Question

2. Suppose there are 2 firms in a market. They face an aggregate demand curve, P=400-.75Q. Each firm has a Cost Function, TC=750+4q (MC=4).

c. Suppose instead that firm A is a Stackelberg leader and gets to choose quantity first. Calculate each firm's best-response function. What is the Nash equilibrium level of production for each firm? What is the equilibrium price? What are the profits of each firm?

Answer 1

Demand function is P = 400 - 0.75(q1 + q2)

In Stackelberg model, firm 1 is a first mover, it must take the reaction function of firm 2 in its computation of marginal revenue.

Derivation of firm 2’s reaction function/best response function

Total revenue of firm 2 = P*(q2) = (400 – 0.75(q1 + q2))q2 = 400q2 – 0.75q2² – 0.75q1q2

Marginal revenue = 400 – 1.50q2 – 0.75q1

Marginal cost = 4

Solve for the reaction function

400 – 1.50q2 – 0.75q1 = 4

396 - 0.75q1 = 1.5q2

This gives q2 = 264 - 0.5q1

Incorporate this in the reaction function of firm 1

Total revenue for firm 1 = P*(q1) = (400 – 0.75(q1 + q2))q1

TR = 400q1 - 0.75q1^2 - 0.75q1q2

= 400q1 - 0.75q1^2 - 0.75q1*(264 - 0.5q1)

= 400q1 - 0.75q1^2 - 198q1 + 0.375q1^2

= 202q1 - 0.375q1^2

MR = MC

202 - 0.75q1 = 4

q1 = 264 and so q2 = 264 - 0.5*264 = 132 units.

(1) Nash equilibrium price,= 400 - 0.75(264 + 132) = $103

(2) Nash equilibrium quantity, of leader = 264 units, of follower = 132 units

(3) Profits of leader = 264*103 - 750 - 264*4 = 25386. Profits of follower = 132*103 - 750 - 132*4 = 12318