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

Introduction function has the property that it can approximate an arbitrary linearly homogeneous function to the sec- 17.1 This chapter and the next cover the econo- ond order; i.e., in terminology, each of these mic approach to index number theory. This chapter three functional forms is flexible. Hence, using the ter- considers the case of a single household, while the fol- minology introduced by Diewert (1976), the Fisher, lowing chapter deals with the case of many households. Walsh and To¨ rnqvist indices are examples of A brief outline of the contents of the present chapter superlative index number formulae. follows. 17.5 In paragraphs 17.50 to 17.54, it is shown that the 17.2 In paragraphs 17.9 to 17.17, the theory of the Fisher, Walsh and To¨ rnqvist price indices approximate cost of living index for a single consumer or household is each other very closely using ‘‘normal’’ time series presented. This theory was originally developed by the data. This is a very convenient result since these three Russian , A.A. Konu¨ s (1924). The relation- index number formulae repeatedly show up as being ship between the (unobservable) true cost of living index ‘‘best’’ in all the approaches to index number theory. and the observable Laspeyres and Paasche indices will Hence this approximation result implies that it normally be explained. It should be noted that, in the economic will not matter which of these three indices is chosen approach to index number theory, it is assumed that as the preferred target index for a consumer price households regard the observed price data as given, index (CPI). while the quantity data are regarded as solutions 17.6 The Paasche and Laspeyres price indices have a to various economic optimization problems. Many very convenient mathematical property: they are con- price statisticians find the assumptions made in the sistent in aggregation. For example, if the Laspeyres economic approach to be somewhat implausible. Per- formula is used to construct sub-indices for, say, food or haps the best way to regard the assumptions made clothing, then these sub-index values can be treated as in the economic approach is that these assumptions sub-aggregate price relatives and, using the expenditure simply formalize the fact that consumers tend to pur- shares on these sub-aggregates, the Laspeyres formula chase more of a if its price falls relative to can be applied again to form a two-stage Laspeyres price other . index. Consistency in aggregation means that this two- 17.3 In paragraphs 17.18 to 17.26, the preferences of stage index is equal to the corresponding single-stage the consumer are restricted compared to the completely index. In paragraphs 17.55 to 17.60, it is shown that the general case treated in paragraphs 17.9 to 17.17. In superlative indices derived in the earlier sections are not paragraphs 17.18 to 17.26, it is assumed that the func- exactly consistent in aggregation but are approximately tion that represents the consumer’s preferences over consistent in aggregation. alternative combinations of is homo- 17.7 In paragraphs 17.61 to 17.64, a very interesting geneous of degree one. This assumption means that each index number formula is derived: the Lloyd (1975) and indifference surface (the set of commodity bundles that Moulton (1996a) . This index number formula give the consumer the same satisfaction or utility) is a makes use of the same information that is required in radial blow-up of a single indifference surface. With this order to calculate a Laspeyres index (namely, base period extra assumption, the theory of the true cost of living expenditure shares, base period prices and current period simplifies, as will be seen. prices), plus one other parameter (the of sub- 17.4 In the sections starting with paragraphs 17.27, stitution between commodities). If information on this 17.33 and 17.44, it is shown that the Fisher, Walsh and extra parameter can be obtained, then the resulting index To¨ rnqvist price indices (which emerge as being ‘‘best’’ can largely eliminate substitution bias and it can be in the various non-economic approaches) are also calculated using basically the same information that is among the ‘‘best’’ in the economic approach to index required to obtain the Laspeyres index. number theory. In these sections, the func- 17.8 The section starting with paragraph 17.65 con- tion of the single household will be further restricted siders the problem of defining a true cost of living index compared to the assumptions on preferences made in the when the consumer has annual preferences over com- previous two sections. Specific functional forms for the modities but faces monthly (or quarterly) prices. This consumer’s utility function are assumed and it turns out section attempts to provide an economic foundation for that, with each of these specific assumptions, the con- the Lowe index studied in Chapter 15. It also provides sumer’s true cost of living index can be exactly calcu- an introduction to the problems associated with the lated using observable price and quantity data. Each of existence of seasonal commodities, which are considered the three specific functional forms for the consumer’s at more length in Chapter 22. The final section deals 313 MANUAL: THEORY AND PRACTICE with situations where there may be a zero price for a The period t price vector for the n commodities under commodity in one period, but where the price is non- consideration that the consumer faces is pt. Note that zero in the other period. 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 The Konu¨ s cost of living index function, C(u, p). The cost function will be used below in order to define the consumer’s cost of living price and observable bounds index. 17.9 This section deals with the theory of the cost 17.11 The Konu¨ s (1924) family of true cost of living of living index for a single consumer (or household) indices pertaining to two periods where the consumer 0 0 0 that was first developed by the Russian economist, faces the strictly positive price vectors p ( p1, ..., pn) 1 1 1 Konu¨ s (1924). This theory relies on the assumption of and p ( p1, ..., pn) in periods 0 and 1, respectively, is optimizing behaviour on the part of economic agents defined as the ratio of the minimum costs of achieving (consumers or producers). Thus, given a vector of com- the same utility level u:f(q), where q:(q1, ..., qn)isa modity prices pt that the household faces in a given time positive reference quantity vector: period t, it is assumed that the corresponding observed 1 quantity vector qt is the solution to a cost minimization 0 1 C( f (q), p ) PK ( p , p , q) (17:2) problem that involves the consumer’s preference C( f (q), p0) or utility function f.1 Thus in contrast to the axiomatic approach to index number theory, the economic Note that definition (17.2) defines a family of price approach does not assume that the two quantity vec- indices, because there is one such index for each refer- tors q0 and q1 are independent of the two price vectors p0 ence quantity vector q chosen. and p1. In the economic approach, the period 0 quantity 17.12 It is natural to choose two specific reference 0 quantity vectors q in definition (17.2): the observed base vector q is determined by the consumer’s preference 0 0 period quantity vector q and the current period quan- function f and the period 0 vector of prices p that the 1 consumer faces, and the period 1 quantity vector q1 is tity vector q . The first of these two choices leads to the determined by the consumer’s preference function f and following Laspeyres–Konu¨s true cost of living index: 1 the period 1 vector of prices p . 0 1 0 1 0 C( f (q ), p ) 17.10 The economic approach to index number PK ( p , p , q ) 0 0 theory assumes that ‘‘the’’ consumer has well-defined C( f (q ), p ) preferences over different combinations of the n con- C( f (q0), p1) 2 = Pn using (17:1) for t=0 sumer commodities or items. Each combination of 0 0 items can be represented by a positive quantity vector pi qi q:[q , ..., q ]. The consumer’s preferences over alter- i=1 1 n Pn native possible consumption vectors, q, are assumed to 1 0 minq pi qi : f (q)=f (q ) be representable by a continuous, non-decreasing and i=1 3 1 0 = Pn (17:3) concave utility function f. Thus if f(q )>f(q ), then the 0 0 consumer prefers the consumption vector q1 to q0.Itis pi qi further assumed that the consumer minimizes the cost of i=1 achieving the period t utility level ut:f(qt) for periods using the definition of the cost minimization problem t=0, 1. Thus we assume that the observed period t that defines C( f(q0), p1) t consumption vector q solves the following period t cost Pn minimization problem: 1 0 pi qi  i=1 Pn t t t t t Pn C(u , p ) minq pi qi : f (q)=u f (q ) p0q0 i=1 i i i=1 Pn = ptqt for t=0, 1 (17:1) 0 0 0 i i since q (q1, ..., qn) is feasible for the minimization i=1 problem 0 1 0 1 =PL( p , p , q , q ) 1 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 where PL is the Laspeyres price index. Thus the (unob- problem involving the output price vector pt. servable) Laspeyres–Konu¨ s true cost of living index is 2 In this chapter, these preferences are assumed to be invariant over bounded from above by the observable Laspeyres price time, while in the following chapter, this assumption is relaxed (one of index.4 the environmental variables could be a time variable that shifts tastes). 17.13 The second of the two natural choices for 3 Note that f is concave if and only if f(lq1+(1l)q2) lf(q1)+ a reference quantity vector q in definition (17.2) leads 2 1 2 (1l) f(q ) for all 0 l 1 and all q 0n and q 0n. Note also that to the following Paasche–Konu¨s true cost of living q 0N means that each component of the N-dimensional vector q is non-negative, q 0n means that each component of q is positive and 4 q>0n means that q 0n but q=0n; i.e., q is non-negative but at least This inequality was first obtained by Konu¨ s (1924; 1939, p. 17). See one component is positive. also Pollak (1983). 314 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE index: Figure 17.1 The Laspeyres and Paasche bounds to the C( f (q1), p1) true cost of living index P ( p0, p1, q1) q K C( f (q1), p0) 2 Pn 1 1 pi qi i=1 1 1 1 = using (17:1) for t=1 q = (q 1 ,q 2 ) C( f (q1), p0) q 0* Pn 1 1 pi qi q 0 = (q 0,q 0) q 1* i=1 1 2 = Pn (17:4) 0 1 minq pi qi : f (q)=f (q ) i=1 q using the definition of the cost minimization problem O A B C D E F 1 that defines

Pn 1 1 pi qi line D. From equation (17.3), the Laspeyres–Konu¨ s true 0 0 i=1 1 1 1 index is C[u0, p1]=[ p0q0+p0q0], while the ordinary Las- Cð f ðq Þ; p Þ Pn since q (q1, ..., qn) 1 1 2 2 0 1 1 0 1 0 0 0 0 0 pi qi peyres index is [ p1q1+p2q2]=[ p1q1+p2q2]. Since the i=1 denominators for these two indices are the same, the is feasible for the minimization problem and thus difference between the indices is attributable to the dif- Pn ferences in their numerators. In Figure 17.1, this differ- 1 0 0 1 C( f (q ), p ) pi qi and hence ence in the numerators is expressed by the fact that the i=1 cost line through A lies below the parallel cost line 1 1 0 1 0 1 through B. Now if the consumer’s indifference curve =P ( p , p , q , q ) 0 C( f (q1), p0) Pn P through the observed period 0 consumption vector q 0 1 0 pi qi were L-shaped with vertex at q , then the consumer i=1 would not change his or her consumption pattern in where PP is the Paasche price index. Thus the (unobser- response to a change in the relative prices of the two vable) Paasche–Konu¨ s true cost of living index is boun- commodities while keeping a fixed standard of living. In 0* ded from below by the observable Paasche price index.5 this case, the hypothetical vector q would coincide with 0 17.14 It is possible to illustrate the two inequalities q , the dashed line through A would coincide with the (17.3) and (17.4) if there are only two commodities; dashed line through B and the true Laspeyres–Konu¨ s see Figure 17.1. The solution to the period 0 cost index would coincide with the ordinary Laspeyres index. minimization problem is the vector q0. The straight However, L-shaped indifference curves are not generally line C represents the consumer’s period 0 budget con- consistent with consumer behaviour; i.e., when the price straint, the set of quantity points q1, q2 such that of a commodity decreases, consumers generally demand 0 0 0 0 0 0 0 more of it. Thus, in the general case, there will be a gap p1q1+p2q2=p1q1+p2q2. The curved line through q is the consumer’s period 0 indifference curve, the set of between the points A and B. The magnitude of this gap 0 0 represents the amount of substitution bias between the points q1, q2 such that f (q1, q2)=f (q1, q2); i.e., it is the set of consumption vectors that give the same utility as the true index and the corresponding Laspeyres index; i.e., observed period 0 consumption vector q0. The solution the Laspeyres index will generally be greater than the 0 1 0 to the period 1 cost minimization problem is the corresponding true cost of living index, PK(p , p , q ). vector q1. The straight line D represents the consumer’s 17.15 Figure 17.1 can also be used to illustrate the period 1 budget constraint, the set of quantity points inequality (17.4). First note that the dashed lines E and F q , q such that p1q +p1q =p1q1+p1q1. The curved are parallel to the period 0 isocost line through 1 2 1 1 2 2 1 1 2 2 1* line through q1 is the consumer’s period 1 indifference C. The point q solves the hypothetical problem of curve, the set of points q1, q2 such that f (q1, q2)= minimizing the cost of achieving the current period utility 1 1 level u1:f(q1) when facing the period 0 price vector f (q1, q2); i.e., it is the set of consumption vectors that 0 0 0 1 0 0 1* 0 1* give the same utility as the observed period 1 con- p =( p1, p2). Thus we have C[u , p ]= p1q1 +p2q2 and sumption vector q1. The point q0* solves the hypothe- the dashed line E is the corresponding isocost line 1 1 0 1 tical problem of minimizing the cost of achieving the p1q1+p2q2=C[u , p ]. From equation (17.4), the 0 0 1 1 1 1 1 0 base period utility level u :f(q ) when facing the period Paasche–Konu¨ s true index is [ p1q1+ p2q2]=C[u , p ], 1 1 1 0 1 1 1 1 1 1 price vector p =(p1, p2). Thus we have C[u , p ]= while the ordinary Paasche index is [ p1q1+p2q2]= 1 0* 1 0* 0 1 0 1 p1q1 +p2q2 and the dashed line A is the corresponding [ p1q1+p2q2]. Since the numerators for these two indices 1 1 0 1 isocost line p1q1+p2q2=C[u , p ]. Note that the hypo- are the same, the difference between the indices is attri- thetical cost line A is parallel to the actual period 1 cost butable to the differences in their denominators. In Figure 17.1, this difference in the denominators is expressed by 5 This inequality is attributable to Konu¨ s (1924; 1939, p. 19); see also the fact that the cost line through E lies below the parallel Pollak (1983). cost line through F. The magnitude of this difference 315 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE represents the amount of substitution bias between the assumption. For the remainder of this section, it is true index and the corresponding Paasche index; i.e., the assumed that f is (positively) linearly homogeneous.7 In Paasche index will generally be less than the correspond- the economics literature, this is known as the assump- 0 1 1 8 ing true cost of living index, PK(p , p , q ). Note that this tion of homothetic preferences. This assumption is not inequality goes in the opposite direction to the previous strictly justified from the viewpoint of actual economic inequality between the two Laspeyres indices. The reason behaviour, but it leads to economic price indices that are for this change in direction is attributable to the fact that independent of the consumer’s standard of living.9 one set of differences between the two indices takes place Under this assumption, the consumer’s expenditure or in the numerators of the indices (the Laspeyres inequal- cost function, C(u, p) defined by equation (17.1), ities), while the other set takes place in the denominators decomposes as follows. For positive commodity prices of the indices (the Paasche inequalities). p 0N and a positive utility level u, then, using the 17.16 The bound (17.3) on the Laspeyres–Konu¨ s definition of C as the minimum cost of achieving the 0 1 0 true cost of living index PK(p , p , q ) using the base given utility level u, the following equalities can be period level of utility as the living standard is one-sided, derived: as is the bound (17.4) on the Paasche–Konu¨ s true cost of  living index P (p0, p1, q1) using the current period level Pn K C(u, p) min p q : f (q , ..., q ) u of utility as the living standard. In a remarkable result, q i i 1 n i=1 Konu¨ s (1924; 1939, p. 20) showed that there exists an Pn 1 intermediate consumption vector q* that is on the = minq piqi : f (q1, ..., qn) 1 straight line joining the base period consumption vector i=1 u q0 and the current period consumption vector q1 such dividing by u > 0  that the corresponding (unobservable) true cost of living Pn  0 1 Ã q1 qn index PK ( p , p , q ) is between the observable Laspeyres = minq piqi : f , ..., 1 6 i=1 u u and Paasche indices, PL and PP. Thus we have the existence of a number l* between 0 and 1 such that using the linear homogeneity of f  Pn  0 1 * 0 * 1 piqi q1 qn PL PK ( p , p , l q +(1l )q ) PP or =u min : f , ..., 1 q u u u P P ( p0, p1, l*q0+(1l*)q1) P (17:5) i=1 P K L Pn The inequalities (17.5) are of some practical importance. =u minz pizi : f (z1, ..., zn) 1 letting If the observable (in principle) Paasche and Laspeyres i=1 qi indices are not too far apart, then taking a symmetric zi= =uC(1, p) using definition (17:1) average of these indices should provide a good approx- u imation to a true cost of living index where the reference =uc( p) (17:6) 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 7 The linear homogeneity property means that f satisfies the following 15.32 in Chapter 15, and the geometric mean of the condition: f(lq)=lf(q) for all l>0 and all q 0n. This assumption is fairly restrictive in the consumer context. It implies that each indif- Paasche and Laspeyres indices can be justified as being ference curve is a radial projection of the unit utility indifference curve. the ‘‘best’’ average, which is the Fisher price index. Thus It also implies that all income elasticities of demand are unity, which is the Fisher ideal price index receives a fairly strong jus- contradicted by empirical evidence. tification as a good approximation to an unobservable 8 More precisely, Shephard (1953) defined a homothetic function to be theoretical cost of living index. a monotonic transformation of a linearly homogeneous function. However, if a consumer’s utility function is homothetic, it can always 17.17 The bounds (17.3)–(17.5) are the best that can be rescaled to be linearly homogeneous without changing consumer be obtained on true cost of living indices without mak- behaviour. Hence, the homothetic preferences assumption can simply ing further assumptions. Further assumptions are made be identified with the linear homogeneity assumption. below on the class of utility functions that describe the 9 This particular branch of the economic approach to index number consumer’s tastes for the n commodities under con- theory is attributable to Shephard (1953; 1970) and Samuelson and sideration. With these extra assumptions, the con- Swamy (1974). Shephard in particular realized the importance of the homotheticity assumption in conjunction with separability assump- sumer’s true cost of living can be determined exactly. tions 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 The true cost of living index when too large, then assuming that the consumer has homothetic pre- preferences are homothetic ferences will lead to a true cost of living index which is very close to Laspeyres–Konu¨ s and Paasche–Konu¨ s true cost of living indices 17.18 Up to now, the consumer’s preference func- defined by equations (17.3) and (17.4). Another way of justifying the tion f did not have to satisfy any particular homogeneity homothetic preferences assumption is to use equation (17.49), which justifies the use of the superlative To¨ rnqvist–Theil index PT in the context of non-homothetic preferences. Since PT is usually numeri- 6 For more recent applications of the Konu¨ s method of proof, see cally close to other superlative indices that are derived using the Diewert (1983a, p. 191) for an application to the consumer context and homothetic preferences assumption, it can be seen that the assumption Diewert (1983b, pp. 1059–1061) for an application to the producer of homotheticity will usually not be empirically misleading in the context. index number context. 316 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE where c(p):C(1, p) is the unit cost function that corre- equal to the ratio) has the following form: sponds to f.10 It can be shown that the unit cost function Pn c(p) satisfies the same regularity conditions that f 1 1 pi qi satisfies; i.e., c(p) is positive, concave and (positively) i=1 11 0 1 0 1 linearly homogeneous for positive price vectors. Sub- Q( p , p , q , q ) Pn t t 0 1 stituting equation (17.6) into equation (17.1) and using pi qi PK ( p , p , q) ut=f(qt) leads to the following equation: i=1 c( p1)f (q1) Pn t t t t = 0 0 0 1 using ð17:7Þ pi qi =c( p )f (q ) for t=0, 1 (17:7) c( p )f (q )PK ( p , p , q) i=1 twice Thus, under the linear homogeneity assumption on the c( p1)f (q1) utility function f, observed period t expenditure on the n = 0 0 1 0 using (17:8) commodities is equal to the period t unit cost c(pt)of c( p )f (q )fgc( p )=c( p ) achieving one unit of utility times the period t utility f (q1) t t = (17:9) level, f(q ). Obviously, the period t unit cost, c(p ), can f (q0) be identified as the period t Pt and the period t level of utility, f(qt), as the period t quantity level Qt.12 Thus, under the homothetic preferences assumption, the 17.19 The linear homogeneity assumption on the implicit quantity index that corresponds to the true cost consumer’s preference function f leads to a simplifica- of living price index c(p1)/c(p0) is the utility ratio f(q1)/ tion for the family of Konu¨ s true cost of living indices, f(q0). Since the utility function is assumed to be homo- 0 1 PK(p , p , q), defined by equation (17.2). Using this geneous of degree one, this is the natural definition for a definition for an arbitrary reference quantity vector q: quantity index.

1 17.21 In subsequent material, two additional results 0 1 C( f (q), p ) from economic theory will be needed: Wold’s Identity PK ( p , p , q) C( f (q), p0) and Shephard’s Lemma. Wold’s (1944, pp. 69–71; 1953, c( p1)f (q) p. 145) Identity is the following result. Assuming that = using (17:6) twice the consumer satisfies the cost minimization assumptions c( p0)f (q) (17.1) for periods 0 and 1 and that the utility function f is c( p1) 0 1 = (17:8) differentiable at the observed quantity vectors q and q , c( p0) it can be shown13 that the following equation holds:

Thus under the homothetic preferences assumption, the @f (qt) entire family of Konu¨ s true cost of living indices collapses pt @q 1 0 i = i for t=0, 1 and k=1, ..., n (17:10) to a single index, c(p )/c(p ), the ratio of the minimum Pn Pn @f (qt) costs of achieving unit utility level when the consumer faces pt qt qt k k k @q period 1 and 0 prices respectively. Put another way, under k=1 k=1 k the homothetic preferences assumption, P (p0, p1, q)is t K where @f(q )/@qi denotes the partial derivative of the independent of the reference quantity vector q. utility function f with respect to the ith quantity qi, 17.20 If the Konu¨ s true cost of living index defined evaluated at the period t quantity vector qt. by the right-hand side of equation (17.8) is used as the 17.22 If the homothetic preferences assumption is price index concept, then the corresponding implicit made and it is assumed that the utility function is quantity index defined using the product test (i.e., the linearly homogeneous, then Wold’s Identity can be product of the price index times the quantity index is simplified into an equation that will prove to be very useful:14 10 will recognize the producer theory counterpart to the result C(u, p)=uc(p): if a producer’s production function f is subject to t t pi @f (q )=@qi constant , then the corresponding total cost function Pn = t for t=0, 1 and k=1, ..., n (17:11) C(u, p) is equal to the product of the output level u times the unit cost t t f (q ) pkqk c(p). k=1 11 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 13 To prove this, consider the first-order necessary conditions for the as C(1, p). Thus f determines c. But we can also use c to determine strictly positive vector qt to solve the period t cost minimization pro- f under appropriate regularity conditions. In the economics literature, blem. The conditions of Lagrange with respect to the vector of q this is known as duality theory. For additional material on duality variables are: pt=lt !f(qt), where lt is the optimal Lagrange multi- theory and the properties of f and c, see Samuelson (1953), Shephard plier and !f(qt) is the vector of first-order partial derivatives of f (1953) and Diewert (1974a; 1993b, pp.107–123). evaluated at qt. Note that this system of equations is the price equals a 12 There is also a producer theory interpretation of the above theory; constant times equations that are familiar to econo- i.e., let f be the producer’s (constant returns to scale) production mists. Now take the inner product of both sides of this equation with respect to the period t quantity vector qt and solve the resulting function, let p be a vector of input prices that the producer faces, let t q be an input vector and let u=f(q) be the maximum outputP that can equation for l . Substitute this solution back into the vector equation n pt=lt !f(qt) and equation (17.10) is obtained. be produced using the input vector q. C(u; p) minq f i=1 piqi: f ðqÞug is the producer’s cost function in this case and c(pt) can be 14 Differentiate both sides of the equation f(lq)=lf(q) with respect to t identified as the period t input price level, while f(q ) is the period t lP, and then evaluate the resulting equation at l=1. The equation n aggregate input level. i=1fi(q)qi=f (q) is obtained where fi(q):@f(q)/@qi. 317 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE

17.23 Shephard’s (1953, p. 11) Lemma is the fol- Superlative indices: The Fisher lowing result. Consider the period t cost minimization problem defined by equation (17.1). If the cost function ideal index C(u, p) is differentiable with respect to the components 17.27 Suppose the consumer has the following uti- of the price vector p, then the period t quantity vector lity function: t q is equal to the vector of first-order partial derivatives sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the cost function with respect to the components Pn Pn of p: f (q1, ..., qn) aikqiqk, i=1k=1 t t t @C(u , p ) qi= for i=1, ..., n and t=0, 1 (17:12) where aik=aki for all i and k (17:17) @pi Differentiating f(q) defined by equation (17.17) with 17.24 To explain why equation (17.12) holds, con- respect to q yields the following equation: sider the following argument. Because it is assumed i t that the observed period t quantity vector q solves the n t t t P cost minimization problem defined by C(u , p ), then q 2 aikqk must be feasible for this problem so it must be the case 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik=1 t t t fi(q)= for i=1, ..., n that f(q )=u . Thus, q is a feasible solution for the 2 Pn Pn following cost minimization problem where the general ajkqjqk price vector p has replaced the specific period t price j=1 k=1 vector pt: Pn  aikqk Pn Pn k=1 t t t = (17:18) C(u , p) minq piqi : f (q1, ..., qn) u piqi f (q) i=1 i=1 (17:13) t where fi(q):@f(q )/@qi. In order to obtain the first equation in (17.18), it is necessary to use the symmetry t where the inequality follows from the fact that q conditions, aik=aki. Now evaluate the second equation t t (q1, ..., qn) is a feasible (but usually not optimal) solu- in (17.18) at the observed period t quantity vector tion for the cost minimization problem in equation t t t q (q1, ..., qn) and divide both sides of the resulting (17.13). Now define for each strictly positive price vector equation by f(qt). The following equations are obtained: p the function g(p) as follows: Pn Pn t t t t aikqk g( p) piqiC(u , p) (17:14) fi(q ) k=1 i=1 = for t=0, 1 and i=1, ..., n (17:19) f (qt) fgf (qt) 2 where, as usual, p:(p1, ..., pn). Using equations (17.13) and (17.1), it can be seen that g(p) is minimized (over all Assume cost minimizing behaviour for the consumer in strictly positive price vectors p)atp=pt. Thus the first- periods 0 and 1. Since the utility function f defined by order necessary conditions for minimizing a differenti- equation (17.17) is linearly homogeneous and differ- able function of n variables hold, which simplify to entiable, equation (17.11) will hold. Now recall the equation (17.12). definition of the Fisher ideal quantity index, QF, defined 17.25 If the homothetic preferences assumption is earlier in Chapter 15: made and it is assumed that the utility function is linearly vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u homogeneous, then using equation (17.6), Shephard’s u Pn u Pn Lemma (17.12) becomes: u p0q1 u p1q1 u i i u i i Q ( p0, p1, q0, q1)=u i=1 u i=1 @c( pt) F t Pn t Pn qt=ut for i=1, ..., n and t=0, 1 (17:15) p0q0 p1q0 i @p k k k k i k=1 k=1vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu Pn Combining equations (17.15) and (17.7), the following u p1q1 Pn q1 u i i equation is obtained: = f (q0) i u i=1 i f (q0)t Pn  i=1 p1q0 qt @c( pt) k k i = c( pt) for i=1, ..., n and t=0, 1 k=1 Pn @p pt qt i using equation (17:11) for t=0 k k vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k=1 u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi,u Pn (17:16) u p1q0 Pn q1 u k k 0 i uk=1 17.26 Note the symmetry of equation (17.16) with = fi(q ) 0 t Pn i=1 f (q ) 1 1 equation (17.11). It is these two equations that will be pi qi used in subsequent material in this chapter. i=1 318 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 18 Pn 1 Pn 0 form) a superlative index number formula. Equation 0 qi 1 qi = fi(q ) 0 fi(q ) 1 (17.20) and the fact that the homogeneous quadratic i=1 f (q ) i=1 f (q ) function f defined by equation (17.17) is a flexible functional form show that the Fisher ideal quantity using equation (17:11) for t=1 index QF defined by equation (15.14) is a superlative sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi,sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi index number formula. Since the Fisher ideal price index Pn Pn 1 Pn Pn 0 P satisfies equation (17.21), where c(p) is the unit cost 0 qi 1 qi F = aikqk 2 aikqk 2 function that is generated by the homogeneous quad- i=1k=1 fgf (q0) i=1k=1 fgf (q1) ratic utility function, PF is also called a superlative index using equation (17:19) number formula. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi,sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 17.30 It is possible to show that the Fisher ideal price 1 1 index is a superlative index number formula by a dif- = fgf (q0) 2 fgf (q1) 2 ferent route. Instead of starting with the assumption that the consumer’s utility function is the homogeneous using equation (17:17) and cancelling terms quadratic function defined by equation (17.17), it is f (q1) possible to start with the assumption that the consumer’s unit cost function is a homogeneous quadratic.19 Thus, = 0 (17:20) f (q ) suppose that the consumer has the following unit cost Thus under the assumption that the consumer engages function: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in cost-minimizing behaviour during periods 0 and 1 Pn Pn and has preferences over the n commodities that corre- c( p1, ..., pn) bik pi pk spond to the utility function defined by equation (17.17), i=1 k=1 the Fisher ideal quantity index QF is exactly equal to the 15 where b =b for all i and k: (17:22) true quantity index, f (q1Þ=f ðq0). ik ki 17.28 As was noted in paragraphs 15.18 to 15.23 of Differentiating c(p) defined by equation (17.22) with Chapter 15, the price index that corresponds to the respect to pi yields the following equations: Fisher quantity index QF using the product test (15.3) is Pn the Fisher price index PF, defined by equation (15.12). Let c(p) be the unit cost function that corresponds to 2 bik pk 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik=1 the homogeneous quadratic utility function f defined by ci( p)= for i=1, ..., n 2 Pn Pn equation (17.17). Then using equations (17.16) and bjk pj pk (17.20), it can be seen that j=1 k=1 Pn c( p1) 0 1 0 1 bik pk PF ( p , p , q , q )= 0 (17:21) c( p ) = k=1 (17:23) c(q) Thus, under the assumption that the consumer engages t in cost-minimizing behaviour during periods 0 and 1 where ci(p):@c(p )/@pi. In order to obtain the first and has preferences over the n commodities that corre- equation in (17.23), it is necessary to use the symmetry spond to the utility function defined by equation (17.17), conditions. Now evaluate the second equation in (17.23) t t t the Fisher ideal price index PF is exactly equal to the at the observed period t price vector p ( p , ..., p ) 1 0 1 n true price index, c(p )/c(p ). and divide both sides of the resulting equation by c(pt). 17.29 A twice continuously differentiable function The following equation is obtained: f(q)ofn variables q:(q1, ..., qn) can provide a second- Pn order approximation to another such function f *(q) t t bikpk around the point q*, if the level and all the first-order ci( p ) k=1 and second-order partial derivatives of the two func- = 2 for t=0, 1 and i=1, ..., n (17:24) c( pt) fgc( pt) tions coincide at q*. It can be shown16 that the homo- geneous quadratic function f defined by equation (17.17) As cost-minimizing behaviour for the consumer in per- can provide a second-order approximation to an arbi- iods 0 and 1 is being assumed and, since the unit cost trary f* around any (strictly positive) point q* in the function c defined by equation (17.22) is differentiable, class of linearly homogeneous functions. Thus the equations (17.16) will hold. Now recall the definition of homogeneous quadratic functional form defined by equation (17.17) is a flexible functional form.17 Diewert 18 Fisher (1922, p. 247) used the term superlative to describe the Fisher (1976, p. 117) termed an index number formula ideal price index. Thus, Diewert adopted Fisher’s terminology but 0 1 0 1 Q(p , p , q , q ) that was exactly equal to the true attempted to give some precision to Fisher’s definition of super- quantity index f(q1)/f(q0) (where f is a flexible functional lativeness. Fisher defined an index number formula to be superlative if it approximated the corresponding Fisher ideal results using his

15 data set. For the early history of this result, see Diewert (1976, p. 184). 19 16 Given the consumer’s unit cost function c(p), Diewert (1974a, p. 112) See Diewert (1976, p. 130) and let the parameter r equal 2. showed that the corresponding utility function f(q) can be defined as 17 Diewert (1974a, p. 133) introduced this term into the economics follows:P for a strictly positive quantity vector q, f (q) 1= maxp n literature. f i=1piqi:c( p)=1g. 319 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE the Fisher ideal price index, PF, given by equation function simplifies as follows: (15.12) in Chapter 15: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn Pn vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u c( p1, ..., pn) bibkpipk u Pn u Pn u p1q0 u p1q1 i=1 k=1 u i i u i i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 0 1 u i=1 u i=1 Pn Pn Pn PF ( p , p , q , q )=t Pn t Pn 0 0 0 1 = bipi bkpk= bipi (17:27) pkqk pkqk i=1 k=1 i=1 k=1 kv=1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu Pn Substituting equation (17.27) into Shephard’s Lemma u p1q1 Pn 0 i i (17.15) yields the following expressions for the period t ci( p )u = p1 u i=1 quantity vectors, qt: i c( p0)t Pn i=1 p0q1 k k t k=1 t t @c( p ) t q =u =biu i=1, ..., n; t=0, 1 (17:28) using equation (17:16) for t=0 i @p vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi,u Pn u p0q1 Thus if the consumer has the preferences that corre- Pn c ( p0) u k k spond to the unit cost function defined by equation = p1 i uk=1 i 0 t Pn (17.22) where the bik satisfy the restrictions (17.26), i=1 c( p ) 1 1 pi qi then the period 0 and 1 quantity vectors are equal i=1 0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi,sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi to a multiple of the vector b:(b1, ..., bn); i.e., q = bu 1 1 Pn 0 Pn 1 and q =bu. Under these assumptions, the Fisher, ci( p ) ci( p ) = p1 p0 Paasche and Laspeyres indices, P , P and P ,all i c( p0) i c( p1) F P L i=1 i=1 coincide. Thepreferenceswhichcorrespondtothe using equation (17:16) for t=1 unit cost function defined by equation (17.27) are, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi,sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi however, not consistent with normal consumer beha- 1 1 = viour since they imply that the consumer will not sub- fgc( p0) 2 fgc( p1) 2 stitute away from more expensive commodities to cheaper commodities if relative prices change going using equation (17:22) and cancelling terms from period 0 to 1. c( p1) = : (17:25) c( p0)

Thus, under the assumption that the consumer engages Quadratic mean of order r in cost-minimizing behaviour during periods 0 and 1 superlative indices and has preferences over the n commodities that corre- spond to the unit cost function defined by equation 17.33 It turns out that there are many other super- (17.22), the Fisher ideal price index PF is exactly equal to lative index number formulae; i.e., there exist many the true price index, c(p1)/c(p0).20 quantity indices Q(p0, p1, q0, q1) that are exactly equal 17.31 Since the homogeneous quadratic unit cost to f(q1)/f(q0) and many price indices P( p0, p1, q0, q1) that function c(p) defined by equation (17.22) is also a are exactly equal to c(p1)/c(p0), where the aggregator flexible functional form, the fact that the Fisher ideal function f or the unit cost function c is a flexible func- price index PF exactly equals the true price index tional form. Two families of superlative indices are 1 0 c(p )/c(p ) means that PF is a superlative index number defined below. formula.21 17.34 Suppose the consumer has the following 22 17.32 Suppose that the bik coefficients in equation quadratic mean of order r utility function. (17.22) satisfy the following restrictions: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn Pn r r r=2 r=2 f (q1, ..., qn) aikqi qk (17:29) bik=bibk for i, k=1, ..., n (17:26) i=1 k=1 where the n numbers bi are non-negative. In this special where the parameters aik satisfy the symmetry condi- case of equation (17.22), it can be seen that the unit cost tions 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 f r defined by equation (17.29) is a flexible functional form; i.e., it can approximate an 20 This result was obtained by Diewert (1976, pp. 133–134). arbitrary twice continuously differentiable linearly 21 Note that it has been shown that the Fisher index PF is exact for the homogeneous functional form to the second order. Note preferences defined by equation (17.17), as well as the preferences that that when r=2, f r equals the homogeneous quadratic are dual to the unit cost function defined by equation (17.22). These function defined by equation (17.17). two classes of preferences do not coincide in general. However, if the n by n symmetric matrix A of the aik has an inverse, then it can be shown 1 22 that the n by n matrix B of the bik will equal A . The terminology is attributable to Diewert (1976, p. 129). 320 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE

Pn Pn pffiffiffiffiffiffiffiffiffi 17.35 Define the quadratic mean of order r quantity 1 1 0 0 1 r pi qi pi qi qi index Q by: i=1 i=1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = Pn Pn pffiffiffiffiffiffiffiffiffi Pn p0q0 p1 q0q1 r 0 1 0 r=2 i i i i i si (qi =qi ) i=1 i=1 i=1 r 0 1 0 1 Pn Pn pffiffiffiffiffiffiffiffiffi Q ( p , p , q , q ) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (17:30) 1 1, 1 0 1 Pn pi qi pi qi qi r s1(q1=q0)r=2 i=1 i=1 i i i =Pn Pn pffiffiffiffiffiffiffiffiffi i=1 0 0 0 0 1 pi qi pi qi qi P i=1 i=1 t t t n t t where si pi qi = k=1pkqk is the period t expenditure Pn share for commodity i as usual. 1 1 pi qi 17.36 Using exactly the same techniques as were i=1 0 1 0 1 = Pn =PW ( p , p , q , q ) (17:33) used in paragraphs 17.27 to 17.32, it can be shown that 0 0 Qr is exact for the aggregator function f r defined by pi qi i=1 equation (17.29); i.e., the following exact relationship r r between the quantity index Q and the utility function f where PW is the Walsh price index defined previously by 1* holds: equation (15.19) in Chapter 15. Thus P is equal to PW, the Walsh price index, and hence it is also a superlative f r(q1) price index. Qr( p0, p1, q0, q1)= (17:31) f r(q0) 17.39 Suppose the consumer has the following quadratic mean of order r unit cost function:24 Thus under the assumption that the consumer engages sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn Pn in cost-minimizing behaviour during periods 0 and 1 r r r=2 r=2 c ( p1, ..., pn) bikpi pk (17:34) and has preferences over the n commodities that corre- i=1k=1 spond to the utility function defined by equation (17.29), the quadratic mean of order r quantity index QF is where the parameters bik satisfy the symmetry condi- r 1 r 0 23 exactly equal to the true quantity index, f (q )/f (q ). tions bik=bki for all i and k, and the parameter r satisfies Since Qr is exact for f r and f r is a flexible functional the restriction r=0. Diewert (1976, p. 130) showed that form, it can be seen that the quadratic mean of order r the unit cost function cr defined by equation (17.34) is quantity index Qr is a superlative index for each r=0. a flexible functional form; i.e., it can approximate an Thus there is an infinite number of superlative quantity arbitrary twice continuously differentiable linearly indices. homogeneous functional form to the second order. Note 17.37 For each quantity index Qr, the product test that when r=2, cr equals the homogeneous quadratic (15.3) in Chapter 15 can be used in order to define the function defined by equation (17.22). corresponding implicit quadratic mean of order r price 17.40 Define the quadratic mean of order r price index Pr*: index Pr by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n  P Pn 1 r=2 1 1 r pi pi qi r* 1 s0 r* 0 1 0 1 i=1 c ( p ) i 0 P ( p , p , q , q ) = r 0 1 0 1 i=1 pi Pn r* 0 P ( p , p , q , q ) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (17:35) 0 0 r 0 1 0 1 c ( p ) r=2 pi qi Q ( p , p , q , q ) r Pn p1 i=1 1 i si 0 i=1 pi (17:32) P where st ptqt= n pt qt is the period t expenditure where cr* is the unit cost function that corresponds to i i i k=1 k k share for commodity i as usual. the aggregator function f r defined by equation (17.29). 17.41 Using exactly the same techniques as were For each r=0, the implicit quadratic mean of order r used in paragraphs 17.27 to 17.32, it can be shown that price index Pr* is also a superlative index. Pr is exact for the aggregator function defined by 17.38 When r=2, Qr defined by equation (17.30) equation (17.34); i.e., the following exact relationship simplifies to Q , the Fisher ideal quantity index, and Pr* F between the index number formula Pr and the unit cost defined by equation (17.32) simplifies to P , the Fisher F function cr holds: ideal price index. When r=1, Qr defined by equation (17.30) simplifies to: cr( p1) Pr( p0, p1, q0, q1)= (17:36) sffiffiffiffiffi sffiffiffiffiffi cr( p0) Pn 1 n Pn 1 qi P qi s0 p1q1 p0q0 Thus, under the assumption that the consumer engages i q0 i i i i q0 1 0 1 0 1 i=1 sffiffiffiffiffii i=1 i=1 sffiffiffiffiffii in cost-minimizing behaviour during periods 0 and 1, Q ( p , p , q , q ) =Pn Pn q0 0 0 Pn q0 and has preferences over the n commodities that cor- 1 i pi qi 1 1 i si 1 i=1 pi qi 1 respond to the unit cost function defined by equation i=1 qi i=1 qi 24 This terminology is attributable to Diewert (1976, p. 130), this unit 23 See Diewert (1976, p. 130). cost function being first defined by Denny (1974). 321 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE

(17.34), the quadratic mean of order r price index PF is Superlative indices: exactly equal to the true price index, cr(p1)/cr(p0).25 Since Pr is exact for cr and cr is a flexible functional The To¨ rnqvist index form, it can be seen that the quadratic mean of order r 17.44 In this section, the same assumptions that price index Pr is a superlative index for each r=0. were made on the consumer in paragraphs 17.9 to 17.17 Thus there are an infinite number of superlative price are made here. In particular, it is not assumed that the indices. consumer’s utility function f is necessarily linearly 17.42 For each price index Pr, the product test (15.3) homogeneous as in paragraphs 17.18 to 17.43. in Chapter 15 can be used in order to define the corre- 17.45 Before the main result is derived, a pre- sponding implicit quadratic mean of order r quantity liminary result is required. Suppose the function of n r index Q *: variables, f(z1, ..., zn):f(z), is quadratic; i.e., Pn 1 Pn Pn Pn 1 1 f (z1, ..., zn) a0+ aizi+ aikzizk pi qi r* 1 i=1 2 i=1k=1 r* 0 1 0 1 i=1 f ( p ) Q ( p , p , q , q ) Pn = r* 0 and aik=aki for all i and k (17:39) 0 0 r 0 1 0 1 f ( p ) pi qi P ( p , p , q , q ) i=1 where the ai and the aik are constants. Let fi(z) denote (17:37) 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 where f r* is the aggregator function that corresponds r respect to zi and zk. Then it is well known that the sec- to the unit cost function c defined by equation ond-order Taylor series approximation to a quadratic (17.34).26 For each r=0, the implicit quadratic mean r function is exact; i.e., if f is defined by equation (17.39), of order r quantity index Q * is also a superlative then for any two points, z0 and z1, the following equa- index. tion holds: 17.43 When r=2, Pr defined by equation (17.35) r Pn ÈÉ simplifies to PF, the Fisher ideal price index, and Q * 1 0 0 1 0 f (z )f (z )= fi(z ) z z defined by equation (17.37) simplifies to QF, the Fisher i i r i=1 ideal quantity index. When r=1, P defined by equation Pn Pn ÈÉÈÉ 1 0 1 0 1 0 (17.35) simplifies to: + fik(z ) zi zi zkzk (17:40) 2 i=1k=1 sffiffiffiffiffi sffiffiffiffiffi It is less well known that an average of two first-order Pn p1 Pn Pn p1 s0 i 1 1 p0q0 i Taylor series approximations to a quadratic function is i 0 pi qi i i 0 i=1 pi i=1 i=1 pi also exact; i.e., if f is defined by equation (17.39) above, P1( p0, p1, q0, q1) sffiffiffiffiffi = sffiffiffiffiffi Pn then for any two points, z0 and z1, the following equa- Pn p0 0 0 Pn p0 1 i pi qi 1 1 i tion holds:27 si 1 i=1 pi qi 1 i=1 pi i=1 pi 1 Pn ÈÉÈÉ Pn Pn pffiffiffiffiffiffiffiffiffi 1 0 0 1 1 0 1 1 0 0 1 f (z )f (z )= fi(z )+fi(z ) zi zi (17:41) pi qi qi pi pi 2 i=1 = i=1 i=1 Pn Pn pffiffiffiffiffiffiffiffiffi Diewert (1976, p. 118) and Lau (1979) showed that 0 0 1 0 1 pi qi qi pi pi equation (17.41) characterized a quadratic function and i=1 i=1 called the equation the quadratic approximation lemma. Pn Pn pffiffiffiffiffiffiffiffiffi 1 1, 1 0 1 In this chapter, equation (17.41) will be called the pi qi qi pi pi i=1 i=1 quadratic identity. =Pn Pn pffiffiffiffiffiffiffiffiffi 17.46 Suppose that the consumer’s cost func- 0 0 0 0 1 28 pi qi qi pi pi tion C(u, p), has the following translog functional i=1 i=1 form:29 Pn 1 1 pi qi Pn 1 Pn Pn i=1 C u p a a p a p p b u = =Q ( p0, p1, q0, q1) (17:38) ln ( , ) 0+ i ln i+ ik ln i ln k+ 0 ln Pn W i=1 2 i=1k=1 0 0 p q Pn i i 1 2 i=1 + bi ln pi ln u+ b00(lnu) (17:42) i=1 2 where QW is the Walsh quantity index defined pre- where ln is the natural logarithm function and the 1* viously in footnote 30 of Chapter 15. Thus Q is equal parameters ai, aik, and bi satisfy the following to QW, the Walsh quantity index, and hence it is also a superlative quantity index. 27 The proof of this and the foregoing relation is by straightforward verification. 25 See Diewert (1976, pp. 133–134). 28 The consumer’s cost function was defined by equation (17.6) above. 26 r* r 29 The functionP f can be defined by using c as follows: f r*(q) Christensen, Jorgenson and Lau (1971) introduced this function into n r 1= maxp f i=1 piqi:c ( p)=1g. the economics literature. 322 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE restrictions: the following equation is obtained: Pn Pn Pn ln C(u*, p1) ln C(u*, p0) aik=aki, ai=1, bi=0 and aik=0  i=1 i=1 k=1 Pn * 0 * 1 ÈÉ 1 @ ln C(u , p ) @ ln C(u , p ) 1 0 for i, k=1, ..., n (17:43) = + ln pi ln pi 2 @ ln pi @ ln pi i=1 These parameter restrictions ensure that C(u, p) defined 1 Pn Pn by equation (17.42) is linearly homogeneous in p,a = a + a ln p0+b ln u*+a 2 i ik k i i property that a cost function must have. It can be shown i=1 k=1  Pn ÀÁ that the translog cost function defined by equation 1 * 1 0 (17.42) can provide a second-order Taylor series approx- + aik ln pk+bi ln u ln pi ln pi imation to an arbitrary cost function.30 k=1 Pn Pn pffiffiffiffiffiffiffiffiffi 17.47 Assume that the consumer has preferences 1 0 0 1 = ai+ aik ln pk+bi ln u u +ai that correspond to the translog cost function and that 2 i=1 k=1 the consumer engages in cost-minimizing behaviour  0 1 Pn pffiffiffiffiffiffiffiffiffi ÀÁ during periods 0 and 1. Let p and p be the period 0 and 1 0 1 1 0 + aik ln p +bi ln u u ln p ln p 0 1 k i i 1 observed price vectors, and let q and q be the period k=1  0 and 1 observed quantity vectors. These assumptions Pn Pn 1 0 0 imply: = ai+ aik ln pk+bi ln u +ai 2 i=1 k=1 Pn Pn  0 0 0 0 1 1 1 1 Pn ÀÁ C(u , p )= pi qi and C(u , p )= pi qi (17:44) 1 1 1 0 i=1 i=1 + aik ln pk+bi ln u ln pi ln pi k=1 where C is the translog cost function defined above.  Pn C u0 p0 C u1 p1 ÀÁ Now apply Shephard’s Lemma, equation (17.12), and 1 @ ln ( , ) @ ln ( , ) 1 0 = + ln pi ln pi the following equation results: 2 i=1 @ ln pi @ ln pi Pn ÀÁÀÁ @C(ut, pt) 1 0 1 1 0 qt= for i=1, ..., n and t=0, 1 = si +si ln pi ln pi using equation (17:47): i 2 i=1 @pi C(ut, pt) @ ln C(ut, pt) (17:49) = (17:45) pt @ ln p i i The last equation in (17.49) can be recognized as the Now use equation (17.44) to replace C(ut, pt) in equation logarithm of the To¨ rnqvist–Theil index number formula (17.45). After some cross multiplication, this becomes PT, defined earlier by equation (15.81) in Chapter 15. the following: Hence, exponentiating both sides of equation (17.49) yields the following equality between the true cost t t t t pi qi t @ ln C(u , p ) of living between periods 0 and 1, evaluated at the Pn si = t t @ ln pi intermediate utility level u* and the observable pkqk 31 k=1 To¨ rnqvist–Theil index PT: for i=1, ..., n and t=0, 1 (17:46) * 1 C(u , p ) 0 1 0 1 =PT ( p , p , q , q ) (17:50) or Cðu*, p0) Pn Since the translog cost function which appears on the t t t si =ai+ aik ln pk+bi ln u for i=1, ..., n and t=0, 1 left-hand side of equation (17.49) is a flexible functional k=1 form, the To¨ rnqvist–Theil price index PT is also a (17:47) superlative index. t 17.49 It is somewhat mysterious how a ratio of where si is the period t expenditure share on commodity i. 17.48 Define the geometric average of the period 0 unobservable cost functions of the form appearing on and 1 utility levels as u*; i.e., define the left-hand side of the above equation can be exactly pffiffiffiffiffiffiffiffiffi estimated by an observable index number formula. The u* u0u1 (17:48) key to this mystery is the assumption of cost-minimizing behaviour and the quadratic identity (17.41), along Now observe that the right-hand side of the equation with the fact that derivatives of cost functions are that defines the natural logarithm of the translog cost equal to quantities, as specified by Shephard’s Lemma. function, equation (17.42), is a quadratic function of the In fact, all the exact index number results derived in variables zi:lnpi if utility is held constant at the level u*. paragraphs 17.27 to 17.43 can be derived using trans- Hence the quadratic identity (17.41) can be applied, and formations of the quadratic identity along with Shep- hard’s Lemma (or Wold’s Identity).32 Fortunately, 30 It can also be shown that, if all the bi=0 and b00=0, then for most empirical applications, assuming that the C(u, p)=uC(1, p):uc(p); i.e., with these additional restrictions on the parameters of the general translog cost function, homothetic pref- 31 erences are the result of these restrictions. Note that it is also assumed This result is attributable to Diewert (1976, p. 122). that utility u is scaled so that u is always positive. 32 See Diewert (2002a). 323 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE

0 1 0 1 r 0 1 0 1 consumer has (transformed) quadratic preferences will @PT ( p , p , q , q ) @P ( p , p , q , q ) be an adequate assumption, so the results presented in t = t @qi @qi paragraphs 17.27 to 17.49 are quite useful to index s* 0 1 0 1 number practitioners who are willing to adopt the eco- @P ( p , p , q , q ) 33 = t nomic approach to index number theory. Essentially, @qi the economic approach to index number theory provides for i=1, ..., n and t=0, 1 (17:53) a strong justification for the use of the Fisher price index PF defined by equation (15.12), the To¨ rnqvist–Theil price index PT defined by equation (15.81), the implicit 2 0 1 0 1 2 r 0 1 0 1 quadratic mean of order r price indices Pr* defined by @ PT ( p , p , q , q ) @ P ( p , p , q , q ) t t = t t equation (17.32) (when r=1, this index is the Walsh price @pi @pk @pi @pk index defined by equation (15.19) in Chapter 15) and the 2 s* 0 1 0 1 r @ P ( p , p , q , q ) quadratic mean of order r price indices P defined by = t t equation (17.35). In the next section, we ask if it matters @pi @pk which one of these formulae is chosen as ‘‘best’’. for i, k=1, ..., n and t=0, 1 (17:54)

@2P ( p0, p1, q0, q1) @2Pr( p0, p1, q0, q1) The approximation properties of T = @pt@qt @pt@qt superlative indices i k i k @2Ps* ( p0, p1, q0, q1) 17.50 The results of paragraphs 17.27 to 17.49 pro- = t t vide price statisticians with a large number of index @pi @qk number formulae which appear to be equally good from for i, k=1, ..., n and t=0, 1 (17:55) the viewpoint of the economic approach to index num- ber theory. Two questions arise as a consequence of these results: 2 0 1 0 1 2 r 0 1 0 1 @ PT ( p , p , q , q ) @ P ( p , p , q , q )  Does it matter which of these formulae is chosen? = @qt@qt @qt@qt  If it does matter, which formula should be chosen? i k i k 2 s* 0 1 0 1 17.51 With respect to the first question, Diewert @ P ( p , p , q , q ) = t t (1978, p. 888) showed that all of the superlative index @qi @qk number formulae listed in paragraphs 17.27 to 17.49 for i, k=1, ..., n for t=0, 1 (17:56) approximate each other to the second order around any point where the two price vectors, p0 and p1, are equal 0 1 where the To¨ rnqvist–Theil price index PT is defined and where the two quantity vectors, q and q , are equal. by equation (15.81), the implicit quadratic mean of In particular, this means that the following equalities are s* 0 1 order r price indices P is defined by equation (17.32) valid for all r and s not equal to 0, provided that p =p r 0 1 34 and the quadratic mean of order r price indices P is and q =q . defined by equation (17.35). Using the results in the P ( p0, p1, q0, q1)=Pr( p0, p1, q0, q1)=Ps*( p0, p1, q0, q1) previous paragraph, Diewert (1978, p. 884) concluded T that ‘‘all superlative indices closely approximate each (17:51) 0 1 0 1 r 0 1 0 1 other’’. @PT ( p , p , q , q ) @P ( p , p , q , q ) 17.52 The above conclusion is, however, not true t = t @pi @pi even though the equations (17.51) to (17.56) are true. @Ps*( p0, p1, q0, q1) The problem is that the quadratic mean of order r price = indices P r and the implicit quadratic mean of order s @pt i price indices P s* are (continuous) functions of the para- for i=1, ..., n and t=0, 1 (17:52) meters r and s respectively. Hence, as r and s become very large in magnitude, the indices P r and P s* can differ substantially from, say, P2=P , the Fisher ideal 33 F If, however, consumer preferences are non-homothetic and the index. In fact, using definition (17.35) and the limiting change in utility is substantial between the two situations being com- properties of means of order r,35 Robert Hill (2002, p. 7) pared, then it may be desirable to compute separately the Laspeyres– r Konu¨ s and Paasche–Konu¨ s true cost of living indices defined by showed that P has the following limit as r approaches equations (17.3) and (17.4), C(u0, p1)/C(u0, p0) and C(u1, p1)/C(u1, p0), plus or minus infinity: respectively. In order to do this, it would be necessary to use econo- metrics and estimate empirically the consumer’s cost or expenditure lim Pr( p0, p1, q0, q1)= lim Pr( p0, p1, q0, q1) function. r!+1 r!1 34 To prove the equalities in equations (17.51) to (17.56), simply dif- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ferentiate the various index number formulae and evaluate the deri- p1 p1 0 1 0 1 i i vatives at p =p and q =q . Actually, equations (17.51) to (17.56) are = mini 0 maxi 0 (17:57) still true provided that p1=lp0 and q1=mq0 for any numbers l>0 and pi pi m>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 pro- portional to the period 0 quantity vector. 35 See Hardy, Littlewood and Po´ lya (1934). 324 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE

Using Hill’s method of analysis, it can be shown that the spective, as well as from the stochastic viewpoint, and implicit quadratic mean of order r price index has the the Walsh index PW defined by equation (15.19) (which following limit as r approaches plus or minus infinity: is equal to the implicit quadratic mean of order r price indices Pr* defined by equation (17.32) when r=1) lim Pr*( p0, p1, q0, q1)= lim Pr*( p0, p1, q0, q1,) seems to be best from the viewpoint of the ‘‘pure’’ price r!+1 r!1 index. The results presented in this section indicate that Pn p1q1 for ‘‘normal’’ time series data, these three indices will i i give virtually the same answer. To determine precisely sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii=1 =  which one of these three indices to use as a theoretical Pn 1 1 0 0 pi pi target or actual index, the statistical agency will have to pi qi mini 0 maxi 0 i=1 pi pi decide which approach to bilateral index number theory is most consistent with its goals. For most practical (17:58) purposes, however, it will not matter which of these three indices is chosen as a theoretical target index for Thus for r large in magnitude, P r and P r* can differ 1 1* making price comparisons between two periods. substantially from PT, P , P =PW (the Walsh price 2 2* 36 index) and P =P =PF (the Fisher ideal index). 17.53 Although Hill’s theoretical and empirical results demonstrate conclusively that not all superlative Superlative indices and indices will necessarily closely approximate each other, two-stage aggregation there is still the question of how well the more commonly 17.55 Most statistical agencies use the Laspeyres used superlative indices will approximate each other. All formula to aggregate prices in two stages. At the first the commonly used superlative indices, P r and P r*, fall 37 stage of aggregation, the Laspeyres formula is used to into the interval 0 r 2. Hill (2002, p. 16) summar- aggregate components of the overall index (e.g., food, ized how far apart the To¨ rnqvist and Fisher indices were, clothing, services); then at the second stage of aggrega- making all possible bilateral comparisons between any tion, these component sub-indices are further combined two data points for his time series data set as follows: into the overall index. The following question then The superlative spread S(0, 2) is also of since, naturally arises: does the index computed in two stages in practice, To¨ rnqvist (r=0) and Fisher (r=2) are by far coincide with the index computed in a single stage? the two most widely used superlative indexes. In all 153 Initially, this question is addressed in the context of the bilateral comparisons, S(0, 2) is less than the Paasche– Laspeyres formula.39 Laspeyres spread and on average, the superlative spread 17.56 Suppose that the price and quantity data for is only 0.1 per cent. It is because attention, until now, has t t focussed almost exclusively on superlative indexes in the period t, p and q , can be written in terms of M sub- range 0 r 2 that a general misperception has persisted vectors as follows:

in the index number literature that all superlative indexes t t1 t2 tM t t1 t2 tM approximate each other closely. p =( p , p , ..., p ) and q =(q , q , ..., q ) Thus, for Hill’s time series data set covering 64 com- for t=0, 1 (17:59) ponents of United States gross domestic product from where the dimensionality of the subvectors ptm and qtm is 1977 to 1994 and making all possible bilateral compar- N for m=1, 2, ..., M with the sum of the dimensions isons between any two years, the Fisher and To¨ rnqvist m N equal to n. These subvectors correspond to the price price indices differed by only 0.1 per cent on average. m and quantity data for subcomponents of the consumer This close correspondence is consistent with the results price index for period t. Now construct sub-indices for of other empirical studies using annual time series each of these components going from period 0 to 1. For data.38 Additional evidence on this topic may be found the base period, set the price for each of these sub- in Chapter 19. components, say P 0 for m=1, 2, ...M, equal to 1 and set 17.54 In the earlier chapters of this manual, it is m the corresponding base period subcomponent quantities, found that several index number formulae seem ‘‘best’’ say Q0 for m=1, 2, ..., M, equal to the base period value when viewed from various perspectives. Thus the Fisher m of consumption for that subcomponent for m= ideal index P =P2=P2* defined by equation (15.12) F 1, 2, ..., M: seemed to be best from one axiomatic viewpoint, the To¨ rnqvist–Theil price index P defined by equation PNm T 0 0 0m 0m (15.81) seems to be best from another axiomatic per- Pm 1 and Qm pi qi for m=1, 2, ..., M i=1

36 Hill (2002) documents this for two data sets. His time series data (17:60) consist of annual expenditure and quantity data for 64 components of Now use the Laspeyres formula in order to construct United States gross domestic product from 1977 to 1994. For this data 1 set, Hill (2002, p. 16) found that ‘‘superlative indexes can differ by a period 1 price for each subcomponent, say Pm for more than a factor of two (i.e., by more than 100 per cent), even m=1, 2, ..., M, of the CPI. Since the dimensionality of though Fisher and To¨ rnqvist never differ by more than 0.6 per cent’’. 37 Diewert (1980, p. 451) showed that the To¨ rnqvist index PT is a 39 r Much of the material in this section is adapted from Diewert (1978) limiting case of P ,asr tends to 0. and Alterman, Diewert and Feenstra (1999). See also Balk (1996b) for 38 See, for example, Diewert (1978, p. 894) or Fisher (1922), which is a discussion of alternative definitions for the two-stage aggregation reproduced in Diewert (1976, p. 135). concept and references to the literature on this topic. 325 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE the subcomponent vectors, ptm and qtm , differs from the index calculated in two stages equals the Laspeyres dimensionality of the complete period t vectors of prices index calculated in one stage. and quantities, pt and qt, it is necessary to use different 17.57 Now suppose that the Fisher or To¨ rnqvist symbols for these subcomponent Laspeyres indices, say formula is used at each stage of the aggregation. That is, m PL for m=1, 2, ..., M. Thus the period 1 subcomponent in equation (17.61), suppose that the Laspeyres formula m 0m 1m 0m 1m prices are defined as follows: PL ( p , p , q , q ) is replaced by the Fisher formula m 0m 1m 0m 1m m PF ( p , p , q , q ) or by the To¨ rnqvist formula PT PNm 0m 1m 0m 1m 1m 0m ( p , p , q , q ); and in equation (17.65), suppose pi qi * 0 1 0 1 * * 1 m 0m 1m 0m 1m i=1 that PL(P , P , Q , Q ) is replaced by PF (or by PT )and Pm PL ( p , p , q , q ) 0 1 0 1 PNm PL (p , p , q , q ) is replaced by PF (or by PT). Then is it 0m 0m pi qi the case that counterparts are obtained to the two- i=1 stage aggregation result for the Laspeyres formula, for m=1, 2, ..., M (17:61) equation (17.65)? The answer is no; it can be shown that, in general, Once the period 1 prices for the M sub-indices have been defined by equation (17.61), then corresponding sub- P* (P0, P1, Q0, Q1) 6¼ P ( p0, p1, q0, q1)and component period 1 quantities Q1 for m=1, 2, ..., M F F m * 0 1 0 1 0 1 0 1 can beP defined by deflating the period 1 subcomponent PT (P , P , Q , Q ) 6¼ PT ( p , p , q , q ) (17:66) values Nm p1m q1m by the prices P1 i=1 i i m Similarly, it can be shown that the quadratic mean of PNm r 1m 1m order r index number formula P defined by equation pi qi (17.35) and the implicit quadratic mean of order r index i=1 Q1 for m=1, 2, ..., M (17:62) number formula Pr* defined by equation (17.32) are also m P1 m not consistent in aggregation. Now define subcomponent price and quantity vectors 17.58 Nevertheless, even though the Fisher and for each period t=0,1 using equations (17.60) to (17.62). To¨ rnqvist formulae are not exactly consistent in Thus define the period 0 and 1 subcomponent price aggregation, it can be shown that these formulae are vectors P0 and P1 as follows: approximately consistent in aggregation. More specifi- cally, it can be shown that the two-stage Fisher formula P0=(P0, P0, ..., P0 ) 1 and P1=(P1, P1, ..., P1 ) * 1 2 M M 1 2 M PF and the single-stage Fisher formula PF in the in- (17:63) equality (17.66), both regarded as functions of the 4n variables in the vectors p0, p1, q0, q1, approximate each where 1M denotes a vector of ones of dimension M and 1 other to the second order around a point where the two the components of P are defined by equation (17.61). price vectors are equal (so that p0=p1) and where the The period 0 and 1 subcomponent quantity vectors Q0 0 1 1 two quantity vectors are equal (so that q =q ), and a and Q are defined as follows: similar result holds for the two-stage and single-stage To¨ rnqvist indices in equation (17.66).40 As was seen in Q0=(Q0, Q0, ..., Q0 )andQ1=(Q1, Q1, ..., Q1 ) 1 2 M 1 2 M the previous section, the single-stage Fisher and (17:64) To¨ rnqvist indices have a similar approximation prop- where the components of Q0 are defined in equation erty, so all four indices in the inequality (17.66) approx- (17.60) and the components of Q1 are defined by equa- imate each other to the second order around an equal tion (17.62). The price and quantity vectors in equations (or proportional) price and quantity point. Thus for (17.63) and (17.64) represent the results of the first-stage normal time series data, single-stage and two-stage aggregation. Now use these vectors as inputs into Fisher and To¨ rnqvist indices will usually be numerically very close. This result is illustrated in Chapter 19 for an the second-stage ; i.e., apply the 41 Laspeyres price index formula, using the information in artificial data set. equations (17.63) and (17.64) as inputs into the index 17.59 Similar approximate consistency in aggrega- number formula. Since the price and quantity vectors tion results (to the results for the Fisher and To¨ rnqvist formulae explained in the previous paragraph) can be that are inputs into this second-stage aggregation pro- r blem have dimension M instead of the single-stage for- derived for the quadratic mean of order r indices, P , and for the implicit quadratic mean of order r indices, mula which utilized vectors of dimension n, a different r symbol is required for the new Laspeyres index: this is P *; see Diewert (1978, p. 889). Nevertheless, the results * of Hill (2002) again imply that the second-order chosen to be PL. Thus the Laspeyres price index com- * 0 1 0 1 puted in two stages can be denoted as PL(P , P , Q , Q ). Now ask whether this two-stage Laspeyres index equals 40 See Diewert (1978, p. 889). In other words, a string of equalities the corresponding single-stage index P that was studied similar to equations (17.51) to (17.56) holds between the two-stage L indices and their single-stage counterparts. In fact, these equalities are in the previous sections of this chapter; i.e., ask whether still true provided that p1=l p0 and q1=mq0 for any numbers l>0 and m>0. * 0 1 0 1 0 1 0 1 PL(P , P , Q , Q )=PL( p , p , q , q ): (17:65) 41 For an empirical comparison of the four indices, see Diewert (1978, pp. 894–895). For the Canadian consumer data considered there, the If the Laspeyres formula is used at each stage of each chained two-stage Fisher in 1971 was 2.3228 and the corresponding aggregation, the answer to the above question is yes: chained two-stage To¨ rnqvist was 2.3230, the same values as for the straightforward calculations show that the Laspeyres corresponding single-stage indices. 326 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE approximation property of the single-stage quadratic 17.62 In this section, the same assumptions about mean of order r index Pr to its two-stage counterpart the consumer are made that were made in paragraphs will break down as r approaches either plus or minus 17.18 to 17.26 above. In particular, it is assumed that the infinity. To see this, consider a simple example where consumer’s utility function f(q) is linearly homo- there are only four commodities in total. Let the first geneous43 and the corresponding unit cost function is 1 0 price ratio p1=p1 be equal to the positive number a, let c(p). It is supposed that the unit cost function has the 1 0 the second two price ratios pi =pi equal b and let the last following functional form: price ratio p1=p0 equal c, where we assume a

These equations can be rewritten as: uate formula (17.71) numerically, it is necessary to have information on: p0q0 a ( p0)r i i i i  base period expenditure shares s0; 0 0 = Pn for i=1, 2, ..., n (17:70) i u c( p ) 0 r  the price relatives p1=p0 between the base period and ak( pk) i i k=1 the current period; and an estimate of the elasticity of substitution between where r:1s. Now consider the following Lloyd (1975)  and Moulton (1996a) index number formula: the commodities in the aggregate, s. The first two pieces of information are the standard ()  1=(1s) Pn 1 1s information sets that statistical agencies use to evaluate 0 1 0 1 0 p i the Laspeyres price index PL (note that PLM reduces to PLM( p , p , q , q ) si 0 for s 6¼ 1 i=1 pi PL if s=0). Hence, if the statistical agency is able to (17:71) estimate the elasticity of substitution s based on past experience,45 then the Lloyd–Moulton price index can 0 be evaluated using essentially the same information where si is the period 0 expenditure share of commodity i, as usual: set that is used in order to evaluate the traditional Laspeyres index. Moreover, the resulting CPI will be 0 0 0 pi qi free of substitution bias to a reasonable degree of si Pn for i=1, 2, ..., n 46 0 0 approximation. Of course, the practical problem with pkqk implementing this methodology is that estimates of the k=1 elasticity of substitution parameter s are bound to be p0q0 somewhat uncertain, and hence the resulting Lloyd– = i i using the assumption of cost minimizing u0c( p0) Moulton index may be subject to charges that it is not objective or reproducible. The statistical agency will behaviour have to balance the benefits of reducing substitution bias 0 r with these possible costs. ai( pi ) = Pn using equation (17:70) (17:72) 0 r ak( pk) k=1 Annual preferences and If equation (17.72) is substituted into equation (17.71), it monthly prices is found that: 17.65 Recall the definition of the Lowe index, r 1=r 0 1 Pn 1 PLo(p , p , q), defined by equation (15.15) in Chapter 15. 0 1 0 1 0 pi PLM( p , p , q , q )= si 0 In paragraphs 15.33 to 15.64 of Chapter 15, it is noted i=1 pi 8 9 that this formula is frequently used by statistical agen- > >1=r cies as a target index for a CPI. It is also noted that, <> => 0 Pn a ( p0)r p1 r while the price vectors p (the base period price vector) = i i i and p1 (the current period price vector) are monthly > Pn 0 > :>i=1 0 r pi ;> or quarterly price vectors, the quantity vector q:(q , ak( pk) 1 k=1 q2, ..., qn) which appears in this basket-type formula is 8 9 Pn 1=r usually taken to be an annual quantity vector that refers > 1 r > < ai( pi ) = to a base year, b say, that is prior to the base period for i=1 the prices, month 0. Thus, typically, a statistical agency => Pn > :> 0 r;> will produce a CPI at a monthly frequency that has the ak( pk) 0 t b 0 k=1 form PLo(p , p , q ), where p is the price vector per-  taining to the base period month for prices, month 0, pt Pn 1=r 1 r a0 ai( pi ) 45 i=1 For the first application of this methodology (in the context of the =  CPI), see Shapiro and Wilcox (1997a, pp. 121–123). They calculated Pn 1=r 0 r superlative To¨ rnqvist indices for the United States for the years 1986– a0 ak( pk) 95 and then calculated the Lloyd–Moulton CES index for the same k=1 period, using various values of s. They then chose the value of s c( p1) (which was 0.7), which caused the CES index to most closely approximate the To¨ rnqvist index. Essentially the same methodology = 0 using r 1s c( p ) was used by Alterman, Diewert and Feenstra (1999) in their study of United States import and export price indices. For alternative methods and definition (17:68): (17:73) for estimating s, see Balk (2000b). 46 What is a ‘‘reasonable’’ degree of approximation depends on the 17.64 Equation (17.73) shows that the Lloyd– context. Assuming that consumers have CES preferences is not a Moulton index number formula PLM is exact for CES reasonable assumption in the context of estimating elasticities of preferences. Lloyd (1975) and Moulton (1996a) inde- demand: at least a second-order approximation to the consumer’s preferences is required in this context. In the context of approximating pendently derived this result, but it was Moulton who changes in a consumer’s expenditures on the n commodities under appreciated the significance of the formula (17.71) for consideration, however, it is usually adequate to assume a CES statistical agency purposes. Note that in order to eval- approximation. 328 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE is the price vector pertaining to the current period holds: month for prices, month t say, and qb is a reference  Pn basket quantity vector that refers to the base year b, b t t b b 47 C( f (q ), p ) minq pi qi : f (q1, ..., qn)=f (q1, ..., qn) which is equal to or prior to month 0. The question to i=1 be addressed in the present section is: can this index be Pn t b related to one based on the economic approach to index pi qi (17:78) number theory? i=1 since the base year quantity vector qb is feasible for the cost minimization problem. Similarly, using the defini- tion of the monthly cost minimization problem that The Lowe index as an approximation to corresponds to the month 0 cost C( f(qb), p0), it can be a true cost of living index seen that the following inequality holds:  17.66 Assume that the consumer has preferences Pn C(f (qb), p0) min p0q : f (q ,...,q )=f (qb,...,qb) defined over consumption vectors q:[q1, ..., qn] that q i i 1 n 1 n can be represented by the continuous increasing utility i=1 1 0 Pn function f(q). Thus if f(q )>f(q ), then the consumer p0qb (17:79) 1 0 b i i prefers the consumption vector q to q . Let q be the i=1 annual consumption vector for the consumer in the base b b since the base year quantity vector q is feasible for the year b. Define the base year utility level u as the utility b cost minimization problem. level that corresponds to f(q) evaluated at q : 17.70 It will prove useful to rewrite the two ub f (qb) (17:74) inequalities (17.78) and (17.79) as equalities. This can be done if non-negative substitution bias terms, et and e0, 17.67 For any vector of positive commodity prices are subtracted from the right-hand sides of these two p:[ p1, ..., pn] and for any feasible utility level u, the inequalities. Thus the inequalities (17.78) and (17.79) consumer’s cost function, C(u, p), can be defined in the can be rewritten as follows: usual way as the minimum expenditure required to Pn achieve the utility level u when facing the prices p: C(ub, pt)= ptqb et (17:80)  i i Pn i=1 C(u, p) minq piqi : f (q1, ..., qn)=u : (17:75) i=1 Pn C(ub, p0)= p0qbe0 (17:81) b b b i i Let p [ p1, ..., pn] be the vector of annual prices that i=1 the consumer faced in the base year b. Assume that the 17.71 Using equations (17.80) and (17.81), and the observed base year consumption vector qb [qb, ..., qb] 1 n definition (15.15) in Chapter 15 of the Lowe index, the solves the following base year cost minimization pro- following approximate equality for the Lowe index blem: results: Pn Pn b b b b b b Pn C(u , p ) minq pi qi : f (q1, ..., qn)=u = pi qi t b i=1 i=1 pi qi b t t 0 t b i=1 fC(u , p )+e g (17:76) PLo( p , p , q ) Pn = b 0 0 0 b fC(u , p )+e g pi qi The cost function will be used below in order to define i=1 the consumer’s cost of living price index. C(ub, pt) 0 t  =P ( p0, pt, qb) (17:82) 17.68 Let p and p be the monthly price vectors that C(ub, p0) K the consumer faces in months 0 and t. Then the Konu¨ s 0 t b 0 t true cost of living index, PK(p , p , q ), between months Thus if the non-negative substitution bias terms e and e 0 and t, using the base year utility level ub=f(qb) as the are small, then the Lowe index between months 0 and t, 0 t b reference standard of living, is defined as the following PLo(p , p , q ), will be an adequate approximation to the ratio of minimum monthly costs of achieving the utility true cost of living index between months 0 and t, b 0 t b level u : PK(p , p , q ). 17.72 A bit of algebraic manipulation shows that the C( f (qb), pt) P ( p0, pt, qb) (17:77) Lowe index will be exactly equal to its cost of living K C( f (qb), p0) counterpart if the substitution bias terms satisfy the following relationship:48 17.69 Using the definition of the monthly cost t b t minimization problem that corresponds to the cost e C(u , p ) 0 t b b t = =P ( p , p , q ) (17:83) C( f(q ), p ), it can be seen that the following inequality e0 C(ub, p0) K

48 This assumes that e0 is greater than zero. If e0 is equal to zero, then 47 t As noted in Chapter 15, month 0 is called the price reference period to have equality of PK and PLo, it must be the case that e is also equal and year b is called the weight reference period. to zero. 329 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE

Equations (17.82) and (17.83) can be interpreted as fol- 17.75 Comparing approximate equation (17.84) lows: if the rate of growth in the amount of substitution with equation (17.80), and comparing approximate bias between months 0 and t is equal to the rate of equation (17.85) with equation (17.81), it can be seen growth in the minimum cost of achieving the base year that, to the accuracy of the first-order approximations utility level ub between months 0 and t, then the obser- used in (17.84) and (17.85), the substitution bias terms et 0 t b 0 vable Lowe index, PLo(p , p , q ), will be exactly equal to and e will be zero. Using these results to reinterpret the 0 t b 49 its true cost of living index counterpart, PK(p , p , q ). approximate equation (17.82), it can be seen that if the 17.73 It is difficult to know whether condition month 0 and month t price vectors, p0 and pt, are not too (17.83) will hold or whether the substitution bias terms different from the base year vector of prices pb, then the 0 t 0 t b e and e will be small. Thus, first-order and second- Lowe index PLo(p , p , q ) will approximate the true cost 0 t b order Taylor series approximations to these substitution of living index PK(p , p , q ) to the accuracy of a first- bias terms are developed in paragraphs 17.74 to 17.83. order approximation. This result is quite useful, since it indicates that if the monthly price vectors p0 and pt are just randomly fluctuating around the base year prices pb A first-order approximation to the (with modest variances), then the Lowe index will serve bias of the Lowe index as an adequate approximation to a theoretical cost of 17.74 The true cost of living index between months 0 living index. However, if there are systematic long-term and t, using the base year utility level ub as the reference trends in prices and month t is fairly distant from month utility level, is the ratio of two unobservable costs, 0 (or the end of year b is quite distant from month 0), C(ub, pt)/C(ub, p0). However, both of these hypothetical then the first-order approximations given by approx- costs can be approximated by first-order Taylor series imate equations (17.84) and (17.85) may no longer be approximations that can be evaluated using observable adequate and the Lowe index may have a considerable information on prices and base year quantities. The bias relative to its cost of living counterpart. The first-order Taylor series approximation to C(ub, pt) hypothesis of long-run trends in prices will be explored around the annual base year price vector pb is given by in paragraphs 17.76 to 17.83. the following approximate equation:50 A second-order approximation to the Pn @C(ub, pb) C(ub, pt)  C(ub, pb)+ [ ptpb] substitution bias of the Lowe index @p i i i=1 i 17.76 A second-order Taylor series approximation Pn b t b b b b t b to C(u , p ) around the base year price vector p is given =C(u , p )+ qi [ pipi ] i=1 by the following approximate equation: using assumption (17:76) and Shephard’s Lemma Pn b b b t b b @C(u , p ) t b (17.12) C(u , p )  C(u , p )+ [ pi pi ] i=1 @pi Pn Pn  b b b t b 1 Pn Pn @2C(ub, pb) = pi qi + qi [ pi pi ] using (17:76) + [ ptpb][ ptpb] i=1 i=1 i i j j 2 i=1j=1 @pi@pj Pn  t b Pn Pn Pn 2 b b = pi qi : (17:84) b b 1 @ C(u , p ) i=1 = pi qi + i=1 2 i=1j=1 @pi@pj Similarly, the first-order Taylor series approximation to t b t b C(ub, p0) around the annual base year price vector pb is [ pi pi ][ pj pj ] (17:86) given by the following approximate equation: where the last equality follows using approximate equa- 51 Pn b b tion (17.84). Similarly, a second-order Taylor series b 0 b b @C(u , p ) 0 b b 0 C(u , p )  C(u , p )+ [ pi pi ] approximation to C(u , p ) around the base year price i=1 @pi vector pb is given by the following approximate equation: Pn b b b b =C(u , p )+ q ½p0p Pn b b i i i b 0 b b @C(u , p ) 0 b i=1 C(u , p )  C(u , p )+ [ pi pi ] Pn Pn i=1 @pi b b b 0 b  = pi qi + qi ½pi pi 1 Pn Pn @2C(ub, pb) i=1 i=1 0 b 0 b + [ pi pi ][ pj pj ] Pn 2 i=1j=1 @pi@pj = p0qb (17:85)  i i Pn Pn Pn 2 b b i=1 0 b 1 @ C(u , p ) = pi qi + i=1 2 i=1j=1 @pi@pj 0 b 0 b 49 It can be seen that, when month t is set equal to month 0, et=e0 and [ pi pi ][ pj pj ] (17:87) b t b 0 C(u , p )=C(u , p ), and thus equation (17.83) is satisfied and PLo=PK. This is not surprising since both indices are equal to unity when t=0. 51 This type of second-order approximation is attributable to Hicks 50 This type of Taylor series approximation was used in Schultze and (1941–42, pp. 133–134; 1946, p. 331). See also Diewert (1992b, p. 568), Mackie (2002, p. 91) in the cost of living index context, but it essen- Hausman (2002, p. 18) and Schultze and Mackie (2002, p. 91). For tially dates back to Hicks (1941–42, p. 134) in the consumer surplus alternative approaches to modelling substitution bias, see Diewert context. See also Diewert (1992b, p. 568) and Hausman (2002, p. 8). (1998a; 2002c, pp. 598–603) and Hausman (2002). 330 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE where the last equality follows using the approximate where g is defined as follows: equation (17.85).  Pn Pn 2 b b 17.77 Comparing approximate equation (17.86) 1 @ C(u , p ) g aiaj 0 (17:93) with equation (17.80), and approximate equation 2 i=1j=1 @pi@pj (17.87) with equation (17.81), it can be seen that, to the accuracy of a second-order approximation, the 17.79 It should be noted that the parameter g will be 56 month 0 and month t substitution bias terms, e0 and et, zero under two sets of conditions: will be equal to the following expressions involving  All the second-order partial derivatives of the con- 2 b b the second-order partial derivatives of the consumer’s sumer’s cost function @ C(u , p )/@pi @pj are equal to 2 b b cost function @ C(u , p )/@pi@pj evaluated at the zero. b base year standard of living u and at the base year  Each commodity price change parameter a is pro- b j prices p : portional to the corresponding commodity j base year b 57  price pj . Pn Pn 2 b b 0 1 @ C(u , p ) 0 b 0 b The first condition is empirically unlikely since it implies e  [ pi pi ][ pj pj ] (17:88) 2 i=1j=1 @pi@pj that the consumer will not substitute away from com- modities of which the relative price has increased. The  second condition is also empirically unlikely, since it Pn Pn 2 b b t 1 @ C(u , p ) t b t b implies that the structure of relative prices remains e  [ pi pi ][ pj pj ] (17:89) 2 i=1j=1 @pi@pj unchanged over time. Thus, in what follows, it will be assumed that g is a positive number. Since the consumer’s cost function C(u, p) is a concave 17.80 In order to simplify the notation in what fol- function in the components of the price vector p,52 it is lows, define the denominator and numerator of the 53 0 t b known that the n by n (symmetric) matrix of second- month t Lowe index, PLo(p , p , q ), as a and b respec- 2 b b order partial derivatives [@ C(u , p )/@pi@pj] is negative tively; i.e., define: 54 b 0 semi-definite. Hence, for arbitrary price vectors p , p Pn t 0 b and p , the right-hand sides of approximations (17.88) a pi qi (17:94) and (17.89) will be non-negative. Thus, to the accuracy i=1 of a second-order approximation, the substitution bias 0 t Pn terms e and e will be non-negative. t b 17.78 Now assume that there are long-run sys- b piqi (17:95) i=1 tematic trends in prices. Assume that the last month of the base year for quantities occurs M months prior to Using equation (17.90) to eliminate the month 0 prices 0 t month 0, the base month for prices, and assume that pi from equation (17.94) and the month t prices pi from prices trend linearly with time, starting with the last equation (17.95) leads to the following expressions for a month of the base year for quantities. Thus, assume the and b: existence of constants aj for j=1, ..., n such that the Pn Pn b b b price of commodity j in month t is given by: a= pi qi + aiqi M (17:96) i=1 i=1 t b pi =pi +aj(M+t) for j=1, ..., n and t=0, 1, ..., T Pn Pn (17:90) b b b b= pi qi + aiqi (M+t) (17:97) i=1 i=1 Substituting equation (17.90) into approximations It is assumed that a and b58 are positive and that (17.88) and (17.89) leads to the following second-order approximations to the two substitution bias terms, e0 Pn t 55 b and e : aiqi 0 (17:98) i=1 e0  gM2 (17:91) 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 t 2 0 t b e  g(M+t) (17:92) index PK(p , p , q ) defined by equation (17.77) and the

56 A more general condition that ensures the positivity of g is that the 52 See Diewert (1993b, pp. 109–110). vector [a1, ..., an] is not an eigenvector of the matrix of second-order 53 2 See Diewert (1993b, p. 149). partial derivatives @ C(u, p)/@pi@pj that corresponds to a zero eigen- 54 value. A symmetric n by n matrix A with ijth element equal to aij is negative 57 semi-definiteP P if, and only if for every vector z:[z1, ..., zn], it is the case It is known that C(u, p) is linearly homogeneous in the components n n that i=1 j=1aij zizj 0. of the price vector p; see Diewert (1993b, p. 109) for example. Hence, 55 Note that the period 0 approximate bias defined by the right-hand using Euler’s Theorem on homogeneous functions, it can be shown that pb is an eigenvector of the matrix of second-order partial deriva- side of approximation (17.91) is fixed, while the period t approximate 2 bias defined by the right-hand side of (17.92) increases quadratically tivesP P@ C(u, p)/@pi@pj that corresponds to a zero eigenvalue and thus n n 2 b b with time t. Hence, the period t approximate bias term will eventually i=1 j=1[@ C(u, p)=@pi@pj ]pi pj =0; see Diewert (1993b, p. 149) for a overwhelm the period 0 approximate bias in this linear time trends detailed proof of this result. case, if t is allowed to become large enough. 58 It is also assumed that agM2 is positive. 331 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE

0 t b corresponding Lowe index PLo(p , p , q ): these assumptions means that: Pn Pn t 0 t b 0 t b b b b B PK ( p , p , q )PLo( p , p , q ) 2= p q =a=b a q =0 M=24; g=0:000008  i i i i C(ub, pt) b i=1 i=1 = (17:100) C(ub, p0) a using equations (17:94) and (17:95) Substituting the parameter values defined in equation  (17.100) into equation (17.99) leads to the following [bet] b = formula for the approximate amount that the Lowe [ae0] a index will exceed the corresponding true cost of living using equations (17:80) and (17:81) index at month t:  2 2 [bg(M+t) ] b t (96t+2t )  B =0:000008 (17:101) agM2 a 2(20:004608) using equations (17:91) and (17:92) Evaluating equation (17.101) at t=12, t=24, t=36, t=48 and t=60 leads to the following estimates for Bt: f(ba)M22aMtat2g 0.0029 (the approximate bias in the Lowe index at the =g 2 simplifying terms fa[agM ]g end of the first year of operation for the index); 0.0069 Pn Pn Pn b 2 b b b 2 (the bias after two years); 0.0121 (the bias after three aiqi t M 2 pi qi + aiqi M Mtat years); 0.0185 (the bias after four years); 0.0260 (the bias i=1 i=1 i=1 =g after five years). Thus, at the end of the first year of the a[a gM2] f g operation, the Lowe index will only be above the cor- using equations (17:96) and (17:97)  responding true cost of living index by approximately a Pn Pn third of a percentage point but, by the end of the fifth b 2 b b 2 aiqi t M 2 pi qi Mt+at year of operation, it will exceed the corresponding cost =g i=1 i=1 < 0 of living index by about 2.6 percentage points, which is fa[agM2]g no longer a negligible amount.60 using equation (17:98): (17:99) 17.83 The numerical results in the previous para- graph are only indicative of the approximate magnitude Thus, for t 1, the Lowe index will have an upward bias of the difference between a cost of living index and the (to the accuracy of a second-order Taylor series corresponding Lowe index. The important point to note approximation) compared to the corresponding true is that, to the accuracy of a second-order approximation, 0 t b cost of living index PK(p , p , q ), since the approximate the Lowe index will generally exceed its cost of living bias defined by the last expression in equation (17.99) is counterpart. The results also indicate, however, that this the sum of one non-positive and two negative terms. difference can be reduced to a negligible amount if: Moreover, this approximate bias will grow quadratically  the lag in obtaining the base year quantity weights is 59 in time t. minimized; and 17.82 In order to give the reader some idea of the  the base year is changed as frequently as possible. magnitude of the approximate bias Bt defined by the last It should also be noted that the numerical results depend line of equation (17.99), a simple special case will be on the assumption that long-run trends in prices exist, considered at this point. Suppose there are only two which may not be true,61 and on elasticity assumptions commodities and that, at the base year, all prices and that may not be justified.62 Statistical agencies should quantitiesP are equal to 1. Thus, pb=qb=1 for i=1, 2 and n i i prepare their own carefully constructed estimates of the pbqb=2. Assume that M=24 so that the base year i=1 i i differences between a Lowe index and a cost of living data on quantities take two years to process before the index in the light of their own particular circumstances. Lowe index can be implemented. Assume that the monthly rate of growth in price for commodity 1 is a1=0.002 so that after one year, the price of commodity 1 rises 0.024 or 2.4 per cent. Assume that commodity 2 The problem of seasonal commodities falls in price each month with a2=0.002 so that the price of commodity 2 falls 2.4 per cent in the first year 17.84 The assumption that the consumer has annual after the base year for quantities. Thus the relative price preferences over commodities purchased in the base year of the two commodities is steadily diverging by about 2 b b 5 per cent per year. Finally, assume that @ C(u , p )/ 60 Note that the relatively large magnitude of M compared to t leads to 2 b b 2 b b @p1@p1=@ C(u , p )/@p2@p2=1 and @ C(u , p )/@p1 a bias that grows approximately linearly with t rather than quad- 2 b b ratically. @p2=@ C(u , p )/@p2@p1=1. These assumptions imply 61 that the own elasticity of demand for each commodity is For mathematical convenience, the trends in prices were assumed to be linear, rather than the more natural assumption of geometric trends 1 at the base year consumer equilibrium. Making all of in prices. 62 Another key assumption that was used to derive the numerical 59 If M is large relative to t, then it can be seen that the first two terms results is the magnitude of the divergent trends in prices. If the price in the last equation of (17.99) can dominate the last term, which is the divergence vector is doubled to a1=0.004 and a2=0.004, then the quadratic in t term. parameter g quadruples and the approximate bias will also quadruple. 332 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE for the quantity weights, and that these annual pref- the prices and quantities of this new non-calendar year erences can be used in the context of making monthly could be compared to the prices and quantities of the purchases of the same commodities, was a key one in base year, where the January prices of the non-calendar relating the economic approach to index number theory year are matched to the January prices of the base year, to the Lowe index. This assumption that annual pref- the February prices of the non-calendar year are erences can be used in a monthly context is, however, matched to the February prices of the base year, and so somewhat questionable because of the seasonal nature on. If further assumptions are made on the macro-utility of some commodity purchases. The problem is that it is function F, then this framework can be used in order to very likely that consumers’ preference functions sys- justify a third type of cost of living index: a moving tematically change as the season of the year changes. year annual index.67 This index compares the cost over National customs and weather changes cause house- the past 12 months of achieving the annual utility holds to purchase certain and services during achieved in the base year with the base year cost, where some months and not at all for other months. the January costs in the current moving year are matched For example, Christmas trees are purchased only in to January costs in the base year, the February costs in December and ski jackets are not usually purchased the current moving year are matched to February costs during summer months. Thus, the assumption that in the base year, and so on. These moving year indices annual preferences are applicable during each month of can be calculated for each month of the current year and the year is only acceptable as a very rough approxima- the resulting series can be interpreted as (uncentred) tion to economic reality. seasonally adjusted (annual) price indices.68 17.85 The economic approach to index number 17.87 It should be noted that none of the three types theory can be adapted to deal with seasonal preferences. of indices described in the previous two paragraphs is The simplest economic approach is to assume that the suitable for describing the movements of prices going consumer has annual preferences over commodities from one month to the following month; i.e., they are classified not only by their characteristics but also by the not suitable for describing short-run movements in month of purchase.63 Thus, instead of assuming that the inflation. This is obvious for the first two types of index. consumer’s annual utility function is f(q) where q is an To see the problem with the moving year indices, con- n-dimensional vector, assume that the consumer’s sider a special case where the bundle of commodities annual utility function is F[ f 1(q1), f 2(q2), ..., f 12(q12)] purchased in each month is entirely specific to each where q1 is an n-dimensional vector of commodity pur- month. Then it is obvious that, even though all the chases made in January, q2 is an n-dimensional vector of above three types of index are well defined, none of them commodity purchases made in February, ..., and q12 is can describe anything useful about month-to-month an n-dimensional vector of commodity purchases made changes in prices, since it is impossible to compare like in December.64 The sub-utility functions f 1, f 2, ..., f 12 with like, going from one month to the next, under the represent the consumer’s preferences when making hypotheses of this special case. It is impossible to com- purchases in January, February, ...., and December, pare the incomparable. respectively. These monthly sub- can then be 17.88 Fortunately, it is not the case that household aggregated using the macro-utility function F in order to purchases in each month are entirely specific to the define overall annual utility. It can be seen that these month of purchase. Thus month-to-month price com- assumptions on preferences can be used to justify two parisons can be made if the commodity space is types of cost of living index: restricted to commodities that are purchased in each  an annual cost of living index that compares the prices month of the year. This observation leads to a fourth in all months of a current year with the corresponding type of cost of living index, a month-to-month index, monthly prices in a base year;65 and defined over commodities that are available in every 69  12 monthly cost of living indices where the index for month of the year. This model can be used to justify month m compares the prices of month m in the the economic approach described in paragraphs 17.66 to current year with the prices of month m in the base 17.83. Commodities that are purchased only in certain year for m=1, 2, ..., 12.66 months of the year, however, must be dropped from the scope of the index. Unfortunately, it is likely that 17.86 The annual Mudgett–Stone indices compare consumers have varying monthly preferences over the costs in a current calendar year with the corresponding commodities that are always available and, if this is costs in a base year. However, any month could be the case, the month-to-month cost of living index (and chosen as the year-ending month of the current year, and the corresponding Lowe index) defined over always- available commodities will generally be subject to sea- 63 This assumption and the resulting annual indices were first proposed sonal fluctuations. This will limit the usefulness of the by Mudgett (1955, p. 97) and Stone (1956, pp. 74–75). 64 If some commodities are not available in certain months m, then those commodities can be dropped from the corresponding monthly 67 See Diewert (1999a, pp. 56–61) for the details of this economic quantity vectors qm. approach. 65 For further details on how to implement this framework, see 68 See Diewert (1999a, pp. 67–68) for an empirical example of this Mudgett (1955, p. 97), Stone (1956, pp. 74–75) and Diewert (1998b, pp. approach applied to quantity indices. An empirical example of this 459 – 460). moving year approach to price indices is presented in Chapter 22. 66 For further details on how to implement this framework, see Diewert 69 See Diewert (1999a, pp. 51–56) for the assumptions on preferences (1999a, pp. 50–51). that are required in order to justify this economic approach. 333 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE index as a short-run indicator of general inflation since it the value of expenditures on commodity i in period 0. will be difficult to distinguish a seasonal movement in Then by the definition of the Laspeyres index defined the index from a systematic general movement in pri- over all n+1 commodities: ces.70 Note also that if the scope of the index is restricted nP+1 to always-available commodities, then the resulting p1q0 month-to-month index will not be comprehensive, i i Pn+1 i=1 whereas the moving year indices will be comprehensive L nP+1 0 0 in the sense of using all the available price information. pi qi 17.89 The above considerations lead to the conclu- i=1 1 0 sion that it may be useful for statistical agencies to n pn+1qn+1 produce at least two consumer price indices: =PL+ Pn (17:103) v0  a moving year index which is comprehensive and i i=1 seasonally adjusted, but which is not necessarily useful 0 for indicating month-to-month changes in general where pn+1=0 was used in order to derive the second inflation; and equation above. Thus the complete Laspeyres index n+1  a month-to-month index which is restricted to non- PL defined over all n+1 commodities is equal to the n seasonal commodities (and hence is not comprehen- incomplete Laspeyres index PL (which can be written in sive), but which is useful for indicating short-run traditional price relative and base period expenditure movements in general inflation. share form), plus the mixed or hybrid expenditure 1 0 pn+1qn+1 divided by theP base period expenditure on n 0 the first n commodities, i=1vi . Thus the complete The problem of a zero price Laspeyres index can be calculated using the usual information available to the price statistician plus two increasing to a positive price additional pieces of information: the new non-zero price 1 17.90 In a recent paper, Haschka (2003) raised the for commodity n+1 in period 1, pn+1, and an estimate problem of what to do when a price which was pre- of consumption of commodity n+1 in period 0 (when it 0 viously zero is increased to a positive level. He gave two was free), qn+1. Since it is often governments that examples for Austria, where parking and hospital fees change the previously zero price to a positive price, the were raised from zero to a positive level. In this situa- decision to do this is usually announced in advance, tion, it turns out that basket-type indices have an which will give the price statistician an opportunity to 0 advantage over indices that are weighted geometric form an estimate for the base period demand, qn+1. averages of price relatives, since basket-type indices are 17.93 Let the Paasche index between periods 0 and n well defined even if some prices are zero. 1, restricted to the first n commodities, be denoted as PP 17.91 The problem can be considered in the context and let the Paasche index, defined over all n+1 com- n+1 1 1 1 of evaluating the Laspeyres and Paasche indices. Suppose modities, be defined as PP . Also let vi pi qi denote t t as usual that the prices pi and quantities qi of the first n the value of expenditures on commodity i in period 1. commodities are positive for periods 0 and 1, but that the Then, by the definition of the Paasche index defined over price of commodity n+1 in period 0 is zero but is positive all n+1 commodities: in period 1. In both periods, the consumption of com- nP+1 modity n+1 is positive. Thus the assumptions on the p1q1 prices and quantities of commodity n+1 in the two per- i i Pn+1 i=1 iods under consideration can be summarized as follows: P nP+1 p0q1 0 1 0 1 i i pn+1=0 pn+1 > 0 qn+1 > 0 qn+1 > 0 (17:102) i=1 1 Typically, the increase in price of commodity n+1 from n vn+1 =PP+ Pn its initial non-zero level will cause consumption to fall so 0 1 1 0 pi qi that qn+1 < qn+1, but this inequality is not required for i=1 the analysis below. 1 n vn+1 17.92 Let the Laspeyres index between periods 0 and =PP+ Pn (17:104) 1, restricted to the first n commodities, be denoted as Pn 1 1 0 L vi =( pi =pi ) and let the Laspeyres index, defined over all n+1 com- i=1 n+1 0 0 0 modities, be defined as PL . Also let vi pi qi denote 0 where pn+1=0 was used in order to derive the second n+1 70 equation above. Thus the complete Paasche index PP One problem with using annual weights in the context of seasonal defined over all n+1 commodities is equal to the movements in prices and quantities is that a change in price when a n commodity is out of season can be greatly magnified by the use of incomplete Paasche index PP (which can be written in annual weights. Baldwin (1990, p. 251) noted this problem with an traditional price relative and current period expenditure annual weights price index: ‘‘But a price index is adversely affected if share form), plus the current period expenditure on any seasonal good has the same basket share for all months of the year; 1 commodity n+1, vn+1, divided by a sum of current the good will have an inappropriately small basket share in its in 1 period expenditures on the first n commodities, vi , divided season months, an inappropriately large share in its off season 1 0 months.’’ Seasonality problems are considered again from a more by the ith price relative for the first n commodities, pi =pi . pragmatic point of view in Chapter 22. Thus the complete Paasche index can be calculated using 334 THE ECONOMIC APPROACH TO INDEX NUMBER THEORY: THE SINGLE-HOUSEHOLD CASE the usual information available to the price statistician It should be noted that the complete Fisher index plus information on current period expenditures. defined by equation (17.105) satisfies the same exact 17.94 Once the complete Laspeyres and Paasche index number results as were demonstrated in para- indices have been calculated using equations (17.103) and graphs 17.27 to 17.32 above; i.e., the Fisher index (17.104), then the complete Fisher index can be calculated remains a superlative index even if prices are zero in one as the square root of the product of these two indices: period but positive in the other. Thus the Fisher price index remains a suitable target index even in the face of n+1 n+1 n+1 1=2 P F =[P L P P ] (17:105) zero prices.

335