Question

The players are two firms in a duopoly, and a set of consumers. The two firms produce a homogeneous good. The firms simultaneously choose their prices. Demand adjusts instantaneously according to the equation Q = 6 − p, Each firm has constant costs per unit of output. Firm 1’s cost per unit is 1, and firm 2’s cost per unit is 2. The firms’ payoffs are their profits. If the two firms’ prices are not equal, consumers will buy, according to the demand function, from the firm with the lower price. If the two firms’ prices are equal, consumers are indifferent between buying from one or the other firm (according to the demand function). a. Write the payoff functions for each firm. b. Find all Nash equilibria in pure strategies.

Answer 1

Q = 6 - p
So, p = 6 - Q
So, p = 6 - (q1 + q2)
So, p = 6 - q1 - q2
where q1 is firm 1's output, and q2 is firm 2's output.

Total revenue of firm 1, TR1 = p*q1 = (6 - q1 - q2)*q1 = 6q1 - q1² - q1q2
Total revenue of firm 2, TR2 = p*q2 = (6 - q1 - q2)*q2 = 6q2 - q1q2 - q2²

Total cost of firm 1, TC1 = 1q1 = q1
Total cost of firm 2, TC2 = 2q2

a. Profit function of firm 1, P1 = TR1 - TC1 = 6q1 - q1² - q1q2 - q1 = 5q1 - q1² - q1q2

ӘР - = 5 — 2q1 — q2 = 0 дqli

So, 2q1 = 5 - q2
So, q1 = (5 - q2)/2
So, q1 = 2.5 - 0.5q2
This is the payoff function of firm 1.

Profit function of firm 2, P2 = TR2 - TC2 = 6q2 - q1q2 - q2² - 2q2 = 4q2 - q1q2 - q2²

OP2 5 = 4- q1 - 2q2 = 0

So, 2q2 = 4 - q1
So, q2 = (4 - q1)/2
So, q2 = 4 - 0.5q1
This is the payoff function of firm 2.

b. Substituting q1 in q2 we get,
q2 = 4 - 0.5(2.5 - 0.5q2)
So, q2 = 4 - 1.25 + 0.25q2
So, q2 - 0.25q2 = 0.75q2 = 2.75
So, q2 = 2.75/0.75 = 3.67

q1 = 2.5 - 0.5q2 = 2.5 - 0.5*(3.67) = 2.5 - 1.835 = 0.67

So, q1 = 0.67; q2 = 3.67