ON THE COLLAPSE OF HISTORICAL CIVILIZATIONS
DAVID H. GOOD AND RAFAEL REUVENY
To explain the collapse of historical civilizations, scholars typically point to suboptimal behaviors
including misunderstanding the natural environment, shortsightedness, or a lack of institutions. We
examine the collapse of four historical societies with a model of endogenous population growth and
renewable resources employing components of optimal resource management, economic growth theory, and the moral philosophy of social welfare function choice. We find that these collapses may have
been socially optimal. Further, we show that the transient behavior of the system is more sensitive to
assumptions than the equilibrium behavior and that focusing solely on equilibria may miss key insights.
Key words: infinite horizon, optimal control, steady state, transient behavior.
Scholars in many disciplines have described
the increase and rapid decrease of population
size associated with collapsed societies such as
the Maya, Anasazi, Rapanui, and Sumerians.
Malthus (1798) connected population fluctuations of this type to natural resource depletion,
yet the issue remains current because of its connections to the sustainability debate (Brander
2007; Pezzey and Anderies 2003).
Diamond (2005) summarizes the extensive
literature on the collapse of historical civilizations and identifies four broad reasons
for societal collapse: (a) damage to the environment due to inadvertent, shortsighted, or
uninformed human activities such as deforestation, land erosion, or overhunting; (b) unpredictable natural disasters such as increased
cooling or droughts; (c) a deterioration in the
relations with neighbors; and (d) cultural responses preventing society from recognizing
the problem or considering potential solutions.
Arguing that the stories of these societies tell
us something relevant today, Diamond (2005,
p. 10) concludes:
The societies that ended up collapsing were
among the most creative and (for a time) advanced and successful of their times, rather
than stupid and primitive. Past people were
neither ignorant bad managers who deserved
David H. Good is associate professor and Rafael Reuveny is professor at School of Public and Environmental Affairs, Indiana University.
The authors thank participants in the Workshop in Political Theory and Policy Analysis, participants at the 2007 meeting of the European Association of Environmental and Resource Economists,
as well as the reviewers and editor of the journal for helpful comments on earlier versions of this manuscript. Melanie Arnold provided invaluable editorial assistance. Any remaining errors belong
to the authors alone.
to be exterminated or dispossessed, nor
all-knowing conscientious environmentalists
who solved problems that we can’t solve today. They were people like us.
While popularly received, Diamond’s thesis
has been criticized by Dasgupta (2005) and
others for not more explicitly incorporating the
role of institutions in alleviating and resolving
environmental problems. This article has little to contribute regarding issues (b) and (c).
Stochastic disasters happen. Strategic interactions between societies are complex and can
mitigate (e.g., trade) or hasten (e.g., war) collapse. However, we notice that many of these
social collapses occurred in isolation or near
isolation. Consequently, we focus on issues (a)
and (d) for these particular cases.
An evaluation of these historical collapses
naturally suggests two counterfactual questions: What would these historical people do
in our circumstances? What would we do in
theirs? Assuming that we are not shortsighted,
ignorant, or act inadvertently, we take Dasgupta’s criticism seriously and examine the
role that social institutions and social preferences might have had on altering the history
of these societies.
Our framework, based on Good and
Reuveny (2006), weds an ecological-economic
model of human-resource interaction with
endogenous population growth (Clark 1990;
Brander and Taylor 1998) to economic growth
theory (Barro and Sala-i-Martin 2004), considering future consequences of current decisions.
We examine the role of endogenous population growth in social welfare from the perspective of moral philosophy (Parfit 1984; Kavka
1982; Cowen 1996; Dasgupta 2000a, 2001a) by
Amer. J. Agr. Econ. xx(x) (xxx 2009): 1–17
Copyright 2009 Agricultural and Applied Economics Association
DOI: 10.1111/j.1467-8276.2009.01312.x
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constructing a family of social welfare functions (SWFs), nesting the utility of the representative individual and aggregate utility as
special cases. Given our attention on historical
societies, our use of economic growth theory
focuses on renewable resources and their socially optimal management (Heal 1998; Mäler
2000; Arrow et al. 2000) rather than capital accumulation or technological change.
We represent institutions with either a decentralized system of enforceable property
rights or a centralized system of social planning. Real-world social institutions are not perfect, so we assess their impacts as bounded
by a best-case scenario (planners possess infinite foresight and have perfect institutions
with costless enforcement) and a worst-case
scenario (planners have neither foresight nor
institutions to implement social plans). Finally,
unlike the bulk of economic growth theory, our
problem demands that we determine population and resource global transition paths in addition to their steady-state solutions.
While it is tempting to conclude that historical civilizations failed because they were
shortsighted, had no resource management institutions, or did not understand their problem,
our results suggest that, even if they somehow
had optimal resource management with costless enforcement and infinite horizons, their
societies still would have collapsed, regardless
of which SWF we consider. We also find that
the principle from moral philosophy called the
Repugnant Conclusion (Parfit 1984), a conjecture that societies that seek to optimize the aggregate SWF are doomed to large populations
living in misery, does not occur unless the discount rate is extremely small.
These results are obtained for historical societies, but, in the spirit of Brander and Taylor
(1998), Brander (2007), Arrow et al. (2000),
and Mäler (2000), they may offer an allegory with which to draw some cautious lessons
about the way we think about the world today.
In the next section we describe four historical civilizations that collapsed, followed by our
modeling approach and mathematical model.
The remaining sections summarize our results
and provide discussion and conclusions.
The Collapse of Four Societies
Diamond (2005), Ponting (1991), Tainter
(1990), and Yoffee and Cowgill (1988) describe
the collapse of several civilizations. Building
on these sources and others cited below, we
Amer. J. Agr. Econ.
provide some detail on the collapses of the
Sumerians, the Maya, the Anasazi, and the Rapanui.
Jacobsen and Adams (1958), Gelburd
(1963), Adams (1981), and Kirch (2005) chronicle the history of the Sumerians, located in
what is now Iraq. The Sumerian civilization increased agricultural production through irrigation, developed a detailed accounting system
that monitored and recorded outputs, and is
generally considered the first literate society.
As their wealth grew, the Sumerians’ prosperous cities became the envy of their neighbors.
To defend against invaders, they maintained
large armies. Faced with the need to feed these
armies, they abandoned their practices of crop
shifting and letting land lie fallow in favor of
intensified irrigation. In the arid climate, this
led to salinization and a sharp decline in output. Sumerian accounting records imply that
crop yield fell 42% in 2400–2100 BC and 65%
in 2100–1700 BC (Ponting 1991). As a result,
health declined, fertility fell, the death rate
rose, and civic order deteriorated. The urban
population peaked at about 200,000 some time
around 2100 BC and fell to about 25,000 by
1500 B.C. (Thompson 2004).
The Maya have been studied by Binford
et al. (1987), Culbert (1988), Abrams and
Rue (1988), and O’Hara, Street-Perrott, and
Burt (1993). Centered in the lowland jungle of
the Yucatan Peninsula, the Mayan population
reached its peak sometime between AD 500
and 800 with 3–14 million in the Peten region,
the society’s social core. The Maya were both
literate and technically sophisticated, creating
the long count calendar. In AD 600 they began building the dazzling pyramids and monuments that they are known for today, but just
200 years later, their society began to deteriorate.
The Mayan society had a hierarchy consisting of nobles, priests, warriors, peasants, and
slaves. Societal elites demanded pyramids and
ornaments, while the warriors, peasants, and
slaves demanded sustenance. These demands,
together with the reliance on agriculture, required a large workforce. As with the Sumerians, the Mayan economy could not keep up
with the demands it faced, so deforestation
was intensified to make way for crops and provide the fuel to make lime for adorning buildings. The resulting soil erosion often ended up
as silt in the rivers and canals, damaging the
irrigation-based agricultural system. As crop
yields fell, health and social order deteriorated, and the population declined. When the
Good and Reuveny
Spanish arrived in the 1500s, they were essentially gone.
Bahn and Flenley (1992), Van Tilberg
(1994), Kirch (1997), and Flenley and Bahn
(2002) study the society on Easter Island. The
inhabitants, the Rapanui, arrived sometime
during the first millennium finding a lush palm
forest. The population of this agrarian civilization grew for several hundred years, peaking
between 7,000 and 20,000, and then declined
rapidly. The Rapanui cleared the forest for
arable land, timber for fishing boats, and logs to
serve as rollers to help move large stone statues
(moai) from their quarries. It required as many
as 500 people to move the moai. As the forest
vanished, the topsoil eroded and with no wood
for boats, the Rapanui lost the ability to fish.
When Europeans arrived on Easter Sunday in
1722 the island was nearly barren. The remaining 2,000 or so people lived in poverty, fighting
over the few remaining resources (Reuveny
and Maxwell 2001).
Finally, Betancourt and Van Devender
(1981), Samuels and Betancourt (1982), Betancourt, Dean, and Hull (1986), and Stuart
(2000) consider the Anasazi. In one major site
in the Chaco Canyon, the Anasazi population peaked between 4,000 and 10,000 (Tainter
1990). In another site located in Montezuma
Valley, the population peaked at about 20,000
(Bureau of Land Management 2006). Beginning around AD 600, they erected some of
the most dazzling and tallest buildings in preColumbian America. As their number grew,
they built outlying settlements where peasants
grew food for the civilization’s core. Seeking
land for food and timber for kivas, the Anasazi
intensified deforestation. By AD 1000, the
trees were gone and soil erosion set in. Facing the dry climate, depleted lands, and low
plant growth, the Anasazi developed gravitypropelled irrigation, but over time the system
cut arroyos into the ground. When the water
levels in the arroyos fell below the field levels,
agriculture ceased. As food production fell, the
population fell. While the Anasazi had withstood droughts before, when a long drought
began in AD 1130, they were already weakened by years of decline. By AD 1200, their
society was gone.
Several factors are shared by these societies. They lived in predominantly isolated
spaces and had at most limited trade (such as
the Anasazi’s small ornamental items). Each
depended on the resources in the environment for their livelihood, facing a finite carrying capacity. Human actions in the form
Social Collapse
3
of deforestation (Maya, Rapanui, Anasazi)
and unsustainable irrigation (Sumerian, Maya,
Anasazi) ultimately led to falling food production and social collapse. Innovations intended
to alleviate these problems were either negligible (Easter Island), allowed society to muddle
through for a while before facing the problem
again (Maya, Anasazi), or created new environmental problems that made things worse
(Sumerian, Anasazi). Finally, exacerbating the
problem, they allocated some of their productive capacities to non-life-sustaining purposes
such as building monuments (Rapanui, Maya,
Anasazi) or maintaining large standing armies
(Sumerians, Maya).
Why would people threaten their existence
by degrading their environments? There are
several common rationales. They did not either perceive or understand their problem because of its complexity; they understood but
they did not care about the future, or they understood and cared but lacked institutions to
enforce corrective policies.
Our analysis differs from the empirical approach taken by Diamond (2005) and others. They describe differences among a small
number of societies, assessing the connection between these differences and societal
collapse. Their sample is arguably nonrepresentative since many collapsing societies left
almost no evidence of their existence and
are consequently excluded from the analysis; the only included societies left monuments to their existence. Our approach avoids
this potential selectivity bias by not making a
cross-society comparison. Indeed, our lack of
discussion of successful societies emphasizes
this point. Instead, we focus on the decisions
made by individual societies and examine how
those decisions might have been made differently. We ask, what would we do if we
were in their shoes? Given our goal, we hold
the society fixed and remove their shortcomings, giving them an understanding of the
future consequences of current actions and
the institutions to implement socially optimal
policies.
Modeling Approach
We begin at the beginning and consider the nature of the social contract from the perspective
of moral philosophy. If we suddenly found ourselves in charge of planning at the initiation
of our society, what would the order of business be? We would first need to know what
4
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parts of our situation cannot be changed. We
assume that technology is static, and the preferences, biological features of resources, and
procreation behavior are not changed. We use
both institutions and the analytical methods
of economic growth theory as tools that form
the mainstay of a modern approach (Smith
1975; Ostrom 1990; World Bank 2003). Using these tools, we focus on the essence of
society’s situation: a lack of understanding,
foresight, and institutions for optimal policy
making.
Our agenda is to identify social goals and
policy instruments and to develop a formal
model. In considering society’s goals, we follow economic norms by measuring social welfare with either an aggregate (sum of utilities of
all individuals) or an individual SWF (utility of
a representative individual). As is common, we
use a representative individual to build these
social welfare measures and ignore distributional issues. While potentially at odds with the
social structures of our four societies, we adopt
this simplification because it allows us to focus
on the dynamic aspects of collapse.
When population is static (Aghion and
Howitt 1998), individual and aggregate SWFs
yield the same results. When population
growth is exogenous, both aggregate utility
(Cass 1965; Lucas 2002) and individual utility (Blanchard and Fischer 1989; Levy 1992;
Turnovsky 2000; Romer 2001) studies ignore
the other function or note the two functions
are similar except for the discount rate. When
the actor is conceptualized as a family whose
size grows exogenously, the individual planner
and the social planner maximize the family’s
aggregate utility (Obstfeld and Rogoff 1998;
Barro and Sala-i-Martin 2004). Since the assumptions of a static population or exogenous
population growth eliminate the possibilities
of population collapse by definition, we reject
them.
When population growth is endogenous, the
controversial issue for the SWF is the relative
importance of population size and individual
utility. Parfit’s (1984) approach uses the perspective of moral philosophy. He considers a
world with limited resources, assumes that social welfare is transitive, and argues that an aggregate SWF implies optimal societies with a
huge number of miserable individuals (the Repugnant Conclusion). In contrast, using an individual SWF implies very small populations
with high individual utility. He rejects both
outcomes as morally unacceptable. He then
searches for something else, a “Theory X,”
Amer. J. Agr. Econ.
which values both life and individual happiness between these two extremes, but he cannot specify it. Taking an axiomatic approach,
Cowen (1996) argues that the importance of
Parfit’s Repugnant Conclusion lies in its implication for comparing outcomes, not in its
establishing a priori objectives. SWFs dealing
with paradoxes more successfully and implying
fewer unpalatable conclusions should be preferred over others. Finally, Dasgupta (2000a,
p. 385) rejects the relevance of the Repugnant Conclusion by writing about an individual
with living standards at or below subsistence
levels:
She is one of the wretches of the earth,
and there are hundreds of millions of such
people alive today, disfranchised, malnourished, prone to illness–but surviving, and
tenaciously displaying that their lives are
worth living by the persistence with which
they continue to wish to live. . .. There is nothing repugnant about a very large imaginable
population, all enjoying positive well-being.
Given our task of managing a historical society, this debate is relevant, but difficult to apply
precisely. However, it is useful to understand
the implications the choice of an SWF has for
our society. In the Rawlsian sense, we seek to
know a little more about what is on the other
side of the veil before making a choice. In deference to Parfit’s Theory X, we employ a family
of SWFs nesting both the individual and total aggregate SWFs in addition to intermediate
cases.
We next identify acceptable policy instruments for our problem. One question is
whether or not to directly alter the population growth process, and if so how. The assignment of fertility and resource harvesting
decisions to individuals or to the State stands
at the heart of this matter. We can see three
potential policy types: direct control (a communitarian approach assigning decisions to the
State), laissez-faire (a libertarian approach assigning decisions to the individual), and a partial or indirect control that is somewhere in the
middle. Under direct control of the resource,
harvesting is determined by a central planner
recognizing the effects on the future. A policy of direct control of human fertility, such
as China’s one-child policy (Tomlinson 1975;
Gigliotti 1983; Decter 1997; Dasgupta 1998),
is more controversial. Other than China, countries do not currently engage in this policy, and
even China is reconsidering it (Attané 2002;
Good and Reuveny
Hemminki et al. 2005). Alternatively, Hardin
(1968, 1998), Ehrlich (1968), and King and Elliott (1997) support direct population control
in order to save the environment.
With no institutions, a laissez-faire resource
policy leads to an overdepletion of common
pool resources. A laissez-faire population policy gives people a right to procreate as they
wish. In this libertarian vein, Nozick (1974) argues that people should be able to do as they
wish so long as they violate no one else’s rights.
Kavka (1982) argues that procreation is a basic human right that supercedes other policy
objectives. Nagel (1981) and Dasgupta (2001a)
reject Nozick’s lack of trade-off between rights,
other benefits, and violation of other rights.
For example, rights to current resource consumption are clearly in conflict with the rights
of future individuals, unless they are taken
into account. Unfettered procreation autonomy may exaggerate this conflict by creating
even more individuals for the future. Procreative rights are not so clearly in conflict with
one another though they may affect the quality of life that progeny face. Dworkin (1978),
Sen (1983), Gewirth (1981), and Parfit (1984)
also reject their claim to absolute rights. For
them, rights are context dependent, reflecting
the prevailing social norms. At any rate, since
a laissez-faire approach for population growth
and resource use is consistent with the rationales that most scholars use to explain social
collapse, they form a natural baseline for comparison.
To the extent that any human rights exist, most would agree that procreation autonomy is a more basic right than access to
a common pool resource. As a consequence,
communitarian management of the resource
and a laissez-faire approach to procreation
form a third possibility where resource management affects population growth indirectly
through the effect of consumption on fertility. This appears to be consistent with current
social norms claiming that procreation autonomy is a basic human right, as stated in the
United Nations Universal Declaration of Human Rights (United Nations 1948) and the Report of the UN World Population Conference
(United Nations 1975). The latter report also
emphasizes that exercise of this right should
take into account the needs of the current and
future generations. We use this moral guidance
in our decision to focus on resource management institutions that affect population size indirectly by altering consumption, rather than
controlling procreation directly.
Social Collapse
5
A Mathematical Model
In light of the previous discussion, we extend
the myopic, no-institutions model of Brander and Taylor (1998), seeking to overcome
limitations that scholars have alleged for historical societies. Others have extended this
model by adding a resource subsistence requirement (Pezzey and Anderies 2003) or innovation (Reuveny and Decker 2000; Dalton, Coats, and Asrabadi 2005; Decker and
Reuveny 2005). While these extensions are interesting, they assume that decision makers are
myopic, eliminating any role for resource management institutions. Since the way that institutions interact with the intrinsic social value
of population is our primary research question, we exclude these extensions for parsimony sake.
Our model gives a social planner the understanding of the effects that current actions
have on the future, institutions with costless
enforcement, and the foresight to implement
socially optimal plans. For our historical societies we assume the environment has a finite
carrying capacity, technology is static, individuals are identical with the same preferences
and no subsistence requirement, and population growth is described by an augmented
Malthusian model with the growth rate a function of consumption. The assumption of a finite
carrying capacity may not apply for modern societies but is appropriate here because our societies depended on closed environments for
their livelihoods and did not trade much with
others. The assumptions of no physical capital and static technology are also appropriate since our societies did not experience any
rapid economic growth such as our industrial
revolution. In fact, throughout the last 5,000
years, economic growth rates typically hovered around zero (Boyden 1987; Johnson 2000;
Dasgupta 2000b, 2001b). The assumption of
Malthusian population growth is also suitable,
since the relationship between consumption
and fertility persisted well into the 19th century (Heerink 1994; Perman et al. 2003).
The model incorporates implications of resource management institutions for future
populations but does not attempt to directly
control their sizes. Mathematically, we solve an
optimal control problem to find the level of resources harvesting each period that maximizes
social welfare. We assume that all the individuals possess the same endowments and preferences, form a production-consumption unit,
and can be characterized by a representative
6
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agent whose utility, u(t), which is a function
of the consumption in time t of a harvested
good, hc (t), and a manufactured good, mc (t).
The harvested good represents a broadly defined composite of renewable resources such
as trees, soil, edible plants, and fisheries. The
manufactured good represents a composite of
everything else.
The production functions of our goods, hp (t)
and mp (t), are defined as linear in labor. The
per capita level of harvest is also linearly related to the resource stock, S(t), representing
the increased ease in harvesting with resource
abundance. The time spent in production is
limited by a constraint: whatever is not spent
in harvesting is spent in producing the manufactured good. The fraction of the individual’s
endowment of one unit of labor spent in harvesting is f (t), the harvesting effort. Assuming
a Cobb-Douglas utility function for the representative agent, we get the following expressions:
(1)
u(t) = h c (t) m c (t)1−
h p (t) = S(t) f (t)
m p (t) = (1 − f (t))
where the units of mp (t) are defined from the
units of f (t). The per capita harvesting rate
is related to f (t) and S(t), with representing harvesting efficiency (Clark 1990). Thus, in
per capita terms our model is neutral to the
production level obtained with the same effort
when people harvest the resource under a centralized direction or a decentralized harvesting
from privately owned resource allotments.
We assume that goods are consumed when
produced and markets clear each period. Thus,
hp (t) = hc (t) and mp (t) = mc (t), which gives the
following expression:
(2)
u(t) = (S(t) f (t)) (1 − f (t))1− .
Following Lotka (1925) and Volterra (1926),
the dynamics of the system arise when population (the predator), L(t), is related to harvesting, and the resource stock (the prey), S(t), is
related to population. Total harvesting is determined by the harvesting of the representative agent times population size. The resource
growth is logistic, with an intrinsic growth rate
r and a carrying capacity K. The change in
the resource stock is determined by the difference between total harvesting and the natural
resource growth. We assume that fertility increases linearly with consumption ( > 0), and
Amer. J. Agr. Econ.
the intrinsic population growth rate, , is negative. Thus, the laws of motion for our system
are as follows:
(3)
d S(t)
= r S(t)[1 − S(t)/K ] − L(t)[f (t)S(t)]
dt
d L(t)
= L(t)[ + f (t)S(t)].
dt
Economic growth studies generally assume
that maximizing either an individual or an
aggregate SWF is the appropriate societal
objective. Parfit’s Theory X lies in between
with flexible value for both the quantity and
the quality of life. Like Parfit and Cowen, we
have no a priori moral grounds to identify the
“correct” weight for population in social welfare. However, we can study this issue by considering a family of SWFs with the utility of
a representative individual ( = 0) at one extreme and aggregate utility ( = 1) at the other,
as well as candidates for Theory X with 0 < <
1. Assuming the social planner faces a discount
rate, , these functions can be nested parametrically by
∞
SWF =
L(t) [f (t)S(t)]
(4)
0
× [1 − f (t)]1− e− t dt.
The decision makers choose an optimal harvesting plan, f (t), that solves the following
problem:
(5)
max
f (t)
∞
L(t) [f (t)S(t)] [1 − f (t)]1− e− t
0
subject to
d S(t)
= r S(t)[1 − S(t)/K ] − L(t)[f (t)S(t)]
dt
and
d L(t)
= L(t)[ + f (t)S(t)].
dt
This model represents a best-case scenario and
relies heavily on costless institutions that implement, monitor, and enforce socially optimal
harvesting plans. When the social discount rate
is infinite, this model coincides with the Brander and Taylor (1998) model, which assumes
a laissez-faire market solution and harvesting
under open access in the absence of resource
management institutions.
Good and Reuveny
Using Pontryagin’s maximum principle,
shadow prices L (t) and S (t) of population
and resource stocks, respectively, we write for
equation (5) the current value Hamiltonian:
Hcv ( f (t), S(t), L(t), L (t), S (t))
(6)
= L(t) [f (t)S(t)] [1 − f (t)]1−
+ S (t)(r S(t)[1 − S(t)/K ]
Social Collapse
7
dS (t)/dt
S (t)
S (t)[r − 2r S(t)/K − L(t) f (t)]
+
S (t)
⇒ =
L(t) f (t) S(t)−1 (1 − f (t))1−
S (t)
L (t) (t)L(t)
+
.
S (t)
+
− L(t)[f (t)S(t)])
+ L (t)(L(t)[ + f (t)S(t)]).
The implied first-order conditions for this
problem are the following:
(7)
max Hcv ( f (t), S(t), L(t), L (t), S (t))
The boundary conditions include the initial state of the system S(0) = S0 and
L(0) = L0 , and the transversality conditions and limt→∞ e− t L (t)L(t) = 0 and
limt→∞ e− t S (t)S(t) = 0. As usual, the maximization of Hcv with respect to f (t) equates
the marginal social value of harvesting with its
marginal costs, and the L (t) and S (t) equations relate social value of population and resource changes to the discount rate.
f (t)
at each point in time
⇒ L(t) S(t) f (t)−1 [1 − f (t)]1−
= S(t) f (t) (1 − )[1 − f (t)]−
− L L(t)S(t) + S S(t)L(t)
d S(t)
∂Hcv
= r S(t)[1 − S(t)/K ]
=
dt
∂S
− L(t)[f (t)S(t)]
d L(t)
∂Hcv
=
= L(t)[ + f (t)S(t)]
dt
∂L
dL (t)
∂Hcv
= L (t) −
dt
∂L
= −L(t)−1 S(t) f (t)
× (1 − f (t))1− + L (t)[ − ]
− [L (t) − S (t)]S(t) f (t)
dL (t)/dt
L (t)( + S(t) f (t))
+
L (t)
L (t)
(S (t)S f )
−
L (t)
⇒ =
L(t)−1 S(t) f (t) (1 − f (t))1−
L (t)
dS (t)
∂Hcv
= S (t) −
dt
∂S
= S (t)[ − r + 2r S(t)/K ] − L(t)
+
× f (t) S(t)−1 (1 − f (t))1−
− [L (t) − S (t)]L(t) f (t)
Results
We consider three scenarios. The first serves
as a reference case where people have no resource management institutions and ignore the
future. In the second, institutions take the form
of a social planner maximizing an aggregate
SWF with an infinite time horizon. The third
scenario differs by maximizing an individual
SWF. In this case, decisions either are centralized through a social planner or could be
decentralized through the assignment of property rights transferable to the extended family, where agents consider the utilities of future
generations as they consider their own. In each
case, we study the interior steady state and the
transition paths leading to it.
The monotonicity of the derivative of the
Hamiltonian with respect to f (t) implies a
unique interior steady state. The system also
has two corner steady states, both involving
L(t) = 0 with either S(t) = 0 or S(t) = K. The
L(t) = 0 and S(t) = 0 steady state describes a
barren environment that cannot support any
people and is not reachable from any initial
conditions with L(t) > 0. The solution with
S(t) = K is only reachable when =
0 (individual utility) and = 0 (no
discounting).
When → ∞, one gets L (t) = 0 and S (t) =
0, and the problem reduces to maximizing utility with respect to f (t) each period, taking L(t)
as given. This leads to a solution of f (t) = .
8
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Amer. J. Agr. Econ.
When 0 < < ∞, the steady-state attributes
are described by
(8)
⎧
⎨
( − r )K + r
4r ( + (1 − − ))
( − r )K + r
−
+
−
4( − )
4r 2
( + (1 − − ))
−
4( − )
1/2 ⎫
⎬
K (r ( − 1) + (r − ) − r + 2 )
−
⎭
2r ( − )
f = −
⎩
−
−
f
r
r
+
L =
f
2 K f 2
S=
u =
−
(1 − f )1− .
An interior steady state (with a positive population size) exists when − < Kf. As figure 1
shows, the situation occurs except when = 0
and = 0, leading to the corner steady state
L(t) = 0 and S(t) = K.
The comparative static analysis of equation
(8) yields unambiguous results only in extreme cases and the transient behavior cannot be solved analytically. We are thus forced
to study the behavior numerically. We use parameters and initial values for population and
the resource calibrated by Brander and Taylor
(1998) for Easter Island, and present effects of
altering the two key parameters of and .1
Figure 1 presents the steady-state solutions
for the resource, population, harvesting effort,
and the contemporary individual utility. In the
no-institutions scenario ( → ∞), the steady
state occurs at S(t) = 6,250 and L(t) = 4,972
and is the same regardless of the value of .
This case is represented by the horizontal lines
in figure 1 and serves as a baseline for solutions
that incorporate foresight. To improve visibility, the vertical scale for population in figure 1
is expanded compared to subsequent figures.
For the individual SWF, when = 0, the
equilibrium is at L(t) = 0 and S(t) = K.
This outcome is consistent with one of Parfit’s
1
The carrying capacity, K = 12,000; resource annual intrinsic
growth rate, r = 0.004; fertility parameter, = 4; intrinsic annual
net birth rate, = −0.01; annual harvesting efficiency rate, =
0.000001; and utility taste parameter, = 0.4. The initial conditions
are given by L(0) = 40 and S(0) = K.
conjectures: a world of bliss with no one to enjoy it. Intuitively, to maximize individual utility
when = 0, it is optimal to maximize S since
harvesting is an increasing function of S. Sustaining S(t) at its maximum value of K requires
that people do not harvest anything since the
resource growth rate is zero when S(t) = K.
These conditions hold only when L(t) = 0.
Compared with the no-institution case, the
harvesting effort for = 0 is lower, the resource stock is larger, population is lower, and
individual utility is higher (as the resource
stock is higher and time not spent on harvesting
is used elsewhere). Unless is less than about
1%, these differences are small, and for a moderate discount rate of 3%, the perfect institutions steady-state harvesting effort is similar to
the no-institution case. Diamond’s contention
that the resource effects occurred so slowly
that they went unnoticed, then, has a counterpart: even if noticed and understood, future
resource effects are too small to matter much
when = 0, unless is extremely low. This
suggests that, in order for policy to have any
effect on steady-state outcomes, planners need
to have a very low discount rate and a long time
horizon.
For the aggregate SWF, when = 0 there
are more people in steady state than for the
individual utility social welfare, and their utilities are lower, consistent with Parfit’s conjecture. As rises, the steady state converges to
the no-institution case, though the differences
from laissez-faire remain large even for high .
Compared with the = 0 case, valuing population explicitly in social welfare also increases
pressure on the resource and raises harvesting efforts. The equilibrium population reflects
tension between a desire to keep population
high, which strains the resource, and a desire to
keep the resource high because this increases
harvesting for a given effort. Since the future
consequences of actions have different dynamics, these forces respond differently to . When
= 0, the tension is resolved optimally at L =
4,697 and S = 6,878, and the harvesting effort
is lower than the no-institutions effort.
As rises above some small value (about
1%), the steady-state population for the aggregate SWF becomes smaller than it is for the
individual SWF. This result contrasts Parfit’s
Repugnant Conclusion, which takes place in
a static environment where resources do not
react to population change. It occurs because
of short-term gains due to a rise in population
resulting from harvesting (higher with an aggregate than an individual SWF) that are more
Good and Reuveny
Social Collapse
9
Figure 1. Equilibrium values of resources stock, population, harvesting effort, and utility for
different and values
than offset by the longer term losses due to resource depletion.
Figure 1 shows that increasing for any
increases the steady-state harvesting effort,
decreases the resource stock, and lowers individual utility. A similar pattern occurs for
these outcomes as rises. Initially, the differences between the steady states obtained
for the alternative values are small. As increases, the differences increase, but as continues to increase above some small level
(about 1%), the differences disappear. The
pattern for the equilibrium population is different. For = 0, population rises from 0 when
= 0 to 4,697 when = 1. If = 0 and
held constant, the equilibrium population increases with , reaches a peak, and then decreases with increasing . Consequently, for a
society that places a positive value on population in social welfare, increasing that value
above some critical level leads to the unintended consequence of decreasing the steadystate population. Values that we hold in a static
sense, ceteris paribus, and that we use to formulate the Hamiltonian do not necessarily follow
through to outcomes that occur in equilibrium,
which consider the dynamics of what keeps the
system there. Attempts to maintain a larger
population than the no-institutions population
level can be futile.
Next, consider Parfit’s Theory X. Figure 1
does not specify Parfit’s Theory X, but it says
what Theory X is not. Figure 1 suggests that
society should prefer low , ceteris paribus,
as this raises individual utility. However, the
relationship between and L suggests that
society should reject values of where population falls with , because for these values
both population and the individual utility decline, and consequently also the aggregate welfare declines. In our application, these values
of depend on . For 0.2% < < 7%, we reject values of larger than about 0.25, as these
values imply equilibria with both smaller population sizes and individual utilities. In practical terms, however, it seems that Theory X is
unnecessary. An individual SWF allows society to enjoy the highest equilibrium individual
utility, for a given . When is larger than 1%
or so, a society maximizing an individual SWF
also ends up with nearly the same population
size as with any of the other SWFs examined
here. The individual SWF also has the added
institutional benefit that can be supported by
private property rights and the presence of
markets.
Figures 2 and 3 present transient solutions for the resource (upper left), population
(lower left), share of labor spent in harvesting
(upper right), and the contemporary individual
10
xxx 2009
Amer. J. Agr. Econ.
Figure 2. Trajectories for resource stock, population, harvesting effort, and utility for different
values
Figure 3. Trajectories for resource stock, population, harvesting effort, and utility for different
values
Good and Reuveny
utility (lower right). Figure 2 presents trajectories with = 0 and = 1 for several discount rates. The trajectories are presented as
a function of years since arrival, because the
actual date when the colonization of Easter
Island began is uncertain. The no-institution
case ( → ∞) is shown by the solid black line.
Serving as a baseline, it presents a rapid population increase followed by a rapid decline
that is roughly consistent with the archeological data on Easter Island (Brander and Taylor
1998). For the aggregate utility social welfare,
we show trajectories for of 1, 4, and 8%. For
the individual utility social welfare, we show
trajectories for = 1 and 4% (the trajectories
for > 4% closely resemble the no-institution
solution).
The aggregate SWF population trajectories
differ from the individual SWF trajectories for
all values, exhibiting more volatility, and
higher booms that climax sooner. For = 0,
harvesting rises with , the resource stock falls
with , and the maximum population rises with
. In this case, foresight moderates the collapse. For = 1, harvesting is higher than for
= 0, and it falls for higher values of . The
resource is lower than for = 0 and rises with
, and the maximum population is higher than
for = 0 and falls with .
Intuitively, with an individual SWF, the planner can raise welfare only by raising individual utility. In contrast, with an aggregate SWF,
the planner can raise welfare by increasing
either individual utility or population. Since
aggregate SWF rises quickly with population,
the harvesting effort is higher for = 1 than
for = 0, increasing the maximum population. This, in turn, lowers the resource stock,
which eventually reduces population; in an effort to get population high quickly, it rises to
an unsustainable level, leading to resource depletion and population collapse. For = 1,
reducing exaggerates the collapse, as measured by the difference between the first peak
and first trough of the population trajectory.
A planner attributing greater importance for
future aggregate utility can increase the objective by promoting fertility. This, however, leads
to a relatively larger fall in resource stock and
eventually population. As rises, the planner
discounts aggregate utility more, and the drive
to raise population moderates.
Figure 2 also shows the effect these dynamics have on individual utility under the two
SWFs. In the case of the individual SWF, as the
discount rate declines, the trajectory of utility
Social Collapse
11
fluctuates less, and the values obtained at each
point in time are higher. In the case of the aggregate SWF, the individual utility also fluctuates less over time, but the values obtained at
each point in time are lower.
Our bounding analyses of the best and worst
cases have different implications depending on
which SWF is used. With individual SWF our
optimal trajectories present a best case in the
sense that they lessen the magnitude of collapse compared to the no-institutions case regardless of the value of . For aggregate SWF
the most stable solution is presented by the
laissez-faire case and the optimal control trajectory aggravates social collapse for any value
of shown in figure 2.
Figure 3 presents trajectories for of 1% and
of 0, 0.1, 0.25, 0.5, and 1. As rises from 0 to
1, the harvesting effort rises and the trajectory
is more volatile. The harvesting trajectories for
alternative values of appear similar, where
the effort first falls and then climbs up, but they
do not converge in the long run. Intermediate
values of have intermediate trajectories. The
resource stock also falls as rises from 0 to
1, more so in the first part of the trajectory,
climbing up to steady-state levels that differ
for the alternative . Clearly, the best value of
at averting social collapse is = 0. The intuition is similar to the one explaining the differences observed between the cases of = 0 and
= 1 in figure 2, hinging on the ability of the
planner to increase aggregate social welfare by
increasing the number of people in the system.
The larger is, the larger is the incentive for
the social planner to do so.
How do figures 2 and 3 compare to the historical record? The estimated date when Polynesians arrived at Easter Island varies from
AD 400 to 1000, and the peak population and
its date vary between 7,000–20,000 and AD
1100–1500, respectively (Ponting 1991; Bahn
and Flenley 1992; Stephens 1969). The island
was deforested by AD 1400–1600. When discovered in the 18th century, there were about
2,000–3,000 Rapanui (Hunt and Lipo 2001;
Flenley et al. 1991; Flenley 1996; Brander and
Taylor 1998). Thus, our population trajectories
are consistent with both an assumption of no
institutions and a broad range of alternative
assumptions about optimal resource management institutions.
Figure 1 shows that the relationship between
steady-state population and for an aggregate
SWF is highly nonlinear, particularly for low
discount rates. Population rises with until a
12
xxx 2009
Amer. J. Agr. Econ.
Figure 4. Trajectories for resource stock, population, harvesting effort, and utility for very low
values
value of about 0.1%, falls until 1%, and then
rises with . Figure 4 presents a detailed examination of trajectories for of 0.5, 0.2, and
0.1%. For the individual SWF, the decline in
continuously dampens the boom-bust cycle.
For of 0.1%, the approach to steady state
does not exhibit a boom-bust cycle. For the aggregate SWF, a decline of from 0.5 to 0.1%
also dampens the boom-bust cycle, reversing
the tendency described in figure 2. Yet even a
of 0.1% does little to mitigate the boom-bust
cycle in this case.
Intuitively, with an aggregate SWF the
planner can raise welfare by increasing individual utility or population. Procedures for
implementing population increases (which depend on harvesting) can be implemented more
quickly than those leading to increases in utility (which rely on the slower process of resource recovery). When is large, only these
population-increasing efforts show up with importance in the SWF. For a low enough , these
dynamic changes and the efforts to increase
SWF by increasing individual utility through
the effect on future values of S(t) become more
meaningful, providing society the incentive to
reduce harvesting now, lower population size,
and increase the resource stock.
Discussion and Implications
Our results show that perfect resource management institutions with an infinite time horizon and a reasonable discount rate by today’s
standards would have done little to avert social collapse. Our focus on these institutions
to avert collapse, rather than on direct population control, may seem controversial. We
have no doubt that direct population control could have averted collapse. For example, the Polynesian society on the island of
Tikopia averted collapse by controlling population through abstinence, abortion, contraception, one-way ocean voyages with no real
chance of success, and infanticide (Kirch 1997).
We could avoid collapse in our model by
including a population-cap constraint or by
seeking optimal population size in order to
maximize utility through controlling consumption and population growth, as in Dasgupta
(1969). Cowen (1989), however, argues optimal population need not maximize some utility, but is what people living sequential lives
would opt for if they would choose how to
live out each life, and some, including Dasgupta (2001a) himself, view this as the wrong
problem to solve. At any rate, both approaches
Good and Reuveny
eliminate the linkage between population and
the environment and imply restricting procreation autonomy, a potential violation of human
rights. Our work, then, has a second interpretation. Finding that any resource management
institution fails to avert social collapse, we must
conclude that some alteration of the population dynamic is necessary if the collapse is to
be averted.
One might argue that our model may not
accurately describe the world today since it
ignores physical capital, innovation, demographic transition, and subsistence-level harvesting and assumes a finite carrying capacity.
In addition, the transient phenomenon of a social crash depends on the starting values being
far from equilibrium. Yet these issues are not
straightforward.
While replacing resources with capital can
alleviate pressures, critical resources may not
have good substitutes. Societies can alleviate pressures by innovation, but progress,
particularly in harvesting productivity, can
also increase degradation and intensify boombust trajectories. Whether innovation can
ultimately alleviate all pressures is unknowable. The overall effect of demographic transition in the model is unclear, depending on
the strengths of the Malthusian and nonMalthusian phases and the discount rate. Additionally, many poor countries still exhibit
Malthusian fertility. Subsistence requirements
are prevalent in developing countries and were
likely prevalent in historical societies, but their
addition to the model will increase the volatility of trajectories, further deviating from our
focus on the best-case scenario.
The assumption of a carrying capacity is appropriate at least in the context of historical
societies. While notable scholars such as Arrow et al. (1995) argue that our planet has
a carrying capacity, the issue is controversial. If this assumption is not relevant for our
planet, our analysis has little applicability today. However, if it is relevant, perpetual economic growth may eventually face problems
of the type faced by historical societies. Countries may raise their effective carrying capacity
through innovation or trade. However, trade
cannot raise planetary carrying capacity, and
innovation may be too infrequent to do it. In
the past, humans went through two major carrying capacity altering events: the switch from
hunter gatherers to an agrarian society and the
industrial/green revolution. With a recent major carrying capacity altering event and with
a population of over 6 billion, doubling every
Social Collapse
13
fifty years or so, our society may be on a trajectory at least qualitatively similar to that of
these historical societies.
While our model may not accurately describe the world today due to its relative simplicity, it may offer insights on the way to
think about population and environment interactions. Beginning with moral philosophy,
what is at issue in Parfit’s view is how production varies with population. If, as some
may argue, production grows proportionately
to, or faster than, population, the Repugnant
Conclusion does not obtain. Yet we bring a
new argument to this debate: it is not only
a matter of how production varies with population, but also how population varies with
production. Parfit’s view is static: increasing
population gives individuals a smaller share of
the resources, leading to a decline in individual utility. However, Parfit creates a world in
which population increases in a vacuum and
additional people exist from nothing. When
population growth depends on production, a
Repugnant Conclusion motivated by Kavka’s
(1982) “obligation to procreate” cannot occur.
The impetus to increase population driven by
an aggregate social welfare is limited by the environment’s ability to support it. Parfit’s expectation that an individual social welfare would
lead to a few people with high utilities is also
prevented in this case, as long as > 0.
Consider next Parfit’s quest to determine the
weight of population in social welfare. We find
that for greater than 0.5%, the maximum
equilibrium population is achieved for of
about 0.25. So, while we cannot say what Theory X is, we can say what Theory X is not. Since
individual utility falls with , increasing beyond this level is counterproductive. However,
on a practical level, Theory X seems unnecessary. An individual SWF produces roughly
the same equilibrium population as any other
function for alternative levels of is associated
with the highest individual utility, and has the
advantage that it is the only optimal solution
that is supportable by a market mechanism.
Our analysis also has implications for the
economic growth literature. Growth studies
typically assume that population growth is exogenous, eliminating collapses by assumption.
Making population growth endogenous introduces potential for cycles in population and
economic variables. Second, the tendency of
growth studies to focus on the steady state
may miss key insights. We find that for even
modest discount rates that equilibrium population levels are roughly similar across welfare
14
xxx 2009
functions, but the transition paths differ dramatically. An equilibrium analysis would mistakenly lead one to conclude that the choice
of SWF did not have much impact on population. Third, the growth literature seems to treat
the choice among the aggregate or individual
SWFs casually, as though the issue is unimportant. We show that the two functions yield different dynamics; the similarity between them
cannot be taken for granted.
Finally, our results underscore the importance of discounting. There has been resurgence of debate on this topic in light of its
relevance to addressing climate change (Stern
2007; Nordhaus 2007; Weitzman 2007). Typically, researchers rule out a zero rate as a matter of mathematical convenience in dynamic
optimizations (Barro and Sala-i-Martin 2004;
Chiang 1992). Others argue that it is fair to use
a zero rate since future generations do not take
part in our decisions, but are affected by them
(Dasgupta, Mäler, and Barrett 1999). Parfit’s
(1984) static logic implies a zero rate. Yet
Arrow (1999) notes that a small rate leads
to large current savings and if future people
do not save much, their profit is at our expense. Still others argue that future generations would likely be more advanced and,
therefore, we need not necessarily sacrifice
for them (Nordhaus 2007). Taking a middle
ground view, Weitzman (1999) suggests using
the lowest plausible future rate as a response
to uncertainty about this rate in the far future.
Which discount rate do people use? Estimates vary widely, including hundreds of
percent per year (Frederick, Loewenstein,
and O’Donoghue 2002). Nordhaus and Boyer
(2000) use 5%. Stern’s (2007) report, which
calls for immediate sharp decline in greenhouse emissions, assumes = 0.1%. Nordhaus
(2007) argues this rate is inconsistent with today’s real interest rate and uses 1.5%, as also
does Cline (1992). Weitzman (2007) argues the
driving factor should be how much insurance
is needed to offset the chance of a calamity due
to climate change and suggests a range of 0.7
to 2.7%.
The above-mentioned debate centers on
whether intergenerational equity drives us to
a moral position for using a very low , or for
basing our decision of on the empirical values
that people seem to use for private decisions.
In the end, the decision is a purely normative
one and we cannot resolve the issue anymore
than those other writers can. We note that our
conclusions have the same flavor as theirs: sacrifices that have long-term benefits are unlikely
Amer. J. Agr. Econ.
to be undertaken unless the social discount
rate is much lower than any rate we can currently motivate empirically.
Conclusion
An emerging theme in studies on the collapse
of historical civilizations attributes the breakdown to anthropogenic environmental degradation. We have described this linkage for
the Sumerians, Maya, Rapanui, and Anasazi.
These societies are atypical because of the levels of greatness they achieved, and since their
collapse was abrupt and spectacular. Rather
than empirically comparing these societies to
others that may have had better institutions
(as Diamond and others do), we construct a
counterfactual case by hypothetically endowing them with perfect resource management
institutions with complete foresight and infinite time horizons. We find that, with a reasonable discount rate by today’s standards, these
best-case institutions would have done little
to avert social collapse. When society maximizes an individual SWF, a discount rate of below 0.5% is necessary to prevent a collapse in
population. Otherwise, like our no-institutions
scenario, it too collapses. When the society
maximizes an aggregate SWF, it collapses
abruptly even for tiny discount rates near
1%.
Having begun our article with a historical
slant, one may wonder which SWF guided the
leaders of our four societies. One might conjecture they chose an aggregate SWF because
it implied the ability to amass armies and build
monuments. An equally plausible conjecture
is that societies that ended up building monuments or amassing armies did so as a consequence of the large workforces they had. Given
the weak archeological evidence, we cannot
draw any conclusions about the direction of
this causality in a Granger sense. However,
in the context of our model, both causal behaviors imply monument building and army
amassing outcomes that are more likely to occur with the larger populations generated by
an aggregate SWF compared with the individual SWF or no-institutions cases. We are more
likely to notice these societies because of it. We
speculate that historical societies that placed
a prime importance on the welfare of the individual may have still collapsed, though less
spectacularly and without our notice.
At any rate, it is tempting to argue that our
societies collapsed because they were myopic,
Good and Reuveny
had no resource management institutions, did
not grasp their problem, did not pass information over time, or did not notice slow
environmental changes. These assertions are
essentially circular because the evidence one
typically uses in supporting them is the collapse
itself. We find that, even if our civilizations
had the institutions, understanding, information records, foresight, and SWFs commonly
used today, they would still have collapsed. It
seems reasonable to assert that these collapses
were socially optimal.
One of the unique features of our approach
is that social collapse is evaluated by the ability
of the political, economic, and environmental
system to avoid drastic fluctuations in the transient solution to the control problem rather
than its effect on the steady state and its neighborhood. Indeed, the choice of SWF, discounting, and the use of institutions have little effect
on the equilibrium population at all but the
lowest discount rates, yet the effects on population dynamics along the transition path are
stark.
[Received January 2008;
accepted February 2009.]
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