4. Uranium Mine Extraction A uranium mining firm hires you as an economist to determine the optimal amount of uranium ore to be extracted from a mine this year (qo) and

next year (q), after which the mine operations cease. The firm would like to extract all of the uranium ore by the end of the second year. There is 50 tons of ore in the mine, the price of uranium ore is $25 per ton this year and estimated to be $30 per ton next year. The firm has a discount rate of .10, and the cost of the mining operation is given by the following function: c(g) 25+5q, +.1q? a. Write down the formula for determining the present value of mining all of the tons of uranium ore between the two periods. b. Write the Lagrangian representing the constrained optimization problem. c. What are the three first order conditions that hold at the optimum where the present value of the uranium mine is maximized? d. In three sentences or less, give a brief economic rationale for each of the three first order conditions./ne. Solve for go and qi, the optimal quantity of ore to be extracted each year. f. How would you calculate the value to the firm of discovering an additional ton of uranium ore? What is this number? How would the optimal allocation of ore extraction change between the two periods in each of the following scenarios? Do go and q, each increase, decrease, or not change? In one sentence, explain your intuition. For each scenario, assume everything else is held constant as described above (that is, changes are not cumulative from one scenario to the next). i. The price in period 1 falls from $25 to $20 per ton. ii. Additional ore reserves are discovered, raising the total to 55.