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ELEPHANTS
Michael Kremer
Charles Morcom
96-17
June 1996
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
ELEPHANTS
Michael Kremer
Charles Morcom
96-17
June 1996
MASSACHUSETTS INSTITUTE
OF Trr
'
lJULlS
1996
UBttnn;u.~
96-17
Elephants'
By Michael
Kremer and Charles Morcom
2
3
Massachusetts Institute of Technology
Cambridge, MA 02139
(617)253-3504
kremer@mit. edu
chasm@mit.edu
June 25, 1996
Existing models of open-access resources are applicable to non-storable resources, such as fish.
Many open-access
and
resources, however, are used to produce storable goods. Elephants, rhinos,
prominent examples. Anticipated future scarcity of these resources will
increase current prices, and current poaching. This implies that, for given initial conditions,
there may be rational expectations equilibria leading both to extinction and to survival.
Governments may be able to eliminate extinction equilibria by promising to implement tough
anti-poaching measures if the population falls below a threshold. Alternatively, they, or private
tigers are three
agents,
may be
able to eliminate extinction equilibria by accumulating a sufficient stockpile of
the storable good.
'We thank Marty Weitzman, Andrew Merrick, John Geanakoplos, and participants at seminars at the JFK School of
Government at Harvard University, the Hoover Institution, Princeton, and Yale for comments, and Ted Miguel for
research assistance. This research
was supported by
the Center for Energy and Environmental Policy Research at
MIT.
Massachusetts Institute of Technology, National Bureau of Economic Research, and Harvard Institute of
International Development.
Massachusetts
Institute
of Technology.
Introduction
I.
Twenty-nine percent of threatened birds worldwide and more than half the threatened
mammals
in Australasia
and the Americas are subject to over-harvesting [Goombridge, 1992].
Most models of open-access resources assume
Gordon, 1954; Schaefer, 1957]. While
inappropriate for
Although
30%
many
that the
may be
this
other species threatened
of threatened
mammals
good
is
non-storable [Clark, 1976;
a reasonable assumption for fish,
by over-harvesting,
are hunted for
as illustrated in Table
presumably non-storable meat,
hunted for fur or hides, which are presumably storable, and approximately
by the
live trade
10%
I.
20%
are
are threatened
[Goombridge, 1992].
African elephants are a prime example of a resource which
to protect as private property,
and
Africa's elephant population fell
et al., 1990].
it is
Dealers in
is
used to produce a storable good.
from approximately
Hong Kong
is
1
.2
technologically difficult
From 1981
to 1989,
million to just over 600,000 [Barbier,
stockpiled large amounts of ivory
[New York Times
Magazine, 1990]. As the elephant population decreased, the constant-dollar price of uncarved
elephant tusks rose from $7.00 a pound in 1969 to $52.00 per pound in 1978, and $66.00 a
pound
in
1989 [Simmons and Freuteo., 1989]. The higher prices increased incentives for
poaching.
Recently, governments have toughened enforcement efforts with a ban on the ivory
trade, shooting of
poachers on sight, strengthened measures against corruption, and the highly
publicized destruction of confiscated ivory.
accompanied by decreases
4
This crackdown on poaching has been
in the price of elephant tusks [Bonner, 1993].
changes reduce short-run ivory supply as well as demand,
4
In
September 1988, Kenya's president ordered
that
it is
not clear that the
poachers be shot on sight, and
took over Kenya's wildlife department.
1
Since these policy
in April
fall in
price
1989 Richard Leakey
would have been predicted under a
this decline.
However, the
fall in
static
price
model, and indeed most economists did not predict
paper, under which improved anti-poaching enforcement
by allowing the elephant population
dynamic model
consistent with the
is
may
set forth in this
increase long-run ivory supply
to recover.
the model, anticipated future scarcity of storable resources leads to higher
Under
current prices, and therefore to
more
intensive current exploitation. For example, elephant
poaching leads to expected future shortages of ivory, and thus raises future ivory prices.
Since ivory
for
a storable good, current ivory prices therefore rise, and this creates incentives
is
more poaching today. Because poaching
rational expectations paths of ivory prices
creates
its
own
incentives, there
may be
multiple
and the elephant population for a range of initial
populations.
In order to gain intuition for
equilibria,
it
is
that
in
xq.
Then
the population at the
model
initial
multiple rational expectations
two period example,
is
for
which we thank
a breeding season during
which population
population of x. Following the breeding season, an
harvested. Denote the elephant population at the beginning of the harvest season
year one as
the
may be
each year there
grows by an amount B(x) given an
is
there
useful to consider the following
Marty Weitzman. Suppose
amount h
why
the population at the
end of the harvest
as simple as possible,
end of the harvest
in year
we assume
two
will
that the
be xq
-
in
h\
world ends
year one will be x
+ B(xq
after
-
two
hi)
-
h2
years.
.
-
h\,
and
To keep
Figure
Time Line
for
II
Two Period Example
Population
Time
Initial
(year 1)
xo
After harvest, hi, in year
xo
1
-
hi
After breeding in year 2
xo-hj + B(x
-
hi)
After harvest, h2 in year 2 (end of world)
xo-hi + B{x
-
hi)
,
-
h2
Let c denote the cost of harvesting an animal, and denote the amount of the good
demanded
assumed
at
to
a price of p as Dip).
be the only cost of storage,
There will be an equilibrium
the initial population
plus
xo
demand during
< D(c) +
is
less than
in
is
denoted
0.
The
interest rate,
which
is
r.
which the animal
enough
to satisfy
is
hunted to extinction
demand during
in year
1
if
the first year at a price of
c,
the second year at a price of (l+r)c. Algebraically, this can be written as:
D((l+r)c).
There will be an equilibrium
minus the amount required
breeding season,
the case if xQ
If
and D(°°) =
Assume D' <
-
is
in
to satisfy first-year
more than enough
D(c) + B(x
which the animal survives
-
D(c))
>
demand
to satisfy
at price c,
if
the initial population,
plus the births in the
second period demand
at price c.
This will be
D(c).
both conditions hold, then there will be two equilibria. In one, the animal survives.
In the other, the price
the breeding that
is
high enough that the population
would have
satisfied second-period
is
eliminated in the
demand never takes
first
place.
period, and
There
will
be
multiple equilibria
£>((1
the initial population
if
+ r)c) + D(c) >x > 2D(c) - B(x Note
is
such that
D(c)).
be an extinction equilibrium for a
that as the interest rate increases, there will
diminishing range of initial population levels. For sufficiently high interest rates, there will
only be a single equilibrium path of population for any
non-storable
initial stock, just as in
fisheries models.
Note
is
consumed.
that the
It
example above
implicitly
assumes
thus applies to goods such as rhino horn,
good
which
is
is
not destroyed
consumed
when
it
in traditional
We will call such goods storable and distinguish them from durable goods,
Asian medicines.
which are not used up when they are consumed. (Ivory
durable good.) In an earlier version of this paper,
equilibria in a two-period
that except
that the
is
often considered an
we showed
that there could
model of durable goods. This paper models
where noted, the
results
would be
we
In the remainder of the paper
example of a
be multiple
storable, but
we
believe
qualitatively similar for durable goods.
use a continuous time, infinite-horizon model, which
allows us to solve for steady-state population and prices, and to examine cases in which
extinction
prices
is
is
not immediate following a shift in expectations, or the path of population and
stochastic.
The model
level
is
carries several policy implications.
steady, so that standard
the species could
still
be vulnerable
providing additional habitat. This
governments have
suggests that even
if
the population
models would predict the continued survival of the species,
eliminate the extinction equilibrium
If
It
to a switch to
would be
is,
an extinction equilibrium.
One way
to increase the population of the animal
to
by
however, likely to be expensive.
credibility, they
may be
able to eliminate the extinction
equilibrium, and coordinate on the high population equilibrium, merely by promising to
implement tough anti-poaching measures
suggests that laws which provide
little
if
the population falls
protection to non-endangered species, and practically
unlimited protection to endangered species
may be justified
Finally governments or conservation organizations
extinction equilibria
by building
the animal
sell the
stockpile
This
somewhat analogous
is
exchange
if
rate (see, for
below a threshold. This
in
some
may be
cases.
able to eliminate the
sufficient stockpiles of the storable good,
becomes endangered or
to central
the price rises
and threatening
beyond a
banks using foreign exchange reserves
to
threshold.
to
defend an
example, [Obstfeld 1986; 1994]). Stockpiles could be built either by
deliberately harvesting animals, or
by storing confiscated contraband taken from poachers,
rather than either destroying or selling
it.
A number of other papers find multiple equilibria in models of open-access resources
with small numbers of players [Lancaster, 1971; Haurie and Pohjola, 1987; Levhari and
Mirman, 1980; Reinganum and Stokey, 1984; and Benhabib and Radner, 1992]. In these
models, each player prefers to grab resources immediately
leave resources in place, where they will
if
grow more quickly,
others are going to
if
others will not
do
so, but to
consume them
immediately. Tornell and Velasco [1992] introduce the possibility of storage into this type of
model.
The
if
there are
effects
many
examined
in the
previous models are unlikely to lead to multiple equilibria
potential poachers, each of
whom assumes
that his or her actions
have only
an infinitesimal effect on future resource stocks, and on the actions chosen by other players.
This paper argues there
may
nonetheless be multiple equilibria for open-access renewable
resources used in the production of storable goods, because
become
scarce,
and
this will increase the price
if
others poach, the animal will
of the good, making poaching more
attractive.
Because poaching transforms an open-access renewable resource into a private exhaustible
5
resource, this paper can be seen as helping unify the Gordon-Schaefer analysis of open-access
renewable resources with Hotelling's [1931] analysis of optimal extraction of private non-
renewable resources.
The remainder of the paper
Gordon-Schaefer fisheries model,
model can be adapted
organized as follows. Section
is
in
which storage
to allow for storage,
and
Section
some
impossible. Section HI shows
probability the
is
some
probability that the
economy
will coordinate
economy
on
the
IV
will
survival.
V concludes with a discussion of policy implications.
In the standard Gordon-Schaefer model, as set forth
—
where x denotes the population, h
rate of population increase in the
extinct,
population
is
=
0,
The
poachers.
rate
by Clark [1976],
= B(x)-h,
n.i
the harvesting rate, and 5(), the net-births function,
5
absence of harvesting. 5(0) =
1.
0, since if the
We will measure the population in
and B(x)
positive and less than
steady state for the population
is
is
no more animals can be born.
carrying capacity, so 5(1)
This
how
The Standard Gordon-Schaefer Model With No Storage
II.
5
presents the standard
classifies the possible equilibria. Section
discusses equilibria in which people believe there
coordinate on extinction and
is
II
is strictly
negative for x
This implies
that,
>
1.
5 is
population
the
is
is
units of the habitat's
strictly positive if
without harvesting, the unique stable
is 1
of harvest will depend on the demand and the marginal cost faced by
The marginal
cost of poaching,
c, is
a decreasing function of the population x, so
often taken to be the logistic function B(x)
= x(l
-
x).
that c
=
c(x),
with c'(x)<0.
We assume that c'(x) is bounded and that there is a maximum
poaching marginal cost of c m so that c(0) = cm
,
Given
zero
at
price, p,
and above a
pm > cm
so that
,
condition
is
consumer demand
will
which
will restrict ourselves to the case in
.
some poaching
D is continuous, decreasing in p, and
D(p), where
maximum price p m We
be profitable, no matter
how
small the population. This
necessary for extinction to be a stable steady-state.
Since the good
is
open-access, and storage
be equal to the marginal poaching
equal to consumer demand, so h
impossible
is
.
is
=
is
assumed
Algebraically,
cost.
p=
to
c{x).
be impossible,
The
must
harvest must be exactly
The evolution of the system
D(c(x)).
price
its
in
which storage
is
thus described by:
— =B(x)-D(c(x)) = F(x)
H.2
at
We assume that B, D,
D(c(0)) > 0, so that F(0)
II.2.
at
F(1)<0
some
since fl(l)
and c are differentiable. Since 5(0) =
<
0, as illustrated in
=
0, and£>(c(l))
point in (0,1), so that extinction
there will generically be points
positive
on (Xu,
and Xu,
state,
Xs
.
it
will
Thus,
Xs
),
equilibrium given
initial
is
0.
we
is
whereas
not inevitable.
if
it
and p m > c m
is
,
a stable steady state of
We will consider the case in which F is positive
Assuming
Xs and X v so that F is negative
extinct,
storage
For most of the sequel,
and the notation, and
is
Figure n. 1. Thus, zero
and negative and decreasing on (Xs
become
if
>
0,
starts
,
1].
F is
single-peaked
and increasing on
Hence,
above Xu,
that
it
if
(0,
population
is
6
,
Xu),
between
will tend to the high steady
impossible, there will be multiple steady states, but a unique
population.
don't strictly need
not too restrictive.
F to be
single peaked, but this requirement simplifies the analysis
Equilibria with Storage
HI.
This section introduces the possibility of storage into a Gordon-Schaefer type model.
We assume that storage is competitive, that there is no intertemporal substitution in demand
for the good,
and
that the cost of storage is
an interest cost, with rate
r.
We will look for rational expectations equilibria, or paths of population, stores,
which poachers, consumers, and
price in
storers are
section considers perfect foresight equilibria, in
considers equilibria in which the path
storage are the
same
as those in the
(i.e.
there are
which the path
stochastic.
model without
The steady
storage, as
is
no cycles or chaotic
at all times. This
deterministic; Section
states
IV
of the model with
we show below.
comprise the entire stable limit
stable steady states. of the last section
storage
is
behaving rationally
and
set of the
Indeed, the
system with
attractors).
We analyze the fairly general model
introduced in the
last section
with two stable
steady states, one at zero and the other at Xs. In fact, the propositions of this section can be
easily generalized to cover
states, or in
Our
much more
which extinction
is
general models in which there are
many
stable steady
not stable.
strategy for finding equilibria
absence of arbitrage opportunities
in
is
as follows.
Simple accounting arithmetic and the
poaching and storage yield local equilibrium conditions
on the possible equilibrium paths. Because there may or may not be storage or poaching,
turns out that there are three possible different
dynamic regimes: no storage, storage, and no
poaching. Using the local equilibrium conditions,
equilibrium paths in each regime.
We will
make
a distinction
The steady
between a steady
state,
8
we
derive differential equations for the
states give terminal or
which
equilibrium.
it
is
boundary conditions
a stationary value of population and stores, and an
which allow us completely
to determine the equilibrium paths,
which we represent using
phase diagrams in population-stores space. The steady states provide a terminal condition that
allow us completely characterize the equilibrium paths.
LOCAL EQUILIBRIUM AND FEASIBILITY CONDITIONS
The
local equilibrium conditions are
determined by the absence of arbitrage
opportunities for both poachers and storers of the good:
The
The Storage Condition
As
prices.
possibility of storage introduces constraints
on the path of
in [Hotelling 1931], in order to rule out arbitrage,
—=
rp,
if
s
m.l
>0,
dt
where
s
denotes the amount of the good that
people would
sell their stores,
to their stores or
case, dpldt
<
rp,
which
the price
Because poaching
the population
is
rising
slack
if
there
is
is x.
Thus
is
is
it
were
quickly, people
slack
when
no poaching,
in
would hold on
stores are zero. In this
profitable to hold stores.
if
there
is
poaching
at all,
poaching another unit of the good,
the "poaching condition"
c(x), if there is
rising less quickly,
more
competitive,
to the marginal cost of
p=
This condition
were
because otherwise people would find
good must be equal
is c(x) if
if
stored. If the price
poach more. This "storage condition"
The Poaching Condition
price of the
and
is
is
poaching.
which case p <
that
HI.2
c(x).
the
Note
that, in addition to the local
equilibrium conditions above, there are
some
feasibility conditions:
At
"Conservation of Elephants"
all
times, the increase in stores plus the increase in
population must equal the net births minus the amount consumed, or
s
Note
that, as
mentioned
consumed. Note also
good.
earlier,
that animals
+ x = B(x)-D(p).
we assume
that the
m.3
good
is
destroyed
which die naturally cannot be turned
when
into the storable
and
stores, s,
must be non-negative
at all times.
The above conditions imply that, once on an equilibrium path, population,
must be a continuous function of time. This
would be anticipated and
As we
arbitraged.
discuss below, there
may be
These conditions must be
path.
is
8
Finally, both population, x,
price,
it
is
stores,
and
because, with perfect foresight, jumps
See Appendix A, proposition A. 1 for a more formal proof.
an
initial
jump
to get to the equilibrium path.
satisfied at all points
on a
rational expectations equilibrium
There are four conceivable dynamic regimes for the system, depending on which of the
storage and poaching conditions (in. 1 and
m.2)
are binding at
any time, but only three of
these potential regimes are actually possible:
No
Storage Regime
Stores are zero, but there
poaching implies that p =
The
poaching. The zero profit condition for
storage condition restricts the rate at
and not induce storage (p<rp). Because the price
rise
g
c(x).
is
We write the conservation condition as an equality.
away
is
inversely related to the population,
Because the price
voluntarily.
10
which the price can
is
positive,
no one would throw the good
it
is
may
possible to translate this condition that prices
the population
may
not
fall
too
fast: differentiating
p-
not rise too fast into a condition that
c{x), the
condition that no one wants to
hold positive stores becomes
dx
—
>r-r^.
c\x)
c(x)
m.4
dt
In the
No
Storage Regime, the dynamics are the same as Section
II,
model with no
the
storage:
x = B(x) - D(c(x))
s=0
P=
Storage Regime
p=
c(x).
c(x)
Stores are positive and there
is
poaching, so dpldt
-
and
rp,
Here, the exponential path of the price translates into a differential equation for
population: differentiating
for the
m.5
.
No
p=
c(x) gives the
same expression, but with
equality, that
we had
Storage regime (HL4). Given the path of population and, hence, price and
consumption, the dynamics of stores are determined by "conservation of elephants" (IH.3), and
we can
express
all
the local equilibrium
dynamics
x=
in
terms of the population,
x:
c(x)
r~
c\x)
s
= B(x) - D(c(x)) - x
P=
No Poaching Regime
the rate of
stores, so stores
must be
must be
rc(x)
Stores are positive, but there
change of population
is
is
just the net birth rate. All
falling at the
rising exponentially at rate
r.
m.6
.
same
rate as
demand
is
demand. For stores
The dynamics can
11
no poaching. Without poaching,
being satisfied from
to
be positive, price
thus be summarized by:
x = B(x)
s
= -D(p)
p=
Note
No
that since there is
No Poaching
Storage,
that there is
maximum
no poaching,
This
it
is
is
m.7
rp
not possible to substitute c(x) for p.
impossible
if
population
no consumption, so the price must be pm but p m
,
is
is
positive, since
greater than c m
To be
in
steady state, stores must be zero because,
which population and
No
which
is
the
was no poaching.
when
they are positive, price must
be rising exponentially. This means that there are only two stable steady
in the
would imply
marginal cost of poaching, so there would have to be poaching, which contradicts
the assumption that there
population
,
it
is
stores are zero,
and what
Xs, stores are zero, and price
is
we
states:
extinction, in
will call the "high steady state", in
c(Xs). If stores are zero, then the
which
system must be
Storage regime, and will thus have the same steady states as the model with no
storage in Section
II, i.e.
x=
orXj.
DYNAMICS WITHIN THE STORAGE AND NO STORAGE REGIMES
We shall begin by looking at the two regimes in which there is poaching: No Storage
and Storage.
Equilibrium Paths in the
No
Storage Regime.
For the system to be in the no storage regime
in equilibrium,
people must not want to
hold positive stores, so price must not be rising faster than rp. Since the price
12
is
determined by
the population,
p=
c(x), storage implies that the
population cannot
fall
too
fast.
Specifically,
from m.4 and m.5,
rc(x)
B(x)-D(c(x))>—-.
ffl.8
c (x)
As
x e [Xu,
is
clear
Xs*],
just binding,
If the
to follow the
from Figure HJ.1,
where
i.e.
B
-
system
No
Xs
'
D=
and
Xv
*
for small
are the
two
r,
m.8
critical points at
< Xj
rc/c'. Moreover,
starts
enough
will hold if
and only
which the storage condition
<X V <XS < Xs
Storage Regime dynamics to Xs, the stable steady
with no stores and a population of exactiy Xu, the unstable steady
Here, as elsewhere, for the sake of clarity
this in
any
we
is
*.
with population in (XuJCs*] and no stores, then
there.
if
state.
it is
If the
state, the
shall not discuss
an equilibrium
system
starts
system will stay
measure zero cases
like
detail.
s
If the
system
starts
with no stores and with population in [Xu, Xu), then the
Storage dynamics will eventually take population to a point less than
therefore, the
system must leave the
discuss this after
we have found
No
Xu
At some
No
point,
Storage Regime and enter the Storage Regime.
We
the equilibrium paths in the Storage Regime.
Equilibrium Paths in the Storage Regime
The dynamics of population
The dynamics of stores
not
consumed must be
of stores,
s,
in terms
are determined
are determined
stored.
by the
price,
which
is
rising exponentially.
by "conservation of elephants": what
is
harvested and
We may rewrite m.6 as a differential equation for the trajectory
of x:
dx
rc(x)
c (*)J
{
13
dx/dl
is still
just rc(x)/c'(x),
Equation
EII.9
which
implies that rational expectations trajectories in population-stores space
must have stores decreasing with population,
increasing function of population
a
minimum
at Xs*.
To
it
would
fall
if
x, if x
< Xv
,
x e (Xu, Xs). There
see the intuition for this, note that
low, population would tend to
III.l,
bounded above.
negative, and
is strictly
rapidly
fall
enough
or x
a
is
if
> Xs
would be
prevent population from falling too rapidly, part of
Stores must be an
maximum of stores
population
rapidly without stores, and as
that price
.
',
and
seen from Figure
rising faster than rate
demand must be
Xj
very high or very
is
may be
at
In order to
r.
satisfied out of stores,
and
so stores must decrease with time. Xu* and Xs* are the points at which, in the absence of
stores, the population
would
and Xs*, the price would
stores, therefore,
to
make
more than
m.9
is
population-stores space.
the
is
enough
more slowly than
current
the population fall fast
Equation
One
rise
fall just fast
that price
rate r with
enough so
Xs
x e [Xu ,XS
.
].
To
The
now
to
at
which
this
still
must increase
r.
positive,
No
Because population, stores and price are continuous
at the
same point
at
x e (Xu,
Xs *),
and
at a
point of transition from the Storage to
and the system enters
can possibly happen
which
it
strictly
is
where population
Storage Regime,
in equilibrium, the
enters the
explained above, stores are decreasing as a function of x, so
at the
stores
be determined by boundary conditions.
see why, consider the following: to be in the
is falling) if
For an equilibrium with
that price rises at exactly rate
equilibria are
must leave the Storage regime
of time (x
storage.
Between Xu*
r.
the differential equation for the trajectories of equilibria in
Storage Regime. The only place
exactly
no
rise at rate
demand must be being harvested and
possibility is that stores run out while population is
No
would
No
Storage regime.
14
As
increasing as a function
maximum at x = Xu*. But stores
No
system
have to run out
Storage Regime, so stores must have been
falling, (or at least not increasing or at
only point
at
which
a
stores could run out
maximum) immediately
is,
therefore,
The other possible boundary condition
run out. Since x
is
population must
be consumed
when
become
until they
maximum
To
when
p
which
reach zero as well.
is
extinct
is
It
becomes
bounded below while
do not run
extinct before stores
stores are positive, the
out. After that, stores will
turns out that the quantity of stores remaining
uniquely determined in a rational expectations
see this, note that the price charged for the last unit of stores must be
sell his
or her stock.
the population
becomes
while stores are positive, so
is
a rate
price people are willing to
momentarily to
price
at
.
that population
extinct in finite time if stores
becomes
the population
equilibrium.
price
decreasing
is
Xs
until price
reaches
we can
pm
The
before the transition.
pay for the good, or a storer would
The zero
extinct
profit condition in
,
the
by waiting
poaching implies that the
must be c(0) = cm Price
.
calculate the amount, U(p),
profit
pm
is
rising exponentially
consumed from
the time
when
:
Iln'-
U(p) =
The amount of stores remaining
We
at the
\D(pe")dt.
m.10
moment of extinction
must, therefore, be U(c m ).
have shown that there can only be two equilibrium paths
in the
Storage Regime
(See Figure m.2)
1
High Steady State Storage Equilibrium
x>
Xs
.
The system evolves
enters the
No
until stores
In this equilibrium, population starts at
run out
Storage Regime. The equations
when population
p=
c(x),
path of population and price. Stores are given by s
15
=
is
Xs*,
and then
and dpldt = rp determine the
s+(x),
where
sAx)=]^\B(q)-D(c(q))-r^p-\dq.
ifc{q)
Extinction Storage Equilibrium. In this equilibrium, population
2
at that
moment,
= U(cm ). The equations p =
stores
path of population and price. Stores are given by s
s e (x)
For
this to
stores
s e (x),
to
If there is
become negative
is
ever negative,
none such, we say
some
at
we
that
extinct,
and
where
= U(cJ + ]^\B(q)-D(c(q))-r-^f-\dq.
equilibrium. If se (x)
s e (x).
=
becomes
and dpldt = rp determine the
c{x)
be an equilibrium, stores must stay positive
would have
m.ll
c{q))
[
at all
HI.12
times along this path. If
point in the future, this path
is
not an
define Xmax to be the smallest positive root of
Xmax = °°. To be an equilibrium, the starting
population must be less than Xmax.
s e (x)
that
Xmax
and
Xs
and
s+(x) are parallel.
is finite if
and only
Both have a minimum
if s e (x) lies
below
s+(x).
Xs
at
.
It is
If Xmax is finite,
clear
it
from Figure
must
lie
III.2
between
Xu
'.
Transitions Between Storage
and No Storage Regimes
We now examine under which circumstances an equilibrium path can move from the
No
Storage to the Storage Regime. If the
path can
Xu >
move
to the Storage
and the system
initial
population
regime and, thence,
starts in the
No
is
to extinction.
enough
It
may have
to
do
this:
if
Storage regime with population less than Xu, then the
system must eventually move to the Storage regime because
fall fast
small enough, an equilibrium
to violate the storage condition
16
once
it
had
if
it
didn't, the
fallen past Xu*.
population would
On
an equilibrium
path, the
system must move to the Storage regime before
may
move
also
the system
=
s e (x)
move
0,
must make the
i.e.
to the
at
regime when
to the Storage
X^.
transition
from the
path in the
If the
it
No
doesn't strictly have
No
The system
that point is reached.
to.
By
continuity of stores,
Storage to the Storage regime where
Storage regime crosses
X^, then the system can
Storage regime path se leading to extinction. At such a transition, the rates of
change of population,
stores,
and price will jump, but the storage and poaching conditions are
not violated because the levels will not jump.
We may thus define two sets of points on equilibrium paths in the Storage and No
Storage regimes:
A
e,
A+
the set of points leading to extinction, and
,
the set of points leading to
the high steady state, as illustrated in Figure III.3.
The system must end up on one of these
of population and stores (x
when
there
is
,so),
paths,
A+ or A e Given
.
arbitrary initial values
there can either be an initial cull, or there can be an interlude
no poaching, as discussed below.
MOVING TO EQUILIBRIUM
If the initial
population and stores are not on one of the equilibria identified above,
then one of two things will happen. If the
initial
point in population-stores space
equilibrium paths described above, then the system
equilibrium paths via a
satisfied
The
cull.
If the initial point is
from stores with no poaching for a while
transition cannot
happen
at
Xs
',
may jump
17
below
the
instantaneously to one of the
above an equilibrium, demand may be
until the
path meets
A e or A+
because the population would be falling there in the
the system could never reach that point.
is
No
Storage regime, so
Culling
If the
system
starts
below an equilibrium path
an instantaneous harvest, which
enough
that
it
is
10
jump
will
poaching up to the point
at
initial
allowed
make
if it is
is
is
and there
,
equal to the marginal
required by rationality, such a
unanticipated, or at the "beginning of time", as
it
is in this
case.
We
a distinction between "initial" values of population and stores and "starting" values,
which are the values just
after the initial cull.
for initial population
(xq, so)
population, c(xq)
which the price
Although continuity of price, population, and stores
is
may be
shall call a "cull". In this case, the price starts high
above the marginal cost implied by the
will be instantaneous
cost.
we
in population stores space, there
and
stores,
and
When we
need
(x(0), s(0)) to
to indicate this,
denote starting
we
(i.e. at
will write
time
on the
equilibrium path) values.
In a cull, live elephants are killed
means
that, in
and the
population-stores space, the system
total quantity
Q = x + s.
For a
must
lie
below
may
s
we
=
be
rational,
it
moves up a downward sloping
is
conserved.
must take the system
identified above,
line s
=
s+(x),
We call this quantity
to a point
on one of the
by
culling, initial population
and stores
and x < Xs*.
get to the path leading to extinction, if
s e (x). If Xmox is finite, points
also be other points
diagonal,
A e or A+.
get to the high steady state equilibrium path
below the
To
of elephants, dead or alive,
cull to
equilibrium paths
To
and turned into dead elephants one-to-one. This
from which
below
X^
s e (x)
is infinite,
the initial point
must
lie
can also cull to the equilibrium, but there
this is feasible. In particular, if the
curve s = s e (x) has a
If the marginal cost of poaching became sufficiently great as the instantaneous rate of poaching became great
enough, the harvest would take place over time, rather than instantaneously. Structurally, though, there is little
two approaches: the rational expectations equilibria are determined by the boundary
conditions (where people anticipate the system must end up), and these are essentially the same in both cases.
real difference in the
18
tangent of gradient -1, then, as illustrated in Figure m.4, points above the curve, but below the
tangent can also reach the extinction equilibrium by culling.
the points at
which
s e (x) has gradient -1 are
The value of Q
the curve.
the extinction equilibrium
exist,
at this
Xu and Xs
tangency, Qmax,
may be
and only points below se can
is
A quick look at III.9 shows that
but only the tangent at
,
the
maximum
reached via culling. If
Xu lies above
Q may have so that
value
X^ < Xu, then
this
tangency doesn't
cull to the extinction equilibrium.
No Poaching
If there are sufficient initial stores, there will
below
c(x),
and there
is
be equilibria
no poaching for a time while demand
Eventually poaching must resume, at a point on
in
which the
is satisfied
A e or A+. While
there
is
population will be rising, and stores falling as they are consumed. Price
exponentially, at rate
downward
r.
In population-stores space, trajectories with
are
all
it.
up on one of the A„ the
In order for
initial
to the path leading to the
boundary of the
curve s
We
=
no poaching,
is rising
no poaching must be
s + (x).
include a
is
no poaching
point must
high steady
lie
a unique,
to
be
downward
rational,
on one of these
state, the initial
and
sloping no poaching
for an initial point to
point must
get
we denote L+, and above
the
lie to
the right of the
of points on trajectories leading to A+, which
To
get to the path leading to extinction, the initial point
set
of points on trajectories leading to points on
more formal treatment of this
in the
19
end
To
trajectories (Figure III.5).
set
boundary of the
than one.
is less
a point on one of the A, paths, price, population, and stores
determined. Given the end point, there
trajectory leading to
the
at
is
out of stores.
sloping and population must be increasing so long as population
When poaching resumes
starting price
Ae
,
must
lie to
the left of
which we denote
Appendix A, proposition A.3.
Le
.
We have now found all the possible equilibria of the model with storage.
illustrated in Figure HJ.6, population-stores
may be
space
depending on whether there exist equilibria leading
both. In the first region, there
case
if
would
lose
would mean
no equilibrium path leading
that stores
steady state. This
were temporarily
is
is
to
most three regions
to extinction.
that killing
This will be the
and storing enough
no equilibrium path leading
and demand were to be
low enough
satisfied
from
that,
to the high
even
equilibria,
some
to extinction,
In this deterministic, perfect foresight model,
determined by exogenously formed,
be no region
to the extinction set
poaching
If,
if it lies
poaching
stores until they should run
Ae
,
in
which equilibrium
is
chosen
is
self-fulfilling expectations.
which survival
is
either through a cull if
above se
is
and some to the high steady
Depending on parameter values, some of these regions may be empty.
that there will
if
cannot recover enough to guarantee species survival. The third region
where there are multiple possible
to
be held long enough that the storers
the case if population and stores are
to cease
out, the population
state.
would have
money. In the second region, there
at
to extinction, the high steady state, or
and stores are high enough, so
the initial population
get to extinction
is
divided into
As
assured. If Xm^
it
lies
below
is infinite,
s e or
,
It is
possible
any point can get
by an interlude with no
.
on the other hand,
X^
is
small enough (less than XJ), then there will be no region
of multiple equilibria, and the fate of the system will be entirely determined by
its initial
point,
and not by expectations.
Note
start
that, if there is
with even
shooting with
if
No
an
initial
no poaching
the eventual fate of the system
Poaching
equilibria,
is
interlude, the population will
extinction.
and one should
20
be rising to
There will often be over-
not, therefore,
become complacent
if
elephant populations are increasing.
It
see
turns out that
Appendix A, proposition A.2. This should not come
largest population
The
Xmax and Q max are both decreasing in r, the storage cost.
can be and
still
For proofs,
as a surprise, Qmax tells us the
reach extinction via culling and a storage equilibrium path.
larger the population, the longer stores
have to be held before extinction. This
is
clearly
going to be less desirable with higher storage costs. Increasing the storage cost thus always
reduces the region of phase space from which extinction
increase storage costs by threatening prosecution of
The
international
ban on ivory trade
For sufficiently large
multiple equilibria at
all;
r,
may have had
is
possible.
anybody found
to
be storing the good.
this effect.
Xmax will be less than Xu, and there will be no region of
the ultimate fate of the species is the
storage possible, given the
Governments could
same
initial
same
conditions. In this sense, our
as in the
model with no
model converges
to the
standard Gordon-Schaefer model as storage cost rises.
If
Qmax > Xs, then even starting from the high steady state with no stores, the
population will be vulnerable to coordination on the extinction equilibrium. This highlights
another possible policy response to limit the possibility of extinction: the government or
private conservation organizations
may
Increasing the habitat, while leaving
population,
Xs
,
more than
increase the size of habitat available to the species.
demand unchanged,
proportionally. At the
Appendix A, proposition A.4
same
will increase the steady state
time, Qmax will
that, for sufficient habitat,
21
We
Xs will be above Qmax,
species will then be safe from speculative attacks leading to extinction
steady state.
fall.
when
show,
in
and the
it is
in the
high
Non-Deterministic Equilibria
IV.
So
the
far,
economy
we have focused on
future possible paths of the
all
agents believe that
which agents may attach positive probability
economy. One reason
to a
number of
to consider this broader class of equilibria
the perfect foresight equilibrium concept has the uncomfortable property that there
be a path from
greater than
steady
which
will follow a deterministic path. This section considers a broader class of
rational expectations equilibria in
is that
perfect foresight equilibria, in
state.
Xs
A to B,
,
and from
B
then for sufficient
For a system
to C, but not
initial
from
A to C.
To
see this, note that
if
may
Q max is
population, the only equilibrium will lead to the high
that starts in the high steady state,
however, an extinction storage
equilibrium would also be possible.
Note also
that the concept of
no poaching regimes
is
also
much more
stochastic paths are admissible, since in order to have an equilibrium with
must be
stores,
and the only way stores can be generated within the model
relevant
when
no poaching, there
is
through a storage
equilibrium. However, within the limited class of perfect foresight equilibria, people must
assign zero weight to the possibility that there might be a switch from an storage regime to a
no poaching regime.
While we have not
fully categorized the extremely
stochastic rational expectations paths,
which we conjecture
equilibria in
that,
when
illustrates
we have been
some more general
which agents believe there
this
is
broad class of equilibria with
able to describe a subclass of equilibria,
aspects of behavior.
We consider
a constant hazard that a sunspot will appear and
happens, the economy will switch to the extinction storage equilibrium, with
22
no
possibility of
any other switches.
11
Thus
all
agents
know that
the
economy
the extinction equilibrium eventually with probability one, but they are unsure
divide time into
two
section. In this section,
when.
before the sunspot (B.S.), and after the sunspot (A.S.).
parts:
equilibrium behavior A.S.
will switch to
is
we
simple:
it
is
just the extinction equilibrium
we found
We
The
in the last
look for equilibria B.S. in which the population does not become
extinct.
We
first
derive a stochastic analogue of the storage condition.
We then show that
there are equilibria with a small switching hazard in
which the behavior
is
similar to that seen
in section HI: the B.S. steady state population is Xs,
and no stores are held
in this steady state.
There are also equilibria with a higher switching hazard
in
which positive
stores are held in
B.S. steady-state equilibrium in anticipation of a switch to the extinction equilibrium. There
cannot be equilibria with a hazard rate above a certain threshold, because in this case
extinction
would become so
jump immediately
likely that
it
would become
certain
and the system would have
to the extinction equilibrium.
We also show that while the
species can survive a series of small increases in the
hazard rate of switching by building up stores after each increase,
sustain the
same increase
in the hazard rate if
it
it
might not be able
to
took place in a single jump, because the
required increase in storage would be so great as to drive the species into extinction.
In this section,
it
is
mathematically more convenient to work with the
and population, Q, rather than
working with
stores, s. Since
Q = s + x,
some
is
of stores
equivalent to
(x,s).
For the sake of clarity and brevity, we relegate
In
working with (x,Q)
total
cases, extinction
is
all
instantaneous after the switch, so there
ascribe positive probability to any further switch.
23
proofs to Appendix B, where
is
no way
we
that agents could rationally
to
derive the results of this section
more formally.
B.S. Equilibrium Conditions
Let the B.S. state be
(x,
Q, p), and the A.S. state be (xe Q,
,
there are positive stores, the expected profit
from storage must be zero, so
that, if
we
If
denote
rate that the sunspot will appear:
p + (p t - p)n =
If stores are zero,
it
must be because expected
ifs>0.
rp
profits
IV.l
from storage are not
p + (p e -p)n<rp
ifs
=
positive,
and so
rv.2
0.
& IV.2 are just generalizations of the storage condition when the price change is
IV. 1
stochastic,
We
there
is
no
and not necessarily continuous.
shall consider equilibria in
which, once the system
possibility of further change. (xe ,
equilibrium path derived in section HI. In
this
that, since the
Q doesn't change when the sunspot appears.
switch happens instantaneously by culling,
by n the hazard
p e ). Note
some
path to which the system could jump.
jumps
to the lowest possible population
function of Q: xe {Q)
is
Q) must,
on
is in
therefore,
cases, there
he on the extinction storage
may be more
this path.
Thus the population
after the switch is a
+ xe (Q) =
Q. This
is
where
.
The system must,
Qm^ >Q>x> xe {Q).
assumption, the system jumps to the extinction equilibrium path
24
to
we must have x > xe (Q).
We will also only formally consider cases in which Qnuu is bigger than Xs
By
system
Because harvesting cannot increase the population, for people
believe in the possibility of a switch to the extinction equilibrium
therefore, be in the region
than one point on
We will consider equilibria in which the
the smallest population such that s e (xe (Q))
illustrated in Figure IV.l.
the extinction equilibrium,
if there is
a
sunspot.
We may, therefore, determine the A.S price as a function of Q, the total stores plus
at
function of
Q (as we prove
When Q <
U(cm ), the system jumps
the population
We
Q is conserved during the
the time of the sunspot.
population
jumps
first
to
in
Appendix B, proposition
B.l),
straight to extinction,
switch.
and
it is
p
t
is
a decreasing
continuous on
andp e = U~\Q). When
[0,
Q>
gj.
U(cm ),
xe (Q), and/?, = c(xe (Q)).
consider the system dynamics
when
there are positive stores, so that IV. 1
holds.
By
Q > x),
the system evolves before the sunspot according to the differential equations:
an argument analogous to that used in Section HI, while stores are positive
(i.e.
Q = B(x)-D(c(x)) = F(x)
x=
——
1
r
[(/
IV.3
+ K)c(x) - KPe (Q)]v
C \X)
s
B.S. Steady States
It is
rather easy to solve for the steady states of IV.3.
for total stores
Xy), just as
and population
was
to
The
first
equation
tells
us that,
be constant, population must be either Xs, or zero (we ignore
the case in the deterministic case in the last section.
We are interested in the
steady state at Xs. Given the population Xs, the B.S. price will be c(Xs).
To determine
the
steady state level of stores, note that the more stores, the lower the A.S. price will be, and so
the less profitable
it
will be to speculate
on the sunspot' s appearance. There
unique level of stores plus population, Qs, that
The second equation of IV.3 allows us
rate, the
satisfies the storage condition
will thus
be a
with equality.
to solve for this level of stores in terms of the interest
sunspot hazard, and the characteristics of the extinction equilibrium after the sunspot:
Qs =p:
Jr + n
——c(x
25
\
s
)\.
IV.4
Because stores must be
This will be the case for
7t
>
positive, this is only a feasible B.S. steady state if
71/,
<
rc(X s
=
'
7t
)
rv.5
.
p e (X s )-c(X s )
the storage condition cannot be satisfied with equality at
71/,
-
where
7i,
If
Q s > Xs
Xs
,
and there must be zero
stores in steady state.
Because
case that
Qmax-
it
must be possible
we must have
Even
if
Qmax
Qs < s
e (Xu)
is infinite, this
to reach the A.S. equilibrium path via culling,
+ Xy.
If
equation
Qmax
still
the right
is finite,
hand side of IY.5
holds, as if stores are too large at
cannot cull to a point on the extinction equilibrium. For
is
it
this to hold,
7t
Xs
also the
is
just
one
,
must be below % h
,
where
*&*
nh =
In
stores.
summary,
In this case,
then, if
Xs
is
Tt
<
71/,
then there
stores
outweigh the expected
If
71/
<
7C
<
71/,,
is
no B.S. steady
a steady state with no stores, just as
case of section IE: the sunspot probability
profit
then there
is
when
rv.6
.
c(Xy )-c(Xs )
is
low enough
it
state
was
equilibrium with positive
in the perfect-foresight
that the costs
of holding positive
the sunspot happens.
a B.S. steady state with positive stores, (Xs
,
Qs
).
possibility of the sunspot causes agents to hold positive stores in anticipation, so the
likely the sunspot's occurrence, the higher the stores held in anticipation of
it
(see
The
more
Appendix
B, proposition B.2).
71
>
71/,,
probability
is
high enough, extinction becomes self-fulfilling even before the sunspot happens,
If
there exists
no B.S. steady
state at Xs-
26
This
is
because
if the
sunspot
and there
is
B.S. Equilibrium
We now
this are in
diagram
Dynamics
summarize the main features of the B.S. equilibrium dynamics. Details of
Appendix B, propositions B.2
in
x
-
Along
is
n<
is
critical
and run out
<
7t
<
The system continues
ith,
*,
and X"'
the B.S. equilibrium path
probability of a sunspot
just at the steady state.
is
71.
is
s+*(x)
when population
where the storage condition
x = Xs,
s
=
in the
No
Appendix B, proposition B.4, Xs*
.
the saddle path of the fixed point (Xs Qs),
This means, as
is
not surprising, that
higher, then higher stores will be held for
all
if
the
population levels, not
We illustrate the dynamics in Figure IV.3.
consider unanticipated changes in
n decreases, of course,
in the
= Xs
Comparative Statics and Unanticipated Changes in
If
,
to the B.S. steady state
As we show
Xs° = Xs
This saddle path rises with increasing
Now we
ignore them.
as for the perfect foresight case.
analogue of Xs
at Xs', the stochastic
decreasing with n. Obviously
71/
same
we
value Xs". In this case, population decreases towards Xs over time, and
just binding.
If
totally unstable, so
be some level of positive stores
Storage regime exactly as in section in.
is
always
then the dynamics are basically the
7i/,
stores are falling
IY.2
B.7. Figure IV.2 illustrates a possible phase
Xu are
the equilibrium path, there will
above a
-
Q space.
Steady states for population
If
sunspot apart from extinction.
state equilibrium before the
no steady
the price
would have
71,
to fall to get
it
the transition hazard. If
back
to the stable manifold,
stop for a while in a non-deterministic version of a no poaching equilibrium.
27
12
71
increases
,
and poaching would
the B.S. equilibrium path
moves
up, and there will be a cull to get to the
from the old one, so long as the increase
required cull will be so large that the
will switch
in
new
7t is
new
not too big. If the increase in
equilibrium path
n
is
equilibrium path cannot be reached, and the system
immediately to the extinction equilibrium. Note that a large increase
necessarily lead to extinction if
If the increase in
n
is
it
happens gradually,
Thus,
if
a policy
maker knew
higher probability of extinction, and
would be best
to
in a series
slow enough, then the equilibrium
the equilibrium ceases to exist altogether
make
is
in
sustainable
up
to the point at
which
(i.e. 7t/,).
that there
had
somehow had
to
be a
shift in
expectations towards a
control over the timing of that shift,
have access to continuously changing information about the
this
k need not
of small, unanticipated steps.
the shift gradual, rather than rapid. This hints that if policy
be best for them to release
too big the
state
it
makers
of the population,
it
might
information on a regular basis, rather than simply trying to
cover up bad news about the availability of the resource and hope that the situation repairs
itself
before people find out. This can only be conjectured, however, because in the model,
there
is
no uncertainty about the population, only about what other agents are thinking.
We conjecture that if there was a chance of switching to a no poaching equilibrium at
any point on the A.S. trajectory, the
have
to
rate
of growth of prices
would
be higher. Similarly, the possibility of switching back to an extinction equilibrium
from a no poaching equilibrium would mean
lower
in the extinction equilibria
in the
no poaching equilibrium. Note
equilibrium makes
it
that the rate of
growth of prices would have
that the possibility of switching to a
28
be
no poaching
harder to have a storage equilibrium, just as increasing storage costs
would.
to
V.
Conclusions and Policy Implications
This paper has argued that there
may
be multiple rational expectation paths of
population and prices for open-access resources used in the production of storable goods.
Expectation of future poaching will increase future prices, and this will increase current
Note
prices, thus rationalizing the initial increase in poaching.
apply to non-storable goods, such as
fish,
because the price of fish depends only on current
supply and demand, and not on expectations of prices.
goods, such as
oil,
argument does not
that this
It
also does not apply to privately held
since anticipation of higher prices will lead people to postpone extracting
the resource.
It is
becoming
areas of Africa
cost-effective for people to assert property rights to elephants in a
[Simmons and Kreuteo,
Most
1989J.
few
elephants, however, continue to live in
open-access areas, and only a fraction of the elephant population can profitably be protected
as private property. (It is expensive to protect elephants as private property, since they
naturally range over
If
huge
territories
and ordinary fences cannot contain them [Bonner, 1993].)
elephants can only be supported as private property above a certain price, then there
one equilibrium
in
which they are a
plentiful open-access resource at a
equilibrium in which they are a scarce private resource
The
it
may be
equilibrium with a higher level of poaching. If
in fact
be vulnerable to a switch
extinction equilibrium
is to
and another
it
indicates that in order to
necessary to preserve a large enough herd not only
to allow the species to survive at current equilibrium
may
price,
be
high price.
analysis carries several policy implications. First,
assure the survival of a species,
safe, but
at a
low
may
poaching
Q max > Xs
,
levels, but also to prevent
then the population
in equilibrium.
One way
may
appear
to rule out the
increase the habitat for the animal, so that the steady-state
29
an
population becomes greater than Qmax-
It
may be
possible for governments and international organizations to avoid the
extinction equilibrium
could keep prices
if
they can
commit
down and reduce
to drastic
the incentive to poach. If a
organization could credibly announce that
protection
the
if
below a
the herd fell
money, whereas
if
the
measures to prevent extinction. This
it
would spend a
certain critical size,
it
government or international
large
amount on elephant
might never actually have
to
spend
same government spent a moderate amount on elephant protection
each year, the herd might become extinct. The model thus suggests a rationale for
conservation laws that extend
little
protection to a species until
then provide extensive protection without regard to cost.
who hold
which the species becomes
stores will prefer to coordinate
extinct. In fact, although
is
declared endangered, and
1
While conservationists and governments may wish
equilibria, people
it
game
to coordinate
on low-poaching
on a high-poaching equilibrium,
officials in
Zimbabwe removed
in
the
horns of some rhinos in order to protect them from poaching, poachers killed the rhinos
anyway. The
their
New
York Times [July
behavior by saying
"If
1 1
Zimbabwe
or 12, 1994], quotes a wildlife official as explaining
is
to lose
increase the values of stockpiles internationally."
Note
that if there
were not subject
its
entire rhino population,
such news would
14
were a "George Soros" of elephants who had
to credit constraints, he or she could use his or her
sufficient resources, or
market power
to
coordinate on the extinction equilibrium simply by offering to buy enough of the good
high enough price.
On
It is
a
A speculator who already owned some of the good would make
the other hand, the
model suggests
that if the
enforcement that the long-run harvest will decline,
because this may lead to a rush to poach.
government plans to impose such strong anti-poaching
should announce these regulations before they are imposed,
it
also possible that the poachers killed the rhinos to obtain the stumps of their horns, or to
poaching easier
at
in the future.
30
make rhino
substantial profits
may be more
by inducing coordination on the extinction equilibrium, so
likely in the
absence of government intervention, assuming the parameters are
such that both extinction and survival equilibria
The model
by accumulating a
also suggests that
it
may be
more
is
likely to
possible to eliminate the extinction equilibrium
becomes
and threatening
to release
it
onto
sufficiently endangered. (Note that this
be time consistent than policies which promise to spend arbitrary
to preserve an animal. If the animal is already
amounts of resources
reason not to
exist.
sufficient stockpile of the storable good,
the market if the animal goes extinct or
policy
this equilibrium
sell the stockpile.)
As
illustrated in Figure HI. 6, if
going extinct, there
Qmax
is finite,
no
is
but greater than
the high steady state, Xs, then an extinction equilibrium will exist in steady state if the
government does not stockpile
stores. If the
government or a conservation organization holds
stores greater than the
boundary of the region where the extinction equilibria cease
and credibly promises
to release
them onto
the market if the population falls
threshold, the extinction equilibrium will be eliminated.
would have
to
pay the
the price might be
interest costs
worth paying
if
on the
stores,
and
below a
The organization holding
this
would
to exist,
stores
entail a financial loss, but
the organization valued conservation,
and the stores
eliminated the extinction equilibrium.
If
(2™
is infinite,
stores cannot eliminate the extinction equilibrium, but they can
extend the range of the survival equilibrium. For example, suppose that the
and the
due
initial stores are zero,
to disease. If there are
no
but that there
demand
is
if there
satisfied
Bergstrom [1990] has suggested
by
stock
is
X
s,
an exogenous shock to population, for example
stores, then the species will
population dips below Xj. However,
equilibrium in which
is
initial
be driven to extinction
are sufficient stores, there will be a
stores
the
no poaching
and the population can recover.
that confiscated
31
if
contraband should be sold onto the
market. This analysis suggests that confiscated supplies of goods such as rhino horn should
be held and released onto the market only
if
it
appears that the market
is
coordinating on the
extinction equilibrium. For example, a rule might be adopted that confiscated rhino horn
would be sold only
above a certain
if
the rhino population dipped
below a
certain level, or the price rose
level.
Stores could be built up not only
by confiscating contraband, but also by harvesting.
Sick animals could be harvested, and animals could be harvested during periods
population
is
temporarily above
its
when
steady state level, due, for example, to a run of good
weather.
Building up stores will reduce the population, but only temporarily. Once the target
stockpile has been accumulated, harvesting to build
the live population will return to the
same
up the stockpiles can be discontinued, and
level as in the absence of stockpiling.
The presence
of the stockpiles, however, will permanently eliminate or reduce the chance of a switch to the
extinction equilibrium. If Qmax
were
less than
Xs
it is
stockpiles gradually, so as to prevent the population
particularly important to build
from
falling
up
below Qmax, and thus
creating an opportunity for coordination on the extinction equilibrium.
Many
conservationists oppose selling confiscated ivory on the market, for fear that
would legitimize
the ivory trade. Building stores achieves the
same goal of depressing
it
prices,
but without the disadvantage of legitimizing the ivory trade. Stores could potentially be held
until scientists
develop cheap and reliable ways of marking or identifying "legitimately" sold
animal products so they can be distinguished from illegitimate products.
While stockpiles may help promote conservation of animals which are
which are storable but not durable, such as rhino horn,
durable goods,
utility
from them. Elephant ivory
32
is
this analysis
does not
killed for
strictly
goods
apply to
often considered an example of such a
durable good.
The government has no reason
to wait before selling confiscated durable
goods, since in any case, private agents will store any durable goods sold on the market. In
practice,
however, there are few completely durable goods. Even ivory
durable, since
due
to
it
depreciates,
and uncarved ivory
is
in the high steady-state.
seals.
model of Section HI, no private
However,
not perfectly
not perfectly substitutable for carved ivory,
changing styles and demand for personalized ivory
In the perfect foresight
is
stores
if the price is stochastic, either
were held by speculators
due
to sunspot
coordination, as in Section IV, or to exogenous shocks, such as weather or disease, then
speculators
may
hold stores, and government stores
may crowd
these out. In the example
considered in Section IV, government stores would crowd out private stores one for one, until
the
government accumulated greater
stores than
would be held by private
agents.
Any
further
accumulation by the government would reduce the range of equilibria in which agents could
anticipate extinction.
Finally,
it is
worth noting that
this
model reinforces the case for
coordination of conservation policy by suggesting that,
ivory by protecting
its
if
international
one country reduces the price of
elephants, this reduce the incentive to poach in other countries. In
conventional models of non-storable resources, increased anti-poaching efforts
will initially drive
Under
this
up the price of the good, encouraging extra poaching
model, increased anti-poaching efforts
in
one country
other countries, both in the short run and in the long run.
33
may
in
one country
in other countries.
reduce poaching in
REFERENCES
Joanne C. Burgess, Timothy M. Swanson, and David W. Pearce.
Elephants, Economics and Ivory^ London: Earthscan Publications, 1990.
Barbier,
Edward
B.,
Benhabib, Jess and Roy Radner, "Joint Exploitation of a Productive Asset:
Approach," Economic Theory, 2(2), April 1992, pp 155-90.
A Game Theoretic
Bergstrom, Ted, "Puzzles On The Economics of Crime and Confiscation," Journal of
Economic Perspectives, 4: 3(1990).
Bonner, Raymond. At The Hand of Man.
Clark, Colin
New
York: Vintage Books, 1993.
W. Mathematical Bioeconomics: The Optimal Management of Renewable
New York: John Wiley and Sons, 1976.
Resources.
Goombridge, B.
(ed.),
& Hall, New York,
Global Biodiversity: Status of the Earth's Living Resources, Chapman
1992.
Gordon, H.S. "Economic Theory of A Common Property Resource: The
Political Economy. 62(1954): 121-142.
Fishery." Journal of
Haurie, Alain and Matti Pohjola, "Efficient Equilibria
Game
Capitalism," Journal of Economic
in a Differential
Dynamics and Control
1 1
of
(1987) 65-78.
Hotelling, Harold, "The Economics of Exhaustible Resources," Journal of Political
Economy, 39
(April, 1931), 137-175.
Lancaster, Kelvin. "The Dynamic Inefficiency of Capitalism," Journal of Political Economy
11(1)1980,322-34
Levhari, David and Leonard J. Mirman, "The Great Fish War: An Example Using a
Dynamic Coumot-Nash Solution," The Bell Journal of Economics 81(5) 1973, 1092-109
Life, "Saving the
Endangered 100" September 1994, pp. 51-63.
New York Times,
"Zimbabwe's Rhino Efforts Faulted." July
New York Times Magazine, "Devaluing the Tusk" January
12, 1994.
1,
1990, pg. 30 and January 7,
1990. pp. 28-33.
Obstfeld, Maurice. "Rational and Self-Fulfilling Balance-of-Payments Crises". American
Economic Review 76(1) 1986, pp. 72-81
Obstfeld, Maurice. "The Logic of Currency Crises",
34
NBER
Working Paper No. 4640 1994
Property
and Nancy L. Stokey. "Oligopoly Extraction of a Common
Natural Resource: The Importance of the Period of Commitment in Dynamic
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International
Reinganum, Jennifer
F.
Economic Review,
vol. 26, no.l,
February 1985. pp. 161-173.
Schaefer, M.B. "Some Considerations of population dynamics and economics in relation to
the
management of marine
fisheries."
Journal of the Fisheries Research Board of Canada.
14(1957): pp. 669-681.
Simmons, Randy T. and Urs
P. Kreuteo. "Herd Mentality." Journal of Policy Review, Fall
1989.
Tornell,
Aaron and Andres Velasco. "The Tragedy of the Commons and Economic
Growth:
Why
Economy, Vol
Does Capital Flow from Poor
to
100, no. 6, 1992. pp. 1208-1231.
Wall Street Journal, November
15, 1994,
pg
1.
35
Rich Countries?" Journal of Political
Appendix
A
Proofs for Section
HI
The path of population,
Proposition A.l
x, stores, s,
and
price, p, is continuous
on an
equilibrium path.
Proof Together,
the storage
which
down
is
infinite
growth
rate of the population,
impossible.
While there
x,
would imply an instantaneous
in price
that the equilibrium price
A jump up in price would violate the storage condition, and
path must be continuous in time.
a jump
and poaching conditions imply
is
poaching,
p = c(x), which
is
continuous and monotonic, so population,
must be continuous. In the no poaching regime, population develops as HL7, so
is
continuous. Population cannot jump suddenly across regime changes, either, as that would
require a
jump
an instantaneous harvest. Population
in price so there is
is
thus continuous.
Stores are differentiable within regimes, and so are continuous. For there to be a jump in
stores across regimes, there
jump
in price,
which
is
would have
impossible.
to
Hence
be an instantaneous harvest, which would require a
stores are continuous.
Proposition A.2
(a)
The maximum
initial
value of population plus stores the system
the storage equilibrium path s e (x)
is
{
Qmax
is
decreasing in storage cost,
Proof
(a)
2mor must either be
tangent to se (x) of gradient
gives dsldx
=
-1).
s e (x)
is
-1
.
e (Xu)
get to
+ Xu).
A.l
r.
X^, or the point lying on the s =
These tangencies occur
concave
still
Qmax, where
Gmax = max X^, s
(b)
may have and
at Xs,
at
Xv or Xs (F(x) =
axis and the
in
so £?max cannot be associated with Xs.
Al
equation
m.9
(b) If
Qmax = X„ax, then
s,(£?ma*)
=
Differentiating with respect to
0.
r,
*;(2™)%^(0=o
or
or
s e '(x) is just (ITI.9),
and 9s/dr
is:
rc(u)
F(u)
ar| ^rc( M )
= -±(s
= --(s
since *
> Xs', and
.s e (;c)
storage condition
Because
s e (x)
is
t
Qmax = s(Xu)
+ Xu,
F(M)
must be non-negative. The second term
strictly
and
decreasing at
then, because
c ^77
("),
dr rc(u)
A.3
«=*.
+ x-X s ')<0
(x)
satisfied with equality,
must be
—
a*; c(M)
+ x-Xs ')-
(x)
Xu is
that is precisely
is
zero, because at
what
is in
,
the
the parentheses.
[
m"
J
<0.
A.4
,;(X_)
independent of
r,
same way,
the result follows in the
but then the equivalent of A.3 holds because the second term vanishes because
independent of
Xs
X^,
-~or
dr
If
c(u)
.x
e
du\ =
;
J
3r
A.2
Xu is
r.
Proposition A.3
If initial population
{x ,s
and stores are
)e\Jp<p_ (A )=E
l
then
(xq, sq)
i
i
if,
and only
if
A.5
,
r=0
where P
is
the projection operator P(x,s,p)
operator mapping [x(0), s(0), p(0)} to
resumption time
t
p
so that §t(xo,so,po)
=
(x,s),
i
= + or
{x(t), s(t), p(t)},
is
e,
there
and
is
()),
is
the time evolution
a starting price
p and poaching
an no poaching equilibrium leading to the point
A2
(x, Si(x), c(x)) at
time
equilibria leading to
PA+
or
(xq, so),
PA e on
x.
These equilibria are
not, in general, unique.
to another point. If there are multiple equilibria
from the same point
then the one with the lower starting price must have a steeper trajectory in s-x space,
consumed
since stores will be
In other words, there
Xs if and only if L+(x
)
<
faster with a
is
and
so
lower price.
a no poaching equilibrium ultimately leading to the steady state
sq
>
s+(xo),
where L+
is
the left
boundary of the
in A.5. Likewise, there is
an no poaching equilibrium leading to extinction
L e (xo) < so, and
See Figure
same
line if
so
>
s e (xo).
III.5.
L, are
downward
EH.5. L, are
downward
Proposition A.4
is
will be the
>
Thus x = §IK.
We
assume
poaching marginal
that
demand, measured
total population, in real units
in real units, is
is
by
(j>.
independent of habitat, and
cost, c, is a function only of population relative to habitat.
The dynamics of the population
<j>
which implies
it
Qmax-
Denote the available habitat by K, and the
c(ty/K).
no poaching
increasing.
Proof
=
if
If the habitat available to the population is increased sufficiently,
always possible to make Xs
c(x)
£+ defined
and only
Le and L+
sloping, because they are possible
and so stores are decreasing, while population
that the
sloping.
if
set
X^ < Xu-
Proof By Figure
paths,
There may be
A, and A+. There may also be cases where the equilibrium passes through
way
its
some
p for
t
in real units
without storage will be:
= KB{<plK)-D{c{<!)IK)),
that
A3
A.6
Thus
x = B(x)--D(c(x)).
The steady
is
and
states Xs,
zero. Differentiating,
functions of habitat, K, and are such that the
Xu will be
we
A.7
RHS
of A.7
find that
dXr
-- 4
A-*
dXrj
-**
A8
"
"
This means that increasing the habitat more than proportionally increases the population in the
high steady
state.
This
In the region
is
not unsurprising, given that
where
Q^^ is
close to Xs, Qmax
X
v(
F(-
u
U
r
s
demand has
= Xy +
nr^r^w
U{c{q))
D{c{q))
not changed.
s e (Xu).
„(„\\^'
re
rc{q)
C
Xi))
When we
W dqA
A Q
A.9
.
rc(q)
differentiate this expression with respect to K,
dQ^
Thus Qmax
We may write this as:
is falling
therefore, that
with AT
we can
find
at
dX v
a rate
U
X
?
D(c(q))c'(q)
bounded away from
K large enough that Qn^ < Xs
A4
A.10
zero.
.
Xs is rising with K.
It
must
be,
Appendix B
Proofs for Section IV
Proposition B.l
pe
Qe
[0,
a decreasing function of Q, continuous on
U(cm )], pe(Q) = IT\Q). For
Proof
Q<
is
IfQ>
[U(cm ),
U(cm ), the system jumps
l
U(c m ), the price will be U~ (Q). xe (Q)
l
is
Qe
decreasing in Q. U~ (U(cm ))
Proposition B.2
= cm =
c(0)
to
Q^}, Pe(Q) =
increasing in x, and increasing in
7t.
For n
Qmax]. For
c(x e (Q)).
xe (Q), and the price
is
then c(xe (Q)). If
is
decreasing in Q, and continuous as required. IT
=
c(xe (U(c m )),
when x = Xu or Xs- The
dQIdt =
[0,
sop e
line
continuous
is
where dxldt =
at
U(cm ).
0, Qo(x), is
in a suitable region (see below), there are
two
steady states, (Xu, Qu), and (Xs Qs), and
,
g+n
-.1
,
n
Both Qi
are increasing with
7t.
The
line
x=
where
i
is
U or 5.
B.l
)
*,(£)) is a trajectory
of the system. See Figure
rv.2.
Proof Since dQ/dt =
dxldt
=
F(x),
dQ/dt =
iff
x=
Xs ov X v From IV.3,
.
the line
where
satisfies
(r+K)c(x)
= np e (Q
B.2
(x)).
Differentiating with respect to x,
(r
c'
and pe are both negative, so
'
same equation with respect
Q
+ 7t)c'(x) = KP :(Q (x))-^.
must be increasing with
x, for
given
B.3
7t.
Differentiating the
to n,
c(x)
=
Pe (Q
)
+ KP :(Qo )^.
Bl
B.4
1
So
that,
on rearranging,
c(x)-pe (Q
dfio
dn
)
B.5
np t
When
the system jumps, population cannot rise, so price cannot
B.5
positive.
is
Steady states are where dQIdt = dxldt
=
0.
fall.
Hence, c(x)
Substituting
<p
e,
and
Xs or Xu into B.2
quickly yields B.l
Proposition B.3
{Xu,
behavior. (Xs, Qs)
is
Qu)
is
totally unstable; there
hyperbolic for
all
The
71.
may
may
or
not be oscillatory
stable manifold is thus a line,
upward
sloping,
and passing through (Xs,Qs)- See Figure IV.3
Proof Consider x and
are small.
Q near the steady states,
Using Taylor's theorem on IV.3 (and assuming
f
K\
8
The eigen-values of this
c are always negative.
strictly larger
than (r
negative. This
+
means
this fixed point.
+ £,
(X,
that we're
Qi
+
9),
allowed
where ^ and
to), to first order,
-IxxJ
B.6
\*.
t
)
linearized system are:
(g
'
=
np'AQ.)
iw
+%
F'(X
p e and
(x,Q)
2
tc)
,
+ n)±.
F(Xu) >
0,
f
(8
^„,2
+ n)
and
At (Xu, Qu), the discriminant
-
c\X
F(XS ) < 0. Thus
and so one eigen-value
that, locally, there exist
47tF(X,) P ;(e,)
1
is strictly
t
B.7
)
(Xs Qs) the discriminant
at
,
positive, the other
is strictly
dimensional stable and unstable manifolds for
is less
than (r
+ tc)
2
,
and
may be
eigen-values have, therefore, stricdy positive real parts, and the steady state
B2
is
negative.
is totally
Both
unstable.
Xs
'
Proposition B.4
which
Xs* = Xs
,
decreasing with K.
is
<
so that for
n< %i, Xs
'
> Xs, and
is
for
> Xs There always
*
.
K>K
Xs < Xs
*
t,
exists
pe
s
s
)
the point at which:
)
C (A s
A(x,k)
=
[
0, this is the
same
+ n)c(x)
Tip e (x)
(r
below, dA(Xsydx
-
as the relation defining
]
/ c'(x).
Xs
Figure IV.4 shows
*.
Since the line A(x,ti) for constant
,
then
n
Xs
*.
Let
crosses F(x) from
> F(XS"). Hence,
/
3A
X/ = Xs
B.9
e
)
—*-— + — = —-F'^
If
at
.
F(X sK = —^-rA(r + 7t)c(X"s )-7tp (X's )]
When 7t =
7C/
E iXlL_
_J
(X )-c(X
1
Proof Xs*
Xs° = Xs
ax; dA
dA
ax;
OK OX
dK
OK
F(XS ) =
0, so
it/,
if
it
=>
^r/ dA n
-= d%.,_—
<0
r
ax;
J
—OK
'
B.10
ox
= Kp e (Xs ). Because p e >
exists, satisfies (r+7t)c(X5 )
c,
solution does exist.
Proposition B.5
Xs*
> Xs,
in
which case
Proof The
Xs*
< Xs,
If stores
rate
then A{Xs, k)
maximum,
it is
a
run out, they must do so
minimum
of stores. If Xs*
of change of stores goes from
>
0,
and so x
is
-
at Xs*.
This
< Xs
is
to
it
to
+
a
is
only possible
if
maximum.
as x falls across Xs*. But if
increasing, not decreasing.
Thus
stores are at a
as stated.
Proposition B.6
For given parameters, there
see Proposition IV.8) in the storage regime. If K
B3
< 7ti,
is
there
only one equilibrium path (but
is
a path
Q = s+*(x) + x where
a
stores run out at
Xs
equilibrium path
is
Proof
If
*
> Xs and
no storage regime.
the system reverts to the
K<
71/,
then stores
may
run out
at Xs',
hazard of switching
(r+K)c(Xs)
The system cannot go
is
not? Proposition IV.3 proves that itp e (Qs)
the stable manifold,
it
is
it
to extinction before the switch,
In that case, assuming a
to.
would have
Qs- This
to tend to a point
=
(r
+
would mean
with
not run out at
Xs
*,
but the system
<
it)c(Xs ). If 7t
strictly
that, if the
n
Kt then
system were on
negative stores, which
not rational ever to be on the stable manifold. If
may
happens: stores
the
meaningless. The system cannot follow the stable manifold of
< Kp e (Xs), so we must have Xs >
allowed. Thus
7t/,
and we get exactly the same
because that path would be the one the system would switch
Why
k>
the stable manifold of the fixed point (Xs , Qs). See Figure IV.5
equilibrium structure as in section HI.
(Xs, Qs)-
If
may move
it
>
7C/,
is
not
then the opposite
along the stable manifold in
equilibrium.
n>
system must be
Proposition B.7
If
%h
which Qs = Qmax, or
is
the hazard rate at
Kh, then the
in the extinction equilibrium,
where
c(X u )-c(X s )
Proof Qs
is
increasing in
before the switch to extinction.
before the switch. Thus
If there is a
7t
at
which
if
n
As
and, once past
7t/,
discussed above,
(Xs,
Qs)
is
Q < Qm^ for all points
Q > Qn^, the stable manifold to (Xs
Qs =
Qm^, then
it
satisfies:
npe(Q max) =
just c(Xu) (recall Figure IV. 1). Solving this for k, such a
B4
the only stable steady state
,
in
equilibrium
Qs) cannot be an equilibrium.
(r
n h does
+ n)c(Xs ). But p e (Qmax)
exist,
and
is
as claimed.
is
Table
I:
Some Species Used
Sources:
[Goombndge
for Storable
Goods or by Collectors
1992], (Life 1994], [Wall Street Journal 1994]
and others
Bears
Lizards
Medicinal Plants
Other Plants
Giant Panda
Horned Lizard
Latin American
Spectacled Caiman
Caiman crocodilus
Tegus Lizard
Monitor Lizard
Varanus niloticus
species of Dioscorea
Himalayan
Asiatic Black Bear
Grizzly Bear
South American
Spectacled Bear
Malayan Sun Bear
Himalayan Sloth Bear
Cats
Tiger
.
Ocelot Felis partialis
F. wiedii
Rauvolfia serpentina
Chisos Mt.
Curcuma
spp.
Parkia roxburghii
Voacanga gradifolia
Key Tree Cactus
Nellie
V.
salvator
Rauvolfia
V.
bengalensis
species of Aconitum
V.
flavenscens
Python reticulatus
molurus
C
manan
C.
optimus
P. curtus
P.
geoffroyi
Eunectes spp.
Orchids
Dendrobium aphyllum
Boa
D. bellatulum
Leopard Cat
F.
sebae
Constrictor
Rat snake Ptyas
Other Mammals
Black Rhino
Amur Leopard
Caucasian Leopard
Markhor Goat
Cory Cactus
Rattan
Calamus caesius
Snakes
Geoffrey's Cat F.
bengalensis
Hedgehog's
Cactus
Orthosiphon aristasus
P.
tigrina
Margay
Dioscorea deltoidea
exanthematicus
Lynx canadensis
Little spotted cat F.
Yew
V.
Cheetah
Lynxfelis
Ephedra
Green Pitcher Plant
Sm. Begonia
species of
mucosus
D. chrysotoxum
D. farmeri
Dog-faced Water Snake
D. scabrilingue
Cerberus rhynchops
D. senile
Sea snakes (genus
D. thrysiflorum
Lapemis and Homalopsis)
D. unicum
Saiga Antelope
Cape Fur Bull Seal
Sea Otter
Toads
Trees
Colorado River Toad
Astronium urundeuva
Aspidosperma polyneuron
African Elephant
Chimpanzees
Turtles
Ilex paraguaiensis
Hawsbill Sea Turtle
Didymopanax morotoni
Egyptian Tortoise
Araucaria hunsteinii
American Box Turtle
Zeyhera tuberculose
Butterflies
Atriplex repanda
Schaus Swallowtail
Cupressus atlantica
Cordia milleni
Homerus Swallowtail
Cupressus dupreziana
Bird wing
Diospyros hemiteles
Queen Alexandra's
Aniba duckei
Birdwing Ornithoptera
Ocotea porosa
alexandrae
Bertholetia excelsa
Birds
Abies guatemalensis
Red and Blue Lorry
Tectona hamiltoniana
Parrots
Mahogany
Quetzal Pharomachrus
Teak
Dipterix alata
I
|
|
I
mocinno
1
Roseate Spoonbill Ajaia
I
ajaia
P
Macaws Ara
'
Hyacinth
spp.
1
Macaw
1
K&Jl
u-n/VU_v*o
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