Question

In class we considered the solution to a duopoly model with a linear demand curve P=a-b(q1+q2) and constant marginal cost c. Consider now a market with three firms instead of two. In this case the demand curve becomes P=a-b(q1+q2+q3). Assume all three firms have the same (constant) marginal cost c. (a) State the profit maximization problem for each firm and find the corresponding first order condition for each. (b) Write the 3 first order conditions in (a) as a system of 3 linear equations in 3 unknowns. What are the unknowns? (c) Write the system in (b) in matrix form and use Cramer’s rule to solve for the equilibrium value of . Be sure to simplify your answer as much as possible. (d) Because the firms all have the same costs, all will produce the same quantities in equilibrium. Will each firm produce more or less when there are 3 firms in the market than when there were only two? (e) Will aggregate output be more or less when there are 3 firms in the market than when there were only two?

Answer 1

SOL;

Given P =a - b(q1 + q2 + q3)

Marginal cost = c

a

b

c

d

Each firm will produce less than when there were only two firms.