- 1 AN INTRODUCTION TO CONSUMER PRICE INDEX METHODOLOGY
- 2 USES OF CONSUMER PRICE INDICES
- 3 CONCEPTS AND SCOPE
- 4 EXPENDITURE WEIGHTS AND THEIR SOURCES
- 5 SAMPLING
- 6 PRICE COLLECTION
- 7 ADJUSTING FOR QUALITY CHANGE
- 8 ITEM SUBSTITUTION, SAMPLE SPACE AND NEW PRODUCTS
- 9 CALCULATING CONSUMER PRICE INDICES IN PRACTICE
- 10 SOME SPECIAL CASES
- 11 ERRORS AND BIAS
- 12 ORGANIZATION AND MANAGEMENT
- 13 PUBLICATION, DISSEMINATION AND USER RELATIONS
- 14 THE SYSTEM OF PRICE STATISTICS
- 15 BASIC INDEX NUMBER THEORY
- 16 THE AXIOMATIC AND STOCHASTIC APPROACHES TO INDEX NUMBER THEORY
- 17 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE
- 18 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE MANY-HOUSEHOLD CASE
- 19 PRICE INDICES USING AN ARTIFICIAL DATA SET
- 20 ELEMENTARY INDICES
- 21 QUALITY CHANGE AND HEDONICS
- 22 THE TREATMENT OF SEASONAL PRODUCTS
- 23 DURABLES AND USER COSTS

# 17 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE

- International Monetary Fund
- Published Date:
- August 2004

## Introduction

**17.1** This chapter and the next cover the economic approach to index number theory. This chapter considers the case of a *single* household, while the following chapter deals with the case of *many* households. A brief outline of the contents of the present chapter follows.

**17.2** In paragraphs 17.9 to 17.17, the theory of the cost of living index for a single consumer or household is presented. This theory was originally developed by the Russian economist, A.A. Konüs (1924). The relationship between the (unobservable) true cost of living index and the observable Laspeyres and Paasche indices will be explained. It should be noted that, in the economic approach to index number theory, it is assumed that households regard the observed price data as given, while the quantity data are regarded as solutions to various economic optimization problems. Many price statisticians find the assumptions made in the economic approach to be somewhat implausible. Perhaps the best way to regard the assumptions made in the economic approach is that these assumptions simply formalize the fact that consumers tend to purchase more of a commodity if its price falls relative to other prices.

**17.3** In paragraphs 17.18 to 17.26, the preferences of the consumer are restricted compared to the completely general case treated in paragraphs 17.9 to 17.17. In paragraphs 17.18 to 17.26, it is assumed that the function that represents the consumer’s preferences over alternative combinations of commodities is homogeneous of degree one. This assumption means that each indifference surface (the set of commodity bundles that give the consumer the same satisfaction or utility) is a radial blow-up of a single indifference surface. With this extra assumption, the theory of the true cost of living simplifies, as will be seen.

**17.4** In the sections starting with paragraphs 17.27, 17.33 and 17.44, it is shown that the Fisher, Walsh and Törnqvist price indices (which emerge as being “best” in the various non-economic approaches) are also among the “best” in the economic approach to index number theory. In these sections, the preference function of the single household will be further restricted compared to the assumptions on preferences made in the previous two sections. Specific functional forms for the consumer’s utility function are assumed and it turns out that, with each of these specific assumptions, the consumer’s true cost of living index can be exactly calculated using observable price and quantity data. Each of the three specific functional forms for the consumer’s utility function has the property that it can approximate an arbitrary linearly homogeneous function to the second order; i.e., in economics terminology, each of these three functional forms is *flexible*. Hence, using the terminology introduced by Diewert (1976), the Fisher, Walsh and Törnqvist price indices are examples of *superlative* index number formulae.

**17.5** In paragraphs 17.50 to 17.54, it is shown that the Fisher, Walsh and Törnqvist price indices approximate each other very closely using “normal” time series data. This is a very convenient result since these three index number formulae repeatedly show up as being “best” in all the approaches to index number theory. Hence this approximation result implies that it normally will not matter which of these three indices is chosen as the preferred target index for a consumer price index (CPI).

**17.6** The Paasche and Laspeyres price indices have a very convenient mathematical property: they are *consistent in aggregation*. For example, if the Laspeyres formula is used to construct sub-indices for, say, food or clothing, then these sub-index values can be treated as sub-aggregate price relatives and, using the expenditure shares on these sub-aggregates, the Laspeyres formula can be applied again to form a two-stage Laspeyres price index. Consistency in aggregation means that this two-stage index is equal to the corresponding single-stage index. In paragraphs 17.55 to 17.60, it is shown that the superlative indices derived in the earlier sections are not exactly consistent in aggregation but are approximately consistent in aggregation.

**17.7** In paragraphs 17.61 to 17.64, a very interesting index number formula is derived: the Lloyd (1975) and Moulton (1996a) price index. This index number formula makes use of the same information that is required in order to calculate a Laspeyres index (namely, base period expenditure shares, base period prices and current period prices), plus one other parameter (the elasticity of substitution between commodities). If information on this extra parameter can be obtained, then the resulting index can largely eliminate substitution bias and it can be calculated using basically the same information that is required to obtain the Laspeyres index.

**17.8** The section starting with paragraph 17.65 considers the problem of defining a true cost of living index when the consumer has annual preferences over commodities but faces monthly (or quarterly) prices. This section attempts to provide an economic foundation for the Lowe index studied in Chapter 15. It also provides an introduction to the problems associated with the existence of seasonal commodities, which are considered at more length in Chapter 22. The final section deals with situations where there may be a zero price for a commodity in one period, but where the price is nonzero in the other period.

## The Konüs cost of living index and observable bounds

**17.9** This section deals with the theory of the cost of living index for a single consumer (or household) that was first developed by the Russian economist, Konüs (1924). This theory relies on the assumption of *optimizing behaviour* on the part of economic agents (consumers or producers). Thus, given a vector of commodity prices *p ^{t}* that the household faces in a given time period

*t*, it is assumed that the corresponding observed quantity vector

*q*is the solution to a cost minimization problem that involves the consumer’s preference or utility function f

^{t}^{1}. Thus in contrast to the axiomatic approach to index number theory, the economic approach does not assume that the two quantity vectors

*q*and

^{0}*q*are independent of the two price vectors p

^{1}^{0}and p

^{1}. In the economic approach, the period 0 quantity vector

*q*is determined by the consumer’s preference function

^{0}*f*and the period 0 vector of prices

*p*that the consumer faces, and the period 1 quantity vector

^{0}*q*is determined by the consumer’s preference function

^{1}*f*and the period 1 vector of prices

*p*

^{1.}**17.10** The economic approach to index number theory assumes that “the” consumer has well-defined *preferences* over different combinations of the *n* consumer commodities or items.^{2} Each combination of items can be represented by a positive quantity vector *q ≡ [q _{1},…,q_{n}]*. The consumer’s preferences over alternative possible consumption vectors,

*q*, are assumed to be representable by a continuous, non-decreasing and concave

^{3}utility function f Thus if

*f(q*(q

^{1})>f^{0}), then the consumer prefers the consumption vector

*q*to

^{1}*q*. It is further assumed that the consumer minimizes the cost of achieving the period

^{0}*t*utility level

*u*for periods

^{t}≡ f(q^{1})*t =*0.1, Thus we assume that the observed period

*t*consumption vector

*q*solves the following period

^{t}*t*cost minimization problem:

The period *t* price vector for the *n* commodities under consideration that the consumer faces is *p ^{t}*. Note that the solution to the cost or expenditure minimization problem (17.1) for a general utility level

*u*and general vector of commodity prices

*p*defines the

*consumer’s cost function, C(u,p)*. The cost function will be used below in order to define the consumer’s

*cost of living price index*.

**17.11** The Konüs (1924) family of *true cost of living indices* pertaining to two periods where the consumer faces the strictly positive price vectors *q≡(q _{1},…,q_{n})* is a positive reference quantity vector:

Note that definition (17.2) defines a family of price indices, because there is one such index for each reference quantity vector *q* chosen.

**17.12** It is natural to choose two specific reference quantity vectors *q* in definition (17.2): the observed base period quantity vector *q ^{0}* and the current period quantity vector q

^{1}. The first of these two choices leads to the following

*Laspeyres-Konüs true cost of living index*:

using the definition of the cost minimization problem that defines *C(f(q ^{0}),p^{1})*

since

where *P _{L}* is the Laspeyres price index. Thus the (unobservable) Laspeyres-Konüs true cost of living index is bounded from above by the observable Laspeyres price index.

^{4}

**17.13** The second of the two natural choices for a reference quantity vector *q* in definition (17.2) leads to the following *Paasche-Konüs true cost of living index*:

using the definition of the cost minimization problem that defines

is feasible for the minimization problem and thus

where *P _{P}* is the Paasche price index. Thus the (unobservable) Paasche-Konüs true cost of living index is bounded from below by the observable Paasche price index.

^{5}

**17.14** It is possible to illustrate the two inequalities (17.3) and (17.4) if there are only two commodities; see Figure 17.1. The solution to the period 0 cost minimization problem is the vector *q ^{0}*. The straight line C represents the consumer’s period 0 budget constraint, the set of quantity points

*q*,

_{1}*q*such that

_{2}*q*is the consumer’s period 0 indifference curve, the set of points

^{0}*q*such that

_{1},q_{2}*q*. The solution to the period 1 cost minimization problem is the vector

^{0}*q*. The straight line D represents the consumer’s period 1 budget constraint, the set of quantity points q

^{1}_{1},q

_{2}such that

*q*is the consumer’s period 1 indifference curve, the set of points

^{1}*q*such that

_{1},q_{2}*q*. The point

^{1}*q*solves the hypothetical problem of minimizing the cost of achieving the base period utility level

^{0*}*u*when facing the period 1 price vector

^{1}Ξf(q^{0})*q*were L-shaped with vertex at

^{0}*q*, then the consumer would not change his or her consumption pattern in response to a change in the relative prices of the two commodities while keeping a fixed standard of living. In this case, the hypothetical vector

^{0}*q*would coincide with

^{0*}*q*, the dashed line through A would coincide with the dashed line through B and the true Laspeyres-Konüs index would coincide with the ordinary Laspeyres index. However, L-shaped indifference curves are not generally consistent with consumer behaviour; i.e., when the price of a commodity decreases, consumers generally demand more of it. Thus, in the general case, there will be a gap between the points A and B. The magnitude of this gap represents the amount of

^{0}*substitution bias*between the true index and the corresponding Laspeyres index; i.e., the Laspeyres index will generally be greater than the corresponding true cost of living index,

*P*.

_{K}(p^{0},p^{1},p^{0})Figure 17.1 The Laspeyres and Paasche bounds to the true cost of living index

**17.15** Figure 17.1 can also be used to illustrate the inequality (17.4). First note that the dashed lines E and F are parallel to the period 0 isocost line through C. The point *q ^{1*}* solves the hypothetical problem of minimizing the cost of achieving the current period utility level

*P*. Note that this inequality goes in the opposite direction to the previous inequality between the two Laspeyres indices. The reason for this change in direction is attributable to the fact that one set of differences between the two indices takes place in the numerators of the indices (the Laspeyres inequalities), while the other set takes place in the denominators of the indices (the Paasche inequalities).

_{k}(P^{0},P^{1},q^{1})**17.16** The bound (17.3) on the Laspeyres-Konüs true cost of living index *P _{K}(p^{0},p^{1,}q^{0})* using the base period level of utility as the living standard is

*one-sided*, as is the bound (17.4) on the Paasche-Konüs true cost of living index

*P*using the

_{K}(p^{0},p^{1},q^{1})*current period*level of utility as the living standard. In a remarkable result, Konüs (1924; 1939, p. 20) showed that there exists an intermediate consumption vector

*q*that is on the straight line joining the base period consumption vector

^{*}*q*and the current period consumption vector

^{0}*q*such that the corresponding (unobservable) true cost of living index P

^{1}_{k}(

*P*,

^{0}*P*

^{1,}*q*) is between the observable Laspeyres and Paasche indices,

^{*}*P*and

_{L}*P*Thus we have the existence of a number

_{P}.^{6}*λ**between 0 and 1 such that

The inequalities (17.5) are of some practical importance. If the observable (in principle) Paasche and Laspeyres indices are not too far apart, then taking a symmetric average of these indices should provide a good approximation to a true cost of living index where the reference standard of living is somewhere between the base and current period living standards. To determine the precise symmetric average of the Paasche and Laspeyres indices, appeal can be made to the results in paragraphs 15.18 to 15.32 in Chapter 15, and the geometric mean of the Paasche and Laspeyres indices can be justified as being the “best” average, which is the Fisher price index. Thus the Fisher ideal price index receives a fairly strong justification as a good approximation to an unobservable theoretical cost of living index.

17.17 The bounds (17.3)–(17.5) are the best that can be obtained on true cost of living indices without making further assumptions. Further assumptions are made below on the class of utility functions that describe the consumer’s tastes for the *n* commodities under consideration. With these extra assumptions, the consumer’s true cost of living can be determined exactly.

## The true cost of living index when preferences are homothetic

**17.18** Up to now, the consumer’s preference function *f* did not have to satisfy any particular homogeneity assumption. For the remainder of this section, it is assumed that *f* is (positively) *linearly homogeneous*. ^{7} In the economics literature, this is known as the assumption of homothetic preferences. ^{8} This assumption is not strictly justified from the viewpoint of actual economic behaviour, but it leads to economic price indices that are independent of the consumer’s standard of living.^{9} Under this assumption, the consumer’s expenditure or cost function, *C(u,p)* defined by equation (17.1), decomposes as follows. For positive commodity prices *P>>0 _{N}* and a positive utility level

*u*, then, using the definition of

*C*as the minimum cost of achieving the given utility level u, the following equalities can be derived:

where *c(p)*≡*C(1,p)* is the *unit cost function* that corresponds to f. ^{10} It can be shown that the unit cost function *c(p)* satisfies the same regularity conditions that f satisfies; i.e., *c(p)* is positive, concave and (positively) linearly homogeneous for positive price vectors.^{11} Substituting equation (17.6) into equation (17.1) and using *u ^{t}= f(q^{t})* leads to the following equation:

Thus, under the linear homogeneity assumption on the utility function f, observed period t expenditure on the *n* commodities is equal to the period *t* unit cost *c(p ^{t})* of achieving one unit of utility times the period

*t*utility level, f(

*q*. Obviously, the period

^{t})*t*unit cost,

*c(p*, can be identified as the period

^{t})*t*price level

*P*and the period t level of utility, f

^{t}*(q*, as the period

^{t})*t*quantity level

*Q*.

^{t}^{12}

**17.19** The linear homogeneity assumption on the consumer’s preference function f leads to a simplification for the family of Konüs true cost of living indices, *p _{k}(P^{0},P^{1},q*), defined by equation (17.2). Using this definition for an arbitrary reference quantity vector

*q*:

Thus under the homothetic preferences assumption, the entire family of Konüs true cost of living indices collapses to a single index, *c(p ^{1})/c(p^{0})*, the ratio of the minimum costs of achieving unit utility level when the consumer faces period 1 and 0 prices respectively. Put another way, under the homothetic preferences assumption,

*p*is independent of the reference quantity vector

_{k}(p^{0},p^{1},q)*q*.

**17.20** If the Konüs true cost of living index defined by the right-hand side of equation (17.8) is used as the price index concept, then the corresponding implicit quantity index defined using the product test (i.e., the product of the price index times the quantity index is equal to the value ratio) has the following form:

Thus, under the homothetic preferences assumption, the implicit quantity index that corresponds to the true cost of living price index *c(p ^{1})/c(p^{0})* is the utility ratio f

*(q*. Since the utility function is assumed to be homogeneous of degree one, this is the natural definition for a quantity index.

^{1})/f(q^{0})**17.21** In subsequent material, two additional results from economic theory will be needed: Wold’s Identity and Shephard’s Lemma. Wold’s (1944, pp. 69–71;1953, p. 145) Identity is the following result. Assuming that the consumer satisfies the cost minimization assumptions (17.1) for periods 0 and 1 and that the utility function f is differentiable at the observed quantity vectors *q ^{0}* and

*q*it can be shown

^{1}^{13}that the following equation holds:

where *f* with respect to the *i*th quantity *q _{i}* evaluated at the period

*t*quantity vector

*q*

^{t}.

**17.22** If the homothetic preferences assumption is made and it is assumed that the utility function is linearly homogeneous, then Wold’s Identity can be simplified into an equation that will prove to be very useful:^{14}

**17.23**Shephard’s (1953, p. 11) Lemma is the following result. Consider the period t cost minimization problem defined by equation (17.1). If the cost function *C(u,p)* is differentiable with respect to the components of the price vector *p*, then the period t quantity vector *q ^{t}* is equal to the vector of first-order partial derivatives of the cost function with respect to the components of

*p*:

**17.24** To explain why equation (17.12) holds, consider the following argument. Because it is assumed that the observed period t quantity vector *q ^{t}* solves the cost minimization problem defined by

*C(u*, then

^{t},p^{t})*q*must be feasible for this problem so it must be the case that f

^{t}*(q*Thus,

^{t}) = u^{t}*q*is a feasible solution for the following cost minimization problem where the general price vector

^{t}*p*has replaced the specific period t price vector

*p*

^{t}:

where the inequality follows from the fact that *p* the function *g(p)* as follows:

where, as usual, *p* ≡*(p _{1},…,p_{n})*. Using equations (17.13) and (17.1), it can be seen that

*g(p)*is minimized (over all strictly positive price vectors

*p)*at p=p

^{t}. Thus the first-order necessary conditions for minimizing a differentiable function of

*n*variables hold, which simplify to equation (17.12).

**17.25** If the homothetic preferences assumption is made and it is assumed that the utility function is linearly homogeneous, then using equation (17.6), Shephard’s Lemma (17.12) becomes:

Combining equations (17.15) and (17.7), the following equation is obtained:

**17.26** Note the symmetry of equation (17.16) with equation (17.11). It is these two equations that will be used in subsequent material in this chapter.

## Superlative indices: The Fisher ideal index

**17.27** Suppose the consumer has the following utility function:

Differentiating *f(q)* defined by equation (17.17) with respect to *q _{i}* yields the following equation:

where *a*_{ik}=a_{ki} Now evaluate the second equation in (17.18) at the observed period t quantity vector ^{t}). The following equations are obtained:

Assume cost minimizing behaviour for the consumer in periods 0 and 1. Since the utility function f defined by equation (17.17) is linearly homogeneous and differentiable, equation (17.11) will hold. Now recall the definition of the Fisher ideal quantity index, *Q _{F}*, defined earlier in Chapter 15:

Thus under the assumption that the consumer engages in cost-minimizing behaviour during periods 0 and 1 and has preferences over the n commodities that correspond to the utility function defined by equation (17.17), the Fisher ideal quantity index *Q _{F}* is exactly equal to the true quantity index, f

*(q*)/f(q

^{1}^{0}).

^{15}

**17.28** As was noted in paragraphs 15.18 to 15.23 of Chapter 15, the price index that corresponds to the Fisher quantity index *Q _{F}* using the product test (15.3) is the Fisher price index

*P*, defined by equation (15.12). Let

_{F}*c(p)*be the unit cost function that corresponds to the homogeneous quadratic utility function f defined by equation (17.17). Then using equations (17.16) and (17.20), it can be seen that

Thus, under the assumption that the consumer engages in cost-minimizing behaviour during periods 0 and 1 and has preferences over the *n* commodities that correspond to the utility function defined by equation (17.17), the Fisher ideal price index *P _{F}* is exactly equal to the true price index,

*c*(p

^{1})/c(p

^{0}).

**17.29** A twice continuously differentiable function *f(q)* of *n* variables *q* ≡*(q _{1},…,q_{n})* can provide a

*second-order approximation*to another such function

*f*(q)*around the point

*q*if the level and all the first-order and second-order partial derivatives of the two functions coincide at

^{*,}*q*. It can be shown

^{*}^{16}that the homogeneous quadratic function f defined by equation (17.17) can provide a second-order approximation to an arbitrary f

^{*}around any (strictly positive) point

*q*in the class of linearly homogeneous functions. Thus the homogeneous quadratic functional form defined by equation (17.17) is a

^{*}*flexible functional form.*Diewert (1976, p. 117) termed an index number formula Q(p

^{17}^{0},p

^{1},q

^{0},q

^{1}) that was exactly equal to the true quantity index f(

*q*(where/is a flexible functional form)

^{1})/f(q^{0})*a superlative index number formula.*Equation (17.20) and the fact that the homogeneous quadratic function f defined by equation (17.17) is a flexible functional form show that the Fisher ideal quantity index

^{18}*Q*defined by equation (15.14) is a superlative index number formula. Since the Fisher ideal price index

_{F}*P*satisfies equation (17.21), where

_{F}*c(p)*is the unit cost function that is generated by the homogeneous quadratic utility function,

*P*also called a superlative index number formula.

_{F}is**17.30** It is possible to show that the Fisher ideal price index is a superlative index number formula by a different route. Instead of starting with the assumption that the consumer’s utility function is the homogeneous quadratic function defined by equation (17.17), it is possible to start with the assumption that the consumer’s unit cost function is a homogeneous quadratic.^{19} Thus, suppose that the consumer has the following unit cost function:

Differentiating *c(p)* defined by equation (17.22) with respect to *p _{i}* yields the following equations:

where *t* price vector *c(p ^{t})*. The following equation is obtained:

As cost-minimizing behaviour for the consumer in periods 0 and 1 is being assumed and, since the unit cost function *C* defined by equation (17.22) is differentiable, equations (17.16) will hold. Now recall the definition of the Fisher ideal price index, *P _{F}*, given by equation function simplifies as follows: (15.12) in Chapter 15:

Thus, under the assumption that the consumer engages in cost-minimizing behaviour during periods 0 and 1 and has preferences over the *n* commodities that correspond to the unit cost function defined by equation (17.22), the Fisher ideal price index * P_{F}* is exactly equal to the true price index,

*c(p*

^{1})/c(p^{0}).^{20}**17.31** Since the homogeneous quadratic unit cost function *c(p)* defined by equation (17.22) is also a flexible functional form, the fact that the Fisher ideal price index * P_{F}* exactly equals the true price index

*c(p*means that

^{1})/c(p^{0})*a superlative index number formula.*

**P**_{F}is^{21}

**17.32** Suppose that the *b _{ik}* coefficients in equation (17.22) satisfy the following restrictions:

where the *n* numbers *b _{i}* are non-negative. In this special case of equation (17.22), it can be seen that the unit cost function simplifies as follows:

Substituting equation (17.27) into Shephard’s Lemma (17.15) yields the following expressions for the period t quantity vectors, q^{t}:

Thus if the consumer has the preferences that correspond to the unit cost function defined by equation (17.22) where the *b _{ik}* satisfy the restrictions (17.26), then the period 0 and 1 quantity vectors are equal to a multiple of the vector

*b*≡

*(b*and

_{1},…,b_{n}); i.e.,q^{0}= b u^{0}*q*u

^{1}=b^{1}. Under these assumptions, the Fisher, Paasche and Laspeyres indices,

*P*and

_{F}, P_{P}*P*, all coincide. The preferences which correspond to the unit cost function defined by equation (17.27) are, however, not consistent with normal consumer behaviour since they imply that the consumer will not substitute away from more expensive commodities to cheaper commodities if relative prices change going from period 0 to 1.

_{L}## Quadratic mean of order *r* superlative indices

**17.33** It turns out that there are many other superlative index number formulae; i.e., there exist many quantity indices *Q(p ^{0},p^{1}q^{0},q^{1})* that are exactly equal tof(q

^{1})/f(q

^{0}) and many price indices

*P(P*that are exactly equal to

^{0},P^{1},q^{0}, q^{1})*C(P*, where the aggregator function f or the unit cost function f is a flexible functional form. Two families of superlative indices are defined below.

^{1})/c(P^{0})**17.34** Suppose the consumer has the following quadratic mean of order r utility function.^{22}

where the parameters *a _{ik}* satisfy the symmetry conditions

*a*for all

_{ik}—a_{ki}*i*and

*k*and the parameter r satisfies the restriction r≡0. Diewert (1976, p. 130) showed that the utility function f

^{r}defined by equation (17.29) is a flexible functional form; i.e., it can approximate an arbitrary twice continuously differentiable linearly homogeneous functional form to the second order. Note that when r = 2, f

^{r}equals the homogeneous quadratic function defined by equation (17.17).

**17.35** Define the quadratic mean of order r quantity index *Q ^{r}* by:

where *t* expenditure share for commodity *i* as usual.

**17.36** Using exactly the same techniques as were used in paragraphs 17.27 to 17.32, it can be shown that *Q ^{r}* is exact for the aggregator function f

^{r}defined by equation (17.29); i.e., the following exact relationship between the quantity index

*Q*and the utility function f

^{r}^{r}holds:

Thus under the assumption that the consumer engages in cost-minimizing behaviour during periods 0 and 1 and has preferences over the *n * commodities that correspond to the utility function defined by equation (17.29), the quadratic mean of order *r* quantity index *Q _{F}* is exactly equal to the true quantity index, f

^{r}

*(q*

^{1})/f^{r}(q^{0}

*).*Since

^{23}*Q*is exact for f

^{r}^{r}and f

^{r}is a flexible functional form, it can be seen that the quadratic mean of order

*r*quantity index

*Q*is a superlative index for each r≠0. Thus there is an infinite number of superlative quantity indices.

^{r}**17.37** For each quantity index *Q ^{r}*, the product test (15.3) in Chapter 15 can be used in order to define the corresponding implicit quadratic mean of order r price index

*P*:

^{r*}where *c ^{r}** is the unit cost function that corresponds to the aggregator function f

^{r}defined by equation (17.29). For each r≠0, the implicit quadratic mean of order

*r*price index

*P*is also a superlative index.

^{r*}**17.38** When r = 2, *Q ^{r}* defined by equation (17.30) simplifies to

*Q*, the Fisher ideal quantity index, and

_{F}*P*defined by equation (17.32) simplifies to

^{r*}*P*, the Fisher ideal price index. When r=1,

_{F}*Q*defined by equation (17.30) simplifies to:

^{r}where *P _{w}* is the Walsh price index defined previously by equation (15.19) in Chapter 15. Thus P

^{1*}is equal to

*P*, the Walsh price index, and hence it is also a superlative price index.

_{w}**17.39** Suppose the consumer has the following quadratic mean of order r unit cost function:^{24}

where the parameters b_{ik} satisfy the symmetry conditions *b _{ik} = b_{ki}* for all

*i*and

*k*, and the parameter

*r*satisfies the restriction

*r*≠

*O*. Diewert (1976, p. 130) showed that the unit cost function

*c*defined by equation (17.34) is a flexible functional form; i.e., it can approximate an arbitrary twice continuously differentiable linearly homogeneous functional form to the second order. Note that when r = 2,

^{r}*c*equals the homogeneous quadratic function defined by equation (17.22).

^{r}**17.40** Define the quadratic mean of order *r* price index *P ^{r}* by:

where *i* as usual.

**17.41** Using exactly the same techniques as were used in paragraphs 17.27 to 17.32, it can be shown that *P ^{T}* is exact for the aggregator function defined by equation (17.34); i.e., the following exact relationship between the index number formula

*P*and the unit cost function

^{r}*c*holds:

^{r}Thus, under the assumption that the consumer engages in cost-minimizing behaviour during periods 0 and 1, and has preferences over the *n* commodities that correspond to the unit cost function defined by equation (17.34), the quadratic mean of order *r* price index *P _{F} is* exactly equal to the true price index,

*c*Since

^{r}(p^{l})/c^{r}(p^{0}).^{25}*P*is exact for

^{R}*c*and

^{r}*c*is a flexible functional form, it can be seen that the quadratic mean of order r price index

^{r}*P*is a superlative index for each r=0. Thus there are an infinite number of superlative price indices.

^{R}**17.42** For each price index *P ^{R}*, the product test (15.3) in Chapter 15 can be used in order to define the corresponding implicit quadratic mean of order

*r*quantity index

*Q*

^{r*}:

where f^{r*} is the aggregator function that corresponds to the unit cost function *c ^{r}* defined by equation (17.34).

^{26}For each r≠0, the implicit quadratic mean of order

*r*quantity index

*Q*is also a superlative index.

^{r}***17.43** When r = 2, *P ^{R}* defined by equation (17.35) simplifies to

*P*, the Fisher ideal price index, and

_{F}*Q*defined by equation (17.37) simplifies to

^{r}**Q*, the Fisher ideal quantity index. When

_{F}*r =*1, P

^{r}defined by equation (17.35) simplifies to:

where *Q _{W}* is the Walsh quantity index defined previously in footnote 30 of Chapter 15. Thus

*Q*is equal to

^{1*}*Qw*, the Walsh quantity index, and hence it is also a superlative quantity index.

## Superlative indices: The Törnqvist index

**17.44** In this section, the same assumptions that were made on the consumer in paragraphs 17.9 to 17.17 are made here. In particular, it is not assumed that the consumers utility function f is necessarily linearly homogeneous as in paragraphs 17.18 to 17.43.

**17.45** Before the main result is derived, a preliminary result is required. Suppose the function of *n* variables,f(z_{1},…,z_{n})≡ f(z), is quadratic; i.e.,

where the *a _{i}* and the

*a*are constants. Let

_{ik}*f*denote the first-order partial derivative of f evaluated at z with respect to the

_{i}(z)*i*th component of z,

*z*. Let f

_{i}_{ik}(z) denote the second-order partial derivative of f with respect to

*z*and

_{i}*z*. Then it is well known that the second-order Taylor series approximation to a quadratic function is exact; i.e., if

_{k}*f*is defined by equation (17.39), then for any two points, z

^{0}and z

^{1}, the following equation holds:

It is less well known that an average of two first-order Taylor series approximations to a quadratic function is also exact; i.e., if *f* is defined by equation (17.39) above, then for any two points, *z ^{0}* and z

^{1}, the following equation holds:

^{27}

Diewert (1976, p. 118) and Lau (1979) showed that equation (17.41) characterized a quadratic function and called the equation the *quadratic approximation lemma*. In this chapter, equation (17.41) will be called the *quadratic identity*.

**17.46** Suppose that the consumer’s *cost function ^{28} C(u,p)*, has the following

*translog functional form:*

^{29}where In is the natural logarithm function and the parameters *a _{i}, a_{ik}*, and

*b*satisfy the following restrictions:

_{i}These parameter restrictions ensure that *C(u,p)* defined by equation (17.42) is linearly homogeneous in *p*, a property that a cost function must have. It can be shown that the translog cost function defined by equation (17.42) can provide a second-order Taylor series approximation to an arbitrary cost function.^{30}

**17.47** Assume that the consumer has preferences that correspond to the translog cost function and that the consumer engages in cost-minimizing behaviour during periods 0 and 1. Let *p ^{0}* and

*p*be the period 0 and I observed price vectors, and let

^{l}*q*and

^{0}*q*be the period 0 and 1 observed quantity vectors. These assumptions imply:

^{l}where *C* is the translog cost function defined above. Now apply Shephard’s Lemma, equation (17.12), and the following equation results:

Now use equation (17.44) to replace *C(u ^{t}, p^{t})* in equation (17.45). After some cross multiplication, this becomes the following:

or

where S^{t}_{i}is the period *t* expenditure share on commodity *i*.

**17.48** Define the geometric average of the period 0 and 1 utility levels as u^{*}; i.e., define

Now observe that the right-hand side of the equation that defines the natural logarithm of the translog cost function, equation (17.42), is a quadratic function of the variables z_{i} ≡ 1np_{i} if utility is held constant at the level *u*^{*}. Hence the quadratic identity (17.41) can be applied, and the following equation is obtained:

The last equation in (17.49) can be recognized as the logarithm of the Törnqvist-Theil index number formula *P _{T}*, defined earlier by equation (15.81) in Chapter 15. Hence, exponentiating both sides of equation (17.49) yields the following equality between the true cost of living between periods 0 and 1, evaluated at the intermediate utility level

*u*and the observable Törnqvist-Theil index

^{*}*P*

_{T}:

^{31}

Since the translog cost function which appears on the left-hand side of equation (17.49) is a flexible functional form, the Törnqvist-Theil price index *P _{T}* is also a superlative index.

**17.49** It is somewhat mysterious how a ratio of unobservable cost functions of the form appearing on the left-hand side of the above equation can be exactly estimated by an observable index number formula. The key to this mystery is the assumption of cost-minimizing behaviour and the quadratic identity (17.41), along with the fact that derivatives of cost functions are equal to quantities, as specified by Shephard’s Lemma. In fact, all the exact index number results derived in paragraphs 17.27 to 17.43 can be derived using transformations of the quadratic identity along with Shephard’s Lemma (or Wold’s Identity).^{32} Fortunately, for most empirical applications, assuming that the consumer has (transformed) quadratic preferences will be an adequate assumption, so the results presented in paragraphs 17.27 to 17.49 are quite useful to index number practitioners who are willing to adopt the economic approach to index number theory.^{33} Essentially, the economic approach to index number theory provides a strong justification for the use of the Fisher price index *P _{F}* defined by equation (15.12), the Törnqvist-Theil price index

*P*defined by equation (15.81), the implicit quadratic mean of order

_{T}*r*price indices

*P*defined by equation (17.32) (when

^{r}**r=*1, this index is the Walsh price index defined by equation (15.19) in Chapter 15) and the quadratic mean of order

*r*price indices

*P*defined by equation (17.35). In the next section, we ask if it matters which one of these formulae is chosen as “best”.

^{r}## The approximation properties of superlative indices

**17.50** The results of paragraphs 17.27 to 17.49 provide price statisticians with a large number of index number formulae which appear to be equally good from the viewpoint of the economic approach to index number theory. Two questions arise as a consequence of these results:

Does it matter which of these formulae is chosen?

If it does matter, which formula should be chosen?

**17.51** With respect to the first question, Diewert (1978, p. 888) showed that all of the superlative index number formulae listed in paragraphs 17.27 to 17.49 approximate each other to the second order around any point where the two price vectors, p^{0} and *p*^{1}, are equal and where the two quantity vectors, q^{0} and q^{1} are equal. In particular, this means that the following equalities are valid for all *r* and *s* not equal to 0, provided that *p*^{0}*=p ^{1} and q*

^{0}

*= q*.

^{1}^{34}where the Törnqvist-Theil price index *P _{T}* is defined by equation (15.81), the implicit quadratic mean of order

*r*price indices

*P*is defined by equation (17.32) and the quadratic mean of order

^{s*}*r*price indices

*P*is defined by equation (17.35). Using the results in the previous paragraph, Diewert (1978, p. 884) concluded that “all superlative indices closely approximate each other”.

^{r}**17.52** The above conclusion is, however, not true even though the equations (17.51) to (17.56) are true. The problem is that the quadratic mean of order *r* price indices *P ^{r}* and the implicit quadratic mean of order

*s*price indices

*P*are (continuous) functions of the parameters

^{s*}*r*and

*s*respectively. Hence, as r and

*s*become very large in magnitude, the indices

*P*and

^{r}*P*can differ substantially from, say,

^{s}**P*, the Fisher ideal index. In fact, using definition (17.35) and the limiting properties of means of order r,

^{2}= P_{F}^{35}Robert Hill (2002, p. 7) showed that

*P*has the following limit as r approaches plus or minus infinity:

^{r}Using Hill’s method of analysis, it can be shown that the implicit quadratic mean of order *r* price index has the following limit as *r* approaches plus or minus infinity:

Thus for *r* large in magnitude, p^{r} and *P ^{r*}* can differ substantially from

*P*

_{T}, P^{1}

*P*

^{1*}

*= P*(the Walsh price index) and

_{w}*P*

^{2=}

*P*(the Fisher ideal index).

^{2*}= P_{F}^{36}

**17.53** Although Hill’s theoretical and empirical results demonstrate conclusively that not all superlative indices will necessarily closely approximate each other, there is still the question of how well the more commonly used superlative indices will approximate each other. All the commonly used superlative indices, *P ^{r}* and

*P*, fall into the interval 0≤r≤2

^{r*}^{37}Hill (2002, p. 16) summarized how far apart the Törnqvist and Fisher indices were, making all possible bilateral comparisons between any two data points for his time series data set as follows:

The superlative spread S(0,2) is also of interest since, in practice, Törnqvist

(r= 0) and Fisher(r= 2) are by far the two most widely used superlative indexes. In all 153 bilateral comparisons, s(0,2) is less than the Paasche-Laspeyres spread and on average, the superlative spread is only 0.1 per cent. It is because attention, until now, has focussed almost exclusively on superlative indexes in the range 0≤r≤2 that a general misperception has persisted in the index number literature that all superlative indexes approximate each other closely.

Thus, for Hill’s time series data set covering 64 components of United States gross domestic product from 1977 to 1994 and making all possible bilateral comparisons between any two years, the Fisher and Törnqvist price indices differed by only 0.1 per cent on average. This close correspondence is consistent with the results of other empirical studies using annual time series data.^{38} Additional evidence on this topic may be found in Chapter 19.

**17.54** In the earlier chapters of this manual, it is found that several index number formulae seem “best” when viewed from various perspectives. Thus the Fisher ideal index *P _{F}=P^{2}=P^{2*}* defined by equation (15.12) seemed to be best from one axiomatic viewpoint, the Törnqvist-Theil price index

*P*defined by equation (15.81) seems to be best from another axiomatic perspective, as well as from the stochastic viewpoint, and the Walsh index

_{T}*P*defined by equation (15.19) (which is equal to the implicit quadratic mean of order r price indices

_{w}*P*defined by equation (17.32) when r=1) seems to be best from the viewpoint of the “pure” price index. The results presented in this section indicate that for “normal” time series data, these three indices will give virtually the same answer. To determine precisely which one of these three indices to use as a theoretical target or actual index, the statistical agency will have to decide which approach to bilateral index number theory is most consistent with its goals. For most practical purposes, however, it will not matter which of these three indices is chosen as a theoretical target index for making price comparisons between two periods.

^{r}*## Superlative indices and two-stage aggregation

**17.55** Most statistical agencies use the Laspeyres formula to aggregate prices in two stages. At the first stage of aggregation, the Laspeyres formula is used to aggregate components of the overall index (e.g., food, clothing, services); then at the second stage of aggregation, these component sub-indices are further combined into the overall index. The following question then naturally arises: does the index computed in two stages coincide with the index computed in a single stage? Initially, this question is addressed in the context of the Laspeyres formula.^{39}

**17.56** Suppose that the price and quantity data for period *t*, *p ^{t}* and q

^{t}, can be written in terms of

*M*sub-vectors as follows:

where the dimensionality of the subvectors *p ^{tm} and q^{tm}*

*N*for

_{m}*m=*1,2,…,

*M*with the sum of the dimensions

*N,„*equal to n. These subvectors correspond to the price and quantity data for subcomponents of the consumer price index for period t. Now construct sub-indices for each of these components going from period 0 to 1. For the base period, set the price for each of these subcomponents, say P

^{0}

_{m}for

*m*= 1,2,... M, equal to 1 and set the corresponding base period subcomponent quantities, say for

*m =*1,2,…,

*M*, equal to the base period value of consumption for that subcomponent for

*m =*1,2.....

*M*:

Now use the Laspeyres formula in order to construct a period 1 price for each subcomponent, say P^{1}_{m} for *m =* 1,2,…, *M*, of the CPI. Since the dimensionality of the subcomponent vectors, *p ^{t}_{m}* and

*q*differs from the dimensionality of the complete period

^{t}_{m}*t*vectors of prices and quantities,

*p*and

^{t}*q*it is necessary to use different symbols for these subcomponent Laspeyres indices, say P

^{t}^{m}

_{L}for

*m*= 1,2,....

*M*. Thus the period 1 subcomponent

Once the period 1 prices for the *M* sub-indices have been defined by equation (17.61), then corresponding subcomponent period 1 quantities Q^{1}_{m} for *m=* 1,2,…,*M* can be defined by deflating the period 1 subcomponent values ^{1}_{m}

Now define subcomponent price and quantity vectors for each period *t* = 0,1 using equations (17.60) to (17.62). Thus define the period 0 and 1 subcomponent price vectors *P ^{0}* and

*P*as follows:

^{L}where 1_{M} denotes a vector of ones of dimension *M* and the components of *P ^{L}* are defined by equation (17.61). The period 0 and 1 subcomponent quantity vectors

*Q*and

^{0}*Q*are defined as follows:

^{1}where the components of *Q ^{0}* are defined in equation (17.60) and the components of

*Q*are defined by equation (17.62). The price and quantity vectors in equations (17.63) and (17.64) represent the results of the first-stage aggregation. Now use these vectors as inputs into the second-stage aggregation problem; i.e., apply the Laspeyres price index formula, using the information in equations (17.63) and (17.64) as inputs into the index number formula. Since the price and quantity vectors that are inputs into this second-stage aggregation problem have dimension

^{L}*M*instead of the single-stage formula which utilized vectors of dimension

*n*, a different symbol is required for the new Laspeyres index: this is chosen to be P

^{*}

_{L}. Thus the Laspeyres price index computed in two stages can be denoted as

*P*that was studied in the previous sections of this chapter; i.e., ask whether

_{L}If the Laspeyres formula is used at each stage of each aggregation, the answer to the above question is yes: straightforward calculations show that the Laspeyres index calculated in two stages equals the Laspeyres index calculated in one stage.

**17.57** Now suppose that the Fisher or Törnqvist formula is used at each stage of the aggregation. That is, in equation (17.61), suppose that the Laspeyres formula *P _{L} (p^{0},p^{L},q^{0},q*

^{1}) is replaced by

*P*(or by

_{F}*P*Then is it the case that counterparts are obtained to the two-stage aggregation result for the Laspeyres formula, equation (17.65)? The answer is no; it can be shown that, in general,

_{T})-Similarly, it can be shown that the quadratic mean of order *r* index number formula *P ^{r}* defined by equation (17.35) and the implicit quadratic mean of order r index number formula

*P*defined by equation (17.32) are also not consistent in aggregation.

^{r}***17.58** Nevertheless, even though the Fisher and Törnqvist formulae are not exactly consistent in aggregation, it can be shown that these formulae are approximately consistent in aggregation. More specifically, it can be shown that the two-stage Fisher formula P^{*}_{F} and the single-stage Fisher formula *P _{F}* in the inequality (17.66), both regarded as functions of the 4

*n*variables in the vectors

*p*, approximate each other to the second order around a point where the two price vectors are equal (so that

^{0}, p^{1}, q^{0}, q^{1}*p*and where the two quantity vectors are equal (so that

^{0}=p^{1})*q*,and a similar result holds for the two-stage and single-stage Törnqvist indices in equation (17.66).

^{0}= q^{1})^{40}As was seen in the previous section, the single-stage Fisher and Törnqvist indices have a similar approximation property, so all four indices in the inequality (17.66) approximate each other to the second order around an equal (or proportional) price and quantity point. Thus for normal time series data, single-stage and two-stage Fisher and Törnqvist indices will usually be numerically very close. This result is illustrated in Chapter 19 for an artificial data set.

^{41}

**17.59** Similar approximate consistency in aggregation results (to the results for the Fisher and Törnqvist formulae explained in the previous paragraph) can be derived for the quadratic mean of order *r* indices, *P ^{r}*, and for the implicit quadratic mean of order

*r*indices,

*P*; see Diewert (1978, p. 889). Nevertheless, the results of Hill (2002) again imply that the second-order approximation property of the single-stage quadratic mean of order

^{r*}*r*index

*P*to its two-stage counterpart will break down as

^{r}*r*approaches either plus or minus infinity. To see this, consider a simple example where there are only four commodities in total. Let the first price ratio

*b*and let the last price ratio

*c*, where we assume

*a ≤ b and ≤ c*. Using Hill’s result (17.57), the limiting value of the single-stage index is:

Now aggregate commodities 1 and 2 into a sub-aggregate and commodities 3 and 4 into another sub-aggregate. Using Hill’s result (17.57) again, it is found that the limiting price index for the first sub-aggregate is [*ab*]^{½} and the limiting price index for the second sub-aggregate is [*bc*]^{½}. Now apply the second stage of aggregation and use Hill’s result once again to conclude that the limiting value of the two-stage aggregation using *P ^{r}* as the index number formula is [

*ab*]

^{2}c^{¼}. Thus the limiting value as

*r*tends to plus or minus infinity of the single-stage aggregate over the two-stage aggregate is

*b*can take on any value between

*a*and

*c*, and so the ratio of the single-stage limiting

*P*to its two-stage’counterpart can take on any value between [

^{r}*c/a*]

^{¼}and

*[*

*a/c*]^{¼}. Since

*c/a*is less than 1 and

*a/c*is greater than 1, it can be seen that the ratio of the single-stage to the two-stage index can be arbitrarily far from 1 as

*r*becomes large in magnitude with an appropriate choice of the numbers

*a, b*and

*c*.

**17.60** The results in the previous paragraph show that some caution is required in assuming that *all* superlative indices will be approximately consistent in aggregation. However, for the three most commonly used superlative indices (the Fisher ideal *P _{F}*, the Törnqvist Theil

*P*and the Walsh

_{T}*P*, the available empirical evidence indicates that these indices satisfy the consistency in aggregation property to a sufficiently high degree of approximation that users will not be unduly troubled by any inconsistencies.

_{w})^{42}

## The Lloyd-Moulton index number formula

**17.61** The index number formula that will be discussed in this section on the single-household economic approach to index number theory is a potentially very useful one for statistical agencies that are faced with the problem of producing a CPI in a timely manner. The Lloyd-Moulton formula that will be discussed in this section makes use of the same information that is required in order to implement a Laspeyres index except for one additional piece of information.

**17.62** In this section, the same assumptions about the consumer are made that were made in paragraphs 17.18 to 17.26 above. In particular, it is assumed that the consumer’s utility function f*(q)* is linearly homogeneous^{43} and the corresponding unit cost function is *c(p)*. It is supposed that the unit cost function has the following functional form:

where the α_{i}, and σ are non-negative parameters with ^{44} The parameter σ is the elasticity of substitution; when σ = 0, the unit cost function defined by equation (17.68) becomes linear in prices and hence corresponds to a fixed coefficients aggregator function which exhibits zero substitutability between all commodities. When σ = 1, the corresponding aggregator or utility function is a Cobb-Douglas function. When σ approaches plus infinity, the corresponding aggregator function f approaches a linear aggregator function which exhibits infinite substitutability between each pair of inputs. The CES unit cost function defined by equation (17.68) is not a fully flexible functional form (unless the number of commodities *n* being aggregated is 2), but it is considerably more flexible than the zero substitutability aggregator function (this is the special case of equation (17.68) where σ is set equal to zero) that is exact for the Laspeyres and Paasche price indices.

**17.63** Under the assumption of cost minimizing behaviour in period 0, Shephard’s Lemma (17.12), tells us that the observed first period consumption of commodity *i,q ^{0}_{i}* will be equal to

^{0}

*=f(q*

^{0}

*) is*the aggregate (unobservable) level of period 0 utility. Using the CES functional form defined by equation (17.68) and assuming that σ ≠ l, the following equations are obtained:

These equations can be rewritten as:

where r ≡ 1 -σ. Now consider the following Lloyd (1975) and Moulton (1996a) index number formula:

where s^{0}_{t} is the period 0 expenditure share of commodity *i*, as usual:

If equation (17.72) is substituted into equation (17.71), it is found that:

**17.64** Equation (17.73) shows that the Lloyd-Moulton index number formula *P _{LM}* is exact for CES preferences. Lloyd (1975) and Moulton (1996a) independently derived this result, but it was Moulton who appreciated the significance of the formula (17.71) for statistical agency purposes. Note that in order to evaluate formula (17.71) numerically, it is necessary to have information on:

base period expenditure shares s

^{0}_{t};the price relatives

${p}_{i}^{1}/{p}_{i}^{0}$ between the base period and the current period; andan estimate of the elasticity of substitution between the commodities in the aggregate, σ

The first two pieces of information are the standard information sets that statistical agencies use to evaluate the Laspeyres price index *P _{L}* (note that

*P*reduces to

_{LM}*P*if σ = 0). Hence, if the statistical agency is able to estimate the elasticity of substitution σ based on past experience,

_{L}^{45}then the Lloyd-Moulton price index can be evaluated using essentially the same information set that is used in order to evaluate the traditional Laspeyres index. Moreover, the resulting CPI will be free of substitution bias to a reasonable degree of approximation.

^{46}Of course, the practical problem with implementing this methodology is that estimates of the elasticity of substitution parameter σ are bound to be somewhat uncertain, and hence the resulting Lloyd-Moulton index may be subject to charges that it is not objective or reproducible. The statistical agency will have to balance the benefits of reducing substitution bias with these possible costs.

## Annual preferences and monthly prices

**17.65** Recall the definition of the Lowe index, *P _{L0}(p^{0}*,

*p*,q), defined by equation (15.15) in Chapter 15. In paragraphs 15.33 to 15.64 of Chapter 15, it is noted that this formula is frequently used by statistical agencies as a target index for a CPI. It is also noted that, while the price vectors

^{1}*p*(the base period price vector) and

^{0}*p*(the current period price vector) are monthly or quarterly price vectors, the quantity vector

^{1}*q*≡

*(q*,

_{1}, q_{2},…*q*which appears in this basket-type formula is usually taken to be an annual quantity vector that refers to a base year,

_{n})*b*say, that is prior to the base period for the prices, month 0. Thus, typically, a statistical agency will produce a CPI at a monthly frequency that has the form

*P*, where

_{L0}(p^{0},p^{t},q^{b})*p*is the price vector pertaining to the base period month for prices, month 0,

^{0}*p*is the price vector pertaining to the current period month for prices, month

^{t}*t*say, and

*q*is a reference basket quantity vector that refers to the base year

^{b}*b*, which is equal to or prior to month 0.

^{47}The question to be addressed in the present section is: can this index be related to one based on the economic approach to index number theory?

### The Lowe index as an approximation to a true cost of living index

**17.66** Assume that the consumer has preferences defined over consumption vectors *q* ≡*[q _{1},…,q_{n}]* that can be represented by the continuous increasing utility function

*f(q)*. Thus if

*f(q*, then the consumer prefers the consumption vector

^{1})>f(q^{0})*q*to

^{1}*q*. Let

^{0}*q*be the annual consumption vector for the consumer in the base year

^{b}*b*. Define the base year utility level

*u*as the utility level that corresponds to

^{b}*f(q)*evaluated at

*q*

^{b}:

**17.67** For any vector of positive commodity prices p≡ *[p _{1},…,p_{n}]* and for any feasible utility level u, the consumer’s cost function,

*C(u,p)*, can be defined in the usual way as the minimum expenditure required to achieve the utility level u when facing the prices

*p*:

Let *b*. Assume that the observed base year consumption vector

The cost function will be used below in order to define the consumer’s cost of living price index.

**17.68** Let p^{0} and p^{t} be the monthly price vectors that the consumer faces in months 0 and t. Then the Konüs true cost of living index, *P _{K}(p^{Q},p^{t},q^{b})*, between months 0 and t, using the base year utility level

*u*as the reference standard of living, is defined as the following ratio of minimum monthly costs of achieving the utility level

^{b}—f(q^{b})*u*

^{b}:

**17.69** Using the definition of the monthly cost minimization problem that corresponds to the cost *C(f(q ^{b}),p^{t})*, it can be seen that the following inequality holds:

since the base year quantity vector *q ^{b}* is feasible for the cost minimization problem. Similarly, using the definition of the monthly cost minimization problem that corresponds to the month 0 cost

*C(f(q*, it can be seen that the following inequality holds:

^{b}),p^{Q})since the base year quantity vector *q ^{b}* is feasible for the cost minimization problem.

**17.70** It will prove useful to rewrite the two inequalities (17.78) and (17.79) as equalities. This can be done if non-negative substitution bias terms, *e ^{t}* and

*e*, are subtracted from the right-hand sides of these two inequalities. Thus the inequalities (17.78) and (17.79) can be rewritten as follows:

^{Q}**17.71** Using equations (17.80) and (17.81), and the definition (15.15) in Chapter 15 of the Lowe index, the following approximate equality for the Lowe index results:

Thus if the non-negative substitution bias terms *e ^{Q}* and e

^{t}are small, then the Lowe index between months 0 and t,

*P*, will be an adequate approximation to the true cost of living index between months 0 and t,

_{Lo}(p^{Q},p^{t},q^{b})*P*(p

_{k}^{Q},p

^{t},q

^{b}).

**17.72** A bit of algebraic manipulation shows that the Lowe index will be exactly equal to its cost of living counterpart if the substitution bias terms satisfy the following relationship:^{48}.

Equations (17.82) and (17.83) can be interpreted as follows: if the rate of growth in the amount of substitution bias between months 0 and t is equal lo the rate of growth in the minimum cost of achieving the base year utility level *u ^{b}* between months 0 and t, then the observable Lowe index,

*P*be exactly equal to its true cost of living index counterpart,

_{L0}(P^{0},p^{t},q^{b}), will*P*Í7.73 It is difficult to know whether condition (17.83) will hold or whether the substitution bias terms

_{K}(p^{Q},p^{t},q^{b}).^{49}*e*and e

^{Q}^{t}will be small. Thus, first-order and second-order Taylor series approximations to these substitution bias terms are developed in paragraphs 17.74 to 17.83.

### A first-order approximation to the bias of the Lowe index

**17.74** The true cost of living index between months 0 and t, using the base year utility level *u ^{b}* as the reference utility level, is the ratio of two unobservable costs,

^{b},

*p*) around the annual base year price vector

^{t}*p*is given by the following approximate equation:

^{b}^{50}

Similarly, the first-order Taylor series approximation to *C(u ^{b}, p^{Q})* around the annual base year price vector

*p*is given by the following approximate equation:

^{b}**17.75** Comparing approximate equation (17.84) with equation (17.80), and comparing approximate equation (17.85) with equation (17.81), it can be seen that, to the accuracy of the first-order approximations used in (17.84) and (17.85), the substitution bias terms *e ^{t}* and

*e*will be zero. Using these results to reinterpret the approximate equation (17.82), it can be seen that if the month 0 and month

^{Q}*t*price vectors,

*p*and

^{0}*p*, are not too different from the base year vector of prices

^{t}*p*, then the Lowe index

^{b}*P*will approximate the true cost of living index

_{LO}(p^{Q},p^{t},q^{b})*P*to the accuracy of a first-order approximation. This result is quite useful, since it indicates that if the monthly price vectors

_{K}(p^{0},p^{t},q^{b})*p*and

^{Q}*p*are just randomly fluctuating around the base year prices

^{t}*p*(with modest variances), then the Lowe index will serve as an adequate approximation to a theoretical cost of living index. However, if there are systematic long-term trends in prices and month t is fairly distant from month 0 (or the end of year

^{b}*b*is quite distant from month 0), then the first-order approximations given by approximate equations (17.84) and (17.85) may no longer be adequate and the Lowe index may have a considerable bias relative to its cost of living counterpart. The hypothesis of long-run trends in prices will be explored in paragraphs 17.76 to 17.83.

### A second-order approximation to the substitution bias of the Lowe index

**17.76** A second-order Taylor series approximation to *C(u ^{b},p^{t})* around the base year price vector

*p*is given by the following approximate equation:

^{b}where the last equality follows using approximate equation (17.84).^{51} Similarly, a second-order Taylor series approximation to C(u^{b},*p ^{Q}*) around the base year price vector

*p*is given by the following approximate equation:

^{b}where the last equality follows using the approximate equation (17.85).

**17.77** Comparing approximate equation (17.86) with equation (17.80), and approximate equation (17.87) with equation (17.81), it can be seen that, to the accuracy of a second-order approximation, the month 0 and month t substitution bias terms, e^{0} and *e*^{1}, will be equal to the following expressions involving the second-order partial derivatives of the consumer’s cost function *u ^{h}* and at the base year prices

*p*:

^{h}Since the consumer’s cost function *C(u,p)* is a concave function in the components of the price vector *p, ^{52}* it is known

^{53}that the

*n by n*(symmetric) matrix of second-order partial derivatives

^{54}Hence, for arbitrary price vectors

*p*and

^{h}, p^{0}*p*, the right-hand sides of approximations (17.88) and (17.89) will be non-negative. Thus, to the accuracy of a second-order approximation, the substitution bias terms

^{l}*e*and e

^{0}^{1}will be non-negative.

**17.78** Now assume that there are long-run systematic trends in prices. Assume that the last month of the base year for quantities occurs *M* months prior to month 0, the base month for prices, and assume that prices trend linearly with time, starting with the last month of the base year for quantities. Thus, assume the existence of constants α_{j} for *j=l,…,n* such that the price of commodity *j* in month *t* is given by:

Substituting equation (17.90) into approximations (17.88) and (17.89) leads to the following second-order approximations to the two substitution bias terms, e^{0} and *e*^{t}:^{55}

where γ is defined as follows:

**17.79** It should be noted that the parameter γ will be zero under two sets of conditions:^{56}

All the second-order partial derivatives of the consumer’s cost function P

_{k}(p^{0},p^{t},p^{b}) are equal to zero.Each commodity price change parameter α

_{j}is proportional to the corresponding commodity*j*base year price*p*.^{b}_{j}^{57}

The first condition is empirically unlikely since it implies that the consumer will not substitute away from commodities of which the relative price has increased. The second condition is also empirically unlikely, since it implies that the structure of relative prices remains unchanged over time. Thus, in what follows, it will be assumed that γ is a positive number.

**17.80** In order to simplify the notation in what follows, define the denominator and numerator of the month *t* Lowe index, *P _{Lo}(p^{0},p^{t},p^{b})* as

*a*and

*b*respectively; i.e., define:

Using equation (17.90) to eliminate the month 0 prices *P _{i}^{0}* from equation (17.94) and the month

*t*prices

*P*from equation (17.95) leads to the following expressions for

_{i}^{t}*a*and

*b*:

It is assumed that *a* and b^{58} are positive and that

Assumption (17.98) rules out a general deflation in prices.

**17.81** Define the bias in the month *t* Lowe index, B^{t}, as the difference between the true cost of living index *P _{k}(p^{0},p^{t},q^{b})* defined by equation (17.77) and the

Thus, for t ≥ 1, the Lowe index will have an upward bias (to the accuracy of a second-order Taylor series approximation) compared to the corresponding true cost of living index ^{59}

**17.82** In order to give the reader some idea of the magnitude of the approximate bias *B ^{t}* defined by the last line of equation (17.99), a simple special case will be considered at this point. Suppose there are only two commodities and that, at the base year, all prices and quantities are equal to 1. Thus,

*M*= 24 so that the base year data on quantities take two years to process before the Lowe index can be implemented. Assume that the monthly rate of growth in price for commodity 1 is α

_{1}=0.002 so that after one year, the price of commodity 1 rises 0.024 or 2.4 per cent. Assume that commodity 2 falls in price each month with α

_{2}=–0.002 so that the price of commodity 2 falls 2.4 per cent in the first year after the base year for quantities. Thus the relative price of the two commodities is steadily diverging by about 5 per cent per year. Finally, assume that

Substituting the parameter values defined in equation (17.100) into equation (17.99) leads to the following formula for the approximate amount that the Lowe index will exceed the corresponding true cost of living index at month t:

Evaluating equation (17.101) at t=12, t = 24, t = 36, t = 48 and t = 60 leads to the following estimates for - *B ^{t}*: 0.0029 (the approximate bias in the Lowe index at the end of the first year of operation for the index); 0.0069 (the bias after two years); 0.0121 (the bias after three years); 0.0185 (the bias after four years); 0.0260 (the bias after five years). Thus, at the end of the first year of the operation, the Lowe index will only be above the corresponding true cost of living index by approximately a third of a percentage point but, by the end of the fifth year of operation, it will exceed the corresponding cost of living index by about 2.6 percentage points, which is no longer a negligible amount.

^{60}

**17.83** The numerical results in the previous paragraph are only indicative of the approximate magnitude of the difference between a cost of living index and the corresponding Lowe index. The important point to note is that, to the accuracy of a second-order approximation, the Lowe index will generally exceed its cost of living counterpart. The results also indicate, however, that this difference can be reduced to a negligible amount if:

the lag in obtaining the base year quantity weights is minimized; and

the base year is changed as frequently as possible.

It should also be noted that the numerical results depend on the assumption that long-run trends in prices exist, which may not be true,^{61} and on elasticity assumptions that may not be justified.^{62} Statistical agencies should prepare their own carefully constructed estimates of the differences between a Lowe index and a cost of living index in the light of their own particular circumstances.

### The problem of seasonal commodities

**17.84** The assumption that the consumer has annual preferences over commodities purchased in the base year for the quantity weights, and that these annual preferences can be used in the context of making monthly purchases of the same commodities, was a key one in relating the economic approach to index number theory to the Lowe index. This assumption that annual preferences can be used in a monthly context is, however, somewhat questionable because of the seasonal nature of some commodity purchases. The problem is that it is very likely that consumers’ preference functions systematically change as the season of the year changes. National customs and weather changes cause households to purchase certain goods and services during some months and not at all for other months. For example, Christmas trees are purchased only in December and ski jackets are not usually purchased during summer months. Thus, the assumption that annual preferences are applicable during each month of the year is only acceptable as a very rough approximation to economic reality.

**17.85** The economic approach to index number theory can be adapted to deal with seasonal preferences. The simplest economic approach is to assume that the consumer has annual preferences over commodities classified not only by their characteristics but also by the month of purchase.^{63} Thus, instead of assuming that the consumer’s annual utility function is *f(q)* where *q ^{1}* is an n-dimensional vector of commodity purchases made in January,

*q*is an n-dimensional vector of commodity purchases made in February, …, and

^{2}*q*is an

^{12}*n-*dimensional vector of commodity purchases made in December.

^{64}The sub-utility functions

*f*, f

^{1},f^{2}…^{12}represent the consumer’s preferences when making purchases in January, February,…., and December, respectively. These monthly sub-utilities can then be aggregated using the macro-utility function

*F*in order to define overall annual utility. It can be seen that these assumptions on preferences can be used to justify two types of cost of living index:

an annual cost of living index that compares the prices in all months of a current year with the corresponding monthly prices in a base year;

^{65}and12 monthly cost of living indices where the index for month

*m*compares the prices of month*m*in the current year with the prices of month*m*in the base year for*m*=1,2,…12.^{66}

**17.86** The annual Mudgett Stone indices compare costs in a current calendar year with the corresponding costs in a base year. However, any month could be chosen as the year-ending month of the current year, and the prices and quantities of this new non-calendar year could be compared to the prices and quantities of the base year, where the January prices of the non-calendar year are matched to the January prices of the base year, the February prices of the non-calendar year are matched to the February prices of the base year, and so on. If further assumptions are made on the macro-utility function *F*, then this framework can be used in order to justify a third type of cost of living index: a moving year annual index.^{67} This index compares the cost over the past 12 months of achieving the annual utility achieved in the base year with the base year cost, where the January costs in the current moving year are matched to January costs in the base year, the February costs in the current moving year are matched to February costs in the base year, and so on. These moving year indices can be calculated for each month of the current year and the resulting series can be interpreted as (uncentred) seasonally adjusted (annual) price indices.^{68}

**17.87** It should be noted that none of the three types of indices described in the previous two paragraphs is suitable for describing the movements of prices going from one month to the following month; i.e., they are not suitable for describing short-run movements in inflation. This is obvious for the first two types of index. To see the problem with the moving year indices, consider a special case where the bundle of commodities purchased in each month is entirely specific to each month. Then it is obvious that, even though all the above three types of index are well defined, none of them can describe anything useful about month-to-month changes in prices, since it is impossible to compare like with like, going from one month to the next, under the hypotheses of this special case. It is impossible to compare the incomparable.

**17.88** Fortunately, it is not the case that household purchases in each month are entirely specific to the month of purchase. Thus month-to-month price comparisons can be made if the commodity space is restricted to commodities that are purchased in each month of the year. This observation leads to a fourth type of cost of living index, a month-to-month index, defined over commodities that are available in every month of the year.^{69} This model can be used to justify the economic approach described in paragraphs 17.66 to 17.83. Commodities that are purchased only in certain months of the year, however, must be dropped from the scope of the index. Unfortunately, it is likely that consumers have varying monthly preferences over the commodities that are always available and, if this is the case, the month-to-month cost of living index (and the corresponding Lowe index) defined over always-available commodities will generally be subject to seasonal fluctuations. This will limit the usefulness of the index as a short-run indicator of general inflation since it will be difficult to distinguish a seasonal movement in the index from a systematic general movement in prices.^{70} Note also that if the scope of the index is restricted to always-available commodities, then the resulting month-to-month index will not be comprehensive, whereas the moving year indices will be comprehensive in the sense of using all the available price information.

**17.89** The above considerations lead to the conclusion that it may be useful for statistical agencies to produce at least two consumer price indices:

a moving year index which is comprehensive and seasonally adjusted, but which is not necessarily useful for indicating month-to-month changes in general inflation; and

a month-to-month index which is restricted to non-seasonal commodities (and hence is not comprehensive), but which is useful for indicating short-run movements in general inflation.

## The problem of a zero price increasing to a positive price

**17.90** In a recent paper, Haschka (2003) raised the problem of what to do when a price which was previously zero is increased to a positive level. He gave two examples for Austria, where parking and hospital fees were raised from zero to a positive level. In this situation, it turns out that basket-type indices have an advantage over indices that are weighted geometric averages of price relatives, since basket-type indices are well defined even if some prices are zero.

**17.91** The problem can be considered in the context of evaluating the Laspeyres and Paasche indices. Suppose as usual that the prices *p ^{t}_{i}* and quantities

*q*of the first n commodities are positive for periods 0 and 1, but that the price of commodity

^{t}_{i}*n +*1 in period 0 is zero but is positive in period 1. In both periods, the consumption of commodity

*n+*1 is positive. Thus the assumptions on the prices and quantities of commodity

*n+*1 in the two periods under consideration can be summarized as follows:

Typically, the increase in price of commodity *n +* 1 from its initial non-zero level will cause consumption to fall so that

**17.92** Let the Laspeyres index between periods 0 and 1, restricted to the first *n* commodities, be denoted as *p ^{n}_{L}* and let the Laspeyres index, defined over all

*n*+ 1 commodities, be defined as

*i*in period 0. Then by the definition of the Laspeyres index defined over all

*n*+ 1 commodities:

where *n + 1* commodities is equal to the incomplete Laspeyres index *P ^{n}_{L}* (which can be written in traditional price relative and base period expenditure share form), plus the mixed or hybrid expenditure

*n+*1 in period

*n+*1 in period 0 (when it was free),

**17.93** Let the Paasche index between periods 0 and 1, restricted to the first *n* commodities, be denoted as * ^{n}_{p}* and let the Paasche index, defined over all n+1 commodities, be defined as

*i*in period 1. Then, by the definition of the Paasche index defined over all

*n +*1 commodities:

where *p ^{n}_{p}* (which can be written in traditional price relative and current period expenditure share form), plus the current period expenditure on commodity

*v*, divided by the ith price relative for the first

^{I}_{i}*n*commodities,

**17.94** Once the complete Laspeyres and Paasche indices have been calculated using equations (17.103) and (17.104), then the complete Fisher index can be calculated as the square root of the product of these two indices:

It should be noted that the complete Fisher index defined by equation (17.105) satisfies the same exact index number results as were demonstrated in paragraphs 17.27 to 17.32 above; i.e., the Fisher index remains a superlative index even if prices are zero in one period but positive in the other. Thus the Fisher price index remains a suitable target index even in the face of zero prices.

For a description of the economic theory of the input and output price indices, see Balk (1998a). In the economic theory of the output price index, *q ^{t}* is assumed to be the solution to a revenue maximization problem involving the output price vector p

^{t}.

In this chapter, these preferences are assumed to be invariant over time, while in the following chapter, this assumption is relaxed (one of the environmental variables could be a time variable that shifts tastes).

Note that *f* is concave if and only if *q≥0 _{N}* means that each component of the N-dimensional vector

*q*is non-negative,

*q »*0

_{N}means that each component of

*q*is positive and

*q>0*means that

_{n}*q ≥*0

_{n}but q≠0

_{n}; i.e.,

*q*is non-negative but at least one component is positive.

This inequality was first obtained by Konüs (1924; 1939, p. 17). See also Pollak (1983).

This inequality is attributable to Konus (1924; 1939, p. 19); see also Pollak (1983).

For more recent applications of the Konüs method of proof, see Diewert (1983a, p. 191) for an application to the consumer context and Diewert (1983b, pp, 1059–1061) for an application to the producer context.

The linear homogeneity property means that *f* satisfies the following condition: *q >>* 0_{n}. This assumption is fairly restrictive in the consumer context. It implies that each indifference curve is a radial projection of the unit utility indifference curve. It also implies that all income elasticities of demand are unity, which is contradicted by empirical evidence.

More precisely, Shephard (1953) defined a homothetic function to be a monotonic transformation of a linearly homogeneous function. However, if a consumer’s utility function is homothetic, it can always be rescaled to be linearly homogeneous without changing consumer behaviour. Hence, the homothetic preferences assumption can simply be identified with the linear homogeneity assumption.

This particular branch of the economic approach to index number theory is attributable to Shephard (1953; 1970) and Samuelson and Swamy (1974). Shephard in particular realized the importance of the homotheticity assumption in conjunction with separability assumptions in justifying the existence of sub-indices of the overall cost of living index. It should be noted that, if the consumer’s change in real income or utility between the two periods under consideration is not too large, then assuming that the consumer has homothetic preferences will lead to a true cost of living index which is very close to Laspeyres-Konüs and Paasche-Konus true cost of living indices defined by equations (17.3) and (17.4). Another way of justifying the homothetic preferences assumption is to use equation (17.49), which justifies the use of the superlative Törnqvist-Theil index *P _{T}* in the context of non-homothetic preferences. Since

*P*is usually numerically close to other superlative indices that are derived using the homothetic preferences assumption, it can be seen that the assumption of homotheticity will usually not be empirically misleading in the index number context.

_{T}Economists will recognize the producer theory counterpart to the result *C*(*u,p*) = *uc*(*p*)): if a producer’s production function *f* is subject to constant returns to scale, then the corresponding total cost function *C*(*u,p*) is equal to the product of the output level *u* times the unit cost *c(p)*.

Obviously, the utility function *f* determines the consumer’s cost function *C(u,p)* as the solution to the cost minimization problem in the first line of equation (17.6). Then the unit cost function *c(p)* is defined as *C(1,p)*. Thus f determines *c*. But we can also use *c* to determine funder appropriate regularity conditions. In the economics literature, this is known as *duality theory*. For additional material on duality theory and the properties of *f* and *c*, see Samuelson (1953), Shephard (1953) and Diewert (1974a; 1993b, pp.107–123).

There is also a producer theory interpretation of the above theory; i.e., let *f* be the producer’s (constant returns to scale) production function, let *p* be a vector of input prices that the producer faces, let *q* be an input vector and let *u=f(q)* be the maximum output that can be produced using the input vector *q*. *c(p ^{t})* can be identified as the period

*t*input price level, while

*f(q*is the period

^{t})*t*aggregate input level.

To prove this, consider the first-order necessary conditions for the strictly positive vector *q* to solve the period *t* cost minimization problem. The conditions of Lagrange with respect to the vector of *q* variables are: *t* quantity vector *q* and solve the resulting equation for X’. Substitute this solution back into the vector equation

Differentiate both sides of the equation *f(*λ*.q) =* λf*(q)* with respect to λ, and then evaluate the resulting equation at λ=1, The equation

For the early history of this result, see Diewert (1976, p. 184).

See Diewert (1976, p. 130) and let the parameter *r* equal 2.

Diewert (1974a, p. 133) introduced this term into the economics literature.

Fisher (1922, p. 247) used the term superlative to describe the Fisher ideal price index. Thus, Diewert adopted Fisher’s terminology but attempted to give some precision to Fisher’s definition of superlativeness. Fisher defined an index number formula to be superlative if it approximated the corresponding Fisher ideal results using his data set.

Given the consumer’s unit cost function *c*(p), Diewert (1974a, p. 112) showed that the corresponding utility function *f(q)* can be defined as follows: for a strictly positive quantity vector *q*,

This result was obtained by Diewert (1976, pp. 133–134).

Note that it has been shown that the Fisher index *P _{F}* is exact for the preferences defined by equation (17.17), as well as the preferences that are dual to the unit cost function defined by equation (17.22). These two classes of preferences do not coincide in general. However, if the

*n*by

*n*symmetric matrix

*A*of the

*a*has an inverse, then it can be shown that the

_{ik}*n*by

*n*matrix

*A*of the

*b*will equal A

_{ik}^{-1}.

The terminology is attributable to Diewert (1976, p. 129).

See Diewert (1976, p. 130).

This terminology is attributable to Diewert (1976, p. 130), this unit cost function being first defined by Denny (1974).

The function *f*^{r*} can be defined by using c^{r} as follows:

The proof of this and the foregoing relation is by straightforward verification.

The consumer’s cost function was defined by equation (17.6) above.

Christensen, Jorgcnson and Lau (1971) introduced this function into the economics literature.

It can also be shown that, if all the *b _{i}*=0 and

*b*, then

_{00}=0*C(u,p)*C

*uC(1,p)*≡

*uc(p);*i.e., with these additional restrictions on the parameters of the general translog cost function, homothetic preferences are the result of these restrictions. Note that it is also assumed that utility

*u*is scaled so that

*u*is always positive.

This result is attributable to Diewert (1976, p. 122).

See Diewert (2002a).

If, however, consumer preferences are non-homothetic and the change in utility is substantial between the two situations being compared, then it may be desirable to compute separately the LaspeyresKonüs and Paasche-Konüs true cost of living indices defined by equations (17.3) and (17.4), *C(u ^{0},C^{1})/C(u^{0},C^{0})* and

*C*, respectively. In order to do this, it would be necessary to use econometrics and estimate empirically the consumer’s cost or expenditure function.

**(u**^{1},p^{1})/C(u^{1},p^{0})To prove the equalities in equations (17.51) to (17.56), simply differentiate the various index number formulae and evaluate the derivatives at p^{0}= p^{1} and q^{0}=q^{1}. Actually, equations (17.51) to (17.56) are still true provided that *p ^{1} =* p

*p*and

^{0}*q*

^{1}*=*µ

*q*for any numbers λ >0 and µ > 0; i.e., provided that the period 1 price vector is proportional to the period 0 price vector and that the period 1 quantity vector is proportional to the period 0 quantity vector.

^{0}Hill (2002) documents this for two data sets. His time series data consist of annual expenditure and quantity data for 64 components of United States gross domestic product from 1977 to 1994. For this data set, Hill (2002, p. 16) found that “superlative indexes can differ by more than a factor of two (i.e., by more than 100 per cent), even though Fisher and Törnqvist never differ by more than 0.6 per cent”.

Diewert (1980, p. 451) showed that the Törnqvist index *P _{T}* is a limiting case of

*P*as

^{r}*r*tends to 0.

See, for example, Diewert (1978, p. 894) or Fisher (1922), which is reproduced in Diewert (1976, p. 135).

Much of the material in this section is adapted from Diewert (1978) and Alterman, Diewert and Feenstra (1999). See also Balk (1996b) for a discussion of alternative definitions for the two-stage aggregation concept and references to the literature on this topic.

See Diewert (1978, p. 889). In other words, a string of equalities similar to equations (17.51) to (17.56) holds between the two-stage indices and their single-stage counterparts. In fact, these equalities are still true provided that p^{1} =λ*p ^{0}* and

*q*µq

^{1}=^{0}for any numbers µ>0 and µ>0.

For an empirical comparison of the four indices, see Diewert (1978,pp. 894–895). For the Canadian consumer data considered there, the chained two-stage Fisher in 1971 was 2.3228 and the corresponding chained two-stage Törnqvist was 2.3230, the same values as for the corresponding single-stage indices.

See Chapter 19 for some additional evidence on this topic.

Thus homothetic preferences are assumed in this section.

In the mathematics literature, this aggregator function or utility function is known as a mean of order *r*, see Hardy, Littlewood and Pólya (1934, pp. 12–13).

For the first application of this methodology (in the context of the CPI), see Shapiro and Wilcox (1997a, pp. 121-123). They calculated superlative Törnqvist indices for the United States for the years 1986—95 and then calculated the Lloyd-Moulton CES index for the same period, using various values of *σ*. They then chose the value of (which was 0.7), which caused the CES index to most closely approximate the Törnqvist index. Essentially the same methodology was used by Alterman, Diewert and Feenstra (1999) in their study of United Slates import and export price indices. For alternative methods for estimating *σ*, see Balk (2000b).

What is a “reasonable” degree of approximation depends on the context. Assuming that consumers have CES preferences is not a reasonable assumption in the context of estimating elasticities of demand: at least a second-order approximation to the consumer’s preferences is required in this context. In the context of approximating changes in a consumer’s expenditures on the *n* commodities under consideration, however, it is usually adequate to assume a CES approximation.

As noted in Chapter 15, month 0 is called the price reference period and year *b* is called the weight reference period.

This assumes that *e ^{0}* is greater than zero. If

*e*is equal to zero, then to have equality of

^{Q}*P*

_{K}and

*P*, it must be the case that

_{L0}*e*is also equal to zero.

^{t}It can be seen that, when month *t* is set equal to month 0, *e ^{t} = e^{Q}* and

*C(u*C(

^{b},p^{t}) =*u*,

^{b}*p*), and thus equation (17.83) is satisfied and

^{Q}*P*. This is not surprising since both indices are equal to unity when

_{L0}=P_{k}*t*= 0.

This type of Taylor series approximation was used in Schultze and Mackie (2002, p, 91) in the cost of living index context, but it essentially dates back to Hicks (1941-42, p, 134) in the consumer surplus context. See also Diewert (1992b, p. 568) and Hausman (2002, p. 8).

This type of second-order approximation is attributable to Hicks (1941-42, pp. 133-134;1946, p. 331). See also Diewert (1992b, p. 568), Hausman (2002, p. 18) and Schultze and Mackie (2002. p. 91). For alternative approaches to modelling substitution bias, see Diewert (1998a;2002c, pp. 598-603) and Hausman (2002).

See Diewert (1993b, p. 149).

A symmetric *n* by n matrix *A* with *ij*th element equal to *a, _{ij}* is negative semi-definite if, and only if for every vector

Note that the period 0 approximate bias defined by the right-hand side of approximation (17.91) is fixed, while the period t approximate bias defined by the right-hand side of (17.92) increases quadratically with time *t*. Hence, the period t approximate bias term will eventually overwhelm the period 0 approximate bias in this linear time trends case, if t is allowed to become large enough.

A more general condition that ensures the positivity of γ is that the vector γ is not an eigenvector of the matrix of second-order partial derivatives

It is known that *C*(u.p) is linearly homogeneous in the components of the price vector *p;* see Diewert (1993b, p. 109) for example. Hence, using Euler’s Theorem on homogeneous functions, it can be shown that *p ^{h}* is an eigenvector of the matrix of second-order partial derivatives

It is also assumed that *a-γ ^{2}* is positive.

If *M* is large relative to *r*, then it can be seen that the first two terms

Note that the relatively large magnitude of *M* compared to *t* leads to a bias that grows approximately linearly with *t* rather than quadratically.

For mathematical convenience, the trends in prices were assumed to be linear, rather than the more natural assumption of geometric trends in prices.

Another key assumption that was used to derive the numerical results is the magnitude of the divergent trends in prices. If the price divergence vector is doubled to α_{1} = 0.004 and α_{2} = -0.004, then the parameter *γ* quadruples and the approximate bias will also quadruple.

This assumption and the resulting annual indices were first proposed by Mudgett (1955, p. 97) and Stone (1956, pp. 74-75).

If some commodities are not available in certain months *m*, then those commodities can be dropped from the corresponding monthly quantity vectors *q ^{m}*.

For further details on how to implement this framework, see Mudgett (1955. p. 97), Stone (1956, pp. 74-75) and Diewert (1998b, pp. 459–460).

For further details on how to implement this framework, see Diewert (1999a, pp. 50-51).

See Diewert (1999a, pp. 56-61) for the details of this economic approach.

See Diewert (1999a, pp. 67-68) for an empirical example of this approach applied to quantity indices. An empirical example of this moving year approach to price indices is presented in Chapter 22.

See Diewert (1999a, pp. 51-56) for the assumptions on preferences that are required in order to justify this economic approach.

One problem with using annual weights in the context of seasonal movements in prices and quantities is that a change in price when a commodity is out of season can be greatly magnified by the use of annual weights. Baldwin (1990, p. 251) noted this problem with an annual weights price index: “But a price index is adversely affected if any seasonal good has the same basket share for all months of the year; the good will have an inappropriately small basket share in its in season months, an inappropriately large share in its off season months.” Seasonality problems are considered again from a more pragmatic point of view in Chapter 22.