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JEL cklssification: D43. This paper is about properties of an n-firm Cournot equilibrium in a market for a single homogeneous commodity. It is well known that ...
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Economics Letters 54 11997) 155-158

Cournot equilibrium with convex demand Serge Svizzero* Universitd du Littoral, Ddpartement d'Economie. 21 rue Saint Louis. BP 774, 62321 Boulogne sur Met', France Received 15 October 1996; accepted 5 December 1996

Abstract We recall that Cournot equilibrium may exist even with convex demand and demonstrate that it may not tend to the competitive outcome when the number of competitors increases. The result depends on the degree of curvature of the inverse demand.

Keywords: Oligopoly JEL cklssification: D43

This paper is about properties of an n-firm Cournot equilibrium in a market for a single homogeneous commodity. It is well known that Cournot equilibrium may exist when the demand is not too convex. In such a case, we demonstrate that a larger market size leads to a larger price-cost margin. So, the competitive outcome ceases to be the limit case of Cournot competition when the number of competitors increases. As Szidarovszky and Yakowitz (1977) stated, convex costs and concave (inverse) demand are sufficient conditions for having an equilibrium. However, the assumption of concave demand has been removed by McManus (1964) for the symmetric case and Novshek (1985) for the asymmetric case. The inverse demand's degree of curvature determines the slope of the reaction function. With convex demand, the products may be strategic complements so the reaction functions are upward sloping. If the demand is sufficiently convex, less market power implies a higher price. We consider the market of a single homogeneous commodity. The global inverse demand P(Y) is twice continuously differentiable and convex, so P ' ( Y ) < 0 , P"(Y) > 0 . Let us denote Y = y + ¢, so that total output is the sum of the production of one firm and that of all the others. Under the Cournot-Nash assumption, each firm chooses its quantity setting and takes as given the quantities ot" the others. Thus it maximizes the following profit function: ¢r(y, y ) = P ( y + y ) . y - ¢ ' y . For simplicity the costs are assumed to be continuous, increasing and linear. The first-order condition is: p ( y + y ) = y . P ' ( y + )7) - c = 0 *Tel.: +33 3 21 99 41 50; lax: +33 3 21 99 41 52. 0165-1765/97/$17.00 © 1997 Published by Elsevier Science S.A. All rights reserved PIi S0165- i 765(97)00023-2

(i)

S. Svizzero / Econmnics Letters 54 (1997) 155-158

156

and it defines the reaction function y = R(f). Eq. (1) possesses a maximum if the profit function is concave. From Eq. (i) it follows: 2" P ' ( y + ~) + y . P"(y + .~) < 0

(2)

Let us assume identical costs between firms. Thus the equilibrium is symmetric, we get Y = n-y, with n the number of competitors. Hence, Eq. (2) may be rewritten as:

y(Y)
0 or equivalently when n < y(Y), and so are the quantities satisfying the proposition. With strategic complements, the reaction functions are upward sloping. From Eq. (I) and using the implicit function theorem, the slope of the reaction curve y = R(y) is

y(Y)

-

n

2"n - y(Y) At a symmetric equilibrium we must have

Y "

n-

I

the right term of this equality being the reaction function of the ( n - 1) other firms. Two cases arise, depending on the slopes of the reaction functions. When the deniand is not too convex, we get

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S. Svizzero I Economics Letters 54 (1997) 155-158

Y Y- nY_I

Fig. I. Quasi-competitiveness with strategic components.

7(Y)- n 2"n - 7(Y)

1 n-

I

or equivalently l + n > ? ( Y ) . Thus, the competitive equilibrium is the limit case of Cournot competition even with strategic complements (see Fig. 1). When the demand is sufficiently convex we have 1 + n < ? ( Y ) , and so the proposition is verified (see Fig. 2). Therefore, Cournot equilibrium may still exist even with convex demand. However, it may not be viewed as an intermediary case between monopoly and perfect competition. Y

! I

_

Y

=y

Fig. 2. Non-quasi-competitiveness with strategic components.

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S. St'izzero / Ecom,mk's Letters 54 (1997) 155-158

References Bulow, J., Geanakoplos, J., Klemperer, P., 1985. Multimarket oligopoly: strategic substitutes and complements. Journal of Political Economy 93 (3), 488-51 I. McManus, M., 1964. Equilibrium, numbers and size in Cournot oligopoly. Yorkshire Bulletin of Social and Economic Research 16, 68-75. Novshek, W., 1985. On the existence of Cournot equilibrium. Review of Economic Studies 52, 85-98. Szidarovszky, F., Yakowitz, S., 1977. A new proof of the existence and uniqueness of the Cournot equilibrium. International Economic Review 18, 787-789.