Suppose that you are extracting copper in a foreign country under a concession that will expire in 2 years. The current period is t= 0, and you are managing the extraction activities over the interval t =0,1,2.., where T, same as before, rents in period t is given by (t) =p*qt(c/2)qt^2, where qt is the amount of copper extracted in period t, p>0 is the fixed unit price for qt over the life of the concession, and c>0 is a cost parameter. Let St0 denote remaining reserves in period t, which shrink according to St+1=Stqt, where S0 is the initial stock of copper.

A) Suppose your goal is to maximize discounted profits over T years with a discount factor of = 1/(1 +r). Write down the Lagrangian that corresponds to the constrained maximization problem for you as a mine owner. Solve for and interpret the first-order conditionsi.e. what must be true for extraction rates to be optimal?

B) (i) Find the T and qt for t=0,1,..., T that maximize the net present value of the rents from the mine. Plot the optimal extraction path that you found above. Discuss whether your extraction path makes sense.

(ii) Should you extract the entire copper deposit if you can choose the time at which the deposit is depleted? Why or why not?