Question

QUESTION 1

Two firms compete by choosing price. Their demand functions are Q1 = 20 - P1 +P2 and Q2 = 20 - P2 +P1 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 zero. Suppose the two firms set their prices at the same time. Find the resulting equilibrium. What price will each firm charge, and what will its profit be?

P1=
P2=
Profit1=
Profit2=

0.5 points   

QUESTION 2

Two firms compete by choosing price. Their demand functions are Q1 = 112 - 4P1 + P2 and Q2 = 112 - 4P2 + P1 where P1 and P2 are the prices charged by each firm, respectively, and Q1 and Q2 are the resulting demands. Each firm has a marginal constant produce at a zero constant average and marginal cost (AC = MC = $0). Suppose the two firms set their prices at the same time. 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.)

P1=

P2=

Profit1=

Profit2 =

0.5 points=   

Answer 1

Firm 1:

Q1 = 20 - P1 +P2

MC = 0

Total Revenue TR1 = P1*Q1 = P1(20-P1+P2)

Profit = P1(20-P1+P2)

= 20P1 - P1² + P1P2

d/dP1 = 20 - 2P1 + P2

Set d/dP1 = 0 we get,

2P1 = 20 + P2

P1 = 10 + 0.5P2 ..............1

Firm 2

Q1 = 20 - P2 +P1

MC = 0

Total Revenue TR2 = P2*Q2 = P2(20-P2+P1)

Profit = P2(20-P2+P1)

= 20P2 - P2² + P1P2

d/dP2 = 20 - 2P2 + P1

Set d/dP2 = 0 we get,

2P2 = 20 + P1

P2 = 10 + 0.5P1

Put the value of P1 from equation 1 we get,

P2 = 10 + 0.5(10 + 0.5P2)

P2 = 10 + 5 + 0.25P2

0.75P2 = 15

P2* = 20

P1 = 10 + 0.5P2

P1* = 20

Profit of firm 1: 20P1 - P1² + P1P2 = 20*20 - 20² + 20*20 = 400 = Profit of firm 2

Answer 2:

Firm 1:

Q1 = 112 - 4P1 +P2

MC = 0

Total Revenue TR1 = P1*Q1 = P1(112-4P1+P2)

Profit = P1(112-4P1+P2)

= 1120P1 - 4P1² + P1P2

d/dP1 = 112 - 8P1 + P2

Set d/dP1 = 0 we get,

8P1 = 112 + P2

P1 = 14 + 0.125P2 ..............1

Firm 2

Q2 = 112 - 4P2 +P1

MC = 0

Total Revenue TR2 = P2*Q2 = P2(112-4P2+P1)

Profit = P2(112-4P2+P1)

= 1120P2 - 4P2² + P1P2

d/dP2 = 112 - 8P2 + P1

Set d/dP2 = 0 we get,

8P2 = 112 + P1

P2 = 14 + 0.125P1

Put the value of P1 from equation 1

P2 = 14 + 0.125(14 + 0.125P2)

P2 = 14 + 1.75 + 0.015625P2

0.984375P2 = 15.75

P2 = 16

P1 = 14 + 0.125P2

P1 = 16

Profit of firm 1: 112P1 - 4P1² + P1P2 = 112*16 - 4*16² + 16*16 = 1792 - 1024 + 256 = 1024 = Profit of firm 2