Question

The demand curve for luminous socks is given by: Q = 50 – 0.5P And the total cost function for any firm in the industry is: C = 4Q

(i) Assume Stackleberg behavior with firm 1 as the leader and firm 2 as the follower.

(i) Determine equilibrium outputs

(ii) Price (1 mark)

iii) Profit levels for the two firms.

(j) Indicate the Stackleberg equilibrium on a diagram.

(k) Indicate the Cournot, Stackleberg, collusive and competitive outcomes on the market diagram.

(l) Calculate the change in consumer surplus when the industry moves from Cournot to Stackleberg equilibrium. What is the value of the deadweight loss in Stackleberg equilibrium? Show that the combined consumer and producer surplus in the Stackleberg equilibrium is less than the surplus to consumers with a competitive solution. (6 marks

) (m) Can leadership in the Stackleberg model of duopoly ever produce lower profit than the equilibrium profit level of a duopolist in the Cournot model? Explain. ?

Answer 1

Q = 50 - 0.5P

0.5P = 50 - Q

P = 100 - 2Q = 100 - 2Q1 - 2Q2

TC1 = 4Q1, therefore MC1 = dTC1/dQ1 = 4

TC2 = 4Q2, therefore MC2 = dTC2/dQ2 = 4

(d)

(i)

For firm 1,

Total revenue (TR1) = P x Q1 = 100Q1 - 2Q1² - 2Q1Q2

Marginal revenue (MR1) = TR1/Q1 = 100 - 4Q1 - 2Q2

Equating MR1 and MC1,

100 - 4Q1 - 2Q2 = 4

4Q1 + 2Q2 = 96

2Q1 + Q2 = 48...........(1) (Reaction function, firm 1)

For firm 2,

Total revenue (TR2) = P x Q2 = 100Q2 - 2Q1Q2 - 2Q2²

Marginal revenue (MR2) = TR2/Q2 = 100 - 2Q1 - 4Q2

Equating MR2 and MC2,

100 - 2Q1 - 4Q2 = 4

2Q1 + 4Q2 = 96...........(2) (Reaction function, firm 2)

(ii)

Equilibrium is obtained by solving (1) and (2). Subtracting (1) from (2),

3Q2 = 48

Q2 = 16

Q1 = (96 - 4Q2)/2 [From (2)] = [96 - (4 x 16)]/2 = (96 - 64)/2 = 32/2 = 16

(iii)

Q = Q1 + Q2 = 16 + 16 = 32

P = 100 - (2 x 32) = 100 - 64 = 36

(iv)

Profit, firm 1 = Q1 x (P - MC1) = 16 x (36 - 4) = 16 x 32 = 512

Profit, firm 2 = Q2 x (P - MC2) = 16 x (36 - 4) = 16 x 32 = 512

(v)

From (1), When Q1 = 0, Q2 = 48 (Vertical intercept) and when Q2 = 0, Q1 = 48/2 = 24 (Horizontal intercept).

From (2), When Q1 = 0, Q2 = 48/2 = 24 (Vertical intercept) and when Q2 = 0, Q1 = 48 (Horizontal intercept).

In following graph, BR1 and BR2 are reaction functions for firm 1 and firm 2 respectively, intersecting at point A with equilibrium output being Q1* for firm 1 (= 16) and Q2* for firm 2 (= 16).