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
In the past decade, the question of what to conserve has received a great deal of attention from ecologists and economists.
8
Although these studies have certainly furthered our understanding of the many and varied complexities of the conservation
question, these studies have not considered the nexus between ecosystem persistence and species conservation. Consequently, we now study a simple model of optimal
species conservation that incorporates the ecological and the economic dimensions of the question into the analysis.
Let us suppose that the ecosystem of section 2 provides ecological and economic benefits to society. The economic benefits include the flow of services provided by
activities such as fishing, grazing, and hunting. Clearly, the continuance of these benefits depends, in part, on the persistence of the ecosystem. As well, society derives
benefits from the existence of this ecosystem. To this end, let B[x
→
, P] denote society’s benefit function. The vector x
→
5 x
1
, . . . , x
r
denotes the r possible economic activities that society may engage in and P, the upper bound from Eq. 8, is our proxy for
ecosystem persistence. We suppose that the r possible economic activities can be varied continuously, and that the benefit function B[·,·] is concave and increasing in
both its arguments. This means that increasing the level of economic activities and or ecosystem persistence raises social benefits, but at a decreasing rate.
Economic activities are costly to undertake and these activities have varied effects on the n species in our ecosystem. Consequently, there is a cost involved in conserving
these species. Let C[x
→
, n] denote society’s cost function. In this formulation, C[x
→
,·] is the cost of engaging in economic activities and C[·,n] is the cost of species conservation.
We assume that C[·,·] is convex and increasing in its arguments. In other words, an increase in either the level of an economic activity or the number of species will lead
to higher costs, at an increasing rate.
Our social plannerecosystem manager’s problem can now be stated. This individual solves
max
x
→
,n
B [x
→
, P] 2 C[x
→
, n ].
9
74 A.A. Batabyal International Review of Economics and Finance 9 2000 69–77
The reader will note that Eq. 9 is a mixed integer programming problem. This is because x
→
is a continuous decision variable by assumption and n is an integer decision variable.
9
To apply the calculus to this problem, we shall interpret n as a rate of species conservation and we shall suppose that there exists a continuous approximation
of P in n. Then the first order necessary conditions to Eq. 9 are
10
] B
[·,·] ]
x
q
5 ]
C [·,·]
] x
q
, q 5 1, . . . , r, 10
and ]
B [·,·]
] P
] P
] n
5 ]
C [·,·]
] n
. 11
Note that the optimal values of the r 1 1 decision variables are given jointly by Eqs. 10 and 11. Eq. 10, the “economic” first order condition, says that each of
the r economic activities should be pursued to the point where the marginal social benefit from this activity equals its marginal social cost. Of greater interest is Eq.
11, the “ecological” first order condition. This equation has implications for species conservation. The equation says that the social plannerecosystem manager should
conserve species at a rate such that the marginal social cost of conservation equals the marginal social benefit. Note that the marginal social benefit is the product of
two terms. The first term captures the effect of a marginal increase in persistence on social benefit and the second term captures the effect of an incremental increase in
the number of species on ecosystem persistence. The reader will note that because persistence is a function of, inter alia, the number of species in the ecosystem, it is
possible to study the question of optimal species conservation in a way that considers the ecological and the economic aspects of the problem jointly.
It should be noted that our specification of society’s benefit and cost functions implies that the optimal values of the x
q
, q 5 1, . . . , r, depend on the optimal value of n, and vice versa. This means that the optimal level at which a particular economic
activity should be carried out depends on the optimal rate of species conservation. In turn, the optimal rate of species conservation depends on the optimal levels of the
r possible economic activities. As opposed to this, if the societal benefit and the cost
functions were separable in their arguments, then this mutual dependence would disappear. However, because economic activities have varied effects on the species
of an ecosystem and because these generally non-identical effects have different implications for the conservation of species, we believe that it is necessary to account
for this dependence between the optimal levels of the various economic activities and the optimal rate of species conservation.
4. Conclusions
