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The economic value of ecological stability

Communicated by Kenneth J. Arrow, Stanford University, Stanford, CA, April 15, 2003 (received for review January 28, 2003)
Abstract
Seemingly intangible ecosystem characteristics that preoccupy ecologists, like ecosystem stability and the responsiveness of populations to environmental variation, have quantifiable economic values. We show how to derive these values, and how their consideration should change environmental decision making. To illustrate these concepts, we use a simple reserve design model. When resource managers choose a particular landscape configuration, their decision affects both the mean abundance of species and the temporal variation in abundances. Population stability and related phenomena have economic value, because management actions affect the variance of ecosystem components. In our example, a larger reserve size is recommended when accounting for the stability of the managed ecosystem.
The dynamical stability of populations and ecosystems governs their responsiveness to fluctuating environmental conditions and determines with what reliability these natural resources provide lifesustaining services (1) to society. Population and ecosystem stability is thus a major structuring theme in ecology (2–5). An ecosystem that is only weakly stable will vary widely in response to a changing environment, but one that is more stable can be relied on to provide ecosystem services consistently. Weakly stable ecosystems will be prone to species extinctions (6) and thus could sustain less diverse biotas. Therefore, it is important that any impacts to the stability of ecosystems be considered when designing environmental management plans. However, we lack accounting frameworks that can evaluate ecosystem stability and factor it into environmental cost–benefit analyses. In this report, we develop such a framework and use it to illustrate when ecosystem stability has quantifiable economic value, and how consideration of that value should change environmental planning.
To illustrate our approach, we use a minimal model so that the ideas remain as transparent as possible. The model examines an idealized bird species that occupies a terrestrial reserve. The principles illustrated by this model are sufficiently general that they will underlie diverse management decisions ranging from marine reserve design (7) to the management of pollination services (8, 9).
General Theory
Consider an area of habitat (A with areal units chosen so that A = 1), part of which is to be set aside as a reserve (area R), and part of which is to be cleared for timber and then left fallow, with the proceeds from the clearcutting invested elsewhere.‡ Suppose the habitat contains a population of wild birds that can generate an annual revenue in situ (6), perhaps from ecotourism (10, 11), from a contingent or hedonic valuation of society's commitment to conservation (12), or through mitigation banking (13, 14). We assume that the birds cannot survive in cleared lands. Juveniles are born each year according to some densitydependent relationship, f(N_{t}, R, a), where N_{t} is adult abundance, and parameter a describes environmental conditions. Juveniles disperse to find territories, and fraction m(R) disperse into the cleared habitat where they die. Adults die each year at densityindependent rate μ. The population dynamics are given by [1] We restrict attention to functional forms for f and values of a for which the population has a single positive stable equilibrium, N̂ (R), for any reserve size greater than the minimum needed to conserve it (15). The stability of the equilibrium is measured by the slope of g, λ = ∂g/∂N. The larger the reserve, the larger and, typically, the more stable the equilibrial population.§ Derivations can be found in the Appendix.
A social planner must decide what size a reserve should be. One basis for such a decision would be to maximize the total longterm economic yield both from clearcutting and from the in situ value of the birds under some social discount rate, δ (16, 17). Suppose the price of timber is L per unit area clearcut, and the annual revenue from the birds is π(N_{t}). Then, the optimal reserve size satisfies [2] By this condition, the return from clearing 1 more acre and investing the revenue elsewhere equals the return from placing that acre into the reserve and allowing a larger population of birds to flourish.
To examine the importance of stability, we extend the model to include environmental variability. We assume the number of offspring that survive to join the adult population varies in response to environmental conditions. We could model these fluctuations by perturbing population size N directly, by perturbing parameter a in f each year, or both (5).
The stochastic population dynamics are characterized by a stationary distribution in which the population fluctuates about a mean abundance N̂ that is set by the reserve size. The variance of population fluctuations, σ^{2}, is always determined by the stability level, λ. The stability level determines how much of an initial impact the population will “remember” in subsequent years. If the perturbations are made to model parameters, then the variance is also determined by the sensitivity of the population to the environmental conditions, ν = ∂g/∂a. The environmental sensitivity of the population determines how large an impact a given fluctuation in environmental conditions has on N. Both the stability and environmental sensitivity of the population are determined by the size of the reserve.
Our condition for the optimal reserve size now becomes [3] where ε is the expectation operator (Appendix), or, in words, [4]
The first term on the righthand side of the expanded form of Eq. 3 represents the marginal expected value of adding another acre to the reserve that arises from the change to the mean population size. A similar term appears on the righthand side of Eq. 2. However, the remaining two contributions are new and reflect the change to the variance of the population. The second term represents the marginal value of stability; it is the value from adding 1 more acre to the reserve that arises because the stability of the bird population changes and results in a change in the population variance. Likewise, the third term represents the value of the change in the environmental sensitivity of the population from conserving another acre.¶
Economic theory that ignored environmental variation would recommend a reserve size that satisfies Eq. 2. However, in a fluctuating environment, the economically optimal reserve size is given by Eq. 3. This equation is new and different to the deterministic recommendation and often will result in larger reserves being recommended. Larger reserves are recommended because of the additional value associated with variance components like stability.
Our method contrasts with the conventional economic approach to resource management in fluctuating environments. The conventional approach would assume that managers could control natural variation and respond to future environmental conditions by shifting reserve boundaries once those conditions have been revealed in annual population counts (18–21). However, if it is excessively costly to adjust reserve boundaries each year, then an approach is required that considers an ecosystem's intrinsic dynamics. Eq. 3 provides such an approach by accounting for the additional value offered by stable ecosystems through their reduced susceptibility to environmental variation. To obtain Eq. 3, we developed the opposing extreme of noninterventionist control to emphasize the difference between the two situations. However, we anticipate a continuum of partial control scenarios that spans the poles from perfectly responsive to unresponsive management (22, 23). The importance of maintaining stable populations will increase when traversing the continuum from responsive to unresponsive management by consideration of different ecological and institutional contexts.
An important intermediate case between the two extremes of responsive and unresponsive management arises when reserve boundaries can be shifted incrementally but unidirectionally, because habitat destruction is irreversible. Our reserve design problem is reminiscent of Arrow and Fisher's classic treatment of quasioption value in such a development context (24) and of subsequent generalizations of their work (25). Our model makes clear, however, that stability and option values are distinct from one another, because, although stability value can be important here, there is no option value. There is no option value in our particular model, because the uncertainty is assumed irreducible, and the development rule is inflexible. However, in general, stability and option values will be complementary, and both will need to be considered in a full accounting framework. It would be interesting to explore any interactions between the two value sources along approach paths to longrun development equilibria.
Example
To illustrate the consequences of valuing the variance components, we specify functional forms in the model. For what follows, we assume that per capita fecundity decreases linearly with population density as resources, such as nest sites, become limiting, f = aN(1  N/(RK)). Moreover, we assume that intrinsic fecundity fluctuates in response to average regional temperature by perturbing a each year. We assume that juveniles disperse smoothly over the whole area, m = 1  R. We consider two in situ revenue functions: (i) a linear function in which there is a fixed price per bird; and (ii) a function in which each additional bird is worth progressively less, either in terms of conservation value or from tourism revenues; [5]
If the in situ revenue depends linearly on the population size (case i), it is optimal either to designate the entire area a reserve (R = 1) or to clearcut it (R = 0). Which of these alternatives is preferable depends on the relative prices of birds and timber and on the discount rate. The reason that one of the two extreme solutions is optimal is that for every acre considered, either the timber value or the in situ value of the birds is greater, and the linearity of π ensures that the rank ordering of these values is always the same. When environmental conditions vary, the expected overall revenue is identical to the overall revenue obtained in a constant environment. Therefore, there is no economic value to stability in this case, and the optimal reserve size is unchanged when environmental variability is included. No change occurs, because the linear social objective is sensitive not to the variance in population sizes but only to the mean.
If investments in the reserve yield a diminishing return (case ii), one must first examine whether any such investment is warranted. As environmental variability is increased, the conditions that must be met for investment in the reserve to prove worthwhile become more stringent, and clearcutting becomes an increasingly attractive option. The conditions for investment in conservation become more restrictive, because the in situ value of the population diminishes when it is often scarce due to natural fluctuations in abundance (Fig. 1).
Provided that some investment in the population is worthwhile, an intermediate reserve size is often optimal in the case of diminishing returns (Eq. 5, case ii) (Fig. 1). The optimal reserve size balances potential increases from the two revenue streams. So long as the bird population continues to offer a competitive investment, it is usually optimal to set aside a larger reserve in variable environments than is desirable in the special case of a constant environment (Figs. 1 and 2). In part, the increase in the optimal reserve size can be attributed to the increase in stability in larger reserves. However, stability cannot be increased in isolation, and one must account for the full suite of changes in ecosystem attributes that accompany an increase in reserve size.
When choosing among particular landscape configurations, managers are choosing among different bundles of ecosystem attributes. These attributes include a mean population size, a stability level, and an associated degree of sensitivity to the environment. At the optimum, the marginal value of these attributes when summed across the entire bundle must equal the marginal value of clearing 1 more acre (Eq. 3). Optimal bundles of attributes are illustrated in Fig. 3 as the variability in the environment is increased. Note that in our particular example, the marginal values of stability and environmental sensitivity are of opposing sign. The stability of the population increases with increasing reserve size, which enhances the value of larger reserves in fluctuating environments. However, increasing the size of the reserve exposes a larger population of birds to fluctuating climatic conditions, and thereby any particular environmental fluctuation initially induces a greater change in population size; i.e., the environmental sensitivity of the population also increases. Even when accounting for the increase in environmental sensitivity, a larger reserve size is optimal in fluctuating environments than is optimal in a constant environment, provided that we are below the threshold level of environmental variability above which no reserve is worthwhile (Fig. 2). However, the relationship between environmental variability and optimal reserve size is not monotonic, and the downturn in optimal reserve size in highly variable environments (Fig. 2) is caused by the interplay between the different ecosystem attributes.
Conclusion
The suite of ecosystem characteristics chosen with some management decision determines the quality of ecosystem services provided to society (1). For efficient decision making, managers must account for the full bundle of ecosystem attributes that are affected by any management actions. Often, economic studies have focused only on the consequences of changes to mean population size and not on additional attributes like population variances. In our example, the variance components inherit their value via the nonlinear revenue function. Alternatively, variance might be valued directly, if society prefers environmental portfolios that ensure reliable provision of ecosystem services (26). We obtain a value for stability and environmental sensitivity under an assumption of risk neutrality. Stability values could be even more important if social planners were risk averse. It would be interesting to examine the values of variance components for other ecosystem services, such as the provision of fresh water, for which there may be few readily available substitutes and for which one would need to generalize stability concepts to more complex ecosystems.
In the specific context of reserves, we argue that the impact of management actions on the variation in ecosystem components should be accounted for in reserve design. The management philosophy embodied in this approach is one of “living with” variation (7) and contrasts with an alternative tradition in natural resource management of trying to minimize and control natural variation (18). Because managers must always confront some residual natural variability, approaches that value natural variation and account for it explicitly in environmental decision making are urgently needed.
Acknowledgments
We thank K. Arrow, W. Brock, L. Buckley, G. Daily, P. Ehrlich, P. Higgins, M. James, C. Kappel, F. Micheli, P. Munday, H. Pereira, and two anonymous reviewers for useful suggestions for improving the manuscript. P.R.A. thanks the U.K.–U.S. Fulbright Commission for support.
Appendix
Here we provide details of the derivations.
Deterministic Model. The deterministic dynamics are given by Eq. 1 and, by assumption, are characterized by the existence of two equilibria N = 0 and the solution, N̂, of The former is stable for reserve sizes below some critical threshold, and the latter is stable and positive above the threshold. The positive equilibrium is stable whenever the magnitude of eigenvalue is less than 1.
The revenue from clearcutting area (1  R) at time zero is L(1  R). The annual revenue from birds is π(N_{t}). Therefore, the present value of both ecosystem services is where for brevity we have assumed rapid convergence to the equilibrium population size after clearcutting. Therefore, maximizing V with the choice of R requires that Eq. 2 be satisfied.
Stochastic Model. If parameter a in f is perturbed each year, then the stochastic dynamics are where n_{t} = N_{t}  N̂ describes the deviation of the population size at time t from equilibrium N̂. By linearizing about the equilibrium, we can approximate the dynamics with the firstorder autoregressive process (27). If N is perturbed directly, then ν = 1 in this equation. This approximation performs best in the vicinity of the equilibrium and deteriorates as perturbations from equilibrium become larger. We assume that the environmental fluctuations ε_{t} are independent identically distributed normal random variables, which implies that the distribution of population sizes will converge to a normal distribution with mean N̂ and variance (27). For brevity, we assume that convergence to this stationary distribution occurs rapidly after clearcutting.
By assuming rapid convergence, we focus on the longrun equilibrium outcome of development and neglect considerations of the approach path taken to this equilibrium. For local reserve design decisions, our assumption of rapid population decline after clearcutting seems reasonable. However, when considering largerscale environmental planning decisions and sustainability questions, it will be important to extend the current work to consider dynamics during the development process itself (25).
We assume that the objective is to maximize the expected present value from both services, where we have taken the expectation of a linear combination of random variables and relied on our assumption of rapid convergence. Maximizing ε [V] with the choice of reserve size requires satisfying Eq. 3. If π is linear in N (Eq. 5, case i), then ε [π(N)] = π(ε [N]), and the present value of the land parcel is the same for the deterministic and stochastic environments. However, this equivalence breaks down if π depends nonlinearly on N (Eq. 5, case ii). If π is nonlinear, then the present value will depend on both the expectation and variance of the population abundance for the stochastic environment (28).
When performing the optimization, we assume that the environment varies over shorter time scales than those over which reserve design decisions are made. Specifically, we assume that managers meet to agree and legislate on a landscape plan, and that once that plan is enforced, it is difficult or expensive to reconvene the legislature to redefine reserve boundaries should more information on the bird population become available. If management were sufficiently flexible that reserve boundaries could be altered over similar time scales to those on which the environment varies, then a more sophisticated dynamic programming approach would be required (20, 25).
Example. Suppose that f = aN(1  N/(RK)), m = 1  R, and parameter a is perturbed. Then for a reserve of size R, Below minimum reserve size μ/a, the population of birds goes extinct, and above it, a stable population can be maintained.
In case ii (Eq. 5), the present value of the two services in a constant environment is A local maximum of V is given by the solution of Eq. 2, For an investment in the reserve to be worthwhile, we require that V(R^{*}) ≥ L for R^{*} ≤ 1 and V(1) ≥ L for R^{*} > 1. Provided an intermediate reserve size is optimal, then it is given by R^{*}, and if R^{*} ≥ 1 and V(1) > L, then it is optimal to conserve the entire area.
When environmental conditions fluctuate, the expected present value of the land parcel is It can be seen from this expression that the overall value of a reserve decreases as environmental variability, σ_{ε}, increases and results in an increase in σ. This finding results from the concavity of function π in case ii.
A local maximum of ε [V] is given by the solution, R^{*}, of Eq. 3, which becomes The criterion that must be satisfied for investment in a reserve to be worthwhile is that ε [V(R^{*})] ≥ L for R^{*} ≤ 1 or ε [V(1)] ≥
L for R^{*} > 1. Provided that an intermediate reserve size is optimal, then it is given by R^{*}, or again, the extreme solution, R = 1, could be optimal.
The first term on the righthand side of the above equation is the marginal value of the change to the mean population size due to an increase in reserve size; the second term is the value of the change to population stability; and the third is the value of the change to environmental sensitivity (Fig. 3, solid, dashed, and dot–dash curves, respectively). The marginal value of clearing 1 more acre appears on the lefthand side of the equation (Fig. 3, crosses).
Footnotes

↵† To whom correspondence should be addressed. Email: armsworth{at}stanford.edu.

↵‡ Here, we examine a tradeoff between an initial liquidation of natural capital and a single sustainable use of land, but our approach could be adapted to compare alternative sustainable land uses.

↵§ The equilibrium is said to be stable if λ < 1 and unstable if λ > 1. Population stability increases as λ → 0, and the population becomes progressively less stable as λ increases toward 1.

↵¶ If N is perturbed directly, then this third term is zero.
 Received January 28, 2003.
 Copyright © 2003, The National Academy of Sciences
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