Question

1. Consider a Stackleberg game of quantity competition between two firms. Firm 1 is the leader and Firm 2is the follower. Market demand is described by the inverse demand function P=1000-4Q. Each firm has a constant unit cost of production equal to 20.

1. Solve for the Nash equilibrium outcome in quantities in this sequential game. What is the equilibrium price? What are the profits for each firm?
2. Suppose that firm 2’s unit cost of production is c < 20. Explain and show why there is no non-negative value of “c” that would allow for both firms, leader and follower, to have the same market share

Answer 1

a. The leader acts as a monopolist and maximizes its profits not considering the reaction of firm 2. Firm 1's optimization is:

Taking the derivative with respect to Q and setting it equal to zero:

Firm 2 decides its quantity taking into consideration firm 1's quantity:

Maximizing profit with respect to firm 2's output:

Therefore, the best response function for firm 2 is:

Since quantity produced by firm 1 is less than 245, firm 2 will produce:

At these quantities, market price is:

Profits of both firms are:

b. Now, firm 2's marginal cost is c. Firm 2's profit is maximized as follows:

For both the firms to have the same market share,

Hence, marginal cost of firm 2 will have to be -470 for the two firms to have the same market share. Therefore, there is no non negative value of c that would allow for both firms to have the same market share.