Includes bibliographical references (leaf 16)

Simaan, Marwan, 1946-

Takayama, Takashi, 1929-

English

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OFDigitized by the Internet Archivein 2011 with funding fromUniversity of Illinois Urbana-Champaignhttp://www.archive.org/details/optimalharvestin466takaFaculty Working PapersCollege of Commerce and Business AdministrationUniversity of Illinois at U r b a n a - C h a m p a i g nFACULTY WORKING PAPERSCollege of Commerce and Business AdministrationUniversity of Illinois at Urbana-ChampaignFebruary 10, 1978OPTIMAL HARVESTING FOLICIES AND REGULATIONS FORRENEWABLE RESOURCEST. Takayama, Professor of Economics, and M. Simaan,Associate Professor of Electrical Engineering atthe University of Pittsburgh#466Summary ;Renewable resources constitute an increasingly important source of foodand material necessary for meeting the needs for survival of the increasingpopulation of human beings in the world. Continuous, unplanned consumption ofthese resources could very dangerously lecd to their extinction.In this paper, we develop a dynamic theory for optimally managing suchresources in order to prevent their extinction, while at the same time insuringan adequate level of supply for human consumption. We derive a policy forharvesting which meets several requirements: (i) does not exceed a certain levelset by technological or conservatiunal considerations, (ii) results in a certainlevel of population of the resource conserved at end of a given time horizon,and (ill) maximizes the total quantity harvested. A discussion of the propertiesof this policy and the effects of enforced regulations on it are also presented..r';IntroductionRenewable resources such as fish, whale, deer, forest, etc.,constitute an increasingly important class of economic resources forthe sustenance and improvement of human welfare on the planet Earth.The common characteristic of these resources are that they are fordirect human consumption and that they can reproduce themselves witha specific speed of renewal given a specific environment.In order to prevent extinction of these resources, regulatoryagencies have been set up to regulate and limit consumption. At thesame time in order to maintain a certain degree of economic growth,consumption must be allowed to be at an adequate level.Before any regulations were intorduced, the history of consump-tion of many such resources (such as whale, deer, etc.) was at a levelhigh enough to make it impossible for sustained growth to take place.When consumption regulations were introduced, the problems that werefaced centered around what policies should the regulatory agenciesimpose in order to insure that an adequate supply of the resources isavailable at all times. Similarly, from the producer 's point of view,the problem remains that of determining production policies in orderto maximize production without violating the regulatory agency's require-ments.In this paper we develop a dynamic theory of renewable resourceseconomics that takes these common characteristics into consideration toestablish:a) an adequate policy. of consumption, andb) a principle of conservation for this class of resources."TJe assume that for these resources, the market is such that consumptionis always equal to production.Since this class of resources etr.braces a large number of species,in this paper we will deal with only one species (for example fish) with-out loss of substance. The problem that regulatory agencies face couldbe summarized as that of determining and recommending the maximum allowableintensity of withdrawal (harvesting) that will sustain an adequate levelof the species for it to grow. The problem that the producing sectorfaces could be summarized as that of determining an optimal withdrawalpolicy that will maximize production without violating the requirementsof the regulatory agencies.Stated differently, renewable resource economists consider thefollowing questions as theoretically and practically important:a) What is the optimal rate at which the species should beharvested?b) Why might the maximum sustainable yield not be optimal?c) Under what conditions will extinction of the speciesoccur? (Peterson and Fisher [1])d) Is withdrawal regulation necessary and if so at what level?The Model2While the literature on renewable resources is extensive, inthis paper we avoid our own review. However, we should mention thatthe model considered in this paper is novel in that it allows for aformulation of the problem in its most natural dynamic optimizationframework and takes into consideration the interplay, or interdepen-dence, of the policies of both the production sector and the regulatoryagencies. Furthermore it is implicitly assumed that this analysis is2We avoid our own review of the existing literature in this field. Thereader is referred to a comprehensive review in [1], and interestingresults in [2] and [3].justified mostly in .cases where the resources are limited relative totechnology and human demand.We conceive of a renewable resource, or species, populationwhich follows dynamics over time described by the differential equation:&& x(t) = f(x(t)) - u(t)wherex(t) denotes the recruitable species population at timet, (more clearly, the unit must be expressed in number,pounds or tons of the resource at or older than therecruitable age);i(t) denotes the rate of change of the population at time t;u(t) denotes the intensity or rate of withdrawal (catch, orharvest) of the resource at time t; andt denotes the real time over which the population andwithdrawal are moving and measured.We assume .that policies are to be determined over a time horizon10, T] and that at time t 0, the species population is known and equalto x(0).3The differential equationx - f (x) (2)itself, is called the biological growth law [1]. It represents the law,such as exponential, quadratic, Volterra, etc., that governs growth ifno harvesting takes place. The function f(-) naturally depends on thetype of species and the environment. From (2), it is easy to see that2Hereafter, unless otherwise stated or special emphasis needed, thetime t in x(t) and u(t) (and other variables that may be introducedlater) will be omitted.;.the population x(t) at time t satisfies:r-x(t) - [ f(x(T)) dx + x(0) (3)and is a function of the initial population x(0) and the populationhistory x(t) over the interval of time [0, t]. For many species suchas fish, deer, etc., growth follows an exponential law. In this caseequation (2) is linear and takes the form:& ax (A)The population at time t, (3) , can be easily shown to be of the form:x(t) - x(0) eat (5)The constant a, assumed to be nonnegative, is called the rate of growth.If withdrawal at the rate u(t) takes place, the population at time t canbe computed to be:x(t) - x(0) eat -" Jea<trT> U<and the total quantitity of species harvested over the time interval[0, t] is •h(t) - u(t) dt (7)If the producing sector's technology permits a maximum rate ofharvesting equal to u , and if no regulations exist to limit har-max' °vesting, then it is easy to determine the time t at which extinctionmaxof the species will occur if the producing sector harvests at the rateu „. This is:maxt - i In [Umax/m ] (8)max a u ax(0) JmaxAThe total quantity harvested is obtained as4Naturally here we assume that u - ax(0) >^ 0.max) '.:•... .'-:. .•u u. max , r max , /oxh » In [ tttt-J (9)max a u ax(0)maxwe note that, from (8) and (9) we have3t3u - u (u - ax(0)) ° U0;max max maxandmax -x(0),1, max ^ rt ,,.,vlu 5 ax(0) + a m u ^IxT0) < ° <U)max max maxwhich essentially confirms well known intuitive conclusions that thebetter the technology (i.e., the higher u ), the faster extinctionwill occur and the smaller the total harvest.From a conservational point of view, as well as from an economi-cal point of view, this of course is not a desirable policy. Unfor-tunately, the production industry, which in many cases is interested inshort term profit maximization, may not worry about these effects. Thiswill necessitate regulation which can be applied by imposing:i) a maximum limit of harvesting rate, u, which is generallysmaller than the maximum technologically feasible rateu (i.e.,u<u ), andmax max11) a required species population , x, left over at the endof the harvesting period T. (In short, the problem that the production sector faces is:given thatx ax - u, x(0) = x given, (12)odetermine a harvesting policy u(t) , which meets <_ u(t) < u, over thetime horizon [0, T], such that the total harvest5 — aXNaturally x must not exceed x(T) « x(0) e which is the species. population if no harvesting takes place..''.''ni',-,f.- r.Jh - ( u(t) dt (13)is maximized; and such that the population at T meets the requirementx(T) = x. In (12), we have chosen a species with exponential growth,(f(x) ax) simply because the analysis in this case is tractableanalytically. The results in general, however, are not limited tothis case and can account for any other form of growth such as quad-ratic, Volterra, etc. In subsequent papers we will show how this canbe done.The problem that the regulatory agency faces is how to chooseu and x in order to insure that extinction of the species will notoccur while at the same time helping the producing sector to maximizeits harvest.Optimal Harvesting Policy and PropertiesIn this section we determine an optimal harvesting policy which:i) satisfies <_ u(t) <_ uii) maximizes the harvest (13) , andill) result in x(T) = x.The standard solution procedure, as well known in optimal controltheory [4], is to define a Hamiltonian function:H(x,u,X) - u + A (ax - u) (14)where X is a Lagrange multiplier. The necessary conditions for opti-mality are [4]:and* ~ I? = ~Xa > x <°) cnd X <T > free6 :' <15)H(x*,u*,X*) > H(x*,u,X*) (16)«Note that X(0) and X(T) depend implicitly on x(0) and are not par-ameters that we choose.If we assume that possibility 2 holds with the switching timet satisfying <^ t £ T, then the other two possibilities will becomes sa special case with t * T for possibility 1 and t for possibility 3.s sIn order to meet the population requirement x(T) = x, we musthaveT— aT , a(T-t) - , /in .x «= x e - . / e udt (19)o t -sor* - x eaT+ I (1 - •*<**>) (20)o aThis gives a harvest period of- u + a(x e - x)I-t »Jl» [ 2 j (21)s a —uThe total harvest resulting from this policy is equal to:— u + a(x e - x)h* - - In [ ] (22)a —uAnd the species population over the time horizon [0, T] can be computedto follow:atx e for <_ t <^ t (no harvesting period)u -a(T-t) u (23)(x ) e v +7 for t < t < T (harvesting period)a as ""It is interesting to note that from (21) and (22) we have:3(T-t ) -(x eaT-x)_£_ „ __o ___ < Q (24)3u u(u + a(x e -x))oand-,< (25)a(T-tg ) -13x u + a(x e -x)oand thatand— j , aT —. aT —01,* i u + a ^x^e~x > x„e ~xfflfi. . I ta { _2 j _^ > o (26).— a — — . , ai —»3u u u + a(x e -x)o3h*_ -u_ <q (2?)3x u + a(x e -x)oThus, the larger u, the smaller the harvesting period but thelarger the total harvest; and the larger x, the smaller the harvestingperiod and the smaller the total harvest. Figures 4 and 5 illustratethe population trajectories according to (23) for different values ofx and u«The optimal harvesting policy (17) can be expressed in termsof the population level x(t) (i.e., in feedback form) by making use of(21). This gives:u*(t) = u St(xh - x(t)) (28)where_ aT —u + a(x e -x)*h-—=TS xo . <29 >u eThe interpretation of this law is that in order to maximize the totalcatch, harvesting should not start until the population reaches a certainlevel x, which we will refer to as the harvesting level.— aTThe special case of possibility 1 (Fig. 1) occurs if x x eIn this case harvesting is restricted at zero level throughout [0, T],and the species is left to growth according to its biological growth law(2). 'The special case of possibility 3 (Fig. 3) occurs if x xand u = ax . In this case harvesting will take place during all ofthe time horizon and is equal to the quantity of species in excess of/.. I'.••:• \10x(t)ii•ST. / it e////u fixed/x2> XlI11J«1x2Fig. 4. Population trajectories with u fixed and x_ > x..x(t)x fixedu2>ux/////Sx eoaTFig. 5.'1 °2Population trajectories with x fixed and u- > u, .,*,\11x . Thus the species will not grow with time and the total harvest isoequal to h* » ax T.Thus from the point of view of the regulatory agency, it wouldseem that in order to help the producing sector maximize its harvest,it should allow for the maximum possible harvesting rate to take place,i.e., u = u . This in essence means that it should not impose any* max r 'regulation on the harvesting rate and allow the producing sector toharvest at its technologically maximum rate. However, from a conserva-tional point of view, x may be selected so as to allow a certain "target"growth rate to take place. In other words, if x is chosen according tox - x eBT (with 6 < a) (30)o*it would mean that at the end of the time horizon the population wouldhave grown according to the target growth law (Fig. 6):x - Sx, x(0) - x (31)oThis would insure conservation of the species. Stiff penalties howevershould be imposed If the requirement x(T) = x is not met by the producer.Conclusions and CommentsIn this paper we have developed a dynamic model for renewableresource economics. We have determined an optimal harvesting policywhich meets two conservational requirements. First, a maximum allowablerate of harvest and second, a terminal constraint on the species popula-tion at the end of the time horizon. It was shown that the optimalpolicy, which maximizes the total harvest, consists of a no harvestperiod followed by a period during which harvesting is done at themaximum allowable rate. The length of each of these periods is afunction of the maximum allowable rate of harvest u and the terminalrequirement on the species population x. The effects of variations•12*(t)KActual No HarvestingGrowth TrajeoioryTarget GrowthTraj ectorActual Trajectory withMaximum Harvestingover [t ,T]5_-/ v.x eoaT"Vx eo8TNo Harvesting Period s Harvesting PeriodTFig. 6. Actual vs. target growth population trajectories.13of u and x on the harvesting period and the maximum harvest have beendetermined. It was shown that a larger x will cause a decrease in thelength of the harvest period and in the total maximum harvest, and thata larger u will cause a decrease in the length of the harvest periodbut an increase in the total maximum allowable harvest. Answers tothe four questions raised in the intorduction have also been given.In summary, the optimal harvesting policy has been given in (28); themaximum sustainable yield is not optimal because it leads to earlyextinction of the species and does not result in maximum total harvest;Extinction of the species will occur at a time t given by (8) ; andfinally it was shown that regulation on the rate of withdrawal may notbe necessary as long as regulations on the terminal population areimposed. The rate of withdrawal is then set by the maximum technologi-gcally feasible rate. This will automatically limit the harvestingperiod according to the expression given in (21) . Thus an easy way ofmonitoring this policy would be to forbid harvesting over the periodA final comment, which is of interest, is with regards to caseswhere the time horizon [0, T] is very large such that the resulting noharvest period may be too long for the optimal policy to be economicallydesirable. In this case, a long no-harvest period may be undersirablefrom the consumer point of view. A possible implementation of theoptimal policy derived in this paper would be to devide the interval[0, T] into several smaller intervals [t., t....] for i » 0,...,N-1 withIf such rate does not exist, or is infinitely large, then naturallya maximum rate u" should be imposed.••.->•*•-.11l.!14t " and t„ T. The population requirement at the end of each ofthese intervals may be imposed to be in the formx. x e * , i 1,. .. ,Ni owhere is the desired "target growth rate". The harvest period isthen spread over [0, I], and the species trajectory will follow a pathas shown in figure (7) . This policy corresponds to a seasonal harvest-.ing law, (such as the case of shrimp harvesting in the Gulf of Mexico),where for instance each year, harvesting is allowed only during a certainseason known as the "hunting or fishing" season.15aoN•HMO£25(0•o mo 00•H MM <e0J t-iP-iM4-> o(0 «H<uB us t-4Oa.00c•H«n0)>b9JXi<Mo« c"d oo Hi-i uu ffl0) 4-1(U aa>u iCD •oV i-lE 8"« MSCO *5B r^60.16References1 F.M. Peterson and A.C. Fisher, "The Economics of NaturalResources," Univ. of Maryland, Feb. 1976, tnimeod.2 J. P. Quirk and V.L. Smith, "Dynamic Models of Fishing ,"in Economics of Fisheries Management, A.D. Scott (ed.),Univ. of British Columbia, 1970, pp. 3-32.3 V.L. Smith, "Economics of Production from Natural Resources,"Amer. Econ. Rev ., June 1968, Vol. 58, pp. 727-755.4 M. Athans and P. Falb, Optimal Control , N.Y. , McGraw-Hill,1966.

Optimal harvesting policies and regulations for renewable resources / BEBR No. 466

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