Strike when the force is with you: optimal stopping with application to resource equilibria.

For the forest sector of the economy, there is a market algorithm for price that is comparable to Hotelling's algorithm. The age distribution of forests, for example, is a feature of the market's portfolio of forest projects.

Conclusion

Because of the heavy sunk investments characteristic of extractive industries, the market must aggregate decisions about unwieldy projects in the determination of the equilibrium price path. Hotelling's insights apply to features of that path rather than the rents of individual units of mineral. Given the price path, extractive firms' decisions apply to projects as well as to units of ore. By intertemporal arbitrage, a mineral deposit is brought onstream when its discounted forward value rises at the force of interest. Quality is a characteristic of the entire project and is defined by the rate of growth of forward value, not by mineral grade, marginal cost, or present value.

The forward-value rule (3) holds more generally for all investment decisions. It is of some significance to observe that option values and strike times apply under conditions of certainty. The theory of investment under uncertainty is an extension of, not a qualitative break from, the theory of investment under certainty.

The authors thank Rodney Beard, Nancy Bergeron, Harry Campbell, Margaret Insley, William Moore, Nguyen Van Quyen, and three referees for helpful comments. Cairns was supported by FCAR and SSHRCC. Davis acknowledges the financial support of CIREQ.

[Received March 2005; accepted May 2006.]

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(1) Under certainty, the force of interest is the instantaneous cost of capital, a more general term used in finance.

(2) Where postponement does not create value the quasi-option value associated with deferring the decision whether or not to invest may still warrant a delay (Mensink and Requate 2005). We emphasize postponement value since there is no quasi-option value under certainty.

(3) A main point of models under uncertainty is that even when there is such pessimism it can be optimal to wait to extract a resource due to quasi-option value.

(4) These conditions are very general: one could also let the initial investment I or the closing cost X be identically zero, or let the firm face a negative cash flow early in the mine's life, from either a flow of investment or variable losses while production ramps up.

(5) Herfindahl's and Hartwick's models raise the question of what would happen if the firm did not enter at the optimal time. With a finite number of firms, one would have to model the implications of the firm's changing its date of entry as a dynamic game. This game, and more fundamentally the nonconvexity inherent in sunk investment, calls into question the common simplification that firms are price takers, both under certainty and in the theory of real options. Under certainty, condition (3) always holds.

(6) With no investment we lose the concept of to, but the purpose is served by the time at which the high-cost ore is initially exploited, [t.sub.H].

(7) The discrepancy is due to the shadow value of the capacity constraint and not to discreteness of time or to rounding.

(8) Condition (3) is a local condition. There could conceivably be examples in which W(t)/W(t) = r(t) (and the second-order condition holds) at more than one point. As usual, the appropriate value of [t.sub.0] would have to be determined by direct comparison.

(9) For a more general analysis see Cairns (2001).

(10) We have alluded to the fact that determination of the force of interest as well as the price is endogenous in general equilibrium. Determining equilibrium quality levels is even more complicated than in our discussion, in which (as elsewhere in financial economics) the force of interest is treated as exogenous.

(11) Managers of growing biological assets are attuned to interior stopping times that compare the rate of growth of asset value against an opportunity cost of delayed harvest (e.g., Clarke and Reed 1990). Evidence to this effect is also found in mining. After discussing an r-percent rule for harvesting trees on p. 44, Torries notes that "... in some cases it may be preferable to base investment and operating decisions on the rate of growth of wealth rather than on the amount of wealth itself."