- Email: [email protected]

Sustainable global economic growth E.N. Chukwu∗ Mathematics Department, North Carolina State University, Box 8205, Raleigh, NC 27695-8205, USA

Abstract This survey highlights some recent advances on modeling the economic state of some nations of the world which are linked up in solidarity and competition or which are isolated. The problem of controllability, convergence, and permanence are explored. Some practical recent policy implications are deduced. Earth revolves around the sun in approximately circular orbit with radius r = a, completing a revolution in the time T = 2(a 3 /GM)1/2 which is one earth year; here M is the mass of the sun and G is the universal gravitational constant. The gravitational force of the sun on the earth is given by GM m/r 2 where m is the mass of the earth. Therefore, if the earth “stood still” losing its orbital velocity, it will fall straight on a line into the sun in accordance with Newton’s second law. Md2 r −GM M = . dt 2 r2 If this calamity occurred what fraction of the normal year T will it take for the earth to splash into the sun?” (Nagle, Saff, Snider) “Why did the earth stand still?” (Ndu Udo) “The symbol of Love and the earth’s lover is the moon. With joy it dances around the earth in a circular orbit. When the inhabitants of the earth prefer hate and revenge to love, the moon stops to love, and stands still, losing its orbital velocity. It falls straight on a line into the earth in accordance with Newton’s second law. The crash makes the earth stand still precipitating the Armageddon.” (Wisdom Goodness) 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Global economic growth; Economic state; MATLAB; Private and government strategies; Interaction term; Solidarity; Competition matrix; Controllability; Omega problem

∗ Tel.: 1+919 5151 7442.

E-mail address: [email protected] (E.N. Chukwu) URL: http://www4.ncsu.edu/∼chukwu. 0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.09.021

786

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

1. Introduction In this survey we highlight recent advances in modeling realistic economic state of several countries. They include Austria, Australia, Nigeria, South Africa, USA, UK, China, Egypt, Jordan, Israel, Japan, Germany and Italy. See [1,2,4–7]. The economic state of each nation is a vector of six things–the gross domestic product (GDP) (y); interest rate (R); employment (or unemployment) (L); value of capital stock (K), prices p(t) (and therefore inﬂation (p(t))); ˙ and cumulative balance of payment (E). Each economic state is isolated except the impact of export function X on the aggregate demand z = I + C + X + G. We use government and private (stroboscopic) ﬁrms’ strategy to control the economic state from an initial state to a desired target. Government strategy is a vector ˙ f0 ], g = [T , g0 , e, , d, M1, M1, where T represent taxes and g0 the autonomous government outlay, e the exchange rate, the tariffs, d the transportation, and distance between trading nations or trade policies, ˙ and f0 = foreign credit and equalization taxes. money supply and its ﬂows are M1, M1, The representative private ﬁrm reacts to government control with the strategy p = [C0 , X0 , I0 , M0 , n, w, x0 , y0 , p0 ], where n is productivity, w is wages, C0 the autonomous consumption, I0 the autonomous investment, X0 the autonomous net export and M0 the autonomous money demand. An ordinary differential game of pursuit is ﬁrst derived from the differential supply and demand principle dz = 1 (z − y); dt and carefully deﬁned economic variables, and equations. The outcome is the model x(t) ˙ = Ax(t) + B1 p(t) + B2 g,

(1)

where x = [y, R, L, K, P , E], p representing the private ﬁrms’ strategy and g the government strategy. Doing some regression executed with MATLAB code the matrices A, B1 , B2 are identiﬁed. The data are extracted from International Financial Statistic Yearbook and UN: National Accounts Statistics. For further details consult http://www4.ncsu.edu/∼chukwu and http://www.math.buffalo. edu/Mad/PEEPS/∼chukwu/ethelbert/nwakuche.html.

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

787

2. Hereditary model If we appropriate the rational expectations principle of Fair [8] as well as the principle of supply and demand, a hereditary economic model, a linear neutral game of pursuit is the outcome: ˙ − h) = A0 x(t) + A1 x(t − h) + B1 g + B2 p. x(t) ˙ − A−1 x(t

(2)

The matrix coefﬁcients A−1 , A0 , A1 , B1 , B2 are identiﬁed using programs in MATLAB, which are available in the cited referencesChukwu’s books [1,2,4,6,7]. This full hereditary system has the economic state, x = [y, R, L, K, p, E]. Here y is the GDP, R the interest rate, L the employment, K the value of capital stock, p the prices, and E the cumulative balance of payment. For control function the government strategy is g = [T1 , g0 , e, , d, M1, f0 ], where T1 is the generalized taxes = − z14 T (t) + z19 T (t − h) − T˙20 (t) − z21 T˙ (t − h); g0 is the autonomous government outlay, e the exchange rate, the tariffs, d the transportation, trade policy, or distance between trading nations, M1 the money supply and its ﬂows M˙ 1 , f0 the foreign credit and equalization taxes. The representative private ﬁrms’ control strategy and reaction to government intervention g is p = [C0 , I0 , X0 , M0 , n, w, x0 , y0 , p0 ], where x0 is the intercept associated with net export, y0 the intercept associated with supply, p0 the intercept associated with prices, all on the supply side with aggregate demand z, z=C+I +X+G ˙ + z4 y(t ˙ − h) + z5 R(t) = z0 + a1 y(t) + z2 y(t − h) + z3 y(t) ˙ − h) + z11 K(t) + z6 (t − h) + z8 L(t) + z9 L(t − h) + z10 L(t ˙ − h) + z18 p(t) − z14 T (t) − z15 e(t) + z16 (t) + z17 d(t) z13 R(t − z19 T (t − h) − z20 T˙ (t) − z21 T˙ (t − h).

788

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

The matrices

⎡

a−11

0

a−13

0

0

0

⎤

⎥ ⎢ a−22 0 0 0 0⎥ ⎢ 0 ⎥ ⎢ 0 −l−01 0 0 0⎥ , l A−1 = ⎢ ⎥ ⎢ −03 0 a6 −a−1 0 0 ⎥ ⎢ a3 ⎦ ⎣ 0 −M7 p6 p(t) 0 0 0 0 0 0 0 b4 b8 b12 ⎤ ⎡ A15 0 0 0 A13 a11 A24 0 0 0 0 ⎥ ⎢ A22 ⎥ ⎢ l l (l − l ) 0 0 0 ⎥ ⎢ 4 1 2 5 A1 = ⎢ ⎥, a4 a5 −a1 0 0 ⎥ ⎢ a2 ⎦ ⎣ −p6 p(t) −M4 p6 p(t) 0 0 0 0 b6 b10 0 0 b17 b2 ⎡ 1 −z15 1 1 z16 1 z17 1 z18 −1 0 0 0 0 0 −2 ⎢ ⎢ ⎢ −m(w)zs14 m(w) m(w)zs15 mzs16 mzs17 mzs13 B1 = ⎢ 1 zs15 zs16 zs17 zs13 ⎢ −zs14 ⎣ 0 0 p6 0 0 p1 p f (t) 0 0 0 b7 b8 b15 ⎡ 1 1 −1 (I13 + C7 ) 0 0 0 −1 0 0 − 2 0 0 0 0 ⎢ 0 ⎢ m(w) 0 0 0 0 m ⎢ 0 B2 = ⎢ 1 0 0 0 0 1 ⎢ 0 ⎣ 0 0 0 −p6 −p3 p2 0 0 0 0 −1 0 0 1 ⎡ ⎤ a12 a14 a16 −a18 0 a01 a23 0 0 a25 0⎥ ⎢ a21 ⎢ ⎥ 0 Ma 8 0⎥ 0 1 l0 ⎢ A0 = ⎢ ⎥. 0 0 0 a0 a8 0⎥ ⎢ ⎣ ⎦ −p6 M1 p −p6 M3 p 0 0 (M6 + p4 )p 0 b1 b5 b9 0 0 0

0 0 0 0 p5 0

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ ⎦ 0 −1 ⎤

0 0 0 ⎥ ⎥ m 0⎥ ⎥, 1 0⎥ ⎦ 0 1 0 0

The equations we derived can be put in matrix form as follows: x(t) ˙ − A−1 x(t ˙ − h) = A0 x(t) + A1 x(t − h) + (t) + q(t).

(3)

This is the equation we are looking for, a dynamic game of pursuit [2, 1.10]. The theory and details are contained in [6]. We have confronted the appropriate formulae with data and used regression methods and MATLAB code to identify the matrices for each of the following countries: USA, Italy, India, Germany, UK, Austria, Australia, Japan, Canada, China, Brazil, and Nigeria. For each country the differential game of pursuit (1) or (2) can be seen to be equivalent (using Hájek’s idea) to a control system x(t) ˙ − A−1 x(t ˙ − h) = A0 x(t) + A1 x(t − h) − w(t),

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

789

where w(t) ∈ W , and W = (P + ker U (t1 − t))∗− Q, with P representing the possibilities of private strategies (P = {p ∈ P ⊂ E 9 } and Q the possibilities of government controls (Q = {g ∈ Q ⊂ E 8 }). Here U (t1 , t) = U (t1 − t) is the fundamental matrix solution of x˙ (t) − A−1 x(t ˙ − h) = A0 x(t) + A1 x(t − h). Recalling the various concepts of controllability of linear neutral systems in Chapter 10 of [2], we prove that the models so identiﬁed are controllable. With hard limits on the control data it can also be proved that the models are constrained controllable when stabilized, and time optimal controllable. Thus from a poor economic state one can steer toward a very good target in minimum time, or with minimum investment using government and representative ﬁrms’ strategies. Implicit in this analysis is the availability of resources from the universe and the earth.

3. Interacting nations’ GDP In the last section, we considered individual models of the economic state of several nations where we used only the net export function, X(t) = x0 + x1 y(t) + x2 y(t − h) + x3 y(t) ˙ + x4 y(t ˙ − h) + x5 R(t) + x8 L(t) ˙ + x11 L(t − h) + x10 L(t) + x12 p(t) + x16 (t) + x15 e(t) + x17 d(t),

(4)

as the link with other nations. Here is the tariff, e(t) is exchange rate. We insert the effects of interaction between one country whose GDP is y1 , and other nations 2, 3, 4 with GDP’s y2 , y3 , y4 into the net export equation for country 1. Thus we must add y1 (b11 y2 + c11 y3 + d11 y4 ) = f (t),

(5)

to the export function. This is the inﬂow of “wealth from outside.” It could be negative. Mathematically, this is part of net export, a function of GDP, tariff, exchange rate, trade policies, etc. Consider the ordinary differential models ﬁrst. There are two groups of four nations in each group-Nigeria, USA, UK and China; and Egypt, USA, Jordan, and Israel. This model includes the interaction term, a matrix A1 (x(t)), a “solidarity function” or a “competition matrix.” This system of equations is written out here in greater detail. The equations for y1 , y2 , y3 , y4 can be written in matrix form as follows. If x = [y1 , y2 , y3 , y4 ] ,

(6)

790

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

then x(t) ˙ = A0 x(t) + A1 (x(t))x(t) + B1 p + B2 g,

(7)

where ⎤ ⎡ 0 0 0 a111 0 0 0 ⎥ a211 ⎢ 0 ⎢ a22y2 A0 = ⎣ ⎦ , A1 (x(t))= ⎣ 0 0 0 a311 a31y3 0 0 0 a411 a41y4 ⎡ ⎤ ⎡ ⎤ 1 0 0 0 1 0 0 0 ⎢0 1 0 0⎥ ⎢0 1 0 0⎥ B1 = ⎣ ⎦ , B2 = ⎣ ⎦. 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 ⎡

a12y1 0 a32y3 a42y4

a13y1 a23y2 0 a43y4

⎤ a14y1 a24y2 ⎥ ⎦, a34y3 0

That is, x(t) ˙ = A0 x(t) + A1 (x(t))x(t) + B1 p + B2 g,

(8)

where ⎤ p1 ⎢p ⎥ p = ⎣ 2⎦, p3 p4

⎡

⎤ g1 ⎢g ⎥ g = ⎣ 2⎦. g3 g4

⎡

Let Bu = B1p + B2q = [B1 If B is continuous and [p Rank

p

B2 ][q ].

(9)

q] does not vanish in any set of positive measure, then

B = 4,

(10)

and system (8) is controllable if ﬁrms and government are cooperating since Rank

[B

AB

A2 B

A3 B] = 4.

(11)

Let pi = 1 (C0i + (I0i + X0i )), gi = i (g0i + C1i Ti + C2 T˙i (t) − I2 Ti (t) − I2 T˙i (t)) X16i (t) + X15i ei (t), i = 1, 2, 3, 4.

(12)

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

791

⎤ C01 ⎢ I01 ⎥ ⎥ ⎢ ⎢ X01 ⎥ ⎥ ⎢ ⎢ C02 ⎥ ⎥ ⎤⎢ 0 ⎢ I02 ⎥ ⎥ ⎢ 0 ⎥ ⎢ X02 ⎥ ⎥. ⎦⎢ 0 ⎢ C03 ⎥ ⎥ ⎢ 4 ⎢ I03 ⎥ ⎥ ⎢ ⎢ X03 ⎥ ⎥ ⎢ ⎢ C04 ⎥ ⎦ ⎣ I04 X04 ⎡

⎡

1 ⎢0 B1p = ⎣ 0 0

1 0 0 0

1 0 0 0

0 2 0 0

0 2 0 0

0 2 0 0

0 0 3 0

0 0 3 0

0 0 3 0

0 0 0 4

0 0 0 4

Let Tai = C1i Ti + C2i T˙i (t) − i2i Ti − I2 T˙i (t)

(13)

and ⎡

B1g

1 1 1 x161 1 x151 0 0 ⎢0 0 =⎣ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 x163 3 x153 0 0 0 0 ⎡ ⎤ g01 ⎢ Ta1 ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ e1 ⎥ ⎢ ⎥ ⎢ g02 ⎥ ⎢ ⎥ ⎢ Ta2 ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢e ⎥ × ⎢ 2 ⎥. ⎢ g03 ⎥ ⎢ ⎥ ⎢ Ta3 ⎥ ⎢ ⎥ ⎢ 3 ⎥ ⎢ ⎥ ⎢ e3 ⎥ ⎢ ⎥ ⎢ g04 ⎥ ⎢ ⎥ ⎢ Ta4 ⎥ ⎣ ⎦ 4 e4

0 2 0 0 0 0 0 4

0 0 0 2 2 x162 2 x152 0 0 0 0 0 0 ⎤ 0 0 0 0 0 0 ⎥ ⎦ 0 0 0 4 4 x164 4 x154

(14)

Clearly, the rank of B1 is 4 if i , i = 1, . . . , 4, is never zero. Also the rank of B2 is 4 if i , i X15i , i = 1, . . . , 4, is not zero. From the numerical work i X16i , i X15i , i = 1, . . . , 4, is usually not zero.

792

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

We earlier derived the dynamics (8). For the private control strategies, p, of the representative ﬁrms one uses autonomous consumption, autonomous investment and autonomous net export of the four interacting countries. The government strategy, g, is a four component vector, [g01 , Tai , i , ei ], of autonomous government outlay, generalized taxes and their ﬂows, tariff, and exchange rate. To study the controllability question we proceed as follows (in the spirit of Hájek). Let U = {u : Bu + B2 Q ⊂ B1 P } = B1 P−∗ B2 G,

(15)

(Pontragin difference of sets). Then x(t) ˙ = A0 x(t) + A1 (x(t))x(t) + Bu is the equivalent control system. We use the game theoretic formulation of Hájek (see the cited Ref. [9]) and study the controllability of this equation. We now gather the dynamical systems of gross-domestic products dy1 (t) dt dy2 (t) dt dy3 (t) dt dy4 (t) dt

= y1 (a111 + a12 y2 (t) + a13 y3 (t) + a14 y4 (t)) + p1 + g1 , = y2 (a211 + a22 y1 (t) + a23 y3 (t) + a24 y4 (t)) + p2 + g2 , = y3 (a311 + a31 y1 (t) + a32 y2 (t) + a34 y4 (t)) + p3 + g3 , = y4 (a411 + a41 y1 (t) + a42 y2 (t) + a43 y3 (t)) + p4 + g4 ,

The system can be studied for controllability from earlier research, where it is identiﬁed in US22.m, UK22.m, Nigeria.m, China22.m in [2,4,6]. There it is demonstrated that the economic states of each of the four nations are controllable. By the economic state we mean the vector x = [y, R, L, K, p, E], with components y the GDP, R the interest rate, L the employment, K the capital stock, p the prices, E the balance of payment, see [1,2,6]. In [4] the ordinary Differential Systems Model of Economy of Austria, South Africa, United Kingdom, USA and Nigeria are studied. The economic data from the International Statistic Yearbook 1994 are adopted. The dynamics are generated in US22.m, Austria22.m, UK22.m, SouthA22.m ([6], pp. 155–182). The output and graphs are displayed. For the ordinary interacting systems-Nigeria, US, UK, China, analogous programs are written in MATLAB and MAPLE and controllability deduced. Also inferred is the possibility of nondecreasing or increasing growth of the gross-domestic product.

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

793

Indeed, suppose the following model is constructed for a group of four interacting countries: dx1 dt dx2 dt dx3 dt dx4 dt

= x1 (−a1 + b1 x2 + c1 x3 + d1 x4 ) + e1 , = x2 (−a2 + b2 x2 + c2 x3 + d2 x4 ) + e2 , = x3 (−a3 + b3 x2 + c3 x3 + d3 x4 ) + e3 ,

(16)

= x4 (−a4 + b4 x2 + c4 x3 + d4 x4 ) + e4 ,

If bi , ci , di , i = 1, . . . , 4, are all positive the system is cooperative. If the natural decay rates ai , i = 1, . . . , 4 are sufﬁciently small or negative, dxi /dt, i = 1, . . . , 4 are positive and x(t) → ∞ as t → ∞. Since xi represents the GDP, it is ever increasing: sustained economic growth is obtained. If bi is negative, x1 > 0, x3 > 0 then bi x1 x2 < 0. This has a bad effect on the growth of x1 . Because the economic model for x1 is controllable we can drive x1 (0), to x1 (t) in time t; also x2 (0) can be steered to x2 (t) in time t. Using control strategies, x1 , x2 can be steered to bigger values. Citizens of nations i = 1, 2 can tolerate transferring positive 2b1 x1 x2 to dx1 /dt to yield dx1 = x1 (−a1 + m1 x2 + c1 x3 + d1 x4 ) + e1 , dt m1 x1 x2 = 2b1 x1 x2 − b1 x1 x2 > 0, etc. Thus dx1 /dt can be made positive and x1 increasing. Sustained growth of GDP can be obtained. Cooperation can sustain economic growth in the region. Observe that implicit in the controls are enough natural resources for the countries. There is enough energy from the sun for humans to exploit! 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 illustrate 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 the third and drive it to extinction. The remaining two will compete it out, and the ﬁrst will drive the second to extinction. The jubilant, triumphant number one will think he controls the universe, but weather and natural calamities can wipe number one off the planet earth. There is no “love” from another to help him. The others are extinct. Recall the following theorem of Ahmad and Lazer [12]. Theorem 3.1. Consider the system x˙1 (t) = x1 (t)[a1 (t) − b11 (t)xI (t) − b12 (t)],

(J)

x˙2 (t) = x2 (t)[a2 (t) − b21 (t)xI (t) − b22 (t)x2 (t)],

(H)

794

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

where a1 (t), b1 (t) are continuous and bounded above and below by positive constants. Suppose fL = inf f (t), fM = sup f (t), respectively, and a1L > b1M a2M /b22L

and

a2M < b21L a1L /b11M ,

(I)

then any solution (x1 (t), x2 (t)) with x1 (0) > 0, x2 (0) > 0 has the property that x2 (t) → 0

as t → ∞,

x1 (t) → x1∗ (t),

where x1∗ is a solution of (J). After a long time, x2 all but vanishes x1 (t) executes the dynamics x˙1 (t) = x1 (t)[a1 (t) − b11 (t)x1 (t)] + f (t),

(17)

where f (t) may represent the effects of weather and other calamities which can help drive x1 to zero. In [5] we derived the full hereditary model. ˙ − h) = A0 x(t) + A1 x(t − h) + A2 (x(t))x(t − h) + B1 p + B2 g x(t) ˙ − A−1 x(t (18)

as the dynamics of the four GDP’s of the four nations which are linked up together in solidarity or competition. We deduced that rank(B1 , 0) = 4, rank(B2 , 0) = 4 since ⎤ C10 ⎢ I10 ⎥ ⎥ ⎢ ⎢ X10 ⎥ ⎥ ⎢ ⎢ C20 ⎥ ⎥ ⎢ ⎤ 0 ⎢ I20 ⎥ ⎥ ⎢ 0 ⎥ ⎢ X20 ⎥ ⎥, ⎦⎢ 0 ⎢ C30 ⎥ ⎥ ⎢ 4 ⎢ I30 ⎥ ⎥ ⎢ ⎢ X30 ⎥ ⎥ ⎢ ⎢ C40 ⎥ ⎦ ⎣ I40 X40 ⎡

⎡

1 ⎢0 B1p = ⎣ 0 0

1 0 0 0

1 0 0 0

0 2 0 0

0 2 0 0

0 2 0 0

0 0 3 0

0 0 3 0

0 0 3 0

0 0 0 4

0 0 0 4

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

⎡

B2g

1 1 X151 1 X161 0 0 ⎢0 =⎣ 0 0 0 0 0 0 0 0 0 0 0 0 3 3 X153 3 X163 0 0 0 ⎡ ⎤ g01 ⎢ e1 ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ Ta1 ⎥ ⎢ ⎥ ⎢ g02 ⎥ ⎢ ⎥ ⎢ e2 ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢T ⎥ × ⎢ a2 ⎥ . ⎢ g03 ⎥ ⎢ ⎥ ⎢ e3 ⎥ ⎢ ⎥ ⎢ 3 ⎥ ⎢ ⎥ ⎢ Ta3 ⎥ ⎢ ⎥ ⎢ g04 ⎥ ⎢ ⎥ ⎢ e4 ⎥ ⎣ ⎦ 4 Ta4

1 0 0 0 0 0 3 0

0 0 0 2 2 X152 2 X162 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 X154 4 X164

795

0 2 0 0 ⎤ 0 0⎥ ⎦ 0 4

The differential game of pursuit [2, 10.4] is equivalent to the control system x(t) ˙ − A−1 x(t ˙ − h) = A0 x(t) + A1 x(t − h) + A2 (x(t))x(t − h) + Bv,

(19)

where v ∈ V,

V = {v : Bv + B2 Q ⊂ B1 P }.

This is true because det(A−1 ) = 0. Because rank(B) = 4 with rank(B1 ) = 4, rank(B2 ) = 4 the control system is function space controllable (see [2], p. 394). Some interesting economic policies can be made from our analysis and theory.

796

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

4. Policy We extract the following data generated for United States’ Economy Relative to other countries. ∗ y2 (t − h) dy2(t)/dt − a22 = a02∗ y2(t) + a12∗ y(t − h) + y2(t)∗ (a21∗ y1(t − h) + a23∗ .y3(t − h) + a24 ∗ y4(t − h)) + p2(t) + g2(t),

p2 = 11∗ (CC(1) + I I (1) + XX(1))/(1 − 11∗ Z4) = 632.12, g2(t) = (1387063249.13 − 397726.95.e(t) + 0.ta(t) + 0d(t) + 279.16T (t) − 73506.69∗ T (t − h) + 120485.89T (t) − 1.7624916∗ T (t − h)/(4851492.752),

dy2(t)/dt + 1.250∗ y2 (t − h) = −9.277x10 − 5∗ y2(t) + 0.91661∗ y2(t − h) + y2(t)∗ (−0.002687∗ y1(t − h) + 4.9979x10−4∗ .y3(t − h) − 7.3097x10−4∗ .y4(t − h)) + p2(t) + g2(t). The above equations were Chukwu’s model of the gross domestic product of US interacting with China, Nigeria, and UK. The impact of China on the rate of growth of US GDP is negative. From the equation one can remedy this by imposing tariffs (with its consequences and other countries’ reaction), or by reducing exchange rate. Senator Elizabeth Dole said in Winston–Salem that she plans to push a bill to slap a 27.5% tariff on Chinese imports if China persists in what she calls unfair trade practices. It is easy to see how tempting this can be. Mathematically it can increase the value of dy2 (t)/dt, the growth rate of GDP. The centerpiece of President Bush’s Asia-Paciﬁc trip (Friday, October 17, 2003, News and Observer: Nation) is a two-day summit of Asia-Paciﬁc Economic Cooperation organizations in Bangkok. Bush said he will urge the leaders of Japan and China to stop manipulating currency markets to keep the value of their currencies low in relation to the dollar. This makes American-made goods expensive abroad. American manufacturers say the strong dollar has badly hurt their foreign sales and brought cuts in jobs at home. The US Economy lost nearly 3 million jobs recently. For competitive hereditary systems without control strategies we cite some theoretical results on competition. The research on interacting species was done by Dr. Pao [13], who proves that it is possible to have convergence to a positive steady state even if there is diffusion and delay provided competition is restricted, i.e., competition coefﬁcients are “bounded,” “regulated,” competition is not too big.

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

797

Theorem 4.1. Consider u(t) ˙ = 1 u(1 − u(t) − 1 v(t − 1 ) − 1 w(t − t3 )), v(t) ˙ = 2 v(1 − v(t) − 2 u(t − 1 ) − 2 w(t − t3 )), w(t) ˙ = 3 w(1 − w(t) − 3 u(t − 1 ) − 3 v(t − t2 )), u(t) = 1 (t)(t ∈ I1 ), w(t) = 3 (t)(t ∈ I3 ).

(20) t >0

v(t) = 2 (t)(t ∈ I2 ), (21)

For each i = 1, 2, 3, i , i , and i are positive constants, i 0 and Ii = [−i , 0]. Let 0 1 1 −1 1 1 A0 = 2 0 2 , A1 = 2 −1 2 . 3 3 0 3 3 −1 Assume A1 is nonsingular. The condition i + i < 1

for i = 1, 2, 3,

(22)

holds. Then (20)–(21) has a unique equilibrium solution ∗ M for some constant vector M = (M1 , M2 , M3 )T , and for any nonnegative (t), and i (0) > 0 the corresponding solution (u(t), v(t), w(t)) of (20) converges to ∗ as t → ∞. Pao’s theorem can be interpreted thus. If competition is “regulated,” “limited,” or “held in check”, i.e. (22), then the solution does not suffer extinction provided A1 is nonsingular. There is persistence for the three competing GDP’s, regardless of the delay. The model can be confronted with data and validated using data in [10,14]. 5. The Omega problem of a United Nations of the world The author postulates that the most important applied mathematics problem since the world began is connected with the worldwide conquest of scarcity subject to the values of love and goodness (see Genesis 3:17–18, Matthew 25: 31–41). This is a mathematics applied problem in the sense that the mathematical economic state of all nations is realistically mathematically created as a dynamical system with interacting solidarity matrix and with government and private strategies. The model is tested for controllability from a state of low growth of GDP, high interest rate, low employment, low value of capital stock, high inﬂation, and negative cumulative balance of payment to very high growth rate of GDP, low interest rate, full employment, low prices (or small inﬂation), and great cumulative balance of payment—the state of paradise. Is it possible to steer to this target with minimum investment in minimum time? The world then uses this model to implement a strategy and usher in universal abundance and prosperity. A working concept of goodness and love is: feed the hungry, give drink to the thirsty, welcome strangers in our homes, clothe the naked, heal and take care of the sick, visit and care for prisoners. Implicit in this is a sustained state of high GDP and full employment, etc. We begin our modeling by considering the system x(t) ˙ − A−1 x(t ˙ − h) = A0 x(t) + A1 x(t − h) + A2 (x)x(t − h) + B1 p + B2 q,

798

E.N. Chukwu / Nonlinear Analysis 63 (2005) 785 – 798

where x = [x1 , . . . , xN ], N of all nations of the world, and use our analysis above as a guide. The econometric meter described in Nonlinear Analysis: Real World Applications ([3], 75–84) is seen to be a powerful tool to promote “goodness through optimal dynamics of the wealth of nations” as reported in Nonlinear Analysis: Real World Applications (4: 2003, 653–666). One can initiate the study of the Omega problem by consulting the recent article by Jiandong Zhao, Jiffa Jiang, Alan C. Lazer, [11], in Chapter 8 of the book by Yang Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering (Vol. 191, Academic Press, 1993), [15]. References [1] E.N. Chukwu, Stability and Time Optimal Control of Hereditary Systems, Academic Press, Boston, 1992. [2] E.N. Chukwu, Stability and Time Optimal Control of Hereditary Systems with Applications to the Economic Dynamics of the US, second ed., World Scientiﬁc, Singapore, 2001. [3] E.N. Chukwu, International economic models as surveillance for the optimal control of the growth of wealth of nations, Nonlinear Anal.: Real World Applications 3 (2002) 75–86. [4] E.N. Chukwu, Optimal Control of the Wealth of Nations, Taylor & Francis, London and New York, 2003. [5] E.N. Chukwu, Cooperation and competition in modeling the dynamics of gross-domestic product of nations, Applied Mathematics and Computation, Elsevier, Amsterdam, 2004 www.elsevier.com/locate/amc. [6] E.N. Chukwu, Differential models and neutral systems for controlling the wealth of nations, Advances in Mathematics for Applied Sciences, vol. 54, World Scientiﬁc, Singapore. [7] E.N. Chukwu, A Mathematical Treatment of Economic Cooperation and Competition among Nations with Nigeria, USA, UK, China and Middle East Examples, Academic Press, Elsevier, in preparation. [8] R.C. Fair, Speciﬁcation, Estimation, and Analysis of Macroeconometric Models, Harvard University Press, Cambridge, MA, 1984. [9] O. Hájek, Pursuit Games, Academic Press, New York, 1975. [10] International Financial Statistics, Yearbook, 1994. [11] Jiandong Zhao, Jifa Jiang, Alan C. Lazer, The permanence and global attractivity in a nonautonomous Lotka–Volterra System, Nonlinear Anal.: Real World Applications 5 (2004), 265–276. www.elsevier.com/locate/na. [12] H. Lazer, S. Ahmad, On the autonomous Lotka competition equation, Proceedings of the American Mathematical Society 1 (1993) 117. [13] C.V. Pao, Global Asymptotic stability of Lotka–Volterra competition systems with diffusion and time delays, Nonlinear Anal. 5 (2004) 91–104. [14] UN National Accounts Statistics. The Economists Book of Vital World Statistics, M. Smith-Morris (Ed), Butler and Tanner, Frome, England, 1990. [15] Yang Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.