FREC 424: Two-Period Resource Extraction Example

Suppose demand QD = 200 - 2P and supply QS = -10 + P, the discount rate r = 0.10 and the total resource stock to be depleted is X = 50 units.

Inverting the demand and supply schedules yields inverse demand PD = 100 - 0.5Q and supply (marginal cost)
PS = MC = 10 + Q.   These are shown in the upper panel of the figure below.

Assume the resource owner’s objective is to allocate 50 units of the resource between two time periods so as to maximize the present value of the total resource rent (TRR) from this stock. TRR is the gray shaded area between the demand and supply schedules between 0 and Q in each of the panels below. Mathematically, TRR is the integral of the marginal resource rent function

MRR = PD – MC = 90 – 1.5Q.

so for any quantity extracted Q > 0,
TRR = 90Q – 0.75Q2.

So the problem is to maximize the sum of the discounted total resource rents TRR0 + TRR1/(1+r) with respect to the two extraction quantities Q0 and Q1, where Q0 + Q1 = 50.

The resource constraint is incorporated into the problem as a zero-valued expression times a multiplier S:

max L =   90Q0 – 0.75Q02   +   [90Q1 – 0.75Q12]/1.10   +   S[50 – Q0 – Q1]

Solving for the first partial derivatives with respect to Q0, Q1 and S and setting these equal to zero yields:

90-1.5Q0 – S = 0

[90-1.5Q1]/1.10 – S = 0

50 – Q0 – Q1 = 0

Combining the first two equations, canceling the S’s, multiplying through by 1.10 and substituting 50-Q0 for Q1 yields:

99-1.65Q0 = 90-1.5(50-Q0)       or       99-90+75 = (1.65+1.5) Q0       or       84 = 3.15Q0

so
  Q0 = 26.67       P0 = $86.67       Q1 = 23.33       and       P1 = $88.33.

The multiplier S turns out to be the present value of the marginal resource rent: MRRt = Pt - MC.

Solving for MRRt = 90 - 1.5Qt in each time period demonstrates that resource rent maximization implies the marginal resource rents should rise at the rate of discount over time:

MRR0 = $50       and       MRR1 = $55       so       MRR0 = MRR1/1.10

This step rule characterizes the optimal time trajectory of MRR between any two time periods. It is known as "Hotelling's Rule:"

MRR0   =   MRR1/(1+r)   =   MRR2/(1+r)2   =   ...   =  MRRT/(1+r)T

We will use this principle to solve more complex resource depletion problems over indeterminate time horizons.