Suppose the factory Afro-Puffs Inc. produces wigs. As a by-product of this wig p

Question

Suppose the factory Afro-Puffs Inc. produces wigs. As a by-product of this wig production, they also produce dangerous emissions of toxic gases (as a result of the strong glue used to hold the hair in place). The De-Lite car factory, down the road, experiences a negative externality from this production process. Suppose that the supply curve (private marginal costs) for the wig factory is X = (2/5)P - 2, and it faces a market demand of Xd = 15 - P/2. The marginal damages caused by the production of wigs can be written as X = P - 1/2. (a) Find the equilibrium price and quantity in the market for wigs. (b) Find the socially optimal level of wigs and the corresponding price. (c) How much should the wig factory be taxed per wig?

Explanation / Answer

Market demand: X = 15 - 0.5P

0.5P = 15 - X

P = 30 - 2X

Total revenue, TR = P. X = 30X - 2X2

Marginal revenue, MR = 30 - 4X

Private marginal cost, PMC: X = 0.4P - 2

0.4P = X + 2

P = 2.5X + 5

Marginal damage, MD: X = P - 0.5

P = X + 0.5

(a) In private market equilibrium, MR = PMC

30 - 4X = 2.5X + 5

6.5X = 25

X = 3.85

P = 30 - (2 x 3.85) = 30 - 7.7 = 22.3

(b)

Social marginal cost, SMC = PMC + MD

P = 2.5X + 5 + X + 0.5 = 3.5X + 5.5

Socially optimal production is obtained by equating MR with SMC:

30 - 4X = 3.5X + 5.5

7.5X = 24.5

X = 3.27

P = 30 - (2 x 3.27) = 30 - 6.54 = 23.46

(c) Tax per unit should be the difference in price under private equilibrium and social equilibrium:

Tax per unit = 23.46 - 22.3 = 1.16

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