Neoclassical natural capital theory and "weak" indicators for sustainability.
The classical economists, in the 18th and 19th centuries, tended to regard the primary environmental supports for economic production activity as either non-scarce (such as air) or nondepletable (such as arable land). From this point of view, a growth in the volume of economic output from one year to the next was not only a gain from the point of view of immediate consumption prospects, it was also a net improvement to the resource base upon which future economic output could be achieved. Success in the short term was thus synonymous with augmentation of potentials for consumption and capital accumulation in the long term. Sustainable growth was simply the continuation of short-term growth.
But, if natural resources are depletable, and essential environmental services can be irreversibly impaired through pollution or ecosystem change, then present-day economic activity can have very high intertemporal opportunity costs. The existence of such irreversibilities ruptures the consonance between short-run performance (GNP-growth) and long-run prospects for (a) economic output and hence consumption levels; and (b) the sustaining of the cycles of resource renewal and the environmental life-support functions that underpin economic activity. It is, thus, no longer possible to regard GNP-growth as a signpost pointing in the direction of long-run economic progress. The short run and the long run may be in conflict; the pursuit of rapid growth as a short-term objective may impair durably the economic welfare prospects of future generations.
A wide range of proposals and practices have emerged over recent years, seeking to define and estimate an "environmentally adjusted" national product, national savings, or national income figure. Measures of "net savings" taking natural capital depreciation into account, and of an "environmentally corrected net national product" (or "green NNP", henceforth gNNP) have widely been proposed as sustainability indicators. The usual recipes involve making subtractions from the conventional GNP. (We do not discuss alternative modeling approaches in this paper; but see de Boer, de Haan, and Voogt 1994; Brouwer, O'Connor, and Radermacher 1996.) The key issue is how, in theory and in measurement practice, one can make the jump from GNP as a measure of this period's output level to "net savings" or gNNP as an indicator of prospects for sustainable future welfare levels relative to the current level of consumption.(1) The usual response in the economics literature relies on theoretical results from neoclassical growth-with-natural-capital theory.
Part II of the paper reviews key elements of the neoclassical theory of economic growth with natural capital. Part III then presents a simple overlapping generations general equilibrium model with a depletable natural resource, which is used to highlight the significance for "socially optimum" consumption time-paths of axiomatic assumptions relating to model structure and parametrization, in particular presumptions about substitutability, time-discounting, and relative prices as measures of opportunity costs. On this foundation we assess the ambition and limitations of recent work, both theoretical and applied, that has sought to assess by the "weak" criteria the "sustainability" or nonsustainability of national economies. Part IV gives a synthetic overview of the search for macroeconomic indicators of sustainable development based on the neoclassical natural capital theoretical approaches. Part V assesses the combined impact of the theoretical and empirical limits to validity of the "weak" sustainability indicators. We will conclude, in Part VI, with the assessment that insurmountable problems of aggregation and measurement at both theoretical and empirical levels mean that the neoclassical theory is not robust - and cannot ever be made robust - as a basis for deriving reliable indicators for sustainability.
II. NEOCLASSICAL NATURAL CAPITAL THEORY AND SUSTAINABILITY
Growth with Natural Capital: The Neoclassical Conventions
It has, by now, become commonplace to refer to ecological goods and services as deriving from existing stocks of "natural capital" (cf. Daly 1994). The biosphere as a habitat and life support system is a finite, and in many respects destructible, reservoir of natural capital. Estimating the severity of trade-offs, and the redistributions of economic opportunities, access to environmental benefits, financial and ecological costs, and burdens of risks, becomes a major task of economics as a policy science.
Our concern in this paper is with one class of theoretical and empirical problems in the estimation of such trade-offs. We focus on the definition, estimation, and interpretation of "indicators for sustainability" pertaining to neoclassical models that assume substitutability between natural and produced capitals as (i) inputs for economic capital accumulation and/or (ii) elements of consumption. These models characterize sustainability as nondecreasing social welfare over time, the social welfare being defined by an aggregate utility function or consumption level. The mathematical models are of two main forms. On the one hand are those in the lineage of growth theory, with an aggregate output that can be used in consumption or invested in economic capital accumulation (see Pezzey 1989, 1992, 1996; Toman, Pezzey, and Krautkraemer 1995; and papers in this volume). On the other hand are intertemporal equilibrium models that consider utility as a function of consumption levels and agents' preferences (as pioneered by Howarth and Norgaard 1990, 1992, 1993; also Muir 1995 gives a good overview).
We can think of these models as expressing "social choices," as signified by population growth, individuals' preferences and institutional arrangements governing endowment or income distribution, subject to the defined technical and resource constraints. In modeling, population change is usually treated as exogenous, so the emphasis is placed on production feasibility (the intertemporal production possibility frontier) and on the social determinants of investment and consumption over time.
On the feasibility side, the growth and/or sustainability potentials for a model economy depend on the specific assumptions made about natural capital renewal rates, about elasticities of substitution between natural and produced capitals, and about technical progress augmenting productivity of capitals. Where "technical progress" and/or elasticities of substitution between natural and produced capitals are made high enough, the value of the economy's capital stock may grow without limit, and thus the "sustainable national income" that is attainable "in the long run" is correspondingly unbounded. In such instances, achieving sustainability appears as a problem of savings. In any particular period there is a trade-off between consumption and capital accumulation. High consumption in a given period means "living off capital" during the period in question, but no permanent damage to "sustainable growth" prospects if this is just a transitory phenomenon.
The feature of the modeling work in the 1970s was the introduction of depletable "natural capital." Analyses focused on the importance of substitutability and technical progress for relieving growth constraints due to the depletability of the natural capital. Three articles appearing just after the 1973/74 OPEC oil crisis, by Dasgupta and Heal (1974), Solow (1974), and Stiglitz (1974), are among the seminal contributions; much of the more recent work follows directly in their line. As an illustration, we outline features of the model developed by Stiglitz (1974), some of which are preserved in our own model to be presented in Part III.
Suppose a production function with labor, economic capital, and natural capital (a depletable resource) as inputs, and a single homogenous output:
Q = f(M, L, R, t) = [M.sup.[Alpha]1][L.sup.[Alpha]2][R.sup.[Alpha]3][e.sup.[Lambda]t],
where Q is the output; M is manufactured capital, L is labor, R is natural capital used in production; and [Lambda] is the rate of technical progress (assumed to be time-invariant). The output can be used either for consumption (C) or for investment to augment the stock of manufactured capital (dM/dt), so: Q = C + dM/dt. Dividing by Q, and writing x = C/Q for the fraction of output consumed, and [Sigma] = (dM/dt)/Q for the fraction saved, we obtain:
1 = C/Q + (dM/dt)/Q [implies] x = 1 - [Sigma].
Assume that labor supply grows at a constant rate, n = (dL/dt)/L, and that there are constant returns to scale, so [[Alpha].sub.1] + [[Alpha].sub.2] + [[Alpha].sub.3] = 1. The parameters [[Alpha].sub.1], [[Alpha].sub.2], [[Alpha].sub.3] then designate the respective output elasticities of manufactured capital, labor, and natural capital. With all these assumptions, it is shown by Stiglitz (1974) that a timepath with a positive rate of growth in per capita consumption, that is, [g.sub.c] = (L/C)d/dt(C/L) = (dC/dt)/C - n [greater than or equal to] 0, requires that [Lambda] [greater than] [[Alpha].sub.3]n. The rate of technical progress must be relatively high compared with the population growth rate and the factor share of natural capital.
For the Cobb-Douglas form of production function, the elasticity of substitution for natural capital by other inputs is always equal to unity. While some input of natural capital is necessary for nonzero production, both the marginal and the average productivity of natural capital are unbounded, and indeed will rise without limit as the ratio of natural to economic capital input tends towards zero. This substitution property is thus a key underlying condition - also identified by Solow (1974) - under which a nondeclining positive level of output/consumption can be sustained indefinitely despite dependency of production on the nonrenewable natural capital.
Consider now the case where population growth and technical progress are zero ([Lambda] = n = 0), so that the significance of substitutability is isolated. Stiglitz showed that a timepath with constant consumption is obtained by setting [Sigma] = [[Alpha].sub.3]. We recall that with Cobb-Douglas production functions, the output elasticities [[Alpha].sub.1], [[Alpha].sub.2], [[Alpha].sub.3] indicate, respectively, the relative importance of economic capital, labor, and natural capital in production, as measured by factor share. So, this condition for nonnegative change in per capita consumption when technical progress and population growth are zero, is that the share of economic product that is saved ([Sigma]) is at least as large as the natural capital factor share ([[Alpha].sub.1]). This illustrates the result now known as "Hartwick's Rule" (Hartwick 1977, 1978). Necessary conditions for a nonnegative per capita consumption in a Cobb-Douglas (unit elasticity of substitution) economy with a constant population are:
* that manufactured capital is relatively more important than natural capital in production ([[Alpha].sub.1] [greater than] [[Alpha].sub.3]), meaning that the factor share received by economic capital (share to profits, [[Alpha].sub.1]) is larger than that going to natural capital (share to rents, [[Alpha].sub.3]); and
* that savings are sufficiently high - more particularly, that for each moment in time there is investment in manufactured capital stock formation (savings) of at least the equivalent of the value of the resource rents.
Sustainability and Intertemporal Distribution Rules
By now a variety of models have been constructed in which there exists the technological capability for unlimited growth in the value of economic capital over time by substituting away from a renewable or nonrenewable natural capital, but where achievement or not of consumption sustainability is a social choice. Typically, solutions are obtained in these models using the criterion of maximizing the present value of "society's utility" as defined by some intertemporal "social welfare function." The generic result is now well known: where there is a sufficiently high time-preference for present consumption over future consumption, the intertemporal equilibrium path will be characterized, from the outset or after a peak is obtained, by monotonically declining values for total capital stock and, correspondingly, per capital utility or consumption levels (see notably Howarth and Norgaard 1990, 1992, 1993; Norgaard and Howarth 1991; Mourmouras 1993; Asheim 1994; Toman, Pezzey, and Krautkraemer 1995; Pezzey 1994, 1997).
The determinants of the distribution over time of consumption include consumers' preferences in two respects, along with the "social distribution rule" that is applied. First, where more than one good enters into individuals' utility functions at a given moment and these goods have differing natural capital requirements for their production or supply, the relative intensity of preferences for one good over another influences the pressure on natural capital. We will not further consider this feature here. Second, individuals' and society's consumption are distributed over time, and this is partly a time-preference phenomenon. We use the term "subjective time preference" to mean the way that a consumer compares the value (in welfare terms to her or himself) of consumption at one moment (or period) in time compared with other moments (or periods).
In an overlapping generations model, each class and generation of consumers will have a distinctive "preference function," and each consumer's rate of time discounting is determined by their particular preference function in conjunction with the consumption opportunity set. In an intertemporal general equilibrium, each individual's subjective rate of time discounting will, at a given moment in time or period, be equalized to the interest rate (which may itself be a function of time) characteristic of the model solution timepath.(2) Consumers' preferences may be taken as a "datum" (or, alternatively, specified as a sociological variable) while the interest rate measuring opportunity cost of capital is influenced by the "social distribution rule" that determines (in conjunction with technological parameters, initial stock levels, and preferences) the particular model equilibrium attained.
The role of savings is now seen to be one of influencing the distribution across consumers (successive generations, for example) of endowments and of consumption opportunities. Thus, as Dixit, Hammond, and Hoel (1980) observed, following a program of investment respecting the Hartwick Rule amounts to a policy choice in favor of intertemporal equity. The problem of reconciling criteria of economic efficiency with intertemporal equity concerns had already been identified by Solow (1974), who considered the timepath of natural resource depletion under the assumption of a maximin social choice function. Dasgupta and Heal (1979) noted that a timepath for present-value-of-utility maximization with either nonrenewable or renewable natural resources (for example, coal or fish, respectively) could be strongly inequitable towards future generations.(3) The issues involved here, discussed early by Page (1977), have been most clearly brought out by models reframing the optimal resource use problem as one of intertemporal general equilibrium with utility maximizing consumers, notably by Howarth (1991, 1992, 1996), and Howarth and Norgaard (1990, 1992, 1993). These authors' usual model form is a closed economy, and the question of time preference is structured by assuming overlapping generations. Our model in Part III is adapted directly from Howarth (1991), so we give the formulation in detail. Each generation lives for two time periods (say n and n + 1), and the nth generation maximizes utility [U.sub.n] = [U.sub.n]([C.sub.n, y], [C.sub.n + 1, o]), where [C.sub.n, y] is consumption during period n when the nth generation is young, and [C.sub.n+ 1, o] is consumption during period n + 1 when the generation is old. Within each generation, all individuals are identical so we can treat them as one, and the emphasis is on aggregate consumption each period; no questions of intragenerational equity are addressed.(4) Markets for natural capital (resources or environmental amenity), consumer goods, and labor are assumed to be "competitive" in the sense of equalization of opportunity costs on all margins. Labor is an initial endowment distributed equally across all generations; each generation "owns" (and thus supplies) labor only while young. Intergenerational transfers are possible through exchange of income for natural capital held as initial endowments. Technical parameters and initial stock levels determine the intertemporal production possibilities frontier (IPPF) for the economy, and the "optimal" point on this frontier is then selected as either:
(i) the equilibrium outcome of utility-maximizing consumers' choices subject to a specified endowment distribution, or
(ii) the optimum of a social welfare function, the latter being formulated in terms of consumption or utility levels through time.
A number of important results emerge. First, sustainability in the sense of indefinitely nondeclining consumption from one generation to the next, is not guaranteed by the "competitive" rule of maximizing present-value of total consumption over time. On the contrary, when the property rights over natural capital are tipped in favor of the "present" generation (still able to be exchanged between generations to enable the old of each period to consume optimally), the typical result is monotonically declining utility levels beyond some period into the future.(5)
Second and conversely, achieving an equilibrium with nondecreasing consumption levels requires that, one way or another, present generations "care enough" about future generations. This "caring for the future" can be expressed through a variety of mechanisms, notably:
* the imposition of a maximin social welfare function; or imposition of a criterion of intertemporal social-welfare maximization subject to nonnegative change in representative individuals' welfare from one period to next;
* the assumption of a sufficiently high level of individual altruism of each generation towards the generation immediately following;
* the assumption of an "obligation" on the part of each generation to provide for a utility level of the generation immediately following at least as high as its own, resulting in a "chain of obligation" indefinitely into the future;
* the explicit award of property rights over natural capital or the benefits obtainable from it as initial endowments distributed "equitably" to all generations.
Third, the possible model equilibria for a given model are each characterized by distinctive trajectories, not just for capital stocks and consumption, but also for relative prices including the time discount rate. It is often said that, for intertemporal efficiency, the price of natural capital such as minerals or energy resources or fish or forest products, should "correctly" reflect the intertemporal opportunity cost (viz., the "user cost"). If sustainability is an objective, we must add the condition that this has to be the opportunity costs as evaluated along an intertemporal efficient path that also satisfies the sustainability criterion.(6)
III. A SIMPLE OLG MODEL WITH
Presentation of the Model
We now specify a multi-period overlapping generations (OLG) model which has the same production function as in Stiglitz's (1974) original problem: three inputs - manufactured capital M, human capital L, and nonrenewable natural capital R - to a Cobb-Douglas production function which produces manufactured capital as its sole output. The manufactured capital can be used in consumption C or saved for investment S.
The model problem is to maximize an intertemporal social welfare function, or in other words an intertemporal distribution rule, subject to a number of constraints. As in the earlier Howarth-Norgaard models, each generation lives two periods. The nth generation is young in period n, and old in period n + 1, and obtains utility from consumption specified by the Cobb-Douglas function of the form [U.sub.n] = ln([C.sub.n, y]) + ln([C.sub.n + 1,o]).
We have set N = 20 periods, which is sufficiently "long" to show the range of solution properties in question.(7) There are N - 1 = 19 generations who each live two periods. Also we add an "old" generation who consumes in the first period only; but no new "young" generation is born in the final Nth period. For our exposition, two social welfare distribution rules will be applied.
The first rule is to maximize the present value of utility (henceforth PVU-max), discounting each generation's utility by a constant factor [Delta], where 0 [less than] [Delta] [less than] 1, relative to the previous generation. The problem to be solved is:
where the sum is over n = 1 to n = N, 
[Mathematical Expression Omitted]
[for every]n = 1 ... (N - 1) 
[M.sub.n+1] = [S.sub.n] [for every] n = 1 ... (N - 1) 
[M.sub.1] = [M.sub.init] 
[L.sub.n] = 1 [for every]n = 1 ... N 
[S.sub.N] = 0 
[C.sub.N,y] = 0 
[Sigma][R.sub.n] = [R.sub.TOT]
where the sum is over n = 1 to n = N. 
[U.sub.n] = ln ([C.sub.n,y]) + ln([C.sub.n+1,o]). 
The constraints have the following meanings:  represents the market clearing condition combined with the production function;  is a device to propagate savings, analytically convenient but not strictly necessary;  sets the initial level of manufactured capital in the first period, owned by the first generation;  specifies a constant endowment of labor for each generation in their "young" period;  specifies that the Nth generation young do not save anything because they are not born;  specifies that the Nth generation young do not consume anything because they are not born;  is the natural resource constraint condition stating that the entire stock of nonrenewable natural capital is used up by the Nth period;  is the nth generation's utility. The symbols are as follows:
[R.sub.n] is the amount of natural resources used in period n,
[R.sub.TOT] is the total amount of natural resources,
[C.sub.n,y] is the amount consumed by the young generation in period n,
[C.sub.n,o] is the amount consumed by the old - the (n - 1)th generation - in period n,
[U.sub.n] = In([C.sub.n,y]) + ln ([C.sub.n+1,o]) is the Cobb-Douglas utility function of the nth generation,
[Delta] is the utility discount factor,
[M.sub.n] is the economic capital of the nth generation,
[L.sub.n] is the labor endowment of the nth generation of young,
[S.sub.n] is the savings of the nth generation of young,
[M.sub.init] is the initial level of economic capital in the first period.
From , each generation gives equal weight to its own consumption as "young" as it does to its own consumption as "old." There is thus no "subjective" discounting within a generation's life. The social discount parameter [Delta] dictates the strength of time-preference (impatience) for the economy overall. If [Delta] = 1 all generations count equally; if [Delta] [less than] 1 the successive generations count progressively less. Conventionally, the social discount rate between generations is given by [Rho] defined by 1/(1 + [Rho]) [equivalent to] [Delta] [if and only if] [Rho] [equivalent to] (1 - [Delta])/[Delta]; note that this is not the interest rate.
The second social distribution rule consists of a "sustainability" requirement overlaid on the PVU-maximization objective, representing the intergenerational equity requirement that each generation be at least as well off as the immediately preceding one:
[U.sub.n+1] [greater than or equal to] [U.sub.n], for n = 1 to n = N - 1. 
Further, and more complicated specifications would be possible, but would not change the essential demonstration character of our results. The optimization problems have been solved numerically with a standard solver under EXCEL 5.0. It is necessary to choose with some care the initial "guess" to avoid non-convergence. Once an equilibrium has been obtained it is possible to vary all parameters gradually to obtain comparative equilibria. The rather primitive specification of the boundary conditions, with abrupt cutoff at the N = 20th period with the death of the (N - 1)th generation, is obviously an artifice; what matters is the qualitative behavior of solutions up to around the 18th period.
Qualitative Typology of PVU-maximizing Solutions
The now large body of work on models of growth-with-natural-capital has shown that at least four qualitatively different sorts of PVU-maximizing timepaths may be obtained, depending on initial capital stock levels and renewal properties, the technological determinants of production feasibility, and the social determinants of the distribution through time of consumption. These are:
(a) monotonic decrease in utility over time: path of decline, clearly nonsustained;
(b) increase of utility for a while, then a turning point with monotonic decline after that: nonsustainable economic growth;
(c) exactly constant utility through time: sustainability as intertemporal economic equality;
(d) monotonic increase in utility through time: sustainable economic growth.
Our focus will be on category (b) solutions as compared with categories (a) and (d). A brief comment about category (c) is warranted. In continuous-time growth models, a constant-utility path can be obtained by applying Hartwick's Rule; but as Asheim (1994) and others have discussed (see also Svensson 1986; Pezzey 1997), if this result is to be obtained as an otherwise unconstrained PVU-max solution then the social discount rate [Rho](t) must be positive but decreasing over time, d[Rho]/dt [less than] 0 for t [greater than] 0. A similar result might be expected to hold for an OLG model, but we have not investigated this mathematically.(8) Our OLG model setup provides for direct investigation of the significance of varying three parameters:
* the initial stock of natural capital [R.sub.TOT] in comparison with initial economic stock [M.sub.init] and labor endowment [L.sub.n] = 1;
* the constant intergenerational discount rate parameter [Delta], where we can write [Rho] [equivalent to] (1 - [Delta])/[Delta];
* the relative importance of natural capital in production as indicated by the output elasticity coefficient [Phi] [equivalent to] (1 - [[Alpha].sub.1] - [[Alpha].sub.2]).
In the examples presented below, we have selected capital stock levels compatible with the possibility of obtaining solutions in all of categories (a), (b), and (d). For demonstration purposes:
(a) Let [R.sub.TOT] = 16; [E.sub.init] = 0.3; [L.sub.n] = 1. 
(b) To see the significance of varying the importance of natural capital relative to economic capital in production, we fix [[Alpha].sub.2] = 0.15 for labor, and consider the two cases [[Alpha].sub.1] = 0.7 and [[Alpha].sub.1] = 0.3.
(c) At the same time, we may vary the social discount rate: we take the cases: [Delta] = .10 [if and only if] [Rho] [equivalent to] (1- [Delta])/[Delta] = 1/9 and [Delta] = .30 [if and only if] [Rho] [equivalent to] (1-[Delta])/[Delta] = 3/7.
This gives us four "scenarios," for which we compare the timepaths for consumption, see Table 1 and Figure 1. The graphs show the consumption levels of the "old" and the "young" in each period. The utility of each generation, [U.sub.n] = ln([C.sub.n], y) + ln([C.sub.n + 1, o]), follows the same trend as the consumption curves.(9) Solution [11-A] is sustained growth; the output elasticity of economic capital is high and the society is sufficiently patient to allow future generations to enjoy a progressively greater utility level. Solutions [11-B] and [11-C] are nonsustainable growth paths with a boom-and-decline form. In case (B) the culprit is the high social discount rate [Rho] = 43 percent per generation; in case (C) the main culprit is the heavy dependence of production on depletable natural capital [Phi] = 0.55, notwithstanding the lower [Rho] = 11 percent. Solution [11-D] is monotonic economic decline, due to heavy dependence on the depletable natural capital and high consumption impatience.
Paths with Non-decreasing Utility across Generations
Now look at the significance of the intergenerational equity requirement  as a supplementary constraint. In this case we get the timepaths for consumption shown in Figure 2. The case (A/S) is unaltered, because the growth is already sustainable over the 20-period horizon. At the other extreme, in case (D/S) the requirement to hold future generations' utility at "equitable" levels means that consumption is dramatically reduced for early periods compared with the non-constrained PVU-max case.(10) In cases (B/S) and (C/S) there is slight reduction in early period's consumption levels, and instead of boom-and-decline the consumption rises more slowly than for the PVU-max path and then stays on a plateau until the end of the time horizon. While the PVU-max criterion is still applied for solution purposes, the nondecreasing utility constraint is dominant in determining the allocation through time of natural capital for production and of economic capital savings from one period to the next.
IV. INDICATORS FOR PROSPECTS OF "WEAK" SUSTAINABILITY
Sustainable National Income and "Savings" Rules for Sustainability
We now review the problem of measuring the requirements for "savings" to provide for sustainability. Let us provide some definitions that pertain, in the first instance, to models with an infinite time horizon.
Sustainable national income. The sustainable national income (henceforth SNI) for an economy may be defined, in theory, as the quantity of goods and services, say [C.sup.*], that may be consumed (rather than conserved/reinvested) in a given period while the economy-system still furnishes the capital stock as the basis for providing (at least) the same level of real consumption [C.sup.*] in every period through the future. At least two somewhat different definitions can be offered for a SNI:
(i) Immediately and thereafter perpetually obtainable income: SNI(i). SNI(i), is the highest level of "income" that can be attained immediately, from some given vector of stocks V(t = 0), subject to the constraint that the income level during t [greater than] 0 is permanently nondecreasing. This is a maximin utility path.
(ii) Later but thereafter perpetually obtainable income: SNI(ii). SNI(ii) is the highest level of "income" that the economy can continuously attain at and after a finite time, starting from some given vector of stocks V(t = 0), subject to the constraint that the income level is permanently nondecreasing.
SNI(ii) is an important reference point for any real or model economy that permits growth of the total capital stock through time, as it will be possible to increase the "sustainable national income" for the future, by, provisionally, restraining consumption below the current sustainable income - the SNI(i) - so that the manufactured capital stock is built up. This is, indeed, the presumption behind traditional macroeconomic modeling that discusses the "tradeoff" between current consumption and growth rate - the so-called "golden-rule" literature; and as Pezzey (1994, 1997) discusses, the concept of SNI(ii) seems very pertinent where natural capital scarcity constrains long-run manufactured capital accumulation.(11)
Hicksian income. The Hicksian definition of a person's (or nation's) income is the amount he/she (or it) can consume during a specified period, while ensuring that his/her wealth at the end of the period is no less than his/her wealth at the outset (Hicks 1946). Assume that the value of total capital stocks is K, measured in money units, so let us write:
K [equivalent to] [Pi] [multiplied by] X,
X = (M, L, R) is the vector of stocks in physical units, and
[Pi] = ([p.sub.1], [p.sub.2], [p.sub.3]) is the vector of relative prices.
Then the Hicksian national income will be associated with the rule: dK/dt = 0. The change in value of capital stock may, generally, be written: dK/dt = d/dt([Pi] [multiplied by] X), and this can be split into two parts:
the current value of savings: [Pi] [multiplied by] dX/dt and the "capital gains" term: X [multiplied by] d[Pi]/dt.
Hartwick's savings rule. Using the above notation, Hartwick's rule is written: [Pi] [multiplied by] dX/dt [greater than or equal to] 0. For a model with constant population we have dL/dt = 0, so this becomes [p.sub.1] dM/dt + [p.sub.3] dR/dt [greater than or equal to] 0. The first term refers to the value, in current prices, of the change in manufactured capital stock; the second term refers to the value, in current prices, of the change in natural capital stock.
Green net national product. Suppose the economy maximizes present value of consumption. Then the net national product (henceforth NNP) is defined as value of consumption plus net change in the value of capital stocks. If natural capital stocks are included, we call it a "green NNP," defined as: gNNP = [p.sub.1]C + ([Pi] [multiplied by] dX/dt) where, as before, C is the physical quantity of consumption, [p.sub.1] is the current price of manufactured capital (which can be saved or consumed) and [Pi] [multiplied by] dX/dt is the Hartwick "net savings" measured in current prices (Solow 1986; Maler 1991).
Because Hartwick's rule does not include the "capital gains" term, the respect of Hartwick's rule at any moment in time does not necessarily imply nonnegative change in the value of total capital stocks. So the Hicksian national income and the net national product (gNNP) are not the same thing.(12) Furthermore, the gNNP and the SNI(i) are not the same thing. As the recent work by Asheim (1994) and by Pezzey (1994, 1997) has made plain, the gNNP and SNI(i) will coincide only if highly restrictive theoretical conditions are fulfilled.
The Reasoning for the "Weak" Indicators for Sustainability
The early work by Solow, Hartwick, and others showed that, for a closed economy obeying the PVU-max criterion, a property of the SNI(i) "maximin" consumption path is that Hartwick's rule is satisfied at all times. However it was not initially remarked that respect of Hartwick's rule in this context was a necessary but not a sufficient condition. In effect, the problems of (1) changes in relative prices along a PVU-max path through time - showing up in, among other places, the "capital gains" term - and of (2) different relative prices associated with each distinct PVU-max solution, were not fully appreciated.(13) Consider, with this in mind, the results that we may obtain if we simply neglect price changes.
* As above, write gNNP = [p.sub.1]C + [Pi] [multiplied by] dX/dt.
* According to the Hartwick-Solow results, along a path of constant consumption dC/dt = 0, Hartwick's rule is necessarily respected in equality form: [Pi] [multiplied by] dX/dt = 0 for all t. Under these conditions we obtain: gNNP = [p.sub.1]C, and this is the SNI(i).
* Now, if it were that the prices do not change, the capital gains term would be zero and we could write: dK/dt = [Pi] [multiplied by] dX/dt.
* Thus along a PVU-max path where Hartwick's rule is respected at all times and also there are no capital gains (if such a path can be found), the net national product gNNP is a measure of the immediately and perpetually sustainable welfare delivery potential, the SNI(i), for the economy and its natural capital stock, and this would also be the "Hicksian national income" at all times.
This is the reasoning that has motivated the estimation of ([Pi] [multiplied by] dX/dt) and gNNP as sustainability indicators. If the above reasoning were valid:
* The gNNP = SNI(i) and so the gNNP could be interpreted as an estimate for level of consumption (in money terms) that may, in principle, be maintained from the present onwards, on a long-term basis, while also maintaining intact the value of the total stock of capital.
* A positive value of the Hartwick term ([Pi] [multiplied by] dX/dt [greater than] 0) would signal that the "net savings" of economic plus natural capital, measured in money units, is positive during the period. A negative value ([Pi] [multiplied by] dX/dt [less than] 0) would signal that the "net savings" is negative, or there is "net depreciation" during the period. This yields the Hartwick-Solow "Weak Sustainability Indicator" or "savings rule" as proposed by Solow (1986), quickly followed by others such as Maler (1991) and Pearce and Warford (1993).
There is, or would be, one final step for "operationalizing" the procedures. This is to estimate the components of the formula gNNP = [p.sub.1]C + [Pi] [multiplied by] dX/dt on the basis of current period prices and quantities. The "sustainable national income" SNI(i) is thus estimated by making deductions ([Pi] [multiplied by] dX/dt) from current GNP ([p.sub.1]C) representing depreciation of capital stocks, including manufactured capital and natural capital.
Theoretical Preconditions for Weak Indicator Validity
The procedures described above have widespread appeal, because they seem to resolve objections made on environmental grounds to the use of GNP as an indicator of macroeconomic performance. But these recipes are theoretically flawed as well as being difficult to implement in statistical practice.
The Hicksian income. Consider the notion that sustainability is achieved if the value of the nation's capital stock remains intact from one generation to the next, meaning dK/dt = 0, while providing the Hicksian income as the consumption at any given time. There are two inaccuracies here. First, as seen above, dK/dt = 0 is not the condition for a maximin SNI(i) timepath in a closed economy. Second, and more important, observing dK/dt [greater than or equal to] 0 in a closed economy at a given moment in time does not guarantee that the economy is capable of sustaining a nondecreasing Hicksian income.
We do not prove this second result mathematically. The reason can be seen intuitively from our model results as follows. On a "single-peak" PVU-max timepath, the value of total capital stock rises initially. There is a portion of the timepath where consumption rises beyond the long-run sustainable level while dK/dt [greater than or equal to] 0. For our discrete-time model the "Hicksian indicator" is the change in the value of the total stock from the nth to the (n + 1)th period, say [Delta] [K.sub.n]. Figure 3 shows, for our scenarios (B) and (C) of single-peak PVU-max consumption, the associated timepaths for the value of the stock of each capital, and the value of total capital stock, measured using the prices current at each point along the PVU-max timepath.
* Case (B) has high output elasticity but a boom-and-decline path for PVU-max consumption is obtained because of impatience. The comparison with the corresponding graphs (B) in Figure 1 and (B/S) in Figure 2 shows that from the 4th to the 10th period consumption by young and old in each period is significantly higher than the plateau level attained in the case (B/S) where the sustained-utility constraint is imposed. The rapid exploitation of the depletable natural capital combined with inadequate savings of manufactured capital compromises future generations' economic chances. We observe in graph (B) of Figure 3 that the sign of [Delta] [K.sub.n] is positive until the end of the 5th period. Thus, by the time a negative [Delta] [K.sub.n] is observed, the consumption level has already, for two periods, trespassed beyond the sustainable level. The Hicksian indicator [Delta] K [greater than or equal to] 0, signalling a nonnegative change in value of total capital stocks from one period to the next, provides a too-weak signal as to whether or not the consumption in the period is compatible or not with the sustainability criterion of nondecreasing utility.
* For case (C), the comparison with the sustainability-constrained case (C/S) shows that consumption "overshoots" the sustainable plateau level at the 2nd period. From Figure 3 case (C), we see that the sign of the [Delta] [K.sub.n] is positive only for the change between the 1st and 2nd periods. So the Hicksian indicator [Delta] K [greater than or equal to] 0 tells us, with one period's delay, that the consumption has trespassed beyond the level compatible with the sustainability criterion of nondecreasing utility.
The Hicksian Rule dK/dt [greater than or equal to] 0 is "too weak" as a sustainability indicator. Along a single-peak path, by the time that the sign of the indicator [Delta] [K.sub.n] changes from positive to negative, significant (and possibly irreversible) damage has already been done to sustainability prospects.
The Hartwick rule. Now consider the proposition that a nondecreasing consumption is assured by respect of the Hartwick rule, [Pi] [multiplied by] dX/dt = 0. The reasoning is that a positive sign of the Hartwickian "net savings" ([Pi] [multiplied by] dX/dt) [greater than] 0 for a PVU-max economy means that [p.sub.1]C [less than] gNNP. If it is assumed that gNNP = SNI(i), then the current consumption would be lower than the maximin income and hence the economy is not violating requirements for sustainable consumption. But as Ashelm (1994) and Pezzey (1994, 1997) have demonstrated:
* the "Hartwick income" defined by the gNNP when [Pi] [multiplied by] dX/dt = 0, is not (generally) the SNI(i);
* the equality between gNNP and SNI(i) holds only for a PVU-max path where the Hartwick rule is respected at every point in time;
* the gNNP for a PVU-max timepath therefore will not, in general, coincide with SNI(i), nor for that matter with SNI(ii). This statement holds even if, for a particular moment in time, it happens that C = gNNP is the PVU-max consumption;
* therefore a positive sign of the Hartwickian "net savings" ([Pi] [multiplied by] dX/dt) for a PVU-max economy is not a reliable indicator that the current consumption [p.sub.1]C is lower than the SNI(i), and hence that the economy is not violating requirements for sustainable consumption.
The measurement problems in theory and in practice. Figures that have, in recent years, been actually produced as putative estimates for a "green GNP" in this perspective, are generally admitted to involve "incomplete" adjustments (e.g., Repetto 1989; El Serafy 1989; Peskin 1991; Pearce and Atkinson 1993; Pearce and Warford 1993; and others since). Yet, there has been a tendency to let it be presumed that these "preliminary" calculations can somehow function as "first approximations," serving the same policy relevance as the theoretically specified measures.(14) This presumption is difficult to defend, partly because of the restricted validity in theory, and partly because of problems of indeterminacy, incompleteness, and systematic measurement biases at the empirical level.
The monetization of environmental deterioration, in neoclassical perspective, relies on the ability to estimate opportunity costs associated with resource use alternatives in economic production, pollution treatment, waste disposal, and environmental management. Strictly speaking, these opportunity costs are definable only within the theoretical framework of an intertemporal general equilibrium model. For valid indicator specification estimation, three related theoretical points thus arise. First, the role of capital gains in indicator definition and measurement must be dealt with correctly. Second, the measurements of dK/dt, or of the gNNP and the related Hartwickian savings ([Pi] [multiplied by] dX/dt), must be specified in terms of prices (or, as the case may be) shadow prices for the particular moment (or period) in time along the particular equilibrium path being considered. But this leads to a third problem, of "chicken and egg." Take the case of the closed economy. The relevant indicators are gNNP and Hartwick "net savings." Estimates will have reliable and transparent "sustainability" indicator properties - those corresponding to a maximin SNI(i) timepath - only if the calculations use the consumption levels, prices, and stock variations corresponding to an economy on a PVU-max intertemporal equilibrium path characterized by constant national. But, as Norgaard (1990) has observed, if the purpose of indicator construction is to learn whether or not an economy is far from a "sustainable" trajectory, we cannot assume the properties of a sustainable trajectory in the process of making the calculations.(15)
V. EMPIRICAL MEASUREMENTS AND THE INTERMEDIATION OF MODELS
The Arrow-Debreu equilibrium is very useful when for instance one comes to argue with someone who maintains that we need not worry about exhaustible resources because they will always have prices which ensure their "proper" use. Of course there are many things wrong with this contention but a quick way of disposing of the claim is to note that an Arrow-Debreu equilibrium must be the assumption he is making for the economy, and then to show why the economy cannot be in this state. (Hahn 1973, 14)
Getting the Model Right?
The basic idea of the "savings" indicator - not generally valid, as we have seen - was that if Hartwick's rule is respected, then the economy is "operating within the bounds of sustainability" in the sense that current consumption [p.sub.1]C is less than the sustainable national income (presumed, somewhat erroneously, to be indicated by gNNP).
To appreciate fully the theoretical limitations of this procedure, we must pose again the question: what is involved, theoretically, in defining the passage from an actual GNP to an estimate for an SNI? Far more than just some arithmetic with some categories of the national accounts and monetized satellite accounts. Both in theory and in fact, the specification of a "sustainable national income" is highly speculative. It depends on the underlying model, and also on assumptions about investment and consumption choices made through time. First, for a given model:
* The maximin SNI(i) and the Pezzeypath SNI(ii) will generally not coincide. As our own model results show, in Figure 2 cases (B/S) and (C/S), there may be the possibility that a judicious investment program in the directions of natural resource conservation, anti-pollution, environmental quality improvements, etc., could permit the economy to attain a value for SNI(ii) that is higher than the SNI(i) that would be immediately feasible.
* Attainment of SNI(ii) does not generally coincide with a PVU-max path. Results presented by Pezzey (1997) and also with our own model, as shown in Figure 2, suggest that paths that attain SNI(ii) will usually not be PVU-max paths - except in cases where C(t) increases without bounds and thus SNI(ii) is also unbounded in the long run.
* The values for SNI(i) and for SNI(ii) will each be a function of (inter alia) the initial capital stock vector. That is, values for SNI(i) and SNI(ii) can, in principle, be determined as functions of an initial stock vector within the analytical framework of a particular model (subject to algebraic tractability and numerical solution convergence obstacles).
So the mathematical passage from a value of gNNP obtained empirically or for a point on a model timepath, to either version of SNI, is model specific and depends on scenario parameters.
* The transformation from GNP to gNNP for the current period can be made on the basis of current prices, neglecting for the moment whether or not these prices are PVU-optimal (but see below).
* The transformation from current gNNP to SNI(i) or SNI(ii) is, by contrast, strongly model-specific. It requires a complicated algebraic transformation that assumes knowledge, for the entire time-horizon of interest, of the social discount rate or rates - the [Rho](t) or the [[Rho].sub.n] - that characterize the existing economic equilibrium, of the substitution and productivity parameters for all sectors (including any so-called technological progress), of demographic trends, and of consumer preferences.
The values obtainable for SNI(i) and SNI(ii) depend more heavily on the choice of model and the parameter specifications than on the empirical price/quantity data obtainable from the real world (see also Common 1993 and Vanoli 1996). But since we do not know what the "right model" is, and we cannot deduce this reliably from empirically available information, we are in a "chicken and egg" situation.
"Let Us Suppose a PVU-max Equilibrium . . ."
We can highlight the significance of model uncertainty for indicator estimation, with the help of our OLG model. Our sustainability criterion leads to solutions that are finite-time-horizon analogues of SNI(ii). It is already shown in Figure 2 that the level of consumption per period attainable in the long run (that is, through until at least the 20th period) under the sustainability constraint varies considerably - from 0.4 to [greater than] 1.5 units for each generation living in a given period, depending on our choice of scenario parameters. The initial stock levels are identical in all cases. Our problem of indicator reliability can be phrased:
* Is it possible to deduce which of these timepaths (A/S, B/S, C/S, or D/S) is the best representation of the economy's welfare-delivery prospects, based solely on past and current price information?
* Can we use indicator measures to gauge how far the economy is from sustainability if, at the same time, we need to know how far the economy is from sustainabilility in order to gauge the validity of the indicators?
The short answers are, no and no. This is serious, because, even in our (probably fictitious) PVU-max world, applying the indicator recipes blindly can give perverse results. Take the situations, such as our scenarios (B) and (C), where a model economy has a nonsustainable PVU-max equilibrium path along which national consumption first rises to nonsustainable levels then falls monotonically. As discussed by Pezzey (1994) and Asheim (1994), such "single peak" consumption paths will necessarily have a portion along which the aggregate wealth - the value of total capital stocks - is rising, prior to a subsequent monotonic decline. Along the rising-aggregate-stock portion of such a path, the weak sustainability indicators will fail to signal the consumption "overshoot" and thus do not signal that the resource use and savings regime is impairing durably the economy's sustainability prospects. These authors' results establish that:
* the use of equilibrium prices to estimate the natural capital depreciation is a systematic underestimate, in the sense that a positive Hartwick "net savings" [Pi] [multiplied by] dX/dt [greater than or equal to] 0 can be obtained despite the fact that the current consumption level is [p.sub.1] C [greater than] SNI(i) and, as such, cannot be sustained indefinitely.
* the gNNP obtained from the formula gNNP = [p.sub.1]C + [Pi] [multiplied by] dX/dt is higher than the SNI(i), which means that an estimate for gNNP obtained by deducting net capital depreciation from GNP will not correctly indicate the extent to which current consumption overshoots sustainability. By corollary, a hypothetical reallocation of economic resources away from consumption to investment equal in magnitude to the current value of capital depreciation ([Pi] [multiplied by] dX/dt) would not be sufficient to reduce the consumption to the SNI(i) as would be required to put the economy onto a sustainable path.
By the time that aggregate wealth has stopped rising (that is, dK/dt becomes nonpositive), and by the time the sign of the Hartwick "net savings" [Pi] [multiplied by] dX/dt changes from positive to negative, it is too late. Significant, and perhaps irreversible, damage to sustainability prospects has already been done. (We have already seen the same sort of defect with application of the "Hicksian" indicator.)
The source of the systematic errors can, indeed, readily be identified in the framework of our model. On the "boom" portion of a single-peak PVU-max path, the price/quantity information is giving "wrong signals" from the point of view of resource allocation for sustainability. In Figure 4, we show the flows of resource inputs to production for the scenarios (C) and (C/S). These correspond respectively to Figure 1(C) with boom-and-decline path for PVU-max consumption due to the time-discounting of future generations' utility, and Figure 2(C/S) with sustained-utility, close to the "maximin" form. Comparison shows that high early consumption under PVU-max involving rapid exploitation of the depletable natural capital compromises future generations' economic chances. From the 2nd through to the 9th period consumption by young and old in each period is significantly higher than the plateau level maintained in the case (C/S) where the sustainability constraint is imposed.
The corollary is that natural capital is, in early periods, being used relatively more quickly for the nonsustainable growth than is the case along the sustainability-constrained path, and also the manufactured capital stock grows relatively more quickly in the unconstrained PVU-max case than in the sustainability-constrained case. This immediately implies mis-valuation. Recall that with a Cobb-Douglas production function, the factor shares are constant and so the relative price p3/p1 is inversely proportional to the input proportions. So, relative to the sustainability-constrained path, the early periods of the PVU-max path is characterized by lower p3/p1 than along the sustained-utility path. These prices "undervalue" each unit of natural capital depleted and "overvalue" each unit of the savings of manufactured capital compared with the sustained-utility path prices.(16)
Empirical Measurements Using Incomplete Inventories and "Strongly Wrong" Prices
Real trends of economic activity are ex hypothesi far from sustainability. Even if one allows the doubtful proposition of a PVU-max interpretation of economic reality, the far-from-sustainability prices and quantities for capital stock variations are systematically wrong for the estimation purposes wanted of them. Worse, there is no reason to believe that current prices and patterns of resource utilization conform to a PVU-max path, and a lot of reasons exist to believe the contrary, notably the prevalence of market power, force majeure, and other forms of non-Pareto-efficient competition (e.g., oligopoly market power, high commercial discount rates, strategic behavior, gratuitous and cynical disposal of toxic wastes, state interventions to furnish low-cost access by commercial interests to forest, water, fisheries, and agricultural resources, and so on). Many environmental services (including waste disposal) and scarce natural resources (including fish, water, forests) are obtained virtually gratis simply because of market power and outright coercion, even when it is known that high opportunity costs (including uncompensated environmental damages) are involved.
Most non-commodified natural capitals such as the atmosphere and oceans, and the great diversity of marine, freshwater, and terrestrial ecosystems, are, in effect, treated by users as "free gifts of nature." Access is determined by social power relations, with or without regard for the future (Arnoux, Dawson, and O'Connor 1993; Martinez-Alier and O'Connor 1996). In the case of irreversibilities, the marginal cost of degradation becomes very high or infinite (Pearce 1976). Sometimes the use of market prices has been defended as pragmatism (citing, among other reasons, the difficulty of "correct" valuation!), and as a matter of making a step in the right direction. This is hardly a convincing defense of systematic error. Moreover, as Victor, Hanna, and Kubursi have commented,
By emphasizing in their empirical work those aspects of natural capital for which economic measures are more readily available (i.e., for resources sold through the market and a few measures of pollution damages), far more has been left out than has been included.(17) (Victor, Hanna, and Kubursi 1997, forthcoming).
The errors and omissions are thus twofold - first of all the exclusion altogether of many ecological assets and services from the accounting scheme, and second the use of market prices that are probably systematically "wrong" from the point of intertemporal opportunity costs. These impose systematic biases in the same direction (being due, generally speaking, to the habit of "self-interested" producers and consumers to treat nature as a "free gift" and to the predominance of present-day market power and purchasing power over, inter alia, future generations' interests). The omissions - tantamount to employing a zero-price in the correction calculations - are sufficiently large that it becomes preposterous to suggest that these empirical figures give any indication of sustainability potential. The "pragmatic" estimation procedures currently in use are prone to give, in the situations of greatest policy need, a positive sign for the putative indicator (whether this is Hartwickian net savings or the Hicksian change in value of capital stock) while in fact the economy is moving on a nonsustainable trajectory. This is not a reliable sort of policy indicator.
In reality, we can only observe past and present quantity-price variations. In the absence of independent knowledge of the economy's key technical and social parameters (stock levels, output and substitution elasticities, social discount rate) it is not possible to infer by how much the prices are "wrong." If we do not know what the "correct" parameter specifications of, inter alia, substitution and output elasticities should be used as the basis for estimating opportunity costs, we are not justified in presuming that they are "revealed" in market prices! This is why we are not justified to infer from a positive sign of net savings ([Pi] [multiplied by] dX/dt) or [Delta] [K.sub.n] whether or not consumption in the nth period is respectful of a sustainability intergenerational equity norm.
For example, the striking difference in position of the curve for value of natural capital stock in Figure 3(B) compared with Figure 3(C) shows how significantly the estimate of the relative importance of natural capital compared with economic capital and labor depends on model parameters - in this case the relative output elasticities. There is no reliable basis for deducing the "correct" value of such parameters from price and quantity information furnished by "the market."
Which model should be selected as the "right" one for the purpose of inferring shadow prices? Because there is no consensus on this, any indicator estimation result will be controversial. Underlying disagreements on scientific and political matters will end up reframed in the arcane language of modeling, without necessarily being resolved.
The Impossibility of Estimation of Model Parameters
Knowing the "correct" shadow relative prices and the associated quantities for calculating the gNNP depends on knowing the intertemporal production possibility frontier for economic production and environmental function (ipso facto the elasticities of substitution and the technological changes for the timespan of analysis), and knowing - or, at least, placing bounds around - the pattern of "demand" for economic goods and environmental functions on the part of future generations.
Norgaard (1990) has pointed out that the use of existing market prices in order to gauge resource scarcity (and, by implication, gauge substitution elasticities) and, thus, the critical values for model price variables, would involve a fallacy of circular reasoning. Similarly, Cabeza (1996) concludes that, in models of endogenous technological change, in order for market forces to induce natural resource productivity-augmenting technological change, the relative input prices have to signal the relative scarcity of these inputs correctly. If prices fail in this role, then the theory itself tells us we can hardly expect the technological change to follow the "appropriate" path.
This brings us to the final question: what empirical evidence might we be able to amass concerning key model parameters such as output elasticities or elasticities of substitution? The short answer is: ambiguous at best. A number of studies have been carried out to obtain estimates of elasticities of substitution for inputs to manufacturing. These have yielded widely varying results. Artus and Perroux (1981), making use of a Translog production function form, obtained elasticity estimates varying from - 6.9 to +1.8. Brown and Field (1979) report elasticities of substitution for several mineral inputs (iron, aluminium, copper) and wood pulp, in relation to manufactured capital inputs, ranging from +0.6 to +15. Kummel (1989) has investigated the estimation of productivity improvements based on a production function with manufactured capital, labor, and primary energy, but the results are inconclusive. Although these sorts of econometric estimations may have worth for individual sectoral analyses, there are several reasons to doubt their relevance as indications for overall sustainability prospects. First, these estimates are sensitive to the form of production function assumed and to the estimation techniques employed (see Faucheux 1993; Faucheux and Noel 1995; Stern 1994a, 1994b). Second, and more important, these results pertain to very specific production inputs, not to the broad spectrum of environmental goods and services. We cannot infer much about substitutability between life support, pollution assimilation, and biological stock renewal functions, and economic capital, from examination of elasticities for a few minerals. As authors such as Daly (1994), Victor (1991), and Victor, Hanna, and Kubursi (1997) argue, knowledge from physical and life sciences suggests that ready substitutability between natural and manufactured capitals should not be presumed, and that casual aggregation is a rather chancy business. For example, thermodynamic irreversibility implies the impossibility of substituting, beyond certain well-defined limits, away from environmental sources of "free energy" as production inputs. Substitution may be reasonably easy between energy types, but this relative ease applies only within the class of energy sources, not between energy and other production inputs (Slesser 1978; Peet 1992). Ecological systems have complex spatial structures, and are interlocked with geophysical processes (such as hydrological cycles) that extend over large (sometimes planetary) distances. These systems cannot be fragmented and transported in the same way as minerals and manufactured capital inputs. There is a strong complementarity of "inputs" in the processes of reproduction and renewal of ecological systems that works against the application of the concept of substitution on the margin.
VI. CONCLUSIONS: UNSUSTAINABILITY OF THE "WEAK" INDICATORS
Estimates for SNI(i), presumed equal to gNNP, and for Hartwick "net savings" as sustainability indicators obtained by deductions of natural capital "depreciation" from conventional GNP using current prices, are both logically invalid and empirically suspect for the situations in which they are most urgently needed. The estimates of changes in total capital stock are often made, in practice, on the basis of current prices without enquiring into the conditions for "correctness" of these valuations. But also - and more crucially - there is no way of being able to gauge, from market price and quantity information, the extent to which they might be "incorrect." The validity of the estimates cannot be gauged without knowing in advance the values of all key technical and social parameters determining actual and possible economic performance, and this is impossible.
So the real value of the neoclassical natural capital theory, properly understood, is merely didactic. It allows the construction of parables to alert us:
* first, to the likelihood of "failure" of existing prices to signal intertemporal opportunity costs of natural resource use and environmental degradation;
* second, to the fact that, even if prices are assumed to be PVU-optimal, they almost certainly do not correspond to anything near a sustainable resource use timepath; and if not, then the "weak criterion" for sustainability (nonnegative "net savings") is logically invalid and wholly unreliable as an indicator of long-run economic performance prospects.
All is not lost, however. The explicit use of scenario-type analyses introducing differing hypotheses about key parameters (e.g., complementarily or substitutability, savings rules, technological change) can, we suggest, serve to help decision makers understand better the difficulties inherent in evaluating sustainability prospects, and the nature of the judgments about uncertainty and burden-sharing involved in resource conservation and investment policies (see also da Motta 1997). But, going beyond the didactic value, empirical quantification of the severity of the risks and "trade-offs" associated with natural capital use requires quite different forms of decision-support analysis from what the neoclassical optimization analysis can provide.
1 The distinctions between GDP and GNP, and between NDP and NNP, do not matter for this paper. Most (though not all) of the models in this domain are "closed" economies, so it is most convenient to write GNP and gNNP.
2 The consumer's marginal rate of intertemporal substitution is made equal to the marginal product of capital.
3 The converse is also true. Dasgupta and Heal (1979), for example, showed that under some conditions an investment program assuring nondeclining per capita consumption while natural capital was depleted could be achieved through an income tax combined with government investment - a policy regime involving a "trade-off" between efficiency and intergenerational equity.
4 Work by Muir (1996) draws attention to the significance of groups having divergent preferences within a given generation. If income distribution is shifted towards groups whose preferences are for goods that are less demanding of natural capital exploitation or who "care more about the future," this will tend to favor sustainability of the model equilibrium. This result is implicit in Howarth and Norgaard's work, and can be inferred from Pezzey's results, but was not systematically brought out.
5 Thus, Pareto-efficiency and sustainability (and equity considerations more generally) must be considered as distinct, and complementary, criteria for characterizing model solutions and policy possibilities (see also Dasgupta and Mitra 1983; Dubourg and Pearce 1996; Toman, Pezzey, and Krautkraemer 1995; Baranzini and Bourguignon 1995).
6 This result also applies to the "correct" valuation of environmental amenity and to the size of a "corrective tax" for internalizing an external effect related to economic production or natural capital exploitation. Howarth and Norgaard (1992) show that, for a situation of cumulative pollution, both the "efficient" tax level and the interest rate are functions of the income distribution between generations (as determined in their model by the choice of social welfare function). For further theoretical discussion of "endowment effects" on valuation, see Martinez-Alier and O'Connor (1996), O'Connor and Muir (1995), and Muir (1996).
7 The number of periods can be varied between N = 3 and N = any large (finite) number, limited only by computational capacity of the software and hardware.
8 An analogue to a "Hartwick path" having [U.sub.n + 1] = [U.sub.n], for n = 1 ... N - 1, could presumably be obtained as a PVU-max solution for only one particular set of intergenerational time-discount parameters [[Delta].sub.n] which varied from one generation to the next. It might be thought that this set of parameters should satisfy [[Delta].sub.1] [less than] ... [less than] [[Delta].sub.n] [less than] [[Delta].sub.n + 1] ... [[Delta].sub.N] [less than] 1;but the overlapping generations and finite period character of the model can complicate the specification (see Lupi and Tomasi 1994 for some indications); this could be explored in our model but we have not done it here.
9 Extension of the model to a larger number of periods (say N = 50) does not alter the qualitative features of the four cases chosen. Note in particular that cases (A) and (B) have the substitution elasticity and output elasticity properties identified by Stiglitz, Solow, Hartwick, and others as capable of supporting a finite nondecreasing consumption indefinitely.
10 For case (D/S), the "old" generation in period 1 has a curiously high consumption. This seems to be an artifact of the way the solution conditions were specified for this "old," in conjunction with the nondeclining utility condition. We have not smoothed the blip away; it is a reminder that inevitably there are some arbitrary elements in the way that initial and final periods are dealt with in an OLG model.
11 Apart from Pezzey's work and our own results reported here, relatively little neoclassical modeling work along these lines seems to have been done. This reflects past computational obstacles (which are no longer as severe), and perhaps also modelers' addiction to PVU-max criteria?
12 See also Johansson and Lofgren (1996), who look at the way that price changes make welfare comparisons for alternative investment/consumption timepaths a hazardous business.
13 This still seems to be the case with some recent literature, for example, Hartwick (1991) and Pearce, Hamilton, and Atkinson (1996). On the other hand, several recent authors correctly expressed misgivings about the correct interpretation and robustness of the deduction-based indicators, without identifying the theoretical "wrong prices" problems as such (e.g., Faucheux and Froger 1994; Aaheim and Nyborg 1995).
14 In work reporting estimations of performance according to the "weak sustainability" criterion, Pearce and Atkinson (1993, 104) stated: "We begin with an intuitive rule for determining whether a country is on or off a sustainable development path. To do this, we adopt a neoclassical stance and assume the possibility of substitution between 'natural' and 'man-made' capital...." They presented empirical calculations supposedly as estimates of the value of the "weak sustainability indicator" for 22 countries, of which only 8 do not fulfill the nonnegativity condition. Proops and Atkinson (1997) present similar sets of results, making adjustments for international trade such that natural capital depreciation is attributed to the country of final goods consumption rather than production. Some changes in magnitudes are observable, but the basic pattern remains the same, and the underlying assumptions of the approach are the same whether or not the figures are "corrected" for international trade. The evident defect of this work is that the depreciation covers only a very small number of categories of marketed natural resources and environmental degradation, using current prices or inferences from current prices.
15 In the case of an open economy, the specifications for sustained consumption will be somewhat different (see the contribution by Brekke 1997, in this issue), and under certain assumptions the Hicks criterion dK/dt = 0 is the relevant one. But still, the same "chicken-and-egg" problem will remain, that the indicator's validity is not assured unless one is already on a sustainable path, and this is precisely what one wants the indicator for, in order to find out.
16 The Hartwick net savings for a period can be written: [p.sub.1] [Delta]M + [p.sub.3] [Delta]R. In each of the early periods, the price and quantity biases for manufactured capital savings reinforce each other, so as to bias upwards the first term in the Hartwick "net savings" measure compared with the sustained-utility path. For the natural capital "depreciation", the price bias (-) is offset by the quantity bias (+) due to the more rapid depletion in each period, but this offsetting is not enough to "correct" the indicator properly. The mathematics gets very messy; see the seminal work by Asheim and Pezzey.
17 The results reported by Pearce and Atkinson (1993) and by Proops and Atkinson (1997), involve estimates of natural capital depreciation using market values for a very limited range of items such as forest products, petroleum, and minerals.
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Faucheux is professor of economic sciences at the Universite de Versailles and director of the Centre d'Economie et d'Ethique pour l'Environnement et le Developpement (C3ED) in France; Muir is a freelance analyst and consultant and formerly research officer at the University of Auckland in New Zealand; O'Connor is professor-associe of economic sciences at the Universite de Versailles and formerly lecturer in economics at the University of Auckland. The preparation of this paper was in part supported by contract EV5V-CT940363 on "Methodological Problems in the Calculation of Environmentally Adjusted National Income Figures" financed by the DG-XII of the European Commission. Thanks to Andrea Baranzini and Richard Howarth for help and advice; responsibility for any remaining errors is with the authors.
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|Title Annotation:||Special Issue: Defining Sustainability|
|Author:||Faucheux, Sylvie; Muir, Eliot; O'Connor, Martin|
|Date:||Nov 1, 1997|
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