Economic growth a unified approach, second edition

Cambridge University Press978-1-107-11523-1 — Economic Growth 2nd Edition Frontmatter More Information Praise for Economic Growth, Second Edition ‘Olivier de La Grandville has written a

In the second edition of this user-friendly book, Olivier de La Grandville provides

a clear and original introduction to the theory of economic growth, its mechanismsand its challenges The book has been fully updated to incorporate several impor-tant new results and proofs since the irst edition In addition to a progressive treat-ment of dynamic optimization, readers will ind intuitive derivations of all centralequations of the calculus of variations and of optimal control theory It offers anew solution to the fundamental question: How much should a nation save andinvest? La Grandville shows that the optimal savings rule he suggests not onlycorresponds to the maximization of future welfare lows for society, but alsomaximizes the value of society’s activity, as well as the total remuneration oflabour The rule offers a fresh alternative to dire current predictions about anever-increasing capital–output ratio and a decrease of the labour share in nationalincome

Olivier de La Grandville is Senior Professor at Frankfurt University andVisiting Professor in the Management Science and Engineering Department atStanford University He was Professor of Economics at the University of Genevafrom 1978 to 2007 and held visiting positions at the Massachusetts Institute ofTechnology, at École Polytechnique Fédérale de Lausanne, at the University ofNeuchâtel and at the University of Western Australia He is the author of sevenbooks on topics ranging from microeconomics to macroeconomics and inance,and his research work has been published in international journals such as the

American Economic Review and Econometrica.

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Praise for Economic Growth, Second Edition

‘Olivier de La Grandville has written a sparkling, wide-ranging and provocative analysis

of economic growth models The work is marked by a large number of novel speciic analytic results which will be of wide use.’

Kenneth J Arrow, Stanford University, Nobel Laureate

‘This is a very useful book It covers in extensive detail the neoclassical perspective on optimal economic growth, going beyond what is available in the current textbook expo-

sitions Especially noteworthy are the new results in Theorem 16.1 Researchers working

on the topic will greatly beneit from the attention that the book pays to the analytical

foundations of the approach and its numerical exploration of speciications of the main model that often do not get the attention they deserve Indeed, the quantitative analysis is

what makes this book especially useful for fully understanding what the standard model

of capital accumulation really teaches us about economic growth.’

Pietro F Peretto, Duke University

‘What strikes in this book is that the author, when confronted with a dificult problem in economic growth, irst checks for the validity of the standard theoretical solution to such a

problem, and, if he inds it wrong, offers his own solution, which turns out to be the correct

one An example is the new chapter 3 on poverty traps I repeat what I already wrote on

the irst edition: this is an important book that every economist should read.’

Giancarlo Gandolfo, Accademia Nazionale dei Lincei, Rome

‘Now in a new edition, this book combines rigorous analysis with a keen attention to its practical implications This applies – for instance – to the implausibly high level of the saving rate required by standard growth theory, for which the author offers an innovative

solution, and to the diagnosis provided for poverty traps, which also suggests how they

can be escaped.’

Graziella Bertocchi, University of Modena and Reggio Emilia

‘Olivier de La Grandville takes the reader on a fascinating tour through neoclassical growth

theory Idiosyncratic in scope and style, the tour stops at major intellectual sights In tion, the author guides us to new and important places of interest that emanate from his

addi-own research All this is accomplished in a formidable self-contained manner.’

Andreas Irmen, University of Luxembourg

‘Economists need a better understanding of the Euler and Pontryagin dynamic equations,

both from an analytical and a computational point of view They also need a new, sonable solution to the crucial problem of optimal growth They will ind both in this

rea-remarkable book by Olivier de La Grandville.’

Bjarne S Jensen, University of Southern Denmark

incor-Daniela Federici, University of Cassino and Southern Lazio-Italy

‘Olivier de La Grandville continues his profound research on economic growth and opment in the second edition of his fascinating book His argument that a realistic model

devel-of economic growth requires competitive equilibrium warrants widespread recognition in

a graduate courses Economic policy makers should heed his indings on the critical role of the elasticity of substitution between input factors for economic growh and the distribution

of factor incomes.’

E Juerg Weber, University of Western Australia

‘This book is much more than an excellent textbook on growth economics: it examines some fundamental questions in the neoclassical growth theory that have thus far not been fully articulated Olivier de La Grandville’s penetrating discussion on the role of factor substitutability and the relation between positive and normative growth theories are par- ticularly insightful I highly recommend this book for anyone interested in the theories of growth and development.’

Kazuo Mino, Doshisha University and Kyoto University

‘A remarkable and masterfully written text De La Grandville’s approach to growth theory

is insightful and reveals how much more there is to learn from the workhorse neoclassical growth model The new edition incorporates substantive and original new material It is a thought provoking combination of a textbook and original essays Essential material for researchers and graduate students interested in growth and development theory.’

Miguel León-Ledesma, University of Kent

‘Olivier de La Grandville presents a sound and stimulating introduction to modern growth theory His analysis is at once rigorous and intuitive, opening new perspectives along the Solovian growth model tradition His approach to the problem of optimal growth leads to

a deeper understanding of main theoretical results of current growth literature, and also uncovers some of its more serious drawbacks His solution is convincing, always leading to reasonable time paths for the economy This book should be read by all scholars interested

in growth theory.’

Davide Fiaschi, University of Pisa

‘This book is a truly delightful revisiting of the theory of economic growth starting with the very essential foundations of the theory: preferences of consumers and technology in the hands of producers Olivier de La Grandville digs deeper into these foundations compared

to other textbooks, and provides us with an original light on the non-trivial role of several key assumptions inherent in neoclassical growth In particular, the systematic analysis of

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the implications strictly concave utility functions for sustainable growth and underlying

competitive equilibria is a very stimulating contribution La Grandville’s determination to

take the neoclassical model to the data and his incomparable intuitive use of optimal

con-trol theory are other remarkable features of this otherwise highly pedagogical and

infor-mative textbook.’

Raouf Boucekkine, University of Louvain, Center of Center of Operations Research and

Econometrics, and Aix-Marseille University

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Title: Economic growth : a uniied approach / Olivier de La Grandville ; with a

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to our children Diane, Isabelle and Henri,

and to their own children Ferdinand and Théodore, Elọse, Maxime and Margaux

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2 Is income per person a fair gauge of society’s welfare? 12

2 A more precise approach: a simple model of economic growth 28

5 The CES production function as a general mean (in collaboration

1 The concept of the general mean of order p, and its fundamental

3 The qualitative behaviour of the CES function as σ changes 98

6 Capital–labour substitution and economic growth (in collaboration

1 Further analytics of the CES function in a growth model 115

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5 The broader signiicance of the elasticity of substitution in the context of

7 Why has the elasticity of substitution most often been observed as

2 The unsustainability of competitive equilibrium with σ> 1 157

3 A vivid contrast: the sustainability of competitive equilibrium and its

8 The long-term growth rate as a random variable, with an

2 The irst moments of the long-term yearly growth rate 177

3 Application to the long-term growth rates of the US economy 183

9 Optimal growth theory: an introduction to the calculus of

4 Functionals depending on n functions y1(x), , yn (x) 199

5 A necessary and suficient condition for y(x) to maximize the functionalb

7 The case of improper integrals  ∞

0 F (y, y′ ,t )dt and transversality

10 Deriving the central equations of the calculus of variations with a

1 A one-line derivation of the Euler equation through economic reasoning:

the case of an extremum for x n

2 Extending this reasoning to the derivation of the Ostrogradski equation:

the case of an extremum for 

R F (x, y, z,∂z

4 End-point and transversality conditions: derivations through direct

11 Other major tools for optimal growth theory: the Pontryagin

2 The relationship between the Pontryagin maximum principle and the

1 The mainstream problem of optimal growth: a simpliied presentation 244

3 The consequences of using exponential utility functions 281

14 Preliminaries: interest rates and capital valuation 297

2 The various types of interest rates and their fundamental properties 301

16 Why is traditional optimal growth theory mute? Restoring its

2 Three papers that should have rung alarm bells: Ramsey (1928), Goodwin

4 How the strict concavity of utility functions makes competitive

17 Problems in growth: common traits between planned economies

2 Common traits of centrally planned economies and poor countries 388

18 From Ibn Khaldun to Adam Smith; a proof of their conjecture 392

2 Two illustrations of the message of Ibn Khaldun and Adam Smith 400

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In conclusion: on the convergence of ideas and values through

to understand the effects, on inequality, say, of social and economic institutionsand their changes Or, in still other hands, it can serve as a vehicle for an abstractpicture of a whole economy Each of these angles could give rise to a different sort

of book

Olivier de La Grandville’s remarkable text is not any of those It is, of course, acareful and clear exposition of the theory itself, with a transparent proof of every-thing that needs proving Yes, but a reader is struck – at least this reader wasstruck – by the number of tables and graphs that amount to numerical experi-ments They depict in great detail the consequences for the message of the theory

of choosing different values for the main parameters The book is written in thespirit of a master model-builder: it tells you how to do it, and why to do it Andinterestingly, the text is more self-contained than the usual: for example, some cen-tral results in the calculus of variations are derived from economic irst principles.And why not? Economic logic is also logic

This instinct of calculation (I intend the echo of Veblen’s “instinct of manship”) is a useful habit of thought It is always comforting to build a modelfrom irst principles, using assumptions that one learned in school and that havebecome internalized Then the model can just be left to do whatever it does Olivier

work-de La Grandville (and I) think that this is not such a good iwork-dea Sometimes whatseems intuitively inevitable seems that way only because it is customary Tryingout alternative parameter-values is a good way to discover whether a model actu-ally gives sensible results Peculiar results may be a warning that some traditionalassumptions may not be so reasonable, or not any more

The main example of this process in the book is concerned with the standing Ramsey problem: How much should a society save? That, of course,depends on the society’s objective (and on the presumption, famously denied

long-by Margaret Thatcher, that there is such a thing as “society”) The traditionalassumption is that at any time there is a social value of current consumption (or

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consumption per person) and this social value is subject to the principle of

dimin-ishing marginal utility In the jargon, there is a strictly concave social utility

func-tion with consumpfunc-tion at any time as its argument These have to be added up or

otherwise integrated over time, and then optimized This way of thinking is by

now second nature

Alas, the text shows, in considerable detail, that proceeding in this way

always – yes, always – gives answers that are simply unacceptable In lar, the proposed initial saving rate is always ridiculously high There is no doubt

particu-that the calculation is giving us the “correct” answer to the problem as posed The

whole point is that if the answer is ridiculous the problem must be badly posed or

formulated

The author’s solution to this impasse is to abandon the presumption of

dimin-ishing marginal social utility of consumption per person He reformulates theRamsey problem as simply optimizing the present value of the lows of consump-tion itself (or an afine function of it) Then the recommended saving rate turns out

to be reasonable, something one can imagine happening Moreover, the solution

to this reformulated problem turns out to have other desirable features

As the text (chapter 16, section 4) shows in a comprehensive theorem, this

formulation of the Ramsey problem yields a path corresponding to competitiveequilibrium; furthermore this path also maximizes society’s net product and itscompensation of labour both in the short run and in the discounted long run It

is a fresh window on the whole problem of optimal growth Just as interestingly

it points the way to what I think is a major open issue in empirically groundedmacroeconomics: the magnitude and role of monopoly (and other) rents in moderncapitalist economies

To take another example of the constructive, calculational approach, the text

explores meticulously the inluence on growth paths of changes in the elasticity of

substitution between labour and capital, breaking some new ground in the process

Of course the effect of capital accumulation on the return to new investment hasbeen a central topic in economics at least since the days of David Ricardo and JohnStuart Mill It has taken on renewed interest recently as the spectre of increasing

inequality in the distribution of income and wealth has come to public attention.Only part of the needed analytical apparatus appears in this book, but it is an

important part The many calculations add measurably to the force of the analysis.Any economist or other social scientist who wants a working knowledge of

growth theory, and a feel for what matters more and what matters less, will ind

Olivier de La Grandville’s book a useful, faithful and stimulating guide It should

be read with a pad of paper, a pencil, and an eraser in hand I know from ence

experi-Robert M Solow

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PREFAC E TO THE SECOND EDITION

It is hard to imagine a more important and dificult economic challenge than todeine a policy that would shape society’s welfare in an optimal way, both forthe present day and for our future generations This is precisely the challengethat Frank Ramsey took up nearly a century ago when he asked the question:

“How much should a nation save?” In trying to answer it, he founded the theory

of optimal growth

It turns out that neither Ramsey nor subsequent theorists came up with a sonable, convincing answer to that famous question Ramsey’s disappointment ispalpable when the answer he reached – an “optimal” savings rate equal to 60% –was, in his own words “greatly in excess of that which anyone would suggest”,adding that the utility function he used was “put forward merely as an illustra-tion” Subsequent essays either remained in the realm of theory or produced thestrangest results: some authors gave short shrift to excessive savings rates or, whenthey worried about those, they had recourse to utility functions that would hardly

rea-be recognizable by anybody, and even less by a whole society; or they resorted tochanging the values and the very signiicance of the parameters they used Despitesuch bold moves, they could never prevent at least one central variable of theeconomy from taking a time path that was never observed historically or that wassimply unacceptable

Until now, the main problem was to deine a model that would lead to

reason-able trajectories for all central varireason-ables of the economy: not only the savings rate,

but the marginal productivity of capital, the growth rate of income per person orthe capital–output ratio, to name a few

In the irst edition of this book, we started unveiling the reason for the ure of the theory to yield consistent, sensible results: we showed that it hinged onthe systematic use of a strictly concave utility function This second edition willshow that whenever we try to modify the utility function to obtain a more accept-able savings rate, we inevitably induce trajectories for other central variables ofthe economy that either do not it with the capabilities of the economy or areinconceivable

fail-In the new chapter 16 we also demonstrate that the concavity of the utility tions impedes any possibility of a sustained competitive equilibrium; any economy

func-xv

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initially in such equilibrium will always veer off from that situation into unwanted

trajectories if it is governed by the standard model We then propose the following

solution to the problem of optimal growth: optimal trajectories of the economy,and irst and foremost the optimal savings rate, should be determined by the Euler

equation resulting from competitive equilibrium

While traditional theory aims at a single objective, the maximization of the

sum of discounted utility lows, the model we offer here leads to the simultaneous,intertemporal maximization of three magnitudes; indeed, by saving and investingalong lines deined by such an equilibrium, along with minimizing productioncosts society is able to maximize:

r the sum of discounted consumption lows;

r the total value of society’s activity, deined by the sum of consumption and therate of increase in the value of the capital stock;

r last but not least, the total remuneration of labour

We show that for all parameters in the range of observed or predictable values, aswell as for quite different hypotheses regarding the future evolution of population

or technological progress, we are always led to very reasonable time paths for all

central variables of the economy The model we propose is also highly robust to

very different hypotheses regarding the future of societies, in the form of S-shaped

evolutions of population and technological progress

Furthermore, the model brings comforting news: contrary to contemporary

gloomy predictions about the inevitability of an increase in the capital–outputratio and a decrease in the share of labour in national income, we show that if

economic policy can bring a society close to competitive equilibrium, then the

capital–output ratio will decrease while the share of labour will increase

We hope that you will enjoy this second edition and its ive entirely new

chap-ters, not only for its results – surprising, and at the same time reassuring and lenging for our future – but also for the methodological novelties it offers While

chal-the basic principles of chal-the calculus of variations were discussed in a standard,

classical way in the irst edition, in this one you will discover (in the new chapter

10) how the Euler equation, its generalization to multiple integrals as well as the

Beltrami equation and the transversality conditions can be derived intuitively with

a single stroke of the pen, rendering those beautiful but apparently dificult to prehend equations almost evident – in fact you will not even need a pen to explain

com-the Euler equation to your neighbour at com-the ballpark; and you will see: he will be

interested

It is with great pleasure that I express my gratitude to a number of individuals

First I want to acknowledge the invaluable help given to me by my colleagueErnst Hairer, whose mastery at solving numerically differential equations is at

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the origin of the spectacular diagrams in chapters 12, 13 and 16 Without hishelp I would not have been able to put the utility functions to the test of com-petitive equilibrium in chapter 16 And of course I want to heartily thank RobertSolow, the co-author of two chapters of the irst edition – now chapters 5 and

6 – who was also kind enough to write a foreword for this second edition I

am also indebted to Kenneth Arrow, Michael Binder, Giuseppe De lis, Giovanni Di Bartolomeo, Maria Dimakopoulou, Robert Chirinko, DanielaFederici, Robert Feicht, Giancarlo Gandolfo, Jean-Marie Grether, Erich Gund-lach, Andreas Irmen, Bjarne Jensen, Mathias Jonsson, Rainer Klump, AnastasiaLitina, Miguel León-Ledesma, Henri Loubergé, Peter McAdam, Bernardo Maggi,Scott Murff, Srinivasan Muthukrishnan, Elisabeth Paté-Cornell, Enrico Saltari,Wolfgang Stummer, Jim Sweeney, Richard Waswo, Juerg Weber and Milad Zarin-Nejadan, as well as to participants in seminars at Stanford, Frankfurt, Luxembourgand Rome (La Sapienza) for their highly constructive remarks My warmest thanks

Archange-go also to Mary Catherine Bongiovi, our very eficient production editor, to nis Starling, who copy-edited this new edition with remarkable acumen and superbskill, and to William Jack who prepared a detailed index

Glen-Olivier de La Grandville

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INTRODUCTION TO THE FIRST E DITION

Why should you read a book on economic growth? Because the subject is tant: it is about the well-being of our societies today and in the future; and

impor-because it is beautiful It carries wonderful ideas, some exposed more than 2000

years ago, spanning all civilizations You will certainly marvel at Ibn Khaldun’sprescience, at Mo Tzu’s wisdom, at Solow’s depiction of transition phases, at

Dorfman’s incredible intuition in solving variational problems

This book is not quite the same as other books Economic growth has attracted,

particularly in the last hundred years, countless, excellent writers who have oped the ield into an immense array of topics, from theoretical to empirical.Rather than trying to cover all developments – of which you can have an idea

devel-through the bibliography – I have wanted to tell you what I found fascinating in

the subject But my hope is also that you will ind here a useful introduction tothis wide area of research, because a lot of the book is not only on ideas but on

methodology as well

A further reason for me to write this text was to submit personal views and

present new results For too many years I have expounded growth theory by

divid-ing the subject, as many did, into two main strands of thought: positive, or

descrip-tive theory on the one hand, normadescrip-tive on the other I am now convinced that thosetwo strands should be uniied – hence the title of the book For clarity’s sake, I

think however that both approaches to the theory should be irst presented arately (parts I and II), and then uniied (part III) Such uniication proceeds not

sep-from any personal whim, but sep-from logical reasons: the results of both strands of

thought mutually imply each other, as will be shown

Let me now underline the new results you will ind in this book:

rA proof of one of the most important, daring conjectures ever made in economics

or social sciences We owe this conjecture, known as “the invisible hand”, to Adam

Smith who wrote, in his Inquiry into the nature and the causes of the wealth of

nations(1776):

Every individual is continually exerting himself to ind out the most

advan-tageous employment for whatever capital he can command It is his own

advantage, indeed, and not that of the society, which he has in view But

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the study of his own advantage naturally, or rather necessarily, leads him toprefer that employment which is most advantageous to the society

He generally, indeed, neither intends to promote the public interest, norknows how much he is promoting it He intends only his own gain, and

he is in this, as in many other cases, led by an invisible hand to promote anend which was no part of his intention.1

Note how far reaching Adam Smith’s conjecture is Not only does the author holdthat the search by individuals for the most advantageous employment of their cap-ital stock is advantageous to society; he takes one step further by stating that thisadvantage is maximized

What is then the exact beneit that Smith could have referred to in his ture? How is it to be measured? The answer is quite surprising It will be shown

conjec-that it is not only one, but two magnitudes conjec-that are simultaneously maximized for

society:

(1) the sum of the discounted consumption lows society can acquire from now

to ininity

(2) the beneits of society’s activity at any point of time t – including today;

those beneits are the sum of the consumption lows received at time t and the rate

of increase in the value of the capital stock at that time

The proof of this theorem, which you will ind in the last chapter, uses themethodology and results expounded throughout this text, and draws both on posi-tive and normative theory If only for this very proof, there would be ample reason

to justify the uniied approach I am advocating

r A thorough analysis of the importance of the elasticity of substitution in thegrowth process Too often do we see growth models carrying the convenient,beloved hypothesis of an elasticity of substitution equal to 1, equivalent to theCobb–Douglas function We now have evidence, however, that in any economy

we might consider the elasticity of substitution is signiicantly different from 1,with a tendency of growing – which is good news, as the reader will discover Asurprise is in store: an increase in the elasticity of substitution will be shown tohave far more importance for society’s future welfare than a similar increase inthe rate of technical progress

rA detailed examination of the consequences of using utility functions in mal growth theory Traditional treatment of the theory usually leads to a system ofdifferential equations which does not possess analytical solutions In my opinion,this issue, involving solving numerically such a system, has been taken too lightly,

opti-1 Adam Smith (1776), An inquiry into the nature and causes of the wealth of nations, 1975 edition

by Dent & Sons, pp 398–400.

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and short shrift has been given to results taking the form of exceedingly high

“optimal” savings rates It can even be read in the literature that some family ofutility functions can be used, although it turns out that no equilibrium point exists.Great care will be taken in analysing the optimal time-paths of the economy, both

in terms of their associated initial values and their ultimate evolution

rA formula for the optimal savings rate of an economy Until now optimal savings

rates could be calculated numerically only – no closed form was available and, as

mentioned, those values often made little sense The formula I submit is expressed

in terms of the fundamental characteristics of the economy and society’s rate ofpreference for the present, and yields reasonable, very reachable values

rApplications and extensions of Dorfman’s modiied Hamiltonian With able insight, Robert Dorfman had introduced a new Hamiltonian to tackle thevariational problems encountered in optimal growth theory To honour Profes-

remark-sor Dorfman’s memory, I propose to call his concept a Dorfmanian It will play afundamental role in the proof of Smith’s conjecture The reader will also ind hereextensions of the Dorfmanian which can yield all high-order equations of the cal-culus of variations – including the Euler–Poisson and the Ostrogradski equations

rThe inal reason why you should read this book is that Robert Solow and myselfneed your help: you will be invited to exercise your sagacity and try to prove aconjecture we are offering at the end of chapter 3 The conjecture, of a mathe-matical nature, is as formidable a challenge to prove as it is easy to express: thegeneral mean of two numbers, considered as a function of its order, has one andonly one inlection point Why is it important? Because as a result, income per

person behaves exactly like a function of production whose dependent variable isthe elasticity of substitution, with a irst phase of increasing returns, followed by

decreasing returns, and our economies seem to be in the very neighbourhood ofthis point of inlection

It is my pleasure to thank a number of persons whose role has been essential in

the realization of this book First and foremost, I would like to express my deepestgratitude to Robert Solow, who not only co-authored a large chapter (chapter 6)and the appendix of chapter 5, but also gave me invaluable advice on many other

important parts of the book Needless to say, I alone remain responsible for anyremaining shortcomings, and the personal views expressed here are not necessarily

condoned by him

My colleague Ernst Hairer, of the Department of Mathematics at the University

of Geneva, has used his program DOPRI for solving numerically the differential

equations of chapters 12, 13 and 16; the stunning phase diagrams 12, 13 and 16 are

his work Ernst Hairer’s generosity led him also to write the program yielding the

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initial values leading to equilibrium for any system of parameters characterizingthe economy I am in great debt to him

Claudio Sfreddo let us use for our regressions in chapter 8 the data he haddeveloped on a common basis for 16 OECD countries in his PhD thesis; we arevery grateful to him

My thanks go inally to those persons who read parts of the manuscript andoffered corrections or very useful remarks I would like to thank in particularKenneth Arrow, Eunyi Chung, Jean-Marie Grether, Bjarne Sloth Jensen, MingyunJoo, Rainer Klump, Patrick de Laubier, Hing-Man Leung, Edmond Malinvaud,Amin Nikoozadeh, Mario Piacentini, Mathias Thoenig, Brigitte Van Baalen, JuergWeber, and Milad Zarin-Nejadan Jon Bilam for Cambridge University Press did amarvellous job in revising the whole text, and I am extremely grateful to him DaveTyler prepared the index in a masterly way; my warmest thanks to him I wouldalso like to express my deep appreciation to Daniel Dunlavey, senior productioneditor at Cambridge University Press, who supervised the whole project with greatexpertise Last but not least, I want to thank heartily Huong Nguyen for her beau-tiful, dedicated work at typing my manuscript It was a joy working with her

F O R E WO R D

by Robert M Solow

There are several different directions from which one can approach the theory ofeconomic growth, even with a fairly strict definition of “theory” The theory canhelp to account for particular historical episodes: Europe after the War, China afterthe reforms, for example More particularly, the theory may help to codify, thoughnot explain, the story of technological progress Alternatively it can be used to try

to understand the effects, on inequality, say, of social and economic institutionsand their changes Or, in still other hands, it can serve as a vehicle for an abstractpicture of a whole economy Each of these angles could give rise to a different sort

of book

Olivier de La Grandville’s remarkable text is not any of those It is, of course, acareful and clear exposition of the theory itself, with a transparent proof of every-thing that needs proving Yes, but a reader is struck – at least this reader wasstruck – by the number of tables and graphs that amount to numerical experi-ments They depict in great detail the consequences for the message of the theory

of choosing different values for the main parameters The book is written in thespirit of a master model-builder: it tells you how to do it, and why to do it Andinterestingly, the text is more self-contained than the usual: for example, some cen-tral results in the calculus of variations are derived from economic first principles.And why not? Economic logic is also logic

This instinct of calculation (I intend the echo of Veblen’s “instinct of manship”) is a useful habit of thought It is always comforting to build a modelfrom first principles, using assumptions that one learned in school and that havebecome internalized Then the model can just be left to do whatever it does Olivier

work-de La Grandville (and I) think that this is not such a good iwork-dea Sometimes whatseems intuitively inevitable seems that way only because it is customary Tryingout alternative parameter-values is a good way to discover whether a model actu-ally gives sensible results Peculiar results may be a warning that some traditionalassumptions may not be so reasonable, or not any more

The main example of this process in the book is concerned with the standing Ramsey problem: How much should a society save? That, of course,depends on the society’s objective (and on the presumption, famously denied

long-by Margaret Thatcher, that there is such a thing as “society”) The traditionalassumption is that at any time there is a social value of current consumption (or

consumption per person) and this social value is subject to the principle of ishing marginal utility In the jargon, there is a strictly concave social utility func-tion with consumption at any time as its argument These have to be added up orotherwise integrated over time, and then optimized This way of thinking is bynow second nature.

dimin-Alas, the text shows, in considerable detail, that proceeding in this wayalways – yes, always – gives answers that are simply unacceptable In particu-lar, the proposed initial saving rate is always ridiculously high There is no doubtthat the calculation is giving us the “correct” answer to the problem as posed Thewhole point is that if the answer is ridiculous the problem must be badly posed orformulated

The author’s solution to this impasse is to abandon the presumption of ishing marginal social utility of consumption per person He reformulates theRamsey problem as simply optimizing the present value of the flows of consump-tion itself (or an affine function of it) Then the recommended saving rate turns out

dimin-to be reasonable, something one can imagine happening Moreover, the solution

to this reformulated problem turns out to have other desirable features

As the text (chapter 16, section 4) shows in a comprehensive theorem, thisformulation of the Ramsey problem yields a path corresponding to competitiveequilibrium; furthermore this path also maximizes society’s net product and itscompensation of labour both in the short run and in the discounted long run It

is a fresh window on the whole problem of optimal growth Just as interestingly

it points the way to what I think is a major open issue in empirically groundedmacroeconomics: the magnitude and role of monopoly (and other) rents in moderncapitalist economies

To take another example of the constructive, calculational approach, the textexplores meticulously the influence on growth paths of changes in the elasticity ofsubstitution between labour and capital, breaking some new ground in the process

Of course the effect of capital accumulation on the return to new investment hasbeen a central topic in economics at least since the days of David Ricardo and JohnStuart Mill It has taken on renewed interest recently as the spectre of increasinginequality in the distribution of income and wealth has come to public attention.Only part of the needed analytical apparatus appears in this book, but it is animportant part The many calculations add measurably to the force of the analysis.Any economist or other social scientist who wants a working knowledge ofgrowth theory, and a feel for what matters more and what matters less, will findOlivier de La Grandville’s book a useful, faithful and stimulating guide It should

be read with a pad of paper, a pencil, and an eraser in hand I know from ence

experi-Robert M Solow

P R E FAC E TO T H E S E C O N D E D I T I O N

It is hard to imagine a more important and difficult economic challenge than todefine a policy that would shape society’s welfare in an optimal way, both forthe present day and for our future generations This is precisely the challengethat Frank Ramsey took up nearly a century ago when he asked the question:

“How much should a nation save?” In trying to answer it, he founded the theory

of optimal growth

It turns out that neither Ramsey nor subsequent theorists came up with a sonable, convincing answer to that famous question Ramsey’s disappointment ispalpable when the answer he reached – an “optimal” savings rate equal to 60% –was, in his own words “greatly in excess of that which anyone would suggest”,adding that the utility function he used was “put forward merely as an illustra-tion” Subsequent essays either remained in the realm of theory or produced thestrangest results: some authors gave short shrift to excessive savings rates or, whenthey worried about those, they had recourse to utility functions that would hardly

rea-be recognizable by anybody, and even less by a whole society; or they resorted tochanging the values and the very significance of the parameters they used Despitesuch bold moves, they could never prevent at least one central variable of theeconomy from taking a time path that was never observed historically or that wassimply unacceptable

Until now, the main problem was to define a model that would lead to

reason-able trajectories for all central varireason-ables of the economy: not only the savings rate,

but the marginal productivity of capital, the growth rate of income per person orthe capital–output ratio, to name a few

In the first edition of this book, we started unveiling the reason for the ure of the theory to yield consistent, sensible results: we showed that it hinged onthe systematic use of a strictly concave utility function This second edition willshow that whenever we try to modify the utility function to obtain a more accept-able savings rate, we inevitably induce trajectories for other central variables ofthe economy that either do not fit with the capabilities of the economy or areinconceivable

fail-In the new chapter 16 we also demonstrate that the concavity of the utility tions impedes any possibility of a sustained competitive equilibrium; any economy

func-initially in such equilibrium will always veer off from that situation into unwanted

trajectories if it is governed by the standard model We then propose the followingsolution to the problem of optimal growth: optimal trajectories of the economy,and first and foremost the optimal savings rate, should be determined by the Eulerequation resulting from competitive equilibrium

While traditional theory aims at a single objective, the maximization of thesum of discounted utility flows, the model we offer here leads to the simultaneous,intertemporal maximization of three magnitudes; indeed, by saving and investingalong lines defined by such an equilibrium, along with minimizing productioncosts society is able to maximize:

r the sum of discounted consumption flows;

r the total value of society’s activity, defined by the sum of consumption and therate of increase in the value of the capital stock;

r last but not least, the total remuneration of labour.

We show that for all parameters in the range of observed or predictable values, aswell as for quite different hypotheses regarding the future evolution of population

or technological progress, we are always led to very reasonable time paths for allcentral variables of the economy The model we propose is also highly robust to

very different hypotheses regarding the future of societies, in the form of S-shaped

evolutions of population and technological progress

Furthermore, the model brings comforting news: contrary to contemporarygloomy predictions about the inevitability of an increase in the capital–outputratio and a decrease in the share of labour in national income, we show that ifeconomic policy can bring a society close to competitive equilibrium, then thecapital–output ratio will decrease while the share of labour will increase

We hope that you will enjoy this second edition and its five entirely new ters, not only for its results – surprising, and at the same time reassuring and chal-lenging for our future – but also for the methodological novelties it offers Whilethe basic principles of the calculus of variations were discussed in a standard,classical way in the first edition, in this one you will discover (in the new chapter10) how the Euler equation, its generalization to multiple integrals as well as theBeltrami equation and the transversality conditions can be derived intuitively with

chap-a single stroke of the pen, rendering those bechap-autiful but chap-appchap-arently difficult to prehend equations almost evident – in fact you will not even need a pen to explainthe Euler equation to your neighbour at the ballpark; and you will see: he will beinterested

com-It is with great pleasure that I express my gratitude to a number of individuals.First I want to acknowledge the invaluable help given to me by my colleagueErnst Hairer, whose mastery at solving numerically differential equations is at

Preface to the second edition xvii

the origin of the spectacular diagrams in chapters 12, 13 and 16 Without hishelp I would not have been able to put the utility functions to the test of com-petitive equilibrium in chapter 16 And of course I want to heartily thank RobertSolow, the co-author of two chapters of the first edition – now chapters 5 and

6 – who was also kind enough to write a foreword for this second edition I

am also indebted to Kenneth Arrow, Michael Binder, Giuseppe De lis, Giovanni Di Bartolomeo, Maria Dimakopoulou, Robert Chirinko, DanielaFederici, Robert Feicht, Giancarlo Gandolfo, Jean-Marie Grether, Erich Gund-lach, Andreas Irmen, Bjarne Jensen, Mathias Jonsson, Rainer Klump, AnastasiaLitina, Miguel León-Ledesma, Henri Loubergé, Peter McAdam, Bernardo Maggi,Scott Murff, Srinivasan Muthukrishnan, Elisabeth Paté-Cornell, Enrico Saltari,Wolfgang Stummer, Jim Sweeney, Richard Waswo, Juerg Weber and Milad Zarin-Nejadan, as well as to participants in seminars at Stanford, Frankfurt, Luxembourgand Rome (La Sapienza) for their highly constructive remarks My warmest thanks

Archange-go also to Mary Catherine Bongiovi, our very efficient production editor, to nis Starling, who copy-edited this new edition with remarkable acumen and superbskill, and to William Jack who prepared a detailed index

Glen-Olivier de La Grandville

Why should you read a book on economic growth? Because the subject is tant: it is about the well-being of our societies today and in the future; andbecause it is beautiful It carries wonderful ideas, some exposed more than 2000years ago, spanning all civilizations You will certainly marvel at Ibn Khaldun’sprescience, at Mo Tzu’s wisdom, at Solow’s depiction of transition phases, atDorfman’s incredible intuition in solving variational problems.

impor-This book is not quite the same as other books Economic growth has attracted,particularly in the last hundred years, countless, excellent writers who have devel-oped the field into an immense array of topics, from theoretical to empirical.Rather than trying to cover all developments – of which you can have an ideathrough the bibliography – I have wanted to tell you what I found fascinating inthe subject But my hope is also that you will find here a useful introduction tothis wide area of research, because a lot of the book is not only on ideas but onmethodology as well

A further reason for me to write this text was to submit personal views andpresent new results For too many years I have expounded growth theory by divid-ing the subject, as many did, into two main strands of thought: positive, or descrip-tive theory on the one hand, normative on the other I am now convinced that thosetwo strands should be unified – hence the title of the book For clarity’s sake, Ithink however that both approaches to the theory should be first presented sep-arately (parts I and II), and then unified (part III) Such unification proceeds notfrom any personal whim, but from logical reasons: the results of both strands ofthought mutually imply each other, as will be shown

Let me now underline the new results you will find in this book:

rA proof of one of the most important, daring conjectures ever made in economics

or social sciences We owe this conjecture, known as “the invisible hand”, to Adam

Smith who wrote, in his Inquiry into the nature and the causes of the wealth of

nations (1776):

Every individual is continually exerting himself to find out the most tageous employment for whatever capital he can command It is his ownadvantage, indeed, and not that of the society, which he has in view But

advan-xviii

Introduction to the first edition xix

the study of his own advantage naturally, or rather necessarily, leads him toprefer that employment which is most advantageous to the society

He generally, indeed, neither intends to promote the public interest, norknows how much he is promoting it He intends only his own gain, and

he is in this, as in many other cases, led by an invisible hand to promote anend which was no part of his intention.1

Note how far reaching Adam Smith’s conjecture is Not only does the author holdthat the search by individuals for the most advantageous employment of their cap-ital stock is advantageous to society; he takes one step further by stating that thisadvantage is maximized

What is then the exact benefit that Smith could have referred to in his ture? How is it to be measured? The answer is quite surprising It will be shown

conjec-that it is not only one, but two magnitudes conjec-that are simultaneously maximized for

society:

(1) the sum of the discounted consumption flows society can acquire from now

to infinity

(2) the benefits of society’s activity at any point of time t – including today;

those benefits are the sum of the consumption flows received at time t and the rate

of increase in the value of the capital stock at that time.

The proof of this theorem, which you will find in the last chapter, uses themethodology and results expounded throughout this text, and draws both on posi-tive and normative theory If only for this very proof, there would be ample reason

to justify the unified approach I am advocating

r A thorough analysis of the importance of the elasticity of substitution in thegrowth process Too often do we see growth models carrying the convenient,beloved hypothesis of an elasticity of substitution equal to 1, equivalent to theCobb–Douglas function We now have evidence, however, that in any economy

we might consider the elasticity of substitution is significantly different from 1,with a tendency of growing – which is good news, as the reader will discover Asurprise is in store: an increase in the elasticity of substitution will be shown tohave far more importance for society’s future welfare than a similar increase inthe rate of technical progress

r A detailed examination of the consequences of using utility functions in mal growth theory Traditional treatment of the theory usually leads to a system ofdifferential equations which does not possess analytical solutions In my opinion,this issue, involving solving numerically such a system, has been taken too lightly,

opti-1 Adam Smith (1776), An inquiry into the nature and causes of the wealth of nations, 1975 edition

by Dent & Sons, pp 398–400.

and short shrift has been given to results taking the form of exceedingly high

“optimal” savings rates It can even be read in the literature that some family ofutility functions can be used, although it turns out that no equilibrium point exists.Great care will be taken in analysing the optimal time-paths of the economy, both

in terms of their associated initial values and their ultimate evolution

rA formula for the optimal savings rate of an economy Until now optimal savingsrates could be calculated numerically only – no closed form was available and, asmentioned, those values often made little sense The formula I submit is expressed

in terms of the fundamental characteristics of the economy and society’s rate ofpreference for the present, and yields reasonable, very reachable values

rApplications and extensions of Dorfman’s modified Hamiltonian With able insight, Robert Dorfman had introduced a new Hamiltonian to tackle thevariational problems encountered in optimal growth theory To honour Profes-sor Dorfman’s memory, I propose to call his concept a Dorfmanian It will play afundamental role in the proof of Smith’s conjecture The reader will also find hereextensions of the Dorfmanian which can yield all high-order equations of the cal-culus of variations – including the Euler–Poisson and the Ostrogradski equations

remark-rThe final reason why you should read this book is that Robert Solow and myselfneed your help: you will be invited to exercise your sagacity and try to prove aconjecture we are offering at the end of chapter 3 The conjecture, of a mathe-matical nature, is as formidable a challenge to prove as it is easy to express: thegeneral mean of two numbers, considered as a function of its order, has one andonly one inflection point Why is it important? Because as a result, income perperson behaves exactly like a function of production whose dependent variable isthe elasticity of substitution, with a first phase of increasing returns, followed bydecreasing returns, and our economies seem to be in the very neighbourhood ofthis point of inflection

It is my pleasure to thank a number of persons whose role has been essential inthe realization of this book First and foremost, I would like to express my deepestgratitude to Robert Solow, who not only co-authored a large chapter (chapter 6)and the appendix of chapter 5, but also gave me invaluable advice on many otherimportant parts of the book Needless to say, I alone remain responsible for anyremaining shortcomings, and the personal views expressed here are not necessarilycondoned by him

My colleague Ernst Hairer, of the Department of Mathematics at the University

of Geneva, has used his program DOPRI for solving numerically the differentialequations of chapters 12, 13 and 16; the stunning phase diagrams 12, 13 and 16 arehis work Ernst Hairer’s generosity led him also to write the program yielding the

Introduction to the first edition xxi

initial values leading to equilibrium for any system of parameters characterizingthe economy I am in great debt to him

Claudio Sfreddo let us use for our regressions in chapter 8 the data he haddeveloped on a common basis for 16 OECD countries in his PhD thesis; we arevery grateful to him

My thanks go finally to those persons who read parts of the manuscript andoffered corrections or very useful remarks I would like to thank in particularKenneth Arrow, Eunyi Chung, Jean-Marie Grether, Bjarne Sloth Jensen, MingyunJoo, Rainer Klump, Patrick de Laubier, Hing-Man Leung, Edmond Malinvaud,Amin Nikoozadeh, Mario Piacentini, Mathias Thoenig, Brigitte Van Baalen, JuergWeber, and Milad Zarin-Nejadan Jon Bilam for Cambridge University Press did amarvellous job in revising the whole text, and I am extremely grateful to him DaveTyler prepared the index in a masterly way; my warmest thanks to him I wouldalso like to express my deep appreciation to Daniel Dunlavey, senior productioneditor at Cambridge University Press, who supervised the whole project with greatexpertise Last but not least, I want to thank heartily Huong Nguyen for her beau-tiful, dedicated work at typing my manuscript It was a joy working with her

P A R T I

Positive growth theory

Economics was born when some people asked the questions: How can societyimprove its living standards? What are the conditions of prosperity? Those ques-tions define the essence of growth theory It is no wonder that in the Westernworld they were raised – rather timidly – at the end of the Renaissance only, byBotero (1589, 1598), and then by Sully (1639) and Child (1668) Why not earlier?Because economics, as any science, is founded on observation; no sign of growthhad ever appeared in the Middle Ages – and how could it have after the end-less wars and plagues that ravaged Europe, particularly in the fourteenth century?Those times carried no hope, and it would have taken a bold thinker to entreat thequestion of development As the French historian Pierre Gaxotte very well put it,

“the man of the Middle Ages does not know of time and numbers”

It is not surprising therefore that we owe to Arab civilization the first

com-prehensive description of the fundamental causes of growth In his Muqaddimah

(1377), the Arab historian Ibn Khaldun went even further and set himself toexplain the causes of the rise and decline of civilizations This is a good place

to recall what Arnold Toynbee (1934, p 321) had to say about Ibn Khaldun’smagnum opus:

He appears to have been inspired by no predecessors, and to have found nokindred souls among his contemporaries, and to have kindled no answeringspark of inspiration in any successors; and yet, in the Prolegomena (Muqad-dimah) to his Universal History he has conceived and formulated a philos-ophy of history which is undoubtedly the greatest work of its kind that hasever yet been created by any mind in any time or place

Ibn Khaldun’s contribution encompasses not only the description of the growthprocess, but also what needs to be done to set it in motion His essay therefore isnot only positive or descriptive; it is normative as well, and it shares that charac-

teristic with Adam Smith’s “Inquiry into the Nature and the Causes of the Wealth

of Nations” that came four centuries later In consequence, it is only fitting that

these works will be presented in the final part of this book where the positive andthe normative approaches are brought together.

In this first part, we will be content to describe the growth process; our first aimwill be to analyse the mechanisms which make income per person grow from oneperiod to another In particular, we will want to answer the following question:from what we know about the growth process, is income per person bound by anupper limit?

To answer this question, we will build a simple, dynamic model of the omy The motion of income per person – the central variable of growth theory –will be shown to be governed by a differential equation whose solution dependsessentially on three sets of hypotheses The first one summarizes the way capi-tal, labour and technical progress define a precise production process The sec-ond translates an important facet of society’s behaviour towards its future, i.e theamount it is willing to save and invest in order to enhance its capital stock A finalset of hypotheses will be made regarding the evolution of population and its labourforce

econ-At the end of part I, we will deal with properties of random growth rates Inparticular, we will want to determine the expected long-term growth rate of aneconomy as a function of the first two moments of the annual growth rate

C H A P T E R 1

The welfare of society and

economic growth

When I have pluck’d the rose,

I cannot give it vital growth again.

Othello

In the history of mankind, attempts to improve living conditions have only veryrecently superseded the struggle for survival In all civilizations, progress has beenexceedingly slow, with abrupt, unexpected downfalls These were concomitant tonatural disasters, epidemics and wars Today, we can estimate that only one fifth ofthe world population enjoys a standard of life that can be considered acceptable.The yardstick commonly used to measure standards of living is “incomeper person” We first show how income reflects the result of economic activity(section 1).1 We then discuss whether income per person constitutes a propergauge for the measurement of society’s welfare (sections 2 and 3)

Fundamentally, nations can benefit, in the long run, only from what they havebeen able to produce In turn, the amount produced within a given time span (forinstance one year) can be measured from at least three perspectives

First, we can consider the types of produced goods and services; these canbroadly be distributed between consumption goods or services on the one hand,and investment goods on the other Consumption goods and services are producedfor the direct use of consumers Investment goods (machines, factories, trans-portation infrastructure, etc.) are produced in order to provide ultimately, at somelater date, consumption goods or services Adding consumption to investmentwould measure a nation’s total activity if the nation had no relations with the

1 This first section is intended for the reader who has had no introduction to national accounts.

outside world If this is not so, two major corrections should be introduced: first,one must add to consumption and investment all exports of goods and services;second, all imports should be deducted: indeed, consumption, investment andeven exports include imports of goods and services, and imports are no part of anation’s activity.

From a second perspective, we may consider the output generated by thenation’s various sectors of activity We then want to determine how much agri-culture or industry, or services have contributed to the nation’s global activity

This can be measured by the net production of each sector Net production is the

sector’s total production net of all purchases from other sectors Indeed we want

to avoid counting twice, or a multiple number of times, the same output Considertwo sectors: the automobile and the aluminium industries The automobile indus-try uses, among many other things, aluminium Simply adding up the production

of each sector would amount to counting twice the value of the aluminium The

net production or net output is appropriately called the value added of the sector.

A third and final approach is to consider the income generated and distributedthroughout the economy Clearly, this income can be generated only by the valueadded of each sector of the economy Three broad categories can be distinguished:labour income, capital income, and profits Labour income is self-explanatory.Capital income is the remuneration of the (physical) capital stock that has beenused in the production process, plus rentals by private individuals Suppose that, in

a given firm, this capital has been rented: the rentals are the capital remuneration

If a firm owns its capital stock, two possibilities arise: first, the firm may haveborrowed in order to buy the capital it owns The remuneration of capital is thenthe interest payments it pays If the firm has financed the capital with its ownresources, the remuneration of capital is the interest payments it has forgone byacquiring the capital stock instead of lending out its resources

We will now summarize in an example each of these three approaches to themeasure of economic activity within the boundaries of a given country The global

result is called the gross domestic product The adjective “domestic” refers to the

fact that the activity is measured within the boundaries of the country considered

The adjective “gross” reflects the fact that investment expenditures include

amor-tization – or depreciation – of capital We will explain in section 2 how to obtainother aggregates also commonly referred to, such as the gross national productand national income Before that we will illustrate with a numerical example thethree approaches to measuring the gross domestic product

1.1.1 The expenditure approach

This is the most natural and arguably the most useful approach from an tive point of view because we are interested in the nature of goods and services that

informa-Income as a measure of economic activity 5

will be available to society It breaks down the gross domestic product into its stitutive parts: consumption, investment, plus exports minus imports One point ofdetail: during a given year, inventories may have accumulated in the producers’hands; by definition, these have not yet been sold They are then considered part

con-of investment

Consumption is usually separated into private consumption and public sumption; private consumption refers to the consumption of individuals; publicconsumption is that of the state (at the national or at the local level) The samedistinction applies to investment We thus have, for example:

Total: gross domestic product: 100

1.1.2 The output (value added) approach

The economy can be divided into a few – or many – sectors If we consider thetraditional three sectors (agriculture, industry, services), we may have:

rvalue added of agriculture: 4

rvalue added of industry: 30

rvalue added of services: 66Total: gross domestic product: 100

1.1.3 The income approach

Retaining the classification referred to in the beginning of this section could give:

Total: gross domestic product: 100

It is possible and highly useful to present in one table those three approaches.Indeed, it is by no means obvious that the three approaches amount to the samenumber In fact, statisticians establishing those national accounts have to introducesometimes large corrections to obtain the desired result

Table 1.1 A simplified input–output table

Income as a measure of economic activity 7

This synthetic, global view was first proposed by Wassily Leontief2in a called “input–output” table The table can be considered as made of three parts

so-In the first part (the upper-left part of table 1.1), only intermediate transactions arerecorded Those are the sales of products or services from one sector to another (forinstance the aluminium sold to the automobile sector, the wheat sold by agriculture

to the food industry, or the insurance premiums sold to the farmers) The tablepresents in its upper-left 9 squares all possible sales between the three sectors wehave considered (there are 6 possible inter-sector sales and 3 possible intra-sectorsales: for instance the energy sales to the aluminium industry are sales within theindustry sector) The first 3 figures of column 4 (2, 9, 20) sum the sales of eachsector to other sectors as well as to itself The first 3 figures of line 4 (5, 20, 6)are the purchases of each sector from other sectors and from itself The total ofintermediate sales and purchases (31) is the 4th figure of line 4 and column 4.The second part of the table (columns 5 to 10) breaks down the expenditureside of the gross domestic product into its components (column 5 indicates con-sumption goods aggregated as the sum of private and public consumption, for a

total of 80; column 6 gives private and public investment, etc.) X and M nate exports and imports respectively The total C + I + X − M = 80 + 15 +

desig-40 − 35 = 100 corresponds to the gross domestic product from the expenditureapproach and to the example we have given in section 1.1

It is now possible to determine the total output generated by each sector byadding the total of intermediate goods and services (col 4) to the column of finaldemand (col 9) We thus get column 10 This will be useful to determine the valueadded of each sector Indeed, just translate the numbers of column 10 into thehorizontal line 9 at the bottom of the table Just by taking the difference betweenthe total output (line 9) and the total purchases of each sector (line 4) gives thevalue added by each sector (line 8) This line is the gross domestic product brokendown according to value added by each sector (4 + 30 + 66 = 100) In turn,each of these net contributions from each sector can be split into their components(labour income, capital income and profits) These are indicated in lines 5, 6 and

7 respectively Their total in column 4 gives the gross domestic product from theincome point of view (68+ 26 + 6 = 100)

Thus the three approaches to the gross domestic product can be viewed in asingle table Two conclusions emerge The first is that if we had to choose fromthose three approaches that which best reflects the welfare of society, we wouldprobably choose the first, i.e the expenditure approach, because we are most inter-ested to know what kind of consumption or investment goods will be available to

2 Wassily Leontief (1906–99) was born in St Petersburg and received his PhD from Humboldt versity in Berlin He taught at Harvard and New York University One of his first input–output tables divided the US economy into 500 sectors He received the Nobel Prize in Economics in

Uni-1974 for his pioneering work on input–output economics.

society We do not care about whatever nominal income is generated if it sponds to huge investments which will be of little use to society.

corre-The second lesson we draw is the one-to-one correspondence between net put and income Income will increase in an economy if and only if net production –

out-or value added – increases

The gross domestic product (GDP) measures the activity taking place inside the

borders of a given nation However, it may be desirable to have a measurementleading to the revenue accrued to people actually residing in a country Thisimplies, in particular, adding to GDP the income received from abroad by resi-dents and subtracting the income paid out to non-residents who are working withinthe boundaries of the country These corrections concern both labour and capitalincome Thus we have:

Gross domestic product(GDP)

+ labour and capital income generatedabroad and received by residents

− labour and capital income generateddomestically paid out to non-residents

= gross national product  GNP Observe that the adjective “national” is most unfortunate since it does not refer to the activity of national economic agents; it only refers to the activity of residents.

Furthermore, within one year investments may have taken place, but somedepreciation of the existing stock of capital has occurred It is important (espe-cially in the context of economic growth) to know by exactly what amount thecapital stock in existence in the economy may have increased This is why depre-

ciation of capital will be subtracted from (gross) investment to obtain net ment, and consequently net national product:

invest-Gross national product (GNP)− depreciation = Net national product (NNP).

Finally, it may be interesting to know exactly what amount of income will be tributed to residents before income taxes For that purpose, two corrections need to

dis-be introduced First, we must deduce from the net national product indirect taxes

– this is the sales tax paid by consumers to producers, which the latter transfer tothe state These taxes form no part of the value added that is the basis of incomedistribution by firms, and therefore must be deducted from net national product

On the other hand, firms may receive subsidies from the state Together with valueadded these may be used to remunerate production factors such as labour or cap-ital Thus we have:

Net national product (NNP)− indirect taxes + subsidies = National income

Income as a measure of economic activity 9

The following summarizes the various steps transforming gross domestic product(GDP) into net national income

Gross domestic product (GDP)

+ labour and capital income received from abroad by residents

− labour and capital income transferred abroad to non-residents

= Gross national product (GNP)

− capital depreciation

= Net national product

− indirect taxes+ subsidies

= National income

Suppose that from one year to another we observe an increase in the gross domesticproduct, or an increase in national income Can we conclude from this observa-tion that the standards of living have increased? Clearly such a conclusion is notwarranted since that increase might be due to the sole increase of prices It is thenimportant to be able to define in a precise way what part of a change in nationalincome is attributable to a change in prices, and what part reflects an increase inoutput available to society For that purpose, price indexes are constructed Wewill succinctly describe here the consumers’ price index

We will use the following definitions and notations:

r pi = price of good i at time t (measurement units: $unit of the good considered)

r p i −p i−1

p i−1 = relative rate of increase in the price of good i

r p i

p i−1 ≡ X i ≡ growth factor of the price of good i  partial price index of good i

We can note that this growth factor reflects also the growth factor of the

expendi-ture necessary to buy one unit, or any fixed quantity, of good i.

Suppose now that between time 0 and time 1 the prices of two goods haveundergone different relative increases (perhaps 10% and 30% respectively) Isthere a way to define a meaningful “average” of those increases? It is obviousthat considering the simple average of those rates (20%) would be of little signifi-cance if, for instance, consumers used to spend a large part of their income on one

of those items Therefore, it is natural to try to measure the consequences of suchincreases for a given consumer in the following way

Consider the growth factor of the expenditure necessary for this consumer to

be able to buy again the same basket of both goods which he or she had initiallypurchased at time 0, before the price increases