## The Dynamic Systems of Basic Economic Growth ModelsTwo central problems in the pure theory of economic growth are analysed in this monograph: 1) the dynamic laws governing the economic growth processes, 2) the kinematic and geometric properties of the set of solutions to the dynamic systems. With allegiance to rigor and the emphasis on the theoretical fundamentals of prototype mathematical growth models, the treatise is written in the theorem-proof style. To keep the exposition orderly and as smooth as possible, the economic analysis has been separated from the purely mathematical issues, and hence the monograph is organized in two books. Regarding the scope and content of the two books, an "Introduction and Over view" has been prepared to offer both motivation and a brief account. The introduc tion is especially designed to give a recapitulation of the mathematical theory and results presented in Book II, which are used as the unifying mathematical framework in the analysis and exposition of the different economic growth models in Book I. Economists would probably prefer to go directly to Book I and proceed by consult ing the mathematical theorems of Book II in confirming the economic theorems in Book I. Thereby, both the independence and interdependence of the economic and mathematical argumentations are respected. |

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### Contents

Basic laws of production | 19 |

12 Homogeneity and the laws of variable proportions | 20 |

13 Technical substitution possibilities | 22 |

14 Factor shares and production elasticities | 25 |

15 Isomarginal isoaverage productivity curves | 26 |

16 Special production functions | 28 |

17 Literature comments | 29 |

Basic OneSector Growth Models | 31 |

104 The geometry of history curves phase curves and trajectories | 176 |

105 Miscellanea on homogeneous differential equations | 184 |

106 The stability properties of ratio solutions | 193 |

107 The stability properties of coordinate solutions | 200 |

108 Synopsis | 208 |

109 Concluding comments | 209 |

Linear and Affine Dynamics in the Plane | 211 |

111 The linear system director function and directrices | 212 |

Classical growth models and homogeneity | 33 |

22 Ratio and coordinate solutions | 34 |

23 Stability of ratio and coordinate solutions | 38 |

24 The geometry of the phase portrait | 39 |

25 Output distribution and factor prices | 40 |

26 CD and CES technologies | 42 |

27 Constant returns to scale | 47 |

28 Directrix solutions steady states and constant ratios | 48 |

29 Classical comparative dynamics | 52 |

Classical growth models and minimal factor rewards | 57 |

32 Decreasing returns to scale and minimal factor rewards | 58 |

33 Constant returns to scale and minimal factor rewards | 63 |

Aggregate endogenous growth models | 67 |

42 Aggregate growth and critical factor productivities | 69 |

Synopsis of endogenous growth models | 73 |

Neoclassical growth models | 75 |

52 The phase portrait and the steadystate path | 77 |

53 Neoclassical growth and special technologies | 80 |

54 Trajectory geometry and kinematics | 86 |

Keynesian growth models | 89 |

61 Harrodian growth models | 90 |

62 Economic aspects and stability issues | 96 |

Basic TwoSector Growth Models | 99 |

Leontief technology and efficient factor utilization | 101 |

72 The dynamic system solutions stability and trajectory geometry | 105 |

73 The economic rationale of the stability conditions | 109 |

74 Endogenous labor supply | 117 |

75 Literature comments | 121 |

Flexible technologies and proportional saving | 123 |

82 The dynamic system solutions stability and trajectory geometry | 136 |

83 The economic rationale of the stability conditions | 140 |

Flexible technologies and classical saving | 142 |

92 Dynamics solutions stability and trajectory geometry | 148 |

93 Comparative twosector dynamics | 152 |

Synopsis of twosector growth models | 155 |

Final comments | 159 |

Basic Dynamic Systems | 161 |

Homogeneous Dynamics in the Plane | 163 |

101 Basic framework assumptions and definitions | 164 |

102 Ratio solutions coordinate solutions and trajectories | 170 |

103 The ratio and coordinate solutions on directrices | 174 |

112 The general ratio solution | 214 |

113 The general coordinate solution | 216 |

114 Ratio and coordinate solutions of triangular systems | 218 |

115 The calculus approach and techniques of integration | 219 |

116 General coordinate solutions and initial value problems | 221 |

117 Initial value problems and coordinate transformations | 224 |

118 Stability of the ratio and coordinate solutions | 227 |

119 The geometry of the phase portrait | 232 |

1110 Parameter space and the hyperconic phase portrait | 235 |

1111 Structural stability and bifurcations | 237 |

1112 Parameter space and kinematic stability properties | 251 |

1113 Classification of linear dynamic systems | 252 |

1114 Affine dynamics in the plane | 254 |

1115 Concluding comments | 258 |

QuasiHomogeneous Dynamics in the Plane | 259 |

122 Quasihomogeneous differential equations | 262 |

123 Quasihomogeneous autonomous dynamic systems | 268 |

124 Concluding comments | 274 |

Discrete Linear Dynamics in the Plane | 276 |

131 Basic framework of the discrete linear system | 278 |

132 Phase plane decomposition and ratios | 279 |

133 The general solution and initial value problems | 282 |

134 Discrete and continuous time solutions | 288 |

135 The phase portrait of discrete linear dynamics | 291 |

136 Final Comment | 303 |

The exponential and logarithmic matrices of regular two by two matrices | 304 |

Growth and LongRun Stability | 311 |

A1 Introduction | 313 |

A2 Stability criteria and degrees of stability | 314 |

A3 Causality and differential equations | 315 |

A4 Stability conditions for autonomous differential equations | 316 |

A5 Rapidity of growth | 320 |

A6 Growth versus stability of autonomous differential equations | 321 |

A7 Economic applications | 322 |

A8 Concluding comments | 328 |

A9 Appendix on the logarithmic derivative | 329 |

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351 | |

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### Common terms and phrases

alternative analysis assumptions asymptotic stability autonomous becomes bounded capital chapter classical comparative configurations consider constant continuous converging coordinate solutions corresponding course curves defined Definition derivative determined differential equations director function director root directrix values discrete dynamic system economic elasticity equilibrium establish existence expressions factor follows formulas geometric given gives global governing functions growth models Hence homogeneous dynamic homogeneous functions immediately imply income increasing initial values integral labor latter Lemma linear linear system logarithmic mathematical matrix Monotone obtain origin output parameter particular paths phase portrait plane positive problems production production functions Proof Proposition quasihomogeneous ratio solutions ratio stability relative stability Remark represent respectively satisfied saving sector solving stability properties stationary strong substitution technologies Theorem theory trajectories two-sector economy types uniqueness variables weak relative zero