the theory of exact and superlative index numbers revisited
TRANSCRIPT
The Theory of Exact and Superlative IndexNumbers Revisited
By Carlo Milana0
March 2005
Abstract
This paper proposes to clarify some important questions that arestill open in the field of index number theory. The main results are thefollowing: (i) the so-called Quadratic Identity on which the superla-tive index numbers are based can be applied in more general cases thanthose traditionally considered; (ii) it is not only the Törnqvist indexnumber that does not rely on separability restrictions, but also someother indicators of absolute or relative changes are not based on suchrestrictions; (iii) in practice, however, all the index numbers or indi-cators that are considered to be superlative in Diewert’s (1976) sensegenerally fail by construction to be really "superlative"; (iv) thesehybrid index numbers may be far from providing the expected second-order approximation to the true index and may be found beyond theLaspeyres-Paasche interval even in the homothetic case. In conclusion,it would be more appropriate to construct a range of alternative in-dex numbers (including even those that are not "superlative") ratherthan follow the common practice of searching for only one "optimal"formula.
J.E.L. Classification C43, E31, O47KEYWORDS: Index Numbers, Separability, Aggregation
0 Istituto di Studi e Analisi Economica, Piazza dell’Indipendenza, no. 4, I-00185 Rome,Italy. Phone (office): +39-06-4448-2300; Mobile: +39-347-8000984; FAX: +39-06-4448-2249; E-mail: [email protected] paper has been prepared for the Specific Targeted Research Project "EUKLEMS-
2003. Productivity in the European Union: A Comparative Industry Approach" sup-ported by the European Commission, Research Directorate-General, within the SixthFramework Programme, Contract No. 502049 (SCS8). Comments received from PirkkoAulin-Ahmavaara, Alberto Heimler, and Dale W. Jorgenson on a previous version of thispaper are thankfully acknowledged. The usual disclaimers apply and the views of theauthor do not necessarily reflect those of the mentioned persons and institutions.
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"This is, I think, a case in which the mathematics itself is impor-tant, independently of the particular economic problem to which it isexplicitly related".
William M. (Terence) Gorman (1959b)
1. Introduction
This paper revisits the theory of "exact" and "superlative" index numbersthat was developed in a seminal article by Diewert (1976) and is still dom-inating the field of economic index numbers. An index number is said tobe "exact" for a function if it is identically equal to the ratio of numericalvalues of that function at any pairs of points taken into comparison. Thefirst mention of the term "exact" in the context of index numbers appearedin the Foundations of Paul A. Samuelson (1947, p. 155). In the introductionto the later enlarged edition of this work, Samuelson (1983, p. xx) conciselydescribed the meaning of his contribution: "Index number theory is shownto be merely an aspect of the theory of revealed preference. [...] this is thepoint of revealed preference–knowledge of but two (P,Q) situations (or ofa limited number of situations) can at best put bounds on each one of oursought-for ratios1."
In the same paragraph, he also wrote: "Thirty-five years after thatanalysis appeared there has been but one major advance in index num-ber theory–namely W. E. Diewert’s formalizing concept of a ’superlativeindex number,’ which is a formula based upon two periods (pj , qj) data thatwill be exactly correct as an ordinal indicator of utility for some specifiedfamily of indifference contours. (Only a few different ’superlative’ formulasare known; perhaps the set of simple superlative formulas is a limited set.)"Index numbers were, in fact, defined to be "superlative" by Diewert (1976,p. 117) (who cited Fisher, 1922, p. 247 for the use of an undefined notionof this term), if they are exact for a function that provides a second-orderdifferential approximation to the unknown true function.
The present paper proposes to clarify the meaning of the purely math-ematical Quadratic Identity on which the superlative numbers are based.Moreover, it generalizes and extends the cases of parameter changes whereit is possible to apply the Quadratic Identity by relaxing, in particular, some
1In subsequent papers, Samuelson (1950, p. 24) and Samuelson and Swamy (1974, p.585) had stressed that the underlying functions and their changes must be homothetic asa necessary condition for the definition of two-sided bounds for the unknown true indexnumber.
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unnecessary restrictions. It is also aimed at reformulating other basic propo-sitions of the theory and finding solutions to some of the problems that stillremain open.
Using a very general transformed quadratic function (the quadratic Box-Cox function) that encompasses well-known special cases, a unified treat-ment of the decomposition of functional value changes is given for the su-perlative index numbers. In this framework, it is shown that the Törnqvist-type index number, which is expressed in terms of relative log-changes, isnot the only index number that does not rely on restrictive separability andhomogeneity conditions. The Fisher "ideal" and the implicit Walsh indexnumbers, which are constructed (in terms of ratios) under separability andhomogeneity restrictions, have counterpart indicators of relative or absolutechanges that are compatible also with cases where these restrictions do nothold.
It is shown, however, that the superlative index numbers that are tra-ditionally constructed using the observed data are not really superlativebecause the weights relative to at least one point of observation are notderivable from the approximating quadratic function for which they are in-tended to be exact. As a consequence, the traditional formulae used toconstruct superlative index numbers are, in fact, hybrid index numbers thatmay be far from providing the expected second-order approximation.
The results obtained have various consequences in the theory and prac-tice of index numbers. Firstly, they help us to explain the empirical resultsrecently obtained by Hill (2002, 2005) (and mentioned by Diewert, 2004,pp. 450-451) concerning the large spread in numerical values of alterna-tive superlative index numbers. In another previous seminal paper, Diewert(1978) had shown that they are expected to approximate each other up tothe second-order and to be numerically very close if the two points undercomparison do not vary very much2. However, it has been surprising to findempirically that the spread between the largest and the smallest superla-tive index numbers often exceeds that between the Laspeyres and Paascheindexes, which are usually considered to be the bounds of the interval ofpossible values of economic index numbers, at least in homothetic cases. Inthe Hill’s (2002, 2005) empirical applications, the largest and the smallestsuperlative index numbers have resulted to differ by more than 100 per centfor a standard US national data set and by about 300 per cent in a cross-
2See also Vartia (1978). In a subsequent paper, Allen and Diewert (1981) indicatedtheoretical and numerical bounds for these index numbers and stated that the choice ofthe index number formula will not matter much if these bounds are narrow. However,Diewert (1978, p. 890, fn. 8), citing the discussion in Lau (1974, p. 183), had alreadyrecognized that, without performing extensive computations involving the third orderpartial derivatives of the index number formulae, we cannot specify exactly how smallshould be the change between the two compared points in order for the superlative indexesto be all very close to each other.
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section comparison of countries based on an OECD data set3. The presentpaper provides an additional explanation of this result.
It has been shown that the only index number that always falls nu-merically between the Laspeyres and Paasche indexes is the Fisher "ideal"index number. This always falls between the former two index numbers justbecause it consists of their geometric average. Summarizing our results, itturns out that: (i) the so-called Quadratic Identity on which the superlativeindex numbers are based can be applied in more general cases than thosetraditionally considered; (ii) it is not only the Törnqvist index number thatdoes not rely on separability restrictions, but also some other indicators ofabsolute or relative changes are not based on such restrictions; (iii) in prac-tice, however, all the index numbers or indicators that are considered tobe superlative in Diewert’s (1976) sense generally fail by construction to bereally "superlative"; (iv) these hybrid index numbers may be far from pro-viding a second-order approximation to the true index number and may befound beyond the Laspeyres-Paasche interval even in the homothetic case.Since the degree of approximation of the available index number formulaecannot be assessed, all of these are equally valid candidates as a good ap-proximation to the "true" unknown index number.
The paper is organized as follows. Section 2 re-examines the QuadraticIdentity within a general framework of accounting for functional value changesof an arbitrary differentiable function. Section 3 extends this analytical ap-proach to the case where the parameters or even the functional forms oftwo functions under comparison may differ and generalizes the results ob-tained in the literature up till now. Section 4 extends the analysis furtherby using transformed functions. Section 5 provides a unified approach toindex numbers by using a general transformed quadratic function with noa priori separability and homogeneity restrictions. Section 6 deals withthe approximation properties of index numbers and establishes conditionsfor constructing truly superlative index numbers. Section 7 concludes withremarks and suggestions on the use of index numbers.
2. General formulation of the quadratic approximation lemma
In the general case of an arbitrary differentiable function, the followingresult is obtained:
3Allen and Diewert (1981, p. 430) had clearly recognized that "in many applicationsinvolving the use of cross section data or decennial census data, there can be a tremendousamount of variation in prices or in quantities between the two periods so that alternativesuperlative index number can generate quite different results".
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LEMMA 2.1. Accounting for Functional Value Differences. Let z be avector of N real valued variables and let us assume that an arbitrary functionf(z) is continuously differentiable at least once, then, for all z0 and z1,
f(z1)− f(z0) =£(1− θ)∇zf(z
0) + θ∇zf(z1)¤T(z1 − z0) (2.1)
where ∇zf(zt) is the gradient vector of f evaluated at zt; and, denoting with
R01(z0, z1) and R11(z
0, z1) the remainder terms associated with the polyno-mials of order one in the Taylor series expansion for f around z0 and z1,respectively, θ takes the particular value θ∗(z0, z1) ≡ R01(z
0,z1)
R01(z0,z1)+R11(z
0,z1)when
R01(z0, z1) + R11(z
0, z1) 6= 0, or θ takes any real number as a value if f islinear in z (R01(z
0, z1) = R11(z0, z1)).
Proofs of propositions are given in Appendix B.
It is straightforward to show that the weight θ in (2.1) falls within theinterval 0 ≤ θ ≤ 1 if f (z) is quasiconcave or quasiconvex. Lemma (2.1) canbe complemented with the following corollaries.
COROLLARY 2.1. Accounting for Functional Value Ratios. If an ar-bitrary function f(z) is homothetic, so that f(z) = F [φ(z)], where F (·)is a well behaved transformation function (real valued, continuously differ-entiable, monotonically increasing and quasiconcave) and φ(z) is also wellbehaved and linearly homogeneous4, then, for all z0 and z1,
f(z1)
f(z0)= IZ · IY (2.2)
where
IZ ≡θ + (1− θ)
PNi=1 s
0iz1iz0i
(1− θ) + θPN
i=1 s1iz0iz1i
=φ(z1)
φ(z0)(2.3)
IY ≡ f(z1)
f(z0)/φ(z1)
φ(z0)(2.4)
4The concept of homotheticity was explicitly spelled out by Shephard (1953) andMalmquist (1953), although earlier researchers as Frisch (1936, p. 25) and Samuelson(1950, p. 24) had dealt with it implicitly.
5
with θ being defined by Lemma (2.1) and sti ≡ ∂f(zt)∂zti
· zti/PN
i=1∂f(zt)∂zti
· zti(t = 0, 1).
IZ and IY represent, respectively, the aggregator index of z and the indexof scale effects. The index IZ is exact for the aggregator or index functionφ, which means that it is identically equal to the ratio φ(z1)/φ(z0)5. Itis homogeneous of degree one in z and is equal to a Laspeyres-type indexnumber if θ = 0, whereas it is equal to a Paasche-type index number if θ = 1.It is straightforward to show that, if also the function φ is well behaved (inparticular, quasiconvex or quasiconcave), then the Laspeyres- and Paasche-type index numbers mentioned above are the bounds of the interval of allpossible numerical values taken by the ratio φ(z1)/φ(z0) for all z0 and z1.
Let us define the quadratic function:
fQ(z) ≡ a0 + aT z +1
2zTA z
= a0 +NPi=1
aizi +1
2
NPi=1
NPj=1
aijzizj , (2.5)
where the ai, aij are constant parameters and aij = aji for all i, j. .The following well-known result can be derived as another corollary of
Lemma 2.1:
COROLLARY 2.2. Diewert’s (1976, p. 117) Quadratic Identity. If andonly if f(z) has the functional form of the quadratic function fQ(z) definedby (2.5) where A 6= 0N×N , then the weight θ in (2.1) is equal to 1/2 for allz0 and z1, so that
fQ(z1)− fQ(z
0) =1
2[∇zfQ(z
0) +∇zfQ(z1)]T (z1 − z0) (2.6)
Some observations concerning this result are in order:(i) The inequality ∇zfQ(z
0) 6= ∇zfQ(z1) is sufficient to infer that the
function is non-linear in z (the converse is, however, not true since a non-linear function may have equal first derivatives at some different points).
(ii) Diewert (1976, p. 117) formulated the Quadratic Identity, whichhe called "Quadratic approximation lemma", under the assumption that
5The terms "index function" and "aggregator function" can be used here interchange-ably. The former was used by Shephard (1953, pp. 47-49) and Solow (1956, pp. 102-106),whereas the latter was used by Diewert (1976).
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the examined function is thrice continuously differentiable. This implies, Inparticular, that f is not linear in z.
(iii) Lau (1979), noted that "generalizing Diewert’s proof to the twicecontinuously differentiable case is straightforward" (p. 74, fn. 1). He alsogave an alternative proof of the Quadratic Identity by assuming a oncedifferentiable function and observed that this "widens considerably the ap-plicability of the lemma and consequently of the results which depend on itsvalidity" (p. 74).
(iv) Under the general once differentiability assumption and no addi-tional non-linearity condition, the quadratic function (2.5) can in fact beregarded as having the most general functional form with which equation(2.6) exactly holds. This equation still holds in special (limit) cases includingthe linear function fL(z) ≡ a0+ aT z, corresponding to fQ with A = 0N×N ,and the constant-value fC(z) ≡ a0, corresponding to an fQ with a = 0Nand A = 0N×N .
(v) The necessity part of Corollary (2.2) means that, when a functionis indeed non-linear (as implied by the thrice differentiability condition),then equation (2.6) is compatible only with a "strictly" quadratic functionalform, corresponding to an fQ with A 6= 0N×N (in this case, fQ can be said tobelong to the class of "strongly" concave or convex functions, using Avriel’set. al., 1988, pp. 1-2, terminology).
The rationale of the Quadratic Identity established by Corollary (2.2)can be explained geometrically as in Figure 1, where a quadratic functionin one single variable is represented. Let the functions f 0
L (z) and f1L (z) be,
respectively, the first-order polynomials of the Taylor series expansions forfQ(z) around z0 and z1, that is
f tL (z) ≡ at + btz
= fQ(zt) + f 0Q(z
t)(z − zt) for t = 0, 1 (2.7)
where prime means differentiation, at ≡ fQ(zt)−f 0Q(zt) zt, and bt ≡ f 0Q (z
t).Taking the first differences yields
f tL (z)− f t
L (zt) = f 0Q(z
t)(z − zt) for t = 0, 1 (2.8)
In Figure 1, f 0L (z1)− f 0
L (z0) = AB0 and f 1L (z1)− f 1
L (z0) = A0B.The first differences of fQ(z) can be derived in terms of the Taylor series
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expansion for f around z0 or z1 as follows:
fQ(z1)− fQ(z
0) = f 0Q (z
0)(z1 − z0) +1
2f 00Q (z0)(z1 − z0)2
= f 0L (z
1)− f 0L (z
0) +1
2f 00Q (z0)(z1 − z0)2 (2.9)
using (2.8)
fQ(z0)− fQ(z
1) = f 0Q(z
1)(z0 − z1) +1
2f 00Q (z1)(z0 − z1)2
= f 1L (z
0)− f 1L (z
1) +1
2f 00Q (z0)(z1 − z0)2 (2.10)
using (2.8)
Multiplying through equation (2.10) by −1 and rearranging terms yield
fQ(z1)− fQ(z
0) = f0Q(z
1)(z1 − z0)− 12f00Q(z
1)(z0 − z1)2
= f 1L (z
1)− f 1L (z
0)− 12f00Q(z
0)(z1 − z0)2 (2.11)
The second derivative f00Q(z) is a constant and has a negative (positive)
algebraic sign if fQ is concave (convex). Hence,
1
2f00Q(z
0)(z1 − z0)2 (BB0 in Figure 1)
=1
2f00Q(z
1)(z0 − z1)2 (AA0 in Figure 1) (2.12)
In view of (2.12), the arithmetic average of (2.9) and (2.11) leads us to
fQ(z1)− fQ(z
0) =1
2[f
0Q(z
1) + f0Q(z
0)] (z1 − z0)
=1
2
©£f 0L (z
1)− f 0L (z
0)¤− £f 1
L (z1)− f 1
L (z0)¤ªusing (2.8)
=1
2
©£AB0
¤+£A0B
¤ªin Figure 1
=1
2
©£AB +AA0
¤+£AB −BB0
¤ª(BB0 < 0 and AA0 < 0 since fQ is concave)
= AB in view of (2.12) (2.13)
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A
A’
B
B’..y=fQ(z)
y=f1L(z) y=f0
L(z)
Figure 1. Geometrical representation of Diewert’s (1976) Quadratic Identity
z0 z1O
y
z
Figure 1:
Since the function is quadratic, the remainder terms AA0 and BB0 of first-order approximations are equal and therefore they completely offset eachother. In the general case of a non-quadratic function f , these remainderterms may differ substantially and f(z1)− f(z0) = AB+ (AA0−BB0) mayturn out to be very different from AB.
The following useful result is obtained in terms of ratios rather than dif-ferences by imposing more restrictive conditions on the quadratic function:
COROLLARY 2.3. Accounting for Functional Value Ratios of a QuadraticHomothetic Function (Byushgens, 1925; Konüs and Byushgens, 1926; Frisch,1936, p. 30; Wald, 1939, p. 331; Pollak, 1971; Afriat, 1972, p. 45; 1977, pp.141-143; 2005, pp. 177-178). If a continuously once-differentiable quadraticfunction fQ(z) defined by (2.5) is a homothetic transformation of a linearlyhomogeneous function, so that ai = 0 for i = 0, 1, ..., N and fQ(z) = [φ(z)]
2,where φ(z) ≡ [12zTAz]1/2, then, for all z0 and z1,
fQ(z1)
fQ(z0)= IZ · IY (2.14)
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where
IY = IZ ≡PN
i=1 s0Q,i
z1iz0iPN
i=1 s1Q,i
z0iz1i
1/2 (2.15)
with stQ,i ≡ ∂fQ(zt)
∂zti· zti/
PNi=1
∂fQ(zt)
∂zti· zti ( t = 0, 1).
We note that IY and IZ represent, respectively, the aggregator index of zand the index of scale effects. they correspond to an ideal Fisher-type indexnumber, which is "exact" (identically equal to) the ratio φ(z1)/φ(z0) =[12z
1TAz1]1/2/[12z0TAz0]1/2 and is linearly homogeneous in z.
The accounting framework of the Quadratic Identity is usually seen asparticularly convenient for a practical approximation of the change in f(z)when this function is unknown but its first derivatives with respect to z aresomehow observable or measurable. The Bernstein-Weierstrauss approxima-tion theorem states that, on a closed bounded domain, a continuous func-tion can be uniformly approximated by polynomials. The function fQ(z)defined by (2.5) can be seen as providing a second-order approximationto an arbitrary true function f(z) around the point z∗ when its parame-ters are "calibrated" to certain numerical values, so that fQ(z∗) = f(z∗),∇fQ(z∗) = ∇f(z∗), and ∇2fQ(z∗) = ∇2f(z∗)67. However, the error of ap-proximation that is obtained by applying formula (2.6) using the "observed"weights ∇zf(z
0) and ∇zf(z1) is different from the error of second-order ap-
proximation obtainable using the weights ∇zfQ(z0) and ∇zfQ(z
1). This canbe assessed more clearly by means of the following result:
LEMMA 2.2. General Quadratic Approximation Lemma. Let f(z) be anarbitrary once differentiable function. If the value change of the arbitrary
6Lau (1974, pp. 183-184) distinguished the concept of "second-order differential ap-proximation" from that of "second-order numerical approximation" and claimed that poli-nomial expressions like (2.5), which can be interpreted as Taylor’s series expansions up tothe second order, provide both types of approximation (see also Barnett, 1983, pp. 19-20for further discussion).
7A "problem of accuracy of approximation" arises with expressions like (2.5). Fuss,McFadden, and Mundlak (1978, pp. 233-234) clearly stated this problem in the followingterms: "If a flexible form is calibrated to provide a second-order approximation at apoint, then the approximation is of this order only in a small neighborhood of this point.In other regions of interest, the form may be a poor approximation to the true function.[...] Further, the qualitative implications of the calibrated approximation may dependon the point of approximation; this is true, for example, of separability, which involvesproperties of the true function beyond second-order".
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f(z) is accounted for by using (2.1) where θ is set equal to 1/2 as in (2.6),then the obtained approximation error is equal to the difference of two first-order approximations multiplied by (0.5− θ), that is
f(z1)− f(z0) =1
2[∇zf(z
0) +∇zf(z1)]T (z1 − z0)
+Error of approximation (2.16)
where
Error of approximation
≡µ1
2− θ
¶©£f 0L (z
1)− f 0L (z
0)¤− £f 1
L (z1)− f 1
L (z0)¤ª
(2.17)
and, for r = 0, 1, f rL (z) is a first-order approximating (linear) function that
is tangent to f(z) at zr, that is f rL (z) ≡ ar+brz = f(zr)+∇zf (z
r)T (z−zr),with ar = f(xr)−∇zf (x
r) xr and br = ∇zf (xr).
The error of approximation represented by (2.17) is made of the differ-ence between two linear approximations to f(z1) − f(z0), which are con-structed, respectively, around z0 and z1. If f is linear in z, then f 0
L (z) =f 1L (z), whereas the term in brace brackets is equal to 0; if f is quadratic,then 1
2 − θ = 0; if f has any other functional form, then, in general, θ 6= 12
and the error of approximation may turn out to be non-negligible. Its sizedepends on the functional form of f and the distance of z1 from z0. Thisis shown in the following numerical example.
EXAMPLE 2.1. Let f(z) be the following cubic function of one singlevariable:
f(z) ≡ a+ b z +1
2c z2 +
1
6d z3 (2.18)
where a = 15.5008794, b = −38.2764683, c = 65.4734033, d = −53.7666768so that f(z) = 1.0 with z0 = 1.0 and f(z) = 1.5 with z1 = 1.5. The firstderivative is given by
f0(z) = b+ c z +
1
2d z2 (2.19)
A second order differential approximation to f(z) around z0 = 1.0 isgiven by the following quadratic function
fQ(z) ≡ α+ β z +1
2γ z2 (2.20)
11
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Quadratic approximation
"true" (cubic) function
Hybrid approximation
Figure 2 - Quadratic approximation versus "hybrid" approximation
1 1.5
f(z¹)-f(z?)
z
Value change of the
Figure 2:
where α = 6.5397667, β = −11.3931299, and γ = 11.7067265 so that
f(z0) = fQ(z0) (2.21)
f 0(z0) = f0Q(z
0) (2.22)
f 00(z0) = f00Q(z
0) (2.23)
In general, these equalities do not hold at z1 6= z0. Table 1 compares, fordifferent changes in z, the actual value changes in f which can be computedexactly using formula (2.1), that is 1
2 [(1 − θ)f 0(z0) + θf 0(z1)]T (z1 − z0)(see column 6) with their "hybrid" approximation obtained by imposingθ = 1/2 in (2.1), that is 12 [f
0(z0) + f 0(z1)]T (z1− z0) (see column 7). Figure2 compares this "hybrid" approximation and the quadratic approximationgiven by identity (2.6), that is 1
2 [f0Q(z
0) + f 0Q(z1)]T (z1 − z0) for different
values of z1.
We note, at the end of this section, that an alternative procedure can beset up using a Divisia index in order to account for value changes in thefunctions taken into exam here. Under the hypotheses defined above, inprinciple, this procedure should lead us to the same results that we haveobtained thus far8.
8See Milana (1993) for alternative index numbers implementing the Divisia index inthe discrete.
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Table 1 – Functional value change computed using (2.1) with alternative weights
0z
1z
)(' 0zf
)(' 1zf
*θθ =
)()( 01 zfzf −(actual value, computable using (2.1)
with *θθ = )
)()( 01 zfzf −(approximated
using (2.1) with θ = 1/2)
Error of
approximation
(1) (2) (3) (4) (5) (6) (7) (8) = (7)-(6) 1.0 1.125 0.313597 1.356885 0.567104 0.113156 0.044051 0.069105 1.0 1.250 0.313597 1.560070 0.724664 0.304217 0.235982 0.068235 1.0 1.375 0.313597 0.923150 1.533673 0.468169 0.231890 0.236279 1.0 1.5 0.313597 -0.553875 -0.791270 0.5 -0.060070 0.560070
Figure 3:
3. Accounting for differences in functional values when parame-ters or functional forms also differ
Let us consider two functions differing in functional forms or in parameters,some of which being functions of other variables, that is, for t = 0, 1,
f t(z) ≡ ht(z, kt) (3.1)
where some parameters of f t are functions of anM -dimensional vector k andothers may differ autonomously. Special cases of (3.1) are of some interestfor the following discussion. If ht is separable in z, then, in the case of"weak" ("non-linear") separability, (3.1) can be rewritten as:
ht(z, kt) = hts£ζ(z), kt
¤(3.2)
If the function ζ is homothetic (homogeneous), then ht is said to be ho-mothetically (homogenously) weakly separable. If all functions in (3.2) arehomothetic (homogeneous), then ht can be expressed as:
ht(z, k) = κt(k) · ζ(z) (3.3)
which is equivalent to
ht(z, k) = Ht · [κ∗t(k) + ζ∗(z)], (3.4)
13
a strongly (additively) separable function, where the starred functions differin parameter values from the respective counterparts in (3.3). More specialcases can be defined where the function (3.1) is separable also in k, so thatκt(k) ≡ σt · κ(k)9.
We recall that separability is a necessary but not a sufficient conditionfor constructing aggregates. An aggregate of separable variables exists if theindex function of those variables is also homogeneous of degree one. In ourcase, we should have λζ(z) = ζ(λz) for all z and λκ(k) = κ(λk) for all k inorder for ζ and κ to be aggregator functions. In the producer context, Shep-hard (1953, p. 63) was the first to realize the necessity of imposing the linearhomogeneity property on the index functions of separable quantity variablesin order for the respective dual price index functions to be independent ofother prices and quantities10.
The concept of homothetic separability has been introduced by Shephard(1953, p. 43) and Blackorby, Lady, Nissen, and Russell (1970) in the pro-ducer and the consumer contexts, respectively. They assumed homotheticfunctions within weakly separable functions. This definition is related tothe concept of homogeneous separability, which is obtained under linear ho-mogeneity restrictions on separable functions, regarded also as aggregabilityconditions (see, for example, Green, 1964, p. 25)11.
Let us now consider the following results:
LEMMA 3.1. Accounting for Functional Value Differences when Parame-ters or Functional Forms Differ. If f0(z) ≡ h0(z, k0) and f1(z) ≡ h1(z, k1)are two arbitrary functions differentiable at least once and characterized by
9The concept of separability was independently proposed by Sono (1945) and Leon-tief (1947a, 1947b) in the consumer and the producer contexts, respectively. The ter-minology of "weak" and "strong" separability was introduced by Strotz (1959). Morespecifically, a function F (x) = F (x1, x2, ..., xM ) is said to be "weakly" separable inthe partition (x1, x2, ..., xM ) if there exist functions F ∗, F 1,...,FM such that F (x) =F ∗[F 1(x1), F 2(x2), ...FM (xM )], and it said to be "strongly" separable if there exist func-tions F ∗∗, F 1,...,FM such that F (x) = F ∗∗[F 1(x1) + F 2(x2) + ...+ FM(xM )].10Solow (1956, p. 104n) attributed to Samuelson the proof that, in a two stage max-
imization procedure, the aggregating quantity index functions must be linearly homoge-nous. However, Gorman (1959a, pp. 476-478) found, in some special cases, conditionsweaker than linear homogeneity for a function of separable variables to be an aggregatorfunction (see also Blackorby, Primont, and Russell, 1978, Ch. 5, Blackorby, Schworm, andFisher, 1986, and Blackorby and Schworm, 1988).11More specifically, this condition is referred to as homogeneous weak separability if the
function is weakly separable as in (3.2) (see, for example, Diewert, 1993, pp. 12-13 andp. 28, and Diewert and Wales, 1995, pp. 260-261). Similarly, we may use the termhomogeneous strong separability if the function is linearly (or log-linearly) separable as in(3.3)-(3.4).
14
different parameter values or functional forms, then,
f1(z1)− f0(z0) =£(1− θ)∇zf
0(z0) + θ∇zf1(z1)
¤(z1 − z0) (3.5)
+"technical" change component (TC)
where
TC ≡ θ£f1(z0)− f0(z0)
¤+ (1− θ)
£f1(z1)− f0(z1)
¤(3.6)
and, by defining R01 and R11 as the remainder terms associated with thefirst-order polinomials in the Taylor series expansion of f1(z1) and f0(z0)
around z1 and z0, respectively, so that θ = θ∗ ≡ R01R01+R
11when R01 + R11 6= 0
or θ is a real number that takes any value if R01 = R11 = 0 (if both f0(z)and f1(z) are linear in z).
It is worth noting that the weight θ in (3.5) falls within the interval0 ≤ θ ≤ 1 if f (z) is quasiconcave or quasiconvex.
Lemma (3.1) can be complemented with the following corollaries.
COROLLARY 3.1. Accounting for Functional Value Ratios between Ar-bitrary Differentiable Functions. If two arbitrary continuously differentiablefunctions f0(z) and f1(z) are homothetically separable in z (so that f t(z) ≡σtF [φ(z)], for t = 0, 1, where F (·) is a well behaved transformation functionand φ(z) is linearly homogeneous, then, for all z0 and z1,
f1(z1)
f0(z0)= IZ · IY · IT (3.7)
where
IZ =θ + (1− θ)
PNi=1 s
0iz1iz0i
(1− θ) + θPN
i=1 s1iz0iz1i
, (3.8)
with θ defined in Lemma (2.1), and sti ≡ ∂f t(zt)∂zt · zti/
PNi=1
∂ft(zt)∂zt · zti , for
t = 0.1,
IT ≡ σ1
σ0(3.9)
15
IY =f1(z1)
f0(z0)/(IZ · IT ) (3.10)
The index numbers IZ , IY , and IT represent, respectively, the aggregatorindex of z, the index of scale effects, and the index of changes in parametersor functional form. In particular, the index IZ is homogeneous of degree onein z and is exact for the function φ, since it is identically equal to the ratioφ(z1)/φ(z0). It is a Laspeyres-type index number if θ = 0, whereas it is aPaasche-type index number if θ = 1. It is straightforward to show that, ifthe function φ(z) is quasiconvex or quasiconcave, then the Laspeyres- andPaasche-type index numbers are the bounds of the interval of all possiblenumerical values taken by the ratio φ(z1)/φ(z0) for all z0 and z1.
If f1 ≡ h1 and f0 ≡ h0 are defined under the homothetic separabilityas in (3.3), then the ratio f1/f0 can be decomposed into an index numberof weighted ratios z1i /z
0i , an index of scale effects, and an index number of
weighted changes in parameters or functional form, as shown in (3.7)-(3.8).By contrast, no separability restriction, however, is required to decomposethe difference f1 − f0 (or h1− h0).
Let us define the quadratic function:
f tQ(z) ≡ at0 + at z +
1
2z Atz
= at0 +NPi=1
atizi +1
2
NPi=1
NPj=1
atijzizj (3.11)
where all parameters are variable. Moreover, the parameters of f tQ are func-
tion themselves of other variables as follows:
at0 ≡ αt0 + ktT · βt + 12k
t T ·Bt · kt,at ≡ αt + ktT · Γt
so thatf tQ(z) ≡ htQ(z, k
t)
where
htQ(z, k) ≡ αt0 + αtT z +1
2zTAtz
+kTβt +1
2kTBtk
+kTΓtz. (3.12)
16
The function htQ(zt, k) can be expressed also as a quadratic function in k:
htQ(zt, k) ≡
ψtQ(k) ≡ bt0 + kT bt +
1
2kTBtk (3.13)
where bt0 ≡ αt0 + αtT zt + 12z
tTAtzt, bt ≡ βt + Γtzt.The quadratic functional form (3.11)-(3.12) may be reformulated in order
to represent special separability cases. The early literature on separabilityof the so-called "flexible" functions, which provide a second-order approx-imation to an arbitrary function, has indicated the required parameter re-strictions12. The "weak" separability of f tQ in z with respect to parameterchanges occurs if (3.11) can be rewritten as:
f tQ(z) ≡ f t
Q[ζ(z)]
≡ at0 + at1ζ(z) +1
2ζ(z) at2ζ(z) (3.14)
where at1 and at2 are variable scalar parameters and ζ(z) must be the linearfunction αT z in order for the function f tQ to have the functional form (3.11).In this case, htQ can be rewritten as
htQ(z, k) ≡ htQ [ζ(z), k]
= αt0 + αt1ζ(z) +1
2ζ(z)at2ζ(z)
+kTβt +1
2kTBtk +
1
2kTγtζ(z) (3.15)
where γt is an M -dimensional (column) vector of variable parameters. Asimilar condition can be imposed on the parameters of the first- and second-order terms in k in order to represent the case of "non-linear" separability ofhtQ in k, by defining the linear function κ(k) ≡ βTk. Therefore, by imposingthe "weak" separability conditions for z and k, htQ becomes a non-linear(quadratic) function whose arguments are the linear functions ζ(z) and κ(k),that is htQ [ζ(z), κ(k)] ≡ αt0 +α
t1ζ(z) +
12ζ(z)a
t2ζ(z) +κ(k)β
t1 +
12κ(k)β
t2κ(k)
+12κ(k)γ
tζ(z).The "strong" or "linear" separability of f tQ ≡ htQ in z, k, and parameter
12See Diewert (1993, p. 15) for references to this literature.
17
changes occurs when Γt = 0M×N in (3.12) so that
f tQ(z) ≡ htQ(z, k
t) = σt[ζQ(z) + κQ(k)] (3.16)
where ζQ(z) = α0 + αz +12z Az, κQ(k) = kTβ + 1
2kTBk, and σt is a factor
of proportionality. We shall see that, under the conditions of homogeneityand the necessary restrictions on parameter values, the form of (3.16) isequivalent to
f tQ(z) ≡ htQ(z, k
t) = σt · ζ∗Q(z) · κ∗Q(k) (3.17)
where which is indistinguishable from the "weak separability" case with onlytwo separable groups of variables13.
The following results regarding the accounting for functional value dif-ferences can be obtained:
COROLLARY 3.2. Accounting for Numerical Value Differences of TwoQuadratic Functions Differing in Parameters. If two quadratic functionsf 1Q (z
1) and f 0Q (z
0) defined by (3.11)-(3.12) differ in parameters, then, forall z0 and z1,
f 1Q (z
1)− f 0Q (z
0) =1
2[∇zf
0Q (z
0) +∇zf1Q (z
1)](z1 − z0)
+Parameter change component (3.18)
where
Parameter change component
≡ (a10 − a00) + (a1 − a0)T
1
2(z1 + z0) +
1
2z0(A1 −A0)z1 (3.19)
or
h 1Q(z
1, k1)− h 0Q(z
0, k0) =1
2[∇zh
0Q(z
0, k0) +∇zh1Q(z
1, k1)](z1 − z0)
+Residual component (3.20)
13This is a well known result (see Gorman, 1959a, p. 471).
18
the residual component is equal to the parameter change component definedby (3.19), in which the sum of the "zero-" and "first-order" terms can be,in turn, decomposed as follows
(a10 − a00) + (a1 − a0)T
1
2(z0 + z1)
=1
2
£∇kh0(z0, k0) +∇kh
1(z1, k1)¤(k1 − k0)
+(α10 − α00) + (α1 − α0)
1
2(z0 + z1)
+1
2(k0 + k1)(β1 − β0) +
1
2k0(B1 −B0)k1 (3.21)
Differently from the Quadratic Identity established by Corollary (2.2),Corollary (3.2) is not an "if and only if" result: the decomposition of valuedifference between two quadratic functions imply (3.18)-(3.20), but the con-verse is not true if they have different parameters. The equations (3.18)-(3.20) may hold, as a particular case, also with functional forms that arelinear in z. Note that the inequality ∇zf
0(z0) 6= ∇zf1(z1) for z0 6= z1 may
occur with linear functions when these have different parameters.Furthermore, we observe that the theory of exact and superlative index
numbers has traditionally considered quadratic functional forms where onlythe parameters of the zero- and first-order terms in z may differ. By contrast,the result obtained by Corollary (3.2) shows that, also when the parametersof second-order terms are different, the difference in functional values of twoquadratic functions can be still split into two separate components due todifferences in z and differences in parameters.
LEMMA 3.2. Quadratic Approximation of a Value Difference of TwoArbitrary Functions with Different Parameters or Functional Forms. Letf1(z) ≡ h1(z, k1) and f0(z) ≡ h0(z, k0) be two arbitrary once differentiablefunctions with different parameters or functional forms. If their value differ-ence is accounted for by using (3.5) where θ is set equal to 1/2 as in (3.16),then , for all z0 and z1,
f1(z1)− f0(z0) =1
2[∇zf
0(z0) +∇zf1(z1)](z1 − z0)
+ Parameter change component
+Error of approximation (3.22)
19
where
Parameter change component
≡ θ[f1(z0)− f0(z0)] + (1− θ)[f1(z1)− f0(z1)] (3.23)
and
Error of approximation ≡µ1
2− θ
¶©£f 0L (z
1)− f 0L (z
0)¤− £f 1
L (z1)− f 1
L (z0)¤ª
(3.24)
with θ = θ∗(z0, z1) ≡ R01(z0,z1)
R01(z0,z1)+R11(z
0,z1)if R01(z
0, z1) +R11(z0, z1) 6= 0, or θ
may take any value if R01(z0, z1) = R11(z
0, z1) = 0, and f tL (z) is a first-
order (linear) approximating function that is tangent to f t(z) at zt (that isf tL (z) ≡ at+ btz = f t(zt)+∇zf
t(zt)(z− zt), with at = f t(xt)−∇zft(xt)xt
and bt = ∇zft(xt), with t = 0, 1).
Let us, now, construct the following general decomposition of the ab-solute value difference between h1(z1, k1) and h0(z0, k0):
h1(z1, k1)− h0(z0, k0)
= (1− λ) [h0(z1, k0)− h0(z0, k0)] + λ [h1(z1, k1)− h1(z0, k1)]
+λ [h1(z0, k1)− h0(z0, k0)] + (1− λ) [h1(z1, k1)− h0(z1, k0)](3.25)
The following technical result is also obtained:
COROLLARY 3.3. Accounting for the Sum of Value Differences between TwoQuadratic Functions with Different "Zero-order" and "First-order" Parame-ters (Caves, Christensen, and Diewert’s, 1982, pp.1412-1413 Translog Iden-tity). If two quadratic function defined by (3.12) have different parametersof the "zero-order" and "first-order" terms but equal parameters the same"second-order" terms in z (that is A1 = A0 = A), then, for all z0, z1, k0, k1,
20
£f0Q(z
1)− f0Q(z0)¤+£f1Q(z
1)− f1Q(z0)¤
(3.26)
or, equivalently,£h0Q(z
1, k0)− h0Q(z0, k0)
¤+£h1Q(z
1, k1)− h1Q(z0, k1)
¤
=£∇zh
0Q(z
0, k0) +∇zh1Q(z
1, k1)¤ T(z1 − z0) (3.27)
This result is known under the name of ’Translog’ Identity, because it wasformulated by Caves, Christensen, and Diewert (1982, pp. 1412-1413) interms of translog functions.
In the more general case of quadratic functions with no restrictions onvariable parameters, we have the following result:
COROLLARY 3.4. Accounting for the Sum of Value Differences betweenTwo Quadratic Functions with Different Parameters (Caves, Christensen,and Diewert, 1982, pp.1412-1413). If two quadratic functions are definedby (3.11)-(3.12), differing in all parameters, including those of the "second-order" terms in z (A1 may differ from A0), then, for all z0, z1, k0, k1,
£f0Q(z
1)− f0Q(z0)¤+£f1Q(z
1)− f1Q(z0)¤
(3.28)
or, equivalently,£h0Q(z
1, k0)− h0Q(z0, k0)
¤+£h1Q(z
1, k1)− h1Q(z0, k1)
¤
=£∇zh
0Q(z
0, k0) +∇zh1Q(z
1, k1)¤(z1 − z0)
−12(z1 − z0)(A1 −A0)(z1 − z0) (3.29)
If A1 = A0, then the term 12(z
1−z0)(A1−A0)(z1−z0) is equal to zero andthe result obtained with Corollary (3.4) is the same of that obtained withCorollary (3.3). It is evident that Caves, Christensen, and Diewert (1982,pp. 1412-1413) introduced the restriction A1 = A0 in order to obtain theequivalence between an arithmetic average of the two functions (evaluatedat z0 and z1) and the component of weighted changes in z. This equivalenceno longer holds when A1 6= A0, but the following result is useful to show
21
that, even in this general case, the overall functional value difference canbe still decomposed into separate components of differences in z, k, andvariable parameters.
COROLLARY 3.5: If two functions f tQ(z) ≡ htQ(z, kt) are defined by
(3.11)-(3.12), then, for all z0, z1, k0, k1,
£f1Q(z
0)− f0Q(z0)¤+£f1Q(z
1)− f0Q(z1)¤
= (a1 − a0)T (z0 + z1) + 2(a10 − a00)
+z0(A1 −A0)z1 +1
2(z1 − z0)(A1 −A0)(z1 − z0) (3.30)
or, equivalently,£h1Q(z
0, k1)− h0Q(z0, k0)
¤+£h1Q(z
1, k1)− h0Q(z1, k0)
¤
=£h1Q(z
1, k1)− h1Q(z1, k0)
¤+£h0Q(z
0, k1)− h0Q(z0, k0)
¤+£h1Q(z
1, k1)− h0Q(z1, k1)
¤+£h1Q(z
0, k0)− h0Q(z0, k0)
¤(3.31)
where
£h1Q(z
1, k1)− h1Q(z1, k0)
¤+£h0Q(z
0, k1)− h0Q(z0, k0)
¤= [∇kh
0Q(z
0, k0) +∇kh1Q(z
1, k1)]T (k1 − k0)
−12(k1 − k0)(B1 −B0)(k1 − k0) (3.32)
and
£h1Q(z
1, k1)− h0Q(z1, k1)
¤+£h1Q(z
0, k0)− h0Q(z0, k0)
¤= 2(α10 − α00) + (α
1 − α0)T (z1 + z0)
+z0(A1 −A0)z1 +1
2(z1 − z0)(A1 −A0)(z1 − z0)
+(k1 + k0)T (β1 − β0)
+k0(B1 −B0)k1 +1
2(k1 − k0)(B1 −B0)(k1 − k0) (3.33)
22
Adding (3.29) to (3.31)-(3.33) and dividing the resulting expression throughby 2 yield the same result obtained by Corollary (3.2).
It is straightforward to show that, in order to obtain 12
h∇zh
0Q(z
0, k0) +∇zh1Q(z
1, k1)iT
·(z1 − z0) = (1 − λ) [f0(z1) − f0(z0)] + λ[f1(z1) − f1(z0)], λ must takethe particular value
λ∗(z0, z1) ≡12
¡F 1 − F 0
¢+ 14(z
1 − z0)(A1 −A0)(z1 − z0)
(F 1 − F 0)(3.34)
where F t ≡ [f t(z1) − f t(z0)] for t = 0, 1. If A1 = A0, then λ∗(z0, z1) = 1/2).This is the case traditionally considered in this context. If A1 = A0 andB1 = B0, then equations (3.20) and (3.21) reduce to the decompositionprocedure used, for example, by Diewert and Morrison (1986, Theorem 1,p. 663) in terms of translog functions with variable "zero-order" and "first-order" parameters and constant "second-order" parameters.
From our results, it is evident that in the more general case, where A1 6=A0, it is still possible to decompose the difference between two quadraticfunctions defined by (3.11) into a component due to (weighted) changesin z and a component due to (weighted) differences in parameters. Thesetwo components turn out to be a weighted average of two functional valuechanges evaluated at the respective reference variables, with weights thatmay differ from 1/2. Similarly, it is still possible to decompose the func-tional value changes in htQ(z, k) defined by (3.12) into three separate com-ponents, respectively due to (weighted) changes in z, k, and parameters.This remarkable result widens considerably the scope of applicability of thedecomposition of differences in functional values of quadratic functions.
COROLLARY 3.6. Accounting for Functional Value Ratios of Two DifferentQuadratic Homothetic Functions (Byushgens, 1925; Konüs and Byushgens,1926; Frisch, 1936, p. 30; Wald, 1939, p. 331; Pollak, 1971; Afriat, 1972,p. 45; 1977, pp. 141-143; 2005, pp. 177-178). If two continuously once-differentiable quadratic functions f0Q(z) and f1Q(z) defined by (3.11) withdifferent parameters are homothetic transformation of a linearly homoge-neous function14, so that ai = 0 for i = 0, 1, ..., N , and f tQ(z) = σt[φ(z)]2
for t = 0, 1, where φ(z) ≡ [12zTAz]1/2, then, for all z0 and z1,
f1Q(z1)
f0Q(z0)
= IZ · IY · IT (3.35)
14 In a linearly homogeneous function, if all the variables change proportionally, also theresulting functional value changes by the same factor of proportionality.
23
where
IY = IZ ≡PN
i=1 s0Q,i
z1iz0iPN
i=1 s1Q,i
z0iz1i
1/2 (3.36)
with stQ,i ≡∂ftQ(z
t)
∂zti· zti/
PNi=1
∂ftQ(zt)
∂zti· zti ( t = 0, 1),
IT ≡ σ1
σ0(3.37)
We note that (3.36) is an ideal Fisher-type index number, which is "ex-act" for (identically equal to) the ratio φ(z1)/φ(z0) = [12z
1TAz1]1/2/[12z0TAz0]1/2.
We finally observe that an alternative procedure can be set up usingDivisia indexes in order to account for value changes in each of the twofunctions taken into exam here. This procedure should lead us to the sameresults that we have obtained in this section.
4. Accounting for value changes of transformed functions
In the previous sections, we have seen that the Quadratic Identity, in itsoriginal formulation, is useful only in the very restrictive case of quadraticfunctions and cannot be applied to arbitrary functions without incurring apossible non-negligible error. In the general case, a way to reduce this erroris to use a transformed quadratic function. With appropriate parametervalues, it is this function rather than a quadratic function that provides asecond-order differential approximation to an arbitrary function.
Suppose that the arbitrary function f t(z) considered in section 3 can bedefined as f t[Z(q)], where Z(q) = z, with the ith element zi = z(qi), so thatf t[Z(q)] = f t(q). Furthermore, suppose that a general quadratic functionf tGQ(q) can be transformed into a quadratic function as follows:
g£f tGQ(q)
¤= at0 + atZt(q) +
1
2Zt(q)AtZt(q) (4.1)
24
where Zt(q) ≡ [zt(q1) zt(q2) ... zt(qN)]T , so that
f tGQ(q) = g−1·at0 + atZt(q) +
1
2Zt(q)AtZt(q)
¸(4.2)
Let us define at0 ≡ αt0+βtK(xt)+ 12K(x
t)BtK(xt) and at ≡ αt+K(xt)Γt
where K(x) ≡ [k(x1) k(x2) ... k(xM)]T so that ghf tGQ(q)
i≡ g
hhtGQ(q, x
t)i,
with
g£htGQ(q, x)
¤= αt0 + αtiZ(q) +
1
2Z(q)AtZ(q)
+K(x)βt +1
2K(x)BtK(x)
+K(x)ΓtZ(q) (4.3)
where zi ≡ z(qi) and km ≡ K(xm), or in vector notation z ≡ Z(q) andk ≡ K(x), so that
htGQ(q, x) = g−1[αt0 + αtiZ(q) +1
2Z(q)AtZ(q)
+K(x)βt +1
2K(x)BtK(x)
+K(x)ΓtZ(q)] (4.4)
The values of all parameters may change as t changes, and g, z, and k arecontinuous and monotonic functions of one single variable with non-zeroderivatives.
Since the functions z and k are continuous and monotonic, it is possibleto invert them in order to obtain
qi = z−1(zi) or, in vector form, q = Z−1(z) (4.5)
xm = k−1(km) or, in vector form, x = K−1(k) (4.6)
so that
g[f tGQ(q)] = g©f tGQ[Z
−1(z)]ª
≡ f tQ(z) defined by (3.11) (4.7)
25
and
g[htGQ(q, x)] = g©htGQ[Z
−1(z),K−1(k)]ª
≡ htQ(z, k) defined by (3.12) (4.8)
If the functions f tQ and htQ have specific parameter values such thatf tGQ(q
∗) = f t(q∗), ∇f tGQ(q∗) = ∇f t(q∗), and ∇2f tGQ(q∗) = ∇2f t(q∗), thenthe function f tGQ = g−1(f tQ) provides a second-order differential approxi-mation to f t around q∗. This should be contrasted with the assumptionin section 3 that it is f tQ rather than g−1(f tQ) that provides a second-orderdifferential approximation to f t around q∗.
We can now obtain the following result:
LEMMA 4.1. Accounting for Value Differences of Two General QuadraticFunctions. If the function f tGQ(q) is defined by (4.2), then
g[f1GQ(q1)]− g[f0GQ(q
0)]
=1
2
ng0[f0GQ(q
0)] · [ bZ 0(q0)]−1 ·∇qf0GQ(q
0) + g0[f1GQ(q1)] · [ bZ 0(q1)]−1 ·∇qf
1GQ(q
1)oT
·[(Z(q1)− Z(q0)] + T (4.9)
where
T ≡ (a10 − a00) + (a1 − a0)
1
2[Z(q0) + Z(q1]
+Z(q0)(A1 −A0)Z(q0) (4.10)
[ bZ 0(q)]−1 is a diagonal matrix in which the (i, i)th element is equal todqidzi
=dz−1(zi)
dzi= 1/z0(qi) (4.11)
(with z0(qi) 6= 0, by assumption).
26
5. Full specification of a general transformed quadratic function
The transformed quadratic function (4.1)-(4.2) can be fully specified bychoosing the functional form of the transformation functions g, z, and k.Among the many candidates, if we define the following functions as suggestedin the original work of Box and Cox (1964):
g(y) ≡ yρ − 1ρ
(5.1)
z(qi) ≡ qλi − 1λ
(5.2)
k(xi) ≡ xλi − 1λ
(5.3)
and replace the functions (5.1) and (5.2) in (4.1), then we obtain the follow-ing quadratic Box-Cox function15:
hf tQρ,λ(q)
iρ − 1ρ
= at0 +NPi=1
atiqλi − 1λ
+1
2
NPi=1
NPj=1
atijqλi − 1λ
qλj − 1λ
(5.4)
which, by replacing (5.1)-(5.3) in (4.3), corresponds to
hhtQρ,λ(q, x)
iρ − 1ρ
= αt0 +NPi=1
αtiqλi − 1λ
+1
2
NPi=1
NPj=1
atijqλi − 1λ
qλj − 1λ
+MP
m=1βtm
xλm − 1λ
+1
2
MPm=1
MPn=1
btmn
xλm − 1λ
xλn − 1λ
NPi=1
MPm=1
γtmi
xλm − 1λ
qλi − 1λ
(5.5)
15The explicit use of the quadratic Box-Cox function can be dated back at least tothe works of Khaled (1977), Kiefer (1977), Appelbaum (1979), and Berndt and Khaled(1979). The special case of a quadratic Box-Cox aggregator function has also been derivedby Diewert (1980, pp. 450-451) starting from a quadratic mean-of-order-r aggregatorfunction.
27
by setting
at0 ≡ αt0 +MP
m=1βtm
xλm − 1λ
+1
2
MPm=1
MPn=1
btmn
xλm − 1λ
xλn − 1λ
(5.6)
ati ≡ αti +MP
m=1γtmi
xλm − 1λ
(5.7)
When ρ→ 0 and λ→ 0, the function (5.4) reduces to the following translogfunctional form:
ln f tTrg(q) = at0 +NPi=1
ati ln qi +1
2
NPi=1
NPj=1
atij ln qi ln qj (5.8)
so that
f tTrg(q) ≡ exp at0 · exp(NPi=1
ati ln qi +1
2
NPi=1
NPj=1
atij ln qi ln qj) (5.9)
or, equivalently, the function (5.5) reduces to
lnhtTrg(q, x) = αt0 +NPi=1
αti ln qi +1
2
NPi=1
NPj=1
atij ln qi ln qj
+MPm=1
βtm lnxm +1
2
MPm=1
MPn=1
btmn lnxm lnxn
+MPm=1
NPi=1
γtmi lnxm ln qi (5.10)
so that
htTrg(q, x) ≡ expαt0 · exp(NPi=1
αti ln qi +1
2
NPi=1
NPj=1
atij ln qi ln qj)
· exp(MP
m=1βtm lnxm +
1
2
MPm=1
MPn=1
btmn lnxm lnxn)
· exp(MP
m=1
NPi=1
γtmi lnxm ln qi) (5.11)
The function f tTrg is homogeneous of degree one in q under the followingconditions:
(i)NPi=1
αti = 1
(ii)NPi=1
atij = 0 for j = 1,2,. . . , N,
28
(iii)NPj=1
atij = 0, for i = 1,2,. . . , N
(condition (ii) implies (iii) under the symmetry of matrix A), and
(iv)NPi=1
γmi = 0, for m = 1,. . . ,M
Similarly, the function htTrg is homogeneous of degree one in x under thefollowing conditions:
(v)MP
m=1βtm = 1,
(vi)MP
m=1bmn = 0, for j = 1,2,. . . , N,
(vii)MPn=1
bmn = 0, for i = 1,2,. . . , N
(condition (vi) implies (vii) under the symmetry of matrix B), and
(viii)MP
m=1γmi = 0, for n = 1,2,. . . , N.
When ρ 6= 0 and λ 6= 0, then the function (5.4) has the following func-tional form:
f tQρ,λ(q) =
"at0 +
NPi=1
atiqλi +
1
2
NPi=1
NPj=1
atijqλi q
λj
# 1ρ µ1
λ
¶ 1ρ
(5.12)
where at0 ≡Ãλ + ρλat0 − ρ
NPi=1
ati +12ρλ
NPi=1
NPj=1
atij
!, ati ≡
Ãρati − ρ
λ
NPj=1
atij
!,
and atij ≡ ρλa
tij . The function f t
Qρ,λ is homogeneous of degree 2λ/ρ in q if
at0 = 0 and ati = 0 for all i ’s, that is if the following conditions are satisfied:
(1 + ρat0) =1
2
ρ
λ
NPi=1
ati (5.13)
λati =NPj=1
atij (5.14)
These two conditions imply
ρ
2λ2=
(1 + ρat0)PNi=1
PNj=1 a
tij
(5.15)
In this case, the function f tQρ,λ is homogeneous of degree 2λ/ρ in q and
can be re-expressed as
29
f tQρ,λ(q) =
"1
2
r
λ
NPi=1
NPj=1
atijqλi q
λj
# 1ρ µ1
λ
¶ 1ρ
=
"NPi=1
NPj=1
atijqλi q
λj
# 1ρ
1 + rat0NPi=1
NPj=1
atij
1ρ
(5.16)
since 1λ =2λρ (1+ρa
t0)/
NPi=1
NPj=1
atij because of the restrictions (5.13) and (5.14)
or (5.15) for the homogeneity of degree 2λ/ρ of f tQρ,λ . Hence
f tQρ,λ(q) = (1 + ρat0)1r
"NPi=1
NPj=1
αtijqλi q
λj
# 1ρ
(5.17)
where αtij ≡atij
NPi=1
NPj=1
atij
. We note thatNPi=1
NPj=1
αtijqλi q
λj = 1 when all qi’s are
equal to 1.Moreover, under the degree 2λ/ρ homogeneity conditions (5.13)-(5.14),
if all parameters atij in (5.16) change proportionally, so that αtij in (5.17)
remain constant (αtij = αij), then the effects of parameter changes are sep-arable from q. This function can be seen as a homothetic transformation ofa linearly homogeneous function, that is
f tQρ,λ(q) = σt[fQr(q)]r/ρ (5.18)
where r = 2λ, σt ≡ (1 + ρat0)1ρ , and
fQr(q) ≡"
NPi=1
NPj=1
αijqr2i q
r2j
# 1r
(5.19)
which is the quadratic mean-of-order-r aggregator function used by Diewert’s(1976, pp. 129-130)16. This function is separable from parameter changesand homogenous of degree one in q.16This functional form is due to McCarthy (1967), Kadiyala (1972), Denny (1972, 1974),
and Hasenkamp (1973).
30
Equation (5.19) reduces to well-known functions for particular values ofr. Denny (1972, 1974) noted that, if r = 1, then it reduces to the generalizedlinear functional form proposed by Diewert (1969, 1971). In an unpublishedmemorandum, Lau (1973) showed that, at the limit as r tends to zero, itreduces to the homogeneous translog aggregator function (Lau’s proof isreported in Diewert, 1980, p. 451). Diewert (1976, p. 130) also noted that,if r = 2, then it reduces to the Konüs-Byushgens (1926) functional form.Furthermore, if all αij = 0 for i 6= j, then it reduces to a CES functionalform.
If the parameters of the first-order terms of the function (5.12) are func-tions of x as follows
at0 ≡ αt0 +MP
m=1βtmx
λm +
1
2
MPm=1
MPn=1
btmnxλmx
λn (5.20)
ati ≡ αti +MP
m=1γtmix
λm, (5.21)
then (5.12) is identically equal to
htQρ,λ(q, x) = [αt0 +NPi=1
αtiqλi +
1
2
NPi=1
NPj=1
atijqλi q
λj
+MP
m=1βtmx
λm +
1
2
MPm=1
MPn=1
btmnxλmx
λn
+NPi=1
MPm=1
γtmixλmq
λi ]
1ρ
µ1
λ
¶ 1ρ
(5.22)
where
αt0 ≡ (λ +ρλαt0 −ρNPi=1
αti +12ρλ
NPi=1
NPj=1
atij − ρMPm=1
βtm +12ρλ
MPm=1
MPn=1
btmn
+ ρλ
MPm=1
NPi=1
γtmi);
αti ≡ (ραti − ρλ
NPj=1
atij − ρλ
MPm=1
γtmi);
βtm ≡ (ρβtm − ρ
λ
MPn=1
btmn − ρλ
NPi=1
γtmi)
Under the parameter restrictions (5.13) and (5.14) imposed on (5.12) forthe homogeneity of degree 2λ/ρ in q, using the definitions (5.6) and (5.7),
31
equation (5.22) becomes
htQρ,λ(q, x) =
"NPi=1
NPj=1
αtijqλi q
λj
# 1ρ
··βt0 +
MPm=1
β∗tmx
λm +
1
2
MPm=1
MPn=1
btmnx
λmx
λn
¸ 1ρµ1
λ
¶ 1ρ
(5.23)
where βt0 ≡ (λ +ρλαt0 −ρ
MPm=1
βtm +12ρλ
MPm=1
MPn=1
btmn);
β∗tm ≡ (ρβtm − ρ
λ
MPn=1
btmn) ≡ βtm +
ρλ
NPi=1
γtmi = βtm since
NPi=1
γtmi = 0 as
a condition imposed for the homogeneity of degree 2λ/ρ in q;btmn =
ρλb
tmn.
Moreover, if
(1 + ραt0) =1
2
ρ
λ
MPm=1
βtm (5.24)
λβtm =MPn=1
btmn (5.25)
NPm=1
γtmi = 0, (5.26)
then βt0 = β
∗tm = 0 (and, therefore, β
tm = 0), and ht
Qρ,λ is homogeneous ofdegree 2λ/ρ in q as well as x. In this case, (5.23) reduces to
htQρ,λ(q, x) = (1 + ραt0)1ρ ·Ã NP
i=1
NPj=1
αtijqr2i q
r2j
! 1r
rρ
·"µ
MPm=1
MPn=1
βtmnxr2mx
r2n
¶ 1r
# rρ
with r = 2λ (5.27)
where βtij ≡btij
NPi=1
NPj=1
btij
. Note, thatMP
m=1
MPn=1
βtmnxr2mx
r2n = 1 if all xi’s and xj ’s
are equal to 117.Note that, with αtij and βtmn constant over t, for all i, j, m, and n, the
function htQρ,λ(q, x) is homothetically separable in q, x, and from parameter
17We may note that, with r = ρ = 2 and all "second-order" parameters being constant(αtij = αij and βtmn = βmn for every value of t), (5.27) reduces to the quadratic mean-of-order-2 functional form used by Diewert (1992, p. 231, eq. (56)).
32
changes. The last ones are captured by changes in the factor of proportional-
ity (1+ραt0)1ρ . The resulting homogeneous functions
"NPi=1
NPj=1
αijqr2i q
r2j
# 1r
and·MP
m=1
MPn=1
βmnxr2mx
r2n
¸ 1r
can be considered as "aggregators" of q and x, respec-
tively18.
6. Accounting for value changes of translog and quadratic mean-of-order-r functional forms
The general quadratic mean-of-order-r function, which is algebraically deriv-able from the quadratic Box-Cox function defined by (5.4)-(5.5), can be usedas a second-order approximation to the an arbitrary unknown function. Us-ing the results obtained thus far, it is possible to account for differences infunctional values, either in terms of differences or in terms of ratios, intoaggregating components of changes in the arguments and parameters of thefunction. We shall follow the tradition of calling indicators the aggregatingcomponents defined in terms of differences and index numbers those thatare defined in terms of ratios (see, for example, Diewert, 1998, 2000). Inorder to save space, we shall deal with only the function f t
Qρ,λ(q) definedby (5.12) and leave to the reader the exercise of deriving the correspondingresults with the more explicit function ht
Qρ,λ(q, x) defined by (5.22). Thefollowing theorems are in order.
THEOREM 6.1. If the general quadratic Box-Cox function with variableparameters defined by (5.4) reduces to the "translog" function (5.6), withρ→ 0 and λ→ 0, then
ln f1Trg − ln f0Trg =1
2
NXi=1
¡s0Trg,i + s1Trg,i
¢ ¡ln q1i − ln q0i
¢+Parameter-change component (6.1)
where
stTrg,i ≡qtif
tTrgi
f tTrgwith t = 0, 1 and f tTrgi ≡
∂f tTrg∂qti
(6.2)
18 In section 3, we have recalled that an aggregator function must be degree-one homo-geneous as well as separable.
33
We note that, in general, stTrg,i 6= sti, where sti ≡ qtif
ti /f
t and f ti ≡ ∂f t/∂qti ,
except at the point of approximation. Moreover,PN
i=1 stTrg,i = 1 if f0Trg
and f1Trg are homogeneous of degree one in q, since in this case f tTrg =PNi=1 q
tif
tTrg,i by Euler’s theorem. If
PNi=1 s
tTrg,i 6= 1, then we can define
s∗tTrg,i ≡ stTrg,i/PN
i=1 stTrg,i and rewrite equation (6.1) as follows:
ln f1Trg − ln f0Trg =1
2
NXi=1
¡s∗0Trg,i + s∗1Trg,i
¢ ¡ln q1i − ln q0i
¢+1
2
NXi=1
[s∗0Trg,i(ξ − 1) + s∗1Trg,i(ξ − 1)]¡ln q1i − ln q0i
¢+Parameter-change component (6.3)
where ξ ≡ PNi=1 q
tif
tTrg,i/f
tTrg, which represents the degree of returns to
scale. The first additive term in the right-hand side of (6.3) is the contribu-tion of the change in q to the functional value difference, whereas the secondadditive term is the contribution of the scale effects.
Taking the antilogarithms of (6.3) yields:
f1Trgf0Trg
= exp[lnf1Trgf0Trg
] = exp£ln f1Trg − ln f0Trg
¤=
(exp[
1
2
NXi=1
¡s∗0Trg,i + s∗1Trg,i
¢ ¡ln q1i − ln q0i
¢]
)
·(exp[
1
2
NXi=1
[s∗0Trg,i(ξ − 1) + s∗1Trg,i(ξ − 1)]¡ln q1i − ln q0i
¢]
)· exp(Parameter-change component)
=NYi=1
µq1iq0i
¶ 12(s
∗0Trg,i+s
∗1Trg,i)
·NYi=1
µq1iq0i
¶ 12[s∗0Trg,i(ξ−1)+s∗1Trg,i(ξ−1)]
· exp(Parameter-change component) (6.4)
whereNYi=1
³q1iq0i
´ 12(s
∗0Trg,i+s
∗1Trg,i)
is the Törnqvist index number of q. The Törn-
qvist index number is said to be "exact" for the translog function becauseit gives the same result as the ratio of two translog functions (net of thescale factor and parameter-change component). This index number belongsto the class of the so-called "superlative" index numbers, which are "exact"for a quadratic function (the "translog"), which can provide a second-orderdifferential approximation to an arbitrary function.
This result does not require separability and homogeneity restrictions. Itis commonly believed that only the case of the translog-based indicators of
34
relative change does not rely on this restrictions (see, for example, Diewert,2004, p. 450). This should be contrasted with the following results, whereother indicators are constructed without imposing these restrictions.
THEOREM 6.2. If a quadratic Box-Cox function with variable parametersis defined by (5.4) with ρ 6= 0 and λ 6= 0, then
(f1Qρ,λ)ρ − (f0Qρ,λ)
ρ =1
2
ρ
λ
NXi=1
(s0Qρ,λi
(f0Qρ,λ)
ρ
(q0i )λ+ s1Qρ,λi
(f1Qρ,λ)
ρ
(q1i )λ
)·[(q1i )λ − (q0i )λ]+Parameter change component (6.5)
where
stQρ,λi ≡qtif
tQρ,λi
f tQρ
with t = 0, 1 and f tQρ,λi ≡∂f t
Qρ,λ
qti(6.6)
We note that, in general, stQρ,λi
6= sti, where sti ≡ qtif
ti /f
t and f ti ≡ ∂f t/∂qti ,
except at the point of approximation. Moreover,PN
i=1 stQρ,λ,i
= 1 if f0Qρ,λ
and f1Qρ,λ are homogeneous of degree one in q, since in this case f t
Qρ,λ =PNi=1 q
tif
tQρ,λ,i
by Euler’s theorem. IfPN
i=1 stQρ,λ,i
6= 1, then we can defines∗tQρ,λ,i
≡ stQρ,λ,i
/PN
i=1 stQρ,λ,i
and rewrite equation (6.5) as follows:
f1Qρ,λ)ρ − (f0Qρ,λ)
ρ =1
2
ρ
λ
NXi=1
(s∗0Qρ,λi
(f0Qρ,λ)
ρ
(q0i )λ+ s∗1Qρ,λi
(f1Qρ,λ)
ρ
(q1i )λ
)·[(q1i )λ − (q0i )λ]
+1
2
ρ
λ
NXi=1
(s∗0Qρ,λi(ξ − 1)
(f0Qρ,λ)
ρ
(q0i )λ+ s∗1Qρ,λi(ξ − 1)
(f1Qρ,λ)
ρ
(q1i )λ
)·[(q1i )λ − (q0i )λ]+Parameter change component (6.7)
where ξ ≡ PNi=1 q
tif
tQρ,λ,i
/f tQρ,λ , which represents, by Euler’s theorem, the
degree of returns to scale. The first additive term in the right-hand side of
35
(6.7) is the contribution of the change in q to the functional value difference,whereas the second additive term is the contribution of the scale effects.
Dividing through (6.5) by³f0Qρ,λ
´ρyields
(f1Qρ,λ)
ρ − (f0Qρ,λ)
ρ
(f0Qρ,λ)ρ
=1
2
ρ
λ
NXi=1
(s0Qρ,λi
1
(q0i )λ+ s1Qρ,λi
(f1Qρ)ρ
(f0Qρ)ρ1
(q1i )λ
)(q1i )
λ − (q0i )λ(q0i )
λ(q0i )
λ
+Parameter change component/(f0)r
=1
2
ρ
λ
NXi=1
(s0Qρ,λi + s1Qρi
(f1Qρ,λ)
ρ
(f0Qρ,λ)ρ
(q0i )λ
(q1i )λ
)(q1i )
λ − (q0i )λ(q0i )
λ
+Parameter change component/(f0Qρ,λ)ρ (6.8)
Linear homogeneity and separability restrictions are not imposed, so that(6.5), (6.7), and (6.8) hold under very general conditions on the parametervalues of f t
Qρ,λ . We leave to the reader a further decomposition of (6.8)similar to (6.7). Let us consider, in particular, the following two specialcases.
First case: General quadratic linear indicators, which are exact for ageneral quadratic linear function given by (5.12) with λ = ρ/2 and ρ = 1.
Using the decomposition
(q1i )12 − (q0i )
12 = (q1i − q0i ) ·
1
(q0i )12 + (q1i )
12
, (6.9)
from (6.5), by defining f tQ1 ≡ f tQρ,λ with λ = ρ/2 and ρ = 1, we obtain
f1Q1 − f0Q1 =NXi=1
(q0i )12 f0Q1
[(q0i )12 + (q1i )
12 ]+
(q1i )12 f1Q1
[(q0i )12 + (q1i )
12 ]
(q1i − q0i )
+Parameter-change component (6.10)
36
Dividing through (6.10) by f0Q1 and replacing (q1i −q0i ) with [(q1i −q0i )/q0i ] ·q0i
yields
f1Q1 − f0Q1
f0Q1
=NXi=1
(s0Q1i
(q0i )12
(q0i )12 + (q1i )
12
+ s1Q1if1Q1
f0Q1
q0i
(q1i )12 [(q0i )
12 + (q1i )
12 ]
)q1i − q0iq0i
+Parameter-change component/f0Q1 (6.11)
(where stQ1i ≡ qtiftQ1i/f
tQ1 with f t1i ≡ ∂f tQ1/∂q
ti)
We can call general quadratic linear indicators of absolute and relative func-tional value differences the indicators given, respectively, by the right-handside of (6.10) and (6.11), which are exact for a general quadratic linear func-tion corresponding to (5.12), where ρ = 1 and λ = 1/2, and do not relyon homogeneity and separability restrictions. This is a rather useful result,since it widens the applicability of this type of indicators. This can be con-trasted with the same case where r = 1 and λ = 1/2 that was examinedunder linear homogeneity and separability restrictions by Diewert (2002,pp. 77-80), who showed that the counterpart index number, expressed inratio terms is the implicit Walsh index number, which in turn is exact for ahomogeneous quadratic linear function.
Second case: General Konüs-Byushgens indicator, which is exact for ageneral Konüs-Byushgens function given by (5.12) with λ = ρ/2 and ρ = 2.
Let us define f1Q2 ≡ f tQρ,λ with λ = ρ/2 and ρ = 2. Using the general
decomposition
[f1Q2 ]2 − [f0Q2 ]2 = (f1Q2 − f0Q2) · (f1Q2 + f0Q2), (6.12)
by dividing through (6.5) by (f1Q2 + f0Q2) and rearranging terms, we obtain
f1Q2 − f0Q2 =NXi=1
(f0Q2i · f0Q2f1Q2+ f0
Q2+
f1Q2i · f1Q2f1Q2+ f0
Q2)(q1i − q0i )
+Parameter-change component/(f1Q2 + f0Q2) (6.13)
where f tQ2i ≡ ∂f tQ2/∂qti for t = 0, 1.
37
Dividing through (6.13) by f0Q2 and replacing (q1i −q0i ) with [(q1i −q0i )/q0i ] ·q0i
yield
f1Q2 − f0Q2
f0Q2
=NXi=1
(s0Q2i
f0Q2
f1Q2+ f0
Q2+ s1Q2i
q0iq1i
(f1Q2)2
f0Q2(f1
Q2+ f0
Q2)
)q1i − q0iq0i
+Parameter-change component/(f1Q2 + f0Q2) (6.14)
where stQ2i ≡ qtift2i/f
tQ2 with f t2i ≡ ∂f tQ2/∂q
ti .
We can call general Konüs-Byushgens indicators of absolute and relativefunctional value differences the indicators given, respectively, by the right-hand side of (6.13) and (6.14), which are exact for a general Konüs-Byushgensquadratic function corresponding to (5.12) where ρ = 2 and λ = 1 and do notrely on homogeneity and separability restrictions. Also this result is usefulto widen the applicability of this type of indicators. This can be contrastedwith the same case where ρ = 2 and λ = 1, that was examined under linearhomogeneity and separability restrictions by Reinsdorf, Diewert, and Ehe-mann (2000, pp. 4-6) and Diewert (2002, pp. 72-76), who showed that thecounterpart index number (expressed in ratio terms) is the Fisher "ideal"index number, which in turn is exact for a homogeneous Konüs-Byushgensquadratic function.
THEOREM 6.3. If the homothetic (homogeneous of degree r/ρ) functionfQρ,λ is defined by (5.18)-(5.19), where ρ 6= 0 and λ 6= 0, then, for all q0iand q1i ,
f1Qρ,λ
f0Qρ,λ
= Iq · IY · IT (6.15)
where, setting r = 2λ and σt = (1 + ρat0)1ρ for t = 0, 1,
Iq ≡
NXi=1
s0Qri(q1i )
r/2
(q0i )r/2
NXi=1
s1Qr,i(q1i )
−r/2
(q0i )−r/2
1r
=
NXi=1
s0Qρ,λi
(q1i )r/2
(q0i )r/2
NXi=1
s1Qρ,λi
(q1i )−r/2
(q0i )−r/2
1r
(6.16)
38
IY ≡
NXi=1
s0Qρ,λi
(q1i )r/2
(q0i )r/2
NXi=1
s1Qρ,λi
(q1i )−r/2
(q0i )−r/2
1ρ− 1r
(6.17)
IT ≡ σ1
σ0(6.18)
with
stQri ≡ qti ·∂fQr
∂qti/fQr (6.19)
= stQρ,λi ≡ qti ·(∂fQρ,λ/∂qti)PN
i=1 qti · (∂fQρ,λ/∂qti)
(6.20)
The index Iq defined by (6.16) is a quadratic mean-of-order-r type indexnumber of q, which is "exact" for the quadratic mean-of-order-r aggregatorfunction (5.19) because it is identically equal to the ratio between the twofunctional values of this function at t = 0 and t = 1, whereas IY definedby (6.17) is the index number of scale effects, and IT defined by (6.18)is the index number of the effects of parameter changes. Being "exact"for a function that can provide a second-order differential approximationto an arbitrary function, the index Iq belongs to the class of the so-called"superlative" index numbers.
Note that the quadratic mean-of-order-r index number defined by (6.16)can be contrasted with the following quadratic mean-of-order-r index num-ber defined by Diewert (1976, pp. 130-131):
IDq ≡
NXi=1
s0i(q1i )
r2
(q0i )r2
NXi=1
s1i(q1i )
− r2
(q0i )− r2
1r
(6.21)
where s0 and s1 are actually "observed". By defining these weights as valueshares, equation (6.21) reduces to well-known index numbers with particularvalues of r. Diewert (1976, p. 135) noted that, if r = 1, then (6.21) reducesto the implicit Walsh (1901, p. 105) index number, and, if r = 2, then (6.21)reduces to the Fisher (1922) "ideal" index number.
39
The indexes Iq and IDq differ in the weights used. Even if these twoweight systems are numerically equal at the point of approximation, say att = 0, so that s0Qr = s0, they may be substantially different at other pointsunder comparison. This is sufficient to make the two index numbers (6.16)and (6.21) rather different. In order to ensure that s0Qr must be equal tos0 at all values of r, the parameters of the underlying aggregator functionshould adjust to the changes in r. Consequently, s1Qr is also a function of r.This means that, using the same s1 with different values of r contradicts theassumption of second-order approximation that is supposed to be providedby IDq . It is, therefore, Iq, rather than I
Dq , that must be called "superlative"
according to the meaning assigned by Diewert (1976, p. 117) to this term.This will be shown more analytically and illustrated by a numerical examplein the remainder of this section.
The function fQr(q) defined by (5.19) provides a second-order differentialapproximation to an arbitrary function f(q) around q0 if
f(q0) = fQr(q0) (6.22)
∇zf(q0) = ∇zfQr(q0) (6.23)
∇2zf(q0) = ∇2zfQr(q0) (6.24)
The equations (6.22)-(6.24) are satisfied for certain values of the parametersof fQr for a given r, as established by the following result:
THEOREM 6.4. If the aggregator function fQr(q) defined by (5.19) pro-vides a second-order approximation to the arbitrary aggregator function f(q)around q∗, then its parameters must be set equal to:
αij =f∗ij − 1−r
f∗ f∗i f∗j
r2 [f
∗]1−r (q∗i q∗j )
r2−1 1 ≤ i < j ≤ N (6.25)
αii =
f∗i − (q∗i )r2−1 NP
j 6=i[fQr(q∗)](1−r)α∗ij(q
∗j )
r2−1
[fQr(q∗)](1−r)(q∗i )r−21 ≤ i ≤ N (6.26)
40
From the above result it is evident that the parameters of a second-order approximating quadratic function fQr(q) are univocally determinedas functions of r.
Furthermore, note that in general,
f(q1) 6= fQr(q1) (6.27)
∇f(q1) 6= ∇fQr(q1) (6.28)
∇2f(q1) 6= ∇2fQr(q1) (6.29)
Therefore, given the definition (6.19), in general
s1 6= s1Qr (6.30)
where s1i ≡ q1i∂f(q)∂q1i
/f(q).
When r changes, say from r∗ to r∗∗, then in general
∇fQr∗ (z1) 6= ∇fQr∗∗ (z1) (6.31)
and, consequently,
s1 6= s1Qr∗ 6= s1Qr∗∗ (6.32)
For each value r = r∗, there is a set of particular parameters α∗ij = α∗ij(r∗)
such that the function defined by
fQr∗ (q) ≡ [ΣNi=1ΣNj=1α
∗ijq
r/2i q
r/2j ]1/r (6.33)
satisfies (6.22)-(6.24). It is immediate to note that all the functions fQr∗ (q)defined over the domain of r∗ approximate each other up to the second order.Consequently, the corresponding "exact" index numbers defined by (6.16),where r = r∗, also approximate each other up to the second order. This isensured by the adjustment of the shares stQr to the value of r. If, instead,the same observed shares st are used for different values or r, as in (6.21),then hybrid formulae are used which cannot be interpreted as superlative
41
index numbers providing second-order differential approximations to an ar-bitrary function and to each other. The following example is a numericalrepresentation of the differences between the two types of index numbers.
EXAMPLE 6.1.
Let us consider the case of three elements, N = 3, with q0i = 1.0 fori = 1, 2, 3 so that , in the case of a linearly homogeneous function,
f(q0) = 1.0 (6.34)
and q11 = 2.0, q12 = 1.7, and q13 = 1.05. Moreover, let assume that, at theobservation point t = 0, the first derivatives of the unknown function f(q0)are the following:
f1 ≡ ∂f
∂q1= 0.53 (6.35)
f2 ≡ ∂f
∂q2= 0.37 (6.36)
f3 ≡ ∂f
∂q3= 0.10 (6.37)
At the same observation point, let us assume that the Hessian matrix ofthe function f(q0) is the following:
∇2f(q0) =
0.0376 −0.0088 −0.0288−0.0088 0.0344 −0.0256−0.0288 −0.0256 0.0544
(6.38)
Therefore, at the point t = 0 and t = 1, the weights sti are
s01 ≡ (q01 · f1)/f = 0.53 s11 ≡ (q11 · f1)/f = 0.69 (6.39)
s02 ≡ (q02 · f2)/f = 0.37 s12 ≡ (q12 · f2)/f = 0.30 (6.40)
s03 ≡ (q03 · f3)/f = 0.10 s13 ≡ (q13 · f3)/f = 0.01 (6.41)
Let us now assume that a quadratic mean-of-order-r aggregator func-tion of the type defined by (5.19) has such parameters α∗ij for a given r∗
42
Table 2 – Parameters, weights, and numerical values of the quadratic mean-of-order r function r = 1 r = 2 r = 4 r = 10 r = 20 r = 40 r = 100 r =1000 r = ?
α 11 0.6052 0.3185 0.1751 0.0891 0.0605 0.0461 0.0375 0.0324 0.0318
α 22 0.4388 0.1713 0.0376 -0.0427 -0.0694 -0.0828 -0.0908 -0.0957 -0.0962
α 33 0.2088 0.0644 -0.0078 -0.0511 -0.0656 -0.0728 -0.0771 -0.0797 -0.0800
α12 = α21 -0.0176 0.1873 0.2897 0.3512 0.3717 0.3820 0.3881 0.3918 0.3922
α13 = α31 -0.0576 0.0242 0.0651 0.0896 0.0978 0.1019 0.1044 0.1058 0.1060
α 23 = α32 -0.0512 0.0114 0.0427 0.0615 0.0677 0.0709 0.0727 0.0739 0.0740
s1
Qr,1 0.6057 0.6025 0.6045 0.6188 0.6544 0.8058 0.9970 1.0000 1.0000
s1Q
r,2 0.3570 0.3539 0.3567 0.3698 0.3448 0.1942 0.0030 0.0000 0.0000
s1Q
r,3 0.0373 0.0437 0.0388 0.0114 0.0008 0.0000 0.0000 0.0000 0.0000
Qr 1.8069 1.8044 1.8066 1.8256 1.8474 1.8750 1.9355 1.9932 2.0000
QDr 1.8336 1.8381 1.8440 1.8389 1.7522 1.6013 1.5078 1.4549 1.44914
Figure 4:
that it provides a second-order differential approximation to f at q0. Table6.1 tabulates the values of these parameters in correspondence of certainvalues of r, so that the quadratic mean-of-order-r aggregator functions fQr
approximate f up to the second order. The shares s0Qr and s1Qr are also cal-culated. The corresponding "exact" and "superlative" index numbers Ir arecompared with the hybrid index numbers IDr , which are constructed usingthe same "observed" shares for different values of r. From the numericalexample, it is evident that the two index numbers diverge increasingly fromone another as r increases. The "really" superlative index numbers tend toincrease as r increases after a certain turning point, whereas the "hybrid"index numbers tend to decrease as r increases after another turning point.Both the "really" and "hybrid" superlative index numbers have asymptoticvalues as r → ∞. In the case considered in this numerical example, theformer tends to a value equal to 2.0, which is higher than that obtained atr = 1 by more than 10 per cent, whereas the latter tends to a value equalto 1.44914, which is lower than that obtained at r = 1 by more than 20 percent (see Table 2).
These results show that the theory of exact and superlative index num-bers has to be widely reconsidered and shed a new light to the apparentlyparadoxical findings recently obtained by Hill (2002, 2005).
43
8. Conclusion
This paper has reached new and more general results in the field of exactand superlative index numbers. It has widen the scope of applicability ofthese index numbers by showing that they can be justified under weakerassumptions than those traditionally made. However, it is also shown thatthese approximating index numbers are rarely applicable in practice. Infact, a superlative index number is defined by using weights derived fromthe function for which it is "exact". If, instead, the weights are derivedfrom observed data, which in turn are assumed to be consistent with the"true" unknown arbitrary function, then the resulting index number is ahybrid formula that may turn out to be very different from a second-orderapproximating index number.
The decomposition procedures described in this paper require that theweights used to aggregate absolute or relative changes in the observed vari-ables be known or "approximated" in some way. In particular, with thequadratic mean-of-order-r index numbers, for the given value of r, knowl-edge of the weights is essential for the computation of index numbers. Ifthe corresponding function for which these index numbers or indicators are"exact" is not the "true" function, then the required weights remain un-known. It is remarkable that these may be far from being approximated bythe weights derived from the observed data. Therefore, these index numbersvary widely from the "true" value and from each other if wide changes inr are not accompanied by the necessary parameter adjustments. This hasconsequences that are particularly relevant in common practice where ob-served or estimated weights are used. These index numbers turn out to behybrid and far from being really "superlative" in Diewert’s (1976) sense.
We conclude that, since using the "observed" data normally available itis not possible to rely on the second-order differential approximation par-adigm, it would be more appropriate to construct a range of alternativeindex numbers (including those that are not superlative), which are to beconsidered all equally valid candidates as good approximation to the trueunknown index number, rather than follow the traditional search for onlyone optimal formula. In intertemporal comparisons dealing with time se-ries data, statistical agencies might further reduce in some degree the spreadbetween alternative index numbers by adopting, for each of them, the chain-ing principle and changing the base period more frequently. However, theyhave to take into account that chained indexes do not satisfy, in general,certain required properties. In intertemporal comparisons dealing with de-cennial census data and interspatial comparisons the reduction of the spreadis more problematic.
44
Appendix A
Observable economic variables
The propositions presented thus far are theorems in numerical analysisrather than economics. The numerical values of the variables involved areassumed to be either known or derivable from some sources of information.In economics, the contexts where index numbers and indicators are usuallyapplied are those regarding production and consumption activities. Theindex numbers and indicators are therefore defined with reference to pro-duction or transformation and utility functions and their dual counterpartsrepresented by value functions such as cost, revenue and profit functions.All these functions are characterized by certain properties and the variablesconcerned are not always "observable". The following cases are examined:
(i) The case of the production function
If the function f t(x) is homogeneous of degree 1 or less than 1 andrepresent a transformation or production function characterized by the usualregularity properties so that
y = f t(x) (A.1)
where y is a scalar measure of the output quantity, and x is a vector of in-put quantities, then the first-order conditions for profit maximization imply(dropping the superscript t to simplify notation)
∂f
∂xi=
wi
pfor all i’s (A.2)
where p is a scalar measure of the output price. Therefore,
si ≡ ∂f
∂xi
xiy=
wi · xip · y (A.3)
45
Under constant returns to scale and perfect competition, p ·y =PNi=1wi ·xi,
so that si = wi · xi/PN
i=1wi · xi andPN
i=1 si = 1.In the general case where p · y R PN
i=1wi · xi, if we want to constructseparately index numbers or indicators of differences in quantities and re-turns to scale, then it might be useful to decompose the foregoing weightsas follows:
si =wi · xiPNi=1wi · xi
+wi · xiPNi=1wi · xi
· (ξ − 1) (A.4)
where ξ ≡PNi=1wi ·xi)/p·y represents the degree of the returns to scale. It is
obtained using the additional information on output prices and quantities.The second term of the right-hand side of equation (A.4) represents theweight capturing the effect of returns to scale (see Caves, Christensen, andDiewert, 1982, pp. 1405-1406 and p. 1408 for a similar decomposition19).
Since prices and quantities of inputs and outputs are usually observable,all the necessary information for the calculation of index numbers and relatedindicators is available.
(ii) The case of the utility function
Let the function f t(x) represent a utility function characterized by theusual regularity properties so that
u = f t(x) (A.5)
where u represents utility, and x is a vector of the quantities consumed.Utility is typically unobserved, so that at least this important variable isnot available for the index number construction. However, if the consumerhas a utility-maximizing behavior subject to a budget constraint, then the
19The variable parameter ξ (in our notation) is obtained with equation (47) in Caves et.al. (1982, p. 1406), where it is, however, misprinted: a simple mathematical derivationreveals that it is incorrectly inverted (the correct form is reported at page 1408 of thesame article).
46
first-order conditions imply (dropping the superscript t to simplify notation)
∂f
∂xi= λwi for all i’s (A.6)
We sum these conditions multiplied by quantities in order to obtain:
NXi=1
∂f
∂xixi = λ
NXi=1
wixi (A.7)
which can be solved for the Lagrange multiplier as follows:
λ =
NXi=1
∂f∂xi
xi
NXi=1
wixi
(A.8)
Substituting the Lagrange multiplier back into the first-order conditionsyields:
∂f
∂xi=
NXi=1
∂f∂xi
xi
NXi=1
wixi
wi for all i’s (A.9)
from which we derive the Hotelling (1935, p.71)-Wold (1944, pp. 69-71;1953, p. 145) identity
∂f
∂xi
1PNi=1
∂f∂xi
xi=
wiPNi=1wixi
for all i’s (A.10)
If we assume linear homogeneity of the utility function (implying marginalutility of income equal to 1), then, by Euler’s theorem,
PNi=1
∂f∂xi
xi = f(x)and, consequently, the foregoing equation becomes
si ≡∂f∂xi
xi
f(x)=
wixiPNi=1wixi
for all i’s (A.11)
47
In this last case, the weights si can be calculated using observed data onprices and quantities.
(iii) The case of the value functions (cost, revenue, and profit functions)
Let a value function (a cost, or revenue, or profit function) be representedby a differentiable function et(p), where p is a vector of prices. By Hicks(1946, p. 331)-Samuelson (1947, p. 68)-Shephard (1953, p. 11)-Hotelling’slemma (1932, p. 594), we obtain directly the optimal levels of quantitiesthrough differentiation (dropping the superscript t to simplify notation):
qi =∂e
∂pi(A.12)
where qi is the ith element of an N -dimensional vector of quantities.Irrespective of the technology of production or utility function, the value
function is always linearly homogeneous in prices. By Euler’s theorem,e(p) =
PNi=1 piqi. Therefore, by dividing both sides of the equation (A.12)
by e(p) we obtain
qiPNi=1 piqi
=
∂e∂pi
e(p)for all i’s (A.13)
Multiplying the foregoing equations by the respective prices pi’s yields
piqiPNi=1 piqi
=pi
∂e∂pi
e(p)= si for all i’s (A.14)
Here, again, si and the numerical value of e(p) can be calculated usingobserved data on prices and quantities.
48
Appendix B
Proofs of theorems
Proof of Lemma (2.1) (Accounting for Functional Value Differences).
Let us consider an arbitrary function f(z) of one single variable. Fromthe Taylor series expansion for f around z0, the following equation can beobtained:
f(z1)− f(z0) = f 0(z0)(z1 − z0) +1
2!f 00(z0)(z1 − z0)2 + ...
+1
n!f (n)(z0)(z1 − z0)n +R0n(z
0, z1)
=nP
m=1
1
m!f (m)(z0)(z1 − z0)m +R0n(z
0, z1) (B.1)
where R0n(z0, z1) is the remainder term. Similarly, from the Taylor series
expansion for f around z1, the following equation can be obtained:
f(z0)− f(z1) = f 0(z1)(z0 − z1) +1
2!f 00(z1)(z0 − z1)2 + ...
+1
n!f (n)(z1)(z0 − z1)n +R1n(z
0, z1)
=nP
m=1
1
m!f (m)(z1)(z0 − z1)m +R1n(z
0, z1) (B.2)
Multiplying through the foregoing equation by −1 and rearranging termsyield
f(z1)− f(z0) = −nP
m=1(−1)m 1
m!f (m)(z1)(z1 − z0)m −R1n(z
0, z1)(B.3)
Using (1 − θ) and θ as weights, the weighted average of (B.1) and (B.3) isgiven by
f(z1)− f(z0) =nX
m=1
·(1− θ)
1
m!f (m)(z0)− (−1)mθ 1
m!f (m)(z1)
¸(z1 − z0)m
+(1− θ)R0n(z0, z1)− θR1n(z
0, z1) (B.4)
49
From (B.4), it follows that
f(z1)− f(z0) =£(1− θ) f 0(z0) + θf 0(z1)
¤T(z1 − z0)
+ (1− θ) R(z0)1 − θ R
(z1)1 (B.5)
Let us choose a value of θ, say θ∗, that minimizes the squared term[(1− θ)R01(z
0, z1) − θR11(z0, z1)]2. Since this term is convex in θ, the nec-
essary and sufficient condition for its minimization is that its first derivativewith respect to θ vanishes, so that
−2 £(1− θ)R01(z0, z1)− θR11(z
0, z1)¤[R01(z
0, z1) +R11(z0, z1)]
= 0 (B.6)
from which, provided that R01(z0, z1) +R11(z
0, z1) 6= 0,
θ = θ∗(z0, z1) ≡ R01(z0, z1)
R01(z0, z1) +R11(z
0, z1)(B.7)
The case of many variables follows in a similar manner using the directionalderivatives. In particular,
f(z1)− f(z0) =£(1− θ)∇f(z0) + θ∇f(z1)¤T (z1 − z0) (B.8)
where, if R01(z0, z1) + R11(z
0, z1) 6= 0, then θ = θ∗ (so that (1− θ∗)R01(z0, z1)− θ∗R11(z0, z1) = 0), or, if R
(z0)1 = R
(z1)1 = 0, then θ may take any value as
a number.
Proof of Corollary (2.1) (Accounting for Functional Value Ratios).
Applying the accounting procedure (2.1) to φ(z) and dividing through
50
both sides by φ(z0) yield
φ(z1)− φ(z0)
φ(z0)=
NXi=1
·(1− θ)
∂φ(z0)
∂zi· 1
φ(z0)+ θ
∂φ(z1)
∂zi· 1
φ(z0)
¸·z1i − z0iz0i
· z0i (B.9)
hence
φ(z1)
φ(z0)− 1 = (1− θ)
NXi=1
s0iz1iz0i− (1− θ)
NXi=1
s0i
+θφ(z1)
φ(z0)
NXi=1
s1i − θφ(z1)
φ(z0)
NXi=1
s1iz0iz1i
(B.10)
(where sti ≡ [∂φ(zt)/∂zi] · zti/φ(zt) = ∂F−1[f(z)]/∂ziF−1[f(z)] · zi = [∂f(zt)/∂zi]·zti
Ni=1[∂f(z
t)/∂zi]·zti,
since φ(zt) = F−1[f(z)])
= (1− θ)NXi=1
s0iz1iz0i− (1− θ)
+θφ(z1)
φ(z0)− θ
φ(z1)
φ(z0)
NXi=1
s1iz0iz1i
(B.11)
Rearranging (B.11) and solving for φ(z1)/φ(z0) yield
φ(z1)
φ(z0)=
θ + (1− θ)PN
i=1 s0iz1iz0i
(1− θ) + θPN
i=1 s1iz0iz1i
(B.12)
Proof of Corollary (2.2) (Diewert’s, 1976, p. 117, Quadratic Identity).
Sufficiency: From the definition (2.4) of the quadratic function fQ, the
51
first difference of this function is obtained as follows:
fQ(z1)− fQ(z
0) = a0 + aT z1 +1
2z1TAz1
−a0 − aT z0 − 12z0TAz0 (B.13)
By adding 12z0TAz1 and subtracting 1
2z1TAz0 (recall that z0TAz1 =
z1TAz0 since AT = A) and substituting 12z0TAz1 with (z0TAz1− 1
2z0TAz1)
and 12z0TAz0 with (z0TAz0− 1
2z0TAz0), equation (B.13) can be rearranged
into the following
fQ(z1)− fQ(z
0) = (a+ z0A)T (z1 − z0)
+1
2(z1 − z0)TA(z1 − z0) (B.14)
which can also be derived directly from the Taylor series expansion of fQ(z)around the point z0, since
∇zfQ(z0) = a+ z0A (B.15)
∇2zfQ(z0) = A (B.16)
Similarly, from
fQ(z0)− fQ(z
1)
= a0 + aT z0 +1
2z0TAz0 − a0 − aT z1 − 1
2z1TAz1 (B.17)
by adding 12z1TAz0 and subtracting 1
2z0TAz1 (recall, again, that z1TAz0 =
z0TAz1 since AT = A) and substituting 12z1TAz0 with (z1TAz0− 1
2z1TAz0)
and 12z1TAz1 with (z1TAz1− 1
2z1TAz1), equation (B.17) can be rearranged
into the following
fQ(z0)− fQ(z
1)
= (a+ z1A)T (z0 − z1) +1
2(z0 − z1)TA(z0 − z1) (B.18)
52
which can also be derived directly from the Taylor series expansion of fQ(z)around the point z1, since
∇zfQ(z1) = a+ z1A (B.19)
∇2zfQ(z1) = A (B.20)
Multiplying through (B.18) by −1 and rearranging terms yields
fQ(z1)− fQ(z
0)
= (a+ z1A)T (z1 − z0)− 12(z1 − z0)TA(z1 − z0) (B.21)
The arithmetic average of (B.14) and (B.21) is given by
fQ(z1)− fQ(z
0) =1
2[(a+ z0A) + (a+ z1A)]T (z1 − z0)
=1
2[∇zfQ(z
0) +∇zfQ(z1)]T (z1 − z0) (B.22)
Necessity : let us start from the following equation,
f(z1) = f(z0) +1
2
£∇zf(z0) +∇zf(z
1)¤T(z1 − z0) (B.23)
where ∇f(z0) 6= ∇f(z1) if z0 6= z1 when f is thrice differentiable. Considerthe polynomial of degree two fQ(z) ≡ a0 + a z + z A z for all z. Thefunction fQ(z) is assumed to be tangent to f(z) in correspondence to z0
so that ∇fQ(z0) = ∇f(z0) and fQ(z0) = f(z0). Let us assume also that
the numerical values of parameters of fQ(z) are such that fQ(z1) = f(z1) issatisfied. From the sufficiency condition,
fQ(z1) = fQ(z
0) +1
2
£∇fQ(z0) +∇fQ(z1)¤T (z1 − z0) (B.24)
53
Substituting f(z0) and f(z1) to fQ(z0) and fQ(z1), respectively, the forego-ing equation becomes
f(z1) = f(z0) +1
2
£∇f(z0) +∇fQ(z1)¤T (z1 − z0) (B.25)
Since both (B.23) and (B.25) must hold simultaneously at z0 and z1, then∇f(z1) = ∇fQ(z1) (6= ∇f(z0) = ∇fQ(z0)). Moreover, f(z1) = fQ(z
1) byconstruction, therefore fQ(z) is tangent to f(z) in correspondence of z1 aswell as of z0. Since this result is to be valid for all the admissible pairs z0
and z1, f(z) must have continuously the same gradients of a polynomial ofdegree at all z0 and z1. Hence, f(z) itself must have the same quadraticfunctional form as that of fQ(z).
Proof of Corollary (2.3) (Accounting for Functional Value Ratios of aQuadratic Homothetic Function)
The proof is given in section 6, Theorem (6.3), within a more generalcontext.
Proof of Lemma (2.2) (General Quadratic Approximation Lemma).
From (2.14) we have
Error of approximation
=£(1− θ)[∇zf(z
0) + θ∇zf(z1)¤T(z1 − z0)
− 12[∇zf(z
0) +∇zf(z1)]T (z1 − z0)
=
µ1
2− θ
¶[∇zf(z
0)−∇zf(z1)](z1 − z0) (B.26)
The first-order approximating (linear) function that is tangent to f(z) at zt
is given by f tL (z) = f(zt)+∇zf(z
t)T (z−zt) = at+btz (with at = f(xt)−∇zf(xt) xt and bt = ∇zf (x
t)). Therefore
f tL (z)− f t
L (zt) = ∇zf(z
t)T (z − zt) (B.27)
Using (B.27) with t = 0, 1 and z = z0, z1, equation (B.26) becomes
Error of approximation
=
µ1
2− θ
¶©£f0L(z
1)− f0L(z0)¤− £f1L(z1)− f1L(z
0)¤ª
(B.28)
54
Proof of Lemma (3.1) (Accounting for Functional Value Differences whenParameters or Functional Forms Differ)
Let us two arbitrary functions of one single variable, f0(z) and f1(z).These functions may differ in parameter values or even in their functionalforms. From the Taylor series expansion for f0(z1) around z0, the followingequation may be obtained:
f0(z1)− f0(z0) = f00(z0)(z1 − z0) +R01 (B.29)
where R01 = R01(z0, z1) is the remainder term of the first-order approxima-
tion.Adding and subtracting f1(z1) and rearranging terms, the foregoing
equation becomes:
f1(z1)− f0(z0) = f00(z0)(z1 − z0) +£f1(z1)− f0(z1)
¤+R01(B.30)
Similarly, from the Taylor series expansion for f1(z0) around z1, the follow-ing equation may be obtained:
f1(z0)− f1(z1) = f10(z1)(z0 − z1) +R11 (B.31)
where R11 = R11(z0, z1) is the remainder term of the first-order approxima-
tion. By multiplying both sides by −1 and rearranging terms, equation(B.31) becomes:
f1(z1)− f1(z0) = f10(z1)(z1 − z0)−R11 (B.32)
Adding and subtracting f0(z0) and rearranging terms, the foregoing equa-tion becomes:
f1(z1)− f0(z0) = f10(z0)(z1 − z0) +£f1(z0)− f0(z0)
¤−R11(B.33)
55
Using (1−θ) and θ as weights, where θ may take any real number as a value,the weighted average of (B.30) and (B.33) is given by
f1(z1)− f0(z0) =£(1− θ)f00(z0) + θf10(z0)
¤(z1 − z0)
+©θ£f1(z0)− f0(z0)
¤+ (1− θ)
£f1(z1)− f0(z1)
¤ª+£(1− θ)R01 − θR11
¤(B.34)
If θ = θ∗, with θ∗ = θ∗(z0, z1) ≡ R01R01+R
11, then
f1(z1)− f0(z0) =£(1− θ)f00(z0) + θf10(z0)
¤(z1 − z0)
+ "Technical" change component (TC) (B.35)
where
TC ≡ θ∗£f1(z0)− f0(z0)
¤+ (1− θ∗)
£f1(z1)− f0(z1)
¤(B.36)
The case of many variables follows in a similar manner using the directionalderivatives, thus obtaining (3.5)-(3.6).
Proof of Corollary (3.1) (Accounting for Functional Value Ratios be-tween Arbitrary Differentiable Functions).
The proof of Corollary (3.1) is analogous to that of Corollary (2.1).
Proof of Corollary (3.2) (Accounting for Numerical Value Differences ofTwo Quadratic Functions Differing in Parameters).
The proof of Corollary (3.2) follows the footsteps of the proof of the "suf-ficiency" part of Corollary (2.2). From the definition (3.11) of the quadraticfunction f t
Q, it is straightforward to obtain:
f 1Q (z
1)− f 0Q (z
0) = (a0 + z0A0)(z1 − z0)
+(a10 − a00) + (a1 − a0)z1
−z0A0z1 + 12(z0A0z0 + z1A1z1) (B.37)
since
f 1Q (z
1)− f 0Q (z
0) =£f 0Q (z
1)− f 0Q (z
0)¤+£f 1Q (z
1)− f 0Q (z
1)¤
(B.38)
56
where
f 0Q (z
1)− f 0Q (z
0) = (a0 + z0A0)(z1 − z0) +1
2(z1 − z0)A0(z1 − z0),(B.39)
using the Taylor series expansion for f0 around z0,
f 1Q (z
1)− f 0Q (z
1) = (a10 − a00) + (a1 − a0)T z1 +
1
2z1(A1 −A0)z1 (B.40)
and
1
2(z1 − z0)A0(z1 − z0) +
1
2z1(A1 −A0)z1
= −z0A0z1 + 12(z0A0z0 + z1A1z1) (B.41)
Similarly,
f 0Q (z
0)− f 1Q (z
1) = (a1 + z1A1)(z0 − z1)
+(a00 − a10) + (a0 − a1)z0
−z0A1z1 + 12(z0A0z0 + z1A1z1) (B.42)
Since
f 0Q (z
0)− f 1Q (z
1) =£f 1Q (z
0)− f 1Q (z
1)¤+£f 0Q (z
0)− f 1Q (z
0)¤
(B.43)
where
f 1Q (z
0)− f 1Q (z
1) = (a1 + z1A1)(z0 − z1) +1
2(z0 − z1)A1(z0 − z1)(B.44)
using the Taylor series expansion for f1 around z1
f 0Q (z
0)− f 1Q (z
0) = (a00 − a10) + (a0 − a1)z0 +
1
2z0(A0 −A1)z0 (B.45)
and
1
2(z1 − z0)A0(z1 − z0) +
1
2z1(A1 −A0)z1
= −z0A0z1 + 12(z0A0z0 + z1A1z1) (B.46)
Multiplying both sides of (B.42) by −1 and rearranging terms yield
f 1Q (z
1)− f 0Q (z
0) = (a1 + z1A1)(z1 − z0)
+(a10 − a00) + (a1 − a0)z0
+z0A1z1 − 12(z0A0z0 + z1A1z1) (B.47)
57
The arithmetic average of (B.37) and (B.47) is therefore
f 1Q (z
1)− f 0Q (z
0) =1
2
£∇f0Q(z0) +∇f1Q(z1)¤ (z1 − z0)
+(a00 − a10) + (a0 − a1)T
1
2(z0 + z1)
+1
2z0(A1 −A0)z1 (B.48)
since ∇f0Q(z0) = a0 + z0A0, ∇f1Q(z1) = a1 + z1A1.Moreover, by definition,
a00 − a10 = (α10 + k1β1 +
1
2k1B1k1)− (α00 + k0β0 +
1
2k0B0k0) (B.49)
(a0 − a1)1
2(z0 + z1) = (α1 + k1Γ1 − α0 − k0Γ0)
1
2(z0 + z1) (B.50)
The sum of (B.49) and (B.50) is¡a00 − a10
¢+ (a0 − a1)
1
2(z0 + z1) =
(α00 + α10) + (α0 − α1)
1
2(z0 + z1)
+1
2(k1β1 + k1B1k1 + k1Γ1z1 + k1Γ1z0
−k0β0 − k0B0k0 − k0Γ0z1 − k0Γ0z0) +1
2(k1β1 − k0β0) (B.51)
Adding and subtracting 12k0β1, 12k
1β0, 12k0B1k1, and 1
2k0B0k1 to the right-
hand side of (B.51) yield (3.21).
Proof of Lemma (3.2) (Quadratic Approximation of Value Differencesbetween Two Arbitrary Functions with Different Parameters or FunctionalForms). The proof of lemma (3.2) follows the proof of Lemma (2.2).
Proof of Corollary (3.3) (Accounting for the Sum of Value Differencesbetween Two Quadratic Functions with Different "Zero-order" and "First-order" Parameters) (Caves, Christensen, and Diewert’s, 1982, pp.1412-1413)Translog Identity).
Let the quadratic function htQ(z, kt) be defined by (3.12). By the
Quadratic Identity (Corollary (2.2),
h tQ (z
1, kt)− h tQ (z
0, kt) =1
2[∇zh
tQ (z
0, kt) +∇zht
Q (z1, kt)]
·(z1 − z0) (B.52)
58
and, therefore, setting λ = 1/2, for t = 0 and t = 1, the first line of theright-hand side of equation (3.25) becomes
1
2[h0Q(z
1, k0)− h0Q(z0, k0)] +
1
2[h1Q(z
1, k1)− h1Q(z0, k1)]
=1
2
£∇zh0Q(z
0, k0) +∇zh0Q(z
1, k0)¤(z1 − z0)
+1
2
£∇zh1Q(z
0, k1) +∇zh1Q(z
1, k1)¤(z1 − z0) (B.53)
Since
∇zh0Q(z
0, k0) = a0 + z0A (B.54)
∇zh0Q(z
1, k0) = a0 + z1A (B.55)
∇zh1Q(z
0, k1) = a1 + z0A (B.56)
∇zh1Q(z
1, k1) = a1 + z1A (B.57)
where at = αt + ktΓt, for t = 0, 1,
∇zh0Q(z
1, k0) +∇zh1Q(z
0, k1)
= (a0 + z1A) + (a1 + z0A)
= (a0 + z0A) + (a1 + z1A)
= ∇zh0Q(z
0, k0) +∇zh1Q(z
1, k1) (B.58)
Using (B.58), equation (B.53) becomes
1
2[h0Q(z
1, k0)− h0Q(z0, k0)] +
1
2[h1Q(z
1, k1)− h1Q(z0, k1)]
=1
2
£∇zh0Q(z
0, k0) +∇zh1Q(z
1, k1)¤(z1 − z0) (B.59)
which is equation (3.27).
Proof of Corollary (3.4) (Accounting for the Sum of Value Differencesbetween Two Quadratic Functions with Different Parameters) (Caves, Chris-tensen, and Diewert, 1982, pp.1412-1413).
By the Quadratic Identity,
£f0Q(z
1)− f0Q(z0)¤+£f1Q(z
1)− f1Q(z0)¤
=1
2
£∇zf0Q(z
0) +∇zf0Q(z
1)¤(z1 − z0)
+1
2
£∇zf1Q(z
0) +∇zf1Q(z
1)¤(z1 − z0) (B.60)
59
Adding and subtracting 12
h∇zf
0Q(z
0) +∇zf1Q(z
1)i(z1−z0) to the foregoing
equation and taking into account the definition of f tQ yield£f0Q(z
1)− f0Q(z0)¤+£f1Q(z
1)− f1Q(z0)¤
=£∇zf
0Q(z
0) +∇zf0Q(z
1)¤(z1 − z0)
+1
2(a0 + z1A0 + a1 + z0A1 − a0 − z0A0 − a1 − z1A1)(z1 − z0)
=£∇zf
0Q(z
0) +∇zf0Q(z
1)¤(z1 − z0)
+1
2(z1A0z1 + z0A1z1 − z0A0z1 − z1A1z1
−z1A0z0 − z0A1z0 + z0A0z0 + z1A1z0)
=£∇zf
0Q(z
0) +∇zf0Q(z
1)¤(z1 − z0)
−12(z1 − z0)(A1 −A0)(z1 − z0) (B.61)
or £h0Q(z
1, k0)− h0Q(z0, k0)
¤+£h1Q(z
1, k1)− h1Q(z0, k1)
¤=
£∇zh0Q(z
0, k0) +∇zh1Q(z
1, k1)¤(z1 − z0)
−12(z1 − z0)(A1 −A0)(z1 − z0) (B.62)
which is equation (3.29).
Proof of Corollary (3.5)
By definition,£f1Q(z
0)− f0Q(z0)¤+£f1Q(z
1)− f0Q(z1)¤
= [a10 + a1 z0 +1
2z0A1z0 − a00 − a0 z0 − 1
2z0A0z0]
+[a10 + a1 z1 +1
2z1A1z1 − a00 − a0 z1 − 1
2z1A0z1] (B.63)
Adding and subtracting 12z0A0z1 and 1
2z0A1z1 and noting that the sym-
metry of At (that is At = AtT ) implies z0Atz1 = z1Atz0, the foregoingequation becomes£
f1Q(z0)− f0Q(z
0)¤+£f1Q(z
1)− f0Q(z1)¤
= 2(a10 − a00)− (a0 − a0)(z0 + z1) + z0(A1 −A0)z1
+1
2(z1 − z0)(A1 −A0)(z1 − z0) (B.64)
60
which corresponds to (3.30).The derivation of equations (3.32) and (3.33) follows the footsteps of
that of equations (3.29), (3.30), and (3.21).
Proof of Corollary (3.6). (Accounting for Functional Value Ratios ofTwo Different Quadratic Homothetic Functions).
The proof follows directly that of Corollary (2.3).
Proof of Lemma (4.1). (Accounting for Value Differences between TwoGeneral Quadratic Functions.)
The difference g[f1GQ(q1)]− g[f0GQ(q
0)] and g[h1GQ(q1, x1)]−g[h0GQ(q0, x0)]
can be decomposed by applying Lemma (3.1) and Corollary (3.1), respec-tively, thus obtaining:
g[f1GQ(q1)]− g[f0GQ(q
0)]
= g©f1GQ[Z
−1(z1)]ª− g
©f0GQ[Z
−1(z0)]ª
= f 1Q (z
1)− f 0Q (z
0) =1
2[∇zf
0Q (z
0) +∇zf1Q (z
1)](z1 − z0)
+ Pc
=1
2
ng0[f0GQ(q
0)] · [ bZ 0(q0)]−1 ·∇qf0GQ(q
0) + g0[f1GQ(q1)] · [ bZ 0(q1)]−1 ·∇qf
1GQ(q
1)oT
·[(Z(q1)− Z(q0)] + Pc (B.65)
where
PC component ≡ (a10 − a00) + (a1 − a0)
1
2[Z(q0) + Z(q1]
+Z(q0)(A1 −A0)Z(q0), (B.66)
the (i, i)th element of the diagonal matrix [ bZ 0(q)]−1is equal todqidzi
=dz−1(zi)
dzi= 1/z0(qi) (B.67)
(with z0(qi) 6= 0, by assumption).
61
Since fQ(z) ≡ g[f tGQ(q)] = gnf tGQ[Z
−1(z)]o,
∇zfQ(z) = g0©f tGQ[Z
−1(z)]ª · bZ 0−1(z) ·∇zf
tGQ[Z
−1(z)]
where the (i, i)th element of the diagonal
matrix bZ 0−1(z) is given by dz−1(zi)dzi
= g0[f tGQ(q)] · bZ 0−1(z) ·∇zftGQ(q) using (4.3)
= g0[f tGQ(q)] · [ bZ 0(q)]−1 ·∇zftGQ(q) using (4.9) (B.68)
equation (B65) can be rewritten as
g[f1GQ(q1)]− g[f0GQ(q
0)]
=1
2
ng0[f0GQ(q
0)] · [ bZ 0(q0)]−1 ·∇qf0GQ(q
0) + g0[f1GQ(q1)] · [ bZ 0(q1)]−1 ·∇qf
1GQ(q
1)oT
·[(Z(q1)− Z(q0)] + Pc (B.69)
Proof of Theorem (6.1):If function g and z in (4.1) are, respectively, logarithmic transformation
of functional values and variables, that is g(y) ≡ ln y and Z(q) ≡ ln q, then(6.1) follows directly from (4.9).
Proof of Theorem (6.2).Let us apply the decomposition procedure (4.9) to the quadratic Box-Cox
function (5.4), thus obtaining:
[(f1
Qr,λ)ρ − 1
ρ]− [
(f0Qr,λ)
ρ − 1ρ
] =1
2
NPi=1[(q0i )
1−λ
(f0Qρ,λ)1−ρ
f0Qρ,λ,i
+
¡q1i¢1−λ
(f1Qρ,λ)1−r
f1Qρ,λ,i]
½(q1i )
λ − 1λ
− (q0i )
λ − 1λ
¾+Parameter-change component
=1
2
NPi=1[q0i f
0Qρ,λ,i
f0Qρ,λ
(f0Qρ,λ)ρ 1
(q0)λ+
+q1i f
1Qρ,λ,i
f1Qρ,λ
(f1Qρ,λ)ρ 1
(q1)λ][(q1i )
λ
λ− (q
0i )
λ
λ]
+Parameter-change component (B.70)
where f tQρ,λ ≡ f t
Qρ,λ(qt), f t
Qρ,λ,i≡ ∂f t
Qρ,λ/∂qti .
62
By defining stQρ,λ,i
≡ qtiftQρ,λ,i
f tQρ,λ
with t = 0, 1, multiplying equation (B.70)
by ρ and rearranging terms we obtain
(f1Qρ,λ)ρ − (f0Qρ,λ)
ρ =1
2
ρ
λ
NXi=1
(s0Qρ,λi
(f0Qρ,λ)
ρ
(q0i )λ+ s1Qρ,λi
(f1Qρ,λ)
ρ
(q1i )λ
)[(q1i )
λ − (q0i )λ]
+Parameter-change component · ρλ
(B.71)
which is equation (6.5).
Proof of Theorem (6.3)
Using the definitions (5.18) and (5.19), we have
[f tQρ,λ(q)/σt]ρ = [fQr(q)]r where λ =
r
2(B.72)
The functions [f tQρ,λ(q)/σ
t]ρ and [fQr(q)]r are homogenous of degree 2λ = r.Using equation (B.70), and denoting f t
Qρ,λ(qt) with f t
Qρ,λ to simplifynotation, it is straightforward to obtain
(f1Qρ,λ/σ
1)ρ
(f0Qρ,λ/σ0)ρ
− 1 =ρ
r
NXi=1
(q0i f
0Qρ,λi
/σ0
f0Qρ,λ/σ0
1
(q0i )r2
+q1i f
1Qρ,λi
/σ1
f1Qρ,λ/σ1
(f1Qρ,λ/σ
1)ρ
(f0Qρ,λ/σ0)ρ
1
(q1i )r2
)· (q1i )
r2 − (q0i )
r2
(where f tQρ,λi ≡ ∂f tQρ,λ/∂qti and r = 2λ)
=NXi=1
s0Qρ,λ
(q1i )r2
(q0i )r2
−NXi=1
s0Qρ,λ +(f1
Qρ,λ/σ1)ρ
(f0Qρ,λ/σ0)ρ
NXi=1
s1Qρ,λ −(f1
Qρ,λ/σ1)ρ
(f0Qρ,λ/σ0)ρ
NXi=1
s1Qρ,λ
(q0i )r2
(q1i )r2
where stQρ,λi
= qtiftQρ,λi
/PN
i=1 qtif
tQρ,λi
since, by Euler’s theorem, ρr =f tQρ,λ
Ni=1 q
tif
tQρ,λi
,
for t = 0, 1.
=NXi=1
s0Qρ,λi
(q1i )r2
(q0i )r2
− 1 +(f1
Qρ,λ/σ1)ρ
(f0Qρ,λ/σ0)ρ
−(f1
Qρ,λ/σ1)ρ
(f0Qρ,λ/σ0)ρ
NXi=1
s1Qρ,λi
(q1i )− r2
(q0i )− r2
(B.73)
By rearranging (B.73) and solving for f1Qρ,λ/f
0Qρ,λ , the following decomposi-
tion is obtained
f1Qρ,λ
f0Qρ,λ
=
NXi=1
s0Qρ,λi
(q1i )r2
(q0i )r2
NXi=1
s1Qρ,λi
(q1i )− r2
(q0i )− r2
1ρ
· σ1
σ0(B.74)
63
Moreover, using the definition (5.19) for fQr and denoting fQr(qt) with f tQr
to simplify notation, from the accounting equation (B.70) we derive:
(f1Qr)r
(f0Qr)r− 1 =
NXi=1
(q0i f
0Qri
f0Qr
1
(q0i )r2
+q1i f
1Qri
f1Qr
(f1Qr)r
(f0Qr)r1
(q1i )r2
)(q1i )
r2 − (q0i )
r2
(where f tQri ≡ ∂f tQr/∂qti)
=NXi=1
s0Qri
(q1i )r2
(q0i )r2
−NXi=1
s0Qri +(f1Qr)r
(f0Qr)r
NXi=1
s1Qri −(f1Qr)r
(f0Qr)r
NXi=1
s1Qri
(q0i )r2
(q1i )r2
=NXi=1
s0Qρ,λi
(q1i )r2
(q0i )r2
− 1 + (f1Qr)r
(f0Qr)r− (f
1Qr)r
(f0Qr)r
NXi=1
s1Qρ,λi
(q1i )− r2
(q0i )− r2
,
where stQρ,λi
= stQri = qtiftQri/f
tQr = qtif
tQρ,λi
/ rρftQρ,λ = qtif
tQρ,λi
/PN
i=1 qtif
tQρ,λi
,
since f tQr = 1σt (f
tQρ,λ)
ρr and f tQri =
1σt
ρr (f
tQρ,λ)
ρr−1f t
Qρ,λi) and, by Euler’s the-
orem, rρf
tQρ,λ =
PNi=1 q
tif
tQρ,λi
,
=NXi=1
s0Qρ,λi
(q1i )r2
(q0i )r2
− 1 + (f1Qr)r
(f0Qr)r− (f
1Qr)r
(f0Qr)r
NXi=1
s1Qρ,λi
(q1i )− r2
(q0i )− r2
(B.75)
since f tQr is homogeneous of degree one in q and, by Euler’s theorem,NXi=1
stQri =
1. By rearranging (B.75) and solving for f1Qr/f0Qr , the following index isobtained:
f1Qr
f0Qr
= Iq ≡
NXi=1
s0Qρ,λi
(q1i )r2
(q0i )r2
NXi=1
s1Qρ,λi
(q1i )− r2
(q0i )− r2
1r
(B.76)
Let us define IT ≡ σ1/σ0, and, taking account of (B.74),
(f1Qρ,λ/f0Qρ,λ)/(Iq · IT ) =
IY ≡
NXi=1
s0Qρ,λi
(q1i )r2
(q0i )r2
NXi=1
s1Qρ,λi
(q1i )− r2
(q0i )− r2
1ρ− 1r
(B.77)
Proof of Theorem (6.4)
64
Following Diewert (1976, p. 140), we note that, if both f and fQr arehomogeneous of degree one, then by Euler’s theorem f(q∗) = q∗∇f(q∗)and fQr(q∗) = q∗∇fQr(q∗). Moreover, since the partial derivative functions∂f/∂qi and ∂fQr/∂qi are homogeneous of degree zero, application of Euler’stheorem on homogenous functions yields, for i = 1, 2, ..., N ,
NXj=1
q∗j∂2f(q∗)/∂qi∂qj = 0 =
NXj=1
q∗j∂2fQr(q∗)/∂qi∂qj (B.78)
These results imply that (6.22), (6.23), and (6.24) will be satisfied under thehypotheses adopted here if and only if
∂fQr(q∗)/∂qi = f∗i ≡ ∂f(q∗)/∂qi, i = 1, 2, ..., N (B.79)
∂2fQr(q∗)/∂qi∂qj = f∗ij ≡ ∂2f(q∗)/∂qi∂qj , 1 ≤ i < j ≤ N . (B.80)
Thus, the N(N + 1)/2 independent parameters αij (for 1 ≤ i ≤ j ≤ N)can be chosen so that the N +N(N − 1)/2 = N(N +1)/2 equations (B.79)and (B.80) are satisfied. Assuming that f is positive over its domain ofdefinition, that q∗i > 0, and r 6= 0, the coefficients α∗ij , for 1 ≤ i ≤ j ≤ N ,can be found by solving the following equations
f∗ij =1− r
fQr(q∗)f∗i f
∗j +
r
2[fQr(q∗)](1−r)α∗ij(q
∗i )
r2−1(q∗j )
r2−1, (B.81)
1 ≤ i < j ≤ N .
thus obtaining
αij =f∗ij − 1−r
f∗ f∗i f∗j
r2 [f
∗]1−r (q∗i q∗j )
r2−1 , 1 ≤ i < j ≤ N (B.82)
Equation (B.81) is equivalent to (B.80) if also (B.79) holds. In this case,taking into account that α∗ij = α∗ji for i 6= j, α∗ii is found as the solution tothe following equation:
NPj=1[fQr(q∗)](1−r)α∗ij(q
∗i )
r2−1(q∗j )
r2−1 = f∗i (B.83)
thus obtaining
αii =
f∗i − (q∗i )r2−1 NP
j 6=i[fQr(q∗)](1−r)α∗ij(q
∗j )
r2−1
[fQr(q∗)](1−r)(q∗i )r−2(B.84)
65
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Papers issued in the series of the EU KLEMS project
All papers are available in pdf-format on the internet: http://www.euklems.net/
Nr.1 Mas, Matilde and Javier Quesada, ICT and Economic Growth in Spain 1985-2002 (January 2005)
Nr.2 Jalava, Jukka, Growth and Productivity in the Finnish Trade Industry, 1975-2003: A National Comparative Perspective (January 2005)
Nr. 3 Milana, Carlo, The Theory of Exact and Superlative Index Numbers Revisited (March 2005)