Externalities and Public Goods – Study Questions for ECMC02H3

Answers:

1. (a) If fishers receive the average amount caught on each lake (remembering that there are no costs to fishing) fishers will fish in each lake until the average amount caught in each lake is equal. In Lake Y, the average amount is constant; however, in Lake X, the average amount declines with additional fishing. The average in Lake X is FX/LX = 10 – 1/2LX. The average in Lake Y is FY/LY = 5. These averages will be equal at 10 – 1/2LX = 5 or LX = 10. In other words, the number of fishers on Lake X will be 10, and the number of fish caught will be 10(10) – ½(10 x 10) = 50. On Lake Y, the remaining 10 fishers will fish and they will catch 5 x 10 = 50 fish. The total number of fish caught will be 100. The externality here is that each fisher in Lake X reduces the amount that others can catch, but no account is taken of this externality.

(b) The optimal number of fishers that maximizes the number of fish caught will be found where the marginal product of the two lakes (not the average product) is equal. The marginal product in Lake X is 10 – LX. The marginal product in Lake Y is 5. These marginal products are equal where 10 – LX = 5, or where LX = 5 (and LY = 15). The total number of fish caught will be (10 x 5) – ½ (5 x 5) + (5 x 15) = 112.5.

(c) In part (a) there is a misallocation because the fishers make decisions according to the average product of their labour rather than the marginal product of their labour. The correct amount of the licence fee will exactly close this gap at the optimum amount of labour allocated to Lake X. The optimum amount of fishers is 5. At 5 fishers, the average product is 10 – ½LX = 7.5 fish. At 5 fishers, the marginal product is 10 – LX = 5 fish. The correct amount of the licence fee is 2.5 fish.

2. (a) Firms will enter this perfectly competitive industry until profit is driven down to zero, in other words until 10 x (500 – N) = 1000, or when N = 400. There will be 400 oil wells and 500(400) – (400 x 400) = 40,000 barrels of oil. In this industry, the private marginal cost of an additional well is $1000. However, the social marginal cost needs to take into account the reduction in output for all wells when each new well is drilled (output of each well is 500 – N).

(b) If the government nationalizes the oil wells and seeks to run them in the public interest, the government should open oil wells until the value of the marginal product of those wells is just equal to the cost of operating a well, or 10(500 – 2N) = 1000. In other words, there should be 200 oil wells in operation. In this case, total output will be 500(200) – (200 x 200) = 60,000 barrels of oil. The output per well will be 500 – 200 or 300 barrels of oil (compared to 100 barrels in part (a)).

(c) In order to give private firms the right incentives, the licence fee has to be sufficient to encourage them to drive profit down to zero at the right number of firms, which is 200. The firm’s revenue is P x q = 10(500 – N) = 5000 – 10N. If the licence fee is F, the firm’s costs have to be sufficient so that profit is zero at N = 200, so 5000 – 2000 = 1000 + F. In other words, F must be $2000.

3. (a) For the beekeeper, the marginal revenue from one hive is $50 and the marginal cost is 30 + .5Q. The beekeeper will profit maximize such that 50 = 30 + .5Q or Q = 40 hives. Each hive pollinates ¼ of an acre, so 10 acres will be pollinated. Without bargaining, therefore, only ½ of the apple orchard will be pollinated.

(b) The value of each hive to the orchard owner is $25, so this is the maximum subsidy the orchard owner would pay to the beekeeper for each hive. However, if this subsidy were paid, the beekeeper’s marginal revenue per hive would be $75. In this case, the beekeeper would want to install 75 = 30 + .5Q or Q = 90 hives. Since each hive permits the pollination of ¼ acre, this would be sufficient to pollinate 22.5 acres, more than the total size of the apple orchard. In fact, a subsidy of $20 will be just sufficient to generate 80 hives.

4. (a) The socially optimal level of methane production will come where marginal benefits and marginal costs of this pollution reduction are equal or where 100 – R = 20 + R. The socially optimal level of R is 40%.

(b) At R = 40%, the marginal cost to farmers of methane reduction is 20 + 40 = $60. If governments were to adopt a fee of $60 for each percent of methane not reduced, farmers would not pay the fee for the first 40% of methane, because their marginal costs of eliminating it would be less than the fee. However, the marginal cost of getting rid of the remaining 60% would exceed the fee, so they would prefer to pay the fee.

(c)The two farmers have costs of MC1 = 20 + 2/3R1 and MC2 = 20 + 2R2. Total methane reduction is the average from these two farms. If the government mandates that each farm reduce methane by 40%, then the overall reduction in methane will be 40%, obviously, and the cost will be the total area under the marginal cost curves which is (20 x 40) + [(46.67 – 20) x 40]/2 = $1,333.40 for Farmer #1. For Farmer #2, the total cost of methane reduction is (20 x 40) + [(100 – 20) x 40]/2 = $2,400. The total cost of methane reduction is $3,733.40.

(d)If, instead, government adopts the methane fee of $60, these farmers will reduce methane by adopting better feed until the marginal costs are equal to the fee. In the case of Farmer #1, that means 60 = 20 + 2/3R1 or R1 = 60%. In the case of Farmer #2, this means 60 = 20 + 2R2 or R2 = 20%. The total amount of methane reduction is the average of these two farms which is 40%. The cost of reduction is (20 x 60) + [(60 – 20) x 60]/2 = $2,400 in the case of Farmer #1. In the case of Farmer #2, the cost of reduction is (20 x 20) + [(60 – 20) x 20]/2 = $800. The total cost of methane reduction is $3,200.

(e) The two farmers have different marginal costs of reducing methane pollution. The first farmer’s marginal cost of reduction does not rise quickly, but the second farmer’s costs do. The fee encourages both farmers to reduce methane production, but the first farmer is able to do so at lower cost and has the incentive to avoid paying the fee as a result. The second farmer pays a bigger fee, because it is uneconomic for him/her to reduce methane production by as much as Farmer #1. The result of the differential treatment of these farmers is a lower overall cost of reaching the pollution reduction target.

5. (a) qA = 100 – P and qB = 200 – P. If the good is a nonexcludable public good, then the marginal benefit is the sum of the value to each person of each unit of the public good. In other words, the sum of the values is (100 – q) + (200 – q) = 300 – 2q, where q stands for the amount of the public good (mosquito control) available to all (this calculation is a bit misleading, since the sum of the values actually produces a non-linear function – try graphing this and see). At a marginal cost of $150, the optimum amount of the public good would be where total marginal benefit equals marginal cost or 300 – 2q = 150 or Q = 75.

(b) In a private market, both of these people would hope to free-ride on the purchase of this good by the other. As a result, it is likely that neither would purchase the good, and no mosquito control would be available on a private market. If this good were offered at a price of $150, it is possible that person B would purchase it (would purchase 50 units), but person A would find that $150 exceeded his/her valuation of even one unit of mosquito control.

(c) Mosquito control would cost $150 x 75 = $11,250. The benefits to each person would be measured as the area under each of their “demand” curves up to 75 units of mosquito control. For person A, this would be [(100 – 25) x 75]/2 + (25 x 75) = $4,687.50. For person B, this would be [(200 – 125) x 75]/2 + (125 x 75) = $12,187.50. This is approximately 28% for person A and 72% for person B.