Question

Suppose there are only two individuals in society. Person \(A\) 's demand curve for mosquito control is given by \\[ q_{a}=100-p \\] for person \(B,\) 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 proportionally to the benefits each individual receives, resulting in each person paying 600 in taxes.

Step-by-step solution

Find the combined demand curve

To find the combined demand curve, we need to sum the individual demand curves for person A and person B: Combined Demand Curve: \(q_{c}=q_{a}+q_{b}\) Plugging in their demand curves, we get: \(q_{c} = (100-p) + (200-p)\) Simplify the equation: \(q_{c}=300-2p\)

Find the optimal level of mosquito control

Now we need to find the optimal level of activity by setting the combined demand equal to the constant marginal cost (\(MC=120\)): \(300 - 2p = 120\) Solve for p: \(2p = 180\) \(p = 90\) We have the price that corresponds to the optimal level of mosquito control. Now, plug the price into either person A or person B's demand curve to find the optimal quantity. Using person A's demand curve: \(q_{a} = 100 - p\) \(q_{a} = 100 - 90\) \(q_{a} = 10\) Since mosquito control is a public good, both individual A and B will consume 10 units of this good. So, the optimal level of mosquito control is 10 units.

Analyze private market production

If mosquito control were left to the private market, it's possible to assume that each person acts independently and decides on their consumption level. Also, both individuals know that once mosquito control is produced, they will both benefit from it. Hence, each person will take into account the other's action in consuming mosquito control. The outcome, in this case, will depend on each person's assumption about the other's action. Therefore, it is not possible to give a specific quantity that would be produced by the private market.

Calculate the government production cost

To find the cost of producing the optimal amount of mosquito control by the government, multiply the optimal quantity by the marginal cost: Total Cost: \(q_{optimal} * MC\) Total Cost: \(10 * 120\) Total Cost: \(1200\)

Allocate the tax bill

To allocate the tax bill between individuals, we can do so proportionally to the benefits they receive from mosquito control. First, let's find the total benefit from mosquito control: Total benefit: \(q_{a}+q_{b}\) Total benefit: \(10+10\) Total benefit: \(20\) Now, we can allocate the tax bill proportionally to their benefits: Person A's tax share: \(Tax_A = \frac{q_{a}}{Total\,benefit} * Total\,Cost\) Person A's tax share: \(\frac{10}{20} * 1200\) Person A's tax share: \(600\) Person B's tax share: \(Tax_B = \frac{q_{b}}{Total\,benefit} * Total\,Cost\) Person B's tax share = \(\frac{10}{20} * 1200\) Person B's tax share = \(600\) So each individual should pay \(600\) in taxes to cover the cost of producing the optimal amount of mosquito control, which is proportional to the benefits they receive.