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REGULATION BY OPTIMAL TAXATION OF AN OPENACCESS SINGLE-SPECIES FISHERY CONSIDERING

ALLEE EFFECT ON RENEWABLE RESOURCE�

ALEJANDRO ROJAS-PALMA

Instituto de Matemáticas, Universidad Austral de Chile.alejandro.rojas.p@mail.ucv.cl.

EDUARDO GONZÁLEZ-OLIVARES

Grupo de Ecología Matemática, Instituto de Matemáticas,Ponti�cia Universidad Católica de Valparaíso, Chile.

ejgonzal@ucv.cl

In this work, a bioeconomic model of an open access single-species �shery is ana-lyzed, using a catch-rate function suggested by W. C. Clark and considering Alleee¤ect in the exploited resource. The harvesting e¤ort is considered to be a dynamicvariable (a function of time) and also it is assumed that the exploitation of the �sh-ery is regulated by an agency by imposing a tax per unit of landed biomass. Themain objectivesare to establish the maximization of the monetary social bene�t aswell as to prevent the extinction of the resource. i.e., a optimal control problemsis obtained, which is solved by means of the Pontryagin�s Maximum Pinciple.

1. Introduction

If harvesting by individuals of a region is causing severe damage of theecosystem of a determined region, in particular when a exploited populationcan become to extinction, then the governing authority of this region shouldplan a regulating policy which would keep the damage to the ecosystemminimal1.To avoid this the regulating authority levies a tax on the catch of the

harvesting agency. This acts as a deterrent to the �sher and helps therenewable resource to grow, which can be an incentive to the �sher, whenthe tax takes the form of a subsidy15.

�This work was partially supported by DII-PUCV project number124720/2009

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In this work, a bioeconomic model of an open access single-species �sh-ery is analyzed using a catch-rate function suggested by Clark4. This catch-rate function is more �exible and realistic than the usual catch-per-unit-e¤ort-hypothes´s used in Schaefer�s model4.The growth of the exploited population is a¤ected by the Allee e¤ect

5;18 or depensation16. The harvesting e¤ort E is considered to be a dynamicvariable (a function of time), i.e., E = E (t) and also it is assumed that theexploitation of the �shery is regulated by an agency by imposing a tax �per unit biomass of landed �sh.The net economic revenue to the �shermen (perceived rent) is the di¤er-

ence between of incomes and the cost4 and the gross rate at which capitalis invested at any time is assumed to be proportional to the "perceivedrent" at that time9.Moreover, the net economic revenue to the society can be considered as

the sum of the net economic revenue to the �shermen (perceived rent) plusthe economic revenue to the regulatory agency.To establish the optimal taxation, �rstly the positive steady state of

this system is determined and conditions for its existence and stability areobtained. Lately, we �nd the proper taxation policy which would give thebest possible bene�t through harvesting to the community while preventingthe extinction of the predator, which is studied invoking the Pontryagin�sMaximum Principle of Control Theory6;8. This form of control is di¤erentfrom the usual optimal harvesting policy trying the maximization of thenet economic revenue to the �shermen.Economists are particularly attracted to taxation because a competi-

tive �sheries can be better maintained under taxation rather than otherregulatory methods. However, little attention has been paid to study thedynamics of �shery resources using taxation as a control instrument2;9;13.

2. Model construction

The populational dynamics of the �shing resource is modeled by the equa-tion

dxdt = F (x)� h (x;E) (1)

where x = x(t) represents the populational biomass in the time , F (x) isthe natural growth function and h (x;E) represents the harvesting rate inthe time.In this work we consider

F (x) = r�1� x

K

�(x�m)x (2)

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a modi�cation of the logistical equation with the factor (x�m) represent-ing the Allee e¤ect. If m > 0, we have the strong Allee e¤ect, meanwhile ifm = 0, it has the weak Allee e¤ect20.The harvesting function is of the sigmoid form19 given by

h (x;E) = qx2Ex2+a2 (3)

where E = E (t) represents the �shing e¤ort in time and q is the catchabilitycoe¢ cient. This function marks a di¤erence with the function commonlyused in the Schaefer hypothesis4, expressed by

h (x;E) = qEx

since it implies that the capture is limited although the biomass is increased,whenever the e¤ort and is limited, that is to say, h ! qE when x ! 1,this function expresses e¤ects of saturation with respect to the abundanceof the stock. Any function of realistic harvest must exhibit this behavior.The harvesting agency�s aim is to obtain as much revenue as possible

through its activity, whereas the community needs the food through har-vesting and is also keen on protecting the resource from extinction. Thus,the bene�t to the community consists of the revenue through the harvestand the retained resource population.Then, the problem of optimization of the community�s bene�t is a con-

ditional optimal control problem in the sense that the revenue is to bemaximized subject to the condition that the population is larger than apositive quantity as t ! 1. In order to achieve this goal a regulatingagency has to curb arbitrary growth of harvesting. This is done by levyinga tax on the catch (which can also be a subsidy).Tax (or subsidy) makes the harvesting e¤ort a dynamic variable. The

net bene�t, from the harvesting, to the society is the revenue before the de-duction of tax, obtained by the harvesting agency. The controlling agency,like the government, levies a tax � on the harvesting agency. The purposeof the tax (which may be a subsidy) is to regulate the harvesting e¤ort.The �shing e¤ort E is governed by the equations

E (t) = �Q (t) 0 < � � 1 (4)

dQ (t)

dt= I (t)� Q (t) (5)

where I (t) represents the rate of gross investment, Q (t) represents theamount of capital inverted in the �shery in the time and represents therate of depreciation of the capital, the �shing e¤ort E at any time t is

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assumed like a proportion of the instantaneous amount of inverted capital,following the idea described in9.If Q (t) is measured in standardized units of �shing [SDF ], for example,

amount of vessels available by the �shery, then, is reasonable to considerthat E (t) must be a proportion of Q (t) and the maximum capacity of e¤ortmust be equal to the number of vessels available (� = 1).The case in that � = 0 indicates the situation in which a �shing ef-

fort is not developed, although vessels available exist. This can re�ect asituation of �shing over-exploitation, the �shing �eet are due to maintainnon-operative because the �shing stock has been exhausted.The economic revenue of the �sherman (perceived rent) can be inter-

preted as the di¤erence between the incomes of the harvest, and the costof realizing the e¤ort

G (x;E) =�(p��)qx2Ex2+a2 � c

�E

where p represents the price by biomass unit, which will consider constant,c represents the cost of harvest by biomass unit and the price by unit oflanded biomass is punished by the tax of the regulatory agency.If it is assumed that the rate of gross investment is proportional to the

perceived rents, is obtained

I = �G (x;E) 0 � � � 1 (6)

Equation (6) says that the rate of maximum investment at any time agreeswith the rents perceived for the case in that � = 1.The case � = 0 can only happen when a situation of perceived rent

exists negative, that is to say, when he is not feasible to stop investing in�xed actives4. If the �shery lost importance and the socials capitals aremanageable, the sole owner of the �shery is bene�ciary, allowing a lost ofcontinuous investment of �xed actives. in this case I < 0 and � > 0 wherethe negative values of investment are possible to be interpreted like lost ofinvestment.Replacing the equation (6) in the equation (5) and deriving the equation

(4) the di¤erential equation obtained is

dE

dt= �

��� (p��)qx

2

x2+a2 � c��

�E (7)

The �sherman and the regulatory agency is two components di¤erentfrom the society. Therefore the income obtained by them are income gainedfrom the society of the �shery. The �shery revenue, also call social revenue,is the sum of the revenue of the �sherman and the revenue obtained by