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}, where rents in period t is given by (t) =pq_t(c/2)q²_t,where q_t is the amount of copper extracted in period t, p>0 is the fixed unit price for q_t over the life of the concession, and c >0 is a cost parameter. Let S_t0 denote remaining reserves in period t, which shrink according to S_t+1=S_tq_t, where S₀ is the initial stock of copper.

A. Suppose your goal is to maximize discounted profits over 2 years with a discount factor of = 1/(1+r). Write down the Lagrangian that corresponds to the constrained maximization problem. Solve for the first-order conditions and interpret them.

B. Suppose we have the following parameter values:p= 1;c= 4;r= 0.05; and S₀= 1.

i) Solve for the optimal extraction path over the two periods. Your answer should be numbers, not parameters.

ii) Should you extract the entire copper deposit if you only have two periods?Why or why not?