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Optimal escapement levels in stochastic and deterministic harvesting models
JOURNAL
OF ENVIRONMENTAL
ECONOMICS
AND
MANAGEMENT
6, 350463
(
1979)
Optimal Escapement Levels in Stochastic and Deterministic Harvesting Models * WILLIAM
J. REED
Department of Mathematics, University of Victoria, Victoria, British Columbia V8W SY.3, Canada Received January 22, 1979; revised September 6, 1979 A constantiscapement feedback policy is shown to be optimal in maximising expected discounted net revenue from an animal resource whose dynamics are described by a stochastic stock-recruitment model, provided that unit harvesting costs satisfy certain conditions. The optimal escapement in thii model is compared with that in the corresponding deterministic model and it is shown how the way in which unit harvesting costs vary with population abundance can be important in determining the relative sizes of the optimal escapements. In most cases, the optimal stochastic escapement is no less than the optimal deterministic escapement.
1. INTRODUCTION The optimal economic exploitation of renewable animal resources is a subject that has received considerable attention in recent years, and much theoretical insight into the problem has been gained through the application of the techniques of optimal control theory using both continuous-time and discrete-time population models (see, for example, Clark [3]). To date, however, the theory has been to a large extent confined to simple deterministic population models. Sjnce the dynamics of most real animal populations appear to exhibit considerable random fluctuation, it is an important question to determine to what extent the results of the deterministic theory carry over to an analysis of analogous stochastic population models. In discrete time, using a deterministic “stock-recruitment” model and a net revenue function linear in the harvest, Clark [2, 3 3 and Spence [14] have shown that, under fairly general conditions, the present value of the economic rent of the resource (total discounted value of the revenue stream) is maximized by a “bang-bang” policy which forces the population to some optimal level of escapement as rapidly as possible. Clark [3, p. 2451 shows how this is analogous to the optimal policy in the corresponding continuous-time model. Reed [ll] and Jaquette [7] have extended this analysis to discrete-time stochastic population models and have shown that an optimal policy to maximize expected present value is given by a feedback control which forces the population back to some optimal escapement level as rapidly as possible. However, in order to prove this * Research for this paper was carried out at the Institute versity of British Columbia. 350 0095~0696/79/04035414$02.00/0
of Animal
Resource Ecology, Uni-
OPTIMAL
ESCAPEMENT
LEVELS
351
analogous result, both of the above authors imposed much more restrictive conditions on the cost structure of the model than did Clark [3) and Spence [14) in their deterministic analyses. In this paper, we first prove the optimality of a feedback “constant-escapement” policy for the stochastic population model of Reed [ll] under more general cost assumptions, which can be derived from a production function model of the harvest. This gives a closer parallel to the deterministic results of Clark [3]. Second, we undertake a comparison of the optimal escapement levels in the stochastic and deterministic models. It will ht> shown how the form of the unit cost function can play an important role in determining the relative optimal levels of escapement. 2. THE
MODEL
The population model we shall employ is the stochastic stock-recruitment model discussed in Reed [ll, 121 and is a simple stochastic version of the model employed by Clark [3]. If X, is a random variable (T.v.) denoting the size of the population in year n, then we suppose that the process (X, l is Markovian with transitions governed by the equation X
a+1
=
(2.1)
z?Lwn>,
where f(x) is the expected or average stock-recruitment function, which we shall assume is differentiable, strictly concave, and nondecreasing,’ i.e., exhibits “normal compensation” ; and (Z,} is a sequence of independent identically distributed (i.i.d.) r.v.‘s with unit mean. Thus the deterministic stock-recruitment relationship of Clark [3] will hold in this model in terms of conditional expectations; i.e., E(X,+l(X, = x) = j(x). W e f ur th er assume that the random multipliers 2, are distributed on some finite interval [a, b], where 0 < a < 1 < b < 30 , with a common distribution function (d.f.) a(z) and that bf(0) > 1 and lim, +oDa!‘(z) < 1. It then follows (Reed [12]) that there exist finite population levels m = max(bf(x): bf(x) 2 2, x > 01 and r = max{af(z):af(x) 2 x,x 2 0) such that with probability 1 (w.p. I), the population will eventually enter and thenceforth stay within the interval [r, ml and on which it will, in the limit, attain a stationary probability distribution (Fig. 1). Note that r will be zero if af’(0) < 1. We shall call any population level 2 for which pr(X,+l
1 x(X,
= x/ = 1
self-sustaining. Clearly, in our model all population levels in the interval [0, r] are self-sustaining (see Fig. 1). We suppose that harvesting can take place each year during a short harvest season throughout which natural mortality in the population is insignificant. If X, denotes the size of the population immediately prior to harvesting in year n, 1 Most of the reauita can be proved for the case of “overcompensation,” i.e., f concave and dome-shaped, provided that certain additional reasonable conditions are imposed on the cost structure. However, to avoid these complications (see Clark [3, p. 24811, we here restrict discu&sion to the case of normal compensation.
352
WILLIAM
J. REED
FIGURE 1
and a harvest of size h, is taken that year, then X+1
= Z,f(X,
- h,).
(2.2)
We make the following economic assumptions which are the same as those made by Clark [3, p. 2431 in his deterministic analysis : (a) Demand is infinitely elastic; i.e., each unit of the population captured can be sold for a fixed price, p, say. (b) Harvesting costs vary with population abundance. We suppose that the unit cost of harvesting is independent of harvest size but increases as the size of the population decreases. Specifically we assume that the unit cost when the population is at the level x is c(x), where c is some nonincreasing function. A cost model of this form can, under certain assumptions, be derived from a production function model of the harvest. If it is assumed that the instantaneous harvest rate h is related to the fishing (or, more generally, harvest) effort E, and to the population abundance x, by a production function of the form h = q(z)Ex,
(2.3)
where q is a coefficient of harvest mortality per unit of effort (catchability coeficient) which in general depends on the population abundance, then the aggregated effort in a time increment At needed to catch an amount Ah when the population is at level 2 is inversely proportional to zq(x). Thus if it is assumed that harvest costs are proportional to total harvest effort and that natural mortality is insignificant during the harvest season, then the unit cost function is of the form 0) = klh7(x)). (2.4) If it is assumed that the catch per unit of effort (c.p.u.e.), xq(x) increases (or at least does not decrease) with population abundance, then c(x) will be nonincreasing. A special case of (2.3) is when the catchability coefficient q is constant. This is the well-known Schaefer production function in which c.p.u.e. is assumed proportional to population size. In this case the unit cost function is of the form c(x) = k/x.
(2.5)
OPTIMAL
ESCAPEMENT
LEVELS
333
Clark [3, p. 2381 gives another justification for a cost function of this form in terms of the expected time taken to locate randomly distributed schools of fish. Although the Schaefer production function has been widely used there is good reason to believe that in many fisheries the catchability coefficient changes with population abundance. Gulland [5) adduces a number of mechanisms whereby the catchability coefficient might be expected to increase with a decrease in population abundance. Among these are a reductiofi in the area inhabited by the population at low levels of abundance and the phenomenon of gear saturation at high levels of abundance (see also Clark [3, p. 2351). Gulland models the catchability coefficient as proportional to z@for some p < 0. This leads to a production function h = KExB,
where 0 = 1 + p 5 1 and K is a constant. In this case the unit cost function is
This is of the Cobb-Douglas
c(x) = k/x@.
(2.6)
form. (2.7)
A limiting case is when /3 = - 1. In this case the c.p.u.e. is independent of abundance and the unit harvesting cost is constant. As can be seen from the above discussion the artifact, of a nonincreasing unit harvest cost, can summarize quite complex responses of the catchability to variations in population abundance. However it cannot adequately describe the phenomenon of diminishing marginal returns on effort, caused, for instance, by congestion on the fishing grounds. For a discussion of the cost implications of a production function which is nonlinear in the effort variable see Clark [3, p. 236-J. We let x, denote the zero-pro$t level; i.e., if c(0) 2 p then x, is the solution to c(x) = p (more strictly the supremum of such solutions). If c(0) < p then we define x, as zero. (c) There is time-discounting of future revenues with an annual discount factor OL< 1, assumed constant. Thus a revenue earned in year 11is discounted by a factor OL~to its “present” value in some brtse year zero. For a sequence of harvests {h,} the expected total discounted revenue is (2.8)
By letting (2.9)
(2.8) can be expressed as
Ein-l I? @CQ(Xn>-
Q(X, - hn)]).
(2.10)
For a given preharvest abundance x the quantity Q(x) represents the maximum revenue that can be realized in a single harvest. We shall call Q(X) the immediate harvest vcslueat the resource. The control problem which we shall consider is the maximizastion of the expected total discounted revenue (or expected present value (E.P.V.)). Mathematically the problem is to find a feedback control law (i.e., a sequence of func-
354
WILLIAM
J. REED
tions (h,(x,) }) to maximize constraints
(2.10) subject
to the state equation
(2.2), the
0 5 hn I xv,,
(2.11)
Xl = Xl.
(2.12)
and an initial condition such as
3. AN OPTIMAL
SOLUTION
We prove in this section that under certain conditions on the cost function c(x) and on the state equation (2.1) there is an optimal policy of the “constantescapement” type; i.e., there is an escapement level S with the optimal harvest h, in any year n being given by h, = 0,
= X, - S,
if if
xrz 5 s, x, 2 s.
(3.1)
We shall assume that the zero-profit level x, is self-sustaining (i.e., x* 2 r). Clearly it will never be optimal to harvest to a level below 2, since this will not only cost money, but also reduce the size of the population. Under optimal exploitation no harvesting will take place until the population is in excess of z=, and once that has occurred the population will never again drop below x,. If V(x) denotes the optimal E.P.V. then it is clear that for x 2 x,, V(x) satisfies the Bellman equation of Dynamic Programming (see, e.g., Kaufmann and Cruon
co b V(x)
=
01
max
vEb,.zl
Q(4
1
-
Q(Y)+
Ia
~r~.fw~@)l
i
*
(3.2)
Now the function Q defined by (2.9) is monotone increasing on the interval (x,, m) since p > c(z) for such 2. Thus it has an inverse, Q-l, well defined on [O, Q(m) J. Letting u = Q(Z) and s = Q(g), and letting W denote the composition IroQ-l, (3.2) can be expressed as W(u)
= (Y max
#~[o.ul(U-V+~
' W(Q(rfCQ'(v)ll)do(z)}.
(3.3)
Here W(u) represents the optimal E.P.V. of the resource as a function of its immediate harvest value u, and (3.3) is the Bellman equation for the maximization problem with state variable u. In this transformed problem the state transitions are governed by v -+ Q(zf[Q-1(v)]1, z E [a, b], rather than by y + z!(y) as in the original problem. If we assume that the function Q (zfC&-l(v) 31 is concave and nondecreasing in v for every z E [a, b] then the transformed problem is similar to the problem of maximizing the E.P.V. with a constant unit harvesting cost, and a concave stock-recruitment function-a problem treated by Reed [ll, Theorem 3, p. 3291. The condition that Q~~f[&-~(v)]) b e concave and nondecreasing in v for all v E [a, b] can be interpreted as meaning that over the full range of possible environmental conditions there be positive but decreasing marginal annual growth in the ,immediate harvest value of the resource. Since Q and j are both differentiable and
since Q-1 is increasing, it follows by differentiating
and using (2.9) that the above
OPTIMAL ESCAPEMENT LEVELS condition
is equivalent
Condition
(A)
to :
P - cCzfb)l .f’b) P - c(x) is nonnegative
(3.4)
and decreasing in x on [x,, nz], for every
This condition resembles condition 3 of Clark [3, p. 245) in his analysis of the deterministic control problem. We shall show that if condition (A) holds then a constant-escapement feedback policy maximizes E.P.V. Then we shall consider some special forms of the cost function c(x) for which condition (A) holds. We shall need a lemma. LEMBIA 1. If both I$ and # are continuous, nonnegative, and concave junctions and if fi is also nondecreasing then the composition Go+ is also continuous and concave. The proof follows readily from the definition of concavity (see e.g., Intrilligator [S, p. 4621 and we omit details. We now use a standard induction technique of Dynamic Programming to prove THEOREM 1. If (1) the zero-profit level x, is self-sustaining, (2) the average stockrecruitment junction j is concave and increasing and (S) the unit cost function satisfies condition (A), then E.P. V. is maximized by a constant-escapement feedback policy of the type (3.1). Proof. We let W,(u) represent the maximum E.P.V. over an N period timehorizon when the initial immediate harvest value of the resource is u (N = 1, 2, . . .). Also we define W,(u) = 0. It follows that the functions W, satisfy the Bellman equation
W,+l(u)
= a max “Elo..]/u-v+~
*K4M3F1(v~I\~d~~
i
(3.5)
for N = 0, 1, 2, . . . . We now proceed by induction and assume that W,(u) is continuous, nondecreasing, and concave on [0, Q(m)]. From Lemma 1 and condition (A) it follows that for each z E [a, b], W,(Q(z j[Q-l(v)]}) is continuous and concave in v on [O, Q(m)]. Since a convex combination of concave functions is also concave it follows that
/ is also continuous
*w,(Q(zfC&'(v)ll)d~(z) a
and concave on [0, Q(m)], and thence that
H,+1(v) = --v +
* w,(Q(zfC&-‘(v)l))d’P(z> / I3
(3.6)
356
WILLIAM
J. REED
is continuous and concave in v on [O, Q(m)]. Now if H,+l attains its maximum on [O, Q(m)] at T,+1, say, it follows from the concavity that it attains its maximum on 10, u] at u if 0 5 u 5 !!‘,+I and at T,+l if !Z’,+l < u _< Q(m). Thus from (3.5) Wn+l(U) = (Y[u + H7&+1(u)l,
= a[~ + Hn+~(Tn+dl,
if
u I T,+I,
if
u > Tnel.
(3.7)
We now show that W,,+l satisfies the induction hypothesis. That it is continuous and nondecreasing is obvious. Its concavity follows from the fact that Hnfl is concave and attains its maximum on [O, Q(m)] at T,+l. Since the induction hypothesis holds for W,(x) = 0, we conclude that it holds for all N. Now from the general theory of Dynamic Programming (see, e.g., Kauffman and Cruon [SJ) we have that W,(u) converges uniformly in u on [0, Q(m)] to W(u). Thus W(u) is also continuous, nondecreasing, and concave and b
H(v) = -v + / (I
~(Q~dCQ-l(~>ll)~W)
(3.3)
is continuous and concave with a maximum on [O, Q(m)] at T, say. It follows from (3.3) that a policy which prescribes a reduction of the immediate harvest value of the population to the level T whenever it is in excess of T, but otherwise prescribes no harvest, is optimal for the maximization of E.P.V. Letting S = Q-l(T), this means that a constant-escapement policy of the type (3.1) at the n level S is optimal. We now show that condition (A) holds for the cost function c(x) = k/x provided that j satisfies a weak additional hypothesis. In this case expression (3.4) in condition (A) can be written as zf(x) X-X
- xc3 xf’(x> .-. 00
fO>
Now if j is concave and increasing and if zj(z,) the case if x, is self-sustaining, then the ratio
zf(zJ - Gl x - x.x
(3.9) > x, for all z E [a, b], which is
(3.10)
is positive and decreasing in z on [x,, m]. This can be seen readily in Fig. 2, in which the slope of the ray P&i joining the points (z,, x,) and (xi, zf (xi)) represents ratio (3.10) at x = 5i. A formal proof can be obtained by differentiating (3.10) and using the properties of differentiable concave functions. Now if we assume that xj’(x)/j(x) is nonincreasing in x then (3.9) will be the product of two positive terms, one decreasing and the other nonincreasing on [x,, m], and will thus itself be positive and decreasing on [x-, ml-i.e., condition (A) will hold. The condition that xf’(x)/f (x) be nonincreasing is a technical one. It is satisfied for the Beverton and Holt Cl] form of the stock-recruitment function j(x) = Ax/(1 + Bx), for the Cushing [4] form j(x) = AxB (0 < B < l), and also for the Ricker [13] form, j(x) = Axe-== on its increasing branch. We have then the following corollary to Theorem 1.
OPTIMAL
ESCAPEMENT
LEVELS
x
FIGURE2 COROLLARY 1. If the average stock-recruitment junction f is concave, is increasing, be nonincreasing and if the unit cost and satis$es the condition that xf’(x)/f(x) junction is c(x) = k/x with the zero-profit level self-sustaining then E.P.V. is maximized by a constant-escapement feedback policy. For a cost function c(z) = k/xc, expression (3.4) in condition
(A) can be written (3.11)
Provided that f satisfies the conditions given in Corollary 1 it is fa.irly easy to show that the first factor in (3.11) is positive and decreasing in x on [x,, m] for all positive 8. The second factor will also be positive and nonincreasing for 0 5 0 5 1, but may not be so for 0 > 1. We thus have COROLLARY 2. Corollary 1 remains true if the unit cost function c(x) = k/xc, where 0 < 8 5 1.
is of the form
Corollary 1 tells us that a constant escapement policy is optimal when the production function is of the Schaefer form-i.e., when the cat&ability coefficient is constant at all levels of abundance. Corollary 2 gives the same result for a catchability coefficient that decreases with population abundance like x8 for -1_ 1. For example, for the Beverton-Holt form f(z) = As/l + Bx it is equal to (1 + Bx)O-~/A~, which is nonincreasing for 0 5 0 < 2. It follows that in this case Corollary 2 holds also for 1 < 0 2 2. However a value of B greater than 1 would imply that the catchability coefficient increases with increased abundance-something that could be expected to hold, at most, over a limited range of population sizes. In this section we have looked for conditions which might reasonably be satisfied and which are sufficient for the optimality of a constant-escapement policy. It is of interest to inquire as to what sort of policy might be optimal when a constant-escapement policy is not optimal. If the expression in braces in (3.2) were not concave in y, but rather biomodal as in Fig. 3, then a policy with three “switching points” would be optimal. In this case an optimal policy would be
WILLIAM
358
J. REED
the feedback policy. h, = X, - SB
= 0 = X, - St = 0
if
x?a > As1
if if
s3 2 xn 2 sz sz 2 x, 2 St
if
St 1 x,.
Such a situation might arise, for instance, if there were depensation present in the stock-recruitment model, or if the unit cost of harvesting decreased rapidly with the population size. Optimal policies with more switching points could result if the expression in braces in (3.2), as a function of y, had more than two local maxima. If the assumption of an infinitely elastic demand function for fish were relaxed the economic yield in any year would not be separable as in (2.10). In this case in general the optimal escapement in any year would depend on the size of the population before harvest. The same would be true if the production function (2.3) were nonlinear in the input variable E. 4. OPTIMAL
ESCAPEMENT LEVELS AND DETERMINISTIC
IN THE STOCHASTIC MODELS
In this section we obtain some results on the optimal stochastic escapement level S of Theorem 1 and compare it with the optimal escapement level D in the corresponding deterministic model. We shall need the following lemma : LEMMA 2. If SN is the optimal escapement level for the N-period problem in Theorem 1, then the sequence {SN) is nondecreasing and converges to S.
The proof follows from the fact that the concave functions HN defined in (3.6) are differentiable with H’N+I (u) 2 HIN(u). We omit details. We now define the function b
l(x) = a!
I0
QCzfWld@@- &Cd.
(4.0
This represents the expected discounted annual growth in the immediate harvest value of the population as a junction of its size. If condition (A) holds then 1 will be
FIGURE 3
OPTIMAL
ESCAPEMENT
359
LEVELS
either monotone or unimodal on [x,, m], since in this case l’(x) can have at most one zero on this interval. Let u be the point at which I attains its maximum on cxcc.7m]. We first show that u is a lower bound for the optimal escapement level S. LEnmA 3. Under the conditions oj Theorem 1, S > CJ.
Proof. We first observe that the functions W,(u) and HN(u) are differentiable for N = 0, 1, . . . . This can easily be proved by induction using (3.7) and (3.8). Now from (3.7), for N 2 1, W’,(u)
= a[1 + H’N(U)l, = a,
U 5
(4.2)
u > TN,
at TN it follows that H’N(u)
Since HN is concave and attains its maximum for u 5 TN and thence that W’,(u)
TN,
2 0
(4.3)
2 a
for 0 < u < Q(m). Differentiating (3.6) and using (4.3) it follows that
H'N+I(u) 2 - 1 + ______
zQ'(zfCQ-'(U)l~d~(z),
which can be written as 1 H’
AI+1 (4
2
---
Q'CQ-1
(u),
(4.4)
“cgl(u)l*
Now HN+l attains its maximum on [0, Q(m)] at Z’N+~and I attains its maximum on [x,, m] at u. We shall show that TN+, > Q(U) and from this it will follow that s N+I = Q-‘(TN+I) 1 Q. First, if c E (x,, m), then from (4.4) H ‘N+l[Q(u)] 2 0. From the concavity of HN+l it follows that TN+~ 2 Q(a). Second, if c = x,, then TN+~ 1 Q(a), since Q(U) = 0. Third, if u = m, then from (4.4) H’N+l[Q(m)-] 1 Z’(m-)/Q’(m) 2 0 (where the superscript ” - ” indicates a left derivative), and therefore from the concavity of HN.+.l it follows that TN+~ 2 Q(m) = Q(u). Thus we can conclude that for fi = 1, 2, . . ., SN+l > u and thence that 8 = lim SN 1 U. N-r-
The next result gives conditions level U.
n
under which the optimal escapement is at the
THEOREM 2. Under the conditions of Theorem 1 if the level u (at which the expected discounted annual growth in the immediate harvest value of the population is maximized) is selj-sustaining then the optimal escapement is at u (i.e., S = u). If on the other hand u is not self-sustaining then S > u, (except in the trivial exceptional case when S = u = m). Proof. In the case when u is self-sustaining we proceed by induction and assume that SN = U. It follows that TN = Q(u). Now from Lemma 3 we know that T N+~ 1 Q(U). We shall show that HN+I attains its maximum on [Q(U), Q(m)] at Q(u)-i.e., that TN+~ = Q(u). From (3.6), for u 2 Q(u) we have
W,(u) = 4~ +EvCQ(~)lI.
(4.5)
WILLIAM
360
J. REED
Furthermore if u is self-sustaining then zf(g) 2 u for all z E [a, 61, and therefore for u 1 Q(u) it follows that Q(zf[Q-l(u)]) 2 Q(a). From (XFj) and (4.5) it then follows that for u 2 Q(u) b HN+l(U)
=
--u
+
a
= I[&-i(u)]
(Q~zfCQ-'(411+ HN(Q)W+)
a
/
+ constant.
(4.6)
Since 1 attains its maximum on [x-, m] at u it follows that HN+I attains its maximum on [Q(u), Q(m)] and therefore on [0, Q(m) J at Q(u)-i.e., TN+1 = Q(U) and SN+~ = u. To complete the induction we show that Sz = u. We have from (3.6) that HI(u) = --u with T1 = 0. From (3.7) it follows that 6
Hz(u)
~Q~~fCQ-'(4lIW4
= --u + I
=
lc&-l(u);.
(4.7)
From this it follows that Tz = Q(u) and X2 = u. We have thus proved that S, = u for N = 2, 3, . . . . It follows that S = limN,, SN = u. In the case when u is neither self-sustaining nor equal to m, u will belong to the interval (x,, m), since by hypothesis z= is self-sustaining. It follows that Z’(U) = 0. In order to prove that S > u in this case it is sufficient to prove that S3 > u and then use Lemma 2. From (4.7) we have that
W,(u) = 4~ + IL-Q-'(411, = crcu + Uu)J, and hence that from (3.6)
W’,(u)
u < Q(u), u 2 Q(u),
> a! for u < Q(u) and W’,(u)
(4.3)
= (Y for u 2 Q(u). Now
.Q'(zfCQ-'(~)l)zd~(z). (4.9) If u is not self-sustaining then the probability measure of the set (z: Q[zf(a)] < Q(U)) is positive. It follows then from (4.9) and the observation following (4.8) that
H’s[QWl
4(u) b Q'Czfb)WW, Q'(4 / a
> -1 + -
i.e., that
H’&&(u)]
> Z’(u)/&‘(u)
= 0.
From the fact that H3 is concave it follows that Ta > Q(u) and S3 > u. Using Lemma 2 completes the proof. n We now compare the optimal escapement levels in the deterministic and stochastic models. The deterministic model of Clark [3] and Spence [14] is obtained from our model by letting the distribution of the r.v.‘s 2, reduce to a degenerate distribution with all its probability at the point 1. It is clear from
OPTIMAL ESCAPEMENT LEVELS
Theorem 2 that the optimal
deterministic
361
escapement is at the level D at which
g(x) = ~QCf(~)l - Q(x)
(4.10)
attains its maximum on [x,, 21, where 2 is the undisturbed equilibrium population level [i.e., 2 = f(g)]. This result has been obtained by Clark [Z]. The next lemma compares u and D. LEMMA 4. Provided D < i (i.e., the optimal harvesting), then
deterministic
policy involves some
(a) c < D or u = D = x, if the junction xc(x) is strictly convex, (b) u > D or u = D = x, if xc(x) is strictly concave, and (c) u = D if xc(x) is linear. Proof. Consider l’(x) - g’(x) = aj’(x)
= d’(x)
[/* Q'Czfb)WW - QUWI] D [c[f(x)l
- [
c[zj(x),zdQ(a)].
(4.11)
Let the r.v. Y be defined as Y = 21(x). It follows that
s b
a
c[.zj(x)]zd$
(z) =
1
--W'cU')I "f(x)
(4.12)
and from Jensen’s inequality (see, e.g., Rae [lo, p. 461) it then follows that the r.h.s. of (4.12) is, respectively, greater than, less than, or equal to
Wf(x)W(Y)4E(Y)I
= cCf(~)l
(4.13)
if the function xc(x) is strictly convex, strictly concave, or linear. Thus since f’(x) > 0 it follows from (4.11), (4.12), and (4.13) that Z’(x) is, respectively, greater than, less than, or equal to g’(x) on [x,, $1 if xc(x) is strictly convex, strictly concave, or linear on [x,, m]. Since by hypothesis both 1’ and g’ are monotone on [x,, $1 the result follows. w Using the results of Theorem 2 and Lemma 4 we arrive at the following theorem in which the optimal escapement levels in the stochastic and deterministic models are compared. THEOREM 3. Under the conditions of Theorem 1 provided that the optimal policies in both the stochastic and deterministic models involve some harvesting, then the optimal stochastic escapement S compares with the optimal deterministic escapement D in the following way: (i) If the unit harvest cost c(x) is such that xc(x) is strict2y concave then S > D or S = D = x,. (ii) If xc(x) is linear then S = D if D is self-sustaining and S > D if D is not self-sustaining. (iii) If xc(x) is strictly convex and S is self-sustaining then S < D or S = D = x-.
362
WILLIAM
J. REED
We note that in (i) and (iii) equality can occur only when it is optimal to maintain the escapement at the zero-profit level. This will occur only for suitably high values of the discount rate (Clark [2], Reed [12]). We also observe that the case when xc(x) is strictly convex and S is not self-sustaining has not been resolved. Under the assumption of Section 2 that the instantaneous harvest at any level of population size is proportional to the effort, it follows from (2.4) that XC(X) is inversely proportional to the catchability coefficient p(z). In other words, XC(Z) can be thought of as the cost associated with the realization of a given per capita rate of harvest mortality when the population is at level x. Under most circumstances, this cost can be expected to stay constant (Schaefer production function) or increase (Gulland [5]) with an increase in population size, although it is conceivable that it could decrease. Condition (i) of Theorem 3 can be interpreted as being that the marginal cost of realizing a given per capita mortality rate resulting from a unit increase in population size at level 2 decreases as x increases. Conditions (ii) and (iii), respectively, are that this marginal cost is constant or increases with x. An intuitive explanation of why S > D under condition (i) even when D is self-sustaining follows from the fact that the marginal average annual harvest cost in the stochastic model resulting from an increase in the escapement level from D to D + 1 is, because of the averaging process, less than the corresponding marginal cost in the deterministic model. This implies that the corresponding marginal increase in E.P.V. is positive in the stochastic case, while zero in the deterministic case. Thus, the optimal stochastic escapement level S is greater than D. Similar interpretations can be given for cases (ii) and (iii) of Theorem 3. We now consider the cost function c(x) = k/xc, which results from the CobbDouglas production function (2.6). COROLLARY 1. If c(x) = k/x (which is the case for the Schafer production junction with constant catchability coeficient) and if the other conditions of Corollary 1, Theorem 1 hold, then S = D if D is self-sustaining and S > D if D is not seljsustaining. COROLLARY 2. If the unit harvest cost is constant (which is the case when the abundance) then S = D if D is self-sustaining and S > D if D is not self-sustaining.
c.p.u.e. is independent of population
COROLLARY 3. If c(x) = k/xc, with 0 < 0 < 1 (which is the case for the Gulland model in which catchability decreases like x@-I), and the other conditions of Corollary 1, Theorem 1 hold then S > D or S = D = x,.
In all of the above examples S has been greater than or equal to D. A somewhat contrived example in which S < D is when the cost function is c(x) = k/xc with 1 < 0 _< 2 and the average stock-recruitment function is the Beverton-Holt form f(x) = Ax/l + Bx. As has already been observed in Section 3 a constantescapement policy is optimal in this case if x, is self-sustaining. Since xc(x) = k/xc-l is convex it follows from Theorem 3 (iii) that S < D if S is self-sustaining. However, as observed in Section 2, these parameter values imply that the catchability coefhcient increases with x like xc-l at least over [x-, ml. In most realistic cases it seems that S would be no less than D. The presence of environmental fluctuation can cause an upward shift in the optimal escapement through two distinct mechanisms. The first, stated in Lemma 4, is that u (and
OPTIMAL
ESCAPEMENT
LEVELS
363
therefore S) will be shifted up from D for certain forms of the cost function. The second mechanism, stated in Theorem 2, is that S will shift up from (Tand therefore from D when u is not self-sustaining. While it appears difficult to establish a precise relationship it seems that the upward shift of S will increase with the degree of environmental fluctuation. This can definitely be established for a shift attributable to the first mechanism when the cost function is of thrl form c(x) = k/xc,
e < 1.
This tentative conclusion agrees with the numerical results of Ludwig and Varah [9] who have considered the optimal control problem for a continuous-time> model with added noise. REFERENCES 1. R. J. H. Beverton and S. J. Holt, On the dynamics of exploited fish populations, Fish. Invest., Ser. 2 (19), London (1957). 2. C. W. Clark, Profit maximization and the extinction of animal species, J. Pal. Econ. 81, 950-961 (1973). 3. C!. W. Clark, “Mathematical Bioeconomics,” Wiley, New York (1976). 4. D. H. Cushing, The dependence of recruitment on parent stock in different groups of fishes, J. Cons. Iti. Ezpbr. Mm. 33(3) 340-362 (1971). 5. J. A. Gullsnd, The stability of fish stocks, J. Cons. Znt. Exph-. Mm. 37(3), 199-204 (1977). 6. M. D. Intrilligator, “Mathematical Optimization and Economic Theory,” Prentice-Hall, E&$ewood Cliffs, N.J. (1971). 7. D. L. Jaquette, A discrete time population control model, Math. B&X%. 231-252 (1972). 8. A. Kaufmann and R. Cruon, “Dynamic Programming,” Academic Press, New York (1967). 9. D. Ludwig and J. M. Varah, “Optimal Harvesting of a Randomly Fluctuating Resource. II. Numerical Methods and Results,” Technical Report 77-12, Inst. Applied Mathematics and Statistics, University of British Columbia, Vancouver (1977). 10. C. R. Rao, “Linear Statistical Inference and its Applications,” Wiley, New York (1965). 11. W. J. Reed, A stochastic model for the economic management of a renewable animal resource, Math. Biosci. 22, 313-337 (1974). 12. W. J. Reed, The steady-state of a stochastic harvesting model, Math. Biosci. 41, 273-307 (1978). 13. W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Canada 11(5), 559-623 (1954). 14. M. Spence, “Blue Whales and Applied Control Theory,” Technical Report No. 108, Inst. for Mathematical Studies in Social Sciences, Stanford University (1973).
OF ENVIRONMENTAL
ECONOMICS
AND
MANAGEMENT
6, 350463
(
1979)
Optimal Escapement Levels in Stochastic and Deterministic Harvesting Models * WILLIAM
J. REED
Department of Mathematics, University of Victoria, Victoria, British Columbia V8W SY.3, Canada Received January 22, 1979; revised September 6, 1979 A constantiscapement feedback policy is shown to be optimal in maximising expected discounted net revenue from an animal resource whose dynamics are described by a stochastic stock-recruitment model, provided that unit harvesting costs satisfy certain conditions. The optimal escapement in thii model is compared with that in the corresponding deterministic model and it is shown how the way in which unit harvesting costs vary with population abundance can be important in determining the relative sizes of the optimal escapements. In most cases, the optimal stochastic escapement is no less than the optimal deterministic escapement.
1. INTRODUCTION The optimal economic exploitation of renewable animal resources is a subject that has received considerable attention in recent years, and much theoretical insight into the problem has been gained through the application of the techniques of optimal control theory using both continuous-time and discrete-time population models (see, for example, Clark [3]). To date, however, the theory has been to a large extent confined to simple deterministic population models. Sjnce the dynamics of most real animal populations appear to exhibit considerable random fluctuation, it is an important question to determine to what extent the results of the deterministic theory carry over to an analysis of analogous stochastic population models. In discrete time, using a deterministic “stock-recruitment” model and a net revenue function linear in the harvest, Clark [2, 3 3 and Spence [14] have shown that, under fairly general conditions, the present value of the economic rent of the resource (total discounted value of the revenue stream) is maximized by a “bang-bang” policy which forces the population to some optimal level of escapement as rapidly as possible. Clark [3, p. 2451 shows how this is analogous to the optimal policy in the corresponding continuous-time model. Reed [ll] and Jaquette [7] have extended this analysis to discrete-time stochastic population models and have shown that an optimal policy to maximize expected present value is given by a feedback control which forces the population back to some optimal escapement level as rapidly as possible. However, in order to prove this * Research for this paper was carried out at the Institute versity of British Columbia. 350 0095~0696/79/04035414$02.00/0
of Animal
Resource Ecology, Uni-
OPTIMAL
ESCAPEMENT
LEVELS
351
analogous result, both of the above authors imposed much more restrictive conditions on the cost structure of the model than did Clark [3) and Spence [14) in their deterministic analyses. In this paper, we first prove the optimality of a feedback “constant-escapement” policy for the stochastic population model of Reed [ll] under more general cost assumptions, which can be derived from a production function model of the harvest. This gives a closer parallel to the deterministic results of Clark [3]. Second, we undertake a comparison of the optimal escapement levels in the stochastic and deterministic models. It will ht> shown how the form of the unit cost function can play an important role in determining the relative optimal levels of escapement. 2. THE
MODEL
The population model we shall employ is the stochastic stock-recruitment model discussed in Reed [ll, 121 and is a simple stochastic version of the model employed by Clark [3]. If X, is a random variable (T.v.) denoting the size of the population in year n, then we suppose that the process (X, l is Markovian with transitions governed by the equation X
a+1
=
(2.1)
z?Lwn>,
where f(x) is the expected or average stock-recruitment function, which we shall assume is differentiable, strictly concave, and nondecreasing,’ i.e., exhibits “normal compensation” ; and (Z,} is a sequence of independent identically distributed (i.i.d.) r.v.‘s with unit mean. Thus the deterministic stock-recruitment relationship of Clark [3] will hold in this model in terms of conditional expectations; i.e., E(X,+l(X, = x) = j(x). W e f ur th er assume that the random multipliers 2, are distributed on some finite interval [a, b], where 0 < a < 1 < b < 30 , with a common distribution function (d.f.) a(z) and that bf(0) > 1 and lim, +oDa!‘(z) < 1. It then follows (Reed [12]) that there exist finite population levels m = max(bf(x): bf(x) 2 2, x > 01 and r = max{af(z):af(x) 2 x,x 2 0) such that with probability 1 (w.p. I), the population will eventually enter and thenceforth stay within the interval [r, ml and on which it will, in the limit, attain a stationary probability distribution (Fig. 1). Note that r will be zero if af’(0) < 1. We shall call any population level 2 for which pr(X,+l
1 x(X,
= x/ = 1
self-sustaining. Clearly, in our model all population levels in the interval [0, r] are self-sustaining (see Fig. 1). We suppose that harvesting can take place each year during a short harvest season throughout which natural mortality in the population is insignificant. If X, denotes the size of the population immediately prior to harvesting in year n, 1 Most of the reauita can be proved for the case of “overcompensation,” i.e., f concave and dome-shaped, provided that certain additional reasonable conditions are imposed on the cost structure. However, to avoid these complications (see Clark [3, p. 24811, we here restrict discu&sion to the case of normal compensation.
352
WILLIAM
J. REED
FIGURE 1
and a harvest of size h, is taken that year, then X+1
= Z,f(X,
- h,).
(2.2)
We make the following economic assumptions which are the same as those made by Clark [3, p. 2431 in his deterministic analysis : (a) Demand is infinitely elastic; i.e., each unit of the population captured can be sold for a fixed price, p, say. (b) Harvesting costs vary with population abundance. We suppose that the unit cost of harvesting is independent of harvest size but increases as the size of the population decreases. Specifically we assume that the unit cost when the population is at the level x is c(x), where c is some nonincreasing function. A cost model of this form can, under certain assumptions, be derived from a production function model of the harvest. If it is assumed that the instantaneous harvest rate h is related to the fishing (or, more generally, harvest) effort E, and to the population abundance x, by a production function of the form h = q(z)Ex,
(2.3)
where q is a coefficient of harvest mortality per unit of effort (catchability coeficient) which in general depends on the population abundance, then the aggregated effort in a time increment At needed to catch an amount Ah when the population is at level 2 is inversely proportional to zq(x). Thus if it is assumed that harvest costs are proportional to total harvest effort and that natural mortality is insignificant during the harvest season, then the unit cost function is of the form 0) = klh7(x)). (2.4) If it is assumed that the catch per unit of effort (c.p.u.e.), xq(x) increases (or at least does not decrease) with population abundance, then c(x) will be nonincreasing. A special case of (2.3) is when the catchability coefficient q is constant. This is the well-known Schaefer production function in which c.p.u.e. is assumed proportional to population size. In this case the unit cost function is of the form c(x) = k/x.
(2.5)
OPTIMAL
ESCAPEMENT
LEVELS
333
Clark [3, p. 2381 gives another justification for a cost function of this form in terms of the expected time taken to locate randomly distributed schools of fish. Although the Schaefer production function has been widely used there is good reason to believe that in many fisheries the catchability coefficient changes with population abundance. Gulland [5) adduces a number of mechanisms whereby the catchability coefficient might be expected to increase with a decrease in population abundance. Among these are a reductiofi in the area inhabited by the population at low levels of abundance and the phenomenon of gear saturation at high levels of abundance (see also Clark [3, p. 2351). Gulland models the catchability coefficient as proportional to z@for some p < 0. This leads to a production function h = KExB,
where 0 = 1 + p 5 1 and K is a constant. In this case the unit cost function is
This is of the Cobb-Douglas
c(x) = k/x@.
(2.6)
form. (2.7)
A limiting case is when /3 = - 1. In this case the c.p.u.e. is independent of abundance and the unit harvesting cost is constant. As can be seen from the above discussion the artifact, of a nonincreasing unit harvest cost, can summarize quite complex responses of the catchability to variations in population abundance. However it cannot adequately describe the phenomenon of diminishing marginal returns on effort, caused, for instance, by congestion on the fishing grounds. For a discussion of the cost implications of a production function which is nonlinear in the effort variable see Clark [3, p. 236-J. We let x, denote the zero-pro$t level; i.e., if c(0) 2 p then x, is the solution to c(x) = p (more strictly the supremum of such solutions). If c(0) < p then we define x, as zero. (c) There is time-discounting of future revenues with an annual discount factor OL< 1, assumed constant. Thus a revenue earned in year 11is discounted by a factor OL~to its “present” value in some brtse year zero. For a sequence of harvests {h,} the expected total discounted revenue is (2.8)
By letting (2.9)
(2.8) can be expressed as
Ein-l I? @CQ(Xn>-
Q(X, - hn)]).
(2.10)
For a given preharvest abundance x the quantity Q(x) represents the maximum revenue that can be realized in a single harvest. We shall call Q(X) the immediate harvest vcslueat the resource. The control problem which we shall consider is the maximizastion of the expected total discounted revenue (or expected present value (E.P.V.)). Mathematically the problem is to find a feedback control law (i.e., a sequence of func-
354
WILLIAM
J. REED
tions (h,(x,) }) to maximize constraints
(2.10) subject
to the state equation
(2.2), the
0 5 hn I xv,,
(2.11)
Xl = Xl.
(2.12)
and an initial condition such as
3. AN OPTIMAL
SOLUTION
We prove in this section that under certain conditions on the cost function c(x) and on the state equation (2.1) there is an optimal policy of the “constantescapement” type; i.e., there is an escapement level S with the optimal harvest h, in any year n being given by h, = 0,
= X, - S,
if if
xrz 5 s, x, 2 s.
(3.1)
We shall assume that the zero-profit level x, is self-sustaining (i.e., x* 2 r). Clearly it will never be optimal to harvest to a level below 2, since this will not only cost money, but also reduce the size of the population. Under optimal exploitation no harvesting will take place until the population is in excess of z=, and once that has occurred the population will never again drop below x,. If V(x) denotes the optimal E.P.V. then it is clear that for x 2 x,, V(x) satisfies the Bellman equation of Dynamic Programming (see, e.g., Kaufmann and Cruon
co b V(x)
=
01
max
vEb,.zl
Q(4
1
-
Q(Y)+
Ia
~r~.fw~@)l
i
*
(3.2)
Now the function Q defined by (2.9) is monotone increasing on the interval (x,, m) since p > c(z) for such 2. Thus it has an inverse, Q-l, well defined on [O, Q(m) J. Letting u = Q(Z) and s = Q(g), and letting W denote the composition IroQ-l, (3.2) can be expressed as W(u)
= (Y max
#~[o.ul(U-V+~
' W(Q(rfCQ'(v)ll)do(z)}.
(3.3)
Here W(u) represents the optimal E.P.V. of the resource as a function of its immediate harvest value u, and (3.3) is the Bellman equation for the maximization problem with state variable u. In this transformed problem the state transitions are governed by v -+ Q(zf[Q-1(v)]1, z E [a, b], rather than by y + z!(y) as in the original problem. If we assume that the function Q (zfC&-l(v) 31 is concave and nondecreasing in v for every z E [a, b] then the transformed problem is similar to the problem of maximizing the E.P.V. with a constant unit harvesting cost, and a concave stock-recruitment function-a problem treated by Reed [ll, Theorem 3, p. 3291. The condition that Q~~f[&-~(v)]) b e concave and nondecreasing in v for all v E [a, b] can be interpreted as meaning that over the full range of possible environmental conditions there be positive but decreasing marginal annual growth in the ,immediate harvest value of the resource. Since Q and j are both differentiable and
since Q-1 is increasing, it follows by differentiating
and using (2.9) that the above
OPTIMAL ESCAPEMENT LEVELS condition
is equivalent
Condition
(A)
to :
P - cCzfb)l .f’b) P - c(x) is nonnegative
(3.4)
and decreasing in x on [x,, nz], for every
This condition resembles condition 3 of Clark [3, p. 245) in his analysis of the deterministic control problem. We shall show that if condition (A) holds then a constant-escapement feedback policy maximizes E.P.V. Then we shall consider some special forms of the cost function c(x) for which condition (A) holds. We shall need a lemma. LEMBIA 1. If both I$ and # are continuous, nonnegative, and concave junctions and if fi is also nondecreasing then the composition Go+ is also continuous and concave. The proof follows readily from the definition of concavity (see e.g., Intrilligator [S, p. 4621 and we omit details. We now use a standard induction technique of Dynamic Programming to prove THEOREM 1. If (1) the zero-profit level x, is self-sustaining, (2) the average stockrecruitment junction j is concave and increasing and (S) the unit cost function satisfies condition (A), then E.P. V. is maximized by a constant-escapement feedback policy of the type (3.1). Proof. We let W,(u) represent the maximum E.P.V. over an N period timehorizon when the initial immediate harvest value of the resource is u (N = 1, 2, . . .). Also we define W,(u) = 0. It follows that the functions W, satisfy the Bellman equation
W,+l(u)
= a max “Elo..]/u-v+~
*K4M3F1(v~I\~d~~
i
(3.5)
for N = 0, 1, 2, . . . . We now proceed by induction and assume that W,(u) is continuous, nondecreasing, and concave on [0, Q(m)]. From Lemma 1 and condition (A) it follows that for each z E [a, b], W,(Q(z j[Q-l(v)]}) is continuous and concave in v on [O, Q(m)]. Since a convex combination of concave functions is also concave it follows that
/ is also continuous
*w,(Q(zfC&'(v)ll)d~(z) a
and concave on [0, Q(m)], and thence that
H,+1(v) = --v +
* w,(Q(zfC&-‘(v)l))d’P(z> / I3
(3.6)
356
WILLIAM
J. REED
is continuous and concave in v on [O, Q(m)]. Now if H,+l attains its maximum on [O, Q(m)] at T,+1, say, it follows from the concavity that it attains its maximum on 10, u] at u if 0 5 u 5 !!‘,+I and at T,+l if !Z’,+l < u _< Q(m). Thus from (3.5) Wn+l(U) = (Y[u + H7&+1(u)l,
= a[~ + Hn+~(Tn+dl,
if
u I T,+I,
if
u > Tnel.
(3.7)
We now show that W,,+l satisfies the induction hypothesis. That it is continuous and nondecreasing is obvious. Its concavity follows from the fact that Hnfl is concave and attains its maximum on [O, Q(m)] at T,+l. Since the induction hypothesis holds for W,(x) = 0, we conclude that it holds for all N. Now from the general theory of Dynamic Programming (see, e.g., Kauffman and Cruon [SJ) we have that W,(u) converges uniformly in u on [0, Q(m)] to W(u). Thus W(u) is also continuous, nondecreasing, and concave and b
H(v) = -v + / (I
~(Q~dCQ-l(~>ll)~W)
(3.3)
is continuous and concave with a maximum on [O, Q(m)] at T, say. It follows from (3.3) that a policy which prescribes a reduction of the immediate harvest value of the population to the level T whenever it is in excess of T, but otherwise prescribes no harvest, is optimal for the maximization of E.P.V. Letting S = Q-l(T), this means that a constant-escapement policy of the type (3.1) at the n level S is optimal. We now show that condition (A) holds for the cost function c(x) = k/x provided that j satisfies a weak additional hypothesis. In this case expression (3.4) in condition (A) can be written as zf(x) X-X
- xc3 xf’(x> .-. 00
fO>
Now if j is concave and increasing and if zj(z,) the case if x, is self-sustaining, then the ratio
zf(zJ - Gl x - x.x
(3.9) > x, for all z E [a, b], which is
(3.10)
is positive and decreasing in z on [x,, m]. This can be seen readily in Fig. 2, in which the slope of the ray P&i joining the points (z,, x,) and (xi, zf (xi)) represents ratio (3.10) at x = 5i. A formal proof can be obtained by differentiating (3.10) and using the properties of differentiable concave functions. Now if we assume that xj’(x)/j(x) is nonincreasing in x then (3.9) will be the product of two positive terms, one decreasing and the other nonincreasing on [x,, m], and will thus itself be positive and decreasing on [x-, ml-i.e., condition (A) will hold. The condition that xf’(x)/f (x) be nonincreasing is a technical one. It is satisfied for the Beverton and Holt Cl] form of the stock-recruitment function j(x) = Ax/(1 + Bx), for the Cushing [4] form j(x) = AxB (0 < B < l), and also for the Ricker [13] form, j(x) = Axe-== on its increasing branch. We have then the following corollary to Theorem 1.
OPTIMAL
ESCAPEMENT
LEVELS
x
FIGURE2 COROLLARY 1. If the average stock-recruitment junction f is concave, is increasing, be nonincreasing and if the unit cost and satis$es the condition that xf’(x)/f(x) junction is c(x) = k/x with the zero-profit level self-sustaining then E.P.V. is maximized by a constant-escapement feedback policy. For a cost function c(z) = k/xc, expression (3.4) in condition
(A) can be written (3.11)
Provided that f satisfies the conditions given in Corollary 1 it is fa.irly easy to show that the first factor in (3.11) is positive and decreasing in x on [x,, m] for all positive 8. The second factor will also be positive and nonincreasing for 0 5 0 5 1, but may not be so for 0 > 1. We thus have COROLLARY 2. Corollary 1 remains true if the unit cost function c(x) = k/xc, where 0 < 8 5 1.
is of the form
Corollary 1 tells us that a constant escapement policy is optimal when the production function is of the Schaefer form-i.e., when the cat&ability coefficient is constant at all levels of abundance. Corollary 2 gives the same result for a catchability coefficient that decreases with population abundance like x8 for -1_
WILLIAM
358
J. REED
the feedback policy. h, = X, - SB
= 0 = X, - St = 0
if
x?a > As1
if if
s3 2 xn 2 sz sz 2 x, 2 St
if
St 1 x,.
Such a situation might arise, for instance, if there were depensation present in the stock-recruitment model, or if the unit cost of harvesting decreased rapidly with the population size. Optimal policies with more switching points could result if the expression in braces in (3.2), as a function of y, had more than two local maxima. If the assumption of an infinitely elastic demand function for fish were relaxed the economic yield in any year would not be separable as in (2.10). In this case in general the optimal escapement in any year would depend on the size of the population before harvest. The same would be true if the production function (2.3) were nonlinear in the input variable E. 4. OPTIMAL
ESCAPEMENT LEVELS AND DETERMINISTIC
IN THE STOCHASTIC MODELS
In this section we obtain some results on the optimal stochastic escapement level S of Theorem 1 and compare it with the optimal escapement level D in the corresponding deterministic model. We shall need the following lemma : LEMMA 2. If SN is the optimal escapement level for the N-period problem in Theorem 1, then the sequence {SN) is nondecreasing and converges to S.
The proof follows from the fact that the concave functions HN defined in (3.6) are differentiable with H’N+I (u) 2 HIN(u). We omit details. We now define the function b
l(x) = a!
I0
QCzfWld@@- &Cd.
(4.0
This represents the expected discounted annual growth in the immediate harvest value of the population as a junction of its size. If condition (A) holds then 1 will be
FIGURE 3
OPTIMAL
ESCAPEMENT
359
LEVELS
either monotone or unimodal on [x,, m], since in this case l’(x) can have at most one zero on this interval. Let u be the point at which I attains its maximum on cxcc.7m]. We first show that u is a lower bound for the optimal escapement level S. LEnmA 3. Under the conditions oj Theorem 1, S > CJ.
Proof. We first observe that the functions W,(u) and HN(u) are differentiable for N = 0, 1, . . . . This can easily be proved by induction using (3.7) and (3.8). Now from (3.7), for N 2 1, W’,(u)
= a[1 + H’N(U)l, = a,
U 5
(4.2)
u > TN,
at TN it follows that H’N(u)
Since HN is concave and attains its maximum for u 5 TN and thence that W’,(u)
TN,
2 0
(4.3)
2 a
for 0 < u < Q(m). Differentiating (3.6) and using (4.3) it follows that
H'N+I(u) 2 - 1 + ______
zQ'(zfCQ-'(U)l~d~(z),
which can be written as 1 H’
AI+1 (4
2
---
Q'CQ-1
(u),
(4.4)
“cgl(u)l*
Now HN+l attains its maximum on [0, Q(m)] at Z’N+~and I attains its maximum on [x,, m] at u. We shall show that TN+, > Q(U) and from this it will follow that s N+I = Q-‘(TN+I) 1 Q. First, if c E (x,, m), then from (4.4) H ‘N+l[Q(u)] 2 0. From the concavity of HN+l it follows that TN+~ 2 Q(a). Second, if c = x,, then TN+~ 1 Q(a), since Q(U) = 0. Third, if u = m, then from (4.4) H’N+l[Q(m)-] 1 Z’(m-)/Q’(m) 2 0 (where the superscript ” - ” indicates a left derivative), and therefore from the concavity of HN.+.l it follows that TN+~ 2 Q(m) = Q(u). Thus we can conclude that for fi = 1, 2, . . ., SN+l > u and thence that 8 = lim SN 1 U. N-r-
The next result gives conditions level U.
n
under which the optimal escapement is at the
THEOREM 2. Under the conditions of Theorem 1 if the level u (at which the expected discounted annual growth in the immediate harvest value of the population is maximized) is selj-sustaining then the optimal escapement is at u (i.e., S = u). If on the other hand u is not self-sustaining then S > u, (except in the trivial exceptional case when S = u = m). Proof. In the case when u is self-sustaining we proceed by induction and assume that SN = U. It follows that TN = Q(u). Now from Lemma 3 we know that T N+~ 1 Q(U). We shall show that HN+I attains its maximum on [Q(U), Q(m)] at Q(u)-i.e., that TN+~ = Q(u). From (3.6), for u 2 Q(u) we have
W,(u) = 4~ +EvCQ(~)lI.
(4.5)
WILLIAM
360
J. REED
Furthermore if u is self-sustaining then zf(g) 2 u for all z E [a, 61, and therefore for u 1 Q(u) it follows that Q(zf[Q-l(u)]) 2 Q(a). From (XFj) and (4.5) it then follows that for u 2 Q(u) b HN+l(U)
=
--u
+
a
= I[&-i(u)]
(Q~zfCQ-'(411+ HN(Q)W+)
a
/
+ constant.
(4.6)
Since 1 attains its maximum on [x-, m] at u it follows that HN+I attains its maximum on [Q(u), Q(m)] and therefore on [0, Q(m) J at Q(u)-i.e., TN+1 = Q(U) and SN+~ = u. To complete the induction we show that Sz = u. We have from (3.6) that HI(u) = --u with T1 = 0. From (3.7) it follows that 6
Hz(u)
~Q~~fCQ-'(4lIW4
= --u + I
=
lc&-l(u);.
(4.7)
From this it follows that Tz = Q(u) and X2 = u. We have thus proved that S, = u for N = 2, 3, . . . . It follows that S = limN,, SN = u. In the case when u is neither self-sustaining nor equal to m, u will belong to the interval (x,, m), since by hypothesis z= is self-sustaining. It follows that Z’(U) = 0. In order to prove that S > u in this case it is sufficient to prove that S3 > u and then use Lemma 2. From (4.7) we have that
W,(u) = 4~ + IL-Q-'(411, = crcu + Uu)J, and hence that from (3.6)
W’,(u)
u < Q(u), u 2 Q(u),
> a! for u < Q(u) and W’,(u)
(4.3)
= (Y for u 2 Q(u). Now
.Q'(zfCQ-'(~)l)zd~(z). (4.9) If u is not self-sustaining then the probability measure of the set (z: Q[zf(a)] < Q(U)) is positive. It follows then from (4.9) and the observation following (4.8) that
H’s[QWl
4(u) b Q'Czfb)WW, Q'(4 / a
> -1 + -
i.e., that
H’&&(u)]
> Z’(u)/&‘(u)
= 0.
From the fact that H3 is concave it follows that Ta > Q(u) and S3 > u. Using Lemma 2 completes the proof. n We now compare the optimal escapement levels in the deterministic and stochastic models. The deterministic model of Clark [3] and Spence [14] is obtained from our model by letting the distribution of the r.v.‘s 2, reduce to a degenerate distribution with all its probability at the point 1. It is clear from
OPTIMAL ESCAPEMENT LEVELS
Theorem 2 that the optimal
deterministic
361
escapement is at the level D at which
g(x) = ~QCf(~)l - Q(x)
(4.10)
attains its maximum on [x,, 21, where 2 is the undisturbed equilibrium population level [i.e., 2 = f(g)]. This result has been obtained by Clark [Z]. The next lemma compares u and D. LEMMA 4. Provided D < i (i.e., the optimal harvesting), then
deterministic
policy involves some
(a) c < D or u = D = x, if the junction xc(x) is strictly convex, (b) u > D or u = D = x, if xc(x) is strictly concave, and (c) u = D if xc(x) is linear. Proof. Consider l’(x) - g’(x) = aj’(x)
= d’(x)
[/* Q'Czfb)WW - QUWI] D [c[f(x)l
- [
c[zj(x),zdQ(a)].
(4.11)
Let the r.v. Y be defined as Y = 21(x). It follows that
s b
a
c[.zj(x)]zd$
(z) =
1
--W'cU')I "f(x)
(4.12)
and from Jensen’s inequality (see, e.g., Rae [lo, p. 461) it then follows that the r.h.s. of (4.12) is, respectively, greater than, less than, or equal to
Wf(x)W(Y)4E(Y)I
= cCf(~)l
(4.13)
if the function xc(x) is strictly convex, strictly concave, or linear. Thus since f’(x) > 0 it follows from (4.11), (4.12), and (4.13) that Z’(x) is, respectively, greater than, less than, or equal to g’(x) on [x,, $1 if xc(x) is strictly convex, strictly concave, or linear on [x,, m]. Since by hypothesis both 1’ and g’ are monotone on [x,, $1 the result follows. w Using the results of Theorem 2 and Lemma 4 we arrive at the following theorem in which the optimal escapement levels in the stochastic and deterministic models are compared. THEOREM 3. Under the conditions of Theorem 1 provided that the optimal policies in both the stochastic and deterministic models involve some harvesting, then the optimal stochastic escapement S compares with the optimal deterministic escapement D in the following way: (i) If the unit harvest cost c(x) is such that xc(x) is strict2y concave then S > D or S = D = x,. (ii) If xc(x) is linear then S = D if D is self-sustaining and S > D if D is not self-sustaining. (iii) If xc(x) is strictly convex and S is self-sustaining then S < D or S = D = x-.
362
WILLIAM
J. REED
We note that in (i) and (iii) equality can occur only when it is optimal to maintain the escapement at the zero-profit level. This will occur only for suitably high values of the discount rate (Clark [2], Reed [12]). We also observe that the case when xc(x) is strictly convex and S is not self-sustaining has not been resolved. Under the assumption of Section 2 that the instantaneous harvest at any level of population size is proportional to the effort, it follows from (2.4) that XC(X) is inversely proportional to the catchability coefficient p(z). In other words, XC(Z) can be thought of as the cost associated with the realization of a given per capita rate of harvest mortality when the population is at level x. Under most circumstances, this cost can be expected to stay constant (Schaefer production function) or increase (Gulland [5]) with an increase in population size, although it is conceivable that it could decrease. Condition (i) of Theorem 3 can be interpreted as being that the marginal cost of realizing a given per capita mortality rate resulting from a unit increase in population size at level 2 decreases as x increases. Conditions (ii) and (iii), respectively, are that this marginal cost is constant or increases with x. An intuitive explanation of why S > D under condition (i) even when D is self-sustaining follows from the fact that the marginal average annual harvest cost in the stochastic model resulting from an increase in the escapement level from D to D + 1 is, because of the averaging process, less than the corresponding marginal cost in the deterministic model. This implies that the corresponding marginal increase in E.P.V. is positive in the stochastic case, while zero in the deterministic case. Thus, the optimal stochastic escapement level S is greater than D. Similar interpretations can be given for cases (ii) and (iii) of Theorem 3. We now consider the cost function c(x) = k/xc, which results from the CobbDouglas production function (2.6). COROLLARY 1. If c(x) = k/x (which is the case for the Schafer production junction with constant catchability coeficient) and if the other conditions of Corollary 1, Theorem 1 hold, then S = D if D is self-sustaining and S > D if D is not seljsustaining. COROLLARY 2. If the unit harvest cost is constant (which is the case when the abundance) then S = D if D is self-sustaining and S > D if D is not self-sustaining.
c.p.u.e. is independent of population
COROLLARY 3. If c(x) = k/xc, with 0 < 0 < 1 (which is the case for the Gulland model in which catchability decreases like x@-I), and the other conditions of Corollary 1, Theorem 1 hold then S > D or S = D = x,.
In all of the above examples S has been greater than or equal to D. A somewhat contrived example in which S < D is when the cost function is c(x) = k/xc with 1 < 0 _< 2 and the average stock-recruitment function is the Beverton-Holt form f(x) = Ax/l + Bx. As has already been observed in Section 3 a constantescapement policy is optimal in this case if x, is self-sustaining. Since xc(x) = k/xc-l is convex it follows from Theorem 3 (iii) that S < D if S is self-sustaining. However, as observed in Section 2, these parameter values imply that the catchability coefhcient increases with x like xc-l at least over [x-, ml. In most realistic cases it seems that S would be no less than D. The presence of environmental fluctuation can cause an upward shift in the optimal escapement through two distinct mechanisms. The first, stated in Lemma 4, is that u (and
OPTIMAL
ESCAPEMENT
LEVELS
363
therefore S) will be shifted up from D for certain forms of the cost function. The second mechanism, stated in Theorem 2, is that S will shift up from (Tand therefore from D when u is not self-sustaining. While it appears difficult to establish a precise relationship it seems that the upward shift of S will increase with the degree of environmental fluctuation. This can definitely be established for a shift attributable to the first mechanism when the cost function is of thrl form c(x) = k/xc,
e < 1.
This tentative conclusion agrees with the numerical results of Ludwig and Varah [9] who have considered the optimal control problem for a continuous-time> model with added noise. REFERENCES 1. R. J. H. Beverton and S. J. Holt, On the dynamics of exploited fish populations, Fish. Invest., Ser. 2 (19), London (1957). 2. C. W. Clark, Profit maximization and the extinction of animal species, J. Pal. Econ. 81, 950-961 (1973). 3. C!. W. Clark, “Mathematical Bioeconomics,” Wiley, New York (1976). 4. D. H. Cushing, The dependence of recruitment on parent stock in different groups of fishes, J. Cons. Iti. Ezpbr. Mm. 33(3) 340-362 (1971). 5. J. A. Gullsnd, The stability of fish stocks, J. Cons. Znt. Exph-. Mm. 37(3), 199-204 (1977). 6. M. D. Intrilligator, “Mathematical Optimization and Economic Theory,” Prentice-Hall, E&$ewood Cliffs, N.J. (1971). 7. D. L. Jaquette, A discrete time population control model, Math. B&X%. 231-252 (1972). 8. A. Kaufmann and R. Cruon, “Dynamic Programming,” Academic Press, New York (1967). 9. D. Ludwig and J. M. Varah, “Optimal Harvesting of a Randomly Fluctuating Resource. II. Numerical Methods and Results,” Technical Report 77-12, Inst. Applied Mathematics and Statistics, University of British Columbia, Vancouver (1977). 10. C. R. Rao, “Linear Statistical Inference and its Applications,” Wiley, New York (1965). 11. W. J. Reed, A stochastic model for the economic management of a renewable animal resource, Math. Biosci. 22, 313-337 (1974). 12. W. J. Reed, The steady-state of a stochastic harvesting model, Math. Biosci. 41, 273-307 (1978). 13. W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Canada 11(5), 559-623 (1954). 14. M. Spence, “Blue Whales and Applied Control Theory,” Technical Report No. 108, Inst. for Mathematical Studies in Social Sciences, Stanford University (1973).