Non-Renewable Resource Allocation Problems


1.      To model the optimal depletion trajectory for a non-renewable resource in a competitive market, we begin by analyzing how the market "steps" from any one time period to the next.  Assume the demand for a non-renewable resource in any time period is Qt = 250 - 25Pt, the competitive industry supply is Qt = -40 + 20Pt, the industry stock X = 50 units and the discount rate r = 0.10.

a)      Invert the demand and supply functions to obtain the marginal rent function R(Q) = P(Q) – MC(Q).  Integrate this with respect to Q to obtain the total rent function.

b)      Set up the Lagrangean to maximize total discounted resource rents over two years t = 0,1 where Q0 + Q1 = 50.  Solve the first-order conditions for the optimal values of Q0 and Q1.   Determine the corresponding prices, marginal costs and marginal rents in each year. 
Check your algebra by verifying that R0 = R1/(1+r).   [This step rule was first articulated by Hotelling (JPE, 1931): 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).]

c)      In the first column of an Excel spreadsheet enter values for Q0 ranging from 0 to 50.  Calculate the corresponding values for P0, MC0, and R0 in adjacent columns.  Since Q1 = 50 – Q0, you can calculate the associated values of R1 and R1/(1+r), which are marginal opportunity costs of Q0.  Create an XY plot of Q0 versus P0, MC0, R0 and  R1/(1+r) that illustrates the rent-maxmizing intersection of R0 and R1/(1+r). occurring at the optimal value of Q0.

  1. Having derived and verified the step rule R0 = R1/(1+r) in the two-period context, you can apply it to solve the optimal resource allocation trajectory for a competitive market over an indeterminate time horizon.  Suppose market demand in time period t is Qt = 10 – Pt .  The solution algorithm involves characterizing the terminal market conditions when the resource is depleted, and applying the step rule backward in time to determine the optimal marginal rent trajectory and the associated trajectories of Q, P and MC, and the depletion trajectory of stock X.

    Base model: Assume initial stock X0 = 50 units, discount rate r = 0.06 and constant MC = $3. 
    Enter the discount rate value in the first row of an Excel spreadsheet.  In the second row enter the following column headings: time, P, MC, R, Q and X.  Let the top data row represent terminal time T.  Enter the demand choke price PT, MC and RT for QT = 0 and XT = 0.
    Working down the spreadsheet from the future toward the present, following the step rule, calculate optimal values for RT-n for previous time periods. 
    Calculate corresponding values for PT-n , QT-n and XT-n.  [X is calculated as the cumulative sum of the current and future values of Q.]
    In which row does the value of X correspond to "today?" What are today’s R0, P0 and Q0?  Counting upward from t = 0, how many years off is depletion year T? 
    Create an XY graph showing the time trajectories of R, P and Q from today until time T.
  2. Variations:  Now compare each of the following scenarios (a) through (k) against this base model.  Copy and paste your base model into successive tabbed worksheets. In each worksheet use the adjacent columns to model an alternative scenario, working backward from T toward the present as before:

    (i)   Calculate the time trajectories of R, P and Q under the alternative scenario

    (ii)  Identify the new values for R0, P0 and Q0 and T,

    (iii) Create an XY plot comparing the time trajectories of R, P and Q for the base scenario versus the alternative scenario. (Align your two models to the same starting point (t = 0).

    (iv) Briefly explain why the model changes the way it does.

    a)      Unanticipated new discovery: The global stock increases to 75 units.  Which row is “today” now, and what are R0, P0 and Q0 and T?

    b)      Higher discount rate: Compare R, P and Q trajectories under a discount rate r = 0.10, reflecting larger exogenous risk. 

    c)      Introduction of a back-stop technology:  A new substitute resource or technology truncates the upper portion of the demand schedule at a choke price of $6/unit; demand below $6/unit is unchanged. 

    d)      Reduction in constant MC:  Reduce MC to $2. 

    e)      MC a function of Q: Let MC = 1 + 0.1Q. 

    f)        MC rises exogenously over time:  Let MC rise at a rate of 3% per year so that MCT = $6. 

    g)      Technology reduces MC over time:  Let MC decline 20% per year from MC0 = $3. 

    h)      Differential extraction costs:  Suppose there are two stocks with different extraction costs: Xa = 30 with MCa = $2, and Xb = 30 with MCb = $5.  (Hint: the cheap stock gets used up before depletion of the expensive stock starts.)

    i)        Per-unit severance tax:  The government imposes a $2/unit severance tax.

    j)        Ad-valorem severance tax:  The government imposes an ad-valorem tax of 20% of market price.

    k)      Monopoly:  The resource is monopolized.  (A monopolist maximizes [P(Q) – MC(Q)]Q so the monopolist’s marginal resource rent function is R(Q) = P(Q)/2 – MC(Q) = MR(Q) – MC(Q).  This is less than the competitive marginal rent function, because the monopolist is sacrificing some resource rents to obtain monopoly profits.)