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 8
EXACT AND SUPERLATIVE INDEX NUMBERS*
W.E. Diewert
1. Introduction
One of the most troublesome problems facing national income accountants and
econometricians who are forced to construct some data series, is the question
of which functional form for an index number should be used. In the present
paper, we consider this question and relate functional forms for the underlying
production or utility function (or aggregator function, to use a neutral
terminology) to functional forms for index numbers.
First, define a quantity index between periods 0 and 1, Q(p0
, p1
, x0
, x1
),
as a function of the prices in periods 0 and 1, p0
> 0N and p1
> 0N (where 0N
is an N dimensional vector of zeros), and the corresponding quantity vectors,
x0
> 0N and x1
> 0N . A price index between periods 0 and 1, P(p0
, p1
, x0
, x1
),
is a function 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):
(1.1) 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. (We indicate the inner product of
two vectors as p · x or pT
x.)
Examples of price indexes are
PLa(p0
, p1
, x0
, x1
) ≡ p1
· x0
/p0
· x0
(Laspeyres price index),
PP a(p0
, p1
, x0
, x1
) ≡ p1
· x1
/p0
· x1
(Paasche price index).
*This article was first published in the Journal of Econometrics 4(2), 1976, pp.
115–145. A preliminary version was presented at Stanford in August 1973. The
author is indebted to L.J. Lau, D. Aigner, K.J. Arrow, E.R. Berndt, C. Blackorby,
L.R. Christensen and K. Lovell for helpful comments. This research was
partially supported by National Science Foundation Grant GS-3269-A2 at the
Institute for Mathematical Studies in the Social Sciences at Stanford University,
and by the Canada Council.
224 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 225
The geometric mean of the Paasche and Laspeyres indexes has been suggested
as a price index by Bowley [1928] and Pigou [1912], but it is Irving
Fisher [1922] who termed the resulting index ideal:
(1.2) PId(p0
, p1
, x0
, x1
) ≡ (p1
· x0
p1
· x1
/p0
· x0
p0
· x1
)1/2
.
The Laspeyres, Paasche and ideal quantity indexes are defined in a similar
manner — quantities and prices are interchanged in the above formulae. In
particular, the ideal quantity index is defined as
(1.3) QId(p0
, p1
, x0
, x1
) ≡ (p1
· x1
p0
· x1
/p1
· x0
p0
· x0
)1/2
.
Notice that PIdQId = p1
· x1
/p0
· x0
; i.e., the ideal price and quantity
indexes satisfy the ‘adding up’ property (1.1). The following theorem shows
that the ideal quantity index may be used to compute the quantity aggregates
f(xr
) provided that the aggregator function f has a certain functional form.
Theorem 1.4. (Byushgens [1925], Kon¨us and Byushgens [1926], Frisch [1936;
30], Wald [1939; 331], Afriat [1972b; 45] and Pollak [1971a]): Let pr
0N for periods r = 0, 1, 2, . . ., R, and suppose that xr
> 0N is a solution
to maxx{f(x) : pr
· x ≤ pr
· xr
, x ≥ 0N }, where f(x) ≡ (xT
Ax)1/2
≡
N
j=1
N
k=1 xjajkxk)1/2
, ajk = akj, and the maximization takes place over a
region where f(x) is concave and positive (which means A must have N − 1
zero or negative eigenvalues and one positive eigenvalue). Then
(1.5) f(xr
)/f(x0
) = QId(p0
, pr
, x0
, xr
), r = 1, 2, . . . , R.
Thus given the base period normalization f(x0
) = 1, the ideal quantity
index may be used to calculate the aggregate f(xr
) = (xrT
Axr
)1/2
for r =
1, 2, . . . , R, and we do not have to estimate the unknown coefficients in the
A matrix. This is the major advantage of this method for determining the
aggregates f(xr
) (as opposed to the econometric methods suggested by Arrow
[1974]), and it is particularly important when N (the number of goods to be
aggregated) is large compared to R (the number of observations in addition to
the base period observation p0
, x0
).
If a quantity index Q(p0
, pr
, x0
, xr
) and a functional form for the aggregator
function f satisfy (1.5) then we say that Q is exact for f. Kon¨us and
Byushgens [1926] show that the geometric quantity index
N
i=1(x1
i /x0
i )si
(where
si ≡ p0
i ·x0
i /p0
·x0
) is exact for a Cobb–Douglas aggregator function, while Afriat
[1972b], Pollak [1971a] and Samuelson–Swamy [1974] present other examples
of exact index numbers. However, it appears that out of all the exact index
numbers thus far exhibited, only the ideal index corresponds to a functional
form for f which is capable of providing a second order approximation to an
arbitrary twice differentiable linearly homogeneous function. For a proof that
the functional form (xT
Ax)1/2
can provide such a second order approximation,
see Diewert [1974b].
Let us call a quantity index Q ‘superlative’ (see Fisher [1922; 247] for an
undefined notion of a superlative index number) if it is exact for an f which
can provide a second order approximation to a linearly homogeneous function.
In the following section, we show that the T¨ornqvist [1936], Theil [1965]
[1967] and Kloek [1966] [1967] quantity index (which has been used by Christensen
and Jorgenson [1970], Star [1974], Jorgenson and Griliches [1972; 83],
Star and Hall [1976] as a discrete approximation to the Divisia [1926] index)
is also a superlative index number. In Section 3, we use the results of Section
2 to provide a rigorous interpretation of the Jorgenson–Griliches method
of measuring technical progress for discrete data.
In Section 4, we introduce an entire family of superlative index numbers.
Section 5 presents some conclusions which tend to support the use of Fisher’s
ideal quantity index in empirical applications and the final section is an appendix
which sketches the proofs of various theorems developed in the following
sections.
2. The T¨ornqvist–Theil ‘Divisia’ Index and the Translog Function
Before stating our main results, it will be necessary to state a preliminary result
which is extremely useful in its own right. Let z be an N dimensional vector
and define the quadratic function f(z) as
f(z) ≡ a0 + aT
z +
1
2
zT
Az(2.1)
= a0 +
N
j=1
aizi +
1
2
N
i=1
N
j=1
aij zizj,
where the ai, aij are constants and aij = aji for all i, j.
The following lemma is a global version of the Theil [1967; 222–223] and
Kloek [1966] local result.
Lemma 2.2. Quadratic Approximation Lemma. If and only if the quadratic
function f is defined by (2.1), then
(2.3) f(z1
) − f(z0
) =
1
2
[ f(z1
) + f(z0
)]T
(z1
− z0
),
where f(zr
) is the gradient vector of f evaluated at zr
.
226 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 227
The above result should be contrasted with the usual Taylor series expansion
for a quadratic function,
f(z1
) − f(z0
) = [ f(z0
)]T
(z1
− z0
) +
1
2
(z1
− z0
)T 2
f(z0
)(z1
− z0
),
where 2
f(z0
) is the matrix of second order partial derivatives of f evaluated
at an initial point z0
. In the expansion (2.3), a knowledge of 2
f(z0
) is not
required, but a knowledge of f(z1
) is required. It must be emphasized that
(2.3) holds as an equality for all z1
, z0
belonging to an open set if and only if
f is a quadratic function.
Actually, the Quadratic Approximation Lemma (2.2) is closely related to
the following result which we will prove as a corollary to (2.2):
Lemma 2.4. (Bowley [1928; 224–225]): If a consumer’s preferences can be
represented by a quadratic function f defined by (2.1), if x1
0N is a solution
to the utility maximization problem
(2.5) max
z
{f(z) : p1
· z = Y 1
, z ≥ 0N },
where p1
0N and Y 1
≡ p1
· x1
, and if x0
0N (i.e., each component of x0
is positive) is a solution to the utility maximization problem
(2.6) max
z
{f(z) : p0
· z = Y, z ≥ 0N },
where p0
0N and Y 0
≡ p0
· x0
, then the change in utility between periods 0
and 1 is
(2.7) f(x1
) − f(x0
) =
1
2
(λ∗
1p1
+ λ∗
0p0
) · (x1
− x0
),
where λ∗
i is the marginal utility of income in period i for i = 0, 1. That is, λ∗
i
is the optimal value of the Lagrange multiplier for the maximization problem
(2.5), and λ∗
0 is the Lagrange multiplier for (2.6).
Bowley’s Lemma is frequently used in applied welfare economics and costbenefit
analysis, while the Quadratic Approximation Lemma is frequently used
in index number theory, which indicates the close connection between the two
fields.
Suppose that we are given a homogeneous translog aggregator function
(Christensen, Jorgenson and Lau [1971]) defined by
ln f(x) ≡ α0 +
N
n=1
αn ln xn +
1
2
N
j=1
N
k=1
γjk ln xj ln xk,
where
N
n=1 αn = 1, γjk = γkj and
N
k=1 γjk = 0 for j = 1, 2, . . ., N.
Jorgenson and Lau have shown that the homogeneous translog function
can provide a second order approximation to an arbitrary twice continuously
differentiable linearly homogeneous function. Let us use the parameters which
occur in the translog functional form in order to define the following function,
f∗
:
(2.8) f∗
(z) ≡ α0 +
N
j=1
αizi +
1
2
N
i=1
N
j=1
γij zizj.
Since the function f∗
is quadratic, we can apply the Quadratic Approximation
Lemma (2.2), and we obtain
(2.9) f∗
(z1
) − f∗
(z0
) =
1
2
[ f∗
(z1
) + f∗
(z0
)] · (z1
− z0
).
Now we relate f∗
to the translog function f. We have
∂f∗
(zr
)/∂zj = ∂ ln f(xr
)/∂ ln xj(2.10)
= [∂f(xr
)/∂xj][xr
j /f(xr
)],
f∗
(zr
) = ln f(xr
),
zr
j = ln xr
j , for r = 0, 1 and j = 1, 2, . . ., N.
If we substitute relation (2.10) into (2.9), we obtain
(2.11) ln f(x1
) − ln f(x0
) =
1
2
ˆx1 f(x1
)
f(x1)
+ ˆx0 f(x0
)
f(x0)
· (ln x1
− ln x0
),
where ln x1
≡ (ln x1
1, ln x1
2, . . . , ln x1
N ), ln x0
≡ (ln x0
1, ln x0
2, . . . , ln x0
N ), ˆx1
≡ the
vector x1
diagonalized into a matrix, and ˆx0
≡ the vector x0
diagonalized into
a matrix.
Assume that xr
0N is a solution to the aggregator maximization problem
maxx{f(x) : pr
· x = pr
· xr
, x ≥ 0N }, where pr
0N for r = 0, 1, and f is
the homogeneous translog function. The first order conditions for the two maximization
problems, after elimination of the Lagrange multipliers (Kon¨us and
Byushgens [1926; 155], Hotelling [1935; 71–74], Wold [1944; 69–71] and Pearce
[1964; 59]), yield the relations pr
/pr
· xr
= f(xr
)/xr
· f(xr
) for r = 0, 1.
Since f is linearly homogeneous, xr
· f(xr
) may be replaced by f(xr
) in the
above, and substitution of these last two relations into (2.11) yields
ln[f(x1
)/f(x0
)] =
1
2
ˆx1
p1
p1T x1
+
ˆx0
p0
p0T x0
· (ln x1
− ln x0
)
=
N
n=1
1
2
(s1
n + s0
n) ln(x1
n/x0
n),
228 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 229
or
f(x1
)/f(x0
) =
N
n=1
(x1
n/x0
n)(s1
n+s0
n)/2
≡ Q0(p0
, p1
, x0
, x1
),(2.12)
where sr
n ≡ pr
nxr
n/pr
· xr
, the nth share of cost in period r.
The right hand side of (2.12) is the quantity index which corresponds to
Irving Fisher’s [1922] price index number 124, using (1.1). It has also been
advocated as a quantity index by T¨ornqvist [1936] and Theil [1965] [1967]
[1968]. It has been utilized empirically by Christensen and Jorgenson [1969]
[1970] as a discrete approximation to the Divisia [1926] index and by Star
[1974] and Star and Hall [1976] in the context of productivity measurement.
The above argument shows that this quantity index is exact for a homogeneous
translog aggregator function, and in view of the second order approximation
property of the homogeneous translog function, we see that the right hand side
of (2.12) is a superlative quantity index.
It can also be seen (using the if and only if nature of the Quadratic Approximation
Lemma (2.2)) that the homogeneous translog function is the only
differentiable linear homogeneous function which is exact for the T¨ornqvist–
Theil quantity index.
The above argument can be repeated (with some changes in notation)
if the unit cost function for the aggregator function is the translog unit cost
function defined by
ln c(p) ≡ α∗
0 +
N
j=1
α∗
n ln pn +
1
2
N
j=1
N
k=1
γ∗
jk ln pj ln pk,
where N
n=1 α∗
n = 1, γ∗
jk = γ∗
kj and N
k=1 γ∗
jk = 0 for j = 1, 2, . . . , N. We also
need the following results.
Lemma 2.13. (Shephard [1953; 11], Samuelson [1947]): If f is positive, linearly
homogeneous and concave; if
pr
· xr
= min
x
{pr
· x : f(x) ≥ f(xr
)} ≡ c(pr
)f(xr
) for r = 0, 1,
and if the unit cost function c is differentiable at pr
, then
xr
= c(pr
)f(xr
) for r = 0, 1.
Corollary1
2.14.
xr
/pr
· xr
= c(pr
)/c(pr
) for r = 0, 1.
Now under the assumption of cost minimizing behavior in periods 0 and 1
(which implies (2.14)), we have upon applying the Quadratic Approximation
Lemma (2.2) to the translog unit cost function,
ln c(p1
) − ln c(p0
) =
1
2
ˆp1 c(p1
)
c(p1)
+ ˆp0 c(p0
)
c(p0)
· (ln p1
− ln p0
)
=
1
2
ˆp1 x1
p1 · x1
+ ˆp0 x0
p0 · x0
· (ln p1
− ln p0
) (using (2.14)
=
N
n=1
(s1
n + s0
n) ln(p1
n/p0
n),
or
(2.15) c(p1
)/c(p0
) =
N
n=1
(p1
n/p0
n)(s1
n+s0
n)/2
,
where sr
n = pr
nxr
n/pr
· xr
(the nth share of cost in period r), pr
0N (period r
prices, r = 0, 1), xr
≥ 0N (period i quantities, i = 0, 1), and c(p) = the translog
unit cost function.2
The right hand side of (2.15) corresponds to Irving Fisher’s [1922] price
index 123. The above argument shows that this price index is exact for a
translog unit cost function, and that this is the only differentiable unit cost
function which is exact for this price index.
Let us denote the right hand side of (2.15) as the price index function
P0(p0
, p1
, x0
, x1
), and denote the right hand side of (2.12) as the quantity
index Q0(p0
, p1
, x0
, x1
). It can be verified that P0(p0
, p1
, x0
, x1
) Q0(p0
, p1
,
x0
, x1
) = p1
· x1
/p0
· x0
in general; i.e., the price index P0 and the quantity
index Q0 do not satisfy the weak factor reversal test (1.1). This is perfectly
reasonable, since the quantity index Q0 is consistent with a homogeneous
translog (direct) aggregator function, while the price index P0 is consistent
with an aggregator function which is dual to the translog unit cost
function, and the two aggregator functions do not in general coincide; i.e., they
correspond to different (aggregation) technologies. Thus, given Q0, the corresponding
price index, which satisfies (1.1), is defined by P0(p0
, p1
, x0
, x1
) ≡
p1
· x1
/[p0
· x0
Q0
(p0
, p1
, x0
, x1
)]. The quantity index Q0 and the corresponding
(implicit) price index P0 were used by Christensen and Jorgenson [1969] [1970]
1
Proof: divide the equation in Lemma 2.13 by pr
· xr
= c(pr
)f(xr
).
2
Note that the validity of (2.15) depends crucially on the validity of (2.14),
which will be valid if p0
and p1
belong to an open convex set of prices P, such
that the translog c(p) satisfies the regularity conditions of positivity, linear
homogeneity and concavity over P.
230 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 231
in order to measure U.S. real input and output. On the other hand, given P0,
the corresponding (implicit) quantity index, which satisfies (1.1), is defined by
Q0(p0
, p1
, x0
, x1
) ≡ p1
· x1
/[p0
· x0
P0(p0
, p1
, x0
, x1
)]. The price-quantity index
pair (P0, Q0) was advocated by Kloek [1967; 2] over the pair (P0, Q0) on the
following grounds: as we disaggregate more and more, we can expect the individual
consumer or producer to utilize positive amounts of fewer and fewer
goods (i.e., as N grows, components of the vectors x0
and x1
will tend to become
zero), but the prices which the producer or consumer faces are generally
positive irrespective of the degree of disaggregation. Since the logarithm of zero
is not finite, Q0 will tend to be indeterminate as the degree of disaggregation
increases, but P0 will still be well defined (provided that all prices are positive).
Theil [1968] and Kloek [1967] provided a somewhat different interpretation
of the indexes P0 and Q0, an interpretation which does not require the aggregator
function to be linear homogeneous. Let the aggregate u be defined by
u = f(x), where f is a not necessarily homogeneous aggregator function which
satisfies for example the Shephard [1970] or Diewert [1971a] regularity conditions
for a production function. For p 0N , Y > 0, define the total cost function
by C(u, p) ≡ minx{p · x : f(x) ≥ u; x ≥ 0N } and the indirect utility function
by g(p/Y ) ≡ maxx{f(x) : p · x ≤ Y, x ≤ 0N }. The true cost of living price
index evaluated at ‘utility’ level u is defined as P(p0
, p1
; u) ≡ C(u, p1
)/C(u, p0
)
and the Theil index of quantity (or ‘real income’) evaluated at prices p is defined
as QT (p; u0
, u1
) ≡ C(u1
, p)/C(u0
, p). The Theil-Kloek results are that:
(i) P0(p0
, p1
, x0
, x1
) is a second order approximation to P(p0
, p1
; g(v∗
)), where
the nth component of v∗
is v∗
n ≡ (p0
np1
n/p0
·x0
p1
·x1
)
1
2 , for n = 1, 2, . . ., N, and
(ii) Q0(p0
, p1
, x0
, x1
) is a second order approximation to QT [p∗
, g(p0
/p0
· x0
),
g(p1
/p1
· x1
)], where the nth component of p∗
is p∗
n ≡ (p0
np1
n)1/2
.
In view of the Theil–Kloek approximation results, we might be led to
ask whether the index number P0 is exact for any general (nonhomothetic)
functional forms for the cost function C(u, p). The following theorem answers
this question in the affirmative:
Theorem 2.16. Let the functional form for the cost function be a general
translog of the form
ln C(u, p) ≡ α∗
0 +
N
i=1
α∗
i ln pi +
1
2
N
j=1
N
k=1
γ∗
jk ln pj ln pk
+ β∗
ln u +
1
2
δ∗
(ln u)2
+
N
k=1
ε∗
i ln u ln pi,
where
N
i=1 α∗
i = 1, γ∗
jk = γ∗
kj ,
N
k=1 γ∗
jk = 0, for j = 1, 2, . . ., N, and
N
i=1 ε∗
j = 0.3
Let (u0
, p0
) and (u1
, p1
) belong to a (u, p) region where C(u, p)
3
These restrictions ensure the linear homogeneity of C(u, p) in p.
satisfies the appropriate regularity conditions for a cost function (e.g., see Shephard
[1970], Hanoch [1970] or Diewert [1971a]) and define the quantity vectors
x0
≡ pC(u0
, p0
) and x1
≡ pC(u1
, p1
). Then
P0(p0
, p1
, x0
, x1
) = C(u∗
, p1
)/C(u∗
, p0
),
where u∗
≡ (u0
u1
)1/2
and P0 is defined by the right hand side of (2.15).
In contrast to the case of a linearly homogeneous aggregator function
where the cost function takes the simple form C(u, p) = c(p)u, Theorem (2.16)
is not an if and only if result; that is, the index number P0(p0
, p1
; x0
, x1
) is
exact for functional forms for C(u, p) other than the translog. In fact, Theorem
(2.16) remains true if: (i) we define C as ln C(u, p) ≡ α0(u) +
N
i=1[αi +
εih(u)] ln pi + 1
2
N
j=1
N
k=1 γjk ln pj ln pk, where N
i=1 αi = 1, N
i=1 εi = 0,
γjk = γkj,
N
k=1 γjk = 0, for j = 1, 2, . . . , N, and h is a monotonically increasing
function of one variable, and (ii) define the reference utility level u∗
as the solution to the equation 2h(u∗
) = h(u1
) + h(u0
). (In the translog case,
h(u) ≡ ln u.)
Thus the same price index P0 is exact for more than one functional form
(and reference utility level) for the true cost of living.
We can also provide a justification for the quantity index Q0 in the context
of an aggregator function f which is not necessarily linearly homogeneous. In
order to provide this justification, it is necessary to define the quantity index
which has been proposed by Malmquist [1953] and Pollak [1971a] in the context
of consumer theory, and by Bergson [1961] and Moorsteen [1961] in the context
of producer theory.
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}. To use the language
of utility theory, the distance function tells us by what proportion one has
to deflate (or inflate) the given consumption vector x in order to obtain a
point on the utility surface indexed by u. It can be shown that if f satisfies
certain regularity conditions, then f is completely characterized by D (see
Shephard [1970], Hanoch [1970] and McFadden [1970]). In particular, D(u, x)
is linearly homogeneous, nondecreasing and concave in the vector of variables
x and nonincreasing in u in Hanoch’s formulation.
Now define the Malmquist quantity index as QM (x0
, x1
, u) ≡ D(u, x1
)/
D(u, x0
). Note that the index depends on x0
(the base period quantities), x1
(the current period quantities) and on the base indifference surface (which is
indexed by u) onto which the points x0
and x1
are deflated. The following
theorem relates the translog functional form to the Malmquist quantity index.
Theorem 2.17. Let an aggregator function f satisfying the Hanoch [1970] and
Diewert [1971a] regularity conditions be given such that f’s distance function D
232 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 233
is a general translog of the form
ln D(u, x) = α0 +
N
i=1
αi ln xi +
1
2
N
j=1
N
k=1
γjk ln xj + β ln u
+
1
2
δ(ln u)2
+
N
i=1
εi ln u ln xi,
where
N
i=1 αi = 1, γjk = γkj ,
N
k=1 γjk = 0, for j = 1, 2, . . . , N, and
N
i=1 εi = 0. Suppose that the quantity vector x0
is a solution to the aggregator
maximization problem maxx{f(x) : p0
· x = p0
· x0
}, while x1
is a
solution to maxx{f(x) : p1
· x = p1
· x1
} and u0
≡ f(x0
), u1
≡ f(x1
). Then
Q0(p0
, p1
, x0
, x1
) = D(u∗
, x1
)/D(u∗
, x0
) ≡ QM (x0
, x1
, u∗
),
where u∗
≡ (u0
u1
)1/2
and Q0 is defined in (2.12).
As was the case with the price index P0, the quantity index Q0 is equal
to Malmquist quantity indexes which are defined by nontranslog distance functions;
i.e., Theorem (2.17) is not an if and only if result.
However, Theorems (2.16) and (2.17) do provide a rather strong justification
for the use of P0 or Q0 since the translog functional form provides a
second order approximation to a general cost or distance function (which in
turn are dual to a general nonhomothetic aggregator function).
Finally, note that Theorems (2.16) and (2.17) have a ‘global’ character to
them in contrast to the Theil–Kloek ‘local’ results.
3. Productivity Measurement and ‘Divisia’ Indexes
Jorgenson and Griliches [1972; 83–84] have advocated the use of the indexes
P0, Q0 in the context of productivity measurement. It is perhaps appropriate
to review their procedure in the light of the results of the previous section.
First, we note (by a straightforward computation) that it is not in general
true that ‘a discrete Divisia index of discrete Divisia indexes is a discrete Divisia
index of the components’ (Jorgenson and Griliches [1972; 83]), where the
‘Divisia’ quantity index is defined to be Q0. In view of the one-to-one nature
of the index number Q0 with the translog functional form for the aggregator
function f in the linearly homogeneous case, it can be seen that the Jorgenson–
Griliches assertion will be true if the producer or consumer is maximizing an
aggregator function f subject to an expenditure constraint, where f is both
a homogeneous translog function and a translog of micro translog aggregator
functions. The set of such translog functions is not empty since it contains the
set of Cobb–Douglas functions. Thus if cost shares are approximately constant
(which corresponds to the Cobb–Douglas case), then the Jorgenson–Griliches
assertion will be approximately true.
It can be similarly shown that, in general, it is not true that a discrete
‘Divisia’ price index of discrete ‘Divisia’ indexes is a discrete ‘Divisia’ price
index of the components, where the ‘Divisia’ price index is defined to be P0: the
first method of constructing a price index is justified if the aggregator function
has a unit cost function dual of the form ˆc[c1
(p1
), c2
(p2
), . . . , cJ
(pJ
)], where
(p1
, p2
, . . . , pJ
) represents a partition of the price vector p and the functions ˆc,
c1
, c2
, . . . , cJ
are all translog unit cost functions, while the second method of
constructing a price index is justified if the aggregator function has a unit cost
function dual, c(p), which is translog.
Jorgenson and Griliches [1972] use the index number formula Q0(p0
, p1
,
x0
, x1
) defined by the right hand side of (2.12) not only to form an index of
real input, but also to form an index of real output. Just as the aggregation
of inputs into a composite input rests on the duality between unit cost and
homogeneous production functions, the aggregation of outputs into a composite
output can be based on the duality between unit revenue and homogeneous
factor requirements functions.4
We briefly outline this latter duality.
Suppose that a producer is producing M outputs, (y1, y2, . . . , yM ) ≡ y,
and the technology of the producer can be described by a factor requirements
function, g, where g(y) = the minimum amount of aggregate input required
to produce the vector of outputs y.5
The producer’s unit (aggregate input)
revenue function6
is defined for each price vector w ≥ 0M by
(3.1) r(w) ≡ max
y
{w · y : g(y) ≤ 1, y ≥ 0M }.
Thus given a factor requirements function g, (3.1) may be used to define
a unit revenue function. On the other hand, given a unit revenue function
r(w) which is a positive, linearly homogeneous, convex function for w 0M , a
factor requirements functions g∗
consistent with r may be defined for y 0M
4
See Diewert [1969] [1974a], Fisher and Shell [1972a] and Samuelson and Swamy
[1974] on this topic.
5
Assume g is defined for y ≥ 0M , and has the following properties: (i) g(y) > 0
for y 0M (positivity), (ii) g(λy) = λg(y) for λ ≥ 0, y ≥ 0 (linear homogeneity),
and (iii) g[λy1
+(1−λ)y2
] ≤ λg(y1
)+(1−λ)g(y2
) for 0 ≤ λ ≤ 1, y1
≥ 0M ,
y2
≥ 0M (convexity).
6
If g satisfies the three properties listed in footnote 5, then r also has those
three properties.
234 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 235
by7
g∗
(y) ≡ min
λ
{λ : w · y ≤ r(w)λ for every w ≥ 0M }
(3.2)
= min
λ
{λ : 1 ≤ r(w)λ for every w ≥ 0M such that w · y = 1}
= 1/ min
w
{r(w) : w · y = 1, w ≥ 0M }.
The translog functional form may be used to provide a second order approximation
to an arbitrary twice differentiable factor requirements function.
Thus assume that g is defined (at least over the relevant range of y’s) by
(3.3) ln g(yr
) ≡ a0 +
M
m=1
am ln yr
m +
1
2
M
j=1
M
k=1
cjk ln yr
j ln yr
k, for r = 0, 1,
where
M
m=1
am = 1, cjk = ckj,
M
k=1
cjk = 0, for j = 1, 2, . . ., M.
Now assume that yr
0M is a solution to the aggregate input minimization
problem miny{g(y) : wr
· y = wr
· yr
, y ≥ 0M }, where wr
0M
for r = 0, 1, and g is the translog function defined by (3.3). Then the first
order necessary conditions for the minimization problems along with the linear
homogeneity of g yield the relations wr
/wr
· yr
= g(yr
)/g(yr
), for r = 0, 1,
and using these two relations in Lemma (2.2) applied to (3.3), we deduce that
g(y1
)/g(y0
) =
M
m=1
(y1
m/y0
m)[(w1
my1
m/w1
·y1
)+(w0
my0
m/w0
·y0
)]/2
(3.4)
≡ Q∗
0(w0
, w1
; y0
, y1
).
Thus again the T¨ornqvist formula can be used to aggregate quantities
consistently, provided that the underlying aggregator function is a homogeneous
translog.
Similarly if the revenue function r(w) is translog over the relevant range of
data and if the producer is in fact maximizing revenue, then we can show that
r(w1
)/r(w0
) = P∗
(w0
, w1
, y0
, y1
) ≡ Q∗
0(y0
, y1
, w0
, w1
), the T¨ornqvist price
7
The proof is analogous to the proof of the Samuelson–Shephard duality theorem
presented in Diewert [1974a]; alternatively, see Samuelson and Swamy
[1974].
index. (Note that in Q∗
0(y0
, y1
, w0
, w1
), prices and quantities are interchanged
compared to Q∗
0(w0
, w1
, y0
, y1
) which appeared in (3.4) above.)
Using the above material, we may now justify the Jorgenson–Griliches
[1972] method of measuring technical progress. Assume that the production
possibilities efficient set can be represented as a set of outputs y and inputs x
such that
(3.5) g(y) = f(x),
where g is the homogeneous translog factor requirements function defined by
(3.3), and f is the homogeneous translog production function defined in Section
2. Let wr
0M , pr
0N , r = 0, 1 be vectors of output and input prices
during periods 0 and 1, and assume that y0
0M and x0
0N is a solution
to the period 0 profit maximization problem,
(3.6) max
y,x
{w0
· y − p0
· x : g(y) = f(x)}.
Suppose that ‘technical progress’ occurs between periods 0 and 1 which
we assume to be a parallel outward shift of the ‘isoquants’ of the aggregator
function f; i.e., we assume that the equation which defines the efficient set
of outputs and inputs in period 1 is g(y) = (1 + τ)f(x) where τ represents
the amount of ‘technical progress’ if τ > 0 or ‘technical regress’ if τ < 0.
Finally, assume that y1
0M and x1
0N is a solution to the period 1 profit
maximization problem,
(3.7) max
y,x
{w1
· y − p1
· x : g(y) = (1 + τ)f(x)}.
Thus we have g(y0
) = f(x0
) and g(y1
) = (1+τ)f(x1
). It is easy to see that
yr
0M is a solution to the aggregate input minimization problem miny{g(y) :
wr
·y = wr
·yr
, y ≥ 0M }, for r = 0, 1, and thus (3.4) holds. Similarly, xr
0N
is a solution to the aggregator maximization problem maxx{f(x) : pr
· x =
pr
· xr
, x ≥ 0N }, for r = 0, 1, and thus (2.12) holds. Substitution of (2.12) and
(3.4) into the identity g(y1
)/g(y0
) = (1 + τ)f(x1
)/f(x0
) yields the following
expression for (1 + τ) in terms of observable prices and quantities:
(1 + τ) =
M
m=1
y1
m/y0
m
[(w1
my1
m/w1
·y1
)+(w0
my0
m/w0
·y0
)]/2
(3.8)
n
n=1
x1
n/x0
n
[(p1
nx1
n/p1
·x1
)+(p0
nx0
n/p0
·x0
)]/2
.
Thus the Jorgenson–Griliches method of measuring technical progress
can be justified if: (i) the economy’s production possibilities set can be represented
by a separable transformation surface defined by g(y) = f(x), where
236 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 237
the input aggregator function f and the output aggregator function g are both
homogeneous translog functions, (ii) producers are maximizing profits, and (iii)
technical progress takes place in the ‘neutral’ manner postulated above.
Since the separability assumption g(y) = f(x) is somewhat restrictive
from an a priori theoretical point of view, it would be of some interest to
devise a measure of technical progress which did not depend on this separability
assumption. This can be done, but only at a cost as we shall see below.
Suppose that technology can be represented by a transformation func-
tion,8
where y1 = t(y2, y3, . . . , yM ; x1, x2, . . . , xN ) ≡ t(˜y; x) ≡ t(z) is the
maximum amount of output one that can be produced given that the vector
of other outputs ˜y = (y2, y3, . . . , yM ) is to be produced by the vector of inputs
x = (x1, x2, . . . , xN ). Assume that t is a positive, linearly homogeneous, concave
function over a convex set of the nonnegative orthant S in K ≡ M −1+N
dimensional space. Assume also that t(˜y; x) is nonincreasing in the components
of the other outputs vector ˜y and nondecreasing in the components of the input
vector x. Suppose that the transformation function t is defined for z belonging
to S by
(3.9) ln t(z) ≡ α0 +
K
k=1
ln zk +
1
2
K
j=1
γjk ln zj ln zk,
where
K
k=1 αk = 1, γjk = γkj and
K
k=1 γjk = 0, for j = 1, 2, . . ., K; i.e., t is
a translog transformation function over the set S.
Suppose that yr
≡ (yr
1, yr
2, . . . , yr
M ) 0M (output vectors), xr
≡ (xr
1, xr
2,
. . . , xr
N ) 0N (input vectors), wr
0M (output price vectors), pr
0N (input
price vectors), and wr
·yr
= pr
·xr
(value of outputs equals value of inputs)
for periods r = 0, 1. Assume that z0
≡ (y0
2, . . . , y0
m, x0
1, . . . , x0
N ) ≡ (˜y0
, x0
)
is a solution to the following output maximization subject to an expenditure
constraint problem in period 0:
(3.10) max
z
{t(z) : q0
· z = q0
· z0
, z belongs to S}
where t is the translog transformation function defined by (3.9), q0
≡ (−w0
2, −w0
3,
. . . , −w0
M ; p0
1, p0
2, . . . , p0
N ) ≡ (− ˜w0
; p0
), and
(3.11) y0
1 = t(z0
).9
8
For a more detailed discussion of transformation functions and their properties,
see Diewert [1973a].
9
Note that q0
· z0
= w1
0
· y1
0
> 0, since w0
· y0
= p0
· x0
.
The first order conditions for the maximization problem (3.10) imply that
(Kon¨us–Byushgens [1926] Lemma)
(3.12) q0
/q0
· z0
= t(z0
)/t(z0
).
As before, we assume that ‘neutral’ input augmenting technical progress
takes place between periods 0 and 1; i.e., if (y; x) was an efficient vector of
outputs and inputs in period 0, then [y; (1 + τ)−1
x] is on the efficiency surface
in period 1. Thus the efficiency surface in period 1 can be defined as the set of
(y1, y2, . . . , yM ; x1, x2, . . . , xN ) which satisfy the following equation:
(3.13) y1 = t[y2, y3, . . . , yM ; (1 + τ)x1, (1 + τ)x2, . . . , (1 + τ)xN ].
Assume that (y1
2, y1
3, . . . , y1
M ; x1
1, x1
2, . . . , x1
N ) ≡ (˜y1
; x1
) is a solution to the
period 1 output maximization subject to an expenditure constraint problem
maxy,x{t[˜y; (1 + τ)x] : − ˜w1
· y + p1
· x = − ˜w1
· y1
+ p1
· x1
; [˜y : (1 + τ)x]
belongs to S}. Then z1
≡ [˜y1
; (1 + τ)x1
] is a solution to the following output
maximization problem:
(3.14) max
z
{t(z) : q1
· z = q1
· z1
, z belongs to S},
where t is the translog function defined by (3.9), q1
≡ (− ˜w1
; p1
), and
(3.15) y1
1 = t(z1
) = t[˜y1
; (1 + τ)x1
].
Again, the Kon¨us-Byushgens-Hotelling Lemma applied to the maximization
problem (3.14), using the linear homogeneity of t, implies that10
(3.16) q1
/q1
· z1
= t(z1
)/t(z1
).
Now substitute (3.12) and (3.16) into the identity (2.11), except that t
replaces f and z replaces x, and we obtain
t(z1
)/t(z0
) =
K
k=1
(z1
k/z0
k)[(q1
kz1
k/q1
·z1
)+(q0
kz0
k/q0
·z0
)]/2
(3.17)
= t[˜y1
; (1 + τ)x1
]/t(y0
; x0
).
Combining (3.13), (3.15) and (3.17), we obtain the following equation
in τ:
y1
1/y0
1 =
N
n=1
[(1 + τ)(x1
n/x0
n)][p1
n(1+τ)x1
n/V 1
(τ)+p0
nx0
n/V 0
]/2
(3.18)
M
m=2
(y1
m/y0
m)[w1
my1
m/V 1
(τ)+w0
my0
m/V 0
]/2
,
10
We assume that τ is small so that q1
· z1
≡ − ˜w1
· ˜y1
+ p1
· (1 + τ)x1
> 0.
238 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 239
where V 0
≡ −
M
m=2 w0
my0
m +
N
n=1 p0
nx0
n = net cost of producing output y1
in period 0, and V 1
(τ) ≡ −
M
m=2 w1
my1
m +
N
n=1 p1
n(1 + τ)x1
n.
Given data on outputs, inputs and prices, equation (3.18) can be solved
for the unknown rate of technical progress τ. Note that equation (3.18) is quite
different from the Jorgenson–Griliches equation for τ defined by (3.8) (except
that the two equations are equivalent when M = 1; i.e., when there is only one
output).
However, it should be pointed out that our more general measure of technical
progress, which is obtained by solving (3.18) for τ, suffers from some
disadvantages: (i) our procedure is computationally more difficult,11
and (ii)
our procedure is not symmetric in the outputs (that is, the first output y1 is
asymmetrically singled out in (3.18)). Thus different orderings of the outputs
could give rise to different measures of technical progress. This is because each
ordering of the outputs corresponds to a different translog assumption about
the underlying technology and thus different measures of τ can be obtained.
However, all of these measures should be close in empirical applications since
the different translog functions are all approximating the same technology to
the second order.
4. Quadratic Means of Order r and Exact Index Numbers
For r = 0, the (homogeneous) quadratic mean of order r aggregator function is
defined by
(4.1) fr(x) ≡


N
i=1
N
j=1
aij x
r/2
i x
r/2
j


1/r
,
where aij = aji, 1 ≤ i, j ≤ N, are parameters, and the domain of definition
of fr, is restricted to x ≡ (x1, x2, . . . , xN ) 0N such that aijx
r/2
i x
r/2
j >
0, and fr is concave. The above functional form is due to McCarthy [1967],
Kadiyala [1971–72], Denny [1972] [1974] and Hasenkamp [1973]. Denny also
defined the quadratic mean of order r unit cost function,
(4.2) cr(p) ≡


N
i=1
N
j=1
bijp
r/2
i p
r/2
j


1/r
, bij = bji, r = 0.
11
Furthermore, we cannot a priori rule out the possibility that equation (3.18)
will have either multiple solutions for τ or no solutions at all. The Fisher
measure of technical progress, to be introduced in Section 5, overcomes these
difficulties.
Denny noted that if r = 1, then (4.1) reduces to the generalized linear
functional form (Diewert [1969] [1971a]), (4.2) reduces to the generalized Leontief
functional form (Diewert [1969] [1971a]), and if all aij = 0 for i = j, then
(4.1) reduces to the CES functional form (Arrow, Chenery, Minhas and Solow
[1961]), while if all bij = 0 for i = j, then (4.2) reduces to the CES unit cost
function.
We may also note that when r = 2, (4.1) reduces to the Kon¨us–Byushgens
[1926] homogeneous quadratic production or utility function, while (4.2) reduces
to the Kon¨us–Byushgens unit cost function. This functional form has
also been considered by Afriat [1972b; 72] and Pollak [1971a] in the context of
utility functions and by Diewert [1969] [1974b] in the context of revenue and
factor requirements functions.
Lau [1973] has shown that the limit as r tends to zero of the quadratic
mean of order r aggregator function (4.1) is the homogeneous translog aggregator
function and similarly that the limit as r tends to zero of (4.2) is the
translog unit cost function.
This completes our discussion of special cases of the above family of functional
forms. The following theorem shows that the functional form is ‘flexible’.
Theorem 4.3. Let f be any linearly homogeneous, twice continuously differentiable,
positive function defined over an open subset of the positive orthant
in N dimensional space. Then for any r = 0, fr defined by (4.1) can provide a
second order differential approximation to f.
By a second order differential approximation to f at a point x∗
0N ,12
we mean that there exists a set of aij parameters for fr defined by (4.1), such
that fr(x∗
) = f(x∗
), fr(x∗
) = f(x∗
), and 2
fr(x∗
) = 2
fr(x∗
); i.e., the
values of fr and f and their first and second order partial derivatives at x∗
all
coincide.
Define the quadratic mean of order r quantity index Qr for x0
0N ,
x1
0N , p0
> 0N , p1
> 0N , for r ≥ 0, as
Qr(p0
, p1
; x0
, x1
) ≡
N
i=1(x1
i /x0
i )r/2
(p0
i x0
i /p0
· x0
)
N
k=1(x0
k/x1
k)r/2(p1
kx1
k/p1 · x1)
1/r
(4.4)
=
N
i=1
(x1
i /x0
i )r/2
s0
i
1/r N
k=1
(x1
k/x0
k)−r/2
s1
k
−1/r
.
Thus for any r = 0, Qr may be calculated as a function of observable
prices and quantities in two periods. Note that Qr can be expressed as the
12
This terminology follows Lau [1974].
240 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 241
product of a mean of order r13
in the square roots of the quantity relatives
(x1
i /x0
i )1/2
(using base period cost shares as weights) times a mean of order −r
in the square roots of the quantity relatives (x1
i /x0
i )1/2
(using period one cost
shares as weights).
It is perhaps of some interest to note which of Irving Fisher’s [1911] [1922]
tests are satisfied by the quantity index Qr. It can be verified that Qr satisfies:
(i) the commodity reversal test (i.e., the value of the index number does
not change if the ordering of the commodities is changed); (ii) the identity
test (i.e., Qr(p0
, p0
; x0
, x0
) ≡ 1); (in fact Qr(p0
, p1
; x0
, x0
) ≡ 1 with the quantity
index equal to one so long as all quantities remain unchanged); (iii) the
commensurability test (i.e., Qr(D−1
p0
, D−1
p1
, Dx0
, Dx1
) = Qr(p0
, p1
, x0
, x1
)
where D is a diagonal matrix with positive elements down the main diagonal
so that the quantity index remains invariant to changes in units of measurement);
(iv) the determinateness test (i.e., Qr(p0
, p1
, x0
, x1
) does not become
zero, infinite or indeterminate if an individual price becomes zero for any r = 0
and Qr(p0
, p1
; x0
, x1
) does not become zero, infinite or indeterminate if an individual
quantity becomes zero if 0 < r ≤ 2);14
(v) the proportionality test (i.e.,
Qr(p0
, p1
; x0
, λx0
) = λ for every λ > 0); and (vi) the time or point reversal test
(i.e., Qr(p0
, p1
; x0
, x1
)Qr(p1
, p0
; x1
, x0
) ≡ 1).
Define the quadratic mean of order r price index Pr for p0
0N , p1
0N ,
x0
> 0N , x1
> 0N , for r = 0, as
Pr(p0
, p1
, x0
, x1
) ≡
N
i=1(p1
i /p0
i )r/2
(p0
i x0
i /p0
· x0
)
N
k=1(p0
k/p1
k)r/2(p1
kx1
k/p1 · x1)
1/r
(4.5)
= Qr(x0
, x1
; p0
, p1
).
It is easy to see that Pr will also satisfy Fisher’s tests (i) to (vi). The only
Fisher tests not satisfied by the indexes Pr and Qr are: (vii) the circularity test
(i.e., Pr(p0
, p1
; x0
, x1
)Pr(p1
, p2
; x1
, x2
) ≡ P(p0
, p2
; x0
, x2
)); and (viii) the factor
reversal test (i.e., Pr(p0
, p1
; x0
, x1
)Qr(p0
, p1
; x0
, x1
) ≡ p1
·x1
/p0
·x0
except that
P2 and Q2, the ‘ideal’ price and quantity indexes, satisfy the factor reversal
test).
For r = 0 define the implicit quadratic mean of order r price index Pr as
(4.6) Pr(p0
, p1
, x0
, x1
) ≡ p1
· x1
/[p0
· x0
Qr(p0
, p1
, x0
, x1
)],
and define the implicit quadratic mean of order r quantity index Qr as
(4.7) Qr(p0
, p1
, x0
, x1
) ≡ p1
· x1
/[p0
· x0
Pr(p0
, p1
, x0
, x1
)].
13
Ordinary, as opposed to quadratic, means of order r were defined by Hardy,
Littlewood and Polya [1934].
14
Thus the quantity indexes Qr, for 0 < r ≤ 2, are somewhat more satisfactory
than the T¨ornqvist–Theil index Q0 defined by (2.12).
Thus the two pairs of indexes (Qr, Pr) and (Qr, Pr) will satisfy the weak
factor reversal test (1.1).
The following theorem relates the aggregator function fr to the quantity
index Qr:
Theorem 4.8. Suppose that (i) fr(x) is defined by (4.1), where r = 0; (ii)
x0
0N is a solution to the maximization problem maxx{fr(x) : p0
·x ≤ p0
·x0
,
x belongs to S}, where S is a convex subset of the nonnegative orthant in RN
,
fr(x0
) > 0 and the price vector p0
is such that p0
· x0
> 0; and (iii) x1
0N is
a solution to the maximization problem maxx{fr(x) : p1
·x ≤ p1
·x1
, x belongs
to S}, fr(x1
) > 0 and the price vector p1
is such that p1
· x1
> 0; then
(4.9) fr(x1
)/fr(x0
) = Qr(p0
, p1
, x0
, x1
).
Thus the quadratic mean of order r quantity index Qr is exact for the
quadratic mean of order r aggregator function, which in view of Theorem (4.3)
implies that Qr is a superlative index number.
Suppose that xs
0N is a solution to maxx{fr(x) : ps
· x ≤ ps
· xs
,
x belongs to S}, where fr(x) > 0, ps
· xs
> 0 for s = 0, 1, 2. Then using (4.9)
three times, we find that
Qr(p0
, p1
, x0
, x1
)Qr(p1
, p2
, x1
, x2
) = fr(x1
)[fr(x0
)]−1
fr(x2
)[fr(x1
)]−1
= fr(x2
)/fr(x0
)
= Qr(p0
, p2
, x0
, x2
).
Thus under the assumption that the producer or consumer is maximizing
fr(x) subject to an expenditure constraint each period, we find that Qr will
satisfy the circularity test in addition to the other Fisher tests which it satisfies.
A similar proposition is true for any exact index number, a fact which was first
noted by Samuelson and Swamy [1974]. Since the circularity test is capable of
empirical refutation, we see that we can empirically refute the hypothesis that
an economic agent is maximizing fr(x) subject to an expenditure constraint.
Thus violations of the circularity test could mean either that the economic
agent was not engaging in maximizing behavior or that his aggregator function
was not fr(x).
We note that Theorem (4.8) did not require that all prices be nonnegative;
only that quantities be positive. fr can also be a transformation function
(recall Section 2) which is nondecreasing in inputs and nonincreasing in outputs.
Theorem (4.8) will still hold except that prices of other outputs must be indexed
negatively while prices of inputs are taken to be positive. The quantity index
Qr may be used in the context of productivity measurement just as we used
the index Q0 in Section 3. We will return to this topic in Section 5.
242 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 243
Theorem (4.8) tells us that fr defined by (4.1) is exact for Qr defined
by (4.4). However, could there exist a linearly homogeneous functional form f
different from fr which is also exact for Qr? The answer is no, as the following
theorem shows:
Theorem 4.10. (Generalization of Byushgens [1925], Kon¨us and Byushgens
[1926]): Let S be an open subset of the positive orthant in RN
which is also
a convex cone. Suppose f is defined over S and is (i) positive, (ii) oncedifferentiable,
(iii) linearly homogeneous, and (iv) concave. Suppose that f
is exact for the quantity index Qr defined by (4.4) for r = 0 (i.e., if xs
is a
solution to maxx{f(x) : ps
· x ≤ ps
· xs
, x belongs to S} for s = 0, 1, then
Qr(p0
, p1
; x0
, x1
) = f(x1
)/f(x0
)). Then f is a quadratic mean of order r
defined by (4.1) for some aij, 1 ≤ i ≤ j ≤ N.
We note that the functional form fr defined by (4.1) may also be used
as a factor requirements function, and that the quantity index Qr defined by
(4.4) will still be exact for fr; i.e., Theorems (4.8) and (4.10) will still hold
except that the maximization problems maxx{fr(x) : ps
· x ≤ ps
· xs
, x belongs
to S} are replaced by the minimization problems minx{fr(x) : ps
· x ≥ ps
· xs
,
x belongs to S} for s = 0, 1, and condition (iv) is changed from concavity to
convexity. Thus the quadratic mean of order r quantity indexes Qr can be used
to aggregate either inputs or outputs provided that the functional form for the
aggregator function is a quadratic mean of order r.
The above theorems have their counterparts in the dual space.
Theorem 4.11. Suppose that (i) cr(p) ≡ i j bij p
r/2
i p
r/2
j
1/r
, where bij =
bji for all i, j, r = 0 and (p1, p2, . . . , pN ) ≡ p belongs to S where S is an open,
convex cone which is a subset of the positive orthant in RN
; (ii) cr(p) is positive,
linearly homogeneous and concave over S; (iii) x0
/p0
· x0
= cr(p0
)/cr(p),
where p0
0N so that (using the corollary (2.14) to Shephard’s lemma) x0
is a solution to the aggregator maximization problem maxx{ ˜fr(x) : p0
· x ≤
p0
· x0
, x ≥ 0N }, where ˜fr
15
is the direct aggregator function which is dual to
cr(p); and (iv) x1
/p1
·x1
= cr(p1
)/cr(p1
), p1
0N so that x1
is a solution to
the aggregator maximization problem maxx{ ˜fr(x) : p1
· x ≤ p1
· x1
, x ≥ 0N }.
Then
(4.12) cr(p1
)/cr(p0
) = Pr(p0
, p1
, x0
, x1
),
where Pr is the quadratic mean of order r price index defined by (4.5).
The proof of Theorem (4.11) is analogous to the proof of Theorem (4.8),
except that p replaces x, cr replaces fr, and Corollary (2.14) is used instead of
the Kon¨us–Byushgens–Hotelling Lemma.
15
Define ˜fr(x) ≡ 1/ maxp{cr(p) : p · x = 1, p belongs to S}, where S is the
closure of S.
Thus the quadratic mean of order r unit cost function cr is exact for
the price index Pr. Since, by Theorem (4.3), cr can provide a second order
approximation to an arbitrary twice differentiable unit cost function, we see
that Pr is a superlative price index for each r = 0. We note also that there is
an analogue to Theorem (4.10) for Pr; i.e., cr is essentially the only functional
form which is exact for the price index function Pr.
However, if we relax the assumption that the underlying aggregator function
be linearly homogeneous, then the index numbers Pr and Qr can be exact
for a number of true cost of living price indexes and Malmquist quantity indexes,
respectively; i.e., analogues to Theorems (2.16) and (2.17) hold.
We have obtained two families of price and quantity indexes: Pr, Qr and
Pr, Qr defined by (4.6) and (4.4) for any r = 0. The first price-quantity family
corresponds to an aggregator function ˜fr which has the unit cost function cr
defined by (4.2) as its dual, and the second price-quantity family corresponds to
an aggregator function fr defined by (4.1). Recall also that the price-quantity
indexes P0, Q0 correspond to a translog unit cost function, while P0, Q0 correspond
to a homogeneous translog aggregator function.
For various values of r, some of the indexes Pr or Pr have been considered
in the literature. For r = 2, P2 ≡ P2 becomes the Pigou [1912] and Fisher
[1922] ideal price index which corresponds to the Kon¨us–Byushgens [1926] homogeneous
quadratic aggregator function f2(x) ≡ (xT
Ax)1/2
, where A = AT
is a symmetric matrix of coefficients and it also corresponds to the unit cost
function c2(p) ≡ (pT
Bp)1/2
, where B = BT
is a symmetric matrix of coefficients.
If A−1
exists, then it is easy to show that the unit cost function which
is dual to f2 is ˜c2(p) = (pT
A−1
p)1/2
(at least for a range of prices). However,
if f2(x) ≡ (xT
aaT
x)1/2
= aT
x, where a 0N is a vector of coefficients (linear
aggregator function), then ˜c2(p) = ˜c2(p1, p2, . . . , pN ) ≡ mini{pi/ai : i =
1, 2, . . ., N} which is not a member of the family of unit cost functions defined
by c2(p) ≡ (pT
Bp)1/2
. On the other hand, if c2(p) = (pT
bbT
p)1/2
= bT
p, where
b 0N is a vector of coefficients (Leontief unit cost function), then the dual
aggregator function is ˜f2(x1, x2, . . . , xN ) = mini{xi/bi : i = 1, 2, . . ., N}, which
is a Leontief aggregator function. Thus P2 is exact for a Leontief aggregator
function (a fact which was noted by Pollak [1971a]) and, since P2 ≡ P2, it is
also exact for a linear aggregator function. This is an extremely useful property
for an index number formula, since the two types of aggregator functions
correspond to zero substitutability between the commodities to be aggregated
and infinite substitutability, respectively.
For r = 1, the price index P1 has been recommended by Walsh [1901;
105]. P1 is exact for the unit cost function c1, whose dual ˜f1 is the generalized
Leontief aggregator function (see Diewert [1971a] for the definition of this
function) which has the Leontief aggregator function as a special case. Walsh
also recommended the price index P1, which is exact for the generalized linear
244 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 245
aggregator function f1 (see Diewert [1971a] for the definition of this function)
which of course has the linear aggregator function as a special case.
If, in fact, a producer or consumer were maximizing a linear homogeneous
function subject to an expenditure constraint for a number of time periods, we
would expect (in view of the Approximation Theorem (4.3)) that the price
indexes Pr and Pr should more or less coincide, particularly if the variation in
relative prices were small. However, since real world data are not necessarily
consistent with this maximization hypothesis, let us consider some empirical
evidence on this point.
Table 1. Comparison of Some Index Numbers Tabulated by Fisher
Price Fisher
index number 1913 1914 1915 1916 1917 1918
PP a 54 100 100.3 100.1 114.4 161.1 177.4
PLa 53 100 99.9 99.7 114.1 162.1 177.9
P0 123 100 100.1 99.9 113.8 162.1 177.8
P0 124 100 100.16 99.85 114.25 161.74 178.16
P1 1153 100 100.13 99.89 114.20 161.70 177.83
P1 1154 100 100.12 99.90 114.24 161.73 177.76
P2 353 100 100.12 99.89 114.21 161.56 177.65
Irving Fisher [1922; 489] tabled the wholesale prices and the quantities
marketed for 36 primary commodities in the U.S. during the war years (1913–
1918), a time of very rapid price and quantity changes. Fisher calculated
and compared 134 different price indexes using this data. Table 1 reproduces
Fisher’s [1922; 244–247] computations for the Paasche and Laspeyres price
indexes, PP a and PLa, as well as for P0, P0, P1, P1 and P2 ≡ P2 ≡ PId.
Fisher’s identification number is given in column 2 of the table; e.g. P2 or the
‘ideal’ price index was identified as number 353 by Fisher. All index numbers
were calculated using 1913 as a base.
Note that the Paasche and Laspeyres indexes coincide to about two significant
figures, while the last four indexes mostly lie between the Paasche and
Laspeyres indexes and coincide to three significant figures. Fisher [1922; 278]
also calculated P2 (and some of the other ‘very good’ index numbers) using
different years as the base year and then he compared how the various series
differed; that is, he tested for ‘circularity’. Fisher found that the average discrepancy
was only about 1/3 percent between any two bases. Thus as far as
Fisher’s time series data are concerned, it appears that any one of the price
indexes, Pr or Pr, gives the same answer to three significant figures, and that
violations of circularity are only about 1/3 percent so that the choice of base
year is not too important.16
To determine how the price indexes Pr compare for different r’s in the
context of cross section data, one may look at Ruggles’ [1967; 189–190] paper
which compares the consumer price indexes PP a, PLa, P0 and P2 for 19 Latin
American countries for the year 1961. The indexes P0 and P2, using Argentina
as a base, differed by about one percent per observation, while P0 and P2, using
Venezuela as a base (the relative prices in the two countries differed markedly),
differed by about 1.5 percent per observation. P2 failed the circularity test
(comparing values with Venezuela and Argentina as the base country) by an
average of about two percent per observation, while P0 failed the circularity test
by about three percent per observation.17
Thus it appears that the indexes Pr
differ more and violate circularity more in the context of cross section analysis
than in time series analysis. However, the agreement between P0 and P2 in
the cross section context is still remarkable since the Paasche and Laspeyres
indexes differed by about 50 percent per observation.
5. Concluding Remarks
We have obtained two families of superlative price and quantity indexes, (Pr, Qr)
and (Pr, Qr); that is, each of these index numbers is exact for a homogeneous
aggregator function which is capable of providing a second order approximation
to an arbitrary twice continuously differentiable aggregator function (or its
dual unit cost function). Moreover, (Pr, Qr) and (Pr, Qr) satisfy many of the
Irving Fisher tests for index numbers in addition to their being consistent with
a homogeneous aggregator function. Note also that if prices are varying proportionately,
then the aggregates Qr and Qr are consistent with Hicks’ [1946]
aggregation theorem.
16
However, as a matter of general principle, it would seem that the chain
method of calculating index numbers would be preferable, since over longer
periods of time, the underlying functional form for the aggregator function
may gradually change, so that, for example, (1.5) will only be approximately
satisfied, with the degree of approximation becoming better as r approaches 0.
17
This failure of the circularity test should not be too surprising from the viewpoint
of economic theory since we do not expect the aggregator function for the
270 consumer goods and services to be representable as a linearly homogeneous
function; that is, we do not expect all ‘income’ or expenditure elasticities to be
unitary.
246 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 247
Although any one of the index number pairs, (Pr, Qr) or (Pr, Qr) could
be used in empirical applications, we would recommend the use of
(P2, Q2) ≡ (P2, Q2) ≡ (p1
·x1
p1
·x0
/p0
·x1
p0
·x0
)1/2
, (x1
·p1
x1
·p0
/x0
·p1
x0
·p0
)1/2
,
Irving Fisher’s [1922] ideal index numbers, as the preferred pair of index numbers.
There are at least three reasons for this selection.
(i) The functional form for the Fisher-Kon¨us-Byushgens ideal index number
is particularly simple and this leads to certain simplifications in applications.
For example, recall equation (3.18) which we used in order to measure
technical progress, (1+τ), in an economy whose transformation function could
be represented by a (nonseparable) homogeneous translog transformation function
t. If we assume t(z) = f2(z), where f2 is defined by (4.1) for r = 2, then
the analogue to (3.18) is
(5.1)
y1
1
y0
1
=
[− ˜w1
· ˜y1
+ p1
· (1 + τ)x1
][− ˜w0
· ˜y1
+ p0
· (1 + τ)x1
]1/2
(− ˜w1 · ˜y0 + p1 · x0)(− ˜w0 · y0 + p0 · x0)
.
If we square both sides of (5.1), the resulting quadratic equation in (1 + τ) can
easily be solved, given market data.
(ii) The indexes P2(p0
, p1
; x0
, x1
) and Q2(p0
, p1
; x0
, x1
) are functions of
p0
· x1
/p0
· x0
and p1
· x0
/p1
· x1
, which are ‘sufficient statistics’ for revealed
preference theory,18
and moreover Q2 is consistent with revealed preference theory
in the following sense: (a) if p0
· x1
< p0
· x0
and p1
· x0
≥ p1
· x1
(i.e., x0
revealed preferred to x1
), then Q2(p0
, p1
; x0
, x1
) < 1; (b) if p0
· x1
≥ p0
· x0
and
p1
· x0
< p1
· x1
(i.e., x1
revealed preferred to x0
), then Q2(p0
, p1
; x0
, x1
) > 1
(i.e., the quantity index indicates an increase in the aggregate); and (c) if
p0
· x1
= p0
· x0
and p1
· x0
= p1
· x1
(i.e., x0
and x1
revealed to be equivalent
or indifferent), then Q2(p0
, p1
; x0
, x1
) = 1 (i.e., the quantity index remains unchanged).
Thus even if the true aggregator function f is nonhomothetic, the
quantity index Q2 will correctly indicate the direction of change in the aggregate
when revealed preference theory tells us that the aggregate is decreasing,
increasing or remaining constant.
(iii) The index number pair (P2, Q2) is consistent with both a linear aggregator
function (infinite substitutability between the goods to be aggregated)
18
See Samuelson [1947], Houthakker [1950] and Afriat [1967]. We should also
mention the nonparametric method of price and quantity index number determination
pioneered by Afriat [1967] which depends only on the R2
inner
products of the rth price vector pr
and the sth quantity vector xs
, pr
· xs
, if
there are R observations. See Diewert [1973b; 424, footnote 2] for an algorithm
which would enable one to calculate a polyhedral approximation φ(x) to the
‘true’ linearly homogeneous aggregator function f(x).
and a Leontief aggregator function (zero substitutability between the commodities
to be aggregated). No other (Pr, Qr) or (Pr, Qr) has this very useful prop-
erty.
6. Proofs of Theorems
Proof of (2.2).
f(z1
) − f(z0
) = aT
z1
+
1
2
z1T
Az1
− aT
z0
−
1
2
z0T
Az0
= aT
(z1
− z0
) +
1
2
z1T
A(z1
− z0
) +
1
2
z0T
A(z1
− z0
)
=
1
2
(a + Az1
+ a + Az0
)T
(z1
− z0
), since A = AT
=
1
2
[ f(z1
) + f(z0
)]T
(z1
− z0
).
Assume f is thrice differentiable and satisfies the functional equation
f(x) − f(y) = 1
2 [ f(x) + f(y)]T
(x − y), for all x and y, in an open neighborhood.
We wish to find the function that is characterized by the fact that
its average slope between any two points equals the average of the endpoint
slopes in the direction defined by the difference between the two points. If f
is a function of one variable, the functional equation becomes f(x) − f(y) =
1
2 [f (x) + f (y)](x − y). If we differentiate this last equation twice with respect
to x, we obtain the differential equation 1
2 f (x)(x−y) = 0, which implies that
f(x) is a polynomial of degree two. The general case follows in an analogous
manner using the directional derivative concept.
Proof of (2.4). λ∗
1 and x1
will satisfy the first order necessary conditions
for an interior maximum for the maximization problem (2.5),
(6.1) f(x1
) = λ∗
1p1
; p1 · x1 = Y 1
.
Similarly, λ∗
0 and x0
will satisfy the first order conditions for the constrained
maximization problem (2.6),
f(x0
) = λ∗
0p0
; p0 · x0 = Y 0
.
Now substitute the first part of (6.1) into the right hand side of the identity
(2.3), and obtain (2.7).
Proof of (2.16). For a fixed u∗
, ln C(u∗
; p) is quadratic in the vector of
variables ln p and we may apply the quadratic approximation Lemma (2.2) to
248 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 249
obtain
ln C(u∗
, p1
) − ln C(u∗
, p0
)
=
1
2
ˆp1 pC(u∗
, p1
)
C(u∗, p1)
+ ˆp0 pC(u∗
, p0
)
C(u∗, p0)
· (ln p1
− ln p0
)
=
1
2
ˆp1 pC(u1
, p1
)
C(u1, p1)
+ ˆp0 pC(u0
, p0
)
C(u0, p0)
· (ln p1
− ln p1
)
where the equality follows upon evaluating the derivatives of C and noting that
2 ln u∗
= ln u1
+ ln u0
,
= ln P0(p0
, p1
, x0
, x1
)
using the definitions of x0
≡ pC(u0
, p0
), x1
≡ pC(u1
, p1
) and P0.
Proof of (2.17). It is first necessary to express the partial derivatives of
D with respect to the components of x, xD(ur
, xr
), r = 0, 1, in terms of the
partial derivatives of f. We have D(ur
, xr
) ≡ maxk{k : f(xr
/k) ≥ ur
} = 1,
for r = 0, 1, since each xr
is on the ur
‘utility’ surface. To find out how the
distance D(u0
, x0
) changes as the components of x0
change, apply the implicit
function theorem to the equation f(x0
/k) = u0
(where k = 1 initially). We
find that
∂k/∂xj ≡ ∂D(u0
, x0
)/∂xj = fj(x0
)
N
k=1
x0
kfk(x0
), j = 1, 2, . . ., N.
Similarly
∂D(u1
, x1
)/∂xj = fj(x1
)
N
k=1
x1
kfk(x1
), j = 1, 2, . . ., N.
Furthermore, the first order conditions for the two aggregator maximization
problems after elimination of the Lagrange multipliers yield the relations
p0
j /p0
· x0
= fj(x0
)
N
k=1
x0
kfk(x0
), j = 1, 2, . . ., N,
and
p1
j /p1
· x1
= fj(x1
)
N
k=1
x1
kfk(x1
), j = 1, 2, . . ., N.
Upon noting that the right hand sides of the last set of relations are identical
to the right hand sides of the earlier relations, we obtain
(6.2) xD(u0
, x0
) = p0
/p0
· x0
and xD(u1
, x1
) = p1
/p1
· x1
.
Now for a fixed u∗
, ln D(u∗
, x) is quadratic in the vector of variables ln x
and we may again apply the Quadratic Approximation Lemma (2.2) to obtain
the following equality:
ln D(u∗
, x1
) − ln D(u∗
, x0
)
=
1
2
ˆx1 xD(u∗
, x1
)
D(u∗, x1)
+ ˆx0 xD(u∗
, x0
)
D(u∗, x0)
· (ln x1
− ln x0
)
= ln Q0(p0
, p1
, x0
, x1
),
where the equality follows upon evaluating the derivatives of D, noting that
2 ln u∗
= ln u1
+ ln u0
, using (6.2), the equalities D(u1
, x1
) = 1, D(u0
, x0
) = 1
and the definition of Q0.
Proof of (4.3). Since both f and fr are twice continuously differentiable,
their Hessian matrices evaluated at x∗
, 2
f(x∗
) and 2
fr(x∗
), are both symmetric.
Thus we need only show that ∂2
f(x∗
)/∂xi∂xj = ∂2
fr(x∗
)/∂xi∂xj, for
1 ≤ i ≤ j ≤ N. Furthermore, by Euler’s theorem on linearly homogeneous
functions, f(x∗
) = x∗T
f(x∗
) and fr(x∗
) = x∗T
fr(x∗
). Since the partial
derivative functions ∂f(x)/∂xi are homogeneous of degree zero, application of
Euler’s theorem on homogeneous functions yields, for i = 1, 2, . . . , N,
(6.3)
N
j=1
x∗
j ∂2
f(x∗
)/∂xi∂xj = 0 =
N
j=1
x∗
j ∂2
fr(x∗
)/∂xi∂xj .
Thus the above material implies that fr(x∗
) = f(x∗
), fr(x∗
) = f(x∗
)
and 2
fr(x∗
) = 2
f(x∗
) will be satisfied under our present hypotheses if and
only if
(6.4) ∂fr(x∗
)/∂xi = f∗
i ≡ ∂f(x∗
)/∂xi, for i = 1, 2, . . . , N,
(6.5) ∂2
fr(x∗
)/∂xi∂xj = f∗
ij ≡ ∂2
f(x∗
)/∂xi∂xj, for 1 ≤ i < j ≤ N.
Thus we need to choose the N(N + 1)/2 independent parameters aij(1 ≤
i ≤ j ≤ N), so that the N + N(N − 1)/2 = N(N + 1)2 equations (6.4)
and (6.5) are satisfied. Recall that x∗
≡ (x∗
1, x∗
2, . . . , x∗
N ) 0N and that
y∗
≡ x∗T
f(x∗
) = f(x∗
) > 0, since f is assumed to be positive over its
250 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 251
domain of definition. Thus since y∗
> 0, x∗
i > 0 and r = 0, the numbers a∗
ij ,
for 1 ≤ i < j ≤ N, can be defined by solving the following equations for a∗
ij:
(6.6) f∗
ij =
1 − r
y∗
f∗
i f∗
j +
r
2
y∗(1−r)
a∗
ijx∗r/2−1
i x∗r/2−1
j , 1 ≤ i < j ≤ N.
The system of equation (6.6) is equivalent to (6.5) if we also make use of
(6.4). Now define a∗
ji = a∗
ij, for i = j, and then a∗
ii is defined as the solution to
the following equation:
(6.7)
N
j=1
a∗
ijx∗
i
r/2−1
x∗
j
r/2
y∗(1−r)
= f∗
i , i = 1, 2, . . . , N.
Now define fr(x) ≡
N
i=1
N
j=1 a∗
ij x
r/2
i x
r/2
j
1/r
, and it can be verified
readily that equations (6.4) and (6.5) are satisfied by fr as defined.
Proof of (4.8). Using assumptions (ii) and (iii) of (4.8) yields
v0
≡ p0
/p0
· x0
= fr(x0
)/fr(x0
),(6.8)
v1
≡ p1
/p1
· x1
= fr(x1
)/fr(x1
).(6.9)
Upon differentiating fr(x0
), the ith equation in (6.8) becomes
v0
i ≡ p0
i /p0
· x0
= (x0
i )(r/2)−1
N
j=1
aijx0
j
r/2
N
k=1
N
m=1
akmx0
k
r/2
x0
m
r/2
,
and therefore
(6.10)
N
i=1
x1
i
r/2
v0
i x0
i
1−(r/2)
=
i j
x1
i
r/2
aij x0
j
r/2
k m
akmx0
k
r/2
x0
m
r/2
.
Similarly, using equation (6.9), we obtain
(6.11)
N
i=1
x0
i
r/2
v1
i
1−(r/2)
=
i j
x0
i
r/2
aijx1
j
r/2
k m
akmx1
k
r/2
x1
m
r/2
.
Upon noting that aij = aji, take the ratio of (6.10) to (6.11),
(6.12) i(x1
i /x0
i )r/2
v0
i x0
i
j(x0
j /x1
j )r/2v1
j x1
j
= k m akmx1
k
r/2
x1
m
r/2
k m akmx0
k
r/2
x0
m
r/2
=
fr(x1
)
fr(x0)
r
.
Take the rth root of both sides of (6.12) and obtain (4.9).
Proof of (4.10). Let x, y be any two points belonging to S such that
(6.13) 1 = f(x) = f(y) = x · f(x) = y · f(y),
where the last two equalities follow from the linear homogeneity of f. Since
f is a concave function over S, for every z belonging to S, f(z) ≤ f(x) +
f(x)·(z−x) = f(x)+ f(x)·z−f(x) = f(x)·z, and similarly f(z) ≤ f(y)·z.
Thus x is a solution to maxz{f(z) : f(x)·z ≤ f(x)·x, z belongs to S}, and
y is a solution to maxz{f(z) : f(y) · z ≤ f(y) · y, z belongs to S}. Since f
is exact for Qr for some r = 0 by assumption, we must have, using (6.13),
Qr( f(x), f(y); x, y) = f(y)/f(x) = 1,
or
N
i=1
(xi/yi)r/2
fi(y)yi/ f(y) · y =
N
n=1
(yn/xn)r/2
fn(x)xn/ f(x) · x,
or
(6.14)
N
n=1
yr/2
n [fn(x)x1−(r/2)
n ] =
N
n=1
y1−(r/2)
n fn(y)xr/2
n ,
where fn(y) ≡ ∂f(y)/∂yn, fn(x) ≡ ∂f(x)/∂xn, and x · f(x) = 1 = y · f(y).
Replace the vector y ≡ (y1, y2, . . . , yN ) which occurs in (6.14) with the vector
yn
belonging to S, where f(yn
) = 1, for n = 1, 2, . . ., N. Regard the resulting
system of N equations as N linear equations in the N unknowns,
f1(x)x
1−(r/2)
1 , f2(x)x
1−(r/2)
2 , . . . , fN (x)x
1−(r/2)
N ,
and since we can choose the vectors y1
, y2
, . . . , yN
to be such that the coefficient
matrix on the left hand side of the system of N equations is nonsingular, we
may invert the coefficient matrix and obtain the solution
(6.15) fn(x)x1−(r/2)
n =
N
j=1
Anjx
r/2
j , n = 1, 2, . . . , N,
for some constants, Aij , 1 ≤ i, j ≤ N. Equation (6.15) is valid for any x
belonging to S, such that f(x) = 1; in particular, (6.15) is true for x = y,
(6.16) fn(y)y1−(r/2)
n =
N
j=1
Anj y
r/2
j , n = 1, 2, . . ., N.
252 Essays in Index Number Theory 8. Exact and Superlative Index Numbers 253
Now substituting (6.15) into the left hand side of (6.14) and (6.16) into
the right hand side of (6.14), we obtain
N
n=1
yr/2
n
N
j=1
Anjx
r/2
j =
N
n=1
xr/2
n
N
j=1
Anjyr/2
n ,
or
(6.17)
n j
yr/2
n Anjx
r/2
j =
n j
xr/2
n Anjy
r/2
j .
Since (6.17) is true for every x, y, such that f(x) = 1 = f(y), we must
have
(6.18) Anj = Ajn, for 1 ≤ n, j ≤ N.
Now take x
r/2
n times (6.15) and sum over n,
N
n=1
xr/2
n fn(x)x1−(r/2)
n =
N
n=1
N
j=1
Anjxr/2
n x
r/2
j = 1,
since x · f(x) = f(x) = 1.
Thus if f(x) = 1, then x satisfies the equation n j Anjx
r/2
n x
r/2
j = 1,
where Anj = Ajn. Since f is linearly homogeneous by assumption, we must
have for x belonging to S,
(6.19) f(x) =


N
n=1
N
j=1
Anjxr/2
n x
r/2
j


1/r
.
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