A Mathematical Treatment of Economic Cooperation and Competition Among Nations, with Nigeria, USA, UK, China, and the Middle East Examples

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Table of Contents for Problems Section Page Computer Code in Modeling Economic Dynamics of Interacting Nations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C1

0

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C2

1

Cooperation and Competition for DATA Program and Equation Identification to Model ..................................

C2

International Cooperation an Competition Models with MATLAB . . .

C4

Program Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Program and Run, UScom.m, Egyptcom.m, Jordancom.m, and Israelcom.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C11 C19

Generation of Ordinary Differential Models: US22.M, UK22.M, Chinacu.M, Nigeria.M (MATLAB) . . . . . . . . . . . . . . . . . . . . . . . . . . .

C33

2.1 Nigeria and the Four Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C74

Computer Programs and Results for Hereditary Models of Nigeria, US, UK, and China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C79

4

Program DATA for US, UK, Egypt, Jordan and Israel . . . . . . . . . . . .

C99

5

Problems and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C121

6

MATLAB Programs and Graphs for Economic Models with Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C127

7

MATLAB EXECUTION of Program by Emeka Chukwu . . . . . . . . . C208

2

3

Table of Contents for Appendices Appendix 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A1

Appendix I Nigeria, U.S.A., U.K., China: Cooperation and Competition Programs and Results MAPLE and MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A113 Appendix II Nigeria – Regional Cooperation and Competition: Programs and Results MAPLE and MATLAB . . . . . . . . . . . A115 Appendix III Middle East Competition and Cooperation: Programs and Results MAPLE and MATLAB . . . . . . . . . . . A128 Appendix IV Hereditary Model with Maple by Norris . . . . . . . . . . . . . . . A139 Appendix V - User Friendly Program for the Full Model for 4 Countries by L. K. Norris . . . . . . . . . . . . . . . . . . . . . . . . .

B1

A1

APPENDIX 0

A2

US3.M

A3

A4

A5

A6

A7

A8

A9

A10

A11

A12

A13

A14

A15

y 4 = y 40 ÷ 0.241255 = [

]´;

A16

A17

A18

y2 = y ÷ 0.257281 = [

]′;

A19

A20

y3 = y30 × 1.40787 = [

]′

A21

A22

y 4 = y 40 ÷ 0.241255 = [

]´;

A23

A24

A25

A26

A27

APPENDIX to Section 1 MATLAB Programs and Graphs For Economic Models with Delay

A28

Program US2.M 1.

Definition of Economic Variables and Terms, pp. A.2-A.3. In this section the economic variables extracted from the International Financial Statistic Yearbook are defined by Symbols.

2.

The U.S.A. data, pp. A.1-A.2. Displaced here are data extracted from the International Financial Statistic Yearbook 1994, UN Financial Statistic 1974, UN National Accounts Statistics.

3.

Equations and Formulae postulated for economic variables in the body of the book are now identified, pp. A.3-A.5. ML = L – M L I C X T G Z

A.3 A.3 (1.5) (1.3) (1.11) (1.2) (1.7) (1.2), (1.12), (1.13)

R& (t ) p& (t )

(1.20), (1.21), (1.23)

B = Balance of Payment E = Cumulative Balance of Payment y = National Income D = deliveries of new equipment dk = flow of capital stock dt L = employment y = income Gy = income government Xy = income export

(1.35), inflation (1.38) (1.37) (1.44) (1.48) (1.53) (1.60) (1.52) (1.50) (1.50)

4.

MATLAB Regression Programs for Economic Variables, pp. A.3-A.6. See “System Indentification Toolbox” for use with MATLAB, Lenhart Ljung, The Math Works, Inc. I.30-I.32, 14.8, pp. 1.79, 2.16.

5.

MATLAB Plot, Subplot programs, pp. A.5-A.6.

6.

Identification

of

the

economic

x& (t ) − A−1 x& (t − h) = A0 x(t ) + A1 x(t − h) + B * u (t )

dynamics:

A29

with coefficients A−1 , A0 , A1 , B, B1 , and given in pp. A.6-A.7 and identified in A.8-A.9 using (Ndu.M, US3.M) program. In A.8-A.13. 7.

Diagrams and plots, Fig. U.S.1 → Fig. U.S.7.

8.

We use the rank condition of Salamon and the full rank of B to deduce the (t )

controllability in W p

of the economic state variables of the dynamics

x& (t ) − A−1 x& (t − h) = A0 x(t ) + A1 x(t − h) + Bu (t )

If γ is any complex number, rank [∆ ( γ ), B ] = n = 6 and rank [ γI − A−1 , B] = 6, are the required conditions (pp, 157, D. Salamon, “Control and Observation of Neutral Sysems”, Pitman Advanced Publishing Program, Boston, MA). This rank condition is satisfied by our linear model, pp. A.22. See [7].

A30

Programs 1.

U.S.A.2.M Jordan2.m Data STATE: x = [ y, R, L, k , p, E ]′ Control Private firms: σ = [C 0 , I 0 , X 0 , M 0 , n, w, x0 , y 0 , p0 ]′ government q = [T1 , g 0 , e, τ, d , M 1, M& 1, t 0 ]′ Program Code Ndu.m Output Diagrams

A31 US3.M

A32

A33

A34

A35

A36

A37

A38

A39

A40

A41

A42

A43

A44

A45

A46

A47

A48

A49

y3 = y30 × 1.40787 = [

]´

A50

A51

A52

A53

A54

A55

A56

%rank (B) = 6 = rank (B, 0) %rank (G) = rank (G, 0) = 6

A57

A58

A59

A60

A61

A62

A63

A64

A65

A66

A67

A68

A69

A70

A71

A72

%G = [B AB A2B A3B A4B A5B] %rank (B, 0) = rank (B) %rank (B) = 6 %rank (B) = 6

A73

A74

A75

A76

A77

A78

A79

Israel’s Results

A80

A81

A82

A83

A84

A85

A86

%rank (B, 0) = rank (B) %rank (G, 0) = rank (G) %rank (B) = 5 %rank (G) = 6

A87

A88

A89

A90

A91

A92

A93

A94

A95

A96

APPENDIX O

UK22.M US22.M

A97

A98

A99

A100

A101

A102

A103

A104

A105

y2 = y ÷ 0.257281 = [

]′;

A106

A107

A108

y3 = y30 × 1.40787 = [

]′

A109

A110

y 4 = y 40 ÷ 0.241255 = [

]´;

A111

A112

A113

APPENDIX I Nigeria, U.S.A., U.K., China: Cooperation and Competition Programs and Results MAPLE and MATLAB

A114

MAPLE EXECUTION

By Professor Larry Norris

A115

APPENDIX II Nigeria – Regional Cooperation and Competition: Programs and Results

MAPLE and MATLAB by Professor Norris

A116

A117

A118

A119

A120

A121

A122

A123

A124

A125

A126

A127

A128

APPENDIX III Middle East Competition and Cooperation: Programs and Results

MAPLE and MATLAB

A129

A130

A131

A132

A133

A134

A135

A136

A137

A138

A139

APPENDIX IV Hereditary Model with Maple by Norris

A140

A141

A142

A143

A144

A145

A146

A147

A148

A149

A150

A151

A152

A153

A154

A155

A156

Set up the coefficients for the differential equations We use the coefficients computed in the last section to define the coefficients for the differential equations.

A157

The Differential Equations

Must add in terms for g[countryX]

> EQN[US]:= D(Y[US])(t)- a[US][-1]*D(Y[US])(t-1) = a[US][0]*Y[US](t) > + a[US][1]*Y[US](t-1) > + Y[US](t)*(coeff_aa[US][Egypt]*Y[Egypt](t-1) + > coeff_aa[US][Jordan]*Y[Jordan](t-1)+ > coeff_aa[US][Israel]*Y[Israel](t-1) + p[US] + g[US]); EQNUS US := D(YUS )(t) - 2.194203600 D(YUS)(t - 1) = - 1.735004395 YUS (t) −

+ 2.317222048 YUS(t - 1) + YUS(t)(-.1456557517 10 7 YEgypt (t - 1) 10-5 YJordan (t - 1) + .1026864688 10-6 YIsrael (t - 1) + 4632.985406 + 858.6987962 eUS (t) + .02694357803 τUS (t) + 1.245437531 TUS (t) - 1.628373557 TUS(t - 1) - .09843320801 D(TUS )(t) - 1.543489422 D(TUS )(t - 1) ) - .7086660721

> EQN[Egypt] := D(Y[Egypt])(t)- a[Egypt][-1]*D(Y[Egypt])(t-1) = > a[Egypt][0]*Y[Egypt](t) + a[Egypt][1]*Y[Egypt](t-1) > + Y[Egypt](t)*(coeff_aa[Egypt][US]*Y[US](t-1) + > coeff_aa[Egypt][Jordan]*Y[Jordan](t-1)+ > coeff_aa[Egypt][Israel]*Y[Israel](t-1) + p[Egypt] + g[Egypt]); EQN Egypt := D(YEgypt )(t) + 3.506494781 D(YEgypt )(t - 1) = 8.327857170 YEgypt (t) - .2917088061 YEgypt (t - 1) + YEgypt (t)(- .1118126676 10−5 YUS (t - 1) - .00004412652028 YJordan (t - 1) - .1232539606 10−6 YIsrael (t - 1) + 26122.94421 + 23577.06585 eEgypt (t) - 3.397252849 τEgypt (t) - 5.016633990 TEgypt (t) + .6904502995 TEgypt (t - 1) - 3.372799571 D(TEgypt )(t) + 1.568556948 D(TEgypt )(t - 1)) > EQN[Jordan]:= D(Y[Jordan])(t)- a[Jordan][-1]*D(Y[Jordan])(t-1) = > a[Jordan][0]*Y[Jordan](t) + a[Jordan][1]*Y[Jordan](t-1) + Y[Jordan](t)*(coeff_aa[Jordan][US]*Y[US](t-1) + > coeff_aa[Jordan][Egypt]*Y[Egypt](t-1) + > coeff_aa[Jordan][Israel]*Y[Israel](t-1) + p[Jordan] + g[Jordan]);

A158

EQN Jordan := D(YJordan )(t) + 6.065252893 D(YJordan )(t - 1) = - .8074004946 YJordan (t)

- 1.779420225 YJordan (t - 1) + YJordan (t)(- .3532328762 10−5 YUS (t - 1) 5

+ .1103555320 10− YEgypt (t - 1) - .9925734476 10−6 YIsrael (t - 1) - 14415.97999 - 1119.769687 eJordan (t) - 6.667560030 τJordan (t) - 1.897597661 TJordan (t) + 1.737812654 TJordan (t - 1) + .3986898661 D(TJordan )(t) + 2.201118565 D(TJordan )(t - 1))

> EQN[Israel]:= D(Y[Israel])(t)- a[Israel][-1]*D(Y[Israel])(t-1) = > a[Israel][0]*Y[Israel](t) + a[Israel][1]*Y[Israel](t-1) + > Y[Israel](t)*(coeff_aa[Israel][US]*Y[US](t-1) + > coeff_aa[Israel][Egypt]*Y[Egypt](t-1) + > coeff_aa[Israel][Jordan]*Y[Jordan](t-1) + p[Israel] + g[Israel]); EQN

Israel :=

D(YIsrael )(t) + 1.027644760 D(YIsrael )(t - 1) = - 3.639595360 YIsrael (t)

- 3.969069969 YIsrael (t - 1) + YIsrael (t)(- .6575146123 10−6 YUS (t - 1) - .6790730561 10−7 YEgypt (t - 1) + .00006435701183 YJordan (t - 1) - 77345.80770 + 92.64006350 eIsrael (t) - 15.67604974 τIsrael (t) + .9243013213 TIsrael (t) + 4.760577195 TIsrael (t - 1) + 2.196499327 D(TIsrael )(t) + 2.380591892 D(TIsrael )(t - 1))

B1

APPENDIX V User Friendly Program for The Full Model for 4 Countries L. K. Norris

B2

4. Countries_CHUKWU.html

4Countries_CHUKWU.mws •

• •

•

• •

Original raw data o US Data o Egypt Data o Jordan Data o Israel Data Prepare the Data o Define the y_values, e_values, tau_values, T_values, T_values, I_values, C_values, G_values and the appropriate variables Correct data for exchange rates o Modify all y_values o Modify all T_values o Modify all tau_values o Modify all II_values o Modify all G_values o Modify all X_values o Modify all C_values Regressions for the coefficients in II, C, X and G for US, Egypt, Jordan and Israel ONLY o Regression for II (I, but Maple considers I = sqrt(-1)) o Regression for G o Regression for C – OK o Regression for X Set up the coefficients for the differential equations The Differential equations

Chukwu’s International Economic Model The Full Model for 4 Countries L.K. Norris June 8, 2005

1

Original raw data

YEARS:= [1966 ,1967 ,1968 ,1969 ,1970 ,1971 ,1972 ,1973 ,1974 ,1975 > ,1976 ,1977 ,1978 ,1979 ,1980 ,1981 ,1982 ,1983 ,1984 ,1985 ,1986,1987 > ,1988 ,1989 ,1990 ,1991 ,1992 ]; > nops(YEARS); >

YEARS := [1966, 1967, 1968, 1969, 1970, 1971, 1972, 1973, 1974, 1975, 1976, 1977, 1978, 1979, 1980, 1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988, 1989, 1990, 1991, 1992] 27 Note: US data runs over the years 1966 - 1992. However, the data for Jordan , Egypt and Israel runs over the period 1976-1996. I will have to adjust for this in the various calculations.

1.1

> > > > >

Us Data

Export[US]:= [39,41.4,45.3,49.3,57,59.3,66.2,91.8,124.3,136.3,148.9,158.8,186.2,228 .9,279.2,303,282.6,276.7,302.4,302.1,319.2,364,444.2,508,557,601.5,640 .5]; nops(Export[US]);

Export US := [39, 41.4, 45.3, 49.3, 57, 59.3, 66.2, 91.8, 124.3, 136.3, 148.9, 158.8, 186.2, 228.9, 279.2, 303, 282.6, 276.7, 302.4, 302.1, 319.2, 364, 444.2, 508, 557, 601.5, 640.5] 27

> > > > >

Import[US]:= [-37.1,39.9,-46.6,-50.5,-55.8,-62.4,-74.2,-91.2,-127.5,-122.7,-151.2,182.5,-212.3,-252.7,-293.9,-317.7,-303.2,-328.1,-405.1,-417.6,-451.7,507.1,-552.2,-587.7,-625.9,-621.1,-670.1]; nops(Import[US]);

Import US := [−37.1, 39.9, −46.6, −50.5, −55.8, −62.4, −74.2, −91.2, −127.5, −122.7, −151.2, −182.5, −212.3, −252.7, −293.9, −317.7, −303.2, −328.1, −405.1, −417.6, −451.7, −507.1, −552.2, −587.7, −625.9, −621.1, −670.1] > > > > > > > > >

27 ## Net export #X[US]:= Export[US] - Import[US]; X[US]:= eval([39,41.4,45.3,49.3,57,59.3,66.2,91.8,124.3,136.3,148.9,158.8,186. 2,228.9,279.2,303,282.6,276.7,302.4,302.1,319.2,364,444.2,508,557,601. 5,640.5] [-37.1,39.9,-46.6,-50.5,-55.8,-62.4,-74.2,-91.2,-127.5,-122.7,-151.2,182.5,-212.3,-252.7,-293.9,-317.7,-303.2,-328.1,-405.1,-417.6,-451.7,507.1,-552.2,-587.7,-625.9,-621.1,-670.1]) ;

XUS := [76.1, 1.5, 91.9, 99.8, 112.8, 121.7, 140.4, 183.0, 251.8, 259.0, 300.1, 341.3, 398.5, 481.6, 573.1, 620.7, 585.8, 604.8, 707.5, 719.7, 770.9, 871.1, 996.4, 1095.7, 1182.9, 1222.6, 1310.6] > ##Private consumption > C[US]:= > [481.6,509.3,559.1,603.7,646.5,700.3,767.8,848.2,927.7,1024.9,1143.1,1 > 271.5,1421.3,1583.7,1748.1,1926.3,2059.2,2257.6,2460.3,2667.4,2850.6,3 > 052.2,3296.1,3523.1,3748.4,3906.4,4139.9] ; > nops(C[US]); CUS := [481.6, 509.3, 559.1, 603.7, 646.5, 700.3, 767.8, 848.2, 927.7, 1024.9, 1143.1, 1271.5, 1421.3, 1583.7, 1748.1, 1926.3, 2059.2, 2257.6, 2460.3, 2667.4, 2850.6, 3052.2, 3296.1, 3523.1, 3748.4, 3906.4, 4139.9] > > > > >

27 II[US] := [130.4,128.1,139.9,155.2,150,175.3,205.2,242.5,245.1,225,285,358.2,434 ,480.3,467.6,558.1,503.4,546.7,718.9,714.6,717.7,749.3,793.7,832.2,799 .5,736.8,796.4]; nops(II[US]);

II US := [130.4, 128.1, 139.9, 155.2, 150, 175.3, 205.2, 242.5, 245.1, 225, 285, 358.2, 434, 480.3, 467.6, 558.1, 503.4, 546.7, 718.9, 714.6, 717.7, 749.3, 793.7, 832.2, 799.5, 736.8, 796.4] > > > > > >

27 ##Government expenditures G[US] := [622.4,667.9,686.8,682,665.8,652.4,653,644.2,655.4,663.5,659.2,664.1,6 77,689.3,704.2,713.2,723.6,743.8,766.9,813.4,855.4,881.5,886.8,904.4,9 29.9,941.1,943] ; nops(G[US]);

GUS := [622.4, 667.9, 686.8, 682, 665.8, 652.4, 653, 644.2, 655.4, 663.5, 659.2, 664.1, 677, 689.3, 704.2, 713.2, 723.6, 743.8, 766.9, 813.4, 855.4, 881.5, 886.8, 904.4, 929.9, 941.1, 943] > > > > > >

27 #exchange rate e[US]:= [1,1,1,1,1,1.0857,1.0857,1.2064,1.2244,1.1707,1.1618,1.2147,1.3028,1.3 173,1.2754,1.1640,1.1031,1.0470,0.9802,1.0984,1.2232,1.4187,1.3457,1.3 142,1.4227,1.4304,1.3750]; nops(e[US]);

eUS := [1, 1, 1, 1, 1, 1.0857, 1.0857, 1.2064, 1.2244, 1.1707, 1.1618, 1.2147, 1.3028, 1.3173, 1.2754, 1.1640, 1.1031, 1.0470, .9802, 1.0984, 1.2232, 1.4187, 1.3457, 1.3142, 1.4227, 1.4304, 1.3750] > > > > > >

27 # Tarrif tau[US]:= [1900,2000,2300,2400,2584,2767,3124,3620,3771,3780,4675,5485,7162,7202 ,7535,9993,8688,9430,12042,13067,13312,13923,15054,16096,16339,16197,1 7164]; nops(tau[US]);

τUS := [1900, 2000, 2300, 2400, 2584, 2767, 3124, 3620, 3771, 3780, 4675, 5485, 7162, 7202, 7535, 9993, 8688, 9430, 12042, 13067, 13312, 13923, 15054, 16096, 16339, 16197, 17164] 27 # trade # In the following u[1] through u[8] represents unknown data from > 1977-1984 > # Will need to estimate these numbers > # > # Comment out d. Data not available for Egypt, Jordan and Israel > # > #d[US]:=[112,113,117,117,100,102,106,108,104,105,108,u[1],u[2],u[3],u[ > 4],u[5],u[6],u[7],u[8],4.04,#4.7,5.3,3.9,1.3,1.9,1.8,4.5]; > #nops(d[US]); > d[US]:=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; > nops(d[US]); > >

dUS := [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] 27 The next entries will be B and F, and then T= X[0] - B[0] - F[0] where X[0] is given above. > # unknown NAME > B[US] := > [2.83,1.16,4.12,8.25,-7.26,-9.88,1.25,-1.28,-3.47,-12.89,-10.39,-4.73, > -17.53,14.75,30.53,35.99,-42.27,15.66,19,33.3,22.1,46.5,16.07,11.69,17 > .8,-6.38,41.22]; > nops(B[US]);

BUS := [2.83, 1.16, 4.12, 8.25, −7.26, −9.88, 1.25, −1.28, −3.47, −12.89, −10.39, −4.73, −17.53, 14.75, 30.53, 35.99, −42.27, 15.66, 19, 33.3, 22.1, 46.5, 16.07, 11.69, 17.8, −6.38, 41.22] > > > > > > > > > > >

27 # unknown NAME # # Comment out d. Data not available for Egypt, Jordan and Israel # #F[0][US]:= #[-0.65,-0.88,-0.84,-0.94,-1.10,-1.11,-1.11,-1.25,-1.01,-0.92,-0.91,-0 .82,-0.86,-0.91,-1.03,-4.51,#-8.73,-9.06,-9.75,-9.56,-10.12,-10.55,-11 .96,-12.32,-12.39,-14.04,-14.46]; #nops(F[0][US]); F[US]:= [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; nops(F[US]);

FUS := [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] > > >

27 # Taxes T[US]:= X[US] - B[US] - F[US] ; nops(T[US]);

TUS := [73.27, .34, 87.78, 91.55, 120.06, 131.58, 139.15, 184.28, 255.27, 271.89, 310.49, 346.03, 416.03, 466.85, 542.57, 584.71, 628.07, 589.14, 688.5, 686.4, 748.8, 824.6, 980.33, 1084.01, 1165.1, 1228.98, 1269.38] > > > > > > > >

27 # GDP y[US] := [769.8,814.3,889.3,959.5,1010.4,1096.8,1206.5,1349.1,1458.8,1584.8,167 6.1,1974.1,21232.7,2488.7,2808.1,3030.6,3149.6,3405.1,3777.2,4038.7,42 68.6,4539.9,4900.4,5250.8,5522.2,5722.9,6038.5]; nops(y[US]); y_at_t_minus_1[US]:= [seq(y[US][k],k=1..26)]; nops(y_at_t_minus_1[US]);

yUS := [769.8, 814.3, 889.3, 959.5, 1010.4, 1096.8, 1206.5, 1349.1, 1458.8, 1584.8, 1676.1, 1974.1, 21232.7, 2488.7, 2808.1, 3030.6, 3149.6, 3405.1, 3777.2, 4038.7, 4268.6, 4539.9, 4900.4, 5250.8, 5522.2, 5722.9, 6038.5] 27 y at t minus 1 US := [769.8, 814.3, 889.3, 959.5, 1010.4, 1096.8, 1206.5, 1349.1, 1458.8, 1584.8, 1676.1, 1974.1, 21232.7, 2488.7, 2808.1, 3030.6, 3149.6, 3405.1, 3777.2, 4038.7, 4268.6, 4539.9, 4900.4, 5250.8, 5522.2, 5722.9] > > > >

26 y_dot[US]:= [seq(y[US][k+1] - y[US][k],k=1..26)]; nops(%); y_dot_at_t_minus_1[US]:= [seq(y_dot[US][k],k=2..26)]; nops(y_dot_at_t_minus_1[US]);

y dot US := [44.5, 75.0, 70.2, 50.9, 86.4, 109.7, 142.6, 109.7, 126.0, 91.3, 298.0, 19258.6, −18744.0, 319.4, 222.5, 119.0, 255.5, 372.1, 261.5, 229.9, 271.3, 360.5, 350.4, 271.4, 200.7, 315.6] 26 y dot at t minus 1 US := [75.0, 70.2, 50.9, 86.4, 109.7, 142.6, 109.7, 126.0, 91.3, 298.0, 19258.6, −18744.0, 319.4, 222.5, 119.0, 255.5, 372.1, 261.5, 229.9, 271.3, 360.5, 350.4, 271.4, 200.7, 315.6] 25

1.2

Egypt Data

Years are 1976 - 1996 ===> only 21 entries

> > > >

Export[Egypt]:= [1034,1470,1945,3251,4322,5307,5810,6159,6387,6598,6034,6500,10700,138 00,19400,31000,40400,43500,40100,45100,48450]; nops(Export[Egypt]);

Export Egypt := [1034, 1470, 1945, 3251, 4322, 5307, 5810, 6159, 6387, 6598, 6034, 6500, 10700, 13800, 19400, 31000, 40400, 43500, 40100, 45100, 48450] > > > > >

21 Import[Egypt]:= [-1772,-2260,-3316,-5254,-6410,-7361,-8176,-8981,-10357,-10638,-10568, -11700,-21700,-24800,-31400,-39800,-44300,-48200,-49200,-49800,-59100] ; nops(Import[Egypt]);

Import Egypt := [−1772, −2260, −3316, −5254, −6410, −7361, −8176, −8981, −10357, −10638, −10568, −11700, −21700, −24800, −31400, −39800, −44300, −48200, −49200, −49800, −59100] 21

> >

## Net export X[Egypt]:= Export[Egypt] - Import[Egypt];

XEgypt := [2806, 3730, 5261, 8505, 10732, 12668, 13986, 15140, 16744, 17236, 16602, 18200, 32400, 38600, 50800, 70800, 84700, 91700, 89300, 94900, 107550]

> > > > >

##Private consumption C[Egypt]:= [3863,4917,6279,8623,11023,11155,13285,16224,21356,24927,27895,35900,4 3550,54100,68950,80900,101000,115000,130500,148900,171700] ; nops(C[Egypt]);

CEgypt := [3863, 4917, 6279, 8623, 11023, 11155, 13285, 16224, 21356, 24927, 27895, 35900, 43550, 54100, 68950, 80900, 101000, 115000, 130500, 148900, 171700] > > > > >

21 II[Egypt] := [1385+195,1825+561,2618+416,3346+450,4062+266,5108+100,6150+100,8233+1 00,9084+120,10555+130,12928+140,14100-650,20150+300,23100+900,26500+18 00,27850-1200,28700-1200,31000+12000,35600-1200,38600+700,42100+1550]; nops(II[Egypt]);

II Egypt := [1580, 2386, 3034, 3796, 4328, 5208, 6250, 8333, 9204, 10685, 13068, 13450, 20450, 24000, 28300, 26650, 27500, 43000, 34400, 39300, 43650] > > > > >

21 ##Government expenditures G[Egypt] := [1571,1697,1841,2059,2549,2841,3584,4160,4957,5668,6134,7350,8600,9700 ,10850,12450,14500,16000,18000,21500,23600] ; nops(G[Egypt]);

GEgypt := [1571, 1697, 1841, 2059, 2549, 2841, 3584, 4160, 4957, 5668, 6134, 7350, 8600, 9700, 10850, 12450, 14500, 16000, 18000, 21500, 23600] > > > > > >

21 #exchange rate e[Egypt]:= [0.4546,0.4753,0.5098,0.9221,0.8928,0.8148,0.7722,0.7329,0.6861,0.7689 ,0.8562,0.9931,0.9420,1.4456,2.8456,4.7665,4.5906,4.6314,4.9504,5.0392 ,4.8718]; nops(e[Egypt]);

eEgypt := [.4546, .4753, .5098, .9221, .8928, .8148, .7722, .7329, .6861, .7689, .8562, .9931, .9420, 1.4456, 2.8456, 4.7665, 4.5906, 4.6314, 4.9504, 5.0392, 4.8718] > > > > >

21 # Tarrif tau[Egypt]:= [399.4,557.4,767.6,958,1435,1707,1719,2031,2073,1927,1929,2463,2981,30 37,3422,4941,5428,6585,7352,8255,8460]; nops(tau[Egypt]);

τEgypt := [399.4, 557.4, 767.6, 958, 1435, 1707, 1719, 2031, 2073, 1927, 1929, 2463, 2981, 3037, 3422, 4941, 5428, 6585, 7352, 8255, 8460] 21 # trade # In the following u[1] through u[8] represents unknown data from > 1977-1984 > # Will need to estimate these numbers > d[Egypt]:=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; > nops(d[Egypt]); > >

dEgypt := [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] 21 The next entries will be B and F, and then T= X[0] - B[0] - F[0] where X[0] is given above. > # unknown NAME > B[Egypt] := > [-1200,-1200,-1200,-1542,-438,-2136,-1851,-330,-1988,-2166,-1811,-246, > -1048,-1309,185,1903,2812,2299,31,-254,-192]; > nops(B[Egypt]); BEgypt := [−1200, −1200, −1200, −1542, −438, −2136, −1851, −330, −1988, −2166, −1811, −246, −1048, −1309, 185, 1903, 2812, 2299, 31, −254, −192] > > >

21 # Net Private Outflow of Capital F[Egypt]:= [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; nops(F[Egypt]); FEgypt := [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

> > >

21 # Taxes T[Egypt]:= X[Egypt] - B[Egypt] - F[Egypt] ; nops(T[Egypt]);

TEgypt := [4006, 4930, 6461, 10047, 11170, 14804, 15837, 15470, 18732, 19402, 18413, 18446, 33448, 39909, 50615, 68897, 81888, 89401, 89269, 95154, 107742] > > > > >

21 # GDP y[Egypt] := [6276,8210,9783,12475,15470,17150,20753,45895,31547,37240,42563,51500, 61600,76800,96100,111200,139100,157300,175000,205000,228300]; nops(y[Egypt]);

yEgypt := [6276, 8210, 9783, 12475, 15470, 17150, 20753, 45895, 31547, 37240, 42563, 51500, 61600, 76800, 96100, 111200, 139100, 157300, 175000, 205000, 228300] 21

> >

y_dot[Egypt]:= [seq(y[Egypt][k+1] - y[Egypt][k],k=2..20)]; nops(%);

y dot Egypt := [1573, 2692, 2995, 1680, 3603, 25142, −14348, 5693, 5323, 8937, 10100, 15200, 19300, 15100, 27900, 18200, 17700, 30000, 23300] 19

1.3 > > > >

Jordan Data Export[Jordan]:= [192,242,264.3,339.5,448.0,588.5,670.2,639.6,746.8,781.5,634.1,756.2,1 020.8,1359.5,1652.1,1697.6,1819.9,1962.1,2093.4,2438.5,2597.2]; nops(Export[Jordan]);

Export Jordan := [192, 242, 264.3, 339.5, 448.0, 588.5, 670.2, 639.6, 746.8, 781.5, 634.1, 756.2, 1020.8, 1359.5, 1652.1, 1697.6, 1819.9, 1962.1, 2093.4, 2438.5, 2597.2] > > > > >

21 Import[Jordan]:= [-422,-540.3,-605.6,-824.5,-961.7,-1392.7,-1555.7,-1453.2,-1519.1,-150 2.7,-1199.5,-1319.7,-1519.7,-1804.5,-2474.3,-2362.6,-2974.7,-3151.7,-3 107.6,-3435.2,-3839.9]; nops(Import[Jordan]);

Import Jordan := [−422, −540.3, −605.6, −824.5, −961.7, −1392.7, −1555.7, −1453.2, −1519.1, −1502.7, −1199.5, −1319.7, −1519.7, −1804.5, −2474.3, −2362.6, −2974.7, −3151.7, −3107.6, −3435.2, −3839.9] > >

21 ## Net export X[Jordan]:= Export[Jordan] - Import[Jordan];

XJordan := [614, 782.3, 869.9, 1164.0, 1409.7, 1981.2, 2225.9, 2092.8, 2265.9, 2284.2, 1833.6, 2075.9, 2540.5, 3164.0, 4126.4, 4060.2, 4794.6, 5113.8, 5201.0, 5873.7, 6437.1] > _orginal_X_values[Jordan]:= > [seq(Export[Jordan][i]-Import[Jordan][i],i=1..19)]; orginal X values Jordan := [614, 782.3, 869.9, 1164.0, 1409.7, 1981.2, 2225.9, 2092.8, 2265.9, 2284.2, 1833.6, 2075.9, 2540.5, 3164.0, 4126.4, 4060.2, 4794.6, 5113.8, 5201.0]

> > > > > >

##Private consumption C[Jordan]:= [325.5,412.8,517.4,736.8,858.3,1074.5,1457.9,1579.1,1648.4,1794.8,1718 .2,1669.8,1626.5,1635.1,1976.5,2039.6,2648.4,2710.7,2774.3,3023.2,3393 .5] ; nops(C[Jordan]);

CJordan := [325.5, 412.8, 517.4, 736.8, 858.3, 1074.5, 1457.9, 1579.1, 1648.4, 1794.8, 1718.2, 1669.8, 1626.5, 1635.1, 1976.5, 2039.6, 2648.4, 2710.7, 2774.3, 3023.2, 3393.5] > > > > > >

21 II[Jordan] := [138+12.2,201+5.5,229.1-6.1,294.5-14.5,452.9+11,672.6+28.4,626.9+23.9, 535.9+53.9,526.8+44.4,384.8+30.1,409.3+35,448.5+67.1,513.4+19.1,554.1+ 9.1,694.0+156.1,678+60.5,1049.2+159.6,1303.5+119.2,1391+60,1395+159.3, 1445.3+52.1]; nops(II[Jordan]);

II Jordan := [150.2, 206.5, 223.0, 280.0, 463.9, 701.0, 650.8, 589.8, 571.2, 414.9, 444.3, 515.6, 532.5, 563.2, 850.1, 738.5, 1208.8, 1422.7, 1451, 1554.3, 1497.4] > > > > >

21 ##Government expenditures G[Jordan] := [155.9,156.6,190,235.3,342.7,455.5,477.9,473.4,534.6,531.7,566.5,586.7 ,604.3,618.8,663.9,742,790.6,857.9,990.2,1081.2,1194] ; nops(G[Jordan]);

GJordan := [155.9, 156.6, 190, 235.3, 342.7, 455.5, 477.9, 473.4, 534.6, 531.7, 566.5, 586.7, 604.3, 618.8, 663.9, 742, 790.6, 857.9, 990.2, 1081.2, 1194] > > > > >

21 #exchange rate e[Jordan]:= [2.579,2.579,2.579,2.579,2.579,2.579,2.579,2.579,2.579,2.579,2.579,2.5 79,1.5579,1.1743,1.0570,1.0357,1.0525,1.0341,0.9772,0.9488,09809]; nops(e[Jordan]);

eJordan := [2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 1.5579, 1.1743, 1.0570, 1.0357, 1.0525, 1.0341, .9772, .9488, 9809] > > > > >

21 # Tarrif tau[Jordan]:= [52.07,66.4,83.2,91.43,101.17,123.41,140.67,140.2,138.06,136.11,130.28 ,143.62,154.67,138.77,172.7,210.2,389.3,337.7,324.2,318.7,336.3]; nops(tau[Jordan]);

τJordan := [52.07, 66.4, 83.2, 91.43, 101.17, 123.41, 140.67, 140.2, 138.06, 136.11, 130.28, 143.62, 154.67, 138.77, 172.7, 210.2, 389.3, 337.7, 324.2, 318.7, 336.3]

21 # trade # In the following u[1] through u[8] represents unknown data from > 1977-1984 > # Will need to estimate these numbers > d[Jordan]:=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; > nops(d[Jordan]); > >

dJordan := [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] 21 The next entries will be B and F, and then T= X[0] - B[0] - F[0] where X[0] is given above. > # unknown NAME > B[Jordan] := > [36.1,-16.5,-288.4,-6.5,373.9,-38.9,-332.7,-390.7,-264.7,-260.5,-39.8, > -351.8,-293.7,384.9,-227.1,-393.5,-835.2,-629.1,-398,-258.6,-221.9]; > nops(B[Jordan]); BJordan := [36.1, −16.5, −288.4, −6.5, 373.9, −38.9, −332.7, −390.7, −264.7, −260.5, −39.8, −351.8, −293.7, 384.9, −227.1, −393.5, −835.2, −629.1, −398, −258.6, −221.9] > > >

21 # unknown NAME F[Jordan]:= [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; nops(F[Jordan]); FJordan := [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

> > >

21 # Taxes T[Jordan]:= X[Jordan] - B[Jordan] - F[Jordan] ; nops(T[Jordan]);

TJordan := [577.9, 798.8, 1158.3, 1170.5, 1035.8, 2020.1, 2558.6, 2483.5, 2530.6, 2544.7, 1873.4, 2427.7, 2834.2, 2779.1, 4353.5, 4453.7, 5629.8, 5742.9, 5599.0, 6132.3, 6659.0] > > > > > >

21 # GDP y[Jordan] := [512.1,624.6,767.9,914.6,1151.2,1426.7,1701.1,1828.7,1981.4,2020.2,216 3.6,2208.6,2264.4,2372.1,2668.3,2868.3,3648.2,3925.6,4400,4773.6,4982. 4]; nops(y[Jordan]);

yJordan := [512.1, 624.6, 767.9, 914.6, 1151.2, 1426.7, 1701.1, 1828.7, 1981.4, 2020.2, 2163.6, 2208.6, 2264.4, 2372.1, 2668.3, 2868.3, 3648.2, 3925.6, 4400, 4773.6, 4982.4] 21 > >

y_dot[Jordan]:= [seq(y[Jordan][k+1] - y[Jordan][k],k=2..20)]; nops(%);

y dot Jordan := [143.3, 146.7, 236.6, 275.5, 274.4, 127.6, 152.7, 38.8, 143.4, 45.0, 55.8, 107.7, 296.2, 200.0, 779.9, 277.4, 474.4, 373.6, 208.8]

19

1.4 > > > >

Israel Data Export[Israel]:= [3.851,6.413,11.771,21.104,48.863,112.684,227.483,546,3099,12700,17507 ,22302,24767,31584,36715,40291,49351,60610,73292,82919,93660]; nops(Export[Israel]);

Export Israel := [3.851, 6.413, 11.771, 21.104, 48.863, 112.684, 227.483, 546, 3099, 12700, 17507, 22302, 24767, 31584, 36715, 40291, 49351, 60610, 73292, 82919, 93660] > > > > >

21 Import[Israel]:= [-6.816,-9.431,-18.047,-31.813,-66.226,-158.993,-343.990,-824,-4210,-1 6570,-23136,-32302,-33803,-39082,-48220,-61183,-71714,-90616,-107750,126074,-141634]; nops(Import[Israel]);

Import Israel := [−6.816, −9.431, −18.047, −31.813, −66.226, −158.993, −343.990, −824, −4210, −16570, −23136, −32302, −33803, −39082, −48220, −61183, −71714, −90616, −107750, −126074, −141634] > >

21 ## Net export X[Israel]:= Export[Israel] - Import[Israel];

XIsrael := [10.667, 15.844, 29.818, 52.917, 115.089, 271.677, 571.473, 1370, 7309, 29270, 40643, 54604, 58570, 70666, 84935, 101474, 121065, 151226, 181042, 208993, 235294] > ##Private consumption > C[Israel]:= > [6.192,8.795,14.576,27.956,58.891,145.250,342.832,910,4160,16490,27800 > ,36391,44255,53168,64889,81668,98049,116564,142768,161831,187831] ; > nops(C[Israel]); CIsrael := [6.192, 8.795, 14.576, 27.956, 58.891, 145.250, 342.832, 910, 4160, 16490, 27800, 36391, 44255, 53168, 64889, 81668, 98049, 116564, 142768, 161831, 187831]

> > > > > >

21 II[Israel] := [2.550+0.140,3.100+0.320,5.540+0.660,11.51+0.75,22.543+0.84,59.002-2.7 68,136.006+4.968,360+5,1568+100,5338+117,8051+401,11294-204,12776+113, 14554+258,20234+422,32280+1960,38068+2138,42485+4505,52014+2220,63213+ 3950,72721+2462]; nops(II[Israel]);

II Israel := [2.690, 3.420, 6.200, 12.26, 23.383, 56.234, 140.974, 365, 1668, 5455, 8452, 11090, 12889, 14812, 20656, 34240, 40206, 46990, 54234, 67163, 75183] > > > >

21 ##Government expenditures G[Israel] := [4.027,5.101,9.022,15.187,44.701,109.973,228.133,545,2919,10313,13645, 19358,22376,25274,31743,40106,45162,52431,61790,77695,90774]

; >

nops(G[Israel]);

GIsrael := [4.027, 5.101, 9.022, 15.187, 44.701, 109.973, 228.133, 545, 2919, 10313, 13645, 19358, 22376, 25274, 31743, 40106, 45162, 52431, 61790, 77695, 90774] > > > > > >

21 #exchange rate e[Israel]:= [1.0158,1.8672,2.4672,4.6517,9.6268,0.0182,0.0371,0.1128,0.6261,1.471, 1.8181,2.1828,2.2675,2.5797,2.9136,3.2657,3.8005,4.1015,4.4058,4.6601, 4.6748]; nops(e[Israel]);

eIsrael := [1.0158, 1.8672, 2.4672, 4.6517, 9.6268, .0182, .0371, .1128, .6261, 1.471, 1.8181, 2.1828, 2.2675, 2.5797, 2.9136, 3.2657, 3.8005, 4.1015, 4.4058, 4.6601, 4.6748] > > > > >

21 # Tarrif tau[Israel]:= [5.69,6.05,6.15,10.54,13,16.6,17.1,37,137.1,538.2,838.7,760.2,679.7,45 5.6,515.8,571,942,748,544,502,523]; nops(tau[Israel]);

τIsrael := [5.69, 6.05, 6.15, 10.54, 13, 16.6, 17.1, 37, 137.1, 538.2, 838.7, 760.2, 679.7, 455.6, 515.8, 571, 942, 748, 544, 502, 523] 21 # trade # In the following u[1] through u[8] represents unknown data from > 1977-1984 > # Will need to estimate these numbers > d[Israel]:=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; > nops(d[Israel]); > >

dIsrael := [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] 21

The next entries will be B and F, and then T= X[0] - B[0] - F[0] where X[0] is given above. > # unknown NAME > B[Israel] := > [-676,-356,-1009,-920,-871,-1361,-2125,-2099,-1423,1157,1497,-1227,-63 > 8,573,-89,-1203,-1238,-2558,-4162,-6339,-7057]; > nops(B[Israel]); BIsrael := [−676, −356, −1009, −920, −871, −1361, −2125, −2099, −1423, 1157, 1497, −1227, −638, 573, −89, −1203, −1238, −2558, −4162, −6339, −7057] > > >

21 # unknown NAME F[Israel]:= [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; nops(F[Israel]); FIsrael := [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

> > >

21 # Taxes T[Israel]:= X[Israel] - B[Israel] - F[Israel] ; nops(T[Israel]);

TIsrael := [686.667, 371.844, 1038.818, 972.917, 986.089, 1632.677, 2696.473, 3469, 8732, 28113, 39146, 55831, 59208, 70093, 85024, 102677, 122303, 153784, 185204, 215332, 242351] > > > > >

21 # GDP y[Israel] := [9.888,14.39,23.745,44.49,111.612,265.148,596.306,1542,7636,28437,4419 1,56572,70181,85471,105805,134855,161738,186576,224838,265701,310429]; nops(y[Israel]);

yIsrael := [9.888, 14.39, 23.745, 44.49, 111.612, 265.148, 596.306, 1542, 7636, 28437, 44191, 56572, 70181, 85471, 105805, 134855, 161738, 186576, 224838, 265701, 310429] 21 > >

y_dot[Israel]:= [seq(y[Israel][k+1] - y[Israel][k],k=2..20)]; nops(%);

y dot Israel := [9.355, 20.745, 67.122, 153.536, 331.158, 945.694, 6094, 20801, 15754, 12381, 13609, 15290, 20334, 29050, 26883, 24838, 38262, 40863, 44728] 19

2

Prepare the Data

In this section we set up the procedures to adjust the values by the exchange rates, and arrange the various lists to have the correct number of entries and starting points Consider for example y (GDP) and y dot (rate of change of GDP) values. The basic info is the list y = [y[1]..y[N]] . In the table we see how the various values y values at t , y dot values at t , y values at t minus 1 and y dot values at t minus 1 are defined in terms of the values y[1] .. y[N] . variable and t y(t)

1 y1

2 y2

3 y3

4 y4

5 y5

N −1 yN

N yN −1

y dot(t)

y 2 − y1

y3 − y2

y4 − y 3

y5 − y4

y 6 − y5

yN − yN −1

y(t − 1)

undefined

y1

y2

y3

y4

yN −2

undefined yN −1

y dot(t − 1)

undefined

y2 − y1

y3 − y 2

y4 − y3

yN −1 − yN −2

yN − yN −1

From this info we make our definitions as follows. We assume the US economy. 1. y values at t[US] := [seq(y[k], k = 2 .. N-1)] so first entry is y[2]. 2. y values at t minus 1[US]:= [seq(y[k],k = 1 .. N - 2)] so first entry is y values(1) at t = 2 . 3. y dot values at t[US] := [seq(y[k+1] - y[k], k = 2..N - 1 )] so first entry is y dot(2) at t = 2

4. y dot values at t minus 1[US] := [seq( y[k+1] -y[k], k = 1 .. N-2)] so first entry is y dot(1) at t = 2 5. e values at t[US] := [seq(y[k], k = 2 .. N-1)] so first entry is e[2] at t = 2. 6. e values at t minus 1 := [seq(e[k],k = 1 .. N - 2)] so first entry is e values(1) at t = 2 . All lists will then have N - 2 entries.

2.1

>

Define the y values, e values, tau values, T values, I values, C values, G values and the dot values of the appropriate variables TheList:= [US,Egypt,Jordan,Israel]; TheList := [US , Egypt, Jordan, Israel ]

> > > > > > > > > > > > > >

## y_values_at_t ## for CountryX in TheList do N:= nops(y[CountryX]); y_values_at_t[CountryX]:= [seq(y[CountryX][k], k = 2 .. N-1)]; end do; ## ## y_values_at_t_minus_1 ## for CountryX in TheList do N:= nops(y[CountryX]); y_values_at_t_minus_1[CountryX]:= [seq(y[CountryX][k],k = 1 ..

N > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >

- 2)]; end do; ## ## y_dot_values_at_t ## for CountryX in TheList do N:= nops(y[CountryX]); y_dot_values_at_t[CountryX]:= [seq(y[CountryX][k+1] y[CountryX][k], k = 2..N - 1 )]; end do; ## ## y_dot_values_at_t_minus_1 ## for CountryX in TheList do N:= nops(y[CountryX]); y_dot_values_at_t_minus_1[CountryX]:= [seq(y[CountryX][k+1] y[CountryX][k], k = 1..N - 2 )]; end do; ##################################################### ## ## e_values## ## ##################################################### for CountryX in TheList do N:= nops(e[CountryX]); e_values_at_t[CountryX]:= [seq(e[CountryX][k], k = 2..N - 1 )]; end do; ## ## e_values_at_t_minus_1 ## for CountryX in TheList do N:= nops(e[CountryX]); e_values_at_t_minus_1[CountryX]:= [seq(e[CountryX][k], k = 1..N

> > > > > > > > > > >

2 )]; end do; ##################################################### ## ## tau_values ## ##################################################### for CountryX in TheList do N:= nops(tau[CountryX]); tau_values_at_t[CountryX]:= [seq(tau[CountryX][k], k = 2..N -

1 > > > > > > > > >

)]; end do; ## ## tau_values_at_t_minus_1 ## for CountryX in TheList do N:= nops(tau[CountryX]); tau_values_at_t_minus_1[CountryX]:= [seq(tau[CountryX][k], k

= > > > > > > > > > > > >

1..N - 2 )]; end do; ##################################################### ## ## T values ## ##################################################### ## T_values_at_t ## for CountryX in TheList do N:= nops(T[CountryX]);

N := 27 y values at t US := [814.3, 889.3, 959.5, 1010.4, 1096.8, 1206.5, 1349.1, 1458.8, 1584.8, 1676.1, 1974.1, 21232.7, 2488.7, 2808.1, 3030.6, 3149.6, 3405.1, 3777.2, 4038.7, 4268.6, 4539.9, 4900.4, 5250.8, 5522.2, 5722.9] N := 21 y values at t Egypt := [8210, 9783, 12475, 15470, 17150, 20753, 45895, 31547, 37240, 42563, 51500, 61600, 76800, 96100, 111200, 139100, 157300, 175000, 205000] N := 21 y values at t Jordan := [624.6, 767.9, 914.6, 1151.2, 1426.7, 1701.1, 1828.7, 1981.4, 2020.2, 2163.6, 2208.6, 2264.4, 2372.1, 2668.3, 2868.3, 3648.2, 3925.6, 4400, 4773.6] N := 21 y values at t Israel := [14.39, 23.745, 44.49, 111.612, 265.148, 596.306, 1542, 7636, 28437, 44191, 56572, 70181, 85471, 105805, 134855, 161738, 186576, 224838, 265701] N := 27 y values at t minus 1 US := [769.8, 814.3, 889.3, 959.5, 1010.4, 1096.8, 1206.5, 1349.1, 1458.8, 1584.8, 1676.1, 1974.1, 21232.7, 2488.7, 2808.1, 3030.6, 3149.6, 3405.1, 3777.2, 4038.7, 4268.6, 4539.9, 4900.4, 5250.8, 5522.2] N := 21 y values at t minus 1 Egypt := [6276, 8210, 9783, 12475, 15470, 17150, 20753, 45895, 31547, 37240, 42563, 51500, 61600, 76800, 96100, 111200, 139100, 157300, 175000] N := 21 y values at t minus 1 Jordan := [512.1, 624.6, 767.9, 914.6, 1151.2, 1426.7, 1701.1, 1828.7, 1981.4, 2020.2, 2163.6, 2208.6, 2264.4, 2372.1, 2668.3, 2868.3, 3648.2, 3925.6, 4400] N := 21 y values at t minus 1 Israel := [9.888, 14.39, 23.745, 44.49, 111.612, 265.148, 596.306, 1542, 7636, 28437, 44191, 56572, 70181, 85471, 105805, 134855, 161738, 186576, 224838] N := 27 y dot values at t US := [75.0, 70.2, 50.9, 86.4, 109.7, 142.6, 109.7, 126.0, 91.3, 298.0, 19258.6, −18744.0, 319.4, 222.5, 119.0, 255.5, 372.1, 261.5, 229.9, 271.3, 360.5, 350.4, 271.4, 200.7, 315.6]

N := 21 y dot values at t Egypt := [1573, 2692, 2995, 1680, 3603, 25142, −14348, 5693, 5323, 8937, 10100, 15200, 19300, 15100, 27900, 18200, 17700, 30000, 23300] N := 21 y dot values at t Jordan := [143.3, 146.7, 236.6, 275.5, 274.4, 127.6, 152.7, 38.8, 143.4, 45.0, 55.8, 107.7, 296.2, 200.0, 779.9, 277.4, 474.4, 373.6, 208.8] N := 21 y dot values at t Israel := [9.355, 20.745, 67.122, 153.536, 331.158, 945.694, 6094, 20801, 15754, 12381, 13609, 15290, 20334, 29050, 26883, 24838, 38262, 40863, 44728] N := 27 y dot values at t minus 1 US := [44.5, 75.0, 70.2, 50.9, 86.4, 109.7, 142.6, 109.7, 126.0, 91.3, 298.0, 19258.6, −18744.0, 319.4, 222.5, 119.0, 255.5, 372.1, 261.5, 229.9, 271.3, 360.5, 350.4, 271.4, 200.7] N := 21 y dot values at t minus 1 Egypt := [1934, 1573, 2692, 2995, 1680, 3603, 25142, −14348, 5693, 5323, 8937, 10100, 15200, 19300, 15100, 27900, 18200, 17700, 30000] N := 21 y dot values at t minus 1 Jordan := [112.5, 143.3, 146.7, 236.6, 275.5, 274.4, 127.6, 152.7, 38.8, 143.4, 45.0, 55.8, 107.7, 296.2, 200.0, 779.9, 277.4, 474.4, 373.6] N := 21 y dot values at t minus 1 Israel := [4.502, 9.355, 20.745, 67.122, 153.536, 331.158, 945.694, 6094, 20801, 15754, 12381, 13609, 15290, 20334, 29050, 26883, 24838, 38262, 40863] N := 27 e values at t US := [1, 1, 1, 1, 1.0857, 1.0857, 1.2064, 1.2244, 1.1707, 1.1618, 1.2147, 1.3028, 1.3173, 1.2754, 1.1640, 1.1031, 1.0470, .9802, 1.0984, 1.2232, 1.4187, 1.3457, 1.3142, 1.4227, 1.4304] N := 21 e values at t Egypt := [.4753, .5098, .9221, .8928, .8148, .7722, .7329, .6861, .7689, .8562, .9931, .9420, 1.4456, 2.8456, 4.7665, 4.5906, 4.6314, 4.9504, 5.0392] N := 21 e values at t Jordan := [2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 1.5579, 1.1743, 1.0570, 1.0357, 1.0525, 1.0341, .9772, .9488]

N := 21 e values at t Israel := [1.8672, 2.4672, 4.6517, 9.6268, .0182, .0371, .1128, .6261, 1.471, 1.8181, 2.1828, 2.2675, 2.5797, 2.9136, 3.2657, 3.8005, 4.1015, 4.4058, 4.6601] N := 27 e values at t minus 1 US := [1, 1, 1, 1, 1, 1.0857, 1.0857, 1.2064, 1.2244, 1.1707, 1.1618, 1.2147, 1.3028, 1.3173, 1.2754, 1.1640, 1.1031, 1.0470, .9802, 1.0984, 1.2232, 1.4187, 1.3457, 1.3142, 1.4227] N := 21 e values at t minus 1 Egypt := [.4546, .4753, .5098, .9221, .8928, .8148, .7722, .7329, .6861, .7689, .8562, .9931, .9420, 1.4456, 2.8456, 4.7665, 4.5906, 4.6314, 4.9504 ] N := 21 e values at t minus 1 Jordan := [2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 2.579, 1.5579, 1.1743, 1.0570, 1.0357, 1.0525, 1.0341, .9772 ] N := 21 e values at t minus 1 Israel := [1.0158, 1.8672, 2.4672, 4.6517, 9.6268, .0182, .0371, .1128, .6261, 1.471, 1.8181, 2.1828, 2.2675, 2.5797, 2.9136, 3.2657, 3.8005, 4.1015, 4.4058] N := 27 tau values at t US := [2000, 2300, 2400, 2584, 2767, 3124, 3620, 3771, 3780, 4675, 5485, 7162, 7202, 7535, 9993, 8688, 9430, 12042, 13067, 13312, 13923, 15054, 16096, 16339, 16197] N := 21 tau values at t Egypt := [557.4, 767.6, 958, 1435, 1707, 1719, 2031, 2073, 1927, 1929, 2463, 2981, 3037, 3422, 4941, 5428, 6585, 7352, 8255] N := 21 tau values at t Jordan := [66.4, 83.2, 91.43, 101.17, 123.41, 140.67, 140.2, 138.06, 136.11, 130.28, 143.62, 154.67, 138.77, 172.7, 210.2, 389.3, 337.7, 324.2, 318.7] N := 21 tau values at t Israel := [6.05, 6.15, 10.54, 13, 16.6, 17.1, 37, 137.1, 538.2, 838.7, 760.2, 679.7, 455.6, 515.8, 571, 942, 748, 544, 502] N := 27

tau values at t minus 1 US := [1900, 2000, 2300, 2400, 2584, 2767, 3124, 3620, 3771, 3780, 4675, 5485, 7162, 7202, 7535, 9993, 8688, 9430, 12042, 13067, 13312, 13923, 15054, 16096, 16339] N := 21 tau values at t minus 1 Egypt := [399.4, 557.4, 767.6, 958, 1435, 1707, 1719, 2031, 2073, 1927, 1929, 2463, 2981, 3037, 3422, 4941, 5428, 6585, 7352] N := 21 tau values at t minus 1 Jordan := [52.07, 66.4, 83.2, 91.43, 101.17, 123.41, 140.67, 140.2, 138.06, 136.11, 130.28, 143.62, 154.67, 138.77, 172.7, 210.2, 389.3, 337.7, 324.2 ] N := 21 tau values at t minus 1 Israel := [5.69, 6.05, 6.15, 10.54, 13, 16.6, 17.1, 37, 137.1, 538.2, 838.7, 760.2, 679.7, 455.6, 515.8, 571, 942, 748, 544] N := 27 T values at t US := [.34, 87.78, 91.55, 120.06, 131.58, 139.15, 184.28, 255.27, 271.89, 310.49, 346.03, 416.03, 466.85, 542.57, 584.71, 628.07, 589.14, 688.5, 686.4, 748.8, 824.6, 980.33, 1084.01, 1165.1, 1228.98] N := 21 T values at t Egypt := [4930, 6461, 10047, 11170, 14804, 15837, 15470, 18732, 19402, 18413, 18446, 33448, 39909, 50615, 68897, 81888, 89401, 89269, 95154] N := 21 T values at t Jordan := [798.8, 1158.3, 1170.5, 1035.8, 2020.1, 2558.6, 2483.5, 2530.6, 2544.7, 1873.4, 2427.7, 2834.2, 2779.1, 4353.5, 4453.7, 5629.8, 5742.9, 5599.0, 6132.3] N := 21 T values at t Israel := [371.844, 1038.818, 972.917, 986.089, 1632.677, 2696.473, 3469, 8732, 28113, 39146, 55831, 59208, 70093, 85024, 102677, 122303, 153784, 185204, 215332] N := 27 T values at t minus 1 US := [73.27, .34, 87.78, 91.55, 120.06, 131.58, 139.15, 184.28, 255.27, 271.89, 310.49, 346.03, 416.03, 466.85, 542.57, 584.71, 628.07, 589.14, 688.5, 686.4, 748.8, 824.6, 980.33, 1084.01, 1165.1] N := 21

T values at t minus 1 Egypt := [4006, 4930, 6461, 10047, 11170, 14804, 15837, 15470, 18732, 19402, 18413, 18446, 33448, 39909, 50615, 68897, 81888, 89401, 89269] N := 21 T values at t minus 1 Jordan := [577.9, 798.8, 1158.3, 1170.5, 1035.8, 2020.1, 2558.6, 2483.5, 2530.6, 2544.7, 1873.4, 2427.7, 2834.2, 2779.1, 4353.5, 4453.7, 5629.8, 5742.9, 5599.0] N := 21 T values at t minus 1 Israel := [686.667, 371.844, 1038.818, 972.917, 986.089, 1632.677, 2696.473, 3469, 8732, 28113, 39146, 55831, 59208, 70093, 85024, 102677, 122303, 153784, 185204] N := 27 T dot values at t US := [87.44, 3.77, 28.51, 11.52, 7.57, 45.13, 70.99, 16.62, 38.60, 35.54, 70.00, 50.82, 75.72, 42.14, 43.36, −38.93, 99.36, −2.1, 62.4, 75.8, 155.73, 103.68, 81.09, 63.88, 40.40] N := 21 T dot values at t Egypt := [1531, 3586, 1123, 3634, 1033, −367, 3262, 670, −989, 33, 15002, 6461, 10706, 18282, 12991, 7513, −132, 5885, 12588] N := 21 T dot values at t Jordan := [359.5, 12.2, −134.7, 984.3, 538.5, −75.1, 47.1, 14.1, −671.3, 554.3, 406.5, −55.1, 1574.4, 100.2, 1176.1, 113.1, −143.9, 533.3, 526.7] N := 21 T dot values at t Israel := [666.974, −65.901, 13.172, 646.588, 1063.796, 772.527, 5263, 19381, 11033, 16685, 3377, 10885, 14931, 17653, 19626, 31481, 31420, 30128, 27019] N := 27 T dot values at t minus 1 US := [−72.93, 87.44, 3.77, 28.51, 11.52, 7.57, 45.13, 70.99, 16.62, 38.60, 35.54, 70.00, 50.82, 75.72, 42.14, 43.36, −38.93, 99.36, −2.1, 62.4, 75.8, 155.73, 103.68, 81.09, 63.88] N := 21 T dot values at t minus 1 Egypt := [924, 1531, 3586, 1123, 3634, 1033, −367, 3262, 670, −989, 33, 15002, 6461, 10706, 18282, 12991, 7513, −132, 5885] N := 21 T dot values at t minus 1 Jordan := [220.9, 359.5, 12.2, −134.7, 984.3, 538.5, −75.1, 47.1, 14.1, −671.3, 554.3, 406.5, −55.1, 1574.4, 100.2, 1176.1, 113.1, −143.9, 533.3]

N := 21 T dot values at t minus 1 Israel := [−314.823, 666.974, −65.901, 13.172, 646.588, 1063.796, 772.527, 5263, 19381, 11033, 16685, 3377, 10885, 14931, 17653, 19626, 31481, 31420, 30128] N := 27 II values at t US := [128.1, 139.9, 155.2, 150, 175.3, 205.2, 242.5, 245.1, 225, 285, 358.2, 434, 480.3, 467.6, 558.1, 503.4, 546.7, 718.9, 714.6, 717.7, 749.3, 793.7, 832.2, 799.5, 736.8] N := 21 II values at t Egypt := [2386, 3034, 3796, 4328, 5208, 6250, 8333, 9204, 10685, 13068, 13450, 20450, 24000, 28300, 26650, 27500, 43000, 34400, 39300] N := 21 II values at t Jordan := [206.5, 223.0, 280.0, 463.9, 701.0, 650.8, 589.8, 571.2, 414.9, 444.3, 515.6, 532.5, 563.2, 850.1, 738.5, 1208.8, 1422.7, 1451, 1554.3] N := 21 II values at t Israel := [3.420, 6.200, 12.26, 23.383, 56.234, 140.974, 365, 1668, 5455, 8452, 11090, 12889, 14812, 20656, 34240, 40206, 46990, 54234, 67163] N := 27 G values at t US := [667.9, 686.8, 682, 665.8, 652.4, 653, 644.2, 655.4, 663.5, 659.2, 664.1, 677, 689.3, 704.2, 713.2, 723.6, 743.8, 766.9, 813.4, 855.4, 881.5, 886.8, 904.4, 929.9, 941.1] N := 21 G values at t Egypt := [1697, 1841, 2059, 2549, 2841, 3584, 4160, 4957, 5668, 6134, 7350, 8600, 9700, 10850, 12450, 14500, 16000, 18000, 21500] N := 21 G values at t Jordan := [156.6, 190, 235.3, 342.7, 455.5, 477.9, 473.4, 534.6, 531.7, 566.5, 586.7, 604.3, 618.8, 663.9, 742, 790.6, 857.9, 990.2, 1081.2] N := 21 G values at t Israel := [5.101, 9.022, 15.187, 44.701, 109.973, 228.133, 545, 2919, 10313, 13645, 19358, 22376, 25274, 31743, 40106, 45162, 52431, 61790, 77695] N := 27 X values at t US := [1.5, 91.9, 99.8, 112.8, 121.7, 140.4, 183.0, 251.8, 259.0, 300.1, 341.3, 398.5, 481.6, 573.1, 620.7, 585.8, 604.8, 707.5, 719.7, 770.9, 871.1, 996.4, 1095.7, 1182.9, 1222.6]

N := 21 X values at t Egypt := [3730, 5261, 8505, 10732, 12668, 13986, 15140, 16744, 17236, 16602, 18200, 32400, 38600, 50800, 70800, 84700, 91700, 89300, 94900] N := 21 X values at t Jordan := [782.3, 869.9, 1164.0, 1409.7, 1981.2, 2225.9, 2092.8, 2265.9, 2284.2, 1833.6, 2075.9, 2540.5, 3164.0, 4126.4, 4060.2, 4794.6, 5113.8, 5201.0, 5873.7] N := 21 X values at t Israel := [15.844, 29.818, 52.917, 115.089, 271.677, 571.473, 1370, 7309, 29270, 40643, 54604, 58570, 70666, 84935, 101474, 121065, 151226, 181042, 208993] N := 27 C values at t US := [509.3, 559.1, 603.7, 646.5, 700.3, 767.8, 848.2, 927.7, 1024.9, 1143.1, 1271.5, 1421.3, 1583.7, 1748.1, 1926.3, 2059.2, 2257.6, 2460.3, 2667.4, 2850.6, 3052.2, 3296.1, 3523.1, 3748.4, 3906.4] N := 21 C values at t Egypt := [4917, 6279, 8623, 11023, 11155, 13285, 16224, 21356, 24927, 27895, 35900, 43550, 54100, 68950, 80900, 101000, 115000, 130500, 148900] N := 21 C values at t Jordan := [412.8, 517.4, 736.8, 858.3, 1074.5, 1457.9, 1579.1, 1648.4, 1794.8, 1718.2, 1669.8, 1626.5, 1635.1, 1976.5, 2039.6, 2648.4, 2710.7, 2774.3, 3023.2] N := 21 C values at t Israel := [8.795, 14.576, 27.956, 58.891, 145.250, 342.832, 910, 4160, 16490, 27800, 36391, 44255, 53168, 64889, 81668, 98049, 116564, 142768, 161831]

3

Correct data forexchange rates

3.1

Modify all y values

>

TheList:= [US,Egypt,Jordan,Israel];

> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >

TheList := [US , Egypt, Jordan, Israel ] for i in TheList do TheNum:= min(nops(y_values_at_t[i]),nops(e_values_at_t[i])); correct_y_values_at_t[i]:= [seq(y_values_at_t[i][k]*e_values_at_t[i][k],k=1..TheNum)]; end do; # for i in TheList do TheNum:= min(nops(y_values_at_t_minus_1[i]),nops(e_values_at_t_minus_1[i])); correct_y_values_at_t_minus_1[i]:= [seq(y_values_at_t_minus_1[i][k]*e_values_at_t_minus_1[i][k],k=1..TheN um)]; end do; # for i in TheList do TheNum:= min(nops(y_dot_values_at_t[i]),nops(e_values_at_t[i])); correct_y_dot_values_at_t[i]:= [seq(y_dot_values_at_t[i][k]*e_values_at_t[i][k],k=1..TheNum)]; end do; # for i in TheList do TheNum:= min(nops(y_dot_values_at_t_minus_1[i]),nops(e_values_at_t_minus_1[i])) ; correct_y_dot_values_at_t_minus_1[i]:= [seq(y_dot_values_at_t_minus_1[i][k]*e_values_at_t_minus_1[i][k],k=1.. TheNum)]; end do; TheNum := 25

correct y values at t US := [814.3, 889.3, 959.5, 1010.4, 1190.79576, 1309.89705, 1627.55424, 1786.15472, 1855.32536, 1947.29298, 2397.93927, 27661.96156, 3278.36451, 3581.45074, 3527.61840, 3474.32376, 3565.13970, 3702.41144, 4436.10808, 5221.35152, 6440.75613, 6594.46828, 6900.60136, 7856.43394, 8186.03616] TheNum := 19

correct y values at t Egypt := [3902.2130, 4987.3734, 11503.1975, 13811.6160, 13973.8200, 16025.4666, 33636.4455, 21644.3967, 28633.8360, 36442.4406, 51144.6500, 58027.2000, 111022.0800, 273462.1600, 530034.8000, 638552.4600, 728519.2200, 866320.0000, .1033036000 107 ] TheNum := 19 correct y values at t Jordan := [1610.8434, 1980.4141, 2358.7534, 2968.9448, 3679.4593, 4387.1369, 4716.2173, 5110.0306, 5210.0958, 5579.9244, 5695.9794, 3527.70876, 2785.55703, 2820.39310, 2970.69831, 3839.73050, 4059.46296, 4299.6800, 4529.19168] TheNum := 19 correct y values at t Israel := [26.869008, 58.5836640, 206.954133, 1074.466402, 4.8256936, 22.1229526, 173.9376, 4780.8996, 41830.827, 80343.6571, 123485.3616, 159135.4175, 220489.5387, 308273.4480, 440395.9735, 614685.2690, 765241.4640, 990591.2604, .1238193230 107 ] TheNum := 25 correct y values at t minus 1 US := [769.8, 814.3, 889.3, 959.5, 1010.4, 1190.79576, 1309.89705, 1627.55424, 1786.15472, 1855.32536, 1947.29298, 2397.93927, 27661.96156, 3278.36451, 3581.45074, 3527.61840, 3474.32376, 3565.13970, 3702.41144, 4436.10808, 5221.35152, 6440.75613, 6594.46828, 6900.60136, 7856.43394] TheNum := 19 correct y values at t minus 1 Egypt := [2853.0696, 3902.2130, 4987.3734, 11503.1975, 13811.6160, 13973.8200, 16025.4666, 33636.4455, 21644.3967, 28633.8360, 36442.4406, 51144.6500, 58027.2000, 111022.0800, 273462.1600, 530034.8000, 638552.4600, 728519.2200, 866320.0000] TheNum := 19 correct y values at t minus 1 Jordan := [1320.7059, 1610.8434, 1980.4141, 2358.7534, 2968.9448, 3679.4593, 4387.1369, 4716.2173, 5110.0306, 5210.0958, 5579.9244, 5695.9794, 3527.70876, 2785.55703, 2820.39310, 2970.69831, 3839.73050, 4059.46296, 4299.6800] TheNum := 19 correct y values at t minus 1 Israel := [10.0442304, 26.869008, 58.5836640, 206.954133, 1074.466402, 4.8256936, 22.1229526, 173.9376, 4780.8996, 41830.827, 80343.6571, 123485.3616, 159135.4175, 220489.5387, 308273.4480, 440395.9735, 614685.2690, 765241.4640, 990591.2604] TheNum := 25

correct y dot values at t US := [75.0, 70.2, 50.9, 86.4, 119.10129, 154.82082, 132.34208, 154.27440, 106.88491, 346.21640, 23393.42142, −24419.68320, 420.74562, 283.77650, 138.51600, 281.84205, 389.58870, 256.32230, 252.52216, 331.85416, 511.44135, 471.53328, 356.67388, 285.53589, 451.43424] TheNum := 19 correct y dot values at t Egypt := [747.6469, 1372.3816, 2761.6895, 1499.9040, 2935.7244, 19414.6524, −10515.6492, 3905.9673, 4092.8547, 7651.8594, 10030.3100, 14318.4000, 27900.0800, 42968.5600, 132985.3500, 83548.9200, 81975.7800, 148512.0000, 117413.3600] TheNum := 19 correct y dot values at t Jordan := [369.5707, 378.3393, 610.1914, 710.5145, 707.6776, 329.0804, 393.8133, 100.0652, 369.8286, 116.0550, 143.9082, 167.78583, 347.82766, 211.40000, 807.74243, 291.96350, 490.57704, 365.08192, 198.10944 ] TheNum := 19 correct y dot values at t Israel := [17.4676560, 51.1820640, 312.2314074, 1478.060365, 6.0270756, 35.0852474, 687.4032, 13023.5061, 23174.134, 22509.8961, 29705.7252, 34670.0750, 52455.6198, 84640.0800, 87791.8131, 94396.8190, 156931.5930, 180034.2054, 208436.9528] TheNum := 25 correct y dot values at t minus 1 US := [44.5, 75.0, 70.2, 50.9, 86.4, 119.10129, 154.82082, 132.34208, 154.27440, 106.88491, 346.21640, 23393.42142, −24419.68320, 420.74562, 283.77650, 138.51600, 281.84205, 389.58870, 256.32230, 252.52216, 331.85416, 511.44135, 471.53328, 356.67388, 285.53589 ] TheNum := 19 correct y dot values at t minus 1 Egypt := [879.1964, 747.6469, 1372.3816, 2761.6895, 1499.9040, 2935.7244, 19414.6524, −10515.6492, 3905.9673, 4092.8547, 7651.8594, 10030.3100, 14318.4000, 27900.0800, 42968.5600, 132985.3500, 83548.9200, 81975.7800, 148512.0000] TheNum := 19 correct y dot values at t minus 1 Jordan := [290.1375, 369.5707, 378.3393, 610.1914, 710.5145, 707.6776, 329.0804, 393.8133, 100.0652, 369.8286, 116.0550, 143.9082, 167.78583, 347.82766, 211.40000, 807.74243, 291.96350, 490.57704, 365.08192] TheNum := 19

correct y dot values at t minus 1 Israel := [4.5731316, 17.4676560, 51.1820640, 312.2314074, 1478.060365, 6.0270756, 35.0852474, 687.4032, 13023.5061, 23174.134, 22509.8961, 29705.7252, 34670.0750, 52455.6198, 84640.0800, 87791.8131, 94396.8190, 156931.5930, 180034.2054]

3.2

Modify all T values

>

TheList:= [US,Egypt,Jordan,Israel];

> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >

TheList := [US , Egypt, Jordan, Israel ] for i in TheList do TheNum:= min(nops(T_values_at_t[i]),nops(e_values_at_t[i])); correct_T_values_at_t[i]:= [seq(T_values_at_t[i][k]*e_values_at_t[i][k],k=1..TheNum)]; end do; # for i in TheList do TheNum:= min(nops(T_values_at_t_minus_1[i]),nops(e_values_at_t_minus_1[i])); correct_T_values_at_t_minus_1[i]:= [seq(T_values_at_t_minus_1[i][k]*e_values_at_t_minus_1[i][k],k=1..TheN um)]; end do; # for i in TheList do TheNum:= min(nops(T_dot_values_at_t[i]),nops(e_values_at_t[i])); correct_T_dot_values_at_t[i]:= [seq(T_dot_values_at_t[i][k]*e_values_at_t[i][k],k=1..TheNum)]; end do; # for i in TheList do TheNum:= min(nops(T_dot_values_at_t_minus_1[i]),nops(e_values_at_t_minus_1[i])) ; correct_T_dot_values_at_t_minus_1[i]:= [seq(T_dot_values_at_t_minus_1[i][k]*e_values_at_t_minus_1[i][k],k=1.. TheNum)]; end do; TheNum := 25

correct T values at t US := [.34, 87.78, 91.55, 120.06, 142.856406, 151.075155, 222.315392, 312.552588, 318.301623, 360.727282, 420.322641, 542.003884, 614.981505, 691.993778, 680.602440, 692.824017, 616.829580, 674.86770, 753.94176, 915.93216, 1169.86002, 1319.230081, 1424.605942, 1657.58777, 1757.932992] TheNum := 19 correct T values at t Egypt := [2343.2290, 3293.8178, 9264.3387, 9972.5760, 12062.2992, 12229.3314, 11337.9630, 12852.0252, 14918.1978, 15765.2106, 18318.7226, 31508.0160, 57692.4504, 144030.0440, 328397.5505, 375915.0528, 414051.7914, 441917.2576, 479500.0368] TheNum := 19 correct T values at t Jordan := [2060.1052, 2987.2557, 3018.7195, 2671.3282, 5209.8379, 6598.6294, 6404.9465, 6526.4174, 6562.7813, 4831.4986, 6261.0383, 4415.40018, 3263.49713, 4601.64950, 4612.69709, 5925.36450, 5938.73289, 5471.34280, 5818.32624] TheNum := 19 correct T values at t Israel := [694.3071168, 2562.971770, 4525.718009, 9492.881585, 29.7147214, 100.0391483, 391.3032, 5467.1052, 41354.223, 71171.3426, 121867.9068, 134254.1400, 180818.9121, 247725.9264, 335312.2789, 464812.5515, 630745.0760, 815971.7832, .1003468653 107 ] TheNum := 25 correct T values at t minus 1 US := [73.27, .34, 87.78, 91.55, 120.06, 142.856406, 151.075155, 222.315392, 312.552588, 318.301623, 360.727282, 420.322641, 542.003884, 614.981505, 691.993778, 680.602440, 692.824017, 616.829580, 674.86770, 753.94176, 915.93216, 1169.86002, 1319.230081, 1424.605942, 1657.58777] TheNum := 19 correct T values at t minus 1 Egypt := [1821.1276, 2343.2290, 3293.8178, 9264.3387, 9972.5760, 12062.2992, 12229.3314, 11337.9630, 12852.0252, 14918.1978, 15765.2106, 18318.7226, 31508.0160, 57692.4504, 144030.0440, 328397.5505, 375915.0528, 414051.7914, 441917.2576] TheNum := 19 correct T values at t minus 1 Jordan := [1490.4041, 2060.1052, 2987.2557, 3018.7195, 2671.3282, 5209.8379, 6598.6294, 6404.9465, 6526.4174, 6562.7813, 4831.4986, 6261.0383, 4415.40018, 3263.49713, 4601.64950, 4612.69709, 5925.36450, 5938.73289, 5471.34280]

TheNum := 19 correct T values at t minus 1 Israel := [697.5163386, 694.3071168, 2562.971770, 4525.718009, 9492.881585, 29.7147214, 100.0391483, 391.3032, 5467.1052, 41354.223, 71171.3426, 121867.9068, 134254.1400, 180818.9121, 247725.9264, 335312.2789, 464812.5515, 630745.0760, 815971.7832] TheNum := 25 correct T dot values at t US := [87.44, 3.77, 28.51, 11.52, 8.218749, 48.997641, 85.642336, 20.349528, 45.189020, 41.290372, 85.029000, 66.208296, 99.745956, 53.745356, 50.471040, −42.943683, 104.029920, −2.05842, 68.54016, 92.71856, 220.934151, 139.522176, 106.568478, 90.882076, 57.788160] TheNum := 19 correct T dot values at t Egypt := [727.6843, 1828.1428, 1035.5183, 3244.4352, 841.6884, −283.3974, 2390.7198, 459.6870, −760.4421, 28.2546, 14898.4862, 6086.2620, 15476.5936, 52023.2592, 61921.6015, 34489.1778, −611.3448, 29133.1040, 63433.4496] TheNum := 19 correct T dot values at t Jordan := [927.1505, 31.4638, −347.3913, 2538.5097, 1388.7915, −193.6829, 121.4709, 36.3639, −1731.2827, 1429.5397, 1048.3635, −85.84029, 1848.81792, 105.91140, 1218.08677, 119.03775, −148.80699, 521.14076, 499.73296] TheNum := 19 correct T dot values at t Israel := [1245.373853, −162.5909472, 61.2721924, 6224.573358, 19.3610872, 28.6607517, 593.6664, 12134.4441, 16229.543, 30334.9985, 7371.3156, 24681.7375, 38517.5007, 51433.7808, 64092.6282, 119643.5405, 128869.1300, 132737.9424, 125911.2419] TheNum := 25 correct T dot values at t minus 1 US := [−72.93, 87.44, 3.77, 28.51, 11.52, 8.218749, 48.997641, 85.642336, 20.349528, 45.189020, 41.290372, 85.029000, 66.208296, 99.745956, 53.745356, 50.471040, −42.943683, 104.029920, −2.05842, 68.54016, 92.71856, 220.934151, 139.522176, 106.568478, 90.882076] TheNum := 19 correct T dot values at t minus 1 Egypt := [420.0504, 727.6843, 1828.1428, 1035.5183, 3244.4352, 841.6884, −283.3974, 2390.7198, 459.6870, −760.4421, 28.2546, 14898.4862, 6086.2620, 15476.5936, 52023.2592, 61921.6015, 34489.1778, −611.3448, 29133.1040] TheNum := 19

correct T dot values at t minus 1 Jordan := [569.7011, 927.1505, 31.4638, −347.3913, 2538.5097, 1388.7915, −193.6829, 121.4709, 36.3639, −1731.2827, 1429.5397, 1048.3635, −85.84029, 1848.81792, 105.91140, 1218.08677, 119.03775, −148.80699, 521.14076] TheNum := 19 correct T dot values at t minus 1 Israel := [−319.7972034, 1245.373853, −162.5909472, 61.2721924, 6224.573358, 19.3610872, 28.6607517, 593.6664, 12134.4441, 16229.543, 30334.9985, 7371.3156, 24681.7375, 38517.5007, 51433.7808, 64092.6282, 119643.5405, 128869.1300, 132737.9424]

3.3 > > > > > > > > >

Modify all tau values TheList:= [US,Egypt,Jordan,Israel]; for CountryX in TheList do TheNum:= min(nops(tau_values_at_t[CountryX]),nops(e_values_at_t[CountryX])); correct_tau_values_at_t[CountryX]:= [seq(tau_values_at_t[CountryX][k]*e_values_at_t[CountryX][k],k=1..TheN um)]; end do; TheList := [US , Egypt, Jordan, Israel ] TheNum := 25

correct tau values at t US := [2000, 2300, 2400, 2584, 3004.1319, 3391.7268, 4367.1680, 4617.2124, 4425.2460, 5431.4150, 6662.6295, 9330.6536, 9487.1946, 9610.1390, 11631.8520, 9583.7328, 9873.2100, 11803.5684, 14352.7928, 16283.2384, 19752.5601, 20258.1678, 21153.3632, 23245.4953, 23168.1888] TheNum := 19 correct tau values at t Egypt := [264.93222, 391.32248, 883.3718, 1281.1680, 1390.8636, 1327.4118, 1488.5199, 1422.2853, 1481.6703, 1651.6098, 2446.0053, 2808.1020, 4390.2872, 9737.6432, 23551.2765, 24917.7768, 30497.7690, 36395.3408, 41598.5960] TheNum := 19 correct tau values at t Jordan := [171.2456, 214.5728, 235.79797, 260.91743, 318.27439, 362.78793, 361.5758, 356.05674, 351.02769, 335.99212, 370.39598, 240.960393, 162.957611, 182.54390, 217.70414, 409.73825, 349.21557, 316.80824, 302.38256] TheNum := 19

correct tau values at t Israel := [11.296560, 15.173280, 49.028918, 125.1484, .30212, .63441, 4.1736, 85.83831, 791.6922, 1524.84047, 1659.36456, 1541.21975, 1175.31132, 1502.83488, 1864.7147, 3580.0710, 3067.9220, 2396.7552, 2339.3702]

3.4

> > > > > > > > >

Modify all II values

TheList:= [US,Egypt,Jordan,Israel]; for CountryX in TheList do TheNum:= min(nops(II_values_at_t[CountryX]),nops(e_values_at_t[CountryX])); correct_II_values_at_t[CountryX]:= [seq(II_values_at_t[CountryX][k]*e_values_at_t[CountryX][k],k=1..TheNu m)]; end do; TheList := [US , Egypt, Jordan, Israel ] TheNum := 25

correct II values at t US := [128.1, 139.9, 155.2, 150, 190.32321, 222.78564, 292.55200, 300.10044, 263.4075, 331.1130, 435.10554, 565.4152, 632.69919, 596.37704, 649.62840, 555.30054, 572.39490, 704.66578, 784.91664, 877.89064, 1063.03191, 1068.08209, 1093.67724, 1137.44865, 1053.91872] TheNum := 19 correct II values at t Egypt := [1134.0658, 1546.7332, 3500.2916, 3864.0384, 4243.4784, 4826.2500, 6107.2557, 6314.8644, 8215.6965, 11188.8216, 13357.1950, 19263.9000, 34694.4000, 80530.4800, 127027.2250, 126241.5000, 199150.2000, 170293.7600, 198040.5600] TheNum := 19 correct II values at t Jordan := [532.5635, 575.1170, 722.1200, 1196.3981, 1807.8790, 1678.4132, 1521.0942, 1473.1248, 1070.0271, 1145.8497, 1329.7324, 829.58175, 661.36576, 898.55570, 764.86445, 1272.26200, 1471.21407, 1417.9172, 1474.71984] TheNum := 19

correct II values at t Israel := [6.3858240, 15.2966400, 57.029842, 225.1034644, 1.0234588, 5.2301354, 41.1720, 1044.3348, 8024.305, 15366.5812, 24207.2520, 29225.8075, 38210.5164, 60183.3216, 111817.5680, 152802.9030, 192729.4850, 238944.1572, 312986.2963]

3.5 > > > > > > > > >

Modify all G values TheList:= [US,Egypt,Jordan,Israel]; for CountryX in TheList do TheNum:= min(nops(G_values_at_t[CountryX]),nops(e_values_at_t[CountryX])); correct_G_values_at_t[CountryX]:= [seq(G_values_at_t[CountryX][k]*e_values_at_t[CountryX][k],k=1..TheNum )]; end do; TheList := [US , Egypt, Jordan, Israel ] TheNum := 25

correct G values at t US := [667.9, 686.8, 682, 665.8, 708.31068, 708.9621, 777.16288, 802.47176, 776.75945, 765.85856, 806.68227, 881.9956, 908.01489, 898.13668, 830.16480, 798.20316, 778.75860, 751.71538, 893.43856, 1046.32528, 1250.58405, 1193.36676, 1188.56248, 1322.96873, 1346.14944] TheNum := 19 correct G values at t Egypt := [806.5841, 938.5418, 1898.6039, 2275.7472, 2314.8468, 2767.5648, 3048.8640, 3400.9977, 4358.1252, 5251.9308, 7299.2850, 8101.2000, 14022.3200, 30874.7600, 59342.9250, 66563.7000, 74102.4000, 89107.2000, 108342.8000] TheNum := 19

correct G values at t Jordan := [403.8714, 490.010, 606.8387, 883.8233, 1174.7345, 1232.5041, 1220.8986, 1378.7334, 1371.2543, 1461.0035, 1513.0993, 941.43897, 726.65684, 701.74230, 768.4894, 832.10650, 887.15439, 967.62344, 1025.84256 ] TheNum := 19 correct G values at t Israel := [9.5245872, 22.2590784, 70.6453679, 430.3275868, 2.0015086, 8.4637343, 61.4760, 1827.5859, 15170.423, 24807.9745, 42254.6424, 50737.5800, 65199.3378, 92486.4048, 130974.1642, 171638.1810, 215045.7465, 272234.3820, 362066.4695]

3.6 > > > > > > > > >

Modify all X values TheList:= [US,Egypt,Jordan,Israel]; for CountryX in TheList do TheNum:= min(nops(X_values_at_t[CountryX]),nops(e_values_at_t[CountryX])); correct_X_values_at_t[CountryX]:= [seq(X_values_at_t[CountryX][k]*e_values_at_t[CountryX][k],k=1..TheNum )]; end do; TheList := [US , Egypt, Jordan, Israel ] TheNum := 25

correct X values at t US := [1.5, 91.9, 99.8, 112.8, 132.12969, 152.43228, 220.77120, 308.30392, 303.21130, 348.65618, 414.57711, 519.16580, 634.41168, 730.93174, 722.49480, 646.19598, 633.22560, 693.49150, 790.51848, 942.96488, 1235.82957, 1340.85548, 1439.96894, 1682.91183, 1748.80704] TheNum := 19

correct X values at t Egypt := [1772.8690, 2682.0578, 7842.4605, 9581.5296, 10321.8864, 10799.9892, 11096.1060, 11488.0584, 13252.7604, 14214.6324, 18074.4200, 30520.8000, 55800.1600, 144556.4800, 337468.2000, 388823.8200, 424699.3800, 442070.7200, 478220.0800] TheNum := 19 correct X values at t Jordan := [2017.5517, 2243.4721, 3001.9560, 3635.6163, 5109.5148, 5740.5961, 5397.3312, 5843.7561, 5890.9518, 4728.8544, 5353.7461, 3957.84495, 3715.48520, 4361.60480, 4205.14914, 5046.31650, 5288.18058, 5082.41720, 5572.96656] TheNum := 19 correct X values at t Israel := [29.5839168, 73.5669696, 246.1540089, 1107.938785, 4.9445214, 21.2016483, 154.5360, 4576.1649, 43056.170, 73893.0383, 119189.6112, 132807.4750, 182297.0802, 247466.6160, 331383.6418, 460107.5325, 620253.4390, 797634.8436, 973928.2793]

3.7 > > > > > > > > >

Modify all C values TheList:= [US,Egypt,Jordan,Israel]; for CountryX in TheList do TheNum:= min(nops(C_values_at_t[CountryX]),nops(e_values_at_t[CountryX])); correct_C_values_at_t[CountryX]:= [seq(C_values_at_t[CountryX][k]*e_values_at_t[CountryX][k],k=1..TheNum )]; end do; TheList := [US , Egypt, Jordan, Israel ] TheNum := 25

correct C values at t US 1023.26848, 1135.87588, 2086.20801, 2229.52674, 2929.87216, 3486.85392, 5587.71456]

:= [509.3, 559.1, 603.7, 646.5, 760.31571, 833.60046, 1199.85043, 1328.05358, 1544.49105, 1851.66964, 2242.21320, 2271.50352, 2363.70720, 2411.58606, 4330.15614, 4435.56177, 4630.05802, 5332.84868,

TheNum := 19 correct C values at t Egypt := [2337.0501, 3201.0342, 7951.2683, 9841.3344, 9089.0940, 10258.6770, 11890.5696, 14652.3516, 19166.3703, 23883.6990, 35652.2900, 41024.1000, 78206.9600, 196204.1200, 385609.8500, 463650.6000, 532611.0000, 646027.2000, 750336.8800] TheNum := 19 correct C values at t Jordan := [1064.6112, 1334.3746, 1900.2072, 2213.5557, 2771.1355, 3759.9241, 4072.4989, 4251.2236, 4628.7892, 4431.2378, 4306.4142, 2533.92435, 1920.09793, 2089.16050, 2112.41372, 2787.44100, 2803.13487, 2711.04596, 2868.41216] TheNum := 19 correct C values at t Israel := [16.4220240, 35.9619072, 130.0429252, 566.9318788, 2.6435500, 12.7190672, 102.6480, 2604.5760, 24256.790, 50543.1800, 79434.2748, 100348.2125, 137157.4896, 189060.5904, 266703.1876, 372635.2245, 478087.2460, 629007.2544, 754148.6431]

4

4.1

Regressions for the coefficients in II, C, X and G for US, Egypt, Jordan and Israel ONLY

Regression for II (I, but Maple considers I = sqrt(-1))

The formula for I(t) is I(t) = I[0] + I[1]*y(t) + I[2]*y(t-1) + I[3]*y dot(t) + I[4]*y dot(t-1) . We apply linear regression to this formula for each of the 5 countries US, Egypt, Jordan and Israel. We will use the symbol II (two Is) because Maple understands I to be the complex unit. > with(stats):with(fit); > TheList:= [US,Egypt,Jordan,Israel];

[leastmediansquare, leastsquare] > > > > > >

TheList := [US , Egypt, Jordan, Israel ] for kk in TheList do [nops(correct_II_values_at_t[kk]),nops(correct_y_values_at_t[kk]),nops (correct_y_values_at_t_minus_1[kk]),nops(correct_y_dot_values_at_t[kk] )] end do; [25, 25, 25, 25] [19, 19, 19, 19] [19, 19, 19, 19]

> > >

[19, 19, 19, 19] for jj in TheList do II_Eqn[jj] := fit[leastsquare[[xx,yy,zz,ww,vv],xx = _II[jj][0]

+ > > > > > > > > > > >

_II[jj][1]*yy +_II[jj][2]*zz + _II[jj][3]*ww + _II[jj][4]*vv,{_II[jj][0],_II[jj][1],_II[jj][2],_II[jj][3],_II[jj][4] }]]([correct_II_values_at_t[jj],correct_y_values_at_t[jj],correct_y_v alues_at_t_minus_1[jj],correct_y_dot_values_at_t[jj],correct_y_dot_val ues_at_t_minus_1[jj]]); coeff_II[jj][0]:= tcoeff(rhs(II_Eqn[jj] )); coeff_II[jj][1]:=coeff(rhs(II_Eqn[jj] ),yy); coeff_II[jj][2]:=coeff(rhs(II_Eqn[jj] ),zz); coeff_II[jj][3]:=coeff(rhs(II_Eqn[jj] ),ww); coeff_II[jj][4]:=coeff(rhs(II_Eqn[jj] ),vv); end do;

II Eqn US := xx = 239.1606017 − .1443733719 yy + .2230408032 zz + .02144511929 ww + .2020955107 vv coeff II US 0 := 239.1606017 coeff II US 1 := −.1443733719 coeff II US 2 := .2230408032 coeff II US 3 := .02144511929 coeff II US 4 := .2020955107 II Eqn Egypt := xx = 4506.055790 + .4409145528 yy − .1452863646 zz − .4225832839 ww − .4255172945 vv coeff II Egypt 0 := 4506.055790 coeff II Egypt 1 := .4409145528 coeff II Egypt 2 := −.1452863646 coeff II Egypt 3 := −.4225832839 coeff II Egypt 4 := −.4255172945

II Eqn Jordan := xx = −438.7716302 + .2192900363 yy + .05273427600 zz + .4425362510 ww + 1.052724441 vv coeff II Jordan 0 := −438.7716302 coeff II Jordan 1 := .2192900363 coeff II Jordan 2 := .05273427600 coeff II Jordan 3 := .4425362510 coeff II Jordan 4 := 1.052724441 II Eqn Israel := xx = −522.8044136 + .2902638528 yy + .04275299661 zz − .2929226782 ww − .1567468804 vv coeff II Israel 0 := −522.8044136 coeff II Israel 1 := .2902638528 coeff II Israel 2 := .04275299661 coeff II Israel 3 := −.2929226782 coeff II Israel 4 := −.1567468804

4.2

> > > > > >

Regression for G

for kk in TheList do [nops(correct_G_values_at_t[kk]),nops(correct_y_values_at_t[kk]),nops( correct_y_values_at_t_minus_1[kk]),nops(correct_y_dot_values_at_t[kk]) ] end do; [25, 25, 25, 25] [19, 19, 19, 19] [19, 19, 19, 19] [19, 19, 19, 19]

> > >

for jj in TheList do G_Eqn[jj] := fit[leastsquare[[xx,yy,zz,ww,vv],xx = _G[jj][0]

+ > > > > > > > > > > >

_G[jj][1]*yy +_G[jj][2]*zz + _G[jj][3]*ww + _G[jj][4]*vv,{_G[jj][0],_G[jj][1],_G[jj][2],_G[jj][3],_G[jj][4] }]]([correct_G_values_at_t[jj],correct_y_values_at_t[jj],correct_y_va lues_at_t_minus_1[jj],correct_y_dot_values_at_t[jj],correct_y_dot_valu es_at_t_minus_1[jj]]); coeff_G[jj][0]:= tcoeff(rhs(G_Eqn[jj] )); coeff_G[jj][1]:=coeff(rhs(G_Eqn[jj] ),yy); coeff_G[jj][2]:=coeff(rhs(G_Eqn[jj] ),zz); coeff_G[jj][3]:=coeff(rhs(G_Eqn[jj] ),ww); coeff_G[jj][4]:=coeff(rhs(G_Eqn[jj] ),vv); end do; G Eqn US := xx = 707.7333918 − .009552954199 yy + .05125119922 zz + .01311846094 ww + .03873990077 vv coeff G US 0 := 707.7333918 coeff G US 1 := −.009552954199 coeff G US 2 := .05125119922 coeff G US 3 := .01311846094 coeff G US 4 := .03873990077 G Eqn Egypt := xx = 969.8153198 + .1146709860 yy − .01856143994 zz + .01552476882 ww + .01116854308 vv coeff G Egypt 0 := 969.8153198 coeff G Egypt 1 := .1146709860 coeff G Egypt 2 := −.01856143994 coeff G Egypt 3 := .01552476882 coeff G Egypt 4 := .01116854308

G Eqn Jordan := xx = −171.7686620 + .2423377028 yy + .03488012324 zz + .1879245652 ww + .08768981602 vv coeff G Jordan 0 := −171.7686620 coeff G Jordan 1 := .2423377028 coeff G Jordan 2 := .03488012324 coeff G Jordan 3 := .1879245652 coeff G Jordan 4 := .08768981602 G Eqn Israel := xx = 1232.246585 + .1965581699 yy + .06486483366 zz + .04300166561 ww + .1983812534 vv coeff G Israel 0 := 1232.246585 coeff G Israel 1 := .1965581699

coeff G Israel 2 := .06486483366 coeff G Israel 3 := .04300166561 coeff G Israel 4 := .1983812534

4.3

Regression for C - OK

In the following ATI = after tax income > for kk in TheList > do > N_at_t[kk]:= nops(correct_y_values_at_t[kk]); > ATI_at_t[kk]:= > [seq(correct_y_values_at_t[kk][i]-correct_T_values_at_t[kk][i],i=1..N_ > at_t[kk])]; > # > N_at_t_minus_1[kk]:= nops(correct_y_values_at_t_minus_1[kk]); > ATI_at_t_minus_1[kk]:= > [seq(correct_y_values_at_t_minus_1[kk][i]-correct_T_values_at_t_minus_ > 1[kk][i],i=1..N_at_t[kk])]; > # > N_dot_at_t[kk]:= nops(correct_y_dot_values_at_t[kk]); > ATI_dot_at_t[kk]:= > [seq(correct_y_dot_values_at_t[kk][i]-correct_T_dot_values_at_t[kk][i] > ,i=1..N_at_t[kk])]; > # > N_dot_at_t_minus_1[kk]:= > nops(correct_y_dot_values_at_t_minus_1[kk]); > ATI_dot_at_t_minus_1[kk]:= > [seq(correct_y_dot_values_at_t_minus_1[kk][i]-correct_T_dot_values_at_ > t_minus_1[kk][i],i=1..N_at_t[kk])]; > end do; N at t US := 25 ATI at t US := [813.96, 801.52, 867.95, 890.34, 1047.939354, 1158.821895, 1405.238848, 1473.602132, 1537.023737, 1586.565698, 1977.616629, 27119.95768, 2663.383005, 2889.456962, 2847.015960, 2781.499743, 2948.310120, 3027.54374, 3682.16632, 4305.41936, 5270.89611, 5275.238199, 5475.995418, 6198.84617, 6428.103168] N at t minus 1 US := 25 ATI at t minus 1 US := [696.53, 813.96, 801.52, 867.95, 890.34, 1047.939354, 1158.821895, 1405.238848, 1473.602132, 1537.023737, 1586.565698, 1977.616629, 27119.95768, 2663.383005, 2889.456962, 2847.015960, 2781.499743, 2948.310120, 3027.54374, 3682.16632, 4305.41936, 5270.89611, 5275.238199, 5475.995418, 6198.84617]

N dot at t US := 25 ATI dot at t US := [−12.44, 66.43, 22.39, 74.88, 110.882541, 105.823179, 46.699744, 133.924872, 61.695890, 304.926028, 23308.39242, −24485.89150, 320.999664, 230.031144, 88.044960, 324.785733, 285.558780, 258.38072, 183.98200, 239.13560, 290.507199, 332.011104, 250.105402, 194.653814, 393.646080] N dot at t minus 1 US := 25 ATI dot at t minus 1 US := [117.43, −12.44, 66.43, 22.39, 74.88, 110.882541, 105.823179, 46.699744, 133.924872, 61.695890, 304.926028, 23308.39242, −24485.89150, 320.999664, 230.031144, 88.044960, 324.785733, 285.558780, 258.38072, 183.98200, 239.13560, 290.507199, 332.011104, 250.105402, 194.653814] N at t Egypt := 19 ATI at t Egypt := [1558.9840, 1693.5556, 2238.8588, 3839.0400, 1911.5208, 3796.1352, 22298.4825, 8792.3715, 13715.6382, 20677.2300, 32825.9274, 26519.1840, 53329.6296, 129432.1160, 201637.2495, 262637.4072, 314467.4286, 424402.7424, 553535.9632] N at t minus 1 Egypt := 19 ATI at t minus 1 Egypt := [1031.9420, 1558.9840, 1693.5556, 2238.8588, 3839.0400, 1911.5208, 3796.1352, 22298.4825, 8792.3715, 13715.6382, 20677.2300, 32825.9274, 26519.1840, 53329.6296, 129432.1160, 201637.2495, 262637.4072, 314467.4286, 424402.7424] N dot at t Egypt := 19 ATI dot at t Egypt := [19.9626, −455.7612, 1726.1712, −1744.5312, 2094.0360, 19698.0498, −12906.3690, 3446.2803, 4853.2968, 7623.6048, −4868.1762, 8232.1380, 12423.4864, −9054.6992, 71063.7485, 49059.7422, 82587.1248, 119378.8960, 53979.9104] N dot at t minus 1 Egypt := 19 ATI dot at t minus 1 Egypt := [459.1460, 19.9626, −455.7612, 1726.1712, −1744.5312, 2094.0360, 19698.0498, −12906.3690, 3446.2803, 4853.2968, 7623.6048, −4868.1762, 8232.1380, 12423.4864, −9054.6992, 71063.7485, 49059.7422, 82587.1248, 119378.8960] N at t Jordan := 19 ATI at t Jordan := [−449.2618, −1006.8416, −659.9661, 297.6166, −1530.3786, −2211.4925, −1688.7292, −1416.3868, −1352.6855, 748.4258, −565.0589, −887.69142, −477.94010, −1781.25640, −1641.99878, −2085.63400, −1879.26993, −1171.66280, −1289.13456]

N at t minus 1 Jordan := 19 ATI at t minus 1 Jordan := [−169.6982, −449.2618, −1006.8416, −659.9661, 297.6166, −1530.3786, −2211.4925, −1688.7292, −1416.3868, −1352.6855, 748.4258, −565.0589, −887.69142, −477.94010, −1781.25640, −1641.99878, −2085.63400, −1879.26993, −1171.66280] N dot at t Jordan := 19 ATI dot at t Jordan := [−557.5798, 346.8755, 957.5827, −1827.9952, −681.1139, 522.7633, 272.3424, 63.7013, 2101.1113, −1313.4847, −904.4553, 253.62612, −1500.99026, 105.48860, −410.34434, 172.92575, 639.38403, −156.05884, −301.62352] N dot at t minus 1 Jordan := 19 ATI dot at t minus 1 Jordan := [−279.5636, −557.5798, 346.8755, 957.5827, −1827.9952, −681.1139, 522.7633, 272.3424, 63.7013, 2101.1113, −1313.4847, −904.4553, 253.62612, −1500.99026, 105.48860, −410.34434, 172.92575, 639.38403, −156.05884] N at t Israel := 19 ATI at t Israel := [−667.4381088, −2504.388106, −4318.763876, −8418.415183, −24.8890278, −77.9161957, −217.3656, −686.2056, 476.604, 9172.3145, 1617.4548, 24881.2775, 39670.6266, 60547.5216, 105083.6946, 149872.7175, 134496.3880, 174619.4772, 234724.577] N at t minus 1 Israel := 19 ATI at t minus 1 Israel := [−687.4721082, −667.4381088, −2504.388106, −4318.763876, −8418.415183, −24.8890278, −77.9161957, −217.3656, −686.2056, 476.604, 9172.3145, 1617.4548, 24881.2775, 39670.6266, 60547.5216, 105083.6946, 149872.7175, 134496.3880, 174619.4772] N dot at t Israel := 19 ATI dot at t Israel := [−1227.906197, 213.7730112, 250.9592150, −4746.512993, −13.3340116, 6.4244957, 93.7368, 889.0620, 6944.591, −7825.1024, 22334.4096, 9988.3375, 13938.1191, 33206.2992, 23699.1849, −25246.7215, 28062.4630, 47296.2630, 82525.7109] N dot at t minus 1 Israel := 19 ATI dot at t minus 1 Israel := [324.3703350, −1227.906197, 213.7730112, 250.9592150, −4746.512993, −13.3340116, 6.4244957, 93.7368, 889.0620, 6944.591, −7825.1024, 22334.4096, 9988.3375, 13938.1191, 33206.2992, 23699.1849, −25246.7215, 28062.4630, 47296.2630]

> > >

for jj in TheList do C_Eqn[jj] := fit[leastsquare[[xx,yy,zz,ww,vv],xx = _C[jj][0]

+ > > > > > > > > > >

_C[jj][1]*yy +_C[jj][2]*zz + _C[jj][3]*ww + _C[jj][4]*vv,{_C[jj][0],_C[jj][1],_C[jj][2],_C[jj][3],_C[jj][4] }]]([correct_C_values_at_t[jj],ATI_at_t[jj],ATI_at_t_minus_1[jj],ATI_ dot_at_t[jj],ATI_dot_at_t_minus_1[jj]]); coeff_C[jj][0]:= tcoeff(rhs(C_Eqn[jj] )); coeff_C[jj][1]:=coeff(rhs(C_Eqn[jj] ),yy); coeff_C[jj][2]:=coeff(rhs(C_Eqn[jj] ),zz); coeff_C[jj][3]:=coeff(rhs(C_Eqn[jj] ),ww); coeff_C[jj][4]:=coeff(rhs(C_Eqn[jj] ),vv); end do;

C Eqn US := xx = 1146.897507 − 1.091692351 yy + 1.427356180 zz + .08628195119 ww + 1.352950713 vv coeff C US 0 := 1146.897507 coeff C US 1 := −1.091692351 coeff C US 2 := 1.427356180 coeff C US 3 := .08628195119 coeff C US 4 := 1.352950713 C Eqn Egypt := xx = 2624.744975 + 1.570466829 yy − .2161467818 zz + 1.055861332 ww − .4910397412 vv coeff C Egypt 0 := 2624.744975 coeff C Egypt 1 := 1.570466829 coeff C Egypt 2 := −.2161467818 coeff C Egypt 3 := 1.055861332 coeff C Egypt 4 := −.4910397412 C Eqn Jordan := xx = 2884.625643 − 1.152859659 yy + 1.055784451 zz + .2422186076 ww + 1.337259658 vv coeff C Jordan 0 := 2884.625643 coeff C Jordan 1 := −1.152859659 coeff C Jordan 2 := 1.055784451 coeff C Jordan 3 := .2422186076 coeff C Jordan 4 := 1.337259658 C Eqn Israel := xx = 14809.72643 + .5226134637 yy + 2.691699860 zz + 1.241932793 ww + 1.346021417 vv coeff C Israel 0 := 14809.72643 coeff C Israel 1 := .5226134637 coeff C Israel 2 := 2.691699860

coeff C Israel 3 := 1.241932793 coeff C Israel 4 := 1.346021417

4.4

>

Regression for X

TheList:= [US,Egypt,Jordan,Israel]; TheList := [US , Egypt, Jordan, Israel ]

nops(correct_y_values_at_t[US]); nops(correct_y_values_at_t_minus_1[Jordan]); nops(correct_y_values_at_t_minus_1[Egypt]); nops(correct_y_values_at_t_minus_1[Israel]); 25 19 19 19 We must reduce the number of entries for all the US values to 19. > > > >

> > > > > > > > > > > > > > > > > > > >

modified_y_values_at_t[US]:= [seq(correct_y_values_at_t[US][k],k=1..19)]; nops(%); modified_y_values_at_t_minus_1[US]:= [seq(correct_y_values_at_t_minus_1[US][k],k=1..19)]; nops(%); modified_y_dot_values_at_t[US]:= [seq(correct_y_dot_values_at_t[US][k],k=1..19)]; nops(%); modified_y_dot_values_at_t_minus_1[US]:= [seq(correct_y_dot_values_at_t_minus_1[US][k],k=1..19)]; nops(%); modified_e_values_at_t[US]:= [seq(e_values_at_t[US][k],k=1..19)]; nops(%); modified_tau_values_at_t[US]:= [seq(correct_tau_values_at_t[US][k],k=1..19)]; nops(%); modified_X_values_at_t[US]:= [seq(correct_X_values_at_t[US][k],k=1..19)]; nops(%);

modified y values at t US := [814.3, 889.3, 959.5, 1010.4, 1190.79576, 1309.89705, 1627.55424, 1786.15472, 1855.32536, 1947.29298, 2397.93927, 27661.96156, 3278.36451, 3581.45074, 3527.61840, 3474.32376, 3565.13970, 3702.41144, 4436.10808] 19 modified y values at t minus 1 US := [769.8, 814.3, 889.3, 959.5, 1010.4, 1190.79576, 1309.89705, 1627.55424, 1786.15472, 1855.32536, 1947.29298, 2397.93927, 27661.96156, 3278.36451, 3581.45074, 3527.61840, 3474.32376, 3565.13970, 3702.41144] 19 modified y dot values at t US := [75.0, 70.2, 50.9, 86.4, 119.10129, 154.82082, 132.34208, 154.27440, 106.88491, 346.21640, 23393.42142, −24419.68320, 420.74562, 283.77650, 138.51600, 281.84205, 389.58870, 256.32230, 252.52216 ] 19 modified y dot values at t minus 1 US := [44.5, 75.0, 70.2, 50.9, 86.4, 119.10129, 154.82082, 132.34208, 154.27440, 106.88491, 346.21640, 23393.42142, −24419.68320, 420.74562, 283.77650, 138.51600, 281.84205, 389.58870, 256.32230] 19 modified e values at t US := [1, 1, 1, 1, 1.0857, 1.0857, 1.2064, 1.2244, 1.1707, 1.1618, 1.2147, 1.3028, 1.3173, 1.2754, 1.1640, 1.1031, 1.0470, .9802, 1.0984] 19

modified tau values at t US := [2000, 2300, 2400, 2584, 3004.1319, 3391.7268, 4367.1680, 4617.2124, 4425.2460, 5431.4150, 6662.6295, 9330.6536, 9487.1946, 9610.1390, 11631.8520, 9583.7328, 9873.2100, 11803.5684, 14352.7928] 19 modified X values at t US := [1.5, 91.9, 99.8, 112.8, 132.12969, 152.43228, 220.77120, 308.30392, 303.21130, 348.65618, 414.57711, 519.16580, 634.41168, 730.93174, 722.49480, 646.19598, 633.22560, 693.49150, 790.51848] 19

> > > > > > > > > >

nops(modified_X_values_at_t[US]); nops(modified_y_values_at_t[US]); nops(modified_y_values_at_t_minus_1[US]); nops(modified_y_dot_values_at_t[US]); nops(modified_y_dot_values_at_t_minus_1[US]); nops(modified_e_values_at_t[US]); nops(modified_tau_values_at_t[US]); nops(correct_y_values_at_t_minus_1[Egypt]); nops(correct_y_values_at_t_minus_1[Jordan]); nops(correct_y_values_at_t_minus_1[Israel]); 19 19 19 19 19 19 19 19 19 19

REGRESSION FOR X(t) FOR US

> > > >

X_Eqn[US] := fit[leastsquare[[xx_US_at_t,yy_US_at_t,yy_US_at_t_minus_1,yy_US_dot_at _t,yy_US_dot_at_t_minus_1,ee_US_at_t,tautau_US_at_t,yy_Egypt_at_t_minu s_1,yy_Jordan_at_t_minus_1,yy_Israel_at_t_minus_1],xx_US_at_t

= >

_X[US][0] + _X[US][1]*yy_US_at_t +_X[US][2]*yy_US_at_t_minus_1

+ > > > > > > > > > > > > > > > > > > > > > > > > >

_X[US][3]*yy_US_dot_at_t + _X[US][4]*yy_US_dot_at_t_minus_1 + _X[US][15]*ee_US_at_t + _X[US][16]*tautau_US_at_t + yy_US_at_t*(_aa[US][Egypt]*yy_Egypt_at_t_minus_1 + _aa[US][Jordan]*yy_Jordan_at_t_minus_1 + _aa[US][Israel]*yy_Israel_at_t_minus_1), {_X[US][0],_X[US][1],_X[US][2],_X[US][3],_X[US][4],_X[US][15],_X[US][ 16],_aa[US][Egypt],_aa[US][Jordan],_aa[US][Israel] }]]([modified_X_values_at_t[US],modified_y_values_at_t[US],modified_y _values_at_t_minus_1[US],modified_y_dot_values_at_t[US],modified_y_dot _values_at_t_minus_1[US],modified_e_values_at_t[US],modified_tau_value s_at_t[US],correct_y_values_at_t_minus_1[Egypt],correct_y_values_at_t_ minus_1[Jordan],correct_y_values_at_t_minus_1[Israel]]); coeff_X[US][0]:= tcoeff(rhs(X_Eqn[US] )); coeff_X[US][1]:= tcoeff(coeff(rhs(X_Eqn[US] ),yy_US_at_t)); coeff_X[US][2]:=coeff(rhs(X_Eqn[US] ),yy_US_at_t_minus_1); coeff_X[US][3]:=coeff(rhs(X_Eqn[US] ),yy_US_dot_at_t); coeff_X[US][4]:=coeff(rhs(X_Eqn[US] ),yy_US_dot_at_t_minus_1); coeff_X[US][15]:=coeff(rhs(X_Eqn[US] ),ee_US_at_t); coeff_X[US][16]:=coeff(rhs(X_Eqn[US] ),tautau_US_at_t); coeff_aa[US][Egypt]:= coeff(coeff(rhs(X_Eqn[US] ),yy_Egypt_at_t_minus_1),yy_US_at_t); coeff_aa[US][Jordan]:= coeff(coeff(rhs(X_Eqn[US] ),yy_Jordan_at_t_minus_1),yy_US_at_t); coeff_aa[US][Israel]:= coeff(coeff(rhs(X_Eqn[US] ),yy_Israel_at_t_minus_1),yy_US_at_t); X Eqn US := xx US at t = −886.7286925 − .2752051129 yy US at t + .3295204000 yy US at t minus 1 + .002601190450 yy US dot at t + .3295502327 yy US dot at t minus 1 + 858.6987962 ee US at t + .02694357803 tautau US at t + yy US at t( −.1456557517 10−7 yy Egypt at t minus 1 − .7086660721 10−5 yy Jordan at t minus 1 + .1026864688 10−6 yy Israel at t minus 1 ) coeff X US 0 := −886.7286925 coeff X US 1 := −.2752051129 coeff X US 2 := .3295204000 coeff X US 3 := .002601190450 coeff X US 4 := .3295502327 coeff X US 15 := 858.6987962 coeff X US 16 := .02694357803 coeff aa US Egypt := −.1456557517 10−7

coeff aa US Jordan := −.7086660721 10−5 coeff aa US Israel := .1026864688 10−6

REGRESSION FOR X(t) FOR Egypt

> > > > > >

X_Eqn[Egypt] := fit[leastsquare[[xx_Egypt_at_t,yy_Egypt_at_t,yy_Egypt_at_t_minus_1,yy_ Egypt_dot_at_t,yy_Egypt_dot_at_t_minus_1,ee_Egypt_at_t,tautau_Egypt_at _t,yy_US_at_t_minus_1,yy_Jordan_at_t_minus_1,yy_Israel_at_t_minus_1],x x_Egypt_at_t = _X[Egypt][0] + _X[Egypt][1]*yy_Egypt_at_t +_X[Egypt][2]*yy_Egypt_at_t_minus_1 + _X[Egypt][3]*yy_Egypt_dot_at_t

+ >

_X[Egypt][4]*yy_Egypt_dot_at_t_minus_1 + _X[Egypt][15]*ee_Egypt_at_t

+ > > > > > > > > > > > > > > > > > > > > > > > > > > >

_X[Egypt][16]*tautau_Egypt_at_t + yy_Egypt_at_t*( _aa[Egypt][US]*yy_US_at_t_minus_1 + _aa[Egypt][Jordan]*yy_Jordan_at_t_minus_1 + _aa[Egypt][Israel]*yy_Israel_at_t_minus_1), {_X[Egypt][0],_X[Egypt][1],_X[Egypt][2],_X[Egypt][3],_X[Egypt][4],_X[ Egypt][15],_X[Egypt][16],_aa[Egypt][US],_aa[Egypt][Jordan],_aa[Egypt][ Israel] }]]([correct_X_values_at_t[Egypt],correct_y_values_at_t[Egypt],correc t_y_values_at_t_minus_1[Egypt],correct_y_dot_values_at_t[Egypt],correc t_y_dot_values_at_t_minus_1[Egypt],e_values_at_t[Egypt],tau_values_at_ t[Egypt],modified_y_values_at_t_minus_1[US],correct_y_values_at_t_minu s_1[Jordan],correct_y_values_at_t_minus_1[Israel]]); coeff_X[Egypt][0]:= tcoeff(rhs(X_Eqn[Egypt] )); coeff_X[Egypt][1]:= tcoeff(coeff(rhs(X_Eqn[Egypt] ),yy_Egypt_at_t)); coeff_X[Egypt][2]:=coeff(rhs(X_Eqn[Egypt] ),yy_Egypt_at_t_minus_1); coeff_X[Egypt][3]:=coeff(rhs(X_Eqn[Egypt] ),yy_Egypt_dot_at_t); coeff_X[Egypt][4]:=coeff(rhs(X_Eqn[Egypt] ),yy_Egypt_dot_at_t_minus_1); coeff_X[Egypt][15]:=coeff(rhs(X_Eqn[Egypt] ),ee_Egypt_at_t); coeff_X[Egypt][16]:=coeff(rhs(X_Eqn[Egypt] ),tautau_Egypt_at_t); coeff_aa[Egypt][US]:=coeff(coeff(rhs(X_Eqn[Egypt]),yy_US_at_t_minus_1) ,yy_Egypt_at_t); coeff_aa[Egypt][Jordan]:= coeff(coeff(rhs(X_Eqn[Egypt] ),yy_Jordan_at_t_minus_1),yy_Egypt_at_t); coeff_aa[Egypt][Israel]:= coeff(coeff(rhs(X_Eqn[Egypt] ),yy_Israel_at_t_minus_1),yy_Egypt_at_t);

X Eqn Egypt := xx Egypt at t = −13100.79975 + .4809991856 yy Egypt at t + .2886745888 yy Egypt at t minus 1 + .03814527782 yy Egypt dot at t − .1923263793 yy Egypt dot at t minus 1 + 23577.06585 ee Egypt at t − 3.397252849 tautau Egypt at t + yy Egypt at t( −.1118126676 10−5 yy US at t minus 1 − .00004412652028 yy Jordan at t minus 1 − .1232539606 10−6 yy Israel at t minus 1 ) coeff X Egypt 0 := −13100.79975 coeff X Egypt 1 := .4809991856 coeff X Egypt 2 := .2886745888 coeff X Egypt 3 := .03814527782 coeff X Egypt 4 := −.1923263793

coeff X Egypt 15 := 23577.06585 coeff X Egypt 16 := −3.397252849 coeff aa Egypt US := −.1118126676 10−5 coeff aa Egypt Jordan := −.00004412652028 coeff aa Egypt Israel := −.1232539606 10−6

REGRESSION FOR X(t) FOR Jordan

> > > > > > > > >

X_Eqn[Jordan] := fit[leastsquare[[xx_Jordan_at_t,yy_Jordan_at_t,yy_Jordan_at_t_minus_1, yy_Jordan_dot_at_t,yy_Jordan_dot_at_t_minus_1,ee_Jordan_at_t,tautau_Jo rdan_at_t,yy_US_at_t_minus_1,yy_Egypt_at_t_minus_1,yy_Israel_at_t_minu s_1],xx_Jordan_at_t = _X[Jordan][0] + _X[Jordan][1]*yy_Jordan_at_t +_X[Jordan][2]*yy_Jordan_at_t_minus_1 + _X[Jordan][3]*yy_Jordan_dot_at_t + _X[Jordan][4]*yy_Jordan_dot_at_t_minus_1 + _X[Jordan][15]*ee_Jordan_at_t + _X[Jordan][16]*tautau_Jordan_at_t

+ > > > > > > > > > > > > > > > > > > > > > > > > > > >

yy_Jordan_at_t*( _aa[Jordan][US]*yy_US_at_t_minus_1 + _aa[Jordan][Egypt]*yy_Egypt_at_t_minus_1 + _aa[Jordan][Israel]*yy_Israel_at_t_minus_1), {_X[Jordan][0],_X[Jordan][1],_X[Jordan][2],_X[Jordan][3],_X[Jordan][4 ],_X[Jordan][15],_X[Jordan][16],_aa[Jordan][US],_aa[Jordan][Egypt],_aa [Jordan][Israel] }]]([correct_X_values_at_t[Jordan],correct_y_values_at_t[Jordan],corr ect_y_values_at_t_minus_1[Jordan],correct_y_dot_values_at_t[Jordan],co rrect_y_dot_values_at_t_minus_1[Jordan],e_values_at_t[Jordan],tau_valu es_at_t[Jordan],modified_y_values_at_t_minus_1[US],correct_y_values_at _t_minus_1[Egypt],correct_y_values_at_t_minus_1[Israel]]); coeff_X[Jordan][0]:= tcoeff(rhs(X_Eqn[Jordan] )); coeff_X[Jordan][1]:= tcoeff(coeff(rhs(X_Eqn[Jordan] ),yy_Jordan_at_t)); coeff_X[Jordan][2]:=coeff(rhs(X_Eqn[Jordan] ),yy_Jordan_at_t_minus_1); coeff_X[Jordan][3]:=coeff(rhs(X_Eqn[Jordan] ),yy_Jordan_dot_at_t); coeff_X[Jordan][4]:=coeff(rhs(X_Eqn[Jordan] ),yy_Jordan_dot_at_t_minus_1); coeff_X[Jordan][15]:=coeff(rhs(X_Eqn[Jordan] ),ee_Jordan_at_t); coeff_X[Jordan][16]:=coeff(rhs(X_Eqn[Jordan] ),tautau_Jordan_at_t); coeff_aa[Jordan][US]:=coeff(coeff(rhs(X_Eqn[Jordan]),yy_US_at_t_minus_ 1),yy_Jordan_at_t); coeff_aa[Jordan][Egypt]:= coeff(coeff(rhs(X_Eqn[Jordan] ),yy_Egypt_at_t_minus_1),yy_Jordan_at_t); coeff_aa[Jordan][Israel]:= coeff(coeff(rhs(X_Eqn[Jordan] ),yy_Israel_at_t_minus_1),yy_Jordan_at_t);

X Eqn Jordan := xx Jordan at t = 2924.026293 + 1.181757113 yy Jordan at t − .06233628481 yy Jordan at t minus 1 + .7348569807 yy Jordan dot at t + 1.207188017 yy Jordan dot at t minus 1 − 1119.769687 ee Jordan at t − 6.667560030 tautau Jordan at t + yy Jordan at t( −.3532328762 10−5 yy US at t minus 1 + .1103555320 10−5 yy Egypt at t minus 1 − .9925734476 10−6 yy Israel at t minus 1 ) coeff X Jordan 0 := 2924.026293 coeff X Jordan 1 := 1.181757113 coeff X Jordan 2 := −.06233628481 coeff X Jordan 3 := .7348569807 coeff X Jordan 4 := 1.207188017 coeff X Jordan 15 := −1119.769687 coeff X Jordan 16 := −6.667560030 coeff aa Jordan US := −.3532328762 10−5 coeff aa Jordan Egypt := .1103555320 10−5 coeff aa Jordan Israel := −.9925734476 10−6

REGRESSION FOR X(t) FOR Israel

> > > > > > > > >

X_Eqn[Israel] := fit[leastsquare[[xx_Israel_at_t,yy_Israel_at_t,yy_Israel_at_t_minus_1, yy_Israel_dot_at_t,yy_Israel_dot_at_t_minus_1,ee_Israel_at_t,tautau_Is rael_at_t,yy_US_at_t_minus_1,yy_Egypt_at_t_minus_1,yy_Jordan_at_t_minu s_1],xx_Israel_at_t = _X[Israel][0] + _X[Israel][1]*yy_Israel_at_t +_X[Israel][2]*yy_Israel_at_t_minus_1 + _X[Israel][3]*yy_Israel_dot_at_t + _X[Israel][4]*yy_Israel_dot_at_t_minus_1 + _X[Israel][15]*ee_Israel_at_t + _X[Israel][16]*tautau_Israel_at_t

+ > > > > > > > > > > > > > > > > > > > > > > > > > > >

yy_Israel_at_t*( _aa[Israel][US]*yy_US_at_t_minus_1 + _aa[Israel][Egypt]*yy_Egypt_at_t_minus_1 + _aa[Israel][Jordan]*yy_Jordan_at_t_minus_1), {_X[Israel][0],_X[Israel][1],_X[Israel][2],_X[Israel][3],_X[Israel][4 ],_X[Israel][15],_X[Israel][16],_aa[Israel][US],_aa[Israel][Egypt],_aa [Israel][Jordan] }]]([correct_X_values_at_t[Israel],correct_y_values_at_t[Israel],corr ect_y_values_at_t_minus_1[Israel],correct_y_dot_values_at_t[Israel],co rrect_y_dot_values_at_t_minus_1[Israel],e_values_at_t[Israel],tau_valu es_at_t[Israel],modified_y_values_at_t_minus_1[US],correct_y_values_at _t_minus_1[Egypt],correct_y_values_at_t_minus_1[Jordan]]); coeff_X[Israel][0]:= tcoeff(rhs(X_Eqn[Israel] )); coeff_X[Israel][1]:= tcoeff(coeff(rhs(X_Eqn[Israel] ),yy_Israel_at_t)); coeff_X[Israel][2]:=coeff(rhs(X_Eqn[Israel] ),yy_Israel_at_t_minus_1); coeff_X[Israel][3]:=coeff(rhs(X_Eqn[Israel] ),yy_Israel_dot_at_t); coeff_X[Israel][4]:=coeff(rhs(X_Eqn[Israel] ),yy_Israel_dot_at_t_minus_1); coeff_X[Israel][15]:=coeff(rhs(X_Eqn[Israel] ),ee_Israel_at_t); coeff_X[Israel][16]:=coeff(rhs(X_Eqn[Israel] ),tautau_Israel_at_t); coeff_aa[Israel][US]:=coeff(coeff(rhs(X_Eqn[Israel]),yy_US_at_t_minus_ 1),yy_Israel_at_t); coeff_aa[Israel][Egypt]:= coeff(coeff(rhs(X_Eqn[Israel] ),yy_Egypt_at_t_minus_1),yy_Israel_at_t); coeff_aa[Israel][Jordan]:= coeff(coeff(rhs(X_Eqn[Israel] ),yy_Jordan_at_t_minus_1),yy_Israel_at_t);

X Eqn Israel := xx Israel at t = −1116.546868 + 1.048444876 yy Israel at t − .5551475739 yy Israel at t minus 1 + .5734028207 yy Israel dot at t − .8066104375 yy Israel dot at t minus 1 + 92.64006350 ee Israel at t − 15.67604974 tautau Israel at t + yy Israel at t( −.6575146123 10−6 yy US at t minus 1 − .6790730561 10−7 yy Egypt at t minus 1 + .00006435701183 yy Jordan at t minus 1 ) coeff X Israel 0 := −1116.546868 coeff X Israel 1 := 1.048444876 coeff X Israel 2 := −.5551475739 coeff X Israel 3 := .5734028207 coeff X Israel 4 := −.8066104375

coeff X Israel 15 := 92.64006350 coeff X Israel 16 := −15.67604974 coeff aa Israel US := −.6575146123 10−6 coeff aa Israel Egypt := −.6790730561 10−7 coeff aa Israel Jordan := .00006435701183

5

Set up the coefficients for the differential equations

We use the coefficients computed in the last section to define the coefficients for the diffential equations. >

_The_List:= [US,Egypt,Jordan,Israel]; The List := [US , Egypt, Jordan, Israel ]

First compute the z[country][i=0..4]

> > > >

# US for i from 0 to 4 do z[US][i]:= coeff_C[US][i] + coeff_II[US][i] + coeff_X[US][i]

+ > > > > > > > > > > > > > > > > > > > >

coeff_G[US][i]; od; # Egypt for i from 0 to 4 do z[Egypt][i]:= coeff_C[Egypt][i] + coeff_II[Egypt][i] + coeff_X[Egypt][i] + coeff_G[Egypt][i]; od; # Jordan for i from 0 to 4 do z[Jordan][i]:= coeff_C[Jordan][i] + coeff_II[Jordan][i] + coeff_X[Jordan][i] + coeff_G[Jordan][i]; od; # Israel for i from 0 to 4 do z[Israel][i]:= coeff_C[Israel][i] + coeff_II[Israel][i] + coeff_X[Israel][i] + coeff_G[Israel][i]; od; zUS 0 := 1207.062808 zUS 1 := −1.520823790 zUS 2 := 2.031168582 zUS 3 := .1234467219 zUS 4 := 1.923336358 zEgypt 0 := −5000.183665 zEgypt 1 := 2.607051554 zEgypt 2 := −.09131999754 zEgypt 3 := .6869480947 zEgypt 4 := −1.097714872 zJordan 0 := 5198.111644 zJordan 1 := .4905251931 zJordan 2 := 1.081062565 zJordan 3 := 1.607536404 zJordan 4 := 3.684861932 zIsrael 0 := 14402.62174 zIsrael 1 := 2.057880362 zIsrael 2 := 2.244170117 zIsrael 3 := 1.565414602 zIsrael 4 := .5810453529

Now compute p[country]

> > > > >

for countryX in _The_List do p[countryX] := (coeff_C[countryX][0] + coeff_C[countryX][0] + coeff_C[countryX][0])/(1-z[countryX][3]); end do; pUS := 3925.252014 pEgypt := 25153.12889 pJordan := −14244.21133 pIsrael := −78578.05428

Compute the formulas for g[countryX], namely g[countryX](t) = coeff G[countryX][0] + coeff X[countryX][15]*e[countryX](t) + coeff X[countryX][16]*tau[countryX](t) - ( coeff C[countryX][1]*T[countryX](t) + coeff C[countryX][2]*T[countryX](t1) + coeff C[countryX][3]*D(T)[countryX](t) + coeff C[countryX][4]*D(T)[countryX](t1) )/(1 - z[countryX][3]); First unassign the letters ”e”, ”T”, and Greek letter ”tau” so that they may be used as variables in the equations. > > > > > > > > >

;

>

unassign(’e’); unassign(’tau’); unassign(’T’); for countryX in _The_List do g[countryX]:= coeff_G[countryX][0] + coeff_X[countryX][15]*e[countryX](t) + coeff_X[countryX][16]* tau[countryX](t) - ( coeff_C[countryX][1]*T[countryX](t) + coeff_C[countryX][2]*T[countryX](t-1) + coeff_C[countryX][3]*D(T[countryX])(t) + coeff_C[countryX][4]*D(T[countryX])(t-1) )/(1-z[countryX][3]) end do;

gUS := 707.7333918 + 858.6987962 eUS (t) + .02694357803 τUS (t) + 1.245437531 TUS (t) − 1.628373557 TUS (t − 1) − .09843320801 D(TUS )(t) − 1.543489422 D(TUS )(t − 1) gEgypt := 969.8153198 + 23577.06585 eEgypt (t) − 3.397252849 τEgypt (t) − 5.016633990 TEgypt (t) + .6904502995 TEgypt (t − 1) − 3.372799571 D(TEgypt )(t) + 1.568556948 D(TEgypt )(t − 1) gJordan := −171.7686620 − 1119.769687 eJordan (t) − 6.667560030 τJordan (t) − 1.897597661 TJordan (t) + 1.737812654 TJordan (t − 1) + .3986898661 D(TJordan )(t) + 2.201118565 D(TJordan )(t − 1)

gIsrael := 1232.246585 + 92.64006350 eIsrael (t) − 15.67604974 τIsrael (t) + .9243013213 TIsrael (t) + 4.760577195 TIsrael (t − 1) + 2.196499327 D(TIsrael )(t) + 2.380591892 D(TIsrael )(t − 1)

Compute the coefficients a[countryX][-1]. > > > >

for countryX in _The_List do a[countryX][-1]:= z[countryX][4]/(1-z[countryX][3]); end do; aUS −1 := 2.194203600 aEgypt −1 := −3.506494781 aJordan −1 := −6.065252893

> > > >

> > > >

aIsrael −1 := −1.027644760 for countryX in _The_List do a[countryX][0]:= z[countryX][1]/(1-z[countryX][3]); end do; aUS 0 := −1.735004395 aEgypt 0 := 8.327857170 aJordan 0 := −.8074004946 aIsrael 0 := −3.639595360 for countryX in _The_List do a[countryX][1]:= z[countryX][2]/(1-z[countryX][3]); end do; aUS 1 := 2.317222048 aEgypt 1 := −.2917088061 aJordan 1 := −1.779420225 aIsrael 1 := −3.969069969

C1

Computer Code in Modeling Economic Dynamics of Interacting Nations

E. N. Chukwu Mathematics Department North Carolina State University Raleigh, North Carolina 27695-8205

C2

0 Introduction In Chapter 1 of the main book the methods of MAPLE and MATLAB codes and programs are introduced and explained. Simple examples are worked out and used to generate the dynamics of the economic state. In these sections the data program and equations are displayead.

1 Cooperation and Competition from DATA Program and Equation Identification to Model NIGERIA Data is in billions of Naira Adopted from International Statistics Yearbook 1994 YEAR = [1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992]´; y11 = [3614 2951 2878 3851 5621 7098 7703 11199 18811 21779 27572 32747 36084 43151 50849 50749 51709 57142 63608 72355 73062 108888 145243 224797 260637 324011]´; In dollars per pound 1966-1967 2.80; Niara per US dollar 1990: 19..646 y1 = y11 ÷ (19.646) = [ ]´; U.S.A. Data is in billions of US dollars Adopted from International Financial Statistic Yearbook 1994 YEAR = [1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992]´; y 21 = [769.8 814.3 889.3 959.5 1010.4 1096.8 1206.5 1349.1 1458.8 1584.8 1767.1 1974.1 2232.7 2488.7 2708.1 3030.6 3149.6 3405.1 3777.2 4038.7 4268.6 4539.9 4900.4 5250.8 5522.2 5722.9 6038.5]; y 2 = y 21 × 1 =[

]´;

UK Data is in billions of pounds and it is adopted from International Financial Statistics Yearbook 1994 YEAR = [1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992]´;

C3

y 31 = [38.37 40.4 43.81 47.15 51.77 57.75 64.66 74.26 83.86 105.85 125.25 148.98 168.53 198.22 231.77 254.93 279.04 304.46 325.85 357.34 384.84 423.38 471.43 515.96 551.12 573.65 595.26]´; In dollar y 3 = y 31 ×(2.4075) = [ ]´;

CHINA: GDP in billions of NT dollars Adopted from International Financial Statistics Yearbook YEAR = [1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992]´; % y 41 = GDP in billions of NT dollars 1966-1970 y 41 = [125.88 143.26 168.43 191.10 217.78]; % y 412 = GDP in billions of Yuan [5994 7488 10155 15141 16275 249 2644 301.0 335.0 366.0 388.0 424.7 467.3 548.5 676.0 633.0 946.4 1,117.9 1,401.5 1,600.8 1,768.0 20,190.0 2,402]´; % 1966 – 1970 y 4 = y 41 ÷ 40 = [ ]´; % 1971 – 1992 y 4 = y 412÷5.2804 = [ ]´; *(national income) W2 vals = X1 vals* X2 vals = [ W3 vals = X1 vals* X3 vals = [ W4 vals = X1 vals* X4 vals = [ dX1 = X1(t + 1) – X1(t) = [ t = 0, …, 10 W5 = X2 vals* X3 vals = [ W6 = X2 vals* X4 vals = [ W7 vals = X3 vals* X4 vals = [ With (plots): with (stats): with (fit):

]; ]; ]; ]; ]; ]; ];

Eq1: = Least square [x1 w2 w3 w4 dx1], dx1 = e1 – a1x1 + b1w2 + c1w3 + d1w4, {e1 a1 b1 c1 d1} (X1 vals, X2 vals, X3 vals, X4 vals, W2 vals, W3 vals, W4 vals, dX1 vals] ); dx2 = x2(t + 1) – x2(t) = [ dx3 = x3(t + 1) – X3(t) = [ dx4 = x4(t + 1) – x4(t) = [ t = 0, …, 10 with (plots): with (stats): with (fit):

] ] ]

C4

Eq2: = Least square [x2 w2 w5 w6 dx2] dx2 = e2 – a2x2 + c2w5 + d2w6 {e2 a2 b2 c2 d2} ([X2 vals W5 vals W6 vals dX2 vals] ); Eq3: = Least square [x3 w3 w5 w7 dx3], dx3 = e3 – a3x3 + b3w3 + c3w5+ d3w7, {e3 a3 b3 c3 d3} ([X3 vals W3 vals W5 vals W7 vals dX3 vals] ); Eq4: = Least square [x4 w4 w6 w7 dx4], dx4=e4 – a4x4 + b4w4 c4w6 + d4w7, {e4, a4, b4, c4, d4}] ([X4 vals W4 vals W6 vals W7 vals dX4 vals]); % We have now found the coefficients of eqi i = 1. :4. We can now find the solutions of eqi, i = 1, …, 4. With plots deq:=diff(x1,t) = x1(t)*(-a1min) + w2(t)*b1min + c1min*w3(t) + d1min*w4(t) + e1min; diff(x2(t), t) = e2min) + x2(t)*(-a2min) + c2min*w5(t) + d2min*w6(t); diff(x3(t), t) = e3min + x3(t)*(-a3min) + b3min*w(t) + c3*w5(t) + d3*(w7(t)); diff(x4(t), t)=e4min + x4(t)*(-a4min) + b4min*w4(t) + c4min*w6(t) + d4min*w7(t)); odeplot(desolve(deq, x1(0)=(3614) (19.646) x2(0) = 769.8 x3(0) = (38.37)(2.4075) x4(0) = 125), {x1(t), x2(t), x3(t), x4(t)} numeric), [t, x1(t)], 0 ..1.76); [t, x2(t)], 0..1.76); [t, x3(t)], 0..1.76); [t, x4(t)], 0..1.76); International Cooperation an Competition Model s with MATLAB x1 = Nigeria GDP= y1 = x x2 = USA GDP = y2 = y x3 = UK GDP = y3 = z x4 = China GDP or national income = y4 = w GDP for all countries in dollars y1 = [ y2 = [ y3 = [ y4 = [ code to convert all values to equivalent in US dollars for k = 1, …, 26 x1 (k ) = y1 (k ) . x 21 (k ) = y 2 (k ) , x3 (k ) = y3 (k ) x 4 (k ) = y 4 (k ) % Assume x1 (1966) = x1 (0) = x(0) = 3614 ÷ (19.646) x 2 (1966) = x 2 (0) = y (0) = 769.8 x 3 (1966) = x 3 (0) = z (0) = 38.37 × (2.4075) x 4 (1966) = x 4 (0) = w(0) = 125.88 ÷ 4.0

]´; ]´; ]´; ]´;

C5

Initial condition % x = [ x1, x2 , x3 , x4 ] x(0) = [3614 ÷ (19.646), 769.8, 38.37 × (2.4075), 125.88 ÷ (40)]′; Code to find dx dy dz dw for k = 1, 26 dx=x(k+1) – x(k) end dx=[ for k = 1 26 dy=y(k+1) – y(k); end dy=[ for k = 1 : 26 dz=z(k+1) – z(k) end dz=[ ffor k=1 : 26 dw=w(k+1) – w(k) end dw=[ dx = dy1 = [ dy = dy2 = [ dz = dy3 = [ dw = dy 4 = [ x=[ y=[ z=[ w=[ for k=1: 26 x y(k)=x(k) ⋅ y(k); end xy=[ for k=1 : 26 x z(k)=x(k) ⋅ z(k); end xz=[ for k=1 26 x w(k)=x(k) ⋅ w(k); end x w=[ for k=1 26 y z(k)=y(k) ⋅ z(k); end

]´;

]´;

]´;

]´; ]´; ]´; ]´; ]´; ]´; ]´; ]´; ]´;

]´;

]´;

]´;

C6

y z=[ for k=1 : 26 y w(k)=y(k) ⋅ w(k) end y w=[ for k=1 : 26 z w(k)=z(k) ⋅ w(k) end zw=[ DATA yw(k) = y(k) × w(k); end yw = [ for k = 1 : 26 zw(k) = z(k) × w(k); end zw = [ DATA y1 = [ y2 = [ y3 = [ y4 = [ y1 y 2 = y1 × y 2 = [ y1 y 3 = y1 × y3 = [ y1 y 4 = y1 × y 4 = [ y 2 y3 = y 2 × y3 = [ y2 y4 = y2 × y4 = [ y 3 y 4 = y3 × y 4 = [ x& = e1 + x(− a1 + b1 y + c1 z + d1w) , y& = e2 + y (− a 2 + b2 x + c2 z + d 2 w) , z& = e3 + z (− a3 + b3 x + c3 y + d 3 w) , w& = e4 + w(− a 4 + b4 x + c4 y + d 4 z ) , dx = x(t + 1) − x(t ) ; dy = y (t + 1) − y (t ) ; dz = z (t + 1) − z (t ) ; dw = w(t + 1) − w(t ) ; dx = e1 + x(− a1 + b1 y + c1 z + d1w) ; dy = e2 + y (− a 2 + b2 x + c 2 z + d 2 w) ;

]´;

]´;

]´;

]´;

]´; ]´; ]´; ]´; ]´; ]´; ]´; ]´; ]´; ]´; ]´;

C7

dz = e3 + z (− a3 + b3 x + c3 y + d 3 w) ; dw = e4 + w(− a 4 + b4 x + c 4 y + d 4 z ) ; ∫ Use Regression to find

∫ ai , bi , ci , d i , ei ,

i = 1, ..., 4.

temp = [dx, ones(size ( x)), x, xy, xz, xw] , thx = arx( temp, [0 1 1 1 1 1 0 0 0 0 0] ) ; xp = predict([dx ones(size (x)) x, xy, xz, xw], thx 4); xx = thdx(4 1 : 5); global e1 xx(1) = e1 global a1 xx(2) = a1 global b1 xx(3) = b1 global c1 xx(4) = c1 global d1 xx(5) = d1 dx = xx(1) + x(− xx(2) + xx(3) y + xx(4) z + xx(5) w) . temp = [dy ones(size( y ), y, xy, zy , wy ] ; thdy = arx( temp, [0 1 1 1 1 1 0 0 0 0 0]) ; y p = predict([dy ones(size ( y )), y, xy, yz, yw], th 4) ; yy = thy (4

1 : 5);

global e2 yy(1) = e2 global a 2 yy(2) = a 2 global b2 yy(3) = b2 global c3 yy(4) = c3 global d 2 yy(5) = d 2 dy = yy (1) − yy (2) y + yy (3) xy + yy (4) yz + yy (5) yw

C8

temp = [dy1 , ones(size( y1 )), y1 , y1 y 2 , y1 y 3 , y1 y 4 ] ; thdy1 = arx ( temp, [0 1 1 1 1 1 0 0 0 0 0] ) ; dy1 p = predict([dy1 , ones(size( y1 )), y1 , y1 y 2 , y1 y 3 , y1 y 4 ] thy1 p 4) ; dy1 y1 = th dy1 (4, 1 : 5) ; global e1 e1 = dy1 y1 (1) global a1 a1 = dy1 y1 (2) global b1 b1 = dy1 y1 (3) global c1 c1 = dy1 y1 (4) global d1 d1 = dy1 y1 (5) thy1 = e1 + y1 (a1 + b2 y 2 + c1 y3 + d1 y 4 ) temp = [dz ones(size( y )), z , zx, zy, zw] ; thz = arx ([temp, [0 1 1 1 1 1 0 0 0 0 0] ) ; zp = predict([dz ones(size( y)), z, zx, zy, zw ] thz, 4) ; zz = thz (4, 1 : 5) ; global e3 zz (1) = e3 global a3 zz (2) = a3 global b3 zz (3) = b3 global c3 zz (4) = c3 global d 3 zz (5) = d 3 dz = zz (1) − zz (2) z + zz (3) xz + zz (4) yz + zz (5) zw temp = [dw ones(size( y )), w, xw, yw, zw] ; thw = arx ([ temp, [0 1 1 1 1 1 0 0 0 0 0] ) ; wp = predict([dw ones(size(w)), xw, wy , wz ], thw, 4) ; ww = thw(4 , 1 : 5) ; global e4

C9

ww(1) = e4 global a 4 ww(2) = a 4 global = b4 ww(3) = b4 global c 4 ww(4) = c 4 global = d 4 ww(5) = d 4 dw = e4 − a 4 w+ b4 wx + c 4 wy + d 4 wz ww(1) − ww(2) w + ww(3) wx + ww(4) yw + ww(5) zw

% Use least square to get the best fit for % g = [dx, dy, dz , dw] dx coeffinit = [ xx(1), xx(2), xx(3), xx(4), xx(5)] ; dx coeff = leastsq(leastdx, dx coeffinit); dy coeffinit = [ yy (1), yy (20, yy (3), yy (4), yy (5)] ; dy coeff = leastsq(leastdy, dy coeffinit); dz coeffinit = [ zz (1), zz (2), zz (3), zz (4), zz (5)] ; dz coeff = leastsq(leastdz, dz coeffinit); dw = coeffinit [ ww(1), ww(2), ww9(3), ww(4), ww(5)] ; dw = coeffinit = leastsq(leastdw, dwcoeffinit); dw found = dw(1) + w * [− dw(2) + dw(3) * x + dw(4) * y + dw(5) * z ] ; dz found = dz(1) + z * [− dz(2) + dz(3) * x + dz(4) * y + dz(5) w ] ; dy found = dy(1) + y * [− dy(2) + dy(3) * x + dz(4) * z + dz(5) * w ] ; dx found = dx(1) + x * [− dx(2) + dx(3) * y + dx(4) * z + dx(5) * w] ; NUUC.m Function xdot = nuuc(t, x) xdot = [dx found; dy found; dz found; dw found]; % Solve over the interval 1984 1999 0 ≤ t ≤ 26 % Invoke ode 45: t 0 = 0 , t f = 26 x0 = [ x (0), y (0), z (0), w(0)]

%Initial condition

x0 = [ x (1)(0) x(2)(0) x(3)(0) x(4)(0)] [t, x] = ode 45(‘nuuc’, t 0 , t f , x0 ) ;

% (maybe)

C10

plot (t, x(1)) plot(t, x(:, 1)) plot(t, x(:, 2)) plot(t, x(:,3)) plot(t, x(:, 4)) mideast.m function xdot = nuuc(t, x) xdot = [dxfound; dyfound; dzfound; dwfound] x(1) = x, x(2) = y, x(3) = z , x(4) = w; %Solve over the interval % 1984 1999 0 ≤ t ≤ 26 . Invoke ode 45: x 0 = [ x 0 (1), x 0 (2), x 0 (3), x 0 (4)]′ % Initial conditions x0 = [ x(0), y (0), z (0), w(0)]′ ; x0 = [3614×(19.646), 769.8×38.37×(1.512), 1125.88×40]´, [t, x] = ode 45(‘nuuc’, t 0 , t f , x 0 ) ; plot(x(:, 1), x(:, 2)); plot(x(:, 1), x(:, 3)); plot(x(:, 1), x(:, 4)); plot(x(:, 2), x(:, 3)); plot(x(:, 2), x(:, 4)); plot(x(:, 3), x(:, 4)); plot(t, x(:, 1)); plot(t, x(:, 2)); plot(t, x(:, 3)); plot(t, x(:, 4)); MAPLE With (plots) deqs : = [diff ( x(t ), t ) = x (t ) * (− a1 + b1* y (t ) + c1* z (t ) + d1* w(t )) , diff ( y (t ), t ) = y (t ) * (− a 2 + b2 * x(t ) + c 2 * z (t ) + d 2 * w(t )) , diff ( z (t ), t ) = z (t ) * (− a3 + b3 * x(t ) + c3 * y (t ) + d 3 * w(t )) , diff ( w(t ), t ) = w(t ) * (− a 4 + b4 * x(t ) + c 4 * y (t ) + d 4 * z (t )) ] ; With (Detools): Deplot(deqs, [t, x, y, z, w], 0..26, {0, x0 }, scene = [x, y], scene = [x, z], scene = [x, w], scene = [y, z],

0 ≤ t ≤ 15

C11

scene = [y, w], scene = [z, w]. stepsize = 1.0, arrows ‘none’]; % a i , bi , c i , d i % i = 1, 2, 3, 4

% best fit coefficients

%Initial conditions x(1966) y(1966) z(1966) w(1966) odeplot(dsolve({deqs, x(0) = 3614÷(19.646), y(0) = 769.8, z(0) = 38.37×1.512, w(0) = 125.88÷40]′ {x(t), y(t), z(t), w(t)} numeric), [t, y(t)] 0..26); [t, x(t)] 0..26); [t, z(t)], 0..26); [t, w(t)]. 0..26); dsolve proc(rkf45 – x) … end numeric odeplot(dsolve({deqs, x(1966)=3614÷(19.646), y(1966)=769.8, z(1966)=38.37×(1.512), w(1966) = 125.88÷40) We can now investigate the interaction coefficients by evaluating all the countries relationships with each other; Nigeria vs U.K. Nigeria vs U.S.A China vs U.S.A. China vs U.K. U.K. vs Jordan We use the model x& = x (−1 + b1 y ) y& = y (−1 + b1 x) b1 = .3, .5, .6, 1, and its graphs Fig. 1-Fig. 8 as insights. Judgement is then made on the consequences of competition and cooperation. One deduces that cooperation pays. Also increased growth can facilitate cooperation. Furthermore increased growth can be simulated by the economic growth model and a judicious choice of strategies.

Program Example DATA U.S. GDP : u Nigeria GDP : v du = u(t + 1) – u(t), dv = v(t + 1) – v(t). u=[ v=[

] ]

C12

uv = [

]

du = − a1*u + b1*uv dv = − a *2 u + b2*uv

temp = [du, u, uv] thdu = arx(temp 0 1 1 0 0) dupredict = predict([du, u, v, thdu, 4); temp = [dv v uv] thdv = arx(temp. 0 1 1 0 0) dvpredict = predict(dv, u, v, thdv, 4); dvdv = thdv(2, 1:2) dudu = thdu(2, 1:2) dudu(1) = - a1 dudu(2) = b1 dvdv(1) = - b1 dvdv(1) = b2 . Use least squre to get the best fit for w = [du, dv] ducoeffinit = [dudu(1), dudu(2)]; ducoeff = leastsq(leastdu, ducoeffinit); dvcoff = leastsq(leastdv, dvcoeffinit); dufound = uduff(1) + duff(2)uv dvfound = vdvff(1) + uvdvff(2) xdot=[dufoun dvfound] coop.m = coop(t, x) xdot = zeros(2, 1); xdot(1) = dufound; xdot(2) = dvfound; x0 = [u(1), v(1)]´ [u(0)=769.8, v(0)=3614÷19.646]= x 0 %initial condition t 0 = 0, t f = 26 , x 0 = [u (0)v(0)] [t, x]=ode45(‘coop’, t 0 , f f , x 0 ) ; plot(t, u) plot(u, v) plot(t, v)

The coefficients of equations (0.8) – (0.15) are identified by using the programs outlined below: Nigeriacom.m USAcom.m Ukcom.m China.m

C13

Matlab programs with data from IMF and World Bank are used to identify the coefficients and then generate a numerical solution of the ordinary differential equations. The parameter identification of the nonlinear models by time series is definitely interesting. It differs from the author’s earlier contributions. It brings good insight to bear in some national policy decisions on ways to raise the growth rate of GDP described by ordinary differential equations. MATLAB and MAPLE programs are available in the literature. For teaching and learning purposes the programs are written out in detail. The conclusions on competition and cooperation are then easily deduced.

C14

NIGERIACOM.m DATA FOR NIGERIA C1 = Y1 = T1 = Y1T1 =y1 – T1 = y1T1. (yT) = yT(t + 1) – yT(t) = yTP I1 = X1 = e1 = τ1 = d1 y2 y3 y4 G1 = % C1 = C11 + c12yT + c3(yT)´ % Consumption is programmed as follows temp = [C1 ones(size(y)), yT, (yT)´]; thc = arx(temp, [0 1 1 1 0 0 0]); C1p = predict([C1, ones(size(y)), yT, yTP], thc, 4); CC1 = thc(2, 1:3); CC1(1) = C11 CC1(2) = C12 CC1(3) = C13 % I1 = I11 + I12 yT – I3(yTP) % Investment is programmed as follows: temp = [I1 ones(size(yT)), yT, Ytp]; thI1 = arx(temp, [0 1 1 1 0 0 0]); I1p = predict([I1, ones(size(yT)), yT, yTP], thI, 4); I I = thi(2, 1:3) I II(1) = I11 I I1(2) = I12 I II(3) = - I13 % XI = XII + X12y1 _ X115τ1 + X116e1 a111 y1 + b111 y y 2 + c111 y y 3 + d111 y y 4 y1 y 2 = y1 y 2 = y12 = [ ]´; y1 y 3 = y1 y 3 = y13 = [ ]´; y1 y 4 = y1 y 4 = y14 = [ ]´; temp = [X1 ones(size(y1)), y, ta, e d y12 , y13 , y14 ] ; thx = arx(temp, [0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0] ) y1 p = predict ([ y1 , ones(size( y1)), y1, τ1, e1, y1, y1y 2, y1 y3, y1y 4], thx, 4); XX1 = thx(7, 1:8);

+

C15

XX1(1) = X11 XX1(7) = c111 XX1(2) = X12 XX1(8) = d111 XX1(3) = X115 XX1(4) = X116 XX1(5) = a111 XX1(6) = b111 % G1 = g11 + g12 y + g13 yp temp = [G1 ones(size(y1)), y1, y1p]; thg1 = arx(temp, [0 1 1 1 0 0 0] ); Gp = predict([G1, ones(size(y1)), y1, y1p], thg1, 4); GG1 = thg(2:1:3); GG1(1) = g11 GG1(2) = g12 GG1(3) = g13 Z1 = (C11 + I11 + X11 ) = CC1(1) + II1(1) + XX1(1) Z2 = (C12 + C13 + X12 + g12 ) = CC1(2) + CC1(3) + XX1(2) + GG1(2) Z3 = (C13 + I13 + g13 + X13 ) Ζ3 = CC1(3) + II1(3) + GG1(3) + XX1(3) λ1 = 1 dy1 = λ1[( Z12 − 1) y1 + Z11 + C12T 1 + C13T1 p − I1 3T1 (t ) − I13T1 p + X 116τ 1a + X 115e1 dt + y1 (a11 + a12 y 2 + a13 y 3 + a14 y 4 ) Let p1 = λ1 (C1 + I1 + X 1 ) = λ1 Z1 q = λ ( g + c T 1 + c T&1(t ) − I T 1(t ) − I T&1 p + X τ+ X e 1

1

1

2

3

3

3

116

115 1

λ1 = 1 dy1 = a1 y + p1 + q1 + y1 (a11 + a12 y 2 + a13 y 3 + a14 y 4 ) dt where: a1 = λ1 ( Z 2 − 1) = λ1 (CC1(2) + CC1(3) + XX 1(2) + GG1(2) − 1) p1 = λ1 Z1 = λ1 (C11 + I11 + X 11) q1 = λ1 (GG1(1) + CC1(2)T 1 + CC1(3)T 1 p − II1(3)T 1 _ II1(3)T 1 p + XX 1(3)e1 + XX 1(4)τa1 + XX 1(5)d1 ) a111 = XX 1(5) b111 = XX 1(6) c111 = XX 1(7) d111 = XX 1(8)

C16

dy1 = a1 y1 + p1 + q1 + y1 (a11 + a12 y 2 + a13 y 3 + a14 y 4 ) dt dy1 = y1 (a111 + a12 y 2 + a13 y 3 + a14 y 4 ) + p1 + q1 dt a111 = a1 + a11 p1 = λ1 (C11 + I11 + X 11 + g11 ) q1 = λ1 ( g1 + C11T + C12T 1 p + C 21T 1 p − I 21T1 p + X 161τ1 (t ) + X 151e1

USA CO.M DATA y 2 GDP of USA C 2 Consumption of USA T2 Taxes of USA I2 Investment of USA X 2 Net export of USA τa 2 tarriff e2 exchange rate d 2 distance between nations G 2 US government consumption y1 DGP of NIGERIA y 3 GDP of UK y 4 GDP of China y 2T2 y1 – T2 y 2T2 P (Y2 – T2)(t + 1) – (y –T)(t) y1 y 2 = y12 = [ y1 y 3 = y1 y 3 = y13 = [ y1 y 4 = y1 y 4 = y14 = [ for k = 1:15 y1 y 2 = y1 (k ) ⋅ y (k ) end for k = 1:15 y1 y 2 = y1 (k ) ⋅ y 3 (k ) end for k = 1:15 y 2 y1 = y12 = y 2 (k ) y1 (k ) end

] ] ]

C17

for k = 1:15 y 2 y 3 = y 2 (k ) y 3 (k ) for k = 1:15 y 2 y 4 = y 24 = y 2 (k ) y 4 (k ) % C 2 = C 21 + C 22 y 2T2 + C 23 y 2T2 p % Consumption is programmed as follows: temp = [ C 2 ones(size ( y 2 )) y 2T2 y 2T2 p] ; th c 2 = arx(temp, [0 1 1 1 0 0 0]); C 2 p = predict([ C 2 , ones(size ( y 2 ), y 2T2 , y 2T2 p ], thc 2 , 4) ; C 2 C 2 = thc 2 (2, 1 : 3) ; C 2 C 2 (1) = C 21 C 2 C 2 (2) = C 22 C 2 C 2 (3) = C 23 % I 2 = I 21 + I 22 y 2T2 + I 23 yTp % Investment is programmed as follows temp = [ I 2 ones(size y 2T2 )), y 2T2 , y 2T2 p ] ; thI 2 = arx(temp, [0 1 1 1 0 0 0] ), I 2 p = predict([ I 2 , ones(size ( y 2T2 ), y 2T2 , y 2T2 p ], thI 2 ,4) ; % X 2 = X 21 + X 22 y 2 + X 215 τ 2 + X 216 e 2 + a 211 y 2 + b211 y 2 y1 + c 211 y 2 y 3 + d 211 y 2 y 4 temp = [ X 2 ones(size ( y 2 )), y 2 , τa 2 , e 2 , y 2 , y 2 y1 , y 2 y 3 , y 2 y 4 ] ; thx 2 = arx(temp, [0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0] ); x 2 p = predict ([ X 2 , ones(size ( y 2 )), τa 2 , e 2 , y 2 , y 2 y1 , y 2 y 3 , y 2 y 4 ]; thx 2 , 4) ; XX 2 = thx 2 (7, 1 : 8) ; XX 2 (1) = X 21 XX 2 (6) = b211 XX 2 (2) = X 22 XX 2 (7) = c 211 XX 2 (3) = X 215 XX 2 (8) = d 211 XX 2 (4) = X 216 XX 2 (5) = a 211 % G 2 = g 21 + g 22 y 2 + g 23 y 2 p temp = [G 2 ones(size ( y 2 )), y 2 y 2 p ] ; thg2 = arx(temp, [0 1 1 1 0 0 0] ); G p = predict ([G 2 , ones(size ( y 2 ), y 2 , y 2 p], thg 2 , 4) ; GG 2 = thg 2(2, 1 : 3); thg 2(2, 1 : 3); GG 2 (1) = g 21

C18

GG 2 (2) = g 22 GG 2 (3) = g 23 z 21 = C 21 + I 21 + X 21 = CC 2 (1) + II 2 (1) + XX 2 (1) ; z 22 = C 22 + I 23 + x 22 + G 22 = CC 2 (2) + II 2 (3) + XX 2 (2) + GG 2 (2) z 23 = CC 2 (3) + II (3) + XX 2 (3) + GG 2 (3) λ1 = 1 % (= 1 · 2?)

We have written Program1: Nigeriacom.m USAcom.m We repeat this for UK and China and write out the ordinary equations generated with all the controls. We write out the model with the coefficients obtained. We make a better approximation using the initial data obtained in program 1 and the “Least Square” command in maple and MATLAB. After ode solve command we display the solution and the diagrams of the real thing together. We now gather the dynamical systems of gross domestic products: dy1 dt dy 2 dt dy 3 dt dy 4 dt Let

= y1 (a111 + a12 y 2 (t ) + a13 y 3 (t ) + a14 y 4 (t )) + g1 + p1 = y 2 (a 211 + a 22 y1 (t ) + a 23 y 3 (t ) + a 24 y 4 (t )) + p1 + g 2 = y 3 (a 311 + a 31 y1 (t ) + a 32 y 2 (t ) + a 34 y 4 (t )) + p 3 + g 3 = y 4 (t )(a 411 + a 41 y1 (t ) + a 42 y 2 (t ) + a 43 y 3 (t )) + p 4 + g 4 x1 = y1 = GDP of Nigeria x 2 = y 2 = GDP of USA x3 = y 3 = GDP of UK x 4 = y 4 = GDP of Chin

Let ei = p i + g i , i = 1, 2, 3, 4. This can be taken to be the average of p i (t ) + g i (t ) in the interval [a, b] under consideration. We now analyze the problem of cooperation and competition.

C19

Program and Run UScom.m Egyptcom.m Jordancom.m Israelcom.m Use the same code as Nigeriacom.m or USAcom.m with data of Egypt, Jordan, Israel in IMF, statistics or from Egypt2.m, Jordan2.m, Israel2.m, US3.m in the sequel. The program US3.m is available in E. N. Chukwu , Differential Models and Neutral Systems for Controlling the Wealth of Nations, World Scientific, Singapore, New Jersey, 2001.

C20

C21

C22

C23

C24

Diagrams Competiton

Diagrams Competition

C25 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % International Cooperation and Competition Models for U.S.A, % % Nigeria, United Kingdom and China is outlined follows % % (statistics used are from International financial yearbook % % statistics, 1965-1992): Step 1 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% year=[1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991]'; GDP_usa=[814.3 889.3 959.5 1010.4 1096.8 1206.5 1349.1 1458.8 1584.8 1767.1 1974.1 2232.7 2488.7 2708.1 3030.6 3149.6 3405.1 3777.2 4038.7 4268.6 4539.9 4900.4 5250.8 5522.2 5722.9]'; GDP_nigeria=[2951 2878 3851 5621 7098 7703 11199 18811 21779 27572 32747 36084 43151 50849 50749 51709 57142 63608 72355 73062 108888 145243 224797 260637 324011]'; GDP_uk=[40.4 43.81 47.15 51.77 57.75 64.66 74.26 83.86 105.85 125.25 148.98 168.53 198.22 231.77 254.93 279.04 304.46 325.85 357 384.84 423.38 471.43 515.96 551.12 573.65]'; GDP_china=[143.26 168.43 190.81 218.43 249.28 287.27 307.36 388.58 524.48 547.00 655.90 264.4 301.0 335.0 455.1 490.2 548.9 607.6 716.4 879.2 1013.3 1178.4 1470.4 1646.6 1832.0]'; exchangeR_usa = 1; exchangeR_nigeria = 19.646; exchangeR_uk = 1.980; exchangeR_china = 5.2804; y1p =[3614 2951 2878 3851 5621 7098 7703 11199 18811 21779 27572 32747 36084 43151 50849 50749 51709 57142 63608 72355 73062 108888 145243 224797 260637 324011]'; y2p =[769.8 814.3 889.3 959.5 1010.4 1096.8 1206.5 1349.1 1458.8 1584.8 1767.1 1974.1 2232.7 2488.7 2708.1 3030.6 3149.6 3405.1 3777.2 4038.7 4268.6 4539.9 4900.4 5250.8 5522.2 5722.9]'; y3p =[38.37 40.4 43.81 47.15 51.77 57.75 64.66 74.26 83.86 105.85 125.25 148.98 168.53 198.22 231.77 254.93 279.04 304.46 325.85 357 384.84 423.38 471.43 515.96 551.12 573.65]'; y4p =[125.88 143.26 168.43 190.81 218.43 249.28 287.27 307.36 388.58 524.48 547.00 655.90 264.4 301.0 335.0 455.1 490.2 548.9 607.6 716.4 879.2 1013.3 1178.4 1470.4 1646.6 1832.0]'; y2 =GDP_usa;

%GDP of U.S.A, Data is is in %billions of US dollars y1 =GDP_nigeria./(exchangeR_nigeria); %y1 for Nigeria data is is %in billions of US dollars y3 =GDP_uk.*(exchangeR_uk); %y3 for UK data is is in %billions of US dollars. y4 =GDP_china./(exchangeR_china); %y4 for China data is is in %billions of US dollars ex1p=exchangeR_nigeria; ex2p=exchangeR_usa; ex3p=exchangeR_uk; ex4p=exchangeR_china;

C26 % The above data may be used for the regression and % refined to yield the systems equations: % % % %

dy1(t)/dt dy2(t)/dt dy3(t)/dt dy4(t)/dt

= = = =

y1(-a1 y2(-a2 y3(-a3 y4(-a4

+ + + +

b1*y2 b2*y2 b3*y2 b4*y2

+ + + +

c1*y3 c2*y3 c3*y3 c4*y3

+ + + +

d1*y4) d2*y4) d3*y4) d4*y4)

+ + + +

e1, e2, e3, e4,

% Where ei, i=1,...,4 are present as windfalls %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % We First use MATLAB System Identification Algorithm to find % % the constant coefficients for: ai, bi, ci, di, ei, % % % % where i = 1, 2, 3,4. The assumptions that initial % % conditions are setup at 1975: Step 2 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y2_1966 y1_1966 y3_1966 y4_1966

= = = =

1.*(769.8); 3614./(19.646); 38.37.*(1.980); 125.88./(5.2804);

% % % %

y0 = [183.956 769.8 75.9726 23.8391];

y2_1966 y1_1966 y3_1966 y4_1966

= = = =

y2(0)=y(0) y1(0)=x(0) y4(0)=w(0) y3(0)=z(0)

% Initial guesses % for time t=1975.

% y = [y1, y2, y3, y4]; Let x=y1; y=y2; z=y3; w=y4 x=y1; y=y2; z=y3; w=y4; xp=y1p./ex2p; yp=ex2p.*y2p; zp=ex3p.*y3p; wp=y4p./ex4p; for t = 1:25 dX(t) = xp(t+1)-xp(t); dY(t) = yp(t+1)-yp(t); dZ(t) = zp(t+1)-zp(t); dW(t) = wp(t+1)-wp(t); end % And xy yz dx

% derivatives of x,y,z and w.

also, = x.*y; xz = x.*z; xw = x.*w; = y.*z; yw = y.*w; wz = z.*w; = dX'; dy = dY'; dz = dZ'; dw = dW';

% Once again % dx = e1 + % dy = e2 + % dz = e3 + % dw = e4 +

we re-state the system equation in another way: x(-a1 + b1*y + c1*z + d1*w); y(-a2 + b2*x + c2*z + d2*w); z(-a3 + b3*x + c3*y + d3*w); w(-a4 + b4*x + c4*y + d4*z);

% For dx = e1 + x(-a1 + b1y + c1z + d1w); temp = [dx, ones(size(x)),x,xy,xz,xw]; thdx = arx(temp,[0 1 1 1 1 1 0 0 0 0 0]); XXp = predict([dx ones(size(x)),x,xy,xz,xw],thdx,4);

C27 xx = thdx(4,1:5); global e1; global a1; global b1; global c1; global d1; e1=xx(1); a1=xx(2); b1=xx(3); c1=xx(4); d1=xx(5); % % Hence, % dx/dt = xx(1) + x.*(-xx(2) + xx(3).*y + xx(4).*x + xx(5).*w); % For dy = e2 + y(-a2 + b2x + c2z + d2w); temp = [dy, ones(size(y)),y,xy,yz,yw]; thdy = arx(temp,[0 1 1 1 1 1 0 0 0 0 0]); YYp = predict([dy, ones(size(y)),y,xy,yz,yw],thdy,4); yy = thdy(4,1:5); global e2; global a2; global b2; global c2; global d2; e2=yy(1); a2=yy(2); b2=yy(3); c2=yy(4); d2=yy(5); % % Hence, % dy/dt = yy(1) + y.*(-yy(2) + yy(3).*x + yy(4).*z + yy(5).*w); % For dz = e3 + z(-a3 + b3x + c3y + d3w); temp = [dz, ones(size(z)),z,xz,yz,wz]; thdz = arx(temp,[0 1 1 1 1 1 0 0 0 0 0]); ZZp = predict([dz ones(size(z)),z,xz,yz,wz],thdz,4); zz = thdz(4,1:5); global e3; global a3; global b3; global c3; global d3; e3=zz(1); a3=zz(2); b3=zz(3); c3=zz(4); d3=zz(5); % % Hence, % dz/dt = zz(1) + z.*(-xx(2) + xx(3).*x + xx(4).*y + xx(5).*w); % For dw = e4 + w(-a4 + b4x + c4y + d4z); temp = [dw, ones(size(w)),w,xw,yw,wz]; thdw = arx(temp,[0 1 1 1 1 1 0 0 0 0 0]); WWp = predict([dw ones(size(w)),w,xw,yw,wz],thdw,4); ww = thdw(4,1:5); global e4; global a4; global b4; global c4; global d4; e4=ww(1); a4=ww(2); b4=ww(3); c4=ww(4); d4=ww(5); % % Hence, % dw/dt = ww(1) + w.*(-ww(2) + ww(3).*x + ww(4).*y + ww(5).*z); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % For a non linear least square we use LSQNONLIN to get the best % % fit for: g = [dx, dy, dz, dw]; Let: X=y1';Y=y2';Z=y3';W=y4'; % Step 3 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X=y1;Y=y2;Z=y3;W=y4; %----------------------------------------------------------------% Fx = leastdx_ndu = (dx - dx_approximate); % dxcoeff_init = [xx(1), xx(2), xx(3), xx(4), xx(5)]; % dxcoeff = lsqnonlin(leastdx_ndu, dxcoeff_init);

C28 function Fx = leastdx_ndu(x) Fx = (x(1) + X.*(-x(2) + x(3).*Y + x(4).*Z + x(5).*W))(20168883.3468455 + X.*(-21022.0945828 + 1.5997696.*Y + 20.2375768.*Z - 35.8843930.*W)); % dxcoeff_init is the starting guess dxcoeff_init = [20168883.3468455 -21022.0945828 1.5997696 20.2375768 -35.8843930]'; %Invoke optimizer: [x,resnorm] = lsqnonlin('leastdx_ndu',dxcoeff_init); %The solution of least square: dxcoeff = [20119293.9740717 -16523.1683606 -43.9263547 230.2238853 -81.9038237] % Therefore, dxfound = 20119293.9740717 + X.*(-16523.1683606 - 43.9263547.*Y + 230.2238853.*Z - 81.9038237.*W); %----------------------------------------------------------------% Fy = leastdy_ndu = (dy - dy_approximated); % dycoeff_init = [yy(1), yy(2), yy(3), yy(4), yy(5)]; % dycoeff = lsqnonlin(leastdy_ndu, dycoeff_init); function Fy = leastdy_ndu(y) Fy = (y(1) + Y.*(-y(2) + y(3).*X + y(4).*Z + y(5).*W))(2983.45292081683 + Y.*(-2.25644986129 - 0.00000695669.*X + 0.00319748140.*Z - 0.00528420661.*W)); % dycoeff_init is the starting guess dycoeff_init = [2983.45292081683 -2.25644986129 -0.00000695669 0.00319748140 -0.00528420661]'; %Invoke optimizer: [y,resnorm] = lsqnonlin('leastdy_ndu',dycoeff_init); %The solution of least square: dycoeff = [2967.37329780199 2.12966222741 0.00000969503 0.00296346340 -0.00558163475]; % Therefore, dyfound = 2967.37329780199 + Y.*(2.12966222741 + 0.00000969503.*X + 0.00296346340.*Z - 0.00558163475.*W); %----------------------------------------------------------------% Fz = leastdz_ndu = (dz - dz_approximate); % dzcoeff_init = [zz(1), zz(2), zz(3), zz(4), zz(5)]; % dzcoeff = lsqnonlin(leastdz_ndu, dzcoeff_init); function Fz = leastdz_ndu(z) Fz = (z(1) + Z.*(-z(2) + z(3).*X + z(4).*Y + z(5).*W))(56.97777315500337 + Z.*(-0.37578186565907 + 0.00000486979517.*X + 0.00010748826334.*Y - 0.00107208220308.*W));

C29 % dzcoeff_init is the starting guess dzcoeff_init = [56.97777315500337 -0.37578186565907 0.00000486979517 0.00010748826334 -0.00107208220308]'; %Invoke optimizer: [z,resnorm] = lsqnonlin('leastdz_ndu',dzcoeff_init); %The solution of least square: dzcoeff = [52.91212804143637 0.30769828428727 0.00000578440732 0.00008580594951 -0.00093605991521]; % Therefore, dzfound = 52.91212804143637 + Z.*(0.30769828428727 + 0.00000578440732.*X + 0.00008580594951.*Y - 0.00093605991521.*W); %----------------------------------------------------------------% Fw = leastdw_ndu = (dw - dw_approximate); % dwcoeff_init = [ww(1), ww(2), ww(3), ww(4), ww(5)]; % dwcoeff = lsqnonlin(leastdw_ndu, dwcoeff_init); % function Fw = leastdw_ndu(w) % Fw =(w(1) + W.*(-w(2) + w(3).*X - w(4).*Y + w(5).*Z))(125.319396364394 + W.*(-2.642157021484 - 0.000000338299.*X + 0.000779397869.*Y - 0.001892163136.*Z)); % %

dwcoeff_init is the starting guess dwcoeff_init = [125.319396364394 -2.642157021484 0.000000338299 0.000779397869 -0.001892163136]';

%Invoke optimizer: %[w,resnorm] = lsqnonlin('leastdw_ndu',dwcoeff_init); %The solution of least square: %dwcoeff = [123.071777707351 -2.197404887312 -0.000153771054 0.006319712139 0.031681966455]; % Therefore, dwfound = 123.071777707351 + W.*(-2.197404887312 0.000153771054.*X + 0.006319712139.*Y + 0.031681966455.*Z); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % To simulate the system of international cooperation and % % competition model % % defined by dxfound, dyfound, dzfound and dwfound using Matlab % % ODE45 leads to the following: Step 4 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function dy = ngusukc_Intern_Coop_Compet(t,y) dy = zeros(4,1); %A colume vector of initial conditions dy(1) = dxfound; dy(2) = dyfound; % As computed above. dy(3) = dzfound; dy(4) = dwfound;

C30 % Solve on a time interval [0 20] with an % initial condition vector y0 = [183.956 769.8 75.9726 23.8391]; % at time equal to 1966. t0 = 1; tf = 1.25; y0 = [183.956 769.8 75.9726 23.8391]; % options = odeset('RelTol',1e-4,'AbsTol',[1e-4 1e-4 1e-5 1e-4]); [T,Y] = ode45('ngusukc_Intern_Coop_compet',[t0 tf],y0);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This section graphically depicts the results of step 1 through % step 4 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% subplot (2,1,1),plot(year,y1) title('Nigerian Gross Domestic Product,1966...1991.'); ylabel('GDP'); xlabel('year'); subplot(2,1,2),plot(year,XXp) title(' Nigeria-in Cooperation with U.S.A,UK and China after Regression-'); ylabel('GDP'); xlabel('year'); pause clf subplot(2,1,1),plot(year,dxfound,'--') title('Nigeria--in Cooperation with U.S.A,UK and China after Leastsq--'); ylabel('GDP'); xlabel('year'); subplot (2,1,2),plot(Y(:,1)) title('Nigeria--in Cooperation with U.S.A,UK and China after ODE45--'); ylabel('GDP'); xlabel('time, t'); pause clf %---------------------------------------------------------subplot (2,1,2),plot(year,y2) title('United States Gross Domestic Product,1961...1991.'); ylabel('GDP'); xlabel('year'); subplot(2,1,1),plot(year,YYp)

C31 title('U.S.A--in Cooperation with Nigeria,UK and China after regression--'); ylabel('GDP'); xlabel('year'); pause clf subplot(2,1,1),plot(year,dyfound) title('U.S.A--in Cooperation with Nigeria,UK and China after Leastsq--'); ylabel('GDP'); xlabel('year'); subplot (2,1,2),plot(Y(:,2)) title('U.S.A--in Cooperation with Nigeria,UK and China after ODE45--'); ylabel('GDP'); xlabel('time, t'); pause clf %----------------------------------------------------------subplot (2,1,1),plot(year,y3) title('United Kingdom Gross Domestic Product,1961...1991.'); ylabel('GDP'); xlabel('year'); subplot(2,1,2),plot(year,ZZp) title('UK --in Cooperation with Nigeria, U.S.A and China after regression-'); ylabel('GDP'); xlabel('year'); pause clf subplot(2,1,1),plot(year,dzfound,'--') title('UK --in Cooperation with Nigeria, U.S.A and China after Leastsq-'); ylabel('GDP'); xlabel('year'); subplot (2,1,2),plot(Y(:,3)) title('UK --in Cooperation with Nigeria, U.S.A and China after ODE45--'); ylabel('GDP'); xlabel('time, t'); pause clf %-----------------------------------------------------------subplot (2,1,1),plot(year,y4) title('China Gross Domestic Product,1961...1991.'); ylabel('GDP'); xlabel('year'); subplot(2,1,2),plot(year,WWp)

C32 title('China-in Cooperation with Nigeria, U.S.A and UK after regression-'); ylabel('GDP'); xlabel('year'); pause clf subplot(2,1,1),plot(year,dwfound) title('China-in Cooperation with Nigeria, U.S.A and UK after Leastsq-'); ylabel('GDP'); xlabel('year'); subplot (2,1,2),plot(Y(:,4)) title('China in Cooperation with Nigeria, U.S.A and UK after ODE45--'); ylabel('GDP'); xlabel('time, t'); pause clf subplot(2,1,1),plot(T,Y(:,1),T,Y(:,2),'--') title('Nigeria and U.S.A Growth Performance after ODE45--'); ylabel('GDP'); xlabel('time, t'); subplot(2,1,2),plot(T,Y(:,1),T,Y(:,3),'--') title('Nigeria and United Kingdom Growth Performance after ODE45-'); ylabel('GDP'); xlabel('time, t'); pause clf subplot(2,1,1),plot(T,Y(:,1),T,Y(:,4),'--') title('Nigeria and China Growth Performance after ODE45--'); ylabel('GDP'); xlabel('time, t'); subplot(2,1,2),plot(T,Y(:,2),T,Y(:,3),'--') title('U.S.A and United Kingdom Growth Performance after ODE45-'); ylabel('GDP'); xlabel('time, t'); pause clf subplot(2,1,1),plot(T,Y(:,2),T,Y(:,4),'--') title('U.S.A and China rowth Performance after ODE45--'); ylabel('GDP'); xlabel('time, t'); subplot(2,1,2),plot(T,Y(:,3),T,Y(:,4),'--') title('United Kingdom and China Growth Performance after ODE45-'); ylabel('GDP'); xlabel('time, t');

C33

2 Generation of Ordinary Differential Models: US22.M, UK22.M, Chinacu.M, Nigeria.M (MATLAB)

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nduo aaa11 = a -24.0752 aaa12 = a -4.6325e+003 aaa13 = a -153.4415 aaa14 = a 0.0594 aaa15 = a 0.2138 aaa16 = a 7.5873e+003 aaa21 = a 128.8538 aaa22 = a -6.3512 aaa25 = a -2.4681 aaa33 = a 0.5561 aaa44 = a 0.5561 aaa45 = a -0.0026 aaa46 = a -154.1025 aaa51 = a -2.1576 aaa56 = a 0.0834

C48

aaa61 = a -0.0373 aaa62 = a -1.6929 aaa63 = a 0.0115 aaa66 = a 27.5839 bbb11 = -7.1006e-006 bbb13 = 154.1025 bbb14 = -0.0177 bbb26 = 4.9127e-005 bbb33 = 0.0834 bbb36 = 0.0151 bbb37 = 0.0374 bbb43 = 27.5839 bbb44 = 0 bbb45 =BB(7)= 0 bbb54 = -0.0177 bbb55 = 0 bbb56 = 6.1542e-004 bbb61 = -6.2732e-010 bbb62 = 8.8347e-005 bbb63 = -0.0136 bbb64 =

C49

-1.5608e-006 bbb65 = 0 bbb66 = 6.1542e-004 ccc24 = 4.9127e-005 ccc32 = 8.8347e-005 ccc55 = -642.1701 ccc56 = 2.5444 ccc0 = 483.5857 I0 =II(1) x0 = -55.5034 p0 = 143.1709 m10 = -0.1225 p0p6m0 = -57.6609 g0 =GG(1) 550.5708 f0 =BB(1)-XX(1) -37.1813 g2 = -128.8538

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aaa56 = 1.4799e-006 aaa61 = -2.5345e+005 aaa62 = -3.1094e+005 aaa63 = 1.2552e+005 aaa66 = 1.1014e+003 bbb11 = 1.3295e-004 bbb13 = 0.0045 bbb14 = 4.6759e-005 bbb26 = -9.4315e-006 bbb33 = 1.4799e-006 bbb36 = 0.0260 bbb37 = -0.0302 bbb43 = 1.1014e+003 bbb44 = 0 bbb54 = 4.6759e-005 bbb55 = 0 bbb56 = 1.3297e-004 bbb61 = 2.6985e-008 bbb62 = 2.0298e-004 bbb63 = -9.2173e-007 bbb64 = 9.4910e-009 bbb65 = 0 bbb66 = 1.3297e-004

C63

ccc24 = -9.4315e-006 ccc32 = 2.0298e-004 ccc55 = -0.2701 ccc56 = 0.3038 c0 = 425.3730 I0 = 0.1518 0= 27.3050 y10 = 0.0047 p0p6m0 = -0.5573 g0 = 4.8036 f0 = -3.7129e+006 g2 = -24.5745

C64

Chinacu.m % china’s data is in billions of yen % taken from the international financial statistics yearbook 1994 & 199 %QM = Quasi money %X = Export (net) %I = Investment (Gross capital formation) %YD=Y(t – 1) YEAR = 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 93 1994 1995 1996 1997]′; X=[-1.5 1.1 9.1 5.1 0.1 -36.7 -25.5 1.1 -15.1 -18.6 51.0 61.8 27.6 -68 63.4 .4 99.9 145.9 274.5]′; G=[65.9 70.5 77.0 83.8 102.0 118.4 136.7 149.0 172.7 203.3 225.2 283.0 9.2 49.7 598.6 669.1 785.2 865.0]′; I=[131.8 125.3 149.3 170.9 212.6 264.1 309.8 374.2 462.4 433.9 473.2 5 0 813.7 1298.0 1685.6 2030.1 2333.6 2569.8] ′; C=[231.7 260.4 286.4 318.3 367.5 458.9 517.5 596.1 763.3 852.4 911.3 1 .6 1246.0 1568.2 2123.0 2783.9 3318.8 3611.8]′; GDP=[455.1 490.1 548.9 607.6 716.4 879.2 1013.3 1178.4 1470.4 1646.6 1 .0 2128.0 2586.4 3450.1 4711.1 5940.5 6936.6 7607.7] ′; PGDP=[47.7 35.0 58.8 8.7 108.8 162.8 134.1 165.1 292.0 176.2 185.4 296 458.4 863.7 1261.0 1229.4 996.1 671.1]′; QM=[52.23 63.25 78.96 96.39 114.91 185.76 248.96 338.34 411.47 555.89 .24 961.11 1261.30 1797.88 2538.04 3514.67 4543.27 5352.45]′; M=[114.88 134.52 148.9 184.89 244.94 301.73 385.9 457.4 548.74 583.42 .95 898.78 1171.43 1676.11 2153.99 2559.68 3066.26 3834.33]′; R=[5.4 5.4 5.20 5.76 5.76 7.20 7.20 7.20 8.64 11.34 8.64 7.56 7.56 10. 10.98 10.98 7.47 5.67]′; Y=GDP YD=[407.4 455.1 490.1 548.9 607.6 716.4 879.2 1013.3 1178.4 1470.4 164 1832.0 2128.0 2586.4 3450.1 4711.1 5940.5 6936.6]′; Yp=Y–YD; DGDP = YD;

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* present (thm) This matrix was created by the command ARX on 11;7 1995 at 21.42 Loss fcn: 3.308e + 006 Akaike’s PFE: 4.135e+006 Sampling interval 1 The polynomial coefficients and their standard deviations are B-polynomial from input # 1 B= 1.0e+003 1.20450290050368 0.99364687858576 B-polynomial from input # 2 B= 0.14662286702664 0.01092539175370 B-polynomial from input # 3 B= 1.0e+002 * -1.45853643949263 2.02496872075137 * present (thg) This matrix was created by the command ARX on 11/7 1995 at 21.42 Loss fcn: 2.222e+006 Akaike’s FPE: 2.995e+006 Sampling interval 1 The polynomial coefficients and their standard deviations are B=polynomial from input # 1 B= 1.0e+002 * -1.52502433147511 8.45512298375709 B =polynomial from input # 2 B=

C71

0.03667340060243 0.01044589449395 B=polynomial from input # 3 B= -0.08221972496287 0.03276266153079 B=polynomial from input # 4 B= 1.0e+002 * 4.44745233591565 1.69950346006197 * present (thx) This matrix was created by the command ARX on 11/7 1995 at 21.42 Less fcn: 1.1963+008 Akaike’s FPE: 1.611e+008 Sampling interval 1 The polynomial coefficients and their standard deviations are B=polynomial from input # 1 B= 1.0e+003 5.70900120430075 6.20155595191171 B= 0.80872057157587 0.07661721691857 B=polynomial from input # 3 B= -0.06664546378586 0.24030339831489 B-polynomial from input # 4

C72

B= 1.0e+003 * -3.21230843129220 1.24653015908688 * present (thi) This matrix was created by the command ARX on 11/7 1995 at 21.42 Loss fcn: 4.553+006 Akaike’s FPE: 6.133e+006 Sampling interval 1 The polynomial coefficients and their standard deviations are B-polynomial from input # 1 B= 1.0e+003 * 6.23079202954826 1.20983700545005 B-polynomial from input # 2 B= 0.17798814165885 0.01494694960449 B-polynomial from input # 3 B= -0.13603017674513 0.04687983887771 B-polynomial from input # 4 B= 1.0e+003 * -1.04251740005487 0.24318063508045 * present (thc) This matrix was created by the command ARX on 11/7 1995 at 21:42 Loss fcn: 4.769e+007 Akaike’s FPE: 6.427e+007 Sampling interval 1

C73

The polynomial coefficients and their standard deviations are B-polynomial from input # 1 B= 1.0e+003 * 1.65149779020011 3.91656610177684 B-polynomial from input # 2 B= 0.63498903067544 0.04838727521329 B-polynomial from input # 3 0.05493832130047 0.15176258204874 B-polynomial from input # 4 B= 1.0e+002 * -0.15709857827455 7.87240783406493 present (tha) This matrix was created by the command PEM on 1/7 1995 at 21.43 Loss fcn: 1.369e+012 Akaike’s FPE: 2.328e+012 Continuous time model estimate The state space matrices with their standard deviations given as imaginary pa a= 1.0e+008 * 0.00000000051123 0.00000256401577i 0.00000004875981 2.64239370991672i b=

+

0.00000000314391i

–

0.00000136981185

+

+

0.00236261720091i

–

0.00005453467510

+

C74

1.0e+009 * 0 -0.00000224970739 + 0.00000611157994i -0.00000004028159 + 0.00195182803735i 0.00004646130480 2.25136896896984i c= 1 0

0 1

Strike a key to continue d= 0 0

0 0

0 1 0

0 0

k=

x0 = 3361 5 lambda = 1.0e+009 * 5.54921488645498 0.00447757617338

0.00447757617338 0.00000385963125

2.1 Nigeria and the Four Regions Gross Domestic Product at Market Prices = y in £ millions YEAR = [1950 1951 1952 1953 1054 1955 1956 1957]´; Y = [524.3 587.0 630.8 682.9 794.8 851.2 900.0 938.7]´; % to = 0 = 1950, tf = 8 = 1957 %postulate y = a + bt2 + ct2 + dt3 + et4

+

C75

temp = [y ones(size(t)) t, t^2 t^3 t^4]´; thy = arx(temp[0 1 1 1 1 1 0 0 0 0 0]); yp = predict([y ones(size(t)) t, t^2 t^3 t^4], thy 4); % t2 = t2, t^3 = t3, t^ 4 = t4. yy = 1hy(4, 1:5) yy(1) = d yy(2) = b yy(3) = c yy(4) = d yy(5) = e. subplot(2.1.1), plot(YEAR, y, YEAR, yp, ‘ + ’) title(National GDP …); ylabel(‘y’); xlabel(‘year’) subplot(2.1.2) plot(year, t year, yp ‘ + ’) % Let x1 = GDP of Northern N x2 = GDP of West, x3 = GDP of East; Data Year = [19.50 1951 1952 1953 1954 1955 1956 1957]; x1 = [7.4 8.0 11.4 10.8 17.5 16.8 18.8 19.9]; x2 = [6.0 7.2 7.9 6.7 7.3 13.7 17.5 17.7]; x3 = [4.4 4.8 5.2 6.2 12.3 11.0 12.5 13.8]; y = [[-524.3 587.0 630.8 682.9 794.8 851.2 900.0 938.7]´; yp = yy(1) + yy(2)t + yy(3)t2 + yy(4)t3 + yy(5)t4 x1 = [ x2 = [ x3 = [

]´; ]´; ]´;

% y = GDP of Nigeria obtained as the function yp. dx1 = x1(t + 1) – x1(t); dx2 = x2(t + 1) – x2(t); dx3 = x3(t + 1) – x3(t). d1y = d1.*[ d2y = d2.*[ d3y = d3.*[ dx1 = e1 + x1.*(a1 + d1*y) dx2 = e2 + x2.*(a2 + d2*y) dx3 = e3 + x3.*(a3 + d3*y) % Use regression code to find a1 d1 d2 d3 a2 a3 e1 e2 e3. temp = [dx1 ones(size(x1)), x1 x1y]

]; ]; ];

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Thx1 = arx(temp, [0 1 1 1 0 0 0]; x1p = predict([dx1 ones(size(x1), x1 x1y thx1 4); x1x1 = thdx1(2, 1 : 3); Global e1 x1x1(1) = e1 Global = a1 x1x1(2) = a2 Global = d1 x1x1(3) = d1 dx1 = x1x1(1) + x1(-x1x1(2) + x1x1(3)y temp = [dx2 ones(size(x2) x2 x2y]; thx2 = arc(temp, [0 1 1 1 0 0 0]; x2p = predict([dx2 ones(size(x2) x2 x2y] thx2 4}; x2x2 = thdx2(2, 1 : 3); global = e2 x2x2(1) = e2 global = a2 x2x2(2) = a2 global = d2 x2x2(3) = d2 dx2 = x2x2(1) + x2(x2x2(2) + x2x2(3)y temp = [dx3 ones(size(x3)) x3 x3y]; thx3 = arx (temp, [0 1 1 1 0 0 0]); x3p = predict([dx3 one(size(x3) x3 x3y] thx3 4); x3x3 = thdx3(2, 1 : 3); global = e3 x3x3(1) = e3 global = a3 x3x3(2) = a3 global = d3 z3x3(3) = d3 dx3 = x3x3(1) + x3(x3x3(2) + x3x3(3)y) dx1 = dx1 = x p (1) dt dx 2 = dx 2 = x p (2) dt dx3 = dx3 = x p (3) dt

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Create an m.file, region.m function xp = region (t, x) % REGION North, west, east interaction. Global d1 d2 d3 xp = [e1 + x(1).*(a1 + d1*y); e2 + x(2).*(a2 + d2*y); e3 + x(3).*(a3 + d3*y)] % to = 1950 % tf = 1957 x0 = [7.4 6 4.4]; % initial conditions to = 0, tf = 7 [t, x] = ode23(‘region’, tot, to, tf, x0); x(1) = x1, x(2) = x2 x(3) = x3; plot (t, x) subplot(2, 1, 1), plot(YEAR, x1, Year, x1p; ..’) title(“NorthGDP’); ylabel(‘x1’); xlabel(‘YEAR’); subplot(2, 1, 2), plot(YEAR, x1, Yearx2, YEAR x1p; …’) subplot(2, 1, 2), plot(YEAR x2, YEARx2p’ ..’) subplot(2,1,1), plot(YEAR, x2 YEAR x2p (..)) title (‘WestGDP’); ylabel(‘x2’); xlabel(‘YEAR’); subplot(2, 1, 1) plot(YEAR x3, YEAR x3p ‘..’); title(‘EastGDP’); ylabel(‘x3’); xlabel(‘YEAR’); subplot(2, 1, 2), plot(YEAR, x3, YEAR x3p, ‘..’)

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Introduction. In chapter 4 we derived a hereditary model-of-interacting Gross Domestic products of nations. The interaction is mirrored in the function which represents net export, a function of contributions of the GDP of other nations at an earlier time to the current GDP. The equation is a differential game of pursuit. It reflects growth through trade. Growth of GDP can be promoted by the choices made by government and private firms. More general nonlinear models are explored. But first we confront the earlier interacting hereditary model with data. We use MATLAB 2com.m programs of the derived delay systems to identify the dynamics of GDP: Nigeria2com.m USA2com.m UK2com.m Chinacom.m Also US2com.m Egypt2com.m Jordan2com.m Israel2com.m DATA Nigeria US UK CHINA DATA US Egypt Jordan Israel

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3 Computer Programs and Results for Hereditary Models of Nigeria, US, UK, and China. (Chapter 4.6).

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It has been suggested that it is unnecessary to explicitly program the models since the methods can be cited from existing literature. But decisions can only be attempted when the systems are identified. We give an example in a resulting theory. A. Theory % % % %

Data generated for United States’ Economy relative to the other countries interacting yields the following: dy2(t)/dt – a_22*y2’ (t-h) = 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) + II(1) + XX(1))/(1-11*Z4) = 632.12 g2(t) = (1387063249.13 – 397726.95 . *e(t) + 0. *ta(t) + 0*d(t) + 179.16*T(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) + 0.01661*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 following 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 percent 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 organization in Bangkok. President Bush that 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 ha badly hurt their foreign sales and brought cuts in jobs at home. The U.S. economy has lost nearly 3 million jobs since President Bush took office. President Bush has chosen the way of exchange rate as a way favorable to the USA – the center piece of his Asia-Pacific trip. This can also help. It may be easier to

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use it to increase the growth rate of GDP. But the most effective way to reduce or make positive the draining of

dy2 dt

by − 6 y 2 (t ) y 4 (t − h) is to add +6.1y 2 (t ) y 4 (t − h) to it.

This is the way of cooperation. The US economic state is controllable, so is that of China. The interacting countries studied, Nigeria, US, UK, China are controllable. A high GDP can be attained by a judicious choice of all admissible controls, 3 – 9 of the representative firm, and 4 – 8 of the government. From the additional growth we can invest effective 7 y 2 (t ) y 4 (t − h) in “cooperative” ventures, making

dy2 dt

positive, and y 2 (t ) increasing.

Overall since

controllability has been proved and cooperation is possible, this is one good way to go. Selecting only one policy can hurt. It requires all the control strategies. For example, the USA secretary of commerce can negotiate a trade agreement with China. As a consequence General Motors can set up factories in China to produce low cost but very good cars in China. Because of the big market GM will make big profit by volume which China will not tax much because of the trade agreement and the economic improvement GM is bringing. In the USA, the huge profit can be taxed a little to create jobs or used by GM to create new jobs and new wealth in the US. Thus by cooperation y 2 , y 4 are increased and + 6.1 y 2 (t ) y 4 (t − h) 2 added to

dy 2 . dt

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4 Program DATA for US, UK, Egypt, Jordan and Israel These data are now displayed for ease of reference and for tutorial reasons. They are useful for suggested exercises. US3.M

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y2 = y ÷ 0.257281 = [

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y3 = y30 × 1.40787 = [

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y 4 = y 40 ÷ 0.241255 = [

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y3 = y30 × 1.40787 = [

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5 Problems and Exercises The following data which can be supplemented from as many sources as are needed are to be utilized for building the “wealth of nation” dynamics for Israel, Egypt, and Jordan. By simulation using internally generated wealth and inflow of wealth from another country show how wealth can be made to grow and grow. Israel2.M % Israel data in Thousands of New Shegalin (New Sheqalim per Thousand 50Q’s 1980) % Adopted from International Financial Statistic Year Book where Noted % Use Ndu.m to obtain matrix entries Year = [1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994]; XX=[3,099 12,700 17,434 22,179 24,635 31,461 36,715 40,299 49,544 60,536 73,334]; Imp=[-4,210 -16,570 -23,136 -32,302 -33,799 -39,078 -48,030 -61,183 -71,709 90,613 -107,843]; X = XX – Imp. %T=government revenue, taxes T=[4,490 15,503 23,094 27,862 29,582 34,355 42,286 39,758 63,295 73,482 88,559]; % G=Government Consumption G=[2,919 10,313 13,641 19,214 22,199 25,108 31,745 40,015 45,783 53,188 62,634]; % C=[16,490 27,801 36,393 44,257 53,168 64,745 81,484 97,915 116,475 142,478]; y=GDP y=[7,636 28,437 44,191 56,572 70,181 85,471 105,805 134,855 161,738 186,576 224,838]; GNI=Gross National Income GNI=[27,318 42,630 54,768 8,402 83,267 103,335 132,365 158,529 183,624 221,509]; e=0.6361 1.6471 1.8181 2.1828 2.2675 2.5797 2.9136 3.2657 1.8005 4.1015 4.4058]; M1=Money Supply (millions of new Sheqalim) M1=[304 1,052 2,238 3,346 3,723 5,376 7,022 7,988 10,541 13,486 14,523]; k=Capital Stock (Gross Fixed Capital formation) k=[1,568 5,338 8,051 11,254 12,776 14,554 20,234 32,280 38,068, 42,485 52,014];

% D=Increase in stock D=[100 117 401 -204 113 258 422 1,960 2,138 4,505 2,220]; I = Investment = Gross Capital Formation I=k + D. C=Private Consumption

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Househ.Cons.Expend., Ind NPISH % Quasi-Money = Md Md=[9,884 26,309 30,800 38,691 47,668 56,842 67,280 79,486 100,085 121,441 153,587]; R=Interest Rate Percent perannum Discount Rate % f=Net Primary Income from Abroad f=[7,303 -1,119 -1,561 -1,804 -1,779 -2,204 -2,496 -2,490 -3,209 -2,952 -3,329]; %NNI=Net National Income NNI=[6,093 22,575, 35,288, 45,791 57,946 70,296 87,k674 113,176 136,468 157,343 190,035]; P=Population P=[4.23 4.30 4.37 4.44 4.52 4.66 4.95 5.12 5.26 5.40]; e=exchange rate (New Sheqalim per thousand SDRS thr 1980) J2=[53 99 137 233 235 146 129 301 577 596 432]; R=[690.3 79.6 31.4 26.8 30.9 15.0 13.0 14.2 10.4 9.8 17.0 14.2]; p=Consumer Prices p=[4.7 18.8 27.9 33.4 38.8 46.7 54.7 65.1 72.9 80.9 90.9]; %wages=w (daily) w=[4.3 15.3 24.7 32.4 39.4 47.7 56.0 63.2 71.5 78.8 87.1]; %Employment=Emp Emp=[1,349 1,368 1,404 1,453 1,461 1,492 1,583 1,650 1,751 1,871]; % Unemployment=UnEmp UnEmp=[97 104 90 100 143 158 187 208 195 158]; J2=Direct Investment in Republic Economy RD=R(t – 1) RP=R(t + 1) – R(t) = R´(t) RPD=R´(t – 1) yD=y(t – 1) yP=y´(t) = y(t + 1) – y(t) yPD=y´(t-1) LD=L(t – 1) LPD=L´(t – 1) MP=M´(t) MPD=M´(t – 1) ML= MD=M(t – 1) yT=y – T yTD=yT(t – 1) yTPD=(yT)´(t – 1)

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Egypt2.m % Use US3.m or % Germany2.m % (GDP = y 2 = y y(0) = 36,618) Egypt2.m Data STATE: x = [y, R, L, k, p, E]; Control: Private Firm: σ = [C0 , I 0 , X 0 , M 0 , n, w, x0 , y10 , p 0 ]′ ;

government: q = [T1 , g 0 , e, τ, d , M 1, M& 1, f 0 ]′ ;

Code program Ndu.m Output Diagrams

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Egypt2.M Egypt DATA Year=[1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994]´; XX=[6,387 6,597 6,034 6,476 10,700 13,800 19,400 13,000 40,400 43,500 40,100]´; Imp=[-10,357 -10,636 -9,837 -11,740 -21,700 -24,800 -31,400 -39,800 -44,300 48,200 -49,200]´; X=XX – Imp; % TX=revenue (p. 469: 81) TX=[12,345 13,681 15,508 16,764 19,916 22,601 23,435 35,430 49,678 59,443 67,828]´; % G=Government Consumption G=[4,957 5,668 6,462 7,350 8,600 9,700 10,850 12,450 14,500 16,000 18,000]´; % k=Capital stock (Gross Fixed Capital Formation) k=[8,921 10,389 12,753 14,100 20,150 23,100 26,500 27,850 28,700 31,000 35,600]´; % I=Investment=Gross Capital Formation + D % I=k + D % D=Increase (Decrease in stock) D=[500 600 340 -650 300 900 1,800 -1,200 -1,200 NA NA]´; I=k + D C=[20,684 24,076 28,338 35,900 43,550 54,100 68,950 80,900 101,000 115,000 130,500]´; % y=GDP in millions of Pounds y=[31,693 37,451 44,131 51,526 61,600 76,800 96,100 111,200 139,100 157,300 175,000]´; e=Exchange Rate e=[0.6861 0.7689 0.8562 0.9931 0.9420 1.4456 2.8453 4.7665 4.5906 4.6314 4.9504]´; M1=Money Supply M=[12,443 14,696 15,973 18,241 20,579 22,471 26,205 28,337 30,832 34,571 38,275]´; Md=Money demand≈QuasiMoney Md=[13,486 15,980 21,129 26,637 33,970 41,623 56,303 70,127 86,762 98,602 109,834]; %R=Interest Rate (Discount) R=[13.00 13.00 13.00 13.00 14.00 14.00 20.00 18.40 16.55 14.00]´; %p=Consumer prices (64) p=[18.9 21.2 29.4 33.4 42.1 53.6 62.6 73.9 82.8 89.9 94.1]´; % Emp=Employment=L L=[NAN NAN NAN NAN NAN 14,926 14,361 13,827 14,399 14,703 15,241]; % UnEmployment=Unemp (67c) Unemp=[NAN NAN NAN NAN NAN 1,108 1,347 1,463 1,416 1,801 1,877]´;

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% Wages=w w=[34,132 39,060 40,832 58,386 67,254 81,341 103,344 125,485 NAN NAN NAN]´; % population=P P=[45.23 46.47 47.81 49.05 50.27 50.86 51.91 52.99 54.08 55.20 56.34] GROSS NATIONAL INCOME GNI=[30,605 35,892 39,397 46,818 NAN NAN NAN NAN NAN NAN NAN]´; % Tariff=Suez Canal dues in millions of Pounds τ=[665.4 654.2 769.0 844.5 904.6 1,506.8 3,177.6 5,707.9 6,187.9 6,628.4 6,998.1]´; % J=[729 1,178 1,217 948 1,190 1,250 734 253 459 493 1,256]´; Jordan2.M % Jordans data in millions of Jordanian dinars % Adopted from International Financial Statistic Yearbook 1994, except where noted % Use ndu.m to obtain the matrix entires Year=[1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995]´; XX=[746.8 781.5 634.1 756.2 1,020.8 1,359.5 1,652.1 1,697.6 1,819.9 1,962.1 2,093.4]; Imp=[-1,519.1 -1,502.7 -1,1995.0 -1,219.7 -1,804.5 -2,474.3 -2362.6 -2,974.7 3,151.7 -3.107.6]; X=XX-Imp; TX=[415.01 440.81 514.39 531.53 544.34 565.40 744.07 828.78 1,168.90 1,191.5 1,306.4] % G=Government Consumption G=[534.6 531.7 566.5 586.7 604.3 618.8 663.9 742.0 790.6 857.9 985.6]´; % k=[Capital Stock (Gross Fixed Capital Formation) k=[526.8 384.8 409.3 448.5 513.4 554.1 694.0 678.0 1,049.2 1,303.5 1,391.0]´; % I=Investment = Gross Capital Formation % I=K + D % D=Increase in Stock D=[44.4 30.1 35.0 67.1 19.1 9.1 156.1 60.5 159.6 119.2 60.0]´; I=k + D C=[1,648.4 1,794.8 1,718.2 1,669.8 1,626.5 1,635.1 1,976.5 2,052.8 2,692.5 2,767.7 2,824.5]; % y=GDP y=[1,981.4 2,020.2 2,163.6 2,208.6 2,264.4 2,372.1 3,668.3 2,868.3 3,537.1 3,858.7 4,246.9]; e=Exchange Rate e=[2.5795 2.5790 2.5790 1.5579 1.1743 1.0570 1.0357 1.0525 1.0341 0.9772]´;

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M1=Money M1=[878.4 848.2 897.1 979.8 1,166.8 1,302.3 1,425.3 1,646.6 1,716.0 1,719.4 1,741.6]´; % Quasi-Money, Money demand = Md Md=[883 1,030.9 1,181.7 1,424.4 1,611.2 1,935.3 2,080.1 2,412.8 2,479.0 2,664.7 2,788.6]; % R=Interest Rate (Discount) R=[6.25 6.25 6.25 6.25 8.00 8.50 8.50 8.50 8.50]´; p=Consumer prices p=[50.5 52.3 52.3 52.2 55.6 69.9 81.2 87.8 91.4 94.4 97.7]´; % Unemployment = UnEmp UnEmp=[97 104 90 100 143 158 187 208 195 158 145 144 170]´; % wages=W (daily) W=1.6 7.7 27.3 44.0 57.9 70.4 85.2 100.0 112.8 127.6 140.6 155.6 178.5 203.3 233.5]´; % J3=Direct Investment in Republic Economy 78bed J3=[77.5 24.9 22.8 39.5 23.7 -1.3 37.6 -11.9 40.7 -33.5 2.9]´; % Balance of payment = B % E=Cumulative Balance of payment (80) dE % =B dt B=[-141.99 -111.74 -153.13 -198.21 -204.57 -137.10 -94.42 12.42 181.00 69.70 105.00]; %GN1=Gross National Income GNI=[1,995.0 2,015.5 2,146.3 2,158.4 2,175.9 2,180.7 2,428.8 2,647.2 3,350.9 3,709.6 4,095.5]´; % P=population P=[-3.83 3.94 4.00 4.06 4.50 4.62 4.80 5.02 5.26 5.51]´; % Import duties = τ τ=[225 174 167 158 173 157 188 195 346 380 363]´; % T=Taxes (Direct) T=[348 348 352 348 489 483 478 449 600 681 966]´; % d=transportation d=[268 297 284 277 255 179 226 256 288 331 397]´; % L=Labor (Industrial Production) L=[68.2 69.2 75.6 69.4 72.9 73.3 72.5 78.1 84.3 89.0]´;

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6 MATLAB Programs and Graphs for Economic Models with Delay Program US2.M 1.

Definition of Economic Variables and Terms, pp. A.2-A.3. In this section the economic variables extracted from the International Financial Statistic Yearbook are defined by Symbols.

2.

The U.S.A. data, pp. A.1-A.2. Displaced here are data extracted from the International Financial Statistic Yearbook 1994, UN Financial Statistic 1974, UN National Accounts Statistics.

3.

Equations and Formulae postulated for economic variables in the body of the book are now identified, pp. A.3-A.5. ML = L – M L I C X T G Z R& (t ) p& (t ) B = Balance of Payment E = Cumulative Balance of Payment y = National Income D = deliveries of new equipment dk = flow of capital stock dt L = employment y = income Gy = income government Xy = income export

A.3 A.3 (1.5) (1.3) (1.11) (1.2) (1.7) (1.2), (1.12), (1.13) (1.20), (1.21), (1.23) (1.35), inflation (1.38) (1.37) (1.44) (1.48) (1.53) (1.60) (1.52) (1.50) (1.50)

4.

MATLAB Regression Programs for Economic Variables, pp. A.3-A.6. See “System Identification Toolbox” for use with MATLAB, Lenhart Ljung, The Math Works, Inc. I.30-I.32, 14.8, pp. 1.79, 2.16.

5.

MATLAB Plot, Subplot programs, pp. A.5-A.6.

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6.

Identification of the economic dynamics: x& (t ) − A−1 x& (t − h) = A0 x(t ) + A1 x(t − h) + B * u (t )

with coefficients A−1 , A0 , A1 , B, B1 , and given in pp. A.6-A.7 and identified in A.8-A.9 using (Ndu.M, US3.M) program. In A.8-A.13. 7.

Diagrams and plots, Fig. U.S.1 → Fig. U.S.7.

8.

We use the rank condition of Salamon and the full rank of B to deduce the (t ) controllability in W p of the economic state variables of the dynamics

x& (t ) − A−1 x& (t − h) = A0 x(t ) + A1 x(t − h) + Bu (t ) If γ is any complex number, rank [ ∆ ( γ ), B ] = n = 6 and rank [ γI − A−1 , B ] = 6, are the required conditions (pp, 157, D. Salamon, “Control and Observation of Neutral Sysems”, Pitman Advanced Publishing Program, Boston, MA). This rank condition is satisfied by our linear model, pp. A.22. See [7].

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Programs 1.

U.S.A.2.M Jordan2.m Data STATE: x = [ y, R, L, k , p, E ]′ Control Private firms: σ = [C 0 , I 0 , X 0 , M 0 , n, w, x0 , y 0 , p0 ]′ government q = [T1 , g 0 , e, τ, d , M 1, M& 1, t 0 ]′ Program Code Ndu.m Output Diagrams

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US3.M

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Jordon2.m

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%rank (B) = 6 = rank (B, 0) %rank (G) = rank (G, 0) = 6

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Egypt2.m

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%G = [B AB A2B A3B A4B A5B] %rank (B, 0) = rank (B) %rank (B) = 6 %rank (B) = 6

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Israel’s Results

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%rank (B, 0) = rank (B) %rank (G, 0) = rank (G) %rank (B) = 5 %rank (G) = 6

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A Gross Domestic Product in Dollars for all Countries % U.S.A. Data in dollars y1 = x1 = x Year=[1984 1985 1986 1987 1988 1990 1991 1992 1993 1994]’; y1 =[3777.2 4038.7 4268.6 4539.9 4900.4 5250.8 5522.2 5722.9 6038.5]’; % Egypt Data in dollars y 2 = x 2 = y % y 2 =[1970 1971 1972 1993 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983]’; % y 2 =[764.3819 824.0170 879.1292 942.4203 1079.8084 1257.0750 1614.6956 2112.2770 2516.9800 3209.5805 3980.1371 4412.3692 5779.8177 6798.3931]’; [1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000]’; y 2 =[8154.0007 9635.4307 11354.0678 13256.6608 15848.5096 19759.1868 24724.7041 28.6096 35786.7871 40470.3013 40024.175 52742.605 58732.2523 65928.2563 72095.2818 77767.0463]’; % Jordan Data in dollars y3 = x3 = z y3 =[2625.0332 2841.8774 3049.4667 3107.7758 3185.1537 3337.0936 3753.5292 4616.6.39 4914.1969 5348.9197 5910.2609 6548.9799]’; %Israel Data in Dollars y 4 = x 4 =%[1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983]’; % y 4 =[453.5594 564.7780 723.5237 435.3456 1349.8217 1885.803 2385.5294 3417.6594 3471.6595 5728.6000 10733.4360 26926.9531 63968.2807 143861.804 372.0152]’; [1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000]’; y 4 =[1842.2232 6860.5684 1061.30010 32534.4430 39020.1021 45012.3929 54243.2971 65764.6615 74303.4037 83218.0172 90737.6943 98941.3242]’;

Jordan J=Direct.Invest. in Republic Econ. 1984 1985 J 3 =[77.5 24.9 22.8 29.5 23.7 –1.3 37.6 –11.9 40.7 –33.5 2.9 13.3 15.5 360.9 310.0]´; U.S.A. GDP(billions of US dollars) 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 y1 =[3,932.7 4,213.0 4,452.9 4,742.5 5,108.3 5,489.1 5,803.2, 5.986.2, 6,318.9 6,642.3 7,504.3 7,400.5 7,813.2 8,300.8 8,759.9 9,256.1]´; Eqypt (1984-1999) GDP In Egyptian pound y 21 =[8,154.0007 9,635.4307 11,354.0678 13,256.6608 15,848.5096 19,759.1868 24,724.7041 28,6096 35,786.7871 40,470.3013 40,024.175 52,742.605 58,732.2523 65,928.2563 72,095.2818 77,767.0463]´; y 2 = y 21 ÷ 0.257281=[ ]´;

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1984 U.S.A. GDP= y1 y1 =[

]´;

%Jordan GDP 1984-1999 y 31 =[1,981.4 2,20.2 2,163.6 2,2086 2,264.4 2,372.1 2,668.3 2,868.3 3,537.1 3,858.7 4,246.9 4,560.8 4,711.0 4,945.8 5, 180.0] Jordan GDP in dollars y 3 = y 31 × 1.40787=[ ]´; % y 41 =Israel GDP in Shekar 1970-1984 % y 41 =[1,880 2,342 2,999 3,877 5,595 5,817 9,888 14,890 23,745 44,490 111,612 265,148 596,306 1,546M 7,636M]´; Israel GDP in shekar [1984-1999]´; y 41 =[7,636 28,437 44,191 56,,572 70181 85,471 105,805 134,855 161,738 186,576 224,838 264,304 307,987 344,938 376,107 410,111]´; y 4 = y 41 ÷ 0.241255=[ ]´; Regional Cooperation and Competition Models Ceuji.m % % % %

x1 = x2 =

U.S.A. GDP = y1 = x Egypt GDP = y 2 = y

x3 = x4 =

GDP for all countries in dollars 1984 USA GDP values in dollars y1 = [ GDP values for Israel in US dollars (1 Israel new shekar = 0.241255 US dollars) (multiply Israel GDP by 0.241255) y4 = [ GDP values for Jordan in US dollars. (1 Jordan dinar = 1.40687 US dollars) (1 multiply Jordan GDP by 1.40787) y3 = [ GDP values for Egypt in dollars (1 Egyptian pound = 0.257281 US dollars) (Multiply Egypt GDP by 0.257281)

Jordan GDP = y3 = z Israel GDP = y 4 = w

]´

]´

]´

C198

y2 = [ Code to convert all values to equivalent in US dollars for k = 1: 1.5 x 2 (k ) = y 2 (k ) * 2.57281 x3 (k ) = y 3 (k ) * 1.40687 x 4 (k ) = y 4 (k ) * 2.31255 x1 (k ) = y1 (k )

]´

% Assume x1 (1984) = x1 (0) = x(0) = 3,9327 % x 2 (1984) = y 2 (1984) = y (1984) = y (0) = (8,154.0007)÷(.25829) % x3 (1984) = z 3 (1984) = z (1984) = z (0) = (1,981.4) × (1.40787) % x 4 (1984) = y 4 (1984) = w(0) = (7,636) ÷ (0.241255) %Assume x1 (1984) = x1 (0) = 3,9327 x 2 (1984) = x 2 (0) = (8154.0007) ÷ (2.5728)) x3 (1984) = x3 (0) = 1,868 × (1.40687) x 4 (1984) = x 4 (0) = 7,636 ÷ (0.241255) %Initial condition x0 =[39327,(8154.0007)÷2.5728, (1.981.4)×(1.40787), (7,636)÷(0.241255)]´; Code to find dx, dy, dz, dw for k = 1:15 dx = x(k + 1) – x(k) end dx = [ for k = 1 : 15 dy = y(k + 1) – y(k); end dy = [ for k = 1 : 15 dz = z(k + 1) – z(k); dx = [ dw = w(k + 1) – w(k) end dw = [ dx = dy1 = [ ]´;

]´

]´

]´

]´

dy = dy 2 = [

]´;

dz = dy3 = [ ]´; dw = dy 4 = [ x=[ y=[

]´; ]´; ]´;

C199

z=[ w=[ for k = 1 : 15 xy(k) = x(k) · y(k); end xy = [ for k = 1 : 15 xz(k) = x(k) ⋅ z(k) for k = 1 : 15 end xz = [ xw(k) = x(k) ⋅ w(k); end xw = [ for k = 1 :15 yz(k) = y(k) × z(k); end yz = [ for k = 1 : 15 yw(k) = y(k) × w(k); end yw = [ for k = 1 : 15 zw(k) = z(k) × w(k); end zw = [ DATA y1 = [ y2 = [ y3 = [ y4 = [ y1 y 2 = y1 × y 2 = [ y1 y 3 = y1 × y3 = [ y1 y 4 = y1 × y 4 = [ y 2 y3 = y 2 × y3 = [ y2 y4 = y2 × y4 = [ y 3 y 4 = y3 × y 4 = [ x& = e1 + x(− a1 + b1 y + c1 z + d1w) , y& = e2 + y (− a 2 + b2 x + c2 z + d 2 w) , z& = e3 + z (− a3 + b3 x + c3 y + d 3 w) , w& = e4 + w(− a 4 + b4 x + c4 y + d 4 z ) ,

]´; ]´;

]´

]´;

]´;

]´;

]´;

]´;

]´; ]´; ]´; ]´; ]´; ]´; ]´; ]´; ]´; ]´;

C200

dx = x(t + 1) − x(t ) ; dy = y (t + 1) − y (t ) ; dz = z (t + 1) − z (t ) ; dw = w(t + 1) − w(t ) ; dx = e1 + x(− a1 + b1 y + c1 z + d1w) ; dy = e2 + y (− a 2 + b2 x + c 2 z + d 2 w) ; dz = e3 + z (− a3 + b3 x + c3 y + d 3 w) ; dw = e4 + w(− a 4 + b4 x + c 4 y + d 4 z ) ;

∫ Use Regression to find ∫ ai , bi , ci , d i , ei ,

i = 1, ..., 4.

temp = [dx, ones(size ( x)), x, xy, xz, xw] , thdx = arx( temp, [0 1 1 1 1 1 0 0 0 0 0] ) ; xp = predict([dx ones(size (x)) x, xy, xz, xw], thx 4); xx = thdx(4 1 : 5); global e1 xx(1) = e1 global a1 xx(2) = a1 global b1 xx(3) = b1 global c1 xx(4) = c1 global d1 xx(5) = d1 dx = xx(1) + x(− xx(2) + xx(3) y + xx(4) z + xx(5) w) . temp = [dy ones(size( y ), y, xy, zy , wy ] ; thdy = arx( temp, [0 1 1 1 1 1 0 0 0 0 0]) ; y p = predict([dy ones(size ( y )), y, xy, yz, yw], th 4) ; yy = thy (4

global e2 yy(1) = e2 global a 2 yy(2) = a 2 global b2

1 : 5);

C201

yy(3) = b2 global c3 yy(4) = c3 global d 2 yy(5) = d 2 dy = yy (1) − yy (2) y + yy (3) xy + yy (4) yz + yy (5) yw temp = [dy1 , ones(size( y1 )), y1 , y1 y 2 , y1 y 3 , y1 y 4 ] ; thdy1 = arx ( temp, [0 1 1 1 1 1 0 0 0 0 0] ) ; dy1 p = predict([dy1 , ones(size( y1 )), y1 , y1 y 2 , y1 y 3 , y1 y 4 ] thy1 p 4) ; dy1 y1 = th dy1 (4, 1 : 5) ; global e1 e1 = dy1 y1 (1) global a1 a1 = dy1 y1 (2) global b1 b1 = dy1 y1 (3) global c1 c1 = dy1 y1 (4) global d1 d1 = dy1 y1 (5) thy1 = e1 + y1 (a1 + b2 y 2 + c1 y3 + d1 y 4 ) temp = [dz ones(size( y )), z , zx, zy, zw] ; thz = arx ([temp, [0 1 1 1 1 1 0 0 0 0 0] ) ; zp = predict([dz ones(size( y)), z, zx, zy, zw ] thz, 4) ; zz = thz (4, 1 : 5) ; global e3 zz (1) = e3 global a3 zz (2) = a3 global b3 zz (3) = b3 global c3 zz (4) = c3 global d 3 zz (5) = d 3

C202

dz = zz (1) − zz (2) z + zz (3) xz + zz (4) yz + zz (5) zw temp = [dw ones(size( y )), w, xw, yw, zw] ; thw = arx ([ temp, [0 1 1 1 1 1 0 0 0 0 0] ) ;

wp = predict([dw ones(size(w)), xw, wy , wz ], thw, 4) ; ww = thw(4 1 : 5) ; global e4 ww(1) = e4 global a 4 ww(2) = a 4 global = b4 ww(3) = b4 global c 4 ww(4) = c 4 global = d 4 ww(5) = d 4 dw = e4 − a 4 w+ b4 wx + c 4 wy + d 4 wz ww(1) − ww(2) w + ww(3) wx + ww(4) yw + ww(5) zw

% Use least square to get the best fit for % g = [dx, dy, dz, dw] dx coeffinit = [ xx(1), xx(2), xx(3), xx(4), xx(5)] ; dx coeff = leastsq(leastdx, dx coeffinit); dy coeffinit = [ yy (1), yy (20, yy (3), yy (4), yy (5)] ; dy coeff = leastsq(leastdy, dy coeffinit); dz coeffinit = [ zz (1), zz (2), zz (3), zz (4), zz (5)] ; dz coeff = leastsq(leastdz, dz coeffinit); dw = coeffinit [ ww(1), ww(2), ww9(3), ww(4), ww(5)] ; dw = coeffinit = leastsq(leastdw, dwcoeffinit); dw found = dw(1) + w * [− dw(2) + dw(3) * x + dw(4) * y + dw(5) * z ] ; dz found = dz(1) + z * [− dz(2) + dz(3) * x + dz(4) * y + dz(5) w ] ; dy found = dy(1) + y * [− dy(2) + dy(3) * x + dz(4) * z + dz(5) * w ] ; dx found = dx(1) + x * [− dx(2) + dx(3) * y + dx(4) * z + dx(5) * w] ; Mideast.m Function xdot = mideast(t, x) xdot = [dx found; dy found; dz found; dw found]; % Solve over the interval 1984 1999

% (maybe)

C203

0 ≤ t ≤ 15 % Invoke ode 45: t 0 = 0 , t f = 15

x0 = [ x (0), y (0), z (0), w(0)]

%Initial condition

x0 = [ x (1)(0) x(2)(0) x(3)(0) x(4)(0)] [t, x] = ode 45(‘mideast’, t 0 , t f , x0 ) ; plot (t, x(1)) plot(t, x(:, 1)) plot(t, x(:, 2)) plot(t, x(:,3)) plot(t, x(:, 4)) mideast.m function xdot = mideast(t, x) xdot = [dxfound; dyfound; dzfound; dwfound] x(1) = x, x(2) = y, x(3) = z , x(4) = w; %Solve over the interval % 1984 1999 0 ≤ t ≤ 15 . Invoke ode 45: x0 = [ x0 (1), x0 (2), x0 (3), x0 (4)]′ % Initial conditions x0 = [ x(0), y (0), z (0), w(0)]′ ; x0 = [1,010.4 36,618 1,868 7,636]´, [t, x] = ode 45(‘mideast’, t 0 , t f , x 0 ) ;

plot(x(:, 1), x(:, 2)); plot(x(:, 1), x(:, 3)); plot(x(:, 1), x(:, 4)); plot(x(:, 2), x(:, 3)); plot(x(:, 2), x(:, 4)); plot(x(:, 3), x(:, 4)); plot(t, x(:, 1)); plot(t, x(:, 2)); plot(t, x(:, 3)); plot(t, x(:, 4)); MAPLE With (plots) deqs : = [diff ( x(t ), t ) = x (t ) * (− a1 + b1* y (t ) + c1* z (t ) + d1* w(t )) , diff ( y (t ), t ) = y (t ) * (− a 2 + b2 * x(t ) + c 2 * z (t ) + d 2 * w(t )) ,

0 ≤ t ≤ 15

C204

diff ( z (t ), t ) = z (t ) * (− a3 + b3 * x(t ) + c3 * y (t ) + d 3 * w(t )) , diff ( w(t ), t ) = w(t ) * (− a 4 + b4 * x(t ) + c 4 * y (t ) + d 4 * z (t )) ] ; With (Detools): Deplot(deqs, [t, x, y, z, w], 0..15, {0, x0 }, scene = [x, y], scene = [x, z], scene = [x, w], scene = [y, z], scene = [y, w], scene = [z, w]. stepsize = 1.0, arrows ‘none’]; % a i , bi , c i , d i % i = 1, 2, 3, 4

% best fit coefficients

%Initial conditions x(1984) y(1984) z(1984) w(1984) odeplot(dsolve({deqs, x(0) = 1,010.4, y(0) = 36,618, z(0) = 1,868, w(0) = 7,636]′ {x(t), y(t), z(t), w(t)} numeric), [t, y(t)] 0..15); [t, x(t)] 0..15); [t, z(t)], 0..15); [t, w(t)]. 0..15); dsolve proc(rkf45 – x) … end numeric odeplot(dsolve({deq1, x(1984) = 10,104, y(1984) = 36,618, z(1984) = 1,868, w(1984) = 7,636) We can now investigate the interaction coefficients by evaluating all the countries relationships with each other; Egypt vs Jordan Egypt vs U.S.A Israel vs Jordan Israel vs U.S.A. Jordan vs U.S.A. We use the model x& = x (−1 + b1 y ) y& = y (−1 + b1 x) b1 = .3, .5, .6, 1, and its graphs Fig. 1-Fig. 8 as insights. Judgment is then made on the consequences of competition and cooperation. One deduces that cooperation pays. Also increased growth can facilitate cooperation. Furthermore increased growth can be simulated by the economic growth model and a judicious choice of strategies.

C205

Program Example DATA

U.S. GDP : u Egypt GDP : v du = u(t + 1) – u(t), dv = v(t + 1) – v(t). u=[ v=[ uv = [ du = − a1*u + b1*uv dv = − a *2 u + b2*uv temp = [du, u, uv] thdu = arx(temp 0 1 1 0 0) dupredict = predict([du, u, uv, thdu, 4); temp = [dv v uv] thdv = arx(temp. 0 1 1 0 0) dvpredict = predict(dv, u, v, thdv, 4); dvdv = thdv(2, 1:2) dudu = thdu(2, 1:2) dudu(1) = - a1 dudu(2) = b1 dvdv(1) = - b1 dvdv(1) = b2 .

Use least squre to get the best fit for w = [du, dv] ducoeffinit = [dudu(1), dudu(2)]; ducoeff = leastsq(leastdu, ducoeffinit); dvcoff = leastsq(leastdv, dvcoeffinit); dufound = uduff(1) + duff(2)uv dvfound = vdvff(1) + uvdvff(2) plot(t, u) plot(t, v) plot(u, v) coop.m functionxdot = coop(t, x) xdot = zeros(2, 1); xdot(1) = dufound; xdot(2) = dvfound; x0 = [u(1), v(1)]´ %initial condition

] ] ]

C206

USA GDP (billions of US dollars) Year=[1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994]; y1=[3,932.7 4,213.0 4,452.9 4,742.5 5,108.3 5,489.1 5,803.2 5,986.2 6,318.9 6,6642.3 7,504.3]; Y1 Vals = y1 Egypt GDP in pounds = y2 Year=[1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994]; y21=[8,154.0007 9,635.4307 11,354.0678 13,256.6608 15,848.5096 19,759.1868 24,724.7041 28.6096 35,786.7871 40,470.3013 52,742.605]; y2=y21 x 0.257281 = [ ] (% in dollars) Y2 Vals= y2 Jordan Year

GDP y31= Year

GDP y31=

1984 1,981.4

1985 2,020.2

1986 2,163.6

1987 2,208.6

1988 2,64.4

1990 2,668.3

1991 2,868.3

1992 3,537.1

1993 3,858.7

1994 4,246.9

GDP in dollars Jordan y3=y31 × 1.40787 = [ Y3 Vals = y3

1989 2,372.1

]

Israel DATA GDP in Shekar Year=[1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994]´; Year

1984 7,636

1985 28,437

1986 44,191

1987 56,572

1988 70,181

1990 105,805

1991 134,855

1992 161,738

1993 186,576

1994 244,838

GDP y41= Year

GDP y41=

GDP in dollars = y4 y4 = y41 × 0.241,255 = [ Y4 Vals = y4 dx1=dx=[dx(1) dx(12) dx(13) dx2=dy=[dy(1) dy(12) dy(13) dx3=dz=[dz(1) dx(12) dz(13)

1989 85,471

]

dx(2) dx(3) dx(4) dx(5) dx(6) dx(7) dx(8) d(9) dx(10) dx(11) dx(14) dx(15)]; dy(2) dy(3) dy(4) dy(5) dy(6) dy(7) dy(8) dy(9) dy(10) dy(11) dy(14) dy(15)]; dz(2) dz(3) dz(4) dz(5) dz(6) dz(7) dz(8) dz(9) dz(10) dz(11) dz(14) dz(15)];

C207

dx4=dw=[dw(1) dw(2) dw(3) dw(4) dw(5) dw(6) dw(7) dw(8) dw(9) dw(10) dw(11) dw(12) dw(13) dw(14) dw(15)]; Data: dX1 values, dX2 values, dX3 values, dX4 values dX1vals=[dx1] dX2vals=[dx2] dX3vals=[dx3] dX4vals=[dx4] dX1vals=[ dX2vals=[ dX3vals=[ dX4vals=[

]; ]; ]; ];

DATA X1values=y1=[ X2values=y2=[ X3values=y3=[ X4values=y4=[

]; ]; ]; ];

W2vals=X1vals.*X2vals=[ W3vals=X1vals.*X3vals=[ W4vals=X1vals.*X4vals=[ dX1=X1(t + 1) – X1(t) = [

]; ]; ]; ];

Similarly, t = 0, …, 10 dX2, dX3, dX4 W5vals=X2vals.*X3vals=[ W6vals=X2vals.*X4vals=[ W7vals=X3vals.*X4vals=[

]; ]; ];

C208

7 MATLAB EXECUTION of Program by EMEKA CHUKWU %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % International Cooperation and Competition Models for U.S.A, % % Egypt, Jordan and Israel is outlined as follows(statistics % % used are from International financial yearbook statistics, % % 1965-1992): Step 1 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% year=[1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996]'; GDP_usa=[1767.1 1974.1 2232.7 2488.7 2708.1 3030.6 3149.6 3405.1 3777.2 4038.7 4268.6 4539.9 4900.4 5250.8 5522.2 5722.9 6038.5 6642.3 7504.3 7400.5 7813.2]'; GDP_egypt=[6.276 8.210 9.783 12.475 15.470 17.150 20.753 25.895 31.547 37.240 42.563 51.500 61.600 76.800 96.100 111.200 139.100 157.300 175.000 205.000 228.300]'; GDP_jordan=[0.5121 0.6246 0.7679 0.9146 1.1512 1.4267 1.7011 1.8287 1.9814 2.0202 2.1636 2.2086 2.2644 2.3721 2.6683 2.8683 3.6482 3.9256 4.4000 4.7736 4.9824]'; GDP_israel=[0.009888 0.014390 0.023745 0.044490 0.111612 0.265148 0.596306 1.542 7.636 28.437 44.191 56.572 70.181 85.471 105.805 134.855 161.738 186.576 224.838 265.701 310.429]'; exchangeR_egypt=[0.4546 0.4753 0.5098 0.9221 0.8928 0.8148 0.7722 0.7329 0.6861 0.7689 0.8562 0.9931 0.9420 1.4456 2.8453 4.7665 4.5906 4.6314 4.9504 5.0392 4.8718]'; exchangeR_jordan=[2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 1.5579 1.1743 1.0570 1.0357 1.0525 1.0341 0.9772 0.9488 0.9809]'; exchangeR_israel=[1.0158 1.8672 2.4672 4.6517 9.6268 0.0182 0.0371 0.1128 0.6261 1.471 1.8181 2.1828 2.2675 2.5797 2.9136 3.2657 3.8005 4.1015 4.4058 4.6601 4.6748]'; y1p =[1584.8 1767.1 1974.1 2232.7 2488.7 2708.1 3030.6 3149.6 3405.1 3777.2 4038.7 4268.6 4539.9 4900.4 5250.8 5522.2 5722.9 6038.5 6642.3 7504.3 7400.5 7813.2]'; y2p =[4.886 6.276 8.210 9.783 12.475 15.470 17.150 20.753 25.895 31.547 37.240 42.563 51.500 61.600 76.800 96.100 111.200 139.100 157.300 175.000 205.000 228.300]'; y3p=[0.3791 0.5121 0.6246 0.7679 0.9146 1.1512 1.4267 1.7011 1.8287 1.9814 2.0202 2.1636 2.2086 2.2644 2.3721 2.6683 2.8683 3.6482 3.9256 4.4000 4.7736 4.9824]'; y4p=[0.007817 0.009888 0.014390 0.023745 0.044490 0.111612 0.265148 0.596306 1.542 7.636 28.437 44.191 56.572 70.181 85.471 105.805 134.855 161.738 186.576 224.838 265.701 310.429]'; ex2p =[0.4581 0.4546 0.4753 0.5098 0.9221 0.8928 0.8148 0.7722 0.7329 0.6861 0.7689 0.8562 0.9931 0.9420 1.4456 2.8453 4.7665 4.5906 4.6314 4.9504 5.0392 4.8718]'; ex3p=[2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 2.5790 1.5579 1.1743 1.0570 1.0357 1.0525 1.0341 0.9772 0.9488 0.9809]';

C209 ex4p=[0.8303 1.0158 1.8672 2.4672 4.6517 9.6268 0.0182 0.0371 0.1128 0.6261 1.471 1.8181 2.1828 2.2675 2.5797 2.9136 3.2657 3.8005 4.1015 4.4058 4.6601 4.6748]'; y1 =GDP_usa; % billions of US % dollars. y2 =GDP_egypt./(exchangeR_egypt);

% GDP of U.S.A, Data is is in

% y2 for Egypt data is is in % billions of US dollars; % Egypt's exchange rate % is given in pounds/US dollar y3 =GDP_jordan.*(exchangeR_jordan); % y4 for Jordan data is is in % billions of US dollars. % Exchange rate is reported in % US dollars/Dinar. y4 =GDP_israel./(exchangeR_israel); % y3 for Israel data is is in % billions of US dollars New % Sheqalim/thousand US dollars. % The above data may be used for the regression and % refined to yield the following systems equations: % % % %

dy1(t)/dt dy2(t)/dt dy3(t)/dt dy4(t)/dt

= = = =

y1(-a1 y2(-a2 y3(-a3 y4(-a4

+ + + +

b1*y2 b2*y2 b3*y2 b4*y2

+ + + +

c1*y3 c2*y3 c3*y3 c4*y3

+ + + +

d1*y4) d2*y4) d3*y4) d4*y4)

+ + + +

e1, e2, e3, e4,

% Where ei, i=1,...,4 are present as windfalls %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % We first use MATLAB System Identification Algorithm to %find % the constant coefficients for: ai, bi, ci, di, ei, % % where i = 1, 2, 3,4. The assumptions is that initial % % conditions are setup at 1975: Step 2 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y1_1975 y2_1975 y3_1975 y4_1975 y0

= = = =

1.*(1584.8); (4.886)./0.4581; 2.5790.*(0.3791); (0.007817)./0.8303;

= [1584.8 10.71 0.9777 0.01];

% % % %

y1_1975 y2_1975 y3_1975 y4_1975

= = = =

y1(0)=x(0) y2(0)=y(0) y4(0)=w(0) y3(0)=z(0)

% Initial guesses % for time t = 1975.

% y = [y1, y2, y3, y4]; Let x=y1; y=y2; z=y3; w=y4 x=y1; y=y2; z=y3; w=y4; xp=y1p; yp=y2p./ex2p; zp=ex3p.*y3p; wp=y4p./ex4p; for t = 1:21 dX(t) = xp(t+1)-xp(t); dY(t) = yp(t+1)-yp(t); dZ(t) = zp(t+1)-zp(t); dW(t) = wp(t+1)-wp(t); end

% derivatives of x,y,z and w.

C210 % And xy yz dx

also, = x.*y; xz = x.*z; xw = x.*w; = y.*z; yw = y.*w; wz = z.*w; =dX'; dy=dY'; dz=dZ'; dw=dW';

% Once again % dx = e1 + % dy = e2 + % dz = e3 + % dw = e4 +

we re-state the system equation in another way: x(-a1 + b1*y + c1*z + d1*w); y(-a2 + b2*x + c2*z + d2*w); z(-a3 + b3*x + c3*y + d3*w); w(-a4 + b4*x + c4*y + d4*z);

% For dx = e1 + x(-a1 + b1y + c1z + d1w); temp = [dx, ones(size(x)),x,xy,xz,xw]; thdx = arx(temp,[0 1 1 1 1 1 0 0 0 0 0]); XXp = predict([dx ones(size(x)),x,xy,xz,xw],thdx,4); xx = thdx(4,1:5); global e1; global a1; global b1; global c1; global d1; e1=xx(1); a1=xx(2); b1=xx(3); c1=xx(4); d1=xx(5); % % Hence, % dx/dt = xx(1) + x.*(-xx(2) + xx(3).*y + xx(4).*x + xx(5).*w); % For dy = e2 + y(-a2 + b2x + c2z + d2w); temp = [dy, ones(size(y)),y,xy,yz,yw]; thdy = arx(temp,[0 1 1 1 1 1 0 0 0 0 0]); Yyp = predict([dy, ones(size(y)),y,xy,yz,yw],thdy,4); yy = thdy(4,1:5); global e2; global a2; global b2; global c2; global d2; e2=yy(1); a2=yy(2); b2=yy(3); c2=yy(4); d2=yy(5); % % Hence, % dy/dt = yy(1) + y.*(-yy(2) + yy(3).*x + yy(4).*z + yy(5).*w); % For dz = e3 + z(-a3 + b3x + c3y + d3w); temp = [dz, ones(size(z)),z,xz,yz,wz]; thdz = arx(temp,[0 1 1 1 1 1 0 0 0 0 0]); ZZp = predict([dz ones(size(z)),z,xz,yz,wz],thdz,4); zz = thdz(4,1:5); global e3; global a3; global b3; global c3; global d3; e3=zz(1); a3=zz(2); b3=zz(3); c3=zz(4); d3=zz(5); % % Hence, % dz/dt = zz(1) + z.*(-xx(2) + xx(3).*x + xx(4).*y + xx(5).*w); % For dw = e4 + w(-a4 + b4x + c4y + d4z); temp = [dw, ones(size(w)),w,xw,yw,wz]; thdw = arx(temp,[0 1 1 1 1 1 0 0 0 0 0]); WWp = predict([dw ones(size(w)),w,xw,yw,wz],thdw,4); ww = thdw(4,1:5); global e4; global a4; global b4; global c4; global d4; e4=ww(1); a4=ww(2); b4=ww(3); c4=ww(4); d4=ww(5); %

C211 % Hence, % dw/dt = ww(1) + w.*(-ww(2) + ww(3).*x + ww(4).*y + ww(5).*z); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % For a non linear least square we use LSQNONLIN to get the best % % fit for: % % g = [dx, dy, dz, dw]; Let: X=y1';Y=y2';Z=y3';W=y4'; Step 3 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X=y1;Y=y2;Z=y3;W=y4; %---------------------------------------------------------------% Fx = leastdx = (dx - dx_approximate); % dxcoeff_init = [xx(1), xx(2), xx(3), xx(4), xx(5)]; % dxcoeff = lsqnonlin(leastdx, dxcoeff_init); function Fx = leastdx(x) Fx = (x(1) + X.*(-x(2) + x(3).*Y + x(4).*Z + x(5).*W))(50521.1066511944 + X.*(-19.4688756513 + 0.0481905675.*Y + 0.0824401719.*Z + 0.1909661775.*W)); % dxcoeff_init is the starting guess dxcoeff_init = [50521.1066511944 -19.4688756513 0.0481905675 0.0824401719 0.1909661775]'; [x,resnorm] = lsqnonlin('leastdx',dxcoeff_init);%Invoke optimizer %The solution of least square dxcoeff = [50437.2381935296 19.4365512459 0.0481114469 0.0823037280 0.1906482006] % Therefore, dxfound = 50437.2381935296 + X.*(19.4365512459 + 0.0481114469.*Y + 0.0823037280.*Z + 0.1906482006.*W); %---------------------------------------------------------------% Fy = leastdy = (dy - dy_approximated); % dycoeff_init = [yy(1), yy(2), yy(3), yy(4), yy(5)]; % dycoeff = lsqnonlin(leastdy, dycoeff_init); function Fy = leastdy(y) Fy = (y(1) + Y.*(-y(2) + y(3).*X + y(4).*Z + y(5).*W))(19.38905883158665 + Y.*(-0.96710027460505 + 0.00002680155452.*X + 0.07612802680479.*Z + 0.00091121672068.*W)); % dycoeff_init is the starting guess dycoeff_init = [19.38905883158665 -0.96710027460505 0.00002680155452 0.07612802680479 0.00091121672068]'; [y,resnorm] = lsqnonlin('leastdy',dycoeff_init);%Invoke optimizer %The solution of least square:

C212 dycoeff = [17.10516754781700 0.66340825723475 -0.00003964533238 0.06008253158577 0.00599884912641]; % Therefore, dyfound = 17.10516754781700 + Y.*(0.66340825723475 0.00003964533238.*X + 0.06008253158577.*Z + 0.00599884912641.*W); %---------------------------------------------------------------% Fz = leastdz = (dz - dz_approximate); % dzcoeff_init = [zz(1), zz(2), zz(3), zz(4), zz(5)]; % dzcoeff = lsqnonlin(leastdz, dzcoeff_init); function Fz = leastdz(z) Fz = (z(1) + Z.*(-z(2) + z(3).*X + z(4).*Y + z(5).*W))(0.08096632578135 + Z.*(-0.03295175682699 + 0.00000028585227.*X +0.00029713572378.*Y + 0.00000647173976.*W)); % dzcoeff_init is the starting guess dzcoeff_init = [0.08096632578135 -0.03295175682699 0.00000028585227 0.00029713572378 0.00000647173976]'; [z,resnorm] = lsqnonlin('leastdz',dzcoeff_init);%Invoke optimizer %The solution of least square: dzcoeff = [0.05050958347340 -0.00604413905119 -0.00000959476644 0.00010602650664 0.00084480523961]; % Therefore, dzfound = 0.05050958347340 + Z.*(-0.00604413905119 0.00000959476644 .*X + 0.00010602650664.*Y + 0.00084480523961.*W); %----------------------------------------------------------------% Fw = leastdw = (dw - dw_approximate); % dwcoeff_init = [ww(1), ww(2), ww(3), ww(4), ww(5)]; % dwcoeff = lsqnonlin(leastdw, dwcoeff_init); function Fw = leastdw(w) Fw =(w(1) + W.*(-w(2) + w(3).*X - w(4).*Y + w(5).*Z))(2.07861918353940 + W.*(-0.14281502297956 + 0.00001868813900.*X + 0.00021807782306.*Y - 0.00896729418813.*Z)); % dwcoeff_init is the starting guess dwcoeff_init = [2.07861918353940 -0.14281502297956 0.00001868813900 0.00021807782306 -0.00896729418813]'; [w,resnorm] = lsqnonlin('leastdw',dwcoeff_init);%Invoke optimizer %The solution of least square dwcoeff = [2.07597310915128 0.14250531775070 0.00001866120246 0.00021562327873 -0.00896049232283]; % Therefore,

C213 dwfound = 2.07597310915128 + W.*(0.14250531775070 + 0.00001866120246.*X - 0.00021562327873.*Y - 0.00896049232283.*Z); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % To simulate the system of international cooperation and % % competition model % % defined by dxfound, dyfound, dzfound and dwfound using Matlab % % ODE45 leads to the following: % % Step 4 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function dy =Intern_Coop_Compet(t,y) dy = zeros(4,1); %A colume vector of initial conditions % % % %

dy(1) dy(2) dy(3) dy(4)

= = = =

dxfound; dyfound; dzfound; dwfound;

% As computed above.

% Solve on a time interval [1 1.25] with an % initial condition vector y0 = [1584.8 10.71 0.9777 0.01] % at time equal to 1975. t0 = 1; tf = 1.25; y0 = [1584.8 10.71 0.9777 0.01]; %options = odeset('RelTol',1e-4,'AbsTol',[1e-4 1e-4 1e-5 1e-4]); [T,Y] = ode45('mideast_Intern_Coop_compet',[t0 tf],y0); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This section graphically depicts the results of step 1 through % % step 4 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% subplot (2,1,1),plot(year,y1) title('U.S.A Gross Domestic Product,1976...1996.'); ylabel('GDP'); xlabel('year'); subplot(2,1,2),plot(year,XXp) title('U.S.A-in Cooperation with Egypt,Jordan and Israel after Regression-'); ylabel('GDP'); xlabel('year'); pause clf subplot(2,1,1),plot(year,dxfound,'--') title('U.S.A--in Cooperation with Egypt,Jordan and Israel after Leastsq--'); ylabel('GDP'); xlabel('year'); subplot (2,1,2),plot(T,Y(:,1)) title('U.S.A--in Cooperation with Egypt,Jordan and Israel after ODE45--'); ylabel('GDP');

C214 xlabel('time, t'); pause clf %---------------------------------------------------------subplot (2,1,1),plot(year,y2) title('Egypt Gross Domestic Product,1976...1996.'); ylabel('GDP'); xlabel('year'); subplot(2,1,2),plot(year,YYp) title('Egypt--in Cooperation with U.S.A,Jordan and Israel after regression--'); ylabel('GDP'); xlabel('year'); pause clf subplot(2,1,1),plot(year,dyfound) title('Egypt--in Cooperation with U.Sl.A,Jordan and Israel after Leastsq--'); ylabel('GDP'); xlabel('year'); subplot (2,1,2),plot(T,Y(:,2)) title('Egypt--in Cooperation with U.S.A,Jordan and Israe after ODE45--'); ylabel('GDP'); xlabel('time, t'); pause clf %----------------------------------------------------------subplot (2,1,1),plot(year,y3) title('Jordan Gross Domestic Product,1976...1996.'); ylabel('GDP'); xlabel('year'); subplot(2,1,2),plot(year,ZZp) title('Jordan --in Cooperation with U.S.A,Egypt and Israel after regression-'); ylabel('GDP'); xlabel('year'); pause clf subplot(2,1,1),plot(year,dzfound,'--') title('Jordan--in Cooperation with U.S.A,Egypt and Israel after Leastsq-'); ylabel('GDP'); xlabel('year'); subplot (2,1,2),plot(T,Y(:,3)) title('Jordan --in Cooperation with U.S.A,Egypt and Israel after ODE45--'); ylabel('GDP'); xlabel('time, t'); pause clf

C215 %-----------------------------------------------------------subplot (2,1,1),plot(year,y4) title('Israel Gross Domestic Product,1976...1996.'); ylabel('GDP'); xlabel('year'); subplot(2,1,2),plot(year,WWp) title('Israel-in Cooperation with U.S.A,Egypt and Jordan after regression-'); ylabel('GDP'); xlabel('year'); pause clf subplot(2,1,1),plot(year,dwfound) title('Israel-in Cooperation with U.S.A,Egypt and Jordan after Leastsq-'); ylabel('GDP'); xlabel('year'); subplot (2,1,2),plot(T,Y(:,4)) title('Israel in Cooperation with U.S.A,Egypt and Jordan after ODE45--'); ylabel('GDP'); xlabel('time, t'); pause clf subplot(2,1,1),plot(T,Y(:,1),T,Y(:,2),'--') title(' U.S.A and Egypt Growth Performance after ODE45--'); ylabel('GDP'); xlabel('time, t'); subplot(2,1,2),plot(T,Y(:,1),T,Y(:,3),'--') title('U.S.A and Jordan Growth Performance after ODE45--'); ylabel('GDP'); xlabel('time, t'); pause clf subplot(2,1,1),plot(T,Y(:,1),T,Y(:,4),'--') title('U.S.A and Israel Growth Performance after ODE45--'); ylabel('GDP'); xlabel('time, t'); subplot(2,1,2),plot(T,Y(:,2),T,Y(:,3),'--') title('Egypt and Jordan Growth Performance after ODE45--'); ylabel('GDP'); xlabel('time, t'); pause clf subplot(2,1,1),plot(T,Y(:,2),T,Y(:,4),'--') title('Egypt and Israel Growth Performance after ODE45--'); ylabel('GDP'); xlabel('time, t'); subplot(2,1,2),plot(T,Y(:,3),T,Y(:,4),'--') title('Jordan and Israel Growth Performance after ODE45--'); ylabel('GDP'); xlabel('time, t');