Table Of ContentADVANCED TEXTBOOKS
IN ECONOMICS
VOLUME 5
Editors:
C.J. BLISS
M.D. INTRILIGATOR
Advisory Editors:
D.W. JORGENSON
M.C. KEMP
J.-J. LAFFONT
J.-F. RICHARD
NORTH-HOLLAND
AMSTERDAM · NEW YORK · OXFORD · TOKYO
APPLIED CONSUMPTION
ANALYSIS
REVISED AND ENLARGED EDITION
LOUIS PHLIPS
C.O.R.E.
NORTH-HOLLAND
AMSTERDAM · NEW YORK · OXFORD · TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V.
Sara Burgerhartstraat 25
P.O. Box 211, 1000 AE Amsterdam, The Netherlands
Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY, INC.
655 Avenue of the Americas
New York, N.Y. 10010, U.S.A.
First edition: 1974
Revised edition: 1983
Second impression: 1990
Library of Congress Cataloging in Publication Data
Phlips, Louis.
Applied consumption in analysis.
(Advanced textbooks in economics; v. 5)
Bibliography: p.
Includes index.
1. Demand (Economic theory) 2. Consumption
(Economics) 3. Statics and dynamics (Social
sciences). I. Title. II. Series.
HB801.P47 1982 338.5'21 82-18834
ISBN 0-444-86531-4 (U.S.)
ISBN: 0 444 86531 4
© ELSEVIER SCIENCE PUBLISHERS B.V., 1983
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Printed in The Netherlands
Introduction to the series
The aim of the series is to cover topics in economics, mathematical economics
and econometrics, at a level suitable for graduate students or final year
undergraduates specializing in economics. There is at any time much
material that has become well established in journal papers and discussion
series which still awaits a clear, self-contained treatment that can easily be
mastered by students without considerable preparation or extra reading.
Leading specialists will be invited to contribute volumes to fill such gaps.
Primary emphasis will be placed on clarity, comprehensive coverage of
sensibly defined areas, and insight into fundamentals, but original ideas will
not be excluded. Certain volumes will therefore add to existing knowledge,
while others will serve as a means of communicating both known and new
ideas in a way that will inspire and attract students not already familiar
with the subject matter concerned.
The Editors
To Lucie
Preface
This is not a book on demand theory. Nor is it a book on the eco-
nometrics of demand systems. Rather, what I have tried to write is a
text on the subject matter between theory and estimation, that is on
the art of reformulating pure theory to make it suitable for econometric
applications. In my experience, it is not sufficient to offer a price theory
course and an econometrics course: most students find it very difficult
to bridge the gap between the two without appropriate training.
As in the art of piano playing, the art of consumption analysis requires
a lot of practising. That is why many exercises are offered to the reader
while only a few answers are given (for the more important ones). The
reader is strongly requested to work out all answers for himself. His
background is assumed to include some knowledge of calculus and of
elementary textbook economics. Purely econometric considerations
have been avoided. The starred sections are more advanced and some-
times require some matrix algebra: they can be used for a 'special
topics' course.
My intellectual debt to Hendrik S. Houthakker, one of the pioneers
in the field, should be obvious. His work on additive functions fascinated
me and is at the core of the static part of this book. His more recent
contributions, with Lester D. Taylor, to the dynamization of demand
systems provided the framework for my own work, which would never
have come from the ground without their help and encouragement.
I am most heavily indebted for research assistance and intellectual
support to E. de Souza, J. Garcia dos Santos, Ph. Rouzier, R. Sanz
Ferrer and D. Weiserbs, whose personal contributions have been
indicated - I hope - at the appropriate places in the next. The editors
of this Series, C. Bliss and M.D. Intriligator, offered many suggestions
and constructive criticism during the various stages of elaboration.
VII
VIII Preface
Thanks are also due to A.P. Barten and A. Deaton for comments on
the final manuscript.
Competent programming assistance was given by M. Vuylsteke-
Wauters over a number of years, and more recently by J.P. Lemaître.
Ph. Rouzier and P. Rousseaux. Last but not least, I wish to thank
B. de Schaetzen, B. de Rochelée and F. De Beir for cheerful secretarial
assistance.
De Haan, July 1973 LOUIS PHLIPS
Preface to the second edition
This second edition is also an enlarged one. Part I now offers a brief
discussion of duality and flexible forms, in particular of Deaton and
Muellbauer's 'almost ideal demand system'. There is also an introduction to
Muellbauer's treatment of the 'aggregation problem'. Part II includes my
work on the demand for leisure and money and on true wage indexes (a new
Chapter X). The second half of the final chapter, on intertemporal utility
maximization, is completely rewritten and largely draws from my work with
Frans Spinnewyn. Throughout, I have done my best to update the refer-
ences. The list of references now extends till 1981.
I wish to thank Simone Clemhout, Wouter Keller, Wilhelm Krelle,
A. H. Q. M. Merkies and Ricardo Sanz-Ferrer for pointing out errors in the
first edition.
After correcting a distressingly large number of printing errors, Γ11 bet
none appear in this edition. Or am I too optimistic?
De Haan, April 1982 LOUIS PHLIPS
VIII Preface
Thanks are also due to A.P. Barten and A. Deaton for comments on
the final manuscript.
Competent programming assistance was given by M. Vuylsteke-
Wauters over a number of years, and more recently by J.P. Lemaître.
Ph. Rouzier and P. Rousseaux. Last but not least, I wish to thank
B. de Schaetzen, B. de Rochelée and F. De Beir for cheerful secretarial
assistance.
De Haan, July 1973 LOUIS PHLIPS
Preface to the second edition
This second edition is also an enlarged one. Part I now offers a brief
discussion of duality and flexible forms, in particular of Deaton and
Muellbauer's 'almost ideal demand system'. There is also an introduction to
Muellbauer's treatment of the 'aggregation problem'. Part II includes my
work on the demand for leisure and money and on true wage indexes (a new
Chapter X). The second half of the final chapter, on intertemporal utility
maximization, is completely rewritten and largely draws from my work with
Frans Spinnewyn. Throughout, I have done my best to update the refer-
ences. The list of references now extends till 1981.
I wish to thank Simone Clemhout, Wouter Keller, Wilhelm Krelle,
A. H. Q. M. Merkies and Ricardo Sanz-Ferrer for pointing out errors in the
first edition.
After correcting a distressingly large number of printing errors, Γ11 bet
none appear in this edition. Or am I too optimistic?
De Haan, April 1982 LOUIS PHLIPS
I
Utility functions
In order to analyse statistical data on consumption we need to start
from some hypothesis. The one that is both natural and fruitful is to
assume the existence, for each (or for the average) consumer, of a
continuous utility function
u =f(xx, . ..,xj
u 2
and to give it certain properties that make it easy to maximize with the
usual mathematical tools.
It will be assumed that there are only a finite number of commodities,
say n: and that each commodity can be characterized numerically.
It will further be assumed that there exists a unit of measure for each
commodity, for example in kilogrammes or in litres or simply in
quantity. This being so, any assortment or bundle or packet of the η
commodities can be numerically expressed as a vector with η com-
ponents: (x x , x„). Each component always refers to the same
1? 2
commodity.
It is useful to consider each commodity as being linked with an
axis in the space of dimension, n, of real numbers, Rn. In this way
all the properties of this space will be at our service. However, the
commodity set will not be considered as covering the entire space, Rn.
We shall limit it by imposing 3 properties which define the commod-
ity set:
(1) A commodity may not be characterized by strictly negative
numbers. Hence the commodity vector, χ = (x ..., x„) can have no
l5
negative components.
(2) Let x° = (χ?, x°, ..., x£) be a bundle (or M-tuple) available to
the consumer. Then any bundle of the form αχ0 = (αχ?, ..., αχ°),
0 ^ α ^ 1, can be extracted from this bundle. This is known as the
property of divisibility.
3
4 Utility functions
(3) The commodity set contains the bundle, (0, 0 0); moreover,
if a bundle x1 belongs to the set then any bundle, x2, where xf ^ x/,
/ = 1, /?, belongs to the set. The commodity set is, therefore, un-
bounded from above.
The symbol V can be interpreted as measuring the 'satisfaction'
derived from the consumption of alternative consumption bundles.
More precisely, V is a number that facilitates the ordering of the
commodity bundles according to the preferences of the consumer.
The utility function is indeed a numerical representation of a pre-
ference ordering, in the sense that V is a number associated with
each possible consumption bundle, such that if one bundle, say x° =
(x?, x°,..., x°), is preferred to another bundle, say x' = (x\, x' ,..., xj,
2
the number associated with x° is greater than the number associated
with x'.
1.1. Basic axioms on the preference relation
The first question to ask is: what are the conditions that make such
a representation possible? In other words, we want to know what
properties the consumer's preferences should have to make it possible
to associate a number with each commodity bundle in the way de-
scribed above.
It is intuitively clear that restrictive assumptions will have to be
made: some possible irregular properties will have to be excluded from
the analysis. The reader is requested to ask himself whether he is ready
to consider the assumptions introduced below as acceptable, i.e. whether
the inevitable loss in realism is not exaggerated.
Economic theory formulates these assumptions as axioms, i.e. as
statements that are accepted as being true without proof. This means
that theorists consider these as intuitively plausible foundations on
which a scientific theory can be built. All sciences start from a set of
axioms, as one has to start somewhere. There is no reason why economic
science should be an exception to this rule.
Some definitions and a minimum of notation are required. The basic
concepts are those of preference and indifference, which are supposed
to have an immediate intuitive meaning. Let x° and χ be two vectors
representing two commodity bundles. We say that
€ x° ^ x ^ means <^ x° is not preferred to x' ^>
