2 .1 . Sustainable Use of Renewable Resources - Chichilnisky

Sustainability : Dynamics and Uncertainty

Graciela Chichilnisy,Geoffrey Heal, and AllessandroVercelliEditors

Kluwer Academic Publishers, 1998

ANDREA BELTRATTI, GRACIELA CHICHILNISKY AND GEOFFREY HEAL

2 .1 . Sustainable Use of Renewable Resources

1. Introduction

We consider here optimal use patterns for renewable resources . Many impor-tant resources are in this category : obvious_,ones are fisheries and forests.Soils, clean water, landscapes, and the capacities of ecosystems to assimi-late and degrade wastes are other less obvious examples .' All of these havethe capacity to renew themselves, but in addition all can be overused to thepoint where they are irreversibly damaged. Picking a time-path for the use ofsuch resources is clearly important: indeed, it seems to lie at the heart of anyconceptof sustainable economic management .We address the problem of optimal use of renewable resources under a

variety of assumptions both about the nature of the economy in which theseresources are embedded and about the objective ofthat economy. In this sec-ond respect, we are particularly interested in investigating the consequencesof a definition of sustainability as a form of intertemporal optimality recent-ly introduced by Chichilnisky [7], and comparing these consequences withthose arising from earlier definitions of intertemporal optimality. In termsof the structure of the economy considered, we review the problem initiallyin the context of a model where a renewable resource is the only good inthe economy, and then subsequently we extend the analysis to include theaccumulation ofcapital and the existence of a productive sector to which theresource is an input.

Although we focus here on the technical economic issues of defining andcharacterizing paths which are optimal, in various senses, in the presence ofrenewable resources, one should not loose sight of the very real motivationunderlying these exercises: many ofthe earth's most important biological andecological resources are renewable, so that in their managementwe confront

' This paper draws heavily on earlier research by one or more of the three authors, namelyBeltratti, Chichilnisky andHeal [1-3], Chichilnisky [7,81, and inparticular many ofthe resultshere were presented in Heal [18] .

49G. Chichilnisky et al. (eds .). Sustainability: Dynamics and Uncertainty, 49-76.© 1998 Kluwer Academic Publishers. Printed in the Netherlands .

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the fundamental choice which underlies this paper, namely their extinction,or their preservation as viable species. In this context the recent discussion ofsustainability or sustainable management of the earth's resources is closelyrelated to the issues ofconcern to us . (For amore comprehensive discussionof issues relating to sustainability and its interpretation in economic terms,see [18] . For a review of the basic theory of optimal intertemporal use ofresources, see [10, 11, 15].)We assume, as in [19] and in earlier work by some or all of us [1-3] that

the renewable resource is valued not only as a source ofconsumption but alsoas a source of utility in its own right: this means that the existing stock ofthe resource is an argument of the utility function . The instantaneous utilityfunction is therefore u (c, s), where c is consumption and s the remainingstock ofthe resource . This is clearly the case for forests, which can be usedto generate aflow of consumption via timber, and whose stock is a source ofpleasure . Similarly, it is true for fisheries, for landscapes, and probably formany more resources. Indeed, in so far as we are dealing with a living entity,there is a moral argument, which we will not evaluate here, that we shouldvalue the stock to attribute importance to its existence in its own right andnotjust instrumentally as a source ofconsumption.

2. The Utilitarian Case without Production

We begin by considering the simplest case, that of a conventional utilitarianobjective withnoproduction : the resource is the only good in the economy. Forthis framework we characterize the utilitarian optimum, and then extend theseresults to other frameworks . Themaximand is the discounted integral ofutil-ities from consumptionand from the existence ofa stock, fo u (c, s) e-btdt,where S > 0 is a discount rate . As the resource is renewable, its dynamics aredescribed by

ht = r (st) - ct .Here r is the growth rate of the resource, assumed to depend only on itscurrent stock. More complex models are of course possible, in which severalsuch systems interact : a well-known example is the predator-prey system . Ingeneral, r is a concave function which attains a maximum at a finite value ofs, and declines thereafter. This formulation has a long and classical history,which is reviewed in [11] . In the field ofpopulation biology, r (st) is oftentaken to be quadratic, in whichcase an unexploitedpopulation (i .e ., ct = 0 dt)grows logistically. Here we assume that r (0) = 0, that there exists a positivestock levels at whichr (s) = 0 Vs >_ s, and that r (s) is strictly concave andtwice continuously differentiable for s E (0, s) . The overall problem cannowbe specified as

maxf00 u (c, s) e-atdt s.t . ht = r (s t) - ct, so given.0

The Hamiltonian in this case

H = u(ct, st) e-it

Maximization with respect tcmarginal utility ofconsumptilevels :

uC (ct, SO = Xt

and the rate of change ofthe

(,Xte-it) = -[u

To simplify matters we shaland s : u (c, s) = ul (c) + u2

differentiable . In this case a

ui(ct) =st = r (st)

~t - JAt = -u2(StIn studying these equations,then examine the dynamics c

2.1 . Stationary Solutions

At a stationary solution, byaddition, the shadow price is

Hence:Sui (ct) = u2 (st) +

PROPOSITION 1 . A stationc(2) satisfies

r (st) = ctu~(st) - J - r' (st)u; (t=t )

The first equation in (3) justhe curve on which consunthis is obviously a prerequisrelationship between the slo:the slope of the renewal fiucurve cuts the renewal functiFigure 1 . This is just the restequal the discount rate if r'[17, 18].

Japer, namely their extinction,ontext the recent discussion ofie earth's resources is closelyore comprehensive discussionrpretation in economic terms,optimal intertemporal use of

~y some or all of us [1-3] thatource ofconsumption but alsoans that the existing stock ofion. The instantaneous utilityzmption and s the remainingor forests, which can be usednd whose stock is a source oflandscapes, and probably for-e dealing with a living entity,-valuate here, that we should>tence in its ownright and not

t of a conventional utilitarian)nlygood in the economy. Fortimum, and then extend theseie discounted integral ofutil-)fa stock, Jo` u (c, s) e-6tdt,renewable, its dynamics are

zmed to depend only on itsse possible, in which severalthe predator-prey system . Innaximumat a finite value ofa long and classical history,ation biology, r (st) is often:d population (i .e., ct = 0 b't)0, that there exists a positiver (s) is strictly concave and'he overall problem cannow

ct, so given.

The Hamiltonian in this case is

H = u(ct, st) e-at + Ate-6t [r (St) - ct] .

Sustainable Use ofRenewable Resources

51

Maximization with respect to consumption gives as usual the equality of themarginal utility ofconsumption to the shadow price for positive consumptionlevels :

uc (ct, SO = Atand the rate of change ofthe shadow price is determined by

(,\te-5t) = - [us (ct, st) e-bt + ate-btr (st)] .

To simplify matters we shall take the utility function to be separable in cand s : u (c, s) = ui (c) + u2 (s), each taken to be strictly concave and twicedifferentiable . In this case a solution to the problem (1) is characterized by

ui (ct) = Xtst = r (st) - ct

(2)at - b,\t = -u2 (st) - Atr' (St)

In studying these equations, we first analyze their stationary solution, andthen examine the dynamics ofthis system away from the stationary solution.


At a stationary solution, by definition s is constant so that r (St) = ct : inaddition, the shadow price is constant so that

Hence:8ui (ct) = u2 (St) + ui (ct) r (st)

PROPOSITION 1 . A stationary solution to the utilitarian optimal usepattern(2) satisfies

r (st) = ct

(3)u'

=b-r'(St) ~ .u, (at)

The first equation in (3) just tells us that a stationary solution must lie onthe curve on which consumption of the resource equals its renewal rate :this is obviously a prerequisite for a stationary stock. The second gives us arelationship between the slope of an indifference curve in the c-s plane andthe slope of the renewal function at a stationary solution : the indifferencecurve cuts the renewal function from above. Such a configuration is shown inFigure 1 . This is just the result that the slope of an indifference curve shouldequal the discount rate if r' (s) = 0 Vs, i.e ., ifthe resource is non-renewable[17,18].

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There is a straightforward intuitive interpretation to the second equationin (3). Consider reducing consumption by an amount Ac and increasing thestock by the same amount . The welfare loss is Ocui : there is a gain fromincreasing the stock of Acu', which continues for ever, so that we have tocompute its present value. But we also have to recognize that the incrementto the stock will grow at the rate r' : hence the gain from the increase in stockis the present value of an increment which compounds at rate r' . Hence thetotal gain is

Ac J 00 u2er'te-atdt = u20c/(r' - 8) .o

When gains and lossesjust balance out, we haveui+u2/(r'-S)=0

which is just the second equation of (3). So (3) is a very natural and intuitivecharacterization of optimality.

2.2 . Dynamic Behavior

Figure 1.

Dynamics ofthe utilitarian solution.

What are the dynamics ofthis system outside of a stationary solution? Theseare also shown in Figure 1 . They are derived by noting the following facts:

1 . beneath the curve r (s) = c, s is rising as consumption is less than thegrowth of the resource .

2. above the curve r (s) = c, s is falling as consumption is greater than thegrowth of the resource .

S

3 . on the curve r (s) = c, s is4. from (2), the rate of chang.

ui (c)6 = ui (c) [b - r'

The first term here is neganegative and large for smac is rising for small s and ,when the rate of change oofpositive slope containin

5 . by linearizing the system

ui (c)

=ui (c) [E - sht = r (st)

around the stationary solutpoint. The determinant of

r(s){b - r(s)} - uiwhich is negative for an)sustainable yield.

Hence the dynamics of pathmality are as shown in figure I

PROPOSITION 2.2 For small ,the derivatives r', r" or ui, al,tend to the stationary solutionorder conditions (2), andfolloiFigure 1 leading to the stationaso, there is a corresponding v.ofthe stable branches leadingstationary solution depends or,ofthe stationary stock as thistends to a point satisfying u2/an indiference curve ofu (c, sgraph ofthe renewalfunction .

This result characterizes optinthe existence of such paths. Tlishes that an optimal path exicharacterized in this paper.Note that ifthe initial resow

consumption, stock and utilitythat because the resource is re

Utilitarian stationarysolution

rental stock s

rian solution .

tation to the second equationtmount Ac and increasing the.s Ocui : there is a gain froms for ever, so that we have torecognize that the increment

;sin from the increase in stockupounds at rate r' . Hence the

re

is a very natural and intuitive

a stationary solution? Thesey noting the following facts:consumption is less than the

)nsumption is greater than the

3. on the curve r (s) = c, s is constant .4. from (2), the rate ofchange of c is given by

ui (c) c = ui (c) [~ -r(s)] - u2 (s) .

Sustainable UseofRenewableResources 53

The first term here is negative for small s and vice versa: the second isnegative and large for small s andnegative andsmall for large s. Hencec is rising for small s and vice versa: its rate of change is zero preciselywhen the rate of change of the shadow price is zero, which is on a lineofpositive slope containing the stationary solution.

5 . by linearizing the system

ui (c) c= ui (c) [b - r' (s)] - u2 (s)ht = r (st) - ct

around the stationary solution, one can show that this solution is a saddlepoint. The determinant of the matrix ofthe linearized system is

r (s)ta - r(s)} - u, {uir"+ u2}

I

which is negative for any stationary stock in excess of the maximumsustainable yield.

Hence the dynamics of paths satisfying the necessary conditions for opti-mality are as shown in figure 1, and we can establish the following result :

PROPOSITION 2.2 Forsmall values ofthe discount rate E or large values ofthe derivatives r', r" or ui, all optimalpaths for the utilitarian problem (1)tend to the stationary solution (3). They do so alongapath satisfying thefirstorder conditions (2), andfollow one ofthe two branches ofthe stablepath inFigure I leading to the stationary solution. Givenany initial value ofthe stockso, there is a corresponding value of co which will place the system on oneofthe stable branches leading to the stationary solution . Theposition ofthestationary solution depends on the discount rate, andmoves to higher valuesofthe stationary stock as this decreases. As b -+ 0, the stationary solutiontends to a point satisfying u2/ui = r', which means in geometric terms thatan indifference curve ofu (c, s) is tangent to the curve c = r(s) given by thegraph ofthe renewalfunction .

This result characterizes optimal paths for the problem (1). It does not provethe existence of such paths. The Appendix gives an argument which estab-lishes that an optimal path exists for all of the problems whose solutions arecharacterized in this paper.Note that ifthe initial resource stock is low, the optimalpolicy requires that

consumption, stock and utility all rise monotonically over time . The point isthat because the resource is renewable, both stocks and flows canbe built up

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overtime provided that consumption is less than the rate ofregeneration, i.e.,the system is inside the curve given by the graph ofthe renewal function r (s) .In practice, unfortunately, many renewable resources are being consumed ata rate greatly in excess oftheir rates ofregeneration : in terms of Figure 1, thecurrentconsumptionrate ct is muchgreaterthan r (st) . So taking advantageofthe regeneration possibilities ofthese resources would in many cases requiresharp limitation of current consumption. Fisheries are a widely-publicizedexample: another is tropical hardwoods and tropical forests in general. Soil isa more subtle example: there are processes whichrenew soil, so that even ifitsuffers a certain amount oferosion or ofdepletion ofits valuable components,it can be replaced. But typically humanuse of soils is depleting them at ratesfar in excess of their replenishment rates .

Proposition 2 gives conditions necessary for a path to be optimal fromproblem (1). Given the concavity of u (c, s) and of r (s), one can invokestandard arguments to show that these conditions are also sufficient (see, forexample, [22]).

3. Renewable Resources and the Green Golden Rule

We can use the renewable framework to ask the question: what configurationof the economy gives the maximum sustainable utility level? 3 There is asimple answer.

First, note that a sustainable utility level must be associated with a sus-tainable configuration of the economy, i.e ., with sustainable values of con-sumption and of the stock. But these are precisely the values that satisfy theequation

ct = r (st)for these are the values which are feasible and at which the stock and theconsumption levels are constant. Hence in Figure 1, we are looking for valueswhich lie on the curve ct = r (s t). Of these values, we need the one whichlies on the highest indifference curve of the utility function u(c, s) : this pointoftangency is shown in the figure. At this point, the slope of an indifferencecurve equals that of the renewal function, so that the marginal rate of sub-stitution between stock and flow equals the marginal rate of transformationalong the curve r(s) . Hence:

PROPOSITION 3.4 The maximum sustainable utility level (the green goldenrule) satisfies

U, (80ui (ct)

Recall from (3) that as the discount rate goes to zero, the stationary solu-tion to the utilitarian case tends to such a point. Note also that any path

which approaches the tangetion function, is optimal acc(or long-run utility. In othermthe limiting behavior of theapproached. This clearly isthe green golden rule, somewould like to know which ofa best . It transpires that in g

3.1 . Ecological Stability

An interesting fact is thatdiscount rates the utilitarianin excess of that giving thethe stock at which the maxonly resource stocks in excare stable under the naturaare ecologically stable . Tothe resource dynamics is ju

s=r(s)-d.

For d < max, r (s), there ato this equation, as shownClearly for s > s2, s < 0.as shown in Figure 2. Onlyyield is stable under the narates, and utilitarian optinan argument of the utilit;maximum sustainable yiel

4. The Rawlsian Solutic

Consider the initial stockthis is to follow the path tconsumption, stock anduleast well off, is the firstpresent model, with initisetting c = r (s 1) for evehighest utility level for thbeing no lower. This rerrassociatedwith the greenen rule is a Rawlsian opti

the rate ofregeneration, i.e .,)fthe renewal function r(s) .irces are being consumed ation: in terms ofFigure 1, the(st) . So taking advantage ofwould in many cases requireies are a widely-publicizedcal forests in general. Soil isi renew soil, so that even ifiti o fits valuable components,ils is depleting them at rates

I a path to be optimal fromid of r (s), one can invokes are also sufficient (see, for

en Rule

luestion : what configurationle utility level? 3 There is a

st be associated with a sus-s sustainable values of con-ly the values that satisfy the

at which the stock and the;1, we are looking for valuesaes, we need the one whichy function u (c, s) : this pointthe slope of an indifferenceat the marginal rate of sub-ginal rate of transformation

tility level (the green golden

o zero, the stationary solu-nt . Note also that any path

which approaches the tangency of an indifference curve with the reproduc-tion function, is optimal according to the criterion ofmaximizing sustainableor long-runutility. In other words, this criterion ofoptimality only determinesthe limiting behavior of the economy: it does not determine how the limit isapproached. This clearly is a weakness : of the many paths which approachthe green golden rule, some will accumulate far more utility than others . Onewould like to know whichofthese is the best, or indeed whether there is sucha best . It transpires that in general there is not. We return to this later.

3 .1 . Ecological Stability

An interesting fact is that the green golden rule, and also for low enoughdiscount rates the utilitarian solution, require stocks ofthe resource which arein excess of that giving the maximum sustainable yield, which is of coursethe stock at which the maximum of r (s) occurs . This is important becauseonly resource stocks in excess of that giving the maximum sustainable yieldare stable under the natural population dynamics of the resource [21] : theyare ecologically stable. To see this, consider a fixed depletion rate d, so thatthe resource dynamics is just

s=r(s)-d.

4. The Rawlsian Solution

Sustainable Use ofRenewableResources 5 5

For d < max, r (s), there are two values of s which give stationary solutionsto this equation, as shown in Figure 2. Call the smaller s 1 and the larger s2 .Clearly for s > 82, s < 0, for sl < s < s2, s > 0, and for s < s I, s < 0,as shown in Figure 2. Only the stock to the right ofthe maximum sustainableyield is stable under the natural population adjustmentprocess: high discountrates, and utilitarian optimal policies when the stock of the resource is notan argument of the utility function, will give stationary stocks below themaximum sustainable yield.

Consider the initial stock level sl in Figure 1 : the utilitarian optimum fromthis is to follow the path that leads to the saddle point. In this case, as noted,consumption, stock and utility are all increasing . So the generation which isleast well off, is the first generation . What is the Rawlsian solution in thepresent model, with initial stock sl? It is easy to verify that this involvessetting c = r (s 1) for ever : this gives a constant utility level, and gives thehighest utility level for the first generation compatible with subsequent levelsbeing no lower. This remains true for any initial stock no greater than thatassociated with the green golden rule : for larger initial stocks, the green gold-en rule is a Rawlsian optimum. Formally,

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Figure 2. The dynamics ofthe renewable resource under a constant depletion rate .

PROPOSITION 4. Foran initial resource stock sl less than or equal to thatassociatedwith the green golden rule, the Rawlsian optimum involves settingc = r (s )for ever. For s 1 greater than the green golden rule stock, the greengolden rule is a Rawlsian optimum.

5. Chichilnisky's Criterion

Next, we askhowthe Chichilnisky criterion (7, 8) alters matters when appliedto an analysis ofthe optimal management ofrenewable resources . Recall thatChichilnisky's criterion ranks paths according to the sum of two terms, onean integral of utilities against a finite countably additive measure and onea purely finitely additive measure defined on the utility stream of the path .The former is just a generalization of the discounted integral of utilities(generalized in the sense that the finite countably additive measure neednot be an exponential discount factor). The latter term can be interpreted as asustainable utility level: Chichilnisky shows that anyranking ofintertemp.oralpaths which satisfies certain basic axioms must be representable in this way.Theproblemnow is to pick paths of consumptionandresource accumulationover time to :

maxaf

u(ct, st)0

s.t . At = I

where f (t) is a finite countab'The change in optimal poli

optimality is quite dramatic . Nf (t) given by an exponentialsolution to the overall optimiztakes a different, non-expontrate which tends asymptoticEin an unexpected way with rethe future : there is empiricalchoices act as if they have ntime . Formally:

PROPOSITION 5.6 Theprobpattern ofuse ofa renewablea constant discount rate.

Proof. Consider first the p00

max

0v. (ct, st) e-

The dynamics of the solutio:ure 3 .

It differs from the problein limiting utility in the maFigure 3. Pick an initial vasaddle-point, and follow theconditions given above:

ui (c) b = ui (c) (6At = r (St) -

Denote by vo the 2-vector c{'Et, st} (vo) . Follow this pat]the green golden rule, i.e .,9t, = s*, and then at t = t'to the green golden rule, i.(because ct < r(st ) alongFormally, this path is (ct , sst , = s*, and ct = r (s*), stAnysuchpath will satisf3

up to time t' and will lea(therefore attain amaximumHowever, the utility integra

Utilitarian stationaryiolution

armgown ruk

ader a constant depletion rate.

sl less than or equal to thatian optimum involves settinggolden rule stock, the green

alters matters when applied;wable resources. Recall thato the sum of two terms, oney additive measure and one.e utility stream of the path.;ounted integral of utilitiesably additive measure needterm canbe interpreted as aany ranking ofintertemporal)e representable in this way.i and resource accumulation

Sustainable Use ofRenewable Resources 5700

maxa~

u(ct, st) f (t) dt + (I - a) slim u (ct, st)

(4)s.t . ht = r (st) - ct, so given.

where f (t) is a finite countably additive measure.The change in optimal policy resulting from the change in the criterion of

optimality is quite dramatic . With the Chichilnisky criterion andthe measuref (t) given by an exponential discount factor, i.e ., f (t) = e-bt , there is nosolution to the overall optimizationproblems There is a solution only iff (t)takes a different, non-exponential form, implying a non-constant discountrate which tends asymptotically to zero . Chichilnisky's criterion thus linksin an unexpected way with recent discussions of individual attitudes towardsthe future : there is empirical evidence that individuals making intertemporalchoices act as if they have non-constant discount rates which decline overtime . Formally :

PROPOSITION 5 .6 Theproblem (4) has no solution, i.e., there is no optimalpattern ofuse ofa renewable resource using the Chichilnisky criterion witha constant discount rate.Proof. Consider first the problem

maxf0" u (ct, st) e-btdt s.t . st = r (s t) - ct, so given.0

The dynamics of the solution is shown in Figure 1, reproduced here as Fig-ure 3 .

It differs from the problem under consideration by the lack of the termin limiting utility in the maximand. Suppose that the initial stock is so inFigure 3 . Pick an initial value of c, say co, below the path leading to thesaddle-point, and follow the path from co satisfying the utilitarian necessaryconditions given above:

u" (c) c = ui (c) [b - r' (s)] - u' (s),

St = r (st) - ct-Denote by vo the 2-vector ofinitial conditions : v0 = (co, so). Call this path{ct, st} (vo) . Follow this path until it leads to the resource stock correspondingthe green golden rule, i.e., until the t' such that on the path {ct, st}(vo),st, = s*, and then at t = t' increase consumption to the level correspondingto the green golden rule, i.e ., set ct = r (s*) for all t > t' . This is feasiblebecause ct < r(st) along such a path. Such a path is shown in Figure 3.Formally, this path is (ct, st) = {at, st} (vo) dt <_ t' where t' is defined byst, =s*, andct=r(s*),st=s*`dt>t'.Anysuch path will satisfy the necessary conditions forutilitarian optimality

up to time t' and will lead to the green golden rule in finite time. It willtherefore attain amaximum ofthe term limt,u (ct , st) over feasible paths.However, the utility integral which constitutes the first part ofthe maximand

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Figure 3.

Asequence of consumption paths with initial stock so and initial consumptionlevel below that leading to the utilitarian stationary solution and converging to it. Once thestock reaches s* consumption is set equal to r (s*) . The limit is a path which approaches theutilitarian stationary solution and not the green golden rule .

can be improved by picking a slightly higher initial value co for consumption,again following the first orderconditions for optimality and reaching the greengolden rule slightly later than t' . This does not detract from the second term inthe maximand. By this process it will be possible to increase the integral termin the maximand without reducing the limiting term and thus to approximatethe independent maximization ofboth terms in the maximand : the discountedutilitarian term, by staying long enough close to the stable manifold leadingto the utilitarian stationary solution, and the limit (purely finitely additive)term by moving to the green golden rule very far into the future .

Although it is possible to approximate the maximization ofboth terms inthe maximand independently by postponing further and further the jump tothe green golden rule, there is no feasible path that actually achieves thismaximum. The supremum ofthe values ofthe maximand over feasible pathsis approximated arbitrarily closely by paths which reach the green goldenrule at later and later dates, but the limit of these paths never reaches thegreen golden rule and so does not achieve the supremum . More formally,consider the limit of paths (ct, st) _ {ct, st} (vo) b't _< t' where t' is definedby st, = s*, and ct = r (s*), st = s* bt > t' as co approaches the stablemanifold ofthe utilitarian optimal solution. On this limiting path st < s* Vt .

m u~i

Hence there is no solution to

Intuitively, the non-existenceto postpone further into the fcost in terms oflimiting utilitofutilities. This is possible beis no equivalent phenomenon

5.1 . Declining DiscountRate

With the Chichilnisky criteric

f000 'u (ct, st) e -qtr,

there is no solution to the piresource . In fact as noted the dfunction oftime . The criterionis still consistent with Chichilsolving the renewable resourcthat we have noted before, n,-the discount rate goes to zero,rule . We shall, therefore, con

00af

u (ct, SO A(t)0

where A (t) is the discount farate q (t) at time t is the prop

and we assume that the disco

t~~q(t) = 0.

So the overall problem is noN00

maxaf

u (ct, st)0

s.t. ht =where the discount factor Arate goes to zero in the limitsolution: in fact, it is the soluthe first term in the above ma:the utility function to be sepaFormally,

ivn sutionsry

imI*

stock so and initial consumptionin and converging to it. Once thenit is a path which approaches the

d value co for consumption,Lality and reaching the greenract from the second term into increase the integral termrm and thus to approximatemaximand : the discountedthe stable manifold leadingit (purely finitely additive)into the future .dmization of both terms inier and further the jump tothat actually achieves thisximand over feasible pathsch reach the green goldense paths never reaches the;upremum . More formally,b't < t' where t' is definedus co approaches the stableis limiting path st < s* Vt .

Hence there is no solution to (4).

o

Intuitively, the non-existenceproblem arises here because it is always possibleto postpone further into the future moving to the green golden rule, with nocost in terms of limiting utility values but with a gain in terms of the integralofutilities. This is possible because ofthe renewability ofthe resource . Thereis no equivalent phenomenon for an exhaustible resource [18] .

5 .1 . Declining Discount Rates

With the Chichilnisky criterion formulated as


af00 u(ct, st) e-atdt + (1 - a) tlim u (ct, st),0

there is no solution to the problem of optimal management of a renewableresource . In fact as notedthe discount factor does not have to be an exponentialfunctionoftime. The criterion can be stated slightly differently, in a waywhichis still consistent with Chichilnisky's axioms andwhich is also consistent withsolving the renewable resource problem. This reformulation builds on a pointthat we have noted before, namely that for the discounted utilitarian case, asthe discount rate goes to zero, the stationary solution goes to the green goldenrule . We shall, therefore, consider amodified objective function

af00u(ct, st) A(t) dt + (1 - a) tlim u (ct, st),

owhere A (t) is the discount factor at time t, fo A (t) dt is finite, the discountrate q (t) at time t is the proportional rate of change of the discount factor :

q (t) _ -

(t)0 (t)

and we assume that the discount rate goes to zero with t in the limit:

taioq (t) = 0.

(5)

So the overall problem is now

max af00 u(ct, st)A(t) dt + (1 - a) tliio u(et, st)0

s.t. st = r (st) - ct, so given,where the discount factor A (t) satisfies the condition (5) that the discountrate goes to zero in the limit. We will show that for this problem, there is asolution :? infact, it is the solution to the utilitarian problem ofmaximizingjustthe fast term in the above maximand, fo u(ct, st)A(t) dt . As before we takethe utility function to be separable in its arguments : u (c, s) = ul (c)+u2 (s).Formally,

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PROPOSITION 6.8 Consider theproblem

maxa f00 {u1 (c) + u2 (s)}A (t) dt + (1 - a) tl yim {ul (c) + u2 (s)},

0 < a < 1, s.t. ht = r (st) - ct, so given,

where q(t) = -(0(t)/0(t)) andlimt, q (t) = 0.Asolution tothisproblemis identical to the solution of "max f'{u1(c) +u2 (s)}0 (t) dt subject to thesame constraint". In words, the conditions characterizing a solution to theutilitarian problem with the variable discount rate which goes to zero alsocharacterize asolution to the overallproblem.

Proof. Consider first the problem maxa f°° {ul (c) + u2 (s)JA (t)dt s.t .ht = r (st) - ct, so given. We shall show that any solution to this problemapproaches and attains the green golden rule asymptotically, which is theconfiguration of the economy which gives the maximum of the term (1 -a) limt-4.0 u (ct, st). Hence this solution solves the overall problem. TheHamiltonian for the integral problem is now

H = {u1(c) + u2 (s)}A (t) + AtA (t) [r (st) - Ct]

andmaximization with respect to consumption gives as before

u 1 (ct) = At .The rate of change of the shadow price At is determined by

d (ato(t)) _ - [u2 (st)0(t) + AtA (t) r (st)] .The rate of change of the shadow price is, therefore,

atA (t) + At,& (t) = -u2 (st) A(t) - AtA (t) r' (st) .

(6)

As ,& (t) depends on time, this'equation is not autonomous, i.e ., time appearsexplicitly as a variable. For such an equation, we cannotuse the phase portraitsand associated linearization techniques used before, because the rates ofchange of c and s depend not only on the point in the c-s plane but also onthe date . Rearranging and noting that A (t)/A (t) = q (t), we have

at+ Atq (t) = -u2 (st) - ui (ct) r' (st)Butin the limitq = 0, so in the limitthis equation is autonomous : this equationandthe stock growth equation form what has recently been called in dynam-ical systems theory an asymptotically autonomous system [4]. According toproposition 1 .2 of [4], the asymptotic phase portrait of this non-autonomoussystem

~t + Atq (t)' = -u2 (st) - ui (ct) r' (st)ht = r (st) - ct

is the same as that of the a

~t = -u2 (St) -ht = r(st.

which differs only in that tzero .9 The pair ofEquatio:stability properties of orilassociated limiting autonoby the standard technique:and ct = r (s t), so that

u2 = -r'

andu1

which isjust the definitionmentsused abovewe canofthe system (8), as sho)N

00"maximize f {

0subject to At

is for any given initial stethat (co, so) is on the styapproaches the Green Gcto the maximum possibl,therefore leads to a soluti

Figure 4 shows the behalcan see what drives this 1constant discount rate arof the path that maximizepath that maximizes theto zero in the limit, thatresolved only in this case

PROPOSITION 7. 1o Cot00

maxa ( {u1(c)J0

0<~

where q(t) = -(0(t)/L= 0. In this case, the soleacterize the solution to "constraint".

- a) liM JU1 (c) + u2 (S)},t +00ct, so given,

0.14 solution to thisproblem12 (s)}A(t) dt subject to the-acterizing a solution to the2te which goes to zero also

.2c1 (c) + u2 (s)}0 (t) dt s.t .ny solution to this problemsymptotically, which is thenaximum of the term (1 -; the overall problem. The

(st) - ct]ives as before

rmined by

r' (st)] .

)re,

(t) r' (st) .

(6)

:onomous, i.e., time appearsannot use the phase portraitsMore, because the rates ofin the c-s plane but also oni = q (t), we have

is autonomous : this equationently been called in dynam-as system [4]. According torait ofthis non-autonomous

is the same as that ofthe autonomous system


at = -ua (st) - ui (ct) r' (st)

(8)St = r (St) - Ct

which differs only in that the non-autonomous term q(t) has been set equal tozero .9 The pair of Equations (8) is an autonomous system and the asymptoticstability properties of original system (7) will be the same as those of theassociated limiting autonomous system (8). This latter system canbe analyzedby the standard techniquesused before. At a stationary solution of (8), .fit = 0and ct = r (st), so that

U2

and

ct =r(st)u l

which is just the definition ofthe green golden rule . Furthermore, by the argu-ments used abovewe can establish that the green golden rule is a saddlepointof the system (8), as shown in Figure 3. So the optimal path for the problem

00"maximize J

{ul (c) + u2 (s)}0 (t) dt0

subject to st = r (st) - ct, so given"

is for any given initial stock so to select an initial consumption level co suchthat (co, so) is on the stable path of the saddle point configuration whichapproaches the Green Golden Rule asymptotically. But this path also leadsto the maximum possible value of the term limt,,,. {u l (c) + u2 (s)}, andtherefore leads to a solution to the overall maximization problem.

o

Figure 4 shows the behavior of an optimal path in this case . Intuitively, onecan see what drives this result. The non-existence of an optimal path with aconstant discount rate arose from a conflict between the long-run behaviorof the path that maximizes the integral of discounted utilities, and that of thepath that maximizes the long-run utility level. When the discount rate goesto zero in the limit, that conflict is resolved. In fact, one can show that it isresolved only in this case, as stated by the following proposition.

PROPOSITION 7. 10 Consider theproblem

maxa J 00{u l (c) + u2 (s)}0 (t) dt + (1 - a) tliM {ul (c) + u2 (s)},0

_+00

0 < a < 1, s.t. §t = r (st) - ct, so given,

whereq (t) = -(,&(t)/A (t) ) . Thisproblem has a solution only iflimt,"', q (t)= 0. In this case, the solution is characterizedby the conditions which char-acterize the solution to "maxfo{ui (c) + u2 (s)}0 (t) dt subject to the sameconstraint".

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A. Beltratti et al.

Figure 4.

Asymptotic dynamics of the utilitarian solution for the case in which the discountrate falls to zero .

Proof. The "if'part ofthis was proven in the previous proposition, Propo-sition 6. The "only if' part can be proven by an extension of the argumentsin Proposition 5, which established the non-existence of solutions in the caseofa constant discount rate . To apply the arguments there, assume contrary tothe proposition that lim inft_.~,,. q (t) = q > 0, and then apply the argumentsofProposition 5 .

p

Existence of a solution to this problem is established in the Appendix .

5 .2 . ExamplesTo complete this discussion, we review some examples of discount factorswhich satisfy the condition that the limiting discount rate goes to zero . Themost obvious is

O (t) = e-a(t) t ,

with

li,1

b (t) = 0.tAnother examples 1 is

A(t) = t-a,

a > 1 .Taking the starting date to be t = 1,12 we have

f,00t`dt =

1a-1

and

5.3 . Empirical Evidence onD

Proposition 7 has substantialmality with a criterion sensitiwith non-renewable resource :behavior of the discount rate :utilities symmetrically in the esense, the treatment ofpresensistent with the presence of trpositive weight on the very los

There is a growing body oflike this in evaluating the fuh:more comprehensive discussiowhich people apply to futurefuturity of the project. Over rethey use discount rates which ;region of 15% or more. For pidiscount rates are closer to stextends the implied discount ryears anddownto ofthe order,framework for intertemporal offuture generates an implicationpersonal behavior that hitherto

This empirically-identified l;sciences which find that humarlinear, and are inversely proporlis an exampleofthe Weber-Fecthat human response to achangpre-existing stimulus . In symbc

_dr _ _Kds s

where r is a response, s a stimto apply to human responses toWe noted that the empirical re ;something similar is happeningof an event: a given change isleads to a smaller response in tothe event already is in the futureapplied to responses to distance

or

r =K

Ifimal paths

a for the case in which the discount

previous proposition, Propo-n extension of the arguments,tence of solutions in the case:nts there, assume contrary toind then apply the arguments

0

ished in the Appendix .

I-xamples of discount factors;count rate goes to zero . The

and

0 =t -+ 0

as

t -+ oo .


5.3 . Empirical Evidence on Declining Discount Rates

Proposition 7 has substantial implications . It says that when we seek opti-mality with a criterion sensitive to the present and the long-run future, thenwith non-renewable resources existence of a solution is tied to the limitingbehavior of the discount rate: in the limit, we have to treat present and futureutilities symmetrically in the evaluation ofthe integral ofutilities . In a certainsense, the treatment ofpresent and future in the integral has to be made con-sistent with the presence of the term limt,,,,, {ul (c) + u2 (s)} which placespositive weight on the very long run.

There is a growing body ofempirical evidence that people actually behavelike this in evaluating the future (see, for example, [20] ; see also [18] for amore comprehensive discussion) . Theevidence suggeststhat the discount ratewhich people apply to future projects depends upon, and declines with, thefuturity ofthe project. Over relatively short periods up to perhaps five years,they use discount rates whichare higher even than commercial rates - in theregion of 15% or more. For projects extending about ten years, the implieddiscount rates are closer to standard rates - perhaps 10%. As the horizonextends the implied discount rates drops, to in the region of 5% for 30 to 50years anddown to of the order of2% for 100 years. It is ofgreat interest that aframework for intertemporal optimization that is sensitive to both present andfuture generates an implication for discounting that mayrationalize a form ofpersonal behavior that hitherto has been found irrational .

This empirically-identified behavior is consistent with results from naturalsciences which find that human responses to a change in a stimulus are non-linear, and are inversely proportional to the existing level ofthe stimulus . Thisis an example ofthe Weber-Fechner law, which is formalized in the statementthat humanresponse to a change in a stimulus is inversely proportional to thepre-existing stimulus . In symbols,

_dr _ _Kds s or

r =Klog s,

where r is a response, s a stimulus and K a constant . This has been foundto apply to human responses to the intensity ofboth light and sound signals .We noted that the empirical results on discounting cited above suggest thatsomething similar is happening in human responses to changes in the futurityof an event: a given change in futurity (e.g., postponement by one year)leads to a smaller response in terms of the decrease in weighting, the furtherthe event already is in the future . In this case, the Weber-Fechner lawcanbeapplied to responses to distance in time, as well as to soundand light intensity,

64

A. Beltratti et al,

with the result that the discount rate is inversely proportional to distance intothe future . Recalling that the discount factor is A (t) and the discount rateq (t) = -A (t) /A (t), we can formalize this as

q (t) =

1 dO

KA dt

t

5.4. Time Consistency

or

A(t)=eK log t = tK

forK a positive constant . Such adiscountfactor can meet all ofthe conditionswe required above: the discount rate q goes to zero in the limit, the discountfactor A(t) goes to zero and the integral fl'A (t) dt = fl' eK log tdt =fl, tK dt converges forKpositive, as it always is . In fact, this interpretationgives rise to the second example of a non-constant discount rate consideredin the previous section. A discount factor A (t) = eK 109' has an interestinginterpretation : the replacement of t by log t implies that we are measuringtime differently, i.e . by equal proportional increments rather than by equalabsolute increments .

An issue which is raised by the previous propositions is that of time consis-tency. Consider a solution to an intertemporal optimization problem which iscomputed today and is to be carried out over some future period oftime start-ing today. Suppose that the agent formulating it - an individual or a society-mayat a future date recompute an optimal plan, using the same objective andthe same constraints as initially but with initial conditions and starting datecorresponding to those obtaining when the recomputation is done. Then wesay that the initial solution is time consistent ifthis leads the agent to continuewith the implementation ofthe initial solution. Anotherway of saying this isthat a plan is time consistent if the passage of time alone gives no reason tochange it. The important point is that the solution to the problem of optimalmanagement of the renewable resource with a time-varying discount rate,stated in Proposition 7, is not time-consistent . A formal definition of timeconsistency is :13

DEFINITION 8. Let (ct , st )t=o,00 be the solution to the problem

maxaf00{ul (c) + u2 (s)}A (t) dt + (1 - a) thin {ul (c) + u2 (s)},o

t-+00

0 < a < 1, s .t. At = r (st) - ct, so given.Let (ct , 9't)t=T,Oo be the solutionto the problem ofoptimizing from Ton, giventhat the path (ct , st)t=o,o0 has been followed up to date T., i.e ., (Ft, st)t=T,msolves

maxa ,f'{ul (c) + u2 (s)}0 (t - T) dt + (1 - a) tlim {ul (c) + u2 (s)},1'

-roo0 < a < 1, s.t. At = r (st) - ct, sT given.

Then the original problem so(Et, St)t=T,. = (Ct, st)t=T,ocperiod [T, oo] is also a soluticstock sT, for any T.

It is shown in [14] that the scin general time consistent onllowing result is an illustration

PROPOSITION 9.14 The solu,renewable resource with a distime consistent, i.e ., the soluti

00

maxaf

{ul (c) + u20

0 < a < l, s.t. At = r (st)is not time consistent.Proof. Consider the first o

which are given in (7) and refu1 (ct) ct + ui (ct) q i

AtLet (ct, st)t=o,. be a solutionconsumption on this at a date 7be a solution to the problemconditions at T given by (crstarting date T, the value ofvalue of the discount factor. HA (T - T), while it is A (T),the two paths will have differtin excess of T . This establishcT > 0, then the initial plan wi

These are interesting and swoptimal path which balancesChichilnisky's axioms, we hatent. Of course, the empirica:behavior must also be inconsing what individuals apparenalways regarded time consisteral choice. More recently, thisand psychologists have notedor his life can reasonably beperspectives on life and differwith inconsistent choices clea

proportional to distance intoA(t) and the discount rate

)gt =tK

can meet all ofthe conditionsero in the limit, the discountA (t) dt = floo eK to g tdt =

is . In fact, this interpretationrant discount rate considered= eK iog t has an interestingplies that we are measuringements rather than by equal

;itions is that of time consis-Aimization problem which isie future period oftime start-- an individual or a society -using the same objective andconditions and starting dateimputation is done . Then weis leads the agent to continuemother way of saying this isme alone gives no reason tom to the problem of optimaltime-varying discount rate,A formal definition of time

nto the problem

- a) lim 1ul (C) + u2 (S)J,. ct, so given.

'optimizing from Ton, givento date T., i.e., (Et, st)t=T,oo

- a) tlun{u1 (c) + u2 (s)},t, sT given.


Then the original problem solved at t = 0 is time consistent if and only if(Ft, St)t=T,oo = (ci st)t=T,,, i.e., if the original solution restricted to theperiod [T, oc] is also a solution to the problem with initial time T and initialstock sT, for any T.

It is shown in [14] that the solutions to dynamic optimization problems arein general time consistent only if the discount factor is exponential. The fol-lowing result is an illustration of this fact.

PROPOSITION 9. 14 The solution to theproblemofoptimal management ofarenewable resource with a discount ratefalling asymptotically to zero is nottime consistent, i.e., the solution to

maxa ,f00 {ul (c) + u2 (s)}0 (t) dt + (1 - a) thin

y {ul (c) + uZ (s)},

0 < a < 1, s.t. st = r (st) - ct, so given, q (t) _ -

, limt-+00 q (t) = 0.is not time consistent.Proof. Consider the first order condition for a solution to this problem,

which are given in (7) and repeated here with the substitution A = ui :ui (ct) 6t + ui (ct) q (t) = -ui (st) - ui (ct) r' (st)

.

(9)st = r (st) - ctLet (ct , st)t=o 0o be a solution computed at date t = 0. The rate of change ofconsumptiononthis at a dateT > 0will be given by (9). Nowlet (Et, st)t=T, o0be a solution to the problem with starting date T, 0 < T < T, and initialconditions at T given by (cr, sT) . When the problem is solved again withstarting date T, the value of A(t) at calendar time T is A(0), the initialvalue of the discount factor. Hence on this path the value of A (t) at date T isA (T - T), while it is A (T) on the initial path. Hence q (T) will differ, andthe two paths will have different rates of change of consumption for all datesin excess of T. This establishes that ifthe optimum is recomputed at any dateT > 0, then the initial plan will no longer be followed.

These are interesting and surprising results: to ensure the existence of anoptimal path which balances present and future "correctly" according toChichilnisky's axioms, we have to accept paths which are not time consis-tent. Of course, the empirical evidence cited above implies that individualbehavior must also be inconsistent, so society in this case is only replicat-ing what individuals apparently do. Traditionally, welfare economists havealways regarded time consistency as a very desirable property of intertempo-ral choice . More recently, this presumptionhas been questioned : philosophersand psychologists have noted that the same person at different stages of heror his life can reasonably be thought of as different people with differentperspectives on life and different experiences . 15 The implications ofworkingwith inconsistent choices clearly need further research .

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A. Beltratti et al.

6. Capital and Renewable Resources

k=F(k,v)-c

Nowwe consider the most challenging, and perhaps most realistic and reward-ing, of all cases: an economy in which a resource which is renewable and sohas its owndynamics can be used together with produced capital goods as aninput to the production ofan output. Theoutput in turn can as usual in growthmodels be reinvested in capital formation or consumed. The stock of theresource is also a source ofutility to the population. So capital accumulationoccurs according to

and the resource stock evolves according to

where k is the current capital stock, Q the rate of use of the resource inproduction, and F(k, v) the production function . As before r (s) is a growthfunction for the renewable resource, indicating the rate ofgrowth ofthis whenthe stock is s.

As before, we shall consider the optimum according to the utilitarian cri-terion, then characterize the green golden rule, and finally draw on the resultsof these two cases to characterize optimality according to Chichilnisky'scriterion.

7. The Utilitarian Optimum

The utilitarian optimum in this framework is the solution to

maxf~ u (Ct) St) e-dtdt subject to

.

(10)k=F(k,a)-cands=r(s)-a

We proceed in the by-now standard manner, constructing the Hamiltonian

H = u(c, s) e-6t + Ae-6t {F (k, Q) - c} + me-6t {r (s) - o}

and deriving the following conditions which are necessary for a solution to(10) :

where rs is the derivative of r with respect to the stock s.

7.1 . Stationary SolutionsA little algebra shows thunderlying differential ecsolution :

S=Fk(1

c=F(k

us (c+ s) = Fa (IUc (c, s)

This system of four equalthe variables k, s, v and c.

It is important to undermodel, and in particular thstock s across alternative ,

First, consider this relalthe capital stock k : in this

and so we havec = F(k, Q (s)),

C9C

-F,198 k fixed

As FQ is always positive,and then switches to neg;between c and s for fixe,across stationary states fgrowth function r (s) and

In general, however, kon v via Equation. (15) . Tas an implicit function, wacross stationary states :

do Fk_ _

_ds -

rs (-bFkkwhich, maintaining the asagain inherits the shape of1090S)k fixed I and that tlzero . The various curves rein Figure 5: for Fk, > 0 tlrises and falls more sharp.from below while increasistationary solution with a

uC = ~, (11)

AF, = ~~ (12)

- Sa = -AFk, (13)

- 8I-L = -'us - hrs 1 (14)

s most realistic andreward-which is renewable and sooduced capital goods as anturn can as usual in growthasumed. The stock of then. So capital accumulation

of use of the resource inAs before r (s) is a growthrate ofgrowth ofthis when

irding to the utilitarian cri-i finally draw on the results;cording to Chichilnisky's

olution to

ructing the Hamiltonian+

Fee-8t

fr (s) _ Q}

necessary for a solution to

,tock s.

(10)

(11)

(12)

(13)(14)


A little algebra shows that the system (11) to (14), together with the twounderlying differential equations in (10), admits the following stationarysolution :

and so we have


This system of four equations suffices to determine the stationary values ofthe variables k, s, o, and c.

It is important to understand fully the structure of stationary states in thismodel, and in particular the trade-offbetween consumptioncand the resourcestock s across alternative stationary states .

First, consider this relationship across stationary states for a given value ofthe capital stock k : in this case we can write

c = F(k, v(s)),

ac Fr..

(19)(7S k fixed -

Qrs.

As FQ is always positive, this has the sign of rs, which is initially positiveand then switches to negative : hence we have a single-peaked relationshipbetween c and s for fixed k across stationary states . The c-s relationshipacross stationary states for a fixed value of k replicates the shape of thegrowth function r (s) and so has amaximum for the same value of s.

In general, however, k is not fixed across stationary states, but dependson Q via Equation (15) . Taking account of this dependence and treating (15)as an implicit function, we obtain the total derivative of c with respect to sacross stationary states :

TS =rs (-b

Fk

+FQ) ,

(20)kk

which, maintaining the assumption that Fk, > 0, also has the sign ofr,s andagain inherits the shape ofr (s). Note that for a given value ofs: I (dclds) ( >1 *149s)k &ed+ and that the two are equal only if the cross derivative Fk, iszero . Thevarious curves relating c and s across stationary solutions are shownin Figure 5 : forFkQ > 0 the curve corresponding to k fully adjusted to s bothrises and falls more sharply than the others, and crosses each ofthese twice,from below while increasing andfrom above while decreasing, as shown. Astationary solution with a capital stock k must lie on the intersection of the

b = F'k (k, ~) (15)Q = r(s), (16)

c = F (k, v) , (17)us (c' s) = FQ (k, o) (5 - r.,) (18)u, (c, s)

68

A. Beltratti et al.

Fully-adjusted relationshipbetween c and s acrossstationary states .

Figure S.

Autilitarian stationary solution occurs where the fully-adjusted c-s relationshipcrosses the same relationship for the fixed value ofk corresponding to the stationary solution .

curve corresponding to a capital stock fixed at k with the curve representingthe fully-adjusted relationship . At this point, c, s and k are all fully adjustedto each other. (In the case of Fk, = 0 the curves relating c and s for k fixedand fully adjusted are identical, so that in the case of a separable productionfunction the dynamics are simpler, although qualitatively similar.)

The stationary first order condition (18) relates most closely to the curveconnecting c and s for a fixed value ofk (the only relevant curve for Fka = 0),and would indicate a tangency between this curve and an indifference curveif the discount rate 5 were equal to zero . For positive J, the case we areconsidering now, the stationary solution lies at the point where the c-s curvefor the fixed value of k associated with the stationary solution crosses the c-scurve along which k varies with s. At this point, an indifference curve crossesthe fixed-k c-s curve from above: this is shown in Figure 5 . Note that as wevary the discount rate b, the capital stock associated with a stationary solutionwill alter via Equation (15), so that in particular lowering the discount ratewill lead to a stationary solution on a fixed-k c-s curve corresponding to alarger value of k and therefore outside the curve corresponding to the initiallower discount rate .

7.2 . Dynamics ofthe UtilitThe four differential equat

k=F(k,a)-c,s=r(s)-a(pt ;

~_b11 =-us .The matrix of the linearize

6 -Fo ), Fok

Fov

-AFkk + Fk,a0

To establish clear genematrix, we have to make simarginal productivity ofththen the eigenvalues of thtu2CJZ + 4u cAFkk . Ther

stationary solution . In thissaddle point.

PROPOSITION 10 . 16 A .solution to be locally asacproductivity ofthe resourc

There are other cases inpoint, involving additive ssolution to the utilitarian f

8. The Green Golden R

Across stationarystates, thstock satisfies the equation

c =F (k, r (s)),so that at the green goldelevel with respect to the in

maxu(F (k, r (s);,Maximization with respec

rian stationary solution

ie fully-adjusted c-s relationshipionding to the stationary solution.

with the curve representingand k are all fully adjustedrelating c and s for k fixede of a separable productionitatively similar.)s most closely to the curve-elevant curve for Fk, = 0),e and an indifference curve)ositive b, the case we aree point where the c-s curveary solution crosses the c-sa indifference curve crossesn Figure 5. Note that as we-d with a stationary solutionlowering the discount rates curve corresponding to acorresponding to the initial

7.2 . Dynamics ofthe Utilitarian SolutionThe four differential equations governing a utilitarian solution are

k = F(k, v) - c (st, At)s = r (s) - a(pt, At, kt)

a-SA=-AFk~_JM= -us - prs

The matrix of the linearized system is

c =F(k, r (s)),

maxu(F (k, r (s)), s) .s,k

Sustainable Use ofRenewableResources 69

To establish clear general results on the signs of the eigenvalues of thismatrix, we have to make simplifying assumptions. If FQQ is large, so that themarginal productivity ofthe resource drops rapidly as more ofit is employed,then the eigenvalues of the above matrix are : rs , b - rs, 1/(2uo,) (2u,6 fu.'J2 +4uccAFkk. There are two negative roots in this case, as rs < 0 at a

stationary solution. In this case the utilitarian stationary solution is locally asaddle point.

PROPOSITION 10.16 A sufficient condition for the utilitarian stationarysolution to be locally a saddle point is that Faa is large, so that the marginalproductivity ofthe resource diminishes rapidly in production.

There are other cases in which the stationary solution is locally a saddlepoint, involving additive separability of the utility function.17 Existence of asolution to the utilitarian problem is established in the Appendix.

8. The Green Golden Rule with Production and Renewable Resources

Across stationary states, the relationship between consumptionand the resourcestock satisfies the equation

so that at the green golden rule we seek to maximize the sustainable utilitylevel with respect to the inputs of capital k and the resource stock s :

Maximization with respect to the resource stock gives

(21)

b - F°AFoassUCC

FaFo 1_ _AFaa ucc

FaAF,,

Faa rs FaAFaa

1_AFaa

- AFkk + Fkva 0 6 + teaFaa

0 USCUcs - u _ss l-~rs -U (~ -r,ucc ucc

70

A. Beltratti et al.

which is precisely the condition (18) characterizing the stationary solution tothe utilitarian conditions for the case in which the discount rate 8 is equal tozero . So, as before, the utilitarian solution with a zero discount rate meets thefirst order conditions for maximization of sustainable utility with respect tothe resource stock. Of course, in general the utilitarian problem mayhave nosolution when the discount rate is zero . Note that the condition (21) is quiteintuitive and in keeping with earlier results. It requires that an indifferencecurve be tangent to the curve relating c to s across stationary states for kfixed at the level k defined below: in other words, it again requires equality ofmarginal rates oftransformation and substitution between stocks and flows.

The capital stock k in the maximand here is independent of s. How is thecapital stock chosen? In a utilitarian solution the discount rate plays a role inthis through the equality ofthe marginal product ofcapital with the discountrate (15) : at the green golden rule there is no equivalent relationship .We close the system in the present case by supposing that the production

technologyultimately displays satiation with respectto the capital input alone:for each level ofthe resource input o there is a level ofcapital stock at whichthe marginal product ofcapital is zero . Precisely,

k(c) =mink :ak

(k, o) = 0.

(22)

We assume that T (o,) exists for all o >_ 0, is finite, continuous and non-decreasing in c. Essentially assumption (22) says that there is a limit to theextent to which capital can be substituted for resources : as we apply moreand more capital to a fixed input of resources output reaches a maximumabove which it cannot be increased for that level of resource input. In thecase in which the resource is an energy source, this assumption was shownbyBerry et al. [5] to be implied by the second law ofthermodynamics : this issueis also discussed by Dasgupta and Heal [11] . In general, this seems a verymild and reasonable assumption. Given this assumption, the maximization ofstationary utility with respect to the capital stock at a given resource input,

mkax u(F (k, r (s)), s)

requires that we_pick the capital stock at which satiation occurs at this resourceinput, i.e ., k = k (r (s)) . Note that

ak (r (s))

ak(c)_ as

aoso that k is increasing and then decreasing in s across stationary states : thederivative has the sign ofrs . In this case, the green goldenrule is the solutionto the following problem:

maxu (F (k (r (s)) , r (s)) , s) ,swhere at each value of the resource stock s the input and the resource andthe capital stock are adjusted so that the resource stock is stationary and

the capital stock maximizrelationship between consistates when at each resourk(r (s)) has the following

ds

dc= F~ ac rs + .

and by the definition ofk tfthe curve relating c and s vthe slope when the capital softhe curves for fixed valuThe total derivative of

resource is nowdu 8k_ds

_- ucFk

_ao rs

By assumption (22) and tlthis to zero for a maximunsion (21) . The green goldeindifference curve and thestationary states for fixed c

PROPOSITION 11 .18 In a)able resources, under thefunction with respect to calcondition us/uc = -Fr,curve and the outer envekcapital stock of k (_o (s*)),resource stock andk denoteofcapitalfirst becomeszen

What ifproduction does notIn this case there is no maya given resource flow and ssatiation of preferences witnot well defined. l9

9. Optimality for the Chi

Nowwe seek to solve the p00

max af

u(ct,0

s.t. k = F(kt, at)

:ing the stationary solution tohe discount rate b is equal to3 zero discount rate meets thefinable utility with respect tolitarian problem may have nopat the condition (21) is quiterequires that an indifferenceaross stationary states for ki, it again requires equality ofn between stocks and flows.independent of s. How is thediscount rate plays a role in

Aofcapital with the discountuivalent relationship .upposing that the productionpect to the capital input alone:evel of capital stock at whichY,

(22)

finite, continuous and non-ys that there is a limit to theresources : as we apply moreoutput reaches a maximumvel of resource input. In theits assumption was shownbyifthermodynamics : this issuen general, this seems a verynnption, the maximization ofk at a given resource input,

.tiation occurs at this resource

across stationary states : theen golden rule is the solution

input and the resource andarce stock is stationary and

the capital stock maximizes output for that stationary resource input. Therelationship between consumption and the resource stock across stationarystates when at each resource stock the capital stock is adjusted to the levelk (r (s)) has the following slope:

dc akds

Fk8o rs + Far.,


andby the definition ofk the first term on the right is zero, so that the slope ofthe curve relating c and s when the capital stock is given by Jc is the same asthe slope when the capital stock is fixed. This curve is thus the outer envelopeof the curves for fixed values of the capital stock.

The total derivative of the utility level with respect to the stock of theresource is now

du o9kds - u,Fk- rs + ucFars + us .

By assumption (22) and the definition of k, Fk = 0 here : hence equatingthis to zero for a maximum sustainable utility level gives the earlier expres-sion (21) . The green golden rule is characterized by a tangency between anindifference curve and the outer envelope of all curves relating c to s acrossstationary states for fixed capital stocks .

PROPOSITION 11 . 18 In an economy with capital accumulation and renew-able resources, under the assumption (22) of satiation of the productionfunction with respect to capital, the green golden rule satisfies thefirst ordercondition us /u, = -Far., which defines a tangency between an indifferencecurve and the outer envelope of c-s curves forfixed values of k. It has acapital stock ofk (o, (s*)), where s* is the green golden rule value of theresource stock andk denotes the capital stock at which the marginalproductofcapitalfirst becomes zerofor a resource input ofv (s*) .

What ifproduction does not display satiation with respect to the capital stock?In this case there is no maximum to the output which can be obtained froma given resource flow and so from a given resource stock. Unless we assumesatiation ofpreferences with respect to consumption, the green golden rule isnot well defined. 19

9. Optimality for the Chichilnisky Criterion

Now we seek to solve the problem

maxaf00 u(ct, st) e-atdt + (I - a) limoo u(ct, st)

(23)s.t. k = F(k t , at) - ct & it = r (s) - o't, st > 0

`dt.

72

A. Beltratti et al.

In the case of satiation of the production process with respect to capi-tal, as captured by assumption (22), the situation resembles that with theChichilnisky criterion with renewable resources above: there is no solutionunless the discount rate declines to zero . Formally:

PROPOSITION 12 .2° Assume that condition (22) is satisfied. Then problem(23) has no solution .Proof. The structure ofthe proofis the same as that usedabove. The integral

term is maximizedby the utilitarian solution, which requires an asymptoticapproach to the utilitarian stationary state. The limit term is maximizedon anypath which asymptotes to the green golden rule . Given any fraction ,0 E [0,1]we can find a path which attains the fraction,Q of the payoff to the utilitarianoptimumandthen approaches the green golden rule . This is true for any valueofQ < 1, but not true for B = 1 . Henceany path can be dominated by anothercorresponding to a higher value of Q.

o

Wenowconsiderinstead optimization with respect to Chichilnisky's criterionwith a discount rate which declines to zero over time :

PROPOSITION 13 .21 Consider theproblem00

maxaf

ul (c, s) A (t) dt + (1 - a) tlim ul (c, s), 0 < a < 1,0 400

s.t. k = F(kt, ut) - ct & ht = r (st) - ct, so given.

where q (t) = -(0(t)/A(t)) and limt,c) q (t) = 0. Any solution to thisproblem is also a solution to the problem ofmaximizing fo ul (c, s) A (t) dtsubject to the same constraint. In words, solving the utilitarian problem withthe variable discount rate which goes to zero solves the overallproblem.

Proof. The proof is a straightforward adaptation of the proof of Proposi-tion 6, and is omitted.

O

As before, the existence ofa solution is established in the Appendix .What does the Chichilnisky-optimal path look like in this case? It is similar

in general terms to the set of paths shown in Figure 4, except that the graphof the growth function r (s) is replaced by the outer envelope of the curvesrelating c and s for fixed values of k. The optimal path moves towards thegreen golden rule, which is a point of tangency between an indifference curveand the outer envelope ofthe curves relating c and s for fixed values ofk. Thispoint is the limit of utilitarian stationary solutions as the associated discountrate goes to zero .

10 . Conclusions

A review of optimal patterrinteresting conclusions. Theit gives the highest sustainalitive discount rate will accuassociated with the green gothe discount rate usedin the ia zero discount rate, there isChichilnisky's criterion in s4

concepts of optimality : a solintegral term of Chichilnisk;which case maximization ofutilities - leads one to the ¬with the inclusion of producnot qualitatively different. Irple display declining discowbehavior is quite consistenthuman choice and summarb

11. Appendix

In this appendix we establis:tions to the various intertemysitions 2, 6, 7, 10 and 13 ofdeveloped initially by Chichenwald [9] . This is a very diof feasible solutions to the ccfunction is a continuous funtinuous function on a compais to find a topology in whiclsonable assumptions about tas introduced in ChichilniskWe consider the utilitaria

this is the most complex ofthe paper are special cases cimplies the existence of soilproblem is :

max

00 u(ct , st) e0

k=F(k,v)-car

We make the following assu

ocess with respect to capi-on resembles that with theabove: there is no solutionly:

?) is satisfied. Thenproblem

that used above. The integralhich requires an asymptoticnit term is maximizedon any3iven any fraction ,3 E [0,1]f the payoff to the utilitarianAe . This is true for any valueSan be dominated by another

ct to Chichilnisky's criteriontime :

lim ul (c, S),0 < a < 1,

(st) - ct, so given.

= 0. Any solution to thiscimizing fool, ul (c, s) A (t) dtthe utilitarian problem with

Ives the overallproblem.lion of the proof of Proposi-

0

hed in the Appendix .like in this case? It is similargure 4, except that the graphouter envelope of the curvesmal path moves towards theBetween an indifference curveLd s for fixed values ofk. Thisins as the associated discount

10 . Conclusions

A review of optimal patterns of use of renewable resources has suggestedinteresting conclusions . Thegreen golden rule is an attractive configuration:it gives the highest sustainable utility level. Utilitarian solutions with a pos-itive discount rate will accumulate a smaller stock of the resource than thatassociated with the green golden rule, although the difference goes to zero asthe discount rate used in the utilitarian formulation gets smaller. Of course, fora zero discount rate, there is typically no utilitarian optimum. Investigation ofChichilnisky's criterion in some measure bridges the gapbetween these twoconcepts of optimality: a solution exists if and only ifthe discount rate in theintegral term ofChichilnisky's maximand declines asymptotically to zero, inwhich case maximization of the integral term alone - the sum of discountedutilities - leads one to the green golden rule . This result remains true evenwith the inclusion of production : matters are more complex in that case, butnot qualitatively different . Interestingly, there is empirical evidence that peo-ple display declining discount rates in their behavior towards the future . Suchbehavior is quite consistent with behavior patterns found in other aspects ofhuman choice and summarized as the Weber-Fechner law.

11 . Appendix

00maxfo

u(ct, st) e-atdt subject to

.

(24)k=F(k,v)-cands=r(s)-v

We make the following assumptions:


In this appendix we establish conditions sufficient for the existence of solu-tions to the various intertemporal optimization problems considered in Propo-sitions 2, 6, 7, 10 and 13 of the text . We use an approach and a set of resultsdeveloped initially by Chichilnisky [6] and applied by Chichilnisky and Gru-enwald [9] . This is a very direct and intuitive approach : we show that the setof feasible solutions to the constraints is a compact set, and that the objectivefunction is a continuous function, and invoke the standard result that a con-tinuous function on a compact set attains amaximum. The delicate step hereis to find a topology in whichwe have compactness and continuity under rea-sonable assumptions about the problem: for this we use weighted Lr spaces,as introduced in Chichilnisky [6].We consider the utilitarian optimality problem analyzed in Section 7, as

this is the most complex of the problem in the paper. Earlier problems inthe paper are special cases of this, so that the existence of a solution to thisimplies the existence of solutions to the earlier problems. The optimizationproblem is :

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A. Beltratti et al.

1 . u (c, s)

is

concave, increasing and differentiable .

It satisfies theCaratheodory condition, namely it is continuous with respect to c ands for almost all t and measurable with respect to t for all values of c ands .

2 . r (0) = 0, 3s > 0 s.t . r (s) = 0 Vs >_ s, max s r (s) _< b1 < oo, and r (s)is concave for s E [0, s] .

3 . For any a3b2 (Q) < oo s.t . F(k, er) < b2 (Q).4. 3b3 < oo s.t . Js) < b3 .5. 3b4< oo s.t . IkI

< b4 .The first two conditions are conventional. The third implies that bounded

resource availability implies bounded output : it is a form of the assumptionmade by Dasgupta and Heal [10] that the resource is essential to production.It is a restatement of assumption (22) in the text. The final two assumptionsimply that it is not possible for either the resource stock or the capital stockto change infinitely rapidly. These seem to be very reasonable assumptions .However, we shall in the end not require them : we shall prove the existenceof an optimal path under these assumptions, and then note that a path whichis optimal without these assumptions is still feasible and optimal with them.

PROPOSITION 14 . Underassumptions (1) to (S) above, the utilitarian opti-mization problem (24) has a solution .

Proof. Under the above assumptions, the set of feasible time paths of theresource stock s and consumption c are uniformly bounded above . (Note thats is bounded by (2), and c by (3) and (5).) They are non-negative and sobounded below. Hence the paths ofs and c are integrable against some finitemeasure and so are elements ofa weighted L1 space . Denote by P the set offeasible paths st and ct, 0 _< t <_ oo : as a subset ofL1, P is closed and normbounded, so that by the Banach-Alaoglu theorem it is weak-* compact. ByLebesgue's bounded convergence theorem, it is also compact in the norm ofLl .

The objective U =fo u (ct, SO e-btdt maps P to the real line fit . To com-plete the proofwe need to show that U is continuous in the norm of L1 . Thisfollows immediately from the characterization ofLP continuity given in [6] :

LEMMA 15 (Chichilnisky) . Let W = fitu (ct , t) dv (t) for a finite measurev (t), with u (ct , t) satisfying the Caratheodory condition. Then Wdefines anorm-continuousfunction from LP to Rfor some coordinate system ofLp ifand only if Iu(ct, t)~ < a (t) + b Ict JP, where a (t) > 0, fit a (t) dv (t) < ooand b > 0.

In the case of our objective the role ofu (ct , t) is played by u (ct, st) e-Jt . Anextension of Chichilnisky's lemma to functions u defined on R2 is straight-forward. As u is defined only on R2 , concavity implies that Chichilnisky's

inequality is satisfied for poptimum.

We have now proven the exthe optimization problemsthe simpler problems can(4) and (5) above, which bthe resource and capital sto,ofthe paper. However, notesolutions to the problemsdo in fact have bounded r,of capital . Hence for sufficthe rates ofchange ofstockproblems . It follows that wfor the unbounded optimiz,

Notes

1 .

See [12] for a detailed listing2.

This proposition, which was3.

Elsewhere we have called thi4.

This result, and the associateand [3] .

5.

We are grateful to Kenn Jud6.

This result was introduced in7.

We are grateful to Harl Ryde .it.

8 .

This result was first proven ft9.

This equality is not alwaysautonomous system to the au

10. This result was first proven it11 . Due to Harl Ryder.12. This discount factor is infinit,13 . Further discussions oftime a14. This result was first proven it15 . For a further discussion, see [16 . This result was first proven it17 .

See [18] for details .18 . This result was first proven it19 .

See[18] for details .20 . This result was first proven ir21 . This result was first proven in

;rentiable . It satisfies theuuous with respect to c andct to t for all values of c and

is r (s) < b1 < oo, and r (s)

e third implies that boundedis a form of the assumptionce is essential to production .t. The final two assumptions-ce stock or the capital stockery reasonable assumptions .we shall prove the existencei then note that a path which;ible andoptimal with them.

i) above, the utilitarian opti-

of feasible time paths of the[y bounded above. (Note thatiey are non-negative and soategrable against some finitepace . Denote byP the set ofofL1, P is closed and norm;m it is weak-* compact. Byalso compact in the norm of

P to the real line R. To com-uous in the norm of L1 . ThisLP continuity given in [6] :

t) dv (t)for afinite measurecondition. Then Wdefines ato coordinate system ofLp if(t) > 0, fgt a (t) dv (t) < oc

played by u(ct , st) e-bt. Anu defined on R2 is straight-,implies that Chichilnisky's


inequality is satisfied for p = 1 . This completes the proof of existence of anoptimum.

El

Wehave nowproven the existence ofan optimal path for the most complex ofthe optimization problems discussed in the paper: existence ofan optimum forthe simpler problems can be deduced from this . Ourproof used assumptions(4) and (5) above, which bound respectively s and k, the rates of change ofthe resource and capital stocks . These assumptions were not made in the bodyofthe paper. However, note from the characterization results in the paper thatsolutions to the problems without bounds on the rates of change of stocksdo in fact have bounded rates of change of the stocks of the resource andof capital. Hence for sufficiently large bounds, the imposition of bounds onthe rates of change of stocks cannot change the solutions to the optimizationproblems . It follows that we have also established the existence of solutionsfor the unbounded optimization problems .

Notes

1 .

See [12] for a detailed listing of many more examples .2.

This proposition, which was first proved in [18], is a strengthening ofresults in [3] .3.

Elsewhere we have called this the green golden rule [3].4.

This result, and the associated concept ofthe green golden rule, were introduced in [1]and [3].

5.

Weare grateful to Kenn Judd for this observation .6.

This result was introduced in [ 18] .7.

Weare grateful to Harl Ryder for suggesting this result and outlining the intuition behindit.

8.

This result was first proven in [18] .9.

This equality is not always true : it requires locally uniform convergence of the non-autonomous system to the autonomous system. For details, see [4].

10 . This result was first proven in [18].11 . Due to Harl Ryder.12 . This discount factor is infinite when t = 0: hence the need to start from t = 1 .13 . Further discussions of time consistency can be found in [14] .14 . This result was first proven in [18].15 . For a further discussion, see [13] and references therein .16 . This result was first proven in [18] .17 . See [18] for details.18 . This result was first proven in [18] .19 .

See [18] for details.20 . This result was first proven in [18] .21 . This result was first proven in [18] .

76

A. Beltratti et al.

References

1 .

Beltratti, A., G. Chichilnisky and G.M. Heal. "Sustainable growth and the green goldenrule", inApproaches to SustainableEconomicDevelopment, IanGoldinand Alan Winters(eds .), Paris, Cambridge University Press for the OECD, 1993, pp . 147-172.

2.

Beltratti, A., G. Chichilnisky and G. M. Heal . "The Environment and the Long Run: AComparison of Different Criteria", Ricerche Economiche 48, 1994, 319-340.

3.

Beltratti, A., G. Chichilnisky and G. M. Heal. "The Green Golden Rule", EconomicsLetters 49, 1995, 175-179.

4.

Bena-im, M. and M. W. Hirsch . "Asymptotic Pseudotrajectories, Chain Recurrent Flowsand Stochastic Approximations", WorkingPaper, DepartmentofMathematics, UniversityofCalifornia at Berkeley, 1994.

5.

Berry, S., G. M. Heal andP. Salomon. "On the Relation between Economic and Ther-modynamic Concepts of Efficiency in Resource Use", Resources and Energy 1, 1978,125-137. (also reprinted in [16]).

6.

Chichilnisky, G. "Nonlinear Functional Analysis and Optimal Economic Growth", Jour-nal ofOptimization Theory andApplications 61(2), 1977, 504-520.

7.

Chichilnisky, G. "WhatIs Sustainable Development?", WorkingPaper, Stanford Institutefor Theoretical Economics, 1993 .

8.

Chichilnisky, G. "Sustainable Development : An Axiomatic Approach", Social Choiceand Welfare 13(2), 1996, 219-248.

9.

Chichilnisky, G. and P. F. Gruenwald. "Existence of an Optimal Growth Path withEndogenous Technical Change", Economics Letters 48, 1995, 433-439.

10 .

Dasgupta, P. S. and G. M. Heal . "The Optimal Depletion of Exhaustible Resources",Review ofEconomic Studies, Special Issue on Exhaustible Resources, 1974, 3-28 .

11 .

Dasgupta, P. S. andG. M. Heal. Economic TheoryandExhaustibleResources, CambridgeUniversity Press, 1979 .

12 .

Daily, G. Nature's Services, Societal Dependence on Natural Ecosystems, Island Press,Washington DC, 1997.

13 .

Harvey, C. "The Reasonableness of Non-Constant Discounting", Journal ofPublic Eco-nomics 53, 1994, 31-51.

14 .

Heal,G. M. The Theory ofEconomic Planning, Advanced Texts in Economics, Amster-dam, North-Holland, 1973 .

15 .

Heal, G.M. "TheOptimal UseofExhaustible Resources" Handbook ofNatural Resourceand Energy Economics, Vol. III, Alan Kneese and James Sweeney (eds.), Amsterdam,

. NewYork and Oxford, North-Holland, 1993, pp. 855-880.16.

Heal, G. M. The Economics ofExhaustible Resources, International Library of CriticalWritings in Economics, Edward Elgar, 1993 .

17.

Heal, G. M. "Interpreting Sustainability", in Social Sciences and the Environment, L.Quesnel (ed.), University ofOttawa Press, 1995, pp. 119-143.

18 .

Heal,G. M. Lectures on Sustainability, LiefJohansen Lectures, University ofOslo, 1995.Circulated as a working paper ofthe Department ofEconomics, University ofOslo, andforthcoming as Valuing the Future: Economic Theory and Sustainability, ColumbiaUniversity Press.

19 .

Krautkramer, J. A. "Optimal Growth, Resource Amenities and the Preservation ofNaturalEnvironments", Review ofEconomic Studies 52, 1985, 153-170 (also reprinted in [16]).

20 .

Lowenstein, G. andR. Thaler. "Intertemporal Choice", JournalofEconomic Perspectives3,1989,181-193 .

21 .

Roughgarden, J. and F. Smith. "Why Fisheries Collapse and What to Do about It", inProceedings ofthe National Academy ofSciences, forthcoming .

22 .

Seierstad, A. and K. Sydsxter. Optimal Control Theory with Economic Applications,Advanced Texts in Economics, Amsterdam, North-Holland, 1987 .

RALPH ABRAHAM, GRAC

2.2 . North-South Tryof the Environm

1 . Introduction

This paper develops a dyrenvironment plays an impcNorth-South model for theof the world economy. TheWe introduce dynamics inendogenous accumulation cduce here a variable whiclenvironmental asset whichcould represent, for examplis extracted to be used as aproperty rights on water vgoods for export.

The paper explains mathof a two-region world . Thproduction. Capital is onetime as a function ofprofit,the environment the dynamare the property rights, the

The models which resultNeumann in 1932 and exterWe establish, in a sequenccoupled logistic maps studioidea is to alter [I] to allow cthe approach to equilibriumour model, which are not fevolution of capital stock th

G. Chichilnisky et al (eds), Sustainat© 1998 Kluwer Academic Publishers

2 .1 . Sustainable Use of Renewable Resources - Chichilnisky

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