THE AXIOMATIC AND STOCHASTIC APPROACHES TO INDEX NUMBER THEORY

THE AXIOMATIC AND STOCHASTIC APPROACHES TO INDEX NUMBER THEORY 6 Introducton 6. As was seen n Chapter 5, t s useful to be able to evaluate varous ndex number formulae that have been proposed n terms of ther propertes. If a formula turns out to have rather undesrable propertes, ths casts doubts on ts sutablty as an ndex that could be used by a statstcal agency as a target ndex. Lookng at the mathematcal propertes of ndex number formulae leads to the test or axomatc approach to ndex number theory. In ths approach, desrable propertes for an ndex number formula are proposed, and t s then attempted to determne whether any formula s consstent wth these propertes or tests. An deal outcome s the stuaton where the proposed tests are both desrable and completely determne the functonal form for the formula. 6.2 The axomatc approach to ndex number theory s not completely straghtforward, snce choces have to be made n two dmensons: The ndex number framework must be determned. Once the framework has been decded upon, t must be decded what tests or propertes should be mposed on the ndex number. The second pont s straghtforward: dfferent prce statstcans may have dfferent deas about whch tests are mportant, and alternatve sets of axoms can lead to alternatve best ndex number functonal forms. Ths pont must be kept n mnd whle readng ths chapter, snce there s no unversal agreement on what the best set of reasonable axoms s. Hence the axomatc approach can lead to more than one best ndex number formula. 6.3 The frst pont about choces lsted above requres further dscusson. In the prevous chapter, for the most part, the focus was on blateral ndex number theory;.e., t was assumed that prces and quanttes for the same n commodtes were gven for two perods and the object of the ndex number formula was to compare the overall level of prces n one perod wth the other perod. In ths framework, both sets of prce and quantty vectors were regarded as varables whch could be ndependently vared so that, for example, varatons n the prces of one perod dd not affect the prces of the other perod or the quanttes n ether perod. The emphass was on comparng the overall cost of a fxed basket of quanttes n the two perods or takng averages of such fxed basket ndces. Ths s an example of an ndex number framework. 6.4 However, other ndex number frameworks are possble. For example, nstead of decomposng a value rato nto a term that represents prce change between the two perods tmes another term that represents quantty change, an attempt could be made to decompose a value aggregate for one perod nto a sngle number that represents the prce level n the perod tmes another number that represents the quantty level n the perod. In the frst varant of ths approach, the prce ndex number s supposed to be a functon of the n commodty prces pertanng to that aggregate n the perod under consderaton, whle the quantty ndex number s supposed to be a functon of the n commodty quanttes pertanng to the aggregate n the perod. The resultng prce ndex functon was called an absolute ndex number by Frsch (930, p. 397), a prce level by Echhorn (978, p. 4) and a unlateral prce ndex by Anderson, Jones and Nesmth (997, p. 75). In a second varant of ths approach, the prce and quantty functons are allowed to depend on both the prce and quantty vectors pertanng to the perod under consderaton. These two varants of unlateral ndex number theory wll be consdered n paragraphs 6. to 6.29. 2 6.5 The remanng approaches n ths chapter are largely blateral approaches;.e., the prces and quanttes n an aggregate are compared for two perods. In paragraphs 6.30 to 6.73 and 6.94 to 6.29, the value rato decomposton approach s taken. 3 In paragraphs 6.30 to 6.73, the blateral prce and quantty ndces, P( p 0, p, q 0, q ) and Q( p 0, p, q 0, q ), are regarded as functons of the prce vectors pertanng to the two perods, p 0 and p, and the two quantty vectors, q 0 and q. Not only do the axoms or tests that are placed on the prce ndex P( p 0, p, q 0, q ) reflect reasonable prce ndex propertes, but some tests have ther orgn as reasonable tests on the quantty ndex Q( p 0, p, q 0, q ). The approach n paragraphs 6.30 to 6.73 smultaneously determnes the best prce and quantty ndces. 6.6 In paragraphs 6.74 to 6.93, attenton s shfted to the prce ratos for the n commodtes between Echhorn (978, p. 44) and Dewert (993d, p. 9) consdered ths approach. 2 In these unlateral ndex number approaches, the prce and quantty vectors are allowed to vary ndependently. In yet another ndex number framework, prces are allowed to vary freely but quanttes are regarded as functons of the prces. Ths leads to the economc approach to ndex number theory, whch s consdered brefly n Appendx 5.4 of Chapter 5, and n more depth n Chapters 7 and 8. 3 Recall paragraphs 5.7 to 5.7 of Chapter 5 for an explanaton of ths approach. 289

CONSUMER PRICE INDEX: THEORY AND PRACTICE perods 0 and, r p =p0 for,..., n. In the unweghted stochastc approach to ndex number theory, the prce ndex s regarded as an evenly weghted average of the n prce relatves or ratos, r. Carl (764) and Jevons (863; 865) were the earler poneers n ths approach to ndex number theory, wth Carl usng the arthmetc average of the prce relatves and Jevons endorsng the geometrc average (but also consderng the harmonc average). Ths approach to ndex number theory wll be covered n paragraphs 6.74 to 6.79. Ths approach s consstent wth a statstcal approach that regards each prce rato r as a random varable wth mean equal to the underlyng prce ndex. 6.7 A major problem wth the unweghted average of prce relatves approach to ndex number theory s that ths approach does not take nto account the economc mportance of the ndvdual commodtes n the aggregate. Young (82) dd advocate some form of rough weghtng of the prce relatves accordng to ther relatve value over the perod beng consdered, but the precse form of the requred value weghtng was not ndcated. 4 It was Walsh (90, pp. 83 2; 92a, pp. 8 90), however, who stressed the mportance of weghtng the ndvdual prce ratos, where the weghts are functons of the assocated values for the commodtes n each perod and each perod s to be treated symmetrcally n the resultng formula: What we are seekng s to average the varatons n the exchange value of one gven total sum of money n relaton to the several classes of goods, to whch several varatons [prce ratos] must be assgned weghts proportonal to the relatve szes of the classes. Hence the relatve szes of the classes at both the perods must be consdered (Walsh (90, p. 04)). Commodtes are to be weghted accordng to ther mportance, or ther full values. But the problem of axometry always nvolves at least two perods. There s a frst perod and there s a second perod whch s compared wth t. Prce varatons 5 have taken place between the two, and these are to be averaged to get the amount of ther varaton as a whole. But the weghts of the commodtes at the second perod are apt to be dfferent from ther weghts at the frst perod. Whch weghts, then, are the rght ones those of the frst perod or those of the second? Or should there be a combnaton of the two sets? There s no reason for preferrng ether the frst or the second. Then the combnaton of both would seem to be the proper answer. And ths combnaton tself nvolves an averagng of the weghts of the two perods (Walsh (92a, p. 90)). 4 Walsh (90, p. 84) refers to Young s contrbutons as follows: Stll, although few of the practcal nvestgators have actually employed anythng but even weghtng, they have almost always recognzed the theoretcal need of allowng for the relatve mportance of the dfferent classes ever snce ths need was frst ponted out, near the commencement of the century just ended, by Arthur Young.... Arthur Young advsed smply that the classes should be weghted accordng to ther mportance. 5 A prce varaton s a prce rato or prce relatve n Walsh s termnology. 6.8 Thus Walsh was the frst to examne n some detal the rather ntrcate problems 6 nvolved n decdng how to weght the prce relatves pertanng to an aggregate, takng nto account the economc mportance of the commodtes n the two perods beng consdered. Note that the type of ndex number formula that Walsh was consderng was of the form P(r, v 0, v ), where r s the vector of prce relatves whch has th component r =p =p0 and v t s the perod t value vector whch has th component v t =pt qt for t=0,. Hs suggested soluton to ths weghtng problem was not completely satsfactory but he dd at least suggest a very useful framework for a prce ndex, as a value-weghted average of the n prce relatves. The frst satsfactory soluton to the weghtng problem was obtaned by Thel (967, pp. 36 37) and hs soluton s explaned n paragraphs 6.79 to 6.93. 6.9 It can be seen that one of Walsh s approaches to ndex number theory 7 was an attempt to determne the best weghted average of the prce relatves, r. Ths s equvalent to usng an axomatc approach to try to determne the best ndex of the form P(r, v 0, v ). Ths approach s consdered n paragraphs 6.94 to 6.29. 8 6.0 The Young and Lowe ndces, dscussed n Chapter 5, do not ft precsely nto the blateral framework snce the value or quantty weghts used n these ndces do not necessarly correspond to the values or quanttes that pertan to ether of the perods that correspond to the prce vectors p 0 and p. The axomatc propertes of these two ndces wth respect to ther prce varables are studed n paragraphs 6.30 to 6.34. 6 Walsh (90, pp. 04 05) realzed that t would not do smply to take the arthmetc average of the values n the two perods, [v 0 +v ]=2, as the correct weght for the th prce relatve r snce, n a perod of rapd nflaton, ths would gve too much mportance to the perod that had the hghest prces and he wanted to treat each perod symmetrcally: But such an operaton s manfestly wrong. In the frst place, the szes of the classes at each perod are reckoned n the money of the perod, and f t happens that the exchange value of money has fallen, or prces n general have rsen, greater nfluence upon the result would be gven to the weghtng of the second perod; or f prces n general have fallen, greater nfluence would be gven to the weghtng of the second perod. Or n a comparson between two countres greater nfluence would be gven to the weghtng of the country wth the hgher level of prces. But t s plan that the one perod, or the one country, s as mportant, n our comparson between them, as the other, and the weghtng n the averagng of ther weghts should really be even. However, Walsh was unable to come up wth Thel s (967) soluton to the weghtng problem, whch was to use the average expendture share [s 0 +s ]=2, as the correct weght for the th prce relatve n the context of usng a weghted geometrc mean of the prce relatves. 7 Walsh also consdered basket-type approaches to ndex number theory, as was seen n Chapter 5. 8 In paragraphs 6.94 to 6.29, rather than startng wth ndces of the form P(r, v 0, v ), ndces of the form P( p 0, p, v 0, v ) are consdered. However, f the test of nvarance to changes n the unts of measurement s mposed on ths ndex, t s equvalent to studyng ndces of the form P(r, v 0, v ). Varta (976) also used a varaton of ths approach to ndex number theory. 290

THE AXIOMATIC AND STOCHASTIC APPROACHES TO INDEX NUMBER THEORY The levels approach to ndex number theory An axomatc approach to unlateral prce ndces 6. Denote the prce and quantty of commodty n n perod t by p t and q t respectvely for, 2,..., n and t=0,,..., T. The varable q t s nterpreted as the total amount of commodty transacted wthn perod t. In order to conserve the value of transactons, t s necessary that p t be defned as a unt value;.e., p t must be equal to the value of transactons n commodty for perod t dvded by the total quantty transacted, q t. In prncple, the perod of tme should be chosen so that varatons n commodty prces wthn a perod are very small compared to ther varatons between perods. 9 For t=0,,..., T, and,..., n, defne the value of transactons n commodty as v t pt qt and defne the total value of transactons n perod t as: V t Pn v t =Pn p t qt t=0,,..., T (6:) 6.2 Usng the above notaton, the followng levels verson of the ndex number problem s defned as follows: for t=0,,..., T, fnd scalar numbers P t and Q t such that V t =P t Q t t=0,,..., T (6:2) The number P t s nterpreted as an aggregate perod t prce level, whle the number Q t s nterpreted as an aggregate perod t quantty level. The aggregate prce level P t s allowed to be a functon of the perod t prce vector, p t, whle the aggregate perod t quantty level Q t s allowed to be a functon of the perod t 9 Ths treatment of prces as unt values over tme follows Walsh (90, p. 96; 92a, p. 88) and Fsher (922, p. 38). Fsher and Hcks both had the dea that the length of the perod should be short enough so that varatons n prce wthn the perod could be gnored, as the followng quotatons ndcate: Throughout ths book the prce of any commodty or the quantty of t for any one year was assumed gven. But what s such a prce or quantty? Sometmes t s a sngle quotaton for January or July, but usually t s an average of several quotatons scattered throughout the year. The queston arses: On what prncple should ths average be constructed? The practcal answer s any knd of average snce, ordnarly, the varatons durng a year, so far, at least, as prces are concerned, are too lttle to make any perceptble dfference n the result, whatever knd of average s used. Otherwse, there would be ground for subdvdng the year nto quarters or months untl we reach a small enough perod to be consdered practcally a pont. The quanttes sold wll, of course, vary wdely. What s needed s ther sum for the year (whch, of course, s the same thng as the smple arthmetc average of the per annum rates for the separate months or other subdvsons). In short, the smple arthmetc average, both of prces and of quanttes, may be used. Or, f t s worth whle to put any fner pont on t, we may take the weghted arthmetc average for the prces, the weghts beng the quanttes sold (Fsher (922, p. 38)). I shall defne a week as that perod of tme durng whch varatons n prces can be neglected. For theoretcal purposes ths means that prces wll be supposed to change, not contnuously, but at short ntervals. The calendar length of the week s of course qute arbtrary; by takng t to be very short, our theoretcal scheme can be ftted as closely as we lke to that ceaseless oscllaton whch s a characterstc of prces n certan markets (Hcks (946, p. 22)). quantty vector, q t ; hence: P t =c( p t ) and Q t =f (q t ) t=0,,..., T (6:3) 6.3 The functons c and f are to be determned somehow. Note that equaton (6.3) requres that the functonal forms for the prce aggregaton functon c and for the quantty aggregaton functon f be ndependent of tme. Ths s a reasonable requrement snce there s no reason to change the method of aggregaton as tme changes. 6.4 Substtutng equatons (6.3) and (6.2) nto equaton (6.) and droppng the superscrpts t means that c and f must satsfy the followng functonal equaton for all strctly postve prce and quantty vectors: c( p) f (q)= Pn p q for all p > 0 and for all q > 0 (6:4) 6.5 It s natural to assume that the functons c( p) and f(q) are postve f all prces and quanttes are postve: c( p,..., p n ) > 0; f (q,..., q n ) > 0 f all p > 0 and all q > 0 (6:5) 6.6 Let n denote an n-dmensonal vector of ones. Then (6.5) mples that when p= n, c( n ) s a postve number, a for example, and when q= n, then f( n ) s also a postve number, b for example;.e., (6.5) mples that c and f satsfy: c( n )=a > 0; f ( n )=b > 0 (6:6) 6.7 Let p= n and substtute the frst equaton n (6.6) nto equaton (6.4) n order to obtan the followng equaton: f (q)= Pn q for all q > 0 (6:7) a 6.8 Now let q= n and substtute the second equaton n (6.6) nto equaton (6.4) n order to obtan the followng equaton: c( p)= Pn p for all p > 0 (6:8) b 6.9 Fnally substtute equatons (6.7) and (6.8) nto the left-hand sde of equaton (6.4) to obtan the followng equaton: p b q a = Pn p q for all p > 0 and for all q > 0 (6:9) If n s greater than one, t s obvous that equaton (6.9) cannot be satsfed for all strctly postve p and q vectors. Thus f the number of commodtes n exceeds one, then there do not exst any functons c and f that satsfy equatons (6.4) and (6.5). 0 0 Echhorn (978, p. 44) establshed ths result. 29

CONSUMER PRICE INDEX: THEORY AND PRACTICE 6.20 Thus ths levels test approach to ndex number theory comes to an abrupt halt; t s frutless to look for prce and quantty level functons, P t =c( p t )andq t =f(q t ), that satsfy equatons (6.2) or (6.4) and also satsfy the very reasonable postvty requrements (6.5). 6.2 Note that the levels prce ndex functon, c( p t ), dd not depend on the correspondng quantty vector q t and the levels quantty ndex functon, f (q t ), dd not depend on the prce vector p t. Perhaps ths s the reason for the rather negatve result obtaned above. Hence, n the next secton, the prce and quantty functons are allowed to be functons of both p t and q t. A second axomatc approach to unlateral prce ndces 6.22 In ths secton, the goal s to fnd functons of 2n varables, c( p, q) andf( p, q), such that the followng counterpart to equaton (6.4) holds: c( p, q)f ( p, q)= Pn p q for all p > 0 and for all q > 0 (6:0) 6.23 Agan, t s natural to assume that the functons c( p, q) and f( p, q) are postve f all prces and quanttes are postve: c(p,...,p n ;q,...,q n ) > 0; f (p,...,p n ;q,...,q n ) > 0 f all p > 0 and all q > 0 (6:) 6.24 The present framework does not dstngush between the functons c and f, so t s necessary to requre that these functons satsfy some reasonable propertes. The frst property mposed on c s that ths functon be homogeneous of degree one n ts prce components: c(lp, q)=lc( p, q) for all l > 0 (6:2) Thus, f all prces are multpled by the postve number l, then the resultng prce ndex s l tmes the ntal prce ndex. A smlar lnear homogenety property s mposed on the quantty ndex f;.e., f s to be homogeneous of degree one n ts quantty components: f ( p, lq)=l f ( p, q) for all l > 0 (6:3) 6.25 Note that propertes (6.0), (6.) and (6.3) mply that the prce ndex c( p, q) has the followng homogenety property wth respect to the components of q: c( p, lq)= Pn p lq f ( p, lq) where l > 0 = Pn p lq lf ( p, q) usng (6:3) = Pn p q f ( p, q) =c( p, q) usng (6:0) and (6:) (6:4) Thus c( p, q) s homogeneous of degree zero n ts q components. 6.26 A fnal property that s mposed on the levels prce ndex c( p, q) s the followng one. Let the postve numbers d be gven. Then t s asked that the prce ndex be nvarant to changes n the unts of measurement for the n commodtes so that the functon c( p, q) has the followng property: c(d p,...,d n p n ;q =d,...,q n =d n ) =c(p,...,p n ; q,...,q n ) (6:5) 6.27 It s now possble to show that propertes (6.0), (6.), (6.2), (6.4) and (6.5) on the prce levels functon c( p, q) are nconsstent;.e., there does not exst a functon of 2n varables c( p, q) that satsfes these very reasonable propertes. 6.28 To see why ths s so, apply the equaton (6.5), settng d =q for each, to obtan the followng equaton: c( p,..., p n ; q,..., q n )=c( p q,..., p n q n ;,...,) (6:6) If c( p, q) satsfes the lnear homogenety property (6.2) so that c(lp, q)=lc( p, q), then equaton (6.6) mples that c( p, q) s also lnearly homogeneous n q so that c( p, lq)=lc( p, q). But ths last equaton contradcts equaton (6.4), whch establshes the mpossblty result. 6.29 The rather negatve results obtaned n paragraphs 6.3 to 6.2 ndcate that t s frutless to pursue the axomatc approach to the determnaton of prce and quantty levels, where both the prce and quantty vector are regarded as ndependent varables. 2 Hence, n the followng sectons of ths chapter, the axomatc approach to the determnaton of a blateral prce ndex of the form P( p 0, p, q 0, q ) wll be pursued. The frst axomatc approach to blateral prce ndces Blateral ndces and some early tests 6.30 In ths secton, the strategy wll be to assume that the blateral prce ndex formula, P( p 0, p, q 0, q ), satsfes a suffcent number of reasonable tests or propertes so that the functonal form for P s determned. 3 The word blateral 4 refers to the assumpton that the functon P depends only on the data pertanng to the two stuatons or perods beng compared;.e., P s regarded as a functon of the two sets of prce and quantty vectors, p 0, p, q 0, q, that Ths proposton s due to Dewert (993d, p. 9), but hs proof s an adaptaton of a closely related result due to Echhorn (978, pp. 44 45). 2 Recall that n the economc approach, the prce vector p s allowed to vary ndependently, but the correspondng quantty vector q s regarded as beng determned by p. 3 Much of the materal n ths secton s drawn from sectons 2 and 3 of Dewert (992a). For more recent surveys of the axomatc approach see Balk (995) and von Auer (200). 4 Multlateral ndex number theory refers to the case where there are more than two stuatons whose prces and quanttes need to be aggregated. 292

THE AXIOMATIC AND STOCHASTIC APPROACHES TO INDEX NUMBER THEORY are to be aggregated nto a sngle number that summarzes the overall change n the n prce ratos, p =p0,..., p n =p0 n. 6.3 In ths secton, the value rato decomposton approach to ndex number theory wll be taken;.e., along wth the prce ndex P( p 0, p, q 0, q ), there s a companon quantty ndex Q(p 0, p, q 0, q ) such that the product of these two ndces equals the value rato between the two perods. 5 Thus, throughout ths secton, t s assumed that P and Q satsfy the followng product test: V =V 0 =P( p 0, p, q 0, q ) Q( p 0, p, q 0, q ): (6:7) The perod t values, V t, for t=0, are defned by equaton (6.). As soon as the functonal form for the prce ndex P s determned, then equaton (6.7) can be used to determne the functonal form for the quantty ndex Q. A further advantage of assumng that the product test holds s that, f a reasonable test s mposed on the quantty ndex Q, then equaton (6.7) can be used to translate ths test on the quantty ndex nto a correspondng test on the prce ndex P. 6 6.32 If n=, so that there s only one prce and quantty to be aggregated, then a natural canddate for P s p =p0, the sngle prce rato, and a natural canddate for Q s q =q0, the sngle quantty rato. When the number of commodtes or tems to be aggregated s greater than, then what ndex number theorsts have done over the years s propose propertes or tests that the prce ndex P should satsfy. These propertes are generally multdmensonal analogues to the one good prce ndex formula, p =p0. Below, some 20 tests are lsted that turn out to characterze the Fsher deal prce ndex. 6.33 It wll be assumed that every component of each prce and quantty vector s postve;.e., p t 0 n and q t 0 7 n for t=0,. If t s desred to set q 0 =q, the common quantty vector s denoted by q; fts desred to set p 0 =p, the common prce vector s denoted by p. 6.34 The frst two tests, denoted T and T2, are not very controversal, so they wll not be dscussed n detal. T: Postvty: 8 P( p 0, p, q 0, q ) > 0 T2: Contnuty: 9 P( p 0, p, q 0, q ) s a contnuous functon of ts arguments 6.35 The next two tests, T3 and T4, are somewhat more controversal. 5 See paragraphs 5.7 to 5.25 of Chapter 5 for more on ths approach, whch was ntally due to Fsher (9, p. 403; 922). 6 Ths observaton was frst made by Fsher (9, pp. 400 406), and the dea was pursued by Vogt (980) and Dewert (992a). 7 The notaton q 0 n means that each component of the vector q s postve; q 0 n means each component of q s non-negatve and q>0 n means q 0 n and q=0 n. 8 Echhorn and Voeller (976, p. 23) suggested ths test. 9 Fsher (922, pp. 207 25) nformally suggested the essence of ths test. T3: Identty or constant prces test: 20 P( p, p, q 0, q )= That s, f the prce of every good s dentcal durng the two perods, then the prce ndex should equal unty, no matter what the quantty vectors are. The controversal aspect of ths test s that the two quantty vectors are allowed to be dfferent n the test. 2 T4: Fxed basket or constant quanttes test: 22 P( p 0, p, q, q)= p q q That s, f quanttes are constant durng the two perods so that q 0 =q :q, then the prce ndex should equal the P expendture on the constant basket n perod, n p q, dvded by the expendture on the basket n perod 0, q. 6.36 If the prce ndex P satsfes Test T4 and P and Q jontly satsfy the product test (6.7) above, then t s easy to show 23 that Q must satsfy the dentty test Q( p 0, p, q, q)= for all strctly postve vectors p 0, p, q. Ths constant quanttes test for Q s also somewhat controversal snce p 0 and p are allowed to be dfferent. Homogenety tests 6.37 The followng four tests, T5 T8, restrct the behavour of the prce ndex P as the scale of any one of the four vectors p 0, p, q 0, q changes. T5: Proportonalty n current prces: 24 P( p 0, lp, q 0, q )=lp( p 0, p, q 0, q ) for l > 0 That s, f all perod prces are multpled by the postve number l, then the new prce ndex s l tmes 20 Laspeyres (87, p. 308), Walsh (90, p. 308) and Echhorn and Voeller (976, p. 24) have all suggested ths test. Laspeyres came up wth ths test or property to dscredt the rato of unt values ndex of Drobsch (87a), whch does not satsfy ths test. Ths test s also a specal case of Fsher s (9, pp. 409 40) prce proportonalty test. 2 Usually, economsts assume that, gven a prce vector p, the correspondng quantty vector q s unquely determned. Here, the same prce vector s used but the correspondng quantty vectors are allowed to be dfferent. 22 The orgns of ths test go back at least 200 years to the Massachusetts legslature, whch used a constant basket of goods to ndex the pay of Massachusetts solders fghtng n the Amercan Revoluton; see Wllard Fsher (93). Other researchers who have suggested the test over the years nclude: Lowe (823, Appendx, p. 95), Scrope (833, p. 406), Jevons (865), Sdgwck (883, pp. 67 68), Edgeworth (925, p. 25) orgnally publshed n 887, Marshall (887, p. 363), Person (895, p. 332), Walsh (90, p. 540; 92b, pp. 543 544), and Bowley (90, p. 227). Vogt and Barta (997, p. 49) correctly observe that ths test s a specal case of Fsher s (9, p. 4) proportonalty test for quantty ndexes whch Fsher (9, p. 405) translated nto a test for the prce ndex usng the product test (5.3). 23 See Vogt (980, p. 70). 24 Ths test was proposed by Walsh (90, p. 385), Echhorn and Voeller (976, p. 24) and Vogt (980, p. 68). 293

CONSUMER PRICE INDEX: THEORY AND PRACTICE the old prce ndex. Put another way, the prce ndex functon P( p 0, p, q 0, q ) s (postvely) homogeneous of degree one n the components of the perod prce vector p. Most ndex number theorsts regard ths property as a very fundamental one that the ndex number formula should satsfy. 6.38 Walsh (90) and Fsher (9, p. 48; 922, p. 420) proposed the related proportonalty test P( p, lp, q 0, q )=l. Ths last test s a combnaton of T3 and T5; n fact Walsh (90, p. 385) noted that ths last test mples the dentty test, T3. 6.39 In the next test, nstead of multplyng all perod prces by the same number, all perod 0 prces are multpled by the number l. T6: Inverse proportonalty n base perod prces: 25 P(lp 0, p, q 0, q )=l P( p 0, p, q 0, q ) for l > 0 That s, f all perod 0 prces are multpled by the postve number l, then the new prce ndex s /l tmes the old prce ndex. Put another way, the prce ndex functon P( p 0, p, q 0, q ) s (postvely) homogeneous of degree mnus one n the components of the perod 0 prce vector p 0. 6.40 The followng two homogenety tests can also be regarded as nvarance tests. T7: Invarance to proportonal changes n current quanttes: P( p 0, p, q 0, lq )=P( p 0, p, q 0, q ) for all l > 0 That s, f current perod quanttes are all multpled by the number l, then the prce ndex remans unchanged. Put another way, the prce ndex functon P( p 0, p, q 0, q ) s (postvely) homogeneous of degree zero n the components of the perod quantty vector q. Vogt (980, p. 70) was the frst to propose ths test 26 and hs dervaton of the test s of some nterest. Suppose the quantty ndex Q satsfes the quantty analogue to the prce test T5;.e., suppose Q satsfes Q( p 0, p, q 0, lq )=lq( p 0, p, q 0, q ) for l>0. Then, usng the product test (6.7), t can be seen that P must satsfy T7. T8: Invarance to proportonal changes n base quanttes: 27 P( p 0, p, lq 0, q )=P( p 0, p, q 0, q ) for all l > 0 That s, f base perod quanttes are all multpled by the number l, then the prce ndex remans unchanged. Put another way, the prce ndex functon P( p 0, p, q 0, q )s (postvely) homogeneous of degree zero n the components of the perod 0 quantty vector q 0. If the quantty ndex Q satsfes the followng counterpart to T8: Q( p 0, p, lq 0, q )=l Q( p 0, p, q 0, q ) for all l>0, then usng 25 Echhorn and Voeller (976, p. 28) suggested ths test. 26 Fsher (9, p. 405) proposed the related test P( p 0, p, q 0, lq 0 )=P( p 0, p, q 0, q 0 )= p q0 = q0 : 27 Ths test was proposed by Dewert (992a, p. 26). equaton (6.7), the correspondng prce ndex P must satsfy T8. Ths argument provdes some addtonal justfcaton for assumng the valdty of T8 for the prce ndex functon P. 6.4 T7 and T8 together mpose the property that the prce ndex P does not depend on the absolute magntudes of the quantty vectors q 0 and q. Invarance and symmetry tests 6.42 The next fve tests, T9 T3, are nvarance or symmetry tests. Fsher (922, pp. 62 63, 458 460) and Walsh (90, p. 05; 92b, p. 542) seem to have been the frst researchers to apprecate the sgnfcance of these knds of tests. Fsher (922, pp. 62 63) spoke of farness but t s clear that he had symmetry propertes n mnd. It s perhaps unfortunate that he dd not realze that there were more symmetry and nvarance propertes than the ones he proposed; f he had, t s lkely that he would have been able to provde an axomatc characterzaton for hs deal prce ndex, as s done n paragraphs 6.53 to 6.56. The frst nvarance test s that the prce ndex should reman unchanged f the orderng of the commodtes s changed: T9: Commodty reversal test (or nvarance to changes n the orderng of commodtes): P( p 0 *, p *, q 0 *, q *)=P( p 0, p, q 0, q ) where p t * denotes a permutaton of the components of the vector p t and q t * denotes the same permutaton of the components of q t for t=0,. Ths test s attrbutable to Fsher (922, p. 63) 28 and t s one of hs three famous reversal tests. The other two are the tme reversal test and the factor reversal test, whch are consdered below. 6.43 The next test asks that the ndex be nvarant to changes n the unts of measurement. T0: Invarance to changes n the unts of measurement (commensurablty test): P(a p 0,..., a np 0 n ; a p,..., a np n ; a q0,..., a n q0 n ; a q,..., a n q n )=P( p0,..., p0 n ; p,..., p n ; q 0,..., q0 n ; q,..., q n ) for all a > 0,..., a n > 0 That s, the prce ndex does not change f the unts of measurement for each commodty are changed. The concept of ths test s attrbutable to Jevons (863, p. 23) and the Dutch economst Person (896, p. 3), who crtczed several ndex number formulae for not satsfyng ths fundamental test. Fsher (9, p. 4) frst called ths test the change of unts test; later, Fsher (922, p. 420) called t the commensurablty test. 28 Ths [test] s so smple as never to have been formulated. It s merely taken for granted and observed nstnctvely. Any rule for averagng the commodtes must be so general as to apply nterchangeably to all of the terms averaged (Fsher (922, p. 63)). 294

THE AXIOMATIC AND STOCHASTIC APPROACHES TO INDEX NUMBER THEORY 6.44 The next test asks that the formula be nvarant to the perod chosen as the base perod. T: Tme reversal test: P( p 0, p, q 0, q )==P( p, p 0, q, q 0 ) That s, f the data for perods 0 and are nterchanged, then the resultng prce ndex should equal the recprocal of the orgnal prce ndex. Obvously, n the one good case when the prce ndex s smply the sngle prce rato, ths test wll be satsfed (as are all the other tests lsted n ths secton). When the number of goods s greater than one, many commonly used prce ndces fal ths test; e.g., the Laspeyres (87) prce ndex, P L defned by equaton (5.5) n Chapter 5, and the Paasche (874) prce ndex, P P defned by equaton (5.6) n Chapter 5, both fal ths fundamental test. The concept of the test s attrbutable to Person (896, p. 28), who was so upset by the fact that many of the commonly used ndex number formulae dd not satsfy ths test that he proposed that the entre concept of an ndex number should be abandoned. More formal statements of the test were made by Walsh (90, p. 368; 92b, p. 54) and Fsher (9, p. 534; 922, p. 64). 6.45 The next two tests are more controversal, snce they are not necessarly consstent wth the economc approach to ndex number theory. These tests are, however, qute consstent wth the weghted stochastc approach to ndex number theory, dscussed later n ths chapter. T2: Quantty reversal test (quantty weghts symmetry test): P( p 0, p, q 0, q )=P( p 0, p, q, q 0 ) That s, f the quantty vectors for the two perods are nterchanged, then the prce ndex remans nvarant. Ths property means that f quanttes are used to weght the prces n the ndex number formula, then the perod 0 quanttes q 0 and the perod quanttes q must enter the formula n a symmetrc or even-handed manner. Funke and Voeller (978, p. 3) ntroduced ths test; they called t the weght property. 6.46 The next test s the analogue to T2 appled to quantty ndces: 0 B @ T3: Prce reversal test (prce weghts symmetry test): 29 p q q0 0, C A P(p 0,p,q 0,q )= B @ q p q0, C A P(p, p 0, q 0, q ) (6:8) Thus f we use equaton (6.7) to defne the quantty ndex Q n terms of the prce ndex P, then t can be seen that T3 s equvalent to the followng property for the assocated quantty ndex Q: Q( p 0, p, q 0, q )=Q( p, p 0, q 0, q ) (6:9) That s, f the prce vectors for the two perods are nterchanged, then the quantty ndex remans nvarant. Thus f prces for the same good n the two perods are used to weght quanttes n the constructon of the quantty ndex, then property T3 mples that these prces enter the quantty ndex n a symmetrc manner. Mean value tests 6.47 The next three tests, T4 T6, are mean value tests. T4: Mean value test for prces: 30 mn ( p =p0 :,..., n) P( p 0, p, q 0, q ) max ( p =p0 :,..., n) (6:20) That s, the prce ndex les between the mnmum prce rato and the maxmum prce rato. Snce the prce ndex s supposed to be nterpreted as some sort of an average of the n prce ratos, p =p0, t seems essental that the prce ndex P satsfy ths test. 6.48 The next test s the analogue to T4 appled to quantty ndces: T5: Mean value test for quanttes: 3 mn (q =q0 :,..., n) (V =V 0) P( p 0, p, q 0, q ) max (q =q0 :,..., n) (6:2) where V t s the perod t value for the aggregate defned by equaton (6.). Usng the product test (6.7) to defne the quantty ndex Q n terms of the prce ndex P, t can be seen that T5 s equvalent to the followng property for the assocated quantty ndex Q: mn (q =q0 :,..., n) Q( p0, p, q 0, q ) max (q =q0 :,..., n) (6:22) That s, the mplct quantty ndex Q defned by P les between the mnmum and maxmum rates of growth q =q0 of the ndvdual quanttes. 6.49 In paragraphs 5.8 to 5.32 of Chapter 5, t was argued that t s very reasonable to take an average of the Laspeyres and Paasche prce ndces as a sngle best measure of overall prce change. Ths pont of 29 Ths test was proposed by Dewert (992a, p. 28). 30 Ths test seems to have been frst proposed by Echhorn and Voeller (976, p. 0). 3 Ths test was proposed by Dewert (992a, p. 29). 295

CONSUMER PRICE INDEX: THEORY AND PRACTICE vew can be turned nto a test: T6: Paasche and Laspeyres boundng test: 32 The prce ndex P les between the Laspeyres and Paasche ndces, P L and P P, defned by equatons (5.5) and (5.6) n Chapter 5. A test could be proposed where the mplct quantty ndex Q that corresponds to P va equaton (6.7) s to le between the Laspeyres and Paasche quantty ndces, Q P and Q L, defned by equatons (5.0) and (5.) n Chapter 5. However, the resultng test turns out to be equvalent to test T6. Monotoncty tests 6.50 The fnal four tests, T7 T20, are monotoncty tests;.e., how should the prce ndex P( p 0, p, q 0, q ) change as any component of the two prce vectors p 0 and p ncreases or as any component of the two quantty vectors q 0 and q ncreases? T7: Monotoncty n current prces: P( p 0, p, q 0, q ) < P( p 0, p 2, q 0, q )fp < p 2 That s, f some perod prce ncreases, then the prce ndex must ncrease, so that P( p 0, p, q 0, q ) s ncreasng n the components of p. Ths property was proposed by Echhorn and Voeller (976, p. 23) and t s a very reasonable property for a prce ndex to satsfy. T8: Monotoncty n base prces: P( p 0, p, q 0, q ) > P( p 2, p, q 0, q )fp 0 < p 2 That s, f any perod 0 prce ncreases, then the prce ndex must decrease, so that P( p 0, p, q 0, q ) s decreasng n the components of p 0. Ths very reasonable property was also proposed by Echhorn and Voeller (976, p. 23). T9: Monotoncty n current quanttes: f q < q 2, then 0 B @ 0 < B @ p q q0 p q2 q0 C A P(p 0,p,q 0,q ) C A P(p 0,p,q 0,q 2 ) (6:23) T20: Monotoncty n base quanttes: fq 0 < q 2, then 0 p q B C @ A P( p 0, p, q 0, q ) q0 0 > B @ p q q2 C A P( p 0, p, q 2, q ) (6:24) 6.5 Let Q be the mplct quantty ndex that corresponds to P usng equaton (6.7). Then t s found that T9 translates nto the followng nequalty nvolvng Q: Q( p 0, p, q 0, q ) < Q( p 0, p, q 0, q 2 ) f q < q 2 (6:25) That s, f any perod quantty ncreases, then the mplct quantty ndex Q that corresponds to the prce ndex P must ncrease. Smlarly, we fnd that T20 translates nto: Q( p 0, p, q 0, q ) > Q( p 0, p, q 2, q ) f q 0 < q 2 (6:26) That s, f any perod 0 quantty ncreases, then the mplct quantty ndex Q must decrease. Tests T9 and T20 are attrbutable to Vogt (980, p. 70). 6.52 Ths concludes the lstng of tests. The next secton offers an answer to the queston of whether any ndex number formula P( p 0, p, q 0, q ) exsts that can satsfy all 20 tests. The Fsher deal ndex and the test approach 6.53 It can be shown that the only ndex number formula P( p 0, p, q 0, q ) whch satsfes tests T T20 s the Fsher deal prce ndex P F defned as the geometrc mean of the Laspeyres and Paasche ndces: 33 P F ( p 0, p, q 0, q ) fp L ( p 0, p, q 0, q )P p ( p 0, p, q 0, q )g =2 (6:27) 6.54 It s relatvely straghtforward to show that the Fsher ndex satsfes all 20 tests. The more dffcult part of the proof s to show that the Fsher ndex s the only ndex number formula that satsfes these tests. Ths part of the proof follows from the fact that, f P satsfes the postvty test T and the three reversal tests, T T3, then P must equal P F. To see ths, rearrange the terms n the statement of test T3 nto the 32 Bowley (90, p. 227) and Fsher (922, p. 403) both endorsed ths property for a prce ndex. 33 See Dewert (992a, p. 22). 296

THE AXIOMATIC AND STOCHASTIC APPROACHES TO INDEX NUMBER THEORY followng equaton: p q q q0 p q0 = P( p0, p, q 0, q ) P( p, p 0, q 0, q ) = P( p0, p, q 0, q ) P( p, p 0, q, q 0 ) usng T2, the quantty reversal test =P( p 0, p, q 0, q )P( p 0, p, q 0, q ) usng T, the tme reversal test (6:28) Now take postve square roots of both sdes of equaton (6.28). It can be seen that the left-hand sde of the equaton s the Fsher ndex P F ( p 0, p, q 0, q ) defned by equaton (6.27) and the rght-hand sde s P( p 0, p, q 0, q ). Thus f P satsfes T, T, T2 and T3, t must equal the Fsher deal ndex P F. 6.55 The quantty ndex that corresponds to the Fsher prce ndex usng the product test (6.7) s Q F, the Fsher quantty ndex, defned by equaton (5.4) n Chapter 5. 6.56 It turns out that P F satsfes yet another test, T2, whch was Fsher s (92, p. 534; 922, pp. 72 8) thrd reversal test (the other two beng T9 and T): T2: Factor reversal test (functonal form symmetry test): P( p 0, p, q 0, q )P(q 0, q, p 0, p )= p q q0 (6:29) A justfcaton for ths test s the followng: f P( p 0, p, q 0, q ) s a good functonal form for the prce ndex, then, f the roles of prces and quanttes are reversed, P(q 0, q, p 0, p ) ought to be a good functonal form for a quantty ndex (whch seems to be a correct argument) and thus the product of the prce ndex P( p 0, p, q 0, q ) and the quantty ndex Q( p 0, p, q 0, q )= P(q 0, q, p 0, p ) ought to equal the value rato, V /V 0. The second part of ths argument does not seem to be vald, and thus many researchers over the years have objected to the factor reversal test. Nevertheless, f T2 s accepted as a basc test, Funke and Voeller (978, p. 80) showed that the only ndex number functon P( p 0, p, q 0, q ) whch satsfes T (postvty), T (tme reversal test), T2 (quantty reversal test) and T2 (factor reversal test) s the Fsher deal ndex P F defned by equaton (6.27). Thus the prce reversal test T3 can be replaced by the factor reversal test n order to obtan a mnmal set of four tests that lead to the Fsher prce ndex. 34 34 Other characterzatons of the Fsher prce ndex can be found n Funke and Voeller (978) and Balk (985; 995). The test performance of other ndces 6.57 The Fsher prce ndex P F satsfes all 20 of the tests T T20 lsted above. Whch tests do other commonly used prce ndces satsfy? Recall the Laspeyres ndex P L defned by equaton (5.5), the Paasche ndex P P defned by equaton (5.6), the Walsh ndex P W defned by equaton (5.9) and the To rnqvst ndex P T defned by equaton (5.8) n Chapter 5. 6.58 Straghtforward computatons show that the Paasche and Laspeyres prce ndces, P L and P P, fal only the three reversal tests, T, T2 and T3. Snce the quantty and prce reversal tests, T2 and T3, are somewhat controversal and hence can be dscounted, the test performance of P L and P P seems at frst sght to be qute good. The falure of the tme reversal test, T, s nevertheless a severe lmtaton assocated wth the use of these ndces. 6.59 The Walsh prce ndex, P W, fals four tests: T3, the prce reversal test; T6, the Paasche and Laspeyres boundng test; T9, the monotoncty n current quanttes test; and T20, the monotoncty n base quanttes test. 6.60 Fnally, the To rnqvst prce ndex P T fals nne tests: T4 (the fxed basket test), the quantty and prce reversal tests T2 and T3, T5 (the mean value test for quanttes), T6 (the Paasche and Laspeyres boundng test) and the four monotoncty tests T7 to T20. Thus the To rnqvst ndex s subject to a rather hgh falure rate from the vewpont of ths axomatc approach to ndex number theory. 35 6.6 The tentatve concluson that can be drawn from the above results s that, from the vewpont of ths partcular blateral test approach to ndex numbers, the Fsher deal prce ndex P F appears to be best snce t satsfes all 20 tests. The Paasche and Laspeyres ndces are next best f we treat each test as beng equally mportant. Both of these ndces, however, fal the very mportant tme reversal test. The remanng two ndces, the Walsh and To rnqvst prce ndces, both satsfy the tme reversal test but the Walsh ndex emerges as beng better snce t passes 6 of the 20 tests whereas the To rnqvst only satsfes tests. 36 The addtvty test 6.62 There s an addtonal test that many natonal ncome accountants regard as very mportant: the addtvty test. Ths s a test or property that s placed on the mplct quantty ndex Q( p 0, p, q 0, q ) that corresponds to the prce ndex P( p 0, p, q 0, q ) usng the product test (6.7). Ths test states that the mplct quantty ndex has the 35 It s shown n Chapter 9, however, that the To rnqvst ndex approxmates the Fsher ndex qute closely usng normal tme seres data that are subject to relatvely smooth trends. Hence, under these crcumstances, the To rnqvst ndex can be regarded as passng the 20 tests to a reasonably hgh degree of approxmaton. 36 Ths asserton needs to be qualfed: there are many other tests that we have not dscussed, and prce statstcans mght hold dfferent opnons regardng the mportance of satsfyng varous sets of tests. Other tests are dscussed by von Auer (200; 2002), Echhorn and Voeller (976), Balk (995) and Vogt and Barta (997), among others. It s shown n paragraphs 6.0 to 6.35 that the To rnqvst ndex s deal when consdered under a dfferent set of axoms. 297

CONSUMER PRICE INDEX: THEORY AND PRACTICE followng form: Q( p 0, p, q 0, q )= p * q p * m q0 m m= (6:30) where the common across-perods prce for commodty, p * for,..., n, can be a functon of all 4n prces and quanttes pertanng to the two perods or stuatons under consderaton, p 0, p, q 0, q. In the lterature on makng multlateral comparsons (.e., comparsons between more than two stuatons), t s qute common to assume that the quantty comparson between any two regons can be made usng the two regonal quantty vectors, q 0 and q, and a common reference prce vector, P * ( p *,..., p* n ):37 6.63 Obvously, dfferent versons of the addtvty test can be obtaned f further restrctons are placed on precsely whch varables each reference prce p * depends. The smplest such restrcton s to assume that each p * depends only on the commodty prces pertanng to each of the two stuatons under consderaton, and p. If t s further assumed that the functonal form for the weghtng functon s the same for each commodty, so that p * =m( p0, p ) for,..., n, then we are led to the unequvocal quantty ndex postulated by Knbbs (924, p. 44). 6.64 The theory of the unequvocal quantty ndex (or the pure quantty ndex) 38 parallels the theory of the pure prce ndex outlned n paragraphs 5.24 to 5.32 of Chapter 5. An outlne of ths theory s gven here. Let the pure quantty ndex Q K have the followng functonal form: Q K ( p 0, p, q 0, q ) k= q m( p0, p ) q 0 k m( p0 k, p k ) (6:3) It s assumed that the prce vectors p 0 and p are strctly postve and the quantty vectors q 0 and q are non-negatve but have at least one postve component. 39 The problem s to determne the functonal form for the averagng functon m f possble. To do ths, t s necessary to mpose some tests or propertes on the pure quantty ndex Q K.As was the case wth the pure prce ndex, t s very reasonable to ask that the quantty ndex satsfy the tme reversal test: Q K ( p, p 0, q, q 0 )= Q K ( p 0, p, q 0, q (6:32) ) 37 Hll (993, p. 395 397) termed such multlateral methods the block approach whle Dewert (996a, pp. 250 25) used the term average prce approaches. Dewert (999b, p. 9) used the term addtve multlateral system. For axomatc approaches to multlateral ndex number theory, see Balk (996a; 200) and Dewert (999b). 38 Dewert (200) used ths term. 39 It s assumed that m(a, b) has the followng two propertes: m(a, b) s a postve and contnuous functon, defned for all postve numbers a and b, and m(a, a)=a for all a>0. 6.65 As was the case wth the theory of the unequvocal prce ndex, t can be seen that f the unequvocal quantty ndex Q K s to satsfy the tme reversal test (6.32), the mean functon n equaton (6.3) must be symmetrc. It s also asked that Q K satsfy the followng nvarance to proportonal changes n current prces test. Q K ( p 0, lp, q 0, q )=Q K ( p 0, p, q 0, q ) for all p 0, p, q 0, q and all l > 0 (6:33) 6.66 The dea behnd ths nvarance test s ths: the quantty ndex Q K ( p 0, p, q 0, q ) should depend only on the relatve prces n each perod and t should not depend on the amount of nflaton between the two perods. Another way to nterpret test (6.33) s to look at what the test mples for the correspondng mplct prce ndex, P IK, defned usng the product test (6.7). It can be shown that f Q K satsfes equaton (6.33), then the correspondng mplct prce ndex P IK wll satsfy test T5 above, the proportonalty n current prces test. The two tests, (6.32) and (6.33), determne the precse functonal form for the pure quantty ndex Q K defned by equaton (6.3): the pure quantty ndex or Knbbs unequvocal quantty ndex Q K must be the Walsh quantty ndex Q 40 W defned by: Q W ( p 0, p, q 0, q ) q q 0 k k= pffffffffffffffffff p q ffffffffffffffffff (6:34) p 0 k p k 6.67 Thus wth the addton of two tests, the pure prce ndex P K must be the Walsh prce ndex P W defned by equaton (5.9) n Chapter 5 and wth the addton of the same two tests (but appled to quantty ndces nstead of prce ndces), the pure quantty ndex Q K must be the Walsh quantty ndex Q W defned by equaton (6.34). Note, however, that the product of the Walsh prce and quantty ndces s not equal to the expendture rato, V /V 0. Thus belevers n the pure or unequvocal prce and quantty ndex concepts have to choose one of these two concepts; they cannot both apply smultaneously. 4 6.68 If the quantty ndex Q( p 0, p, q 0, q ) satsfes the addtvty test (6.30) for some prce weghts p *, then the percentage change n the quantty aggregate, Q( p 0, p, q 0, q ), can be rewrtten as follows: Q( p 0, p, q 0, q ) = p * q p * m q0 m m= = Pn = p * Pn q p * m q0 m m= p * m q0 m m= w (q q0 ) (6:35) 40 Ths s the quantty ndex that corresponds to the prce ndex 8 defned by Walsh (92a, p. 0). 4 Knbbs (924) dd not notce ths pont. 298

THE AXIOMATIC AND STOCHASTIC APPROACHES TO INDEX NUMBER THEORY where the weght for commodty, w, s defned as w p * p * m q0 m m= ;,..., n (6:36) Note that the change n commodty gong from stuaton 0 to stuaton s q q0.thustheth term on the rght-hand sde of equaton (6.35) s the contrbuton of the change n commodty to the overall percentage change n the aggregate gong from perod 0 to. Busness analysts often want statstcal agences to provde decompostons such as equaton (6.35) so that they can decompose the overall change n an aggregate nto sectorspecfc components of change. 42 Thus there s a demand on the part of users for addtve quantty ndces. 6.69 For the Walsh quantty ndex defned by equaton (6.34), the th weght s pffffffffffffffffff w W p pffffffffffffffffffffff ;,..., n (6:37) q 0 m m= p 0 m p m Thus the Walsh quantty ndex Q W has a percentage decomposton nto component changes of the form of equaton (6.35), where the weghts are defned by equaton (6.37). 6.70 It turns out that the Fsher quantty ndex Q F, defned by equaton (5.4) n Chapter 5, also has an addtve percentage change decomposton of the form gven by equaton (6.35). 43 The th weght w F for ths Fsher decomposton s rather complcated and depends on the Fsher quantty ndex Q F ( p 0, p, q 0, q ) as follows: 44 w F w0 +(Q F) 2 w ;,..., n (6:38) +Q F where Q F s the value of the Fsher quantty ndex, Q F ( p 0, p, q 0, q ), and the perod t normalzed prce for commodty, w t, s defned as the perod prce pt dvded by the perod t expendture on the aggregate: w t p t p t m qt m m= ; t=0, ;,..., n (6:39) 6.7 Usng the weghts w F defned by equatons (6.38) and (6.39), the followng exact decomposton s 42 Busness and government analysts also often demand an analogous decomposton of the change n prce aggregate nto sector-specfc components that add up. 43 The Fsher quantty ndex also has an addtve decomposton of the type defned by equaton (6.30) attrbutable to Van Ijzeren (987, p. 6). The th reference prce p * s defned as p * (=2) +(=2)p = P F ( p 0, p, q 0, q ) for,..., n and where P F s the Fsher prce ndex. Ths decomposton was also ndependently derved by Dkhanov (997). The Van Ijzeren decomposton for the Fsher quantty ndex s currently beng used by the US Bureau of Economc Analyss; see Moulton and Seskn (999, p. 6) and Ehemann, Katz and Moulton (2002). 44 Ths decomposton was obtaned by Dewert (2002a) and Rensdorf, Dewert and Ehemann (2002). For an economc nterpretaton of ths decomposton, see Dewert (2002a). obtaned for the Fsher deal quantty ndex: Q F ( p 0, p, q 0, q ) = Pn w F (q q0 ) (6:40) Thus the Fsher quantty ndex has an addtve percentage change decomposton. 45 6.72 Because of the symmetrc nature of the Fsher prce and quantty ndces, t can be seen that the Fsher prce ndex P F defned by equaton (6.27) also has the followng addtve percentage change decomposton: P F ðp 0, p, q 0, q Þ = Pn v F ( p p0 ) (6:4) where the commodty weght v F s defned as v F v0 +(P F) 2 v ;,..., n (6:42) +P F where P F s the value of the Fsher prce ndex, P F ( p 0, p, q 0, q ), and the perod t normalzed quantty for commodty, v t, s defned as the perod quantty qt dvded by the perod t expendture on the aggregate: v t q t p t m qt m m= ; t=0, ;,..., n (6:43) 6.73 The above results show that the Fsher prce and quantty ndces have exact addtve decompostons nto components that gve the contrbuton to the overall change n the prce (or quantty) ndex of the change n each prce (or quantty). The stochastc approach to prce ndces The early unweghted stochastc approach 6.74 The stochastc approach to the determnaton of the prce ndex can be traced back to the work of Jevons (863; 865) and Edgeworth (888) over 00 years ago. 46 The basc dea behnd the (unweghted) stochastc approach s that each prce relatve, p =p0 for, 2,..., n, can be regarded as an estmate of a common nflaton rate a between perods 0 and. 47 It s assumed that p =a+e ;, 2,..., n (6:44) where a s the common nflaton rate and the e are random varables wth mean 0 and varance s 2. The least 45 To verfy the exactness of the decomposton, substtute equaton (6.38) nto equaton (6.40) and solve the resultng equaton for Q F.It s found that the soluton s equal to Q F defned by equaton (5.4) n Chapter 5. 46 For references to the lterature, see Dewert (993a, pp. 37 38; 995a; 995b). 47 In drawng our averages the ndependent fluctuatons wll more or less destroy each other; the one requred varaton of gold wll reman undmnshed (Jevons (863, p. 26)). 299