# The owner of a uranium mine hires you as an economist and asks you to determine the optimal number of uranium ore which should be extracted from the

The owner of a uranium mine hires you as an economist and asks you to determine the optimal number of uranium ore which should be extracted from the mine this year (q0) and next year (q1), after which time no mining will be possible. The owner would therefore like to extract all the uranium ore by the end of the second year. There are 60 tons of uranium ore in the ground. The price this year is \$30 a ton and the price next year is known to be \$35 a ton. The cost of mining is given by the following function: c(qi) = 25 + 4.5qi + .1q2. The owner’s discount rate is .10.

(a) Write down the formula for determining the present value of mining all of the tons of uranium ore between the two periods.

(b) What three first order conditions hold at the optimum where the present value of the uranium mine is maximized? (hint: use the Lagrange multiplier approach)

(c) IN THREE SENTENCES OR LESS give a brief economic rationale for each of the three first order conditions. Solve for q0 and q1.

(d) How would you calculate the value to the owner of discovering an extra ton of uranium ore? What is this number?

(e) How would the allocation of the optimal amount of uranium ore to be mined in each of the two time periods change in a qualitative sense (e.g. q0 increases and q1 decreases) under each of the following situations? For each question assume that each of the other parameters remains fixed at values given at beginning of problem.

• Second period price increases to \$40

• Fixed cost part of cost function increases from 25 to 35

Parameters in second period cost function decrease with linear term going from 4.5 to 3.5 and quadratic term going from 0.1 to 0.09. (First period cost function remains unchanged)

• Price in the first period increases from \$30 to \$35

• Discount rate decreases from .10 to .06