Refer to Problem 18. It seems realistic to assume that the game could continue indefinitely - after all, the world will need otters 50 years from now just as badly as it does today. But time is money, so a dollar of profit a year from now is only valued as much as $d received today, where d (the discount rate) is some amount less than $1.
Assume that you adopt a grim trigger strategy whereby your country pledges to produce 1,000 otters as long as the other country produces 1,000 otters. But should the other country ever produce 2,000 otters, you will respond by producing 2,000 otters, forever and ever. After you announce your strategy, the other country pledges to abide by the grim trigger strategy.
a. What stream of profits will you generate if both countries adopt such a grim strategy? (Don't forget to discount each year's future profits by the appropriate multiple of d.) Express your answer as a sum.
b. Suppose your country decides to take advantage of the other country's pledge to reduce output by expanding its output in the first year. What stream of profits can you expect to generate? Again, be sure to discount appropriately, and express your answer as a sum.
c. Suppose d = 0.5. Are you better off cheating on your agreement, or abiding by it? How does your answer change if d = 0.99? What about d = 0.01?
d. At what level of d are you indifferent between cheating on or abiding by the agreement?