Question

Consider a market with n ≥ 2 firms, where the demand for the product of firm i is given by qi = a − bpi + c times summation of pj, where a, b > 0 while c can be either positive or negative. Firms compete in prices.

1. Suppose that all the firms have the same constant marginal costs. Find the equilibrium. How does it change with c and n? Under what condition(s) does this equilibrium exist?

2. Suppose now that the firms have constant but di§erent marginal costs. Find the equilib- rium. What are the conditions that determine the number of firms in the market? Do the most e¢cient firms operate in the market, or are there equilibria in which less e¢cient firms operate while some more e¢cient ones do not?

Answer 1

In Stackelberg model where firm 1 is a first mover, it must take the reaction function of firm 2 in its computation of marginal revenue.

Derivation of firm 2’s reaction function

Total revenue of firm 2 = P*(q2) = (42 – 2(q1 + q2))q2 = 42q2 – 2q22 – 2q1q2

Marginal revenue = 42 – 4q2 – 2q1

Marginal cost = 4

Solve for the reaction function

42 – 4q2 – 2q1 = 4

4q2 = 38 – 2q1

q2 = (9.5 – 0.5q1)

Incorporate this in the reaction function of firm 2

Total revenue for firm 2 = P*(q1) = (42 – 2(q1 + q2))q1

= (42 – 2q1 – 2(9.5 – 0.5q1))q1

= (23 – q1)q1

MR = 23 – 2q1

Equate MR = MC

23 – 2q1 = 9

2q1 = 14

q1 = 7

q2 = 9.5 – 0.5*7 = 6

Price = 42 – 2(6 + 7) = $16

Profit for firm 1 = 16*7 – 9*7 = $49

Profit for firm 2 = 6*16 – 4*6 = $60

Firm 2 is better off as its profits are higher.