## The Cournot Model

We will begin with a simple model of duopoly-two firms competing with each other-first introduced by the French economist Augustin Cournot in 1838. Suppose the firms produce a homogeneous good and know the market demand curve. Each firm must decide how much to produce, and the two firms make their decisions at the same time. When making its production decision, each firm takes its competitor into account. It knows that its competitor is also deciding how much to produce, and the price it receives will depend on the total output of both firms.

The essence of the Cournot model is that each firm treats the output level of its competitor as fixed, and then decides how much to produce. To see how this works/ let's consider the output decision of Firm 1. Suppose Firm 1 thinks that Firm 2 will produce nothing. Then Firm I's demand curve is the market demand curve. In Figure 12.3 this is shown as £h(0), which means the demand curve for Firm 1, assuming Firm 2 produces zero. Figure 12.3 also shows the corresponding marginal revenue curve MRi(O). We have assumed that Firm I's marginal cost MCi is constant. As shown in the figure. Firm I's profit-maximizing output is 50 units, the point where MRi(O) intersects MCi. So if Firm 2 produces zero. Firm 1 should produce 50.

Suppose, instead, that Firm 1 thinks Firm 2 will produce 50 units. Then Firm I's demand curve is the market demand curve shifted to the left by 50. In Figure 12.3 this is labeled £>i(50), and the corresponding marginal revenue curve is labeled MRi(50). Firm I's profit-maximizing output is now 25 units, FIGURE 12.3 Firm I's Output Decision. Firm I's profit-maximizing output depends on how much it thinks Firm 2 will produce. If it thinks Firm 2 will produce nothing, its demand curve, labeled Z)i(0), is the market demand curve. The corresponding marginal revenue curve, labeled MRi(O), intersects Firm I's marginal cost curve MCi at an output of 50 units. If Firm 1 thinks Firm 2 will produce 50 units, its demand curve, £>i(50), is shifted to the left by this amount. Profit maximization now implies an output of 25 units. Finally, if Firm 1 thinks Firm 2 will produce 75 units. Firm 1 will produce only 12.5 units.

the point where MRi(50) = MCi. Now, suppose Firm 1 thinks Firm 2 will produce 75 units. Then Firm I's demand curve is the market demand curve shifted to the left by 75. It is labeled Dt(75) in Figure 12.3, and the corresponding marginal revenue curve is labeled MRi(75). Firm I's profit-maximizing output is now 125 units, the point where MRi(75) = MCi. Finally, suppose Firm 1 thinks Firm 2 will produce 100 units. Then Firm I's demand and marginal revenue curves (not shown in the figure) would intersect its marginal cost curve on the vertical axis; if Firm 1 thinks that Firm 2 will produce 100 units or more, it should produce nothing.

To summarize: If Firm 1 thinks Firm 2 will produce nothing, it will produce 50; if it thinks Firm 2 will produce 50, it will produce 25; if it thinks Firm 2 will produce 75, it will produce 125; and if it thinks Firm 2 will produce 100, then it will produce nothing. Firm 1 's profit-maximizing output is thus a decreasing schedule of how much it thinks Firm 2 will produce. We call this schedule Firm I's reaction curve and denote it by Q*\(Qi). This curve is plotted in Figure 12.4, where each of the four output combinations we found above is shown as an x FIGURE 12.4 Reaction Curves and Cournot Equilibrium. Firm I's reaction curve shows how much it will produce as a function of how much it thinks Firm 2 will produce. (The xs, at Qi = 0,50, and 75, correspond to the examples shown in Figure 123.) Firm 2's reaction curve shows its output as a function of how much it thinks Firm 1 will produce. In Cournot equilibrium, each firm correctly assumes how much its competitor will produce,and thereby maximizes its own profits. Therefore, neither firm will move from this equilibrium.

We can go through the same kind of analysis for Firm 2 (i.e., determine Firm 2's profit-maximizing quantity given various assumptions about how much Firm 1 will produce). The result will be a reaction curve for Firm 2, i.e., a schedule Q*2(Qi) that relates its output to the output it thinks Firm 1 will produce. If Firm 2's marginal cost curve is different from that of Firm 1, its reaction curve will also differ in form from that of Firm 1. For example. Firm 2's reaction curve might look like the one drawn in Figure 12.4.

How much will each firm produce? Each firm's reaction curve tells it how much to produce, given the output of its competitor. In equilibrium, each firm sets output according to its own reaction curve, so the equilibrium output levels are found at the intersection of the two reaction curves. We call the resulting set of output levels a Cournot equilibrium. In this equilibrium, each firm correctly assumes how much its competitor will produce, and it maximizes its profit accordingly.

Note that this Cournot equilibrium is an example of a Nash equilibrium.2 Remember that in a Nash equilibrium, each firm is doing the best it can given what its competitors are doing. As a result, no firm has any incentive to change its behavior. In the Cournot equilibrium, each duopolist is producing an amount that maximizes its profit given what its competitor is producing, so neither duopolist has any incentive to change its output.

Suppose the firms are initially producing output levels that differ from the Cournot equilibrium.-Will they adjust their outputs until the Cournot equilibrium is reached? Unfortunately, the Cournot model says nothing about the dynamics of the adjustment process. In fact, during any adjustment process, the model's central assumption that each firm can assume that its competitor's output is .fixed would not hold. Neither firm's output would be fixed, because both firms would be adjusting their outputs. We need different models to understand dynamic adjustment, and we will examine some in Chapter 13.

When is it rational for each firm to assume that its competitor's output is fixed? It is rational if the two firms are choosing their outputs only once because then their outputs cannot change. It is also rational once they are in the Cournot equilibrium because then neither firm would have any incentive to change its output. When using the Cournot model, we must therefore confine ourselves to the behavior of firms in equilibrium. 