1.  Suppose you own 20 units of a depletable resource (Q).   The cost of extracting and selling the resource is a constant $5 per unit.  The (inverse) demand for the resource is P_t = 20 - 0.2Q_t where Q_t is the amount you extract and sell in time period t.  Your discount rate is r = 0.10. 

If you want to maximize the present value of your profits from this resource over two time periods, you would ...
     maximize   Profit = P₀Q₀ + P₁Q₁/(1+r) = (20 - 0.2Q₀)Q₀ + (20 - 0.2Q₁)Q₁/1.10
     subject to Q₀ + Q₁ = 20

Set up the Lagrangean form of this maximization problem, take the first partial derivatives with respect to Q0, Q1 and the Lagrangean multiplier, set these equal to zero, and solve these three equations for the optimal values of Q₀, Q₁ and the multiplier.  
Plugging Q₀ and Q₁ into the inverse demand function, solve for P₀ and P₁. 
Calculate (the present value of) your total profit over the two time periods.


2.  In this context, the Lagrangean multiplier R = [P_t - C]/(1+r)^t  represents the present value of the marginal profit (P_t - C) the owner gets from selling an extra unit of the resource in any time period t.   The economist Harold Hotelling observed that in competitive markets for depletable resources, the marginal profit tends to rise at the rate of discount over time.   Explain why this is likely to happen. 


3.  Suppose the total stock of Q is 1,000 units, distributed among many sellers.  If the market for this resource follows Hotelling's rule, then the rent trajectory from today (t=0) to some future time of depletion (t=T) looks like...
      P₀ - C  =  [P₁ - C]/(1+r)  =  [P₂ - C]/(1+r)²  =  ...  =  [P_T-1 - C]/(1+r)^T-1  =  [P_T - C]/(1+r)^T

If the demand is P_t = 20 - 0.2Q_t, C = $5 and r = 0.10, then in the final time period, at the point of depletion when Q_T = 0 and P_T = $20, the marginal profit will be $20 - $5 = $15.   If the market follows Hotelling's rule, then the marginal profit in the previous period T-1 should be [P_T-1 - $5] = $15/(1.10) from which you can solve for P_T-1 (= $18.64) and obtain Q_T-1 from the demand schedule (= 6.818 units).  And in the next previous period T-2 the marginal profit will be [P_T-2 - $5] = $15/(1.10)² from which you can solve for P_T-2 (= $17.40) and obtain Q_T-2 (= 19.835 units).  And so on....

In the first column of a spreadsheet, calculate this rent trajectory backwards from $15 in terminal time period T toward the present.  In the adjacent columns, calculate the corresponding P_t and Q_t values.  In the fourth column, calculate the cumulative stock of Q required to supply Q_T + Q_T-1 + Q_T-2 + ...  from that time period up to time T.  Work your way down the rows of the spreadsheet (backward in time) until cumulative stock requirement most closely matches today's total stock of 1,000--this row represents "today."   What is today's marginal profit, P and Q?  If you denote this row as time period 0, how many years off is time period T?  Create an XY graph of the price and marginal profit trajectories through time. 


4.  Suppose there is a new discovery of 500 additional units of the resource.  Work your way further down the spreadsheet until the cumulative stock requirement most closely matches a stock of 1,500 units.  What happens to "today's" marginal profit, P and Q?  How many years off is time period T now?