stable trading structures in bilateral oligopolies

File: 642J 226601 . By:BV . Date:23:05:97 . Time:14:49 LOP8M. V8.0. Page 01:01Codes: 4173 Signs: 2347 . Length: 50 pic 3 pts, 212 mm

Journal of Economic Theory � ET2266

journal of economic theory 74, 368�384 (1997)

Stable Trading Structures in Bilateral Oligopolies*

Francis Bloch

CORE and Groupe HEC, 1 rue de la Libe� ration,78350 Jouy-en-Josas, France

and

Sayantan Ghosal

CORE and Queen Mary and Westfield College, University of London,Mile End, London E1 4NS, England

Received November 15, 1994; revised September 17, 1996

This paper analyzes the formation of trading groups in a bilateral market whereagents trade according to a Shapley�Shubik (J. Polit. Econ. 85 (1977), 937�968.)trading mechanism. The only strongly stable trading structure is the grand coalition,where all agents trade on the same market. Other weakly stable trading structuresexist and are characterized by an ordering property: trading groups can be ranked bysize and cannot contain very different numbers of traders of the two types. Journal ofEconomic Literature Classification Numbers: D43, D51. � 1997 Academic Press

1. INTRODUCTION

In the traditional representation of markets, traders cannot choose theset of agents with whom they trade. Goods are exchanged on a single,anonymous market in which all the agents participate. In a competitiveeconomy, the assumption that all agents trade on the same market caneasily be justified. Since any competitive equilibrium belongs to the core,no coalition of traders could gain by trading on a smaller submarket. In the

article no. ET962266

3680022-0531�97 �25.00Copyright � 1997 by Academic PressAll rights of reproduction in any form reserved.

* We have greatly benefitted from discussions with Franc� oise Forges, Jean JaskoldGabszewicz, Roger Guesnerie, and Heracles Polemarchakis. We are also grateful to seminaraudiences at CORE, Erasmus University, Valencia, University Paris-I, the E� cole Nationaledes Ponts et Chausse� es, the SITE 1995 Summer Workshop on Coalition Formation atStanford, the Second Economic Theory Conference in Kephalonia, and the World Congressof the Econometric Society in Tokyo for their comments. The suggestions of an associateeditor and an anonymous referee greatly improved the quality of the paper. Finally, we grate-fully acknowledge financial support from the European Union (Human Capital and MobilityFellowship, Contract ERBCHBCT920167) for the first author and from a CORE Ph.D.Fellowship for the second author.


terminology we adopt in this paper, the trading structure where all agentstrade on the same market is ``stable'' since it cannot be disrupted by theactions of any coalition of traders. However, in small economies where traderscan enjoy market power, the presence of a single market grouping all traderscannot be so easily defended. In fact, the classical model of Cournot oligopolysuggests that traders have an incentive to set up distinct, separate markets onwhich they behave as monopolists. Hence, when traders behave strategically,coalitions of traders may benefit by forming small submarkets, and thestructure in which all agents trade on a single market may fail to be stable.

In this paper, we analyze the incentives to form separate markets in asimple finite economy where all agents behave strategically. We considerthe framework of bilateral oligopoly, with two types of traders, where eachtype of trader has a corner in a different commodity.1 Each trader submitsan offer to the market and, as in the noncooperative model of exchange ofShapley and Shubik [13], prices are determined by the ratio of aggregateoffers of the two commodities. We first show that the trading structurewhere all agents trade on the same market is stable. As in the competitivemodel, coalitions of traders cannot benefit from forming smaller marketsand the intuition from the partial equilibrium analysis of Cournotoligopoly fails in a general equilibrium setting where traders take intoaccount the reaction of traders on the other side of the market.

The incentives to form trading groups in a bilateral oligopoly can bederived from two simple comparative statics results on the equilibrium ofthe noncooperative model of exchange. First, traders on one side of themarket always benefit from an increase in the number of traders on theother side. In fact, as the number of traders on one side of the marketincreases, the total quantity they offer rises while the structure of competi-tion between traders on the other side of the market remains unchanged.Second, on any symmetric market, where the number of traders of the twotypes is equal, the traders' utilities increase with the size of the market. Thisis due to the fact that on symmetric markets, the relative price of the twogoods is equal to one, and as the number of traders increases, the equi-librium approaches the efficient competitive equilibrium allocation.2 Giventhese two results, it is clear that no coalition of traders can deviate from thestructure where all agents trade on the same market. Traders obtain lowerutility on any symmetric submarket and, on asymmetric submarketstraders who are more numerous on the market must obtain a lower utilitythan on some symmetric market.

369STABLE TRADING STRUCTURES

1 This model of bilateral oligopoly, which is a special version of the Shapley�Shubikstrategic market games [13], was introduced by Gabszewicz and Michel [4].

2 This result was already noted by Postlewaite and Schmeidler [11] who show, in a generalsetting, that the equilibrium outcomes of a Shapley�Shubik market game become arbitrarilyclose to a Pareto-efficient allocation as the number of traders goes to infinity.


To further the analysis of stable trading structures, we distinguishbetween two concepts of stability. In the strong version of stability, coali-tions of traders can deviate if the utility its members obtain is at least ashigh as before, and strictly higher for some member of the coalition. In theweak version of stability, the deviating coalition must make all its membersstrictly better off. A striking result we derive is that the only strongly stabletrading structure is the one in which all agents meet on the samemarket. To understand this result, note that any other weakly stable tradingstructure must be asymmetric and treat symmetric traders differently. As inthe core, an allocation which treats similar players differently can bedisrupted by some coalition of traders. If the single market trading struc-ture is the only strongly stable trading structure, other weakly stablestructures may exist. We show that they must satisfy a strong orderingproperty: if one trading group has more members of one type it must havemore members of the other type as well. Hence, in a weakly stable tradingstructure trading groups can be ranked according to size and, furthermore,no market can contain widely different numbers of traders of the two types.

It should be noted at the outset that our characterization of stabletrading structure relies on the particular restrictions we place on theactions of traders. We first assume that each agent only trades on onemarket and exclude the possibility that the same agent be present ondifferent markets.3 Second, once a trading group is formed, we constrain allagents to trade according to a Shapley�Shubik strategic market game. Theuse of Shapley�Shubik trading mechanisms poses well-known difficulties: theequilibrium outcome is typically not Pareto-optimal, a commodity which isowned by less than two agents cannot be traded at equilibrium. Furthermore,the properties of equilibria are not robust to changes in the specification of thetrading mechanism. In spite of these difficulties, we have adopted theframework of a Shapley�Shubik model of noncooperative exchange becauseit can be viewed as a natural generalization of Cournot oligopolies andembodies in a simple way market power on the two sides of the market.

While, to the best of our knowledge, the formation of trading groups(coalitions of traders of different types) has not been analyzed before, theresults we obtain are related to the study of cartels (coalitions of traders ofthe same type) in a general equilibrium framework. Most of the literatureon market power in general equilibrium has focused on cooperativeconcepts such as the core [3, 1, 14] the von Neumann Morgenstern stablesets [6], the Shapley value [5] or the nucleolus [7]. An important excep-tion is the study of formation of cartels in strategic market games due toOkuno, Postlewaite and Roberts [9]. While the objectives of these works

370 BLOCH AND GHOSAL

3 A richer model, where agents could fragment their trades and participate to differentmarkets deserves further investigation, but it is beyond the scope of the present paper.


and the emphasis of our paper are different, one of the results of thisliterature applies to the formation of trading groups. Examples have beenprovided to show that, in a general equilibrium context, an increase inmarket power does not necessarily lead to an increase in utility. The exist-ence of ``disadvantageous monopolies'' has been noted by Aumann [1] inthe context of the core (see also [10]) and by Okuno, Postlewaite andRoberts [9] in the context of strategic market games. In our setting, thepresence of ``disadvantageous monopolies'' implies that traders' utilitiesmay increase with the number of traders of the same type.

The rest of the paper is organized as follows. In the next section, weanalyze the model of bilateral oligopoly and derive comparative staticsresults by varying the number of traders on the market. Section 3 presentsour main results on stable trading structures. We conclude and give somedirections for further research in the last section.

2. BILATERAL OLIGOPOLIES

In order to analyze the formation of trading groups, we introduce in thissection a simple model of bilateral oligopolies. Following the general defini-tion proposed by Gabszewicz and Michel [4], a bilateral oligopoly isrepresented by a Shapley�Shubik strategic market game with two types oftraders, where each type of trader has a corner in a different commodity.More precisely, we consider an economy with two commodities labeled xand y and two types of traders. Each trader of type I owns one unit of thefirst commodity whereas each trader of type II is endowed with one unit ofthe second commodity. The set of traders of types I and II are denoted Kand L respectively with cardinality k and l.4

We denote the utility functions of the two types of traders by u(x, y) fortraders of type I and v(x, y) for traders of type II. We suppose that themarket is perfectly symmetric between both types of traders so that theutility functions satisfy u(x, y)=v( y, x) for all x and y.

In the bilateral oligopoly model, all traders behave strategically. Eachtrader i of the first type offers an amount bi of the first commodity on themarket and each trader j of the second type offers an amount gj of thesecond commodity. The strategy spaces of the two types of traders aregiven by

SI =[bi # R | 0�bi�1] for traders of type I

SII =[gj # R | 0�gj�1] for traders of type II.


4 Throughout the paper, we adopt the following convention: sets of traders are denoted bycapital letters and their cardinality by the corresponding lower case letter.


For any vector of offers (b1 , ..., bk , g1 , ..., gl), the final allocationsobtained by the traders are given by

(xi , yi)=\1&bi , bi� gj

� bi+(xj , yj)=\gj

� bi

� gj, 1& gj+ if : gj>0, : bi>0.

Otherwise, each trader consumes his endowment.5

Definition 2.1. A market equilibrium is a vector of offers (b1* , ..., bk* ,g1* , ..., gl*) such that

v For any trader i in K, bi* maximizes

u \1&bi , bi� gj*

�t{i bt*+bi+ .

v For any trader j in L, gj* maximizes

v \gj� bi*

�t{ j gt*+ gj, 1& gj+ .

A market equilibrium is thus defined as a Nash equilibrium of the par-ticular strategic market game we analyze. In order to characterize themarket equilibrium of the bilateral oligopoly model, we put the followingrestrictions on the utility functions.

Assumption 2.1. The utility function u is twice continuously differen-tiable, increasing and strictly concave.

Assumption 2.2. The utility function u satisfies the boundary conditions

limx � 0

&ux+uyy<0

limx � 1

&ux+uyy>0.

Assumption 2.3. The two goods x and y and y are complements, i.e.uxy�0.


5 The model we analyze is only one of the possible variants of the Shapley�Shubik tradingmechanisms. (See [12] for different versions of strategic market games.) We have ruled outthe possibility of ``wash sales'' by not allowing traders to offer the commodity they don't own.We have also assumed away the existence of a ``nume� raire commodity'' and suppose that thereis only one trading post where the two goods are exchanged.


Assumption 2.4. The utility function satisfies uy+ yuyy&2uxy>0.

While Assumption 2.1 is a classical regularity assumption on the utilityfunctions, the other three assumptions form a set of sufficient conditions toguarantee the existence and uniqueness of equilibrium. The boundaryconditions of Assumption 2.2 are needed to show that traders are alwayswilling to trade, but will only offer a fraction of their entire endowment.Together with the restrictions on the agents' strategy spaces, this assump-tion guarantees that the equilibrium is interior. Assumption 2.3 excludesfrom the analysis situations where the marginal utility of consuming one ofthe goods is decreasing in the consumption of the other. This assumptionof complementarity of the two goods is a sufficient (but by no meansnecessary) condition for the traders' maximization problems to be wellbehaved. Finally, Assumption 2.4 guarantees that the reaction function ofa trader is increasing in the bids of traders on the other side of the market.6

Proposition 2.1. Under Assumptions 2.1�2.4, if k�2 and l�2, thereexists a unique nontrivial equilibrium.7 The equilibrium is symmetric andgiven by (b*, g*) where b* and g* are implicitly defined by the equations:

&ux \1&b*,lg*k ++uy \1&b*,

lg*k + lg*

kk&1kb*

=0

&vy \kb*l

, 1& g*++vx \kb*l

, 1& g*+ kb*l

l&1lg*

=0.

Proof. See the Appendix.

Proposition 2.1 indicates that for any choice (k, l ) of the numbers oftraders of both types, there exists a unique market equilibrium that issymmetric. Hence, we may define, for any pair (k, l ) the equilibrium utilitylevels of agents of type I and II as

U(k, l )=u \1&b*(k, l ),lg*(k, l )

k +V(k, l )=v \kb*(k, l )

l, 1& g*(k, l)+ .


6 When the utility function u is additively separable, u(x, y)=u1(x)+u2( y), Assumption 2.4reads: u$2( y)+ yu"2( y)>0. This expression is reminiscent of Novshek [8]'s condition for theexistence of a Cournot equilibrium. However, Novshek's condition is the opposite of Assump-tion 2.4: it guarantees that each oligopolist's reaction function is decreasing in the choice ofthe other oligopolists whereas we require that each trader's reaction function is increasing inthe choice of the traders on the other side of the market.

7 The trivial equilibrium, where all traders offer 0 always exists. If either k=1 or l=1, itis in fact the only market equilibrium.


Since the market is symmetric, clearly, b*(k, l )= g*(l, k) and hence,U(k, l )=V(l, k). The next two propositions analyze the effects of changesin the size of the market on the utility levels of the traders.

Proposition 2.2. Consider two symmetric markets (k, k) and (k$, k$)where k$>k. Then U(k$, k$)=V(k$, k$)>U(k, k)=V(k, k).

Proof. If k=1, there is no trade on the market (k, k) and, givenAssumption 2.2, U(k, k)<U(k$, k$). Next suppose k�2. On the symmetricmarket (k, k), the equilibrium is characterized by the equation

&ux(1&b*, b*)+uy(1&b*, b*)k&1

k=0. (1)

Now observe that dU�dk=dU�db* db*�dk and dU�db*=&ux+uy . ByEq. (1), ux=uy(k&1)�k<uy so that dU�db*>0. Hence, to show that theutility increases with the number of traders on a symmetric market (k, k),it suffices to show that each trader's bid b* is an increasing function of k.

A simple computation shows that

db*dk

=uy

k2(&uxx+uxy&((k&1)�k)(uyy&uxy))>0.

Hence each trader's bid increases with the number of traders in the market,and the proof of the Proposition is complete. K

Proposition 2.2 asserts that, on symmetric markets, the utility of eachtrader increases with the number of traders. In the specific model of non-cooperative exchange we analyze, as we replicate the number of traders onthe market, the equilibrium outcomes converge monotonically to thePareto-efficient competitive equilibrium. This result can be interpreted inthe following simple way: on a symmetric market, traders of the two typesmake identical offers and the relative price at equilibrium is equal to 1.However, the presence of market power leads all traders to offer less thanthey would at a competitive equilibrium. As the number of tradersincreases, each trader's market power is reduced and the quantities offeredincrease, leading to an allocation closer to the efficient competitive equi-librium allocation.

Proposition 2.3. Consider two markets (k, l ) and (k, l $) with l $>l. Forany k�2, U(k, l )<U(k, l $).



Proof. See the Appendix.

Proposition 2.3 states that, as the number of traders on the other side ofthe market increases, the utility of each trader increases. This result relieson the fact that, as the number of traders of one type increases, the totalquantity offered by these traders increases, even though each individualoffer might be reduced. Hence, traders on the other side of the market,whose number remains unchanged, face a more favorable situation. In theterminology of partial equilibrium oligopoly theory, an increase in thenumber of traders on one side of the market leads to an ``expansion ofdemand'' without affecting the structure of competition on the other side ofthe market.

However, one of the most intuitive results obtained in partial equi-librium oligopoly analysis does not carry over to the case of a bilateraloligopoly. In the classical Cournot oligopoly, as the number of firmsincreases, the profit of each firm is reduced. In the context of a bilateraloligopoly, an increase in the number of traders of one type may induceeither positive or adverse effects on the traders of this type. This ambiguityresults from two opposite effects arising as the number of traders increases.On the one hand, this increase reduces the market power of the traderswhich leads to a decrease in utility. On the other hand, an increase in thenumber of traders induces traders of the other type to increase their offers,yielding an increase in utility. In general, the balance between these twoeffects cannot be ascertained.8

3. STABLE TRADING STRUCTURES

In the previous section, we have characterized the market equilibrium forany fixed sets K and L of traders of both types. We now analyze the forma-tion of groups of traders and study the emergence of stable tradingstructures. For this purpose we now let M and N denote the universal setsof traders. We assume that there is an equal number of potential traders ofboth types so that m=n.

Definition 3.1. A trading group is a pair (K, L) where K/M andL/N.

A trading group is a collection of traders of both types who agree totrade with one another. We assume that agents can only trade inside the


8 These two effects also arise when one considers, rather than markets with variablenumbers of agents, markets where traders can ``organize'' and form syndicates. There, asOkuno, Postlewaite and Roberts [9] show through two examples, the formation of asyndicate can either increase or decrease the utility of the traders.


trading group, so that different trading groups correspond to separate,exclusive markets.

Definition 3.2. A trading structure is a collection of trading groups,B=[(K1 , L1), . . .(KT , LT)] such that �T

t=1 Kt=M and �Tt=1 Lt=N and

for all t, Kt {< and Lt {<.

A trading structure B corresponds to a collection of separate markets onwhich agents trade. We focus on trading structures where all tradersparticipate to a market and do not consider the possibility of excludingsome agents from trading.9

Before defining stable trading structures, a last piece of notation isneeded. For any trading structure B and any subset S of traders, we denoteby B(S) those trading groups to which members of S belong. Moreprecisely, we define

B(S)=[(Kt , Lt) | Kt & S{< or Lt & S{<].

A trading structure B is stable if no coalition of traders can form atrading group in which they all obtain higher utility. In other words, in astable trading structure, traders have no incentive to leave the market theyare engaged in and form a new trading group. Depending on the meaningattached to ``higher utility'', two different concepts of stability are obtained:in the strong version of stability, coalitions of traders can deviate if all itsmembers obtain at least as high a utility as before and some traders obtaina strictly higher utility; in the weak version of stability, coalitions of tradersonly deviate if all its members can be made strictly better off.10

Definition 3.3. A trading structure B is strongly stable if there doesnot exist a trading group (K, L) such that

v For all coalitions (Kt , Lt) in B(K), U(k, l )�U(kt , lt).

v For all coalitions (Kt , Lt) in B(L), V(k, l )�V(kt , lt).

v Either for some (Kt , Lt) in B(K), U(k, l )>U(kt , lt) or for some(Kt , Lt) in B(L), V(k, l)>V(kt , lt).


9 In the conclusion, we discuss the possibility of having trading structures with exclusion,namely structures where Kt=< or Lt=< for some t.

10 The distinction between strong and weak stability is meaningful because utility is nottransferable among agents: once a trading group is formed, its members are constrained to usea Shapley�Shubik trading mechanism. If utility (or goods) were transferable, traders whoobtain a higher utility could compensate other traders, and the distinction between strong andweak stability would vanish.


Definition 3.4. A trading structure B is weakly stable if there does notexist a trading group (K, L) such that

v For all coalitions (Kt , Lt) in B(K), U(k, l )>U(kt , lt).

v For all coalitions (Kt , Lt) in B(L), V(k, l )>V(kt , lt).

The next Proposition provides an important necessary condition for atrading structure to be weakly stable.

Proposition 3.1. Let B be a weakly stable trading structure. Then thereexists an ordering of the trading groups in B, (K1 , L1), ..., (Kt , Lt), ...,(KT , LT) such that for all t, kt�kt+1 and lt�lt+1 .

Proof. Let B be a weakly stable trading structure and suppose bycontradiction that there exist two trading groups (Kr , Lr) and (Ks , Ls)such that kr>ks�1 and ls>lr�1. Then consider the following deviation(K, L)=(Kr , Ls). Since ls>lr�1, by Proposition 2.3, U(kr , ls)>U(kr , lr).Similarly, since kr>ks�1, V(kr , ls)>V(ks , ls). K

Proposition 3.1 shows that all weakly stable trading structures mustsatisfy a strong ordering property. For any two trading groups (Kr , Lr)and (Ks , Ls), if the second trading group contains more members of thefirst type, kr<ks , then it must also contain more members of the secondtype, namely lr�ls . In other words, markets can be ranked by size, frombig markets containing a large number of traders of both types to smallmarkets with few traders of each type. In particular, widely asymmetricmarkets where a large number of agents of one type face a small numberof participants of the other type cannot be part of a weakly stable tradingstructure.

Proposition 3.2. The only strongly stable trading structure is thestructure in which all agents trade on the same market, B=[(M, N)].

Proof. We first show that the trading structure [(M, N)] is stronglystable. Suppose, by contradiction that there exists a trading group (K, L)such that U(k, l )�U(m, n), V(k, l )�V(m, n) and either U(k, l )>U(m, n)or V(k, l )>V(m, n). First, by Proposition 2.2, k{l. Without loss ofgenerality, assume k>l. We then have, by Proposition 2.3

U(k, l )<U(l, l )�U(m, n),

a contradiction.Next, suppose that there exists another strongly stable trading structure

B$. We first show that B$ necessarily contains two trading groups (Kr , Lr),(Ks , Ls) such that either U(kr , lr){U(ks , ls) or V(kr , lr){V(ks , ls).

First note that, since the number of traders of the two types are equal,there must exist two trading groups (Kr , Lr) and (Ks , Ls) such that kr>lr



and ls>ks . Furthermore, since B$ is strongly stable, it is also weakly stableand the ordering property derived in Proposition 3.1 must hold. Firstsuppose that kr�ks and lr�ls . We then have kr>lr�ls>ks . ByProposition 2.3, V(ks , ls)<V(ls , ls). By Proposition 2.2, V(ls , ls)�V(lr , lr)and, by Proposition 2.3 again, V(lr , lr)<V(kr , lr). Hence V(ks , ls)<V(kr , lr). Next, if ks�kr and ls�lr , a similar reasoning shows that U(kr , lr)<U(ks , ls).

Now, without loss of generality, pick two tradings groups (Kr , Lr),(Ks , Ls) in B$ such that U(kr , lr)<U(ks , ls) and construct the followingdeviation (K, L). Let K be a set of cardinality k=ks consisting of a subsetK$r of members of Kr and a subset K$s of members of Ks . Furthermore, letL=Ls . Since k=ks , U(k, ls)=U(ks , ls) and V(k, ls)=V(ks , ls). Further-more, U(k, ls)=U(ks , ls)>U(kr , lr). Hence the deviation is profitable tomembers of K$r and the trading structure B$ cannot be stable. K

Proposition 3.2 establishes that the unique strongly stable trading struc-ture is the grand coalition, containing all the traders. Hence, in spite of thetrader's incentives to form different markets in order to increase theirmarket power, the only structure which is immune to deviations by coali-tions of traders is the structure where all agents meet on the same market.This striking result has the following interpretation. First of all, the grandcoalition is strongly stable since traders cannot increase their utility eitherby forming smaller, symmetric structures (Proposition 2.2) or by formingasymmetric structures (Proposition 2.3). No other trading structure can bestrongly stable because in any weakly stable structure, some traders of thesame type must receive different utilities, and badly treated agents canbuild profitable deviations.11

Proposition 3.2 also shows that weakly stable trading structures alwaysexist. Unfortunately, no complete characterization of the set of weaklystable trading structures can be obtained in general. However, examplesshow that it is possible to sustain other market structures as weakly stabletrading structures.12

4. CONCLUSION

This paper analyzes the formation of trading groups in a simple strategicmarket game with two types of traders where each type of trader has acorner in a different commodity. Our main result shows that the trading


11 This result is related to the equal treatment property of the core. As in the core, amechanism giving different utilities to symmetric agents can be blocked by some coalition.

12 These examples are available from the authors.


structure in which all agents trade is the unique strongly stable tradingstructure. Other weakly stable trading structures can exist and are charac-terized by a strong ordering property: trading groups can be ranked by sizeand cannot contain very different numbers of traders of the two types. Ouranalysis is based on two general properties of the noncooperative model ofexchange. First, in a symmetric market, the utility of the traders increaseswith the size of the market. Second, the utility of traders increases with thenumber of traders of the other type.

In the course of our analysis we have focused our attention on tradingstructures where all agents participate in active markets. Stable tradingstructures where some agents are excluded from all markets also exist anddeserve further analysis. In the context of our model, no trading structurewith exclusion can be strongly stable, since the excluded players can easilybe integrated into a trading group in a way which increases their utilitywhile not affecting the other members of the trading group. However,weakly stable trading structures with exclusion do exist, and their charac-terization poses new problems which need to be solved.

Finally, our characterization of stable trading structures depends cruciallyon several assumptions that we made on endowments, utility functions andthe institutions in which trade actually takes place. Whether or not ouranalysis extends to more general settings, with a general specification ofpreferences or endowments, or to models where traders can choosenot only the set of agents with whom they trade but also the rules thatgovern exchange within a trading group, is an important topic for furtherresearch.

APPENDIX

Proof of Proposition 2.1. We first show that the equilibrium must besymmetric. Consider a nontrivial equilibrium where � gj* {0 and� bi* {0. The maximization problem of a trader i of type I can bewritten

Max0�bi�1

u \1&bi , bi� gj

�t{i bt+bi+ .

Taking the derivative of the utility with respect to the bid bi , we obtain

�u�bi

=&ux+uy�t{i bt � gj

(�t{i bt+bi)2 .



Furthermore, note that

�2u�b2

i

=uxx&2uxy�t{i bt � gj

(�t{i bt+bi)2

+uyy(�t{i bt � gj)

2

(�t{i bt+bi)4 &uy

�t{i bt � gj

(�t{i bt+bi)4

<0.

Hence, under Assumption 2.3, the maximization problem faced by eachtrader is strictly concave. Note finally that Assumption 2.2 implies thatlimbi � 1 �u��bi<0 and limbi � 0 �u��bi>0 so that the maximization problemhas a unique interior solution bi # (0, 1) characterized by the first ordercondition

�u�bi

=&ux+uy�t{i bt � gj

(�t{i bt+bi)2=0.

Now suppose by contradiction that for two traders i and k of type I,bi {bk . Without loss of generality, let bk<bi . Since bi and bk are part ofthe equilibrium profile, the following two equations must hold

ux \1&bi ,bi � gj

�t{i bt+bi+uy \1&bi ,

bi � gj

�t{i bt+bi+=

�t{i bt � gj

(�t{i bt+bi)2

ux \1&bk ,bk � gj

�t{k bt+bk+uy \1&bk ,

bk � gj

�t{k bt+bi+=

�t{k bt � gj

(�t{k bt+bk)2 .

By Assumptions 2.1 and 2.3, the function

h(b)=ux \1&b,

b � gj

� bt +uy \1&b,

b � gj

� bt +is strictly increasing, so that h(bk)<h(bi). But observe that �t{i bt<�t{k bt , contradicting the equilibrium conditions.



Since the equilibrium is symmetric, we may denote by b* and g* the bidsof traders of type I and type II respectively. The equilibrium is then charac-terized by the solution (b*, g*) of the system of equations

&ux \1&b,lgk ++uy \1&b,

lgk+

lgk

k&1kb

=0

&vy \kbl

, 1& g++vx \kbl

, 1& g+ kbl

l&1lg

=0.

Next define the total bids on the two sides of the market as B=kb andG=lg. After this change of variables the equilibrium conditions become

f (G, B)=&ux \1&Bk

,Gk++uy \1&

Bk

,Gk+

GB

k&1k

=0 (2)

g(G, B)= &vy \Bl

, 1&Gl ++vx \B

l, 1&

Gl +

BG

l&1l

=0. (3)

We now show that the system of Eqs. (2) and (3) has a unique solution(B*, G*). First note that, by Assumption 2.3, �f��B<0 and �g��G<0.Hence, Eq. (2) defines implicitly the reaction function of traders of type I,B=,(G) whereas Eq. (3) defines implicitly the reaction function of tradersof type II, G=�(B).

By implicit differentiation we obtain

,$(G)=&�f��G�f��B

=&uxy+uyy(G�k)((k&1)�B)+uy((k&1)�B)&uxx+uxy(G(k&1)�B)+uy(G(k&1)�B2)

.

By Assumption 2.4, ,$(G)>0, so that the reaction function is upward slop-ing. Furthermore note that

,$(G)GB

=&uxy(G�B)+uyy(G2�k)((k&1)�B2)+uy(G(k&1)�B2)

&uxx+uxy(G(k&1)�B)+uy(G(k&1)�B2).

Since &uxx>0, we have

,$(G)GB

<&uxy(G�B)+uyy(G2�k)((k&1)�B2)+uy(G(k&1)�B2)

uxy(G(k&1)�B)+uy(G(k&1)�B2)<1.

The elasticity of the reaction function , is thus everywhere smaller thanone. By a similar argument, it is easy to show that the reaction function ofthe traders of type II, � is increasing and has an elasticity �$(B)(B�G) less



than one. These properties of the reaction functions indicate that, if anequilibrium exists, it must be unique and globally stable. More precisely, inorder to prove existence and uniqueness of the equilibrium, we consider thefunction

H(B)=B&,(�(B)).

It is clear that (B*, �(B*)) is a market equilibrium if and only ifH(B*)=0. We now determine the sign of the function H at the two bounds0 and k.

We first show that limB � 0 H(B)<0. To see this, note that

,(�(B))B

=,(�(B))

�(B)�(B)

B.

Now, obviously, limB � 0 �(B)=0. The boundary conditions (Assump-tion 2.2) ensure that limB � 0(�(B)�B)=+� and limG � 0(,(G)�G)=+�.Hence limB � 0(,(�(B))�B)=+� and limB � 0 H(B)<0.

Next we show that limB � k H(B)>0. To this end, we prove that, for anyG # (0, l ), ,(G)<k. Suppose to the contrary that there exists a value G forwhich ,(G)=k. This implies that limB � k uyy(k&1)�k&ux>0, contradictingthe boundary conditions. Finally note that

H$(B)=1&,$(G) �$(B)>0.

Hence, the function H is strictly increasing over the interval (0, k) andthere exists a unique solution to the equation H(B)=0. K

Proof of Proposition 2.3. We assume without loss of generality thatl�2 since for l=1 the Proposition is immediate. First recall that the equi-librium utility level of a trader of type I is given by

U(k, l )=u \1&B*(k, l )

k,G*(k, l )

k +=u \1&,(G*(k, l ))

k,

G*(k, l )k + .

Hence,

dUdl

=1k

�G*�l

(&ux,$(G*)+uy).

From the equilibrium condition (Eq. (2)), we obtain

uy=uxk

k&1B*G*

.



We thus have

&ux�$(G*)+uy =ux \ kk&1

B*G*

&,$(G*)+=ux

B*G* \

kk&1

&,$(G*)G*B*+ .

Now, since ,$(G*)(G*�B*)<1, &ux�$(G*)+uy>0, and sign(dU�dl )=sign(�G*��l ). In order to compute the sign of dU�dl, it thus suffices toanalyze whether the total bids of traders of type II, G*, increase ordecrease with a change in l.13

By total differentiation of the equilibrium conditions (Eqs. (2) and (3))with respect to l, we obtain

dfdB*

dB*dl

+df

dG*dG*

dl=0

dgdB*

dB*dl

+dg

dG*dG*

dl=&

dgdl

.

After some manipulations we obtain

dG*dl

=&(dg�dl )(df�dB*)

(df�dB*)(dg�dG*)&(df�dG*)(dg�dB*).

To compute the sign of this expression, first not that df�dB*<0 and

dgdl

=vyxB*l2 &vyy

G*l2 &vxx

B*l2 +vxy

G*l2 +vx

B*G*l2>0.

Next observe that

dfdB*

dgdG*

&df

dG*dg

dB*=1&

(df�dG*)(dg�dB*)(df�dB*)(dg�dG*)

=1&,$(G*) �$(B*)

>0.

The total bids of traders of type II are thus increasing in l, showing that theequilibrium utility level of traders of type I increases with the number oftraders of type II. K


13 Our method of proof, linking the effect of changes in exogenous parameters to thestability properties of the equilibrium was inspired by Dixit [2]'s analysis of comparativestatics in oligopoly.


REFERENCES

1. R. Aumann, Disadvantageous monopolies, J. Econ. Theory 6 (1973), 1�11.2. A. Dixit, Comparative statics in oligopoly, Int. Econ. Rev. 27 (1986), 107�122.3. J. J. Gabszewicz and J. Dre� ze, Syndicates of traders in an exchange economy, in ``Dif-

ferential Games and Related Topics'' (H. Kuhn and G. Szego, Eds.), North-Holland,Amsterdam, 1971.

4. J. J. Gabszewicz and P. Michel, ``Oligopoly Equilibrium in Exchange Economies,'' COREDiscussion Paper 9247, CORE, Universite� Catholique de Louvain, 1992.

5. R. Guesnerie, Monopoly, syndicate and Shapley value: About some conjectures, J. Econ.Theory 15 (1977), 235�251.

6. S. Hart, Formation of cartels in large markets, J. Econ. Theory 7 (1974), 453�466.7. P. Legros, Disadvantageous syndicates and stable cartels: The case of the nucleolus,

J. Econ. Theory 42 (1987), 30�49.8. W. Novshek, On the existence of Cournot equilibrium, Rev. Econ. Stud. 52 (1985), 85�98.9. M. Okuno, A. Postlewaite, and J. Roberts, Oligopoly and competition in large markets,

Amer. Econ. Rev. 70 (1980), 22�31.10. A. Postlewaite and R. Rosenthal, Disadvantageous syndicates, J. Econ. Theory 9 (1974),

324�326.11. A. Postlewaite and D. Schmeidler, Approximate efficiency and non-Walrasian equilibria,

Econometrica 46 (1978), 127�135.12. L. Shapley, Noncooperative general exchange, in ``Theory and Measurement of Economic

Externalities'' (S. Lin, Ed.), Academic Press, New York, 1976.13. L. Shapley and M. Shubik, Trade using one commodity as a means of payment, J. Polit.

Econ. 85 (1977), 937�968.14. B. Shitovitz, Oligopoly in markets with a continuum of traders, Econometrica 41 (1973),

467�502.


stable trading structures in bilateral oligopolies

Documents