Question

The inverse demand for a homogeneous-product Stackelberg duopoly is P = 12,000 -5Q. The cost structures for the leader and the follower, respectively, are C_L(Q_L) = 3,000Q_L and C_F (Q_F) = 6,000Q_F..

a. What is the follower’s reaction function?
Q_F= ____ - _____ Q_L

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

Answer a : P = 12000-5Q

Q =Q_L+Q_F

P = 12000-5 (Q_L+Q_F)

where

TC_L = 3000Q_L

TC _F = 6000Q_F

Profit = TR -TC

Profit of follower = (12000-5Q_L- 5Q_F)QF- 6000Q_F

_{Profit of the follower =} 12000QF -5Q_LQ_F -5Q²_F -6000Q_F

6000-5QL =10QF

Follower reaction curve :

QF = 600-0.5QL

Answer b : Leader should be produced that amount of output in which profit is maximum

Profit maximisation condition of a leader = 12000QL-5QL² -5(600-0.5QL)QL -3000QL

Profit maximisation condition = 6000-9QL

6000=9QL

QL = 6000/9 = 667 units

Putting the value in ereaction function we get

QF = 600-0.5*667 = 267 units

Leader output =667 units

Follower output = 267 units

Answer c : Equiliburm price =

P = 12000-5(QL+QF)

P = 12000-5*934

P = $7330

Answer d : Profit of the follower :

Profit = 6000QF-5QF QL- 5QF²

Puting the value of Ql =667 units anf QF = 267

Profit of the follower = 6000*667-5*667*267-5*(667)²

Profit of the follower = $3,55,555

Where as Profit of the leader = 6000QL - 9*QL²

Profit of the leader = 6000*267- 9*667/2

Profit of the leader =$ 2,000,000 (Approx)

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