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.