Theory of the Price Index: Fisher’s Test Approach and Generalizations

103 Pages·1976·2.862 MB·English

by  Wolfgang Eichhorn, Joachim Voeller (auth.)

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continuation on page 97
Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. KUnzi
Mathematical Economics
140
Wolfgang Eichhorn
Joachim Voeller
Theory of the Price·lndex
Fisher's Test Approach and Generalizations
Spri nger-Verlag
Berlin· Heidelberg· New York 1976
Editorial Board
H. Albach . A. V. Balakrishnan . M. Beckmann (Managing Editor)
P. Dhrymes . J. Green' W. H~denbrand . W. Krelle
H. P. Kunzi (Managing Editor) . K. Ritter' R. Sato . H. Schelbert
P. Schonfeld
Managing Editors
Prof. Dr. M. Beckmann Prof. Dr. H. P. Kunzi
Brown University Universitat Zurich
Providence, RI 02912/USA 8090 Zurich/Schweiz
Authors
Wolfgang Eichhorn
Joachim Voeller
Institut fUr Wirtschaftstheorie
und Operations Research
Universitat Karlsruhe
7500 Karlsruhe/BRD
Library of Congress Cataloging in Publication Data
Eichhorn, Wolfgang.
Theory of the price index.
(r.e.:ture notes in economics and mathematical. systems ;
~40 : Mathematical. economics)
Bibllogra.pby: p.
~. Price indexes. I. Voeller, Joa.chim., ~946-
joint author. II. Tille. III. Series: Lecture notes
in economics and mathematical. systems ; ~40.
HB225.E37 339.4'2'0~ 76-57252
AMS Subject Classifications (1970): 90A99, 39A30
ISBN-13: 978-3-540-08059-6 e-ISBN-13: 978-3-642-45492-9
001: 10.1007/978-3-642-45492-9
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation, re
printing, re-use of illustrations, broadcasting, reproduction by photocopying
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Under § 54 of the German Copyright Law where copies are made for other
than private use, a fee is payable to the publisher, the amount of the fee to
be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1976
Dedicated
to the memory of Etienne Las p e y res
who taught Economics at the University of
Karlsruhe in the academic year 1873/1874
and to Wilhelm K r ell e on the occasion
of his 60~h birthday.
Preface
We wish to thank Georg Hasenkamp for
valuable comments on an earlier draft
of the manuscript and Steven Diamond
for his kindness in reading the manu
script and providing advice regarding
the style of the exposition. We are
also grateful to Miss Ingeborg Kasper
for her careful typing of the manu
script.
Contents
1. Introduction
2. Price Indices Depending only on Prices 4
2.1 Definition, Examples, Implications 4
2.2 Characterizations of Pri~e Indices 15
3. Price Indices Depending on Prices and Quantities 22
3.1 Definition, Examples 23
3.2 Fisher's System of Tests 29
3.3 Implications and Characterizations 35
3.4 Independence and Inconsistency of Fisher's Tests 44
3.5 General Solution of the Inconsistency Problem 54
4. Price Levels, Price Indices, and Fisher's Equation 59
of Exchange
4.1 Definition, Examples, Implications 60
4.2 Characterizations of Price Levels 64
4.3 Fisher's Equation of Exchange Reconsidered 72
5. Bibliography 83
6. Index 88
1. Introduction
In the face of the economic, political, and social problems resul
ting from world-wide inflation, theories of the price index have
gained new attention. This newfound interest in price indices stems
from the fact that all such indices are designed to serve as yardsticks
for measuring the price behavior of goods and services. That is, all
price indices relate to the concept of the 'purchasing power of money'.
If prices increase, then the value of the unit of money declines, i.e.,
the purchasing power of money shrinks. If the prices of certain goods
and services fall, however, the purchasing power of money increases
in relation to these commodities.
Usually, the price behavior of goods and services is analysed for a
specified period of time. However, the same methods of analysis may
also be applied to interspatial price comparisons. Regional differen
ces in the purchasing power of money or so-called purchasing power
parities between two countries are determined in this manner.
This monograph focuses mainly on the theory of the price index starting
with Irving Fisher's contributions in the early 'twenties. Our purpose
is
to develop consistent sets of properties in order to define
price indices and so-called price levels;
to characterize special classes of price indices as well as
price levels (including well-known indices such as Laspeyres'
and Paasche' s index or Fisher's "ideal index");
to derive the general solution of the inconsistency problem
of Fisher's tests;
to evaluate the validity of Fisher's famous equation of ex
change.
In this monograph, 'axioms' or 'tests' are used to develop price
indices and to examine their characteristics. These axioms or tests
- 2 -
serve as criteria which each price index should reasonably satisfy.
R. Frisch [1936, p.5] refers to this method as the 'test approach'
and R.G.D. Allen [1975, p.47] uses the term 'statistical approach'.
P.A. Samuelson and S. Swamy ·[1974, p. 567J speak of 'mechanical tests'
applied to index number formulae.
In his book "The Making of Index Numbers" published first in 1922,
I. Fisher stated that it was possible to reduce the number of concei
vable price indices by subjecting them to tests he originally developed.
Using this method, Fisher claimed he could identify one "best" index
number formula. Even though Fisher considered his "ideal index" the
best all around index number, the determination of a price index
satisfying all of his tests has failed.
As a result of this failure, Fisher eliminated one specific test - the
Circular Test (see section 3.2) - from the test system for, what he
calls, 'economic reasons'. Yet, R. Frisch [1930J, A. Wald [1937J,
S. Swamy [1965J, W. Eichhorn [1973J, [1976J, J. Voeller [1974J and
W. Eichhorn/J.Voeller [1976J have evaluated Fisher's original system
of tests, selectively loosening certain test requirements. Their
primary objective was the proof that certain subsets of these tests
are inconsistent.
R. Frisch [1930J was the first to publish an inconsistency theorem for
a true subset of these tests. He assumed the differentiability of the
index function. His theorem, however, is not valid as W. Eichhorn
[] 973J has shown. In 1937, A. Wald succeeded in formulating a non
existence theorem for another subset of Fisher's tests witho"ut making
any additional regularity assumptions. Convinced that Wald's proof was
wrong, S. Swamy [] 96~ proved a non-existence theorem for still another
subset of tests. However, Swamy assumed the differentiability of the
price index. W. Eichhorn 0973J, 097~, J. Voeller 097{J and
W. Eichhorn/J. Voeller [197€] further contributed to the solution of
the inconsistency problem of Fisher's tests by proposing several new
theorems of independence and non-existence for different subsets of
Fisher's tests. These new theorems provide the complete solution to the
inconsisteney problem.
- 3 -
Since their introduction, the economic relevance of Fisher's tests
has been questioned. For example, Fisher himself ultimately questioned
the usefulness of the Circular Test and he even objected to the devel
lopment of any price index conforming to that test. G. v. Haberler
[1927J disapproved of the Factor Reversal Test, while R. Frisch [193q]
preferred to discard the Commensurability Test rather than the Deter
minateness Test (see section 3.2 for description of the tests).
P.A. Samuelson and S. Swamy differ from Frisch's preference for the
Determinateness Test and write [1974, p. 572J: "This condition, it
seems to us, is an odd one and not at all a desirable one".
Because of these basic differences of opinion, a common approach toward
the theory of the price index, as presented in this monograph, could
not develop. The test approach differs significantly from the 'economic
theoretic' view of index numbers. The economic-theoretic school takes
the taste or the preference structure of a consumer into account in or
der to define a 'cost-of-living index'. This line of thought treats the
prices and quantities of goods as functions of each other, whereas
in our approach prices and quantities are treated as independent vari
ables.The theory of the cost-of-living index has long been established
in the literature and it is permanently enriched (see, for instance,
S.N. Afriat [1967J, [197~ K.S. Banerjee [197'D; C. Blackorby and R.R.
Russell [1976J i W.E. Diewert [1976J; F.M. Fisher and K. Shell [197:[J;
R. Frisch [l93~, [1954]; G. Hasenkamp [1976J; L.R. Klein and H. Rubin
[1948J; J. Muellbauer [197![J; L. Phlips [1 97{]; L. Phlips and R. Sanz
Ferrer [197~; R.A. Pollak [197"0, [i97~; P.A. Samuelson and S. Swamy
[1974]; !>1.J. Ulmer [1949J; J. Voeller [197{l, and A. Wald 0932], [J93~).
This monograph presents a theory of the price index which follows from
J. Fisher'S original line of reasoning in sections 2 and 3. Section 2
introduces price indices depending only on prices. In section 3 price
indices depending on both prices and quantities are defined and
characterized. Fisher's system of tests is thoroughly analysed with
respect to the independence and inconsistency of the subsets of these
tests. Section 4 contains axiomatic theories that refer to the concept
of a price level. The relationship between price levels and price
indices is also examined. The final section concludes with a review
of I. Fisher's well-known equation of exchange.
2. Price Indices Depending only on Prices
In this section,we only consider price indices which depend on the
prices of a 'base year' and a 'comparison year', respectively. For
technical reasons, we further assume that the domain of defini tionof all
price indices is always restricted to the positive orthant, i.e., only
positive components of the various vectors are admitted. This require
ment is especially necessary for price indices which depend on both
prices and quantities, since even classical price indices such as
Laspeyres' or Paasche's index (see (3.1.7), (3.1.8» cannot properly
be defined for certain points of the non-negative orthant. This
restriction can be eased by imposing a condition stating the behavior
of a price index as any scalar argument (price or quantity of a
commodity) tends to zero. This condition is introduced in section 3.2
(see the Determinateness Test (3.2.3».
2.1 Definition, Examples, Implications
Let
{xlxe:JR, JR the set of all real numbers, x> a},
{~
and let
be the price vectors of n commodities in a base year and in a compari
son year, respectively.