Dynamic theory of fisheries economics: II; differential game theoretic approach

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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<x> 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. rfji<!<j-t '*') %■' ^ar/'^^:Ji ,4.i3,-'.' atfj

,io') );■* i^.'iz •=>s6ft'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*