Question

- 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.

- Solve for the Nash equilibrium outcome in quantities in this sequential game. What is the equilibrium price? What are the profits for each firm?
- 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.

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