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

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.