Question

Consider the market for cars. Cars provide private transportation benefits but also contribute to highway congestion and increase the risk of death for pedestrians. Cars provide no external benefits. Let Q be the quantity of cars. The private marginal cost (PMC), social marginal cost (SMC), and the consumer demand curve (Q) are given as,

PMC = Q SMC = 1.5Q Q = 18 – 2P.

a. What is the competitive market equilibrium quantity and price of cars?

b. What is the socially efficient quantity?

c. Calculate the deadweight loss under the competitive market equilibrium?

d. Design a Pigouvian tax to correct the failure and restore the efficient outcome in this market.

Answer 1

(a) In competitive equilibrium, Demand = PMC.

Demand: Q = 18 - 2P

2P = 18 - Q

P = 9 - 0.5Q

9 - 0.5Q = Q

1.5Q = 9

Q = 6

P = 6

(b) Socially efficient outcome is obtained when Demand = SMC.

9 - 0.5Q = 1.5Q

2Q = 9

Q = 4.5

P = 1.5 x 4.5 = 6.75

(c) When Q = 6,

PMC = 6

SMC = 1.5 x 6 = 9

Deadweight loss = (1/2) x (SMC - PMC) x Change in quantity = (1/2) x (9 - 6) x (6 - 4.5) = (1/2) x 3 x 1.5 = 2.25

(d) The efficient Pigouvian tax is the difference between SMC and PMC at efficient output level. When Q = 4.5,

Unit Pigouvian tax = SMC - PMC = 1.5Q - Q = Q = 4.5.