Question

Suppose there are only two firms, firm1 and firm2, in the market. They both choose
a quantity to produce simultaneously. The market price is determined by the market
demand:
p =130-(Q1+Q2)
where Q1 is the output quantity of firm1 and Q2 is the output quantity of firm2. Firm1’s
total cost of production is 10Q1 and firm2’s total cost of production is 10Q2. That is, both
firms have a constant marginal cost of 10.

Task 1. What’s firm 1’s best response function? Plot it with Firm 1’s choice on the horizontal axis and firm2’s choice on the vertical axis.

Task 2. What’s firm2’s best response function? Plot it in the figure from Task 2. Clearly
label which curve is which firm’s best response function.

Task 3. What’s the Nash equilibrium outcome? Howmuch is the profit of firm1 at the
Nash equilibrium?

Task 4. If the two firms collude and jointly decide on the total output quantity that
maximizes the joint profit, what is this total quantity? Howmuch profit does firm1 make if
each produces half of this quantity?

Answer 1

A typical firm will maximize profit by equating Marginal revenue with Marginal Cost.

Task 1:

p = 130 - (q1+q2)

Firm 1:

Total Revenue = p*q1 = (130 - (q1+q2))*q1

MR = dTR/dq1 = 130 - 2q1 - q2

MC = 10

MR = MC

130 - 2q1 - q2 = 10

q1 = 65 - 0.5q2

This firm 1's best response function

Task 2:

Total Revenue = p*q2 = (130 - (q1+q2))*q2

MR = dTR/dq1 = 130 - 2q2 - q1

MC = 10

MR = MC

130 - 2q2 - q1 = 10

q2 = 65 - 0.5q1

Task 3:

Put the value of q2 in q1 we get,

q1 = 65 - 0.5(65 - 0.5q1)

q1 = 65 - 32.5 + 0.25q1

0.75q1 = 32.5

q1 = 43.33

q2 = 65 - 0.5q1 = 43.33

P = 130-(43.33+43.33) = 43.34

Firm 1 Profit = TR - TC = 43.33-43.34 - 43.33*10 = 1444.6 = Firm 2 profit.

Task 4:

If the two firms collude, we treat q1+q2 = q

p = 130 - q

Total Revenue = (130-q)*q

MR = 130 - 2q

MC= 10

MR = MC

130 - 2q = 10

2q = 120

q = 60

p = 130 - 60 = 70

Profit = TR - TC = 60*70 = 60*10 = 3600

Each firm's Profit = 3600/2 = 1800