UNIVERSITY Of •^ STACKS Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/dynamictheoryoff441taka coe- ^ Faculty Working Papers DYNAMIC THEORY OF FISHERIES ECONOMICS — II; DIFFERENTIAL GAME THEORETIC APPROACH T. Takayama #441 College of Commerce and Business Administration University of Illinois at U r ba n a - C ha m p a ig n FACULTY WORKING PAPERS College of Commerce and Business Admlnistratix>n University of Illinois at Urbana-Champaign October 12, 1977 DYNAMIC THEORY OF FISHERIES ECONOMICS — II; DIFFERENTIAL GAME THEORETIC APPROACH T. Takayama #441 ■i l-i ;•.,:,»... •: '•'■•■:',■ ■i.'.iK October 2nd, 1977 DYNAMIC THEORY OF FISHERIES ECONOMICS— II; DIFFERENTIAL GAME THEORETIC APPROACH T. Takayama University of Illinois Abstract A two countries-single speriss dynamic optimizacion problem in fisheries is formulated in differential game theoretic framework. The Cournot-Nash solution concept is introduced and two strategies in arriving at the Cournot-Nash solutions; are discussed. These strategies are (1) closed-loop catch strategy and (2) open- loop catch strategy. A quadratic benefit function is then used as the objective function of each country that is to be maximized subject to a linear population-catch dynamics. Th'^ solutions are then derived for each catch strategy and their implications are d^''- cussed. The major conclusions are as follows: (i) The stable optimal catches show that each country must regulate the tonal catch of her fleet, (ii) The mash-size snould also be regulated. (XXI) The opeu-locp optimal catch strategies exist only when two countries' future discount rates are the same, (iv) The higher the future discount rates, the more fish vri.ll be caught now and in the near future, and the slower the conver- gence of this fish population to the desired level. October 2nd, 1977 DYNAMIC THEORY OF FISHERIES ECONOMICS— II: DIFFERENTIAL GAME THEORETIC APPROACH* T. Takayama** Introduction Even after the imposition of the two hundred mile territorial waters limit by many countries, there are many fish species being caught in the open international waters. The two hundred mile territorial waters reg;ulation is a revelation of inherent conflicts among a number of countries whose: fishing fleets are after one or more fish species of their common interest and need. Of course, the regulation does not solve all the problems of the host country in relation to the desired catch o? fish species, but alleviates the intermediate long-run burden of driving the fich population to an undesirably low level close to extinction [2]. The cod war between the United Kingdom and Iceland is another example of international conflicts in the field of fisheries economies. Can they resolve this type of conflict in any manner whatsoever? This is an important question in both theory and politics. In this paper we plan to partially answer this question from a theoretical point of view. As a natural axtension of the dynamic optimization approach related to fisheries economics [1, 10, 11], we apply the differential game theory to a two country-one fish species situation. In section 1, the fish population dynamics and general properties of the differential game model applicable to fisheries economics are discussed. Qua- dratic objective fuiictionals and linear fish population dynamics cases are then solved for the Cournotr-Nash equilibrium and various implications of the solution are discussed in section 2. In the last section we discussed future research *This project is partially supported by the Ford Foundation Grant, lEO #750-0111. **Professor of Economics at University of Illinois, Champaign-Urbana, Illi- nois 61801. He is grateful to his colleagues, professors. Royal Brandis, H. Brems, T. Sawa, M. Simaan for their encouragement in this work. He alone is responsible for any error in this paper. ■'-;..! rJfe •J li! directions in this field and conclude with some positive notes for future research in the near future. 1. Differential Game Formulation of Fisheries Economic (Conflict) Problem The problem the fisheries economicsts face is that of determining and recommending the fishing intensity that will maximize the economic value to the consuming societies and also maximize the producers' surplus at a level of pro- duction in perpetuity. In the previous paper, we took the stand that the con- suming societies can be looked upon as one society with one market for a fish species and also the producers' supply response can be represented by a single market supply function. In this paper, we deal with two countries or consuming societies pursuing their individual economic value or goal such as a maximum wel- fare or profit or whatnot. The two producer societies are also separated in the sense that the cost function of each country is expressed by its own currency unit. We also assume in this paper that there is no trade of fresh fish or pro- cessed fish products between the two countries. Thus, country 1 tries to maximize its welfare from the consumption and production (catch) of a fish species. The same or different objective may hold true for country 2. For simplicity's sake, let us write this objective function as a function of the (recruitable) fish population, p(t), at time t (^ 0) and the intensity of catch by the ith country fleet x.(t) at time t (^0); that is, 1 /-S (1.1) W^ = / e'''i^W^(p(t),x^(t))dt where r denotes the rate of the future discount by country i, i = 1,2. W^(p(t), x(t)) denotes the welfare or benefit or profit or any other objective that the ith country tries to maximize, which depends on the fish popula- tion (to a lesser extent) and the catch of fish by the country's fleet for its own consuming society. Thus far, the objective function of each country is completely indivi- dualized. This is a clear-cut departure that the differential game theoretical approach makes from the ordinary optimization approach. There is, however, one common object between them that is to be carefully observed; that is, the ocean in which the population of a fish species of their interest grows or declines in relation to the fishing intensities of fishermen of the two countries. The fish population in this paper is considered as a directly observable state variable, and the catch by the fishermen in each country is considered as a control vari- able; there are naturally two control variables in a two countries-one fish species case. The fish population (and catch interaction) dynamics is written as (1.2) p(t) = f(p(t), Xj^(t),t), te[0, T) where p(t) is defined as before, (recruitable) fish and population at time t, state variable; x(t) denotes total intensity of catch by two countries; t denotes actual time; and T is the terminal time greater than zero and can be finite or infinite. In our two countries model, the total catch intensity x(t) can be divided into two parts as follows : (1.3) x(t) = Xj^(t) + X2(t) where x (t) is defined in relation to (1.1) and denotes the Intensity X of catch by the ith country fishermen, at time t, i = 1,2. The simplest dynamics we can consider is the linear first-order dynamics such as (1.4) p = a + bp - X where we assume that a and b are constants and most likely (1.5) a ^ 0 and b > 0 . The Figure 1 below shows this catch fish population dynamics. -a/b p Figure 1: Catch-fish population djmamics; linear case. A nonlinear dynamic case is also of great theoretical interest. For instance, the Lotka type or Voltera type quadratic dynamics [ 5 ] may be important in application. However, we confine ourselves, in deriving some practical infer- ences, to the linear dynamics case, (1.4), in this paper. In Figure 1, the steady-state locus of (p, x) combinations is shown by p=0=a+bp-x equation. When catch is zero, the minimum sustainable popula- tion (below which the species become extinct) is -(a/b). To the right of the "^The dynamic identification, t of p(t), x(t), etc., will be omitted, unless otherwise stated in the following development. ii.1 ;e">lj3r d •>:{/■.) .) I'x . q) ^o line, the population (stock) to catch relationship is favorable to increase in fish population. To the left of the line, due to relatively heavy catches, the fish population tends to be zero. In formulating the differential game problem within the framework of two countries and one fish species, let us employ as general a formulation as possible. The problem is clearly divided into the following two subproblems: Fisheries Differential Game Problem (F D G P) ; (i) Find x (t) that maximizes (ii) Find x (t) that maximizes W in tl.l) subject to (1.2) W, in II. 1) subject to (1.2) for t£ [0, T). for t£[0, T) . This problem looks very much like a djmamic duopoly problem formulated in differential game theoretic framework [7, 8]. The substantial differences between these formulations are: (a) the duopolists are assumed to pursue the same kind of objective, profit in case of [7, 8], while in this case the objec- tives can be (i) profit in different monetary terms, or (ii) satisfaction, or (iii) any combination thereof, or (iv) any other objective one country claims to attain, and (b) the duopolist control their own supply quantity looking at the market dynamics represented by the price change, while in this case the intensity of catch is controlled by the fishing industry in each country looking at the fish population d5mamics. By carefully checking the objective function, W , (1.1), the fish population dynamics, (1,2), and the total and individual country catch rela- tionship, (1.3), we can conclude that the objective function depends only on Xj^(t) and X2(t); (1.6) W^ = W^(x^(t), X2(t)), i = 1, 2. (n As in static duopoly situations there are many different strategies the two parties can employ; there may be equally many or more strategies our dynamic controllers (nations) can assume. In this paper, we confine ourselves to the following two cases: (i) closed-loop strategy, and (ii) open-loop strategy. The close-loop strategy is the one that takes the fish population information p(t) as time goes on and adjusts its catch to it. On the other hand, the open-loop strategy is the one that looks at the initial fish population and then decides the whole course of catch intensity decisions as a function of time t only. Of course, in optimal control theory, where there is only one controller or decision maker, there is no difference between the outcomes of these two optimally (in some sense) chosen strategies. However, these two strategies are known to bring about two quite different catch intensity histories for these two countries [ 9], as will be revealed later in the next section. From the game point of view, we deal only with the so-called "Coumot- Nash" equilirim game [3] , that can be defined as a pair of catch histories (x*^(t, p(t)), x*(t, p(t)) for the closed-loop strategy, or (x*Q(t), x^^Ct)) for the open-loop strategy that satisfies the following conditions: W^(x*^(t, p(t)), x*^(t, p(t)) ^ (1.7.1) (1.7.2) W^(x^^(t, p(t)), x*^(t, p(t)) and W2(x*^(t, p(t)), x*^(t, p(t)) ^ W2(x*^(t, p(t)), X2^(t, p(t)). or W^(x*Q(t), x*Q(t)) ^ \^^10^^>' ^20(^>) W2(x*Q(t), x*Q(t)) ^ W2(x*Q(t), X2Q(t)) , and (1.7.1) states that the pair of catch histories using the closed-loop strategy by both countries, (x* (t, p(t)), X2c^*^' P(t))) is better or equally as good as any other closed-loop catch strategies that this country can assume when another country sticks to the star -marked closed-loop strategy. (1.7.2) similarly states that the pair of catch strategies using the open- loop strategy by both countries, (x* (t), x* (t)) is better or equally as good as any other open-loop catch strategies that this country can employ when another country sticks to the star-marked open-loop strategy. In the sense stated above (1.7.1) and (1.7.2) are similar to static Cournot or Nash equilibrium. There are other dynamic games such as the dynamic Stackelberg game [3 ] , but due to its operational difficulty we confine ourselves to this Cournot-Nash differential game in this paper. The necessary conditions for optimality of dynamic Cournot-Nash (hereafter "Cournot" only) catch strategy in closed-loop form, can be derived as follows. The Hamil onian of the Fisheries Differential Game Problem (FDGP) can be written as (1.8) H^ = W^(x^(t, p(t)), X2(t, p(t))) + A^(i:)f(p(t), x^(t, p(t)) + X2,(t, p(t)). for i = 1,2. Following Ho [3], Starr and Ho [8], and Simaan and Takayama [7], the following coupled necessary conditions are derived: Country 1 ■ p = f(p(t), x^(t,p(t)) + X2(t, p(t))), p(o) = p^ (1.9) (i) ^1 = ^iH(^> - h^^^^i ^ M, vi) ^ + X, (t) |i =0 3x. 1 dx^ X (T) = 0 or lim e"^^X(t) = 0. •^ ^-4 1 (1.9) -continued (1.9) (ii) Country 2 p = f(p(t), x^(t, p(t)) + x^it, p(t)), p(o) = P, / 3f _^ 9f 3xA ^2 " ^2'*'2^^^ " '''2^^^ f2 = A2(t)||=0 9x„ 2 dx- X (T) = 0 or lim e"^2^ X(t) = 0. t^co 2 9f ^^1 Here In the second condition for each country we find a term — for the dX, dp first country and ^ — -r^l for the second country (these are like "conjectural 9x^ 9p variation term" in static Cournat case) representing "interdependence" or "inter- connectedness" of their actions. Of course, each country can follow their strategies that ignores this interdependence; that is, each assumes that the other country's strategy is deter- mined by the initial fish population, p , and time, t, only. Then, by constructing the Hamiltonian, the following necessary conditions result (1.10) H°= W.(x-(t), x_(t)) + Y.(t)f(p(t), x/t) + X (t)) 1 1 1 .id X J- ^ for i = 1,2. Country 1 (1.11) (1) p = f(p(t), x^(t) + X2(t)), p(0), p(o) = p^ 9f V '^i \^'^ "■ \^'^ 9F |^l + Y,(t)f =0 dx^ 1 ox^ Y(T) = 0 or lim e"^l^Y'^(t) = 0 (1.11) — continued (1.11) (ii) p = f(p(t), x^(t) + X2(t)), p(o) 3f = P, Y2 = r^ Y2(t) + Y2(t) 3p ?2 + Y2 (t) If =0 ax„ I OX- Y^(T) = 0 or lim e ^l^Xit) = 0 9f 3x Here in this open-loop catch strategy case, the interaction term -r— —^r^ is ox, dp missing from the second condition for each country. In the next section, we will trace out the consequence of this difference in detail. 2. Quadratic Benefit — Linear Population Dynamics Case The main purpose of explicibly formulating FDGP in the quadratic benefit- linear fish population dynamics framework is to firmly quantify optimal strategies and resultant fish population dynamics, and consequently to derive policy impli- cations from the results. In this section we deal with the following two cases: Case 1: (Competitive Markets — Close-Loop Game) The social benefit function is defined as the over-time integral of the instantaneous (we omit this term hereafter) consumers' and producers' surplus accruing from the fish market , and the two countries use the closed-loop catch strategy. Case 2 : (Competitive Markets — Open-Loop Game) The social benefit is the same as defined in Case 1, but the two countries use the open-loop catch strategy. Another pair of cases dealing with a monopolist controlling the domestic Tiarket in each country can be solved easily, but we will not go into them in this paper. 10 There are other combinatorial possibilities for our case studies of the Coumot game, and we plan to do some exhaustive work later in this area. 2.1. Competititve Markets — Closed-Loop Catch Strategy Differential Game Following Sarauelson [6] and Takayaraa [10], let us express the consumers' benefit functional as the time integral over [0, T) , where T could be infinity, of the integral of the instantaneous demand (felicity) function: (2.1) Consumers' Benefit in the ith country = T -r t 12 CB. S / e i [a.x, - tt 3. x. ]dt 1 / 1 i 2 1 1 ./ex [a.x^ - 2 *'o for i = 1,2. where the instantaneous demand function of the ith country for the fish species is given as (2.2) P^, = a. = 6. X., i = 1,2 di X XX We assume in the two cases we handle in this section, that the catch x. is consumed without any wastage in the process or the consumption is measured in the fresh fish unit. The total cost of catching x, units of fish is expressed as the over- time integral of the integral instantaneous producers' supply function. T (2.3) TC. E f e'^-i^Lyx., + ^ e,x/]dt /-r .t r , 1 rN 2i e X [^x.. + - e.x. ] where the instantenous producers' supply function is expressed as 11 (2.4) P . = y. + 0. x.(t) SX 1 11 for i = 1,2, The social benefit function of the individual countries can be expressed as the difference between the consumers' benefit, CB , and the total cost of catch, TC. ; that is i SB.(x (p(t)) = CB. - TC E /* e"''i''[(a. - y.)x. - y (6. + 0,)x? (2.5) SB^.(x, (p(t)) = CB, - TC, E / e ^i [ (a, - y,)x, - ^ (6, + 0,)x, ]dt o for 1 = 1,2. Now, let us define our competitive markets — closed-loop catch strategy differential game as follows: CMCLDG: Find the d5mamic paths of the pair (x* (p(t)), x* (p(t))) that satisfy the following Cournot conditions SB^(x*^(p(t)) ^ SB^(x^^(p(t))) and SB2(x*^(p(t)) ^ SB2(x2^(p(t))) subject to: (1.4.C) p = a + b p -[x^^(p(t)) + X2^(p(t))]. The Hamiltonianfor this problem is given as (2.6) Hf = (a. - ii.)x. -^(3. +0.)x^ 1 1 1 ic 2 1 1 ic + A.^ (a+bp - K^^ -x^^). The necessary conditions for optimality of the pair of catches are: 12 (2.7)-, (i) Cii) (a) p = 2 + bp - x^^ - x^^, p(o) = p^ (b) \, — (b - r^ - |f2c)X^^ (c) (Ci^ - ti^) (3, +0i)xie-\c = ° (d) A (T) = C or lim e ^1*^ ^0^,^^) = 0 (a) 3 + bp - X Ic ^'2c' P ^°^ = Po ^(b) A^^ = (-b -r^ -|fl)A2^ (c) (a„ - ^2) - (^2 + Q2)^'2c - ^2c = ° (d) A^ (T) = 0 or lira e"^2^A^ (t) = 0. lor the siipjlicity of Cv^n^pucation, in the following we assume ti at (2.8) a^ = a. - a Bt^ = 02 s 6 'i = h = y e, = G. 5 0 whicli is ^i^aivaien^ to siaCriig that i:be t^wo contries possess the same demand and cost structures. In i;a::.s case, aftar scxe calculations involving the solving of 2 Ricatti equations, we get the following solutions. The first pair are (2.9) x^*(pu)) = x|^(p(t)) = 1^ whicn are not df,sirable or stable solutions for these two countries unless the fish population at the initial period was already (2.10) -,* = L 2 + e - a The detailed d'?rivation of the solutions is given in the Appendix at the end of this paper. 13 (2.11) - a A pair of optimal stable catch strategies exist and are (a) X* (D(t)^= g - y _ (2b-2r^+r?) 2 (a - y) , 2b-2ri+r9 .. + 2 — "P^^J (b) X* (p(t) = " - ^ - (2b-2r^+rj) l(a - P) ^2b:ir2±rip(t) . - a The fish population moves along the following converging path. (2.12) (2.13) = (b-ri -r2) 3b -{^^} 2(a - u) 3 + e P(t) , - a where we assume (i) b - r^ - r^ > 0 (U, H|^.,,„ The fish population converges to (2.14) 1 r2(a - I b L B + G - P) 0 - a > 0 Due to (2.12), we find the ultimate fish population as t tencfg infinity as to (2 •^» PJ - I ["^^ -] > 0- Thus, the fish population history accompanied by the optimal catch (2.11) is given as 14 (2.16) P^(t) = (p^ p*)e + p* ^c ' c as From (2.11) and (2.1A) we get the ultimate optimal catch intensity (2.17) -!e = ^fc=fTi • The following comparative d3mamics conclusions can be derived from the above results. The discount rates do not affect the optimal target (t -*<») catches or the target population; that is (2.18) .^_^^o ^,^=0, 1 = 1, 2. 9r. 1 3r, dr, Given the social discount rate of country 1, r , an increase of the discount rate of country 2, r , will decrease the catch intensity of country 1 at any level of the fish population less than p* , and increase that of country 2. Similarly, an increase of the discount rate of r^ , given that of country 2, r„ , will increase the catch intensity of country 1, and decrease that of country 2 at any p(t) < p*. (2.19) In other words, we have 9x (p(t)) 9r. = ¥^^'^ - P^> 3x^^(p(t)) 2 3r. 1 3^ p(t)) for i, j = 1,2. As a result, if covmtry 1 becomes more conscientious about the future generations than country 2 in relation to fish consumption and production, r^ ' > r^ , country 1 becomes more conservation-oriented than before in order to attain J .0 J. rfjis6ft'j';i.'li t:nti . ' 15 the maximum pcosible social baneflt in the long future and in perpetuity. Thi.5 conclusion establishes the principle of conservation for renewable resources including fish, deer, forestry, and others. Also this sfTiie prirclple has a potential element that makes one country, say country 1, legislate the two hundred mile territorial waters limit if she sees foreign vessels exploit her territorial fishing beds to an undesirable extent . Traditional contro/ersies over whether or not some catch control regula- tion is necessary can be answered in this two country-one fish species framework. As in the single country-single species case [10], we can conclude that the total catch tor each country stipulated by (2.il) is necessary for both countries to en^^oy ultimate maximum social benefit. The reason is rather straight- forward. Since, hy (2.7) (i,c) and (ii.c), the marginal social value of a unit of fisti sold, in ta-- narket always excaads the marginal individual cost of catching the same by A. 'D(t/)> 0 cr X. (t) > 0 for p < p* or p < p^ , individual fish- j -, f. .. V . / ro "^o c o o ericin wou; d like ro fish moxe, if left to follow their own individual profit maximisation p-rinclples, thar. ths socially opti&al (2.11). This overcatch rep- resented by v.2.9) at each mom.ir t of tiKe. when p(t)< p* or p* , must be stopped. An internatiCiiax regulatici-s controlling the catches by both countries must be worked out in thl^ ca^e. Kow to acc-omplish this lies outside the scope of this 3 paper . Mesh-cizt regulation arguments have been carried out in a static frame- work in [1, 11], and in dynamic one-ccuntry on3-fish species framework in [10]. Following [10], one can develop similar arguments for some kind of mesh regulations. The reason is simple enough: the iniormatd on on the optimum ri.esh-size for the industry and the consuroing societies, if it exists, is external to the individual For seme arguments in this direction, the reader is referred to [1, 11]. 16 fishermen and the industry concerned in each country. 2.2. Competitive Markets-Open-Loop Catch Strategy Differential Game If the two countries decide to take the open-loop strategy in the compe- titive market within their national boundary, the objective fimctions remain to be the same as (2.5) for i = 1,2. The population dynamis is the same as before, (1,4), with (2.20) X = x^Q(t) +X20 (t) The problem can be defined as follows : CMOLDG (Competitive-Markets-Open-Loop Catch Strategy Differential Game) Find the dynamic paths of the pair (x* (t), x*^(t)) that satisfy the 20 following conditions SB^(x*Q(t)) ^ SB^(x^Q(t)) and SB2(x*Q(t)) ^ SB2(x2Q(t)) subject to (1.4.0) p = a + bp -C^^qM + X2Q(t)) The Hamiltonian for this problem is given as (2.21) H. = (a. - y.)x. 1 X 1 xo + A. (a + bp xo for i = 1,2. v(B. + G.)x^ X xo ""lO ~ ""20^ (2.22) The necessary conditions for optimality of the pair of catches are: (i) (a) p = a + bp - x^Q - x^Q, p(o) = p^ (b) X 10 •(b - r)A 10 (c) (a^ - u-,^) = (3-1^ + G^) x^Q - \o ^ ^ (d) A^q(T) = 0, lim e -" X^QCt) = 0 17 (2 . 22) — continued (a) p = a + bp - x^^ - x^Q, p(o) = p^ (ii) (b) A^g = -(b-r)X2o (c) (a^ - ■.^■) - (B2 + 0,)x,o - X^o = 0 -r t (d) X^q(T) = 0, lim c ^ X^^(t) = 0 I As in the previous case. Case 1, wt assume for simplicity's sak^, that (2.8) holds; the two countries are the same in their demand and supply structure. In this case, hov/ever, there is no optimal stable (convergent) open- loop catch strategies unless zhe following condition holds: (2.23) r = r„ (-■ v herearter) . The implicaticn oj; this condition is quite Interesting and suggestive. If the two countries with the same demand and supply (cost) structure engaging fisiiing the -jana fish ^p-acies in the common ocean (fishing banks), and if one country's future disccunt rate is different from the other, these countries can- not find any reasonable catch strategy that will eventually bring them to their target catch and fish population defined and discussed in the previous subsection. Wrat luakes the opef-loo^.^ catch stratrgy case so diagonally different from the closec'-locp ca^ch strategy game? In the case of the closed-loop case, the parties involved observe carefully the history of fish population over time, p(t). This makes ons party's catch responsive to the other party's catch through the observed fish population. Thus, each party responds to each other sensitively, and this interaction, brings about an optimum catch strategy to each party (as long as (2.13) holds). However, in the case of the open-loop strategy game, each party decides that she can determine her o',jn optimal strategy on the basis of the initial fish population and time. Thus, unless complete symmetry exists 18 in their environment, they cannot come up with any optimal catch strategy. After superimposing (2.8) and (2.23), we get the following results. A pair of singular solutions exist and are: (2.24) x^*(t) = X2*(t) = f^ . They are not desirable or stable solutions unless the fish population is already at the level represented by (2.10). Optimal and stable, open-loop catch strategies exist and are (2.25) x*Q(t) = x*Q, f^\ ct - y /2b-r \ g - y | , /2b - r ^ /. >, The fish population dynamics using the open-loop catch strategies is then (2.26) P =\-^ I'a a. q' - a - (b-r)p ?, ,/b-rY2(a - v) P \ b ^ 6 + 0 (2.27) (i) b-r > 0 v... 2(a - y) , . n This population converges to (2.28) p; = i plVe^ - a] which is exactly the same as the closed-loop desired or target population, (2.14). (2.29) The ultimate optimal catch in perpetuity is given by 10 ^20 S + 0 as expected . The optimal time profile of the fish population in this case is given as •(b-r)t (2.30) p^(t) = (p^ - pj )e + P*, 19 As a consequence, the optimal open-loop catch strategy brings a much faster convergence to the common target fish population ifzca -y) _ J. b |_ B + 0 J At the same time, for a given initial fish population, the optimal open- loop catch is always smaller than the optimal closed-loop counterpart as the following computation shows: (2.31) x*^(p(0))- x*^(0) (2b - r)(D ■ -o- ^H 0, for p(0) < p* = p* = p* which is shown in Figure 2 below. X * X = c X* o ( X* ^ c = X* + ic ■X* ■^ic 2 "" ~ ^ B + y 0 = ■ X* + lO X* = xo X* O ) p = 0 Zx*^'tp(t))// Ac Zx?_(t) xo 'Ao p« Figure 2. Optimal catch strategies^x* (p(t)) andJxf-Ct), As a consequence, the open-loop catch strategy, assuming of course that the two countries use the same future discount rate, brings a much faster con- vergence to the target fish population. <«1i Ob .3-iKt,-i; I .»ic 0 = (< A,; , -^.q X 51 (' \i'. \ 20 i U(a -p) _ 1 b L B + © J" Thus, declaring that both countries employ the same future discount rate and stipulate and monitor the optimal history of catch each country is to follow, may be a faster way to reach the target population and catch. It is also easy to conclude that along the stable convergent linear paths, A E and A E, the catches and the fish population increase till they reach the c o long-run equilibrium point, the Cournot-Nash equilibrium point, if these two countries employ the closed-loop and open-loop catch strategy, respectively (provided p < p* = p* ) . The future discount rate has no effect on the target catch or fish popu- lation. However, as r increases, the optimal intial catch will increase due to the fact that (2.32) ^'^io'^"'' 1 r * r ^^s^ • i -y T^ = 2 (P* - P(o))>0, 1 = 1,2, as long as p(o) < p*. This is consistent with the larger future discount rate or the assertion of the "more now, less later" attitude of a society. The same conclusions as those for the closed-loop case apply to this case as to the catch control and mesh-size issues. Conclusion In this paper, we have formulated the fisheries economies problem involving two countries and one fish species in a differential game framework. A quadratic social benefit and linear fish-catch dynamics problem is then solved for the Cournot-Nash differential game solutions. Obviously, there are at least two strategies to play within this game framework, and the solutions are derived 21 for the closed-loop catch and open-loop catch strategies. We find that, if the demand and (cost) supply structures of the two countries are exactly the same, the closed-loop catch strategy generates a pair of optimal catch paths to follow for any future discount rate of indivi- dual countries as long as b - r^ ~ ^9 '**-'> while the open-loop counterpart pro- duces a pair of twin catch paths to follow only if the future discount rates of individual countries are exactly the Game. This is an outstanding feature of this renewable resource differential game. In both cases, the increasing appreciation of the present over the future increases the optimal catch at present, slows down the process of con- vergence of the fish population to the target. In the case of the closed-loop catch strategy, an increase of country I's future discount rate, r , increases the initial optimal catch of country 1, while country 2 accepts the fact that r has increased and curtails the optimal catch accordingly (see (2.19)). VThether country 2 is willing to follow this course of action may not be an issue at all. Rather, that r. and r„ are histori- cally given and accepted as such must be the basic framework of our model. Of course, if the sum of the two countries' discount rates, r ■*" r„ , exceed the rate of increase of the fish population, b, there will not be any economically meaningful solution for our closed-loop strategy game. While the open-loop strategy game is much less restrictive in this respect since that the common rate of future discount, r, must be less than the growth rate of the fish population, b, is the only requirement. However, since the condition for the existence of optimal catches depends on the equality of the two rates, the open-loop catch strategy may be considered too restrictive. A dynamic cheory of fisheries economies we have developed so far has estab- lished a conservation principle that a scarce renewable resource such as a fish 22 species for human consumption requires a total catch control or a control over the catch by the fishing fleets of every individual country. Also, we argue that some workable mesh-size regulation must be enforced since individual fishermen have no way of knowing what mesh-size is optimal from the consuming societies' point of view (as well as their own in the long-run), that is, this information is external to both consumers and fishermen. There are many directions to go and many topics to cover in the future research in the area of renewable resources economics. A natural extension of our two countries-single species formulation is a two countries-single species formulation with different demand and supply structures. Also, finite time horizon problems should be solved for more practical problems. A system of matrix Ricatti equations must be solved to obtain any meaningful quantitative results from a multiple species-two countries formulation. In this direction, computer-based algorithms are absolutely necessary, and many diverse and practi- cal problems may be solved efficiently. Another extension is to solve more than two countries-one species model by effectively utilizing our model developed here along with the concept of coalitions. An ultimate extension is to formulate and solve multiple countries- multiple species problems in generality, but quantitatively. Fish population-catch djmamics can be made nonlinear to attain generality. One observation dealing with a quadratic population dynamics case is already dis- cussed elsewhere [10], and may shed some new light on the predator-prey dynamics [5] widely accepted by the economics profession for some time. Econoraetrically, estimation of parameters in population-catch (or forestry growth and harvesting) dynamics is a challenging field. This is a field in which various interdisciplinary activities will prove most productive. Cooperative 23 >f forts in these directions and topics stated above by fisheries specialists, )ptimal control and differential game specialists, economists, and environ- lentalists will make regulations and control economically and politically viable md sound . 24 References [1] J. H. Boyd; "Optimization and Suboptimization in Fishery Regulation: Comment," American Economic Review, June 1966, vol. 56, pp. 511-517. [2] H. E. Crowther; "Our Fishing Industry in the World Race: A Strategy or an Awakening Challenge?" in Recent Developments and Research in Fisheries Economics (Eds, F. W. Bell and J. E. Hazleton) , 1967, pp. 19-27. [3] Y. C. Ho; "Differential Games, Dynamic Optimization and Generalized Control Theory." Journal Optimization Theory and Application, September 1970, Vol. 6, pp. 179-209. [4] P. A. Samuelson; "A Universal Cycle?" The Collected Scientific Papers of Paul A. Samuelson, Vol. 3, 1972, pp. 473-486. [5] ; "Generalized Predator-Prey Oscillations in Ecological and Economic Equilibrium" ibid., pp. 487-490. [6] ; "Intertemporal Price Equilibrium: A Prologue to the Theory of Speculation," ibid., Vol. 2, 1966, pp. 946-984. [7] M. Simaan and T. Takayama; Dynamic Duopoly Game: Differential Game Theoretic Approach, Faculty Working Paper #155, February 1974. [8] and ; "An Application of Differential Game Theory to a Dynamic Duopoly Problem with Maximum Production Constraints," to appear in" Automat ica, March , 1978. [9] A. W. Starr and Y. C. Ho;"Nonzero-Sum Differential Games." Journal of Optimization THeory and Application, March 1969, Vol. 3, pp. 184-206. [10] T. Takayama; Dynamic Theory of Fisheries Economics - 1: Optimal Control Theoretical Approach, Faculty Working Paper #437 , College of Commerce and Business Administration, September, 1977. [11] R. Turvey; "Optimization and Suboptimization in Fishery Regulation," American Economic Review, March 1964, Vol. 54, pp. 64-76. 25 Appendix In this appendix only Case 1 problem is solved in detail. The reader irf.ll find it easy to solve Case 2 problem by following the development below. From (2.7) and assuming (2.8) we have the following necessary condi- tions (A.l) (i) (ii) (a) p = a + bp -X, - X , p(o) = p ic Zc o (b) \, —i^-r,- -^ )\, (c) (a -u) - (B + 0)x^ - X ^ = 0 Ic i^c -r t (d) lim^ "■ X^^(t) = 0 (a) p = a + bp - x^^ - x^^, p(o) = p^ (b) L = -(b - r. - ^^Ic (c) (a - y) - (3 + 0)X2^ " ^2c " ° (d) lim e"'^2'^X„ (t) = 0 zc By setting (A. 2) X^ = K.p + E., i= 1,2, ic X X we have, from (A.l) (i,c) and (ii,c). (A. 3) X q - y K,- ic e + Q B + e*^ B+0 ^P - Ei r, i = 1,2. By differentiating (A. 2) with respect to t, and equating the results to (A.l) (i,b) or (ii,b) respectively, one gets 26 K K K (A. A) {K. + K^(b + g^ g-fg) + (b -r. + ^)\} P + {K,(a--f^ 2(a - y) e F F K (A. 5) for i = 1,2. Equality in (A. 4) should hold for all variations of p. Thus, we get K K K ^i -^ ^i (b + sTT -^ 3^^ + (b - r, + 3^) K^ = 0 ^i<- -'r^ ^ FT^-^3^> + (b - r. + 3^)E^ + E^ = 0 Assuming that K and E converge to zero as t tends to infinity, we get K. K^ (A. 6) (A. 7) for i = 1,2 that K and (2b - r, + -34-^; + K^(a - K. 1 iB+OB+ee+o'^i 2(a - y) ;) K, = 0 E E K + ^r^-^ + (b - r. + ^ B+0B+0 B+0 for i = 1,2. One set of signular solutions is obtained as; ^1 " ^2 " ° i 6 + 0' i •)E. = 0 \= E2=0. The other set of stable solutions is obtained after some algebraic manipulations (A. 8) = (2b - 2r + rg)(B + 0) 1 3 _ _ (2b - 2r^+ r,)(B + 0) E = (2b - 2r;^ + r,) [2(a - p) - a(B + 0)1 ■"• 3b E = (2b - 2r^ + r.^)[2(a - u) -a(e + 0) ] ^ 3b By substituting (A. 8) into (A. 2), (A. 3) and (A.l) (ii,a) or (ii,a) in that order, we get the desired results. • «tJ-'- i 2 ■ J Of - i^^riir' t^--^ ^ e^T- T c; vy •,6P l":";iBj' ^l&:id jglifl i.-n. a 195"' G faD«?±r>t' 0 .0 (B.i±) 27 The two other unstable pairs of strategies as other solutions of the Ricatti equations are (A.9) (a) X* g - U ic B + 0 (b) x*^(p(t)) = f-ri {^{ffi -a) ^ ^= and r \ * - ot - U t 2b-r Vg - y ) (A. 10) J. 2b-r 1 /, >, jt _ g - y Corresponding to these strategies, the fish population moves along the paths stipulated by the following dynamics (A. 11) • _ r. (2(a - y) \ r p - - iS Vb + 0 - 7 "^ 2-" p for i = 1,2, which are unstable due to the fact that r. > 0 for i = 1,2. It is obvious that these two solutions do not really satisfy (2.7) (i, d) or (ii, d) . Therefore, even though these are solutions of the Ricatti equations, they are not those of (2.7). similar results as above can be obtained for the open-loop strategy case. 3-9*