Optimal Allocation of an Exhaustible Resource Over Time

PART ONE: Two-Period Model

The demand equation for a resource is QD = 100 - 4P and the competitive industry supply equation is QS = -20 + 8P.

  1. Calculate the market-clearing equilibrium values for P and Q where QS = QD.   (You can graph these functions if you want to verify this equilibrium point visually.)
    Calculate the point elasticities of demand and supply with respect to price at this equilibrium point.
    Calculate the producer and consumer surpluses accruing in this market.

    Now solve the demand function for P in terms of Q to obtain the inverse demand (willingness-to-pay) function.
    Solve the supply function for P in terms of Q to obtain the inverse supply (willingness-to-sell) function.
    Subtract the inverse supply function from the inverse demand function to obtain marginal resource rent
    (R = PD - PS) as a function of Q.

  2. Suppose the total stock of the resource is only 40 units, to be allocated between two time periods (years 0 and 1).
    Using a discount rate r = 0.06, calculate the quantities to sell in each year (Q0 and Q1) where R0 = R1/(1+r) and
    Q0 + Q1 = 40. 
    Once you have solved for Q0 and Q1, determine P0, P1, R0 and R1.
    [Maximizing the discounted stream of total rents over time implies the present value of the marginal resource rent is the same in each time period.  This rule was first articulated by Hotelling (JPE, 1931): in an efficient, competitive market, an exhaustible resource’s marginal rents will rise at the rate of discount through time. 
    This step rule applies for any two consecutive time periods t and t+1, since Rt/(1+r)t = Rt+1/(1+r)t+1 simplifies trivially to Rt = Rt+1/(1+r).]

  3. Calculate the optimal values of Q0, Q1, P0, P1, R0 and R1 as in Question 2, but using a discount rate r = 0.10.

PART TWO: Multi-Period Model

Given the same supply and demand schedules Qt = 100 - 4Pt (demand) and Qt = -20 + 8Pt (supply), and initial discount rate r = 0.06, suppose the total stock of the resource is 500 units to be extracted over multiple years.  

The optimal extraction schedule follows Hotelling's Rule: resource rents in a competitive market tend to follow a time path which makes resource owners indifferent between selling in any one time period and selling in any other. This implies that the marginal rents increase at the rate of discount through time, so that

R0 = R1/(1+r) = R2/(1+r)2 = R3/(1+r)3 = . . . = RT/(1+r)T

Assume the total stock of the resource is depleted in time T, so that QT = 0. Looking backward from time period T, this optimal rent trajectory can be rewritten as

RT = (1+r) RT-1 = (1+r)2RT-2 = (1+r)3RT-3 = . . . = (1+r)TR0
  1. Although you don't yet know how far off T is, you can solve the marginal rent function to determine RT where QT=0 at the demand choke price. What is RT, the terminal rent in the final year when the stock is depleted?

  2. Knowing RT, you can work backward through time and determine Rt for any earlier time period t<T. Use a spreadsheet to calculate Rt for each of the 30 time periods prior to T.

  3. Knowing Rt for any time period, you can use the inverse rent function to calculate the amount Qt a competitive resource market will provide in that time period. In an adjacent column of the spreadsheet, determine Qt for each of the 30 time periods prior to T.

  4. Given Qt, you can use the inverse demand schedule to calculate the price Pt of the resource in each time period. In an adjacent column, determine Pt for each of the 30 time periods prior to T.

  5. In an adjacent column, determine the cumulative amounts of the resource extracted between periods t and T. In what time period does the cumulative amount of resource extracted match the current stock? If this time period is today, how many years away is T? What are P, Q and R today?

  6. Create an XY plot showing how P, Q and R move through time from today to depletion time T.

  7. Suppose an unanticipated new discovery increases the total resource stock to 600 units, so "today" is a different time period in your spreadsheet. Now how many years off is T? How does this new discovery change today's P, Q and R?

  8. Create a graph showing how the time paths of P, Q and R differ under the two stock sizes.

  9. Given the same supply and demand schedules and a total stock of 500 units, solve out Rt, Qt and Pt and cumulative extraction trajectories under a discount rate of 0.03. Now how many years off from today is time period T? Now what will today's P, Q and R be?

  10. Show graphically how the time paths of P, Q and R differ under the two discount rates (total stock is 500 units in either case).