A Dynamic Cournot-Nash Model for the Fish War under
Stochastic Circumstance
Chen Bo (205870340)
The term “”Fish War” is frequently cited in the media for the conflicts
between/among countries which share the same water for fishery. The model developed
in the paper sheds light on the economic implications of these conflicts especially
between two countries. Though it is too simple to accommodate all the important aspects
of such conflicts, my model actually has two basic features. First, I assume a continuous
stochastic growth model for the dynamic fish stock (population), which captures the
randomness characteristic witnessed in reality. Second, a Cournot-Nash game allows us
to explore the basic strategy likely adopted by the participant in which it should also take
into account the actions of the other participant. I will also prove that under a continuous
harvesting pattern, the Cournot-Nash situation will not exist.
1. Introduction
The term “Fish War” discribes the severe fishery conflicts between/among countries
which share the same water for fishery. One of the most well-known conflicts is the “Cod
War”, which took place between Iceland and the United Kingdom. Canada and the
United States also frequently encounter “Fish War”. Thus I develop a model to shed
some light on the economic implications inherent in the fishing conflicts especially under
the duopolistic situation, in which, as we observe in the realistic “Fish War”, the
decisions of the participants have an effect on the evolution of the stock (population) of
fish.
1
My model has two basic features: First, there exists a duopolistic competition in
which each of the participants should take into account the action of the other when it
tries to maximize its total discounted utility. Second, the fish stock growth will follow a
continuous stochastic evolution path; that is, the fish stock will change at a certain rate
but also will be disturbed by a “white noise”. Thus the fish population in the future
periods is always uncertain.
Lethari & Mirman (1980) applied the dynamic Cournot-Nash model to analyze the
fish war. I will follow a similar method so as to capture the characteristic that each of the
two Cournot duopolists will try to maximize its total discounted utility given the action of
the other in a dynamic framework. However, it is worth noting that although it may be
interpreted in a dynamic context, the original Cournot solution is static in nature, as is
made clear by the Nash analysis in the theory of Games. Hence I will also explore the
situation under a continuous harvesting pattern in the extension which is allowed in a
Cournot situation (Part 3).
Concerning the fish growth model, Smith(1968) assumed the recruitment

rate(growth rate) function is given by: X t  X t  X t2 , that is, the growth rate at time t,

X t , is a function of current stock X t , given the birth rate  and the death rate  . While
Levhari & Mirman (1980) used an exponential growth function as: X t 1  X t , which
stated that the biomass growth is following an exponential rule. However, both of the
growth models ignore the randomness in the nature. Plourde & Bodell(1984) stressed that
in the natural world, it is straightforward to understand that many unexpected factors, like
anomaly climate, oil spill, etc., will affect the growth rate of the fish stock. Thus, I will
2
employ a continuous stochastic growth model for dynamic fish stock in my analysis
below.
2. The Model
First of all, to capture the randomness in the dynamic fish stock, I assume the
change of fish stock is following a continuous Brownian Motion as:

X t = X t    dwt  X t    dt
(1)
where wt is the Brownian Motion process such that E( wt )= 0 and Var( wt )= t;  is the
oscillation, and  is the expected net growth rate, and both of these two parameters are
positive. The integration of equation (1) is given by:
X t  X 0  ewt (  1 / 2
2
)t
(2)
Now let us focus on the Cournot-Nash process. As in the duopolistic situation, I
assume that there are only two countries sharing the same water for fishery and both of
them have the same instantaneous utility function for current consumption. For
convenience, I assume that the utility function for country i is natural logarithmic 1 ,
i.e. U i (Ci )  ln Ci , (i=1,2). Meanwhile, let 0   i  1 be the discounted rate of utility for
country i. Furthermore, I suppose the objective of each country is to maximize the
expected sum of discounted utility of fish consumption. Finally, as required by the
Cournot –Nash analysis, I need assume that the harvesting will only occur once at the
beginning of a single period and it will not affect the parameters of  and  . That is, the
1
As shown in equation (2), the dynamic fish stock is an exponential function. And the assumption of
natural logarithmic utility will easily convert the exponential function into a linear one so as to easily derive
the critical result for my analysis.
3
harvesting just affects the stock of fish, and it is discrete compared with the continuous
growth of fish population.
Next, I start to consider the problem in the context of a one-horizon case. I will also
assume that at the end of the last horizon, each country will just get half of the remaining
stock of fish. Let the initial fish stock be X 0 , then if there is only one horizon, country 1
will try to maximize its expected discounted utility taking into account the action of
country 2. Specifically,
Max ln c
1
0  c1  X 0  c2
=
Max ln c
1
0  c1  X 0  c2
 E1 ln 1 / 2( X 0  c1  c2 ) ewt (  1 / 2

2
t)

 1 ln 1 / 2  ln( X 0  c1  c 2 )    1 / 2 2

(3)
where c1 is the optimal present consumption of country 1, given the consumption of
country 2. X 0  c1  c2 is the remaining fish stock just after the harvesting at the very
beginning(t=0), then the fish population will grow to be X 1  X 0 ew1 (  1 / 2
2
)1
at the
beginning of the second horizon(or at the end of the first horizon). Moreover,
E ln ew1   1 / 2 = E( w1    1 / 2 2 )=   1 / 2 2 ( E(w1 ) =0).
2
Then the first-order condition is given by:
( 1  1 ) c1  c2  X 0
(4)
Similarly, the optimal response of country 2will be:
c1  (1   2 )c2  X 0
(5)
Thus the Cournot-Nash equilibrium is given by the simultaneous solution of equation (4)
and (5), which is:
4
2
X0
1   2  1  2
c1* 
(6)
and
1
c2* 
1   2  1  2
(7)
X0
Furthermore, the remaining fish stock just after harvesting will be :
X 0  c1*  c2* 
1  2
X0
1   2  1  2
(8)
Therefore, the optimal utility of country 1 is given by:
ln c1*  E ln 1 ln(
= ln
1  2
X 0 )ew (  1 / 2
1   2  1  2
1
2
)
1


1  2
X 0  1 ln
 ln X 0  u  1 / 2 2 
1   2  1  2
 1   2  1  2

= (1  1 ) ln X 0  A1
where A1  ln
(9)
 2 ( 1  2 ) 
 1 (   1 / 2 2 ) , which is a constant and independent
( 1   2  1  2 ) 2
1
of X 0 , so it will have no effect on the optimal policy choice.
Looking one step further, let us consider a 2-horizon case. Because these two
countries will repeat the same Cournot-Nash game in the second horizon, so the total
utility in the second horizon will be the same as what we discussed in the one-horizon
case except that the fish stock at the beginning of the second horizon will
be X 1  X 0 ew1 (  1 / 2 ) . Thus we can easily form the objective function for country 1 in
2
the 2-horizon case. Specifically,

ln c1  1 E (1  1 ) ln( X 0  c1  c2 )ew1   1 / 2  A1
2

5
= ln c1  1 (1  1 ) ln( X 0  c1  c2 )  A1
(10)
and the first-order condition is given by:
(1  1  12 )c1  c 2  X 0
(11)
Similarly, the optimal response for country 2 is:
c1  (1   2   22 )c 2  X 0
(12)
Again, the simultaneous solution for equation (10) and (11) will give the Cournot-Nash
equilibrium value of consumption for the 2-horizon case, which is:
 2   22
c1 
X0
(1  1  12 )(1   2   22 )  1
(13)
and
c2 
1  12
X0
(1  1  12 )(1   2   22 )  1
(14)
Because the process will be repeated for a n-horizon case( n  1), by the induction method,
the solution for c1 is given by:
c1 
 2   22  ......   2n
X0
(1  1    ......  1n )(1   2   2n  ......   2n )  1
2
1
(15)
and
c2 
1  12  ......  1n
X0
(1  1  12  ......  1n )(1   2   2n  ......   2n )  1
(16)
Then as n   ,
c1* 
2
 2 (1  1 )
X0
1   2  1  2
as n   ,
2
(17)
2
1  12  ......  1n
1  2
 2 (1  1 )
lim c1  X 0 lim
 X0

X0
n 
n   (1     2  ......   n )(1     n  ......   n )  1
1
1

1
1
1
2
2
2
1   2  1  2
0
(
)(
) 1
1  1 1   2
6
and
c2* 
1 (1   2 )
X0
1   2  1  2
(18)
Furthermore, the remaining stock is given by,
12   22  1  2
X0  c  c 
X0
1   2  1  2
*
1
*
2
(19)
Equation (17) and (18) represent the consumption policies for country 1 and country 2
respectively in an infinite horizon case, while equation (19) represents both countries’
combined investment policy for fish.
However, as I mentioned before, the Nash-Cournot solution for the infinite horizon
case may not necessarily be equal to the steady-state value, i.e. the remaining fish stock
may not sustain at a certain (expected) level. Actually, the steady state condition is given
by,
X t  E ( X t  c1  c 2 )e
1
2
w1  (    2 )
3
(20)
which means that before the next harvesting, the expected remaining stock of fish should
equal the current fish stock. Employing Eq.(19) , Eq.(20) can be simplified as(for details,
see appendix 1),
1   2  1  2
   e
2
2
1   2  1  2
(21)
It is interesting to consider the case of a social planner whose objective is to
maximize the total utility of both countries rather than each separately. For simplicity, I
assume that these two countries have the same discounted rate, say, 1   2   , and
analogically, we can derive the value of
3
Note,
c2 in the infinity-horizon case
X t here is the realized current stock.
7
they equally share the harvesting and consumption, ceteris paribus. The objective is then
to maximize the total discounted sum of expected utility, which will yield (for details, see
appendix 2),
'
c* 
1 
X0
2
(22)
and
X * = X 0
'
(23)
Compared with the result of the Cournot-Nash solution (Eq.(17)~(19)) for an infinite
horizon (let 1   2   ), we get,
'
c* 
1 
1 
X 0  c* 
X0
2
2
(24)
and
X * = X 0  X * 
'

2
X0
(25)
The inequity of (22) and (23) show clearly that
countries under Cournot
competition will tend to consume more and invest less then those implied by Pareto
Optimum.
3. Extensions
In part 2, I assume that each country just harvests once in a period, which is the prerequirement of Cournot-Nash model. However, the discrete harvesting assumption might
be too restrictive. For instance, some tropical countries may harvest fish at any time.
Therefore, I will discuss the optimal policy under continuous harvesting pattern.
8
First of all, because like fish growth, the harvesting is also continuous, the change of
fish stock will then turn to be,

X t = X t    dwt  X t  (  R1  R2 )  dt
(26)
where R1 and R2 are the harvesting rate of country 1 and country 2 respectively.
Moreover,
X t  X 0  ewt (  1 / 2
2
 R1  R2 ) t
(27)
Then country 1’s objective is to maximize the integral of continuous utility within a
certain duration (say, T). Specifically,

Max E 
T
0
(e  1t ln R1 X t )dt  Ee  1T ln
R1
1

XT 
2

(28)
Substitute Eq.(25) into (26) and the first-order condition is given by(for details, see
Appendix 3),
1e   T T  Te   T 
1
1

R1
1
1 e
1
1
 1T

1
1
e  1T
(29)
thus,
lim R1   1
T 
(30)
Similarly, when T   ,
R2   2
(31)
Not surprisingly, the solution shows that the Cournot-Nash situation will not exist in
the continuous harvesting case since it is impossible to take into account the reaction of
the other country and modify its action in each piece of time.
9
The steady-state here requires the expected dynamic fish stock change will be zero,
i.e.

E X t  Ewt  (   R1  R2 )dt  0
 R  R1  R2  
(32)
where R is the total harvesting rate. Thus in the continuous harvesting case, where
R1  1 and R2   2 , the steady-state condition will be achieved only if 1   2   .
4.
Summary and Conclusion
In part 2, I formed a model accommodating both dynamic and strategic aspects
under a stochastic fish growth circumstance. The dynamics enters through the stochastic
growth of fish population, and the strategic aspect enters since there are only two
countries and both of them have to take into account the action of the other participant
when it tries to maximize its expected total discounted utility. The solution for CournotNash equilibrium gives us the optimal consumption and investment policies. Furthermore,
it is also proved that countries in Cournot-Nash situation will tend to consume more
while investing less than the Pareto Optimum level.
In part 3, I extend the model into a continuous harvesting-continuous growth
situation. Not surprisingly, the result proves that under a continuous harvesting pattern,
Cournot competition will not happen, instead, each country will only maximize its own
expected total discounted utility without considering the reaction of the other’s for there
is no time to consider and modify its policy to incorporate the other participant’s reaction.
10
Reference
Levhari, D., L., and J. Mirman. 1980.The Great Fish War: An Example Using a
Dynamic Cournot-Nash Solution. The Bell Journal of Economics 11(1): 322-334
Smith, V.L. 1974. General Equilibrium with a Replenishable Natural Resource. Review
of Economic Studies. 41:105-115
Plourde, C., and R. Bodell. 1984. Uncertainty in Fisheries Economics: The Role of the
Discount Rate. 1(2): 155-170
11
Appendix 1
As Eq(20), the steady-state condition is given by,
1
2
w1  (    2 )
X t  E ( X t  c1  c 2 )e
= ( X t  c1  c 2 )e
1
2
  2
 E (e w1 )
(i)
As defined, w1 ~ N(0,1), thus the right hand of Eq(i) will be,
( X t  c1  c 2 )e
= ( X t  c1  c 2 )e
1
2
  2
1
2
  2
= ( X t  c1  c 2 )e
1
2
  2






2
e
2
e
w1
e

e

w12
2
dw1
w12  2w1  2  2
2



e

( w1  ) 2
2
dw1
dw1
2
=
( X t  c1  c2 )e
2
Therefore, substitute Eq(19) into Eq.(i), I get

 2   22  1  2
Xt  ( 1
)Xt  e 2
1   2  1  2
2

1   2  1  2
   e
2
2
1   2  1  2
which is Eq.(21)
12
Appendix 2
Assuming that these two countries have the same discounted rate, say, 1   2   , and
they equally share the harvesting and consumption. And the objective is to maximize the
total discounted sum of expected utility. Then in one-horizon case,
1
w1    

2
ln
2
c



E
ln(
X

2
c
)
e


0
Max
c


FOC is given by,
(1   )2c  X 0
which will yield,
c
1
X0
2(1   )
and
X 0  2c 

1 
X0
Then the one-horizon Cournot-Nash Value is,
w1     2
1

2
X 0    E ln(
X0 e
)
2(1   )
1 
1
ln
= (1   ) ln X 0  A1
where A1 is the sum of the rest term excluding X 0
Furthermore, in a 2-horizontal case, the objective will be,
Max { ln 2c    E ((1   ) ln( X 0  2c)e
1
2
w1     2
 A1 ) }
c
13
FOC is given by,
(1     2 )2c  X 0
So we get,
c
1
2(1     2 )
X0
Using Induction method, and let n (number of total horizon) approaches to infinity, we
can obtain,
c* 
1
1 
X0 
X0
1
2
2(
)
1 
(22)
and
X 0  2c  X 0
(23)
14
Appendix 3

Max E 
T
0
(e  1t ln R1 X t )dt  Ee  1T ln
R1
1

XT 
2

1 2
 T  1t

E
e
[ln
R

ln
X


w

(


  R1  R2 )t ]dt 
0
1
0
t



2
= Max 

R1
 Ee  1T [ln 1  ln X  w  (   1  2  R  R )T ] 
0
T
2
1


2
2
=
Max 
T
0
R1

e  1t ln R1dt   T0 e  1t R1tdt  e  1T R1T  A
where A is the rest of the term excluding R1 .
Therefore, FOC is given by,

T
0
e  1t dt
  T0 e  1t tdt  e  1T T
R1


1
1
R1
1

e  1t |T0
1

 R1 
1
1

1
1
e  1T

R1
[te  1t |T0   T0 e  1t dt ]  e  1T T
1
1
Te  1T 
1

2
1
e  1T 
1

2
1
 e  1T T
1  e  1T
1
1
 1e  1T T  Te  1T 
1
1
(29)
e  1T
Thus,
lim R1 
T 
1
1
1
000
 1
(30)
15