[Alessandro Citanna, John Donaldson, Herakles Pole

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Studies in Economic Theory

Editors

Charalambos D.AliprantisPurdue UniversityDepartment of EconomicsWest Lafayette, in 47907-2076USA

Nicholas C. YannelisUniversity of IllinoisDepartment of Economics

Champaign, il 61820USA


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M. A. Khan and N. C. Yannelis (Eds.)Equilibrium Theoryin Infinite Dimensional Spaces

C. D. Aliprantis, K. C. Borderand W. A. J. Luxemburg(Eds.)Positive Operators, Riesz Spaces,and Economics

D. G. SaariGeometry of Voting

C. D. Aliprantis and K. C. BorderInfinite Dimensional Analysis

J.-P. AubinDynamic Economic Theory

M. Kurz(Ed.)Endogenous Economic Fluctuations

J.-F. LaslierTournament Solutions and Majority Voting

A. Alkan, C. D. Aliprantis and N. C. Yannelis(Eds.)Theory and Applications

J. C. MooreMathematical Methodsfor Economic Theory 1

J. C. Moore

Mathematical Methodsfor Economic Theory 2

M. Majumdar, T. Mitra and K. NishimuraOptimization and Chaos

K. K. SiebergCriminal Dilemmas

M. Florenzano and C. Le VanFinite Dimensional Convexityand Optimization

K. VindIndependence,Additivity, Uncertainty

T. Cason and C. Noussair(Eds.)Advances in Experimental Markets

F. Aleskerovand B. MonjardetUtility Maximization. Choice and Preference

N. SchofieldMathematical Methods in Economicsand Social Choice

C. D. Aliprantis, K. J. Arrow, P. Hammond,F. Kubler, H.-M. Wu and N. C. Yannelis (Eds.)Assets, Beliefs, and Equilibriain Economic Dynamics

D. Glycopantis andN. C. Yannelis (Eds.)Differential Information Economies

A. Citanna, J. Donaldson, H. M. Polemarchakis,P. Siconolfi and S. E. Spear(Eds.)Essays in DynamicGeneral Equilibrium Theory

M. KanekoGame Theory and Mutual Misunderstanding

Titles in the Series


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Alessandro Citanna John DonaldsonHerakles Polemarchakis Paolo Siconolfi

Stephen E. SpearEditors

Essays in DynamicGeneral Equilibrium TheoryFestschrift for David Cass

With 23 Figures

and 3 Tables

1 3


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Professor Alessandro Citanna

Department of Economics and FinanceHEC, ParisFrance

E-mail: [email protected] Finance and Economics DivisionGraduate School of BusinessColumbia UniversityNew York, NY 10027USAE-mail: [email protected]

Professor John DonaldsonGraduate School of BusinessColumbia UniversityFinance and Economics DivisionNew York, NY 10027USAE-mail: [email protected]

Professor Herakles Polemarchakis

Brown UniversityDepartment of EconomicsProvidence, RI 02912USAE-mail: [email protected]

Professor Paolo Siconolfi

Graduate School of BusinessColumbia UniversityFinance and Economics Division

New York, NY 10027USAE-mail: [email protected]

Professor Stephan E. Spear

Tepper School of BusinessCarnegie Mellon UniversityPittsburgh, PA 15213

USAE-mail: [email protected]

Library of Congress Control Number: 2004114983

ISBN 3-540-22267-7 Springer Berlin Heidelberg New York

This work is subject to copyright.All rights are reserved, whether the whole or part of the material is concerned,specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction onmicrofilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof ispermitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version,and permission for use must always be obtained from Springer-Verlag.Violations are liable for prosecution underthe German Copyright Law.

Springer is a part of Springer Science+Business Media

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Essays in General Equilibrium TheoryFestschrift for David Cass

Guest Editors Introduction

Alessandro Citanna1, John Donaldson2, Herakles Polemarchakis3,

Paolo Siconolfi4

, and Stephen Spear5

1 Department of Economics and Finance, HEC, Paris, France(email: [email protected])and Finance and Economics Division, Graduate School of Business, ColumbiaUniversity, New York, NY 10027 (email: [email protected])

2 Finance and Economics Division, Graduate School of Business, ColumbiaUniversity, New York, NY 10027 (email: [email protected])

3 Department of Economics, Brown University, Providence, RI 02912

(email: Herakles [email protected])4 Finance and Economics Division, Graduate School of Business, ColumbiaUniversity, New York, NY 10027 (email: [email protected])

5 Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213(email: [email protected])

This book is a supplement to the special edition ofEconomic Theory hon-oring David Cass on the 30th anniversary of his joining the faculty at theUniversity of Pennsylvanias Economics Department. As with the ET is-sue, the contributions to this volume are, for the most part, from Davesstudents or co-authors, and we hope they communicate both to Dave andto the economics profession generally the high regard those of us who havetrained under Daves tutelage or worked with him on research have for him.

Most scientists would be happy to have had one major, influential ideaover the course of their careers. Dave Cass has had three, and is still goingstrong.

His first major contribution to economics was the characterization ofoptimal growth trajectories in his thesis work under Hirofumi Uzawas su-pervision. The celebrated Cass criterion for optimal time paths in the onegood growth model quickly followed. The essence of this work is the searchfor price characterizations of efficiency for dynamic time paths, an effortthat directly pointed the way to the subsequent full dynamic decentraliza-tion of the neoclassical optimal growth model, a fact that permits its use


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VI Guest Editors Introduction

for modeling a wide range of business cycle and other macroeconomic phe-nomena. Accordingly, Dave is rightly honored, in conjunction with TjallingKoopmans, as one of the fathers of dynamic macroeconomic analysis.

Daves second contribution the notion of a so-called sunspot equilib-

rium in dynamic economies which he developed jointly with Karl Shell isalso the stuff of legend, and grew out of his long and productive collabora-tion with Karl at Penn. The early impetus for Daves interest in this topicstemmed from work he did with Manny Yaari on overlapping generationsmodels, and from his early acquaintance with Bob Lucas at Carnegie Mel-lon and Lucass seminal work on rational expectations in dynamic economicmodels. To quote from the interview with Dave by Spear and Wright inMacroeconomic Dynamics

I wasnt so interested in macro, but what struck me, and this is relatedto some of my later work, was the assumption that [Lucas] made to solve forequilibrium, that the state variables were obvious. . . . Bob and I had somelong discussions, and I would say, Well Bob, why is this the actual statespace in this model? That question came up . . . after I came to Penn. Atsome point Karl [Shell] and I started talking about that and we developedwhat we called the idea of sunspots. (Spear and Wright [8])

In addition to raising troubling questions about what the right state

space was for dynamic stochastic economies, the notion of sunspot equilib-rium raised a number of deep questions about the overall determinacy ofeconomic equilibria and the role of the welfare theorems in the occurrenceor non-occurrence of sunspot equilibria. These questions spawned a largeliterature on determinacy in dynamic economies in which the welfare the-orems broke down. These include overlapping generations models, growthmodels with externalities or taxes, and models in which asset markets wereincomplete. All were shown to allow the existence of sunspot equilibria.

And, in a suitable twist of intellectual fate, macroeconomists have morerecently begun to explore the question of whether sunspots can provide amore plausible source of fluctuations in dynamic equilibrium models thanthe conventional aggregate productivity disturbances.

Daves third major contribution to economic theory was his work ongeneral equilibrium with incomplete markets, work which grew out of hisexploration of the question of existence of sunspot equilibria in models withincomplete asset markets. Daves follow-on work on existence and deter-minacy of general equilibrium in models with incomplete asset marketsspawned another large literature which has come to be known simply asGEI.

The earliest work on market incompleteness goes back to Arrow in the1950s, Diamond in the mid-60s and a number of related papers in thefinance literature between the late 1950s and early 70s (Geanakoplos [4]provides an excellent survey of this literature). The canonical GEI model wasformulated by Radner in the early 1970s (Radner [7]) in a paper which also


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Guest Editors Introduction VII

pointed up one of the fundamental puzzles about models with incompletemarkets: the possible loss of dimensionality in the span of the asset payoffsas prices vary.

This potential for non-existence of equilibrium (which was formally de-

veloped in Harts [6] counterexamples to existence of equilibrium) left theliterature in limbo for almost a decade, until Daves work on existence ineconomies with purely financial assets pointed the way out. As Geanakoplosnotes

Suddenly in the middle 1980s the pure theory of GEI fell into place.In two provocative and influential papers, Cass [1,2] showed that theexistence of equilibrium could be guaranteed if all the assets promise

delivery in fiat money, and he gave an example showing that withsuch financial assets there could be a multiplicity of equilibrium. Al-most simultaneously Werner [9] also gave a proof of existence of equi-librium with financial assets, and Geanakoplos and Polemarchakis[5] showed the same for economies with real assets that promisedelivery in the same consumption good. (Geanakoplos [4])

This work was followed very quickly by results showing that the non-existence problem pointed out by Hart was not generic, and led ultimatelyto the generic existence results of Duffie and Shafer [3], and again spawneda new literature looking positively at the welfare implication of market in-completeness, and normatively at issues of asset engineering.

In the course of making these contributions, Dave has worked with alarge group of coauthors, including (to date): Y. Balasko, L. Benveniste,G. Chichilnisky, A. Citanna, R. Green, M. Majumdar, T. Mitra, M. Okuno,A. Pavlova, H. Polemarchakis, K. Shell, P. Siconolfi, S. Spear, J. Stiglitz,A. Villanacci, H.-M. Wu, M. Yaari, and I. Zilcha. Daves graduate students

(to date) include S. Chae, A. Citanna, J. Donaldson, R. Forsythe, F. Kyd-land, Y. W. Lee, M. Lisboa, A. Pavlova, T. Pietra, P. Siconolfi, S. Spear,S. Suda, J-M. Tallon, and A. Villanacci. Those of us who have workedwith Dave and/or under his tutelage as graduate students have benefitedtremendously from his razor-sharp analytic mind, from his willingness towork at understanding problems we have posed to him, or new methodolog-ical techniques we have discovered, and (perhaps most importantly) fromhis no-nonsense approach to doing science.

References

1. Cass, D., Competitive equilibria with incomplete financial markets, CARESSWorking Paper, University of Pennsylvania, April 1984

2. Cass D., On the number of equilibrium allocations with incomplete financialmarkets, CARESS Working Paper, University of Pennsylvania, May 1985


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VIII Guest Editors Introduction

3. Duffie, D. and W. Shafer, Equilibrium in incomplete markets, I, Journal ofMathematical Economics, 14, 285-300, 1985

4. Geanakoplos, J., An introduction to general equilibrium with incomplete as-set markets, Journal of Mathematical Economics, 19, 1990

5. Geanakoplos, J. and H. Polemarchakis, Existence, regularity and constrainedsuboptimality of competitive equilibrium when markets are incomplete, in W.Heller, R. Starr and D. Starrett, eds., Essays in Honor of Kenneth Arrow,Vol. 3, Cambridge University Press, Cambridge, MA, 1986

6. Hart, O., On the optimality of equilibrium when the market structure isincomplete, Journal of Economic Theory, 11,3, 418-443, 1975

7. Radner, R., Existence of equilibrium of plans, prices and price expectations,Econometrica, 40, 2, 289-303, 1972

8. Spear, S. and R. Wright, An interview with David Cass, Macroeconomic

Dynamics, 2, 533-558, 19989. Werner, J., Equilibrium in economies with incomplete financial markets,

Journal of Economic Theory, 36, 110-119, 1985.


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Table of Contents

Suleyman Basak and Anna PavlovaMonopoly Power and the Firms Valuation: A Dynamic Analysisof Short versus Long-Term Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Christian BidardRevealed Intertemporal Inefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Graciela Chichilnisky and Olga GorbachevVolatility and Job Creation in the Knowledge Economy. . . . . . . . . . . . . 45

Thorsten Hens, Janos Mayer, and Beate PilgrimExistence of Sunspot Equilibria and Uniqueness of Spot Market

Equilibria: The Case of Intrinsically Complete Markets . . . . . . . . . . . . . 75

Mukul Majumdar and Nigar HashimzadeSurvival, Uncertainty, and Equilibrium Theory: An Exposition . . . . . . . 107

Richard McLean, James Peck, and Andrew PostlewaiteOn Price-Taking Behavior in Asymmetric Information Economies . . . . 129

Sujoy Mukerji and Jean-Marc Tallon

Ambiguity Aversion and the Absence of Indexed Debt . . . . . . . . . . . . . . 143Tito PietraSunspots, Indeterminacy and Pareto Inefficiency in EconomiesWith Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Aldo Rustichini and Paolo SiconolfiGrowth in Economies With Non Convexities:Sunspots and Lottery Equilibria, Theory and Examples. . . . . . . . . . . . . 203

Jamsheed Shorish and Stephen SpearShaking the Tree:An Agency-Theoretic Model of Asset Pricing . . . . . . . . . . . . . . . . . . . . . . 243

Neil WallaceCentral-Bank Interest-Rate Controlin a Cashless, Arrow-Debreu Economy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 267


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Monopoly Power and the Firms Valuation:

A Dynamic Analysis of Short versus

Long-Term Policies

SuleymanBasak1 andAnna Pavlova2

1 London Business School and CEPR, Institute ofFinance and Accounting,London NW1 4SA, UK [email protected]

2 Sloan School of Management, Massachusetts Institute of Technology, Cambridge,MA 02142-1347, USA [email protected]

Summary. Recent anti-trust cases exacerbated the concerns of investors regard-ing the effects of a firms monopoly power on its production choice, shareholdervalue, and the overall economy. We address this issue within a dynamic equilibriummodel featuring a large monopolistic firm whose actions not only affect the price

of its output, but also effectively influence the valuation of its stock. The latterrenders time-inconsistency to the firms dynamic production choice. When the firmis required to pre-commit to its strategy, the ensuing equilibrium is largely in linewith the predictions of the textbook monopoly model. When the firm behaves in atime-consistent manner, however, the predictions are strikingly atodds. The trade-off between current profits and the valuation of future profits induces the firm toincrease production beyond the competitive benchmark and cut prices. This policymay result in destroying shareholder value,and does indeed fully wipe out the firmsprofit in the limit of the decision-making interval shrinking to zero, in line with the

Coase conjecture.

1 Introduction

The pervasiveness of monopoly power, across many product categories inalmost every part of the world economy, has long pre-occupied economistsand lawmakers. There is also undisputed evidence that monopoly power is

widespread amongst firms dominant enough to matter at an economy-wide

We thank Steve Spear and the anonymous referees for helpful suggestions.We are also grateful to Franklin Allen, Dave Cass, Peter DeMarzo, Bernard Du-mas, Ron Giammarino, Rich Kihlstrom, Leonid Kogan, Branko Urosevic, DimitriVayanos, seminar participantsat Boston University,Universityof Coloradoat Boul-der, Columbia University, MIT, University of Pennsylvania, Princeton University,American Finance Association Meetings, and European Finance Association Meet-ings for valuable comments. All errors are solely our responsibility.


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2 Suleyman Basak and Anna Pavlova

level, the so-called bellwethers. This is evidenced by the series ofongoinganti-trust cases brought about by the U.S. government against, for exam-ple, IBM (1969 - 1982), AT&T (1974- 1982), Microsoft (1994 - 1995, 1998 -present), and Bell Atlantic (1996 - 1997). Apparently, there is concern that

these large monopolists may not just exert power in their own product mar-kets, but their actions mayimpact theoveralleconomy. For example, as arguedin the Microsoft case, US consumers and businesses alike feared to becomecritically dependent on Microsoft products. There are well-known dominantplayersoutsidetheUS,too. For instance, OPECs production decisions appearto affect not only theoil producing countries economies, but also theoverallworld economy. While the surveys of imperfect competition by Hart (1985)andBonanno (1990)have highlightedthe importanceof a generalequilibrium

analysis of such large monopolists, most studies of monopoly behavior havebeen undertaken at a partial equilibrium level assuming no impact on othermarkets. A true general equilibrium approach must face up to the challengeof accounting forthe so-called feedbackor Ford effect, asonce argued byHenry Ford:throughthe influence of itsown actions, a non-price-taking firmmay affect the wealth of its customers, and hence the demand for the firmsproduct. Finally, althoughthere is growingworkon general equilibrium assetpricingwith marketimperfections(e.g., seethe surveyby Sundaresan(2000)),

the consideration of monopoly power is still missing in thisliterature.Our primary objective inthis paper istoinvestigatetheoptimal behavior

of a monopolist who has sufficient power to impact economy-wide pricing.We model the extreme case of an economy containing a single monopolisticfirmwho then, ata generalequilibriumlevel, impactstheoverallpriceof con-sumption inthe economy. Beyondthetraditional assumptionthat its actionsimpact the priceof itsown good,the firms actions alsoinfluencethevaluationof its stock. Aswillbecome evidentfromour analysis,the dynamic settingweemp

loy leads

toa dis

tinc

tio

n betw

een sho

rt-t

erm andlo

ng-t

erm pol

icieso

ft

he

monopolistic firm. (For a competitive firm,there is no such distinction.) Partofour emphasiswillbeto study the differences betweenthe ensuing equilibriaunder short versuslong-term policies.3

We adopt a familiar Robinson-Crusoe formulation of a discrete-timeproduction finite-horizon economy populated by a representative consumer-investor-worker and a representative firm. The consumer derives utility from

3Firms short-termism isreceivingincreasingattention from both academicsand

practitioners. For example, Business Week (09/13/1999) voices the often-quotedcomplaint that Wall Street exerts increasing pressure on many firms for short-termresults due to a shrinking of investors time horizons. The article points out thata share in AT&T is held for an average of 1.1 years, down from 3 years in 1990;a share in General Motors is held for an average of 1 year, down from 2 years in1990. Firms short-termism is also frequently blamed on ever increasing managerialturnover.Anecdotal evidenceand recent academic literature (Allen and Gale (2000),Palley (1997)) seem to agree that revolving management may be damaging for along-lived firm.


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Monopoly Power and the Firms Valuation 3

consumption and leisure, simultaneously invests in financial markets(includ-ing the firms stock), and earns a labor income. The consumers labor is de-manded by the firm as the sole input to a stochastic non-constant-returns-to-scale production technology, producingthe only good in the economy. We

assumetheobjectiveofthis firm istomaximize its market value,orthe presentvalueof its expected profits. The firm has monopolypower inthe good market,in that it takes account ofthe impact (via market clearing) of its productionplanonthe price of itsoutput. Inour setting, the monopoly power manifestsitself as an impact ofthe firmslabor demandonthe state prices(orthe pric-ing kernel). This manipulation of state prices results intime-inconsistency ofthe firms production strategy, inthat it has an incentiveto deviate fromtheinitialplan atalater date. Toruleout time-inconsistency,we focusontwodis-

tinct types of monopolistic strategies: a pre-commitment strategy inwhichthe firm initially chooses a plan to maximize its initial value, and thereaftercannot deviate fromthat plan; and a time-consistent strategy inwhichthefirm chooses a plan each period, maximizingthe value at that period, takinginto account the re-adjustments that it will make in the future. Consistentlywith the literature (e.g., Blanchard and Fischer (1989, 11.4)), we interpretthetime-consistentstrategyas a short-sightedor short-term strategy, andthepre-commitment one aslong-term.

Solving for the pre-committed monopolists strategy reveals his optimalplan and the extent of his monopoly power to be driven by the concurrentprofit,the marginalproduct oflabor, andthe consumers attitudetowards riskover consumption. The most immediate implicationsonthe ensuing dynamicequilibrium are consistent with the predictions oftextbook static monopolymodels:lower goodoutputandlowerlabor input, and higher priceof consump-tion than the competitive counterpart. However, in contrast to the textbookcase, profits and the firms value can be either higher or lower. This arisesbeca

use i

tis

the firms ini

tia

l va

lue

the m

on

op

olis

tma

ximizes, n

ot the pr

ofi

ts

nor the value at later times. By restricting production, he moves away fromthe competitive, profit maximizing, production plan.

Theoptimal behaviorofthetime-consistent monopolist contrasts sharplywiththat ofthe pre-committed monopolist. Hisoptimal production plan andhis monopolypower are driven by the negativeofthe ex-dividend stock price,in placeof concurrentprofits. In attemptingtomaximizethe currentstockval-uation,this monopolist tradesoff betweentodays profit andthe ex-dividendstock price; hence the appearance of the ex-dividend price. In direct oppo-sition to the pre-commitment case, the equilibrium good output and labordemand are higherthanthe competitive case,whilethe priceof consumptionis lower. It is in this short-term monopolists interest to depress the currentpriceof consumption, so asto boost todays stock price. Yet more strikingly,the profits in everyperiod are decreased inthe monopolists presence, and mayeven go negative,while the firmsvalue can be eitherlower or higher than inthe competitive benchmark. We also arguethat our main conclusions regard-ing the time-consistent equilibrium hold true in the infinite horizon limit of


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our economy within a parametric example. To further explorethe robustnessofthese results, we provide an alternative, albeit extreme, form of a produc-tionopportunity (constant-returns-to-scale) underwhichthe monopolypowervanishes, and the solution coincideswith the competitive.

It iswell-recognized in monopolistic modelsof durable goods,that absentcommitment the monopolistic firm mayin a sense competewith itself acrossdifferent time periods, and weaken its monopoly power. The Coase (1972)conjecture proposes that as the time interval between successive decisions isreduced to an infinitesimal length, this intertemporal competition will drivethe firms profitsto zero. Sinceourlong-lived firms stock resembles a durablegood, it is of interest to explore the limiting case of our economies wherethe decision-making time interval shrinksto zero. While the competitive and

the pre-commitment monopolistic equilibrium retaintheir basic discrete-timestructure and implications,we findthetime-consistent equilibriumto tendtothe limit of zero profits and hence zero firms value at all times. Hence, un-derthetime-consistent scenario, monopoly power destroys shareholdervalue.However, within our framework, this zero profit limit does not coincidewiththe competitive solution.

The importance of imperfect competition is, of course, well-recognized inmany areasoftheoretical and applied research. The standard textbooktreat-

ment of monopoly (Mas-Colell, Whinston and Green (1995, Chapter 12), Ti-role (1988, Chapter 1)) considers a profit-maximizing firm which is the onlyproducerof a good(and has noinfluenceonother markets), in a static partialequilibrium setting. The main implications of the textbook monopolist arerestriction of production and a raising of the price ofoutput in the marketfor its product. While this behavior is consistent with the equilibrium im-plications of our pre-committed monopolist, it is at odds with those of thetime-consistent monopolist. The distinction is due to our accounting for theec

on

om

y-wide impac

t oft

he firms decisio

n and in part

icul

ar fo

rt

he impacton its stock price valuation.

Inthat regard,the most closely relatedwork in a general equilibrium set-ting where the valuation of financial securities is affected by market powerarethe worksofBasak (1997) andKihlstrom(1998). These authors considera singlelarge consumer-investor acting as a non-price-taker in securities mar-kets, in aLucas(1978)-type pure-exchange setting.4 Whilethe monopolists inKihlstrom andBasak select a consumption-investment plantomaximizeutil-ity, our monopolist selects a production plan to maximize the stock price ofhis firm. Inthis respect,ours is much closerto the standard textbook monop-olist. Basak demonstratesthat the non-price-taker acting as a price-leader inall markets manifests itself as a dependence ofthe state priceson the agentsconsumption choice. Kihlstrom relates the dynamic security price choice ofthe monopolist to the Coase (1972) conjecture, and shows that the inability

4See also Lindenberg (1979) and Grinblatt and Ross (1985) for related analysiswithin a static mean-variance framework.


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ofthe monopolist tocommit to the future(second period)pricequote reduceshis monopoly rents. Consequently, the first period security price is less thanin the commitment scenario (but is still higher than the competitive price).In the spirit ofKihlstrom, but with an additional moral hazard problem of

the monopolist, isthe dynamic model ofDeMarzo andUrosevic(2001).5 Ab-sent commitment,they demonstrate an analogofthe Coase conjecture inthecontinuous-timelimit oftheir economy.

The rest ofthe paper isorganized as follows. Section 2 describesthe econ-omy. Section 3 characterizes equilibrium in the economy with a monopolisticfirm for the cases when the firm can commit to its future production planand when it cannot. It also presents comparison ofthe resulting equilibriumquantitiesto thoseof a benchmark competitive economy. InSection 4,wetake

the economy to its continuous-timelimit and explorethe Coase conjecture inthe context ofour economy. Section 5 concludes and the Appendix providesall proofs as well as discussions of alternative choices of the numeraire andthe case of a monopolistic-monopsonistic firm.

2 The Economy

We consider a simple Robinson-Crusoe production economy with a repre-sentative firm and a representative consumer-investor-worker. We make thestandard assumption that the consumer-investor-worker represents a con-tinuum of identical atomistic agents who take prices as given and cannotact strategically. The economy has a finite horizon [0, T], in which trad-ing takes place at discrete times t = 0, . . . , T . There is a single consump-tion good serving as the numeraire (other choices of the numeraire are dis-cussed in Remark1). Uncertaintyis represented bya filtered probabilityspace

(, F, {Ft; t= 0, 1, . . . , T }, P)generated bya production shock process. Allstochastic processes are assumed adapted to {Ft; t= 0, 1, . . . , T }, all stated(in)equalities involving random variables hold P-almost surely. We assumeall processes and expectations are well-defined,without explicitly statingtherequired regularityconditions.

The financialinvestment opportunities are represented by: a riskystockofthe firm in constant net supply of1 share that pays out dividends (t), t =1, . . . , T ; and enough zero net supply securities to dynamically complete themarket. is endogenously determined via the firms optimization problem.Dynamic market completeness allows the construction of a unique system ofArrow-Debreu securities consistent with no arbitrage. Accordingly, we maydefinethestate pricedensity process(orthe pricing kernel(s)/(t), s t),where (t, ) is interpreted as the Arrow-Debreu price (per unit probability

5See also DeMarzo and Bizer(1993), who were the first to pointout the connec-tion between durable goods and securities markets. DeMarzo and Urosevic (2001)also provide acomprehensive review and classification of the literature.


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P) of a unit of consumption good in state at time t. The process istobe determined in equilibrium. Thetime-t(cum-dividend) valueV(t) ofthestock(ofthe firm) is then given by

V(t) =E

Ts=t

(s)(t)

(s)Ft . (1)

As evident from the subsequent analysis, our main results are valid in a de-terministic version ofour model, where has only one value in each period.We introduce uncertainty to make our formulation comparable to financialmarkets models, standard in the literature.

2.1 The Consumers Preferences and Optimization Problem

A representative consumer-investor-worker is endowed at time0 with1 shareof the stock, and at each time t with units of available labor, to allocatebetweenleisureh(t), and labor (t), for which he is paid awage at ratew(t).The consumer intertemporallychooses a non-negative consumption processc,labor process, and portfolio (of securities)process soasto maximize hislife-time utility.6 The consumer derives a separable von Neumann-Morgenstern

time-additive, state-independent utility u(c(t)) +v(h(t)) from consumptionand leisure in [1, T]. The functions u and v are assumed three times contin-uously differentiable, strictly increasing and strictly concave, and to satisfylimc0 u

(c) =, limh0 v(h) =.7

Under the complete markets assumption, the consumer-workers dynamicoptimization problem can be cast in itsArrow-Debreuformulation as a staticvariational problemwith a single budget constraint:

maxc,

E T

t=1

u(c(t)) + v( (t))

(2)

subject to E

Tt=1

(t)

c(t) w(t) (t)

E

Tt=1

(t)(t)

. (3)

We do not explicitly apply the nonnegativity constraints c(t) 0, (t) , (t) 0, because the conditions limc0 u

(c) = and limh0 v(h) =

6We introduce labor as one of the simplest, most commonly-adopted choices ofinput to the firms production function; employing labor does not introduce theintertemporal complexity ofemploying capital.

7We make the assumption ofseparable utility for simplicity, given our intentionis to focus on a comparison of monopolistic behavior in a particular good market.Our analysis readily extends to the general non-separable case u(c, h). Our majorcomparative staticsresults (Propositions 2 and 4)remain valid under theassumptionuch 0 (which includes separable utility and Cobb-Douglas u(c, h) =

1

(ch1),for (0, 1).)


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guarantee c(t) > 0, (t) < , while (in the equilibrium provided) the firmsproduction technology (Section 2.2) guarantees(t) 0.

The first-order conditionsofthe static problem (2)(3) are

u(c(t)) =y (t), (4)v( (t)) =y (t) w(t), (5)

wherethe Lagrangian multipliery satisfies

E

Tt=1

(t)

c(t) w(t) (t)

= E

Tt=1

(t)(t)

. (6)

Consequently,v( (t))

u(c(t)) =w(t). (7)

2.2 The Firms Production and Optimization

The representative firm inthis economy facesthe same information structureand set of securities asthe consumer. Ateachtimet = 1, . . . , T ,the firmuses

labor, D

(t), as its only input to a production technology, f, which providesconsumption good asoutput.8 Thetechnologyis stochastic, driven bya shockprocess , assumed (without loss of generality) to be strictly positive. Thefirms output at time t is given by f(D(t), (t)). We assume f is increasingand strictlyconcave in its first argument andthat limD0 f(

D, ) = andlimD0 f(

D, ) 0. The firm pays out a wagew(t) for each unit oflabor itutilizes, so itstime-t profit is

(t) =f(D(t), (t)) w(t) D(t), (8)

all ofwhich it pays out as dividendsto its shareholders. The firms objectiveis to maximize its market value, or the present value of its expected profitsundervarious market structures.

The firms behavior is the main focus of this work. For the purpose ofcomparison,we firstconsidertheoptimalchoiceof a competitive firm, andtheresulting competitive equilibrium(Section 2.3), andthenturnto an economywhere a firm exercises monopoly power in the market for the consumption

good(Section3). Inthelatter set-upour assumptionofthe firms maximizingits value is prone to the criticismthat appliesto all equilibrium models withimperfect competition. To this day, the issue whether value maximizationis the appropriate objective is still open (see Remark 1). Our viewpoint inthis paper is to simply adopt the most well-understood equilibrium concept,

8Forsimplicity,we donot modelthe time-0 consumption/leisureand productionchoice. All our results for t = 1, . . . , T quantities, in the propositions of the paper,would remain valid if the time-0 choice were additionally modeled.


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the Cournot-Walrasian equilibrium, and despite its possible criticisms, focuson the implications. In particular, we interpret our representative consumer-shareholder as a standardWalrasian price-taking agent. As such, he perceiveshis decisions as having no effect on the valuation of his consumption/labor

stream, given by theleft-hand sideofthe budget constraint (3). Ontheotherhand, as a shareholder, he can affect the quantity on the right-hand side of(3), his initial wealth,which is equal to thevalueofthe firm. Monotonicity ofhis indirect utility ofwealth, then, justifies the firms value maximization asa desired objective of its shareholder.

2.3 Benchmark Competitive Equilibrium

Theobjectiveofthe competitive firm is choosethe inputD

soas to maximizeits time-0 value, V(0), taking the state prices and wage as given. The first-order conditions for a competitive firms problem are given by

f(D(t), (t)) w(t) = 0 t= 1, . . . , T . (9)

Definition 1 (Competitive Equilibrium).Anequilibrium in aneconomyof one competitive firm and one representative consumer-worker is a s et ofprices (c, wc) and choices (cc, c, D

c) such that (i) the consumer chooses

hisoptimal consumption/labor policy giventhestate priceand wage processes,

(ii) the firm makes itsoptimal labor choice given the state prices and wage,and(iii) the consumption and labormarketsclearatall times:

cc(t) =f(Dc

(t), (t)) and c(t) =Dc

(t) . (10)

It is straightforward to show from (7)(10) that in the competitive equi-librium,the equilibrium labor c =D

cis given by

f(c(t), (t))

v( c(t))

u(f(c(t), (t)))= 0 (11)

andthe equilibrium consumption and profit processes by

cc(t) =f(c(t), (t)) , c(t) =f(c(t), (t)) f(c(t), (t)) c(t). (12)

The equilibrium state price density, wage and firm value processes are givenby

c(t) =u(f(c(t), (t))) , wc(t) =f(c(t), (t)) , (13)

Vc(t) =

E

Ts=t

u(f(c(s), (s)))

u(f(c(t), (t)))

f(c(s), (s)) f(

c(s), (s))c(s) Ft

. (14)

The above conditions present a fully analytical characterization of theequilibrium,with(11) determining thelabor as a functionofthe shock andthen(12)(14) determining all otherquantities. Equation(11) statesthat themarginal rate of substitution between consumption and leisure is equated tothe marginal product oflabor.


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3 Monopolistic Equilibrium

In this section, we assume the consumption good market to be imperfect,in that the firm has monopoly power therein. We take the firm to act as a

non-price-taker in its output market, taking into account the impact of itsproduction plan choice on the price ofoutput. The firm is still a price-takerin its input/labor market,takingthewagesw as given. (The extensionto thecase where the firm is additionally a non-price-taker in the labor market isstraightforward and is discussed in Appendix C.)

Wewill observethat the firms production strategyis time-inconsistent, inthe sensethata monopolisthas an incentiveto deviate from histime-0plan atalater date. Whenthe monopolistgetsto timet, he no longer cares about the

time-0 valueofthe firm;rather, hewouldliketo revisethe production plan soasto instead maximizethetime-tvalueofthe firm. InSection3.1,we considerthe optimal choice of a monopolistic firm that chooses an initial strategyso as to maximize its time-0 value and pre-commits to that strategy, notdeviating atsubsequent times. InSection3.2,we solve forthetime-consistentstrategy of a monopolist who re-optimizes his production plan to maximizethe firms current value at each datet,taking into account the fact that he isnot restricted from revisingthis plan at the future datess = t + 1, . . . , T . The

former can be thought of as a long-term strategy, providing the first-bestsolution to the problemof maximizingthe firms initial value. The latter canbe interpreted as a short-term or short-sighted strategy since the firmcontinually re-optimizes every period to boost current value. We present theequilibrium for both cases.

3.1 The Pre-Commitment Case

We n

owf

orm

ula

te

the m

on

op

olis

ts

op

timiza

tio

n pro

blem

.Reca

lling

the price

-taking consumers demand (4), clearing in the consumption good market im-plies

(t) =u

f(D(t), (t))

/y . (15)

The monopolists influence in the good market manifests itself, via (15), asa (non-linear) impact of its input demand on state prices. Accordingly, thepre-committed monopolist solvesthe followingoptimization problem at time0:

maxD,

V(0) subject to(t) =u

f(D(t), (t))

/y , t= 1, . . . , T , (16)

wherey satisfies(6).Proposition 1 presents the optimal solution to this problem, assuming it

exists.9

9It is well-known that the objective function i n models with monopolisticfirms may not necessarily be concave and hence not satisfy the second-order


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Proposition 1.The pre-commitment monopoly optimal labor demand D,t= 1, . . . , T satisfies

where

f(t) w(t) =A(t) f(t)(t) > 0 , (17)

A(t) u(t)

u(t) ,

and f(t), u(t) and their derivatives are shorthand for f(D(t), (t)),u(f(D(t), (t))) andtheirderivatives.

The structure of the first-order conditions bears resemblance to that ofthetextbook single-period monopolist. Anydirectincrease in profitdueto an

extra unit of input must be counterbalanced by an indirect decrease via theimpact ofthat extra unit on the concurrent price system. In our set-up, theextent of monopolypower is driven by the currentprofit,the marginalproductoflabor andthe(positive) quantityA,which capturesthe consumers attitudetoward riskover consumption. (Thequantity A can also be restated in termsofthe textbook monopoly markup.) The higher the marginal product themore responsive is output to an extra unit of input. The more risk-aversethe consumer, the less his consumption reacts to changes in the state price,or conversely, the more the state price reacts to changes in his consumption,and so the more incentive the monopolist has to deviate from competitivebehavior. In thelimit of a risk-neutral investor(preferencesquasi-linearwithrespect to consumption), the monopolist cannot affect the state price at alland so the best he can do is behave competitively.

We now turnto an analysisof equilibrium in this economy.

Definition 2 (Monopolistic Pre-Commitment Equilibrium).Anequi-librium in an economy o f one monopolistic firm and one representative

consumer-worker i s a s et of prices (

, w

) and choices (c

,

,

D

) suchthat (i) the consumer chooseshisoptimal consumption/labor policy giventhestate priceand wage processes, (ii) thefirm makes itsoptimal labor choice in(16) given the wage, and taking into account that the price system respondsto clear the consumptiongood market, and(iii) the price system is such thatthe consumption and labormarketsclearatall times:

c(t) =f(D(t), (t)) and (t) =D(t).

The fullyanalyticalcharacterizationofthe equilibrium inthe monopolisticpre-commitmenteconomyis given by (18)(21).Equilibrium is determined by

conditions without additional assumptions (e.g., Tirole (1988, Chapter 1)). Inthe pre-commitment case a sufficient condition for concavity is u(c)f(, ) +2u(c)f(, )(f(, )w)+u

(c)f(, )(f(, )w)+u(c)f2(, )(f(, )w) c(t), w(t)< wc(t). (23)

Thefirms initialvalue satisfies,

V(0)> Vc(0).

However, the time-t profit (t) > 0 and firms value V(t) can be eitherhigher or lower thanc(t) andVc(t), respectively, t= 1, . . . , T .

10See Lemma A.1 of Appendix A for the existence of an interior solution (t) (0,) to equation (18).


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The monopolisthas an incentivetomanipulatethe priceof consumption byproducing less than his competitive counterpart. Reducing the supply raisesthe price at which clearing occurs, thereby increasing the valuation of themonopolists intertemporal profits alongwith his stock price. The firm needs

to use less input and, accordingly, the wage rate decreases. These results areconsistent with the textbook static monopoly analysis. A difference in ourmodelisthat theflow of profits,(t), inthe monopolistic equilibrium maybelowerthanthat inthe competitive benchmark(as illustrated in Example1 ofSection3.2); this is because it isthe present value of profitsthat our monop-olist is strivingtomaximize, not profits per se. By restricting production,themonopolist moves away from the competitive, profit maximizing, productionplan. Since the monopolist is maximizingthe firms initial value and since he

has market power, it is intuitive that the firms initial value will be higherthan in the competitive case. However, we have not forced him to maximizelater values ofthe firm, so it is not surprisingthat these later values may liehigher or lower thanthe competitive case (as illustrated in Example 1).

Theoptimal policies and equilibrium inthis subsection rely onthe abilityofthe monopolist topre-commit tohistime-0plan. Without this commitment,the monopolist would later want to deviate from his initial plan; that is, atany date t >0, the solution to the firms problem

maxD(s), (s), st

V(t) subject to(s) =u

f(D(s), (s))

/y , s= t, . . . , T

is different from his time-0 plan, unless V(t) = 0. Due to the intertemporaldependenceofV onthe priceof consumption, ifthe monopolistsolves his prob-lem at the initial period, hewould change his mind at alater stage about theoptimal process. Inotherwords, his production plan is not time-consistent.A similar problem arises inthe context of a durable good monopoly (e.g.,Ti-

role(1988, Chapter 1)), or in the context of dynamic securities marketswithnon-price-taking investors(Basak(1997),Kihlstrom(1998)). The firms stockin our model is similar to a durable good since it provides value over manyperiods. Ifthere were a credible mechanism for the monopolist to commit tohistime-0 plan,the time-0 share price ofthe firm that he ownswould be ce-terisparibusthe highest possible; if however such commitment is impossible,we will show that the monopolist might be better off behaving competitively(Example 1). Possible pre-commitment mechanisms may include (i) employ-

ing a productiontechnology with atime-to-build feature, and followingTirole(1988), (ii) handing the firm over to a third party instructed to implementthe optimal strategy with a penaltyfor deviating from it (although renegoti-ation would be a potential issue), (iii) long-term relationships/contracts, (iv)money-back guarantee penalizing deviations from atargetedlabor demand.Modeling additionally a pre-commitment mechanism is beyond the scope ofour analysis. However,we providethe pre-commitmentsolution sinceweviewit as avaluable yardstick against whichwe compare thetime-consistent solu-


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tion, reported below. Moreover, the pre-commitment solution also resemblesthe standard textbook monopoly behavior.

3.2 The Time-Consistent Case

We now turn to the time-consistent monopolistsoptimization problem. For-mally, the time-consistency requirement imposes an additional restriction onthe firms production choice: at no time t can the monopolist be willing todeviate from his time-0 strategy. If the firm is not restricted from revisingits strategy at datest= 1, . . . , T , it should optimally choose the intertempo-ral production plan taking into account that it will always act optimally inthe future. We proceedto findthe monopolists subgame perfect strategy by

backward induction; this is a time-consistent strategy. Specifically, the mo-nopolist choosesthe current input and price of consumptionto maximizethefirms current value given the firms future maximized value t = 0, . . . , T ,i.e., solvesthe dynamic program:

V(t) maxD(t), (t)

V(t)

subject to(t) =uf(D(t), (t))/y , (24)V(s) =V(s) , (D(s), (s)) =argmax V(s) , s= t + 1, . . . , T ,

wherey satisfies(6).Proposition 3 presents the optimal solution to this problem, assuming it

exists.11

Proposition 3.The time-consistentmonopoly optimal labordemandD, t=1, . . . , T 1 satisfies

f(t) w(t) =A(t) f(t) Vex(t)


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The ex-dividend value of the firms stock at time T is zero; hence theterminal first-order condition and optimal choice of the firm coincide withthose ofthe competitive firm. The first-order condition at time T is the ter-minal condition for the backward induction: the remaining labor choices are

determined by solving(25) backwards.Similarly to the pre-commitment case, at the optimum for the time-

consistentmonopolist the increase in profitdueto an extraunit of input usedmustcounteract the indirectdecreaseviathe impact ofthatextraunit ontheprice system. However, the extent of this monopoly power is now driven bythe negativeofthe ex-dividend stock price in placeofthe currentprofit. (Theconsumers attitudetoward risk for consumption andthe marginalproduct oflabor appear as in the pre-commitment case.) The short-sighted monopolist

onlycares about (andtriesto manipulate) the current valuationofthe stock,which is made up of current profit plus the ex-dividend value of the firm.Giventhetime-consistency restriction,this monopolist takesthe futurevalueof profits as given and canonlyboost the firms ex-dividendvalue bydepress-ing the current price of consumption. So, while it was in the pre-committedmonopolists interest to boost the current price of consumption, it is in thetime-consistent monopolists interest to depress it; hence the negative termin (25). The ex-dividend value of the firm appears because this is what he

manipulates; the higher isVex(t),the strongerthe incentiveto cutconcurrentprofit tomanipulatethe firmsvaluation. Hence,the firmsvalue serves as anextra variable in determining the optimal policy ofthe firm.

We now define an equilibrium in a monopolistic economy containing thetime-consistent firm.

Definition 3 (Monopolistic Time-Consistent Equilibrium). An equi-librium in an economy o f one monopolistic firm and one representativeconsumer-worker i s a s et of prices (, w) and choices (c, , D) such that

(i) the consumer chooseshisoptimal consumption/labor policy giventhestateprice and wage processes, (ii) the firm chooses a time-consistent strategy soas to maximize itsobjective in (24) given the wage, and taking into accountthat the pricesystemrespondsto clear the consumptiongood market, and(iii)the consumption and labormarketsclearatall times:

c(t) =f(D(t), (t)) and (t) =D(t).

The fully analytical characterization of the monopolistic time-consistent

equilibrium is presented in (26)(29). Again, from labor market clearing, wefirst determine thelabor as a function ofthe shock (26).12 Then,(27)(29)give the remaining equilibrium quantities. The equilibrium labor is given by

f((t), (t)) v( (t))

u(f((t), (t)))= A(t) f((t), (t)) Vex(t) (26)

12See Lemma A.3 of the Appendix for the existence of an interior solution (t) (0,) to equation (18).


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andthe equilibrium consumption and profit processes by

c(t) =f((t), (t)) , (t) =f((t), (t)) v( (t))

u(f((t), (t)))(t). (27)

The equilibrium state price density, wage and firm value processes are givenby

(t) =u(f((t), (t))) , w(t) = v( (t))

u(f((t), (t))), (28)

V(t) =E T

s=t

u(f((s), (s)))

u

(f((t), (t)))

f((s), (s)) v( (s))

u

(f((s), (s)))

(s) Ft .(29)Proposition 4 presents a comparison of the monopolistic time-consistent

andthe competitive economies.

Proposition 4.The equilibrium labor, output, state price and profit in thetime-consistent monopolistic and the competitive economies satisfy, for allt= 1, . . . T 1, all(t):

(t)>

c

(t), f((t), (t))> f(

c

(t), (t)), c(t)> c

c

(t) (30)(t)< c(t), w(t)> wc(t), (t)(t)< c(t)c(t), (t)< c(t). (31)

Thefirms initialvaluesatisfies,

V(0)< Vc(0).

At timeT, all the equilibriumquantitiescoincidewith those of the competitivebenchmark. Thefirms value, V(t), canbe either higher or lower than inthe

competitive economy, Vc(t), t= 0, . . . , T 1.

The short-sighted monopolists behavior is exactly oppositeto that ofthemonopolist who pre-commits. He desires to depress the current price of con-sumption (to boost the current stock price); hence the lower level of stateprice density. He achieves this by increasing output, hence demanding morelabor input, and in return pushing up the wage rate. Although this monop-olist moves his labor demand in the opposite direction to the pre-committedmonopolist, he also moves away fromthe maximum profit point and so prof-its are reduced relativeto the competitive case. In his attempt tomanipulatethe stock price, concurrent profit and its value are cut. This type of behav-ior may overall result in a higher or lower value ofthe firms stock price. Byconsistently reducingthe value of concurrent profit, the short-sighted ownersof the monopolistic firm may, then, cause damage to the firm and its stockprice. This is so even though the time-consistent monopolist is restricted tomaximize the firms value at each point in time. For the pre-committed mo-nopolist, we could explainthis result by his not attemptingto maximizethis


26/278


quantity, but here the explanation must be more complex (see Examples 1and 2).

We nowpresent three exampleswhich deliver additional insights into theequilibriumquantities. The first example demonstratesthat the monopolistic

firms profits can go negative in equilibrium. The second example allows aninvestigationofthe relationship betweenthe monopolistic andthe competitivefirm values, as well as the infinite horizon limit. The third example showshow under an alternative, albeit extreme, form of a production opportunitythe time-consistent monopolists profits go to zero, while the pre-committedmonopolists profits do not.

Example 1 (Negative Profits) Here, we provide a numerical example inwhich the time-consistent monopolistic firms equilibrium profits (t) aresometimes negative, and its stock price is always lower than that of a com-petitive firm. Considerthe following parameterization:

u(c) + v( ) = c

+

( )

, (0, 1),


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tion 4, depending on the parameterization, nothing prevents the sign of(1)from becoming negative. This is a notable feature ofour solution. Althoughreal-world firms frequently announcelosses,this phenomenon cannot be gen-erated in a competitive model where production occurs within the decision

period (provided the firm has an option to costlessly shut down during thatperiod). Negative profits can obtain in models of predation, where rivalrousproducers are competing for market share(e.g., Tirole (1988, Chapter9)); inour model, in contrast, the monopolist controls the entire market and doesnot fear entry, yet his profit may still go negative.

A further striking conclusion revealed by Table 1 isthat non-competitivebehavior ofthe firm does not necessarily result in a higher value ofthe firm;in fact, the time-consistent monopolistic firms stock price is lower than its

competitive counterpart at all times. The recognition of his monopoly poweractually damages the time-consistent monopolist at all points in time, andhe would be better off behaving competitively. For completeness and com-parison we also provide the equilibrium quantities for the monopolistic pre-commitment economy.Example 2, having alongertime horizon, andyieldingan explicitsolution, allowsustoinvestigatethis stock price effectmore closely.

Example 2 (Firm Value Comparison and Time Horizon Effects)Since it appears that the intertemporal nature of the problem is critical, inthis examplewe extendthe horizon beyondT =2. It would alsobe importantto discuss the limiting case of our model as T , since in some models(e.g., repeated principal-agent),the equilibrium changes dramatically ontran-sitioning from a finiteto infinite horizon. Inorder to obtain explicit formulaefor all equilibrium quantities, we simplify beyond the general specificationof the utility function employed so far. The utility we adopt has the formu(c(t), t) v((t), t) = t(log c(t) (t)/), (0, 1), > 1, where isthe time discount factor, required to make pertinent quantities well-defined

inthelimiting case ofT , and the secondterm isthe disutility oflabor.We show equilibrium labor to still be bounded from above by = 1. Thefirms technology is represented by f((t), (t)) = (t) (t), (0, 1). Toensure that the equilibriumquantities are well-defined in the infinite horizonlimit, we assumethat is bounded, as well as assume away speculative bub-bles. The resulting exact formulae for the equilibrium labor choices, profitsand stock prices are reported in Table 2, with the comparative statics as inProposition 4.13 The time-consistent labor choice monotonically increases as

T t increases, convergingto a limit (and diverging away from the competi-tive labor, which is constant). As T t increases, time-consistent profits fall,

13Wedonot report themonopolisticpre-commitmentequilibrium quantities sincefor logarithmic utility over consumption, the monopolists problem is not well-defined, as is the casein the standard textbook treatmentof monopoly(Mas-Colell,Whinston and Green (1995, p.429)). Moreover, one may extend our earlier analy-sis of the monopolistic pre-commitment and competitive equilibria to the case ofinfinite horizon without major difficulties.


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although do not vanish. At this point, the parallel with to the Coase (1972)conjecture is unavoidable: ceteris paribus, the more future production deci-sionsthetime-consistent firm can make, thelower its profits today. Doesthefirm erode its profits by allowing for more future decisions? We revisit this

issue inSection 4.

Table 2. Equilibrium labor choice, profits, and firm value and their lim-

its as T t in the competitive and monopolistic time-consistenteconomies. The reported formulae are for the economies parameterized byu(c(t), t) v((t), t) = t(log c(t) (t)/), (0, 1), > 1; f((t), (t)) =(t) (t) , (0, 1)

CompetitiveEquilibrium

Monopolistic Time-ConsistentEquilibrium

(t) 1/

1((1))Tt+1

1(1)

1/(t) (t) / (1 ) (t)

1((1))

Tt+1

1(1)

/

(1)(1)+((1))Tt+1

1(1)

V(t) (t) / (1 ) (t) / (1 )1

((1

))

Tt+1

1(1)/+1

1Tt+1

1

limTt

(t) 1/

1(1)

1/lim

Tt(t) (t) / (1 ) (t)

1(1)

/(1)(1)1(1)

limTt

V(t) (t) / (1)1

(t) / (1)

[1(1)]/+1

Of special interest is the value of the firms stock. In Figure 1, we plotthe firmvalue inthe competitive and monopolistic equilibria as a functionoftime. The monopolistic firm value can be either higher or lower thanthat ofthe competitive firm, depending on the age of the firm. At first blush, thisresult seems surprising, because thetime-consistent monopolist isoptimizingthe firmsvalue at each pointintime. Since he has more marketpowerthan acompetitive firm, howcan his firmsvalue comeout lower? At timet he solves

the problem

V(t) = maxD(t),(t)

f(D(t), (t))w(t)D(t)

+ 1

u(f(D(t), (t)))E

(t + 1)V(t + 1) |Ft

.

The competitive firm solves the same problem except that: it has no powerover (t); and that w(t), V(t+ 1), (t+ 1) are replaced by their respective


29/278


values in the competitive equilibrium. This latter distinction explains whythe monopolists firm value may come out lower: he faces both a differentequilibrium wage and different choices of the future state prices and firmvalue.

0

0.5

1

1.5

2

2.5

1 2 3 4 5t

Vc(t)

V(t)

V(t)

Fig. 1. Equilibrium firm value versus time in the competitive and mo-

nopolistic time-consistent economies. The dotted plot is for the competitiveeconomy and the solid plot is for the monopolistic time-consistent. The economiesare parameterized by u(c(t), t) v((t), t) = t(log c(t) (t)/), = 0.9, =1.05; f((t), (t)) =(t) (t) , = 0.2, (t) =1, t; T =5

Figure 1 also revealsthat in this example the monopolistic firm increasesthe firms value later in its lifetime and decreases it earlier in life. This isbecause the monopolistic firm competes with itself in f uture time periods;earlier in life it faces more competition, which adversely affects its value. Inthe limit ofT t , the competitive firm value is unambiguously higher

(Table 2).

Example 3 (Constant Returns to Scale Technology) A constant ret-urns to scale technology is widely employed in both the monopoly literature(e.g., Tirole (1988)), and asset pricingliterature (e.g., theworkhorse produc-tion assetpricing model of Cox,Ingersolland Ross(1985)). Thus far,we havebeen assuming strictlydecreasing returnsto labor;in this example,we extendour analysis to the case of constant returns, f(, ) =. The firms profit isnowgiven by(t) =(t)(t) w(t)(t).


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For the competitive firmto demand a finite positive amount oflabor, theequilibrium wage wc(t) must equal (t); the equilibrium labor c(t) is thendetermined fromthe consumersoptimization (7):

v( c(t))u(f((t), c(t)))

=(t). (32)

The remaining equilibriumquantities, for all t= 1, . . . , T , are

cc(t) =f(c(t), (t)), c(t) = 0,

c(t) =u(f(c(t), (t)), Vc(0) =Vc(t) = 0.

Adap

ta

tio

no

fou

r analy

sis inS

ect

io

n3.1

show

st

hat t

he pre-c

ommi

tmen

tmonopolistic equilibrium labor demand(t) (assuming it exists and yields amaximum in the firms problem) solves

u(f((t), (t)))

u(f((t), (t)))(t)(t) = 1 , (33)

implying (t) < c(t). The remaining equilibrium quantities are obtainedfrom (19)(21). The pre-commitment equilibrium, then, retains the main

implications derived in Section 3.1; in particular, the firm cuts productionand raises the price ofoutput. The main difference from Section 3.1 is thatmonopolyprofits are nowalways higherthanthe competitiveones(which arezero), consistent with the prediction of the textbook monopoly model. Thefirmsvalue is also higher.

The monopolistic time-consistent equilibrium coincideswith the compet-itive one. To see why, recall from Proposition 4, that at the final time,(T) = c(T). Since c(T) = 0, the time-(T-1) ex-dividend value of the

monopolistic firm, Vex(T 1), is zero; hence by backward induction it can beshown that (t) =c(t)t, and the competitive equilibriumobtains:

c(t) =f(c(t), (t)), (t) = 0,

(t) =u(f(c(t), (t)), V(0) = V(t) = 0, t= 1, . . . T .

In contrast to the decreasing returns to scale case, the time-consistent mo-nopolist loses all his monopoly power. Accordingly, the value of his stock, as

well as profit, are strictly lower than in the pre-commitment equilibrium. Itis quite clear that the short-sighted behavior has a devastating effect on thefirms valuation. Ironically, it benefits the representative consumer, who, ineffect, setsthe labor demand ofthe firm so asto maximize his own expectedlifetimeutility.


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Remark1 (Price Normalization inMonopolistic Models).Ours is a general equilibrium model with imperfect competition; accord-

ingly, it is not immuneto the general criticismof all such models initiated byGabszewicz and Vial (1972) (see also Mas-Colell (1982)). The criticism has

to do with the so-called price-normalization problem which arises in thepresenceof multiple goods becauseWalrasian(and Cournot-Walrasian) equi-librium theory determines relative prices but haslittle to say about nominalprice formation. Inthe perfectlycompetitive benchmark,this deficiencyis notan issue, sincethe nominalprice indeterminacydoes notaffectreal quantities.In the monopolistic (imperfectly competitive) equilibrium, however, the realquantities become dependent onthe choiceof price normalization. Indeed, forsome choicesof normalization,Dierker and Grodal (1986) show that no equi-

librium exists, while for others it does exist. This prompted us to considerthe robustness of our results to alternative specifications of the numeraire.Our model contains two commodities, the consumption good and labor, ateach time and state. Instead ofthe consumption good, we could have speci-fied a basket of commodities consisting of units of the consumption goodand (1 ) units of labor as the numeraire, (0, 1).14 Our main quali-tative results and corresponding intuitions would then still remain valid, asdemonstrated inAppendix B.

Relatedto the price-normalization problem, is the issue ofwhether maxi-mizingthe firmsvaluethroughtime isthe correct objective. Fromthe point-of-view of realism, it is generally acceptedthat this iswhat a firm should do,so the problem we solve is consistent with conventional thinking. However,theliterature such asDierker and Grodal (1986) (see also Dierker and Grodal(1999)), clearlycasts some doubt onthis approach. Hart (1985) suggeststhatprofit maximization may be inappropriate in a monopolistic world becausethe owners of the firm are interested not in monetary profit per se but in

wha

t tha

tpr

ofi

tcan b

uy

. Alth

ough

Har

tdisc

usses ma

ximiza

tio

no

fow

ners

utilityas an alternative objective, he points out that the question of how toaggregate is not resolvedwhenthere are multipleowners. Similarly,Bonanno(1990) (in his survey)remarks, unfortunately we are stillfar from a satisfac-tory theory of general equilibrium with imperfect competition. To this day,the price normalization and the objective of the firm under imperfect com-petition in general equilibrium remain open issues. Based on this, we havesimply adopted the most commonly employed price normalization, as wellasthe most conventionally consideredobjective function asthe best startingpoint, and we focus on the implications.

14Such a b asket is a natural and realistic numeraire to adopt, as argued, forexample, by Pavlova and Rigobon (2003) (for more discussion of prices indexes, seeSchultze (2003)).


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4 Connection to the Coase Conjecture

Aswe discussed in Section 3, the time-inconsistency ofthe firms productionplan in our monopolistic pre-commitment equilibrium is similar to the time-

inconsistency arising in the context of a durable good monopoly (e.g., Tirole(1988, Chapter 1) and references therein). The long-lived firms stock in ourmodel is similar to a durable good in that its value is durable over manyperiods. The durability of goods implies that a monopolistic firm in a sensecompeteswith itself in pricing goodsthatitproduces at othertimes and musttake this into account in its production behavior. Similarly, in our context,current owners of the firm can be thought of a s competing with futureowners in maximizing the value of the firms stock; they must account for

possible revisionsto todays intertemporalproduction plan by the futureown-ers. This effective competition between past and future sales in the durablegood problem tends to weaken the monopoly power and may ultimately en-force competitive behavior. This observation is the Coase (1972) conjecture:asthetime intervalbetween successive decision points shorten,the priceofthedurable good charged by the monopolist converges to the competitive price(marginal cost), and the monopoly profits are driven to zero.

To examine the extension of Coases intuition to the case of a long-lived

monopolistic firm manipulatingthe price of its stock,we explorethelimitingcase ofour economy as the decision-making interval shrinks.15 We partitionthe time horizon [0, T] in our economy into n small intervals oflength , sothat t= 0, , 2 , . . . , n = T. All flow variables in the economy ((t), c(t),f(t) and(t)) are now interpreted as flowsoverthe intervalttot + . Wethen takethe limit as n ( 0). Proposition 5 reports the resultingoptimality conditions and equilibrium characterization for the competitive,monopolistic pre-commitment and monopolistic time-consistent economies,ass

uming appr

opria

te reg

ulari

tyc

ondi

tio

ns are sat

isfied.

Proposition 5.Inthe continuous-time limit, asn , t:

(i) The competitivefirmsoptimal labordemandD satisfies

f(D(t), (t)) w(t) = 0. (34)

Consequently, the competitive equilibriumcharacterization isgivenby theconti-nuous-timeanalogsof equations (11)(14).

15Our goal here is to merely explore what happens to equilibrium quantities asthe firms decisions become more frequent, and to not present a continuous-timeextensionofour model. Similar discrete-time models with T = have been arguedto exhibit multiple equilibria as the decision-making interval shrinks, according tothe Folk Theorem (see also Ausubel and Deneckere (1989)), which motivated us tokeep the horizon T finite.


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(ii) The pre-commitmentmonopolisticfirmsoptimal labordemandD satis-fies

f(D(t), (t)) w(t) =A(t) f(t) (t) >0. (35)

Consequently, the monopolistic pre-commitment equilibriumcharacteriza-tion isgivenby the continuous-timeanalogsof equations (18)(21).

(iii) Assume further that (t) [k, K], 0 < k < K < . Then thecontinuous-time limit exists for the monopolistic time-consistent equilib-rium. The time-consistent monopolistic firms optimal labor demand D

satisfies

(D(t), (t)) =f(D(t), (t)) w(t)D(t) = 0, t. (36)

Consequently, the equilibrium labor fort isgivenby

f((t), (t)) v( (t))

u(f((t), (t)))(t) = 0. (37)

Furthermore, fort,

c(t) =f((t), (t)), (t) = 0, (t) =u(f((t), (t)),

w(t) = v( (t))

u(f(, (t)))and V(t) = 0.

The equilibrium characterizationofthe competitive andthe monopolisticpre-commitmenteconomies are exactlyanalogousto the discrete-time charac-terizations.Our main focus isonthe monopolistictime-consistentequilibrium.There, the firms profit and value indeed shrink to zero, as the monopolists

decision interval becomes arbitrarily small in line with the Coase conjec-ture. In other words, the competing forces between current profits and thevaluationof future profits may diminish shareholdervalue, fully destroying itinthe continuous-timelimit. Note, however,that our monopolists profits arenot equated to the competitive ones. This is simply because, in contrast tothetextbookMarshallian framework adopted byCoasewhere competition im-plies free entry of firms, competitive behavior inour setting does not yield zeroprofits in equilibrium. This distinction is dueto ourArrow-Debreu-McKenzie

setting;in particular,to the assumptionsthat there is a fixed numberof firms,andthat the technology of each firm is convex.

5 Conclusion

In this paper, we model a production economy which, along with a standardrepresentative consumer, includes alargevalue-maximizing monopolistic firm.


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The firm manipulates itsvaluation aswellasthe priceofthe goodthatitpro-duces. This feature makes its time-0 production plan time-inconsistent. Weaddressthetime consistency problem intwo polarways: first, we assumethefirm can crediblycommit tonever revoking itstime-0decision(long-term pol-

icy);and second,we assumethat the firmtakes intoaccount thatit willreviseits production plan at each decision point (short-term policy). At a generalequilibriumlevel,we show that thelong-term policyislargely consistent withthe implications ofthe textbook static monopoly model; as compared to thecompetitive economy, the output is decreased and the price of consumptionis increased, yet the profits and the firms value can be either increased ordecreased. The short-term policy, however, is at odds with the static model;the output is increased while the price is decreased. More strikingly, under

the short-term policy,the profits in everyperiod are decreased, and mayevengo negative, while the firms value can drop below than in the competitivebenchmark. The distinction between the long- and short-term policies be-comes even sharper in the continuous-time limit of our economy: while thepre-commitment equilibrium retains its basic discrete-time structure and im-plications, the time-consistent equilibrium tends to the limit of zero profitsand hence zerofirmsvalue atall times. Sinceour primaryfocus is comparisonbetweenthelong- and short-term approaches, and comparisonwith thetext-

book model,we haveomitted an analysisofthevolatilityand risk premiumofthe monopolistic firms stock price, and ofthe impact of monopoly power onthe equilibrium interest rate, market price of risk, output and consumptionvariabilities. This analysis would be straightforward in the continuous-timelimit ofour economy, but lesstractable in discrete-time.

Our analysis involving a single monopolistic firm, a single consumptiongood, and a representative consumer is, admittedly, simplistic. Our goal inthis paper has been to develop the minimal setting possible capturing themechanism

thr

ough

which

the firms marke

t p

ower ma

y impac

t va

lua

tio

n

in the economy, and not to produce the most empirically plausible model.Throughout the paper, we also discuss the robustness ofour implications toalternative modeling approaches. For realism, however, one would need toextendthisworkto include multiple imperfectly competitive firms producinghomogeneous, or differentiated, goods, or multiple consumers. The former(multiple firms) would drawfromthe resultsoftheoligopoly literature,whilethe latter (multiple consumers) would involve aggregation ofthe consumerspreferences into a representative agent.

Appendix A. Proofs

Proof of Proposition1.

Sincethere is no consumptionor production at time0,we can set(0) = 1/yw.l.o.g. Upon substitution of (15) into (1), and using the definition (8), we


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obtain an equivalent representationof(16):

max

E T

t=1

u(f((t), (t))f((t), (t)) w(t)(t) , (A.1)where the firmsobjective is nowa function of{(t); t= 1, . . . , T } only. Thefirst-order conditions for (A.1) are given by

u(f((t), (t)))f((t), (t))(f((t), (t)) w(t)(t))

+ u(f((t), (t)))(f((t), (t)) w(t)) = 0, t= 1, . . . , T .

Usingthe definition ofA(t) and rearranging above yields(17). The sufficient

condition for concavity in footnote 9 guarantees that the labor solving (17)is the maximizer for (16). Since the firm has the option to shut down (setD(t) = 0) during any period [t, t+ 1] in this static optimization, (t) 0and u(f((t), (t))(t) 0. Thistogether with strict concavity of each termin (A.1) (by assumption) guarantees that u(f(D(t), (t))(t) > 0 at theoptimum, and hence(t)> 0. (t)> 0 togetherwithA(t)> 0,f(, (t))> 0(by assumptionson preferences and technology), ensuresthat the expressionon the right-hand side of(17) is strictly positive.

The following Lemmas A.1 and A.2 are employed in the proofs below.Lemma A.1 shows that under regularity conditions (satisfied, for example,by power preferences over consumption u(c) = c/ with (0, 1) andpower production f(, ) = , (0, 1); no additional restriction on v(h)is required) both equilibriumc and belongto the interiorof[0,]. LemmaA.2 is employed in the proofs of Propositions 2, 4 and 5.

LemmaA.1.

Under the standard assumptionsonpreferences andproduction(see Sections2.1 and 2.2), there exists a u n i que solution, c(t) (0,), to (11). As-sume further that lim0 u

(f(, )) f(, )) < , lim0 u(f(, )) < ,

lim0 u(f(, )) f(, )/u(f(, ))< 1 andlim0(u

(f(, ))) f(, ) +u(f(, ))< . Thenthere exists a solution, (t) (0,), to (18).

Proof of Lemma A.1. Since lim0 u(f(, ))f(, ) = > lim0 v

( ), and lim v

( ) = > lim u(f(, ))f(, ), and since u

f

is decreasing in while v

is increasing in , there exists a unique so-lution, c (0,), to (11). Since lim0 u(f(, ))f(, ) v

( ) A(t)f(, )(u

(f(, )f(, ) v( )) = andlim u(f(, ))f(, )

v()A(t)f(, )(u(f(, ))f(, )v()) = , continuity ofu(f(, ))f(, ) v ( ) A(t)f(, )(u(f(, ))f(, ) v ( )) on(0,) together with the boundary behavior at 0 and ensures exis-tenceof a solution, (0,), to (18).


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LemmaA.2.

f(, ) v()u(f(,)) is decreasing in forall.

Proof of Lemma A.2. Assumptions on preferences and production andstraightforward differentiation deliver the result.


Sincethe right-hand sideof(18)is strictlygreaterthan zeroduetoProposition1, and the right-hand side of (11) is zero, it follows from Lemma A.2 that(t)< c(t). The remaining inequalities in(22)(23)are then straightforwardtoderive:theyare dueto f(, )and w =v ( )/u(f(, ))being increasing

in , , the good market clearing, and = u

(f(, )) being decreasing in, . Equation(6)is automaticallysatisfied in equilibrium dueto clearing inthe good and labor markets, hence y is indeterminate and we can normalizey= 1. Togetherwithour earlier normalization(0) = 1/y,thisyields(0) = 1in boththe competitive and monopolistic pre-commitmentequilibria. Forw =w, the function (,,w)u(f(, ))(f(, ) w) achieves its maximumat. is strictlydecreasing inw, hence for anyw > w,u(f(, ))(f(, )w)< u(f(, ))(f(, ) w), for all. Thistogetherwithwc(t)> w(t)impliesthatfor eachterm inthe expression forV(0) (1),we have(t)(t)>c(t)c(t), t. HenceV(0)> Vc(0).Example1 provides evidence forthelastassertion ofthe Proposition.


Substituting (15) into (1) and solving (24) at time t = T, we obtain theoptimality condition f(T) w(T) = 0, identical to that of the competitivefirm. Consequently, V(T) =f(T) w(T)D(T) >0 andVex(T) = 0. At time

t= T 1, the first-order condition for (24) isf((T 1), (T 1)) w(T 1)

u(f((T 1), (T 1))) f((T 1), (T 1))

u(f((T 1), (T 1)))2

E[u(f((T), (T))) {f(T) w(T)D(T)}|FT1] = 0 .

Using the definition ofVex and rearranging above yields (25) at time T 1.Continuing the backward induction, we obtain (25) for all t = 1, . . . T 1.

The sufficientcondition for concavityin footnote11 guaranteesthat thelaborsolving (25) is the maximizer for (24). V(t), obtained on each step of thebackward induction, is strictly positive because the firms objective in (24)in strictly concave and V(t) 0 (V(t) < 0 is ruled out since the firm canshut down at any time and thus increase its value to zero). Consequently,

Vex(t) =E[(t+1)(t) V(t + 1) |Ft]is strictlypositive. ThistogetherwithA(t)> 0

andf(, (t))> 0 impliesthat the right-hand sideof(25) is strictlynegative.


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LemmaA.3.

Under the standard assumptionsonpreferences andproduction(see Sections

2.1, 2.2), and the regularity condition 2 u(c)

u(c) + u(c)u(c) <

f(,)f2(,)

, c =

f(, ); , , (footnote 11), there exists a unique solution, (t) (0,), to(26).

Proof of Lemma A.3. Since on each step of the backward induction,lim0 u

(f(, ))f(, ) v( ) + A(t)u(f(, ))f(, )Vex = andlim u

(f(, ))f(, )v()+A(t)u(f(, ))f(, )Vex = and sinceu(f(, ))f(, )v

()+A(t)u(f(, ))f(, )Vex is decreasing in,there

exists aunique solution, (0,), to (26).


Sincethe right-hand sideof(26)is strictly lessthan zerodueto Proposition3,

andthe right-hand sideof(11) is zero, it follows fromLemmaA.2that(t)>c(t). Consequently, f(, ) and w = v( )/u(f(, )) being increasing in, , and the good market clearing yield the comparisons on f, c, w. Thecomparisonon follows from =u(f(, )) being decreasing in , .

To prove the remaining statements, define the equilibrium profit function((t), (t)) f((t), (t)) v((t))

u(f((t),(t)))(t). The first-order condition for

maximization of((t), (t)) with respect to is satisfied by (t) suchthat

f((t), (t)) v( (t))

u(f((t), (t)))

= v( (t))

u

(f((t), (t)))

+A(t) f((t), (t))

u

(f((t), (t)))

v( (t))

>0 . (A.2)

Due to Lemma A.2, (t) < c(t), t, (t). It is straightforward to verify that

for any such that f v()u(f(,)) 0, , < 0. This together with (26)

implies that (c(t)) < 0 and ((t)) < 0 t, (t). Continuous function

( ; ) does not change signon[c(t),(t)], because if it did,there hadto be

a point(t) on[c(t), (t)]satisfying((t)) = 0,which is notpossible forwe

have shown that any such point has to lie to the left ofc(t). Consequently,( ; )is monotonicallydecreasingon[c(t),(t)], and hence(t)< c(t), t.

It then follows that since (t) < c(t), (t)(t)< c(t)c(t), and hence from

(1) that (t)V(t) < c(t)Vc(t). This together with the argument behind thenormalization adapted fromthe proofof Proposition 2,yields V(0)< Vc(0).

Since at timeT the equilibrium conditions(11)and(26) ofthe competitive

andthe monopolistictime-consistenteconomies coincide,(T) =c(T),yield-ing equalizationofthe remaining equilibriumquantities at timeT. Finally, an


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example can be constructed to verify that V(t) can be lower or higher thanVc(t). Example 2 of Section 3.2 demonstrates this under a modified set ofassumptionson v(h).


Consider an arbitrary partition t = 0, , 2 , . . . , n = T of [0, T]. Theconsumer-workers problem is nowgiven by

maxc,

E

Tt=1

u(c(t)) + v( (t))

subject to E

Tt=1

(t)

c(t) w(t) (t)

E

Tt=1

(s)(s)

,

wheretheflow ofutilityis definedover a rateof consumption andleisure. Theabove yieldsthe first order conditions

u(c(t))=y (t) , (A.3)

v( (t))=y (t) w(t) .

For any , the first-order conditions are equivalent to (4)(5), consequently,the consumer demand facing the firm is the same as in the discrete case.The value ofthe firm is given by

V(t) =E

Ts=t

(s)

(t)(f((s), (s)) w(s)(s))

Ft

. (A.4)

The competitive firm is choosing D to maximize V(0) in (A.4) taking asgiven; accordingly, its labor demandD satisfies for all t= , . . . , T

f(t) w(t) = 0.

The pre-committed monopolist is choosing D and so asto maximizeV(0)in (A.4) subject to (t) =u(f((t), (t)))/y (as implied by (A.3)). His labordemand D satisfies

f(t) w(t) = A(t)f(t)(t).

Since the equations above are independent of , equations (34) and (35)obtain;in addition,the continuous-timelimitsofthe discrete-time competitiveand monopolistic pre-commitmentequilibria are nowgiven by the continuous-time analogsof(11)(14) and(18)(21), respectively, for all t [0, T].

Similarly, the problem ofthe time-consistent monopolist is given by (24)with V(t) specified in (A.4). The backward induction solution yields the fol-lowing for the firmslabor demand D


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(f(t) w(t)) = A(t) f(t) Vex(t) 0 , t= , . . . , T . (A.5)

(t)is bounded from above by, and, anticipating equilibrium, sincethe right-hand of(A.5) is nonpositive, it is bounded from below by c(t) > 0 due to

Lemma A.2. (t) is bounded by assumption, hence f(, ),f(, ), u(f(, ))and A(t) are bounded. As we take the limit as 0 in (A.5), given theboundedness, its left-hand side tends to zero, hence, so must the right-handside. Giventhe boundednessof all quantities exceptVex(t) onthe right-handside of(A.5), we must have Vex(t) 0. Consequently, (t) 0; in addition,givenour boundedness argument,V(t) 0, andthe firmslabor demand hasto be suchthat D(t) yields zero profit (t) =f(D(t), (t)) w(t)D(t) = 0,as is stated in(36). In the resulting equilibrium, (t) = 0 aswell, hence(37).

(37) yields the equilibrium labor, which in turn determines the remainingequilibriumquantities.

Appendix B. Alternative Specifications of the Numeraire

Supposethat insteadofthe consumption good,we specified a basket of com-modities consisting of units ofthe consumption good and (1 ) units oflabor as the numeraire, (0, 1). In other words, the price ofthis basket isnormalizedto 1:

1 =p(t) + (1 )w(t), t= 1, . . . , T , (A.6)

where p(t), w(t) are prices ofthe consumption good and labor, respectively,in units ofthe numeraire.

Underthis numeraire,the price-taking consumer-workers problem is givenby

maxc,

E

Tt=1

u(c(t)) +

[Alessandro Citanna, John Donaldson, Herakles Pole

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