The inverse demand for a homogeneous-product Stackelberg duopoly is P = 20,000 -
ID: 1220992 • Letter: T
Question
The inverse demand for a homogeneous-product Stackelberg duopoly is P = 20,000 -4Q. The cost structures for the leader and the follower, respectively, are CL(QL) = 2,000QL and CF (QF) = 4,000QF.
a.) What is the follower’s reaction function? QF =___?____ - ____?____ QL (Fill in the blank & show equation)
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: $ ?
Explanation / Answer
(a)
Leaders reaction function:
P= 20000 - 4(QL+QF)
P=20000 - 4QL - 4QF
TRL = P x QL = 20000QL - 4QL2 - 4QLQF
MRL = 20000 - 8QL - 4QF
MCL = 2000
For equilibrium
MRL = MCL
20000 - 8QL - 4QF = 2000
8QL = 18000 - 4QF
QL = 2250 - 0.5QF (Leader's reaction function)
Similarly, Followers reaction function:
P= 20000 - 4(QL+QF)
P=20000 - 4QL - 4QF
TRF = P x QF = 20000QF - 4QLQF - 4QF2
MRF = 20000 - 4QL - 8QF
MCF = 4000
For equilibrium
MRF = MCF
20000 - 4QL - 8QF = 4000
8QF = 16000 - 4QL
QF = 2000 - 0.5QL (Leader's reaction function)
(b)
Let, Leader : first mover
Follower's reaction function is: QF = 2000 - 0.5QL
Then, P = 20000 - 4QL - 4QF
= 20000 - 4QL - 4(2000 - 0.5QL)
= 20000 - 4QL - 8000 + 2QL
P = 12000 - 2QL
TRL = 12000QL - 2QL2
MRL = 12000 - 4QL
For equilibrium
MRL = MCL
12000 - 4QL = 2000
4QL = 10000
QL = 10000/4 = 2500 units
QF = 2000 - 0.5QL = 2000 - 0.5(2500) = 750 units
(c)
P= 20000 - 4(QL+QF) = 20000 - 4(2500+750) = $3750
(d)
Leader's profit = P x QL - TC = ($3750 * 2500) - (2000 * 2500) = $4,375,000
Follower's profit = P x QF - TC = ($3750 * 750) - (4000 * 750) = -$187,500
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