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

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.