Question

Two firms compete by choosing price. Their demand functions are Q1 = 20 - P1 + P2 and Q2 = 20 +P1 -P2 where P1 and P2 are the prices charged by each firm, respectively, and Q1 and Q2 are the resulting demands. Note that the demand for each good depends only on the difference in prices; if the two firms colluded and set the same price, they could make that price as high as they wanted, and earn infinite profits. Marginal costs are 2.

a. Suppose the two firms set their prices at the same time. Find the resulting Nash equilibrium. What price will each firm charge, how much will it sell, and what will its profit be? (Hint: Maximize the profit of each firm with respect to its price.) To determine the Nash equilibrium in prices, first calculate the reaction function for each firm, then solve for price. With zero marginal cost, profit for Firm 1 is:

Please show all calculations! I need help with understanding how marginal costs fit into all of this as well. Thank you so much!

Answer 1

Bertrand competitors compete for prices. Given that marginal costs are 2, total cost is 2Q assuming FC = 0, Profit functions are

?1 = p1q1 – c1

= (20 – p1 + p2)p1 – 2(20 – p1 + p2)

= 20p1 – p1^2 + p1p2 – 40 + 2p1 – 2p2

= 22p1 – p1^2 + p1p2 – 40 – 2p2

?2 = p2q2 – c2

= (20 – p2 + p1)p2 – 2(20 – p2 + p1)

= 20p2 – p2^2 + p1p2 – 40 + 2p2 – 2p1

= 22p2 – p2^2 + p1p2 – 40 – 2p1

Profit is maximized for prices

?1’(p1) = 0  

22 – 2p1 + p2 = 0 or p1 = 11 + 0.5p2

?2’(p2) = 0

22 – 2p2 + p1 = 0 or p2 = 11 + 0.5p1

Solve these BRFs to get

p1 = 11 + 0.5*(11 + 0.5p1)

p1 = 16.5 + 0.25p1

p1 = 22

p2 = 11 + 0.5*22 = 22

q1 = 20

q2 = 20

profits are

?1 = ?2 = 20*22 – 2*20 = 400