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

A central technological feature of extractive industry is that resource projects are lumpy and can be brought onstream at a time chosen by an investor. Traditional project analysis has presumed that the investor invests if the current net present value (NPV) of a project is positive (Dixit and Pindyck 1994, pp. 4-5, 144-47). The theory of investment under uncertainty shows that a positive NPV is necessary but not sufficient for optimal immediate investment (optimal stopping); the option of waiting to invest must also be evaluated.

Under certainty, the case examined herein, there is a similar incentive to wait to invest in a lumpy project until the time at which the project's value is maximized. Again, a positive NPV is necessary but not sufficient for optimal immediate investment. We derive an "r-percent" rule for optimal stopping that applies to the rate of growth of the NPV of lumpy projects. At any time in an extractive industry, for example, there are many known properties that have not yet been developed. Some would produce a positive discounted cash flow if opened immediately. The firm faces a choice of the time to develop its property. At the optimal time, the time derivative of the NPV of the project, as of any reference point in time, is zero. This condition implies that at the optimal time of investment the current NPV (in financial terminology the forward value) of the entire project is rising at the force of interest. (1) Prior to that time the forward value is rising at a rate that exceeds the force of interest, and investment is optimally postponed. (2)

The idea of postponing the time to cut trees, to develop rural land or public infrastructure, or to serve wine is not new. But the applicability of postponement considerations to all lumpy investments has not been stressed. Nor has there been explicit recognition that a simple, intuitive, r-percent stopping rule applies generally. Marglin (1963) seems to have been the first to argue that postponement should be considered for any irreversible, lumpy investment. He cites Bain (1960) as the first to raise the point in print. After two decades, Porter (1982, 1984) revives the argument for a "new" way of evaluating nonrenewable resource projects under certainty, taking the option to postpone into account. He speculates that pessimism as to future commodity prices may have been a factor in limiting attention to now-or-never decisions in models of petroleum and minerals. (3) Neither Marglin nor Porter, however, discerns an r-percent rule. Arnott and Lewis (1979), Chiang (1984), Mishan (1988), Clarke and Reed (1990), Brealey and Myers (2003), and Hands (2004) derive a timing rule similar to the r-percent rule of the present article, but do not emphasize its generality beyond selected land and natural resource investments. In finance, postponement has gained importance only recently, as an outcome of analysis under uncertainty. The solution is presented as a calculation of certain critical values, namely, the optimal time of investment or the value of the investment at that time (Dixit and Pindyck 1994, pp. 138-49), or else as a rule for a "present-value index" (Moore 2000) or modified NPV or internal rate of return (Capozza and Li 2002), rather than as an r-percent rule.

In the literature on nonrenewable resources no one has expressed the investment decision using the rate of change of the entire program value. Rather, r-percent rules have been expressed as changes in the value of a unit of reserves. Application of the r-percent optimal-stopping rule orders equilibrium investments in a way that is not necessarily in order of NPV nor even of physical quality. As a result, the "least-cost-last" anomaly is resolved without resort to stochastic explanations. Consistently with the rule's applicability to whole projects, quality and rent are seen to be characteristics of projects rather than individual units of reserves. Hotelling's r-percent rule for units of reserves is recast as an equilibrium pricing algorithm in light of the stopping rule.

In a section on sequential projects we discuss how literature on setup costs and optimal extraction initiated by Hartwick, Kemp and Long (1986) derives ostensibly different conditions for the timing of new investment. We discuss how the findings of this literature relate to ours and link the two back to the canonical problem of forestry.

Nonrenewable Resources

A nonrenewable resource project involves an irreversible, lumpy capital expenditure on capacity and an extraction plan which specifies outputs in future time periods, a shut-down time, and possibly other choices (Cairns 2001). We assume that investment is instantaneous and continuous in intensity. By a lumpy investment we mean a capital expenditure that extinguishes the option to invest at a later date for either economic or technical reasons. A profit-maximizing firm chooses its time of investment, [t.sub.0] [greater than or equal to] 0, such that the project's net present value, as of time t = 0, is maximized.

For a given choice of [t.sub.0], let the optimal level of capacity be represented by K([t.sub.0]) and its installed cost by I(K([t.sub.0]), [t.sub.0]). In the equilibrium into which the project fits, let the optimal closing time be represented by T([t.sub.0]), the optimized net cash flow at any time t [member of] [[t.sub.0], T([t.sub.0])] by [phi] (t, [t.sub.0], K([t.sub.0])), and the optimal closing expense by X(T([t.sub.0]), K([t.sub.0])). In general, the force of interest may vary through time and is written r(s). At the present date, t = 0, the value of a property developed at time [t.sub.0] [greater than or equal to] 0 is

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We call V([t.sub.0]) the discounted forward value of the mine and assume that it is twice differentiable.

If [t.sup.*.sub.0] [member of](0, [infinity]) is the optimal choice (an interior solution) of [t.sub.0] then V([t.sup.*.sub.0]) > 0, V([t.sup.*.sub.0]) > V(0) and V([t.sup.sub.0]) = 0. Let W([t.sub.0]) = V([t.sub.0])exp[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the value of the project at the time of opening, [t.sub.0]. This value, which we call the forward value, is traditionally identified as the NPV at [t.sub.0]. By direct differentiation,

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

or

(3) W([t.sup.*.sub.0])/W([t.sup.*.sub.0]) = r([t.sup.*.sub.0]).

At the optimal investment time [t.sup.*.sub.0], the discounted (to time t = 0) forward value is nonnegative and stationary, and the forward value (current NPV at time t = [t.sub.0]) is nonnegative and growing at exactly the force of interest. (4)

Condition (3), an "r-percent" rule, is a necessary condition for a (local, interior) maximum of the value of the investment opportunity. In any equilibrium of the sector, even a noncompetitive one, condition (3) must hold, regardless of the characteristics of the resource or the extraction technology.

Condition (3) applies to the present value of the entire project, and not to the net price of individual units of the resource. Even though it is an r-percent rule, then, it is not Hotelling's rule. Rather, it is an optimal-stopping rule.

The second-order condition for a maximum stipulates that, in a neighborhood of [t.sup.*.sub.0], [d.sup.2]V([t.sub.0])/[dt.sup.2.sub.0] [less than or equal to] 0. Let that neighborhood be ([t.sub.1], [t.sub.2]).Then V([t.sub.0]) [greater than or equal to] 0 when [t.sub.0] [member of] ([t.sub.1], [t.sup.*.sub.0]). Therefore,

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

or

(5) W([t.sub.0])/W([t.sub.0]) [greater than or equal] r([t.sub.0]) when [t.sub.0] [member of] ([t.sub.1], [t.sup.*.sub.0]).

At a time before the optimal time of investment, the forward value of the investment, conditional on having to invest at that time, is rising faster than at the force of interest. Similarly, just after the optimal time, the forward value is rising more slowly than at the force of interest:

(6) W([t.sub.0])/W([t.sub.0]) [less than or equal to] ([t.sub.0]) when [t.sub.0] [member of] ([t.sup.*.sub.0], [t.sub.2]).

The market value of the investment opportunity (presuming optimal timing, i.e., striking at time [t.sup.*.sub.0]) at any time t < [t.sup.*.sub.0] is

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(Recall that W(t) is the forward value of developing the project at the time in question, t < [t.sup.*.sub.0].) By condition (7),

(8) [??](t) = r(t)[PI](t):

the market value of the whole project, not of individual units of the resource, is rising at rate r(t). As in Hotelling's rule, the total return on the investment opportunity is a capital gain from not developing until the optimal time, [t.sup.*.sub.0]. That is to say, under certainty, at any time t < [t.sup.*.sub.0], there is a value to the option to wait, namely, [PI](t) - W(t) > 0. The time paths for these various expressions of value are illustrated in figure 1.

[FIGURE 1 OMITTED]

The option of waiting to invest until time [t.sup.*.sub.0] provides value similar to that demonstrated in the real-options literature. Even a mineral deposit whose present value is negative if developed now (V(0) < 0) has a positive market value if there exists a to such that W([t.sub.0]) > 0. This positive value is often attributed entirely to uncertainty, but in the present model it arises from optimal management under certainty.

The optimal time of investment (the strike point) occurs when the value of the option to wait to invest is zero. Condition (3) is a hitting boundary condition, where the state variable is the rate of change of forward asset value and r is the (possibly time-varying) critical value, below which the option should be exercised. Typically, the critical value is calculated from value-matching and smooth-pasting conditions (e.g., Dixit 1993), whereas here it is derived directly as the outcome of a first-order condition. Following the typical presentation, let the market value be expressed as a function of the forward value, i.e., as Y(W), rather than as a function of time, i.e., as [PI](t) as we have. Then a value-matching condition at W([t.sup.*.sub.0]) = [W.sup.*] is that Y([W.sup.*]) = [W.sup.*], and a marginal (smooth-pasting) condition is that Y'([W.sup.*]) = 1. In the analysis of the present article, which takes place in the temporal domain, value matching is expressed as

(9) [PI]([t.sup.*.sub.0]) = W([t.sup.*.sub.0]).

As with analysis under uncertainty, value matching is not sufficient for finding the optimal stopping time, for it yields an infinite number of possible values of [t.sub.0], only one of which is the optimal value [t.sup.*.sub.0]. An additional, smooth-pasting condition is typically specified, which takes the derivative of each side of the value-matching condition with respect to the choice variable and sets them equal. In the temporal domain, the smooth-pasting condition is

(10) [??]([t.sup.*.sub.0]) = [??] ([t.sup.*.sub.0])

Combining conditions (8)-(10) gives

(11) [??]([t.sup.*.sub.0]/W([t.sup.*.sub.0]) = [??]([t.sup.*.sub.0]/[PI]([t.sup.*.sub.0]) = r(t.sup.*.sub.0]),

which is another way of expressing optimality conditions (3) and (7).

The smooth-pasting condition is often perceived in the real-options literature as being subtle and somewhat technical (Dixit and Pindyck 1994, p. 109) and there is ongoing effort to explain it in more intuitive terms (e.g., Sodal 1998). Here, the smooth-pasting and value-matching conditions follow directly from condition (7) for an optimally managed investment opportunity. Together they produce the r-percent stopping rule (3), which tells decision makers to strike when the force is with them.

Implications of the Stopping Rule for Nonrenewable Resources

The stopping (entry) decisions of extractive firms are usually considered to be dependent on the quality of their reserves. Quality is identified intuitively as being a function of the grade of reserves, or more generally of marginal extraction costs, and has come to be defined in terms of the order of extraction. Some models involve two or more reserves with constant but different grades (Herfindahl 1967; Hartwick 1977). (5) It is optimal for the higher-grade deposits to be exploited first. Other models have a continuum of grades. In equilibrium, sectorial extraction costs increase through time as recourse is taken to lower grades (Levhari and Liviatan 1977; Solow and Wan 1977). It is implicitly assumed that any given unit of the resource can be extracted at any date. Some units that are not currently being extracted could produce a positive current cash flow but it is optimal to wait to extract those units.

The rule of exploiting the highest grade first applies when there is no lumpy investment or clean-up cost. Price and the flow of the resource are such that condition (3) is satisfied for each (infinitesimal) unit of reserve. Even in such simple models, optimal stopping has some bite. The simplest example is Herfindahl's model, in which unit cost c is constant and there is no capacity constraint. Let the forward value of exploiting a unit of low-cost (high-quality) ore at time t [greater than or equal to] 0 be represented by [w.sub.L](t) = p(t) - [c.sub.L] and the market value of that unit by [[pi].sub.L](t). Price is rising such that the net price of the low-cost deposit, p(t) - [c.sub.L], is rising at rate r while it is in production. At time t = 0, the unit has value

(12) [w.sub.L](0) = [p(0) - [c.sub.L) = [[pi].sub.L](0) > 0.

Now consider a higher-cost (lower-quality) unit, with cost [c.sub.H] > [c.sub.L], and for simplicity of exposition let p(0) > [c.sub.H]. By condition (5), the high-cost ore is not optimally exploited until the low-cost ore is exhausted, beginning at some time [t.sup.*.sub.H] > 0. For t < [t.sup.*.sub.H], (6) a low-cost unit has market value of [[pi].sub.L](t) = p(t) - [c.sub.L]. A high-cost unit has market value of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that

(13) [w.sub.H](0) = [p(0) - [c.sub.H]] < [[pi].sub.H](0) < [[pi].sub.L](0),

with [w.sub.H](t) rising at greater than r%.

Even in this simple case, the traditional NPV rule can fail in both timing and ordering of extraction. Let the low-cost (high-cost) unit be embedded within a mine with [R.sub.L]([R.sub.H]) units of reserves. The Hotelling valuation principle (Miller and Upton 1985) holds in this simple case: the present, forward and market values ([V.sub.L], [W.sub.L] and [[PI].sub.L], respectively) of the low-cost mine at time 0 are

(14) [V.sub.L](0) = [W.sub.L](0) = [R.sub.L][W.sub.L](0) = [R.sub.L][p(0) - [c.sub.L]]

= [R.sub.L][[PI].sub.L](0) = [[PI].sub.L](0).

Once the high-cost deposit is in production (beginning at [t.sup.*.sub.H] > 0), p(t) - [c.sub.H] rises at rate r and the principle again applies:

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As of time t = 0, however, [V.sub.H](0) = [W.sub.H](0) = [R.sub.H][p(0) - [c.sub.H]] and

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

With p(t) - [c.sub.H] rising at greater than r percent, the forward value of the mine is rising at greater than r percent, and its value is increased by delaying extraction. The traditional net-present-value rule, to invest if [R.sub.H][p(0) - [c.sub.H]] > 0, is an incorrect timing rule.

Now let reserves at the high-cost mine be greater than at the low-cost mine to the extent that [R.sub.H] > [R.sub.L][p(0) - [c.sub.L]]/[p(0) - [c.sub.H]]. Then, [R.sub.H][w.sub.H](0) > [R.sub.L][w.sub.L](0). Ranking the two projects by net present value at time zero is also incorrect for establishing the order of extraction: it is still optimal to extract the low-cost deposit first.

There has been a tendency to invoke uncertainty to explain observations that the highest grade is not always developed first, and to hold to the assumption that the only costs are variable costs. Slade (1988), for example, uses uncertain forward prices to explain what she considers to be anomalous behavior, where some deposits with lower extraction cost are extracted later than deposits with higher extraction cost. In reality, mines have capacity constraints, long lives and complicated costs. Cairns and Lasserre (1986, 1991), Gaudet, Moreaux, and Salant (2001), Holland (2003) and others have derived, under certainty, exceptions to the rule of exploitation in order of decreasing physical quality such as ore grade.

Hartwick (1989, p. 56) observes that, under certainty, a mine with a low variable cost may be exploited later than one with a high variable cost if the former has a higher set-up cost. This observation is an implicit recognition of conditions (3) and (5). The next example confirms Hartwick's intuition.

Example 1. Suppose there are two gold mines. Mine L has two units (million ounces) of reserves and unit operating cost of [C.sub.L] = 200. Mine H has six units of reserves and unit operating cost of [c.sub.H] = 300. Suppose that the mines require investment in fixed capacity in order to produce. For ease of computation we assume that capacity must be installed in unit quantities. Mine L, an underground mine, has an investment cost of [C.sub.L] = 200 per unit of capacity and mine H, an open pit, has [C.sub.H] = 30. Since marginal cost is constant, subject to considerations of discreteness the mines operate at capacity through their lives (Crabbe 1982). Let the interest rate be 5% per period.

The forward value, W, is found by optimizing investment I and production q looking forward from the strike point to. We use the following formula for the forward value:

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

such that [q.sub.i](t) [less than or equal to] [K.sub.i]([t.sub.0]) = [I.sub.i]([K.sub.i]([t.sub.0]), [t.sub.0])/[C.sub.i] (since investment cost per unit is [C.sub.i]). Values of the variables in the formula for [t.sub.0] = 1, 2, 3, 4 are obtained by trying integral values of [K.sub.i]([t.sub.0]), computing a net present value, and taking the highest present value as the forward value. Results are reported in table 1. For mine L, the optimal level of investment, [I.sub.L]([K.sub.L]([t.sub.0]), [t.sub.0]), is always 200 (one unit of capacity since [C.sub.L] = 200). The forward value of mine L rises by 8% between periods 1 and 2 and then falls. Therefore, it is optimal to wait until period 2 to invest in mine L. Production is one unit in periods 2 and 3; then the reserves are exhausted.

For mine H the optimal level of investment varies according to its timing. The forward value of mine H rises by only 2% between periods 1 and 2 and then falls. It is optimal to open mine H in period 1, with two (60/30) units of capacity. (The choice of strike time for mine H is a corner solution.) Production from mine H is at capacity in periods 1, 2, and 3.

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

The usual approach to the analysis of sequential development when there are setup costs masks the r-percent stopping rule (3) for investment decisions. Fischer and Laxminarayan's (2005) study of pesticides and antibiotics, for example, derives necessary conditions for the dates of transition from one variety to another which appear to differ from rule (3). To examine the implications of sequentiality for the stopping rule, we explicitly adopt their assumption that development is in sequence. We show that the optimality conditions that they find and stopping rule (3) are consistent. It is also of interest that, if the "utility" function u is interpreted as monopoly profit, their model is an example of a noncompetitive market in which rule (3) holds. Their notation is different from ours above; for ease of comparison we adhere to their notation as closely as possible.

At time t = 0, a firm makes an investment K that gives rise to an integral of discounted net benefits, U(S, T) = max [[integral].sup.T.sub.0] u(q(t))[e.sup.-rt]dt, during the exploitation of the first variety, which acts as a nonrenewable resource with stock S (so that [[integral].sup.T.sub.0] q(t)dt [less than or equal to] S). After the stock is depleted at some time T, a new investment, having forward value V (recall that our notation is W), is made. The present value of the firm is

(18) F(V) = max[-K + U(S, T) + [Ve.sup.-rT]].

At time T (the optimal time of transition to the next variety), the following holds:

(19) u(q) - qu'(q) = rV.

A particular case in which u' (q) = [q.sup.-1/[eta] receives special attention.

A more elaborate rendering of the problem allows for nonstationarity. There may be natural variation in the incidence of the pest or disease. Let the marginal utility function vary sinusoidally but "on average" be the same as in the original problem:

(20) u'(q, t) = [1 + a cos([omega]t + [theta])][q.sup.-1/[eta], where a < 1.

Also let the set-up cost be a function of time, represented by [kappa](t). We abstract from the possible nonstationarity of the force of interest to emphasize these sources. Depending on the phase, [theta], and the levels of a and [kappa](t), it may pay to wait to make the first investment or to wait between exhaustion of the first variety and investment in the second. The problem is now the optimal management of a compound option over various types of investment. Let the point of initial investment be represented by [T.sub.0] [greater than or equal to] 0; the time of exhaustion of the first variety by [T.sub.1] > [T.sub.0]; and the time of investment in the second by [T.sub.2] [greater than or equal to] [T.sub.1]. We expand notation in the obvious way and write the objective as

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

subject to the exhaustibility of each variety and the conditions on [T.sub.0], [T.sub.1], and [T.sub.2], which enter the problem as constraints. We also note that

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The problem can be solved as an optimal control in two stages (Tomiyama 1985; Amit 1986; Makris 2001). We depart from this literature by representing present values as forward values at the strike point discounted to the present (as above) and admitting the possibility that investments may not be made immediately. Given an initial "stock" S of variety no. 1, the Lagrangian for the problem of finding the decision times is

(23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where V([T.sub.2]) is the optimally controlled continuation value after investment at [T.sub.2] in the second (and subsequent, if any) stages.

The first-order condition for the choice of the initial investment date is

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [T.sub.0] > 0 (an interior solution) then [[mu].sub.1] = 0 and the Hamiltonian [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also zero. Furthermore, the first-order condition for the optimization with respect to q on the interval ([T.sub.0], [T.sub.1]) is that [lambda] = [u.sub.q]. Therefore, [u--[qu.sub.q]] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since u(q, t) is a strictly concave function of q, we have q([T.sub.0]) = 0. Therefore,

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the r-percent rule (3) holds for the choice of [T.sub.0]. If [[mu].sub.1] > 0 then [T.sub.0] = 0 and

(26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The condition that [[mu].sub.1] > 0 implies that [T.sub.0] = 0 and is not freely chosen. Condition (3) becomes an inequality whenever the strike time is constrained.

The first-order condition for the time of exhaustion of the first deposit is

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also, the time of investment in the second deposit obeys the condition

(28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [[mu].sub.2] > then [T.sub.2] = [T.sub.1] and

(29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This equation generalizes the transition condition (19) by including the term -l)([T.sub.1]), which Fischer and Laxminarayan implicitly assume to be zero. (12) Their condition is equivalent to the one stressed by Tomiyama, Amit and Makris, of equality of the Hamiltonians of the two stages at the transition. The Hamiltonian of the first stage, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is equal to the LHS of equation (29). In the maximization in equation (18), the term rV (with I? = 0) is the derivative of the value function of the second stage with respect to [T.sub.2]: rV = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the Hamiltonian in the second stage. The interpretation also holds for equation (29).

Equation (29) implies that V([T.sub.2])/V([T.sub.2]) < r when the constraint that [T.sub.2] = T1 is effective, (13) so that the rate of change of net benefits from the second variety falls below r. The reason is that net benefits from the second variety are not maximized freely since it is not developed until immediately after the first is exhausted. This masking of rule (3) bears comparison with the theory of the mine under capacity constraints and the masking of the r-percent rule that applies to individual units of mineral (cf., Example 1).

On the other hand, condition (28) for the choice of [T.sub.2] expresses the r-percent stopping rule (3) when the constraint is not effective, i.e., when [[mu].sub.2] = 0. Since a transversality condition is that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 0 and V([T.sub.2]) - rV([T.sub.2]) = 0, equation (29) holds at [T.sub.2]. On the interval ([T.sub.1], [T.sub.2]), V(t) - rV(t) > 0.

A special case of sequential or compound options is the exploitation of a forest. Planted land provides an option to harvest. In the ith rotation, let the cost of planting at the optimal (strike) time [t.sup.*.sub.i] be represented by P([t.sup.*.sub.i]) and the forward harvest value at (strike) time [T.sub.i] by [w.sub.i]([T.sub.i]|[t.sup.*.sub.i]). Faustmann's rule determines the optimal harvest time [T.sup.*.sub.i] and the market value of bare land, L(t).14 It is commonly expressed using first-order conditions and rates of growth within a rotation but also implies that

(30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or that

(31) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Equation (31) recognizes the possibility of an option to plant, i.e., of a gap between harvesting at [T.sup.*.sub.i] and planting at [t.sup.*.sub.i+1]. The market value of bare land for t [member of] [[T.sup.*.sub.i], [t.sup.*.sub.i+1]] is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], consistently with rule (8) for market value and with rule (5) for forward value. (15) Also consistently with rule (8), the market value of trees and land at the end of rotation i is equal to the accumulated value of newly planted land:

(32) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If planting is suboptimally delayed until [T.sub.i+1] > [t.sup.*.sub.i+1] and the land is managed optimally there-after, the forward and market values at [t.sub.i+1] are equal to the value of bare land L([t.sub.i+1]). Once the forest is replanted, the forward value, W([T.sub.i+1]| [t.sub.i+1]) = [w.sub.i+1]([T.sub.i+1]| [t.sub.i+1]) + L([T.sub.i+1]), obeys rules (3), (5), and (6) for harvests at, before, and after [T.sup.*.sub.i+1].

In the stationary conditions usually assumed, L([T.sup.*.sub.i]) = L([t.sup.*.sub.i]): bare land has a constant value because the forward value is stationary. Also, [t.sup.*.sub.i+1]) = [T.sup.*.sub.i]: replanting is immediate by condition (6). For any planting time [t.sub.i], figure 2 illustrates (a) an option value to letting a forest grow until the optimal harvest time, [T.sup.*.sub.i], (b) the smooth-pasting condition at [T.sup.*.sub.i], and (c) the forward-value rules (3), (5), and (6).

[FIGURE 2 OMITTED]

Faustmann's rule has been known for the point-input, point-output problem since 1849, and has historically been cited by many as a particular case of the r-percent stopping rule (3) without making the generalization to other assets. Our analysis of sequential development shows that rule (3), sometimes constrained, holds for the time of planting as well as the time of harvest. Furthermore, these properties hold mutatis mutandis if the land is taken out of forestry and put to another use, provided that that use is incorporated into an appropriately modified value function.

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."

(12) Their condition arises because, under the stationary conditions frequently studied in intertemporal models, waiting to invest has a cost, viz., interest on the optimized value of continuing the sequence immediately.

(13) It is obviously true under stationary conditions, when V([T.sub.2]) - 0.

(14) The market value L(t) is the discounted forward value assuming optimal choices of future planting and harvest times. It is the value of the option to plant.

(15) The forward value of planting at [t.sub.i+1] < [t.sup.*.sub.i+1]) is less than the market value:[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and that forward value is rising at a rate greater than r percent.

Robert D. Cairns is professor in the Department of Economics, McGill University, Montreal, QC, Canada. Graham A. Davis is professor in the Division of Economics and Business, Colorado School of Mines, Golden, CO. Table 1 Decision Inputs Mine L L H H t p I W I W 1 400 200 200 60 * 560 * 2 410 200 * 216 * 90 * 571 * 3 416 200 * 159 * 180 * 516 * 4+ 350 200 93 30 236 * indicates numbers refer to a special set of numbers.