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)