The Dynamic Systems of Basic Economic Growth Models

Front Cover
Springer Science & Business Media, Aug 31, 1994 - Mathematics - 356 pages
Two 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
Book 1
333
Book 2
345
Index
351
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