PART ONE: Static (single-period) model

Suppose you own some large stock of an exhaustible resource that is sold in a competitive market for a constant price P = MR = $10 per unit.
Your marginal extraction cost for any year's production is MC = $4 + $0.05Q.
Your marginal resource rent is MRR = MR - MC = $6 - $0.05Q.

1. You would integrate (reverse-derivitate) this marginal resource rent function to obtain the total resource rent function. What is this integral?
   (NOTE: The unknown constant C that you would add to this indefinite integral would be your fixed costs, which you would deduct from your total rent to determine your final profit. Resource rents are related to profit, but are generally not the same as profit!)

2. To determine the value of Q that would maximize your profit, re-derivitate this sucker with respect to Q, the fixed cost drops out, and you should recover the same marginal rent function MRR = $6 - $0.05Q.
   Set this derivative equal to zero, and solve for the value of Q that maximizes (or minimizes?) your profit in a single time period. (When MRR = 0, MR = MC, which is the usual static profit-maximization rule for a firm.)
   Show your work, and explain how you know this is a rent-maximizing (not -minimizing) solution.

PART TWO: Two-period model

Okay, suppose your stock of this resource is X = 100 units, and your objective is to maximize your profits (and resource rents) from it over a two-year time period. Graphically, you could plot MRR₀ left-to-right and MRR₁ right-to-left with the horizontal axis restricted to 100 units total.

You can probably intuit that maximizing your profit over two time periods means equating marginal profit or MRR in each time period. Without discounting, you would split the resource equally between the two time periods, but this is America, and big natural resource extraction companies like yours discount the money they have to wait for. Graphically, this discounting shifts the MRR₁ schedule downward, which moves the intersection of MRR₀ and MRR₁ rightward.

Using discount rate r = 0.05, mathematically determine the quantities Q₀ and Q₁ to extract and sell in each year that maximize the sum of this year's total rent plus next year's discounted total rent, where Q₀ + Q₁ = 100.
Solve this problem two ways, showing your work:

3. Solve by simple substitution:   maximize TRR(Q₀) + TRR(100-Q₀)/(1+r) with respect to Q₀, set the single derivative to zero and solve it (caution: messy algebra ahead!) for the optimal values of Q₀ and Q₁ = 100 - Q₀.

4. Solve by Langrangean:   maximize TRR(Q₀) + TRR(Q₁)/(1+r) + R[100 - Q₀ - Q₁]
   with respect to Q₀, Q₁ and R.
   Set all three partial derivatives equal to zero and solve for the optimal values of Q₀, Q₁ and R.

(You should get the same answers either way: the correct value of Q₀ is between 50 and 52.) The Lagrangean method is the same maximization problem with the resource constraint tacked on at the end as a zero-value term: 100 - Q₀ - Q₁ = 0 and/or R = 0. This maximization should yield the same optimal values for Q₀ and Q₁ as the substitution method, but it also yields a value for R, which turns out to be the present value of MRR in either time period. This is the shadow price or marginal opportunity cost of extracing an additional unit of the resource today instead of next year.

The first partial derivatives of the Lagrangean with respect to Q₀ and Q₁, set equal to zero, yield

So the rent-maximizing extraction schedule requires that (a) the present values of MRR must be the same in each time period, which implies that (b) the marginal rent will rise at the rate of discount over time.

This "step rule" between time periods was first articulated by Harold Hotelling (JPE, 1931), and is known as "Hotelling's Rule." It defines the rent-maximizing strategy for any two adjacent time periods:

and thus across all time periods until (asymptotic or actual) depletion:

5. Now solve for the optimal values of Q₀, Q₁ and R when the discount rate r = 0.08.
   

PART THREE: Optimal extraction over an indeterminate time horizon

The two-period problem proves the validity of Hotelling's Rule, but it's unlikely that you would arbitrarily decide to deplete your stock in just two periods. More realistically, your objective would be to maximize the present value of your total resource rent stream over some optimal number of years. Ordinarily these dynamic (multi-time-period) optimization problems involve some pretty fancy math, but you can apply the optimal step rule obtained in the two-period case to solve this problem fairly straightforwardly.

Suppose your initial stock of the resource is X₀ = 500 units, and the marginal resource rent function is the same as before:
MRR_t = $6 - $0.05Q_t.

Hotelling's Rule says the optimal MRR trajectory should follow the path

So from this final time period you can use Hotelling's Rule to solve the optimal trajectory for MRR working backward in time:

If Q_T = 0 and MRR_T = $6, then MRR_T-1 = MRR_T/(1+r) = $6/(1+r), and more generally, MRR_t-1 = (1+r)MRR_t

The implied optimal extraction quantities for any value of MRR are Q_t = 20[6-MRR_t]

As you solve the optimal trajectories of MRR and Q backward in time, you can recognize which time period is "today" by comparing your current stock to the stocks X_t that would be required in each time period t to supply all the quantities from time t until depletion time T.

The time period when the accumulating stock value Q_T + Q_T-1 + Q_T-2 + Q_T-3 + ... matches or exceeds your current stock level is the present.

6. Work this out for yourself using an Excel spreadsheet.

   Create four column headings titled YEAR, MRR, Q, and STOCK.  
   In the first row of the MRR column, enter the number 6, which is the final MRR as the last bit of stock is depleted.  
   In the next cell down, enter the formula to discount the above cell by one year, using a discount rate r = 0.05.   (It will simplify things later if the discount rate is a fixed reference cell in this formula.)  
   Copy this formula downward so you obtain an MRR trajectory for about 20 years.

   For each MRR_t, calculate the implied value Q_t in the adjacent column.

   In the STOCK column, calculate the resource stock X_t that would be required to supply the cumulative quantities of Q_t + Q_t+1 + ... + Q_T from each time t upward to depletion time T.

   In what time period does this cumulative stock match your current stock of 500 units?
   If this spreadsheet row corresponds to "today" (t = 0) how many years away is T?
   What are the optimal values of Q, MC and MRR today?

   Create XY-plots showing the optimal time trajectories of Q, MRR and STOCK from today forward to depletion time T.

Calculate Q, MRR and STOCK trajectories for each of the following alternative scenarios, and compare each alternative model to the "base" model you calculated above.

7. Suppose your stock of the resource is located in Berzerkistan, and your analysts have calculated that there is a 10% likelihood in any given year that the Berzerki government will seize your operation and nationalize it without paying you any compensation. Incorporating this risk into your discounting, solve the optimal trajectories for Q, MRR and STOCK under a discount rate r = 0.15.
   Create XY-plots comparing the optimal trajectories of Q, MRR and STOCK from today forward under discount rates of r = 0.05 versus r = 0.15.
   (The graphs should show each pair of trajectories starting at the same time and ending at different times.)
   EXTRA CREDIT: See if you can calculate the economic cost of this risk--the difference between the present values of the two total rent streams. (In other words, how much would it be worth to you if the Berzerki prime minster who's creating this nationalization risk just happened to get assassinated?)

8. Suppose a civil war in some other country destroys its production of this resource, so the world price is now $12 per unit for the forseeable future.
   Calculate and graphically compare the optimal time trajectories for Q, MRR and STOCK under resource prices
   P = $12 versus P = $10.

9. Suppose there is a marginal pollution damage cost of $1 for each unit of Q that you extract. Calculate and graphically compare your optimal time trajectories for Q, MRR and STOCK under each of the following pollution control regimes:
   (a.) The government controls the pollution from your resource extraction operation by limiting your production to a maximum of 40 units per year.
   (b.) The government charges you a pollution tax of $1 for each unit of Q that you extract.
   Which of these pollution control regimes is more profitable for you? Which is more socially efficient?

10. Suppose the resource price rises as you and your competitors sell less in each successive time period, so your expected price in each time period is P = MR = $15 - $0.10Q. This implies a marginal resource rent function
    MRR = $15 - $0.10Q - [$4 + $0.05Q] = $11 - $0.15Q.
    Calculate and graphically compare the optimal time trajectories of P, Q, MRR and STOCK under this demand scenario versus the fixed price scenario of the base case.

11. Suppose an unanticipated new recovery technology increases the extractable quantity of your particular resource stock to X₀ = 750 units, but leaves your MC schedule and the overall market price unchanged.
    Calculate and graphically compare the optimal time trajectories of Q, MRR and STOCK for stock sizes of 500 versus 750 units.

(I am hoping that analyzing these alternative scenarios gives you some insight into the economic incentives of competitive resource extraction firms!)