Cournot Competition - Graphically Finding The Cournot Duopoly Equilibrium

Graphically Finding The Cournot Duopoly Equilibrium

This section presents an analysis of the model with 2 firms and constant marginal cost.

= firm 1 price, = firm 2 price

= firm 1 quantity, = firm 2 quantity

= marginal cost, identical for both firms

Equilibrium prices will be:

This implies that firm 1’s profit is given by

• Calculate firm 1’s residual demand: Suppose firm 1 believes firm 2 is producing quantity . What is firm 1's optimal quantity? Consider the diagram 1. If firm 1 decides not to produce anything, then price is given by . If firm 1 produces then price is given by . More generally, for each quantity that firm 1 might decide to set, price is given by the curve . The curve is called firm 1’s residual demand; it gives all possible combinations of firm 1’s quantity and price for a given value of .

• Determine firm 1’s optimum output: To do this we must find where marginal revenue equals marginal cost. Marginal cost (c) is assumed to be constant. Marginal revenue is a curve - - with twice the slope of and with the same vertical intercept. The point at which the two curves ( and ) intersect corresponds to quantity . Firm 1’s optimum, depends on what it believes firm 2 is doing. To find an equilibrium, we derive firm 1’s optimum for other possible values of . Diagram 2 considers two possible values of . If, then the first firm's residual demand is effectively the market demand, . The optimal solution is for firm 1 to choose the monopoly quantity; ( is monopoly quantity). If firm 2 were to choose the quantity corresponding to perfect competition, such that, then firm 1’s optimum would be to produce nil: . This is the point at which marginal cost intercepts the marginal revenue corresponding to .

• It can be shown that, given the linear demand and constant marginal cost, the function is also linear. Because we have two points, we can draw the entire function, see diagram 3. Note the axis of the graphs has changed, The function is firm 1’s reaction function, it gives firm 1’s optimal choice for each possible choice by firm 2. In other words, it gives firm 1’s choice given what it believes firm 2 is doing.

• The last stage in finding the Cournot equilibrium is to find firm 2’s reaction function. In this case it is symmetrical to firm 1’s as they have the same cost function. The equilibrium is the intersection point of the reaction curves. See diagram 4.

• The prediction of the model is that the firms will choose Nash equilibrium output levels.