Commercial fisheries under price uncertainty

JOURNAL

OF ENVIRONMENTAL

ECONOMICS

AND MANAGEMENT

9, 1l-28 (1982)

Commercial Fisheries under Price Uncertainty’ PEDER ANDERSEN Institute of Economics, University ojAarhus, Aarhus, Denmark Received September 18, 1980; revised February 15, 1981 The deterministic models applied in economics of fisheries are extended to comprise price uncertainty and risk aversion among the fishing units. It is proved that in the open-access fishery both the total fishing effort and the number of fishing units are reduced as the variance of the price increases; that the total fishing effort may be smaller in the open-access fishery than in the optimal fishery at a high variance; that only a fixed producer price system can create a first-best optimum, and that a tax on revenue is more efficient than both fishing unit quotas or tax on catch.

1. INTRODUCTION

Traditionally, deterministic models are applied in comparing an open-access fishery to an optimal, rent maximizing, fishery as well as in analyzing advantages and disadvantages from using different methods of regulation.2 However, unpredictable fluctuations in fish prices, in factor prices, and in the level of fish populations are normal phenomena. Therefore, the traditional models are insufficient. It is necessary to develop models which include uncertainty and clarify if and how the results from the traditional analyses have to be modified. In this article only a single aspect of exploitation of fish resources under uncertainty is discussed. By use of the theory of the firm under price uncertainty, e.g., Sandmo [ 131 the deterministic model presented in Clark [6] is extended to include an uncertain price and fishing units with risk aversion. A fishing unit is a vessel with equipment and crew, producing fishing effort. The analysis is partial, distributional aspects are ignored, and administrative costs and political problems of regulating the fishery are assumed not to exist. The analysis is restricted to one fishery with a single homogeneous fish population which is uniformly distributed on the fishing ground. Furthermore, it is assumed that the fishing units are capable of entering alternative fisheries immediately. The article is organized as follows. In Section 2 the behavior of the fishing units under price uncertainty is presented. The open-access fishery and the optimal fishery ‘This article is a revised version of a paper presented at a seminar held at University of Tromso, Norway, April, 1980. I am indebted to the participants in the seminar, to friends and colleagues, and to two anonymous reviewers for their valuable comments on earlier versions of the paper. Particularly, I want to thank Jorgen Sandergaard and Kirsten Stentoft, University of Aarhus, and Jon Sutinen and Jim Opaluch, University of Rhode Island. The final version was finished during my stay in the Department of Resource Economics, University of Rhode Island. R.I.AES. No. 1998. I gratefully acknowledge the support by the Department of Resource Economics, and the financial support by the Reinholdt W. Jorck & Hustru Foundation, the Andelsbank Jubilee Foundation, the Carlsberg Foundation, and the Knud I-Iojgaard Foundation, all Denmark. ‘See, for instance, Warming, [ 181, Gordon [9], Turvey [ 171, Christy [4], Clark [5, 61, Anderson [3], Hamresson [I I], Sinclair [16], Andersen [I, 21, Crutchfield [8], Scott [IS]. 11 0095~0696/82/010000-18$02.00/O Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form reserved

12

PEDER ANDERSEN

under price uncertainty are presented and compared in Sections 3 and 4. In Section 5 the consequences of using tax on catch, tax on revenue, nontransferable quotas, and transferable quotas are compared, and in the following section some consequences of assuming heterogeneous fishing units are discussed. Finally, there are some concluding remarks.

2. THE BEHAVIOR

OF ,THE FISHING

UNITS

The fishing units are assumed to have perfect knowledge of the population and of the cost function for fishing effort. The catch rate hi of fishing unit i at time t will be hi = q . x - Ei,

level

(1)

‘where q is the catchability coefficient which is assumed to be constant and known. x is the biomass of the population, i.e., the total weight of the population, and Ei denotes the standardized fishing effort produced by unit i.3 q . Ei denotes the fishing mortality caused on the population by unit i. The fishing mortality is equivalent to the depreciation rate of the population caused by fishing. In this model only the stock externalities are considered. The population x is influenced by the total fishing mortality produced by all fishing units, but it is assumed that the fishing mortality produced by unit i is infinitely small compared to the total fishing mortality. Consequently, the fishing unit can regard the population level as exogeneously given. It is assumed that the fishing unit considers the price of fish p stochastic with a subjective density function A(p) and an expected price [[p] = F. 5 is the expectations operator. f;.(p) is considered independent of the catch rate h, and also of the total catch rate from all fishing units in this and in the alternative fisheries. The assumption of the existence of alternative fisheries, into which the fishing unit can enter immediately, implies a cost function, ci( Ei), comprising only variable costs and the opportunity cost corresponding to the best alternative fishery. Furthermore, it is assumed that the level of fishing effort must be determined ex ante, i.e., before the price of fish is known, and only on the basis of the knowledge of the price summarized in the density function. The fishing unit is assumed to maximize the expected utility of the profit, II,, from fishing. Utility is represented by a concave, continuous, and twice differentiable function of profit, Q( IIi)4, where

y(rJ)

> 0,

U”(l-Ii) < 0,

(2)

so that the fishing unit has risk aversion. ‘q is constant when the fishing effort is measured in standardized units. The alternative is to make q dependent on -E and x. The fishing effort produced by a fishing unit is the result of the use of a number of ‘production factors as, for instance, the size of the fishing vessel, engine power, equipment of gears and k&&xl equipment, crew and its ability, and the number of fishing hours. 4The transitivity axiom is assumed to be valid, although the crew of a fishing unit consists of more than one fisherman.

13

FISHERIES UNDER PRICE UNCERTAINTY

The expected utility at a given population

level is

=@(pq.x.E,-ci(Ei))].

(3)

To obtain necessary and sufficient conditions for a maximum, (3) is differentiated with respect to fishing effort. 5[ Vtni)(P IIVtni)(P

* 4

* 4 ’

. x - c;(E,))]

x - c;(E,))*

= 0,

- lJ:(I&)cj’(E,)]

(4 < 0.

(5)

By use of the results from Sandmo [ 13, p. 671 it followss (6) and, since L$‘(II,) > 0 c;(E,)
(7)

Equation (7) shows that the utility m aximizmg level of fishing effort is characterized by expected value of marginal catch, p - q . x, being larger than the marginal cost of fishing effort. At a deterministic price the fishing effort will be chosen according to cj(Ei) =p.q.x Ei = 0

ifp-q-x2mi,

(84

ifp - q. x < mi,

(8.2)

where mi is the minimum average cost of effort. Thus, at a given population level the fishing effort will be smaller at a stochastic price than at a deterministic price equal to the expected price p, provided p * q . x 2 mi. By introduction of a risk element, yi, the determination of the effort under price uncertainty can be formalized in a simple way. The risk element yi is defined as the expected marginal profit, which just compensates for the disadvantage of taking over the risk at a marginal extension of the fishing effort. yi is the difference between the expected marginal revenue, F - q +x, and the marginal costs, cf( E,), at the optimal fishing effort, and covers the marginal risk cost of the fishing unit. If yi is inserted into (7) we get cj(Ei)

+yi=F.q.x.

‘From (4) follows that [[U;(IIi) .p . 9. x] = I[&‘(&) . ci(E,)] or [[r/i(II,)(p = [[L$‘(Hi)(cj(Ei) -j. 9. x)]. Since lTi = [[l-f,] + (p -jF) . 9. x . E, we get p >p and then Q’(IIi)(p 9 . x -j. q . x) < U;(t[TI,])(p . q . x -j. q. x). holds for all p because ( p * q . x - j . q . x) -C 0 if p < jJ. Therefore, it turns out - j 9 . x)] < U;([[II,])l[p . q . x - ji . q . x] = 0. From this inequality and equation in this footnote we obtain Eq. (6).

(9) 9. x -j.

9. x)]

r/,‘(II,) < c/l’(C[II,]) if But the last inequality that I[ Q’( I&)( p q . x by use of the second

PEDER ANDERSEN

14

The risk element depends on the variance of the price, up’, and on the degree of risk aversion. Since an increase in variance and in risk aversion will both increase yi only changes in the variance are analyzed in the following. The variation of yi with the profit Iii depends on the type of risk aversion. However, this variation is unimportant for the qualitative results that follow because yi is nonnegative. Therefore, analytically y, will be considered a constant marginal cost in Ei and progressively rising with increasing variance. y;( u;) > 0, y;‘( u;) > 0, and yi(0) = 0. The optimal fishing effort under price uncertainty is therefore determined by c;(E,) + yi(u;)

=p * q * x Ei = 0

ifp.q.x?mi+yi,

(10.1)

ifjj*q*x
(10.2)

Let t!$denote the inverse of the extended marginal cost function, i.e., including the risk element. The fishing effort is then determined directly by q - x > mi + yi,

(11.1)

Ei = Eim

if jj * q * x = mi + yi,

(11.2)

Ei = 0

ifp.

(11.3)

Ei = l?,( p - q . x) > Eim

ifp.

q * x < mi + yi.

Ei” is the effort level corresponding to minimum average cost. From (11) it follows that if the fishing effort is positive when the variance is zero, the fishing effort at a given population level will be smaller when the price is stochastic. If the fishing effort is positive and the price is stochastic, then the expected total profit as well as the expected marginal profit will be positive. The conditions of exit and entry to the fishery follow from (11) fishing unit i leaves the fishery, if j . q - x < m, + yi

(12.1)

fishing unit i enters the fishery, if jJ . q . x > m, + yi

(12.2)

The expected price must be higher than a given deterministic price to make the fishing unit participate in the fishery. The fishing unit will leave the fishery if the expected profit is smaller than yi - Eim and will enter the fishery if the expected profit is higher than yi . Eim. For the fishing unit characterized by (11) an increase in the expected price j or an increase in the population biomass x will increase the fishing effort, provided p . q . x L mi + yi. A sufficient condition is, cf. Sandmo [13, p. 691, that the fishing unit exhibits decreasing absolute risk aversion, RJII), with increasing profit, i.e., R’#I) < 0, where Jw-o

UyII) = - U(H)

‘0.

An assumption of decreasing absolute risk aversion implies that the risk element y, is nonincreasing in profit.

15

FISHERIES UNDER PRICE UNCERTAINTY

Intuitively, the fishing effort is expected to be reduced at an increase in the variance of price, ceteris paribus. A sufficient condition is, cf. Ishii [ 12, pp. 768-7691, that R’JII) 5 0, i.e., a decreasing or constant, absolute risk aversion. Qualitatively, the effects of an increase in variance are equivalent to the effects of an increase in risk aversion, provided that the price of fish is constant in the alternative fisheries. Obviously, an increase in the opportunity cost will reduce the profit and increase RA(II). Thus, a fishing unit will reduce the effort, provided it has a decreasing absolute risk aversion which is both a necessary and sufficient condition, cf. Sandmo [13, p. 781. 3. THE OPEN-ACCESS

FISHERY

A population of fish is a renewable resource, and in a fishery with many fishing units stock externalities will occur. A production function for a fishery must therefore include stock externalities and the population dynamic elements. The growth rate of the population, including fishing, is

(14)

$ = F(x) - ; hi, i=l

where F(x) is the net growth rate, excluding fishing, hi is the catch rate of fishing unit i, and N is the number of fishing units. For simplicity we apply the Schaefer mode16, in which F(x) is a logistic function F(x)

= r * X(1 - x/X).

r is a population specific constant indicating the proportional for x -+ 0 and X is the natural equilibrium population level. F(x) in (15) has the following properties:

(15)

growth rate, F(x)/x,

F(O) = F(Z) = 0,

(16.1)

F(x)

> 0

for 0 < x < X,

(16.2)

F’(x)

s 0

for x f xMsy, where 0 < xMsy c X,

(16.3)

F”(x)

< 0.

(16.4)

xMsy is the population level at which the growth rate of the biomass, excluding fishing, is at maximum, and at which the maximum sustainable yield (MSY) can be obtained. The population is biologically overfished if x < xMsy. The biological limits of the fishery are described by (15) while the relationship between the change in the population and the activity from N fishing units is given by (14). The behavior of fishing unit i is described by (11) and (12). In the following all fishing units are assumed to be identical with decreasing absolute risk aversion. 6The Schaefer model has found widespread use in economic analyses of fisheries. For details of the model, see, e.g., Schaefer [ 141, Clark [5], Hannesson [I I], or Andersen [I, 21.

PEDER ANDERSEN

16

l x

FIG. 1. The logistic growth function.

The total supply of fishing effort, E,, from N fishing units is given by, cf. (11) ET=6’&.q-x(t))

= ; t$(p.q.x(t)).

(17)

i=l

8, is an increasing function of the biomass x, as Mi/gx > 0. If p - q - x > m, + yi, the fishing units obtain an expected net profit, and new units will enter the fishery. The total fishing mortality q - 0, will then increase, the population will be reduced, and the effort of the individual fishing units will be reduced to Eim. The properties of the growth function and the behavior of the fishing units ensure that the fishery will converge to an equilibrium characterized by’ F(2)

= Q(p

* q * 3) * q * 2,

08)

where 3 denotes the equilibrium population in the open-access fishery, 6 denotes the equilibrium number of fishing units, and F( 3) is equal to the equilibrium catch rate. For each of the fi fishing units the following holds p - q. ,t = mi + yi = mj + yj, Ei =&“‘=Ei”,

i,j=l

,..a, fi,

(19.1) (19.2)

and thus ji - q . .f - Eim - # = (m, + yi) - Eim - 6.

(20)

In the open-access fishery with identical fishing units the expected revenue is equal to the cost inclusive risk cost. This holds both in the aggregate and for each of the # units, cf. (19). The total expected profits correspond to the total risk cost, and the expected net profit calculated as the expected profit less the risk cost will therefore be zero. From (19) it follows that the fishing effort is produced in the most efficient way. With (18) and (19) simultaneously fulfilled the open-access fishery is in bioeconomic equilibrium. ‘The model is assumed to be stable even though the units are identical and the presumed entry-exit mechanism does not predict which units will be in. the given fishery. However, the presumed entry-exit system together with the Schaefer model ensure a unique equilibrium, see, e.g., Anderson [3, p. 1431, or Andersen [ 1, p. 1081. See Section 6 for the case of nonidentical fishing units.

FISHERIES

UNDER

PRICE

UNCERTAINTY

17

A decrease in costs will increase the catch rate, as the fishing effort is increased, and F’(x) c 0, provided x > xMsv. If x < xwsy a decrease in costs will decrease the catch rate in the long run as the fishing effort is increased, and F’(x) > 0. As the total effort increases, the number of fishing units will increase too. A reduction of the risk element yi and/or an increase in the expected price have the same effects as a decrease in costs. With a stochastic price the individual fishing units obtain an expected profit of yi - Eim and an expected marginal revenue of jj . q - i. With a deterministic price, j, there is no profit in open-access equilibrium, and the marginal revenue is j - q - 5.2 is the equilibrium population in the deterministic open-access fishery. Hence it follows that

2 >f.

(21)

The equilibrium population in an open-access fishery with a deterministic price is smaller than the equilibrium population in an open-access fishery with a stochastic price. Therefore, the catch rate of the individual fishing units is higher when the price is stochastic, cf. (19.1). As 2 > 2, it turns out that $ Ei” < .$ ET,

(22)

i=l

where ti is the equilibrium deterministic price. Consequently,

number of fishing units in the open-access fishery with a

A < 15.

(23)

The number of fishing units and the total fishing effort are smaller in an open-access fishery with a stochastic price than in an open-access fishery with a determiuistic price. If (21) and (22) are applied the following rules of the total catch rate in the open-access fishery at a deterministic price and at a stochastic price can be established: iq-*l-Eim>iq-.t-Ei”’ i=l

i=l

3

A

~q+Ei”‘<~q+Eim i=l

i=l

(24.1)

if xMuIsy>J?>.f,

(24.2)

if i > xMsY > 2.

(24.3)

i=l

A

~q+Ei”’

if3>>>xx,,,

A

Z

xq+Eim i=l

If the population is not biologically overfished, the catch rate will be smaller if the price is stochastic, cf. (24.1), whereas the catch rate will be higher at a stochastic mice if the population is biologically overfished, cf. (24.2). If the population is

18

PEDER ANDERSEN

biologically overfished at a deterministic price, but not biologically overfished at a stochastic price, it cannot be determined a priori whether the catch rate is higher at a deterministic or at a stochastic price, cf. (24.3). We now apply the results from Section 2 on the effects of an increase in variance which increases the risk element, yi, and one of the results in this section, namely that the effort of the fishing units staying in the fishery will be Eim. It follows immediately from these results that an increase in variance will reduce the number of fishing units, increase the equilibrium population, and increase the expected profit per fishing unit by an amount corresponding to the increased cost of risk.

J!?l!< 0, dYi

d”>o, dyi

d(t[nil) , o dYi *

(25)

The catch rate increases with an increase in variance if the population is biologically overfished, and decreases if the population is not biologically overfished, cf. (24.1-24.3). Finally, an increase in the expected price will, ceteris paribus, increase the total fishing effort but will not change the effort of the individual fishing units in bioeconomic equilibrium as the expected marginal revenue of the individual fishing units will be unchanged on account of the decrease in the population. An increase in the expected price will therefore increase the number of fishing units.

4. OPTIMAL

FISHERY:

FIRST-BEST

In deterministic models an optimal fishery is defined by rent maximization. Similarly, in this context an optimal fishery under price uncertainty is defined by maximization of the expected rent above risk costs of the individuals. This definition of optimality implies that society is assumed to be risk neutral in the sense that society attaches importance to risk only through the costs of risk borne by the individuals. Furthermore, it is assumed that the price of fish is an appropriate measure of the social benefit of fish, that there is no difference between private and social cost, and that the social rate of discount is zero.8 If-without costs or distortions-it is possible to establish a situation in which the fishing units do not carry any costs of risk, the optimal fishery under price uncertainty coincides with the optimal fishery at a deterministic price. Such an optimum is here designated a first-best optimum. It is assumed that the regulating authority is able to vary the variance of the producer price without costs and has an expected price equal to the expected price of the fishing units. The maximization problem is (26)

‘The analysis is restricted to the static case but can easily be extended to the dynamic case. The cmsequence of an increase in the social rate of discount will be as in the deterministic case. As shown in, e.g., Clark [5] an increase in the social rate of discount increases the optimal level of fishing effort.

19

FISHERIES UNDER PRICE UNCERTAINTY

subject to F(x)

= q * x ; Ei.

(27)

i=l

The first-order conditions are i=

(p-x)*qYx=c;(Ei)+yi,

(p-h)

.q.x’

1,

(28.1)

CN(EN)-yN,

(28.2)

N

x=

j,F*q*

Ei

(28.3) 5 q . E, - F’(x)

’

i=l y;(q)

=

(28.4)

0.

X is the shadow price of the biomass. As the fishing units are identical it follows from (28.1) and (28.2) that all units operate at minimum average cost, m,. As9 yi(u,‘) = 0 if and only if up’ = 0 it is obtained from (28) that a first-best optimum can be realized only by use of a fixed producer price system where the fishing units sell the catch at a price pa, which is known with certainty. pa is fixed by the authority according to

(29)

i= l,...,N*.

p&i-A=-$&

x* is the first-best optimal biomass and N* is the first-best optimal number of fishing units. As the variance on pa, up,, * is zero the risk element, yi, is zero, too. A first-best optimum is realized by use of a fixed producer price system under price uncertainty because society which is here assumed to be risk neutral has taken over all the risk. From the first-order conditions it follows that a first-best optimal fishery is characterized by, cf. also Clark [6, p. 11151 p * q * x* * F’(x*) F’(x*) - I;(x*)

= mi’

whereF(x*)

F(x*)

= 7,

(30)

and F(x*)

= q. x* 5 E/.

(31)

i=l

Equations (29)-(31) 9Recall the assumptions

uniquely determine the first-best optimal number of fishing y: > 0,~:’

> 0, and vAO) = 0.

PEDER ANDERSEN

20

units N*, the first-best optimal population x*, and the optimal effort of the individual fishing units ET = E,!“, (i, j) = 1, . . . , N*. In the Schaefer model F(x) > F’(x) for all x. From (30) it appears that F/(x*) < 0 if m, > 0 which means that x* > xMvIsy.As F’(x*)/(F’(x*) - F(x*)) < 1 we get that the shadow price of fish, X, as expected is positive. Now it is possible to compare the first-best optimal fishery to the open-access fishery. The condition of equilibrium for the fishing units in an optimal fishery and an open-access fishery with stochastic price respectively is, cf. (29) and (19) jT.q-x*=mi+X*q-x*, fl.q.2=mi+yi,

i=

l,...,

N*,

(32.1)

i=

l,...,

A.

(32.2)

From these conditions it follows that

zi,t[nil N*

ifX.q.x*

5 tini1 > 7

$yi.

(33)

i=l fi

’ i

It turns out that if the risk element is smaller (higher) than the shadow value of the marginal catch, the population is smaller (higher), the number of fishing units is higher (smaller), and the expected profit per fishing unit is smaller (higher) in the open-access fishery than in the first-best optimal fishery. Thus it is proved that the fishing effort in an open-access fishery under price uncertainty may be smaller than in the optimal fishery, and if so the population will be economically under-exploited. lo The condition is a large variance of the price and/or a high degree of risk aversion. The basic conclusion in the theory of economics of fishery-that the fishing effort will always be higher, and the population will always be smaller in an open-access fishery than in an optimal fishery-can therefore not be generalized to the case of uncertainty.” 5. METHODS

OF REGULATION:

SECOND-BEST

The methods of regulation analyzed here include tax on catch and revenue and catch quotas for individual fishing units. With no uncertainty, these methods are theoretically equivalent as to efficiency and are capable of creating an optimal fishery, see Clark [6]. Taxes and quotas on fishing effort are not analyzed here, as these are analytically identical to taxes and quotas on catch since there is a unique relationship between fishing effort and quantity of catch at a given level of the population, cf. (1). “This happens easier at a positive social rate of discount, see Footnote 8. “It should be mentioned that the fishing effort can also be smaller than the optimal fishing effort if the population has the Allee effect, and the fishery is caught in the “low catch trap,” cf. Hannesson [IO, pp. 24-261, but in this case the population is always smaller than in the optimal fishery.

21

FISHERIES UNDER PRICE UNCERTAINTY

Other methods. of regulation are not discussed since it is well knowni that these cannot ensure optimal exploitation even under complete certainty. For instance, the use of total catch quotas will result in excess-capacity and inappropriate utilization of the fishing units and the processing industry, and licenses per vessel and/or per fisherman will cause factor distortions. Restrictions on the type of fishing gear and periods of fishing also cannot solve the problem of stock externalities, but they may cause an increase in the catches. Equilibrium conditions are found for each of the following methods of regulation (1) a tax on catch (2) a tax on revenue, i.e., an ad valorem tax (3) nontransferable fishing unit quotas, i.e., a permission to catch a given quantity of fish (4) transferable fishing unit quotas, i.e., trading of quotas between fishing units is allowed. These four methods of regulation are compared with the fixed producer price system mentioned in Section 4. We notice that a fixed producer price system is equivalent to an optimal variable tax on catch, r*, cf. (29), where P=p-pa

and

5[7*]

ET* ‘+p.

Similarly, there exists an equivalent optimal variable tax on revenue, p*, such that

Equilibrium ing:

conditions at these first-best optimal tax systems will be the follow(p-5*)*q.x*=mi, p(l

- p*) . q. x* = mi,

i= l,...,

N*,

(36)

i = l,...,

N*.

(37)

In order to limit the analysis, the adjustment problem is ignored although this of course is an important aspect of the implementation of any form of regulation. Tax on Catch Suppose for the moment that the regulating authority is unable to vary the tax to follow the variation in price but only is capable of using a constant tax on catch, calculated on the basis of the expected price, p. Therefore, the costs of risk, yi, cannot be eliminated. From (28) it can be deduced that the conditions of equilibrium in a fishery regulated by an optimal constant tax on catch, r’, are (jJ-7’)-q.xS=mi+yi, p - q * x; * F’(x;) Ft(x:) _ Ftx:)

i = l,..., = mi + yig

Ns,

(38) (39)

‘*See, for instance, Crutchfield (7, 81, Christy [4], Anderson [3], Sinclair [16], Andersen [I], Scott [ 151, and Clark [6].

22

PEDER ANDERSEN

where x: is the equilibrium population, and N,” is the equilibrium number of fishing units.‘3 By comparing the conditions of the first-best optimum, (29) and (30), to the conditions of the second-best optimum obtained by a constant tax on catch, it follows that x* -= x:,

ww

N* > N,“.

W2)

The population will be larger and the total fishing effort as well as the number of fishing units will be smaller in the fishery regulated by a constant tax on catch. The N,” fishing units produce at minimum average cost, mi, obtaining an expected profit of yi . Eim, which exactly corresponds to the risk costs. Finally, total catches and tax revenue are smaller than at first-best optimum. An increase in variance increases the risk element, y;(u,‘) > 0, as well as (xf - x*) and (N* - N,“), i.e., the difference between the first-best optimum and the secondbest optimum obtained by a constant tax on catch will be increased. Tax on Revenue Now suppose instead that the authority can regulate only by a constant tax on revenue which differs from a constant tax on catch as it effects the risk costs. A constant tax on revenue, p, 0 I ~1I 1 reduces the expected producer price to p( 1 - CL)and the variance to up’<1 - P)~. As the risk element yi is dependent on the variance, we get y, > y/, where yj‘ is the risk element of fishing unit i at a constant tax on revenue, CL. The second-best optimal conditions of equilibrium in a fishery regulated by an optimal constant tax on revenue, @, are cf. (28). p(1 - @) . q . xi = m, + yr, p. q * x; - F’( x;) Fqx;) _ Qxe)

_ - “i + yi”,

i=

l,...,

NSP’

(41) (42)

where xi is the equilibrium population, and N; is the equilibrium number of fishing units. By comparing the conditions of the first-best optimum, (29) and (30), of the second-best optimum obtained by a constant tax on catch, (38) and (39), and of the second-best optimum obtained by a constant tax on revenue, (41) and (42), it follows that (43.1) x* -c x; -c x;, N*>N;

>Nf.

(43.2)

From (43) it appears that the equilibrium population at an optimal constant tax on revenue is larger than at the first-best optimum, but smaller than at an optimal 130f course, there exists a constant 7 < 5* so that x = x*. But it will not be a first-best nor a second-best optimum since the rent (tax revenue) will be less than otherwise.

23

FISHERIES UNDER PRICE UNCERTAINTY

constant tax on catch. Thus, the total fishing effort and the total catches are larger at an optimal constant tax on revenue than at an optimal constant tax on catch, but smaller than at first-best optimum. Since all fishing units operate at minimum average costs in all three cases, the number of fishing units at an optimal tax on revenue is smaller than at first-best optimum, but larger than at an optimal constant tax on catch. The expected profit of the individual fishing units is smaller at an optimal tax on revenue than at an optimal tax on catch, as the social loss in the form of individual risk costs is reduced by use of a tax on revenue instead of a tax on catch. The higher the optimal constant tax on revenue, the smaller the difference between the optimum in a system of tax on revenue and the first-best optimum will be, because the variance on p( 1 - CL)is reduced. $‘ --, 0 if u,‘( 1 - p)2 + 0. Accordingly, we may conclude that a system of tax on revenue is more efficient than a system of tax on catch because the risk costs are reduced when society takes over part of the risk by taxation of revenue.

Nontransferable Quotas If the same cost function of the fishing units is assumed as above, i.e., that minimum average costs including risk costs are m, + yi, the optimal total quota Q’ will be Q’ = F( xb) = F(x:),

where xi = xs.

w

The optimal total quota Q’ is equal to the growth rate of the population, excluding fishing, at the optimal population x6 which is identical to the optimal population at an optimal tax on catch, x’, as the marginal risk costs of the fishing units are identical under these two forms of regulation. The optimal allocation of Q’ is obtained by allocating to Ni fishing units a quota of Qi” = 4. x; - Ei”,

i=

l,...,

N”Q’

(49

where N; is determined by % Q' = 2 Q;.

(46)

i=l

From (44), (45), and (46) it follows that the population, the number of fishing units, the total catch, and the fishing effort of the individual fishing units are identical at optimal regulation by nontransferable quotas and by an optimal constant tax on catch. Clearly, this equivalence depends critically on the assumption of identical fishing units. As to efficiency, the use of nontransferable quotas therefore is identical to the use of an optimal tax on catch provided an appropriate allocation to N; fishing

PEDER ANDERSEN

24

units.‘4Y Is This result indicates that when yi > 0, first-best optimum cannot be obtained by use of nontransferable fishing unit quotas and is less efficient than a system of tax on revenue. Transferable Quotas

Assume now that there exists a perfect competitive market for quota units. A quota unit is a permission to catch one unit of fish. The level of the population, the price of quota units, and the risk element will be determinants in the demand function Di of fishing unit i for quota units, given a constant expected price of fish.

Di = Di(X, 0, Yi), where u is the price of quota units. We assume i3Di z ’ 0,

ao,
’

!A&<(), aYi

where i3Di/ayi < 0 indicates a reduction of the demand for quota units by an increase of the risk element, for instance, caused by an increase in the variance of the pliCe. The aggregate demand for quota units becomes

D(x, 0, Y) = i Di(X, 0, Yi)*

(49)

i=l

In an optimal quota system the regulating authority supplies a total quota of Qs which is equal to the growth rate of the biomass, excluding fishing, at the optimal population x6. RecaIl that this is determined in considering the risk element yi and the minimum average costs mi. At the population x6 and the risk element yi the price of quota units is determined by

D(Ulx~,Yi)= Q’* The conditions of equilibrium

(50)

for fishing unit i will be

Di = Qi =q-x6.

Ei,

i = l,...,

N,

(51)

and (~-U).q.X~=Cj(Ei)+yi,

i=

l,...,

N.

(52)

14As to income distribution, the two methods will be identical if the expected net profit is paid as a license fee per fishing unit. ‘5Possible long-run investment problems and the possibility of decreasing efficiency of the fishing units are ignored.

25

FISHERIES UNDER PRICE UNCERTAINTY Expected rt?"ellUe

total

Total

cost

Total

/’

FIG.

total

I

t1 First-best

Cost

Expected revenue

! I

Second-best; tax on catch

incl.

I

open-access; uncertainty

open-access; certainty

Total effort

2. Open-access, first-best, and second-best fisheries.

If, for instance, (p-u)

*q*x;>m,

+yi,

i = l,...,

N,

(53)

there is an expected net profit for each of the N fishing units. Therefore the price of quota units will increase until (p-d)~q~x~=m,+y,,

i=

l,...,

Ni,

(54

where 0’ is the equilibrium price of quota units, and N; is the number of fishing units each producing Eim units of fishing effort. The optimal population in a system of tax on catch is identical to the optimal population x6, and thus the total quota, Q’, is equal to the total catch in a system of tax on catch. A comparison between (38) and (54) leads to the conclusion that the price of quota units, oS,is equal to the optimal tax on catch, rS, and that the number of fishing units will be equal too. Thus, the two methods of regulation are equivalent under both a stochastic and a deterministic price. Above it was concluded that a system of tax on revenue is more efficient than a system of tax on catch. As a quota system in respect to economic efficiency is equivalent to a system of tax on catch, it can be concluded that whether the quota units are transferable or not, a quota system is less efficient than a system of tax on revenue under price uncertainty. 6. HETEROGENEOUS

FISHING

UNITS

In this section some of the consequences of removing the assumption of identical fishing units are briefly discussed. The fishing units can be nonhomogeneous both in terms of the subjective price distribution/the degree of risk aversion and in efficiency. The former is indicated by risk elements varying across units while the latter is indicated by specific individual cost functions. This gives three different cases to consider.

26

PEDER ANDERSEN

Varying Risk Elements-Identical

Efficiency

We assume that the fishing units can be uniquely ranked by the magnitude of the risk element, yi, i.e., y, Iy*l...

‘ylv.

(55)

The influence of these differences in risk elements on the fishing effort of the units follows immediately from the analysis in Section 2. Ei =- Ej

foryi
(56.1)

E; > E;”

and

p. q. x > mi + yi,

E,=E,?

and

p. q. x = m, + y,.

i=

l,...,

(r-

l),

(56.2) (56.3)

With efficiency being the same for all units, effort is decreasing with increasing degree of risk aversion and/or increasing subjective variance. Only the marginal fishing unit r operates at the minimum average costs, mi, while all inframarginal fishing units accordingly obtain positive expected net profits. It is easily verified that nonhomogeneity in terms of risk elements leaves our previous conclusions on regulation unaffected. Thus, only a system of fixed producer price can ensure a first-best optimum, and regulation by use of tax on revenue will also in the case of nonhomogeneity be more efficient than regulation by quotas or tax on catch. Identical Risk Elements-

Varying Efficiency

It is assumed that differences in efficiency are appropriately measured by the minimum average costs. Again, we assume that a unique ranking exists m, Im,

1...

Sm,.

(57)

By application of the results in Section 2 the implication units is easily derived. Ei > Ej

form, Cmjifp*q.xlmi

E, > Eim

and

jF.q.x>rn,

E,=E,”

and

jY.q.x=m,+y,.

for effort of the fishing

+yi, +yi,

(58.1) i= l,...,

(e - l),

(58.2) (58.3)

These results are obvious. Thus, as long as all y’s are identical, varying efficiency among fishing units does not affect the relative efficiency of the different methods of regulation found in Section 5. Varying Risk Elements-Varying

Efficiency

The conditions of equilibrium for the fishing units correspond to (56) and (58). With different y’s and m’s it is impossible a priori to rank the fishing units according to the level of fishing effort.

FISHERIES UNDER PRICE UNCERTAINTY

27

If mi + yi < mj + yj and m, > mj, fishing unit i, being the less efficient, produces the larger effort. Thus, if efficiency and the degree of risk aversion are positively correlated, i.e., that high efficiency and high risk aversion occur together, the fishery does not in general consist of the most efficient units. Again, a fixed producer price system may create a first-best optimum, and as yi >y;,i= l,..., N a tax on revenue will reduce the risk costs as well as the cost of production, c,(E,), of each unit compared to a fishery regulated by quotas or tax on catch. Nontransferable quotas allocated as to efficiency may reduce the cost of production but in that case the increase in risk costs will exceed the reduction in cost of production. 7. CONCLUDING

REMARKS

The main purpose of this article has been to extend the traditional economic models of fishery to comprise uncertainty, here limited to an analysis of price uncertainty and risk aversion among the fishing units. The analysis modifies the traditional results in several respects. Firstly, in the open-access fishery both the total fishing effort and the number of fishing units are reduced as the variance of the price increases. Secondly, it is shown that the total fishing effort may be smaller in the open-access fishery than in the optimal fishery at a high variance and/or at a high degree of risk aversion. Thirdly, it is proved that a fixed producer price system is the only method of regulation creating a first-best optimum under price uncertainty. Furthermore, a tax on revenue is proved to be a more efficient method as to economic efficiency than both fishing unit quotas or tax on catch. In order to obtain a more realistic description of a fishery under uncertainty the analysis should be extended in several ways. Factor price uncertainty and especially population uncertainty seem quite as important as price uncertainty. Also, it is crucial for practical application of the analysis to take into consideration that the information and the expectations of the authority and of the fishing units may differ. REFERENCES I. P. Andersen, “Fiskeriokonomi” (Economics of Fisheries), Sydjysk Universitetsforlag, Esbjerg (1979). 2. P. Andersen, Nogle grundtrzk i fiskeriokonomi (Some fundamentals in economics of fisheries), Nationabkon. Tidrskr. 119, I-20 (1981). 3. L. G. Anderson, “The Economics of Fisheries Management,” The Johns Hopkins Univ. Press, Baltimore and London ( 1977). 4. F. T. Christy, Jr., “Alternative Arrangements for Marine Fisheries: An Overview,” Resources for the Future, Washington, DC. (1973). 5. C. W. Clark, “Mathematical Bioeconomics: The Optimal Management of Renewable Resources,” Wiley, New York ( 1976). 6. C. W. Clark, Towards a predictive model for the economic regulation of commercial fisheries, Canad. J. Fish: Aquat. Sci. 37, 1111-1129 (1980). 7. J. A. Crutchfield, Economic aspects of international fishing conventions, in “Economics of Fisheries Management-A Symposium” (A. D. Scott, Ed.), pp. 63-78, University of British Columbia, Institute of Animal Resource Ecology, Vancouver (1970). 8. J. A. Crutchfield, Economic and social implications of the main policy alternatives for controlling fishing effort, J. Fish. Res. Board Canada 36, 742-752 (1979). 9. H. S. Gordon, Economic theory of a common-property resource: The fishery, J. Pol. Econ. 62, 124-142

(1954).

28

PEDER ANDERSEN

10. R. Hatmesson, “Economics of Fisheries: Some Problems of Efficiency,” Lund Economic Studies, Lund (1974). 11. R. Hannesson, “Economics of Fisheries,” Universitetsforlaget, Oslo (1978). 12. Y. Ishii, On the theory of the competitive firm under price uncertainty: Note., Amer. Econ. Rew. 67, 768-769 (1977). 13. A. Sandmo, On the theory of the competitive firm under price uncertainty, Amer. Econ. Rev. 61, 65-73 (1971). 14. M. B. Schaefer, Some considerations of population dynamics and economics in relation to the management of marine fisheries, J. Fish. Res. Board Canada 14,669-681 (1957). 15. A. Scott, Development of economic theory on fisheries regulation, .I. Fish. Res. Board Canada 36, 725-741 (1979). 16. W. F. Sinclair, Management alternatives and strategic planning for Canada’s fisheries, J. Fish. Res. Board Canada 35, 1017-1030 (1978). 17. R. Turvey, Optimization and suboptimization in fishery regulation, Amer. Econ. Rev. 54, 64-76 (1964). 18. J. Warming, Om grundrente af fiskegnmde (On rent of fishing grounds), Nationabkon. Tidpskr. 49, 499-505 (1911).