Cournot Duopoly

Duopolies

We will begin our discussion with an investigation of duopolies. For the following duopoly examples, we will assume the following:

The two firms produce homogeneous and indistinguishable goods.

There are no other firms in the market who produce the same or substitute goods.

No other firms can or will enter the market.

Collusive behavior is prohibited. Firms cannot act together to form a cartel.

There exists one market for the produced goods.

In 1838, Augustin Cournot introduced a simple model of duopolies that remains the standard model for oligopolistic competition. In addition to the assumptions stated above, the Cournot duopoly model relies on the following:

Each firm chooses a quantity to produce.

All firms make this choice simultaneously.

The model is restricted to a one-stage game. Firms choose their quantities only once.

The cost structures of the firms are public information.

In the Cournot model, the strategic variable is the output quantity. Each firm decides how much of a good to produce. Both firms know the market demand curve, and each firm knows the cost structures of the other firm. The essence of the model is this: each firm takes the other firm's choice of output level as fixed and then sets its own production quantities.

The best way to explain the Cournot model is by walking through examples. Before we begin, we will define the reaction curve, the key to understanding the Cournot model (and elementary game theory as well).

A reaction curve for Firm 1 is a function Q1*() that takes as input the quantity produced by Firm 2 and returns the optimal output for Firm 1 given Firm 2's production decisions. In other words, Q1*(Q2) is Firm 1's best response to Firm 2's choice of Q2. Likewise, Q2*(Q1) is Firm 2's best response to Firm 1's choice of Q1.

Let's assume the two firms face a single market demand curve as follows:

Q = 100 - P

where P is the single market price and Q is the total quantity of output in the market. For simplicity's sake, let's assume that both firms face cost structures as follows:

MC_1 = 10

MC_2 = 12

Given this market demand curve and cost structure, we want to find the reaction curve for Firm 1. In the Cournot model, we assume Q2 is fixed and proceed. Firm 1's reaction curve will satisfy its profit maximizing condition, MR = MC. In order to find Firm 1's marginal revenue, we first determine its total revenue, which can be described as follows

TR = P * Q1 = (100 - Q) * Q1= (100 - (Q1 + Q2)) * Q1= 100Q1 - Q1 ^ 2 - Q2 * Q1

The marginal revenue is simply the first derivative of the total revenue with respect to Q1 (recall that we assume Q2 is fixed). The marginal revenue for Firm 1 is thus:

MR1 = 100 - 2 * Q1 - Q2\

Imposing the profit maximizing condition of MR = MC, we conclude that Firm 1's reaction curve is:

100 - 2 * Q1* - Q2 = 10 => Q1* = 45 - Q2/2

That is, for every choice of Q2, Q1* is Firm 1's optimal choice of output. We can perform analogous analysis for Firm 2 (which differs only in that its marginal costs are 12 rather than 10) to determine its reaction curve, but we leave the process as a simple exercise for the reader. We find Firm 2's reaction curve to be:

Q2* = 44 - Q1/2

The solution to the Cournot model lies at the intersection of the two reaction curves. We solve now for Q1*. Note that we substitute Q2* for Q2 because we are looking for a point which lies on Firm 2's reaction curve as well.

Q1* = 45 - Q2*/2 = 45 - (44 - Q1*/2)/2 = 45 - 22 + Q1*/4 = 23 + Q1*/4 => Q1* = 92/3

By the same logic, we find:

Q2* = 86/3

Again, we leave the actual computation of Q2* as an exercise for the reader. Note that Q1* and Q2* differ due to the difference in marginal costs. In a perfectly competitive market, only firms with the lowest marginal cost would survive. In this case, however, Firm 2 still produces a significant quantity of goods, even though its marginal cost is 20% higher than Firm 1's.

An equilibrium cannot occur at a point not in the intersection of the two reaction curves. If such an equilibrium existed, at least one firm would not be on its reaction curve and would therefore not be playing its optimal strategy. It has incentive to move elsewhere, thus invalidating the equilibrium.

The Cournot equilibrium is a best response made in reaction to a best response and, by definition, is therefore a Nash equilibrium. Unfortunately, the Cournot model does not describe the dynamics behind reaching equilibrium from a non-equilibrium state. If the two firms began out of equilibrium, at least one would have an incentive to move, thus violating our assumption that the quantities chosen are fixed. Rest assured that for the examples we have seen, the firms would tend towards equilibrium. However, we would require more advanced mathematics to adequately model this movement.

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