1 Introduction
The study of market power in imperfectly competitive markets has commanded much attention from economists. The workhorse model of industrial organization economists—that of Cournot competition—takes a partial equilibrium perspective and makes the assumption that only firms have market power. Bilateral oligopoly is a natural generalization of Cournot competition to consider markets in which both sellers and buyers can have market power and so behave strategically in manipulating prices to be more favourable to them. The purpose of this exposition is to introduce the reader to the study of bilateral oligopoly by leading them through worked examples that illustrate some of the main ideas to emerge from the literature.
Bilateral oligopoly was first introduced by Gabszewicz and Michel (1997). It consists of an economy in which there are two commodities and agents have “corner endowments”, i.e. they are endowed with only one of these two commodities. In a noncooperative game, agents can then choose the amount of their initial endowment to put up in exchange for the other commodity. It is often convenient to think of the first of the two commodities as a consumption good and the second as commodity money (which can be viewed as a numeraire), in which case the agents endowed with the consumption good are called sellers, and those endowed with the commodity money are called buyers. Despite the simplicity of the model, the structure is rich enough to study many interesting phenomena that may arise in imperfectly competitive economies. A fruitful line of research focuses on testing the robustness of the partial equilibrium analysis of Cournot oligopolies to general equilibrium models without production where all agents, sellers and buyers, have market power. For instance, Bloch and Ghosal (1997) and Bloch and Ferrer (2001a) study agents’ incentives to create submarkets as the Cournot game suggests that firms could gain market power by forming smaller submarkets. Dickson and Hartley (2008) and Amir and Bloch (2009) focus on the existence of Nash equilibrium and the comparative static properties of the equilibrium. Dickson (2013a) pays particular attention to the effect of entry of additional sellers in bilateral oligopoly, comparing the results to the conventional wisdom from the Cournot model.
The bilateral oligopoly model belongs to the line of research on strategic market games initiated by the seminal papers of Shubik (1973), Shapley (1976), and Shapley and Shubik (1977). There are many types of strategic market games (see Giraud (2003) and Levando (2012) for a survey). Here we mention only the “trading post model” and the “window model” which can be seen as different institutional mechanisms through which prices are determined. In the first model, trade is decentralised through a system of trading posts where commodities are exchanged. Dubey and Shubik (1978) studied the trading post model where only commodity money is used to buy other commodities while Amir, Sahi, Shubik, and Yao (1990) considered the case in which any commodity can be used to buy other commodities.^{1}^{1}1A strategic market game with trading post and fiat money was considered by Peck, Shell, and Spear (1992). Differently, Sahi and Yao (1989) studied the window model where trade is centralised by a clearing house in which there is a “window” for each commodity. This model was first proposed informally by Lloyd S. Shapley, and we call it the “Shapley window model”. It is important to stress that when there are only two commodities, as in bilateral oligopoly, the three models coincide.
The aim of this paper is threefold. First, we provide a unified framework, by means of nontrivial examples,^{2}^{2}2Dickson (2013b) shows that examples with CobbDouglas utility functions, very common in the literature, have very peculiar features which do not hold in general. to compare the results obtained in simultaneousmove and sequentialmove versions of bilateral oligopoly with the Cournot model and the CournotWalras approach. We also underline the advantages and disadvantages of each approach and we survey the main contributions on imperfect competition and strategic market games that are linked with our analysis. Secondly, we show how the bilateral oligopoly model can be used to study different kinds of oligopoly: symmetric oligopoly, where all agents act strategically; asymmetric oligopoly where only some agents act strategically; simultaneous oligopoly, where sellers and buyers make their choices without knowledge of others’ decisions; and sequential oligopoly, where some agents move first. Thirdly, we focus on how the bilateral oligopoly model can clarify how either strategic or competitive behaviours may emerge endogenously depending on the fundamentals of the economy. In other words, we show how the bilateral oligopoly model can provide “a foundation for perfect competition” and “a foundation for Cournot oligopoly” in twocommodity exchange economies. These are research programs whose aim is to define models where perfect competition and oligopoly are not assumed a priori but they are an equilibrium outcome.^{3}^{3}3See Gale (2000) for a discussion on why the study of foundations is important.
We start our analysis by recasting the classical Cournot game as an exchange game where firms are replaced by sellers characterised by initial endowments and utility functions. Next, we consider the CournotWalras equilibrium concept in exchange economies, introduced by Codognato and Gabszewicz (1991) and Gabszewicz and Michel (1997). This latter model describes, in a general equilibrium framework, the same kind of imperfect competition as the Cournot game. The exchange versions of these cornerstone models to study imperfect competition allow us to compare them with the bilateral oligopoly model in a clear way. In fact all examples in the paper are based on the same exchange economy where sellers have quasi–linear utility functions and buyers have quadratic utility functions that generate a linear demand. By comparing the different approaches we find that there are three main differences between the CournotWalras approach and the bilateral oligopoly model. First, the latter model, with a finite number of agents, describes a symmetric oligopoly, since all agents are allowed to act strategically, whereas the CournotWalras approach describes an asymmetric oligopoly where only the sellers are allowed to act strategically—they can manipulate prices by changing their actions— while the buyers are assumed to treat prices as given and beyond their control. Second, the CournotWalras approach has an intrinsic twostage nature which cannot be reconciled with the CournotNash equilibrium of simultaneousmove bilateral oligopoly where all agents act together. In order to capture the twostage structure considered in CournotWalras, we define a sequential bilateral oligopoly, where sellers move in the first stage and buyers move in the second stage, and we adopt a subgameperfect Nash equilibrium as the equilibrium concept. Third, in bilateral oligopoly all agents are treated symmetrically and no a priori assumption is made on agents’ behaviour while in CournotWalras sellers are assumed to act strategically and buyers are assumed to behaved competitively. Okuno, Postwaite and Roberts (1981) stressed the fact that in those kind of models no explanation is given as to why a particular agent should behave strategically rather than competitively.
A natural question to ask is whether it is possible to use the bilateral oligopoly model to study an asymmetric oligopoly where no assumptions on agents’ behaviours are made. This is the socalled problem of the “foundations of oligopoly”. We answer positively to that question by showing how a partial replica of the exchange economy –we replicate only buyers– can be used to obtain different agents’ behaviours endogenously. In fact, even if all agents have a priori the same strategic position, at the limit of the partial replica sellers turn out to have influence on prices whereas buyers turn out to be competitive. This is due to the fact that buyers lose their ability to influence prices when their number increases. We also show an example where, when only buyers are replicated, sellers’ and buyers’ commodity bundles at the limit of the subgameperfect Nash equilibrium of a sequentialmove game in which the sellers move first are the same commodity bundles at the CournotWalras equilibrium. Therefore, we conclude that the bilateral oligopoly model is useful to study asymmetric oligopoly by using the partial replica. Furthermore, the partial replica in the sequential bilateral oligopoly can provide a foundation of the CournotWalras approach.
In the last part of our paper, we study the relationship among the Walras (competitive) equilibrium and the other equilibrium concepts examined. To this end, we show an example where, when all agents in the economy are replicated, sellers’ and buyers’ commodity bundles at the limit of the CournotNash equilibrium correspond to the commodity bundles at the Walras equilibrium. This is not surprising because, as in the partial replica, when an agent is replicated they lose market power and in the limit he behaves competitively. Since all agents are replicated, in the limit everyone behaves competitively and then the equilibrium outcome must be Walrasian. Therefore, the bilateral oligopoly model can be used to study the foundations of perfect competition as competitive behaviours can be obtained endogenously in equilibrium.
We conclude our analysis by making some welfare considerations on the different types of competition analysed. By the First Welfare Theorem, the Walras equilibrium is Pareto efficient while the allocations obtained with the other equilibrium concepts are Pareto inefficient. This can easily be seen because there exists an allocation that Pareto dominates the one obtained at any equilibrium different from the Walras equilibrium. In other words, when markets are imperfectly competitive the equilibrium outcome is Pareto inefficient.
The rest of the paper has the following structure. In Section 2 we introduce the Cournot exchange game. In Section 3 we describe the CournotWalras approach. In Section 4 after having defined the simultaneous bilateral oligopoly model and the CournotNash equilibrium, we give an example with a finite number of agents and an example where we partially replicate the exchange economy concluding there may be differences between a simultaneous bilateral oligopoly and the CournotWalras approach even when the number of buyers is large. In Section 5 we introduce the sequential bilateral oligopoly model, we partially replicate the exchange economy, and we compare the limit of the subgameperfect Nash equilibrium with the CournotWalras equilibrium. In Section 6 we compare our previous results with the Walras equilibrium and we make some welfare considerations. In section 7 we draw some conclusions and we outline some open problems.
2 Cournot exchange game (partial analysis)
“Let us now imagine two proprietors and two springs of which the qualities are identical, and which, on account of their similar positions, supply the same market in competition.”
A. A. Cournot (1838)
In this section, by taking a slightly different approach from the one proposed by Cournot (1838), we consider the two proprietors as sellers of water and not as producers. This means that they are characterised by utility functions instead of cost functions. The interpretation, however, is very similar: supplying some of the good to the market reduces the seller’s holdings of it that provides some disutility, i.e. a cost. The consumption good (water in Cournot’s example) is denoted by and it is exchanged for money which is denoted by .^{4}^{4}4Commodity should be called commodity money because it enters in the utility function. However, for simplicity, we simply refer to it as money. A commodity bundle is a point in which is the set of all feasible commodity bundles. As in Cournot (1838), we assume that buyers are represented by the demand for the consumption good which is a downward sloping function of the price, i.e., . Each seller is characterised by a utility function, , which represents their preferences, and by an initial endowment, , such that and , i.e., sellers hold only the consumption good. Sellers face the demand and we assume that they choose a supply of the consumption good in order to obtain a commodity bundle which maximises their utility. We suppose that there are sellers.
We now introduce the Cournot exchange game . The strategy set of seller is
(1) 
with the offer of the consumption good that seller puts up in exchange for money. Let and . Let and be elements of and respectively. We denote by the total amount of the consumption good sold by sellers. We assume that the market for the consumption good clears, which means that the total supply is equal to the demand. Therefore, for each , the price of the consumption good is which implies and is normalized to 1. For each , the commodity bundle of a seller is given by
(2)  
for . The payoff function of a seller is
(3) 
for . A Cournot exchange game is then a set . We now introduce the definition of a Cournot equilibrium.
Definition 1.
The strategy profile is a Cournot equilibrium for the game if for each seller we have , for each .
We now consider an example to illustrate the Cournot equilibrium concept.
Example 1.
Consider a market with two sellers such that and , for . Buyers are represented by the demand function . Since we assume that the market for the consumption good clears, , we obtain . To find the Cournot equilibrium we need to find the strategies and which maximise the sellers’ payoffs. Consider, without loss of generality, the maximization problem for seller 1
(4)  
subject to 
This payoff function is strictly concave in , and an easy way to find the maximum is to solve the problem as an unconstrained maximisation problem then check the constraints are satisfied. The firstorder condition is
Since sellers are identical, we consider the symmetric Cournot equilibrium where . We then find the following Cournot equilibrium
At the Cournot equilibrium the price is , demand for the consumption good is and the sellers’ commodity bundles are
∎
While we consider a slight variation on the original Cournot model as we treat firms as sellers with utility functions, the underlying principles are exactly the same. Shubik (1973) raised the following critique to this type of oligopoly model:
“The Cournot duopoly model is an open market model involving money. After trade has taken place neither the amount of goods nor the amount of money in the system is conserved. Goods flow out into the market and money flows in from the market.”
Shubik’s paper advocates a closed market model, where commodities flow within the system and after trade the total amount of each commodity does not change. In our example due to our treatment of the sellers the amount of the consumption good is preserved, but it is immediate to see that the final amount of money is coming from outside the model. This is a typical feature of partial equilibrium models where each market is considered in isolation from the rest of the economy. Therefore, M. Shubik is suggesting to study oligopoly within a general equilibrium model. Furthermore, another critical feature of the Cournot exchange game is that agents are modelled in different ways: sellers are represented by utility functions and initial endowments while buyers are represented by a demand function.
In the next section we address these two issues by introducing a twocommodity exchange economy with corner initial endowments and by using the CournotWalras equilibrium as solution concept.
3 CournotWalras approach
We start by defining the buyers who have been represented by a demand function in the previous model. A buyer is characterised by a utility function which represents their preferences and by an initial endowment, , such that and , i.e., buyers hold only money. We suppose that there are buyers in the exchange economy indexed . The difference between sellers and buyers in this model lies in the initial endowments: sellers hold only the consumption good while buyers hold only money. We can now formally define an exchange economy which is the set containing all the pairs
describing sellers and buyers. In twocommodity exchange economies the price vector is simply
.We now describe the agents’ behaviours in the CournotWalras approach. In the Cournot game sellers were permitted to act strategically while the buyers behind the demand function are implicitly assumed to act as price takers. The CournotWalras approach preserves this asymmetry but derives the Walrasian (competitive) demands of the buyers from their characteristics. The Walrasian demands of a buyer are the functions and that associate to each positive price vector an amount of commodity and that maximises the utility function in ’s budget set .^{5}^{5}5More generally, Walrasian demands can be correspondences which associate to each price the set of commodity bundles which maximise the utility function in the budget set. For the sake of simplicity, we just consider the case in which Walrasian demands are functions. In other words, for any positive price vector , the amounts and solve the following maximization problem^{6}^{6}6Note that this maximisation problem has a solution only for continuous utility functions and positive prices (see Varian (1992) p. 98).
subject to 
for any buyer . We assume again that the market of each commodity clears. Therefore, for each , the price vector at a CournotWalras equilibrium must solve
(5) 
i.e., buyers’ total demand of the consumption good must be equal to sellers’ total supply of the consumption good. We denote by the price vector that solves equation (5).^{7}^{7}7To avoid cumbersome notation, but without confusion, in the paper we denote the same elements of different models with the same symbol. If equation (5) holds then the market for money also clears. By having described the price formation rule, which is purely Walrasian, we can now define the rules to calculate the commodity bundle of each agent. The buyers’ commodity bundles are given by the Walrasian demands calculated at the price vector . Differently, since sellers’ strategies are the same of the Cournot exchange game, their commodity bundles are given according to the equations in (2) at the price vector . Consequently sellers’ payoff functions are defined as in (3). We finally introduce the notion of an allocation which is a specification of a commodity bundle for each seller and buyer.
We now have all the elements to define the CournotWalras equilibrium for the exchange economy .
Definition 2.
A CournotWalras equilibrium for the exchange economy is a vector and an allocation such that

for each seller we have , for each ;

for the sellers and for the buyers .
We now consider an example to illustrate the CournotWalras equilibrium concept.
Example 2.
Consider an exchange economy with 2 sellers and 2 buyers such that and , for , and and , for . To calculate a CournotWalras equilibrium, the first step is to finding buyers’ Walrasian demands. We then solve the constrained maximization problem
subject to 
for both buyers. By using the Lagrange multiplier method, we can obtain the Walrasian demands
for . Study of the secondorder conditions reveals the solution corresponds to a maximum.^{8}^{8}8It is worth noting that, since the utility functions are quasilinear, there can be corner solutions. However, for the sake of simplicity and since this does not affect our analysis, we just consider interior solutions. Since we normalise , the price of the consumption good which solves equation (5) is
By construction, this is the same inverse demand function as in Example 1, but derived from buyers’ characteristics rather than assumed. Therefore, it is immediate to see that sellers’ maximisation problem is equivalent to the maximisation problem (4) in Example 1, so and . Finally, the allocation at the CournotWalras equilibrium is
∎
It is immediate to see that sellers’ commodity bundles and the total demand of the consumption good from buyers correspond to the ones at the Cournot equilibrium. Therefore, the CournotWalras equilibrium captures, in a general equilibrium framework, the same kind of competition as the Cournot exchange game but where the total quantities of the two commodities in the system are preserved. In other words, the CournotWalras approach allows us to study asymmetric oligopoly in a general equilibrium setting, with sellers acting strategically and buyers assumed to treat prices as fixed. Therefore, oligopoly models based on exchange economies can address Shubik (1973)’s critique.
The CournotWalras equilibrium concept was introduced by Gabszewicz and Vial (1972) in a production economy and it was recast in exchange economies by Codognato and Gabszewicz (1991) and subsequently by Gabszewicz and Michel (1997). These contributions consider exchange economies characterised by few oligopolists (sellers in our case) and many small agents (buyers in our case). Furthermore, following the Cournotian spirit, the oligopolists are allowed to act strategically as they are few, while small agents are assumed to act competitively as price takers as they are many. However this kind of assumption was criticised by Okuno, Postlewaite and Roberts (1980):
“Traditional general equilibrium treatments of such situations [in which some but not all agents have market power] have been deficient in that they have simply assumed a priori that certain agents behave as price takers while others act noncompetitively, with no formal explanation being given as to why a particular agent should behave one way or the other.”
Our Example 2 supports this view as it shows that it is possible to use the CournotWalras equilibrium in odd cases where buyers are assumed to behave competitively even if there are only two of them. In other words, the CournotWalras approach does not endogenously derive agents’ behaviours but it assumes
a priori that some agents can influence prices while others are price takers. Another shortcoming of the CournotWalras approach is that it is extremely challenging to obtain a general existence result because oligopolists influence prices by manipulating the Walras price correspondence which may fail to be continuous unless strict assumptions are imposed on the set of pricetaking agents. Nevertheless, some existence results have been obtained in particular frameworks: Bonisseau and Florig (2003) prove the existence of a CournotWalras equilibrium in linear exchange economies; and Codognato and Julien (2013) proved the existence in mixed exchange economies where agents on the continuum have CobbDouglas utility functions.^{9}^{9}9Shirai (2010) prove the existence of a CournotWalras equilibrium in production economies.The issue of asymmetrically imposing behavioural assumptions on some agents is overcome in bilateral oligopoly models in which both sellers and buyers are treated symmetrically in that all agents are allowed to behave strategically—no pricetaking assumptions are imposed a priori—and departing from the Walrasian tradition the price is constructed from agents strategic decisions.
4 Simultaneous bilateral oligopoly
As stressed in the introduction, the bilateral oligopoly model is a strategic market game based on a twocommodity exchange economy with corner initial endowments.^{10}^{10}10Bloch and Ferrer (2001a) and Dickson and Hartley (2013b) also consider the case in which agents are endowed with both commodities, i.e. have “interior” initial endowments, and can choose whether they become sellers or buyers. We define the exchange economy as as in the previous section. The strategy set of a seller is defined as in (1). The strategy set of a buyer is
is the bid of money that buyer makes on the consumption good. Let and . Let and be elements of and respectively. For each , the price vector is determined such that the price of the consumption good is given by the ratio of the total money bids made for the good to the total amount of good made available by the sellers:
(6) 
with . By having defined the price formation rule, which is nonWalrasian, we can now define the rules to calculate the commodity bundles of each agent. For each , the commodity bundle of each seller is given by
(7)  
and the commodity bundle of each buyer is given by
As such, each agent’s amount of the commodity they are endowed with reduces by the quantity of that commodity they offer to the market, and the amount of the other commodity is given by their proportional share of the aggregate amount of that commodity offered by the other side of the market, so sellers receive a share of the aggregate bid , and buyers receive a share or the aggregate offer . The payoff function of agent is , for . The simultaneous bilateral oligopoly model is then a set .
The definition of a CournotNash equilibrium is as follows.
Definition 3.
The strategy profile is a CournotNash equilibrium for the game if for each seller we have , for each , and for each buyer we have , for each .
We now consider an example to illustrate the CournotNash equilibrium concept.
Example 3.
Consider the exchange economy defined in Example 2. To find the CournotNash equilibrium, we have to solve the payoff maximization problems for all agents. We first consider sellers and then buyers. Consider the maximisation problem of seller 1:
subject to 
This payoff function is concave in ; we solve as an unconstrained problem then check the solution satisfies the constraints. The firstorder condition is
Since all sellers are identical and all buyers are identical, we consider a symmetric CournotNash equilibrium where and . Then the previous equation becomes,
and we obtain
(8) 
Consider next the maximisation problem of buyer 3:
subject to 
Again this is concave in own strategy and we solve by considering the unconstrained maximisation problem. The firstorder condition is
Since we consider the symmetric CournotNash equilibrium, we then obtain
(9) 
By combining equations (8) and (9), we find that the CournotNash equilibrium for the bilateral oligopoly model is
At the CournotNash equilibrium the price vector is and the allocation is
∎
This example clarifies that the simultaneous bilateral oligopoly model allows us to study oligopoly where all agents are treated symmetrically and no assumptions on their behaviours are made a priori. Indeed, given the structure of the game, all agents act strategically because they can manipulate the price by changing their actions. Therefore bilateral oligopoly addresses the critique raised by Okuno et al. (1980). The fact that buyers act strategically is one of the reasons why sellers’ and buyers’ commodity bundles at the CournotNash equilibrium are different from those obtained at the CournotWalras equilibrium, even if the exchange economies in Examples 2 and 3 are the same. We stress again that buyers are assumed to behave as pricetakers in the CournotWalras approach while they are permitted to act strategically in bilateral oligopoly. Furthermore, while the CournotWalras model suffers from issues because of the potential discontinuity of the Walras price correspondence, this is not the case in bilateral oligopoly because the price formation rule is nonWalrasian. The existence of a CournotNash equilibrium was studied by Bloch and Ferrer (2001a) and Dickson and Hartley (2008), who also studied uniqueness of equilibrium. Additionally, Bloch and Ferrer (2001b) showed the existence when all agents have a constant elasticity of substitution utility function.
At this point of the analysis a natural question arises: is it possible to employ the bilateral oligopoly model to study asymmetric oligopoly where no a priori assumptions on agents behaviours are made? In other words, is it possible to develop a framework where the differences in agents’ behaviours arise endogenously in equilibrium? This is also known as the problem of providing a strategic foundation for oligopoly.
In the literature, two main approaches are used to this end. The first approach consists in using mixed exchange economies where large agents (oligopolists) are represented by atoms and small agents (competitive agents) by an atomless continuum. An atom is an agent whose initial endowment is nonnegligible compared to the total endowment of the economy while a agent in the continuum holds only a negligible part of it. The model was proposed by Gabszewicz and Mertens (1971) and Shitovitz (1973) in cooperative game theory. Shitovitz (1973) stated:
“The main point in our treatment is that the small and the large traders are not segregated into different groups a priori; they are treated on exactly the same basis. The distinctions we have found between them are an outcome of the analysis; they have not been artificially introduced in the beginning, as is the case in the classical approach.”
This approach based on mixed exchange economies was also extended to noncooperative game theory by Okuno et al. (1980), in order to address their critique, and further generalised by Busetto, Codognato, and Ghosal (2011) and Busetto, Codognato, Ghosal, Julien, and Tonin (2017). In this setting, while all agents have a priori the same strategic position, they found that in equilibrium large agents represented by atoms have market power while small agents on the continuum behave as if they are pricetakers.
The second approach is based on considering partial replicas of exchange economies where the number of some agents is increased while proportionally reducing their weights. This method was introduce by Drèze, Gabszewicz, and Gepts (1969) in cooperative game theory and it was adapted to bilateral oligopoly by Dickson and Hartley (2013). The main result of this approach is that agents who are replicated lose their ability to influence the prices and at the limit they behave competitively whereas the others keep market power.
In the last part of the section, we consider this second approach and we study the limit of a partial replica where we replicate only the set of buyers. We study such limit because it represents an asymmetric oligopoly where buyers are asymptotically negligible and so should behave as pricetakers. Consider an exchange economy where the first agents are sellers and all other agents are buyers. We associate to each buyer a weight which is used to scale their initial endowment when compared to the whole economy. Therefore, the total initial endowment of money is give by . When increases, the number of buyers increases but the total initial endowment of money remains the same as in the initial economy. Given the price formation rule (6) and since , if increases, then the buyers’ ability to influence prices decreases and at the limit, for , totally disappears. It is important to stress that the limit of a sequence of CournotNash equilibria, which is denoted by , is not a CournotNash equilibrium of the limit economy as the underlying game in the limit is not welldefined. Nevertheless, the limit of the sequence gives us an object that can be compared to the other solution concepts in finite economies.
We now consider an example to illustrate how to partially replicate an exchange economy and how to calculate its limit.
Example 4.
Consider an exchange economy with 2 sellers and buyers such that and , for , and and , for . As such, it is as if there are copies of both the buyers 3 and 4 defined in Example 2. To each buyer is associated a weight and then . To simplify the analysis, we consider a symmetric CournotNash equilibrium, where all buyers play the same strategy, and then . But then, if we consider the sellers’ payoff maximisation problems, it is straightforward to verify that sellers’ best responses are as in the previous example
(10) 
for . On the contrary, buyers’ maximisation problems are different from the previous example because now there are buyers instead of two. Consider the maximisation problem of a typical buyer 3:
(11)  
subject to 
To simplify the notation, let . The first order condition of the buyer’s unconstrained maximisation problem can be written as
(12) 
Since we consider a symmetric CournotNash equilibrium where and , from the previous equation we obtain
(13) 
Let . By combining equations (10) and (13), we obtain the following CournotNash equilibrium for the bilateral oligopoly model
The limit of this CournotNash equilibrium for is for sellers and for buyers. At the limit the price vector is and the sellers’ and buyers’ commodity bundles are
∎
It is immediate to see that the commodity bundles at the limit of the CournotNash equilibrium are different from the ones at the CournotWalras equilibrium of Example 2. This result is somewhat surprising as at the limit of the CournotNash equilibrium buyers have no influence on the price, as is assumed in the CournotWalras approach, yet the equilibrium allocations differ. This is due to the fact that the CournotWalras approach has an intrinsic twostage structure—sellers decide their supply in the first stage taking as given the Walrasian demands of the buyers—whereas when considering a CournotNash equilibrium in the bilateral oligopoly model all agents choose their actions simultaneously so sellers must form beliefs about how the buyers will behave. This fact was stressed by Dickson and Hartley (2013a) who considered the conditions under which the two models coincide in the limit, and when they do not. In mixed exchange economies, Codognato (1995) and Busetto, Codognato, and Ghosal (2008) showed that the allocation at a CournotWalras equilibrium is different from the one at a CournotNash equilibrium of the Shapley window model. Therefore, we can conclude that the bilateral oligopoly model can be useful to study simultaneous asymmetric oligopoly in a closed market model by partially replicating the underlying exchange economy but it does not provide a foundation of the CournotWalras approach. In the next section, we address this last issue by studying a sequential bilateral oligopoly model.
5 Sequential bilateral oligopoly
In this section we consider a sequentialmove bilateral oligopoly model with a twostage structure where the timing of the model is exogenously specified as follows: in the first stage sellers simultaneously choose the quantities of the consumption good to put up in exchange for money; in the second stage buyers simultaneously choose the quantity of money to bid for the consumption good. At the end of the second stage bids and offers are aggregated and price is determined according to the usual rule in (6). Groh (1999) first considered a sequential reformulation of bilateral oligopoly in the context of an example, that was extended to general settings in Dickson (2006).^{11}^{11}11Dickson and Hartley (2013) consider a sequential market share game and Busetto et al. (2008) consider a sequential reformulation of the Shapley window model.
We first consider an exchange economy and we then define the sequential bilateral oligopoly. The strategy set of a seller is defined as in (1). The strategy set of a buyer is
Let and be an element of . For each , the price vector is such that
(14) 
with . For each , the commodity bundle of a seller is given by
for , and the commodity bundle of a buyer is given by
for . The payoff function of an agent is , for . The sequential bilateral oligopoly model is then a set .
We now define a subgame perfect Nash equilibrium (hereafter SPNE) which is the equilibrium concept we use in this dynamic game.
Definition 4.
A strategy profile is a SPNE for if an only if it is a CournotNash equilibrium in every subgame of the game .
Note that in the sequential bilateral oligopoly the subgames are the whole game and the subgame in which buyers choose their optimal bids following any vector of offers from the sellers. We now consider an example to illustrate how to find the SPNE and how to find the limit of a SPNE when we partially replicate the exchange economy. We stress again that the limit is not a SPNE as the underlying game is not welldefined at the limit.
Example 5.
Consider the same exchange economy defined in Example 4. As before, to simplify the analysis, we consider a symmetric SPNE that can be computed as follows. We first calculate the optimal buyers’ strategy for any feasible strategies of sellers. This guarantees to find a CournotNash equilibrium in the buyers’ subgame. We then substitute the optimal buyers’ strategy in the sellers’ payoff functions to determine their reaction functions given the responses of the buyers, after which we can find mutually consistent best responses. The strategy profile obtained in such way is a SPNE. Consider a typical buyer 3. It is immediate to verify that the buyer’s maximisation problem is as in (11). Therefore the best response of buyer 3 solves the first order condition in (12). Since we consider a symmetric SPNE where all buyers play the same strategy, equation (12) can be rewritten as
We then obtain
(15) 
with . Consider next the maximisation problem of seller 1. If we substitute the strategy in the seller’s payoff function we obtain
(16)  
subject to 
The first order condition of the seller’s unconstrained maximisation problem is
Since we consider a symmetric SPNE where , the previous equation becomes
(17) 
Therefore, by combining equations (15) and (17), we obtain the following SPNE for the sequential bilateral oligopoly
The limit of the SPNE for is for sellers and for buyers. At the limit the price vector is and the sellers’ and buyers commodity bundles are
∎
It is immediate to see that the commodity bundles at the limit of the SPNE are different to those at the limit of the CournotNash equilibrium in which moves are simultaneous, but are the same as those at the CournotWalras equilibrium. We stress again that in this framework all agents are treated symmetrically and the different behaviours of sellers and buyers are obtained endogenously at the limit of the SPNE. With this last example, we have shown that sequential bilateral oligopoly in which it is exogenously specified that sellers move first and buyers move second can provide a closed market model to study sequential asymmetric oligopoly by partially replicating the underlying exchange economy. Furthermore this model can provide a foundation for the CournotWalras approach. It is worth stressing that the price equation (14) in the limit, when , corresponds to the inverse demand function in Example 1. In mixed exchange economies, Busetto et al. (2008) showed that the set of the CournotWalras equilibrium allocations coincides with a particular subset of SPNE allocations of the twostage reformulation of the Shapley window model.
6 Walras equilibrium and welfare
The Walrasian analysis in the synthesis reached in the contributions of Debreu (1959), Arrow and Hahn (1971) and Mckenzie (2002) crucially relies on the pricetaking assumption, i.e., all agents are assumed to behaved competitively. It is then important to study under which conditions on the fundamentals of an economy agents consider prices as given endogenously in equilibrium, without making ad hoc assumptions. This is the problem of providing a strategic foundation for perfect competition, which is similar in spirit to the problem of finding a foundation for models of oligopoly. In fact, a strategic foundation for perfect competition can also be provided by considering continuum exchange economies or replicated exchange economies.^{12}^{12}12These different approaches are illustrated by the distinction between limit theorems and theorems in the limit (see Gale 2000). See MasColell (1982) for a study on the links between the two approaches. In cooperative game theory Aumann (1964) provided a foundation for perfect competition in economies with a continuum of agents by showing an equivalence result between the core and the Walras allocation. Subsequently, in noncooperative game theory, Dubey and Shapley (1994) and Codognato and Ghosal (2000) considered strategic market games with a continuum of agents and they show equivalence results between the CournotNash and Walras allocations. Those results are based on the fact that when there is a continuum of agents everyone is negligible and cannot influence prices. The second approach based on replicated exchange economies was pioneered in cooperative game theory by Edgeworth (1881) and further analysed by Debreu and Scarf (1963). Subsequently, Dubey and Shubik (1978), Sahi and Yao (1989), and Amir et al. (1990) applied this technique to strategic market games. They show that when all agents in the economy are replicated the allocation at the limit of the CournotNash equilibrium corresponds to the allocation at the Walras equilibrium.^{13}^{13}13Note that when all agents in the economy are replicated it is not necessary to decrease their weight.
Those results are heuristically based on the fact that when the number of agents increases their influence on prices decreases and at the limit totally disappears.
^{14}^{14}14MasColell (1980) surveyed the main contributions on the foundations of perfect competition by considering other noncooperative approaches that do not rely on strategic market games.We now show how the bilateral oligopoly model can be employed to provide a foundation for perfect competition. Let . We first define a Walras equilibrium concept.
Definition 5.
A Walras equilibrium is pair of a price vector and an allocation such that each commodity bundle maximises agent ’s utility function within his budget set, for , and all markets clear, i.e., and .
From the analysis made in Section 4, we can conjecture that if both the number of sellers and buyers are replicated while proportionally reducing their weights, all agents lose market power and the sellers’ and buyers’ commodity bundles at the limit of the CournotNash equilibrium will correspond to ones at the Walras equilibrium. We now consider an example to illustrate this point.
Example 6.
Consider an exchange economy with sellers and buyers such that and , for , and and , for . It is basically as the exchange economy defined in Example 2 with copies of both sellers 1 and 2 and buyers 3 and 4. The Walras equilibrium of the exchange economy is^{15}^{15}15It is straightforward to verify that sellers’ and buyers’ commodity bundles are the same for any .
We now compute the CournotNash equilibrium. As before, we first consider the maximisation problem of seller .
subject to 
The first order condition of the seller’s unconstrained maximisation problem is
Since we consider a symmetric CournotNash equilibrium where all sellers play the same strategy, we obtain
(18) 
with .
Consider now the maximisation problem of buyer .
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