Question

Suppose there are only two individuals in society. Person \(A\) 's demand curve for mosquito control is given by \\[ q_{n}=100-p \\] for person \(\mathrm{B}\), the demand curve for mosquito control is given by \\[ q_{b}=200-p \\] a. Suppose mosquito control is a pure public good; that is, once it is produced, everyone benefits from it. What would be the optimal level of this activity if it could be produced at a constant marginal cost of \(\$ 120\) per unit? b. If mosquito control were left to the private market, how much might be produced? Does your answer depend on what each person assumes the other will do? c. If the government were to produce the optimal amount of mosquito control, how much will this cost? How should the tax bill for this amount be allocated between the individuals if they are to share it in proportion to benefits received from mosquito control?

Short answer

Answer: The optimal level of mosquito control is 10 units. The tax bill should be allocated as follows: Person A would be responsible for $92.31, and Person B would be responsible for $1107.69.

Step-by-step solution

a. Optimal Level of Activity with Constant Marginal Cost

First, let's add up the individual demand curves for Person A and Person B to form the aggregate demand curve. \(Q_D = q_a + q_b = (100 - p) + (200 - p) =\) \( 300 - 2p\) The aggregate demand curve is given by: \(Q_D = 300 - 2p\) Since it is a pure public good, the optimal production occurs when the marginal benefit (MB) equals the marginal cost (MC): \(MB = 120\) To find the price where MB = MC, we can solve for p: \(Q_D = 300 - 2p\) \(300 - 2p = 120\) \(2p = 180\) \(p = 90\) Now, substitute the optimal price back into either individual demand curve to find the optimal quantity: \(q_a = 100 - p\) \(q_a = 100 - 90\) \(q_a = 10\) Both individuals benefit from the same quantity, so the optimal level of mosquito control is 10 units.

b. Private Market Outcome

In a private market, each individual would only consume mosquito control if their individual demand curves met the marginal cost of production. For Person A: \(q_a = 100 - p\) \(100 - p = 120\) \(p = -20\) Since the price cannot be negative, Person A would not consume any mosquito control in the private market. For Person B: \(q_b = 200 - p\) \(200 - p = 120\) \(p = 80\) Now, we have the quantity consumed by Person B as: \(q_b = 120\) However, since mosquito control is a public good, both individuals would benefit from it. The problem then is that both individuals may base their decision on how much the other person consumes. This uncertainty leads to the possibility of underproduction or overproduction of mosquito control.

c. Government Production Cost and Tax Bill Allocation

If the government were to produce the optimal amount of mosquito control (10 units) at a constant marginal cost of $120, the total production cost would be: \( Cost = 10 * 120 =\) \(1200\) Now, to allocate the tax bill proportionately to the benefits received by each individual, we can find the proportion of the contribution of each individual to the total benefit: Person A's contribution: \(Proportion_A = \frac{q_a}{q_a + q_b} =\) \(\frac{10}{10 + 120} =\) \(\frac{1}{13}\) Person B's contribution: \(Proportion_B = 1 - Proportion_A =\) \(\frac{12}{13}\) Now, we can allocate the tax bill accordingly: Tax Bill for Person A: \(Tax_A = Cost * Proportion_A =\) \(1200 * \frac{1}{13} =\) \(92.31\) Tax Bill for Person B: \(Tax_B = Cost * Proportion_B =\) \(1200 * \frac{12}{13} =\) \(1107.69\) Therefore, the tax bill should be allocated as follows: Person A would be responsible for \(92.31, and Person B would be responsible for \)1107.69.