Question

Consider a duopolistic market with demand Q(p) = 500 - 50p. Firm 1 produces using a constant marginal cost c1 = 8; firm 2’s marginal cost is c2 = 6 and has a production limit of 25 units. Compute the equilibrium.

Answer 1

Inverse demand is P = 500/50 - Q/50 or P = 10 - 0.02(q1 + q2)

Profit functions are

?1 = 10q1 - 0.02q1^2 - 0.02q1q2 - 8q1 and ?2 = 10q2 - 0.02q1q2 - 0.02q2^2 - 6q2

= 2q1 - 0.02q1^2 - 0.02q1q2 = 4q2 - 0.02q1q2 - 0.02q2^2

Maximize profit to get best response functions

2 - 0.04q1 - 0.02q2 = 0 and 4 - 0.04q2 - 0.02q1 = 0

q1 = 50 - 0.5q2 and q2 = 100 - 0.5q1

Solve them to get

q1 = 50 - 0.5*(100 - 0.5q1)

q1 = 50 - 50 + 0.25q1

q1 = 0.

Then q2 = 100 but it is constrained by q2 = 25 so equilibrium has q2 = 25 and then q1 = 50 - 0.5*25 = 37.5.