Essays in Index Number Theory, Volume I
W.E. Diewert and A.O. Nakamura (Editors)
c 1993 Elsevier Science Publishers B.V. All rights reserved.
Chapter 1
OVERVIEW OF VOLUME I
W. Erwin Diewert*
1. Introduction
This volume and a subsequent one gather together most of my published papers
on index number and aggregation theory. In addition, these volumes
contain some new papers which have not been published before. In Volume I,
Chapters 1, 2 and 14 are new. Also proofs have been added to the previously
published version of Chapter 6. The previously published papers have not been
changed in content but minor errors have been corrected and the references to
unpublished papers have been replaced by references to the published versions
in most cases.
This volume is divided into six parts.
Part One provides an introduction to this volume.
Part Two is devoted to the early history of index number theory along
with biographical comments on two of the giants of index number theory: A.A.
Kon¨us (who sadly died recently) and E.L.E. Laspeyres.
Part Three presents two surveys of index number theory. The first one is
Chapter 5 and the second is Chapter 7. As the second survey is more mathematical
in nature, the underlying economic theory that is used in Chapter 7 is
presented in Chapter 6.
Part Four is concerned mainly with the concept of a superlative index number
formula. This concept will be defined below in Section 4 of this overview.
Part Five is devoted to axiomatic or test approaches to index number
theory and some closely related measurement problems involving the axiomatic
characterization of symmetric means. The axiomatic approach to index number
theory will be explained in Section 2 of this overview.
Part Six is concerned with one particular mechanism for aggregating over
commodities, namely aggregation when prices move proportionally. The original
result in this area is known as Hicks’ [1946] Composite Commodity Theo-
rem.
*This research was supported by a Strategic Grant from the Social Science and
Humanities Research Council of Canada. Thanks are due to Shelley Hey and
Louise Hebert for typing assistance.
2 Essays in Index Number Theory 1. Overview 3
Looking ahead to Volume II in this series, it will deal with five topics: (i)
the measurement of input, output and productivity; (ii) the measurement of
price and welfare change using consumer theory; (iii) alternative approaches to
the measurement of change (using differences instead of ratios); (iv) multilateral
comparisons; and (v) nonparametric approaches to measurement.
This chapter provides a brief overview of index number theory. In addition
to serving as an introduction to the subject, this overview should be helpful
to readers who are interested in locating material on specific topics covered in
this volume. Those readers who want a short course on index number theory
should read this overview plus Chapters 2 and 5. For a more in depth course,
the reader should add Chapters 7 and 8 and Sections 1–3 of Chapter 13 (the
axiomatic part). Readers who are primarily interested in the measurement of
inequality or in functional forms for social welfare functions or in the theory of
choice under uncertainty can restrict themselves to Chapter 14.
Several chapters could be used as readings for a topics course in economic
theory which would cover duality theory and the economic theory of index
numbers. The relevant chapters are 6 (which provides the duality theory background
for Chapter 7), 7, 8, 11, 14, 15 and 16.
In these volumes, we shall study aggregation problems in economics. Economic
theory is for the most part concerned with modeling the demand and
supply for individual goods and services (commodities) by individual economic
agents (producers or consumers). However, due to the truly enormous numbers
of both commodities and agents in real life economies, empirical economics uses
data that are always aggregated over commodities and often aggregated over
agents. How should this aggregation over goods and agents be accomplished?
More specifically, the aggregation over goods problem asks: how can we
aggregate or summarize individual microeconomic data on prices into a single
aggregate price level and individual data on quantities into a single aggregate
quantity level so that the product of the price level times the quantity level
equals the sum of the individual prices times the quantities for the commodities
to be aggregated? How exactly to construct these aggregate levels is the index
number problem in economics.
Although these volumes are primarily concerned with the aggregation over
goods problem, some of the chapters touch on aspects of the aggregation over
agents problem. The aggregation over agents problem asks the following question:
under what conditions will price and quantity data, which are constructed
by summing over economic agents, behave as if the aggregate data were the
solution to a microeconomic optimization problem involving a single consumer
or producer? On the aggregation over producers problem, the reader is referred
to Gorman [1968b], F.M. Fisher [1965], Diewert [1980] and Blackorby
and Schworm [1984]. On the aggregation over consumers problem, see Gorman
[1953], Muellbauer [1975] [1976], Berndt, Darrough and Diewert [1977],
Lau [1977a] [1977b] [1982], Diewert [1983a], Jorgenson and Slesnick [1983] and
Chapters 6 (Section 10) and 11 in this volume.
Returning to the aggregation over goods problem, it should be noted that
this problem also encompasses two other aggregation problems: (i) the aggregation
over time problem and (ii) the aggregation over space problem.1
As
Debreu [1959] noted many years ago, the definition of a commodity is flexible
enough to encompass not only the “physical” characteristics of a good or service,
but also its time and spatial characteristics; i.e., the same good sold at
a different place or time can be regarded as a distinct commodity.2
Triplett
[1990a; 11–13] also stressed that different terms of sale can serve to make the
same physical good into different commodities. However, for practical measurement
purposes, we cannot take the “fundamental” unit of time or space to
be too small, since the smaller we make the unit of time or space within which
production or consumption takes place, the less actual production or consumption
there will be to observe, and comparisons between these tiny units will
become meaningless.3
Thus for normal economic data, the time period under
consideration is usually: (i) a shift (a part of a working day), (ii) a day, (iii)
a week, (iv) a month, (v) a quarter, or (vi) a year. A normal “spatial” unit
is usually: (i) an enterprise4
or a household at a specific address or (ii) an
aggregate of enterprises or households over a region. The region could be: (i)
a county or municipality, (ii) a metropolitan region, (iii) a state or province,
(iv) a country, or (v) a group of countries.
Once the fundamental units of time and space have been chosen, we typically
aggregate production or consumption data on an individual “physical”
1
This problem could be an aggregation over commodities problem or an aggregation
over agents problem. If the same physical good is being sold by a
single firm at several locations, then we have an aggregation over commodities
problem.
2
Alfred Marshall [1887; 373–374] seems to have been the first to appreciate that
strawberries being made available at different times of the year or at different
locations were separate commodities; see Chapter 2, Section 10 below.
3
Thus if the fundamental unit of time or space is too small, production or
consumption of most goods will be zero and comparisons between quantities
and prices of the same good between adjacent periods will not be informative
(this can be viewed as an example of the “new good” problem to be discussed
later).
4
An enterprise is usually defined to be the smallest production unit with a
specific geographical address that can provide basic statistics on its inputs and
outputs. The Standard Industrial Classification (SIC) attempts to develop
criteria for grouping together enterprises. Triplett [1990a] [1991] criticized the
“theory” behind these grouping attempts and provided his own criteria for
grouping based on economic approaches to aggregation theory.
4 Essays in Index Number Theory 1. Overview 5
commodity i as follows: calculate the aggregate value V t
i and the total number
of units xt
i of the good produced or consumed within period t. Then the
microeconomic quantity of this good is taken to be xt
i and the corresponding
microeconomic price is defined to be pt
i ≡ V t
i /xt
i. If we are given microeconomic
price and quantity data (pt
i, xt
i) for T periods and the N commodities,
so that t = 1, 2, . . . , T and i = 1, 2, . . ., N, then to solve this particular aggregation
over goods problem, we want T aggregate prices P1
, P2
, . . . , PT
and T
aggregate quantities Q1
, Q2
, . . . , QT
such that
(1)
N
i=1
pt
ixt
i = Pt
Qt
for t = 1, 2, . . . , T.
Thus we want the aggregate value in period t, Pt
Qt
, to equal the corresponding
microeconomic value,
N
i=1 pt
ixt
i, for each time period t.5
The aggregate period t
price Pt
is supposed to represent all of the period t microeconomic prices
pt
1, pt
2, . . . , pt
N in some sense and the aggregate period t quantity Qt
is supposed
to represent all of the period t microeconomic quantities xt
1, xt
2, . . . , xt
N in some
sense. The index number problem is: how exactly are these aggregates Pt
and
Qt
to be constructed?6
There are two main approaches to index number theory: (i) the axiomatic
approach and (ii) the economic approach. The difference between the two
approaches can be explained as follows. Referring back to equation (1), denote
the period t microeconomic price and quantity vectors as pt
≡ (pt
1, . . . , pt
N )
and xt
≡ (xt
1, . . . , xt
N ) respectively. In the axiomatic or test approach, the
period t price and quantity levels, Pt
and Qt
, are regarded as functions of the
microeconomic price and quantity vectors, pt
and xt
, where pt
and xt
are both
free to vary independently. Thus we have
(2) Pt
≡ P(pt
, xt
); Qt
≡ Q(pt
, xt
) for t = 1, 2, . . . , T
where P(p, x) and Q(p, x) are each functions of 2N variables and (p, x) ≡
(p1, . . . , pN , x1, . . . , xN ). In the economic approach, the microeconomic price
vectors pt
are regarded as independent variables, but the quantity vectors xt
are regarded as dependent variables; i.e., xt
is determined as a solution to some
microeconomic optimization problem involving the observed price vector pt
.
5
Of course, t could index locations instead of time periods, in which case we
are attempting to construct comparable aggregates Pt
, Qt
over space rather
than time.
6
Note that we do not attempt to determine how the goods to be aggregated
are chosen. Triplett [1990a] [1991] addresses this question and suggests a number
of economic approaches that could be used to choose the goods that are
to be aggregated. We shall discuss three of his suggested approaches below
(separability, Leontief aggregation, and Hicksian aggregation).
As a concrete example of the economic approach, suppose a consumer
has preferences over differing amounts of N goods that can be represented by
the linearly homogeneous and increasing utility function f where u = f(x) is
the utility or satisfaction level associated with the nonnegative consumption
vector x ≡ (x1, . . . , xN ) ≥ 0N ≡ (0, . . . , 0). Given a strictly positive vector of
consumer prices pt
0N for period t and an observed period t consumption
vector xt
≥ 0N , if the consumer is maximizing utility subject to a budget
constraint or is minimizing the cost of achieving the utility level ut
≡ f(xt
)
then xt
will solve the following cost (or expenditure) minimization problem:
(3) min
x
{pt
· x : f(x) ≥ ut
} = c(pt
)f(xt
) for t = 1, 2, . . . , T,
where pt
·x ≡
N
i=1 pt
ixi and c(pt
) is the minimum cost of achieving one unit of
utility; i.e., c is the unit cost function that corresponds to the linearly homogenous
utility function f. The equality in (3) follows from the linear homogeneity
of f; see equation (2.17) in Chapter 6 below. Since xt
solves the minimization
problem in (3) by assumption, we have
(4) pt
· xt
=
N
i=1
pt
ixt
i = c(pt
)f(xt
) for t = 1, 2, . . . , T.
Comparing equations (1) and (4), it is reasonable to identify the period t unit
cost c(pt
) with the price level aggregate Pt
and the period t level of utility
f(xt
) with the quantity level Qt
; i.e., in the economic approach, we have:
(5) Pt
≡ c(pt
); Qt
≡ f(xt
) for t = 1, 2, . . . , T.
We are now in a position to give an outline of this overview.
In Section 2, we pursue the axiomatic approach in a bit more depth. In
particular, we indicate how the functional forms for the functions P and Q
which occur in (2) are determined.
Section 3 explains the theory behind the economic approach in more detail.
Unfortunately, as one can see by inspecting equations (3)–(5) above, the
economic theory of index numbers is often of limited use due to the unobservable
nature of the functions which crop up; e.g., in (5), the utility function f
and the corresponding unit cost function c cannot be observed directly. Thus
in Section 4, we discuss how the basic theory behind the economic approach
is modified and extended to yield “good” observable approximations to the
unobservable theoretical constructs. It is a rather remarkable fact that at this
point, the two major approaches to index number theory converge; i.e., it will
turn out that “good” functional forms for P and Q in the axiomatic approach
are also “good” functional forms from the viewpoint of the economic approach
to index number theory.
6 Essays in Index Number Theory 1. Overview 7
Section 5 gives a brief overview of other approaches to index number theory
while Section 6 explains the new good problem.
Section 7 discusses the problem of choosing a functional form for taking
an average. For example, Irving Fisher [1922] used arithmetic, geometric and
harmonic means as well as some other types of averages to construct an average
price ratio. Which type of average should we use and what are the mathematical
properties of these different kinds of averages?
Section 8 discusses the economic and mathematical tools that the various
chapters in the book require.
Section 9 concludes with some personal observations.
2. The Axiomatic or Test Approach
Recall from the previous section that the vectors of period t microeconomic
prices and quantities are pt
≡ (pt
, . . . , pt
N ) and xt
≡ (xt
1, . . . , xt
N ) for t =
1, 2, . . . , T. In the axiomatic approach, the prices and quantities, pt
i and xt
i,
are regarded as independent variables and our index number problem is to
somehow construct period t price and quantity aggregates, Pt
and Qt
, such
that pt
· xt
≡
N
i=1 pt
ixt
i = Pt
Qt
for t = 1, 2, . . . , T; i.e., such that equations
(1) hold. It is (at first sight) natural to require that the price level Pt
be a
function of the components of pt
and that the quantity level Qt
be a function
of the components of xt
; i.e., we define Pt
and Qt
by
(6) Pt
≡ P(pt
); Qt
≡ Q(xt
) for t = 1, 2, . . . , T
where P(p1, . . . , pN ) and Q(x1, . . . , xN ) are functions of N variables that are
to be determined somehow. Note that in equations (6), the functions P and
Q are assumed to be the same for all T periods (but, of course, the individual
price and quantity vectors pt
and xt
usually change from period to period).
At this point, we have to distinguish a number of separate branches of the
axiomatic approach to index number theory. The reader will note that equations
(6) are different from our earlier equations (2); in (6), the price function
P(pt
) depends only on the period t price vector pt
while in (2), the price function
P(pt
, xt
) depended on the period t price and quantity vectors, (pt
, xt
).
Thus equations (2) and (6) give rise to different branches of the axiomatic
approach to index number theory.
Yet another branch is obtained by setting T = 2 and by asking that the
price aggregates for periods 1 and 2, P1
and P2
, and the quantity aggregates,
Q1
and Q2
, satisfy the following equations:
(7) P2
/P1
= P(p1
, p2
, x1
, x2
); Q2
/Q1
= Q(p1
, p2
, x1
, x2
).
The functions P and Q which occur in (7) are now functions of the 4N micro
prices and quantities that pertain to periods 1 and 2. Note that P2
/P1
is to be
interpreted as (one plus) the aggregate growth rate in prices going from period 1
to 2 and Q2
/Q1
is to be interpreted as (one plus) the aggregate growth rate in
quantities going from period 1 to 2. The forms for the functions P and Q which
occur in (7) are to be determined by this branch of the axiomatic approach to
index number theory. This branch is known as bilateral index number theory
since it aggregates price and quantity data over only two periods.
Finally, in multilateral index number theory, we ask that the price aggregates
Pt
and quantity aggregates Qt
satisfy the following equations:
(8)
Pt
= Ft
(p1
, . . . , pT
, x1
, . . . , xT
) t = 1, 2, . . . , T
Qt
= Gt
(p1
, . . . , pT
, x1
, . . . , xT
) t = 1, 2, . . . , T
where now the 2T functions Ft
and Gt
to be determined depend in principle
on all of the micro prices and quantities over all of the periods.
Thus there are separate axiomatic approaches to index number theory that
are centered around each of the equations (2), (6), (7) and (8). Most of the
analysis of the axiomatic approach to index number theory to be presented
in this book will be on the bilateral approach (7) with some coverage of the
multilateral approach (8).
For material on the bilateral approach, see Chapter 2, Section 4; Chapter 5,
Section 2; Chapter 12, Section 3; and Chapter 13, Sections 2 and 3.
For material on the multilateral approach, see Chapter 2, Sections 6.3
and 8; Chapter 5, Sections 6 and 9; and Chapter 12.
None of the chapters in this volume utilize the approaches given by equations
(2) and (6). In this introduction, we explain why this is so.
First consider the case represented by equations (6). Recall that we want
our aggregates Pt
and Qt
to satisfy equations (1) as well. Substitution of (6)
into (1) yields:
(9) P(pt
)Q(xt
) =
N
i=1
pt
ixt
i, t = 1, 2, . . ., T.
Simply by examining equations (9), it can be seen that if the number of
goods N is equal to or greater than two, then it is impossible for functions P
and Q to exist which will satisfy (9) for all pt
≡ (pt
1, . . . , pt
N ) ≥ 0N and
xt
≡ (xt
1, . . . , xt
N ) ≥ 0N .7
Thus our analysis of this particular branch of the
axiomatic approach to index number theory comes to an abrupt end.
Now consider the branch of the axiomatic approach represented by equations
(1) and (2). Substitution of (2) into (1) yields:
(10) P(pt
, xt
)Q(pt
, xt
) =
N
i=1
pt
ixt
i ≡ pt
· xt
for t = 1, . . . , T.
7
This result is due to Eichhorn [1978b; 144].
8 Essays in Index Number Theory 1. Overview 9
What properties should the price level function P(p, x) and the quantity
level function Q(p, x) have? It seems reasonable to demand that P(p, x) be
linearly homogeneous in the components of the price vector p ≡ (p1, . . . , pN )
so that if all prices increase by a common factor λ, then so will the price level;
i.e., we assume that the function P satisfies the following test or axiom:
(11) P(λp, x) = λP(p, x) for all λ > 0, p 0N and x 0N .
Similarly, it seems reasonable to demand that Q(p, x) be linearly homogeneous
in its quantity variables x; i.e., we assume that Q satisfies the following axiom:
(12) Q(p, λx) = λQ(p, x) for all λ > 0, p 0N and x 0N .
It is also reasonable to ask that the aggregate price and quantity levels be
positive if all microeconomic prices and quantities are positive. Thus we assume
that the functions P and Q satisfy:
(13) P(p, x) > 0; Q(p, x) > 0 if p 0N and x 0N .
In order that (10) hold for all possible price and quantity vectors p and x,
we require that P and Q jointly satisfy the following axiom (the product test):
(14) P(p, x)Q(p, x) = p · x for p 0N and x 0N .
Note that (13) and (14) imply that the functions P and Q cannot be determined
independently; e.g., if we have determined Q, then P is determined as P(p, x) ≡
p·x/Q(p, x). Thus property (12) for Q implies that P(p, x) will be homogeneous
of degree 0 in the elements of x; i.e., if P and Q satisfy (12), (13) and (14),
then for p 0N , x 0N and λ > 0, we have:
P(p, λx) = p · λx/Q(p, λx) using (13) and (14)(15)
= λp · x/λQ(p, x) by (12)
= p · x/Q(p, x)
= P(p, x) using (13) and (14).
Since units of measurement in economics are quite arbitrary (e.g., purchases
of gasoline could be measured in litres or gallons), it would be very
useful if our price and quantity aggregates were invariant to changes in the
units of measurement. Thus our last test that we impose on the price function
P(p, x) is the following one:
(16) P(d1p1, . . . , dN pN , x1/d1, . . . , xN /dN ) = P(p1, . . . , pN , x1, . . . , xN )
for all pi > 0, xi > 0 and di > 0 for i = 1, . . . , N.
Test or axiom (16) can be written more compactly using matrix notation as
P(Dp, D−1
x) = P(p, x) for all diagonal matrices D with positive elements on
the main diagonal where D−1
denotes the inverse of the matrix D.
We can now show that the above properties on the functions P and Q are
inconsistent.
Proposition 1. Properties (11), (13), (15) and (16) on P are inconsistent;
i.e., there does not exist a function of 2N variables, P(p1, . . . , pN , x1, . . . , xN ),
which satisfies these properties.
Proof. Applying (16) with di = xi for i = 1, . . . , N, we obtain the following
equation:
(17) P(p1, . . . , pN , x1, . . . , xN ) = P(p1x1, . . . , pN xN , 1, . . . , 1).
If P satisfies the linear homogeneity in prices property (11) so that P(λp, x) =
λP(p, x), then (17) implies that P is also linearly homogeneous in x; i.e., (11)
and (17) imply that P(p, λx) = λP(p, x) and hence (15) cannot hold.qed
An impossibility result similar to that given in Proposition 1 was obtained
by Eichhorn [1978b; 144–145] who used monotonicity axioms on P and Q in
place of our positivity axiom (13).8
In view of the impossibility theorem in Proposition 1, it does not seem
fruitful to pursue the axiomatic model defined by equations (2) any further.
Thus we have eliminated our first two axiomatic models from further consideration
and can now turn to the bilateral model defined by equations (7).
In order for (7) to be consistent with (1), we shall require that the functions
P and Q satisfy the following equation (the product test or the weak factor
reversal test):
(18) P(p1
, p2
, x1
, x2
)Q(p1
, p2
, x1
, x2
) = p2
· x2
/p1
· x1
.
If (18) holds (and if P and Q are nonzero), then it can be seen that once we
have determined either P or Q, then the remaining function is determined by
rearranging (18). For example, suppose we have pinned down the functional
form for P; then Q(p1
, p2
, x1
, x2
) is determined as the function p2
· x2
/ p1
·
x1
P(p1
, p2
, x1
, x2
). Thus, given that the product test (18) holds, if we require
that P(p1
, p2
, x1
, x2
) satisfies enough tests or properties so that the functional
form for P is determined, then the functional form for Q will also be determined.
How are these properties for P determined; i.e., what are “reasonable”
properties that P should possess? One way of finding such reasonable properties
is to consider what the mathematical properties of P are when N = 1;
i.e., when there is only one good and there is no aggregation over commodities
problem. In the one good case, P(p1
1, p2
1, x1
1, x2
1) must equal the ratio of the
period 2 price to the period 1 price, p2
1/p1
1. Note that this last function has the
following homogeneity properties: it is homogeneous of degree 1 in p2
1, homogeneous
of degree −1 in p1
1, homogeneous of degree 0 in x1
1, and homogeneous of
8
Eichhorn assumed that P(p, x) was increasing in each component of p and
Q(p, x) was increasing in each component of x. For more on the history of this
axiomatic approach, see footnote 15 in Chapter 13.
10 Essays in Index Number Theory 1. Overview 11
degree 0 in x2
1. Thus it seems “reasonable” to impose these same homogeneity
properties on P when we are aggregating over N goods. This leads to the
following homogeneity tests or axioms for P: for p1
0N , p2
0N , x1
0N ,
x2
0N and λ > 0:
P(p1
, λp2
, x1
, x2
) = λP(p1
, p2
, x1
, x2
);(19)
P(λp1
, p2
, x1
, x2
) = λ−1
P(p1
, p2
, x1
, x2
);(20)
P(p1
, p2
, λx1
, x2
) = P(p1
, p2
, x1
, x2
);(21)
P(p1
, p2
, x1
, λx2
) = P(p1
, p2
, x1
, x2
).(22)
Note that when N = 1, the single price ratio p2
1/p1
1 is invariant to changes
in the units of measurement. In the case of a general N, it seems “reasonable”
to ask that P also be invariant to changes in the units of measurement, so that
(23) P(Dp1
, Dp2
, D−1
x1
, D−1
x2
) = P(p1
, p2
, x1
, x2
)
where D is any diagonal matrix with positive elements on the main diagonal.
At this point, the reader should be able to get the gist of the bilateral
test approach to index number theory. The task of this approach is to specify
a number of “reasonable” tests or axioms which are sufficient to determine a
unique functional form for P. At the same time, we cannot be too ambitious
and specify too many desirable properties so that an impossibility theorem
results of the type that was derived for the axiomatic models (2) and (6). The
longest set of mutually consistent axioms for the bilateral test approach that
we have been able to find appears in Sections 2 and 3 of Chapter 13 below.
There we find that the Fisher [1922] ideal price index defined as
(24) PF (p1
, p2
, x1
, x2
) ≡ (p2
· x1
p2
· x2
/p1
· x1
p1
· x2
)
1
2
satisfies some 20 “reasonable” tests. However, before turning to these sections,
the reader should read Section 4 of Chapter 2 which lays out the early history
of the test approach.
The multilateral test approach has not been as well developed as the bilateral
approach. A dominant set of tests which uniquely determines the functions
Ft
and Gt
which appear in (8) has not yet been presented in the literature. In
this volume, aspects of the multilateral approach to index number theory are
developed in Chapter 5, Sections 6 and 9, and in Chapter 12.
3. The Economic Approach: Theory
The economic theory of index numbers is based on the definitions of two constrained
optimization problems. One of these optimization problems leads to
the Kon¨us [1924] price index and the other leads to the Malmquist [1953] quantity
index.
We first consider the economic theory of price indexes. Consider a consumer
who wants to minimize the cost of achieving a certain utility level or a
producer who wants to minimize the cost of achieving a certain output level.
For the sake of definiteness, we will use the language of utility theory in what
follows.9
Thus let x ≡ (x1, . . . , xN ) ≥ 0N be a nonnegative consumption vector,
let y ≡ (y1, . . . , yM ) be a vector of “other” consumer variables that matter,
and let F be the consumer’s utility or preference function so that u = F(x, y)
denotes the level of utility that the consumer achieves if he or she consumes x
and y. Let p ≡ (p1, . . . , pN ) 0N be a positive vector of prices for the goods in
the x vector and consider the following (conditional on x, y) cost minimization
problem:
(25) C(p, x, y) ≡ min
x≥0N
{p · x : F(x, y) ≥ F(x, y)}.
Thus in (25), the consumer minimizes the commodity cost p · x ≡
N
i=1 pixi
of the goods in the x vector required to achieve the reference utility level
u ≡ F(x, y), given that the consumer’s y vector is fixed at y. Note that
C(p, x, y) is linearly homogeneous in the elements of p; i.e., for λ > 0, we have:
(26) C(λp, x, y) = λC(p, x, y).
If pt
0N is the observed vector of prices for x goods that the consumer
faces in period t for t = 1, 2, then the Kon¨us [1924] conditional price index10
for the x goods for period 2 relative to period 1 is defined as
(27) PK(p1
, p2
, x, y) ≡ C(p2
, x, y)/C(p1
, x, y).
9
To get the producer interpretation in what follows, let F be the producer’s
production function, let x be an input quantity vector, let y be a vector of
“other” variables, and let u be the output target.
10
In the original formulation of Kon¨us, the y vector did not appear in F, so that
F(x, y) was replaced by F(x) and C(p, x, y) was replaced by C(p, x). The more
general formulation that we are outlining now is due to Pollak [1975] at this
level of generality. However, a special case of the present theory was worked
out by Shephard [1953; 61–71] [1970; 145–146] (the homogeneously separable
case). See also Solow [1955–56], Green [1964], Geary and Morishima [1973] and
Arrow [1974].
12 Essays in Index Number Theory 1. Overview 13
By using the homogeneity equation (26), it can be seen that PK(p1
, p2
,
x, y) defined by (27) is homogeneous of degree 1 in p2
and homogeneous of
degree −1 in p1
, which are counterparts to the homogeneity properties (19)
and (20) in the test approach to index number theory.
If the y vector does not actually appear in the consumer’s utility function,
then the reference vectors x and y in (27) can be replaced by the reference
utility level u ≡ F(x), C(p, x, y) can be replaced by C∗
(p, u) ≡ minx≥0N {p · x :
F(x) ≥ u},11
and PK(p1
, p2
, x, y) can be replaced by
(28) P∗
K(p1
, p2
, u) ≡ C∗
(p2
, u)/C∗
(p1
, u).
If, in addition, F is linearly homogeneous so that F(λx) = λF(x) for x ≥
0N and λ > 0, then C∗
(pt
, u) = C∗
(pt
, 1)u ≡ c(pt
)u and (28) reduces to
P∗
K(p1
, p2
, u) = c(p2
)/c(p1
). Thus in this case, we end up with the same economic
indexes that we had earlier in equations (4) and (5).
Another way that we can end up with (4) and (5) from the general model
(27) is to assume that the consumer’s utility function has the following struc-
ture:
(29) F(x, y) = F∗
[f(x), y] where f(λx) = λf(x) for λ > 0.
We also require that F∗
be increasing in its first argument. If F has the structure
given by (29), then we say that there is a homogeneously separable aggregate
in x; note that the micro aggregator function f is linearly homogeneous.
With F having the structure given by (29), the conditional cost minimization
problem (25) becomes:
C(p, x, y) ≡ min
x≥0N
{p · x : F∗
[f(x), y] ≥ F∗
[f(x), y]}
= min
x≥0N
{p · x : f(x) ≥ f(x)}
since F∗
is increasing in its first variable
= c(p)f(x)(30)
where c(p) ≡ minx{p·x : f(x) ≥ 1} is the unit cost function corresponding to f
and (30) follows from the linear homogeneity of f. Thus under (29), the Kon¨us
price index (27) again reduces to c(p2
)/c(p1
); this follows by substituting (30)
into (27). This implication of (29) is essentially due to Shephard [1953; 61–71]
who made the same assumptions in the context of production theory.
11
The mathematical properties of C∗
are laid out in Chapter 6, Section 2.
Since the conditional cost function C can be regarded as the negative of a
conditional profit function, the mathematical properties of C can be determined
by studying Section 11 of Chapter 6.
Thus under a variety of assumptions, the rather complicated theoretical
price index defined by (27) reduces to the unit cost ratio c(p2
)/c(p1
) which is
consistent with our initial economic approach given by equations (4) and (5).
We turn now to the economic theory of quantity indexes. Again, let the
consumer’s utility be defined by u = F(x, y). Consider first the following
constrained maximization problem involving a single positive variable λ:
(31) D(x, x, y) = max
λ≥0
{λ : F(λ−1
x, y) ≥ F(x, y)}.
In the maximization problem (31), we take a given quantity vector x ≡ (x1,
. . . , xN ) and deflate each component by the positive number λ, obtaining the λ
deflated vector λ−1
x ≡ (x1/λ, x2/λ, . . . , xN /λ). We choose this deflation factor
λ to be as large as possible subject to the constraint that the deflated vector
λ−1
x when combined with the fixed reference vector y yields just as much
utility as that yielded by the fixed reference vectors x and y. This maximal
deflation factor defines the distance or deflation function D(x, x, y).
Note that D(x, x, y) is linearly homogeneous in the components of x; i.e.,
we have
(32) D(λx, x, y) = λD(x, x, y) for λ > 0.
If F(x, y) is increasing in the components of x, then by inspecting (31),
it can be seen that D(x, x, y) will also be increasing in the components of x.
This monotonicity property and the homogeneity property (32) lead us to use
the deflation function D to define the following conditional Malmquist [1953]
quantity index for the x goods:
(33) QM (x1
, x2
, x, y) ≡ D(x2
, x, y)/D(x1
, x, y)
where xt
is the consumer’s observed period t quantity vector for the x goods
for t = 1, 2.
By using the homogeneity property (32), it can be seen that QM (λx1
, x2
,
x, y) = λ−1
QM (x1
, x2
, x, y) and QM (x1
, λx2
, x, y) = λQM (x1
, x2
, x, y) for λ >
0. Thus our economic quantity index has homogeneity properties with respect
to x1
and x2
that correspond to the homogeneity properties (21) and (22) in
the test approach.
If the y vector does not actually appear in the consumer’s utility function
(this is the case considered by Malmquist [1953]), then the reference vectors x
and y can be replaced by the reference utility level and u ≡ F(x), D(x, x, y)
can be replaced by D∗
(x, u) ≡ maxλ≥0{λ : F(λ−1
x) ≥ u}, and QM (x1
, x2
, x, y)
can be replaced by
(34) Q∗
M (x1
, x2
, u) ≡ D∗
(x2
, u)/D∗
(x1
, u).
14 Essays in Index Number Theory 1. Overview 15
This is the definition of the Malmquist quantity index that we utilize in Chapters
5, 7 and 11 below.12
If, in addition to not depending on y, F(x) is linearly homogeneous so that
F(λx) = λF(x) for λ > 0, then for u > 0 we have:
D∗
(x, u) ≡ max
λ≥0
{λ : F(λ−1
x) ≥ u}
= max
λ≥0
{λ : λ−1
F(x) ≥ u}
= max
λ≥0
{λ : λ ≤ F(x)/u}
= F(x)/u.(35)
Thus under the assumptions that F(x, y) does not depend on y and F(x) is
linearly homogeneous in x, the Malmquist quantity index (33) or (34) reduces
to
(36) Q∗
K(x1
, x2
, u) = [F(x2
)/u]/[F(x1
)/u] = F(x2
)/F(x1
)
which is again consistent with our earlier economic model defined by (4) and
(5).
Another way that we can end up with (4) and (5) is to assume that F(x, y)
is homogeneously separable in x; i.e., assume F satisfies (29). Under these
conditions, the distance function D defined by (31) becomes:
D∗
(x, x, y) ≡ max
λ≥0
{λ : F∗
[f(λ−1
x), y] ≥ F∗
[f(x), y]}
= max
λ≥0
{λ : f(λ−1
x) ≥ f(x)}
since F∗
is increasing in its first variable
= max
λ≥0
{λ : λ−1
f(x) ≥ f(x)} using the linear homogeneity of f
= max
λ≥0
{λ : λ ≤ f(x)/f(x)} assuming that f(x) > 0
= f(x)/f(x).(37)
Substituting (37) into (33) yields QM (x1
, x2
, x, y) = f(x2
)/f(x1
) so that, in
this homogenously separable case, the Malmquist quantity index is independent
of the reference vectors x and y and is consistent with our preliminary economic
model defined by equations (4) and (5) above.
Note that the optimization problem (31) that defined the distance function
D did not involve consumer prices. However, prices soon make their
appearance when we attempt to find observable bounds on the theoretical
12
The mathematical properties of D∗
are studied in Section 5 of Chapter 6.
Malmquist indexes defined by (33) or (34); see Chapter 5, Section 4; Chapter
7, Section 3; and Chapter 11, Section 6.
We have not specified how the reference vectors x and y, which appear
in the definition of the Kon¨us price index (27) and the Malmquist quantity
index (33), are chosen. In practical situations, we generally choose x, y to
be either (x1
, y1
) or (x2
, y2
) or an average of these two vectors which are the
observed data pertaining to periods 1 and 2. In the following section, we
shall indicate how these choices lead to observable bounds on the theoretical
economic indexes.
Triplett [1990a] [1991] has recently suggested that economic theory could
be helpful in deciding how industrial, commodity and occupational classifications
could be set up. One of the methods he suggested was essentially based on
(29), the assumption that a homogeneously separable aggregate exists on the
producer side of the economy13
(two of the other methods that he suggested,
Hicksian and Leontief aggregation, we shall discuss shortly). In order to implement
Triplett’s suggestion, we need methods for determining whether separable
aggregates exist; i.e., we need methods for testing whether assumption (29) is
valid. In the early literature on testing for the existence of a separable ag-
gregate,14
researchers generally assumed that under the alternative hypothesis
of nonseparability, the function F(x, y) on the left hand side of (29) was a
flexible functional form15
and they then found restrictions on the parameters
of F which collapsed F(x, y) down into the null hypothesis form, F∗
[f(x), y].
However, Blackorby, Primont and Russell [1977c] [1978] showed that for commonly
used flexible functional forms for F (such as the translog), under the
null hypothesis of homogeneous separability, the resulting F∗
or f (or both)
would necessarily be inflexible. This problem with the early separability tests
has been overcome; see Hall [1973], Woodland [1978], Blackorby, Schworm and
Fisher [1986], and Diewert and Wales [1991]. However, the results of this later
separability testing literature are not terribly encouraging from the viewpoint of
finding an effective economic classification mechanism: the available empirical
results generally reject the assumption that a separable aggregate exists.
13
Actually Triplett [1990a] assumed only a separable structure (not a homogeneously
separable structure) so that (29) holds but f(x) is not necessarily
linearly homogeneous. In this case, the economic price index defined by (27)
does simplify to (28) but it does not simplify to a ratio of unit cost functions,
c(p2
)/c(p1
). The type of separability assumed by Triplett is studied at great
length in Blackorby, Primont and Russell [1978].
14
See Berndt and Christensen [1973b] [1974], Berndt and Wood [1975], Jorgenson
and Lau [1975], and Denny and Fuss [1977].
15
A flexible functional form is one that can provide a second order approximation
to an arbitrary twice continuously differentiable functional form; see
Chapter 6, Section 10.
16 Essays in Index Number Theory 1. Overview 17
There is one additional theoretical topic which arises in the economic approach
to index numbers: namely the role of price or quantity proportionality
as an aggregating mechanism.
Consider first the role of price proportionality. Suppose that the consumer
or producer is solving a cost minimization problem like (25) for T periods.
Assume that the observed x vector in period t, xt
≡ (xt
1, . . . , xt
N ), solves the
following cost minimization problem:
(38) min
x
{pt
· x : F(x, yt
) ≥ F(xt
, yt
) = ut
} ≡ C(pt
, yt
, ut
)
for t = 1, 2, . . . , T where (pt
, xt
, yt
) is the observed price and quantity data
pertaining to the consumer or producer in period t. Suppose that in addition to
the above assumption of cost minimizing behavior on the part of the consumer
or producer, the prices pt
≡ (pt
1, . . . , pt
N ) vary in strict proportion over time so
that
(39) pt
= λt
α, t = 1, 2, . . . , T,
where λt
> 0 is the period t proportionality factor and α ≡ (α1, . . . , αN ) is a
fixed vector. Under these conditions, it is natural to choose the period t price
level for the x goods Pt
to be λt
and, of course, the corresponding aggregate
quantity level Qt
will have to equal pt
· xt
/λt
so that the adding up relations
(1) hold. Thus define
(40) Pt
≡ λt
; Qt
≡ pt
· xt
/λt
for t = 1, 2, . . ., T.
Substituting (39) into (38) yields for t = 1, . . . , T:
(41) pt
· xt
= C(pt
, yt
, ut
) = C(λt
α, yt
, ut
) = λt
C(α, yt
, ut
)
where the last equality follows from the linear homogeneity of C(p, y, u) in p.
Comparing (40) and (41), we have
(42) Qt
= C(α, yt
, ut
), t = 1, 2, . . . , T.
Now consider the following sequence of cost minimization problems involving
the scalar aggregate Q:
(43) min
Q
{Pt
Q : Q = F(yt
, ut
)}, t = 1, 2, . . . , T,
where the original micro function F has now been replaced by the aggregate
commodity requirements function F defined by
(44) F(y, u) ≡ C(α, y, u).
Using (42) and (44), it can be seen that the Qt
defined in (40) solves the
tth minimization problem in (43) for t = 1, 2, . . . , T and we have Pt
Qt
=
pt
· xt
for t = 1, . . . , T. Thus if the prices of a group of goods vary in strict
proportion over time, then there exist aggregate prices and quantities, Pt
and
Qt
, defined by (40) and the aggregate quantity Qt
can be treated as if it were
an actual microeconomic good. This is a rough and ready version of Hicks’
[1946; 312–313] Aggregation Theorem. For a more detailed exposition of this
result in both the consumer and producer contexts, see Chapter 15 below.
Chapter 16 below spells out some implications of Hicks’ Aggregation Theorem
for elasticities of substitution. Chapter 16 also considers what happens if prices
vary only approximately in fixed proportions rather than exactly proportionally.
Now consider the role of quantity proportionality. Suppose that F(x, y) is
the consumer’s utility function and that the observed quantity data pertaining
to the consumer for period t are (xt
, yt
) for t = 1, 2, . . ., T. The period t utility
level is ut
defined as
(45) ut
≡ F(xt
, yt
), t = 1, . . . , T.
Suppose that the quantity vectors xt
≡ (xt
1, . . . , xt
N ) vary in strict proportion
over time so that we have
(46) xt
≡ λt
β, t = 1, . . . , T,
where β ≡ (β1, . . . , βN ) is a fixed vector and λt
> 0 is the period t factor of
proportionality. Define an aggregate utility function F for an aggregate Q of
the x goods and an arbitrary y vector as follows:
(47) F(Q, y) ≡ F(Qβ, y).
Under suitable regularity conditions on F, the aggregate function F will inherit
its properties. If we define the period t aggregate Qt
to be
(48) Qt
≡ λt
, t = 1, . . . , T,
then the aggregated data (Qt
, yt
) will be consistent with the micro data (xt
, yt
)
since
ut
= F(xt
, yt
) by (45)(49)
= F(λt
β, yt
) by (46)
= F(λt
, yt
) by (47)
= F(Qt
, yt
) by (48).
18 Essays in Index Number Theory 1. Overview 19
This is Leontief’s [1936] Aggregation Theorem in the consumer context. Of
course, a similar result holds in the producer context. If pt
is the micro period t
price vector for the x goods, then the aggregate period t price Pt
for the x
aggregate can be defined as
(50) Pt
≡ pt
· xt
/Qt
, t = 1, . . . , T.
Leontief’s Aggregation Theorem makes an appearance in Chapter 10 below.
In that chapter, Allen and Diewert attempt to determine empirically
whether proportional quantity variation or proportional price variation is more
justified. They suggest that if quantities vary proportionally, then the aggregates
should be constructed using equations (48) and (50), while if prices vary
proportionally, then equations (40) should be used to construct the aggregates.
Leontief and Hicksian aggregation can be regarded as special cases of the
homogeneous separability model (29) above. For Leontief type aggregation,
the micro aggregator function f has the following functional form:
(51) f(x1, . . . , xN ) ≡ min
i
{xi/βi : i = 1, 2, . . ., N}.
For Hicksian aggregation, the micro aggregator function f has the following
functional form:
(52) f(x1, . . . , xN ) ≡ α1x1 + · · · + αN xN .
Thus the aggregation theorems of Leontief and Hicks are consistent with the
assumption of homogeneous separability where the micro aggregator function
is either a fixed coefficients, no substitution Leontief aggregator function or it is
a linear, infinite substitution aggregator function. However, these aggregation
theorems can be given a broader interpretation: (i) in planned economies, some
groups of commodities may be produced in fixed proportions by central decree
irrespective of the technology, and (ii) in market economies, some groups of
prices may move proportionally due to regulation or due to union wage policies.
In this section, we have given a brief outline of how economic theory can be
used in order to define a theoretical Kon¨us price index of the form (27) or (28)
and to define a theoretical Malmquist quantity index of the form (33) or (34).
However elegant the theory of these indexes may be, there is a practical problem
associated with the empirical use of these indexes. The problem is that the
theoretical price index depends on knowing the consumer’s or producer’s cost
function and the theoretical quantity index depends on knowing the economic
agent’s distance or deflation function and we, as outside observers, do not have
this knowledge. In the following section, we turn to a description of some of
the methods which have been suggested to remedy this knowledge deficiency.
4. The Economic Approach: Empirical Approximations
There are at least three different ways to operationalize the theoretical indexes
defined in the previous section: (i) econometric estimation; (ii) nonparametric
bounds; and (iii) the theory of exact index numbers. We shall consider each of
these possibilities in turn in this section.
Consider first an approach that relies on econometric estimation. This
approach is relatively straightforward: given time series or cross section data on
production units or households, we can postulate a functional form for the cost
function C or the aggregator function F and estimate the unknown parameters
which appear in the functional form by regression analysis. Typically, we choose
functional forms for C or F that are flexible; i.e., the functional form has a
sufficient number of free parameters so that it can provide a second order
approximation to an arbitrary cost or aggregator function (with the appropriate
regularity conditions).16
This econometric approach is discussed in Section 6.4
of Chapter 2 and in Sections 10 and 11 of Chapter 6 below.
The problem with the econometric approach that relies on flexible functional
forms is that it becomes unwieldy or impossible17
as the number of
goods N to be aggregated becomes large, since the number of unknown parameters
to be estimated grows at a rate approximately equal to (1/2)N2
. This
leads us to discuss the second approach for implementing the economic theory
of index numbers, the method of bounds.
To show how the bounds method works, consider the tth cost minimization
problem in (38) above and suppose that the observed x vector for period t, xt
,
solves this problem for t = 1, 2. We suppose that we can observe the data
(pt
, xt
, yt
) for t = 1, 2. We shall use the vectors (xt
, yt
) for t = 1, 2 as the
reference vectors (x, y) in (27) which defines the theoretical Kon¨us price index
PK(p1
, p2
, x, y). Consider the following cost minimization problem:
(53) C(p2
, x1
, y1
) ≡ min
x
{p2
· x : F(x, y1
) ≥ F(x1
, y1
} ≤ p2
· x1
16
Recent work on functional forms has moved beyond the local approximation
idea that is inherent in the definition of a flexible functional form and
has attempted to provide some global approximations; e.g., see the seminonparametric
functional form literature started by Gallant [1981] [1982], Barnett
and Jonas [1983], Barnett and Yue [1988], and Barnett, Geweke and Wolfe
[1991]. See also the attempts by Diewert and Wales [1992a] [1992b] to provide
functional forms that are flexible but at the same time can approximate arbitrary
Engel curves (in the consumer context) or input expansion paths (in the
producer context).
17
This impossibility result will be discussed in more detail in Volume II; see
Diewert [1992a].
20 Essays in Index Number Theory 1. Overview 21
where the inequality in (53) follows since x1
is a feasible (but not necessarily
optimal) solution to the cost minimization problem in (53). Similarly, we have:
(54) C(p1
, x2
, y2
) ≡ min
x
{p1
· x : F(x, y2
) ≥ F(x2
, y2
} ≤ p1
· x2
where the inequality in (54) follows since x2
is a feasible solution for the cost
minimization problem in (54). If C(p1
, x2
, y2
) > 0 and p1
· x2
> 0, then (54)
may be rewritten as
(55) 1/C(p1
, x2
, y2
) ≥ 1/p1
· x2
.
We can now substitute the inequalities (53) and (55) into (27) for (x, y) ≡
(x1
, y1
) and (x, y) ≡ (x2
, y2
) respectively to deduce inequalities or bounds on
the following Kon¨us price indexes PK (assuming that all costs are positive):
PK(p1
, p2
, x1
, y1
) ≡ C(p2
, x1
, y1
)/C(p1
, x1
, y1
)(56)
= C(p2
, x1
, y1
)/p1
· x1
≤ p2
· x1
/p1
· x1
;
PK(p1
, p2
, x2
, y2
) ≡ C(p2
, x2
, y2
)/C(p1
, x2
, y2
)(57)
= p2
· x2
/C(p1
, x2
, y2
)
≥ p2
· x2
/p1
· x2
.
Thus (56) says that the (unobservable) theoretical Kon¨us price index P(p1
, p2
,
x1
, y1
) (which uses the period 1 level of utility u1
≡ F(x1
, y1
) as the reference
indifference surface) is bounded from above by the (observable) Laspeyres price
index p2
· x1
/p1
· x1
and (57) says that the (unobservable) Kon¨us price index
P(p1
, p2
, x2
, y2
) (which uses the period 2 level of utility u2
≡ F(x2
, y2
) as
the reference indifference surface) is bounded from below by the (observable)
Paasche price index p2
· x2
/p1
· x2
.
Kon¨us [1924] worked out a brilliant technique which enables us to go beyond
the one sided bounds (56) and (57) to deduce two sided bounds for a theoretical
Kon¨us price index: under suitable regularity conditions, we can take a
weighted average of the two reference vectors (x1
, y1
) and (x2
, y2
) and deduce
that the resulting theoretical price index P[p1
, p2
, λx1
+(1−λ)x2
, λy1
+(1−λ)y2
]
lies between the Laspeyres and Paasche price indexes for some λ such that
0 < λ < 1. This technique of proof is used repeatedly in this volume; e.g., see
Section 3 of Chapter 5, Sections 2 and 3 of Chapter 7 and Sections 3 and 4 of
Chapter 11.
The above argument shows that the gap between the Paasche and Laspeyres
price indexes will include the value of a theoretical economic index. This suggests
that taking some sort of average or symmetric mean of the Paasche
and Laspeyres price indexes should yield an empirically observable price index
which is “close” to an unobservable theoretical price index. If we take
the geometric mean of the Laspeyres and Paasche price indexes, we obtain the
following index:
(58) (p2
· x1
/p1
· x1
)
1
2 (p2
· x2
/p1
· x2
)
1
2 = PF (p1
, p2
, x1
, x2
)
where PF is the Fisher [1922] ideal price index defined earlier by (24). Recall
that the Fisher ideal index seemed best from the viewpoint of the test or
axiomatic approach to index number theory. The above argument suggests
that it is also very good from the viewpoint of the economic approach.
The theory of bounds can be extended to cover the case where there are
more than two observations. This leads us into revealed preference theory and
Afriat’s [1967] nonparametric approach to preference estimation in the context
of consumer theory. There is also a corresponding nonparametric approach to
technology estimation in the context of producer theory; see the discussion at
the end of Chapter 7. We pursue this topic in Volume II.
We turn now to the exact index number approach to approximating the
unobservable economic indexes defined in the previous section. In this approach,
an explicit functional form for the aggregator function or the dual cost
function is chosen. With a strategic choice of functional form plus the assumption
of optimizing behavior, it turns out that we can determine the value
of an economic price or quantity index using only the observable price and
quantity data (p1
, p2
, x1
, x2
) that pertain to the goods to be aggregated for
observations 1 and 2. An example will illustrate how the method works.
Suppose that the cost function C∗
which appears in (28) has the following
explicit functional form:
(59) C∗
(p, u) ≡ (pT
Bp)
1
2 u
where B ≡ [bij] is an N × N symmetric matrix of parameters which satisfy
certain regularity conditions.18
Assume that the consumer or producer is cost minimizing in periods 1
and 2. Using (59), we have:
p1
· x1
= C∗
(p1
, u1
) = (p1T
Bp1
)
1
2 u1
;(60)
p2
· x2
= C∗
(p2
, u2
) = (p2T
Bp2
)
1
2 u2
.(61)
Cost minimizing input demand functions can be obtained by differentiating
C∗
(pt
, ut
) with respect to the components of the price vector (see Lemma 4
18
These regularity conditions ensure that C∗
(p, u) is nondecreasing and concave
in p over a domain of prices which includes the observed price vectors p1
and
p2
.
22 Essays in Index Number Theory 1. Overview 23
in Section 2 of Chapter 6, Shephard’s Lemma). Thus differentiating (59), the
cost minimizing observed vectors x1
and x2
satisfy:
x1
= pC∗
(p1
, u1
) = (p1T
Bp1
)− 1
2 Bp1
u1
;(62)
x2
= pC∗
(p2
, u2
) = (p2T
Bp2
)− 1
2 Bp2
u2
(63)
where pC∗
(pt
, ut
) ≡ [∂C∗
(pt
, ut
)/∂p1, . . . , ∂C∗
(pt
, ut
)/∂pN ]T
for t = 1, 2.
Now let us evaluate the Kon¨us price index (28) for an arbitrary u > 0:
P∗
K(p1
, p2
, u) ≡ C∗
(p2
, u)/C∗
(p1
, u)(64)
= (p2T
Bp2
)
1
2 /(p1T
Bp1
)
1
2 using (59).
Consider the formula for the Fisher ideal price index (24). Replace pt
· xt
by
(ptT
Bpt
)
1
2 ut
for t = 1, 2 (see (60) and (61) above). Then
PF (p1
, p2
, x1
, x2
) =
p2
· x1
(p2T
Bp2
)
1
2 u2
p1 · x2(p1T Bp1)
1
2 u1
1
2
=
(p1T
Bp1
)− 1
2 p2T
Bp1
u1
(p2T
Bp2
)
1
2 u2
(p2T Bp2)− 1
2 p1T Bp2u2(p1T Bp1)
1
2 u1
1
2
using (62) and (63) to eliminate x1
and x2
= (p2T
Bp2
)
1
2 /(p1T
Bp1
)
1
2(65)
where (65) follows from the line above using the symmetry of B which implies
that p2T
Bp1
= p1T
Bp2
. Since (65) equals (64), we have for any u > 0:
(66) P∗
K (p1
, p2
, u) = PF (p1
, p2
, x1
, x2
).
Thus the theoretical Kon¨us price index P∗
K(p1
, p2
, u) is exactly equal to the
empirically observable Fisher ideal price index PF (p1
, p2
, x1
, x2
), provided that
there is cost minimizing behavior in the two periods and that the functional
form for the cost function C∗
is given by (59). In this case, the Fisher ideal
price index PF is said to be exact for the cost function defined by (59).
The theory of exact index numbers was started by Kon¨us and Byushgens
[1926] and resurrected by Pollak [1971a], Afriat [1972b] and Samuelson and
Swamy [1974]. Diewert noticed (see Chapter 8 below) that the cost function
defined by (59) is flexible; i.e., it can provide a second order approximation
to an arbitrary twice continuously differentiable cost function that is dual to
a linearly homogeneous aggregator function. Thus Diewert called the Fisher
price index PF a superlative index number formula since it is exact for a flexible
functional form. Many additional superlative index number formulae are
derived in Chapter 8 below.
Note that we have now provided three separate justifications for the use
of the Fisher ideal price index.
5. Other Approaches to Index Number Theory
The earliest approach to index number theory was the fixed basket approach. In
this approach, a representative basket of quantities x ≡ (x1, . . . , xN ) is chosen.
The price level Pt
for period t is defined to be the cost of purchasing this fixed
basket at the period t prices, pt
≡ (pt
1, . . . , pT
N ):
(67) Pt
≡ pt
· x, t = 1, . . . , T.
This approach eventually evolved into the axiomatic approach to index number
theory as researchers attempted to be more precise about the basket x; see
Chapter 2, Section 2.
Another early approach to index number theory was the statistical approach
initiated by Jevons [1865] [1884] and pushed forward by Edgeworth
[1888] [1901] [1923]. In this approach, ratios of the price of each good i in
period 2 to its price in period 1, p2
i /p1
i , for i = 1, . . . , N are regarded as independent
random variables which have a common mean: (one plus) the inflation
rate going from period 1 to period 2. Thus by the law of large numbers, an
average of these price ratios such as
N
i=1(1/N)(p2
i /p1
i ) should approach (one
plus) the inflation rate as N (the number of goods) becomes large. This approach
was effectively criticized by Fisher [1911], Walsh [1924], Bowley [1928]
and Keynes [1930; 71–81] and eventually fell out of favor; see Section 3 of
Chapter 2. However, more recently “neostatistical” approaches that rely on
minimizing some sort of statistical error have been suggested; see Section 10
of Chapter 5 and Section 4 of Chapter 7 for descriptions of the neostatistical
approaches of Stuvel [1957], van Yzeren [1956], Theil [1960], Kloek and deWit
[1961] and Banerjee [1975]. Finally, a few papers have looked at the statistical
sampling problems involved in the construction of a conventional Laspeyres
price index under conditions of incomplete information; see Szulc [1989] and
Balk [1991].
Yet another approach to index number theory is the continuous time approach
initiated by Divisia [1926]. Up to this point, we have regarded the
basic microeconomic price and quantity data pt
i, xt
i as functions of a discrete
time indicator, t = 1, 2, . . . , T. In the Divisia approach, prices and quantities
pi(t), xi(t) are regarded as functions of a continuous time variable t where
0 ≤ t ≤ T. The problem with this approach is that economic data are almost
never available as continuous time variables. Hence for empirical purposes,
it is necessary to approximate the continuous time Divisia price and quantity
indexes by discrete time data. Since there are many ways of performing these
approximations,19
the Divisia approach does not seem to lead to a definitive result.
The Divisia approach is discussed in Section 5 of Chapter 2 and Section 7
of Chapter 7; it will be discussed further in Volume II.
19
See Diewert [1980; 443–446].
24 Essays in Index Number Theory 1. Overview 25
A final approach to index number theory might be termed the difference
approach. The traditional (bilateral) approach to index number theory is based
on a ratio approach. For example, the Kon¨us price index defined by (28) is a
ratio of cost functions C∗
[p2
, F(x)]/C∗
[p1
, F(x)] evaluated at different prices
p1
and p2
, holding the quantity vector x fixed. As another example, the Allen
[1949] quantity index20
QA is defined as a ratio of cost functions evaluated at
the period 1 and 2 quantity vectors, x1
and x2
, and at a common reference
price vector p:
(68) QA(x1
, x2
, p) ≡ C∗
[p, F(x2
)]/C∗
[p, F(x1
)].
However, some early index number researchers such as Bennet [1920] and Montgomery
[1929] [1937] used differences instead of ratios.21
Thus the difference
counterparts to the Kon¨us price index and the Allen quantity index can be
defined as follows:
PD(p1
, p2
, x) ≡ C∗
[p2
, F(x)] − C∗
[p1
, F(x)];(69)
QD(x1
, x2
, p) ≡ C∗
[p, F(x2
)] − C∗
[p, F(x1
)].(70)
The difference price index PD has been forgotten in modern economic analysis
but the difference quantity index QD lives on and is now known as a consumer
surplus measure. In fact, if we set the reference prices in (70) equal to p1
and p2
, we obtain the equivalent and compensating variations defined by Hicks
[1941–42; 128] [1946; 331]; see Section 7 of Chapter 7.
For more material on Bennet’s contributions, see Sections 5 and 6 of Chapter
2 and Section 7 of Chapter 7. The difference quantity index defined by (70)
will be discussed more fully in Volume II; see Diewert [1992b].
6. The New Good Problem
In both the economic and axiomatic approaches to index number theory, we
require price and quantity information on the same list of commodities for
two or more observations. How can we apply these theories when the list of
commodities changes?
20
For more on the Allen quantity index, see Section 3 of Chapter 7.
21
Bennet [1920] developed an economic approach while Montgomery [1929]
[1937] developed a test approach using differences. Bennet [1920; 461] also
developed Divisia’s [1926] mechanical differentiation approach (before Divisia’s
publications appeared) as well as a mechanical differencing approach.
The list of commodities utilized or produced by a consumer or producer
could change for a large number of reasons: (i) the consumer’s wealth or educational
status could change from period to period, inducing a demand for different
goods and services; (ii) advertising could make consumers or producers
aware of a product and trigger a demand for it; (iii) a transportation improvement
could lower shipping costs and lead to an increased availability of goods
along the route of the improvement; (iv) a prohibitive tariff could be lowered
opening up the domestic market to previously unavailable imported goods; (v)
government regulations prohibiting the consumption of certain goods could be
relaxed;22
(vi) population growth in a region could allow new firms offering specialized
goods or services to locate in the region (e.g., a speciality bookstore,
a used machinery dealer, the delivery of cable television, a sports franchise,
etc.); and (vii) technological progress creates thousands of new products every
year (e.g., video recorders, video cameras, more powerful computers, robots,
thousands of new food products, etc.). Thus the “new” good problem is a
tremendously pervasive phenomenon.
Obviously, the quantity of a “new” good produced or consumed in the
period before its introduction is zero. However, the economic theory of index
numbers requires a price to go along with this zero quantity — what should
this price be? Hicks [1940; 114] provided a satisfactory theoretical solution to
this problem many years ago: (i) from the viewpoint of a consumer or producer
buying units of the “new” good in the first period that it makes its appearance,
the price in the previous period should be that price which would have been
just high enough to have driven the purchaser’s demand down to zero and
(ii) from the viewpoint of the producer of the “new” good, the price in the
previous period should be that price which would have been just low enough
to have induced the producer to supply zero units. Although this Hicksian
solution to the pricing problem for new goods is theoretically satisfactory, it
is not so satisfactory from a practical point of view since the determination
of the appropriate shadow prices will generally require rather sophisticated
econometric analysis and statistical agencies are not usually in a position to
undertake this.23
In view of the increasing pace of the introduction of new
goods and services in most economies, we are led to conclude that the relative
22
Examples of this phenomenon include: (i) the repeal of prohibition in the
U.S.; (ii) the recent striking down of the German purity of beer laws which
were used to exclude imports; (iii) countless local procurement rules which are
gradually being whittled down by national and international agreements; and
(iv) privatization of previous government monopolies which often leads to an
increased choice set.
23
Statistical agencies in Canada and the U.S. are currently doing some econometric
analysis to deal with the new good problem (or the problem of quality
change as it is sometimes called) in the areas of housing and computers.
26 Essays in Index Number Theory 1. Overview 27
neglect of the new good problem has probably led to an upward bias in the
measurement of inflation in most market economies. This topic requires a great
deal of additional research.
The new good problem is discussed in more detail below in Section 10 of
Chapter 2 and in Section 11 of Chapter 5.
7. Symmetric Means
A mean or an average of N numbers x1, x2, . . . , xN can be defined as a function
M(x1, . . . , xN ) of the N numbers which has the following property:
(71) M(k, k, . . . , k) = k;
i.e., if all N numbers are equal to the same number k, then the mean of these
numbers is also equal to k. A symmetric mean M(x1, . . . , xN ) has the additional
property that the function treats each variable xn in a symmetric manner,
so that we can interchange the order in which the xn appear in the function
but the function value remains unchanged.
Symmetric means are used in index number theory fairly frequently: recall
our earlier discussion around equation (58) where we took a particular symmetric
mean (the geometric average) of the Paasche and Laspeyres price indexes
to obtain an approximation to a theoretical price index. Symmetric means are
used in this volume in Appendix 3 of Chapter 9.
In addition to being used in index number theory, symmetric means are
used in two other areas of economics: (i) the measurement of welfare and
inequality and (ii) in modeling choice under uncertainty.
In view of the importance of symmetric means in measurement economics,
a systematic survey of symmetric means from an axiomatic perspective is presented
in Chapter 14 below. In Chapter 14, we also develop a model of choice
under uncertainty, due originally to Dekel [1986], Chew [1989] and Chew and
Epstein [1989a], from a new axiomatic perspective. Our Implicit Expected
Utility Model turns out to be much more flexible than the traditional Expected
Utility Model. We illustrate this by applying our model to some simple
problems involving insurance, gambling and investing.
8. Mathematical and Economic Prerequisites
A substantial portion of the proofs in this volume are quite elementary in
nature. As in Section 4 of this overview, quite often proofs rest on feasibility
arguments; i.e., we need only show that a certain set of quantities is feasible
(but not necessarily optimal) for some optimization problem. Throughout the
book, calculus and some matrix algebra will enable the reader to follow the
majority of the proofs.
Occasionally, some specialized mathematical material is used which is not
covered in the usual mathematics for economists course which is given to advanced
undergraduates or beginning graduate students.
In Chapters 6 and 15, some use is made of the Maximum Theorem of
Debreu [1952; 889-890][1959; 19] and Berge [1963; 116], and also of specialized
material on convex and concave functions such as the Fenchel [1953] closure
operation. These somewhat complex results are required only to establish various
continuity properties; the reader who is primarily interested in economics
can simply ignore the material on continuity problems. Good references for
the results on convex sets and concave functions that are used in Chapters 6
and 15 are Rockafellar [1970] and Roberts and Varberg [1973]. However, we
use only a small fraction of the material presented in these references.
The theory of functional equations is used when developing the axiomatic
approach to index number theory but, as we saw in Section 2 of this chapter,
the functional equations that arise are usually so simple that no specialized
preparation is required. An exception to this rule occurs in Chapter 14 where
Aczel’s [1966] monograph on functional equations is used in a few places.
Chapter 14 also uses some material on two specialized topics which are not
usually covered in standard mathematics for economists courses: (i) inequalities
and (ii) generalized concave functions. The main inequality used is the CauchySchwarz
inequality. Proofs of it may be found in Hardy, Littlewood and Polya
[1934; 16] and many other places. Two useful references for the material on
generalized concave functions are Diewert, Avriel and Zang [1981] and Avriel,
Diewert, Schaible and Zang [1988].
9. Personal Notes
It seems appropriate to conclude this introductory chapter by indicating how
I got into the index number business in the first place.
As a student at Berkeley during the years 1964–1968, I had become interested
in index number and aggregation problems by reading some of Richard
Stone’s work. However, I did not immediately pursue this interest. While I
was an Assistant Professor at the University of Chicago during the years 1968–
1970, I was asked to referee a paper by Sydney Afriat for the Journal of Political
Economy. I had to work on this paper for two or three weeks before I finally
understood it. I wrote up a lengthy referee’s report with a recommendation to
28 Essays in Index Number Theory 1. Overview 29
accept subject to the author revising the paper to make it more understandable.
Sydney did not choose to revise the paper. Instead he resubmitted it to
the International Economic Review where it was published as Afriat [1973d].24
In any case, I had become acquainted with Afriat’s research, and he eventually
sent me his classic work on index numbers, Afriat [1972b], which was originally
written in 1970.
Afriat [1972b] developed the theory of exact index numbers; i.e., he showed
how certain functional forms for a utility function were consistent with certain
index number formulae.25
I became very interested in this approach to index
number theory since my Ph.D. thesis was concerned with functional form problems
for utility and production functions. Thus while visiting Stanford during
the summer of 1973,26
I decided to look up some of the more obscure articles
that were listed in Afriat’s [1972b] references, including Kon¨us [1924] and
Byushgens [1925].27
Fortunately, the Hoover Institution Library at Stanford
had all of the old issues of the Russian journal Voprosi Konyunkturi and so I
stumbled onto the classic article on exact index numbers and duality theory
by Kon¨us and Byushgens [1926] which had not been translated into English
or referred to in the English language literature on index numbers.28
I studied
the technique of proof used by Kon¨us and Byushgens, and I was able to utilize
their techniques to establish a large class of exact index number formulae. This
led me to write a Stanford Technical Report, “Homogeneous Weak Separability
and Exact Index Numbers,” during the summer of 1973. I submitted this
paper to the Review of Economic Statistics and the Quarterly Journal of Economics
but they both rejected the paper. I then submitted it to the Journal
of Econometrics because I thought that I might get a better reception there,
since I was an Associate Editor at the time. Dennis Aigner accepted the paper
except that he asked me to drop the material on homogeneous weak separability
(recall our discussion around equation (30) above) which was mostly due to
24
Several years later, I dug out this old referee’s report, revised it a bit and
published it as Diewert [1973b]. I also gave a seminar on this material when
I visited Harvard during the summer of 1970. I shared an office with Giora
Hanoch who immediately saw that Afriat’s revealed preference techniques could
also be applied to production theory; see Hanoch and Rothschild [1972]. My
thanks to Zvi Griliches and Dale Jorgenson for arranging that visit to Harvard.
25
Pollak [1971a] picked up on Afriat’s work and provided a beautiful exposition
of the economic approach to index number theory.
26
My thanks to Larry Lau for arranging this and other summer visits to
Stanford.
27
I believe Afriat found these references in Schultz [1939].
28
Actually Schultz [1939; 9] quoted parts of a letter from Kon¨us in which Kon¨us
alluded to his joint article with Byushgens (Buscheguennce) but Kon¨us did not
provide a title or place of publication of his joint article.
Shephard [1953]. I agreed and the paper was published as Diewert [1976a] and
is reprinted as Chapter 8 in the present volume.
Although the Review of Economics and Statistics had rejected my paper,
they remembered that I had submitted a paper on index number theory to
them. Thus in 1975, I was asked to referee Vartia [1976a]. I was very enthusiastic
about the paper and recommended publication, but the Review rejected it
despite my positive report. However, my refereeing experience proved fruitful
since it led to a friendship with Vartia, and it also stimulated me to write a
paper in the summer of 1975 entitled “Ideal Log Change Index Numbers and
Consistency in Aggregation.” This paper was eventually published as Diewert
[1978b] and is reprinted as Chapter 9 in this volume.
My first two papers on index numbers, Chapters 8 and 9 below, were concerned
with the properties of superlative index numbers. Irving Fisher [1922;
244] called an index number formula “superlative” if it agreed (numerically for
his chosen data set of 36 commodities for the years 1913–1918) very closely
with the Fisher ideal index numbers for the same data set. Since the Fisher
ideal index is exact for a flexible functional form, I decided to use Fisher’s
term “superlative” to describe any index number formula that was exact for a
flexible functional form (for either a unit cost function or an aggregator function).
Somewhat surprisingly, my classification of superlative index number
formulae turns out to be roughly equivalent to Fisher’s classification since all
known superlative index number formulae (according to my definition) closely
approximate each other numerically; see Theorems 5 and 6 in Chapter 9.
Over the years, I have been gratified to see that the superlative index
number concept has received some recognition in statistical agency circles; see
the Bureau of Labor Statistics [1983], Hill [1988] and Triplett [1992].
It seems appropriate to note here that my attitude towards the test approach
to index numbers has changed over the years. Initially, I was very much
influenced by Frisch’s [1936] survey article on index numbers (which can still
be read with profit) where he criticized the test approach for its indeterminacy:
no index number formula satisfied all “reasonable” tests and it was difficult to
determine which tests should be dropped in order to end up with a consistent
set of tests leading to a unique index number formula. Thus Frisch (and I) criticized
the test approach for not leading to a definite result. My early papers
on index number theory favored the economic approach (which I still generally
favor).
My early enthusiasm for the economic approach to index number theory
has been tempered by two considerations. The first is that it is sometimes
very difficult to implement the economic approach empirically. For example,
in regulated industries, it is usually unrealistic to assume that firms are competitively
maximizing profits subject to the constraints of technology and fixed
exogenous prices. Under these circumstances, the economic approach can be
30 Essays in Index Number Theory 1. Overview 31
rescued if observed output prices are replaced by appropriate shadow prices,29
which are usually equal or proportional to marginal costs. The problem with
this solution is that many regulated firms (such as telecommunications firms)
generally produce thousands of outputs and it is virtually impossible to determine
the marginal cost of each product with any degree of accuracy. Thus the
economic approach can break down from the viewpoint of practical measurement
problems and hence must be replaced by another approach. The second
consideration which increased my appreciation for the test approach was the
discovery that the Fisher ideal index number formula seems to have emerged
as the “best” alternative from the viewpoint of the axiomatic approach, since,
as I show in Chapter 13, it satisfies some 20 “reasonable” tests. None of the
other leading index number formulae satisfy as many “reasonable” tests. Thus
the test approach does seem to lead to a determinate choice of functional form
for an index number formula.
A final perspective on the test or axiomatic approach is due to Wolfgang
Eichhorn. A decade ago, he pointed out to me that the economic approach is in
fact an axiomatic approach: it just uses different axioms (optimizing behavior
with prices being independent variables generating quantities as endogenous
dependent variables whereas the traditional test approach to index number
theory regards both prices and quantities as exogenous independent variables).
Eichhorn’s insightful comment led me to write the paper reprinted here as
Chapter 11, which develops the axioms of the economic approach to index
number theory. In Chapter 14, an additional axiomatic approach to another
economic problem is developed: I attempt to axiomatize a decision maker’s
choices in an uncertain environment.
To conclude this review, I would like to thank my co-author, Robert Allen,
for his permission to include our joint work in this volume. A very special
thanks is due to my co-editor, Alice Nakamura, and also to Dale Jorgenson
who made these volumes possible. As well, I would like to thank the following
people for fruitful discussions or correspondence over the years on index number
and related measurement problems: Sydney Afriat, Dennis Aigner, Maurice
Allais, Bob Allen, Keir Armstrong, Ken Arrow, B.K. Atrostic, Bert Balk,
Bill Barnett, Ernst Berndt, Jeff Bernstein, Chuck Blackorby (Blackie), Walter
Bossert, John Bossons, Alexandra Cass, Doug Caves, Peter Chinloy, Lau
Christensen, Dianne Cummings, Rob Danielson, Masako Darrough, Ed Dean,
Angus Deaton, Gerard Debreu, Mike Denny, David Donaldson, Lorraine Eden,
Wolfgang Eichhorn, Larry Epstein, Rolf F¨are, Rob Feenstra, Ivan Fellegi, Frank
Fisher, Kevin Fox, Mel Fuss, Frank Gallop, Terrence Gorman, Zvi Griliches,
Shauna Grosskopf, Tom Gussman, Shirley Hahn, Peter Hammond, Diana Hancock,
Arnold Harberger, Rick Harris, Peter Hill, Chuck Hulten, Dale Jorgenson,
Frank Kiss, Ulrich Kohli, Serge Kolm, Alexander Kon¨us, Larry Lau, Dennis
29
This point was originally made by Frisch [1936].
Lawrence, Peter Lawrence, Bernie Lefebvre, Tracy Lewis, Ramon Lopez, Marilyn
Manser, Doug May, Jim Melvin, Nimfa Mendoza, Claude Montmarquette,
Cathy Morrison, Alice Nakamura, Dale Orr, Tae Oum, Celik Parkan, Don
Patterson, Andreas Pfingsten, Bob Pollak, Dan Primont, Craig Riddell, Alan
Russell, Bob Russell (R Cubed), Tom Rymes, Jacob Ryten, Paul Samuelson,
Bohdan Schultz (Szulc), Bill Schworm, Karl Shell, Ron Shephard, Margaret
Slade, Barbara Slater, Dunc Smeaton, Spenser Star, Genio Staranczak, Frank
Stehling, Leo T¨ornqvist, Mike Tretheway, Jack Triplett, Yasuko Tsurusaki,
Arja Turunen-Red, Ralph Turvey, Dan Usher, Jan van Yzeren, Yrj¨o Vartia,
Arthur Vogt, Terry Wales, Bill Waters, John Weymark, Ken White, Frank
Wykoff, and Kim Zieschang. To all of you, thank you for the many hours of
thought on measurement problems that you have stimulated.
References for Chapter 1
Afriat, S.N., 1967. “The Construction of Utility Functions from Expenditure
Data,” International Economic Review 8, 67–77.
Afriat, S.N., 1972b. “The Theory of International Comparisons of Real Income
and Prices.” In International Comparisons of Prices and Output, D.J.
Daly (ed.), National Bureau of Economic Research, New York: Columbia
University Press, 13–69.
Afriat, S.N., 1973d. “On a System of Inequalities in Demand Analysis: An
Extension of the Classical Method.” International Economic Review 14,
460–472.
Allen, R.G.D., 1949. “The Economic Theory of Index Numbers,” Economica
N.S. 16, 197–203.
Arrow, K.J., 1974. “The Measurement of Real Value Added.” In Nations and
Households in Economic Growth: Essays in Honor of Moses Abramovitz,
P. David and M. Reder (eds.), New York: Academic Press.
Avriel, M., W.E. Diewert, S. Schaible and I. Zang, 1988. Generalized Concavity,
New York: Plenum Press.
Balk, B.M. 1991. “On the First Step in the Calculation of a Consumer Price
Index,” Discussion paper, Netherlands Central Bureau of Statistics, P.O.
Box 959, 2270 AZ Voorburg, The Netherlands.
Banerjee, K.S., 1975. Cost of Living Index Numbers: Practice, Precision and
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Barnett, W.A. and P. Yue, 1988. “Semiparametric Estimation of the Asymptotically
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32 Essays in Index Number Theory 1. Overview 33
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Berge, C., 1963. Topological Spaces, New York: Macmillan.
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