Consider a lake stocked with fish. The total value of fish caught at the lake depends on the number of fishers, as shown in the accompanying table. As the table indicates, 1 fisher can catch $36 worth of fish in a day, 2 fishers can catch a total of $66 worth of fish, 3 fishers can catch a total of $90 worth of fish, and so forth. The fishers are identical, and the opportunity cost of a day at the lake is $18 for each fisher.

(i) Complete the table by calculating the value of each fisher's catch, on average, and the social marginal benefit of the lake.

(ii) If use of the lake is nonexcludable, how many fishers come to the lake? What is the total value of their catch? What is their total cost? What is the social gain from the existence of the lake?

(iii) What is the optimal number of fishers at the lake? What is the social gain if this optimum is achieved?

(iv) What entrance fee would lead to the optimal outcome?

(i) The average value of each fisher's catch is the total value of their catch divided by the number of fishers. The social marginal benefit of the lake is the value of the total catch attributable to the last fisher. These calculations are summarized in the accompanying table.



(ii) When the lake is common property, fishers continue to use the lake until the value received equals the marginal cost. Thus, 7 fishers use the lake. Each fisher catches $18 worth of fish, for a total value of $126. Each fisher incurs a cost of $18, for a total cost of $126. Thus, social gain equals $126 - $126 = $0. The rents from the lake are totally dissipated.

(iii) For use of the lake to be optimal, the social marginal benefit must equal the marginal cost. Therefore, the optimal number of fishers is 4. Their total catch is worth $108, and their total cost is 4 ? $18 = $72, for a social gain of $108 - $72 = $36.

(iv) The entrance fee equals the difference between the value per fisher and the social marginal benefit at the optimum. Thus, the entrance fee is $27 - $18 = $9 per fisher.