Economic Growth: A Unified Approach

Cover
Front Matter
Praise for Economic Growth, Second Edition
Title
Copyright
Table of Contents
Foreword by Robert M. Solow
Preface to the second edition
Introduction to the first edition
Part I: Positive growth theory
1 The welfare of society and economic growth
1 Income as a measure of economic activity
1.1 Three approaches to measuring economic activity: a simple example
1.1.1 The expenditure approach
1.1.2 The output (value added) approach
1.1.3 The income approach
1.2 A global view of the three approaches: the input–output table
1.3 From gross domestic product to national income
1.4 National income at current prices and at constant prices
2 Is income per person a fair gauge of society’s welfare?
2.1 Expenditures that should be excluded from GDP
2.1.1 Expenditures decided without individuals’ consent
2.1.2 Expenses decided with individuals’ consent
2.2 Fundamental factors neglected in the calculation of GDP
2.3 Should we, finall, rely on income per person?
3 A major caveat
2 The growth process
1 The growth process: an intuitive approach
2 A more precise approach: a simple model of economic growth
2.1 Hypotheses
2.1.1 Hypothesis regarding the production function
2.1.2 Hypothesis regarding the investment behaviour of society
2.1.3 Growth of population
2.1.4 Technical progress
2.2 The equation of motion of the economy
2.2.1 A fundamental property
2.2.2 Deriving the equation of motion of the economy
2.3 An economic interpretation of the equation of motion
2.4 The speed of convergence toward equilibrium
2.5 Alternate production functions and stability analysis
2.5.1 The Walras–Leontief, or Harrod–Domar case
2.5.2 The “third” case
3 Introducing technical progress
3 Poverty traps
1 Introduction
2 The bare facts
3 Correcting a serious mistake
3.1 Hypotheses
3.2 Stability analysis
4 Escaping poverty traps
4 A production function of central importance
1 Motivation
1.1 Definitionof the elasticity of substitution
1.2 Geometrical representation
1.3 Properties
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
1.3.2 If the production function is homogeneous of degree one, the elasticity of
substitution depends solely upon the capital–labour ratio
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
2 The links between the elasticity of substitution and income distribution
3 Determining the constant elasticity of substitution production function
3.1 Identifying constant α
3.2 Identifying constant β
5 The CES production function as a general mean
1 The concept of the general mean of order p, and its fundamental properties
1.1 Definitio
1.2 Important particular cases
1.3 The fundamental property of general means
2 Applications to the CES production function
3 The qualitative behaviour of the CES function as σ changes
3.1 The opening up of the surface F_σ(K, L) when σ increases
3.2 Income per person as a function of r
3.2.1 The case 0 < σ < 1
3.2.2 The case σ > 1
3.3 Income per person as a function of σ
6 Capital–labour substitution and economic growth
1 Further analytics of the CES function in a growth model
1.1 Simple proofs of theorems 1 and 2, and additional results
1.2 Equilibrium and disequilibrium, and the cases of ever-increasing or decreasing income per person
1.2.1 Introducing the Pitchford constant
1.2.2 The threshold value of the elasticity of substitution generating permanent growth
1.2.3 The case of disequilibrium with ever decreasing income per person
2 The elasticity of substitution at work
2.1 How σ can boost an economy at various stages of its development
2.2 The consequences of an increasing elasticity of substitution on equilibrium income per person
3 Introducing technical progress
3.1 General specificationsof technical progress and results
3.2 Asymptotic growth with labour-augmenting technical progress
3.2.1 Asymptotic growth when σ ≥ 1
3.2.2 Asymptotic growth when σ < 1
3.3 How does a change in the elasticity of substitution compare to a change in technological progress?
4 Time-series and cross-section estimates
5 The broader significanceof the elasticity of substitution in the context of economic growth
7 Why has the elasticity of substitution most often been observed as smaller than 1? And why is it of importance?
1 Introduction
2 The unsustainability of competitive equilibrium with σ > 1
3 A vivid contrast: the sustainability of competitive equilibrium and its associated growth paths with σ < 1
3.1 The optimal evolution of income per person
3.2 The time path of the capital–output ratio
3.3 The optimal savings rate
3.4 The share of labour in total income
8 The long-term growth rate as a random variable, with an application to the US economy
1 From daily to yearly growth rates
2 The first moments of the long-term yearly growth rate
2.1 First method: supposing that the log X(t−1,t) variables are normal N(μ, σ^2) and independent
2.2 Second method: the variables log X(t−1,t) are i.i.d. with mean μ and variance σ^2; no probability distribution is inferred about them
2.3 The expected value and variance of the long-term growth rate in terms of E(Xt−1,t) and VAR(Xt−1,t)
2.4 Determining probabilities for intervals of the n-horizon growth rate
2.5 The convergence toward the geometric mean
3 Application to the long-term growth rates of the US economy
Part II: Optimal growth theory
9 Optimal growth theory: an introduction to the calculus of variations
1 The Euler equation
2 Fundamental properties of the Euler equation
3 Particular cases of the Euler equation
4 Functionals depending on n functions y_1(x), . . . , y_n(x)
5 A necessary and sufficient condition for y(x) to maximize the functional ∫F(x, y, y′)dx
6 End-point and transversality conditions
7 The case of improper integrals ∫F(y, y′, t)dt and transversality conditions at infinity
10 Deriving the central equations of the calculus of variations with a single stroke of the pen
1 A one-line derivation of the Euler equation through economic reasoning: the case of an extremum for ∫F(x, y, y′)dx
2 Extending this reasoning to the derivation of the Ostrogradski equation: the case of an extremum for ∫∫F(x, y, z, ∂z/∂x, ∂z/∂y)dxdy
3 An intuitive derivation of the Beltrami equation
4 End-point and transversality conditions: derivations through direct reasoning
4.1 A reminder of terminal point conditions
4.2 The logic behind those conditions
5 Conclusions
11 Other major tools for optimal growth theory: the Pontryagin maximum principle and the Dorfmanian
1 The maximum principle in its simplest form
2 The relationship between the Pontryagin maximum principle and the calculus of variations
3 An economic derivation of the maximum principle
3.1 First derivation
3.2 A beautiful idea
4 First application: deriving the Euler equation from economic reasoning
5 Further applications: deriving high-order equations of the calculus of variations
5.1 First extension
5.2 Second extension
5.3 Further extensions
12 First applications to optimal growth
1 The mainstream problem of optimal growth: a simplified presentation
2 The calculus of variations approach
2.1 Applying the Euler equation
2.2 Economic derivations of the Ramsey equation
2.2.1 A fist, intuitive, approach with a very short time interval
2.2.2 Infinite horizon
2.2.3 Finite horizon
2.2.4 Alternative derivations in future value
3 The Pontryagin maximum principle approach
3.1 The Hamiltonian approach
3.2 The Dorfmanian approach
4 Optimal paths
4.1 The second-order differential equation in K_t
4.2 The system of first-order differential equations in K_t and C_t and its phase diagram
4.3 A fist experiment with utility functions
13 Optimal growth and the optimal savings rate
1 The central model of optimal growth theory
2 The consequences of using power utility functions
2.1 A close look at optimal growth paths
2.2 The initial value of c_0 leading the economy to the steady state as a function of α
2.3 Questioning the relevance of power utility functions
3 The consequences of using exponential utility functions
Part III: A unified approach
14 Preliminaries: interest rates and capital valuation
1 The reason for the existence of interest rates
2 The various types of interest rates and their fundamental properties
2.1 The forward rate with discrete and continuous compounding
2.2 The continuously compounded spot rate
2.3 Introducing the missing link: the continuously compounded total return
2.4 An economic interpretation of e
2.5 An economic interpretation of log x
2.6 Fundamental properties
3 Applications to the model of economic growth
3.1 The future value of a dollar invested in a capital good
3.2 Evaluating one unit of capital
3.3 Deriving the Fisher equation
3.4 Deriving the value of an asset from the Fisher equation
3.4.1 First solution: identificationof
with
and
with
3.4.2 Second solution: identificationof
with
and
with
15 From arbitrage to equilibrium
1 The case of risk-free transactions
1.1 The case of an undervalued asset
1.1.1 Arbitrageurs
1.1.2 Investors’ behaviour
1.2 The case of an overvalued asset
1.2.1 Arbitrageurs’ actions
1.2.2 Investors’ behaviour
2 Introducing uncertainty and a risk premium
16 Why is traditional optimal growth theory mute? Restoring its rightful voice
1 Overview
2 Three papers that should have rung alarm bells: Ramsey (1928), Goodwin (1961), King and Rebelo (1993)
2.1 Ramsey: the first difficulties
2.1.1 Ramsey’s reaction to his result
2.1.2 A neglected, important question
2.2 The second alarm: Goodwin (1961)
2.2.1 Context and results
2.2.2 Goodwin’s reaction
2.3 The paper that should have been the final alarm: King and Rebelo (1993)
3 The ill-fated role of utility functions
3.1 A fist analysis
3.2 A further examination
4 How the strict concavity of utility functions makes competitive equilibrium unsustainable
4.1 Initial conditions as determined from competitive equilibrium
4.2 Planning with strictly concave utility functions from an initial situation of competitive equilibrium: a disaster in the making
4.3 The incompatibility of the traditional approach and competitive equilibrium: an analytic explanation
5 A suggested solution
5.1 The intertemporal optimality of competitive equilibrium: its multiple facets in one theorem
5.2 The optimal evolution of the economy under competitive equilibrium
5.2.1 The optimal time path of the savings rate
5.2.2 The optimal growth rate of income per person
5.2.3 The optimal time path of the capital–output ratio
5.2.4 The optimal evolution of the labour share in competitive equilibrium
6 The robustness of the optimal savings rate: the normal impact of different scenarios
7 Conclusion
17 Problems in growth: common traits between planned economies and poor countries
1 The consequences of planning
1.1 First possibility: the selling price is fixed at p̂
1.2 Second possibility: the selling price is fixed below p̂
1.2.1 Consequences of rationing
1.2.2 The black market
2 Common traits of centrally planned economies and poor countries
2.1 A closed economy
2.2 A broken growth process; the potential of the economy is not achieved
2.3 Absence of equality of chances
2.4 Stability of the system
18 From Ibn Khaldun to Adam Smith; a proof of their conjecture
1 Ibn Khaldun’s message
1.1 First factor: demographic growth
1.2 Second factor: technical progress
1.3 Third factor: the search for individual profit
1.4 Fourth factor: the principle of private property
1.4.1 The fist transgression: slavery
1.4.2 The second transgression: private and public monopolies
1.4.3 The third transgression: excessive taxation
1.4.4 Consequences
1.5 Fifth factor: soundness of political and legal institutions
1.5.1 The all-important letter from Tâhir b. al-Husayn (821)
1.5.2 Ibn Khaldun’s valuation of the Tâhir b. al-Husayn letter
2 Two illustrations of the message of Ibn Khaldun and Adam Smith
2.1 First illustration: the consequences of suppressing popular communes and (partially) liberalizing markets
2.2 Second illustration: the advantages derived by society from technical progress introduced by firms for their own profit
3 A proof of their conjecture
19 Capital and economic growth in the coming century
1 What are the hypotheses we can agree upon?
2 What can we conclude?
In conclusion: on the convergence of ideas and values through civilizations
Further reading, data on growth and references
Index