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

Despite the very substantial differences in characteristics of the mines, their periods of exploitation overlap significantly. The NPV rule would also see overlap: both would open in period 1 and thereby 5.7 (216/1.05 - 200) units of present value would be lost at mine L. In this example, entry in order of period 1 forward value is not an unambiguous criterion, and in any case is clearly spurious. During production from mine L (in periods 2 and 3), net price, p(t) - 200, rises at 2.9% (216/210 - 1). During production from mine H, net price, p(t) - 300, rises first at 10% (110/100 - 1) and then at 5.5% (116/110 - 1). Net price never rises at r = 5%.

At both mines, the shadow values of the capacity constraint and of the mineral sum to the net price. For example, if the capacity constraint for mine L is relaxed by one unit in period 2, the gain to the program is 4.3 ((410-200) -(416-200)/1.05) units of forward value when the unit is transferred from period 3 to period 2. This is the shadow value of the constraint in period 2. In period 3 the shadow value is zero, since relaxing the constraint would lead to no change: increasing production in period 3 at the expense of period 2 would reduce value. The shadow value of the mineral rises at r = 5%: it is 205.7 (210 - 4.3) in period 2 and 216 in period 3.

At mine H, capacity is constrained in periods 2 and 3. Forward shadow values of capacity are 5 (110 - 100 x 1.05) in period 2 and 5.75 (116 - 100 x [1.05.sup.2]) in period 3. The shadow value of the mineral again rises at 5%, from 100 to 105 to 110.25.

If mine L had 1/100 of a unit more initial reserves, it would operate in the same way except to produce that 1/100 of a unit in period 4, yielding a net cash flow of 1.5 and a contribution of 1.4 to forward value. The time of closing also depends on all parameters and not just marginal cost or grade.

Example 1 demonstrates that, with the constrained capacity, the Hotelling valuation principle is not valid, even once the mines are in operation (cf., Cairns and Davis 1998, 2001; Davis and Cairns 1999). For example, for mine H in period 2, remaining reserves are 4. Present value is not 4(410 - 300) = 440; it is 2(410 - 300) + 2(416 - 300)/1.05 = 441. (7) Moreover, net price can rise faster than at rate r during the exploitation of a mine. If so, the shadow value of capacity may be zero at an internal point and not monotone decreasing to zero.

Because of the different times at which the various sorts of cost are incurred, marginal cost is not a sufficient statistic for ordering extraction. In general, present value and its rate of growth depend on the whole schedule of extraction costs, including general inflation, transportation costs, investment and closing costs, as well as on the initial reserves, etc. For any unidimensional, physical measure of quality there are bound to be apparent anomalies.

In fact, there exists no technologically based measure of quality: comparative-statics-style changes in the force of interest over a sufficiently long interval could lead to a switch in the order of extraction. Given conditions (3) and (5), the highest available quality of a reserve is not defined exogenously in terms of highest ore grade or lowest extraction costs, nor even in terms of the present value of the mine or present value per unit of reserves. Rather, it is expressed endogenously in terms of the rate of growth of the forward value of the mine, [??](t)/W(t). (8) A higher-quality mine is one whose rate of growth of forward value falls earlier to the force of interest, for whatever reason. In financial-economic terminology, a higher-quality mine has a higher opportunity cost of delay at any given moment.

This definition of quality may seem discomfitingly vague when compared with the intuitive, but incomplete, physical or technological measures (grade or marginal cost) on which much theory has been based. The apparent vagueness is due to the endogeneity, in intertemporal equilibrium, of price paths, interest rates and rates of growth of forward value, given other economic variables such as cost inflation and technological change. The place of each mine in this equilibrium explicitly recognizes the opportunity cost of delay and the role of the force of interest in determining the optimal delay. Such ideas have to date only been forcefully put forward in models of uncertainty (e.g., Litzenberger and Rabinowitz 1995).

In the next section we discuss, in general terms, how the market price of a mineral may be determined over time.

Hotelling's Rule or Hotelling's Algorithm?

Example 1 illustrates an r-percent rule by which, at a producing project, the shadow value of a unit of reserves within the mine rises at the rate of interest. (9) Another r-percent rule, equation (8), applies to the market value of the investment opportunity, [PI](t), at any time prior to exercise. In addition, the main rule studied in the present article, rule (3), applies to the forward value of the mine (or more generally of any lumpy decision) at the optimal stopping time, W([t.sup.*.sub.0]). Example 1 makes it clear that the rule for the value of units of reserves is nested within rule (3) for investment timing in that the former rule is incorporated into the optimal choices of timing and the level of investment. More generally, in equation (1), the optimized values of output and capacity are entered into the value function. Contrary to the usual expression of Hotelling's rule when marginal cost is constant, it is inconsistent with these three rules for net price to rise at the force of interest. The difference between price and marginal extraction cost is the sum of the shadow values of capacity and of the mineral, and only the latter rises at rate r(t).

The nesting of the r-percent rules has the implication that, if a model has very simple assumptions (if I(K, [t.sub.0]) and X(T, K, [t.sub.0]) are identically zero), the rules collapse into a single rule. Each (infinitesimal) unit of stock is, in effect, a separate project that can be realized at any time. The net prices of like units rise at the rate of interest in equilibrium. Hotelling's rule, as customarily applied to individual units of reserves, holds only when there is no sunk cost.

In this context Hotelling's insight needs re-interpretation. Hotelling's intent was to explain the sectorial equilibrium of a nonrenewable resource, especially its price movements. In realistic examples the equilibrium is far more complicated than can be characterized by Hotelling's rule. To distinguish this more complicated equilibrium we call the method of computing it Hotelling's market algorithm: "The market" (as sometimes personified) aggregates the characteristics of all deposits, the assumed behavior of operators, the pattern of demand, etc. to determine the price path into which the development of each deposit fits. (10) In its most general form the algorithm finds the equilibrium of a very complicated dynamic game among present and future operators of mines. Rule (3) is nested within this equilibrium, and hence within Hotelling's algorithm.

Hotelling's algorithm differs from one proposed by Gaudet, Moreaux, and Salant (2001, p. 1153) in that market prices, rather than Hotelling rents, are the underlying basis for equilibrium allocations. As in Example 1, when there are sunk costs the shadow value of a unit of the resource is observationally confounded with the shadow value of capacity, which does not behave according to an easily predicted pattern. Indeed, in the model giving rise to condition (3), "Hotelling rent" is not readily defined. The only natural unit of analysis is the entire project, and the only way to define rent is as the market value of the project, [PI] (t).

This perception of rent undermines the distinction between Hotelling and Ricardian rents current in the literature on green accounting and elsewhere. Hotelling rent is held to be related to scarcity of the resource in an aggregate sense (the difference between price and marginal cost), and Ricardian rent to differences in quality (differences between marginal cost and the costs of inframarginal units). In a fuller analysis, the quality of a mineral deposit is expressed endogenously by its rate of change of forward value in equilibrium, which in turn is determined by Hotelling's algorithm. The forward value of the project, W(t), and its rent, [PI](t), incorporate both quality and scarcity.

Over short time intervals there is no need for net (of marginal cost) price to conform to intertemporal conditions such as not rising at a rate greater than r(t). There is also no need for a firm to heed Hotelling's algorithm consciously, or to be able to perceive the equilibrium process, or to base investment timing on unobservable shadow values, but only to make decisions based on observable project parameters and the forward price path decentralized by the market equilibrium, as in other branches of micro theory. Rule (3), because of its observability and intuitiveness, would be readily accepted by practitioners, who routinely calculate the NPV of their projects and who occasionally withhold from development projects with positive NPVs, especially in rising price environments (Torries 1998, p. 75). (11)

Sequential Projects