Question

A firm has the total variable cost function: ??? = 2? + 2?^2 and total fixed cost of $20. Suppose the market demand is: ? ? = 20 − ?.

a. What are the firm’s profit maximizing output and price if it is a monopolist? How much is its profit at this output level? How much is consumer and producer surplus? (Hint: use a diagram to illustrate the market),

b. Suppose the government implements a tax of $3 tax unit sold. What are the firm’s profit maximizing output and price? How much is its profit at this output level? Will the firm continue to operate in the short run? In the long run?

c. What would the equilibrium price and quantity be in a competitive industry (pre-tax)? How much is consumer and producer surplus? What is the deadweight loss from the monopoly?

Answer 1

MC = dTVC/dQ = 2 + 4Q

TC = TFC + TVC = 20 + 2Q + 2Q²

Since Q = 20 - P,

P = 20 - Q

(a)

Monopolist maximizes profit when MR = MC.

TR = P x Q = 20Q - Q²

MR = dTR/dQ = 20 - 2Q

20 - 2Q = 2 + 4Q

6Q = 18

Q = 3

P = 20 - 3 = 17

TR = 17 x 3 = 51

TC = 20 + 2 x 3 + 2 x 3 x 3 = 20 + 6 + 18 = 44

Profit = TR - TC = 51 - 44 = 6

From demand function, when Q = 0, P = 20 (vertical intercept of demand curve)

Consumer surplus (CS) = Area between demand curve & price = (1/2) x (20 - 17) x 3 = 1.5 x 3 = 4.5

From MC function, when Q = 0, MC = 2 (vertical intercept of MC curve)

When Q = 3, MC = 2 + 4 x 3 = 2 + 12 = 14

Producer surplus (PS) = Area between MC curve & price = (1/2) x [(17 - 14) + (17 - 2)] x 3 = 1.5 x 18 = 27

(b)

The tax will increase MC by $3 and new MC is: MC1 = 2 + 4Q + 3 = 5 + 4Q

Equating MR and MC1,

20 - 2Q = 5 + 4Q

6Q = 15

Q = 2.5

P = 20 - 2.5 = 17.5 (price paid by buyers)

Price received by sellers = 17.5 - 3 = 14.5

TR = 14.5 x 2.5 = 36.25

TC = 20 + 2 x 2.5 + 2 x 2.5 x 2.5 = 20 + 5 + 12.5 = 37.5

Profit = 36.25 - 37.5 = - 1.25 (loss)

Since firm makes a loss, it will exit market in long run.

TVC = 2 x 2.5 + 2 x 2.5 x 2.5 = 17.5

Since TR > TVC, firm will continue in short run.

(c)

In competitive industry, P = MC.

20 - Q = 2 + 4Q

5Q = 18

Q = 3.6

P = 20 - 3.6 = 16.4

CS = (1/2) x (20 - 16.4) x 3.6 = 1.8 x 3.6 = 6.48

PS = (1/2) x (16.4 - 2) x 3.6 = 1.8 x 14.4 = 25.92

Deadweight loss = (1/2) x Change in P x Change in Q = (1/2) x (17 - 16.4) x (3.6 - 3) = (1/2) x 0.6 x 0.6 = 0.18