Consider the market for gin, in perfect competition, where market demand is given by P=20-4Q and supply is given by P=Q. Suppose that alcoholism is a major problem in this community. Not only does it affect the health of the people who consume the gin, but it also affects community members who do not consume or sell alcohol. Thus, while the marginal private cost of gin is given by P=Q, the marginal social cost is given by P=5Q. What is the magnitude of the per unit tax that the local government should impose on suppliers of gin to bring the economy to the socially optimal allocation? [Round your answer to 2 decimal places when needed.]

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BlackTom AI · Accepted Answer

To begin, restate the setup: the market demand is P = 20 − 4Q, the private supply (without external effects) is P = Q, and the social marginal cost is MSC = 5Q due to externalities.
Next, determine the socially optimal quantity by equating the price consumers are willing to pay to the social marginal cost: P_d = MSC.
From the demand equation, P_d = 20 − 4Q. Setting 20 − 4Q = 5Q gives 20 = 9Q, so the socially optimal quantity is Q* = 20/9 ≈ 2.222.
Compute the corresponding price consumers would pay at this Q* using the demand curve: P* = 20 − 4Q* = 20 − 4*(20/9) = 20 − 80/9 = (180 − 80)/9 = 100/9 ≈ 11.111.
Now incorporate a per-unit tax t on suppliers. The private price received by suppliers (after tax) must satisfy the private supply relation P_private = Q, because the supply curve is P = Q without externalities. The price paid by consumers is P_consumer = P_private + t, or equivalently P_private = P_consumer − t. Using P_consumer = P_d and P_private = Q, we have Q = P_d − t.
Express P_d in terms of Q from the demand curve: P_d = 20 − 4Q. Substitute into Q = (20 − 4Q) − t to obtain 5Q = 20 − t, i.e., Q = (20 − t)/5.
Set this equal to the socially optimal quantity Q* to find the required tax: (20 − t)/5 = 20/9. Multiply both sides by 45 to clear denominators: 9*(20 − t) = 100, which yields 180 − 9t = 100. Solve for t: −9t = −80, so t = 80/9 ≈ 8.89.
Thus, the per-unit tax that brings the economy to the socially optimal allocation is approximately 8.89 (rounded to two decimals).

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