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.
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 equilibrium (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.
2. A Mixed Markets Exchange Economy2.1. The Framework
The space of commodites is. The economy thus includes a finite set of divisible consumption goods, indexed by. Let be a complete measure space of agents, where denotes the set of traders, a -algebra of all measurable subsets of, and a real valued (with), 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, where is the set of atoms, while is the set of small traders. The set of atoms embodies two subsets: the subset of leaders and the subset of followers, so. The measure space
is purely atomic, while the measure space
is atomless. Therefore, is the counting measure on, when restricted to, and the Lebesgue measure, when restricted to.
2.2. Assumptions
Any trader is represented by his initial endowments, his utility function which represents his preferences among the commodity bundles, and his strategy set (see thereafter). A commodity bundle is a point in, where (a closed convex set). An assignment (of commodity bundles to traders) is an integrable function from to. All integrals are with respect to. We consider the following set of assumptions regarding utility and endowments.
Assumption 1. For all, , is continuous, strictly monotone in and concave for, and strictly quasi-concave on for. In addition, is measurable.
Assumption 2. The distribution of initial endowments among traders satisfies:
Traders will exchange some amounts of their endowments in order to reach their final allocations. A feasible allocation is an assignment for which
. The price vector is given by
.
2.3. 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 matrix, where bkl 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 is a function, defined on such that for all, and such that, are real valued integrable functions on. Therefore, from (1) a strategy selection for is a function defined on such that, with for all, , and such that, are real valued integrable functions on,. Similarly, a strategy selection for is a function defined on such that, with for all, and such that, are real valued integrable functions on. Given (resp.) for all (resp.), , one can define a strategy profile as the aggregate matrix (resp.
),. In addition, we define B as the aggregate matrix. We also denote by a strategy profile obtained by replacing in by,. The definition of a CE is given in Codognato [7] for mixed exchange economies. We now characterize and define the SCE.
3. The Stackelberg-Cournot Equilibrium3.1. 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 (resp.) a strategy profile which coincides with (resp.) for all (resp.) except for with (resp.), (resp.). The strategic plan of follower (resp.) may be written:
The solution to these programs yields the best response functions of follower and of follower. Let (resp.) be the real valued integrable function on (resp.) with values in defined by
for all (resp.
for all). In the symmetric equilibrium, (resp. ) for, (resp.,). The resulting best responses are
for all, and for all. Thenthe 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 a strategy profile which coincides with for all except for with,.
Leader, then solves the following program:
(7)
The solution to this program yields the best response function of leader. Let be the real valued integrable function on with values in defined by for all. In the symmetric SCE, for, , so one gets the strategy profile, from which we deduce and. The vector of equilibrium relative prices is, where. The equilibrium allocation for any corresponds to the assignment:
3.2. 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 such that:
i. for all
ii.
iii.,
iv.,
v.
, ,.
4. 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, (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:
We first determine the SCE. Given strategy profiles and the market clearing condition given by (3) leads to:
The first strategic step consists in determining the best-response functions of follower, , and follower, , which are the solutions to:
Assuming symmetry, i.e., for, , one obtains:
The second equation defines implicitly the best-response of follower i, where is the vector of leaders’ strategies,
is the vector of all followers strategies but, while represents the strategy profile of the small traders. Note that from (A1) depends neither of nor on. In the symmetric SCE, one gets for all and all, with, so,.
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 for all and all, with, , so:
where, , represents the elasticity of the best response function of follower, , (correctly) perceived by,.
Equations (C1) yield the equilibrium strategy
of leader, , and then, , and, , and,.
Let’s now determine the CE. Given strategy profiles and one deduces:
The first strategic step consists in determing the bestresponses of the followers, , and, , which are the solutions to:
Assuming symmetry, i.e., , one gets:
and:
The second equation defines implicitly the best-response of trader,. Assuming symmetry among the atoms for all and all, with, , one deduces:
where represents the equilibrium strategy of atom,.
If for any, ,
,
, then,. In addition, if, , then (C1) and (C2) are equivalent. So, one concludes that (C1) and (C2) lead to the same equilibrium strategies, prices and allocations. QED.
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.
5. 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,.
Assume the following specification for endowments:
The preferences of any trader are represented by the following utility functions:
The strategy sets are given by:
5.1. The SCE
Given and, the Sahi and Yao [9] price mechanism yields
, so one deduces the price system:
The best-responses and of the followers and are solutions to the following system of equations:
Assuming symmetry among the small traders, i.e., , , one gets the bestresponses functions:
The former best response satisfies: when, reflecting strategic complementarities, while when, reflecting strategic substituabilities. In addition,.
The program of the leader becomes:
Little algebra lead to the SCE strategy for the leader:
From (14), the equilibrium strategies of the followers are given by:
where.
The SCE price system and allocations are given by:
5.2. The CE
Given and, the same price mechanism yields:
The best-response functions of any atom, and of any trader are the solutions to:
Little algebra lead to the CE strategies:
The CE equilibrium price system and allocations are then:
Consider (16)-(19) with (22)-(24). The SCE and the CE relative price and allocations coincide if and only if. In addition, note that (15) may be written as
which leads to,. From
(14), one gets
, so zero conjectures are consistent.
6. Conclusion
In this paper, we consider a general equilibrium concept - the 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.
7. Acknowledgements
I am grateful to an anonymous referee for her/his remarks and suggestions. All remaining deficiencies are mine.
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doi:10.1007/s00712-010-0187-3
. The economy thus includes a finite set 
of divisible consumption goods, indexed by
. Let 
be a complete measure space of agents, where 
denotes the set of traders, 
a 
-algebra of all measurable subsets of
, and 
a real valued (with
), 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
, where 
is the set of atoms, while 
is the set of small traders. The set of atoms embodies two subsets: the subset of leaders 
and the subset of followers
, so
. The measure space
is purely atomic, while the measure space
is atomless. Therefore, 
is the counting measure on
, when restricted to
, and the Lebesgue measure, when restricted to
.
, his utility function 
which represents his preferences among the commodity bundles
, and his strategy set (see thereafter). A commodity bundle is a point in
, where 
(a closed convex set). An assignment (of commodity bundles to traders) is an integrable function 
from 
to
. All integrals are with respect to
. We consider the following set of assumptions regarding utility and endowments.
, 
, 
is continuous, strictly monotone in 
and concave for
, and strictly quasi-concave on 
for
. In addition, 
is measurable.
for which
. The price vector is given by
.
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 
matrix
, where bkl represents the amount of commodity k any trader t offers in exchange for commodity
. A strategy set for any trader 
may be written:
is the set of all matrices 
satisfying
. A strategy selection for 
is a function
, defined on 
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for all
, and such that
, 
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. Therefore, from (1) a strategy selection for 
is a function 
defined on 
such that
, with 
for all
, 
, and such that
, 
are real valued integrable functions on
,
. Similarly, a strategy selection for 
is a function 
defined on 
such that
, with 
for all
, and such that
, 
are real valued integrable functions on
. Given 
(resp.
) for all 
(resp.
), 
, one can define a strategy profile 
as the aggregate matrix 
(resp.
),
. In addition, we define B as the aggregate matrix
. We also denote by 
a strategy profile obtained by replacing 
in 
by
,
. The definition of a CE is given in Codognato [
, 
, with
, is the solution to:
exists and is unique.
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 
(resp.
) a strategy profile which coincides with 
(resp.
) for all 
(resp.
) except for 
with 
(resp.
), 
(resp.
). The strategic plan of follower 
(resp.
) may be written:
of follower 
and 
of follower
. Let 
(resp.
) be the real valued integrable function on 
(resp.
) with values in 
defined by
for all 
(resp. 
for all
). In the symmetric equilibrium, 
(resp. 
) for
, 
(resp.
,
). The resulting best responses are
for all
, and 
for all
. Thenthe system of aggregate best response functions may be written:
exists and is unique. We denote 
a strategy profile which coincides with 
for all 
except for 
with
,
.
, then solves the following program:
(7)
of leader
. Let 
be the real valued integrable function on 
with values in 
defined by 
for all
. In the symmetric SCE, 
for
, 
, so one gets the strategy profile
, from which we deduce 
and
. The vector of equilibrium relative prices is
, where
. The equilibrium allocation for any 
corresponds to the assignment:
, consistent prices 
and an allocation 
such that:
for all 
, 
, 
, 
,
.
(
leaders and 
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. To simplify, suppose
. Assume (A1) and (A2):
and 
the market clearing condition given by (3) leads to:
, 
, and follower
, 
, which are the solutions to:
, for
, 
, one obtains:
of follower i, 
where 
is the vector of leaders’ strategies,
is the vector of all followers strategies but
, while 
represents the strategy profile of the small traders. Note that from (A1) 
depends neither of 
nor on
. In the symmetric SCE, one gets 
for all 
and all
, with
, so
,
.
, 
, whom program may then be written:
for all 
and all
, with
, 
, so:
, 
, represents the elasticity of the best response function of follower
, 
, (correctly) perceived by
,
.
, 
, and then
, 
, 
and
, 
, and
,
.
and 
one deduces:
, 
, and
, 
, which are the solutions to:
, 
, one gets:
,
. Assuming symmetry among the atoms 
for all 
and all
, with
, 
, one deduces:
represents the equilibrium strategy of atom
,
.
, 
,
,
, then
,
. In addition, if
, 
, then (C1) and (C2) are equivalent. So, one concludes that (C1) and (C2) lead to the same equilibrium strategies, prices and allocations. QED.
. 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
,
.
and
, the Sahi and Yao [
, so one deduces the price system:
and 
of the followers 
and 
are solutions to the following system of equations:
, 
, 
, one gets the bestresponses functions:
when
, reflecting strategic complementarities, while 
when
, reflecting strategic substituabilities. In addition,
.
.
and
, the same price mechanism yields:
, 
and of any trader 
are the solutions to:
. In addition, note that (15) may be written as
which leads to
,
. From
, so zero conjectures are consistent.