Stackelberg-Cournot and Cournot Equilibria in a Mixed Markets Exchange Economy

In this note, we compare two strategic general equilibrium concepts: the Stackelberg-Cournot equilibrium and the Cournot equilibrium. We thus consider a market exchange economy including atoms and a continuum of traders, who behave strategically. We show that, when the preferences of the small traders are represented by Cobb-Douglas utility functions and the atoms have the same utility functions and endowments, the Stackelberg-Cournot and the Cournot equilibrium equilibria coincide if and only if the followers’ best responses functions have a zero slope at the SCE.


Introduction
Oligopolistic competition in general equilibrium has been developed in two main directions.The first is the Cournot-Walras equilibrium approach, which is modeled by Gabszewicz and Vial [1] in an economy with production, and in exchange economies by Codognato and Gabszewicz [2,3], Gabszewicz and Michel [4], and Busetto, Codognato and Ghosal [5,6].This class of models includes agents who behave strategically (the atoms), while other agents behave competitively (the atomless continuum of traders).The second is the Cournot equilibrium (CE) based on strategic market games as notably modeled by Shapley and Shubik [7], Dubey and Shubik [8], Sahi and Yao [9], and Amir, Sahi, Shubik and Yao [10].In this approach, all traders always behave strategically and can send quantity signals indicating how much of any commodity they are willing to buy and/or sell.Some contributions aim at comparing the CE with other strategic equilibria.Codognato [11] studies the equivalence between the Cournot-Walras equilibrium and the CE, while Codognato [12] compares two Cournot-Nash equilibrium models.In this note, we compare the CE and the Stackelberg-Cournot equili-brium (SCE) defined for finite economies in Julien and Tricou [13,14].From the benchmark of strategic market games, the SCE concept inserts Stackelberg competition into interrelated markets.We determine the conditions under which the CE and the SCE are equivalent.
The equivalence is studied in an economy embodying atoms and a continuum of traders.We thus consider a mixed exchange economy a la Shitovitz [15] and Codognato [11], in which the traders who are endowed with a corner endowments are atoms, while the traders endowed with all other commodities are represented by an atomless continuum.Markets are complete and prices are consistent.We assume the individual positions and the timing of moves as given.In addition, existence and uniqueness of oligopoly equilibrium are deleted.We rather focus on the case for which both sets of strategic equilibria can have a nonempty intersection.Indeed, when the preferences of the small traders are represented by Cobb-Douglas utility functions, and when the atoms have the same endowments and utility functions, the SCE and the CE coincide if and only if the followers' best responses functions have a zero slope at the SCE.We so spread the result obtained by Codognato [11] for Cournotian economies to a class of exchange economies in which the strategic interactions recover from sequential decisions.We also provide a generalization of Julien [16] because henceforth all the traders behave strategically.
The paper is organized as follows.Section 2 specifies the mixed markets exchange economy.Section 3 provides a characterization and a definition of the SCE.Section 4 is devoted to the statement and the proof of the proposition.In Section 5, an example is given.In Section 6, we conclude.

The Framework
The space of commodites is  .The economy thus includes a finite set of divisible consumption goods, indexed by .
  be a complete measure space of agents, where T denotes the set of traders, a   -algebra of all measurable subsets of , and ), non negative and additive measure defined on .The space of agents embodies large traders, represented by atoms, and small traders, represented by an atomless continuum.So, let 0 1 , where 0 is the set of atoms, while 1 is the set of small traders.The set of atoms embodies two subsets: the subset of leaders 0 T and the subset of followers , so .The measure space , , T T is purely atomic, while the measure space   is atomless.Therefore,  is the counting measure on , when restricted to 0 T , and the Lebesgue measure, when restricted to .

Assumptions
Any trader is represented by his initial endowments Traders will exchange some amounts of their endowments in order to reach their final allocations.A feasible allocation is an assignment for which ( )d ( ) . The price vector is given by .

Strategy Sets
Each trader uses fractions of his initial endowment to trade them for the  commodities.The strategic behavior then involves all the amounts of the owned good(s) that are engaged in exchange of all commodities.A strategy for a trader ,  , may be represented by an kl , where b kl represents the amount of commodity k any trader t offers in exchange for commodity .A strategy set for any trader  may be written: The strategy set of any trader is the set of all matrices satisfying .A strategy selection for , defined on such that t t T  for all  , and such that kl , are real valued integrable functions on .Therefore, from (1) a strategy selection for 0 ( ) defined on such that t  , with for all 0 , , and such that kl , are real valued integrable functions on 0 T , .Similarly, a strategy selection for ), , one can define a strategy profile as the aggregate . In addition, we define B as the aggregate matrix .We also denote by a strategy profile obtained by replacing in by t , .The definition of a CE is given in Codognato [7] for mixed exchange economies.We now characterize and define the SCE.

The SCE: Characterization
A SCE can be modeled as a sequential game in two steps, which is solved by backward induction.The characterization of the SCE relies on the strategic market game mechanism provided by Sahi and Yao [9], since it generates consistent relative prices.Thus, given a strategy profile , , with , is the solution to: These conditions stipulate that the aggregate value of all goods supplied to buy any commodity l must be equal to the aggregate value of this good l supplied to buy any other commodity.From Sahi and Yao [9], we know that when the matrix B is irreducible, the market price exists and is unique.
The strategic plan of follower t, is determined by two elements: he manipulates the   consistent relative prices, and he takes as given the matrices of bids of all leaders and all other followers.We thus denote by 2 2  \ ( ) ) a strategy profile which coincides with (resp.
may be written: ( The solution to these programs yields the best response functions of follower ., , \ ( ), ), ).In the symmetric equilibrium, (resp. 2 ( ) The resulting best responses are for all t .Then, the system of aggregate best response functions may be written:   The system of equations given by ( 6) determines a consistency among the followers' best response functions.
We assume that the solution  exists and is unique.We denote 1 The solution to this program yields the best response function ., \ ( )  ,    0 , so one gets the strategy profile , from which we deduce and .The vector of equilibrium relative prices is . The equilibrium allocation for any corresponds to the assignment: , .ĥ

The SCE: Definition
A SCE is a noncooperative equilibrium of a game where the players are the traders, the strategies are their supply decisions and the payoffs are their utility levels.

Definition. (SCE) A Stackelberg-Cournot equilibrium is given by a matrix
, consistent prices and an allocation for all t T i. ii.

Equivalence between the SCE and the CE Proposition. Assume that the preferences of the small traders are represented by Cobb-Douglas utility functions, and the atoms have the same endowments and utility functions. Then, the Stackelberg-Cournot and the Cournot equilibria coincide if and only if the followers' best responses functions have a zero slope at the Stackelberg-Cournot equilibrium.
Proof.Consider n atoms, each being indexed by i, ( m leaders and followers), and a continuum of traders, each being indexed by t, . To simplify, suppose .Assume (A1) and (A2): ( ) = ( ), ( ) , , The strategy profiles are given by: , one obtains: , , The second equation defines implicitly the best-response of follower i, , where is the vector of leaders' strategies, for all and all  , with The second strategic step consists in determining the equilibrium strategy of any leader , , whom program may then be written: , .

  
At the symmetric SCE, we get ) Let's now determine the CE.Given strategy profiles and b one deduces: The first strategic step consists in determing the best- , and , , which are the solutions to: The economy embodies two atoms 1 (the leader) and (the follower), each of measure , and an atomless continuum of traders represented by the unit interval   where represents the equilibrium strategy of atom , .
The equivalence result stipulates that Stackelberg competition in interrelated markets can lead to Cournot outcomes.Provided symmetry assumptions regarding the primitives, this equivalence holds if and only if the consistent conjectures are zero.Why?Any leader rationally expects that a change in his strategy will elicit no reaction from the followers.Consequently, it mimics the case where the traders take the decisions of their rivals as given when optimizing, thereby behaving as if they played a simultaneous move game (with the belief that their rivals behave following a Cournotian reaction function).In such a case, the value of the elasticity of the best response functions coincides with the true slope of the best response functions (here zero): conjectures are fulfilled and are thus consistent.This means that the strategies are neither substitutes nor complements in equilibrium.This result may be explained as follows.The shape of the reaction functions and their slopes at equilibrium depend notably on the market demand function.The Cobb-Douglas specification leads to an isoelastic aggregate market demand function (constant unitary price elasticity).So, when all atoms have the same endowments and preferences, their market powers are equal, which implies that their (Cournotian) equilibrium strategies are identical.Our proposition extends a result obtained in partial equilibrium by Julien [17] to cover a general equilibrium framework.In addition, it spreads the result obtained in Julien [16] to cover mixed markets exchange in which all traders behave strategically.

An Example
Consider the case for which .The price system is Assume the following specification for endowments: The preferences of any trader are represented by the following utility functions: The strategy sets are given by: Assuming symmetry among the small traders, i.e.
, one gets the bestresponses functions: (  are the solutions to:

Conclusion
In this paper, we consider a general equilibrium conceptthe Stackelberg-Cournot equilibrium-where all traders behave strategically.One side of the market includes negligible traders, while the other side embodies atoms.In this economy, the strategic interactions recover from sequential decisions.The framework used belongs to the class of mixed markets exchange models.Traders have not the same "weight": this idea is captured with a mixed measure space of traders.Such a specification notably enables to model asymmetries in the working of market power in interrelated markets.It also gives some insights regarding the consequences of market power in a general equilibrium perspective.Finally, it facilitates comparisons between general equilibrium concepts in economies where all agents behave strategically.
Within this framework, it is shown that the set of Stackelberg-Cournot equilibria and the set of Cournot equilibria can have a nonempty intersection.When the preferences of the small traders are represented by Cobb-Douglas utility functions, and when the atoms have the same endowments and utility functions, the SCE and the CE coincide if and only if the followers' best responses functions have a zero slope at the SCE.Provided conjectures of atoms are consistent, the traders behave as if they played a simultaneous move game.So, the equivalence result stems from consistent conjectures formed by leaders.
ψ ) be the real valued integrable function on (resp.T ) with values in defined by step consists in determining the best-response functions of follower i ,

b
is the vector of all followers strategies but , while t b represents the strategy profile of the small traders.Note that from (A1) depends neither of nor on i  b .In the symmetric SCE, one gets then solves the following program: Leader