Chapter

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

Author(s):
International Monetary Fund
Published Date:
August 2004
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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 pt that the household faces in a given time period t, it is assumed that the corresponding observed quantity vector qt is the solution to a cost minimization problem that involves the consumer’s preference or utility function f 1. Thus in contrast to the axiomatic approach to index number theory, the economic approach does not assume that the two quantity vectors q 0 and q 1 are independent of the two price vectors p0 and p1. In the economic approach, the period 0 quantity vector q 0 is determined by the consumer’s preference function f and the period 0 vector of prices p0 that the consumer faces, and the period 1 quantity vector q1 is determined by the consumer’s preference function f and the period 1 vector of prices p1.

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 ≡ [q1,…,qn]. The consumer’s preferences over alternative possible consumption vectors, q, are assumed to be representable by a continuous, non-decreasing and concave3 utility function f Thus if f(q1)>f(q0), then the consumer prefers the consumption vector q1 to q0. It is further assumed that the consumer minimizes the cost of achieving the period t utility level ut ≡ f(q1) for periods t = 0.1, Thus we assume that the observed period t consumption vector qt solves the following period t cost minimization problem:

The period t price vector for the n commodities under consideration that the consumer faces is pt. 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 p0(p10,,pn0) and p1(p11,,pn1) in periods 0 and 1, respectively, is defined as the ratio of the minimum costs of achieving the same utility level utf(q), where q≡(q1,…,qn) 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 q0 and the current period quantity vector q1. 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(q0),p1)

since q0(q10,,qn0) is feasible for the minimization problem

where PL 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 C(f(q1),p0)Σi=1npi0qi1 and hence

where PP 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 q0. The straight line C represents the consumer’s period 0 budget constraint, the set of quantity points q1,q2 such that p10q1+p20q2=p10q10+p20q20. The curved line through q0 is the consumer’s period 0 indifference curve, the set of points q1,q2 such that f(q1,q2)=f(q10,q20); i.e., it is the set of consumption vectors that give the same utility as the observed period 0 consumption vector q0. The solution to the period 1 cost minimization problem is the vector q1. The straight line D represents the consumer’s period 1 budget constraint, the set of quantity points q1,q2 such that p11q1+p21q2=p11q11+p21q21.. The curved line through q1 is the consumer’s period 1 indifference curve, the set of points q1,q2 such that f(q1,q2)=f(q11,q21) i.e., it is the set of consumption vectors that give the same utility as the observed period 1 consumption vector q1. The point q0* solves the hypothetical problem of minimizing the cost of achieving the base period utility level u1Ξf(q0) when facing the period 1 price vector p1=(p11,p21).. Thus we have C[u0,p1]=p11q10*+p21q20* and the dashed line A is the corresponding isocost line. Note that the hypothetical cost line A is parallel to the actual period 1 cost line D. From equation (17.3), the Laspeyres-Konüs true index is C[u0,p1]/[p10q10+p20q20], while the ordinary Laspeyres index is [p11q10+p21q20]/[p10q10+p20q20]. Since the denominators for these two indices are the same, the difference between the indices is attributable to the differences in their numerators. In Figure 17.1, this difference in the numerators is expressed by the fact that the cost line through A lies below the parallel cost line through B. Now if the consumer’s indifference curve through the observed period 0 consumption vector q0 were L-shaped with vertex at q0, 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 q0* would coincide with q0, 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 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, PK(p0,p1,p0).

Figure 17.1The 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 q1* solves the hypothetical problem of minimizing the cost of achieving the current period utility level u1f(q1) when facing the period 0 price vector p0-(p10,p20). Thus we have C[u1,p0]=p10q11*+p20q21* and the dashed line E is the corresponding isocost line p11q1+p21q2=C[u0,p1]. From equation (17.4), the Paasche-Konüs true index is [p11q11+p21q21]/C[u1,p0], while the ordinary Paasche index is [p11q11+p21q21]/[p10q11+p20q21]. Since the numerators for these two indices are the same, the difference between the indices is attributable to the differences in their denominators. In Figure 17.1, this difference in the denominators is expressed by the fact that the cost line through E lies below the parallel cost line through F. The magnitude of this difference represents the amount of substitution bias between the true index and the corresponding Paasche index: i.e., the Paasche index will generally be less than the corresponding true cost of living index, Pk(P0,P1,q1). 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).

17.16 The bound (17.3) on the Laspeyres-Konüs true cost of living index PK(p0,p1,q0) 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 PK(p0,p1,q1) using the 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 q0 and the current period consumption vector q1 such that the corresponding (unobservable) true cost of living index Pk(P0,P1,q*) is between the observable Laspeyres and Paasche indices, PL and PP.6 Thus we have the existence of a number 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>>0N 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 ut= f(qt) 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(pt) of achieving one unit of utility times the period t utility level, f(qt). Obviously, the period t unit cost, c(pt), can be identified as the period t price level Pt and the period t level of utility, f(qt), as the period t quantity level Qt.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, pk(P0,P1,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(p1)/c(p0), 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, pk(p0,p1,q) is independent of the reference quantity vector 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(p1)/c(p0) is the utility ratio f(q1)/f(q0). Since the utility function is assumed to be homogeneous of degree one, this is the natural definition for a quantity index.

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 q0 and q1 it can be shown13 that the following equation holds:

where f(q1)/qi denotes the partial derivative of the utility function f with respect to the ith quantity qi evaluated at the period t quantity vector qt.

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

17.23Shephard’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 qt 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 qt solves the cost minimization problem defined by C(ut,pt), then qt must be feasible for this problem so it must be the case that f(qt) = ut Thus, qt is a feasible solution for the following cost minimization problem where the general price vector p has replaced the specific period t price vector pt:

where the inequality follows from the fact that qt(q1t,,qnt) is a feasible (but usually not optimal) solution for the cost minimization problem in equation (17.13). Now define for each strictly positive price vector p the function g(p) as follows:

where, as usual, p(p1,…,pn). 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=pt. 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 qi yields the following equation:

where fi(q)f(qt)/qi In order to obtain the first equation in (17.18), it is necessary to use the symmetry conditions, aik=aki Now evaluate the second equation in (17.18) at the observed period t quantity vector qt(q1t,,qnt) and divide both sides of the resulting equation by f(qt). 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, QF, 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 QF is exactly equal to the true quantity index, f(q1)/f(q0).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 QF using the product test (15.3) is the Fisher price index PF, defined by equation (15.12). Let 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 PF is exactly equal to the true price index, c(p1)/c(p0).

17.29 A twice continuously differentiable function f(q) of n variables q(q1,…,qn) 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 shown16 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.17Diewert (1976, p. 117) termed an index number formula Q(p0,p1,q0,q1) that was exactly equal to the true quantity index f(q1)/f(q0) (where/is a flexible functional form) a superlative index number formula.18 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 QF defined by equation (15.14) is a superlative index number formula. Since the Fisher ideal price index PF satisfies equation (17.21), where c(p) is the unit cost function that is generated by the homogeneous quadratic utility function, PF is also called a superlative index number formula.

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 pi yields the following equations:

where ci(p)c(pt)/pi. In order to obtain the first equation in (17.23), it is necessary to use the symmetry conditions. Now evaluate the second equation in (17.23) at the observed period t price vector pt(p1t,,pnt) and divide both sides of the resulting equation by c(pt). 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, PF, 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 PF is exactly equal to the true price index, c(p1)/c(p0).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 PF exactly equals the true price index c(p1)/c(p0) means that PF is a superlative index number formula.21

17.32 Suppose that the bik coefficients in equation (17.22) satisfy the following restrictions:

where the n numbers bi 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, qt:

Thus if the consumer has the preferences that correspond to the unit cost function defined by equation (17.22) where the bik satisfy the restrictions (17.26), then the period 0 and 1 quantity vectors are equal to a multiple of the vector b(b1,…,bn); i.e.,q0= b u0 and q1=b u1. Under these assumptions, the Fisher, Paasche and Laspeyres indices, PF, PP and PL, 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.

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(p0,p1q0,q1) that are exactly equal tof(q1)/f(q0) and many price indices P(P0,P1,q0, q1) that are exactly equal to C(P1)/c(P0), where the aggregator function f or the unit cost function f is a flexible functional form. Two families of superlative indices are defined below.

17.34 Suppose the consumer has the following quadratic mean of order r utility function.22

where the parameters aik satisfy the symmetry conditions aik—aki for all i and k and the parameter r satisfies the restriction r≡0. Diewert (1976, p. 130) showed that the utility function fr 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, fr equals the homogeneous quadratic function defined by equation (17.17).

17.35 Define the quadratic mean of order r quantity index Qr by:

where sitpitqit/Σk=1npktqkt is the period 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 Qr is exact for the aggregator function fr defined by equation (17.29); i.e., the following exact relationship between the quantity index Qr and the utility function fr 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 QF is exactly equal to the true quantity index, fr(q1)/fr(q0).23 Since Qr is exact for fr and fr is a flexible functional form, it can be seen that the quadratic mean of order r quantity index Qr is a superlative index for each r≠0. Thus there is an infinite number of superlative quantity indices.

17.37 For each quantity index Qr, 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 Pr*:

where cr* is the unit cost function that corresponds to the aggregator function fr defined by equation (17.29). For each r≠0, the implicit quadratic mean of order r price index Pr* is also a superlative index.

17.38 When r = 2, Qr defined by equation (17.30) simplifies to QF, the Fisher ideal quantity index, and Pr* defined by equation (17.32) simplifies to PF, the Fisher ideal price index. When r=1, Qr defined by equation (17.30) simplifies to:

where Pw is the Walsh price index defined previously by equation (15.19) in Chapter 15. Thus P1* is equal to Pw, the Walsh price index, and hence it is also a superlative price index.

17.39 Suppose the consumer has the following quadratic mean of order r unit cost function:24

where the parameters bik satisfy the symmetry conditions bik = bki for all i and k, and the parameter r satisfies the restriction rO. Diewert (1976, p. 130) showed that the unit cost function cr 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, cr equals the homogeneous quadratic function defined by equation (17.22).

17.40 Define the quadratic mean of order r price index Pr by:

where sitpitqit/Σk=1npktqkt is the period t expenditure share for commodity 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 PT is exact for the aggregator function defined by equation (17.34); i.e., the following exact relationship between the index number formula Pr and the unit cost function cr 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 unit cost function defined by equation (17.34), the quadratic mean of order r price index PF is exactly equal to the true price index, cr(pl)/cr(p0).25 Since PR is exact for cr and cr is a flexible functional form, it can be seen that the quadratic mean of order r price index PR is a superlative index for each r=0. Thus there are an infinite number of superlative price indices.

17.42 For each price index PR, 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 Qr*:

where fr* is the aggregator function that corresponds to the unit cost function cr defined by equation (17.34).26 For each r≠0, the implicit quadratic mean of order r quantity index Qr* is also a superlative index.

17.43 When r = 2, PR defined by equation (17.35) simplifies to PF, the Fisher ideal price index, and Qr* defined by equation (17.37) simplifies to QF, the Fisher ideal quantity index. When r = 1, Pr defined by equation (17.35) simplifies to:

where QW is the Walsh quantity index defined previously in footnote 30 of Chapter 15. Thus Q1* is equal to 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(z1,…,zn)≡ f(z), is quadratic; i.e.,

where the ai and the aik are constants. Let fi(z) denote the first-order partial derivative of f evaluated at z with respect to the ith component of z, zi. Let fik(z) denote the second-order partial derivative of f with respect to zi and zk. Then it is well known that the second-order Taylor series approximation to a quadratic function is exact; i.e., if f is defined by equation (17.39), then for any two points, z0 and z1, 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, z0 and z1, 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 function28 C(u,p), has the following translog functional form:29

where In is the natural logarithm function and the parameters ai, aik, and bi satisfy the following restrictions:

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 p0 and pl be the period 0 and I observed price vectors, and let q0 and ql be the period 0 and 1 observed quantity vectors. These assumptions imply:

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(ut, pt) in equation (17.45). After some cross multiplication, this becomes the following:

or

where Stiis 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 zi ≡ 1npi 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 PT, 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 PT: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 PT 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 PF defined by equation (15.12), the Törnqvist-Theil price index PT defined by equation (15.81), the implicit quadratic mean of order r price indices Pr* defined by equation (17.32) (when 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 Pr defined by equation (17.35). In the next section, we ask if it matters which one of these formulae is chosen as “best”.

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, p0 and p1, are equal and where the two quantity vectors, q0 and q1 are equal. In particular, this means that the following equalities are valid for all r and s not equal to 0, provided that p0=p1 and q0= q134.

where the Törnqvist-Theil price index PT is defined by equation (15.81), the implicit quadratic mean of order r price indices Ps* is defined by equation (17.32) and the quadratic mean of order r price indices Pr 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”.

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 Pr and the implicit quadratic mean of order s price indices Ps* are (continuous) functions of the parameters r and s respectively. Hence, as r and s become very large in magnitude, the indices Pr and Ps* can differ substantially from, say, P2 = PF, the Fisher ideal index. In fact, using definition (17.35) and the limiting properties of means of order r,35 Robert Hill (2002, p. 7) showed that Pr has the following limit as r approaches plus or minus infinity:

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, pr and Pr* can differ substantially from PT, P1P1*= Pw (the Walsh price index) and P2=P2*= PF (the Fisher ideal index).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, Pr and Pr*, fall into the interval 0≤r≤2 37Hill (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 PF=P2=P2* defined by equation (15.12) seemed to be best from one axiomatic viewpoint, the Törnqvist-Theil price index PT defined by equation (15.81) seems to be best from another axiomatic perspective, as well as from the stochastic viewpoint, and the Walsh index Pw defined by equation (15.19) (which is equal to the implicit quadratic mean of order r price indices Pr* 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.

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, pt and qt, can be written in terms of M sub-vectors as follows:

where the dimensionality of the subvectors ptm and qtmNm for 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 P0m 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 P1m for m = 1,2,…, M, of the CPI. Since the dimensionality of the subcomponent vectors, ptm and qtm differs from the dimensionality of the complete period t vectors of prices and quantities, pt and qt it is necessary to use different symbols for these subcomponent Laspeyres indices, say PmL 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 Q1m for m= 1,2,…,M can be defined by deflating the period 1 subcomponent values PL*(p0,p1,Q0,Q1) by the prices P1m

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 P0 and PL as follows:

where 1M denotes a vector of ones of dimension M and the components of PL are defined by equation (17.61). The period 0 and 1 subcomponent quantity vectors Q0 and Q1 are defined as follows:

where the components of Q0 are defined in equation (17.60) and the components of QL 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 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 PF*(orbyPT*) Now ask whether this two-stage Laspeyres index equals the corresponding single-stage index PL that was studied in the previous sections of this chapter; i.e., ask whether

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 PLm(p0m,p1m,q0m,q1m) is replaced by the Fisher formula PFm(p0m,p1m,q0m,q1m) or by the Törnqvist formula PTm(p0m,p1m,q0m,q1m) and in equation (17.65), suppose that PL*(P0,P1,Q0,Q1). is replaced by PLm(p0m,p1m,q0m,q1m)and PL (p0,pL,q0,q1) is replaced by PF (or by PT)- 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,

Similarly, it can be shown that the quadratic mean of order r index number formula Pr defined by equation (17.35) and the implicit quadratic mean of order r index number formula Pr* defined by equation (17.32) are also not consistent in aggregation.

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 PF in the inequality (17.66), both regarded as functions of the 4n variables in the vectors p0, p1, q0, q1, approximate each other to the second order around a point where the two price vectors are equal (so that p0=p1) and where the two quantity vectors are equal (so that q0 = q1),and a similar result holds for the two-stage and single-stage Törnqvist indices in equation (17.66).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, Pr, and for the implicit quadratic mean of order r indices, Pr*; 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 index Pr to its two-stage counterpart will break down as 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 p11/p10 be equal to the positive number a, let the second two price ratios pi1/pi0 equal b and let the last price ratio p41/p40 equal 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 Pr as the index number formula is [ab2c]¼. Thus the limiting value as r tends to plus or minus infinity of the single-stage aggregate over the two-stage aggregate is [ac]1/2/[ab2c]1/4=[ac/b2]1/4. Now b can take on any value between a and c, and so the ratio of the single-stage limiting Pr to its two-stage’counterpart can take on any value between [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 PF, the Törnqvist Theil PT and the Walsh Pw), 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.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 homogeneous43 and the corresponding unit cost function is c(p). It is supposed that the unit cost function has the following functional form:

where the αi, and σ are non-negative parameters with Σj=1nαi=1. The unit cost function defined by equation (17.68) corresponds to a constant elasticity of substitution (CES) aggregator function, which was introduced into the economics literature by Arrow, Chenery, Minhas and Solow (1961).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,q0i will be equal to u0c(p0)/pi, where c(p0)/pi is the first-order partial derivative of the unit cost function with respect to the ith commodity price evaluated at the period 0 prices and u0=f(q0) 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 s0t 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 PLM 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 s0t ;

  • the price relatives pi1/pi0 between the base period and the current period; and

  • an 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 PL (note that PLM reduces to PL if σ = 0). Hence, if the statistical agency is able to estimate the elasticity of substitution σ based on past experience,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, PL0(p0,p1,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 p0 (the base period price vector) and p1 (the current period price vector) are monthly or quarterly price vectors, the quantity vector q(q1, q2,…, qn) which appears in this basket-type formula is usually taken to be an annual quantity vector that refers to a base year, 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 PL0(p0,pt,qb), where p0 is the price vector pertaining to the base period month for prices, month 0, pt is the price vector pertaining to the current period month for prices, month t say, and qb is a reference basket quantity vector that refers to the base year 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[q1,…,qn] that can be represented by the continuous increasing utility function f(q). Thus if f(q1)>f(q0), then the consumer prefers the consumption vector q1 to q0. Let qb be the annual consumption vector for the consumer in the base year b. Define the base year utility level ub as the utility level that corresponds to f(q) evaluated at qb:

17.67 For any vector of positive commodity prices p≡ [p1,…,pn] 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 pb[p1b,,pnb] be the vector of annual prices that the consumer faced in the base year b. Assume that the observed base year consumption vector qb[q1b,,qnb] solves the following base year cost minimization problem:

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

17.68 Let p0 and pt be the monthly price vectors that the consumer faces in months 0 and t. Then the Konüs true cost of living index, PK(pQ,pt,qb), between months 0 and t, using the base year utility level ub—f(qb) as the reference standard of living, is defined as the following ratio of minimum monthly costs of achieving the utility level ub:

17.69 Using the definition of the monthly cost minimization problem that corresponds to the cost C(f(qb),pt), it can be seen that the following inequality holds:

since the base year quantity vector qb 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(qb),pQ), it can be seen that the following inequality holds:

since the base year quantity vector qb 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, et and eQ, are subtracted from the right-hand sides of these two inequalities. Thus the inequalities (17.78) and (17.79) can be rewritten as follows:

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 eQ and et are small, then the Lowe index between months 0 and t, PLo(pQ,pt,qb), will be an adequate approximation to the true cost of living index between months 0 and t, Pk(pQ,pt,qb).

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 ub between months 0 and t, then the observable Lowe index, PL0(P0,pt,qb), will be exactly equal to its true cost of living index counterpart, PK(pQ,pt,qb).49 Í7.73 It is difficult to know whether condition (17.83) will hold or whether the substitution bias terms eQ and et 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 ub as the reference utility level, is the ratio of two unobservable costs, C(ub,pt)/C(ub,p0). However, both of these hypothetical costs can be approximated by first-order Taylor series approximations that can be evaluated using observable information on prices and base year quantities. The first-order Taylor series approximation to C(ub,pt) around the annual base year price vector pb is given by the following approximate equation:50

Similarly, the first-order Taylor series approximation to C(ub, pQ) around the annual base year price vector pb is given by the following approximate equation:

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 et and eQ will be zero. Using these results to reinterpret the approximate equation (17.82), it can be seen that if the month 0 and month t price vectors, p0 and pt, are not too different from the base year vector of prices pb, then the Lowe index PLO(pQ,pt,qb) will approximate the true cost of living index PK(p0,pt,qb) to the accuracy of a first-order approximation. This result is quite useful, since it indicates that if the monthly price vectors pQ and pt are just randomly fluctuating around the base year prices pb (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 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(ub,pt) around the base year price vector pb is given by the following approximate equation:

where the last equality follows using approximate equation (17.84).51 Similarly, a second-order Taylor series approximation to C(ub,pQ) around the base year price vector pb is given by the following approximate equation:

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, e0 and e1, will be equal to the following expressions involving the second-order partial derivatives of the consumer’s cost function 2C(ub,pb)/pipj evaluated at the base year standard of living uh and at the base year prices ph:

Since the consumer’s cost function C(u,p) is a concave function in the components of the price vector p,52 it is known53 that the n by n (symmetric) matrix of second-order partial derivatives i=1npibqib=2. is negative semi-definite.54 Hence, for arbitrary price vectors ph, p0 and pl, 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 e0 and e1 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 αj 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, e0 and et: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 Pk(p0,pt,pb) are equal to zero.

  • Each commodity price change parameter αj is proportional to the corresponding commodity j base year price pbj.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, PLo(p0,pt,pb) as a and b respectively; i.e., define:

Using equation (17.90) to eliminate the month 0 prices Pi0 from equation (17.94) and the month t prices Pit from equation (17.95) leads to the following expressions for a and b:

It is assumed that a and b58 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, Bt, as the difference between the true cost of living index Pk(p0,pt,qb) 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 pib=qib=1, since the approximate bias defined by the last expression in equation (17.99) is the sum of one non-positive and two negative terms. Moreover, this approximate bias will grow quadratically in time t.59

17.82 In order to give the reader some idea of the magnitude of the approximate bias Bt 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, Σi1npibqib=2 Assume that 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 α1 =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 α2 =–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 2C(ub,pb)/p1p1=2C(ub,pb)/p2p2=1 and 2C(ub,pb)/p1p2=2C(ub,pb)/p2p1=1. These assumptions imply that the own elasticity of demand for each commodity is -1 at the base year consumer equilibrium. Making all of these assumptions means 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 - Bt: 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 q1 is an n-dimensional vector of commodity purchases made in January, q2 is an n-dimensional vector of commodity purchases made in February, …, and q12 is an n-dimensional vector of commodity purchases made in December.64 The sub-utility functions f1,f2, f12 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 and

  • 12 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 pti and quantities qti of the first n commodities are positive for periods 0 and 1, but that the price of commodity 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 qn+11<qn+10, but this inequality is not required for the analysis below.

17.92 Let the Laspeyres index between periods 0 and 1, restricted to the first n commodities, be denoted as pnL and let the Laspeyres index, defined over all n + 1 commodities, be defined as PLn+1. Also let vi0pi0qi0 denote the value of expenditures on commodity i in period 0. Then by the definition of the Laspeyres index defined over all n + 1 commodities:

where pn+10=0 was used in order to derive the second equation above. Thus the complete Laspeyres index PLn+1 defined over all n + 1 commodities is equal to the incomplete Laspeyres index PnL (which can be written in traditional price relative and base period expenditure share form), plus the mixed or hybrid expenditure pn+11qn+10 divided by the base period expenditure on the first n commodities, Σi=1nvi0. Thus the complete Laspeyres index can be calculated using the usual information available to the price statistician plus two additional pieces of information: the new non-zero price for commodity n+ 1 in period pn+11, and an estimate of consumption of commodity n+ 1 in period 0 (when it was free), qn+10. Since it is often governments that change the previously zero price to a positive price, the decision to do this is usually announced in advance, which will give the price statistician an opportunity to form an estimate for the base period demand, qn+10.

17.93 Let the Paasche index between periods 0 and 1, restricted to the first n commodities, be denoted as np and let the Paasche index, defined over all n+1 commodities, be defined as Ppn+1. Also let vi1pi1qi1 denote the value of expenditures on commodity i in period 1. Then, by the definition of the Paasche index defined over all n + 1 commodities:

where pn+10=0 was used in order to derive the second equation above. Thus the complete Paasche index Ppn+1 defined over all n+1 commodities is equal to the incomplete Paasche index pnp (which can be written in traditional price relative and current period expenditure share form), plus the current period expenditure on commodity i=1nvi0., divided by a sum of current period expenditures on the first n commodities, vIi, divided by the ith price relative for the first n commodities, 1,pn+1,1. Thus the complete Paasche index can be calculated using the usual information available to the price statistician plus information on current period expenditures.

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, qt is assumed to be the solution to a revenue maximization problem involving the output price vector pt.

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 f(λq1+(1λ)q2)λf(q1)+(1λ)f(q2)forall0λ1andallq10nandq20n. Note also that q&#x2265;0N means that each component of the N-dimensional vector q is non-negative, q &#x00BB; 0N means that each component of q is positive and q>0n means that q &#x2265; 0n but q&#x2260;0n; 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: f(λq)=λf(q) for all λ>0 and all q >> 0n. 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 PT in the context of non-homothetic preferences. Since PT 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.

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(u,p){Σi=1npiqi:f(q)u} is the producer’s cost function in this case and c(pt) can be identified as the period t input price level, while f(qt) is the period 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: pt=λtf(qt), where λ’ is the optimal Lagrange multiplier and f(qt) is the vector of first-order partial derivatives of evaluated at q’. Note that this system of equations is the price equals a constant times marginal utility equations that are familiar to economists. Now take the inner product of both sides of this equation with respect to the period t quantity vector q and solve the resulting equation for X’. Substitute this solution back into the vector equation pt=λtf(qt) and equation (17.10) is obtained.

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 i=1nf(q)qi=f(q) is obtained where fi(q)f(q)/qi.

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,f(q)1/maxp{i=1npiqi:c(p)=1}.

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 ik has an inverse, then it can be shown that the n by n matrix A of the bik will equal A-1.

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

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

The function fr* can be defined by using cr as follows: fr*(q)1/maxp{i=1npiqi:cr(p)=1}

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 bi=0 and b00=0, then C(u,p) CuC(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).

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(u0,C1)/C(u0,C0) and C(u1,p1)/C(u1,p0), respectively. In order to do this, it would be necessary to use econometrics and estimate empirically the consumer’s cost or expenditure function.

To prove the equalities in equations (17.51) to (17.56), simply differentiate the various index number formulae and evaluate the derivatives at p0= p1 and q0=q1. Actually, equations (17.51) to (17.56) are still true provided that p 1 = p p0 and q1=µq0 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.

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 r as 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 p1p0 and q1 = µq 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 eQ is equal to zero, then to have equality of PK and PL0, it must be the case that et is also equal to zero.

It can be seen that, when month t is set equal to month 0, et = eQ and C(ub,pt) = C(ub,pQ), and thus equation (17.83) is satisfied and PL0=Pk. This is not surprising since both indices are equal to unity when 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).

A symmetric n by n matrix A with ijth element equal to a, ij is negative semi-definite if, and only if for every vector z[z1,,zn], it is the case that Σi=1nj=1naijzizj0.

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 2C(u,p)/pipj that corresponds to a zero eigenvalue.

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 2C(u,p)/pipjthat corresponds to a zero eigenvalue and thus Σi=1nΣj=1n[2C(u,p)/pipj]pibpjb=0; see Diewert (1993b, p. 149) for a detailed proof of this result.

It is also assumed that a-γ2 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 α1 = 0.004 and α2 = -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 qm.

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.

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