Question

Two firms, A and B, compete as duopolists in an industry. The firms produce a homogeneous good. Each firm has a cost function given by: C(q) = 30q + 1.5q² The (inverse) market demand for the product can be written as: P = 300 − 3Q , where Q = q1 + q2, total output.

Question: If each firm acts to maximize its profits, taking its rival’s output as given (i.e., the firms behave as Cournot oligopolists), what will be the equilibrium quantities selected by each firm? What is total output, and what is the market price? What are the profits for each firm?

Answer 1

Two firms, A and B, compete as duopolists in an industry. The firms produce a homogeneous good. Each firm has a cost function given by:

The (inverse) market demand for the product can be written as: P = 300 − 3Q , where Q = q1 + q2, total output. The firms play a Cournot Duopoly game here.

Now we will answer the following questions.

First we will write down the profit functions of firm A and firm B. Then will differentiate them with respect to their productions to get the Reaction Functions. The calculations are shown below.

From the above calculations we can see the reaction functions are,

RF1: 3q1+q2 = 90........(1) and

RF2: q1+3q2 = 90........(2).

Adding the equations we get,

4q1+4q2 = 180

or, q1+q2 = 45.........(3)

Hence, the total quantity produced by the firms

Q* = q1+q2 = 45

Now, we will solve the equations (1) and (2) by subtracting them from equation (3) one by one. After solving the equations we get,

q1* = 45/2 = 22.5 and

q2* = 45/2 = 22.5

Hence each firm will produce 22.5 units of output.

The total output is 45 units.

From the demand curve we see

P = 300-3Q. Here Q*= 45.

Hence, P* = 300-3×45 = $165

Hence, the market price is $165.

Now, the total costs of the firms are,

or, C(q1) = 1434.375

This is same for firm 2 as q1*=q2*=22.5.

Hence, C(q2) = 1434.375

Now, the profit of firm A

π1* = P*.q1* - C(q1)

​ = 165×22.5 - 1434.375

= $2278.125

The profit is same for firm B as the profit function is same for firm B.

Hence,

π2* = $2278.125

Hence, the profit of both firm A and firm B is $2278.125.

Hope the solution is clear to you my friend.