Solve for equilibrium prices in the following differentiated-product Bertrand
model. (20 points, 4 points each)
Q1 = 300 – 12P1 + 4P2 + 3P3
Q2 = 275 – 10P2 + 2P1 + P3
Q3 = 250 – 8P3 + 2P1 + P2
Assume that each firm has a marginal cost of 10.
a. Write down each firm’s profit function.
b. Write down the profit-maximization conditions.
c. Use an equation solver to get the equilibrium prices.
d. Insert the computed equilibrium value of P1 into the best-response functions
for P2 and P3. Then, show the equilibrium for P2 and P3 graphically.
e. Suppose that Firm 1’s marginal cost rises from 10 to 15, but Firm 2 and
Firm 3’s marginal costs remain at 10. Do Firms 2 and 3 undercut Firm 1 and
take all its business away? Explain.