Marine Resource Economiis. Volume 8. pp. 89-99 0738-1360/93 $3.00 + .00Primed in the USA. AU rights reserved. Copyright C 1993 Marine Resources Foundation

The Optimal Feeding of Farmed FishTERRY HEAPSDepartment of EconomicsSimon Fraser UniversityBurnaby, B.C.Canada

Abstract Optimal feeding schedtdes and harvesting policies for a fish farmare examined. It is shown that if fish growth is density independent, then theculling offish before the final slaughter date need not be part of the optimalharvest plan. As well, a number of results are derived concerning the effect ofchanges in discount rates, mortality rates, various types of cost and fish priceson the optimal slaughter weight and date. These results are compared to thosein Bj0rndal (1988) where feeding rates were exogenous.

Keywords Aquaculture, optimization, fish feeding.

Introduction

This paper extends v 'ork of Bj0rndal (1988) and Arnason (1992) on the optimalharvesting and feeding of farmed fish. Bj0rndal (1988) analyses extensively theoptimal slaughtering date for farmed fish under the assumption that the rate ofgrowth of average fish weight is an exogenous function of age. In particular, hedetermines the comparative static effects of changes in discount rates, harvestingcosts and feed costs on the optimal slaughter age. He notes, however, a need togeneralize the model so that feeding rates which control the rate of growth ofweight become choices to be made by the farmer.

Arnason (1992) derives necessary conditions that optimal feeding rate patternsmust satisfy. He does not, however, analyze how endogenizing feed rates affectsBj0mdars results. This paper extends Bj0rndars analysis to the model wherefeeding rates are choice variables. In this case final slaughter age and final slaugh-ter weight are independent choices so the impacts on both these variables areexamined. In Bj0rndal, these variables change in the same direction. The resultshere confirm Bj0rndars results with respect to fmal slaughter weight but notalways with respect to final slaughter age. For example, an increase in feedingcosts induces the farmer to feed the fish less and to spread the feeding over alonger period of time. However, the farmer is also induced to grow the fish to asmaller final weight which requires a shorter growing time. The total effect onfinal slaughter age thus depends on actual parameter values.

Arnason (1992) states but does not show that in his model a culling of the fishstock before the final slaughter date is not economically optimal. The other pur-pose of this paper is to provide a demonstration of this result. As noted byArnason, this result is due to the assumptions that mortality and growth aredensity independent.

89

90 Heaps

The ModelThe model described here is that of Amason (1992) with a few modifications toachieve more generality. Therefore, the reader is referred to that paper for a moredetailed discussion of the model. The state of the fish stock in the enclosure attime t after the setting out of smolts is described by n(t) the number of fish and w(t)the average weight of these fish. The options facing the fish farm manager arefeeding, culling and the time at which all fish remaining in the pen will be slaugh-tered. These choices are represented by f(t) - amount of feed per fish, h(t) - thenumber offish culled and T - the final slaughter time.

The dynamics of the growth of the fish stock are then described by the differ-ential equations

h = -mn - h (1)

w = G{w,f,t) (2)

The first equation assumes that m - the instantaneous rate of natural mor-tality of the stock is constant and is independent of h - the rate of culling. Thisequation is the simplest form in which mortality is depicted in the literature.

The second equation should incorporate the ideas that individual weight gainfollows a logistic pattern, is enhanced by feeding but with diminishing returns.Thus the partial derivatives of G have the signs (G/w)^ < 0, Gf > 0 and G - < 0.It will be assumed also that there must be feeding in order for the fish to gainweight. Thus G(w,f) = 0 has a solution for some f > 0 for all weights w. Anexample of several weight growth curves corresponding to different rates of feed-ing is shown in Figure 1.3.'

Equation (2) omits the possibility that the rate of weight gain may also dependon the number offish in the pen. This simplification is necessary in order to do thetypes of analyses that are done below.

The economic side offish farming is modelled in a slightly more general waythan in Arnason (1992). The value of a fish is allowed to vary with the weight ofthe fish and is denoted R(w). R(w) is net of the costs of capturing and slaughteringa fish (either as part of a cull or of the final harvest). The costs of feeding, s perkg. of feed, are assumed to be constant. An allowance is also made for fixed costsk which must be paid in every period that fish are being held.

The fish farm owner should be interested in determining the options which willmaximize the present value of the returns from the harvested fish minus thepresent value of the feeding and fixed costs incurred. This calculation will bebased on the time (t = 0) smolts are introduced into the pen and only one harvestcycle will be considered." The owner's problem can then be represented mathe-matically as

MAX R(w(T))n(T)e-''^ + P(R(w)h - sfn)e-Mt - k[l - e^-' l/i (3)<h).(0 - '"

n(T),W(T).T

' The example is a variant of the logistic equation which is a special case of the generalgrowth model in Schnute (1981). It is G - w(.O4 - .0006w /f) where w is in kg. and f is inkg. of feed per week. The solution to this example is given in the last section of this paper.^ The start of a harvest cycle is determined by the time at which smolts are available andhence is assumed not to influence the timing of the previous harvest.

optimal Feeding 91

(fi

XOUJ

40 60 80TIME (wks.)

120 160

Figure 1. Weight Growth Paths for Constant Feed Rates.

subject to h S: 0, f ^ 0, the state equations (1) and (2) and n(0) = n ,. w(0) = w^given. Here, i is the owner's discount rate and the first term in (3) is the presentvalue of the return from the final slaughter. The terms under the integral signrepresent the returns from culling and the costs of feeding. The final term in (3) isthe present value of fixed costs over the harvest cycle.

This optimization problem will be broken into two parts. First, n(T), w(T) andT will be taken to be fixed and the integral part of (3) will be maximized. Thentransversality conditions will be used to characterize the optimal choices of n(T),w(T) and T.

Conditions for Optimal Culling and Feeding

Necessary conditions that the solution to maximizing the integral part of (3) mustsatisfy are derived in the appendix. They include the two state equations (1) and(2) plus the equations

Gf = Gw - r + (R'(w)Gf - s) —

R'(w)G - rR(w) - sf = 0 if h > 0

(4)

(5)

92 Heaps

where (jf - G/Gf and r = j + m. Equation (4) with h = 0 is equation (5) inAmason (1992). A direct economic interpretation of (4) is difficult to produce, butit is a condition to ensure that the present value of feeding costs, to meet a desiredtarget weight, is kept to a minimum. Equation (5) can be interpreted as follows.The net gain per fish to culling fish at fime t is R(w). If this fish was not culled itsvalue would grow by R' (w)w = R' (w)G net of the cost of feeding this fish; thiscost is growing at the rate sf. It is worthwhile not culling this fish provided that therate of value gain (that is, (R' (w)w - sf)/R(w)) exceeds the opportunity cost ofthe fish capital. That opportunity cost Is i the return to capital invested in themarket plus the rate of loss offish through natural mortality m. It should be notedthat this interpretation can be given because the only gains to culling are the netvalue of the fish harvested plus the reduction in future costs. If weight growth isdensity dependent however, then the rate of weight growth gains to culling mustalso be considered.

Analysis of Culling

It is shown here that there is an optimal policy which does not involve any culling.The intuition for this result is that for an opfimal culling program, the marginal netpresent value of benefits from culling should be the same at any time at whichthere is culling. Shifting culling from one time to another has no effect on fishweight growth and hence no effect on fish value. Moreover the marginal netpresent value of benefits from culling at any time are constant with respect to thescale of the culling. Thus there can be no losses associated with reducing allcuUings to zero and doing all harvesting at the final slaughter age.

To formally get this result, note that from (1) h = -mn - h. This equationallows one to eliminate the h from the maximand in (3) which from now on will bereferred to as J. Then integration by parts show that

Tr'^R{w)ne-"dt = R(w)ne-" - P (R'(w)G - iR(w))ne-"dt

Jo 0 Jo

Using this expression one then obtains

J = R(w(O))n(O) + r i(R'iw)G - rR(w) - sf)ne-''dt - k[l - e-'^/i (6)

In the previous section, (5) shows that if h > 0 the expression under theintegral sign is zero. Thus, an optimal culling which ends at time T* yields thesame J as an impulse culling at time T* which leads to the same n(T*).

Now suppose that there is an optimal culling - feeding policy which involvesan impulse culling of An = n(T*)" - n(T*)" > 0 at time T* < T. Consider aculling - feeding policy which involves the same feeding policy as the optimalpolicy. The culling policy is changed, however, to increase the impulse at time T*

^ The control problem here is linear in h. (5) is the singular path condition for this controlvariable.

optimal Feeding 93

by (1 - e)n(T*)^ and to change culling to eh(t) for times t > T*.** The averagefish weight will not be changed by this modified policy but the numbers offish willbechanged to6n(t)fort >T*. Let JO) be the value of J with this modified policy(where 0 = 1 gives the value of the original policy). Then

J(e) = J(l) + (6 - 1) P (R'(w)G - rR(w) - sf)ne-Mt

J(6) has a maximum at 9 = 1 only if the integral part of this expression is zero.However, if this is the case then by (6) above reducing the final slaughter datefrom T to T* would not reduce J. Thus an impulse culling before the final slaughterdate is not better than the optimal no culling policy.

In sum, it has been shown that an optimal culling program is no culling at all,i.e. h = 0. Equation (4) then simplifies to Arnason's equation

Gf = G^ - r (7)

The optimal feeding curves are described by equations (7) and (2) and from nowon will be referred to as the 3- curves. In the case that G does not depend on age,these equations describe a family of nonintersecting curves in (w, F) space whoseslope is given below.

+ Gfw = GKGw - r)/G (8)

Three of these curves are illustrated in Figure 2 by dashed lines.^

Transversality Conditions

Since it has been shown that no culling is the optimal policy, it follows that n(t) =n^e"""'. The fish farmer's problem (3) simplifies then essentially to the problemconsidered by Arnason (1992). The conditions he obtained for the optimal w(T)and T can be written as:

(T) = (l/Gf)(R'(w)Gf - s)e-^'^ = 0 (9)

a(J/nJ/3T = (-rR(w) - sf + s(G/Gf) - k,e'"^)e-^"' = 0 (10)

where k^ = k/n^ and all variables are evaluated at time T.The nature of the optimal policy can now be illustrated in the case where G

does not depend on t by rewriting the transversality conditions in the form

*(w,f) = R'(w)G - rR(w) -s f = k^e""^ (11)

^ The total impulse becomes n(T*)" - en(T*)'*' so 9 is restricted to lie between 0 andn(T*)-/n(T*) + .^ The example uses i = .001% and m = .0023% per week (i.e., 5.2 and 12.0% per year).

94 Heaps

0.38

"1a.

(kgs

.FE

ED

/ / /

. • ^ • • ' ' . - • ' ' '

Legend

tr .

1WEIGHT (kgs.)

Figure 2. Optimal Feeding Trajectories.

[w.O = R'(w)Gf - s = 0

3.3

(12)

Since f = R'(w)Gn^ < 0 by assumption, the curve 'f = 0 has at most onesolution for f for every value of w. This curve will be called the termination curve9" as the final harvest occurs when one of the optimal feeding curves crosses thiscurve. The optimal S* curve is the one which starts at w^ and takes time T to reacha point ((w(T).f(T)) on the termination curve where T, w(T) and f(T) satisfy (11).In fact many of the points on the termination curve will be optimal ending pointsfor some value of k. The relationship between the optimal feeding curves and thetermination curve is illustrated in Figure 2.^ Differentiation of (12) shows that theslope of the termination curve is given by

R'(w)Gffdf\- = -R"(w)Gf - R'(w)Gfw

This expression can then be rewritten in the form

" The economic parameters used in this example are R(w) - pw - c where p ^ $7.50 perkg.,c = $2ands = $1.15 per kg. of feed. The termination curve does not necessarily havea positive slope as shown in Figure 2.

Optimal Feeding 95

df\— j + Gfw + Gf*w/(R'(w)G) = GKGw - r)/G (13)

Thus (see (8)), the 9' trajectories intersect the termination curve from above when^ ^ > 0, are tangent to it where ^ ^ = 0 (i.e., at the maximum of ^ ) and intersectit from below when ^^ < 0. It will be shown in the next section that '^^ < 0, isa sufficient (second order) condition for the fish farmer's problem to have asolution. Hence all points on the termination curve to the right of the point oftangency represent a maximum for some value of k.

Comparative Dynamics

This section will try to identify the qualitative effects of changes in parameters inthe model on the optimal w(T). T and f(T). The method is to totally differentiatethe first order conditions for a maximum and then to solve for the relevant de-rivatives. To begin, the expressions that will be obtained are simplified if the firstorder conditions (9) and (10) are rewritten in the form

= 0 (14)

:'"' )e- ' = 0 (15)

Letting 0 be a parameter, then total differentiation gives

+ G^fe + G^fl(5f(T)/6e) + GfZ ^'^'

where A is Gfe'"' times the Hessian matrix of the maximization problem and henceis negative definite. In doing this differentiation, it must be remembered that theoptimal (0 and particularly f(T) are functions of w(T), T and 6 as described insection 3. Changes occuring in f(T) due to changes here are denoted by 8 ratherthan by d; the latter differentiation includes endogenous changes in the terminalvalues as well as direct changes due to changes in 6. The term Z in (16) is equalto a(k«e'"'^)/3e. Finally the formula for A is

A = (17)

It is of interest to check that the second order conditions for maximizing J/n^with respect to w(T) and T are actually satisfied. It is easy to see in Figure 2 thatif T increases and w(T) is fixed then a lower S-'trajectory becomes optimal. Thatis less feeding is done so that it takes a longer time to get the desired increase inweight. Thus 8f(T)/8T < 0. On the other hand, if w(T) increases and T is fixed thena higher 3^-trajectory becomes optimal. However, the direction of change of f(T)cannot be determined. Nevertheless, since ^ff = R'(w)Gn- < 0 and ^ ^ 0 by (15)the (2,2) entry of A is negative. By Young's theorem the two off diagonal elementsof A must be equal. Using this fact to eliminate 5f(T)/6w(T), then

96 Heaps

(18)

This is positive if ^ ^ ^ 0 so that all points on the termination curve to the rightof the point of tangency and some to its left represent solutions to the fish farmer'sproblem for some value of k.

Comparative dynamic results are obtained by solving (16) for the two deriva-tives and using |A| > 0 and other conditions to try and sign the results. It followsthat aw(T)/ae has the same sign as

(19)

and dT/dO has the same sign as

( I / G ) ^ f / - Z) (20)

The results listed in Table I can now be obtained.'' The details are given in theappendix. The harvesting cost parameter c is introduced by letting R = R(w) -c and a fish price index p is introduced by letting R = R(w,p) where Rp > 0 andR'p > 0. The results coincode with the results of Bj0mdal (1988) where the feedingrate was specified exogenously except for p and his results dJ/ds < 0. aT/ai < 0and 3T/am < 0. The first of these results does not generalize because here anincrease in s causes the farmer to reduce feeding rates and this reduction tends tolengthen the amount of time required to grow the fish to the desired weight. Theother two results similarly do not generalize because an increase in either i or mwill also reduce feeding rates as a cost cutting measure in response to a fall in thevalue of the farm.

Example

Suppose that R(w) = pw - c and G(w,f) = w(a - bwf ""O where e > CT > 0. Fora fixed f. w would tend to the limit of (aP/b)"^*' which increases with f. Theoptimal feeding trajectories given by (7) and (2) can now be written as:

f = (ea + r)/(I + CT) = a (21)

w = a - bw^^" (22)

The following explicit solution for f and w can then be obtained.

f(t) = Ae"' (23)

-"^' + e ""•'') (24)

where p = (ae - aCT)/(b€) and B is another constant of integration.The termination curve which has the formula R' (w(T))Gf = s becomes here;

"^ ' ' ' = s (25)

' An expression for df(T)/d& can be obtained but it can never be signed.

optimal Feeding 97

The other transversality condition (11) becomes

pw(T) (a - bw(T)*f(T)-' ) - r(pw(T) - c) - sf(T) = M'"'^ (26)

No explicit solution of these equations is possible. However, (23) and (25)imply

'e '* ' " ' " ° ' ' (27)

and using (24) with t = T and the initial condition w^ = pA*7(B + 1) gives:

B = ((w(T)/wJ*e-°'^ - 1)/(1 - (w(T)/w^)^e-^*^) (28)

These two equations can be substituted in (24) (with t = T) and with somerearrangement one can obtain

J^) (29)

Then using (25) again the other transversality condition (26) becomes

pw(T)(a - r) - (1 + a)(s/a)'^'"^'''(pb)"*'-^"'w(T)''-^'>'*'^'^' + re = M'"'^

(30)

These last two equations can be solved numerically for w(T) and T. The middledashed curve in figure 2 shows the optimal solution to the fish farmer's problemfor the parameters previously given, e = 2, o- = 1, k = $.05 per smolt and w^ =.05 kg.. The optimal final slaughter weight and age for this example are w(T) =3.17 kg. andT = 131 weeks. The net present value of the fish farm is J/n , = $1.19per smolt. In the special case k = 0 and either e = 0 or r = 0, (30) can be solvedexplicitly for w(T) and (25) used to determine f(T). The results are

(31)

(32)

The other transversality condition (29) can also be solved explicitly in the subcaser = 0. This is the case where the integration constant B = 0.** The solution for Tis T = ((I + o-)/(a(r))ln(w(T)/w ,). The optimal feeding trajectory also has thefollowing explicit solution in this case.

w(t) = w^e'^'''"^''^'

f(t) = ((1 + a)bWoVa)>'''e'^''*'^''»'

« From (23) w(T) - (s/(frpb))f(T)' - "wfT)"*. Using (23) and (24) with t - T then gives w(T)= (s/((rpbp))A(Be''^ + e"'^). Substituting this in (26) and some manipulation gives(a - r)sABe''^/(b3) - rs(l + a) (c - CT)Ae°'^/(a€ - ra) + rccr - k^CTe'" Thus when r =0, AB = k^a/(s{l + a)).

98 Heaps

This example is admittedly hypothetical but has been of some help in understand-ing the nature of the optimal feeding paths in the general model.

Appendix

Pontryagin's maximum principle provides a set of necessary conditions that thesolution to maximizing the integral part of (3) must satisfy (see Clark (1990) forexample). The problem has two state variables w and n and two control variablesh and f. The Hamiltonian H for the problem can be written as

H = (R(w)h - sfn)e"'* - X(mn + h) + qG(f,w,t)

The conditions for maximizing H with respect to h and F are:

Hh - R(w)e^" - X = 0 i f h > 0 (a.l)

otherwise

Hh = R(w)e-" - \ < 0 and h - 0.

Hf = - s n e " " + qGf = 0 (a.2)

The costate equations are;

X = -Hn = sfe"'* + mX (a.3)

-'* - qG^ (a.4)

Equation (4) is obtained by solving (a.2) for q and then substituting this togetherwith h = - mn - h into (a.4). Equation (5) is obtained in a similar way from (a. I)and (a.3).

The comparative dynamic results in Table 1 are obtained by making the fol-lowing substitutions into (19) and (20).

k: Ir, = 0, f' . = 0> 8f(T)/5k = 0 and Z = e'"' /n(O).

aw(T)/dk a -^n<Sf(T)/ST)e'"' /n(O) < 0

aT/dka -(I/G)(^ff<8f(T)/8T) - GfM' )e"'' /n(O) < 0

For the second result note that (1/G) ( fl<Bf(T)/8T) - Gf*^) equals the negativeof the (1,1) entry of A (by the equality of the off diagonal elements of J) and hencemust be positive.

s: % = -f, %^ = - 1 , 8f(T)/8s = 0 and Z = 0.

aw(T)/ds a - m^ - f^n(8f(T)/8T) < 0

i: % = -R(w), ^fi = 0, 6f(T)/8i > 0 and Z = 0.

aw(T)/ai a m^^(^8f(T)/8i) - ^ff(8f(T)/8T)R(w) < 0

Optimal Feeding 99

Table 1Comparative Dynamic Results"

Optimal Variable

w(T)

m — / —c + + +p either + or - ' ' ?

a. The table gives the sign of the changes inthe optimal variables resulting from increases ine. A ? indicates that no general result can beobtained.

b. This column gives the results of Bj0rndal(1988). In his model. T and w(T) must change inthe same direction.

c. These cases were not actually consideredby Bj0rndal (1988) but hold for his model.

d. One of these results must hold.

m: ^I'^ = -R(w) , %„, = 0, 8F(T)/8m > 0 and Z = Tk^e'"'^.

aw(T)/am a mH'^n<8f(T)/8m) - ^ff<8f(T)/8T)(R(w) + Tk je""" ) < 0

c: where R = R(w) - c; i.e., c represents harvesting costs which are proportionalto the number of fish slaughtered but do not vary with fish size.

y]r^ = r, ^f^ = 0, 8f(T)/5c = 0 and Z = 0.

aw(T)/ac a \l'fl<8f(T)/8T)r > 0

a (1/G) (>I'fi<8f(T)/8T) - Gf>I' )r > 0

p: where R = R(w,p) with Rp > 0 and R'p > 0; i.e., p is an index of fish price.

^ p = R'pG - rRp, 4fp = R'pGf, 8f(T)/5p = 0 and Z = 0.

It can be shown that R'p (aw(T)/ap) - rRp {BT/dp) > 0 so either aw(T)/ap > 0 oraT/ap < 0.

References

Arnason, R. 1992. Optimal Feeding Schedules and Harvesting Time in Aquaculture. Ma-rine Resource Economics 7(1): 15-35.

Bj0rndal, T. 1988. Optimal Harvest Time in Fish Fanning. Marine Resource Economics5(2):139-159.

Clark, C. W. 1990. Mathematical Bioeconomics. New York: Wiley, 2nd. edn..Schnute. J. 1981. A Versatile Growth Model with Statistically Stable Parameters. Cana-

dian Journal of Fisheries and Aquatic Sciences 38:1128-1140.