large market games with demand uncertainty

Journal of Economic Theory 109 (2003) 283–299

Large market games with demand uncertainty

James Peck�

Department of Economics, The Ohio State University, 1945 North High Street, Columbus, OH 43210, USA

Received 25 February 2002; final version received 9 October 2002

Abstract

We consider a market game with a continuum of consumers, where the measure of each type

is stochastic. Nature selects the set of active consumers, who make bids and offers on c� 1

spot market trading posts. Existence of type-symmetric Nash equilibrium is proven. When

facing price uncertainty, best responses are unique, and a Nash equilibrium to the sell-all game

is typically not a Nash equilibrium to the original game. Under plausible circumstances,

consumers strictly prefer to be on one side of the market.

r 2003 Elsevier Science (USA). All rights reserved.

JEL classification: C7; D4; D8

Keywords: Demand uncertainty; Market game; Trading post

1. Introduction

The Shapley–Shubik market game model has been and will continue to be animportant tool for understanding competitive markets. When information issymmetric and the number of consumers is large, open-market Nash equilibriumapproximates a competitive equilibrium.1 Because prices are determined explicitlyfrom the bids and offers of consumers, there is a potential distinction between aplayer being too small to influence the price and a player being a price-taker. Withasymmetric information, a player could be uncertain of the bids and offers of others,and therefore not know the price, while being unable to influence the price with her

�Fax: +614-292-3906.

E-mail address: [email protected] [3,9,14]. Peck and Shell [11] show that the Nash equilibrium allocation can be arbitrarily close to a

competitive equilibrium allocation when short sales are allowed, even with a small number of consumers.

Ghosal and Morelli [6] look at myopic retrading, and show convergence to a Pareto optimum.

0022-0531/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.

doi:10.1016/S0022-0531(03)00022-X

own bids and offers.2 As a result, the model avoids the paradoxical conclusions ofcompetitive rational expectations models when information is asymmetric. Thus, themarket game model with asymmetric information might be useful in areas, such asfinance and macroeconomics, where the price-taking assumption has led to paradox.The purpose of this paper is to study the existence of equilibrium, and to

characterize the equilibrium strategies as far as possible, in a market game with acontinuum of consumers, and where the measure of each type of active consumer isstochastic. Think of nature as selecting the set of active consumers out of a larger setof potentially active consumers. Then the active consumers choose how much ofcommodities 1 to c� 1 to offer on the c� 1 trading posts; bids are made with thenumeraire commodity, c: The price of commodity j is a spot market price, specifyingnoncontingent delivery of commodity j: The realization of spot market pricesdepends on the realization of demand.3 The usual formulation specifies uncertaintyabout endowments, preferences, or technology, rather than uncertainty about whowill show up at the market. We specify uncertainty about the measure of activeconsumers, because it is a major concern to participants on real world markets, andbecause it arises naturally if we take seriously the notion that traders must arrive at a‘‘trading post’’ in order to be counted.4

The existence of type-symmetric Nash equilibrium is shown in Proposition 4.2.Although existence of equilibrium, with a continuum of consumers, is hardlysurprising, a number of obstacles must be overcome. The approach adopted by Peck,Shell, and Spear [13], for the model without uncertainty, does not work whenconsumers face price uncertainty. When prices are known, there is a continuum ofbest-responses for a given player yielding the same consumption bundle,characterized by the amount of ‘‘wash sales’’. It follows that an equilibrium to thegame in which all endowments are offered for sale, for which existence is relativelyeasy to prove, is also an equilibrium to the original game. With demand uncertainty,however, an equilibrium to the sell-all game is not an equilibrium to the originalgame, in general. The approach adopted here is to put e bids and offers on eachmarket, and take limits as e approaches zero, as in [3]. Their analysis must beadapted to the current model and extended, in order to show that the limiting bidsand offers cannot be converging to zero.We derive the following results about best-responses and equilibrium strategies.

Unless the equilibrium distribution of prices is degenerate, a consumer’s best-response is uniquely determined. Thus, the multiplicity of best-responses in thecertainty model is not robust to the introduction of (even) small amounts of demand

2See [1,7]. In [10], players are small but do not know the price, because the endowments supplied to the

market are not known at the time players bid.3There are missing markets, in the sense that consumers have no opportunity to insure themselves

against the risk of being on the long side of the (spot) market. Demand uncertainty implies asymmetric

information, because an active consumer knows that she is active, but the rest of the market does not know

that she is active.4However, uncertainty about endowments or preferences can be accommodated in the present model,

by interpreting a type to include the realization of one’s endowment or preferences. See the concluding

remarks for a discussion.

J. Peck / Journal of Economic Theory 109 (2003) 283–299284

uncertainty. Under very plausible circumstances, consumers respond to theequilibrium distribution of prices by restricting themselves to one side of themarket. When there are only two commodities, Proposition 5.1 provides a simplesufficient condition to guarantee that a consumer will be on one side of the market.This condition is satisfied for any equilibrium if the consumer is sufficiently riskaverse, the demand uncertainty is nontrivial, and the initial allocation is not Paretooptimal.

2. Demand uncertainty

We consider an exchange economy with n types of consumers and c commodities.Consumers are competitive, in the sense of being unable to influence prices.However, as we shall see, consumers are not price takers, because consumers do notknow the aggregate bids and offers of others when they submit their own bids andoffers. To capture these ideas, we represent the set of consumers of each type as acontinuum. For i ¼ 1;y; n; let the set of ‘‘potentially active’’ consumers of type i berepresented by the interval, ½i � 1; i�: From the set of potentially active consumers,½0; n�; nature selects a Lebesgue measurable subset of active consumers, who play themarket game. Since our analysis centers exclusively on type-symmetric Nashequilibrium, and potential deviations therefrom, the only relevant uncertaintyconcerns the aggregate measure of each type of consumer. Let ai denote theLebesgue measure of active type-i consumers, and let a be given by a ¼ ða1;y; anÞ:We assume the existence of density functions for a; conditional on being an activeconsumer of type i; denoted by fið�Þ; which are assumed to be strictly positive overthe support, fa :

%aipaip%ai for i ¼ 1;y; ng: We assume that %ai4

%ai40 holds for

i ¼ 1;y; n:5

For tA½0; 1�; let xi;tðaÞARcþ denote the consumption vector of consumer t of type i

in aggregate demand state a: The consumption of commodity j by a consumer ði; tÞin state a is denoted as x

ji;tðaÞ: The endowment vector of each consumer of type i is

denoted by oiARcþþ: The Bernoulli utility function of a type-i consumer is denoted

by ui; and each consumer is a von Neumann–Morgenstern expected utilitymaximizer. We make the following assumptions on ui:

Assumption 1. For each i; the Bernoulli utility function ui is strictly increasing,strictly concave, and twice continuously differentiable.6 For all j ¼ 1;y; c and all

5Because there is aggregate uncertainty, being active conveys information, so different types might have

different beliefs about a; even if they started with common priors. We have in mind the following process.

First, nature draws a from an unconditional density, f : Next, for i ¼ 1;y; n; nature chooses a set of active

consumers, whose measure is ai: This selection is done such that each ‘‘potential’’ type-i consumer is

equally likely to be chosen.6Our results continue to hold for utility functions defined only on Rc

þ; where utility converges to

negative infinity as some xji approaches zero (for example, log utility). We can always truncate the utility

function at a sufficiently low enough level that the truncation does not affect the equilibrium.

J. Peck / Journal of Economic Theory 109 (2003) 283–299 285

%x40; there exists%U40 such that

@uiðxiÞ@x

ji

X%U ð1Þ

holds for all xi satisfying xjip %x: For all j ¼ 1;y; c and all

%x40; there exists %UoN

such that

@uiðxiÞ@x

ji

p %U ð2Þ

holds for all xi satisfying xjiX

%x:

Notice that Assumption 1 is trivially satisfied if the utility function is separable.

3. The market game, C

There are c� 1 trading posts. On trading post j; active consumers offerto sell commodity j and bid with commodity c: Inactive consumers are assumedto offer and bid zero. We denote consumer ði; tÞ’s offer of commodity j and

bid for commodity j; conditional on being active, as qji;t and b

ji;t; respectively. Also, let

qi;t ¼ ðq1i;t;y; qc�1i;t Þ and bi;t ¼ ðb1i;t;y; bc�1

i;t Þ denote consumer ði; tÞ’s offers and

bids, respectively. A consumer’s offer cannot exceed her endowment of thecommodity in question, and the total amount of commodity c bid on all postscannot exceed her endowment of commodity c: The strategy set of consumer ði; tÞ isthen given by

Si;t ¼ ðqi;t; bi;tÞAR2ðc�1Þþ : q

ji;tpoj

i for joc andXc�1j¼1

bji;tpoc

i

( ):

We will sometimes denote the strategy of consumer ði; tÞ by si;t:When strategies are type-symmetric, the total amount of commodity j offered on

trading post j ðjacÞ and the total amount of commodity c bid on trading post j in

state a; QjðaÞ and BjðaÞ; are given by

QjðaÞ ¼Xn

i¼1aiq

ji ð3Þ

and

BjðaÞ ¼Xn

i¼1aib

ji: ð4Þ

We define the price of commodity j in state a to be

pjðaÞ ¼ BjðaÞQjðaÞ: ð5Þ


Thus, the price pjðaÞ is the spot market price of commodity j; in terms of commodity c:7

The outcome function or allocation rule is determined as follows. The allocationfor active consumer ði; tÞ in state a is given by

xji;tðaÞ ¼ oj

i � qji;t þ

bji;t

pjðaÞ for joc; ð6Þ

xci;tðaÞ ¼ oc

i �Xc�1j¼1

bji;t þ

Xc�1j¼1

pjðaÞqji;t ð7Þ

with the convention that 0=0 ¼ 0:8

Expected utility, while concave in consumption, is not necessarily jointly concavein bids and offers. Dubey and Shubik [3] and Peck, Shell, and Spear [13] solve thisproblem by observing that the consumption possibilities set is convex, which impliesa unique consumption bundle that is a best-response to the bids and offers of otherconsumers. Convexity of the consumption possibilities set, for the present modelwith demand uncertainty, does not follow immediately from previous papers. Here, abid or offer is placed on many spot-market trading posts simultaneously. For

example, a bid bji;t is made for all state-contingent commodities ðj; aÞ such that

consumer ði; tÞ is active in state a: However, due to the fact that an individual cannotaffect prices, the consumption possibilities set is indeed convex.

Lemma 3.1. Fix the (type-symmetric) offers and bids of all consumers except ði; tÞ;which determines pjðaÞ for each j and a: Assume that pjðaÞ is bounded above zero and

below infinity for each j and a: Suppose bssi;t � ðbqqi;t; bbbi;tÞASi;t yields consumption bxxi;tðaÞin state a; and essi;t � ðeqqi;t; ebbi;tÞASi;t yields consumption exxi;tðaÞ in state a: Then for all l;0plp1; the strategy lbssi;t þ ð1� lÞ essi;t is contained in Si;t and yields consumption

lbxxi;tðaÞ þ ð1� lÞexxi;tðaÞ in state a:

Proof. The fact that lbssi;t þ ð1� lÞ essi;t is contained in Si;t is obvious. From (6), the

resulting consumption for joc is

oji � ½lbqqj

i;t þ ð1� lÞeqqji;t� þ

lbbbji;t þ ð1� lÞebbj

i;t

pjðaÞ

¼ l oji � bqqj

i;t þbbbj

i;t

pjðaÞ

" #þ ð1� lÞ oj

i � eqqji;t þ

ebbji;t

pjðaÞ

" #¼ lbxxj

i;tðaÞ þ ð1� lÞexxji;tðaÞ:

7Considering strategies that are not type-symmetric might allow the price to depend on the identities of

the active traders, and not just the aggregate demand state, a: We would need a sigma-algebra over the

subsets of ½0; n�: The simplest way to ensure that the game is well defined would be to cancel trade if the bidsor offers are not integrable. Introducing this machinery would not affect our analysis of type-symmetric

Nash equilibrium. See [2] for a discussion and analysis of a market game with a continuum of players.8See [13] for an interpretation.


From (7), the resulting consumption of commodity ðc; aÞ is given by

oci �

Xc�1j¼1

½lbbbji;t þ ð1� lÞebbj

i;t� þXc�1j¼1

pjðaÞ½lbqqji;t þ ð1� lÞeqqj

i;t�

¼ l oci �

Xc�1j¼1

bbbji;t þ

Xc�1j¼1

pjðaÞbqqji;t

" #þ ð1� lÞ oc

i �Xc�1j¼1

ebbji;t þ

Xc�1j¼1

pjðaÞeqqji;t

" #¼ lbxxc

i;tðaÞ þ ð1� lÞexxci;tðaÞ: &

Proposition 3.2. Fix the (type-symmetric) offers and bids of all consumers except ði; tÞ;which determines pjðaÞ for each j and a: Assume that pjðaÞ is bounded above zero and

below infinity for each j and a: Then the set of best-responses for consumer ði; tÞ is

convex. Furthermore, suppose that for each joc; pjðaÞ depends nontrivially on a: Then

the best-response is unique.

Proof. Lemma 3.1 shows that the consumption possibilities set for consumer ði; tÞ isconvex. Since the utility function,

Vi;tðxi;tÞ ¼Za

ui;tðxi;tðaÞÞfiðaÞ da

is strictly concave in state-contingent consumption, it follows that the expectedutility maximizing state-contingent consumption is uniquely determined for almost

all states, a: Let bssi;t � ðbqqi;t; bbbi;tÞASi;t and essi;t � ðeqqi;t; ebbi;tÞASi;t be arbitrary best-

responses for consumer ði; tÞ: Lemma 3.1 shows that a convex combination of bssi;t

and essi;t yields a convex combination of the consumption bundles, bxxi;t and exxi;t; andtherefore is weakly preferred to at least one of the bundles. Therefore, the convexcombination of bssi;t and essi;t must also be a best-response.

For almost all states, a; the optimal consumption bundle is the same for bssi;t andessi;t: From (6), we havebbbji;t

pjðaÞ � bqqji;t ¼

ebbji;t

pjðaÞ � eqqji;t ð8Þ

for all joc: Rearranging (8), we have for almost all states, a; and all joc;

pjðaÞ½bqqji;t � eqqj

i;t� ¼ ½bbbji;t � ebbj

i;t�: ð9Þ

From (9), we see that there are two possibilities. One possibility is that bqqji;t ¼ eqqj

i;t andbbbji;t ¼ ebbj

i;t hold for all joc; which implies bssi;t ¼ essi;t; so that the best response is

unique. The other possibility is that, for some joc; pjðaÞ depends trivially on a: Thatis, for almost all states, a; pjðaÞ is constant. &

Proposition 3.2 indicates that, unless consumers face no uncertainty about prices,best-responses are uniquely determined. On the other hand, for the case of certainty,consumers can choose from a continuum of best-responses by augmenting net trades


with wash sales. Thus, the indeterminacy of best-responses is not robust to theintroduction of demand uncertainty, even in small amounts. The intuition is thatconsumers face a distribution of spot-market prices for a particular physicalcommodity, so that a wash sale that maintains consumption for one price realizationchanges consumption for other price realizations. Without knowing the price, it isimpossible to augment bids and offers so that they always cancel each other out.

4. Existence of type-symmetric Bayes-Nash equilibrium

Our symmetry assumptions imply that all type-i consumers have the same beliefsabout aggregate demand, conditional on being active. It is natural to ask whether anontrivial type-symmetric Bayes–Nash equilibrium necessarily exists. One problemto be overcome is establishing that the appropriate best-response mapping is welldefined, and that the fixed point is nontrivial (with positive bids and offers). Peck,Shell, and Spear [13] first show existence for the ‘‘sell-all’’ game in which consumersare forced to offer their entire endowments, and then show that an equilibrium of thesell-all game is also an equilibrium of the original game. To guarantee that bestresponses are well defined, the mapping is adjusted to require bids to be positive, butthis constraint is shown not to bind at the fixed point. We cannot use this approachhere, because a best-response that is constrained to offer one’s entire endowment isnot necessarily a best-response when the consumer is unconstrained, as indicated byProposition 3.2 above.The existence argument adopted here is similar to the one of Dubey and Shubik

[3]. Artificial bids and offers of e are placed on each market, and a type-symmetricBayes–Nash equilibrium is shown to exist for the game, Ge: Corresponding prices areshown to be uniformly bounded above zero and below infinity, for a sequence ofequilibria as e approaches zero. From the maximum theorem, the limiting strategiesform a type-symmetric Bayes–Nash equilibrium for G: It is then shown that thelimiting bids and offers cannot all be zero, unless autarky is Pareto optimal.9

Definition 4.1. The game, Ge; is defined to be exactly the game, G; with the exceptionthat an offer and bid of e40 are exogenously placed on each trading post. That is,Eq. (5) is replaced with

pj;eðaÞ ¼ BjðaÞ þ eQjðaÞ þ e

: ð10Þ

We insert the superscript, e; into the notation for offers, bids, prices, andconsumptions to clarify that the game under consideration is Ge rather than G: Ourfocus is on type-symmetric Bayes–Nash equilibrium. When, for i ¼ 1;y; n; allconsumers of type i choose the strategy sei ¼ ðqe

i ; bei Þ; only the aggregate realizations

9Dubey and Shubik [3] omit the argument that the limiting equilibrium is not the no-trade equilibrium.

Our argument relies on the fact that consumers have no effect on prices, so that any positive e is largerelative to the size of a consumer. See [8] for a different context in which this argument can be applied.


of demand in each state are important, and Eq. (10) can be rewritten as

pj;eðaÞ ¼Pn

i¼1 aibj;ei þ ePn

i¼1 aiqj;ei þ e

: ð11Þ

The expected utility of consumer ði; tÞ; as a function of the type-symmetric strategieschosen by everyone else, se; and her own strategy, %sei;t; is denoted by V e

i;tð %sei;t; seÞ;given by

V ei;tð %sei;t; seÞ

¼Za

ui;t y;oji � %q

j;ei;t þ

%bj;ei;t

pj;eðaÞ;y;oci �

Xc�1j¼1

%bj;ei;t þ

Xc�1j¼1

pj;eðaÞ %qj;ei;t

! fiðaÞ da: ð12Þ

Notice that Lemma 3.1 and Proposition 3.2 continue to hold for Ge; where pricesare now written as a function of a; given by Eq. (11). In particular, we know that thebest-response correspondence is convex valued.

Proposition 4.2. Under assumption 1 and our other maintained assumptions, there

exists a nontrivial type-symmetric Bayes–Nash equilibrium for G:

Proof. We first show that there exists a nontrivial type-symmetric Bayes–Nashequilibrium for Ge: Denote the best-response correspondence for consumer ði; tÞ asgei;tðseÞ; given by

gei;tðseÞ ¼ arg max V ei;tð %sei;t; seÞ:

Given our symmetry assumptions, V ei;tð %sei;t; seÞ and gei;tðseÞ are independent of t; so we

denote the best-response correspondence of type-i consumers by gei ðseÞ: For eachpositive e; prices are bounded above zero and below infinity, so V e

i ð %sei ; seÞ is

continuous. Since the strategy sets are compact, it follows that gei ðseÞ is well definedfor each i; e: It follows from Proposition 3.2 that gei ðseÞ is convex valued. Let geðseÞ ¼ðge1ðseÞ;y; genðseÞÞ hold.We know that the strategy sets are compact and convex, and that ge is convex

valued. The conditions for Kakutani’s fixed point theorem will be satisfied if we candemonstrate that ge is upper hemicontinuous. Suppose that we have a sequence oftype-symmetric strategies, se;m-se;� and (for some i) a sequence of best responses,

%se;mi - %se;�i ; such that we have

%se;mi Agei ðse;mÞ for all m; and

%se;�i egei ðse;�Þ:

Clearly, %se;�i is a feasible strategy for type i: Since it is not a best-response to se;�; wehave, for some feasible bssi;

V ei ðbssi; se;�Þ4V e

i ð %se;�i ; se;�Þ:


By continuity of V ei ; we have, for sufficiently large m;

V ei ðbssi; se;mÞ4V e

i ð %se;mi ; se;mÞ;

contradicting the fact that %se;mi Agei ðse;mÞ must hold. This establishes upper

hemicontinuity.Applying Kakutani’s fixed point theorem, there exists se ¼ ðse1;y; senÞ such that

seiAgei ðseÞ for i ¼ 1;y; n: It follows that se is a Nash equilibrium for Ge:We now consider a convergent subsequence of equilibria, fseg; where se is a Nash

equilibrium of Ge and e-0:10. Let us show that all limiting prices are uniformlybounded above zero and below infinity.

Case 1: Suppose we have pj;eðaÞ-0 for some ðj; aÞ:Then it follows from (11), and the fact that the lower support for a is strictly

positive, that bj;ei -0 for all i and pj;eðaÞ-0 for all a: Fix a type, i:

Case 1a: For some a; we have xc;ei ðaÞ-0:

From the allocation rule, (7), we haveXj0aj;c

bj0;ei -oc

i ð13Þ

and Xj0ac

pj0;eðaÞqj0;ei -0:

For qj0;ei to be a best-response, we must have q

j0;ei -0 for all j0ac: Therefore, for

sufficiently small e and all j0ac; we have

xj0;ei ðaÞ4oj0

i

2: ð14Þ

From (13), there must be some commodity j00 such that bj00;ei 4

oci

c�2 holds. Thus, we

have pj00;eðaÞ is uniformly bounded above zero for all a: Differentiating V ei ð %sei ; seÞ

with respect to bj;ei and b

j00;ei ; we have

@V ei ð %sei ; seÞ@b

j;ei

¼Za

@uiðxei ðaÞÞ

@xji

1

pj;eðaÞ �@uiðxe

i ðaÞÞ@xc

i

" #fiðaÞ da ð15Þ

and

@V ei ð %sei ; seÞ@b

j00;ei

¼Za

@uiðxei ðaÞÞ

@xj00i

1

pj00;eðaÞ �@uiðxe

i ðaÞÞ@xc

i

" #fiðaÞ da: ð16Þ

Since (14) holds for j00; from Assumption 1 and Eqs. (15) and (16), it follows that a

type-i consumer could increase utility by reducing bj00;ei and increasing b

j;ei by the same

small amount. Thus, case 1a is impossible.

10This is a slight abuse of notation. We should define a sequence indexed by n; fseng; where en-0 as

n-N:


Case 1b: There exists%x such that lime-0 x

c;ei ðaÞX

%x holds for all a:

If the liquidity constraint is not binding, so increasing bj;ei is feasible, then

Assumption 1 and Eq. (15) imply that utility would be increased. If the liquidity

constraint is binding, then there must be some commodity j00 such that bj00;ei 4

oci

c�2

holds, and a type-i consumer could increase utility by reducing bj00;ei and increasing

bj;ei by the same small amount. Thus, case 1b is impossible.

Case 2: Suppose we have pj;eðaÞ-N for some ðj; aÞ:Since bids are bounded, we must have q

j;ei -0: Because the lower support for a is

strictly positive, we have pj;eðaÞ-N for all a: Differentiating V ei ð %sei ; seÞ with respect

to qj;ei ; we have

@V ei ð %sei ; seÞ@q

j;ei

¼Za

pj;eðaÞ@uiðxei ðaÞÞ

@xci

� @uiðxei ðaÞÞ

@xji

" #fiðaÞ da: ð17Þ

For sufficiently small e; xj;ei ðaÞ4oj

i2must hold for all a: Feasibility imposes a uniform

upper bound on xc;ei ðaÞ; based on allocating all of commodity c to type-i consumers.

By Eq. (17) and Assumption 1, it follows that utility can be increased by increasing

qj;ei : Thus, case 2 is impossible.

Since prices can be bounded above zero and below infinity, uniformly in a; payoffsare continuous in the type-symmetric strategies se: From the maximum theorem, itfollows that the limiting strategies, s� form a Nash equilibrium to the limiting game,which is G:To complete the proof, we must show that the limiting equilibrium is nontrivial.

Suppose that all bids and offers are zero at the equilibrium to G; so that we haves� ¼ 0: For ðb�

i ; q�i Þ ¼ 0 to be the limit of the sequence of type i consumers’ Nash

equilibrium strategies to Ge; ðb�i ; q�

i Þ must be a best-response to the limiting

distribution of prices, p�ðaÞ: The reason is that for any positive e; no matter howsmall, an individual consumer cannot affect the distribution of prices, and thedistribution of prices is converging to p�ðaÞ: Thus, we have

@Við0; 0Þ@b

ji

¼Za

@uiðojiÞ

@xji

1

pj;�ðaÞ �@uiðoc

i Þ@xc

i

" #fiðaÞ dap0 ð18Þ

and

@Við0; 0Þ@q

ji

¼Za

pj;�ðaÞ @uiðoci Þ

@xci

� @uiðojiÞ

@xji

" #fiðaÞ dap0: ð19Þ

Rewriting (18) and (19), we have

@uiðojiÞ

@xji

@uiðociÞ

@xci

p1R

a1

pj;�ðaÞ

h ifiðaÞ da

ð20Þ


and

@uiðojiÞ

@xji

@uiðociÞ

@xci

X

Za

pj;�ðaÞfiðaÞ da: ð21Þ

Combining (20) and (21) yieldsZa

1

pj;�ðaÞ

� �fiðaÞ da

Za

pj;�ðaÞfiðaÞ dap1: ð22Þ

However, a strict version of Jensen’s inequality implies that the left side of (22) isstrictly greater than one unless the price distribution is degenerate. Therefore, there is

a constant, pj;�; such that pj;�ðaÞ ¼ pj;�; for almost all a: Since this argument holds forall types i and commodities j; it follows from (20) and (21) that all consumers havethe same marginal rate of substitution at their endowments, so the initial allocationis Pareto optimal.We have established the following. If the initial allocation is not Pareto optimal,

the supposition that s� ¼ 0 is contradicted, so s� is nontrivial. If the initial allocationis Pareto optimal, then s� ¼ 0 is possible, but we also know that there are othernontrivial equilibria to G: Simply let everyone choose positive bids and offers oneach market, such that we have

bji

qji

¼

@uiðojiÞ

@xji

@uiðociÞ

@xci

:

Clearly, offers can be chosen to be positive, but small enough so that a type-i

consumer’s bids do not exceed oci : &

Proposition 4.3. In a type-symmetric Nash equilibrium with positive aggregate bids

and offers on each market, the distribution of prices is degenerate (i.e., for each j; there

is a constant, pj;�; such that pj;�ðaÞ ¼ pj;�; for almost all a) if and only if the initial

endowments are Pareto optimal.

Proof. Let the initial endowments be Pareto optimal. Since each consumer receivesat least as much utility at the Nash equilibrium allocation as the utility of herendowment, concavity of the utility functions implies that the Nash equilibriumallocation must be the endowment allocation. Otherwise, the expected allocationis feasible and Pareto dominates the initial allocation. From the allocation Eq. (6),it follows that

qj;�i ¼ b

j;�i

pj;�ðaÞ


holds for all i; j; a: Since aggregate bids and offers are positive there must be some

type i for which we have qj;�i 40: Therefore,

pj;�ðaÞ ¼ bj;�i

qj;�i

holds for all a: Thus, the distribution of prices is degenerate.Let the initial endowments not be Pareto optimal, and suppose that the

distribution of prices is degenerate. Then, in equilibrium, each consumer receives aconsumption bundle that is independent of a: There must be at least one typeof consumer, i; and commodity, j; for which consumer i is a net seller of commodityj: [If, instead, each consumer’s N.E. bundle equals her endowment, then herliquidity constraint is not binding. Therefore, all utility gradients at the endowmentare proportional to the degenerate price vector, contradicting the fact thatendowments are not Pareto optimal.] Since consumer i is a net seller ofcommodity j; we have

pj;�ðaÞ4bj;�i

qj;�i

for all a:

Given a�i; straightforward algebra yields pj;�ð%ai; a�iÞopj;�ð%ai; a�iÞ: Because f ð�Þ is

strictly positive over its support, this contradicts the supposition that the distributionof prices is degenerate. &

5. Properties of the equilibrium

Proposition 3.2 establishes that, when the underlying demand uncertainty createsa non-degenerate distribution of equilibrium prices for each commodity, thenconsumers’ optimal strategies are uniquely determined. The optimal strategy forconsumer ði; tÞ could be strictly in the interior of her strategy set, Si;t; or it could be

on the boundary. Since it is unusual to find consumers on both sides of a given real-world market simultaneously,11 of particular interest is the case in which consumersstrictly prefer to trade on one side of the market. To be sure, in the real world,consumers often have an endowment of zero for the commodities that they wish tobuy, which would require them to be on one side of the market in G: However, it isinteresting to ask whether the risk of price fluctuations, by itself, is enough topartition the traders into buyers and sellers.Modifying our previous notation, denote type-i consumption as a function of

the price vector, p; and the consumer’s strategy (which may or may not be

an equilibrium strategy), si; as xiðsi; pÞ ¼ ðx1i ðsi; pÞ;y; xc

i ðsi; pÞÞ: Also, for

11This statement excludes financial intermediaries and market makers, who attempt to buy low and sell

high. In the current model, buying and selling must take place at the same price.


j ¼ 1;y; c� 1; let Hji ðp; siÞ be given by

Hji ðp; siÞ ¼

@uiðxiðsi; pÞÞ@x

ji

� pj@uiðxiðsi; pÞÞ@xc

i

: ð23Þ

We can interpret Hji ðp;siÞ as a type-i consumer’s marginal utility of commodity j;

given that increasing consumption of j will occur at the expense of reducing

consumption of c: There are two effects of pj on Hji ðp; siÞ; making the overall effect

ambiguous. The direct effect of an increase in pj (holding consumption constant) isto increase the opportunity cost of commodity j: The second term in (23) increases,

lowering Hji ðp; siÞ: The net-trade effect of an increase in pj reduces x

ji and increases

xci : The reason is that the fixed b

ji purchases less of commodity j; and the fixed q

ji is

sold for more of commodity c: Thus, the net-trade effect of an increase in pj tends toincrease the marginal utility of j and reduce the marginal utility of c; raising

Hji ðp; siÞ: Sharp results are available for the case of two commodities, and therefore,

one trading post.From the type-symmetric equilibrium strategies, s�; and the density function for

active consumers of each type, fiðaÞ; a distribution of prices is determined, whosedensity function (when it exists) we denote by hiðpÞ:

Proposition 5.1. Assume that there are two commodities, c ¼ 2; and that the initial

allocation is not Pareto optimal. If we have12

@Hiðp; s�i Þ@p

40 ð24Þ

for all p in the support of hi (i.e., for all prices consistent with equilibrium), then

consumer i will be on one side of market one. That is, we have either b�i ¼ 0 or q�

i ¼ 0:

Proof. Expected utility of a type-i consumer can be written as a function of herstrategy, si; and the density of prices (if such a density function exists), given by

Viðsi; hiÞ ¼Z

p

ui o1i � qi þ

bi

p;o2

i � bi þ pqi

� �hiðpÞ dp: ð25Þ

The proof of Proposition 4.2 demonstrates that equilibrium prices must be boundedaway from zero and infinity, so the best-response correspondence is well defined.Furthermore, since the initial allocation is not Pareto optimal, there must exist astrictly positive density function, hiðpÞ; defined over a non-degenerate support, ½

%p; %p�:

From Proposition 3.2, the best-response, s�i ; is uniquely determined. Suppose that

b�i 40 and q�

i 40 hold.

12Since there are two commodities, all bids and offers are on market one, so we eliminate the

commodity superscript.


We claim that there exists bpp; such that%pobppo %p and Hiðbpp; s�i Þ ¼ 0 hold. If we have

Hiðp; s�i Þ40 for all pA½%p; %p�; this implies

@Viðs�i ; hiÞ@qi

¼ �Z

p

Hiðp; s�i ÞhiðpÞ dpo0:

It follows that utility can be increased by reducing q�i ; contradicting the fact that s

�i is

the best-response of type-i consumers. If we have Hiðp; s�i Þo0 for all pA½%p; %p�; this

implies

@Viðs�i ; hiÞ@bi

¼Z

p

Hiðp; s�i Þp

hiðpÞ dpo0:

It follows that utility can be increased by reducing b�i ; contradicting the fact that s

�i is

the best-response of type-i consumers. This establishes the claim that there exists bpp;such that

%pobppo %p and Hiðbpp; s�i Þ ¼ 0 hold.

Consider the following deviation for a type-i consumer:

%qi ¼ q�i � d;

%bi ¼ b�i � dbpp:

Evaluated at d ¼ 0; we have

@Viðs�i ; hiÞ@d

¼Z

p

Hiðp; s�i Þp � bpp

phiðpÞ dp: ð26Þ

Inequality (24), and the fact that we have Hiðbpp; s�i Þ ¼ 0; imply

Hiðp; s�i Þo0 for pobpp;and

Hiðp; s�i Þ40 for p4bpp:Therefore, the right side of (26) is strictly positive, so reducing bids and offers withd40 increases utility. This contradicts the fact that s�i is a best response, so our

supposition that b�i 40 and q�

i 40 hold is false. &

Remark 5.2. The conclusion of Proposition 5.1, that a type-i consumer will be onone side of the market, continues to hold when the other consumers choose strategiesthat are out of equilibrium. We must interpret s�i as the type-i consumer’s best-

response to the strategies played by everyone else, and replace the condition aboutPareto optimality with the condition that the consumer faces a strictly positivedensity function, hiðpÞ; defined over a non-degenerate support, ½

%p; %p�:

Here is some intuition for why a consumer strictly prefers to be on one side of themarket. Suppose that the consumer has strictly positive bids and offers. There mustbe a price, bpp; at which her marginal rate of substitution equals bpp: Reducing bids andoffers at the ratio bpp reduces her net purchase of commodity 1 when pobpp holds.Under inequality (24), we have Hiðp; s�i Þo0 for pobpp; so the net marginal utility of

commodity 1 is negative in these states, and the consumer is better off reducing her


net purchase of commodity 1. Reducing bids and offers at the ratio bpp increases hernet purchase of commodity 1 when p4bpp holds. Under inequality (24), we haveHiðp; s�i Þ40 for p4bpp; so the net marginal utility of commodity 1 is positive in these

states, and the consumer is better off increasing her net purchase of commodity 1.Thus, when the consumer reduces her bids and offers at the ratio bpp; she is happy tohave her net purchase of commodity 1 reduced when the actual price is below bpp(when she is already consuming a lot), and she is happy to have her net purchase ofcommodity 1 increased when the actual price is above bpp (when she is not consumingvery much). The only way to avoid this beneficial deviation is to have either her bidor offer be zero, so that the reduction in bids and offers is not feasible.

Remark 5.3. Other things equal, a more risk averse consumer is more likely to be onone side of the market. To see this, we can compute

@Hiðp; s�i Þ@p

¼ �@2uiðxiðs�i ; pÞÞ@ðx1

i Þ2

b�i

ðpÞ2� @2uiðxiðs�i ; pÞÞ

@ðx2i Þ

2pq�

i �@uiðxiðs�i ; pÞÞ

@ðx2i Þ

: ð27Þ

The first two terms on the right side of (27) are positive, and the last term is negative.Let a particular consumer ði; tÞ become more and more risk averse, holding constantthe rest of the economy with a non-degenerate equilibrium price distribution. The

entire expression for@Hiðp;s�i Þ

@peventually becomes positive, as long as ðb�

i ; q�i Þ remains

bounded away from ð0; 0Þ: This occurs if consumer ði; tÞ has a marginal rate ofsubstitution (at her endowment) that is either always less than or always greater thanany equilibrium price realization.

When there is more than one trading post, c42; similar forces are at work, butinteraction effects complicate the condition for strictly preferring to be on one side of

the market. Even if the utility function is separable in the c commodities, Hji ðp; siÞ

depends on prices other than pj; because all prices affect xci : In general, it will be

impossible to find bppj; such that%pjobppjo %pj and H

ji ðbppj; p�j ; s�i Þ ¼ 0 hold for all prices of

other commodities, p�j:The intuition for being on one side of the market does not depend on the

specification of a continuum of consumers, and the result that the consumer isunable to affect the price. My conjecture is that, for economies with a finite numberof consumers and nontrivial demand uncertainty, we have (1) best-responses areuniquely determined when the price distribution is non-degenerate, and (2) plausibleparameter specifications (involving sufficiently high risk aversion) will haveconsumers on one side of the market in all Nash equilibria.

6. Concluding remarks

Our specification of uncertainty can be interpreted as uncertainty overendowments. Obviously, the case in which the endowment of consumer ði; tÞ is


either 0 or oi fits directly into our framework. Because the number of types isarbitrary, the model also includes general endowment uncertainty and preferenceuncertainty as special cases, at least when the number of realizations is finite.Consumers take no actions until they learn their endowments, so the endowmentrealization can be considered to be part of what determines one’s type. For example,suppose that we want to model n ‘‘initial’’ types of consumers, with k nonzerorealizations of endowments for each initial type. This is equivalent to nk types ofconsumers in our specification.13

Our game, G; is a variant of the Shapley–Shubik market game with commoditymoney, since bids are made with commodity c: This variant has been studied byShapley–Shubik [15], Dubey and Shubik [3], Ghosal and Morelli [6], and manyothers. We have avoided the variant studied by Postlewaite and Schmeidler [14],Peck and Shell [12], Peck, Shell, and Spear [13], Jaynes et al. [9], and others. In thosemodels, bids are made with a fiat money, and consumers must satisfy a budgetconstraint or else face punishment. In this literature, severe punishments are imposedto ensure that the budget constraints hold in equilibrium. The problem with severepunishments when there is demand uncertainty is that, to ensure that her budgetconstraint holds in all states of nature, a consumer will typically allow slack in somestates. However, if one consumer’s budget constraint holds as a strict inequality, itfollows that another consumer’s budget constraint must be violated.14 To focus onthe issue of demand uncertainty, rather than bankruptcy and punishments, we haveadopted a model without budget constraints.15

Because bids and offers cannot be made contingent on price, there is nomechanism in place to ensure that marginal rates of substitution are equated acrossconsumers. Only the most extreme coincidence would allow the Nash equilibrium tobe ex post efficient. Other mechanisms involving the submission of demand schedulesor limit orders could be more efficient.16 On the other hand, prices in G can becomputed easily by a clearing house. Perhaps more importantly, consumers cantransmit their strategies easily in G; while submission of demand schedules can betedious and time consuming. It would be interesting to explore these tradeoffs. Onemight expect to observe Shapley–Shubik trading posts on markets where traders facerelatively little price uncertainty, at least at the moment a trader places a bid or offer.

13The assumption that fiðaÞ is strictly positive over its support might be violated if we model uncertaintyover (strictly positive) endowments, but certainty over the measure of active consumers. However, this

assumption is only used to ensure that there is actual demand uncertainty in Propositions 4.3 and 5.1. The

existence argument and characterization of best-responses under price uncertainty does not depend on this

full-support assumption.14See the discussion in [12, Section 5].15The problem is not fiat money, per se, but the specification that consumers can create arbitrary

amounts of ‘‘inside’’ fiat money, limited only by the threat of punishments if they violate their budget

constraint. In a dynamic setting, the acquisition of fiat money can be modeled more carefully, as in

[4,5,10].16A strategy in G is similar to a market order, and can be interpreted as the submission of a special two-

parameter demand function. See [11]. Submission of arbitrary demand schedules would not allow ex ante

efficiency, in general, because there is no way to insure fully against price risk with spot markets only.


Acknowledgments

I thank Matt Jackson and David Schmeidler for helpful conversations. Any errorsare my responsibility.

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large market games with demand uncertainty

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