Cournot Model

Cournot Model Assumptions

Let's go over the assumptions in the Cournot model!

• There are several assumptions in Cournot's model:
  1. Firms are rational, and their objective is to maximize their profits;
  2. Firms produce homogeneous products;
  3. Firms compete by setting output quantities;
  4. Firms make decisions simultaneously;
  5. Firms treat their competitor's output as fixed;
  6. There is no cooperation between the firms;
  7. Firms have enough market power such that their output decision can affect the market price.

Keep these at the back of your head, as everything will become more apparent in the next section, where we will look at the model mathematically!

Cournot Model of Duopoly

Let's look at the Cournot model of a duopoly in terms of some mathematical equations and graphs!As economists love to have fun, let's give our firms names: 'The Happy Firm' and 'The Lucky Firm.'We assume that the products that the firms produce are homogeneous. The two firms will decide to set their quantities simultaneously. Each firm will first consider what its competitor would do and then set its own output to maximize its profits.The Happy Firm is thinking about how to tackle this challenge and decides to create a schedule of all the possible quantities that the Lucky Firm could produce.The Happy Firm had plotted a line representing how much output it should produce given the Lucky Firm's decision. This function is called Happy Firm's reaction function in a duopoly.

Imagine that the Lucky Firm goes through the same exercise and finds its reaction function. We can now plot these two reaction functions on one graph, as shown in Figure 1 below.

Fig. 1 - Reaction functions example

Figure 1 above shows the two reaction functions; one for the Happy Firm and one for the Lucky Firm. The two curves have the same form because the two firms in our example are the same. The reaction curves look different because they show one firm's profit-maximizing output given the other firm's output. Where the two reaction functions intersect is known as Cournot equilibrium. Why is this an equilibrium?Think about it more generally from the Nash equilibrium point of view. It is an equilibrium because, at this point, no firm has an incentive to deviate from its strategy. Or in other words, each firm is doing the best it possibly can considering what the other firm is doing.

Suppose the firms initially start producing quantities that differ from the Cournot equilibrium. In that case, the model cannot predict any of the dynamics of quantity adjustments, which is the limitation of this model.

Cournot Model Example

Let's look at an example of a Cournot model with equations and graphs!

Let's revisit our Happy Firm and Lucky Firm. Imagine the market demand curve is:\(P=300-Q=300-(Q_1+Q_2)\)

Where:\(Q=Q_1+Q_2\)\(Q_1 - \hbox{the production of the Happy Firm}\)\(Q_2 - \hbox{the production of the Lucky Firm}\)\(Q - \hbox{the total production of both firms}\)Let's set the marginal costs to zero for simplicity:\(MC_1=MC_2=0\)

How can we find the reaction function of the Happy Firm?Remember the profit-maximizing rule:\(MC=MR\)

We need to find the total revenue of the Happy Firm:

\(TR_1=P\times Q_1=(300-Q)\times Q_1=\)\(=300Q_1-(Q_1+Q_2)Q_1=\)\(=300Q_1-Q_1^2-Q_2Q_1\)Marginal revenue is then the first derivative with respect to Q1:

\(MR_1=\frac{\Delta TR_1}{\Delta Q_1}=300-2Q_1-Q_2\)

We know that:

\(MC_1=0\)

For the profit-maximizing rule to hold:\(MC_1=MR_1=0\)\(MR_1=300-2Q_1-Q_2=0\)

Rearrange to find Q1:\(2Q_1=300-Q_2\)\(Q_1=150-\frac{1}{2}Q_2\) (1)We found the reaction function for the Happy Firm!

We don't need to go over all these calculations for the Lucky Firm as we know that its reaction function is symmetric and is:

\(Q_2=150-\frac{1}{2}Q_1\) (2)

We know that the Cournot equilibrium occurs when the two functions intersect. We can then plug the value of Q2 into the equation for Q1 (1) to get:

\(Q_1=150-\frac{1}{2}\times(150-\frac{1}{2}Q_1)\)\(Q_1=150-75+\frac{1}{4}Q_1\)

\(\frac{3}{4}Q_1=75\)

\(Q_1=100\)

We have found Q1! Now we can plug the value of Q1 into (2):

\(Q_2=150-\frac{1}{2}Q_1=150-\frac{100}{2}=100\)

We have now found Q2 as well!

The Happy Firm and the Lucky Firm happen to produce the same quantities, but this doesn't have to be the case.

The total quantity produced in the market is:

\(Q=Q_1+Q_2=100+100=200\)

We can now find the equilibrium market price from the original demand equation:

\(P=300-Q=300-200=100\)

This means that each of the two firms earns a profit equivalent to their total revenue, as the marginal costs are zero:

\(\pi_1=\pi_2=TR_1=TR_2=(300-Q)\times Q_i=(300-200)\times 100=10,000\)

We can now plot our Cournot equilibrium on a diagram! Take a look at Figure 2 below.

Fig. 2 - Cournot equilibrium example

Figure 2 shows a Cournot equilibrium for the duopoly consisting of the Happy and Lucky firms. Note that this equilibrium occurs at the intersection of the two reaction functions. The reaction function of each firm represents its output given its competitor's output.

Cournot model collusion

Let's imagine for a moment that the two firms decided to collude. How would the Cournot equilibrium look, then? It would be rational for the Happy Firm and the Lucky Firm to maximize their total profits and then split those however they agree.

Recall the market demand equation:

\(P=300-Q\)

The total combined revenue for the two firms is then:

\(TR=P \times Q=(300-Q) \times Q =300Q-Q^2\)

Let's find the marginal revenue of the joint production:

\(MR=\frac{\Delta TR}{\Delta Q}=300-2Q\)

Setting MR equal to zero and solving for Q yields:

\(Q=150\)

Now the two firms can produce whatever quantities they want. Still, to jointly profit-maximize, they need the total quantities to add up to 150.

The Happy Firm and the Lucky Firm owners are friends, so they decide to split the profit evenly. Therefore, they produce the same quantities:

\(Q_1=Q_2=75\)

What is interesting to see is something called a collision curve. A collision curve would show all the possible output combinations that the firms can produce. These outputs would inevitably add up to 150 and thus maximize joint profits.

The equation of the collusion curve is:

\(Q_1=Q_2=75\)

Take a look at Figure 3 below for a visualization.

Fig. 3 - The collision curve

Figure 3 shows the collusion curve in yellow, which has some very important insights. Without cooperation, firms can make less profit and have to produce higher output. With cooperation, they can restrict their joint output and enjoy higher profits.

The amount of profit that the firms were making jointly before cooperation was:

\(\pi_1+\pi_2=10,000+10,000=20,000\)

By colluding, they can enjoy higher profits of:

\(\pi_1+\pi_2=P \times Q = (300-150) \times 150 = 22,500\)

Cournot Model vs. Bertrand Model

What is the difference between the Cournot model vs. the Bertrand model? The main difference is that in the Cournot model, firms compete in quantities. In contrast, in the Bertrand model, firms compete in prices. This has a few significant implications.

Cournot saw a colluding duopoly acting akin to a monopoly in terms of price and quantity setting. In contrast, Bertrand saw price competition in a duopoly leading to a similar outcome as in the perfect competition.

A Cournot equilibrium is stable, and there is no incentive for the two firms to engage in price wars. However, in the Bertrand model, firms are likely to go through a price war, bidding down prices to their marginal costs until no firm has an incentive to deviate. That is, raising the price either above or lowering it below the marginal cost would be worse for the firm. This is an outcome that similarly occurs in the perfect competition model.