Ludovic Julien

Olivier Musy

Aurélien Saidi

The Stackelberg equilibrium as a consistent conjectural equilibrium

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

The Stackelberg (1934) model is a sequential model embodying two …rms, one leader and one follower. They both have perfect information about the market demand function. The leader …rm moves …rst and makes its decision taking into account the reaction of the follower, which is correctly perceived [START_REF] Negishi | A model of duopoly with Stackelberg equilibrium[END_REF]). The follower rationally sets its own output level according to any quantity set by the leader, with the expectation that the leader will not counter-react. Similarly, the leader may expect the follower to conform to his best strategy. In a Stackelberg equilibrium, beliefs are self-ful…lled, and are therefore an essential feature of such a kind of economies.

In this note, we provide a conjectural interpretation of the Stackelberg equilibrium concept. We consider conjectural variations in a static deterministic model with many …rms. These conjectures capture the way a …rm anticipates the reactions of its rivals when it decides to increase its supply by one unit, in a simultaneous game [START_REF] Bowley | The mathematical groundwork of economics[END_REF], [START_REF] Friedman | Bounded rationality, dynamic oligopoly and conjectural variations[END_REF], Figuières et al. ( 2004)). In our model, we assume an asymmetry in the formation of beliefs: some …rms (type 1) make conjectures while others (type 2) do not. The asymmetry could be justi…ed for instance by some di¤erences on costs that would put …rms in asymmetric positions, and thus cause them to formulate some asymmetric conjectures. 1 But, the foundations of such an asymmetry is out of scope in this paper. The distribution of …rms between each group is assumed to be exogenous. In our model, no sequential structure is assumed regarding the timing of decisions.

In the conjectural variations literature, signi…cant research has been devoted to local consistency of static conjectures [START_REF] Figuières | On the e¤ects of conjectures in a symmetric strategic setting[END_REF]). In our framework, a locally consistent equilibrium is such that each type 1 …rm is able to correctly perceive the slopes of the reaction functions of both type 2 …rms and their competitors. This means that conjectures coincide with the true values of the slopes of reaction functions [START_REF] Bresnahan | Duopoly models with consistent conjectures[END_REF], [START_REF] Perry | Oligopoly and consistent conjectural variations[END_REF], [START_REF] Ulph | Rational conjectures in the theory of oligopoly[END_REF]). We consider symmetric (within a cohort) constant conjectural variations (Bowley (1924), [START_REF] Figuières | On the e¤ects of conjectures in a symmetric strategic setting[END_REF], [START_REF] Perry | Oligopoly and consistent conjectural variations[END_REF]). 2 It is known that under constant marginal costs, the competitive equilibrium turns out to be the only consistent equilibrium [START_REF] Bresnahan | Duopoly models with consistent conjectures[END_REF], [START_REF] Perry | Oligopoly and consistent conjectural variations[END_REF]). The existence of an asymmetry in the formation of beliefs complicates the modelling of conjectures since it introduces indirect e¤ects. Hence, the preceding result still holds in our model.

Within the same economy, we also study a sequential Stackelberg model à la [START_REF] Daughety | Bene…cial concentration[END_REF], without assuming any speci…c form for the demand function and for the cost functions. We consider the relationship between the sequential Stackelberg equilibrium and the equilibrium of the simultaneous move game for 1 Introducing an asymmetry in costs between …rms could allow to endogenize their type. Similarly, the timing of positions is often exogenous in Stackelberg competition, while it can be grounded on costs asymmetries (Van Damme and Hurkens (1999)).

2 By constant conjectural variations we mean that the formation of conjectures do not vary with a change in the strategy of a leader.

The Stackelberg equilibrium as a consistent conjectural equilibrium Ludovic A. Julien, Olivier Musy, Aurélien W. Saïdi this same basic economy in case of asymmetric conjectural variations. We then extend a literature which considers conjectural variations as useful tools to easily capture in a general model various oligopoly con…gurations (perfect, Cournotian and collusive competitions). A synthesis of this approach has been provided by [START_REF] Dixit | Comparative statics for oligopoly[END_REF], in which the Stackelberg equilibrium has not been studied. Two propositions are proved. We …rst show that there exists a continuum of conjectural variations such that the conjectural equilibrium locally coincides with the Stackelberg equilibrium (Proposition 1). Second, we de…ne the conditions under which a conjectural equilibrium is a locally consistent equilibrium (i.e. such that conjectures are ful…lled). The concept of (local) consistency is restricted to …rms making conjectures. Two conditions on consistency are featured: consistency within a cohort and consistency among cohorts. The Stackelberg equilibrium ful…lls only the latter condition (Proposition 2). We provide an example illustrating these two propositions. Studying a linear economy, we notably show that the conjectural equilibrium may coincide with the multiple leader-follower Stackelberg equilibrium model developed by [START_REF] Daughety | Bene…cial concentration[END_REF]. In addition, in case of asymmetry, the competitive equilibrium is locally consistent. The paper is organized as follows. In section 2 the model is featured. Section 3 provides a de…nition and a characterization of the conjectural equilibrium. Section 4 states two propositions reagarding the Stackelberg equilibrium. In section 5, we give an example for a linear economy to illustrate both propositions. In section 6, we conclude.

The model

Consider an oligopoly industry with n …rms which produce an homogeneous good. There are two types of …rms, labeled 1 and 2, so the industry includes n 1 …rms of type 1 and n 2 …rms of type 2, with n 1 + n 2 = n. Type 1 …rms make conjectures regarding the reaction of other type 1 …rms as well as type 2 …rms to a change in their strategy, while type 2 …rms are assumed not not to make any conjecture (see thereafter).

Let p(X) be the price function for the industry (the inverse of the market demand function), where X denotes industry output. We assume dp(X) dX < 0 and

d 2 p(X) dX 2
0. Let x i 1 and x j 2 represent respectively the amounts of good produced by …rm i, i = 1; :::; n 1 , and j, j = 1; :::; n 2 . Assume X = X 1 + X 2 , where X 1 = P n1 i=1 x i 1 and X 2 = P n2 j=1 x j 2 , denote respectively aggregate output of type 1 …rms and type 2 …rms. In what follows, we also denote X i

1 = P i x i 1 (resp. X j 2 = P j x j 2 )
the output of all type 1 (resp. 2) …rms but i (resp. j). In addition, it is assumed that p(X) is continuous.

The cost function of any …rm i or j is denoted by c i 1 (x i 1 ), i = 1; :::; n 1 or c j 2 (x j 2 ), j = 1; :::; n 2 . We assume c i 1 (:) = c 1 (:)for all i and c j 2 (:) = c 2 (:) for all j. In addition,

dc i 1 (x i 1 ) dx i 1 > 0 ( dc j 2 (x j 2 ) dx j 2 > 0) and d 2 c i 1 (x i 1 ) d(x i 1 ) 2 0 ( d 2 c j 2 (x j 2 ) d(x j 2 ) 2 0
). The pro…ts functions of …rms i and j may be written:

i 1 = p(X)x i 1 c 1 (x i 1 ), 8i = 1; :::; n 1 , j 2 = p(X)x j 2 c 2 (x j
2 ), 8j = 1; :::; n 2 .

(

) 1
Conjectural variations formed by …rm i are denoted by i 1 (respectively i 2 ) for i = 1; :::; n 1 , and characterize the beliefs of …rm i as for the reaction of type 1 …rms (respectively type 2 …rms), to a one unit increase of its output [START_REF] Bresnahan | Duopoly models with consistent conjectures[END_REF], [START_REF] Perry | Oligopoly and consistent conjectural variations[END_REF]). Following [START_REF] Perry | Oligopoly and consistent conjectural variations[END_REF], we only consider constant conjectures, so beliefs of a …rm are independent of the supply of the others. Other speci…cations are conceivable [START_REF] Figuières | On the e¤ects of conjectures in a symmetric strategic setting[END_REF]).

Formally, the conjectural variations may be de…ned as follows:

i 1 = dX i 1 dx i 1 (2) i 2 = dX 2 dx i 1 , i = 1; :::; n 1 , where i 1 2 [ 1; n 1 1] and i 2 2 [ 1; n 2 ]
represent intracohort and intercohort conjectures respectively, that is conjectures formed by …rm i regarding the slope of the aggregate reaction functions of type 1 …rms and type 2 …rms. 3 In the following, we will only focus on the symmetric case, i.e. i 1 = 1 and i 2 = 2 for i = 1; :::; n 1 .

We must consider 1 + 2 1. Among the market outcomes usually featured, three cases are of particular interest. Perfect competition corresponds to 1 + 2 = 1. In this case, each …rm of type 1 expects their direct rivals to absorb exactly its supply expansion by a corresponding supply reduction, so as to leave the price unchanged. When 1 = 0 and 2 = 0, …rms expect no reaction to its change in supply: this is the Cournotian case. Finally, when 1 = n 1 1 and 2 = n 2 , the market is collusive. Among the possible con…gurations, it is worth considering the cases for which type 1 …rms form correct expectations.

In this model, …rms -whether of type 1 or 2 -play simultaneously. The players are the …rms, the strategies are their supply decisions and the payo¤s are their pro…ts.

De…nition 1 An economy is a game corresponding to a vector of 2n components, including n strategies and n payo¤ s

x i 1 ; x j 2 ; i 1 ; j 2 j=1;:::;n2 i=1;:::;n1 .

Conjectural equilibrium: de…nition and characterization

Before dealing with the characterization of the conjectural equilibrium (CE), we provide a de…nition of it. such that for any i = 1; :::; n 1 and any j = 1; :::; n 2 :

De…nition

(i) i 1 (x i 1 ( ); x i 1 ( ); xj 2 ( )) i 1 (x i 1 ( ); x i 1 ( ); xj 2 ( )) for all x i 1 6 = xi 1 (ii) j 2 (x i 1 ( ); xj 2 ( ); x j 2 ( )) j 2 (x i 1 ( ); x j 2 ( ); x j 2 ( )) for all x j 2 6 = xj 2 .
A conjectural equilibrium is a non cooperative equilibrium of game . In the equilibrium, each …rm i determines a strategy xi 1 according to the conjectures 1 and 2 , such that no deviation is able to increase its pro…t i 1 when the strategies of the others remain unchanged. In addition, each …rm j, j = 1; :::; n 2 determines its optimal strategy xj 2 by taking as given the strategies of all other …rms. Firms of type 2 do not make conjectures regarding unilateral deviations. This corresponds to a Nash equilibrium conditional on expectations formation. We now provide a full characterization of a CE.

Characterization of the CE

Given (1), programs of …rms i and j may be written:

x i 1 2 arg max p(X)x i 1 c 1 (x i 1
), i = 1; :::; n 1 , x j 2 2 arg max p(X)x j 2 c 2 (x j 2 ), j = 1; :::; n 2 .

(

) 3
The …rst-order conditions are given by:

p(X) + (1 + 1 + ) dp(X) dX x i 1 dc 1 (x i 1 ) dx i 1 = 0, (4)
p(X) + dp(X) dX x j 2 dc 2 (x j 2 ) dx j 2 = 0, (5)
where

@X2 @x i 1 and 1 = dX i 1 dx i 1 = @X i 1 @x i 1 . It di¤ers from dX2 dx i 1 because dX2 dx i 1 = @X2 @x i 1 + @X2 @X i 1 dX i 1 dx i 1 = (1 + 1 ), with @X2 @X i 1 = @X2 @x i
1 from the symmetry of x i 1 and X i 1 in the reaction function of type 2 …rms (equation 5). By rearranging the equality, it comes that = 2 1+ 1 . Finally, as 1 + 2 1, 2 [ 1; 0]. These optimal conditions above implicitly de…ne the best response function i 1 ; X 2 ; )

x j 2 = 2 (X 1 ; X j 2 of both types of …rms:

x i 1 = 1 (X and
). In the symmetric equilibrium, x i 1 = x 1 for i = 1; :::; n 1 and x j 2 = x 2 for j = 1; :::; n 2 , so one gets x 1 = 1 (n 1 ; X 2 ; ) and x 2 = 2 (n 2 ; X 1 ). The expected value of can be deduced from 1 and 2 by di¤erentiating X 2 = n 2 2 (n 2 ; X 1 ). By substituting X 2 in 1 , we therefore obtain the equilibrium strategy of a type 1 …rm: x1 ( ) = ~ 1 (n 1 ; n 2 ; ). The equilibrium strategy of a type 2 …rm is then deduced: x2 = ~ 2 (n 1 ; n 2 ; ). And the equilibrium pro…ts can be derived: ~ i 1 = ~ 1 (n 1 ; n 2 ; ) for any i = 1; :::; n 1 and ~ j 2 = ~ 2 (n 1 ; n 2 ; ) for any j = 1; :::; n 2 .

Conjectural and Stackelberg equilibria

We now prove under which conditions the CE (locally) coincides with the Stackelberg equilibrium (SE). We …rst de…ne a SE. Consider that type 1 …rms denote leaders, while type 2 are the followers. So, as in [START_REF] Daughety | Bene…cial concentration[END_REF] such that for any i = 1; :::; n 1 and for any j = 1; :::; n 2

(i) i 1 (x i 1 ; x i 1 ; xj 2 (x i 1 ; x i 1 )) i 1 (x i 1 ; x i 1 ; x j 2 (x i 1 ; x i 1 )
), for all x j 2 (x i 1 ; x i 1 ) and

x i 1 6 = xi 1 , (ii) j 2 (x j 2 ; x j 2 ) j 2 (x j 2 ; x j 2 ) for all x j 2 6 = xj 2 .
In a Stackelberg equilibrium, all …rms optimize their pro…t function, and beliefs are self-ful…lled. Leaders and followers play a Cournot game within their respective cohort. The game is played under complete but imperfect information among leaders and among followers. However, leaders have perfect information about the reaction function of the followers. Existence and uniqueness of the SE are skipped (see DeMiguel and Xu (2009), [START_REF] Sherali | A multiple leader Stackelberg model and analysis[END_REF], Sherali et

al. (1983)).

Proposition 1 There exists a continuum of conjectural variations such that the conjectural equilibrium locally coincides with the Stackelberg equilibrium.

Proof. Equations ( 4)-( 5) implicitly determine the equilibrium strategies when …rm i, i = 1; :::; n 1 , conjectures the slopes of the aggregate reaction functions.

In a SE, the reaction function of …rm j is obtained from pro…t maximization, for given strategies of the leader X 1 , and de…ned by x j 2 = 2 (X 1 ; X j 2 ). In the symmetric case, the reaction function becomes: x 2 = 2 (n 2 ; X 1 ). The program of leader i is:

x i 1 2 arg max p(x i 1 + X i 1 + X n2 j=1 2 (n 2 ; X 1 ))x i 1 c 1 (x i 1
), i = 1; :::; n 1 .

For any i = 1; :::; n 1 and j = 1; :::; n 2 , the two …rst-order conditions sequentially obtained are given by:

p(X) + 1 + i dp(X) dX x i 1 dc 1 (x i 1 ) dx i 1 = 0, (6)
p(X) + dp(X) dX x j 2 dc 2 (x j 2 ) dx j 2 = 0. (7)
where i = n 2 @ 2 @x i 1 (n 2 ; X 1 ) is the actual slope of type 2 …rms'aggregate reaction function: In equilibrium, one gets i = n 2 @ 2 @x i 1 (n 2 ; X1 ) for i = 1; :::; n 1 . Equation [START_REF] Friedman | Bounded rationality, dynamic oligopoly and conjectural variations[END_REF] above is identitical to equation [START_REF] Dixit | Comparative statics for oligopoly[END_REF]. Let 1 + = , that is

1 + 2
1+ 1 = , be the conjecture of type 1 …rms in a CE. The preceding system collapses to the system de…ned by equations ( 4)- [START_REF] Dixit | Comparative statics for oligopoly[END_REF]. Thus, the CE locally coincides with the SE. QED.

Remark 1 Notice that while the two equilibria coincide for 1 = 0 and 2 = = , they also coincide for an in…nite number of values for 1 and 2 such that 1 + = or 2 = + ( 1) 1 .

Proposition 1 states that the equilibrium outcome of a simultaneous move game collapses to the equilibrium outcome a Stackelberg sequential game in which a cohort of agents make expectations regarding the impact of their decisions on the choices of another cohort of agents. We then complete the analysis of [START_REF] Dixit | Comparative statics for oligopoly[END_REF] by showing that conjectural variations can also be useful to represent the Stackelberg market outcome in a static game. When embodying intercohort conjectural variations, the framework exhibits a large number of solutions corresponding to the Stackelberg market outcome.

We now focus on locally consistent CE. 4De…nition 4 A locally intercohort-consistent conjectural equilibrium for is a CE with 2 = n 2

d 2 dx i 1 (n 2 ; X1 ) and = 2 1+ 1 ,
where n 2 2 is the aggregate reaction function of type 2 …rms.

A locally intercohort-consistent CE restricts the consistency to intercohort conjectures, i.e. to type 1 …rms'conjectures regarding type 2 …rms'reactions. But it presumes that 2 must be correctly expected, without implying ful…lled conjectures on . It de…nes a partially consistent equilibrium.

De…nition 5 A locally intracohort-consistent conjectural equilibrium for is a CE with 1 = d i 1 dx i 1 (n 1 ; xi 1 ; X2 )
, where i 1 is the aggregate reaction function of type 1 …rms but i.

A locally intracohort-consistent CE restricts the consistency to intracohort conjectures; i.e. to any type 1 …rm's conjectures regarding other type 1 …rms' reactions. It also de…nes a partially consistent equilibrium.

De…nition 6 A locally consistent conjectural equilibrium for is an intracohort and intercohort consistent CE.

A locally consistent CE is an equilibrium strategy x ~i for each i such that no …rm perceives an incentive to change its supply, which is based on conjectural variations, assumed to be a correct assessment of type 1 and type 2 …rms. Each …rm i is then able to correctly perceive the equilibrium values of the slopes of the two aggregate reaction functions. Proposition 2 Within the set of CE which coincide with the SE, a necessary condition for a SE to be a locally consistent CE is 1 = 0. In that case, 2 = = n 2 @ 2 @x i 1 (n 2 ; X1 ) = 1. Then the SE converges toward the competitive equilibrium.

Proof.

According to def. 4 and 5, a locally consistent CE satis…es 1 =

@ i 1 @x i 1 (n 1 ; xi 1 ; X2 ) and 2 = n 2 @ 2 @x i 1
(n 2 ; X1 ). Equations ( 5) and ( 7) being identical, the aggregate reaction function of type 2 …rms must be equivalent in both the CE and the SE. As a consequence, in the equilibrium, their slopes must be equal; i.e. 2 = n 2

@ 2 @x i 1 (n 2 ; X1 ) = n 2 @ 2 @x i 1 (n 2 ; X1 ) = .
To construct the locally consistent CE, we must determine the best responses of all type 1 …rms but i and of all type 2 …rms. We follow a procedure given by [START_REF] Perry | Oligopoly and consistent conjectural variations[END_REF] for the oligopoly case. The aggregate reaction functions of type 1 …rms but i and type 2 …rms are implicitly obtained from the …rst-order conditions ( 4) and ( 5) of …rms i and j respectively:

p(X) + (1 + 1 + ) dp(X) dX X i 1 n 1 1 dc 1 X i 1 n1 1 dx i 1 = 0, p(X) + dp(X) dX X 2 n 2 dc 2 X2 n2 dx 2 = 0,
where X 2 = n 2 2 (n 2 ; X 1 ), and X i

1 i 1 (n 1 ; x i 1 ; X 2
). Di¤erentiating implicitly the preceding equations, one gets in the symmetric equilibrium:

dX i 1 dx i 1 = (1 + ) h dp(X) dX + (1 + 1 + ) d 2 p(X) dX 2 X i 1 n1 1 i n1(1+ )+ 1 n1 1 dp(X) dX + (1 + )(1 + + 1 ) d 2 p(X) dX 2 X i 1 n1 1 1 n1 1 d 2 c1 d(x i 1 ) 2 , dX 2 dx i 1 = h dp(X) dX + d 2 p(X) dX 2 X2 n2 i 1 + dX i 1 dx i 1 n2+1 n2 dp(X) dX + d 2 p(X) dX 2 X2 n2 1 n2 d 2 c2 d(x j 2 ) 2 .
According to De…nitions 4 and 5, a consistent CE must satisfy the next three conditions:

(C1) dX i 1 dx i 1 = 1 (C2) dX 2 dx i 1 = 2 = (C3) 2 = (1 + 1 ).
According to Proposition 1, the conjectural equilibria which coincide with the SE, are de ned by 1 + = . This equality and condition (C2) are jointly satis…ed provided 1 = 0 or 1 = 1+ < 1. As 1 2 [ 1; n 1 1], a consistent SE must satisfy:

1 = 0 2 = = .
From Def. 4 and 5, consistent conjectural equilibria are …xed points of:

dX i 1 dx i 1 = 1 dX 2 dx i 1 = 2 .
Since d 2 P (X) dX 2 0, the condition

dX i 1 dx i 1 = 1 = 0 requires = 1.
This corresponds to a speci…c case of perfect competition in which type 1 …rms do not react to any move of another type 1 …rm, while type 2 …rms determine their strategy without modifying the equilibrium price. QED.

Proposition 2 puts into light the consistency of the CE and its correspondence with the SE: consistency of the SE requires that the …rms to correctly perceive the true slope of the aggregate reaction function emanating from the …rms which do not form conjectures. For 6 = 1, none of the intercohortconsistent CE that coincide with the SE can intracohort-consistent.

An example

It is assumed that p(X) is continuous, linear and decreasing with X and that it may be written:

p(X) = max f0; a bXg , a; b > 0. (8)
The cost functions of any …rm i and j are assumed to be linear, i.e. c 1 x i 1 , 8i = 1; :::; n 1 and c 2 x j 2 , 8j = 1; :::; n 2 . We assume c 1 = c 2 = c, 8i, 8j. The pro…ts of any …rms i and any …rm j may be written

i 1 = a c b(x i 1 + X i 1 + X 2 ) x i 1 for i = 1; :::; n 1 and j 2 = (a c b(X 1 + x j 2 + X j 2
))x j 2 for j = 1; :::; n 2 . This economy has a unique symmetric competitive equilibrium given by x

i 1 = x j 2 = a c
bn , i = 1; :::; n 1 and j = 1; :::; n 2 , with i 1 = 0 for i = 1; :::; n 1 and j 2 = 0 for j = 1; :::; n 2 .

Assume symmetry among all …rms of each type, i.e. x i 1 = x 1 for any i = 1; :::; n 1 and x j 2 = x 2 for any j = 1; :::; n 2 . The best response function of all type 1 …rms but i may be deduced from (4) and the best response of any type 2 …rm is given by (5): 5(a bX) b(1

+ 1 + ) X i 1 n 1 1 c = 0, (9)
x 2 = a c b(n 2 + 1)

X 1 n 2 + 1 . (10)
From [START_REF] Perry | Oligopoly and consistent conjectural variations[END_REF], one deduces the slope of the reaction function of type 2 …rms: 6

dX 2 dx i 1 = 1 + dX i 1 dx i 1 , with = n 2 n 2 + 1 . ( 11
)
In addition, di¤erentiating (9) leads to:

dX i 1 dx i 1 = (n 1 1) 1 + dX2 dx i 1 n 1 + 1 + . ( 12
)
The intracohort-consistency condition

dX i 1 dx i 1 = 1 yields: 1 + 2 + n 1 + 1 + n 1 1 1 = 0. (13)
From the second consistency condition 2 = (1 + 1 ), one deduces after rearrangement:

2 1 + (1 + )n 1 1 + (1 + )(n 1 1) = 0. (14)
From equation ( 14), the locally consistent conjectural equilibrium is unique and de…ned by: 7

= n 2 n 2 + 1 , (15) 1
= (1 + )n 1 2 + q [(1 + )n 1 ] 2 4(n 1 1)(1 + ) 2 , 2 = (1 + 1 ).
It has been shown in proposition 2 that within the set of CE equilibria that coincide with the SE, a consistent equilibrium must satisfy 1 = 0. In the above equations, there are only two cases for which 1 = 0: either n 1 = 1 (the leader is alone and behaves as a monopolist) or = 1 (perfectly competitive behaviors are expected for type 2 agents, which corresponds to n 2 ! 6 When n 2 = 1, one has 2 = 1 2 , which is the true slope of the reaction function in case of a linear Stackelberg duopoly. 7 , for any feasible values of 2 and . It is worth noting that proposition 1 is also satis…ed for 1 = 0 and 2 = n2 n2+1 .

Finally, from ( 9)- [START_REF] Perry | Oligopoly and consistent conjectural variations[END_REF], the equilibrium strategies are:

x1 ( 1 ; 2 ) = a c b[n 1 + (n 2 + 1)(1 + 1 + )] , (16)
These equilibrium strategies coincide with the strategies obtained in the multiple leader-follower Stackelberg equilibrium developed by [START_REF] Daughety | Bene…cial concentration[END_REF].

In addition, when 1 = 1 and 2 = = 0, (11)-( 12) yields:

dX i 1 dx i 1 = 1, (20)
dX 2 dx i 1 = 0.
So, the asymmetric competitive equilibrium is locally consistent. Note that it does not imply that is correctly expected (since = n2 1+n2 ). It corresponds to competitive equilibrium strategies, where type 1 …rms share the market. This result con…rms that perfect competitive equilibrium is consistent when agents form competitive conjectural variations under constant marginal costs [START_REF] Bresnahan | Duopoly models with consistent conjectures[END_REF], [START_REF] Perry | Oligopoly and consistent conjectural variations[END_REF]).

Conclusion

We determine the conditions under which the Stackelberg equilibrium coincides with the equilibrium of a simultaneous move game in which …rms form asymmetric conjectures. We also precise the de…nitions of consistent conjectural variations in an asymmetric framework.

x2 ( 1

 1 ; 2 ) = (a c)(1 + 1 + ) b[n 1 + (n 2 + 1)(1 + 1 + )] . (17)When 1 = 0 and = n2 n2+1 , these equilibrium strategies become: 1 + 1)(n 2 + 1) .

  , the industry now includes n 1 leaders and n 2 followers, with n 1 + n 2 = n.

		j=1;:::;n2
	gies xi 1 ; xj 2	i=1;:::;n1

De…nition 3 A Stackelberg equilibrium is given by a (n 1 + n 2 )-tuple of strate-

  As 1 + 2 1 one gets 1 (1 + ). Thus for n 1 2, one has The SE is only partially consistent, according to de…nitions 4 and 5. The SE is a locally intercohort-consistent CE if 2 = n2 n2+1 for any feasible values of 1 and , while it is a locally intracohort-consistent CE when

	1 = (1+ )n1 2	+	p	[(1+ )n1] 2 4(n1 1)(1+ ) 2
					(1+ )n 1
	p is 1 = [(1+ )n 1 ] 2 4(n 1 1)(1+ ) 2 (1+ )n 1 2 + p	<	(1+ )n 1 2	2 (1 + ). One deduces that the only possible value

[(1+ )n 1 ] 2 4(n 1 1)(1+ ) 2 .

1).

This property holds in case on symmetry across …rms (see[START_REF] Dixit | Comparative statics for oligopoly[END_REF]).

In an oligopoly framework, local consistency has been de…ned by[START_REF] Bresnahan | Duopoly models with consistent conjectures[END_REF],[START_REF] Perry | Oligopoly and consistent conjectural variations[END_REF] and[START_REF] Figuières | On the e¤ects of conjectures in a symmetric strategic setting[END_REF].

One cannot use the …rst-order condition to …rm i's pro…t maximization problem to de…ne how all …rms but i respond to a one-unit increase in i's output, because doing so would ignore the indirect e¤ects of a one-unit increase in i's output.On this point, see notably[START_REF] Kamien | Conjectural Variations[END_REF] and[START_REF] Perry | Oligopoly and consistent conjectural variations[END_REF]).


Ludovic A Julien

Olivier Musy

Aurélien Saidi

Aurélien W Saïdi

On hierarchical competition in oligopoly

Introduction

Stackelberg competition has been widely studied in the literature. Several extensions have been added to the original duopoly model (Stackelberg 1934), including a larger number of stages (Boyer and Moreaux 1986; Daughety 1990), and/or a larger num-ber of players (Sherali 1984; Heywood and McGinty 2008; Watt 2002). The overall sequential structure of a Stackelberg model is usually called a hierarchy (Boyer and Moreaux 1986; Watt 2002). In this context, a puzzling result has often been noticed: despite the hierarchical structure, firms behave as Cournotian oligopolists on the resid-ual demand. In the specific case of a single firm per stage, the standard Stackelberg duopoly leader produces exactly the same quantity as a monopolist. More generally, for a larger number of firms per stage and a homogeneous product, the strategies of leaders at any stage depend neither on the number of followers who play after, nor on the number of remaining stages. While this property is a striking feature of the Stack-elberg literature, neither its foundations, nor its implications have been fully formally studied. 1 In this paper, we go a step further by investigating the conditions under which such a property holds. To do this, we determine the conditions under which a Stackelberg leader firm behaves as a Cournotian oligopolist. We then consider the maximiza-tion program of a Cournot oligopolist and we analyze the assumptions under which the program of a Stackelberg leader is equivalent. While the literature only focused on the equivalence between the equilibrium strategies, our approach allows us to study the impact on profit functions of alternative assumptions on market demand, costs and the hierarchical structure. 2 The equivalence between the best response functions and the equilibrium strategies follows.

We show that the property holds whatever the number of cohorts and firms in the economy, as long as we conjointly assume the three following assumptions: a linear demand, constant and identical marginal costs. These three assumptions, unlike the structure and size of the economy, are critical features. Assuming the linearity for market demand and costs is not sufficient since individual marginal costs have also to be equal for all firms. We fill a gap in the existing literature by showing that these assumptions are not only sufficient but also necessary conditions for a T -stage hierarchical oligopoly model to reduce to a succession of Cournot games. Any departure from one of these assumptions rules out the equivalence between the Cournot and the Stackelberg strategies. We display several examples to illustrate our results.

The paper is organized as follows. In Sect. 2, we present the hierarchical model, state the assumptions and provide a graphical interpretation of the property. The next two sections deal with the formal proof that the hierarchical model reduces to a succession of Cournot games. Section 3 considers sufficient conditions, while Sect. 4 deals with necessary conditions. In Sect. 5, we provide an illustration with three cohorts. Section 6 concludes.

The hierarchical model under linearity assumptions

General framework

Consider one homogeneous good produced by n firms which oligopolistically compete in a hierarchical framework. There are T stages of decisions indexed by t, t ∈ 1, T . Each stage embodies one cohort and is associated with a level of decision. The whole set of cohorts represents a hierarchy. Cohort t is populated by n t firms, with t n t = n. The distribution of the firms within each cohort is assumed to be observable and exogenous. 3 This latter assumption notably implies that position of firms and timing of moves are given. 4 A firm i which belongs to cohort t has to decide strategically (simultaneously with firms of the same cohort, and sequentially among the hierarchy) its level of output denoted by x i t . The aggregate output of cohort t is denoted X t = n t i=1 x i t , where x i t stands for firm i's output within cohort t. In addition, X -i t = -i x -i t denotes the production of all firms belonging to cohort t but i.

The n t firms which belong to cohort t, behave as followers with respect to all firms of cohort τ , τ ∈ 1, t -1 , whose strategies are taken as given. However, they behave as Stackelberg leaders toward all firms of cohort τ , τ ∈ t + 1, T . They consider the best-response functions of all firms belonging to these cohorts as functions of their strategies.

The inverse market demand function for the homogeneous good specifies the market price p as a function of aggregate output X , and is denoted by p(X ). We assume that p(.) is continuous and twice differentiable, with dp(X ) d X < 0. For the equilibrium aggregate output X * , the equilibrium price is unique and equal to p(X * ).

The cost function of any firm i which belongs to cohort t is denoted by φ t (.). It is a continuous and twice differentiable function with

dφ t (x i t ) dx i t > 0 and d 2 φ t (x i t ) d(x i t ) 2 ≥ 0.

The linearity assumptions

In this section, we assume

• an inverse market demand function which is linear

p(X ) = a -bX, ( H1
)
where a, b > 0 and X = T t=1 X t , • constant marginal costs

φ t (x i t ) = c t x i t , i = 1, . . . , n t and t = 1, . . . , T (H2)
• and identical marginal costs

c t = c, t = 1, . . . , T. (H3)
These three assumptions are standard in the literature on oligopoly analysis (see Daughety 1990; Carlton and Perloff 1994; Vives 1999, among others).

The puzzling result of the hierarchical model stressed in the introduction has been regularly stated in the literature (e.g. Boyer and Moreaux 1986; Watt 2002 ). It can be captured by the property below:

Property 1 In the Stackelberg linear economy with an homogeneous product, firms in cohort t, t ∈ 1, T , behave as Cournotian oligopolists on the residual demand left by firms of cohorts τ (1 ≤ τ < t).

While this property has been quoted in the standard literature, it has been understood in a very restrictive way: the equilibrium strategies of firms in the hierarchical model coincide with those of a multistage Cournot model. In this paper, we enrich the meaning and implications of Property 1, presented as the equivalence of the profit functions in the two models up to a linear transformation. This implies the equivalence of the reaction functions in both models and thereafter of the equilibrium strategies (which is then a consequence rather than a definition of the Cournotian behavior).

Property 1 is usually observed under assumptions (H1)-(H3). We intend to investigate the sufficiency and necessity of these assumptions in the occurrence of this property as it has been defined above. Sufficiency has been considered in the literature for the restrictive definition of Property 1 but can be formally established in our enlarged framework. Necessity of these assumptions has never been studied.

Implications: a graphical interpretation

In this subsection, we illustrate Property 1 by a graphical approach, which is based on demand and costs rather than on best response functions. This property will be formally studied in the next section.

Consider two successive stages, say t -1 and t, with t = T . At stage t -1, the aggregate output resulting from the individual productions of firms of cohorts 1 to t -1 is given by t-1 τ =1 X τ . No particular value is given to X τ , except that it must generate a non-negative profit and is decided once for all. Under Property 1, the aggregate production of cohort t is computed as if it were the last cohort. In this case, the aggregate output of cohort t (for t-1 τ =1 X τ ) would establish to X t = t τ =1 X τ -t-1 τ =1 X τ . The subsequent price p t τ =1 X τ is the market price that would be effective if there were only t cohorts in the hierarchy. However, since cohort t is not the last cohort, the (unique) equilibrium price of the economy is lower and equal to p( T τ =1 X τ ). Nevertheless, the aggregate output of cohort t remains unchanged when the price falls down to p T τ =1 X τ . We assume that any leading cohort τ < t expects firms of cohort t (or more) to act symmetrically while firms of cohort t maximize their profit for any given quantity t-1 τ =1 X τ produced by their predecessors. The overall situation for cohort t is depicted in Fig. 1.

In this figure, we illustrate the behavior of cohort-t firms when acting as Cournotian oligopolists on the residual demand left by firms of cohort τ < t, as if they did not take into consideration firms playing after. In other words, Property 1 implies that leaders choose their optimal quantities so as to equalize the Cournotian marginal revenue of the cohort (dashed line) to the marginal cost (full line). As an immediate consequence, strategies of the leaders at any stage depend neither on the number of followers who play after nor on the remaining number of stages. This consequence corresponds to the standard result about the linear model usually expressed in the literature: given that the strategies of leaders do not depend on the number of followers, leaders behave as if they played a Cournot game on residual demand.

Sufficiency

The linearity assumptions as sufficiency conditions

Demonstrating Property 1 requires to exhibit the link between leaders and followers' profits.

Lemma 1 Let γ t ≡ T τ =t+1 1
1+n τ be the leader's markup discount factor. Under (H1)-(H3), the markup earned by a cohort t firm, t < T , in a T -cohort economy is a constant share γ t < 1 of the markup it earns in a t-cohort economy for any given vector of outputs (X 1 , . . . , X t-1 ) produced by the previous cohorts:

p T τ =1 X τ -c = γ t p t τ =1 X τ -c for t < T. ( 1
)
Proof See Appendix A.

Notice that under conditions H2 and H3, the markup is always equal across cohorts. The discount factor γ t differs from one cohort to another and measures the impact on market power of the number of followers. More specifically, it represents the reduction of a leader's markup due to the presence of the additional cohorts t + 1 to T . It affects less intensively the market power of the last cohorts in the sequence since they face a reduced number of followers. Market power shrinks as t tends to infinity.

The existence of cohort τ equally impacts by a coefficient 1/(1 + n τ ) the markup expected by a leader t (t < τ) in a t-stage economy, whatever the quantities produced by the first t cohorts (this results directly from H1-H3).

Corollary 1 For any strategy x i t , the profit obtained by a cohort-t firm in the sequential T -stage structure is a constant share of the profit of a t-stage economy (where cohort t would be the last cohort):

π i t (x i t ) = p T τ =1 X τ -c x i t = γ t p t τ =1 X τ -c x i t .
(2)

Proof This corollary directly results from Lemma 1.

In other words, each cohort can behave as if there were no following cohorts since it earns a constant share of the profit realized in an oligopoly structure market where it represents the last cohort, whatever the aggregate output t τ -= 1 1 X τ produced by the leaders. Provided that cohort-t firms maximize their profit for any vector of strategies (X 1 ,..., X t-1 ), cohort-τ leaders (τ<t) act as Cournot players who choose simultaneously and assume their direct rivals' output as given, ignoring the following cohorts, that only discount the value of their profits without changing the nature of the maximization program.

In the next sections, the right-hand side of Eq. ( 2) divided by γ t will be called the myopic profit (see Appendix A). By contrast with the sequential structure of our economy (left-hand side of Eq. ( 2)), the associated myopic program would result from the behavior of a firm that would observe the strategies of the leaders and would ignore the supplies of its followers.

Lemma 2 Let η t-h,t ≡ t τ t h 1 1 1 n τ be the follower's output discount factor. Under (H1)-(H3), the output = of - a fi + rm + i in cohort t ≤ T can
be expressed as a share of the output produced by a firm playing previously and belonging to cohort thfor h ∈ 1, t -1 , that is:

x t = η t-h,t x t-h . Proof See Appendix B.
The follower's output discount represents the decrease in optimal quantities for a follower resulting from a contraction of the residual demand when playing latter in the hierarchy. It is a share of cohort th's output, whether optimal or not, which is optimal for cohort t to produce. For any follower, this share decreases when going further in the sequence. Its value depends negatively on the number of leading cohorts and on the number of corresponding players (because residual demand decreases with respect to this parameter).

From the previous lemmas, the following proposition can be stated:

Proposition 1 Assumptions (H1)-(H3) are sufficient conditions for Property 1 to hold: the T -stage Stackelberg linear economy reduces to a collection of Cournot games in which firms compete oligopolistically on residual demand.

Proof The proposition directly ensues from Lemmas 1 and 2.

Characterization of the linear hierarchical equilibrium

Maximizing the sequential profit (left-hand side of Eq. ( 2)) is tantamount to maximize the myopic profit (right-hand side of Eq. ( 2)) since γ t is a constant term. In the linear economy, strategies of firms do not depend on the number of firms playing after, which equally impact the profit associated to each strategy. As a consequence the optimal strategies and the equilibrium strategies remain unchanged whatever the number of stages and the number of followers. 5 The literature only covers the similarity of the equilibrium strategies in both the T -stage Stackelberg linear model and the collection of staggered static problems but does not provide any explanation for this coincidence (see Boyer and Moreaux 1986; Anderson and Engers 1992; Watt 2002).

Corollary 2

The equilibrium strategy of cohort-1 firms may be obtained either from the sequential profit maximization or from the myopic profit maximization:

x 1 = 1 1 + n 1 a -c b ≡ η 1 X * ,
where X * = (ac)/b is equal to the perfect competition aggregate output. We then deduce the equilibrium strategy of any firm i in cohort t, t ∈ 1, T :

x t = η 1,t 1 + n 1 a -c b ≡ η t X * with η t ≡ t τ =1 1 1 + n τ .
Notice that in the equilibrium, each firm of cohort t produces a share η t of the perfect competition equilibrium output.

Corollary 3

The equilibrium price and profits are given by:

p = c + (a -c) T τ =1 1 1 + n τ , π i t = (a -c) 2 b t τ =1 1 (1 + n τ ) 2 T τ =t+1 1 1 + n τ , t = 1, . . . , T -1 π i T = (a -c) 2 b T τ =1 1 (1 + n τ ) 2 .
De Quinto and Watt (2003) use a term similar to η t to analyze welfare through market power and mergers.

Remark 1 Leaders of any cohort t, t ∈ 1, T -1 , do not ignore the true number of followers in cohort τ>t or do not form incorrect estimates about it.

Remark 2 The leader's markup discount factor is equivalent to a taxation of profits. The tax rate τ t = 1 -γ t differs from one firm to another. In the literature, neither the necessity of these conditions nor the fact that the leaders effectively behave as Cournotian oligopolists have ever been demonstrated. The only results feature that the equilibrium strategies in a T -stage model coincides with the equilibrium strategies that would be obtained with staggered static problems where firms either compete monopolistically (for n t = 1) or oligopolistically on the residual demand.

Necessity of assumption (H1)

We have to show that the linearity of the inverse demand function is a necessary condition for firms to behave as Cournotian oligopolists in each cohort, provided that the constant marginal cost is different from zero.

Proof

Under assumptions (H2) and (H3), firms of cohort T -1 act as Cournotian competitors on residual demand if and only if:

123 p T -1 τ=1 X τ + X T -c x t = γ t p T -1 τ=1 X τ -c x t , (3) for a coefficient γ t ≡ τ T =t+1 1 1+n τ > 0, or equivalently: p T -1 τ=1 X τ + X T -c = γ t p T -1 τ=1 X τ -c . ( 4
)
The conditions on the inverse demand function under which the equality in ( 4) is satisfied must be established. For a strictly positive marginal cost, the only inverse demand function satisfying (4) is linear. The proof is derived in two steps. We first need to demonstrate that Eq. ( 4) requires the reaction function of the last cohort to be linear. Second, we check that this property occurs only if the inverse demand function is linear.

Denote by g the function defined by g(.) = p(.)c and by f the reaction function of the last cohort:

X T = f ( T 1 τ= - 1 X τ ).Let O g and O f be the orders of the terms g( T 1 τ= - 1 X τ ) and f ( τ T = - 1 1 X τ ). Since γ t is a constant, it is an order-0 term and γ t g( T 1 τ= - 1 X τ ) has the same order as g( T 1 τ= - 1 X τ ), that is O g . The left-hand side of Eq. (4) is a composed function. The term g[ X T -1 τ 1 + f ( T 1 τ= - 1 X τ )] is a term of order O g O f for O f = 0 and of order O g for O f = = 0.
In the former case (with O g = 0), the equality in (4) requires that O f = 1, which means that f be a linear function. 6 Consequently, the aggregate reaction function of cohort T must be linear in

T 1 τ= - 1 X τ . Formally: 7 X T = α 1 + α 2 T -1 τ=1 X τ , (5)
where α 1 > 0 is the aggregate output of cohort T when the other cohorts do not produce, and -1 <α 2 ≤ 0 is the slope of the reaction function of cohort T .The values of those parameters are determined on the ground of economic plausibility.

Strategies are said to be substitute for α 2 < 0 and independent for α 2 = 0. Adding α 2 X T to each side of the equation above and rearranging yields:

X T = α 1 1 + α 2 + α 2 1 + α 2 T τ=1 X τ . ( 6
)
Given Eq. ( 2), the first-order condition for profit maximization of firm i belonging to cohort T is:

p T τ=1 X τ + dp dX T τ=1 X τ x i T -c = 0, i = 1,...,n T . ( 7
)
The n T equations in (7) implicitly define the reaction functions of cohort-T firms. Summing those equations for all i, i = 1,...,n T , leads to the reaction function of cohort T . This function must be consistent with the form determined in (6). Summing Eqs. ( 7) for all firms i, i = 1,...,n T , that belong to cohort T and using Eq. ( 6) to substitute for X T yields:

- p(X) -c dp/dX = μ 1 + μ 2 X, (8)
with:

μ 1 = α 1 n T (1 + α 2 ) > 0, 2 μ= α 2 n T (1 + α 2 ) ≤ 0 and X = T τ=1 X τ .
Finding the condition(s) on the inverse demand function p under which Eq. ( 4) is satisfied requires to solve Eq. ( 8). By inverting and integrating both sides of Eq. ( 8), we get:

dp/dX p(X) -c dX =- 1 μ 2 μ 2 μ 1 + μ 2 X dX, ( 9
)
whose solution is:

ln[μ 3 (μ 1 + μ 2 X)] for μ 2 = 0 ln[ p(X) -c]=- 1 μ 2 ln[ p(X) -c]= -1 μ 1 X + ln μ 4 for μ 2 = 0, (10)
where μ 3 > 0 and μ 4 > 0 stand for the constants of integration. The inverse demand function is finally obtained by identification. There is no economic rationale to consider that the inverse demand function is defined with respect to c. Thus, for c = 0, all terms in c must disappear from Eq. (10). Two cases must be considered.

Case 1: c = 0 In that case, terms in c disappear from Eq. ( 10) and the values of the parameters μ 1 to μ 4 may be freely chosen to determine the inverse demand function.

For μ 2 = 0, Eq. ( 10) becomes:

p(X) =[μ 3 (μ 1 + μ 2 X)] -1/μ 2 ≡[α -β X] ρ for any α, β, ρ > 0.
A specific case is the linear demand function with ρ = 1. It is worth noting however that for ρ = 1, another class of non-linear functions is compatible with the property that firms behave a la Cournot on residual demand when c = 0.8

For μ 2 = 0, Eq. ( 10) collapses to: p(X ) = μ 4 e -X/μ 1 ≡ αe -ρ X for any α, ρ > 0 which defines a last class of non-linear demand functions.

Case 2: c = 0 For μ 2 = 0, Eq. ( 10) collapses to:

p(X ) = μ 4 e -X/μ 1 + c.
Such a solution implies that the inverse demand function directly depends on the marginal cost c, which has no economic meaning. So this case can be ruled out.

For μ 2 = 0, Eq. ( 10) becomes:

p(X ) -c = [μ 3 (μ 1 + μ 2 X )] -1/μ 2 ,
We isolate the class of demand functions independent of c. Therefore, for sake of tractability, we use a first-order Taylor expansion around X to approximate the different inverse demand functions.

p(X ) -c p( X ) -c -μ 3 [μ 3 (μ 1 + μ 2 X )] -1/μ 2 -1 (X -X ) p(X ) p( X ) -μ 3 [μ 3 (μ 1 + μ 2 X )] -1/μ 2 -1 (X -X ).
(11)

In the above equation, only parameters μ 1 and μ 2 may depend on c (μ 3 being a constant of integration). We now show that μ 1 necessarily depends on c for α 1 = κ(1 + α 2 ), with κ ∈ R ++ , a case that will be dealt with later. Notice first that α 1 must depend on c since Eq. ( 7) must be satisfied, notably for T -1 τ =1 X τ = 0. Since p is independent of c, x T (and then X T ) must depend on c. For T -1 τ =1 X τ = 0, recall that X T = α 1 . According to (8), parameter μ 1 also depends on c provided α 1 = κ(1+α 2 ), with κ ∈ R ++ , which implies that μ 1 must disappear from the previous equation.

When μ 2 does not embody terms in c, the only possible value for μ 1 to vanish is μ 2 = -1. In that case: p(X ) p( X ) -μ 3 (X -X ), where the value for μ 3 can be freely chosen. Set, for instance, μ 3 = β > 0. Then, the inverse demand function is:

p(X ) = α -β X for any α, β > 0, ( 12
)
where α and β are parameters independent of c, and μ 1 = (αc)/β > 0 according to Eq. ( 8). Equation ( 12) corresponds to assumption (H1). It satisfies Property 1 for any (α, β) ∈ R 2 ++ . When μ 2 includes terms in c, the right-hand side of the Taylor expansion is indepen- 8), the first case corresponds to perfect competition: p = c. The second means that price is constant for any value of the aggregate output: p = c + 1. In both cases, the inverse demand function is dependent of c. These cases can be ruled out.

dent of c if μ 1 + μ 2 X = 0 or μ 3 (μ 1 + μ 2 X ) = 1. From (
Thus, the linear demand function is the only class of functions consistent with the property that firms act as Cournotian oligopolist on residual demand when c = 0.

Example for assumption (H1)

Consider T = 2 and n 1 = n 2 = 1. The inverse market demand function is given by P(X ) = e -X 2 . One can derive the best response function of the follower x 2 (x 1 ) =

x 2 1 + 2 -x 1 /2. The equilibrium strategy of the leader is then x1 = √ 2 -1.
These strategies differ from the equilibrium outputs that would be obtained if there were only one monopolistic firm, i.e.

x m = √ 2/2.
4.2 Necessity of assumption (H3)

Proof

Under assumptions (H1) and (H2), the reaction function of a cohort-T firm (assuming symmetry of strategies across cohort-T ) can now be explicitly defined:

x T = a -c T -b T -1 τ =1 X τ b(1 + n T ) , ( 13
)
and the profit function of cohort-(T -1) firms is:

π T -1 (x T -1 ) = 1 1 + n T āT -1 -bX T -1 -c T -1 -n T (c T -1 -c T ) x T -1 , ( 14
)
where āt ≡ ab t τ =1

X τ is the residual demand left by the first t cohorts.

For cohort-(T -1) firms to behave as Cournotian oligopolists, the profit function must be such that:

π T -1 (x T -1 ) = 1 1 + n T πT -1 (x T -1 ). ( 15
)
This is the case if and only if:

c T -1 = c T .
Using backward induction, it can be proved that for cohort-t firms, t < T -1, to behave as Cournotian oligopolists, the following property must hold:

c 1 = c 2 = • • • = c T -1 = c T . ( 16
)
Thus, when the inverse demand function is linear and cost functions are constant, a necessary condition for firms of any cohort to behave as Cournotian competitors on residual demand requires assumption (H3).

Example for assumption (H3)

Consider T = 2, n 1 leaders and n 2 followers. The inverse market demand function is

P(X ) = a -bX. Assume φ t (x i t ) = c t x i t , t = 1, 2. One easily gets x1 = (a-c 1 )+(c 2 -c 1 )n 2 b(1+n 1 )
. For c 1 = c 2 , the equilibrium strategy of leaders depends on the number of followers. It is no longer the case for c 1 = c 2 since x1 = a-c b(1+n 1 ) .

Necessity of assumption (H2)

Proof

Assume that assumptions (H1) and (H3) are satisfied. The firm j of cohort t then determines x j t such that it maximizes its myopic profit function π j t :

π j t (x j t ) = āt-1 -b n t i=1 x i t x j t -φ(x j t ), ( 17
)
where φ is a non-decreasing function and is identical across firms.

The reaction function of cohort-t firms, when behaving symmetrically, is implicitly defined by:

āt-1 -b(1 + n t )x t - dφ dx t (x t ) = 0 for any t > 1. (18)
When maximizing its profit, a cohort-τ firm (τ < t) substitutes to X t the aggregate reaction function of cohort t. Strategy of a cohort-τ firm is then independent of n t provided X t does not contain any n t term. If so x t is a function of n t , i.e. x t = X t /(1 + n t ).

The aggregate reaction function of cohort t can be rewritten as:

X t = āt-1 -dφ dx t (x t ) b for any t > 1. ( 19
)
It does not depend on n t provided dφ t dx t (x t ) does not contain any n t term. Since x t is a function of n t , the marginal cost is independent of x t , that is constant:

φ(x t ) = cx t for anyt ∈ 1, T .
(20)

Example for assumption (H2)

Consider T = 2, n 1 leaders and n 2 followers. The inverse market function is P(X ) = a -bX. Assume φ t (x i t ) = c(x i t ) 2 /2 for i = 1, . . . , n t and t = 1, 2. One gets x1 = Consider as an illustration an economy with T = 3 under assumptions (H1)-(H3). In this three-stage game, cohorts 1 to 3 are populated by respectively n 1 , n 2 and n 3 firms.

Cohort 1 plays first but we solve the game by backward induction, starting from the last cohort.

The program of firm k, k = 1, . . . , n 3 , which belongs to cohort 3 may be written:

Arg max x k 3 π k 3 = a -b X 1 + X 2 + x k 3 + X -k 3 -c x k 3 ,
that is:

Arg max x k 3 π k 3 = ⎡ ⎣ a -b n 1 i=1 x i 1 -b n 2 j=1 x j 2 -b(x k 3 + X -k 3 ) -c ⎤ ⎦ x k 3 , ( 21
)
where x i 1 (resp. x j 2 ) is the strategic supply of firm i (resp. j ) in cohort 1 (resp. 2), observed by firm k when choosing its own strategy. Each firm k considers as given the strategies of its n 3 -1 competitors in cohort 3. In the symmetric equilibrium:

x k 3 = x -k 3 and X -k 3 = (n 3 -1)x -k 3 .
The first-order condition leads to the reaction function of any firm k:

x k 3 (X 1 , X 2 ) = 1 1 + n 3 a -c b - X 1 + X 2 1 + n 3 . ( 22
)
When choosing its own strategy, each firm j belonging to cohort 2 observes the strategic supplies of the n 1 leaders and considers as given the strategies of its n 2 -1 competitors. It also takes into account the reaction function of cohort-3 firms. Then, the program of the jth cohort-2 firm may thus be written:

Arg max x j 2 π j 2 = a -b X 1 + x j 2 + X -j 2 + n 3 k=1 x k 3 (X 1 , X 2 ) -c x j 2 ,
that is:

Arg max x j 2 π j 2 = 1 1 + n 3 a -b n 1 i=1 x i 1 -b(x j 2 + X -j 2 ) -c x j 2 . ( 23
)
In the symmetric equilibrium:

x j 2 = x -j 2 and X -j 2 = (n 2 -1)x -j
2 . The first-order condition leads to the reaction function of any firm j:

x j 2 (X 1 ) = 1 1 + n 2 a -c b - X 1 1 + n 2 j = 1, . . . , n 2 (24)
Finally, firm i belonging to cohort 1 takes into account the reaction functions of all its followers and considers as given the strategies of its n 1 -1 competitors. Its program may be written:

Arg max x i 1 π i 1 = ⎡ ⎣ a -b ⎛ ⎝ x i 1 + X -i 1 + n 2 j=1 x j 2 (X 1 ) + n 3 k=1 x k 3 (X 1 , X 2 ) ⎞ ⎠ -c ⎤ ⎦ x i 1 ,
that is:

Arg max x i 1 π i 1 = 1 (1 + n 2 )(1 + n 3 ) [a -b(x i 1 + X -i 1 ) -c]x i 1 . ( 25
)
In the symmetric equilibrium, one gets the equilibrium strategy of leader i:

xi 1 = 1 1 + n 1 a -c b i = 1, . . . , n 1 . ( 26
)
We deduce the equilibrium outputs of cohort-1 and cohort-2 firms:

x j 2 = 1 (1 + n 1 )(1 + n 2 ) a -c b j = 1, . . . , n 2 (27) xk 3 = 1 (1 + n 1 )(1 + n 2 )(1 + n 3 ) a -c b k = 1, . . . , n 3 (28)
As specified in Property 1, the equilibrium strategy of any leader in the first stage depends neither on the number of followers who play after, nor on the number of remaining stages. This property also holds for firms in stage 2.

The corresponding aggregate outputs in the three stages are:

X1 = n 1 1 + n 1 a -c b X2 = n 2 (1 + n 1 )(1 + n 2 ) a -c b (29) X3 = n 3 (1 + n 1 )(1 + n 2 )(1 + n 3 ) a -c b
The corresponding equilibrium market price and profits are given by:

p = c + a -c (1 + n 1 )(1 + n 2 )(1 + n 3 ) , ( 30
)
and

πi 1 = (a -c) 2 b 1 (1 + n 1 ) 2 (1 + n 2 )(1 + n 3 ) π j 2 = (a -c) 2 b 1 (1 + n 1 ) 2 (1 + n 2 ) 2 (1 + n 3 ) (31) πk 3 = (a -c) 2 b 1 (1 + n 1 ) 2 (1 + n 2 ) 2 (1 + n 3 ) 2
So the equilibrium values coincide with those given in Corollary 3.

A three-stage Cournot economy

Consider now a three-stage Cournotian economy.

In stage 1, any firm i, i = 1, . . . , n 1 , solves:

Arg max x i 1 π i 1 = [a -b(x i 1 + X -i 1 ) -c]x i 1 . ( 32
)
Notice that this program is identical to the program of Eq. ( 25) up to a factor η 1,3 = 1 (1+n 2 )(1+n 3 ) . The equilibrium strategy xi 1 of firm i must then be identical to the solution of Eq. ( 25):

xi 1 = 1 1 + n 1 a -c b = xi 1 , i = 1, . . . , n 1 . ( 33
)
In stage 2, any firm j, j = 1, . . . n 2 , determines its supply from the residual demand left by cohort-1 firms:

Arg max x j 2 π j 2 = a -b n 1 i=1 xi 1 -b(x j 2 + X -j 2 ) -c x j 2 , that is: Arg max x j 2 π j 2 = 1 1 + n 1 a -b n 1 i=1 xi 1 -b(x j 2 + X -j 2 ) -c x j 2 . ( 34
)
Provided the aggregate outputs of cohort 1 in both economies (whether optimal or not) are equal, that is X1 = X1 , this program is identical to the program of Eq. ( 23) up to a factor η 2,3 = 1 1+n 3 . The equilibrium strategy x j 2 of firm j must then be identical to the solution of Eq. ( 23):

x j 2 = 1 (1 + n 1 )(1 + n 2 ) a -c b = x j 2 , j = 1, . . . , n 2 . ( 35
)
Finally, any firm k, k = 1, . . . , n 3 , belonging to cohort 3 determines its supply from the residual demand left by cohort-1 and -2 firms:

Arg max x k 3 π k 3 = ⎡ ⎣ a -b n 1 i=1 xi 1 -b n 2 j=1 x j 2 -b(x k 3 + X -k 3 ) -c ⎤ ⎦ x k 3 . ( 36
)
Provided the joint aggregate outputs of cohorts 1 and 2 in both economies (whether optimal or not) are equal, that is X1 + X2 = X1 + X2 , this program is identical to the program of Eq. ( 21). The equilibrium strategy xk 3 of firm k must then be identical to the solution of Eq. ( 21):

xk 3 = 1 (1 + n 1 )(1 + n 2 )(1 + n 3 ) a -c b = xk 3 , k = 1, . . . , n 3 . ( 37
)
The concordance of the Stackelberg and Cournotian maximization programs (up to a factor) and the resulting equilibrium strategies illustrates Property 1.

Conclusion

The paper explored a general hierarchical linear model in which firms compete in quantities. A key property has been studied: under the assumptions of linear demand, constant and identical marginal costs, each firm behaves at any stage as a Cournotian oligopolist on residual demand. We formally established that these assumptions are not only sufficient but also necessary for strictly positive marginal costs.

This property of the linear hierarchical T -stage model means that a firm in cohort t maximizes its profit when it chooses its output as if it were the last follower of the economy. The associated maximization program of the firm is equivalent up to a discount factor to the maximization program it would face in a similar t-stage hierarchical economy. The sooner a firm plays within the T -stage hierarchy, the more discounted its profit compared with the t-stage economy.

Finally, applying Thales' theorem to triangles ABC and AD F leads to:

BC = 1 1 + n T D F = E F. (43)
Actually, E F is the markup of a leader after the entrance of the last cohort, while D F is the markup of a leader before the entrance of cohort T . Equation ( 43) can be rewritten as:

p T τ =1 X τ -c = 1 1 + n T p T -1 τ =1 X τ -c . ( 44
)
Step 2 assume property (1) is true for any cohort t = Th (1 ≤ h ≤ T -2) then it is true for cohort Th -1.

If property (1) holds for cohort Th then:

p T τ =1 X τ -c x i T -h = γ T -h p T -h τ =1 X τ -c x i T -h , ( 45
)
with γ T -h ≡ T τ =T -h+1 1 1+n τ . Thus, maximizing firm i's profit is tantamount to maximize the myopic profit defined as follows:

max x i T -h p T -h τ =1 X τ -c x i T -h . ( 46
)
When firms of cohort-(Th) act symmetrically, the myopic marginal revenue (dashed line) is defined by: 10

Rm (X

T -h ) = āT -h-1 -b 1 + n T -h n T -h X T -h . ( 47
)
In the same way as in step 1, it can be shown that:

p T -h τ =1 X τ -c = 1 1 + n T -h p T -h-1 τ =1 X τ -c . ( 48
)
By assumption, the following property is satisfied: )

p T τ =1 X τ -c = γ T -h p T -h τ =1 X τ -c . ( 49

Fig. 1

 1 Fig. 1 Graphical representation of (H1)-(H3)

4

  Beyond the sufficiency of the linearity assumptions This section establishes the following proposition and provides some examples. Proposition 2 Assumptions (H1)-(H3) are necessary conditions for Property 1 to hold when costs are strictly positive: the T -stage Stackelberg linear economy reduces to a collection of Cournot games in which firms compete oligopolistically on the residual demands.

  equilibrium strategy of any leader depends on the number of followers. When firms behave a la Cournot in each stage, one has x1 = a b(1+n 1 )+c = x1 . 5 An illustration with three cohorts 5.1 A three-stage Stackelberg economy

The standard Stackelberg duopoly prevails when T = 2andn 1 = n 2 = 1.

We therefore do not question the way a specific firm could or should become a leader (see[START_REF] Anderson | Stackelberg versus Cournot equilibrium[END_REF][START_REF] Amir | Stackelberg versus Cournot equilibrium[END_REF][START_REF] Matsumura | Quantity-setting oligopoly with endogenous sequencing[END_REF]).

The case O g = 0 means that price is constant, that is independent of aggregate output.

The case O f = 0 is a subcase of Eq. (5) for which α 2 = 0.

This result was pointed out by[START_REF] Anderson | Stackelberg versus Cournot equilibrium[END_REF] in footnote 5 on page 129.

Acknowledgments

We are grateful to C. Bidard for his comments and remarks on a previous version of the paper. Two anonymous referees are acknowledged for their remarks and suggestions. Remaining errors are ours.

Appendix A: Proof of Lemma 1

The proof is by backward induction and structured in three steps.

Step 1 property 1 is true for cohort t = T -1 (with T > 1).

The inverse demand function faced by firms is defined by:

where X τ = n τ i=1 x i τ ≥ 0 is the aggregate production of cohort τ . For any quantity of output X T -1 produced by cohort T -1, the resulting residual demand faced by followers of cohort T is:

where āT -1 is considered as given by followers. Geometrically (see Fig. 1), followers must select a couple (X, p) on the segment [D, A].

When acting symmetrically, the associated marginal revenue of cohort-T firms (dashed line on Fig. 1) is defined by: 9

Considering the following derivatives:

it comes that:

We deduce from the two previous equations that:

Step 3 from steps 1 and 2 we conclude by backward induction that Property 1 is true for any cohort t (with 1 ≤ t ≤ T -1).

Appendix B: Proof of Lemma 2

Applying Thales' theorem to triangles ABC and ADF leads to:

Actually, C F is the optimal output produced by cohort t, that is X t , while AF is the maximal quantities cohort t can produce to generate non-negative profit (equal to the difference between the perfect competition equilibrium supply and the output already produced by the previous cohorts). The property above can be rewritten as:

Notice that AC is also the maximal quantities cohort t + 1 can produce to generate non-negative profit. Then, equality (52) applied to cohorts t and t + 1 becomes:

By backward induction, it turns out that: