Elephants and Mammoths
John R. Boyce
Naima Farah
September 2014
Abstract
Open access resources may face the threat of extinction if over-exploited. We
show how the presence of an exhaustible substitute, mammoth ivory, alters the
exploitation path of another resource, elephant ivory, given that both of the resources
are open access. If the initial population of the species is sufficiently high, presence
of a substitute reduces the harvest rate, and the population reaches the stable steady
state. For significantly low level of population, the extinction of the species is delayed
when there is a substitute. This result suggests that for any endangered open access
resource, policies which promote trading of any alternative resource may save an
species from extinction.
1 Introduction
While elephants have been harvested for their ivory since antiquity, the rapid decline
in their population in the 1980s lead to trade in ivory being declared illegal under the
Convention on International Trade in Endangered Species (CITES) in 1989. Yet continuous poaching of elephants threatens their extinction. Many species have become
extinct by human actions in the past. The population of passenger pigeon (Ectopistes
migratorius), a wild North American bird, was 3 to 5 billion1 at the time of arrival of
1
According to Smithsonian Institution, passenger pigeon once constituted 25 to 40 percent of the
total bird population of the United States.
1
the Europeans in North America. With mass deforestation due to European settlement
and commercial exploitation of pigeon meat, by the mid 1880s, the passenger pigeon
completely disappeared. Other examples of extinction due to over-exploitation include
the Dodo bird (Raphus cucullatus) and the Tasmanian tiger (Thylacinus cynocephalus).
This paper examines the effect a substitute has upon open access harvesting of a renewable resource. Most bioeconomic models for open access renewable resources assume
that only one particular resource is harvested for a specific good.2 This is a reasonable
assumption if no close substitute exists for that good. For example, prior to kerosene,
in the late 18th and early 19th century, Sperm whale oil was a choice for illumination
for many households over tallow candles and other types of whale oil since Sperm whale
oil provided brighter, odourless, and higher quality illumination, leading to a no-closesubstitute situation (Fouquet and Pearson 2006). Often however, there exists one or
more close substitutes for a particular renewable resource: Viagra for Rhinoceros horn,
petroleum for whale oil, and other poultry for passenger pigeon. After the ban on elephant ivory trade by CITES in 1989 and fall of Soviet Union in 1991, mammoth (Mammuthus primigenius) ivory has risen as a substitute for elephant (Loxodonta africana)
ivory (Martin et al. 2010, 2011).
Using a dynamic general equilibrium bioeconomic model, this paper analyses how
poaching of elephants is affected when mammoth ivory is consumed as a close substitute
to elephant ivory. Because mammoths are extinct, mammoth ivory is a non-renewable
resource which is depleted over time; elephant ivory, in contrast, is renewable and produced by poachers. Considering elephant and mammoth ivory as imperfect substitutes,
we develop a simple theoretical model to calculate the equilibrium elephant harvest
rate, elephant population level, mammoth tusk collection rate, and price of elephant
2
For example, see Clark (1973), Cropper (1988), Swanson (1994), and Kremer and Morcom (2000).
2
and mammoth ivories.
We find, not surprisingly, that the presence of a mammoth ivory substitute lowers
the demand for elephant ivory. This is important because extinction occurs only if the
elephant population is reduced below the unstable steady state.3 We show that the
presence of a substitute causes the unstable steady state, the minimum viable population, to occur at a smaller population level. Thus a substitute can save an endangered
species from extinction. We also show that the evidence on mammoth and elephant
ivory production implies that these two goods are not perfect substitutes since demand
has remained positive for both types of ivories, even though the prices of elephant ivories
has been changing over time.
This paper contributes to the existing resource management literature by considering explicitly the role a substitute plays on the extraction of an open access renewable
resource. The resource management literature, where an extensive part is devoted to
fisheries, is generally based on some form of bioeconomic model. Also a branch of
this literature examined the economics of open access resource. One of the first works
done on open access resources was by Gordon (1954) who analysed the consequences of
open access resource exploitation. He demonstrated the conditions under which overexploitation of an open access resource might generate non-viable stock level. Clark
(1973) established that an open access resource, which is also a special case of presentvalue maximization, has an infinite discount rate. Clark concluded that open access
resource exploitation results in extinction of the species if, and only if, an immediate
profit can be made by harvesting the last unit of the resource. Swanson (1994) has
characterized the main factors leading to extinction as: (1) open access to resource, (2)
3
In bioeconomics, a steady state is defined as the point where the growth rate of population is zero.
A steady state is stable when for population levels larger or smaller than the steady state, the population
goes back to the steady state. On the other hand, a steady state is unstable if the population moves
away from the steady state when the population becomes higher or lower than the steady state level.
3
high price relative to cost of harvesting and (3) low growth rate relative to harvesting
of the resource. Like rhinoceros (Ceratotherium simum) and many other endangered
species, the African elephant fulfils all of these characteristics.
The case of African elephant is interesting for main two reasons.
4
First, poaching of
African elephants did not stop completely even after the ban by CITES, and second, this
is an endangered resource for which there exists a close substitute. Barbier et al. (1990)
examined the pre-ban ivory trade, existing regulation policies, and its implications for
elephant management from economic perspective. They analysed for the pre-CITESban era the decline in African elephants, the increase in ivory exports, trends in African
elephant ivory export quantity and quality, and ivory demand trend of the consumer
countries. Although Barbier et al.’s work discussed the flaws of the pre-ban era’s regulation policies, and explained why poaching will not stop completely after CITES ban,
their study does not examine the impact of the trade of mammoth ivory on the elephant
population.
This paper is the first paper to examine how the extraction of a renewable resource
changes in the presence of a non-renewable substitute when both substitutes are openaccess resources. In the absence of a substitute resource our model evolves as Kremer
and Morocom’s (2000) no-storage scenario. Kremer and Morocom (2000) focused upon
how the possibility of storage of elephant ivory effects the poaching equilibrium. They
showed that future scarcity of resources increases current poaching, which can cause the
species to go extinct. We do not consider ivory as a storable resource in our model
4
The Asian elephant (Elephas maximus) is also endangered. According to Inernational Organization
for Conservation of Nature (IUCN) from the last estimate, Asian elephant population is 41,410 to
52,345 (Sukumar 2003). The major threats to them are habitat loss, degradation, and fragmentation
(Leimgruber et al., 2003; Sukumar, 2003; Hedges, 2006). It has been argued that relative to the African
elephant, poaching is less severe for the Asian elephant because some males and all females lack tusks
(Dawson and Blackburn, 1991). However, elephants are poached for other products as well, like meat
and leather, and poaching is a threat for the long term survival of some Asian elephants ( Kemf and
Santiapillai, 2000; Menon, 2002).
4
since Kremer and Morocom’s (2000) condition for storage, that the price of ivory grows
at a rate equal to the market interest rate, can not hold if a continuous supply of an
alternative substitute resource prohibits the price from rising.
Our model suggests that policies to promote the trading of the substitute resource
help to diminish the exploitation of an endangered renewable resource and, in some circumstances, can save the resource from extinction. Since We model the effect of trading
a substitute resource on the exploitation rate of an open-access resource, our model can
similarly be applied for other renewable open access resources whose population are vulnerable to extinction but which have a natural or synthetic substitute.
The remainder of the paper is organized as follows. Section 2 provides a brief background to the consumption of elephants and mammoth ivory production. Section 3
presents the general equilibrium model and in sections 4 and 5 we discuss the dynamics
of elephant population both for imperfect and perfect substitute cases. Section 7 concludes.
2 Background
Elephants are harvested mainly for their ivory; although in some parts of Africa, elephant
meat is also widely consumed (Stiles 2012). In spite of the ivory trade ban in 1989 by
CITES, ivory seizure data from The Elephant Trade Information System (ETIS) show
in 2011 the total weight of ivory seizure was 24,300 kg, up from 9,505 kg in 2000. The
rising trend for ivory seizure indicates that the demand for ivory might be increasing;
this rise can also imply that law and order is being more enforced.
5
Mammoth ivory, from the large woolly elephant, which became extinct around 10,000
years ago, has contributed to meet the rising demand for ivory. Mammoth ivory, which
can be crafted in the same way as elephant ivory, competes with the elephant ivorycrafted artefacts and is demanded by ivory customers. A large stock of mammoth tusks,
from around 156 million mammoth carcasses, lies beneath the permafrost in the Arctic
tundra. These are exploited by the mammoth tusk hunters every summer. It is estimated that nearly 50,000 mammoths have been excavated since Siberia became a part
of Russia in the 17th century. After the collapse of Soviet Union in 1991, Russia has
been exporting on average 60 tonnes of mammoth ivory each year to Hong Kong, the
prime buyer of mammoth ivory (Martin et al 2010, 2011).
Like African elephants, mammoth ivory is an open access resource. Although mammoth tusks have been extracted since the 19th century, this business has flourished after
the collapse of the Soviet Union in 1991. While the Russian government issues permit
to hunt for mammoth tusk in the Arctic nations, any Russian, with a valid permit, can
collect and do mammoth tusk trading. However, many Russian mammoth hunters look
for mammoth tusk in Siberia without any valid permit, this again supports the claim of
mammoth tusk being an open access resource ( Larmer 2013).
Statistical Background
This section presents some statistics on the consumption of elephant and mammoth
ivory.
6
Figure 1: African Elephant Population, 1979-2012
Source: African Elephant Specialist Group (AfESG), Daniel Stiles (2004), and Barbier et al (1990)
Figure 1 shows that elephant population was declining very sharply during the preCITES-ban era. From 1979 to 1989, the elephant population more than halved from
1,343,100 to 622,700. Since early 2000, the African elephant population has started to
rise and currently has an increasing trend. According to IUCN Red List of Threatened
Species, although elephant populations may at present be declining in parts of their
range, major populations in Eastern and Southern Africa are currently increasing at a
rate of 4 percent annually (Blanc et al. 2005, 2007). Eastern and Southern Africa’s
elephant population accounts for over two thirds of the total known population of the
continent. According to IUCN Red List analysis, the magnitude of ongoing increases in
7
Southern and Eastern Africa are likely to outweigh the magnitude of any likely declines
in northern and western regions, so the African elephant population may continue increasing.
Figure 2: Estimated Weight of Total Raw Ivory in Ivory Seizures (kg), 1989-2010
(Source: The Elephant Trade Information System (ETIS). For 1989-2006 total
weight of seized ivory includes raw, semi-worked and worked ivory, and for 2007-2011
total weight of seized ivory includes raw and worked ivory.
Figure 2 shows estimated weight of raw ivory seized has a decreasing trend. Although
the estimated weight of seized ivory is increasing since 2002. However, a large amount of
ivory seizure may not only indicate high rate of elephant poaching and increased demand
for ivory but also improved and more stringent law and order enforcement.
8
Figure 3: Total Number of Ivory Seizures, 1989-2011
Source: Monitoring the Illegal Killing of Elephants (MIKE)
Figure 3 shows the total number of ivory seizures by year. Number of ivory seizures
declined for a while after CITES ban, though it started to rise slowly since mid 90s, and
there was a sharp increase around 2008.
Mammoth and Elephant Ivory Trade
Since 2007, each year Russia exports on average 60 tonnes of mammoth ivory around
the world (Martin et al. 2010, 2011), most of which go to Hong Kong. Since mammoth
9
tusks are more expensive than elephant ivory, objects made from mammoth tusk cost
more than elephant tusk made objects.
Figure 4: Russian Mammoth Ivory Exports to Hong Kong
Source: Hong Kong Trade Statistics, Census and Statistics Department, Hong Kong. Since
statistics on export of mammoth ivory from Russia was not available, we present the import
statistics of mammoth ivory of Hong Kong. Hong Kong is the prime buyer of mammoth ivory,
since it is the hub of ivory carving industry.
Figure 4 shows the increasing trend of mammoth ivory import of Hong Kong from
Russia. The statistics suggests that mammoth ivory consumption is also increasing,
like elephant ivory. According to Martin et al (2011), from 2000 to 2009 205,830 kg of
mammoth tusks were imported into Hong Kong, which suggests an annual average of
20,583 kg.
10
Table 1: Retail Prices of Elephant and Mammoth Ivory Items in Hong Kong, December
2010-January 2011
Elephant
Mammoth
Item
Size (cm)
Av. price (USD)
Av. price (USD)
Pendant
5
50
82
Animal, Human (Figures)
4
144
277
15
5,231
6,518
30
15,256
6,923
25,128
33,206
Magic Ball
Source: Martin et al. (2011)
Table 2: Wholesale Price of Elephant and Mammoth Tusk, (2004)
Site
Elephant (USD)
Mammoth (USD/kg)
1-3 kg tusks
10 kg tusks
Grade A
Grade B
Hong Kong
200
320
275
225
Fuzhou,China
316
-
364
243
Source: Martin (2006)
Tables 2 and 3 show mammoth ivory items are more expensive than elephant ivory
items, which suggest mammoth ivory and elephant ivory are not perfect substitutes. To
explain how mammoth ivory and elephant ivory differ in prices, Martin et al. (2010,
2011) point out that the average price for all grades of mammoth ivory in 2009 was USD
350 per kg, whereas raw elephant ivory can be purchased in some parts of central Africa
for less than USD 50 per kg. Moreover, ivory traders paid USD 144 per kg for elephant
ivory at the 2008 auctions in Southern Africa, which was organized under the permission
of CITES.
11
3 Model Assumptions
Assume mammoth ivory is an imperfect substitute to elephant ivory. Since mammoth
hunters, with or without permit, can go to Arctic Siberia and hunt for mammoth ivory,
we also consider mammoth ivory as an open access resource. But as mammoths are extinct, it is a non-renewable resource. We ignore the possibility of storage since elephant
ivory and mammoth ivory are substitutes, storing one good does not increase the price
as ivory demand can be fulfilled by the alternative available source. So the storage condition, which says the growth rate of ivory price is greater than or equal to the market
interest rate, can not hold.
The elephant population growth rate is the difference between the net birth function
and harvest rate.
dx
= G(x) − h,
dt
x(0) = x0 ,
(1)
Where x(t) denotes the stock of elephant population at time t, x0 is the initial stock of
elephants, G(x) is the net birth function in absence of harvesting, and h is the harvest
rate. The carrying capacity of the population is K which is the stable steady state level
of elephants when h = 0, i.e., G(K) = 0. Also, the population does not grow when there
is no initial population to begin with, that is G(0) = 0. In other words, given h = 0,
G(x) > 0,
∀ 0<x<K
This assumes that,without harvesting, the population is viable for all x > 0. Alternatively, if the population exhibits critical decline, then G(x) < 0 for some 0 < x < x,
where x represents the minimum viable population.
The stock of mammoth ivory at time t is S(t). Since the stock is exhaustible with
12
mammoth tusk collected at rate y, the stock S declines according to
dS
= −y,
dt
S(0) = S0
(2)
Production
Let us consider that there are two goods in the economy, ivory I, and all other goods
z. There are, however, two types of ivory, elephant ivory h, and mammoth ivory y. For
each good, there is only one factor of production, labor. The total labor endowment L
is allocated among the three sectors. So the labor market clearing condition is:
L = Lh + Ly + Lz
(3)
The production functions for the harvest rate h, mammoth tusk collection rate y, and all
other goods production z are each constant returns to scale. These production functions
are:
h=
Lh
,
c(x)
y=
Ly
,
m
z = Lz
(4)
From the production functions, one unit of h requires c(x) units of labor, one unit of
y requires m units of labor, and one unit of z requires 1 unit of labor. c(x), the unit
labor requirement for elephant ivory harvesting, has two sorts of cost embedded in it.
First is the cost of bearing the risk of extracting a resource illegally, and second is the
production cost. We assume c(x) is decreasing with x, that is c0 (x) < 0, is continuously
differentiable, and c(x) is bounded so that the maximum marginal cost of poaching elephant is c(0) = cm < ∞. For simplicity, we assume finding one mammoth tusk does
not increase the probability of finding another tusk. Thus the labour requirement for
mammoth tusk collection rate m is constant.
13
Firms pay their workers wage w per unit of labor and earn ph for elephant ivory, pm
for mammoth ivory, and pz for all other goods. Thus profits for each sector are given
by:
Lh
− wLh
c(x)
Ly
Πy = py y − wLy = py
− wLy
m
Πh = ph h − wLh = ph
(5)
(6)
Πz = pz z − wLz = pz Lz − wLz
(7)
Where values of h, y, and z are substituted from equation (4). Firms take their prices
as given. Free entry implies that for wage w = 1, the equilibrium prices are,
ph = c(x),
py = m,
and
pz = 1
(8)
4 General Equilibrium
Consumption
There are L identical individuals, each supplying one unit of labor to the firms and
earning w for each unit of labor. Each representative consumer’s utility is given by
α z 1−α
I
Ui =
L
L
(9)
Since ivory sector consists of elephant and mammoth ivory and they are close substitutes,
the ivory sector can be written as:
σ
" σ−1 σ−1 # σ−1
h σ
y σ
I
=
+
,
L
L
L
14
σ>1
(10)
In equation (10), the parameter σ indicates the elasticity of substitution between elephant and mammoth ivory. When σ > 1, elephant and mammoth ivory are substitutes,
and when σ < 1, they are complements.
Each individual’s budget constraint is
1 = c(x)
h
y
z
+m +
L
L L
Where we substitute the prices as, ph = c(x),
py = m,
(11)
and pz = 1
Because the individual has Cobb-Douglas preferences, she will spend α amount of her
total income for purchasing ivory and (1 − α) for buying other goods. That is pI I ∗ = αL
and pz z ∗ = (1 − α)L. Here pz = 1 and I ∗ and z ∗ are the equilibrium levels of ivory and
other goods to be consumed. Since we do not know pI yet, we write the optimization
problem separately for ivory and other goods sector. Optimization for other goods sector
implies that the optimal amount of other goods is z ∗ = L(1 − α).
The optimization problem for ivory sector can be written as,
max
h,y
s.t.
# σ
" σ−1
y σ−1 σ−1
h σ
σ
+
L
L
α = c(x)
h
y
+m
L
L
Solving the optimization problem for ivory we get the equilibrium amount of elephant
harvest rate h∗ and mammoth tusk collecting rate y ∗ as functions of the prices c(x) and
m for each individual5 :
5
Therefore the equilibrium amount of labor used in each sector can be calculated from the production
(1−σ)
∗
∗
αLm(1−σ)
functions from (4): L∗h = c(x)αLc(x)
(1−σ) +m(1−σ) , Ly = c(x)(1−σ) +m(1−σ) , and Lz = (1 − σ)L.
∗
∗
∗
Summing Lh , Ly , and Lz leads to L which shows that all individuals are employed and each sector uses
a portion of the total labor, L.
15
h∗ (x)
αc(x)−σ
=
,
L
c(x)1−σ + m1−σ
y ∗ (x)
αm−σ
=
,
1−σ
L
c(x)
+ m1−σ
I ∗ (x)
α[c(x)−σ + m−σ ]
=
L
c(x)1−σ + m1−σ
(12)
From equation (12) when elephant and mammoth ivory are perfect substitutes, that
is, σ = ∞ then:
∗
h (x) =



h0 (x) ≡
α
c(x)
if c(x) < m
(13)


0
if c(x) > m
Characterization of Equilibrium Harvest
The first result shows both types of ivory demands are decreasing in their own prices.
Proposition 1. Both demands are downward sloping in their own prices.
Proof. From equation (12), h∗ (x) =
αLc(x)−σ
.
c(x)1−σ +m1−σ
Differentiating with respect to c(x)
gives:
dh∗ (x)
dc(x)
αLc(x)−σ
(1
[c(x)1−σ +m1−σ ]2
=
−σαLc(x)−σ−1
c(x)1−σ +m1−σ
=
−αL
[σc(x)−σ−1 [c(x)1−σ
[c(x)1−σ +m1−σ ]2
=
−αL
[σc(x)−2σ
[c(x)1−σ +m1−σ ]2
=
−αL
[σc(x)−σ−1 m1−σ
[c(x)1−σ +m1−σ ]2
−
− σ)c(x)σ
+ m1−σ ] + (1 − σ)c(x)−2σ ]
+ σc(x)−σ−1 m1−σ + c(x)−2σ − σc(x)−2σ ]
+ c(x)−2σ ] < 0
Similarly, differentiating the mammoth ivory collection rate with respect to collection
cost gives:
dy ∗ (x)
−αL
=
[σc(x)−σ−1 m1−σ + m−2σ ] < 0
dm
[c(x)1−σ + m1−σ ]2
Proposition 1 say ivory production is a decreasing function of their own production
16
cost. In other words, elephant harvesting decreases when harvesting cost increases, and
mammoth ivory production decreases when collection cost increases.
The next result shows ivory demand are deceasing in other ivory’s price when they are
substitutes.
Proposition 2. When elephant and mammoth ivory are substitutes, both ivory demand
is increasing in other price.
Proof. Differentiating the harvest rate with respect to the cost of mammoth tusk collection rate gives,
dh∗ (x)
dm
−σ
−σ
c(x)
= − α(1−σ)Lm
m(1−σ)2
T 0
as
σ T 1, which says if σ > 1
elephant harvesting goes up when mammoth ivory collection cost is high. Similarly, to
see the effect of the change in cost of harvesting c(x) on equilibrium mammoth ivory
collection rate, differentiating y ∗ with respect to c(x) gives,
0
as
dy ∗
dc(x)
σ
−σ ]
c(x)
= − αL[(1−σ)m
[mσ c(x)1−σ ]−2
T
σ T 1, which says an increase in the cost of harvesting elephants will increase
mammoth ivory collection.
Results from proposition 2 imply that the harvest rate of a resource increases when
the cost of producing the substitute good rises.
The below proposition shows how the substitutability of elephant and mammoth
ivory affects demand of ivory.
Proposition 3. When elephants and mammoth ivory are complements in consumption
(σ < 1), demand of both types of ivory approach zero as the other resource is exhausted
(or price of either resource converges to infinity); and when they are substitutes (σ > 1),
an exhausted stock (or infinite price) of the substitute causes demand to be positive.
Proof. When σ 6 1, and S = 0 or m → ∞ (x = 0 or c(x) → ∞) imply that
h∗ (x)
=0
m→∞
L
lim
and
17
y ∗ (x)
=0
c(x)→∞ L
lim
When σ > 1, and S = 0 or m → ∞ (x = 0 or c(x) → ∞) imply that
h∗ (x)
α
=
≡ h0 (x)
m→∞
L
c(x)
lim
and
y ∗ (x)
α
=
≡ y0
m
c(x)→∞ L
lim
.
Proposition 4. When elephant and mammoth ivory are substitutes (σ > 1), the demand
for both ivory becomes zero when the cost of producing the other resource approaches zero.
Proof. When σ > 1, and m → 0 (c(x) → 0) imply that
h∗ (x)
=0
m→0
L
lim
y ∗ (x)
=0
c(x)→0 L
and
lim
Proposition 3 and 4 imply that the consumers always consume the relatively cheaper
ivory.
The next result shows how demand absent a substitute compares with the demand
in the presence of a substitute.
Proposition 5. Demand for both types ivory is lower when a substitute exists.
Proof.
h∗ (x)
=
h0 (x)
=
αc(x)−σ
c(x)1−σ +m1−σ
α
c(x)
c(x)1−σ
c(x)1−σ + m1−σ
<1
∀ 0<m<∞
and
y∗
m1−σ
=
<1
y0
c(x)1−σ + m1−σ
18
∀ 0 < c(x) < ∞
Thus in the presence of a substitute, the exploitation rate of any particular resource
is smaller than the exploitation rate when there is no substitute.
Comparing I ∗ (x) and h0 (x)
Now, consider how aggregate ivory demand is affected by the presence of a substitute.
Proposition 6. When there exists a substitute for elephant ivory, then the aggregate
demand for ivory I ∗ (x) is greater than the demand for ivory without substitute h0 (x) iff
c(x) < m.
Proof. let c(x) > m. Then multiplying with m−σ and adding c(x)1−σ in both sides give
us c(x)1−σ + m−σ c(x) > c(x)1−σ + m1−σ . Then again multiplying both sides by αL:
αLc(x)1−σ +m−σ c(x) > αLc(x)1−σ +m1−σ . This can also be written as,
αL
c(x) .
αL[c(x)−σ +m−σ ]
c(x)1−σ +m1−σ
>
In other words it says I ∗ (x) > h0 (x). This also implies when c(x) > m, then
I ∗ (x) > h0 (x) and when c(x) < m, then I ∗ (x) < h0 (x).
In other words, when both mammoth and elephant ivory are consumed and m < c(x),
that is mammoth ivory is cheaper than elephant ivory, then equilibrium demand for ivory
is higher than it would be absent in substitutes. Conversely, when c(x) < m, that is
elephant ivory is cheaper than mammoth ivory, then I ∗ (x) < h0 (x).
Effect of Population Size on h∗ (x) and y ∗
The below results show the effect of the elephant population size on the harvest rate and
mammoth tusk collection rate.
Proposition 7. Elephant harvest rate is increasing in it’s population stock
Proof. Differentiating the equilibrium harvest rate with respect to population stock x(t)
19
gives:
dh∗ (x)
dh∗ (x) 0
=
c (x) > 0
dx
dc(x) | {z }
| {z } <0
<0
where
dh∗ (x)
dc(x)
< 0 from proposition 1.
The above result says the harvest rate h(x) is an increasing function of the population
stock x(t) regardless of the presence of substitute. The larger is the stock, the higher
∗
dy ∗ 0
is the harvest rate. Contrarily, dydx(x) =
c (x) < 0 which means mammoth ivory
dc(x) | {z }
| {z } <0
>0
collection rate is a decreasing function of elephant population stock, here
proposition 2.
20
dy ∗ (x)
dc(x)
> 0 from
5 Dynamics of elephant population level x
Figure 5: Dynamics of elephant population x
Now, we can turn to an analysis of how the elephant population dynamics are affected
by the presence of a substitute. Figure 4 shows the relevant functions, G(x) the net
birth function, and the harvest rates h0 (x) and h∗ (x).
In the absence of mammoth tusk collection, that is, when y = 0, the elephant harvest
rate is h0 (x) and the system evolves as described in Kremer-Morcom (2000) poachingwithout-storage scenario. When there is no mammoth tusk collection, then the demand
for ivory is satisfied only by elephant harvesting. With no storage of elephant ivory, the
elephant population growth rate is the difference between net growth function G(x) and
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the harvest rate h0 (x),
ẋ = G(x) − h0 (x)
There are three possible steady-state levels of the elephant population. If the initial
level of population x0 is less than x0u , then the population will become extinct since
h0 (x) > G(x) in this interval causing ẋ < 0. If x0 is in the interval (x0u , x0s ], the growth
rate of the elephant population is positive (ẋ > 0) and when x > x0s , the growth rate of
the elephant population is negative (ẋ < 0). Thus for x0 > x0u population will always
move towards the stable steady state x0s . If x0 is exactly at x0u , the elephant population
remains there forever since at x = x0u , ẋ = 0.
Dynamics of x in presence of mammoth tusk
In the presence the of mammoth ivory substitute, we consider the equilibrium level of
harvest rate of elephants is h∗ (x) and the equilibrium level of mammoth tusk collecting
rate is y ∗ (x). While the stock of mammoth ivory S(t) is positive, h∗ (x) and y ∗ are
functions of c(x) and m. Once mammoth ivory is exhausted, for each h∗ (x), the time
path of x, x(t), depends on whether G(x) S h∗ (x), that is ẋ S 0.
At time t = 0 the initial elephant population is x0 and mammoth ivory stock is S0 . If
the mammoth ivory stock is exhausted at time t = Ts , then x(Ts ) represents the steady
stock from where the elephant population starts to approaches it’s final equilibrium. The
steady state level of x after mammoth tusk stock is exhausted depends on the position
of x(Ts ) relative to x0u , if x(Ts ) > x0u , then xs is the steady state; if x(Ts ) = x0u then x0u
is the steady state; and if x(Ts ) < x0u , then elephant population goes extinct.
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When x0 < x∗u
If x0 < x∗u , then h∗ (x) > G(x), which implies that ẋ < 0. With this level of initial
elephant population might reach extinction before mammoth tusk stock is exhausted
and since x∗u < x0u , x(ts ) < x0u , so elephants will become extinct, all else constant.
When x0 > x∗u
If x0 > x∗u , then h∗ (x) < G(x), which implies that ẋ > 0. With this condition the
elephant population can face multiple scenarios. If at time t = Ts , then mammoth ivory
stock is exhausted, that is S(Ts ) = 0, and x(Ts ) is in between (x0u , ∞], then x will move
to the stable steady state x0s . If at t = Ts and x(Ts ) = x0u , then ẋ = 0 and the system
will stay at x0u . If at t = Ts and x(Ts ) < x0u , then ẋ < 0 and the elephant population
eventually becomes extinct.
When x0 = x∗u
If x0 = x∗u , then h∗ (x) = G(x). But at t = Ts , when S(Ts ) = 0, the harvest rate become
h0 and thus at t = Ts , ẋ < 0 and the elephant population eventually goes extinct.
Analysis of the above steady states leads to propositions 4 and 5:
Proposition 8. If there exists an exhaustible substitute resource, then for initial population stock x0 (x∗u , x0u ] and for sufficiently high substitute resource stock S0 , the elephant
population will rise above x0u avoiding extinction.
Proof. For x0 (x∗u , x0u ], ẋ = G(x) − h∗ (x) > 0. If S0 is large enough so that x(Ts ) > x0u ,
RT
where S0 = 0 s y ∗ (x)dt implicitly defines Ts . Then since ẋ > 0, dc(x)
dt < 0 implying that
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dy ∗ (x)
dt
> 0. For a given x ∈ (x∗u , x0u ], a higher S0 implies a larger time t, which implies a
larger x(Ts ).
The above analysis also gives corollary 1 and 2:
Corollary 1. If x0 < x∗u < x0u , extinction occurs even when there is a substitute
Corollary 2. If x0 > x0u > x∗u , population move to the stable steady state xs
From the statistics in section 2, the slowly increasing trend of elephant population
may imply that, the population stock is in a point in between x∗u and x∗s . If so, then
the theory here predicts that the elephant population will approach the steady state
will approach the steady state x∗s and then if mammoth ivory is exhausted, the elphnat
population will fall back to x0s .
6 Elephant and Mammoth Ivory as Perfect Substitutes
If elephant and mammoth ivory are perfect substitutes, there can be three outcome
depending on the relative costs of production. If the cost of mammoth ivory collection is
greater than elephant harvesting m > c(x), then only elephant ivory fulfils all demands
which implies y = 0, h > 0 and ẋ = G(x) − h∗ where h∗ (x) =
m < c(x), then h∗ = 0, y ∗ =
αL
m
αL
c(x) .
Contrarily, if
and ẋ = G(x). Finally the only scenario, when both
elephant and mammoth ivory are consumed is when m = c(x), here y ∗ + h∗ =
αL
m
and
h∗ = G(xm ) where xm satisfies c(xm ) = m.
In the presence of a perfect substitute, no harvesting will occur when c(x) > m and
harvesting of elephants will only occur when c(x) < m. This, the value of xm such that
c(xm ) = m will be a steady state if
αL
m
≥ G(xm ) as long as the mammoth ivory exists.
We call this a temporary steady state equilibrium. The condition that
αL
m
> G(x) is
required since otherwise the ivory demand will be less than the net births of elephants.
The temporary steady state may not be reached if the initial elephant population is
24
insufficient. As in the imperfect substitute case, the ultimate steady state elephant
population will depend upon x(Ts ) relative to xu .
Figure 6: Dynamics of elephant population x when elephant and mammoth ivory are
perfect substitutes
From figure 6, mu and ms indicate the mammoth ivory collection costs associated
with the unstable steady state xu and stable steady state xs respectively. Now consider
there can be 3 mammoth ivory collection costs m1 , m2 and m3 , where m3 < ms ≤ m2 <
mu ≤ m1 .
(i) m3 < ms
When mammoth ivory collection cost is as low as m3 , the steady state occurs where the
cost of harvesting elephants is c(x3 ) = m3 . If initial elephant population x0 > x3 , then
c(x0 ) < m3 , and thus h = h0 (x), y = 0 which implies that ẋ < 0 and so x will move
to x3 . At x3 , h(x3 ) + y = h0 (x) where h(x3 ) = G(x3 ). So at x3 , ẋ = 0. Thus, once
the population reaches at x3 it will stay there until mammoth ivory is exhausted. If
25
0 < x0 < x3 , then c(x0 ) > m3 which implies that h = 0, y =
αL
m.
Therefore ẋ > 0, which
causes x to approach x3 . If initial elephant population x0 = x3 , then, c(x0 ) = m3 , which
implies that h(x3 ) + y =
αL
m
and h(x3 ) = G(x3 ) so that ẋ = 0, and thus if the population
starts at x3 , it will also stay there until mammoth ivory is exhausted. Therefore for all
m < ms , for all initial elephant population stock, the population moves to x3 , which
makes x3 a temporary equilibrium.
(ii) ms ≤ m = m2 < mu
When mammoth ivory collection cost is ms ≤ m = m2 < ms , then at x2 the cost
of harvesting elephants is c(x2 ) = m2 . If initial elephant population x0 > x2 , then
c(x0 ) < m2 which implies that h = h0 (x) and y = 0. Now either x0 > xs or x2 < x0 < xs .
If x0 > xs , then ẋ < 0 which implies that x approaches xs . And if, x2 < x0 < xs , then
ẋ > 0 and so x approaches xs . Now contrarily, If x0 < x2 , then c(x0 ) > m2 and thus
h = 0 and y = h0 (x), which implies ẋ > 0 and so x approaches xs . If x0 = x2 , then,
c(x0 ) = m2 and h(x2 ) + y =
αL
m
so h(x) < G(x). Thus ẋ > 0 and x approaches xs . Also
for m2 = ms , at xs , c(xs ) = ms .So in the similar process stated above for all x, x < xs
and x > xs elephant population approaches xs , and at x = xs , c(xs ) = ms , which implies
h(xs ) + y =
αL
m,
thus x stays at xs . So when ms ≤ m < mu , for all positive elephant
population stock, x approaches to xs ; that means xs is a temporary equilibrium when
m = m2 .
(iii) m = m1 ≥ mu
When mammoth ivory collection cost is m = m1 > mu , at x1 the cost of harvesting
elephants is c(x1 ) = m1 . If initial elephant population x0 > x1 , c(x0 ) < m1 then
h = h0 (x) and y = 0. When x0 > x1 , then either x0 > xu or x0 < xu . If x0 > xu , then
h0 (x) < G(x) which implies ẋ > 0 and so x approaches xs . If x0 < xu , then ẋ < 0,
which implies x approaches x1 . If initial elephant population x0 < x1 , c(x0 ) > m1
26
which implies that h = 0 and thus x approaches x1 . If x0 = x1 , then c(x0 ) = m1 and
h(x1 ) + y =
αL
m,
here h(x1 ) = G(x1 ) and thus ẋ = 0 and thus x = x1 . Finally, when
x0 = xu , then c(xu ) < m1 , so h = h0 (x) and y = 0. Here h(xu ) = G(xu ) and so x = xu .
In the similar process stated above, when m = mu , at xu , c(xu ) = mu . So for any
m = mu , if initial elephant population x0 > xu , then x approaches xs , and if x0 < xu ,
then x approaches x1 . Thus for all m ≥ mu , for 0 < x0 < xu , the temporary steady
state is x1 ; for x0 = xu , the temporary steady state is xu ; for x0 > xu , the temporary
steady state is xs .
Definition 1. A temporary equilibrium is the temporary stable steady state of the elephant population as long as mammoth ivory stock is positive, S > 0.
Proposition 9. When elephant and mammoth ivory are perfect substitutes (i) for m 6
ms all positive initial elephant populations lead to temporary steady state x3 6 xs , (ii)
for ms < m < mu , all positive initial elephant populations lead to temporary steady
state xs , (iii) for m ≥ mu (a) all positive initial elephant populations x0 ≤ xu lead to
temporary steady state x1 ≤ xu , (b) all positive initial elephant populations x0 > xu lead
to temporary steady state xs .
Proposition 10. After mammoth ivory is exhausted (1) if x0 < xu and m > mu , then
elephant population goes extinct, x → 0, (2) If x0 > xu , for all m, then the elephant
population moves to xs , (3) If x0 < xu and m < mu and if the substitute resource stock
S0 is large, then elephant population moves to xs , avoiding extinction.
For perfect substitute case, elephant population only reach extinction if cost of collecting mammoth ivory is very high like m > mu and initial elephant population is very
low, x0 < xu . Other than this, the population either stays at xu or reaches the stable steady state xs . Most importantly, these results do not change with the substitute
resource’s stock size.
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7 Concluding Remarks and Policy Suggestions
This paper examines the change in the extraction path of a renewable natural resource
occurs for the presence of an exhaustible substitute resource in the market. Given that
both of the resources are open access, this paper shows that upon the presence of a
substitute, the extinction of a species can be delayed only if the species has a stock level
greater than a threshold level. Although the threshold level, below which the species
goes to extinction with continuous harvesting, becomes small when there is a substitute
in the market. For a safe and high level of population that species reaches a steady
stock level, both when the substitute is perfect and imperfect. Presence of a substitute
for that species, changes the extraction path significantly when the substitute is perfect.
If the cost of extraction of the perfect substitute is lower than the poaching cost of the
species, the species is not harvested at all. In case of imperfect substitute which has
lower production cost than the poaching cost, the harvest rate of that species never
becomes zero, but smaller than the harvest rate without a substitute for each level of
population. Thus presence of a substitute for a renewable natural resource, prevents
extinction of a species, given that the species had a safe large initial population from
the begin with. For small initial level of population, extinction could not be prevented
but be delayed. This extra time is important since it let the government to take steps
to prevent over-exploitation or poaching. Upon the presence of a relatively cheaper perfect substitute, a species never reaches extinction, given that the substitute stock is big
enough.
This paper suggests to form policies that would make the substitutes of a particular
renewable resource available to the consumers so that the demand for a resource could be
mitigated partially or completely with the other substitute and a species can be saved
from being extinct. This paper also suggests supporting the trading of a substitute
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resource, by lowering it’s extraction cost, might slow down or stop the exploitation of
any endangered species or renewable resource.
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