Stackelberg Duopoly: Two firms both have the same constant marginal cost of $20

Question

Stackelberg Duopoly:

Two firms both have the same constant marginal cost of $20 0 and zero fixed cost; market price P = 140 2(q1 + q2). Both firms choose outputs to compete.

(a) Find the subgame perfect equilibrium outcome of the Stackelberg Duopoly game with Firm 1 moving first. First, solve for the follower’s (Firm 2’s) best response function. Then solve for the leader’s optimal strategy.

(b) Find the Nash equilibrium of the Cournot duopoly under the same assumptions on costs and demand.

(c) Compare the two equilibria. Discuss the differences.

Explanation / Answer

(a) STACKELBERG

P = 140 – 2Q where Q = q1 + q2

P = 140 – 2q1 – 2q2

So,

Total revenue of firm 1, TR1 = P x q1 = 140q1 – 2q12 – 2q1q2

Total revenue of firm 2, TR2 = P x q2 = 140q2 – 2q1q2 – 2q22

So, Marginal revenue of firm 2, MR2 = dTR2 / dq2 = 140 – 2q1 – 4q2

MC2 = 20

Equating MR2 = MC2,

140 – 2q1 – 4q2 = 20

Or,

2q1 + 4q2 = 120

Dividing both sides by 2:

q1 + 2q2 = 60

Or,   q2 = (60 – q1) /2 = 30 – 0.5q1    ..... (1)

This is firm 2’s response function. Substituting (1) in TR1:

TR1 = 140q1 – 2q12 – 2q1q2 = 140q1 – 2q12 – 2q1 (30 – 0.5q1)

= 140q1 – 2q12 – 60q1 + q12

= 80q1 – q12

So, MR1 = dTR1 / dq1 = 80 – 2q1

Equating MR1 = MC1 [Where MC1 = 20]

80 - 2q1 = 20

2q1 = 60

q1 = 30

therefore, q2 = 30 – 0.5q1    [From (1)]

= 30 – 0.5 x 30 = 15

Q = q1 + q2 = 30 + 15 = 45

P = 140 – 2Q = 140 – (2 x 45) = 140 - 90 = 50

(b) COURNOT

P = 140 – 2Q where Q = q1 + q2

P = 140 – 2q1 – 2q2

So,

Total revenue of firm 1, TR1 = P x q1 = 140q1 – 2q12 – 2q1q2

Total revenue of firm 2, TR2 = P x q2 = 140q2 – 2q1q2 – 2q22

So,

Marginal revenue of firm 1, MR1 = dTR1 / dq1 = 140 - 4q1 - 2q2

Equating with MC1:

140 - 4q1 - 2q2 = 20

4q1 + 2q2 = 120

2q1 + q2 = 60 ......(1) [Reaction function, firm 1]

Marginal revenue of firm 2, MR2 = dTR2 / dq2 = 140 – 2q1 – 4q2

MC2 = 20

Equating MR2 = MC2,

140 – 2q1 – 4q2 = 20

Or,

2q1 + 4q2 = 120

q1 + 2q2 = 60 .....(2) [Reaction function, firm 2]

Equilibrium is obtained by solving (1) & (2).

2q1 + q2 = 60 ......(1)

(2) x 2:

2q1 + 4q2 = 120 .....(3)

(3) - (1): 3q2 = 60

q2 = 20

q1 = 60 - 2q2 = 60 - (2 x 20) = 60 - 40 = 20

Q = q1 + q2 = 20 + 20 = 40

P = 140 - 2Q = 140 - (2 x 40) = 140 - 80 = 60

(c) Market quantity is higher in Stackelberg than under Cournot (45 > 40) and market price is lower under Stackelber than under Cournot (50 < 60).

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