• The original stock of an exhaustible natural resource isx = 100.
• For simplicity, extraction costs are zero.
• There are two generations, one now, and one next period.
• Each generation chooses an amountc_t of the resource to use, and the marginal value of that use atMU(y_t) = 500-5*y_t. (Of course, the second generation doesn't really have much choice, it just takes what's left over after the first generation is done.)
• Suppose that the relevant interest rate is 5%.
• The price in each period is the marginal utility of consumption in that period. Suppose that the interest rate is 5%, and there exists a "backstop technology" that produces a perfect substitute for the exhaustible natural resource at a price of $253 per unit. How much of the resource would be used in Period 1?

1. With the backstop at P = $253, what would be the price in Period 1? (r = 5%)
2. With the backstop at P = $253, what would be the percentage change in the price of the resource from Period 1 to Period 2? (r = 5%)