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Copyright @ 1994 by Princeton University Press Publlshed by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex A1l Rights Reserved Library of Congress Cataloging-in-ablication Data Dixit, Avinash K. Investment under uncertainty / by Avinash K. Dixit and Robert S. Pindyck P. Cn1. Includes bibliographical references and index. ISBN 0-691-03410-9 making. 1. Pindyck, Robert S. 1. Capital investments-Decision II. Aritle. 11G4028.C4D58 1993
658.15'54-dc20
93-26321 CIP
This book has been composed in Times Roman Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources http://pup.princetcm.edu Printed in the United States of Merica 10 9 8 7 6 5 4 3
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Contents
Preface
Part 1. Introduction A New View
l 2 3 4 5
()f
Investment
3 4 6 8 10 23
The Orthodox Theory The Option Approach Irrcversibility and the Ability to N'Vllit An Ovcrviewof the B()()k Noneconomic Applications
2 Developing the Concepts Through Simple Examples I
2 3 4 5 6
Price Uncertainty Lasting Two Periods Extending thu Example to Three Pcriods Uncertainty over Cost Uncertainty over Interest Rates Scale versus Flexibil ity G uide to the Literature Part II. Mathematical
26 27 41 46 48 51 54
Background
3 Stochastic Processes and lto's Lemma l Stochastic Processes 2 The Wiener Process 3 Generalized Browninn Motion Ito Processes 4 Ito's Lemma
59 60 63 70 79
5 Barriers and Long-lkun Distribkltion 6 Jump Processes 7 Guide to the Literature Appendix
The Kolmogorov Equations
Dynamic
Optimization under Uncertainty Dynamic Programming l Contingent Claims Analysis 2 Relationship between the Two Approaches 3 4 Guide to the Literature Appendix A Recursive Dynamic Progralnming Appendix B-optimal Stopping Regions Appendix C Smooth Pasting
83 85 87 88 93 95 114 l20 l25 l26 128 130
Part 111.A Firm's Decisions
5
Investment
1 2 3 4
Opportunities and Investment Timing The Basic Model Solution by Dynamic Programming Solution by Contingent Claims Analysis Characteristics of the Optimal Investment Rule Alternative Stochastic Processes Guide to the Literature
l35 l36 140 147 l52
6 The Value of a Project and the Decision to Invest 1 The Simplest Case: No Operating Costs 2 Operating Costs and Temporary Suspension 3 A Project with Variable Output 4 Depreciation 5 Price and Cost Uncertaipty 6 Guide to the Literature
175 l77 l86 195 l99 207 21 l
7 Entry, Exit, Lay-up, 1 Combined Entry
213 215 229 242
and Scrapping
and Exit Strategies Lay-up, Reactivation, and Scrapping
Guidc to the, Literature
Part lV. Industry
Equilibriunl
8 Dynamic Equilibrium in 11 Competitive l The Basic Iattlition 2 Aggregate Uncertainty 3 I ndtlstry Eqtlilibrium with Exit 4 Firm-specific Uncertainty 5 A Gene ral Mode I 6 Guide to the Literature 9
Policy Intenrention
lndustry
247 Ozttl
252 'R
6l 2(:7 277 280
and Imperfect Ctpmpetition
2 82 283 ?)(b
1 Social Optimality 2 Analyses of Some Commonly Used Policit!s 3 Example of an Oligopolistic lndtlstry 4 G u id e to t he Lit e ra tu re
3t)i) 3 l4
Appendix
3 I5
Stlme Expected Present Valut!s Part V. E xtensitlns
and
Applictltilylls
10 Sequential Investment l Decisions to Start ltnd Ctlmplete :t Nltlltistagd Pnlject 2 Contilluous Investment and Timc to Btlild 3 The Lutarning Curve anl Optilnltl Prlldtlction Dccisons 4 Cost Uncertainty and Learning 5 Gu ide to the Literatu re Appendix Numerical Solution D iffe r (., n t ia l E q u a t io 1-1
()f
328 339 345 35l
Pltrtial
Incremental
1 2 3 4
3l9 32l
Investment and Capacity Choice G radual Capacity Exptnsion with Diminish ing Returns Increasing Rttturns and Lumpy Additions to Capacity Adj ustment Costs Guide to the Literature
12 Applications and Empirical Research 1 Investments in Offshore Oil Reserves 2 Electric Utilities' Compliance with the Clean 5 Tilt: Timing of Environmentai Poiicy
357 359 377 38l 39I 394 396
Air Act
4()5 4 I2
(?ontents 4 Explaining Aggregate Investment Behavior 5 Guide to the Literature
4 19 425
References
429
Symbol Glossary
445
Author Index
449
Subject Index
455
Preface
Bool provides a systematic treatment of a new theoretical approach capital investment decisions of firms. stressing the irreversibility of most to and decisions the ongoing uncertainty of the economic environinvestment which in those decisions are made. This new approach recognizes the ment value waiting of better for (but never complete) information. It option analogy with options in financial markets, which thc theory of exploits an much richer dynamic framework than was pllssible with the permits a of theory investment. traditional The new view of investment opportunities as optitlns is the product of over a decade of research by many economists, and is still an active topic in journal articles. It has led to some dramatic deptrtures from the orthodox theory. It has shown that thc traditional present value'' rule. which is taught to virtually every business school student and student of economics, can give very' wrong answers. The reason is that this rule ignores irreversibility and the option of delaying an investment. For the same reason, the new theoly also contradicts the orthodox tcxtbook view of production and supply going back to Marshall, according to which firms enter or expand when the price excceds long-run average cost, and exit or contract when price falls below average variable cost. Policy prescriptions based on the traditional theory, for example. the use of interest rate cuts to stimulate investment and antitrust policies based on price-cost margins, are also called into question. In this book we have tried to present the new theory in a clear and systematic way, and to consolidate, synthesize, and in some places extend the various strands of this growing body of research. While there is a Iarge and burgeoning Iiterature of journal articles, including a survey article by THls
ttnet
xii
/5-(:/l(.'t.
each of us, :1 book l'ormat has d istinct advan tages. l t tll/fe rs tIs the space t() develop the different themes in more detail and better order. and to place them in relation to one another. It also gives us the opportunity to introduce and explain the new techniques that underlie much of this work, to economists. We hope that the result is 11 but that are often unfamiliar better pedagogic treatment, of use to students, researcllers, and practitioners. However, perhaps even more important than pedagogy is the ability of the book tbrmat to provide a broad vision of the subject, and of the mechanisms of the dynamic, uncertain economic world. Our main aim is to clarify and explain the theory, but we think this is often best done by applying it to the real world. Therefore we often obtain numerical solutions tbr our theoretical models using data that pertain to some specific industries or products. We believe that the cumulative weight of these calculations constitutes strong prima facie evidence for the validity and the quantitative significance of the new theory of investment. However, more rigorous econometric testing, and the detailed work that is needed to devise improved decisionmaking tools for managers, await further research. We believe that this is an exciting and potentially important subject, and hope that our book will stimulate and aid such
research.
Who Should Read This Book? The first audience This book is intended for three broad audiences. consists of economists with an interest in the theory of investment, and in its policy implications. This includes graduate students engaged in the study of micro- and macroeconomic theory as well as industrial organization, and researchers at universities and other institutions with an interest in problems relating to investment. The second broad audience is students and researchers of financial economics, with an interest in corporate finance generally, and capital budgeting in particular. This would includc graduate students studying problems in capital budgeting (that is, how firms should evaluate projects and make capital investment decisions), as well as anyone doing research in finance with an interest in investment is finance decisions and investment behavior. Finally, the third audiece practitioners. This includes people working in financial institutions and concerned with the evaluation of companies and their assets, as well as corporate managers who must evaluate and declde whether to go ahead with large-scale investments for their firms. Some parts of this book are fairly technical, but that should not deter
r/-(!/c(.z the interestetl rellder. Tllc first twt) cllapters provide 11 lirly brie ( kllltl self-con t:linetl introtluct itdn to thtr tlltltlr.y of irreve rsible investnl e 1) t u ntlc r uncertainty. Tllese chapters convey many of tllc basic ideas, wllile llvtlitling technical details anl any l'nathenliltical tbrmalisln. Rttading these trhapters that is a low-risk investment we can alnlost gtlaralltee will have a high those practitioners whose knoWledge ot- econol-nics and fireturn. Even nance textbook's is vefy rtlsty should be able to follow tllese two chapters without too m uch effort t)r di fficul ty. %9eanticipate that many read ers will want to go into tlle tlltlory in mtre detail and address some of the technical issues. but Iack sonle ()f the necessary mathematical tools. With those readers in mind we hllvtl included two chapters (Chapters 3 and 4) that provide a self-contained introduction to the mathematical ctlncepts and tools that underlie this work. (These tools have applicltbility tllat goes well beyond the theo!z of investment under uncertainty, and so we anticipllte thltt some readcrs will find these chapters useful even if their applied interests lie elsewhc re, for example, macroeconomics, international trade, ()r labor economics.) .
However. wtl think that techniques are best Iearned by usi ng thcnl. Theretbre we do not attempt to bc very rigorllus t)r thorough in the mathematics as such. Nv'e rcly on intuition as far as possiblc. sktz tch stlmtl simple formal arguments in appendiccs, and reikr the reatlers whl) wish greater mathematical rigor or depth to mllre advanced treatises. Ftlr mtlst readers. we rccommend reading Chapters 3 and 4 onct:, and prllceeding t() the later chapters where thtt techniques arc used. Nve tllink tlltly will ()f thtl mathematics in this way tllan l7y emergc with a mtlch better grasp trying to master it first and in the abstract. Finally, we expect that many rcaders will wilnt to sec the ncw view t)l' investment develtlped in detail, including as many of its ram ificatitlns as possible. along with examples and applications. Chapters 5 through l2 provide just that. building up slowly from a basic and fairly simple model of irreversible investment in Chapter 5, to more complete modcls in Cilapters 6 and 7 that acctlunt lbr decisions to start or stop producing. t() the modklls in Chapters 8 and 9 that account tbr the interactions ()f firms within industries, and finally to the more ad vtnced extensions of thc theory and its applications in Chapters 10, l 1, and l 2.
Acknowledgments This book is an outgrllwth of the researcll that cach of us hlts been doing and over the past several years on the theory of investment. That research
xiv
Prelce
ormously from our interactions with hence this book-have benefitted colleagues and friends at our home institutions and elsewhere. To try to list all of the people from whom we received insights. ideas. and encouragement would greatly lengthen the book. However, there are some individuals who have been especially helpful through the comments, criticisms, and suggestions they provided aer reading our research papers and draft chapters of this book, and they deserve special mention: Giuseppe Bertola, Olivier Blanchard, Alan Blinder, Ricardo Caballero, Andrew Caplin, John Cox, Bernard Dumas, Gene Grossman, Sandy Grossman, John Leahy, Gilbert Metcalf, Marcus Miller, Julio Rotemberg, and Jiang Wang. In addition, we want to thank Lead Wey for his outstanding research assistance throughout the development of this book. Our thanks. too, to Lynn Steele for editorial help in preparing the hnal draft of the
manuscript.
The second printing of the book gave us the opportunity to correct several errors. We thank al1 the readers who brought these errors to our attention, most particularly Marco Dias of PUC, Rio de Janeiro, and David Nachman of Georgia State University. We are sure that more errors remains but we practice the preaching of capital theory: manuscripts should not be improved to the point of pertkction. but only to the point where the rate of further improvement equals the rate of interest. Finally, we want to thank Peter Dougherty, our editor at Princeton University Press, for his encouragement and advice as we prepared manuscript, and his guiding hand in seeing the book through to the
production. Both of us also received hnancial support for which we are very grateful. Avinash Dixit acknowledges the support of the National Science Foundation and the Guggenheim Foundation. Robert Pindyck acknowledges support from the National Science Foundation and from MIT's Center for Energy and Environmental Policy Research. Avinash K. Dixit Robert S. Pindyck
l part
lntroduction
Chapter
1
A New View of lnvestment
investnlent lts tllc ltct tlf illctlrring an inlnlcd itte cost in thc expectation o' ftlture rewitrtls. Fi rms thllt cllllstruct plants and instkyll equipment. mercllllnts whll Ilty in 11 stock of gllllds lr sltle. llnd persons whl) spend time ()n voclltitllll cdtlclltitln klrtr tll investors in tllis sense. Stln-lewhlt less obviotlsly. :t firm that slluts dlpwn lt ltlss-lnltking plant is :lls() the p:tyments it mtlst mllk-tl t() extract itself l'rllm cllntratrttlltl ctllmmitnlents. incltltlingseverance pllynlents t() I:tl7()r.ltre tlle initiltl expcllditu rtl. lnd tlle prllspcctive rcward is the retltlctitll, il1 l'uttlre lllsses. Viewed frllm tllis perspective. investnlent detrisilllls lre tllliqui tlltls. Ylltlr purchase ()f this tntltlk' wlls an investment. Tlle rewltrd, we I1()pe. will be 1111 ()f understanding invcstment decisitlns if ytlu are 1kn ccontlmist. and imprtlved ()1' lbility to makd sucll decisilllls in the imprllved cklu rse ytltlr l'tlture carecr an dtEtIines
E(.-oNtlMlcs
--illvesting'':
if ytlu artt
a business schotll student. investment decisillns sllare
in three imptlrtant characteristics comple the In partilll Iy degrees. Fi i is Iy te rst, nvestmen t varying or other wllrds, the initial cost ()f investment is at least partially sunk; yotl cannot recover it alI should you change ytltlr m ind. Sectlnd, t he rut is ttlt.'ertll'v over the futtlre rewards from thtt invdstment. Tlle best y()u can do is t() assess the probabilities of the alternative outcomes that can me:tn greater or smaller profit (or loss) for yllur vcnture. Third, you have somc leeway of your investment. Yotl can postpone action to get about the tnlg ()f about the certainty) coursc, complete more intbrmation (but never, Most
-rcb'o-sible.
future.
'Tllese th rct: cllari.ti-iit.: ritii itisill ic I itt;i ((? dcfcl I l 111lc i.llc t?I?(ilI lul dcttisioldsol' investors. is interacton is the focus ()f t his bf.)tlk.bl'.t dklvclllpth e theof'y ()f 'l-h
The orthodox theory of investment has ntt recognized
the important
qualitative and quantitative implications of the interaction between irreversibility, uncertainty, and the choice of timing. We will argue that this neglect explains some of the failures of that theory. For example. compared to the predictions of most earlier moels, real world investment seems much less sensitive to interest rate changes and tax policy changes, and much more sensitive to volatility and uncertainty over the economic environment. We will show how the new view resolves these anomalies, and in the process offers some guidance for designing more effective public policies concerning investment. Some seemingly noneconomic
personal decisions also have the charac-
teristics of an investment. To give just one example, marriage involves an up-front cost of courtship, with uncertain tbture happiness or misery. It may be reversed by divorce, but only at a substantial cost. Many public policy decisions also have similar features. For instance. public opinion about the relative importance of civil rights of the accused and of social order lluctuates through time, and it is costly to make or change Iaws that embody a particular relative weight for the two. Of course the costs and benehts of such noneconomic decisions are difhcult or even impossible to quantify, but our general theory will offer some qualitative insights for them, too.
1 The Orthodox Theory How should a tirm, facing unccrtainty over future market conditions, decide whether to invest in a new factory'? Most economics and business school students are taught a simple rule to apply to problems of this sort. First, calculate the present value of the expected stream of prohts lhat this facto:y will generate. Second, calculate the present value of the stream of expenditures required to build the factory. Finally, determine whether the difference between the two the net present value (NPV) of the investment-is greater than zero. If it is, go ahead and invest.
' Some dccisions that are the opposite of investment gettingan immediate benefit in return for an uncertain future cost- are also irreversible. Prominent examples include the exhaustion uf Ilaiural rain forcsts. 0ur methods appjy to thcse resuurces and (he dcstruclion of lropicai decisions, too.
/k?T-t!.y/??Ic/,/ .,z1lfebv lzkzw'0J*
Of course, there are issuds that arise in calculating this net present value. Just how should the expected stream of profits from a new factory be estimated'? How should intlation be treated'? And what discount rate (or rates) should be used in calculating the present values'? Resolving issues like these are important topics in courses in corporate finance, and especially capital the NPV of an budgeting. but the basic principle is fairly simple--calculate whether and it is positive. investment project see The net present value rule is also the basis for the neoclassical theory of investment as taught to undergraduate and graduate students of economics. Here we lind the rule expressed using the standard incremental or marginal approach of the economist: invest until the value of an incremental unit of capital is just equal to its cost. Again, issues arise in determining the vaiue of an incremental unit of capital, and in determining its cost. For example. what production structure should be posited? How should taxes and depreciation be treated? Much of the theoretical and empirical literature on the economics of investment deals with issues of this sort. We ind two essentially equivalent approaches. One, following Jorgenson ( l 963). compares the per-period value ofan incremental unit ofcapital (its marginal product) and an percost'' that can be computed from the purchase pcriod rental cost'' ()r price- thc intercst and depreciation rates, and applicable taxes. The (irm's desired stock ofcapital is tbund by equating the marginal product and the user cost. The actual stock is assumed to adjust to the ideal. either as an ad hoc Iag process.or as the optimal response to an explicit cost of adjustment.The book by Nickell ( 1978) provides a particularly good exposition of developments of 'bequivalent
'&user
this approach.
The other tbrmulation. due to Tobin ( l 969), compares the capitalized value of the marginal invcstment to its purchase cost. The value can be obscrved directly if the ownership ()f the investment can be traded in a secondary market', otherwise it is an imputed value computed as the expected present value of the stream of proGts it would yield. 71e ratio of this to the cost) of the unit. called Tobin's ty, governs the purchase price (replacement should be undertaken or expanded if q exdecision. lnvestment investment should undertaken. and existing capital should be reduced, if it 1 be not ceeds ; < 1 The optimal rate of expansion or contraction is found by equating the q adjustment which depends on the difference of benefit, marginal cost to its T:IX rules alter somewhat, and this but the basic principle 1. between ty can of investis similar. Abel (1990) offers an excellent sulwey of this fy-theory mcnt. In alI of this, the underlying principle is the basic net presen: valuc rule. .
Ilttrollltctioll
The Option Approach
2
llusiness practice lil'td tllltt stltrl'l Iltlrtlltl tes ()r Iltlrdle--- rattl. (lbstt l-vers Illes l 11 t)t Iltlr Nvortls. lil-llls tlt) typical lu i ly llree tlle ctlst of capittl rt t or are not invest until price ristls substantially above Itlng-rtln average cllst. (311tlle downside, lirlns stay in busi ness for lengthy periotls svllile absorbi ng opktryting t)f
k-
I-:
.-'
Yh
net present value rule, however. is based on some implicit assumptions that are often overlooked. Most impllrtants it assumes that either the investment is reversible, that is, it can somehow be undone and the expenditures recovered should market conditions turn out to be worse than anticipated.
or, if the investment is irreversible. it is :1 now or never proposition, that is, if the firm does not undertake the investment now, it will not be able to in the future. Although some investments meet these conditions, most do not. lrreversibility and the possibility of delay are very important characteristics of most investments in reality. As a rapidly growing Iiterature has shown, the ability to delay an irreversible investment expenditure can profoundly affect the decision to invest. It also undermines the simple net presentvalue rule, and hence the theoretical lbundation ofstandard neoclassical investment models. The reason is that a firm with an opportunity to invest is holding an analogous to a hnancial call option it has the right but not the obligation to buy an asset at some future time of its choosing. When 11 firm makes an irreversible investment expenditure, it exercises, or its option to invest. It gives up the possibility of waiting for new information to arrive that might affect the desirability or timing of the expenditure; it cannot disinvest should market conditions change adversely. This Iost option value is an opportunity cost that must be included as part of the cost of the investment. As a result. when the value of a unit of capital is at least as large the NPV rule and its purchase installation cost'' must be modised. The value of the unit as must exceed the purchase and installation cost, by an amount equal to the value of keeping the investment option alive. Recent studies have shown that this opportunity cost of investing can be large, and investment rules that ignore it can be grossly in error. Also, this opportunity cost is highly sensitive to uncertainty over the future value of the project, so that changing economic conditions that affect the perceived riskinessof future cash tlowscan have a large impact on investment spending, Iarger than, say, a change in interest rates. This may help to explain why neoclassical investment theory has so far failed to provide good empirical models of investment behavior, and has led to overly optimistic and tax policies in stimulating forecasts of effectiveness of interest itoption''
'kills,''
''invest
lll substantially below average vllriable cost without in(ar alsla exit. Tllis disinvestmen! ducing seenls tt) conllicl vit 11sttllltlfkrd tlltloryNvill tlnctt explained irrcversibility ltntl optioll valtle arc be see. it can but as sve
Iosses. and price can
accotlnted tbr.
Of course, one can always redehne NPV by subtracting frol'n the convtl ntional calculation the opportunity cost of exercising the option to invest- and if NPV is positivo-' holds once this ct'rrection has then say t hat t he rule made. However. to do so is to accept otlr criticisnA. To highlight the inlbeen values, in this book we prefer tt) keep them separate fron) option portance of NPV'- terthe conventional NPV I f others prefer to contintle to use u-arcful option nclude all rclevllnt they tt) are minology, that is 5ne as long as ()f who readi tllat ly PV. N Readdrs usage can values in their deti nition pre fer translate our statenlents into tllat language. ()f In this book we develop the basic theory irreversible investlnent under Ginvest
'-positive
uncertainty, emphasizing the optitln-like charllctdristics ()f investnltlnt ()pptlr(lbtained fronl nlethods tunities. Nv show how optimal investment rtlles can bt2 options financial nlark in ets. Skc also dethat have been developed for pricing ()1* optimal matllematical theor,y velop an eqtlivalent approach based on the tlncertainty illustrate dynamic programming. sequentiai decisions under the optimal investment decisions of Iirms in :1 v:triety of sittlatiolls nt!w entry. determination of the initial scale of the hrm and futtlrc talstly changes of scale, choicc between diilk re nt forms of investment that offer d iftkrent dc01* succttssive stages grees of llexibility to meet future condititlns. completion of a complcx multistage project. temptlrary shutdown and rcstart. pcrmanent exit. and so forth. Nv'ealso analyze how the actions ofsuch f'irmsare aggregated to determine the dynamic eqtlilibrium of an industry. 'bgc
Iinancial assets. thtt (lppllrtunities t() Therefore optitlns.'' called this book sometimes real assets are acquire could be titled S'The Real Options Approach to lnvestment-''
To stress the analogy with options
()n
-treai
'rate
investment.
The option insight also helps explain why the actual investment behavior wisdom taught ln business schools. Firms to yield a return in excess ()f a required.
ol' firms differs from the received invest in projects that are ttxpected
zsummcrs ( l987. p. 31)())fllund hurdle nlltts ranging frtlm 8 !t) 3f) pcrcent, wlh a median :tl of 15 and a mcan of l 7 pcrcent. 7-he cost of riskless capital was mucll lllwer; ltlwing ftlr the and rcal rate cltlse tlltt 4 intcrest the nominal rate was percellt, deductibility of interest cxpenscs. i'tlr investments with a!. (s!9911 p f. I Thd. bllrlltt rilttt Jlpprtlpri:lle tc zere. See alse Dertuuztts systematic risk will exceed tht: risk Iess rate. bu t not by enough t() jusliIk the n umlltrrs used by many companies. .ct
Illtrziltctizlll
3
lrreversibility and the Ability to Wait
Before proceeding, it is important to clarify the notions of irreversibility, the ability to delay an investment, and the option to invest. Most important, what makes an investment expenditure a sunk cost and thus
irreversible'?
Investment expenditures are sunk costs when they are firm or industry specific. For example, most investments in marketing and advertising are firm specihc and cannot be recovered. Hence they are clearly sunk costs. A steel plant, on the other hand. is industry specihc it can only be used to produce steel. One might think that because in principlethe plant couid be sold to another steel company, the investment expenditure is recoverable and is not a sunk cost. This is incorrect. If the industry is reasonably competitive. the value of the plant will be about the same for all lirms in the industly so there would be little to gain from selling it. For example, if the price of steel falls so that a plant turns outa expost, to have been a ttbad'' investment for the tirmthat built it, itwill also be viewed as a bad investment by othersteel companies, and the ability to sell the plant will not be worth much. As a result, an investment in a steel plant (or any other industrpspecific capital) should be viewed as largely a sunk cost. Even investments that are not hrm or industly specific are often partly irrcversible because buyers in markets for used machines, unable to evaluate the quality of an item, will offer a price that corresponds to the average quality in the market. Sellers, who know the quality of the item they are selling, will be reluctant to sell an above-average item. This will lower the market average quality, and therefore the market price. This problem (see Akerlof, 1970) plagues many such markcts. For example, office equipment, cars, trucks, and computers are not industry specific. and although they can be sold to companies in other industries, their resale value will be well below their purchase cost, even if they are aimost new. lrreversibility can also arise because Of government regulations or institutionai arrangements. For exampie. capital controls may make it impossible for foreign (or domestic) investors to sell assets and reallocate their funds, and investments in new workers may be partly irreversible because of high costs of hiring, training, and Iiring. Hence most major capital investments are in large part irreversible. Let us turn next to the possibilities for delaying investments. Of course, srms do not always have the opportunity to delay their investments. For example, there can be occasions in which strategic cons' iderations make it imperative for a 5rm to invest quickly and thereby preempt investment by '%lemons''
zdNev Pcw
t#'
Illj-(:-lltleltt'
existingor potential competitors.3 However. in most cases. delay is at Ieast feasible. There may be a cost to delay tlle risk of entry by other tirms, or simplyforegone cash tlows but this cost must be weigled against the benehts of waiting for new intbrmation. Those benetits are oen large. As we said earlier. an irreversible investment opportunity is much like a
financialcall option. A call option gives the holder the right, forsome specified amount of time, to pay an exercise price and in return receive an asset (e.g.. a share of stock) that has some value. Exercising the option is irreversible: although the asset can be soid to another investor, one cannot retrieve the option or the money that was paid to exercise it. A lirm with an investment opportunity Iikwise has the option to spend money (the price''). in the future. in of for value. project) return Agains asset an or some now (e.g..a the asset can be sold to another tirm, but the investment is irreversible. As with the financial call option. this option to invest is valuable in part because the tkture value of the asset obtained by investing is uncertain. lf the asset rises in value. the net payoff from investing rises. If it falls in value, the tirm need not invcst, and will tnly lose what it spent to obtain the investment opportunity. The models of irreversible investment that will be developed in texercise
Chapter 2 and in later chapters will help to clarity' the optionlike nature of an investment opportunity. Finally. one might ask how firms obtain their invcstment opportunities. that is. options to invest. in the first place. Sllmetimes investment opportunities result from patents, or ownership of land or naturai resourccs. More generally, they arise from a firm's managerial resources. technological knowledge, reputation, market posititn, and possible scale, alI of which may have been built up over time. and which enable the tirm to productively undertake investments that individuals or other lirms cannot undertake. Most important. these options to invest arc valuable. Indeed, for most firms, a substantial part of their market value is attributable to thefr options to invest and grow in the future, as opposed to the capital they already have in place.'l Most of the economic and financial theory of investment has focused on how hrms should (and do) exercise their options to invest. To better understand investment behavior it may be just as important to develop better models of how firms obtain investment opportunities, a point that we will return to in Iater
chapters.
3See Gilbert ( 1989) and Tirolc ( 1988, Chapter 8) for surveys of the literature on such strategi aspects of investlnegt. 4For discussions of growth options as sources of firm value, sec Mycrs ( 1977), Kester ( 1984). and Pindyck ( 1988b).
Illtrotlltctioll
4
An Overview of the Book
In the rest of this chapter, we outline the plan of the book and give a flavor of some of the important ideas and results that emerge from the analysis. 4.A
A Few Introductory
Examples
The general ideas about treal options'' expounded above are simple and intuitive, but they must be translated into more precise models betbre their quantitative signiEcance can be assessed and their implications for firms, industries, and public policy can be obtained. Chapter 2 starts this program in a simple and gentle way. We examine a firm with a single discrete investment of two decision periopportunity that can be implemented within a of undergoes ods. Between the two periods, the price the output a permanent shift up or down. Suppose the investment would be profitable at the average price, and therefore a tbrtiori at the higher price. but not at the Iower price. By postponing its decision to the second period, the firm can make it having observed the actual price movemcnt. It invests if the price has gone up. but not if it has gone down. Thus it avoids the loss it would hjwe made if it had invested in the hrst period and then seen the price go down. This value of waiting must be traded off against the loss of the period- l profit flow. The result-the decision to invest or to wait-depends on the parameters that specify the model. most importantly the extent of the uncertainty (whichdetermines the downside risk avoided by waiting) and the discount rate (which measures the relative importance of the future versus the present). We carry out several numerical calculations to illustrate these effects and build intuition for real options. We also explore the analogy to financial options more closely. We introduce markets that allow individuals to shift the risk of the price going up or down, namely contingent claims that have different payoffs in the two eventualities. Then we construct a portfolio of these contingent claims that can exactly replicate the risk and return characteristics of the hrm's real option to invest. The imputed value of the real option must equal that of the replicating portfolio, because othemise there would be an arbitrage opportunity an investor could make a pure profit by buying the cheaper and selling the dearer of the two identical assets. We also examine some variants of the basic example. First, we expand window of investment opportunity to three periods, where the price can the down between periods 2 and 3 just as it could between periods 1 go up or and 2. We show how this changes the value of the Option. Next, we examine uncertainty in the costs of the project and in the interest rate that is used to 'window''
discotlnt futtt re p rtlh-t Ilows. F inallv. &ve consiler clloictl bc tNveel) prtlittcts (-81* differen t scltles. 11 Iarce r oroiect llllvil1c hicller Iixed u'tlsts btlt ltlvver()7e rtttilltt costs. Even with tllese extensions
antl variations,
the analysis remaills
at the
level of an illustrative and very simple example rather than that of a theory with some claim to general ity. In Iater chapters we proctrtld t() ulevelop a broader theoretical framework. Btlt the example does yield some valuable insights that survive the generalization- and we summarize tllcm ht, re. First, the example shows that the opporttlnity cost of thk! option to invest
is a significant component of the firnl's investment decision. The option value increases with the sunk cost of the investment and with the degree of uncertainty over the future price, the downside component of the risk btling the most important aspect. These restllts are conlirmcd in more general models in Chapters 5-7. Second- we wiIl see that the optilln va Itle is not affected if t he firm is able the risk by trading in forward or futtlres marke ts. In t2flicien t markets hedge to risk is fairly priced, so any decrease in risk is offset by the decretstl in such The fonvard transaction is a financial operation that Ilas nt) eflkct ()n return. firm's real decisions. (Tl1is is another tlxample the of the Nlodigliani-M iIlt!r
theorem at
work
.
)
Third, whcn future costs arc tlncertltin- thoir dflktzt on the investment decision depends ()n thc particular llrm t)f the tlnccrtainty. l 1*tlle uncertainty pertains to tlle pricc the Iirm must plty for :111 inptlt. the eftkct is jtlst like that of output price risk. The freedom not to invest if the input price turns out t() have gone up is valuabld, st) immediat.e investment is Iess readily matle. l-lowever, instcad suppose that the project consists of several steps. the uncertainty pertains to the total cost of investment. and intbrmation about it will l7e rttvealed only as the first few steps of the project are undertaken. Then these steps have information value twer and above their contribution t() the c()nventionally calculated NPV Thus it may be desirable to start the project evcn if orthodox NPV is somewhat negative. Nvereturn to this issue in Chapter l () and model it in a more general theoretical framewllrk. Fourth, we will see that investment t)n a smaller scale. by increpsing future flexibility,may have a value that offsets to some dcgree the advantage that a larger investment may enjoy due to economies t)f scale. 4.B
Some Mathematical Tools
In reaiity, investment prqects can have ditlerent windows ol opportunity, and of the future can be uncertain in different ways. Thereforc the
various aspects
Itltrltittf:tit)tj
simple two-period examples of Chapter 2 must be generalized greatly before tlley can be applied. Chapters 3 and 4 develop the mathematical tools that are needed for such a generalization. Chapter 3 develops more general models of uncertainty.
We start by
explaining the nature and properties of stochastic processes. These processes combine dynamics with uncertainty. In a dynamic model without uncertainty, the current state of a system will determine its future state. When uncertainty is added. the current state determines only the probability distribution of future states, not the actual value. The specifcation in Chapter 2, where the current price could go either up or down by a Iixed percentage with known probabilities, is but the simplest example. We describe two other processes that prove especially useful in the theory of investment Brownian motion and Poisson processes-and examine some of their properties. Chapter 4 concerns optimal sequential decisions under uncertainty. We begin with some basic ideas of the general mathematical technique for such optimization: dynamic programming. We introduce this by recapitulating the two-period example of Chapter 2. and showing how the basic ideas extend to more general multiperiod choice problems where the uncertainty takes the form of the kinds of stochastic processes introduccd in Chapter 3. We establish the fundamental equation of dynamic programming, and indicate methods of solving it for the applications of special interest here. Then wc turn to a market setting, where the risk generated by the stochastic process an be traded by continuous trading of contingent claims. We show how the sequential decisions can be equivalently handled by constructing a dynamic hedging strategy a portfolio whose composition changes over time to replicate the return and risk characteristics of the real investment. Readers who are already familiar with these techniques can skip these chaptersaexcept perhaps for a quick glance to get used to our notation. Others not familiarwith the techniquescan use the chapters as a self-contained introduction to stochastic processes and stochastic dynamic optimization-even if their interests are in applications other than those discussed in the rest of the
book.
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uncertainty. Then the investment decision is simply the decision to pay the sunk cost and in return get an asset whose value can lluctuate. This is dxactly analogous to the financial theory of call options the right but ntlt the oblivalue tbr a preset exercise price. gation to purchase an asset of ouctuating
Theretbre the problem can be solved directly using the techniques developed in Chapter 4. The result is also familiar from financial theory. The option can is the money'' . when the value of the asset rises be profitably exercised above the exercise price. However, exercise is not optimal when the option is onlyjust in the money, because by exercising it the lirm gives up the opportunity to wait and avoid the Ioss it would suffer should the value fall. Only when the value of the asset rises sufsciently above the exercise price the option is in the money''--does sufhciently its exercise become optimal. An alternative tbrmulation of this idea would help the intuition of economists who think of investment in terms of Tobin's q. the ratio of the value of a capital asset to its replacement cost. In its usual interpretation in the investment literature. the value of a capital asset is measured as the expected present value of the prolit tlow it will generate. Then the conventional criterion for the firm is to invcst when (/ equals or exceeds unity. Our option value criterion is more stringent', q must cxceed unity by a sumcient margin. It must equal or excced a critical or threshold value (?*. which itself excceds unity, beibre investment becomcs optimal. We perform several numerical simulations to calculate the value of the option and the optimal exercise rule, and examine how these vary with the amount of the uncertainty, the discount rate. and other parameters. We (ind that tbr plausible ranges of these parameters, the option value effect is quantitativelyvery important. Waiting remains optimal even though the expected rate of return on immediate investment is substantially above the intcrest rate ()k' return on capital. Return multiples of as much as rate or the two or three times the normal rate are typically needed before the 5rm will exercise its option and make the investment. 'in
Gdeep
wtnormal''
4.D The Firm's
lnvestment
Decision
These techniques are put to use in the chapters that follow. Chapters 5-7 constitute the core theory of a hrm's investment decision. We begin in Chapter 5 by supposing that investment is totally rreversible. Then the value of the project in place is simply the expcctcd prcscnt valuc of th stream of prolits (or losses) itwouid generate.This can be computed in terms of the underlying
Interest Rates and Investment
Once we understand why and how (irms should be cautious when deciding whether to exercise their investment options, we can also understand why interest rates seem to have so Iittle effect on investment. Econometric tests of the orthodox theory generally 5nd that interest rates are only a weak or insignihcant determinant of investment demand. Recent history also shows that inlerest rate cuts tetld to llavr unly a limited stimuiative effect on investment', the experience of 1991-1992 bears the latest witness to that. The
IntroflLtctiolk
'
options approach offers a simple explanation. A reduction in the interest rate inakes the ftlture generally more important relative to the present, but this increases the value of investing (the expected present valtle of the stream of proEts) and the value of waiting (the ability to reduce t)r avoid the prospect
of tkture losses) alike. The
net effect is weak and sometimes even ambiguous. real apprtach also suggests that various sources of unThe options
certainty about future prolits quctuations in product prices, input costs, exchange rates, tax and regulatory policies have much more important effects on investment than does the overall Ievel of interest rates. Uncertainty about the future path of interest rates may also affect investment more than the general level of the rates. Reduction or elimination of unnecessary uncertainty may be the best kind of public policy to stimulate investment. And the uncertainty generated by the very process of a lengthy policy debate on alternatives may be a serious deterrent to investment. Later, in Chapter 9, we construct specisc exaniples that show how policy uncertainty can have a major negative effect on investment. 4.E
Suspension and Abandonment
Chapters 6 and 7 extend the simple model. As future prices and/or costs Iluctuate, the operating profit of a project in place may turn negative. In Chapter 5 we assumed that the investment was totally irreversible, and the hrm was compelled to go on operating the project despite Iosses. This may be true of some public selwices, but most firms have some escape routes available. Chapters 6 and 7 examine some of these. In Chapter 6 we suppose that a lossmaking project may be temporarily suspended, and its operation resumed Iater if and when it becomes profitable again. Now a project in place is a sequence of operating options; its value must be found by using the methods of Chapter 4 to value all these operating options. and then discount and add them. Then the investment opportunity is itself an option to acquire this compound asset. In Chapter 7 we begin by ruling out temporary suspension, but we allow permanent abandonment. This is realistic if a live project has some tangible ()r intangible capital that disappears quickly if the project is not kept in operation-mines Ilood, machines rust, teams of skilled workers disband, and brand recognition is Iost. If the hrm decides to restart, it has to reinvest in alI these assets. Abandonment may have a direct cost; tbr example, workers may have to be given severance payments. More importantly, however, it often has aIl uppurtunity cusi-ihe loss uf the optiun to preserve the capitai so it can be used prohtably should future circumstances improve. Theretbre a firm with
Nvill
tlllt!ratc some Iosses to k'ektp th is opt ion a proiect in place sufficientlyextren'le losses will il'ldtlctl it to abandon.
al ive, and ollly
have he re an interl i1) ketl pa ir of options. svlltlll t he fi 1-11-1exe rcises its option ttl invest, it gets :1. proiet:t i11 place antl an opt il'n to abandtln. If it exercises the option to abandllll. it gets the option to invest again. The two options must htl priced simultltndously to determine tlltl opti nlal investment and abandonment policies. Tlle link age has important implitrations', for examples11 higher cost t)f abandonment makes the firm even more cautious about investing, and vice versa. 'We illtlstrate tlle theory using numbers that are typical of the copper industry. and tind a very wide range of fluctuation of the price of copper between the investment and abandonment thresholds. 'We also consider an intermediate situation. where both suspension and at di fferen t costs. A project in suspension requ ires abandonment are llvailable expenditurt't. ongoing for exannplc, keeping a ship Iaid up. but restarting some is cheap. Abandonment saves on the rnaintenltnce cost. and may even bring some immediate scrap value. but then the l'ull investment cost must be incurred again if proti t potential recovers. Now we must determine the optimal switches between tllrce altcrnlltives an idle jirm, an operating project. and project. suspended We tllis. antl dk) illustrate it lbr the case ()f crudd oil a I n tct
&ve
tankers. 4.F
Temporary
versus
Permanent
Employment
While most ofour attention in this book is k)n tirms-capital investment choices. similar considcrations apply to their hiring and Iiring decisions. Each of these choices entails sunk costs, each decision must be made in an uncertain environment, and each allows some freedom 01' timing. Thereforc the above ideas and results can be applied. For exampltt, a new worker will not be hired until the value of the marginal product of labor is sufliciently above the wage rate, and the required margin or multiple above the wage will be higher tlle grcater the sunk costs or the greater the unccrtainty. The U.S. Iabor market in mid- 1993 offered a vivid illustration of this theory in action. As the economy emerged from the rccession of the early 199()s.hrms increased production by using overtime and hiring temporary workers even for highly skilled positions. But permanent new full-time hiring was very slow to increase. The current Ievel of prolitability must have been high, since firms were willing to pay wage premiums of 5() percent for overtime alld io usc ugeflcies that suppiied temporar.ywgrkors and charged fees Of wufk 25 percent or more of the wktge. But these same firms were not willing to make
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the commitment involved in hiring regular new workers.s Our theory offers 11 natural explanation for these observations. A high level of uncertainty about future demand and costs prevailed at that time. The robustness and durability of the recovery were unclear-, it was feared that inllation would return and lead the Federal Reserve to raise interest rates. Future tax policy was very uncertain, as was the level of future health care costs that employers would have to bear. Therefore we should have expected firms to be very cautious, and wait for greater assurance of a continued prospect of high Ievels ofprofitability before adding to their regular full-time labor force. in the meantime, they would prefer to exploit the current proit opportunities using less irreversible (even if more costly) methods of production, namely overtime and temporary work. That is exactly what we saw. 4.G
IcpTf
The path dependence can leatl to the lbilowing kind of sequence ot' events. When the tirm (irst arrives on the scene and contemplates investment. the current protit is in the intermediate range between the two thrcsholds.
Theretbre the lirm decides to wait. Then proiit rises past the upper thresllold.
so the lirm invests. Finally. profit falls back to its old intermediate Ievcl. btlt that does not take it down to the Iower threshold where abandonment would occur. Thus the underlying cause (currentproiitability) has been restored to its old level. but its effect (investment)has not. Similar eftkcts have Iong been known in physics and other sciences. Tlle most familiar example comes from electromagnetism. Take an ironsar andt' loop an insulated wire around it. Pass an electric current throtlgh the wire--k the iron will become magnetized. Now switch the current off. The magnetism is not completely lost; some residual effect remains. The cause (the current) was temporary, but it Ieaves a longer-lasting eftkct (the magnetized bar). This phenomenon is called Ilvsresis. and by analogy the failure of investment decisions to reverse themselvcs when the underlying causes are fully reversed can be called cconomic hysteresis. A striking example occurred during the l 980s. From l98() to I984. thtt dollar rose sharply against other currencies. The cost advantage of foreign lirms in US markets became very substantial. and ultimately Ied to a Iarge rise in US imports. Then the dollar fell sharply. and by l987 was back to its l t)8() Ievel. However, the imptlrt penetration was not fully reversed: in fact it hardly decreased at :111. It took :1 largcr fall in the dollar to achieve any signihcant rcduction in imports.
''
Hysteresis
When we consider investment and abandonment (orentry and exit) together. a firm's optimal decision is characterized by two thresholds. A suffciently high current level of proEt, corresponding to an above-normal rate of return on the sunk cost, justilies investment or entry, while a sufficiently large Ievel of current loss Jeads to abandonment or exit. Now suppose the current level of protit is somewhere between thcse two thresholds. Will we see an active firm? That depends on the recent history of protit iuctuations. If the protit is at its current intermediate level having most recently descended from a high level that induced entry. then there will be an active firm. However, if the intermediate Ievel was most recently preceded by a 1owIevel that induced exit, then there will not. In other words, the current state of the underlying stochastic variable is not enough to determine the outcome in the economy'. longer histor.y is needed.a-he economy isgathdevendent. ideaofpathdependence and illustrated, hasbeen recentlyexplored rominently and by Artlaur 19aa). p (l9a6) oavid(19as, vheyallow an possibility: evenverylong-run propertiesof theirsystems more extreme altered by slight differences in initial conditions. uere we have a more kind of path dependence.-rhe long-run distribution of the possible o t economy is unaltered, but ti:e short- and medium-n,n evolution still be dramatically affected by initial conditions.
4.H 1 .1
a 'rhe t mos even are moderate stateshe can
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.
Industry Equilibrium
In Chapters 8 and 9 the focus turns from a hrm's jnvcst mtl n t decisit) ttl ns the equilibrium ot a wbolu industry composcd ot many such urms. o ne.s srstreaction might be tat tlae competition among urmswill destroy anyone firm's option towait- eliminating the efocts ofirreversibilty and uncertainty that we found in 5-7. does each destroy firm's option chaptcrs competition to wait, but this does not restore the present value approach and results of the orthodox theoe. on tlae contrary, caution when making an irreversible decision remains important- but different reasons. orsomewhat Iirmcontemplating its investment-knowingthat the considerone oture of induste demand and its own costs are uncertain, and knowing that thereuqre many other firms facing similar decisions with similar uncertainty. The Iirrlais ultimatelyconcerned with tue.eonsequences of its decision for its Own profift, but it must recognizc htok't'he similar-dcisions of otllrr Iirms will affect it. In this respect, two types ofcertainty must 6'v distinguished because q j j
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they can have different implications for investment: aggregate uncertainty that affects all hrms in the industly and Iirm-specilic or idiosyncratic uncertainty facing each Iirm. To see this, tirst suppose that investment is totally irrcversible, and consider an industrrwide increase in demand. Any one Iirm expects this to lead to a higher price, and so improve its own prolit prospects, making investment firms are making more attractive. However. it also knows that several other effect of the dea similar calculation. Their supply response will dampen the mand shift on the industry price. Therefore the upward shift of its own protit potential will not be quite as high as in the case where it is the only rm and has a monopoly on the investment opportunity. However, with investment being irreversible, a downward shift of industry demand has just as unfavorable an eftkct in the competitive case as in the monopoly case. Even though other competitive hrms are just as badly affected, they cannot exit to cushion the fall in price. Thus the competitive response to uncertainty has an inherent asymmetry: the downside exerts a more potent effect than the upside. This asymmetry makes each tirm cautious in making the irreversible investment.
The ultimate effect is very similar, and in some models identical, to that ofthe option value for a firm possessing a monopoly on the investment opportunity and waiting to exercise it. In fact, the theory of a competitive industry can be formulated by giving each tirm the option to invest, valuing these real options the competitive as in Chapters 5-7, and fnally imposing the condition that in should value be option this zero. equilibrium If we allow some reversibility, the exit of other lirms does cushion the effect of adverse demand shocks on price. But then each firm*s exit decision recognizes this asymmetric effect of demand shocks in an initially poor situation: their upside effect is more potent than the downside. Thus competitive firms are not quick to leave when they start to make operating losses; they wait a while to see if things improve or if their rivals leave. The overall effect is just like that for a single monopoly firm's abandonment decision that we found in Chapter 7. ln fact, the competitive equilibrium model of joint entry and exit decisionswith aggregate demand shocks thatwe analyze in Chapter 8 has exactly the same critical Ievels of high and Iow prices to trigger entry and exit as the corresponding monopoly model of Chapter 7. Firm-specific uncertainty does not lead to this kind of asymmetly. lf just experiences a favorable shift of its demand, say some idiosyncratic hrm one switch of fashion, then it knows that this good fortune is not systematically shared by other Erms, and therefore does not fear tht entry ofother firmswill erode its prol'it potentiai in the same way. However, then llc vuluc uf waitillg reemerges in the older familiar form. The lucky hrm does have a monopoly
./4 Nebv P'c! $.' (
)j-
11l l
'(j'.b'tlt
ltv l
t
on the opporttlnity to enter witll its Iovvcost. Tl'lcrt.lt'oreit alstl llas 1111 optitln value Of waiting-v it can thereby avoid a Itlss if its Itnv cost sllould ttlrn otlt to be transitory. Tllus tirm-specitictlncertainty in industry equilibritlnl also Ieads to investment decisions si milar lo those fotl ntl in Cllaptc rs 5-7 l'or :111 isol a tetl srrn. 4.I
Policy Toward Investment
Some readers might interpret the rcsult that uncertainty makes li rnls Iess eager to invest as indicating a need for government policy intervention to stimulate investment. That would be a hasty reaction. A social planner also gcts information by waiting, and therefore should also recognize tlle opportunity cost of sinking resources into 11 project. A castl rorpolicy inte rvention will arise only if firms face :1 diftkrcnt value (.)f waiting than does society as :1 whole. in other words, if some market failure is associated with the decision
process.
Chapter 9 focuses on these issues. Our lirst result is :1 confirmatitln
01'
the standard theory of gene ral eq uil ibrium. lf markots for risk are complctc, and if Iirms behave as competitive price- takers (in this stochastic dynanlic context this must be interpreted to mean that ttach firm takes as given the stochastic proccss of thc pricc and has ratillnal expectations about it). then the equilibrium evolution ol' the industry is stlcially cfficient. A stlcial planner would show the same degree of llesitancy in making the investment dccision. If markets for risk are incompletc, bcnelicial plllicy intervtlntions dt) exthe correct policy needs some carcful calculation and implemttntlltion. The biunt tools that are often used for handling unccrtainty can havc adverse ()1* . effects. We illustrate this by uxamining the consequences prce ccilings and floors. For example, price supports promote investment by reducing the downside risk. However, the resulting rightward shift of the industry supply function implies lower prices in good times. Averaging over gotld times and bad, we tind that the overall rcsult is a Iower long-run averagtt price. In othllr words, the polic-yis harming the very group it sets out t() hclp. Pricc tll.ltlror ceiling policies. for example, urban rent controls and agricultural prictl stlpports. are usually criticized because they reduce overall econtlmic efficiency. Our finding is perhaps a politically more potent argument against thtm: their distributional effect can be pelwerse, too. We also study the effect of uncertainty concerning future policy itself. For example, if an investment tax credit is being discusscd, firms will recognize iiiot'e kalue 11'1wl'tl'llg, bcuausu llt;lc i: u pl ubabiliiy tilat the ctlst t)iinvestment to the firm will fall. We hnd that such policy uncertainty can have
ist, but
llttroelltctiolt
,,4New Iztl'v of
wish to a powerful deterrent eftkct on immediate investment. lf gtwernments spend a long stimulate investment, perhaps the worst thing they can do is to right way to do so. time discussing the
4.J Antitrust
Nrmscontinue their export operations at :1 loss, domestic firms allege predatory dumping and call for the standard trade policy response of countervailing import duties. However. our analysis suggests that the foreign firms may be simply and rationally keeping alive their option of future operation in our market, with no predatory intent whatever. Only a sufficiently Iong time series of data will allow us to test whether the supposed collusive or predatory actions are merely natural phases in the evolution of a competitive industry or genuine failures of competition.
and Trade Policies
Chapters 8 and 9 paint a very diftkrent picture of competitive equilibrium than the one familiar from intermediate microeconomics textbooks. There lhe industry if the price rises to equal the we are told that firms will enter long-run average cost, and they will exit if the price falls as low as the average variable cost. Our theory implies a wider range of price variation on either side. For example. in the face of aggregate uncertainty, firms' entry as soon will not constitute an industry as the price rises to the long-run average cost equilibrium. Each tirm knows that entry of other similar firms will stop the price from ever rising any higher. while future unfavorable shifts can push the price below this Ievel. Also, a future price path that sometimes touches the long-run average cost and otherwise stays below this level can never offer normal return on the firm's investment. Only if the price ceiling imposed
a by entry is strictly above the Iong-run average cost can the mix of intervltls of supernormal proit and ones of subnormal profit average out to a normal return. Similarly, tirms will exit only when the price ills sufhciently far below the average variable cost. They will tolerate some losses, knowing that thc cxit of other firms puts a Iower bound on the price. The equilibrium level of this qoor is determined by averaging out the prospects of future losses and profits to Zero.
Thus we hnd that competitive equilibrium under uncertainty is not a stationary state even in the long run, but a dynamic process wherc prices can luctuate qute wfdely. Periods of supernormal profits can alternate with perfods of losses. A similarviewofdynamic equilibrium as astochastic process has become quite common in macroeconomics, but is sumrisingly uncommon in microeconomics. particularly with regard to its implications or antitrust polic'yor international trade policy. The conceptual framework of such policies in practice are based on obseris generally static, and the recommendations particular of of industry instant. We :nd that the at a vations an rethinking of view substantial both the theory and the calls for a dynamic tsnapshots''
practice.
For example, in industrial organization theory, excess prohts suggest collusion or entry barriers, calling for antitnlst action. ln our dynamic perspective, substantial prriods of supernormal prohts w'ithout ncw cntry can occur cvcn though al1 hrms are small price takers. In international trade, when foreign
J?l1.t;'./?'?7t.vIl
4.K
. :: ..
Sequential and Incremental
Investment
In Chapters 10 and l 1 we return to a single tirm-s investment decision and examine some other aspects of it that are important in applications. Chapwhich must ter 10 deals with investments that consist of several stages, aIl of be completed in seuence before any output or prolit flow can commence. The 5rm can constantly obselwe some indicator of the future protit potential, and this fluctuates stochastically. At any stage. the firm may decide to continue immediately, or wait for conditillns to imprtwc. At an early stage ()f the investment sequence, most of the cost remains to be sunk. Therefore the will go ahead with the program only if it sees a sufficiently high threshold 517)1 level of the indicator of profitability. Gradually. as more steps are completcd and less cost rcmains to be sunk, the next step is justified by ap ever smaller threshold. In this sense, bygones affect the decisions to come. In Chapter 10 we examine another effect of current decisions on the f'uture, namely the learning curve. According to this theory, the cost ofproduction at any instant is a decreasing function of cumulated output experience. Thus the current output t1()wcontributes to a reduction in alI future production costs. This additional value must be added to the current revenues before comparing tem to the current costs of production to decide on the optimal level of production. We examine the dynamic output path under these conditions. We 5nd that greater uncertainty lowers the value of future cost reductions, and that Ieads to a reduction in the pace of investment. In Chapter 11 we turn to the study of incremental investment, where outand profit flow are available al1 the time as a I'unction ot' the installed stock put capital. The aim is to characterize the optimal policy for capacity expansion. of When production shows diminishing returns to capital, we can regard each new unit of capacity as a fresh project, which begins to contribute its marginal product from the date of its installation. Then the criterion derived in ChapiIl tel 6 for illvrstlllellt suel'l it pfojet contknues to apply. If pi'oduction shows increasing returns to capital over an interval, then all the units of capacity in
22
Iltlrodttctiolt
:tn appropriately constructed range must be regarded as a single project. and the criterion tor its installation is again a natural generalization of thpt tbr a .j, single project described in Chapter 6. When the Iirm can choose its rate of capacity expansion, we must specify'
how the costs of this expansion depend on its volume and pace. Different assumptions in this respect imply different optimal policies. We construct a general model that places the alternatives in context. and in particular shows the relationship between the adjustment costs models that have been the mainstay of theoretical and empirical work over the last decade and the irreversibility approach that has been the focus of our book. Empirical
and Applied Research
In Chapter 12 we turn to some examples that illustrate applications and extensions of the techniques developed throughout the book. We also discuss the relevance of the theory for empirical work on investment behavior. We begin Chapter 12 with a problem of great interest to oil companes how to value an undeveloped offshore oil reserve, and how to decide when to invest in development and production. Aswewill see, an undeveloped reserve is essentially an option; it gives the owner the right to invest in development of the reserve and then produce the oil. By valuing this option we can value the reserve and determine when it should be developed. Oil companies regularly spend hundreds of millions of dollars for offshore reserves, so it is clearly important to determine how to value and best exploit them. We then turn to an investment timing problem in the electric utility industry. The Clean Air Act calls for reductions in overall emissions of sulfur dioxide, but to minimize the cost of these reductions, it gives utilities a choice. They can invest in expensive to reduce emissions to mandated Ievels, or they can buy tradeable that Iet them pollute. There is considerable uncertainty over the future prices of allowances, and an investmcnt in scrubbers is irreversible. The utility must decide whether to maintain Pexibility by relying on allowances or invest in scrubbers. We show how this problem can be addressed using the options approach of this book. To show how the principles and tools developed in this book have relevance beyond hrms' investment decisions, we address a problem in public policy when should the government adopt a policy in response to a threat to the environment, given that the future costs and benehts of the policy arc uncertain? We will argue that the standard cost-benefit framework that cconomists havc traditionally uscd to uvaluate envii-onnpental policies is dellcient. The reason is that thcre are usually important irreversibilities associated with environmental policy. These irreversibilities can arise with respect to '<scrubbers''
'allowances''
environlnental dnlnage itself. btlt also with respect to tlld costs of adapting t() policies to reduce the daluage. Since tlltr atloption of ::n envinlnnlental policy is rarely a now or ntwer proposition. the sanle tdchniques used to sttldy tllt.l to the optinlal tim ing of an optimal ti m ing ot' an investment can be lpplied environmental policy. At the end of Chapter 12. we disctlss some of the enlpirical inlplicat itlns
of irreversilnility nnd unccrtainty for investl-nent behavior. Tllere is trollsitltlrable anecdotal evidence that firms make investment decisions in :1 way tllat is at least roughly consistent with the theory developed in this book. tbr tlxalmple, the use of hurdle rates that are much larger than the opportunity txlst of capital as predicted tnythe capital asset pricing model. Some ntlmdrical simulations based on plausible parameters for costs and tlncertainty are found to replicate features of reality. However, more systematic econolnetric testing of the theo!y is still at a very early stage. Nve review some work of this kind. point out its difficulties, and suggest ideas for ftlture research.
5
Noneconomic Applications
Our tbcus in the book will be ()n investment decisions implications for industry equilibrium. Tllese matlers are
()f
lirms antl their
01'
primary interest
to economists. their conditions (technologyand rcstlurcc avktilability) antl of the value t)f the firm) arc gcnerally agrced uplln, antl criteria (maximization theory leads to quantiliable predictions about them. lnvestment decisions of consumers (purchas:ofdurablcs) and workers (educationand human capital) have obvious parallels. Research on these issues already cxists. and wtl will sketch some of this literature in Chapter I 2. Many other personal and societal choices are made undcr the same basic
conditionss namely irreversibility, ongoing uncertainty, and some leeway in timing. Therefore we can think about them in terms of some tlf the models and results outlined above, and offer some qualitative speculations. We believe there is scope for more seriotls research along these Iines. This would amount to extending work such as that of Becker ( 1975. l 98()) on econtlmic approaches to social phenomena by incorporation of option values. Indecd, some recognition Qf this aspect can be read nto Becker's own remarks on the subject, for example, in Becker (1962,pp. 22-23), but a more thorough and formal analysis along these lines remains a potentially fruitful project tor the future. Although we cannot spare much splce in an already lengthy book, we believe that readers will Iind some speculations along tllese Iines both interesting and thought-provoking.
Introdttciotl
5.A Marriage and Suicide Marriage entails signilicant costs of courtship, and divorce has its own mon-
etary and emotional costs. Happiness or misery within the marriage can be only impertkctly tbrecast in advance. and continues to lluctuate stochastically even aler the event. Therefore waiting tbr a better match has an option value, and we should expect prospective partners to look for a sufhciently high initial threshold of compatibility to justifygetting married. We should expect the option value to be higher in religions or cultures where marriage is less reversible. Therefore we should expect that. other things being equal, individuals in such societies will search more carefully (andon the average, search Ionger), and will insist on a higher threshold of the quality of the match. On the contrary. when divorce is easy. we should see couples entering into marriage (or equivalent arrangements) more readily. Of course. other things are not equal. Societies that make divorce more
difficult presumably attach higher value to marriage. Therefore we should expect them to counteract the rational search and delay on the part of individuals technologies.'' Emby social pressure. and provision of bettcr of these will carefully ideas designed have be tests to pirical to distinguish the and only raises the interest and opposing often influences. but that separate research. such of challenge ''matchmaking
Perhaps the most extreme example of economics applied to sociological phenomena is the Beckerian theory of suicide developed by Hamermesh and Soss (1974). According to them. an individual will end his or her own life when expected the present value of the utility of the rest of Iife falls short ofa benchstandard cutoff Most people react by saying that the model mark or view rational of what is an inherently irrational action. gives an exccssively opposite. Whatever its merits or demerits as the Our theory suggests exactly descriptive theofy, the Hamermesh-soss model is not rational tr/ltll/q/l from the prescriptive viewpoint. because it forgets the option value of staying alive. Suicide is the ultimate irreversible act, and the future has a lot of ongoing will uncertainty. Therefore the option value of waiting to see if improve'' should be very large. The circumstances must be far more bleak than the cutoff standard of the Hamermesh-soss model to justifypulling the trigger. This is true even if the expected direction of life is still downward; all that is needed is some positive probability on the upside. Now return to the argument that most suicides are irrational, and ask exactly how they fail to be rational. There are' several possibilities, but one seems especiaiiy pertinent. Suicidesproject the bieak present into an equaiiy bleak future, ignoring uncertainty, and thereby ignoring the option value of 'tzero.''
ttsomething
,4 tge.v Piku,
tp-/' lllb
'tl-//'?lt.vl/
life. Then religious or social proscriptions against suicide selwe a useftll ftlllction as measurcs to compensate for this tiltlre of rationality. By raising the perceived cost of the act. these taboos lower the threshold of quality of litk that leads to suicide when option value is ignored. This can correct the failure of the individuals' forethought, and bring their threshold in contbrmity with the optimal rule that recognizes the option value. 5.B
Legal Reform and Constitutions
Finally, consider law reform. Different t-undamentalconstitutional or legal principles often conllict with one another in specific contexts', for examplethe civil rights of the accused often stand in the way of better enforcement of law and order. Public opinion as to the appropriate rdlative weight to be attached to these contlicting principles can shift over time. How quickly sllould the 1aw respond to the latest swing of opinion'? G iven some legislative antl administrative costs of changing laws. our theory suggests that the option to wait and see if the trend ()f opinion will reverse itself has some valutl. Reform should be delayed until the forcc of current opinion is sufticicntly overwhelming to offset this value. But there is good reason to believe that market failure.-' At the proccss is subject to a kind of myopia or all timess people and politicians tend to believe that at last they have got it right; the balance ()f opinion that has just been reached will prevail tbrcvor. Thereforc they will ignorc futurc unccrtainty and (lption values, nnd change the laws too frequently. The time to anticipate and defcat this tendency is when the cllnstitution is framed. Knowing that futurc generations will be too trigger-happy in changing laws. the founding fathers can artilicially raisc the cost ()f change, thereby making the politically flawed thresholds for Iegal changc coincide with the truly optimal thresholds. Thus various supermajority requirements for c()nstitutional change can be seen as a commitment device that corrccts for the shortsightedness of future generations. These are just a few examples of how the theory and ttlchniques ddveloped in this book are applicable to some fairly far-reaching problems. Wk have deliberately stated them in speculative and provocative ways. hoping to attract interest and research along these Iines from a broader spectrum of social scientists. In te rest of the book, we will focus largely on investmenta but ask the reader to keep in mind the much broader potential applicability of the thcory. t&ptllitical
Illttlttllillg
Chapter
2
Developing the Concepts Through Simple Examples
illvestnldllt Ilrl'jl'llt'tllls il1clllltintllptls tillle. Fillktlly. ve hvill stage for ilxlhn-lille 01* t l1eillvestlllellt vllell tlle illvestlnent tlecisitlns brielly (:lst'p:ll'!tl tlnccrttti posed to its payllff) is n- when futtl 1.t? i1-1ttt rest rates 1.1re tlllce rt :1 111.().t
when one must choose the scllle
01-
an investnlent.
1 Price Uncertainty Lasting 'lWo Periods Consider a firm thllt is trying to ddcide whether to invest in :1 widget factory. The investment is colnpletely irreversible the facto!y can only l7e tlsed to make widgets, and Should the nlarket for wdgets evaporate. t he firlu cannot teuninvest''and recover its expenditure. To keep Inattors as silnple :'s possible. we will assume that the factory can be built instantly, at a cllst / and will produce one witlget per year forever. with zero operating cost. Currently the price of a widget is $2()()-but ncxt year the price will change. Witll prtlbltbility t/it will rise to $3()().and with probability ( l t/ ). it will 1*1111 to $ 1()(). Tlpe price will then remain at this new level tbrever. (stltlFigtlre 2. l Again. to keep things simple, we will assunle that the risk ('ver tlle futtlre price of widgcts is fully divcrsitiable (that is. it is tlnrdlatcd tt) wllat llappens with the overall economy). I-lence the Ii rln slltltlltl tliscoullt futtl rc cash sows tlsing the risk-frce ratc ()f interest. wI1ich wtl will tlk'e t() be l () -
-
.)
FlltMs
MAKE,
implement, and sometimes revise their investment decisions con-
tinuously through time. Hence much of this book is devoted to the analysis problems. Idowevdr, it is best to of investment decisions as continuous-time begin with some simpl examples, involving a minimal amount of mathematics, in which investment decisions are made at two or three discrete points in time. In this way, we can convey at tle outset an intuitive understanding of tlf the basic concepts. In particular, we want to show how the irreversibility and how it requires an investment expenditure affects the decision to invest, modihcation of the standard net present value (NPV) rule that is commonly taught in business schools. We also want to show how an opportunity to invest is much Iike a hnancial option, and can be valued and analyzed according.
This chapterbegins by discussing these ideas in the context of a simple example in which an investment decision can be made in onlyone oftwo possible periods now or next year. We will use this example to see how irreversibility creates an opportunity cost of investing when the future value of the project is uncertain, and how this opportunity cost can be accounted for when making the investment decision. We will also see how the investment decision can be analyzed usng basc option pricing techniques. We will examine the characteristics of the firm's option to invest in some detail, and see how the value of that option, and the investment decision, depend on the degree of uncertainty over the future value of the project. Extending the example to three periods will provide more insight into the problem o investment timing, and set the
26
percent.
For the time being we will set / ().5. (Lalcr we will slltl 1($1)()and t/ how the investment decision depends ()11 / and t/ G ivcn tlltlstl vlllutls Iklr / and q, is this a gool.l invcstment'? Should the firm invest nllw, or wtluld it lptt better to wait a year and sce whether the price ()1' widgets gtles up ()r tlown? Suppose we invcst now. Calculating the net present value ()f th is investnlent in the standard way (and noting that the expeaetl future price of widgets is =
=
.)
l
1
=
l
P,
=
$300
P:
=
$10 O
%
----'-
q Po
=
=
2
=
30O
=
1 O0
$200 (1
-
q)
2b*i;: Iz p-tl
.33.
j/
pz lJt?bz't.
.
ryj4'
je'itlltk.t.
'--'*-
Illtrlltitt:til';
always S200), we get NPV
=
-
1600 +
x
x() V (1.1)
=
t
J=()
-
1600 + 2200
=
$600.
(l)
It appears that the NPV of this project is positive. The current value of a widget factory, which we will denote by f.$l, is equal to $2200,which exceeds the $ 1600 cost of the factory. Hence it would seem that we should go ahead with the investment. This conclusion is incorrect, however. because the calculations above ignore a cost the opportunity cost of investing now. rather than waiting and keeping open the possibility of not investing should the price fall. To see this. let us calculate the NPV of this project a second time, this time assuming that instead of investing now. we will wait one year and then invest only if the price of widgets goes up. (Invsting only if the price goes up is in fact ex post optimal.) In this case the NPV is given byl NPV
=
(0.5) -
1600
l.1
'''Z Y'N
+
''
I
=
l
300 (1.1)/
=
850 1. j
=
Developing ?/lc Cllll()epls F/l/-f?/(/l
f-rtl/?l/p/t'-
How much is it worth to llave the flexibility to makd the investment decision next year, rather than llaving to invest either now or never'? (NVc know that having this tlexibilityis of some value, because wtt would pretkr option-f is easy to wait rather than invest now-) The value of this to calculate; it is just the difference between the two NPV-S. that isp$773 $600 173. In other words, we should be willing to pay $ 173 more tbr an investment opportunity that is llexible than one that only allows us to invest biflexibility
-
=$
n0W.
Another way to Iook at the value of tlexibility is to ask the following question: How high an investment cost l would we be willing to accept to have a llexible investment opportunity rather than an inllexible or never'' one? To answer this, we tind the value of 1, which we denote by 7, that makds the NPV ()f the project whtln we wait eqtllkl to the N PV whe n I $ I (A()() and we invest now. that is. equal to $6t)().Substituting l for the $ l )()() and substituting $600 for the $773 in etluation (2): --now
=
N PV
$773.
(Note that in year (), there is no expenditurc and no revenue. In year 1. the $1600 is spent only if the price rises to $30(),which will happen with probability 0.5.) If we wait a year betbre dcciding whether to invest in the factory, the project's NPV today is $773.whereas it is only $600 if we invest in the factory now. Clearly it is better to wait than to invest right away. Note that if our only choices were to invest today or never invest. we would invest today. In that case there is no option to wait a year, and hence applies. no opportunity cost to killing such an optionv so the standard NPV rule We would likewise invest today if next yearwe could disinvest and recover our $ 1600 should the price of widgets fall. Two things are needed to introduce an irreversibility, and the ability to opportunity cost into the NPV calculation invest in the future as an alternative to investing today. There are, of course, situations in which a firm cannot wait, or cannot wait very long, to invest. (One example is the anticipated entry of a competitor into a market that is only large enough for one hrm. Another example is a patent or mineral resource lease that is about to expire.) The less time there is to delay, and the greater the cost of delaying, the less will irreversibility affect the investment decision. We will explore this point later in this book as we develop more general models of investment.
vb-llplc
(().5)
=
-7 l l
x
-/
3(j()
+
.
=
l
( l l )l
=
$()1)
.
.
-1
$ l t.)8(). In tlther Solvingthis for yields (l/lf/ ()lllv ltobv llow :tt a ctlst factory widgct the widget llctory llob' build opportunity to
vvords.
the (lpportunity to build :t v:tlue 1 the $ t)()()has sanle as an (,1, fJr llext S 1t?8(). yctlr at a ctlst with the fulr widgets. Finally, suppose therc exists 11 l'utures market tlxpected (ne equal thu price for delivery futtlrc sptlt t() tures year thlm ntlw price. that is. $2()().2N'Wluldthtt ability t() hetlge on the futurcs market changc =
tlt-
our investmcnt decision'? Specilically, wlluld it Iead us to invest now, rlther than waiting a year? The answer is n(). see this. consider investing now ttnd hedging the price risk with futures cllntracts. X) hedge all price risk. we wtluld need to sell short futures for l 1 widgets; this would exactly offset any fluctuations in the NPV of our project next year. (If the price of widgets rises to $3()(). our project would be worth T33()().but we would lose Sl l ()() on thc futurcs contract. If the price tkll to $ 1()t),our project would be worth only $ 1l ()()- but we would earn an additional $ l l()()from the futures. Eithcr way, wll end up *1'il
:ln this exampie. the futures pric? woultl equai thc expcctcd future pricc because wc assumed that the risk is fully diversiliablt!. ( If the price ()1* widgets were ptlsitivttly ctlrrclatcd wilh the markct portfolio, thc futures price wtluld be less than the expected future sptlt price.) Ntpte that if widgets were storablc and aggrcgllte storage is pllsitive. the marginal ctlnveniencc yield from holding inventory would havc to be I 1) percent. The rcason is that since thc futurus price equals the current sptlt price. thc nct htllding cllst (thc intcrest ctlst ()f I t) pcrcvnt Icss thc marginal conveniencuyield) lnust be zerl).
Llevelqalg
Illl-ltlllcitlll
3()
with 11 net project value of $220().)But this would also mean that the NPV of less the 1&)0 cst ()f the investment), exactly our project today is Slt) ($2200 withotlt hedging.s what it is Hence there is no gain from hedging. and we are still better off waiting until next year to make our investment decision. This result is a variant of the .$
Modigliani-Miller ( l 958) theorem. Operating in te futures market is only policy. and barring the possibility of bankruptcys it has a form of snancial value of the tirm's no real consequences for investment decisions or for the investment opportunities.
to Financial
Analor
1.A
Options
Our investment opportunity is analogous to a call option on a common stock. It gives us the right (whichwe need not exercise) to make an investment expenditure (theexercise price of the option) and receive a project (a share of stock) the value of which lluctuates stochastically. ln the case of our simple the lz?/n(.,y,'' meaning that if it were example, we have an option that is option is said to be payoff. (An would yield positive net a exercised today it negative exercising of tlte lnoney'' yields net payoff.) We today if it a it is money,'' better the to wait rather found that even though the option is will exercise our rises to $300. we than exercise it now. Next year if the price will worth fz'l which receive be $33()() option by paying $1600and an asset will be worth only $ l 1()(), asset falls 10(),this If the price 300/( to S 1 )f Ej'''j and so we will not exercise the option. that We found that the value of our investment opportunity (assuming the actual decision to invest can indeed be made next year) is $773. It will be helpful to recalculate this value using standard option pricing methods, because laterwe will use such methods to analyze other investment problems. To do this, let F'tl denote the value today of the investment opportunity, that is, what we should be willing to pay today to have the option to invest in the widget factory, and 1et Fl denote tbe value of this investment opportunity nextyear. Note that Fl is a random variable', it depends onwhat happens to )Jthe price of widgets. lf the price rises to $300,then Fk will equal S'') 300/( 1 1600 $1700.On the other hand, if the price falls to $100,the option to invest ''f??
Lkout
nin
=
=
.1
.
.1
-
=
aexnl.q., tc horizens of a uea.r or so. If tb.ere wf.wre f' .3Ntzst futures markets r' r . for widgets over the indeh' nite future, an equivalent hedge wou ld be to sell short one each futu re year, that is, to sell forward tht! enti re output stream. The resul t wou Id be tznlv
.
J
:k
Itl,res
4,nrd rk'et
widget
in
the same.
//1tJ
oll(.'t:ilts F/l?'t'pI/g/?
'?llp&,
E--b-tllllil..b'
willgo unexercised. and in that case /--1will equal (). Tllus wtl know al I ptlssible values for F3 The problem is to Iind 6-). the value of thd optitln today.i .
To solve this problem, wc will creattl a portlbl io that has two tron-lptlntt 1-1ts: the investment opportuni ty itsel anl :1 certain Iltllnbe r of widgets. NVe wil I pick this number of widgets so that the portfolio is risk-fret!, that is- so that its value next year is independent of wllether the price ()t' widgets gtes up t)r down. Since the porttblio will be risk-frce. we know that tlle rate of rettlrn (lntt can earn from holding it must be the risk-gree ratc of interest. (I f the rettlrll on the portfolio were Iligher than the risk-frec rate. nrbitragetlrs cotlld earn unlimited amounts of money by borrowing at the risk-frt!tl rnte and btlying the portfolio. If the portfolio's return were Iess than the risk-free rate. arbitrageurs could earn money by selling short the porttblio and investing tlle funds at the risk-free rate-) By setting the portfolio's return equal to the risk-free rate, we will be able to calculate the current value of thc investment opporttlnity. Specitically. consider a portfol io in wllich one l:()ItIs the investment (3pportunity, and sells short n widgets. ( Ifwidgets were a traded commodity. such as oil, one could obtain a short position by borrowing frllm nnllther prodtlceror by going short in the futures market. For tlle moment, howevera sve need not be concerned with the actual implementation of this pllrtfolio-) The value of this porttblio today is m4) /$) tl J.1) /$) 2()()??. The value of the ptlrtF 1 lt /$ This depends ()n l9b I1' Pl turns llut to be folio next year is * I that Fi l 7()0, then * l l 7()() 3t)()??. If /RI turns out t() be $ l ()(). $300. so 1t. that then F ()() * l (). N()w, let us chotlse 11 s() that the ptlrtfolil) I so risk-free, that that * I is independent t)f what hitppens t() tlle price. is, s() is do this, just set =
=
=
-
.
=
.
=
=
-
-
=
-
-
-l')
)() 3()()11 177( --
=
--
l ()()l l
.
whether the price ()t' 8.5. With /1 (lhosen in this way, tp I widgets rises to $3()()or falls to $ 100. Now let us calculate the return from holding this portfolio. That return is the capital gain, *1 *), minus any payments that must bc made t() hold tht.l short position. Since the expected rate of capital gain ()n a widget is zero (the expected price next year is $200, the same as this year's price). n() rational investor would hold a Iong position unless he or she could expect to earn at least 10 percent. Hence selling widgets short will require a payment of -85()-
or, n
=
=
-
4In this example. aII uncertainty is resolved next year. Therefllre an ()p(i()n to wllit has Ilt) convttntillnal NI'V critcrilln at that timc. Waiting has relevance only this year. ln Iatt!r chapters we will ctlnsider mllre gencnll sl'tuatitlns to wait rtltllins value al :111 times. where uncertainty is never fully resolved. and the tlption
value next year. and investment is made accordina t() the
/?l/p-t'pt/t..li'f..)/l 0. l J1) $20 perwidget peryear. (This is analogous to selling short a dividendpaying stock; the short position requires payment of the dividend. because no rational investor will hold the offsetting Iong position without receiving that dividend.) Our portfolio has a short position of 8.5 widgets, so it will have to pay out a total of $ 170. The return from holding this porttblio over the year is theretbre =
-850
= = 680 Because this return is risk-free.
free rate, which we have assumed is portfolio, *4) 6) 11 C3: =
6,
-
-
+ 1700
-
l70
Fi).
we know that it must equal the risk10 percent. times the initial value of the
-
680
-
6)
=
0. 1 (6)
l700).
-
=
Ektllllples
dynamic programming approach. In this case it gives exactly the same answer. because all price risk is diversitiable. Later in this book we will explore this connection between option pricing and dynamic programming in more detail. 1.B Characteristics of the Option to Invest We have seen how our investment decision is analogous to the tlecision to exercise an option. and we were able to value the option to invest in much the same way that financial call options are valued. To get more insight into the nature of the investment option. Iet tls see how its value depends on various parameters. ln particular, we will determine how the value of the option and the decision to invest depend on the direct cost of the investment. /. on the initial price of widgets, Jl,, on the magnitudes of the up and down movements in price next period. and on the probability t/ that the price will rise next period.
Changink /e Cost ofthe Investment So far
we have Iixcd the cost of the
investment. 1, at $ 16()(). I-low much would were above or below this number'? We can Nnd out by going through the same steps that we did before. Doing so, it is easy (lbtain a risk..frce porttblio depends to see that the short position nacded to
the option to invest be worth if /
on l
as follows--s
11 16.5
=
-
up??/.'?/c
(5)
We can thus determine that 6) $773. Note that this is the same value that we obtained before by calculating the NPV of the investment opportunity under the assumption that we follow the optimal stratebry of waiting a year before deciding whether to invest. We have found that the value of our investment opportunity, that is, the value of the option to invest in this project. is $773. The payoff from investing (exercising the option) today is $2200 $ 1600 $600. But once we invest. our option is gone, so the 5773 is an opportunity cost of investing. Hencc the /1// cost of investing today is $ 1600 + $773 $2373 > $2200. As a results we should wait and kecp our option alive. rather than invest today. We have thus come to the same conclusion as we did by comparing NPV's. This time, however, we calculated the value of the option to invest, and explicitly took it into account as one of the costs of investing. Our calculation of the value of the option to invest was btsed on the construction of a risk-free portfolio, which requires that one can trade (hold a long or short position in) widgets. Of course, we could just as well have constructed our portfolio using some other asset, or combination of assets, the price of which is pertkctly correlated with the price of widgets. But what if one cannot trade widgets. and there are no other assets that span'' the risk in a widget's price'? In this case one could still calculate the value of the option to invest the way we did at the outset by computing the NPV for each today versus Mzait a year and invest if the price goes investment strategy (invest up), and picking the strategy that yields the highest NPV That is essentially the =
Deb'elopil'g Ille Ctp/lctz/p/-Tllrottgll
=
The currcnt
value
of the option to invcst is thcn given by
6) Equation
0.4)05 1.
-
(7)gives the value
=
l5t)() t).455 1. -
of the investment opportunity as a tnction of $16()(),it is of / for which
the direct cost of the investment, 1. We saw earlier that if l better to wait a year rather than invest today. Are there values investing today is the pretrred strategy'? To answer
=
this. recall that we should invest today as Iong as the payoff
from investing is at least as Iarge as thel// costs that is, the direct cost, 1, plus the opportunity cost 6). Since the payoff from investing today is ), $22()0. =
5As before. the valud of the portfolio next year is * j = F, t If Pl = $3(1(),F = 336#)- /, = 33t)t) l 31)4)l1. lf Pb = 51()1).and 1 is not so low that we wlluld invcst anyway, Fb = (). l0() ?l. Setting the 4)1 for each price scenario equal gives the equation above for tt and *1 = -n
,
so tl)h
-
-
's
-
.
34
Iltlrodllction
we should invest today if 2200 we should invest as Iong as
6,. Substituting in equation
I +
>
I + 1500
0.455 /
-
(7) for FI.).
22()().
<
Thus, if I < $ l 284- one should invesl today rather than wait. The reason is that waiting means giving up revenue in thtl tirst year, and in this case the lost revenue exceeds the opportunity cost of committing resources rather than keeping the investment option opcn. However, if l $ 1284, Ph) 1. and one would be indifferent between investing $9 16 6) today and waiting until next year. (This can also be seen by comparing the project's NPV if we invest today with the NPV if we wait until next year-, in either case the NPV is $9 16.) And if J > $1284. one is better off =
=
=
-
waiting.
Ljt..'b-ellpbq//?t; C-()?1(..(,J7/A. Tlllzllqlt 5??lJ?&-/.i2rt???lJ)&: l 11u'l'ltllntpfili vtit illg (l 11c (the sacridice t)f tl'le irllrnediilte prtplit) Nvhtltlle (lpti k)f lnal ly :t fter tlbservillg ttJ decide ;.l17iI tlle ity r t ilc prictl l'ltts value I1l;.tl. and i beconltls exercise down). ilnnlediate t)I)t gone up t)r
Price Cltanging the .l'z1lWt# btlt now vary the initial Let us again fix the cost of the investment- l at $ l (,11)(), price of vvidgets. J1,. To do this, we will assulne tllat whatever /1) happens t(.' Nvith be, with probability 0.5 the pricc next year will be f () percent higher. antl probability ().5 it will be 5t) percent lower. (See Figtlre 2.3.) .
portfolio in vvllich To value the option to invest, we agai n stl t up :1 risk and sell short some ntlluber of widgets. The value of this hold the option we porttblio today is r.p4/ 6) 11 J1,. lts valtle next yttar tlepends ()11 l Tht, value of a Nvidget t'actorynext year is 1,-1 )7:-')/31/ ( l l )r l l li but we wil l l )t)t). the cost ()f tllc invcsfnlent. only fnvest in the tcqlr.y if its value ttxcfceds l l P, l (:)()1. Stlppllse /:1)is in the range wllere if thc pritrtl Hence max 11). that is, if 1) = l /1)) it wil 1be worthwhi le to invest, bu t if goes up next year ( (ltht!r pllssibilitics sI1llrtly.) will down it not. (We will consitler price the goes klntl t'p 11 /1) 11 thd = tlp, f1, 6.5 l (,1)() if prictl * l J1) l l Then l glltls sctlnltritls the these * gives Equating thtt down. prictl 1 twt) tlr if the goes nAak' risk I'retz : the portfolil) /1 of t hat cs value -free
=
The dependence of 6) on I is illustrated in Figure 2.2. This graph shows the value of the option. 6), and the net payoff from investing today, lz't) /, both as functions of 1. For I > $1284, 6) 150() ().455 l > l$j /, so the option should be kept alivea that is. we should wait until next year betbre deciding whether to invest. However. if I < $ l284. 6) 1500 ().455 / < 1$) /. s() the option should be exercised now- and hence the valtle ()f the option is jtlst its net payoff, K) I In the terminology of options. when / is small. tht, nttt payllff thlm imin the mtlney.'' At mediate investment becomes large. ()r the (lption is critical threshold where point or it is stltliciently deep. the ctlst of waiting a -
=
-
-
=
-
-
-
.
(?1%
tllltyvtriglls
-
.
=
=
.
.
,$
-l
-
=
.5
-.().5
.5
=
-
's
-
K&dtlep
/1)+ 81)1)whtlther tlle price gtles Note that with 11 chosen this way. tp I up t)r down. Ntlw let us calculate the return ()n tilis ptlrtllitl- rel-nembcring that the short position will require a paymdnt ()1- (). 1 ?? J1) I /1) l ()t). That return 6:1(). Since the return is risk-free, i t mtlst cqtlal (). l *t) is 6.6() /-1) -8.25
=
2000
W l
.4,5
-
.-.1 r
tt
1600
=
ww
-
--w
12Oo 80O
y-1500
-
--.
'.0.455 / ..-
.-.
--
I I
1 l I '
400
80O
=
-
Fz l )
--
400
O O
-
.6)
l
xw
N
x. xv
N
1600 12O0N 1284 /
Pa
'-'->' Nw
N
N
N
N
2000
=
2
=
1.58
=
o
--*'
Pz 1 2
r
t
n r E) o2 0 1 f'-izure 2.3. Prdrt?f (' IVlqcL =
N.z
.
%.J I
'
.
%''
'
q p
.h#'
.
t)
36
llltrodttclioll
0. 1 /7i) 1
.Jl)+ l60. Solving
.65
-
tbr F() gives the
Fll
=
7.5 fl)
value
Table 2. 1-
of the option to invest:
727.
-
We have calculated thevalue of the investment option assuming thatwe would only want to invest if the price goes up next year. Howevera if Jl) is Iow enough we might never want to invest, and if C) is high enotlgh it might be better to invest now rather than waiting. Below what price would we never invest'? From the equation above, we see that 15) owhen7.5 C)= 727, or /$ = $97. If /l) is less than $97, 1Z1 will be Iess than the $1600cost of the investment even if the price rises by 50 percent next year. Forwhat values of /$ should we invest now rather than wait? Once again, should invest now if the current value of a widget factory, K),exceeds its we . total cost, $1600 + &). Hence the critical price, which we will denote by C*) satisfies F4) 1600 + 6), that is. 11 #t*) 1600 + 7.5 C7 727, or #(*j= $249. If f1) exceeds $249. we are better off investing today rather than waiting. The option is sufficiently deep in the money that the cost of waiting (the sacrifice of period-o profit) outweighs the benefit.: We obtained this critical price by tinding the value of the investment option, but we also could have found it by calculating (as a function of /1)) the NPV of the project assuming we invest today. and equating it to thtt NPV assuming we wait until next ycar and then decide whether to invcst based on the outcome for price. (We leave this for you as an exercise.) The value of the investment option is thus a piecewise-linear function of the current price, lj, and the optimal investment rule Iikewise depends on Jl). If C) s $97. 6) 0. and one should never invest in the factory. If $97 < C) S $249, then 6) = 7.5 J$) 727, and one should wait a year and invest if the price goes up. lf J1) > $249. then 6) = 11 /1) 1600, and one should invest immediately. This solution is summarized in Table 2.1. In Figure 2.4 we have plotted 6) as a function of Jl,. Suppose our only choice were to invest today or else never invest. Then the option to invest would be worth the maximum of zero and 11 C) 1600, that is. the value
of a widget factory today, Pi), less the $ 1600 cost of building the factory. (In Enancialjargon.this is the ltrinsic pflftl of the option the maximum of zero and its value if it were exercised immediately-) Also, we would nvest today > > 4 as long as 11C) > $ l600, that is. as Iong as /1) $ 1 6. When Jl) $249. the lhen it is f)(), value l worth f1) I because of l intrinsic its invest optil)n to shown l 60() l wait. is today rather than 1 fl) Hence a solid invest as optimal to < $249, however- the option valucs of J1) When than /!) $249. for grcater line unexorcised- at to invest is worth lnore than I l J$) 1()()1),and should be Ie until next least year. '
-
-
-
y'
1800 1600
=
1400
-
1 1 Pz
1200
-
ur
z
1000
.
.
800
,'
7.5/% 727 -
-
'Ybu might think that immediate investment woultl be justifie only if we wouitl invest ncxt
year irrespective of whether the price went up or down. In fact. thc critical price for immediatll investment is Iower. Assuming we wait, we will invest next year if f''j 16t)0 1l 1$ 1600 > (). 0.5 Jl,, we would invcst ()nly if 5.5 /l) l6t)0 > (). t)r so if the price gocs down. that is. P, /1) > $291 which exceeds Jl; $249.Tlle problem with the faulty intuition is that it ignorcs thc protit we could elkrn thiq year by inves4 ir!g now lndt-txd f. a .!; of. say, $260. it is better tt) invust today even though if we did wait, we would ?7:J/ invest next year if it turned klut that tlle price wen down, -
=
:
=
=
for
,
.--..w.
.
6O0
.'
.
40O
0
40
80
120
160 ra
Cz
1 1 l j l l
I I l l
,'
'
O
'
,
1
.'
200
.
.
l
' ''
-
-
1600
-
200
24O
280
320
38
Developtj; J/rf.! C-llllw.'pts7W/'(Jf
Il3rotLtctl'llll
Note from Figure 2.4 that 6) is a convex function k)f /1), and that 6) is greater than or equal to the net payoff from exercising the option today. 1$)- 1, up to the optimal exercise point ($249in this examplc). As we will see, the value of an option to invest typically has these charactcristics.
Changing the Probabilitiesfor Price We can also determine how the valtle of the option to invest depends on
t/,
the probability that the price of widgets will rise next year. (So far we have assumed that this probability is 0.5.) To do this. we will let the initial price, J1,, be some arbitrary value, but we will fixthe cost of the investment, /, at $ 1600.
We can then follow the same steps as before to find the value of the option and the optimal investment rule. The reader can verify that the short position in widgets needed to construct a risk-free portfolio is n 8.5, and is independent of (The reason is that n depends on the possible values tbr the portfolio in period 1, * 1 and not on the probabilities that the portfolio will take on those values.) However, the payment required for this short position does depend on /, because the expected capital gain from holding a widget depends on t/. X) calculate thisa !et 7()4#1 ) denote the expected price of widgets next year. calculated conditional on the knowledge of the period-l) price.? Thus rt)(Pb) ((/ + 0.5) f'1). Therefore the expectcd rate of capital gain on 11 widget is g-()( P3) Jl)J/C) q 0.5. Hence the required payment per widget in the short position is 0.5)1 /l) (0.6 f/ ) J1).Setting * 1 *t) ((1.6 t/ ) l /1) t).I *()s t0. l ( and setting 11 8.5, we find that if Jl) > $97, the value ()f the option is =
.
,
=
=
-
-
-
=
-
-
-
-
=
-
=
8)
=
15 (? Jl)
l455 q
-
.
Note that 6) increases as q increases (aslong as f1) > 97). This is as we would expect, because a higher q means a higher probability that the price will go up and the option to invest will be exercised. If C) S $97,we will never invest. whether or not the price goes up, and 6) 0. How does the decision to invest depend on :/? Recall that it is better wait rather than invest today as long as /-1) > 1)) 1. In this case. )'() = to /X'4 f1)/( l (6 + 10 q) /$. Hence it is better to wait as fl) + S l q + 0 5) =
-
.1)1
.
Igh
t'trl//lp/t.,, LL-
-t'//'l/a#(.z
long as I 5 /1) l 455 q > (t) + l () t/ ) /1) - l (',(2)(),t llat is. as Iong as /1) < :1..; t/ increasds. tllat is- :t ( l ()()() 1455 (1 )/((' - 5 t/ ). >Nlso-l'*11 tlecreases #* () higher probability of a price increase intluces the lirnl t() invest nlortl reatlily. Why'?The cost of waiting is the revenue foregone by not selling a widget this with /''(). Since a Iligher ty makes a batl otltcollle next Period. which increases period less likely. it rctluces the valtle of vvaiting. Tllus :1 higher q implies tllat of svaiting exceed the value of waiting. a smaller 3-1)suffices to make tlle cost (/
-
-
Increasing the Uncertaint.y over f'Wt'c When we changed the probability t/ wllile keeping alI tlle othttr parametors flxed.we changed the expected price in period l Suppose the expectttd price in period l remains fixed at the initial price Ievel. #(), btlt we increase the size of the up and down changes so that the variance of tlle period- 1 price spreatl in tlle distribtltion increases. %vhat e ffect wotlld such 11?zI:*tl?l-J??-c'.c'?7.'/Ig PI have of value the investment option. 6-). and on the critical price. for on the svhich rather t han wait? optimal it is to invest inAmedilltely #*, above 0 will (/ is ().5. btlt now the pricc wi 11t2ithe r rise 'We assume as before that rather than l7y5() percent as before. Thus or fall in ptlriod 1 by 75 percent, expected value is st ill fl1. To lind /$ ). the variance of l>b is grcate r. but its port f(llio and we go t hrllugh the sanle steps as be tbre. creat ing a risk equating its return to the risk-free return. Again. let the ptlrttblio bc Iong the 1$) 11 /!). I11 investment option and short 11 witlgets. s() its vClltltl ntlw is thl will bc worth period l thtl price will rise t() l /!), in which cilse the prlject (h wil ?? /1). ()r l tltl ual l 9.25 /.:1)- l (k)() ) 1zr1 l 1 141 19.25 J1l so that I which will equal -().25 11 J:1y.Equating case t' l the price will fall to ().25 P(,.in these two possible values ()f tp 1 and solving for ?? gives .
.-free
=
-
.75
,
.75
=
=
-
n (Then, *
-
C) + 267 irrespective
-3.21
1
l 2.83
=
=
1()f)7/ /!)
.
t)f 141 Remembcring .)
/'$) position will require a payment of (). l n J'll l portfolio is 8.34 Jl) 8) 693. Setting tllis equal to t).l 107 and solving for 6) gives .28
=
-
-
-
that thtl shtlrt l ()7, the return ()n the /El) (). 1 1$) - l + *4) .28
=
=
/$) 8.75 /-1) 727. =
7We will use C to denote the expectation
(mean)of a random
variable,
on C will indicate that the expectation is conditional on information
and a time subscript
available
as of that time. Similarly, we will use b' to denote variances. These symbols are used consistently throughtlut thc book. Other notation is specific to each chapter ur even section. A symblll glossary at the end of the book collects the signiticant symbtlls and states what they denote.
-
If Jl) is $200, 6) is $1023, which is substantially larger than the value of 5773 that we found before when the price could only rise or l'all by 5() percent. Why does an increase in uncertainty increase the value of the option to invcst'?
Because it increases the upside potential payoff from the option. leaving the
lntroductioll
40
downside payoff unchanged at zero (sincewe will not exercise the option if the price falls). We can also calculate the critical initial price. 131. that is sufhcient to warrant investing now rather than waiting. Again, just equate the current value of the widget factoly Ftl 11 J$, to its total cost, $1600 + ). Utilizing equation (12)for F(), this gives Pl $388,which is much larger than the value of $249 that we found before. Because the value of the option is larger, the opportunity cost of investing now rather than waiting is larger, so there is a greater incentive to wait.
Developing
Ille
Cf-nlcz/pf.Tllrollgll .ki7lJ?/t3Erflp?lp/p-j'
It is easy to show (andshould be intuitively clear) that the point of indift-erence between investing now and waiting occurs in the range of Jl) where investment in period l is warranted if the price goes tlp but not if it goes down. In this case the NPV in equation ( 14) simplifies to
=
NPV
=
tbad
'ibad
Suppose that the initial price is J$, but in period l the price becomes
/1
(1 +
ul/$
with
probability q,
(1
J) J
with
probability l
= -
=
-
/ + #; + q
''= (1 + !/)/1) (1.1) f l ,
(1
-
(1
q)
=
JIIJ.
I/)
t
=
j
J) J1) (1.1) , -
On the oiher hand, if we wait the NPV is
Equation
I
=
( j( 0. 1
t'. 1 + ( I
0. l + ( l
l.1
-
t/)
(F)( l
-
-
J)
t'bad
srstspelled
out by Bernanke
j
.
(16) has one detail that is important
#(*jdoes not depend $ to note upward the size of It only depends in any way on an move. on the size of (/ and of 11 downward d, the probability downward l ( ) move. move. Also. a the critical the magnitude of the is ds larger is the prices Jl'j it is Iarger the ; wait-g news'' that drives the incentive to Possible spread in l in a We can also examine the effect of a mean-preserving J/( lt + J). Then more general way than we did betbre. Suppose we set q (/ #(2j lt and tt J S /5 ) Jl), and k'4P ) Hcnce if we increase proportionally. we keep fy and 7( P3) unchanged while increasing the variance of 1) Observe from cquation ( 16) that if J is largdr but (y is unchanged. C; incrcases. Again. we hnd that a mean-preserving spread increascs the incentive to wait. tt.
-
ttbad
=
=
=
.
.
Extending the Example to Three Periods In our example, we made the unrealistic
assumption
that there is no uncer-
tainty over the price of widgets after the first year. in most markets. future prices are always uncertain, and the amount of uncertainty increases with the time horizon. In otherwords, while the expected future price of widgets might always equal the current price. the variance of the future price will typically be greater the farther into the future we look. Later in this book we will model 9If current proht can be negative and the 5rm is contemplating
B'Fhis news principle'' was be found in Cukierman (1980).
( l5)
.
'>D
+
+ 1l ( 1 +
-
To keep things general, we will 1et the cost of the investment be 1. In this case, the NPV if we invest now is NPV
(- l
Equating the NPV of equation ( l3) tbr investing now with the NPV of equation ( l5) for waiting and solving for J1)gives Jl;
tgood
q l l .
X ''Bad News Principle'' We can take this one step t'urther by allowing both the probability q of an upward price move as well as the sizes of the upward and downward moves news'' (an upward move) to vary. ln so doing, we can determine how news'' (a downward move) separately affect the critical price, C7, and that warrants immediate investment (inthe above calculation the upward and downward moves had to increase or decrease together). We will see that /l; depends only on the size of the downward move, not the size of the upward of it is the ability to avoid the consequences move. The reason is that wait-S news'' that leads us to
=
( 1983), and
the ideas can also
a costly disinvestmcnt
or
abandonment of a project, the bad news principle turns into a good news principle: the size and probability of an upturn are the driving forces behind the incentive to wait. Rccalling our discussion of suicide in Chapter 1. we should emphasizc this point. If the potential bad outcomes become even worse. that does not increase the incentive for immediate abandonment. However, if potential good outcomes become better, that increascs the value of staying alive.
Illtr(;tltt'tl'tlll
the stochastic evolution of price in just this way. At this point, however. we dan obtain additional insight into the nature of the investment problem by extending our example so that there arc three periods in which the investment decision might be made. We will asstlme some level Jl1,and at /71 l C) or /71 again either increase .5
=
('!
h
1
l i .
=
() the price of widgets begins at as before that at ? 1 it will either increase t)r dccrease by 50 percent (to 2. it will 0.5 J1,), each with probability 0.5. Then, at t =
/
=
=
or decrease by 5() percent with equal probability. Hence values tbr Pj-.2.25 f1)-0.75 C), and 0.25 C).The price three possible there are remains this level for :111 l t 2. (See Figure 2.5.) We will again fix the at then of 1, at $ 1600. investment, the direct cost adding of price uncertainty, our investment problem period By one more becomes quite a bit more complicated. One reason is that there are now Iive different possible investment strategies that might make sense and must be considered. In particular. it might be optimal to (i) invest immediately; (ii)wait a year and then invest if the price has gone up. but never invest if the price has gone down; (iii)wait 11 year and invest if the price has gone up, but if it went down wait another year and invest if it then goes up; (iv)wait two years and only invest if the price has gone up both times; or (v)never invest. Which rule is optimal will depend on the initial price and the cost of the invdstment. and the value of the investment option must be calculated for eat:h possible rule. The second complicating factor is that while we can still compute the value of the investment option by constructing a risk-free portfolio, the makeup of that portfolio will not be constant over the two years; we will have to
We wil l again approach this problenl tlsi ng option prici ng n'let hotls. %Ve ()- /7().as :1 l'tlnction want to obtain the value of tlle option to invest at / of the initial price. /1,, as well as the optinaal investnient rtlle. Tllk) trick is to work backwards. We will solve tsvl) separate investment prllblenls looking l first for l ().5 /-1),and then for l fonvard from t l /1)-asstl lning in both cases that Nve have not yet invested. In each case we will tltlttlrlnine Fb, the value of the investment option at t = 1 by constructing a risk-free portfolo and calculating its rttttlrn. Given the tsvo possble values for (one for PL = t).5 1) and one for J''! = l /,1,(),we then back up to / () and determine 6) by again constrtlcting a risk-free portfol io and calculat ing its =
.5
=
=
.
=
,
-l
.5
=
return().5 J'l). and that /1) is such tllat we wtptlld Suppose that at / l PI invest in period 2 if the price goes up, but not if it goes down. Now construct a portfglio that incltldes the option to invest and is sllort slln-le ntlmber /1 I t'f J-.1 11 l /31 I f tlle price goes up widgets. The value ()f ths portlblio is * 1 in period 2 (to (3.75 /l), we wi lt invest. so Fz wiII ttqual (3.75 Jl)/ ( l l JJ (p: will cqual 8.25 C) lt()() ().75 11 1 1)p. lf t htlprice 1600 8.25 f-1) I()()()-and ().25 down in pcriod 2 /1) t() ), we will not invest. fC':will eqtlll ()allntl *2 will ( goes /1 ! f$,. Eqtlating the expressillns )r (pa under thc twl) sccnltrills. equal llio the wi be find hat risk t d' 21 11 free I6 )/ /$)-, t Ilen *2 i 324)1 ptprt , we will equal 8(3() 4. l 25 /l, whether tl3e prit:e gtes up t)r tltJwn. Caltrulnting tlle portfolio's return (ma t'p I (), l 11 ! 1) ) and se tt ing it eq tlal t() thc risk return (().l ml ) gives us the valtle of the investment optitln: /th 3.75 /1) 727.3. Also. note from this tllat F 1 () Nvhen J1) l 93.94. I-ldnce if the price ' has gone down in pcritld l and /$) < 193.94, we will never invest. We must now repeat this exeruise assuming that priue has gone up in period l that is, that /71 = l /1). Y()u can verify that in this casc the riskfree portfolio requires a short position ()f n 1 = 16.5 1()67/ J'1)widgets, and the value of the investment optilln is F I lI J$) 727.3. In this casc Fl = () when J'l) = 64.65. I-lence if /$) < 64.65 wtt will never invest at 1111, even if the price goes up in both periods. Als(?. suppllse we invest in peritld 1 rather than waiting until period 2. Then &ve would obtain a net value 1z'1- l 11 ( 1 #()) 1600. Setting this equal to h and solving for /1)gives /'!) l 66.23. =
=
,
=
-
.
)-N't7
-
.
=
-
-
-
-().25
.5
=
-
-
-
-fr(2tl
-
-
=
E)
=
l 0
l= 2
=
l
=
3
-
=
.5
J2 J2
Pj
=
1
Pa
=
2.25/%
-'--'>'
Pz = 2.25P
-'-'''>-
.
-
.25
=
.5/%
L 2
Pv
Pz = 0.75/%
'-''-'>'
Pz
=
Q.75Pz
'--'->'
=
..
J2
2
%
=
-
.5
-
=
0.5/%
L 2
%
=
0.25/%
-''-*'
Pz = 0.25/%
'--'*-
''l''l''hcmethod of kecping a ptlrtl'lllitl riskless by changing its ctlmpllsititln through repeatcd hedging str:ltetzy.'' llnd is of considerable irnpl3rtanctt in financial trading is called :1 economics. We will devktlopit in a gtlneral setting of continutlus tirnt in Chaptttrs 4 and 5. bsdynamic
Figure 2.5.
Prc: of Widgets
Itllrodttctioll
44
Hence if J1) > 166.23, we should invest in period 1 if the price has gone up, rather than wait another year. we have We now know Fl and the optimal investment strategy (assuming /1 of possible we can the tbr each so outcomes two invested) for not yet and calculating portfolio risk-free constructing again Fi) by a determine once its return. Since the optimal investment strategy depends on Jl,, this must be which done for diterent ranges of Jl), that is. for 64.65 < & S 166.23 (in periods), for both tbe in price only if period in 2 up invest goes case we would the price would if 1 which invest in period 193.94 < we case 166.23 C) :jE (in would never invest), and for J'1)> 193.94 (in goes up, but if it goes down we whichcase we wouid invest in period l if the price goes up, but if it goes down wait and invest if it goes up in period 2). The solution, which the reader might
E-b'tllllplf.n.
up/?/l/tr
Dcvc/trp/s/lt#?c Collt'.'pt-b-Tlll-otqqll
2800 ?400 11Pz
,
want to verify, is summarized in Table 2.2. Figure 2.6 shows 6) plotted as a function of f'1).As in the two-period model, it is a piecewise-linear function. but now therc are more pieces. each corresponding to an optimal investment strategy. Also, suppose the only choice were to invest today or else never invest. Then the option to invcst would be worth l 1 /1) l60t and we would invest today as long as 1l f1) > $1600,that is. as long as /l) > $146.When l) > 301.19, the option is worth this much, because thcn Fu 9.2 C) 1057.9 l 1 /1) l 600. and it is optimal to invest today rather than wait. Hence 11 Jl) 1600 is shown as a solid
2000 u!:'
1600
-
-1
1600
' .
.* -
1 200 ..
400
'
*
0
40
80
120
. I 160
1'
. ,
l
-1 I
j
I
I
200
I 1 1 l
.'
.*
l 1, ,1
j
1 )
e
.*
800
O
--.hw
240
280
320
360
400
-
=
=
-
-
-
Iine for J$) > 3()l 19. l7ut is shown as 11 dotted line for l 46 < /1) < 3()l l 9. Finally, notc once again that /$) is 11 convex function of Jl,, and that f;;.)ls grcater than or equal to thtl nct payoff from exercising the (lption today, l$) /, up l l 9 in this case). to the optimal exercise point ($3() As butbrc. we could examine the dependenc: of 6) and the optimal exercisc point #*() on the cost of the investment. /, or on the variance of price ' changes. (Going through these caculations is no'w a bit more laborious. howexample, we ever. because three periods are involved instead of twt).) For could show that if the variance of the price changes increases, while the expected price changes rcmain the same. the value of the option /-1)will increase, reader might want as will the critical exercise price C*)As an exercise. the ()r decrease by 75 percent in each period. check P this Ietting increase by to instead of by 50 percent as in Figure 2.5. If we wanted to, we could now extend our example to four periods. allowing the price to again increase or decrease by 50 percent at t = 3. We could then work backwards. sndingFz for each possible value of Pz, then hnding Fi for each possible value of P3 and hnally hnding Ff). In Iike manner, we could then extend the example to five periods, to six periods, and so on. As we did this, we would tind that the curve for 8) would have more and more kinks. We will see in Chapter 5 that as the number ot- periods becomes Iarge, the .
.
-
Table 2.2.
ofoption 1//1/14:
/kvt!Jl
to
Optimal Investment Rule
Option Value
Region
Never invest.
=
0
=
5.11 Jl)
166.23 < C) :; 193.94 5)
=
7.5C) 727.3
301.19 Fo
=
9.2 C)
C) S 64.65 64.65 < C) :; 166.23
8) 6)
attd lnvestment Rttle
-
330.6 Invest in period 2 only if price goes up in period 1 and in
period 2.
193.94 < C)
.s
-
-
Invest in period l if price goes up. If price goes down in period 1, then never in-
vest.
1057.9 Invest in period l if price goes up. If price goes down in period 1, then wait and invest in period 2 if price
goes up.
/% 301 19 >
.
Fi)
=
11 J$
-
1600
Invest in period 0.
.
.
,
46
llltl-oflttcioll
curve fklr 6) will approllcll a smootll culwe tllat starts at zert) and rises to nleet the curve showing tlle net payoff from irnlnedilte investnlent ( fz$) I ). In ltct th two curves meet tangentially, and the point where they meet dehnes the threshold #4,*where immediate investment is optimal. -
Adding more and more periods
will make
our example
tlnreasonably
complicated. howevcr, and in any case wtluld be less than satisfactory because ultimately we would like to allow the price to increase or decrease at every future time 1. Thus we need a better approach to solving investment problems of this sort. In Chapter 5 of this book we will extcnd otlr example by allowing the payofffrom the investment to tluctuatecoltiltllottsly over time. As we will see, this continuous-time approach is quite powerful, and ultimately quite simple. However, it will require some understanding of stochastic processes, as well as Ito's Lemma (whichis essentially a rule for differentiating and integrating functions of stochastic processes). These tools, which are becoming more and more widely used in economics and qnance. provide a convenient way of analyzinga broad range of investment timing and option valuation problems. In the next two chapters we provide an introduction to these tools for readers who are unfamiliar with them.
Uncertainty over Cost We now return to our simple two-period example and examine some alternative sources of uncertainty. In this sectiona we will consider uncertainty over the cost of the investment. Urkertainty over cost can be especialy important for Iarge projects that take time to build. Examples include nuclear power plants (where total construction costs are very hard to predict due to engineering and regulatory uncertainties). large petrochemical complexes, the development of a new line of aircra, and Iarge urban construction projects. Also, large size is not a requisite. Most R&D projects involve considerable cost uncertainty', the development of a new drug by a pharmaceutical company
is an
example.
In the context of our rwo-period example, suppose that the price of a widget is now $200, and we know that it will always remain $200. However, the direct cost of building a widget factory, 1, is uncertain. We will consider two different sources of uncertainty regarding 1. The first, which we will call input cost lfactrrftzfn/y, arises because a widget factory requires steel. copper, and labor to build, and the prices of these construction inputs fluctuate stochastically over time. ln addition. government regulations
jleveloping
F/l/'f'.?l/t/l
//?c Qhlll-)lL
vsitttill'
C-b'tlll'lillfu'vb'
t llis call cllltllgtr tl'le reqtlil'trtl tltltlltititrs change unpredictably (lver tilntl, llesv salkty regtllllt illlls of one or more constrtlctitlll inputs. ( Ft'r tlxlllllple. add to labor requirenzents. cllallging envirtlllnltlntal rcgtllltilllls I'lltty or may require more capital.) Tllus. altlltlugll I l'nigllt be kntlNvn today- its vllltltl llext year is uncertain. kllltl
As one might expect- this kind of
tlncertainty
has the sal'ne eflct
()11
the
investment decision as uncertainty over the l'tlturc value of the paytpff fronl the investment, IZ it creates an opportunity cost of investing now rather tllan waitingfor new intbrmation. As :1 result- :1 projoct could have :1 conventitlllally measured NPV that is positive, but it might still be tlneconomical to begin investing. As an example. suppose that I is $ l6()()today. but next year it will increase $2400 or decrease to $8t)().each vvith probability ().5. As before. the interest to is l0 percent. Should we invest today, or wait until next year? lf wtl invest rate NPV is again given by equation ( l ). that is- l ()() + 22()t) the $()()(). today. =
-
This NPV is positive. but once again it ignores an (lpportunity cost. X) stle tlntil &ve recalculate this time the assulming wait NPV.but let next year. this, us in which case it will be ex post optimal to invest tlnly if / falls to $81)().l n this case the NPV is given by N PV
...-.84)() (().5) + l l t
'-<'
.
(1n year () therc is no
expenditure
S8()(),which
=
l
2()t) l ( l 1) .
7()4) I l
$()3t).
.
and n() revenue. I 11 ye:yr l wc invest (lnly happcn with probabil ity ().5.) I1*wtl wait a year to investp the project-s N PV ttlday is $4)36.s() wtiting ,
will
. if l falls to before deciding whether is clearly better than investing now. At this point it might appear that uncertainty alwltys lellds llne t() ptlstpone investmentss t)r at Ieast incrcase the hurdle rat: that the investment must meet. but this is not the case. If investing provides inlrmatitln- tlncttrtainty can lower the hurdle ratc for a project. Consider uncertainty ovklr the physical diffculty of completing a project. g'V'ewill call this lccllnical I//?tz?-?tI??/p-. assuming factor costs are known, how much time. tlffort, and materials will ultimately be required to complete a project'? This k ind ()f uncertainty tr:ln ' only be resolved by actually undertaking and completing the project-l One ' l This is a simplilication in that, for some projects, cost u ncertainty ca n btl rtld uced by Ii rst performing additional engineering studies. Thc investmcnt prtlblenl is tllcn more compl icated because one has three choices instead of two: start constructilln n()w, undcrtakc :ln engineering study and then begin construction only if thc study indicates costs are likely t() be 1()w.()r abandlln the project com pletely.
48
llltrotlltctioll
then observes actual costs (andconstruction time) untbld as the project proceeds. These costs may from time to time turtl out to be greater or less than anticipated (as impediments arise or instead the work moves ahead faster than planned), but the total cost of the investment is only known tbr certain when the project is complete. With this kind of uncertainty, a project can have an expected cost that makes its NPV negative. but if the variance of the cost is sufticiently high, it can still be economical to begin investing. The reason is that investing reveals information about cost, and thus about the expected net payoff from investing further. It therefore has a value beyond its direct contribution to the completion of the project. This additional value (calleda shadow valtte because it is not a directly measurable cash tlow) lowers the full expected cost of the
investment.
The following simple example will help to illustrate this. Suppose that the
price of a widget is and always will be $20(),but the cost of building a widget factory is uncertain. To build a factoly one must initially spend $10()0:With probability 0.5 the factory will then be complete, but with probability 0.5 an additional $30()0willbe required to complete it. Since the expected cost of the factory is 1000 + (0.5)(3000)$2500.and its value is $220().the NPV of the investment seems to be negative. suggesting that we should not invest. But this ignores the value of the information obtained from complcting thc (irst stage of the project. and the fact that we can abandon the project should a second phase costing $3000be required. The correct NPV is 1000 + (().5)(2200) $ 100. This is positive, so one should invest in the first phase of the project. =
-
=
We see, then, that the uncertainty ovcr the cost of a project can Iead to one postpone investing or to speed it up. If the resolution of uncertainty is independent ot what the firm does. it has much the same effect as uncertainty over the payoff from investing, and creates an incentivc to wait. But if the uncertainty can be partly resolved by investinp ir has the opposite effect. We will return to uncertainty over cost Iater in this book, and examine its implications in more detail.
This makes the investment more attractive, 2 invest.l
Z1 =
Next suppose that the cost of the investment and the payoff from investing are both known with certainty, but the interest rate used to discount future cash flows changes in an unpredictable manner. Howwill uncertainty over the interest rate affect the decision to invest'?
x
c()()
( l 1)t
=
$2200.
.
t=()
ln ourexample.the expected value of the interest rate nextyear and thereatker is 10 percent. but the actual value is unknown. So, thc value ofthe factory next year will be with probability 0.5. q'''=1) 200/( l 15)? $ 1533 l
=
.
=
=
S=()200/4 l f
Uncertainty over lnterest Rates
and increases the incentive to
Nonetheless, uncertainty over future interest rates can still Iead to a postponement of investment. The reason is that the second effect of interest rate uncertainty works in the opposite direction .- it creates :1 value ofwaiting (to see whether interest rates rise or fall). This effect works in much the same way as uncertainty over the payoff from the investment. To clarify this, let us again return to our two-period example. This time we will assume that the price of a widget is xed at $2()0aand that the cost of building the widget factory is (ixed at $2000.The only uncertainty is over interest rates. Today the interest ratc is l0 percent. but next year it will change. There is a 0.5 probability that itwill increase to 15 percent. and a ().5 probability that it will decrease to 5 percent. It will then rcmain at this new level. What will a widget factory be worth ndxt ycar? lf there wcre no uncertainty over interest rates, that is, if we knew that the interest ratc will rcmain 10 percent. then the value of the factory next year would be
Zl
4
can llavc two eftkcts on an investment decision.
First, unpredictable lluctuations in interest rates can increase the npectetl value of a future payoff from investing. For example. suppose the investment yields a perpetuity that pays $ l per year forever. The present value of this l () percent, the value Perpetuity is $1/r, where r is the interest rate. lf r 1- is uncertain. and But $10. $1/0.10 equal either 5 percent is suppose can each with probability 0.5. so that 7(r) 15 percent, 10 percent. Then the or expected value of the perpetuity is 0.5( 1/0.05) +0.54 1/0. l5) $ 13.33 > $ l 0.
=
.05)f
=
$42()t)
with probability 0.5.
l2ln technical languagev this result is an implication of Jensen lnequality. combined with the fact that the prcsent value of a future cash tlow is a convex function of the interest rate. 'x
Jensen's lnequality says that if x is a random variable and 7'(-r)is a convcx function of then 7(./-4x)1> 7'(-(.,r1). Thus if the cxpected value of remains the same but (he variance of increases.ft/t-rllwill increase. In the case at hand. if the expectedvalue of nextyear's interest rate remains fixed but the uncertaine around that value increases, the expected present discounted value of a payoff reccived next year will increastt. .v&
.r
.r
Illtrothloqioll vltlue
Llence the expected
(.)1'
(().5)(42()()) $2867,wllicll
1.z')is (().5)(l 533) +
=
rdtes are certain. the future interest rate (holding the increases the expected value of the constant) value ot rate interest expected uncertainty the investment decison? First. affect l4ow project. But dlles the that if there were no uncertainty over interest rates, we wotlld clearly
is higher than it is when ftlture intcrest We sec. thens that
uncertainty
(lver
notc want to invest today. The N PV of the 5:/
! 13
=
:!()()()
-
project if we invest today is 7e ky ()()
-#-
:::::1
l
whereas the N PV today if $2000 $0.
l
J ( l l)
zuz
$:!1)() .
.
we wait until next year is only
$2200/ l 1 .
-
N PV
If we invest now.
is different if interest rates are uncertain.
The situation the NPV is
'
+ 20() +
-20()()
=
C Z1)
=
-
1.1
l 800 +
2867 I.1
=
=
(0.5)
1 -2()()() + l 1 l 1 .
'M
200 .05)J
.
l
=1)
(1
=
Scale versus Flexibility
t)l' scllltt trltll As students of ectlnonlics or btlsiness IklltI-I1 etlrly 0n- tltztllloluit?s be an important sotlrce of cost sltvings. By btlilding 11 Iarge plant instelttl tlf I'nigllt btt :tl7I(l to retltlt!e its avtlrltge tnlst lllltl two or three smaller ones, a lirm increase its prolitability. This stlggests tllat lirlus sllotlld respond tt) groNvtll iI1 demand for their prodtlcts by btlnching their investments, that is. investing in neNv capacity only infrequently, btlt atltling ICrgll ltntl efticient plllnts eatlll
time. Wllllt sllotlltl firnls dlla Ilowevtlr. when t Ilcre is untrertlkin ty
$806.
( I f we invest today we can mak: and sell :1 widget now tbr $2()().and we will have a factory whose expected value next year is $2867.)This NPV is positive, but suppose we wait until next year before deciding whether to invest. lf the interest rate rises to 15 perccnt, thtl value of the factory wfll be only $ 1533which is less than the $2000cost 01- the investment. Hence we will only invest if the interest rate falls to 5 percent. Since there is a t).5 probability that this will happen, the NPV assuming wtl wait is NPV
5
$ 1()()().
Tllis NPV is higher, so it is better to wait than invest now. This is a simple analysis of interest rate uncertainty. but it has some important implications. First, mean-preserving volatility in interest rates will value ofa project. but will also create an incentive to wait ncrease the expected rather than invest now. The reason is that, as with unccrtainty over future cash flows.uncertainty over future interest rates creates a value to waiting for new information. Second. if an objective of public polic.yis to stimulate investment, the stability of interest rates may be more important than the Ievel of interest rates. Policies that Iead to lower but more volatile interest rates could end up depressing aggregate investment spendingal; Aswe will illustrate in Chapter 9, with
tlelll:tntl
'Tllis prlllllttln is an inlpllrtant tlI1e 1-4.)1' l I'le e lect ric ut ility intltlst ry. I t is cheaper per unit (lf cllp:lcity t() btlild :1 Iargc cllal-li red plhver plltnt tlllkn it is tt) add capacity in srnall anltltln ts. Istlt ltt the sllnle tin-ltl. tltilities litce (.rtlnlr tlleir electricity siderable uncertainty over the rrlte Iy vvllicll tlle denAClnd gives the utility Ilexilpil ity. lltlt will gro:v.l4 jNtlding capacity in slnall ilnltlunts is also frlortt ctlstly. I-lence it is inlptlrtant t() be able tl) vllltle this llexillil ity. 7-heoptions approach used in t Ilis bt)()k'is sve II suited t() tI() t 11is. I Ie rc Tvt.t vi II illustrate the basic idtla with a sinlple exllnlple in wllich lemantl gnlwtll is certain. but there is uncertainty (lver relative fuel prices. 15 Consider a uti lity that faces constant dc mand grllwt 11()f I()() n'lcgawrt tts (&l'W) per year. l-ldncc the utili ty mtlst add to capacity; thc qtlestilln is l)(3w. There arc tsvo altcrnatives. The utility can build a 2t)()-MSV ctllll-lirtld pllnt (enough capacity for two ycars worth ()1- additional demand) :tt a capital ctlst of $190 million (Plant A), t)r it can build a l())-MW (lil-fired plant at a capital Inuch
14The reason is nllt justthat theril is ullcertainty tlvcr llle grtlwtll
'3lngersoll and Ross ( l992) have developed a continuous-time model of invejtment interest rate uncertainty, and it leads to more or Iess the same cllnclusions.
(lver
growth (as therc tlstlally isl'? I1*tlltl Iirln irrevklrsibly invests in :t lltrge adtlit i(1l1 llnly sltlvly ()1- even sh ri 11ks. it &v Il IiI'ltl itse If to capaci ty- and tlenlltnd grow,s holding capital it dllcs ntlt neetl. I Ience svllell t l1e grllwtl'l ()f tlel-nlntl is tlncertain. tllere is a tradeofl' between scltle ectlnlll'n ies antl the llexilnility tllktt is gained by invcsting I'ntlre frcqtlcnt ly iI1 slnilll increnlents t() ctptcity lls tlley are neetled.
tlj'
ttltal eluctricity denlltlltl. ric pllwk!r
but also bccause utilitics now (lftcn Iind thcnlselvcs ctlmpet ing with (lther slltlrct:s flftllecl (such as ctlgenerlltion). ls-rhis is 11 nltldified vcrsitln t)f an example prcsented in liwhill ( It?89).
Illtrltlla
frlrl/p
Annual Fuel Costs
Plant A
t 1
t 0 19
l= 2
=
=
19
--*
--*'
19
'''-*'
the second year. To decide which choice is best. let us calculate for each the P resent value of the expected cost of generating 100 MW of power per year forever starting this year, and an additional 100 MW per year starting next year.
First, suppose we commit the full 200 MW to either coal or oil. Then, if coal. the present value of the tlow of cost is choose we
200 MVV
(21) Plant B l= 2
l= 1
l 0 =
I
30
--*-
30
--+-
Note that 180 is the capital cost for the full 200 MW and that 19 is the annual operating cost tbr each 100 MW the hrst of wbich begins now and the second next year. Next. suppose we choose oil. Since te expected operating cost for oil is $20 million per year, the present value is
2
100 MVV
20
PVn
l00 +
=
10() + 1l .
L 2
Figure 2.7.
=
lo
Cloosing zlrltlllg Eleclric
-->
t
---
PIants r()w't.r
cost of $100 million (Plant B). At current coal and oiI prices. the coal-fired plant is not only more economical in terms of its capital cost ($90million per 100 MW of capacity), but also in terms of its operating cost; operating Plant A will cost $ 19 million per year for each 100 MW of power, whereas Plant B will cost $20 million per year. We will assume that the discount rate for the utility is 10 percent per year. and that each plant lasts forever. In this case, if fuel prices remain constant, Plant A is clearly the preferred choice. Fuel prices are unlikely to remain constant, however. Since what matters is the relative price of oil compared to coal (andsince the price of coal is in fact much less volatile than the price of oil), we will assume that the price of coal will remain fixed, but the price of oil will either rise or fall next year, with equal probability, and then remain constant. If it rises, the operating cost for Plant B will rise to $30 million per year, but if it falls, the operating cost will fall to $10 million per year. (See Figure 2.7.) The choice of plant is now more complicated. Although Plant As capital cost is lower because of its scale, and its operating cost is lower at the current oil price, Plant B affords the utility more tlexibility because it only requires a commitment to one year's worth of demand growth if the price of oiI falls, the utility will not be stuck with the extra l00 MW of coal-burning capacity in
20 + )J ( 1.1 .
)
=(
'X'
l=
l
20 ( l 1)l'
=
$61 1.
.
(22)
Thus it would seem that Plant A is preferred. afforded by the smaller oilBut this calculation ignores the oexibility sred plant. Suppose we install 10() MW t)f oil-red capacity now. but then if the price of oiI goes up next year, we install 20() MW of coal-fired capacity. rather than another oil-hred plant. This would give us a total of 30() MW of capacity, so to make the cost comparison meaningful, we must net out the present value of the additional l ()()MW. which would be utilized starting two ' years from now: PV' 1.1=
100 +
'x'
2()
(j j 1............::1) .
+
1
''
1 +i
l ()0
1..1 180 l.l
;
I
'X'
+ I
=
l
10 (l.1)f
9() -
(1.1)2
-'*
19
+ l
=
l
( 1.l)f
=
$555.
Note that the second line in equation (23)is the present value of the capital and operating cost fqr the second 10()-MW oil plant (whichis built only if the price of oil goes down), and the third Iine is the present value of the capital and operating cost of thehrst 100 A'/1zIZ of a 200-MW coal plant. This present value turns out to be $555 million, so installing the smaljer oil-hred plant and thereby retaining flexibility is the preferred choice.
Illtl-lltt-tilltl
One way to value th is llexibility is to ask how much lower wou ltl the capital cost of Plant A have to be to n-lnke it t he prc ferred choice. Let l be the capital cost of Plant A. Then the gresent value t)f the costs of installing and operating Plant A is .1
C*
+ /-1 '
/
=
()
''M
l9 + 1 ( 1)t
19
I + 399. .1
.
/
=
l
(' I l )l
'
.
The present value of the cost of providing the 2()()MW of power by installing Plant B now and then next year installing either Plant A or B (dependingon whether the price of oiI goes up or down) is
*
10()+
(.1 .1)1
f (1 =
+
20
1
i
I
.1 ' -
1.l
+
'M
10()
1 ''i
1.1
0.5 IA + (1.1)2
(=
-.:%D
t
=
l ()
+
l
l
(1.1.),
l9 (1.1)/
To find the capital cost that makes the utility indiffcrent betwllen thdse choices. just equate these present valtles and solve for /..1:
/z1+ 399
=
5 I 0.5 + 0.248 /..1.
$148.3 million. Hence thkleconomies
of scale wlluld
have to be quite Iarge (so that a 2()()-MW coal plant was less than 75 perccnt ()f the cost of two 100-MW oiI plants) to make giving up the flexibility of the smaller plant
or,
1)
=
economical.
6
Guide to the Literature
The net present value criterion and its application to investment decisions is an important topic in corporate Nnance courses, and is the starting point for much of what we do in this book. Readers unfamiliar with NPV calculations, including the use of the capital asset pricing model to determine risk-adjusted discount rates, may want to review a standard textbook in corporate hnance. A good choice is Brealey and Myers (1992). Although we largely ignore the implications of taxes in this book, they affect the choice ot discount rate for NPV calculations. Taggart ( 1991) can
provitles a rtzvievv of tlltl varitltls apprllacllk!s t(.)c:llctllating tliscotlnt rlttes (adtaxes) Ibr tlstl in tlle st tntlct rtl NI7V Illtpde l Rtlbltck' ( l 9S()) J'ustetl for risk antl Il()l'l1inlll tnlsll I1()&vs llfttlr-tax sllolyld ltlsvilys 17t)dist-tluntctl shows that riskless I'y at the after-tax risk-free rate (l'or cxanlple. tlle Treasu bill rate tinlt!s 1 Iu illtls ttnd Nlytl rs antl iuback l 992) derive t siluple ltlltl ratd tax ( ). the corporate lloNvs in N PV c:llculations. robust rule for discounting risky cash .
betNvettn
invcstThroughout this book Nve vvill enlphasize the tronnectiolls valuatitln exercising of linancial opt itns. Al t htlugll and the and decisions ment ('ptitlns itnd option pricing tecllccrtainly not necessary, stlnle l'anlliarity vvitll niqtles will be helpful svhen reading tllis botlk. Brealtly and N'lyers ( l9t)2) provide :1 simple introduction: so do thtl expository stllweys by Rubinstein tlctIlilttd treatnlonts, sec Ctlx and Rtlbin( 1987) and Varian ( 1987). F()r more JarrtlNv ltnd Rudtl l t)83). Nlthotlgh soluewhat 989'). and 1 l I-lull ( stein ( 985). ( also nally. for hdtl ristic 97f) ftl1. Fi article is tlse Sn1it11 1 by ( datcd. the survey ()pl illns. ;ts Nlllslln and N'jttrttln Kester of 1984 ( see )discussions investmenls ltnd t)f Clpe land. K()lIe ltnd ltstln l Chapter 2 l 987), :1 r. ( ( 1985), Trigeorgis and N1u rrin ( l gg I ).
part11
Mathematical Background
3 chapter
Stochastic Processes and lto's
Lemma
THls
t()()Is stllcllastic c:llctlcl lAIvI'IIl and thc next prtwide the mathematical lussdynamic prtlgranlming. and cllntingcnt claims an:tlysis that will be ttsed throughout the rest t)f this b()()k. Witll these t()llls. wc clln study investment decisions using a continuous-time apprtlach. which is both intuitiveiy tppealing and quite ptlwcrful. I11 addition, thtl concepts and techniqtles that we introduce herc are becoming widely used in 1: 11tlmbcr ()f arcas ()' ccllnomitrs and finance. and so are worth learning even ap:trt fnlm tlleir ttpplicatilln t() investment problems. This chapter begins with a discussitln of stllchastic proeesses. We wiil begin with simplc discrctc-time processesp and thu!n ttlrn t() the Svicnt!r prllcess (or Brownian motion), an important ctlntinuous-time process that is a fundamental building block Ibr many of the models that we will develllp in this book. Nvewill cxplain the mcaning and properties of the Wiener pnlcessa and shtlw how it can be derived as the continullus Iimit of a discrete-time random walk. Wc will then see how the Wiener process can be generalized to a broad class of continuous-time stochastic processes, callo,d Ito processes. lto processes can be used to represent the dynamics of the value ()f a prtlject, output prices, input costs. and other variables that evolvc stochastically over time and that affect the decision to invest. As we will see. these processes do not have a tmc derivative in thtl conventional sense. and as a result, cannot always be manipulated using the ordinary rules of calculus. To work with these processes. we must make use
slalllelllalictll
Xtz-/fyrcv//lz/ t)f
t
of Ito's Lemma. This lemma, sometimes called the Fundamental Thcorem stochastic calculus, is an important result that will allow us tt7 diftkrentiate and integrate functions of stochastic processes. We will provide a heuristic derivation of Ito's Lemma and then, through a variety of examples. show how it can be used to perform simple operations on functions of Wiener processes. We will also show how it can be used to derive and solve stochastic differenthat make tial equations. Next, we will introduce jumpproccsses processes continuously and show infrequent but discrete jumps.rather than tluctuate the ApFinally, in how they can be analyzed using a version of lto's Lemma. equations.which describe pendix to this chapterwe introduce the Kolmogorov the dynamics of the probabilitydensity function tbr a stochastic process. and show how they can be applied.
1 Stochastic Processes A stochastic process is a variable that evolves over time in a way that is at Ieast in part random. The temperature in downtown Boston is an example: its variation through time is partly deterministic (risingduring the day and falling winter). and partly at night, and rising towards summer and falling towards examplc: it anther unpredictable.l of is stock lBM The price and
random suctuates randomly. but over the Iong haul has had a positivc expected of growth that compensated investors for risk in holding the stock.
rate
Somewhat more formally, a stochastic process is detined by a probability over time t Thus. for given times law for the evolution xt of a variable < < t, etc., we are givcn. or can calculate. the probability that the h fl etc.. lie in some specitied range, for example corresponding values .r
.
-r2.
.rl
Stochaslic f's-tpcc--c-and Ito
'.
Lemlna
The temperature in Boston and the price of IBM stock are processes differ in an important respect. The temperature in Boston is a slatiolull' that This means, roughly, that the statistical properties of this variable process. long periods of time.'s For example. although the expected twer are constant temperature tomorrow may depend in part on today's temperature. the expectation and variance of the temperature on January 1 of next year is Iargely independ'ent of today's temperature, and is equal to the expectation and variance of the temperature on January 1 two years from now. three years from llonsuliolltlty now,etc. The price of IBM stock, on the other hand, is a process. value of this price can grow without bound- and. as we will soon The expected see, the variance of price F years from now increases with F. The temperature in Boston and the price of IBM stock are both contlltotts-tinle stochastic processes, in the sense that the time index / is a continuous variable. (Even though we might only measure the temperature or stock price at particular points in time, these variables vary continuously through time-) Although we will work mostly with continuous-time processes in this book, it is easiest to begin with some examples of t/-crc/c-/?ne processes. that is, variables whose values can change only at discrete points in time. Similarly, the set of aII Iogically conceivable values for (often called the states) can be continuous or discrete. Our dehnition above is general enough to allow alI these possibilities. One ()f the simplest examples of a stochastic process is the discrete-ltle t-crc/c-'/trl/crandom walk. Herea xt is a random variable that begins at a known takes a jump of size l cither up or down. value and at times t l 2. 3. each with probability 1. Since the jumps arc independent of ttach other, we 2 can describe thc dynamics of with the following equation-. -p
-r(),
=
.
.
.
.,
.
,p
.r3.
,
arrives and we observe the actual value xl we can condition the of probability future events on this information.z
When time
fl
r/
where 6/ is a
random variable
with
xt
I
-
+
6t
.
probability distribution
.
probte/
.!
1)
=
probt6t
=
=
-
1)
=
2
(J
1 2.
=
,
.).
.
.
We call a discrete-state process because it can only take on discrete values. For example, set m) = ()- Then for odd values ()f t, possible values /), and for even values of ?, possible values of of xt are (-f l l 0, 2. t ). The probability distribution for xt is found xt are (-/, -r?
' One might argue that the randomness is a rcdcction of (he limitations uf metcortlltlgy, and that in principlc it could be eliminatcd if we could build sufhcicntly complete and accurate meteorological models. Perhaps. but from an operational point of view, ncxt wcek's tempcraturc is indeed a random variable. In this book we will not attempt any detailed or rigorous treatment of stochastic processes. offering instead the minimal explanations and intuitions that sufhce for our applications. R)r detailed and general treatments. see Cox and Miller ( 19fi5), Feller ( 197 l ). and Karlin and Taylllr ( 1975).
-
.
.
,
.
-
,
,
.
.
.
.
.
.
.
,
-2.
.
.
.
,
3'I-his ignores the very Iong-run possibilitics of giobal warming
()r
cooling.
/3t?-/kj(?-J?/7?/ ibltltllzltltlliz-tll
fronl the binornial distribu tion. Fllr / steps. t he prllbllbi Iity tllat there downward
Jklmps
ltlld
/
-
11
(1
tl()Wll
Jtlmps is
are
1/
,
.
.
-r/
.t'f
--a
-
11
will take t7n the val ue J
.I
-r,
.
(')--.
Therelbres the probability that
for xt+ l depdntls on Iy (711 antl lltlt lttltl 1t itpnklIly (31! vvh:lt llappelletl bt.?l'(l1,t.? tirrleI For exalnplt!- in t htl cllse t)f t 11esi l'nple rlt 11t1tlnl hval k give 11 l)y eq uat it)11 ( 1), if xt zzz t''. t he 11 xt .y.I ca 11 etI tli.t I 5 (lr 7- (2 ltcll v it 11p rlllplb iIity J.a Tl) t, va ltlcs of xt I -Fl1eN'llll'kt'v prtlperty is e tc.. ltrtl irrelevltllt (lnce we k Ilow becausd it cllll greatly silnplify tlle :tl'lktlysis (41*:1 sttpcllttstic prklctlss. j-mportant svill see tllis shortly lts we tu rn tk) ctntilltltltls-t inlt, prtlcesses. e
2ll 11t time
-
J
is
.
-
.
2 The Wiener Process 'Wk will use this probability distribution in thtl next section when we derive the Wiencr process as the continuous limit of the discrete-time random walk. At this points however. note that the range of possible values that can take .%-t
t)n increases
with
?,
t,11'
as does the variance
Hence
-r,
.
is a nonstational'y
-r,
PVOCCSS.
Because the probability of an upward or downward jtlmp is 1. at time 2 value of is zero for alI t. (Likewise, at time tn the = () the expected for T > / is xt One way to generalize this process is expected value of by changing the probabilities for an upward or downward jump. Let p be the probability of an upward jump and t/ ( l #) the probability ()f tt downward //n7 f), the i; 1lt time J jump. with p > t/. Now we have a randonl ww/ wilh with and is incrcasing t expected value of xt for t > () is greater than zero. Another way t() generalize this process is to let thc size of the jumpat each time t b e :1 continuous random variable. For example. we might let thc sizd ol' each jump be normally distributed with mean zero and standard deviation fr. Then we rc fer to m as a discrcte-tinle ctpn/nlttxf'-xltl/f.l stocllaslic prfpctzw. stochastic process Anothcr example of a discrete-time continuous-state is the (zt-order ta//tr/rcgrt.'.s'.s'/r ll/-fpcc.-.abbreviatcd as AR( 1). l t is given by lhe -r,
.)
-rr
=
-
=
.
equation x?
=
t5
+ pxt
-
I
(t
+
(3)
,
where J and p are constants, with l < p < l and ts is a normally distributed hs the random variable with zero mean. This process is stationary, and value. of g'I-his value p), expected its current irrespective long-run /( 1 equation value setting in expected is found by (3) xt- l x; long-run and solving for -r.JThe AR( 1) process is also referred to as a mean-reverting expected value. We process, because xt tends to revert back to this long-run version of this process Iater in this chapter. will examine a continuous-time -
,
.r/
-
=
=
.r
Both the random walk (withdiscrete or continuous states, and with drift or without) and the AR( I ) process satisty' the Nlarkovproperty, and are thereThis property is that the probability distribution fore called &Ial-k-ov,;;l')ces-.e.b..
A
svienerprocess
also called 1.: Bl-ovltitlll ??lt)/?')?? is :t colltilltltltls-tilutt stochastic process t hree important prllpe rt ies.4 Fi rst- it is :1 :Il-l()b.' //?wcess. As explained abllve. th is rneans that t hc prtlbllbil ity tl ist ribtltitln llr :111 future values tlf tlltl prtlccss depends (Jnly t'I1 its ctlrrent vltle. llntl is tlI)llIL fected by pllst valtles 01' thc prtlcess ()r l)y ltlly tltller ctlrrent il'll'tlrllllttitlll. As :1 result. the current value ()f the prtlcess is ltIIt'lntl needs t() Illtlk'tl :1 Ilest lklrttcrtst of its futu rut vllltlt). Sectlnd, tlle Wie l'le r prllcess l'lrlsd'/lt/t.z/?t.vit(?/// l('l''lll,ll'. This means tllllt tlle probllbil ity distribtltitln jklr t 11tlcllktnge il) tlle prllcess (lveI()1tinle illte 1ar111. tny ('t Iler ( ntlnllverllppillg) any time interval is independent ()!' Third. cllllllges in tlle prtlctlss (lvtlr ally fin ite intervkll t ill'le llre ll()l'l3l(lIh' distribttted,with a variance that increuses Iinettrly vvith the tinle intef-val. The Nlarkov propcrty is particularly iI'npllrtltnt. Agltin. it impIies t I11tlt'Illy current inlbrm:ttilln is tlscll I )r lrec:lst ing the l'tlttlril plttl) ()f thc pnlcdss. Stock prices arc (llktln nltldtzlltld as Nlltrktlv prllcesses. ()11 tlle gnltl nds tllttt public in fllrmatilln is quickly incorptlrated in thc ctl rrent prictt ()f t 11estllck'- s() vllltle. (Tl1is is called tl'!c vveltk' that the past pattcrn of prices has n() tklreclksting forrn of market dfliciency. I1*it did not h()Id. investlprs clltlltl in principle the market'' through technical analysis. that is- by tlsing the pltst prkttel-n ()1* prices to forecast the futurtl-) The fact that a viener pnlcess has independen t increments means that wc can think of it as a ctpntintltltls-tinle versilln ()f 11 random vvalk. a point tllat we will return t() belosv. The three conditions discussed above tht.t Mtrk'tlv pftlperty. independent increments, and changes that are normltlly distributetl may seem tltlite restrictive, and might suggest that there arc vefy ft)w real-wllrld variallles Nvith
--lpeltt
41n 1827. the blltanist Rtlbtzrt Brllwn lirst (lbscrvetl and dttscribcd thtt nplltitln tll' snpltll particles suspendud in a Iitluid. resllting I'r()n1tlle Clppllrent successive llnd r:lntltlnl inlplcls ()1' neighbllring pilrticles; hcnce thc term Brllwniltn mtplitln. In l t?()5, Alltzrt l inslcin pr( )pt lsctl lt mathematical thet'r.y ()1' Brtpwnian mtltilln. which was furthtlr develllpetl illld nl:tdtt nltlrt.l rigt )r( 3tls by Ntlrbert Wiener in l923.
f'ltzc/cgrol//lf
Matllematical
64
while
that can be realistically modelled with Wiener processes. For example, it probably seems reasonable that stock prices satisfy the Marktw property and have independent increments, it is not reasonable to assume that price stock ehanges are normally distributed; after all, we know that the price of a changes in that can never fall below zero. It s more reasonable to assume logarithm stock prices are Iognormally distributed. that is, that changes in the of the price are normally distributed.s But thisjust means modelling the logarithm of price as a Wiener process, rather than the price itself. As we will see. used through the use of suitable transformations, the Wiener process can be extremely broad range of variables that vary as a building block to model an continuously) almost and stochastically through time. continuously (or of a Wiener process somewhat more useful properties the restate lt is to z(J) then Wiener is any change in z, Az, corresponding process, formally. If a conditions: following satishes the A/, interval to a time l The .
relationship
between Lz and Lt is given by Az
=
where Et is a normally distributed and a standard deviation of 1.
random variable
with a mean of zero
1 0 for 2. The random variable Et is serially uncorrelated, that is, t # s. Thus the values of Lz for any two different intervals of time with independent are independent. (Thus z(J) follows a Markov process increments.l -(6t6.&.
=
Let us examine what these two conditions imply for the change in z over this interval up into n units of some hnite interval of time F. We can break change the = in z over this intelwal is given length Lt each, with n F/f.Then
by z(. + r)
z(.)
Ei
=
1lo 'J Lelttlna
We will make considerable use of this property later. Also. note that the Over the long run its variance will go to Wiener process is nonstationary.
insnity.
By Ietting ht become intinitesimally small, we can represent ment of a Wiener process, dz, in continuous time as Jz
dt.
6J
=
the incre-
0- and Fttcj since et has zero mean and unit standard deviation. :((z)2) dt. Note, however, that a Wiener process has no time erivatiye, i Et (11)-1/2 which becomes inh ite as in a conventional sense; hz/zt . XZCUCS 20 FO@ app -ltz)
v''-f.
=l
The 6's are independent of each other. Therefore we can apply the Central is normally LimitTheorem to theirsum, and say that the change z(J+ F) which point, Iast tt fhis and variance F. with n distributed mean zero particularly and Lt, is Lt Lz depends fact not on the that on follows from important; the variance ofthe change n a Wienerprocessgrows Iinearly with the time horizon. -z(.)
=
=
=
'$
At timeswe maywant to workwith two or more Wienerprocesses. and we be will interested in their covariances. Suppose that zl (J) and ca(;) are Wiener h .. dz?) p1: JJ, where pIc is the coejhcieltt processes. Then we can write of ctprrtl/zlfon between the two processes. Because a Wiener process has a (z-(((/z)21/JJ 1)a variance and standard deviation per unit of time equal to 1 also the covariance per unit of time tbr the two processes.b is p!c -ttzl
=
2.A
Brownian Motion with Drift
We mentioned
earlier
that the Wiener process can easily be generalized into generalization of equation (5) is the
more complex processes. The simplest Brownian motion w#7 tryi: dx
a
=
(t
+ c dz.
where Jz is the increment of a Wiener process as defined above. In equatfon (6), a is called thtt drift parameter, and f.r the variance parameter. Note ihat is normally disover any time intelwal L, the change in denoted by a2 Lt. tributed. and has expected value CL-%.b a AJ and variance )J(-r ) ..r,
-t7,
=
Figure 3.1 shows three sample paths of equation (6),with trend a = 0.2 per year, and standard deviation t'z = 1.0 per year. Although the graph is shown in annual terms (overthe time period 1950 to 2000). each sample path wasgenerated by taking a time intelwal, Lt, of one month. and then calculating a trajectory for x(/) using the equation
'' Rccall that if and F' are random l CovfzrF )/(cx (.v ). In this case t'z r Jv
variables.
zr
5we always use natural logarilhms. that is, those with base e.
=
=
=
n -
(??lt/
=
l.
Et
-//c/ltlyffc Processes
=
=
.
their coefscient
of correiatitln
is pxt.
=
6(t
l Jltlt.'q?'r/tlplt/
lltlthelltatictl
Stochasic /''/'tzt:'cxxtts.(1,1,-1
'.$'
//tJ
l-cltlllla
>'! 20
!'
66% Confidence Intewal
15
-,----
-
10
z e t -
'
,,.'
e
-
-
.
-
-
-
.
-
-
-
.
.
.
L
..
-
/
(*- r >' l C 1-xk -
jl
'J
.
k
#
-
-'
-
-
#
!l
'
->--.-ys
?..
w ..
-
.
-jj ..
.
.
''
-
--
-
--'
-
-
--
-
-
-w
4 'hr $
5
.e
.j
( ,
..-. .,.. -
''
,.
...
-
..-
-
...1
.'
/
l
'- Reallzatlon .
.
O -5
55
50
60
70
65
Time
=
80
85
90
95
1O0
Time
(). In equation (7), at eah time ? 6, is drawn fnlm a normal with with distribution zero mean atld tlnit standard deviation. (AIs() note that the t'z have been put in monthly terms. A trend of ().2 and parameters a per ytlllr of ().()!67 per mllnth. A standard deviation of l per year implies a trend (1.()833 per implies a variance of l per year- and hence 11 variancc of i1 month, so that thc standard deviation in monthly tcrms is ().()833 ().2887.) -t1()5()
75
,
t ikln prllcess grtlvvs linea rIy wi tl1 t l1e t inle Iltlriztlll. ! Ile stlplldlt rtl (41' tlle ?-r?f?/ fitltl Iltle i Iltlrizlpll.l'1 he ellctl t f')(')-I7e rck!n t ctpn t rne grows t I1esqllfllv ltllelttl lrtlcklst is F ;t given I)y t I-ls interval t'llr nn1111
tlevii.t
Wiener
;.s
.()
A' 1 ) 74
+ () () l t''l()7 F + () 2887 .
.
r
.
.()
=
=
Also shown is a trend Iine, that is. equation (7) with (qt (). Figure 3.2 shows an optimal forecast of the same stochastic process. Here, a sample path was generated from l 95() t() the end of l 974, again using b)-15 equation (7), and then forecasts of to 2()()(). ) were constructcd tbr l (For comparison, a rcalization, that is. continuation of the sample path, was also generated.) Recall that because of the Markov property. oll-%' the value of -r(J ) for December l 974 is needed to construct his forecast. The forecasted value of for a time T months beyond December 1974 is given by + 0.t)1667 F. -l974+r =
.r(/
-r
=
-rlt974
The graph also shows a 66-percent forecast conndence interval, tllat is, plus or minus one standard deviation. (A the forecasted trajectory for gs-percent conhdflnce interval would be given by the forecasted trajector.y plus or minus 1 standard deviations.) Recall that since the variance of the -r(/)
.96
One can sinlilarly clpnstrtlct 9()- t)r tls-percent ctln litltrllce illtervltls. (ll-lselwe klld alsl' l'nlnl 3.2 tllat. in tlle ltlng rtl n. tlle On(l can Figures 3. 1 ()f nltltitlnvvllereas in the shtlrt the Brllwnian dominant dcternlinilnt ' trend is Atgain. t'lf volatility the pnlcess dtlminates. t 11is is :tn inlplicat itpl) t)I' run. the ()t' (.p ) is a t and the stand:krd tleviat ikln is f'.r ? the fact that the mean hlllds. for large / that is- thc lklng run. / <<::I btlt I'(lr small I tlle (lppllsite when f.x > t). Another way t() see th is is to ctlnsider the prllbabil ity that xf < This probability is ver.y snal I Ibr large ? btlt is abllut .!. I'(lr snlall ? 2 The Brllwnian mtltilln with dri is a fai rly simple stllcllastic prtlccss. but it is impllrtant that it be clearly understolld. At this ptlint- t he delinititln of t Wiener process and tlle properties of' its general izatilln in ctltl:ttitln (f) ) nlay t/-r dtlpend t)n thut square seem somewhat arbitrary. svhy.for examplc- should (// changes in ? And is i reastlnable expect of and t() uve r dt t ust not j on root mlltivatc be ntlrmally bcttcr distributed'? One Fnite interval t() vvty t() any equation (6) and its propcrties is to show llovvit rdlates t() a random walk in discrcttl ti me We turn to th is next. -,
.r()
-
.
.
-
-t)l
.
.
-t'
,
.
68 2.B
Nlillllenlatical
Random Walk Representation
of Brownian
l'lt/c/cqrr..lldll/
Motion
stocllastic
J'sptrtr--tzx tlpz(/ lto
'.
(')9
-crrl/?ifl
Let us examine the distribution for future values ofx. First. observe that ) Lh. Also, the mean of l-r is I (p -gl-r
Here we show how equation
(6)can be derived as the continuous limit of :1 discrete-time random walt.7 To do this. we will divide time up into discrete periods of length Lt, and we will assume that in each period the variable A' either moves up or down by an amount h. Let the probability that it moves 1 p. Figure 3.3 up be p, and the probability that it moves down be q shows the possible values of x in each of three periods. assuming it begins at For each possible combination of t and the point the probability of it being reached is also shown. Note that from each period to the next, is a random variable that can take on the values id./l. Also note that follows a Markov process with independcnt increments the probability distribution for its future value depends only on where it is now. and the probability that it will move up or down in each period is indepedent of what happened in previous periods. =
.v().
=
(t,((
2
)
-r
-
1
=
2 2 p ( t h ) + q ( A/; ) -
.r
.r
l
3ih
xc 2h
p2
''-'-'---
MN
Vt-rl
-.
=
((
A-)21
-
(tt;(A.tj2)
tl
=
p
.M
Nw
.<
3/)2(7---.Nw
2pq
q xc 2h -
AAv 'w
'N
M
NM
-
qa
Fkure 3.3-
l= 1
t= 2
4 pq (A/;)2.
=
(8) -.r())
/? ( p and variance
l (l
-
(p
h
q)
-
-
t/
/
=
(#
)2) ( Lh )2
q ) z /7lt
-
,
4pq t ( A/1)2/
=
tt
.
To interpret this. consider a series of n independent trials.where a success in any one trial counts as 1 and occurs with probability p, while a failure counts with probability of successes in n 1 t/ p. The number as 0 and occurs independent trials has a binomial distribution with mean np and variance npq'' see Feller ( 1968. pp. 223. 228). The expressions above are analogous. Now 11 variance, for example. is success counts as L11 and a failure as -A/?, so the 4Lh )2 times that ()f the usual binomial exprcssion. So far the probabilities p and q and the increments /1 and Lt have been chosen arbitrarily. and shortly we will want to Iet Lt go to zero. As it does. variance of (m m)) to remain unchanged, and we would Iike the mcan and choice of p, :?. Lll, and Lt. In addition, particular independent of thc be to would Iike to reach equation (6) in the limit. We can ensure that this will we indeed be the case by setting
t
---
'''w
I != 0
q )2j(A/2 j2
-
-
-
3pq2--
2
xo 3h
(p
-
A time interval of length t has 11 = t/ht discrete steps. Since the successivesteps of the random walk are independent, the cumulated change (xt is a binomial random variable with mean
=
MNM h
xc +
.
Thus the variance of A..r is
...<
#3
+
2
-
-r,
Xa +
(Aj, )
=
l= 3
and P
Random W/k Reprasenlalpn of Brownian Motion
Then
-
-
g
-'
l +
p ''rhis approach is developed in detail by Cox and Miller ( 1965). Our exposition is based on Dixit ( 1993a).
h
4
-
J'l-f q
=
-
-v()),
c
J''''-l t ,
q
.
a
(T
Substitute these expressions for the mean and variance of (-p -
j
=
Af
!:
-
(z =
U (T
g I
-
,/-f
-
(T
j
.
Lh.
and p q into the formulas above for and Iet Lt approach zero. For any snite -
Backyroltllzl
A./tl//?t'??It?/(.-(l/
/, tlle nulmber of steps, ?l. then gtes converges to lt nornlal dist ribtltion, t
a tr
/
1
2h
(y)
-J -
''i:
A/1
2
and variance
to intinity. and the bintlnliltl distribtltitll) witll I'neltn z
/?
af
=
t :
where. again. tlz is tlle increment (.)f a Wiener process- and (.r. J') antl /?(.r- t ) the d l-i ltl'ld variare knovvn (nonrandonl ) functions. Tlle new feattlreltntlis t hat 111: current stattt tilne. Tlltt tnllltnutltlsance coeflicicnts artl functions of represented stochastic equation by a-(t't ( 1l ) is tralled :111 lto process time p r(7cc*' ()f ()1* 11
(,.2
t
.(j
t
2 o. t
-.-s
At
.
These are exactly the values we need for Brownian motion-, a is the drift. and (,2 the variances per unit time. ln the Iimit as ti .-..> (). both the mean and are independent of Lll and Lt. variance of (-p We see, then, that Brownian motion is the Iimit of :1 random walk, when the time interval and step Iength go to zero together while preserving the relationship of equation (9). Furthermore, this relationship between Lll and (-rI ) Lt is not an arbitrary one: it is the Qnly way to make the variance of should now be clear depend on t and not on the number of steps. Thus it why dx in equation (f))depends (via l.zj on the sqtlare root of tl and not on dt. And it should also be clear why changes in over inite periods are normally distributed; as the number ()t- steps becomcs very Iarge. the binomial distribution approaches a normal distribtltion. (), thc total An interesting property of Brownian motion is that as Lt distance travelled over any finite interval of time becomes inhnite. This lbllows A/7 with certainty. t'rom the relationship between Lh and t Since 12-rl 7(1A.,r1) Lh. Then the total expected length of the path ovcr a time interval ' of length t is .r1))
-
.r()
-
.
the incrernellts this palcess. Sillcc Consider the mean and variancc t// equal variance of t-t' is to tl (), The 12) (-r t ) -t#zl - t-(J-t-) l ( ilt )2, antl (lt, tlf orkler dt )5/2 vllich is (tz)contains in ((lt in ) ternls in which For dt infinitesinnally snlall. terms in ((// )2 and ((/J )'1/2 can be ignored. and to order tlt the variance is Flt/-t-) b? ( t ) dt -ltt'z
--lt/-t'
-
.
.
,
.
.t-
.
.
tll-lj
iIlstl ntrllleotls
'We re l'br to t? (-r l ) ls the expected and to /?2(.r l ) lls tlle instantanellus .
'tln'tlll.-(:
i ?-tl/tz
(')f'
t lle Itf.) prtlcess.
/-41//-...
.
.r
-...
=
.
Geomctric Brtlwnian Nlotion ltlotiolt
(')j' tlqtllttitln An ilnpllrtant speci:tl cllse ( l l ) is t llc gctlttetric /..;D-r)l$???tl?? 11t.1 tryi lt I'1c rc tl ( / ) tt /?(a- / ) f z'.r vvllt.trtl a lt 11tl f'r :1re ct')llst lt 11ts. witll In this clkse eqtllttitln ( I l ) bectlrnes -t'
.r
.
.
=
112 /?
=
J
2 11 . 2t
I fz =
,
(l
x
.
-
.
-r
x
-r
t lt
+
t'r
-t- t / z
.
From (lur discussion ()1*the simple Brownian I'notitln of eqtlatillll (f)).we k nllw are ntlrn-lally distribtlted. Since tllcse that perccntage changes in a'a absolute ehanges in logarith m of naturktl are tlle changt!s in are Iognormallv distributed. The relation between and its logarithm is stlmewhat mtlre compl icatttd in this context. In thtt nttxt section we will show that if ( ) is givt!n l7yequation . (12), thcn F(-r ) log is the lollowing sinnple Brownian motilln wth drif t: -t-/-r.
t
t
-t'.
.r,
+x, depending which goes to inlinity as tt goes to zero. Likewise, Lx/tt th motion of thd sample whether l-r Brownian +A/l paths Thus a or on will and such and and not paths downs, look very jagged, must have many ups will speak of and exist, we cannot not be differentiable. The derivative t/-r/J/ /t/I will exist. will and (1 ) 8 t#-v/Jf ). However, -W-r) in general so --,
=
-
.
(t7t#-rl.
.v'
-r
(
-t'
=
fl /--
=
2 (a - .!a f'.r ) d t + c t Jz
.
()f-r
3
Generalized Brownian Motion-lto
Processes
*
;
broad range of alI of which lre
nllrmally
is so that over a finite time interval2 t the change in the2 logarithm .!. variance tscl 1k) with and f, it can l)e r.r t As r .r i dstributed menn (c - rz ) t 2 val / of expected the ( ) is give n I)y ue shown that if currently ( ()) .
.
.t-
The Wiener process can selwe as a building block to model stochastic variables. We will examine t number of extmples,
-r,
=
.t-
.r4),
-gr (/ ) I
=
at
(-JU''
)
klthematical
72
and the
Stslci-#l-tXl?lf
stocltastic '/-tlc'c--t...
'.
of x(/) is given by8
variance
F(-r(?) 1
-vk)
=
2
e
7a I
e
t.y a
f
' -
jj
15000
.
This result for the expectation of a geometric Brownian motion can be used to calculate the expected present discounted value of -r(I) over some period of time. For example, note that G)
(r
-
alld /'/t'p Len,ln:a
x(1) e-r' dt
e-qr-lt
()
()
dt
m)
=
m)/(r
=
-
( 14)
(z)-
provided the discount rate r exceeds the growth rate a. This will prove useful in later chapters when we will need to calculate the discounted present value of a prost flow that follows a geometric Brownian motion. Geometric Brownian motion is t'requentlyused to model securities prices, well as interest rates. wage rates. output prices, and other economic and as tinancial variables. Figure 3.4 shows three sample paths of equation ( 12)a 0.2, that 0.09, that is, 9 percent per year, and fr with a drift rate of a numbers particular chosen because they were is, 20 percent per year. These
#
are approximately equal to the annual expected rate of growth and standard deviation of the New York Stock Exchange lndex, expressed in rea. (constantdollar) terms. As with Figure 3. 1. the sample paths were generated by taking is calculated using the equation a time interval. Lt. of one month. Then,
',t
: ff:l :,$pC l; j,. l
1OOO0 7500
''
I
)' 5000
,
''*b
,)' l ; . ? $, ; , I
0
' ,
'
..-
50
55
/'
'N
,.
.-
...
.>
60
65
.wd
..-
o
w
# =
<'..J >
/
1J $
f
.
85
90
.z
.)
jj
r s)I j
,
,,,,%r
. .)
a-+wk /xa
.
%,
t%Mjz
e'
otcf.w
*
.
l
1*
-
.
,
.
2500
'
'
k /
,
.)
k
=
=
l; 4$ l'1
12 5O0
,-
j
.
N
.* .-.z'
-
70
75
80
95
10O
Time
.r(?)
!i I
'
xt
! '
( .
!
kl-.
1
=
.0075
x?-
l
+ 0.0577 xt-
l
t
,
(15)
100. (Again, at each time 1, 61 is drawn from a normal diwtribution with with zero mean and unit standard deviation.) Also shown is the trend line. that is, equation (15)with Et 0. Note that in one of these sample paths the d'stockmarket'' outperformed its expected rate ofgrowt, but in the other two sample paths it clearly underpedormed. .r195:
15000
/ / J /
12 500
66% Confidence lnterval
=
z'
10 0OO
=
Figure 3.5 shows an optimal forecast of this process. As before, a sample path was generated from 1950 to the end of 1974, and then forecasts of x(/) were constructed for 1975 to 2000. For comparison, a realization. that is, a continuation of the sample path. is also shown. Once again, because of the Markov property, only the value of -r(I) for December 1974 is needed to construct the forecast. The forecasted value of is given by -r
-iI974+r
=
(1.0075)r
z'
7500 z'
5000
.,e
A'
Aitchison and Brown and its properties.
( 1957)
A
/'
z'
z
z
/
Z
Z
#'
z'e
Forecas
A
-**'
2500 'e .e .-
.
0
1970
1975
1980
1985
-
.-
-
--
1990
1995
-rl974,
Time Figure 3.5.
Bsee
z'
/ /
for a dctailed discussion
()f
the Iognllrmal distribution
Opt?Tfz/
Forecast
t?JGeometnc
Brownian ifcft'pn
+
fytlc'/t--prrcpff/lt/
/k/tI//!tz,?ItI//t'fI/
utoclltls ic
l1lLl Ito r/-tzlt-t.tk(.'-
'.j.
Lclnllltl
the tigwhere F is moasured in months starting in January l975. Als() sllllwn in stantlllrd leviation the ctlnfidence interval. Since ure is a (nti-pereent ibrecast in changes grows with tlltl square root of the til'e horizon. percentage ot the upper and lower botlnds of this contidonctl intelwal are given by .r
(l
F
.0075)
.
.
YF ( l 0577) rlty7g and
F
.()()75)
(1
.
.
.0577)
(1
--V-U -
'
, tn4
.
.i
realNote that this contidence intelwal becomes qtlite wide. In this particular tlnderpertbrmed its forecast. market'' ization, the Hstock
3.B
Processes
slean-Reverting
gsee the Appendix to this chapter for a derivation of equations ( l 7) and ( 18). I Observe frol'u thesc equations that the expectcd value ot' xt converges to as t becomes Iarge. and the variance converges to c.r2/2??. Also. as ) cxa. l,7l)-r/l 0, vvhich means that can never deviate frolu even l'nolnent:lrily. -....
-....
-i:,
-t'
(7.2 ()- becomes t simple Brownian motion. and F1-: Finally. as ?? / ) Figure 3.6 sllows four sample paths of equation ( l 6) Ib r di ffe rtln t valtlcs 1 and of n. ln each case, t:r 0.(5 in nlontlllv terms, J l ) begins at The first is for 4 (), wllich corresponds to a simplt) Broqvnian motion Nvithtlut drift. Note that (/ ) tends t() dri ft l-ar fronl its initial val tle tlf I Sam ple pat hs are also sllown for q = 0.0 l 0.02, and ().5. Nllte that tlle larger t) is. thc Itlss (3.5, ( I ) rnakk!s only smal I and When 1) x (f) tends to dri away from short-lived exctlrsions away from Figure 3.7 shows an optimal forecast 01* this process. tbr t) = ().t)2. I 11 this case :1 sample path was generuted frol'n l t.)5() to the tlnd (lt' l t.)8(). and for l 98 I (t) 2()()4).I'--t'rcllmpltristlna 11 then tbrecasts of ) wcre ctlnstructed realization, that is. a contintlatitln of the sannple pllth. is ltIs() slltlvn. as is :1 .r
-..+
...-+
.
-r(/
=
-t'()
=
,
=
As the sample paths in Figures 3. 1 and 3.4 illustrate. Brownian motions tend realistic for some economic to wander far from their starting points. This is Others. Ctanvariables for examplc. speculative asset prices btlt not lbr such as copper or oil. sider, for example. the priccs of raw commodities motions, Brownian Although such prices are often modelled as geometric marginal related somehow should to long-run be one could argue that they !he pritx of oi1 migll! production costs. ln other word s, while in the shtlrt run revtplutions in oiltluctuate randomly up and down (in response to wars t)r weakening ()i- the producing countries, or in response to the strengthening or marginal OPEC cartel), in the longer run it ought to be drawn back towards th tlil should of btl of producing oil. Thus one might argue that the price
.r
-
,
-i
=
.
-i.
-t-(?
-.
cost modelled as a metll-reknet'tillgpt-ocess.
also known as an Onlsleill-
The simplest mean-reverting process is thc following: Uhlenbeck prcew
J-r
n (.i
=
.v -
) lt + c dz.
Ievel of x, that is. is the Here, n is the speed of revcrsion, and commodity price, then might the Ievel to which tends to revert. ( If x is a commodity.) Note that this be the long-run marginal cost of production of and If x is the expected change in depends on the difference between short interval greater (less)than it is more likely to fall (rise)over the next of time. Hence this process, although satisfying the Markov property, does not have independent increments. and x follows equation ( 16)s then its lf the value of is currently ttnormal''
-i
-i
-r
-i.
-:
-r
-i,
m)
-r
expected value at any future '
(.rI
time f is
)
.i
=
+ (m)
Time -i
-
-'lf
)e
.&-
=
.
76
Matllelnatical ftzc-/tvrtxf?lt-/ <j' '
'v' %'
t'1
and tben
.75
).
1 50
,1:
.
1.25 p
/
I
, I
(5
1#
9
,
I
$
I
%t
I9 1
l
$
I
$
l l
,'.e
.
.
# k
, I
1
l I
w
z
z
9%
/
1d1 J ' 1,
,.
?
'& ,
11
.
; f
,'
bb ze .>
q
.
I
p 9
,
j''
'
*
't..
J'
'
> 11
j
1
t
j
1
z
w '
'
-
'
1 ''
)1 h $ l?
1980
j
$
Forecast -
..*
-
-
-
---
-
---
-
'-
-
-
---
-
-
-
t-v
*
.
j
1985
1990
1995
=
=
(16)is the Iimiting case
-.+
(19) 6/
is normally distributed with mean zero and standard deviation
fnk,
Thus one could estimate the parameters of equation (16) using discrete-time data (theonly data ever available) by r'unning the regression
xt
-
xt-t
=
.r) -
dt + rz dz. .v'
(20)
a + b xt-
l
+
t
,
-r
-
.x'
We will examine the implications of different mean-reverting processes for investment decisions Iater in this book.
2000
version of the first-order autore-
gressiveprocess in discrete time. Specihcally, equation 0 of the following AR(1) process: as Lt
where and
z? (.i
Alternatively. proportional changes in a variable might be modelled as a simp1emean-reverting process.This is equivalent to describing-rt/) by the process J-x ?? (-i x) dt + a Jz.
43%
continuous-time
=
.
OptimalForecast of Mtlfzn-flzt'r/z?!l Process
(16)is the
.
1
-
-r.
t-percent conhdence interval for the forecast. Note that after four or tive c2/24 0.0()25/0.04 0.()65. years. the variance of the forecast converges to conhdence the 66-percent 1 interval so (+ standard deviation) convergcs to the forecast +0.25. Equation
+ b)
-
( l + /7)2
-i
I
Time Figure 3.7.
log ( l
,
-r(f)
,
=
1975
log (1 + bj, and
-
, is the standard error of the regression. It is easy to generalize equation ( 16). For example, one might expect to revert to as in ( 16), but the variance rate to grow with Then one could use the following process:
'
-
...
..,
--
=
=
where
l
)'
j
?'i
=
l
2 lz'f ? ---
j
--r---
IJ
.
-t1//7,
-i
Ii
)...f
' '
Ltlvllzlt.l
1
rj 1
I j
's
J'
h
' .
1 I '
'
calculating
' %.
)
1,
I
a'-
#
, e
,.
-
lk'I
:
i
t
1
11 j
0 50
I
.yj?
I
%
,
I
$1
,
l '
.
,
/%.', j
'j
'
,
'''
.
''
'1
,1
j
l
R6
j
'
'
''
-
/
.
'
*
,,.
,e''
l
0.25 1970
z
/
l
$.
11.;
js
..
'''''
..e
'
,
'
n te r'va
l
.
I f
.a'-'''--z-'-.----'-
confidence j I
66oz
'
)
*;!
, 1 '
JI
/1
()
klscti-lic Processes tl?1l Io
Betbre closing this subsection. let us return to question we asked earlier: a are the prices ofraw commodities and othergoods best modelled as geometric Brownian motions or as mean-reverting processes'? One way to answer this is to examine the data for the price variable in question, and in particular to estimate equation ( 19) and test whether the coefficient of xt- I on the righthand side is significantly different from zero. There are two problems with this. First, under the null hypothesis that this coefficient is indeed zero (so that xt follows a random walk), its ordinary least squares estimator is biased towards zero, so one cannot use a standard J-test to determine whether the estimate is signiticantly different from zero. However, there are alternative ' tests, called ttnit root Jc'a, that can easily be applied instead. () Second and more serious. it usually rcquires many years of data to determine with any degree of contidence whether a variable is indeed mean reverting. As an illustration, Figures 3.8 and 3.9 show the prices of crude oil and copper, in constant 1967 dollars. over the past 120 years-lt' A cursory Io()k at these hgures suggests that these prices are mean reverting, but that the rate of mean reversion is very slow. This is indeed confirmed by running unit root tests on the data. Running these tests on the full 120 years of data, tnne can ''The tcsts wcre originally developed by Dickcy and Fuller 1981), and have since been ( extended and rehned. See Chaptcr 15 of Pindyck and Rubinfcid 1991) for ( an introduction to thesc tcsts. ld'''l''he data for 1870 to 1973 are frt)m Manthy ( 1978); data after 1973 are from publications of the U.S. Encrgy Information Enttrgy ald U.S. Bureau of Mines. Prices were etlated by the Wholesale Price Indcx (now the Producer Price lndex).
/bltltllt'ltltltik'tll pltlc'/lrflll/lt
80
15.0
70
12.5
60
10.0
50 7.5
40
5.0
30
2.5 O
20 10 1870
.0
1870
1890
1 910
1930
1950
1970
1990
1910
1890
1930
1950
1970
1990
Tim Iziqtttp-tr 1?.?l.
())'
/:rl't'c,
lt/t'
('r7
(lil ilt l t/f) ;r /)(,IItt?:j' J)t'r l/ztln-.l
easily reject the random walk hypothesis: that is, the data conlirm that thd prices are mean reverting. However, if ont, performs unit root tests using data for only the past 30 or 40 years. one fails to rtject the random walk hypothdsis.
This seems to be the case for many other economic variables as well; using 3() random or so years of data. it is difscult to statistcally distinguish between a walk and a mean-reverting proccss. As a result, one must often rely on thcoretical consideratitlns (for example, intuition concerning the operation of equilibrating mechanisms) more than statistical tests when deciding whether or not to model a price or other variable as a mean-reverting proccss. Another criterion that may enter into this modelling decision is analytical tractltbility. As we will see in Iater chapters, valuing a project and solving for an optimal investment rule is oen much simpler when the underlying stochastic variables are modelled as geometric Brownian motions.l l
4
Ito's Lemma
'We have seen that the I t() prllcess ()f cqtlatitln ( I l ) is ctlnt inutlus iI1 time- btlt is not diflkren tiable. l-lowever. we wiIl oen nced to wllrk wi t 11functions o1* Ito processes. and s'vtr will want to take the diffcrenti:tls ()f stlch functions. thtt vllltle of an tlptillll t() invest in it ctppper For example. we might tltlscrilltt mine as :1 function of thc price ()ir ctlpper- which in turn might be rttpresentild by a geometric Brownian motion. In this case we wlluld wrtnt tt) determine the stochastic process that the value of the (lption fkllltlws.-R)do this. and in general tt) differentiate or integrate ftlnctions of I to processes, we will ntted to make use of 110 I-clllllla Ito-s Lemma is easiest to understand as a Xtyltlr sdries expansion. Sup) follows the process of equation ( l I ), and consider a function pose that Fx, t) that is at Ieast twice differentiable in and once in t We would like to hnd the total differential ()f this function, J F The usual rulcs of calctllus dehne this differential in tc rms of first-order cllanges in and t : '.
.
-r(/
-r
.
.
ll It is usually impossible to obtain analytical solutions for optim:ll investment rules when the underlying stochastic variables are modelled as mean reverting, so that one must instead usc numerical solution techniques. For a strong argument ugainst modtllling the prices ()f oil and other exhaustiblc resources as gcometric Brownian mlltions. sec Lund ( l99 lb).
)3 F (J-r)3 + (J.r)2 + 1 J? + 12 + t: i).r3 ().:-2 ?t ;): F
JF
) F Jx 0r .
dt and cannot be ignored when writing the differential of in equation (25) captures exactly this offect.
in x.-
changes But suppose that we also include higher-order tcrms for
We can easily extend this Taylor serics expansion to functions of several t ) is a function Ito processes. For example. suppose that F = F-txl x,,,, where of time and of the ??l lto proeesses .v,,,
.
.
. .
dx ()2
=
J2(.r
(J/ )2 +
j'j
.
-
,
with C(#-n dzy.)
dF
=
bl (x t ) it ,
=
=
.
.
.
.
.
.
.
.#-
t
..'.'=
l
=
??l
pij /? Then Ito's Lemma gives the differentia! J F as .
.)F
pF tt + lt
Again. we can substitute
infinitesimally Terms in (Jf)3/2 and dtlz go to zero faster than J/ as it becomes small, so we can ignore these terms and write
(t-v)2
.
i
(:/1).1/2 + /p2(x, t) dt.
a (.r, t ) gpt-r.f )
.
.
-v)
vanish in the limit. To see In ordinary calculus, these higher-order terms all the whether that is also the case here, expand the third and tburth terms on for 11 to equation r ( ) right-hand side of equation (22).First. substitute determine (Jx)2: '-!
F. The extra ttlrm
;)F
p.v'i
fl i
pt
( .r
l
,
.
.
.
/
:
i
(26) for J.rf
.+
p2 F
.!.
dxi +
.
dxi
px p-r./
./
t-r/
and write this in expanded
t)
2)F
.+
!z.
bz ( r I ;
a
)
.j-i
'
-
.
.
.
.
.
'
form as t)
,
i
C)2F
)
.,r
.
ci
.
(28)
equation (22). every term As for the fourth term on the right-hand side of raised to a power greater than l in the expansion of (t.r)3 will include dt the case will to zero faster than (J in thc Iimit. This is likcwise .
and so go for any higher-order differential d F as
the terms. such as (#.r)4, etc. Hence Ito's Lemma gives
pF dbt =
.
(lt
+
;) F pr
.!.
dx +
2
Example: Geometric Brownian Motion. Let us return to the geometric Brownian motion of equation ( 12). We will use Ito's Lemma to show that the process followed by Ftx ) log x is indeed given by equation ( l 3). Since 1/.r2,we have from (24): (), ) F/)-v l /.v and p2 >-/J).r2 )Flt
p: F (r)2. i) :.2
=
=
equation We can also write this in expanded form by substituting
dF
=
DF
at
&1F
)F
a + alx t ) . + 1. 2 b (.r. t ) px a ;)x .
1)x
F/0-2 > 0), and negative if F is a F is a convex function of (that s, 02F/p.2 < 0). For Ito processes, dx behaves concave function of (thatis, effect of convexity or concavity is of order the and (J.r)2 Jl, like so like dt, er
.x
-
(29)
C)FJz. dt + :(-t-, t )
=
just
=
.
( 11) for J-r:
Compared to the chain rule for differentiation in ordinary calculus. equation simplicity that the (25) has one extra term. For some intuition, suppose for 0. Now Cdxl = 0, but CdF # 0. drift rate J(.x. 1) (), and that ) Fjt will be positive if This is an implication of Jensen's Inequality. t(dF) =
=
Hence over any hnite time interval F, the change in log is normally distz2) F and variance c2 F. tributed with mean (c log less than (x? The reason is that log x Why is the drift rate of F(x) is a concave function of so with x uncertain, the expected value of log changes by less than the logarithm of the expected value of (Once again this isjust a consequence of Jensen's Inequality.) Uncertainty over is greater the longer the time horizon, so the expected value of log x is reduced by an amount that increases with time; hence the drift rate is reduced. -v
-
=
-v
.v
-t.
.v.
.r
G.:ltltrll.rrcv//pJ
klalhonatiall
82
Example: Correlated Bnlwnian Motions. As a sectlnd examplc. consider and v each follow gcome t ric Brownian = where the tunction F(.' motio ns: .1,)
f'/t'p Stocltastic
5
/7?'(?(..t!-t$t'A'
tllltl
b. /..-c'??sF?lt/
kjl t 3 '>
Barriers and Long-Run Distribution
.r
.r.!',
-
t/a-
(/J
-x'
a
=
..
+ a'.
-r
.r(),
.1.tz
t/.7,.
-'t-
and 1'(.)1 Suppostl starts at Iosvs t I1e si nlple Brlqnvniann'lotiol: (lt' eqtl:tt ion (6). Left to itse 1$at ti rfle I it will llave a norlnal distributitln &vi t 11I'nttan (.:) ?) and variance t'r: l'. However, now wtt constrain the process 017 not to an upper rellecting barrier at T. Using our random walk representation- t llis means that starti ng at Jf 2. 11 if tf ies to taktt 7111 upward step it is mttvetl right back to T th (A step down to T 2 h is allowed to occtlr witlltlut interference.) Similarly, we place tt lower retlecting barrier at &. In economic applications, stlcll barriers often occur because tlf tlquil ibratmechanisms in the market. For examplc. if x is the price of a commolity. ing subject to an upper barrier where new tirms enter, and :1 Iower barrier it is where incumbent Iirms will exit. gv'ewill encounter stlch mllddls in Cllaptdrs 8 and 9. Here we want to consitler what llappens when follows sucll a process for a long time. The intluence o 1-the initial .rll wi II tl isappear when one tlf the barriers is hita because of thtl Markov prllperty. As tlle motitln goes back and forth trapped betweutn the twt) barriers- we cxpect it will settle lown t() a stationary long-run prouess. Nve want to Sind the probability density of this distribution. which we denote by / ( ) Once again we use thc rltndtlm walk representatitln. Clnsitler any three Lll I n :t ny llI1e sma lI t inle i17te 1-::11 adjacent ptli nts. say a nd x + th Lt, the probability mass 4 (-r Lll ) mtlves tlp witl) Ilrlll'habil ity /?- lntl tlle mass / (-r + ll ) movt!s tlllwn with pnlbability /, where p knd f/ Ctrtt given I7y equation ( I()). These mtlves const itutc the probability mass ltt ilfter thc entl of the interval. If the probability distribution is tt) be statillnary. this must be - t $ (.r) Thus Jus
,
'ross
-'
(1v'
a
=:
,
: tl ;
.
+
t'.rl
.
.v f l
--
,,
.
with Jz,.J p dt 'We will tind the proccss followed by F (-4the process followed by G log F. .3/),
'd-z.v
=
.
.
and
=
;)2 FjL) dF
for
Now substitute dF
f/x
.r
=
t/
v+
!'
tlx
+
1 we have from
=
,
(27):
d)'.
f
a
F J/ + lt7'xJzx +
f.r).)
+ p qv
:.
F.
t'r,. tzr)
(3l )
Hence F also follows a geometric Brownian motion. What about G = log F? Going through the same steps as in the previous example, we hnd that
i
(av. +
=
er
! o.z
-
2
.
-
2
j.
.
c,2) dt + r,v .
.
dzv +
(32)
cz,, dzv. .
.
.
From equation (32)we see that over any time interval T. the change in Iog F (ybl. 1 fyjz, ) T and variance is normally distributed with mean (ex + r 1: 2 . 2 + o'l + 2p tyv crl T. lt7'r -
-
,
.
xf', where x follows Example: Present Discounted Value. Suppose F (-v') the geometric Brownian motion of equation ( l2). We will show how to calculate the expected present discounted value =
C ()
tz -r
db-
=
(?tz +
! 00 2
Observe from equation Hence we can use equation motion to write
dt +
t'.r
JJj
.r
c2j F dt + p - 1)
-
(r
1)
/)j
.rP-2
a2 r2 ''
dt
F du.
=
xejfr ()
.
.r
.r.
-
,
-
.
-
.t'
4( (33)
(33)that Ffollows ageometric Brownian motion. (14)for the expectation of a geometric Brownian
F(m)) exp ((pa + 1.0 (# 2 and the present discounted value is
The omitted terms all go to zero faster than (kA/7)2 (or AJ
from both sides, divide by lLh )2, and takc Iimits as
.
.
.
-
Lll
,
.
.
) .
.
)
. ,
). S() we canccl 4 (-r ) --,.
(). This gives the
Nlatherrtaictll
84
diftk re nt iaI eq ua tion where y
4'' (-r)
y
=
4' (-r)
Sflc/fgprll/l'
The general solution of this equation
4(.r)
6 Jump Processes
is easily seen to be B,
zl eYx +
=
consider where and B are constants to be determined. For that we mtlst what happens at the barriers 7 and Consider the upper barrier. The mass rellected right of probability 4 (;f' A) moves up with probability p, but is probabitity the balancing equation masses is the hh. Therefore back to rf modised to ..4
:.
-
-
Expanding this and simplifying as before. we Iind 4'(T)
=
y
z,'1
/(2)
lz expty
.t')
/
7) (expt)z
-
exply
'tsudden
-r
we
-
.
other words
:.)1.
The long-run stationary probability density is thus a simple exponential. It is natural that if the x process has a positive drift rate (a > 0 and therefore exponential should be rising toward the upper barrier. and if it y > 0), the negative drift rate, the density should be falling to the right. If y. > t), has a :!. and consider a process with we can in fact Iet the lower barrier go to stationary distribution whose will have long-run 2) it a only an uppcr barrier at < (), if left the of 7. exponentially Likewise, we can Iet the y to density falls Ef upper barrier go to x. We will have occasion to make use of this distribution at severl points in Chapters 8 and 9. On a couple of occasions we will generalize the argument to death.'' More substantial extensions include allow ajump process of general Ito process, and examination of the acfollows a the case where distribution o x rather than only its Iong-run tual dynnmics of the probabilily require extensions the development of a differstationary state. Both these ential equation for the probability density the Kolmogorov equation and treat this in an Appendix to this chapter. -cx)
So far we have considered only diffusion processes. that is. stochastic processes that are everpvhere continuous. Often. however, it is more realistic to model an economic variable as a process that makes infrequent but discrete jumps. An example would be entry by a new competitor in a market with tkw value of :1 rms. so that price suddenly drops. Likewise. one might model the patent as subject to unpredictable but sizable drops in response to competitors' success in developing related patents. Or, one might view the price of oil as a mixed Brownian motion-jump process', during normal times the price tluctuates continuously, but the price can also takd Iarge jumps or falls if a war or revolution begins or ends. In this section. we discuss Poisson (>lpllp) processcs' and we introduce a version of Ito's Lemma that will help us to work with them. A Poisson process is a process subject to jumpsof (ixed or random size. which the arrival times follow a Poisson distribution. We call these jumps for lrrrfl/ rate of an event. during a time Letting denote the lneall the dt, probability that an event will occtlr inhnitesimal length of interval will and probability that not occur is given by by dt, the an event is given which l/, of size itself be is dl The event can a random variable. a jump 1 with analogy the Wiener by denote Poisson Let q a proccss; in process 'Ievents.''
4(7).
Substituting the general solution into this, we hnd B = 0. The same conclusion could have been obtaincd by considering the Iower barrier &. must be choscn to cnsure that the whole probaFinally, the constant bility mass between ,. and T must sum to one. This givcs =
'.j'
.
2 ajo'l.
=
11t1 //('? Lcntttla t'f Stochaslic farrpcctswfzt'
t Then
we write
with probability l with probability
() lf
=
tlt,
-
tlt
.
the stochastic process for the variable x as a Poisson differential corresponds to the Ito process of equation ( 1l ) as follows:
equation, which
Jx
=
J'x. t ) dt
+
,g(-r.
t ) dq
(38)
.
t ) and gt-r, f ) are known (nonrandom)functions. function of and t, and Suppose that S(-r. f ) is some (differentiable) expression expected change in H, that would for the like derive that we an to expand follows: d H as is, CdHl. To do this.
where fx,
-r
6IH
=
1)H )J
dt +
pH = )t dt
+
/?H 3 -):
pH
px
dx
(39)
g.t.r. /)
J/ + g(-t. t) dqj.
(Note that higher-order terms go to zero faster than dt because. unlike for the Ito process. dx does not depend on dt Thus changes in cause changes in S in two ways. First, Hx. t ) will change continuously and deterministically .)
-r
sltlllletnatical /.?fl()'/v-l?-t'Jd/'?ltl
86
! ;L.
.
in rcsponse to the dri in Second, thertt is 11 possibility that a Poisson event willoccur; if it does, will change by the random amoullt d/ gt-r. f ), and Hx. t ) will change accordingly. Since the probability that :1 Poisson event will occtlr over the interval dt is dt, we have .r.
-r
(40) where the expectation to the size of the jump lt. given by ()n
'tT
Hj
'H
=
)t
+
.
J (x. / )
side of the equation is wth respect Hence the expectation of the differcntial of H is
-tlz'
)
=
(J Iz-/f 11/3
=
ps J-
dt
-r
+
u (( Hx
+ g(-v'.t4 l/. /)
H (-r. t )1l (lt
-
dt
tl (.r t ) .
H (x +
pH
+ 1? /J a (x t ) .
ar ,g(.v .
/)
11 -
t)
-
;)2 H
lU
a-ra
Hfx t ) 1) tl .
P
(42)
Example: Present Value of Wages. Suppose an individual who lives forever receives a wage F41) which rises by a constant amount 6 at random points in time. If l is the mean arrival rate of raises, we can write the differential
-
t//',
.
so that
E:
+
#2
.
wage
Suppose a nlachi ne prt'd tlcos 11 kxlnstan t tltnv Exanlple: Value ef a Nlachine. :' it It reqtli res no ma i11tenance. btl t at son'le plli nt Iong operates. t of pro as as will will and have to be discartldd. I1' is the arrival rate time it break down in I'nach breakdown. and the discotl intl va Itle'? is /? of a n t ratd. what is t he Tht value of the machine follows the prtlcess 's
where an becomes
%eevcnt''
is lt
p P'
.
Note that the second-order derivative is relevant only for the variance contributed by the continuous part of the process. The jump part contributcs the last term on the right-hand side involving a difference in values of H at discretely different points'.
/) tt' tlle
Thus l'' is equivalent to 11 perpettlity that pays out forever the current 11/plus the capitalized valtle of the averagtl raise per tlnit tinle.
f
(lt
Stl I )
(1 6/p)
=
.
(4 1)
i)H + i)/' + Cu((
equation for the individual's
E:
fz-(Izl'''l
value of the change in the function Hfx, / ) is given by
' (J H 1
1/-
-
lz' J(/
.
=
l with probabil ity l Tllen the asset return equatilln
t/l
=
.
zr (/J + C(JJ,'' )
Thus, F=
7r
=
dt -
1/ J/
.
X .
p+
Hence we can treat the profit flow as a perpetuity, and valuc it by increasing This is a very goneral idea: if a profit llow can the discount rate by an amount when with arrival rate occursa then wll can calculate the a Poisson event stop value of the stream as if it never stops, but adding t() the expected present will discount rate. We come across this in many applications in later chapters. .
wage as
J )F 1 with probability l with tt earnings stream? expected =
In this case.
IZUIJ) t// +
=
=
.
tlse
then the expected
JJ
p 1ze
the right-hand
equation (4l ) in the same way that we used Ito's Lemma when with continuous processes. working Sometimes we meet a combination of an lto process and :1 jump process. The tbrmer goes on aII the time; the Iatter occurs infrequently. Then the appropriate version of Ito's Lemma also combines the two effects. Thuss if We can
We call treat 1..-as an llsset and cquate the nllrnnktl rettlrn on it at rate sum of the dividelld (currentwage) and tlle expected capital gttin:
.
=
(F (lq
,
(43)
What is the present value of the individual's
Guide to the Literature Our treatment of stochastic processes and Ito's Lemma has been at an introductory and heuristic Ievel. For a more in-depth development of stochastic
their properties. see Cox and Miller ( 1965), Feller ( 1971), and
and XtyKarlin and Taylor ( 1975. 1981). Cox and Miller ( l 965) and Karlln equations. of Kolmogorov the nice treatments lor (1981)provide particularly fora discussion ofthe use ofcontinuous Also, see Chapter3of Merton (1990) modelling. For a much more and economic and jump processes in snancial and Shreve ( 1988). Karatzas stochastic processes, see rigorous approach to of Ito's Lemma and discussions For more detailed but still introductory and Brock ( 1982), Malliaris Chow (1979), its application, see Merton (1971), of Poisson discussion nice also and Hull (1989).(Merton (1971) prtwides a Kushner rigorous treatments, see (1967), processes,with examples.) For more Also, for and Chapter 4 of Harrison (1985). Arnold ( 1974), Dothan (1990), of expected present values, see more detailed discussion of the calculation
a
(1993a).
Dixit
point
+ tll
-r
.
Hence
We recognize this as a dynamic generalization of the stationary-state computation in the text; see equation (35). Lll t Now expand $ (m),f(); x t ) in a Xaylor series around t) (-'E0 r(1 -'E J 1: -
-
.
.-
'.
.
/
(.r()/() ,
.'
-t -
lXh /'
t
-
.
t)
$ (-r()tt ;
=
,
-r
.
/)
$
2. /
-
Lt
-
2:./7
2)4 p-r
+
al
(t h )
'z
Note that third- and higher-order terms are of order (Al)3/2
hence will go to zero faster than Lt. Expand 4 (.r4).?()'. and substitutc these expressions into equation (44):
+ Lh
-r
:24
2
px2
(AJ)2, .
/
-
+
.
.
.
etc.. and
t.t ) Iikewise.
Appendix - (p
The Kolmogorov Equations
A
follows a At timeswe will want to answer questions of the following sort: If what probability the value is is x), particular stochastic process and its current that it will be within a certain range at time t later'? Or, what is the probability will have reached a point xl within a time f :G F? To answer questions that like these we will need to describe the probability distribution tbr x and its evolution over time. This can be done using the Kolmogorov eqttations. We will derive the Kolmogorov equation for the simple Brownian motion .x(l)
Finally, we use p + q
also substitute
Lll
=
=
o'
-
(1
;)4
) L lt
i)t.
+
,
2.
2
(p +
ty
) (25./?)
a
/924
-
.
2)v..'.!
1 and from equation ( 1()), p f/ ) Lt. We t(y/t7' J, divide thrllugh by Lt, and rearrange'. -
.
=
.,:(1)
representation with drift of equation (6).using the discrete-tme random that we introduced in Section 2.B of this chapter. Recall that we broke a with time interval of length t into n tjLt discrete steps, and in each step, probability p, would increase by an amount h, and with probability q 1 p, it would decrease by hh. Finally, to keep the variance of (m x()) f o. independent of the particular choice of Af. we set Lh Let /(m), /0; J) be the probability density function for -(I), given that walk
Equation (45)is called the Koltnogorovl-ward ctzllt?ltpn for the Brownian motion with drift. and it describes the evolution over time of the probability density function f) (m)- /4)', x. f ). In a similar manner. one can derive the Kolmogorov fonvar d e q uation for the gene raI Ito process of equation ( l 1): l 2
=
=
-r
-
-
=
.
.z.
at an earlier time
f() we
have x(0)
=
-v().
Thus
(46) Equations (45)and (46)are called t'forward'' equations because they have at time /#), and are solved forward as bounda:y conditions the initial value for the density function for future values of One can Iikewise describe the evolution of the density function backward in time, that is. taking at time t as the boundaq conditions and solving for the density function for previous .rt)
-r.
.r(/)
Over the interval of time t A/ to J, the process can reach the pont in one Lh, or by decreasing from the of two ways. by increasing from the point -x'
-
.r
-
I2Sce Karlin and Tayltlr ( l98 1) for dcrivations mogorov forward and backward cquatitlns.
.
and more dctailed discussion!i
()1'
thc Kt)l-
t'
'' .
i
l
!
1
:7
I 2
.4:
1
, '
. .
1i y t !
@ ) ). :j l ' kE ' jt : '! $
:k
Iue S
va
Of
PI-OCCSS
time
1t
-r()
lxlckbllll r. The K///?rltl,l.lt//-&t'
f4) <
.
.
r
'C !.' ,'r
,j
g,
.! b 2 (.-r()/()) ,
z
) y.a u
(-r() ?t)',
/
t ) -F.t'l (-rt)/0)
-r-
.
-
i) A () -
.
p ::::z
-
) l()
4 (-vi)/()
/)
-r
',
-
-
()
(-r() f() ,
',
-r
.
j
6F
'' ,
x tf +
. .
sy
jy c,
..y.
g.
j.a
j
.,yj
2
.
2
.y
o
.
t)
u)t)
c .a
jjy
,y
(y.y y
.
( g- (.yy )
((), t ) a jyqj t.-(.ry)
.jv)
..
(l 8 )
..
y,lto ( () t j, ( jao .
qjo
y,e rj
y. (ya j.y
.
.
....'
.
'
.
.r#
''
'
.
'
,
.
j,
.
,
..
,
.,
-
,..
.
.
.,
-
.,j
.
.
,'
.p),
,
,
. ') ..;.. t
',
'
uz
i)/
'x
vgz.
.....
ihi;k t-
-,,-:::-
I
t
.Ltz 2
...
;)a4
:
-
:
)..r
n
,t
-
;)() il-r
+
. 2 .j
(j ! )
n 4.
'jjjt
....'j
r---
y' j,
.-.!,. b'i
a
1.
o.l ()2 M
-
,
.
3A'/
)()
=
0A'J
(52)
.
t)t
v,: ...
,
.
y
...
. j .y q
,;,.'
'''.=
1
.-
.
ikl'v
(0 ()) .
=
m),
and
V/(())1
=
kV&f? (0, 0)
-
-r-()
a
=
0.
.
.,. f
.
1.,
'%
'
.. '
r!
jyj
g
'
'j
q q
f'.
tt
c.,'. .?
'
.1 .;1
(y
(.,r) Jdt +
rr (-r) dz.
(54)
tnc/tpn
'
j (-min)
j (-max) $). .
=
=
for the resource, ' 3nd
.
j (a )
>
0
and is concave. with fOr
-rnlin
<
-r
<
-rm:lx
.
)..j
'
',
-
.
'-
.
!
is the growth
.
'.
j ir
Here,
.(-r)
t.
.
)?
,j.
.
*,4
t,
'
gy(-r)
=
j..
:
'
t/-r
,
..,
'.
This partial differential equation must be solved subject to the boundary conditions:
Now consider a more complicated stochastic proccss. which clltlld btt used to describe the evolution of a rencwabie resllurce stock. )- sublect t() -r(?
:r
v..t
the text. and the solution pnlceeds as
.
.
<:'
,t.
ry9
(36) in
,.
?,,: ,,
J '(t,
'i
tlquation
-.
,
.!.
Ttjjtirsnis the same as
...,
.
t his for ls/bt in equation (50),and integrate by parts to get tj-j: substitute following equation for Ivle- 1):
,
.
'
=
,,
.'''
;
The Kolmogorov forward equation for this process is 0/
g k!
q.
.
,.')
(..5())
-?x
=
.
.
;!.).
)'. 6l
,yr
..
pM k,lL,-
A'f((), )
.
.
;
,
p
Then
.
..,,
.r(l
-N
! 4 l
j
,
.
..2j
-,
In the textwe asserte d thatequations (17)and (18)givethe mean andvariance claim. x(/). we can use the Kolmogorov forward equation to prove this aS ) 'Write the momen t-generating function f0r
.
k
y.yyj
Exatnple.. The Steadpstate Distribution lor a Renewable Resllurce. Tlle Kolomogorov cquations are partial difterential eqtlations. antl so (tre usulllly $.2 dificult to solve. However, ;k. we are often interestcd in the Iong-run steady-state ;'. . jaaracteristics variables. Not all stochastic of stochastic c t t processes will have . j bablity distributions that pro converge to some steady-state function (tlle getric Brownian motion for examplea will not but the Ornstdin-ulllenbeck ome .;. ., k' exists. in most cases it can process will). However. if :1 steady-state tlistribution '-' '. readily found using the Kolmogorov gorward equation-wllich be then redtlces rLb to an ordinary dierential equation. :, As a simplc first exercise-we begin by rcdcriving the negativc uxpt,nentiltl g, k..: distribution tor a Iong-run stationa:y Brownian motion between reqecting barriers dfrectly trom eqtlation (45).Tllis tlistributitln is independent ol thll i 1. pt' initial /(). and the currcnt t so we get an ordinafy difikrcntifll equltilln t')r
w
w
i i1
gyj g
( l 7) and
.
tlq
1
solutiol...
.
:
! :
yy
(a'
=
yt
,'
:,
' . 1. 9.' t 7 h'
lllosving
',.
of
''
)
,y..,,ty.
?
(.48)
dz.
tr
ve rify equa tions
-)
,
'; ! '
f
In this example we will return to the ornstein-uh ing) process of equation (16).For simple Ornstein-uhlenbeck (mean-revert equation becomes 0, so t h a t the simplicity,set =
/
.
li J #' jt t
lenbeck Process.
t/ v
(0 .
; ,,..vijjt
j
''
.
- -
,(. ',
-:
'
.
4:
--
u
.
.
il
Llsing t h e l'act t jla
i), )LL
qjq. vl. 4
.
I
,
.
,
I,as tl..c
(.:
:$,
jzt )v: i
it
t)
'
...
.
'.
J .t'ln:l'tlJ
can verj-y that the equation
jjyy,t;
).:'
(4.y)
.
-n
vhereadcr
:.4 .;
:
f)
'.
r.
1 j,;' j;, k, t1)$;..
this point it wewill use tbe Kolmogtrov eqtlat iof:sj tater in this btltlk, but at is useful to consider a couple ot oxamp eS. Example:
;
Ito
t//lt
tl'
''
)
lAl'll(:esse.%
.
ib.tt 'i7''
uftpc/lfl-lt;
.,;na .
.tt
is iiz
.
,
:j .. q.
.
-
=
i iJj'
:
.)
-i
;
'
.
.
..
. the lto iott tbr
tlf/ldf/f
; ,. .
'
:k
''1
:
'
l.
;
'
..,g5. (), . jjtkvi,
'fvk
B(lL..kjIr(-IlI' lt
/,.JtI//lt.,??ltlJ'f;l/
Q()
'.
' 5'..
t'
i
.
. '
rf. ;@' .'.tl. ;gk
.
t''.j. .,L6
' .?. ! .
Much of the literature on renewable rcsources deals with detcrministic versions of equation (54,),and compares the socially optimal t/(-r) with that resulting lrom a competitive market. However, as biologists and population ecologists have long recognzed. most renewable resource stocks ttvolve .
MathematicillSf-lckgrtplt?l Although tlucstochastically, and (54)is a natural for density probability x in study the tuate stochastically, it is of interest to steady-state equilibrium. /(), or / in the forIn steady-state equilibrium, 4 does not depend on x(), function density as 4co(a').Then the ward equation (46).Write the stationary once): equation becomes (afterintegrating representation.i3
'l
J
a (A')
(-t') /x (.:)1 L.1 .
J-r (c
=
This can be rcwritten as
#ltrz (x) 4x(x )1 (7.2 (x) 4. (x) =
-
2
.l
o.c (.r)
.
t?
(-r)1px
(-r)
will
-r(l)
(-E)
4
Dynamic Optimization under
.
Uncertainty (55)
q (.x)) dx.
-
Chapter
steady-state This can be integrated to give the followingequation for the
density:l'l
$x (-K) where ln is a constant
=
q (p) dv a a (p )
'r
m
o'a(-r)
yw )
2
eXp
-
of integration, chosen so that
As an example, suppose that
.f(.r)
=
JX
.
Sxxldx
fx, is the logistic tunction. (r
x (1
x/ K4
-
1
=
.
that is, TIME
,
that
capacity'' of the resourcc stock. Also, suppose wherc K is the fz Then equation there is no harvesting, that iss q (-r) 0, and that (.r) c-r. v2 < 2(z: when that applies density steady-statc (56) yields the following tcarrying
=
=
2a/frS
4...(-)
2 (2a/tz K4
=
-
I
za/rrs 2 e
2.r/rr
-
-
.r
:
&-
/
F-(2(z/rx 2
-
1)
,
(57)
determine that the where r' denotes the gamma function. From this we can expected value of in steady-state equilibrium is -x'
1
2 2a
a particularly important role for investment decisions. The payoffs
In this chaptcr
CT ..-
I>l-.kys
to a tirm-s investment made tllday accrue as a strcam over the future, and are affectcd by uncertainty as well as by tlther decisiuns that the firm or its rivals will make later. The (irm must look ahead to aIl these dtwelopments when making its currcnt dtlcision. As we emphasized in Chapter 2. one aspect ()f this future is an opportunity to make thtl same decision later: theretbre the option t)f postponement should be included in today's menu of choiccs. The mathematical techniques we employ to model investment decisions must be capable 01' handling :111 tese considerations.
.
reduce the steady-state expected value Note that stochastic suctuations &2 2e. Also, if of x, and as c2 approaches 2a, Cl-%.xlapproaches zero. is, stochastic Nuctuations will drive the resource stock to extinction, that 4x 0 with probability l jlzora more detailed discussion of collpses and well as a derivation of the this and related models of renewable resources, as optimal stochastic harvesting rule q' (-v),see Pindyck (1984).1 .x(;)
-.+
.
For a 13Sec, for example. Beddington and May ( 1977) and Goel and Richter-Dyn ( 1974). economics in a deterministic context. sec Clark ( l976). rcncwable of overview resource good applied '4Mertt)n ( 1975) also provides a erivation of this equation. and shows how it can be populatien. evolving with stochastically a to a neoclassical model of growth
we
devclop two such techniques-.
dynamic programming
and contingent claims analysis. They are in fact closely related to each others and Iead to identical rcsults in many applications. However, they make different assumptions about financial markets, and the discount rates that hrms use to value future cash tlows. Dynamic programming is a
vely
general tool for dynamic optimization,
and is particularly useful in treating uncertainty. It breaks a whole sequence of decisions into just two components: the immediate decisions and a valuation function that encapsulates the consequences of aIl subsequent decisions, startingwith the position that results from the immediate decision. If the planning horizon is finitea the very last decision at its end has nothing following it, and can therefore be found using standard static optimization methods. This solution then provides the valuation function appropriate to the penultimate 93
?J(/
ltltlltntlttictllStlckgrtplf and
so decision. That. in turn, serves for the decision two stages from the end. condition. This sequencd initial on. One can work backwardsall the way to the of computations might seem difficult, tlut advances in computing hardware obtan soluand software have made it quite kasible. and in this book we will what inlinite. is horizon tions to several problems of this kind. If the planning simplilied by its recuris calculation might seem like an even more diflicult exactly like the sive nature: each decision leads t() anotllcr problem that looks but also often numericttl computation, original one. This not only facilitates solution, and characterization of the makes it possible to obtain a theoretical sometimes an analytical solution itselt.
Contingent claims analysis builds on idcas from financial economics. Beof costs gin by observing that an investment project is dehned by a stream uncertain of and benefits that vary through time and depend on the unfolding the right to an investment Opportuevents. The lirm or individual thar owns completed project. owns an nity, or to the stream of operating prolits from a markets for quite a rich menu asset that has a value. A modern economy has be of assets of all kinds. If our investment project or opportunity happens to However, price. market even will known have it traded assets. a one of these value for it by relating if it is not directly tradeda one can compute an implicit traded. other that asscts are it to will All one needs is some combinatit)n tar portfolio of traded assets that at exactly replicate the pattern of returns from our investment project. every composition of (The eventuality. unccrtain future and in every future date the of prices thtl comchange cotlld it fixed; be as this portfolio need ntat project must equal ponent asscts change-) Then the value of the investment would present an discrepancy because of portfolio. any that the total value the two assets or of cheaper buying the profit by arbitrage opportunity'.a sure this calculation in Implicit valuable one. combinations, and selling the more opportunity in the its fnvestment should firm use is the requirement that the could inthe buy arbitrager did not, if an most efhcient way, again because it value of the know Once profit. and make positive we a vestment opportunity and timing of inthe investment opportunity, we can find the best form, size, optimal investment the determine value, and thus vestment that achieves this
policy.
In the first section of this chapter we develop the basic ideas of dynamic that programming. For continuity we start with the same two-period example time it Ionger extend to spans, in was the mainstay of Chapter 2, and then consider the two special discrete periods and in continuous time. There we namely 3, lto and Poisforms of uncertainty that were introduced in Chapter claims to of contingent tbe general ideas son processes. Section 2 develops
uncertain future payoffs. and tlle detcrnlination of tlleir prices by llrbi trtgt!. Again the focus is on lto and Ilisson prllcesses. Section 3 explains tlltt r(2lationsbips between tlle tsvo pproacbes. slc illsl) tlevelop tllis relationsllip to obtain a simple nlethod lr discounting and valtling paytlffs vith vtlrying
degrees of
risk.
Throughotlt, vd enlphasize the intuition bellintl the nltltllods ratlpcr tll:lll forlnal proofs. A sketcll of the techniclll detllils is placctl in an appentlix.
1 Dynamic Programming In this section we introduce the basic ideas of dynanpic progntnlming. We start with the two-period exanlple t'ron-lCh:hpter2. thtls provitling a sinlple concrete setting for the ideas and contilltlity with the previotls analysis. Tllen we extend the itleas and tlevelop the general thetlf.y of nlultiperiotl decision strategies. Finally. we Iet time be continuous- antl reprtlstlnt the tlntlerlying uncertainty with either Ito or Poissiln processes. That is the stttting for nAost of the applications that will appear in later chapters.
1.A The Two-period Example An extended
example
in Chapter 2 intnltltlced the ideas
()1*
(lptimal
timing of
investment ddcisions and ofoption val ues. We Ilegan by cllmparing thc present values that result from immediate investment and from wltiting. Dyntnlic programming is in cssence a systematic method of making stlch comparisons for more general dynamic decisions. In trder to illustrate this. Iet us prtlcettf.l to generalize our example. As the first step. in this section we rework the simple
twlJ-pcrilld
dxample
from Chapter 2 in a more general wty. Belrc. we chose specific numerical values for most of the parametcrs in the example; now we will Iet them have arbitrary values. Let I denotc the sunk cost ()f investment in the factory tllat then produces one widget per period forcver, ltnd Iet 1- be the intcrest rate.
Suppose the price of a widget in the current period () is J.!). From period 1 onward, it will be ( l + tt) /1)wth probabi Iity t/, and ( l t/) /$) with probability (1 q ) First, suppose that the invcstment opportunity is available only in period 0,-if the firm decides not to invest in period (), it cannot change its mind in period 1 Let lz'i)denote the expected present valuc of thd revenues tht! firm gets if it invests. Weigllting the two altcrnative possibilities )r widget prices -
-
.
.
Matllonatical Sltrl.-qlrr-pld/lt/
by their
respective
llallic
t7J7?/?l.::tlJI'tr-???
ltlltlel-
t7/lt.r(,'/'/tliil/p
This outcome of future optimal dccisions is sometimes called the cltlttatioll p//l/tr. From the perspective of period 0, the period- l price /)1 and therefore the values Pr1and $ are alI random variables. Let &)denote the expectation (probability-weightedaverage) calculated using the information available at period 0. Then we have
probabilities, discounting, and adding, we have
,
,
the invest(Note that we need r > 0 for convergence of the sum.) If K) > 1, not made is investment the 1, < l,. if l, ment is made and the firm gets Kj and investing between indiftkrent = firm is 1, )$l the and the 5rm gets 0'. if of payoff the net the f2() denote Let not investing and gets zero in either case. whether t)n invest, decide to a () period in to project to the firm, if it is forced that now-or-never basis. Thus we have shown -
m:ux ( r$) 1. () ).
f2()
-
=
Now consider the aclual situation, where the investment opportunity remains available in future periods. Here the period-t) decision involves a when period l different tradeoff'. invest now, or wait and do what is best actions in different ahead to its own arrives. To assess this. the tirmmust Iook will change. so not the conditilans future eventualities. From perigd l onward 1 Hence period beyond there is no point postponing any prohtable projects period 1 we need look ahead only as far as Suppose the irm does not invest in period 0, but instead waits. In period l .
.
the price
will
be
/1
Jl) with probability q'
(1 +
tt)
(1
J) Jl) with probability 1
= -
-
The present lt will stay at this level for periods 2. 3, is period 1 back to of revenues, discounted .
.
.
.
(/
.
value of this stream
,
periods For each of the two possibilities (theprice going up or down between 0 and 1), the firm will invest if 1Z1 > 1, realizing a net payoff Fj
=
max
gP'j
-
l 0) ,
.
This could be called the expected continuation value. orjust the continuation value- with the expectation being understood. Now return to the decision at period 0. The irm has two choices. If it invests immediately, it gets the expected present value of the revenues minus the cost of invcstment. 1z$) 1. If it does not, it gets the continuation value &)() Iderived above, but that starts in period 1 and must be discounted by the factor l /( 1 + r) to express it in period-t) units-The optimal choice is obviously the one that yields the larger value. Therefore the net present value of the whole investment opportunity optimally deployed, which we denote by &, is -
1;.)
=
m ax
l.$)
-
l
l ,
l
+ r
'.)-4 ) (Fb1
.
The firm's optimal decision is the one that maximizes this net present value. This capturcs thc essential idea of dynamic programming. We split the whole sequence of decisions into two parts: the immediate choice. and the remaining decisions. aII of whose effects are summarized in the continuation value. To tind the optimal sequence of decisions we work backward. At the last relevant decision point we can make the best choice and thereby tind the continuation value (h in our example). Then at the decision point betbre that one, we know the cxpected continuation value and therefore can optimize the current choice. In our example there were just two periods and that was the end of the story. When there are more periods. the same procedure applies
repeatedly. The decision where the investment opportunity remains available at period 1is Iess constrained than the one where it must be made on a now-or-never basis in period (). Equation ( 1) shows the net payoff f24)for this latter case; since that situation terminates the decision process at time 0, Iet us call it the termination vtrl/lleut time 0. N()w we have the net worth 6) of the less constrained decision problem from equation (3).The difference (Fi) f2()) is just the value of the extra freedom, namely the option to postpone the decision. ln Chapter 2 we calculated the value of the investment opportunity, Fi), and the termination value, Q(), for some specihc cases of this general model, -
khltllenlaticalStlcqrtplfnt/
98
where the parameters C), etc., were given numerical values. Readers can now refer back to those examples and place them in the context of th general theory. Hcre we point otlt one fcature of those results. by reference to Figure 2.4, which shows these values as functions of the initial price, C).When C) exceeds the critical level of 249, the tirm finds it optimal to invest at opce. Then the option to postpone is worthless, and 6) coincides with f2f), which equals 1-r() I in this range of the price. Nvllen 1) < 249, it is optimal to wait; then the graph of 6) lies above that of ftl. A similar property holds as other parameters are varied in other figures in Chapter 2.The idea is that the critical point where immediate investment just becomes optimal is tbund where the lines representing the value of the full opportunity, ), and the termination f2(), meet. value, ,
-
To get a better idea of the factors that affect the value of the option to postpone, let us examine more closely the sources of the dit-ferences between -()and Q(). First, by postponing the decision the firm gives up the period-o revenue /-1).This difference favors immediate action. Second, postponing the decision also means postponing the cost of investment; this favors waiting since the interest rate is positive. (More generally, the cost of investment could itself be changing over time, and that would bring new considerations; for example, if the firm expects capital equipment to get cheaper over time. that is an additional reason for waiting-) Third. and most important, waiting allowsa separate optimization in each of tht: contingencies of a price rise and a price fall, whereas immediate action must be based on only the avcrage of the two. This ability to tailor action to contingency, specifically to refrain from investment if the price goes down, gives value to the extra freedom to wait-l 1.B
Many Periods
1)t)
they can be regarded as limits ()f randlln) each time period and ()f each stcp 01' the W2y.
Nvith our application
I In technical terms- the mltximum is a convex function. so by Jensen's Inequality the averof age the separate maxima in equation (2) is greatttr than the maximum o1' the corresponding
avera gCS,
to investment in mind. we will retkr to the decisitllls
of a firm, but the theory is ofcotlrse perfectly general. The firm's curront stattls as it affects its operation and expansion opportunities is described by a state variable F()rsimplicity ofexposition we taktt this to be 11scalar (realntlmberlp but the theofy extends readily to vector states of any dimension. At any date is known. btlt ftlttlrc or period t the current value of this variable random variables. We values + l are suppose that the process is .r.
.rf
.
erf
-p+a,
,
.
.
.
Markov. that is. aIl the intbrmation relevant to the determination of the probability distribution of ftlture values is summarized in the ctlrrent sta t e xt At each period t, some choices tre Ctvltilable to the firm. and we represent them by the control variablets) l/. In the Ill3llve cxample where the only clloice was whether to invest at once or wait. we could let lt btl a scalar binary variable, whose value () reprcsents waiting and l represents investing at once. In other applications, for example. if the scale of inveslment is a matter of choicc. lt can be a continuous variable. I f the firm has choices in :ltldition to those bearing on invcstment, lbr cxample. hiring labtlr at time / then lt can be a vectllr. Thtl value lIt of the control at time / must be chllsen tlsing (lnly thll inlbrmation that is available at that time, namely The state and the contrlll :lt time / aftkct the firm-s immttdiate prtllit I1()wwhich we denote by n.t (m lIt ). I'Iere thtt relevant contnll variable s/, migllt be the quantity of labor hired t)r raw matcrials purchased. The and ltI ()f period t also affect the probability distribution ()f future states. I-lere ltt can be the amount of investment or Rct D. or evcn a decision to abandon the enterprise. Let *? ttt ) denote the cumulative probability distributilln l'unction ! 1 of the state next period conditional upon the current information (state and .
,
-rf
.
.
-rf
.%-t+
We now generalize the two-period example above. 0ur applications in subsequent chapters are mostly to situations where time is continuous and thc uncertainty takes the form of Wiener processes or more general diffusion processes for the state variables. However, in this subsection we develop the theory of dynamic programming in a setting where uncertainty is modelled using discrete-time Markov processes. Some general properties are easier to demonstrate in this format. Also, the setting ofthe rest of the book is a limiting case. Diffusion processes are Markov processes, and as we saw in Chapter 3,
walks in discrete tinne :ls tlle ltlngt11ol' snlall in a stlitClble t'leta'lllle
walk both
-rf
.
control variables).
The discount factor between any two periods is I /( l + p). where p is the discount rate. The aim is to choose the sequence tli' controls jlff) ovflr time so as to maximize the expected net present value of thc payoffs. Sllmetimes we will force the decision process to end at some periotl T, with a hnal payoff that depends on the state reached; we de note th is termittation J,vl.vf.#/' function by Q r fx.r ) We are ready to apply the basic dynamic programming technique. Rcmember the idea is to split the decision sequence into two parts, the immediate period and the whole continuation beyond that. Suppose the current date is t and the state is xt Let us denote by F)(.p ) the outcome - the expected net '
.
.
s'latllel:lilticalJ'ltkckqrtpl/llt/ present value of a1lof the firm's cash flows when the Iirm makes all decisions optimally from this point onwards. When the 5rm chooses the control variables
ltt,
it gets an immediate Optimal
profit Ilow n.t xt, ur). At thtz next period (1+ l). the state will be decisions thereafterwill yield, in the notation we have establisheds
l)y,/lfl??lcOplilnizaion I/nJt!?-Ullcertall' lf the many-period problem has a tixed finite time horizon T. we similarly start at the end and work backward. At the end of the horizon firm gets a termination payoff f2rl-rrl. Then the period before,
can the
-p+1
.
(-rf+l).
';+1
This is random from the perspective of period /. so we must take its expected value ',(F;+l (m+l)).That iswhatwe called the continuation value.2 Discounting back to period t, the sum of the immediate payoff and the continuation value is (-rI
n't The frm will choose #) xt ). Thus
ut
ttt
,
)+
1
1+
-,
p
(&+l (-r?+ l )1.
to maximize this, and the result will be
just the
value
Thus we know the value function at F 1. That in turn allows us to solve the maximization problem for llr-c, leading to the value function Fr-al-rr-a), and so on. At one time this was thought to be too complex a procedure to be practicable. and aIl kinds of indirect methods were devised. However, advances in computing have made the backward calculation remarkably usable, and several of our numerical simulations in later chapters use it. Later in this chapter we will oftkr an example of it. -
lnhnite Horizon
1.C The idea behind this decomposition is formally stated in Bellman*s Principle lil/fl/fltffl/l, Of Optimality-.adl Optnalpolicy has J/l(?J7rt@t!r@l/lflJ. Wlliletrtlle Ihe
rtrrrlflrlrl#
ChOiCC.% Conlitute
(ln
Optimulpolic'y
114//1
rlMpo':t
10 J/lfJ SlbproblClll
starting at the state that results rrrl //lfJ initial fltW(?ll.. Here the optimality of the remaining choices uf+I l/+2, etc., is subsumed in the continuation value, so only the immediate control ttt rcmains to be chosen optimally. ,
The result of this decomposition, namely equation
(4), is called
the Bell-
Ltjelndamental equation sfpplfrnl/l-y. Tt) reiterate, the hrst man eqttation. or right-hand side is the immediate proht, the second term constithe term on continuation value. and the optimum action this period is the one the tutes
maximizes the sum of these two components. In the two-period example, immediate investment gave P'4) /, waiting & )/ had no period-o payout but only a discounted co'ntinuation value (1 + r), and the optimal bina:y choice between these alternatives yielded
that
-
'()(
the larger of these two. Thus our earlier equation the general Bellman equation (4).
(3) is a special case
tf
lf there is no tixed finite time horizon for the decision problem, there is no known final value function from which we can work backward. Instead, the problem gets a rccursive structure that facilitatcs theoretical analysis as well as numerical computation. The crucial simplilication that an intinite horizon brings to equation (4) is independence l'rllm timc t as such. Of course the current state xt matters. but the calcndar date / by itself has no effect. This works provided the Ilow profit function :., the transition probability distribution function *. and thu discount rate p arc themselves al1 independcnt of the actual Iabel of the date. a condition that is satisied or assumed in many economic applications. In this setting, the problem one period hence looks cxactly like the probIem now, except of course for thc new starting state. Therefore the value hmction is common to all periods, although of course it will be evaluated at Therefore we writc the function as Fxt ) without any time different points symbol. The Bellman equation for any f becomes the function label on -rf
.
Flxt ) tn71eexpectation notation is generally clear enough. However, to make it precise that the information at time t includes the state and the control at that time. we slatc it formally once for reference'.
Since
=
max ,,,
n' (.,p u: ,
)+
1 l + p
(
I
-(.rI+
l
)l
and xt-vbcould be any of the possible states: write Then, for all we get form as and -rr
-r'.
.,r
.r,
.
them in general
rlfz,?? ic.
!()2 wherc we have now denoted the expectation as conditioncd ()n tlltl kpowledge Illlinitely ot. the current period s and ll. This is the Beliman eqtlation Ior the repeating, or recursive. dynamic programming problem. ?
.
;
,
-t-
.
,
-r,
equations
,
as the
hmciollul
Despite superticial appearances, this equation is not Iinear. The optimal choice of lt depends on all the values Flx'j that appeara weighted by the appropriate probabilities. in the expectation on the right-hand side. When this optimal control is substituted back. the result can be nonlinear in the Fx') values. In general we do not know whether nonlinear functional eqtlations have solutions, let alone unique ones. Fortunatcly. the recursive Bcllman equalion has a very special structure that allows one to prove existence and uniqtleconditions typical of our economic ness of a solution function F(-r) under side technical issue for our more practical concerns. applications. This is a sketch the ideas in Appendix A to this chapter. and of We include a brief excellent readers treatments in more theoretical books, can find interested with and example, Lucas Prescott ( 1989, Chapters 4,9). But the Stokey for technical argument does have an indirect payoff: it is essentially a practical solution method. This takes the form of an iterative procedure. Start with any guess tbr the l' side of equation (5) true value function, say Ft (-r). Use it on the right-hand l which rule choice tt can now be expressed and hnd the corresponding optimal right-hand side becomes the as a function of x alone. Substituting it back()f Ft2)(-r). Now use it as the next guess t)f the true call it a new function value function, and repeat the procedure. Then the successive guesses F3 (x), F4b ( v), etc., will converge to the true function. Convergence is guaranteed with a good initial guess no matter how bad the initial guess, but of course the procedure will reach the desired accuracy of the approximation in fewer ,
.r;
steps.
lil ? li--tl
t rl/ l
:tl
Jt/t/?-
Ivllrc't','/:, ilI
b'
Tlle key lies in tlle ltcttlr l / ( l + /)) 017 tlle rigllt-hand siule. Tllis beillg less than l it scales down, or contracts, any errors i11 the guess from one stcp to the next. As Iong as the profit llows are botllldetl. any errors in tlle clloice of soltltion is Ieft. u cannot blow up. Gradually. only the correct .
Now that we have no fixed terminal date from which to work backwarda we value function F seem to have Iost an explicit or constructive way to tind the And without knowing the function F we cannot find the optimal control tt by solving the maximization problem on the right-hand side of the Bellman equation. Thus we need assurance that a solution actually exists. and a way to find it. Luckily, neither question is vey diflicult. The recursive Bellman equation (5) can be thought of ts a whole list of equations. one for each possible value of with 71 whole Iist of unknowns, namely all the values F(.r). If x took on only a tinite number of discrete values
-rf this would be a simultaneous system with exactly as many number of unknowns Fxi ). More generally. we can regard (5)as F as its unknown. equation, with the wholehtltction
t';I/J
This procedtlre is very easy to
understand-
progranx- and conlpute.
I t can
take :1 long tilne on tlle computer. especially if tlle discount rate p is very small doNvn. I-losvever. that is no Ionger so that each step does only a little scaling consideration when individtlals can Ieave their (nvn personal prohibitivd a computers running for days without tying up scarce mainframe resources.
Theret-ore this method is increasingly used in many applications- and even in econometric work 'Wewill givtt a nunnerical exanlple of this method, too. Iater in the chapter. .
1.D Optimal Stopping One partictllar class of dynamic progrilnlnling problems is very importltnt for our applications. Here the choico in any period is binary. One alternative corresponds t() stopping the prllcess to taktl the termination paytll-- and the other entails contintlation for onc periotl. when ttnother similar binaf'y choice will be available. In the investment nlodel tlf Chapter 2 and the tirst setrtitln of this chapter. stopping correspllnds t(' mllking tlle investment. ltnd continuation correspllnds to waiting. l-lere continuatitln dtles not generate any prlllit flow within the period. Howevera in other contexts thtlre may I7e sonle such flow. Ftlr example- for a firm in bad ttconom ic ctlndi titlns t hat is con te nl piatltnd ing shutdown, continuation nlay givc a prtltit flow (positive ()r negative). eqtlipment. vltlue minus ol' tllc and plant termination may yicld some scrap tlther any severancc payments thc lirm is required t() make to its workers and restorlltitln, of site breaking contracts. etc. costs Let zr (-r) denotd the Ilow prolits antl t'''z(.r ) thc tcrnlination payoff. Then the Bellman equation becomes F( ) -r
=
m ax
t' (x )
zr ( ) + .r
,
l
For some rangc ofvalues of-r. the maximum
,
1. (F ( ) ..
I+ p
()n
-r
i
-r
1
.
the right-lland
(f)) side of this
will be achieved by termination. and for other values ()f it will be achieved through continuation. In general this division could be arbitrary; intervals where termination is optimal could alternate with ones wherc contintlation is optimal. However, most econtlmic applications wili have more structure. -r
with torminatitln optimal on one side and There will be a single cutoff continuation on the other. For example, in the invttstment problem ()f Chapter 2 we had a critical Ievcl of thc initial price. /1) 249, such that in perilld (), -r*,
=
Matllcmatical Stzc/grlfnt
104
investmentwas optimal to its right butwaitingwas optimal to its Ieft. All ofour applications will have a similar property, and in each case it will be intuitively clear which action is optimal on which side of the threshold or cl:toff point. To complete the reader's understanding of this result. we should explain the general conditions that lead to it. We stat these intuitively here, and explain them somewhat more formally in Appendix B. For sake of dehniteness we concentrate on the case where continuation for x > x' and stopping is optimal for x < x*. Let us examine the forces that will make continuation more attractive relative to termination for higher values of First, the immediate protit from continuation should become larger relative to the termination payoff. Since the former is a :ow and the latter is a stock, we need to express them in comparable terms. The precise condition turns out to be that
ttlltler naltlic O/p/l'l?lzt7fibpl
f.7?,(7n-Jt.2j??(v
1()5
time is continuotls. We Lt goes to zero n.(a- lf 1) for t he rate of the profit llow, so that the actual protit over the time period of Iengtll AJ is gx. l/. /) Lt. Similarly, Iet p be the discount rate per unit time. so the total discounting over an interval of length L is by the factor 1,/(l + p lf ). and
write
The Bellman equation
(5) now
becomes
and rearrange
to write
,
,
is optimal
Multiply by ( l + p
1)
-r.
n.(x) + ( 1 +
p4
-
l
-
(7)
(Q (x lx1 t'-z(-v) ')
-
should be increasing as increases. Second. any current advantage should not be likely to be reversed in the near future. For this, we need positive serial To correlation, or persistence, in the stochastic process of evolution of be more precise, if this period's rises. the conditional distribution *(.r'1-r) of next period's values x' should give greater weight to larger values, that is. it should shift evelywhere to the right. (In technical language, this is the dominance.'') These two conditions together concept of are sufEcient to ensure the desired result. lf the expression in (7) is decreasing as x increases, then continuation will Note that the second be optimal to the left and termination to the right of require negative persistence condition stays unchanged', we do not switch it to of the stochastic process.
Divide by
l and
let it go to zero. We get
-r
p F(
-r
-r.
-r
thrst-orderstochastic
-r*.
that both of these properties will always bd satished for our applications. The srstis easily veried in each instance', the second is tnle for random walks, Brownian motion. mean-reverting autoregressive processes, and indeed in almost all economic applications we can think of. For ease of notation in conveying te concept, we did not allow the protit payoff to depend on time t as such, but that extension presents no terminal or difhculty.The threshold simply becomes a function of time, (/). This will be the case in much o the rest o the chapter and in many o Our appliations. We repeat
-r*
1.E
Continuous Time
Now return to the general control problem of Section 1.B, but suppose each time period is of length Lt. Ultimately we are interested in the limit where
.
t)
=
m ax 1/
;'r
(
-r
.
1/
.
t)+
l tlt
t.-gtF j
,
where ( 1/Jl ) JII(IFl is the limit of 7qty Fj l We must remember that the expectation is conditioned on the current and II, and we must rcmember to include the inlluence of changes in both x and / when we calculate the change in Fx, 1) ovcr the interval Jl. .
.r
This form of the Bellman equation makes explicit the idea that the en-
titlement to the Ilow of prolits is an asset, and that t) is its value. On the left-hand side we havc the normal return per unit time that a decision maker. using p as the discount rate. would require for holding this assct. On -(-r.
' thv right-hand side. the first term is the immediate payout or dividend from the asset, while the second term is its expected rate of capital gain (loss if negative). Thus the right-hand side is the expected total rcturn per unit time from holding the asset. The equality becomes a no-arbitrage or equilibrium condition, expressing the investor's willingness to hold the asset. The maximization with respect to tt means that the current operation of the asset is being managed optimaily, of course bearing in mind not only the immediate payout but also the consequences for future values. The limit on the right-hand
side depends on the expectation
correspond-
ing to the random f a time Lt later. There are two classes of stochastic processes in continuous time that allow such limits in a form conducive to further analysis and solution of the function Flx. t) in the continuation region.
Luckily they are particularly useful for many economic applications. In fact and Poisson processes we discussed in Chapter 3. We will
they are just the Ito
develop the theory of dynam ic programmi next two subsections.
ng in the ir spet:i Iic contxts
i1,1the (
valtle function F
,
we Illlve
.
The above analysis is local to the sllort time interval (/ t + dt ), and the resulting equation holds for any 1. We can complete the analysis by choosing :1 terminal payoff, gr lctting the horizon a finite time horizon T and imposing using rccursive the structurea or some other way. ln any of be infinite and of existence and uniqueness of solutions mathematical proofs these. rigorous continuous Since the details are immaterial ftlr time. become quite hard in applications, we omit them and refer the reuder to Fleming and Rishel our ( 1975) or Krylov ( 1980). Our mathematics has been simplilied in another respect. We have treated will conthe limit to continuous time in a very casual and heuristic way, and tricky reader quite that some tintle to do so. However, it is fair to warn the rigorous carcfully treatments. in more issuesare hidden, and must be handled In discrete time, we stipuiated that the action llt taken in the current period t but not on the random could depend on the knowledge of the current state the two coalesce. We have to be careful l In continuous time future state about not to allow choices to depend on intbrmation abotlt the futtlre. evcn with of would hindsight. benefit the acting be tthe next instant-'' Otherwise we requiring and could make insnite profits. Technically this can be avoiddd by strategies while the time right'' in from the the uncertainty to be in thc stochastic processu!s left-'' Then the from any jumps are change until//-/ alter Jf:' installt. occur at an n-s-fclnf,while the actions cannot 39-44)1. For a discussion and rigorous analysis, See Duffie ( 1988. pp. l -
-k
,
-p+
.
t'continuous
''continuous
t-. !
1.F
Ito Processes
stochastic process that yields a simple form for (8) The lirst continuous-time is the Ito process we discussed n Chapter 3. Equation ( 11) of that chapter dehned the formula for its increment. which we recapitulate here, but now allowing the drift and the diffusion parameters to depend on the control variable as well as the state variable:
dx
=
a (-r ,
1/
.
t ) dt + bx
.
11
,
t ) Jz-
where dz is the increment of a standard Wiener process. As before, we the proht llow as zrt-r, tt. t) and the value of the hrm (asset)as Fx. t ). ='
.x
.
tt
.
t ) b;. ,. (a. t ) .
) .
We can express the optinal If as function of Fl a (.t f )a Fx(.v ). F a. (.' / ) well as x. f and thc various paramete rs t hat govtlrn t Ilc functional lbrm ()t' Jr as s a, and b. Substituting this expression for the ('ptimal 1/ back into the right-hand side ofequation ( l ())- vve get a partial diftrcntial eqtlltion of the second tlrder. with F as the dependent variable and x tnd / as tlle independcnt variables. In general this equation is very complieated. However. in many applicatillns we can develop wasys to solve it analytically or nunaerically. . The solution methods are gencrally clllltl(.lglltls to thf lse I'(lr discrete time. lf there is a (ixed time Iimit F when a terminatilln payllff f2 fxv F ) is tlnlklrctttlthen the eqtllltion has 11 boundllry contlititln -
,
.
,
.
.
-.( r .
We can start at timc
r
.
r)
and work
f2 (
.r
()u
.
F)
r w:ly backward to find F (-r J ) for al I .
earlier times. In fact. in practice we have to chollse 11 discretc grid t)I' values of A. and t on which to calcuiate the slllu tion. We will oflkr twt) examples ()f this procedurca onc later in this chapter when we directly solve the underlying dynamic programming problem. and one in Cllapter l t) where we solvc the partial differcntial eq uation itsel f. If thc time horizon is inhnite and the l'tlnctions
zr t/, ttnd ,
/?d() not deptlnd
explicitlyt)n time the neither does the value i'unction depend equation ( 10) becomcs an ordinary differential equation with independent variable'. ,
()n .r
time, and as its only
write
+ dx the Let x be the known starting position at time f and x' ftar Lemma A/. lto's of time of small interval end the a random position at the of Applying it to 3. equation Chapter stated in (25) such a process was ,
4..2.! bl (-r
Note that we have followed standard calculus notation and used primes tt) denote total derivatives of a function ofone independcnt variable, and subscripts to denote partial derivatives of a function 01' several independent variables.
Mathematical fkckgrtxlnf,l
108
We will generally adhere to this, but will occasionally use subscripts even for superscript for total derivatives, for example if the function symbol needs a some other reason. will ln most of our models of investment throughout Chapters 5-9, we will and develop have occasion to fofmulate and solve equations like (11), appropriate solution methods for them gradually. We turn to one special kind of control, namely optimal stopping of an Ito process. that is of particular
importance in all of our applications. 1.G
Ullcertatty
Now the Bellman equation
l ()9
for the optimal
stopping problem,
(6),
becomes Fx
.
t)
=
( 2(.r. t )
max
zr (.,r t ) +
,
-
l ( 1 + p dt )-
z.-(Ft-r
+ dx t + ,
In the continuation region, the second term on the right-hand
of the two. Expanding it by Ito's Lemma and simplity'ing as partial differential equation satished by the value function: 12 bl ( v' t ) Fra.(.v t j + 9
Optimal Stopping and Smooth Pasting
,
.
tl
tx I ) F).(.x t ) + .
.
/7)(.: t ) ,
-
JJ
) I A'l
)
.
side is the larger above, we get the
p F(x t ) + zr j..r t ) ,
,
=
0
.
(l 3)
where subscripts denote partial derivatives. > This holds for (/), and we must Iook for boundary conditions From the Bellman equation, we know that in the that hold along stopping region we have F(x, / ) Q (x. /). so by continuity we can impose the condition F(.r* (J ) J ) (-r*(/ ) t ) for alI t (l4) .r*
.r
Here we consider a binary decision problem. At every instant, the 5rm can either continue its current situation to get a proft Cow, or stop and get a termination payoff. Both the prost flow xx. t) and the termination payoff Q (x. t) can depend on a state variable x and on time f where x followsan Ito ,
process
Optilllizatiolt I/lltcr ??t'???lc
-r*(/).
.r
=
=
''z
.
=
,
.
This is oten called thc condition'' because it matches the values of the unknown function F(.r. / ) to those of the known termination pyoff function f2 (a'. t ). But the boundary itself is an unknown'. the region in (-r. tq space over which the partial differential equation ( 13) is valid is itself endogenous. The boundary of tllat region. namely the culwe -E*(/), is called a boundary,'' and the whole problem of solving the equation and determining its region of validity is called a free-boundary problem. It is clear that we need a second condition in addition to ( l4) if we are to find x*(1) jointlywith the function F(x, I). The general mathematical theory of partial differential equations is of Iittle help in this regard; the conditions applicable to free boundaries are specific to each application and must come from economic torphysical or biological. as the case may be) considerations. For us the right condition turns out to require that for each t, the values F@. t) and 62(-t, t), regarded as functions o'f x, should meet tangentially at etvalue-matching
dx
=
a (x t dt + bx ,
.
(12)
t ) #z.
The most obvious example is of a 5rm deciding whether to cease operation and sell its equipment for its scrap value. lnvestment decisions can also be put in this form: continuation means waiting, and the flow payoff is zero; stopping expected present value means investing. and the termination payoff isjust the minus the of investment. cost of future protits from the project Intuition suggests that for each t there will be a critical value x*(f ), with continuation optimal if xt lies on one side of x. (J), and stopping optimal on the other side. We saw in Section 1.D that some conditions must be imposed on the proht flow and termination payoff functions to ensure this. Continuation will be relatively more attractive for larger values of if the expression in (7) is an increasing function of x. In continuous time, if x follows the Ito process (12), then the analysis of Appendix B shows that .x'
xtxl -
p
f2(x t4 + ,
(1(.x,
t)
tt + t'-2xt.x.
lj bx. J)2 fzxxtx. tj + fzf(x, f )
should be increasing in x for each t. Similarly, continuation will be less attractive relative to stopping for larger if the expression is decreasing. In each of will hold. For sake of exposition we our applications, one of these conditions the former takc up case. (/) for various Given such conditions, we can regard the critical values ?) space into two regions, with contindivides frming the that (-r, a curve t as uation optimal above the curve and termination optimal below it. Of course = x.(t) in advance, but must hnd we do not know te equation of the curve x problem. programming solution of of the dynamic the it out as a part .x
.x*
ttfree
the boundary
-r*
(J),
or b (x*(/ ) t ) ,
=
Q.r(-x*( I ) t ) .
for all
I
.
( l5 )
This is called the contact'' or condition'' because it requires not just the values but also the derivatives or slopes of the two functions to match at the boundaly. t'high-order
t'smooth-pasting
While continuity is very intuitive, continuity of slopes or smooth pasting
is more subtle and remarkable. However, the argument for it is somewhat technical, so we relegate it to Appendix C. Here we merely illustrate how it works.
Nlathenlatical f'ltlc/v-qrot//?t/ 1.H
Optimal Abandonment
Example
1;h.p?7:l??1/c(jlilliltli--flkl'llllI/??r/cr
of a Machine
IvlllQ'Q,t.t(lillI.
x
that follow are full t)f applications of dynamic programming and contingent claims analysis. Here. however. having developed the theory at a general and rather dry Ievel. it wtptlld be useful to offer a concfete example. We do not solve it in detail, but merely state the solution backed by some intuition. We hope this illustrates the various steps in a specihc context, and prepares the reader for more detailed agplications to come. Suppose the asset is a machine used to make widgets, with a total physical life of T years. Its profitability declines over this Iife, both because it wears out gradually and thus produces less output or rcquires more maintenance, and because technical progress elsewhere in the economy makes this machine less competitivewith newer ones.rrhere are also random shocks to its productivity, because of the general business cycle, or because o idiosyncratic variations in the demand for widgets. Let the state variable x be the current operating and suppose it evolves according to profit The chapters
0 05 .
0
2
4
6
-0.05
10
8
x
-0.1 0 -0.15
now,
J-r
=
t? dt +
b t/z.
0 to rellect the gradual decline over the Iifetime. At any time during the physical life. the firm can abandon the machine. If the current prost tlowbecomes negative. this may seem an attractive alternative. Once abandoned, however. the machine will rust quickly and be very costly to restart should the protit llow recover. Therefore the abandonment decision will have to Iook ahead to such future possibilities. The tirm will condition. Of course, acccpt some losses to keep the machine in operating Iife of the machine is physical accepting losses will be less compelling if the variables, the currcnt drawing to a close. Thus we must keep track of two there will that profit, x, and the age of the machine, t. Intuitively, we know this curve the be a threshold culwe x't ) such that if the current x falls below machine will be abandoned. 10 The actual parameters used in the calculation are F = l() years. p which = standard implies a 0.2, l per year, and b percent per year, a deviation of 0.2 over one year, or a variance of 0.04 per year. To obtain a numerical solution, we use a discrete approximation to Brownian motion, 0.01, or 3.65 days. Correspondingly,each discrete step of the profit with Lt
where a
<
=
-0.
=
=
0.02. b Al We solve the dynamic programming problem directly using the method outlined at the end ofsection 1.B, starting at F when there is no future consideration, and marching backward one time step at a time. Figure 4.1 shows the solution. Part (a) shows the optimal threshold curve the free boundary
variable x is th
=
=
Figure 4 1. .
f-lfzrrtrca/fp?'land
zl/;fJ?'lt/t'prl??lc/J/
*(1)
l
/V/'tl//lt:'/?lflfl'cfl/Backgl-oltlltl
-v*(J) in (1 space. At each Js if the current is above this curve. the machine is kept in operation and has a value Fx. t ) that satisfies the differential eqtlation corresponding to (13),namely, .r)
-v'
,
Below the curve. the machine is abandoned tbr the termination value 0. (We value could have also obtained a solution by specifying some other scrap approaches machine the the of age function.) Note how the culwe goes to 0 as
the physical limit of 10 years there is no reason to keep alive :1 loss-making project in the hope of a future turnaround if the machine is going to die remaining physical Iife of the machine, the very soon anpvay. The greater the effect ultimately greater is the willingness to absorb some losses. However, this will be kept operating Ievels off. With a lo-year horizon. a brand-newmachine
loss is about 0.15. f ) as :1 of figure shows graphs of the value function the (b) the values would greater of t. As we expect, function of x for some particular machine has valutl thc of the value machine. However. the is the greater is that we assumed even if = 0, because, given the simple Brownian motion that the valuethe in Note will risc that there is t'uture. possibility A', the for matching and smooth-pasting conditions hold; for cach of the chosen /, both (/ ). Fx. t) and l (.r, t) fall to zcro as approaches What if the machine has a very long physical life that we can regard as effectively inlinite? Now we can leave calendar time out of the picture and solve a recursive functional equation for F(x), or we can remove time from the partial differential equation ( l3) and writc it as an ordinary differential
even if the
current
-(-r.
Part
.r.
-v
fihvllf7llltp(lillit'liztllilll'
lt?lf/c?-
Illtoz>l-ltlilllbt
called the mean arrival rate. I1*a jumpoccu rs- it is ofsize gt-r / ) l/. vvlle re t) is a known tknction and lt is a random variable. Witll probability ( 1 - t// ) there is no randon: jump. and moves by :1 deterministic anlount /'(.r. t ) d. -(-v'
,
.
-t-
We write this compactly by analogy with the notation for Brownian motion as J-r
where iIq is othenvise.
11
random
./'(-r
=
t)
-
variable
iit
+ g(.r t ) tlq .
that equals
lt
.
with probability
tit
and 0
Now consider dynamic programming when the state variable tbllows a Poisson process. We illustrate this for the optimal stopping problem. since this will be our most common application in this context. Once again the values of x will fall into two subsets, one where immediate stopping is optimal with the termination value f2 (a-. t ), and the (lther where continuation is optimal for at least the next short interval JJ, with the llow payoff n' (.r. / ). What happens to the asset value return equation (8) in this case? First suppose lt is :1 known nonrandom number. Then thdre arut two possibilities for the change in vatue tl F depending on whether a jump from J lt takds us into the stopping rcgion or not. I1'it does. then ..rto + ,
.g(.'.
.-
.r
.r*
.t.
equation
! b: F'' ( r ) +
2
a F' ( r ) ''
*
p F( ) + x -r
-
=
() .
The solution and the threshold can be found using the value-matching and 0. In Chapters 5-7 we will smooth-pasting conditions F(x.) = t). F'(.r*) develop this procedure in some detail, so here we merely invite interestcd readers to tly it out. The result in our numerical example is that lhe critical 17. Thus a lo-year life is already quite close to for abandonment equals innity as far as the effect of the future on the willingness to absorb current .t*
=
-r*
-0.
losses is concerned. 1.I
Using this in (8) and Ietting
If
g() tt) zero- wll have
region. we get a similar equation t )tt remains in the continuation ) (-r+ g(.t. J )tt) replaced by Fx + g(.r, / )11).More generally, when lf is random, we must allow both of thesc cases, and obtain a combined equatilln by taking the expectation over the distribution of u. ct-r.
+ with but -r
Note a new feature'. unlike the partial differential equation ( 13) for the Ito process case. the equation ( l7) is not 1)a11to the continuation region. We cannot hope to solve it separately and then paste it to the terminal payoff at the stopping boundary. Therefore the problem for the general Poisson proccss is quite hard. However, there are some simple cases. Suppose the jump. if one occurs, is always to the same known point, say Then we have -r().
(p
Poisson Processes
We introduced Poisson ump) processes in Chapter 3. Over a short interval dt of time. the probability of ajump in the random variable x is J/, where is
tlt
+
! F (-r
,
t)
=
n. (x t ) + ,
F (x ) / ) .
,
if is in the continuation region, and a similar equation but with f2 (-r().1) if it is in the stopping region. For example, if corresponds to a sudden stoppage vr(l
.zkl
l )/ ( /) + ) 1-Ie re t he 7r ( () :1 ntl 1-.(-r t ) f2 (.1..(4/ ) ltddition rllte /). disctltlnt the tt) Poisstln arrival rate sinlply acts Iikklall ulltlcrlying Mvill applicatitlns vvllere thtl develtlp stlme In Iater chapters we ltnalysis. I an-lenable n a couple simpltt Poisson process to uncertainty follows a untlergoes variable the wllere a ot' instances we will have a combined model, there is if lto proctlss Poisson jurnpwith :1 specified hazard rate- and follows an (.)tltl ined tions conlbine thtl featu res and equations the slllu ir the Then nlaj
of t he I'low payof then
.r
.
-
.
-
.
.r
ump.
here.
wllere a is t Ile grtlqvtll rltttl p:trllllleterand t/c tlle incre lnt!l: t ()1' t Ile stttlltlartl generlll prtlcess latklr.
tll:tt tlltl firnl-s (ltltptlt call itself bt't traded as :ln asset in Tllis Nvotl Id litc raIIy be t 11etrase i1-t l1eotl tptlt is :1 conp nltltl ity Iike oil ()r ctlpper. Il1 tllc Ilext section we Nvill slltlsv thltt it is stlfficient that tlle risk i11 t ht.)dyllttn-licsof I'lanlo Iy t he te rn1 tll.'l(.lve. clln l)e rtlpl icllted l7ysol-ne port folio tlt' t radetl assets. Like any llsset. the (ltltptlt is Ileld l7y investtlrs tlll ly il' it prtlvitles l stlll ficiently high rettlrn. Pltrt ()1* the rtlttlrn ctlllles il) tlltl I'(lrnl t)l' tlle expccttltl (.)f 11 divitlelltl. u. Flltltller prictl ltpprtltzikltitll'l. pltrt nllly ktlst.) ctlrne in t 11e lrll'1 l'nigllt 1')e tllat tl'le t'lr 11 intlirectly &v()()tI prlldtlct tretl dircctly ( grlqlvs n-lllre ) (the holtler ()1' tlil t'lr cllpper I-n igllt be lt firll'l tlltkt !7lilllstt) tlse tllese kts illptlts and jintls it ctlnvellient tt) I)()ltI its tlvvn illvellttlry rtttller tlllll'l rely t)I1 tlle spllt yvi rnarket t l'lcn t 11edivitle nd is t llt2ilnpl icit n itlnce yieltl-- ). Nve ll tl isctlss kkl-ltl (11' ().' Ilerc we tl1is ctlnvelliellce yicld in slln-le tletltil in Cllllptklrs 5 the rllle tlividentl ltlltl tlelltlttz its rktte l)y (. Le( stiptllltte tllttt tilcrk.t lt is pt a ..hJ sifnply (11* cxpected l1e r:lte retu rll. denllte t t()t1lI Tllis expcctcd rcttlrn nltlst l')eentltlgll t() ctln-lpellsrtte tlle Illlltlers lklr risk ()f ctltl rse it is nllt risk' :ls stlcll t l'lat nlllt ters. btl t ()11 ly 11()11t1 iversi Iiltlnlc risk 'rlle wl1()Ie rnltrket pllrtl'(llit) prllvides tlle nlllxirrltlnl ltvltilltlAle tliversificlttillll. ()1* the rlte ()t' rettl rn on tlle rtsset Nvitll tlltt ()n tllc &vl1()lt! so it is the cllvllrillnce tlt 1*4.4)1 rnark ptlrt it) t illtt detc rnl ines t l1e risk prenl itl 111 tllat tlltl riskless rltttt ()1* rcttl rn ''Fhrlltlgllllut (lur ltnalysis we will ltssunle lied. nously speci l'or exarnple thc lts re tu rn ()n gllvern rnen t b()I1tIs.3 r is extlge (.)f tlll cllndititln R-ht'tnt he I'undln-len equ ilibriu n'1frtlrll t i-lecapi tll tsset p rici ng rnodel (C.A PN1) sltys tllltt Nosv Nve
llsstlnle
finallcialIllllrktlts.
2
Contingent Claims Analysis
'lz
-r.
Wllen wc studied the optinAal stopping problem in dynanpic progrtmnling. we interpreted Fx. l ) as the nlarket valu of an asset that entitles the owner to dquatilln (8) expressetl the condithe Iirm's futurc prolit lltlws x(.-. t ). The short interval of til-neathe who asset tklra holds this tion that tbr an investor prtlvide :1 totlll eapital together gain and thc expecttld imnzedittteproht flow exogcnouslybtl t in pritctice specified this disctlunt rate rate of return p. bkk tllerefore it and of capital. opportunity cost it has the interpretation as the darned ()n other investment have should equal the return thc investor could make thtl ide:l characteristics. N()w risk we opportunities with comparable t)f risk. :1 more cxplicit, and extend it to providtt better trcatment economics has develtlped sllphisticatetl theories describing the
Financial
aggregation
decisions of investors. the markct equilibfia restllting i'rom the ()1* assets. Tlle basic sctting is of such decisionss Ilnd the equilibrium prices with dilkrcnt rcturn and risk' rich menu of tradt!d assets an economy with a replica its return and risk characteristics. 7b value :1 new asset, we try to ()f the ()f traded existing assets. Thc price characteristics through a portfolio equal the market value of this portfolio. Any discrepnew asset must then by arbitrageurs who Iook lr sure protits by buying would exploited be ancy repackaging it, and selling it in the mtlre valuable form. is cheaper, whichever Thercfore price discrepancies for equivalent assets or portfolios could not analpersist in equilibrium. The asset held in the continuation region of our assumed that thd ysis can be analyzed in ttlis way. Much of this theory has and wc shall begin Ito described by be an process, uncertainty underlying
w-k.rlll'lvc
',
.
.
.
'can
likewise. Replicating Portfolio We begin
variable
-r;
the simplest setting. Suppose the profit j1()w depends on a think ()t- it as the tirm's output price. Since we will be dealing
with
where () is an
aggrttgate
market
paramcter
(the
I'nitrkct
3In reltlity even plvernnlent bllntls have sl3nle risk becaustt complicat ifln I'''llra fulle r genc ral ctlu ilibri u In mtldc I t hat klete rn1 ines rates. set! C()x. l ngerstyl 1. antl lttlss ( l 985). .
pricc
(,1'
l Ilc
()1*
risk ) that is
inlllttitln: wc neglcct tllis tklrn1 st ructu re t) 1*i11terttst
Mathematical Jfzc/v-/lrtpaltk
116
between pxp, is the coefficient of correlation and the whole market portfolio ln.4 We hnd the value F'(-r. tj of a firm with proht flow n. @. J ) (reallythe value of the asset that entitles the owner to the protit flow stream) by replicating its
exogenous to our analysis. and returns on the particular asset
(5-r
=
Thus the total return per dollar invested is +
a + J).t
??
1+
nx
dt
+
t:r
nx
Jz.
l + nx
The total return per dollar invested is
; (.r t ) + s
e-r Fx ( r t ) + 1 2 c .
Fx
.
-
t)
2 -
r2 Faa (x t ) .. J/ + ,
f.y
x Fv.(x t ) .
)
/
) thd right-hand
pl-r
side becomes
,
?'
l
A-
l.(-r J )
F.%- )
F.v(.r t ) Fx t )
-r
.
+
(J
+
t
.
)
a
On simpI1h-cation. t he re t tl rn eq u at ion becomes
for the
.
.
al d 1ffe re n tial eq uation
11 parti
value:
This is strik-ingly similar to the corresponding equcttion ( l3) that we derived by dynamic programmng methods. Indeed. the analogy s almost exact if we rewrite the dynarnic programming equation tbr the geometric Brownian f',r motion process using tl (-r ? ) = t.z and /?(-r.t ) The only remaining differktncttis that the risk-less interest rate 1- is used in place of tbe exogenously specihkeddiscount rate pa and the coeflicient of the F.vterm has an '- ( instead of a. NVewill discuss this ctlrresptlndence between the dynamic programming and contingent claims valuatilln approaches in Section 3. An alternative and equivalent way to derive the same result is to construct 01* t he firm and 11 un its of a short pt'W/f-vl in t 11easset a porttkllio that tnlnsists 11 chtlsen Then is t() make this ptlrtllit) riskless. This is algebraically somewhat sinlpler. s() wtt will generally use this methlld in the luture. l'Iowever. the one given ltbllve denlllnst rates the cllncept of ctlnstructing 11 lvplicatlg pf/rr/b/ta morc diructly and clearly. =
.r.
-
-t'
.
we must
tz.
2.B
there-
Even if the risk in x is not tlirectly traded in the market. it suffiees to be able to tradc some (lther assct whtlse risk tracks ()r spans the uncertainty in Now we show how this works. In the process we generalize the above analysis by Ietting follow an arbitrafy Ito proccss t)f equation ( 12) above. We also demonstrate the alternative approach to replication mentioned above. We suppose therc is a traded asset whose stochastic fluctuations are perfectlycorrelated with the stochastic process for x. (This traded asset could be a simple asset such as a stock or futures contract. or a dynamic portfolio of simplttassets, that is. a portlblio of assets whose contents are adjusted continuouslys() that the value of the porttblio is perfectly orrelated with the process To remind ourselves that the traded asset is track-ing or spanning the for risk in we call it the spanning (i.e.,replicating asset. Let A'denote its market price. Then the stochastic proccss of X must take the form
fore choose
The Use of Spanning Asscts
-r.
-r
However, in the market. two assets with identical risk must earn equal return. Therefore this choice must also ensure
(.r. 1)
+
(.r
c2 x2 F Fx (.: J ) + 1. x,r (x t ) 2 .
Fx,
.
J)
.
=
r + n
(a
+ jlar
1 + nx
.
4lD'or more on the capital assct pricing modclv see any standard text on financial economics'. Brealey and Myers ( 1992) is relatively elementary, and Huang and Litzenberger ( 199f)) is more
advanced.
5To be rigorous. in continuous timc r(.r. t ) can ciange even over this short interval, and the is of magnitude and we can ignorc it.
evolution of it is random. However. the differcnce made by this consideration
dIl
(l +
.
Ft-r
If our portfolio is to replicate the risk of owning the firm,
.Tr
lzb'/
.r
-r
,
Substituting for
'ttttlcr
.
Compare this with holding ownership of the 5rm for the same short intelwalof time dt. This costs Fx. J) to buy. The dividend is the prolit zrt-r. /) J/; this involves no uncertainty since is known when the initial decision is being made.s The asset also yields a random capital gain, which we calculate using Ito's Lemma as
z (.,r t ) +
JJ/j???j--t?/jc'?? Ikbvlltl''ki-f'7--
-r
return and risk characteristics using traded assets of known value. Speciticallys consider investing a dollar in the riskless asset and also buying n units of the hrm's output', we will choose n shortly to achieve the desired replication. This portfolio costs (1 + nx4 dollars. Hold it for a short interval of time dt. In this time, the riskless asset pays the sure return r dt, while the other asset pays a dividend n dt and has a random capital gain of /1 dx n e-r dt + /1 tz'-r Jz. r
cwtzlct.zl-/tljli''
-r.)
.r,
llltl Sti/t.-'t/-/4l
Ibltlllleltltlt't'll
Note two points. Firsts the coefficients z1, (.r. t ) and Bfx. t ) llre I-unctions ()1' thd not the price of the replicating asset. X. This is in kecping state variable, with the notion that the state variable summarizes all the intbrmation about the current state of the economy. Second. the coefficients z:ll-r t ) and St-r, t ) of the asset price process (2l ) need bear nt) relation to the t1 (.t-.l ) and /?(-r.t ) of the state evolution process ( 12),.but the two 'Wiener process increments dz must be the same if X is to track the stochastic tluctuations in (Wllen we say same, we mean that the two must have identical realizations, not merely that they have the same probability law.) Suppose the replicating asset also pays a flow dividend at rate D(-r, t ). Then one dollar invested in this asset over the small interval of time from t to (/ + dt ) generates the total return: -r,
.
-r.
This has basically t l1e salme forn) as the earl ier (2()'). The merit of the contingent clainps valuation approach in this context is that all the coefficie nts of these equa tions are e ither k nown frona the specilication of the mode I stlch as (1 (.r t ) t)r can btl obse rved or dstimated from the market- as with ii ( / ) Tllen the pa rtia I d ifferential eqtlation can be solved to obtain the valtle t)f the firnl. .
.r
.r
.
Next we ask what rate of return will make an investor willing to hold this
replicating asset. The capital asset pricing model (CAPM) formula for the required expected return. which we denote by /z,r(-r- t ), is Jz .r (..r-J )
=
r +
4 ;)b St-r t ) ,,,
(22)
.
.
To understand
The market this, compare it to the earlier lbrmula ( l 9) lr is it the in the is aggregatc two cases. an so same parameter, 4 is S(.t--t ). and since tllc changes The standard deviation of the return on tiX and t-r are pertctly corrclated, the correlation coemcient Inetwecn the zzrates of return on and the markct is the same as that between and the market, namely pv.,,,. Finally, note that in an equilibrium wherc thtl asset X is actually held, we must have
price of
-r.
risk
'
.r
.
Srnooth Pasting The above analysis assunles (lnly that the varitlus assets are held dtlring a very short intcrval ol' tinle t/J Wllllt happens lfter til'ntl / + t// is ()f nk) tzllncern. and does nklt alkct tlle validity ()f tlle partial diflkrential eqtlatitlns (24) ()r (2(),).I.lowever. sllltltillns t() these eqtlat ions rctluire bllu ntlary conditions. antl .
there
lre sllme attentit) n tt) Illnger ti nAe spans. I f the firm thltt is l'leing vltltletl abilvt, Iltks: Iixetl tilne Illlriztln T svhen it is forced to take lt te rn4 ination payoff f2 (.t-r F) then &ve can solve t htl pa rtial .
.
d iftk rll nt ia I (lq ua t io n su bject to t he b()tl nda lz c() nd it i()n 1--( F ) f2 (x T ) for aIl Likewise. thtl fi rm may be-/brccf/ t() take t l1e te rm inatilln pityllff at an ( l ) I-!ere the boundary earlier time l if the state variable hits a threshold condition is clearly .t-
.
.
-t.
-r*
.
;; (-r t ) .v
.
=
D(.r t ) + zlt-r ,
,
/
)
(23)
.
Now consider a portfolio that consists of the firm and
11
units of a short
dollars to buy. I-lold it for a position in the asset X. This costs (F(-r. t ) n short interval of time J?. During this time the firm pays a dividend of Jrt-r, t ) dt. z''l
-
Further, since a unit of X pays a dividend of D(-r. / ) Xdt the holder of a short position must pay this to some corresponding holder of the long position. Tlle capital gain on the portfolio is ,
dF
-
6IX
n
=
g ; + a F.v+
bl Fxx
.51 .,z1 -
n
dt +
g!l&
-
?1
B
.'1
Jz.
ctlndition ( l 4) we found in the setztitln ()n This is exactly the value-matching dynamic programming. Sometimes the firm can clloose its tcrmination optimally. knowing its termination payoff function f2 (.,r / ). This decision will be made so as to maximize the firm's value. We know from our dynamic programming analysis that such choicedetermines a threshold or free boundary (/ ), and that the appropriate additional condition ( 15) is the property, .
-r+
where we have used Ito's Lemma, and have omitted the arguments of the functions for brevity. To make this portfolio riskless, we must choose n b F / ( BzY)
tsmooth-pasting''
=
.
#
l.( (/ ) t ) ..r
,
:=
f2 t. (
-'t'
#
(t ) t ) ,
f()r a ll /
.
/ltlthi-jl/'f?l//lf/ 1bf1(llIlL'IlltlliL'(II
120 2.D
Poisson Processes
Suppose the state variable follows the Poissonjump process t)f equation ( l ()). rather than an Ito diffusion process. Can we create a rcplicating portfolio and use it as we did above to obtain an equation analogous to (20)? ln principle, it might be possible to 5nd an asset that duplicates the stochastic dynamics of a'(/). For example. if x(/ ) is the price of oil (andwe believed that it followed a Poisson process), the replicating asset could be a near-term tktures contract on oil. More generally, however. we would have to with a dynamic porttblio of assets. the components ofwhich were replicate follows a diffusion process. this continuously adjusted as ) tluctuates. lf continuous. is itself feasible, because the path of is so the porttblio can be value It is not feasible if another. from time to one adjusted as moves over takes discrete follows and Poisson a jumps. process x This means that when working with Poisson processes, we will tlsually have to make one of tw() assumptions. First. we can assume that stochastic changes in associated with the Poisson process are uncorrelated with market portfolio. Then there is no adjustment tbr risk. and equation (24))will again (Note that this is equivalent to dynamic progranlming hold, but with - r ?-.) Altcrnatively. cqual we can use to thd risk-free ratc. with the discount rate with programming, discount rate, p. dynamic an exogenous .z
-r(f)
.v(l)
-v.(/
-r
.r
-t'
-a.
3
Relationship
between the
rmo
Approaches
By now the reader should begin to see the close parallels between dynamic programming and contingent claims valuation. The value function of dynamic programming and the asset value in contingent claims analysis satisfy very similar partial differential equations. The Bellman equation of dynamic programming has an interpretation in terms of asset value and the willingness of investors to hold the asset. The boundary conditions in the contingent claims approach are based on the idea that investors want to choose the option ex-' ercise date optimally to maximize the value of their assets. However, there are some differences, too. The dynamic programming approach started by specifying the discount rate, p, exogenously as a part of the objective function. In the contingent claims approach the required rate of return on the asset was derived as an implication of the overall equilibrium in capital markets. Only the riskless rate of return, r, was taken to be exogenous (and even that can be endogenized if the theory is taken to an even more
genentl equilibrium lcvtll ()1' anltlysisl. Thus the offers 11 better t reatment ()1' tlle d iscount rate. Balancillg this considdration.
the contingent
contingent
claims approach
claims approach
requires
the existence of :1 sufficiently rich set of markets in risky assets. The crucial requirement is that the stochastic component dz of the return on the asset we are trying to value be exactly replicated by the stochastic comptnent of the return on some traded asset tordynamic portfolig of traded assets). This can be quite demanding we require not only that the stochastic components obey the same probability law. but also that they are perectly correlated, namely that each and every path (realization) of one process is replicated by th other. Dynamic programming makes no such demand: if risk oannot be traded in markcts. the objective fnction can simply reflect the decision maker-s subjectivc valuation t)f risk. Tlle tlbjective function is usually assumed to have the form ()f the present valtle of 11 tlow function calculated tlsing 11 constant discount rate. p. This is restrictive in its own way. but it too can lnt2generalized. Of course we Iave ntl objective or tllnsklrvabltl knowludge t)l' private preferences, so testing the thet'ry can be hardu r--utility--
Thus
we see that the
twl' mcthllds have offsetting advantages and disadtllgether they can handle quitc a large variety of applications. v:tntages. In specific applications ()11e may be mtlre convenient in practice than the (lther. and diftkrunt rcaders mlly develtlp a better feel Ibr one rather than the other. but there is no diflrence ()1' principle between thtl two ()n their common grotlnd. In the chapturs that l'ollllw. we will often track the two approaches in parallel. and switch from one to the other as convenient. For example. in Chapter 5 we will first ustl dynamic pnlgramming and then use contingent Ctnd
clain'ls analysis to solve a vcry basic investment problem when to nnake a sunk expenditurc I in return for 1. factllry tzurrcntly worth )' wherc )' tbllows a getlmetric Brownian motion. This will let us explore the differenctls in the in mllre detail. twl) ltpproaches ,
Equivalent Risk-neutral Valuation A further exploration of the relationship between d ynamic programming and contingent claims valuatilln Ieads to a usetkl way of writing down and interpreting slllutions to partial diftkrential equations for asset values. We will illustrate tllis in the familiar context of a firm whose profit flow ;rrt-r. / ) depends We will also force termination at a finite time F, with on a state variable a terminal payoff Q (x.r F ). We will assume that the state variable foklows -r.
,
/'?JJr?'?lI'.zf?JJ'fpl'?
l-a-hvblttlltlio
Tllis is ust tllkl tlyllllrn ic progranlrnillg t'ttltllttitlll ( I2) 1)1-tlle presellt cltse of geornetric BroNvn iklll nlot i()n. 'Flze boun(.l:tl'ytztpntli itdtlis t
11gttometric Brownian motitln
(1x
ilt
-t-
a
=
+
('r
a'
t/--
.
Nvill allow us to develop the exptlsition in the simThese special assumptions much plest ptlssible way, but the reader will see tha! tlle underlying itleas are more generally valid. valtle of Stlppose at time t the curre nt state is Let 1--(.r t ) denote the the lirm, namely, the title to the stated strealn of profits. We will derive this in each of the two ways developed before in this chapter. Begin with dynamic programming. Here we stipulate an exogenotls disvalue count rate. p. Then F(.r, /) is just the expected present .r.
.
F
F (-r I )
Ct
=:
,
(;'
-/?(r
-1
)
.7r
(
.
r
.
r ) ( j r '+ e
1111(1.,1- /....'5?f'(??-/t'??l/:-
-/'t
r-f
)
)( r -r
.
r)
F a- F )
f? (-r F )
.
('(.)1- l lI
.
.t'
,
which our expression (25) slltisti!s by const rtlctitlll. I11 ('ther pression is the solutitln to the pflrtial tlifferential tlquation. I-lad Nve lnelltlllby tleriving
tlle equntitln nnd thtt bllundal'y
.r
.r
-
-
-
.
.
as of time t If we consider the situation a short time dl later, the state variable will (af tlle asset will have changed to have moved to (-r + J-r). and thc value Fx + dx t dt ). X) express this in time-l equivalent units. we must discount
where t-; denotes the expectation based on the information
+
.
Further, (/.r is a random increment from the perspectve must take an expectation. Thtls
:-P2f
,
(2t4) This idea of splitting the whole time interval from t to F into twl) parts the wholu immcdiatc short interval t/J and the continuation bdyond that is thc is Bellman equation. essence of dynamic programming. Thus euation (26) a during the interval JI, is taken action instance this kind. trivial In no but of a right-hand side. so there is no maximization on the (lmit side terms right-lland of the (26) using Ito-s Lttmma- and Expand yields (). than This dt dt as that go to zero faster --
+ e
,
=
dt
-/'
n.(-r t ) (
7t (F (-r + J-r t + d ) j ,
x (.z-.t ) J/ + ( 1 - p ''6
1k!jF ( r / ) c . t
.
1.'
''
-(.r,J) -
+
''
9
ga .!
=
2 :.2
/7.2
(lt ) ( A'(-r. / ) + Fi (-r, ) + F. (-r t ) a
(lt
,
antl
--3
.
l
it by the factor of time / so we
tronditioll-
tlx-
then started look ing for :1 solution. that woultl have seelmed t fornlidktble task. Howeve r- th is is lt vtt fy lucky and excptional instancc Nvhtlre vvtt kneqv the solution (25) even be 1'()re vvtl t1e rivetl the eq uation. O f cou rse it neetls a Iot of wllrk to eval tlate the expectation. We k novv that given tlle ini tial at time /. the state at any ftlture t inle is llpgntqlrnlalIy d ist ribtl ted. >Xlso-lbr > ) ltntl v:triktnctl r 2 /. the Iogarithnl of a-r has l'nean I()g + (tz ..ty ) ( r Il /' ) tlnd / ). ( f2 tlllctiollltl r have (.r/. ient (.r. ) l x ver.y ctlnvell f'r ttlrnls- tor exllluple.povvers t)r tlxplpnelltialsit is ptpssibte (t) evltlultte thtt expression (25) explicitly.Othenvise tln(2 Inust rtlsllrt t() Iltlnlericlll stpltltitllls. I''Illwever. tlle expression llktssome conceptutl use. as we sot'n see. Tllc result thktt (25) is tlle stlltltilln t() tlle tlil'lkrelltiltl etltlatitpll 27) Nvitll ( the blltlndkry cllllditil'n (28) is :t speci:ll case ()1- :1 ver.y generttl result k nllsvll I'(lr as tllc Fttyllnlltn-lllc lklrmtlllt: see Kltr:ltzlls tlld Sllreve ( l 988. pp. 2671*1*) ()!' it.f' a mtlre detailed Ctnd rigllrlltls tlisctlssilln -
.
svortls- tllat
-r
dt
.
.
'will
N()w cllnsider tlle prlllpleln
()1*
v:ltli ilg
contingent clltinls s'lklltlCltitlll. I Iere sve itlreiltly the partial diftk rll ntial equktitln (2()), svhich
Recall tllat
tlle lirnl l'rtlnl t l'le perspective (11' slltpwetl t I):tt tlle vllltle skttislies we restttte lk,r t! ase klt' re ferencc:
is the riskless in terest rate. C'ntl tti pt a is the dividend t)r conveniencc yield t)n the asset. The boundafy conditikln is again (28). I-lerc we do nllt k now thll stllutilln in advance. I'Illwever- we can write it down immediately by ntlticing the formal analogy between this partial differcntial tlq uat ion and thkl ()ntl (ll7titi netl us ing t he dynam ic progranlm ing 1.
=
-
.r
dt j
.r2
Fvx(.r. t) +
.
p Fx- t) + zr (-v-t)
a
.r
Fv(-r. /) +
) (.r- /)
1J/.
Substituting in (26)and simplifying, we see that Fx. t) satisfies the following partial differential equation:
(27)
'' In qtlantum elcctrtldynlmics tlle result prklvcs t() hllve inlnlense Ilrltctical tltilily. In lct it :111 underlies Fttynnlltn's t I949 ) diclgrammatic tcchnique I'(lr sunlnlillg prt ll'y:ll'lilities (lver ptpssible paths of u particle. l-lis apprllach, develllpetl beforc dynamic pnpgrlmmiflg ltnd l t()'s Lenlnla had ()f. becn thought lhc prtll'lttbility was ;1n amuzing acllievement. Since the deptrlltlcnl vari:lble amplitude in quantllm ttlectrodynanl ics is ctlruplex valued, tht! anilltlgy with dynarrlic prllgrCtrnming ltnd cllnti ngttnt cIa inls val uat illn m:ly nllt extttllf.lbcytlntl l 11emrtt Ilenllll icitI lrrnal isnl. I1*it r( does. then in addititln t() :tII his acllievements in physics. Feyllnl:lll clltlld be claimcd ;ls tlle l'atllt!r tf (inancial econlymics.
s'liltllellaticill
'ltlc/g/-tpl//lt/
L'yllal'lic Op/l'?l?ztzlll ltllder
approach, (27).The exogenously specified diseotlnt rate p tbr the latter is growth rate a of the genow replaced by the riskless market rate r. and the J). replaced In other words. we can ot' motit)n by is (l' ometric Brownian evaluate the future payoff by discounting it at the riskiess rate r. provided we with a different growth rate are willing to pretend that follows a process a' J. r parameter -r
Uplc-cr/t/ll/.y
! 25
-r/
Next suppose f2 (-r) Ibr some given on the lognormal distribution.
p. Now.
=
using standard
results
-
.v
=
-
and Fx. ())
Therefore the solution is
For
C(A,
f)
=
where is an t, but thereaer -r'
(29)
S't that starts at the same initial point follows a new geometric Brownian motion
artificial variable
.r
at time
0.2
xtl exp
=
((1a p(#
l ) + (?.
-
e-& F
J)
-
p
-
r)
r j.
1 this reduces to tk we get just tz-'' F' the as above. For p H riskless and theretbre discounted at the rate r. Finally, if is 1 is payoff p a root of the quadratic equation
p
-t'
=
=
-r0
1 0.2 2 pp
l ) + (r
-
J)
-
p
-
r
().
=
then we get F(.r. 0) Here t I1etbrce of growth is exactly offset by that of discounting; thus we can just evaluate the terminal payoff function at the initial state a- and call that the asset value. Nve will see this used later in Chapter (b and elsewhere. -r#
=
The expectation
t-' is taken l
with respect to this stochastic process, from the (namelythe value of at time ?. .r)
perspective of the information
risk-neutral valuation-'' a proWe have here an instance of economics: see financfal applicability in wider interest and with much cedure Dufhe ( 1988, Section 17) and Huang and Litzenberger ( 19t?(3,Chapter 8) for rigorous and general theory.
3.B
Examples
We illustrate the procedurc with somc simple examples. In each case we Cind the initial value F(.r. t)). First consider the simplest case whcre therc is no protit tlow, and the Then terminal payoff is f2 (-t) =
For a somewhat
Seequivalent
-r.
F (-r ()) -
-
e
=
P F
.
Taking the equivalent risk-neutral perspectivesthe expectation cess is E-Er 1t) 1 ..'
,
=
.E e
4r...
)
r
trickier cxample. consider the case of risk neutrality tl and b. detine
of the
)
1 i1-tl
( ) -r
<
.t-
-
(s
/7 .
() otherwise.
Now F(-t. ()) is simply the probability that the geometric Brownian motilln,
starting at the initial value will aftcr time F end up in the interval (tl. bt. By letting b converge to anwc get the limit of F (-r. 0)/(: as the corresponding probability density. .r.
-fp)
Either of our approaches
12 /7'2
.r'
pro-
.
and
tbr numbers
and
shows that this function satisfies
differential equation
S() (v'r 1 -
no discounting.
.
-r
2
F ,.x ( r t ) + a . .
,
.r
6r (
.r
,
t)+
6.(x t ) ,
=
the partial
()
.
In other words. we have derived the backward Kolmogorov equation Appendix to Chapter 3) as a corollary of our asset-pricing formula.
(seethe
Therefore
4 In other words, we reeognize that grows at rate as and discount its future value at the risk-adjusted rate y.. This is obvious, but serves as a simple way reader's conhdence in it. to clarify the general formula and build up the
Guide to the Literature
-r
Dynamic programming was developed by Richard Bellman and others in the 1950s.It s a standard tool in economic analysis and operations research. and
ail'ned
is treated in several textbooks. For a partictllarly sil-np.Ie exptpsitilln expositions can l7e at economists, see Dixit ( I9t)(), Cllapter 1I ). fltl'ler gilod llntl t)87). Schwartz ( l tlt? l ). and Kanlie n ftlund in Dreyfus ( 1965). Harris ( l
For an outstanding and very thorough treatment. with sevdral applicatitlns economics. and to dynamic general equilibrium theory. growth theory, Iabor 98t)). other topics, see Stokey and Lucas with Prescott ( 1 In this book we use dynamic programming only in the context of optimal control and stopping et- Brllwnian motion. For morc details but still at :,11 intuitive level, see Dixit ( 1993a). For vel'y rigorous treatments. see Fleming and Rishel ( 1975) and Krylov ( 1980). Contingent claims analysis was systematically developed tbllowing the t?73), aIpioneering papers of Black and Scholes ( l 973) and Merton ( I 97 I I though an early contribution by Samuelson ( 1965) is noteworthy in thltt it introduced stochastic calculus and the smotlth-pasting condition. Contingent claims analysis has now become an establishetl part of the literature on linancial economics, and has textbooks devoted to it, for example. Cox and Rubinstcgn ( 1985), Hull ( l 98t)), and Jarrtlw and Rudd ( l983.). A beginner 987) and Varian ( 1987) in can beneht from the expositions of ()lRubinstein ( l Ecllntlmic Perspcctivcs. C()x Ctnd Rtlss special symposium in tlle Jtlurnal a t),79) develop cllntingent claims vll( I 976) and Cox. Ross, and Rubinstein ( l (.)1%Balwnian mllt illn fhat we uation using the random walk rcprcsttntatilln discussed earlier in this chapter. Mortl advanced treatments include Dulie t?9()). ( l 988, l992), Dothan ( I9.)(3), and I-iuang and Litzenbergcr ( I For a pioncering rigorotls treatment of equivalent risk-neutral vltltlation related ideas, see ldarrison and Krcps ( l 97t?). Prccursors of these idcas and and Cox and Ross ( l976). can be tbund in Arrow ( !97()) (lxcellent and vtlry enjoyable history ()f the devel1 Bernstein ( 992) is an opment of thesc ideas. -
Appendix A
Recursive Dynamic Programming
l-qhlttl'tl I'li'l't I1:'tl1tI l1:'t Itlv'v tl-lr'k '''k'e Iltl itltl lls :t 1-1 'We sec k :1 ftl Ilct i(.)11 J'-( ) t 1.1 lt t s:l t is 1ies (5) I'0g;.l rtl t Ile rigll ts jlk:t itpl-l('11* 11 ftI Ilctilll''l. 1-1:t(.'I'- t'yl' :.1 f1.1 Ilt-tit'll (..-.i ivel't :t ftl /7-(.r). t 11c l'igll t tppe side tle li Iltls :.1 l'ltlyv l'tlllct i()I1 ()I'-r. 'T-lles(.)Itl t i()I1 is 11l'tlllct itlll t- vlllll''l()1)e itrlpI te I'11'!s11 (', 1* lle on i11 t 11is Svtty. Ie:tds 1(.) itse If. I11 tec 1111 see k l.t lixetl !7(.'1 t t .r
.
-
'sve
(lpera to r.
Ft')1 :.1141ittlrllt ive prtlcetltl re t st trts t 11 l I'ly i1)it itll clltpictl /St 11 ( r ) NoNv tl1 is () rigllt-llantl sitlc 1'(5). tlse tlle rigll t-ha ntl sitle is ftllly kIltlv.vll. ol'l t he st) t he restl Its k) (''ltpplyi ng t I1e() pt! rt t(')1- tra 11 be ca lctl1:1ttltl lk'lr evtl lz C:lll t 11tl 1 resuIt 11 neqv ftlnct io 11 Ii-b (.r ) Next tlse it (./1-1 t I'ltlrigI1t- l1and s ide to gtlt the next iterate F S1( r ) and so tln. Wllltt Illkppells to t I'le seqtlence t) 1'stlch llntrtitlns F$ l ( t- ) as ??l goes to in fi n ity-? Supptpstl tllllt instekptl of l---l ' ( r ) vve tlse 11tli l't'krtlllt st:lrt ing functitln. sity. l' .z'f ( r 1 F' 11 ( :' ) -h. 1-for sonAe posit ive tztpnstant l Subslituting inlo t he righthand s ide o 1'(5 ). Iirst 114.)t t't t I1:tt 'svi
.
.r
.
-'
.
.
'''
.
''rllus Nvt! ge t :.111 ext r:t te rn1 ( l I'nizi 11g cllllice ntlt ;.tIter tlle I'nltxi
'Fllus. lr ftlnct i(')n
:tl
t I1e restll t
.r.
2
Proceeding in this
t)l'
()1* klpplyi
.Z ) (a ) .
..4
/-
l
1'
t I'le rigll t- I1;lI)tl
sitle. is lldes 11 I-l(')r t I1evlll tlc t)l' Ckny ()1' llt2 (')t I1e t r tt.lrnls. ' ng t 11etlperllttlr t() Z 1 (.r ) &vi ll I')t211 Ilesv -
(111
-1-11
.
2)
l
( . ) + ( l + /) ) ... J. .t
.
vvay.
ln other vvllrds, changes (')r in the initilllly chllscn ftlnctilln tleclly the in l ( proportitln geomctrically / l + /? ) ltt each step ())- our iterlltitln. Then it is intuitively clear, and not too hard tt) provu rigtlrously- that the iteratitln procceds to the same limiting functitln 1-2(.t-) regardless ()f the inititl chllice. In l the Iim it, ( t ) bcctlmes the same as >-t?''' ( r ) the limiti ng functilln F.( ) is of the Iixed pllint iteration step. l t obviotlsly satisfies the functitlnal equatitln a ()f The (5). geometric reductilln in crrors (technicallytlle property mapping tion property'-) has alltlwed us to prtlve existence and un iqueness of the solution. and the iterative procedurd ctlnsti tutes a numcrical algorithm. tettrrllrss-
-t'''
Here we sketch some tcchnical arguments that prove the existent:e and uniqueness of the solution ()f the Bcllman equation (5) for infinitc-horizon ()f reference'. dynamic programming. We restate the equation for ease
I
-h /? )
+
'
'
.r
-bcllntrac-
B
Ffl(XXrf)N?1t/
Nltlletntlilull
128
Here we consder the case of a bina:y choice between continuation and stopwhich we repeat for ease of reference'. The Bellman equation is
(6),
F(x)
=
max
t' (x) z (-v)+ .
1111(1e1.fl/'lc-tz/-&lpl/)' f.?J7???li'-:'t??'&,?
(11:The
Assulnptioll
Optimal Stopping Regions
ping.
ilyllalltic
1 l+p
'
1-(Fx )
I -K!
.
Continuation is optimal for those values of for which the maximum on the right-hand side of (6)is attained at the second argument, that is, .v
region'' Call the corresponding divisions of the range of the ttstopping of the interested respectively. in structure We region.'' are and the &continuation
.r
these regions.
For arbitrary speciscations of zr(.r), f2(.r), and *(.r'l-r), the regions could be any sequence of alternating intelwals. Thus continuation may be optimal for a lowest range of values of stopping optimal for a range above that, then of continuation again. and so on. However, one expects that in many problems and high low into the division of will range be a clean economic interest, there such that continuation is optimal for values separated by a threshold. say, and stopping optimal for x > (orperhaps the other way around). -t' < We hnd some conditions on the payoff and distribution functions that yield such a division. f) (-r) by G (x) Subtract Q (x) from both sides of (6),and denote Fx) for brevity. Then -v.
-v*,
-v'*
-x*
-
expression
n'(a-) + ( l
+ p )-
1
'
'
Q ( r ) d * (a-I ) -r
.
-
f2 (-v)
is a monotonic
function of x; tbr dehniteness. make it increasing. This is just the difference between the value of waiting for exactly one period before stopping, and that of stopping right away. The nice point is that the advantage of waiting for exactly one period translates into an advantage of waiting until an optimally chosen stopping time. If the function is increasing, it, we expect continuation to be optimal for high a' and termination forjow decreasing, the other way around. $. ,'
-r;
.
.'
Assltlnption
and immediate termination is optimal when the opposite inequality holds.
l29
There (21-.
is positive persistence of uncertainty, in the sense that shifts uniprobability distribution *(.v'l.r) of future values right when the current value x increases. If this failed. a larger current relative advantage for high x would be more likely to be reversed in the near future. This assumption will hold for alI the processes we will considcr.
the cumulative formly to the
.v'
Given these two assumptions,
the solution
function G(x) for
(3l) must
bc increasing. To see this. note that thc second argument of the max operator on the right-hand side consists of two parts. The tirstswhich is just the expression in Assumption I has been directly assumed to be increasing. The other. namely the integral, is incrcasing if G(.v) is. To see this, note that by Assumption (2J,a larger x shis the probability weights attached to the increasing set of values G(-r') to the right. and thereforc raises the expected value. Thus. startingwith an increasing function. the right-hand side yields another increasing function. Then the lixed point of the ittlration step, namely the solution of (31), is itself an increasing function.7 ,
We have proved that the second argument
in (31)is increasing. Therefore
there is a unique x' such that the second argument is positive if and only if and stopping is optimal Then continuation is optimal to the right of x> to the Ieft, as we set out to prove. -v*.
.r*,
In continuous time, we must replace p by p dt and zr(.v) by zr(-v) dt. Suppose x' follows the geometric Brownian motion dx + J.r, and =
+ (1 +
p)
-
:
-v
=
J* (x'I.r) G (-x'/)
Now make two assumptions that together sufhce to establish the desired
property.
-v
7To be technically prccisc. the operator
is closed on the convex cone of nondecreasing
functions. and therefore has a fixed point in this subspace. That is also a sxedpoint on f)f space functions. However, wc have already shown in Section A of the Appendix that xed point is uniquc.
the whole the latter
z'
l 3()
t'Yltllllolltltictll /.?tl('/v',r'pll?l(/
Then. on expanding j''z (A') using It()-s Lelnma and sinlplifying. (l I becomes silnply the requirement tlllt
dt + rr Assumption Jz
tlz.
-r
.v'
l 3.l
.n'(-r)
-
p f2 (
.
2
pz S''z(-r) + a t'r
)+
-r
-''
'
.
.
'
f2 (.r)
be monotonic. +'
C
Smooth Pasting
Here we consider the optimal stopping problem with a finite horizon and time
dependence, when the state variable follows an Ito process. We demonstrate somewhat more formally the value-matching and smooth-pasting conditions for (14)and (15)that determine the free boundary that separates the continuation and stopping rcgions. Over a short intcwal Fx
t)
.
=
max
time dt, the Bellman equation
()f
( f2 (.r)
zr (x / ) .
.
+ (l
(//
(6) becomes p dt ) Fx / ) + -(J F'J)
-
.
.
Stopping is optimal if the first term in the braces on the right-hand side is the Iarger of the two. and continuation is optimal if the second dxpression is larger. Fix a particular t. For dehniteness we consider the case where continuation is optimal tbr x > x* t ) and stopping for a- < x* (/ ) First suppose. contrary to the assertion in ( 14), that F'(x*(? ). t ) < f) (.r*(?)- J ). By continuity, we will have Fx, t ) < fz (.v t ) tbr just slightly to the right of (/ ). The second expression on the right-hand side diffdrs from Fx ? ) only by terms of order d so for sufficiently small t//, it. too, will be Iess than 2 (-r t ) over this range. Then immediate stopping wili be optimal tbr such contrary to the definition of (?) as the thrcshold. Next suppose that Ffx' t ) t ) > t'l (-r* (?). / ). By continuity, we will have F(.r / ) > (.v t ) for just slightly to the left of JJ, small continuation will yield a value that is x*(/). Then, for sufhciently /). also greater than f) (.v. so stopping cannot be optimal there, contrary to the definition of x* t ). The argument for the smooth-pasting condition also proceeds by contradiction. We develop it with the aid of Figure 4.2. Again consider just the case where continuation is optimal to the right of ) and stopping to the Ieft. If the functions F(x, /) and f2(.r. / ) do not meet tangentially at x*t ), they must meet at a kink. This cannot be an upward-pointing kink as in part (a) of the hgure; else by continuity f2(-r '/ ) would exceed F(-r / ) for slightly greater than (/), and termination rather than continuation would be optimal tor such x, contrary to the desnition of as the threshold. Next consider a downward-pointingkink as in part (b) of the figure. Here we show that (J) .
-r
.r
*
,
.
,
,
.t-.
.r
indil-l'klrtl'llccI'ltltsvtlel'ltlle twl) cllklices: contintlatilln for cannot be :1 pllint short interval of t in)e A / is tltl li ni tely t he be tter p()Iicy. 'I-he in tui tive idell is 11 thltt l7yNv:iting 11 little l7itItlnger. vtl c:ln (.'ll'lstlla!ll the next step of and cl)lltlse ()1* the twro dlles better tllan ptlsitillns t)l1 tlitht?r sitle (.)1' the k illk 'Xn llverkge tlllltlgh Intlst be discotllltetl kink itself. tllllt ltver:lge ptlint is trtlc the even zN/ tlccurs I'(1r it inle is th:lt lkter. Brllwnirtn t n'llltillll t he bccause rtlllsl')n ()f lntl ltre tlle A/ prtlpllrtitlnal ()n the elkct tlleir t() stltllrtt rollt s() is steps tl vvhilc t)f eflkct tlle A/ isxltln Wllcn ting is pnlpllrtillllal t() tl valtle. is snlltl 1. the prrner effect is rclatively nlucll lilrger. argunlellt needs t() be spelled ou t in :.1 Iitt Itl nlorut tlgebraic dc tail ('lt'
.t'
.
'l-llis
-1-11t:
-
.
''rllis
.
C.s
*
.
'z
.v
.
,
-v*(/
steps
in Chapter 3sizu A/? /)(.r
yvtl
(:,1'
JIl + tl (a-
/7
.
I
)
.
treat the pnlcess ('1* as lt rtntltlnl wltl k t hat cltn take ) A / tl p ()r dtnvll wi t 11respttot ive prtllatbi lities .r
/
zx t
/ /7(
/)
-v.
1
kI n Chapter 3 the drift they are mtlre general
1.2 l
I
q
.
tl
(
/)
-r -
Al
/ b(
-t-
.
/)
l ,
and diffusion ctpefficittnts u and t'r werc cilnstant: httre lnctitlns u (.r / ) and /?(-t. / ) l llltlwed cllnsider thc alternative Ntlw ptll icy: continuatilln j'tlr tin'le tt Clnd stopping t() take by further continuation if the next step of is upward. .
,
,
.r
the ttrmination payoff if it is dllwnward. Witll the appropriate probability weighting and d iscounting, this yields
.r
.
,
.
.x
n' (
-r
*
( ) t ) L J + ( 1+ p L / ),
1
gp F (
*
A-
(t ) +
tf:5.
/1 / + A t ) + .
.v*(/)
-r*
Expand this in a Tayltlr series anlund
condititln just above, and
remtll-nbering
(/
)(
.r
*
(/ )
-
f
/? / + ,
f.
/
)
J
(-r* (/ ) t ), using tlltt val ue-matclling that A/ is ()1* order ( A/; )2. Tl'1tlfirst ,
,
Matllematical Slci-qrt/l/?lr
132 >0
terms are F(x* t ) / ) + .
) !'Fr (-r*t )
,
t)
-
Q v (x t ) J )12./l *
,
.
If the functions meet with a downward-pointing kink as in part (b) of Figure 4.2, then lr > f2.v at -r*(J), so the second term is positive. Therefore the alternative policy does better than the common value of continuation or contradicting its dehnition as the threshold where the termination at optimal policy is a matter of indifference between the two. -r*(f),
partlll
A Firm's Decisions
5
Chapter
lnvestment Opportunities and lnvestment Timing
Wl-rl I -rI IIE nlathematical
preliminidries
bellind
tls, we can
now turn to the anal-
(.)f
I1,1this chapter and tllrotlghtlut ysis investment decisions under tlncertainty. will witll main cxpenditures tlur that have booka be investment this cllncern characte ristics. the expendittlres Ieltst important vttry First. partly are at twt) llther wllrds-.u//lktllltt l7e recovered. coxts in Second. these cllllnllt irrcversible; llle oppllrtunity tllat b(l the I'irnl has delayed. so investments tlan to wait for (ltller arrive abtltlt and nlarket conditions information prices, costs. t() new betbre it commits resourccs. As the simple ktxamples in Chapter 2 suggdsted. the ability to delay an ircxpenditure revcrsible invcstment can prolbundly affect the decision to invest. simple In particular. it invalidates the net present value rule as it is commonly studcnts in busincss schools: Invttst in a project when the prcsent taught to expected cash tlows is at least as large as its ctlst.-' This rule is value of its incorrcct because it ignllrcs thc (lpptlrtunity cost of making a comnlitment waiti ng for new in formation As and the option of thereby givi ng up now, we saw in Chapter 2, that opportunity cost must be included as part of the total cost of investing. In this chapter and those that follow. we will examine this opportunity cost and its implications for investment at a greater Ievcl of generality and in more detail. In this chapter, we will set forth and analyze in considerable detail one models of irreversible investment. In this of the most basic continuous-time model, which was originally developed by McDonald and Siegel ( 1986), a firm t-
.
adFirm l5-Decisiolts must decide when to invest in a single project.The cost of the investments /, is known and fixed, but the value of the project. 1Z, follows a geometric Brownian motion. The simple net present value rule is to invest as Iong as > /, but as '
McDonald and Siegel demonstrated, this is incorrect. Because
htturevalues
of V are unknown, there is an opportunity cost to investing today. Hence the optimal investment rule is to invest when P' is at least as large as a critical value F* that exceeds /. As we will see, for reasonable parameter values. this critical value may be lwt? or three times as Iarge as /. Hence the simple NPV rule is not just wrong; it is often very wrong. After describing the basic model in more detail, we will show how the
optimal investment rule (thatis, the criticalvalue 1Z*) can be found bydynamic programming. An issue that arises, however, is the choice of discount rate. If capital markets are (in a sense that will be made clear). the viewed problem be investment as a problem in option pricing, and solved can the techniquesof contingentclaims analysis. Wewill re-solve the optimal using of the examine the characteristics problem in this and then investment way, option Finally, invest and its dependence key frm's we will to parameters. on value model considering stochastic for the the by alternative extend processes will V. characterize optimal the project, find and the In particular, we of investment rules that apply when P' follows a mean-reverting process, and then when it follows a mixed Brownian motion/poisson jump process. Stcomplete''
1 The Basic Model Our starting point is a model srstdeveloped by McDonald and Siegcl (1986). They considered the following problem: At what point is it optimal to pay a sunk cost l in return for a project whose value is lz', given that V evolves according to the following geometric Brownian motion:
J?l !.,-t?-'???lt?/
1/ t)/'-yJt'p?-?'l/?lities
f llli
Ill !.zt!.???lt..'? l / Til'l ll
costs are positive and managers have the option to shut down the factory temporarily when tlle price of output is below variable cost, and/or the option to abandon the project completely, l'' w'ill not follow a geometric Brownian motion even if the price of widgets does. (We will develop models in which the output price follows :1 geometric Brownian motion and the project can be temporarily shut down and/or abandoned in Chapters 6 and 7.) If variable cost is positive and managers do not have the option to shut down (perhaps because of regulatory constraints). Pr can become negative, which is again in conflict with the assumption of lognormality. In addition, one might believe that a competitive product market will prevent the price from wandering too far from Iong-run industrpwide marginal cost. or that stochastic changes in price are Iikely to be infretluent but large. so that F should follow a meanreverting orjump process. For the time being we ignore these possibilities in order to provide the simplest introduction to the basic ideas and techniques.
We allow exogenously specihed mean rcversion in Section 5(a) of this chaptera and consider industry equilibrium in Chapters 8 and 9. Note that the tirm's investment opportunity is equivalent to a perpetthe right but not the obligation to buy share of stock at uaI call (lption a a prespecified price. Thercfortt the decision to invest is equivalent to deciding when to exercise such an (lption. Thus. the investmcnt decision can be viewed as a problem of optilln vtluation (as we saw in the simple examples presented in Chaptttr 2).1 Altern:tively. it can be viewed as a problem in dynamic programming. Wc will derive the (lptimal investment rule in two ways, first using dynamic programming. and then using option pricing (cntingent claims) methods. This will allow us to compare these two approaches and the assumptions that each rcquires. We will then examine the characteristics of
the solution.
In what follows. we will denote the
value
of the investment opportunity
(that is, the value of tht) option to invest) by F( P'). We want a rule that maximizes this value. Since the payoff from investing at time I is /'1 J, we want to maximize its expected present value: -
dF
=
a )' dt +
.>'
/' Jz.
where dz is the increment of a Wiener process. Equation ( 1) implies that the current value of the project is known, but future values ate lognormally distributed with avariance that grows linearlywith the time horizon; the exact formulas are in Section 3(a) of Chapter 3. Thus although information arrives over time (the hrm observes V changing), the future value of the project is always uncertain. Equation
ample, suppose
(1) is clearly
an abstraction from most real projects. For exthe project is a widget factory with some capacity. If variable
F( p' )
=
max
k6jt
Zr
-
/ )d-/'F
1
where i denotes the cxpectation, T is the (unknown) future time that the investment is made, p is a discount rate, and the maximization is subject to lThe investmcnt opptlrtunity is analogous to a perpetual call option on a dividend-paying stock. (The payout stream from the completcd project is equivalent to the dividend on the stock.) A soI utitln to this option valuatiun and cxtlrcise problem was (irst found by Samuelson ( 1965).
zl Iirln
138
'.
Inecisiolls
eqtlation (1) for P- For this problem to make sense. we mtlst also assume that a < p,. othenvise the integral in equction ( l ) could be matle indehnitely larger by choosing a larger F. Thus waiting Ionger would always be a better policy,and the optimum would not exist. $Ve will let denote the differncc > p e; thus we are assuming J 0. .
Finally- by substituting expression so Iu t ion f0 r F-( 1. ) :
ing
-
(4) illto eqtlatilln (.3).we
obt:lin the
llltlsv-
(
-
1.A The Deterministic Case
(1z
=
euT
14
-
e-PT
(3)
or fall over time- s() it is clearly > and 1, l'' if never invest otherwise. Hence optimal to invest immediately (P/. 01. Fj max What if 0 < a < p? Then F(F ) > () evcn if currcntly lz' < 1, because eventually P' will exceed 1. Also. even if )' now excceds 1, it may I7ebetter to with wait rather than invest now. To see this, maximize F( lz') in equation (3) condition is respect to F. The lirst-order
o--rhenPe(/) will remain constant
=
.
=
-
.
*
=
-
.
=
=
F(Iz') a :s
*
Figtlre 5. l shows F 1/-) as a ftlnct ion of I for / l p ().1()- antl a = (10.03, and 0.()(' ln each ease. the tangency point of F IS ) with the line F I (.r). is at the critical value Pr Note that F F ) incrcases when a p //(p increases. as does the critical valtle lz'e G rowth in lz' crcates 11value to waiting. and increases the valtle of the investnAent opportunity. =
future time T is Suppose
l''
'
Although we will be mostly concerned with the ways in which the investment decisionis affected by uncertainty, it is useful to first examine the case in which there is no uncertainty, that is, t'z in equation ( 1) is zero. As we wiil see, there can still be a value to waiting. P-(()). Thus given a ctlrrent P-, )'0 cal, where P'() 0, P' (/) lf c assuming opportunity we invest at some arbitrary value investment the of the =
for U' >
l.B
The Stochastic Case
'sve will now return tt) the general case in which cr > (). The prl'blelu is t() determine the ptlint at which it is optinlal to invcst / in rettlrn for 1111 asset worth V Since l'' evtllves stochastically. we will not be able to determine 4 time T as we did ltbllve. Instcad. our investment rule will take tlle form of .
-
2
.0
1
.8
.6
# F( )' ) JF
-(p
=
a
-
whichimpliesz
1
) p'
r + P l e
f?-t/l-tz,
-p
r
=
(;
1 .
1
.4
p/
() (p a ) )' a Note that if IZ is not too much larger than J, we will have F* > ().The reason for delaying the investment in this case is that in present value terms. the cost e-PT whereas the payoff is reduced of the investment decreases by a factor of e-sn-x'r of factor smaller the F*
=
max
-
Ioi!
,
%-'
.
-
by
Fo/ what values of )' is it optimal to invest immediately'? By setting T* 0, we see that one should invest immediately if lzr k: P' where :,
As we a critical value P'* such thtt it is optimal to invest tlnce F : value to that /'*, is. will result higher greater value in a of tr a see, a higher growth both genera. mind, that in however, waiting. It is important to keep in thereby waiting and value > uncertainty to (a > 0) and ((z 0) can create a affect investment timing. In the next twosections.we will solve this investment problem in twowayss following the techniques described in Chapter 4. First, we will use dynamic programming, and then we will solve the same problem over again using contingent claims methods. This will enable us to carefully compare these two approaches. F*.
the deterministic case), we assume that a < p, or J > (). With this the Bellman equation becomes the tbllowingdiftkrential equation be satished by F( r' ):
(9) In additions F F) must satisty' the tbllowingboundary conditions: F(0) F( F
*
)
Condition
ln the terminology of Chapter 4. we have an optimal stopping problem in continuous time. Because the investment opportunity, F( 1/,), yields no cash flows up to the time F that the investment is undertaken. the only return from holding it is its capital appreciation. Hence. as we saw in Chapter 4, in the continuation region (valuesof l?' for wllich it is not optimal to invest) the Bellman equation
is
=
C(dFj.
Equation (7)just says that over a timc intelwal t/. the total expected return equal to its expected rate of capital on the investment opportunity. p F f, is
appreciation.
We expand iIF using Ito's Lemma, and we use primes to denote derivatives. for example. F' dl-hlll, F'' = dlb-/dl'' etc. Then =
F'( p-) dv + 12 F''( p-) (J)')2
db-
=
gives
( 1) t'or#lz into this
*)
=
*
1
-
1
.
( 10) arises from the obselwation that if
)' goes to zero. it will stay of the stochastic is implication at zero (this an process ( 1) for P' J.Therefore when will of value option P' invest be 0. The other two conditions to the no consideration optimal from Z* is the price at which it is investment. of come in investp the Ianguage of Chapter 4, the free boundary of the optimal to or region.Then continuation ( 1l ) is the value-matching condition: itjust says that rcceives investing, the lirm a net payoff 1z'* 1. Finally, condition ( l2) is upon condition. discussed in Chapter 4 and its Appendix C. the If F( V ) were not continuous and smooth at the critical exercise point P-'a one could do better by exercising at a different point. Note that equation (9) is a second-order differential equation, but there are three boundary conditions that must be satissed. The reason is that aI()) is known. the position of though the position of the first boundary ( other second words, boundary is not. In the boundary'' /'* must be the determined as part of the solution. That needs the third condition. Equation ( 11) has another useful interpretation. Write it as P'* F( P'*) When the hrm invests, it gets the project valued F, but gives up the oppor1. fu?i/.yor option to invest, which is valued at F( F). Thus its gain, net ()f the F Z). The critical value F* is where this net gain opportunity cost. is P equals the direct or tangible cost of investment, 1. Equivalently, we could write the equation as V* = / + F( F*), setting the value of the project equal to the full cost (directcost plus opportunity cost) of making the investment. We will discuss this point in more detail later. To find F( P), we must solve equation (9)subject to the boundary conditions ( 10)-4 12). ln this case a solution is easy to find; we can guess a functional form, and determine by substitution if it works. We Iirst state the solution and derive some of its properties, and then discuss it in more detail. =
-
'
=
-
*
expression and noting that f'ttfz)
a P' F?( Z ) dt +
Hence the Bellman equation
1 (7.2
becomes
)
(7.2
P' 2 F''( Izr) dt
=
0
.
by dt):
(afterdividing through
72 F'' 1z) + a P' F'/l
-
p F
:::s
=
tefree
-
SLJF')
2
/'
( l0)
ttsmooth-pasting''
p Fdt
Substituting equation
0.
=
=
F' ( P'
Solution by Dynamic Programming
notation. that must
(8)
0.
Itwll be easier to analyze the soiuton and to compare it to that obtained using 6. To ensure p contingent claims analysis if we make the substitution with connection in explained already existence of an optimum (for reasons =
-
=
:'
il-l?l lLlc'isit'-u lzl 1-2 -.
'
'T'osatisfy tlle blltl ndary cond ition ( 1())-tlltl sol u tion
t:tktl t Ile fornl
fntlst
.I J /' l
F ( )' )
a I'Id
( 13 )
'
where z.l is a constant that is yet to be detc rl'n ined, and pl > I is lt k ntlwn constant whose val ue dklpends on the paramete rs fz /)- nnd J o f the di ffe rd ntial equation.
t). so t Ile gel'le ral
.!. a
( /)
*
*
Pr
#l
=
p
l -
l
( l4)
1.
2
g( /)
) yr)'
stll tl t ioll tk) eq tlttt itln F-(
.
The remaining boundary conditions. ( l l ) and ( l 2)- can btl tlsed to solvtz for the two remaining unk nowns the constant z1, and the critical value )' at which it is optimal to invest. By substituting ( l 3) into ( 1 l ) and ( 12) antl rearranging, we Iind that
f5
'
.!. S + a
1
'
/rT
)
-
(9) c:tl'l 14e vri ttel'l
zzl
)
t
I
l''-# ' +
z 1a
'n
-
/)/f'' 2
:ts
l-'01
where Ctnd zla are constants tt) be deterluilltttl. ln otlr pnlbldnl. tlle blltllldary (). leltving tlltl soltltion ( I 3). condition ( I()) implies that zrh Tg answer our econom ic quest ions conctt rn ing t I1eInu It iple p I / (#1 I ) we nAust therefore examine t he qtladratic equation (.l 6) in mtlre dtlta i1.Si ntre this equation- or something closely sinAilarp Nvill appear in ltlnnost every chapter. it he lps to establish 11 standa rd terfn inology and obtain sonle general rcsults at the tlutset.
,41
-
.
vvill general Iydtlnllte tlltt variable in the eq uation by p- and the wllold quadratic expression (thc Ieft-hltnd side) by t'2.Thus O-wist function 017 the vari%9Q
and
able /J, as well as the parameters /.)aand unless it is important to d() so. explicitly (''r,
Equations ( 13)-( l 5) give the value of the investment opportunity and the optimal investment rulep that is, the critical value 1.! at which it is optimal to invest. We will examine the characteristics of this solution in some detail later. For the time being, the most important point is that since pb > l we have /31/4/$1 l ) > l and r' > l Thus the simple N PV rule is incorrect; uncertainty and irreversibility drive a wedge betwecn the critical vltlue P' and 1. The size of the wedge is the I-actor pb/(Jl1- 1). and it becomes important t() examine its magnitude for realistic values of the undcrlying parameters, and its response to changes in these parameters. Tt) do that we must examine the solution (l 3) in more detail. *
,
*
-
.
*
2.A The Fundamental
qwlplp
-
5)p
p
-
=
p,
=
-
< () ren-lklnlbt)r Iy parabtlla that gtlcs to c.'o as p glles t() Als(). C'')( ( we are assuming J > ()) ltnd k?(()) -p < (). Tllerelbre the grilph crllsstls tlle horizlpntalaxis llt llne pllint t(.) thc rigllt t)l' I and ltnlpthtl!- t() t i1e lel't ()1' (). ''I-llat is. (lne rtlot, clll it p, excecds I and the (lther. pz, is Ilegative. sve lcus on tlltl ptpsitive rlult pl I'I()w dtles it changtl il' 11 pllrameter. sty changes'? Tllis is answered l)y standard ctlnlparative stilt ics. Diflrentilte c .,.l-':x).
-
,
.
.
,
f!a
(p
-
.5l/c,.
2+
-
tl/'-..,.
2
-
J. 2
2
l
0.
1 + 2p/r.r
2
>
l
.
$,
1
-p ()(/J
tlependence
.
I
The two roots are .)
sllow this
O
Since the second-order homogeneous differential equation (9) is Inear in the dependent variable F and its derivatives, its general solution can be expressed as a Iinearcombination of any two independent solutions. Ifwe try the function .,zlP'/3, we see by substitution that it satisfies the equation provided p is a root of the quadratic equation 1) + (p
svewill not
Althtlugh the r()()ts ()f a tltlltdratic are k ntlwn in explicit algebraic ftlrm. helps t() show them gellmetrically. F igtlre 5.2 shlnvs kl as lt ftlnctilln ()f p it The coetlicient of p? in k1(p ) is positive. so the graph is lln upsvard-pllinting
Quadratic
-
J.
-8
(3
zl l
-j
Dectxlolls
lll
tlplt/ Ill k'(-z'/p?l (?? It Fj/?litlg
7fJ-J??1cnt f?p/7',)?'/I t ? lities
and tll'' is given by equation long as Vt : 1, or n't : (p the firm should invest when
totally:
exprcssion
the quadratic
Irnl
( l ). The
e) 1.
-
) Q )/$1 + p Jc all derivatives are evaluated
where at pk Also
i) Q
(),
=
i)c
#1. Figure 5.2 shows that ) (? lp
at
>
pp
t'z
=
1)
-
at pj > 1. Therefore p3yl(7'< 0. In otherwords, as tz increases. #! decreases. and therefore /21/(Jl 1) increases. The greater is the amount o uncertainty wedge between 1Z* and 1, that is, the over future values of V, the larger is the
Iarger s the excess return the firm will demand before it is willing to make the irreversible investment. Readers can likewise verify two other properties of this quadratic. First, 1). increases as 6 increases, so a higher means a lower wedge pk/(/1 p, Second, p, decreases as p increases. so a higher p implies a larger wedge. We will discuss these results in greater detail and offer some numerical values in Section 4 of this chapter. Some limiting results concerning p, are also intbrmative. We merely state oa, them; they are easily verised using the algebraic formula. First, as (.. f.r is inlinite. lhe invests if is, firm F* that never and 1 x, we have pt (). We have Next consider what happens as t'z -
-..+
-.+
-.
If a
>
0. then /31
-.+
p,
If a S 0. then
-.+
p/tp (x7
t5)
-
and P'*
and 1!%
(p/)
.-->.
pI pI
Relationship to Neoclassical Investment Theoc
To push this analysis a bit further, suppose that the project itself is an inhnitely lived factory that produces a profit tlow, zt, that follows the process
(1n.
(z
n' dt +
f;r =
dz.
X =
C
Jrs. I
e
--- /,
( 16'
---
l
) jj
t
s
=
e) I
>
p
tz) /.
-
a)
-
=
! 2 p + a o'
x! ,
p-a
to
#1.
2 p + 1a t'z pt ) /
=
p 1.
>
=
JTJ
=
Jr()
e
/'?
max F
p
-
=
r
.7r1)
/
-
a :1
time T n'
can verit
t ,.-
that a
r
=
>
v
-
1 t;p- p-tz
=
)F
/ e
-
p
a
-
-
P
y. .
when
tzr
7r1)C'
=
# /
.
() is the sccond-order
condition
for this
maximization.) Therefore the firm should wait to invest even if there is no uncertainty, because waiting allows the postponement (and thus discounting) of the payment /.4 As equation ( 18) shows. with unccrtainty there is an additional 1 v2 pk term, so that the firm must wait even Ionger betbre investing. 2
This additional
term can be thought of as a correction
investment model.
Hence V is given by
)$
-
Since we have assumed zero depreciation, pl is the Jorgensonian user cost of
(The readcr
=
(p
capital'. the Jorgensonian rule is to invest when n.t p 1. Equation ( l8) says that when t-uturcprolits are uncertain. the threshold 7r* must exceed this user cost of capital. ln the absence of uncertainty. the Jorgensonian investment rule has the firm investing when (p a) 1. As we saw before. this p /. ll() when zt viewed timing rule. agains the firm must choose F be optimal Oncc as an can to maximizc
The solution is to invcst at
earlier.
(p
I
-
zr*
These results conform to those of the deterministic case that we examined
2.B
1
-
Thus the critical profit Ievel x* can be written as
/.
/.
-.>.
p
,
-
-.+
#
=
investment-3From
0
>
A: n'*
Another way to Iook at this is in terms of the Jorgensonian approach the quadratic eqtlation ( l6) satistied by p, we have
.
p Q /pf.r
xt
0
usual Marshallian rule is to invest as I-lowever. equation ( l 4 ) tells us that instead
slorgenson ( I 963) shtlwt!d that absent uncertainty. proht from an extra unit of capital cquals the user cost for suggesting this viewpllint in this cllntcxt.
to the neoclassical
thc tirm should invcst when the marginal to Giuseppe Bertola
of capital. Our thanks
Z-Ii.)our knowledge. this pllint was Iirst noted by Marglin
( 1963,
Chapter 2),
1 Frr/l
146 2.C
Relationship
''
Dccisiot
to Tobin's q tthe
value introduced 11 magnittlde q, defined lts the ratio of reprtduction ctlrrent lo of existing capital goods. or of titles them'' to cost.'' This haj become a central concept in the orthodox theory of investment. The idea is that if this ratio exceeds unity. :1 5rm can incrcase its market value by increasing its capital stock. Hence we should see a firm investing when the value of q tbr the irm is above 1 but not when it is Iess than l Furthermore, market value of f/ we can aggregate by calculating a based on the average firms in an industry (orthe whole economy) and the corresponding average replacement cost of capital. We should then observe at the aggregate level that investment spending is potitively reiated to the value of this (/. and interA number of issues arise with respect to the measurement pretation of q. An important one is that what should matter for investment is lnarginal q, that is, the (2 that applies to the (irm or industry's incremenproject, as opposed to the ab'erage c? that applies to the entire investment tal stock of the firm or industry. We will address the concept of capital and value in marginal q more detail in Chapters l l and l2. Here we want to emphasize a different problem. As stated. the numerator in t/ is the market value of existing the llc.rl assets. What is relevant for thc investment dccision is the eftkct of project on the value of the hrm. We have seen that to get this. the cost of using subtracted from the valud of a project. Thereup the option to invest must be and option value F Pr) is installed. the value value )' when project of a fore, should 1/ by F P' ). not V Correspondingly, q should increase of the firm ratio F 1. 1z') be defined as the (1z' ( 1/ Then the critical value of q that justifies will indeed be 1, as we see from equation ( 1l )swhich dehnes the investment
Tobin
( l969)
fnvc-sl/llcll f
.
-
.
ies
perspective on the decision.
tI?l(/ IlLbestllelz
effect
I
TilkI'??t)
of irreversibility
on the
hrm's
investmeut
In Chapters l 1 and l2, we will have occilsion to discuss the Iiterattlre
-btller
,
l-lpporttLllil
()11
investment that uses the t/ conccpt. Some articles have used the lirst t)f tht! above definitions of q and others the secontl. Thcretbre we will have to bkl careful to distinguish them. We will call the first net of the option valtle. antl of the tirm--concept, and the sotlontl having l as the upper Iimit the omitting the option value, and having pk/(/1 l ) as the critical Ievtll the .
tvalue
-
Gvalue
of assets in place'' concept.
Solution by Contingent Claims Analysis If the firm doing the investing is publicly held and its managers
w:lnt thcir
decisions to reoect the shareholders' interests- they should try to maximize the market value of the firm. Llow do we know that thtl invcstment rtllc derived above will do this'? One problem with this investment rule is that it is based on an arbitrary and constant discount rate. p. It is not clear where this disclltlnt rate should come from. ()r cvcn that it shlltlld be constant over time. As we saw in Chapter 4. we can use contingent claims (optionpricing) methods to derive a slightly modised invdstment rule that will indeed maximize the mllrkct value of the (irm. In this sectilln. we will go thnltlgh the reqtli red steps in detail.
-
critical F*.
However. it is difficult to allocate the portion of a firm's market value that (/ comes from the marginal unit of capital. Partly because of this, has come to namely, differently. measured, somewhat as the ratio be dehned. or at least would flow from value profits that of a completed of the expected present lo the value construction. using This corresponds investment, to the cost of its exercised and been already where has the option to invest of an existing asset. V(I. defining notation, this means q as its opportunity cost is a bygone. In our We can express the correct investment criterion in terms of this notion of q : the threshold q. that justihesinvestment is given by
(/* Jl/(Jl =
-
IZ
Reinterpreting
the Model
The use of contingent claims analysis requires (lne important assumptitln: V must be spanlled l7y existing assets in the ecllntlmy. Specihcally, capital markets must be sufliciently so that, at Ieast in principle, one could Iind an asset or construct a dynamic porttblio of assets (that is, a portfolio whose holdings are adjusted continuotlsly as asset prices change). the price of which is perfectly correlatcd with )z' This is etluivltlent to saying that markets are sutxciently complete that the firm's decisions tlo not affect the opportunit'y set available to investors.s The assumption of spanning should hold for most commodities, which typically traded on both spot and futures markets. and lbr manufactured are
stochastic changes in
etcomplete-'
.
1) > 1.
lluctuates, there will be stretches of time when the conventionally measured q exceeds 1 without attracting investment. This gives a new
As
3.A
5For a more rigorous and deta iled d iscussion oI' spanning and its ifnpl ica titlns. sce 1.Iuang and Litzenberger ( I 9t)()) and Du fhe ( I992). Du ffic and I'juang ( I 985) Iay ()u ( thk! lb 11sc t ()1* conditions needed for spanning.
goods to the extent that prices are correlated with the values of shares or portfolios. However, there may be cases in which this assumptivn will not hold; an example might be a project to develop a new product that is unrclated to any existing ones, or an R&D venture, the results of which may be hard to P redict.
We will assume in this section that spanning holds, that is, that in princiuncertainty over future values of V can be replicated by existing assets. With this assumption, we can determine the investment rule that maximizes the hrm's market value without making any assumptions about risk preferences or discount rates. Also, the use of contingent claims analysis will make it easier to interpret certain properties of the solution. Of course, if spanning does not hold, dynamic programming can still be used to maximize the present value of the hrm's expected llow of profits, subject to an arbitrary discount rate. See the discussion in Chapter 4, Section 3 for more on the relationship between the two approaches. We follow the theoryof contingent claimsvaluation outlined in Chapter4, Section 2, but repeat some details for reinforcement and clarity. Let be the price of an asset or dynamic porttblio of assets perfectly correlated with V and denote by px,,, the correlation of with the market portfolio. Since is perfectly correlated with P' px,?, pf.,,, We will assume that this asset or portfolio pays no dividends, so its entire return is from capital gains. Then
ple the
.,t
.
.r
-
=
,
.
-r
evolves according
to
dx
/.z
=
-r
J/ +
tr
x dz.
where #t, the drift rate, is the expected rate of return from holding this asset or portfolio of assets. According to the Capital Asset Pricing Model (CAPM), risk. As explained in g should reflect the asset's systematic (nondiversihable) Chapter 4. Jz wiil be given by
Jz
=
1-
+
4 pxm t7',
where r is the risk-free interest rate, and 4 is the market price of risk.fi Thus, y. is the risk-adjusted expected rate of return that investors would require if they are to own the project. We will assume that a, the expected percentage rate of change of V, is Iess than this risk-adjusted return p.. (As we will see, the hrm would never invest if this were not the case. No matter what the current where r. is the expected rcturn on the markct. and fr,,, is the b'rhat is. 4 (r?a r/a standard deviation of that return. If we take the New York Stock Exhange lndex as the market. 0.08 and tz. ().2. so / Qz 0.4. For a more detailed discussion of the Capital Asset rm r =
-
rkC
-
,
i:k;
Pricing Model, see Brealey and Myers
( 199 I ) or
Duffie ( 1992).
level of V the firm would always be better off waiting and simply holding on to its option to invest.) We will let J dcnote the diftkrence between pt and a. > (), and this plays the same role that is. a. Thus we are assuming y corresponding assumption the the in dynamic programming tbrmulation as of Section 2. .
(5
=
-
(
The parameter ( plays an important role in this model. We discussed its role as an explicit or implicit dividend in Chapter 4'. here we elaborate on those remarks. It will be helpful to draw upon the analogy with a hnancial call option. If lz'were the price of a share of common stock, J would be the dividend rate on the stock. The total expected return on the stock would be yz = ( + a. that is. the dividend rate plus the expected rate of capital gain. If the dividend rate were zero, a call option on the stock would always be held to maturity. and never exercised prematurely. The reason is that the entire return on the stock is captured in its price movements, and hence by the call option. so there is no cost to keeping the option alive. However. if the dividend rate is positive. there is an opportunity cost to keeping the option alive rather than exercising it. That opportunity cost is the dividend stream that one forgoes by holding the option rather than the stock. Since 6 is a proportional dividend ratc. the higher is the price of the stock. the grcater is the tlow of dividends. At some high enough price, the opportunity cost of foregone dividends becomes great enough to make it worthwhile to exercise the option. For our investment problem./z is the expected rate ()f return from owning the complcted project. It is the equilibrium rate established by the capital market, and includes an appropriate risk premium. If f > (), the expected rate of capital gain on the project is less than p.. /'/zTc, J is an opportttny cto'l of delaying constntction rJ the project. tIITJinstead keep'g /c option to inves alive. If ts were zero. there would be no opportunity cost to keeping the option alive, and one would never invest. no matter how high the NPV of the project. > 0. On the other hand, if J is That is why we assume very large, the valuc of the option will be very small, becausc the opportunity cost of waiting is large. As x, the value of the option goes to zero; in effect, the only choices arc invest now or never, and the standard NPV rule again applies. to Tbe parameter t5 can be interpreted in other ways. For example, it could re:ect the process of entry and capacity expansion by competitors. (However. in Chapter 8 we will discuss more complete models that endogenize the process of rivals' entry, and hnd that the resulting equilibrium cannot be well described by simply raising the J for each hrm.) Or it can simply renect the cash flows from the project. If the project is inhnitely lived, then equation ( 1) can represent the evolution of F during the operation of the project, and P' is the rate of cash flow that the project yields. Since we are assuming that 6 is -..
onjtant. this is consistent wi th future ot the prqect's market value.'
cash llows being
11 constallt
proportion
When some other parameter of tlle model (suchas t'r) varics, wtl must ask what happens to J. Various possibilities can be imagined. We will always suppose that the riskless interest rate ?- is tixed by the larger considerations of the whole capital market. independently of what happens to any one asset (()r hrm or even industry). The aggregate market price of risk $ is likewise held fixed. Now suppose o' increases. This raises the rsk-adjusted discotlnt rate g. To preserve equilibrium in the market for either a or J must change. Two extreme cases are logically possible. First, might be a t'undamental fact about x, so that must respond to the change in Jz (forexample, the dividend Alternatively, j rate might depend on the quantity of the commodity held). change of-r price and the must process might be a basic behavioral parameter, that both a and t take up so that a does the adjusting. A third possibility is part of the adjustment. In our numerical or comparative static exercises we will often regard as a basic parameter independent of o., but will mention alternative possibilities where they make a material difference to the results. -r.
changes. /V'( l'' ) I'nay change h-()I1-l one short illttl 1w:11 ()15 tinle t(.)tlle ntlxt. s() th:tt the conlposition of the portfolio vvill l7e cllangetl. l-losvtlveraovt)r eacll sllol't Id ?, fixcd. interval t)f length t/ vvtt 114.) .
The short position in this portfol io Nvil I requ ire a payl-nellt ol' l F. ( l ) dollars per tirne period-. othenvise no rational invdstor will cnter into thd IilI1g side of the transaction. We discussed tllis in Cllitpter 2, Sectilln I (:t). allt.l recapitulate thtl calculation brietly. An investor holding lt Iong ptlsititln il'!tlltt project will demand the risk-adj usted return p.t lz' Nvh ich eq ua ls t l1e capi ta I gain a Jz-phts the dividencl stream (5 V Since the short position inclutlos li-'$l ) units of the project- it will req uire paying out t P- F' ( 17) 'Tttk'ing t his p:tylne 1)t into account. the total rettlrn from holding the porttblio over a sllort tinle interval dt isS '
'
'
.
-
.
.
To obtain an expression t
lklr t' F, tlse Ito's Lelnlma: J-- ( 1'-) (1 l -F. .!. 17 ( 1,-) (t l lz')2 '
l J'--
'
'
'
?
.
Hencc the total rettlrn on the ptlrttblio s 3.B
Obtaining a Solution
Let us now turn to the valuation of our invcstment opportunity, antl the optimal investment rule. Once again, we will lct F( V ) bc the value of the firm's option to invest. We will determine F(1') in much the same way that we did in the two-period example of Chapter 2 or tbe general theory of Chapter 4, Section 2 by constructing a risk-free portfolio, determining its expected rate ()f return to the risk-free ratc of of return, and equating that expected rate
interest. Consider the following portfolio: Hold the option to invest. which is F'( F) units of the project torequivalently, worth F(V), and go short ,1 of the asset or portfolio that is perfectly correlated with F). The value of F F'( V ) P' Note that this portfolio is dynamic; as 1/this portfolio is *
..!.J'J 2
2 ( l'' ) (t/ P' )
''
(5
.-
2 1. 2 1/-1--.( P' ) t l t 2 f'z ''
17
.12'
( I-' ) t //
.
'
-
J f' /-..( 1..,) tlt
.
Note that this return is risk-free. I-lencc t() avllid arbitragtl pllssibilities, it m ust eq uaI 1- tp t/J 1- gF. F' ( 1z') )' 1(ll : =
-
=
.x
=
-
.
7A constant payout rate, J, and required expected Letting n. denote the Ilow of profit from the project. I
=
1)
Jz,
imply an infinite prllject life.
F
F 1$
returna
'T e -
Ht
/
l/
J 6j e
=
tl
l J'
-1
V
e
-J'l
/t
.
which implies F = x. lf the project has a finite life. equation ( 1) cannot rcprescnt the evlllution of f'' during the operating period. l-lowever, it can represent its evoltltitln prior to ctlnstruction of the project, which is all lhat matters for thc investmcnt decisilln. See Majd and Pindyck ( l 987, t)f th is point. pp. l 1- l3), for a morc detailctl discussion
Dividing through by dt and rdarranging tion that F P' ) must satisfy:
1 fr2 2
1,/2 F ( P' ) + (l ''
gives the following differential equa'
' -
5) P' F ( P' )
-
r f-- =
()
.
Observe that this equation is almost identical to dquation (9) obtained only difference is that the risk-free in terest
using dynamic programming. The
,z1
152
Firnl 'J Decisils
Itlvcstlttent Clpptprfktl llkft.-s
rate r replaces the discount rate p. The same boundary conditions ( 10)-( 12) will also apply here, and for the same reasons as before. Thus the solution for F P' ) again has the tbrm -(Z)
except that now and theretbre
1-
=
zd
Vl$'
replaces p in the quadratic equation for the exponent
pb ,
(24) The critical
value
l'* and the constant
,4
are again given by equations
(14)
and ( 15).
Hence the contingent claims solution to our investment problem is equivalent to a dynamic programming solution. under the assumption of risk neutra lity (thatis the discount rate p is equal to the risk-free ratel.g Thus whether or not spanning holds, we can obtain a solution to the investment problem. but without spanning. the solution will be subject to an assumed discount rate. In either case. the solution will have the same form. and the effects ofchanges in t'z or J will Iikewise be the same. One point is worth noting. however. Withvalue for the out spanning, there is no theory for determining the assumptions about make restrictive investors' or discount rate p (unlesswe managers' utility functions). The CAPM, tbr example. would not holds and so it could not be used to calculate a risk-adjusted discount rate in the usual ,
'Ecorrect''
Way.
tz/l
Itl k't.r.sf lntlllf Tl? tilg
by equations ( 13-). ( 14). (15).and (24).Some ntlmerical solutions will llclp tt) illustrate the results and show how they depend on the values of the variotls parameters. As we will see. these results are qualitatively the same lts those that come out of standard option pricing models. Unless othenvise noted, in what follows we set th cost of the investment.
0.04. 0.04, and (.T 0.2 (at annual rates). (N()te that /, equal to 1, r we do not need to know g or a, but only the difference between them- t5.) Payout rates on projects vary enormously from one project to another. so this valtle of 4 percent for t should be viewed as reasonable, but not nec representative. As for o', the standard deviation ot the rate of return n the 1 stock market as a whole has been about 20 percent on average. Although $ this represents a diversitied portfolio of assets, it also includes the effects of Ieverage on eqtlity returns. and so might be a reasonable number for an average asset. =
=
=
assarily
'
1. Thus Given these parameter values, pj 2. l'- = 2/ 2. and the simple NPV rules which says that the lirm should invest as long 4as l is at least as large as /, is grossly in error. For this reasonable set of parameter values, V must be at least twice as large as l beforc the Iirm should invcst. The value of the firm-s investment tlpptlrtunity is F )' ) 1 1,'2 for l,' < 2. and g V F( 1/-) l for P- > 2 (sincethtl lirm exercises its option t() invest ltnd the #' > wht!n payoff net P' 1 receivcs 2 ). Figure 5.3 plots F ( P' ) as a function of l.' tbr these parameter values. ltnd 0 and c also for ().3. In each case. the tangency point ot' F ( l'' ) with the line P- l gives the critical valutl P-*. The Iigure also shows that thtt simple NPV rule must bc moditied to include the opportunity cost of investillg now rather than waiting. That oppllrtunity cost is exactly F( P' ). When 1'- < 1F( P' ) > IZ l and the re forc lz' < l + F ( l'' ) : the value of the pnlject is less than its htll cost, the direct uost l plus the opportunity cost F P' ). When fr (J. V* = /, and F( P' ) () for V :; l .J Note that F( IZ'Jincreases when o' increases. as does the critical value l-' Thus greater uncertainty increases the vitlue of a rm's investment oppllrtunities, but (tbrthat very reason) decreases the amount of actual investing that the firm will do. As a result, when a firm-s market or economic environment becomes more uncertain. the market value of the tirm can go up, even though the firm does less investing and pcrhaps produces Iess. The dependence of 1z-*on c is also shown more directly in Figure 5.4. Observe that P'* increases sharply with cz Tlltts investment highlv setlxitive &.l volatilityin project vfz/lzd', irrespective ta/l-npes/on or managers risk JJre/rf//7cto'. *
=
z..1
=
=
'
=
-
=
-
-
'z
=
=
-
4
Characteristics
of the Optimal lnvestment
*.
Rule
-
Let us assume that spanning hljlds, and examine the characteristics of the optimal investment rule and the value of the investment opportunity. as given
=
=
.
.
gThis result was (irst demonstrated by Cox and Ross ( 1976). Also. note that equatilln (23) is the Bcllman equation for the maximization (>f the net payoff to the risk-free portfolio that wc constructed. Since the portfolio is risk-frec. the Bellman equation for that problem is
?-* dt
(i)
-J 1/ F' ( F ) dt + 7(J(p ).
=
is that is, the return on the portfolio equals the per-period cash 0ow that it pays out Iwhich negative. since F F'( F) must be paid in to maintain the short positionl. plus thc expccted rate F F'( /' ) P and expanding dF as before. one can see that of capital gain. By substituting * and not r equation (23)follows from (i).Alsov note that in equation (i), Jz so one must still have an estimate of the risk-adjusted expected return that applies to F This is an example risk-neutral valuation'' procedure discussed in Chapter 4. Section 3.A of the .
=
-
(5
=
-a
-a.
.
'*equivalent
.
.
'
and irrespective of the c-v/crlf to the market. Firms can be risk
wltich
neutral.
tlle r'/crlcw
'
of P' is correlated
and stochastic
''p'1k/1
changes in P- can
zzlFintz
'.
Decisiotls
16
2.0 .8
1 1
.2
10
I
'
5- 1 o C'
12
l I l
1.4
1
14
I 1 I
.6
l
'
l I 1
0.8
(.r O.3 =
0.6 (r
O.4
=
1
(r 0 (when ?, > p)
0.2
0.0
1
0.5
0.0
6
l l
4
1
?$= 0.02 ?h= 0.04
s
=
0.08
.
2
1
1
3.0
2.5
2.0
1.5
.0
l l
I I l !
=
8
l
I
0.2
11
will still increase F
*
and hence
be completely diversiliable: an incrcase in t'.r tend to depress investment.it' (5. Obselwe that Figures 5.5 and 5.6 show how F lz') and P' depend ()n ().08 results in a decrease in A'(F ), and hence from to in J t).04 increase an (5 () ftr value V'. (In the Iimit as x, F P' ) a decrease in the critical IZ < J, and 1z'* /, as Figure 5.6 shows.) The reason is that as 6 becomes else constant exccpt for tz), thc cxpected rate of everything Iarger (holding growth of P' falls, and hence the expected appreciation in the value of the wait rather option to invest and acquire Pefalls. ln effect. it becomes costlier to than invest now. To see this, consider an investment in an apartment building, where P' is the net flow of rental income. The total return on the building. this which must equal the risk-adjusted market rate, has two components income flow plus the expected rate of capital gain. Hence the greater the income flow relative to the total return on the building the more one forgoes *
0
0.0 O.1
by holding itself.
O.2
0.3
O.4
O.5
0.6
O.7
an option to invest in the buildi ng rather than
0.8
tlwni
0.9
1.0
ng the builtling
We have treated t'z and (5 as independent parameters. It' insteatl wtt allow adjust t5to as c changes, then each unit increase in f-z requres an incre:se in of 4 p..,n units. because
-.>.
-+
-.>
0. we have P' = ! if J : 0.()4s but P' discussion of the Jorgenstlnian criterion.
dtlhlotethat for our
earlicr
q
*
=
*
>
l if J
<
().()4. This bears
(ltlt
Readers can now specify desired values for these parameters, and tht!n comabove to obtain the effect of changes in cr lr this
bine the two calculations case. If the risk-free
rate. r. is increased, F( P' ) increases, and s() does IZ The is that the present value of an investment expenditure l made at a reason future time T is le-rT but the present value of the project that one rcceives +
.
in return for that expenditure is Ve-'bT Hence if is fixed, an increase in 1reduces the present value of the cost of the invcstment but does not reducc its payoff. However, note that while an incrcase in r raises the value ()f a firm's investment options, it also results in fewcr of those options being exercised. .
Hence higher
(real)interest
rates redtlce investment, but for a different rcason
Figure 5.8 provides another way of seeing how the optimal investment depends rule on the parameter values. It also lets us cast our results in terms tvalue of assets in place'' dehnition that ignores Tobin's of q. Here we use the exercising option, as explained in Section 2(c) of the the opportunity cost q. V*jI above. Then pb/(pl 1) is the critical value of this T/, that is, required to invest. The hgure shows contours of constant q. the multiple of l plotted for different values of the parameter combinations 2r/,2 and 2J/c2. =
(r
2
than in the standard model. In the standard model, an increase in the interest rate reduces investment by raising the cost of capital: in this model, it incrcases the value of the option to invest and hence increases the opportunity cost of t5 investing now. (Figure 5.7 shows the dependence of Z* on r for equal to 0.04 and 0.08.) Once again in this calculation we held fixed as r increased. If instead we hold a fixed. then increases one for one with r. Now a Iower 1. reduces p, and increases the critical Ievel F*. In this sense. a lower interest rate disctrages investment. This is a pure manifcstation t)f the option idea: a low interest rate makes the future relatively more imgortant, therefore it increases the opportunity cost of exercising the option to invest.
=
O.2
=
1 1
1.4
?lil),. /7( 1/ ). jtiyr tll/p/pfzrtll
(r
4
I !
l
0.1
6
I 1
I
0.2
8
*
N
l l I l
0-04
0.3
0.0
10
1
0.8
We have scaled
pl
=
t/
*
/(f/
* -
and 6 by 2/c2 because, as the reader can verify by substituting I ) into equation ( I6). (/ mtlst satisty' 1-
*
2r c
1/ 2 zcc:
*
25
q. ---
t'r
2
.
q
#
-
l
As the figure shows. the multiple is Iargc when t is small )r 1- is large. These comparative statics results are the same as those that apply to hnancial call options. Our option to invcst is analogous to a perpetual call option on a dividend-paying stocks wherc l'' is the price of the stock, is the (proportional) dividend rate, and l is the exercise price of the option. The value of the call option on the stock and the optimal exercise rule will depend on the parameters c, f5, and 1- as illustrated by Figures 5.1-5.7. I I We repeat that it is important to be careful when interpreting comparative statics results. because different parameters are unlikely to be independent of each other. For example, an increase in the risk-free rate, r, is likely to result
-
I1
F()r more detailed discussions ()f linancial call options tnd Hull ( l t?89).
Cox and Rubinstein ( 1985)
and their comparative
statics. see
,4
158
lrrn
.$
el-p'qlf.7ll-
y'tz-//z/cn/ 11.1
4.5
10
4.0
9 8
3.5
=
b V
5 0.08
2.0
(,/?t'/
Ill I
'/7??II.??
'estlnellt
x
q* 4 =
6 ?)= () 04
q.
c* 2
N
=
1
Mxh.
7
3.0 2.5
(-lportttllitics
5
'
=
q*
=
1.5
4
.5
3
q
.
1 aa
=
.
2
q
0.5 0.0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0
0
1
2
3
4
5
6
.
1
=
7
.2
8
9
10
21$/4r2
which.
the drift rate
if return. /z. n an increase in the t5. increase in rz is likely to Likewisea in increase an ct is constant, implies an which again in if implies increase accompanied in an /.z. by an increase be when sllould kept mind in interdependencies be anaa is constant. These lyzing how a change in a market-driven parameter (suchas r) will affect the value of the investment opportunity and the optimal investment rule. risk-adjusted
expected
/5
Another issue that should be kept in mind when performing comparative statics experiments is that our model assumes that the parameters a. tz. etc., are hxed numbers. If a and t'r are changing over time or in response to changes in the state variable F (eitherdeterministically or stochastically) and the firm knows this, it should take this into account when determining the optimal investment rule. For example, it may be that u and tz in equation (1) should be replaced with functions etz, 1) and o' ( 1z',l). This will complicate the problem considerably. If time affects the parameters, the value of the inIZ and vestment opportunity will Iikewise be a function of both and time 1, and equation. Even if a c are equation (23)will become a partial differential functions ()f F alone, as with a mean-reverting process for V, the ordinary
differential equation lklr >-()'' ) wi Il becon-le m() rc cilmpl need numerical sllltltion: wtt will s()()n stlc an tlxanlple ()t'
iclt tcd ankl typical Iy this. l : J.1..'2 I 11 / lnd /;-(P-) 4 (3.1)4-the drift rate Lx is
Figures 5.9 and 5. I() show sample paths lr V both cases svt't assume that pz (J.(18, so that Nvith tbl 0.()4.(As before, 1- ().t)4 and t'r ().2. btlth at anntlal ratcs-) We I'ltlgin cacll 1 Taking l t il'ne i1: te lw:tl ()f (llle set of sampie paths in 1t)8() wi th 14) 1 then calcula th, usi the Ih cquatfon we te ng mon -
.
.
(o.j ) lor. tlne might believc to the mean-rcverting prllcess
that cz Iluctuatt:s
dg
=
r;
(
sttlcllltstically
- r.r ) dl +
&c
tpz?
(lver
t inle. l'(lr (Jxrlmple.
accllrding
,
where dv is the incrcment ()f a Wicner prllcess thllt is tlnctlrrelatttd with 31:. Tlle value ()1* lhe invcstment opportu n ity will tht!n be a I'unctilln ()f two statc variillllcs. k' 11nd Clntl will t isfy s:: a partilll differential cquatiun. Prllblems (,1' this sort llavtl bttt!n sludied by I'IuII lllld Wllitc ( l t.;87), Scott ( I t?87), and sviggi 987 I ). ns ( .
Decisiolls
'u$-
zrlFinn
l60
./?J1Jf.?-J??)f.:vlJ Oppol-tlllliticsf/zl/
hl
Tirntg
zc-j'//nf:!/l/
where at each time /, 6, is drawn from a normal distribution with zero mean and unit standard deviation. (Note that the coefficient 0.0577 0.20/ 12 is the l'lolttllly standard deviation.) =
%'
5
Since 1.$)
=
l
1. the standard NPV rule would call for investing im-
=
mediately. However, F ( Z()) 0.25, so P'() < I + -(fG),and the firm should wait rather than invest. In Figure 5.9, the firm happens to wait approximately fiveyears before I'' reaches F* 2. This waiting time can vary considerably from one sample path to the next. In the sample path shown in Figure 5.10 for example. the lirm must wait much longer nearly 20 years before P' reaches the crtical value of 2.13 =
4
=
3
v-/
-
l N VQ
k. tu
2 1 8 7) t)
1*
#N%
,
S
??
%Z
-
Alternative
Stochastic
Processes
k' / -
The use of a geometric Brownian motion as a model for U is convenient. but in somc cases may not be realistic. In this section we will examine the value of the investment opportunity and the optimal investment rule when P' follows alternative stochastic processes. We will first consider the case of a mean-rcvcrting process. and then a Poisson jump process.
-1 1996 1998 2000 1980 1982 1984 1986 1988 1990 1992 1994 Time Pah of F( /' ) and /'
ufzmp/t'
Figure 5.9.
/
-
#'
1
Process
Mean-Reverting
5.A
.5
Suppose P' follows the mean-reverting proccss
1
.0
dv
1 N
l
.
0.5 '--'-'N w Q
F( Z)
; (!1(21)vt:-: .
*
'd
'
& ,
l
&)
0.5
) ?z .
.
I
!
?
'b e'
*%
s
-
$* w jf
1980 1982 19
. '
J: I & , w
..
,
,
,
l :
1 . ' -
IIl
t
)
I
j$ p
'
/
,
/
v
l :
'w'
'%.x
'
.. '
j '
.
Ik
;
x
;
'
;.: :..J ;j te
,..
:$ '
l I; 1 I
@
.;
.
Y %
1
t/ t
'
ht
#
;
?
?
%
1986 1988 1990 1992 1994 1996 1998 2000 Time
Figure 5.10.
'
'
: ......
J
.
l p
J k :1
, '
.1
e
'
Another Sample Path of F( P ) and l''
=
q (l7
-
p')
l
(26)
+ cz p' Jz.
F) so lhat the expccted perccntage rate of change in F is ( l Idtl fZ), and the expected absolute rate of change is ( l/J/) td/ P ) ( n Z tl 3z'2 a parabola that equals zero at P' and has a maximum and V P 0 n at 1' P /2. As wc will see, an advantage of this particular process is that we will be able to obtain an analytical solution to the investment problem. 1-(JF/ =
-
=
'
=
-
=
.
=
To hnd the optimal investment rule,we will use contingent claims analysis. Let p, be the risk-adjusted discount rate for the project (that is. y. reoects the systematic risk in the stochastic lluctuations in P-). In this case the expected
time'' can be computed analytically. We will '''T'he expectation and variance of this reader to some simple cases in Dixit ( 1993a, refer the intercsted expressions. these but need not and the more rigorous theory in Karlin and Taylor ( 198 1. pp. 242-244) or Harrison pp. 54-57). ( 1985. pp. l 1-14). *twaiting
-
v dt
Dct.'fl'fp?.l-
z'1 I l'rlll
162 rate of growth
S'shortfall,''
(
of P' is not constant, but is instead a ftlnction t)f l-' Hence the 14 Jz ( ljtlt) -((/Iz' )/ 1-' is likewise a l'tlnction of )' :
Tiltlillg
1/
From the second Iine of equation
.
=
alltl Ill l,'t!JJ?nt>/
In pfJ.j'/??1(.!??/ ()pptlt'l d/l/i'(!-
wo have (.3()').
-
p')
=
rt
-
q
( 17 vl.
(27)
-
1tl P'/4.'r-a.Nvtz call tral-s'll-nl eqtlatitlll (32) By making the substitution ?y/cr2 (2 ) a/ (.r ) llnd ) so t h a t 11( 1.--) into 11 standard form. Let h ( P' ) = h'' )' ) (2n/c.r2):g''(-r ). Then (32) beconles -r
(27)substi-
The dift-erential eqtlation (23)will again applya but with eqtlation tuted for Hence F( 1z-)must satist .
=
r g'' (-r) + b
where
le +
b=
=
F l')
z'1P6?11 ( lz'),
=
(29)
(28) and
rcarranging
x ) g' (.t-)
2 (r
t? gt-t- )
-
/.z +
-
=
)/rr2
??
().
.
,
,
15
rep resentation:
3
.r2
H ( ; 0 /7)
where and 0 are constants that will soon be chosen in such a way as to make h( V ) satisfy' a diftrential equation with a known solution. Substituting this equation
=
,
Equation (33) is known as Kummer's Equation. l ts solutitln is the conlltlent hypergeometric function 11(.,r: 0 /?(p) ) wh ich has the following se ries
zd
expression for F( k' ) into equation :
-
'
.g(.t-
=
(28) Also: F F) must satisfy the same boundalz conditions ( 10)-(12) as before, and for the same reasons. gNote that 1,' 0 is an absorbing barrier for the process (26),so F40) 0.1 Finding a solution to equation (28)is a Iittle more complicated than it was for equation (23).We define a new function 11( P' ) by
=
'
-r
.
gives the tbllowing
=
l+
0
+
-r
J
0 (0 + 1)
bb + l ) 2!
Nve have verified that the slllution solut itln is
+
p (f? + l )(f? + 2 )
/?(/?+ l $fl, + ?)
to equatilln
.r
+
.3!
(28)is indeed
.
.
. .
of the forn-l
of equation (29). The
1-2 (Z )
(3())
=
zl )'
'/
2p
lI
J
.'J
l
. '
f?
'
%
*
1)
,
where is a constant that is yet t() be determined. We can find zt :ts wcll as the critical value V at which it is optimal to invest, l'rom the remaining / and F;. ( P' ) l Because two boundary conditions. that is, F ( )' ) P' the conll' uent hypergeometric function is an infinittl seritts. zl and V must be found numerically. ,4
*
of P' so the bracketed terms in Equation (30)must hold for equation the and second lines of must equal zero. First we choose both the Iirst first line of the equation equal to zero: 0 to set the bracketed terms in the any value
1 fz: 0(0 a
-
1) + (r
-
/1 + r? P ) (
,
-
r
=
0.
*
#
*
=
-
=
.
#
We can gain some insight into the effects of mean reversion by looking numerical solutions. Unless otherwise stated. wtl will set / 1 at will vary ?? and P Ntlte, however, that We and 0.2. 0.08. 0.04. c Jz r the dependence of ( l /tl ) C d )' )/ Z on t) depends on thtr scaling 01* P Wc will work with values of P in the range of 0.5 to l so a value of r? of ().5 or ablwe implies a ver.y high rate of mcan reversion. Figure 5.1 1 skows the value of the investment opportunity F tfZ ) and the 0.05 (whichimplies a relatively Iow rate of mean critical value P'* for several
This quadratic equation has two solutions for 0, one of which is positive and 0, we use the other negative. To satisly' the boundar.y condition that F(0) =
the positive solution: (31)
=
=
=
=
.
.5,
n
'4We have not developed a model that explains why l'' is mean reverting, and unless there is
a payout strcam that is mean reverting, P' will have a rate of return that is below the equilibrium rate Jz, See McDonald and Sittgel ( 1984) For a discussion of this ptlint.
,
.
=
l5See Abramtlwitz and Stcgun ( l9f)4), Sectitln 7.9 ()f Pearst'n ( l(J9()), ()r Slater ( 1$9f)1)) for discussions ()f the cllnflucnt hypergeofnetric ftlnction and its prtlperties.
..glFrp'?l Decisiotls 's
0.8
0.8
0.7
0.7
l/- /
0.5
0
1 I 1 I
0.3
;
=
1.5 1
0.2
.
;=
0.1
j
j
.0
O.5
I
I
1
1 1
I I
l I l I
1
0.0
0.2
F igure 5. l l
0.4
0.8
O.6
1
.0
1
.2
1
=
1
t
1
1
.6
2.0
.8
0
I I
I
C
l l
1.0 I
t
V=0.5
0.1
I
1 l p 1
.5
0.2
1
.4
g =
I
I
.5
0.3
1
I I I
0.0
1 1
l j
I I l I l
l
'
I l
.0
0.0 0.2
0.6
0.4
0.8
1
.0
1
.2
1
1
.4
.6
1
2.0
.8
.
reversion) and P 0.5. 1 and l For comparison. note that if ?) were zero, the model would correspond to the basic one in the previous section, with In Figure 0.08,. in that case, l'* would equal l Jz a 0 and hence close each value P' is fairly enough of P, to 1 so that for 5.1 1, r? is small .5.
.0.
=
(5
.39.
=
=
.39.
*
As we would expect, the larger is P. the larger is F )') and the higher is Z*. Other things equal, a larger P implies a higher expected rate o growth ()f so that an option to buy P- will be worth more. Figures 5.12 and 5.13 also show the value of the investment opportunity F( /') and the critical value P'* for P = 0.5, 1.0. and 1 but in Figure 5. l2, ',
.5,
P is 1 n 0.1, and in Figure 5. 13. ?? 0.5. As tese figures show, when P is 0.5 (smaller (larger than /), a larger value of n increases F( Izr). but than /), a larger value of ?? reduces F( 1z').(If l7' < I and n is Iarge. it is unlikely that V will exceed / for very much time, and the optfon to invest will not be worth much. On the other hand, if > I and ?? is large, even if )' is initially small, it is Iikely to quickly rise above / and remain above / most of the time, so that F Z) will be large.) =
Figures 5.12 and 5.13 also show that if J? and n are Iarge. F( P' ) will no longer be uniformly convex; it will be concave for small valtles of P' This .
is a result of the particular stochastic proccss (26)that we used to describe the evolution of lz' That process has an absorbing barrier at 1/ () (sothat -(()) 01. but the absolute rate of mean reversion rises rapidly tbr small but positive values of IZ so that F( V ) likewise rises rapidly. This is most evident (). but the expected rate of in Figure 5. l3 for the case of V- 1 F(0) ()f P' becomes large growth once P' is even slightly greater than zero, so that =
.
=
,
.5-
=
Ff P' ) rises rapidly. Figure 5.1 zlshows
3
=
F* as a function of the mean-reversion Obselwethat 1z'*increases with ?? when parameter ?? for ' is large. but decreases with r; when I is small. This is an implication of what when P s large, a larger ?? increases F F) (and hence F*), we saw before'. I but when is smail, a larger ?? rcduces F( I'' ). Figure 5. 14 suggests that thc critical value 0.5, 1 and l .0,
=
.5.
'-
=
with
is the dividing line, but in on the risk-adjusted expected
also depends
?? V. rises or falls rate of return, 1J.. In Figure 5.14, /.t
fact, whether
=
0.08.
0.04. Other things Figure 5.15 also shows 1z'*as a function of ;, but for p, equal, a lower value of Jz implies a lower value for the expected rate of capital 6 ( l /J/ ) E IJZ )/ 1z',and hence a larger value of FV ). yz gain This increasc in F P' ) will be most pronounced when n is small. (When z? is =
'shortfall,''
=
-
166 0
Illt
l.)ili.x,ulltllilit'vb'p/lt/ lll I 'tM'p??t'??/ lltllg
-t?.j'????t,???
.8
2.2 I 1
0.7
2.0
l l
0.6
7
=
1
V= 1
j
.5
I
1.8
I 1
0.5
.5
ls 1 6
K- o 4 u.
O .3
O.2
v-
.
1
o
O1
0.0
j
j
I l
l
I
1
0.4
0.6
0.8
1.2
1.4
1.6
1.8
.)
'*
.()
=
and Figure 5.17 shows the same. but for declines with y; again. a higher /2 implies
n
,
l Note that in cases. lz' higher capital gain J( F), and hence a Iower F( 1z') and lower )'*. However, the rate of decline depends on P and n-When r7 is small, Z* begins at a higher value (again.if tt 0 were zero, the model would reduce to that of the previous section. with a x), and declines more rapidly (becausethe and 6 Jz so that Iims-t) rate of reversion to P and hence the expected rate of capital gain for V is small). Also, as we would expect, the larger is P the largcr is )' (andF )' )1, whatever the values of n and /z. Our choice of the mcan-reverting process (26) for l'' was convenient in that it led to a quasianalytical solution for the value of the investment opportunity and the optimal investment rule. This should not be viewed as particularly restrictive. We could just as well have specified some alternative mcan-reverting process for V (for example, 0ne in which the absolutes rather .5.
1111
=
=
=
''e
=
*
,
.0
0.1 O.2
0
.3
0
.4
0
.5
0
.6
0.7
0
.8
09
1.0
.
thltn pttrcentage. rate of melln reversitln is Iinear in Iz'). Depentling t)n the I'1tlt hltve prllcess. the restll ting tl il)k rential equa t itln tbr Ii-(lz') nligllt t)r lm igl)t :1 k nknvn series solutikln; in any case it clltlld be slllved numerically, ustlally wi th Iittle di fficulty.
*
t-shortfall''
:1
0
2.0
;z' is in any case expected to revert quickty to Henee if pz is small. V* will decrease with l unless 17 is substantially larger than /. Finally, Figures 5. 16 and 5. l 7 illustrate the dependence tlf )' on /t. Fig= ().()5. (). 1 and ().5, I and ure 5.16 shows )'* as a function of p. for
large.
.0
V=O.5
1.O
1.0
1
.2
7::: 0.5 0.0 0.2
)=
1.4
I l
S.B
Cdlmbined Brownian Motion and Jump Process
basic model in which P' follows a getlmetric Brtlwnian wtl will allow for the pllssia Poisson jtlmp downward. This version (:,1' the model could describe a situatilln in which a conlptny has a patent that gives it the option to invest in a project whtlse value is P' l7ut s'uccessful will allow them ot hcr compan ies are a lso doing research which if to invest in a similar project. If and when one of those other companies is successful. the resulting competition will reduce profits. and hence V. To modity- our basic model, we will assume that V folltlws the mixed Brown ian motion/jump process-. Let us now return to
(lur
motion, and extend it in a diffcront way. This time bilitythat at some random poi nt in time, P' will take
,
.
,
(11,'
=
a Izrt/J +
f'r
Z
(1.z
-
IZ tt/
.
168
Z
j*V tl '
J) PI- 151011.% '
'
nl
S
'
*
2.2
2
1 1
.6
V=1.5
1
l 69
.8
2.4 2.2
*
N
Clllpt?rfltllif ies tz?lt.d Illvestn-tetltTbnillg
2.6
2.0 .8
l/lI,zt.!sfl?lcllf
2
=
.0
1
V=1.O
.4
'r) 0.05
.8
1
.6
1.2
1
V=0.5
1.O
0.0
0.1 O.2
0
.3
0
.4
0.5
0.6
0
.7
0.8
0.9
1.O
T1
=
() j
Tl
=
0
.
.4
1
.2
1
.0
.5
0.02 0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
m
and where dq is the increment of a Poisson process with mean arrival rate ()j. We will assume that if an dq and Jz are independent (sothat t'event'' occurs, q falls by some xed percentage / (with0 4 .'.'.S1) with will equation 1. that P' Thus Nuctuate as a geomctric probability (36)says ,
for Brownian
motion. let us
tbr simplicity set e
=
() and write
-ldzdqq
=
Brownian motion. but over each time interval tlt there is a small probability dt that it will drop to (1 /) times its original value. and it will then continue fluctuating until another event occurs. (Poisson jump processes of this kind are described in Chapter 3.) It is important to be ciear about the meaning of equation (36).First, note that the expected percentage rate of change in P' is not a. but instead is V (1/J/) a 4, because over each interval of time dt there is a that will Pfall by 100 4 percent. Thus incrcases in reduce probability J/ the expected rate of capital gain on /' by increasing the chance of a sudden drop in V. Second. because a Poisson event occurs only infrequently, most of the time the variance of dVj I'' over a short interval of time dt is just that of the Brownian motion part. c2 dt. However. if the event occurs. it contributes a very Iarge deviation, so its contribution to the variance calculated given the information at t cannot be neglected. Using the random walk approximation
JJ
c.rV
.'.'.:.!
with
l probability 1( 2
-
tlt
)
.
ilt
)
,
.!.
tF
=
J/ with probability
P'
-fr
with
-() V
1
2( probability ). dt
-
.
-
Then
b-fdv1
=
'((JP' )2j
=
t//
-
(1
-
/
Iz
,
dt ) v2 F2 dt +
.
t//
42 /2
.ldvjj
=
-
FIJP'J =
=J
-((Jlz')2j
2:2
t-gtlzrjjz 42 ),2 dt
dt +
in (JJ)2 etc. Note that this variance (7.272 Jf is the itlstantaneous
ignoringterms
-
,
has two components.
(or
4&local'')
variance
The first component. of dv which comes from ,
k',t:z.J'/??t,l?/
lll
2 2
.4
'llill'g
't,./??ltvl/
%Vewill now proceed t() solve for the
2.8 .6
t??lt/ Ill ! tll'.lltll-tlttll'ties
tlptilmal
investment rtlle
namic programming. :Ve will ltsstllntl tllat the firm is risk neutr:ll. discount rattt is /? ' 1. Then the BeIIn'llln etltlat ion for 1--( )' ). the investment opporttlnity. is .
t
2.2
We now expand
F tlt
t- (( l f--)
=
s() tll:tt its val tle of the
.
F using thtl vcrsion of Ito's Lemmil tbr combined
and Poisson processes (seeSection 5 of Chaptcr 3):
2.0 1
f/
'
dy-
tlsing
Brownian
.8
m 0.5 =
1
.6
1 1 1
'l
.4
=
Replacing e with
0. 1
,
0.05
0.06
0.08
2
?
'
I,z y ( ;p') .j. l - (
.0
0.02 0.04
this can be rewritten as
-
1. (T 2
.2
=
1-
0.1 O 0.1 2 0.14
0.1 6
0. 1 8 0.20
j)
-
) F (P ) + -
+
'
(t.
r,' y' ( p-) F g( 1
-
4 ) P' 1
=
()
(3t)) .
The same boundary conditions ( 1())-( l 2) apply as betbre. The soI u tion to (39) is again oI-the form F 1z-) zl )' #' bu t now thc positive solution to a slightly more complicated nonlinear equatikln: =
,
pt is (4())
Thc value ()f p that satisfies (4t1)l'nd also satisties the condition /J'(()) t) can be lbund numerically. Then. given /.11P' and zl can agtin be lklund frllm equations ( I4) and ( 15), which in turn follow from boundary conditions ( 1I ) and ( 12).1f' Figure 5. 18 shows the critical value P- as a function ()f rr for / = (), and l (In each case. = 0.4, (). k r = 5 = ().(14. and l = 1 Note that the larger is 4. the smaller is The reason is that a larger value of 4 implies a smaller valuc of thc investment opportunity (whenan event occurs, ' will fall by a Iarger fraction), which means a smaller opportunity cost of investing now rather than waiting. Xlble 5. 1 shows p! P' and /1 for various values of for the case in which / l (so that P' falls to zero when an event occurs). In this table, ( =
the Brownian motion part of the process, and is conditional t)n n() jump occurring. The second component. 42 )'2 dt accounts for the pllssibiiity ()f a jump. Shortly we will want to use Ito's Lemma to 5nd the diftkrential of a function of V. As we saw in Chapter 3, when applying Ito-s Lemma to :1 comtt' this bined Brownian mrion/jump prtlccss, it is t'lnly llle Iirst ctlmptanent variance that contributes to the new term involving second-order derivatives. ,
jump part contributes a different term involving a diftkrence in values at discretely different points. we will want to know Finally, in order to gauge the effects of changing IZ tluctuates continuously the expected value of F, the amount of time that before dropping. To determine &-T), we use the fact that the probability that e-k 7-. Therefore the probability that no event occurs in the interval (0. T) is T /F. Therefore the hrst event occurs in the short interval ( F. F +JF) is e-h until Ptakes is Poisson expected time jump the a
The
,
*
.
*
.)
.
.
'
*
.
*,
,
,
=
l If 4 I (s() that thc event is that b' falls to zero. where it remains forever), equatitln (41)) simpliiies to a quadratic cquatitln. which is just like our ttarlier equatilln except that the Poisson parameter l gcts addcd to thc intercst rate in the constant tcrm. The gositive solution is =
an increasc in would be cquivalent to an increase in the 7 would Iead to an lcrease in F( V ) and )'
risk-free
rate r, and
*.'
0
0.6
0.5
0.4
0.3
0.2
0.0 0.1
0.8
0.7
0.9
1.0
G
Figure 5.18. = ().l
Crifitfl/ Vltle V
*
Flfrlclrlll
flJ tl
(
?T
I'oi.Y.LmlnroWnQn /WfJllWll.Willt JWIXIW
jOr
.
affects the
value
of
of 1.) A positive 0.2. and I 0.04, tz r reduces the expected rate of First, it the investment opportunity in two ways. which F(l'). reduces Second, it increases capital gain on (froma to a ), intervals P' of time, and this changes hnite in over the variance of percentage is effect to reduce F( P'). tends to increase F(V). As Table 5.1 shows, the net value /'*. Furthermore. this net effect is and therefore to reduce the critical quite strong; small increases in lead to a substantial drop in P'*. For example, 0.2, !he expected time for F to remain using equation (37)we know that if positive, '(F), is 5 years, but compared to when is zero, falls by more than half, and V* drops from 2 to 1.33. =
value
=
=
=
'
-
=
,4
As we said earlicr, it is important to be careful when interpreting comparative statics results. In this case we have increased while holding a iixed. expected rate of return on V One could argue that the market-determined Should r) remain constant, so that an (which in this case is the risk-free rate,
The particularjump process given by equation
(36)Ieads
to a differential
equation for F'( 1Z) that is easy to solve. Onc could. of course. specify alternative proccsses for J''. For example. a firm holding a patent might face many potential competitors, each of which is trying to develop its own patent. The success of a competitor might cause V to fall by some randoln, rather than fixed.amount. Ovtlr time. additional competitors may succeed in entering the market. so that P' continues to fall. The calculation of the optimal investment rule for a model of this sort will be more difhcult, however, and would likely require a numerical solution method.
6
Guide to the Literature
Antdcedents of the McDonald-siegel
(1986)model
include Myers
(1977),
who showed that hrms' investment options are a component of their market value, Xlurinho ( 1979), who showed that natural resource reserves can be viewed as options to produce the resource and valued accordingly, and work by
.
increase in should be accompanied by a commensurate increase in a (oth1. Then erwise no investor would choose to hold this project). Suppose / remains constant, we would if a increases exactly as much as so that ce t5) in equation (40)with r + ). In this case have to replace the terms (r =
-
.
-
-
''Mertlln ( 1976) derivcd a formula for the value of a call option on a non-dividend-paying sttlck whose price folltlws a mixcd Bruwnian motion/puissonjump process, and showed that if the '
but w'ith r'
rcplacing the risk-free rate, r. He kept the expected rate of capital gain Es r + on the stt'ck constant, that isvreplaced the drift rate f.z with a + k. Thcn an increase in k will increase the valuc of the option.
..,1F/-/')l Cukierman
(1980) and
Bernanke
( l 983). The Iast two authors
'
Decisiolls
alst) developed
models in which hrms have an incentive to postpone irreversiblc investment decisionsso that they can wait for new inlbrmation to arrive. However, in their models, this information makes the future value ofa project less uncertain; our focus in this chapter has been on situations in which information arrives over time, but the lkture is always uncertain. Brock, Rothschild. and Stiglitz ( 1988) examine capital-theoretic implications of uncertainty. and Demcrs ( 1991) is a more recent contribution to the study of investment when information arrives over time.
To derive optimal investment rules using contingent claims analysis, we had to assume that uncertainty over the payoff from the investment was spanned by existing assets. Dufse and Huang (1985)Iay out the full set of conditions required tbr dynamic spanning', Huang and Litzenberger (1990), Duffie (1992), and Dothan ( 1990) provide detailed discussions at the graduate textbook level. Also, we saw that the basic investment option is analogous to a perpetual call option on a dividend-paying stock, and has the same properties as such a hnancial option. For detailed treatments of the valuation and optimal exercising of financial optionss see Cox and Rubinstein ( 1985), Hull (1989), or Jarrow and Rudd ( 1983), and see Smith ( 1976) tbr an overview of option pricing methods and results. 0ur results for the basic case in which P' follows a geometric Brownian motion are also described and interpreted in the survey articles by Dixit ( 1992) and Pindyck ( 1991b), upon which this chapter draws. Also, Merton ( 1977) and Mason and Merton ( 1985) discuss the connections between financial options and investment decisions. Unless there are bankruptcy costs. the ModiglianiMiller theorem holds, so the tirm'sreal investment decisions are independent of its financial structure. For a general overview of neoclassical investment theory, including implications of adjustment costs, see Nickell (1978).
6 chapter
The Value of a Project and the Decision to lnvest
THE
Imslc
model of irreversible investment in Chapttr 5 demonstrated
a close
analogy between a tirm's option to invest and a financial call option. I 11 the case of :1 call option. the pricc ot- the stock tlnderlying the op. tilln is assumed to follow an exogenously specihed stochastic proccssa usually :1 gcometric Brownian motion. ln
(lur
model of real investmcnt. the corresptlnding state V for which we stipulated an exogenous
variable was the value of the project, stochastic proccss.
,
However, as we cxplained at the beginning of Chapter 5, letting P' folIow an exogenous stochastic process, and particularly a geometric Brownian motion, is an abstraction from reality. First, if the project is 11 factory and there are variable costs of operation, l'' will not follow a geometric Brownian motion. Second and more important. the value of a project depends on future prices of outputs and inputs, interest rates. etc. These in turn can be explained in terms of the underlying demand and technology conditions in various markets. Hence fluctuations in P' can be traced back to uncertainty in these more basic variables. How deep one goes depends on the purpose of the analysis. To understand a firm's behavior, we might be satisfied to work with an exogenous process for the output and input prices. At the industry level, we must make the output price endogenous. At an even more general equilibrium level, the input prices must also be determined simultaneously by considering all industries' factor demands. In this chapter we take some first steps along this route.
..,4Firl''
'.
Decisiolls
For most of the time in this chapter, we consider a tirmthat has the priviIeged opportunity or monopoly right to invest in a single discrete project that will produce a given output llow. The basic uncertainty is over the demand for this output, but given a fixed sale, there is an immediate correspondence between demand and price. Therefore we allow the output price P to be exogenous, and determine the value V of the project, and the value F of the option to invest. in terms of the stipulated stochastic process for P. The methods are the same as those employed in Chapter 5, namely, contingent claims analysis or dynamic programming. We will see that once again, the value of the option to invest includes a holding premium, which Marshallian implies a stiffer test for investment than the traditional
criterion. In Section 1, we begin with the simplest case where production has no
operating costs. Then the value of a completed project is just the discounted present value of the revenue tlow. and the formulas of Chapter 5 that were expressed in terms of Iz- translate immediately into corresponding tbrmulas in terms of #. ln Section 2 we introduce an operating cost C. Hence the project will generate a flow of operating prost equal to (P C) per period. This raises a new issue the output price can go below C from time to time. which would make the operating proht negative. We must specify what happcns then. We will consider two, somewhat extreme, possibilities. The lirst: which is the subject of this chapter. is that the project can be costlessly shut down if P falls below C. and latercostlessly restarted if P rises above C. In effect. this makes the project an inhnite sequence of instantaneous operating options, each of which is exercised if # > C, and can be valued accordingly. The other extreme possibility, which we will consider in Chapter 7, proibits such temporary suspension by supposing that the full investment cost / must be incurred ovcr again if operations are ever rcsumed. Then some Iosses will be sustained to keep the option of future operations alive. but if the losses grow sufhciently large, the project will be abandoned. Of course reality lies somewhere between these two extremes. Ongoing projects generally build up specific assets workers' if opskills, customers' loyalty, etc. that will gradually disappear, or resumption involves a cost, but less than the cost eration is suspended. of starting anew, and the difference depends on the nature of the product and the duration of the suspension. Our analysis of the extreme cases yields results that can be suitably combined to tit particular applications that Iie in between. -
ttrust,''
'fhc Uildc of
(1
Project
fJ/1(1
tlte Dflcl'sftp/l/t) Ilb-est
transient price tluctuations. Now the prolit tlowbecomes a nonlinear of the price. which alters the effect of uncertainty on investment.
t'unction
All ofthe analysis up to this point assumes that the projectp ont:e installed, goes on producing the output llow forevcr. This unrealistic assumption is made only to convey the basic ideas of option values in a simpler Inanner. In Section 4 we relax this assumption by introducing depreciation. We show that the effect on option values depends not k)n the mortality of one projecta but on how we specit'y the opportunities available to the firm after its initial project has reached the end of its Iife. We also show that option values remain of considerable signiticance even with fairly rapid depreciation. In the concluding section of this chapter we consider a situation where variables that affect the firm's investment decision the output price and two . the investment cost .. are both random. Here the value of the option to invest is a function of both of these independent variables, and theretbre it satisties a partial differential equation. In general such equations are difhcult and must be solved numerically. A special feature of homogeneity helps us reducc the problem to an ordinary diftkrential equation and solve it analytically. Now investment is optimal only when the ratio of output price to investment cost exceeds a threshold inlluenced by the option value ()f waiting. Throughout this chaptcr, the insights about option values which we gained from thc analysis of Chaptcr 5 will remain valid and valuable as we introduce new reaturesinto the model. The techniques developed there will also continue to be useful. In future chapters we will continue the program of generalizing the mlldels and posing new issues. In Chaptcr 7 we will consider the possibility of temporarily mothballing or permanently abandoning a project if its cash flow turns negative. Then. in Chapters 8 and 9 we will move to the level of industry equilibrium, where each lirm has the opportunity to invest in a single project. In Chaptcr l () we will return to the perspective of a single lirm. but generalize the nature of the project, letting it consist of a number of investment steps, all of which must be completed before the proqt tlowsbegin. Finally, in Chapter l 1 we consider incremental investment. where each unit of addition to capacity begins to yield its marginal revenue product as soon as it is installed.
'rhus
and
In Section 3 we allow some instantaneous variation of inputs like labor to vary the output flow from the project in response to
raw materials,
1 The Simplest Case: No Operating Costs In this section the hrm's investment project, once completed, will produce a fixed flow of output forever. For convenience, we will choose the units so that the quantity of output from the project is equal to one unit per year.
Suppose the inverse 19 F' D( Q),where ()f costs production )' D 1). Hence. P =
=
the stochastic
dtlmanll function giving price P in terms (.)1' qtlantity Q is )' is 11stocllastic shift vnriablc. ln this section the variltble are assumed t() be zero, so the Iirm-s prolit llow is just without further loss of generality. we can take P itself as
variable.
For mtst of the time in this chapter we will assume the simpldst stochastic process for #, and one closest to the framework- of Chapter 5. namely, tlle geometric Brownian motion:
(1P
P (lt +
a
=
r.r
P dz.
The profit Ilow is # in perpettlity, and its expected value grows at the trend rate a. If future revenues are discounted at the rate lx. then the expected present value V of the project when the currcnt price is # is just given by F #/t/z a). In this case I'r, being a constant multiple of P, also follows a geometric Brownian motion with the same parameters x and t'z Hence the investment problem rcduces to the model we studied in Chapter 5, but we will rework it directly in terms of P to set the stage for the generalizations to =
-
.
COm0.
1.A The Risk-Adjusted Rate of Return --spanned''
=
r +
/
ty
psp,
,
(2)
where r is the discount rate appropriate to riskless cash flows. p/,,,, is the coeffcient of correlation between the asset that tracks P and the whole market portfolio, and 4 is the market price or risk. For the value of the project, P', As in Chapter 5, we will denote the to be bounded, we must have p. > difference Jz a by .
-
.
Investors will hold the output or asset perfectly correlated with P only if they get a total expected rate of return, /z, from it. Of this, a comes in the form of expected capital gain. The rest, must accrue as some kind ()f dividend. If the output of the project is a storable commodity (forexample, oil or copper), ,
lce rese 11t t l'ltlllet p?lfl?-tt:/lfl/ :(211 f5 11re 1.-., t 11:t t is. t 11k,l of benetits ( Iess storagc costs) tllat tlle nltrgillal stored tlnit providcs. l NV:will generally Iet J be 111-1exogentltlsly specilidtl pllranldter. I-lowevcr. in prnctict!(lver tinle and/or in response to the convtrniellce yield can valy (stochllstieally markdt-widevariables stltrh as total sttlrage). ttnd otlr nlodels can be adaptdd to account for this--' .tlll'tlxqc.
Wlxen slll-ntl untlerlying paraneter changes, the equilibritlm relatiilnsllip J nltlst cont intle to hold. btlt which of the tl1ree nlagn ittldes adjust tt) Jz equilibriunl depends on the underlying technology and behavior. We restore
assune that the riskless rate 1- and the market price of risk /, being properties of the whold Iuarke t. are totally exogenous t() our analysis. Nosv. when t 11et-z of the P asset increases, y nlust incrcase. lf is a fundanlental market constante then a must change one for (lne with /.t I-lowever. if a is 11 fundamen tal I'narket constants then must adjtlst'. )r examplea tlle total amtltl nt t)l' storagc n-light change. Wllerl wll sttldy the effects of chltnges in f'r i)n the hrm-s investment decision. our ansqvttr will depend on which of these viewpoints we adopt. (5
.
Generally
another this out. 1.B
The capital asset pricing model allows us to determine the risk-ltdjustcd discount rate y.. For this we need the stochastic tluctuations in P to be by tinancial markets, that is, there is a traded asset, or one could construct a dynamic portfolio of assetss that s pertctly correlated with #. For simplicity of exposition we suppose that the project output is dircctly tradeable. In this case, the discount rate Jz will be the market risk-adjusted expected rate of return on P. As in Chapter 5 we have the CAPM tbrmula p.
l''tz/lt-v !'.t!/f/7h't'???? !1()N'v Nvi
we will tak'kl J to be the blksic paranleter ltnt.l Iet t; adjust. Wllen specification Icads to an impllrtant different rdstllt- we will point
Valuing thtl Prlyjttt!t
Our prlljetrt is 11 cllntingent
()r
derivative
ltsset.
whose pltytlffs depend
t)n
tlle
value ('f thtt nlllre bksic ltsset P. Tllen we can de rivc the vllltle ()1* thc pnect ()1' as a functitln 1/( J'') thc price ()f the basic ltsset. We l'ollllw tlle prllcedure ()f contingent claims valuatitln that was disctlsscd in tliflkrent ctlnttlxts in Cllapters 4 and 5. N'Vconstrtlct l riskless ptlrt f(llio by taki ng sui table cllmbinat ions of the asset t() be valtled (the project) and btsic asstlt ( 134. Since th is pllrtt'tllio is riskless, it must earn the riskless rate ()1* rcturn. Tllat condition yields a diffe ren tial equation f()r thut tl 11known value of (lur project. The equation can then be solved given appropriate boundary conditions. ' Thesc benests can includc ak1 increklsetl :lbility to smllllth prtlductiiln- avllid stllcktyuts- tnd facilitate the scheduling (3f prlptluctitln rknd s:lles. Ct3nvtlnitlnce yield is tlle reastln t hat Iirnls l1()ld inventories in the lirst plactta even when the expected capitlll gain )n tlltlstt invcntlpries is l'lelf tw lr I'ntpst the risk-adjusted ratc. ()r even negative. As (,11e wlluld tlxpcctp clplnnltltlities- mklrginal conveniencc yield varies inverscly with thc ttltal amount (1f sttpragtr. For tlmpirical studics t't' convcnience yield and its rtlle in cllmmodity price ftlrmation, see Pindyck ( !993 c.d ). see. for example, G ibson and Schwartz ( l9t?().l 99 I ). whf.) shllw l1()wtlil-pnlduci ng prtljccts lows 11 geonletric Brownian mflt ilhn, and the ratc ()1' ctlnvecan be valtled when the price t)f ()iI 1*()1 nience yield also llltlws :1 sttlchastic prtlcess. AIs(). Brennan ( l 99 l ) estimlltes and tests altcrnlltivc i'unctionsand stochastic prtlcesses j'llr convenience yieltl and its tlepentlence t)n price tlnd tinlc.
zl Firm
l 8()
''
Decisiolls
Suppose we construct a portfolio at time t that contains one unit of the project, and a short position of n units of output. where we will choose n to small make the portfolio riskless. We consider hclding this portfolio over the intervaj of time (I. f + dt. The holder of the project will get the revenue or profit llow # dt over the time interval of length fX.A1so, a holder of each unit ofthe short position equai to must pay to the holder of the corresponding long position an amount namely, earned. the dividend or convenience yeld that the Iatter would have (5 Pdt. Thus holding our portfolio yields a net dividend CP n P4 l. It also capital gain, which is equal to yields a (stochastic)
Tlle I-'il/ff e tp- tl Proju-t
tl/lt
J/ltz
i/ecisioll
/f.) lllj.'e.b1
importantly for otlr immediate purpose. given our economic and t > (), the two rtlots satisfy pk > l and p? < ().
conditions
1. >
0
Then the general solution of the homogeneous part of the equation is a '/1 and Bz P#:. To it linear combination of the two independent solutions B3 we add any particular solution of the full equation: the easiest to spot is #/J. Theretbre
-
dv
-
n f,I#
=
( et#l P (lr,(,) + #
(1.''(/:) -
-
,,)
nl c(P)
#2 v''P,
apsz
+
) t/
1.C
tz.
(Note that we have used Ito's Lemma to express dv in terms of the price F'(#) so that the terms in dz disappear and the process.) Now choose n risklessv3 The total return to the portfolio is then portfolio becomes =
(P
-
t5
,2 P P'/4 P) + 1
z
Equating this to the riskless return have the differential equation
t-
gI''( #)
192 Ize/'(#) n
-
PIJ/
and collecting terms, we
=
!. 0.2 2
pp
-
1) + (?. J) -
p
-
Fundamentals and Speculative Bubbles
The term P/45in the solution has an immediate interpretation-. it is just the expected present value ofthe revenuc stream Pt when the initial level is J'--f-his tt.-( l is because Pt 1 # tz'' and discounting at the appropriate risk-adjusted rate =
.
# g ives;
j d(.
Simplesubstitution shows that the homogeneous part of the equation has AP'. provided p is a root of the fundamental solutions of the form l'( #) equation quadratic
(2=
where the constants S1 and B1 remain to be determined.
r
=
0.
(4)
This might be callud the fundamental comptlnent of the value of the project, in the sense that it is justihedby the prospective profit tlows.Thtt other two terms must be spcculative components of value. We can eliminate them by invokingeconomic considerations to rule out speculation. First. it makes sense to require that IZ (()) = 0. If the price is ever zero in the geometric Brownian motion ( l ). it will lbrcvcr remain zcro'. in technical terms. zero is an absorbing barrier for the proccss. With no prospect of a profit now, the asset should have zert) value. I-lowever, sincd pz < (), that power of P goes to intinity as P goes to zero. To prevent the value from diverging, we must set the corresponding coeflicient Bz = (). The othcr term, B3 #/J. is not so easy to get rid of. It represents a compox. People might value nent of P' attributable to speculative bubbles as P the asset above its fundamentals if they expected to be able to resell it later at a sufhcient capital gain. That is exactly what this Iirst term ensures. #/J' To see this, we show that an asset that is always valued at yields its risk-adjusted capital alone. expected from its gain By lto's return appropriate ,
This is very familiar from Chapter 5', there we gave a detailed account of f5, and c. Most its roots and their dependence on the three parameters r, composition of the portfolio is held constant over the short intewal (/. t + dt ); thus lt remains equal to V'( #ff )) cven though P' P) itself changes over this interval as P changes. Over hedging strategy'' will adjust the portfolio at successive a Ionger period of time. the t/I. I ldt ) we will sct n at F'( P(t + Jf ) ), and so on, rather like Thus intelvals. (1 + + over short of such strateges in the limiting case of continuous a chained price index. Rigorotls formulation () needs great care; sce I'larrison and Kzeps ( 1979) and Dufhe ( 1988, pp. 138-147). time as J/ A'l''he
'(
'Adynamic
.-..+.
..-.
zl l lrp??
elstons
.j
/'t..) l'zl/t.z
(.)/- (1
Ile
/5't4.jtz(.? tll
?t //1c
I.l,ci-b.illll Ilt !
'(,.b.(
optitln to invest and J.t sllort pllsition ('f 11 Folloyving tl'ltt sanle steps as be fore, t 1ltl reatle r diffe re n t ial tlqtla t io 11
where the third line follows from the fact tllat /$1satisties the quadratic equastandard tion Q (). and the Iast Iine uses the CAPM tbrmula (2).Thtls the of covariance times that of P. The deviation of the return on P#' is exactly /:11 market with the #/' with the market porttblio also becomes pt times that of P portfolio. With the covariance and the variance both multiplied by the same PI$b and the market ptlrtfolio is the factor, the correlation coefficient between namely, ps,, Theretbre same as that between # and the market porttblio, the risk-adjusted rate of return for Pth is (r + 4 p/w, Jl fr ), which is exactly the expected rate of return in the Iast line above-s In the remainder of this chapterwe will rule out such speculative bubbles. (.)f value fotlnd by direct we are Ie with the fundamental component integration above, namely, Pl. P' ( P$
Then
=
'
) 13 ( 1') -
-
'(
l
'
F-( P )
() .
This is jtlst like eqtlat ion (3) for the value of the project, btlt of cllurse the option has no divide nd or prolit llow. This is a homogeneous Iinear eqtlation of second order. so its solution is :1 Iincar combination of any two linearly independent soltltions- say,
=
.
' '
1-- ( /7 ) + (1.
J-/ I') units klt' the otltptlt. trill'1 check t hat xve gttt t he
J--( 13)
zl
Pl'' +
I
:-12
/.:/$2
where z11 and are constants to be determined. This solution is valid over of prices tbr wllich it is optimal t() hold the tlption. Since hightlr the range make investnlent more attractive. the range in question extends from prices threshold investmcnt P* Of course P* is itself an unknown to btt zero to an tlf solution. the :1 Thus we have three unknowns, zll determined' as part conditions and need three to complete the solution. and J''*, zzh
.
,4-a,
,
The Iimiting behavior of F ( l>j near zero gives tls (ll1e condititln. Wllen ()f it rising t() the exercise thresholtl P. is qtlite prllspect option should be almost worthless at this tlxtren-lc. X) remote. Tlltlrctbre the lls 19goes t() zttro. we should stlt the cllefhcient enstlre that F ( P) goes to zero ()f P equal negative of the to zero'- thus zlc (). power ()f cllntlititlns, F( /:') ;lt P* At other F()r thc we consider the behltvior two the option, and thereby Ctcqtl ire optimal exercise thrcshold it becomes this to exercise the of value V ( #) by incurring price (stlnkcost ) an asset (the project) of invcstment /. As in Chapter 5, twl) conditions govttrn this. First. the value of the option must equal tht! nct value obtained by exercising it; this is the condition: valtte-lnatcllilbq P is very small. th
=
1.D
Valuing the Option to Invest
.
Once we know the value IZ of an installed project as a function of thtl current Ito's price P, we can obtain the diftksion process of )' from that of P by using allow would find 5 Chapter us methods to of Lcmma. Then, in principle, the ()f P' I-lowcver, function the in project as a the value F of the option to invest the drift and diffusion parameters of the process of P' are generally quite complicated expressions. making it hard to solve the differential equation linking F and V. An alternative and generally simpler approach is to tindthe value of the option to invest as a function of the price, F( P), using the above solution for ;'' ( P) as the boundary condition that holds at the optimal exercise .
F (P )
*
Seconds the graphs of F P) and is the slnooth-paslb? condi tion:
threshold.
We will now employ this method for our simple project. Once again we followthe steps of contingent claims valuation. Now the porttblio will consist 5The fact that the risk-adjused discount rate for P equals thtt cxpectcd ratc ot' growth of Pt6when p is a root ofthe fundamental quadratic was also shown using the equivalent risk-neutral valuation procedure in Chapter 4, Section 3.A.
IZ
V ( 19 )
*
=
( P)
F' P* )
-
I
.
/ should meet tangentially
-
at #* this ',
Z'( #*).
=
Using the specihc functional forms of F( P) and Z ( #), we ctn write the
value-matchingand
smooth-pasting
conditions
z1l( P. )/', pl
-d1
(
.'*)/$1
-
i
=
#*/J
=
1//5.
as -
/
./z-lI'-uil-lllllecisiolb' '-
l84
1.E
These yield
For reference
The 'Llltteo)' (1 Pl-tljct
we also state the solution for
./4
(#:
=
-
zzll
1)/1:-1 /-6/.-11
; it is
/ (Jpl )/3,
(10)
.
Using the relation (5),we can express the price threshold equivalently in terms of a value threshold,
//?(.zilecisioll t() /,1!
t//lf/
185
-t,.j'?
Dynalnic Programming
If the risk in P cannot be spanned by existing assets. then we eannot construct a riskless portfolio and use it to obtain a differential equation tbr Pr( #). As explained in Chapters 4 and 5. we can instead use dynamic programming with an exogenotlsly specitied discount rate p. although we will not be able to relate this discount rate to the risk-less rate and the market price of risk using CAPM. Here is a quick summary of the steps. The value of the project at time t can be expressed as the sum of the operating protit over the interval (/ l A-dt ) and the continuation value beyond t + dt Thus -
.
1,( #) This is exactly the equation ( l4) of Chapter 5. Thus our approach expressing ()f the option in ternls of the both the value of the project and the value result could have obtained by the as we same underlying price has produced instance, )' is the In directly. value project with present of the the starting approaches is equivalence of the of and multiple #. the two just a constant will We result general. pert-ectly is the easy to demonstrate directly. However. generally find it somewhat more convenient to work in terms of #, as it is economicallythe more basic variable. The important point stressed in Chaptur 5 was that P' > 1', the option value ofwaiting to invest implies an action threshold where the xpected value from investing exceeds the cost. Here the corresponding idea is that P*/ > l /5 t5 the llow-equivalent (per unit of time) cost of or P' > 1. We could call l investment'.that is the Ievel the initial prot flow must have if its subsequent expected value is to cover the cost of investing.f' excceds /, ln Chapter 5 we discussed at length the factor by which magnitude calculated its 1 multiple'' value namely the pk/(/51 ). We /5. We do not need to for a range of variation of the parameters l'. c, and calculations for corresponding will perform repeat those points here, but we chapter. this in models to be developed later the new and more general the Likewise, in Chapter 5 we defined Tebin's q in sense of the ratio of namely V! 1. This allows their replacement cost. the value of assets in place to investment occurs us to interpret the effect ofwaiting as the possibility that no exceeds 1 as long as it remains below p /4#, 1). We can now though q even similarly detine q P/, /') and obtain the same interpretation.
Expanding the right-hand
=
,-l
P J/ +
lz'( # +
tl
#)
JJ f-z-7'
1
side using Ito's Lemma. we have
where odt ) collects terms that go to zero faster than f/. Simplifving, dividing (). we get the differential equation by JJ, and proceeding to the Iimit as t/J -.+.
*
'*
toption
-
This is exactly Iiktt eqtlation (3) that we had earlier. except that 1- is rcplaced by the (arbitrary)discount rate p and ( 1- /5 ) by (x. The equation can be solved by c). similar methods, and ruling out bubbie solutions, we get ' ( P) #/tp For this t() make economic sense we netld p > (x. Then the option to invest can be analyzcd similarly. Start with a P in the range ((). P* ), where the option continues t() bc held. Split the future into the immediate intelwal (l. t + tJ ) and thd continuation beyond that. Expanding and simplil ing as above yiclds the differential equation -
=
12 c2
Pl F'' Pj +
/.' F' /a)
(x
P)
-(
p
-
=
-
().
Now consider the quadratic equation
-
,
Q- )
=
Since p cost
45 6lf we Icave out the uncertainty and the trend in price. we get = r and thc flow-equivalent :he the of opportunity cost amount investeda 1. 1. btcomes just the interest cost or
/,2
pp
-
1) +a J$ p -
=
0.
> a, the Iarger root p, of this exceeds 1 Since p > 0. the other negative. is Then the solution for the option value takes the form root pz F( #) ##' where the constant z1l remains to be determined. =
.
,41
,
!' ./'??
z'l
186
'J
17irl'l Igeciiolls
Finally we use value matching and smooth pasting between F ( #) and l'' ( /2) at #* to complete the soluton. The result is
J'5-(?jf..('/ 'I-lle P/l(.,
2.A
(q1-
t'l??(/ 111e
fl
The Value
()1*
f..ltlta-hf?/lI()
'est
the Project
flnce again sve consitler the porttblio that consists ()1' a unit (.)f tlle prtlject :kl1tI lt )-/, ( f-') units of :1 short position in the assct tllat spans P. Nvhen lleltl for the short tinle interval t t + dt ), the owner of this portfolio can exercise the current operation option. That is prolitable if P > C; the resl.l Iting prof i t llosv rate is just zrt P4 max ( 17 C, ()). The other aspects of the portfolio (capital gains. dividend payment for the short position. etc.) are as before. Tlle re fore the diftkre n tial equation for t he va ltle of the project is .
which is the natural analog of (9)above. For most of the rest of this chapter we will assume that spanning holds and use contingent claims methods, Ieaving to the rcader the obvious modihcations that apply when dynamic programming is used instead. Occasionally. for variety and simplicity of exposition, we will do the opposite.
2
Operating Costs and Temporary Suspension
-
1.
(,,2
P? I.z!'(1o4+ (r
J) Pv' /'.')
-
-
?-
)' ( P4 + zr ( #)
=
().
This is solved by familiar methods. The homogeneous part has two intlependent solutions Pl$' and J4/2 exactly as above. The only new feattlre is tllat tlle nonhomogeneous part. or forcing ftlnction ( P). is defined diflkrently when # < C and when P > C. Tllerelre we solve the equation separ:ttely lbr P < C and P > C. and then stitch together the two solutions at the pllint P C. In the region 17 < C. wc have n. ( #) () and only the homogenellus part of the equation rcmains. Therefore tlle general stlltltion is just a linear combinatilln of the two ptlwer solutions corrcspllnding t() the two .n'
Suppose once again that the output price follows the geometric Brownian (q motion of equation ( 1). Then a. t'r, y.. and > Jt a are all constants. If the option of investing in the project is evergoing to be exercised. we need g, > a, will also assume or > 0, and we will assume that this is indeed the case. We operation can entails that C. but the project the tlow operation of cost a that and costlessly C, when below and suspended 19falls costlessly temporarily be the llow from proht rises Therefore. instant at above C. any resumed later if P -
this project is given by =
max
(#
-
C.
( l 1)
t)J.
McDonald and Siegel (1985)pointed out another useful way to Iook at such a project. It gives the owner an infinite sct of options. The option at time 1, if exercised, means paying C to reccive the P that prevails at that instant. Since each option can only be exercised at its specified instant, these European call options.; They also showed that the project can be valued
are by valuing each of these options (usingthe standard Black-scholes formula), and then summing these values by integrating over t. We will find it easier to value the project as a simple contingent claim that depends on P. We will go through the steps of obtaining P' ( P) in the following subsection, and then afterwards we will turn to the problem of valuing the option to invest.
can
=
roots: )' ( 13)
zr(#)
7A European option can be exercised tlnly at the time of cxpiration. at any time up to and including the time of expiration.
be exercised
=
An American
option
=
K3 P/'i + K? P/:
where the constan ts S! and K? rutm ai n to be determ ined. I11 thtl region P > C, we take another linear combination ()f thtl pllwer stllutillns of the homtlgtlneous parts and add on any particular slllution of the full equation. A sinApltt substitution shows that ( P!( - C-jl-l satisties the equation. Tllerefore the general solution for P > C is ( P) f/z-
where the
=
Bt
Pl$I
+ Bz /.'/$:+ P/ds
-
C/ r.
constants B, and Sa arc to be determined. These solutions have straighttbrward economic interpretatitlns. In the region P < C, operation is suspended and the project yields no current proht tlow. However, there is positive probability that the price process will at some futurc time move into the region P > C, when operation will resume and /J < Q.is jtlst the expectcd presen t prots will accrue. The val ue 1' ( #) wht!n value of such future ows.
I't'-l? l l-?-?';v krllh (-'(-.Lf'-b '--. '-b.
a'il
11k l 119
.j7
Next consider the region P > C. Suppose tbr a moment that tlle hrm is forced to continue operation of the project tbreverseven during those times worth of such a project'? when the risky revenues fall below C. What is the net the rate value of at a, and is discounted the revenues grows The expected expected present value is the risk-adjusted appropriate rate so y' back at the is at the riskless discounted = C #/.The sure constant cost stream Pl;k constitutes worth #/ C/r) value The ( net C/r. rate r, yielding a present impose solution did not that Since the solution above. the last two terms in (he other terms losses, despite two any requirement to continue operating operations the future in the suspend value option to of must be the additional should # fall below C. The constants in the solutions are determined using considerations that apply at the boundaries of the regions. Begin with P < C. As P becomes very small, the event of its rising above C becomes unlikely except perhaps in the vaiue t)f future profits should then very remote future. The expected present value of the project. However, with pz negative, go to zero, and so should the Pth goes to cx:)as P goes to (). Therefore the constant multiplying this term, namely, K:, should be zero. Now turn to # > C. When P becomes vel'y Iarge. in the very the suspensioll option is unlikely to be invoked except perhaps rule should out the remote future. so its value should be zero. Rlr this we ().8 This Ieaves of P, by making Bb = positive
The Wl/llt;'ol' f? p/-p./tzc/t???(/
//lc
llccisioll t() /?ll.t,-/
These are two Iincar cquations the solution
in the unknowns
cl-pl
Sl
pg
=
Jl
pz
-
yy
-
and Bz,'they readily yield
SI
j
-
,
J
?-
-a)
c1-#:
S2
-
r To verif
C
tives of the two component KL
c/f, =
-1
/31K I cll'
=
=
at C, we have
solutions
Bz Cz + C/J
-
>
/31(l-
C/r.
p? B1 c/l:-l+ 1/J.
Q(?-/(r Therefore
le
))
-
tlut
1 c2
=
speculative bubblcs jtlst as wc tlid l7el'tlrc.
j
-
.
t
pz (r
?- >
c
,.t5/4,-
J)
-
.
Qps at p j)2
-
>
=
r/(r
-
t5).
We
().
(5 t-jqr ) must Iie either to the right of the larger root pl or to the of the smaller root pz. First suppose 1- > J, so 1-/(,J) > (). Then -
-
r/(r ar
done. Next, suppose
)
(
-
r
/1 > p?,
>
J, so r/ (r
<
/5
)
<
and multiplying by thc negative number we have the desired rcsult again.
pz (r
-
<
(5
-
pl
)
(). Then
<
.
). which reverses the inequality,
A numerical example will hep to illustrate this solution. Unless otherwisc noted. we set 1- 6 = 0.04, and C l0. Figure 6.1 shows P' ( P4 as a function of # for t'r (). 0.2- and 0.4. When t'z (), there is no possibility that P will rise in the future, so in this case the prcect will never produce (and has no value) for P < 10. lf P > 10. IZ ( P) ( P l0)/().04 25 # - 25t). Hwever. if fr > 0, the project always has some value as long as # > ()',although the firm may not be producing today, it is likely to produce at some point in the future. Also, since the upside potential for future proht is unlimited while the downside is limited to zero. the greater is c, the greater is the expected future flow of proht, and the higher is fzr( #). Figure 6.2 shows IZ ( P) for o' 0.2 and 6 0.02, 0.04. and 0.08. For any fixed risk-adjusted discount rate, a higher value of t5 means a lower expected rate of price appreciation, and hence a Iower value for the project. =
=
=
=
=
=
FlNate that these argunlents oru ruling
/'
and
t)
-
r/tr -
C This still leaves two constants, diffuse where the two regions meet. Since the Brownian motion of P can freely across this boundary, the value function cannot change abruptly across it. In fact the solution Pr( #) must be continuously differentiable across C. For a heuristic argument see Dtxit ( 1993a, Section 3.8)., a rigorous proof is in Karatzas and Shreve 1988. Theorem 4.4.9). Equating the values and deriva-
(
/$2
pj
-
thesexevaluate the quadratic expression
.
for which we consider the point P
/1
have
and we >
-
-
Since the term in A'1 captures the expected profit from the option to resume operations in the futurev and that in Bz the value of future suspension options, both the constants should be positive. For that. we need
power
if P
yj
=
=
-
=
l 9()
400
4OO
350
35O
300
300
(r 0.4 =
250
250
& 200
& 200
r
r
=
$ 0.02 =
Le
0.2
150
150
15 0.04 =
1O0
100 50 0
G
-
0
2
8
6
4
10
14
12
=
16
8
50
0 18
20
0
0
2
6
4
8
10
0.08
=
12
14
18
16
20
P
2.B
smooth-pasting conditions thttn give
The Value of the Option to Invest
Now that we know the value of the project, P' ( P), we can tind the value of tle option to invest in the project. F ( #). as well as the optimal investment rule. Since the price # follows the geometric Brownian motion ( 1), we can go through the same stcps as in the prtwious sdction to establish that the value of the option to invest takes thc form F( #)
=
zd1
#/1 +
.,42
P# =
=
-
.
IZ-
/J 1
l
pb- 1
(p ) .
./1
=
=
/.ya ( p+ )p? + /a*/ t 1 /12/y2 ( # )p? - + j j j
-j
?. -
/
.
.
.
Recal I that l$zis k nown frtlrn ( I4), s() tl'lc pai r 01* equat ions ( I 5 ) :.1ntl ( l ()) be we are Ieft with an equation for can solved for z1j and P* Eliminating the investment threshold: ad1
.
(p I
.
0 is an absorbing barrier. so that F'40) (),we know that zh = ().At smooth-pasting the optimal exercise point #* we have the value-matching and conditions linking F #) with the appropriate P' ( #) from equation ( l 2). Of exercised when # < C; there is no reason to course the option will not be to keep the project idle for some time. This only l investment cost incur the zdl Pts' verified cannot satist value matching and smooth formally'. can be KL J'/$' Theretbre with in equation ( 12) we use the solution fbr I pasting and region. operating that is, for P > C. The value-matching the ( P) in
Since P
X1 ( P +)ts,
-
p?) B? ( P* )X
.
+
t.I
-
I ) P* /t
-
Isb( Q-/?'
+ l)
=
().
Equation ( l 7l.which is easily solvcd numercally. gives the (lptimal investment rule. The reader can check first, that ( I 7) has a unique positivc slllutilln for #* that is Iarger than (C + 1- I ), which is the Marshallian full cost (operating cost plus interest on the capital cost of investment ), and second, that l,' ( P' ) > 1, so that the project must have a N PV that exceeds zerl) beftlrc it is optimltl t()
invest.
Returning to our n ume rical example, th is s()ltl titln lr F ( P ) :, nd 1'. is t5 ().3 f'or ,1).2- and I ().()4 rz I(/).
shown graph ically in Figure
=
=
,
=
=
z1Firm 'J Decisions 400 350 300 l l
7 250
C 2o0 L,..c1s0 %. <
I
l I
FP$
I l
1/(P)- I
l
0
An increase in ( also increases the critical price #* at which the tirm should invest-There are two opposing effects. lf J is Iarger. so that the expected rate of increase of P is smaller, options on future production are worth less, so P' (#) is smaller. At the same time, the opportunity cost of waiting to invest rises (theexpected rate of growth of F( P4 is smallerl.so there is more incentive to exercise the investment option. rather than keep it alive. The hrst eftkct dominates, so that a higher ( results in a higher P*. This is illustrated in Figure 6.5. which shows F( P) and P' ( Pb l for 6 0.04 and 0.08. (In both cases. 1- 0.04, and o. 0.2.) Note that when ( is increased. V ( #) and hence F( P4 fall sharply, and the tangency at #* moves to the right. This result might at first seem to contradict what the simpler model of Chapter 5 told us. Recall that in that model, an increase in J reduces the critical value of the project. P- at which the tirm should invest. However, while in this model P. is higher when 6 is Iarger, the corresponding value of the project, )' ( l>*b.is lower. This can be seen from Figure 6.6. which shows P* IZ as a function of t'z for J 0.04 and 0.08, and Figure 6.7, which shows ( P*). will rise from 23.8 to If, say, fr is 0.2 and t is incrcased from 0.04 to ().08, P* =
=
=
I l
50
'est
-
l
100
The I'zi//t;. ()3*tl Pl'tlj'ect fl?lt tlle Dt-ac-j-fp?l o 1/13
*
,
8*=23.8
-s0
l I I
=
=
-100
0
4 Figure <.J.
Vltte
(Note:
=
J
-
0.434.fz
=
28
Opptvrlllzl(v. F( P ), t?lf/ V ( P ) /
0J'J'?'llzcTzaflal r
24
20
16
12
8
(). 2. and I
=
=
650 600 550
l()())
/. Recall that from the value-matching The hgure plots F( P) and P' ( #) = P' J7*) /. and from the smooth-pasting F #*) satishes ( P. condition, of the two curves. of point tangency is #* condition, at a is changed. these useful how examine curves shift when rz or to lt is P' #) for any P. fz results in in increase in ( increase before. an As we saw an production, future call of options projet:t is the explained, on (As we a set and the greater the volatility of price, the greater the value of these options.) However, although an increase in c raises the value of the project. it also increases the critical price at which it is optimal to invest. that is, p #*/;)c > 0. The reason is that tbr any P, the value of the investment option (andthus the opportunity cost of investing), F( P), increases even more than V ( P4. Hence as with the simpler model developed in Chapter 5, increased uncertainty reduces investment. This is illustrated in Figure 6.4. which shows F #) and 1z'( #) I 0. the critical price is 14, which just makes 0, 0.2, and 0.4. When (z for (r equal to its cost of l 00. As fz is increased, both P' (#) the value of the project and FP) increase; P* is 23.8 for t:z 0.2, and 34.9 for t'r = 0.4.
50O 45O
-
-
-
=
=
=
I
-I
l l I l l l 1 1 l 1 I
tr 0.2 =
400
C 350 L' 300 250
(F
C' 20O
=
CT
0.4
I
I 1
j
X 1 P*
-50
-100
t 0
4
8
12
I l
1
100
50 0
0
1
h
150
=
16
=
14.0
((r
20
=
0) IXP* I
24
I
p-
=
:.p,9 ((r 0.4) I =
N
=
23.8
28
(r
=
0.2)
32
I 1
36
),
1000
500
900 400
--I
800 :
3OO
600
I
C
KVQ
700
=0.04
Y tt-
l
R
i
200
% N .(;.()a
gj
100
'
0 -100
I l
/*=23.8 0
4
8
12
16
20
24
28
4OO
300
I
2O0
1
100
/a*=29.21
0
30
=0.04
500
1 j
l
@$
1$=0.08
0.05 0.10 0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
(F
P
50
40
3
=0.08
$ 30
8=0.04 20
10
0
0.05 0.10 0.15 Figttre 6.6.
0.20
0.25
0.30
0.35
0.40
0.45
0.50
A Project with Variable Output
The basic model ()f this chapter can be cxtended and generalized i11 several ways. One is to allow a more general price prtlcess. for example, a meanreverting process such as the one we discussed in Chapter 5, ()r an even mtlre general Ito process. The (lnly diftkrdncc this makes is that in the diffdrential equations for the value of the project and that ()f thc option- the coefficients become morc complicated ftlnctions of P. In almost 1,11 such cases we mtlst rely on numerical solutions. Since there are n() new general econtlmic insights to be had from such calculations, wc will not develop this Iinc of models here. A different direction of extension is worth some attention. Suppose the single discrete project, oncc installed, allows some I'lexibilityin its operation at any instant, by varying some inputs, such as labor or raw materials, that do not require any irreversible commitments that extend ovdr time. Thttn tht! optimal amounts ()f tllesklltt any instant will dttpend ()n thtt output price at that
./g1F lrm '
l96
'
'
'
'
D CCl%%IOn<%
instant; the project will have an upward-sloping supply curve. The resulting proiit llow will also depend on the price. Looking ahead to this, the firm's investment decision will be affected. We now examine this in more detail. say, labor At each instant,we have the choice of some operatingvariables, the vector r. These generate output or some intermediate inputs, denoted by according to a production function /1(:) and entail a variable cost C(o). The optimal choice of t? maximizes the operating prost. We can capture the result in a reduced-form instantaneous profit function'.
zr(P)
( P (p)
max
-
U
C(p)
-
(18)
1.
In Section 2 we studied a basic model where, at each instant, the firm had a simple binary choice of whether or not to operate. Now we can think of that as a special case of this more general formula. Suppose the variable t) can take on just two values, 0 and 1. where the former corresponds to suspension and the latter to operation. Define
if p
=
1.
max (# C. ()J. Then we have thc model developed in Section 2 with zr(#) function. production Cobb-Douglas example is the Anothervery familiar and Suppose t? is a scalar, =
11( ?J)
by setting the input price Let us concentrate on output price uncertainty example of analyze will joint price and cost uncertainty an constant at c. (We proht maximization chapten) instantaneous Then section of this in the hnal function demand gives the input
b'
(9 #/c1l/t'-1?1
=
t-.#-/7
Project
#?fJ
/77/z/
Decision
/fa
lJl lz'f?-z
The power of P in the expression for the maximized profit zr( #) ex#) is a convex function ot' #. This is a standard duality'' ceeds 1,' thus property', see Dixit ( 1990) or Varian (1991). The intuition is that, without any instantaneous variability in operations, output is constant and revenue and protits change linearly with #. When instantaneous variation is possible, the optimal choice must make prout increase faster when P increases, and decrease slower when P falls. This makes zr( #) convex. This has important consequences regarding the effect of uncertainty on investment. Now Ie! us tind the value of the project, that is, the value of the whole each i stant is $ sequence of operating options. Here, of course, the option at q exercised, but with a different choice of t? as # varies. lf we, had all wed a. flow lixed cost of prgduction at each instant. there would have been a Iower bound on P (the familiar Marshallian minimum average cost) below which the operation would be suspended. Wewill continue to assume that the price follows the geometric Brownian motion of equation (1).Then familiar steps produce the differential equation for the value of the project .c(
-
/f?
=
'Tht?Vltte
.
and the instantaneous supply function for output
The homogeneous part ofthis is thc same as before. but the nonhomogeneous PY Substituting term is differdnt. We tr.ya particular solution of the form K3 solution the solving Iind we for Kt, and .
K
PP
/
-
(r
-
J) y
-
1 c2 y (y
2
-
1)j
.
This may Iook complicatcd, but it has a natural economic intemretation: PY calculated usit is just the expected present value of a proht stream K another be This appropriate risk-adjusted discount rate. as can seen ing the application of the equivalent risk-neutral valuation formula of Chapter 4, Section 3(a). It says that we can use the riskless discount rate provided we change t5). Then for the stochastic process of # to have a different growth rate (r of have the results we using distribution f lognormal for 8, any future -
l
, (1,,)
==
'
I)PX/CIf'/' -f
1 .
,
inputs is straightforward
Generalization to several variable identical expressions. The proht flowwhen the variable
;.r( P )
=
(1
-
0)
and yields almost
input is chosen optimally is
For brevity of notation we write this as
where P is the initial price, and denotes expectation calculated relative to this new process. Multiplying by e-rt and integrating, we get the present value in the prevous equation. -'
#1/t1-#1 (8/c)f?/t1-f?1
(19)
.
7:.( #)=S#P,
where y= 1/( l
-
))> 1.
, '
zl l--il-l'll I..(-'(-'i.%it')ll.% '.s.
198 To see the result more directly, we use Ito's Lemma to PY, which we denote by a'. as
write
the expected
growth rate of
In Section 1.C abovewe calculated the risk-adjusted discount rate appropriate to any powerof #. Using that formula, the rate p,' tbra protit tlowproportional PY is to y.' r + p (/z r). =
-
Then the return shortfall or convenience yield on t
=
y
=/
.
! -
must be
G'option
-
.r
'
F
J
.
(r
-
PY
The left-hantl sitle is tlle cxpected present value of prolits; this nltlst exceetl value I'ntlltiplethe ost of investnlent by the pl/(p1 y ). Untler tlltl > establisllel, conditions we have p1 lz > ()- so the multiple again exceeds tlnity. Now we can examinc the effect of uncertainty on investnlent. ln earlier work. &ve regarded 6 :s a pllran-leter indtlptzndent 01' ry; I(2t tls continue to do that for a while. We already know one effect of an increase in t'r : it lowe rs pj and theretbre increases the n'ltlltiple. pb/(/.11 y ). Tllis contribtltes to incrcasing #*. Greater volatility raises the value of the option to invest. and therefore requires a higher ctlrrent thrcshold of protability to bring forth investment. However, now we have an added eftkct. As increases. holding ( fixed- we see from equation (2l ) that decreases. Then equation (23)shows that this added e ffect leads to a lower P* and tllere fore it con tribtltes :1 grea ter induceme nt to invest. Th is new aspect arises fron'l the ctlnvexity of protit in price. As we look farther ahead into the future, the variance ()f the distribtltion of price increases. By Jensen's Inequality. the expectcd value of :1 convex function increases. Ito-s Lemma makes this precise: there is an added tern; (7.2 y ( y l ) in the expected groqvth ratc t)f tlle prkltit ltpw.Tllereforc :t Iarger c mcans 1 Iarger expetzttldprdsent vltltle ()1' thc prlltit strtlan-l. ltnd tlltls 1t grcater incelltivc tt) Inequttlity effkct'- ()11 nvestment invest. We will come acrtlss tilis again in Cllapter l l
J) y
-
-
1 /.2 p,( p,
a
-
I)
.
With this, our particular solution isjust K #F/J', which is the expected present risk-adjusted using tlle appropriately
value of the protit stream calculated discount rate.
and set the value of the project
Once again we rule out bubble solutions equal to the fundamental'.
t
-
Kilensen's
)' ( P4
K
=
PY
/5
' .
.
For all this to make cconomic sense we need J' > t). That can be lpotcd at from another perspective. We can recognize the expression for 6' asjust the negative of our t'undamentalquadratic evaluated at y. Theretbre by requiring that 6' > (), we are requiring Q(y) < 0. Then y must Iie between the two roots of the quadratic, specihcally y < pl In turn, this amounts to a restriction 0 < pb l )/pL on the permissible power in the Cobb-Douglas production .
-
function.
The solution for the value of the option takes the familiar form F P)
=
yzl1
##'
.
and smooth pasting,
Finally. the usual conditions of value matching
4
Depreciation
We have assumed thus I'ar that thtt investmcnt, once madtt, Iasts forever. I n reality. physical decay or tcchnological obsolescence Iimit the Iife of a project. In other words, capital deprecitttes th rough agc. ()r ustl. or advance of competing technologies. It is not difficult in principle to modify our analysis to takc this aspect into account, although that does complicate the algebra to some extent. In this section we will indicate how stlch modification can be made for relatively simplc patterns of decay that are commtlnly used in economic
theory. options'' apDepreciation also has conceptual relevance for our proach to investment. One would cxpect that an opportunity to invest in a depreciating project would be less valuable, and therefore that allowance for depreciation would reduce the importance of the issues we have stressed. Our analysis will show that this intuititln must be interpreted With carc. The value of an option to tako an action depends on the dcgree of irreversibility of the itreal
F P*j
can be solved to investment:
=
p' ( #*)
characterize
-
F'( #*)
/.
=
1z''4#*),
the threshold of price P* that triggers the
K ( P*4Y J'
= p,
#! -
/ )z
.
z'1
200
Firm
'.
Decisions
action.This depends not only on the life expectancyof one project, but also on the opportunitics that may or may not remain available after the Iirst projcct comes to the end of its life. 4.A
Exponential Decay
We begin with a form of depreciation that is most Often used in economic theory Iargely for its analytical convenience. Here the lifetime is random and follows a Poisson process. At any time F, if the project has lasted that long, there is probability JT that it will die during the next short interval of time dT. Now, from the initial perspective, the probability that the project dies before r, or the cumulative probability distribution function of the random T'. lifetime F, is 1 e-k The corresponding probability density function of F -
is
e-k
T
Suppose that during its lifetime the project will produce a unit flow of output with no variable costs. The initial price is P. and its subsequent path fl follows a geometric Brownian motion with growth rate a. The risk-adjusted discount rate appropriate to P is p.. If the project lasts exactly F years, the expected present value of its proht Eows is
'
r
e-#t /1 dt
()
r
# e'l
=
f,-''
'
= #(1 =
-t/1
-
-
-t'
e
1F
1/(p.
-
a
and 6 is the return shortfall or where as beforewe have abbreviated p. the risk in P. Now we replicates traded convenience asset that yield on a the can use the probability density of the lifetime for a Poisson process to obtain the expected value of the project: -a
=
l P
1
J
y
= #/(
=
(I?If/ //lc oftl F'a/-f.?.jccf Dccisioll to /'?1I't?.J
20 l
This formula has alternative interpretations. First. the project might have infinite lifes btlt it might function less and Iess well as it ages. producing an an output tlowe-kt at time t after installation.-fhat will yield the same discounted present value as the expectcd value calculated above. Second. the machine might need an increasing stream of maintenance expenditures as it ages. We can then regard P e-kl as a prox'y or reduced-form expression for the projit tlow it can produce at time 1. Next we value an option to invest in such a project. We must distinguish possibilities. The option may give the lirm the right to invest in this project two the project dies, the hrm has no further rights. Or the tirm just once; aer right to invest in perpetuity', after one project dies it gets the may have the original opportunity to invest back again. We begin with the hrst case. Let F( P) denote the value of the option. As before. we construct a portfolio consisting of one unit of the option, and F' /7) units of a short position in the asset that spans the risk in #. Note that such a traded asset is quitc exogenous to the project or the firm; it does not die with the project. Therefore its convenience yield is t5 /.1 (z. Then the calculation goes through as betbre, and yields the functional form for the option value =
-
F( #)
=
z-1l
P/1
.
and
/1 is the positive
F P)
(.
1 -
where l is the
=
IZ
( P)
-
1.
F' P)
=
P''( P).
sunk cost of investment. A simple calculation gives
This is almost identical to our earlier formula (9)for the investment threshold for an infinite-lived project-The only difference is that the 6 on the riyht-hand side is replaced by + This can be understood by considering the special t5 r, and r l isjust the case of risk neutrality and zero trend in the price. Then annualized or flow equivalent of the cost of the insnite-lived investment. Then at the threshold the proft flow should be a multiple of this cost sow,to rellect the value of the option that is sacrihced when the investment is made. When we introduce deprcciation, the flow equivalent cost is increased by the Poisson death parameter because the sunk cost of investment must be recouped over a .
+
=
+ J).
Formally. we can regard the project as infinite-lived, but augment
the rate at
which future prohts are discounted by adding the Poisson death parameter, so that the discount rate increases from Jz to p. + This conforms to our general analysis of Poisson processes in Chapters 3 and 4. .
root of the
The investment threshold P* and the constant z1l are jointly determined by solving the value-matching and smooth-pasting conditions
)
T')/.
e-6
I-za/fJ
where z1l is a constant to be determined, familiar qtladratic equation (4).
dt
()
P g1
Tlle
zl 17//'??l
:!():?
'.
flc't.'/'.tpn.
shorter expected Iifetime. However. the same option value multiple applies to this new llow cost: the equation tbr pl is unaffected by depreciation. Tllerefore the intuition that depreciation would reduce the importance of option values is not valid in this context. to invest is available only onccp its exercise isjust as
Since the opportunity
irreversible even though the project has a finite Iife. If we take the alternative perspective where the physical life is infinite but the otltput flow decreases exponentially, then the depreciation has no bearing on the irreversibility of the action. Matters are different if the option to invest is available in pemetuity, so that the hrm regains the right to start another identical project aer the first one expires. Of course the randomly evolving price might be too low to justify investment at the instant the lirst project dies. but the firm once again has the option, and can start a second project when the price rises again to the threshold. We now turn to this case. In this analysis we must consider revenues and values that are various nonlinear functions of Ps and the risk-adjusted discount rate appropriate to each is different. We take the dynamic programming perspective with an exogenously specilied discount rate p. However, a similar analysis can be conducted using the contingent claims approach provided the Poisson risk of the project's death is fully diversifiable; this point was discussed in the context of Poisson processes in Chapter 4. Let #* denote the investment threshold. and Iet F( Pj denote the value the option to invest. While unexercised. that is, in the range () < P < P*, of option merely has an expected capital gain this F( fJ)J CLd =
h
P F'( P4 + 1.c2 /72 F/'4 p) j (J/.
2
Setting this equal to the normal return p F #) (lt, we get a familiar differential equation for F Pj, and an equally familiar solution F #)
is where quadratic ..dl
=
P#'
adl
-
current project F J7). 'Thus
??lt/
tlle /)(.,(--j-&?,l :t.) /?l!-e/
will die and the Iirnx svilI go l7:lck to Iltlltling tlle opt itlll svllI't 11
Expanding the right-lland side using Ito-s Lemma
and simplii'
ing, we havtl
This has the solution J( #)
where #1 is a quadratic
B6 P p'' + #/(p
=
(z)
-
to be determined, and
constant
2!
+
4,.2
p(p
1) +
-
(z
p
l
P p,
.
J'1 is the positive
(/) +
-
.4
+
)
=
root of the
().
(26)
Over the range P
project dies
> P', a similar analysis applies, btlt if the current a new one will be immediately started. Therefore
This becomes
The solution
is J( P)
where Bz is a constant quadratic (25).
=
Bj
JJ/h
+ P( p
-
a
to be determined. and
)
-
). //p
p? is the
.
root of the
negative
#) must meet tangentially at the common Now the two branches of point P. of their ranges of validity. Also, since P* is the investment threshold, F( P) must satisfy value matching and smooth pasting with J( P) / at #*. Thus we have four equations to determine the constants Cl Bb Bn and the threshold #*. This completes the solution. The scemingly complicated procedure yields 11very simple answer. Since the two branches of the function J( #) to the Ieft and right of P* meet and tangentially at P*, we can use either branch for the value-matching smooth-pasting conditions that link F(#) and J P) / at #*. The left-hand branch gives the solution more easily. The value-matching condition is .74
-
a constant to be determined.
12 (7.2 pfp
,
Tlle billtle t7./-l Ih-oj'ct
1) +
a
p is the positive root of the
and
p
-
p
=
0.
(25)
0 to rule out the term with the As usual, we have considered the Iimit as # negative root. Let JP4 denote the value of an installed project along with that of all -+
future replacement options. First consider the range P < P*. Over the next short interval of time Jl, the proht flow is PJl. Then with probability dt, the
.
-
,
204
..dFirln Decisiolts '
Table 6.1.
lz/?eItl
Option
Mltltiples
Depreciatioll
ksf
&
4.B
tl?7t/ Ihe
'lcc-tpll
o lll !
205
't?.y?'
Sudden Death
Next to the exponential or Poisson decay. the tbrm of depreciation most popular in economic analysis is a Iixed hnite life during which (he project continues in perfect health, followed by a sudden instantaneous death.''
4
).
().l
().:!
t).:5
().ul
0.00 0.01 0.05 0.10 0.15
1.4215 1.3706 1.2651 1.2077
2.0000 1.8632 1.5954 1.4561 1.3813
2.7631 2.5000 2.0000 1.7500 1.6193
3.7321 3.2966 2.4868 2.0938 1.8927
1.1759
Tlle Idl/c oj- fl Project
Let T denote the fixed finite life. The prtect
continues to produce a
unit of output, and tllerefore the protit tlow lC), for F years, at which point it suddenly stops functioning. Let the initial price be P. The value of the project at installation is the discounted present value of expected profits over it s life t irne : r
l''( #)
=
= '/'',
B1
The smooth-pasting condition
' Bl pb
+ #/(p +
a)
-
1.
=
a)
().
=
/ J/
X (l
-
-
(?
1F
t 3t -t
e-'s
7-
1/(M
-
(')
jj.
where we have used the risk-adjusted discount rate p, and recognized that the expected value of the price, and thereforc the protit tlow,grows exponentially at rate a.
-1
-
'
:-7. ()
= # (l
is
J'/$', + l/(p +
t-
In equation (5) wc found the value P-( #) P/t5 for an infinite-lived project. The above formula is now seen as a very natural generalization; the inlinite-life case is obtained by taking the limit ps F goes to x. The value ()f an optilln to invest can now be found lbllowing the same steps as in the previous scction. and distinguishing thc cascs ofa one-time and a perpetual option. We Ieave this as an excrcisc. =
Solving these and writing p
-
x
=
J, we find
(27) Contrast this with the formula (24)for the case where the right the to invest in the project just once. The identical, but the option value multiple is different. The root #( an equation (26)different from equation (25)for pj The latter
option gavc cost part is now-equivalent comes from has the depreciation parameter added to the discount rate p. Therefore p'l > pb and 1) < J1/(J1 1), so that in this case depreciation does lower the J'ylp'j option value multiple. If the option to invest is perpetually available, its exercise on any one occasion is a less irreversible act when the project depreciates .
-
4.C
The General Case
We can subsume the above common
forms of depreciation into a very genzr( P. /). The Poisson a profit t1()w
eral analysis. Suppose the project produces
-
faster.
We illustrate in Table 6.1 the numerical signiscance of this effect. As in the central case of ourearliercalculations in thischapter,we take p r 0.04 =
=
0. We then show the option value multiple J'j/(#'j 1) for various and a values of tr and Depreciation has a signihcant effect on the option value multiples. But the multiples rcmafn substantially above 1, particularly when (r is not too Iow. =
-
.
''In economic theory such dcpreciation
by sudden death is slymctimcs labelled the Wllnderful One-hoss '-one-hoss
shay.'' The term comes frllm thc pocm '*The Deacon's Nlasterpiecc: or Shay'' by 01 iver Wcndcll Ilol mes, Sr.: Havc you hcard of thu wonderful onc-hoss shay Which was built in such a logical way That it ran a hundred years to a day And then, of a suddcn, it PiCCCS all at Once ... WC rlt to ...
We give this quotation in an attcmpt to dcmonstrate
culturc.
that economists are not entirely dcvoid of
/ I-'il-t';
l.lpl..t'b.illts
'.
shlly-*case decay case has x declining exponentially. while in the gtlnerlllly. and then :1 More to zr migllt while drops for constant zero. n' stays physical suffcrs gradtltll capital either because time. gradually decline over technology its of conembodying date at the project. the because decline, or the the In latter with and projects. later case, newer struction, must compete when J drops P. endogenotls date will zr ) to ( the suspend operation at project This obsolescence-'' termed zero; this phenomenon is sometimes advance and mlAnotonic was modelled in the framework of a deterministic and Bliss Yaari and l967) Weizsacker, ( in technology by Solow, Tobin. von research uncertainty of extension exerto the case as a ( 1968),. we suggest its cise for interested readers. -tone-hoss
-eeconomic
The value of such a project can be expressed as a present value integral. but for general forms of zr that is hard to evaluate. We describe an alternative approach. lt has the added advantage of yielding a general formula tbr the value of a project at any moment during its lifetime, and not just at the initiai
date of construction. Let Pr( P, t) denote the value of a project as a function of the current price P and the current time f We consider the usual portfolio that holds the project and goes short /1 units of an asset perfectly correlated with P, for dividends a short interval of time dt. Over this interval, the portfolio earns shortfall'' or zCP. t ) dt pz a is thd n J Pdt, where as usual J convenience yield on the P asset. and the second term in the expression for the dividend represents the payment the holder of the short position must make to the holder of the corresponding long position. The porttblio has a capital gain given by .
'return
=
-
-
where we are using subscripts to denote partial derivativcs sincc P' now has 11 1z',4#. /) two independent variables; thus Vp 1)V1L)P, etc. By choosing expected return we can make the portfolio riskless. Then we can set its total riskless the return'. to equal =
=
7r( P t ) ,
=
-
(5
# Ypl P, t ) +
r (I''' ( P, t)
-
c2
/22 Vpp P, t) + P',( P, ; )
P Fpt #, t )1,
Or
This is a partial differential eqtlation that can be solved numerically. If there is a known muimum possible lifetime T, then the solution can be started
at ? r wi t 11t I1e ct'lntlitildn l'' ( I'. T ) () lklr tlIl 1*. y'v'.twill illtlstr:.lttl stlcll stlltllillns il'l:lI)()t Iltlr ctllltext il1Clllptttr lllltl
trtpn-lplctctl
bllckwltrd.
I().
poillt kltlt tllc klctlllklnlikz siglliticltllcc ()f tlle nevv ternl l''f( 13. / ) t Ilklt vltlue (.)t' tllt'tprljetrt i2:lI1 l'1tlw cllllllge l'or tNvt) I'eastllls: ente rs the picttl re. -'F11t., llitittl t)f tlle sttlcllklst ic price, alltl ftkre i Itle v:y nt pure pllssage o1*t in)e be:.1di of cllllnges tlle ftlture proti Ie proli t tltlws.Tlle latter t? l'ftltrt is dxactly caus't tllat econonzists l j ( 13.t ) gives tls a qulklltitanletll by deprecikttioll. 'Tlltls what dtltlnt'jnlilz o1' depreciation. Tllis concept was well eltlcidated tive Ineasure by Sanluelson ( l 964) in the context of trt't rtktillty and pe rftlct l'oresigllt, and tllat illtrorptlrates tlnce rtain ty ktnd rational Ilere &ve Ilave a nattlral klxtonsitlll Ilere
&ve
expectations.
Price and Cost Uncertainty Thus ttr we have allowetieonly one randol'n variable. nllmely. the tlutptlt price (or a demand shi variabltl). ktlcping :111 (lther paralmeters bearing on the investment decision kntlwn lnd constant. Sv tlid this to develop tlle analytical methods in a relatively simple setting. The same methods can be employt!d in more general situations where twl) or more randon) variables affect the firm's decision. F()r exanlplc, if b()tl1the investment cost I lknd the outptlt price 13 (,1' tlle pnjcct lknd the villtle are tlncttrtain. thcn wc have tt) exprtlss the valtlc ()f lxltI) ()1* tllcsc variltblcs, lz'( P, / ) and of thtl klption to invest as functillns whtlle rtlgitln ()f vltlues of ( P. I ) where intlle Nve l find P. Then llavtl t() )-( will (lccur. the whole region wllertl it will nllt llccur. and the critical vestnpent ()r separating threshold the twll regitlns. Needless t() say, this boundary curve the vltlue mathematically Nvith twt) independent variables. is morc diffictllt. eqtllltions. their satisfy and slllution partial diflkrontial functions can require (Jxalmples complexity-lt' with speI-ltlwever. methods of some numerical somc of solved homogencity can be by cial tkatures in particular. some form reducing the problem t() one state variable. We now illustrate tllis. Consider a unit-sized project whostl investment cllst / and the revenue flow P are both uncertain. We can even nllow the uncertainty in these two variables t() be correlated dtle t() sllme commlln macroeconomic shocks. Thus Inotions: we assume that P and I follow the geometric Brown ian
d #/ 1.'
=
ap
dt +
fT/,
dzl,
(1 .
11 I
=
al
J/ +
t'r/
(lzI
,
,-1F?-v? $- dqlt?t7in-itl,7 r;
208 where
CWz2s J =
:J
tti ,
gt/Fy j
dt
=
Sflzp Jz/
.
)
p Jf
=
-
tl//t/ 0J,tl r/-tp-Jct'/
rule. By now. the steps should be very familiar. Let F P. /) be the value of
Note that the (IP and dl on the right-hand side are stochastic. However. we F'p and zl Fl to get rid t7f these terms and make the can choose ??l portfolio riskless. Then the holder of the portfolio over the intelwal (z f + dt will have the sure cagital gain =
for brevity. Considcr a portfolio consisting of one unit m units short in the output. and ?, units short in capital. By Ito's
=
.
He must also make a payment corresponding to the convenience yields on output and capital. (?,1 l, P + 11 t5/ /) t/l, to hold the short position. Equating the sum of these two components to the risk-less return r ( F m P 11 14 JJ and collecting terms. we get the basic equation -
the option. We find a differential equation for it. We assume that both the risks in output price and investment cost are spanned by existing assets, and work with assets whose prices are P and 1. respectively.l l Call these assets and
+ (1-
J l ) I f--I
-
l
-
-
F
=
0
-
.
tcapital''
As there are two independent variables ( P. /). this is a partial differential equation. It applies over the region of the ( P. 1) space where it is optimal to hold the option unexercised. Over the region where the option is immediately
exercised. we have
P
/:7 p=
Invest
Xx
l
Free Boundary
At the boundary between the two regionsp this becomes a value-matching condition. The two functions must also meet tangentially at the boundary, yielding two smooth-pasting conditions Fr ( P. l )
wait Figure 6.8.
lnvrt.p./l?lt'n?with Pnke and Cb'; Unccrtainly
=
a dynamic programming
approach
Ieads to
:1
very similar
differential equation.
lZ
'
( Pj
=
l/
/,
.
F; ( P, l )
=
-
1
.
The differential equationa together with these boundary conditions, should tix the position of the boundary itselt and also yield a solution for the function F in the waiting regign. The fact that the boundary itself is an unknown makes problems of this kind quite difcult. In fact the theory of partial differential equations has little boundary'' problems in general. Analytical soluto say about this class of tions are rarely available. and numerical solution methods are mostly ad hoc, each tailored to iit a particular situation. This is in principle no different from t'free
l1Otherwise
209
.t?-$?
=
=
of the option,
l() /?I )
Lernfna. we have
t'rate-of-return
Etoutput''
tlle lt.zc-.tp/l
.
Once an investment is made, further uncertainty in the evolution of the investment cost is irrelevant. The value of a live project when the current price ap4 P/tp, where p. J> = r + / p/w, a'l, is P/p.p is P is simply P' ( P) risk-adjusted appropriate discount rate to P, and l, the Jz /, - ep s the shortfall'' in #. tconvenience yield'' or The value of the option to invest, however, depends on both P and /. Intuitively, we expect that the option will be held when P is low or l is high, and exercised when P becomes sufsciently high for given 1, or / becomes sufhciently low tbr given #. Figure 6.8 shows the suggested regions in (/. #) space corresponding to waiting and investing, and the boundaryseparating the two. Our aim is to make this intuition more precise. and develop an analytical method to find the boundary and thereby determine the optimal investment =
Tllt: I-'J/l/t.'
g' ?'r?'l .,.,.j ym. .
2 l0
. .j
.
.
.
)L t)IJ 16) j ,
.
.
.
js
tlle problem even whcn only the price is uncertain: the investment threshold P* is itself unknown and becomes the free-boundary point that separates the one-dimensional range of values of P where investment occurs from the much easier to find the range where it does not occur. In one dimension it is analytically in the present case the whether Luckily, solutions or numerically. tIs reduce it to one dimension. allows of to the problem natural homogeneity
If the current values of both P and I are doubled, that will merely double the value of the project and also the cost of investing. The optimal decision should therefore depend only on thc ratio /7 - P/ 1, and theretbre the boundvalue ary in Figtlre 6.8 should be a ray through the origin. Correspondingly, the enabling I us to of the option should be homogeneous of degree l in ( P. ),
write is now the function to be determined. Successive diftrentiation gives
where
Le t pl de n o te t he l:trge r root of t 1)is; i1'J t assume), then JI > l Tllen we tind
:1 1)
d
f,
itrllNve a re b(')t 11pos it ive (svl'l
.
This ray throuyh the origin separates thtt rtlgitlns of w:liting and investl'ption value nltlltiple ment in the ( P, J) space. I ts slope has the stalltlllrd 12 interpretation. Thus if either t'z'/, or (.'l increase. pb will dekrrease. and the multiple pj/(/31 l ) will increase. However. the multiple will decrease if p increases-,holding their variances fixeds :1 greater covariance betwecn cllanges in P and 1 implies less uncertainty twer their ratio. ltnd henctt :1 reduced incentive to wait. -
.
Fl
and
P, /)
Fra/'t P. /)
./',(
p)
=
Fl ( P- l )
,
/'''( p)/
=
1.
-11 ( P
/)
.
=
Guide to the Literature /'(
=
.1''
/J)
F'/ ( #. /) = ,2 /'''( p)/ 1.
-
a
2,p
cp o'l + c/ a )
a
/7
.
p .1''' p)/ 1.
-
.
(28)and grouping terms,
Substituting this in the partial diffcrential equation we find J. (c/a c
?)
p
-
.,,
J (P) + (t/
.1
(/,)
-
.,
p
( JJ)
.
./ (p)
,
-
=
() .
(c;)
This is an ordinary differential equation for the unknown function fp) of the scalar independent variable p. Moreover, it has exactly the same form uncertain. Its boundary as the familiar (6) for the case when only the price is value-matching condition becomes also The similar. conditions are
./*(/7) JJ/J/, =
conditions
The two smooth-pasting
j'p)
=
l/Jp
1 .
become ./*(JJ)
.
-
-
p ./''(P)
=
-
1.
Of these three conditions, any one can be derived from the other two. We can select the value-matching condition and the first smooth-pasting condition uncertainty case, and as the two that are exactly parallel to the pure price complete the solution as before. The fundamental quadratic is Q
=
1
2
(c/1 2p fzp -
o'l +
fT/)
pp
-
1) + 6;
-
p)
p
-
j/
=
0.
This chapter prescnted more complete models t)f investment, and also extended the important idea that the value of a lirm is Iargely the value k)f a set of options. In Chapter 5 we saw that a tirm has vltluable options t() invest. Some of these will be exercised and stlme not, so thltt the value of the tirm will equal the value of its existing projects (that is, its capital in plactl) plus the valuc ()f its options to invcst in new prlljects in thll future. In tllis chapter we to saw that an existing project can also be viewetl as a set of options--options produce and earn profits should pricklbe stlfhciently lligh rulative to operating cost and can be valued accordingly. This idea appears in Marcus and Mtldest ( 1984) in the context of agricultural productitln decisions. btlt was Iirst spellud out in detail by McDonald and Sillgel ( I985). Tlley shllwed that if prit:c folIows a geometric Brownian motion. a unit-output prlject witl) jixe llperating cost can be valued as the sum of an infinite set of European cal I options. The approach used in this chapter is morc general and mllrtt tractable we simply valued the project as a single contingent claim, and thcrtlby derived a ditkrential equation for 1/'( #) in the same wlty that we derivod an equation I'()r the value of the investment option, F( #). in Chapter 5. We then used tlle solution for )' ( P) in the boundary conditions to Iind the solutitln lr F ( /3), ajong with the optimal investment rule. This approach was dutvelllped in Pindyck ( l 988b) in the context of incremental investment. 12It may seem puzzling that thc risklcss interest r:tte Ills disappeared frllm the qtladratic equation and hcncc frtlm thc slllutitln. In fact. ?' rtlmains, cllllcuttllk!din /z /,. j/,, lnd 6: .
A F'r//l
'J
Deci-iolts
We have seen that an important starting point for the valuation of projects and investment opportunities is the underlying stochastic process for the outmotion. put price, #(f ).We assumed tlat ifprice follows a geometric Brownian expected risk-adjusted return. the y.. For than less its expected rate of growth is storable commodity, this diftkrence would have to retlect the convenience a yield accruing to holders of inventory. This point was raised by McDonald and Siegel ( 1984), and the stochastic structure of convenience yield itself is examined in Gibson and Schwartz ( 1990) and Brennan ( l99 l). This chapter introduced a number of free boundae problems, some of which could be solved analytically. and some of which require numerical solution methods. For readers seeking more background on the mathematics of these problems and their solution, we suggest Guenther and Lee ( 1988) and Fasano and Primicerio ( 1983). Bertola ( 1988, Chapter I ) has a very gendral model wfth simultaneous uncertainty in the output price. the price of capital goods. and prices of variable
Chapter
7
Entry, Exit, Lay-up, and Scrapping
inputs.
variables. and Most of these methods rely on discretization of the state solutions of multivariable numerical we have adopted the samc approach for problems in Chapters 4 and 1(). Somc recent work t)n numerical methods is whole based on an alternativep called the finitc element method. Here the small funcsplit cellsthe variables is into region of space of the independcnt in polynllmial approximated low-order by a tion we are trying to obtain is cdgtls the togcthur at each cell. and the diffcrent approximations are pastcd of the cells. This method looks promising for many economic applications. Interested readers should consult books by Johnson ( 199()) and Judd ( 1992).
previous chapter we showed how one can hrst value a project, and then value the option to invest in the project and determine the optimal investment rulc. Our starting point was a stochastic process for the evolution of the price of the project's output. and hence unccrtainty over the future 2ow of operating prolits. This f1()w of profits could sometimes becomc negative, and we assumed that at such times the firm could suspend operation. and resumc it later if the profit flow turned positive. without paying any lump-sum IN THE
restarting costs. For many projects, this assumption of costless suspension and restarting suspend and later restart s unrealistic. In some cases it is almost impossible to the operation of a project. An example is a research Iaboratory engaged in the development of a new pharmaceutical product', suspending the laboratory's operation may mean losing its team of research scientists and hence the ability to resume development ofthe product in the future. In other cases it is possible to suspend and later restart the operation, but only at a substantial cost. For example. if the operation of an underground mine is suspended. a sunk cost and an ongoing fsxed cost must be incurred to prevent the mine frorn flooding with water so that it can be later reopened, and an additional sunk cost must be incurred to actually reopen it. This chapter begins with a model that is the opposite extreme of the will assume that if the operation is one developed in the last chapter. We again jirm whole the investment cost over to must incur ever suspended, the
stopping or
,..1Firltt
'x
Decisiolls
and crtlmbles once it is unused.) Instead restart it. (This is as ifcapital suspension, active of tirm must contemplate outright abandonment. now an i
Of course it will not take this step the moment its operating prolit llow turns negative. Since restarting is costly, there will be an option value ()f keeping the operation alive. and abandonment will be optimal only at a suficiently large threshold level ofoperating Iosses. We will assume that the firm retains whatever market power it had to start with, and does not lose its rlk/ll to invest if it abandons operation. (ln Chapter 8 we will consider the other extreme of a perfectly competitive industry, where any of a large number of potential lirms can invest, and any active (irm can abandon without retaining any special
privilege of
the value () if thc prqject is not operatillg. and tlle valtlc l if it is operktting. tt) ssvitcl) betNveen thesd tyvl). Tlle firm-s strategy ctlnsists of kt pair of tlecisions In each discrete state the lirm hlts a call option on the other. >.Xnitlle lirn; dxercise the option to invest. This gets it the tlow01' operating proht. plus can option to abandon. sv'ennust lind tlle rtlles for optilual exercise of tllese an options simultaneously in terms of the tlnderlying ralldtn-l variable (price). Sinxilarly. when sve consider tlle third alternative of or up,-' there are three tliscrete states. with optilnal ssvitches llnlong then; tt) be --nlothballing-'
-wl:ly-
calculated.
reentry.)
In most cases, reality lies between the extremes of costless temporary suspension and immediate total rusting. The capital sunk in most projects rusts when not used, but does so gradually. Machinery or ships rust Iiterally', other intangibles like customer mines are subject to cave-ins and nooding; loyalty and brand-name recognition fade.l Then restarting is costly. but not quite as costly as new investment. In some cases, the cost of restarting rises with the duration of the suspension.'lb model this.we must consider an optlon to restart (as distinct from the option to start ab initio), and introducc thc elapsed time since the last suspension as an explicit state variable affecting the value of this new option. The resulting partial differential equation can be solved numerically, but we will not treat this case here. Later in this chapter we will consider another intermediate possibility. 'tlkusting'' of idle capital can be prevented by undertaking literal in the case of machinery, ships, and mines, and tigurativein the case of intangibles like customer loyalty. Instead of abandoning, a firm may choose to keep its project alive by maintaining capital but not actively producing output. For example, ships are or ''Iaid up.'' This incurs an ongoing but the maintenance cost, prospect of future reinvestment cost. The saves alternatives depends on the relative magnitudes of betwen these tradeoff and likelihood of the a quick return of favorable operating the two costs, on '.maintenance''
'tmothballed''
conditions. Mathematically, in addition to the state variable (forexample. price) that evolves stochastically and affects the prostability of operation, the possibility of abandonment introduces a second discrete state variable, which takes on 1A scientist undertaking a new research project must invest capital acquiring familiarity with the literature. learning new mthematical tcchniques or Iaboratory skills. etc. Our own experience is that such capital rusts very quickly if we set the project aside for as little as a few weeks.
1 Combined Entry and Exit Strategies We conhne the disctlssion to the case of demand uncertaintya asstlming :1 geometric Brownian motion price process. lnterested readers can develop extensions to other proccsses along the Iines of Section 5 of Cllapter 5. and t)f to uncertainty in other variables along the Iines of Section 5 Chltpter 6. Nv'e the and ntltation wherever possible. The investlnent of Chapter 6 setup retain and abandonment decisions are made l7y a firm that takes price as given. and we again assume that thtl pricc lbllows a geometric Browni:ln motitln,
t/ 1.*
a /3 t/ +
=
f'z
P Jz'.
(thatis, enters the market ), it tlbtains
11prleu't tllltt produces pcr pcriod. and Iasts forcvcr or tlntil abandoned. Variable C are knllwn and constant. Tlle risklcss rate of interest is exogenouslyIixed at r. We will assume that stochastic lluctuations in price are spanned by other assets in the economy (althtlugh.as wll havtl seena if this were not the case, a solution could be obtained by dynamic programming). The appropriately risk-adjusted discount rate for the firm's revenues is
If thc Iirm invests
one unit of output (lperation costs of
pz
=
r +
4 /) tn'f
rr,
where / is the market price of risk, and pJ,,$f is the coeflicient ()f correlatign lx between the price P and the entire market portfolio. As usual, we Iet > (). and that rate-of-return price, shortfall we the assume denote on (
=
-a
(
The firm must incur a Iump-sum cost I to invest in the project. and a lump-sum cost E to abandon it. This latter cost might include legally rcqured termination payments to workers, or costs of rttstoring the site of a mine to its natural condition. It might be the casc that part of the investment cost l is not sunk. so that E is ncgative, reflecting !he portftln of lhc fnvestment that
z4Ffrzn:. Dections can be recouped upon exit. Of course we need l + E 0 to machine'' of rapid cycles of investment and abandonment. >
rule out a
ttmoney
Ently
'-rf,
Lflv-Uz
(l/lt
Scrapplg
of equations and boundary complete the solution.
conditions
contains
just enough information
to
In Chapter 6 we began by hnding the value IZ of a live project, and then went on to the value F of the option to invest. Now that sequence becomes which is an a full circle. The live project is really a composite asset, part of exercised, goesback option 5rm abandon. that the is to the inactive If option to acquires another option words, namely, the asset, it to invest. state. In other project. exercised Thus the live back in turn, it leads When this option is to a determined be interlinked, and must values of a Iive firm and an idle firm are
%Mebegin with the idle firm. To obtain a differential equation for l''(j( P). construct a portfolio with one unit of the optiop to invest, and a short position of Zt1'1#) units of output. The steps that follow are exactly the same as those in Chapters 5 and 6, so we omit them and leave them as an exercise for the reader. The resulting equation is
simultaneously.
This has the general solution
Intuition suggests that an idle hrm will invest when demand conditions become sufsciently favorable. and an active firm will abandon when they become sutxciently adverse. Indeed, we will see that the optimal strategy for investment and abandonment, or for holding or exercising the two options, will take the form of two threshold prices, say, #s and Pg, with #s > Pg. An idle 5rm will hnd it optimal to reman idle as long as P remains below Pu, and will invest as soon as P reaches the threshold Pu. An active lirm will remain PL,. ln active as long as P remains above PL, but it will abandon if P falls to PL and #s, the optimal policy is (he range of prices between the thresholds whether with operation or waiting. We it be active the continue staus quo, to verify this intuition. Of course we must find the values of these proceed to now thresholds in terms of the exogenous data.
1.A Valuing the
K)(#) where z11 and of the quadratic
and
=
pz
=
!
z1l ##' + zh ##2
are constants to be determined. and pt and equation familiar from Chapters 5 and 6:
a4:
p,
=
-
a
(p
-ta (p -
-
-
l/J
2
l/z
2
+
-
((p
t5)/c
g(p
.)//r2
-
Ztlt #)
.
-
-
-
2 1. cl +
!2)
2
the roots
zp/fr
2
>
1.
+ 2p/cz z
<
().
Since the option to invest gets very far out of the money and therefore becomes nearly worthless as P goes to (). the coefficient zh corresponding to the negative root pj must be zcro. That leaves
wo Options
The value of the firm is now a function of the exogenous state variable #, and of the discrete state variable that indicates whether the tirm is currently idle (0) or active (1).To clarify this, we will change the notation slightly, Ietting P'()tP) denote the value of the option to invest (thatis, the value of an idle hrm), and letting 1zh Pj denote the value of an activ frm. Note that lZ1 (#) is the sum of two components. the entitlement to the profit from operation, and the option to abandon should the price fall too far. Over the range of prices (0, #s), an idle 5rm holds on to its option to invest. As in Chapter6, an arbitrage argument tells us that 1z'a(#) satishes adifferential equation over this interval. The boundary conditfons link values and derivatives of P7( #) to those of 1zh( P) at Pg. Similarly, over the range of prices CPL, cx)), (#) an active hrm remains active, holding its option to abandon. 1z'1 conditions corresponding the boundary equation, differential and satisses a link the values and derivatives of lZI ( P4 to those of Pr(,(#) at Pg This system
2
pz are
./zll
=
PD'
.
Remember that this is valid over thc interval ((). #//) of prices. Next consider the value of the active firm. The calculation is similar, except that the live project part of the portfolio pays a net cash flow ( 15 C) tl. Then we get -
The general solution to this equation is
)' l ( #)
=
#1 #/' + Bz #/': + P/6
-
C/r.
As in Chapter 6, we interpret the last two terms as the value of the live project
when the hrm is required to keep it operating forever despite any losses, and the hrst two terms as the value of the option to abandon. The Iikelihood of abandonment in the not-too-distant future becomes extremely small as P
z'l
2 18
Finn
'.
Decisiolts
should go to zero as P goes to cx), so the value of the abandonment option corresponding to the positive becomes very Iarge. Hence the coefficient 1'11 should be zero. Ths leaves root /:11
( P) 1,z'1
Bz PX + P/6
=
C/r.
-
This is valid for P in the rangc ( J. (x)). At the investment threshold P//, the tirm pays the lump-sum cost I to exercise its investment option. giving up this asset ofvalue K)(#/,) to get the live project which has value 1z'1 ( P//). For this we have the conditions of value smooth pasting: and matching ,
Ft1(Pll4
P'1( Ptl )
=
pastng
Z1'(PlI)
=
.
(7) and smooth-
are
F1 ( PL
Using equations written as
Z(/)( Puj
1,
threshold Pg, the value-matching
Likewise, at the abandonment conditions
-
)
l$)( PL
=
)
-
E
(4) and (6) for K,(#) -
zdl
sa p/hy/ +
#y, l$'+ -1
pb -/31 z4l Pl1 yzl1
-
+
z1l
p' %.
+
)
1$;( fi )
=
.
(8)
and lh ( #), these conditions can be lht/
c/r
-
#a s 2 plk-t 1:
Pg/3'+ B: #Xg + Pt /t -!
-#1
F1'( PL
.
L pz y a ,pt-l
+ jyj -
=
1.
(j())
.() ,
C/r
=
+ jyj
=
E,
-
()
.
the thresholds Pll, #/. These tbur equations determine the four unknowns values. and option the Bz in coeffkients and the The equations are very noniinear in the thresholds, so that an analytic solution in closed form is impossible. However, it can be proved that a solution exists, is unique, and has economically intuitive basic properties. The thresholds satisfy 0 < Pt < Ptf < x, and the coefficients of the option value terms, ,4l and Bzn are positive-z Some other important general economic insights results require numerical can be inferred by analytic methods, but further solution. We proceed to these in turn. a/4l
Elttll'.
A-a-J'J,
1.B
Comparison
Ltlv- f-#7,ttlltl
with Myopic Decisions
The theory of investment and abandonment as typically presented in intcrmediate microeconomics textbooks is based on the Marshallian concepts of long-run average cost and short-run variable cost. For our unit-sized firm- the long-run average cost is the sum of the operating cost and the interest 011 the sunk cost of investment. (C + r l). The textbook theory tells a firm to invest if the price exceeds this. Similarly, an active firm should abandon if the price falls short of the variable cost C. When there is an explicit lump-sum cost E the tirm should also take into account the interest on this of abandonment. threshold the becomes (C rE4. cost. so that In otherwords, the traditional Marshallian concept is to compare the rate of return on the investment, ( P C)/ 1, and that on disinvestmenta (C #)/ E' to the normal return r. Implicit in this view is an assumption of static expectations or myopia that is, the current price is assumed to prevail brever. This may be appropriate for analyzing a price change that came as a surprisc. and when the firm knows for sure that it will never happen again. Howevera such price changes are rare. In most real-world situations, the demand (and cost) conditions facing a Iirm change all the time, and the (irm must make its investment and disinvestment dccisions taking into account that the future is and always will be uncertain. Hence a more natural theoretical approach is to assume that the firm has rational expectations ltbout the probabilistic law of motion for its uncttrtain environment. Our model above does just that the Iirm's decisions are optimal given the stochastic process ( l ) lbr the pricc. Let us now ask what difference it makcs to give thtl 5rm rational rather than static expectations. How does the optimal investment threshold PlI compare with the Marshallian threshold (C + 1- I ). and the optimal abandonment threshold P. compare with the Marshallian threshold (C 1- E)'? To answer this, we begin by defining the function -
-
-
-
G ( P)
-
=
Ph(P) -A
I
-
PI$b
1,$,4P) + fa ##? + P/( - C/r.
( 13)
This function can be formally defined for aII #. Note. however, that 1Z1( P) dehnes the value of an active firm only over the intelwal ( Pg x), and P4)(#) defines the value of an idle hrm only over ((). J5/). Therefore over valtte of the range ( Pt. Pu ), we can interpret G (#) as the firm's lcremental worth active, much the active rather than that is, how in becoming more it is ,
,
'The proofs arc lengthy and not in themsclves interesting, so we omit them. Intercsted rcaders shlluld consult Dixit ( l98t?a, Appendix A).
s-t.'rt.rlr?r,vltr
inactive state.
.,4Fim' :$'Decisions
220
Elltty ff-tt Lay- Up. tl/lt/ Scrappillg
For small values of #, the dominant term in G(#) is the one with the
negative power pj of P,' it is decreasing and convex. For large values of #, the dominant term is the one with the power pb > 1,'this term is negative. decreasing, and concave. For intermediate values, the third term contributes to the increasing portion of G(#). T'hus the general form of GP4 is as shown in Figure 7.1. The boundafy conditions that apply at the thresholds can be written terms of GP). The value-matching conditions (9)and (11)become G( Plfj
while the smooth-pasting
1,
=
conditions
G'( Pu )
=
0,
G /2 )
=
-
E
in
=
.
0.
Refering to Figure 7.1, these conditions imply that the graph of G #) should range from Pg to #/./, and should be tangential to the l at the upper end, and tangential to the horizontal end.3 Note that G(#) is concave at Pu and convex -
.
G(8)
I
-E
l I
Ps
I
#
)r
P//, and using the boundary conditions
=
tind -r l + #//
C
-
=
-
1 o. 2 G8'( #j/ ) 2
that must hold
0,
>
1.1.
P
1
PL
-
This is of course :1 manifestation of the option value of the status quo that we discussed in Chapters 5 and 6.4 We will discuss its implications below, after we have examined its quantitative signiscance in numerical simulations.
1.C
Comparative Statics
Although the equations defining thd thresholds are highly nonlinear and do not have closcd-form solutions. the total differcntials corresponding to small changes in exogenous paramcters are, as usual, Iinear. This makes it rclatively straightfomard to obtain qualitative comparative statics results for at Ieast some parameters. On thc other hand, scveral paramcters of interest, notably a. and fT, enter into the quadratic equation whose roots are the p. r. ( and pa,so changus in these parameters can have complicated effects pl powers function G. This makes thc analytical comparative static expressions the on diflicult to interprct, and we must resort to numerical simulations. The other parameters, namely, J, E, and C, have simpler effects, and serve to illustrate the general method. We consider the investment cost I in detail; the other two are similar. Working with te function G rcmains uscful. and it helps to show its dependence on the option value coefficients. Thus we write G( #. zll Bz). The value-matching and smooth-pasting conditions are =
I
0
Evaluating this at lhk. we
< C ?' E. ln other Similarly at the other end, we have or Pu C + words. the optimal tresholds with rational expectations are spre' fartht/ apart than the Marshallian ones with static expectations. When inactive hrrjs take into account the uncertainty over future prices, they are more reluctant to invest, and if they are already active, thy are more reluctant to abandon. j
have an S shape over the horizontal line at height line at E at the lower at PL
Pc
,
>
(10)and (12)can be written G?(PL )
Now consider the upper threshold. Subtracting the differential equation P'()( #) from equation (5)for lz-l( P4 we have
(3) tbr
-
.
Figure 7.1.
Dtlfcrrn/lclfprl ofthe Frey/ltp/t'
Pt und /5/
G( Pli
G ( J$/ :
.'This can be made into a gcometric method for computing the thresholds, given good interactive graphics softwarc. Supcrimpose the horizontal lines /. E on the graph of G( #). and adjust the coefficients z11and % until you get the tangencies. The abscissas of the tangency points are the optimal thresholds Plt. &.
B2)
zd .
l
.
.,4
.
l
,
=
Bz)
G( Pl-
/, =
zd
.
0,
G p ( PI-
l
,
Sa) ,z1l
,
,
=
Bz)
E
-
=
.
(14)
0.
-
some macroccontlmists may lind it expectatitlns implies more incrtia. ntlt less.
a pleasing irony that going from static to rational
$
zdFirm :' Decisions Now suppose that / changes by (//, and consider how the four dogenous variables /1 l Bz, Pl., and Pll respond. Begin by differentiating the value-matching conditions ( l4) totally. Denote the partial derivatives of G by subscripts as usual, and write Gzf( Pll. A l Sa) = G.4(S), etc., for brevity. We get ,
,
G.4(1,) (IA l + G
dB1
3(1)
0.
=
zlA l
=
Pg* #// L
-
where L
which is positive because Pu
=
>
.,
d 52
.
Pup
,
PL
and
j.
=
#2' d //
-
ph ##l
-
tt
p,
>
t
() >
,
#a.
at /5/ in =
(15)to write
0,
which yields -
(pbf'sp'
ppz
-1
g
Since G P) is concave at #//, Gpp(H)
dl > 0. The investment threshold rises expect. Similarly, Pg falls as E rises.
-
pz s/h-l s
) # PL
=
-(Jl
-
pp, t
j d; j
ks
.
is negative and then dPu > 0 when with the investment cost, as we should
Similarly, the lower smooth-pasting condition G p#(
+#a-I
#2) PL/f!
again should the price process ttlrn stllliciently I'avllrable in tlle ftlturc. Tllerefore, the larger is the investlnent cost. tbe larger is this option valuc antl thtt greater is the reluctance to abandon. The nlirror inlltge restllt. namely. that the investment threshold Ptl rises as tlle abantlonnlent cost E increases. is perhaps even clearer. The tirl'nis nlore reluctant to undertake the prlject if it might have to incur a Iarger cost to shut it down in the ftlturt!.
If we evaluate these conlp:trative static derivatives as / antl lL2 both go to zero, %ve find P:l P:- both go to C. btlt tl P// /t/ / cx'aand / 1':./f/ / and lik-ewise with respect to E'- Thus the entry and exit thresholds begin to spread apart very rapidly even for small costs of entq llnd exit. Exp:tnding thtl four equations of the set ( i4) and ( 15) in Tayior series and carrying throtlgh some tedious algebra, Dixit' ( 199 l a) finds that --v
-cx3.
.
lOg(
' /3 l:'/// PL ) = 16* (/ + E)
.
where k is a constant. ln other wtlrds, entr.y and exit costs tllat nre smal I and ()1* third order (proportionalto 63 where 6 is small) produce :1 gap bdtween the entry and exit thresholds that is of lirst order (proportillnalto 6 ). Thtls vttry small sunk costs huve a disproportionately large ekct on the hrm-s dccisions. ()f
smal I costs on the thresholds is entircly sym-
metric: the entry threshold is affectetl by the exit cost just as strongly as l7y the entry cost. For Iarger magnitudcs. each type (.)f cost will affect its threshold more strongly. The reason is intuitively clear. Cilnsidcr :1 firm contemplating entry. Tht: entry cost must be paid immedilktely, while the cxit cost affects the firm's entry decision only through the prospect that it will havd to pay that some time in the future. Because ()f discounting. the immediate effect is the stronger ()ne. Howcver. if the ctlsts are vttr.y small, the thresholds are very close togethcr, and the Brownian motion is almost surc to reucll the other threshold very quick-ly.Therefore the diflkrence madc by discounting is small, and vanishes in the limit. S*()wn-'
G pP S) d f5/ + G ,zl (S) d,'l 1 + G #s( S) dBz
=
.$ft.'rtl////l?
Morcover, the cffect
.
Now differentiate the smooth-pasting condition
G />1, (S) J &/
l
..--...
-
,
atlt
.
Note that the terms Gp(H) J#s and GpL) J#/. have vanished because of the smooth-pasting conditions (lsl.Therefore the general comparative staticsystem in the four endogenous changes dA 1, dBz, J#&, and #J5f in fact separates into a simpler system. First we solve the above two equations for the changes in the option value coefficients 6lA 1, dBz. Then we can totally differentiate the smooth-pasting conditions to gei te changes in the thresholds dpu, t//N. /'' e tc the solution is Noting that Grt ( S) 1% =
E-llttl'. h/, Ltlv- L/JA
gives cjj
ja
.
Since Gppl-, > 0, we have dPL < 0 when dl > 0: the abandonment threshold falls when the investment cost rises. This important interaction between the costs and thresholds should also be intuitive upon renection. The 5rm abandons an ongoingprojectwith some reluctance because of its optionvalue. By keeping the project alive, it avoids having to incur the investment cost once
We leave it as an exercise for the readcr to vcrify thrtt botll IL; and P;. rise as C rises: as one would expect, a prllject with higher operating cost is undertaken more reluctantly and abandoned sooner. 1.D
An Example: Entry and Exit in the Copper Industry
We have seen that uncertainty
over future demand conditions
increases the
firm's zone of inaction; thlt is, t causes the optimal investment and abandonment thresholds to be spread farther apart than the lraditillnal Marshallian ones. How much larger does this zone of inaction become in practice? Is it really necessary to account lbr irreversibility and unccrtainty as we have, t)r
zl f'*il-t't might the simple Marshallian rules provide a good enough most investment and abandonment decisions'? To answer is useful to look at a specihc example. We
will
examine
'-
llecisiolls
approximation
for these questions. it
EllIl'.
Fzit. Ltlv-Up. tl?7f/
tscrappbq
2 25
variable cost at C $0.80per pound, abtlut the average for U.S. producers in 1992.but we will also vary this cost to determine its impact on the ent:y and exit thresholds. (Ft)r comparison, the average price ofcopper was about $1 in 1992 but twer the 1985-1992 period it fell to as low as $0.60per pound a-nd rose to over $ 1.50 per pound-) =
.00
the decision to invest in a new copper
production
facility a combined mine. smeiter, and rehnery and the decision to permanently abandon a facility that is currently operating. The price of copper has historically been quite volatile. (Tl1e standard deviation of annual percentage changes in the price of copper has been 20 to 50 percent over the past two decades.) In addition, opening or closing a mine or smelter involves large sunk costs, so that copper producers need to make these entry and exit decisions very carefully, taking uncertainty into account.
In reality. a producer with an operating copper mine has an alternative option to permanent abandonment or continued operation. A copper mine and Iater reactivated should the price rise. can be temporarily Mothballing and reactivation involve sunk costs (constructionis needed to prevent the mine from llooding or caving in while it is inactive. and an additional expenditure is needed to reactivate the mine). as well as an ongoing fixed cost (to pump out water, prevent unauthorizcd entry, etc-). However, if reactivation in the not-too-distant future is Iikely, this can still be cheaper than abandoning and later building a new mine from scratch. In the next section. we will expand our basic model of entry and exit to include the possibilities of mothballing and reactivation. For the time being. however, we will ignore Likethis additional option and consider only investment and abandonment.s refinery, mine could shut but for down a but not a wise, a producer open or t)f refined will the intcgratcd production simplicity we copper as one treat ttmothballed,g'
operation. We will consider a facility that produces l0 million pllunds of relined copper per year. To keep the analysis simple, we will ignore the fact that the mine's reserves are limited and will eventually run out; we will assumd instead that the mine can operate forever. (This is not too extreme an assumption, since most copper mines can operate for at Ieast 2() or 30 years.) A reasonable number for the cost of building such a mine, smclter. and refinery is I $20 million, and for the cost of abandonment (largelyfor cleanup and environmental restoration) is E $2 million. (These and all other numbers are in 1992constant dollars.) Average variable cost of production varies across srms in the United States, and even more so across different countries. We will set =
=
,
for the real risk-adjusted annual rate of return for a copper mine or refinery is /.1 = 0.06, for the average rate of convenience yield (or return shortfall) is J Jz a 0.04, and for the real risk-free interest rate is r 0.04. Finally, we will take 0.2 as a base value for the volatility parameter, fy,but we will also consider values of that parameter in the range of 0.1 to 0.4, consistent with estimates that differ in different periods of time-t' Given these parametervalues, equations (9)to ( 12) can be solved numericallyfor the constants z1l and Bz and the entry and exit thresholds PH and P:. Figure 7.2 shows tke critical entry and exit thresholds. P$t and #/.. as functions 0.2, these thresholds are of the volatility parameter o'. Observe that for fr about $1.35 and $0.55,respectively. For comparison, if there were no uncertainty over tkture prices (c ()). the thresholds would be $0.88 and $0.79.7 Hence a very moderate amount of unccrtainty causes the zone of inaction ().55 from 0.88 0.79 $0.()9to l $().8(). to increase dramatically Observe that this zllne increases as o' increases; if o. is 0.4. the width of this zone is about $ l Figures 7.3 and 7.4 show the dependence of the entry and exit thresholds the operating cost, C. and on the sunk cost of exiting. E Observe that as on operating cost increases, both Pu and PL increase. A highcr operating the reduces expected tlow of profit from. and hence the value ot the the cost that project, so a higher pricc is required before the firm is willing to invest. In addition. thc firm will abandon at lt higher threshold price. because it will lose more money when C is higher. A reasonable
value
=
=
-
=
.
=
=
.35
=
-
=
-
.3().
.
.
f'See.for examplc. Bodie and Rosanski ( 198(3)and Brcnnan ( 199I ) rcgarding rz The use t)f ts (1.(34for thc rate of ctlnvenienctt yield is close t() its avcrage value twer the past two decades, but tlne shtluld keep in mind that this paramctcr l3as fluctuated widcly ovcr timu. There have been sustained periods when it has been close to zero, and shorter periods (when the total stock ()f inventories has been l()w) during which it has been as high as 3() f)r 4() percent per year. We havc assumcd a constant to simplify the analysis. For discussions of conveniencc yield and its behavillr, in general and for copper. see Brennan ( 199 l ) and Pindyck ( l993c.d). 7sincc the prtlject produces l () million pounds of copper per year. the NPV of investing 8/1).4)4.tThe rcvenue llow is discounted + 1() #/().04 (expressed in millions t)f dollars) is 0.t)2. and the operating cost is discounted at at rate Jz (9.1)6 but is expected t() grow at rate (), the tirm should invest if this NPV > 0, that is, if P > $0.88. the riskless rate 1, ().04.) If fr .
=
f
-2()
-
=
=
=
5Brennan and Schwartz ( 1985) use cuntingent claims methods focus on the options to mothball and later
reactivitte
the mink!.
t() value
a copper minev and
=
Likewise. once the firm is in the market. the NPV of exiting is -2 positive whcn /4 < $().79.
-
f-'/0.()4 + 8/0.()4, which is
t!
.,4F irm Decisiolls '.$'
1.8
1.6
1.6
1.4
Ps
1.4 W
1
.2
u 1 0.
.0
c
r c (Q.
Elltl'. E'a/, Ltlb'-U/7.(111t1 A-crfl//J?/l
1.2
Ps
c = 1.0 o o-
c. 0.8
C
J
o 0.6
(D 0.6 Q
Q
pL
0.4
0.2
0.2 0.O
0.10
0.15
0.20
Figure 7.2.
0.25
0.30
0.35
Cfppptzn'E rl/ry unil Exl Thrcxllolilx (1.$. F Ilzl(.'i#vvl.s'
())'
=
=
=
0.2
O.3
0.4 O.5 0.6 O.7 C (Dollars per pound)
0.8
0.9
1
.0
Fiqure 7.3.
/'z
=
=
=
=
0.0 0.1
1.6 1.4 Ps
.2
K
1
o 1
.0
c.
c. 0.8 o 0.6 Q
Pz
0.4 0.2
=
=
rz'1 K)+
0.0
0.40
Obselwe that when the abandonment cost E increases. the entry threshalso increases. The reason is that tbr any price P, a higher E reduces Pu old value of the option to abandon an active jroject, and hence reduces the the of value the project, which in turn implies that the price must be higher before the 5rm is willing to invest in the hrst place. Likewise, an increase in E reduces the abandonment threshold P:.,.the (irm must pay more to exercise its abandonment option, so the price must fall more before it is willing to abandon. Observe, however, that while #/f rises and Pg falls when E increases. they do not rise and fall by very much. The reason is that the value of the option to abandon is largely determined by tz and by the much larger entry cost 1, and does not change very much as E is varied. Figure 7.5 shows the value of an idle Iirm, F(,4#), and the value of an active firm, F1 (#), both as functions of the price #. (We have used the base case parametervalues'. l $20 million, E $2 million, C $0.80 per pound, 0.04, and o' 0.2.) Also shown are the thresholds Pu and PL.. Note r Pg, fG(#) exceeds 1zrl( P) by the abandonment 2, since cost E that at # option. giving up at that price it is optimal to exercise the abandonment E + Pr1 and receiving 1r). Likewise, at P Pti it is optimal to invest, so =
Pt
0.4
1.
O.O
-5
0
5
s (hzlillitlrl (1()1111rs )
1O
15
228
z4Firm
250
I l 1 1
2O0
K'
l
t
I I
I
l
I
l
1 l
t/1 (8)
I I
I
1
1 I l l l
I I
l
t/o(8)
50
I 1
O
0.0
0.2
0.4 Figure 7.5.
'
1
I I
I
I 1
Pv
0.6 0.8 1 P (Dollars per pound) .0
P'ljf P # und
'l
( P 3 u.
F'l1?!ctt)?I.s
1
.2
JL 1
.4
1.6 Fkure 7.6.
flJP
Finally, Figure 7.6 shows the function G( Pj 1z'1 F()( Pj. Note the ( #) S shape of this curve in the region of inaction between Pt. and Plt, and the tangency with the horizontal line at / at # Pu, and with the horizontal line at E at P JN This example can help us understand the behaviorof copper producers in the United States and elsewherc during the past two decades. During periods of very Iow prices (forexample. in the mid-1980s, when copper prices had fallen to their lowest levels in real terms since the Great Depression), lirms often continued to operate unprohtable mines and smelters that had been opened when prices were high. At other times when prices were high, hrms failed to invest in new mines or reopen seemingly profitable ones that had been closed when prices were low. This response of producers to uncertainty had a feedback effect on the price Ievel itself. The reluctance of lirms to close down mines during the mid-1980s when demand was weak allowed the price of copper to fall even more than it would have otherwise. We have assumed here that the price process of copper is exogenous to the srm.In Chapter 8 we will see how the price of a competitively produced commodity can be made endogenous in an equilibrium model of industry =
=
-
=
s'crtk/p/aaq
I
!
10O
W/?) k/. Lav- U/?,(111t1
Decixiolu-
I l
j
1so &-
''
-
G( P )
=
1.-1( P )
1. ( P ) '4,
-
behavior. At that point we will ruturn to this example of entry and exit in the copper industry. but in thu context of :& competitive equilibrium where price is endogenous.
.
2
Lay-up, Reactivation,
As mentioned above. a copper
and Scrapping producer has other options besides perma-
nently abandoning an operating mine when the price of copper falls. Instead, the mine could be put into a state ()f temporary suspension, allowing it to be reactivated in the future at a sunk cost much less than the cost of building a new mine from scratch. Plants that are or ships that are up'' are examples of tis state of temporary suspension. '4mothballed''
''laid
Mothballing, Iike permanent abandonment, requires a sunk cost, which will by maintaining the denote in addition, E.u. is plant mothballed, once a we capital requires a cost flow /W.The operation can be reactivated in the future
at a further sunk cost R. Mothballing only makes sense if the maintenance cost ij'I is less than te cost C of actual operation, and if the reactivation
4 F.r/l? llccisiolts -.
cost R is Iess than the cost of fresh investment l ; we will assume that these conditions are indeed met. Our objective is t() determine how the value of an operating project, the value of the opportunity to invest in such a prtlject. and the decision rles for investment. mothballing. reactivation, and scrapping are affected by the various costs 1. Est, /W,and S, as well as the volatility of the output price. As before, we will assume that the price follows the geometric Brownmotion ian ( l). The hrm must decide whether and when a plant should be mothballed, taking into account this uncertainty over future prices. Intuition
suggests the fbllowing general scheme. Starting from a state in which it does not have any kind of capital installeda the tirm will make the investment if the price rises to a threshold Pu. The firm will mothball an operating project if the price falls to another threshold Pjt. Given a project in mothballs. the lirm will reactivate it if the price rises to yet a third threshold PR. Since the cost of reactivation is Iess than that of investing from scratcha we expect Pl < f5/ .
If instead the price falls. making reactivation a sufficiently unlikely or remote event, there is a fourth threshold 1%at which the mothballed project will be scrapped altogether to save on maintenance cost. Then the firm will revert to
the original idle state.
Of course all these thresholds P/?, C/, PR, and P.vare endogenous. and must be determined in terms of the basic paralndters. Even more fundamentally, we must ask if the lirm will find it optimal to use the mothballing option at all. If the maintenance cost M is sufciently high. or the reactivation cost R not sufsciently Iess than the full investment cost /, then the tirm might tind it better to scrap an operating project directly if the price hits a Iower threshold PL.; in that case we are back to the model of the previous section. We must determine endogenously whether mothballing figures in thc firm's optimal
Similar consitlerations apply to tlle investment cost l In prillciplc- one imagine installing a project in the n'ltltllballtld state at 11.cost s:ty. and can activating this it later at a cost /?. I-lovvever.we see Iittle reastln then cheaper ltn tllan silnply operating be investing in indirect route shotlld ever project. I-lence the firnl :vi 11never Iind it optinl:ll to take t llis rtltlte-. it &vi11 never invest into a mothballed projcct. By postponing investnlent tlntil tlle instant of operation. it can d elay spending the first tranche J of capital costand save on the llow 54 of maintenance cost.s Th is Ieaves us onc less switch to consider anlong the possible six ssvitches five switches that are conceivable acrtlss the three states. Of tlle I'enlaining idle to live. live to motllballctl. nlllthballed to scrltp. lnothb:ll Ied to Iivcpand live to scrap the first lur are usetl if nlothbltlling s acttlally a part 01' the optilual strategy. Otherwise only the Iirst and thtl last artl tlsed. We proceed for a while on the assumption that when price falls t() :1 ctlrtain point. mothballing is tlsed. and then determine its Iimits 01* vltl idity dtlring thc ctpurstt (1f the analysisWe will contintlc t() denote the idle and tlperating states l)y the Iabels Iabel ??l for the mllth0 and I rcspectively- and we introdtlce the :lddititlnal balled state. We lind the vllue ()1* the lirm in each state as the apprtlpriatc combinations ()f the expected profit ()r ctlst strcams ltnd thtl (lptions tt) switch. The method is exactly the same as that tlsed earler in this chapter, and in Chaptcrs 5 and ('). s() we wi lI sketch out t htt analysis antl om it m any of the .
-l-
'wlly
,
details. The firm can btl in thtl idle state over the intervlll ((). again givcn by equation (4) alxlv(.!:
strategy. 2.A
when done via the stttge ol' mtltllball illg. Cllsts of preparing a sllip and luovillg it to and frol'n the Iay-tlp Iocation n'llty llavtl to btl inctlrred tvvice-but there may be solne saving in Iktbor Iiring ctlsts if tlle Inllre gr:ltlultl route al losvs tlle Iabor tbrce reduction to btl ach ievetl lnyreti rdnaent or qtlits. NVeleave it to the reader to exanline the issues raised I)y stlch nonatltlitivity of cllsts.
its value is once
ILl) t)f prices. Tllen
Rules for Optimal Switches
In the previous section we denoted the cost of abandoning a Iive project by E Now we are letting Eu be the cost of mothballing an operating project, and Es be the cost of scrapping a project already in mothballs. (ln the case tf a mine, the former may be the cost of hring the miners and the latter the cost of site restoration. With a ship, the latter may be negative, representing the scrap valtle.) To keep the exposition simple. we will assume that E,u + Es E, so the cost of directly abandoning an active project is just the sum of the costs of mothballing it rst and then abandoning the mothballed project. In practice, going from an operational project to total scrapping may be more or less costly .
is a constant to bc determined. This is just thc value of the option whcre We invest. have as tlsual eliminated thd term in the negativc power pz by to the fact that Jr)( P) must go t() zero as 13goes t() zero. using z41
=
'iI n an ()Iigtlplll istic ind ust r.y t he re m:ly bt2 strategic rtrasl )ns l'tr ht )ld ing a pr( jcct in n1()t 11 but we d() n()t ctlnsidcr (hat markct structu rc hcrc. balls. -
zz1F'1,lll s Decisiolts '
232
-
Similarly, the operating state can prevail over the interval ( P./. (6) above: the
(x)ls
with
value of the firm again given by equation
1,'1 ( P)
B1 fM2 + P/5
=
where the constant Bz remains to be determined. In the model developed in the previous section, Bz J':2 represented the value of the option to abandon. Now this term is the value of the option to mothball. As before. the other two terms give the expected present value of continuing operations forever. Of course' the mothballing option derives its value from further possibilities of reactivation or scrapping. The mothballed state can continue over some range of prices ( #., PR). Since neitherzero nor insnity is included in this range.we cannot eliminate either the positive or the negative power in the option value part ofthe solution. Therefore the value of the mothballed project is given by
b' ( #) ?ll
=
J-'/31 + Dz P*
DL
M/r.
-
where the constants D! and Da remain to be determined. The first term in equation (17)is the value of the option to reactivate the mothballcd project. The second term is the value of the option to scrap the project. Finaily, the last term is simply the capitalized maintenance cost. assuming the project remains in the mothballed state forever. value-matching and At each switching point, we have appropriate smooth-pasting conditions. For the original investment. these conditions
are
Z()(PH )
Zl ( PlI
=
the conditions
For mothballing,
Z1( P5f)
I'
=
( P'u )
)
-
Z(;( PI1 )
1.
ZI'
=
( Pli )
E AJ
VI( P.u)
.
=
( P.bf) Z,J?
.
For reactivation, Vm( PR
)
=
( P8 )
1Z1
-
1z',;, ( PR )
R-
=
sense. and thereby determine the Iimits on the range mothballing option is actually used.
Z1'(PR )
-
D l J#'R +
(Bz
Dal Pgtk +
-
-#1 D1 PRyl
l
-
+
/32( s 2
PR/6
-
E 5.
(C ) U)J,
.
-#1 Dl Pu/3.-1+ pz ( s 2
lz't( lb' )
.
pp?-
l
+ j/j
R
S,
=
.() .
o2 )
-
=
,
-
-
investment-abandonment system, each expressed as a fgnction of thc lumpsum and llow costs. Remember our comparative static results: both functions H and 1 are increasing in the flow cost argument
PR
=
H R
.
EM
,
C
A/v
'/) -
.
C. the function H is in-
L (R E
=
,
,v
C
,
-
M4
.
Turn now to the remaining four equations in the eight-equation system above.Thevalue-matching and smooth-pastingconditions fornew investment are familiar:
- A 1 l3'+ -p3 Wl Pupb
# c ,/12 + # // /J s
/$ s 2
+
-
pp-t + u
c/r jyj
1,
=
.().
(22) (23)
.
This system of eight equations determines the four thresholds Pu, J''./, Pe, Ps and the four option value coefhcients zdl B?, D1 Da. Wc can solvc the equations formally, but then we must ask whetherxthe solution makes economic ,
/W)/r
-
ptk-3 + j/j (). (c1) st regard of We can this as system four equations in the four unknowns D1 (S2 Dz), PR, and Pu, and solve it on its own. Furthermore. the system has exactly the same form as the one we worked with in the previous section for the case of investment and abandonment without the mothballing option: compare equations ( 18) to (21) with equations (9) to ( 12) above. We need only reinterpret R as the cost of investment, Est as the cost of abandonment. and (C Ms as the cost of operation-Then PR is like the investment tizrcshotd from the previous section. and #,v is like the abandonment threshold. Tocontinuewith the analogy to the earlier model. let usdeline H( 1. E. Cj as the upper threshold and Ll. E. C) as the lower threshold that solve the
')
=
(C
-
oz)
-
-1
li ('Ps)
the
(20)
J's
=
of parameters where
and reactivation. Using the functional tbrmsabove, two thresholds become
Finally. for scrapping, Z,,,( C )
233
creasing in the lump-sum cost arguments I and E, while the function 1 is decreasing in these two. Now the reactivation and mothballing thresholds of our present model can be written in the form
.
are -
EW/, Ltly- Up. tll?# Scrapping
The most promising starting point is the interaction between mothballing the four equations at these
C/?',
-
fl/r-M
Those conditions
at the scrapping threshold become
( D?
zzj
-
,
p l ( Dl
-
2
) Pvpt +
A I ) Psp
o2
pp?
+
s,
.-/vyr
.-
s
-1
pz o2
.
p/32-1 s
=
() .
(24) (25)
234 tlle tbresllolds //, /3$.and tllll ctleffiThese equations hrtve six unknowns z.1l z41 and the soltltion to the Iirst group of D?. Howevera Bj, D ). ( cients 1w() the cotlflicients; w know relations equations abtlve among us lbur gave vzll complete the sol ution. and )a fore Therc Bz can we + Dl D! ( ) ( ) This system is too complicated to grasp analytically. so wtl will present the nature of the some numerical simtllations that provide more insight into of the solution. properties general inttlition suggcsts However, some solution. mothballing tantamount is to costless and then both R First, if M zero. are 1 model ()f 985) McDonald-siegel basic back the and to we are ( suspension, Chapter 6. Then the mothballing and reactivation triggers both converge to C, and the scrapping trigger collapses to zero. Now consider raising R and 5'I gradually, one at a time. When R is raised holding NI constant. the reactivation threshold Pll will rise and the mothballing threshold P5f will fall. just as the investment and abandonment thresholds did earlier in this chapter when thc investment cost increased. The threshold Pll for new investment will rise: when reactivatfon is more costly. the option of mothballing is less useful. and therefore the lirm ttlso rise: is more reltlctant to invest. Finally, the scrappng threshold Ih. will Iaid-up project will h()Id not on to a when reactivation is more costly, the firm as willingly when the price falls. J./ falls and the As we keep on increasing R, the mothballing threshold (.)1' lhe (lptimal scrapping threshold P.s rises. Fllr muthballing to be a part where these twl) strategy, we must have P5t > Jt$..Thercfore the value of R t)f the parameter space where moththresholds meet delines the boundary balling ceases to be relevant. Write F(- for the common value of the two at this boundary. Adding the value-matching conditions (2())and (24)satisfied by the common Pc. and likewise for the smooth-pasting conditions (2l ) and -
,
./11
=
-
-
.
(25), we find
',
There wi 11also be :1 tratle-off between tlle crit itlaI values of 1?:tl1(.l ) l t Illtt I'i de ne the botlndaries of t he tlse 01' nlothbal !ing. Nvlltrn S is large r, tlle cri t ictl vaiue of NI will be smaller, and vice ve rsa. 2.B
Numerical
Results
Now we turn to numerical solutions to verify these intuitions. Tlle paranActers cost 51 and the rcactivation cost R. X) focus on them, we will asstlmtl for the rest of tllis exposition tllat the lay-up and scrapping costs E and f/ are l7oth ztl ro-, t hen so is thei r stlnlthe cost ()f direct abandonment. E. Shortly wtl wil l present another ntlnlerical example that illustratcs the effkcts 017 nlltk' ing tllese parktmeters nonzero. We normalize to C l We asstlme a risk-netltral (irm. with 1().(5. (),2. Then pz = ;The price process has a ().t)5. and (5 ()ltnd f'z pt 0.()5.The Itlmp-sum cost ()t' investing is l 2. antl there is no Itlnlp-sum cost of disinvesting. so E (). Witll these nunlbersa lnd ignoring the ptlssibility of mothball ing. the investment and abltndtlnment threshtllds turn out to be ().7 I 35. f5/ l a nd J.
of greatest intercst are the 9tlw maintcnance ./
=
..
=
.
=
=
=
=
-(x
=
=
=
.5977
=
=
Now allow mllthballing. and consider twt) cases, t%1 ().()I and 54 ().45. Ftr each. we consider a range ()f val ucs ()1' S. Tlle rcsu It ing val ues ()f the four thresholds arc shown in Table 7. 1 The eftkcts ()f vktrying R for fixetl &'fcttn I7e seen separately in each casc detailed therc. In Case I thc maintenance cost is =
=
.
,
low; Nl = ().() l Now mothballing is used tlvcr some range of prices until thll critical Iimit of R ::::: 1 once R excceds this Rs mothballing is never used. rises R As over the range 1) to R, ( l ) hf rises to its ievei in the absence ()f mothbalIing, (2) l>.bttkllsand P. rises unti 1,whe n /? R, the twl) mee t at PL (3) PR rises, only to bttcome irrclevant oncc R reaches R. In Case 2. the maintenance cost is higher'. NI ().05- Now mothballing relevant is over a shorter range; R is only a little largor than 1 The general pattorn is easier to see from Figure 7.7. Fixing 54 at a relatively low value. the various thresholds are shown as functillns of R by thc thicker culwes. Mothballing is part of the optimal strategy when S < R. Nvhen R A: R, the two thresholds P.f and Ps merge into Pl. Svhen a higer vaIue of IV is considered. the curves shift to the positions shown by the thinner lines. The Pu curve sh i s down, the Ih. cu rve sh ifts u p, and thc two meet at .
,76,.
zll Pct' + Bz Jtt: + Pc/
-p3
live prljtlct Iess roatlilys antl reactivate :1 nlot Ilballed projet)t I'ntlrd reatlily. &vi l-losvttve r, 1..'11a nd Ih. Il rise t he li rfn Nvill l'tl 1114.)re rklltl tlt:.k 1)t to iIlvcst lt t lt I1. wi Inot hbrllled prlject nlore reatl ily. Once ngain :1 1':11 and Il scrap a Iing I''bt:1 !ld rising l)v wi lI nleet svhen 54 rises to a crit icaI levc1.'tbr al'ly Iliglle vltltltls :1 r of 51- mgthball ing vil l not be uscd.
-d1
Pc(,
-1
+
pz s 2 ppz-' c
-
C/
,'
+ jyj
=
=
-
( E st + Esj
=
-
E
.
=
().
These are exactly the abandonment equations ( l 1) and ( l 2) that. together with the corresponding pair for investment (9) and ( 10), were satisfied by the thresholds Pu and Pi. earlier in this chapter when mothballing was not available at all. Thus the whole story fits together as it shotlld. For high enough values of R, the firm ignores the possibility of mothballing and switches optimally between idle and active states as before. Next hold R hxed and raise M. This reduces (C M), the flow cost saving from mothballing. Therefore both Pa and Pu fall; the firm will mothball :1 -
a lower value W.The Pu curve shifts up to reach its hnal constant Ievel at this new lower W.The PR curve shifts down, only to end when mothballing ceases to be used. Note also that as R increases starting at zero, the restarting and mothballing thresholds spread apart very rapidly. Since we have set the 0, the sum of the costs of the pair of switches, namely, mothballing cost Eu =
R + Eu, is small, andwe have an instance of the cube root formula (16)above. One further numerical experimeht is of interest, namely, making both R and M small to drive the mothballing model to the limit of otlr model in Chapter 6. This does happen, but the approach to the limit is very slow, in
/7zJ
//?l$$;f tll
.6;t?-tl/7/7f?l.t;
kecping with the general insight that even small sunk costs matter a great deal when thcre is ongoing uncertainty. Thus. keeping C l etc.. as above. even when we reduce the costs associatcd with mothballing t() /W ().()()l and P,u ().9 l9, each about l() perccnt 1 and R ().02, we find PR value. namely l The scrapping threshold from Iimiting their common away becomes C 0.0963, again significantly above its limit of zero. =
,
=
.4)89
=
=
=
.
=
2.C
Example: Building, Mothballing,
and Scrapping OiI Tankers
The numerical solutions presented above help to illustrate the qualitative dependence of the optimal thresholds on the various cost parameters. However, it is useful to also examine optimal investment. mothballing, reactivation, and scrapping decisions for a real-world example. As with our copper industry example, this can give us a better appreciation for the importance of sunk costs and uncertainty, and aso show how the mode! can be applied in practice. We will apply our model to the oi1 tanker industry. Oi1 tankers provide particularly good example since the potential or actual owners of tankers a face considerable profit uncertainty, as well as substantial sunk costs. The uncertainty arises because the market for oiI tankers is very competitive, and tanker rates (therevenue per day for the use of a tanker. that is, the price #
Decisions
L-lltt''
in our model) lluctuatc considerably as oil prices tluctuate, as the geographical distribution of oiI production and consumption change. and as the supply of tankers changes. Also, sunk costs are important because of the considrable expense of building a new tanker and maintaining or reactivating one that is mothballed.
11
.,z1Finn
tankers.with capaities Tllere are fourgeneral sizes of oil tankers-small capacities around tankers,with medium (dWt), 3s.ooodeadweight tons around and, since the with around dwt, large capacities 140,000 dwt, tankers, 85,000 with of about capacities Crude (VLCC's). Large Carriers'' mid-1970s, and other and dwt. operating, Revenue construction, rates costs do 270,000 economics of with investment, increase linearly tanker capacity the not so mothballing, etc., will vary across these different categories of tankers. We will focus on one particular category-medium tankers with an 85,000-dwt Ktvely
capacity.g
The average cost of building a new tanker with an 85,00()-dwt capacity is about I $40 million. (All costs and revenues are expressed in 1992 dollars.) The one-time cost of mothballing a tanker is Eu $200,000, the cost of million mothballed tanker is E.v scrapping a (thatis, the tanker has a reactivating mothballed value), tanker of this and the cost of positive scrap a mothballed maintaining annual tanker is R cost of $790. 000. The size is a and operating M annual 1992 labor costs. the $515. 000. Finally, given fuel cost for this tanker is C = $4.4 million. ln 1992, this tanker would earn a gross revenue of about P $7.3 million per year. We have assumed that P followsa geometric Brownian motion. and the drift (z and volatility c for that process can be estimated from the sample mean and sample variance of an actual time series for gross revenue. Using quarterly data for 1980 through 0.15. Finally, we use a value of 0.05 mid-1992,we found that t:r 0 and c risk-free and the risk-adjusted rate #t (sothat the real interest rate r, for both 10 = 0.05.) Figure 7.8 shows the critical thresholds #s, PR, 8w, and Ps as functions of the one-time reactivation cost R. Note that for our base value of $790,000,
'aic
I .tIv-
t.//
tl//t
tb-clzqlpitlg
l
PH
10
l l
9
j
I I 1
W 8
=(3 c
7
I
I I
= 6
u: =
>
5
Q.
4
l
Pa
I I I 1 I
P.
3
I I
Ps
2
l
=
=
1 O.0
0.5
1
.0
1
2.0
.5
2.5
3.0
-$3.4
=
3.5
R (Million Dollars)
=
Izigure 7.8.
C*r#CfJ/
TllrL.'.b'ltt
?/f
l
tlN
/;il?tt'/f 111.%( )/' Ictlcli
'tl/tl/l
C).%I
/f
=
=
=
=
gFor a general introduction to the oiI tanker industry, see Rawlinson and Porter (1986). Goncalves (1992)has also used contingent claims rnethods to examine optimal investment decisions for tankers, as well as thc relation between spot and long-term contract prices. 'BWeobtained timeseriesdata for revenue,costs, and other industryvariables from Marsoft. Inc., of Boston. Our thanks to Dr. Arlie G. Stcrling, the Prtsident of Marsoft. for making this data availableand giving us advice on a variety of economic issues related to this industry. Thanks are also due to Victor Norman and Siri Pettersen Strandenes of the Nomegian Schot)l of Economics and Business, Bergen, who also provided data and advice.
the 1992 average gross revenue of $7.3 million pcr year wlluld have been sufhciently high to reactivate a mothballed tanker, but well below the threshoId revenue of about $9.5 million per year needed to invest in a ncw tanker.
Consiptent with this result, thcre
was
indeed little
()r
no invcstment in new
tankers during 1992. Observe that PR and Ps rise as R is increased. and P5f falls, but not very fast. Mothballing remains a viable option as Iong as this ctlst of reactivation is below about $2.7 million. The investment threshold P// also rises as R rises (althoughso siowly that it is hard to discern from the graphl; a higher R reduces the value of a tanker. and therefore raises the revenue that I a 5rm must expect to receive before it is willing to invest-l
Figure 7.9 shows the optimal thresholds as functions of the one-time of cost mothballing an operating tanker, Eu. Observe that the qualitative
l1 Unlike Figurc 7.7, the curves for Pg and Pbt do nflt mct!t ()n thc vertical axis in the cubc form. becausc in drawing the earlier sgurewc assumed the cost ()1' mllthballing tt) btt zero, root which is not the case in this numerical example.
J*/777-.Silecisions .,z1
240 11
Fs
1: 18 7 c c uq
>
Pa
5
PM 3
1
0.0 -iqItnl-t;F. J'.
0.5 L2*l?/ic'ttl;r-7lz-t'./lzJ/z/.
2.0
1 1 Eu (Million Dollars) .0
6Ij(
9
.5
tp/c (n6,- ir7zzzty Lntl.t /7l/zlt7rzlzll/
rJ/!'
llltll
*
:F 7
O c =
>
*
I l I
Pa
5
Q.
Pu
I 1 I
3
1 I I 1 1
PS 2.5
1 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
.0
llitl-i;d7'1/ /pzz
dependence of the thresholds on Esf is the same as it is on R. Values of E.u above$1 million are unrealistic (recallthatour base value is $200,000), butwe have included this largerrange for illustrative pumoses. Note that mothballing remains an option as long as Eu is below $2.1 million.
Figure 7.10 shows the optimal thresholds as functions of annual cost of a mothballed tanker, M. As the Iigure makes clear, this is a critical parameter in determining whether mothballing is a viable option for the rm; note that mothballing remains an option only up to an M of about $720.000.and our base value of $515,000is not far from this value. Remember that higher values of i%1 reduce the value of a tanker and thereby raise the threshold for investment. A higher value of 2%/ also makes mothballing less desirable and therefore reduces 'r and Pm,and raises #x. Figure 7.1 1 shows the four critical thresholds as functions of the annual cost of operation, C. As with a higher value of iV, a higher operating cost reduces the value of the tanker. and thereby raises the threshold #s required for investment. As one would expect, however, changes in the operating cost have a much greater effect on the value of a tanker, and hence on J5./, than maintaining
l 1 I l I I I l I
G'
l I 1 I I
PS
I 1 I I
Ps
l I l I I I 1 1 1 I 1 I 1 I 1 1 I
A
we
11
i 1 I l
9
=
k/'r); Erc Lay- Up. tl/lt/ Swapplg
do changes in the maintenance cost M. Also. because a higher operating cost reduces the value of an operating tanker, it raises the threshold PR at which a mothballed tanker is reactivated, and raises thc thresholds P5t and PR, so that the revenue # need not fall as far before the irm is willing to mothball or scrap its tanker. Finally. Figure 7.12 shows the critical thresholds
as functions of c, the
standard deviation of annual percentage changes in the revenue #. Observe that the thresholds. and especially PtI and &, are quite sensitive to rz. As we saw earlier with our copper example. for large values of o', the zone of inaction widens considerably. Also, note that if fy is Iess than about (). 1 #.J and Ps coincide, and mothballing is not an option used by the firm. The reason is that mothballing is useful only if there is a reasonable probability of a substantial increase in revenue in the near future (so that the tanker will be reactivated). With o' < 0. l the probability of a sufhciently Iarge increase in revenue is too small to make mothballing economical, given the costs of mothballing, reactivation, and maintenance. ,
,
12
10 @8
Ps
o c
E 6
X
Jp&
I l
4
1
3.0
3.5
4.0
4.5
5.0
5.5
6.0
10
I
: a
1
, I l l
6
0 0.00
Ps
t
Pp
l
PM l
2
l
2.5
I
4
Ps
I
2.0
j
Y,
I
0
'@' 12
;
Pv
I
2
14
&
l
j
I I
c; O
1 1 I
l l
16
1
=Y
7.
18
I I I I I I
l I
0.05
0.10
pS 0.15
0.20
0.25
Guide to the Literature The pioneering article on the jointtlecisions to invest and abandon is Brennan and Schwartz ( l985). They construct a vcry gdneral model of the dccision to open, c Iose, an d mo thball a mine producing a natural resllurcc whose price fluctuates over time. The finiteness t)f the total stock in the mine introduces addditional complexities beyond a11 the issues considered in this chapter. In fact the vely complexity of the modcl conceals some important concepts. Having obtained the system of equations characterizing the price thresholds Brennan and Schwartz immediately for investment and abandonment, numerical solutions. Thtty to obtain the ratio ()f the entry and exit resort threshold prices, and show that it exceeds 1 for rcasonable parameter '
values.
Dixit (1989a) isolates the entry and exit decision from isstles of Iay-up stocks. This allows some analytical results and insights. ln particular, finite or the ent:y and exit threshold prices can separately be compared to the myopic or Marshallian criteria of full and variable costs. respectivcly. This clarises the role 01- the time value of the stlparate options to invest and to abandon,
or morc generally. the tlption follows Dixit's treatnlent. Some of the earliest
valtle
work
()11
()1'
0.30
0.35
0.40
tlle stktttls qull. Our cxptlsititln
clllsely
C (Millioc Dollars)
optinAal
nllltllballing
decisions
was
by
klossin ( l 968). who developed lt IT1lltltll in whicll (lperating )IIllws reventle a trendlcss random witlk with upper and Itlwt,r retlecting bitrriers. Itnd in which there is no ptlssibility tt' scrapping. I-Ie calculated the optimai rtlvttntle levcls at which it is optimal t() nltlthball and reactivate tlle prtect. The nlore general model of Brennan and Schwartz ( I985) incltldes the possibility ()f and abandllnmtlnt. mothballing as wcll as active tlpttration Howcver. thcy the transition the confuse twt) states frlln :tn ttltive state, t3yusing thc sCtme to Iowcr thrcshold symblll lbr mothballing as
lr
abllndllnment-
Nvhcn they
fnln'll.r
to numerical solutions they specializc l() a model where the mClintttnance cllst is zcro's then scrapping is never used and they consitler switches betwden j ust two states operation and mothballing. Once again (lur approacll follllws the subsequent but somewhat clearer analysis 01* Dixit ( l t?88).
Thc models we studied in this chapter artl instlnces ()f the gcncrnl problem of optimal switching among a number of alternatives in rttspllnse t() changing economic conditions. Each swtch is an exercise of an ()pti()n- ftntl
zdFirm Decisiolls 's
244
each switch yields an asset that combines a payoff flow with the option of switchingagain. Thus we have a stlt of Iinked or compound options. and must price them simultaneously. There is a large body of literature analyzing such compound options, either from a general theoretical perspective or aiming at specic applications. Geske (1979) is an early example of this kind in financial economics;subsequent articles include Geske and Johnson ( l 984) and Carr ( 1988).
Turning to real investment decisions. Kulatilaka and Marcus ( 1988) develop a model of switches between two modes with three time periods, and indicate how it can be extended to many motles and switches. They also survey the early Iiterature on real options by placing the different models within their
framework.
Fine and Freund ( 1990) examine a general two-period model where the hrm must choose its capacity belbre uncertainty is resolved, and it can choose either specific capital (suitablefor a particular kind of output) or capital (which can produce all kinds of output, albeit at greater cost). Triantis and Hodder 1990) havc a similar model in continuous time. He and Pindyck
oexible
partW
lndustry Equilibrium
(
( 1992) go further by allowing subsequent expansion of capacity. ln all these models, option value is twen more important. because by waiting the Iirm preserves the opportunity of making a better investment Iater, and not just that of not investing at all. as was the tlase in the models of Chaptcrs 5 llnd 6. Bentolila and Bertola
( 199()) examine
employment
decisions of firms
when there arc hirfng and firing costs. Dixft ( l 989b,c) considers production and import-export choices when exchange rates tluctuate. This builds on earlier work by Baldwin and Krugman ( 1989).,see also Irugman ( 1989). Dumas ( 1992) and Krugman ( 1988) develop such models at a general equilibrium level to endogenize the exchange rate process. Kogut and Kulatilaka ( 1993) consider multinational hrms' decisions to shift production from one country to another in response to exchange rate movements. Van Wijnbergen ( 1985) constructed a two-period model of capital outflows from less-developed countries in a context of policy uncertainty.
i
l?qC1lzNr-rlus 5 through 7. we examincd a vklriety of investnlent and tlisinvestment decisions )r :1 singlc (irm. Throughout, we asstlnled tllitt the hrm has a montlpoly right to invest in a givcn project. and wc ignored the possibility of other firms entering in competititln. l Tllc profit tlows of an operatitlnal project werc subject to ongoing shocks. Since we were assuming that the project would yield a fixed output Ilow, we could model these shocks as an exogenous price process. We tbund that th0 textbook Marsllall ian present value criteria or comparisons of price nnd cost were very far fnlm the true optimal choices ()f the tirm. With moderate amounts of uncertainty, investment was justihed only when the current price excceded the long-run average cost, or the current rate of return on the sunk cost of investment exceeded tht! cost of capital, usually by a large margin. Similarly, abandonment was triggered not when the pricc fell below average variable cost. or when the currcnt operating proht became negative. but only when the loss became sufficiently Iarge. The reason was the option value of waiting bcfore making irrevcrsible decisions in conditions of evolving uneertainty.
l In Chapttt r 5 we mentioned
t hat the rate tlf return short l'alI ()n the vaI uc ()f a project, wherc l.t is the competitivc expected return ()n an asset with cquivalcnt risk. clluld rcllttct the possibility ()1' entry by other lirms. I-lowtlvcr. we did ntlt explicitly mtldel entry, ntlr did wc offer any justificationftlr why it should Iead to a ctlnstant rate of return shtlrtfall.
J
ing stochastic processes and the decision rules of other tirms. Each lirrri has the capacity to produce the llow of one unit of output, which it can activate by incurring a sunk cost. There are no variable costs of production. and the elasticityof demand is large enough to ensure that each tirm that has paid its sunk cost will in fact want to produce at its capacity level.3 The prices of different hrms' outputs can change unpredictably because of aggregate industry demand shocks, or because of hrm-specific demand shocks that reoect changing relative tastes for the tirms' products. Hence one hrm's output can sell at a price that includes an exogenous premium or zExtension to a CAPM setting is straightlbrward and mercly introduces stlrnc extra nofalign
in this contextv so we leave it to intcrcstud rcaders. 3With
Q
irmss this requires a pricc tlasticity is very weak.
by this assumption
in excess of l / Q, so thtl restriction
entailed
=
pz
-
fz,
downside risk by chllosing not to invcst if thtt price tlls. while preselwing t he upside ptltcntial to invcst if the price rises. Thus the consequence of waiting is that the payoff to the firm is a convex function of zY so the expccted value of waiting ristts with uncertainty in X. This contrast between the effects of tlncfartainty on the expectcd valucs ol- the payllff from investi ng immediately verstls value'' premium waiting explains why greater unccrtainty raistls the waiting. on -soption
Now consider intlustry-widd uncertlinty. The firm k ntlws that if F rises. en try just as attractive for othcr firms as for itsttlf. I-lowever, when
this makes
4Otlr t reatnaent t) f thtt resu lting ctln t in tl u m lf ra ndtlm va riables 11nd l heir lktws (41* largt.r numbers is dttcidedly hetlristic. btlt thc restllts are intuitively clellr. itntl il jilrmltl rigllrtlus trtrltnlent would be far t()() lcngthy lnd di flicult I'(lr thc presellt purpllse. 5tte Judtl ( l 985 ) lr the blsis (,1' the rigorous tlleofy. (
249 In practice, most (irms d() not enjoy mollopoly rights to invest, btlt insteatl must consider the possible entry of new competitors. ()r expltnsion ol' existing ones. This raises 11 fundamental doubt concerning our earlier conclusions. The opportunity to wait. and its value, depend on what the firm-s competitors do. 'With free entry. should this value not be reduced to zdro'? Wtlultl that not restore the Marshallian criteria comparing price to the Iong-rkln average cost in the case of investment- and to the average variable cost for disinvest ment? Thus the reader might suspect that the thcory of an individtlal hrm would not survive an extension of the scope of thtl analysis to thtp level of industry
llttlllstlLy )-t////'/#;v2./?)2
other firms enter, the industry supply curve sllifts to the right, and the price rises less than proportionately with F. Therefore price is a tloncave function of F, and then so is the profit tlow. Greater uncertainty in 3- now reduces the expected value of investing relative to that of not investing. That is why the firm requires a higher current prolitability (in excess of the Marshallian normal return) before it will invest.
We should stress the similarity as well as the diftrence between the two scenarios. In each. the underlying symmetric demand shoek translates into an asym'metric protit llow shock; but this happens in very different ways in the two cases. In the case of firm-specific uncertainty, the downside ofthe profit shock is cushioned by the p'ossibility of waiting. Thus greater uncertainty makes waiting more valuable relative to investing at once. In the case of industrywide uncertainty. any one firm in the mass of competitve potential firms has the a zero value of waiting. However, the upside protit potential is cut off by uncertainty reduces the of Therefore firms. other entry of greater prospect value of investing relative to that of not investing :tt all. In reality there are several other factors that can aftkct the convexity concavity of profit tlowsas a t'unctionof the underlying shock variable. l 1* or adjust some variable inputs instantaneousiy, then its protit flow tirm the can of the price, as we saw in Chapter 6. Setltion 3. In function becomes a convex will sde that when the tirm can add to its capital 1l in Chapter we addition, and is llow given by a producton functilln- thertt can be its output stock, which the marginal protitability of an incremental iilvestment by other ways I-lowever. price. the above intuition still operatt!s similarly. in becomes convex For example, with tirm-spccificuncertainty. the possibility of wlliting cuts off the downside risk and maktls thd profit flow an evml l/?tprf.r convux ftlnction of the underlying shock variable. In most of Chaptcrs 8 and 9, hllwcvcr. we will
and therefore charaeterized by ctll'lsiderablu inertia. should the government attempt to encotlrage investment'? I-low will various policy instruments aftkct investment? In particular, what will be the dffect of governnxent policies to reduce uncertainty tforexample. through the use tlt' price controlsl? 'What will be the effect of uneertainty concernilg the governfnentfs own actions (for example. uncertainty over future tax rates and regulatory changesl'? Such questitlns must be examined at the industry level if they are to be useful guides to policy in many practical situations-, therefore these chapters are the right place tbr ther study.
/Jp??f???1(7 /. tlltilillrilllt
?l tl
Illtltt-stll' trtp/zl/pt?f/rc
:!5 5
in an essential way on the joint presence of the two kinds of uncertainty examined in the context of a simple but general model in Section 5.
are
When all uncertainty is industq-wide, the multiplicativc factor X in tlle general demand curve ( l ) is constant, so we can just set it equal to I Then the industry's inverse demand curve becomes .
P
=
i'
DQt.
The aggregate shock J' will follow the geometric Brownian motion process dY
=
a )' fJ +
ty
1' dz.
On the production side. we assume that there is a Iarge number
of
risk-neutral competitive lirms. Each hrm can undertak'e a single irreversible investment. requiring an initial sunk cost 1. Once this investment is made. it yields a flow of one unit of output forever with no variable cost of production.
We embed such tirms in an industry by supposing that each unit of output is very small relative to the total industry output Q, so that each firm is an innitesimal pricc taker. When Q firms are active. the short-run eqtlilibrium price can be determined from cquation (2) above. As we discussed before, this is the simplest continuation of the model of Chapters 5-7 that serves our present purpose. Latcr and in Chapter 9. we introduce various generaiizations. whcre each hrm has some variable cost. short-run output variabilitya exit possibilities. etc.. where the shocks affect demand in more general ways, and wherc the industry has some imperfect
competition.
To set the stage for the competitive industry equilibrium. think of the static model. The industry price a single number is parafirm. The sum of the individual firms' optimum quantity re-
usual textbook metric to each
I'ltlll-bll7'I-t/l//k'/'lrl/l/? I11 an interval of t inle vvhen 11(.) nesv en try taktls plltce. ? is lixed. s() P is proportional to 1' and equation (3) gives
dP
a
#
tlt
+
f'r
13t/--.
A potential entrant observes this process. and interprets a lligh pricc as a signal of a high level of demand. Intuition suggests that tllere shotlld be an
upper threshold level P which, if reached. will trigger new entry. As soon as any one new firm enters. Q increasesp and price decreases aIong the demand culwe that applies for that instant. Thus. if the price ever climbs to P, it is immediately brought back to a slightly lower level. I n technical terms, the thrcshold # becomes an uppc r rejlectillg /?flv(!r on the pricc process.
Of course firms rationally anticipate aIl this. Thus the price process from the perspective of any one firm is 11 moditication 01' the geometric Brownian motion process of equation (4).That process is valid as Iong as P < /''. I-lowever, the price cannot go any higher; an upper rcflecting barrier is imposed a t #. In Chapters 5-7 we studied a monopoly firm's dntry tlecisilln when it faccd a geometric Brownian pricc proccss. Now we set, that a competitive lirm's entl'y decision is a similar problem- dxcept that thtl price prtlcess has a ceiling at P imposed by this firm's rational expectlttitln ()f (lther firms' entry decisions. So we mtlst rcexamine a firm-s optimal entry decisitlna taking intt) account this new pricc process. The solution is again charactcrizcd l)y a thrcsllllld /7*, stlch that the firm which we focus will choose to enter if the price rises t() 13*.Since a11 firms ()n arc identical. industry equilibrium rcquires this Iirm-s entry threshold t() equal its rational expectation of aIl (lther firms- threshold. So we start with a trial P, and solve one firm-s dccision problem to (ind its P%.Tlpe (ixed point of the process, namely, #* P, determines the industry equilibrium. Note how thtl intuition or guess that the equilibrium price process is a geometric Brownian motion with a reflecting ceiling has simplified the calculation. We need to look for the Iixed point of a chain of reasoniny that takes tls from one number to another, instead of a mapping on a whole function space of general stochastic =
processes.
not restricted upNvard by the rttllect ing hnltl-l-icl-. $ve svtlultl llavc l,( l'4 /. J f.z tll /''/(. is 11esl1() bctyvee rt re t tl t l'ltldiscotl nt rltttt t he risk- free ( intertlst rrtte in this case ) lnt.l tl'ltlexptlctetl lnlttt (,1* trtlvvt 11()1- 17 v'llell it is belknv tlle barrier-s The rellecting llllrrit)r at l'' cuts tlff stll'ntl ()1* t I1e tlpsitle potklntiitl for priot)s antl prolits- so the firnl's val tll.) nl ust 111 ftct l)e less t l1l 1) l'b/fq svil l Iind a forlutl Ia for tlle correctikln. Start a t 111-.inilqial price /2 < 1'. I11 11 stl l'liciently snlll Il i1) lA.'al t)l' inle (Il te t the price process is al rnost stl re to stay bel(.nvt he iliIlg. ve can carry out the usulll dynarnic prograrn rning or trolltillgent clllill'ls Ctnalysis to tll'ltllill the fa l'n iliar diftk re n tia l eq tlat io 11 Nvtlre
yvlltl
's've
.
.
''rllen
.e
where B is a constant fundamelltal quadratic t,-.1 ln
to be 1.f'.r : a
The Value of an Active Firm
The first step in the equilibrium calculatior is to hnd the value ofan established hrm facing a price process of the stipulated type. Since the future price path depends on the current price level #, the expected present value of the firm's future prolits is a function of P, which we denote by u( J.'). I1*the price process
p (p
-
antl
l ) + (1-
-
t
pb is t l1tt ptlsitive )p
-
1.
rtltlt
tll'
tlle
t)
.
p! > 1 tlntler the tlsual clllltlitillns ?' > t) itntl J p. (). This conforn'ls tt1 the intuition stkttetl abt3ve'. /'/J Nvtlultl btl tllc value ot' thc firnl in the absencc ()1 . the prictl ceil illg. s() 1) l ,/ rntlst jAe t jye kytly-rvqytjtju due to the ceiling. Thereforc B slltltlltl be negkttivc. 7-I1eillterpretatitln :l1s() explainswhy wc left out the ternl invlllving the negative rtdtlt ()1' the qulldratic. If the currcnt price 1.' is vcry snall. thtl barrier 19is unlikely tt) bc reached except in the far futtlre. Then the ctlrrcctitln arising frllnl tlle barrier shlluld /J become small. I-lowklver, a llegltivtl pllwttr ()1' gtlcs t() inlinity lls /' gtles t() therelre, shlluld that term not appeltr in the s()ltltion. zertl; To deterrnine the value of the conslan t B we lollk 11t the upper tlnd poi nt. From a starting point vcry close t(.)tl'ttl rellecting barrier P. the price is ktlnAtJst surc to fall during the next small time intcrval. If thtt value functitln lllts Ct negative(rcspectively. positive ) slope at tllis pllint. therc &vi 11be surc arbitrage profits (respectiveiy. losses) to be made. rulc th is ptlssibility (lut. we lmust tct
-rt)
have
, p(I ) .)
2.A
tlutttlrnAilltld.
p1 B P
#1- l
+ 1/t
().
(7)
'
/?ltII.J?y
256
Eqltilibriltnl
This looks like a smooth-pasting condition. but it is not a consequence of any optimization. Such a condition holds at any retlecting barrier for :1 diffusion PI-OCeSS.
the solution more fully. we must gt) into the details ()f tlle hxed-point process for constructing the equilibrium. Consider a firm contemplating entry. Write /'( P) for the value of its option to enter. As in Chapter 6. this takes the form i' X) Z PV' To understand
=
0', as explained earlier, the barrier cuts off some upside price potential, so the correction to the value is a reduction. Substituting for B into (6), We have Note that B
<
where z1 is a constant to be determined. and pbis as above. If the hrm decides to enter when the price is P, it pays the investment cost I and receives in return an asset that we just valued at p( P). The optimal entry threshold #* satisties two familiar conditions. First, value matching'. ./'(#*)
2.B
t7(#*)
=
1.
-
and second, smooth pasting'.
Equilibrium
The quickway to hnd the industry's equilibrium is to use a dynamic zero excess profit condition. At W,the common ent:y threshold for all tirms. each firm is just indifferent between entering and staying out, so the value of being in, u(W),must exactly equal the entry cost /. Using equation (8) above, this gives
W
=
#l
/$1 -
l
5 1.
Most remarkably. this is the same entry price as that for a unit-sized monopolist hrm facing the same demand process; compare this to equation (9) of Chapter 6. The two situations differ in two ways. The monopolist of Chapter 6 was not threatened by ently so there was no upper barrier on the price process', now there is. However. the monopolist had a positive option value of waiting, while any t)f several identical potential firms of this chapter must have zero value of waiting. lt so happens that the two differences exactly offset each other. This coincidence between a competitive hrm's and a monopolist's entry threshold is in the context of a very special example. In Chapter 9 we will hnd a very general result of this kind a competitive firm can make the correct investment decision by acting myopically in the matter of future competitive entry, and acting as if it were going to be the last lirm ever to enter this industry. That result also rests on a similar exact offset of two effects, one on the value of investing and the other on the value of waiting.
Sec Malliaris and Brock ( l 982. p. 2()()) or Dixit ( l993a, Section 3.5).
f' #*)
=
t,'(
#*).
Using the functional forms for the functions
and
# 1 z'1( p. )l1-
1
/'( #)
# 1 s ( p, )p1 -
=
l
and
p(
+ j yj
#). we have
.
Notfz that we have already solved for the constant B in terms of the assumed upper barrierW but some expressions convey more insight when B is retained ,
as such.
can be solved for the threshold
These two equations we have
stant
P. and the con-
a
and
zl
=
B+
1 pl
/5
(P*)
j
-#j
Obselwe two features of the solution'. the barrier T affects the solution only zd in the via the constant B in the value function p( #). and the constant IJ3 rcsponds value function /'( option one for one to changes in B. These have important implications for the equilibrium.
First. eqtlat itll1( 1()) sllows tha t thc orltry t hreslloltl lA% is indepentlent ()f B and therefore of thtl barritt r /4. Of cotl rse. tbr t Iltl s()l tlt i()11 t() nlake econom ic sense. the barrier mtlst not be lower than thtl entry tilresholtl. Given that. the exact Ievel of the barrier is im I'naterial to the e n tl'y decision ol' t he li rm tlsing tlle relation between the This remark-able property can btl tlndcrstood zzl and B. Any shift of the upper barrier Ilas eqtlal effects (41) the twl). constants cffects value of an active firnl antl thd option valtle ()f and thus equal on the 01- tlle upside price potential has a potential tirm. Roughly speaking, :1 cutoff effects equal on the value of ente ring at once and the valtle of Nvaiti ng jtlst Ct little Ionger, so it does not alter the trade-off between the two. Since the barrier does not matter as long as it is high enougha we can let it go to infnity without affecting the entry threshold. That limit corresponds to the decision of a tlnit-sized nlonopoly firnl that faced 1111identical demand culwebut faced no th reat of entry by othtt r lirms alld the retklre nt) upper barrier to the price process. This explains the coincidence of the entry thresholds of the monopoly and conlpetitivc indtlstfy cases. Once again, we alert the rcader to a very general result of this k'ind to come in Cllapter 9. 1' To solve explicitly tbr the competitive industry equilibritlm- we sct P. .?
v (P )
l l l
fpj
-
I
Monopoly
1
#
=
in ( l ()); this reproduces equation (9)awhich was derivcd beforc by quick inttlitive P in equation ( l 1) gives argument. Moreover, setting P* a ..d 0. Theretbre the option value 01- an idle tirnl /'( P) is identically zttrt). equation
=
=
This was the intuitive starting
veritied it using
a more rigonlus
The difference between
point of otlr quick calculation-, theoretical argument.
:1
monopoly
and a competitive
now we have (irm is shown
graphically in Figurc 8. l The range of prices facing a competitivc firm is bounded above because of the entry of others; the monopolist's price range is not restricted. Tlle mllnopfllist has a positive value of waiting, and 11 positive value of investing ovcr :1 range Ieading up to its optimal entry threshold. A competitive hrm always has a zero value ()f waiting. Its value ()f investing is negative for most of its pricc range, and only just climbs to zero tt tlle tlpper end of the range of possible prices. .
The differencc can be understood by rclating the entry threshold to the Marshallian concept of the normal rate of return. This is most simply discussed in the case of u 0, sl) that the process is trendlesss and as we are assuming risk-neutral srms,6 = l.. Now the unit-sized monopoly also faces a trendless price process-, given the current pricc P, the expectatioll of the price at any future point in time is also P. lf such a hrm invests when the price is /1) r 1, =
=
fpj
B0
Competition
lll
!/--/7)/
Etll/ ilillrill)z7
ryl//ib/?rl//?l in Ilyntlpllc L-.
Table 8.1.
it will earn a normal return on its sunk cost. However, we saw in Chapter 6 that it does not invest until the price rises to P*s which is pk/(p1 1) times -
Jl). We explained this in terms of the option value of waiting. Now we see that a unit-sized competitive hrm also waits until the price rises to the same level. even though its option value of waiting is zero. The explanation Iies in the difference between the price processes in the two cases. The competitive firm's price process has an upper barrier, which reduces its expectation of future prices and returns. Specitically. since the tirm knows that all other firms face the same choice and make the same decisions. the prfce will never rise above the level that prevails at i(s instant of entry; the current price when it enters is not the average but the best price it will ever ()f get. If competitive srmsadoptdd the rule entering when the price reached normal would only return at those instants when entry was earn a C9,they they would taking place earn lower returns at all other instants. The average would then be insufficient to justify the initial investment return over time expenditure. On the other hand. when the entry threshold exceeds f1).each firm will experience some period of supernormal returns and some periods of subnormal returns. The equilibrium 7 is exactly the level that ensures a
normal return
on average.
Since the entry threshold u'um industry etluilibrium price coincides with the threshold for a monopoly with the same parameters a, c, r. and J, we need not present detailed numerical calculations for the ctlmpetitive equilibrium case; instead we refer the reader to those in Chapters 5 and 6. However, a few summary numbers are useful. Table 8.1 shows pk and thc current rate of return on investment at the threshold.
75/l
pl/(pl
=
l).
-
lbr r 0.05, a 0 and 0.03. and fr (), ().2.and 0.4. Notc thatwhen a t'z ()a I equals the Marshallian return. that is, the interest rate 1, the return 0.05.This is also the case when a 0.03 and tz 0. (As discussed in Chapter 5, when a is positive there is a value to waiting even if there is no unccrtainty. and indeed in this case p /4p1 1) 2.5. but since J falls as (z increases. the return remains equal to r.J For either valuc of e, as rz is increased to 0.2 and 0.4, p falls and the required return -#/I rises to about two or three times its =
=
=
=
=
-#/
=
=
-
Marshallian
value.
=
=
Hence the general finding from Chapters 5 and 6. that the
hrm's optimal decisions differ substantially from the implications of the textbook present value approach, has an exact parallel for a competitive industry. Its equilibrium
differs substantially from the picture offered by the Marshalelementary and intermediate microeconomics
tJf the monopoly Iirm in ChapThe above basic model cltlscly tbllowcd that rcstlits tbr the competitive industry with ter 6, and gave us very analogous model are left for Chapaggrcgate uncertainty. Most of the extcnsions of this naturally here. $Ve ter 9. In this chapter we take up just one that tits more introducc exit. and construct a model that clsely follows that of the monopoly again. the results for the firm's entry and exit decisions in Chapter 7. Once competitivc industry with aggregate unccrtainty are thoroughly parallei. For exit to be :1 meaningful option. we need two conditions. First. the this possible operating prolit l1()w must somctimes becomc negative; we make unit-sized lirm. Second, temporary by introducing a variable cost C for each penalty must be ruled without rcsumption and opdration cost of a suspension of exit E. As betbre. this out; we do so. We also introduce a lump-sum cost required severancc payments or costs ()f restoring can comprisc any Iegally representing land. It can also be negative (but numcrically less than cost /), any nonsunk portion of the entry cost. Now the intuition is that the exit of other firms will generate a tloor a gencrated Iower rcfiecting barrier--on the price process, just as their ently rational expectareflecting barrier. Each firm will have a ceiling an upper namely, a geometric Brownian motion it faces. the about price process tions wil! again between these two barriers. The firm's own entry and exit decisions equilibrium take the form of upper and lower thresholds on the price. The each levels of the two barriers will be tbund from a hxed-point argument'. thresholds should equal the barriers generated by the behavior of all
firm's hrms in the industry.
Ll'ct
'Tl'le cillculrlt itln is s()I''j''ltl'$.vI'1:t t rnllre c(')I''l1 plict ttlt.l t 11:1 11 il1t I)e 1)l'ttvitltls ciLlstl ()1*1.l')t;tIIy irrevel-silnle t.'rn t ry, llecll tlsc v'vtl c:l1111(.) t first Ii11tl l lle v;l ltl kl ( ) 1'4.111 ivkl llctivc frn! Cntl tlltln t'ttr tlptitll valtltl i)l' :11.1 inldclive tirn). ?X.1'! lirnl-s trlltlictls inc! ude tlle option to exi t. so the tvvo values l'n tlst be lltl nd si tl ltltlleotlsly. n'l An active 5rm faces the price process of eqtlation (4) with barriers J7 and P. Let p! ( P) denote its valud when the ctlrrent price /7 is in the interior (7fthe range (P. 741.Th is satisfies the fam iIiar d iffe rc n tial klq u :1 t ion
l''s kr:'k lt'
Nvi f7()1*givc 11 1):1rrie rs /7 111)tl 11 t 11 dstl l'(ltl1-t.ltl tl :1t'i( ) l'ls. tllge t 11 t.y1. t 1)t 11e t'( I .1tl it it) IN.'t2 tl 1*f..,t 1)t.l l'T'itl t2 (Nv() ') 1) I1 :1 ,1 lltlc It? 1, !'1 ( 1) i r rs. t t t s t cl rtl sl :'t s l 1 1..a/?, /9..., g '/ '/ t-i tl ) il1 I't- t t l i1't..l111t-.11l s / / I 1(. l l I1t- t I) 1*c s l1( Itl :'i ! / / l I ) 1)11 l l -1; ; l I 1t I i l t t :, I y c xtr 1)( (alnlpll.l ()1' 11 te t llt2dtl ttl rnli na t il,lI1 t he etl u iIil')ri tl n1 /. .1
.
',
.
,
Bj P#' + Bz P* + P/f5
C/
-
.
(l1e lltl 1-vrlltlc- 113:1(t:11 ij1g
tls
( B3
z.II ) /7///$1 .+ ( Bz
-
-
' .
FJ
1j ) .
-
,-I! )
/$'
l/ ''
)I
-11 -
# I (B
l-.
.
.
..=
-
.
begin by hnding the thresholds. Note that and snloot h-pasti ng cond ititlns cttn 17tlsvrit ten Let
( Bb
The Iast two terms give the expectcd prcsnt value of prolits corresponding t() 11price proccss without barriers. Thtt slight ncw twist is tllat since tlltt eqtlation holds only over the finite range of prices ( /7. J7).we cannot tlse considerations of limits as P goes to zero or infinity to el iminate Ilitller of the (lther terms. The first term, corresponding to the positive root /:11represen ts the rcduction in value due to the ceiling on the price process. Thc second term, corresponding to the negative root pz, is the value of the (irm's option to exit. At the upper reoecting barrier we have thc arbitrage contlition Isimilrtr t() eqtlatilln (7) lr the entrronly case abovel
.
.
'.,
.
The form of the solution is equally fanxilinr froln Chapter 7: =
.
')
t-l ( B! l?l( P)
.
-
.+.
-F. 2 ( B2
t)
( Ba
/f' - l
IL ..
1
/62
) J5/
.g12 -
+
zzl?
-
al -
,
) lh
p? ( lh
+ J/
2)
'SJ
l.D
l/
/ #:- l
+ lh /J
)I
zzl.a -
(/
-
/?
-
1.
-
/
+ I / (y ' ()
C7/?'
-
'
-
+ l /(
.
(2 l )
.
(
.
/f
-
'n
-
oj
-
(2J1)
.
(24 )
()
.
Ikega rd ( Ilese ;.1s fotl r equkltions in fotlr unk llosvns: t 11et 11resllfpltls l'tt Pl. kltl ttl-- constan ts ( Bt /11 ) ( I)? zzla ) In ( 11is f()1-111l 1'1tl alltl t he tl l t itlls tlf dx:lctly iI1 1. 7 t l'lttt tlle I itlent ic:tl 2) tt') cqtlltttltls systenl (t))-( ltre t'lrtt l) rtl 1.I)e de fined tlle dntry antl exit tllresholds for a n'yollllplply li rn'l. .
-.clll'nposi
-
-
.
.
-llltpte
-1-11
prcscnt systt'trll yields
the sanle thrdsllllltls. intlependklntly of tl'lc llktrl'icrs />. with tt'ltrllly irrilversible tlnt 1-,7. t llt, t');l rriers tlt) l'lt')l affet:ttlltt th rcsholtl. (.)1* cotlrse. for tlle ttxe rcise to bt! Illea n ingfu 1-t Iltl l'):trrie rs IA I'ld rntlst be Tvidtl enotlgh t(') span t he t 11rcsllolds. t hat is- Nve I'ltlel.l u'' /7/. :1
P.
F.s
in tl'le prcviotls
fnodel
()f tlle bllrriers is inlnllltc rill. (')I'1ctl t he exact ptlsitiol) /$/ :G 1'. (Elltlqlcnvistl Clffects erlcll p:lir ( )I' t'll')t il 1I1 v:1l tlc ltgllillt l'le rcrksl')n is t l1:lt t he exrtct pllsilitln I)? etltllllly. 11). t 11e t1i l't'cI'tl Ilctls ( /l1 cllllstttllts zll /.f1 :lI1tI ( B? zla ). governing the trade-llffs betwccn immediltc ltct illI1 antl wltit illg l'or entry ()r exit. Ctrll unaltertld. -l'llerellre
--1-.-
..
.
-
/$/ Ctntl Finally- turn t() the equilibritlm. F()r tllis we sinlply set 19 ()1* eqtlations (21)-424). 7-I1is 11t!illg itlen t iclll with P P1.in tllc abtlve system cqtlltions (9)-( l 2) t)l' C'llktptc r 7. thc corresponding system for 11 monoptllist wc do not nced to repeat tlle numerical calculations conductcd tllttre. Tll remarks there concerning the magnitude ot' the range of incrtia ctlntinue t() apply.The ceiling and lloor on the competitive equil ibrium price process di ffcr stlbstantiallyfrom thd Marshallian long-run ltverage cost ttnd ave rltgc vtriable ctlst. rcspcctively. and are ver.y far apart l'rom each otlle r. We will s(u)I1 t)I'l'er a dramatic numerical example of this. It rcmains to find the .separate values of activc and itlle Iirnls. F('r this. combi ne the tvvo equations at the barriers- ( 14 ) and ( 1()), wi t 11 llt2 tNvt? we smooth-pasting cond tions ( 19) and (2()). to get tj') ( /'' ) () '?(') ( 15) ()r =
where now the first term is the value of the activation optitln, and the second term is the increase in value that results from the Qoor on the price process.
At the lower reEecting barrier we have
' (Z)
U()
=
pj
.,:1l
..Lf' l + pz zla .C#:-' -
=
().
Now suppose the cntry thrcshold is lb and the exit thrcshold satisfythe usual value-matching and smooth-pasting conditions
1,:(.PtI ) l ( P1.) 1-, p(') ( Pll )
=
=
=
'!
v ( b-)
.'.'''''=
p I ( P/, )
-
lA)( IL)
-
u'1( PlI ) p(-,
( Pl.)
.
,
/
1-3..These
( 17)
.
E
( 16)
.
(18) (l 9)
(20)
't
=
p1
.,z1
1
/7p,-
I
+
p.a zl2 /
1 ' p2-
()
.
-
in
Regard these as a pair of linear equations P p'
p p
l
-
l
./,42
z,4l
.
.
.-5p: -
/-- /
l
=
=
=
=
.$0.55,
=
=
.35
.$0.88
.$0.79.
S-fortunate''
--unfortunate''
rttalization.
whktt These figures show particular realizations of price, but we could ask regions different expected in to stay of the time the price should be
percentage
slktrsllallian
r
(),Fl ()./1
(). l ().2
().8 4 ().8l
()./9
t).
13 t).19
().
().
-1
mining indust:y We now return to our example of entry and exit iIA tlle copper the readers results give rical the to nume from Chapter 7, restating some of t)f indtlstry data to the involved. I n fact. use a better feel for the magnitudes sometlling illustrate the stoly o a Iirm having a montlpoly right to illvest wts sdtisfat:numbe have the more a rs same of an anomaly in Chapter 7', now central the equilibrium. ln case tory interpretation in the context of industry mindaverage-sized of building an studied. we assumed that the capital cost 10 million pounds of copper pttr year) was smelter, and refinefy (producing rcstoration was E = upon abandonment / $20 million, and the cost of site itllowed to = ptlund. l3ut variable wkts $().84) per cost was C $2 million. The annual (1.2 volatiiity in rr parltmcter was vary around this ligure. The price risk lttss interest units. and was also allowcd ttl vary arotllld this rangc. Tlle tllese Ilumbdrs. ().()4. With sllortfall was rate was r = 0.()4. and the return the exit thrtlsilold the Marshallian entry threshold price would be $1).88alld exit thresholds and the correct entry 50.792.As shown in Tabie 8.2. however. and the Nlarshallian shows respectively. Thc table also are 1.35 and and o'. the actual thresholds for other values of C $().84)per pound Figures 8.2 and 8.3 show. for the central case ot' C that this price Observe 0.2, sample paths for the price of copper. and c the upper and lower ouctuates as a gcometric Brownian motion between of thresholds S 1 and $().55. exit and re*ecting barriers, which art the entry tlso shown lines'horizontal Iigures as These thresholds are shown in tbe time is the hgurcs, and (In are the Marshallian thresholds of for increments purposes measured in years, and each year was divided into 50 (fromthe point of generating sample paths.) Figure 8.2 shows a of of view of a copper producer) sample path. in which the pricc spends much shows 8.3 while Figure an the tfme at the upper end of its rangc, .$
k:
,42
-P. Then the only solution
Entry Exit, and Price in the Copper Industry
3.A
Ll3wer threshold
Uppcr thrtlshllld
P#'-1 p2 P/'-l
0. is z11 is nonsingular as long as should given be. it identically zcro, as Therefore the value of an idle rm is stllulilln. competitive conditions and identical lirlns. This completes tlle ''I'S >
/t/fn?zj.r E./l//y ulld l-r/ Tllreslloltls ?' Copper (N()te: see text tbr paramete rs)
The coefficient nlatrix
-1
() :9 .-1
-n
().() !
().tl I )
().:!
1
.t
-4
().8 4
-9
.t)
Ctlrrecl
51arshallian
l 1:! l 35 l 7s
() 6):! (). 7t):! ().7 ):!
().ti3 t). ; ; ().a/ 5
().;t) 1 f? 1 ; 1 ()
() 65):! () ):!
() :!ts () () ().15; ().
.
.
.
.()
(). ().
().()
Clprrect
.5
.t1
-/
.
.
.
.21
.5
().$7 6)2 (). ) q):!
-7()
by calculating the
of its range over the long run. We can answer this question observing that P follows long-run stationafy distribution for price. Begin by its reflecting barriers the geometric Brownian motion ot cquation (4) between
Ito's Ldmma. we know that p < 1og P follows :1 1 c2 and the a' a simple Brownian motion with the drift parameter 2 = retlecting barriers corresponding .L? log -C variance paramtlter f'r betwcen ()f Cllapter 3- Sdction 5 to find antl = l()g W. Ntlw we can tlstl thd result with density t)f p. I t is an exponential distribution, the long-run distributioll a'/cz2, and the constant uf proportionality K is chosen = 2 K el''t'. where y unity. With this. it is easy to calculzte to acllievc a total prllbability mass of The various subsets of the range (. 7.8. what proportion tlf time p spends in P. for corresponding the ranges translated into results can then be price will be For our base case parameter values, we find that. on average, and $1.35 about between the upper Marshallian and entry thresholds of $0.88 producers the time. half than time. copper Thus of more the 58.5 percent would call a analysis microeconomc we traditional what in will be earning protit. Price will be between the two Nlarshallian thresholds of P and
T. Therufortl.
tising
=
-
.
-j
supernormal which case we can say $0.79 and $t).88 about 1 l pcrcent of the time (in Finally, price will be prohts). subnormal but aarning positive that firms arc threshold of between the exit threshold of $().55and the lower Nlarshallian .3
S().7Q, so that firms are incurring losses, about 30.2 percent of the time. t)f ;&competitive Tlltlse Iigurcs sllllw vcfy drllmatcally llow the dynamics k)t' uncertainty will differ from the textbook picture. induste undtlr conditions the copper industry to be in a expect thtt time. oiwe For almost 9() percent
:?()tj
N c
state that according to tlle Marshallian theory simply otlght not to exist. Eit lltlr price will be above long-run average cost without attracting new entry, ()1' it will be below average variable cost withotlt inducing exit. X) sollltlolllz wI)() tries to interpret such observations using the perspective of the Nlarshallian textbook, supernormal protits without entry will stlggest dntry barriers- ltntl continued operation of loss-making firms will stlggest cutthroat competition or predation. Our theory says that such episodes, even covering Iong stretches and most of the Iife of the industry, can be consistent with pertkct competition. when the competitive prtlcess is appropriately interpreted in the stocllast ic dynamiccontext. We will return to the policy implications of this in Cllapter 9.
% c
4
* =
=
r-
'c
x1
O
O
Firm-specihc
In this 4
8
12
16
24
20
28
32
36
40
Time lJi:kt
p-r
(.
j2.
?l/7/t.
uhitl?
/'fltlt
Section
Uncertainty
we trcat the other extreme tlastl where
alI
tlncertainty
demand curve ( l ) is purely film-specitic. Now thc industry-wide shllck equal to l and any one Iirm's inverse demand curve becomes
in tlltl F is set
,
t?/'
L'L)I)lJz.r
J'/- (.t,
/4
=
X /.) ( (..l ) .
Rccall the intuition concerning (irm-specific uncdrtaillty t hat we ('tltlined in the jntroduction to Chapter 8. Wllen (lne tirm gets a gotld shtlck' tt) its proitability. it knows that on averagc this ltlck is not shared by its competitors. Thercfore it need not invest at (lnce: it has some leeway to wait lnd see whether this good fortune is transitory. Our first model aims to makd this idtta more
%
precise.
Cl-
l
j'
El
1
l
(R c
j
/
% c
t
!.
This opportunity to wait is relevant
O
o
4
8 Figure 8.3.
,
12
I
16
/1 nolhcr
.
y
$'
Therefore we need anothcr
l
2o 24 Time
only if cach lirm is able to make its
decision after obscrving the current level of its potcntial prolitability. To capture this aspect in its purest form, one would define an indust:y consisting of a sxed population of potential firms. each ofwhom can continuotlsly tlbselwe its firm-specihcshock. and then decide whethcr t() invest. I-lowcver, tllat fails to capture another basic feature of competitive equil ibrium, namely, l'rce entry.
prior stage of decision making, namely, that ()f tlleir currcnt prolit ()f the specilic shock. Therefore we construct a two-stage model as follows. By paying an ently cost R, any Erm can get an initial draw of its demand shock from a known distribution. Thereafter ths variable will follow a geomctric Brownian motion process that is firm-specific, or independent across firms. Each firm can start actual operation by paying a further sunk investment cost /.
becoming one of these potential firms who get to observe potential. This decision must be made whottt knowledge
28
32
Pricc SatnplefklJl oj' t--fy/?/?(.'r
36
40
y
lt:6ltt-YIl7'
268
/5:/!///J?-lf??2
An example will help ix the idea. Consider a pharmaceutical company that can develop a new drug by incurring the research cost R. This yields all initial estimate of its efcacy and prostability. The firm patents the drug, but unless the profit estimate is sufhciently high, it will not incur the additional investment expenditure l that is necessary to begin production. Over time the proht estimate may increase as new uses are found for the drug, or decrease as other drugs to treat the same condition are discovered by other firms. We characterize the competitive equilibrium of such an industry in the long run. There are numerous competitive hrms facing independent shocks. and there is substantial uncertainty and volatility at the firm level. However. different srms'shocks are independent, and the operation of the Iaw of large numbers ensures that industry aggregates are nonrandom.s Thus a nonrandom total volume of output can be produced by firms whose identities change through time but whose aggregate population distribution remains stationary. However, the firm-level uncertainty leaves a mark on the industry equilibrium'. the parameters of the distribution of active tirms, and thcretbre the actual values at which the nonrandom industry quantity and price settle. do depend on the extent of uncertainty faced by each firm. The idea that relatively tranquil industry-wide conditions conceal much firm-level uncertainty has been emphasized in recent empiricai work. Davis and Haltiwanger ( 1990) and others have demonstrated quitu impressively the large gross hirings and firings that underlie small net changes in employment in the U.S. economy. The models that are constructed for applications of this kind generally contain too much context-specihc detail to let the general intuition stand out. Our simple model can help the reader develop a better conceptual understanding and more general intuition lbr such phenomena. We begin by specifying the nature of uncertainty in Xso as to jit with the zr hrm's two stages of decisions. A new entrant gets an nitial draw of from a known distribution. Thereafter its Xevolves as a geometric Brownian motioh 6IX
=
fz
Xdt +
(7'
Xdz.
(26)
We have intemreted #as an idiosyncratic demand shock (random tluctuations of taste shifts giving rise to price premia for slightly diftkrent varieties in the industry). What ultimately matters is the shock to profitability, and we could also think of X as a technology shock that appears in a reduced form in the formula for the hrm's prost flow aer te instantaneously variable choices have been optimized out, as discussed in Chapter 6, Section 3. S
Recall that we arc not giving a formal rigorous treatment of this.
/y/75
/? ltr
f-6111ililnl-il/? z? i,t
z;
i t'tz lllzlll.b'll)' Ltlttlint>lit
There is free entry into the industly and anyone can get the initial draw by paying R. However. there is no obligation to start production at once. A further sunk investment I must be incurred to activate the process, and the hrm can wait to see if A'evolves to a more favorable level before making this
irreversible commitment. We will characterize the long-run stochastic equilibrium of such an industry. In fact we will postulate such an equilibriump and then determine endogenouslythe various stationary magnitudes (price.number of firms, etc.) that constitute the equilibrium. Suppose N is the nonrandom stream of new entrants who pay the fee R and learn their initial X. Then their shocks will evolve independently and stocllastically. A nonrandom flow N1will reach the activation decision. We also want to keep the total number of active lirms, Q, constant. To permit this. we assume that all firms. whether waiting or active, This process face an exogenous Poisson process of death with parameter is also independent across firms. Then in a stationary equilibrium iY1must .
equal
Q.
All uncertainty being idiosyncratic, we specify that each firm is risk neutral and makes its decisions to maximize its expected net worth. Let r denote the risk-free interest rate at which future prolit tlows are discounted. 4.A
The Activation
Decision
In the Iong-run stationary
equilibrium
with a large and constant
number
Q
of active firms. each new entrant or waiting tirm takes this Q as given. Its profit flow is X D( fJ). lt continually obsen'es zr. and decides when to pay its investment cost l and become an active producer. This is formally identical to the basic single-lirm model we studied in Chapter 6, Section I Equation (9) of that section gave us the price threshold P. that triggered investment. In the notation of this chapter, that becomes a threshold A'*on the hrm's shock, and the defining equation becomes .
zr Df Q) where pb is the positive
p1
=
p' -
l
(r +
-
a
(27)
) /,
root of the fundamental quadratic
Q H !a c2 p(p
-
1) + a
p
-
(r +
)
=
0.
The condition that ensures convergence of the expected prolit flow is r+k > a; note that the Poisson death probability acts Iike a discount rate in achieving convergence, as we saw in the discussion of Poisson processes in Chapter 3,
Section 5 and Chapter 4. Section l.I. Asstllning the convklrgencc condition to we have p l > 1 as in the famil iar nlodels t)f Cha pters 5 tnd (). Our intuitive discussion in the introduction tt) th is chapter tells tls tlle reason why equation (27)is exactly analogous to the corresponding tbrmultts lor the monopolist's entfy decision in Chapters 5 and 6. When uncertainty is firm-specihc, a hrm that gets a favorable X does have an edge over its competitors. The favorable ZYis specific to this Iirm; if it does not invest at its zr and jump in. Theretbrc u positive value of once, a rival cannot waiting does survive, and the firm's optimal decision shows familiar inertia. Of coucse this is an incomplete account of the industry equilibrium until show that Q can be determined in a way that is consistent with the above we needs more steps in the development of the model. That story.
be met.
Ssteal''
4.B
I $-1
The Entry Decision
With A'*determincd as above. we can find the value of a potential firm that observes its current zr in an industry with Q active tirms:
I 1'-
'
'ir'
.'
2::.
ae
5k'''* .
This is also familiar from Chapters 5-7. The uppcr line is the optitln value for a waiting tirm,and the lower line is the cxpected present value (.)1-profits that the potential firm will get by immediate activation, net of tlle costs of the activation.Ofcourse the latter formula applies in the region where immediate activation is optimal, and the former in the rcgion wllere waiting is optimal. ln tkct .Y* is determined by starting with these two expressions and value malching and smooth pasting. That also yields
A plltential entrant can then calctllate its cxpected payllITC r! P' ( A'. ()) 1. Free entry conditioned using the knllwn distributitln 0f the initial draw 01zr.
ensures
C.v(V ( X. ? ) l
zr,
the zero expected net value condition ---p3
value function
P' (A3. Figure 8.4 shows a typical value function and its shift as changes. The intuition is that if there are more active firms in the industry, Q
any one new entrant perceives :1 smaller prospective protit flow and therefore requires a higher firm-specilic shock to prolit before committing itself to activation.
(3())
-
where R is the initial entry cost. We just saw that the left-hand side is monotonic in )-, therefore this equation determines the equilibrium Q. As an example, supposc the distribution of the initial draw is uniform zr over the interval ((). ). If the activation thrcshold X* turns out to be larger than
A highcr Q Iowers D( Q).Theretbre from (29) it Iowers z1( (2),and from (27) it raises A'*. Finally, from (28)we see that a higher Q Iowers the whole
R
=
A ( 0-.) X
(30) is
/ (1 + #I )
=
R
.
(but
In simple cases, for instance, if D Q) is isoelastic, this admits an explicit < X, we get solution for Q. 11. 'e<
algebraicallymessy) A ( Q)
1 + pb
(-Y*)v
I+
1
a''*
+
D ( Q)
2 (r +
This must be solved numerically.
- x)
..-.
gX a
-
( A'*)
?
)-
j j
a-y
-y.-
vc
)
.
p
.
dlt/ttilil) rl/? /??f 11I-$./?Jk'
??
Even now the argument is not complete. The number of active Iirms Q arises from a complex chain of initial entry decisions- independent rnndom fluctuations of the tirms' shock variables X, subsequent entry decisions, and independent random deaths. We must show how these interact in a consistent way to produce the industry's equilibrium Q.
4.C
The Distribution of Firms
Recall the actual life history of any one firm that has just paid the entry cost R. It begins with an X randomly drawn from its known distribution. If the initial Xexceeds the threshold zP, the lirm pays the investment cost / and becomes an active producer at once. Otherwise it Iets its Xevolve. and activates if and when A'*is reached. Throughout this process. the hrm faces a constant and exogenous probability rate of death. Such new entrants arrive at rate N. The full stochastic dynamics of each of them the probability that it will be alive and occupy a position X at time f--can be examined using the Kolmogorov equation. which we developed in the Appendix to Chaptcr 3. Here our aim is more Iimited. For industry equiIibrium, only the total numbers of tirms in various states matter I1()wmany what valucs of X. Theretbre the active. and waiting with llow arc many are 1awof large numbers a'llows us to restrict attention to a long-run stationary equilibrium. This means that the rates of Poisson dcath by exit. and of ltctivation, are constant through time. Likewise, the numbers of Iirms with varillus current Ievels of X are constant through tim. Of course thc actual identities of the tirmsoccupying these positions keep changing, but for our purposc any Iirm is like any other with the same zY. The method of calculating this long-run distribution of hrms is the same that of Chapter 3. Section 5, but now we must include two new features. as namely. fresh entry and Poisson dcaths. lt proves more convenient to work 1og X. Let g(-r) denote the density function in terms of the logarithm. of the initial draw of x, and G(x4 the corresponding cumulative distribution. Note that the range of x extends to log zP. Of the to the left. Let newlyentering firms, N (1 G(-r*)Jimmediately get a draw Iarge enough to justify activation. The rest join the mass of firms that do not complete the second step of committing the investment cost at once, but wait to reach the activation threshold. For both groups. continues to evolve. Applying Ito's Lemma to (26), we see that x follows the Brownian motion -t'
=
=
-r
=
t? J
where !?
=
a
-
t?
tlilL,tili tqrtpz
ztr
F?lcllt.b.:l y'
1. t7'?. Also, both groups suffer exogenous
awith
'tdeaths''
under the
Poisson process parameter Begin with the waiting firms. which are distributed over the range Let N $ (.v) denote the density of such tirms at Iocation x; the (-co. factor N just scales this by the rate of entry and Ieads to a simpler equation for 4(.:.). For the density to remain constant through time, the rate at which firms arrive at x (havingreceived positive shocks from below or negative shocks from above) must equal the rate at which firms at x move away (having received shocks of the Brownian motion process or Poisson death). We tlow'' of tirms in a more precise way. express this equation of For this purpose. we use the binomial approximation to Brownian motion that proved so useful in Chapter 3. Section 2(b). Divide time into short intervals of duration dt. and the x space into short segments, each of Iength t'r dt. Of the lirms located in one such segment. in one short time J (/J will die. Of the rests :1 fraction # will move one interval a proportion where will right. and a t'raction q move to the Ieft, segment to the .
.v*).
''balanced
=
la
;, ==
g I
--?
(/?
-#-
fT
j
jl
,
-b'
t
-
fT
tt
j
.
It starts out with N 4 (.r) tlh Now consider the segment centered at ltll ()f (lt, these fnove away with either hrms. In thtl next unit time period Poisson or Brownian shocks. New cntrants. as wcll as firms from the Ieft and right. arrive to take their places. Figure 8.5 sllows these flows schematically. For balancu we need .r.
N
4 (x ) J
=
N
+
t// (/
gx ) dh +
(l
-
/J
J/ ) N
(1
-
t/?
) N / (x
4 (x + J ) dh
-
t/l
) dh
(32)
.
Cancclling the common factor N dh, expandingthe Sxzdh) on the right hand simplifying, we get the differential equation
side by Xtylor's theorem, and
-r*
-c'o
-
Jx
???'c- S-:/? / ililil'il /? ?l il l 11611
t + c Jz
,
1 c2 /?'(x) a
-
w 4'(.v)
-
4(.v) +
g(-t)
=
().
(33)
This equation is slightly different from the one of Chapters 5-7 because it pertains to a simple rather than geometric Brownian motion. However, the method of solution is ve.y similar. It is easy to verify that the general solution has the form
Illtlll-tr)ldsf/l/flprlf//l pN(1
-
(htx -
>.dtj
#(X)d/1
.
Iilillt-itll
?1
il !
tl
(75,tll, cz//ib,e111(l/-/? )'
For the second condition. observe that the firms that hit becomc ltctive this and are lost to distribution. Consider the segnlent of the space just to # the Ieft of x centered at dh For t his segment- we must modi tk equation the balance (32)above. It does not get any incomers from the right. .r*
qNj
dhldh
Dv?lf? ? ?: ic.df7I
-
tx+
J1) dhjdh
.
.j-
*
.r
Y
o
%; s
c,A-
+
x
=
-6)
,jOo
Entry to x
=
*
-
x
dh
+
x
i
* -
dh
x
Exit from x O
o >
qN(1
-
A.dl)
j
c
N. dt
@
4
o R =
.
o
'>
Since these firms activate in the time interval t// the rate of activation is 2 N - 2 t.y 4/ ( v. ) sve illustrate this calculation for a case ttmenable to an analyticul solution. Suppose the distribution ot zr is uniform over thc intelwal ((). Then .r log has an exponential distribution over (-x. where :Iog 7t Thus ,
.!.
*
-
.
eu ,r).
.
=
PN(1
.
bxldh
-
.-*.
'Vo
%o %
cs
*
O
%z &
%
+
=
-'*
x
cyN
Similar simplification now gives 4 (-r) ().Letting (//1 go to zero, we have the condition 4 (-r ) 0. 'We can also calculate the rate at which waiting Iirms hit and so become active. These are just the fraction p ( l l dt ) of the N 4 (.r J/?) J/l tirms located in the segment just to the le of Using Taylor's theorem again, the leading term in this expression is
+
i
dh
.
o oo&
4oo +
-
-
therefore
VJ
R
-r
=
.
.
X d1)
.
fldry'. Slltjt. and Er/
(?f
G (-r)
I'p/ikg Firnts
where the last term is a particular solution of the full equation. and the hrst two terms arise from the general solution of the homogeneous part. The constants Cl and Ca remain to be determined, while pl and p'a are roots of the quadratic
2-
1 r.r2
y
2
2
-
p
y
-
=
0
.,
thus one is positive
and the other is negative. condition for determination of the constants Cl and (:2 is found The from the consideration that the total mass of waiting hrms, that is,
srst
*
.
.-)r
=
,
(')(:?l
txldh
Figltre 8.5.
-
yr
17
$ (A') dx,
-X
must be hnite. That helps rule out the negative root in the solution of the homogeneouspart, as a negative exponential would go to inhnity as-r -..+
-co.
=
g(-r )
exp ( x
=
Ji-)
-
.
.
We will simply assume that the rest of the parametcrs of the problem arc such that the activation threshold x* turns out to be Iess than leavjng the other case,which is simpler because none of the entering hrms activate immediately, to the reader. A particular integral of (33)is easily verihed to be ,
4()(.r)
=
ex-k
/
(
+ w
-
2
v2
) .
T'he denominator of this must be positive for it to make economic sense. This just says that the quadratic expression (2 evaluated at y 1 is negative. Then the positive root yq must in fact exceed unity. We assume this, and for typographic convcnience write that root as y. Then the general solution of (33)is =
$ (A')
=
C
CY''
+
/()(-r)
.
Illdusly
/(a-*)
The constant C is to be determined from the condition
=
ltl/7 I tr f
ffxxrfrl
0. This yields
(34) From this we can calculate some aggregates for the waiting hrms. Their total number is
ililll-i f 11 l il I
The various
Cl
I (II fu5'JJ)7 p'C 11
('fl tllllL'tili
algebraic
expressions
get complicated.
but some general
principles stand out. When the firms in an industry are subject to specitic shocks, the investment decisions of each are importantly intluenced by the option value of waiting for better realizations of its own shock. At the industry level. the shocks and the responses of tirms can aggregate into long-run stationary conditions, so that the industry output and price are nonrandom.
However. the equilibrium Ievels of these variables are aftkcted by the parameters of firm-specilic uncertainty. Also. behind the aggregate certainty lies lluctuations: firms enter, invest. and exit in a great deal of randomness and fortunes. individual shocks their the to response to In reality, industrpwide and hrm-specitic shocks occur together. Therefore we will proceed to combine the models studied thus far in this chapter into a single general model that encompasses both kinds of shocks. Then in Chapter 12 we will consider an application to some real data. '
is
The rate of activation - a
o' a
'
+
N / (-v)
1 o' 2 (y 2
N
=
+
p
1)
-
.
j tz a
-
e.r
f
-.
'
2
We could similarly tind the distribution of the mass of active firms-These G t-x. x). because some firms may will extend over the entire range decline. They are also augmented by their X have activated and then had and activates that its initial JYabtwe finds that part of the new entry Ilow diminished numbers Their x*. at flow are immediately,and by the activation deaths. Poisson by the Ilow of In fact we only need the total numbcr of active firms, not the distribution llow that of their X values, to find the industry equilibrium. The new entry from flow found activation was G'(.v*)1. T he N 1 activates immediately is c2 active of population The above waiting hrms as /'(x*). the solution for 2 numbers constant, we need hrms being Q,the death tlow is l Q. To keep the .r
a'*
-
-1
.Q We can calculate
=
N
gl
*
-
G (x )
Q from earlier
x* and
-
12
2
f:r
$' ( x' ) ) *
<
conditions
.
of equilibrium'. the activaT'herefore this equation
tion condition (27)and the free entry condition (30). delines N. Since new entrants are indifferent lo their choice because of the convention of long-run free entry condition (30), we can adopt the usual them of do enter. Thus the equilibrium analysis and suppose that just enough stationary''
Hstochastically
ln our special (35) becomes
depicted above can bc sustained. has the uniform distribution, the condition
equilibrium
case where
zr
+w + b'
1 tz 2 y 2
-
l Q
=
N
l
-
e
.r* -.
-
1 tz' 2
2
.
This ailows us to calculate the rate of new entry N, either relative to thc mass waiting hrms, of active firms Q, or relative to the total mass of active and
(Q +
M4.
5 A General Model Now we examine industry cquilibrium in a more general model where thc two kinds of uncertainty, firm-specihc and industry-wide, coexist. To allow tbr the added notational and mathematical complexities of the joint uncertainty, we makc some simplifications in cach. 0ur treatment closely follows Caballero and Pindyck ( 1992). We assume that the hrms are risk neutral (or that the industrpwide risk is uncorrelated with the risk of the economy's capital market as a whole), leaving the more general case for the reader. We suppose that the industry demand is isoelastic', thus the dcmand equation ( 1) becomes X
=
YF
where 1/6 is the price elasticity of demand. We also omit the second siagc from the above model of hrm-specihc uncertainty; thus / a'P
(36)
Q-V
activation =
0 and
() As before, each firm has the capacity to produce a unit of output and has no variable costs of production; thus, # is also its profit Ilow. Also as before, we can think of this as the reduced form of a more general structure in which any variable inputs (thosewithout irreversibility or adjustment cost) are chosen at their optimal levels. We continue to treat firms as intinitesimal and their number as a continuous variable', then Q equals this number. =
.
llltlllslryStylt//prl/pn The two shocks Ibllow independent geometric Brownian motion proc-
esses.g Let dzk and Jz, be increments of independent standard Wiener dt and #z,,) 0. Then the firmprocesses; thus )Jt#4r ) V(#z,,) shock follows specific
D3'??f???1i(' iL.f/I li lilll-iltl7l il !
(IX
and the industrpwide
=
=
ea. Xdt + h
a
$,
il :tr
rattl is incrtlascl by t() a IIovvfor Poisson ne r. This sat isties t 11tld iflre 1-1t ial equation
I'nan
svhere
zrtzk ,
F' Jl + o'b. )' Jz$..
(38)
New entrants pay the sunk cost R and get an initial X chosen from the knowndistribution. Let Tdenote the expected value ofthis initial zt.-rhere are no further costs of activation, and the proEt flow is always positive. Theretbre all entrants will become active at once. In the model of Section 2 we had only industry-wide unccrtainty, and the equilibriumwas characterized by an upper reflecting barrier on the price. With added hrm-specific uncertainty, the natural generalization is a similar barrier on the industry-wide factors in the expression (36)for the pricea namely, l/P'
=
J'
Q-'
=
=
(a, +
6)
tleatlls
i1) t he
Wdt + cp Wdzv.
lr ((z',. + 6 ) e,,, is :ln abb reviation This value function wil be homogeneous of degree 1 in A': a firm that is luck'y enough to have twice as Iarge an initial draw of its specilic shock wi11forever have thc expectation of tsvice as Iarge a pro (it lltlw and there fore twice as Iarge a value. However. the value function svill not be honlogeneous of degree l in I'U-.if the indtlstry-wide shock is larger, the threshold that draws fresh entry will be reached sooner and so the period of proportionately larger profit tlow will not Iast as Iong. Thus wtl can write F' ( l.U) ZY,( lz23. Substituting in to the above partial differential eqkjation for l,' yic lds an (rd inary diffe re n tial equat io n for p. nam t.lIy .
'
=
This takes us back to the
tnliliar
ground of Chapters ($and 7. The solution is
(39)
.
The idea is that as long as kk'is below a threshold level kU,no new firms will enter. while existing hrms continue to suffer Poisson deaths. so do Q. The geometric Brownian motion of )' and the Poisson death process of Qwill induce a geometric Brownian motion on 1.f',namely, crP'
l p?t 111.$,/77*
=
shock follows
dY
't'
where t I'le t.liscotl nt noNv-farll
-tgr
=
=
llll'ili (--r??
tl
-
(40)
However, if I'P'rises to the threshold W,enough new firms will enter to prevent it rising any further, thus making W an upper renecting barrier on the process (40). Of course 'V must be determined endogenously as a part of the system of equations for the equilibrium. We now proceed to verify this conjecture. The value of an active tirmis
wherc /1 is qtladratic
a constant to be determined, and : '!-cy, : . p (J$ -
As usual
we
have eliminated
l ) + au
/.$ -
thc term
(r +
p is
the positive n)()t
-
r
)
=
()1*
the
i).
nvolving the negative
root by consid-
ering the limit as 1#' goes to zcro. The interpretation ()f the slllution should also be familiar from our carlier treatment of purely aggregate tlncertainty. The tirst term t)n the right-hand side in the solution represents the expected present value of profit l-lowif IzP'could continue its process without any fear of fresh entry, and the second term is the reduction in value because actual fresh entry places the ceiling on the F proccss. Also as usual. for the solution to make economic sense, we need the dcnominator in the hrst term on the e?,,), to be positive. This in turn yields right-hand side, namely, (r' + ax #l > l -
-
.
The constant is determined, as was done earlier in thischapter in equation condition at the rcllecting barrier: v'( 1,1Z) (). (7), from the This yields Rsmooth-pasting''
=
'i-rhc assumption of independence is natural. Decomposc A-into two components. one in
the direction of F' and
the other ortbogonal to it. The former, being perfectly correlatcd with the have to be industry-widc and so should be includcd in F' itself. That leaves the (irm-specificshot:k independent of F F shock. would
.
/?1411/./?3/
280
t /z? lt ililnl-il
zYp(Hz').A of a firm with a givcn X is VLX )F) value expected the W''. Therefore only observes however, entrant, prospective rI'T is defined by the from the entrant's perspective is -Xp(hF). The threshold value must equal the cost of entry R expected this of free ently condition so at that point. Substituting and simplifying, we tind The actual
value
=
WW
=
/31 (? + .
/:$1 1 -
).
-
ax
-
a.)
R
.
This is a very natural generalization that combines the corresponding equations for the case of pure industrrwide uncertainty - equation (10)of Sec(27)of Section 4 tion 2 above-and pure firm-specific uncertainty--equation need should familiar comment. by be to now too above. Its properties For further theoretical details of this model we refer the reader to Caballero and Pindyck (1992).In Chapter 12 we will discuss an empirical application of it.
6
Guide to the Literature
Lucas and Prescott ( l 971 ) considered the rational expectations equilibrium of investment in a competitive industry using a discrete-time Markov chain model. They established the optimality of the equilibrium. Lippman and Rumelt (1985) combined entry and exit in a similar model. Edleson and Osband ( 1988) showed that a competitive tirm's entry and exit thresholds in equilibrium were not !he Marshallian ones. The coincidence between a monopoly firm's option-value thresholds and a competitive hrm's free entry threshold was first noted by Leahy (1992). Our exposition mostly follows Dixit ( 1993b). Dumas (1992) presented an early model of general equilibrium where each firm made costly switching decisions under Brownian motion uncertainty; this was in the context f international relocation o productive capital, and the aim was to characterize the equilibrium dynamics of real exchange
rates. Caballero (1991 ) attempted to study industry equilibrfum by the shortcut of modellingjust one lirm's decision, parametrically varying the price elasticity of its demand, and interpreting perfect competition as the limit when the elasticitygoes to inlinity. He argued that the traditional present value criterion would be restored in this limit. However, one must recognize that the level of price about which to make the demand more elastic is itself endogenous,
and moreovcr, it acts as a ceiling or rctlecting barrier. These effects can be understood only by conducting a proper industry-levcl analysis. See Pindyck ( 1993a) for a more specihc and detaildd discussion of this point. Caballero and Pindyck ( 1992) laterjoined forces to prgduce the model of industry equilibrium with combined tirm-specificand aggregate uncertainty that forms the basis of our treatment above.
Chapter
9
Policy lntervention and
lmperfect Competition
no inherent distortion or market failure, so the inertia is socially optimal. A planner facing the same uncertainty and contemplating the same irrcversible investment decisions should hesitate jtlst as much. Policy intervcntion is justitied only if some kind of nzarket failure coexists with the uncerttinty and irreversibility.A natural one in this context is the failure of mark-ets to share risk. and we discuss the isstles that it raises. However, we also tind that some policies to reduce risk. for example. price controls, can have counterproductive and even surprising side effects. The mathematical
apparatus
for examining social optimality has a useful
by-product: it allows us to extend the simple model of Chapter 8 i!1:1 variety of useful ways. Earlier we had assumed multiplicative demand shocks. and identical firms each with unit output capacity and zero variable costs of production. Using the new techniques, we can remove 1111 of these restrictions. The only diftkrence this makes is that in industry eqtlilibrium tht! ceiling (rcflecting barrier) on the price process is no longer constant, btlt can change
with the
IN TldE
previous chapter
we developed
a basic model of competitive
industry
equilibrium in which each firm makes its irreversible investment decisions in an environment of ongoing uncertainty. knowing that al1 other (irms are similarlysituated and are making similar decisions. We considered industrywide as well as firm-specific uncertainty, and found that both had the effect of makingfirms less cager to invest, but for different reasons. With industry-wide uncertainty, each hrm knows that others will enter or expand in response to favorable developments just as it does. The resulting incrcase in supply will dampen the price increase. thereby rcducing the firm's upside profit potential, and therefore the expected value of its own investment. With (irm-specihc uncertinty, each firm can take advantage of its own good fortune. However, then it must compare immediate investment against the alternative ofwaiting to reassess the situation. Waiting allows it to See if its good fortune is transient. and thus reduce the downside risk of investment. Thus uncertainty makes waiting more valuable, and discourages immediate investment. ln this chapter we extend the simple model that was used to make thdse points, and pursue its further implications. The srst issue is with respect to the social optimality of this industry equilibrium. Many would argue that if hrms are hesitant to invest in the face of ongoing uncertainty, the government should act to provide additional investment incentives. We find that this argument is often wrong, and at the very least must be severely qualified. The equilibrium
of the basic model displays a great deal
()f
inertia, but has
currently
active numbcr of firms.
Finally, we rellkxthe assumption of perfect competition, and consider the intlustryequilibrium when the number of potential competitors is small. The stochastic choice problem then becomes :1 stochastic game. Such games are quitc difficult with regard to both their general theory and their solution in our speci ic contcxt, but we develop a simple special model that brings out some useful insights. A more gcneral analysis of this kind is 11 promising topic for futurc research. Even more promising arcas of research related to the wtlrk of these chapters comprise econometric work to test these theories. and empiricai work to apply them where appropriatc. In Chapter 12 we provide a brief surveyof the Iittle work of this kind known to tls. and wtt suggest avenues for
further empirical
research.
1 Social Optimality Our industry consists of hrms that operate in a well-ftlnctioning
capital
market where any risk is efliciently priced. The output market is competitive, and the firms have rational expectations. This is just the standard Arrow-
Debreu frtmework where complete markets efficiently coordinate decisions uncertainty. Therefore we should expect the indtlstry socially optimal. We now demonstrate this optimality
involving time and equilibrium to be explicitly.
'f/ldl'//?rllz/l
Indttstry
284
IAlllicb'
ln the process we develop a new method of solving for the industry equilibrium that proves useful in its own right. The social optimum is the soluton market to a simple dynamic optimization problem. When it replicates the optimization problem. equilibriumswe can find the Iatter by solving this same
In all but the simplest of models, this metod of hnding equilibrium is far easier than trying to find a flxed point of the endogenous stochastic process of prices. We will demonstrate this by extending the simple model of Chapter 8 short-run to allow tbr heterogeneous hrms, each having an upward-sloping supply curve, and to handle nonmultiplicative shocks to demand.
Optimum and Equilibrium
The Correspondence be-een
the social optimum in the basic model of Chapter 8 with aggregate uncertainty. Recall that in that model, the industry's inverse demand curve is p = y 0( (2) We begin by characterizing
,
the multiplicative
where motion
shift variable JF
=
a i'
ilt
F tbllows the geometric Brownian +
t:r
)'
(2)
t/z.
The industry output Q equals the number of producing units, which is trcated as a continuous variable. Each unit can be activated by incurring a sunk cost 1. Suppose the decision to add capacity to the industry is taken by a social planner. In this context, we can trcat the productive units as hrms in a decentralized interpretation or implementation of the social optimum. The area under the demand curve at any instant is given by
f?l/tr?3Jc?lt()?l
llltl
/??lJ?c?7L,c-t(->(???1/?tz//t??
285
where the sum is takell ovcr all those instants when capacity additions take place. The uncertllin reventles are discotlnted at the rate y. that includes the systematic risk. while the eertain costs are discounted at the riskless rate r. The planner maximizes this social net present value, given the initial conditions, say. F() F and (.?() Q. =
=
This is a dynamic programming problcm. Here we will solve it somewhat
heuristicallyso as to get quick Iy to the results. In Chapter l 1 we will examine a similar problem of a single firm's capacity expansion in somewhat greater formal detail-l
functisn, thafl' 7 Let 1'P'(Q. F) denote the maximized value of the objective is, the Bellman value function. where the arguments are the initial state ( Q. F'). As usual in dynamic programming, we break up the problem into the immediate future and the continuatitln beyond it. Tlle social planner-s immediate cuncern is whether to expand the capacity by some amount 61 Q. This would (/ cost l Q, and would increase the value function by 1'P( Q. i') J Q. Theretbre ( Q. )') < I We expect intuitively that higher the addition is not justihed if rl'''(? values of F' will justif further capacity additions. Therefore for each Q there will be a threshold. which we denote by /' ( )). such that the marginal addition 6IQwill be optimal only when )' reaches F( (?).Moreover. the curve F( Q) will be upward sloping. Of cotlrstl we must obtain F( Qj as a part of the solution. In the regilln btllow this curve. wherc F < F ( Q)j.tlae Iirm will not add capacitytbr thc next shllrt interval of time dl Thercford the Bellman equation of dynamic programming will become .
.
Expanding the right-hand side using Ito's Lemma and simplifying as we did so often in Chapters 5 and 6. we see that the Bellman function satisties the
differential equation This can be thought of as the flow of total social utilitygenerated
by the output
)' D (.2)is the shift variable takes the value F; then )' &'( Q) llow Q this the implementation, isjust the utility. decentralized marginal social ln the variable maximize aims the social the planner there to Since costs, are no price. capacity expansion. value utility, of social of the net of costs expected present The social utility at any instant t is )'/ b'( Q(). If an amount LNQt is added to when
capacity at this instant. a cost equal to I objective function is therefore
=
2
Qt is incurred. The
social planner's
For each
Q, we can regard this
to F. Again we draw on the down the solution
as an ordinary differential equation linking I'P' experience of work in Chapters 5 and 6 to write
pP'(Q. )')
=
Q) F'/11+ r &(:)/J.
B
1A fter rcading Chaptc r 1 I Section l rcatle rs mkly lind it usefu I tt) retu rn to this section and comparc the nlethllds and the resu Its. .
,
286
lndttstty l-/ldpl'l?ild?n
where S( Q) is a function to be determined,z is dehned the pos it ive root o f the fundamental quadratic (
as f.z
-
a, and
/:31is
,
O <'
Assuming as before,
1- >
we
() and
1.c2
>
pp
2
>
l ) + (r
-
j)
-
p
t-
-
=
().
(), as needed for convergence, we have pk > 1. Also, the limit as i' 0 to eliminate the term with
have considered
--,.
D
xO
licv
,97/t:v3.t:-?7?t-.v?
1.U'(.?( Q. F) in the Bellman function equals the cost / of the marginal unit value-matching condition). Therefore
B'Q) r:,
+ ?' &'( (.?)/J
(the
p B'Q)
3'/',-. +
We can solve these two equations F(
Q4U ,( Qj
l )/' - I
-(p1 -
The precise expression
lfI
&, ((?)
I 1-,,
tti
/ pl p,
.
is not of great interest here. btlt tlle interpretation
of BQ) has considerable importancc. In equation (5)athe second terma )' &( Q)/, is the expected present value of &( Q$ if Q is held fixed at its current level while F follows its Brownian motion process. Theretbre the Iirst term, S( Q) ):#' must be the value of society's ability to invest and thereby increase Q optimally in response to the evolution t)f F. In other wllrds, the first term is the value placed by society on its options to explnd this intlustry. ,
Moreover, the derivatives of the gain /1/: and the cost 1 with respect to )' F) must be zero, must be equal (thesmooth-pasting condition). So 1,#*42jz4.?.
or
B' (2) =
'
1.
=
,'t't
/??l/7t:'?./- C't',)l?lpt://'/t)??
us to look tbr a tixedpoint in the space of l'tlnctillns M1( (?),wllicll can be vttr.y diffcult. The social optimization approacl'l dxtends far nlorkl easily fo sucll contexts-,we will illustrate some of these in the next section. Returning to the simple model, wtt ctn also eliiminate F between equations (6) and (7) to get B' O). and integrate tllis to Iind l)( ?) and thtls complete the soltltion for 1zF'1 Q. F). %ke have
the negative
root pz. To complete the solution we must find the optimal capacity expansion policy.At the boundary where the marginal Joth unit is added, the increase
illlt
&'(Q)/
0.
=
to hnd the threshold =
/3j
/11 -
/5
l
1.
F((.?); it satishes
(8)
The equation has a very natural interpretation. The left-hand side is just the derivative with respect to Q of the llow of total social utility given by equation (3)sand thus equals F D( (.?),the price. We see, then, that the socially optimal threshold is dehned very neatly in terms of a critical level of the price. More remarkably, this critical price is just the competitive equilibrium entry threshold that we found in equation (9)of Chapter 8. The social optimum and the competitive equilibrium are one and the same. Tliis conhrms the intuition tht led us down this avenue in the first place.3 The social optimality perspective. although formally equivalent to the perspective of a competitive equilibrium, has some practical advantages. For example, if the demand shock is not multiplicative. the entry threshold price varies with the current capacity. A pure equilibrium approach would require
When a marginal expansion
is actually carried out- society gains )' &'( Q.)4/(q output. but loses the value
in the form of expected present value of the extra F#' of the marginal option thus exercised - #'( Q)
lnote B'( ?) is negative 1.
Then the value-matching condition (6) equates the balance of these two effects (thenet benefit) of the action to its cost 1. We can use the notation ()f Chapters 5 and ( and write
l( (). F ) Then
=
F U'( Qjlk
j'f Q. F )
.
=
-
B' ( )) F/'
.
Q. F) is the value
of the marginal unit aer it is nstalled- and Q. F') valuc of the option t() install it. Thc (lptimal the threshllld is found from the value-matching and smooth-pasting conditions exactly analogous to those of equations (7) and (8) in Chapter 6, namcly p(
,/'(
-#
?since Q entcrs also depends on Q.
the differential equation
(4)like
a parameter.
the constant of integration
3We have merely verificd the coincidence of the equilibrium and the optimum for a special ktnd rigorous proof. see Lucas and Prescott ( l 97 l ). a general
example. For
/(
Q. Y)
=
p
( Q. F')
-
/
.1''
,
( Q. F')
L,)' (
=
Q )' ) ,
.
Then the value of this whole industry to society, when Q units arc already installed and the current level of the stochastic variable is F, is givcn by 1,F(Q. F)
Q
'x.
F') dq +
./'(/ )' ) Jtf Q that is, the sum of the values in place of aIl the installed unitsa and the sum of the values of options to install aII the future units. Tlle reader can easily check that our solution in equation (5) for k$/'(Q, F) achitwes exactly this. In Chapter l 1 we will consider a firm that can expand its capacity incrementally, and find a similar option value component in the total valuc (a1' the
srm. There
plt?
=
()
,
,
we will recapitulate this intuition and build
,
()n
it.
Illtittstl'
288
/7t)lic:' /?l t crw'c?;f i.l ,1
/J-t/I////??-lI??l
1.B A More General Model The social optimality perspective allows us to generalize the model ot' industry equilibrium, and in the process bring out an important principle that governs competitive firms' irreversible investment decisons. Suppose industry demand takes the very general form
(1 llt
c/7-L'cf (& / /???/?
?7/Jz,t Jl'(J/l
:! 9t)
In other words, the marginal cost is equalized across hrms, and the aggregate marginal cost. That theorem also gives
marginal cost equals each firm's
The aggregate output tlow is also a matter of choice. The social planner of utility over cost), will choose this to mximize the net social surplus (excess whichwe denote by SN. F) When the demand shift variable is at )' and there are N active tirms, this means .
P
=
D( Q, 1'')
.
multiplicaThus we no longer restrict the demand shift variable F to enter becomes then under demand the the expression curve tbr area tively.The
( 10) Differentiating, we have Ul Q. F) = D( Q. F). We also let tirmshave more general and llexible technologies. Each hrm Jlnd hrms differ in their abilities to can vary its rate of output instantaneously. /1), where ty is its rate dt.xso. Suppose Iirm n has the variable cost function cq. total of output. These functions have the standard properties of incrcasing of in order labelled and marginal costs. cq > 0. ckt/ > (). and the hrms are increasing cost, so c,, > 0 and c#,, > 0. Suppose N hrms are currently active; we regard this as a continuous variable-Asocial planner, or the discipline ofa competitive market.will ensure with firm 11 producing that aggregate production Q is allocated auross firms the aggrcgate cost function qn) in such a way as to minimize cost. Thus C( Q, N) will achieve the foliowing minimum: 39
in
m t/ (n )
()
cq (n)-
The lirst-order conditions for this minimization ,
where ts is the Lagrange
)')
=
max
( U Q. F)
The first-order condition for this is &q)(Q. F) D
Q. l')
=
cq q (/1) .
?1)
=
(o
C( Q. N4 1.
-
=
Cbt
Q. N). or
for aIl lt
Thus price equals each irm's marginal cost. which of short-run equilibrium in a competitive market. By the envelope theorem, we have
(().N).
(EE
is the standard condition
This isjust the Iast firm's operating profit tlow,and we abbreviate it as zr( N, F). Now consider the social planner's investment problem. A new firm can be establishcd by making an irreversible investment 1. For notational simplicity valuation prowe assume risk neutrality, althllugh the equivalent risk-neutral cedure of Chapter 4, Section 3 can handle risk aversion. By natural extension of the previous special case, the social objective is
subject to
?l) (113
tl) cz/((2(11)
.( N.
czz (s
multiplier.;
for aIl 11
are EE
when new firms are established. maximization the result of the I#'(N. denote Let F) as a function of the initial value retracing the steps of the namely, function. F), By Bellman the state (N. in the previous subsection, we can multiplicative shock case argument of the write down its form
where the sum is taken over all those instants
It).NJ .
By the envelope theorem. we have
CQ ( Q, /V) = u).
with the largest 1If N is large relative to Q,there may bc a cornt!r solutitln where silme tirms will conlplicale tht algebra sliglltly without initial margillal ctlst c (0. n ) have zcro output. This this consideration. changing anything of economic significance: tllereforc we umit
rk'(,V, F)
=
B(N4 3'/'. + T(N. F).
where F(/V, F) is the expected present value of the surplus that would result if N were kept constant at its initial value forever, and the hrst term is the value of society's expansion
options.
290
lndttstry Eiplilibriunl
The critical Ievel of )' that would make it optimal to establish the marginal Nth firm is then given by the familiar value-matching and smooth-pasting
conditions l'Irv( N, F)
Eliminating B'p
B' N) F/3' + Tx ( /V. F')
<
between these two equations, Tx(N. F)
-
i' C,v)' (N' r) pb
we
=
=
/.
get /.
This is the implicit function dehning the threshold F' as a function of N. To interpret it, observe that *
F(N. i')
'
=
SLN, )$) e-rt dt
We saw above
C
uv
()
?
.
that (N. )') was simply te operating profit n. (N, F) of the Nth firm. Thus TN N, F) is just the expected value of this operating profit, calculated holding N constant while Yt follows its stochastic process starting at the given initial Y. Then T/vr is the marginal effect on tis expected value of starting at a higher level of the stochastic variable F. Since higher F raises the prolit, Tvv is positive. Therefore in equation ( l 1) we must have Tv > 1. In other words, at the threshold that justihesestablishing the marginal 5rm (IN when there are N actual hrms, the marginal expected value of so doing exceeds the cost of the action. The reason is also familiar: the excess is just the opportunity cost to society of exercising the option. Thus the general model in many respects replicates the analysis and the results of the earlier simple case. However, one result does not sulwive; only in very special cases will the threshold level found from (1 1) imply a constant price D( Q, F'). Generally, the threshold price will be a function of the current number of lirms, N. Once again, since there are no distortions or market failures, the social planner's problem yields the same result as would a direct solution for the competitive equilibrium, but now we see te merit of our indirect approach. u/v
If the entry thresllold price is a ftlnction W(N), then tlltt endogenous pricul process of the equilibrium is much mtlre complicated. Its ceiling (reflecting barrier) shifts as new Iirms enter. To tind the equilibrium. we mtlst solve a fixed-point problem in a complicated ftlnction space. The social optimality problem remains :1 simple dynamic optimization calculation. There is more. The tbrmula ( l l ) conceals an important and simple principle that governs the entry decisions of competitive tirms.Suppose N firms are already in the industry, and the next marginal firm is contemplating entry. Suppose it has rational expectations about the stochastic evolution of F, but it assumes itself to be the Iast entrant, ignoring the fact that other tirms will enter after it if and when F rises to suitable Ievels. It will calculate the expected present value of its prolits as X
p( N. F')
zr ( N. F, ) c
C
=
()
enter, and tind
(N. 5 ) e-r dt
29 l
-''
?
Jf
=
Fv( N. F )
.
It will carry out the usual calculation for the value /'( N. 1') of its option to
X =
/??.1/?tp-/t',t'? C-t.ll'tll'tititlll
t?,!t/
'elltio'l
.
Differentiating under the integral sign, we get Fx ( N. l')
fp/cv /rl?t.v7
.l'N
1')
=
bNj F#'
where b N) is determined jointlywith the entry value-matching and smooth-pasting conditions /'( N. F )
=
t? (n
.
1/-)
-
1.
.
( N. F )
threshold
=
t.' y ( N,
F ( N) from the
1/)
.
It is easy to check that. on eliminating /J( N). these reduce to equation ( l l )p exactlythe equation for the threshold in the social optimum cum competitive equilibrium. where each frm recognizes the possibility of futtlre ttntry and of the effect of such entry on the price process and on its own prolit tlow. In other words. each firm can make its entry decision by finding the expected present value of its prohts as if it were the Iast h'nn #ItIJ would enter this industry, and then making the standard option valtle calculation. While the 5rm should entertain rational expectations about the stochastic process other lirl'n.' enlr.v t7trc'.s'l'tlrly. Not of Y, it can be totallv myopic in the matter W* would reach the if it correctly anticipated their only does it same decision as it with a far simpler calculation. entry decisions, but it gets to this answer will When the Nth firm pretends that it be the last one to enter this industry,it is ignoring two things. First, it is thinking its proht Ilowwill be given by the stochastic evolution of n'N. F) as )' changes, holding N fixed. Thus it is ignoring the reduction in the upside of its proht caused by the subsequcnt entry that would occur in response to F rising to new heights. Other things equal, this would make investment seem more attractive to this firm than it '
Ill6ittnntt?lkTylf/llrl/lzy
would if it took entry into account. However, it is also ignoring the fact that the prospect of future ent:y reduces its value of waiting. lt is pretending that it has the ltlxury of postponing its decision, and acting as though there is a positive value to its option to wait. Other things equal. this makes investment seem less attractive than it would othelwise. These two effects exactly offset each other, so that in this case, two wrongs make a right in the irm's optimal choice. We noted a special case of this in Chapter 8, Section 2(b), where the exact level of a price ceiling was immaterial to the calculation of a firm's entry threshold price, as long as the ceiling was at Ieast as high as this threshold. Now we see the general version of the effect. This remarkable was discovered by Leahy (1992).
property of the
competitive equilibrium 1.C
Implications
for Antitrust
and Trade Policy
We have seen that the standard correspondence between social optimality and competitive equilibrium holds in the kind of model we have constructed so far. As long as there are no externalities, and the relevant risks can be traded in efscient markets, dynamics and uncertainty are not by themselves sufhcient reasons for policy intervention. True, the market outcome shows a great deal of inertia ranges of shocks where no investment takes place-and in popular perception the hesitancy of firms to invest might seem reason enough for the government to intervene and speed up the pace of investment. However. the social planner would not want to invest any faster. inertia is optimal-a There are other features of our stochastic dynamic market equilibrium that are often thought to be inetsciencies requiring corrective policies. That is because the conventional textbook view of equilibrium is static, based on the Marshallian long-run picture. In that view, srmsenter an industry when the price rises above the long-run average cost. and exit when price falls below the average variable cost. A price in excess of long-nn average cost is then regarded as evidence of entry barriers. calling for antitrust measures. Similarly. a price below average variable cost is often regarded as a sign of predatory dumping, usually by foreign tirms,and thus ajustitication for trade
sanctions. We want to emphasize that such conclusions are liable to be fundamentally mistaken, because in reality economic conditions are never tranquil. It is essential to think of the industry equilibrium in such a situation as itself changing in response to evolving uncertainty. that is, a stochastic process. The natural competitive dynamics of an industry in the face of ongoing uncertainty of the industry has features that the static will have phases when a theory would interpret as deviations from competitive behavior. This idea of ''snapshot''
/2:7lic'b'/?l tewell
Jp? l t? l l (1
J/??/7t! r1Lzc/
>t)l
??/?
t'/
il
t)? t
regarding competitive equilibrium as a stochastic process has become quite common in macroeconomics', it is time for industrial organization theory and antitrust policy to recognize the same reality. Suppose such an industry comes to the attention of policy authorities at when the price is between the Marshallian long-run Ievel Jl) and instant an the equilibrium threshold W.They see established tirms making supernormal profits. but no new entry taking place. Using conventional microeconomics or industrial organization theory, they would suspect the presence of monopoly power or entry barriers, and might take antitrust action. That would be wrong; the process viewed as a whole is fully competitive, long-run expected returns are normal, and the equilibrium is socially optimal. Likewise, if the price is below thc minimum average variable cost. that need not signal predatory dumping by the hrms that are incurring the losses. Remember that if market conditions are sufsciently volatile, the Iower threshold price at which firms should optimally exit will be well below the minimum average variable cost. Thus in such a situation we may simply be observing hrms rationally riding out a bad period to keep their sunk capital alive.s The numerical example of the copper industry in Chapter 8 showed us that the market price is Iikely to be outside the Marshallian range tbr most of the time. Thus not just snapshots but time series of a tkw years' duration may also be insufticient. Only by observing the evolution of the industry for a very long time can we hope to spot genuine departurus from the competitive norm. Basing policies on snapshots can result in scrious mistakes despite the policymaker-sbest intentions. In the picture presented above. prices above long-run averagc costs and the rcsulting temporafy large profits are merely due to the swings of demand in a competitive industry that permits only a normal protit as a long-run average. However. governments oen t:y to control the supposedly excessive profits of lirms, and protect consumers from these supposedly excessive prices. Urban residential rent controls are a common instance of such policies. In Section 2 we will develop a model that describes the effect of such policies on investment in a true dynamic context. We will find that price controls can depress investment and thereby reduce industry supply to such an extent that the averaj;e price in the long run actually goes IIp. Thus a policy of price
5Of course, cven if there wcrc no uncertainty whlttsocver
over future market conditions,
pricc could be bcltlw the minimum average variable cost if lirms are moving down the steep part of a Iearning curve. Finally, price could bc below the minimum average variable cost because of (nf volatile economic ctlnditions and a learning curve for example. in the case of a combination semiconductors.
Illtlltstl E-t/l//ib/.?l-ii/pl?
294
'
controls can have perverse effects oven from the perspective of the group it is designed to help. Conversely, if the government introdtlces price floors to support (irms in bad periods, Iirms will react accordingly. They will ttnter the industry in greater numbers, and that can then makc the bad times even worse. Ieading to a large drain of government revenues. Agricultural price supports, a Iong-standing policy fixture in the United States and in E urope (throughthe Common Agricultural Policy), often have such eflkcts.
1.D Market Failures and Policy Responses We found above that uncertainty and irreversibility do not by themselves constitute market failures that warrant government intelwention. It is important to emphasize this point, because public policy debates often err on this matter. The existence of adjustment costs is itself often thought to be an economic problem requiring policy action when industries and workers suffer adverse shocks,especially those arising from international competition. Calls for government action in such cases should be based on some other genuine market
t'ailure. This does not mean that markcts always work pcrtctly
and government
intervention is never called tbr. Some forms ()f market failure do arise naturally in a dynamic environment with uncertainty. In particular, markets tbr risk are often incomplete. The rcasons for this hav: to do with asymmetric information or the sheer complexity ofcomplete contracts. Labllr income risk is particularly difficult to insure against. When such separate causes of market failure coexist with our basic issue of irreversible choices under ongoing uncertainty, the two intcract to produce some new and interesting kinds of suboptimal outcomes. We do not have the space here to provide a detailed description of models
that demonstrate such market failures. Therefore we will merely outline the economic intuition behind them, and refer the interested reader to the work on which our discussion is based. Dixit and Rob (1993a,b). The failure of risk markets is most frequent and natural
in the context
of labor income. Now labor supply decisions. for example. educational and occupational choices, also involve substantial sunk costs or irreversibilities, and must be made in an environment with ongoing uncertainty. Thus they are in essence investment decisions, and our general framework is naturally suited to their analysis. That is the setting we use. Consider an economy that offers two alternative occupations, which we call sectors and which may be dlfferent industries or cities. The relative will attractivenessof the two fluctuates over time, for example, because of random
technologic:tlslltlcks. A shil't l'n'nl ontl sector to t he t)t Iler reqtlires solntt stlnk tlirtrtll-nstances) retraining. travei. costs that can inclutle (tlependingo11thtl ptlrchase antl sale of :1 hotlse, moving hotlselloltl effectsa til-ne and eflrt involved in mak' ing nesv friendss and many othtlr tangible and intangible costs. The refore an individual will not make the ssvitcll un less the re lative attractiveness of the other sector is sufticiently higll to offse t not jtlst the normal return on these costs. but also the option value of the stattls qtlo. The relative price betwecn the outptlts of the tsvo sectors is determined in equilibriul'n by the standard equal ity of demand and supply. Call the scctors I'nobi lity. the inconles of worktl rs in 1 and 2. l 11 the absence of any Iabor each sector move in proportion to the tochnological productivi ty shock lbr that sector. This comprises the income risk that people face. Now suppose sector 2 receives a favorable shock. Wllell :11,1individual nloves fron; sector 1 to sector 2 in responsea this raises the output of 2 and so lowtlrs its equilibriunl price relative to 1 That dampens the initial increase (')f incomes in sector 2 relative to I In other words, when one person moves to the favored sectorthat reduces the extent of income risk tced by :111 the othtlrs in both sectors. If the risk were efhciently allocatetl using compltlte markets. this price change would be ()nIy a pecuniary externality. I lowevcr. because risk markets arc incomplete. the redtlction in risk has rea 1(2 ffects: it is lt bencficial cxternality conveyed by the mover ()n aIl (lthers in society. Since the mover does not take this stlcial beneiit intt) acctltlnt in his private calctllatitln. the resulting labtlr mobility is subtlptimally l()w. A governnlent that can ellcourage some extra mobility (perhaps by subsidizing jol7changes and any retraining that might be involved) can achieve a better outcome for sllciety as a wll()le-t' This mcchanism operates via the endtlgenous relative pricc. Thdrefore it is less significant the morc elastic is demand. In the limiting castl t)f a small economythat is open to world trade. the relative price is determined (perhaps as a stochastic process) by world market conditions- and the shi of Iabor across sectors in the economy has no impact on this price. Then the degrce of mobility in equilibrium is the second-best optimal given the Iimitations on the risk markets. Unless the government can devise new ways to share risk, it cannot improve upon the outcome of uncoordinated private choices. Various kinds of taxation do provide indirect risk sharing. In an open economy, trade taxes are such an instrument. Suppose the government levies .
.
'' In thc light of this analysis, it is irollic that govcrnments tlften pursue policies that actually reduce labor mobi Iity. l'tlr exam plc, public housing p()Iicics (hat favllr lllng-t imc residents of a city. I t is then nttccssary tl) u ntle rtake a variety ()1* socilll insu rll ncc meltsu res that amelioratc t hc grca tc r incomc risk t hat t hc Illcked-i n individulkls I'ak7tl,
296
11J( ltt-bntjlf/l
tilill rI d?z:
a tax (or offers a subsidy) so that the domestic relative price is not the same as the world relative price. Now the incomes in the two sectors are tied to the domestic output prices, and are furter affected by the distribution of tax revenues (or the contributions to the subsidy proceeds). This changes the allocation of risk. Then it is not hard in principle to find a policy that will improve risk bearing and thus generate a social improvement. However, the precise nature of the policy is highly situation specific; a simple recipe such as an import tariff cannot be guaranteed to be benelicial across ny broad range of circunstances.
2 Analyses of Some Commonly Used Policies Governments do employ some policies that alter the uncertainty facing firms and consumers in their economic decisions. The policies are usually motivated by some immediate political or economic reason a belief that rms are charging excessive prices, or a perceived need to stimulate investment. Now the simplest economic analysis teaches us that policies implemented with one aim often have other side effects', these are oen surprisingly detrimental. This problem is all the more severe in the stochastic dynamic environment that is the theme of our book. In this section we illustrate this by analyzing two such policies. 2.A
Price Controls
Governments often attempt to reduce price volatility by imposing controls. The immediate aim is usually to protect consumers from excessively high prices, as in the case of urban rent ceilings or the U.S. natural gas and oil price controls of the 1970s, or to protect the incomes of produccrs, as with the European Common Agricultural Policy and the plethora of agricultural price supports in the United States. Economists are generally critical of such measures. They emphasize the harmful side effects, for example, the scarcity and poor quality of rental housing in cities with rent controls, and the wheat mountains and wine lakes in Europe. The economic analysis underlying these arguments is the standard textbook supply-and-demand framework. If the price is kept below the market-clearing Ievel, there will be too little supply, and if it is kept above, too much. However. this is a very static picture, while the effects in reality are mostly dynamic, operating through the investment decisions of landlords or farmers. Our approach to investment under uncertainty permits a richer analysis of the effects of price controls, and yields a deeper understanding of their side effects. '
Policb.f/lfclaf'c/llt?ll
tl/lt/
Ilnperjltck Ct-aipllpfizf itl'll I
We continue to use our basic model. To recapitulate, each firm has the cnpacity to produce one unit output when active. a sunk cost of investment 1, and a variable cost of production C. For simplicity of notation. we assume risk neutrality and the riskless discount rate l'. The industry demand curve is
p where the aggregate shock
=
y DqQj,
F follows the geometric Brownian motion
JF
=
a J' t// +
t:r
F' /z.
The simplest and unihed treatment of price ceiling and lloor policies can be
given if the industry' equilibrium involves both entry and exit. Therefore we adopt the model of Chapter 7. where 11 firm that is not producing must lose its sunk investment. Then we know that the industry equilibrium in the absence of any controls is characterized by an entry threshold and an exit threshold. For reasons that will be evident soon. we write the tbrmer as S and the latter as .
-f5
Now suppose the government imposes kl ceiling and/or a tloorE on the price. We take this to mean that if at the cxisting level of Q and F in the industry the price given by ( l2) that would clear the market exceeds W.then the Iirm can collect only Wand the excess dcmand is rationed. If the price that would clear the market is below -P. thdn the hrms reccive -C,and government becomes the buyer of last resort to absorb the excess supply. We assume in the background that such government purchases are either destroyed, or exported or given to other countriesa with no future feedback on demand or supply in our economy. (This is broadly true for European agricultural policies.) Nor shall we be concerned here with the eftkcts on the government budget. ln Chapter 8 we studied an aspect of a firm's entry and exit choices when such policies are in effect. In fact we found that as long as exceeds the natural entry threshold S, the actual Ievel of > has no effect on S at all. A lower ceiling reduces the value of investing immediately and thc value of waiting by equal amounts, leaving unaffectcd the trade-off that governs the investment threshold. A similar situation holds when the lloor L>is actually lower than the natural exit threshold S. Now we must allow the ceilings and floors to bind, and see how that will affect hrms' choices and the industry's equilibrium. Forsimplicity of exposition we do this separately for ceilings and Iloors; then the two treatments are in principle easy to combine into a joint analysis. First suppose the price lloor is too Iow to affect the picture, say, ..P 0. but ceiling the sometimes binds. That is, sometimes the price that would that -#
=
-15
298 clear tlte nlarket exceetls --#.slv will call this llyptltlletical markct-clearing price the slladow prkt?. sv'easstlnAe that the (irms. lMlt 11actual and potential, can obselwethe degree ofscarcity (landlordssee how many prtpspective tenants are going around looking tbr apartmcnts, or read about it in the newspapers), so that they can calctllate the shadow price. This will inlluence their decisions. Even though :1 unit of output currently gets only the controlldd price P, if the shadow price is mtlch higher, they know that the control is Iikely to stay binding, and therefore the price is unI ikely to fall below M1 for 11 Iong time. Therefore a higher shadow price will make investment more attractive even though it does not alter the actual current protit Ilow. Then there will be a threshold S that will trigger new investment. As the eeiling -1 is gradually -%5 will rise. When the ceiling approaches lowered, the shadow price thresllold the Marshallian Iong-run average cost ( C + 1- /). the shadow price threshold will approach inhnity if the controls are going to allow only a normal return. hrms will invest only when they are assured that this state of affairs will last .
forever.
are the eftkcts ol' reaching the linAits of tlltl rnnge- and the const:lnts rernain Bb 1: to be deterlu ined. To help tlo thtt- %9< httve the value-matching slnootll-pasting conditions at the tlntl'y tllreshold: and
two tcrnls ,
Final Iy, the value function must be con t intlously di ftkrc n tiable lt t t I1e P where the tsvo regi mes nleet. Tllus the expressitlns l'or p ( frol'n ( l 3 ) alld ( l 4) evaluated at P must be equal, and the same must hold for the corresponding expressions for v' ( 5') ln all we have six equations to determine the two thresholds Sns tnd the four constants z1l z1:, B3 Bz. These need numerical solution. which we soon <$')
.
,
,
illustrate. First we develop Some useftll information lbout thc eqtlilibrium. Just Iikul actual price in Chapter 8, the shadow price here tbllows a geometric the Brownian motion witl'l parameters fx and and retlecting barriers at U.S and S. -
'r
Suppose '-P is in this range o- its effectiveness. Now we mtlst determine < -S. For this- we consider the valtle t,' of an actual thresholds such that S < hrm as a function of the prevailing shadow price When the shadow price S is in the rangc s-b--D1). it is also the actual price, and the srm'sprotit flow is (- C). Then. following tmiliar steps. its value is -51
v.
-
Then we can tind the long-run distribution of 5-, using the same methods as we did the re. The Iogarith m of tbllows 11 si mple Brownian motion wi t 11the .!.c2 and thtl variance coefficient o'. Thtlrefore it drift coefhcient a' a 2 exptlnential distribution between its bitrriers l()g and log -?b, witll the has an density proportional tt) eY 't' whure y = 2 a'/rr2 In turn, the density of S can be seen t() l7e proportitlnal to
=
-
.
,
.
The hrst two terms give the expected value of profits if neither exit nor the binding ceiling were ever going to be relevant', note that with rising at the t5. The other expected rate a, the revenue liow is discounted at rate 1- a values of the consequences of reaching two terms are the expected present the exit threshold and the price ceiling in the future. The powers p, and pz are remain to be the roots of the familiar quadratic, and the constants z1l and determined. To aid in this, we have the value-matching and smooth-pasting conditions at the exit threshold:
g'
.b'
-
=
,-12
=
-
pt.l
=
-p
,
0.
and
(W C)/r -
+ /31 S#' + Bz 5'/h .
Again the hrst two terms are the value ()f receiving this profit flow forever; t5. note that the constant revenue -j5must be discounted at rate r, not The other
/
V
j / g7q
V -
.,....%P'
j
,
.
When the ceiling is bindingy that is, the shadow prce Iies in the range
(W.Sj, the prolit flow is ('F C),
-'
-
if y # (). As P decreascs from the ntl-policy en try thrcshold to tlle Marshallian Iong-run average cost. this probability rises j'rom () to I Using the density function we can alst) comptlte the Iong-rtln average P1 -( of the actual price that will prevail in the long run. Since the actual price equals the shadow price S in the interval ( S. DI) and equals the ceiling pricc -# in t he in t e rva I ( 'J) t he ex p ec t a t io n be co m e s .
() these two expressions must be interpreted as limits. taken using If y UHospital's Rule. The probability of the ceiling being binding becomes =
greater presstlre of demand before they act. This dynamic decrease ot' supply in turn implies that the pressure of demand will be high enough to keep the ceiling binding more often. In effect. the policy generates its own ''need.''
glog S
log
-
W)/ (log X log -
...
).
and the expectation of the price becomes
rr#l
=
( g'F
) + -/-(log S
-
-
1og >
J ) / g1og S
-
log
1.
..
Now we turn to the numerical simulation. Since price ceilings are more likely to be applied to an industry in which demand and the free market price 0.05. which are 0.02. We take t.r 0.2 and 1have been rising, we let a used earlier values Finally. of the have in chapters. 1 we set C typical we variable cost), and units that normalizes 2, / the average (this is a choice of which makes the Marshallian long-run average cost 1.1. Table9.1 shows the results. In the absence of anycontrols (orequivalently, when the ceiling is at an irrelevantly high level. in effect at inhnit) the entry threshold is 1.5324 and the exit threshold is 0.6790. As the ceiling is lowered, the most important effect is on the entry thrcshold of the shadow price. lt increases gradually at first, but then more and more rapidly. By the time the ceiling is halfway between the natural thrcshold and the Iong-run average cost, the effect gathers momentum. The long-run avcrage t'requencywith which the ceiling is binding also increases. The exit threshuld also rises, but this is quantitatively much less significant. This analysis gives us a clearer idea of the mechanism by which price controls discourage investment. Under price controls. firms wait to observe a =
=
=
=
=
Table 9.1.
(Parameters: a
=
0.02.
Perhaps the most interesting and surprising result is the eftkct of controls the long-run average price. As the ceiling is lowered. this price tcreases. In on other words. the policy fails to achieve its intended goal of reducing the prices to consumers: its etct in the long-run average sense isjust the opposite! The reason is that while the incidence of prices above the ceiling is cut off. the proportion of times the price stays at the new ceiling is increased because of the reduction in investment and therefore in long-run suppiy. In Figure 9. l we see this shift of the distribution of prices: as > is lowered, the density to its right disappears, but there is a larger point mass of probability at P, and some of the left tail also disappears. The effect on the average price is ambiguous from the picture. but from the numerical simulations we find that the effect j
ti
s perverse.
F'
The analysis of a price lloor follows similar steps. As we raise L from 0 to that the exit threshold S would have in the absence of any policy, the level the Iloor remains nonbinding. Beyllnd this point, it starts to bind for the relatively Iow states of demand. That in turn changes the entry and exit thresholds. As the price floor rises. the downside risk of investment is reduced. Tllerefore firms are more willing to enter and less eager to Ieave, so btlth thresholds fall. When the price lloor rcaches C. the exit threshold s drops to zero-, incumbent firms that have the guarantee of their variable costs being covered will never leave. When the price tloor is between the variable cost C and the Marshallian
Iong-run average cost ( C + r I4, the exit threshold s stays at zero, while the entry t hre shold S remains above (C +r l4. This is because (irms contemplating new investment still need some periods of stlpernormal prolit to balapce the periods of below-normal protits. Finally, as the price lloor rises to (C + r/). the entry threshold falls to that same Ievel. Tt characterize the equilibrium in more detail, we proceed as before. First suppose the price floor -C.is high enough to be sometimes binding, but below C. When the shadow price S Iies in the interval (.t, -P). the Iirm's profit C) and its value is flow is
.2j! 0 0.75 0.80 0.85 0.90 0.95 0.97 0.99
.l..t
-
''!).
ln the interval (Z. the floor does not bind, the actual price equals the shadow price, the protit flow is (S C), and the value is -
f./'ects
t#'
:
j
(1.7462
1
0.7462 0.7398 0.7 l84 0.6722 0.5735 0.4982 0.3543
1.6668
.6667
l
.6665
l
.6645
.6584
1
1.6433 1.6323 I .6090
303
Ihice J--lo0l.
I'rtltnkltxilily
()
0.0 126 (). l 804 0.35 l 1 0.529 I 0.7238 0.8 l 85 0-916()
r7l1' I .()3()9
l 1.()3t)9 .4)34)3
1
1
.0287
l
.0256
1.02()l 1.0l 64 l
.()(')9t)
Then the vaiue-matching and smooth-pasting conditions at and S, and the continuous differentiability at Z, complete the solution. We can also compute the probability that the lloorwill bind in the long run. and the Iong-run average price, as before. Next suppose the price ooorZ Iies in the range (C. C + r/). Here ... (). For S in the range (0. f.), we get =
rlxl
=
jl
-
C)/r
+
S#'
zz!
.
This is just like the above solution except that only the positive root is retained by consideration of the Iimit as S goes to (). For S in the range (Z. X). the solution is exactly as above. Now the value-matching and smooth-pasting
conditions at S and the requirement of continuous four equations to determine the three constants threshold S.
differentiability ./dl
,
at Z give #1 and B? and the ,
Calculation of the Iong-run average price is different in this case. Since
firms never leave once they have entered. the long-run number of lirms goes to inhnity and the distribution of the shadow price has the whole probability piled up at zero. Then in the Iimit the ceiling is always binding, and the average price equals the ceiling price.
We show a numerical simulation in Table 9.2. The parameters are as in the ceijing case, except that since price qoors are more likely to be prevalent in industries with declining demand, we set a Once again the most remarkable feature is the decline in the long-run average price as the lloor is raised from its Iowest nonbinding Ievel up to the variable cost C. The reason -0.02.
=
is the same: the price is higher in the Iowest states of demand. but this is more than offset by the effect of the induced extra entry leading to a lower price in the higher states of demand. The effect is quantitatively somewhat more modest, but it is unmistakable. Thus the Iong-run effect of the price Iloor policy may be harmful even to those it is intended to help. The observation that farmers in many countries are so dependent on farm price supports and yet so dissatished with their outcomes may have this explanation. Of course once the price Iloor rises above the variable cost Ievel, it is always binding.
Then the average price equals the ooorand rises one-for-one with it, but this is a rather pathological state of affairs. 2.B
Policy Uncertainty
Governments can not only deploy measures
to reduce the uncertainty
facing
potential investors. they can also create uncertainty through the prospect of policy change. This feature of the polic'y process is particularly relevant in the United States, where changes in t:tx policy are constantly suggested and
/pltl/-f?)/iltlililzriltll
304
even after a specihc new tax legislation is introduccd in the Congress it goes through months and even years of debate and modilication. It is commonly believed that expectations of shis of policy can have powerful effects on decisions to invest. Our theory of investment under uncertainty can
examine the validity of this belief. However, policy uncertainty is not likely to be well captured by a Brownian motion process; it is more likely to be a Poisson jump. We now develop an example, based on Metcalf and Hassett ( 1993), to show how such policy uncertainty affects investment. We adapt our basic model of Chapter 6. Begin with a firm contemplating discrete investment with sunk cost /. The project will produce one unit of a flow forever after with no variable costs of production. The output output price t'ollowsthe geometric Brownian motion
(1P
u
=
P dt +
tr
P Jz.
1a. Then the The firm is risk neutral and ?- is the discount rate. Let J when the initial completed project. value from of revenues a expected present is #, given P/t5. is by price The policy instrument we consider is nn investment tax credit at a given rate 9. When this policy is in effect, the sunk cost of investment to the 5rm is lowered to ( l 0) 1. However, the government can switch between two policy regimes, one wherc the credit is not available, and the other where it is. We will identify the state without the tax credit by the subscript 0, and that with the tax credit by the subscript l The switches between the two policy regimes are Poisson processes. Starting with a state when the credit is not in effect, the probability that it will be implemented in the next short interval of time dt is l dt and when the credit is initially in etect, the corresponding probability that it will be withdrawn is () dt. Intuition suggests the following form for the tirm's investment policy. Over an interval of low values of #, say, ((). /71), the firm will not invest irrespective of whether the credit is in etect. Over an interval ( J$ f1)). the hrm will invest if the credit is in effect. but if not, the hrm will hnd it preferable to wait in the hope that such a policy will be implemented. Beyond Jl). the prospect of immediate revenues will be so large that the tirm will invest irrespective of the current policy. To determine the thresholds PI and Jl), we procded as usual. Consider the net payoff to the investment opportunity as a function of the price, 1$)(P) ( P) when the credit is in effect. We obin the absence of the credit and 1zr1 tain expressions and equations for these, and conditions they satisfy at the =
-
-
range ( Jl3.x),
3()j
the Iirm always invests at once, so we have
/'()(P)
Pj
=
/.
-
and F1 ( P)
Pj
=
(1
-
0j 1.
-
(18)
Over the range (#l C)),the firm invests at once if the credit is in effect, 1z-1 so ( #) is given by ( l8) as above. However, P'4)(P) is more complicated. Over the next short interval of time dt, with probability l l the credit will be implemented, the 5rm will invest. and its value will become lzrl( P + J P). Othemise its value will be )'(,4 # + t P4. Thus ,
r)( #)
e-rd'
=
(
l
dt
'tp'l
(P +
tI#)J+ ( l
l
-
tlt,
z7(K)(P
Expanding the right-hand side using lto's Lemma. retaining in dt. and simplifying as usual, we have
+ dP4j
) .
leading terms
This is the usual dift-erential equation for a hrm waiting to invest, except for the additional term involving l which captures the expected capital gain from institution of the credit in the immediate future. Note that in the range of prices ( Pl Jl)). we know the expression for 1z'1( P4. Then we can obtain the general solution to the differential equation: ,
,
.
.
.
thresholds.
) ( #) lzh
=
B3 P#t ' '' + Bz 'Vt
where Sl and Sa are positive and negative
1':
+
/5(
lP +
l -
l)
(1
-
()) l
r + 2.1
constants to be determined, and roots of the familiar quadratic
v
J( 1)1 and J( l )a are the
The seemingly complicated notation for the quadratic and the roots serves to distinguish them from some other related expressions and roots that will
appear soon.
Finally we address the region (0, 771 ). Here the firm waits in both policy regimes. and each regime can switch to the ther. Following the same steps as above that 1ed to the lifferential equation (19)for the waiting Iirm, we have a pair of differential equations:
t
-
t;r2
P1 pe'?(#) + (r ()
/7.2
p2
v''p) l
+
(/.
-
-
j) # pr't'j () J) #
-
''p)
l
-
r
K)(#) +
r gj
(#)
+
l
(#) gprj
o gIr)( /a)
-
-
p'()(#)j
=
0,
prj( p)j
=
().
bldltxtryf'tr/l/l.?nb//7
3()6
that can be solved easily. Detine two new
These yield two Iinear combinations
functions, say,
:
l (P )
1z',( P4
=
-
lz))( P4
.
'C'f (L>()l ,1/? t'I ili()ll lic-' /zl/ c,?7 l Jilnltt/ 11(1 11?lJ?(,?7' fa/? .el
smooth-pasting condit ions. F()r 1,$)( P) t 11is is not 11 klecision t llrtlshtlltl. but l Ilc function has to be ctlntintlotlsly different itl7le across it- s() (2l ) and (.2()) slltltlltl have equal values and derivatives tllttre. Finallya for tlle optinltllu clltlice ()t' the threshold /$, the expressions for lz$)( l3j to the Ie (2())and the rigllt ( l 7) should satist the value-matching antl srflooth-pasting contlitions. In all Nvt) have six equations to deternAine the thresllolds /31 C) and the l'otlr collstants ,
1:! 1
:!:
(7.2
#2
v'' :b.
(,.2
P1
P) + (r
v'' tl
-
P) + (?. j) P Iz-?(P) -
?.
-
il
K,( #)
=
0,
St B?, Ca , and Ds We illustrate tlis calculation with a numerical sklltltion. Xlke a (). = tz 0. l and r = 0.05.va1ues that were fairly typical in our earlier calculations. Then J r a = 0.05. Let / = 20: this is just a clloice of tlnits that gives thc Marshallian investment threshold l = l and allows easier interprctation ot' the numbers to come. 'With these numbers. and no tax credit policy at aII, tlle optimal investment threshold would be #* 1 Thus the normal tlption value premium over the Marshallian thresllold is i).37. We consider a l o-percent (). I If this credit were alwfys tax credit. s() 0 in effect, the threshold would be lowered by l() percent to l l I-lowevttrwhen the two regimes can switch back and forth in a Ptlisson process. botll thresholds are affected. 'We examine the effects on the twl) thresholds, /ly when the credit is not currently in effect. and /3l wllen it is currently in effect. as the probability rates of enactment ( ) ) and removal (&)) val'y tlver a range from 0 to ().5 in each case. The results are shown in Tables 9.3 ktnd 9.4 First consider thc situation when the crcdit is not currently in effect. Table 9.3 shows that as the probability of enactmcnt I witllin the next year increases, the threshold /$)increases. This is intuitively klbvious; the prospect of a reduced cost of investment increases the value ofwaiting. The impressive aspect is the magnitude of this effcct. The usual option value premium over the normal return ()f l if the credit had never been mentioned. was ().37., a 3o-percent probability of enactment of the 1(l-percent tax credit more than doubles this premium to 0.8 l Remember that the tax credit had thc aim of lowering the premium to 0.23. However, while the policy is being discussed and enactment is uncertain. the effect is to deprcss investment very .
.
=
)'J( #)
j) #
(?. +
-
.b
()
+
!
) 1Z.$.(P)
=
0.
=
Each of these equations yields a solution with powers of P that are the roots of a familiar quadratic equation. ln each case we have an interval of # that extends to 0, so we include only the positive riot. Thus U ( #)
Ca P /l(t'3!
=
-
.,.
.
( #)
I). #/lt2)l
=
,
.&
-
.
.3702.
=
=
where C:, and D.s.are
constants
Q(0) - 1a
v2
p(0)I is the positive
to be determineda
pp
-
1) + fr
-
j)
p
-
r
=
root of
0,
and /(2) ! is the positive root of
.
.233
.
.
J4())comes from the quadratic 24())where the constant term includes neither () norl p 1) comes from the quadratic k?(l ) whose constant term has l alone, and J42) comes from the quadratic Q(2)whose constant term has both () and l As beforea the subscript l refers to the positive root and 2 to the negative root. Then the relevant roots satisfy the following chain of inequalities: Mnemonically,
,
.
,
.
/3(2)1> /(1)1 > #(.) l
>
1> 0
>
p( l )a.
With this notation, we can write down the solutions for 1.$)(#) and 1zh( #) in the range (0, Pl ) as
(2 1) (22) Now we can relate the expressions for the value in the different regimes. /71 the hrm invests if the credit is in effect, so the expressions P'1 for ( P4 to the left (22)and right ( 18) should satisfy the value-matching and
At the threshold
,
strongly.
Even when the credit is not in place. thtl threshold
J1)is affected by the
probability () of its removal. This is because the lirm calculates that at the random future time when the t;kxcredit is enacted, the economic cllnditions might and the credit might be removed before they irnprove be very unfavorable, it for invest. sufhciently This reduces the value of waiting now. However, to effect negligible. is quantitatively the Next consider thc situation when the credit is currently in effect- Now the relevant threshold is f-'1in X'tble 9.4 We see that it decreases as () increases: .
dTyz//f/l/-fl//hl
/A?tlttstl?l
308
lJllit'
)'
fnvtr-frrlc?,fThreshold w/1(?/lTax Credit is (Parameters: a = 0, tz = 0.1, r = 0.05, I
Table 9.3.
in Effect
,1:)/ =
(JlU
20.)
1 4)
0.0 0. 1 0.2 0.3 0.4 0.5
0.0
0.1
0.2
1.371 1.371 1.371 1.371 1.371 1.371
1.498 1 1.492 1.491 1.491 1.490
1.642 1.641
.494
1.640
1.639 1.638 l .638
0.3
0.4
0.5
1.8 13
2.003
2.201
13 1.813 1.812 1.812 1.812
2.003 2.20 1
l
.8
2.003 2.003 2.003 2.002
2.201 2.201 2.201 2.200
readily now.This
the prospect of Iosing the credit induces hrms to invest more effect is quantitatively not as strong as the delaying effect of
on Jl) above. premium the before 0.5 all the from to 0 way () in need Here we an increase 17. is halved from 0.233 to 0.1 with a higher Moreover, an increase in l increases /:1 The point is that removed, it is effect is in credit if the now tax enactment probability, even immediately likely to be restored fairly quickly. so the imperative to invest = quite becomes but 0, if 4) irrelevant is is less strong. Of course this issue investment-promoting = the then 0.5, If values of (). l signihcant for larger 1()) is very small. credit (higher effect of the prospect of removal of the current l
.
( /$ )
-ccl
lh Investment Threshold wcn Tax Credit is (Parameters: a = 0, o' = 0. 1, r = 0.05, 1 = 20.)
Table 9.4.
I
)
0.0 0.l 0.2 0.3 0.4 0.5
0.0
1.233 1.177 1. l52 1. 135 1.125 1.117
0. l
1.233 1.196 1.176 l l 153 1.145 .162
.
0.2
1.233 1.209 1.193 1.l82 1.174 1.167
1.233 1.216 1.204 1.195 1.188 1.183
0.5
0.4
0.3
1.233 l 1.212 1.204 .221
l
.198
1.194
.233
1
1.224 1.216 1.210 1 1.201 .205
/a J??1k'tr,?Jt?z7
tll 7:/
,?7/7 t'l il 11z7/7t>r/,t-/t7&
These results suggest that uncertainty
fa/
l
about the enactment
309 of stimulus
policies is likely to have a very detrimental effect on investment. In fact, if a government wishes to accelerate investment, the best thing it can do is to enact a tax credit right away, threaten to remove it soon, and swear never to restore it (high () and l()w l ). The credibility of such a policy is. of course, open to doubt. While our analysis s conducted at the tirm level, we can hnd its industry implications using the methods of Chapter 8. If we consider a competitive industryconsisting of numerous such hrms, the price threshold for each firm's investment decision will simply become the ceiling for the indujry's prie process ip the stochastic dynamic equilibrium. Now the increase in J$ will $ be interpreted as the effect of the reduction in industry supply due to the individualfirms' reluctance to invest. While the debate on whether to institute media, firms a tax credit proceeds in the administration. the Congress,and the waiting mplies and to their consumers in the a cost wait upon the outcome tbrm of higher prices. Metcalf and Hassett ( 1993) have a more general model where the scale of investment is itself a matter for choice. They hnd that policy uncertainty not only rases the threshold at which the firm invests, but also lowers the scale ot' its investment. When we develop the theory of investments of varying scale in Chapter l 1, the reader will be able to handle this extension as an exercise.
3
Example of an Oligopolistic Industry
We have thus farconsidered two extreme market structures, namely monopofy in Chapters 5-7 and perfect competition in Chapter 8 and Chapter 9 to this
point. The reason is practical rather than fundamental. Oligopolistic industries in our stochastic dynamic setting present formidable difficulties. The development of stochastic game theory for such applications is quite recent, and tractable models using that theory are rarer still. We will develop a particularlysimple example, based on Smets (1991),that brings out some of the issues. More general and more richly detailed treatments must await further research.
The general point is not difhcult to state. On the one hand, uncertainty imply an option value ofwaiting, and therefore greater hesirreversibility and each investment decisions. The fear of preemption by a rival, hrm's in itancy hand, other the suggests the need to act quickly. Which of these considon is the more important depends on the parameters of the problem, erations
3l 0
/-Llttl'lillrittllj //1(/!/:1r)7
and on the ctlrrent state illustrate this tension.
of the underlying
shock. Our simple model selwes to ' -
We consider two firms, each with the potential t() produce
:1
Iicy /?7t c/3 /7t?
?c,77
?ill ?
l 11??/7c,,7'
fl/lt
','t
C/J?? ?/?('J/
i()1l
%
unit outptlt
llow, which it can activate by incurring the sunk cost /. There are no variable costs of production, and we suppose that the industry demand is sufficiently elastic to ensure capacity production. Thtls industry output is 0, l or 2 depending on the number of active firms. The price is given by the demand function ( 1), which we restate here: #
>'
I
y 1)( Q),
=
Z
and the multiplicative shock
)' follows the geometric Brownian motion (2). For simplicity of notation we assume that the firms are risk neutral, or that the risk in )' has zero correlation with the overall market risk. Thus the discount rate for all future costs and revenues, certain or uncertain, is the riskless
rate
xe...
/2 D (2 ) where
#l
=
pl
-
I
l
and J have the usual meanings. If F zr Fa, the tbllowerwill invest at once, and get the value i' D(2)/J /. If F < i'a, the follower will wait until the threshold is hrst hit, and at that point /. Therefore its expected present value is get /2 D(2)/J -
-
'
J( l'a D(2)/J
-
I
z
F1
z
z'
z
z
z
z
F(
z'
z' zz
z
Z
z'
f
z I I I
FJ
'z
Fa
Fa
F
- /
(23)
.
p,
E gc-r
l
..e-
r.
Dynamic games are usually solved backwards, and this one is no exception. We begin by supposing that one of the firms has already invested. and find the optimal decision of the other, which we now call the tbllower. Then we look at the situation where neither firm has invested. and consider the decision of either as it contemplates whether to go first, knowing that the other will react in the way just calculated. The tbllower'sprot tlowwill be F D(2). Following familiar steps, we can hnd the threshold level Fa that will trigger its investment. lt satisfies
I I I .,,e ..e' l l I I l/'a( ?') l I I l l I I l I I
%( ?)
,
1 ,
hrst time when the stochastic process of the demand where F s the (random) shock reaches L starting at i'. We calculate this expectation in the Appendix. Using it, the follower's value can be stated as
if
F
i'c
The form of this function can be seen in Figure t).2. Nllte that the twl) branches meet tangentially at F:-,this is a smooth-pasting-like property of present values of Brownian motion at points of :1 switch of regimes.? Now suppose neither (irm has invested. and one of them is contemglating becoming the Ieader. gOf course this need not actually arise for some rangets) of values of F, but we must consider thtt hypotheticaf sccnaril) precisely in grder to determine when immediate investment for at Ieast one firm is optimal and so hnd just these ranges of i'.) In making this calculation, this tirm wilt take into ttccotlnt the actitln of the other tirmafter it Sees that its rival has invested. That is just tlle follower's
,
(24)
?In fact this is just an alternative way t)f dcriving rule and the val ut formula in Chapter 5, cquatiun (()).
and expressing
lltlr
very Iirst invcstrllcnt
ltlttililll-itlt't J?lt/!/-f?)7
51;!
decision we studied abtwe. So if )' 1 L, the follower will invest at once, the leader's protit llow will also be F D(2), and its valtle will be the same as the tbllower's. If F' < F2, then the follower will wait until l'a is hit. In the meantime. the leader will have the larger profit tlow 3eD l ). and its expected value will be
skj' ()
where, as before, T is the srsttime the stochastic process ofthe demand shock reaches :2 starting at F'. Again we compute the expectations in the Appendix and state the result here. The leader's vlue is P'a( J'')
1zr1 ( F)
=
i' D (2 )/ (
=
( 1/) F D( 1) + ( Y/
(1
l'a)/1
-
I
-
( Y!
l'a)/J1
r? D(2)/J
'
-
j
(25)
I
-
,
=
B1f pk is large enoughv negative slope. but this does
-
l'l
/''
() () r My a
+
--v
r
2
ya 0(2) -
j
f5
Thus
r
D 1)
>
/+
p'
y' X
D 1)
-
0(2)
wj
/2
-
J 1'' the hrst investment does not occur until the current
profit flow of the first investor provides a supernormal return on the sunk cost. The reason is similar to that in the case of perfect competition. Even though the value of waiting is zero, the firm contemplating being the srstto invest recognizes that future entry by the other hrm will reduce the upper end of the distribution of protit Qows.Therefore it requires enough of a current premium in compensation. Unlike the pertctly competitive case, though, the expected present value of the firm at this point is positive. With just two irms and no free entry, this makes intuitive sense. Extending this analysis to N tirms is simple in principle but messy in practice. However, the results should be evident without doing the formal work. At the smallest )' that triggers investment by the first firm, say, ih we )'.v ( Fl). The common value will will have )'l D( 1) > I and l'1 ( Fl ) N infinity. go to zero as goes to .
.
l
r I 1.,)(1)
111(..311 C(?l?IJ?t!f
(5
This has a more complicated shape, whic is shown in Figure 9.2. It is con8 cave over the range ?' < /, and its slope is discontinuous at Fi. The Iatter that the follower's decision changes is of course a consequence of the fact discontinuously at J'2. For a range of )' to the le of F2, the Ieader's value cxcceds that of the follower because it enjoys a higher prot tlow before the follower invcsts. However, tbr a range of very low values of F, the leader's value is less than the follower's because the leader incurs the investment cost up front but has a low profit flow initially. The two curves cross at the point Fl We have not designated one particular firm as the leader. and each tirm's own proht considerations should govern whether it wants to Iead or tbllow. The outcomes are different depending on the initial circumstances. lf the initial i' is below l'l neitherhrm will invest. When l'1 is reached, one will invest at once and the other will wait until Fa, but the two are indifferent between these two roles. Note that at i'1, we have P'l ( J'l ) P'2(FI ) > 0. Then equation (25)gives
1,' 0(1) j
Policy fllfc/-t'cllfo/l (ll1t Itnpellct
l
>
0
.
1z'1( l') can actually peak to the left of :2 and then approach not affect the qualitative results that follow.
F with
Next suppose
=
)' starts somewhere
.
.
.
=
in the range ( FI F'2). Then each hrm afford the option of .
stands to gain by seizing the Ieader's role. Neither can waitingabecause the other will invest if it does not.
Note that if both hrms invest at once. each will have a value D(2)/J - 1. This is even Iower than the follower's lz'a(F) while )' is in the range ( ih l'a), from their joint perspective. Of so simultaneous investment is a such mistakes equilibria. If the game were played can occur in game course the equilibrium discrete would mixed time, in be in strategies, where each independently chooses a probability of investing at once. Given that the other firm is choosing the equilibrium mix, cach is indifferent between investing and not inveting. However, with positive probability (the product of the two individualinvestment probabilities) the two invest together and get the lower value. In continuous time we would Iike the probability of such coincidences to go to zero. Doing that correctly requires some delicate limitingconsiderations. We omit these details. and refer the interested readers to Fudenberg and Tirole ( 1985). We merely state the outcome in the resulting continuous-time equilibrium. One firm, chosen at random, gets to invest at once, and therefore gets the leader's (larger)value. The other then waits until l'a is hit and so gets the follower's (lesser)value. Tllere are really two equilibria of this kind, where the two roles are interchanged between the frms. Since the firms are economicallyidentical, the two equilibria are indistinguishable for our purpose. '
,
Gmistake''
l//dlrl/'z/ Ill6ltt-tl?'/J* el
ln practicep some minor which one invests first.
tlnspecihed
Qatters are ditferent it the .
!
diftkrence between the lirms can govern '
'
roles of leader and..
.
tollower are
txogenously
preassigned to the firms. Now the follower cannot invest until the Ieader has (aption value done so, and the leader has the ability tt wait and recognizes an V/$. usual for values forms. have the a range of F tf doing so. The option neither zero Y# 11 F/31 includes that for range + B that includes zero, and Bt of the qsual negative and root pz the nor infinity, where pkis the positive root values these option as dashed quadratic in the parameters. Figure 9.2 shows value tlnce. They of investing at meet curves smoothly pasted to the leader's the latter at the points F1:, Fa', and Fn as Fhown. Then the designated Ieader will invest at once if F is in the range ( F,/. FJ)(whenthe follower will wait until /2 is reached), and when F' exceeds F3 (whenthe other will tbllow at once). z41
of J' either below F1/or in the range ( FJ F3), the Ieader will prefer entirely above to wait. For some parameter values. (he option valtle cklrve lies waits for the whole of (0. /3 ) 1/1(F); then the Ieader For
4
values
.
Guide to the Literature
The general idea of the optimality of competitive equilibrium in a dynamic and uncertain environment as long as markets are complete goes back to Arrow ) present the earliest spectic modei of and Debreu. Lucas and Prescott (1971 explicitly. The fact that irreversible costs this that demonstrates investment themselves constitute a market failure shifting by do not capital Iabor of or of economic adjustment to much-needed in emphasis the given context a was conditions Mussa by international trade changing ( 1982), but I1econsidered unexpected shock. terministic single to d a response on1y a e Lucas and Prescott ( 1974) studied a stochastic dynamic equilibrium with costly intersectoral movements of labor. They assumed risk neutrality, so the equilibrium was again socially optimal. Dixit and Rob ( 1993a,b) introduced risk aversion and incomplete risk markets. and examined some policies that can improve upon the suboptimal equilibrium in these circumstances. The fact that competitive hrms would make the optimal entry decision even if they acted myopically and ignored aI1 future entry was discovercd by Leahy (1992). Our treatment of the dynamic effects of price controls follows Dixit 199 lb). See also Newbery and Stiglitz (1981)for a thorough analysis of price ( stabilization policies in agriculture.
Our treatment 01, policy unccrtainty was based on Mettutlf Clnd Hassett 1993). Additional work on tax policy and investment from a real tlpt ions per( spective includes Majd and Myers ( l986), and Maclie-Mason ( I t)9()). Using different technical approaches. Rodrik ( l99 l ) examined the eftkcts of uncertainty over policy reforms designed to stimulate investment (for example. a tax incentive), and showed that if each year there is some probability thrtt the policy will be reversed. the resulting uncartainty can eliminate any stimulative effect that the policy would otherwise have on investment. Aizenman and Marion ( 1991) developed a similar model in which the tax rate can rise or fall, and showed that this uncertainty can reduce irrevcrsible investment in physicaland human capital, and thereby suppress growth. The trade-off between the strategic incentive to invest early in an oligopoly and the value of tlexibility in the face ()f uncertainty has been studied by several writers using a two-period mtxlet'. examples are Appttlbaum and Lim ( l 985), Spenccr and Brander ( l 9t?2), and Kulltil aka and Perotti ( l992). The Iast of these pairs of authors make an interesting new point. In a quantity-setting duopoly the lirst mover gets lk lltrger market sllare. Therefore the leader's profit is a more convex function ()f a demitnd shock variable than is the foliower's. As a resulta an increase in tlncertainty increases the relative value of early investment. Models of duopoly in :1 continuous-time framewllrk' are rarer. because the underlying theory of stochastic games in contintlous time is itsclf an ongoing research topic. A recutnt paper by Dutta llnd Rustichiui ( 199 l ) tlflrs a promising framework, and Smets ( l 99 l ) builds 11 dullpllly model using it. Our treatment follows his.
Appendix
A
Some Expected Present Values
Here we establish the formuas stated in section 3 for F
-r 7gf:r
1
and
(f7
.r
-''
t!
7-
)z'dl
()
when )' follows the geometric Browian motion (2), and T is the random first time the process reaches a fixed level Fa starting from the general initial position F'.For a more general approach tt) the calculation ofsuch exprcssions, see, for example, Harrison ( 1985, p. 42) or Karlin and Taylor ( 1975. p. 362).
tlltilillrittlll Illtltt-tlq' lJ*
Write /'( F) for the first of these expectations. As long as F < L, we l'c in the next short can choose Jf sufficiently small that hitting the level time interval dt is an unlikely event. Then the problem restarts from a new recursion level (l' + dY4. Theretbre we have the dynamic programming-like
expression j '( F')
e-rdl
=
'j
y' + Jy)
)'
).
Expanding the right-hand side, recalling that F follows the process using Ito's Lemma. this becomes
Simplifying and letting dt 2 1. 2 c
/2
(2),and
partV
0, we get the differential equation
-.+
''
+ a y'
f ( y)
.'(
F)
r
-
.J( F)
Extensions and Applications
().
=
This has the general solution
f't
=
z11
F#' +
FX
-d2
.
root and pzthe negative root of the standard quadratic. and are determined by a pairof boundary conditions. T close F':, F is likely to be small and e-'' to l therelbrc As i' approaches 7* close and Iarge e-'' small, F likely is be F' to (); F2) is very When to 1. /( F2/'' l and z1: these, () so 0. Using we get therefore /(0)
where Jl is the positive The constants
.?d!
..42
.'
=
,41
=
=
f This is used in equations Similarly, dehne
F)
=
( F/
=
F2)/J1 .
(24)and (25)in the
text.
This satishes the differential equation !
2
2
r.r
y2
,'
I (F) +
,
y g ( y)
r g(
-
y) + y
.() .
This has the general solution g ( y)
and the boundary
=
conditions
Sa
=
1.11F/$1+ Bz /11 + F/(r are g(F2)
Sj
0
which is used in (25)in the
text.
0, g(0)
=
=
-
=
1-/31
y'c
-
a),
0. Therefore
/(r
-
a).
.
10 chapter
Sequential lnvestment
IN
''rldls
and the following chapters
we return
to the investment decisions of
:1
single firm. In Chapters 5, 6, and 7, we developed a series of models in which the :rm must decidc when (:tndwhether) to invest in a single grojcct. I11 Chaptcr 5 we assumed that the value of that project
dvlllvcd
as an exllgenotls
stochastic process, and we derived thtt optimal investment rule. In Chapter 6 we allowed the price of the project's output to evolve as an cxogenotls stocllastic process. and then, given a variable cost of protltlction. wc derivcd l7tltllthe value of the project and the investmcnt rule. Finally, in Chnptcr 7 we exttlnded the model to allow tbr the mothballing and later reactivation of the prtlject, as well as scrapping. In all three of those chapters we considered a single, discrcte project, 11 single initial investment decision. In hence and many situations, however, investment decisions are made setptenliallv, and in a particular order. R)r example, investing in new oi1 production capacity is a two-stage process. Firsta reserves of oiI must Le obtained, either through exploratitln or otltright purchase. Second, development wells and pipelines must be built s() tllat the oil can be produced from these reservcs. An oiI company might invest in thc Iirst stage (forexample, by buying proved reserves of oil). but decide to wait rather than immediately invest in the second stage. Investing in a new Iine t)f aircraft is also a multistage process that begins with engineering, and continues with prototype production, testing, and the final tooling stages. An investment in a new dnlg by a pharmaceutical company begins with research that (withsome probability) Ieads to a new compound, and continues with cxtcnsivc testing until FDA approval is obtained, and concludes with thc construction ()f a pr()duction facility and the marketing of the product. Both the aircraft company
320
Eztensions
and the pharmaceutical companycould decide to go ahead of these investments, and then wait on the later stages.
tlnt
Applicatils
with the first stages
Even investments that appear to involve only a single decision can turn especially large out to be sequential. The reason is that many projects (and midway and halted ones) take considerable time to complete. and can be think of such temporarily or permanently abandoned. As a result, we can option projects as having many stages; each dollar spent gives the tirm an which it may or may not exercise to go ahead and spend the next dollar. The key characteristic of sequential investments is the ability to temporarily or permanently stop investing if the value of the completed project falls, or if the expected cost of completing the investment rises. (lf one had no started, investing would choice but to complete the project once it had been This possibilty of stopping midonce again involve oniy a single decision.) compoltnd options; each stage stream makes these investments analogous to the firm invested) gives an option to complete the next completed (or dollar problem boils down to hnding investment dollarl.The stage (orinvestthe next sequential these (andirreversible) expenditures. a contingent plan for making kinds of sequential investdifferent examine several In this chapter we analogies, will draw to as we have done try ment problems. In each case we options. of valuation tinancial exercising and with the throughout this book, sequential investments. three-stage and simple twoWe begin with some fairly and use them to illustrate a basic approach to valuing the Iirm's option to inrule. Next, we turn to problems of vest and hnding the optimal investment spends money a dollarat a time, it takes time continuous investment-the iirm investing in response to to complete the projcct, and the lirm can always stop problems, we will investment these conditions. In studying changing market the completed project value of uncertainty the over draw distinctions between and output), the project's the of price uncertainty over (due, tbr example, to project. completing actually the uncertainty over the cost of In each of these investment problems, the Iirm does not earn any cash only the flows from the project until the project is complete. (Investing in sufticient aircraft is not engineering and prototype production stages of a new will also examine for the hrm to actually sell airplanes and earn money.) We the related problem of a tirm moving down a learning curve. Here current llow, and also production serves two functions: it yields an immediate profit Chapter 11 we discuss lowers future costs. The latter is like an investment. In each unit of where usual sense, modelling of incremental investment in the capital contributes to the proht flow as soon as it is instailed. The mathematical models of this chapter and the next involve two state variables. One is the number of stages completed or the amount of capital
Sequetltial
/'?lI.'c-/?7Ic?I/
installed; the other is the price or some other indicator of protitability of investment. Then the diftkrential equations that emerge from dynamic programmingor contingent claims analysis are partial differential equations, with the value of the project or the option as the dependent variable. and the two state variables as the independent ones. Solution of such equations typicaliy requires numerical methods. For the class of models we consider, such solution is relatively easy. In an appendix to this chapter we brietly discuss the numerical procedure, and in the text we apply it to our specihc model.
1 Decisions to Start and Complete a Multistage Project In Chapter 6 we worked baclwards to hnd the value o a project together with the optimal investment rule. Given a stochastic process for the project's output price, P, we derived the value of the project. P' ( P). Then, given )' ( P), in project, F P4, we were able to find the value of the option to invest the that Recall which invest. optimal it is we needed to the critical price P* at and P' ( P4 to 5nd F( #) dnd #* because l'' ( P*) appeared in two of the boundary conditions that accompany the differential equation for F( #). We can use this working baclwards to solve sequential investment problems same approach of which the investment occurs in two or more discrete stages. in this how To see can be done, considr a two-stage investment in new oil capacity. First, reserves of oil must be obtained. through exploproduction at purchase. some cost I Second, development wells (andpossibly ration or built, be at an additional cost /z. Suppose that the price of oil, pipelines) must specihed exogenous stochastic process. Then the tirm begins #, follows some worth option. with an i ( P), to invest in reselves. Making this investment worth Fat #), to invest in development wells. another option, the tirm buys second yields production capacity, worth F ( #). investment Making this work backwards to find the optimal investment rules. First, as we We can P' have shown in Chapter 6, ( P) is the value of the srm'soperating options, and can be calculated accordingly. Nexta >t P4 can be found in exactly the value of a single option to invest in Chapter same way that we obtained the 6,' it wll satisfy a differential equation subject to boundary conditions for an matching'' and example, Fa(0) = 0!. and for endpoint should which make lhis investment the 5rm pasting'' at the critical price 'a* at in development wells. Finally, Fl ( P) can be tbund. It will satisfy a similar differential equatien and boundary conditions, but now there will be another critical price. J$*. at which the hrst-stage investment should be made. Note that because the payoff from the tirst-stageinvestment is t #), that is, the .
tfor
K
tsmooth
322
b'tcllsiolls 2.-
z'Lilplicaliolls
tp/lt
value of the option to invest in the
secontl strlge- we need t() kntlw Fz #) in to solve for F j ( #) and tlle critic:tl price P* Tllat is why we solve the invstment problem by working backwards. Intuitively, we should expect Pj* to be grettter than l3# As we saw in Chapters 5 and 6. the higlldr the sunk cost required to obtain2 :t risky payoff, the higher the critical value of the payoff ( in this trase the higher the critical price) tlat triggers the investment. So. if t projcct has two stages (costing, say, $5 million each). going ahead with the investment will reqtlire a higher critical price if both stages remain to be completed than it will if ontt stagc is already hnisheda making the rcmaining sunk costs smailer. The extension to a three-stage project should be obvious. Now. 1et Fi ( #) denote the value of the option to invest in the first stages F'c( #) the value of the option to invest in the second stage, and lq #) the value of the option to invest in the thrd stage. We work backwards as betbre. First, find the value of the Enished project. k' ( #). Second. use this to Iind J3) and the critical price P). Third, use F5 ( #) to find Fz ( #) and the critical price Finally, use (#) to find Fl (#) and the critical price P.1 at which it is optimal to invest in the srststage. We have assumed that the output price /3 is the (exogenous)stochastic state variable, so that lz' Fk Fz, etc., ktre all functions of P. However. in some cases there is no such thing as an actual output price until acr the project has been completed and the output llow commcnces. Thtl actual price ()f a home videocassette recorder, or of a personal computer, for example. was not observable when no such thing was being sold. The firms invcsting in these technologies and production facilities formed an estimate of this price based on some observabe indicators. for example. what users were willing to pay tbr related services, or from surveys. In our models. P should Iik-ewise be interpreted as just such an obselwable indicator of protitability when the relevant price is not directly observable. For simplicity, we will go on referring to this state variable as the price. At this point, the mechanics of actually carrying out the calculations described above may not be quite as clear as the overall approach. We will therefore work through the mechanics in some detail for the case of a twostage project. Iorder
.r'??
rtlv'v
vcqlttnllitll f(.)lIovs
.
!
'.'sll,tclll
a getlnle tric B
n i;.tn n1(.)t i(.)11:
'
tl P
=
t/J a 27 + cr 15tz z.
.
-t
t* .
,
,
The lnvestment Rule f0r a Two-stage Project Let us examine the two-stage project in more detail. We will assume as in Chapter 6 that the project, once completed, produces one unit of output period at an operating cost C. The output can be sold at a pricc P, which
We will asstlme that the price uncertainty is spanlldd
t7ycapital
nlarkcts.
so
that contingdnt claims nlethotls tran I7e used. Let yt dcnottt tlle risk-lldjustetl (. discount rate that agplies tk) P. and let t As in Chapter (), we witl Jt tnlstlessly the stlspendetl when that opefation be tenlpllrarily and can assunle =
-
.
P falls below C, and costlessly restlnled wllen P rises above C. so tllat the profit Ilow is given by zr ( #) mx gP C, ()1. '$Vewill assume that investing in the first stago of the projtzct reqtlires a sunk cost It and investing in the second stage rcquires :1 sunk cost I?. We can now solve the investment problem by wllrk ing backwards. first finding the value of the completc project V ( Pb. next (inding the value of tlle option to invest in the setrond stage. Fz J''), along with the critical price Pa*tbr that investment, and tinally Iinding the valtle 06 the option t() invcst in tlle tirst stage. F 1 ( #), and the corresponding critcal price /$* =
-
,
.
The
rza/e(?ftlle
Project
Nv'ealrcady know how to find P' ( Pj; we wtlrkcd thrtlugh this in sllme detail in Section 2 of Chapter 6. At this pllin ta we wiIl j tlst sunl mar ize the steps. Recal l that )' ( P) must satisfy the )l I()wing d ifferen tial equtt itln :
subject to P' (()) solution is
t1,and thc continuity
=
and
p'
p1
=
pa
=
1. a -
Bz /'/1: + #/J
-
1 - (r a
C. The
=
pz are
(?
=
z11PY'
V( P)
The constants tion, that is,
tlt' V ( l'4 and 1Z/, (#) at P
-
-
the solutions
J)/c2 + r
-
)/c
2
-
C/
to the fundamental quadratic equa-
g(r j)/f7'2 -
g(r
-
r
(
)/c
J-j: + 2,-/0.2 a
-
2
-
1 a
j2+ zs/g2
and B1 are determined from the continuity of 1z'( P) and The constants and f'- (#) at P C, are equal tt) .dl
'
=
z,=
cl -#, I /l :! /@l ---
p /'-
p t
j
(4)
Extensions atld c l
Bz Equations for any P.
(3).(4),and
-pj
yj
=
/3I
Seqlletltial /?l1'e??)7t;'?7/
zzl/pp/cflftpi.
-
/$
p
-
j
.
.
r
(5)give us the value of the completed
(5) project, V ( #),
The Second-stage Investment We lind the value F 2 #) along with same way that we in Chapter 6. The
t
>
of the option to invest in the second stage of the project, which it is optimal to invest, in the the critical price P* a at solved the investment problem for the single-stage project value of the option must satisfy the differential equation
-L'he First-stage Investment Given Fz P) and Pa', we can now back up to the tirst stage of the project and hnd the value of the option to invest. Fi ( #). and the critical price G*.By going through the usual steps. we can determine that Ft (#) will also satisfy the differential equation (6),but now subject to the boundary conditions =
Fi ( P) )
=
Fz ( PL*)
F'l ( #*I )
=
Fa'( P3*)
subject
Fz ( 'a*) F' ( #*)
2 a
IZ
=
( Pz*)
Iz
-
(8) (9)
.
P-'( #*). 2
=
We can guess and then conhrm that P? > C. so we use the solution for F(#) in the operating region, that is, tbr # > C, in the boundary conditions (8) and (9).We then obtain the solution Fz ( #)
=
D? #/1
From boundary conditions
(8)and (9)we
and that #z*is the
to the equation
solution
( l0)
.
determine that
Recall from Chapter 6 that equation ( l2) must be solved numerically for PJ. The solution given by equation (10) applies for # < #a'. When P t Pa' Iirm exercises its option to invest. and Fa(P) = )' ( #)- /2. This is important the what follows, so we restate it t'ormally: for Fz( #)
Dz ##,
for P
=
Pe( #)
-
Iz
<
#*, 2
(13)
,
.
D1 P#t
=
.
We can now use boundary conditions ( l5) and ( l 6) to 5nd the constant b and the critical price #j*. However, when substituting #( #) on the right-hand side of these boundary conditions, we need to know whether G*is greater or smaller than Pt ; note from equation ( 13) that if /$* < Pa* Fz( #1*) 92 ( h*)/' but if P*l > P*2 Fj ( #; ) P' ( P.1 ) Iz As we explained earlier. intuitively we would expect P3.to be greater than Jt' ; now we demonstrate this formally. Suppose #; (G #J,.. Then Fz ( P% l ) which is condition l6) implies Dz D3, that Jh*)/1 and the boundary Dc ( ( contradicted by the boundary condition ( 15). Hence we must have #; > /$*. Since #J > P', and since in this model the investment can be completed instantaneously (before P has a chance to change), we know that once # reaches h*and the 5rm invests. it will complete tpf stages of fc project. In other words. it will never be the case that the firm will invest in the hrst stage, and then wait rather than also investing in the second stage. This result may wondering why we bothered to seem anticlimactic, and the reader may be simply combine the two stages, did not this Why problem. we solve two-stage solution and the investment problem that was the /2, ten use to 13+ let l relatcd 6? There are two reasons for our choice. First, derived in Chapter firms often do complete te that real world. time. takes investing so in the with the later stages. wait and proceeding then before early stages of a project require technical different Second, the two stages may or managerial skills, countries. advantageously different located in or they may or they may be hrm these may sell a reasons one be subject to diftkrent tax treatments. For method contains another. calculaton completed the Our project to partially of the price of such a partially completed project. We will come back to this point shortly, but hrst let us complete the solution to our problem. =
.
'r
=
.
=
=
,
.
-
.
=
(12)
/1
-
The solution has the usual form:
i ( #) to the boundary conditions
0.
i (0)
326
Extensions and Applications
Seqttelltial //ll'c.f?'?1t'!n/
Because the firm should invest in both stages of the projectqif it should all, we can rewrite boundary conditions : (15)and (16)as
aOO
invest at
Fl ( PL. )
=
Pr( P(4
F'(#*) l 1
=
Pr?(#*).
Substituting the solution for Ypl conditions tell us that
-
(pl /2) -
Sz (#)X
)
+
+
(#l
-
/1
+
-1 Jpl
l (.p.)( l
-#1
1) #k*/
) ,
+ Iz + /!)
0.
=
(21)
Note that these equations are identical to equations (11) and (12),except that #t*replaces P?, and lz + 1L replaces Iz. It can be shown that these equations also imply that Pj* > #c*,and that Dl < %. As we would expect, the option to invest in the second stage of a project (giventhat the Iirst stage has already been completed) is worth more than the option to invest in the first stage (for which the payoff is the option to invest in the second stage). We have found that the solution to the multistage investment problem has exactly the same form as the solution to the single-stage problem that we solved in Chapter 6. The only thing that changes is the Mmount of the investment; the solution forthe srststage uses the total investmentcost h +lz, and the solution for the second stage uses the second-stage cost I1. Figure 10.1 shows Fl (#), Fc(#), and the critical prices G*and #a*,for the example that we worked out in Section 2 of Chapter 6, butwith the total cost of the investment I 100 broken down into two parts, /1 Iz = 50. The other parameters are the same as those used in Chapter 6. that is, rr = 0.20 and r = J 0.04. =
=
=
1.B
Summac
200 p
M Q
8a(P)
150
and Discussion
We have seen how we can work backwards to determine the values of the
options to invest in each stage of a multistage project, togetherwith the critical prices that trigger investment. Extending the steps that we took above to projects with three or more stages should be straightfomard. The idea is to start at the end, and then work bacltwards, using the solution for each stage in the boundary conditions for the previous stage. Note, however, that this just boils down to solving the model of Chapter 6 for different values of the
I
j
j
I I l I
l
1
P1
1,
'
-50 -100
j
1 1 l
1oo
O
#1lC/r
l
C Lq so
equation -
l
250
(l8)
,
in the operating region, these bounda:y
pz (#1)(,a-/l: pk
and that #; is the solution to the
-
I
.82
D3 =
lz
1,
y(p)
0
5
-
y(J5 fa
10
-
Ij
8:
/2
-
15
20
25
P Figure 10.1. Crlctz/ Prices and Option Izil/llf?.)r a Fkvtp-.skfl.cProject ( P3*is critical price for investment in tirst stagc, and P? that for second stage.)
remaining sunk to invest in any
cost required stage
to complete the project. The value of the option the form
j of an N-stage project is of 6 ( #)
=
Dj #/''
and the coetxcient Dj and critical price Pj. ( 11) and ( 12), with #j* rcplacing P? and I). +
.
are found by solving equations (.+l + + IN replacing Iz Of what use is this result, however, if investment in the hrst stage always implies immediate investment in all succeeding stages'? Indeed, if investment .
.
.
.
is instantaneous and there are no other impediments to investing in all stages at once, there is no need to go through these steps we can simply take the total cost, /, of all of the different stages of the investment, and then use the results of Chapter 6, or equivalently, use equations (11) and (12), with / replacing J2. In practice, it is often the case that it is not possible
(or desirable)
to
invest at once in every stage of a project. First, most multistage projects take considerable time to complete. (Investments in new oiI production capacity, in the development of a new line of aircraft, or in the development of a new drug are al1 examples of this.) Hence price might rise above the crtical level
Eztensiolu
328
trl/lt'
Applicatils
that triggers the srststage of investment, and then while that (irst stage is undemay, fall below the critical level needed to trigger the second stage.
Then the hrm shotlld wait rather than proceed with the Iater stages of the project; it is clearly important to know the critical prices for each
stage.
Second, even if investment can proceed very quickly, it may be important nonetheless to know the values of the options to invest in later stages of the project together with the critical prices. The reason is that the 6rm might decide to invest in the early stages, and then sell off the rights to the Iater stages rather than do the investing itself. An example is the development of a new drug by a small biotechnology company. The company might be very adept at doing the R&D needed to develop and patent the drug (the srst stage of the project), but tlzen tind it best to Iiccnse or sell the patent to a Iarge pharmaceutical company that is better able to test, produce, and market the drug (thelater stages of the project). Third, there can be other considerations that lead a company to invest in the hrst stages of a project but then delay the later stages. For example, the 5rm might have a unique opportunity to buy land or mineral rights (the hrst stage), but then want to wait before developing the Iand or exploiting the mineral rights. Or. government regulations might force the company to delay the later stages of an investment (forexample, regulations requiring the testing of new drugs). fact that most multistage We will focus on the tirst of these reasons-the investments take considerable time to complete. In the next section we will turn to a slightly more complicated model in which the investment occurs continuously the firm invests a dollar at a time. so that there are effectively an insnite number of stages to the project-and the investment also takes a minimum amount of time to complete.
2
Continuous lnvestment
and Time to Build
We turn now to a model developed by Majd and Pindyck (1987)in which a 5rm invests continuously (eachdollar spent buys an option to spend the next dollar) until the project is completed. investment can be stopped and later restarted costlessly, and there is a maximum rate at which outlays and to build''). Hence the solution construction can proceed (thatis, it takes of the model provides a nlle for optimal Sequential investment that accounts for the time required to actually undertake the investment. dtime
Inveslnltnl us'f.?tyl/f.??l/ff7l In this model. the firm receives nothing until the project is completed. The payoff upon completion is F, the value of the operating project. Also, it is assumed that lzrfollows an exogenous geometric Brownian motion process, that is,
dv
=
jf
Pr dt +
fr
)' Jz.
(Later we will see how the model can be expanded so that the price of the project'soutputqratherthan P' itself, follows anexogenousstochastic process.) We will assume that spanning applies, and we 1et g equal the market riskadjusted expected rate of return from owning and operating the project. We Jz a. assume as usual that a < p,, and let Like Rome, this project cannot be built in a day; k is the maximum rate at invest. Investment is also irreversible, so the which the firm can (productively) rate of investmenta J(/). has the constraint () :G Itj :Gk. If no investment is made. the previously installed capital does not decay. If lzrfalls to a sufhciently low Ievel the firm can suspend investment, and if F rises Iater, resume at the point it le off. Denote by K the total remaining expenditure required to complete the project. Then the dynamics of K are given by =
(IK
-
-1
=
dt.
We therefore have two state variables that alect the optimal investment decision. The hrst is the remaining investment required to complete the projecq K. which follows equation (23).The second is the current market value of the completed project. Z, which follows equation (22).The problem is to find the optimal investment rule, /*(P, S). Because there are no adjustment costs or other costs associated with changing the rate of investment, the problem will have a solution at any point in time, the optimal rate of investment will be either () or k. As a result, the optimal investment rule reduces to a critical cutoff value for the completed project, Z*(S'), such that when V F*(S) the 5rm invests at the maximum rate 1, and there is no investment otherwise. As always, the firm has an option to invest that it may or may not exercise. will denote by F(V. A-) the value of this option, assuming it is exercised We optimally, that is, assuming that the 5rm follows the optimal investment nlle V', and not investing othemise. Then, as in earlier by investing when IZ will F(F, 5nd K4 and obtain the critical value P'*(S) as part chapters, we of the solution. We can do this using dynamic programming, or, if spanning holds, using contingent claims analysis. We will assume that spanning holds and use contingent claims analysis to derive a partial differential equation ''bang-bang''
Extensiotts
330 that must be satisfied by us F* K).
K)
For review, we will derive the differential equation portfolio containing the option to invest and a short (or Fv units of an asset ordynamicportfolio of assets F).The value of this portfolio is m F(F, K) Fv change in this value is =
=
= =
dF
Fv dv + !2
1z (r
-
for F(F. &). Consider a position in Fv units of F perfectlycorrelatedwith P', and the instantaneous
Fv d Izr
-
2
Appliciltions
A'); the solution to this equation will give
F(IZ,
2.A An Equation for FV,
#*
lnt'/
j
-
FK
+
dt
FK
dK
-
smooth-pasting
conditions that F( F. K4 V*. Condition (25)jtlst says that when K reaches zero, the project is completed, and the lirm receives the payoff I!. Also, as V becomes very Iarge relative to the total required expenditure &, it becomesvery unlikely that the investment will ever be halted prior to comple=
tion. However, the project will still take time K(k to complete, and during this time the expected rate of increase in V is only /,t J. Hence for very Iarge V, a $1 increase in V leads to an increase in F( P-, K4 equal to -
(27)above.
Fv dv
2.B
.
The short position requires a payment flow of 6 Fv P' dt, and, to the extent that investment is taking place, an additional outflow of l f. Hence the total t5 l dt Fv P' dt, and since the portfolio is return on the portfolio is dl risk-free, this must equal r * dt. Substituting for #*, dividing through by dt, and rearranging gives the following partial differential equation for FV. A'): -
along with the value-matching and and &.V, K4 be continuous at V
This is just condition
& v (:p')2
72 F g dt z
Sequctltial Jnvesfmellf
Obtaining a Solution
When 1/
<
Z* and /
=
0, equation
(24)has the
F Z K) .
=
analytical
solution
X h' l3'
-
where This solution might seem troubling in that it does not appear to depend on zzl K. In fact it does depend on K, through the z4. As we will see, be found with the in conjunction boundary V* V*Kj, and hence will must vary with K. When V > F* and l 1, equation (24)is a partial differential equation the parabolic of type that must be solved numerically. To do this, we srst eliminate Wusing equation (28)togetherwith the value-matchingand smoothpasting conditions: t'constant''
Equation (24)is also the Bellman equation from dynamic programmingwhen the discount rate is the risk-free rate, r. As an exercise, the reader might want to conlirm this by rederiving equation (24)using dynamic programming.
Note that equation (24)is linear in /, so that the optimal investment rule is indeed to either invest at the maximum rate kor not invest at all. When there is no investment, that is, 1 = 0, the term in FK disappears, and equation (24) simplises to an ordinary differential equation that can be solved analytically. k, the equation will have to be solved numerically for However, when l both FV, Kj and the critical boundary V*K). The solution to equation (24)must also satisfy the following boundary
=
=
(29)
=
conditions:
FIP. 0) F(0, K4
V,
=
0,
=
(26)
Then we numerically solve equation (24)(settingl 1), along with equation (29) and boundary conditions (25)to (27)using a hnite difference method. This procedure transforms the variables V and K into discrete increments, and the partial differential equation (24)into a snitedifference equation. The resulting equations can be solved algebraically, beginning at the terminal condition K 0 and proceeding backwards in increments of tk K, and hnding the free boundary V*(K) at each value of K alongwith FV. K) foreach value =
=
lim
F-
Fy ( 1Z, K)
=
e-nK/k
btolsiolls of V. The details of this to this chapter.l
numerical
and Applicutions
procedure are discussed in the Appendix Table 10.1. Nttmerical Exatnple of Opfmfl/ Investment Sl//t?
A Numerical Example
2.C
As anexample, considera project that requires a total investment of$6 million (S), which can be spent productively at a rate no tster than $1 million per = 0.20 0.06. and f.r year (). As for the other parameters, we set r 0.02, rates). (a11at annual As explained in the Appendix, the solution procedure requires a discretization of the variables P' and K. For this example. we will assume that investment outlays are made quarterly, that is. K is measured in discrete units of $0.25 million. (The solution procedure uses the logarithm of Pr rather than F itself; we take increments of log P' of 0.15.) The numerical solution is summarized in Table 10.1. Each entry in the table is the value of the investment option, Fb''. Kj, corresponding to a particular level of F and K. (To save space, only values of K in increments of $1 million are shown in the table.) Entries with an asterisk correspond to the critical value V*K4. For example, a project with $4 million of investment expenditures remaining has a critical value Z* $7.03 million. that is, if F k $7.03 million the firm should make the next quarterly installment of investment at once; othelwise it should not. Also, if P' = $7.03 million, the value of the firm's investment option is $ 1.65 million. (Note that this is Iess than 1Z K = $7.03 $4 $3.03.The reason is that the hrm only receives k' upon completion of the project, which will not occur for at least four years.) Table 10.1 can be used to make optimal sequential investment decisions as the construction of this project proceeds, that is, as K falls from $6 million to zero. Unlike the simple model discussed in the previous section, here there is no guarantee that once investment begins it will continue until the project is completed. Because the project takes time, F can fall during construction to a point where investment will at least temporarily stop. Finally. note that this table can be used to evaluate any project requiring a total outlay of $1. $6 million, as long as the same values for r, 6, t7', and k apply. $2, Table 10.2 illustrates how the solution depends on the parameters f:r and J. The table shows the critical cutoff value, Z*(A'), for the initial investment decision (thatis. when K $6 million), for tr 0.10, 0.20. and 0.40, and for the = 0.02, 0.06, and 0.12. (The middle entry in this table corresponds to =
*
-
.
.
.
=
,
=
lHawkins
(1982)
=
developed a model of a revolving credit agreemcnt
structure to this one. but yields analytical solutions in both regions.
(Note.. Entries show value of the investment option. FV. K). Starred entries indicate the optimal investment rule; the value of V corresponding to each starred entry is the cutoff value. F*( &). Parameter values are r 0.02, f.'r 0.20. J 0.06. and maximum rate of investment k $1 miilion per year-) =
(Note: U' is present value of J,' assuming investment prllceeds tt maximunl ratc i- $ l million per yea r. 1z' and P' llre lbr K $6 Iz1ilIion. ) **
*
*
**
furthdr increased to (). l 2. The
reasen is tlat it takes tilntt to btlild tltt project. effect on t ht! incttlltive to invest. Tlle pklyt'lft'1.--is ill)ly reccived tlpon conlplotion, antl is expected tt1grow ollly ;tt thd rattl /t f. Tilue to btlild theretbre retluces tllc presellt vlllutl of the p:lyoff, and redtlces it by :1 largcr alnount as f increases. Tllis in turn l'cducos tlltr incelltive to invest- that l ().2 shows, for higll valtles is- increases the ctlrrent critical value 1.z' zks of this second eftkct can dolninate, so tllat I'- increases when t is inereased. N'v'ecan net out this second e fikt:t by calculating the present vaiue of f.'assunling that thtl investment expentlittlres are mad e at the maximunl rate kNvhicll
has
:11-1(lppositc
-
'lilble
*
.
*
*
.
0.20 '
0.40
Z P'
12.81 7.03
* *
so that it takes T Kj 1-years to colnplettt the project. The disctltln t rate is ;.t btlt lz-has an expected rate of growtl) a. so this present value, )' is silnply **
*
8*
''Il'tt7lk)l ().2
tbase case'' shown in Table 1(). l Observe that tbr any value of J, V increases when o' is increased, and can increase dramatically when t'z is increased above 0.20. This is quite similar to the results we gbtained for the simple modcl P resented in Chapter 5.2 .)
*
The dependence of 1Z* on J, however. is less obvious. In the model ()f Chapter 5, an increase in t always reduced the critical value lz' incrcasing the incentive to invest rather than wait. Rucall that the reason for this is that is the shortfall in the expected rate ofreturn from htllding the option to invest rather than the completed project itsclf, and hen rcpresents an opportunity cost tlf waiting, rather than investing now. In Table 10.2, however, V. tirstdecreases as J goes from 0.02 to 0.06, but then (whenc ().l() or 0.20) incrcases as J is
ltlso
shllyvs vltl ues t)f 1.z- Ntltil t llat lr any va Itle of f'r IZ alvvays decreases as t is increased. I-lowIarge a wedge do irreversibility nntl tlle abil ity to dclftydrive between the payoff of the investment and its cost? One way t() lnswt!r this is to compare P' t() the present value ()f tlle investment expendittlres needed to complete the project. Assum ing that these expendittlres are made tnlntinuously (wer the pcriod F S/ l'. their prcsent valuc is **
.
**
=
&'/ l
*,
=
**
.
K
.It
=
j. t
-
,
()
1,
r
tj r
(j
=
-
c
!'
&/ L
)
k .
1*
For our example, K $5.65 mill ion. Tllus as X'tble I().2 shows. V is higher than &* for any values of rr and J, and is mtlch higher il' c is large and/or (5 is '
**
=
small.
We can also examine
the ways in which
uncertainty
and time to build
interact in affecting the invcstment decision by calculating P' for differcnt values of k. Figure l 0.2 shows $6 million, and as t function of 1:for K 2.3 when (). small. Observe 0.03 and that l is changes in 1: have only = effect k' then small The expected that the is reastln ratc of growth on a of V is close to /z, the risk-adjusted market ratc, s() the ability to speed up construction has little effect on the value of the investment option or on the investment decision. However, if 6 is large, P'* can be fairly sensitive to k. *
:We sbouid also remind rcaders of our discussion in Cbapters 5 and 6 conccrning the parameter As long as somellnc is holding the assct thllt tracks the risk in P. the equilibrium condition t y a must be satisfied. When a' changes, hllw this condition goes ()n being fulfilleddcpends on the hidden assumptitlns conccrning the behavior ()1' the holders of this f.
=
-
assct. It is reasonable to assume that thc risk-frctl rate is r exogcnous. being determined by government policy (r (he behavior of a much wider class of investors. Then an increase in f'z raises /z = r+/ f.r ppm. where 4 is the market pricc ofrisk and p:= tht! cllrrelation coefficient between the asset tracking P and the market portfolio. If the holdcrs of the P-assct have an exogenous convenicnce yield J that remains unchanged, thcn somehow u must change to presel've equilibrium. In this case fz and J can be treated as independent parameters. That is the interpretation we have taken here. However, if (r is exogenously ixcd, thcn the convcnience yicld J must change lo take up the slack. We leave this casc for intercsted rcaders.
'
*
=
(b
*
.
'hAs explained in the Appcnd ix. t htl calctl Illtiflns t)I'
approximation. of the finite l0.3 wlluld lic ()n smtltlth curvcs. tlifference
Abscnt
'
arc stlbjtlct t() nu mcrical tlrrt)r beclluse such errtlrs. the ptlints plotled in Figurcs l (1.2 antl -
t7?'l# sjppliciltions
Ftellsions
2.D
30 28
26 24
z,? s- 20 1a c; .2 16 tt '@
'!
=o UN =
O E w
@ y
s
+
.
N.x
X
+
?p 0.03 =
s
+-----
14
b...w
j2 jc
+ .
+
''= .-=
'--.>x.
8
6 4 2 0 0.
:
0.50
1
=
0.12
1
.00
2.
.50
2.50
3.00
Maximum Investment Rate, k (millionsof dollals per year) llltte.
l7ij;ttre 1(3.33. (7rJcJ/ (?-az
The Value of Construction Time Flexibility
How valuable
'
=
> z:
tetlttelltill /?lv,c-/??,c?l/
r'
*.
().()0-. rr
ils az
/7!l?1cJ/a
())' #/(a??7Ill?T
(1.20va n tl JJ
=
ltlte
is tl4e ability to speed up construction of a project'? Many projects can be built with alternative construction technologies that differ in terms of nexibility over the rate of construction. Technologies offering greater tlexibilitytend to be more costly, so the increased cost must be balanced against the value of the greater construction time tlexibility.We can use this model to determine the value of that greater ilexibility. Since greater construction time tlexibilitycorresponds to a higher k, we simply calculate the value of the investment opportunity, F r''.K4, for different values of k. The change in F corresponding to a change in k then measures the incremental value of extra llexibility. Of course this value of extra tlexibilitywill depend 4 on P' and K. as well as the other parameters. Figure 10.3 shows this calculation for the base case parameter values r 0.06. It plots F 1z'. K.' as a function of holding P0.02. t'r 0.20. and constant (lirstat $ 10 million and then at $ 15 million) and holding K constant (at $6 million). For each value of F, the incrementai value of construction that is. by the slope of the curve. As the time flexibility is given by A sgure shows, the incremental value of llexibility is always positive, but falls as k increases. Alsoa thc horizontal lines in the Iigure show the value of the investment opportunity when there is maximum tlexibility,that is, J' x,. when IZ = 10, this value is 4.(), and whcn P' l5. it is 9.0.5 Consider two different construction technologies with the same 1(), the same total cost K 6. but different maximum rates of investment'. k 0.5 tbr the first. and 1 for the second. As can be seen from the figures the incremental value of the more gexible technology is 2 F/s 0.98/0.5 ;:k: 2. Hence one should be willing to pay up to $2 million to have access to the more oexible technology. The incremental value is higher if V is higher; if 1z' 15, the incremental value is about 5.5. Of course, n general greater flexibility might be accompanied by a different total required investment, K. In that case, we can rank the technologies by comparing F fz'. K., k') for each. =
-)
=
f?/'
Itlb'e.b'tttlz,ttt, k
1ti millio r!)
=
'
-//,
=
Then, lower values of k (thatis. longer minimum construction times) imply a lower present value of the payoff from completing the project. and hence a higher cttrrent critical value F*. As above, this second effect could have been muted by considering F**; we leave this for the readers. We have assumed that k is constant. which may be unrealistic for some projects. where the maximum rate of investment can depend on the stage of the construction. However. this model can easily be modifed to allow the l'unction of the total remaining maximum rate of investment to be a (known) substituting investment, K. Thisjust means a function of K rather than a conequation numerical procedure can be used to stant for 1 in (24).The same maintain the assumption that obtain a solution. Likewise. it is not necessary to 1 1: 0. There might be some positive lower bound on the rate of investment (for example, to maintain a construction site), so that the constraint becomes I S l S 1, with I or k possibly dependent on K. This constraint can be interpreted in two ways: either that the 5rm is forced to proceed at this minimal rate once it embarks on the project, or that any smaller expenditure is tanif not properly tamount to abandonment because existing stages will maintained. The optimal decision rules will be different in the two cases: the initial entry threshold will be stricter if there is no way out Iater. '
=
'
=
=
=
.0
=
=
4Another measure of the incrcmental valuc of
is the change oexibility
in F( 1z',A-;1)/ K
(the
value of the investment opportunity per dollar of total rcquired investment) corresponding to a change in the minimum construction time Kl k. Notc, however. that Ff P', KL k) is not Iinear homoyeneous in K, so that tis rneasurc will also dcpend on K. (he model reduces l Nvhen k to the one we studied in Chaptcr 5. By using the results ().()2, ().2(). t5 ().06, and K 6. the from Chapter 5. the reader can chcck that witb r value F V exceed the value of the investment is Since 8.6. 1() and btlth this. l5 critical opportu n ity is just V A- that is, 4.() (r t.).(). =
,x,
=
*
=
=
-
,
f:r
=
=
=
338
tl,?f/
Fblellsiolls
Ainplicatiolts
Once we know 1/-( #), we can proceed pretty nlucll :ts beforc. We write ol the investl-nent (Jption as F ( P. A'), ankl tllen by going throtlgll tlle usual step obtain the folIowing partiltl di ftrential equatilln:
10 !/
9
2k-
C' E
=
x
U=
7 -u5
>w .G
t/
+
c) -Q >
1
0
,D'
0
0.4
F P. 0)
x
A .....D0.8
(35)
t/' $1O million =
1.2
1.6
:./jt? ?l.?z: lrhlItt ?l tr L-() t
I ;h ()Mvs I:f 1/ J; k ) :ts fu nc tit) n '.
(33) (34)
2.0
2.4
2.8
Maximum Rate of Investment Rate, k (millionsof dollal's per year) -
l'' ( P4.
n
.-'M
A'
l-iffttl-tr / (.-/.
=
.A
...0..-
+
=
tllat 11 replactts
....0--1
.
2
Note that this equation is the same as equation (24)ttxccpt V. Boundary conditions (25)to (27)now become
$1O million,k
=
(32)
+
+
we'+
3
B
+
+
+...A'
uq =
$15 million +
; y 6 cpz 2= 5 *dw m (1)c 4 E M
$15 million.k
the valtle
8
N
-
=
/?l1.'c-/??lt???/ qtz.tyl/tiv/fl/
()
f L' ftlr 1,
ar
?
().()2. rr
oitlt
xz
t,
/7/ttr/)ilily,
().2(1,:tn
tl
JJ
Also, the value-matching and smooth-pasting conditions again apply-. F-( P. S) and Fpl P. K) must be continuous at the critical price P = P. ( K ). A solution lbr F( P. A') along with the critical prit:e P* ( K ) can be obtained numerically in the same manner as before. The main diffcrence is boundary condition (33).Numerical Malutls for P-( Pj must be calculated for thc entire range of # used in the solution proccdurc. Tllen, at thtt terminal boundafy ( K 0), F #. ()) is set eqtlal to 1,Z' ( Pj tbr each vitlue ()f P. Moving backwardsin incremcnts ()f /X.Ks J--( #. Kj and the btlundaly #* ( K ) are found for each K as beforc. The result will be a tablc simil:tr t() l (). 1 except that the left-most column will contain priccs, and tlle right-most colunln will contain projcct values (#) that corrcspllnd to those priccs. =
=
T() rn illit)n I
''r)'1l7lt)
-
2.E
A Simple Extension
ln Chapter 6 we argued that it is often more natural to think of a project's output price, rather than the value of the project. as following an exogenous stochastic process. Also, an advantage of starting with pricc as the stochastic variable is that we could determine how the value of the project depends on price, and hence how it (aswell as the investment decision) is aftkcted by price uncertainty. The same arguments apply to this model of continuous investment with time to build. Fortunately, it is easy to modify the model so that the price of output is the exogenous stochastic state variable. As before, assume that there is a constant operating cost C, and that operation of the project can be temporarily and costlessly suspended when P falls below C, and costlessly resumed when # rises above C. We will also again assume that # follows the geometric Brownian motion of equation ( 1).Then, as shown in Chapter 6 and reviewed at the beginning of this chapter, the value of the project, V ( Pj, is
givenby equation (3).
'r
3
The Learning Curve and Optimal Production
Decisions
Continuous sequential investment problems ()f the sort that we examined in the previous section are actually quitc general. One closely related problem is the decision to produce when a firm faces a learning tmlwe and tlncertainty over future demand. When there is a learning curvc, current production has two benehts. As always, it results in output that can be sold at some market price, but in addition. it moves the tirm down the Iearning curve, reducing its ftttttreproduction costs.t' Hence with a learning curve, part of the firm's ' For thetlrctical d isctlssitlns 01* 1he lttarn ing cu l-ve and its ilnpl ica litlns lr prici ng and prtlduction in a deterministic sctting, scc spencc( l98 l ). Kalish ( 1t)83), and Pindyck ( 1985). A variety of empirical studies have demtlnstrated tht! imptlrtance tlf Icklrning curves lr a wide range ()1' industries; ftaran examplc. sce Lik!bernlan ( I 984).
Fb-tcllsiolls
340
t'lnt
Applicatiols
cost of current production is actually an investment, the payoff from which is the reduction in future costs. Furthermore, it is an irreversible investment', if and recover the firm regrets its production decision, it cannot its costs. Finally, if the price of the Iirm's output lluctuates stochastically, the future payoffs from this investment will be uncertain. ttunproduce''
A hrm's production decisions when it faces a learningculwe and a stochas-
Seqttelltial /klv-p??fswf
Q, and one control variable, in this case subject to 0 :j x .G 1.The problem is to tind the value of the firm. P' ( #. (.2).along with the optimal production -r,
rules x* ( P,
-r,
.
if Q < Qm
(,,2
!. 2
,,
,
(36)
.
As in Chapter 6, we will assume that the iirm can, with no additional cost, shut down when # is low and later rcsume production if and when P rises sufhciently. But unlike our model in Chapter 6. now it may be optimal to continue prodcing even if the current profit llow is negative. The reason is that the value of current production depends both on the current cash llow and on the amount by which future costs are lowered. (In other words, there is a shadow value to current production which measures the beneht of moving down the Iearning curve-) Wewill assume that uncertainty over P isspanned by existingassets in the economy, and that /.1 is the risk-adjusted discount rate for future revenue. As ft-a-hlowwe usual, we let can use ourstandard contingentclaims method operating to value the :rm and determine its optimal (value-maximizing) strategy. The value of the hrm will depend on P and also on how far it has moved down the learning curve (thatis, on cumulative output, Q). As in the model of the previous section, there are two state variables, in this case P and =
#: p'pp + (r
j) #
-
vp +
x
rc
if P
p' +
/.
-
(#
.r
Q4must
c( :))
-
t C( Q4
+ V
=
satisfy the
0.
,
(38)
otherwise.
This formalizes our earlier intuition. A unit of current production yields an immediate beneht #, and a future beneht in the form of lower costs of future production, which is captured in the increment to the value of the tirm. 1%. Production is justifiedif the sum of these benests exceeds the current cost C( Qj. Once the problem is solved and the function IZ ( #. (.?)is known, we have threshold curve or free boundary, #* ( Q4,such that a P
>
P
+
(C,?) if and only if
P + P'(?( P.
(.?)> C( Q).
Then the optimal production rule becomes l if # A #*( Qj and x () if P < P* Q). Also, since cost falls as Q increases, we will have dP*/dQ < 0. We now proceed to derive this solution. The solution to (37)must satisfy the following boundary conditions: -r
,
if Q >- Q
Q).
The reader should be able to easily verify that )' ( #. followingpartial dit-ferential equation:
tically evolving output price are thus closely related to the kind of continuous investment decisions we just examined. At each instant the firm must observe price and decide whether to produce (andthereby invest in future cost reductions). knowing that it can later cease producing should the price fall sufGciently, and then resume producing should the price rise. To show how an optimal production rule can be lbund and to illustrate the close parallel with the investment problem, we make use of a recent model by Majd and Pindyck ( 1989). In this model the tirm sells its output at a price P that follows the geometric Brownian motion ofequation (1).Marginal production cost is constant with respect to the rate of output. up to a capacity constraint arbitrarily sct unity. However, there is a learning curve; marginal cost declines with cuat mulative output, Q, until it reaches a minimum Ievel, Letting c denote the initial marginal cost, and Q,nthe level of cumulative output at which learning ceases, we can write the marginal cost function as
34l
Z (0, Q4
=
Iim Vp P.
Qj
P-w(r
P*(Q4 C(Q4+
=
0. =
(39) 1/.
P:( P.
-
F(#.
=
(.?) 0. =
Qm) F( #), =
(41) (42)
condition that P' ( #, along with the value-matching Qj is continuous at P P* Conditions (39)and (40)are analogous to boundary conditions (26)and (27) in the mbdel of investment with time to build in the preceding section; 0, it remains 0, so that )' (39) just says that if P 0, and (40) says that will P becomes high, the firm almost surely always produce, and then if very the incremental value of a $1 increase in price is just the present value of $1 =
=
=
342
Exellsiolls
J/lt.f
v'.6illllictliolts
Seqlttnltial
Condition (41) follows frol tlpe per period i'orever, discounted at ;L a maximizatfonof value with respect to production it is just the intuitive detl inition of #*( Q) above, and replaccs thc smootll-pasting condition. Finally. condition (42)is analogous to boundary condition (25)in the model ()f investand production cost ment with time to build. It says that once Q reaches becomes constant, cumulative production can no longer affect the value of the firm, so that V is a function only of P. Nvhat is F in this case? We already derived it in Section 2 of Chapter 6 or eqtlation (3) above: =
Entries shoqv vklltle ()1- the tirn1 l,' ( /?. Q)$Starretl ent ries ind ica te t)ptimal prodtlction rule; tlle price correspllnding t() tlacll sta rretl ent ry is tlle 0.05. c critical price P* ( l-'). Paranleter valtles Ctre 1. J 4(). l 0a (...?,a 2() :111d t'z t) 2() j .
=
,
.
.
Curntllative Prodtlct itln in U llits trtlst in dk')lltrs)
(current marginal
Price
if P
(43) >
where Jl and pz are, respectively, the positive and negative roots of damental quadratic, and /1l and Sa are constants given in equations (5) above. When P
<
P* and
-r
=
0, equation Z ( P.
(37)has the
(.?) =
tl PF'
usual analytical
the fun-
(4)and solution
.
3.A Characteristics
.
of the Solution
Table 10.3 shows a solution for the following parametervalues'. initial marginal 40. hnal marginal cost 'J 10. Q,,,= 2() (so that lz cost c (J.t)693), 6 = and 0.20.7 0.05 The table shows. for various amounts of cuo' r mulative production. the value of the firm as a function of price. as wcll as the critical price required for the firm to produce. For exarple. when cumulative production is zero tsothat marginal cost is S40), the firm should produce when the price is $25.53 or more, and at this price the value of the firm is $178.53. When the price is below $25.53 the 5rm does not produce. but still has value =
=
=
=
=
7Recall that fr is the annual standard deviation of percentage price changcs. If thc outptlt of this firm happened to be an industrial commodity (for example, copper, cotton. t)r lumber), rz should in fact be signihcantly larger than (1.20. Also. notc that if all price risk is diversifiable so that y J implies that a r, then setting r 0, but if tbere is systcmatic risk so thltt Jz > t', a > 0. =
220.39 2(5.32 194).97 l 77.33 l 64.38 152. I l 14().5i) 129.55 l l 9-26
238.()6 222.89 2()8.42 l 94.62 l8 l
243.77 228.6)() 2 l4. l 2 2()().32 l 87. l 8 174.67 l 62.69 15 l 14().79
.76
When P > P' and x l (37)does not have an analytical solution and must be solved numerically. Once again.a tinitedifference method is used, whereby the partial differential equation is transformed into a diffcrence equation that can be soived algebraically. This method is essentially the same as that used in the preceding section, and described in the Appendix. =
beeause the priee may rise in the future. As Q increases, the value of the tirm costs have been reduced), and #* falls. The critical price falls to rises (because long-run the cost of $10 as Q reaches 20..at that point the tirmhas reached the of the learning curve and the shadow value of cumulative production bottom is zero. Of course, even if there is no uncertainty, when there is a learning curve a should produce at a price that can be substantiallybelow current marginal lirm because of the shadow value of cumulative production.8 To see how uncost affects the hrm's production decision. we can examine this shadow certainty V, and its dependence on P and (r. Figure 10.4 shows, for Q 0, I''c value, 0, 0.05, 0.1, 0.2, 0.3, and 0.5. Also shown is the as a function of P for a equation line c #; note from (41)that #* satisies P CQ) V, and so, when Q = 0, is given by the intersection of this line with the Vc culwe. When 0, so Vc is zero up to the critical price of $19.00 (if P is J r t'r 0, a below this price the tirm will never produce), and then is constant at $21.00.9 Note from this sgurethat the larger is c. the larger is #*. For example, when r.ris 0.5, #* is about $31. The effect of uncertainty on Vc dcpends on the current price. The possibility of future increases in price raises r%.and the possibility of decreases reduces it. At low priccs. the possibiiity of increases in price dominates, so uncertainty increases Vc. To see this. note that if o. = 0, the price can never increase (because r), so future cost savings have no value when price is low. However, if o' > (), the price may later rise enough to justify production, so reductions in future cost have some value. The greater is tr, the greater is the probability that the firm will begin to produce over some finite horizon. and thus the greater is the present value of reductions in future cost. At high prices the effect is just the opposite. Suppose P is high and the firm is producing. If c > 0. the pric may fall to the point where the firm shuts down, and the higher is o', the sooner this is likely to occur. Thus for high prices, a higher tr implies a lower I/. It is this higher price region that is relevant for the production decision. Thus. as Figure 10.4 shows, an increase in c raises P-. This result can also be understood by remembering that with a learning curve. part of the srm's production cost is actually an irreversibke investment in reduced future costs.
C
'% N
=
=
-
-
BAs Spence
( 1981) has
shown, if the discount rate is zcro
(and there
is no uncertainty). a
competitive firm should produce when price exceeds the long-run marginal cost that will prevail when the lirm rcaches the bottom of its Iearning curve. gWhen fz 0, Pb and P* can be found analytically by integrating the tlow of discounted profit. =
.
16 14
O
(J!
o5
l I I I 1
12
$ l
6
l
4 2
l I 1
I
0
u/l/
l7igttre l (l..6.
(for r
=
J
=
fltllv
0.1)5,
When future price is uncertain.so
fz
bIlllIe =
.
(r l I 1 i
:)?C
o.2
=
G
0.3
=
'
l
I
I l $ 1 I l I I I I l I l I I I I 1 l
$ 1 I I I l l I
20 Price
10
() j
ZQ
I
10
O
=
=
()-
=
=
8
=
=
(y
18
=
-
345
ty l I 1 l I
.
C
-
P
I
1 1
30
littttlitib'z.
0.5
=
40
/3r/a:/l/c/itpll
0x().()5,().l 0.2. (3.3,and 0.5) .
is the payoff from this investment. As usual,
this implies an opportunity cost to investing now rather than waiting to see how price will evolve. The ntlt benefit of this investment which is measured by Vo thus falls, pushing the critical price up closer to where it would be if there were no Icarning curve.
4
Cost Uncertainty and Learning
ln most of the investment problems that we have examined so far. it is the future payoffs from the investment that are uncertain. However. sometimes the cost of an investment is more uncertain than the payoffs. Nuclear power plantssforwhich construction costs are hard to predict due to engineering and regulatory uncertainties. are an example. Although the future value of a completed nuclear plant is also uncertain (becauseelectricity demand and costs of alternative fuels are uncertain), construction cost uncertainty is greater than revenue uncertainty. and has deterred utilities from building new plants. Other examples include the development ofa new line of aircraft, urban con-
struction projects, and many R&D projects, such as the development of a new drug by a pharmaceutical company.
346
-lellsiolls
-,',-
d'//lt
zjpplicaliollx
When projects take time to complettl. the Iirm facuts11 seqtlential investbtlth ment problem that can involve two differcnt kinds of cost tlncertainty, of which were discussed briefly in Chapter 2. Tlle first is lechllical uncertainty, and relates to the physical difficulty of completing the project: Assuming prices of construction inputs are known, how mucll timea effort, and materials will ultimately be required'? Technical uncertainty can only be resolved by tlndertaking the project; actual costs and construction time unfold as the project proceeds. These costs may be greater or less than anticipated if impediments arise or if the work progresses more quickly than planned. but the total cost of the investment is only known for certain when the project is complete. The second kind of uncertainty, which we will call inpllt ccu'l unccrtainty, external is to what the Iirm does. It arises when the prices of Iabor, land, and needed materials to build a project fluctuate unpredictably, or when unprechanges in government regulations change the cost of construction. dictable and regulations change regardless of whether or not the tirm is investPrices and uncertain the farther into the future one Iooks. Hence inptlt ing, are more cost uncertainty may be particularly important for projects that take time to complete or are subject to voluntary or involuntary delays. Both technical and input cost uncertainty increase the value of an investment opportunity for the same reason that uncertainty over future cash tlows increases it the net payoff from the investment is a convex function (Jt' the cost of the investment. I-lowever, as %9e saw in Chapter 2, these two types of uncertainty affect the investment decision differently. Technical unccrtainty makes investing more attractive; a project can have an expectt!d cost that makes its conventional NPV negative. but it can still be economical to begin investing. The reason is that investing reveals intbrmation about cost, and therefore has a shadow value beyond its direct contribution to the completion of the project; this shadow value Iowers the full expected cost of the investment. (In Chapter 2, we illustrated this in the context of a simple two-period
example.)
On the other hands input cost uncertainty
makes
it Iess attractive
to
invest now. A project with a conventional NPV that is positive might still be uneconomical, because costs of construction inputs change whether or not investment is taking place, so there is a value of waiting for new information before committing resources. Also, this effect is magnilied when fluctuations in construction costs are correlated with the economy, or, in the context of of cost is high. The reason the Capital Asset Pricing Model, when the is that a higher beta raises the discount rate applied to expected future costs, which raises the value of the investmect opportunity as well as the benefit from waiting rather than investing now. ttbeta''
Since tcchnical alld inptlt cost tlncertainty have diflkrent eftkcts ()n investment, it can be important to incorporate botll in tbe analysis. l-lere Nve sulnmarize a nlodel developed l7yPindyck ( l 9t)3b) tlltlt yiclds decisiol) rules for irreversible investments subject to both types t)f cost uncertainty. Tllt! basic idea is that thll project in qtlestion has ttn actual cost ()f conapletitln that - only the expected is a random variable, &', cost & t ( Kj is known. As in the mtldel of Section 2, tlle project takcs time t() complete the maximum rate at which the lirm kzan (productively)invest is I Upou completion. the tirm receives an asset (tbr example, a factory or new drug) whose value, l'' is k'nown wi t h ce rtain ty. To allow for both teclnical and factor cost uncertainty. we wil! assume the expected cost K evolvcs according (() that -
-
.
-
=
.
,
lK
=
-
/ dt +
1,
(I K)
l ''
(/::
+ y KJ
t.l
-
where dz and tlw are the increments of uncorrelated 'Wiener processes.ltl I t is unlikelythat the technical difficulty of completing :1 project will have much to do with the state of the overall economy, but tllis may not be the case lbr the evolutien of constrtlction costs. Henk:e we will assume that atl risk associated with Jz is divcrsifiable, (thatis, (/z is uncorrelated with the economy and the stock market ), but (l may be ctlrrelated with the market.
Note that the second term on the right-hantl side of tquation (45) tlescribes tcchnical uncertainty. If y (). K can change only if the hrm is investing, and the instantaneous variance ()1' t/ &/ &-increases Iincarly with // K. When the firm is investing. the expected change in K twer an intervtll Lt is l l/. but the realized change can be grcatt!r or less than this, and K can even increase. As the project proceeds, progress will at times be slower an d 2 t times faster than expected. The variancc of &V falls as K galls but the ? . fl. is only known when the actual total cost of the prolect, / prqect is comJ() pleted. The last term ()n the right-hand side t)f equation (45)describes input cost uncertainty. If w 0, the instantaneous variunce of ts/ K is constant =
-
,
.
=
''C-rhis
is a special casc
()f
the folltlwing ctlntrtlllttd diffusion prtpcess: JK
=
-
I d t + g( f K ) tI .
-.
wherc g; > (). glt :S((), and gs. : 0. R)r this equation to makc cctlntlmic sttnsta ccrtain conditions should hold: (i) F ( K 1-h is homogeneous of degree ! in K 8- ztnd L' ( ii ) F. < (), that iss an s. increase in the expected cost of an investmcnt ilways rcduccs its value: (iii) the instantantlous variance of d K is bounded for all finite K and approachcs zttrl) as & (); and ( iv) if thc (irm ''
'.
',
.
.
,
-.+.
' . at the maxim um ratc k unt il the prqcct is cllm plcte. li f) jb 1' d the expectcd txlst (() ctlmplttitln. Equation (45) mccts thcsc ctlndititlns.
invests
'x
.
=
&- s() that K is indced ,
Etensions TJZIJApplications
Seqttelltial
/'?l:'tM-/?'nt??l/
349
and independent of 1. Now K will lluctuate even when there is no investment; ongoing changes in the cosis of labor and materialswill change K irrespective of what the firm does. The problem is to hnd an investment policy that maximizes the value of investment opportunity, FK4 FCK.. V, /c): the =
F(S)
f
&) Ve ->'
max
=
l (1)
-
/(;)(,-ltl dt
.
()
Iim
A'
along with the value-fnatching condition that FK4 is continuous at K*. Condition (50)says that at completion. the payoff is Condition (51) says that when K is very large. the probability is very small that over some finite time it will drop enough to begin the project. Condition (52)lbllows from (49),and is equivalent to the smooth-pasting condition that F.'( Kj is continuous at K*. ''.
When l
=
Problem
We will assume that spanning holds, so that Jz can be eliminated from the or dynamic porttblio of assets perfectly
problem. Let x be the price of an asset correlated with lz?. so that dx follows J-t
=
axr A' Jf
+ cx
Jlz?
-r
.
Then, the reader should be able to conlirm that F A-) must satisly' the following differential equation: J. 2
:!
p
l KF
''
K4 +
- 4.vy K
.l.
:!
I
-
0s equation
=
'
r
=
( K)
p
1 F K4
-
-(K4 where p? is the negative root mental quadratic equation
F A).
(48)
-
4.B
=
.
Because equation (48)is linear in 1, the rate of investment that maximizes F( K4 is always either zero or the maximum rate k.'
1
2
() otherwise.
.
p
y
'-
-
0.
=
Characteristics of the Solution
When only technical
-
F'K)
-
1 t ().
uncertainty
: P 1. 2
-r-asset
for 1p2 K F''K4
4:
-
(51)) of the funda-
condition
The remaining boundary conditions are used to determine A along with K. and the solution for F S) for K < &*. Tllis is done numerically. which is not difficultonce equation (48)has been appropriatcly transformed-l 1 A family of solutions for K < S* can bc found, each of which satislies condition (50)vbut a unique solution, together with the value of z1, is determined from condition (52) and the co n ti n u ity ()f F ( K ) at K
=
k
l)
-
solution
X SX
=
(fromboundary
?-2 pp .!. a
analytical
.
where 4.r > (r.r rl/qr, and rx is the risk-adjusted expected rate of return on r + 4 pxa, (h, where / is the market price of risk and pxm is ..:. namely, rx and the market portlblio. Thus the correlation coefficient between the yvr / pxm. Since / is an economy-wide parameter, the only project-specific parameter needed to determine y.r is pxm
=
(48) has the simple
*
2
K F ay
F' &)
0,
=
(52)
0. Here Jz is an appropriate subject to equation (45),0 S l (/) k, and K?4 risk-adjusted discount rate, and the time of completion, ?, is stochastic.
Solving the Investment
F( S)
cxl
-+
is present, equation
J K F'' ( K )
'
I F (K)
-
I
-
=
(48) reduccs to Ff K)
1,
.
Then K can change only when investment is occurring, so if K > K* and the firm is not investing, K will never change. and FK4 0. When r 0, equation (54)has an analytical solution'. =
=
(49)
Equation (48)therefore has a free boundary at a point K*, such that lts k 0 othenvise. The value of K. is found along with when K S K. and It) F(S) by solving (48)subject to the following boundary conditions:
l' When /
=
=
(50)
make
=
k. equation (48) has a first-degree singularity at K = 0. To eliminate = F J.( y'). where T' = 1og K Then equation (48)becomes
the subslitutitln
-)
.
''(
./ )') and boundary conditillns
-
/? ()
21./'?( y') t. + y 2 e '
-
p
2
-
2l' + 2r /( ), ) '
= p
-
p
2
.
I'
e- + y
(f()) to (52) are transformcd accordingly.
2
'
this,
350
LL. r/t.,??.t???.$(111t1 z'lpp/cfl/tl/l'
Table 10.4.
Cr/ca/ C'ost /t'p C>oll.llvlleliotl. &-*,tls
tl
F-ttllctiotl()j'
p
(ltltl
y
tNote-. K* is found by solving equation (4S) and its boundar.y contlitions for F( K). The parameters p and y measure the degrees of tcchnical ltnd input 1(), k = 2, cost uncertainty respectively. Other paramcter values are )'' ().1 0.05, and $ r =
=
=
if there vvere a systelnatic conlponent to the inptlt cost tlncertainty. Tlltls for rnany investrnentsa and plt rticu Iarly 'r Iarge indust rial projccts lr Nvllicll input costs tlucttlate. increasing uncertainty is Iiktlly to deprcss investnlellt.
The opposite will be the case only Ibr investlnents Iike Rct D progr:tlus, for which technical uncertainty is tr n'lllre nlportant. The application of this nxodel requircs cstimates of p and y This reqtlires estimating confidence intervals around projectetl cost for tlllch sourcc klf tlncertainty.X) break cost unccrtainty into technical and inptlt cost colnponents. note that the lirst is independent of time. whereas the variance of cost tltle to the second component grows linearly with the time horizon. Thus. t value for p is tbund by estimating the standard tleviation of cost F years into tlle future assuming no investment takes placklprillr to that time. Tllis estimate. 'r, could come from experience with construction costs, or Ihm an accotlnting model of cost combined with varianc: estimates for individual inputs. Tllen. r/ T. To estimate w we can make tlse of the fact that if y)z = (). the variance of the cost to completion is givcn t7712 .
)?
tp
()
t). l
().:!
().5
0 0. 1 0.2 0.3 0.4 ().5 0.6 0.7 0.8 0.9 1.0
8.9257 8.9844 9. 1309 9.3750 9.7168 10.156 10.693 l 1.328 12.05 l 12.861 13.770
Let us return now to the general case, which requires numerical solution. Since increases in p and y have opposite effects on K*, it is useful to determine the net effect for combinations of these parameters. Table 10.4 shows S* as 0, P' = 10. k 2, and t' 0.05. Note a function of both w and y, for / with and K. decreases that increases with p. but is much more sensitive to y changes in y. Whatever the value of p, a y of 0.5 reduces K* to about one(). Also, this drop in K* would be even larger fotlrth of its value when y' =
l.)
--
-.
V( X )
=
2
K
!' 2
.-
-W '
()1' thtl expected 'T-husif one standard deviation (.lf 11project's cost is 357.) (5t)r))) cost. p would be ().343 (0.63).1Using (5t)) tntl ltn initial cstimate of expected cost, S(()), a valud for 17 can be basdd t)n an estinlate ()f thc time-independent standatd deviation t)t &.
.
value of the investment opportunity were there no possibiiity of abandoning the project. The Iast term is the value of the put option. that is. the option to abandon the project should costs turn out to be higher than expected. Note that for w > 0, S* > IZ and K* is increasing in p. The more uncertainty there is, the greater is the value of the investment opportunity, and the larger is the maximtlm expected cost for which beginning to invest is economical.
=
=
=
=
5
Guide to the Litcrature
The model of continuous investment with time to build that was presented in Section 2 has as its antecedent the mtldel of optimal abandonment developed by Myers and Majd ( 1984). They valucd a prtlject when ctlnstruction could be permanently abandoned at any point in time, and. using a similar numerical sblution method, found the optimal abandonment rule. Morc reccn t studies of investment with time to build includc Bar-llan and Strange ( 1992). Time l7 Lc t G ( K ) e qua tion :
: 1 b, 1 K G ( K ) - L'G- ( &-) + 2k #' ''
'
2
subject to tht! boundaf'y page 2()9.j Tlle stll u titln
contli tillns G 4i)) f) :1 nd to tI1is is (;( A-) = 2 K' / ( 2
; t cc )
=
-
,'J
),
=
()
.
ISee Ka rlin and 'Payltlr ( 198 1), st , tlle v:lriance is given by (5t')). =
-x';.
Exellsi..nls fl/lf./ Appll'l-'iltiolls
to build (andrelated delays) is usually ignored in theoretical and empirical models of investment, but as Kydland and Prescott ( 1982) have showns it can have important macroeconomic implications. The model of cost uncertainty presented in Section 4 is related to scveral earlier studies. The value of information gathering was explored by Roberts who developed a similar model of sequential investand Weitzman (1981), which in ment a project can be stopped in midstream. and the process of investing reduces both the expected cost to completion and the variance of that cost. They derive an optimal stopping rule, and show that it may pay to go ahead with the early stages of an investment even though the NPV of the entire project is negative. Weitzman, Newey, and Rabin (1981 ) then used that model to evaluate demonstration plants for synthetic fuels, and showed that learning about costs could justifysome of these investments. MacKie-Mason (1991) extended the Roberts and Weitzman analysis by allowing for investors (who pay the costof a project) and managers (whodecidewhether to continue or abandon the project) to have contlicting interests and asymmetric information. He showed that asymmctric learning about cost Ieads to inefscient overabandonment of projects. Grossman and Shapiro ( 1986) also studit!d invcstments for which the total effort required to reach a payoff is unknown. They modelled the payoff as a Poisson process. with an arrival rate specised as a function ()f the cumplative effort expended. They allowed the rate of progress to be a concave function of effort, and focused on the rate of investment. rather thap on whether one should proceed or not. Also related is the work of Baldwin ( 1982). who analyzed sequential investment decisions when investment opportunities arrive randomly and the firm has limited resources. She valued the sequence of opportunities and showed that a simple NPV rule Ieads to overinvestment, that is. there is a value to waiting for better opportunities. Likewise, ifcost evolves stochastically, it may pay to wait for cost to fa11.Finally, Zeira ( 1987) developed a model in which a firm learns about its payoff function as it accumulates
capitai.
All of these models of Iearning and cost uncertainty
Seqllelltial
/'?lrfz'J/???t,Nl
353
uncertainty), but is more restrictive in that the order in which dollars are spent is predetermined. ln the Appendix, we briefly explain the numerical procedure used to solve the partial differential equation for the model of investment with time to build in Section 2. Readers who wish to develop some expertise in solving partial differential equations trould begin with one of the following textbooks: Haberman (1987), Guenther and Lee ( l988), or Carrier and Pearson 1976). (
Numerical methods forsolving the PDE'S that arise in option pricing problems are discussed in Brennan and Schwartz (1978), Geske and Shastri (1985),and Hull and White ( 1990), among others.
Appendix A
Numerical Solution of Partial Differential Equation
This appendix shows how the partial differential equation (24)and its associated boundary conditions for the model of continullus investment with time to build can be slllved numerically. The procedure used here is a variant of the explicit tbrm ()f the finite difference method. Other numerical procedures could be used. and some might be more efficient or more accurate. Our objective here is simply to illustrate one approach. To improve the accuracy of this proccdure and makc it possible to calcula.te F( 1z'.K4 over a sufhciently Iarge range tbr l', we begin by making the
followng transformation:
F( l'' K ) .
Y
e-
/
'
X/ l
where X Es log )' The partial differential ar.yconditions then become .
G( X. A ) r
equation
.
(for F
>
P'
*
) and bound-
belong to a broad
class of optimal search problems analyzed by Weitzman ( 1979). ln what he characterized as a box'' problem, one must decide how many inwith opportunities uncertain vestment outcomes should be undcrtaken, and in what order. In this case. cach dollarspent on a project is a single investment opportunity, and the uncertain outcome is the amount of progress tllat results. 'tpandora's
The model presented in Section 4 is more general in that expected outcomes can evolve stochastically even when no investment is taking place (inputcost
G(X. ()) -'
lim
.r-.'x
=
e.Y.
e-''&'/k G .t .(X /t-)j
G(zP, K)
.
=
(59) =
c-'5A-/l' .
( I /w) G,r(A'*. K).
Note that the coefhcients of the PDE are no longer functions of V.
354
Extetlsiolv' tl?lJ Applicaions
The hnite difference method transtbrms the continuous variables )' variables, and replaces the partial derivatives in the PDE with tinite differences. The explicit form of this method corresponds to a specific choice of tinite differences for this substitution. Let G(X. A') H Gt LX, j 2. Kj E!e G,.,j where :s i :s nt and 0 S j .'.'.f n. NOw make the
and K into discrete
-b
C7??, + I .)
Now substitute
2L
th is for t7,,,+ l
.).
zr
t:
c,/zl
,'
+ ( ?'
-
in eqtlation
t
1j .'.NK / 1'
+
L, I -
(($2)(set ting i
.j .
/71)..
,
substitutions'.
Gr r
;:::z
(G+
Gx
Qz
EGi+1.y Gi-t.jjl'lhv'.
GK
;4J
(tk.j+l
.j
2Gi,). + G-
-
l
J/(2.
.).
2
,.r)
.
-
Gi.)./hK.
-
Finally, the free-boundary condition becomes
The PDE then becomes the difference equation
where
The solution proceeds back-wards from the terminal boundary, as illus-
trated in Figure 10..5. First, the values of G along the terminal bllundary )' ()) are calculated using equation (63).Stepping back to j 1 equation (64) is used to calculate G,,,.l and equation (62)is used to calculate the valtlcs 2. etc. Each time a value for G is calculated, equation 1 tn n? of G tbr (65) is used to check whether the frce bllundary has been reached. Because =
=
,
,
=
Note that p'b +#?+p-
1.13The terminal boundalycondition
=
g.t
.j
and the upper boundary condition Le--f-rKlk
lim
e
:=
=()
fax
.
K)
=
l )' i
-
G,n-,.jj/2LX
=
e-th'/k
emx-r-bihz'lk
Using the Iinite difference approximation (G m +
upper boundaw Am
=
11
becomes .
G ,!r'(rna-r
i
-
,
(63)
.
G .' CX A')j
X-*n
then becomes
-
.
@ ux I
.
free
boundary
above for Gx, this becomes =
I I I 1
.
zr.
analogous to the two-point form we used in Chapter 3 and elsewhere. Or. as Brennan and Schwartz ( 1978) have shown, one can interpret the difference equation (62) in terms of a jump process with probabilities p+ and p-, respectivcly, of upwards and downwards jumps.
1 i; ..n
.
y -
c-
1l
em'-Y+r-b'i'%Klb
dl-l-hisis a three-point random walk reptesentation of thc Brownianmotion of
I I I l I I I I I l I
i
-b
=
j= n HK/AJI
Iower boundary *- A&--
@ o
q
t
/=0
356
Extensions
of numerical error due to the discretization. equation exactly, so one checks whether it holds to within some
l?l#
Applications
(65)will never hold specitied bound 6'.
Chapter
(66)
lncremental lnvestment and
LX/2 tends to work well.) Once where 6 is chosen arbitrarily. (Setting 6 this check identities the free boundary, equation (28)is used to determine the value of the coetxcient a. as well as the values of G below the free boundary. =
Capacity Chqice
We can summarize this procedure as follows:
1. First, hII in the terminal boundary using equption (63). 2. Then, for )' 1 to j ?;, use equation (64)to calculate G,n.j, and then move down the column, using equation (62) to calculate Gi.). for 1 < m. 3. Finally, at the free boundary. calculate the value of the coefficient a and use equation (28)to till in the values of Gi.). in the lower region. =
11 't) $
=
Note that the solution for P' ( S) will be subject to some numerical error and that is why the points because of the finite difference approximation. plotted in Figures 1().2and 10.3 d() not Iic on smooth curves. In gcneral, this numerical error will be smaller the smaller the size ()f the increments LX *
and
.'y'
(
2. K.
()f individual Iirms' investment decisions that were developed in MonEl-s Chapters 5-7. and the modcls of industfy equilibrium in Chapters 8 and 9, were based ()n a simple discrete unit ofinvestment, namely, a single project ofa given fixed size. In Chapter l (lwe continued to examine a single project. but we accounted for the fact that completion of a project often rcquires a sequence ofsteps. each of which increases the sunk cost that is committed to the project. In this uhapter. wc examine a hrm's investment decisions in a more general context. We allow each 5rm to hold and operate a Iarge number of projects, to add new projects, and perhaps to retire oId ones. Its existing projects continue to produce output and profit tlowseven as the firm contemplates adding new ones. ln other words, we think of the hrm as choosing an investment policy to alter its stock of capital. Its current output flow depends on its installed capital stock, and perhaps on flows of instantaneously variable inputs like labor and raw materials, through a production function. We begin by considering how a firm irreversibly invests in its capital stock. When the production function shows diminishing returns to scale. we capital in decreasing order of can conceptually line up successive units of them and think of as distinct investment projects. their marginal productivity, Now suppose the output price, or the prices of some variable inputs, or some other parameter that affects productivity, is uncertain. For each of our conceptually discrete projects, we find a threshold of prohtability that justihes the investment. This turns out to have the usual option value interpretation'.
TIIE
358 thc expected present valtle of the project's margnal contribution must be :1 multiple #!/4/:/1 I) of the cost ()f investmenta where pk is the tlsual root of the fundamental quadratic that involves the discotlnt rate and thtl drift and volatility eoefficients of the price process. Successive units of tmpital, with successivelylower marginal productsa will require higher and higher thresholds. Thus we will find a cuwe'' linking the stock of capital and the threshold valuc for the stochastic state variable. 'When this state variable is below the curve, no investment is made. while if the variable rises abtwe the curve, just enough is invested to bring us back to the curve. -
S.threshold
Nvedehne a version of Tobin's (/ based on tlle value of assets in place but unit of capital. Then the firm's optimal investment (/'' rises unit when this to p /(p1 l ). If there are regions of increasing returns in the production function, then a threshold that justifies investment of one unit of capital may also justifyseveral additional units that have higher marginal products. Thus the investment policy will yield some sudden jumps in the stock of capital to cross regions of increasing returns. In Section 2 we will find the exact condition that determines the threshold and the size of such jumps;it will turn out to be :1 natural analog of the option value multiplier condition, but now expressed in terms of the average product of the discrete jump in capital. instead of the marginal product of the next small unit. In previous chapters, wherc each Iirm simply had to decide whether to ahead with a single project. the cost of investment go was :1 number, and it did not matter whether wt thought of it as a lixed cost of decision making, or a sunk cost representing the irreversibility of investment, or an adjustment cost that is higher when the firm tries to install a given addition to its capital stock in a shorter span of timc. However, now thc (irm is choosing the time path of its stock of capital, and we must specify n more detail how the cost of investment depends on the scale and rate of investment. The distinctions just mentioned become important; optimal investment policy takes very different forms depending on the naturc of the cost. The models prsented in the first two sections of this chapter treat irreversible investment without any costs of adjustment or Iixed costs of decision making. Thus the sunk cost of any atldition to the stock of capital is proportional to the size of that addition. In Section 3 we consider a more general model that includes all three kinds of costs. We show how these other costs affect the investment decision, and we show the relation to and the differences between our approach and the traditional :-theory of investment. Wewill confine the discussion to asingle firm. lt is not difhcult in principie to take the analysis to a competitive equilibrium Ievel as we did in Chapters 8 ,
applying to the marginal policy is to add a marginal
''marginal
-
and 9. bu t thd
icatitlns qtlickly fnoullt tlp. lpnd it secnls nltlrt.l to fotyuson tlle nklvvissttes raisetl by vnriabltt cnpacity in thtt sinplest setting possible. algebraic
'ol-npl
inRporttnt
1 Gradual Capacity Expansionwith
Diminishing Returns
sve begin vvith 11 nltltlel of a Iirl-n-s capacity expallsion. based on Pintlyck ( 19881)) and Bertola ( l t?89). l t is 11 sinlple generalization of the Inodt?l of Chapte r 6. As t he re.
we assu me that the Iirn1
has a nlonop. oly right to invest
in the indtlstry. Eaeh unit of capital costs &'. and investnAent in the unit is irrevcrsible. Wllen the lirm has & units ol' capital in plact!. the (low of outptlt ? is given by the prodtlction function Q) GqA.-) Tlle indtlst r'y denAanl fu nction vvherc the slli ft v:t riable F ll lows the geonletric Brtnvn ian F D( L.)s, is # motion (ll (lz. t F a 1'' + f'r 3.
For simplicity-
Nve ktssunle
thitt there
ltre
no vltriable
costs. st) the prot
lltnv is
'standard
.?.r
F D ( ( K) ) G ( &-)
,'*
F J/ ( K )
.
and tlle marginal revenue prodtlct ()f cnpitlt is F ll' ( K ). l n this sectilln wtt asstlme that thcre are diminishing returns to capital in the sense that tlltl marginal revenue product is decreasing in S. ()r the revenue Inctilln is concave in K. s() H''i K ) < (). Tllis ctltlld be the casc bccause ()17 pllysical diminishing returns in productitln (G''( &) < ()1. ('r bectuse ('f l dtlwnwardsloping indust:'y dumand culwe ID'f .?) f) j. ()r sllme ctlmbination ()t' the twl). ( In fact. evcn if there werc physical incrcasing rettlrns, wtl could have a decreasing marginal revenue pnlduct :ts Iong as tlle demand ctlrvc llad :1 sufficientlyIarge negative slope.)
More generally,
we can regard
the prlllit llow as the outcome of an in-
stantaneous optimization problem whutre variable inputs such as labor or raw materials are chosen holding the level of carlitlkl fixed. Tlle only new lature is that the stochastic variable need not cnter multiplicatively. Wtl will consider this extension later in this section, and ptlint (lut its impl ications for the effect of uncertainty on investment.
The investmcnt decision can as usual be examined in one t)f twt) esscntially equivalent ways: dynamic programming and contingent claims analysis. N'Vetreated the single-project dccision using both approaches in Chaptors 5 and 6, and showed their similarity. Since tllen we have been using either one or the other to suit expository ctlnvenicnce. I-ltlwever, the present setti ng with
360
Ezellsiolts
incremental capacity choice is somewhat new, both approaches explicitly.
I?l#
Applicatiolts
so we will once again discuss
fl/ltf Itlcrentental fllvtr-sflllc/rlf Ctlptlcfry
-/lt?lk;,
investment problem when the initiai capital stock is 0 &, + ( l t?J Klz. Define an investment policy by the tbllowing rulc: if the stochastic variable evolvds on the path (1$1,expand capacity on the p-averaged path (p Kat + ( i 0) A,? J. -
-
1.A Optimal lnvestment via Dynamic Programming Given the initial capital stock K and the initial level of the stochastic demand
shift variablc F', the hrm wants to choose the path of its capital stock, Kt, to maximize the expected present value of its operating profits, net of the cost of investing. Let p be the rate at which future prohts are discounted. Write I'Ftffa l') for the maximized value function or the Bellman function of the
The separate paths labelled a and b being feasible, at any time each of Kut and Kl,t cannot use any more information than is revealed in the path of the stochastic variable up to the time f Then the same is true of the p-averaged path. so it is lkasible, tet). Because the function H is concave, the policy just defined yields the .
revenue tlow
srm's choice problem. Of course this function is not known in advance but must be determined as part of the solution.! Consider a short interval of time dt. Actual decisions are made continu-
ously and the interval // is an arbitrafy construct. so we will be interested in the Iimit as dt 0. The protit Ilow during this interval is zr #1-,the effect of the interval is of order J: and thus can be ignored. Suppose discounting over firm increases its capital stock from K to K' at the end of this interval. The the demand shift variable changes over the interval from F to i' + dY. Of course, given the firm's information at time t, it does not know what dY will be, but knows its probability distribution from equation ( 1). Therefore it calculates the expected value of this increase in the capital stock as ....-*
i'/ H0 Kut + ( l
1Wc considered a mathematically very similar problem from social plannerfs perspective a
in Chapter 9. Section I.A. There our concern was to establish a correspondence bctween thc social optimum and the cquilibrium of a compelitive industry. We developed the model in the quickest heuristic way to reach tbat goal. Now we want rnore detailed characterization of the firm's investment ptllicy, and will need to examine tlle model in an appropriately grcatcr detail. Readers will grasp the method and the results more quickly by recalling somc of the intuition we developed in Chapter 9, and then going back to rethink that modcl in the Iight of detail that ernerges now.
0 ) Ff S( Kltt)
-
.
The investment cost tlowisjust the p-weighted average ofthe cost liows tbr the two separate policies tbr cases fl and b. Discounting, and taking expectations over the possible realizations (F/ ), we have
or I'Uis concave in K.
This concavity ensures that the maximization of (3) can be characterby iztld the familiar Kuhn-Tucker conditions of calculus. The derivative of the expression in (3) with rcspect to K' is (/I
(3) The hrm will choose K' to maximize the right-hand side of this expression. Then the resulting maximum will be just the initial value 1**4A-. F) of the Bellman function. We assumed that the function H(K) is concave. Then it is easy to show that the Bellman function 11/must also be concave in S. To see why. consider any two starting capital stocks, say, Ka and Kl,, and suppose the optimal investment policy leads to (Kat I and (K.t ) as the respective capacity expansion paths corresponding to a particular realization path of the stochastic variable, l l'/ ). Now let 0 be any number between zero and one, and consider the firm's
04 Kst) = 0 i'/ H ( Kut) + ( 1
-
e-?) ( s Ijjr & ( Kt y
.j.
.
-
(/
y)j
.s
) .
As dt (). this tends to Izlzk ( K'. F) K. Irreversibility requires K' A: S. If CU'C( K. F) is already :; v, the maximand will be decreasing with respect to K' throughout the range K' k K. Then the optimum policy will be to keep K' K, that is.' not to make any addition to capacity. If 111.4S. F) > v, optimum policy will be to set K' at the level desned by the srst-order the condition Izkk( K'. F) K by instantaneously installing the amount of capital ( K' S). Suppose for a moment that the full problem has been solved, and the function $,P'is known. Then the investment policy described above can be visualized as follows. In K. l') space, draw the curve defined by -.+
-
=
=
-
Ws.CK, F')
=
v.
(4)
We show the result in Figure 11.1 It is shown upward-sloping because a higher i' raises the proft from any given Ievel of capital. and therefore should justifz a larger capital stock and hence more investment. For now we leave matters at this intuitive level. Formal conlirmation will follow, and the labels in the hgure anticipate some of that work to come. .
362 X
VIK) =
71
f3,-1 H
Wx
Immediate
investment
=
corrttspondingriglltNvartl n-lllve ()f &. s( the tinle patll ()f &-at such ltlcatitlns is lllsl)Iltlndil't'el'ellti:tlple,' tlle til-nedcrivlltivo (1 &-/t// is in illite. Thus tlltl bllrritlr controi policy does not generate 11 Iini te rltttt of invcstnlent throtlgh tinle. btlt occasional small btlrsts ()1' investnlent :ls thtl barrier is Ilit. Of eourse in lt Inore nlacrlascopicvievv ('f til-ne, we can c:llctllllte tlltt avenlgtl rate t't' invcstllltlntthat is. the average rate of change ()f capitltl (lver a Iong interval of tinle.
''K '(/f)
i
Now we proceed tt) develop the argtlnlent nlld the optimal policy in solne('f begi n wi tI1 the I-egit'n inact ion belosv the ctll've Izp'&. < whc re the po1icy is to be ()110 of zero investnlent. I-lere wtl
Barrer
what grcater detail.
control
'sv
mtlst
.
Ax
< K,
substitute K'
inaction
-
K in
(3) to get
Expanding the right-hand
the initial value
side by Ito's Lenlma and keeping terms to order
t//, wc gtlt
1P(A'. )' ) = 1'' ./( A-) tlt + ( l Below or to the right of the curve, we havc 1'P':- < &- and 11() investment is made. As F Iluctuates stochastically, the point ( K. F) moves vertically up or down. lf it ever rises to meet the curve and tries to cross it.just enllugh investment is made to stop it from crossing; this is shown in Figure l 1 l as lt series ()fsmall steps indicating a phase of gradual investment. In technical terms this contrllls'' because it prevcnts the point policy is sometimes called a the of the controlled state system from crllssing 11 certain barrier representing ()f barrier control 3r) F in :1 the KIf phase t() take us ( space. aer curve until rises the barrier again. F below the investment stops to hit back curve. If the initial point is above or to the Ie of the culwe 1Pk v, thtl stock of capital is instantly increased by a discrete amount Iarge enough to movc the point horizontally to the curve. Thereaer the policy of barrier control takcs over. Thus. except possibly at the initial time (). the point ( K. F) will never be seen above the curve.z Correspondingly, :1 discrete jump in the capital stock can only occur at the initial instant. Thereafter, the capital stock is either constant over a period of time while i' is Iow enough to keep us below the curve, or it responds continuously to small increments of F at the curve. If the derivative of K with respect to t were finite, we would call it the//tpkv the rate of investment I Howevers this is problematic for barrier policy. or As we explained in our discussion of Brownian motion in Chapter 3, the time path of i' is not differentiable. At the barrier any upward move of F leads to a
+ 21.r'r =
,
ttbarrier
.
).f'z
IJ' 1.1( K )
(.,2
.!. ?
:2 ik').). ( K. y ) +
K )' ) + /) $.)z'(
-
.
F': 1,l4,). ( K. F') t// 1
a y ;,p').( K.
y)
(z
F' 1,$.( &. 3-) t/?
j (lt
''Fl'lereftlre 1,P41'. 3-) satisfies the di lren
(.,v'
F 'P').( K )'') .
.
t ial eqtlat illn -
/) kj/zlK. y j + y JJ ( K )
=
().
Since I.U is a functilln ()f two variables- tlls is rcktlly a partial differential equation. Luckily it tloes not invtllve any derivativeswith respect to S--rhuswe can rttgard it as an ordinary dil'lkren tial eq uatitln that Iinks I,Pto F, and th ink ol' K as a parameter that shis the whole functitlnal rclationsllip. Any constants (.)fintegration are also lllkctlld by &, and thus they shtluld be regarded as funct ions o f K. The gttneral solution of (5) is then famiiiar from Chapters 6 and 7: W''4K. F') where
=
/31and pa are,
damental
f'11(K) F'/, + B1( K ) F/f: + F' /-/ ( K )/ (p respectivcly,
(?-
a )
.
thc positive and negative roots of the fun-
!
2
alplp
-
!) +
(x
p
p
-
=
().
of integration to be determincd. are The region of zero investment in Figure l I 1 includes the Iimit as F goes zero. X) keep 1F4K. F) finite there, we must le:tve out the negative power ttconstants''
.
to
-
quadratic
and B3(K), B? Kj ktbllvtt ?At the initial instant the ptlint ( &. i' ) thc curve either because ntlntlptimal may be iI1 ()r icics folltawed thc bccatlsc uncxpcctcd shllt:k jtlst mtlved thtt curvc. wcrc pol past stllnc
B3(&), is determined by considering the other The remaining <. region of zero investment. namely, the curve HQ(S, F) bounda:y of the = F(A-). write of this F the equation curve as Solving for J' in terms of K, Then &.. K. F(S)) = S'I ( K) l'( K4I$' + F(A) H'Kj/$ r'Izr&.t tGconstant''
=
=
where B'l ( S) denotes the derivative of Sl ( K) with respect to K. This is just othercondition for whatwe have been callingavalue-matchingcondition.The requires condition, which that the derivative smooth-pasting optimality is the with of &' with respect i' should equal derivative the F ) respect to of Izj/ArtS', derivative of v is zeros the F(X). Since ir threshold to i' as increases to the we have (8) These two conditions can be combined to determine not only #'j ( K) but also the Iocation of the investment threshold F(X). that is. the free boundary l'(S)
JI
/1
-
The interpretation of equation (6)should now be transpltrent. The term is the expected present value of the protits the firm would get if it kept its capital stock constant at the initial level K forever. Then the optimal future capacity other term. #1 K) F/1 must be the value of the srm-s expansion, that is, the current value of its options to expand capacity in the unit of future. When the firm exercises its option to install the ( K + (slth capital. it gives up the marginal option value, which is why B'1 (A-) in equation ( 10) is negative. Then equation (7) says that at the threshold that justihes the incremental investment dK. its expected contribution to the capitalized profit flow should equal its cost of installmentp/l/.s the opportunity cost of the option to wait. Using the solution for S'l (K4. we can find Bb(X') itself by integration'. .
BL
(A3
jx
E-J't ()1 dk
H (K )
H'gej
g /J1
r.w
=
For this integral to converge, Cobb-Douglas case
1 S'(X) pb-1
have a natural generalization of that result to the case of a marginal
F' S( Sl/
<
=
K?
.
j.
j
-
'-
=
x.
'
pl 1
j. co
-
/ j.;' ( k j
'>
S'( K4 must decrease sufhciently
y'
(K)
.?
/1
) js. j. ) pls'
y(w-l
tfl.
rapidly. The
.
illustrate the point. xowthe integrand is a constant times A'-/',t'-''$ For will of the integral, the power must exceed 1 in numerical value, so
/91
s((s)(,.--) ( ) j
-
.
,
(10)
Obsen that as K increases, H'(K4 dccreases and therefore Y(K4 increases. This conlirms our earlier intuition that the threshold curve should be upward-
sloping.
The formula for the threshold is strikngly familiar. Suppose we install a marginal unit of capital (IK when the existing stock is K and the stochastic shock has the value )'. The contribution ofthis marginal unit to the profit flow and futtlre prohts is 1' H'(K4 dK. Since i' is expected to grow at the rate value of this contribution expected discounted the present at the rate p, are unit marginal is K' dK. Then The installing dK/p of e). cost the is F' H'K4 stock is marginal hrm's capital addition the to equation (9) says that the multiple expected value when the by the cost exceeds the justified present ,
-
we
addition to existing capacity.
=
and -
/31/(p1 l). This is exactly the multiple that rellected the option value of the status quo in the criterion for making a single discrete investment in Chapter 5. -
.
Then the solution becomes =
365
convergence In otherwords. 0 must be suf:ciently Iess than 1, we must have pk > 1/( l which means returns to capital must be decreasing sufhciently quickly. These retums to capital are in revenue terms, and combine the effects coming from thc production function and those from the demand curve; see equation (2). -/?).
We can see, then, that this convergence requirement makes good economic sense. If returns to scale were constant instead of decreasing, a threshold level of the shift variable )' that made a unit of investment desirable would make a larger expansion of capacity proportionately even more desirable, so there would be no linite solution for the firm's optimal
capacty.
We will proceed assuming that the convergence
condition holds. If the
marginal product of capital falls to zero at some sufficiently high level X, then the upper Iimit of integration can be replaced by W and convergence is not
366
Erlcrl'tpn' and Applicatiolls
11problem. This is the case, for example. in the model developed by Pindyck ( l988b), where a linear demand curve is assumed. .
The behavior of the threshold cufve for small K desewes some comment. In growth theory it is common to assume that as K (x); this is 0, H' K4 called the Inada condition. If this condition holds forour firm, then F'( &) 0. This is the case shown in Figure l 1.1. However, if the marginal product S'( K4 stays bounded for smail K, then F(K) will stay positive; installation of even the first unit of capital will require a strictly positive threshold Ievel of the -->
Increlnental
/'?Ik'fzz-/???t,?l?
alltl Ctkplcry Clloice
367
terms of F' along the curve be K F)', this is ust the inverse of F ( A-) Then the firm jumpsfrom ( K 1') to ( K ( F) F') and .
.
lIz-( K. 1')
.->
(K rlz-
=
.
F) F)
&-
-
.
gK ( )')
the function
,
-
SJ.
-->
stochastic variable.
The most noteworthy feature of the solution is that we can regard the successive marginal increments to capital as distinct little projects, each contributing its marginal product independently of the others. Then we can find the optimal investment threshold for each using the standard option pricing approach, and string them together to obtain the solution of the full capcity expansion problem. Why can we treat each unit of capital independently of the others'? The marginal unit of capital labelled Kz can be treated independently of the one labelled Kk < Kz because of diminishing returns. The marginal product of the unit Kz is H'Kz4 only when alI previous units are in place. Diminishing returns ensures this: the threshold for KL is lower than that for K1, so the unit labelled A-! is going to be installed before that labelled Sc.3 Conversely, the unit at K3 can be treated independently of the unit at K > A-1 because of the cancellation of two effects that we saw in Chapter 9. When the decision on the unit at /f1 is made according to the rule (9),the firm is acting as if this is the Iast unit it is going to install. It is thereby ignoring two things. First, it is ignoring the rcduction in the productivity of capital that future expansion willcause. This makes it more willing to invest. Second, it is ignoring the need to have the unit at A'1 in place to pave the w'ay for the next unit to come, thereby pretending that it has a greater leeway to wait. The two effects exactly cancel, allowing the firm to act on the unit at K3 as if there were no further expansion opportunities. Although all the interesting action is below or at the curve F(A'), for the sake of completeness we state the solution to the firm's dynamic programming problem above the curve. Here the optimal policy is to install a lump of capital instantaneously to move horzontally to the culwe. Let the solution for K in
Thtls in fact we have 'vg
=
&'
everywhere
above the curve.
1.B The Contingent Claims Approach Now Iet us see how this investment problem can be solved using contingent claims methods. We will continue to view the stock of capital as a sequence of marginal projects. Each is an asset that contributes a marginal llow of prohts. The steps to be followed when valuing each of these assets, along with the option to acquire it, are very familiar from Chapters 5-7. We assume that the stochastic lluctuations of )' are spanned by assets traded in financial markets. and write /.z for the appropriately risk-adjusted expected rate of return on this porttblio of spanning assets. The return shortfall or convenience yield on l'. namely, Jz e, is denoted by J as before. Now consider te ( K + Jflth marginal unit of investment. Once inunit this will yield the marginal profit flow F H' K4 J& forever. Let stalled, vK. F) tlK denote the value of this installed unit. Now, consider a portfolio consisting of this unit, and a short position of n units ()f the asset (orportfolio of assets) that tracks F. This portfolio yields a cash tlow of tF H'LA') n (5 F) per unit time. (The second term of this cash llow is the amount the holder of the short position must pay to the holder of the corresponding Iong position.) The portfolio also has an expected capital gain or Ioss as )' fluctuates. Using Ito's Lemma we get -
-
The choice
n
=
l?r
makes the portfolio riskless. Of course as F' and K change
through time, the number of units n in the short position must be changed continuously (that is. a dynamic hedging strategy must be employed) to maintain this property. Since a riskless asset must earn the riskless return r, we have Or
12 c 2 ),2 17r r 3In the social planner's problem of Chapter 9, Section l .A, there were constant returns to capacity expansion in the industry. but a downward sloping demand; therefore the planner's objective function (socialsurplus) had diminishing returns.
+
(r
-
J) y
t?j,
-
r r + )' H'fKl
As we have seen in Chapter 6, Section 1, this equation
=
0.
(13)
has the general solution
368
Etensions
Applicatiotls
l/lt.
The only new feature is that the constantsofintegration are nowspecific to the tlnderconsideration, unitof capital so they are functions of K. As in Chapter6, rule corrcsponding the to the negative root pz to ensure a finite out term we 0, and that for the positive root to rule out a bubble as i' value as ?' x, leaving only the fundamental term. or the discounted present value of proht, -.+
369
J?l!?t;..5'J/??c?7J and Cbfpttc/y Cll()i('e lllcrettalelaltal
to the fundamental option that is given
value of having the un it in place minus the value up when the unit is installed:
of the
..-.+
L'(S. F)
F' H'qK'b/6.
=
(14)
Next we value the option to invest in this incremental unit of capital. Denote the value by J'CY. A-). Retracing the steps of Chapter 6, we construct a dynamically rebalanced riskless portfolio consisting of the option and a short position of fv units of the asset that tracks F.setting its return equal to r yields
the equation
,2
/2
fvv + (r
J) )'
-
j
r
-
y'
=
0.
This has the general solution
Therefore the value-matching and smooth-pasting conditions of dynamic programming, l'fk &- and I'fzky 0, coincide with those of option pricing just =
=
above.'l 1.C
Marginal
We interpreted equation (9)as saying that the expected present value of the prosts contributed by a marginal unit of capital had to equal a multiple of its purchase cost to justifyits installation. This can be expressed in terms of Tobin's q. The new feature is that we must detine the q appropriate for a q.'' Recalling the distinction we made marginal unit of capacity, or tvalue-of-assets-in-place'' concept. Applied to the in Chapter 5, we take the marginal unit, this is just the ratio of the expected present value of the protits it contributes to its replacement cost, or Hmarginal
The option is kept alive as F () and must have a finite value there, so we corresponding eliminate the term to the negative root #a. However, for some will exercised and the option value will not be relevant F snite the option get the other J' term. This leavcs x, so we cannot infer as -.+
t/( K4
=
F H' ( K4l (j
&-
)
.
...,+.
J '( K
)')
.
bb ( &) F/.
=
.
Then the constant l)b( Kj and the threshold F(X') for optimal exercise are jointly determined from the familiar conditions of value smooth pasting:
of the option matching and
f
K, F)
=
1/
(K
.
F)
fv ( K. F)
<.
-
=
( K.
t?r
i')
.
Then 41/4/1 1) is the threshold to which (/ ( S) must rise before investment in the marginal unit at K can be justihed. In Chapter 5, Section 2.C we dehned another sense of Tobin's t/, namely. one based on the value of the firm and taking into account the opportunity cost of exercising an option to invest. Similarly here, the marginal effect of capacity expansion on the value of the firm is the contribution to expected present value of profits, u( K. F), minus the value of the option to invest, .fK, F). Then the marginal q in the value-of-the-tirm sense would be defined as (p(A'. )') fK. F))/v, and its critical threshold would be l This is an exact analog of the corresponding observation for a discrete project in Chapter 5. -
-
.
Simple algebra yields F (K )
and
&.
1 S' ( K4
-
1 /,1 -i
b l (K )
These are
pl /31
=
pj
-
H'K)
=
1.D The Effect of Uncertainty /3, .
K
J
expressions for the investment threshold function and the marginai option value coefhcient we found from the dynamic programming approach of the previous section. In fact the marginal increase in the Bellman value function that results upon investing the CK +JA')th unit is exactly equal
just the
Let us reexamine the solution for the investment threshold in equation (9). This formula shows two effects of uncertainty on investment. First, as in Chapters 5-7, if (7' increases, the root Jl decreases and the option value multiple 4A1so compare the somewhat brief discussion of this issue in the social uptimality Chapter 9, Section 1.A.
model
of
370
lxellsiolls
fl/lt
zzLi.plictltiolls
pj /(/31 l ) increases. This raises the threshold of )' for any given S. In this sense. greater uncertainty implies less willingness to nvest. -
I t is important to understand
the precise sense
()f
the result. lt says nlercly
tllat some values of F that would have jtlstitieda given increment of capacity It says notling about tbr a low value of rr may be insuftkient for kl higher how soon ()r Ilow often the threshold will be reached as )' follows its stochastic process. Starting from :1 given initial value. a more volatile prllcess will reach a given target level of F' sooner than will a Iess volatile process. To find overall effect of increasing uncertainty, we must balance the eftkcts of raising the threshold and raising the volatility of movements of F. In the next sction we will see how. in the context of an analytically tractable example, these forces can balance out on average over time. ('r
.
'I-llis &viIl replace thll nla rginal con t ribtl t ion 1' 11 ( &-)/(5il1t 11t!tirnl's ca IctlI:lt ikln 'There fore the va Itltl-lmatching contl ition (7) 5vi Il l'le.oll'le '
.
1,U&. ( Ff
'
l l ( A-) F ( A-)? . +
1 ( &') ) '
.
t:return
-
/.t
=
1.
+
4 pvst
rr.
Here we arc using the notation of Chaptcrs 5 and 6, st 4 is the market pricc of risk. and pviu is the correlation coefhcient between the asset that tracks F and the market portlblio. Now the effect of an increase in c depends on what else is held constant. If r and a are fundamental exogenous constants, then t5 an increase in o' must increase z and 5. On the other hand, if r and are basic adjtlst offset the the Iirst casc t'r then of effect In must a to constants. on /z. will affected will remain second threshold unchanged. in in be the it the (9).
There is a third eftct of uncertainty that does not appear in our formulation so far, because we have made the profit l'low function Iinear in Y. Morc generally, we could have stipulated :1 function n'( K. i'). Now the determination of the threshold proceeds as in the general model of industry equilibrium expansion in Chapter 9. For notational simplicity we will adopt the dynamic programming approach witl an exogenously fixed discount rate p.but parallel results can be obtained in the contingent claims valuation approach using the equivalent risk-neutral valuation procedure of Chapter 4, Section 3. The marginal effect ()f capacity expansion on the profit flow is zrc ( K. F). Holding K fixed, let the stochastic shock evolve along its random path (l'f ) starting at the initial value F. Calculate the discounted present value of the marginal profit flow,
.
.
) ( &-) ) '
v
.
and thll snltlotll-pasting condititln (8) Nvill l'letrtllntl I1/& 1 ( &- F ( K )) .
.
.
pI
l
B'I ( K) F ( &-)/ 1 -
'the
The second effect of uncertainty that is apparent from eqtlation (9)comes shortfall'' or convenience yield from the term 6. Recall that this is thc and the risk-adjusted that /.1 is discount rate #t a.
n & ( &-
+
n &.j ( K .
.
h''( A-))
() .
E-Iin)inati ng J?':( 1*-4between the tsvo. the t hrtlsholtl l'tlnct itlll 1*( K ) is inaplicitly defined by )'( K4 fl &' rl &.( K. i' ( K) ) &. ( &-.F ( K )) .
-
.
p,
Tlle profit tlow is positively related to the stochastic variable F- so F1&.). is ptpsitive.Thercfore F1&.( K. F ( S)) > v ln tltller wortls. tllc expccted present value of the marginal contribution to thtl prolit llow sllotlltl excecd tlle trtlst of the marginal increase in capaci ty in order to jtlstil'ythe invest me nt. Tllt2 required excess is just the usual oppllrtunity cost t)f oxttrcising tllis mlrginal option to expand-s Since F' enters nllnlinearly on the left-h:lnd side- gre:lter tlncertainty in the stochastic process of F has an additionlll effect. Rtlughly speaking. if tlle marginal protit flow xs- ( K. F) is :1 convex tknctitln (')f F, gre:lter uncertlti nty implies a Iarger marginal expected present vlllue n s. ( &'- F') and thcre lre lt Iower threshold. s() that :tn increase in uncertainty enclltlntges investnlent. A simple example of this adnlits a closed-forn) stllutilln tllat Iklrtller clarilit)sthc point. Suppose protluction involves 11variable inptlt (lal7llr) thlt can be chosen optimally at each instant. Suppose tlle production function is CilI)bDouglas. Tht!n zr4K. F') nlllx I F K'' L.' - ?t' L 1. ,
=
1.
where Er denotes the
wage rate and 0 +
1/ <
I to ensurc diminislling returns
'
K''/$'
to scale. Then we 5nd Jr
( K F)
where C is a positive constant P urpose. Therefore
,
=
whose
1/t C )z'
-''
-'''
,
precise vtlue is imm:tterial lbr our present
Etensions tJ/IJ Applications
372
Since the power of )' is 1/(1 t?) > 1, we see that ZKCK. 1') is convex in F. In fact we can explicitly calculate the threshold for this case. A straighttbrward application of Ito's Lemma gives -
l / ( 1-
7(F'f
t'
)
)
)' l / ( l
=
-
k.
)
explta/l
1
2
+ 1c
p)
-
a
w/( j
-
w)
2
J?,crf.???ltz??&?/ lllveslltlell
Capttcily C/ltpcc
flzpt:
policy is to increase & whenever /Wrises to A'/.and this keeps tb'l from rising any further. ln this calculation, we will denote the natural Iogarithms of alI these 1og l', etc. variables by the corresponding lowercase variables; thus y =
1f 1
When
.
ln
Fi, no investment is being made, so that K is constant,
<
and
?n
follows the same Brownian motion as y, namely. an absolute motion with drift parameter a 1 c2 and volatility parameter o'. At times, however, when 2 m is equal to Fi, investment takes place and K rises. Using the theory of -
17c(A'. l')
t* 0 l ( l
=
a/(1
p-
Applying the formula
-
,py) ,
k-(p
-
1. c2
-
p/(1
2
-
F
p)2
j y( l
,,
-
)
jc-
(l
-f? -
:., j/(
l
-
k, ) .
solving for the threshold, we hnd
(15)and
(,, y.(x.) -
w)
-
)
p
.-.
'
1
,,)
---
k?
-
a/
(1
w)
-
).
2 p/ (z
-
C 0 / (1
)
'
=
w.
Long-Run
Average Investment
Following the optimal policy derived above, the firm invests in small bursts whenever the state K. F) hits the thrcshold curve from below. Although each instant of such a hit is random. we can calculate the average rate ofgrowth of the firm's capital stock over a long interval of time. Doing so provides another useful way of looking at the effect of uncertainty on investment. To do this, we consider the special (andanalytically tractable) case of the Cobb-Douglas production function, H(Kb In this case, equation
=
(9)for the F S-t
We will abbreviate
K0 can be written
1-P'
/ =
I
i'
FH)J,
<
-x
n
Fri,
<
as
of this distribution
To determine the long-run averagc growth rate of K, we treat the Brownian motion of nl as a discrete random walk, in the manner discussed in Chapter 2, Section Z.B. Divide time into very short intervals of duration JJ, fy dt. Nowa given the and the range of ln into very small steps of size tlh ?n, above equilibrium. for distribution know that in long-run exponential we of ln probability Iies in the immediately Hi is a' dl. that step the to the Ieft ln the of and that h7, le does happen to lie just to Suppose note that y can either will experience an upward up or down. The probability that y move t/f of size l +a' J is given by p ( /cJ. If that happens- an offsetting move increase in k prevents ln from hitting Fzi. In that case, =
=
dh
-
(1
-
0 ) J-
J
J Kl K
(),
=
=
JJ:
=
dh ( ( 1
-
0)
.
Over a long time span, the average growth rate of capital is therefore given by the product of these probabilities and growth rate: the probability that m will be very close to i (whichis a'dh ), times the probability that y has an upward move p), times the percentage growth in K should an upward move in y occur Ldk dh/l 1 /7)1. To get the growth rate, we divide the product by f:r (J, we 5nd dt The result is e' p dhlljfdt ( 1 - 0) j. Recalling that dh that the long-run average growth rate of K is -
=
K
1 0
by M. The right-hand so nuctuates,does .H. The optimal
the left-hand side of this equation
side is a constant; call it ibl for short. As
-
.
.
/1
-
n1
-
=
threshold
exple'
2 a/o.l l > 0. We will be interested in the part where e' p? is close where to Fzi. very
-
-
a'
-'
)
p
-
(l
p 2 )
-
We see that an increase in c lowers the numerator in the expression within the second set of parentheses on the right-hand side, and therefore contributes to lowering the threshold. Of course the increase in z lowers the root Jl and this contributes therefore raises the option value multiple pb/(/1 1/( 1 w)J; the raising other parameters p, a, the threshold. The balance depends on to
and
Brownian motion. we can characterize the long-run distribution of ln. We will ( 1985, p. 90), or Dixit so we proceed on this density
omit the derivations. and refer the reader to Harrison (1993a. p. 61). The distribution exists only if a > ! tr2, 2 assumption. Then the distribution is exponential. with
/7.2
12
(2e/c2
-
1)/(1
-
()4
=
ge
/7.2 -
-
j/(1
-
p).
' Thus in this Cobb-Douglas case, a Iarger c means a lower long-run average growth rate of the capital stock, and thus Iess investment on average.
Lxtellsiolls
1.F
wd/l/p/ct'lft?/p.
f'l/lt
Depreciation
Now we examine
what
happens ifsome of the capital installed by the firm grad-
ually decays. Most theoretical treatments of the stlbject assume, for analytical convenience. that the decay occurs exponentially as a Poisson process. Thus suppose the firm's capital stock at time / is K. Dtlring the next small interval of time tf each of these existing units will stop functioning with probability //. Deaths of different units are independent events. so we can invoke the law of Iarge numbers and say that exactly K t// units die during this interval. .
If the firm installs dK,: new units (grossinvestment), the change tiK in its capital stock (netinvttstment) is given by JK
=
t//k'g
-
( l6)
K (1?.
Such depreciation complicates our concepttlal device of treating differunits of capital as different projects ranked by their order of installation. ent A unit that was installed as the CK + J/flth will take on a diftrent Iabel if some earlier units decay. Luckily this was only :1 conceptual device to allow us to introduce the concept of option value of a marginal unit in a clearer way. AIl units of capital afe physically homogeneous and interchangeable in
their productive potential, ordering.
so no harm is done by abandoning
their rank
Greater complication arises if each unit of capital has a given hnite Iife, in as our earlier treatment of a single discrete project that depreciates by sudden death. Then, to describe the state of the firm. we must keep track not just of its total stock of capital, but also the age of each unit. However, this is a computational problem rather than a conceptual one, and our exposition of the underlying economic ideas can do without such added complexity. We begin as usual, by finding the value of the firm's stock of initially installed capital K. Take the dynamic programming approach hrst. If the firm K dt, or makes no new investments, the existing stock will decay as tS e-J), and the expected K( K e-ht The profit Ilow at time J will be i'/ HK present value calculated using the discount rate p will be =
=
-
.
This Ilas an interestiny interpretation. First. stlppose only the darl ittr tlllits of capital decay while tlltl marginal o nt! renlai ns alivc. Tllc 11 its rank slitics l down from A-to K e-k and its nlarginal prodtlct rises to /-/ K (., - / ) Secontl. recognize tlle probability that tht) nparginal unit itself l'nay die t 11is raises the discount rate fronl p to p + Then ( l 8) givos us the expectetl presdnt v:tlutt calculated tak ing both of these e ffects in to account. Alternativdly we can use the contingent olnims approach. I'Iere we slltltlltl recognize the decay of capital as :1 negative dividend on the asset. Otller tllan thats Nve follow the ustlal stops to get '(
.
'.
.
1 tz 2 :2 I,' a
.
.
1
(K
capitaI :
the value of the initial marginal
(r
-
j) y j'j' ( K. y)
K Jzk.( K. y )
-
0.
=
( l t.l)
Then ( 17) is the solution to this equation when wc tlse the proper risk-adjusted rate of discount y. in place of p. Next we consider the value ol' the firm-s optitlns to invest in adtlitional capital. when the current stock is K and the ctlrrent value of the stochastic shift variable is F' We wil l denote the value of these options by F( K. ),') A s long as these options go unexerciseda installed capital goes on decaying. Tllus t// later tbe firm will acqtlirc back its option to invest t he Iast a short time (lt 2&. capital. units value K of This has K marginal ( K F); rttcllll that F( K. F) is the value of future expansion options, so an increase in K meltns sacrihcing some of these options at the margin, and hence Fs. is negative. Now wc can set up the usual risk-free portfolio to detcrmine F K- )/ ): Ilold one unit of the option and sell short Fr units of the nsset spanning F. This yields .
.
.
.!J' a
2
)' 2 F 3 r + (r ,
-
j
) F' Fv
K F s.
-
-
r F
=
() .
Differentiating with respect to K, we get an equation for the value )' S. F) FK K. F) of the marginal option : ( /7.2
y2
-
(17) to get
y) +
- r V ( K. F) + (' H( A')
) We can differentiate
.
/'yy.(K, K
y) + (r.
fg (K. F )
-
-
) )'
/.j,( K. y)
(r + k ) ./'(K. F )
=
().
=
(2t.3)
unit of
This is a partial differential equation that generally would need to be solved numerically. However, for an interesting special case we can invoke an economic homogeneity argument to reduce it to an ordinary differen tial
Extensions
376
(lnt'
Applications
equation and solve it analytically. Supposc the dependence of the profit tlow on the capital stock takes the special Cobb-Douglas form HKq
K0
=
0
<
0
<
l
.
e
t
t
Now consider any two situations that differ in their initial values of K and )' but have the same value of i' K9-t Then the above formula will yield exactly the same expected value. The value of the opportunity to invest a marginal unit of capital should also be the same in the two situations. In other words, variable the option value f K. )') should depend on just the single composite y = F. K9-t write.j.CK.i') g(y). Then we can substitute fv(K, )') = Ke-'3 g'y), etc.. in (20)to get .
=
(21) This equation
has a very familiar form, and yields the solution
.g'(.y) Byts' =
where pj is the positive namely,
root of the standard
(22)
.
quadratic associated with
(21),
-
-+.
.
-
i')
=
vlk-.
i')
This yields i'(X')
=
-
fv( K- F)
K.
:1 ( + P) &fy-.1 pj 1 9 K -
lloice
l ) of the tlow cost ( + k0$ &-. Note that the cost is no a multiple J1/(/J1 marginal longer t &' but is increased to retlect the possibility of decay of the depreciation of it the unit. The muitiple is also modified to take into account dehnes quadratic that p, the and earlier installed units. In particular, in the which of increasing effect has the interest rate is increased from 1' to r + -
,
pb/(/31 1). Both these effects have exact parallels namely equation (27)ot' in (he case of a single discrete depreciating project. Chapter 6, Section 4.
pL
2
and thereby lowering
-
Increasing Returns and Lumpy Additions to Capacity
reach diminAs a 5rm expands its capacity. we would expect it to eventually ishng returns, if for no other reason than because of limits to organization and coordination within the tirm. Howevers it is quite common for a firm to For example. a encountdr an initial phase or phases of increasing returns. be produced before needed output be capital can any may sizable amount of units corrediscrete in have expansion to occur may at all. And later. capacity of laws efficient Somctimes geometry plant scale. minimum sponding to some needed of metal a construct when to the amount dictate increasing returns, as while its capacity varies dimension. linear of its varies the containcr square as cxtend our theory of as the cube. For ali these rcasons. it is important to which there are increasin situations expansion to cover incremental capacity features that and the complexities manageable new ing returns. To keep the
=
.
pr
( K- F'').
(23)
slightly modifed to take account of depreThis has the usual interetation, value the requires of the current marginal product 9 F K*- to be ciation. It
will work within
simple
structure
a as as we we intro tluce conceptua 11y depreciation. will ignore possble. Hence fbr now we units of Suppose H' K) is increasing over some range, and consider two < Kz, and K3 them Label and that this Kz in range. so investment located at Sl therefore S'( /f1) < H'Kz). Ifwe use the formula (9)to guess the thresholds clear,
We have eliminated the term corresponding to the negative root by the usual 0. We will have pL > 1 if r + > 1- ( + consideration of the limit as y t5 > 0. this is assured. > k6 Since 0. we are assuming t) 1)), or + unit once installed. marginal value the z?( of K. l') the We alreadyobtained of the option) is installation Then the threshold l'(A-) that justifies (exercise smooth-pasting value-matching conditions and determined from the fK.
lllb'estllellt t//,(/ (hqlaciy
,
Now consider the expected presentvalue vlk l') of the profit flow that would marginal unit of capital. Using (18)above, this is result from the (K + tslth
0
/'?lcrf???It.z?1/fI/'
that will justifyinstallation of these units, we will hnd i'1 > Fz. If F unit Kz should is on a rising path from some low initial levei, this says that the unit of at Kz, and the Iabelling unit K However, our be installed before the units are a1l that earlier H' Sa), presumes assigning it the marginal product dquation thresholds based (9)must on already in place. Thus our guess of the be wrong for this range of increasing returns. Looking at the situation the other way around, a threshold of i' that justihes installation of the unit at K3 is also high enough to justify installation should at Kz, and indeed the whole range in between. Therefore investment units should at installed labelled be once be lumpy; a whole collection of our when a common threshold is reached. It remains to determine the size of FI
and
'a
.
b-telsiolls f2u
the lump. and the appropriate threshold that triggers capacity-t'
I??t/
zjpplicatiol
this Iarge expaqsion
of HK4
A rigorous derivation ()f the optimal investment policy turns out to be mathematically quite complicated. but the basic restllt can be understood intuitively. Theretbre we will merely state the result and explain it', formal proofs are in Dixit ( t993c), and we Ieave it to interested readers to pursue the m Consider the simplest textbook form ()t- increasing returns, where an initial range of increasing marginal product is followed by one of decreasing marginal produd. The upper panel of Figure 1l shows this case for the production function H S). Note that the marginal product is at its maximum when K is at the point of inllection. which is Iabelled K*. The average product is at its maximum when K K**, where a ray from the origin is tangent to
'
.
1 I l I I
.2
=
the curve.
If the Iirmfs initial stock of capital happens to exceed S*, then returns expansion are diminishing. In this case, the basic theory developed further to in Section l appliesa and the threshold is given by the tbrmula (9). We will call this the right-hand branch of the investment threshold function tbr the present situation. and denote it by YR A'). Next, suppose the initial capital stock is S() < &*, s() that there are increasing returns to capacity expansion. Now. draw a Iine from the point on H X') corresponding to K()so that it is tangent to the production function at a point K3 > &*, as shown in the upper panel of Figure l 1 With &-1chosen in this way, the capacity increment from &)to K6 has the property of maximizing the average product it contributes. Therefore it is worth considering as a candidate for the optimal amount of lumpy capacity expansion. In fact, that does turn out to be the optimal amount. The claim is that the threshold of F that justilies the marginal unit at K3 namely, Yft S1 ), is exactly the level that justihes capacity expansion from &) to Kb instantaneously in a discrete lump. Note that by construction, the average product ()f the added capacity is just the marginal product of the final unit:
K0
K
*
l 1 l l l l I I
H A'%) I/(Kb
-
/31
=
H'
!
)
K
*
I
'
FL(K)
1
I I
I 1
K0
K
*
I I I
I I
K1
&
l
*
*
Figure l l 2. .
Substituting in (9).we have .
6In Chapter 2, Section 5 we considcrcd an example that presented :1 trade-tlff btltween scale and llexibility. That does not arise here because scalc economies pertain to the capital stock in the production function. lf Kt already cxits and then f Kz Kt ) is inslallttd. tbe same output llow commences as if the whole K: had been installcd at once. That wis not the case in the example of Chaptcr 2: hvo small plants were not equivalent to (lne large one. -
*
Fn(&)
.
-
1 l 1 l I I i I l
K1 K
.:!.
(>/(K )
j
In other words, at the instant the capacity expansion is madc, the value ()1* tllc additional output equals p.l /(/31 l ) timcs the cost of installation. Ctlmpare cllndition t'tlr the this with equation (9).That equation gave us the lntllyillal invested under conditions of diminishing rklturns; now wc havc an exactly unit -
Extouiu.
380
Increlnental /?1vc.p?l(?n/and Ctlptlcfy Clloice
pZ'zt'/ Applications
sizes. AIl we need is that diminishing returns eventually prevail. generally ensured by rising organizational costs within the tirm.
to the lump of capital invested
analogous total condition that is appropriate under conditions of increaslng returns.
The general idea should now become ciear. Given any point &) in the the corresponding range (0, S*), we can follow this tangency procedure to hnd Yl-( K) by threshold KL and function construct the left-hand branch of the YL (A%) = YRCK, The result is shown in the Iower half of Figure 11.2. ). delining As Kt is gradually decreased starting just to the teftof K*nthe corresponding point KL gradually rises, and so does its F>(Sl ). Therefore the curve L (A-) is downward-sloping. Finally, as A't)falls to zero. /f1 rises to S**, so L(0) JQ(S**). The two branches YL (A') and YRCK,together comprise the investment threshold curve. In the region below this curve, the optimal policy is zero investment. If the initial capital stock Xkjis less than K', and F' rises to hit the curve at F (&)) F>(A'1), then capacity is immediately increased to K3 To the right of K3 the investment policy is the gradual one of bttrrier control the curve. then an immediate as in Section 1. If the initial position is above branch of the curve is right-hand ex'pansion the capacity to move to lumpy ,
=
=
.
,
optimal.
If the lirm starts out without any initial capitala then it waits until F rises to YL (0) = tK**), and then installs capacfty K** at on: it is optimal to jump over the whole range of increasing average product. Thercfore we will observe the tirmwith a capital stock in that range only if previous nonoptimal policies have left it there. We considered a very special form of incrcasing returns. namely. a single initial region of increasing marginal product followed by one of decreasing marginal product. This allowed us to develop the idea in the context of the familiar textbook setting. Howevcr, the idea itself is pertkctly gcneral. Whatever the shape of the production function H K), given the initial &),we are to find a value K3 to its right that maximizes the average incremental product LH X'1) S( A-())1/(S1 Ahp).In more technical Ianguage, we seek the conright of the initial capital vex hull of the production function. but only to the definition, returns at S1 must be irreversible. By is stock because investment employed be of Section to tind the threshprocedure 1 can diminishing. The capital marginal of would at K3 Then the of increment F that old justifya whole from applies &) to K3 This idea the discrete jump to same threshold differentiable; for example, is when not production function the works even amount of minimum if where H K) 0 some it may have an initial portion rise in it and all. thereafter needed may capital is to produce any output at efscient minimum of successive plants discrete-sized corresponding to steps -
-
.
.
=
38 l and this is
We considered incfeasing returns that arise on the output side. However, lumpy capacity expansion might be optimal for cost reasons. Suppose the cost is not proportional to the amount of new capacity installed, but increases less than proportionately. There may be a lumj-sum fixed cost of making any changes at all, or there may be economies of scale in a more general form. ln the next section we wilt consder Iump-sum costs, along with other kinds of costs that may be associated with capacity expansion.
3
Adjustment
Costs
The theory of how firms make intertemporal choices gives us rules for determining the Iirm's desired or optimal capital stock at each point in time. The demand for gross investment during any one time period can then be (he end of calculated: gross investment equals the desired capital stock at the the deprebeginning, plus the period. minus the actual capital stock at that shocks example. tbr occur, ciation that occurs during the period. Any capital In thestock. alter desired the demand shifts or interest rate changes, practice. In shock. reflect this ory, investment demand should immediately the effects of such shocks on investment have been tbund to be more gradual and spread out over many future periods. Economistsrationalized this by positing the existcnce of adjustment co./---costs of changing the capital stock such too rapidly and modifying their theories of investment to account for
costs.
than the mere existence of adjustment In factwe need somethingstronger might respond gradually to a costs to explain why a srm'sinvestment choices strictly convex function of the rate shock. Speciscally,we need costs that are a would have to be the case that the marginal of investment. In other words, it of investment. Then cost of investment is an increasing function of the rate marginal the optimal rate of investment is determined at the point where the equal cost of speeding up the adjustment of capital to its desired level isjust to the marginal beneht of doing so. The assumpton o strictly convex adjustment costs became common in and empirical literature on investment, but not without crittheoretical the earliest days. Rothschild (1971)is worth quoting at Iength on its from icism
this question:
ljktellsiolts tlplt/ z'Lllllli?tltl'tllls
The arguments
given as to wl4y the cost t)f adjustment ftlnction are quite weak. Eisner and Strotz ( l tX,3) give two. The first is that, as the lirm increases its demand lr investment'' goods in a single period, pressure will be put on the supply ()f investment goods. Our model is that of a Iirm whch is a price taker in factor markets; for it such considerations are clearly inappropriate. The second argument is that there are costs associated with integrating new equipment into a going concern: reorganizing production lines, training workers, etc.'' This is simply an assertion, and hardly a compelling one. Decrcasing costs are just as plausible No reason is seen why training necessarily as increasing costs. entails increasing costs. Training involves the use of information, which is a classic case of decreasing costs. Similarly, reorganizing production lines involves both the use of intbrmation as a factor of production and indivfsibilities.
should l7e convex
/?lt.7'(!?'?l(.vl/f?/
/?ll-f?.$'J??lt.vl/ (llltl
(.l/?t!(/p C'Ilt'll' -t?
F()r nllsv Nve will tlcveltlp tlle ideil ('fstrictly lttlitlstnltlnt tronvtlx costs Nvit I1i1) rnllde tll:l (ltlt bri :vi general vt':ry ( I its re Illt ions Ilip t 11( 11eappl4lcll Nvkl 11:ly.kl a ngs
follosved in t his bllok. Classilication
f3f
Adjustment Costs
'increasing
.
.
.
.
.
.
.
.
.
Alternative explanations of the observed gradual response of investment in economic conditions were also offcred. Of particular interest in the context of this book is the distinction between tirm-levelinvestment decisions and obsclwations or data that are almost invariably at more a aggrgate level. Suppose each firm adjusts its capital stock when the shock to its profitability reaches a thrcshold ()f the kind we have been calculating. Different tirms diftkr in their technological or managerial abilitiess local factor market conditions. and many other things. Theretbre different tirms have diftrent action thresholds. Even firms that have equal thresholds may be at different historically determined initial conditions relative to this threshold. An aggregate shock will immediately push some lirms over their threshold and they will invest, while other firms will merely get closer to their thresholds without crossing them. As time passes. some of these hrms will experience other tirm-specitic shocks that will take them across their own thresholds and bring forth their investment responses. Thus the eftkct of the shock on aggregate investment will be sprcad out over time even though the action of each 5rm is concentrated at a point in time. This idea is more intricate to work out in a simple model, because the aggregate response of investment depends on the relationship between aggregate and firm-specihc shocks. and on the original location of iirms relative to their action thresholds. lnteresting theoretical and empirica! work in this area has begun; see particularly Bertola and Caballero 1990, (
to changes
1992).
Let K denote the h-rm's capital stock, and F' tlenote a stochastic slli vrtl-iable that aftkcts its profits. Le t Jr (.&. )z') tIe ntlte t he prtlfit I'Iow:ts 1, function of t llese state variables. Suppose existing capitrtl tlepreciates l't the rate New capit:tl can be added, and ol! capital can be renloved (overantl ltbove tlle nCtttlnll loss through depreciation). but these actions entail certain costs. .
We will divide timc into small intefvals of Iength t. Stlppose tllat tltl ring such interval the tirmtakes deliberate action t() change its capitltl stock one by the amount L K. ln other words. total gross investluent (pllsitive(11- negative as appropriate) during this time interval is &-.so thrtt thtl stock at tlle elltl (11' the period is ( K K / + L K ). There (tre three types ol' ctlsts Ckssociated with this action. kr:.
-
First, there may be lump-sum costs for taking any
llctitln
Tllese n'lly such things. any Call these ustock'' fixed uosts. There may also be ctlsts that aczlzrtle as a givcn rate of tlllwover the time interval Lt whttre the ttctitpn is being taken: trall these tsllow''fixed costs. Each kind mtty also be diflkrent depending on whetht!r the capital stock is being incrcased t)r ddcreased. To 17t:pertkctly general. we Iet thcse costs be
btt managerial decision costs. fixed costs of placing
tp + + (p +
S.v ()-
-
/
1/
i1-L K if f K
orders.
>
()
<
()
Ctt ltll,
()r
.
.
Next, there may be costs that valy linttarly with tlle quanti ()f change in ty capital stock, but do not depend on the the length of the time in terval (smali) over which this change is accomplisled. Tht:y will inclutle primarily the atrtual priccs paid for the capital goods purchased- but mtty lave other components. N'Vcall these costs, and write them as tilinear''
K+
.(
K
v- It A'I
-v-
=
t K
if A K it' 2 &
> <
(). ().
Note the asymmetry; it captures the irrcversibility that Ilas been our central concern in this book. 1f4-- < (), then installetl capital can I'1ereducfad by selling it at a price -K- per unit. lf g -K+, then thc full cost (.)1' purcllase be =
can
Er/encvu
384
Applications
(l?lf.
recovered upon reselling, and investment is fully reversible. lf <+ + v- > 0, < <+, the recovery is partial and there is some irreversibility. (We or possibility K- +v+ < 0',othenvise the hrm could make an infinite the rule out ofmoney by quickly and repeatedly buying and selling large amounts amount capital.) Of of course we can also allow v- > 0', here the tirm must pay to be reduce its capital stock, for example, site restoration costs when allowed to mines are shut down, or hring costs forworkers associated with the machines that are being retired. -g-
Finally, there may be adjustment costs that depend on the rate at which capital ts K/Lt. This stock is being changed, that is. that depend on l the + Ldt adjustment costs.'' (/) denote the is the traditional category of adjustment which the with unit Then these costs accrue. time, or flow, rate per =
Sconvex
cost during our time interval of length
AJ
is
Lt + LK/Ltl.
lllcrelllcntal /?I
=
=
altd
Choic(: Cf//ltvcfky'
stock more rapidly. hold A K fixed and respect to Lt is + which is negative because
vary tt.
t K
LK
-
Lt
l
+
,
The derivative of the cost
with
K .
Lt
+ is convex and satisties +(())
=
0. Thus reducing
Lt increases the cost.
Each article in the voluminous literature on investment usually treated just one of these costs. This made it difficult to compare the results of different models based ()n different assumptions in this regard. Recently, Abcl and Eberly ( 1993) have constructed a very general model that includes almost all of these costs (exceptstock flxed costs *+ and *-), thereby facilitating such comparisons and oftkring an improved understanding of the issues. Our treatment follows them.
3.B
0 and +(/) > 0 when I # 0. The function tp is convex, and satisfies +(0) 0. This If the function + is differentiable at zero. then we must have +'(0) is true in the commonly used case of quadratic adjustment costs; panel (a) of Figure l 1.3 illustrates this case. However. the function could have a kink at zero. as illustrated in panel (b) of the same hgure. Here the right-hand derivative of + at zero, denoted by +'(0+), differs from the left-hand derivative, denoted by +/(0-). There can also be other more general asymmetry in the form of +(/) for positive and negative values of 1. Observe the nature of these costs. The flow rate at which the costs are incurred, namely, +(/), is a nonlinear function of the Ilow rate of gross investment, 1. However, the total cost incurred over a time interval of length Lt is proportional to this length. To see how it costs more to change the capital
vcsncllt
The Bellman Equation
Now we are ready to examine the firm's optimal investment policy. For aI-
gebraic simplicity we will assume risk neutrality or an exogenously specified K, F) discount rate p, and use the dynamic programming approach. Let 1zP'4 denote the value of te Iirm as a function of the state variables. Over the next short interval Lt of time, the irm chooses L K and restarts with a new capital K Lt + A K) and a new random value of the stochastic variable. stock ( K i'). Write f'(A K. A/) for the costs of the action.which may include L F + ( say. all the components discussed above. To statc this explicitly. of any or -
*+ C(A K, A/)
=
+
4+ Lt
+
K+
A/I' + Lt +(
K/ttj
()
*-
+
4-
Lt
Then the Bellman equation
-
lc-
S + Lt +t
&/A/)
if LK if LK if LK
> =
<
(),
0, ().
(25) can be written as
&p(/) Ap(/
) Z
g
N N. N
'P t (0-)
N
w.
-..
Ccl?ll'c.rAdjttsllnol
.'' Ap
l
(c+)
A
I
/ Figure l 1.3.
z'
A
Cosl.%
As in Abel and Eberly ( 1993), it is important to distinguish three cases according to whether the maximum on the right-hand side is achieved for K
positive. zero, or
negative.
First, suppose the optimal LLK is positive, that is, the firm is making positive gross investment. The first-order condition is
386
Exlensiolls tl/lt Applicaliolls
The discrete tinle period L was introduced only lbr
ltnltlytical
conveniences
and our real interest is in tlle continutltls-tin')c nlotlel. Tllerelbrc wkl take the limit as L / (). We assunle tbr the nlonlent that the rate t)f gross investmentp l L /f/Al, is linite; we will soon check wlletller and when this can be valid. ....->
=
Nvith this assumption
the time path of K is continuous; since F folltlws a motion Brownian geometric its time paths are also continuous. Tllere fore the condition in the linlit becomes
Ir,U:( A--F )
v+ + tp
=
'
(l )
.
This determines the rate ot- investment as 11 lbnction of the state variables. Since we have various kinds of fixed costss we must also consider a total condition that ensurcs the global optimality of the solution to the Ioctl firstorder condition- For this. we need to comparc the restllt of setting t. K I @./ where l is defined by (27)above. and that of setting 25.K 0. We need =
=
now somewllat Imore convdn ien t ll()t to expross it in ratio llt7stll utt? nl:tgnittlde a nd ctlI1l pare it to t lltl pu rchase as an of capacity i t (including adjtlsl mtt 1-1t cost ). tln Now express equations
lbl-nl- l7ut to keep it cost of t 11enn arginal
(27) and (28) in this Iangtlage.
A rattt (,1, gross contribtltes tllc nlarginal
investlnent I stlstained over a short tinle intelwnl tt anlount / A t to tlle capital stllck. Tlle lirst-orde r cond 1t ion (27) sets the marginal benefit (increase in the val tle of t ht! tirm) 01' this act illn equal to its marginal cost, vvhile the total cond ition (2S) enstlres that the total be neht exceeds its total cost. sve can illustrate and clari fy these rel ilt ionsh ips using a d iagram o 1'11 kintl that is fam iIiar fronl inte rmed iltte microeconomic t heory, and is a mod ihcat ion of a (igure in Abel and Eberly. In Figtlre l l we show the marginal cost as a function of the rate of investment. For ptlsitive grtlss investment (to the right of t he vertical axis) th is is just the right-lland side ol' (27).The critica I v:ll tld marks the point where the area betwetln the vcrtical :txis. :1 horizontal Iine at height (/ and the nlarginal cost culwe s ttxat:t ly equal to the tlow fixed trost 4+ .4
-q
,
N'Vhen t/ Again our interest is in thc limit 0. as Lt The point to note is that if thdre arc stock ixcd costs of investment (*+ > 0), they constitute the leading term in the total condition and in the > ().That is direct contradiction. Thcrcfore limitwe require if there is a a hxcd that is, stock cost, a one-time cost of making any positive gross addition to the stock of capital, thcn a hnite rate of gross investment cannot be optimal. This makes good intuitive sensc; such a policy would require incurring the stock fixed costs at evc:y instant of the time interval over which the rate of investment is positive, which would be inlinitely costly. The optimal policy will allow the capital stock to jump in discrete stcps at isolatcd instants. It can be
l'ntt rginal
.
the / tlefincd :llllng the nlarginal cost ctllwe satislies l7()tI1the and the total ctlndi tions with eqtlll.l ity. F-tlr t/ > tlte ttltal ctllltli tion -t/.
-t/
.-w
srst
-*+
characterized
using methods
similar to those of Section 2 above. Here 1et us assume that *+ = () and proceed. In this case, both sides of the total condition have leading terms ()f orddr t/. So we must expand the values of 1.1/using Ito's Lemma and simplify. This givcs l'Pk ( K, )') /
>
/+
+
K+
l + +( l).
(28)
To interpret this, we invoke the concept of Tobin's q. At any instant, the current state ( K. F) is known and theretbre lzpk( K. F) can be evaluated. This is just the marginal increase in the value of the hrm that woultl result if it were given an additional marginal unit of capital. We could call it Tobin's marginal q using the value-of-the-firm-sense. The only difference s that it is
(h
+
K+
+Ap
l
(c+)
-x
(l) -
+&1z '
(0-)
q
I
Extensions
388
(1rlt/
Applications
is satisfied as an inequality. Therefore this portion of the marginal cost curve thicker can be regarded as the lirm's investment demand curve. This is shown the in sgure. F()r negative gross investment the first-order condition is lV&.(/f,F)
=
-K-
+ +'(/)'.
(29)
Let g.denote the critical remember that +'(/) horizontal line and the marginal value where the area trapped between the flxed vertical axis equals the flow cost 4- of negative cost curve to the left of the the firm's marginal becomes cost curve gross investment. Then, for q < g.,the policy rule for negative gross investment. If q lies between the two critical levels g.and %,zero investment is optimal. We now proceed to examine various implications of this analysis, and is negative when l is negative.
relate it to our earlier
work.
ltlcremental Jnpc-pncnl
Finally, note that we began by setting the stock fixed costs *+ and *- at powerful reason for zero. If those costs are positive, they are an even more choosing isolated instants of time only and action time. of the most taking no lumps. discrete decreased in capital of increased stock is or when the takes equation Bellman the a particoptimal. is When zero investment using Lemma right-hand side lto's of the Expanding (26) ularly simple form. and simplifying, we get
1 tz 2 y2 J#r. K y) + ? a .(
'
=
and
(2J. ( l J Note
that the right-hand sides of the tirst-orderconditions (27) and (29), corresponding to positive and negative gross investment, do not ap(). and the marginal cost curve in Figure 11.4 has proach the same limit as l this. (1a) a discontinuity at the vertical axis. There are in turn two reasons for emphasiztld throughout &r+> -K-, which is the irreversibility aspect we have this book. glb!A kink in the flow adjustment cost function, +/40+) > +'(()-), which also contributes to the discontinuity. The point here is that the marginal cost of even the tirstsmall amount of investment is positive; therefore positive gross investment is not made until F rises sufficiently high relative to K to generate a compensating positive marginal beneht from having the extra capital. A similar situation holds for gross disinvestment. exceeds the (2) With a positive tlow fixed cost 4+. the critical value the vertical marginal of right-hand the cost branch the curve on of intercept Iess than the left-hand intercept. Both axis. Similarly, a positive /- makes rthe inaction. effects widen of range these A range of inaction exists as soon as any one of these three forces is in operation,' t h e t h ree o p erate in the same direction and reinforce cach other concerned. as far as the optimality of zero gross investment is -.>
K 1#k(K, i')
-
a )' 1lj. K. y)
S. l') + n'K. F) p 1,P'(
-
an equation value-of-the-lirm.
lt is more convenient to convert this into
I'VKCK.)'); recall that this is the marginal. (not ratio) concept of Tobin's q. Diterentiating
The Range of Inaction In the above analysis, zero gross investment is optimal over a range of values l'Io ( K, i'), and therefore over a corresponding region in the space of of q the underlying state variables (K. F').There are three causes of this, and in the general model of Abel and Eberly, each is explaincd in terms of a diffcrent kind of adjustment cost. They appear below under the hcadings (l a1, (lb1
389
and Cfzpflc'l'/.y Clloice
-
47.2/2
t/ y y
( K.
+
zrA.' ( K.
0.
tbr qK.
F)
=
and absolute
with respect to K, we get
y) + a y qv ( K. y)
t? K. F) - (p + ) (
=
-
F')
.
=
K (/. (K. y)
(J())
().
This holds over the rcgion of inaction in ( K. F) space, but the region itself this is a free-boundary must be determined as a part of the solution. Therefore from conditions somewhat one model to the vary problem. The boundary the situation linear. adjustment of are next. For example, wen the only costs conditions at the upper is like that of Section 1 of this chapter. Then at the intinite) fact are q = v+ boundary (whereI is about to become positive, in have = Similarly we q = x.- and (value matching) and qv 0 (smoothpasting). = 0 at the lower boundary.
qt.
3.D
Barrier Control
If convex adjustment costs are relatively unimportant, so that +'(/) is very small for all /, then the marginal cost curve in Figure 11.4 is nearly liat. The critical values of q are approximately given by
and 17/.-4K, F) that came out of our analysis of pure the optimal this chapter. earlier Moreover. as q reaches in irreversibility of and for likewise the rate large, gross disinis investment very rate of gross adjustment if are absent limit costs the in happens What convex at
These are
just the limits on
,
vestment j.. altogether and only Iinear or
costs remain'? that The answer is an inhnite rate of investment operates to increase capital in small but rapid bursts, so as to prevent q stock decrease the
390
1:.jaensiolls
tlp?t
pp/ctl/tp?/.
,4.
from ever rising above i'j-or falling below j!..This is just the policy of control'' that we obtained tbr the pure motlel that considcred irreversibility on its own-? The rigorotls demonstration of the limiting procedure wotlld be mathematically too diffictllt and lengthy to attempt here, but we hope that the result is intuitively appealing, and that it clarifies the relationship between the irreversibilityapproach we have treated in detail, andthe adjustment costview that has been common in much of the previous Iiteratnre on investment. wbarrier
3.17 The
QuadraticCdlst Case
'sve will cornplete tlltt solutilln tt) tl'lc ttlitlstnlent ctlst I'nodel lklI- :1 ptlre ltlltl simple speciltl case-s I.lere we I'legletrt aII t)t Iler llspetrts of costs. llI1tIsupptlse J I '? tllat the adjustmellt cost funct ilqll-lis synan'le t ric alltl qtlad ratic- tp ( / ) there Tlel is no range of inaotion. and thtt investnnent denaantl funet ion is .
sinAply
for al I t? We also suppose that tlle flow prgfit funct ion is linear in the capi tal stock and nlultipl icative in thtl shock, so zr ( K. 1/) /? &' 1'' Tllen the equation for the dynamics of t/ becon-les l
t/
.
.
3.E
The Dynamics of q
When gross investment occurs at 11 nonzero rate, thc optimal rate is delined by the first-order condition (27)or (29)as appropriate. Each dehnes l as a function of and the parameters that affect this relationship are K+ or &'-, and the functional form of + In other words, for :1 given t2 the optimal rate of investment is determined entirely by the properties ()f adjustment costs. The seemingly surprising thing is that the uncertainty. as captured in the parameter tr of the stochastic process of 3', does not enter into the picture at all, nordoes the trend of F, or the discount rate p. Itwould indeed be surprising if these aspects were irrelevant for investment. However. the surprise is only apparent, because this reasoning holds :/ lixed. The evolution of t/ is affected in a significant way by all of these parameters. For example. suppose ( S. F) is in a region where the rate of investment is positive. Let the investment I (t?). Expanding the rfght-hand side of the Bellman equation function be l (26) and simplifying as usual, we have ,
.
=
12 v2
/2 r,F + rr
(y
)' py +
(/
-
K) 1jo
p ;P'
-
-
4+
v+ /
-
+ ( /) + z
-
=
0,
where we have omitted the arguments of functions to save notation. We can dit-ferentiate this with respect to K to gct an cquation for t/, namely,
12 ,2
:2
:
l ?' F' + a )r qv + (/
The hrst-order condition
! a
(,2
simplihes
F2 q ?'F + a )' qv + ( /
The solution.t/f,
.
Kj qs.
-
(p +
)q
this to .
Kj
(yc
-
(p +
.)
(/
+
LZ'K =
1.G 2 2
1'2 Ll '
.
1 .
+ z y
t/ y
+ (g - ). K4 g y
-
(/) +
/1 1'
jg +
().
This looks quite complicated. but econom ic intuition suggests the lrn-l ()f the solution. Since the profit llow is proportional to thd c:lpital stock. and adjustment costs do not depend on this sttlck's the lnaxinlized value 11Z( K. ) should be a Iinear expression in K. so 11z':. (/ should depend t)n F alontl. Then :ve have an ordinary diffcre ntial equttion '
=
!.t'r2 y 2 (/ y., z The solution
.
+ a y
t/ j
,
-
) (/ + /? y
( /) +
=
()
.
is
f2( F )
=
/? Y/ (p
+
-
a)
.
The intuition is that when the capital stock productls prolit flow at ctlnstant returns to scale and adjustment costs d() not depend ()n the installcd stock. we can regard each installed unit of capital quite independently ()f any other. lt produces the profit Ilow /1 F, but is stlbject to deprcciation by a Plpisson process with parametcr 1. Then (32) gives just the expected present val ue of this uncertain profit tlow. That must be the marginal value of this unit of capital to the (irm. Once t:/ is known. the optimal investment policy follllws immediately, alld that is our main concern in this section. If desired. however, thll solution lbr the value function itself can be obtained fairly easily; it is simply t/ K plus the valuc of expansion options. For that. and for a more general version of this model. we refer the reader to Abel ( 1983).
0.
4
F') is affected by alI relevant parameters.
7Barrier control uses inhnite rates of investlncnt at instants of timc that suln to measure zero'. that is how it generates a finite increase in the capital stock ovttr a Iinite intcfval of time. See Harrison and Taksar ( I 983) for details. That is alst) why the tlt'w jixt:d costs / antl 4- llre irrelevant in this case; applied (lver time instants tlltalling t() mttasure zerl), they have zeru effect. .
Guide to the Literature
The model of this chapter has quite close links in some respects with the standard neoclassical theory of investment. That tlleory began by assuming
392
'tellsiolts
F-.-t-
fl/lJ Applictllions
competitive conditions. and ignoring unertainty and irreversibility.The rcsult was as if the firm could choose to rent any desired stock of capital for one period. The optimality condition was the same as that for any variable input: the marginal revenue product of the capital stock over the period was set equal to its rental cost, which equals interest plus depreciation (or minus capital gain, if any). Jorgenson (1963)is probably the most prominent and widely used model of this kind. Such models
implied a very rapid response of investment to changing whereas the response in reality was much more gradual. of the neoclassical model were proposed in response this observation; perhaps the most prominent was to introduce adjustment to models adopted Early this approach include those of Lucas ( 1967) that costs. 1968). and Gould ( Another feature that was introduced to make the model more realistic irreversibility. Here Arrow (1968) is the early and prominent publication. was certainty the In and pertkct foresight. irreversibility implied stopcontext of ping investment a little before the peak of potential prolitability, bearing in mind the worse times to come. Finally, uncertainty was introduced in a relatively simple way. The capital stock had to be chosen knowing only the probability distribution of 11 random variable, and other inputs Iike labor could be chosen after the realization of its actual value. A simple model by Hartman (1972)brought out the cssential idea: the marginal product of capital was often a convex functilln of the random variable', thereforc greater uncertainty implied a higher expeuted marginal product and theretbre greater investment. Similar results in dynamic and stochastic contexts are in Abel ( 1984) and Craine ( 1989). All of these earlier developments are surveyed and expositcd in the excellent book by Nickell (1978). More fully dynamic models combining uncertainty and adjustment costs were introduced by Abel (1983)and Hayashi (1982).Abel ( 1990) provides an excellent sulwey of some of this more recent work. A recent model by Abel and Eberly (1993)integrates many of the features of earlier trcatments. and our exposition of adjustment costs is based on it. . Models that analyze variable capacity choice in the presence of irreversibility and uncertainty can be said to have begun with Pindyck ( 1988b) and Bertola ( 1988, Chapter 1) and ( 1989), although some glimpses of the idea can be found in Brennan and Schwartz ( 1985). Bertola and Caballero (1990) consider aggregation over hrms in this context. Bertola (1988.Chapter 2) discusses the relation with Tobin's f/.
Capacity expansion under increasing returns was studied extensively in is a prominent l96()s in the context of national planning. Weitzman (1970) the work, of this kind. much of this the In tbcus was on tindingthe costmodel time of demand, not on firm value meeting for path minimizingpath a given assumed certainty and pertkct foresight', work also maximization. Most of the exception. Manne (1961) is a rare early Some dynamic programming problemswe have encountered in this chapter have special features that make their rigorous mathematical theory quite diffcult. Either the rate of investment is infinite so the paths of capital stock or the capital stock itself makes discrete jumps,and so are nondifferentiable. control is at times discontinuous. In mathematical jargon,they are problems. We treated them as if they were no different than ordinary dynamic programming. We proceeded heuristically, appealing to intuition and making the results appear reasonable. We hope to have succeeded in meeting the requirements of most readers, but some will demand more rigorous proofs. We can do no better than refer them to two classic articles, Harrison and Takcase (barrieror instantaneous sar (1983), which treats the nondifferentiable control), and Harrison, Sellke. and Taylor (1983). which is about the discontinuous case (impulsecontrol). Dumas (1991) also provides a different and insightful perspective on barrier control. At an intermediate level of rigor, Dixit ( l99 1c) may be consulted. Dixit ( 1993a) is an exposition at about the same standard of heuristics as this book. but provides compact statements of the techniques and the results in fuller detail. ttsingular''
12 chapter
Applications and Empirical Research
payoffs fronl such investnnents. F-tlrtlxltnlplt!. ftlel prices. over the ftlture deluand for dltzct
tllertl is tllltrertaillty twc 1- 1'tlttl rtl ricity- ('ver tllc lttlre ellviri'lllllentaI regtllations that will constrain thil tltility- antl twtlr tlltt ctlsts of ll terllalivtl technologics to conlply with thostl regullltiolls. In Section 2 t)f this chnpter wil foctls on a specihc problenl faced by Inany utilities with coal-btlrning plnver plants. The Clean Air Act calls tbr reductions in overall emissions of stllftlr
dioxide (SO:), but to minimize the cost of these redtlctions, it gives utilities to rcduce enlissiolls to a choice. They can invest in expensive they Ievels, buy trade:ble tllat allow thtzn'l to mandated or can rcduces ot' rcdtltring air polltltitla (This the total social cost pollute. system svith creating utilities enqission retltlcincentive for by thc highcst ckst of an alloNvances buy If the future allowances.) tion to prices of were knosvn. this would be a straightfolward problenp. I-lowever. there is considerabltt tlncertainty over the future prices of allowances- antl an investnlent in scrubbers is irreversible. Tlle tltility Inust decitle whether to maintain llexibility by relying on allowances, or invest in scrubbers. ve will see I)t)w tllis problelu can be addressed using the options approach of this ln(.ltlk' --scrubbers-'
--allowances'-
.
WE
that by this point we have made it quite clear that the approach presented in this book is Ipplicable to a broad range ()f investment problems. Our numerical calctllations were guided by ntlmbers typical ofsome specihc industries, for example. copper mining in Chapters 7 and 8, and oil tankers in Chapter 7. However, for the mllst part otlr mtldels were simple and stylized. In this chapter we prcsent some examples tllat illustrate actual applications of the techniques. and their extension to other prllblems and l.IOPE
'*options''
issues.
We will begin with a problem faced regularly by companies in extractive resource industries how to value an undeveloped resource resclwe, and how to decide when to invest in the development and production of the reserve. As we have explained earlier in this book, an undeveloped resource reserve is best understood as an option. namely, an option to invest in development of the reserve and then produce the resource. By valuing this option we can value the reserve and determine when it should be developed. We will focus on the specific example of offshore oil reserves. Oil companies regularly bid hundreds of millions of dollars for offshore petroleum tracts auctioned off by the United States government, so it is clearly important to be able to correctly value these reserves and determinc how best to exploit them. Valuation and investment timing problems are also very important in the electric utility industry. Electric utilities make large and irreversible investments in new power plants, and face considerable uncertainty over the future
The principlcs antl analytical tllllls dtlveloped in this Inlpok-hltve rc levance that goes well beyond hrms' investment decisions. As tlne exanlple. in Section 3 generCtl problem i11environmental we show how the tools can be appl ied tt) lt when should the government atlopt 11 plllicy in rcsptlnse tt) policy design :1 perceived threat to the environment'? Tlle stantlartl framewtlrk by wllich
economists evaluate environmental policies is cost-bcnelit Ctnalysis. Cllllsitler. for example, a carbon tax to reduce global warnling. By distorting relative priccs. this policy wotlld impose an expected Ilow of costs ()n sllciety in dxcess of the government tax revenues it gencrates. Presumably, it llsl) yields an cxpected flow of benefits to society. I-lotlseholds and firms wlluld btl rn Iess fuel. less carbon dioxide (COa) wlluld accumulat: in the atmospllerca global mean temperatures would not rise as muchs and the damage caused by higher temperatures would be correspondingly smaller. Thc standard framework would recommend this policy if the present value ()f the expected flow of benefits exceeds the present valud of the expected I'Iowof costs. This standard framework ignores three important characteristics
of most
environmental problems and the plllicies designed to respond to them. First, there is considerable uncertainty over the future costs tnd benefits ()f adopting a particular policy. N'Vithglobal warming, for example, there is uncertaintyover how much average temperatures are Iikely to rise with or without reduced CO2 emissions, and there is also uncertainty over the economic impact of high!r temperatures. Second, there are usually important irrcvcrsibilities associated with environmental policy. These irreversibilities can arise with respect to
Ftellsiolls
396
tl/pt
Applicatiolls
environmental damage itself. but also with respect to the costs of adapting to policies to reduce the damage. Third. the adoption of an environmental policy is rarely a now-or-never proposition. In most cases the government can delay action and wait for new information. As a result, the same techniques that we used to determine the optimal timing of an investment can be used to determine the optimal timing of an environmental policy. At the end of this chapter, we discuss some of the empirical implications
of irreversibility and uncertainty for investment spending at the industry or count:y Ievel. Although there is considerable anecdotal evidence that firms make investment decisions in a way that is at least roughly consistent with the theory developed in this book (forexample, the use of hurdle rates that are much larger than the opportunity cost of capital as predicted by the CAPM), the task of more systematic econometric testing and application isstill in its infancy. We review some work that exists. indicate the difficulties it encounters, and suggest topics for future research.
1
lnvestments
in Offshore Oil Reserves
We begin with a model of offshore petroleum leases developed by Paddock, Siegel, and Smith (1988). The U.S. government regularly auctions off leases offshore of for tracts of Iand, and oil companies typically perform valuations
such tracts as part of ther bidding process. Because bids can involve hundreds of millions of dollars. it is important that these valuations be done accurately. In addition, oil companies mustdecide what to dowith tracts that they succeed in leasing. How high should the price of oil be before they spend hundreds of millions of dollars more to develop the reserve and begin production? It should be apparent that the failure to recognize and account for the optionlike nature of an oil reserve can lead to serious errors in its valuation. Suppose, for example, that one tried to value a reserve using a standard net present value approach. Depending on the current price of oil. the expected rate of change of price, and the cost of developing the reserve, one might construct a scenario for the timing of development and hence the timing (and size) of the future annual cash Ilows from production. One would then value the reserve by discounting these numbers forward and summing. In addition, since oil price uncertainty is not completely diversihable, the greater is the perceived volatility of oi1prices, the Iargerwould be the discount rate, and the smaller the estimated value of the undeveloped reserve. However, this would
Applictltions
f//l#
Epl/prcfl/
397
Researcll l
underestimate the value ot- the reserve, and probably by a large amount. The actually reason is that it ignorcs the tlexibilitythat the owner has over when to value. reserve's that Also because option note develop the reserve. that is, the of this option value. the greater fs the volatility of oil prces, the lrqcr is the value of the reserve just the opposite of what a standard NPV calculation would tell us. ()f an offAs we discussed in Chapter l0, the valuation and exploitation shore oil tract can be viewed as part of a multistage investment problem. The seismic and drilling activity to tind out how first stage involves exploration much oi1is present and the cost of extracting it.The second stage (whichwould only occur if the exploration results are favorable) involves development lhe installation of the platforms and production wells that are needed to extract the oil. The last stage involves the extraction of the oil over some period of the largest capital expenyears. Since it is the development stage that involves ditures. it is the stage for which option value is most important.z Hence we will focus on the valuation of an undeveloped (butwell delineated) reserve, and the decision as to when to develop it. ln doing so, we must account for the fact that the option to devclop the reserve is not a perpetual one; offshore which limit the time leases are usually subject to relinqttishment rcgrcrrlw/'. it. developing betbre hold the tract the company can
The close connection between the value of an undeveloped reserve and option on a stock is illustrated in Table 12.1.3 The underlying asset in call a the case of a call option is the stock price; for an undeveloped rcserve it is the as we will see, is in turn a function of value of a developed reserve (which, exercise for pricc the undevcloped reserve is the develof The oil). price the expiration is the relinquishment requirement. and the time to opment cost, The value and optimal exercise rule for a call option dcpend ()n te stock's dividend rate; the higher the dividend. the greater the opportunity cost (in terms offoregone dividends) of holding the option rather than exercising. The analogous variable for the developed reserve is the net production revenue less the rate of depletion; one forgoes this by delaying development.
1This seems to be the way that the government arrives at its own btten systematically too low. Sec Paddock, Siegel, and Smith ( 1988).
valuations,
which
have
zWhen exploration generalcs information pertinen! to Ialer slages of decisons. il has an addilional value. as we saw in Chapter l0. However,exploration expenditurcs are relatively small.
As for the last stage, once a rescrve is developed. extraction almost always proceeds at a more or less hxcd rate. ''''l--lleanalfagy is wilh an American call tlption, lhat is. an option that can be exercisttd at time up to and including the expiration date. A European option can only be exercised on any its expiration date. The table is frl'm Paddock, Siegel. and Smith ( I988).
/. vtellsitllls Table l2. 1
vtdt C'onkpalisoll tp./.(1 Qlll (..)J?;j(?,? .
..&?t/(.,y
.
.
.,1(q:v(t
tpp?
tl/lt'/
zzti'li.'.lict1tilllls
zl/lp/t-fllf-llp.fl/lf l S*1? lP?(.*f11 1t(.M'Pf1?-t.*/1 t
(?// lcsvtsv, .
sve can
alsl.)
llsstlrne
folItlu?s
lt BrllNvniall
Call option
l'nlltit'n
Undcveloped rttservtt
tllat t 11tlra tll
()1*
ret tl rI1
(.)11
tlle tleveltppetl
rese
Iwe
prtlccss: tlt
1t
/.t
Bf lz-l
,.
tlt
+
t-zl. t/--.
'
Stock price
Value of developed rese've
Exercise price
Cost of development
Time to expiration
Relinquishment reqtlirement
Volatility of stock pricc
Volatility of value ot- developed
where /zs is the risk-adjusted espected rate of rulttlrn roquirutl by a competitivu capital lnarket. Ctlnlbin ing equatillns (2)and (3)gives the following eqtl ation for the dynam ics of P' the uni t va lue of a developed reselwe: ,
t/ l,'
reserve Dividend on stock
Net prodtlction revenue from developed reserve less depletion
where 6: the payout by ,
(Jz
that is, a fraction be written as
tzl
-
Bt t//
=
(a)
lt can ,
Bt 17t f/ + J ( Bt r'-J )
(2) where
17, is the after-tax profit from producing and selling a barrel of oil.
4'rhis is a standard asstlmptitln that rcllccts geolllgical constraints on tlltl cxtraction rate.
See Mccray
( 1975).
( rlt
-
1''t)/
.
Note the close analogy Ilertl with otlr model ()f investment in Chapter 5. the valtle of a pnject )IIows a geol'ne tric Brown ian mot i()l1 l n Chapter 5, the value ot-the project was give n by d P' a P' J/ + r.r fz'(/-n wi t 11 (.)1- rettlrn. We called f (z rz j. wllere pz is the risk-atljusted expected rate the rate of return shortfalla and explained that it could retlect the cash llows fnlm the opernting project. Tlle same is trtle heres except tllat now the paytlut rltte is net t)f ddpletion. In fact. the only sign ilicant diffcrence between tl1is mlltlel and the tlne in Chapter 5 is th:tt here the (lptilln to invest (that is. to develop the reserve ) does not Iast brever: there is 11 rcl inquisllment reqtlirement. Also, since the marginal cost t)f production is snlllll (mostof the cost t)f production is the sunk cost of development), and since oil tzan be extracted only slosvly ((,pis typically abllut l () percent per year), r1/ > P) so that 5 > t), and developed reserves will alwltys be prllduced. Before proceeding, it will be uscful to roughly estimatc thc payout rate 6. The per-barrel value of a developed reservc typically ttquals about one-third of the market price of ()iI,per-barrel production costs are about 3() percent of the market price, and the corporate tax rate net of depreciation allow:tnces is about 34 percent. so that the after-tax protit on a barrei of oiI is about 46 percent of the price. Using (0 (). 1 and letting P denote the market price of a barrel of oil, this gives .
=
-
,
.
of the oil is produced each year. Then the return.
R(dt
(o
=
We begin by characterizing the valuc of 11 developed reselwt!. Let Bt be the number of barrels of oil in a developed reserve. be the value per barrel of the developed reserve. and Iet Rt be the return over an instant of time to the the owner of the developed reserve. Tllis return will have two components llow of prohts from production. and the capital gain on tht! remaining oil. As a reasonable approximations we can express production from a developed reserve as an exponential decline.'l so that =
:=
reserve, is given
in which
The Value of a Developed Resen'e
f/ B(
l-' z/--.
f'rr.
rate from a unit of producing leveloped tsl
1.A
6t ) V t/? +
1,
=
Thus holding a developed rcscr've is Iikt! holding a stock that has a dividend
yieId of abou t 4 pe rce n t.
1.B
tzal
Extensions
400
Applications
Given equation (4)tbr thevalue of adeveloped reserve,we can now determine the value of an undeveloped reserve as well as the optimal timing rule for its development. Since there is a variety of hnancial instruments that can be used example, futures contracts. to replicate lluctuations in the price of oil (for spanning clearly holds, companies), oil the of shares and forward contracts, value used be contingent methods to claims an undeveloped reserve. can and undeveloped unit of value J) one-barrel the reserve. of FV. denotc a Let verify reader usual the and through the equation can steps, going Using (4) F(F, t) satisfy must that ..
P
-j.j,
+
(r
j) p' Fv
-
-
r
=
-
;
Gexercise
F40, t) =
F(F*,
40 1
l?e-b-etll-cll
V*
=
*
Fv ( rz'' t ) .
D' r c & O .c c) c =
2 f.r 0.25 =
1.5
(r 0.15 =
X
0.5 20
10
5
Time Figure 12.1 (Shows F
*
Cn'ticul Hll/t,
.
t5 / D tbr
=
().1)4llnd r
=
25
(years)
vr Developtnettt
()f
0iI Sf-urr-'t.
().t)l 25, where 1.)is dcvelopment cost)
0,
=
max j Fr /)
2.5
.
Note that equation (6) is a partial differential equation; since the option to develop the reserve expires at time F, the value of the option depends on the current time t. Equation (6)must be solved subject to boundary conditions. Letting D price'' be the per-barrel cost of developing the reserve (thatis, the conditions the these option), are: of
F )'. F)
/?l/?rcflf
l?1f/
Reserve and the Optimal
The Value of an Undevelope Development Rule
&2:2
zdillnli?iltillls
=
D,
-
(8)
0J,
(9)
D,
-
1
(10)
.
Condition (8)just says that at expiration, the option to develop will be exercised if VT > D. The other boundary conditions are standard. Equaon (6)cannot be solved analytically, but it is not difscult to obtain (See Chapter 10, espea solution numerically using finite difference methods. solutions from Siegel, 12.2 Table and 12.1 Figure show Appendx.) cially the based on the payout rate 6 0.04 discussed above, Smith, and Paddock (1987) values of an after-tax real risk-free rate of interest r of 0.0125, and different value of o'v is an important input to this model, and can be estimated The o'v. in different ways. An estimate could be based on historical datay or obtained from an assessment by oil industry experts of, say, the go-percent conhdence interval for the price of oil at some time in the future. Estimates based on data over the past 30 years would put cq, at about 0.15, but estimates based =
on industry forecasts might be would be from 0. 15 to 0.25.
somewhat
higher. A reasonable
range for rrt,
Figure 12.1 shows the critical ratio V*! D as a function of the ntzmber of IZ tupirations = D l ycars to expiraton, for f.r,, 0. 15 and 0.25. Note that at rule NPV standard condition the that boundary (8)).so (which follows from applies. 'Fhis critical ratio increases to 2 or more, however, when the firm can wait at least a few years before deciding whether to develop the reserve. This option to invest result is similar to those tlbtained f0r the simple ratio sensitive is critical to the time not very in Chapter 5. Also, note that the for many than Hence time is greater one or two years. to expiration if that approximation reasonable to ignore such investments in oi1 reserves, it is a option the to altogether, simply treat and the reiinquishment requirement and the disappears. perpetual/ in equation (6) Then the term develop as *1
=
'tperpetual''
scompare the similar finding tbr (lur example of a depreciating machine in Chapter 4'. a teniniinite as i'ar as thc optirnal abandonment rule was concerned. year physical lifc was effectively
4()2
Ixtellsiolls
Applications
zjpplilultions
tll/t/
Table 12.2. Clilioll l4?/t,.j. pt'l- $/ oj' Dt.!l't?/t???tv?/ Cost (N(.)te: Because option valtles llre honpogeneotls il1tlle develllpl-nent cost. total option value is the entry in the table times the totltl develtlpment cost.) (z,.ac (. I4 2
rr,.
=
t).25
t/plf
E-pllpl'?c.tl/
Resetlr-ll
Suppose that the valtle t)l' the developetl reselwe is $ I 2 por barrcl. the payout rate (that is. net production rcvenues ldss tldpltltion as a I'rltction of the reserve value) is 4 percent- that dcvelopmen t tak uts threc ydars (typical gbr 1,11 o l'fsllorc reselwein the Gulf of Mexico), and thitt the present value t)f the tlevelopment cost is $ 11.79 per barrel. Then we wotlld vltlue the undevoloped reselwe as
tbllows.
/' / /7
7- a= 5
1- z= 1()
1- u= l5
7- a= 5
7- z= l()
0.80 0.85 0.90 0.95
().()18 l t)
().028 12
().()3,3()9
(). l ()3t)2
0.()276l (3.04024 0.05643
().03894 0.(35245 0.06899
0.0443() ().()5S03 ().07458
0.07394 0.09 l 74 0. l l l 69 ().13380
().l2305 0. l4390 0. 16646
l Since development takes three years, we mtlst calculate the present value of the developed reserve. The correct discount rate is thkl payout rates (that is, the difference between the risk-adjusted rattl yz antl the expected t).()4,the rate of growth of the resefve value, which is /z J). Since present value of the developed reserve is V' e'-0.12($ l2) $ I().t44. 2. We next calculate the ratio of this developed reserve valtle to the present ().9(). Tllis value ofthe development cost. That is 1Z// D $ l ().64/$ l l is Iess than l so the devclopment option is actually otlt ()1* tlltt money. 3. We can now ustl Table l 2.2 to calculate the value of the undevelopcd of (). I42. tlle option value per reserve. Assuming a stantlard deviation dollar of development cost is 0.()5245, and the total tlevelllpme nt cost l ()() million) $ I l 79 million- I-lcncc the total value t)l' the is ($1 l 1 I 79 millilln) undeveloped rescrve is (().()5245)($ $6 I million. .
(5
45
-
1 l l l () 1.15
.()()
().()8890
0.0766 l ().1()l 16 0. l3042 0. l 6472
.05
.
0.()943 l (). l l754
(). l 1253 (). 14()25 (). l7242
(). 15804
().l 4464
(). l8438 ().2 1278
0. l 7599
0.2432 l
().l907 l 0.2 l664 0.24424 0.27349
=
=
=
.79
=
=
,
f'rr.
.79)4
=
.
.84
equation can be solved analytically. (Tl1e solution is the same as the one we obtained in Chapter 5.) Table l 2.2 shows the value of an undeveloped reserve (that is, the value of the development option ) per dollar of development cost, for o'v (). l 42 gan estimate obtained by Paddock. Siegel. and Smith ( 1988)1, and for a,, 0.25. When F/D is Iess than 1 that is. the value of the developed rcserve is Iess than the development cost. we would not develop the reserve even using the standard NPV criterion. (In the parlance of financial options, the of the money.'') lf V/ D exceeds 1 (so that the development option is b development the money''). the standard NPV criterion would option is tell us to develop, but as Figure 12. 1 shows. unless V/ D is signihcantly greater than 1, this is not the optimal decision. The entries in the table show the value of the option per unit cost of development, that is, FV. J)/ D. For example. 1, the option value is about nine cents when o'v 0.142, T 10, and V/D for eve:y dollar of development cost. A simple example will help illustrate how these results can be used.f' Consider an undeveloped reserve which, if developed, is expected to yield 100 million barrels of oil, and has a ten-year relinquishment requirement. =
=
.
t
tein
=
=
t'-rlis examplc is discussed
=
iI1
nltlrtt dtrt:til in Sicgel. Smith. antl Ptddock
( 1t)87).
a
=
Thus. although this undeveloped reserve could ntlt be prtllitably developed given current oil prices. it is still worth ltblltlt $62 mi IIitln because of its option value. Furthermore, this valuc ctluld rise substantially if condititlns in world oil mark-ets changed in a way that made the perceivcd volatility ()1' oil prices higher. For example, as we can see from Table I 2.2. if f'r,, were t() rise to 0.25, the valuc of thc undcveloped reserve wlluld rise to (().I43t?())4$ l l rrlillitlrl ) $16b'.(bti rrlillit) :1. -lb)
.
=r
1.C
Mean Reversion in the Price of OiI
We have assumed that the value of a developt!d oil reserve Iike the price of oil follows a geometric Brownian motion. Howcver, as we have mentioned earlier in this book, one might argue that oil prices, and hence rescrve values, followsome different stochastic process. For example, one might believe that (lther commodities) are over Iong periods of time, oil prices (andthe priccs of drawn back towards long-run marginal cost, and thus are mean reverting. Or. one might believe that the price of oil is best rcprcsentttd by a Poisson jump process, rather than a continuous Ito process. Unfortunately, with limited amounts of data it is difhcult to determine whether or not a price process is indeed mttan reverting or has a signilicant
and Applicatils
Etoiolu
404
ttunit
root test'' jump component. For example. in principle one can pertbrm a reverting random walk. However, whether series is is price or a mean a to test this is a weak test that for short time series (tbrexamples 30 years or less) will often fail to reject the hypothesis of a random walk. even if the series is in fact mean reverting.7 The reason is that any mean-reversion is typically very slow, and is therefore hard to dscern in a short time series. ln this case one might ask whether the results of an analysis are likely to change very much depending on whether one begins with a mean-reverting (orPoisson) process for the underlying stoehastic state variable.
With regard to mean reversion. the numerical examples that we developed in Chapter 5 provide some guidance. Additional numerical results, for the oil reserve example of this section, were obtained by Wey (1993). As one would expect, the extent to which the answers change depends on the rate of mean reversion and the specisc level to which the process is reverting. Consider the following mean-reverting process, which we examined in Chapter 5, for the value of a developed oil reserve:
Then the partial differential equation reserve, F 1z',/), becomes
(6) for
the value of the undeveloped
lf the time until relinquishment is Iong enough (morethan Iiveyears), we can ignore the time dependencc of F(F, t), so that the term & in equation (12) disappears and we have an ordinary differential equation identical to the one in Section 5(a) ofchapters. Aswe saw there, the solution can be written using the contluent hypergeometric function,which has aseries representation. One can use this solution. with diffcrent values for q and V, to determine the extent to which mean reversion matters for the valuation of the undeveloped reserve and for the optimal development rule. -
Wey (1993)has shown. using a loo-year series for the real price of crude oil, that a reasonable estimate for ?) is about 0.3, and that using this value (and which mean reversion matters depends on a value of ch of 0.20), the extent to
7see Pindyck and Rubinfeld ( 1991) for an explanation of these tests and their limitations. That book provides an example (on pages 462-465) in which unit root tests are performed tbr the prices of crude oil, copper. and Iumber. The tests fail to reject a random real (inllation-adjusted) walk when 30 years of data arc used. but reject a random walk (in favor of a mean-reverting process) for crude oil and copper when uscd tbr data series covering a l l7-year period.
4()5
vzlpfp/itzfib/i-j-(lz1JEnkpirical Researclt
the value to which )' reverts, V, relative to the development cost. D. As one would expect. if V is much larger than D, accounting tbr mean reversion gives when V < Dbecause )' is expected a largervalue forthe undeveloped reserve about is twice as large as D, ignoring if that F time. Wey shows to rise over reversion can Iead one to undervalue the reserve by 40 percent or more. mean
On the other hand. if V is about as large as D, ignoring mean reversion will matter very little. Paddock, Siegel, and Smith's (1988)original study was intended to illustrate the optionlike nature of an undeveloped oil reserve, and show that the use of standard NPV methods would lead to a substantial undjrvalua-y tion of the reserve as well as premature development. That basic result still 'j holds whether or not one accounts for mean reversion. Of course, tbr an oil valuation company about to bid $500million for offshore Ieases, a lo-percent worthwhile would do be to a careful error adds up to a Iot of money, so it characteristics reversion of the other and any analysisthat accounts for mean relevant. believed be value to process that are price or developed resen
Electric Utilities' Compliance with the Clean Air Act Under the Clean Air Act Amendments of 1990, electric utilities in the Unitcd States are restricted in their emissions of SOz. However. they can comply with the regulations in different ways. First, they can purchase tradeable emission ailowances from other utilities. and then. with these allowances. continue to emit as much SOa as they did before. Alternatively, they can avoid the cost of buying allowances by sufhciently reducing their SO2 emissions. This, too, can be done in different ways. One possibilil is to pay a sunk cost of retrolitting the plant and then switching from high-sulfur to (moreexpensive) low-sulfur coal. The second possibility is to make a (sunk)investment in emitted-B that is, devices that remove the SOa before it is over time. Each alternative involves costs that can change unpredictably In particular. the emission aliowances, which are traded on spot and futures markets,g will fluctuate in value as electricity demand Quctuates, as technologies arrive for removing sulfur more chtaply and effectively, 'scrubbers,''
new
Sscrubbers.
technically rcferrcd to as flue gas dcsulfurizers. are veasels in which the SO? before going up the
released from burning coal reacts with an alkali and is thereby smokestack.
t:captured''
%Atthe time of this writing, futures trading of emission allowanccs had not yet begun but
was planned for the near future.
,,
l-lellsiots
t??lt/
.Ai..1illictltiolls
and as government regulations themselves change. Likewise, tlle cost differential between Iow-stllfur and high-sulfur coal lluctuates over time as supply and demand conditions in coal markets change. Since switching to Iow-sulfur coal and investing in scrtlbbers both involve stlnk costs. the utility faces an irreversible investment problem of the sort that we are now well equipped to solve. Herbelot (1992)recently examined tliis problem in detail, using the options approach. He showed that because of the oexibility they provide, purchasing emission allowances may be the preferred alternative even if the present value of expected compliance cqsts is higher with allowances than it is by installing scrubbers or switching to low-sulfur fuel. He also showed how one can calculate the values of the options to install scrubbers or switch fuels, and how these option values reduce the utility's trtle compliance cost. In this section we use Herbelotfs example to illustrate the solution to this investment
problem.
Under Phase 11of the Clean Air Act Amendments, utilities will be able to emit only l Ibs of SO2 per million BTUS (MMBTU) of ftlel burnt-l'' A utility that is able to emit less than 1.2 lbs of SO2 per MMBTU can sell allowanccs for the difference to other utilities. A utility that emits more than l Ibs t)f SOa per MMBTU must bkly allowances to cover the difference- Of course the utility can reduce its emissions (insteadof buying allowances) by installing scrubbers or switching fuels. Tle utility must determine when to optimally install scrubbers or switch fuels. and must determine the present value of its expected costs of compliance, taking into account these options.
(). l 2. c:lse p:t rtt rne ttl r va Iutls lt re (qol):tl1 klI1I1 tl :t l bas is) : (';' ./ retlso n Ie tltility ().()5, and ().8. alltnvance 4Eatrll permits a to p o .r1) 0. l t..1 hatl not yet begull. but it emit one ton of SO2 In l992. trading in lllltlwances was expected in the industry that they would be worth as mtlch as $5()()each. Wllen trading began in l993. prices were bclow $2()().As lbr the coal price premitlm D. it has averaged about $().45per MMBTU during l 992 and early 1993.
llb
--bClst't
=
-'
=
=
=
=
.
'We will assume that high-sulfu r coal emi ts a-// 3.3 lbs ol' SO2 per I dmits SOa.1 and of coal MMBTU, low-sulfur Nvewill consider l lbs x/. ttxpected existing coal, is to Iast for anpower plant that burns high-sulfur an other 20 years, has a capacity of 536,000 kilowattss has :1 capacity factor of 80 percent, and a heat rate of 0.00898 MMBTU per kilowatt-hour of electricity. Hence the total number of MMBTUS of energy that the plant is expectcd to each burn year is given by =
.0
=
B
=
(536. ()()())(4).80)(365) (24) (().00898)
=
33.7
l ()t' MM B'Tu/year.
x
( I 5)
.2
.2
2.A
The Model
First. suppose this utility had no option to switch to low-stll rurcoal or to install scrubbers. Then its annual cost of complying with the allowed I lbs 1 35,385..1/ per year, 1: of SOa per MMBTU would be S((-r,/ litk ()g the and the expected present value of this tlowof cst (lver the zll-year would be plant .2
.2)/2()()())
We will consider two stochastic state variables. thtt price of an allowance.
measured in dollars per ton, and the price premium per MMBTU of lowsulfur versus high-sulfur coal- D. We will assume that these variables lbllow (correlated) geometric Brownian motions: CIA = azj zl dt + r.r.,1 Jzal, ad
( l 3)
=
-
case-' cost of compliance, and we will compare it to the cost the utility has the options to switch fuels or install scrubbers. when switches to Iow-sulfur coal, it will have t() pay a one-time sunk utility lf the retrofitting cost of $25 per kilowatt of capacity. which, for a 536,0()()-kilowatt This is a
,4,
.4/
itbasc
plant, amounts to a sunk cost of &sw $ 13,400.()00. In addition. it will have to pay on an ongoing basis a coal price premium of D per MMBTU. as well as an increase in nonfuel operating costs of $0.50per MMBTU. However. each allowances, and in addition year it will save having to buy B (-x.// l will able sell of allowances. B (1 be )/2000 these it to so the cost saving will be B (-r// dollars. per year =
.2)/200()
tD
tlt + co D tzp. ao D
=
(14)
with f'tzxzo) p J/. Let ;LA and y.o be the risk-adjusted expected returns allowances and the price premium, respectively, and let the return on i P.A falls'' be denoted by azl and Js * p.n ao. Herbelot shows that =
tshort-
fszf
I'lunder
-
-
.2
-v/-
-
.v/.)/2()()()
,4
-
-
Phase 1. utilities can emit 2.5 Ibs ()f SO2per MM BTU. Phase 11 goes intl) effcct in the year 20()1),but lbr simplicity we will assume it is in elect ntlw.
ll'There are a variety ()1' different grades of Iow-sulfur coal with different sulfur contents, but this is a reasonable approximation. 17Pollution is measurcd in 1bs per MMBTU. and the cost of alltlwanccs is in dollars per t()n.
The 2()()0in thc dcnominator cunverts pounds to tons.
Etolsions
408
and Applications
If the utility installs a scrubber, it will have to pay a one-time sunk cost of $200 per kilowatt of capacity, which. for its 536,000-kilowatt plant, amounts to scrubber requires energy a sunk cost of Kscr $107,200,000.In addition, the is0.73cents per kilowatt-hour and maintenance to operate.rrhe operatingcost the scrubber of annual total cost for operating of electricity, which implies a Scrubbers are above. $27,421,000.The flow cost saving can be calculated as saving 0.9 the is B and cost 0.1 90 percent effective. so =
.vs/2000
,1
-r1.
-7//
=
dollars per year.
Although in principle a utility could switch fuels more than once. in practice this is unlikely, so we will consider only the possibility of a single switch from high-sulfur to low-sulfur coal. Also, once the utility has switched to lowsulfur coal, there is no point to investing in scrubbers. As a result, we can proceed as follows. First, we determine the present value of the cost of compliance for a utility that is burning high-sulfur coal and has the option to switch to Iow-sulfur coal. Then. assuming that this option to switch fuels has not been exercised, we value the option to install scrubbers and determine the present value of the cost of compliance allowing for this option as well.
2.B
Applications
To value the option to switch fuels. we can assume that spanning
F (and The utility can switch to low-sulfur coal at any time /, where 0 .G t that the = the time is, switches payoff, utility present at F t, 20 years). If the value of cost savings from t to T, is T
B
.r / l
-'
L
-
2000
AI e
-
J d (r
bmWvqt
.
s'j2
,12
Substituting the values listed above for S, risk-free rate, r, equal 0.05, we get -1.1.(.5(
,1 ,
t
,
f)
=
E1
-
e
F -1 )
J-SW .4.1
.
-
+ (?-
-
(y2 + 12 D Dl &'w bs + p c
t0)
D
subject to the boundary
FSW
D
+ F* I
zj
ao zl
r F*
-
D 0
=
FSW ,4o
+
,
(r
-
j.j) .
.,1
F'SW
1
(js))
9
conditions #'SW(,4 ,
D.
r)
=
().
(2!) (22)
Note the similarity of equation ( l 9) to the partial differential equations that we examined in Chapter l0. Like those equations, equation ( l9) must be solved numerically.l3There is a critical boundary, z1*(D. tj, that must be found when as part of the solution, and the optimal investment rule is to switch fuels /)/)/ > 0. > and pz1*t pad*(D. /)/J? > this will In D 0 D. have case, we ,4 (If the prcmium for low-sulfur coal rises. the allowance price must increase to make switching economical. Likewise, as time passes and the remaining life of the plant falls. the allowance price must rise to justifythe capital cost (af switching fuels-) In equation (16)we found the present value of the utility's expected costs of compliance assuming that it could not switch fuels or install scrubbers. If value of the utility's option to switch fuels, that present value we include the at time t 0 becomes .,1*.
(
-
j
Jr
( 17)
* sw ( Dt
holds
(23) .'.'.:
=
Elnpirical fkc'fzwrc/l
and use contingent claims methods. The assumption of spanning is qtlite reasonable in this case because emission allowances are (or soon will be) traded on a futures'market, and coal is traded on a variet'y of informal fonvard markets. and its price is partly correlated with the prices of other fuels. By going through the usual steps, the reader should be able to show that the value of' Dt J), must satisty' the the option to switch fuels, which we denote by followingpartial differential equation'.
The Option to Switch Fuels
*SW
flzlt
-vs,
xu,
,4,
and
o. and letting the
1
(18)
=
(24) During 1992-1993, % was about $0.45 per MMBTU and a reasonable range *SW( Dt At /) is for A( was $200 to $500 per ton. For this range of values, money.'' of However, the negative, so the option to switch fuels is clearly tdin rise and D fall sufhciently, and we the money'' should it could become ,
'out
z4
want to determine
its value.
,
13l-lerbelot ( 1992) structures the problcm using the binomial method as a way of obtaining numerical solutions. An alternative method of solution is to derive the partial differcntial equation numcrically using a linite difference method. along the as we have done here, and then solve it lines discussed in the Appcndix to Chapter 11).
E'xtensiolls t'l/lt
4 1()
zz3pplicaiolls
Applicatiot's
Herbelot ( l 992) has calculated Fs%' and shown that for a reasonable rahge ()f parameter values it is quite small (at most a few million dollars). This is of the not surprising-, as we saw above, the option to switch tels is far money'' even for an allowance price. A, as low as $200.
tl?l(/
L-. ??1J?f'n'ctlJ /ktz.tztll-c/l
ttout
F'SC''
..l
(.z.1'(D- t ) D. / ) ,
=
zf mst'r ..I (. ( D t ) D / ) *
.
The Option to Install Scrubbers
2.C
Now consider the option to install scrubbers. The utility can install scrtlbbers at any time /, where 0 :G t .G W. If it installs scrubbers at time J, the payoff. that is, the present value of cost savings from t to F, is F
*
wr
T
0 9 Y /J .
B
=
-
zzlf
2000
l
-
.J
j
( r - / ) (j r
gg y; j ()(j(j jy -
-
,
-
rjr -
f
)
( 1. ()0l t)()0z.,
548. 42(). ()()()).
-
.
=
zz1
=
=
-*W(z'1
-$155
.
uout
'in
To determine the value of the option to install scrubbers. which we denote and the optimal exercise rule, we can proceed as above. As Iong as we by continue to account for the possibiiity of switching fuels, the net payoff from installing scrubbers depends on Dt, as well as At and t, and thus so does &cr. F%C'' r will satisfy a partial differential equation similar to equation As a result, M 0 : f &' ( 19 ) r &'Cr,
r ! JA2 W2 y.%c.. + !2 2 ,4
,1
+
(r
-
6
)
The boundary conditions
l
-
jp
.
.
j
j p .
.
jj
.
r. and shows that Herbelot obtains numerical solutions lbr F%<''' it is nbout million the 15 $ for base case parameter values. This may seem small. but remember that the option is well of the money.'' Once wtt account tbr this option value, as well as the value of the option to switch ftlels- the utility's expected cost of compliance at J = 0 becomes tout
t/ r
Note that scrubbers entail a maintenance cost of $27,421.000each year which is certain and hence is discountcd at the risk-free rate that must be paid annually. If the utility installed scrubberss it would also have to pay a capital cost Kscr $107,200,000and would give up the option to switch fucls. 0, F 20, and If t $500, the net cost savings would be negative; it is D. t ), million (lessthe value of the fuel switching option, about which is fairly smalll. Like the option to switch fuels, the option to install of the money'' for the range of allowance prices we have scrubbers is the money'' if the allowance price rises considered, and only becomes $655. above =
yw (
kj +
t
e-0.05tr-')j
= (1
e
.
*
sw
2
(To
)D
DD + sscr
CF F*3 D +
p
F'SCV I
/7'.,1
-
O'D zj
r
sscr o
p
Ft*C
.j.
.1
.
=
are also similar'. PCTCA D, F) .
F'SG
(() D t )
=
0, 0.
0
'
(r
-
ja
) zj Fscr ..I
Herbelot also shows how this prcsent value and its component depend on the various parameters of the problem. 2.D
option values
An Exercise for the Reader
We have said that the value of the option to switch fuels is quite small. so it would be a reasonable approximation to ignllre the possibility of fuel switching. In addition, the option values and critical investment thresholds for a plant with a Zl-year life are not very diterent from those for a plant with a l 5or 25-year Iife, so we could also simplify the problem by ignoring time. If we eliminate Dt and l as variables. the partial differential equation (26) for the value of the scrubber option becomes an ordinary differential equation that can be solved analytically. As an exercise. the reader can make these simplifications and then calculate the value of the option to install scrubbers and the critical threshold /I* tlat should trigger installation. The reader should also examine how this option value and critical threshold var.y as parameters such J.,I are varied. as c,.I and
2.E
Enerr
Conservation by Homeowners
Residential energy conservation became an important issue in the l 97()s, and U.S. income tax Iaw from 1978 to 1985 offered credits for investments that would promote this aim, for example, better attic insulation, double glazing, and some structural alterations. Empirical studies found that consumers' response to such policies war extremely low. If consumers were making their decisionson the basis of conventional net present value calculations, they must have been using very high discount rates. Various explanations, for example,
Extensiols
fuel price distortions Pa radox.''
fznt
Applications
or lack ot information. were offered for this
uenergy
The options approach suggepts a different explanation. Most of these investments are irreversible, and the price of heating fuels (andtheretbre randomly through time. Therefore the savings from the investment) ouctuate wait until the return to their investment is sufrationally should consumers ciently above their opportunity cost of funds. Hassett and Metcalf ( 1992) hnd support for this explanation. The theory of investment is basically the model of Chapter 5 above. They consider the effect of a ls-percent tax credit. For the parameters of the fuel price process and the conservation technology, they 5nd an option value multiple of 4.23. Thus the current rate of return must be over four times the cost of funds before a homeowner will make the energy-saving investment. They allow heterogeneity among households. so the responses to policy and price changes are distributed through time. They calculate the mean increase in investment by simulating alternative realizations of price paths in a series of 1000 replications. If uncertainty in the price process is ignored and the traditional Marshallian criterion is employed, the eftkct of the tax credit is dramatic. Investment increases from about 27 percent of households to 43 percent immediately, and from 4() to 60 percent after live years. However, when uncertainty about future fuel prices is taken 0.()93). the credit is almost ineffective. Without the into account (witht'z credit, very Iittle fnvestment is made because the price rarely hits the high threshold for action under uncertainty. In typical simulations, after 2() years only 5 percent of households adopt the energy-saving technology. However, the t:tx credit does not improve matters much. Compared to the no-policy baseline. investment incrcases by only 0.2 percent immediately, and by Iess than 3 percent after 2() years. The tindings not only support the options approach, but also suggest that tax credits are not an effective policy to achieve =
Applicfltions (l?1JE.pupirltl/
an environmental policy can also involve Iost option value. but in this case there are two kinds of irreversibilities, and they work in opposite directions.
First, policies aimed at reducing ecological damage impose costs on society. For example, such policies might force coal-burning utilities to install scrubbers. or lirms to scrap existing machines and invest in more fuel-efficicnt ones. This creates an opportunity cost of adopting a policy now, rather than waiting for more intbrmation about ecological impacts and their economic consequences. This opportunity cost biases traditional cost-beneht analysis t favor of policy adoption. As with irreversible investment decisions, the sunk costs associated with policy adoption can make it preferable to wait rather than adopt the policy now. Second, environmental damage can be partially or totally irreversible. F()r example, increases in greenhouse gas (GHG) concentrations are long-lasting. Even if radical policies were adopted in the future to drastically reduce GHG emissions, these concentrations (whichhave a natural decay rate ofabout onehalf percent per year) would take many years to fall. In addition, the damage to various ecosystems from higher global temperatures (or from acid itied lakes and streams, or from the clear-cutting of forests) can be permanent. This means that adopting 11 policy now rather than waiting has a sltttk /7tz?lcJit. that is, a negative opportunity cost. This negativc opportunity cost biases traditional cost-benefit analysis agtlinst policy adoption. Hencc it may bd desrable to adopt a policy now, even though the traditional analysis declares it -l/?pl:
uneconomical. We will illustrate this interaction of uncertainty and irrcversibility using simple model. based on Pindyck ( 1993e). which in turn builds on Nordhaus a (1991). Let NI be the stock of an environmental pollutant example, at-
mosphcric GHG concentration, emission t)f the pollutant. Then
energy conselwation.
div/tlt
The Timing of Environmental
Policy
policy design As explained at the beginning of this chapter, environmental with much policy. design of irreversible investment the in an has common When a firm makes an irreversible investment expenditure, it gives up the possibility of waiting for new information that might affect the desirability or timing of the expenditure. and this lost option value is an opportunity cost that must be included as part of the cost of the investment. The adoption of
or the acidity
tfor
a Iake). and E the rate of that M evolves as we can assumtt =
y E t)
-
()f
tVll )
.
where is the rate of natural decay of 54. (Thus the smaller is the more irreversible are the effects of emissions-) We will represent the flow of social cost (that is. negative benefit) associatcd with the stock of pollutant Mt by a function Bi'lt 0t ), where ot is a variable that shifts over time, perhaps stochastically, to reflect changes in tastes and technologies-l'l For simplicity, .
3
Besetlrch
..
.
14F()r cxample, if ib1 is the GIIG concttntration. shihs in 0 might retlect new agricultural techniques tllat rcducu the cost ()1* a higher &I. or altcrnatively, Jemographic cbanges that raise lhe ctlst.
lLt'/tr/g-j'tp/?- z'zal
we wi lI ma ke
B ( 54t
.
.
-(
=
t???t/
Elllpiri'tll
A.$
,
will assume that 0: follows a geometric Brownian motion'.
d0
IU'V
We will assume that until the policy is adopted, the rate of emissions Et stays at the constant level E Once the policy is adopted, Et falls immediately to adopting zero, where it remains. Finally, wc will assume that the social cost of the time of adoption the policy is completely sunk, and its present value at is K 15 The problem is to tind a rule for policy adoption that maximizes the net present value function'. .
=
.
We solve this problem using dynamic programming by delining the net l#'N(p. M) dcnote the value present value function foreach oftwo regions. Ltlt and region E let l4'zl 0. M) denote function for the ), (sothat Et which region 0). As the reader Et the value function for the (in 'no-adopt''
=
S'adopt''
(as an exercise),
confirm
can e q u a t io n :
=
N
rlzr
0.
,/)
must satisl
the following Bellman
(36) Likewise,
I'F''I
0. )z/) must satisfy the Bellman equation
*
(p
*
(f?
.w)
zl pp
=
*
*
*
.
0 A'J
-
?' +
(,-
and J.P'''ltl* Az/)
=
lz E f? a )( 1- + -
-
a
-
-
ajlow
of sunk costs over time
(42)
.
..y
j.
(y
.
where is a positivo ctlnstant to btl detcrnlined, antl. fnlnl btltlndllr.y ctlndition (38), p is the positive root of the quadratic eqtlCttitln J. (T- p(p I ) +f.z p ()a 2 that iss -r
-
p I''P?V
l =
-
a -
2
a
a
-
o
2
expenditures
for insulation on all new homcs). AIl that matters is that adopting the policy implies a commitment to this llow of costs, so that we can replace the ilow with its present valuc at the time of adoption. Also, if B( /V/, p,) is linear as in equation (33). it is always optimal to redtlce E to zero if is optimal to reduce it at all. .
,
=
.
-1
+
-
2
c
2
l
t.e
+
,
>
G-
the present value l'tlnctilln tllat applies betbre adoptitln
()f
tlle pol icy, Ilas
three components. The first tlrm ()n the right-lland siclkl flf (4 1) is tlle vltlue of the option to adopt the policy at some time in thc l'uture. Tlle secllnd term is the present value of the tlow of social cost resulting frtlm the current stock of pollutant, 54. The third tdrm is the presen t valtle ()1* the l1()w ()f social cost that would result if emissions continued at the rate E tbrever. (Note that this social cost is reduced by the value of the option to reduce emissions, that is, the first term-) Once the policy has been adopted- E t) and 1.P'.,1applies. =
Then the only social cost is from the ctlrrent stock of pollutant. There are still two unknowns. the constant and the critical value 0* at which the policy should be adopted. and they arc determincd from boundary .,d
( for example.
)
z4
.j/'-I
=
o.
(.(,. -
K
.
j-
( ( j j(?'
I -'1 p K ) -
entail
fx
p /1../
-
conditions (39)and (4()): 15Note that the policy might
b(.')tlntilt I'y
&-
-
I-y;'1(f? tbl',
54)
llltlNvil'lg
of 0 at or ttbove Nvhich the ptllicy should be at :1 solution and then tltlte rluintl wllether lirm that the sol ut ions for l UX ltntl 1.1/-'.1 arc
--guess''
CP-'V
l.I.z'''1(p /kJ)
544
where 0 is the critical valtle adopted. In this case we can it works. The reader can con given by
(35) subject to equation (32).Here. ? is the (ingeneral, unknown) time that the policy is adopted, 17()as usual denotes the expectation based on information at time t 0, and r is the discount rate. This is an optimal stopping problem, that is. we must find tht! optimal point at which to commit to spending K to reduce Et to zero. given the (possbly stochastic) dependence of Mt on Et, and given the stochastic evolution of ot
o
I'F;P'V
(34)
0 J/ + c 0 Jz.
=
J?fJ.$.tvl/%-/?
''Flese equations nlust be solved sul7j ect to the condi tions: r'F''V (() A-/) ()
ot) linear: B ( ivl't ( )
and we
Appli'atiol
zlpplications
p
-
-
l
v+ ;.
.)(,.
(x
)(?' +
-
-
y
cvlp.j
-
('
)p
-j?l ,
j .
Extensiolls and Applications We can now see how the optimai timing of policy adoption depends on the deuncertainty twer future costs and benefits, as well as other parameters. gree of First, note that an increase in f.r implies a decrease in p and hence an increase in p*. As we would expect, the greater the uncertainty over the future social cost of the pollutant, the greater the incentive to wait rather than adopt the policy now. and hence the greater must be the current cost in order to trigger adoption. Second. an increase in the discount rate r increases the value of the option to adopt the policy and thus also increases 0*. This is analogous to the effect of a change in the interest rate on the value and optimal exercise point for a hnancial option', an increase in 1. implies a reduction in the present value of the cost K of policy adoption. so that the option to adopt is worth more but it should be exercised Iater. Third, an increase in the rate of ation'' of the stock of pollutant. also increases ?*; a higher implies that the environmental damage from emissons fs Iess rreversible. so that the sunk beneht of adopting the policy now rather than waiting is reduced. Finally. an increase in the initial rate of emissions E reduces t?*;the reason is that if E is higher, the social cost of waiting is higher, and since the cost K of adopting the policy is hxed, it pays to adopt earlier. It is probably unreaiistic. however, to assume that the cost of reducing emissions from E to zcro is independent of E. The cost K should certainly be approximaan increasing function of E and probably convex in E As a first E'. tion, however, we can assume that K is proportional to iedepreci-
,
Applications
ll/lrcf?/
and
Iesetlrcll
7.681.()()(). and 0* $42 per ton. Hence at the current value of zl (b 20, the policy should not be adopted, but the value of society's option to adopt it, 0#, is $0.64billion. The policy should only be adopted when 011 $2. 1 billion, and the reader can check 0 reaches $42 per ton; then that boundary conditions (39)and (40)are satistied. We leave it to the 0.4, and show that it reader to go through the calculations for, say, o' =
=
=
,,1
zzl
=
=
Ieads to a larger critical value 0*. 0 Figure 12.2 shows the solution graphically for 51 0 (sothat lF''1 WN for all values of (?). Note that 0* is found at the point of tangency of with the line 1p..1 K. (If M were greater than zero. we would have 1.P.1 X WN tz), so K (t?)and the line IF we would simply rotate both -9 A'//(r + downwards.) Figure 12.3 shows how the solution in Figure 12.2 changes =
-
-
-
A0
WN
.
.
=
=
NA
In this case, 0* becomes
0*
8*
kr
-
=
a)
(r +
-
a4p '
Then, 0. is independent of E. Also, in this case, we would never want to we would want reduce emissions by anything Iess than l00 percent (assuming value of the a1l). the reduce is option them that to adopt The reason at to WN and I#r'' A0#, E, M that of in and linear is indcpendent are the policy, so linear in M and E. example will illustrate this calculation and indicate the A numerical 0 (so that the expected social cost per magnitudes involved. Suppose a 1, E of 100.000 0.20, p' /W constant), 0.02. o. is r unit = and 1.62, K billion. Then p $2 tons per year. p() $20 per ton.
0
l I j I NA
'?E'O
=
=
=
=
=
=
=
-
(r
=
Figure 12.2.
#Jr sb/u/iflll
tV = 0
-
a)(r
+
?1 -
tx)
K
418
Erfln.lkpn. and Applications
GI
(y2
4
> gj
Z z
z A*
2
V
A
kvq(yjl
z
A
A'
w-
M
w
A'
NN(jy 2
0* j
N N N
N
...
*** ..-
I $ I l
0z *
.ee
,.e
j
/
0
A
,,e .e.
I l
,ee 'e ..-
NA
-
K
'y80 (r Figure IZ.J.
Eflct of
fncrefzJing
(r
fJ?1
fl?lt/ Etltpirical Set-wrc/l zd/lp/ctrlll't7/?'
-
a)(r
+
8
-
(x)
0.
Explaining Aggregate lnvestment
Behavior
The explanation of aggregate and sectoral investment spending has bcen one of the Iess successtkl endeavors in empirical ecllnomics. For the most part, econometric models kave not been vely useful for explaining or prcdicting investment spending. The problem is not just that these models have betln unable to explain and predict more than a small portion of the movemnts of investment. In addition, constructed quantities that in theory should have strong explanatory power such as Tobin's q, or various measures of the cost of capital in practice do not.lt: The irreversibility of most investment expenditures may help to explain neoclassical investment theory has failed to provide good empirical modwhy els of investment behavior. Eftcts of risk are typically handled by assuming that a risk premium (obtained.say, from the CAPM) can be added to the discount rate used to calculate thc present value of a project. However. as we have seen throughout this book the correct discount rate cannot be obtained without actually solving the option valuation problem. the discount rate need not be constant over time. and it need not equal the firm's average cost of capital. As a result. simple cost of capital measures may be poor explanators of investment spending.
4.A
Models Based on Tobin's q
The problems that arise when irreversibility is ignored can be seen in the context of models based on Tobin's q. A good example is the model of Abel and Blanchard ( 1986), which is one of the more sophisticated attempts to framework; it uses a carefully constructed explain investment in a marginal rather than for average q, incorporates delivery lags and measure costs of adjustment, and explicitly models expectations of future values of explanatory variables, However, it does not treat irreversibility and option values. We will briefly review its structure. ttinvest in the marginal The model is based on the standard NPV rule, value of the expected llow of prohts unit of capital if the present discounted resulting from the unit is at ieast equal to the cost of the unit.'' Let z ( Sf It ) be the maximum value of profits at time t, given the capital stock Kt and investment level lt, that is, it is the value of prohts assuming that variable factors are used optimally. It depend on 1: because of costs of adjustment; -theor.y
when c is increased. Note that p falls to a number closer to 1, so that zl 0# becomes less convex, and WN0j Nattens out so that 0* becomes larger. This model is very simple, and is only intended to illustrate how option value can arise in publicpolicyproblems, aswell as srms'investment problems. To make the model more realistic, we want to specify Blivlt, @) as a convex function of Mt, and perhaps the cost of reducing Et, kLE), as a convex function of E. Then it will no longer be the case that if it is optimal to reduce E at all, it is optimal to reduce it to zero, andwe could consider the optimal amount assuming that further reductions could be made in the of reduction, future should ot increase sufsciently. The problem then becomes exactly analogous to the incremental investment problems that we examined in Chapter 11.
,
Ibpbr ovcrviews and comparisons of traditional approaches tt) the ccontlmetric modclling Spcnding, investment t)f sec Chirinko ( 1991) and Kopckc ( 1985, 1993).
420
Extensions and Applications
lzr/p/ < 0, and Llln.l 11 < 0, that is, te more rapidly new capital is purchased and installed, the more costly it is. (See Chapter 11 for a discussion of adjustment costs.) Then the present value of current and future protits is given by 16
=
x
j
./=()
1'=0
Ct
where p is the discount
.
lh-v.j
( Kt+J..
/)+J.)
.
rate. Maximizing this with respect to lt subject to ,
Kt
(where
p,+,.)-1
(1 +
(1
=
-
) Kt
-
l
+ lt
.
is the depreciation rate), gives the following marginal condition:
- t where
qt =
t-t
/
co ./=0
tpa/p ltt
(1 + i
=
qt
n't.v'/
gh+f)-l
pKt..j
=()
(49)
,
(1
-
)
j .
Recall our discussions of f? in Chapter l l What we have here is marginal ; it pertains to the contribution of the marginal unit of capital being installed. However, it does not consider the effect of this unit on the whole value of the hrm. Only the present value of the profit tlows contributed by this unit is computed, and the opportunity cost of exercising the option to make this investment is ignored. Also, it is not expressed as a ratio to the purchase cost. Thus in the terminology used in Chapter 11. this is the marginal, value-ofassets-in-place, absolute concept of Tobin's q. Equation (49)says that investment occurs up to the point where the cost of an additional unit of capital equals the present value of the expected flow of incremental prosts resulting from the unit. Abel and Blanchard estimate both linear and quadratic approximations to qt, and use vector autoregressive representations of pt and Llnbtjk to model expectations of future values. Their conclusion is simple and stark: ttour data are not sympathetic to the basic restrictions imposed by the q theory, even extended to allow for simple delivery lags-'' T'hroughout this book we have emphasized what amounts to a very different extension of the q theory, namely, one that recognizes the opportunity cost of exercising an option to invest. Abel and Blanchard's data for pt are based on a weighted average of the rates of return on equity and debt. However, in our approach the discount rates pt need not equal the average cost of capital. lnstead, they can only be determined as part of the solution to the .
Applicatiotls
/:11#
Empirical Sc-alrc/?
421
firm'soptimal investment problem, and this involvesvaluing the firm's options marginal investments now or in the future. Thus the to make (irreversible) solution to the investment problem is more complicated than the hrst-order condition given by equations (49) and (50). For example. consider a project that has zero systematic risk. The use of a risk-free rate for p would lead to much too large a value for qt, and might suggest that an investment expenditure should be made wken in fact it should be delayed. Furthermore, there is no simple way to adjtlst p properly. The problem is that the calculation ignores the opportunity cost of exercising the option to invest.
Untbrtunately, incorporating irreversibility into econometric models of aggregate investment spending is not simple. As we have seen. the equations describing Optimal investment decisions are nonlinear. even for simple cases. Also, it can be difficult to measure (or statistically identify) the variables or parameters that retlect key components of risk. Additional problems arise if uncertainty on investwe want to explain the long-run equilibrium effects of uncertainty should spending. irreversibility and that Althoug know we ment raise the threshold (for example, the expected rate of return on a project) required for a firm to invest, we can say very Iittle about the effects of uncertainty on the firm's Ionpntn average rate of investment or average capital stock without making rcstrictive functional or parametric assumptions. We will say more about this below. but it means that tests cannot be based on simple equilibrium relationships between rates of investment and measures of risk. whether for hrms. industries. or countries. Finally, there are serious problems of aggregation. Our theory of optimal investment decisions applies most directly to a tirmor similar decisionmaking unit. Different hrms can Lave different technologies or managerial abilities, shocks.Then theywill have and they may be subject to differcnt (idiosyncratic) historical accidents thresholds. action Moreover, may leave different different relative with situated capital stocks differently of that to their action hrms are hrst develop must industry aggregates, If pertain data we thresholds. our to the theory of industry level before we implications at can test them. the our micro decision analysis the level the wish conduct empirical of the at If we to all specilications and that these identify differences. need data can units, we AlI of this is a very difscult task. In the next section we will review some work that takes different approaches to it. 4.B
Empirical Implications
of Option Based Models
The models developed in this book suggest a much greater role for uncertainty as a determinant of investment than do traditional models. We saw in
422
b-tellsiolls f7.-
zklll'-tllkteltiolts
f???t/
Chapters 8 and 9 tllat althotlgh the mechanisms are different. fllr both a competitive industry and for a monopolist firm. tlncertainty affccts invcstment by affecting a critical threshold that triggers investment. That critical threshold can apply to the value of :1 completed project, to the price taf the project's output, or, in the case of incremental investment. to the marginal profitability of capital. Through a variety of models, we saw that this threshold can be very sensitive to the volatility of the underlying stochastic state variable. These modcls
must be developed further to draw
ttlt
their empirically
testable implications. First, it is important to emphasize that the models do not describe investment per se, but rather the critical threshold required to trigger investment. In Chapter l 1, for example. investment is implicitly defined as a stochastic process that will control the value of the marginal product of capital at a certain barricr. The modcls tell us that if volatility increases. the threshold increases. Only to the extent that we can also describe (ormake assumptions about) the distribution across Iirms of the values of potential projects, or of the marginal pro:tability of capital, can we also derive a structural model that relates volatility to actual investment. Without going this far, we can draw some intrences about how investment should respond in the short run to changes in volatility and other parameters. For example. a one-time increase in volatility should reduce investment at least temporarily, as project values that werc abtwe or close to what was a lower critical threshold are now below :1 higher one. Also- we have seen that holding everything else equal, an increase in the drift of, say, the output price lowers the critical threshold, and hence should be accompanied by an increase in investment. again in the short run. In the context of incremental investment at the Iirm or competitive industry Ievel. increases in the volatility of the marginal prolitability of capital, or decreases in its average growth rate, should lead to at least a temporary decrease in investment. There is vely little that can be said, however, about the effects of uncertainty on the long-run equilibrium values of investmcnt, the nvestment-tooutput ratio, or the capital-output ratio. To see this. note that although we know that an increase in volatility raises the ra/ldrct/ retttrn needed to trigger investment. we do not know what it will do to the average realized retttrn. The reason is that the firm requires a higher return to invest when volatility is higher. but it does so exctly because it is more Iikely to encounter periods of vely low returns (whenit will find itselt' holding more capital than it needs). The same indeterminacy applies to the investment-to-output ratio, J/ Q. ln long-run equilibrium, we have 1/ Q LK Px./ Q(Kj P ( Pc/ #) ( K/ Q S)). where is the depreciation rate. If the volatility of the marginal revenue product of capital increases, the required return increases, and =
=
,
investnpent falls for :tI)y given set t)f prices. so tha t the price of otlt ptit 13 rises and P./ /7 falls. Stlppose the protltlction technology is Cobb-Dtltlglas with constant rdturns. Then &/ V) 1/ Ld&--'' rises. These tqvoebcts .)(
-zl
=
Nvork
i1-1opposi te di rections. so wut artt
tlnilbld
to conclude wllat will hagpen to
// Q. Another Nvay to stle this is to note that. as before- :tn increase in velatility restllts in :1 higher threshold bu t also :1 greate r frequency in svh ich th (.l firn1 holds Inore capital than it nlltlds, so thrlt thtl productivity of capital cotlld 1/1111 on average, that is, J/ Q could rise. Hence we cannot ciaim ()n theklretical grotlnds that countries with more volatile or more unstable economies sllotlld have, on average. lower ratios of investment to gross domestic proudct (G D P) or lower capital-output ratios than countries with more stable economies. I n very specific n'lodels- for example the Cobb-Dotlglas Gtse of Chapter l l we do get clear results. but then their robtlstness is subject to question. .
Empirical
E vidence
Given the difhculties outlined abovep it is not surprising that there have been few attempts to statistically test the theor,y of irreversible investment under uncertainty. Here we brielly summarize some t)f the Iimited work tat has been done to date. One approach is to focus ()n the threshold that triggers invcstmcnt. and whether it depends on meastlres of risk in ways that thc theory prcdicts. see This has the advantage that the rellttionship between the threshold and risk is much easier to pin down than thc relationship between invcstment and risk. The disadvantage is that thtl tllrdshold cannot be observed directly. This approach was used in :1 recent study by Caballero and Pindyck ( l992) of U.S. manufacturing industries. Caballero and Pindyck dcrived an cxpression tbr the required return needed to trigger investment in a competitive market with free entry. constant returns to scale, and a Cobb-Douglas production tcchnology.They then framed tests in terms of this required return. Althotlgh the required return cannot be observed directly.one can obtain a proxy.for it by for example. the using extreme values of the marginal protitability ofcapital maximumover some period of time, or an average of the values in the iighest decile or quintile. Caballero and Pindyck showed that for U.S. manufacturing data. such proxies induted show a positive dependence on the volatility of the marginal prohtability of capital. as the theofy predicts. Another approach is to examine how period-to-period movemcnts in volatility affect investment. Rindyck and Solimano (1993)did this for a cross section of 3() countries, calculating a 28-year time series for the marginal prolitability of capital for eltch country (again,assuming a Cobb-Douglas
Exellsils
and Applications
production technology with constant rdturns). They divided the sample up into three subperiods of ninc years each, calculated the mean and sample standard deviation of annual log changes in the marginal prolltability of capital, and the average investment-to-GDp ratio. for each subperiod and each countly. They then ran panel regressions to determine the dependence of the
investment-to-GDp ratio on this mean and standard deviation in each period. They found that the investment ratio indeed varies positively with the mean and negatively with the standard deviation, as the theory predicts.l7 These studies are quite limited in scope. More ambitious t'structural'' models of investment spending, and of other irreversible dccisions like spending on research and development. attempt to dstimate the optimality conditions of the full stochastic dynamic programming problem, using methodof-moments estimators. The work of Rust ( 1987) is among the best known examples of this; pioneering attempts to use such methods in industrial organization include Pakes ( 1986). This is a promising approach, but still in early stages. Serious diffkulties of specihcation and computation are only gradually being resolved. Specific applications often run up against unobserved state variables, and the approach can accommodate only very limited kinds of persistence over time (serialcorrelation) among these. The equilibrium of interactions among different tirmsposes further difhculties. Computational requirements remain formidable, although improvements in that technology are so rapid that this is not likely to remain an obstacle for long. Discussion of these matters would need at least one chapter and probably a monograph of its own-Therefore we merely refer interested readers to the verygood surveys of the state of this art by Rust (1993) and Pakes (1993). Short offull structural models. but more advanced in its attempt to tackle aggregation, is the work of Bertola and Caballero ( 1990, 1992). The first of these papers considers the demand for consumer durables. These are, of course, investment decisions. Grossman and Laroque (1990)had pointed out that a consumer facing transaction costs of purchase and resale would optimallyfollow threshold rules for the decision to purchase or trade durables Iike houses or automobiles. In the economy as a whole, different consumers have different thresholds and different historically determined initial positions
17In related worka Pindyck f 1986. 19t)la) studied aggregate' U.S. invcstment using stock market data. on the grounds thatwhen product markctsbecome morevolatile, sock pricesshould also become more volatile. This was indeed the case, lbr exarnple. during the recessions of 1975 and 1980, and most dramaticallyduring thc Great Depression. He found that investment spending is negatively and significantly related to the variance of stock returns. even after accotlnting for thc effects of mean returns. Iong- and short-term interest ratcs. and the growth rate ol' GNI!
Applications
fznf.
E rlpfrfcl/ Researcll
425
relative to these thresholds. They are also subject to different (idiosyncratic) shocks. Aggregation smooths shocks aswell assome common (economy-wide) ofthe all-or-nothing of decision rules. but leaves the threshold aspect out some substantial serial correlation in the aggregate expenditures on durables. In fact their simulations have such realistic time-series properties that Andrew Caplin, discussing their paper, concluded-. *tBertola and Caballero have left us with the unusual problem of trying to rationalize a surprisingly successful empirical exercise.'' However, this is not a systematic econometric test of the real options hypothesis against specihed alternatives.l8 The gap between teory and empiricism in this area can be viewed in two ways. One would be a mixture of concern and skepticism concern that ahead witout has indeed it often does), and empirical support theory run (as validity. view and skepticism about its The is other opportunity hence as an real The theory has several implications that borne in the challenge. out are world, and throughout this book we have tried to spell them out at every available opportunity. We obtained numerical solutions for many of our models, using values for parameters such as the variance of the stochastic shocks that were representative of specific actual commodities or industries. We found qualitative implications, and several approximate quantitative ones, that conform to experience the high hurdle rates used by business hrms when they judge investment projects, the reiative ineffcacy of interest rate cuts as policies to stimulate investment, the signihcant and detrimental effect of poliu-y uncertainty on investment, and so on. Thus there is much prima facie evidence that attests to the merit and significance of the theory. Then more serious empirical testing becomes a research opportunity. We hope that our book will spread and improve the understanding of the theory, and indirectly contribute to the progress of research on the empirical side.
5
Guide to the Literature
The use of option pricing models to value offshore oil reselwes and determine when they should be developed was pioneered by Paddock, Siegel, and Smith (1988). Siegel, Smith, and Paddock (1987)provide a nontechnical exposition of their model and its application-The hrst section of this chapter draws heavily from these two articles. A predecessor is the study by Tourhinho (1979), 'dother good and promising empirical Lam
( 198t?).
studies
of consumer
durables are Eberly (199 l ) and
426 which Iirst showed that resotlrce reserves can be understoldtl itnd valtlt!d as options. Related sttldies incltlde Ekern ( l985), Stenslantl lllld Tjostlleiln ( l989, 1991). Jacoby and Latlghton ( l 992), and Lund ( l9t?2). Tll(2 book edited by Lund and
Oksendal( l 99 l ) contains
some general theory of real options
in
the context of natural resources, and some empirical sttldies 01' oil. Iisheries, etc. Also, Morck, Schwartz. and Stangeland ( 1989), tlse these techniques t() value forestry resourcess and Brennan and Schwartz ( 198j) tlsd them to valtle a copper mine and determine the rules for temporary closing or permanent abandonment. Applications have also appeared reccntly in tlle petroleum engineering literature (see.forexample. Lohrenz and Dickens ( 1993) and Stibolt and Lehman ( 1993)1 and on urban land valuation gseeQuigg ( 1993)).
When examining an electric utility's problem ofcomplyingwith the Clean Air Act, we did not discuss the numerical solution method. and we reviewed the characteristics of the solution and their dependence ()n various parame-
ters only brieoy. Interested readers can retkr to Herbelot's ( 1992) thesis for more detail. Also, a related study is Martzoukos and Teplitz-sembitzky ( l992), which examines the implications of uncertainty over the future demand for electricity for investments in new transmission lines. In Section 3 we examined
the timing of environmental
policy, and saw
how the irreversibility of environmental damage creates an option value that can bias a traditional cost-beneht analysis against policy adoption. This option value was first noted by Arrow and Fisher ( 1974). Httnry ( 1974asb), and Krutilla and Fisher ( 1975), and has been elaborated upon by Hanemann (1989), Fisher and Hanemann (1987),and Lund ( l99 l ). among others.lg Recent studies that are related to the analysis in Section 3 include Kolstad
(1992) and Hendricks (1992).Kolstad developed a three-period model with learning about the net benefits from a lower stock of pollutant. I-le showed that with no sunk cost of policy adoption, the faster the rate of Iearning,
b the lower the hrst-period emission Ievel should be. but if the costs of adoption policy are partly sunk, the effect of uncertainty is ambiguous. Also, Hendricks (1992)used a continuous-time model of global warming to study the timing of policies to irreversibly reduce emissions, allowing for a (partially)irreversible accumulation of the pollutant. He considered uncertainty over a parameter Iinking the global mean temperature to the atmospheric GHG concentration, allowed for learning by letting uncertainty
lg-l-hisconcept of option value shf'uld be distinguished frtlm that of Schmalensee ( l972), which is more like a risk prernium needed to compensatc risk-avcrse consumers because of uncertainty over future valuations of an environmental tmenity. For a discussion of this latter concept, see Plummer and Hartman ( l 98f)).
ovcr this paramdter ttll (lvt)r tinle. lll(.l sllovvetl htwv tlle speed of learning tffects the tinling ()1* plllicy. Finally, tklr :1 simple nlodel of G I-IG el'nissions and global wclrnling (tlle nlotlel ()n wllich tlttl I-lendricks study is btsed). see
Nordllatls ( l 99 l ). Retrent enlpirical rtlsutarcll 01) irrevtlrsible investnAent includes Bizer and Sichel ( l 988) and Caballcro ( l tlt) I b). Tlle Iattdr, along with Btlrtola alld Caballd ro ( I9t?(), I992) is impo rtan t in its development of aggregat ion vvhen Iirnl-specific and indtlstry-wide shocks coexist. Empirical work tlsing nlicro data, which ain-ls to estilnate the structural dynamic programnling nlotiels of individual choice. is stlrveyed by Rust ( l 993) and Pakes ( l 9t93).
Iteferences
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39-53. and Investment.'' In Handbook fJJMonetary 1990. and Frank Hahn. New York: Northeds. Benjamin Friedman E'cfpntplzlc', Holland. Present Value of Profits and and Olivier J. Blanchard. 1986. , CyclicalMovements in Investment--' Econometrica 54 (March): 249-273. Unihed Mode! of Investment Under and Janice C. Eberly. 1993. , Uncertainty.'' Unpublished working paper, The Wharton School, University of Pennsylvania. Februaly. Abramowitz, Milton, and lrene A. Stegun, eds. 1964.Handbook ofMathematical Fttnctions. Washington D.C.; National Bureau of Standards. ttconsumption
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'tl'he
2%
Aitchison. J., and J.A.C. Brown. 1957. The Lognomlal Disribution. Cambridge, England: Cambridge University Press. Uncertainty, Persistence Aizenman, Joshua, and Nancy Marion. 1991. and Growth.'' NBER Working Paper No. 3848, September. Akerlof, George A. 1970. tThe Market for Lemons: QualitativeUncertainty and the Market Mechanism.'' QttanerlylottmalofEconomics 84 (Nevember): 488-500. Markets Under UncerAppelbaum, Elie. and Chin Lim. 1985. tainty.'' Rand ltpltmfk/of Economics 16, Spring, 28-40. tpolicy
t
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Rejrolces
Arnold, Ludwig. 1974. Stocliastic Dljf'erential f#lItJJD?1'.' Tllelly J??( z'l/p/vcfltions. New York: John Wiley & Sons. Arrow, Kenneth J. 1968. Capital Policy with lrreversible lnvestment-'' ln Vtilte. Capital tpi Growth. Essays in Honor t#' Sir John Hicks, ed. James N. Wolfe. Edinburgh, Scotland: Edinburgh University Press. 1970. Essays in //:t? Theoty t#' Risk Bearing. Amsterdam: North. Holland. 7 and Anthony C. Fisher. 1974. tiEnvironmental Preservation, Uncertainty, and Irreversibility.'' Quarterly 1t?l/mfl/ofEconomics 88, 312-3 l9. Arthur, W Brian. 1986. Location Patterns and the Importance of History.'' Paper No. 84, Center for Economic Policy Research, Stanford University. Baldwin, Carliss Y. 1982. Sequential Investment when Capital is Not Readily Reversible.'' Jtrnal ofFinance 37 (June): 763-782. Baldwin, Richard, and Paul Irugman. 1989. 4persistentTrade Effectsof Large Exchange Rate Shocksa'' Quarterly of Economics l04 (November): 635-654. Bar-llan, Avner, and William C. Strange. 1992. Lags.'' Working Paper, Tel-Aviv University and University of British Columbia. Becker, Gary S. 1962. in Human Capital: A Theoretical Analysis.'' Journal ofpolitical Economy, 70, Supplement (October): 9-49. 1975. Human Capal, second edition. N.ew York: NBER and . Columbia University Press. 1980. leatise on the Familv. Cambridge, MA: Harvard University . Press. Beddington, John R., and Robert M. May. 1977. d'Harvesting Natural Populations in a Randomly Fluctuating Environment.'' Science 197, 463-465. Bentolila, Samuel, and Giuseppe Bertola. 1990. 'tFiring Costs and Labor Demand; How Bad is Eurosclerosis?'' Revicw of Economic Stttdies 57 (JuIy): 381-402. Bernanke, Ben S. 1983. Uncertainty, and Cyclical InvestJottrnal of Economics 98 (February): 85-106. ment.'' Quarterly Bernstein, Peter L. 1992. Capital Ideas: 'J77:Improbable OHgi?uof Modem FiI// Street. New York: Free Press. Bertola, Giuseppe. 1988. Adjustment Costs and Dynamic Factor Demands: lnvestmentandEmployment Under Uacdrltzlfy. PIA.D.Dissertation, Cambridge, MA: Massachusetts Institute of Technology. 1989. Investment.'' Unpublishedworkingpaper, Prince. boptimal
t'lndustry
'ioptimal
./t/amtz/
ttlnvestment
Glnvestment
.,d
'lrreversibility,
ttlrreversible
ton University.
Relrellccs and Ricardo J Caballero. 199(). tlinkcd
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1601-1626. Stensland, Gunnar, and Dag B. Tjgstheim. l989. ''Optimal Investments Using Empirical Dynamic Programming with Application to Natural Re62 (Januaryl: 99-120. sources.'' Journal (l/llf'ntf'l l99 l Decisions with Reduction of Uncer1 and tainty Over Time An Application to Oil Production.'' In Stocllastic Models and Option Izilfc', eds. D. Lund and B. Okscndal. New York: North-Holland. Stibolt, R. D., and John Lehman. 1993. Value of a Seismic Option.'' of Petrolettm Engineers, SPE 25821, 25-32. Proceedings of sbcll.p Stokey, Nancy L., and Robert E. Lucas, Jr., with Edward C. Prescott. 1989. Recursive Models in Economic Dynamics. Cambridge, MA: Harvard University Press. Summers, Lawrence H. 1987. 'Investment Incentives and the Discounting of Depreciation Allowances.'' In The Effect. fp Tmtion on Capital zdccamulation, ed. Martin Feldstein. Chicago, IL: Chicago University Press. Valuation and Cost of Capital ExpresTaggart, Robert A. 199 1. sions with Corporate and Personal Taxes.'' Financial Management 20, Roptimal
.
.
-rhe
dtconsistent
8-20.
Tirole Jean. 1988. The Tcsr.y of l'ntl5.f/tll MIT Press.
Oyanizflion. Cambridge, MA:
Rcjrelces
444
Tobin, James. 1969. CtAGeneral Equilibrim Approach to Monetary Theory.'' Joumal of Money, Credit and sfznln. 1 (February): 15-29. tt-f'he Valuation of Reserves of Natural Resources'. Tourinho, Octavio A. 1979. An Option Pricing Approach-'' Unpublished Ph.D. dissertation, University of California, Berkeley. Flexibility as a Triantis, Alexander J., and James E. Hodder. 1990. Complex Option.'' Jourrtal of Finance 45 (June): 549-565. Managerial Flexibility.'' Trigeorgis, Lenos, and Scott f! Mason. 1987. Midland Comorate Finance Jottmal 5, Spring, 14-21. t-rrade Reform, Aggregate Investment and Van Wijnbergen, Sweder. 1985. Capital Flight-'' Economics Letters 19, 369-372. Varian, Hal S. 1987. Arbitrage Principle in Financial Analysis.'' Joumal ofEconomic Perspectives 1 (FaIl)-.55-72. 1991. Microeconomic zlac#'--, third edition. New York; W W . Norton. Growth with Scale Economies in the Weitzman. Martin L. 1970. Creation of Overhead Capital.'' Review of Economic Stttdies 37 (October): 555-570. Econometricu 47 1979. ''Optimal Search for the Best Alternative.'' . (May): 641-654. tsequential R&D StratWhitney Newey, and Michael Rabin. 198 1. , of l 574-590. BellJournal Economics for Synfuels.'* 2, egy Wey, Lead. 1993. deEffectsof Mean-Reversion on the Valuation of Offshore OiI Reserves and Optimal Investment Rules.'' Unpublished undergraduate thesis, Massachusetts Institute of Technology, May. Wiggins, James B. 1987. t'Option Values Under Stochastic Volatility.''-/txlrrlll of Ffn/nctz/ Econontics 19, 35 1-372. Zeira. Joseph. 1987. as a Process of Search.'' Journal of Political Economy 95 (February); 204-2 l0.
Symbol Glossary
Etvaluing
'Tvaluing
'trr'he
ftoptimal
Rlnvestment
ideas and models, it is ln a book as long as this, combining many different unique notation throughout. and presewecompietelyconsistent impossibleto
central We have tried to maintain consistency for the concepts that are most of couple in there, Even 3-9. a and appear repeatedly in the core chapters example, conflict p different uses (for places demands of standard notation in switch rather than such coefhcient). In correlation cases, as discount rate and have risked some ambiguity. Luckily the of notation uses,we forone strange to proximity, so in the same symbol does not need two different uses in close :' intended. which use is each case. the context clarifies extensions 6f the core theory and which 10-12. treat some Chapters ln notation of the original discuss some applied literature, we take over the has diftkrent notation sometimes earlier articles we review, so some of our . pending the but meaning there. We have tried to minimizc these occurrences, in Kanji characters mathematarrival of standardized notation and the use of ical writing, some such problems are unavoidable. ttglobal'' notation, that is. Here we give a list, for easy reference, of our ttlocal'' symbols book. All other are usage (almost)consistent throughout the each section or subsection-They may denote different to each chapter or even each of these is dehned where parameters or variables in different places',
used.
functions of a We should also clarify our notation for derivatives. For For derivatives. single variable, we generally use primes to denote the (total) of example. F(P') has derivatives F' 1z), F'' Z), etc. For functions two or more appropriate partial derivatives. For variables, we use subscripts to denote the IZ. J), Fvv ( P'. I), /5 ( P'. f ) etc. example. F( F, f ) has derivatives Fv (
445
446
usj'ltll)olC7/t?..fl?)'
Uppercase Roman Letters z4,B Constants of integration in solutions
C F
P
Q
r
P' l'U
Uppercase Greek Letters of differential equation.
Operating cost of 11 project. Value of the opportunity to invest', more generally, the value function in dynamic programming (aisocalled the Belman function). Capital cost of investment (Chapters 2. 5-9)., rate ofinvestment (Chap-
ters
Pre Iix for tinite btlt small incren-lent (e.g.,A / ). Terminal payoff tnction in lynanlic progral-nnling. Value of replicating (riskless)portfolio, ctlnlulntive probability distribution function.
10-12).
Stock of capital, either installed (Chapter 11) or remaining to the completion of the project (Chapter 10). Output price. Industry output quantity. Terminal time. Value of asset in place. Total value of a whole range of capacity expansion opportunities, for the hrm (Chapter 11) or for social optimality (Chapter 9). Firm-specisc stochastic shock. Industry-wide stochastic shock.
Lowercase Greek Letters parameter of simple Brownian motiona or proportional growth rate parameter of geometric Brown ian motion. Variable in fundamental qultdratic. Its positive and negative roots are denoted by /$1and pz, respectively. Rllte of relurn shortfall or convenience yield. Arrival rate of Poisson process: depreciation rate. Mean-reversion rate. Risk-adjusted (CA PM ) discount rate. Prolit tlow. Market price of risk; probability density fttnction of diffusion process Dri
variable.
Lowercase Roman Letters
Exogcnous discount ratc in dynamic programnling; correlation coefhcien t Variance parameter in Brownian mtltion. General tim: variable (mainlyused as variable of integration in expressions for present discountcd values). .
d m, n p, q q dq r l u .x dz
447
Coefscients of diffusion process. Production cost function for a firm with variable output level (Chapter 9). Insnitesimal increment prefix (e.g.,the differential dt). Short positions in replicating riskless portfolio. Probabilities in random waik representation. Tobin's q (Chapters 5, 6, 11. 12)., a competitive srm'soutput Ievel (Chapter 9). Increment of a Poissonjumpprocess (mainlyChapters 3, 4). Riskless interest rate. Current time. Control variable in theory of dynamic programming (Chapter 4),. size of jump in basic Poisson process (Chapters 3, 4). General notation for the state variable of a stochastic process. Increment of standard Wiener (Brownian motion) process.
Calligraphic
Letters
Expectation operator. Variance operator. Expression for the fundamental quadratic.
426 Arthur. W. Brian, 16 Baldwin, Carliss Y., 352 Baldwin. Richard. 244 Bar-llan, Avner, 35 1 Becker, Gary S., 23 Beddington, John R., 92 Bellman. Richard, 125 Bentolila, Samuel, 244 Bernanke. Ben S., 40, l74 Bernslein, Peter L., 126 Bertola, Giuseppe, 145, 212. 244, 359, 382, 392, 424, 427 Bizer, David S.. 427 Black, Fischer, l 26
Blanchard. Olivier J., 19-420 Bliss, Christopher J., 206 Bodie, Zvi. 225 Brander, James. 315 Brcaly, Richard A., 54-55, 1l 6, 148
Brennan, Michael J., 179, 212. 224-225. 242. 353-354, 392, 426
Brock, William A.. 88. 174, 256 Brown, J. A. C., 72 Brown, Robert, 63
Caballero, Ricardo J., 277, 280, 382. 392, 423-424, 427 Caplin. Andrew, 425 Carr, Peter, 244 Carrier, George F., 353 Chirinko, Robert, 4 19 Chow. Gregory C.. 88 Clark, Colin W., 92 Copeland, Tom, 55 Cox, D. R., 60, 68. 88 Cox, John C., 55, l 15, 126, 152, 157, 174
450 Craine. Roger, 392 Cukierman. Alex, 40, 174 David, Paul A., 16 Davis, Steven J., 268 Debreu, Gerard. 314 Demers, Michel, 174 Dertouzas, Michael, 7 Dickens, R. N., 426 Dickey, David A., 77 Dixit, Avinash K., 68, 88. 126, 161, 174, 188, 197, 218. 223, 242-243. 244, 256, 280, 294, 314. 373, 378, 393 Dothan. Michael U., 88, 126. 174 Dreyfus, Stuart E., 126 Duffies Darrell, 106a 124, 126, 147-148, 174. 180 Dumas, Bernard, 244, 280, 393 Dutta, Prajit K., 315 Eberly, Janice C., 385, 387-388,
391-392, 425 Edleson, Michael, 280 Einstein, Alberta 63 Ekern, S., 426 Fasano, A., 212 60, 69- 88 eller,williamFeynman, Richard R, 123 Fine, Charles H., 244 Fisher, Anthony C., 426 Fleming, Wendell H., 106, l26 Freund. Robert M., 244 Fudenberg, Drew, 313 Fuller, Wayne A., 77 Geske. Robert, 244, 353 Gibson, Rajna, 179, 212
Attthor /,llt.za-
Gilbert. Richard J., 9 Goel. S.s 92 Goncalvesa F rank lin D.a 238 Gould. John R, 392 Grossman, Gene M., 352 Grossman. Sanford J., 424 Guenther, Rtnald B., 2 l2. 353
zll///lr
hldex
Kalish. Shlomo, 339 Kanlien. Morton I l26 Karatzas. Ioannis- 88, 123, l 88 Karlin, Samuel, 6()- 88-89. l6 I 3 l5, 35 l Kester. W. Carl. 9. 55 Kogut. B ruce 244 Koller. Tim, 55 Kolstadv Charles D., 426 Kopcke. Richard W., 4 19 Ireps, David, l 26, 18t) Krugman, Paul R.. 244 Krutilla. John M. 426 Kryjov, N. V, 106. 126 Kulatilakap Nalin. 244 Kushner. Harold J.. 88 Kydland. Finn E.. 352
,
,
Haberman. Richard, 353 Haltiwanger, John C., 268 I-lamermesh, Daniel S., 24 Hanemann. W Michael, 426 Harris, Milton, l26 Harrison, J. Michael, 88, 126, l6l 180, 315, 373, 390, 393 Hartman, Richard, 392, 426 Hassett, Kevin A., 304, 309, 315, 4 12 Hawkins, Gregory D., 332 Hayashi. Fumio, 392 He, Huas 244 Idendricks, Darryll, 426 Henry, Claude. 426 Herbelot, Olivier, 4()6, 4()t?-41 l .
426 Hodder, James Holmes, Oliver Lluang, Chi-fu, 174 Hull, Johns 55r l 74s 353
E., 244 Wendell Sr.. 2(5 I 16, 124, l 26, 147, 88, 126, 157, 159.
Ingersoll, Jonathan E., 50. l 15 Jacoby, Henry D., 426 Jarrow, Robert A., 55, 126, l74 Johnson, Claes, 212. 244 Jorgenson, Dale. 5, 145, 392 Judd, Kenneth L., 212, 25 l
,
Lam. Pok-sang. 425 Laroque, Guy, 424 Laughton, David G., 426 Leahy. John, 280. 292, 3l4 Lee. John W, 2 12, 353 Lehman, John. 426 Liebermana Marvin B.. 339 Lim, Chin. 315 Lippman. Steven A., 280 Litzenberger, Robert H., 1 16, 124, 126, 147, 174 Lohrenz, John, 426 Lucas, Robert E., 102, 126, 280, 286, 314, 392 Lund, Diderik, 78, 426 Maclie-Mason,
Jeffrcy
Malliaris, A. G., 88, 256 Alan S.. 393 Manthy, Robert S., 77 Marcus, Alan J.a 21 1, 244 Nlarglin. Stephen, l 45 Nlarion- Nancy, 3 15 Martzoukos. Spiros H.. 426 Mason, Scott. 55, l 74 May. Robert M.. 92 Mccray, Arthur W, 398 NlcDonald, Robert, 135-136, 162. 173, 186, 2 11-212. 234 Nlcrton, Robert C., 55, 88, 92a l 26. l73-1 74 Metcalf. Gilbert, 304. 309, 3 l 5, 4l2 Miller, D. R., 60, 68, 88 Modest. David M., 21 l Manne.
..
K., 315:
352 Majd, Saman, 150, 315, 328, 340, 35 l
Morck.
Randall.
426
Mossin. Jan. 243 Mussa. Michael.
314
Myers, Stewart C.. 9, 54-55, l l6, 148a l 73. 3 l5, 35 1 Newbery,
David, 3l4 Newey, Whitney. 352 Nickell, Stephen J.. 5, 174, 392 Nordhaus, William D., 4 l3, 427 Norman, Victor, 238 Oksendal, Bernt, 426 Osband, Kent, 280 Paddocks James L., 396-397, 402. 405, 425
424 Plummer, Mark L., 426 Porter, Mchael E., 238 Prescott, Edward C., 102. 126. 280, 286. 314, 352 Primicerio, M., 212
Ouigg,Laura. 426 Rabin, Michael, 352 Rawlinson. Richard. 238 Richter-Dyn, N., 92 Rishel, Raymond W., 106. 126 Rob. Rafael, 294, 314 Roberts, Kevin. 352 Rodrik, Dani. 315 Rosanski. Victor, 225 Ross, Stephen A., 50, 115, 126, 152 Rothschild, Michael, 174, 381 Ruback, Richard S., 55 Rubinfeld, Daniel L., 77. 404 Rubinstein. Mark, 55. 126. 157,
174 Rudd, Andrew, 55, 126, l74 Rumelt, R. R, 280 Rust, John, 424, 427 Rustichini, Aldo, 315 Samuelson, Paul A., 126, 137. 207 Sawhill, James W, 51 Schmalensee, Richard, 426 Scholes, Myron, 126
' Schwartz. Eduardo S., 179, 212. 224, 242. 353-354, 392, 426 Schwartz, Nancy, t26 Scott, Louis O., 159 Sellke, Thomas M., 393 Shapiro. Carl, 352 Shastri, Kuldeep, 353 Shreve, Steven E.. 88, 123, 188 Sichel, Daniel E.. 427 Siegel, Daniel R., 135-136, 162, 173, 186, 211-212, 234, 396-397, 400, 402, 405, 425 Slater, L. J.. 163 Smets. Frank. 309, 315 Smith. Clifford W., Jr., 55. 174, 396, 397, 400, 402, 405, 425 Solimano. Andres, 423 Solow, Robert M.. 206 Soss, Neal M., 24 Spence, A. Michael, 339, 344 Spenccr, Barbara, 3l5 Stangeland, David, 426 Stegun. Irene A., 163 Stensland, Gunnar, 426 Sterling, Arlie G., 238 Stiboit, R. D., 426 Stiglitz, Joseph E., 174, 314 Stokey, Nancy L.. 102, 126 Strandenes, Siri Pettersen, 238 Strange. William C., 351 Summers, Lawrence H., 7 Taggart, Robert A.. 54 Taksar, Michael I., 390, 393 Tylor, Allison J., 393 Taylor, Howard M., 60, 88-89, 161, 315, 35 1 Teplitz-sembitzky, Withold. 426 Tirole. Jean. 9, 313
Alttllor /ilt/t.ttTjosthe ima Dag B.. 426 Tobin. James. 5, 146. 2()6 Tourinho, Octavio A., l 73, 425 Triantis. Alexander J.. 244 Trigeorgis, Lenos, 55 Van Wijnbergen, Sweder, 244 Varian, Hal S., 55. 126. 197 Von Weizack-er, Christian, 2()6
Weitzman,
Martin
L.. 352. 393
Nvey. Lead- 404 White.
Alan. 159, 353 Wiener, Norbert. 63 Wiggins, James B., 159
z)) Unit root test, 77, 404 Upper reflecting barrier, 254, 261 Upper threshold on price, 26l Urban land valuations 426
Valuation.
equivalent
risk-neutral,
12 l 124. 126 Value assets in place. 369 Value function, l ()l 270-27 1 279, 285, 299, 360 Valtle matching. l3()s l90, 2 l8. 257. 262- 270. 368 Value of 11 firm, 2 l 1 of 11 machine. 87 ()f 11 prtject, l 75 of active firm, 2 1('), 226, 254 of assets in place, 147 of llexibility. 29. 3 15, 337 ()f idle firm. 226 of inlbrmationa 352 of invcsting, 258 of invcstment opportunity, l 92- t94 337 346 of mothballed prllject, 232 of option to abandon, 232 of option to invest, 32, l 9(), 216, ,
.
,
,
,
321 of option to mothball, 232 of option to reactivate, 232 of option to scrap, 232 of product. 321 of project, 136, 179, 187, 197. 200 of the tirm, 147, 216, 369 of waiting, 14(), 258 Value-matching condition, 109, 141 183, 192, 203, 209. 222, 232, 234, 286-287, 298, 3()7, 331, 341 349, 364 Vtlue-ol--assets-in-placc, 420 Value-of-thc-firm-sensc, 386 ,