- Email: [email protected]
1 Cooperation and competition
1 Cooperation and Competition We now analyze the problem of cooperation as follows: First we determine the current state of a f f a i r s - cooperation or competition using the computer program, ngusukc.m. Let
x I = GDP of Nigeria x 2 = GDP of U.S.A. x 3 = GDP of U.K. x 4 = GDP of China e~. i = 1, ..., 4 as windfalls, or the reaction of the representative firm to government strategy.
The raw data may be used for the regression, and then refined, to yield the systems dxl (t) dt
dx3 (t) dt
dx4 (t) dt
= x l ( - a I + blX 2 + ClX 3 + d l x 4 ) + e I ,
= x 3 ( - a 3 + b 2 x I + c 3 x 2 + d 3 x 4 ) + e 3,
- x 4 (- a 4 + b4x 1 + c4x 2 + d4x 3 ) + e 4.
(1.O)
We now use the MAPLE regression code to find the coefficients ai, hi, ci, d i , e i ,
i = 1, 2, 3, 4
which are constants, and infer the degree of cooperation or competition. These can also be extracted from the earlier regressions.
25
A Mathematical Treatment of Economic Cooperation
26
MATLAB Program The models of gross domestic products (GDP) of interacting countries are built from the historical data available from the World Bank and I MF. In constructing the dynamic model of the GDP the aggregate demand, z was formulated as (1.1)
z=C+I+X+G
where C denotes consumption, I investment, X net export and G government outlay. In the formula for each component the coefficients are made as realistic as possible. We can appropriate the MATLAB Linear Regression Models of the System Identification Toolbox of Ljung [12, pp. 1-13, pp. 1-79]. The arx command of the toolbox allows an arbitrary number of inputs. It also allows nonlinear inputs. It permits delays in the structure both at the output and the input. Suppose, for example, that y(t) = b 0 + b I u(t) + b 2u 2 (t) + b3u
3 (t).
(1.2)
z = [y ones(u), u, u.^2, u.^3];
Let
th=arx(z,[O 1 1 1 1 0 0 0 01).
(1.3)
This is a model with one output and four inputs. Consider the arx model [ 11, p. I- 13] y(t) + aly(t
-
1) + ..-+
anaY(t
- na) = blu(t - nk) + b2u(t - nk -
... + b n b u ( t - n k - n b +
1) +
e(t) .
1) + (1.4)
We determine the best values of the coefficients a I , a 2 ,--., b n b that best fit the data. The function arx implements the least square estimation method using the MATLAB "1" operation for over-determined linear equations, see Lennart Ljung, System Identification Theory for the User [ 12, p. 462]. Let, z = [ y ( t ) , y ( t - 1), . . . , y ( t u = [u(t - nk), u(t - nk -
1), .-.,
na)]
u ( t - n k - bn+ 1 )]
27
Cooperation and Competition
Then z 2 = [z, u] is used to determine a model (1.4). The best values can be computed with the command th = arx(z2, [na, nb, nk] ) .
The numbers in the second argument tell arx to find a model (1.4) with na, a parameters, b, nb parameters, and nk the number of delays from input to output. Equation (1.4) defines the exact model structure. The function arx implements the least square estimation method. Consider, for example C(t) = C O + C l y ( t ) + C 2 y 2 + C3R .
Let
z = (C ones(size(y)), y, y2, R) ; th=arx(z,[O
1 1 1 1 0 0 0 0]).
The theory is treated in L. Ljung [12, p. 462]. Consider the following example; we determine the coefficients C i in C = C O + C l y ( t ) + C2))(t ) + C 3 R .
Temp = [C, ones(size(y)), y, fi(t), R]; thc=arx(temp,[O 1 1 1 1 0 0 0 0]); C p = predict(C, ones(size(y)), y, j~(t), thc, 1) ; CC = thc(3, 1:4) ;
CC(l) = CO,
CC(2) = C1,
CC(3) = C 2 ,
CC(4) = C3.
28
A Mathematical Treatment of Economic Cooperation
Remark. The coefficients are used to construct the identified model dynamics, which is an ordinary differential system. This is solved by (1) ode45 Nonstiff differential equations, method Runge-Kutta or
(2) ode23 Nonstiff differential equations, method Runge-Kutta. We now describe how to use MATLAB to solve initial value problem of ordinary differential equation. The odeplot command yields a time-series plot of the MATLAB solution. Step 1. Rewrite the problem as a system of FIRST order odes of the form dv
- " = f (t, y) dt
yi = fl (t, y) Y2 = f 2 (t, y) P
Y n = f n (t, y ) .
Step 2. Code the system of First-order ODE's in MATLAB dv
- " = odefun(t, y) dt
-:
dt
is a column vector, the derivative ofy.
For example, the Van der Pol equation (second-order) Yi' - (1 - y? ) Yi + Yl = 0
is equivalent to Yi = Y2
Cooperation and Competition
29 y~ = ( 1 - y ? ) y 2 - Y l Yl : y(1),
Y2 : y(2)
y = (y(1), y(2)) a two element vector function dy / dt = v@l(t, y)
dydt = [y(2); ( l - y(1)^2*)y(2)- y(1)]. We can use ode45 on time interval [0, 20] with initial values y(1) = 2, y(2) = 0;
[t, y] = ode45(vdpl, [0 2 0], [2; 0]). This example uses to pass vdp 1 as a function handle to ode45. The output is a t column vector and a solution array y. The first column o f y corresponds to Yl and the second column to Y2 9
1.1
Output
Plot command plot(t, y'(:, l),' ', t, y(:, 2),' ') title(' solution of van der pol Equation, \mu = 1');
x label('time') ylabel('solution y'); legend('y-', 'y-2'). Using Maple to Solve the Equation In the following we use Maple to first fit the system of equations to the historical data of various nation's gross national products, and then to solve the system of non-linear equations numerically and to display the solutions graphically. Below is a brief description of these two parts of the analysis, a brief prepared by Norris.
30
A Mathematical Treatment of Economic Cooperation
Fitting the equations to the data. The various coefficients that describe the interaction of the nations in the system of equations occur linearly in the equations and hence one may use linear regression to fit the equations to data. We used Maple's "leastsquare" function in conjunction with Maple's "fit" subpackage of Maple's statistical package "stats" to do the linear regression. One can obtain a Maple worksheet illustrating these commands from the Applications Center at the URL www.maplesoft.com. The appropriate worksheet is listed under Education --> Statistics in the Applications center. Solving the equations and displaying the solutions. Once the coefficients are found using Maple's statistical tools, we solved the equations numerically using Maple's desolve command with the "type=numeric '' option. Finally the numerical solutions generated in this way are conveniently displayed graphically using Maple's plot[odeplot] command in the plots package. Suppose the following model is constructed for a group of four interacting countries. dr, l / d t = x
l ( - a I + b l x 2 +ClX 3 + d l x 4 ) + e 1
dx 2 / d t = x 2 ( - a
2 +b2x 2 +c2x 3 +d2x4)+e2
dx 3 / dt = x 3 ( - a 3 + b 3 x 2 + c 3 x 3 + d 3 x 4 ) + e 3 dr 4 / dt = x 4 ( - a 4 + b 4 x 2 + c 4 x 3 + d 4 x 4 ) + e 4 .
If bi, ci, d i , i = 1, ..., 4 are all positive, the system is cooperative. If the natural decay rates a i, i = 1,..., 4 are sufficiently small or negative dx i / dt, i = 1, .-., 4 are positive and x(t) ~ oo as t ~ oo.
Since x i represents the GDP, it is ever increasing: sustained economic growth is obtained. If c i is negative (x I > 0, x 3 > 0) then CiXlX 2 < 0. This has a bad effect on the growth of x I . Because the economic model for x I is controllable we can drive x 1(0), to x 1(t) in time t; also x 2 (0) can be steered to, x 2 (t) in time t. Using control strategies, x 1, x 2 can be steered to bigger values. Citizens of nations i = 1,2 can tolerate transferring positive 2 b i x l x 2 to dx I /dt to yield dx l / dt = x l ( - a I + m l x 2 + ClX 3 + d l x 4 ) + e I
Cooperation and Competition
31
m l X l X 2 = 2 b i X l X 2 + C i X l X 2 , m > 0. Thus dr I / d t can be made positive and x 1 increasing. Sustained growth of GDP can be attained. Cooperation can keep economic growth in the region. In practice it is a universal belief that competition pays better than cooperation. The novel argument here is that cooperation is superior. The examples that follow illustrated it. This validates the theorems of Lazer and Ahmed. In a group of four species, three can gang up against one and drive it to extinction. Since three are still competing, two can gang up against no. 3 and drive it to extinction. The remaining two will compete it out, and no. 1 will drive no. 2 to extinction. The jubilant triumphant no. 1 will think he controls the universe, but weather and natural calamities can wipe no. 1 off the planet earth. There is no "love" from another to help him. The others are extinct. Recall the following theorem of S. Ahmad, On the nonautonomous Volterra-Lotka competition equation, Proc. Amer. Math. Soc. 117, No. 1, 1993. T h e o r e m 1. Consider the system
Xl (t) = x 1(t)[a 1(t) - b l l (t)x 1(t) - b12 (t)x 2 (t)],
(J)
J:2 (t) = x 2 (t)[a 2 (t) - b21 (t)x 1(t) - b22 (t)x 2 (t)],
(H)
where a 1 (t), b 1(t), are continuous and bounded above and below by positive constants. Suppose f L = inf(t), f M = sup f(t), respectively and
alL > blM a2M /b22 L and a2M < b21L alL /bllM,
(1)
then any solution ( x I (t), X 2 (t)) with X 1 (0) > O, X 2 (0) > O, has the property that
x 2 (t) ~ 0 as t ~ oo,
x 1(t) ~ x I (t),
where x 1 is a solution of (J) after a long time, x 2 all but vanishes, Xl (t) executes the dynamics Xl (t) = x I (t)[a I (t) - bll(t)x I (t)] + f ( t ) where fit) may represent the effects of weather and other calamities which can help drive x I to zero.
32
A Mathematical Treatment of Economic Cooperation
The four influential countries will soon be seen to have economic states which are function space controllable. That means that with the current state of the economy each country can climb up: with higher GDP, low interest rate, full employment, very good value of capital stock, low prices (i.e., low inflation) and surplus balance of payment. See [5]. The excess wealth created by the technical strategies of the representative firms and the government can be used to fund cooperative interaction between the nations. This will promote sustained growth. It will limit and put a check to competition which can lead to war and destruction. Having got used to prosperity born of cooperation war inclinations will be minimized, friendly relations will be planted and sustained. First we shall test whether the existing economic situation is a competitive or cooperative one. To sharpen our perception we compare with simple models and deduce how much we need to increase the coefficient of cooperation made possible by increase in gross domestic products of pairs of countries.
References
G. Gandolfo, Economic Dynamics Methods and Models, Textbooks in Economics, North-Holland, Amsterdam, 1990. [21 G. Gandolfo, Economic Dynamics, Third Editon, Springer-Verlag, Berlin, 1996. [3] International Financial Statistic Yearbook, 1994, 2000. [4] UN National Account Statistics. [5] E. N. Chukwu, Stability and Time Optimal Control of Hereditary Systems with Applications to the Economic Dynamics of the USA. World Scientific, 2003. [6] E.B. Lee and L. Markus, Foundations of Optimal control theory, John Wiley and Sons, Inc., New York, 1967. [7] R . G . Underwood and D. F. Young, Null Controllability of Nonlinear Functional Differential Equations, SIAM Journal of Control and Optimization, 17(1979) 753-768. [8] M. M. Vainberg, "Some Problems in Differential Calculus in Linear Spaces, Uspehi Mat. Nauk, 7(4):55-102 (in Russian). [9] K. Deimling, Nonlinear Analysis, Springer-Verlag, New York, 1985. [10] S. Ahmad, On the nonautonomous Lotka Competition Equation, Proc. Amer. Math. Soc, 117, No. 1, 1993.
[1]
Cooperation and Competition
33
[11] Lennard Ljung, MATLAB Programs and Theory, Systems Identification Toolbox for use with MATLAB, USERS Guide, The MATH Works, Inc, July 1992. [12] Lennard Ljung, Systems Identification Theory for the user PTR Information and Systems Science Series, Prentice Hall, Englewood Cliffs, New Jersey, 07632, 1987. [13] E.N. Chukwu, Cooperation and Competition in modeling the dynamics of gross-domestic products of nations, Applied Mathematics and Computation 163 (2005) 991-1021. [ 14] With remarks submitted.
x I = GDP of Nigeria x 2 = GDP of U.S.A. x 3 = GDP of U.K. x 4 = GDP of China e~. i = 1, ..., 4 as windfalls, or the reaction of the representative firm to government strategy.
The raw data may be used for the regression, and then refined, to yield the systems dxl (t) dt
dx3 (t) dt
dx4 (t) dt
= x l ( - a I + blX 2 + ClX 3 + d l x 4 ) + e I ,
= x 3 ( - a 3 + b 2 x I + c 3 x 2 + d 3 x 4 ) + e 3,
- x 4 (- a 4 + b4x 1 + c4x 2 + d4x 3 ) + e 4.
(1.O)
We now use the MAPLE regression code to find the coefficients ai, hi, ci, d i , e i ,
i = 1, 2, 3, 4
which are constants, and infer the degree of cooperation or competition. These can also be extracted from the earlier regressions.
25
A Mathematical Treatment of Economic Cooperation
26
MATLAB Program The models of gross domestic products (GDP) of interacting countries are built from the historical data available from the World Bank and I MF. In constructing the dynamic model of the GDP the aggregate demand, z was formulated as (1.1)
z=C+I+X+G
where C denotes consumption, I investment, X net export and G government outlay. In the formula for each component the coefficients are made as realistic as possible. We can appropriate the MATLAB Linear Regression Models of the System Identification Toolbox of Ljung [12, pp. 1-13, pp. 1-79]. The arx command of the toolbox allows an arbitrary number of inputs. It also allows nonlinear inputs. It permits delays in the structure both at the output and the input. Suppose, for example, that y(t) = b 0 + b I u(t) + b 2u 2 (t) + b3u
3 (t).
(1.2)
z = [y ones(u), u, u.^2, u.^3];
Let
th=arx(z,[O 1 1 1 1 0 0 0 01).
(1.3)
This is a model with one output and four inputs. Consider the arx model [ 11, p. I- 13] y(t) + aly(t
-
1) + ..-+
anaY(t
- na) = blu(t - nk) + b2u(t - nk -
... + b n b u ( t - n k - n b +
1) +
e(t) .
1) + (1.4)
We determine the best values of the coefficients a I , a 2 ,--., b n b that best fit the data. The function arx implements the least square estimation method using the MATLAB "1" operation for over-determined linear equations, see Lennart Ljung, System Identification Theory for the User [ 12, p. 462]. Let, z = [ y ( t ) , y ( t - 1), . . . , y ( t u = [u(t - nk), u(t - nk -
1), .-.,
na)]
u ( t - n k - bn+ 1 )]
27
Cooperation and Competition
Then z 2 = [z, u] is used to determine a model (1.4). The best values can be computed with the command th = arx(z2, [na, nb, nk] ) .
The numbers in the second argument tell arx to find a model (1.4) with na, a parameters, b, nb parameters, and nk the number of delays from input to output. Equation (1.4) defines the exact model structure. The function arx implements the least square estimation method. Consider, for example C(t) = C O + C l y ( t ) + C 2 y 2 + C3R .
Let
z = (C ones(size(y)), y, y2, R) ; th=arx(z,[O
1 1 1 1 0 0 0 0]).
The theory is treated in L. Ljung [12, p. 462]. Consider the following example; we determine the coefficients C i in C = C O + C l y ( t ) + C2))(t ) + C 3 R .
Temp = [C, ones(size(y)), y, fi(t), R]; thc=arx(temp,[O 1 1 1 1 0 0 0 0]); C p = predict(C, ones(size(y)), y, j~(t), thc, 1) ; CC = thc(3, 1:4) ;
CC(l) = CO,
CC(2) = C1,
CC(3) = C 2 ,
CC(4) = C3.
28
A Mathematical Treatment of Economic Cooperation
Remark. The coefficients are used to construct the identified model dynamics, which is an ordinary differential system. This is solved by (1) ode45 Nonstiff differential equations, method Runge-Kutta or
(2) ode23 Nonstiff differential equations, method Runge-Kutta. We now describe how to use MATLAB to solve initial value problem of ordinary differential equation. The odeplot command yields a time-series plot of the MATLAB solution. Step 1. Rewrite the problem as a system of FIRST order odes of the form dv
- " = f (t, y) dt
yi = fl (t, y) Y2 = f 2 (t, y) P
Y n = f n (t, y ) .
Step 2. Code the system of First-order ODE's in MATLAB dv
- " = odefun(t, y) dt
-:
dt
is a column vector, the derivative ofy.
For example, the Van der Pol equation (second-order) Yi' - (1 - y? ) Yi + Yl = 0
is equivalent to Yi = Y2
Cooperation and Competition
29 y~ = ( 1 - y ? ) y 2 - Y l Yl : y(1),
Y2 : y(2)
y = (y(1), y(2)) a two element vector function dy / dt = v@l(t, y)
dydt = [y(2); ( l - y(1)^2*)y(2)- y(1)]. We can use ode45 on time interval [0, 20] with initial values y(1) = 2, y(2) = 0;
[t, y] = ode45(vdpl, [0 2 0], [2; 0]). This example uses to pass vdp 1 as a function handle to ode45. The output is a t column vector and a solution array y. The first column o f y corresponds to Yl and the second column to Y2 9
1.1
Output
Plot command plot(t, y'(:, l),' ', t, y(:, 2),' ') title(' solution of van der pol Equation, \mu = 1');
x label('time') ylabel('solution y'); legend('y-', 'y-2'). Using Maple to Solve the Equation In the following we use Maple to first fit the system of equations to the historical data of various nation's gross national products, and then to solve the system of non-linear equations numerically and to display the solutions graphically. Below is a brief description of these two parts of the analysis, a brief prepared by Norris.
30
A Mathematical Treatment of Economic Cooperation
Fitting the equations to the data. The various coefficients that describe the interaction of the nations in the system of equations occur linearly in the equations and hence one may use linear regression to fit the equations to data. We used Maple's "leastsquare" function in conjunction with Maple's "fit" subpackage of Maple's statistical package "stats" to do the linear regression. One can obtain a Maple worksheet illustrating these commands from the Applications Center at the URL www.maplesoft.com. The appropriate worksheet is listed under Education --> Statistics in the Applications center. Solving the equations and displaying the solutions. Once the coefficients are found using Maple's statistical tools, we solved the equations numerically using Maple's desolve command with the "type=numeric '' option. Finally the numerical solutions generated in this way are conveniently displayed graphically using Maple's plot[odeplot] command in the plots package. Suppose the following model is constructed for a group of four interacting countries. dr, l / d t = x
l ( - a I + b l x 2 +ClX 3 + d l x 4 ) + e 1
dx 2 / d t = x 2 ( - a
2 +b2x 2 +c2x 3 +d2x4)+e2
dx 3 / dt = x 3 ( - a 3 + b 3 x 2 + c 3 x 3 + d 3 x 4 ) + e 3 dr 4 / dt = x 4 ( - a 4 + b 4 x 2 + c 4 x 3 + d 4 x 4 ) + e 4 .
If bi, ci, d i , i = 1, ..., 4 are all positive, the system is cooperative. If the natural decay rates a i, i = 1,..., 4 are sufficiently small or negative dx i / dt, i = 1, .-., 4 are positive and x(t) ~ oo as t ~ oo.
Since x i represents the GDP, it is ever increasing: sustained economic growth is obtained. If c i is negative (x I > 0, x 3 > 0) then CiXlX 2 < 0. This has a bad effect on the growth of x I . Because the economic model for x I is controllable we can drive x 1(0), to x 1(t) in time t; also x 2 (0) can be steered to, x 2 (t) in time t. Using control strategies, x 1, x 2 can be steered to bigger values. Citizens of nations i = 1,2 can tolerate transferring positive 2 b i x l x 2 to dx I /dt to yield dx l / dt = x l ( - a I + m l x 2 + ClX 3 + d l x 4 ) + e I
Cooperation and Competition
31
m l X l X 2 = 2 b i X l X 2 + C i X l X 2 , m > 0. Thus dr I / d t can be made positive and x 1 increasing. Sustained growth of GDP can be attained. Cooperation can keep economic growth in the region. In practice it is a universal belief that competition pays better than cooperation. The novel argument here is that cooperation is superior. The examples that follow illustrated it. This validates the theorems of Lazer and Ahmed. In a group of four species, three can gang up against one and drive it to extinction. Since three are still competing, two can gang up against no. 3 and drive it to extinction. The remaining two will compete it out, and no. 1 will drive no. 2 to extinction. The jubilant triumphant no. 1 will think he controls the universe, but weather and natural calamities can wipe no. 1 off the planet earth. There is no "love" from another to help him. The others are extinct. Recall the following theorem of S. Ahmad, On the nonautonomous Volterra-Lotka competition equation, Proc. Amer. Math. Soc. 117, No. 1, 1993. T h e o r e m 1. Consider the system
Xl (t) = x 1(t)[a 1(t) - b l l (t)x 1(t) - b12 (t)x 2 (t)],
(J)
J:2 (t) = x 2 (t)[a 2 (t) - b21 (t)x 1(t) - b22 (t)x 2 (t)],
(H)
where a 1 (t), b 1(t), are continuous and bounded above and below by positive constants. Suppose f L = inf(t), f M = sup f(t), respectively and
alL > blM a2M /b22 L and a2M < b21L alL /bllM,
(1)
then any solution ( x I (t), X 2 (t)) with X 1 (0) > O, X 2 (0) > O, has the property that
x 2 (t) ~ 0 as t ~ oo,
x 1(t) ~ x I (t),
where x 1 is a solution of (J) after a long time, x 2 all but vanishes, Xl (t) executes the dynamics Xl (t) = x I (t)[a I (t) - bll(t)x I (t)] + f ( t ) where fit) may represent the effects of weather and other calamities which can help drive x I to zero.
32
A Mathematical Treatment of Economic Cooperation
The four influential countries will soon be seen to have economic states which are function space controllable. That means that with the current state of the economy each country can climb up: with higher GDP, low interest rate, full employment, very good value of capital stock, low prices (i.e., low inflation) and surplus balance of payment. See [5]. The excess wealth created by the technical strategies of the representative firms and the government can be used to fund cooperative interaction between the nations. This will promote sustained growth. It will limit and put a check to competition which can lead to war and destruction. Having got used to prosperity born of cooperation war inclinations will be minimized, friendly relations will be planted and sustained. First we shall test whether the existing economic situation is a competitive or cooperative one. To sharpen our perception we compare with simple models and deduce how much we need to increase the coefficient of cooperation made possible by increase in gross domestic products of pairs of countries.
References
G. Gandolfo, Economic Dynamics Methods and Models, Textbooks in Economics, North-Holland, Amsterdam, 1990. [21 G. Gandolfo, Economic Dynamics, Third Editon, Springer-Verlag, Berlin, 1996. [3] International Financial Statistic Yearbook, 1994, 2000. [4] UN National Account Statistics. [5] E. N. Chukwu, Stability and Time Optimal Control of Hereditary Systems with Applications to the Economic Dynamics of the USA. World Scientific, 2003. [6] E.B. Lee and L. Markus, Foundations of Optimal control theory, John Wiley and Sons, Inc., New York, 1967. [7] R . G . Underwood and D. F. Young, Null Controllability of Nonlinear Functional Differential Equations, SIAM Journal of Control and Optimization, 17(1979) 753-768. [8] M. M. Vainberg, "Some Problems in Differential Calculus in Linear Spaces, Uspehi Mat. Nauk, 7(4):55-102 (in Russian). [9] K. Deimling, Nonlinear Analysis, Springer-Verlag, New York, 1985. [10] S. Ahmad, On the nonautonomous Lotka Competition Equation, Proc. Amer. Math. Soc, 117, No. 1, 1993.
[1]
Cooperation and Competition
33
[11] Lennard Ljung, MATLAB Programs and Theory, Systems Identification Toolbox for use with MATLAB, USERS Guide, The MATH Works, Inc, July 1992. [12] Lennard Ljung, Systems Identification Theory for the user PTR Information and Systems Science Series, Prentice Hall, Englewood Cliffs, New Jersey, 07632, 1987. [13] E.N. Chukwu, Cooperation and Competition in modeling the dynamics of gross-domestic products of nations, Applied Mathematics and Computation 163 (2005) 991-1021. [ 14] With remarks submitted.