Arnason 1990 Minimum Information Management in Fisheries

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Minimum Information Management in FisheriesAuthor(s): Ragnar ArnasonSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 23, No. 3(Aug., 1990), pp. 630-653Published by: Blackwell Publishing on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/135652 .

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Minimum informationmanagement n

fisheries

RAGNAR ARNAS ON Universityof Iceland

Abstract. This paper deals with problems of optimal managementof common-property ish-eries. It advances the propositionthatmany fisheries managementschemes, which aretheo-retically capable of generatingefficiency, are actually not practicable,owing to their hugeinformationalrequirements.This applies, for instance, to managementby means of correc-tive taxes/subsidies.The paper proceeds to show thatthereexists, underfairly unrestrictiveconditions, market-basedmanagement systems that require minimal information for theiroperationbut lead neverthelessto efficiency in common-property isheries. One such system

is the IndividualTransferableShareQuota system (ITSQ).

La gestion des peches avec un minimumd'information.Ce memoire traite des problemesde gestion optimale des peches qui sont en propriete commune. On suggere que nombred'arrangements our gerer les peches, qui sont theoriquement apables d'assurer 'efficacite,ne sont pas susceptibles d'etre mis en pratiqueparce que les besoins informationnelssontenormes. C'est le cas des methodes de gestion utilisantun ensemble de taxes et subven-tions. On montrequ'il existe des systemes de gestion fondes sur le marchequi, imposantdes conditions vraimentpeu restrictives et requerantune information minimale pour finsd'operations, engendrentune utilisation efficace des peches en proprietecommune. Un telsysteme est ITSQ un systeme de quotasindividuelstransferables.

I. INTRODUCTION

Since the work of Gordon (1954) and Scott (1955) it has been widely recognized

that common property fisheries generally operate in a socially suboptimal manner.

As suggested by Turvey (1964) this state of affairs can be usefully regarded as

stemming from externalities in the harvesting process. Any commercial fishery

A previousversion on this paperwas presentedat the workshopon the Scientific Foundationsfor Rights Based Fishing in Reykjvaik1988. I am gratefulto the participantsn this workshop,especially G.R. Munro and J.E. Wilen, for constructivecomments on the paper.I would like tothank the two anonymous refereesfor theirhelpful comments.

CanadianJournalof Economics Revue canadienned'Economique, XXnI,No. 3August aouit1990. Printed n Canada Imprim6au Canada

0008-4085 / 90 / 630-653 $1.50 ? CanadianEconomics Association





632 RagnarArnason

of taxes and individual transferablequotas are considered in section III. Section

iv presents a variantof the individual transferablequota system that allows the

fisheriesmanager,undercertainconditions,to identifythe optimalmanagementofthe resourcewith very little information.Finally, the main conclusionsof the paper

are summarized n section v.

II. THE BASIC FISHERIES MODEL

Consider a fisheryin which a numberof fishingfirmsexploit a single stock of fish.

Let the fishing industryconsist of N fishing firms,whereN > 0. At a given point

of time some of these N firmsmay not be operating n the industry.Thus N refers

to potentiallyactive fishing firms.The harvestingfunctionsof the fishing firmsare

Y(e(i; t), x(t); i), i = 1, 2, . .N, for e(i; t), x(t) _ 0,

where e(i; t) refers to the fishing effort of firm i at time t and x(t) representsthe

biomass of the fish stock at time t. To simplify the notation,redundant unctional

argumentswill frequentlybe suppressedbelow. The functions Y(., .; . ) are taken

to be twice continuouslydifferentiable, hat is, S2, increasingandjointly concave

in e andx. Moreover,

Y(oi.;.)= Y(.,O;.) = Ye(.,0;.) = Yx(O,;.) =0.

The harvestingcost functions are

C(e(i); i), i 1, 2, . .N, for e(i) ? 0,

where the functions C(.;.) are assumedto be S2, increasing and convex in e(i).

While inoperative ishingfirms do not incurcosts, fishingfirmscurrentlyoperatingin the industryare assumedto experiencecosts even if they do not exertanyfishing

effort. Thus, C(O; ) > 0 with the strictinequalityapplyingto operativefirms.

Growth of the fish stock is definedby the differentialequation

x'-ax(t)/lat = G(x) - Y(e(i), x; i), all x > 0 (1)

The naturalgrowthfunction,G(x), is assumedto be S2 andexhibits the following

properties:

G(xl) = G(x2) = 0, wherex2 > xl ? 0 andG"(x) _a2G(x)/aX2 < 0.

The function G(x) is, in other words, unimodal and concave, and there exists a

biomass level for which growthis positive.



Minimum nformationmanagement 633

Finally, let p and r refer to the marketprice of catch and the discount rate,

respectively. It is assumedthatp is finite andp, r > 0. In what follows it will be

taken for granted hat these prices coincide with social shadow prices.Given these specifications,the instantaneousprofitfunction for a representative

fishing firmi may be writtenas

ir(e(i), x, p; i) =p. Y(e(i), x; i) - C(e(i); i), (2)

where 7r(., , .;.) is S2 and concave in e(i) and x. The present value of firm i's

future profitsfrom the fisheryis definedby

Pv({e()} {x}, p, r; i)-j ir(e(i), x, p) *exp (-r * )dt, (3)

where the braces, {-}, indicate that the time path of the respective variable is

involved. Providedthat the economic prices involved in the profit functions, (2)

and (3), accuratelyreflect the respective social values, these functions may be

takenas measuresof social benefits.Notice, however, thatthey do not necessarilymeasure resourcerents since some of the profits may be intramarginal nes. (Fora discussion of resource rents in fisheries see Copes 1972.)

The efficiency propertiesof this kind of fisheries model have been extensivelyinvestigated n recentyearsand arenow well established(an excellent referenceis

Clarkand Munro 1982). For the purposesof this paper, however, it is helpful to

review the essentials of this theory briefly.The social problemis to find a time path of fishing effort for the fishing firms

that maximizes the presentvalue of industry profits subject to the biological and

technical constraintsof the problem.More formally:

Maximize EPv({e(i)}, {x},p,r;i) (I)

all{e(i)}I

Subject to (a) x' = G(x) - Y(e(i), x),

(b) x, e(i) ->-, all i.

The necessaryconditionsfor a solutionto problem i) include the following (fordetails see appendixA):

(p -) * Ye(i) Ce(i) = 0, for all t and i for which e(i) > 0, (4)

where pLepresents he current hadow value of an additionalunit of biomassalongthe optimal path.Conditions(4) thusstate thatto maximizepresentvalueof profitseach firm'smarginalbenefitsof effort, evaluatedat marketprices less the shadow

value of biomass, shouldequal its marginalcosts of effort.



634 RagnarArnason

The movement of pL long the optimal pathis given by the differentialequation

H 't*(Yx +r-Gx PZpEYx (5)

In bionomic equilibriumx'(t) = e'(i; t) = 0, all i. Hence, the equilibrium, a isgiven by the equation

IL p I Yx(e*, x)/(ZYX(e*, x) + r-GX(x)), (6)

where e* represents he optimalequilibriumeffort level of firm i.

So, in equilibrium,the shadow value of biomass, p, depends directly on the

harvesting unctions of all active firms, the biomass growthfunctionand the eco-nomic prices, p and r. Moreover,since each firm's optimal fishingeffort level, e*,

dependson its cost function, so does 1t.Considernow the behaviourof the fishingfirms. We takeit that each firmseeks

to maximize is own profits. In the fisheries economics literature there is some

ambiguityconcerningthe firms'perceptionof the biomassgrowthconstraint.4The

most reasonableassumption,however,appears o be that of rationality.This means

thatthe firms take the appropriate otice of all variablesandrelationshipsaffectingtheir profit functions including the resource growth constraintand each other's

fishingeffort.Thus,each firm will attempt o maximizeits profitsgiven the fishingeffort exerted by other firms. Since, in practice, the fishing effort of other firms

cannot be instantaneouslyobserved, the fishing firms must form predictionsor

expectations concerning this variable.A certain equilibrium,usually referred to

as Nash-Cournotequilibrium (see Cournot 1897; Nash 1950), is reached when

the firmscorrectly predict each othersfishingeffort. Notice, however, that outside

biological equilibriuma given Nash-Cournot quilibrium s only momentary, ince

changes in biomass require adjustmentsn individualfishing effort.

On these assumptions, he ith firmsattempts o solve the following problem:

Maximize Pv({e(i)}, {x}; p,r;i) (II){e(i)}

Subject to (a) x' = G(x)-E Y(e(i),x),

(b) x, e(i) 0,

(c) e(j), j 7 i given.

Solving thisproblemfor all the firms yields the following necessaryconditions(fordetails see appendixA):

(p - v(i)) Ye(i) Ce(i) = 0, for all t and i for which e(i) > 0, (7)

4 For instance Clark (1976, 1985) assumes that competitive firms ignore the biomass growth con-straintentirely. For the contraryview see Dasgupta and Heal (1979).




where o(i) is firm i's evaluation of the currentshadow value of an additionalunit

of biomass.

The structureof conditions (7) and the socially optimal ones,(4), above, are

identical.The only difference is that private firmsmodify the marketcatch price

by v(i) insteadof the social shadow value, 1t. The key question therefore is howthe o(i)s comparewith p.

The solution to the privateprofitmaximizationproblem,(II), implies the follow-

ing movementof the o(i)s over time:

5'(i) =(i) * ( Yx+ r -G ) -p Yx, all i. (8)

Thus, in bionomic equilibrium,5 r(i),is given by the equations:

o(i) = p *Y,(e(i),x; i)/ (> Y, + r -G, all i. (9)

Therefore,comparingequations (9) and (6) for the same x and e(i)s, it is clearthat in equilibrium,a _ v(i), all i. This means that the social shadow value of

biomass is at least as greatas the privateone. In fact, equality betweenthe social

and privateshadowvalues of biomass is attained only when there is a single firm

operatingin the industry.It follows from conditions (4) and (7) that for a given

equilibriumbiomass, x, the competitive fishing effort, e, will exceed the optimal

one if the numberof active fishing firms exceeds one.This argument s sufficient to establish the fundamentalpropositionof fishery

economics, namely, that competitiveutilization of a common fish stock generallyyields suboptimaleconomic results.

The above results, incidentally,also show that the common assertionthat com-

petitive fishing firmsequatemarginal ncome with marginalcosts (see, e.g., Clark

1976, 1985) is not generally valid. Providedthe firms are rational, in the sensedefined above, their privateevaluation of the shadow value of the resource will

be positive and marginalincome will consequentlyexceed marginal costs. With

rational irmsthe customaryassertionapplies only asymptotically, hat is, when thenumberof active firmsapproaches nfinity. However, if there are fixed harvesting

costs, that is, C(O) > 0, an infinite number of firms is incompatiblewith profitmaximization.6

The relationshipbetween the privateand social shadow value of biomass be-comes particularlysimple if the fishing firms are identical. Clearly, in that caseo(i) oa(j) = C, all i and j, and

I = N1 C, (10)

where N1 denotes the numberof active firmsin the industry.Thus,in this particular

case, we see that private evaluationof the shadow value of biomass decreases

monotonouslywith the numberof firms active in the industry.

5 Notice that in bionomic equilibriumfirms' expectationsof each other's fishingeffort must becorrect.Bionomic equilibrium herefore mplies a Nash-Cournotequilibrium.

6 C(O)> 0 implies that firms have to reacha finite size to breakeven.



636 RagnarArnason

III. FISHERIES MANAGEMENT

Given the inefficiency of competitivefisheries demonstratedabove, it is obviouslydesirable to devise a regulatoryregime that is capable of realizing as much of

the attainable economic benefits as possible. Over the years, many managementsystems have been suggested for this purpose. In this section we briefly considertwo of the more respectableof these systems, a tax on catch and individual catch

quotas.

1. Taxes on catchThe inefficiency of competitive exploitation of a common fish stocks is due to

external diseconomies in production.By reducing the fish stock, each firm's har-vesting activity adversely affects the harvestingpossibilities of other firms in the

fishery.Since the work of Pigou (1912), it has been recognizedthat many external-ities can, at least in principle,be remedied within the marketsystem by imposing

correctivetaxes or subsidies. In the case of fisheries, the appropriateax turnsoutto be analytically elegant but, unfortunately, xtremely difficultto apply.

Comparing he social and private conditions for profitmaximization,equations

(4) and (7), respectively, we see that firm's i imputed net output price is p -

a(i) insteadof the socially appropriate ne, p - tL.It follows that the appropriate

corrective output tax for firmi is

-ri) = A - a(i). (I11)

Equation (11) gives the corrective tax on catch at each point of time. The

developmentof -r(i)over time is defined by the differentialequation

Y i) = ,u' - a'(i), (12)

where tL'and d'(i) are given in equations (5) and (8) above.There are two importanthings to notice about the optimaltax. First,the optimaltax is in generalnot uniform over firms. Only if the firms are identicalwill therebe a single optimal tax. Otherwise, rational firms will have differentevaluations

of the shadow value of biomass7 and this must be reflectedin the correctivetax.8

This result, clearly, has somewhatdisturbingsocio-political implications.9

Second, the informational equirementsor determining he optimaltax are im-

mense. To calculate ji and a(i) for all i, the tax authoritymust solve the social

7 Notice that identical technology does not imply identical firms, since they may be of differentsizes. Given the same technology, the bigger firms will generallyhave a higherevaluationof theshadow value of biomass.

8 Assertions of an identical optimal tax for non-identical irms (see Clark 1985) seem to be basedon the tacit assumptionof non-rational irms, i.e., firms that do not attach any shadowvalue tobiomass left in the sea.

9 Imposing different output tax rates on firms, not to mention higher rates on the smallerfirmsas would normally be required,would tend to contradictwidely held notions about fairnessintaxation.




optimalityproblemas well as each firm'sprofitmaximizationproblem. To be able

to do so, the tax authoritymust have at its command all the data relevantto thefishing firms.In particular, he tax authoritymust have full knowledge of the re-

source growthfunction, and the harvestingand cost functions of all the firms at

all points of time. Moreover,the tax authoritymust continuouslymonitorthe state

of the resourceand the movementof the relevanteconomic prices for the optimal

tax must be continuously adjustedto new conditions. Clearly,these tasks wouldexceed the capabilitiesof most tax authorities.

2. An individual transferablequota system

Let us now consider a fisheryregulatedby means of individualcatchquotas.Many

variantsof this system are conceivable. Here we restrict out attentionto the fol-lowing.The catch quotas stipulatethe maximum rate of catch permitted o each fishing

firm at a point of time. This is quite restrictive.More generally, a catch quota

limits the catch volume over a period of time which may be of any length. A

quota system constraining he rate of catchimplies infinitesimalquota periods and

may be referred o as the continuousquota system.Alternatively,a system of catch

quotaswith finitequotaperiods may be referred o as the discrete quota system.It

is important o notice that fisheriesmanagementon the basis of the discretequota

system is not generallycapableof generatingfull efficiency.10A central authority,which we may refer to as the quota authority, ssues the

catch quotas.The quotas are issued continuouslyat each point of time. The sum

of the catchquotasconstitutes the total quota,Q.The catch quotas aretransferablewithout anyconstraintsand are perfectlydivis-

ible. The quotasthus constitutea homogeneoustradeablecommodity. We assume

that there is a marketfor this commodity and, moreover,that this market s open

to every one interested in trading. Also, to bypass the tedious problems of dis-

equilibriumtrades, we assume that all tradingtakes place at equilibriumprices.

The equilibriumquotaprice is denotedby s.The quota authoritymay allocate quotasto firms free of chargeor through the

quotamarket.Let qO(i, t) > 0 represents ree allocationof catch quotasfrom the

quota authority o firm i at time t. The quotaauthoritysells the remainderof thetotal quota in the quotamarket. Let z(i, t) representfirm's i instantaneousquota

purchasesat time t. Then the total quotaconstraintmay be writtenas

Q(t) = > (qo(i, t) + z(i, t)), all t. (13)

The individualquotaconstraint,on the otherhand, is

Y(e(i), x) < qo(i, t) + z(i, t), all i and t. (14)

10 If the quota period is finite, differentexploitation pathswill satisfy the quota constraint.Gener-ally, however,not all these paths are optimal.



638 RagnarArnason

Let us now consider the behaviour of individual fishing firmswithin this insti-

tutional framework.Theirprofitmaximizationproblem may be writtenas

00Maximize I (p *Y(e, x) - C(e) - s z) *exp (-r 'r)dt ('I')

Subject to (a)qo +zY(e, x),

(b)x' G(X)-Z Y(e, x),

(c) e? 0.

Now, it is easy to check (see appendix B) that the solution to problem (iii)

includes the conditions

s >O=qo+z = Y(e, x), all i, (15)

(p - S) Ye(i) - Ce(i) = 0, for all i for which e(i) > 0, (16)

where s, it will be recalled,is the marketprice of quotas.The messageof (15) is that,providedthat the marketpriceof quotasis positive,

firms will not leave any quotas unused.It follows that total catch will equal total

quotas and x/ = G(x) - Q. Thus, the quota system under discussion effectivelyseparates ndividualfishing decisions from the developmentof the fish stocks. It

follows that the basic stock externality imposed by fishing firmson each other in

competitivefisheriesis eliminated.

Comparingconditions (16) with the socially optimal ones given by equations(4) of the previous section, it is apparent hatprivate harvestingwill be optimalif

s = ,u, that is, if the marketprice for quotas equals the optimal shadow value of

the resource.Now, the marketprice for quotaswill depend, amongotherthings,on

the total supplyof quotas,thatis, Q. To see this, notice thatequation (16) defines

the following set of instantaneousquotademand functionsfor active firms:

s = p -Ce(e(i))/Ye(e(i), x), all active i. (17)

Equilibrium n the quotamarketrequires

Q = Y(e, x). (18)

Finally,solving equations (17) and (18) yields the instantaneousquota price

s S(p, Q). (19)




Therefore,by supplying the appropriateotal quota, the quotaauthority an controlthe quota price andthus ensureoptimalutilization of the fish resource.

This argumentestablishes the important esult that it is possible, at least withinthe frameworkof this particularquotasystem, to generatefull economic efficiencyin the fishery by judicious choice of total quotas. However,just as in the taxa-tion case discussedabove, the volume of informationneeded to accomplishthis isdaunting.First,since the marketpriceof quotas, s, must be set equal to the optimalshadow valueof theresource, i, the lattermust be calculated.This involvessolvingthe social optimalityproblem (i). Second, in orderto select the appropriate otalquota, Q, the marketprice function (19) must be obtained.This involves solvingthe privateprofitmaximizationproblems.Both of these tasks require exhaustive

andcontinuouslyupdatedknowledgeof the biomass growthfunction,the cost andharvestingfunctions of all the firms, and the outputprice, just as in the taxationcase discussed above. Comparedwith the outputtax, however, managementviacatch quotashas one slight advantage.It does not require the calculationof indi-vidual firms' shadow value of biomass, that is, a(i). The reason is that, since thisquotasystem eliminatesthe resource stock externality,c(i) does not influence thebehaviourof the fishing firms.

IV. MINIMUM INFORMATION MANAGEMENT IN FISHERIES

The resultsof the previous section demonstrate he possibility of managing a fishresourceoptimallywith the help of catchquotas.To attainthatobjective,the quotaauthority imply has to pick the appropriateime pathof the total quota.The snagis thatto do this the quotaauthoritymusthave at its commandan immenseamountof informationabout the economics of the fishery.In fact, just as in the taxationcase, the quotaauthorityhas to know in detail the economic conditions of all thefishing firms.

The unilateralselection of total quotasby the quotaauthority,does not, on the

otherhand,exploit the available nformationefficiently.It maybe takenfor grantedthat all information he quota authoritycan possibly obtain in order to determinethe optimaltotalquotais alreadyavailablewithinthe fishingindustry.Afterall, thefishingfirmshave at least as muchknowledgeabout their own cost andharvestingfunctions as the most determined ffort of thequotaauthority ouldpossibly secure.Moreover,since the state of the fish stocks is a majordeterminantof their profitfunction,the fishing firmscan be relied on to make efficient use of the availablebiological data. In fact, given a reasonably competitive environment,only thosefirmsthatefficientlycollect and interpretall the relevantinformationwill survive.It follows that most of the work necessary for the quota authorityto determineoptimal total quotas will merely constitutea duplicationof work alreadycarriedout by privateagents in the fishery.

The question thus naturallyarises whether there exists a way for the quotaauthorityto harness this market informationin order to determine the optimalquota.In this section we shall explore this question.



640 RagnarArnason

1. An individual transferableshare quota system

Considera continuousquotasystem where the quotas are permanent hares in the

total allowable rate of catch. In otherrespects the quota system is as discussed insubsection iii.2 above. More precisely, the essentials of this quota system are asfollows:

1. The individualcatchquotas are sharesin the total allowable rate of catch. Thesequotas are referred o as share quotas.

2. The sharequotas impose an upper limit on the firm's permittedrate of catch.3. The share quotas arepermanent n the sense that they allow the holder the stated

share in the total quota in perpetuity.4. The share quotas are transferableand perfectly divisible.

5. Thereexists a marketfor sharequotas.This market s perfect in the sense thatit is open to everyone interested n trading, all the tradersare price takers, and

the marketequilibratessupplyand demandinstantaneously.6. The quota authority ssues the initial shares and subsequentlydecides on the

total quotaat each point of time.

We refer to this system as the individual transferableshare quota system or, in

short,ITSQ.

The significance of a sharequota system comparedwith a quantityquota system

is primarily n terms of the impact of total quota variationson the economics ofthe firms. Under a sharequota system, changes in total quotas are automaticallyreflectedin uncompensatedquota increasesor decreases for individual firms. Un-der a quantity quota system, on the other hand, total quota adjustmentsmay be

affected by trades in the quota market.11Hence, in this system, individualfirmsare compensatedfor variationsin total quotas, which has important mplicationsfor management,as will become clear in subsection iv.3. Otherwise,the practicaldifference between holding share quotas and quantity quotas, from the point ofview of individualfirms, is rather rivial. Individualshare quotas,for instance, may

still be denominatedand traded n volume terms.Within the institutional rameworkof the ITSQ system, individualquota holdings

at time t are given by

q(i, t) = a(i, t) - Q(t), all i and t, 1 >_a(i, t) > 0, (20)

where q(i, t) standsfor the volume of quotas and a(i t) the share in total quotasor sharequotasheld by firmi at time t. As before, Q(t) represents otal quotas.

The share quotasheld by firm i at time t are given by the equation

rta(i, t) = ((i, O) + z(i, T) - dT, (21)

11 Uncompensatedalterationsof quantityquotas by the quota authorityare of course also conceiv-able. If such adjustmentsare proportional o the quota held, however, the quantityquota systemessentiallyamounts to a sharequotasystem.




where a(i, 0) represents he firm's sharequotas at some arbitrary nitial point oftime and z(i, t) its purchasesof sharequotasat time t. Notice thata negativez(i, t)is interpretedas a sale of share quotas by firm i at time t.

As in subsection iii.2, the instantaneousprofit function of firm i at time t is

7r(i) = p - Y(e(i), x) - C(e(i)) - s - z(i), (22)

where, it may be recalled,p represents he catch price, e(i) the fishing effortexertedby firmi, and x the fish stock biomass. The marketprice for a unit of share quotais S.

To simplify the argument et us make the assumption that firms do not hold

unusedquotas.12 n otherwords:

Y(e(i), x) = q(i), for all i.

Therefore,given the propertiesof the harvesting unction, fishing effort mustsatisfy

e(i) = E(q(i), x), all i andx > 0.

And the profit function can be writtenin a more convenient form as

7r(i) = p *q(i) - C(E(q(i), x)) - s *z. (23)

Now, within this particularquota system, the social problem is to pick totalquotas and allocate individualquotasto firms so as to maximize economic benefitsfrom the fishery.More precisely:

00

Maximize J = (p *a(i) Q - C(E(a(i) - Q, x)) -exp (-r - t)dt (iv)all{ca(i)},{Q}

Subject to (a) x' = G(x) -Q,

(b) Za(i)= 1,

(c) a(i) > O, all i,

(d) Q > 0.

The correspondingcurrent-valueHamiltonian unctionmay be written as

H -E(p *a(i) Q - C(E(a(i) *Q, x)) + 1 * (G(x) - Q), (24)

whereji

is the costate variablefor the resourcegrowth constraint, a).The solutionto problem (iv), if it exists, must satisfy the following conditions:

(P - CE Eq(i))= ,u, for all active firms,(24.1)

(PCE -Eq(i)(O,X)) < t >a(i) =q(i) =0,

12 The validity of this common assumption(see, e.g., Clark 1985) is examined in appendix c.



642 RagnarAmason

'-r * = ZCE 'EX-/ *GX- (24.2)i

Now, p, is the currentshadow value of the resourcealong the optimalpath. Inother words, p, measures the marginalcontributionof additionalbiomass to themaximal level of the objective functional, J. The left-hand side (LHS) of (24.1)is the current instantaneouscontributionof an additional unit of share quota tothe operating profits of firm i. Thus, the message of (24.1) is that firm i shouldreceive additional share quotas while the marginalprofits created by these sharequotasexceed the shadow value of the correspondingresource units. Those firmsfor which ,uexceeds marginalprofitsof quotasat all quotalevels arenot allocatedany sharequotas.

These results basically reiteratethe general optimalityconditions for commonproperty fisheries which were derived in section ii without any reference to theinstitutional organizationof the fisheries. This is as expected; for the maximumeconomic benefits attainablefrom the fishery should not depend on institutionalarrangements.

Within the frameworkof the ITSQ system, firm i's profit maximization problem

is

Maximize j (p *a *Q-C(E(a *Q, x)-s *z) *exp (-r * t)dt, (v)

Subject to (a) ' _= a/alt = z

(b) 1 ? a ? 0.

It is worth noticing that the control variable, z, appears linearly in this prob-lem. Consequently, he optimal controlwill be of a bang-bang character.Since zis unbounded, his means that the actual sharequotasof the firms will be instanta-neously adjusted o desiredlevels. Thus, quota holdingswill at all times be at their

optimallong-run level, given the variablesthat are exogenous to the firms,thatis,

Q, x, s, p, and r. Whenthese variableschange,however, quota holdings must beadjusted.The current-valueHamiltonian or this problemmay be written as

H =p . a. Q-C(E(a . Q, x)-s z +a. z, (25)

where a is the shadow value of an additionalunit of quotato firm i.The necessaryconditions for a solution to (v) include

s a, for active firms,

(25. 1)s _ a for inactive firms,

a' -ra *=-(p-CE Eq)*Q. (25.2)

Accordingto (25.1), firmsshouldpurchaseadditionalunitsof sharequotasin themarket as long as their shadow values exceeds their marketprice and vice versa.




Profit maximizationmay also require some firmsto sell all their share quotas andcome to rest with no quota holdings at a point wheres > a. Equation(25.2) givesthe rule of motion for the shadow value of quotas.

Combining (25.1) and (25.2) yields the time pathof quota prices.

r *s-s' =(P -CE *Eq) *Q (26)

The LHS of this equation may be interpretedas the cost of holding a unit of sharequota. The term r *s representsthe opportunitycost of holding a unit of sharequota comparedwith investingits marketvalue. The terms' measures the capitalgain/loss of holding a sharequotadue to its instantaneousprice changes. The sumof these two terms is the total cost of holding a unit of share quota. The RHS of

equation (26), on the otherhand,is the marginalprofitsof sharequota holdings. Itrepresents he economic benefits to the firmof utilizing an additionalunit of sharequotafor fishing.

Equation (26) may be regardedas the fundamentaldynamic demand functionfor share quotas. Provided that the fishery is pursuedat all, only the price pathdefinedby (26) is compatiblewith private profitmaximizationand, consequently,equilibrium n the quota market.Equation(26) is reminiscent he famous arbitragerule for asset holdings due to Hotelling (1931). According to the Hotelling rule,the rate of asset price increase must equal the marketrateof interest. In this case,however, the assets are potentially productive. Hence the rule is modified by themarginal profits of quota holdings, i.e. by the term (p - CE *Eq) *Q.

Now, let a*(i, t) denote the sharequota holdings by firmi at time t that solvethe private profitmaximizationproblem,that is (v). Similarly,let a**(i, t) be the

sharequota holdings by firm i at time t that solve the social problem, (iv). Thefollowing lemma is now available:

LEMMA 1. For a given initial biomass and time path of total quotas, x(O) and

{Q(t)}, respectively,a*(i, t)o a**(i, t),all i and t.

Proof. Accordingto equations (24.1) for the optimal program:

CE * q(i) a**(i) Q, x) = CE *Eq(q)(a**() *Q, x), for all active i and and all t.

According to equations (25.1) and (25.2) for the privateprofit maximization pro-gram:

CE *Eq(i)(a*(i) Q, x) = CE *Eq(i)(a*(i) *Q, X), for all active i and j and all t.

Addingthe totalquota constraint, ia(i) = 1, which musthold in both cases, theseconditionsyield identical solutions for a**(i, t) and a*(i, t) as functions of x(t)and Q(t).

Now, biomass at time t is determinedby the initial biomass level, x(O) say,and the biomass growth constraint,x' = G(x) - Q. In other words, x(t) =

X(x(O), {Q}). QED



644 RagnarArnason

Lemma1 states the important esult that,within the frameworkof the sharequotasystem defined above, the total quota will always be caught in the most efficient

manner. It follows that it takes a particularly nept quota authority o escape the

generationof some economic rents under this managementsystem.According to lemma 1, the quota authoritycan ensure optimal utilizationof the

fish stock by selectingthe appropriateime pathof total quotas. Writingthe optimalshare quotas as

a*(i) = a**(i) = F(x(O), {Q}, i),

a formal representation f the problem facing the quota authority s

jrOo

Max [p F(x(O), {Q}) Q - C(E(F(x(O),{Q})

x Q, X(x(O), {Q}))] exp (-r *t)dt. (vi)

While this problemhas considerably ewer controlvariables han the general prob-lem, (iv), above, it may not be much easier to solve. Since the solutionrequires, or

instance,full knowledgeof the F(., ., . ) functions, the quota authorityhas to knoweach firm's harvestingand cost functions in detail and solve its individual profit

maximization problem.However, such a procedureruns afoul of the information

problem discussed in subsection iII.2.

2. Minimum nformationmanagement chemeAbove, it has been pointed out that the huge amount of informationrequiredtodetermine he optimal evel of total quotas by solving the social optimalityproblem,renders that approach o the fisheriesmanagementproblem impracticable n most

cases. In this section an alternativeapproach s proposedthat efficiently exploitsmarket nformation,enablingthe quota authority o identifythe optimaltotal quotapathwith minimal information.The fundamentaldea is thatwithinthe framework

of the permanentshare quota system defined above, the prevailingquota marketprice reflects all relevant nformationabout the currentandfuture conditionsin the

fisheryavailableto the fishingfirmsor, moregenerally, heparticipantsn the quota

market.It follows that the quota authorityonly has to monitor the quota marketprice to become privy to the same information.We now proceed to clarify this

idea.The dynamicdemand functionfor sharequotas,equation (26), above, holds for

all active firmsat all times. Multiplyingthis equationfor each firmby its privateprofit maximizing quotalevel, q*(i), we find

(s' - r . s) . q*(i) = -(P - CE . Eq) q*(i) Q, all i and t.

Since liq*(i) - Q, summingover all active firmsyields

(s' -r *s) = P- (p-CE *Eq) *q*(i), all t.




And, solving this differentialequationfor an arbitrarynitial time t -0, we obtain(for details see appendixD):

s(O) (p - CE(Eq(q*(i),X(x(0), {Q})) *q*(i)] exp (-r * t)dt. (27)

Now, s(0) is the marketprice of sharequotas at time 0. The term(p - CE *Eq)on the RHS of (27) is the marginalprofitsof catch quotas. If all quota units heldare as profitableas the marginalone, the expression(p - CE-Eq) q*(i) measuresthe profitsmadeby firmi. This expression,in otherwords,represents he resourcerents13 obtained by firm i at a point of time. More precisely, as the future is

unknown,(p - CE *Eq) q*(i) mustbe expected resource rents.In fact, since (27)is based on private profitmaximizingbehaviour,the variableson the RHS of (27)mustbe the futureexpectationsof the firms in the fishing industry.Thus,equation(27) states that the currentmarketprice of sharequotas equals the presentvalueof expected futureresourcerents in the fishery.In this sense the prevailing quotamarketprice reflects the relevantinformationaboutcurrentand futureconditionsin the fishery.

The share quotas, by definition, sum to unity. It follows that s(0) on the LHS

of (27) is also the currentmarketvalue of all outstandingsharequotas.Thus, the

following fundamentalpropositionhas been established:

PROPOSITION1. Underthe individualtransferableshare quota system,ITSQ, definedabove the marketvalue of outstandingquotas equals thepresent value of expectedfuture resourcerentsgenerated in thefishery.

Now, accordingto lemma 1, q*(i) _ a*(i) Q in (27) is in fact identical to theoptimala(i) *Q, given the total quota.Hence, the RHS of (27) dependsonly on thepathof totalquotas,{Q}, theinitialbiomasslevel, x(0), and theexogenousvariablesp and r. It follows that

adjustingthe total quota so as to maximize the value ofoutstandingquotasis equivalent o maximizingthe presentvalueof expectedfutureresourcerentsin the fishery.This establishesthe following corollaryto proposition1:

COROLLARY. 1

00

Maxs(O)X* Max si (p - C,(Eq(q*(i), x)) *q*(i) exp (-r - t)dt.

Corollary1.1 is formulated n termsof expected rents.Expectedrents,however,are not necessarilyequal to actual rents. On the other hand, futureconditions in

13 The concept of resourcerents was employedby Gordon in his seminalpaperon the fisheriesproblem in 1954. Since then the problemof common-propertyisherieshas generallybeen seen asone of dissipationof resourcerentsand the objectiveof fisheriesmanagement he restorationofthese rents.Fora fuller discussion of resourcerents in fisheriessee Copes (1972).



646 Ragnar Arnason

a fishery are normallyunknown,and consequentlyexpectationshave to be reliedupon. The crucialpoint is that there appears very little reason to give more cre-

dence to the expectationsof the quotaauthority hanthose of the firms. Rathertheopposite. After all, the firms are generally more knowledgeableabout their owneconomic conditions than the quota authority s. Moreover,accuratepredictionsare vital to the profitabilityof the firms and, for that matter, any player in thequotamarket.Thus, appealing to the principleof rationalexpectations,14ndustryexpectationsmay well be the best available predictor of futureconditions in thefishery.Formally we adoptthe following assumption:

ASSUMPTION 1. Theexpectations of thefishing firms are the best availablepredictor

of future conditions in thefishery.

Given assumption 1, corollary 1.1 suggests an especially efficientway of maxi-mizing resourcerentsin the fishery. Providedthe quota market s reasonablycom-petitive, the quota authoritydoes not need to collect any informationabout thefishery.It only has to monitorthe prices in the quotamarketandadjusttotalquotasso that the value of outstandingquotas is maximized.This, accordingto corollary1.1, will automaticallymaximize the resource rentsgenerated n the fishery. Com-paredwith solving the individualoptimizationproblemsdirectly,as the traditional

fisheriesmanagementsystems demand,this task appearsrelativelytractable.The fisheriesmanagementproblem,however, s not themaximizationof resource

rents.The managementproblem, more generally, is to maximize aggregate profitsin the fishery.The difference between profits and resource rents is the so-calledintramarginalents that areearnedby the more efficientfirms. Only when the firmsare equally efficient will profits coincide with resourcerents. On the other hand,at least in competitiveeconomic equilibrium, hereare groundsto expect equalitybetween resourcerents and profits. If, for instance, capital marketswere perfectandall firmshad access to the sametechnology, profitmaximizationwould require

them to be equally efficient and no intramarginal ents would be earned. (Thisargument s developed further n appendixE.)

We proceedby assumingthatresource rents andprofitsareequivalent.In otherwords:

ASSUMPTION 2. (p - CE *Eq(q*,x)) *q*(i) = p *q*(i) - C(E(q*, x)), all i and t.

According to this assumption, marginaloperatingprofits,evaluated at the profit-maximizing level of individual quota holdings, equal average operating profits.

Alternatively,we may say that marginaloperating costs equal averageoperatingcosts. Notice also that assumption2 implies that the firmsoperateat the point ofmaximumaverageprofitsgiven theirquota holding.Alternatively,hey may be saidto minimize the costs of filling their quota. These attributesare often associatedwith the catch quota system.

14 On the principle of rationalexpectations see, for example, Lucas and Sargent (1981, xi-xvi).




On the basis of assumption2, equation(27) may be rewrittenas

s(O) p1 P q*(i)-

C(E(q*(i),X(x(O), {Q}))] *xp (-r *)dt. (28)

Equation(28) states that the marketvalue of all share quotas equals the presentvalue of expected future operating profits in the fishery. This result, it may benoted,is a standardassertion n the theory of productiveassetprices (see, e.g., VanHome 1971, 15-19). Here,however, the propositionhas been derivedon the basisof individualprofitmaximizingbehaviourand assumption2.

The expression on the RHS of equation (28) is very similar to the objectivefunction of the quota authoritypresentedin problem (iv), above. By lemma 1,q*(i) in (28) is identicalto the optimalq(i), given the total quota.Thus, the onlydifferencebetween(28) and the objective function in problem(iv) is thatthe futurevariables in (28) are not the expectations of the quota authority.However, byappealingto assumption1, these expectationsmay be takento constitute the bestpredictorof futureconditionsin the fisheryavailableto the quotaauthority.

These argumentshave establishedthe fundamentalpropositionof this paper.

PROPOSITION2. MINIMUM INFORMATIONMANAGEMENTSCHEME (MIMS). In the individualtransferableshare quotasystem,ITSQ, definedabove, and given

assumptionsI and2, adjusting current total quotas to maximizethe marketvalue of total outstandingquotasat each point of time is equivalentto the maximizationof profitsattainablefrom the resource.Formally:

00

Maxs(O) {Max J (p, q* - C(E(q*, x)) *exp (-r Tr)dr.

Thepractical mplicationsof thispropositionareobvious. To identifythe optimaltotal quota at a point of time, the quota authoritydoes not have to collect dataabout the fish stocks and the economics of the harvestingfirms. It has merely tomonitor the share quota price in the quota market and adjust the total quota soas to maximize the total value of the share quotas. Proposition 2, on the otherhand,dependsheavily on assumptions1 and2. It they do not hold, the propositionmay not be true.Notice, however,that the essence of 1 is thatthe firmsformulaterationalexpectations. If that is not the case, it is hard to imagine any market-based management hatwill attainefficiency in the fishery.Assumption2, on theotherhand, is basically the requirementhat firms are equally efficient. Given theexistence of quotamarkets, t is not obvious why less efficientfirms would chooseto continuefishing operations nstead of selling off theirsharequotasto their moreefficientcompetitors.

3. Alternativetransferablequota systemsIt will be appreciatedhatnot all individualtransferablequotasystems or ITQShavetheconvenientmanagementpropertiesof the sharequotasystemdescribedabove.In



648 Ragnar Amason

particular,quota permanenceseems to be a prerequisite or minimuminformationmanagement schemes of the type described in subsection iv.2. Also, it appears

thata system of quantityquotas,even if permanent,requiresmore information oroptimal management hanthe share quota system.Tuming our attentionfirst to quota permanence, t seems obvious thatthis will

play an importantrole in any efficiency considerations.Clearly, if the firms holdrelatively permanentquotas, they will have a greater interest in maintainingtheresource base thanif they held only transitoryquotas.To examinethis issue let usassume the sharequota system of the previous session with the modificationthatthe quotas may not be permanent.Thus,let the following equationreplace equation(21) of the previous section:

rta(i, t) = a(i, 0) + j[-1 a(i, r) + z(i, r)]dT, (29)

where, as before, a(i, t) representsfirm i's sharequotaand z(i, t) is sharequotapurchasesat time t. / is the quota non-permanenceparameter. 3 = 0 denotes

permanentquotas as in subsectioniv.1 and 3 > 0 quota non-permenence. n fact,/ -+ oo indicates completely transitoryquotasas in subsectioniii.2. Thus (29) is a

fairly general representation f the degree of quotapermanence.Now, repeatingthe exercise of subsectioniv.2, it is easy to derivethe following

equation corresponding o equation (28):

s(O) [p q* C(E(q*,X(x(O), {Q}))] *exp (-(r + / * )dt. (30)

Thus, it emerges that, if quotas are not absolutely permanent,that is, / > 0,the market value of share quotas underestimatesthe present value of expectedfutureoperating profits in the fishing industry.A positive quota non-permanenceparametereads to anexcessive discountingof futureprofits.Infact, if / -* oo, then

s(O) -+ 0. It follows from this that if / > 0 a managementpolicy of maximizing

the total value of sharequotaswill not yield optimalresults.Such a policy wouldoveremphasizepresent profitsat the expense of future ones.15

Let us now turn our attention o quantityquotas.Assumethe same quota systemas defined in subsection iv.1 with the exception that the quotas are now quantityquotas, that is, rights to a certain rate of catch irrespectiveof the total quota.The instantaneousprofitfunction of a representative irm i remainsunchangedas

specified in equation (22) above. The state variable, however, is now q(i) insteadof a(i), and the individualprofitmaximizationproblemis

Maximize j[p q(i) - C(E(q(i), x) - s z] - exp (-r - t)dt,

15 It should be noted thatequation (30) and the ensuing argumentsalso apply to the case of privatediscount ratesdifferingfrom social ones by /.




Subject to (a) q'(i) z(i),

(b) q(i) ? 0.

Proceeding as in the share quota case yields the following quota price equationcorresponding to equation (28) of subsection iv.2:

s *Q(O)= [S -v *- C(E(q*, x))}Q(O)/Q(t)] *exp (-r * )dt. (31)

The currentmarketvalueof outstandingquotasdepends,in this case, on thepresentvalue of future expected profits multipliedby the ratio of currentto future total

quotas. Thus, if total quotas increase over time, the market value of outstanding

quotaswill underestimate he presentvalue of futureexpected profitsin the fisheryand vice versa. This is readily understandable.The quantityquota system impliesthat such quota increases have to be purchasedby the fishing firms. Hence thesocial benefits of increasingQ are underestimated.Only in equilibrium,when thetotal quotaremainsconstant,will the maximization of quota values be equivalentto maximizingthe presentvalue of profitsin the industry.

V. CONCLUSIONS

Fisheries managementsystems suggested in the literaturegenerally call upon thefisheries managerto calculate the optimal level of certain managementcontrols.These controlsmay be tax rates, individualcatchquotas, access licence prices, etc.In most fisheries the data required o performthese calculations are enormous and

exceed, by far, any realistic assessmentof the capacityof the fisheriesmanager.Itfollows that the traditional isheriesmanagementsystems are of limited practicaluse.

A system of individual transferablequotas appearsto alleviate this problemsomewhat.Provided the quotamarketoperatesreasonably smoothly, any total al-

lowable catch will be harvested efficiently in this system. Consequently, someresource rentswill normallybe generatedunder this system.

Moreover,thereappearsto exist a certain variantof the individual transferable

quota system, namely, the individual transferable harequota system, or, in short,ITSQ, that allows the fisheriesmanager,under certain conditions, to determine theoptimal total quota with minimal collection of information.In this system thefisheriesmanageris essentially requiredonly to monitorthe quota marketand toadjustthe totalquotasuntil the current otalquotamarketvalue is maximized. Thisis referred o as minimum informationmanagement.

The class of individualtransferablequota systems allowing minimuminforma-tion management s not specified in the paper. However, it is demonstrated hatcommon variantsof this system may not have this property.Thus, quota perma-nence seems to be a prerequisite or minimuminformationmanagement,and, per-hapsmoresurprisingly,permanentquantityquotasdo not appear o allow minimuminformationmanagementexcept in equilibrium.



650 RagnarAmason

APPENDIX A: BASIC FISHERIES MODEL

1.Thesocial problemThe social optimizationproblemis

Maximize (p - Y(e(i), x) - C(e(i))) exp (-r t)dtall {e(i)}

Subject to (a) x'-G(x)-Z Y(e(i), x),

(b)x,eO ,alli.

The Hamiltonian unctioncorresponding o his problemmay be written

H - [p *Y(e(i), x) - C(e(i)] + ,u (G(x) - E Y(e, x)).

Thenecessaryconditionsfora solution to thisproblem,provided t exists, include:16

(P -A) *Ye(i) - Ce(i) < 0, e(i) ? O, e(i) ((P - ,) *Ye(i) - Ce(i)) 0,

At=t (ZYx +r-Gx) -p. Yx.

The constraints a) and (b), hold.

2. TheprivateproblemThe privateprofit maximizationproblemfor a representative irmi is

(00Maximize / (p - Y(e(i), x) - C(e(i)) *exp (-r * )dt,

{e(i)} Jo

Subject to (a)x' = G(x) - E Y(e(i), x),

(b)x, e ?0 , all i.

The Hamiltonian unction for firm i is

H = p - Y(e(i), x) - C(e(i) + u(i) *(G(x) - E Y(e, x)).

The necessaryconditions for a solutioninclude

(p - 5(i)) - Ye(i) -Ce(i) < 0, e(i) > O, e(i) *((p - a(i)) Ye(i) -Ce(i)) = 0,

cT(i)'=cT(i) (Z Yx+r-Gx)-pYx.

The constraints a) and (b), hold.

16 Forreference on the optimizationtechniquesemployed in this papersee, for example, Takayama(1974).



Minimum information management 651

APPENDIX B: THE CONTINUOUS TRANSITORY QUOTA SYSTEM

The Hamiltoniancorresponding o problem(III)

is

H = p * Y(e, x) - C(e) - s z + uf* (G(x) - Y(e, x)) + -L (qo + z - Y(e, x)).

Necessary conditions for solving (III) include

s = I-

qo + z-Y(e, x) _-O, IL O, At*(qo + z-Y(e, z))-O.

Therefore, if s > 0, qo + z - Y(e, x) =0.

APPENDIX C: DO FIRMS USE ALL THEIR QUOTAS?

Consider the problem facing firmi:00

Max (Y(e, x)-C(e)-s z) *exp (-r *t)dt{e, z} J

Subject to (a) a Q>? Y(e, x),

(b) a' = z.

The presentvalue Hamiltonian unction is

H = (Y(e, x) - C(e) - s -z) exp (-r t) + u (a Q -Y(e, x)) +-A z.

Necessary conditionsfor a solution include

(Ye-Ce)-exp(-rt)= (- Ye, fore >O

s exp (-r * t) = A

1/'- Q

LimAt*a 0 (necessarytransversality ondition)t-00

a Q - Y(e, x) > 0 X: o = 0 (complementary lacknesscondition).

Now, assuming q* = a* Q > Y(e*, x), u,= 0 by complementaryslackness.But

thenAt'= 0 and At s a constant.In the time intervalof unusedquotas,u'= 0 impliesYe- Ce 0; that is, marginaloperatingprofitsof effort are zero. In a productive

fishery,this result occurs for only a relativelylow biomass. In fact Ye Ce = 0 is

not compatiblewith optimalmanagementof the resource.Also, in the time interval

of unusedquotas,s *exp (-rt) A- implies thats' = r s. In otherwords,the quotaprice must increaseexponentiallyat the rateof the interestrate.



652 RagnarArnason

APPENDIX D: SOLUTION TO EQUATION (26)

Consider the differentialequation

s- r *s = -F(q, x),

whereF(q, x) = ( p-CE *Eq) *q.

The solution to this differentialequation is

rt

s(t) - exp (-r *t) = (q, x) - exp (-r -T) *dT + s(O).

Now, since p(t) is finite by assumption,s(t) must also be finite. Therefore t

oo o:>x(t) - exp (-r t) -> 0, and

rts(O) = F(q, x) *exp (-r *T) *dT.

APPENDIX E: ARGUMENTS IN SUPPORT OF ASSUMPTION 2

Assumption2 states that

P -CE (E(q*, x)) *Eq(q*, x) = p -C (E(q*, x))/q*.

Assume that all firmshave access to the sametechnology.Accordingto (26), r-s-s'can be regardedas the cost of holding a unit of quota. The benefit, on the other

hand, of holding a unit of quota is p - CE *Eq. Denote the quota holding cost

by S(q*) and the benefit by MP(q*).Notice that MP(q*) is the marginal operating

profits of quota holdings. Also, let AP(q*)= p - C(E(q*, x))/q*, where AP(q*) isthe average operatingprofitsof quota holdings.

Now, assume that S(q*) = MP(q*) > AP(q*), for q* > 0. Then, clearly, the firm

will be makingan overall loss and will be betteroff by selling its quota. Therefore,since q can be instantaneouslyadjustedat no cost, q* cannot have been optimaltothe firm.

Alternatively assume that S(q*) = MP(q*) < AP(q*). This means that the firmwill be makingoverallprofits.Therefore, since the firm's technology is availableto other firms, these profits are also attainableto them. Consequently,the quota

holdings of inactive, namely q* = 0, cannot be optimal.Therefore,MP(q*) = AP(q*).

REFERENCES

Andersen, P. and J.G. Sutinen (1985) 'The economics of fisheries law enforcement.'LandEconomics 61, 387-97



Minimum information management 653

Clark,C.W. (1976) MathematicalBioeconomics: The OptimalManagementof RenewableResources (Wiley)(1985) Bioeconomic Modelling and Fisheries Management Wiley)

Clark,C.W. and G.R. Munro(1982) 'The economics of fishing andmodem capital the-ory: a simplified approach.'L.J. Mirmanand D.J. Spulber, eds, Essays in the Eco-nomicsof RenewableResources (North-Holland)

Copes, P. (1972) 'Factorrents,sole ownershipand the optimal level of fisheries exploita-tion ManchesterSchool of Social and EconomicStudies 40, 145-63

Coumot A. (1897) Researches into the MathematicalPrinciples of the Theory of Wealth(MacMillan)

Dasgupta,P.S. and G.M. Heal. (1979) Economic Theory and ExhaustibleResources(JamesNisbet)

Gordon,H.S. (1954) 'Economictheoryof a common propertyresource:the fishery.Journalof Political Economy 62, 124-42

Lawson,R.M. (1984) Economicsof Fisheries Development(FrancisPinter)Lucas R.E. and T.J. Sargent,ed. (1981) RationalExpectationsand EconometricPractice

(GeorgeAllen & Unwin)Nash, J., Jr (1950) 'Equilibriumpoints in N-person games.' Proceedings of theAcademy

of Sciences 36, 48-9Pigou, A.C. (1912) The Economicsof Welfare (Macmillan)Scott, A.D. (1955) 'The fishery:the objectivesof sole ownership.'Journal of Political

Economy 63, 116-24Smith, V.L. (1968) 'Economics of production romnaturalresources.' AmericanEco-

nomic Review 58, 409-31

- (1969) 'On models of commercialfishing.' Journal of Political Economy 77, 181-98TakayamaA. (1974) MathematicalEconomics(Dryden Press)Turvey,R. (1964) 'Optimizationand suboptimizationn fishery regulation.'American

Economic Review54, 64-76Van Home, J.C. (1971) Financial Managementand Policy (PrenticeHall)

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Arnason 1990 Minimum Information Management in Fisheries