Problem
1. This question explores a simple example of resources where the cost of mining increases with the total amount of resource extracted. Suppose that the profit on a copper mine at time t is given by pt =pqt - cqt2 - kst , where p, c and k are exogenous constants, qt is the amount of copper extracted in period t, and st is the total amount of silver extracted in all periods before t. The evolution of sis given by st+1- st= qt. The value of sis initially zero. Also, the resourceis infinite supply in the sense that there is no upper limit on s. The firm will own and operate the mine in periods 0 (now) through T-1; at T the firm expects to be nationalized and to have no subsequent revenues or costs. The firm's goal is to maximize the sum of the profits it earns in periods 0 to T-1. You may assume the interest rate is zero.
A. Set up the firm's optimization problem and take first order conditions. Briefly explain what each equation shows.
B. Solve for an expression giving the firm's output (qt) at each point in time from 0 to T - 1. Discuss your result and give an intuitive explanation.
The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.