Economic Growth: A Unified Approach

COVER......Page 1
HALF-TITLE......Page 3
TITLE......Page 5
COPYRIGHT......Page 6
DEDICATION......Page 7
CONTENTS......Page 9
INTRODUCTION......Page 12
PART I Positive growth theory......Page 17
1 Income as a measure of economic activity......Page 19
1.1.1 The expenditure approach......Page 20
1.2 A global view of the three approaches: the input–output table......Page 21
1.3 From gross domestic product to national income......Page 24
1.4 National income at current prices and at constant prices......Page 25
2.1.1 Expenditures decided without individuals’ consent......Page 28
2.3 Should we, finally, rely on income per person?......Page 29
3 A major caveat......Page 31
1 The law of supply and demand, and the equilibrium price......Page 32
3 Fundamental properties of the equilibrium point......Page 34
4 Consumers’ and producers’ surplus......Page 35
Exercises......Page 37
Answers......Page 39
1 The growth process: an intuitive approach......Page 45
2.1.1.1 Geometric construction of production functions......Page 47
2.1.1.2 Properties of homogeneous functions......Page 49
2.2.1 A fundamental property......Page 54
2.2.2 Deriving the equation of motion of the economy......Page 55
2.3 An economic interpretation of the equation of motion......Page 59
2.4 The speed of convergence toward equilibrium......Page 60
2.5 Alternate production functions and stability analysis......Page 61
2.5.1 The Walras–Leontief, or Harrod–Domar case......Page 64
3 Introducing technical progress......Page 70
Appendix 1: The concept of elasticity, its use in economics, and its applications to growth rates......Page 74
Appendix 2: Solution of the first-order differential equation......Page 79
Exercises......Page 81
Answers......Page 83
1 Motivation......Page 90
1.2 Geometrical representation......Page 92
1.3 Properties......Page 94
1.3.1 If the production function is homogeneous of degree 1, the elasticity of substitution is positive if and only if the marginal product of capital is a decreasing function of capital......Page 95
1.3.2 If the production function is homogeneous of degree one, the elasticity of substitution depends solely upon the capital–labour ratio......Page 96
1.3.3 If the production function is homogeneous of degree one, the elasticity of substitution is equal to the elasticity of income per person with respect to the wage rate......Page 97
2 The links between the elasticity of substitution and income distribution......Page 98
3 Determining the constant elasticity of substitution production function......Page 99
3.1 Identifying constant α......Page 101
Exercises......Page 102
Answers......Page 103
CHAPTER 4 The CES production function as a general mean......Page 106
1.2 Important particular cases......Page 108
2 Applications to the CES production function......Page 110
3.1 The opening up of the surface Fσ (K, L) when σ increases......Page 113
3.2 Income per person as a function of r......Page 119
3.2.2 The case σ > 1......Page 120
3.3 An economic interpretation of the coefficient δ......Page 122
3.4 Income per person as a function of σ......Page 124
Appendix: On general means: an alternative proof of a famous theorem and a conjecture......Page 127
1 An alternative proof of a famous theorem......Page 128
2 A conjecture......Page 129
CHAPTER 5 Capital–labour substitution and economic growth......Page 130
1 Further analytics of the CES function in a growth model......Page 131
1.1 Simple proofs of theorems 1 and 2, and additional results......Page 133
1.2 Equilibrium and disequilibrium, and the cases of ever-increasing or decreasing income per person......Page 134
1.2.1 Introducing the Pitchford constant......Page 135
1.2.2 The threshold value of the elasticity of substitution generating permanent growth......Page 140
1.2.3 The case of disequilibrium with ever decreasing income per person......Page 143
2 The elasticity of substitution at work......Page 145
2.1 How σ can boost an economy at various stages of its development......Page 146
2.2 The consequences of an increasing elasticity of substitution on equilibrium income per person......Page 147
3 Introducing technical progress......Page 149
3.1.2 Theorem 5......Page 150
3.2 Asymptotic growth with labour-augmenting technical progress......Page 151
3.2.1 Asymptotic growth when…......Page 153
3.2.2 Asymptotic growth when σ < 1......Page 157
4 Time-series and cross-section estimates......Page 161
5 The broader significance of the elasticity of substitution in the context of economic growth......Page 168
Appendix: Derivation of the cost function with capital and labour-augmenting progress......Page 170
CHAPTER 6 The long-term growth rate as a random variable, with an application to the US economy......Page 172
1 From daily to yearly growth rates......Page 174
2 The first moments of the long-term yearly growth rate......Page 179
2.1 First method: supposing that the…......Page 180
2.2 Second method: the variables…......Page 181
2.3 The expected value and variance of the long-term growth rate in terms of…......Page 182
2.5 The convergence toward the geometric mean......Page 183
3 Application to the long-term growth rates of the US economy......Page 185
Appendix: A reminder on moment generating functions......Page 186
Exercises......Page 187
Answers......Page 188
PART II Optimal growth theory......Page 191
CHAPTER 7 Optimal growth theory: an introduction to the calculus of variations......Page 193
1 The Euler equation......Page 194
2 Fundamental properties of the Euler equation......Page 199
4 A necessary and sufficient condition for y(x) to maximize the functional b F(x, y, y )dx......Page 200
Appendix 1: Differentiating an integral with respect to a parameter – the Leibniz formula; a geometrical interpretation......Page 203
Appendix 2: Solution of the brachistochrone problem......Page 205
Appendix 3: Necessary and sufficient conditions for the integrand of the functional b a F(x, y, y )dx to be concave in y, y......Page 210
Exercises......Page 211
Answers......Page 212
CHAPTER 8 Other major tools for optimal growth theory: the Pontryagin maximum principle and the Dorfmanian......Page 215
1 The maximum principle in its simplest form......Page 216
2 The relationship between the Pontryagin maximum principle and the calculus of variations......Page 217
3.1 First derivation......Page 218
3.2 A beautiful idea......Page 220
4 First application: deriving the Euler equation from economic reasoning......Page 221
5.1 First extension......Page 222
5.2 Second extension......Page 223
5.3 Further extensions......Page 224
CHAPTER 9 First applications to optimal growth......Page 226
2 The calculus of variations approach......Page 227
2.1 Applying the Euler equation......Page 228
2.2 Economic derivations of the Ramsey equation......Page 229
2.2.2 Infinite horizon......Page 230
2.2.3 Finite horizon......Page 231
3 The Pontryagin maximumprinciple approach......Page 232
3.1 The Hamiltonian approach......Page 233
4.1 The second-order differential equation in Kr......Page 234
4.2 The system of first-order differential equations in Kt and Ct and its phase diagram......Page 235
4.3 A first experiment with utility functions......Page 240
Answers......Page 246
CHAPTER 10 Optimal growth and the optimal savings rate......Page 249
1 The central model of optimal growth theory......Page 250
2 The consequences of using power utility functions......Page 252
2.1 A close look at optimal growth paths......Page 255
2.2 The initial value of c0 leading the economy to the steady state as

a function of α......Page 261
2.3 Questioning the relevance of power utility functions......Page 263
3 The consequences of using exponential utility functions......Page 265
Exercises......Page 272
Answers......Page 273
PART III A unified approach......Page 275
CHAPTER 11 Preliminaries: interest rates and capital valuation......Page 279
1 The reason for the existence of interest rates......Page 280
2 The various types of interest rates and their fundamental properties......Page 282
2.1 The forward rate with discrete and continuous compounding......Page 283
2.2 The continuously compounded spot rate......Page 286
2.3.1 Definition of the continuously compounded total return......Page 289
2.4 An economic interpretation of e......Page 291
2.5 An economic interpretation of log x......Page 293
2.6 Fundamental properties......Page 297
3.1 The future value of a dollar invested in a capital good......Page 298
3.2 Evaluating one unit of capital......Page 299
3.3 Deriving the Fisher equation......Page 300
3.4 Deriving the value of an asset from the Fisher equation......Page 301
3.4.1 First solution: identification of t0 with…......Page 302
Exercises......Page 303
Answers......Page 304
1 The case of risk-free transactions......Page 309
1.1.1 Arbitrageurs......Page 311
1.1.2 Investors’ behaviour......Page 312
1.2.1 Arbitrageurs’ actions......Page 313
2 Introducing uncertainty and a risk premium......Page 314
CHAPTER 13 Optimal savings: a general approach......Page 317
1 Competitive equilibrium and its resulting savings rate......Page 318
2 The optimal savings rate: derivation from optimal growth......Page 319
3 A geometric construction of the optimal savings rate......Page 324
4 Values of the optimal savings rate and its sensitivity to parameter changes......Page 325
5 Estimating the risk premium of the economy......Page 327
7 Consequences for our future......Page 328
7.1 The (in)significance of an infinite time horizon......Page 330
7.3 Is a change in the elasticity of substitution preferable to a change in the rate of technical progress?......Page 332
8 Conclusion......Page 335
Answers......Page 336
CHAPTER 14 Problems in growth: common traits between planned economies and poor countries......Page 339
1.1 First possibility: the selling price is fixed at p......Page 340
1.2.1 Consequences of rationing......Page 343
1.2.2 The black market......Page 344
2.2 A broken growth process; the potential of the economy is not achieved......Page 345
Exercises......Page 346
Answers......Page 347
CHAPTER 15 From Ibn Khaldun to Adam Smith, and a proof of Smith’s conjecture......Page 348
1 Ibn Khaldun’s message......Page 349
1.3 Third factor: the search for individual profit......Page 350
1.4 Fourth factor: the principle of private property......Page 351
1.4.2 The second transgression: private and public monopolies......Page 352
1.4.3 The third transgression: excessive taxation......Page 353
1.5 Fifth factor: soundness of political and legal institutions......Page 354
1.5.1 The all-important letter from T  ahir b. al-Husayn (821)......Page 355
2 Two illustrations of the message of Ibn Khaldun and Adam Smith......Page 356
2.1 First illustration: the consequences of suppressing popular communes and (partially) liberalizing markets......Page 357
2.2 Second illustration: the advantages derived by society from technical progress introduced by firms for their own profit......Page 359
3 A proof of Smith’s conjecture......Page 361
IN CONCLUSION: ON THE CONVERGENCE OF IDEAS AND VALUES THROUGH CIVILIZATIONS......Page 365
FURTHER READING, DATA ON GROWTH AND REFERENCES......Page 367
INDEX......Page 372