Question

Suppose the natural gas industr y consisted of only two firms. L et these firms have identical cost functions, C(q) = 40q . Assume the demand curve f or the industry is given by P = 100 ? Q and that each firm expects the othe r to behave as a Cournot competi tor. a. Calculate the Cournot-Nash equi librium for each firm, assuming that each chooses the output level that maximizes its pr ofits when taking its rival’s output as given. What are the profits of each firm? b. What would be the equilibrium quantity if Firm 2 had constant m arginal and average costs of $25 and Firm 1 had constant marginal and average costs of $40? c. Assuming that both firms have t he original cost function, C(q) = 40q , how much should Firm 2 be willing to invest to low er its marginal cost from 40 to 25, assuming that Firm 1 will not follow suit? How muc h should Firm 1 be willing to spen d to reduce its marginal cost to 25, assuming that Firm 2 will have marginal costs of 25 regardless of Firm 1’s actions?

Answer 1

Answer:

A)

P = 100-Q1-Q2

Profit = TR-TC = PQ-TC

Firm 1: Profit = 100Q1-Q1²-Q1Q2 – 40Q1 = 60Q1-Q1²-Q1Q2

Firm 2: Profit = 100Q2-Q2²-Q1Q2 – 40Q2 = 60Q2-Q2²-Q1Q2

Calculate reaction function of each firm by derivating each firm’s production function with respect to its output and equate it equal to 0

Firm 1: d(60Q1-Q1²-Q1Q2)/dQ1= 60-2Q1-Q2 = 0

Firm 2: d(60Q2-Q2²-Q1Q2)/dQ2 = 60-2Q2-Q1 = 0

This gives:

Q1* = 30-0.5Q2

Q2* = 30-0.5Q1

Substitute the reaction function equations in to each other to find Cournot output and price

Upon solving, this gives:

Q1 = Q2 = 20

P = 100-Q1-Q2 = 100-20-20 = $60

Profit of each firm: (P-AC)Q1 = (P-AC)Q2 = (60-40)20 = $400

B)

Calculate new reaction functions:

Firm 1: Profit = 100Q1-Q1²-Q1Q2 – 40Q1 = 60Q1-Q1²-Q1Q2

Firm 2: Profit = 100Q2-Q2²-Q1Q2 – 25Q2 = 75Q2-Q2²-Q1Q2

Calculate reaction function of each firm by derivating each firm’s production function with respect to its output and equate it equal to 0

Firm 1: d(60Q1-Q1²-Q1Q2)/dQ1= 60-2Q1-Q2 = 0

Firm 2: d(75Q2-Q2²-Q1Q2)/dQ2 = 75-2Q2-Q1 = 0

This gives:

Q1* = 30-0.5Q2

Q2* = 37.5-0.5Q1

Substitute the reaction function equations in to each other to find Cournot output and price

Upon solving, this gives:

Q1 = 15; Q2 = 30

P = 100-Q1-Q2 = 100-15-30 = $55

Profit of each firm: (P-AC)Q

Firm 1 = (P-AC)Q1 = (55-40)15 = $225

Firm 2 = (P-AC)Q2 = (55-25)30 = $900

C)

Firm 2 should be willing to invest amount equal to increase in profits.

So, Firm 2 will be willing to invest $900-$400 = $500 maximum