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The Primitive Hunter Culture, Pleistocene Extinction, and the Rise of Agriculture



The hypothesis that megafauna extinction some 10,000 years ago was due to "overkill" by Paleolithic hunters is examined using an economic model of a replenishable resource. The large herding animals that became extinct, such as mammoth, bison, camel, and mastodon, presented low hunting cost and high kill value. The absence of appropriation provided incentives for the wastage killing evident in some kill sites, while the slow growth, long lives, and long maturation of large animals increased their vulnerability to extinction. Free-access hunting is compared with socially optimal hunting and used to interpret the development of conservationist ethics, and controls, in more recent primitive cultures.
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Journal of Political
The Primitive Hunter Culture, Pleistocene Extinction, and the Rise of Agriculture
Author(s): Vernon L. Smith
Journal of Political Economy,
Vol. 83, No. 4 (Aug., 1975), pp. 727-756
Published by: University of Chicago Press
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The Primitive Hunter Culture, Pleistocene
Extinction, and the Rise of Agriculture
Vernon L. Smith
University of Southern
and California Institute
of Technology
The hypothesis that megafauna extinction some 10,000 years ago was
due to "overkill" by Paleolithic hunters is examined using an economic
model of a replenishable resource. The large herding animals that
became extinct, such as mammoth, bison, camel, and mastodon, pre-
sented low hunting cost and high kill value. The absence of appropriation
provided incentives for the wastage killing evident in some kill sites,
while the slow growth, long lives, and long maturation of large animals
increased their vulnerability to extinction. Free-access hunting is com-
pared with socially optimal hunting and used to interpret the develop-
ment of conservationist ethics, and controls, in more recent primitive
1. Introduction
Many archaeologists and other scientists believe that the available
evidence supports the hypothesis-startling to nonspecialists-that the
unusual incidence of large-animal extinctions throughout the world
during the late Pleistocene period was caused, to an important extent,
by Paleolithic hunters. Even if true, the extinction of large animals is
but one of the more dramatic examples of the very substantial impact
that primitive as well as modern man has had on his "natural" environ-
ment (Heizer 1955). The purpose of this essay is threefold: (1) to acquaint
Support from the National Science Foundation, the Center for Advanced Study in
the Behavioral Sciences, and the Fairchild Distinguished Scholar program at Caltech is
gratefully acknowledged. I wish also to express my debt to Robert F. Heizer for a great
deal of help and encouragement in the course of many discussions on the topic of this
paper and for providing me with a guide to the relevant archaeological literature.
If I have been a poor student, he bears no responsibility for the final product.
[Journal of Political Economy, 1975, vol. 83, no. 4]
(C 1975 by The University of Chicago. All rights reserved.
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economists with some of the evidence and hypotheses from other sciences
concerning the role of primitive man as hunter par excellence, (2) to
parameterize some important features of these observations and hypoth-
eses in the context of a model of the primitive hunter culture which, in
the face of animal extinctions, turns increasingly to agriculture, and
(3) to attempt to demonstrate to archaeologists and other scientists the
potential that economic analysis may have in unifying and integrating
this body of evidence and conjecture. In what follows, the reader should
bear in mind that much of the "evidence" to which reference will be
made is subject to dispute; certainly the interpretation of that evidence
is controversial.
It is my belief, in reading the archaeological literature in this area,
that there is a natural economic rationale for "overkill" as a possible
explanation for the large-animal extinctions which has not been ade-
quately or fully articulated. Briefly stated, the large extinct herbivores
hunted by primitive men, such as the mammoth and bison, were
gregarious herd animals, easily located, and apparently easily approached
and struck with crude missile weapons or stampeded into "jumps."
Multiple kills were therefore likely, but because of their large size, even
a single kill represented high value. The combination of low hunting
cost and high value would make large animals the most economical prey.
Furthermore, in the absence of appropriation or other incentives for the
individual hunter to attach value to the live animal stock, wastage killing
was possibly commonplace. Biologically, the larger genera of animals are
characterized by slow growth, long lives, and long periods of maturation,
and are therefore the most vulnerable to hunting pressure. That is, the
hunter harvest is more likely to exceed net biological growth, causing a
decline in biomass.
In Section 2, some of the facts, conjectures, and interpretations of late
Pleistocene extinctions are summarized. In Section 3, a particularly
simple form of existing models of production from common-property
replenishable resources is used to stylize the hunter-agrarian economy.
This permits a comparative-statics treatment of the effect of prey size,
vulnerability, and value, and of predator technology and population, on
prey biomass and extinction potential (Section 4). Socially optimal
hunting is modeled in Sections 5 and 6 and the Appendix, on the assump-
tion that institutional mechanisms of control (property-right systems or
cultural or legal constraints that internalize the social costs of individual
hunter actions) are adequate to support the optimal sustained-yield
harvesting of prey. These optimal patterns are compared with free-
access hunting. This analysis is used to develop the conditions under
which it may be optimal to "conserve" or, alternatively, to destroy a
hunted species and to compare such cases with the corresponding free-
access solutions.
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2. The Hypotheses of Pleistocene Extinction
One of the great scientific puzzles studied extensively by archaeologists,
paleontologists, and geologists is the cause and process of the unprece-
dented wave of large-animal extinction in the late Pleistocene period.
Martin (1967, pp. 75, 82-86) suggests the loss of over 200 genera world-
wide and lists 80 late Pleistocene extinct animals in continental North
America of which 49 had an average adult body weight in excess of
110 pounds (the "megafauna"). These megafauna included camels,
horses, bison, mastodons, llamas, ground sloths (including a giant the
size of an elephant), mammoths (the largest, Mammuthus
imperator, was
13 feet tall and considerably larger than the African elephant), beavers,
short-faced bears, armadillos, several saber-tooth cats (including Smilodon,
the tiger), shrub oxen, moose, tapirs, antelope, and many more. Of the
49 genera in the late Pleistocene, 33 became extinct at a time which
could have roughly coincided with the arrival of the Paleo-Indians in
North America. Of the 31 smaller extinct mammals listed by Martin,
only one could have been associated with man. Those terminal Pleistocene
megafauna apparently found in "direct association" with man include
ground sloths, camels, mastodons, horses, mammoths, shrub oxen, tapirs,
and the extinct bison. Evidence of human predation is clearest in the
case of mammoth and extinct bison. That Clovis fluted-point hunters
killed mammoth around 1,000 years ago is hardly open to question, and
sometime later, perhaps after the sudden disappearance of the mammoth,
the Folsom point was developed and used to kill now-extinct bison
(Haynes 1964). The Clovis, Folsom, and subsequent Scottsbluff point
projectile technologies seem specifically designed for big-game hunting.
Although accelerated extinctions had occurred in periods earlier than
the late Pleistocene, they had affected marine organisms, plants, and the
smaller mammals as well as the larger mammals. Furthermore, the
pattern of worldwide extinction of the larger mammals seems suspiciously
to correlate with the migration chronology of man. This has led Martin
(also see Sauer 1944) to the hypothesis that Pleistocene extinction was
due to overkill by Paleolithic hunters armed with the stone-tipped spear,
fire, and the communal hunting party. Martin (1967, p. 75) states:
Except on islands where smaller animals disappeared, extinc-
tion struck only the large terrestrial herbivores, their ecologically
dependent carnivores, and their scavengers. Although it may
have occurred during times of climatic change, the event is not
clearly related to climatic change. One must seek another cause.
Extinction closely follows the chronology of prehistoric man's
spread and his development as a big-game hunter. No con-
tinents or islands are known in which accelerated extinction
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definitely predates man's arrival. The phenomenon of overkill
alone explains the global extinction pattern.
A scenario built around this hypothesis goes as follows: For 200,000
years prior to the arrival of man, the large herbivores of North America
were increasing in diversity and experienced no shrinkage of range. They
evolved and survived over tens of millions of years in the presence of
numerous environmental changes and predators. Several genera had
emigrated over the Bering land bridge to the hospitable environment of
North America. Consequently, the North America of 15,000 years ago
was comparable to nineteenth-century Africa in terms of the huge,
strange, "unlikely" beasts that grazed the plains and browsed the forests
and brush. Then, approximately 12,000 years ago, the first Paleolithic
men, ancestors of many of the present-day Indians, arrived across the
exposed Bering land bridge. They were hunters, perhaps driven to wider
migration by the dwindling herds of prey in Eurasia. They brought with
them the culture, skill, and technology of big-game hunting-the spear,
perhaps the atlatl (spear thrower), fire, and stone projectile points. At
some time in this migration, they developed the Clovis fluted point-a
work of craftsmanship
in stone carefully adapted to the demands of killing
large animals. These hunters preyed on gregarious herds of mammoth,
bison, and perhaps mastodon, camels, tapirs, horses, and other animals
which were easy to locate and probably showed little fear of the new
predators. By 1
1,000 years ago, this efficient new predator had wiped out
the mammoth and was concentrating on now-extinct species of bison.
The bison may have been killed by jumps (as was common within
historic times by Indians) and perhaps fire drives, and by this time the
Clovis point was giving way to the Folsom projectile point. The pop-
ulation of Paleo-Indians expanded rapidly across North and South
America, appearing at the southern tip of South America by 10,000 years
ago, and, one may conjecture, lived affluently for as long as the game was
plentiful. As the herds disappeared, their predators, the saber-toothed
tiger, dire wolf, and hyena, became extinct. Hunting effort was directed
at smaller, less vulnerable game which produced a relatively meager
existence and was eventually replaced by an agricultural technology in
which subsistence depended on crops of corn (and later beans and
squash) supplemented with small game.
The scenario is plausible but is by no means an established fact. That
man arrived about 12,000 years ago is probable, as there is no firm radio-
carbon dating of any earlier evidence of man (Haynes 1967). That man
hunted mammoth and, later, two species of now-extinct bison is surely a
certainty based on documented kill sites (Haury, Antevs, and Lance 1953;
Gross 1951; Agogino and Frankforter 1960; Leonhardy 1966).1 Hester
' This is conjecture on my part, but it seems plausible that the Bering land bridge
might have acted as a filter through which only the most able hunting tribes could have
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and Wendorf (1962, pp. 166-67) report that the most common hunting
pattern for both mammoth and bison was to stalk and kill animals while
they were drinking in a pond or stream. Killing occurred by means of
spears aimed at the thoracic region, although in one site, the presence of
boulders suggests they were used to kill wounded mammoth. A second
pattern was the stampede, probably present in three kill sites of early man.
The animals (extinct bison) were driven into a stream or over a cliff,
sometimes in numbers as high as several hundred. At the Olsen-Chubbuck
site in Colorado, well-preserved and carefully excavated remains of bones
and artifacts prove that about 8,500 years ago some 200 Bison occidentalis
were stampeded into an arroyo only 5-7 feet deep. The injured animals
were killed by projectile points generally of the Scottsbluff type. About
75 percent of the animals were then systematically butchered (Wheat
1967). The killing of bison in the sixteenth to eighteenth centuries by
stampeding them over carefully selected "jumps" is quite well established
(Kehoe 1967; Butler 1971).
Whether the early Americans stampeded animals by fire drives is not
known, nor is it critical to the overkill hypothesis. The first account of the
southwestern Indians, by Cabeza de Vaca, records that the inhabitants
of what is now southwest Texas burned favorable animal ranges in order
to deprive the animals of forage and force them to frequent areas where
they could be more easily hunted (Covey 1961, p. 81; cited by Jelinek
1967, p. 197). Burning of this type (and there is abundant evidence for
the occurrence of fires in association with man) could have been a more
effective means by which man contributed to Pleistocene extinction than
by the occasional fire drive. In the case of herd animals such as Bison
are easily stampeded, it is not clear that fire drives were even functional
unless it was to ensure that the confused animals would not stampede in
the wrong direction!
That the mammoth was gone by 10,000-1 1,000 years ago is also likely,
based on radiocarbon dating. That there existed a big-game hunting
tradition is also clearly established by the widespread occurrence of the
Clovis projectile point type. It is found from Florida to Nova Scotia, in
the high plains, the Southwest, across the Midwest, and in the South.
It was a large projectile, 7-15 centimeters long and 3-4 centimeters
wide. Bases were concave, and a fluting or channeling extended from the
base up to one-half the length of the point. They were flaked by percussion
and the base edges ground down to prevent cutting of the thongs that
passed. The bridge would not have been a suitable viaduct for a gatherer culture,
"because no likely food sources but game existed for most of the year in the tundra areas
they traversed" (Jelinek 1967, p. 195). Hence, the early North Americans may have
been the product of a selection process that favored only the most mobile, skilled, and
dedicated hunters. This could help explain why megafauna extinction in North America
was more rapid than in Europe, Asia, and Africa.
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secured them to the spear shaft.2 Early American points were probably
too heavy, large, and ill-designed for arrow tips. Spears, thrust or thrown
as a javelin, or darts-perhaps propelled by the atlatl-were the main
tool of the hunt. Clovis points date back to 12,000 years ago and appar-
ently evolved into the Folsom. The Folsom point dates from about
0,000- 11,000 years ago and is much less widely dispersed than the
Clovis. The Folsom point is more delicately made, with fine-edge flaking,
and is associated with the hunting of the extinct Bison antiques. The
Scottsbluff and several similar points date from about 9,000 years ago
and are associated with the killing of the slightly smaller extinct Bison
occidentalis. By 7,000-10,000 years ago, projectile points had been adapted
to the killing of modern smaller game such as sheep, the so-called
American bison, deer, and antelope. A primitive maize, perhaps in the
early stages of domestication, has been dated by radiocarbon to around
5,000-6,000 years ago (Mangelsdorf and Smith 1949).
Until recently, the commonly accepted cause of late Pleistocene
extinctions was climatic change and a reduction in grassland areas. This
view probably still predominates. Thus, according to Guilday (1967,
p. 121), "the fact that the late Pleistocene extinctions were so widespread
and geographically almost simultaneous does call for a major overlying
cause, however. I suggest that the prime mover was post-Pleistocene
desiccation. Evidence for such an episode is present on all continents, and
its effects would have been both swift and lethal. It may have been the
spur to turn man from hunting to a life centered around animal husbandry
and agriculture." This states the climatic as against the "overkill"
extinction hypothesis. The desiccation referred to is associated with a
drier climate following recession of the last great ice sheet.3 A variant of
the climate hypothesis attributes extinction to the effect of more severe
seasonal fluctuations (colder winters, warmer summers) on those mammals
with longer gestation periods (Slaughter 1967). But here we have an
identification problem, for it is the mammals with longer gestation
periods, longer periods of maternal care, and longer lives that are most
vulnerable to hunting pressure.
2Jelinek (1967, p. 196) notes the significance of this design technology for the hypoth-
esis of a vulnerable fauna (mammoth): "Grinding would prevent the edges of the point
from cutting the lashing that bound it to a shaft if the point was subjected to repeated
lateral stress" as would occur "in a point on a thrusting spear or lance whose shaft
remained in the hand of the hunter after it penetrated the animal-a technique that
would be most effective against a relatively easy quarry and of little use against a skittish
and fearful prey."
3 However, desiccation followed the three previous glaciation periods and in one
instance was probably more severe. "Recent pollen evidence from western America
seems to indicate that in at least some areas occupied by the extinct fauna the conditions
following the retreat of an earlier glaciation (Illinoian) were probably more arid and as
warm or warmer than at present. Thus conditions of temperature and aridity do not
appear likely as direct causes of extinction" (Jelinek 1967, p. 194).
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Finally, of course, there is the multiple-cause, or combination, hypoth-
esis, here defined by Hester (1967, p. 170): "I take the view that Pleisto-
cene man could not have caused the extinction of the North American
megafauna until after natural causes had greatly reduced the population
of each species." However, Hester's study of historical extinction (extinc-
tion or near extinction of species since European settlement) in North
America lists (1) hunting by primitive man and (2) hunting by civilized
man with firearms as the first and second major factors in order of
Much of the earlier skepticism surrounding the "overkill" hypothesis
stems from a disbelief (to some extent perhaps a romantic disbelief) in the
ability of primitive men to accomplish, with primitive instruments, the
destruction of such huge creatures, already the prey of the formidable
saber-toothed tiger and dire wolf. Yet these men were not genetically,
in terms of intelligence and skill capacity, that different from modern
man. Also, modern studies of predation by the timber wolf on Isle Royale
indicate that moose stock may be strengthened by the killing of old,
weak, and diseased animals (Mech 1970). The large kill sites of mammoth
and bison suggest wastage-killing beyond immediate butchering require-
ments-so that there is some reason to believe that man was orders of
magnitude more effective in predation than his animal competitors.
Overfishing in historic times is well known. The demise of great whales
(and recently the Alaskan king crab) is well known, and the capacity of
man for wholesale rivalrous killing, even with the most primitive of
weapons, is dramatically documented in the following Palo Alto Times
(March 13, 1973) account:
In the course of a few hours early Sunday a shoal of 637 pilot
whales were driven into a narrow fjord on the island of Vaga
(Faeroe Islands, Denmark) by stone-throwing islanders in an
armada of small boats. Then they were slaughtered with long
spears and knives in a gruesome spectacle that has been part
of Faeroese life for centuries. The whales churned their tails
furiously in shallow water ... The shoal of whales was one of
the biggest since more than 2000 pilot whales were killed in
one day east of here 20 years ago.
Some time between 12,000 and 3,000 years ago the early Americans
turned from an exclusively hunting and gathering culture to one based
more and more on agriculture. I assume that men found it to their
economic advantage to make this change. It is perhaps significant to the
overkill hypothesis that man did not turn from big game to smaller game
except as a supplement to agriculture, as a result of the large-animal
extinctions. Even the plentiful American bison apparently was hunted
only incidentally until after the introduction of riding horses by the
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Spanish in historic times.4 It may have been the case that only the largest
herding animals were economical to hunt with the tools of Paleolithic man.
Perhaps the weakest element in the overkill hypothesis is the scant
direct evidence that early North Americans hunted extensively any of
the extinct animals other than mammoth and bison. 5 Extinct horse,
camel, tapir, mastodon, and antelope have been found in mammoth and
bison kill sites but not so as to clearly demonstrate death by the spear6 (or
other means attributable to paleohunters). At the Lehner Ranch site in
Arizona (Haury, Sayles, and Wasley 1959) nine mammoths and at least
one each of horse, bison, and tapir occur in a single bone bed. Directly
associated with these bones was evidence of man's destruction of at least
some of the animals. Thus, Clovis spear points were found in situ among
ribs of a mammoth and a bison. However, Irwin-Williams (1967, p. 346)
reports that at a kill site near Puebla, Mexico, "the character of the
assemblage (bones and projectile points) indicates hunting and butchering
activities involving mastodon, mammoth, horse, camel, four-horned
antelope, etc." But the fact that such evidence is not firm or more wide-
spread is puzzling in view of the fact that there is much evidence for the
hunting of the horse as well as mammoth at an earlier date in Europe.
Sites such as Solutr6 in France contain the remains of an estimated
100,000 horses (MacCurdy 1933, p. 173; cited byJelinek 1967, p. 195).
But for the purposes of this paper, it is enough to assume that the earliest
Americans subsisted primarily on mammoth and bison, turning to
gathering, agriculture, and supplemental game as these animals became
The abandonment of agriculture and the return to the hunting of
4 When the horse was reintroduced to the New World by the Spanish in the sixteenth
century, Equus had been extinct throughout the Americas for only about 8,000 years.
In North America horse bones are among the most common Pleistocene fossils (Martin
and Guilday 1967, pp. 41-42). Upon reintroduction, the horse reproduced and spread
rapidly and thrives today i n the wild, as does the burro, under extremely arid conditions
in Nevada, Arizona, and Utah. This development makes it the more puzzling that late
Pleistocene extinction of the horse would have been due to desiccation.
5 The survival of African megafauna is often cited as negative evidence for the overkill
hypothesis. But this view has been challenged by several writers. Martin (1967, pp.
110-11) notes that some 50 genera (about 30 percent) disappeared in Africa during the
Pleistocene. Most of this extinction occurred before 40,000-50,000 years ago and "seems
to coincide with the maximum development of the most advanced early Stone Age
hunting cultures.... The case of Africa neither refutes the hypothesis of overkill nor
supports the hypothesis of worldwide climatic change as a cause of extinction." Jelinek
(1967, p. 194) also suggests that Africa is not comparable to those areas of the northern
hemisphere where extinction occurred, because the African flora was more favorable
for gathering. Thus, gathering may have been sufficiently economical to have reduced
the hunting stress to which the African megafauna was exposed.
6 Martin (1973, pp. 969-74) explained the absence of kill sites for horse, camel, and
ground sloths by the hypothesis that they were killed too quickly and easily to leave
extensive fossil traces. The idea is thatt vulnerability to overkill and archaeological
visibility are inversely related.
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bison by some American Indian groups in historic times (seventeenth
and eighteenth centuries) is well established. According to Wedel (1936),
"the introduction of the horse deeply affected the Pawnee, as it did most
of the plains tribes... [leading to] the tendency toward a nomadic,
bison-hunting mode of life made possible by the horse.... From a
sedentary tribe the Pawnee became one in which the chase and maize
culture shared almost equally."
The more revolutionary effect of the horse on the "fighting Cheyenne"
of the northern plains is reported by Strong (1940, pp. 359, 370, 375-76).
Wedel (1940, p. 327) reports that the Cheyenne and Arapahoe abandoned
their villages, pottery arts, and horticulture to become bison hunters,
while the Plains Apache (already subsisting on bison herds in 1541 as
reported by the Spanish explorer Coronado) merely adapted the horse
to a preexisting bison culture. Apparently, the vast encampments with
large tepees of bison hide familiar to later European settlers depended on
a substantial increase in the bison harvest made possible by the riding
3. A Model of the Primitive Hunter-Agrarian Economy
I have characterized the Paleo-Indian as a big-game hunter who turned
to agriculture as his chief prey became extinct but whose descendants
returned to a more nomadic hunting economy after the introduction of
the riding horse. This stark representation will be stylized in an economic
model of subsistence based on free-access hunting and/or agriculture in
which the biomass of game is determined by biological growth con-
siderations that are autonomous but are affected by the harvest product
of the hunt.
Consider an economy of population n, each member of which is free
to engage in hunting or agriculture7 as a productive activity. Hunting
activity is applied to a single homogeneous species of biomass, M, such as
mastodon, mammoth, or bison, and yields a per capita output of m per
unit of time. Agricultural activity is applied to the production of a single
homogeneous crop, such as corn or beans, and yields a per capita output
of c per unit of time. Then H units of hunting labor per capita, and A
units of agricultural labor per capita, are employed, with L = H + A,
the total per capita labor available. The production function for corn is
c = g(yA) and for meat m = f (fpH,
Mln), in which it is assumed that
increasing the stock of game and of hunters by the same proportion has
no effect on the per capita output of meat. The parameters P and y are
efficiency parameters for labor in hunting and farming, respectively.
7 I shall refer to the alternative to hunting as "agriculture," but it could just as well
be gathering. To the early North Americans, the only viable alternative to hunting prior
to 5,000-6,000 years ago would seem to have been gathering.
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nf, F
nf [,3H(M), M/n]
F(M) kG(M)
FIG. 1
Thus, an increase in /3
increases the hunting efficiency of labor. The
effect of a technological change in weapons, or the introduction of the
horse into the Plains Indian culture, is assumed to be captured by an
appropriate increase in /3.
It will be understood, without always making it explicit, that all of the
variables are dated, that is, M = M(t), H = H(t), and so on with n,
L, fl, and y given constants.
The hunted resource is assumed to be subject to a biological growth
law (see, e.g., Lotka 1956; Smith 1968; Plourde 1971), which, in the
absence of predation by man, is given by M'(t) = F[M(t)]. It will be
assumed that F(M) can be written in the form F(M) = kG(M),
k > 0, G"(M) < 0, M > 0; G'(M0) = 0, G(M) ? 0, 0 < M < M,
where 2M is the maximum naturally sustainable stock of the hunted
biomass (see fig. 1). The parameter k expresses the biotic growth potential
and therefore the predator vulnerability of the species in the sense that
an increase in k increases the growth rate of the stock for 0 < M < A?
but does not increase the natural equilibrium stock, R. It may be sup-
posed that a great many factors influence k. A change in climate affecting
reproduction, infection by disease, or a change in food availability would
alter k for a given genus. Among different genera, k would vary with
growth characteristics such as feeding habits, energy requirements,
gestation period, age of maturation, life span, and efficiency of food
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conversion. Generally, the larger genera of mammals are slower growing,
have longer gestation periods, require longer periods of maternal care,
and live longer. These factors spell greater vulnerability to hunting
pressure in that a given tonnage reduction in biomass will require longer
to regenerate. In this regard, the modern forms of those extinct species
hunted by or found in association with primitive man have relatively
long lives (and gestation periods): llama, 20 years (10-1 1 months
gestation); camel, 45 years (10-14 months); horse, 25-30 years (11-12
months); elephant, 60-80 years (22 months); and bison, 18-22 years
(9 months). It is perhaps significant that the American bison (Bison
familiar to nineteenth-century European settlers dates no further back
than Paleo-Indian times and is believed to represent a dwarfed form
of the extinct Bison occidentalis (Edwards 1967). Also, mammoth kill sites
commonly contain the remains of the giants of the species (Mammuthus
imperator, 4.0 m high; Mammuthus
columbi, 3.6 m high), both larger than
the largest African elephant (3.52 m high) (Martin and Guilday 1967).
On the matter of size, age of maturity, and speed of growth, Hammond
(1961, p. 321) observes that a considerable reduction in size has occurred
over the last 50 years in the major beef breeds of cattle due to deliberate
selection for early maturation in body proportions. Hence, under common-
property conditions we have Paleolithic hunters selectively harvesting the
larger, slower-growing mammals due to emphasis on consuming the stock.
But under appropriation with domesticated animals and emphasis on
sustained-yield harvesting, investment favors the smaller, earlier-maturing
animals which provide a higher biomass growth rate.
When each member of a population of size n applies H units of labor
to hunting, this yields a harvest of nm = nf (flH, Mmn)
units of the
replenishable resource, and the net growth rate of the resource is given by
M'(t) = F(M) - nf (/H, Mln). (1)
Each of n individuals is assumed to choose (H, A) so as to maximize a
utility function u(c, m) subject to L = H + A and the above production
constraints. By substituting c = g[y(L - H)] and m = f (/3H, Mmn),
the problem can be expressed
max u{g[y(L -H)], f (JH, M/n)
If u(g, f) is concave in H for given M, an interior maximum is defined
by the condition
-U1yg' + U2/fl = 0? 0 < H< L. (2)
Equation (2), requiring the marginal rate of transformation
between corn
and meat to be equal to their marginal rate of substitution, can be
regarded as determining an economic equilibrium between H and M.
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That is, for each resource stock, M, there is a corresponding economical
expenditure of hunting labor so that (1) and (2) together determine a
differential equation in M'(t) and M(t) only. An illustrative solution to
(1) and (2) is shown graphically in figure 1. The curve at the bottom
of figure 1 represents growth in the biomass of the hunted resource net
of harvested units. The equilibrium harvest function, nf (/H, M/n), is
shown intersecting the growth function, F (M), at two points Mu and MS,
both of which satisfy (1) and (2) when M'(t) = 0. Stability in the
neighborhood of a point M' where M'(t) = 0 requires dM'/dM < 0.
Differentiating (1) and (2), one can easily verify that the sign of dM'/dM
is ambiguous, even where F'(M) < 0, given only the concavity of u,
and g. In the illustration of figure 1, point MS is shown as locally stable
while Mu is not.
More specific results will be derived and illustrated graphically, while
those parameters essential to the subsequent applications of the model
are retained, by introducing the following simplifying assumptions:
1. u(cm) = c + vm, u1 = 1, u2 = v. Corn and meat are perfect
substitutes, and value is measured in subjective corn-equivalent units.
The parameter v is the society's subjective value of meat relative to corn.
Thus, u is the per capita income (welfare) of the society.
2. f (flH, M/n) and g[y(L - H)] are increasing, concave, and
homogeneous of degree 1, with f (0, M) = f (JH, 0) = g(O) = 0.
Hence, letting x = flHn/M be hunting intensity, that is, total hunting
labor per unit biomass, we can write
f (TH, Mmn)
= (M/n) )(x), Ob'
> 0, 4" < O0
4(0) = 0,
A = 0'(X) > 0, f2 = - x+' > 0,
g[y(L - H)] = y(L - H),g' = 1.
Applying these assumptions, (2) becomes flv4'(x) = y, or
=/fl __ - +b'(r/v), (3)
where r is a relative efficiency parameter, that is, the efficiency of labor
in agriculture relative to hunting, and r/v is the real wage or the oppor-
tunity cost of hunting (the value of the corn forgone). Under these
assumptions, (1) becomes
M'(t) = F(M) - M4[4(4-'1(r/v)]. (4)
At an equilibrium point, M*,
M'(t) = F(M*) - M*0[0'(-l)(r/v)] = 0, (5)
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and the stability condition, near M*, is
dM' F'(M*) -04'(-l(r/v)] < . (6)
Equilibrium can also be expressed as occurring at M* where the average
biological growth rate, a(M*; k), is equal to the harvest intensity (output
per unit biomass), 4. That is, rewriting (5) and substituting F(M) =
(M), kG(M*) a(M*; k) = 4[4'(1)(r/v)] (7)
4. Comparative Statics of Hunting
The marginal effect of any of the parameters k, v, n, and /3
on equilibrium
hunting effort, H*, and equilibrium biomass, M*, can be deduced by
differentiating (3) and (7). By differentiating u* = y(L - H*) +
(vM*1n)4(x*), the effect of such parameters on equilibrium per capita
income can also be ascertained. In determining such effects, any factor
which reduces the equilibrium stock of the resource may also, in the
limiting case, produce extinction (see Gould [1972] and Clarke [1973]
for analyses of extinction). From (7), it is clear that extinction of a species
due to hunting pressure will occur if a(O; k) < 4)[4)'(
')(r/v)], that is,
if rnv
< o'{0 -1[a(0; k)
}. It follows that unless + (x) has an upper bound
below a(O; k) there is always a real wage rate small enough (i.e., a return
on agricultural labor that is small enough) to produce extinction.
The effect of changes in the indicated parameters on (M*, H*, u*)
is summarized below:
1. dM* a dH* H- dM*
dk ka' dk M* dk
d- = (v/n)(0 - x4') dM* > 0,
where a' = [8a(M*; k)]/OM* < 0. Consequently, if larger animal
species have a lower biotic growth potential (i.e., smaller k), this will
tend to (i) reduce the equilibrium stock (and increase the possibility of
extinction), (ii) encourage agricultural effort at the expense of hunting,
and (iii) decrease per capita income.
2 dM* _ (0 )2 dH* H* dM* M*+' H*
d/= a'4)"f df3 M dfl n#2o" f
du* v dM*
--V n+ qX~}dfl 0
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The greater the efficiency of hunting labor, the smaller the equilibrium
stock of animals. One implication is that larger animals and/or gregarious
animals that tend to congregate in herds, such as mammoth, bison,
antelope, camels, and llamas (among the extinct genera), would have
comparatively high visibility and low search cost, making them easier
prey and increasing ft.
Thus, Edwards (1967, p. 149) argues that "human
technology, including use of missile weapons, greatly reduces the counter-
attacking defensive advantages of large size and emphasizes concealment
and speed of flight. At this point ... the genetically selected optimum
body size of many forms declines sharply." Also, the introduction of the
riding horse into the Plains Indians culture by the Spanish would have
had the effect of increasing bison hunting efficiency and reducing the
stock of bison. Armed with the horse and the bow, the Plains Indians in
another 200 years could possibly have depleted the stock of bison as
effectively as did Buffalo Bill and the U.S. cavalry.
However, changes in hunting efficiency could either increase or
decrease hunting effort and per capita income (dH*/dp 2 0, du*/d/3
0). Greater hunting efficiency could release labor for agricultural employ-
ment or so reduce the animal stock that the society is made poorer. It
would appear that this was not the effect of the horse on the Plains
Indians, many of whom were uprooted from their agrarian activities but
who achieved greater affluence as bison hunters. This affluence could
have been a temporary phenomenon; that is, in the short run, given the
animal stock, we have dM*/dfl = 0 and du*/d/3
= vH*O' > 0. The
short-run effect of an increase in hunting efficiency is always to increase
per capita income.
dM* (0,)2 dH* M*+' H* dM*
3. = - SO, ~~~~ ~
-~ + 20
dv va'<" dv /3vn0" M* dv <
du* M*q5 v dM*
dv n + n(OVn dv
The greater the consumption value of the hunted resource, the smaller
will be the equilibrium biomass. Hunting labor and per capita income
could also be smaller depending on how much the biomass of animals
is depleted (if the species becomes extinct, then, obviously, hunting will
dM* dH* H
dn dn n
du* - (vM*l/n2) (O -x') < 0.
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In a mixed economy, increasing the human population causes no change
in the stock of animals but reduces hunting effort and per capita income.
With constant returns in agriculture, equilibrium requires total hunting
intensity to be constant. Consequently, any increase in population will be
offset by a corresponding decrease in each individual's hunting labor.
This is a very strong empirical implication of the model, for it asserts that
(under our technological assumptions) once a hunting society diversifies
into agriculture (or gathering), the pressure of increasing population on
animal stocks disappears. Of course, the moment decreasing returns occur
in agriculture, this result no longer holds.
It should be noted that the assumption of a mixed economy is presumed
not to apply to the Paleo-Indians. For a pure hunter culture we have
H = L, and the equilibrium animal stock is defined by F(M) -
iIb(flLn/M) = 0. Hence,
dM (fL/M)V < 0
dn D
if D = a' + (f3Ln/M2)0' < 0, which is required for stability.
5. Institutional and Analytical Aspects of Optimal versus
Free-Access Hunting
Economists have long been familiar with the proposition that uncon-
strained nonpriced access to any common-property resource such as a
fishing or hunting ground (Gordon 1954; Scott 1955; Smith 1968;
Plourde 1971) leads to the inefficient use of such resources. This in-
efficiency takes the form of a reduction of the natural biological stock
of the resource below the optimal stock required for sustained-yield
harvesting. The phenomenon can be described as an instance of market
(or price mechanism) failure after Bator (1958) or of property-right
failure after Demsetz (1967). It is perhaps more accurately described
as an instance of incentive failure caused by cultural or institutional
inadequacies. What fails is the private incentive of the individual to
harvest (and "conserve" the stock) at socially optimal levels over time.
In principle, optimality can be achieved by (1) simulating the market
that has failed, for example, by instituting a user charge-somewhat
erroneously called a "tax"-for the resource, thereby inducing the
individual to economize user payments by conserving his use of the
resource; (2) instituting a property-right system which induces the indi-
vidual to conserve his use of the resource as a means of maximizing
the return on his property; (3) constraining individual hunting activity
by social or legal restrictions such as quotas, sharing rules, licensing, or
prohibitions; and (4) limiting the hunting harvest by enculturating
voluntary conservationist values or behaviour.
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Property rights, social or legal restrictions on individual harvesting,
and the enculturation of conservationist behaviour have all been used
extensively and ingeniously by primitive peoples at one time or another.
However, the evidence is recent, for there appears to be no evidence to
suppose that Paleolithic practices exhibited such sophistication. It is the
hypothesis of this section that sometime after the extinction of the mega-
fauna, human culture developed a sensitivity to the need to prevent
overharvesting. Whether man as a superpredator was in fact the principal
agent in the extinction of the large herbivores and their dependent
carnivores and scavengers, it is plausible to assume that men saw parallels
between hunting and the loss of the valued prey, from which arose oral
and religious traditions, myths, and superstitions which had the effect
and perhaps the intention of conserving common-property resources. At
some point the ancestral message became a directive to "take sparingly
of the bounty of nature."
Heizer (1955) provides pages of documented examples of primitive
strictures on the harvesting of replenishable resources. One of the most
common techniques was the private ownership of land-fishing holes,
hunting grounds, nut-bearing trees, and grass seed areas (see Heizer
[1955, p. 4] for numerous reference summaries on land ownership).
Constraints on harvesting from common-property lands took many forms.
Great Lakes Indians stripped only a portion of the fiber off basswood
trees in order that the wound would heal and the tree live. Vancouver
Island Indians "never fully strip the bark from a cedar tree lest the tree
die and its spirit curse the man who peeled the bark and he die also"
(Heizer 1955, p. 4). The Choctaw had laws regulating the game that
could be killed by one family, with strict accounting by the captain of
each band. The Kaska trap marten in a given area only every 2 or 3
years. The Iroquois spared the females of hunted species during the
breeding season; the sparing of pregnant females was widespread. The
Yurok had "game laws" the violation of which would cause loss of
"hunting luck" (Heizer 1955, pp. 4-5). The Naskapi of Labrador are
cited as typical of numerous tribes that believe animals and plants were
created to help man (Heizer 1955, p. 6). In return for killing an animal,
the hunter must protect it from profane treatment, such as wasting the
animal or letting dogs gnaw its bones, lest the animal take offense and
spoil the success of the hunter. Certain species may be hunted by some
tribes but avoided by others in the belief that the tribe's ancestry traces
to such species. Many tribes believe that game is watched over by super-
natural authorities who become angry with men if too many deer are
killed or if they merely wound the animals (Heizer 1955, p. 7).
Many more such examples could be cited, but evidence for con-
servationist ethics and institutions (defined as any set of strictures, laws,
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or practices which limit the harvesting of common-property resources)
is widespread among primitive peoples in historic or near historic times.
Such primitive practices may appear to be exceedingly crude rationing
devices. But every efficient price system has its dual equivalent quota
system, and modern legislators no less than primitive peoples find it more
natural to think in terms of quota restrictions on external effects than in
terms of prices.
Among primitive peoples who have invented property-right systems,
there are instances of sophistication that would rival the modern property
deed. Thus, among the Karok private ownership of a particular fishing
ground did not mean owning the land along the river but owning the
right to fish a given stretch of the river regardless of who owned the land
(Kroeber and Barrett 1960, pp. 3-4). A fishing right might entitle the
owner to use the spot every third night and day, with the right transferable
by sale or inheritance. Similarly, the right to hunt or share in the hunting
of sea lions on a particular rock was owned, and each person on a par-
ticular stretch of coast had rights to some cut of a beached whale, with
residents of other areas denied such rights except by inheritance or
purchase (Kroeber and Barrett 1960, p. 115).
The possibility of the existence of property rights or quota regulations
governing hunted resources raises the issue of optimal versus free-access
harvesting of species. In the following analysis, the assumptions of the
previous simple model of production from a common-property hunted
resource and an appropriated agricultural resource will be used to state
an optimal control model. Primarily, the model will be used to study the
conditions for optimal versus free-access species extinction.
If 3 is the time preference discount rate for an individual and we
assume that instantaneous utility, u = c(t) + vm(t), is additive over
time, then total welfare for the economy is
lim ue
T- oo
to be maximized subject to the production function and resource con-
straints and the resource growth equation (1). Making the substitutions
c = y(L - H) and m = (M/n)4)(x), the current-value Hamilton-
Lagrange criterion is (Arrow 1968)
= y(L - H) + ( 0
)p(fln) + giM[a(M;k) - Go(fM )]
to be maximized with respect to the control variable H, where 0 <
H < L. Necessary conditions, according to the maximum principle, are
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that < O. then H = 0,
if = O then 0 < H < L (8)
OH > 0, then H L=
= - (9)
M'(t) = I[a((M; k) - 4(x)], (10)
and the transversality conditions
lim e-t u(t)AM(t) = 0, lim e-tMu(t) > 0.
Letting 4 = nit, condition (8) implies the following:
> 0j fHf), then H = 0,
r H
if = ,then 0 < H < L, (11)
< (/iJHn) then H = L.
Consequently, if we define 40 = 0'(0), 40 = (v - r/4') if 4/ > r/v,
with 40 = 0 otherwise, and let the separating boundary between a pure
hunter economy and a hunter agrarian economy be ~ (M) v -
r/0'(flLn/M), then (9) and (10) can be written8
[ - F'(M)], if4o < ,or 4 > do;
[ - F'(M)] - (v- ,) (12a)
4 (t) = I X r0 r
(Vr]( -) (-)
if 4(M) < <
? 0; (12b)
- F'(M)] -(v -L[q(fln ) - (kiln)q(I3Ln)]
if fl> n > r
- or 4 < ,(M); (12c)
8 These conditions are sufficient as well as necessary if T is concave in M for given t
and t and for H set at its maximizing level. Under the assumptions in the text, T is
concave in M for v 2 4 2 0.
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Ma(M; k), if > 4o; (13a)
M {a(M; k) - ( r
M' (t) if=M ~0 13b
M a(M; k) - M if
4 < (M). (13c)
Note that the equality condition in (11) reduces to (3) for free-access
harvesting when 4 = 0. The 4 is interpreted as the market value of a
live unit of the animal stock. Ordinarily we would think of 4 as being
nonnegative. A negative value would correspond to animals that are a
public nuisance requiring a bounty for optimal social control. This value
is zero to the individual in the absence either of adequate property rights
in live animals or of harvesting restrictions which impute value to live
animals. Thus, enculturated limitations on free-access harvesting, such as
lead hunters to believe they will receive supernatural punishment if they
harvest too much game, impute a positive value, 4, to live animals. One
does not need to pass judgment on the merits of such devices for social
control over the chase to appreciate their behavioral (and imputed price)
Since 4 is the social marginal value of a live animal, the quantity
v - 4 in (11)-(13) is the net marginal value of a harvested animal.
Since 0 - (f3Hn/M) b' for H < L is the marginal physical product of
the biomass of game (biological capital), equations (1 2b) and (1 2c)
require the net marginal value productivity of the game stock,
(v - 4)
[0 - (fiHn/M) '] for H < L, to equal net interest on investment
in a live animal less capital gains, 4[b - F'(M)] - 4', where the interest
rate, 6 - F'(M), is reckoned net of the biological "own" rate, F'(M).
The biological rate, F'(M), is analogous to a capital depreciation rate
when F'(M) < 0 and a capital appreciation rate when F'(M) > 0.
Equations (12) and (13) provide two first-order autonomous differential
equations in [4(t), M(t)] which, together with the transversality con-
ditions and initial conditions, must be satisfied along an optimal bionomic
development path. Paths satisfying (12) and (13) will be characterized
by the usual phase diagram representation in (d, M) space.
In figures 2-4 the set of points E(4) is defined by the condition 4'(t) = 0
in (12) and represents the stationary state asset demand for the animal
stock. The set of points B(4) is defined by the condition M'(t) = 0 in
(13) and represents
the stationary state asset supply of animals. Properties
of these functions and the phase diagrams in figures 2-4 are derived in
the Appendix.
Along E(4) the value of the marginal product of the game stock is
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sto 7~~~-----
X (M***) UP:
FIG. 2
equal to interest net of biological growth. Hence, an increase in the
animal stock reduces biological growth, increases net interest, and reduces
the price of live animals. Along B(4), the harvest intensity is equal to
average biological growth. For a mixed economy, increases in the live
animal price reduce the harvest per unit of biomass and increase the
game stock.
An optimal equilibrium path yielding a mixed economy in the station-
ary state is shown in figure 2. Starting with initial conditions (Al, 4), it is
socially desirable for the culture to specialize in hunting. As the game
stock is depleted and its value rises, optimality requires the economy
eventually (beginning at P) to begin agricultural production. In long-run
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X (M)
\E, ..~~~~~~~~~~~~
a/ ~
p~~ ~~~ 4
/|X 1'''
61 4 Ms a M*-
I4 /1
FIG. 3
equilibrium at (M**, *),both hunting and agriculture reach stable
levels of production. Figure 3 illustrates an optimal path for an economy
that begins as, and remains to the end, a culture of specialized hunters.
In this case, the free-access economy produces a socially optimal equilib-
Agricultural specialization may occur following a period of hunting
that causes extinction. This is illustrated in figure 4 in which 40 > 40,
where B() -Eb4) = O
. At all prices 4 ab, the harvest rate
exceeds the growth rate for every live animal stock, causing extinction.
At all prices 4 2~ 40, interest on investment in animals exceeds the net
value of the marginal product of live animals for every animal stock.
, Qbo
> XeO
implies that, along an optimal development path, the
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A F 4
economy perpetually consumes capital and must eventually wipe out the
stock of live animals and specialize thereafter in agriculture. Along an
optimal path beginning at (M, ~) one would at first observe a pure
hunter culture, then a mixed economy, and ultimately an agrarian
The effect of free-access harvesting and its contrast with an optimal
development path are obtained by setting ~ = 0 for all t. This condition
replaces those stated in (12) and corresponds to the nonexistence of a
market in live animals or of any equivalent valuation system for expressing
the opportunity costs of the current harvest. In effect, the "demand" for
biological capital is perfectly elastic at =0. If ~' > 0, the free-access
economy eventually harvests to extinction as in figure 4. If ~' < 0, such
an economy harvests short of extinction and conserves an animal stock
M*= B(0) > 0 as in figures 2 and 3.
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Figures 2 and 4 apply to "high-impatience" economies, 6 > Fo. For
low-impatience economies, 6 < Fo, extinction of an animal species will
never be optimal since the net cost of biological capital 6 - F'(M) -) 0
as M Ma, making it optimal to stop biological capital consumption
at M- Ma for all capital prices 4 > 40. This means that the static asset
demand for biological capital becomes completely inelastic at M = Ma
4 2 40, as illustrated by M = E6(Q) in figure 3. However, the concept
of "high" or "low" impatience must be measured relative to the biotic
growth potential of the hunted species. Using the simple parameterization
F(M) = kG (M), a relatively high (low) impatience economy is defined
by 6/k > G' (6/k < Go). If Fo = kG' is finite for any given species, it is
clear that there always exists a cultural impatience rate, 6, high enough
that it may be optimal tQ harvest the species to extinction.
6. Comparative Stationary States of Hunting
For interior solutions 0 < H < L, the effect of the parameters (6, r, v, k)
on the optimal stationary-state level M** is obtained by implicit differen-
tiation of the following equations:
-[6 F'(M**)] - (v - )( - x') = 0, (14)
a(M**;k) -?(x*) = 0, where = r/(l)( r) (15)
Since 6 -F'P+q -F"
D = (?0,) a < 0,
"(V -4
we deduce dM** _ _ +
)_ a
>_ 0
dk kD
dM** O'(6 -F') + 1'(k - x') > 0
dr /"(v - n4)D
dM** (0l)2(6 -PF) <O
dv O"
(v - D
dM** (h,) 24 0.
d6 b"(v - D
The optimal stationary-state animal stock is smaller (and the prospect of
extinction greater) the lower the biotic potential of the species, the lower
the efficiency of labor in agriculture relative to hunting, the higher the
cultural value placed on meat, and the higher the culture's preference
for present over future consumption. Certain features of the prey stock
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may affect both r and k. Thus, if larger animals have a lower biotic
potential and are easier to hunt, this implies lower r and lower k, yielding
a magnified decrease in M**.
These results and the similar conclusions of Section 4 for free-access
hunting do not provide any new evidence on the causes of Pleistocene
extinction. They are offered in an attempt to demonstrate the use of a
coherent economic framework for the study and evaluation of extinction
or other hypotheses concerning the primitive hunter culture.9 It is hoped
that the framework of this paper will enhance the possibility of a more
comprehensive evaluation of the limited qualitative cross-cultural,
chronological, and regional data on hunting-gathering-agricultural
activities in primitive societies.
In this Appendix,
of the differential equations (12) (13) and their
will be developed
in greater
detail. First
1. From (12a) we have 4'(t) j 0 according
as a
F'(M) j 0 for 4 > o0.
Let 3
F'(M) ? 0 for 0 < M < Ma, and a
F'(M) > 0 for M > Ma.
Then 4'(t) < 0 in the region (4 > 40, 0 < M 5 Ma), and ('(t) > 0 in
(4 > X M > Ma).
This is illustrated
by the vertical
arrow above
40 in figure
for the case in which Ma = 0, that is, 3
F'(O) 2 0. It should be noted that
40 = 0 in the event that q0 < rv, that is, hunting
is uneconomical relative
at all hunting intensities
even when live animals have zero value.
This degenerate case leads to agricultural
from time zero, which
must be distinguished from agricultural
the hunting
prey to extinction.
2. From (12b)-(12c), let M = E(4) be defined by the set of points
[(4, M) | ,'(t) = 0, <
? < 'o M 2 0]. The condition
{'(t) = 0 requires
3-F'(M) = < :(M) <
[ < Boa
-(4) M (M~) qY (M)] , 4 < (M), (16b)
is, the net rate of interest must
the relative
value marginal
of the biomass of prey. The function
E (4) implied by the interior
is derived
graphically in figure 5 for two distinguishing
cases: (1) if Fo < 3, the
curve labeled E (4) is obtained; (2) if Fo > 3, the curve labeled E6(4) is the
result. The curve in quadrant
I of figure
5 is the relative value marginal
ductivity of the biomass.
II shows net interest as a function
of the
of game. For each price of live game 4 such that net interest
9 For example, if the environment was economically more favorable for gathering in
Africa than it was in North America, then the overkill hypothesis is not inconsistent
with the greater survival of megafauna in Africa.
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I[ \I )X (H+
I I~ ~
- - - -- - - - - - - - -
I I 8
-Fo X
M__ I I
____ _ X_/
FIG. 5
relative value marginal productivity, we associate a biomass M 2 0 in quadrant
IV. Curve E(> or E,(4) represents economic equilibrium in the capital market
where capital gains (losses) vanish. Quadrant IV also illustrates the boundary
,(M) =_
[v - r/'(/flLnIM)] separating the interior region {(M) < st < do,
representing a mixed hunter-agrarian economy, from the region 4 < 4(M),
representing the specialized hunter economy. This boundary is monotone in-
creasing, ,'(M) = - (r0'j8Ln)1[( 0') 2M1] > O.
with lim ,(M) = oo if lim 0'(x) = O. and lim s(M) = ,0.
M0 E
O+ x0 M - X0
Some key properties of E (4) are summarized below.
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a) At M= O, if a-Fo 2 O, then = e < 40 < v satisfies (16a). At
M = Ma, if
c - Fo < 0 and 3 = F'(M8), then 4 = 40.
b) lim E-(M) - 0
since - (flLn/M) O' -* 0 in (16b) while 3 - F' > 0.
c) The function E(4) is monotone decreasing; that is, differentiating (16a)
and (16b), it can be verified that dM/dX < 0 and from (12b) and (12c) that
@4'(t)/a, > 0, 4 ? 40, as illustrated by the vertical arrows in figure 2.
Now consider (13):
1. From (13a), M'(t) j 0 according as 0 < M g YE,
if > 40 as indicated
by the horizontal arrows above 40 in figure 2.
2. From (13b) and (13c), let M = B(4) be defined by the set of points
[(4,M) I
M'(t) = 0, <
?c , M ? 0]. This implies
(Hn) if Hn ,- (r)
a (M; k) = f( 1 7)
A) (M )n if flS < r/'-)( 4
For each <
?0, this equation provides a corresponding M 2 0. Points on
M = B(4) represent biological equilibrium in the prey stock. In figure 6, let
k = k1, giving the monotone decreasing percentage average rate of biomass
growth a(M; k1). Since this function is everywhere below the harvest per unit
of biomass when the economy specializes in hunting, it follows that we have an
interior maximum 0 < H < L for every 4 and a corresponding M determined
by the intersection a(M; kl) - 0{0'(ql)[r/(v - I)]}. At 4 = I"' equilibrium
is at Q't, while at 4 = r" < f 's equilibrium is at Q". For this case the function
B(4) will be in the region 4(M) < c <
?0 as shown in figure 2. But if k = k2
the constraining boundary set defined by 0(fiLn/M) intersects a(M; k2) at two
points corresponding to a biomass M1 and M2 as shown in figure 6. Consequently,
for any 4 such that 4l < 4 < 42, where M1 = B(41), M2 = B(42) (e.g., 4 = 4
in fig. 6), hunting intensity is constrained at the level flLn/M and the harvest
intensity is 0(fiLn/M) < a(M; k2), with M'(t) > 0 for all M1 < M < M2.
Thus, at 4, an increase in the stock above M1 reduces harvest intensity by more
than growth intensity because hunting effort cannot be increased. The stock
rises until M2 is reached, where the growth rate is depressed to the level of the
harvest. Hence, equilibrium on M = B(4), just above 41, is reached because of
naturally occurring diminishing returns to biomass growth and cannot be
influenced by the control H. In figure 3 this means that the function B(4) intersects
the boundary g(M) and is discontinuous at M1 = B(41). This phenomenon is a
property of labor-scarce economies, since for L large enough B(d) will be entirely
in the interior set for a mixed economy.
Some key properties of B(4) in the interior [E(M) < 4 < 40, M ? 0] are:
a) At M = 0 4 =
-b j 0 satisfies (17).
b) AtM = Ma(?a; k) = 0 and Xo=
c) The function B(4) is monotone increasing, that is, from (17), dM/dX > 0.
From (1
3b) and (1
3c), referring to figures 3 and 6, if:
i) 4 < 1, then (a) M'(t) > 0 if M < B(4); (b) M'(t) < 0 if B(4) < M <
M1; (c) M'(t) > 0 if M1 < M < M2; and (d) M'(t) < 0 if M > M2.
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\4a(m; kQ \
L~~~~~~~~~~~~~~~~~~~~~~~~~~~"-) r-r
FIG. 6
ii) 1< < 42 then (a) M'(t) > 0 if M < M2 and (b) M'(t) < 0 if
M> M2.
iii) 2 < 4 < don then (a) M'(t) > 0 if M < B(4) and (b) M'(t) < 0 if
M > B(4). Each of these directions of motion is illustrated by the horizontal
arrows in figure 3.
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... My interest is in explaining the origins of this historically important event. Specifically, I test the hypothesis, put forth by Vernon L. Smith (1975), that megaherbivore extinction in prehistoric times caused the Neolithic Revolution. 1 Smith links the emergence of agriculture with the famous overkill hypothesis by Martin (1967a), who insists that mass extinction of large mammals in the Late Pleistocene is attributed to over-hunt by primitive men. Without property rights over available resources, hunters kept hunting prey mammals and some went extinct, reducing available biological resources. ...
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... the very distant past. Moreover, larger mammals are biologically more vulnerable because they tend to have longer periods of gestation and maternal care, older age at sexual maturity and the first parturition and lower birth rate, which lowers reproductive success (McDonald, 1984;Smith, 1975Smith, , 1992Johnson, 2002;Brook and Bowman, 2004). These biological restrictions are underlying elements that give rise to the variation in extinction. ...
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