With the establishment of 200-mile territorial zones in the ocean for most countries having coastlines, the control of fishing in these zones has become highly regulated by these countries. In this sense, fishing in territorial waters can be considered as a sole owner fishery problem.
On the other hand, if the citizens and commercial fishermen of a given country are permitted to fish freely in their territorial waters, the prob- lem becomes that of an open access fishery. The solutions of these two extreme problems are quite different, as will be shown in this section.
10.1.1 The Dynamics of Fishery Models
We introduce the following notation and terminology which is due to Clark (1976):
ρ = the discount rate,
x(t) = the biomass of fish population at timet, g(x) = the natural growth function,
u(t) = the rate of fishing effort at time t; 0≤u≤U, q = the catchability coefficient,
p = the unit price of landed fish, c = the unit cost of effort.
Assume that the growth functiongis differentiable and concave, and it satisfies
g(0) = 0, g(X) = 0, g(x)>0 for 0< x < X, (10.1) where X denotes the carrying capacity, i.e., the maximum sustainable fish biomass.
The state equation due to Gordon (1954) and Schaefer (1957) is
˙
x=g(x)−qux, x(0) =x0, (10.2)
10.1. The Sole-Owner Fishery Resource Model 313 where qux is the catch rate assumed to be proportional to the biomass as well as the rate of fishing effort. The instantaneous profit rate is
π(x, u) =pqux−cu= (pqx−c)u. (10.3) From (10.1) and (10.2), it follows that x will stay in the closed interval 0≤x≤X providedx0 is in the same interval.
An open access fishery is one in which exploitation is completely uncontrolled. Gordon (1954) analyzed this model, also known as the Gordon-Schaefer model, and showed that the fishing effort tends to reach an equilibrium, called a bionomic equilibrium, at the level where total revenue equals total cost. In other words, the so-called economic rent is completely dissipated. From (10.3) and (10.2), this level is simply
xb = c
pq and ub = g(xb)p
c . (10.4)
LetU > g(c/pq)p/cso thatubis in the interior of [0, U].The economic basis for (10.4) is as follows: If the fishing effort u > ub is made, then total costs exceed total revenues so that at least some fishermen will lose money, and eventually some will drop out, thus reducing the level of the fishing effort. On the other hand, if the fishing effortu < ubis made, then total revenues exceed total costs, thereby attracting additional fishermen, and increasing the fishing effort.
The Gordon-Schaefer model does not maximize the present value of the total profits that can be obtained from the fish resources. This is done next.
10.1.2 The Sole Owner Model
The bionomic equilibrium solution obtained from the open access fishery model usually implies severe biological overfishing. Suppose a fishing regulatory agency is established to improve the operation of the fishing industry. In determining the objective of the agency, it is convenient to think of it as a sole owner who has complete rights to exploit the fishing resource. It is reasonable to assume that the agency attempts to maximize
J = ∞
0 e−ρt(pqx−c)udt (10.5) subject to (10.2). This is the optimal control problem to be solved.
314 10. Applications to Natural Resources 10.1.3 Solution by Green’s Theorem
The solution method presented in this section generalizes the one based on Green’s theorem used in Sect.7.2.2. Solving (10.2) for u we obtain
u= g(x)−x˙
qx , (10.6)
which we substitute into (10.3), giving J =
∞
0 e−ρt(pqx−c)g(x)−x˙
qx dt. (10.7)
Rewriting, we have J =
∞
0 e−ρt[M(x) +N(x) ˙x]dt, (10.8) where
N(x) =−p+ c
qx and M(x) = (p− c
qx)g(x). (10.9) We note that we can write ˙xdt=dxso that (10.8) becomes the following line integral
JB =
B
[e−ρtM(x)dt+e−ρtN(x)dx], (10.10) where B is a state trajectory in (x, t) space,t∈[0,∞).
In this section we are only interested in the infinite horizon solution.
The Green’s theorem method achieves such a solution by first solving a finite horizon problem as in Sect.7.2.2, and then determining the infinite horizon solution for which you are asked to verify that the maximum principle holds in Exercise 10.1. See also Sethi (1977b).
In order to apply Green’s Theorem to (10.10), let Γ denote a simple closed curve in the (x, t) space surrounding a region R in the space.
Then,
JΓ = ,
Γ[e−ρtM(x)dt+e−ρtN(x)dx]
=
R
∂
∂t[e−ρtN(x)]− ∂
∂x[e−ρtM(x)]
dtdx
=
R
−e−ρt[ρN(x) +M(x)]dtdx. (10.11)
10.1. The Sole-Owner Fishery Resource Model 315 If we let
I(x) = −[ρN(x) +M(x)]
= (ρ−g(x))(p− c
qx)−cg(x) qx2 , we can rewrite (10.11) as
JΓ=
R
e−ρtI(x)dtdx.
We can now conclude, as we did in Sects.7.2.2and 7.2.4, that the turn- pike level ¯xis given by setting the integrand of (10.11) to zero. That is,
−I(x) = [g(x)−ρ](p− c
qx) +cg(x)
qx2 = 0. (10.12) In addition, a second-order condition must be satisfied for the solution ¯x of (10.12) to be a turnpike solution; see Lemma7.1 and the subsequent discussion there. The required second-order condition can be stated as
I(x)<0 forx <x¯and I(x)>0 forx >x.¯
Let ¯xbe the unique solution to (10.12) satisfying the second-order condi- tion. The procedure can be extended to the case of nonunique solutions as in Sethi (1977b); see Appendix D.8 on the Sethi-Skiba points.
The corresponding value ¯u of the control which would maintain the fish stock level at ¯x is g(¯x)/qx.¯ In Exercise 10.2 you are asked to show that ¯x ∈ (xb, X) and also that ¯u < U. In Fig.10.1 optimal trajectories are shown for two different initial values: x0<x¯ and x0 >x.¯
Let
π(x) = g(x)(pqx−c)
qx . (10.13)
Withπ(x) obtained from (10.13), condition (10.12) can be rewritten as dπ(x)
dx =ρ
pqx−c qx
, (10.14)
which facilitates the following economic interpretations.
The interpretation ofπ(x) is that it is the sustainable economic rent at fish stock level x. This can be seen by substituting u=g(x)/qx into (10.3), where u = g(x)/qx, obtained using (10.2), is the fishing effort required to maintain the fish stock at level x.Suppose we have attained
316 10. Applications to Natural Resources
Figure 10.1: Optimal policy for the sole owner fishery model the equilibrium level ¯xgiven by (10.12), and suppose we reduce this level to ¯x−εby removing ε amount of fish instantaneously from the fishery, which can be accomplished by an impulse fishing effort of ε/qx.¯ The immediate marginal revenue MR from this action is
MR= (pqx¯−c) ε qx¯.
However, this causes a decrease in the sustainable economic rent which equals
π(¯x)ε.
Over the infinite future, the present value of this stream is ∞
0 e−ρtπ(¯x)εdt= π(¯x)ε ρ .
Adding to this the cost cε/qx¯ of the additional fishing effort ε/qx,¯ we get the marginal cost
M C = π(¯x)ε ρ + cε
qx¯.
Equating MR and MC, we obtain (10.14), which is also (10.12).
When the discount rateρ= 0,Eq. (10.14) reduces to π(x) = 0,