Question

Consider an industry where firms have a production function with private marginal cost ??? (?) = 40 + 2? and an additional (external) marginal cost to society of ??? (?) = ?/4. The market is characterized by the inverse demand function ?(?) = 400 − 2?. 1. Find the equilibrium in the competitive market. Calculate the (private) producer and consumer surplus in this case. 2. Find the equilibrium if the market is monopolized. Calculate the (private) producer and consumer surplus in this case. 3. Find the social optimum, and calculate the (private) producer surplus, consumer surplus, and the externality cost in this case. 4. Compare the three outcomes (price, output, total surplus) in the three cases (1-3).

Answer 1

TR = p x Q = 400Q - 2Q²

MR = dTR/dQ = 400 - 4Q

From demand function, when Q = 0, P = 400

From MCp function, when Q = 0, MCp = 40

(1)

Setting p = MCp,

400 - 2Q = 40 + 2Q

4Q = 360

Q = 90

p = 400 - 2 x 90 = 400 - 180 = 220

CS = (1/2) x (400 - 220) x 90 = 45 x 180 = 8100

PS = (1/2) x (220 - 40) x 90 = 45 x 180 = 8100

TS = CS + PS = 8100 + 8100 = 16200

(2)

Setting MR = MCp,

400 - 4Q = 40 + 2Q

6Q = 360

Q = 60

p = 400 - 2 x 60 = 400 - 120 = 280

CS = (1/2) x (400 - 280) x 60 = 30 x 120 = 3600

When Q = 60, MR = 400 - 4 x 60 = 400 - 240 = 160

PS = (1/2) x [(280 - 40) + (280 - 160)] x 60 = 30 x 360 = 10800

TS = CS + PS = 3600 + 10800 = 14400

(3)

In social optimal, p = MCp + MCe

400 - 2Q = 40 + 2Q + 0.25Q

4.25Q = 360

Q = 84.71

p = 400 - 2 x 84.71 = 400 - 169.42 = 230.58

CS = (1/2) x (400 - 230.58) x 84.71 = (1/2) x 169.42 x 84.71 = 7175.78

PS = (1/2) x (230.58 - 40) x 84.71 = (1/2) x 190.58 x 84.71 = 8072.02

MCe = 0.25 x 84.71 = 21.18

TS = CS + PS = 7175.78 + 8072.02 = 15247.8

(4)

Price is maximum with monopoly (case 2).

Output is maximum in competitive market (case 1).

TS (total surplus) is maximum in competitive market (case 1).