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3. Equivalence between the SWE and the CWE
PROPOSITION: Suppose the preferences of traders in the atomless continuum are represented by Cobb-Douglas utility functions. The Stackelberg-Walras and the Cournot-Walras equilibria coincide only if 1) the atoms have the same endowments and preferences and 2) the elasticity of the best-response functions is equal to zero in equilibrium.
Proof. Consider commodities indexed by h, There are n atoms indexed by i, , with m leaders and followers. There is also a continuum of traders, each being indexed by t,. We make the following assumptions:
We first determine the SWE.
The competitive step is determined before the strategic steps. Given a price vector, trader solves:
It leads to
. Given, the program of any trader i may be written:
When, it leads to
. Given a strategy profile
and from Walras’ law, the price system is the solution to:
It leads to the equilibrium relative prices:
In the first strategic step, follower i, , determines his best response function, which is the solution to:
The first-order condition is:
The preceding equation yields the best-response of follower i, , where and represent respectively the vector of leaders’ strategies and the vector of all followers strategies but i. In the symmetric case, one gets, which requires and for all i and all, with, so,.
The second strategic step consists in determining the equilibrium strategy of any leader i,. The program of any leader i, , may then be written:
At the symmetric SWE, one has, which requires and, , so the first-order conditions may be written:
, represents the elasticity of the best response function of any follower i, , correctly perceived by any i,. Equations (C1) yield the equilibrium strategy of any leader i, , from which, , and, are deduced.
Let us now proceed to the characterization of the CWE. There is only one strategic step: the atoms plays a simultaneous move game between themselves. Any trader i, solves:
In the symmetric equilibrium, the first-order condition for i, , leads to the equilibrium strategy of trader i, which is the solution to:
If for any i, ,
, then, and,. One concludes that the SWE and CWE allocations coincide. QED.
When the best response functions have a zero elasticity in equilibrium, any leader rationally expects that a change in his strategy will elicit no reaction from the followers. The elasticity thus coincides with the true slope of the best response functions (here zero): conjectures are fulfilled and are thus consistent. It is as if the leaders made no expectations regarding the reactions of the followers to a change in their decisions. Consequently, the traders take the decisions of their rivals as given when optimizing, and thus behave as if they played a simultaneous move game, believing in the same way their rivals behave following a Cournotian reaction function. This condition on consistent conjectures is necessary but not sufficient. It may also hold when both equilibria do not coincide. In addition, the shape of the reaction functions and their slopes at equilibrium depend notably on the market demand function. The CobbDouglas specification leads to an isoelastic aggregate market demand function. Thus, the market demand which addresses to the atoms has a constant unitary price elasticity. So, when all atoms have the same endowments and preferences, their market shares are equal, which implies that their (Cournotian) equilibrium strategies are identical. If strategic traders did not have the same endowments and preferences, their equilibrium strategies would differ and could not correspond to the Cournotian ones (the same result can hold in industrial organization when firms have not the same marginal costs). Therefore, we extend a result obtained in partial equilibrium by Julien  to cover exchange economies.
4. An Example
Consider the case for which. The price system is. The economy embodies two atoms (the leader) and (the follower), each of measure , , and an atomless continuum of traders represented by the unit interval with the Lebesgue measure,. The following assumptions are made:
The strategy set of i is given by
. Let us determine the competitive step. Given, t solves:
It leads to.
Given and, , one gets,. Thus,. Given a strategy profile, and from Walras’ law, the relative price is the solution to
The equilibrium price system is then:
The first strategic step may be written:
This leads to the best response function of the follower:
One has when, and.
The program of the leader may then be written:
The SWE strategies are:
The equilibrium price and allocations are given by:
We finally determine the CWE. Given
andthe program of any trader may be written:
It leads to the equilibrium strategies:
The equilibrium price system and allocations are then:
Then the SWE and the CWE coincide. We also check that from (13).
In mixed markets exchange economies the CWE and the SWE can coincide: one condition stems from the fundamentals (endowments and preferences), while the other concerns consistent expectations formed by the atomic part of the economy.
I am grateful to an anonymous referee for her/his remarks and suggestions. All remaining deficiencies are mine.