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 9
SUPERLATIVE INDEX NUMBERS AND
CONSISTENCY IN AGGREGATION*
W.E. Diewert
1. Introduction
Recently, Vartia [1974] [1976a] proposed a discrete approximation to the continuous
time Divisia1
price or quantity index which has the following two remarkable
properties: (i) the price index and the corresponding quantity index
(which is defined by the same formula except that prices and quantities are
interchanged) satisfy Fisher’s [1922] factor reversal test (i.e., the product of
the price and quantity indexes equals the expenditure ratio for the two periods
under consideration) and (ii) the Vartia I price or quantity index has the
property of consistency in aggregation.
Vartia defines an index number formula to be consistent in aggregation
if the value of the index calculated in two stages necessarily coincides with the
value of the index as calculated in an ordinary way; i.e., in a single stage.
The economic theory of index numbers (see Afriat [1972b], Pollak [1971a],
Samuelson and Swamy [1974]) is concerned with rationalizing functional forms
for index numbers with functional forms for the underlying aggregator function.
In Section 2, we show that the Vartia I price and quantity indexes are
consistent only with a Cobb–Douglas aggregator function. This is perhaps not
a surprising result, since thus far, the only way in which the two stage method
of calculating index numbers has been justified from the viewpoint of the economic
theory of index numbers, is to assume that the underlying aggregator
function is weakly separable in the same partition that corresponds to the two
stages.2
Thus to justify the two stage method of constructing index numbers
*This article was first published in 1978 in Econometrica 46(4), pp. 883–900.
The author wishes to thank C. Blackorby, Z. Griliches, C. Sims, and a referee
for helpful comments and Keith Wales for programming help. Correspondence
with P.A. Samuelson has also been very helpful. This research was supported
by a Canada Council grant.
1
See Divisia [1926], Wold [1953; Ch. 8], Jorgenson and Griliches [1967], and
Hulten [1973].
2
See Shephard [1953; 61–71], Solow [1955–56], Gorman [1959], Blackorby, Pri-
254 Essays in Index Number Theory 9. Consistency in Aggregation 255
for any partition of variables, one thus far has had to assume that the aggregator
function is weakly separable in any partition of its variables, but then
the results of Leontief [1947] and Gorman [1968a] imply that the aggregator
function is strongly separable in the coordinate-wise partition of its variables.
If we also assume that the aggregator function is linearly homogeneous, then
using Bergson’s [1936] results, it can be seen that the aggregator function must
be a mean of order r (Hardy, Littlewood and Polya [1934]); i.e., a CES function.
The mean of order r functions that are consistent in aggregation include
the Cobb–Douglas aggregator function (i.e., r = 0) and the linear aggregator
function (i.e., r = 1). For a more comprehensive discussion of the difficulties
involved in combining subindexes to form a complete index, see Pollak [1975]
and Blackorby, Primont, and Russell [1978].
In spite of the rather negative result that the Vartia I price and quantity
indexes are exact only for a Cobb–Douglas aggregator function, we show
in Section 3 that the Vartia index approximates to the second order any superlative
index, and that all superlative indexes closely approximate each other.
Moreover, using the consistency in aggregation property of the Vartia index,
we can show that all superlative indexes are approximately consistent in aggregation.
A superlative quantity index is an index number formula which
is consistent with a consumer or producer maximizing a “flexible” aggregator
function subject to a budget constraint. A “flexible” functional form is one
which can provide a second order approximation to an arbitrary function (c.f.
Diewert [1976a]). Thus a practical objection to the use of superlative index
number formulae (i.e., they are not consistent in aggregation) loses its force,
since they will be approximately consistent in aggregation. Moreover, the degree
of approximation will become closer if the chain principle for constructing
indexes is used rather than the fixed base method (if we are dealing with time
series data where changes in prices and quantities between successive periods
are generally smaller than changes relative to a fixed base).
In Section 4, we apply the results of Section 3 to some issues in the theory
of aggregation, to the measurement of productivity change, and to applied
welfare economics. Proofs of our theorems are relegated to an appendix.
A second appendix numerically compares various index number formulae,
constructed in one and two stages and also constructed using the chain principle
versus a fixed base. This may help to illustrate more concretely some of the
rather abstract approximation theorems presented in the paper.
An index number formula like Vartia’s could be termed pseudo superlative
since it approximates a superlative index number formulae to the second order.
A final appendix proves some theorems about pseudo superlative index numbers
and relates these results to some results due to Samuelson and Swamy [1974].
mont, and Russell [1978], and Geary and Morishima [1973].
2. Exact Index Numbers and the Vartia Indexes
Define an aggregator function f(x) as a function of N nonnegative variables
x ≥ 0N which has the following three properties: (i) f is positive for positive
arguments (i.e., f(x) > 0 for x 0N ); (ii) f is linearly homogeneous (i.e.,
f(λx) = λf(x) for λ ≥ 0, x ≥ 0N ); and (iii) f is concave (i.e., f[λx1
+ (1 −
λ)x2
] ≥ λf(x1
) + (1 − λ)f(x2
) for 0 ≤ λ ≤ 1, x1
≥ 0N , x2
≥ 0N ).
Generally, aggregator functions are taken to be either utility functions in
the consumer context or production functions in the producer context.
An aggregator function f satisfying the above conditions has a total cost
function defined by C(u, p) ≡ min{p · x : f(x) ≥ u} = u minx/u{p · x/u :
f(x/u) ≥ 1} (using the linear homogeneity of f) = u minz{p · z : f(z) ≥ 1} ≡
uc(p) where p · x ≡ N
i=1 pixi and c(p) denotes the unit cost function which
corresponds to f.
It turns out that the unit cost function c(p) satisfies the same regularity
conditions as f; i.e., c(p) is positively linearly homogeneous and concave for p
0N . Moreover, given a unit cost function satisfying these regularity conditions,
its aggregator function dual can be defined by f(x) ≡ 1/[maxp{c(p) : p · x =
1, p ≥ 0N }].3
Thus if the functional form for the aggregator function is known (or the
functional form for its unit cost function dual c(p) is known) and the economic
agent is engaging in cost minimizing behavior, then the quantity aggregate can
be defined as u ≡ f(x) (or u ≡ p · x/c(p)) and the price of the aggregate may
be defined as p0 ≡ p · x/f(x) (or p0 ≡ c(p)). Since the functional form for f
(or c) is not generally known, the above results are not particularly useful in
empirical applications. However, as Afriat [1972b], Pollak [1971a], Samuelson
and Swamy [1974], and Diewert [1976a] have shown, it is sometimes possible
to relate known functional forms for index numbers to functional forms for
aggregator functions. We now outline this theory of exact index numbers.
First define a quantity index between periods 0 and 1 as a function,
Q(p0
, p1
; x0
, x1
), of the price vectors in periods 0 and 1, p0
and p1
respectively,
and the corresponding quantity vectors x0
and x1
while a price index
between periods 0 and 1 is a function, P(p0
, p1
; x0
, x1
), of the same price and
quantity vectors. Given either a price index or a quantity index, the other
function can be defined implicitly by the following equation (Fisher’s [1922]
weak factor reversal test):
(1) P(p0
, p1
, x0
, x1
)Q(p0
, p1
, x0
, x1
) = p1
· x1
/p0
· x0
;
i.e., the product of the price index times the quantity index should yield the
expenditure ratio between the two periods.
3
See Shephard [1953], Samuelson [1953–54] [1972], Chipman [1970], and Diewert
[1974a].
256 Essays in Index Number Theory 9. Consistency in Aggregation 257
The positive linearly homogeneous, concave aggregator function f is defined
to be exact for the quantity index Q(p0
, p1
, x0
, x1
) if for every p0
0N ,
p1
0N , x0
a solution to the aggregator maximization problem maxx{f(x) :
p0
· x ≤ p0
· x ≤ p0
· x0
, x ≥ 0N }, and x1
a solution to maxx{f(x) : p1
· x ≤
p1
· x1
, x ≥ 0N }, we have
(2) Q(p0
, p1
, x0
, x1
) = f(x1
)/f(x0
).
Similarly, the positive, linearly homogeneous, concave aggregator function
f with unit cost function dual c is defined to be exact for the price index
P(p0
, p1
, x0
, x1
) if for every p0
0N , p1
0N , x0
a solution to maxx{f(x) :
p0
·x ≤ p0
·x0
, x ≥ 0N }, and x1
a solution to maxx{f(x) : p1
·x ≤ p1
·x1
, x ≥ 0N },
we have
(3) P(p0
, p1
; x0
, x1
) = c(p1
)/c(p0
).
With the above preliminaries disposed of, define the Vartia4
[1974] [1976a]
price index PV (p0
, p1
; x0
, x1
) as
(4) ln PV (p0
, p1
; x0
, x1
) ≡
N
i=1
[L(p1
i x1
i , p0
i x0
i )/L(p1
· x1
, p0
· x0
)] ln(p1
i /p0
i )
where the logarithmic mean function L, introduced into the economics literature
by Vartia [1974] and Sato [1976a], is defined by L(a, b) ≡ (a−b)/(ln a−ln b)
for a = b and L(a, a) ≡ a.
The Vartia quantity index QV (p0
, p1
, x0
, x1
) is defined by
(5)
ln QV (p0
, p1
, x0
, x1
) ≡
N
i=1
[L(p1
i x1
i , p0
i x0
i )/L(p1
· x1
, p0
· x0
)] ln(x1
i /x0
i )
= ln PV (x0
, x1
, p0
, p1
);
i.e., the price and quantity indexes have the same functional form except that
the role of prices and quantities are interchanged. Vartia shows that PV and
QV satisfy the factor reversal test (1) and have the property of consistency in
aggregation.
4
We consider only the Vartia I indexes since they are the indexes which have
the consistency in aggregation property. Sato [1976a] showed that the Vartia II
indexes were exact for a CES aggregator function.
Theorem 1. The only once differentiable, positive, linearly homogeneous and
concave unit cost function which is exact for the Vartia price index defined by
(4) is the Cobb–Douglas unit cost function.
Theorem 2. The only once differentiable, positively linearly homogeneous
and concave aggregator function which is exact for the Vartia quantity index
defined by (5) is the Cobb–Douglas aggregator function.
Since the Cobb–Douglas aggregator function is extremely restrictive, one
might suppose that the Vartia indexes would not be very useful. However,
the results of the next section indicate that the Vartia indexes are useful in
establishing certain general theorems about superlative indexes.
3. Approximation Properties of the Vartia Indexes
A frequently5
used discrete approximation to the Divisia price index is defined
by
(6) ln P0(p0
, p1
, x0
, x1
) ≡
1
2
N
i=1
[(p1
i x1
i /p1
· x1
) + (p0
i x0
i /p0
· x0
)] ln(p1
i /p0
i )
while a quantity index Q0 is defined by
ln Q0(p0
, p1
, x0
, x1
) ≡
1
2
N
i=1
[(p1
i x1
i /p1
· x1
) + (p0
i x0
i /p0
· x0
)] ln(x1
i /x0
i )
(7)
= ln P0(x0
, x1
, p0
, p1
).
T¨ornqvist [1936] [1937] and [1971; 47] urged the use of the above indexes,
while Kloek [1966] [1967] and Theil [1967] [1968] showed that the indexes had
some good local approximation properties. In addition, Diewert [1976a] showed
that the translog6
unit cost function is exact for P0 defined by (6) and that a
linearly homogeneous translog aggregator function is exact for Q0 defined by
(7).
Diewert [1976a] defined a price index (quantity index) to be superlative
if a unit cost function c (aggregator function f), capable of providing a second
order differential approximation to an arbitrary twice differentiable linearly
5
See, for example, the empirical work by Christensen and Jorgenson [1969]
[1970].
6
See Christensen, Jorgenson and Lau [1971].
258 Essays in Index Number Theory 9. Consistency in Aggregation 259
homogeneous function, is exact for it. Since a linearly homogeneous translog
function can provide a second order approximation to an arbitrary twice differentiable
linearly homogeneous function (see Lau [1974]), it can be seen that P0
defined by (6) is a superlative price index and Q0 defined by (7) is a superlative
quantity index. In general, “superlative” indexes are consistent with “flexible”
functional forms for the underlying aggregator function.
Since the price index P0 defined by (6) resembles somewhat the Vartia
price index PV defined by (4), the following result may not be too surprising.
Theorem 3. The Vartia price index differentially approximates7
the superlative
price index P0 to the second order at any point where the prices and quantities
for the two periods are equal; i.e., PV (p0
, p1
, x0
, x1
) = P0(p0
, p1
, x0
, x1
)
and the first and second order partial derivatives of the two functions coincide
provided that p0
= p1
0N and x0
= x1
0N .
The proof of the above theorem makes use of the following two lemmas
which are also proved in an appendix.
Lemma 1. Define the functions f and g of the two variables λ and γ by
f(λ, γ) ≡ 1
2 [(λ/γ)+1] and g(λ, γ) ≡ L(λ, 1)/L(γ, 1) where L is the Vartia mean
function. Then f differentially approximates g to the first order at the point
λ = 1, γ = 1; i.e., f(1, 1) = g(1, 1), f1(1, 1) = g1(1, 1), and f2(1, 1) = g2(1, 1).
Lemma 2. Suppose that: (i) the function of n variables g differentially approximates
the function g∗
to the second order at the point y∗
; i.e., g(y∗
) = g∗
(y∗
),
yg(y∗
) = yg∗
(y∗
), and 2
yyg(y∗
) = 2
yyg∗
(y∗
); (ii) the function of n variables
f differentially approximates the function f∗
to the first order at y∗
; i.e.,
f(y∗
) = f∗
(y∗
) and yf(y∗
) = yf∗
(y∗
) and (iii) g(y∗
) = g∗
(y∗
) = 0. Then
h(y) ≡ f(y)g(y) differentially approximates h∗
(y) ≡ f∗
(y)g∗
(y) to the second
order at the point y∗
.
The following theorem may be proved in a manner analogous to the proof
of Theorem 3.
Theorem 4. The Vartia quantity index differentially approximates the superlative
quantity index Q0 to the second order at any point where the prices
and quantities for the two periods are equal.
Thus PV (p0
, p1
, x0
, x1
) will be close to P0(p0
, p1
, x0
, x1
) provided that p0
is close to p1
and x0
is close to x1
. If we call an index which can approximate a
superlative index differentially to the second order at any point where p0
= p1
7
The term is due to Lau [1974]: it means that the approximating function has
the same level and first and second order partial derivatives at a point as the
function that it is approximating: i.e., c(p∗
) = c∗
(p∗
), pc(p∗
) = pc∗
(p∗
),
and 2
ppc(p∗
) = 2
ppc∗
(p∗
) for some p∗
.
and x0
= x1
a pseudo superlative index, it can be seen that the Vartia price
and quantity indexes are pseudo superlative.
It turns out that the Vartia price and quantity indexes closely approximate
other superlative indexes and that all of the superlative indexes that are
known thus far approximate each other to the second order for small changes
in prices and quantities.
For r = 0, define the quadratic mean of order r price index Pr as
(8) Pr(p0
, p1
, x0
, x1
) ≡
N
i=1(p0
i x0
i /p0
· x0
)(p1
i /p0
i )r/2
N
k=1(p1
kx1
k/p1 · x1)(p0
k/p1
k)r/2
1/r
.
It can be shown (Diewert [1976a]) that Pr is exact for the quadratic
mean of order r unit cost function, cr(p) ≡ N
i=1
N
j=1 bijp
r/2
i p
r/2
j
1/r
. Since
cr can approximate an arbitrary unit cost function to the second order, Pr is
a superlative price index.
For r = 0, define the quadratic mean of order r quantity index Qr as
(9) Qr(p0
, p1
, x0
, x1
) ≡
N
i=1(p0
i x0
i /p0
· x0
)(x1
i /x0
i )r/2
N
j=1(p1
j x1
j /p1 · x1)(x0
j /x1
j )r/2
1/r
.
It can similarly be shown that Qr is exact for the quadratic mean of
order r aggregator function, fr(x) ≡ N
i=1
N
j=1 aijx
r/2
i x
r/2
j )1/r
, and that Qr
is a superlative quantity index.
Theorem 5. For any r = 0, Pr(p0
, p1
, x0
, x1
) = P0(p0
, p1
, x0
, x1
) and the first
and second order partial derivatives of the two functions coincide, provided
that p0
= p1
0N (all price components are positive) and x0
= x1
> 0N (at
least one quantity component is positive).
Theorem 6. For any r = 0, the quantity index Qr differentially approximates
Q0 to the second order at any point where the prices and quantities for the two
periods are equal; i.e., Qr(p0
, p1
, x0
, x1
) = Q0(p0
, p1
, x0
, x1
) and the first and
second order partial derivatives of the two functions coincide, provided that
p0
= p1
> 0N and x0
= x1
0N .
Theorems 3 and 5 imply that the Vartia price index PV approximates
all of the superlative indexes P0 and Pr while Theorems 4 and 6 imply that
the Vartia quantity index QV differentially approximates all of the superlative
indexes Q0 and Qr, provided that price and quantity changes are small between
the two periods.8
It is worth emphasizing that these theorems hold without the
8
Unfortunately, we cannot specify exactly how small is “small” without performing
extensive computations involving the third order partial derivatives of
the index number formulae; cf. the discussion in Lau [1974; 183].
260 Essays in Index Number Theory 9. Consistency in Aggregation 261
assumption of optimizing behavior on the part of economic agents; i.e., they
are theorems in numerical analysis rather than economics.
4. Applications and Conclusions
For many years, it was thought that the indexes P0 and Q0 had the property of
consistency in aggregation;9
i.e., it was thought that a discrete “Divisia” index
of discrete “Divisia” indexes was the discrete “Divisia” index of the components.
Although P0 and Q0 are not consistent in aggregation, the results of
the previous section show why they are approximately consistent in aggregation:
each P0 subindex is approximated to the second order by a Vartia index
of the same size, while the “macro” P0 index is approximated to the second
order by a “macro” Vartia index. Thus the macro index of the subindexes is
approximated to the second order by a Vartia macro index of Vartia subindexes
which is identically equal to a Vartia index of the original micro components,
which in turn approximates to the second order a P0 index in the micro components.
Therefore, given time series data where indexes are constructed by
chaining observations in successive periods, we would expect P0 and Q0 to be
approximately consistent in aggregation.
The same conclusion holds for the quadratic mean of order r price indexes
Pr and quantity indexes Qr: they will be approximately consistent in
aggregation since each Pr approximates PV and each Qr approximates QV .
Some empirical evidence which tends to support the theoretical results
above is available. Parkan [1975a] compared the price indexes P0, P2 (Fisher’s
[1922] ideal price index) and P0 (defined implicitly by (1), the weak factor
reversal test, using Q0 as the quantity index) and the quantity indexes Q0, Q2
(Fisher’s ideal quantity index) and Q0 (defined implicitly by (1) using P0 as
the price index) using some Canadian post war consumption data on 13 goods
constructed by Gussman [1972] and Cummings and Meduna [1973]. He also
calculated the nonparametric price and quantity indexes defined by Diewert
[1973b; 424]. Parkan [1975a] then computed all four of the price indexes and
all four of the quantity indexes in two stages, calculating subaggregates in each
case and then aggregating these subaggregates using the same index number
formula. It was found that the resulting total of eight price indexes generally
coincided to three significant figures, and the eight quantity indexes similarly
closely approximated each other. The theoretical results above provide an
explanation for this rather puzzling empirical phenomenon. In Appendix 2, we
replicate portions of Parkan’s computations.
9
See, for example, Jorgenson and Griliches [1972; 83] and Theil [1973; 498].
The above results also have an application to the measurement of total
factor productivity. Let u be the quantity of some output, let x be a vector of
other outputs and inputs (all indexed positively), let p be the corresponding
vector of output and input prices (where price components which correspond to
outputs are indexed negatively while components which correspond to inputs
are indexed with a plus sign), and suppose that the technology of the economy
can be represented by a homogeneous translog transformation function t, where
u = t(x). Then if producers are maximizing t(x) subject to an expenditure
constraint for periods 0 and 1, it can be shown that u1
/u0
= Q0(p0
, p1
, x0
, x1
)
provided that no technical change has taken place between the two periods.
If technical change takes place in a factor augmenting fashion, then the equation
u1
/u0
= Q0(p0
, p1
, x0
, x1
) (which depends only on observable prices and
quantities) may be modified in a manner which will enable one to compute
the amount of technical progress (cf. Diewert [1976a]). It turns out that Theorem
4 above can be extended to cover the case where a subset of the prices
are negative, so that the Vartia quantity index will still approximate Q0. Thus
we may use the same argument as before to show that the “Divisia” index
Q0(p0
, p1
, x0
, x1
) may be approximated to the second order by taking a “Divisia”
index of “Divisia” subindexes or in fact taking any superlative quantity
index of superlative subindexes.10
The results of the previous section may also be applied to the problem of
measuring changes in the welfare of a consumer. Let u be utility, x a consumption
vector, and p the corresponding vector of rental prices. If the consumer is
maximizing a homogeneous translog utility function subject to an expenditure
constraint for periods 0 and 1, then since the homogeneous translog is exact
for Q0,
(10) u1
/u0
= Q0(p0
, p1
, x0
, x1
);
i.e., the relative change in utility is equal to the value of the “Divisia” quantity
index Q0. If changes in prices and quantities are small between the two periods,
then the theorems in the previous section show that the change in welfare can be
10
This argument still does not quite provide a rigorous justification for the
Jorgenson–Griliches [1967] [1972] method of measuring technical progress in
the case of discrete data. Their method can be justified if we assume that
the economy’s transformation surface can be represented by an equation like
g(u, x1, . . . , xM ) = f(xM+1, xM+2, . . . , xN ) where u, x1, . . . , xM are outputs
and xM+1, . . . , xN are inputs so that outputs are separable from inputs which
is more restrictive than assuming u = t(x1, . . . , xN ). However, it turns out that
the Jorgenson–Griliches method of measuring total factor productivity can be
rigorously justified, at least approximately, even in the general (nonseparable)
case; cf. Diewert [1980].
262 Essays in Index Number Theory 9. Consistency in Aggregation 263
approximated to the second order by evaluating the “Divisia” index Q0, or by
the Vartia index QV , or by taking a “Divisia” index of “Divisia” subindexes, or
by taking any known superlative index of superlative indexes. This same string
of equivalences can be applied in a more general situation, since it is possible to
justify the quantity index Q0 in the context of an aggregator function f which
is not necessarily linearly homogeneous. Such justifications for Q0 have been
provided by Kloek [1967] and Theil [1968].
In order to provide a somewhat different justification for Q0, it is necessary
to define the Malmquist [1953] quantity index. Given an aggregator
function f and an aggregate u ≡ f(x), define f’s distance function as
D(u, x) ≡ maxk{k : f(x/k) ≥ u}. The distance function tells us by what proportion
one has to deflate the given consumption vector x in order to obtain
a point on the indifference surface indexed by u. Now define the Malmquist
quantity index as QM (x0
, x1
, u) ≡ D(u, x1
)/D(u, x0
) and note that it depends
on the two quantity vectors x0
and x1
, and the base indifference surface (indexed
by u) onto which the points x0
and x1
are deflated. Then it can be
shown (Diewert [1976a]) that if the consumer’s preferences can be represented
by a translog distance function (which can approximate general nonhomothetic
preferences to the second order) and the consumer is maximizing utility subject
to a budget constraint for the two periods, then
(11) QM (x0
, x1
, u∗
) = Q0(p0
, p1
, x0
, x1
),
where the reference utility level u∗
≡ (u0
u1
)1/2
is the square root of the product
of the base and current period utility levels. As before, the right hand side of
(11) can be approximated for small changes in prices and quantities by any of
the indexes Qr or by the Vartia index QV .
To summarize, the above arguments show that constructing aggregate
price and quantity indexes by aggregating two (or more) stages will give approximately
the same answer that a one stage index would, provided that either
a superlative index or the Vartia index is used.11
The present author has argued elsewhere (Diewert [1976a]) that the Fisher
price and quantity indexes, P2 and Q2, are probably the best of the superlative
indexes to use in empirical applications. But what about using the pseudo
superlative Vartia indexes, which have the very attractive property of consistency
in aggregation (whereas the superlative indexes are only approximately
consistent in aggregation)? Unfortunately, the Vartia quantity index has the
property that rescaling the prices in either period will generally change the
index (i.e., in general QV (λp0
, p1
, x0
, x1
) = QV (p0
, p1
, x0
, x1
) for λ = 1) while
11
However, note that our arguments do not justify the existence of well behaved
subaggregates which satisfy the usual regularity conditions of microeconomic
theory.
the Vartia price index has the property that rescaling the period 1 prices does
not in general change the value of the price index by the same scale factor (i.e.,
in general PV (p0
, λp1
, x0
, x1
) = λPV (p0
, p1
, x0
, x1
) for λ = 1). These defects
of the Vartia indexes will probably preclude their use in empirical situations,
but as we have seen, the Vartia indexes have proven to be very useful from a
theoretical point of view.
Appendix 1: Proofs of Theorems
Proof of Theorem 1. If xs
is a solution to maxx{f(x) : ps
·x ≤ ps
·xs
, x ≥
0N } for s = 0, 1 where f is positively linearly homogeneous and concave over
the positive orthant, then it is easy to see that xs
is also a solution to the
expenditure minimization problem minx{ps
· x : f(x) ≥ f(xs
)} = c(ps
)us
for
s = 0, 1 where c is the unit cost function which corresponds to f and us
≡ f(xs
)
for s = 0, 1. Moreover, if c(p) is once differentiable, then by Shephard’s Lemma
[1953; 11], xs
= pc(ps
)us
for s = 0, 1 where pc(ps
) is the gradient vector of
c evaluated at the price vector ps
.
Thus using Shephard’s Lemma to eliminate quantities in the index number
formula, if c is once differentiable, positively linearly homogeneous, concave
and exact for the Vartia price index defined by (4), we must have for every
p0
0N , p1
0N , xs
a solution to maxx{f(x) : ps
· x ≤ ps
· xs
, x ≥ 0N } for
s = 0, 1 where f is dual to c,
(12) ln
c(p1
)
c(p0)
≡
N
i=1
L[p1
i ci(p1
)u1
, p0
i ci(p0
)u0
]
L[p1 · pc(p1)u1, p0 · pc(p0)u0]
ln
p1
i
p0
i
,
where u0
≡ f(x0
), u1
≡ f(x1
), and ci(ps
) ≡ ∂c(ps
)/∂pi for i = 1, 2, . . . , N.
Thus (12) is to hold for some functional form for c with the appropriate regularity
properties for all ps
0N and scalars us
> 0 for s = 0, 1.
Let the last N − 1 components of the vectors p0
and p1
be identical; then
(12) becomes
(13) ln
c(p1
)
c(p0)
=
L[p1
1c1(p1
)u1
, p0
1c1(p0
)u0
]
L[p1 · 1
pc(p1)u1, p0 · pc(p0)u0]
ln
p1
1
p0
1
.
Since (13) must hold for all positive u1
and u0
, and the left hand side
is independent of u0
and u1
, the right hand side must also be independent
of u0
and u1
. This will be the case only if p1
1c1(p1
) = α1p1
· pc(p1
) and
p1
0c1(p0
) = α1p0
· pc(p0
) where α1 is a constant; i.e., the expenditure share
on the first good must be constant. We can similarly show that expenditure
264 Essays in Index Number Theory 9. Consistency in Aggregation 265
shares on all goods must be constant; i.e., we must have for j = 1, 2, . . . , N and
all p0
0N ,
(14) αj = p0
j cj(p0
)/p0
· pc(p0
) = p0
j cj(p0
)/c(p0
),
where the second equality follows from the assumption that c is linearly homogeneous.
From (14) and the linear homogeneity of c(p), it follows that c(p)
must be proportional to
N
j=1 p
αj
j where
N
j=1 αj = 1. Since we require c(p)
to be concave, we must also have αj ≥ 0. Thus in order to be exact for the
Vartia price index, c(p) must be of the Cobb–Douglas functional form.
Proof of Theorem 2. If xs
is a solution to maxx{f(x) : ps
·x ≤ ps
·xs
, x ≥
0N } for s = 1, 0 where f is positively linearly homogeneous and concave over
the positive orthant, then it is well known that if f is once differentiable,
elimination of the Lagrange multipliers for the maximization problems yields
the identities ps
/ps
· xs
= xf(xs
)/xs
· xf(xs
) = xf(xs
)/f(xs
) for s = 0, 1
where the second equality follows from the linear homogeneity of f. Thus
(15) ps
= ps
· xs
xf(xs
)/f(xs
), s = 0, 1.
Thus using (15) to eliminate prices from the index number formula, if f
is once differentiable, positively linearly homogeneous, concave, and exact for
the Vartia quantity index QV defined by (5), we must have for every x0
0N ,
x1
0N , scalars e0
> 0, e1
> 0 and price vectors ps
≡ es
xf(xs
)/f(xs
) for
s = 0, 1,
(16) ln
f(x1
)
f(x0)
=
N
i=1
L[e1
fi(x1
)x1
i /f(x1
), e0
fi(x0
)x0
i /f(x0
)]
L(e1, e0)
ln
x1
i
x0
i
.
Since the left hand side of (16) is independent of the scalars e0
and e1
, the
right hand side must also be, and this will only be the case if for all x1
0N
and x0
0N , there exist constants αj such that
(17) αj = fj(x1
)x1
j /f(x1
) = fj(x0
)x0
j /f(x0
) (j = 1, 2, . . . , N).
Upon integrating the partial differential equations (17), we find that f(x)
must be proportional to N
j=1 x
αj
j and in order for f to be linearly homogeneous
and concave, we further require that αj ≥ 0 and N
j=1 αj = 1. Thus in order to
be exact for the Vartia quantity index, f must be of Cobb–Douglas functional
form.
Proof of Lemma 1. The proof is a straightforward computation if one
makes repeated use of l’Hospital’s rule (see Rudin [1953; 82–83]) and the definition
of the Vartia mean function L.
Proof of Lemma 2. Obviously h(y∗
) = h∗
(y∗
). We have
yh(y∗
) = f(y∗
) yg(y∗
) + yf(y∗
)g(y∗
)
= f∗
(y∗
) yg∗
(y∗
) + yf∗
(y∗
)g∗
(y∗
)
= yh∗
(y∗
)
and
2
yyh(y∗
) = f(y∗
) 2
yyg(y∗
) + yf(y∗
) T
y g(y∗
)
+ yg(y∗
) T
y f(y∗
) + 2
yyf(y∗
)g(y∗
)
= 2
yyh∗
(y∗
),
since g(y∗
) = g∗
(y∗
) = 0 where T
y f(y∗
) is the transpose of the column vector
yf(y∗
), etc.
Proof of Theorem 3. We show that ln PV differentially approximates ln P0.
Define the values vs
j = ps
i xs
j for s = 0, 1 and j = 1, 2, . . ., N. Then it is easy to
see that our result will be true if the first and second order partial derivatives
of the log of the Vartia price index regarded as a function of p0
, p1
, v0
, and v1
,
N
i=1
v1
i − v0
i
ln v1
i − ln v0
i
ln( N
j=1 v1
j ) − ln( N
j=1 v0
j )
N
j=1 v1
j −
N
j=1 v0
j
ln
p1
i
p0
i
,
are equal to the first and second order partial derivatives of the log of P0
regarded as a function of p0
, p1
, v0
, and v1
,
N
i=1
1
2
v1
i
N
j=1 v1
j
+
v0
i
N
j=1 v0
j
ln
p1
i
p0
i
,
evaluated at any point such that p1
= p0
0N , v1
= v0
0N . It turns
out that matters are simplified if we introduce the additional variables V 1
≡
N
j=1 v1
j and V 0
≡
N
j=1 v0
j and regard the index numbers as functions of p1
,
p0
, v1
, v0
, V 1
, and V 0
. Let us also define the jth component λj of the vector of
variables λ by v1
j = λjv0
j , and the jth component δj of the vector of variables δ
by p1
j = δjp0
j , j = 1, 2, . . . , N. Define the scalar γ by V 1
= γV 0
. It can be seen
that our theorem will be true if the first and second order partial derivatives
of the log of the Vartia price index regarded as a function of p0
, δ, v0
, λ, V 0
,
and γ,
N
i=1
v0
i
V 0
λi − 1
ln λi
ln γ
γ − 1
ln δi,
266 Essays in Index Number Theory 9. Consistency in Aggregation 267
are equal to the first and second order partial derivatives of the log of P0
regarded as a function of p0
, δ, v0
, λ, V 0
, and γ,
N
i=1
1
2
λi
γ
+ 1
v0
i
V 0
ln δi,
evaluated at any point such that p0
0N , v0
0N , V 0
> 0, and where
λ = 1N (a vector of ones), δ = 1N , and γ = 1. Thus we need only show that
for each i, the level, the first order partial derivatives and the second order
partial derivatives of the function hi
(λi, γ, δi) ≡ fi
(λi, γ, δi)gi
(λi, γ, δi) where
fi
(λ, γ, δi) ≡ (λi −1)(ln λi)−1
(ln γ)(γ−1)−1
and gi
(λi, γ, δi) ≡ ln δi are equal to
the corresponding level, first and second order partial derivatives of the function
hi∗
(λi, γ, δi) ≡ fi∗
(λi, γ, δi)gi∗
(λi, γ, δi) where fi∗
(λi, γ, δi) ≡ 1
2 [(λi/γ)+1] and
gi∗
(λi, γ, δi) ≡ ln δi, evaluated at λi = 1, γ = 1, and δi = 1.
By Lemma 1, the level and the first order partial derivatives of fi
and fi∗
coincide at λi = 1, γ = 1, and δi = 1. Since gi
≡ gi∗
, the levels and all partial
derivatives of gi
and gi∗
coincide. Moreover, gi
(1, 1, 1) = gi∗
(1, 1, 1) = 0 and
thus our result follows using Lemma 2.
Proof of Theorem 5. Straightforward but tedious calculations show that
if p0
= p1
≡ p 0N , x0
= x1
≡ x > 0N , then:
ln P0(p0
, p1
, x0
, x1
) = ln Pr(p0
, p1
, x0
, x1
) = 0;(i)
∂ ln P0(p0
, p1
, x0
, x1
)/∂x1
i = ∂ ln Pr(p0
, p1
, x0
, x1
)/∂x1
i = 0,(ii)
∂ ln P0/∂x0
i = ∂ ln Pr/∂x0
i = 0
dropping the arguments p0
, p1
, x0
, x1
for brevity,
∂ ln P0/∂p1
i = ∂ ln Pr/∂p1
i = xi/p · x,
∂ ln P0/∂p0
i = ∂ ln Pr/∂p0
i = −xi/p · x
for i = 1, 2, . . . , N;
∂2
ln P0/∂x1
i ∂x1
j = ∂2
ln Pr/∂x1
i ∂x1
j = 0,(iii)
∂2
ln P0/∂x1
i ∂x0
j = ∂2
ln Pr/∂x1
i ∂x0
j = 0,
∂2
ln P0/∂x1
i ∂p1
j = ∂2
ln Pr/∂x1
i ∂p1
j
= 1
2 δij/p · x − 1
2 pixi/(p · x)2
where δij equals 1 if i = j and is zero otherwise,
∂2
ln P0/∂x1
i ∂p0
j = ∂2
ln Pr/∂x1
i ∂p0
i
= − 1
2 δij/p · x + 1
2 pixj(p · x)2
,
∂2
ln P0/∂x0
i ∂x0
j = ∂2
ln Pr/∂x0
i ∂x0
j = 0,
∂2
ln P0/∂x0
i ∂p1
j = ∂2
ln Pr/∂x0
i ∂p0
j
= 1
2 δij /p · x − 1
2 pixj /(p · x)2
,
∂ ln P0/∂x0
i ∂p0
j = ∂2
ln Pr/∂x0
i ∂p0
j
= − 1
2 δij/p · x + 1
2 pixj/(p · x)2
,
∂2
ln P0/∂p1
i ∂p1
j = ∂2
ln Pr/∂p1
i ∂p1
j
= −xixj /(p · x)2
,
∂2
ln P0/∂p1
i ∂p0
j = ∂2
ln Pr/∂p1
i ∂p0
j = 0,
and
∂2
ln P0/∂p0
i ∂p0
j = ∂2
ln Pr/∂p0
i ∂p0
j
= xixj/(p · x)2
for 1 ≤ i, j ≤ N.
Proof of Theorem 6. Proof is the same as Theorem 5 except that prices
and quantities are interchanged.
Appendix 2: Empirical Comparison of Index Numbers
In this appendix, we use the Canadian consumer data constructed by Cummings
and Meduna [1973] in order to compare empirically various index number
formulae.
The primary data are price indexes for 13 components of Canadian consumer
expenditures for the years 1947–1971 and the corresponding per capita
quantity series. These data were constructed by Cummings and Meduna by
aggregating (using T¨ornqvist price indexes) Canadian national accounts data
pertaining to expenditure on over 40 consumer goods categories plus some series
constructed by Gussman [1972]. Rental prices for each category of consumer
durables were constructed. These data are available on request.
Table 1 lists Vartia (PV ), T¨ornqvist (P0
), implicit T¨ornqvist (P0), Fisher
(P2), Laspeyres (PL(p0
, p1
, x0
, x1
) ≡ (x0
· p1
/x0
· p0
), and Paasche (PP (p0
, p1
,
268 Essays in Index Number Theory 9. Consistency in Aggregation 269
Table 1. Comparison of Single State Chained Index Numbers
Implicit Fisher
Year Vartia T¨ornqvist T¨ornqvist Ideal Laspeyres Paasche
PV P0 P0 P2 PL PP
1947 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1951 1.3851 1.3851 1.3850 1.3851 1.3857 1.3845
1956 1.4874 1.4875 1.4874 1.4875 1.4888 1.4861
1961 1.6556 1.6558 1.6556 1.6557 1.6578 1.6536
1966 1.8795 1.8797 1.8795 1.8796 1.8831 1.8761
1971 2.3228 2.3230 2.3227 2.3228 2.3285 2.3172
x0
, x1
) ≡ x1
·p1
/x1
·p0
) price indexes for the 13 goods using the chain principle.
Only selected years are reported in order to conserve space. Note that all of the
indexes coincide to three significant figures and that PV , P0, P0, P2, and PL
coincide to four significant figures (after rounding off) over the entire period.
The close correspondence of PV , P0, P0, and P2 is not unexpected since all
of these indexes differentially approximate each other to the second order at
a point where prices and quantities are equal and, of course, using the chain
principle, prices and quantities are approximately equal going from one year
to the next. What is somewhat unexpected is the close correspondence of PL
and PP to the other indexes. However, in the following appendix we show
that PL and PP differentially approximate the other four indexes to the first
order at any point where prices and quantities are equal. Thus, for our data, it
appears that annual changes in prices and quantities are small enough so that
chained Paasche and Laspeyres price indexes approximate reasonably closely
any chained superlative price index.
The index numbers in Table 1 were constructed using a double precision
Fortran program. As a check on our computations, we calculated a chained
Vartia quantity index, used the weak factor reversal test (1) to construct an
implicit price index PV , and we found PV coincided with PV to five decimal
places.
In order to ascertain the effect of chaining, see Table 2 which compares
the six index number formulae using a fixed base (1947=1) throughout. The
fixed base Vartia, T¨ornqvist, and Fisher indexes turn out to be very close to
each other as well as to their chained counterparts (all coincide to within one
half percent). However, the general effect of using a fixed base is to increase
the differences between the various index number formulae: in the fixed base
Table 2. Comparison of Single Stage Fixed Base Index Number
Year PV P0 P0 P2 PL PP
1947 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1951 1.3825 1.3833 1.3822 1.3834 1.3833 1.3834
1956 1.4814 1.4834 1.4804 1.4831 1.4847 1.4815
1961 1.6487 1.6532 1.6483 1.6518 1.6564 1.6472
1966 1.8772 1.8843 1.8782 1.8818 1.9070 1.8568
1971 2.3121 2.3224 2.3158 2.3188 2.3621 2.2763
case, PL differs from PV by about two percent in 1971, while PL differs from
PP by four percent in 1971.
The relatively large divergence between the Paasche and Laspeyres indexes
when a fixed base year is used is a source for some concern. In many
national accounting systems (such as Canada’s), the consumer price index is
constructed using a fixed base Laspeyres index, while the implicit deflator for
the consumption expenditures component of GNP is a fixed base Paasche index.
Given the importance of consumer price indexes in indexing wages and cost of
living supplements, it is important that official price indexes be consistent with
each other.
The above considerations suggest that government agencies should use
chained rather than fixed base indexes in order to deflate expenditures into
constant dollar quantities. More formally, there are at least three reasons why
chained rather than fixed base indexes should be used in the context of time
series data where period to period changes in prices and quantities are small:
(i) All superlative, pseudo superlative, Paasche, and Laspeyres index
numbers should coincide quite closely if they are constructed using the chain
principle. (The degree of coincidence should be somewhat greater for the superlative
and pseudo superlative indexes, since their index number formula
derivatives coincide to the second order, but only to the first order with the
Paasche and Laspeyres formulae.)
(ii) The Paasche, Laspeyres, or any superlative index number can be regarded
as discrete approximations to the continuous line integral Divisia index,
which has some useful optimality properties from the viewpoint of economic
theory.12
These discrete approximations will be closer to the Divisia index if
the chain principle is used.
(iii) The use of chained indexes avoids the vexing problems which arise
when the base year in the fixed base indexes is changed. For example, the base
12
See Malmquist [1953], Wold [1953], and Hulten [1973].
270 Essays in Index Number Theory 9. Consistency in Aggregation 271
year in the Canadian system of national accounts has just been changed from
1961 to 1971, and new Paasche price indexes for the various components of
GNP using 1971 as the base year have been computed for the years 1971–1974.
Unfortunately the new price and quantity indexes are not proportional to the
old indexes which used 1961 as the base year. Thus most econometric models
using national accounts data will have to be reestimated using the new indexes.
The use of chained indexes would avoid the discontinuities introduced by these
periodic changes in the base year.
We now turn to a comparison of index numbers constructed in one stage
versus two stages. We formed the subaggregate “nondurables” by aggregating
the following five goods: food, alcohol, tobacco, energy, and other nondurables.
The subaggregate “services” consisted of two goods: medical services, and other
services; the subaggregate “semidurables” consisted of two goods: clothing, and
other semidurables; and finally the subaggregate “durables” consisted of the
following four goods: motor vehicles, housing, land, and other durables. Two
stage price indexes were constructed by using a price index number formula
(along with its corresponding quantity index defined by the weak factor reversal
test) in order to construct price and quantity indexes for the four subaggregates
listed above. The same index number formula was then used in order
to construct an aggregate price index. Of course, for the Vartia, Laspeyres,
and Paasche indexes, the two stage procedure gave rise to precisely the same
aggregate indexes as the usual one stage procedure; i.e., PV , PL, and PP are
consistent in aggregation.
Table 3. Comparison of Two Stage Chained Index Numbers
Year P0 P0 P2
1947 1.0000 1.0000 1.0000
1951 1.3851 1.3851 1.3851
1956 1.4875 1.4874 1.4875
1961 1.6558 1.6556 1.6557
1966 1.8797 1.8796 1.8796
1971 2.3230 2.3228 2.3228
Table 3 lists the aggregate price indexes which resulted when the two stage
procedure was applied using the chain principle with the index number formulae
P0, P0, and P2. Comparison of Tables 1 and 3 shows that the T¨ornqvist
indexes, P0 and P0, and the Fisher index, P2, are approximately consistent in
aggregation to a very high degree of approximation indeed.
Recall that in equation (4) of the text, the logarithm of the ith price relative,
ln(p1
i /p0
i ), is weighted by wi(p0
, p1
, x0
, x1
) ≡ L(p1
i x1
i , p1
i x0
i )/L(p1
·x1
, p0
·x0
)
where L is the logarithmic mean function. Vartia [1974; 91–93] shows that
N
i=1 wi(p0
, p1
, x0
, x1
) < 1 (unless p1
i x1
i /p0
i x0
i = constant for i = 1, 2, . . . , N
in which case the weights wi sum to precisely 1). The sum of the weights for
selected years is tabled in Table 4; the first column of Table 4 is the sum of
the chained weights,
13
i=1 wi(pt−1
, pt
, xt−1
, xt
), while the second column is the
sum of the fixed base weights, 13
i=1 wi(p1947
, pt
, x1947
, xt
).
Table 4. Sum of Vartia Weights
Year Chained Sum Fixed Base Sum
1948 .99979 .99979
1951 .99977 .99746
1956 .99995 .99419
1961 .99994 .99112
1966 .99997 .98929
1971 .99998 .98631
Note that it is not correct to say that the Vartia price and quantity indexes
are biased downwards due to the fact that the weights wi(p0
, p1
, x0
, x1
)
generally sum to a number less than one. This would be the case if the
weights were constant, but they are not. Moreover, the Vartia quantity index
has the same weights as the Vartia price index, and since PV (p0
, p1
, x0
, x1
)
QV (p0
, p1
, x0
, x1
) = p1
· x1
/p0
· x0
, it cannot be the case that both the Vartia
price and quantity indexes are biased downwards.
Appendix 3: More on Pseudo Superlative Index Numbers
In this section, we prove some additional theorems about pseudo superlative
index numbers.
Theorem 7. If P(p0
, p1
, x0
, x1
) is a pseudo superlative price index, then the
corresponding quantity index defined by using the weak factor reversal test,
Q(p0
, p1
, x0
, x1
) ≡ p1
· x1
/[p0
· x0
P(p0
, p1
, x0
, x1
)] is a pseudo superlative quantity
index.
Proof. The Vartia quantity index can be defined as
(18) QV (p0
, p1
, x0
, x1
) ≡ p1
· x1
/[p0
· x0
PV (p0
, p1
, x0
, x1
)]
272 Essays in Index Number Theory 9. Consistency in Aggregation 273
where PV is the Vartia price index. Since P is pseudo superlative, P and PV
have the same first and second order partial derivatives when evaluated at an
equal price and equal quantity point, and thus comparison of the definition of
Q with the definition of QV given by (18) shows that Q and QV will have the
same first and second order partial derivatives.qed
Of course, a similar theorem holds if the roles of P and Q are interchanged
in Theorem 7.
The following theorem shows that the Paasche and Laspeyres price indexes
are not pseudo superlative.
Theorem 8. Let P(p0
, p1
, x0
, x1
) be any pseudo superlative price index (i.e., P
has the first and second order partial derivatives defined in Theorem 5). Then
P(p0
, p1
, x0
, x1
) = PL(p0
, p1
, x0
, x1
) ≡ p1
· x0
/p0
· x0
= PP (p0
, p1
, x0
, x1
) ≡
p1
· x1
/p0
· x1
and the first order partial derivatives of the pseudo superlative
index P, the Laspeyres index PL, and the Paasche index PP coincide, provided
that p0
= p1
≡ p 0N and x0
= x1
≡ x > 0N . However, the second order
partial derivatives of P, PL, and PP evaluated under the same conditions do
not coincide.
Proof. Again, straightforward but tedious calculations show that if p0
=
p1
≡ p 0N , x0
= x1
≡ x > 0N , then ln PL = ln PP = 0 = ln P, and
the first order partial derivatives of ln PL(p0
, p1
, x0
, x1
), ln PP (p0
, p1
, x0
, x1
)
and ln P(p0
, p1
, x0
, x1
) all coincide. The second order partial derivatives of
ln PL(p0
, p1
, x0
, x1
) are (dropping the arguments p0
, p1
, x0
, x1
for brevity):
∂2
ln PL/∂x1
i ∂x1
j = 0,
∂2
ln PL/∂x1
i ∂x0
j = 0,
∂2
ln PL/∂x1
i ∂p1
j = 0, = ∂2
ln P/∂x1
i ∂p1
j ,
∂2
ln PL/∂x1
i ∂p0
j = 0, = ∂2
ln P/∂x1
i ∂p0
j ,
∂2
ln PL/∂x0
i ∂x0
j = 0,
∂2
ln PL/∂x0
i ∂p1
j = (δij/p · x) − [pixj /(p · x)2
] = ∂2
ln P/∂x0
i ∂p1
j ,
∂2
ln PL/∂x0
i ∂p0
j = −(δij/p · x) + [pixj/(p · x)2
] = ∂2
ln P/∂x0
i ∂p0
j ,
∂2
ln PL/∂p1
i ∂p1
j = −xixj/(p · x)2
,
∂2
ln PL/∂p1
i ∂p0
j = 0,
∂2
ln PL/∂p0
i ∂p0
j = xixj/(p · x)2
,
where δij = 1 if i = j and is 0 otherwise.
The second order partial derivatives of ln PP (p0
, p1
, x0
, x1
) can be obtained
from the second order partial derivatives of ln PL(p0
, p1
, x0
, x1
) listed
above if x0
and x1
are interchanged.qed
A symmetric mean m(x, y) of two nonnegative numbers x and y can be
defined as any function which satisfies the following three conditions:
m(x, y) = m(y, x),(19-i)
m(x, x) = x,(19-ii)
min(x, y) ≤ m(x, y) ≤ max(x, y).(19-iii)
Samuelson and Swamy [1974; 582] assert a theorem which states that
in the case of a homothetic aggregator function, any symmetric mean of the
Laspeyres and Paasche index numbers will approximate the true index number
up to the third order in accuracy. Although the following theorem does
not make the assumptions about optimizing behavior on the part of economic
agents that Samuelson and Swamy make in their theorem, the following theorem
in numerical analysis does provide a counterpart to the Samuelson–Swamy
result.
Theorem 9. Any twice continuously differentiable symmetric mean of the
Laspeyres and Paasche price indexes, m[PL(p0
, p1
, x0
, x1
), PP (p0
, p1
, x0
, x1
)],
is a pseudo superlative price index.
Proof. If m(PL, PP ) = P
1
2
L P
1
2
P , then the resulting index is P2, Fisher’s
ideal index, which is pseudo superlative by Theorem 5. Note that when
p0
= p1
and x0
= x1
, m(PL, PP ) = m(1, 1) = 1 = P
1
2
L P
1
2
P where we have
used property (19-ii) of m. Thus we need only show that if p0
= p1
≡
p 0N , x0
= x1
≡ x > 0N , then the first and second order partials of
m[PL(p0
, p1
, x0
, x1
), PP (p0
, p1
, x0
, x1
)] equal the corresponding partial derivatives
of P2(p0
, p1
, x0
, x1
) for any twice differentiable symmetric mean function
m. Thus we need to know the first and second order partials of m(x, y)
evaluated at (x, y) = (1, 1).
By partially differentiating (19-i) with respect to x, we obtain
(20) m1(x, y) = m2(y, x)
where mi denotes partial differentiation with respect to the ith argument of m,
i = 1, 2. Partial differentiation of (19-ii) with respect to x yields the following
identity:
(21) m1(x, x) + m2(x, x) = 1.
When x = y, (20) and (21) imply the following relations:
(22) m1(x, x) = m2(x, x) =
1
2
.
274 Essays in Index Number Theory 9. Consistency in Aggregation 275
By partially differentiating (21) with respect to x, we obtain the following
identity (using also m12 = m21 which follows from the assumption that m is
twice continuously differentiable):
(23) m11(x, x) + 2m12(x, x) + m22(x, x) = 0.
Now partially differentiate (20) with respect to x, set x = y, and obtain the
following identity:
(24) m11(x, x) = m22(x, x).
Relations (23), (24), and m12 = m21 imply that:
(25) m11(x, x) + m12(x, x) = 0 = m21(x, x) + m22(x, x).
The magnitude of m11(x, x) = m22(x, x) = −m12(x, x) cannot be determined
in general, but it turns out that we only need equations (22) and
(25) in order to evaluate the first and second order partial derivatives of
m[PL(p0
, p1
, x0
, x1
), PP (p0
, p1
, x0
, x1
)] and to show that they are equal to
the corresponding partial derivatives of P2(p0
, p1
, x0
, x1
) when p0
= p1
and
x0
= x1
. (Recall that the first order partials of PL and PP are equal using
Theorem 8.)qed
Note that the proof of the above theorem did not require property (19-iii)
on the symmetric mean m. The proof of the following theorem is analogous to
the proof of Theorem 9.
Theorem 10. Any twice continuously differentiable symmetric mean of two
pseudo superlative indexes is also a pseudo superlative index.
Vartia [1974] [1976a] presents a geometric proof that the logarithmic mean
L(x, y) ≡ (x−y)(ln x−ln y) lies between the geometric mean and the arithmetic
mean of the nonnegative numbers x and y; i.e., he shows that M0(x, y) ≡
x
1
2 y
1
2 ≤ L(x, y) ≤ M1(x, y) ≡ 1
2 x + 1
2 y.
We conclude this appendix by stating that
(26) M0(x, y) ≤ L(x, y) ≤ Mr∗ (x, y), x > 0, y > 0,
where r∗
≡ 1/3 and the mean of order r for r = 0 is defined by Mr(x, y) ≡
(1
2 xr
+ 1
2 yr
)1/r
. Since
min(x, y) ≤ M0(x, y) ≤ M1/3(x, y) ≤ M1(x, y) ≤ max(x, y)
(see Hardy, Littlewood and Polya [1934; Theorems 5 and 16]), it can be seen
that L(x, y) is a symmetric mean (recall definition (19)) and that the bounds
(26) on L(x, y) are tighter than Vartia’s bounds.
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