Arrow and General Equilibrium Theory - Institute for Advanced Studies

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American Economic Association
Arrow and General Equilibrium Theory
Author(s): Darrell Duffie and Hugo Sonnenschein
Source: Journal of Economic Literature, Vol. 27, No. 2 (Jun., 1989), pp. 565-598
Published by: American Economic Association
Stable URL: http://www.jstor.org/stable/2726689
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Journal of Economic Literature
Vol. XXVII (June 1989), pp. 565-598
Arrow and General Equilibrium
Theory*
By DARRELL DUFFIE
Stanford University
and
HUGO SONNENSCHEIN
University of Pennsylvania
We are pleased to acknowledget he helpful commentso fJohn Roberts,
Faruk Gul, Salvador Barbera, Dilip Abreu, Vijay Krishna, and two
referees. We are also grateful for bibliographic assistancef rom Mat-
thew Jackson and Remy Chaudhuri and for typing by Jill Fukuhara.
I. Introduction sponse to contemporarye quilibriumt heory: To
what extent has the achievement been primar-
THE GREATEST ACHIEVEMENTS of economic ily technical? What have been the major con-
theory concern the determinationo f value ceptual breakthroughs?W hat is the importance
in competitive marketsa nd the extent to which of the welfare theorems? Do they make a sub-
competitive markets lead to an efficient alloca- stantive contribution?W hat is the logical status
tion of resources. Kenneth J. Arrow'sc ontribu- of a theory of value that admits the possibility
tion has been central. The purpose of this essay of a large number of equilibriump rice vectors?
is to place Arrow'sw ork on general equilibrium Has modern general equilibriumt heory helped
theory into perspective and, in particular, to or hindered the creation of more dynamict heo-
evaluate criticallyh is three majorc ontributions ries of price determination?W hat are the uses
to the theory: his proof, with Gerard Debreu, of the theory? Has Arrow's emphasis on the
of the existence of general economic equilib- mathematical and axiomatic model served us
rium; his analysis of the relationship between well? What are the unifyingt hemes in his work?
equilibriaa nd optima;a nd his extension of equi- Finally, as a tribute to the enduring quality of
librium theory to cover the case of uncertainty. Arrow'sc ontributionw e will discuss where his
After explaining the contents of Volume 2 of ideas have taken us and consider some ques-
the Collected Papers (hence-forth referred to tions on the frontiers that appear especially
as CPII) and reviewing the genesis of the gen- compelling.
eral equilibrium model, we will take these up
Contents of the Book
in turn. Our discussion will emphasize some
of the most contentious questions raised in re- In our review of this volume, General Equi-
librium, the second of six1 comprising Arrow's
* This is a review of Kenneth J. Arrow. Collected
Papers of KennethJ . Arrow. Vol. 2. GeneralE quilib- 1 These are as follows: Volume 1-Social Choice
rium. Cambridge, MA: Harvard U. Press, Belknap and Justice, Volume 2-General Equilibrium, Vol-
Press, 1983. ume 3-Individual Choice Under Certainty and Un-
565
566 Journal of Economic Literature, Vol. XXVII (June 1989)
CollectedP apers, we focus on Arrow'sc ontribu- Each chapter is actually a lightly edited ver-
tions in three areas:( a) the existence of equilib- sion of the paper as originally published, with
ria, (b) the relationship between equilibrium new headnotes "to give the reader some insight
and Pareto optimality, and (c) general equilib- into the circumstancest hat motivated the writ-
rium under uncertainty;t hese contributionso c- ing" (CPII, p. vi). While these circumstances
cur principallyi n chapters 4, 2, and 3, respec- are usually interesting, it is sometimes discon-
tively. Chapters 1 and 5 complete the segment certing not to know (from this volume alone)
of Arrow'sw ork from this volume that originally where the headnotes end and the paper itself
appeared during 1951-54; these are important begins.
papers which deal with substitution in the
The Development of General Equilibrium
Leontief model of production. The remaining
Theory
chapters, numbered 6 through 14, were origi-
nally published during 1968-81. Chapter 11 is What are the origins of general equilibrium
more an excursion into the theory of difference theory? The classical economists (and we have
equations than a piece of general equilibrium particularly in mind Adam Smith, Ricardo,
theory (althoughi t may be argued that the class J. S. Mill, and Marx) had a theory of value
of difference equations under consideration is that is driven by the cost of production and a
of special interest in the study of dynamic eco- zero profit condition. Insofara s they took mar-
nomic models). A large portion of Arrow's re- kets to be related, their work had an aspect of
search on general equilibrium theory between general equilibrium;h owever, they ignored the
1954 and 1968 is collected, along with some influence of demand on value. Arrowa nd Star-
of the work of his collaboratorL eonid Hurwicz, rett (1973), with the classicists in mind and
in Studies in ResourceA llocation Processes (Ar- Leontief to lean on, provide the following ac-
row and Leonid Hurwicz 1977). The best- count of how value can be explained without
known work from Arrow and Hurwicz (1977) reference to demand.
deals with the stability of economic equilib-
Therei s one primaryfa ctoro f productionA. ll
rium; an example of Arrow'sw ork on stability other goodsa re producedu nderc onditionso f
can also be found in chapter 12 of the volume fixed coefficientsw ith one output,t he inputs
under review here. While our review does not beingt he primaryfa ctora ndp ossiblyo therp ro-
deal in detail with the work contained in the duced goods. . . . In symbolsl et pi be the
Arrow-Hurwiczv olume, we believe that the priceo f producedc ommodityi, v the priceo f
research program set forth there is important the primaryfa ctora, yt he amounot f commodity
for understandingA rrow'sc ontribution, so we j usedi n the productionof one unito f commod-
ity i, and b the amounto f the primaryfa ctor
will give it some attention. The bulk of the
in the productiono f one unit of commodityi.
material in the remaining chapters (6, 7, 8, 9,
Thent he conditiono f zero profitsi s
10, and 13) is expository, and illustrates well
Arrow's strength in this dimension. (Chapter
Pi aijpj+ biv,
10 is joint work with David Starrett.) Notable
among these chapters is the published version
since the right-handsi de is the cost of produc-
of Arrow's Nobel lecture of 1972, reprinted as ing one unit of commodityi. As we let i vary
chapter 9. Chapter 14, the last, studies the effi- overt he producedc ommoditiesw, e havea sys-
ciency of allocations with costly transfers. Be- tem of equationsi n the unknownp rices, pi,
yond Arrow's work in general equilibrium in v. . . . The classicael conomistism plicitlya nd
the CollectedP apers and in Arrowa nd Hurwicz Leontiefe xplicitlys olved for these prices in
(1977), there are the monographsG eneral Com- termso f v; thus the price of each commodity
petitive Analysis, coauthoredw ith Frank Hahn is a constantm ultipleo f v and the multiplei s
completelyd eterminedb y the input-output
in 1971, and Essays in the Theory of Risk Bear-
coefficientas ija nd bi. (p. 228, CPII)
ing (Arrow1 970).
Antoine Cournot (1838) saw clearly the role
of demand in the determinationo f equilibrium
certainty,V olume 4-The Economicso f Information,
Volume 5-Production and Capital, and Volume 6- in a single market; not until the neoclassicists
Applied Economics. arrived on the scene, however, does one have
Duffie and Sonnenschein: Arrow and General Equilibrium Theory 567
a fully integrated multimarkett heory of value. Small general equilibrium models, for example
The role of Leon Walras in both incorporating with one set of indifferencec urves to represent
demandi nto the explanationo f value and simul- the utility levels in each of two countries, play
taneously taking into account the relationship a significant role in the analysis of trade and
of marketsi s central. Joseph Schumpeter (1954) taxation issues. Also, the usefulness of general
put it most strongly: ". . . the discovery [of equilibriumt heory is not restrictedt o situations
economic theory] was not fully made until Wal- in which we can determine the data of an econ-
ras, whose system of equations, defining (static) omy and solve for equilibrium prices. Some of
equilibriai n a system of interdependent quanti- the most important lessons to be learned from
ties, is the Magna Carta of economic theory" the theory are qualitative, such as the condi-
(Schumpeter 1954, p. 242). tions under which free markets and exchange
A refined version of the Walrasiant heory sur- lead to an efficient allocation of resources, or
vives today as our best expression of the forces which commodities to tax in order to raise reve-
that determine relative value. In general, price nue most efficiently. Today, the general equi-
is not determined by technology alone: A librium model is not the exclusive province of
change in tastes will influence the price of a the high-tech theorist; rather, it is a basic part
pound of salmon relative to the price of a pound of the professional economist's tool bag, and
of calf's liver; this will influence the quantity one that is increasingly used.
produced of certain wines as well as their price.
The price of a bottle of 1934 Yquem is influ- II. The Existence of General Economic
enced by the weather in that year; it is also Equilibrium
influenced by the distribution of wealth. The
A Modern Synthesis of the Walrasian Model
Walrasian theory has the capacity to explain
and Existence of Equilibria
the influence of taste, technology, and the dis-
tribution of wealth and resources on the deter- The existence2 problem is stated in a simple
mination of value. Nothing that came before form as follows. The data of a private ownership
the Walrasiant heory had this capacity. Neither economy are tastes, technology, the initial hold-
partiale quilibrium theory nor theories that de- ings of commodities by consumers, and the
pend on technology and resources alone pro- ownership of firms. All agents are assumed to
vide as strong an explanation of value. Al- take prices as given. The supply function of
though, for certain markets, it is possible to each firm is assumed to be single-valued (this
explain how price responds to small parameter means that for each vector of input and output
changes with partiale quilibriumr easoning, few prices there is a unique profit-maximizingp ro-
economists would contend that this method is duction plan), and the net aggregate supply of
adequate when economies are disturbed in a firms is obtained by adding the supplies of indi-
major way. (We have in mind, for example, a vidual firms. (By convention, inputs are de-
major disruption in the capacity to move oil noted by negative quantities and outputs by
or a large increase in tariffs.)M oreover, if one positive quantities.) Each household's income
is to start an investigation into relative values is determined by the value of the household's
without beginning from given prices, as might initial endowment and the value of the profit-
be appropriatef or the analysis of an economy maximizing actions of the firms in which the
some decades in the future, then it is quite household holds stock; these in turn are both
clear that the Walrasiant heory is the most use- determined by prices. Household aggregated e-
ful conceptual frameworka vailable. mand, which depends on prices and the distri-
Before we turn to the existence of general bution of income, is thus seen to depend on
equilibrium and Arrow's contribution, some prices alone. Finally, household aggregate sup-
further words about the usefulness of general
equilibrium theory are in order. The essence
of general equilibriumd oes'not preclude aggre- 2Arrow's work on the welfare theorems and equi-
librium under uncertaintyp receded his work on the
gation;w hat is essential is an emphasis on inter-
existence theorem. We reverse the ordero f presenta-
marketr elationsa nd the requirement that vari- tion in order to put the more foundationalr esult in
ables are not held fixed in an ad hoc manner. its proper place.
568 Journal of Economic Literature, Vol. XXVII (June 1989)
ply is the sum of initial endowments. The condi- Existence of Equilibrium Before the Early
tion that the difference between aggregate de- Fifties
mand and aggregate supply is zero, applied to
It is usually stated that Walrast ook the obser-
each commodity, yields the familiare xcess-de-
vation that his system contains as many equa-
mand system of f equations in the f price varia-
tions as unknowns as a proof of the existence
bles:
of equilibrium, and this is probably correct.4
zi(pl, pe) = O, i = 1,2,. e. In any case, an adequate proof of the existence
of equilibrium requires more than counting
The forces of supply and demand, which are equations, and Walras had nothing past this
defined by the data of the economy, will be to offer. Indeed, the requisite mathematicsf or
in balance at prices p if and only if p solves the existence of solutions to nonlineare quations
In Walrasianth eory,v alue is determined
(1).3 such as (1) were not available to Walras; this
by a solution to (1): General equilibrium re- came only later with Luitzen Brouwer's( 1912)
quires that all markets clear. Such an equilib- fixed-point theorem.
rium price vector p depends, through the de- The formal existence theory begins with the
mand functions of consumers and the supply formulation of the Casselian system (Gustav
functions of firms, on the primitive data:t astes, Cassel 1924)a nd its refinementb y Hans Neisser
technology, and endowments. (1932), Heinrich von Stackelberg( 1933), Fred-
The role of an existence theorem is to insure
erick Zeuthen (1933), and Karl Schlesinger
that, for all economies from a broad class, there
(1933-34). This led to the first rigorous proofs
will be at least one solution to (1) in nonnegative
of the existence of equilibrium, by Abraham
prices. In the absence of a solution the model Wald (1933-34, 1934-35, 1936).5 Wald's pub-
does not offer a theory of value. lished proofso f existence require that aggregate
One sometimes hears that the existence of demand be independent of the distribution of
equilibrium follows from the observation that, income and satisfy the weak axiom of revealed
because the total quantity of any good sold is preference in the aggregate.6 In this Wald is
necessarily the total quantity purchased, the close to the case in which aggregate demand
prices we observe in the actual world are equi- is of the class generated by a single consumer.
librium prices. General equilibrium theory is The requirement that demand has such a spe-
an attempt to explain the prices that we ob- cial form is extremely restrictive;i t has the ad-
serve. The equilibrium prices of our model are vantage, however, that it guarantees unique-
taken to correspond with the prices that we ness of equilibrium, and Wald appreciatedt his
observe in actual economies, but they are not point. Furthermore,i t greatly simplifiest he ex-
the same objects as the prices in actual econo- istence proof by allowing one to replace fixed-
mies. In particular, it is not necessarily true
that observed prices are market-clearing.I n or-
4 Leon Walrasu nderstood the possibility of multi-
der to understandt he implicationso f assuming
ple solutions. His example (Walras1 874-77), which
that observed prices are market-clearing,o ne is interpreted by Jaffe (in his notes that accompany
needs at least a well-articulated theory with the English translation,1 954, of Walras)a nd Schum-
sharp criteria for the existence of market-clear- peter (1954) to demonstratet hat he fully appreciated
the possibility of nonexistence, does not make the
ing prices. A model may or may not have at
point in a satisfactorym anner (see Takashi Negishi
least one equilibrium price, and when it does 1987).
not produce at least one equilibrium price it 'Arrow reviews the history of existence theorems
will not serve to explain prices as devices that in his paper "EconomicE quilibrium"f or the Interna-
tional Encyclopediao f the Social Studies (1968). This
clear markets.
is chapter 6 in CPII. See also Weintraub (1983).
6 The weak axiom of revealed preference requires
3Because the decisions of households and firms that there is no pair of distinct commodity bundles
depend only on relative prices, and because for an x and y such that when x is demanded y can be
arbitraryp rice vector (Pl, P2, . . , Pe) the value affordeda nd when y is demanded x can be afforded.
of excess demand is zero (this is Walras' law: Wald speaks of a result in which demand is generated
IiPAi(PbP2, * , Pe) = 0), the system( 1)i s properly by utility-maximizingc onsumers (this does not imply
regarded as composed of f - 1 equations in f - 1 the weak axiom in the aggregate), but the proof was
unknowns. never published.
Duffie and Sonnenschein: Arrow and General Equilibrium Theory 569
point theorems by a maximization argument McKenzie begins with a continuous aggregate-
and a separatingh yperplanet heorem. (The sep- demand function that is hypothesized to satisfy
arating hyperplane theorem is stated in Foot- a boundary condition that cannot be shown,
note 28.) But, againt here is no reasont o believe in general, to follow from utility maximization.
that aggregate demand takes such a special This is indeed a weakness, or at least an incom-
form. pleteness, in McKenzie's approach; however,
it should not obscure the importance of his
The Modern Existence Theorems
contribution8.
In 1954, Arrow and Gerard Debreu pub- In their search for conditions on the charac-
lished their proofo f the existence of equilibrium teristics of household preferences that are suffi-
for a competitive economy (this is chapter 4 cient to ensure that the excess-demand func-
in CPII). Lionel McKenzie (1954) provided a tions in (1) are sufficiently continuous, Arrow
proof at approximatelyt he same time as Arrow and Debreu discovered an important demand
and Debreu. David Gale (1955) and Hukukane discontinuityp roblem. With monotonic prefer-
Nikaido (1956) had their own versions of an ences (more of a commodity is better), when-
existence theorem. We take the point of view ever the price of a commodity is zero, the
that these proofs are primarily technical amount of the commodity demanded will be
achievements for which the required mathe- unbounded (or more precisely, undefined).
matical machinery had recently been put in Furthermore, if the initial endowment is not
place.7 We will argue further that the Arrow interior to the consumption set, the quantity
and Debreu proof involved a substantial demanded can vary continuously with positive
amount of technical innovation and that the prices, but be undefined precisely at a zero
method of proof has been especially fruitful. price. Even with a one-consumere conomy hav-
The McKenzie and Arrow-Debreu results ing smooth and strictly convex preferences and
were presented at the 1952 Winter Meeting no production, equilibrium need not exist if
of the Econometric Society. For a model with the consumer's initial endowment is on the
general demand, these are the first equilibrium boundaryo f the consumption set.
existence results communicated to large audi- The basic difficulty is illustrated in Figure
ences and out in print. They are rightfully ac- 1. Indifference curves over nonnegative com-
corded a special place, but one should keep modity pairs are labeled ID, I2, and I3; their
in mind the independence of the Gale and Ni- slopes approach zero as the quantity x2 of the
kaido treatments and the fact that publication second good approachesz ero. The endowment
of the Nikaido paper was delayed. Also, the vector x contains a positive amount of only the
argument used by Nikaido and Gale plays a first commodity. The excess demand for the
central role in the classical treatment of exis- first commodity is negative when Pi =A0 and
tence provided later by Debreu (see Debreu excess demand approaches zero as Pi ap-
1959, p. 82). McKenzie, unlike Arrowa nd De- proaches zero. The equilibrium price "wants
breu, assumed constant-returns-to-scalein pro- to be" Pi = 0, in the sense that excess demand
duction. This difference vis-a-vis the Arrow- approaches 0 as Pi approaches 0; however, at
Debreu formulation is superficial, however,
because each Arrow-Debreu economy (with
8 For the record, it must be stated that Arrowa nd
possibly regions of strictly decreasing returns
Hahn's reading of McKenzie's contribution, that
to scale) can be embedded in an equivalent if specialized to the case of exchange, it is
constant-returns-to-scalee conomy (McKenzie identical to Wald's"( Arrowa nd Hahn 1971, p. 51),
1959). A more substantial difference between does McKenzie an injustice. To be specific, McKen-
zie did not assume, nor do his conditions imply, that
the Arrow-Debreua nd McKenzie formulations
aggregate demand satisfies the weak axiom of re-
is that Arrow and Debreu start with individual
vealed preference. In a letter to Roy Weintraub( the
consumers and a distribution of wealth, while relevant portion of which is published in Weintraub
1983, pp. 36-37), Arrowl ater recognizes this distinc-
tion. Although McKenzie does limit himself in his
7See Roy Weintraub (1983), Arrow (1987), and existence proof to the case of special (Cobb-Douglas)
Debreu (1987) for more detailed and personal ac- demand functions, he points out (p. 155) that the
counts. techniques are adequate for the general case.
570 Journal of Economic Literature, Vol. XXVII (June 1989)
require, in theorem 1 of their paper, that the
X2 initial endowment of each household be interior
to its consumptions et. (Arrowh ad faced a diffi-
culty much related to the demand discontinuity
problem in his earlier work on the second wel-
fare theorem. More about this appears in Sec-
tion III.)
'2
budget line at
The Arrow-Debreu Method
price p*
By their own account, the work of Wald had
1 0 s n = o
x pi#O very little direct influence on Arrow and De-
breu. It would appear that they each had a
13 feeling for the intrinsic importance of the
problem.9A n appropriatet ool was needed, and
this was provided by Shizuo Kakutani's( 1941)
x generalization of the classic Brouwer (1912)
Figure1 . Discontinuity of Demand at a Zero Price fixed-point theorem. In Arrow'sv iew, he, De-
breu, and McKenzie were simultaneously
primed by the work of John Nash (1950) for
p,= 0 excess demand is not defined (because the use of that theorem. He writes, "It was
more of good 1 is always preferred, and budget the paper of John F. Nash, Jr., showing the
feasible). Equilibrium in a one-consumer econ- existence of equilibriump oints to games by the
omy requires that the consumer demands his use of Kakutani's (1941) fixed-point theorem
initial endowment, but in Figure 1 there are [equivalentt o von Neumann's (1937)], that sug-
no prices at which this is the case. The problem gested to several of us the correspondinga naly-
is similar in economies with more than one con- sis for the concept of general competitive equi-
sumer, and can be resolved by requiring that librium. Gerard Debreu, Lionel McKenzie,
the initial endowment of each household be and I all followed up this lead independently,
interior to the consumption set (see, for exam- each in his own way" (CPII, p. 58).10T he influ-
ple, Negishi 1987, p. 371). (To see why, observe
that if the consumer's initial endowment is x,
9 There are currents of indirect influence, and of
then a low (high) relative price for x1 leads to particulari nterest is one in which OskarM orgenstern
a positive (negative) demand for xl, and that plays a major role. Morgenstern was familiar with
demand varies continuously with price. At the the existence question early on. In the mid thirties,
Morgensternp ut the economist Schlesinger in touch
budget line that is drawn, supply equals de-
with the mathematicianW ald, taking pride in his
mand.)
institute's role in the birth of the first rigorous exis-
One of the nice contributions of the Collected tence theorem. After he moved to the United States
Papers is the accounts that precede each paper. in 1938, Morgensternw orked to keep the importance
The "demand discontinuity problem" we have of the existence question alive. In a review of Hicks'
Value and Capital Oskar Morgenstern( 1941)w rites:
just discussed is of historic interest because of
its role in the Arrow-Debreu collaboration. Ar- Hicks . . . is systematically incorrect [when he counts
equations in order to argue the existence of equilibrium
row and Debreu began their work on an exis- prices] because the determination of a system of equations
tence theorem independently. As Arrow writes, does not necessarily depend only upon the equality of the
"Debreu and I sent our manuscripts to each number of unknowns with the number of equations. ...
We have as yet such [existence] proofs, only for two systems
other and so discovered our common purpose. of equations, those of von Neumann and of Wald.
We also detected the same flaw in each other's Finally, although Morgenstern'sc ollaborationw ith
work; we had ignored the possibility of discon- von Neumann did not result from his interest in the
tinuity when prices vary in such a way that Walrasiane xistence theorem, it led to John Nash's
equilibrium result for games, which is the technical
some consumers' incomes approach zero. We
starting point for Arrow and Debreu's independent
then collaborated, mostly by correspondence,
attack on the existence theorem.
until we had come to some resolution of this 10B ecause Arrow and Debreu began their work
problem" (CPII, p. 59). This resolution was to independently, a substantiala mounto f reconciliation
Duffie and Sonnenschein: Arrow and General Equilibrium Theory 571
ence of Nash on McKenzie is perhaps less clear proved that generalized games have an equilib-
than what is suggested by Arrow. McKenzie, rium provided that (1) the feasible sets are con-
Gale, and Nikaido approached the existence vex and vary continuously with the actions of
problem through the device of the aggregate all players, (2) the value of the payofff unctions
excess-demandf unction. They made no attempt vary continuouslyw ith the actionso f all players,
to accommodatet he Nash theorem to their pur- and (3) for all fixed choices by other players,
pose and indeed their arguments may be re- choices by a player that raise his own payoff
garded as having more transparent economic form a convex set.11 At the heart of Debreu's
intuition than that provided by Arrow and De- proof is the fixed-point theorem of Kakutani
breu. Arrowa nd Debreu's approach,v ia an ex- (1941), who was motivated by von Neumann's
tension of the Nash theorem, is quite subtle, (1937) work on games.
and we will see in a moment that it is especially The Arrow-Debreu proof of the existence of
well suited for extensions of the existence theo- Walrasiane quilibriumf or an economy proceeds
rem. With this backgroundw e now turn to an by (1) associating a generalized game with the
exposition of the Arrow-Debreu method. economy, (2) proving that there exists at least
An n-person game is defined by specifying one equilibrium of the generalized game, and
for each agent (a) a set of a priori available (3) demonstratingt hat in an equilibrium of the
choices, and (b) a real-valuedf unctiont hat spec- generalized game all markets in the economy
ifies the agent's payoff as a function of the n- clear. The continuity and convexity properties
tuple of choices by all agents. A Nash equilib- required for applicationo f Debreu's result fol-
rium n-tuple of choices has the property that, low from convexity assumptionso n production
given the choices of other agents, each agent's sets and preferences plus (for example) the re-
payoff is maximized. In economic models, quirement that endowments be interior to the
agents may be constrained to pick from sets consumption set.
that depend on the choices of others; for exam- The above approach to the existence theo-
ple, a consumer must choose from the set of rem, because it operates at a level prior to the
bundles that he can afford,w hich indirectly de- aggregation of supply and demand, is funda-
pends on what other agents choose. (For exam- mentally different from looking for a solution
ple, what others choose influences prices and to the excess-demand function system. From
prices determine what an agent can afford.)I n a technical perspective it represents an ex-
order to handle cases in which, for each agent tremely creative step; in addition, it has great
there is a set of feasible choices that depend power as a vehicle for extending the existence
on the choices of others, Debreu formulated theorem. A few examples illustrate the latter
the notion of a generalized n-person game. In point. The Walrasiant heory of value has been
a generalized n-person game, equilibrium re- criticized for failing to take into account the
quires that, given the choices of other agents, fact that an agent may judge quality by price,
each agent's payoff is maximized subject to or be inconsistent in his choices (as when pref-
feasibility; furthermore, what is feasible de- erences are not transitive),o r have preferences
pends on the actions of others. Debreu (1952) that depend on the choices of other agents.
The original Arrow-Debreu proof, unlike the
proofs that work via the construction of an ex-
was essential before they could proceed with their
cess-demand function (such as in McKenzie
collaboration.T his was facilitatedb y the fact that in
their independent startst hey had attempted to incor- 1954; Nikaido 1956; Gale 1955; and Debreu
porate Kakutani'st heorem by extending Nash'se qui- 1959), is relatively easily modified to take into
librium theorem for games. Ultimately, as Arrowe x- account all of these ingredients. The general-
plains, they ". . . followed more closely Debreu's
ized game approacha llows each agent's payoff
more elegant formulation, based on the concept of
generalized games" (CPII, p. 59). Arrow describes to be influenced by the choices of others, and
his own formulationo n page 59 of the Collected Pa- so it applies to economies in which the pre-
pers. It is now known that the equilibriume xistence ferred sets of each consumer vary with the
theorem is equivalent to the Brouwer-Kakutantih eo- choices of other consumers or producers (as
rem. This is a consequence of the fact that the class
of excess-demandf unctionsi s knownt o have no struc-
"
ture beyond homogeneity and Walras'l aw. This is, of course, a loose statement.
572 Journal of Economic Literature, Vol. XXVII (June 1989)
would be the case with externalities) or with model. Furthermore, a case can be made that
prices (as when quality is judged by price). Sim- we have been slow to pay attention to the limi-
ilarly, nontransitivities in preference are ad- tations of the Walrasiant heory as a theory of
mitted by allowing, for each choice by the value. For example, the Walrasiant heory does
consumer, a different set of relevant "quasi- not shed light on when price taking applies,
indifference curves."12A s long as these quasi- or how prices are formed, or which vector of
indifferencec urves satisfy the standardc onvex- prices will prevail when there is more than one
ity requirement and vary continuously with that clears markets. These limitations have
one's own choices and the choices of others, been increasingly apparent as the theory has
the Arrow-Debreu proof (via the Debreu been refined; to our mind their realization
lemma) still goes through. (The result is ob- should be regardeda s one of the majorp roducts
tained in a different manner by Andreu Mas- of modern equilibrium theory. We will close
Colell 1974.) Similarly, the Arrow-Debreu ap- this section on existence by considering some
proach works well to establish an existence of the major triumphs that have been accom-
theorem for economies with taxation (obtained plished since the 1950s as well as some of the
in a different manner by Kevin Sontheimer major challenges that remain.
1971 and John Shoven 1974), or with externali- The existence theorems, while absolutely
ties in production.'3 Although Arrow and De- fundamental,a re primarilyt echnical in nature,
breu do not mention these extensions, it is diffi- and so it is not surprising that many of the
cult to imagine that they did not appreciate major advances since the 1950s have also been
the possibility of at least some of them. technical. Nevertheless, the step forward that
we regard as the most significanth as required
Existence Theorems:T heir Importance and
a reformulation of the basic model. To most
Their Shortcomings
readers the most unrealistic descriptive as-
Here then is the history of value theory that sumptions imposed by Arrow and Debreu in-
emerges. Before the neoclassicists the concep- volve the convexity of preferences and produc-
tion of the determinants of value was seriously tion sets."4 Both of these assumptionsr ule out
flawed. In particular, multimarket models ig- indivisibilities and the latter rules out fixed
nored the influence of demand. Walras made costs and the resulting U-shaped average cost
the major advance by providing a model of curves; as a result they would appear to limit
economies in which value is determined by sup- the applicabilityo f the Walrasianm odel greatly.
ply and demand in all markets simultaneously. But without them household demand and firm
Over a period of 80 years Walras' ideas were supply can be discontinuous and there might
refined, but it was not until Arrow-Debreu, appeart o be little possibility for an equilibrium
McKenzie, and their contemporariesp rovided existence theorem. General equilibrium theo-
sets of conditions under which general equilib- rists now understand that convexity of prefer-
rium must exist that the relevance of the Wal- ences and production sets is less critical than
rasiant heory of value was understood. was believed. Provided that one is willing to
On the positive side, the existence theorems formulate an economy as composed entirely of
of the 1950s have led to a deep understanding agents who are infinitesimalr elative to the mar-
of the conditions under which, for each econ- ket (we will discuss other reasons for favoring
omy of price-takinga gents, there exists at least this formulation in a moment), the convexity
one vector of market-clearingp rices. On the assumption can be dispensed with. Individual
negative side these conditions are seen to be behavior may be discontinuous; in the aggre-
severe, and it can be maintainedt hat they affirm gate, however, there will be continuity. In
the descriptive irrelevance of the Walrasian some form this observationi s very old, but was
12 These are better referredt o as "behaviorc urves"
because indifference loses meaning in the absence
of transitivity. 14 The assumptions that handle the "demand
13 For a dissenting opinion on the success of the discontinuityp roblem"a lluded to before are also not
extension to externalitiesi n production, see McKen- very realistic; however, they have a more technical
zie (1981, pp. 838-39). flavor.
Duffie and Sonnenschein: Arrow and General Equilibrium Theory 573
x22 P1i
P2
=P2
budget
/ at prices ~~~~~~to~ b~e~ ~~~~~~portion
"filled in"
a a= I (x il cX 2)
b=(xb? xb)
endowment
x xi X1 Xi xi X1
a. Single-Agent Demand b. Aggregate Demand
Figure 2. Continuity of Aggregate Demand with a Continuum of Agents Having Nonconvex Preferences
developed by Michael Farrell (1959) and Je- Figures 2a and 2b help us to explain how
rome Rothenberg (1960), and brought to gen- the existence of equilibrium is demonstrated
eral equilibriumt heory in a mannerc ompatible in the absence of the assumptiono f convex pref-
with the Arrow-Debreu-McKenziet heory by erences. We will see how the continuity of de-
Robert Aumann (1966).1 5 Despite Arrow'sc on- mand obtains in the aggregate even when it
tribution to this literature (Arrow-Hahn1 971, fails for individuala gents. In Figure 2a we have
chapter 7), his statement, "The assumption of described the preferences and endowment of
convexity cannot be dispensed with in general a single agent. At prices p* demand is not sin-
equilibriumt heorems concerning the existence gle-valued; as a result, the excess-demand cor-
of equilibrium strictly defined" (Arrow 1968, respondence (depicted in 2b) jumps at this
p. 120, CPII) and his emphasis on the role of price. This discontinuityd isappearsw hen there
convexity in his Nobel lecture (Arrow 1973; is a continuum of infinitesimal agents. To un-
CPII, chapter 9) have suggested a reluctance derstand why, consider an economy in which
to let go of convexity. In fact, the following there is a continuumo f infinitesimala gents with
strong statement is entirely in order: With a the initial endowment and preferences given
continuum of infinitesimal agents, convexity of in 2a. If at p*, 3/4 of the agents are assigned
the aggregatep roductions et and of preferences the bundle a and 1/4 of the agents are assigned
is simply not required, nor is it implied.1 6 the bundle b, then all agents will be maximizing
utility and mean demand will be 3/4 of the way
from b to a. By considering proportions other
15 Of course, global increasing returns to scale (or
than 3 to 1, all the points between a and b
more modestly, situations in which efficient scale is
reached at a level of output which is noninfinitesimal are filled in and the result is an excess-demand
relative to the total size of the market) remains a function (correspondence)w ith sufficientc onti-
problem. Also, we do not deny the descriptive reality nuity to play its part in the existence of equilib-
of the latter situation.
rium. With the realizationt hat convexity need
16 In the absence of external economies, convexity
not be assumed when the Walras model is for-
of the aggregate production set is implied by the
assumptiont hat firms are infinitesimal;h owever, the mulated with a continuum of infinitesimal
fact that convexity of the productions et is not essen- agents, the conditions needed for existence ap-
tial is underscoredb y the fact that with externale con- pear to be quite modest indeed. Moreover, it
omies the aggregate production set will, in general,
is for the case in which agents are small (infini-
not be convex (JohnC hipman 1970)a nd nevertheless
Walrasiane quilibrium is ensured to exist. tesimal) relative to the market that we find the