70 A.A. Batabyal International Review of Economics and Finance 9 2000 69–77
comprehend the functioning of such systems, then we must also understand the many and varied interdependencies between such systems.
1
Once it is recognized that ecological-economic systems are jointly determined, it follows that these systems should be studied as one system.
2
However, because this recognition has been recent, a number of issues pertaining to the functioning of jointly
determined ecosystems remain inadequately understood. As such, this article has two objectives. First, we use reliability theory to characterize and to bound the ecosystem
stability property known as persistence. Next, we use this bound on persistence to study a species conservation problem in which society derives benefits from the persistence of
the underlying ecosystem and from the pursuit of economic activities on this ecosystem.
The health or the well being of an ecosystem can be described by a number of different concepts. In this article, we shall focus on the notion of persistence. Be-
cause persistence refers to “how long a variable lasts before it is changed to another value . . .” Pimm, 1991, p. 14, it is the appropriate concept to focus on whenever
the longevity of the species in an ecosystem is salient. For instance, the work of Costanza et al. 1995 has informed us that in coastal and estuarine ecosystems, the
health of the ecosystem depends on how long the composition of a small number of generalist species—that collectively determine the health of the ecosystem—lasts. As
such, it should be clear to the reader that for such ecosystems, the notion of persistence provides us with an apposite measure of ecosystem health.
Despite the significance of the concept of persistence, there are very few studies of persistence in the economics literature. In particular, we are aware of only one article
that has studied persistence, and linked persistence to the number of species in an ecosystem. Batabyal 1999c has provided an explicit characterization of ecosystem
persistence. However, in his modeling framework, Batabyal makes two key assumptions. First, he supposes that persistence depends only on the keystone
3
species of an ecosys- tem. Second, he assumes that there are no interaction effects between the keystone
species of an ecosystem. These assumptions detract from the generality of his analysis. Although Batabyal’s definition does provide a link between persistence and the
number of species in an ecosystem, this definition does not account for the fact that there will generally exist interaction effects between the different species in an
ecosystem.
4
Given this state of affairs, in this article we use reliability theory
5
to provide a bound on the persistence of a stylized ecosystem. This bound explicitly accounts
for interaction effects between the species of our stylized ecosystem. The rest of this article is organized as follows. Section 2 describes the theoretical framework and then
computes an upper bound on ecosystem persistence. Section 3 uses this bound to study a species conservation problem in which society derives benefits from the persistence of
the underlying ecosystem and from the pursuit of economic activities on this ecosystem. Finally, section 4 concludes and offers suggestions for future research.
2. Theoretical framework
2.1. Preliminaries Consider a stylized, stochastic ecosystem that consists of n species, where n P N.
Economic activities such as hunting and logging, and natural events such as droughts
A.A. Batabyal International Review of Economics and Finance 9 2000 69–77 71
and fires result in shocks of varying magnitudes to this ecosystem. While these shocks are in general detrimental to the various species of the ecosystem, they need not have
the same impact on all the species. For instance, a shock resulting from excessive hunting will generally affect an ecosystem quite differently than will a shock that
results from a fire. The presence of these shocks means that the lifetimes of the different species in our ecosystem are random. Let us denote the lifetime of the ith
species, 1 i n by L
i
. Tilman 1996 has noted that as a result of interspecific competition and other factors, one can generally expect there to be interaction effects
between the various species of an ecosystem. To account for these interaction effects, we note that the shocks make the species lifetimes random variables; moreover, we
suppose that these random variables, that is, the L
i
, are not necessarily statistically independent.
There will generally be some substitutability between species in the performance of ecological functions. Hence, we will need to make an assumption about the degree
of this substitutability. The cases of zero substitutability and partial substitutability have been studied in Batabyal 1998, 1999a. Consequently, in this article, we shall
study the case of perfect substitutability. In other words, we shall say that our ecosystem is functional at time t if and only if at least one of the n species is alive. The reader should
note the precise sense in which we are using the word “functional.” By functional, we are referring to the very minimal condition under which our ecosystem is able to
provide a flow of economic services to society over time. We are not saying that our ecosystem’s persistence depends on the survival of a single species. In fact as we shall
soon see, persistence depends on all the species of the ecosystem.
6
Let us now formally describe and compute an upper bound on the persistence of our ecosystem.
2.2. Persistence By virtue of the previous paragraph’s substitutability assumption, we can now
provide an expression for the lifetime of our ecosystem. That expression is ecosystem lifetime 5 maxL
1
, . . . , L
n
. 1
The persistence of our ecosystem is given by the expected lifetime of our ecosystem i.e., by the expectation of the left hand side of Eq. 1. To the best of our knowledge,
for the case in which the L
i
are not necessarily statistically independent, an exact expression for E[ecosystem lifetime] cannot be computed. Consequently, we now follow
Ross 1997, pp. 509–510 and compute an upper bound for the persistence of our ecosystem. First note that
ecosystem lifetime k 1
o
i
5
n i
5
1
L
i
2 k
1
, 2
where k P R
1
is a constant and L
1
5 maxL,0. Our goal now is to compute E[ecosys- tem lifetime
]. Taking expectations of both sides of Eq. 2, we get Persistence
; E[ecosystem lifetime] k 1
o
i
5
n i
5
1
E [L
i
2 k
1
]. 3
Because L
i
2 k
1
is a non-negative random variable, the expectation on the right
72 A.A. Batabyal International Review of Economics and Finance 9 2000 69–77
hand side of Eq. 3 can be simplified. This simplification yields E
[L
i
2 k
1
] 5
∞ k
Prob {L
i
. s}ds. 4
Using Eq. 4, the upper bound for ecosystem persistence becomes Persistence k 1
o
i
5
n i
5
1 ∞
k
Prob {L
i
. s}ds. 5
Note that Eq. 5 holds for every value of the constant k. Consequently, to make the above bound as tight as possible, it will be necessary to choose k optimally. In
other words, our goal now, shown in Eq. 6, is to solve min
kP
R1
[k 1
o
i
5
n i
5
1
∞ k
Prob {L
t
. s}ds]. 6
The first order necessary condition to this problem is
7
o
i
5
n i
5
1
Prob {L
i
. k} 5 1. 7
We can now rewrite Eq. 5. The best upper bound for the persistence of our ecosystem is given by
Persistence k 1
o
i
5
n i
5
1 ∞
k
Prob {L
i
. s}ds, 8
where k solves Eq. 7. Eq. 8 is the upper bound for the persistence of our stylized ecosystem. Now let
the indicator variable V
i
5 1 when L
i
. k, and let V
i
5 0 when L
i
k. Then we see that E[R
i
5
n i
5
1
V
i
] 5 R
i
5
n i
5
1
Prob {L
i
. k} 5 1. This tells us that if k is chosen optimally, then the expected number of species lifetimes that will exceed k equals one. This
result is noteworthy because if in fact exactly one of the L
i
exceeds k, then we can replace the inequality in Eq. 8 with an equality. In other words, in some circumstances,
the expression in Eq. 8 will give us an exact characterization of ecosystem persistence. In this article we have not focused on the computation of a lower bound for ecosystem
persistence. However, it should be clear to the reader that we can always choose zero to be the lower bound on the persistence of our stylized ecosystem.
Eq. 8 tells us that the upper bound on persistence depends on the number of species in the ecosystem n and on the probabilities that the lifetimes of the various
species in this ecosystem will exceed an optimally chosen constant k. From this we conclude that if the number of ecosystem species can be counted and if the above
probabilities can be determined, then we will be able to compute a numerical measure of persistence. Precise measures of ecosystem persistence will generally be difficult
to estimate empirically. In such instances, numerical measures of persistence that are based on the best scientific evidence can be useful in the design of ecosystem manage-
ment policies.
It is well known in ecology that the structure and the dynamics of an ecosystem often depend on the existence of certain key species. These “keystone species” can
A.A. Batabyal International Review of Economics and Finance 9 2000 69–77 73
be very important to the functioning of an ecosystem. Consequently, before we end this section, let us briefly discuss the way in which our previous analysis can be
modified to account for the keystone species of an ecosystem. Suppose that of the n species in an ecosystem, m—where m P N and m , n—are keystone species. Then,
one possible way to study ecosystem persistence would be to focus on the m keystone species and abstract away from the remaining n 2 m species. In our theoretical
framework, this would involve replacing n with m in all the previous equations.
We now use the upper bound in Eq. 8 as a proxy for ecosystem persistence and we study an aspect of the optimal conservation of species. The reader should note
that in our framework, a social plannerecosystem manager takes the ecological and the economic aspects of the conservation question into account.
3. Ecosystem persistence and species conservation
