Consider a country with a non-renewable resource. There are 150 units of the resource. Suppose there are two periods [you can think of them as the present and the future, but analytically, we will call them period 0 and period 1]. Suppose both the private and social discount rates are 10% (i.e. r = 0.1). There is no international trade and suppose that the inverse demand curve for the resource (in each period) is p = 200 - x, where x is the amount of the resource available for use in the relevant period. The marginal cost of extracting q units of the resource in each period is c(q) = q2/2. Note that this means that marginal cost is increasing in q. Let q0 be the amount of the resource extracted in period 0, and let q1 be the amount of the resource extracted in period 1. Suppose the resource is durable and is fully reusable (think for example of diamonds for jewelry - once dug up and cut and polished they can be used in both periods). So in the demand curve above, x is the total amount ever extracted up to that time. In period 0, x0 = q0, but in period 1, it is the sum of the amount extracted in both periods: x1 = q0 + q1. Recall that if the total benefit of consuming x in any period is B(x), then the marginal benefit, B'(x), is the price in that period. So B'(x) = 200 - x, where x is consumption in that period. (a) How much would a benevolent social planner choose to extract in each period? Does the price rise over time? Why? (b) Suppose that instead of having a benevolent planner, there is a competitive market.
Consider a country with a non-renewable resource. There are 150 units of the resource. Suppose there are two periods [you can think of them as the present and the future, but analytically, we will call them period 0 and period 1]. Suppose both the private and social discount rates are 10% (i.e. r = 0.1). There is no international trade and suppose that the inverse demand curve for the resource (in each period) is p = 200 - x, where x is the amount of the resource available for use in the relevant period. The marginal cost of extracting q units of the resource in each period is c(q) = q2/2. Note that this means that marginal cost is increasing in q.
Let q0 be the amount of the resource extracted in period 0, and let q1 be the amount of the resource extracted in period 1.
Suppose the resource is durable and is fully reusable (think for example of diamonds for jewelry - once dug up and cut and polished they can be used in both periods). So in the demand curve above, x is the total amount ever extracted up to that time. In period 0, x0 = q0, but in period 1, it is the sum of the amount extracted in both periods: x1 = q0 + q1.
Recall that if the total benefit of consuming x in any period is B(x), then the marginal benefit, B'(x), is the price in that period. So B'(x) = 200 - x, where x is consumption in that period.
(a) How much would a benevolent social planner choose to extract in each period? Does the price rise over time? Why?
(b) Suppose that instead of having a benevolent planner, there is a competitive market. Demonstrate how intertemporal arbitrage will yield the efficient solution that you found in (1).
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