## Resources for the Future

Krautkraemer and Toman

With cost rising as the resource is depleted, the assumption of complete physical exhaustion from extraction—which is not empirically realistic anyway—no longer makes sense in the theoretical model either. Instead, what drives extraction behavior is economic exhaustion of the resource—at some point, cost becomes so high relative to the price that buyers are willing to pay that a particular mine or field is abandoned.

With resource heterogeneity and incomplete physical exhaustion, the simple Hotelling model summarized by the efficiency conditions (1)–(3) no longer applies. Instead, we must modify the equi-marginal principle as in the following equation:

T

(4)

∑ + − + = + + = + = − t s s t R s q t t t q t t C C C P 1 ) 1 ) ( ( ) 1 ( δ λ δ

In (4), subscripts q and R on cost C signify rates of change of cost with respect to the extraction rate and the depletion of reserves, respectively. The term λ_{t }is a generalized measure of user cost that reflects the rate at which future extraction cost is increased as a consequence of future depletion. This stock effect is part of the full opportunity cost of current extraction.

# Manipulating (4) a bit algebraically yields:

(5)

t q t q t t C P C P − + = − − − ) [ 1 ( 1 , 1 δ

] + C_{Rt }.

Equation (5) shows how with increased stock providing a kind of “dividend” in terms of holding down extraction cost over time (C_{Rt }< 0), the net return from depletion no longer has to rise at the rate of discount rate to induce resource owners to retain their holdings.

Application of this framework to describe the depletion over time of different energy deposits is frequently done but somewhat more problematic (Livernois and Uhler 1987). In the simplest case where resource quality varies across deposits but is homogeneous within a deposit, the optimal extraction pattern generally requires exploiting those deposits in a sequence from low cost to high cost. The user cost for a lower-cost deposit is greater than the user cost for a higher-cost deposit. At the time of transition from one deposit to the next-most-costly deposit, the marginal extraction cost plus user cost is the same at each deposit. This implies that the resource price rises at a rate slower than the rate of interest during the transition. Simultaneous extraction from different deposits can be optimal when marginal extraction cost at a deposit

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