Question

Consider an oligopolist market with demand: Q = 18 – P. There are two firms A and B. The cost function of each firm is given by C(q) = 8 + 6q. The firms compete by simultaneously choosing quantities.

a. Write down firm A’s profit function and derive firm A’s reaction function.

b. Plot the reaction functions of both firms in a diagram.

c. What is the optimal quantity produced by firm A and firm B?

d. Now suppose firm B invests in a technology which doubles its fixed costs and lowers its marginal costs by half. Find Firm A and Firm B’s new optimal quantities and profits

Answer 1

Market inverse demand function:

Total cost function for each firm is given as:

Simultaneous quantity setting by two firms: Cournot duopoly model

a)

Profit function for A (assuming B sticks to its quantity selected)

where is the expected fixed quantity already set by B

Firm A's reaction firm is to choose optimal quantity production which maximizes firm A's profit for any level of quantity chosen by firm B.

Best response function is calculated as:

b)

Due to the symmetry of cost functions:

Best response function for firm B is given by:

Both of these response functions are plotted as:

c)

Optimal quantity produced by firm A and firm B in the equilibrium is given as intersection of their best response functions-

As can be seen from the intersection in graph,

Numerically also it can be found by solving the best response functions-

Plugging it back into BR for firm B:

d)

Now suppose firm B invests in a technology which doubles its fixed costs and lowers its marginal costs by half.

Hence total cost function for firm B is:

Best response function for firm A remains same. Let us calculate best response function for firm B.

Profit function for B (assuming A sticks to its quantity selected)

Best response function is calculated as:

Solving the best response functions, we get:

Plugging it back into BR for firm B:

This makes sense since firm B has now improved technology which less marginal cost per unit production, thus quantity produced is more.

Optimal profits are calculated using profit function: