Question

3. Consider an oligopoly in which firms choose quantities. The inverse market demand curve is given by P = 280 - 2(X + Y ), where X is the quantity of Firm 1, and Y is the quantity of Firm 2. Each firm has a marginal cost equal to 40. What is the Stackelberg equilibrium, when Firm 1 acts as the leader? What is the market price at the Stackelberg equilibrium? What is the profit of each firm?

Answer 1

In Stackelberg model where 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

Total revenue of firm 2 = P*(Y) = (280 – 2(X + Y))Y = 280Y – 2Y² – 2XY

Marginal revenue = 280 – 4Y – 2X

Marginal cost = 40

Solve for the reaction function

280 – 4Y – 2X = 40

240 – 2X = 4Y

This gives Y = 60 – 0.5X

Incorporate this in the reaction function of firm 1

Total revenue for firm 1 = P*(X) = (280 – 2(X + Y))X

TR = 280X – 2X^2 – 2XY

= 280X – 2X^2 – 2X*(60 – 0.5X)

= 280X – 2X^2 – 120X + X^2

= 160X – X^2

MR = MC

160 – 2X = 40

X = 60 and so Y =  60 – 0.5*60 = 30 units.

Price = 280 - 2(60 + 30) = $100

Hence, Stackelberg equilibrium, when Firm 1 acts as the leader, is X = 60 units and Y = 30 units.

Market price at the Stackelberg equilibrium is $100 per unit

Profit (leader) = 100*60 - 40*60 = $3600

Profit (follower) = 100*30 - 40*30 = $1800