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Suppose the factory Afro-Puffs Inc
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=(25)P−2X=(25)P−2, and it faces a market demand of Xd=15−P2Xd=15−P2. The marginal damages caused by the production of wigs can be written as X=P−12X=P−12.
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?
Expert Solution
a) The equilibrium price and quantity in this market is found by equating marginal benefit (MB) or demand function to marginal private cost (MPC) or supply function of the firm.
MB=MPC15−P2=(25)P−215+2=25P+P2910P=170.9P=17P=18.89MB=MPC15−P2=(25)P−215+2=25P+P2910P=170.9P=17P=18.89
Equilibrium price is $18.89. Equilibrium quantity (Q) is calculated by substituting P with 18.89 in the demand or firm's supply function.
Q=15−18.892Q=5.56Q=15−18.892Q=5.56
b) In cases of negative externality, production creates costs for people who are not part of the production or consumption. To find the socially optimal level of wigs production, the firms should take into account the cost of production to society, or marginal damage. Hence, marginal cost is the vertical sum of the supply (MPC) and the marginal damage (MD). Marginal cost to society (MSC) is calculated as follows:
MSC=MPC+MDMSC=25P−2+P−12MSC=75P−52MSC=MPC+MDMSC=25P−2+P−12MSC=75P−52
The socially optimal level of production is the level of output where the marginal cost to society of producing wigs equals the market demand or marginal benefit.
MB=MSC15−P2=75P−5275P+P2=15+521910P=3521.9P=17.5P=9.21MB=MSC15−P2=75P−5275P+P2=15+521910P=3521.9P=17.5P=9.21
Socially optimal price of wigs is $9.21. We calculate the socially optimal level of wigs (QsQs) by substituting P with $9.21 in the demand function.
Qs=15−9.212=10.40Qs=15−9.212=10.40
c) If the government chooses to impose a tax, the tax per wig should be equal to the difference between the market equilibrium price and socially optimal level of price.
Tax=18.89−9.21=9.68
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