# Solving the Index-Number Problem in a Historical Perspective

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1 Solvng the Index-Number roblem n a Hstorcal erspectve Carlo Mlana * January 2009 "The fundamental and well-known theorem for the exstence of a prce ndex that s nvarant under change n level of lvng s that each dollar of ncome be spent n the same way by rch or poor, wth all ncome elastctes exactly unty (the homothetc case). Otherwse, a prce change n luxures could affect only the prce ndex of the rch whle leavng that of the poor relatvely unchanged. Ths basc theorem was well known already n the 930's, but s often forgotten and s repeatedly beng redscovered". "* + Although most attenton n the lterature s devoted to prce ndexes, when you analze the use to whch prce ndexes are generally put, you realze that quantty ndexes are actually most mportant. Once somehow estmated, prce ndexes are n fact used, f at all, prmarly to 'deflate' nomnal or monetary totals n order to arrve at estmates of underlyng 'real magntudes' (whch s to say, quantty ndexes!)". "* + The fundamental pont about an economc quantty ndex, whch s too lttle stressed by wrters, Leontef and Afrat beng exceptons, s that t must tself be a cardnal ndcator of ordnal utlty"..a. Samuelson and S. Swamy (974, pp ) Introducton The ndex-number problem s typcally a problem of aggregaton of changes n heterogeneous elements. Mathematcally, t conssts n reducng the relatve change of the elements of a vector nto changes n one sngle numercal value, a scalar. In hs famous Econometrca survey of general economc theory dedcated to the problem of ndex numbers, Ragnar Frsch (936, p. ) descrbed t n these terms: The ndex-number problem arses whenever we want a * Isttuto d Stud e Anals Economca, azza dell Indpendenza, 4, 0085 Roma, Tel , e-mal: personal e-mal: paper has been prepared for the EUKLEMS project funded by the European Commsson, Research Drectorate General as part of the 6 th Framework rogramme.

2 quanttatve expresson for a complex that s made up of ndvdual measurements for whch non common physcal unt exsts. The desre to unte such measurements and the fact that ths cannot be done by usng physcal or techncal prncples of comparson only, consttute the essence of the ndex-number problem and all the dffcultes center here. In economcs, the soluton of ths problem s necessary n every decomposton of changes of total nomnal values nto meanngful aggregate prce and quantty components. The natonal accountants are asked to provde a splt of the changes of nomnal economc aggregates nto a deflator and a volume component. Smlarly, montorng monetary polces usually entals a decomposton of the ndex of money supply nto an nflaton ndex and a volume representng the purchasng power of crculatng money. At frm level, changes n nomnal profts can be accounted for by decomposng them nto a productvty component (a volume ndex) and market prce condtons (a deflator or prce ndex). It turns out that ths s possble only under very restrctve condtons. In the general case, every attempt of forcng the applcaton of ndex number formulas s doomed to yeld msleadng results (see, e.g., McCusker, 200, Derks, 2004, Offcer and Wllamson, 2006 on ntertemporal comparsons of the purchasng power of money and Leontef, 936 and Samuelson, 947, p. 62, who warned us aganst the tendency to attach sgnfcance to the numercal value of the ndex computed ). Even when the aggregaton condtons are not rejected on the bass of the observed data, there stll remans a certan degree of uncertanty regardng the pont estmate of the ndex number. Followng the truly constructve method establshed by Afrat (98), we can bypass ths uncertanty by revertng the problem and askng: () whether the avalable data can be ratonalzed by well-behaved true ndex functons, () f yes, what are the upper and lower bounds of the regon contanng the numercal values of possble ndex functons? () f the data cannot be ratonalzed by well behaved ndex functons, then ether the data are not generated by a ratonal behavour (and a correcton for neffcency may be attempted), or else the data are generated wthn a dfferent set of varables to be consdered n an alternatve or extended accountng framework. Snce well-behaved true ndex functons respect, by constructon, all Fsher s tests (see Samuelson and Swamy, 974), also the reconstructed upper and lower bounds of the set of possble values of the true ndex respect those tests, and so does a geometrc mean of those bounds, whch may be requred for practcal needs of pont estmaton. Ths soluton s 2

3 purely constructve and s obtanable rrespectve of the actual exstence or non-exstence of the underlyng utlty of producton functons. The purpose of ths paper s to present a soluton of the ndex number problem n the perspectve of the theoretcal developments occurred durng the last century. It represents a further step forward wth respect to Afrat s (98)(2005) method used n Afrat and Mlana (2009) wth the defnton of approprate consstent tght bounds of the true ndex number. Further references to the current state of the theory and applcatons of ndex numbers can be found n Vogt and Barta (997), von der Lppe (200)(2007), Balk (2008), and the manuals on consumer prce ndces (CIs), producer prce ndces (Is), and mport-export prce ndces (XMIs) publshed jontly by ILO, IMF, OECD, UN, Eurostat, and The World Bank (2004a)(2004b)(2008). Although, for brevty reasons, we shall concentrate manly on the prce ndex, mportant mplcatons for the quantty ndex wll be also consdered. Irvng Fsher and the deal ndex number formula In Fsher's (9) book The urchasng ower of Money. Its Determnaton to Credt, Interest and Crss, the theory of the prce level was related to the quantty theory of money. Let M = stock of money, V = the velocty of crculaton of money; p = prce level of the th transacton, T = volume of the th transacton carred out usng money. The startng (nfamous) equaton of exchange s () MV = p T + p 2 T p n T n, In order to make the foregong equaton workable, the followng verson s usually consdered (2) MV = T where s the aggregate prce level and T s the volume of all transactons, whch have been replaced wth the aggregaton Q of real outputs q, q2,..., q n, often measured by real GD, that s MV = Q (see Fsher, 9, Ch. 2). Equaton () does not necessarly mply equaton (2). Whle the former s n prncple based on observable varables, the latter contans nonobservable aggregates and reles on computaton technques n order to correctly construct them. It s n ths ven that Irvng Fsher dedcated energes and efforts n the search of hs deal ndex number formula satsfyng as many desred propertes as possble. Ths search culmnated n hs famous book The Makng of Index Numbers publshed n 922 (3 rd edton 3

4 927), where he recognzed that no ndex number would satsfy all the desred propertes, but he chose the geometrc mean of the Laspeyres and aasche ndces as hs deal ndex number formula. Appled to the prce ndex between the ponts of observaton 0 and, ths deal ndex number s gven by (3) 0, 0, 0, F L where 0, L 0 0 pq pq pq pq and 0, pq 0 0 pq pq pq t t t t t t t t where p [ pp 2... p n ] and q [ qq 2... q n ] are the prce and quantty vectors and, 0, L 0,,, and 0, F are the Laspeyres, aasche, and Fsher s deal prce ndces. Ths formula had been prevously consdered by Bowley and others before 899 (see Bowley, 923, p. 252) and recommended by Walsh and gou, although t does not generally satsfy the transtvty or crcularty property, that s 0,2 0,,2 F F F (whereas, any rato of aggregate prce levels, f any, s transtve by constructon: 2 / 0 = ( 2 / )( / 0 )). Surprsngly, Fsher dropped the requrement of ths property and deemed t as unmportant compared to other propertes whch hs deal formula always satsfes. In ther artcle dedcated to economc ndex numbers, Samuelson and Swamy (974) commented Fsher s choce n these terms: Indeed, so enamoured dd Fsher become wth hs so-called Ideal ndex that, when he dscovered t faled the crcularty test, he had the hubrs to declare, therefore, a perfect fulflment of ths so-called crcular test should really be taken as proof that the formula whch fulfls t s erroneous (922, p. 27). Alas, Homer has nodded; or, more accurately, a great scholar has been detoured on a trp whose purpose was obscure from the begnnng (p. 575). By contrast, n order to avod strong dscrepances n the results obtaned, the subsequent developments n ths feld have been devoted to satsfy, among the other tests, the transtvty property n multlateral comparsons. Constant-utlty ndex numbers Bennet (920) ntroduced a method by whch a change of expendture can be analysed nto two parts, one correspondng to changes n cost of lvng and the other to changes n standard of lvng (p. 455). Ths decomposton was proposed n terms of absolute dfferences. Konüs (924) and Allen (949) have, respectvely, ntroduced the concepts of constant-utlty ndexes 4

5 of prces and quanttes n terms of ratos. Konüs prce ndex s defned as K (, u) p q p 0 0, p q( p, u) whch takes nto account the prce-nduced adjustments n quanttes for a gven level of utlty u. Settng u u yelds the Laspeyres-type Konüs prce ndex p q p q( p, u ), whle settng K pq 0 0, p q( p, u ) where p q p q( p, u ). 0 K 0 (, u ) p q p 0 0, pq where u u yelds the aasche-type Konüs prce ndex It must be noted that the constant-utlty ndex numbers 0 K and K cannot be computed drectly snce the respectve compensated expendtures 0 p q( p, u ) and 0 0 p q( p, u ) cannot be usually observed. Unless the demand functons 0 qp (, u ) and qp (, u ) are somehow estmated and smulated wth prces p and 0 p respectvely (as n the econometrc approach), a way to proceed wth the concept of Konüs constant-utlty ndex numbers s to establsh ther (upper and lower) lmts, when possble. In the general (non-homothetc) case, Konüs had establshed the followng one-sded bounds wth the prce ndex from the pont of vew of demand (on the supply sde, the algebrac sgns are reversed) 0 K 0 pq pq 0 0 L and pq pq 0 K snce 0 0 p q( p, u ) p q and p q( p, u ) p q, because the left-hand sdes of these last nequaltes are those actually consstent wth a cost-mmnzng behavour at the prces p and p 0 respectvely. Konüs (924) also consdered varous stuatons n relaton to the rankng between the Laspeyres and aasche ndces. In summary, from the pont of vew of demand, the followng alternatve cases are possble: Case : Laspeyres < aasche Case 2: Laspeyres aasche 0 K L K K L 5

6 or and or L K 0 K L 0 K L Konüs observed that t s always possble to fnd a reference utlty level, say u *, such that the cost of lvng ndex falls between the Laspeyres and aasche ndexes, that s or * K L n case * K L n case 2. Konus clamed that these results would suggest that we can work wth the Laspeyres and aasche bounds and take an average of the two to approxmate the true prce ndex. Allen (949) observed that the economc (utlty-constant) quantty ndex could be obtaned drectly, for gven reference prces p, as Q A p q( p, u ) 0 p q( p, u ) Settng 0 p p yelds the Laspeyres-type true Allen quantty ndex Q 0 A 0 0 (, u ) p q p 0 0, pq where p q p q( p, u ), and settng p p yelds the aasche-type true Allen quantty ndex Q A pq 0, p q( p, u ) 0 where p q p q p u (, ). The Laspeyres- and aasche-type true Allen quantty ndex numbers can also be obtaned by deflatng the nomnal ncome rato between the two observaton ponts by the aasche- and Laspeyres-type true Konüs prce ndex numbers, that s: Q Q p q( p, u ) p q p q p q A / / K p q p q p q( p, u ) p q 0 p q p q p q( p, u ) p q 0 A / / K p q( p, u ) p q p q p q The theory of bounds wth respect to the quantty ndex numbers s smlar to that of the prce ndex numbers. Followng Konüs suggeston, any pont of the numercal nterval between these two ndex numbers could correspond to the true quantty ndex wth a certan level of relatve prces. 6

7 The ndetermnacy of the numercal value of true ndex wthn the Laspeyres- aasche bounds seemed to be elmnated by another fndng that s descrbed n the followng secton. Exact and superlatve ndex numbers Byushgens (924) and Konüs and Byushgens (926) have ntroduced the concept of exact ndex numbers for the true aggregator functon by showng that the Fsher deal ndex formula (the geometrc mean of the Laspeyres and aasche ndex numbers) may yeld the same numercal value of the ratos of values taken by a quadratc aggregator functon. If the observed data were generated by a demand governed by such functon, then the transtvty o crcularty property would be satsfed by Fsher deal ndex formula. Followng the modern generalzaton of ther proposton, let us assume a utlty functon such that the correspondng mnmum expendture functon has the quadratc mean-of-order-r functonal r / 2 r / 2 / r form C( p, u) c r ( p, u) u, where c r ( p, u) ( p A( u) p ) wth r 0, 0 < r, and the Q Q matrx A ( u) s a normalzed symmetrc matrx of postve coeffcents a ( u) a ( u) satsfyng the restrcton a j ( u ), so that c r ( p, u) f p [...]. j Q j j The functonal form c r can be seen as a generalzaton of a CES functonal form, to Q whch t collapses f all aj 0 for j (see McCarthy, 967 and Kadyala, 972), and t reduces to the Generalzed Leontef functonal form wth r (Denny, 972, 974) and the Konüs- Byushgens (926) functonal form wth r 2 (Dewert, 976, p. 30). Snce the quadratc functonal forms can be seen also as second-order approxmatons to any arbtrary functonal form, they have been called flexble by Dewert (976). We have, n fact, r/ 2 r/ 2 c ( (4) p r, u) ( u ) Q p A p c ( / 2 / 2 r p0, u0) r r Q p0 A( u0) p0 / r / r r / 2 r / 2 r / 2 r / 2 / r r / 2 r / 2 / r ( u) ( u0) 0 0 ( u ) r / 2 r / 2 r / 2 r / 2 r / 2 r / 2 0 ( u) 0 ( u0) 0 0 ( u0) p A p p A p p A p p A p p A p p A p r / 2 r / 2 r / 2 r / 2 snce p A( ut) p0 p0 A( ut) p wth a symmetrc Au ( t ) r / 2 r / 2 r / 2 r / 2 r / 2 r / 2 / r r / 2 r / 2 / r p A( u) p p pˆ0 pˆ0 A( u0) p0 p0 A( u) p r / 2 r / 2 r / 2 r / 2 r / 2 r / 2 r / 2 r / 2 0 ˆ ˆ ( u) 0 ( u0) 0 0 ( u0) p p p A p p A p p A p 7

8 where ^ denotes a dagonal matrx formed wth the elements of a vector where p a ( u ) p r/ 2 r/ 2 pq t t t j j t tj t r/ 2 r/ 2 ptjqtj pt A( ut ) p j t s r /2 p r s0 / 2 / 2 / r r /2 r r p0 p0 A( u) p r /2 r/ 2 r/ 2 p0 p0 A( u0) p s r /2 p, whch s the observed value share of the th quantty r 2 r /2 c( p, ) ( ) t u p t t aj ut p j tj t ( pt, t ) / t t t pt r/ 2 r/ 2 r ( pt A( ut ) pt ) q C u p u u by Shephard s lemma, wth a j beng the (,j) element of matrx A. Thus, the ndex number yelds exactly (s exact for) the same numercal value that would be obtaned as a rato of the values of the underlyng functon n the two compared stuatons. Dewert (976) called superlatve the ndex numbers that are exact for flexble functonal forms and descrbed them as approxmatng each other up to the second order. By contrast, t has been noted that these ndex numbers are far from beng second-order approxmatons to each other (see Mlana, 2005 and Hll, 2006a) and that ths termnology dverges n meanng from that used by Fsher (922), who has defned superlatve those ndex numbers that smply performed very closely to hs deal ndex formula wth hs dataset. Snce all the prce varables and utlty are consdered here at ther current levels, the shares s t are those actually observed. As we shall see also below, n the homothetc case, we have C( p, u) c( p ) u and, consequently, the observed shares s t are equal to the theoretcal weghts that are functons only of prces (wth A( u0) A( u) A ). The frst multplcatve bracketed element of the last lne of (4) can be consdered as a canddate prce ndex number (5) r /2 r /2 p0 0 0 r /2 p0 r /2 p r ( p, p, q, q ) Q p s 0 s r whch corresponds to Dewert s (976, p.3) quadratc mean-of-order-r prce ndex number. As r tends to 0, the prce ndex r tends to the Törnqvst ndex number: Q lm exp[ ( )(ln ln )] 2 (6) r0 r T s Q 0 s p p0 whch s exact for the translog cost functon 8

9 (7) ct ( p, u) u exp( 0 u ln u lnp u ln p ln u j ln p ln p j ) 2 j If r = 2, then the prce ndex r s to the deal Fsher ndex. Q We note that, f the observed data were generated by a demand consstent wth a mnmum quadratc cost functon c r ( p, u) wth specfc parameters values, at least locally, Q then we would have r ( p0, p2, q0, q2) r ( p0, p, q0, q) r ( p, p2, q, q2), that s the exact ndex number Q Q Q r would satsfy the transtvty property as well as all the other Fsher s tests between Q the three observaton ponts. If the transtvty property s not satsfed, then ether the demand s not governed by a ratonal behavour or the ndex number formula s not exact for the actual cost functon or utlty functon consstent wth the data. At tme of the dscovery of Konüs and Byushgens (926), the concept of homothetcty of ndfference curves and ts relatonshp wth exstence of a pure prce (and quantty) ndex was not wdely known. The concept of homothetcty was explctly spelled out by Shephard (953) and Malmqust (953) n the feld of producton technology and ndependently by Afrat (972) under the termnology of concal functons n the feld of consumer utlty. Earler contrbutons datng back at least from Antonell (886) and ncludng Frsch (936, p. 25) and Samuelson (950, p. 24) have dealt wth t mplctly. When the true prce ndex defned by Konüs s not ndependent of the utlty level, as n the general non-homothetc case, the correspondng Allen true quantty ndex fals to be lnearly homogeneous (f all the elementary quanttes are multpled by a factor λ, then the ndex number fals to be proportonal by the same factor λ). In Allen s (949, p. 99) words, *t+he ndex has no meanng unless we make the assumpton that the preference map s the same n the two stuatons. Ths affects, n a way, also the prce ndex: although ths ndex s always lnearly homogeneous by constructon n the non-homothetc case t results to be a spurous prce ndex whose weghts are functons not only of prces but also of the utlty level and, then, of the demanded relatve quanttes. Ths has been usually overlooked also n the current lterature on economc ndex numbers. Wth the quadratc functon consdered above, only f A( u) A( u0) A would the weghts be functons only of prces. In the applcaton of ndexes defned by Dvsa (925), ths s called path ndependence snce the ndex s ndependent of the path taken wth respect to the reference quantty varables. Hulten (973) has shown that the Dvsa ndex s path- 9

10 ndependent f and only f the underlyng functon s homothetc (tastes do not change). Ths can be seen mmedately related to the Törnqvst ndex number n the lmt of nfntesmal changes: hence ( ) (8) ln p ln p dln p d ln Dv lm t0 ( st s( t t) ) st 2 t dt (9) 0, t dln pt Dv exp( s ) t 0 t dt dt t t t t whch s the Dvsa prce ndex. If the weghts s t are not functons of the prces alone (as n the homothetc case), but depend also on relatve levels of the reference quanttes, then the Dvsa prce ndex s not a pure prce ndex. These consderatons were already mplct n the analyss of contrbutors n the early part of last century, who were well aware of the mportance of homothetc tastes for the exstence of economc aggregate ndex numbers. A. L. Bowley, for example, n search of a constant-utlty prce ndex had been among the frst proponent of the geometrc mean of the Laspeyres and aasche ndexes (whch had later become famous as Fsher deal ndex). He also devsed another ndex as an approxmaton to the constant-utlty prce ndex gven by the followng formula, prevously proposed by Edgeworth: (0) E 0 p ( q q ) 0 0 p ( q q ) to be appled under the hypothess of no changes n tastes. He, n fact, wrote: Assume that our records represent the expendture of an average man, and that the satsfacton he derves from hs purchases s a functon of the quanttes bought only, say u(q ), are the numbers of unts bought of the n commodtes. Further, suppose that the form and constants of ths functon are unchanged over the perod consdered. The last condton lmts the measurement to an nterval of tme n whch customs and desres have not changed and to a not very wde range of real ncome. The analyss and conclusons do not apply to comparsons between ctzens of two countres, nor over, say, 60 years n one country (Bowley, 928, pp ). Identcal preferences, mplyng a homothetc utlty functon, have been noted as early as the work of Antonell (886) as a necessary and suffcent condton for aggregaton. Condtons for aggregaton holdng only locally and allowng global preference heterogenety have been studed by Afrat (953-56)(959) and Gorman (953)(96). 0

11 It s remarkable, however, that also the foregong Bowley-Edgeworth ndex number does not satsfy the requrement of transtvty. In general, the lack of transtvty would sgnal the poor approxmaton gven by the formulas chosen. Ths s the stuaton encountered partcularly n nterspatal comparsons, where the alternatve measures could dffer more than 00% even wth superlatve ndex numbers (see, e.g., Hll, 2006a, 2006b). Gven the dscouragng results obtaned wth specfc ndex number formulas, we now turn to the method of lmts by consderng the exercse of testng the data for consstency wth hypothetcal homothetc changes, followng Keynes, Hcks, Samuelson, and Afrat. John Maynard Keynes method of lmts In hs Treatse on Money, Keynes (930, Vol.I, ch. 8) made no explct reference to the dea of a prce ndex. Rather, he compared the purchasng power of money n two stuatons of consumpton dfferng n relatve prces. The comparson was made by usng the so-called method of lmts (p. 98). No change n taste and proportonalty of composte quanttes (and prces) wth respect to total real expendture are assumed. These hypotheses mply monotoncty along a beam lne where, at gven relatve prces, all ndvdual quanttes change proportonally. Two alternatve ratos of real expendtures can be calculated at constant relatve prces of the base and the current stuatons, respectvely. It turns out that these ratos are the upper and lower lmts (bounds) of the ndex of the real expendture. (As shown by Leontef, 936, pp and Afrat, 977, pp. 08-5, 2005, these lmts correspond, respectvely, to the Laspeyres and aasche ndex numbers of real expendture.) Smlar methods were used by other authors. In hs famous revew artcle, Ragnar Frsch (936, p. 7-27) mentoned gou, Haberler, Keynes, Gn, Konüs, Bortkewcz, Bowley, Allen, and Staehle and dscussed them brefly. Keynes (930, p. 99) hmself observed: Ths concluson s not unfamlar * +. It s reached, for example, by rofessor gou (Economcs of Welfare, part I, chapter VI). The matter s also very well treated by Harberler (Der Snn der Indexzahlen, pp ). The dependence of the argument, however, on the assumpton of unformty of tastes, etc., s not always suffcently emphassed (talcs added). He wrtes, here, the followng footnote: Dr. Bowley n hs Notes on Index Numbers publshed n the Economc Journal, June 928, may be mentoned amongst those who have expressly ntroduced ths necessary condton.

12 Keynes method of lmts has not been wdely used, probably because t has not been mmedately understood n ts fundamental reasonng. Frsch (936, p. 26), for example, whle concedng the correctness of Keynes proof, overlooked the real sense of hs proceedng by observng: If we know that q 0 and q are adapted and equvalent, the ndfference-defned [prce] ndex can be computed exactly, namely, as the rato 0 0 p q / p q [snce t s assumed t t that q q( p, u) wth t = 0,]. In these crcumstances, to derve lmts for t s to play hdeand-seek. It was Staehle who frst ponted ths out. In fact, Keynes dd not assume that q 0 and q were necessarly on the same ndfference curve, but on homothetc ndfference curves on the hypothess of unformty of tastes. Ths mples monotoncty along a beam (a lne where all ndvdual quanttes change proportonately) along whch the purchasng power of money can be compared at dfferent prces. Ths reasonng was later recovered and further developed by Afrat (977, pp. 08-5). Hcks Laspeyres-aasche nequalty condton In a chapter enttled The Index-Number Theorem of hs Revson of Demand Theory, John Hcks (956, pp ) establshed a proposton on the Laspeyres-aasche nequalty on the demand sde () Laspeyres (L) aasche () (for both prce or quantty ndexes) (see also Hcks, 958 and the prevous prelmnary analyss contaned n Hcks, 940). The (non-negatve) dfference between Laspeyres and aasche ndcates a substtuton effect (S) n the case the ponts of observaton are on the ndfference curve or the sum of substtuton effect and a certan ncome effect (I) n the case they are not on the same ndfference curve. In the more general case, we have (2) L = I + S where, L and are the Laspeyres and aasche ndexes (we use Hcks orgnal notaton denotng the aasche ndex as ). If the ncome-elastctes of all commodtes are the same (that s the preferences are homothetc), then I s equal to zero. In ths case, the proporton of demanded quanttes do not change as real ncome changes. We have the followng possble results: 2

13 Case : L < 0 (Hcks ndex-number theorem breaks down) meanng ether that demand s not governed by ratonal behavour and/or the preferences are non-homothetc wth a negatve and strong enough ncome effect so that real-ncome change nduces a relatve expanson n demand for those goods whose prces have relatvely rsen. A strong negatve ncome effect offsets a postve substtuton effect (I + S < 0) Case 2: L 0 (Hcks ndex-number theorem holds), meanng ether that preferences are homothetc (so that I = 0 and S 0) or preferences are non-homothetc (wth I 0 and I + S 0). If preferences are homothetc, mplyng that the ncome-elastctes of all commodtes are the same then the proporton of demanded quanttes do not change as real ncome changes. and I s equal to zero. The Hcks ndex-number theorem pontng to a postve L dfference (case 2) s a necessary and suffcent condton for usng the observed data on prces and quanttes to reconstruct true ndex numbers based on hypothetcal homothetc preferences. These, however, do not necessarly concde wth the actual crtera governng the observed behavour. In other words, the L nequalty mght be the result of the concomtant nonproportonal effects of real ncome changes as well as substtuton effects under nonhomothetc preferences (f any), but the observed data could always be ratonalzed by a hypothetcal homothetc preference feld f L 0. Under ths condton we could always reconstruct true prce and quantty ndex numbers that are consstent wth those homothetc preferences and, as such, always respect all Fsher s requrement, ncludng transtvty. Ths s, n fact, (as Keynes had recalled) the only condton under whch t s possble to make such constructon. Samuelson s consderatons on the Laspeyres-aasche nequalty Independently from Hcks (956) and consstently wth hs ndex-number theorem, n ther surveys on the conclusons of the theory of bounds, Samuelson (974)(984), Samuelson and Swamy (974), and Swamy (984) have consdered the followng cases. Case : L < 0, so that the observed relatve prces are not negatvely correlated wth the observed relatve quanttes (as expected wth homothetc changes). In such an anomalous case, we mght obtan the followng rankng (wrtten n matrx notaton, where p t and q t are prce and quantty vectors at tme t): 3

14 0 0 p q( p, u ) p q p q p q p q p q p q p q( p, u ) Konüs Laspeyres aasche Konüs Laspeyres-type aasche-type wth 0 qp (, u ) and 0 qp (, u ) beng the vectors of non-observed (theoretcal) quanttes that would have been demanded at the prce-utlty combnatons 0 ( p, u ) and 0 ( p, u ), respectvely. Ths s a rather problematc case, where aggregaton s not possble. Even Fsher s deal ndex, whch conssts n the geometrc mean of the Laspeyres and aasche ndexes, fallng between these two ndexes, s farther than ths last ones from both true economc ndexes! (See the numercal example gven by Samuelson and Swamy, 974, where the Ideal ndex cannot gve hgh-powered approxmaton to the true ndex n the general, nonhomothetc case, p. 585.) Case 2: L 0. If preferences are homothetc, then I = 0 and S 0. If preferences are non-homothetc wth real-ncome changes nducng a relatve expanson n demand for those goods whose prces have relatvely fallen (a case consdered by Samuelson, 974, 984, Swamy, 984 and others under the name of Engel-Gerschenkron effect ), then I 0, whch renforces the postve substtuton effect S 0. In these two cases, we can rely on the followng rankng 0 p q p q p q p q p q( p, u ) p q aasche Konüs Laspeyres aasche-type and 0 0 p q (, u ) p q p p q p q p q p q aasche Konüs Laspeyres Laspeyres-type The Laspeyres and aasche ndex numbers correspond to alternatve fxed proportons utlty functons (wth zero commodty substtuton). Notng ths fact, Swamy (984, fn. 0) wrote: Ths s not to dssuade scholars from usng the Laspeyres and aasche ndexes, but merely to urge them to restrct the use of these ndexes to local changes n p. These ndexes can be used to determne bounds for the true ndex whch may not be known. 4

15 Afrat s ndex formula: Any pont n the Laspeyres-aasche nterval, f any Along the lnes open by Hcks (956), the jont nformaton gven by the Laspeyres and aasche ndexes could provde us wth an alternatve nformaton concernng two lmtng functons allowng substtuton effects whose dfference s equal to S 0 consdered above. These two lmtng functons are pece-wse lnear boundares of a set of possble homothetc utlty functons, whch can ratonalze the observed data. Even though these data have been actually generated under non-homothetc preferences, the Hcks (956) Laspeyres-aasche nequalty condton s necessary and suffcent for constructng true ndex homothetc functons that can also ratonalze the same data. It s n ths ven that Afrat (977, pp. 08-5) recovered Keynes (930) reasonng on the purchasng power of money under the hypothess of unchanged tastes and translated t nto the constructon of the bounds of a true prce ndex. As recalled by Samuelson and Swamy (974, p. 570), t s possble to nvoke the Shephard-Afrat s factorzaton theorem under the hypothess of homothetcty to separate the expendture functon nto meanngful aggregates of prces and quanttes. As Samuelson and Swamy (974, p. 570) have recognzed, *t+he nvarance of the prce ndex [from the reference quantty base] s seen to mply and to be mpled by the nvarance of the quantty ndex from the reference prce base. Ths concluson was antcpated n Afrat (977, pp. 07-2). A pure prce ndex s consstent wth a concal (homothetc) utlty functon ratonalzng the observed prces and quanttes n dfferent stuatons. The concal (homothetc) utlty condton whch permts ths determnaton, for arbtrary p0 and p, s a non-observatonal object, a purely hypothetcal metaphyscal concept. The correspondng dual mnmum expendture functon admts the factorzaton nto a product of the prce and quantty functons. C( p, u) c( p) u( q ) Defnng the amount of money devoted to total expendture (or ncome) as E, so that E C( p, u), we can obtan the cardnal measure of utlty as a deflated value of ncome, that s n the homothetc case where V( p, E) s the ndrect utlty functon. c ( p) u ( q) u V ( p, E) E c( p) c( p) 5

16 The observed (uncompensated) Marshallan demand functons for each elementary quantty s gven by Roy s dentty q V ( p, E) / p dv ( p, E) / de for =,2, whch, n the homothetc case, becomes q a ( p ) E where c( p) / p a ( p) c( p) for all s The ncome elastcty of the demanded th quantty s thus obtaned q E a ( p) E q a ( p) for all s that s, all ncome elastctes are equal to n the homothetc case. The problem s whether we can recover the prce ndex whch s expressed as a rato of prce levels / 0, 0 c( p ), c( p ) 0 0 whereas, the money-metrc utlty ndex s measured by whch s the rato of volume levels E / E Q0, Q / Q0 0 0, Q E / c( p ), Q E / c( p ) The expendture ndex consstent wth the recovered homothetc utlty con be decomposed as follows: C( p, u) c( p) u c( p) E / c( p) 0, Q0, C( p, u ) c( p ) u c( p ) E / c( p ) Strctly speakng, the nverse of the prce ndex, / 0,, s the ndex of purchasng power of one unt of money and the quantty ndex Q0, ( E / E0) (/ 0, ) s the ndex of purchasng power of monetary ncome, or real ncome. Consstently wth the hypothess of homothetcty, ths last ndex corresponds to the ndex of utlty u( q) / u( q 0). In Afrat (977, p. 0) words: The concluson * + s that the prce ndex s bounded by the aasche and Laspeyres ndces. * + The aasche ndex does not exceed the Laspeyers ndex. * + The set of values *of the true ndex + s n any case dentcal wth the aasche- 6

17 Laspeyres nterval. The true ponts are just the ponts n that nterval and no others; and none s more true than another. There s no sense to a pont n the nterval beng a better approxmaton to the true ndex than others. There s no proper dstncton of constant utlty ndces, snce all these ponts have that dstncton. The same concluson s replcated n Afrat (2005, p. xx): Let us call the L nterval the closed nterval wth L [Laspeyers ndex] and (aasche ndex] as upper and lower lmts, so the L-nequalty s the condton for ths to be non-empty. Whle every true ndex s recognzed to belong to ths nterval, t can stll be asked what ponts n ths nterval are true? The answer s all of them, all equally true, no one more true than another. When I submtted ths theorem to someone notorous n ths subject area t was receved wth complete dsbelef. Here s a formula to add to Fsher s collecton, a bt dfferent from the others. Index Formula: Any pont n the L-nterval, f any. In my revew artcle (Mlana, 2005), t s shown that any prce ndex number that s exact for a contnuous functon can be translated nto the followng form (3) ( ) s p 0 ( ) ( ) p0 N p q0 N p q 0, p 0 p0q 0 p0q ( ) s p where, for t = 0,, C( pt, ut ) C( pt, ut ) st pt / p j tj p p t tj C( pt, ut) pt qt / p j tjqtj usng Shephard s lemma ( q t = ) p and s an approprate parameter whose numercal value depends on the remander terms of the two frst-order approxmatons of C(p,u) around the base and current ponts of observatons. The ndex 0, s lnearly homogeneous n p (that s, f p p0, then 0, ). Wth 0, t reduces to a Laspeyres ndex number, whereas, wth, t reduces to a aasche ndex number. The true exact ndex number, f any, s numercally equvalent to 0,. If the functonal form of C( p, u ) s square root quadratc n p, then 0, can be transformed nto a Fsher t t t 7

18 deal ndex number. In ths case, the ndex 0, s numercally equvalent to a quadratc mean-of-order-2 ndex number. Here, agan, the prce ndex s nvarant wth respect to the reference utlty level f and only f C( p, u) s homothetcally separable and can be wrtten C( p, u) c( p ) u, so that s p q / p q t t t j tj tj p t c( pt) c( pt) / ptj. j p p t tj Moreover, Q0, [ C(, u) / C( 0, u0)]/ 0, p p s the quantty ndex measured mplctly by deflatng the ndex of the functonal value wth the prce ndex 0,. It has the meanng of a pure quantty ndex f and only f 0, s a pure prce ndex. The parameter, however, remans unknown and we cannot rely on the second-order dfferental approxmaton paradgm. For ths reason, t s concluded that t would be more approprate to construct a range of alternatve ndex numbers (ncludng those that are not superlatve), whch are all equally vald canddates to represent the true ndex number, rather than follow the tradtonal search for only one optmal formula (Mlana, 2005, p. 44). revous attempts n ths drecton usng non-parametrc approaches based on revealed preference technques nclude Banker and Mandratta (988), Manser and McDonald (988), Chavas and Cox (990)(997), Dorwck and Quggn (994)(997), but these do not provde, n general, strngent tests for homothetcty and, more mportantly, the derved ndex numbers fal to satsfy the transtvty requrement. An alternatve approach to the Afrat methodology would be that of the econometrc estmaton of the functon c( p) n order to elmnate the ndetermnacy of the true ndex number (see, among the frst attempts, Goldberger and Gamaletsos, 970 and Lloyd (975), and, among the most recent contrbutons, Blundell et al. 2003, Neary, 2004, and Oulton, 2005), but ths mples the mposton of a subjectve choce of a pror functonal forms where stochastc components of the derved demand functons are also ncluded. The theory of bounds becomes more complex wth the addton of the stochastc term to each demand functon (see, e.g., hlps, 983). Crtcal remarks on ths approach could be made regardng the non-dentfablty of the elastctes of substtuton and the bas n changes n technology or consumer tastes f no a pror nformaton s avalable (see, e.g., Damond, McFadden and Rodrguez, 978). 8

19 Consstent prce ndces between several observaton ponts The approach outlned n the prevous secton can be enhanced by consderng more than two observaton ponts smultaneously. Ths dea had been advanced durng the debate on ndex numbers n the early part of last century. Frsch (936, p. 36), commentng the soexpendture method of Staehle (935), wrote: The comparson between two paths wll be more exact f made va an ntermedate path. The closer the ndvdual paths the better. Knowng a very close path-system s equvalent to knowng the ndfference surfaces themselves. In ths case the ndfference ndex can be computed exactly. Smlar statements were wrtten also by Samuelson (947, ch. VI). It s worth quotng Samuelson and Swamy s (974, p. 476) own words: * + Fsher mssed the pont made n Samuelson (947, p. 5) that knowledge of a thrd stuaton can add nformaton relevant to the comparson of two gven stuatons. Thus Fsher contemplates Georga, Egypt, and Norway, n whch the last two each have the same prce ndex relatve to Georga : We mght conclude, snce two thngs equal to the same thng are equal to each other, that, therefore, the prce levels of Egypt and Norway must equal, and ths would be the case f we compare Egypt and Norway va Georga. But, evdently, f we are ntent on gettng the very best comparson between Norway and Egypt, we shall not go to Georga for our weghts *whch are+, so to speak, none of Georga s busness. *922, p. 272]. Ths smply throws away the transtvty of ndfference and has been led astray by Fsher s unwarranted belef that only fxed-weghts lead to the crcular s test s beng satsfed (an asserton contradcted by our / and Q / Q forms. j j One of Afrat s man contrbuton n ndex number theory has been the development an orgnal approach of constructng aggregatng ndex numbers usng all the data smultaneously (see Afrat, 967, 98, 984, 2005). He also has developed an effcent algorthm to fnd the mnmum path of chaned upper lmt ndex numbers (the chaned Laspeyres ndces on the demand sde). In the followng secton ths algorthm s brefly descrbed. From these chaned upper lmt ndex numbers can be derved drectly the chaned lower lmt ndex numbers (the chaned aasche ndces on the demand sde). 9

20 The proposed method In ths secton, for expostonal convenence, some notaton s changed wth respect to the prevous sectons. The matrces of blateral Laspeyres (L) and aasche (K) ndex numbers comparng aggregate prces at the pont of observaton relatve to those at pont j, for,j =, 2,, N, are respectvely L L2... L N L2 L22... L 2N L and LN LN 2... LNN K K... K K K... K K K K... K 2 N N N N 2 NN where j Lj pq j j pq, and Kj pq j pq L j. Obvously, K j and L K. L j The Laspeyres and aasche ndex numbers are usually consdered as two alternatve measures of the unknown true ndex number j whch can be seen as an aggregaton of r j r the elementary prce ratos p / p or, alternatvely, as a rato of aggregate prce levels,.e. /, where and j j j are true aggregate prce levels at the th and jth ponts of observaton. The prce level rato, always respects, by constructon, the base reversal test, that s /, and the crcularty test, that s. By contrast, n the general case j j 20 t tj j where the elementary prce ratos and the relatve quantty weghts change, the Laspeyres and aasche ndces fal to be base- and chan-consstent, that s L / L K, j j j Lt Ltj Lj and Kt Ktj Kj. Even more unacceptable s well-known falure of chaned ndexes to return on the prevous levels f all elementary prces go back to ther older levels (the so-called drft effect ): L L L. and K K K. These falures make the two t t t t ndex number formulas, lke all the other alternatve formulas, unsutable to represent a prce ndex. Nevertheless, as we shall see below, they are useful for testng the exstence of the true prce ndex and constructng ts consstent bounds. ( Lj Kj The so-called L-nequalty condton s that Lj Kj on the purchaser s sde ndex number on the suppler s sde) s necessary and suffcent for the exstence of a true prce j wth a numercal value fallng between the Laspeyres and aasche ndces. If ths condton s not satsfed for all pars of observaton, then a correcton of the data for

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