Ecological Economics 26 (1998) 227 – 242
ANALYSIS
Biological and economic foundations of renewable resource
exploitation
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, Israel
b
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 model
of human harvesting. The model incorporates developing technology and the effects of market forces on the
sustainability of common property resources. The bases of the model are analogies between the economics of resource
harvesting and allocation by firms and adapted organisms in nature. Specifically, the paper makes the following
points: (1) it shows how economic and ecological theories may be unified; (2) it punctuates the importance of time
frame in the two systems (evolutionary versus market); (3) it shows, contrary to prevailing economic wisdom, how
technological progress may be detrimental to resource preservation; (4) it shows how the anticipated effects of high
discount rates on resource use can be catastrophic when synergized by progress in harvesting technology; (5) it
suggests 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 global
implications on the relationship between economic growth and the ability of modern societies to maintain the
environment 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
228
U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
‘‘Man stalks across the landscape, and desert
follows his footsteps’’
Herodotus (5th century B.C.)
1. Introduction
The fate of humankind will be determined by
how sustainable ecosystems and the renewable
resource species in them are managed. Species in
nature evolved via Darwinian processes in response to interactions among members of the
same species, with other species, and their physical environment and its carrying capacity. Human
harvesting of some species is an additional ever
increasing burden of mortality that has only been
recently imposed, often based upon unrealistic
assumption of maximum sustainable yield (Getz
and Haight, 1989; Hilborn et al., 1995). Only the
human predator has escaped regulation of its
numbers, and through ingenuity forestalled
Malthus’ ‘doomsday prediction’ concerning excessive human population. This has been explained
by capital accumulation and technological progress (exogenous or endogenous) including the
discovery of new goods and methods of production since the industrial revolution (Schumpeter,
1934; Solow, 1956, 1970). This enabled increases
in production in agriculture in its various forms,
the harvests of naturally occurring resources, and
increase in resource processing and distribution.
Recent economic literature on growth posits that
given non-ending technological progress, food
production will continue to outpace demand for
several centuries, ignoring natural resource limitations with optimistic views about the role of technology in surmounting resource scarcity and
environmental degradation (Grossman and Helpman, 1994; Barro and Sala-I-Martin, 1995;
Romer, 1990).
In contrast, many ecologists and some
economists recognize limits to human population
growth set by the relative rates of renewable
resource exploitation and regeneration, and by the
increasing degradation our finite world (Hardin,
1993; Solow, 1993; Daly, 1994). Despite this, the
economic literature on renewable resource exploitation largely ignores or oversimplifies the biological basis of the ‘reproductive surplus’ that is
the basis of sustainable yield approaches (Hilborn
et al., 1995). While many technological advances
have produced positive private and societal economic benefits, some have caused disastrous environmental problems (e.g. excessive agronomic
inputs, van den Bosch, 1978) and others have led
to over-exploitation of renewable resource populations (e.g. fisheries worldwide, Hilborn et al.,
1995). The invisible hand of the market has especially failed to prevent the over-exploitation and
destruction of many common property resources
that have free access characteristics (Gordon,
1954; Hardin, 1968).
This paper examines the effects of technological
progress and discount rate on the sustainability of
free access renewable resources. The terms ‘human’ and ‘firm’ are used interchangeably as the
firm is single owner. A physiologically-based
predator-prey (i.e. firm-resource) population dynamics model is extended to include humans with
their associated technology as the top predator in
the food chain (e.g. alga “ krill“ whale“
whalers). The model incorporates the realism of
hierarchical energy flow between feeding (i.e.
trophic) levels (Gutierrez and Baumgärtner, 1984)
and captures the essence of the competition between harvesting units (Gutierrez et al., 1994;
Schreiber and Gutierrez, 1997). Our analysis has
parallels in the bio-economic study of adapted
species in nature using the same model (Gutierrez
et al., 1997). That paper should be considered
dual to this one.
2. The biological basis of renewable resource
harvesting
2.1. A common model for humans and other
species
At the dawn of time, primitive humans were
scavenger-gatherers buffeted by the vagaries of
the environment. In their primal state, humans
differed little from all other animal species in
nature 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 (decreasing resources) and top-down (natural enemies including diseases) factors which limited
their capacities to overexploit their environment.
They were an integral part of the food chain and
their capacities and demands for resources were
constrained by genetics molded by natural selection in an ecosystem context (Gutierrez et al.,
1997). However, evolving increased mental capacities enabled humans to escape some of the uncertainty of their environment and the regulation of
their numbers, and to develop technologies that
enabled them to exploit their environment at increasingly higher rates. As human societies developed, their demand rates for resources also
increased and larger investments were made in
social organization that further enhanced their
capacity to harvest resources. The model is flexible enough to capture these different stages of
human evolution.
Despite their progress, humans, like all organisms, must acquire resources and allocate them
(Gutierrez and Curry, 1989), and this is the basis
for extending the biological model to human
economies. Analogies between biological and
modern economic processes were proposed by
Winter (1971) and extended by Gutierrez and
Regev (1983). Nature (or the ecosystem) is
analogous to the economic system, energy is the
currency in biology, individual organisms are
equated to single-owner firms (individuals), fitness
is profit (i.e. what can be invested in the next
cycle), organism genetics are akin to firm decision
rules, adaptivity of individual species to long-term
firm survival strategies, and markets are
analogous to Darwinian processes, etc. Profit
maximization is the assumed goal of individuals
(fitness in other organisms), and selection for or
against strategies occurs at the level of the individual in both systems.
Fig. 1 shows the energy flow in a specific food
chain where humans are the top consumer
(Gutierrez and Curry, 1989), but harvesting of
more than one resource level (say, krill and whale)
is also possible. In nature, the source of energy
used by most life forms is the sun which is captured via photosynthesis by primary producers
(plants) in the chemical bonds of simple sugars.
Some of this energy is used to acquire other
essential nutrients required to form complex
molecules for growth. Plants, be they simple single celled alga or large trees, are ultimately eaten
by herbivores as the energy travels up the food
chain to other biological and economic consumers. At some point in human evolution, resources were valued by price so that monetary
U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
230
value replaced energy. However, despite the different units of flow, the acquisition and allocation
functions per individual at all levels have the same
form, and the allocations of resources within levels uses the same priority scheme (Gutierrez and
Wang, 1977): first to unassimilated wastes (excretion in animals or wastage in human production
systems), respiration (maintenance costs), reproduction (profits) and somatic growth (capital investment). The analogies between the biological
and economic systems are consistent in most respects including the notion of the discount rate,
analogous in biology to the genetically based expectation of environmental hazard (Gutierrez et
al., 1997). Unlike most other species, the transition from primal humans (true animals) to modern humans has had both Darwinian (genetic) and
quasi-Lamarkian (non-genetic learning) aspects.
Further important differences are that modern
human economic rationale for resource acquisition is driven by hedonistic lust for material
goods, and allocations may be made to consumption that does not contribute to growth. In biology, demands for resources are genetically based
and finite, and allocations are invested in strategies that increase adaptivity to the environment
but may not contribute directly to growth
(Gutierrez et al., 1997). This conceptual model of
energy flow between trophic levels (i.e. the currency of biological systems) is the foundation for
our economic model of resource depletion
(Gutierrez et al., 1994).
2.2. The biological model
Assume a food chain with primal humans as
the top predator (e.g. Fig. 1). Let Mi (i =1,…, n)
denote the mass of the ith trophic level, where n is
the top predator, then following Gutierrez et al.
(1994) and Schreiber and Gutierrez (1997), the
dynamics of any trophic level (except for the top
predator) is governed by the following equation
of motion:
dMi (t)
= ui Mi Di h(si ) −ni (Di )Mi
dt
− Mi + 1Di + 1h(si + 1)
(1)
where h is concave with si = (ai Mi − 1)/(Di Mi ) and
h%(0)= h() = 1. A discussion of alternative preypredator models is given in Yodzis (1994). The
concavity of h is easily demonstrated in biology
by simple enzyme kinetics and animal feeding
experiments (Holling, 1966) or yield effort relationship of human harvesting. Since, d/dx[Dh(ax/
D)]x = 0 = a, and limx “ Dh(ax/D)=D the
conditions h%(0)= h() = 1 are not restrictive.
The top predator obeys the same relation, except
that the rightmost term in Eq. (1) is missing. The
lowest level resource (M0) in the food chain is
incident solar energy, and it is considered fixed.
The parameters of the model are: ai, i= 1,…, n is
the proportion of the i− 1 trophic level accessible
to the i-th trophic level (0B ai B 1); Di is the
maximal rate per unit mass of the i-th trophic
level extractable from level i −1; ui is the conversion efficiency of the i-th trophic level, and (1−
ui) is the proportion of the resource lost through
wastage; ni (Di ) is respiration or cost rate per unit
mass as a function of the potential extraction rate;
and Ci is per unit mass cost spent on adaptedness
to meet expected environmental hazards.
The function h(si ) incorporates the biology of
resource acquisition by individuals, and represents
the probability of achieving resource catch rate
Di. Specifically, the supply-demand ratio (Arditi
and Ginzburg, 1989) is included in the model
using the form h(si )= 1−exp(−ai Mi − 1/Di Mi )
(Gutierrez, 1992; Gutierrez et al., 1994). This
model captures the effects of random search, variable resource availability and demand for the
resource, as well as intra-trophic level competition
for resource acquisition. Thus h(si ) is the proportion of the demand acquired (i.e. the supply-demand ratio). The rate of resource (Mi − 1)
depletion by all members of the i-th trophic level
is DiMih(si ), and it is readily verified that
Di Mi h(si )5 ai Mi − 1 with ai 5 1 setting the limit
on the extraction from the lower trophic level.
When ai is sufficiently low compared to the maximum biomass reproductive rate of level i− 1
(ai 5 ui − 1Di − 1 − ni − 1(Di − 1)), the lower topic
level will survive any population size and demand
rate of its predator (Gutierrez et al., 1994). Thus
(1−ai ) can be viewed as a safe refuge of the i-1
trophic level from predation guaranteeing its sur-
U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
vival. This point will be revisited later because of
its importance to harvesting policy for modern
humans. The biological model has been widely
applied to natural resource problems (Gutierrez,
1996).
231
benefit. The notation in Eq. (1) is simplified by
substituting y for Mn, the human predator that
harvests a lower resource level (x=Mn − 1). Using
this notation, Eq. (1) is rewritten as two differential equations for state variables x and y:
dx/dt x; = g(x)−yDh(s)
(2)
3. The economic model
dy/dt y; = uyDh(s)−(nD + C)y
(3)
An endogenous growth model based on Eq. (1)
for exploitive firms is used to examine the economic optimization of owner consumption of a
renewable resource. A single-owner firm or harvesting unit (e.g. a fishing boat per person) is
assumed. As in the biological model, the per
capita harvest rate is D · h(s), where h(s) = h(ax/
Dy) is the proportion of the potential harvest
demand D acquired of which the fraction 05 u 5
1 is usable. The extraction cost is nD, and the
remainder is either consumed (C) or reinvested in
the firm that grows at a rate u · D · h(s) − nD− C.
The new parameter c may be viewed as consumption by an individual or dividend of a modern
firm and has a biological analog. We assume that
each individual is interested in maximizing the
long-term utility or benefits of consumption
U(C). In other words, maximizing the present
value of benefits obtained from the consumption
stream: e − dt U(C) dt where d is an instantaneous discount rate. Consumption level depends,
of course, on firm size and its dynamics as described below. We argue that the differences between the biological (Gutierrez et al., 1997) and
economic objectives are due to combinations of
forces that drive the economic parameters as well
as the time scale (market versus evolutionary
time) within which human societies and biological
systems operate. The economic interpretation of
the biological model is facilitated by reducing the
notation.
where g(x) is the renewal rate of the resource and
assumed to be concave, yDh(s) is the harvest by
all firms and 05 1− u51 is the proportion
wasted. Increases in y imply recruitment of new
individuals to the industry. An important difference between Eq. (1) and Eq. (3) is the consumption term C. Potential harvest capacity (D) may
also be interpreted as capital, and cost n(D) is
assumed for simplicity to be linear in D, i.e.
n(D)= nD.
3.1. A dynamic model of human har6esting of a
renewable resource
Assume our system considers only the two top
trophic levels, while the third or base trophic level
(M0) is considered fixed. Further, assume that the
renewable resource is managed for societal
3.2. The economic har6esting model
Potential harvesting capacity (D) and consumption (C) are control variables determined by the
quest for profit or utility maximizing of individual
humans in economics, and fitness and adapted
maximization in biology (Gutierrez et al., 1997).
The parameter D in primal humans was small,
and became a control variable in modern societies. D may increase as firms seek to satisfy
expanding markets. The parameter a in h(s) is a
technology parameter that for primal humans was
small but may approach unity for highly efficient
modern harvesters. Thus, D · h(s) is the individual’s production function for search capacity D.
Our assumptions imply that h(s) satisfies the concavity and positive marginal productivity of the
control variables (c,D) as it is increasing with the
resource level (x), and decreases with competition
from the other users (y).
3.3. Societal optimization
The economic model Eq. (3) differs from the
original biological model Eq. (1) in two ways; first
consumption (C) is introduced as an additional
form of expending energy, and second, a positive
discount rate (d) is incorporated into the human
U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
232
maximization process. Both terms have biological
analogs and are important components of the
biological (or primal human) model (Gutierrez et
al., 1997). In economics, dividends or consumption (C) rewards the individual agent, firm or end
user, and this reward (benefits) is conventionally
denoted by a monotone increasing concave utility
function U(C) that saturates. Thus the far-sighted
individual seeks to maximize the present value of
utility over an infinite time horizon, and hence the
long-run objective function of the society consisting of y individuals is assumed to be:
max
D,c
&
e − d · t yU(C) dt
(4)
where G is defined in the Appendix A, and C*(d)
is an increasing function of d. Since x; and l: 1
do not depend on y, we focus our attention on
them. As equations are defined only for l1 5
U%(C*)(u − n), we restrict attention to [0,] ×
[0,U%(C*)(u −x 2)] in the (x, l1) phase space. The
results summarized in Fig. 2a are derived in Appendix A. The definitions of the landmarks in this
figure are:
(a) xd is the level of x where its regeneration
rate g%(xd ) equals the discount rate d.
(b) xu is unexploited carrying capacity defined
by g(xu)= 0, and xu \ 0.
(c) xs is the optimal steady state resource level.
0
subject to Eq. (2) and Eq. (3), where e − dt is a
discounting factor. By Pontryagin’s maximum
principle, the maximization of Eq. (4) subject to
Eq. (2) and Eq. (3) is equivalent to maximization
of 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, associated with the constraints Eq. (2) and Eq. (3),
respectively. Necessary conditions for an optimal
solution (if one exists) are1:
(i ) HD = 0, HC = 0
(ii ) l: 1 =dl1 − Hx
(iii ) l: 2 =dl2 −Hy
(6)
(i6) tlim
e − dt li (t)= 0
“
Using Eq. (6), Appendix A shows the optimal
solution 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)
1
Subscripted functions denote partial derivatives, e.g.
Hx =(H( · )/(x, and prime denotes the first derivative of a
single variable function, e.g. g%(x) =dg(x)/dx, or l: i (t)= dli /
dt.
Fig. 2. Phase diagrams of the bio-economic model: (a) in the
(X,l1) space; (b) the effect of increasing the discount rate d on
the general results in (a) (dotted lines denote isoclines with
higher a); (c) the effect of increasing the technology parameter
a (dotted lines denote isoclines with higher a); (d) the effect of
increasing the efficiency of processing parameter u; (e) the
effect of increasing the cost parameter n; and (f) the effect of
eroding the resource base.
U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
Fig. 3. Steady state of human population.
(d) xc is the competitive solution for the resource level x.
(e) l *1 is shadow price of the resource x (i.e., the
marginal 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 level
satisfying g(xc)= −axc[h(G(0))/G(0)] (see below).
The resource level xd is the lower bound on the
steady state value of x obtained in the societal
solution. Notice that xu \xec \xd, and xs \xc
always holds, but xd \xc holds only if d is sufficiently small and a is sufficiently large.
The implication of the above solution for the
optimal paths of the resource x and population of
harvesting units y is illustrated in Fig. 3 (Appendix B). This equilibrium solution for xs and
y*, denoted as the societal solution, can be viewed
as an optimization process for a human society
regulated by a ‘benevolent dictator’ whose objective is the long-run maximization of utilitarian
welfare, and is in sharp contrast to the solution
regulated exclusively by the competitive market.
3.4. The competiti6e market solution
The institutional framework of competitive
markets implies that the market solution will
force resource levels to xc as the long-run equilibrium (Appendix B). In a competitive framework, individuals maximize their own utility
function (related to consumption), disregarding
233
all harvesting effects of depletion of the free access resource. The competitive equilibrium solution implies that the price l1 = 0, and the optimal
solution is xc (Fig. 2a). This ignores biological
reality that there may be a critical level of x that
leads 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 and
competitive solutions xs and xc, and hence the
larger is the chance that a competitive solution
will lead to extinction. Examination of Eq. (B1)
and Eq. (B2) in Appendix B shows that the slope
of x; = 0 in the (x···l1) space is lower when g(x) is
less elastic and h(G) is more elastic; that is when
resource regeneration is relatively slow, and the
proportion of the demand satisfied is close to one.
Below we examine the sensitivity of the solution
to changes in parameter values. The proofs of
these results are given in Appendix C.
4. Sensitivity analysis
4.1. Increasing discount rate d
Fig. 2b shows that keeping all other parameters
constant, increasing the discount rate d reduces
the societal steady state solution xs but the level of
xc is unaffected. The solid lines indicate the original isoclines and the dashed lines indicate the
isoclines for a larger value of d (Appendix C). As
we have noted before, xc equals the limiting societal solution when the discount rate increases
without 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 y
decrease because C*(d) increases and the payoff
decreases. The decrease in the population of human firms is mitigated by the effect of increases in
the discount rate that raise the individual’s rate of
resource exploitation. The lower bound xd on the
societal solution suggests that if the discount rate
(d) is sufficiently small, technological improvements should not drive a publicly regulated resource to extinction. Conversely, if the discount
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U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
rate becomes too large, the resource may be
driven to extinction even under a socially optimal
policy. It is generally recognized that the discount
rate is determined in the macro economy and
ignores its potential impact on a specific renewable resource. This decoupling may be an important factor in depleting resources, possibly leading
to their extinction.
Gutierrez et al. (1997) argue that the biological
discount rate is based upon the uncertainty of the
environment (i.e. expected hazards). It may be
large or small, and its effects on resource exploitation rate are in the same direction as the discount
rate in economics. So an obvious question is why
don’t primitive humans that live in a precarious
environment over exploit their resources or do
they? The answer lies in part on technical
progress.
4.2. Technological progress
The technology parameter (a) determines what
proportion of the resource can be exploited. In
general, if a is sufficiently low compared to the
maximum biomass regeneration rate of the resource, the resource will survive any size predator
population and demand rate (Gutierrez et al.,
1994). This parameter may also include social
constraints where individuals limit their capacity
to harvest (a form of private ownership). The
technical capabilities (a) of primitive societies
were small, and in our whale example the impact
was also small. However, as technology progresses, 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 on
the two isoclines in the phase diagram in the
(x···l1) space. The solid lines indicate the original
isoclines and the dashed lines indicate the isoclines
for a larger value of a (Appendix C). Under a
competitive market structure, technology that increases A reduce xc and increase the likelihood of
driving the resource x to xe. This result is in sharp
contrast to the conventional view that technological improvements are the driving force behind
increased income per capita and supports population growth. However, in our model when public
institutions take control of an endangered re-
source, increasing A increases the steady state
resource value (l1), and the resource equilibrium
level is reduced. This occurs because an increase
in harvesters results due to technological
improvements.
In a competitive economy, technological progress alone is potentially sufficient to drive the
resource population to extinction, independent of
the well-known effects of increasing the discount
rate. As discussed above, increasing the technology of harvesting (a) can also drive a resource
under public control to extinction, but this also
depends on the discount rate d. Thus, a synergistic combination of high technology (a) and a high
discount rate (d) may greatly increase the risk of
over-exploitation and resource extinction even under a socially optimal policy. As shown in Appendix C, the payoff and the steady state
population of y increase with a. The increase in y
has an additional negative effect on xs of increasing a.
4.3. Lower wastage
A higher u implies lowers wastage, and for the
competitive solution may drive the resource population quickly to extinction (Fig. 2d). Increasing u
does not affect xd, which is the lower bound on
the societal solution. The competitive solution
moves to the left, so that reducing waste has a
similar effect to that of improving technology. In
a competitive framework, the resource level may
be driven closer to extinction. The change in the
optimal societal solution xs is undetermined, but
l1 increases as wastage is reduced (u increases).
However, increasing u may reduce xs. If the social
discount rate d is sufficiently low, xd will remain
sufficiently distant from the extinction level. If,
however, d is sufficiently high, lowering wastage
may in fact put the resource at the risk of extinction as xc “ xe. u like a is a parameter of efficiency
and operates in the same direction, and hence
increasing either raises the risk of over exploitation and possibly extinction if the discount rate is
high enough.
However, the effects of u are constrained by the
value of a, and are best seen by reference to
primitive societies which are thought to waste
U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
little of their harvests. Despite a higher u, their a
is typically low, hence regardless of their perception of environmental uncertainty, efficient utilization of resources may have little impact on the
renewable resource.
4.4. Higher maintenance costs n
An increase of n reduces l1 and the payoff and
consequently reduces the optimal population density. 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 approaches
u, xs approaches xu and marginal gains fall (i.e. it
is no longer cost effective to harvest) and suggests
that the societal level xs increases with n. From a
policy point of view, one may interpret an increase in n as a tax levied on harvesting capacity,
suggesting effective policy implications for preserving resources.
4.5. Eroding the resource base
The resource base for x is often eroded by
human activities including those used to harvest
the resource x. Decreasing the resource base
parameter (M0) has the obvious effect of shifting
x; to the left and l: 1 downward, implying lower
competitive and societal solutions. In addition,
the payoff and optimal population density increase (Fig. 2f). Clearly, there are synergistic effects of reduced environmental carrying capacity
and factors enhancing over exploitation increase
the likelihood of resource extinction.
5. Conclusion
5.1. Species 6ersus human optimal beha6ior
The time evolution of an ecosystem is driven by
Darwinian selection processes that determine
which individuals and species survive. All organisms in all trophic levels are part of the ecosystem,
and each has demand capacity for resource acquisition that operate within the bounds of its genetic
code — its objective is to perpetuate the survival
of its DNA (sensu Dawkins, 1995). As argued in
235
Gutierrez et al. (1997), individual organisms in
nature behave as if they are driven by a quest for
utility maximization to increases individual
fitness. Although the resource has free access,
there is a biological price to it implied by its
effects on marginal growth that leads to resource
sustainability. Similar arguments could be made
for humans in their primal state.
However as modern humans became the top
predators, their objective became more than simply perpetuating the survival of DNA. In economic terms, the objective function became
maximization of the present value of a non decreasing utility of consumption by all individuals
subject to the resource constraints. In contrast to
the ecosystem, competitive market forces are
driven by individual quest for utility maximization disregarding their individual effects on natural resource depletion, and this determines zero
prices for free access (common property) resources. These differences strike at the heart of
the renewable resource exploitation problem as
practiced by humans.
5.2. Analogies between economic and biological
systems
Analogies between biological and economic systems enabled the development of a common
model for both systems (this paper and Gutierrez
et al., 1997), but some very important differences
between economic and biological systems exist.
Among these are the fact that the extraction
capacity (D), the transformation parameter (u),
the maintenance cost (n) and the resource apparency parameter a may change rapidly in response
to economic factors. In contrast, their biological
analogues are relatively ‘fixed’ on an evolutionary
time scale. In fact, the demand in economics
might exceed all of the resource that is available.
The parameter a is interpreted in economics as
the technology parameter of resource harvesting
and plays an important role in the dynamics
leading to resource extinction. In the economic
model, a higher value of a means that the technology for exploiting previously inaccessible portions
of the population is increased. A higher value of u
means that wastage is reduced and increasing n
236
U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
means that the per firm maintenance cost of harvesting capacity increases.
The concept of the discount rate (d) introduced
in the long-term objective function in the discount
factor (e − dt) also has a biological analog. In
economics, the discount rate reflects time preference, partly because of uncertainty about the future, and it is determined by market forces in the
whole economy. In biology, the analogous concept is the likelihood that investments in progeny
will survive to contribute to the genetics of future
generations (Gutierrez et al., 1997) and is implemented in the biological model as an ecological
discount factor in the objective function.
The concept of economic consumption also has
a biological analogue. In biology, some species
invest heavily in excess reproductive capacity and
other produce few progeny and invest more in
their care but these investments do not contribute
directly to growth (so called r- and K-selected
strategies, respectively, see Southwood and
Comins, 1976). These costs are, by analogy, consumption as defined in economics, but this allocation in biology is used to increase adaptivity and
not for hedonistic purposes.
5.3. The sustainable har6esting problem
Renewable resources (biological populations)
dynamics have spatial and temporal characteristics that affect their dynamics and those of populations that harvest them (i.e. firms).
Furthermore, if the environment of species is degraded, species may not be restored to its prior
abundance, and of course once a species has been
driven to extinction it cannot be brought back.
For these reasons, management policies for renewable resources must resolve questions that affect resource sustainability before the damage
becomes irreparable. Among these questions are
the optimal extraction rate, optimal human and
resource population levels, the appropriateness of
harvesting technology, and how market structure
affects harvesting behavior.
Competitive markets have failed to provide an
appropriate mechanism for pricing resources with
free access (Gordon, 1954; Hilborn et al., 1995),
and consequently, over exploitation of resources
has been a common practice in forestry and
fisheries because the cost of renewable resources is
largely neglected by harvesters. The consequences
of harvesting explored here using an micro-economic model that incorporated the dynamics of
both the resource being exploited and the exploiting population produced some conflicts with conventional economic wisdom.
5.4. Ecological conflicts with con6entional
economic wisdom
Economic growth theory has largely ignored
the relation between growth and natural renewable resources, and assumed population growth to
be exogenous (Barro and Sala-I-Martin, 1995).
Consequently, conventional economic models
have not incorporated the realistic biology of the
harvesting units extracting renewable resources
(Hilborn et al., 1995). Instead, neoclassical economic growth theory suggested that technological
improvements were the source of increasing per
capita income (Solow, 1956). The conventional
wisdom that follows is that increases in technology (including the discovery of new renewable
resources) enhance productivity and this maintains growth. The underlying biological realism of
our model, however, identified increases in harvesting and utilization technology as reducing
steady state resource levels and at the same time
increasing the number of users, in what would
appear to be a vicious cycle leading to over-exploitation of the resources and the collapse of the
industry.
The golden rule of economically balanced
growth (that maximizes the steady state per-capita
consumption) stipulates that the rate of saving
associated capital accumulation is obtained when
marginal productivity of capital is equated to
population growth and capital depreciation rates
(Phelps, 1966). Marginal productivity of capital
equals the interest (discount) rate, and this occurs
in our model as the lower bound of the societal
solution for resource exploitation xd. It is well
known that high discount rates increase the rate
of exploitation of natural resources. Since the
discount rate is determined by the market in the
U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
whole economy, this value may be higher than the
socially optimal level for the environmental
preservation (Weitzman, 1994), and specifically
for maintaining a particular renewable resource at
a sustainable level. The socially optimal level of
the resource (xs) is that which is sustainable for
the optimal population of firms (y*). It has been
shown here that there is a synergistic effect between improvements in harvesting technology and
high discount rates that encourage faster exploitation of the resource, possibly leading to its
extinction.
The conventional wisdom for reducing the
wastage of harvest is that resources are preserved
if their utilization is improved. However, our
model shows that as wastage is reduced, the optimal steady state level of the resource is reduced,
again contradicting common wisdom. This effect
increases with increasing discount rate, and is
similar to the synergistic effect between technology and discount rate. This occurs because the
payoff for the firms increases following better
technology and lower wastage. Increases in the
cost to firms detract from growth, countering
gains from decreases in wastage, and thus leading
to higher resource levels. A way of increasing cost
is the use of Pigouvian taxes on firm capacity (D)
suggesting interesting policy implications. Of
course, restricting harvests may be efficient but
often hard to enforce as harvesters find ways to
circumvent regulations.
In modern economies, the resource base may be
increased, as is done in agriculture by improved
production methods, new varieties or breeds of
animals, more agronomic inputs, etc. but these
may also lead to unforeseen adverse consequences
(van den Bosch, 1978; Kenmore et al., 1985).
While privately owned agricultural systems are
highly managed, free access renewable resources
are overused inefficiently and productivity may be
lowered or destroyed as competing firms seek to
maximize profits while ignoring renewable resource depletion and environmental costs. This
is verified by our model where all of the steady
state levels in our system are reduced as the
base resources for the exploited population are
eroded.
237
5.5. Policy implications
All of the above results accrued via the dynamics of the biological model, rather than by a priori
assumptions. Our model points to the need to
simultaneously control technology and the discount rate. It is obviously impossible, nor is it
desirable to regulate the advance of technology,
hence the major option left is to reduce the discount rate below the market equilibrium and also
to regulate the harvest. If a society considers the
preservation of an environmental resource important, the social discount rate should be lower than
the private one (Weitzman, 1994). Regulation of
harvest can be accomplished via a Pigouvian tax
on the capacity of the firm which in our model
drives down firm numbers, but leaves open the
question of increasing size of the remaining firms.
In all cases, what is absolutely clear is that total
harvest by all firms should be only that level that
assures the sustainability of the resource at equilibrium density. This could be done without taxes,
but would require strong enforcement and sound
notions about the maximum sustainable yields
(MSY) — weak assumptions about carrying capacity will only lead to still more resource exploitation disasters. As pointed out by Hilborn et
al. (1995), the notion of MSY is unrealistic as
natural populations fluctuate (often widely) in
response to drastic changes in biotic and abiotic
factors. The possibility of over exploitation leading to extinction is more likely when stochastic
perturbations affect the resource (El Niño effects
on Pacific fisheries); however, this issue has not
been tackled in the present paper.
The dual goals of economic growth and ecosystem sustainability are often in conflict (Goodland,
1995). Impetus for resolving some of these issues
comes as the standard of living improves; this
despite the apparent contradiction that the improvement in a large part may have resulted from
resource depletion. Increasingly affluent societies
demand improved environmental quality leading
to public pressures for environmental regulation
via market and non-market mechanisms. On a
larger global scale, however, the difficulty lies in
the recognition that improvement in the standard
of living in heavily populated less developed coun-
238
U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
tries would certainly lead to over exploitation
of fragile natural resources as technology and
the market interact to satisfy ever increasing consumer demands (Goodland, 1995). Sustainable
development must include viable environmental, social and economic sustainability but
not necessarily the sustained economic throughput growth that is the basis of the common worldwide economic paradigms (sensu Goodland,
1995).
5.6. Epilogue
In this paper we examined harvesting from a
food chain, but humans may harvest more than
one resource in the food chain (or web). For
example, humans harvest both whales and krill,
and we might ponder what the impact of interacting economic and technological parameters might
be on the system — will whales be placed in
greater danger by krill harvesters than whalers.
Schreiber and Gutierrez (1997) use the underlying
basis of this biological model to examine biological interactions in food webs, and demonstrate
how species displacements have occurred in several systems. This model can be used as the basis
for examining human harvesting of several
trophic levels in a renewable resource systems.
The questions of physical and human capital are
Lamarkian like processes that also impact renewable resource problems, but they were not
addressed in this paper, nor do we address
how big should firms be or how human capital
drives technology, and in what direction?
Can human capital be the basis for developing
viable renewable resource management schemes,
or will it simply contribute more to over exploitation?
Clearly, ecology and economics are at a crossroads of conflict: the alternatives are sustainable
renewable resource management based on sound
biology, or will over exploitation and mutual
annihilation result as we scramble for ever decreasing resources. If the latter is our fate, then
‘‘… as a final bit of irony, it will be insects that
polish the bones of the last of us that fall.’’
(Robert van den Bosch, 1978).
Appendix A
In this appendix, we derive equations of motion
for our maximization problem using the Pontragin maximum principle. Recall, the current
value Hamiltonian for this problem is given by
H= U(C)y+ l1(g(x)− F(x,y,D))
+ l2(uF(x,y,D)− (nD + C)y) where l1 and
l2 are the costates associated with x and y and
F(x,y,D)= Dyh(ax/Dy). The optimal solution
must 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, consequently, 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),
l2n
ax
= Z−1
l2u− l1
Dy
(A1)
Furthermore
HDD = FDD (x,y,D)(ul2 − l1)5 0
implies l2u] l1 on the optimal path.
The equation of motion for x’s costate is given
by,
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’s
costate 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) and
Fy (x,y,D)= DZ(ax/Dy). Putting this all together,
the equation of motion for candidate solution to
our maximization problem are given by,
x; = g(x)− ax
y; =
h(G)
G
(A2)
ax
(uh(G)− n)− Cy
G
(A3)
1
((d + C)U%− U)
U %%
(A4)
C: =
U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
l: 1 = l1(d −g%)−ah%(G)(uU% −l1)
where
G =Z − 1
(A5)
U%n
U%u −l1
(A6)
(x;
h%(G)G −h(G)
= − ax
G%(l1) \0
(l1 x; = 0
G2
(B1)
)
(x;
h(G)
g(x)
=g%(x)−a
=g%(x) −
B0
(x x; = 0
G x; = 0
x
(B2)
where the second equality in Eq. (B2) follows
from the concavity of g. Therefore, (l1/(xx; = 0 \
0 and the x null isocline is a strictly increasing
function of x.
To prove the monotonocity of the l1 null-isocline, note that
(B3)
is greater than zero whenever (l1,x)[0,U%(u −
n)]×(xd,) where xd is the unique solution to
g%(x)= d if it exists else zero. On the other hand,
(l1/(x = −l1g%%(x)\ 0 by convexity of g.
Therefore
)
To find the equilibrium of Eq. (A2), Eq. (A3),
Eq. (A4), Eq. (A5), Eq. (A6), we first solve for
C: = 0. We begin with three observations: (/
(C (U%(C)(d +C)− U(C)) =U%%(C)(d +C) B0 as
U is concave; limC “ 0 U%(C)(d +C) −U(C) \0 as
U(0) = 0 and U%(0)\0. limC “ U%(C)(d +C) −
U(C)B0 (this follows from our assumption that
limC “ U%(C)=0). These observations imply
that there is a well defined differentiable function
C*(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 on
y, the remainder of our isocline analysis is restricted to the x −l1 plane with C =C*(%). As
Z − 1 is only well defined on the interval [0,1], we
further restrict our analysis to (x,l1) [0,) ×
[0,U%(C*)(u −n)).
To prove monotnicity of the x nullcline, we first
observe that G%(l1)=Z − 1( · )(U%n)/(U%u− l1)2 \0
and h%(G)G−h(G)B 0 by convexity of h.
Therefore,
)
(l: 1
= d−g%(x)
(l1
− a[h%%(G)G%(l1)(uU%− l1)− h%(G)]
Appendix B
)
239
(l1
B0
(x l: 1 = 0
(B4)
whenever x\ xd. As x; B0 for any x\ xd, it follows that the entire l1 null isocline lies to the right
of x= xd and is strictly decreasing in x.
Putting this all together, we get the phase diagram shown in Fig. 2a. In this figure, xc is the
largest value of x such that g(x)= ax[h(Z − 1(n/
u))]/[Z − 1(n/u)] and xu is the largest value of x
such that g(x)= 0 (the equilibrium achieved by
the resource in the absence of harvesting).
Note: The level xc is also obtained as the solution of individual decision making in a competitive 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
and
LD = 0 imply that ax/Dy=Z − 1(n/u). Inserting
this feedback rule into x; and solving for its largest
equilibrium determines xc.
To determine the stability of (x*,y*,C*), the
optimal equilibrium, we evaluate the variational
matrix of the equations of motion at this point.
Using Eq. (A4) and Eq. (B1), Eq. (B2), Eq. (B3),
Eq. (B4), we get
Á (x;
(x; Â
(x;
à (x l
(C Ã
1
à :
à Á−
(l (l: 1 (l: 1
à 1
Ã=Ã+
(x (l1 (C
Ã
à Ä0
:
:
:
(C
(C
(C
Ã
Ã
Ä (x (l 1 (C Å
+
+
0
Â
Ã
+Å
where − /+ indicate the sign of the entry and indicates that the sign is irrelevant. Since (0,0,1) is
an unstable eigenvector for this matrix, along the
optimal path C(t) C*. Furthermore, as
240
det
−
+
U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
+
B0
+
there is an unstable and stable eigenvector in
x − l1 space. Hence along the optimal path l1(t)
and x(t) are monotonic and l1 is determined by a
feedback rule l1(x) that satisfies l %1(x) B0.
Using Eq. (A2) and Eq. (A3), we examine the
x − y phase for the optimal paths. The y null
isocline is defined by
cy =
ax
(h(G)− n).
g
(B5)
Taking the derivative of the right hand side of Eq.
(B5), we get
a
(uh(G)−n)+
G
(B6)
ax
(uh%(G)G− uh(G) +n)Gx
G2
(B7)
On the y nullcline, Eq. (B6) is strictly positive and
Eq. (B7) equals
U%(C*)n
ax
n−u
2
U%(C*)u −l1(x)
G
Gx.
Since l1(x)[0,U%(C*)(u −n)] and Gx B0 this
term is also strictly positive. Therefore, the y
nullcline is stricltly increasing with respect to x.
Notice that certain initial conditions of x and y
produce a ‘snowballing’ effect such that y is not
monotonic in time.
Appendix C
In this section, we examine how parameters
effect the isocline structure in x − l1 space, the
payoff, and the optimal equilibrium value of y.
Unfortunately, our conclusion with respect to x*
can be only speculative and are based on numerical simulations.
First, consider a. Notice that (x; /(a = −
x(h(G))/G B0 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 with
a, consider the optimization problem defined by
Eq. (A1) subject to the constraints Eq. (A2) and
Eq. (A3) plus the addition constraint a; = 0. Let
v3 be the present value costate associated with
a.The equation of motion for this costate is given
by − v; 3 = xh%(ax/Dy)(uv2 − v1) where v1 and
v2 are present value costates associated with x
and y, respectively. Integrating with respect to t
and using the transverslity condition on v3, we
get v3(0)= 0 xh%(ax/Dy)(uv2 − v1) dt\ 0 where
x, D, C and y are evaluated on the optimal path.
Since (under the assumption that the payoff differentiable with respect to a) v3(0) equals the
derivative of the payoff as function of the a, the
payoff increases with a.
Consider the parameter u. Notice that Gu B 0
and therefore (x; /(u = −ax(h%(G)G−h(G))Gu /
G 2 and (l: 1/(u = −ah%%(G)Gu (uU%(C*)−l1)−
ah%(G)U%(C*) are strictly negative (Fig. 2d). As
with the payoff analysis for a, the payoff increases
as a function of u.
Consider the discount rate, d. C*(d) and xd are
increasing/decreasing functions of d and Gd B 0.
Therefore (x; /(d = − ax(h%(G)G− h(G))Gd /G 2 is
strictly positive. Unfortunately, we are unable to
draw a conclusion about the motion of the l1
nullcline. However, the motion of the xd and
simulations suggest that it is as indicated in Fig.
2b. A routine calculation shows that the payoff
decreases as a function of d.
Consider the parameter, n. We have G%(n)=
Z − 1%( · )(U%(C*))/(U%(C*)u− l1) is strictly positive. From this it follows that
(x;
h%(G(n))G(n)− G%(n)h(G)
= − axG%(n)
(n
G(n)2
and
(l: 1
= − ah%%(G(n))G%(n)(uU%(C*)− l1)
(n
are strictly positive by concavity of h (Fig. 2e).
These facts imply l1 at the optimal equilibrium
decreases as a function of n. The payoff also
decreases with n. Although we can draw no general conclusions about the effect of n on the
equilibrium resource, we can show that as n approaches u the equilibrium resource density approaches xu. This suggests that resource density
increases with n.
Eroding the resource base is equivalent to replacing the function g(x) with another function
U. Rege6 et al. / Ecological Economics 26 (1998) 227–242
g̃(x) such that g̃%(x)B g(x), g̃(0) = 0 and g̃ is
concave. Such a change only effects the x zero
growth isocline by shifting it left (Fig. 2f). Hence,
decreasing the resource base decreases the resource density and increases l1 at the optimal
equilibrium. It is easy to check that it also decreases the payoff.
To conclude, we want to understand how y*
changes with respect to the parameters. To do
this, we need the following lemma.
Lemma. Let h be one of the parameters of the
model (i.e. a,u,n or d). For any h̃ sufficiently close
to h, there exists an initial condition, (x0,y0), such
that the optimal paths y(t) and ỹ(t) associated with
h and h̃ that satisfy this initial condition are both
monotonically decreasing.
Proof. Given h, define the sector Sh =
{(x,y):x; (x,y)\0 and y; (x,y) B 0}. Since the x
nullcline is vertical and the y nullcline is
monotonically increasing with respect to x,y(t)
for all optimal paths with initial condition in S
are monotonically decreasing. By continuity there
exits a e\ 0 such that for all h̃ e-close to h, the
sectors, Sh and Sh̃ have a nonempty intersection. Choosing a point, (x0,y0), in this intersection
provides the desired initial condition.
Now, consider a. When a increases, the payoff
for any initial condition of x and y increases but
C(t)C* remains constant. Assume we are given
ã \a \0 such that ã −a is sufficiently small. The
lemma gives optimal paths ỹ and y with the same
initial conditions such that both paths are
monotonic in time. Since the payoff associated
with ã is larger than the payoff associated with
a, ỹ(t)]y(t) for all t ]0. Consequently, y* increases with a. Similarly, we can conclude that y*
increases with u and increasing resource base, but
decreases with d and n.
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