Question

The inverse demand for a homogeneous-product Stackelberg duopoly is P = 10,000 -5Q. The cost structures for the leader and the follower, respectively, are CL(QL) = 3,000QL and CF (QF) = 5,000QF..

a. What is the follower’s reaction function?

QF = - QL

b. Determine the equilibrium output level for both the leader and the follower.

Leader output:

Follower output:   

c. Determine the equilibrium market price.

$   

d. Determine the profits of the leader and the follower.

Leader profits: $

Follower profits: $

Answer 1

Solution A

Total revenue (TRF) = P x QF = 10,000QF - 5QFQL - 5QF2

Marginal revenue (MRF) = TRF / QF = 10,000 - 5QL - 10QF

Equating MRF & MCF,

10,000 - 5QL - 10QF = 5,000

10QF = 5,000 - 5QL

QF = 500 - 0.5QL [Reaction function of Follower]

Solution B

The leader will know the reaction function of the follower. For the Leader,

TRL = P x QL = 10,000QL - 5QL2 - 5QLQF = 10,000QL - 5QL2 - 5QL x (500 - 0.5QL)

= 10,000QL - 5QL2 - 2,500QL + 2.5QL2 = 7,500QL - 2.5QL2

MRL = TRL / QL = 7,500 - 5QL

Equating MRL & MCL,

7,500 - 5QL = 3,000

5QL = 4,500

QL = 900

QF = 500 - (0.5 x 900) = 500 - 450 = 50

Solution - C

QL + QF = 900 + 50 = 950

P = 10,000 - (5 x 950) = 10,000 - 4,750 = 5,250

Solution - D

Profit = Q x (P - MC)

Leader profit = QL x (P - MCL) = 900 x (5,250 - 3,000) = 900 x 2,250 = 2,025,000

Follower profit = QF x (P - MCF) = 50 x (5,250 - 5,000) = 50 x 250 = 12,500