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Biological and economic foundations of renewable resource exploitation

ANALYSIS

Filippo ColonnaMatr. N° 525709Natural Resources EconomicsA.A. 2018/2019

MODELLO DINAMICO DELLE POPOLAZIONI E MODELLO DI RACCOLTA DELLE RISORSE

1. Relazione tra teorie ecologiche ed economiche

2. Importanza del tempo (Evoluzionismo vs Mercati)

3. Progresso tecnologico come determinante per la conservazione delle risorse

4. Effetti disastrosi della sinergia tra alto tasso di sconto e tecnologia di raccolta

5. Effetti di una maggiore efficienza sulla raccolta delle risorse

6. Effetti del degrado ambientale su consumatori e risorse

2 05/12/2018 Natural Resources Economics

ANALOGIE DI ALLOCAZIONE DELLE RISORSE IN BIOLOGIA ED ECONOMIA


MODELLO BIOLOGICO

Le funzioni ecologiche di sviluppo si basano sulla dinamica temporale della biomassa tra i diversi livelli trofici:

𝑑𝑑𝑀𝑀𝑖𝑖(𝑡𝑡)𝑑𝑑𝑡𝑡

= 𝜃𝜃𝑀𝑀𝑖𝑖𝐷𝐷𝑖𝑖ℎ 𝑠𝑠𝑖𝑖 − 𝑣𝑣𝑖𝑖 𝐷𝐷𝑖𝑖 𝑀𝑀𝑖𝑖 − 𝑀𝑀𝑖𝑖+1𝐷𝐷𝑖𝑖+1ℎ(𝑠𝑠𝑖𝑖+1)


MODELLO ECONOMICO

Un modello di crescita endogeno di imprese che utilizzano risorse è basato sull’equazione precedente, utilizzata per esaminare l’ottimizzazione economica di imprese (soggetti) che sono consumatori di risorse rinnovabili.


MODELLO ECONOMICO

𝑑𝑑𝑑𝑑𝑑𝑑𝑡𝑡

≡ ��𝑑 = 𝑔𝑔 𝑑𝑑 − 𝑦𝑦𝐷𝐷ℎ 𝑠𝑠

𝑑𝑑𝑦𝑦𝑑𝑑𝑡𝑡

≡ ��𝑦 = θ𝑦𝑦𝐷𝐷ℎ 𝑠𝑠 − 𝑣𝑣𝐷𝐷 + 𝐶𝐶 𝑦𝑦


OTTIMIZZAZIONE SOCIALEMassimizzazione della funzione obiettivo di lungo periodo

𝑀𝑀𝑀𝑀𝑑𝑑𝐷𝐷,𝑐𝑐

�0

∞𝑒𝑒−𝛿𝛿𝛿𝛿𝑦𝑦𝑦𝑦 𝐶𝐶 𝑑𝑑𝑡𝑡

Dalle proprietà del principio del massimo…è equivalente massimizzare l’Hamiltoniana a valore corrente.

𝐻𝐻 = 𝑦𝑦 𝐶𝐶 𝑦𝑦 + 𝜇𝜇1 𝑔𝑔 − 𝐹𝐹 + 𝜇𝜇2[𝜃𝜃𝐹𝐹 − 𝑣𝑣𝐷𝐷 + 𝐶𝐶 𝑦𝑦]


CONDIZIONI DEL PRIMO ORDINE

• 𝐻𝐻𝐷𝐷 = 0; 𝐻𝐻𝐶𝐶 = 0 Principio del massimo• 𝜇𝜇1 = 𝛿𝛿𝜇𝜇1 − 𝐻𝐻𝑥𝑥 Equazione di costato 1• 𝜇𝜇2 = 𝛿𝛿𝜇𝜇2 − 𝐻𝐻𝑦𝑦 Equazione di costato 2• lim𝛿𝛿→∞

𝑒𝑒−𝛿𝛿𝛿𝛿𝜇𝜇𝑖𝑖 𝑡𝑡 = 0 Condizione di trasversalità


SOLUZIONI OTTIME

• ��𝑑 = 𝑔𝑔 𝑑𝑑 − 𝛼𝛼𝑑𝑑 ℎ[𝐺𝐺]𝐺𝐺

• ��𝑦 = θℎ 𝐺𝐺 − 𝑣𝑣 𝛼𝛼𝑥𝑥𝐺𝐺− 𝐶𝐶𝑦𝑦

• 𝜇𝜇1 = 𝜇𝜇1 𝛿𝛿 − 𝑔𝑔′ 𝑑𝑑 − 𝛼𝛼ℎ′(𝐺𝐺)(𝜃𝜃𝑦𝑦′ − 𝜇𝜇1) e 𝐶𝐶 = 𝐶𝐶∗(𝛿𝛿)


DIAGRAMMA DI FASE DELLO SPAZIO (x,µ)


DIAGRAMMA DI FASE DELLO SPAZIO (x,µ)incremento di δ


DIAGRAMMA DI FASE DELLO SPAZIO (x,µ)incremento di α


DIAGRAMMA DI FASE DELLO SPAZIO (x,µ)incremento di θ


DIAGRAMMA DI FASE DELLO SPAZIO (x,µ)incremento di v


DIAGRAMMA DI FASE DELLO SPAZIO (x,µ)Erosione dello stock della risorsa di base 𝑀𝑀0


STATO STAZIONARIO DELLA POPOLAZIONE y


COMPARAZIONE TRA SISTEMI ECONOMICI E BIOLOGICI

1. Θ, v, D sono variabili nei sistemi economici e relativamente fissi in biologia (la domanda economica di risorsa potrebbe essere maggiore dello stock disponibile per questo motivo)

2. α – progresso tecnologico – gioca un ruolo fondamentale per la preservazione della risorsa

3. δ – fattore di sconto – ha una analogia in biologia

4. C – fattore di consumo – in biologia ha il ruolo di investimento energetico concentrato nella progenie (popolazioni r e K strategiche), in economia è legato soltanto al piacere morale


PROBLEMI DI RACCOLTA SOSTENIBILE

1. Dinamiche spazio/temporali delle risorse che influenzano le popolazioni che ne fanno uso

2. Il degrado ambientale influenza quelle che sono le caratteristiche di base della funzione di crescita e sviluppo delle popolazioni

3. SOSTENIBILITA’ = non è soltanto legata alle generazioni future, ma anche ai problemi legati alle decisioni politiche da attuare per non danneggiare quelle che sono le condizioni ottime attuali

4. Il livello ottimo sostenibile di raccolta per la popolazione non è sempre (quasi mai) il livello ottimo sostenibile di sfruttamento per lo stock


IRREALISMO DELLA MSY

La Massima Resa Sostenibile viene considerata irrealistica dal fatto che l’andamento delle popolazioni non è un andamento stabile e prevedibile, ma fluttua in risposta a quelli che sono i cambiamento biotici e abiotici dell’ambiente circostante.

La probabilità di un sovrasfruttamento di una risorsa è molto più alta quando le perturbazioni stocastiche influenzano la risorsa.


«Man stalks across the landscape, and desertfollows his footsteps»Erodoto – Quinto Secolo a.C.


Ecological Economics 26 (1998) 227–242

ANALYSIS

Biological and economic foundations of renewable resourceexploitation

U. Regev a,*, A.P. Gutierrez b, S.J. Schreiber b, D. Zilberman c

a Department of Economics and the Monaster Center for Economic Research, Ben Gurion Uni6ersity, 84105 Beer-She6a, Israelb Di6ision of Ecosystem Science, College of Natural Resources, Uni6ersity of California, Berkeley CA 94720, USA

c Department of Agricultural Economics, College of Natural Resources, Uni6ersity of California, Berkeley CA 94720, USA

Received 10 December 1996; received in revised form 2 June 1997; accepted 12 June 1997

Abstract

A physiologically based population dynamics model of a renewable resource is used as the basis to develop a modelof human harvesting. The model incorporates developing technology and the effects of market forces on thesustainability of common property resources. The bases of the model are analogies between the economics of resourceharvesting and allocation by firms and adapted organisms in nature. Specifically, the paper makes the followingpoints: (1) it shows how economic and ecological theories may be unified; (2) it punctuates the importance of timeframe in the two systems (evolutionary versus market); (3) it shows, contrary to prevailing economic wisdom, howtechnological progress may be detrimental to resource preservation; (4) it shows how the anticipated effects of highdiscount rates on resource use can be catastrophic when synergized by progress in harvesting technology; (5) itsuggests that increases in efficiency of utilization of the harvest encourages higher levels of resource exploitation; and(6) it shows the effects of environmental degradation on consumer and resource dynamics. The model leads to globalimplications on the relationship between economic growth and the ability of modern societies to maintain theenvironment at a sustainable level. © 1998 Elsevier Science B.V. All rights reserved.

Keywords: Population dynamics; Fitness; Adaptedness; Energy flow; Technological progress; Resource utilization

* Corresponding author.

0921-8009/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved.

PII S0921-8009(97)00103-1

U. Rege6 et al. / Ecological Economics 26 (1998) 227–242228

‘‘Man stalks across the landscape, and desertfollows his footsteps’’

Herodotus (5th century B.C.)

1. Introduction

The fate of humankind will be determined byhow sustainable ecosystems and the renewableresource species in them are managed. Species innature evolved via Darwinian processes in re-sponse to interactions among members of thesame species, with other species, and their physi-cal environment and its carrying capacity. Humanharvesting of some species is an additional everincreasing burden of mortality that has only beenrecently imposed, often based upon unrealisticassumption of maximum sustainable yield (Getzand Haight, 1989; Hilborn et al., 1995). Only thehuman predator has escaped regulation of itsnumbers, and through ingenuity forestalledMalthus’ ‘doomsday prediction’ concerning exces-sive human population. This has been explainedby capital accumulation and technological pro-gress (exogenous or endogenous) including thediscovery of new goods and methods of produc-tion since the industrial revolution (Schumpeter,1934; Solow, 1956, 1970). This enabled increasesin production in agriculture in its various forms,the harvests of naturally occurring resources, andincrease in resource processing and distribution.Recent economic literature on growth posits thatgiven non-ending technological progress, foodproduction will continue to outpace demand forseveral centuries, ignoring natural resource limita-tions with optimistic views about the role of tech-nology in surmounting resource scarcity andenvironmental degradation (Grossman and Help-man, 1994; Barro and Sala-I-Martin, 1995;Romer, 1990).

In contrast, many ecologists and someeconomists recognize limits to human populationgrowth set by the relative rates of renewableresource exploitation and regeneration, and by theincreasing degradation our finite world (Hardin,1993; Solow, 1993; Daly, 1994). Despite this, the

economic literature on renewable resource ex-ploitation largely ignores or oversimplifies the bi-ological basis of the ‘reproductive surplus’ that isthe basis of sustainable yield approaches (Hilbornet al., 1995). While many technological advanceshave produced positive private and societal eco-nomic benefits, some have caused disastrous envi-ronmental problems (e.g. excessive agronomicinputs, van den Bosch, 1978) and others have ledto over-exploitation of renewable resource popu-lations (e.g. fisheries worldwide, Hilborn et al.,1995). The invisible hand of the market has espe-cially failed to prevent the over-exploitation anddestruction of many common property resourcesthat have free access characteristics (Gordon,1954; Hardin, 1968).

This paper examines the effects of technologicalprogress and discount rate on the sustainability offree access renewable resources. The terms ‘hu-man’ and ‘firm’ are used interchangeably as thefirm is single owner. A physiologically-basedpredator-prey (i.e. firm-resource) population dy-namics model is extended to include humans withtheir associated technology as the top predator inthe food chain (e.g. alga�krill�whale�whalers). The model incorporates the realism ofhierarchical energy flow between feeding (i.e.trophic) levels (Gutierrez and Baumgartner, 1984)and captures the essence of the competition be-tween harvesting units (Gutierrez et al., 1994;Schreiber and Gutierrez, 1997). Our analysis hasparallels in the bio-economic study of adaptedspecies in nature using the same model (Gutierrezet al., 1997). That paper should be considereddual to this one.

2. The biological basis of renewable resourceharvesting

2.1. A common model for humans and otherspecies

At the dawn of time, primitive humans werescavenger-gatherers buffeted by the vagaries ofthe environment. In their primal state, humansdiffered little from all other animal species innature having finite demand for resources, and

U. Rege6 et al. / Ecological Economics 26 (1998) 227–242 229

Fig. 1. Analogous allocation of resources in biology and human economies.

their numbers were regulated by bottom-up (de-creasing resources) and top-down (natural ene-mies including diseases) factors which limitedtheir capacities to overexploit their environment.They were an integral part of the food chain andtheir capacities and demands for resources wereconstrained by genetics molded by natural selec-tion in an ecosystem context (Gutierrez et al.,1997). However, evolving increased mental capac-ities enabled humans to escape some of the uncer-tainty of their environment and the regulation oftheir numbers, and to develop technologies thatenabled them to exploit their environment at in-creasingly higher rates. As human societies devel-oped, their demand rates for resources alsoincreased and larger investments were made insocial organization that further enhanced theircapacity to harvest resources. The model is flex-ible enough to capture these different stages ofhuman evolution.

Despite their progress, humans, like all organ-isms, must acquire resources and allocate them(Gutierrez and Curry, 1989), and this is the basisfor extending the biological model to humaneconomies. Analogies between biological andmodern economic processes were proposed byWinter (1971) and extended by Gutierrez andRegev (1983). Nature (or the ecosystem) is

analogous to the economic system, energy is thecurrency in biology, individual organisms areequated to single-owner firms (individuals), fitnessis profit (i.e. what can be invested in the nextcycle), organism genetics are akin to firm decisionrules, adaptivity of individual species to long-termfirm survival strategies, and markets areanalogous to Darwinian processes, etc. Profitmaximization is the assumed goal of individuals(fitness in other organisms), and selection for oragainst strategies occurs at the level of the individ-ual in both systems.

Fig. 1 shows the energy flow in a specific foodchain where humans are the top consumer(Gutierrez and Curry, 1989), but harvesting ofmore than one resource level (say, krill and whale)is also possible. In nature, the source of energyused by most life forms is the sun which is cap-tured via photosynthesis by primary producers(plants) in the chemical bonds of simple sugars.Some of this energy is used to acquire otheressential nutrients required to form complexmolecules for growth. Plants, be they simple sin-gle celled alga or large trees, are ultimately eatenby herbivores as the energy travels up the foodchain to other biological and economic con-sumers. At some point in human evolution, re-sources were valued by price so that monetary


value replaced energy. However, despite the dif-ferent units of flow, the acquisition and allocationfunctions per individual at all levels have the sameform, and the allocations of resources within lev-els uses the same priority scheme (Gutierrez andWang, 1977): first to unassimilated wastes (excre-tion in animals or wastage in human productionsystems), respiration (maintenance costs), repro-duction (profits) and somatic growth (capital in-vestment). The analogies between the biologicaland economic systems are consistent in most re-spects including the notion of the discount rate,analogous in biology to the genetically based ex-pectation of environmental hazard (Gutierrez etal., 1997). Unlike most other species, the transi-tion from primal humans (true animals) to mod-ern humans has had both Darwinian (genetic) andquasi-Lamarkian (non-genetic learning) aspects.Further important differences are that modernhuman economic rationale for resource acquisi-tion is driven by hedonistic lust for materialgoods, and allocations may be made to consump-tion that does not contribute to growth. In biol-ogy, demands for resources are genetically basedand finite, and allocations are invested in strate-gies that increase adaptivity to the environmentbut may not contribute directly to growth(Gutierrez et al., 1997). This conceptual model ofenergy flow between trophic levels (i.e. the cur-rency of biological systems) is the foundation forour economic model of resource depletion(Gutierrez et al., 1994).

2.2. The biological model

Assume a food chain with primal humans asthe top predator (e.g. Fig. 1). Let Mi (i=1,…, n)denote the mass of the ith trophic level, where n isthe top predator, then following Gutierrez et al.(1994) and Schreiber and Gutierrez (1997), thedynamics of any trophic level (except for the toppredator) is governed by the following equationof motion:

dMi(t)dt

=ui Mi Di h(si)−ni(Di)Mi

−Mi+1Di+1h(si+1) (1)

where h is concave with si= (ai Mi−1)/(Di Mi) andh %(0)=h(�)=1. A discussion of alternative prey-predator models is given in Yodzis (1994). Theconcavity of h is easily demonstrated in biologyby simple enzyme kinetics and animal feedingexperiments (Holling, 1966) or yield effort rela-tionship of human harvesting. Since, d/dx [Dh(ax/D)]�x=0=a, and limx�� Dh(ax/D)=D theconditions h %(0)=h(�)=1 are not restrictive.The top predator obeys the same relation, exceptthat the rightmost term in Eq. (1) is missing. Thelowest level resource (M0) in the food chain isincident solar energy, and it is considered fixed.The parameters of the model are: ai, i=1,…, n isthe proportion of the i−1 trophic level accessibleto the i-th trophic level (0BaiB1); Di is themaximal rate per unit mass of the i-th trophiclevel extractable from level i−1; ui is the conver-sion efficiency of the i-th trophic level, and (1−ui) is the proportion of the resource lost throughwastage; ni(Di) is respiration or cost rate per unitmass as a function of the potential extraction rate;and Ci is per unit mass cost spent on adaptednessto meet expected environmental hazards.

The function h(si) incorporates the biology ofresource acquisition by individuals, and representsthe probability of achieving resource catch rateDi. Specifically, the supply-demand ratio (Arditiand Ginzburg, 1989) is included in the modelusing the form h(si)=1−exp(−ai Mi−1/Di Mi)(Gutierrez, 1992; Gutierrez et al., 1994). Thismodel captures the effects of random search, vari-able resource availability and demand for theresource, as well as intra-trophic level competitionfor resource acquisition. Thus h(si) is the propor-tion of the demand acquired (i.e. the supply-de-mand ratio). The rate of resource (Mi−1)depletion by all members of the i-th trophic levelis DiMih(si), and it is readily verified thatDi Mi h(si)5ai Mi−1 with ai51 setting the limiton the extraction from the lower trophic level.When ai is sufficiently low compared to the maxi-mum biomass reproductive rate of level i−1(ai5ui−1Di−1−ni−1(Di−1)), the lower topiclevel will survive any population size and demandrate of its predator (Gutierrez et al., 1994). Thus(1−ai) can be viewed as a safe refuge of the i-1trophic level from predation guaranteeing its sur-


vival. This point will be revisited later because ofits importance to harvesting policy for modernhumans. The biological model has been widelyapplied to natural resource problems (Gutierrez,1996).

3. The economic model

An endogenous growth model based on Eq. (1)for exploitive firms is used to examine the eco-nomic optimization of owner consumption of arenewable resource. A single-owner firm or har-vesting unit (e.g. a fishing boat per person) isassumed. As in the biological model, the percapita harvest rate is D ·h(s), where h(s)=h(ax/Dy) is the proportion of the potential harvestdemand D acquired of which the fraction 05u51 is usable. The extraction cost is nD, and theremainder is either consumed (C) or reinvested inthe firm that grows at a rate u ·D ·h(s)−nD−C.The new parameter c may be viewed as consump-tion by an individual or dividend of a modernfirm and has a biological analog. We assume thateach individual is interested in maximizing thelong-term utility or benefits of consumptionU(C). In other words, maximizing the presentvalue of benefits obtained from the consumptionstream: e−dt U(C) dt where d is an instanta-neous discount rate. Consumption level depends,of course, on firm size and its dynamics as de-scribed below. We argue that the differences be-tween the biological (Gutierrez et al., 1997) andeconomic objectives are due to combinations offorces that drive the economic parameters as wellas the time scale (market versus evolutionarytime) within which human societies and biologicalsystems operate. The economic interpretation ofthe biological model is facilitated by reducing thenotation.

3.1. A dynamic model of human har6esting of arenewable resource

Assume our system considers only the two toptrophic levels, while the third or base trophic level(M0) is considered fixed. Further, assume that therenewable resource is managed for societal

benefit. The notation in Eq. (1) is simplified bysubstituting y for Mn, the human predator thatharvests a lower resource level (x=Mn−1). Usingthis notation, Eq. (1) is rewritten as two differen-tial equations for state variables x and y :

dx/dt x; =g(x)−yDh(s) (2)

dy/dt y; =uyDh(s)− (nD+C)y (3)

where g(x) is the renewal rate of the resource andassumed to be concave, yDh(s) is the harvest byall firms and 051−u51 is the proportionwasted. Increases in y imply recruitment of newindividuals to the industry. An important differ-ence between Eq. (1) and Eq. (3) is the consump-tion term C. Potential harvest capacity (D) mayalso be interpreted as capital, and cost n(D) isassumed for simplicity to be linear in D, i.e.n(D)=nD.

3.2. The economic har6esting model

Potential harvesting capacity (D) and consump-tion (C) are control variables determined by thequest for profit or utility maximizing of individualhumans in economics, and fitness and adaptedmaximization in biology (Gutierrez et al., 1997).The parameter D in primal humans was small,and became a control variable in modern soci-eties. D may increase as firms seek to satisfyexpanding markets. The parameter a in h(s) is atechnology parameter that for primal humans wassmall but may approach unity for highly efficientmodern harvesters. Thus, D ·h(s) is the individu-al’s production function for search capacity D.Our assumptions imply that h(s) satisfies the con-cavity and positive marginal productivity of thecontrol variables (c,D) as it is increasing with theresource level (x), and decreases with competitionfrom the other users (y).

3.3. Societal optimization

The economic model Eq. (3) differs from theoriginal biological model Eq. (1) in two ways; firstconsumption (C) is introduced as an additionalform of expending energy, and second, a positivediscount rate (d) is incorporated into the human


maximization process. Both terms have biologicalanalogs and are important components of thebiological (or primal human) model (Gutierrez etal., 1997). In economics, dividends or consump-tion (C) rewards the individual agent, firm or enduser, and this reward (benefits) is conventionallydenoted by a monotone increasing concave utilityfunction U(C) that saturates. Thus the far-sightedindividual seeks to maximize the present value ofutility over an infinite time horizon, and hence thelong-run objective function of the society consist-ing of y individuals is assumed to be:

maxD,c

&�0

e−d · t yU(C) dt (4)

subject to Eq. (2) and Eq. (3), where e−dt is adiscounting factor. By Pontryagin’s maximumprinciple, the maximization of Eq. (4) subject toEq. (2) and Eq. (3) is equivalent to maximizationof the current value Hamiltonian:

H=U(C)y+l1(g−F)+l2[uF− (nD+C)y ](5)

where l1(t) and l2(t) are current value multipliers,known also as costate or auxiliary variables, asso-ciated with the constraints Eq. (2) and Eq. (3),respectively. Necessary conditions for an optimalsolution (if one exists) are1:

(i ) HD=0, HC=0(ii ) l: 1=dl1−Hx

(iii ) l: 2=dl2−Hy

(i6) limt��

e−dt li(t)=0

(6)

Using Eq. (6), Appendix A shows the optimalsolution must satisfy

(i ) x; =g(x)−a x ·h [G ]/G(ii ) y; = (u h(G)−n)ax/G−Cy(iii ) l: 1=l1(d−g %(x))−a h %(G)(u U %−l1)and C=C*(d)

(7)

where G is defined in the Appendix A, and C*(d)is an increasing function of d. Since x; and l: 1do not depend on y, we focus our attention onthem. As equations are defined only for l15U %(C*)(u−n), we restrict attention to [0,�]×[0,U %(C*)(u−x2)] in the (x, l1) phase space. Theresults summarized in Fig. 2a are derived in Ap-pendix A. The definitions of the landmarks in thisfigure are:

(a) xd is the level of x where its regenerationrate g %(xd) equals the discount rate d.(b) xu is unexploited carrying capacity definedby g(xu)=0, and xu\0.(c) xs is the optimal steady state resource level.

Fig. 2. Phase diagrams of the bio-economic model: (a) in the(X,l1) space; (b) the effect of increasing the discount rate d onthe general results in (a) (dotted lines denote isoclines withhigher a); (c) the effect of increasing the technology parametera (dotted lines denote isoclines with higher a); (d) the effect ofincreasing the efficiency of processing parameter u ; (e) theeffect of increasing the cost parameter n ; and (f) the effect oferoding the resource base.

1 Subscripted functions denote partial derivatives, e.g.Hx=(H( · )/(x, and prime denotes the first derivative of asingle variable function, e.g. g %(x)=dg(x)/dx, or l: i(t)=dli/dt.


Fig. 3. Steady state of human population.

all harvesting effects of depletion of the free ac-cess resource. The competitive equilibrium solu-tion implies that the price l1=0, and the optimalsolution is xc (Fig. 2a). This ignores biologicalreality that there may be a critical level of x thatleads to extinction of the resource (i.e. xc�xe).The lower the slope of x; =0 in the (x−l1) space,the larger is the distance between societal andcompetitive solutions xs and xc, and hence thelarger is the chance that a competitive solutionwill lead to extinction. Examination of Eq. (B1)and Eq. (B2) in Appendix B shows that the slopeof x; =0 in the (x ···l1) space is lower when g(x) isless elastic and h(G) is more elastic; that is whenresource regeneration is relatively slow, and theproportion of the demand satisfied is close to one.Below we examine the sensitivity of the solutionto changes in parameter values. The proofs ofthese results are given in Appendix C.

4. Sensitivity analysis

4.1. Increasing discount rate d

Fig. 2b shows that keeping all other parametersconstant, increasing the discount rate d reducesthe societal steady state solution xs but the level ofxc is unaffected. The solid lines indicate the origi-nal isoclines and the dashed lines indicate theisoclines for a larger value of d (Appendix C). Aswe have noted before, xc equals the limiting soci-etal solution when the discount rate increaseswithout bound. The other effects of increasing d

are: the value of xd is also pushed to the left,becoming the limiting resource level when A rises;the shadow price of the resource (l1) decreases(Appendix C); and C* increases with increasing d.When d increases, the steady state levels of ydecrease because C*(d) increases and the payoffdecreases. The decrease in the population of hu-man firms is mitigated by the effect of increases inthe discount rate that raise the individual’s rate ofresource exploitation. The lower bound xd on thesocietal solution suggests that if the discount rate(d) is sufficiently small, technological improve-ments should not drive a publicly regulated re-source to extinction. Conversely, if the discount

(d) xc is the competitive solution for the re-source level x.(e) l1* is shadow price of the resource x (i.e., themarginal gains of y at the optimal solution).(f) xe is the level of x that assures its extinction.The resource level xc is the ultimate myopic

solution as d��, and is the largest resource levelsatisfying g(xc)= −axc[h(G(0))/G(0)] (see below).The resource level xd is the lower bound on thesteady state value of x obtained in the societalsolution. Notice that xu\xec\xd, and xs\xc

always holds, but xd\xc holds only if d is suffi-ciently small and a is sufficiently large.

The implication of the above solution for theoptimal paths of the resource x and population ofharvesting units y is illustrated in Fig. 3 (Ap-pendix B). This equilibrium solution for xs andy*, denoted as the societal solution, can be viewedas an optimization process for a human societyregulated by a ‘benevolent dictator’ whose objec-tive is the long-run maximization of utilitarianwelfare, and is in sharp contrast to the solutionregulated exclusively by the competitive market.

3.4. The competiti6e market solution

The institutional framework of competitivemarkets implies that the market solution willforce resource levels to xc as the long-run equi-librium (Appendix B). In a competitive frame-work, individuals maximize their own utilityfunction (related to consumption), disregarding


rate becomes too large, the resource may bedriven to extinction even under a socially optimalpolicy. It is generally recognized that the discountrate is determined in the macro economy andignores its potential impact on a specific renew-able resource. This decoupling may be an impor-tant factor in depleting resources, possibly leadingto their extinction.

Gutierrez et al. (1997) argue that the biologicaldiscount rate is based upon the uncertainty of theenvironment (i.e. expected hazards). It may belarge or small, and its effects on resource exploita-tion rate are in the same direction as the discountrate in economics. So an obvious question is whydon’t primitive humans that live in a precariousenvironment over exploit their resources or dothey? The answer lies in part on technicalprogress.

4.2. Technological progress

The technology parameter (a) determines whatproportion of the resource can be exploited. Ingeneral, if a is sufficiently low compared to themaximum biomass regeneration rate of the re-source, the resource will survive any size predatorpopulation and demand rate (Gutierrez et al.,1994). This parameter may also include socialconstraints where individuals limit their capacityto harvest (a form of private ownership). Thetechnical capabilities (a) of primitive societieswere small, and in our whale example the impactwas also small. However, as technology pro-gresses, A may potentially be larger than unity(i.e. using satellites we can find the last whale).

Fig. 2c illustrates the effects of increasing A onthe two isoclines in the phase diagram in the(x ···l1) space. The solid lines indicate the originalisoclines and the dashed lines indicate the isoclinesfor a larger value of a (Appendix C). Under acompetitive market structure, technology that in-creases A reduce xc and increase the likelihood ofdriving the resource x to xe. This result is in sharpcontrast to the conventional view that technologi-cal improvements are the driving force behindincreased income per capita and supports popula-tion growth. However, in our model when publicinstitutions take control of an endangered re-

source, increasing A increases the steady stateresource value (l1), and the resource equilibriumlevel is reduced. This occurs because an increasein harvesters results due to technologicalimprovements.

In a competitive economy, technological pro-gress alone is potentially sufficient to drive theresource population to extinction, independent ofthe well-known effects of increasing the discountrate. As discussed above, increasing the technol-ogy of harvesting (a) can also drive a resourceunder public control to extinction, but this alsodepends on the discount rate d. Thus, a synergis-tic combination of high technology (a) and a highdiscount rate (d) may greatly increase the risk ofover-exploitation and resource extinction even un-der a socially optimal policy. As shown in Ap-pendix C, the payoff and the steady statepopulation of y increase with a. The increase in yhas an additional negative effect on xs of increas-ing a.

4.3. Lower wastage

A higher u implies lowers wastage, and for thecompetitive solution may drive the resource popu-lation quickly to extinction (Fig. 2d). Increasing u

does not affect xd, which is the lower bound onthe societal solution. The competitive solutionmoves to the left, so that reducing waste has asimilar effect to that of improving technology. Ina competitive framework, the resource level maybe driven closer to extinction. The change in theoptimal societal solution xs is undetermined, butl1 increases as wastage is reduced (u increases).However, increasing u may reduce xs. If the socialdiscount rate d is sufficiently low, xd will remainsufficiently distant from the extinction level. If,however, d is sufficiently high, lowering wastagemay in fact put the resource at the risk of extinc-tion as xc�xe. u like a is a parameter of efficiencyand operates in the same direction, and henceincreasing either raises the risk of over exploita-tion and possibly extinction if the discount rate ishigh enough.

However, the effects of u are constrained by thevalue of a, and are best seen by reference toprimitive societies which are thought to waste


little of their harvests. Despite a higher u, their a

is typically low, hence regardless of their percep-tion of environmental uncertainty, efficient uti-lization of resources may have little impact on therenewable resource.

4.4. Higher maintenance costs n

An increase of n reduces l1 and the payoff andconsequently reduces the optimal population den-sity. Note also that the competitive solution xc

moves to the right and away from extinction level(Fig. 2e), but xd is not affected. As n approachesu, xs approaches xu and marginal gains fall (i.e. itis no longer cost effective to harvest) and suggeststhat the societal level xs increases with n. From apolicy point of view, one may interpret an in-crease in n as a tax levied on harvesting capacity,suggesting effective policy implications for pre-serving resources.

4.5. Eroding the resource base

The resource base for x is often eroded byhuman activities including those used to harvestthe resource x. Decreasing the resource baseparameter (M0) has the obvious effect of shiftingx; to the left and l: 1 downward, implying lowercompetitive and societal solutions. In addition,the payoff and optimal population density in-crease (Fig. 2f). Clearly, there are synergistic ef-fects of reduced environmental carrying capacityand factors enhancing over exploitation increasethe likelihood of resource extinction.

5. Conclusion

5.1. Species 6ersus human optimal beha6ior

The time evolution of an ecosystem is driven byDarwinian selection processes that determinewhich individuals and species survive. All organ-isms in all trophic levels are part of the ecosystem,and each has demand capacity for resource acqui-sition that operate within the bounds of its geneticcode — its objective is to perpetuate the survivalof its DNA (sensu Dawkins, 1995). As argued in

Gutierrez et al. (1997), individual organisms innature behave as if they are driven by a quest forutility maximization to increases individualfitness. Although the resource has free access,there is a biological price to it implied by itseffects on marginal growth that leads to resourcesustainability. Similar arguments could be madefor humans in their primal state.

However as modern humans became the toppredators, their objective became more than sim-ply perpetuating the survival of DNA. In eco-nomic terms, the objective function becamemaximization of the present value of a non de-creasing utility of consumption by all individualssubject to the resource constraints. In contrast tothe ecosystem, competitive market forces aredriven by individual quest for utility maximiza-tion disregarding their individual effects on natu-ral resource depletion, and this determines zeroprices for free access (common property) re-sources. These differences strike at the heart ofthe renewable resource exploitation problem aspracticed by humans.

5.2. Analogies between economic and biologicalsystems

Analogies between biological and economic sys-tems enabled the development of a commonmodel for both systems (this paper and Gutierrezet al., 1997), but some very important differencesbetween economic and biological systems exist.Among these are the fact that the extractioncapacity (D), the transformation parameter (u),the maintenance cost (n) and the resource appar-ency parameter a may change rapidly in responseto economic factors. In contrast, their biologicalanalogues are relatively ‘fixed’ on an evolutionarytime scale. In fact, the demand in economicsmight exceed all of the resource that is available.The parameter a is interpreted in economics asthe technology parameter of resource harvestingand plays an important role in the dynamicsleading to resource extinction. In the economicmodel, a higher value of a means that the technol-ogy for exploiting previously inaccessible portionsof the population is increased. A higher value of u

means that wastage is reduced and increasing n


means that the per firm maintenance cost of har-vesting capacity increases.

The concept of the discount rate (d) introducedin the long-term objective function in the discountfactor (e−dt) also has a biological analog. Ineconomics, the discount rate reflects time prefer-ence, partly because of uncertainty about the fu-ture, and it is determined by market forces in thewhole economy. In biology, the analogous con-cept is the likelihood that investments in progenywill survive to contribute to the genetics of futuregenerations (Gutierrez et al., 1997) and is imple-mented in the biological model as an ecologicaldiscount factor in the objective function.

The concept of economic consumption also hasa biological analogue. In biology, some speciesinvest heavily in excess reproductive capacity andother produce few progeny and invest more intheir care but these investments do not contributedirectly to growth (so called r- and K-selectedstrategies, respectively, see Southwood andComins, 1976). These costs are, by analogy, con-sumption as defined in economics, but this alloca-tion in biology is used to increase adaptivity andnot for hedonistic purposes.

5.3. The sustainable har6esting problem

Renewable resources (biological populations)dynamics have spatial and temporal characteris-tics that affect their dynamics and those of popu-lations that harvest them (i.e. firms).Furthermore, if the environment of species is de-graded, species may not be restored to its priorabundance, and of course once a species has beendriven to extinction it cannot be brought back.For these reasons, management policies for re-newable resources must resolve questions that af-fect resource sustainability before the damagebecomes irreparable. Among these questions arethe optimal extraction rate, optimal human andresource population levels, the appropriateness ofharvesting technology, and how market structureaffects harvesting behavior.

Competitive markets have failed to provide anappropriate mechanism for pricing resources withfree access (Gordon, 1954; Hilborn et al., 1995),and consequently, over exploitation of resources

has been a common practice in forestry andfisheries because the cost of renewable resources islargely neglected by harvesters. The consequencesof harvesting explored here using an micro-eco-nomic model that incorporated the dynamics ofboth the resource being exploited and the exploit-ing population produced some conflicts with con-ventional economic wisdom.

5.4. Ecological conflicts with con6entionaleconomic wisdom

Economic growth theory has largely ignoredthe relation between growth and natural renew-able resources, and assumed population growth tobe exogenous (Barro and Sala-I-Martin, 1995).Consequently, conventional economic modelshave not incorporated the realistic biology of theharvesting units extracting renewable resources(Hilborn et al., 1995). Instead, neoclassical eco-nomic growth theory suggested that technologicalimprovements were the source of increasing percapita income (Solow, 1956). The conventionalwisdom that follows is that increases in technol-ogy (including the discovery of new renewableresources) enhance productivity and this main-tains growth. The underlying biological realism ofour model, however, identified increases in har-vesting and utilization technology as reducingsteady state resource levels and at the same timeincreasing the number of users, in what wouldappear to be a vicious cycle leading to over-ex-ploitation of the resources and the collapse of theindustry.

The golden rule of economically balancedgrowth (that maximizes the steady state per-capitaconsumption) stipulates that the rate of savingassociated capital accumulation is obtained whenmarginal productivity of capital is equated topopulation growth and capital depreciation rates(Phelps, 1966). Marginal productivity of capitalequals the interest (discount) rate, and this occursin our model as the lower bound of the societalsolution for resource exploitation xd. It is wellknown that high discount rates increase the rateof exploitation of natural resources. Since thediscount rate is determined by the market in the


whole economy, this value may be higher than thesocially optimal level for the environmentalpreservation (Weitzman, 1994), and specificallyfor maintaining a particular renewable resource ata sustainable level. The socially optimal level ofthe resource (xs) is that which is sustainable forthe optimal population of firms (y*). It has beenshown here that there is a synergistic effect be-tween improvements in harvesting technology andhigh discount rates that encourage faster exploita-tion of the resource, possibly leading to itsextinction.

The conventional wisdom for reducing thewastage of harvest is that resources are preservedif their utilization is improved. However, ourmodel shows that as wastage is reduced, the opti-mal steady state level of the resource is reduced,again contradicting common wisdom. This effectincreases with increasing discount rate, and issimilar to the synergistic effect between technol-ogy and discount rate. This occurs because thepayoff for the firms increases following bettertechnology and lower wastage. Increases in thecost to firms detract from growth, counteringgains from decreases in wastage, and thus leadingto higher resource levels. A way of increasing costis the use of Pigouvian taxes on firm capacity (D)suggesting interesting policy implications. Ofcourse, restricting harvests may be efficient butoften hard to enforce as harvesters find ways tocircumvent regulations.

In modern economies, the resource base may beincreased, as is done in agriculture by improvedproduction methods, new varieties or breeds ofanimals, more agronomic inputs, etc. but thesemay also lead to unforeseen adverse consequences(van den Bosch, 1978; Kenmore et al., 1985).While privately owned agricultural systems arehighly managed, free access renewable resourcesare overused inefficiently and productivity may belowered or destroyed as competing firms seek tomaximize profits while ignoring renewable re-source depletion and environmental costs. Thisis verified by our model where all of the steadystate levels in our system are reduced as thebase resources for the exploited population areeroded.

5.5. Policy implications

All of the above results accrued via the dynam-ics of the biological model, rather than by a prioriassumptions. Our model points to the need tosimultaneously control technology and the dis-count rate. It is obviously impossible, nor is itdesirable to regulate the advance of technology,hence the major option left is to reduce the dis-count rate below the market equilibrium and alsoto regulate the harvest. If a society considers thepreservation of an environmental resource impor-tant, the social discount rate should be lower thanthe private one (Weitzman, 1994). Regulation ofharvest can be accomplished via a Pigouvian taxon the capacity of the firm which in our modeldrives down firm numbers, but leaves open thequestion of increasing size of the remaining firms.In all cases, what is absolutely clear is that totalharvest by all firms should be only that level thatassures the sustainability of the resource at equi-librium density. This could be done without taxes,but would require strong enforcement and soundnotions about the maximum sustainable yields(MSY) — weak assumptions about carrying ca-pacity will only lead to still more resource ex-ploitation disasters. As pointed out by Hilborn etal. (1995), the notion of MSY is unrealistic asnatural populations fluctuate (often widely) inresponse to drastic changes in biotic and abioticfactors. The possibility of over exploitation lead-ing to extinction is more likely when stochasticperturbations affect the resource (El Nino effectson Pacific fisheries); however, this issue has notbeen tackled in the present paper.

The dual goals of economic growth and ecosys-tem sustainability are often in conflict (Goodland,1995). Impetus for resolving some of these issuescomes as the standard of living improves; thisdespite the apparent contradiction that the im-provement in a large part may have resulted fromresource depletion. Increasingly affluent societiesdemand improved environmental quality leadingto public pressures for environmental regulationvia market and non-market mechanisms. On alarger global scale, however, the difficulty lies inthe recognition that improvement in the standardof living in heavily populated less developed coun-


tries would certainly lead to over exploitationof fragile natural resources as technology andthe market interact to satisfy ever increasing con-sumer demands (Goodland, 1995). Sustainabledevelopment must include viable environmen-tal, social and economic sustainability butnot necessarily the sustained economic through-put growth that is the basis of the common world-wide economic paradigms (sensu Goodland,1995).

5.6. Epilogue

In this paper we examined harvesting from afood chain, but humans may harvest more thanone resource in the food chain (or web). Forexample, humans harvest both whales and krill,and we might ponder what the impact of interact-ing economic and technological parameters mightbe on the system — will whales be placed ingreater danger by krill harvesters than whalers.Schreiber and Gutierrez (1997) use the underlyingbasis of this biological model to examine biologi-cal interactions in food webs, and demonstratehow species displacements have occurred in sev-eral systems. This model can be used as the basisfor examining human harvesting of severaltrophic levels in a renewable resource systems.The questions of physical and human capital areLamarkian like processes that also impact re-newable resource problems, but they were notaddressed in this paper, nor do we addresshow big should firms be or how human capitaldrives technology, and in what direction?Can human capital be the basis for developingviable renewable resource management schemes,or will it simply contribute more to over exploita-tion?

Clearly, ecology and economics are at a cross-roads of conflict: the alternatives are sustainablerenewable resource management based on soundbiology, or will over exploitation and mutualannihilation result as we scramble for ever de-creasing resources. If the latter is our fate, then‘‘… as a final bit of irony, it will be insects thatpolish the bones of the last of us that fall.’’(Robert van den Bosch, 1978).

Appendix A

In this appendix, we derive equations of motionfor our maximization problem using the Pon-tragin maximum principle. Recall, the currentvalue Hamiltonian for this problem is given byH=U(C)y+l1(g(x)−F(x,y,D))

+l2(uF(x,y,D)− (nD+C)y) where l1 andl2 are the costates associated with x and y andF(x,y,D)=Dyh(ax/Dy). The optimal solutionmust satisfy HD=HC= limt�� e−dt li(t)=0, l: 1=dl1−Hx and l: 2=dl2−Hy.

HC=0 implies that l2=U %(C) and, conse-quently, l: 2=U¦(C)C: . HD=0 implies that(l2u−l1)FD(x,y,D)=l2ny. Since FD(x,y,D)=yZ(ax/Dy) where Z(s)=h(s)−sh %(s),

axDy

=Z−1� l2n

l2u−l1

�(A1)

Furthermore HDD=FDD(x,y,D)(ul2−l1)50implies l2u]l1 on the optimal path.

The equation of motion for x ’s costate is givenby,

l: 1=dl1−Hx=l1(d−g %(x))

−ah %�

Z−1� l2n

l2u−l1

��(ul2−l1)

where the second equality follows from Eq. (A1).On the other hand, the equation of motion for y ’scostate is given by l: 2=dl2−Hy. Hence

l: 2=dU %−U+Fy(x,y,D)(l1−uU %)+ (nD+C)U %

= (d+C)U %−U

where the last line follows from Eq. (A1) andFy(x,y,D)=DZ(ax/Dy). Putting this all together,the equation of motion for candidate solution toour maximization problem are given by,

x; =g(x)−axh(G)

G(A2)

y; =axG

(uh(G)−n)−Cy (A3)

C: = 1U %%

((d+C)U %−U) (A4)


l: 1=l1(d−g %)−ah %(G)(uU %−l1) (A5)

where

G=Z−1� U %nU %u−l1

�(A6)

Appendix B

To find the equilibrium of Eq. (A2), Eq. (A3),Eq. (A4), Eq. (A5), Eq. (A6), we first solve forC: =0. We begin with three observations: (/(C (U %(C)(d+C)−U(C))=U %%(C)(d+C)B0 asU is concave; limC�0 U %(C)(d+C)−U(C)\0 asU(0)=0 and U %(0)\0. limC�� U %(C)(d+C)−U(C)B0 (this follows from our assumption thatlimC�� U %(C)=0). These observations implythat there is a well defined differentiable functionC*(d) such that U %(C*(d))(d+C*(d))−U(C*(d))=0. Furthermore, C*%(d)\0, C*(0)=0 and limd�� C*(d)=�.

Since Eq. (A2) and Eq. (A5) do not depend ony, the remainder of our isocline analysis is re-stricted to the x−l1 plane with C=C*(%). AsZ−1 is only well defined on the interval [0,1], wefurther restrict our analysis to (x,l1)� [0,�)×[0,U %(C*)(u−n)).

To prove monotnicity of the x nullcline, we firstobserve that G %(l1)=Z−1( · )(U %n)/(U %u−l1)2\0and h %(G)G−h(G)B0 by convexity of h.Therefore,

(x;(l1

)x; =0

= −ax�h %(G)G−h(G)

G2

�G %(l1)\0 (B1)

(x;(x)x; =0

=g %(x)−ah(G)

G)x; =0

=g %(x)−g(x)

xB0

(B2)

where the second equality in Eq. (B2) followsfrom the concavity of g. Therefore, (l1/(x �x; =0\0 and the x null isocline is a strictly increasingfunction of x.

To prove the monotonocity of the l1 null-iso-cline, note that

(l: 1(l1

=d−g %(x)

−a [h %%(G)G %(l1)(uU %−l1)−h %(G)] (B3)

is greater than zero whenever (l1,x)� [0,U %(u−n)]× (xd,�) where xd is the unique solution tog %(x)=d if it exists else zero. On the other hand,(l1/(x= −l1g %%(x)\0 by convexity of g.Therefore

(l1

(x)l: 1=0

B0 (B4)

whenever x\xd. As x; B0 for any x\xd, it fol-lows that the entire l1 null isocline lies to the rightof x=xd and is strictly decreasing in x.

Putting this all together, we get the phase dia-gram shown in Fig. 2a. In this figure, xc is thelargest value of x such that g(x)=ax [h(Z−1(n/u))]/[Z−1(n/u)] and xu is the largest value of xsuch that g(x)=0 (the equilibrium achieved bythe resource in the absence of harvesting).

Note: The level xc is also obtained as the solu-tion of individual decision making in a competi-tive markets. The individual problem is then:maxC,DU(C) subject to the constraint uF/y−(nD+C)]0. Using the Lagrangean L=U(C)+vC)uF/y− (nD+C)), LC=0 andLD=0 imply that ax/Dy=Z−1(n/u). Insertingthis feedback rule into x; and solving for its largestequilibrium determines xc.

To determine the stability of (x*,y*,C*), theoptimal equilibrium, we evaluate the variationalmatrix of the equations of motion at this point.Using Eq. (A4) and Eq. (B1), Eq. (B2), Eq. (B3),Eq. (B4), we get

ÃÃ

Ã

Ã

Ã

Á

Ä

(x;(x(l: 1

(x(C:(x

(x;l1

(l: 1

(l1

(C:(l1

(x;(C(l: 1

(C(C:(C

ÃÃ

Ã

Ã

Ã

Â

Å

=ÃÁ

Ä

−+0

++0

�

�

+ÃÂ

Å

where −/+ indicate the sign of the entry and �indicates that the sign is irrelevant. Since (0,0,1) isan unstable eigenvector for this matrix, along theoptimal path C(t) C*. Furthermore, as


det�−

+++�B0

there is an unstable and stable eigenvector inx−l1 space. Hence along the optimal path l1(t)and x(t) are monotonic and l1 is determined by afeedback rule l1(x) that satisfies l1% (x)B0.

Using Eq. (A2) and Eq. (A3), we examine thex−y phase for the optimal paths. The y nullisocline is defined by

cy=axg

(h(G)−n). (B5)

Taking the derivative of the right hand side of Eq.(B5), we get

a

G(uh(G)−n)+ (B6)

axG2(uh %(G)G−uh(G)+n)Gx (B7)

On the y nullcline, Eq. (B6) is strictly positive andEq. (B7) equals

axG2

�n−u

� U %(C*)nU %(C*)u−l1(x)

��Gx.

Since l1(x)� [0,U %(C*)(u−n)] and GxB0 thisterm is also strictly positive. Therefore, the ynullcline is stricltly increasing with respect to x.Notice that certain initial conditions of x and yproduce a ‘snowballing’ effect such that y is notmonotonic in time.

Appendix C

In this section, we examine how parameterseffect the isocline structure in x−l1 space, thepayoff, and the optimal equilibrium value of y.Unfortunately, our conclusion with respect to x*can be only speculative and are based on numeri-cal simulations.

First, consider a. Notice that (x; /(a= −x(h(G))/GB0 and (l: 1/(a= −h %(G)(U %(C)u−l1)is strictly negative for (x,l1)�R+ × [0,U %(C*)(u−n)] (Fig. 2c). To see how the payoff changes witha, consider the optimization problem defined byEq. (A1) subject to the constraints Eq. (A2) andEq. (A3) plus the addition constraint a; =0. Let

v3 be the present value costate associated witha.The equation of motion for this costate is givenby −v; 3=xh %(ax/Dy)(uv2−v1) where v1 andv2 are present value costates associated with xand y, respectively. Integrating with respect to tand using the transverslity condition on v3, weget v3(0)=�0 xh %(ax/Dy)(uv2−v1) dt\0 wherex, D, C and y are evaluated on the optimal path.Since (under the assumption that the payoff dif-ferentiable with respect to a) v3(0) equals thederivative of the payoff as function of the a, thepayoff increases with a.

Consider the parameter u. Notice that GuB0and therefore (x; /(u= −ax(h %(G)G−h(G))Gu/G2 and (l: 1/(u= −ah %%(G)Gu(uU %(C*)−l1)−ah %(G)U %(C*) are strictly negative (Fig. 2d). Aswith the payoff analysis for a, the payoff increasesas a function of u.

Consider the discount rate, d. C*(d) and xd areincreasing/decreasing functions of d and GdB0.Therefore (x; /(d= −ax(h %(G)G−h(G))Gd/G2 isstrictly positive. Unfortunately, we are unable todraw a conclusion about the motion of the l1

nullcline. However, the motion of the xd andsimulations suggest that it is as indicated in Fig.2b. A routine calculation shows that the payoffdecreases as a function of d.

Consider the parameter, n. We have G %(n)=Z−1%( · )(U %(C*))/(U %(C*)u−l1) is strictly posi-tive. From this it follows that

(x;(n

= −axG %(n)h %(G(n))G(n)−G %(n)h(G)

G(n)2

and

(l: 1(n

= −ah %%(G(n))G %(n)(uU %(C*)−l1)

are strictly positive by concavity of h (Fig. 2e).These facts imply l1 at the optimal equilibriumdecreases as a function of n. The payoff alsodecreases with n. Although we can draw no gen-eral conclusions about the effect of n on theequilibrium resource, we can show that as n ap-proaches u the equilibrium resource density ap-proaches xu. This suggests that resource densityincreases with n.

Eroding the resource base is equivalent to re-placing the function g(x) with another function


g(x) such that g %(x)Bg(x), g(0)=0 and g isconcave. Such a change only effects the x zerogrowth isocline by shifting it left (Fig. 2f). Hence,decreasing the resource base decreases the re-source density and increases l1 at the optimalequilibrium. It is easy to check that it also de-creases the payoff.

To conclude, we want to understand how y*changes with respect to the parameters. To dothis, we need the following lemma.

Lemma. Let h be one of the parameters of themodel (i.e. a,u,n or d). For any h sufficiently closeto h, there exists an initial condition, (x0,y0), suchthat the optimal paths y(t) and y(t) associated withh and h that satisfy this initial condition are bothmonotonically decreasing.

Proof. Given h, define the sector Sh={(x,y):x; (x,y)\0 and y; (x,y)B0}. Since the xnullcline is vertical and the y nullcline ismonotonically increasing with respect to x,y(t)for all optimal paths with initial condition in Sare monotonically decreasing. By continuity thereexits a e\0 such that for all h e-close to h, thesectors, Sh and Sh have a nonempty intersec-tion. Choosing a point, (x0,y0), in this intersectionprovides the desired initial condition.

Now, consider a. When a increases, the payofffor any initial condition of x and y increases butC(t) C* remains constant. Assume we are givena\a\0 such that a−a is sufficiently small. Thelemma gives optimal paths y and y with the sameinitial conditions such that both paths aremonotonic in time. Since the payoff associatedwith a is larger than the payoff associated witha, y(t)]y(t) for all t]0. Consequently, y* in-creases with a. Similarly, we can conclude that y*increases with u and increasing resource base, butdecreases with d and n.

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