On the uniqueness of Cournot equilibrium in case of concave integrated price flexibility

We consider a class of homogeneous Cournot oligopolies with concave integrated price flexibility and convex cost functions. We provide new results about the semi-uniqueness and uniqueness of (Cournot) equilibria for the oligopolies that satisfy these conditions. The condition of concave integrated price flexibility is implied by (but does not imply) the log-concavity of a continuous decreasing price function. So, our results generalize previous results for decreasing log-concave price functions and convex cost functions. We also discuss the particular type of quasi-concavity that characterizes the conditional revenue and profit functions of the firms in these oligopolies and we point out an error of the literature on the equilibrium uniqueness in oligopolies with log-concave price functions. Finally, we explain how the condition of concave integrated price flexibility relates to other conditions on the price and aggregate revenue functions usually considered in the literature, e.g., their concavity.


Introduction
Cournot equilibrium is one of the oldest equilibrium notions in mathematical economics. Its definition refers to an abstract representation of an oligopoly, which can be described as follows. A set N := {1, . . . , n} of firms produce a homogeneous good. Each firm i ∈ N chooses a quantity of production x i from its own feasible production set X i , where X i ⊆ R, facing a production cost c i (x i ), where c i : X i → R. The aggregate production y = i∈N x i , which is an element of (the Minkowski-sum) Y := l∈N X l , is sold to the market at the price p(y), given by a function p : Y → R. Each firm i ∈ N receives a profit π i (x) := p( l∈N x l )x i − c i (x i ). So this defines firm i's profit function π i : X 1 × · · · × X n → R. We refer to p as the price function and to c i as the cost function of firm i. Given this oligopolistic structure, x is called a (Cournot) equilibrium if, for all i ∈ N , x i ∈ argmax z i ∈X i π i (x 1 , . . . , x i−1 , z i , x i+1 , . . . , x n ).
The conceptual definition of a Cournot equilibrium implies neither its existence nor its uniqueness, however both the problem of the existence and that of the uniqueness of an equilibrium are of interest in economics. The two problems are not unrelated to one another. In particular, when the existence problem is solved the uniqueness problem reduces to the problem of equilibrium semi-uniqueness, that is, the problem of the existence of at most one equilibrium. For a review of many results on these topics see also [3,7].
In this article we are concerned with equilibrium semi-uniqueness in Cournot oligopolies with convex cost functions and concave integrated price flexibility. Henceforth, we shall assume that either X i = R + or X i = [0, m i ] where m i > 0. In case X i = [0, m i ], we say that firm i has a capacity constraint. Our main concern is semi-uniqueness and not uniqueness because in our case each conditional profit function of each firm 1 is quasi-concave and it is not difficult to see that under additional continuity and compactness conditions various wellknown theorems (as, e.g., that of Nikaido-Isoda or Kakutani fixed point theorem) guarantee the existence of an equilibrium.
To the best of our knowledge there are no general semi-uniqueness results for Cournot oligopolies with convex cost functions and concave integrated price flexibility. Important results, such as those in [5] for concave aggregate revenue and those in [1] for log-concave price functions do not cover our results because there are price functions with strictly concave integrated price flexibility (like p(y) = e − √ 1+y ) which are not log-concave (as in [1]) and to which there is not associated a concave aggregate revenue function (as in [5]). Our results are complementary to those in [5] because there exist price functions without concave integrated price flexibility, to which there is associated a strictly concave aggregate revenue function. However, we shall prove that all decreasing continuous log-concave price functions have concave integrated price flexibility.
The organisation of the article is as follows. In Sect. 2 the notion of concave integrated price flexibility is formally defined and compared with other convexity notions. In Sect. 3 we study the Nash-sum, i.e., the function σ : E → R defined by σ (e) := n l=1 e l where E is the of equilibria. There we provide sufficient conditions for the Nash-sum to be injective and sufficient conditions for it to be constant. In Sect. 4 we identify a class of oligopolies with concave integrated price flexibility for which the Nash-sum is injective and constant, which implies that an oligopoly in this class has at most one equilibrium; also the existence problem will be considered. Section 5 reconsiders the case of log-concave price functions. Section 6 concludes and an Appendix contains some results on (generalized) convexity.

Integrated price flexibility
Consider a price function p : Y → R. If p is differentiable at y ∈ Y and p(y) = 0, the price flexibility of p at y is defined by i.e., as the elasticity of p at y. 2 For a positive continuous price function p : Y → R, the function L p : Y → R is defined by We call this function the integrated price flexibility of p. We use this terminology because for each y ∈ Y at which p is differentiable, also L p is differentiable and The concavity of the integrated price flexibility is the property we address in this article. Fact F (in the Appendix) guarantees that a (strictly) decreasing continuous log-concave 3 price function p has a (strictly) concave integrated price flexibility L p . Facts A and B1 guarantee that if p is positive and continuous, then the (strict) concavity of L p implies the (strict) decreasingness of p. Just to provide an example, consider the price function given by p(y) = δe −(α+βy) γ with positive parameters. For this function Dη p (y) = − βγ (α+βγ y) (α+βy) 2−γ < 0 holds and therefore p has a strictly concave integrated price flexibility.
It is good to compare the property of the concavity of the integrated price flexibility with three other properties of p that play an important role in oligopoly theory: the concavity of p, the log-concavity of p and the concavity of the aggregate revenue function r associated to p, where r : Y → R is defined by r (y) := p(y)y. For simplicity, we shall do this for the context where the price function p is positive, decreasing and differentiable; the domain Y of p may be compact or not.
First note that η p ≤ 0, Dr = p · (η p + 1) and that the (strict) concavity of the integrated price flexibility of p is equivalent to the (strict) decreasingness of η p . Historically, oligopolies with a concave price function belong to the first types studied. A positive concave price function p is log-concave and has a concave associated aggregate revenue r (and a strictly concave associated aggregate revenue r if p is strictly concave). The log-concavity of p and the concavity of r have received special attention in the literature (e.g., [1,5,7]). Concave aggregate revenue functions are compatible with other types of price functions, like the strictly convex ones. However, Fact H shows that the log-concavity of p and the concavity of r are quite incompatible when Y = R + . As the log-concavity of p implies the decreasingness of η p , the log-concavity of p implies the concavity of the integrated price flexibility. 4 Therefore, also concave price functions have concave integrated price flexibility. Table 1 illustrates.
It turns out that with our method we can obtain equilibrium semi-uniqueness results also for price functions that are unbounded at 0, like p(y) = 1/y (y = 0) (and, for example, 5 2 In this article D f is the symbol we use for the derivative of a real function f of one real variable; D + f (D − f ) will be used for right (left) derivatives. Furthermore when f is a real function of more than one real variable, then D i f (D + i f, D − i f ) will be used for the (right, left) derivative with respect to the ith variable. 3 Log-concavity of a real-valued function presupposes that this function is positive.
Yes Yes p(0) = 137). In order to handle them we consider the following variant of L p . For q ∈ Y \{0} and a price function p : Also this function will be called integrated price flexibility of p (with respect to q). This function depends on q, but various of its properties, e.g., its (strict) concavity, do not depend on q. We say that p has concave (resp. strictly concave) integrated price flexibility if there exists a q such thatL p is concave (resp. strictly concave).

Nash-sums
The result in the following lemma gives very weak sufficient conditions in order that 0 be not an equilibrium when there exist at least two equilibria. In this lemma we assume that each cost function c i is convex. We recall that this implies that c i is left and right differentiable at each interior point of its domain and that c i may not be right differentiable at 0, but its right derivative exists at 0 as an element of R ∪ {−∞}. If firm i has a capacity constraint m i , then c i may not be left differentiable at m i but its left derivative exists as an element of R ∪ {+∞}. We write R := R ∪ {−∞, +∞}.

Lemma 1 Suppose each cost function is convex, and -p is left and right differentiable at each point of Int(Y ),
Also suppose p is strictly decreasing on Y \ {0} or each cost function is strictly convex. If there exists more than one equilibrium, then 0 is not an equilibrium.
Proof By way of contradiction, suppose a = 0 and b are two distinct equilibria. Let follows: the first (second) inequality holds as a (b) is an equilibrium, 7 the third holds as D − p(y b ) ≤ 0 and the fourth as p is decreasing on Y \ {0}, c i is convex, and either p is even strictly decreasing on Y \ {0} or c i is strictly convex.

Theorem 1 Suppose each cost function is convex, and
Each of the following conditions is sufficient for the Nash-sum σ to be injective: By Fact B2 we see that p ∈ D ≤ and that in case I even p ∈ D < . So Lemma 1 applies and therefore y a = 0.
As a and b are equilibria, We now prove that If D − p(y b ) = −∞, then (5) holds. Now suppose D − p(y b ) = −∞ and hence p is left differentiable at y b . First we prove that Well, as b l > 0 for every l ∈ J and b is an equilibrium, we have for every l ∈ J from which it follows (as cost functions are increasing), As D + p(y a ) ≤ 0 and y a − (y b −ỹ b ) ≥ỹ a , (5) follows.
As each c i is convex (and strictly convex in case II), (5) implies which in turn implies the existence of i ∈ J such that which is in contradiction with (4). Proof By Theorem 2 the Nash-sum is constant. By Theorem 1 the Nash-sum is injective. It follows that there exists at most one equilibrium.

Semi-uniqueness
Concerning the explicit monotonicity property of p in Theorem 3 it is interesting to note that the concavity ofL p does not imply the decreasingness of p (consider for example p(y) = ln(y + 1)). But by Facts A and B1, a positive continuous p with concave integrated price flexibility L p is decreasing.

Existence and uniqueness
In order to avoid further technicalities we now investigate the problem of the existence of a unique Cournot equilibrium assuming the differentiability of p everywhere.
For i ∈ N and k ∈ l∈N \{i} X l , firm i's conditional revenue function r

Theorem 4 Suppose cost functions are convex, increasing and continuous, p is positive, differentiable and has concave integrated price flexibility L p . Then profit functions are continuous and conditional profit functions are quasi-concave.
Proof The continuity of the profit functions is clear. In order to prove the quasi-concavity of the conditional profit functions we fix i ∈ N and k ∈ l∈N \{i} X l . Note that r (k) i is differentiable and Dr i − c i is concave and the proof is complete. Now suppose I = X i . As I is convex, it follows that I is bounded; let d := sup I . As the restriction (r

Log-concave price functions
Consider a continuous (strictly) decreasing log-concave price function p. So L p has (strictly) concave integrated price flexibility L p by fact F and Theorem 3 implies:

Corollary 2 Suppose the price function is log-concave, decreasing and differentiable and each cost function is convex and increasing. Then each of the following conditions is sufficient for the existence of at most one equilibrium: (I) The price function is strictly decreasing. (II)
All cost functions are strictly convex.

Related to Corollary 2 is Theorem 2.3 in [1] for a duopoly:
Suppose no duopolist has a capacity constraint, the price function is log-concave and strictly decreasing and each cost function is convex and strictly increasing. If for each duopolist i it holds that p(x)x − c i (x) < 0 for x large enough, then there exists a unique equilibrium.
Note that in this 'theorem' it is not assumed that the price function is differentiable. The next example, where all conditions of Theorem 2.3 in [1] hold, shows that this theorem is false: Example 1 Consider the following duopoly: It is straightforward (using the symmetry of the situation) to verify that ( 9 20 , 11 20 ) and ( 11 20 , 9 20 ) are equilibria.
As in Example 1 also all conditions of Corollary 2(I) hold, with the exception of the differentiability of p, it follows that Corollary 2 does not hold anymore when we omit the assumption that the price function is differentiable.

Concluding remarks
As we have explained in the Introduction, and shown in the body of the article, our results extend (and correct) a result in [1]. It should be clear that, e.g., the results in [1] are not sufficient to conclude that the triopoly given by 100 for each firm i ∈ N , has a unique Cournot equilibrium; also, the results in [3] (in particular Theorem 6.1) does not prove this fact. 8 So, to the best of our knowledge, the results presented in this article are new.
Our results build on-and extend-a technique developed in [6]. We conjecture that this technique can be also used to simplify and shorten the proof of important equilibrium semiuniqueness results for oligopolies with convex cost functions and aggregate revenue functions such as that in Lemma 5 of [5] (which neither implies nor is implied by the results of this article, as it is clear from the table in Sect. 2).
It is good to remark that in this article, unlike in [5], we cannot guarantee equilibrium semi-uniqueness when the cost functions are not increasing. The reason is that in the proof of Theorem 2 we utilize a particular type of quasi-concavity of the conditional revenue functions which implies the crucial inequality (6). As it can be seen from the proof of Theorem 4 this particular quasi-concavity requires that, at each Cournot equilibrium e, the conditional revenue of a firm i that produces e i > 0 must be either concave and increasing, or concave and increasing up to some point x i ≥ e i where the conditional revenue is maximized and decreasing beyond that point, whence (6) follows easily. But this does not keep on holding-and hence inequality (6) does not generally hold true-if the cost function of firm i is convex but not increasing. It is good to note that also the conditional profit function of firm i is concave and increasing up to e i and decreasing beyond that point, thus inheriting the same type of quasi-concavity of the conditional revenue function; also this fact does not keep on holding when the cost functions are convex but not increasing.
This particular type of quasi-concavity has a strong connection with the notion of semiconvexity provided in [4]. We think that this type of generalized convexity could have many natural applications in economics and, to the best of our knowledge, this is the first article dealing with economic applications that explicitly refers to such a generalized convexity notion.
Finally, a large part of the literature on the semi-uniqueness and uniqueness of Cournot equilibrium takes into consideration oligopolies without capacity constraints (or better, firms' feasible production sets are assumed to be upper unbounded). As explained in [2], the assumption of upper unboundedness of the feasible productions sets of the firms is not an inconsequential assumption and influences some results of the literature on Cournot equilibrium, in particular when one considers the dynamics. Clearly, the equilibrium semiuniqueness and uniqueness results of this article do not depend on the upper unboundedness of the feasible productions sets of the firms because we allow both upper bounded and upper unbounded feasible productions sets (note that when we provide sufficient conditions for the existence of a Cournot equilibrium we impose the upper boundedness of the profit functions, but the feasible production sets can be either bounded or not).
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
G. If r is (strictly) concave, then p ∈ D ≤ (D < ). H. Suppose Y = R + . If p is log-concave on R ++ and non-constant on Y \ {0}, then r is not concave.

Proof of Fact A As p is right and left differentiable on Int(Y ), p is continuous on Int(Y ).
Together with D + p(y) ≤ (<) 0 (y ∈ Int(Y )), this implies that p is (strictly) decreasing on Proof of Fact B1 With the exception of D − p ≤ (<) 0 this Fact is a direct consequence of the (strict) concavity of L p . The important observation here is to realize that D + L p (0) = 0 which can be proved with the rule of de l'Hôpital: Proof of Fact B2 Having the proof of Fact B1, we only have to prove that D − p < 0 in caseL p is strictly concave. So suppose there exists y 0 ∈ Y \ {0} with D − p(y 0 ) ≥ 0. As p is decreasing on Y \ {0}, it follows that D − p ≤ 0. So D − p(y 0 ) = 0 and therefore D −L p (y 0 ) = 0. But D −L p is strictly decreasing and D −L p ≤ 0, which is impossible.
Proof of Fact C By contradiction, suppose there exist y 1 , y 2 , k, s as above and Since p is positive and decreasing on [y 1 , y 2 ], 0 < 1 p(y 1 ) ≤ 1 p(y 2 ) ; so and hence AsL p is (strictly) concave, we have D +L p (y 1 ) ≥ (>) D −L p (y 2 ). As p is decreasing on Y \ {0}, this leads to .
. It is sufficient to prove that H k \ {0} is convex. We provide the proof by contradiction. So suppose . As g is a derivative, g is also a Darboux-function. Therefore there exists By Facts A and B1, p is decreasing. Fact C implies that Dp(x 4 + k)x 4 + p(x 4 + k) ≥ Dp(x 2 + k)x 2 + p(x 2 + k). It follows that 0 < g(x 2 ) ≤ g(x 4 ) = 1 5 g(x 2 ), which is impossible.
Proof of Fact E By way of contradiction, assume that there exists y 0 ∈ Y \ {0} with D − p(y 0 ) ≥ 0. As p is strictly decreasing, D − p/ p ≤ 0 holds. This implies D − p(y 0 ) ≤ 0. So D − p(y 0 ) = 0. As p is log-concave, D − p/ p is decreasing. As D − p(y 0 )/ p(y 0 ) = 0, it follows that D − p/ p vanishes on the proper interval ]0, y 0 ], and so does also D − p. As p is also continuous on this interval, it follows that p is constant on this interval and therefore not strictly decreasing on this interval, which is a contradiction with the strict decreasingness of p.
Proof of Fact F First we note that sufficient for the (strict) concavity of a function f defined on a real interval is that f is continuous, its left derivative D − f is well defined and (strictly) decreasing, and real-valued on the interior of its domain. Well, L p has all these properties. In fact, we only have to prove that D − L p is (strictly) decreasing. As p is (strictly) decreasing (using Fact E) we have D − p p ≤ (<) 0 , so D − L p ≤ (<) 0. As p is log-concave, D − p p is decreasing and therefore also D − L p is (strictly) decreasing.
Proof of Fact G As r is a (strictly) concave function it holds that r is left and right differentiable at each interior point of its domain. As p(y) = r (y)/y (y = 0), this implies also that p is left and right differentiable at each point of Int(Y ). As r is (strictly) concave, it holds for all y ∈ Int(Y ) that D + r (y) ≤ D − r (y) ≤ (<) r (y)−r (0) y−0 . So D + p(y)y + p(y) ≤ D − p(y)y + p(y) ≤ (<) p(y) and thus D + p(y) ≤ D − p(y) ≤ (<) 0. In case Y = [0, m], the proof of D − p(m) ≤ (<) 0 is similar.
Proof of Fact H By way of contradiction, assume that r is concave. As r is concave, r (0) = 0, r ≥ 0 and r is somewhere positive, it follows that lim y→+∞ r (y) exists as an element of R and lim y→+∞ r (y) > 0.