2. Consider 2 firms selling fertilizer competing as Cournot duopolists. The inve
ID: 1119920 • Letter: 2
Question
2. Consider 2 firms selling fertilizer competing as Cournot duopolists. The inverse demand function facing the fertilizer market is P = 1-Q, where Q = qA + qB. For simplicity, assume that the long run marginal cost for each firm is equal to ¼, i.e. c(q)=%q for each firm. a) Find the Cournot Nash equilibrium where the firms choose output simultaneously. b) Depict the outcome of (a) graphically with a diagram of the reaction fiunctions. Identify the Nash equilibrium, and clearly label your graph c) Find the Stackelberg Nash Equilibrium where firm A as the Stackelberg leader.Explanation / Answer
Given inverse demand function, P = 1- Q, Q= qA + qB
And c (q) = 1/4q.
a. In cournot competition, firms compete in outputs simultaneously. Hence the objective function of firm A is to maximize profit, A
Max. A = P qA – c(qA)
= (1- qA - qB) qA – 1/4 qA
= qA - qA2 - qA qB - 1/4 qA
= 3/4 qA - qA2 - qA qB
Differentiating with respect to qA,
dA/ dqA = ¾ - 2 qA - qB
Putting dA/ dqA = 0 by first order condition,
¾ - 2 qA - qB = 0
qA = ½(3/4 - qB) .......................................(1)
Similarly objective function of firm B is to maximize profit, B
Max. B = P qB – c(qB)
= (1- qA - qB) qB – 1/4 qB
= qB – qB2 - qA qB - 1/4 qB
= 3/4 qB – qB2 - qA qB
Differentiating with respect to qB,
dB/ dqB = 3/4 – 2qB – qA
Putting dB/ dqB = 0 by first order condition,
¾ - 2 qB – qA = 0
qB = ½(3/4 – qA) .......................................(2)
= ½ (3/4 - ½(3/4 - qB)) Substituting from equation (1)
= ½(3/4 -3/8 + 1/2 qB)
= 3/8 -3/16 + 1/4 qB
3/4 qB = 3/16
qB* = ¼
qA* = ½(3/4 – 1/4)
= 1/4
C. Under stackelberg competiton firms compete in outputs where one firm is a leader and the other is a follower. The follower firm B cannot observe firm A’s output and maximizes it’s profit function. Firm A being the leader can observe firm B’s reaction function and use it to maximize his own profit. Hence the problem under stackelberg is,
Max. B = P qB – c(qB)
= (1- qA - qB) qB – 1/4 qB
= qB – qB2 - qA qB - 1/4 qB
= 3/4 qB – qB2 - qA qB
Differentiating with respect to qB,
dB/ dqB = ¾ – 2qB - qA
Putting dB/ dqB = 0 by first order condition,
qB = ½ (3/4 – qA) .......................................(2)
The leader firm A now maximizes it’s profit,
Max. A = P qA – c(qA)
= (1- qA - ½ (3/4 – qA)) qA – 1/4 qA
= (qA - qA2 – 3/8 qA + ½ qA2) - 1/4 qA
= 3/8 qA - ½ qA2
Differentiating with respect to qA,
dA/ dqA = 3/8 - qA
Putting dA/ dqA = 0 by first order condition,
qA* = 3/8
qB* = ½ (3/4 – 3/8)
= 3/16
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