation of monetary mass, it occurred to me that the notion is not a speed (from a dimensional analysis approach) but a frequency. It is defined as the ratio of (i) the price (p) multiplied by the number of transactions (T) done in one cycle, to (ii) the monetary mass (M) available during that cycle. If we count the number of times the monetary mass is used to make transactions we obtain a frequency not a speed (vM = pT), In what follows we will denote the number of transactions with N, that has a production component Np and a transaction component Nt. Another thing to consider is the fact that there is a space dimension to the process, i.e., products are manufactured in certain points in space and transported to others where they are consumed, and the cycle is repeated. Thus there is a price of production and a price of transportation. Each one has components depending on the technologies available, the cost of labour, taxes, debt service, etc. So we may consider the price p as having a production component pp, and a transaction component pt. Hence, the relation of the speed of monetary mass rotation may be written as:

νM = ppNp + ptNt

(1)

Approaching the economic system behaviour from the cyclical viewpoint, we may consider the findings of scientists such as Kondratiev, Marchetti, etc., that point toward the existence of cycles in the evolution of prices, penetration of technologies, etc. Let us take another step along this line of reasoning. The economic production brings together various sub-products into a single more complex product. If each sub-product is considered to be characterized by a cycle with a given associated wave having a phase velocity of propagation in the economic space, then a complex product made of various sub-products may be considered as being a wave packet having a group velocity (V), As a wave the packet has also a frequency (ν) and a wavelength (λ), Keeping at the level of the product scale of magnitude we may assess that a relation similar to 1 will apply, i.e., the associated part of the total monetary mass to the given product is rotated to cover the production and the transaction of that product. Thus νιMi = ppiNpi + ptiNti where i denotes the product. For all products in a given economy we may sum (integrate) over the previous relation:

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Oscillatory Processes in Economic Systems

ΣινMι = Σι (ppιNpι + ptιNtι) = Σι ppιNpι + Σι ptιNtι

(2)

We note that: W(ν) = ΣνM, total amount of money used for the given economic cycle. U(x) = Σ ppNp, total amount of money used for production – this depends on the place of production associated with the costs of production (a discussion on Cobb–Douglas and other production functions may be done here) and the distributed cost of labour in the space. E = Σ ptNt, total amount of money for transaction of products to the consumption location. Consider the need to define a transaction function as we have defined production functions. To keep it simple but still close to reality, consider that both pt and Nt increase with the speed (v) with which products are moved between the production and consumption points. Also, we introduce a parameter (m) that indicates the inertia of transport, such as the amount of import taxes on the product that increases the amount of money needed to bringing the product to consumption. Thus, E = mv2. (We will write it as E = mv2/2 to maintain the physical analogy.) Finally, W = U + E; or, for what will follow, E = W-U. 9.2. Optimality Conditions and Associated Equations In order to discuss the relations among the above parameters we consider two elements of optimality. The economy operates such that products reach consumption in a minimum trajectory, i.e., with a minimum amount of money. Thus the money associated with transactions is minimum (this may be linked to general equilibrium considerations and to efficiency ones), In our notation above

∫

W − U ds = min

we may consider that the movement of products due to transactions is done in an environment having a dispersive influence on the products transaction, depending on the specific frequency (ν) of the product (described as a wave) and on the position (x), The trajectory of the product will vary inversely proportional to this parameter which we denote by V(ν,x), The consideration we make refers to the fact that there should be positive interference of waves leading to a stationary number of waves (hence, of product cycles), i.e., Number of waves = min. This implies

ds

∫λ

= min ; ν ∫

ds

λ

ds

= min and, finally,

∫ V (v, x) = min .

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Nonlinear Models for Economic Decision Processes

The first consideration above results (see Fermi, 1961) in: δ

∫



∫ 

W − U ds = 0 ;

and, with some work, in m

W − U δ ds −

 ds  = 0  2 W −U

δU

d 2x ∂U , which is a typical motion equation =− 2 ∂x dt

saying that the larger the difference between the production spots, the faster the products will pass from one place to another, obviously depending also on the inertia parameter described above. The consideration made above together with the assumption that

1 = f (ν ) W (ν ) − U ( x ) V (ν , x )

(3)

result in the need to determine f(ν) and W(ν) that are, so far, arbitrary. Considering that the group velocity of the complex product  d 1 ν  equals the  ( )   dν V 

(

transfer velocity of the product therefore

2 m

)

W − U and that U is independent of ν,

W (ν ) −U ( x) is also considered independent of ν, we have νf =

constant and dW/dν = constant, i.e., W = Mν. This last constant is considered from Equation (1) to be the monetary mass M (assumed constant over a cycle), Also, f = 2m / M ν . This way, the equation below (Equation 3 above with expressions introduced) determines the dispersive properties of the economic space:

V (ν , x) =

Mν 2m

1 Mν − U ( x)

Now we write the monochromatic wave equation to describe the evolution of products:

1 ∂ 2ψ ∇ψ − 2 2 =0 V ∂t 2

The solution of this equation is (where ω = 2πν and M/2π = M\)

ψ = ue

− iω t

= ue

−i Wt M\

(we assume ϖ fixed)

Oscillatory Processes in Economic Systems

145

To determine u, we have:

∇ 2u +

ω 2 ∂ 2u V 2 ∂t 2

=0

Introducing V(ν,x), we get:

2m

∇ 2u −

M\ We write ωu =

2

( M \ ω − U )u = 0

− 1 ∂ψ and obtain: i ∂t

iM \

M2 ∂ψ = − \ ∇ 2ψ + Uψ ∂t 2m

Due to the fact that ψ is complex more comments will be made on, for example, the normalization of ψ. This equation gives the probability to find the product in a given space, depending on the available monetary mass and the specific frequency of the product. If, for a cycle, M is constant, then for the states of fixed frequencies (W = Μν = const → ν = const), we may get a time-dependent equation valid only for these cases, i.e., for those specific products. 9.3. Production Potential and Quantization There are various possible ways of considering the production potentials. We give here only an example where the production is described by the squared natural frequency (similar to eigenfrequency) of the product, multiplied by the square of the distance from the point of production and by the inertial parameter m/2. At infinite distance from the production the potential becomes infinite, so the product can only have finite movements. What may lead to such a potential is the fact that the cost of production varies with the product of labour and capital. Spacewise, both parameters are functions of distance x, i.e., if coming from a greater distance the cost is higher. So, U = Kx2 /2 and the equation above reads: ∇2 u −

 2m  1 E − Kx 2  u = 0  2   2 M \

To solve the equation we introduce the independent variable ξ = (mK/M\2)x; and an eigenvalue λ = 2E/ M\2(m/k)1/2 = 2E/(M\2ω).

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Nonlinear Models for Economic Decision Processes

We have noted ω2 = K/m and it is associated with the natural frequency of the product. This gives the formula of the potential U = (m/2)ω2x2, which is similar to the linear oscillator potential. The resulting equation becomes:

∇ 2 u − (λ − ξ 2 )u = 0 We refer the reader to various texts where the solution of such an equation is found based on Hermite polynomials, and it is shown that the eigenvalue λ = 2n + 1. The values of E are situated in the negative region since this is a payment. Thus En = (n + 1/2) M\ω. It would be interesting to discuss also the situation where E is positive. Then the eigenvalues are continuous and the stability of the system behaviour is under scrutiny. Since E are the values of the costs of transiting products, one may see that to bring a product to consumption one cannot avoid paying, i.e., even if n = 0 there is a nonzero value of the cost E. Moreover, Equation (1) becomes M\ω = U + (n + 1/2) M\ω. Thus, for a cycle, U = - (n-1/2) M\ω. This shows that when production takes place, i.e., n > 0, the potential is negative, suggesting a stable regime for that product. Could we infer that production is the stable dynamical state of the economic world and the lack of it leads to instability? 9.3.1. Eigenvalues of Planet Earth’s economy One important comment to make here concerns the fact that all economic activity takes place in a limited resource environment that encompasses the planet Earth. If we consider that the production potential is given by the Earth’s resources, then we may have, at a totally different dimension from the case we have analysed above, a potential that has a special form to attract resources ‘from infinity’. We may consider that production on the whole planet attracts its components from +/- infinity on the x-axis considered above (in the most complex approach, a spherical geometry may be considered), Thus a production cost potential function of the form U(x) = - U0/ch2(ax) may be considered. The equation for negative transfer costs E is written as: ∇2ψ −

 U 2m  E − 2 0 ψ = 0 2  M \  ch (ax) 

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Oscillatory Processes in Economic Systems

Fig. 9.1. Potential function

With the change of variable ζ = th(ax) and introducing the notations: ε=

2mE 2mU 0 , = s ( s + 1), s = M \ a a 2 M \2

8mU 0 1  −1 + 1 + 2  a 2 M \2

  

we have d dζ

 ε 2  dψ    (1− ζ 2 )  +  s( s + 1) − ψ =0 d ζ   1− ζ 2  

This is the generalized Legendre functions’ equation. It is brought to the hypergeometric form by the substitution ψ = (1− ζ 2 )ε /2 ω (ζ ) and temporarily changing the variable (1/2)(1-ζ) = u: u(1-u)ω’’ + (ε+1)(1-2u)ω’-(ε-s)(ε+s+1)ω = 0 The finite solution for ζ = -1, i.e., x = ∞ is

ψ = (1−ζ2)ε/2F(ε-s, ε + s + 1, ε + 1, (1-ζ)/2) In order that ψ remains finite for ζ = -1, i.e., for x = ∞, we need to have ε-s = -n, where n = 0,1,2,... (then F is an nth-order polynomial finite if ζ = -1), This way the levels of E are determined by the condition s-ε = n to be: M 2a2 En = − \ 8m

 −(1 + 2n) + 1− 8mU 0  a 2 M \2 

2   

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Nonlinear Models for Economic Decision Processes

Where there is a finite number of levels determined by the condition, ε > 0, i.e., n < s. A final statement here draws attention to the fact that Legendre polynomials provide a basic representation of the behaviour of the planet’s economy. Further analysis of the meaning of this result is needed. 9.4. Oscillatory Behaviour – Some Numerical Results The situation in Romania after the perturbation of late 1989, that triggered a transient in the following years, resulted in some values showing an oscillatory behaviour. What we do next is to try some reverse engineering on this behaviour (exponentially damped sinusoidal) to extract the parameters of interest for the process, i.e., the natural frequency, the time constant and the damping constant. The evolution of the data is given in the figure below:

Industrial Production 0.20000 0.15000 0.10000 0.05000 0.00000

Forecast N

-0.05000 1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166

cycle ind prod

-0.10000 -0.15000 -0.20000 -0.25000

Fig. 9.2. Oscillatory behaviour of industrial production

Further on, we give the characteristic function for such a behaviour with the parameters resulting from fitting the real data and the calculation of the main parameters.

Oscillatory Processes in Economic Systems

A

φ

Td

1/a

0.18

1.52

45.3

130

ωd = 0.138701662

149

– parameters from fitted data above

Td = 45.3 months (years d = 3.775)

φ = 1.515393705 exp(-ax)*A*sin(2*π/Td*x+φ) – characteristic function where

ωn = A*ωd φ = atan(ωd/a) ωd = 2*π/Td ζi = a/ωn

undamped natural frequency phase damped natural frequency damping ratio

Deduction of the values of interest is given below: a = ζi*ωn ωd = ωn*√(1-ζi^2) a = ωd(ζi/(1-ζi^2)) ζi = √(1/(1+(ωd/a)^2)) ζi = 0.055374283 ωn = a/ζi = 0.138914804 Tn = 45.23049473 months (years n = 3.769207894) The formulae and values above suggest that the evolution of industrial production has a cyclic behaviour with a natural frequency ωn = 0.138 (i.e., a period of 45.2 months), and also has a transition period with a time constant, 1/a, of 130 months. Also there is a damping constant, ξi = 0.055, whose value is positive and smaller than 1. Thus, the associated differential equation of the process is given below with the notation above, and with y = industrial production.

d2y dy − 0.015 + 0.019 y = 0.019u 2 dt dt We give in Appendix 9.1 the theory associated with this behaviour. Since the damping constant is positive subunitary, the poles of the transfer function in the Laplace transform space are complex with negative real parts. This implies a damped oscillatory behaviour stabilizing to a natural frequency.

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Nonlinear Models for Economic Decision Processes

9.5. Conclusion The specific equation determined above for the evolution of industrial production in Romania after the change in 1990 represents an alternative way of considering the evolution of main economic parameters. Starting with the data that shows behaviour associated with a specific secondorder oscillatory model, and working our way back to identify the coefficients of the associated differential equation, has led us to describe the observed behaviour in a coherent way. The applications presented in this paper open up the possibility to represent an economy as a general oscillating system where various characteristic parameters are associated with specific dynamic equations. This is a first step for a potential systematic widespread analysis. Appendix 9.1. Second-order Systems The behaviour of many oscillatory systems can be approximated by the differential equation:

d2y dy − 2ζω n + ω n2 y = ω n2 u 2 dt dt

(a1)

The positive coefficient ωn is the natural (undamped) frequency while ζ is the damping ratio. The Laplace transform (L) of y(t) when the initial conditions are zero is

  ω n2 Y (s) =  2 U ( s) 2   s − 2ζω n s + ω n  where U(s) = L(u(t)), The poles of the transfer function Y(s)/U(s) = [ωn/ (s2-2ζωns+ωn2)] are:

s = −ζω n ± ω n ζ 2 − 1 Note that: 1. If ζ > 1 both poles are negative and real 2. If ζ = 1 both poles are equal, negative and real (s = -ωn)

Oscillatory Processes in Economic Systems

151

3. If -1 < ζ < 1 the poles are complex-conjugate with negative real parts s = −ζω n ± jω n 1− ζ 2

4. If ζ = -1 the poles are imaginary and complex-conjugate ( s = ± jω n ) 5. If ζ < -1 the poles are in the right half of the s-plane Of particular interest to our work is Case 3, representing an underdamped second-order system. The poles are complex conjugates with negative real parts and are located at

s = −ζω n ± jω n 1 − ζ or

2

s = −a ± jω d

where 1/a = 1/ζωn is the time constant of the system and ω d = ω n 1 − ζ 2 is the damped natural frequency of the system. For fixed ωn the figure below shows these poles as a function of ζ, -1 < ζ < 1. The locus is a semicircle of radius ωn. The angle θ is related to the damping ratio by θ = cos-1ζ.

Fig. A.9.1. Locus of the poles

The characteristic function of such an equation is

y (t ) = (1 / ω d )e − at sin(ω d t + φ ) where φ = tan-1(ωd/a).

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References Gandolfo G. (1997), Economic Dynamics, Springer-Verlag, Berlin. Grubler A., Nowotny H. (1990), Towards the Fifth Kondratiev Upswing: Elements of an Emerging New Growth Phase and Possible Development Trajectories. International Journal of Technological Management 5(4):431– 471. Harvey A., Jaeger A. (1993), Detrending, Stylized Facts and the Business Cycle. Journal of Applied Econometrics 8:231–247. Hodrick R., Edward P. (1980), Post-War US Business Cycles: An Empirical Investigation. Journal of Money, Credit and Banking 29(1):1–16. Kondratiev N. (1981), I Ccicli Economici Maggiori, Biblioteca Capelli, Rome.

A drunk man touches all around a pole of cement, then stops in bewilderment and says, ‘They have put me in cement’. When you think there is no solution, take two steps back and enlarge your vision; maybe the problem is not concave but convex.

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Chapter 10

Final Thoughts on a Different Way of Looking at Economic Processes

We have reviewed in the previous chapters the possibility of using nonlinear models to describe the behaviour of various economic systems and thus take better decisions on their evolution. From the application of bifurcation theory to show that the economies, where child labour is used and the ones where it is not used, are but two equilibriums of a unique reality, we moved on to modelling how migrations may reverse generating cycles. We then continued to describe the city as a reactor of economic interactions, showing that the geography of a city depends of the intensity of the interactions and that the dynamic of the process explains the saving patterns and Zipf’s conjecture. Further on, the change from monopoly to free market in the power sector was shown to have potentially chaotic (deterministic chaos) regimes, and better ways to regulate the markets in order to avoid shocks to the economy were suggested. The allocation of funds between competing technologies that are penetrating the market may be seen in a much better way if the behaviour is represented in association with the solution of a Fokker–Planck equation for the process. Decisions are thus available on how to avoid discontinuities, such as ‘abandon a project’, or how to prepare to absorb their consequences. Further on, considering technology penetration, we showed how a typical non-linear approach may explain the so-called ‘next singularity’ in technological evolution, and the fact that production of goods and of technologies may generate behaviour governed by ‘strange attractors’. We also introduced a measure of the gain of technological information, that associates a multivalent modal logic to the technological research.

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Reaction diffusion models like the Brusselator, generating Benard–Taylor dynamic stability, are the basis for the description of institution structures and their evolution. Finally, we tried to approach the oscillatory nature of economic processes, from product cycles to whole economy cycles, with the oscillatory mechanics instruments showing that discrete (not-continuous) behaviour in economic systems may be associated to some natural quantization that may occur due to the limits of the world we live in. Here, along with the production functions, some transition functions are suggested for the circulation of products from manufacturers to consumers. Moreover, the overall economic cycles are superpositions of product cycles, which may give new insights into the occurrence of economic crises as processes that may be predicted and absorbed. Looking at the applications above one may be tempted to see them as separated from one another. Nevertheless there is a similarity among them, i.e., the logic used to approach each case. Normally an experimenter watches the process and makes observations about it. Then he describes the said process and continues with predictions about its evolution. Finally there is the metalogic of the experimenter–process system. Approaching the description of processes starts with the first stage above: observing the process, trying to understand its basic behaviour. Once you think you have understood how the process behaves you must try to describe this behaviour, which often is not an easy task. For this, one needs to have a basis of models, especially nonlinear ones that provide a richer set of potential scenarios. The stage of prediction actually gives the possibility to refine the approach taken and test the effectiveness of the model used. Finally, it is very important to consider the gain of information and to see whether the effectiveness of the approach is measured by the coherence of the predicted system behaviour. There is a given synergism – in the sense of importing to a domain, such as economic decisions, the models describing the behaviour of a different domain – in the approach we have chosen in this book. There is also a resulting semantics that is exported. One way to do it would be to make a brief equivalence of terms at the beginning, and then use the terminology from the domain where the model was first developed, then interpret the results at the end using the language of the new domain. My feeling, which this book expresses, is that this prevents young students from understanding how to think about the process of using mathematics to describe a given behaviour. To avoid this I have kept the description of the

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models at the ‘manual’ level on purpose. The semantics is different; and giving the economic scientist reader some time to see the structures of process description is more important than, for example, making the nuclear reactor physicist considering the text as an elementary (for him) manual description. The approaches described are all and each leaving space for development. New indicators may be introduced, for example, to predict discontinuities or crises. A different way to make decisions in economic processes may lead to better performance and even to defining performance in a different way through a natural internalization of various externalities, some of them static, others resulting from the process dynamics. Measuring knowledge and introducing the experimenter into the process using a metalogic approach may bring us closer to finding the ‘economics’ that ensures the dynamic stability we need to achieve sustainability in our living environment. Finally, we stress that there is a given modal logic underlying the economic decision processes and that the specific complexity of the processes’ behaviour can be better understood by developing and implementing models that have this type of complexity embedded in their features. As we have seen, the economic systems show a very large spectrum of behaviour that presents in some situations characteristics of quantum systems, such as the change of the system by the operation of measuring its status. This, for example, occurs especially in market-related systems where an inquiry on a certain share price will definitely change the evolution of that price in the future. Moreover, on the same line, the finite nature of the planet’s resources, on one hand, shows that production is the stable regime while the lack of it generates instability; on the other hand, it shows that there are discontinuous levels of financial stability to which the system may go with sudden changes that create shocks which test the system’s resilience. The decisions we take strive to maintain stability (not to always understand static equilibrium but mostly dynamic stability), or to foresee the occurrence of sudden changes and predict the new level to which the system is going and the associated impacts generated in the process. Thinking in these terms, one may either find the shortcut to get out of a ‘crisis’ or to create one. Anyway, if more people know this, there is hope that our world will move from large systems made of linear individual interactions, call them ‘complex’ systems, to large systems made of nonlinear interactions, which

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we may call ‘sophisticated’ systems. And the latter may have more hope of maintaining dynamic stability that ensures predictability. I hope that this book will open up the taste for a process-understanding oriented approach to economic decision, based on a wide spectrum of complex models.

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Index

agents, 6

funds allocation between technologies, 96

Bayes probability, 17 Benard–Taylor dynamic ,129 benefits of competition, 71 bifurcation, 99 Brusselator model, 131

gain of information, 121 Hénon attractor, 119 human infrastructures, 9

chaos deterministic, 81 child labour, 25 complex systems, 116 contestable markets, 9 costs of complexity, 74 cusp catastrophe, 98, 114

income value distribution, 66 indicators aggregated, 8 institutional structures, 136 labour market, 25 Laplace probability, 17 Limits, 5 Lorentz transform, 123

decision theory, 13 diffusion equation for transactions, 52 discontinuous behavior, 116 discrete nature of economic activity, 150 economic cycles, 141 economic interactions, 155 energy markets structure, 74 environmental, 7 epistemic and ontological sense, 129 equilibrium, 65, 99, 103, 113, 135

market freedom vs. Information, 72 master equation of process, 96 mean free path, 51 measuring technological information, 122 migration cycles, 35 migration cycles model, 36 memes (Dawkins), 130 Minkowski space, 10 modal logic multivalent, 157

Fokker–Planck equation, 97

non-linear models, 26

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one over income law, 64 oscillatory behaviour, 148

regulatory agencies, 87 resilience, 80

polarisation of opinion, 91 power law (Zipf), 64 power market dynamic, 81 power monopoly unbundle, 87 privates penetration, 85 production functions, 117 production of knowledge, 119 production of technologies, 118

saturation, 114 saving behaviour, 59 Schrödinger equation, 141 Social, 119 strange attractors (Henon), 117 strategic conduct, 103 sustainability, 11

quadratic phase diagrams, 84 quantization, 145 reaction cross-sections, 64 reaction–diffusion, 132

technical revolution, 114 technological, 6 technologies penetration, 115 transition functions, 143 welfare vs. Poverty, 9