Dynamic Efficiency of Conservation of Renewable Resources under ...

Journal of Economic Theory 95, 186�214 (2000)

Dynamic Efficiency of Conservation of RenewableResources under Uncertainty1

Lars J. Olson

Department of Agricultural and Resource Economics, University of Maryland,College Park, Maryland 20742

and

Santanu Roy

Department of Economics, Florida International University, Miami, Florida 33199

Received May 24, 1999; final version received April 28, 2000

We examine the efficiency of conservation of a renewable resource whose naturalproductivity is influenced by random environmental disturbances. We allow fornon-concave biological production and stock-dependent social welfare. Unlikedeterministic models, conservation may be inefficient no matter how productive theresource growth function is. In addition, improvements in the natural productivityof the resource might increase the possibility of extinction. We characterize theconditions on social welfare, resource growth, the discount rate, and the distribu-tion of environmental disturbances that are sufficient for conservation to beefficient. The productivity of the resource under the worst possible environmentalconditions, the discount rate, and the welfare function are all crucial factorsin determining the efficiency of conservation. Journal of Economic LiteratureClassification Numbers: Q20, O41, D90. � 2000 Academic Press

1. INTRODUCTION

Beginning with the seminal paper by Clark [3], the economics of renewableresource conservation has primarily been studied in the context of deter-ministic models of resource growth. The conventional wisdom from that

doi:10.1006�jeth.2000.2685, available online at http:��www.idealibrary.com on

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1 The authors express their sincere thanks to the Tinbergen Institute, Rotterdam and theCenter for Agricultural and Resource Policy at the University of Maryland, respectively, forgenerous support of research visits that allowed this paper to be completed. We thank RobertBecker, Ngo Van Long and a referee for comments. We have also benefitted from commentsmade by members of the audience during presentations of the paper at the 1998 Wintermeeting of the Econometric Society, Guelph University, Dalhousie University, the Universityof Maryland, and the University of Michigan. This material is based on work supported bythe National Science Foundation under Grant SBR-9515065.

literature is that when the stock of the resource has a negligible effect oncurrent welfare,2 then some form of conservation is efficient so long as theresource is more productive than the rate at which agents discount thefuture. The deterministic models on which this conclusion is based are instark contrast to reality, where environmental disturbances cause varia-tions of considerable magnitude in the productivity of renewable resources.When resource growth is stochastic the conventional wisdom from deter-ministic models does not apply. In particular, conservation may be inef-ficient no matter how productive the resource is and a simple comparisonbetween resource productivity and the discount rate is not sufficient todetermine whether it is economically efficient to conserve a resource.

This paper analyzes the implications of environmental productivity shocksfor the dynamic efficiency of resource conservation in a fairly generalstochastic model of renewable resource allocation. The main purpose is toderive analytical conditions on social welfare, resource growth, the discountrate, and the distribution of environmental disturbances under which it isefficient to conserve the resource.

At a fundamental level, models of renewable resources can be viewed asextensions of the classical growth model that generalize its assumptionsregarding both social welfare and the production technology. This paper isrelated to the literature on optimal growth under uncertainty that originatedin [2] where the main issue relates to the convergence and uniqueness ofthe limiting distribution of the stochastic process of optimal capital stocks.In that literature, social welfare usually depends only on current consump-tion and it is standard to impose the requirement that the capital stockremains bounded away from zero (ruling out the possibility of extinction),either directly, or by assuming strong Inada conditions on the welfare andproduction functions and by assigning strictly positive probability mass tothe lower bound on production.3 A salient property characterizing thenatural production or biological growth of many renewable resources isthat the growth rate is low from small stocks, but it increases as the stockbecomes larger, and then eventually diminishes as the environmental carry-ing capacity is approached. It is therefore common in models of renewableresources to allow non-concave biological production functions (for example,S-shaped production functions).4 A non-concave biological production function

187RESOURCE CONSERVATION UNDER UNCERTAINTY

2 See [14] for the precise meaning of ``negligible,'' in this context.3 This set of assumptions is used in [2] for the case where production is concave and in

[10] for the case of nonconcave production.4 In the optimal growth literature, the qualitative properties of optimal policies with non-

concave production have been characterized��see [8, 5], among others, for the deterministiccase and [10] for the stochastic case. The characterization of the efficiency of conservation in[13] in a stochastic model with stock-independent utility and [14] in a deterministic modelwith stock-dependent utility, also follow for non-concave production functions.

makes it necessary to consider two types of conservation. The first isreferred to as a safe standard of conservation, or a critical stock such thatconservation is always efficient from larger stocks (though it may be inef-ficient to conserve the resource from smaller stocks). The second is globalconservation, where the resource is conserved from any positive initialstock.

In contrast to the classical growth framework, social welfare from theharvest of renewable resources may depend on both resource consumption�harvest and the resource stock. This is because the instantaneous cost ofharvesting a given quantity of the resource may depend on the stock,5 andthere may be amenity or existence values associated with the resource. Thisfeature of the welfare function for renewable resources has importantimplications for the dynamic behavior of optimal resource stocks. In parti-cular, the optimal investment policy can be a non-monotone functionof the current stock and, over time, optimal resource stocks may exhibitcyclical or even chaotic dynamics [9]. Our earlier work on deterministicmodels with a stock-dependent welfare function [14] has shown that, inthe absence of environmental disturbances, conservation can be efficienteven if the marginal productivity of the resource is always less than thediscount rate and that the welfare function plays an important role.Further, with a non-monotonic optimal policy, conservation of the resourcefrom low stocks does not necessarily imply the resource will be conservedfrom high stocks. In the presence of uncertainty this has the strikingimplication that an improvement in the productivity of the resource mightactually increase the range of stocks from which extinction is efficient; andwe illustrate this in an example developed in this paper.

The literature characterizing the dynamic efficiency of conservationunder uncertainty is quite small. In a framework where welfare dependsonly on consumption, [1] and [18] contain some sufficient conditions forthe avoidance of extinction, but these conditions are stated in terms of theMarkov transition equation for optimal capital stocks and are not directlyverifiable from the production and social welfare functions and the distri-bution of environmental disturbances. The conditions in [1] are veryrestrictive in our setting since they assume that the optimal investmentpolicy is a monotone and concave function of the resource stock, propertiesthat cannot be guaranteed in general even when the production and welfarefunctions are concave. A more rigorous examination of conservation and

188 OLSON AND ROY

5 Whether this is true is an empirical issue that depends on the resource under considerationand the technology used to harvest it. Among other factors, harvest costs depend on themarginal concentration of the resource. When this concentration varies with the stock, thenso will harvest costs. Although the concentration may remain constant when stocks are large,as in a schooling fishery (see [4]), it must almost inevitably decline as the stock becomes verysmall.

extinction in this framework is contained in [13]. Our paper generalizesthe conditions for conservation derived in that paper.

A few papers on the economics of renewable resources deal with the ques-tion of conservation when there is uncertainty about resource growth andstock-dependent welfare. [17, Theorem 5] provides sufficient conditions forresource conservation in a model where an (s, S ) investment policy isoptimal.6 When there are no fixed costs these conditions assume that thesocial welfare function is separable and linear in both consumption and theresource stock, and that the resource growth function is strictly concave.[11] analyzes a model with strictly concave growth with assumptions onsocial welfare that ensure the optimal investment policy is monotone. Thepossibility of extinction is then characterized using conditions on the kernelof the optimal stochastic process of resource stocks; however, as in [1] and[18], these conditions are not directly verifiable from the primitives ofthe model. The more recent analysis in [7] assumes that social welfare isindependent of the stock and linear in consumption, and also assumes aspecific parametric form for the resource growth function. The assumptionof linear social welfare function (in both [17] and [7]) implies that theefficiency of conservation is solely determined by the productivity of theresource relative to the discount rate, but we shall demonstrate that thisresult does not hold under more general conditions.

In this paper, we analyze a fairly general model of an optimally managed,single species renewable resource in discrete time. The criterion is the maxi-mization of the expected discounted sum of social welfare over time, wheresocial welfare is a concave function of both consumption and the resourcestock. The evolution of resource stocks is governed by a biological produc-tion function that maps investment (the current stock less consumption)and the outcome of an i.i.d. environmental productivity shock into the stocknext period. Our general framework allows for non-concave production func-tions that exhibit ``bounded growth.'' We develop verifiable conditions onsocial welfare, resource growth, the discount rate, and the distribution ofenvironmental disturbances that are sufficient for conservation to be efficient.

The paper is organized as follows. The basic model and preliminary resultsrelating to the dynamic optimization problem are outlined in Section 2.The concepts of a safe standard of conservation and global conservation andtheir relation to the nature of optimal policy are outlined and illustrated inSection 3. Two examples are provided to illustrate the ways in which thestochastic case differs from the deterministic one. The first example showsthat higher, but uncertain, productivity may reduce the set of initial stocks


6 The (s, S ) terminology comes from the optimal inventory literature and refers to a policywhere consumption is zero if the initial stock is less than S, while if the stock exceeds S thepolicy is to invest s and consume the rest.

from which conservation is efficient. The second example shows that thereare instances where conservation is not efficient no matter how productivethe resource growth function may be. Sections 4 and 5 develop sufficientconditions for a safe standard of conservation and global conservation,respectively. Section 6 presents additional examples that provide insightinto the role played by the welfare function and the lower bound on resourceproductivity in our conditions for conservation. Section 7 summarizes theresults and contains some concluding remarks.

2. THE MODEL

This paper examines the issue of conservation within the context of theproblem of choosing a sequence of resource consumptions (or investments)to maximize the expected discounted sum of social welfare given a stochasticbiological production function for resource growth and a known distributionof environmental disturbances. At each date t, the current resource stockyt # R+ is observed and a harvest or consumption level, ct , is chosen. Theremaining stock represents resource investment or escapement, xt= yt&ct .The feasible set for consumption and investment is denoted by 1( y)=[(x, c) | 0�c, 0�x, c+x�y]. Let [ \t]�

t=1 be an independent and identicallydistributed (i.i.d.) random process taking values in some compact set, 3, asubset of the interval [\

�, \� ] with 0<\

��\�\� <�. 8 denotes the (common)

distribution of the environmental disturbances \t . Growth of the resourceis governed by a biological production or stock-recruitment relationship,f : R+_[\

�, \� ] � R+ , that determines the resource stock next period (gross

output) as a function of current investment in the stock and the environ-mental disturbance such that yt+1=f (xt , \t+1). Resource growth net ofinvestment in the stock is given by f (x, \)&x. It will be useful to referto the upper and lower bounds on production defined by f� (x)=sup\ # 3 f (x, \) and f

�(x)=inf\ # 3 f (x, \). The resource production or growth

function is assumed to satisfy the following assumptions throughout thepaper:

T.1. For all \, f (x, \) is strictly increasing in x.

T.2. For all \, f (0, \)=0.

T.3. f (x, \) is continuous in (x, \) on R+_[ \�, \� ]. For each \ # [ \

�, \� ],

f (x, \) is continuously differentiable in x on R++ .

T.4. There exists x� >0 such that f� (x)<x for all x�x� , and y0 # (0, x� ].

Assumptions T.1�T.3 are standard monotonicity and smoothness restrictionson production. Assumption T.4 is a bounded growth restriction typicallyassociated with a natural carrying capacity for the ecosystem beyond whichthe resource stock cannot grow.

190 OLSON AND ROY

The lower bound on the intrinsic growth rate (i.e., the marginal productat zero investment) is given by the lower right derivative of f, which isdenoted by D+ f (0, \)=lim infx a 0 f $(x, \). Define f

�$(0)=inf\[D+ f (0, \)]

to be the lower bound on the intrinsic growth rate over all possible realizationsof the random shock. We also assume

T.5. f�$(0)>0.

Assumption T.5 ensures that the marginal product is bounded away fromzero no matter how small the investment in the resource stock. It accom-modates cases where the resource production function is not capable ofreplacing investment from small stocks so f (x, \)<x for x close to zero.This is sometimes referred to as critical depensation in resource growth andit implies that extinction occurs from small stocks even if the resourceis never harvested. Of course, T.5 also encompasses cases where theresource is productive enough to sustain itself in the absence of harvesting.

We make no assumption about the concavity of the production functionin general. However, as the resource exhibits bounded growth it isreasonable to assume that eventually diminishing returns must set in. Foreach \ # [ \

�, \� ], if lim infx a 0[ f (x, \)�x]<�, let S(\)=[x̂�0 | [ f (x̂, \)�x̂]

�[ f (x, \)�x] for all x�0]. Otherwise, let S(\)=[0]. Define x̂(\)=sup[x | x # S(\)]. Thus, x̂(\) is the highest investment among the set ofinvestments that maximize average productivity. In the case of a multi-plicative shock x̂(\) is identical for all \. We assume that:

T.6. For each \ # [ \�, \� ], f (x, \) is concave in x on [x̂(\), �).

Define x*=sup\ x̂(\). If x*>0 then the production function is non-con-cave for at least some \. If the production function is concave for all \ thenx*=0. We now state a minimal productivity assumption that insures thelower bound on resource growth is productive from some level of resourceinvestment.

T.7. If x*>0, then f�(x*)>x*; if x*=0, then f

�$(0)>1.

Without an assumption like T.7 there could exist a sequence of produc-tivity shocks such that, from any stock, the resource will become extincteven under a policy of pure accumulation; however, T.7 is slightly strongersince it requires that the resource be sustainable from stocks larger than x*.Figure 1 illustrates a stochastic, non-concave biological production func-tion that satisfies all the assumptions T.1�T.7.

The existing literature on resource allocation with non-concave produc-tion focuses on models where the resource growth function is S-shaped andwhere the resource can always be sustained from low stocks. The model of


FIG. 1. An example of a stochastic biological production function.

resource growth employed here generalizes these two restrictions. First, asillustrated in Fig. 1, it allows for the possibility of critical depensationwhere the resource is incapable of sustaining itself from low stocks. Insuch cases, the important question is whether economic efficiency impliesconservation of the resource from large stocks.7 Second, the model in thispaper considers a broader class of growth functions than those that areS-shaped. Resource growth is allowed to exhibit almost any pattern ofincreasing and decreasing returns on the interval [0, x̂(\)]. The model inthis paper also allows for the possibility that the production functions crossso that one environment is most favorable for resource productivity whenstocks are low, while another environment is most favorable when stocksare high.

Social welfare in each period depends on both consumption and theresource stock and is denoted U(c, y). This welfare function can incorporateconsumer and producer surplus from resource consumption as well asexistence or other non-consumptive values associated with the resource.Even when non-consumptive values are absent, the welfare function willtypically depend on the stock through the effect of the stock on the cost ofharvesting. The objective is to maximize the expected discounted sum ofsocial welfare over time, where 0<$<1 is the discount factor.

Define P=[(c, y): 0�c�y, y�0], P0=[(c, y): 0<c�y, y>0]. Thewelfare function satisfies the following restrictions throughout the paper.

192 OLSON AND ROY

7 A clear exposition of some related fundamental issues involved in optimality of extinctionand conservation is contained in [15].

U.1. U(c, y) is nondecreasing in y.U.2. U(c, y) is concave in (c, y) on P.U.3. U(c, y) is twice continuously differentiable on P0 , and Uy is

bounded above on [(c, y): 0<c�y�x� ].U.4. For any y>0, either Uc(c, y)>0 for all c # (0, y], or there exists

a !( y) # (0, y] such that Uc(c, y)>0 for all c # (0, !( y)) and Uc(c, y)<0 forall c>!( y).

Assumptions U.1�U.3 are standard. U.4 is weaker than the typicalassumption that U is increasing over the domain of c. It implies thatwelfare is either increasing or unimodal in c. This means that for each stockthere is a unique strictly positive consumption that maximizes U over theinterval (0, y]. This allows for the possibility that marginal harvest costsmight exceed marginal benefits at large harvest levels so that excessiveconsumption might decrease instantaneous welfare.

The partial history at date t is given by ht=( y0 , x0 , c0 , ..., xt&1 , ct&1 , yt).A policy ? is a sequence [?0 , ?1 , ...], where ?t is a conditional probabilitymeasure such that ?t(1( yt) | ht)=1. A policy is Markovian if for each t, ?t

depends only on yt . A Markovian policy is stationary if ?t is independentof t. Associated with a policy ? and an initial state y is an expected discountedsum of social welfare, V?( y)=E ��

t=0 $tU( yt , ct), where [ yt , ct] aregenerated by ?, f, and 8 in the obvious manner. The value function V( y)is defined by V( y)=sup[V?( y): ? is a policy]. Assumption T.4 ensuresthat for all y>0, V?( y)<+� for any policy ?. We assume that thereexists a policy ? such that V?( y)>&� from all y>0.8 Thus, the dynamicoptimization problem is well defined and the value is finite from any initialstate. A policy, ?*, is optimal if V?*( y)�V?( y) for all policies ? and all yand V?*( y)=V( y). Standard dynamic programming arguments (e.g.,[19]) imply that there exists an optimal solution such that the value func-tion satisfies the functional equation:

V( y)= supx # 1( y)

U( y&x, y)+$ | V( f (x, \)) d8(\).

Further, V is increasing and continuous. Let X( y) be the set of maximizersof the expression on the right hand side of the functional equation. X( y)is an upper-hemicontinuous correspondence that admits a measurableselection. X( y) shall be referred to as the (stationary) optimal investmentcorrespondence, while C( y)=y&X( y) is the optimal consumption corre-spondence. Every measurable selection from X( y) generates a stationaryoptimal policy and vice-versa. The maximum and minimum selections from


8 If f�$(0)>1 or U(0, y)>&�, then this always holds. If neither of these hold (which is

possible under our assumptions), then this can be insured if the discount factor $ is smallerthan a critical value that depends on the social welfare function and f

�$(0).

X(y) are denoted by Xm(y)=min[x: x # X(y)], and Xm(y)=max[x: x # X(y)].Let h

�( y)=f

�(Xm( y)), and h� ( y)=f� (Xm( y)). The properties of these two

selections are particularly important in characterizing resource conserva-tion.

We now state some well-known results about the monotonicity ofoptimal policies under the assumption:

U.5. Ucc+Ucy�0 on P0 .

Define u(x, y)=U( y&x, y). U.5 implies that u12�0, i.e., u is a super-modular function on the set [(x, y): ( y&x, y) # P0]. For x1>x2 and y1>y2

this is equivalent to [u(x1 , y1)&u(x1 , y2)]�[u(x2 , y1)&u(x2 , y2)] (see[20, Section 3]). The economic interpretation of U.5 is that it is a com-plementarity condition under which an increase in the current resourcestock raises the marginal value of investment in future stocks.

Given this form of complementarity, optimal investment is a monotoniccorrespondence (see e.g., [6, Proposition 2]). The correspondence X( y) isascending if x # X( y) and x$ # X( y$) for y�y$ implies max[x, x$] # X( y)and min[x, x$] # X( y$).

Lemma 1. If U.5 holds then X( y) is an ascending correspondence andXm( y) and XM ( y) are nondecreasing in y.

3. CHARACTERIZING CONSERVATION IN ASTOCHASTIC ENVIRONMENT

This paper defines resource conservation to be an outcome where thestock is strictly bounded away from zero with probability one. This conceptof resource conservation requires that the resource does not become extinctin finite time, nor does the stock size become arbitrarily close to zero, evenif the worst environment is realized in all periods. The paper focuses ontwo types of conservation, global conservation and the existence of a safestandard of conservation.

Definition. A safe standard of conservation exists if there is some ;>0such that lim inf[yt]�; almost surely for all y0 # [;, x� ].

A safe standard of conservation exists if, starting from any initial stocklarger than ;, the optimal sequence of resource stocks on any sample pathis almost surely bounded away from extinction by ;.The stock may becomeextinct or approach extinction from initial stocks smaller than ;.

Definition. Global conservation occurs if lim inf[ yt]>0 almost surelyfor all y0 # (0, x� ].

194 OLSON AND ROY

This definition of global conservation is stronger than requiring that thestock remain positive with probability one. It seems reasonable that allow-ing the resource stock to come arbitrarily close to zero does not constituteconservation, even though the resource may never actually be reduced tozero in finite time. Hence, our definition of conservation rules out asymptoticextinction as well as outcomes where the support of the limiting distributionof stocks has zero as an endpoint with no probability mass.

What must be the nature of optimal investment in order to ensure thetwo kinds of conservation defined above? Global conservation requires thatoptimal investment be positive from all stocks greater than zero. Underpositive investment, if extinction occurs from any stock it must also occurfrom stocks close to zero. Therefore, global conservation is ensured ifinvestment is positive, and if for stocks within a neighborhood of zero, theoptimal stock next period does not decrease below its current level nomatter how adverse the environmental disturbance; i.e., there exists `>0such that h

�( y)�y for all y # (0, `).

Next, consider a safe standard of conservation. Much of the existingliterature considers the existence of safe standard of conservation underconditions analogous to U.5. As noted in Lemma 1, the optimal investmentfunction h

�( y) is then a non-decreasing function of the stock. In this case a

safe standard of conservation exists if there is a level of current stock ;such that the optimal stock next period is at least as large as ; even underthe worst state of nature, i.e., h

�(;)�;. This represents a safe standard since

the monotonicity of optimal investment in current stock implies that theresource remains above ; almost surely from all stocks larger than ;.

While the standard approach is to impose an assumption like U.5, it isconceivable that such an assumption might be too restrictive in certainsettings. This may happen if changes in the stock influence the marginalutility of consumption more than changes in consumption, for example,because the marginal harvest cost falls very sharply with increase in stocksize. Therefore, it also seems useful to study the existence of safe standardof conservation under more general conditions where there may not be anymonotonic selection from the optimal investment correspondence. In thatcase, even if the optimal policy from ; is to accumulate the resource, it ispossible that extinction is optimal from stocks higher than ;. In our earlierpaper [14], we provide a deterministic example where extinction is optimalfrom low and high stocks while conservation is optimal from intermediatestocks. The basic intuition is that if Ucy is positive and very large, then themarginal utility of consumption can be high enough at large stocks towarrant a harvest that leads to extinction, while a more moderate harvestis optimal from intermediate stocks. Thus, in order for ; to be a safestandard of conservation it needs to be shown that the optimal stock next


period is almost surely above ; for all values of the current stock above ;,i.e., h

�( y)�; for all y�;.

When the possibility of a nonmontonic optimal investment function iscombined with stochastic environmental disturbances, the implications forconservation can be somewhat surprising. As the following example shows,uniformly better stochastic resource productivity can reduce the set ofstocks from which conservation is optimal.

Example 1. The example is a stochastic version of Example 3.1 in[14]. The resource growth function is given by f (x, \)=\F(x), where:

x&x(x+0.1)(x&1) if x�0.8

F(x)={4x3&12.1x2+12.3x&3.2 if 0.8<x�1

x0.1 if x>1.

The social welfare function is:

U(c, y)=pc&e:c&;y,

where p=10.0, :=1.0, and ;=2.0. With these parameter values the socialwelfare function satisfies U.1�U.4, but not U.5. The discount rate is givenby $=1�1.47, which corresponds to the higher discounting case consideredin [14]. For parametric examples like this the optimal policy can be foundusing numerical dynamic programming methods. Figure 2(a) shows theresource production function and the mapping from stocks in period t tostocks in period t+1 under an optimal policy when \=1.0 with proba-bility one. This represents the poor productivity case. Extinction is optimalfrom both low and high stocks, while conservation is optimal from inter-mediate stocks. Figure 2(b) depicts resource production and the functionsthat govern the transition of optimal stocks for the case where \=1.0 withprobability 0.999 and \=3.0 with probability 0.001. This represents a firstorder stochastic increase in the distribution of the (multiplicative) environ-mental disturbances. Resource productivity is always at least as good andsometimes better than in the poor productivity case. The surprising outcomeis that the resource fails to survive two consecutive good environmentaldisturbances. Since this happens with probability one along any time pathof disturbances, it is almost surely optimal to harvest the resource toextinction from all initial stocks. This is not simply an artifact of animprovement in resource productivity. If \=3.0 with probability one, thenconservation is optimal from all initial stocks. It is the combined influencesof a nonconcave resource growth function, environmental uncertainty, andthe relation between the stock and consumption in social welfare that arethe driving forces behind the outcome of this example. The example provides

196 OLSON AND ROY

FIG. 2(a). Extinction from low and high stocks, conservation from intermediate stocks.

FIG. 2(b). Increased resource productivity leads to almost sure extinction.


one illustration of why it is important to study these factors together inorder to better understand their role in determining economic incentivesfor the conservation of resources.

Conventional analysis of conservation in deterministic models focuses onthe relationship between the discount rate and the growth rate of theresource as the primary determinant of the efficiency of conservation. If theresource is $-productive in the sense that the discounted maximum averageproductivity of the resource is greater than one, then some form of conser-vation is optimal. In general, when resource growth is stochastic it is notpossible to ensure the efficiency of conservation by focusing solely on theproductivity of the resource relative to the discount rate. This is true evenif social welfare is independent of the stock. The following exampledeveloped by Mirman and Zilcha [12] shows that no matter how productivethe resource growth function may be there are instances where conservationis not efficient.

Example 2. Let the resource production function be defined by f (x, \)=\x1�2, where \ is distributed uniformly on some (suitably chosen)compact interval 0<\

��\�\� <�. Since limx � 0 f $(x, \)=� for all \, the

intrinsic growth rate is infinite for all \. The utility function is not specifieddirectly. Instead, Mirman and Zilcha specify a value function V thatsatisfies $E[V$( f (x, \)) f $(x, \)�V$( f (x, \

�))]<1. This implies that if x is

the optimal investment from a stock y, then h�( y)=f (x, \

�)<y so that the

optimal policy under the worst environment exhausts the resource in thelimit from any initial stock. They show indirectly that there exists a wellbehaved utility function that generates the desired value function. In theirexample, the support of the limiting distribution of optimal resource stockshas zero as an endpoint. This endpoint has no probability mass so thatcomplete exhaustion of the stock occurs with zero probability; however, forany =>0, there is some finite time period when the stock falls below = withstrictly positive probability. The net result is that even though the resourceis infinitely productive for all outcomes of \, optimal resource stocks donot satisfy either definition of conservation given above.

This example clearly shows that when resource productivity is affectedby environmental disturbances then conservation cannot be guaranteed byconditions on resource productivity and discounting alone.9 Instead, it isnecessary to consider the interaction between resource productivity and thesocial welfare function, as well as the role of the lower bound on resourcegrowth when developing conditions that are sufficient for an optimal policy

198 OLSON AND ROY

9 In the concluding section in [12], the authors state, ``it should be noted that in randomgrowth models the utility function in combination with the production function determine theproperties of the steady state.'' See also, [13].

to satisfy the two kinds of conservation as outlined above. That is the mainpurpose of the analysis in Sections 4 and 5.

4. THE EXISTENCE OF A SAFE STANDARD OF CONSERVATION

This section examines the conditions under which there is a safe standardof conservation under an optimal policy. Before we go into aspects of theproblem that involve intertemporal trade-offs, there is one class of readilyidentifiable situations in which conservation is always efficient. These aresituations where even a myopic agent will never harvest the resource to extinc-tion. If the marginal utility from consumption is negative when investmentfalls below the level needed to maintain the stock at its current level under theworst productivity shock, then investment by a myopic agent will be sufficientto at least replenish the stock. Since an optimizing agent never consumesmore than a myopic one, this ensures that conservation is efficient. Proposi-tion 1 outlines the conditions explicitly.

Proposition 1. (a) If there exists some x>0 such that f�(x)>x and

Uc( y&x, y)�0 for all y�f�(x), then x is a safe standard of conservation.

(b) If U.5 holds and if there exists some x such that f�(x)>x and

Uc( f�(x)&x, f

�(x))�0, then x is a safe standard of conservation.

Proof of Proposition 1. (a) Choose any y�f�(x). If Uc( y&x, y)�0 it

cannot be optimal to invest less than x from y, even for a myopic agent.Hence, Xm( y)�x. Since the resource stock next period is at leastf (Xm( y), \)�f

�(x), the proof follows by induction. (b) If Uc( f

�(x)&x,

f�(x))�0 then optimal investment from f

�(x) is at least x. Under U.5, the

optimal investment policy is ascending in x so from any y�x, optimalinvestment is never less than x. K

Observe that Proposition 1 does not apply when social welfare is strictlyincreasing in consumption. In what follows, we derive conditions for a safestandard of conservation that are also applicable to situations where amyopic agent would not conserve the resource. The analysis begins withthe case where the stock and investment are complementary in the sense ofU.5. Then we consider the general case.

To obtain a tight condition for conservation it is necessary to overcomethe technical difficulties caused by the nonconvexity of the feasible set forthe dynamic optimization problem when the resource production functionis not concave. Our methodology is to first consider a convexified resourceallocation problem obtained by taking the convex hull of the production


function for each \. We derive a condition that ensures a safe standard ofconservation for this modified problem and we then show that this is alsoa safe standard of conservation for the original problem.

Define a modified production function

F(x, \)={xf (x̂(\))�x̂(\)f (x, \)

if 0�x<x̂(\)if x�x̂(\),

and consider the dynamic optimization problem where F replaces f. For themodified problem, let W( y) be the value function and /( y) the optimal(investment) policy function. As in the original optimization problem,define F

�(x)=inf\ F(x, \), F� (x)=sup\ F(x, \), /m( y)=min[x: x # /( y)],

/M( y)=max[x: x # /( y)], H�

( y)=F�(/m( y)) and H� ( y)=F� (/M( y)). Since F

is concave for all \, W( y) is concave. Let W$+ denote the right-derivativeof W. For c optimal from y in the modified problem it can be shown thatW$+( y)�Uc(c, y)+Uy(c, y), and if c>0 then W is differentiable at y withW$( y)=Uc(c, y)+Uy(c, y).10 Note that any policy that is feasible in theoriginal problem is also feasible in the modified problem. In addition, if theoptimal policy in the modified problem is such that /( y)�x* for ally�f

�(x*) then / is also feasible in the original problem and so it must be

an optimal policy in the original problem.Consider the class of welfare functions for which U.5 holds, i.e., there is

complementarity between investment and the stock. In this case, the mini-mum selection Xm( y) from the optimal investment policy correspondenceis monotone and, as discussed in the previous section, the existence of asafe standard of conservation is equivalent to the existence of a strictlypositive stock ; such that h

�(;)�;. In deterministic optimal growth models

with nonconcave production function and stock-independent welfare, theexistence of a safe standard of conservation is ensured by a ``$-produc-tivity'' requirement that the maximum discounted average productivity ofthe resource growth function be greater than one (see, [5]). If either theproduction function is stochastic (as in [13]) or the marginal utility fromconsumption is stock-dependent (as in [14]), the existence of a safestandard of conservation also depends on properties of the welfare func-tion. Proposition 2 provides a sufficient condition for a safe standard ofconservation that extends these results.

200 OLSON AND ROY

10 Let x be optimal from y�0. Then x is feasible from y+= for =>0. The principle ofoptimality then yields (W( y+=)&W( y))�=�(U( y+=&x, y+=)&U( y&x, y))�=. Letting= a 0 yields W$+( y)�Uc(c, y)+Uy(c, y), where c=y&x. Similarly, if y&x>0 then x isfeasible from y&= and (W( y)&W( y&=))�=�(U( y&x, y)&U( y&x&=, y&=))�= for =>0,which implies W$&( y)�Uc(c, y)+Uy(c, y). Then it must be that W is differentiable at y sinceconcavity implies W$&( y)�W$+( y).

Given an output, y, and an input x, define the set of economically viableinvestments from y that would lead to a resource stock smaller than x by9( y, x)=[z: y&!( y)�z�x�y].

proposition 2. Under U.5 if there exists some x # (x*, x� ) such thatf�(x)>x and

infz # 9( f

�(x), x)

$EUc( f (x, \)&z, f (x, \))+Uy( f (x, \)&z, f (x, \))

Uc( f�(x)&z, f

�(x))

f $(x, \)>1,

then x is a safe standard of conservation.

Proof of Proposition 2. Consider the modified (convex) problem. Letx # (x*, x� ) and define y$=F

�(x)=f

�(x) and x$=/( y$). We want to show

that x$�x. Suppose not. Since y$&x$>0, the principle of optimality yieldsU( y$&x$, y$)&U( y$&x$&=, y$)�$E[W(F(x$+=, \))&W(F(x$, \))] forsufficiently small =>0. It follows that

Uc( y$&x$, y$)

= lim= a 0 \

U( y$&x$, y$)&U( y$&x$&=, y$)= +

�lim inf= a 0

$E \W(F(x$+=, \))&W(F(x$, \))F(x$+=, \)&F(x$, \) +\F(x$+=, \)&F(x$, \)

= +�$E \lim inf

= a 0 \W(F(x$+=, \))&W(x$, \))F(x$+=, \)&F(x$, \) +\F(x$+=, \)&F(x$, \)

= ++ ,

by Fatou's lemma

=$EW$+(F(x$, \)) F$(x$, \)

�$EW$+(F(x, \)) F$(x, \), as x>x$

�$E(Uc(F(x, \)&/m(F(x, \)), F(x, \))+Uy(F(x, \)&/m(F(x, \)),

F(x, \))) F$(x, \)

�$E(Uc(F(x, \)&x$, F(x, \))+Uy(F(x, \)&x$, F(x, \))) F$(x, \),

by /m(F(x, \))�x$=/m(F�(x)) and U.5.


Since x$ # 9( f�(x), x) the sequence of inequalities contradicts the assumption

in the proposition, hence x$�x in the modified problem. As a consequence,starting from any y0�f

�(x), investments always remain above x in the

convexified problem. But this means that starting from any y0�f�(x), the

optimal stock is at least f�(x) in the original problem. If not, there is some

policy in the original problem that yields greater expected value than thepolicy associated with /. But such a policy is feasible in the modifiedproblem so this would contradict the optimality of the policy / in themodified problem. K

The intuition underlying Proposition 2 is as follows. Given the worstproduction from some investment level consider a policy that depletes theresource below the original level. If all such policies have a marginal valueof consumption strictly less than the expected discounted marginal value ofinvestment, then it must be the case that the optimal investment is one thatsustains the stock. Since optimal policies are monotonic under U.5, thestock is conserved under all productivity shocks and from any larger initialstock. Observe that the condition in Proposition 2 reduces to the classical$-productivity condition when welfare is independent of the stock and thereis no uncertainty.11

Next, we consider the general case where the welfare function does notnecessarily satisfy U.5. In this case, the minimum selection from theoptimal investment correspondence is not necessarily monotonic in currentstock size and showing that ; is a safe standard of conservation requiresnot only that h

�(;)�;, but, in addition that h

�( y)�; for all y�;. Define

A(x)= infz # 9( f� (x), x) \1+

Uy( f� (x)&z, f� (x))

Uc( f� (x)&z, f� (x))+ .

A(x) represents the minimum marginal value of investment normalized bythe marginal welfare from consumption when evaluated at all economicallyviable investments from f� (x) that would reduce the stock below x.

202 OLSON AND ROY

11 It is straight-forward to see how the main assumption of Proposition 2 rules out outcomeslike that in the example contained in [12]. In that example, U depends only on consumption andUc(c)>0 for all c. Under these simplifications the main assumption in Proposition 2 canbe stated as: infz # [0, x] $EUc( f (x, \)&z) f $(x, \)�Uc( f

�(x)&z)>1. Let x$=X( f

�(x)) be the

optimal policy from an initial stock, f�(x), obtained under resource production in the worst

environment. Three important properties of the solution in the example are: (i) it satisfies thestochastic Ramsey�Euler equation Uc( f

�(x)&x$)=$EUc(C( f (x$, \))) f $(x$, \), where x$=

X( f�(x)). (ii) the uniqueoptimal investment and consumption policies, X( y) and C( y), are

both nondecreasing, and (iii) x$<x. From these facts it follows that Uc( f�(x)&x$)=

$EUc(C( f (x$, \))) f $(x$, \)>$EUc(C( f (x, \))) f $(x, \)>$EUc( f (x, \)&x$) f $(x, \), but thisis clearly ruled out by the condition above.

Lemma 2. If there exists an x�x* such that f� (x)>x andA(x) $Ef $(x, \)>1, then Xm( f� (x))�x.

Proof of Lemma 2. Let x$=Xm( f� (x))=Xm(F� (x)) and suppose x$<x.Then it follows that

Uc(F� (x)&x$, F� (x))�$EW$+(F(x$, \)) F$(x$, \)

�$EW$+(F(x, \)) F$(x, \)by x>x$ and concavity of W and F

�$W$+(F� (x)) EF$(x, \) by concavity of W

�$(Uc(F� (x)&x$, F� (x))

+Uy(F� (x)&x$, F� (x))) EF$(x, \).

This contradicts A(x) $Ef $(x, \)>1. K

This lemma characterizes outcomes under the best possible realization ofthe random shock affecting resource production. If the marginal value ofinvestment valued in terms of consumption times the expected internal rateof return on investment exceeds one plus the discount rate for all economi-cally feasible investments that would reduce the stock, then it is notoptimal to undertake such an investment and the optimal investment is onethat enhances the stock. However, this does not rule out the possibility thatin worse states of nature, the optimal policy leads to a reduction in stockand eventual extinction. In other words, it does not guarantee a safestandard of conservation.

Define m( y, x)=inf[Uy( y&z, y): x�z�y]. Given an initial stock y,m( y, x) provides a lower bound on the marginal stock effect over feasibleinvestments larger than x. Using this, define

B(x)=\ Uc( f� (x)&x, f� (x))+m( f� (x), x)Uc( f

�(x)&x, f

�(x))+Uy( f

�(x)&x, f

�(x))+ .

The numerator of the expression defining B(x) is a lower bound on themarginal value of sustainable investment under the best state while thedenominator represents the marginal value of sustainable investment underthe worst state. Thus, B(x) is a lower bound on the ratio of the marginalvalue of sustainable investment under the best realization of the produc-tivity shock to that in the worst case. The concavity of U implies that B(x)is less than 1. In the deterministic case, B(x) differs from unity only to theextent that the lower bound on the marginal stock effect provided bym( f (x), x) is different from the value of the marginal stock effect realizedat (c, y)=( f (x)&x, f (x)).


The next proposition shows that the existence of a safe standard ofconservation can be achieved by replacing A(x) by B(x) in the conditionof Lemma 2.

Proposition 3. If there exists some x�x* such that f�(x)>x, Uc( f

�(x)&x,

f�(x))�0 and B(x) $Ef $(x, \)>1, then x is a safe standard of conservation.

Proof of Proposition 3. Consider the modified (convex) problem. Notethat A(x)�1�B(x) so that Lemma 2 implies x$�x where x$ # /(F� (x)).Next, suppose there exists some y�F

�(x) such that /m( y)<x. This implies

Uc(F�(x)&x, F

�(x))+Uy(F

�(x)&x, F

�(x))

�Uc( y&x, y)+Uy( y&x, y)

�Uc( y&x, y)

�Uc( y&/m( y), y)

�$EW$+(F(/m( y), \)) F$(/m( y), \)

�$EW$+(F(x, \))) F$(x, \)

�$W$+(F� (x)) EF$(x, \)

�$(Uc(F� (x)&x$, F� (x))+Uy(F� (x)&x$, F� (x))) EF$(x, \)

�$(Uc(F� (x)&x, F� (x))+m(F� (x), x)) EF$(x, \). (1)

This contradicts B(x) $EF$(x, \)=B(x) $Ef $(x, \)>1 (since x�x*).Hence, we have shown that from all initial stocks y0�F

�(x)=f

�(x), the

optimal stock in the next period of the convexified problem is at least f�(x).

As in the proof of Proposition 2, this means that starting from anyy0�f

�(x), investment always remains above x in the original problem. K

As indicated earlier, the term B(x) in the condition outlined in Proposi-tion 3, is less than one. This means the expected internal rate of return oninvestment in the resource must be larger than 1�$ for the conditions of theproposition to hold.12 The proposition requires conditions stronger thanexpected $-productivity in order to guarantee that conservation is efficient.In the case when there is no uncertainty and social welfare depends onlyon current consumption, B(x)=1 and the condition outlined in Proposi-tion 3 reduces to the usual delta-productivity condition.13

204 OLSON AND ROY

12 This is not to suggest that using an internal rate of return investment rule yields the expectednet present value optimal choice of actions. It simply shows the relation between the (welfareadjusted) internal rate of return and the interest rate that is sufficient for conservation to beoptimal in terms of the expected discounted sum of social welfare.

13 If U.5 holds there is no need to introduce the lower bound m( y, x) in the inequality (1)or in the statement of Proposition 3.

It is easy to see that the condition for a safe standard of conservation inthe general case as outlined in Proposition 3 is stronger than the one wehave derived in Proposition 2 for the case where stock and investment arecomplementary. Both are conditions of the form $E[�(x) f $(x, \)]>1,where the term �(x) represents welfare effects. These originate from twosources. First, the marginal value of investment in additional stock reflectsthe marginal utility of future consumption as well as the direct marginalwelfare associated with increments to the future stock. Thus, even in adeterministic setting, if the level of investment is just sufficient to sustainthe current stock (so that consumption and the stock are constantacross periods), the marginal welfare gain from investment differs from thesacrifice in marginal welfare from current consumption. Second, the factthat the resource production function is stochastic implies that themarginal value of investment is evaluated over all possible realizations ofthe environmental disturbance. The ratio of the marginal value of invest-ment to the marginal utility of current consumption generally differs acrossstates of nature. In general, �(x) represent a lower bound on the ratio ofthe marginal gain in value from an increase in investment to the marginalwelfare sacrificed by the corresponding reduction in current consumption.In the case where consumption and the stock are complementary (Proposi-tion 2), it is possible to derive a much tighter bound on this ratio com-pared to the general case analyzed in Proposition 3. This is becausecomplementarity implies that the marginal welfare gain from an increase inthe future stock is monotonic in future consumption. In the general case, thisproperty is lost. Further, to account for the potential non-monotonicity ofthe optimal investment policy it is necessary to ensure that there is nostock higher than the (potential) safe standard from which it is optimal toinvest less than the safe level. The net outcome of these considerations isthat a more general approach than expected-delta productivity must betaken in the formulation of any condition for conservation in the stochasticcase.

5. GLOBAL CONSERVATION

In this section we examine the conditions under which the resource isconserved from any initial stock and along any realized path of the randomshock. As discussed in Section 3, this is ensured if we can show that h

�( y)=

f�(Xm( y))�y for all y in some neighborhood of zero. In particular, this

requires that f�( y)�y in a neighborhood of zero. Otherwise, the resource

production function is characterized by critical depensation and, evenunder a policy of pure accumulation, the resource is not productive enough


to conserve itself from low stocks. Therefore, in the rest of this section weassume:

T.8. f�$(0)�1 and if f is concave then f

�$(0)>1.

First, we provide conditions that are sufficient to rule out immediateextinction as an optimal policy. This is equivalent to guaranteeing thatoptimal investment in the resource is strictly positive from all initial stocks.

Lemma 3. (a) Assume either (a) U is increasing in c for all y andlim= a 0 [Uc(=, =)+Uy(=, =)] $ED+ f (0, \)>Uc( y, y), or (b) Uc( y, y)<0 forall y >0. Then Xm( y)>0 for all y>0.

Proof of Lemma 3. Suppose not. Then for some y>0, Xm( y)=0.Consider an alternative action from y where consumption is reduced toy&= and =>0 is invested. Then consume the entire output next period,f (=, \)>0. From the principle of optimality

[U( y, y)+$U(0, 0)+$2V(0)]�=

&[U( y&=, y)+$EU( f (=, \), f (=, \))+$2EV(0)]�=

=[[U( y, y)&U( y&=, y)]�=]

&$[E[U( f (=, \), f (=, \))&U(0, 0)]�=]�0. (2)

The condition in the lemma implies

Uc( y, y)<lim= a 0

$[Uc( f (=, \), f (=, \))+Uy( f (=, \), f (=, \))] D+ f $(0, \).

which, in turn implies

lim= a 0

[U( y, y)&U( y&=, y)]�=

=Uc( y, y)

<E[lim inf= a 0

$[Uc( f (=, \), f (=, \))+Uy( f (=, \), f (=, \))] D+ f (0, \)]

�E[lim inf= a 0

$[Uc( f (=, \), f (=, \))+Uy( f (=, \), f (=, \))]( f (=, \)�=)]

�lim inf= a 0

$E[[Uc( f (=, \), f (=, \))+Uy( f (=, \), f (=, \))]( f (=, \)�=)]

�lim inf= a 0

$E[U( f (=, \), f (=, \))&U(0, 0)]�=,

but this contradicts (2) for = small enough. The proof of (b) is trivial. K

206 OLSON AND ROY

This lemma encompasses the Inada condition used to guarantee interiorinvestment in the standard stochastic growth model (see, for example, [2]).14

As an example of its application in the case of stock-dependent welfare, it issatisfied for the class of exponential welfare functions U(c, y)=c:y ;, where:>0, ;�0, and :+;<1. It is easy to see that strictly positive investment isa necessary requirement for global conservation. The conditions given in thelemma above are sufficient for this to occur. Optimal investment may,however, be positive under weaker conditions. So, the results below simplyassume that optimal investment is strictly positive.

As in the discussion on safe standard of conservation, our first resultprovides conditions under which even a myopic agent will conserve theresource from all initial stocks.

Proposition 4. If optimal investment is strictly positive from all initialstocks and if there exists some =>0 such that Uc( f

�(x)&x, f

�(x))<0 for all

x # (0, =), then global conservation is optimal.

Proof of Proposition 4. Since X( y)>0 for all y>0 and X( y) is upper-semicontinuous it follows that x0#inf[X( y): y # [ f

�(=), x� ]]>0. Let z=

min[=, x0], let y # (0, f�(z)) and define x=f

�&1( y). Since x # (0, z) the

hypothesis of the proposition implies Uc( f�(x)&x, f

�(x))<0, or equiv-

alently, Uc( y&x, y)<0. From this it follows that Xm( y)>x so thatf�(Xm( y))>f

�(x)=y for all y lying in (0, f

�(z)). Hence, from any y0 ,

lim inf[ yt]=lim inf[ f�(Xm( yt&1))]�min[ f

�(x0), f

�(z)] a.s. K

We now proceed to derive conditions for global conservation that applyin situations where a myopic agent might not conserve the stocks. In ouranalysis of a safe standard of conservation, we were able to obtain considerableleverage by taking the convex hull of the production set and studying themodified convex dynamic optimization problem. This was a fruitful approachbecause the best hope for finding a safe standard of conservation is in theregion where average productivity of the resource is maximized. Further,for stocks above this region and for the class of resource production func-tions admissible under T.1�T.7, the convex hull coincides with the originalproduction possibilities for the resource. Unfortunately, this approach isnot useful in analyzing global conservation. Global conservation requiresconservation in a neighborhood of zero which is precisely where resourceproduction possibilities are most likely to exhibit nonconvexities. Thiscomplicates our task considerably.


14 In the classical growth model, the Inada condition also guarantees that optimal con-sumption is strictly positive, which our condition does not do.

The next result examines global conservation when U.5 holds so that theoptimal investment policy is monotone.

Proposition 5. Under U.5 if optimal investment is strictly positive fromall initial stocks, if lim infx a 0 Uc( f

�(x)&x, f

�(x))>0, and if

lim infx a 0

$EUc( f (x, \), f (x, \))+Uy( f (x, \), f (x, \))

Uc( f�(x)&x, f

�(x))

f $(x, \)>1,

then global conservation is optimal.

Proof of Proposition 5. Suppose not. Then, there exists sequences [xn],[ yn] a 0 such that f

�(xn)<yn for all n where xn # X( yn). U.5 implies

Uc( yn&xn , yn)�Uc( f�(xn)&xn , f

�(xn)). From the Ramsey Euler equation

Uc( yn&xn , yn)=$E[[Uc( f (xn , \)&X( f (xn , \)), f (xn , \))

+Uy( f (xn , \)&X( f (xn , \)), f (xn , \))] f $(xn , \)]

�$E[[Uc( f (xn , \), f (xn , \))

+Uy( f (xn , \), f (xn , \))] f $(xn , \)],

which then yields 1�$E[( f $(xn , \))[Uc( f (xn , \), f (xn , \))+Uy( f (xn , \),f (xn , \))]� [Uc( f

�(xn)&xn , f

�(xn))]]. This yields a contradiction for n

large enough. K

Proposition 5 is the natural analogue of Proposition 2 for stocksapproaching zero. If under the worst production from stocks close to zero,a policy that further depletes the stock has a marginal value of consump-tion that is strictly less than the expected discounted marginal value of zeroinvestment, then optimal investment must be one that leads to conservation.

Our final result provides general conditions that are sufficient for globalconservation even if the optimal investment policy is non-monotone andU.5 does not hold.

Proposition 6. If optimal investment is strictly positive from all initialstocks, if lim infx a 0 Uc( f

�(x)&x, f

�(x))>0, and if

lim infx a 0

$EUc( f (x, \), f (x, \))+m( f (x, \), 0))

Uc( f�(x)&x, f (x))+Uy( f (x)&x, f (x))

f $(x, \)>1,

then global conservation is optimal.

Proof of Proposition 6. Suppose not. Then there exists sequences [xn],[ yn] a 0 such that f

�(xn)<yn for all n where xn # X( yn). From the

Ramsey�Euler equation

208 OLSON AND ROY

Uc( yn&xn , yn)=$E[[Uc( f (xn , \)&X( f (xn , \)), f (xn , \))

+Uy( f (xn , \)&X( f (xn , \)), f (xn , \))] f $(xn , \)]

�$E[[Uc( f (xn , \), f (xn , \))+m( f (xn , \), 0))] f $(xn , \)].

From concavity of U it follows that Uc( y&x, y)+Uy( y&x, y) is nonin-creasing in y so that

Uc( f�(xn)&xn , f

�(xn))+Uy( f

�(xn)&xn , f

�(xn))

�Uc( yn&xn , yn)+Uy( yn&xn , yn)

for yn�f�(xn). Using this, Uy�0, and the Ramsey�Euler equation we

obtain

Uc( f�(xn)&xn , f

�(xn))+Uy( f

�(xn)&xn , f

�(xn))

�Uc( yn&xn , yn)+Uy( yn&xn , yn)

�Uc( yn&xn , yn)

=$E[[Uc( f (xn , \)&X( f (xn , \)), f (xn , \))

+Uy( f (xn , \)&x( f (xn , \)), f (xn , \))] f $(xn , \)]

�$E[[Uc( f (xn , \), f (xn , \))+m( f (xn , \), 0))] f $(xn , \)].

This yields a contradiction for n large enough. K

The first condition in Proposition 6 simply requires that, as the stockgoes to zero, sustainable investment is economically viable under the lowerbound on resource productivity. The second condition is the analogue ofthe condition in Proposition 3 in the case where the stock approaches zero,except that the numerators are different. The source of this difference is thatProposition 3 uses information from the convex hull of the productionfunction, whereas in the neighborhood of the origin the production func-tion may be convex.15

6. ILLUSTRATIVE EXAMPLES

The results of this paper clearly show that the relation between socialwelfare and resource productivity is important in determining whether


15 If U.5 holds then in the proof the lower bound on the stock effect defined by m( y, x) canbe replaced by Uy evaluated at ( f

�(x)&x, f

�(x)).

conservation is efficient. It is also apparent that the lower bound onresource productivity has a strong influence on whether or not oursufficient conditions for conservation will be satisfied. The role of theseeffects is illustrated in three examples. The first example illustrates theconditions under a stochastic version of the logistic resource growthfunction that is common in applications, and the class of welfare functionsexhibiting constant relative risk aversion or intertemporal elasticity ofsubstitution. The next example illustrates the conditions for a social welfarefunction of the form U(c, y)=c:y ;+#y and a resource productionfunction characterized by multiplicative environmental disturbances. Thisexample highlights the role played by the contribution of the stock tosocial welfare. In each of these examples the social welfare function satisfiesassumption U.5. The final example is a continuation of Example 1.The social welfare function of that example allows us to contrast the condi-tions for the general case where U.5 is violated with those that are relevantwhen the stock and investment are complementary in welfare in the senseof U.5.

Example 3. Let resource growth be governed by a stochastic version ofthe logistic growth function f (x, \)=x+\x(1&x�k)) with 0<\

��\�

\� �1, and suppose that social welfare is given by U(c, y)=c:�:, where0{:<1. The parameter k represents the natural carrying capacity of theenvironment under all states of nature and it is assumed that x0�k. Notethat the production function satisfies T.1�T.7 on the interval [0, k].Further, f is concave so that the relevant safe standard of conservation, ifit exists, is at x*=0. For this example, the table below summarizes theparametric conditions that are sufficient for each type of conservationunder the propositions of this paper.

This clearly shows the differences between the various sufficient condi-tions for resource conservation. The table also provides a nice illustrationof how social welfare and resource productivity interact to determine theefficiency of conservation under conditions of uncertainty. For instance,when applied to this example, Proposition 3 requires that the discountedaverage rate of return from investing in the resource must exceed one byan amount that depends on the ratio of the worst and best productivityshock and the welfare parameter : that encompasses both risk aversionand the willingness to substitute consumption across time. Ceteris paribus,as the relative difference between the worst and best productivity shocksbecomes larger or as the agent exhibits more risk aversion�less willingnessto substitute consumption across time, the average productivity of theresource must be higher in order for resource conservation to be guaranteedunder Proposition 3. In principle, all three conditions in the table are sufficientfor global conservation since the production function is concave and social

210 OLSON AND ROY

TABLE I

Conservation under Stochastic Logistic Growth and Isoelastic�CRRA Welfare

Sufficient condition for conservation

Proposition 2, safe standard of conservation, under U.5 $\�

1&:E[(1+\)�\1&:]>1Proposition 3, safe standard of conservation $(\

��\� )1&: E(1+\)>1

Propositions 5 and 6, global conservation, with or without U.5 $\�

1&:E(1+\):>1

welfare is independent of the stock and satisfies U.5. It can be seen directlythat conditions implied by Propositions 5 and 6 are equivalent, while thecondition in Proposition 2 is weaker than those of Propositions 3, 5and 6. So for this example the weakest sufficient condition for resourceconservation is provided by Proposition 2, and it involves a joint restric-tion on the discount rate, the welfare function, the lower bound onresource productivity, and the productivity of the resource in all states ofnature.

Example 4 (Safe standard of conservation). Consider the social welfarefunction U(c, y)=c:y ;+#y, where :>0, ;>0, :+;<1, and #�0. Oneinterpretation of this welfare function is that the first term representsconsumptive values that are complementary to the stock, while the secondterm represents any non-consumptive values associated with the resourcestock. This welfare function satisfies U.5, so the optimal investment policyis monotone. Let the resource production function be multiplicativelyseparable in investment and the random shock. In particular, let f (x, \)=\.(x), where \ is uniformly distributed on the interval [\

�, \� ] and

.: R+ � R+ is such that f satisfies T.1�T.7. Define %=.(x*)�x*. Thecondition for a safe standard of conservation in Proposition 2 is equivalentto

E _\\%&1\�%&1+

:&1

\\\�+

;

_1+;: \1&

1\%+&+

#x:+;&1

:( \�%&1):&1 (\

�%) ;& $\%>1.

Terms involving ; embody the stock effect on consumptive welfare.If ;=0, then the first term in square brackets reflects stochastic influence.If ;>0 then stochastic influence is apparent in all four terms in thesquare bracket. In the deterministic case the first two terms drop out andlast two terms in the square brackets capture the effect of stockdependence.


Example 5 (global conservation). Consider the social welfare functionin Example 1, U(c, y)=pc&e:c&;y, with p>0, ;>0 (so that Uy>0), andp>: (so that Uc(0, 0)>0). Assumption U.5 holds if :(:&;)>0. In thatcase, the conditions for global conservation as stated in Proposition 5reduce to:

\1+;

p&:+ $Ef $(0, \)>1.

In the general case of Proposition 6, the conditions for global conservationare that the resource exhibit expected $-productivity at zero, or that$Ef $(0, \)>1. In this example it is easy to see that when the stock andinvestment are complementary the conditions for global conservation areweaker than expected delta-productivity. Note that the ratio ;�( p&:)represents the intrinsic marginal rate of substitution between the stock andconsumption, or the marginal value of stocks near zero relative to themarginal value of consumption. The larger this ratio, the lower is theproductivity requirement that the resource must satisfy for global conserva-tion to be optimal.

7. CONCLUSION

This paper shows that when renewable resources are subject to environ-mental productivity shocks, the efficiency of natural resource conservationdepends on several factors including the discount rate, the social welfarefunction, the marginal productivity of investment in the resource, and thelower bound on resource productivity. The next table summarizes theimplications of the results of this paper for the efficiency of resource conser-vation under various conditions. For ease of comparison the table assumesmarginal social welfare from consumption is positive. The table's first threerows summarize the implications of the results in this paper for resourceconservation in the general case, in the case where there is complementaritybetween the investment and the stock, and in the case where social welfaredepends only on consumption or harvest.16 The last row provides thedeterministic benchmark, again when social welfare is independent of thestock.17

212 OLSON AND ROY

16 This also illustrates how the results in [13] pertaining to sufficient conditions for conser-vation in a stochastic model with stock independent welfare are a special case of the resultsin this paper.

17 Strictly speaking our results on conservation with deterministic production allow moregeneral production functions than those typically assumed in the existing literature.


TA

BL

EII

Suffi

cien

tC

ondi

tion

sfo

rR

esou

rce

Con

serv

atio

n:A

Com

pari

son

Acr

oss

Scen

ario

s

Con

diti

ons

for

asa

fest

anda

rdof

cons

erva

tion

.C

ondi

tion

sfo

rgl

obal

cons

erva

tion

.T

here

exis

tsso

me

x#

(x*,

x�)su

chth

atf �(x

)>x

and.

..X

m(y

)>0,

f �$(0)

�1

and.

..

Gen

eral

mod

el\

Uc(

f� (x)

&x,

f� (x)

)+m

(f� (

x),x

)U

c(f �(x

)&x,

f �(x))

+U

y(f �(x

)&x,

f �(x))+$E

f$(x

,\)>

1.lim

inf

xa

0

$EU

c(f(

x,\)

,f(

x,\)

)+m

(f(x

,\),

0))

Uc(

f �(x)&

x,f(

x))+

Uy(

f(x)

&x,

f(x)

)f$

(x,\

)>1.

Stoc

kan

din

vest

men

tco

mpl

emen

tary

inw

elfa

re

inf

z#

9(

f �(x),

x)

$EU

c(f(

x,\)

&z,

f(x,

\))+

Uy(f

(x,\

)&z,

f(x,

\))

Uc(

f �(x)&

z,f �(x

))f$

(x,\

)>1.

limin

fx

|0

$EU

c(f(

x,\)

,f(

x,\)

)+U

y(f

(x,\

),f(

x,\)

)U

c(f �(x

)&x,

f �(x))

f$(x

,\)>

1.

Wel

fare

inde

pend

ent

ofre

sour

cest

ock

inf

z#

[0

,x

]

$EU

c(f(

x,\)

&z)

Uc(

f �(x)&

z)f$

(x,\

)>1.

limin

fx

a0

$EU

c(f(

x,\)

)U

c(f �(x

)&x)

f$(x

,\)>

1.

Det

erm

inst

icre

sour

cegr

owth

and

stoc

k-in

depe

nden

tw

elfa

re

$f(x

)�x>

1.lim

inf

xa

0

$f$(

x)>

1.

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214 OLSON AND ROY

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