#2. The inverse demand for a homogeneous-product Stackelberg duopoly is P =20000
ID: 1203374 • Letter: #
Question
#2. The inverse demand for a homogeneous-product Stackelberg duopoly is
P =20000 – 5QL – 5QF. The cost of the leader and the follower, respectively, are
CL(QL) = 3000QL, and CF(QF) = 3000QF. F stands for “follower” and L stands for “leader”.
What is the follower’s reaction function?
What is the residual demand for the leader?
(c) Determine the equilibrium output level for the leader.
(d) Determine the equilibrium output level for the follower.
(e) Determine the equilibrium market price.
(f) Obtain the profits for the leader and the follower, assuming average total cost is the same as marginal cost. Which one is higher?
Explanation / Answer
The inverse demand function: P = 20000 - 5QL - 5QF
TRL = P x QL = 20000QL - 5QL2 - 5QFQL
MRL = 20000 - 10QL - 5QF
MCL = 3000
For equilibrium
MRL = MCL
20000 - 10QL - 5QF = 3000
=> 10QL = 17000 - 5QF
QL = 1700 - 0.5QF (Leader firm's reaction function)
Similarly, follower firm's reaction function will be:
P = 20000 - 5QL - 5QF
TRF = P x QF = 20000QF - 5QLQF - 5QF2
MRF = 20000 - 5QL - 10QF
MCF = 3000
For equilibrium
MRF = MCF
20000 - 5QL - 10QF = 3000
=> 10QF = 17000 - 5QL
QF = 1700 - 0.5QL (Follower firm's reaction function)
Let, Leader is : first mover
Follower's reaction function is: QF = 1700 - 0.5QL
Then,P = 20000 - 5QL - 5QF
= 20000 - 5QL - 5(1700 - 0.5QL)
= 20000 - 5QL - 8500 + 2.5QL
P = 11500 - 2.5QL
TRL = 11500QL - 2.5QL2
MRL = 11500 - 5QL
For equilibrium
MRL = MCL
11500 - 5QL = 3000
5QL = 8500
QL = 8500/5 = 1700 units
QF = 1700 - 0.5QL = QF = 1700 - 0.5(1700) = 850 units
P = 20000 - 5QL - 5QF = P = 20000 - 5(1700) - 5(850) = $7250
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