Civilizations and Optimal Social Collapse
David H. GOOD* and Raphael REUVENY*
*School of Public and Environmental Affairs, Indiana University, Bloomington, Indiana, USA
email [email protected]
ABSTACT
Scholars have linked the Malthusian trap to the collapse of historical civilizations such as the Maya and
Easter Island. Others model population-resource dynamics, assuming individuals do not act collectively,
have open access to resources, and lack foresight and resource management institutions. These models
can generate boom-bust cycles representing flourishes and collapses of civilizations. Scholars have also
suggested that institutions could have prevented these collapses. These institutions imply that actors are
forward looking, and consider effects of current actions on future outcomes. We wed the no-foresight
approach to features of economic growth theory, adding endogenous population growth and resource
carrying capacity to a growth model by way of a long lived social planner or a system of property rights.
Unlike the bulk of the growth literature, we focus on the transition path to equilibrium and the
equilibrium, rather than only on the equilibrium. It is easy to say that civilizations collapsed because
they were shortsighted, did not have appropriate institutions, or were primitive. We find that collapses
might be socially “optimal.” Our work compares alternative social welfare functions and indicates that
some are much more prone to collapse.
JEL Classification: Q20, D90, J10 and N57
Conference categories: (1) Commons (2) Political Economy
1. Introduction
Malthus (1798) believed that population growth will eventually lead to man-made natural
resource depletion, conflict, and population decline. Several authors (Ponting, 1991; Diamond, 2005)
have applied Malthus to the decline of historical civilizations such as the Sumerians, Anasazi, Maya, and
Easter Island. Others authors (e.g., Clark, 1990, Brander and Taylor, 1998) have modeled the populationresource nexus in the spirit of the predator-prey model of Lotka (1925)and Volterra (1926). This model
can generate boom-bust cycles, which represent the flourishing and collapse of civilizations. The model
assume that people do not act collectively, have open access to resources, and lack foresight and resource
management institutions. With these assumptions, they fall into the Malthusian trap. Brander and Taylor
suggest that institutions such as property rights and markets, or optimal central planning could have
averted the collapse of Easter island - and other civilizations by implication. These institutions imply that
actors are forward looking, and consider the effects of current actions on future outcomes.
We extends the no-foresight population-resource model by wedding it to modern economic
growth theory. Specifically, we add endogenous population growth and fixed capacity resources to an
economic growth model, or alternatively we add resource management institutions by a social planner or
private individuals with property rights to the story of the “tragedy of the commons” (Hardin ,1968;
Ostrom, 1990, 1999; and Dietz et al. ,2003). We incorporate institutions with infinite foresight through
the use of optimal control techniques, and consider two alternative social welfare functions. One
function is based on the utility of a representative individual. A second function is based on the total
utility of society. Traditional economic growth models do not highlight distinctions in these two
approaches by their use of an assumption that population growth is exogenous. In our situation,
endogenous population growth is key. A second distinction of our paper is that we focus on the time
paths for important variables rather than simply examining steady states values.
It is tempting to say that historical civilizations failed because they were short sighted, failed to
have institutions, or did not understand the forces that doomed them. Our results suggest that the collapse
of these civilizations was inevitable. Even if actors had complete property rights and perfect markets and
utilized optimal resource management with infinite horizon, or even if a benevolent social planner
managed their resources optimally, they would have still exhibited boom-bust cycles. These results hold
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regardless of whether the social welfare used is the utility of a representative individual, or the
civilization’s aggregate utility. The boom bust cycle obtained from using the aggregate utility exhibits
higher peaks and lower valleys compared with the one obtained from the utility of individual agent.
Our paper implies that neoclassical methods of managing population-resource relationships in a
system with a carrying capacity, where population reacts endogenously through Laissez-faire
consumption-induced changes in fertility are likely to generate boom-bust cycles. It should be noted that
the term “resource management institutions” used here captures institutions set to control the resource
stock over time. This could include central planning, assignment and enforcement of property rights to
individuals, or establishment of norms. While we do not study institutions such as population control,
which could have changed the fate of civilizations, we return to this issue in the last section.
The next section tells the stories of four civilizations that collapsed. Section 3 discusses common
threads in these collapses. Section 4 presents our model, and Section 5 presents our solution results.
Section 6 discusses implications for modern societies, and concludes the article.
2. The Collapse of Four Societies
We begin with the collapse of the Easter Island civilization, which is an interesting case to study
since it remained isolated for nearly 1400 years. Scholars believe in the first millennium few Polynesians
arrived there. They created a thriving agrarian civilization. The population thrived for several hundreds
years, peaked at about 7,000-20,000 people, and then declined rapidly. When the Rapanui (islanders),
arrived they found a lush forest. However, when the island was discovered by Europeans in the 18th
century, its civilization had all but disappeared and the island was nearly barren. Its 2,000 or so people
lived in extreme poverty, fighting over the few remaining resources.
Many scholars argue that the collapse of Easter Island exemplifies the Malthusian Trap. The
islander depended on the environmental for livelihood, both as a source of plants for food, and for wood
to build boats for fishing. The islanders used the tree trunks as primitive wheels, on which they rolled
the very heavy huge statues they built to their final locations. The method taxed the forest. Once the
forest disappeared, the top soil eroded and the land degraded. Unable to build boats, the islanders also
lost the ability to fish. As the amount of food available fell, the island’s population declined, and its
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social structure broke down (Flenley and Bahn, 2002; Diamond, 2005; Brown and Flavin,1999; Kirch,
1997; Bahn and Flenly,1992; Ponting, 1991; Weiskel, 1989). The story of Easter Island is probably the
most famous, but it is not unique. Other civilizations also exhibited “gradual emergence, brief flowering
and rapid collapse” (Weiskel, 1989: 104). We also examine the Sumerians, Maya, and Anasazi.
The Sumerian civilization arose in the fertile valley between the Tigris and Euphrates, in area
which today is part of Iraq. It is generally considered to be the world's first literate civilization, having
attained literacy by about 3000 BC (Tainter,1990; Ponting, 1991; and Thompson, 2004). The Sumerian
civilization was comprised of a number of cities that utilized the land separating them for agriculture. In
2500-2100 BS, its urban population size peaked at 200,000-300,000, and then declined to about 25,000 in
1500 BC. The sharp decline was precipitated by environmental decline. To increase the productivity of
the land in the generally arid climate of the region, the Sumerians developed a complex system of
irrigation that brought water from the rivers to their fields. With irrigation, the Sumerian civilization was
able to move from a status of subsistence farming, to a status of growing cash crops traded within the
civilization and with non-Sumerian societies in return for things such as metals and the manufactured
goods of the period.
Seeking to increase wealth by growing more cash crops, partly because they had to maintain
armies to defend this wealth, the Sumerians increased land utilization by constantly irrigating it. They
also abandoned the techniques of crop shifting and allowing lands to lie fallow. In the arid climate, the
constant irrigation led to salinization of most lands, making them unuseable. The decline in output led to
loss of cash crops, weakening the civilization. As the salinization progressed, the society lost essential
harvests, fertility fell, health declined, death rate rose, and civic order deteriorated. The Sumerians
became weak and were conquered in 2370 BC by the Akkadians. The story just told illustrates the role of
short sightedness and inability to understand forces of societal collapse. The intense irrigation increased
wealth in the short-term, but damaged the environment. In 2400-2100 BC, crop yield fell %42. In the
next 400 years, it fell another %65. An observer in 2000 BC writes “the earth turned white (Ponting,
1991: 72),” indicating the extent of the damage; the Sumerians were doomed.
The story of the Maya also is old, dating back from 2500 BC. Situated near and within a lowland
tropical jungle, the Maya world was located in Southern Mexico and Northern Guatemala (Tainter, 1990;
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Ponting, 1991; and Diamond, 2005). At the peak of the Classic Maya period (AD 500-800), the
population of the Central Peten region, the social core, was between 3 and 14 million. When the Spanish
arrived in the 16th century, there were only 30,000 people in the region. The collapse of the Mayans is all
the more amazing, considering that remarkable civilization they created. The Mayans were technical and
literate. Their technology focused on agriculture and food productivity, irrigation and water systems,
astronomy, the long calendar, and large scale architecture. In AD 600, they began building spectacular
pyramids and monuments, but only 200 years later, their civilization began to disintegrate.
Like the Sumerians, the Mayans used land efficiently. The growing food production promoted a
population boom. Eventually, the intensive cultivation system could not keep up with the demands placed
on it by the growing population and the elite wishes for ceremonial buildings. To satisfy both, more
forests were cleared. As a result, the top soil eroded and the land lost nutrients and degraded (Diamond,
2005). Some eroded soil ended up as silt in the rivers and canals, damaging the irrigation system devised
to increase food production. As land degraded, new, more marginal lands came into use, and crop yields
gradually declined. Public health and social order deteriorated and the population began to fall. By the
10th century, the Maya civilization was virtually gone. All that remained was the 1839 lament of its first
external visitor, the American John Stephen: “Nothing ever impressed me more forcibly than the
spectacle of this once great and lovely city, overturned, destroyed, and lost, discovered by accident,
overgrown with trees for miles around, and with out even a name to distinguish it (Stephen, 1841).”
The Anasazi civilization in the southwest US was much smaller than the Maya and Sumerian
civilizations (Betancourt and Van Devender, 1981; Samuels and Betancourt, 1982; Betancourt, Dean,
and Hull, 1986; and Diamond, 2005). Yet it was a resourceful and geographically extensive society,
which erected the largest buildings in pre-Columbian America and controlled a relatively large area in
northwest New Mexico and southwest Colorado. In its three major sites in Chaco Canyon (including the
dazzling Pueblo Bonito), which essentially served as the Anasazi capital, the local population is said to
have peaked at between 4,400 to 10,000 individuals (Tainter, 1990; BLM, 2005).
In Chaco Canyon, the Anasazi prospered beginning at 600 AD. As their number grew, food
pressure was partially relieved by building outlying settlements, where peasants grew food for the center.
Seeking land for food and timber for building, the Anasazi intensified deforestation. By AD 1000, the
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trees were gone and soil erosion set in. Facing erratic rainfall, exhausted land, and low plant growth due
to the dry climate, the Anasazi developed gravity-propelled irrigation. The water cut arroyos in the
ground. When the water levels in the arroyos fell below the field levels, agriculture had to stop. With
declining food, the population fell. Finally, a drought hit in AD 1130. Strife and hunger set in and by AD
1200, Chaco Canyon was virtually abandoned.
3. Sources of Societal Collapse
While the previous section examined only four societies there are many more cases that had
similar fates: the Indus valley, the Norse in Greenland, the Teotihuacan in Mexico, and Mangareva in the
Pacific. A number of scholars have put forward various causes of societal collapse (e.g., Diamond, 2005;
Brander and Taylor, 1998; Ponting, 1991; Tainter, 1990). These causes fall into several groups: (1)
Resource degradation; (2) the pursuit of irrational objectives such as the construction of large temples or
the maoi on Easter Island; (3) ignorance of the difficulty regarding their situation; (4) limited foresight
regarding future outcomes; (5) poor leadership that had a narrow definition of social welfare; (6) a lack of
resource management institutions.
Brander and Taylor (1998) argue that Easter Island was not a favorable place for creating resource
management institutions because the islanders were shortsighted, did not understand their problem, and
did not notice it as the resources disappeared slowly relative to their life-span. They argue that the
collapse could have been averted by using effective institutions to govern the island's resources, but they
do not investigate this claim. Similarly, Diamond (2005) concludes that collapses reflect four failures of
group decision-making: failure to anticipate outcomes, failure to perceive problems, failure to generate
solutions, and failure of the solutions they generate. Civilizations may fail to anticipate outcomes because
they had not encountered the problem before. Even when a problem is not new, people may not anticipate
its outcome, particularly when they are illiterate, like the Anasazi, and cannot pass efficiently data over
time. Literate people also may fail. The Maya’s records consisted only of the king’s actions and
astronomical events. Groups may incorrectly anticipate outcomes. The Vikings in Iceland used European
agricultural techniques, thinking that lush flora was like the one they left. Iceland could not tolerate these
techniques. More primally, short and long-term interests may clash; people tend to care more about the
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present than the future.
Slowly changing situation may be difficult to perceive especially when societies lack the ability to
transmit information across generations. Nutrient-poor soils plagued the US Southwest and other places.
Leaders are often located far from the problem and do not realize it. Other problems trend slowly, hidden
by fluctuations. Gradual changes and the problems that befall them are thus overlooked.
Why would societies fail to solve problems? Rational behavior calls for the advancing of one’s
own interests, regardless of others. People indulge in it when the benefits are large and immediate and the
loss is spread over many people and years, making retaliation unlikely. The “tragedy of the commons”
exemplify this phenomenon. The rush to use the resource before others can be pervasive, harming the
community. In addition, at times, the principal consumer of a resource has no interest in preserving the
resource, but society does. Clashes of interest also can occur between leaders and the masses, as leader
seek to profit themselves even if this hurts society, as occurred in our cases.
Irrational behavior, which harms everybody, also can cause failure. Religion and cultural norms
may create a set of values which are not in the long run best interests of society. The Rapanui deforested
their land, seeking to erect statues. The Christian values of the Greenland Norse and their orthodox
philosophy prevented them from adopting local hunting techniques that might have saved them. Other
reasons stem from psychology. People in a crowd may adopt lines taken by others. Janis (1983) describes
“Groupthink” by which leaders under stress suppress critical thinking. Lastly, people in state of panic or
grief may suppress correct perceptions of reality, seeking to avoid more painful feelings.
Even though civilizations may attempt to solve problems, their solutions may fail. Some solutions
may not be optimal, and some solution may exceed society’s capabilities. Other solutions may be deemed
too expensive as long as the damage is thought to be small. At times, a problem has gone on for too long
that anything done now to eradicate it is futile; it’s “too little, too late.”
These failures imply a failure to deal with open access, which neoclassical economic growth
theory assumes away. Assuming that population grows exogenously and utility in a period depends on
consumption in the period, agents with foresight – a social planner or a representative agent – choose a
consumption path that maximizes the sum of discounted future utilities. Open access, in contrast, leads to
the tragedy of the commons. Scholars suggest solutions to this problem (Smith, 1975; Ostrom, 1990). One
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solution changes preferences toward conservation. Other solutions involve institutions: user charges,
harvesting quotas, and property rights. Contemporary policymakers promote institutions. Observing that
global renewable resources have been depleted, World Development Report (2003) calls for institutions
with a long-run view that utilize all existing information and anticipate future problems.
The origin of institutions is debated (Nelson and Sampat, 2000). Some say they arise as a response
to needs. Others argue their evolution is unplanned and subject for inertia. In reality, institutions may not
be efficient, and the need for institutions may not lead to their emergence. Ostrom (1990) and others
conclude that institutions may not arise when gains and loses are vague, actors try to shift burdens of
adjustment to others and argue over the nature of the problem, some people are not sure they will gain
from change or may lose, people are shortsighted, and monitoring and enforcement are costly.
The literature suggests that optimal resource management institutions would have averted the
collapse of historical civilizations. While we do not extensively study the processes by which these
institutions could have evolved, we entertain the possibility that at least some of our civilizations were
aware of their growing problem, and perhaps tried to avert it. We examine what the effects of these
institutions would have been if they existed. On a naive level, one might argue that their civilizations
would not have failed from a lack of resources if they had resource management institutions. On the other
hand, there might be limits in what resource management institutions can do.
Beginning with Easter Island, since Polynesia consists of discrete civilizations with common
ancestry, it is possible to infer on one society by observing others (Kirch, 1984; Ferdon, 1981). Since
other Polynesian islands developed resource management institutions, it is plausible they also existed on
Easter Island. All over Polynesia, there was a resource management institution called rahui. Put in place
by a chief and supported by religion, it forbade resource harvesting. It could be imposed for ceremonial
purposes, but also for conservation. Anyone who disobeyed was killed (Kirch, 1984; Ferdon, 1981;
Williamson R. W., 1933). We know that the Rapanui had rahui prohibitions on the harvest of birds, eggs,
fish and crops (Lee, 2002; Metraux, 1971). Moreover, contemporary islanders believe that many plants
vanished from the island with the chiefs who controlled their harvesting (Metraux, 1971).
The claim that the islanders’ life span was short relative to the slow growth rate of the forest also
faces difficulties. Recent studies suggest that 15% of the islanders lived above 55 years, and 25% lived
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between 40-50 years (Hunt and Lipo, 2001, Shaw, 2001). With a typical population of about 6000, these
numbers suggest that nearly a thousand people would have been around to tell “when I was a boy...”
stories, even skipping generations from grandparent or even great-grandparents to grandchild. Last,
Stevenson et al. (2002) finds that as the island’s ground lost moisture, the islanders covered planted
grounds with small stones to contain water. Thus, they were aware of their growing problem.
The Sumerians also were aware of their problem. Their records tell the story of declining yields
and rising salinization. As land deteriorated, farmers moved to lands with lower quality, and the state
increased taxes facing a declining output (Tainter, 1990; Ponting, 1991). To circumvent a problem of the
water flows declining lower than the field levels, the Anasazi built dams across canyons, used more fields
that rain could irrigate, and stored rainwater coming down from over cliffs (Kohler and Mathhews, 1988;
Windes and Ford, 1996; Bull, 1997; Diamond, 2005). The Mayans tried to contain their soil erosion
problem by terracing hill sides cleared from trees, and constructing raised fields in swampy areas. Other
methods developed to increase declining food supply included irrigation systems, draining waterlogged
areas, mulching, fertilizing, flood-water farming, and growing fish and turtled in the canals (Sharer, 1977;
Turner, 1974; Turner and Harrison, 1981; Tainter, 1990, Pointing 1991).
Some historical societies succeeded in striking a balance between environmental demands and
damages, employing foresight. By 650 BC, Greece increasingly suffered from land erosion associated
with overgrazing and deforestation. By 600 BC, the Greek leader Solon sought to stop the cultivation of
the hill slopes in order to contain the problem. A few decades later, the Greek leader Peisistratus offered
monetary premiums to farmers switching from cultivation to planting olives trees on the hills, having
determined that only these trees could grow on the rocky ground, keeping the topsoil in place (Ponting,
1991; Beck, 2004). The Pacific island of Tikopia provides a second example. While it was also isolated
like Easter Island, its Polynesian society did not collapse. Kirch (1997) argues that the difference has to do
with society acting to ensure that population would not exceed a size deemed appropriate for the island’s
carrying capacity. The people of Tikopia implemented this policy by prevention of conception, sexual
abstention, abortion, one directional ocean trips of young males leaving the island, forcing out parts of the
population, and infanticide.
Finally, ancient Egypt exploited the natural yearly floods of the Nile, which covered the Nile
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valley, depositing silt. The Egyptians exploited this natural fertilization and did not try to change it by, for
example, building dams upstream. The Egyptian agriculture was a success story for thousands of years,
without suffering from problems such as salinization or land degradation. By the late 19th century, modern
agriculture methods brought by the British increased Egypt’s reliance on irrigation and on flood control by
way of building dams upstream. As a result, the natural land productivity fell, considerably increasing
Egypt’s reliance on manmade fertilizers (Butzer, 1976; Ponting, 1991).
These three historical societies apparently understood some of the connections between their
current actions and future outcomes, as well as implemented some form of resource management
institutions with a long term vision. More broadly, one may assume that humanity generally progresses in
unison. If some societies were able to acquire some understanding of resource-population processes, it is
not impossible that other societies, including those that collapsed, gained similar understanding and even
had some form of resource management institutions in place. Perhaps this was not enough to assure their
long term survival, a point to which we turn next.
4. Analytical Models and Solution Strategies
The previous section suggests that there are some cases where historical societies understood the
nature of their environmental problem, and might have had some form of resource management
institutions. The exact form of these institutions and their ultimate effectiveness is not fully known. The
objective we have in this paper is to determine something about averting societal collapse. Our modeling
approach is to consider two alternative types of social welfare functions. This first is used by the vast
majority of economic growth theory that describes social welfare as the sum of the utility levels for
everyone in society. The second approach represents social welfare with the average utility level or the
utility of a representative individual. Socially optimal policies are identified by solving an optimal control
problem that identifies the level of harvesting effort expended at each point of time in the future.
The crux of the issue we address is what the implications are for endogenous population growth
under alternative social welfare functions. As a limiting case, both these social welfare functions nest the
same solution when the discount rate approaches infinity. In either instance, with an infinite discount rate
the impact of current harvesting effort does not consider the impact on either the future resource or
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population levels. This situation describes open access resource use. It should be noted that we exclude
policies that many might consider to be inappropriate for our civilization today, though they were used at
least the Tikopia society discussed above. That is, we consider only policies of laissez faire population
growth. Our model incorporates the implications of resource management institutions for the future
population, though it does not attempt to directly manipulate or control its size.
When population growth is endogenous, philosophers such as Parfeet (1982) and Kavka (1982)
have identified the laissez faire population level as a morally defensible baseline. We consider the social
welfare function to be constrained by two human rights: to exist (it is inappropriate to end a life as a
matter of public policy for the benefit of others) and to procreate as individually desired (public policies
that explicitly limit family sizes are incompatible with human rights). These issues are intimately
entangled with some of the great debates of our time, including abortion, assisted suicide, manipulated
genetics, cloning, and extraordinary preservation of life. We do not hope to resolve them here. Instead we
take a neutral position that will not consider explicit population controls here. Choices which directly
affect population are the result of individual decisions, not public policy. Policies affecting future
populations are only indirect. Thus, for example, if families have more income and this lowers infant
mortality, the only way to affect future populations is by preventing income from rising too quickly.
To implement this, we solve an optimal control problem to find the level of resources harvesting
each period that maximizes the relevant social welfare. We start with a model that is similar to the one
developed by Brander and Taylor (1998), and reparameterize it in order to facilitate extension. We extend
this model to allow a social planner to have foresight, resource management institutions and the ability to
enforce their decision rules, or alternatively to allow a private agent to have property rights and foresight.
The structure of our model is consistent with the basic model used by the bulk of the economic growth
literature, with two distinctions: (1) the environment has a carrying capacity (an assumption to which we
return in the conclusion section); and (2) we consider endogenous population growth, which depends on
the harvested resource (or, alternatively, the income this harvesting generates).
Of course, we do not argue that historical societies used the mathematical tools employed here.
We contend only that they could have used a set of resource management and conservation institutions
that were transmitted across generations and enforced though customs, norms, beliefs, and or traditions,
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perhaps with the aid of religion or a dynasty of chiefs. We have provided some evidence that it was not
beyond the capability of these historical societies to understand basic human-environment relationships
and consider the future in their decisions. And if they did, the outcomes could have been at least similar
to those generated by optimal control techniques. In other words, in our model societies make decisions
“as if” they use optimal control, in the same way that modern economic agents behave “as if” they
maximize a social welfare function.
As in economic growth theory, we stylize the problem by assuming that all households are
identical with the same endowments and preferences, form a production-consumption unit, and can be
described by a representative agent. The utility of a representative individual at time t,
of the consumption of a harvested good,
, and a manufactured good,
, is a function
. The harvested good
represents a broadly defined composite of natural resources such as trees, soil, edible plants, and fisheries.
The manufactured good represents a composite of everything else, including leisure. The production
functions of these goods,
and
are assumed to be linear in the amount of labor supplied. In
addition, the higher the level of resource stock,
, the easier it is to harvest. Time spent in productive
activity is limited by a constraint: whatever is not spent in harvesting is spent in producing of m.
Population is assumed to be fully employed. The fraction of the individuals’ endowment of one unit of
labor spent in harvesting is
, the level of harvesting effort.
Assuming a Cobb-Douglas utility function for the representative agent:
(1)
where the units of
are defined from the units of
.
Following Clark (1990), our production
function assumes that harvesting per capita is related to the effort,
, and the size of the stock S(t), with
á representing the catachability or harvestability of the resource. Notice that the per capita rate for
harvesting in (1) is not dependent on the number of individuals harvesting,
. Similarly, if ownership
of the resource is divided among individuals, rather than being collective, the same per capita production
relationship would hold as individual harvesting effort is concentrated in a smaller part of the stock.
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We assume that goods are consumed when produced and that markets clear each period. The
implications for the representative individual are that
in
ultimately being a function of
and
and
. This is reflected
:
(2)
Following Lotka (1925) and Volterra (1926), the dynamic nature of the system arises when
population,
, (the predator) is related to the level of harvesting, and the resource stock,
, (the
prey) is related to population. Total harvesting is determined by the harvesting of the representative agent
times population size. The natural resource growth is logistic, with an intrinsic growth rate r and a
carrying capacity
. The change in the resource stock is determined by the difference between total
harvesting and the natural growth of the resource. Human fertility follows a Malthusian behavior, where
the growth rate increases with the harvested good per capita. ä denotes the intrinsic net birth rate of the
population (natural birth rate minus natural death rate) with no harvesting, and ö describes the increase in
the birth rate as harvesting increases. Based on these assumed behaviors, the equations of motion of the
resource and population are:
(3)
We use this setup to consider three institutions. The first is a framework without resource
management institutions. In this case, individuals consume from the stock without consideration for the
future. Alternatively we consider resource management institutions either in the form of a social planner
that cares about all generations within a planning horizon, though the welfare of future generations are
discounted. Third, we may view decision making as decentralized through the assignment of property
rights that can be transferred to the extended family through bequests. Current decisions are made on
behalf of future generations by proxy. The utilities of the future generations are incorporated into the
decision makers’ objective functions as they incorporate their own. That is, they address the intergenerational problem with the attitude “what would I want me to do if I were them?”
Faced with a planning time horizon, T, a representative agent chooses an optimal plan for
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harvesting effort, f(t), that maximizes the following functional:
(4)
where ñ is the discount rate. We use problem 4 as a general case to describe our three special cases.
When
, our problem simplifies to maximizing the discounted welfare of a representative agent.
When
, our problem simplifies to maximizing the discounted sum of the utilities (the number of
individuals times the representative individual). When
, our problem simplifies to maximizing
only the current utility of an individual. In this case, because current population is unaffected by current
decisions, the integral is unnecessary, population is essentially exogenously determined and it does not
matter if we solve the problem with
or
.
Alternative monitoring and enforcement mechanisms could plausibly affect the ability of agents to
optimize this objective. The actual response of this system will be bounded by the worst case (where
institutions are completely ineffective and the resulting time horizon is
or alternatively the
case described above) and the best case where monitoring and enforcement of harvesting rules are costless
and complete. We might view, for example, values of
greater than the optimal social discount rate but
less than infinite to characterize situations where resource management institutions exist but where
monitoring and enforcement are less than perfect.
It is also important to note that while future generations can not be present at the initial
conception of the resource management plan, the solution will be consistent with what future generations
would have chosen for themselves when they take possession of the stock as a result of the principle of
optimality (see, e.g., Bellman, 1957). Further, very different institutional arrangements might be
necessary to support the optimal social policies. For example, a social welfare function which is based on
the representative individual could be supported by either institutions that are centralized (a social
planner) or decentralized (a market system with intergenerationally transferable property rights ). On the
other hand, in our system the social welfare function based on the sum of utilities can only be supported
by a centralized system of decision making and enforcement.
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Our consideration of problem (4) as the objective either of a social planner or private owner
depends heavily on what the decision maker would try to do. We have made two strong normative
assumptions in our description of the social welfare function. First, we have identified a potential role for
discounting. Second, in one version of our model (ù=0), we assume that social welfare is based on the
welfare of the representative individual in society and that no consideration should be paid to the number
of individuals that exist in describing social welfare. While these assumptions are common in the
neoclassical growth literature, we also seek to study the case the considers the aggregate welfare of all
these representative individuals in society (ù=0).
One common solution strategy for this dynamic optimization problem is to use Pontryagin’s
maximum principle. Using shadow prices
and
of population and resource stocks, respectively,
we construct the current value Hamiltonian for our problem, formulated for a general case where ù is a
parameter of the problem.
(5)
with first order and boundary conditions (at t=0 and t = T, respectively):
(6)
Several features of this solution are worth emphasizing. First, the implications of our choice of
which social welfare function to choose manifests itself directly only in one of the equations (the costate
equation for the shadow price of population). This explicitly has implications in the optimization of the
Hamiltonian with respect to
and in the costate equations for
that these shadow prices have on
and
that the optimal trajectories of
It is only through the effects
and
are affected.
Second, the interpretation of these first order conditions is quite sensible. As usual, the point by
point maximization of the
with respect to
implies that one should equate the marginal value of
consumption of the harvested resource with it marginal costs. This marginal cost is the sum of the
reduced production manufactured good (valued at a price of 1) as harvesting rises, the change in the
growth rate of the population (valued at a shadow price of
resource (valued at a shadow price of
) and the change in the growth rate of the
). In the case without foresight, the only cost on the right hand
side of this expression that matters is the reduced production of the manufactured good (using
imposing the transversality conditions that the values of the shadow prices are zero at
and
). Our model
solves the problem of the commons by internalizing the consequences on changing value for the resource
and population to the decision maker at each point in time.
The next two equations simply repeat the resource recovery process and the laissez faire
population growth process. The implications for our differential equations for
and
also imply
a conventional interpretation by relating changes in population and resources to the discount rate. Note
that in each case the numerator is divided by a shadow price and recall that shadow prices are the
incremental utilities from differentially changing the state variables. For
, one equates the discount
rate to the sum of (1) the percent net change in resource growth rate (natural growth rate minus harvesting
rate) due to a change
; (2) the percent change in the growth rate of the resource price (shadow price);
(3) the percent change in marginal utility of the resource; (4) the percent change of the marginal social
value of population growth rate; and (5) when using the aggregate social welfare function a term
representing the utility of the representative individual. For
, one equates the discount rate to the sum
of (1) the percent change in the marginal social value of harvesting due to a population change d; (2) the
percent change in value of population growth rate; and (3) the percent change in the growth rate of the
shadow price of population.
16
Third, the nature of the Hamiltonian guarantees a unique interior solution for harvesting effort
under all circumstances. Also, the boundary conditions of this problem come in two pairs, where the
initial conditions define the values of the stock variables (
conditions define the values of the shadow prices (
and
and
), and the terminal (transversality)
).
Due to its complexity, no analytic solution for the problem based on either the
the
or
social welfare function exists, and we are forced to identify numerical solutions. We consider
using two approaches. The first numerical solution is based on the Pontryagin formulation of our problem
and the first order conditions presented in (6). The difficulty with this formulation is that we must find
initial conditions for all state and costate variables. Once obtained, solving the system of differential
equations and maximizing the Hamiltonian with respect to
is straight forward. Numerically, we
implement this procedure by first guessing starting values for
period, we maximize the Hamiltonian with respect to
of
. Then, for each time
numerically using bisection. We use this value
and Runge-Kutta method to get the next value in the solution for the four differential equations.
Ultimately, these steps lead to values of
searches over alternative starting values of
(
and
and
and
. This “shooting” method (see Judd, 1999),
and
until the transversality conditions
) are satisfied. We use the Davidon-Fletcher-Powell algorithm to implement
this search. Different initial guesses for the values for
and
ultimately lead to the same
solutions, though convergence was sometimes painfully slow.
The second numerical solution approach, based on Kirk (1970), is to maximize the functional
subject to the equations of motion in (4) directly through parametric variation of extremals. Under this
solution approach, the trajectory of the control variable is given a functional form that depends on several
unknown parameters. For our implementation, we allow the trajectory of
to be characterized by local
quadratic approximations for each 10 year period. This implies that the trajectory for a 1000 year time
horizon, for example, would involve 100 time intervals. Requiring that the time intervals splice together
for continuity leads to 201 unknown parameters (a value of
interval). Once a candidate trajectory for
and another two values for each time
is described, the trajectories for
and
are then
determined by solving the differential equations system using the Runge-Kutta method.
Finally, the present value of social welfare is calculated by numerically integrating over the time
17
horizon using the second order Newton-Cotes integration. Alternative trajectories of
are considered
by varying the parameters defining it. The problem is essentially turned into a problem that numerically
maximizes a dependent variable (the integral) as a function of several independent variables (the
parameters describing the trajectory for
optimal trajectory for
). The search for the optimal values of parameters, and the
is found using the Davidon-Fletcher-Powell algorithm.
The infinite horizon element of the solution is implemented by using a rolling window. Consider
first a finite horizon problem, for
, where
is time horizon size. In the next period, the
planner revises the plan by solving a new problem as more information about the future becomes
available. This problem is solved for the same finite horizon, but for
; and so on. As the
discount rate gets larger, the new information about the future (T+1, T+2, etc.) becomes less and less
important and the agent’s behavior approaches the solution to the infinite horizon problem. The overall
trajectory is constructed by combining the first time periods from the particular solutions, respectively.
Intuitively, like the infinite horizon problem, in this setup people behave as though they are at the start of
the planning period. For a given ñ, as T increases, the solution approaches the infinite horizon solution,
and the added future that is being considered is less and less important in determining current decisions.
Very small values of
imply that these revised plans will be close to T=4 solution.
Both approaches yield similar trajectories for all variables, differing at most by about 2 percent.
Convergence occurs much faster using the parametric variation of extremals method particularly for long
time horizons and for the aggregate social welfare function. Because of the number of scenarios we
examine, we focus on the parametric variation of extremals approach.
5. Simulation Results and Discussion
In this section, we describe several formulations of the problem and compare their solutions. This
is followed by evaluating which social welfare was relevant for the historical societies discussed in
Sections 2 and 3. We first consider a case without foresight, and then evaluate the implications of using
different discount rates and conceptual assumptions on the nature of the social welfare. In all our
numerical solutions, we use the same parameter values and initial values for population and the resource
18
stock, as those used by Brander and Taylor (1998).1
The steady state results for different discount rates are presented in Figure 1. The numerical
results for the global trajectories obtained in the optimizations are presented in figures 2. Each of these
figures presents the time paths for the resource (upper left), population (lower left), share of labor spent in
harvesting (upper right), and the contemporary utility level of a representative individual (lower right). In
all figures, the black line indicates the trajectories from the case without foresight.
With no foresight, the time horizon T in the problem stated in (4) is set to zero, and the initial
values of the shadow prices are set identical to the transversality conditions. In this case, the optimization
reduces to choosing a value of f(t) that maximizes current utility, which gives
. This solution is
then substituted into the equations of motion for the population and the resource. Consequently, the
solution obtained is exactly the one obtained by Brander and Taylor (1998).
Almost all studies in the literature on economic growth and its environmental extension assume
that people have an infinite horizon, and that optimally controlled systems converge to a steady state. The
comparative statics of this steady state are typically examined while transition paths are ignored. Infinite
horizon decision making is, of course, not observed in reality. Even so, it is interesting to assume that our
agents have an infinite horizon, and then relax this assumption.
The formulation of the infinite horizon problem is similar to the problem presented in (4),
differing only in the time horizon (T) being set to 4, and the transversality conditions being set to
and
. The steady state solutions are obtained by setting the
time derivatives in (6) to zero, and solving the resulting algebraic system. There are three steady state
solutions for this system: two corner solutions, and one interior for
integral in equation (4)) is unbounded. The corner solutions involve
. For
our functional (the
, and either
or
, and
are not interesting since they can not be reached from our initial conditions.
The interior solutions for alternative values of ñ are described numerically in Figure 1, for two
cases: the individual utility-based social welfare, and the aggregate utility based social welfare. As shown
1
The resource carrying capacity K = 12,000. The resource intrinsic growth rate r = 0.04. The
population fertility parameter ö = 4. The intrinsic human net birth rate ä = -0.1. The harvesting
efficiency á = 0.00001. The utility taste parameter â = 0.4. The initial conditions for the population and
the resource stocks are L(0) = 40, and S(0) = 12,000, respectively.
19
in Figure 1, as the discount rate approaches zero in the individual utility-based welfare, the agent cares as
much about the future as the present and the equilibrium is driven closer to the corner solution (
and
). To maximize the utility of the representative individual, it is optimal to make the value of S as
large as possible. Sustaining
at its maximum (carrying capacity,
) requires that individuals
collectively do not harvest anything since the resource recovery rate is zero when
conditions hold only when
. These two
approaches 0.
For the aggregate utility-based solution, the result obtained for the case in which the discount rate
is approaching zero is markedly different from the one obtained for the individual utility-based solution.
In this case, social welfare rises with both L and S. However, as L rises, S must decline, and vice verse.
When the discount rate is zero, the tension between the two forces is resolved optimally at L of about
4000 and S of about 7000, as shown.
[Figure 1 here: Steady State for Alternative Discount Rates and Social Welfare Functions]
As the discount rate rises for the individual utility-based social welfare, the equilibrium resource
stock, population, utility and the harvesting rate converge to the equilibrium values from the model
without foresight (
). An interesting feature of this system is that any
or
so leads to substantively the same steady state as the model with no foresight. This suggests that unless
the discount rate is unreasonably low, even optimal resource management institutions with infinite
foresight and perfectly functioning enforcement will have little effect on the equilibrium of the system.
As the discount rate rises for the aggregate utility-based social welfare, the equilibrium presents a
different behavior than the one for the individual utility-based social equilibrium. For a discount rate
larger than %2, the differences in the population steady state from the individual utility-based social
equilibrium largely disappears, but the differences in the resource, harvesting rate, and individual utility
remain substantial throughout the %0-10 range of discount rates presented. The steady state harvesting
rate for the aggregate utility-based welfare is consistently larger than the one obtained for the individual
utility-based welfare, and the resource stock for the aggregate case is considerably lower. The steady state
individual utility for the aggregate utility-based welfare is lower than the steady state individual utility for
the individual utility-based welfare. The difference grows up to a discount rate of about %0.9 and then
declines, but remains substantial for a discount rate of %10.
20
On the whole, this suggests that at the steady state of the system, the population level is very
insensitive to either the choice of the social welfare function or the discount rate; the resource stocks are
moderately sensitive to the choice of the social welfare function but not sensitive to discount rates above
2%; and harvesting effort is very sensitive to the choice of social welfare functions and the discount rate.
Note also that while the steady state population is unchanged across the choice of welfare functions, the
steady state individual welfare is markedly lower with the aggregate social welfare function. In effect,
society works to increase population and trades off individual utility in the attempt, but is ultimately
unsuccessful in increasing population.
Turning to the transient solution, Figure 2 presents infinite horizon trajectories for the resource
stock, population, harvesting effort and utility. The equilibrium described in Figure 1 requires 3000 years
to achieve in some cases and, especially for the aggregate social welfare, is not achieved even then.
[Figure 2 here: Infinite Horizon Trajectories for Alternative Discount Rates]
The case with no foresight (and effectively no resource management institutions) is represented
by the solid black line. This trajectory is characterized by a boom-bust cycle, a rapid population increase
followed by a rapid decline. This is roughly consistent with the archeological evidence on Easter Island.
Notice that this forms a limiting case for both the trajectories of the individual-utility based social welfare
function (black dashed lines), as well as for the aggregate based social welfare function (dashed gray
lines) as the discount rate rises.
The dynamic behavior of the system under the individual utility-based social welfare closely
resembles the no foresight solution when the discount rate is %4 (or more), and is still similar to the
trajectory when the discount rate is %1, although the boom-bust cycle is relatively more attenuated. In
contrast, the dynamic behavior of the system under the aggregated utility-based social welfare is markedly
different from the no foresight solution for 1%, 4% and even 8% discount rates.. In the aggregate case,
the boom-bust is much more pronounced, and trajectory is considerably less damped than in the individual
case. The peak in the particularly trajectory occurs more quickly, and register fluctuations with a larger
amplitude over time.
Turning to the welfare of individuals under alternative systems along the transition paths, note
that the utility of an individual in the individual utility case is larger than in the aggregate utility case.
21
While there are more people in the system in the aggregate case on average, they are “less happy” than the
fewer people in the system in the individual utility case. During the initial periods of the aggregate social
welfare models, individual utility is incredibly low as they are essentially just “slaves” working to
increase future population levels.
As we compared the no foresight trajectory to the historical evidence, we would also like to
compare the trajectories with alternative social welfare functions to the archeological and anthropological
record on Easter Island. It is not easy to compare the resource in the model to the real world because our
model (like other models of this type) is stylized since it represents a complex of resources (forests,
fisheries, etc.). Nonetheless, we can discuss the population trajectory and the timing of the resource
trajectory. Period 0 in Figure 2 is in the range AD 400-1000, which is when Polynesian settlers are said to
have arrived on Easter Island.2 The estimated maximum population on the island ranges from 7,000 to
20,000, and the peaked in the range AD 1100-1500.3 The island was largely or overwhelmingly
deforested in AD 1400-1600.4 When Easter Island was discovered in the 18th century, the Dutch admiral
Rogeveen estimated there were about 3,000 people on the island.5
Given the variations in these numbers, we can say that the archeological record is consistent with
both no resource management institutions, and optimal resource management with infinite horizon at
reasonable discount rates, where the social welfare is based on either the individual or the aggregate
utility. The collapse of Easter Island and, by implication, of the other civilizations we discussed cannot be
used to determine that their behaviors were sub optimal. One possibility is that these societies had less
2
Studies provide different dates. For example, Brander and Taylor (1998) use AD 400, Gowdy
(1998) and Bahn and Flenley (1992) use AD 700, while Brown and Flavin (1999) use AD 500.
Skjolsvold (1994) and Martinson-Wallin (1994) provide the range AD 600-1000, and Martinsson-Wallin
and Wallin (2000) and Stevenson (1997) provide the range AD 800-1000.
3
See, e.g., Ponting (1991), Bahn and Flenley (1992), Van Tilberg (1994). For additional sources,
see Brander and Taylor (1998).
4
For the date of the forest vanishing, see Hunt and Lipo (2001); Flenly et al. (1991), Flenly
(1996), and Brander and Taylor (1998).
5
It should be noted that by the 19th century Easter Island stopped being a closed system and thus
our model becomes less applicable. For example, many of the islands’s inhabitants were repatriated to
South America against their will during the 19th century, which is not included in the model.
22
than optimal institutions. However it is also possible that they had infinite foresight, a plausible discount
rate, and a social welfare based on either the aggregate or individual utility in society.
In Section 2, we discussed the collapses of the Sumerians, the Maya, Easter Island, and the
Anasazi. Assuming that these civilizations used resource management institutions, what kind of social
welfare did they have? In general, actors maximizing the individual-based welfare are not willing to
tradeoff utility of an individual for having more people. Actors maximizing the aggregate-based welfare,
are willing to make this trade off, and their welfare rises with L, ceteris paribus. While we cannot be sure
what kind of social welfare our civilizations had, we can turn to history to gain more insight.
The Sumerian society consisted of prosperous, which were the envy of their neighbors and of each
other. To defend against potential predators, they maintained relatively large armies. This increased the
need for food, as the soldiers were not productive. More land was required, forests were cut, and
irrigation was intensified, leading to salinization. The need for large armies suggest that the Sumarian
social planner of each city had an L@u function in mind, giving priority to have more people in society as a
source for soldiers and labor to work in the fields and create and maintain irrigation system.
The Maya also consisted of prosperous and rivalrous cities and maintained armies. It is plausible
that the Mayans had an aggregate social welfare in mind in light of their relatively abrupt collapse over
200-300 years, which characterizes the aggregate welfare trajectories. Their agricultural methods (e.g.,
raised fields, deforestation, irrigation systems) and the construction of their pyramids and wood-plastered
large palaces required many people, which supports our conjecture. The Anasazi employed peasants
producing food for the elites. Given the hard environmental conditions of the southwest US, it is likely
that many people were needed for work in the fields, for digging irrigation canals, and for building dams
across canyons. And like the Maya, the collapse of the Anasazi also was abrupt.
The Rapanui were organized in a few clans that competed over building statues. The movement
of these massive stone structures, maoi, which weighed up to 80 metric tones, required up to 500 people
per monument. The need for people is consistent with an aggregate social welfare. In contrast, Tikopia
controlled population to not exceed some level. This suggests that the Tikopians were willing to tradeoff
people for a higher utility for those living on the island. Hence, the Tikopia civilization might have had an
individual utility-based social welfare in mind.
23
6. Conclusion
This paper seeks to identify what optimal resource management institutions could have done to
the fate of collapsing historical civilization, assuming laissez-faire population growth rate in response to
changes in resource harvesting, or income. Optimal institutions are based on infinite horizon and have no
enforcement and monitoring costs. Practical institutions are based on finite levels of foresight, have
simple monitoring and enforcement, and do not imply large inter-generational inequities. We find some
evidence for the existence of practical institutions in some of our historical civilizations.
While we cannot be sure, we examined the situation from best case and worst case perspectives.
At worst, optimal institutions did not exist in these societies. At best, they did. Real world, practical
institutions, which are based on finite foresight and are plagued by difficulties in assigning property rights
and in enforcing them, will lead to solutions somewhere between the best case and worst case trajectories.
In simulations, we employed parameters for Easter Island, but our analysis directly applies to the other
historical collapses we discussed.
It is important to recognize the circumstances surrounding our model and its assumptions before
generalizing the results to other situations. Our model assumes a finite carrying capacity. Whereas this
assumption seems appropriate for historical civilizations, which in general were isolated and faced finite
environments on which they depended for their livelihoods, contemporary societies can import, effectively
borrowing carrying capacity from others. That said, contemporary developing nations are both agrarian as
well as generally much less integrated to the world economy than developed countries. Their societies do
face in many ways a finite carrying capacity for the local environments, particularly when these
environments are in marginal lands such as in parts of Sub-Saharan Africa. Second, at the global level,
trade obviously cannot increase the carrying capacity of biosphere. Here, the only possibility to increase
carrying capacity is through technological progress, an issue to which we will return shortly.
It is easy to say that institutions fix things. We find that optimal resource management institutions
with infinite time horizon and a reasonable discount rate would have failed to alter the boom-bust
outcome. Our model can be interpreted as describing the behavior of an individual agent or a social
planner. In the first interpretation, the externality of the commons is internalized by setting property
rights. In the second, it is internalized by employing a government. While it is tempting to label this as
24
either a market failure (for a representative agent version) or a governmental failure (for a social planner
version), these deficiencies do not imply social sub-optimality. Both the best case and worst case models
produce outcomes that are consistent with the collapse of civilization. This implies that real institutions
(with all their limitations) would also have led to similar outcomes for our historical societies.
We believe that an infinite time horizon is unreasonably long, and a discount rate of 1% is
unreasonably low by today’s standards. As evidence, consider, for example, contemporary analyses of
global warming, which typically use time horizons of 100 years or less. Furthermore, one widely cited
study by Nordhaus and Boyer (2000) (summarized in Nordhaus, 2001) uses a discount rate of 5%. One of
the lowest discount rates used in the analysis of global warming is 1.5% (Cline, 1992).
Our results underscore the problem associated with the modeling focus on the steady state, which
is typically used in the economic growth literature. Authors rarely, if at all, compute the global transition
trajectory leading to the steady state. At most, they compute the trajectories for the linearized system in
the close vicinity of the steady state. Our results show a small difference for the steady state between the
individual actor-based utility, and the society-based utility. We also see that the steady state itself does
not change much as a function of the discount rate for both social welfare functions. These results seems
to vindicate the standard approach of focusing on the steady state.
However, the transition trajectories, which can take long periods (as they do in our case) are
markedly different for the two social welfare functions. The trajectories obtained for society-based
welfare are much more volatile than those obtained for the individual-based welfare. They are also
presenting a more intense boom-bust nature, with higher peaks and lower valleys. Moreover, the boombust occurs much earlier in the trajectory for the society-based social welfare.
Our results also underscore the importance of discounting for the transition trajectories. In the
economic growth literature, this issue is debated along several considerations. Some scholars rule out the
use of zero discount rate as a matter of mathematical convenience for dynamic optimizations with infinite
time horizons (Barro and Salai-i-Martin, 2004; Chiang, 1992). Other scholars look at it as a moral issue.
Since future generations do not participate in current decisions, but are affected by them, it is fair to
include them with an equal weight in present decision making (Ramsey, 1928; Dasgupta et al., 1999). In
contrast, Arrow (1999) notes that a small or zero discount rate demands large current savings, but there is
25
no guarantee that future generations would not chose lower savings rates, in a sense profiting at the
expense of the present. Taking a middle ground position, Weitzman (1999) advocates using the lowest
plausible future rate of return on capital as a response to uncertainty about the rate in the far future. These
arguments all have merits, although it is not so clear how to choose among them. While our paper does
not resolve this philosophical debate, it suggests that optimal resource management institutions with
reasonable discount rate would have failed to avert the collapse of our societies.
One should be cautious about straightforward application of these results to contemporary
societies, because our model is relatively simple. For example, while appropriate for the historical
societies considered here, our model does not consider the effects of technological change or demographic
transition. Intuitively, technological progress could either exaggerate or alleviate the boom-bust
population cycles observed here. Preliminary numerical results suggest that technology that improves
harvesting efficiency tends to exaggerate the cycles. On the other hand, technology that increases the
natural resource recovery rate tends to alleviate these cycles. When both types of technological change
occur together, the overall effect depends on the parameters employed.
The effect of demographic transition in our model is also unclear. In a model without foresight, if
the demographic transition curve (the relationship between population growth and harvesting per capita) is
never higher than the linear reference case used here (equation 3), demographic transition will decrease
the amplitude of population and resource cycles, increase their period, and lengthen the time to
equilibrium. If the Malthusian phase of the demographic transition curve (when population growth rises
with harvesting per capita) is above the linear reference case at low harvests, and the non-Malthusian
phase (when population growth declines with harvesting per capita) is below it at high harvests, the results
are ambiguous. Initially, population grows slower than in the linear case. Eventually it is pushed into the
Malthusian phase and grows faster. Population crushes are exaggerated, potentially intensifying the
boom-bust cycle. With foresight, outcomes are even harder to predict since harvesting effort changes with
time and the level of foresight.
Taking a broader view, introducing endogenous population growth is an improvement over the
bulk of economic growth models, which assume that population growth is exogenous, but it may also
create normative difficulties. At the heart of the debate is the obligation to future individuals if current
26
actions imply that those individuals will not exist in the future. This brings with it paradoxes concerning
whether it is socially superior to be born to a miserable existence or not to be born at all (Parfit, 1982;
Kavka, 1982). These paradoxes imply the need to extend the definition of Pareto optimality, and to
explicitly incorporate L(t) in the social welfare. While we dealt with the second issue, the first issue is
ultimately normative and beyond the scope of this paper. The neoclassical growth literature interested in
markets uses the individual-based social welfare function (e.g., Blachard and Fischer, 1989; Jones, 1995;
Aghion and Howitt, 1998). It shows that when population growth is exogenous, the use of L@u or u does
not alter the mathematical nature of the problem. This result does not hold when population growth is
endogenous, as here. In our case, the two social welfare functions generate markedly different solutions.
Finally, one can argue that if the goal is to eliminate boom-bust cycles, both we and economic
growth theory are using the wrong social welfare function since it does not explicitly recognize that
society has preferences for population levels and rates of growth and decline. In fact, as noted, scholars
observe that the Tikopia society did not collapse precisely because it had preferences for a certain
population level it deemed appropriate for the island. Incorporating these features into our model are
valuable research extensions but their use in the real world, of course, is normatively problematic.
This paper contributes to the literature on optimal economic growth in that it examines a system in
which there are interdependencies between population and a renewable resource with a finite carrying
capacity. Our model does not take place in “Ramsey’s vacuum,” where the economic system grows
forever and natural resources do not place limits on growth. Natural resources did place limits on
historical civilizations. While substituting away from resources with capital can alleviate some of the
pressures, it likely cannot eliminate all of them. In the end, whether our historical societies were
farsighted and fully understood their environmental problem cannot be fully known. But even if they
would have had infinite horizon with a modest discount rate by today’s standards, this would not have
saved them, particularly so if their social welfare gave preference to have many people in society.
27
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Fig 1. Equilibrium values for Resource, Population, Harvesting Effort and Utility as a function of Discount Rate
Figure 2. Time Trajectories for Resource, Population, Harvesting Effort and Utility for alternative discount rates