Stable Trading Structures in Bilateral Oligopolies

Journal of Economic Theory  ET2266 journal of economic theory 74, 368384 (1997) article no. ET962266

Stable Trading Structures in Bilateral Oligopolies* Francis Bloch CORE and Groupe HEC, 1 rue de la Liberation, 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 where agents trade according to a ShapleyShubik (J. Polit. Econ. 85 (1977), 937968.) trading mechanism. The only strongly stable trading structure is the grand coalition, where all agents trade on the same market. Other weakly stable trading structures exist and are characterized by an ordering property: trading groups can be ranked by size and cannot contain very different numbers of traders of the two types. Journal of Economic Literature Classification Numbers: D43, D51.  1997 Academic Press

1. INTRODUCTION In the traditional representation of markets, traders cannot choose the set of agents with whom they trade. Goods are exchanged on a single, anonymous market in which all the agents participate. In a competitive economy, the assumption that all agents trade on the same market can easily be justified. Since any competitive equilibrium belongs to the core, no coalition of traders could gain by trading on a smaller submarket. In the * We have greatly benefitted from discussions with Francoise Forges, Jean Jaskold Gabszewicz, Roger Guesnerie, and Heracles Polemarchakis. We are also grateful to seminar audiences at CORE, Erasmus University, Valencia, University Paris-I, the Ecole Nationale des Ponts et Chaussees, the SITE 1995 Summer Workshop on Coalition Formation at Stanford, the Second Economic Theory Conference in Kephalonia, and the World Congress of the Econometric Society in Tokyo for their comments. The suggestions of an associate editor and an anonymous referee greatly improved the quality of the paper. Finally, we gratefully acknowledge financial support from the European Union (Human Capital and Mobility Fellowship, Contract ERBCHBCT920167) for the first author and from a CORE Ph.D. Fellowship for the second author.

368 0022-053197 25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.

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terminology we adopt in this paper, the trading structure where all agents trade on the same market is ``stable'' since it cannot be disrupted by the actions of any coalition of traders. However, in small economies where traders can enjoy market power, the presence of a single market grouping all traders cannot be so easily defended. In fact, the classical model of Cournot oligopoly suggests that traders have an incentive to set up distinct, separate markets on which they behave as monopolists. Hence, when traders behave strategically, coalitions of traders may benefit by forming small submarkets, and the structure 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 a simple finite economy where all agents behave strategically. We consider the framework of bilateral oligopoly, with two types of traders, where each type of trader has a corner in a different commodity. 1 Each trader submits an offer to the market and, as in the noncooperative model of exchange of Shapley and Shubik [13], prices are determined by the ratio of aggregate offers of the two commodities. We first show that the trading structure where all agents trade on the same market is stable. As in the competitive model, coalitions of traders cannot benefit from forming smaller markets and the intuition from the partial equilibrium analysis of Cournot oligopoly fails in a general equilibrium setting where traders take into account the reaction of traders on the other side of the market. The incentives to form trading groups in a bilateral oligopoly can be derived from two simple comparative statics results on the equilibrium of the noncooperative model of exchange. First, traders on one side of the market always benefit from an increase in the number of traders on the other side. In fact, as the number of traders on one side of the market increases, the total quantity they offer rises while the structure of competition between traders on the other side of the market remains unchanged. Second, on any symmetric market, where the number of traders of the two types is equal, the traders' utilities increase with the size of the market. This is due to the fact that on symmetric markets, the relative price of the two goods is equal to one, and as the number of traders increases, the equilibrium approaches the efficient competitive equilibrium allocation. 2 Given these two results, it is clear that no coalition of traders can deviate from the structure where all agents trade on the same market. Traders obtain lower utility on any symmetric submarket and, on asymmetric submarkets traders who are more numerous on the market must obtain a lower utility than on some symmetric market. 1 This model of bilateral oligopoly, which is a special version of the ShapleyShubik strategic 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 general setting, that the equilibrium outcomes of a ShapleyShubik market game become arbitrarily close to a Pareto-efficient allocation as the number of traders goes to infinity.

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To further the analysis of stable trading structures, we distinguish between two concepts of stability. In the strong version of stability, coalitions of traders can deviate if the utility its members obtain is at least as high as before, and strictly higher for some member of the coalition. In the weak version of stability, the deviating coalition must make all its members strictly better off. A striking result we derive is that the only strongly stable trading structure is the one in which all agents meet on the same market. To understand this result, note that any other weakly stable trading structure must be asymmetric and treat symmetric traders differently. As in the core, an allocation which treats similar players differently can be disrupted by some coalition of traders. If the single market trading structure is the only strongly stable trading structure, other weakly stable structures may exist. We show that they must satisfy a strong ordering property: if one trading group has more members of one type it must have more members of the other type as well. Hence, in a weakly stable trading structure 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 stable trading structure relies on the particular restrictions we place on the actions of traders. We first assume that each agent only trades on one market and exclude the possibility that the same agent be present on different markets. 3 Second, once a trading group is formed, we constrain all agents to trade according to a ShapleyShubik strategic market game. The use of ShapleyShubik trading mechanisms poses well-known difficulties: the equilibrium outcome is typically not Pareto-optimal, a commodity which is owned by less than two agents cannot be traded at equilibrium. Furthermore, the properties of equilibria are not robust to changes in the specification of the trading mechanism. In spite of these difficulties, we have adopted the framework of a ShapleyShubik model of noncooperative exchange because it can be viewed as a natural generalization of Cournot oligopolies and embodies 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, the results we obtain are related to the study of cartels (coalitions of traders of the same type) in a general equilibrium framework. Most of the literature on market power in general equilibrium has focused on cooperative concepts such as the core [3, 1, 14] the von Neumann Morgenstern stable sets [6], the Shapley value [5] or the nucleolus [7]. An important exception is the study of formation of cartels in strategic market games due to Okuno, Postlewaite and Roberts [9]. While the objectives of these works 3 A richer model, where agents could fragment their trades and participate to different markets deserves further investigation, but it is beyond the scope of the present paper.

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and the emphasis of our paper are different, one of the results of this literature applies to the formation of trading groups. Examples have been provided to show that, in a general equilibrium context, an increase in market power does not necessarily lead to an increase in utility. The existence of ``disadvantageous monopolies'' has been noted by Aumann [1] in the context of the core (see also [10]) and by Okuno, Postlewaite and Roberts [9] in the context of strategic market games. In our setting, the presence of ``disadvantageous monopolies'' implies that traders' utilities may increase with the number of traders of the same type. The rest of the paper is organized as follows. In the next section, we analyze the model of bilateral oligopoly and derive comparative statics results by varying the number of traders on the market. Section 3 presents our main results on stable trading structures. We conclude and give some directions for further research in the last section.

2. BILATERAL OLIGOPOLIES In order to analyze the formation of trading groups, we introduce in this section a simple model of bilateral oligopolies. Following the general definition proposed by Gabszewicz and Michel [4], a bilateral oligopoly is represented by a ShapleyShubik strategic market game with two types of traders, where each type of trader has a corner in a different commodity. More precisely, we consider an economy with two commodities labeled x and y and two types of traders. Each trader of type I owns one unit of the first commodity whereas each trader of type II is endowed with one unit of the second commodity. The set of traders of types I and II are denoted K and L respectively with cardinality k and l. 4 We denote the utility functions of the two types of traders by u(x, y) for traders of type I and v(x, y) for traders of type II. We suppose that the market is perfectly symmetric between both types of traders so that the utility functions satisfy u(x, y)=v( y, x) for all x and y. In the bilateral oligopoly model, all traders behave strategically. Each trader i of the first type offers an amount b i of the first commodity on the market and each trader j of the second type offers an amount g j of the second commodity. The strategy spaces of the two types of traders are given by S I =[b i # R | 0b i 1]

for traders of type I

S II =[ g j # R | 0g j 1]

for traders of type II.

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

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For any vector of offers (b 1 , ..., b k , g 1 , ..., g l ), the final allocations obtained by the traders are given by  gj  bi

\ + b (x , y )= g \  g , 1& g + (x i , y i )= 1&b i , b i i

j

j

j

j

if

: g j >0,

: b i >0.

j

Otherwise, each trader consumes his endowment. 5 Definition 2.1. A market equilibrium is a vector of offers (b * 1 , ..., b * k , , ..., g *) such that g* 1 l v For any trader i in K, b * i maximizes

\

u 1&b i , b i

 g* j .  t{i b *+b t i

+

v For any trader j in L, g * j maximizes

\

v gj

 b* i , 1& g j .  t{ j g t*+ g j

+

A market equilibrium is thus defined as a Nash equilibrium of the particular strategic market game we analyze. In order to characterize the market equilibrium of the bilateral oligopoly model, we put the following restrictions on the utility functions. Assumption 2.1. The utility function u is twice continuously differentiable, increasing and strictly concave. Assumption 2.2.

The utility function u satisfies the boundary conditions lim &u x +u y y<0

x0

lim &u x +u y y>0.

x1

Assumption 2.3. u xy 0.

The two goods x and y and y are complements, i.e.

5 The model we analyze is only one of the possible variants of the ShapleyShubik trading mechanisms. (See [12] for different versions of strategic market games.) We have ruled out the 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 ``numeraire commodity'' and suppose that there is only one trading post where the two goods are exchanged.

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Assumption 2.4.

373

The utility function satisfies u y + yu yy &2u xy >0.

While Assumption 2.1 is a classical regularity assumption on the utility functions, the other three assumptions form a set of sufficient conditions to guarantee the existence and uniqueness of equilibrium. The boundary conditions of Assumption 2.2 are needed to show that traders are always willing to trade, but will only offer a fraction of their entire endowment. Together with the restrictions on the agents' strategy spaces, this assumption guarantees that the equilibrium is interior. Assumption 2.3 excludes from the analysis situations where the marginal utility of consuming one of the goods is decreasing in the consumption of the other. This assumption of complementarity of the two goods is a sufficient (but by no means necessary) condition for the traders' maximization problems to be well behaved. Finally, Assumption 2.4 guarantees that the reaction function of a trader is increasing in the bids of traders on the other side of the market. 6 Proposition 2.1. Under Assumptions 2.12.4, if k2 and l2, there exists a unique nontrivial equilibrium. 7 The equilibrium is symmetric and given by (b*, g*) where b* and g* are implicitly defined by the equations:

\

lg* lg* lg* k&1 =0 +u y 1&b*, k k k kb*

+ \ + kb* kb* kb* l&1 &v \ l , 1& g*+ +v \ l , 1& g*+ l lg* =0. &u x 1&b*,

y

Proof.

x

See the Appendix.

Proposition 2.1 indicates that for any choice (k, l) of the numbers of traders of both types, there exists a unique market equilibrium that is symmetric. Hence, we may define, for any pair (k, l ) the equilibrium utility levels of agents of type I and II as lg*(k, l )

\ k + kb*(k, l) V(k, l )=v \ l , 1& g*(k, l)+ .

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

6

When the utility function u is additively separable, u(x, y)=u 1(x)+u 2( y), Assumption 2.4 reads: u$2( y)+ yu"2( y)>0. This expression is reminiscent of Novshek [8]'s condition for the existence of a Cournot equilibrium. However, Novshek's condition is the opposite of Assumption 2.4: it guarantees that each oligopolist's reaction function is decreasing in the choice of the other oligopolists whereas we require that each trader's reaction function is increasing in the 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, it is in fact the only market equilibrium.

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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 changes in 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, given Assumption 2.2, U(k, k)
k&1 =0. k

(1)

Now observe that dUdk=dUdb* db*dk and dUdb*=&u x +u y . By Eq. (1), u x =u y(k&1)k0. Hence, to show that the utility 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 uy db* = 2 >0. dk k (&u xx +u xy &((k&1)k)(u yy &u xy )) 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 each trader increases with the number of traders. In the specific model of noncooperative exchange we analyze, as we replicate the number of traders on the market, the equilibrium outcomes converge monotonically to the Pareto-efficient competitive equilibrium. This result can be interpreted in the following simple way: on a symmetric market, traders of the two types make identical offers and the relative price at equilibrium is equal to 1. However, the presence of market power leads all traders to offer less than they would at a competitive equilibrium. As the number of traders increases, each trader's market power is reduced and the quantities offered increase, leading to an allocation closer to the efficient competitive equilibrium allocation. Proposition 2.3. Consider two markets (k, l ) and (k, l$) with l$>l. For any k2, U(k, l)
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Proof.

375

See the Appendix.

Proposition 2.3 states that, as the number of traders on the other side of the market increases, the utility of each trader increases. This result relies on the fact that, as the number of traders of one type increases, the total quantity offered by these traders increases, even though each individual offer might be reduced. Hence, traders on the other side of the market, whose number remains unchanged, face a more favorable situation. In the terminology of partial equilibrium oligopoly theory, an increase in the number of traders on one side of the market leads to an ``expansion of demand'' without affecting the structure of competition on the other side of the market. However, one of the most intuitive results obtained in partial equilibrium oligopoly analysis does not carry over to the case of a bilateral oligopoly. In the classical Cournot oligopoly, as the number of firms increases, the profit of each firm is reduced. In the context of a bilateral oligopoly, an increase in the number of traders of one type may induce either positive or adverse effects on the traders of this type. This ambiguity results from two opposite effects arising as the number of traders increases. On the one hand, this increase reduces the market power of the traders which leads to a decrease in utility. On the other hand, an increase in the number of traders induces traders of the other type to increase their offers, yielding an increase in utility. In general, the balance between these two effects cannot be ascertained. 8

3. STABLE TRADING STRUCTURES In the previous section, we have characterized the market equilibrium for any fixed sets K and L of traders of both types. We now analyze the formation of groups of traders and study the emergence of stable trading structures. For this purpose we now let M and N denote the universal sets of traders. We assume that there is an equal number of potential traders of both types so that m=n. Definition 3.1. L/N.

A trading group is a pair (K, L) where K/M and

A trading group is a collection of traders of both types who agree to trade 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 variable numbers of agents, markets where traders can ``organize'' and form syndicates. There, as Okuno, Postlewaite and Roberts [9] show through two examples, the formation of a syndicate can either increase or decrease the utility of the traders.

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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=[(K 1 , L 1 ), . . .(K T , L T )] such that  Tt=1 K t =M and  Tt=1 L t =N and for all t, K t {< and L t {<. A trading structure B corresponds to a collection of separate markets on which agents trade. We focus on trading structures where all traders participate to a market and do not consider the possibility of excluding some agents from trading. 9 Before defining stable trading structures, a last piece of notation is needed. For any trading structure B and any subset S of traders, we denote by B(S) those trading groups to which members of S belong. More precisely, we define B(S)=[(K t , L t ) | K t & S{< or L t & S{<]. A trading structure B is stable if no coalition of traders can form a trading group in which they all obtain higher utility. In other words, in a stable trading structure, traders have no incentive to leave the market they are engaged in and form a new trading group. Depending on the meaning attached to ``higher utility'', two different concepts of stability are obtained: in the strong version of stability, coalitions of traders can deviate if all its members obtain at least as high a utility as before and some traders obtain a strictly higher utility; in the weak version of stability, coalitions of traders only deviate if all its members can be made strictly better off. 10 Definition 3.3. A trading structure B is strongly stable if there does not exist a trading group (K, L) such that v For all coalitions (K t , L t ) in B(K), U(k, l )U(k t , l t ). v For all coalitions (K t , L t ) in B(L), V(k, l)V(k t , l t ). v Either for some (K t , L t ) in B(K), U(k, l)>U(k t , l t ) or for some (K t , L t ) in B(L), V(k, l)>V(k t , l t ). 9 In the conclusion, we discuss the possibility of having trading structures with exclusion, namely structures where K t =< or L t =< for some t. 10 The distinction between strong and weak stability is meaningful because utility is not transferable among agents: once a trading group is formed, its members are constrained to use a ShapleyShubik trading mechanism. If utility (or goods) were transferable, traders who obtain a higher utility could compensate other traders, and the distinction between strong and weak stability would vanish.

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Definition 3.4. A trading structure B is weakly stable if there does not exist a trading group (K, L) such that v For all coalitions (K t , L t ) in B(K), U(k, l )>U(k t , l t ). v For all coalitions (K t , L t ) in B(L), V(k, l)>V(k t , l t ). The next Proposition provides an important necessary condition for a trading structure to be weakly stable. Proposition 3.1. Let B be a weakly stable trading structure. Then there exists an ordering of the trading groups in B, (K 1 , L 1 ), ..., (K t , L t ), ..., (K T , L T ) such that for all t, k t k t+1 and l t l t+1 . Proof. Let B be a weakly stable trading structure and suppose by contradiction that there exist two trading groups (K r , L r ) and (K s , L s ) such that k r >k s 1 and l s >l r 1. Then consider the following deviation (K, L)=(K r , L s ). Since l s >l r 1, by Proposition 2.3, U(k r , l s )>U(k r , l r ). Similarly, since k r >k s 1, V(k r , l s )>V(k s , l s ). K Proposition 3.1 shows that all weakly stable trading structures must satisfy a strong ordering property. For any two trading groups (K r , L r ) and (K s , L s ), if the second trading group contains more members of the first type, k r U(m, n) or V(k, l)>V(m, n). First, by Proposition 2.2, k{l. Without loss of generality, assume k>l. We then have, by Proposition 2.3 U(k, l )l r

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and l s >k s . Furthermore, since B$ is strongly stable, it is also weakly stable and the ordering property derived in Proposition 3.1 must hold. First suppose that k r k s and l r l s . We then have k r >l r l s >k s . By Proposition 2.3, V(k s , l s )U(k r , l r ). Hence the deviation is profitable to members of K$r and the trading structure B$ cannot be stable. K Proposition 3.2 establishes that the unique strongly stable trading structure is the grand coalition, containing all the traders. Hence, in spite of the trader's incentives to form different markets in order to increase their market power, the only structure which is immune to deviations by coalitions of traders is the structure where all agents meet on the same market. This striking result has the following interpretation. First of all, the grand coalition is strongly stable since traders cannot increase their utility either by forming smaller, symmetric structures (Proposition 2.2) or by forming asymmetric structures (Proposition 2.3). No other trading structure can be strongly stable because in any weakly stable structure, some traders of the same type must receive different utilities, and badly treated agents can build profitable deviations. 11 Proposition 3.2 also shows that weakly stable trading structures always exist. Unfortunately, no complete characterization of the set of weakly stable trading structures can be obtained in general. However, examples show that it is possible to sustain other market structures as weakly stable trading structures. 12

4. CONCLUSION This paper analyzes the formation of trading groups in a simple strategic market game with two types of traders where each type of trader has a corner 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, a mechanism giving different utilities to symmetric agents can be blocked by some coalition. 12 These examples are available from the authors.

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structure in which all agents trade is the unique strongly stable trading structure. Other weakly stable trading structures can exist and are characterized by a strong ordering property: trading groups can be ranked by size and cannot contain very different numbers of traders of the two types. Our analysis is based on two general properties of the noncooperative model of exchange. First, in a symmetric market, the utility of the traders increases with the size of the market. Second, the utility of traders increases with the number of traders of the other type. In the course of our analysis we have focused our attention on trading structures where all agents participate in active markets. Stable trading structures where some agents are excluded from all markets also exist and deserve further analysis. In the context of our model, no trading structure with exclusion can be strongly stable, since the excluded players can easily be integrated into a trading group in a way which increases their utility while not affecting the other members of the trading group. However, weakly stable trading structures with exclusion do exist, and their characterization poses new problems which need to be solved. Finally, our characterization of stable trading structures depends crucially on several assumptions that we made on endowments, utility functions and the institutions in which trade actually takes place. Whether or not our analysis extends to more general settings, with a general specification of preferences or endowments, or to models where traders can choose not only the set of agents with whom they trade but also the rules that govern exchange within a trading group, is an important topic for further research.

APPENDIX Proof of Proposition 2.1. We first show that the equilibrium must be symmetric. Consider a nontrivial equilibrium where  g * j {0 and {0. The maximization problem of a trader i of type I can be  b* i written

\

Max u 1&b i , b i

0bi 1

 gj .  t{i b t +b i

+

Taking the derivative of the utility with respect to the bid b i , we obtain u  t{i b t  g j =&u x +u y . b i ( t{i b t +b i ) 2

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Furthermore, note that  2u  t{i b t  g j =u xx &2u xy b 2i ( t{i b t +b i ) 2 +u yy

( t{i b t  g j ) 2  t{i b t  g j &u y 4 ( t{i b t +b i ) ( t{i b t +b i ) 4

<0. Hence, under Assumption 2.3, the maximization problem faced by each trader is strictly concave. Note finally that Assumption 2.2 implies that lim bi  1 ub i <0 and lim bi  0 ub i >0 so that the maximization problem has a unique interior solution b i # (0, 1) characterized by the first order condition u  t{i b t  g j =&u x +u y =0. b i ( t{i b t +b i ) 2 Now suppose by contradiction that for two traders i and k of type I, b i {b k . Without loss of generality, let b k
\  b +b +=  b  g ( u 1&b , \  b +b + b  g u 1&b , \  b +b + =  b  g ( u 1&b , \  b +b + u x 1&b i ,

t

t{i i

y

j

t

k

i

j

k

t

t{k k

y

t{i

i

t{i

x

bt  gj 2 t{i b t +b i )

i

bt  gj . 2 b t{k t +b k )

k

j

t{k

k

t

t{k

i

By Assumptions 2.1 and 2.3, the function b  gj

\ b + h(b)= b g u 1&b, \ b + u x 1&b,

t

j

y

t

is strictly increasing, so that h(b k )
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Since the equilibrium is symmetric, we may denote by b* and g* the bids of traders of type I and type II respectively. The equilibrium is then characterized by the solution (b*, g*) of the system of equations

\

lg lg lg k&1 =0 +u y 1&b, k k k kb

+ \ + kb kb kb l&1 &v \ l , 1& g+ +v \ l , 1& g+ l lg =0. &u x 1&b,

y

x

Next define the total bids on the two sides of the market as B=kb and G=lg. After this change of variables the equilibrium conditions become B G B G G k&1 =0 f (G, B)= &u x 1& , +u y 1& , k k k k B k

\ + \ + G G B l&1 B B , 1& +v , 1& g(G, B)= &v \ l l + \ l l + G l =0. y

x

(2) (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, fB<0 and gG<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 traders of type II, G=(B). By implicit differentiation we obtain ,$(G)=&

fG &u xy +u yy(Gk)((k&1)B)+u y((k&1)B) = . fB &u xx +u xy(G(k&1)B)+u y(G(k&1)B 2 )

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

G &u xy(GB)+u yy(G 2k)((k&1)B 2 )+u y(G(k&1)B 2 ) = . B &u xx +u xy(G(k&1)B)+u y(G(k&1)B 2 )

Since &u xx >0, we have ,$(G)

G &u xy(GB)+u yy(G 2k)((k&1)B 2 )+u y(G(k&1)B 2 ) < <1. B u xy(G(k&1)B)+u y(G(k&1)B 2 )

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

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than one. These properties of the reaction functions indicate that, if an equilibrium exists, it must be unique and globally stable. More precisely, in order to prove existence and uniqueness of the equilibrium, we consider the function H(B)=B&,((B)). It is clear that (B*, (B*)) is a market equilibrium if and only if H(B*)=0. We now determine the sign of the function H at the two bounds 0 and k. We first show that lim B  0 H(B)<0. To see this, note that ,((B)) ,((B)) (B) = . B (B) B Now, obviously, lim B  0 (B)=0. The boundary conditions (Assumption 2.2) ensure that lim B  0((B)B)=+ and lim G  0(,(G)G)=+. Hence lim B  0(,((B))B)=+ and lim B  0 H(B)<0. Next we show that lim B  k H(B)>0. To this end, we prove that, for any G # (0, l ), ,(G)0, contradicting the boundary conditions. Finally note that H$(B)=1&,$(G) $(B)>0. Hence, the function H is strictly increasing over the interval (0, k) and there exists a unique solution to the equation H(B)=0. K Proof of Proposition 2.3. We assume without loss of generality that l2 since for l=1 the Proposition is immediate. First recall that the equilibrium utility level of a trader of type I is given by

\

U(k, l )=u 1&

B*(k, l ) G*(k, l ) ,(G*(k, l )) G*(k, l ) , , =u 1& . k k k k

+ \

Hence, dU 1 G* = (&u x ,$(G*)+u y ). dl k l From the equilibrium condition (Eq. (2)), we obtain u y =u x

k B* . k&1 G*

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+

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We thus have &u x $(G*)+u y =u x =u x

\

k B* &,$(G*) k&1 G*

+

G* B* k &,$(G*) . G* k&1 B*

\

+

Now, since ,$(G*)(G*B*)<1, &u x $(G*)+u y >0, and sign(dUdl )= sign(G*l). In order to compute the sign of dUdl, it thus suffices to analyze whether the total bids of traders of type II, G*, increase or decrease with a change in l. 13 By total differentiation of the equilibrium conditions (Eqs. (2) and (3)) with respect to l, we obtain df dG* df dB* + =0 dB* dl dG* dl dg dg dB* dg dG* + =& . dB* dl dG* dl dl After some manipulations we obtain dG* (dgdl )(dfdB*) =& . dl (dfdB*)(dgdG*)&(dfdG*)(dgdB*) To compute the sign of this expression, first not that dfdB*<0 and dg B* G* B* G* B* =v yx 2 &v yy 2 &v xx 2 +v xy 2 +v x >0. dl l l l l G*l 2 Next observe that df dg df dg (dfdG*)(dgdB*) & =1& dB* dG* dG* dB* (dfdB*)(dgdG*) =1&,$(G*) $(B*) >0. The total bids of traders of type II are thus increasing in l, showing that the equilibrium utility level of traders of type I increases with the number of traders of type II. K 13 Our method of proof, linking the effect of changes in exogenous parameters to the stability properties of the equilibrium was inspired by Dixit [2]'s analysis of comparative statics in oligopoly.

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REFERENCES 1. R. Aumann, Disadvantageous monopolies, J. Econ. Theory 6 (1973), 111. 2. A. Dixit, Comparative statics in oligopoly, Int. Econ. Rev. 27 (1986), 107122. 3. J. J. Gabszewicz and J. Dreze, Syndicates of traders in an exchange economy, in ``Differential 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,'' CORE Discussion Paper 9247, CORE, Universite Catholique de Louvain, 1992. 5. R. Guesnerie, Monopoly, syndicate and Shapley value: About some conjectures, J. Econ. Theory 15 (1977), 235251. 6. S. Hart, Formation of cartels in large markets, J. Econ. Theory 7 (1974), 453466. 7. P. Legros, Disadvantageous syndicates and stable cartels: The case of the nucleolus, J. Econ. Theory 42 (1987), 3049. 8. W. Novshek, On the existence of Cournot equilibrium, Rev. Econ. Stud. 52 (1985), 8598. 9. M. Okuno, A. Postlewaite, and J. Roberts, Oligopoly and competition in large markets, Amer. Econ. Rev. 70 (1980), 2231. 10. A. Postlewaite and R. Rosenthal, Disadvantageous syndicates, J. Econ. Theory 9 (1974), 324326. 11. A. Postlewaite and D. Schmeidler, Approximate efficiency and non-Walrasian equilibria, Econometrica 46 (1978), 127135. 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), 937968. 14. B. Shitovitz, Oligopoly in markets with a continuum of traders, Econometrica 41 (1973), 467502.

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