Problem 1

Suppose there are two firms in an industry, X and Y. Demand for each firm's product is, respectively:

Q^D_X = 90 - 3P_X + 2P_Y

Q^D_Y = 90 - 3P_Y + 2P_X

Both firms also face a constant marginal cost of 10 per unit: MC_X = MC_Y = 10, and there are no fixed costs for either firm.

Using the example above as a guide, find the equations that characterize the "best responses" for each firm, expressing each firm's optimal price in terms of the rival's price:

Firm X's best response:	P*X =
Firm Y's best response:	P*Y =

Now find the numerical values of Nash equilibrium prices, which is characterized by all firms playing best responses simultaneously:

Firm X's equilibrium price:	P*X =
Firm Y's equilibrium price:	P*Y =

Problem 2

This problem is similar to the previous one, but instead of two firms, there are now three firms in the industry. Suppose the three firms in the industry are X, Y, and Z. Demand for each firm's product is, respectively:

Q^D_X = 90 - 3P_X + P_Y + P_Z

Q^D_Y = 90 - 3P_Y + P_X+ P_Z

Q^D_Z = 90 - 3P_Z + P_X+ P_Y

All firms also face a constant marginal cost of 10 per unit: MC_X = MC_Y = MC_Z = 10, and there are no fixed costs for any of the firms.

Again, find the equations that characterize the "best responses" for each firm, expressing each firm's optimal price in terms of the rivals' prices:

Firm X's best response:	P*X =
Firm Y's best response:	P*Y =
Firm Z's best response:	P*Z =

Now find the numerical values of Nash Equilibrium prices, which is characterized by all firms playing best responses simultaneously:

Firm X's equilibrium price:	P*X =
Firm Y's equilibrium price:	P*Y =
Firm Z's equilibrium price:	P*Z =