Resource depletion under uncertainty: implications for mine depreciation, Hartwick’s Rule and national accounting

Resource and Energy Economics 25 (2003) 219–238

Resource depletion under uncertainty: implications for mine depreciation, Hartwick’s Rule and national accounting David W. Butterfield∗ Department of Economics, McMaster University, 1280 Main Street West, Hamilton, Ont., Canada L8S 4M4 Received 20 October 2000; received in revised form 5 March 2002; accepted 5 August 2002

Abstract Lozada’s equation [Resource and Energy Economics 17 (1995) 137] for the change in value of a non-autonomous dynamic program is generalized to stochastic control and applied to the depreciation of a competitive mine facing price, reserve and discount rate uncertainty. Mine depreciation includes the costs of these risks, as well as an adjustment to the ‘net price’ used to value depletion and revisions. The change in value equation also provides the basis for a stochastic version of Hartwick’s Rule [American Economic Review 67 (1977) 972] for sustainable consumption, that the risk adjusted value of net investment equal zero. The analysis has implications for the Weitzman [Quarterly Journal of Economics 90 (1976) 156] welfare measure and the stochasic Hamiltonian. © 2002 Elsevier Science B.V. All rights reserved. JEL classification: C61 Keywords: Stochastic control; Mine depreciation; Hartwick’s Rule; National accounting

1. Introduction Interest in sustainable development and the integration of natural resources into national accounts has stimulated the search for theoretically correct measures of natural resource depletion. Empirical studies (Foy, 1991; Bartelmus et al., 1993; Tengblad, 1993; Smith, 1994; US, 1994; Diaz and Harchaoui, 1997) have experimented with a variety of approaches to measuring the depletion of non-renewable resources. Early theoretical discussions of measuring non-renewable resource depletion include Landefeld and Hines (1985), El Serafy (1989), and Hartwick (1990). Lozada (1995) developed a general equation for the change ∗ Tel.: +1-905-525-9140x23818; fax: +1-905-521-8232. E-mail address: [email protected] (D.W. Butterfield).

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in the value-function of a non-autonomous dynamic program, which he used to derive an equation for mine depreciation. Vincent et al. (1997) derived a special case of Lozada’s result in their analysis of Hartwick’s Rule in a small resource-exporting economy. Hartwick and Long (1999) also derive Lozada’s equation as a part of their analysis of Hartwick’s Rule in non-autonomous models. Cairns (2000) derives a version of the Lozada’s equation for a model of the competitive mine in which the size of the initial capital investment implies a constraint on the capacity of the mine. Lozada (1995) applies his equation to the problem of optimal extraction by a competitive mine, which is a non-autonomous problem because the price of the extracted resource is exogenous and changes over time. Lozada’s equation assumes perfect foresight so that the future price profile of the extracted resource is known with certainty. However, both theoretical studies of mineral extraction and empirical valuation of mineral deposits emphasize the effects of uncertainty about future prices (Pindyck, 1981; Brennan and Schwartz, 1985; Dias and Rocha, 1999) and the size of the remaining resource stock (Pindyck, 1980) on the extraction decision and on the value of the mine. Crabbé (1982) provided an early summary of these sources of uncertainty and their implications for the theory of non-renewable resource extraction. Thus, it is desirable to have an equation for mine depreciation which takes these important uncertainties into account. This paper generalizes Lozada’s results to stochastic non-autonomous control problems and to the theory of the competitive mine under uncertainty. The following section presents the stochastic version of Lozada’s general equation for the change in the value function and this result is applied to the competitive mine in Section 3. Lozada (1995) also points out that his equation implies a general version of Hartwick’s (1977) Rule for sustainable development, which has been prominent in the literature on sustainable development. Vellinga and Withagen (1996) and Asheim (1997) provide succinct summaries of this literature. Here again, the presence of important uncertainties make it desirable to have a stochastic version of Hartwick’s Rule, which is presented in Section 4. Of course, a stochastic Hartwick’s Rule is only one of many ways to incorporate uncertainties into the discussion of sustainable development. Finally, the envelope equation for the current value Hamiltonian, which provides the basis for Lozada’s result, also provides the basis for the theory of national accounting. In Section 5, the implications of the stochastic version of the envelope equation for the theory of national accounting, first considered by Aronsson and Löfgren (1995), are discussed.

2. The change in the value function of a non-autonomous stochastic control problem One formulation of the stochastic control problem (see Fleming and Rishel, 1975; or Fleming and Soner, 1992) may be stated as
 T  maximizeEt f(x(s), u(s), s) ds + F(x(T)) (1) u(s)

t

subject to dx(s) = g(x(s), u(s), s) ds + σ (x(s), u(s), s) dz(s)

(2)

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221

and x(t) given.where x(s) is an n×1 vector of state variables, u(s) an m×1 vector of control variables, dz(s) a d × 1 vector of the increments of independent Brownian processes, f(s) a scalar-valued flow of benefit function, F a scalar valued function of the terminal date state variables, g(s) an n × 1 vector of scalar functions, gi , which describe the deterministic evolution of the state variables, σ (s) an n × d matrix of scalar functions, σ il , which describe the uncertainty associated with the changes in the state variables, and T is a fixed terminal time. Alternative formulations drop F and allow the terminal time, T, to be freely chosen or use an infinite time horizon. The problem can also be modified by adding constraints on the control and/or state variables. The vector of optimal controls, uo , maximizes the stochastic Hamiltonian, H(x(s), u(s), p(s), px (s), s) = f(x(s), u(s), s) + p(x(s), s; T) · g(x(s), u(s), s) + 21 trace(p x (x(s), s; T)Σ(x(s), u(s), s))

(3)

where p(x(s), s; T) = Jx (x(s), s; T) is an n × 1 vector of marginal values of the state variables, px (x(s), s; T) = Jxx (x(s), s; T) is an n × n matrix of second-order partial derivatives of J with respect to the state variables, and Σ(x(s), u(s), s) ≡ σ (x(s), u(s), s)σ (x(s), u(s), s) is the n×n variance–covariance matrix of the state variables; so that uo = uo (x(s), p(x(s), s; T), px (x(s), s; T), s) = uo (x(s), s; T) is the vector of optimal closed-loop controls. Since the arguments of uo evolve stochastically and only include the current values of the state variables and time, it is appropriate to think of uo as a Markov control process (Fleming and Soner, 1992). Fleming and Soner (1992, pp. 158–161) provide sufficient conditions for the existence of the optimal control, uo . These include continuity of f, F, g and σ ; continuous 1st derivatives of f, g and σ with respect to u; boundedness conditions on the t and x derivatives of g and σ ; polynomial growth conditions on f and F; and conditions on the Brownian process, dz. The stochastic Hamiltonian represents the contribution of today’s events (choices of the control variables, u; stochastic events, dz; and passage of time, dt) to present (f) and future (p·g) welfare, adjusted for the cost (if Jxx = px is negative definite) of the risk ((1/2) trace (px Σ)) associated with the changes in the state variables. The optimal value function is
 T  o o o o J(x(t), t; T) = Et f(x (s), u (x (s), s; T), s), s) ds + F(x (T)) (4) t

uo (x o (s),

where s; T) is the vector of optimal closed-loop control variables, and x o (s) the stochastic process implied by the optimal control process, uo (x o (s), s; T). Notice that this implies that J(x o (T), T ; T) = F(x o (T)). The optimal value function is related to the optimized stochastic Hamiltonian by the Hamilton–Jacobi–Bellman (HJB) necessary condition (Fleming and Soner, 1992) −Jt (x(t), t; T)=H(x(t), uo (x(t), p(t), px (t), t; T), p(t), px (t), t; T)=H o (x(t), t; T). (5) Since p and px are equal to Jx and Jxx , respectively, the HJB equation is a second-order partial differential equation, which, together with the terminal condition, J(x(T), T ; T) =

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F(x(T)) must be satisfied by the optimal value function, J. If the partial derivatives Jt , Jx and Jxx exist and are continuous, then the optimal value function may be the unique “classical” solution to the HJB equation. If the value function is not sufficiently smooth, as in many applications in economics, there may be many solutions to the HJB equation, but under some conditions the optimal value function will be the unique “viscosity” solution to the HJB equation (Fleming and Soner, 1992, Chapter 5). Even at points where the value function is not twice continuously differentiable, the HJB equation is “close” to being satisfied. Thus, the HJB necessary condition is relevant to the optimal value function even when the value function is not twice continuously differentiable. A good overview of viscosity solutions to partial differential equations is provided by Crandall et al. (1992). Here, in order to avoid these mathematical complexities, we will assume that the value function, J, has continuous first- and second-order partial derivatives with respect to x(t) and a continuous first derivative with respect to t. Lozada’s equation for the change in the value function is based on a result which is often referred to as the envelope property of the optimized Hamilton (Seierstad and Sydsaetter, 1987, p. 86 and Aronsson and Löfgren, 1993, 1995); namely, that the time derivative of the Hamiltonian along the optimal path equals its partial derivative with respect to time. This should not be confused with the dynamic envelope theorem (Caputo, 1990; LaFrance and Barney, 1991) which refers to an envelope property of the optimized value function. The stochastic control version of the envelope property of the optimized Hamilton is given by Eq. (6).
 ∂H o 1 o o dH = (6) dt + Hx · σ dz = ft + p · g t + trace(p x Σ t ) dt − p t · σ dz ∂t 2 since Hxo = −pt (see appendix Eq. (A.1)). A proof of Eq. (6) is outlined in Appendix A. Letting σ go to zero leads to the certainty result, dH o /dt = ∂H o /∂t, where the partial derivative indicates differentiation before substitution of p(x(t), t) and p x (x(t), t) into Ho (see LaFrance and Barney, 1991). Taking expected values of (6) leads to a result which is analogous to the certainty case, Et [dH o ] = (∂H o /∂t)dt. Every stochastic differential equation such as (6) has an equivalent stochastic integral equation (Fleming and Rishel, 1975), which for (6) is H o (x(T), T ; T) − H o (x(t), t; T)  T  T ∂H o Hxo (x(s), s; T) · σ (x(s), s) dz(s) (x(s), s; T) ds + = ∂s t t Rearranging (7) and taking expected values implies that  T ∂H o o o H (x(t), t; T) = Et H (x(T), T ; T) − Et ((s), s; T) ds ∂s t

(7)

(8)

since the expected value of the last term in (7) is zero. Thus, today’s value of the optimized Hamiltonian equals today’s expected value of the final date Hamiltonian less the expected contribution of deterministic exogenous events to the final date Hamiltonian. If there are no deterministic exogenous events (∂H o /∂s = 0), then today’s Hamiltonian equals the expected value of the final date Hamiltonian.

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Eq. (8) provides the basis for the change in value equation, which is given by Eq. (9). 1 dJ = p · (g dt + σ dz) + trace(p x Σ) dt 2 
 T 1 (fs + p · g s + trace(p x Σ s ))ds dt − [Et H o (T)] dt + Et 2 t

(9)

A proof is outlined in Appendix A. This equation reduces to Lozada’s proposition 1 under perfect foresight, when σ and Σ = 0 and the expectation operators are dropped. The first term in this equation represents the value of deterministic and stochastic changes in the state variables, and the second term represents the cost of the risk associated with the stochastic changes in the state variables. Thus, the first and second terms together represent the value of risk-adjusted net investment for the stochastic program. The third term represents a capital gain equal to today’s expected value of the sum of the future effects of today’s (see the discussion below in Section 3) exogenous deterministic changes on the benefit, growth and risk functions, and the fourth term represents a capital loss equal to today’s expected value of the effect of the horizon drawing nearer. Note that Eq. (9) represents the actual change in the value function and not the expected value of dJ over the interval dt. This change equals the risk-adjusted value of net investment plus net capital gains. A similar result can be established for the discounted version of the above problem. If the time dependence of the benefit function, f, includes a discount factor, A(t), so that f(x(t), u(t), t) = A(t)f c (x(t), u(t), t)

(10)

then an equivalent “current-value” problem can be defined as J c (x(t), t; T) = A(t)−1 J(x(t), t; T) = maximum A(t)−1 Et u




T t

 A(s)f c (x(s), u(s), s) ds + A(T)F(x(T)) (11)

subject to dx(t) = g(x(t), u(t), t)dt + σ (x(t), u(t), t)dz and x(t) given. The associated current-value stochastic Hamiltonian is H c (x(t), u(t), t; T) = A(t)−1 H(x(t), u(t), t; T) = f c (x(t), u(t), t) + p c (x(t), t; T) · g(x(t), u(t), t) + 21 trace(p cx (x(t), t; T)Σ(x(t), u(t), t))

(12)

c . The HJB necessary condition for where pc = A(t)−1 Jx = Jxc and p cx = A(t)−1 Jxx = Jxx the current-value problem is

−Jtc (x(t), t; T) + ρ(t)J c (x(t), t; T) = H c (x(t), uo (x(t), p c (t), pcx (t), t; T), pc , pcx , t; T) = H co (x(t), t; T)

(13)

where ρ(t) = −A(t)−1 A (t) is the instantaneous rate of discount. Note that in the standard case of a constant discount rate, A(t) = e−ρt and −A(t)−1 A (t) = ρ.

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The envelope property of the current-value stochastic Hamiltonian is stated in Eq. (14).


dH

co

 ∂H co co c + ρ(t)(H − f ) dt + Hxco · σ dz = ∂t
  1 1 c c c c c = ft + p · g t + trace(p x Σ t + ρ(t)) p · g + trace(p x Σ) dt 2 2 c c − [pt − ρ(t)p ] · σ dz

(14)

The deterministic component of (14) equals the deterministic exogenous change in Hco plus the subjective interest on the value of the changes in the state variables (net investment) adjusted for risk. Note again that taking expected values of (14) yields a result analogous to the certainty case, Et [dH co ] = [(∂H co /∂t) + ρ(t)(H co − f c )]dt. Rearranging the stochastic integral form of (14) and taking expected values yields an equation for the optimized current-value stochastic Hamiltonian. H co (x(t), t; T) = A(t)−1 A(T)Et H co (x(T), T ; T) − Et  + Et

T t

A(t)−1 A(s)ρ(s)f c (s) ds



T t

A(t)−1 A(s)

∂H co (s) ds ∂s (15)

The additional (third) term in the current-value result (15) as compared to the present-value result (8) represents the effect of moving the base period for discounting ahead by ds at each instant of time. See Appendix A for derivations of Eqs. (14) and (15). Eq. (15) provides the basis for the expression for the change in the current-value function. 1 dJ c = p c · (g dt + σ dz) + trace(p cx Σ) dt 2 
 T 1 −1 c c c A(t) A(s)(fs (s)p (s) · g s (s)) + trace(p x (s)Σ s (s)) ds dt + Et 2 t   −1 co − A(t) A(T)Et H (x(T), T ; T) dt + ρ(t)J c dt 
 T −1 c A(t) A(s)ρ(s)f (s) ds dt (16) − Et t

The additional terms in the current-value Eq. (16), as compared to the present-value Eq. (9), represent the effect of moving the base-year for discounting, t, forward by dt. Note that in Eq. (16)  ρ(t)J c (t) − Et

T t

A(t)−1 A(s)ρ(s)f c (s) ds = Et



T t

A(t)−1 A(s)(ρ(t) − ρ(s))f c (s) ds (17)

so that these two terms cancel if the instantaneous discount rate is constant. A derivation of Eq. (16) is provided in Appendix A.

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3. Depreciation of the competitive mine Lozada (1995) applies his deterministic formula for the change in the value function to the value of the mine and mine depreciation. For a competitive mine, the flow price of the extracted resource, p(t), is exogenous. If q(t) is the extraction flow, R(t) is the level of the remaining stock and r(t) is the exogenous interest rate (which may or may not be constant), then f c (R(t), q(t), p(t)) = π(t) = [p(t)q(t) − C(q(t), R(t))]

(18)

is the current flow of profit, where the derivatives of total extraction cost function, C, are assumed to satisfy Cq > 0, Cqq ≥0, CR < 0, and CqR < 0. In the certainty case, the dynamics of the remaining stock are given by dR/dt = −q(t), where R(0) = R0 is given, so that g(t) = −q(t). In the optimal control interpretation of the problem, q(t) is the single control variable and R(t) is the single state variable, while p(t) and r(t) are uncontrolled exogenous state variables, which some authors (LaFrance and Barney, 1991, p. 373, for example) treat as time-varying parameters. The treatment of p(t) and r(t) as time-varying parameters, which is implicit in Lozada’s and others discussions of non-autonomous models, implies that the current value Hamiltonian for the certainty case is H c (R(t), q(t), t) = f c (t) + JRc (t)g(t) = [p(t)q(t) − C(q(t), R(t))] − JRc (t)q(t)

(19)

and the optimal control, qo (t), satisfies p(t) − Cq (qo p, R(t)) = JRc (t)

(20)

or net price (price less marginal extraction cost) equals the marginal value of a unit of the resource stock, JRc (t). Lozada assumes that the endpoint is freely chosen so that the solution satisfies the terminal condition H c (Ro (T), qo (T), T) = 0. This assumption also implies that the current value Hamiltonian and the current value function do not depend on T. Lozada’s (1995, Eq. (7)) deterministic equation for the change in the current value of the mine is  T  s dJ c (t) dp(s) o e− t r(v)dv = −JRc (t)qo (t) + q (s) ds dt ds t  T  s + e− t r(v)dv (r(t) − r(s))πo (s) ds (21) t

The first term represents physical depletion valued at net price (many authors use the term depletion for the change in value of the resource as distinct from the change in the value of the mine, dJc (t)/d(t), see Cairns (2000) for example); the second term represents the change in the value of the mine due to exogenous price change; while the third term represents the effect of exogenous interest rate changes. For simplicity, we will assume a constant interest rate in what follows, so that the last term disappears and (21) becomes  T dJ c (t) dp(s) o e−r(s−t) (22) = −JRc (t)qo (t) + q (s) ds dt ds t Vincent et al. (1997) also derive (22) (their Eq. (A.19)) in their examination of the resourceexporting small open economy. The resource extraction portion of their model is formally

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equivalent to the competitive mine since both the flow price of the resource and the interest rate are exogenous. Cairns (2000) also derives a version of Lozada’s equation for the competitive mine. Cairns works with a model in which initial fixed capital investment is necessary to develop and open the mine and also places a limit on the capacity of the mine (Campbell, 1980; Crabbé, 1982; Cairns, 1998; Davis and Moore, 1998). While his formula for the change in the value of the open mine is consistent with Lozada’s, he points out that the value of the mine represents both the value of the resource and the fixed capital invested in the mine. Thus, given a depreciation schedule for fixed capital, the change in the value of the resource can be calculated by subtracting the depreciation of the fixed capital from the change in value of the mine. Since there is no unique way to calculate the depreciation of the fixed capital, there is no unique way to calculate the depreciation of the resource. Since Lozada’s model does not have an initial fixed capital investment, all of the value of the mine is attributed to the resource, as implied by his definition of the “stock price”, Jc (t)/R(t). The stochastic model of the competitive mine allows stochastic change in both the resource stock and the exogenous price. We assume that the evolution of price follows dp(t) = αp(t) dt + σ p p(t) dzp (t)

(23)

where α and σ p are constant and dzp is a Brownian motion. Under this assumption, p(t) is log-normal (Pindyck, 1980, 1981). The assumption that the resource price follows the geometric Brownian motion given by (23) is convenient but may not be the best representation of resource prices. Alternative models for oil prices, which include mean reversion and jump processes, are discussed by Dias and Rocha (1999) and Dixit and Pindyck (1994, pp. 403–405). We also assume that the remaining resource stock evolves stochastically according to dR(t) = −q(t) dt + σ R dzR (t)

(24)

where σ R is the constant standard deviation of R(t) and zR is a univariate Brownian motion. In the context of the mine, it is natural to think of the remaining stock, R, as being today’s estimate of the remaining stock and the stochastic term as revisions to those estimates (see Gilbert, 1979). Using this interpretation, we would expect the size of revisions to be larger, the larger is the remaining stock and the larger is the amount extracted so that it would be better to let σ R depend on R and q, with σRR , σqR > 0. However, the assumption that σ R is constant has been used extensively in the literature (Pindyck, 1980, 1981), and we will use it first before adding this complication. Following Pindyck (1980, 1981) in treating p(t) as an exogenous state variable and assuming that the stochastic terms are uncorrelated (a reasonable assumption for a competitive mine), the stochastic current value Hamiltonian becomes H c (R(t), p(t), q(t), t) = [p(t)q(t) − C(q(t), R(t))] − JRc (t)q(t) + Jpc (t)αp(t) c c + 21 JRR (t)(σ R )2 + 21 Jpp (t)(σ p )2 p2 (t)

(25)

The term representing deterministic price change, Jpc (t)αp(t), represents the effect of today’s deterministic price change on the value of the mine (see the discussion below). Since σ p and σ R are constant, the first-order condition for qo , (20), is unchanged. This is because p(t)

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is exogenous, so that its evolution is unaffected by q and R. While it is conventional in the stochastic case to treat p(t) as an exogenous state variable, in the certainty case it has been treated as a time-varying parameter, suggesting the possibility of treating p(t) as a stochastic parameter. This treatment would remove the p(t) and p2 (t) terms from the equation for the current value Hamiltonian (25), but these terms would still appear in the HJB equation and the change in value formulas. The choice affects the interpretation of the Hamiltonian. If p(t) is treated as an exogenous state variable and included in the Hamiltonian, then the Hamiltonian is interpreted as the contribution of today’s events to value, including today’s profit and the effect of today’s physical depletion and today’s price change on future profit. If p(t) is treated as a stochastic parameter and not included in the Hamilton, the Hamiltonian only includes endogenous variables and must be interpreted as the contribution of today’s choices of endogenous variables, namely qo , to value. The formula for the change in the current value of the mine is c (t)(σ R )2 dt dJ c (t) = −JRc (t)qo (t) dt + JRc (t)σ R dzR + 21 JRR c + Jpc (t)[αp(t) dt + σ p p(t) dzp ] + 21 Jpp (t)(σ p )2 p2 (t) dt

(26)

where JRc (t) is given by (20) and equals net price. The first term also appears in Lozada’s Eq. (22) for the certainty case and represents physical depletion, valued at net price. The second term represents revisions which, under these assumptions, are exogenous and stochastic. c < 0), then the third term represents If the current value function is strictly concave in R (JRR the cost of the risk associated with revisions. Even though the flow objective (profit) is risk neutral, if the cost function is convex in R, profit will be concave in R, as will the value function. The fourth term represents the change in value due to the actual (deterministic c < 0) of the risk asand stochastic) price change, and the last term represents the cost (if Jpp R p sociated with the stochastic price change. If σ and σ go to zero, (26) reduces to Lozada’s equation, as Jpc (t)αp(t), is equivalent to the integral in (22). The apparent difference between Lozada’s Eq. (22) and the deterministic version of (26) occurs because Lozada follows the tradition in the certainty model of treating p(s) as a time-varying parameter so that the effects of deterministic changes in p(s) appear in the partial derivative of the current value Hamiltonian with respect to time, ∂Hco (s)/∂s. If p(s) is treated as an exogenous state variable, as is the tradition in stochastic control models of the mine, then the effects of the deterministic changes in p(s) appear explicitly as Jpc (t)αp(t) dt for the log-normal specification in (23). The equivalence of the term representing deterministic price change in (26) and (28), Jpc (t)αp(t), to the integral in Lozada’s Eq. (22) deserves more discussion because it helps to clarify the interpretation of this integral. In order to compare these terms assume that σ R and σ p equal zero, so that price is deterministic and (26) reduces to dJ c (t) = −JRc (t)qo (t) dt + Jpc (t)αp(t) dt

(27)

Since the exogenous price is deterministic in this case, the last term of (27) and the last term of (22) should be equivalent and we should have  T dp(s) o Jpc (t)αp(t) = q (s) ds e−r(s−t) (28) ds t

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Since σ p = 0, dp/dt = αp(t) and p(s) = p(t)eα(s−t) , and the left hand side of (28) becomes
 T  ∂ dp(t) ∂J c (t) dp(t) −r(s−t) = e π(s) ds ∂p(t) dt ∂p(t) t dt 
 T dp(t) dp(s) e−r(s−t) qo (s) ds (29) = dp(t) dt t This equals the right-hand side of (28) only if the dp(s)/ds term in the integral on the right hand side of (28) is equal to (dp(s)/dp(t)) × (dp(t)/dt). Thus, the integral in Lozada’s Eq. (22) and, by implication, the corresponding integral terms in the general results (9) and (16) should be interpreted as the present value of the future marginal effects of today’s price change dp(t), operating through its effect on future prices, dp(s). LaFrance and Barney (1991) use a similar analysis in discussing time-varying parameters in the context of the envelope properties of the deterministic value function (their dynamic envelope theorem). The above argument can also be made in a different way. Suppose that p(s) = p(t)eα(s−t) but that today’s exogenous price change did not occur. Then the new price profile, p∗ , starts at p∗ (t + dt) = p(t), while for the old price profile, p(t + dt) = p(t)eα dt . Thus, for s ≥ t + dt, p(s)/p∗ (s) = eα dt . As dt goes to zero, $p(s) = p(s) − p∗ (s) goes to p(s)α dt for s ≥ t + dt. The effect on Jc is given by  T  T e−r(s−t) $p(s)qo (s) ds = e−r(s−t) αp(s) dt qo (s) ds $J c (t) = t t
 T  dp(s) o = e−r(s−t) (30) q (s) ds dt ds t The logic of the argument is that the change in t, dt, causes a change in p(t), dp(t) = (dp(t)/dt) × dt; the change in p(t), given a profile, p(s), which could be deterministic or stochastic, causes a change in the price profile,dp(s) = (dp(s)/dp(t)) × dp(t) = (dp(s)/dp(t)) × (dp(t)/dt) × dt; the change in the price profile causes a change in the future flow of profit = dp(s)qo (t). It is the change in the future flow of profit which changes the value function, so that “today’s effect” is the change in the value function over the interval (t, t + dt). This logic applies whatever the form of p(s). If we return to our discussion of the change in value function and drop the assumption that the size of the stochastic component of dR is independent of q and R, the stochastic Hamiltonian (25) and the equation for the change in value of the mine (26) are unchanged, but the first order condition which determines qo becomes c p(t) − Cq (qo (t), R(t)) + JRR (t)σ R (qo (t), R(t))σqR (qo (t), R(t)) = JRc (t) (31) q

c (t) < 0 and σ > 0, the new term represents the cost of the risk of revision associated If JRR R with marginal extraction. The marginal value of the resource stock now equals net price adjusted for this risk. Substituting for JRc (t) from (31) in the expression for the change in the value of the mine (26), it becomes c dJ c (t) = [p(t) − Cq (t) + JRR (t)σ R (t)σqR (t)][−qo (t) dt + σ R (t)dzR (t)] c c + 21 JRR (t)(σ R )2 dt + Jpc (t)[αp(t) dt + σ p p(t) dzp ] + 21 Jpp (t)(σ p )2 p2 (t) dt

(32)

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229

The value of both physical depletion and revisions is now reduced by the risk cost of marginal extraction. Eq. (32) implies that, to accurately measure the change in value of the mine, the value of revisions should be reduced by the cost of the risk associated with revisions and, further, that the net price used to value both physical depletion and revisions should be reduced by the marginal risk cost of extraction. Also capital gains or losses associated with changes in the price of the extracted resource should be included, and mine depreciation must include the risk cost of these changes. Brennan and Schwartz (1985), in a model in which price evolves stochastically but the resource stock evolves deterministically, introduce the possibilities that the mine may be closed, reopened or abandoned in response to the stochastic movement of price. An equation for the change in value of the mine based on (32) would only apply when the mine is open. Brennan and Schwartz also assume that the exogenous stochastic movement of the resource price is reflected in a futures market for the extracted resource so that the mine owners can completely hedge against uncertainty. Their theory of finance approach leads to a slightly different necessary condition for the value function than the HJB condition implicit in the above analysis. Because the Brennan and Schwartz theory of finance approach assumes that the risk in future prices can be represented by the relationship between observed spot and futures prices, it has been widely used to value various claims on mineral deposits (see Dias and Rocha, 1999, for a summary). However, since the stochastic component of the change in reserves will be different for each mine, it will not generally be possible for a particular mine to find a set of financial assets which can be used to completely hedge this risk. Thus, the finance approach cannot be used to deal with this source of risk for a single mine. In the context of measuring mine depreciation on national accounts, however, it might be possible to use average data for the mining industry to capture the risk associated with revisions. In the context of the competitive mine, it is natural to introduce a third source of uncertainty by allowing the discount factor, which represents the opportunity cost of capital, to evolve stochastically. Weitzman (1998) examines this assumption in the context of the basic national accounting model, assuming that the discount factor also follows a log-normal process, dA(t) = −rA(t) dt + σ A A(t) dzA (t)

(33)

where r and σ A are constant. This specification implies that the instantaneous discount rate, ρ(t), satisfies ρ(t) dt = −

dA(t) = r dt − σ A dzA (t) A(t)

(34)

Assuming that r is constant rules out deterministic change in the instantaneous discount rate. While it may not be realistic to assume that dzA and dzp are independent, for simplicity we will assume that dzA is independent of both dzp and dzR . The resulting equation for the change in the value of the competitive mine is c dJ c (t) = [p(t) − Cq (t) + JRR (t)σ R σqR (t)][−qo (t) dt + σ R (t) dzR (t)] c c + 21 JRR (t)(σ R )2 dt + Jpc (t)[αp(t) dt + σ p p(t) dzp + 21 Jpp (t)(σ p )2 p2 (t) dt c + JAc (t)[−rA(t) dt + σ A A(t) dzA ] + 21 JAA (t)(σ A )2 A2 (t) dt

(35)

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The additional terms represent the effects of the deterministic and stochastic changes in the discount rate and the cost of the risk associated with the stochastic change. Eq. (35) incorporates the main stochastic variables which affect the depreciation of the competitive mine. 4. A stochastic version of Hartwick’s Rule Hartwick (1977), in the context of a macroeconomic model featuring exhaustible natural resources, established that, in order to maintain consumption constant, the value of exhaustible resource depletion must be offset by the value of increases in reproducible capital, so that net investment for the economy is zero. Hartwick’s “Rule” has been a focus of attention in discussions of sustainable development indicators and has been thoroughly investigated. Vellinga and Withagen (1996) and Asheim (1997) provide succinct summaries of research in the area. Lozada (1995) showed that in the certainty case his equation for the change in the value function leads directly to a generalized version of Hartwick’s Rule. Lozada assumes certainty, a constant geometric discount rate, an infinite horizon, and no deterministic exogenous change (the “autonomous-except-for-discounting” case) so that (16) reduces to dJ c = (pc ·g) dt. Also, totally differentiating the deterministic current value function, using the deterministic version of the current value HJB necessary condition (13) and the deterministic version of the current value Hamiltonian (12) yields dJ c = (ρJ c − f c ) dt. Thus, dJ c = (ρJ c − f c ) dt = (pc · g) dt (36) Since pc ·g is the value of net investment in all the state variables, zero net investment implies both that dJ c = 0 and thatf c = ρJ c . Thus, since Jc is constant, f c must also be constant so that the flow of benefit (consumption in Hartwick’s model) must be constant. Lozada (1995) points out that Hartwick’s Rule does not hold in the presence of exogenous deterministic change (ftc and gt unequal to zero). He does not, however, explore a version of the rule which corrects for this change. Vincent et al. (1997) do provide a corrected version for the case of a small resource-exporting economy. A general version of their result can be obtained from Eq. (16) by assuming no uncertainty and a constant discount rate.
 T  c c −1 c c dJ = p · g dt + A(t) A(s)(fs (s) + p (s) · g s (s)) ds dt (37) t

Thus, in order to keep dJ c = 0, and Jc and f c constant, net investment must equal the negative of the present value of the future effects of today’s exogenous deterministic change. If the present value of these effects is positive (negative) then the value of net investment must be negative (positive) by an equal amount in order to achieve a constant flow of benefit. Hartwick and Long (1999) provide an even more general version for non-autonomous certainty models which includes the case of a variable discount rate, a case which has also been examined by Asheim (1997) and Kemp and Long (1998). Their result can also be obtained from Eq. (16)
 T  dJ c = p c · g dt + A(t)−1 A(s)(fsc (s) + p c (s) · g s (s)) ds dt
 +

T t

t

 A(t)−1 A(s)(ρ(t) − ρ(s))f c (s) ds dt

(38)

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231

Here, in order to keep dJ c = 0, the value of net investment must exactly offset the capital gains or losses due to both the future effects of deterministic exogenous change in f c , g and the instantaneous discount rate, ρ. The equation for the change in the stochastic current value function (16) provides the basis for a general stochastic version of Hartwick’s Rule. Assuming an infinite time horizon, (16) becomes 1 dJ c = pc · (g dt + σ dz) + trace(p cx Σ)dt 2
 ∞  1 −1 c c c + Et A(t) A(s)(fs (s) + p (s) · g s (s)) + trace(px (s)Σ s (s))ds dt 2
t ∞  + Et A(t)−1 A(s)(ρ(t) − ρ(s))f c (s)ds dt (39) t

It also can be shown that dJ c = (ρJ c − f c )dt + pc · σ dz

(40)

Taking expected values of (39) and (40) implies that
1 c Et dJ = p c · g + trace(p cx Σ) 2  ∞ 1 A(t)−1 A(s)(fsc (s) + p c (s) · g s (s)) + trace(p cx (s)Σ s (s)ds + Et 2  t ∞ A(t)−1 A(s)(ρ(t) − ρ(s))f c (s)ds dt + Et t

= (ρJ c − f c )dt

(41)

If Et dJ c (t) = 0, then Et J c (t + dt) = J c (t) and Et f c (t + dt) = ρ Et J c (t + dt) = ρJ c (t) = f c (t) and Et df c = 0, an expected value version of a constant flow of net benefit. The term, pc · g + (1/2)trace(pcx Σ) in (41) represents the expected value of the changes in the state variables adjusted for risk, or the risk adjusted expected value of net investment. If this value always equals the negative of the capital gains due to the expected future effects of today’s exogenous deterministic change plus the capital gains due to the expected effects of future discount rate changes, then the expected value of tomorrow’s flow of benefit will always equal today’s flow of benefit. In the absence of exogenous deterministic change and future discount rate changes, the stochastic version of Hartwick’s Rule requires that the risk adjusted expected value of net investment equal zero. Under these conditions, if the stochastic version of Hartwick’s Rule is followed, (40) implies that the current value function will follow the Brownian motion given by dJ c = pc · (σ dz). 5. Implications for Hamiltonian-based (NNP) measures in national accounting The theory of national accounting is based on the result that in a dynamic optimization of a macroeconomic model, the “interest” at the subjective discount rate on the current-value

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of the economy equals the current-value Hamiltonian (Weitzman, 1976), which is equivalent to Net National Product (NNP) as measured in conventional national accounts. This interest represents the hypothetical constant flow of welfare which has the same present value as the optimal path of the economy (see Asheim, 1997). Hartwick (2000) and Aronsson et al. (1997) provide overviews as well as new contributions to the incorporation of natural resources into national accounts based on this macroeconomic approach. Both Lozada (1995) and Aronsson and Löfgren (1993, 1995) recognize that the envelope equation for the current-value Hamiltonian establishes a connection between the subjective interest on the current value function of a deterministic optimal control problem and the corresponding current-value Hamiltonian. They both assume a constant discount rate, ρ(s) = ρ, and an infinite horizon (which implies that the final period Hamiltonian equals zero (Michel, 1982) in deriving the deterministic equation  ∞ ∂H co ρJ c (x(t), t) = H co (x(t), t) + e−ρ(s−t) (s) ds = [f c (s) + p c (s) · g(s)] ∂s t  ∞ −ρ(s−t) c + e [fs (s) + p c (s) · g s (s)] ds (42) t

Interpreted in national accounting terms, the “interest” at the subjective discount rate on the current-value of the economy equals the current-value Hamiltonian (which represents the value of the effects of today’s choices of the control variables on current and future welfare) plus the value of the future effects of today’s (see the discussion above) deterministic exogenous changes in the current welfare and growth functions (ft and g t ). If there is no deterministic exogenous change (the “autonomous-except-for-discounting” case) this hypothetical constant flow of welfare equals the current value Hamiltonian (Net National Product or NNP). They both point out that, in the deterministic case, the presence of deterministic exogenous change breaks this equality, which is the basis for Weitzman’s flow of welfare measure. Aronsson and Löfgren (1995) also examine the stochastic “autonomous-except-fordiscounting” case with no exogenous deterministic change. In this case, both the envelope equation for the current-value Hamiltonian (15) and the current-value HJB necessary condition (13) reduce to ρJ c ((x(t), t) = H co (x(t), t)   = f c (x, uo ) + p c (x, t) · g(x, uo ) + 21 trace(p cx (x, t)Σ(x, uo ))

(43)

so that the Weitzman welfare measure does equal the stochastic current-value Hamiltonian, which can still be interpreted as NNP if the value of net investment, pc (x(t), t) · g(x(t), uo (t)), is corrected for the cost of risk. A finance approach, of the type used by Brennan and Schwartz (1985) to value mineral deposits, could provide the basis for valuing the risky net investment terms in the stochastic Hamiltonian. However, in the presence of exogenous deterministic change (the “discounted nonautonomous” case), the envelope Eq. (15) becomes

D.W. Butterfield / Resource and Energy Economics 25 (2003) 219–238

 ∞ ∂H co (s) ds ρJ c (x(t), t) = H co (x(t), t) + Et e−ρ(s−t) ∂s t 
1 = f c (t) + p c (t) · g(t) + trace(pcx (t)Σ(t)) 2
  ∞ 1 + Et e−ρ(s−t) fsc (s) + p c (s) · g s + trace(p cx (s)Σ s (s)) ds 2 t

233

(44)

Eqs. (43) and (44) have two important implications. First, it suggests that the value of conventionally measured NDP, f c (t) + pc (t)·g(t), should be adjusted for cost of the risk associated with net investment, (1/2) trace(pcx (t)Σ(t)). Second, it implies that capital gains resulting from deterministic exogenous change should be included in the Weitzman measure of the flow of welfare. The SNA (United Nations, 1993) has resisted including capital gains in the “core” accounts, but has recognized them as “other changes in value” in auxilliary accounts. Eq. (44) implies that risk-adjusted NDP should be augmented by these capital gains in national accounts. Aronsson and Löfgren (1995) argue that measurement of the last term in (44) requires knowledge of “the expected value of future (italics added) technological (deterministic exogenous) change” and, thus, makes it impossible to use a “national product (Hamiltonian) based welfare measure”, presumably either because they believe that it is impossible to form expectations about future deterministic exogenous change or because these expectations would not be reflected in today’s market data. It is not obvious that either of these claims is true, and they provide an area for future research. In fact, the discussion in Section 3 suggests that the integral in (44) represents today’s capital gains due to the future effects of today’s deterministic exogenous change. Of course the theory of stochastic control assumes that the functional forms of f, g and Σ are known at all future times so that only the stochastic properties of the arguments are needed in order to calculate the expected value of the integral in (44). We suggest that there is at least some possibility that current market data could capture the expected value of the effects of deterministic exogenous events and be used to augment the stochastic Hamiltonian in welfare measurement, in a way similar to the use of other changes in value accounts to augment the core accounts in the SNA (United Nations, 1993).

6. Conclusions Following Lozada’s (1995) approach for deterministic dynamic programs, an envelope equation for the stochastic Hamiltonian has been used to develop an expression for the change of the value function in non-autonomous stochastic programs. This expression has been applied to the depreciation of the competitive mine, for stochastic price change, for both exogenous and endogenous stochastic revisions to reserves and for stochastic change in the discount rate. The results imply that mine depreciation should be adjusted for the risks associated with stochastic changes in the price of the resource, with stochastic revisions to reserves and with stochastic changes in the discount rate. They also imply that the net price used to value physical depletion and revisions should be adjusted for endogenous marginal risk cost of revision associated with extraction. Future research should focus

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on using market data to measure these adjustments for risk, drawing on the theory of finance. The general expression for the change in the current value function also implies a stochastic version of Hartwick’s (1977) Rule for sustainable consumption, in the absence of deterministic exogenous change. It has also been used to derive a stochastic version of Hartwick’s Rule which accounts for exogenous deterministic change, which generalizes Hartwick and Long’s (1999) deterministic version. The stochastic version of the envelope property of the current-value Hamiltonian provides a connection between Weitzman’s (1976) welfare measure and the stochastic Hamiltonian, which can be interpreted as “risk-adjusted” Net National Product. Even if there is exogenous deterministic change, there is at least a possibility that an augmented stochastic Hamiltonian, based on today’s market data, could be used as the basis for welfare measurement, contrary to the claim of Aronsson and Löfgren (1995). Again, this possibility provides an area for future research.

Acknowledgements I thank Andy Muller and the referees for very helpful comments.

Appendix A. Proof of the envelope property of the present value Hamiltonian, Eq. (6): The proof begins by differentiating the HJB necessary condition (5) with respect to the vector of state variables, x(t), yielding −Jtx (x(t), t; T) = −pt (x(t), t; T) = Hxo (x(t), t; T)

(A.1)

where the differentiation of Ho is carried out after the substitution of p(x(t), t, T) and px (x(t), t, T) into Ho (LaFrance and Barney, 1991). Differentiating a second time yields o (x(t), t; T) −Jtxx (x(t), t; T) = −pxt (x(t), t; T) = Hxx

(A.2)

We assume that the partial derivative of J on the left side of (A.2) exists, which is equivalent to Ho being twice differentiable in x(t). Applying Ito’s Lemma to Ho (x(t), t, T) gives   o Σ) dt + Hxo · σ dz (A.3) dH o (x(t), t, T) = Hxo · g + Hto + 21 trace(Hxx Finally, direct differentiation of Ho with respect to t yields ∂H o ∂H o ∂H o · p · u + + · p xt + t t ∂uo ∂p ∂p x 1 = g.pt + trace(p xt Σ) + ft + p.g t + 2

Hto =

∂H o ∂t 1 trace(p t Σ t ) 2

(A.4)

D.W. Butterfield / Resource and Energy Economics 25 (2003) 219–238

235

since ∂H o = 0, ∂u

∂H o = g, ∂p

1 ∂H o = Σ ∂p x 2

and

1 ∂H o = ft + p.g t + trace(p x Σ t ) ∂t 2

where the notation ∂Ho /∂t is used to denote differentiation before substitution of p(x(t),t) and px (x(t),t) into Ho (see LaFrance and Barney, 1991). Substituting the HJB necessary condition (5), (A.1), (A.2) and (A.4) into (A.3) yields the envelope property (6). A.1. Derivation of the change in the present value function, Eq. (9): The proof follows Lozada’s (1995) procedure. We first apply Ito’s Lemma to the present value function, yielding dJ = (Jx · g + Jt + 21 trace(Jxx Σ)) dt + Jx · σ dz

(A.5)

Using the present-value HJB Eq. (5) and noting that Jx = p and Jxx = px , we obtain dJ = (p · g − H o + 21 trace(p x Σ)) dt + p · σ dz

(A.6)

Eq. (A.6) can be modified in two different ways. The more conventional modification substitutes the definition of Ho , (4), into (A.6) to obtain dJ = −f dt + p · σ dz

(A.7)

which can also be expressed in stochastic integral form as J(x(T), T ; T) − J(x(t), t; T)   T f(x(s), uo (s), s) ds + =− t

T t

p(x(s), s; T) · σ (x(s), uo (s), s) dz(s)

(A.8)

Note that taking expected values of (A.8) yields the definition of J, since  T Et J(x(T); T) = Et F(x(T), T) and Et p · σ dz = 0 t

Lozada’s approach, however, substitutes the equation for Ho derived from our result, (8), into(A.6) to find Eq. (A.9): 1 dJ = p.(g dt + σ dz) + trace(p x Σ) dt 2 
 T 1 (fs + p.g s + trace(p x Σ s )) ds dt − [Et H o (T)] dt + Et 2 t

(A.9)

A.2. Derivation of the envelope property of the current-value stochastic Hamiltonian, Eqs. (A.10) and (A.12): A formula for the change in the optimized current-value Hamiltonian, dHco , can be derived either by noting that H co (t) = A(t)−1 Ho (t), so that dH co = ρ(t)H co dt +

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A(t)−1 dH o and substituting into (6), or by directly applying Ito’s Lemma to Hco and using the current-value HJB necessary condition (13). The current-value result is
co  ∂H co co c dH = + ρ(t)(H − f ) dt + Hxco · σ dz ∂t
 1 1 c c c c c = ft + p · g t + trace(p x Σ t ) + ρ(t)(p · g + trace(p x Σ)) dt 2 2 c c (A.10) − [pt − ρ(t)p ] · σ dz since Hxco = −p ct + ρ(t)p c as a result of differentiating the current value HJB Eq. (13) with respect to x. The stochastic integral form of (A.9) can be rearranged to give  T  T ∂H co (s) ds A(t)−1 A(s)ρ(s)f c (s) ds − A(t)−1 A(s) H co (x(t), t; T ) = ∂s t t  T − A(t)−1 A(s)(p ct (s) − ρ(s)p c (s)) · σ (s) dz t

+ A(t)−1 A(T )H co (x(T ), T ; T ) and taking expected values of (A.11) gives Eq. (A.12)  T  H co (x(t), t; T ) = Et A(t)−1 A(s)ρ(s)f c (s) ds − Et t

+ A(t)−1 A(T )Et H co (x(T ), T ; T )

(A.11)

T t

A(t)−1 A(s)

∂H co (s) ∂s (A.12)

A.3. Derivation of the equation for the change in the current-value function, Eq. (A.14): This equation can be derived directly from Ito’s Lemma. Alternatively, observing that since J c = A(t)−1 J, dJ c = −A(t)−1 A (t)J c dt + A(t)−1 dJ = ρ(t)J c dt + A(t)−1 dJ

(A.13)

and substituting for dJ from (9) gives the current-value result. 1 dJ c = p c · (g dt + σ dz) + trace(p cx Σ) dt 2
 T  A(t)−1 A(s)ρ(s)f c (s) ds dt + ρ(t)J c dt − Et
 + Et

T t

t

 1 A(t)−1 A(s)(fsc (s) + p c (s) · g s (s)) + trace(p sx (s)Σ s (s)) ds dt 2

− [A(t)−1 A(T )Et H co (x(T ), T ; T )] dt

(A.14)

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