21 Cournot

For more information on these topics, see Allen, Doherty, Weigelt, and Mansfield Chapter 11: Oligopoly.

21.1 Classwork 21: Cournot Competition

Last chapter, we started learning about oligopolies. We learned that if firms collude by restricting output like a monopolist would, they can raise prices and maximize joint profits. But cartels are unstable because each firm has an incentive to cheat by lowering prices or increasing output. On top of being unstable, cartels are also usually illegal. Then we learned about price leadership: if there’s one dominant firm and many small firms, the dominant firm uses price leadership to maximize their profit, understanding that the small firms can sell as much as they want at the price that the dominant firm chooses.

In this chapter and the next, we will continue our study of oligopolies. This chapter focuses on Cournot competition, where firms in an oligopoly make simultaneous decisions about quantities rather than prices. Compared to price competition—which drive prices down to marginal cost—Cournot competition is less aggressive. This model helps us understand how prices decrease as the number of firms in an oligopoly increases.

Next chapter, we’ll learn about Stackelberg Competition, where firms make sequential rather than simultaneous moves. A leader firm moves first and commits to a quantity, and a follower firm observes the leader’s choice and then decides their quantity.

Cournot Competition Basics

In Cournot Competition, firms compete by choosing output quantities simultaneously. Think of it like two gas stations on opposite corners. They could engage in a price war, continuously undercutting each other until neither makes any profit. Or they could each choose their capacity (like how many pumps to install) and then let the market price emerge based on total supply. The capacity decision creates a form of commitment that helps avoid the race to the bottom on price.

Example

Let market demand be given by $P = 100 - Q$ where $Q = Q_1 + Q_2$. For simplicity, assume that both firms have zero costs. Firms 1 and 2 choose $Q_1$ and $Q_2$ simultaneously, with each firm thinking: “What quantity should I produce to maximize my profits, given what my rival might produce?”

For Firm 1:

\[\begin{align} \text{Revenue } &= P \times Q_1\\ &= (100 - Q) Q_1\\ &= (100 - Q_1 - Q_2) Q_1\\ &= 100Q_1 - Q_1^2 - Q_1Q_2\\ \end{align}\]

Because $TC = 0$, to maximize profit, set the derivative with respect to $Q_1$ equal to 0:

\[\begin{align} \frac{\partial \pi}{\partial Q_1} = 0 &= 100 - 2Q_1 - Q_2 \\ 2Q_1 &= 100 - Q_2\\ Q_1 &= 50 - \frac{1}{2} Q_2 \end{align}\]

This gives Firm 1’s “reaction function”: $Q_1 = 50 - \frac{1}{2} Q_2$.

By symmetry, Firm 2 has the same reaction function: $Q_2 = 50 - \frac{1}{2} Q_1$.

The equilibrium occurs where these reaction functions intersect (or by symmetry, $Q_1 = Q_2$):

\[\begin{align} Q_1 = 50 - \frac{1}{2} Q_2 &= 100 - 2Q_2\\ 100 - Q_2 &= 200 - 4Q_2\\ 3Q_2 &= 100\\ Q_2 &= 33.33 \end{align}\]

So total output is $33.33 + 33.33 = 66.67$, and $P = 100 - Q = 100 - 66.67 = 33.33$. Each firm makes $PQ_1 = 33.33^2 = 1,111$ in profit.

The key to understanding Cournot competition is the reaction function - it tells you the profit-maximizing quantity for each firm given what their rival produces. If the firm knows their rival’s cost function, they also know their rival’s reaction function, which tells them all they need to know to select the profit maximizing amount to supply.

Practice Problems

1) Basic Cournot Competition

Let market demand be given by $P = 80 - Q$ and assume there are only two firms, both with a total cost function of $TC = 20Q_1$ or $TC = 20Q_2$. Find each firm’s reaction functions and find their intersection. Show that $Q = 40$, $P = 40$, and show that each firm makes $400 in profit.

2) Comparing Cournot to Bertrand

Price competition is also called Bertrand Competition. Under Bertrand (price) Competition, firms undercut each other’s price to capture a larger market share, driving the price down to marginal cost. Take the same demand and cost functions as in the previous problem and:

Calculate the market price $P$, market supply $Q$, and profit to each firm $\pi$.
Complete the sentence: Compared to Bertrand Competition, under Cournot Competition, firms make (less/more) profit, prices are (lower/higher), and firms supply (less/more).

3) Comparing Cournot, Bertrand, and Monopoly/Perfect Collusion

Take the same demand and cost functions from problem 1 and assume firms 1 and 2 are able to perfectly collude, splitting monopoly profits evenly. Calculate the market price $P$, market supply $Q$, and profit to each firm $\pi$. Then fill in the blanks:

Prices are lowest under (Cournot/Bertrand/Perfect Collusion)
Quantity supplied is highest under (Cournot/Bertrand/Perfect Collusion)
Firm profit is highest under (Cournot/Bertrand/Perfect Collusion)

4) Cournot with Asymmetric Costs

Let market demand be given by $P = 100 - Q$. Suppose Firm 1 has $TC = 20Q$ and Firm 2 has $TC = 30Q$. Calculate their reaction functions and find the intersection. Under Cournot Competition, how much does each firm supply?