Scanner Data, Elementary Price Indexes and the Chain Drift Problem

Transcription

1 1 Scanner Daa, Elemenary Price Indexes and he Chain Drif Problem Erwin Diewer, 1 Revised Ocober 7, Discussion Paper 18-06, Vancouver School of Economics, Universiy of Briish Columbia, Vancouver, B.C., Canada, V6T 1L4. Absrac Saisical agencies increasingly are able o collec deailed price and quaniy informaion from reailers on sales of consumer producs. Thus elemenary price indexes (which are indexes consruced a he firs sage of aggregaion for closely relaed producs) can now be consruced using his price and quaniy informaion, whereas previously, saisical agencies had o consruc elemenary indexes using jus reail oule colleced informaion on prices alone. Thus superlaive indexes can now be consruced a he elemenary level, which in heory, should lead o more accurae Consumer Price Indexes. However, reailers frequenly sell producs a heavily discouned prices, which lead o large increases in purchases of hese producs. This volailiy in prices and quaniies will generally lead o a chain drif problem; i.e., when prices reurn o heir normal levels, quaniies purchased are frequenly below heir normal levels and his leads o a downward drif in a superlaive price index. The paper addresses his problem and looks a he likely bias in various index number formulae ha are commonly used. The bias esimaes are illusraed using some scanner daa on he sales of frozen juice producs ha are available online. Keywords Jevons, Duo, Carli, Uni Value, Laspeyres, Paasche, Consan Elasiciy of Subsiuion (CES), Fisher and Törnqvis price indexes, superlaive indexes, mulilaeral indexes, GEKS, CCDI, Geary-Khamis, Similariy Linked and Time Produc Dummy indexes, elemenary indexes, scanner daa, bias esimaes. JEL Classificaion Numbers C43, C81, E31. 1 W. Erwin Diewer: Vancouver School of Economics, Universiy of Briish Columbia, Vancouver B.C., Canada, V6T 1Z1 and he School of Economics, UNSW Sydney, NSW 2052, Ausralia (erwin.diewer@ubc.ca). The auhor hanks Corinne Becker-Vermeulen, Jan de Haan, Claude Lamboray and Mick Silver for helpful commens and graefully acknowledges he financial suppor of a Trans Alanic Plaform Digging ino Daa gran. The projec ile is: Online Prices for Compuing Sandards of Living Across Counries (OPSLAC).

2 2 Table of Conens 1. Inroducion Page 3 2. Comparing CES Price Levels and Indexes Page 6 3. Using Means of Order r o Aggregae Price Raios Page Relaionships beween Some Share Weighed Price Indexes Page Relaionships beween he Jevons, Geomeric Laspeyres, Geomeric Paasche and Törnqvis Price Indexes Page Relaionships beween Superlaive Fixed Base Indexes and Geomeric Indexes ha use Average Annual Shares as Weighs Page To Chain or No o Chain Page Relaionships beween he Törnqvis Index and he GEKS and CCDI Mulilaeral Indexes Page Uni Value Price and Quaniy Indexes Page Qualiy Adjused Uni Value Price and Quaniy Indexes Page Relaionships beween Lowe and Fisher Indexes Page Geary Khamis Mulilaeral Indexes Page Weighed Time Produc Dummy Mulilaeral Indexes Page Relaive Price Similariy Linked Indexes Page Inflaion Adjused Carry Forward and Backward Impued Prices Page Conclusion Page 60 Appendix: Daa Lising and Index Number Tables and Chars A.1 Daa Lising Page 62 A.2 Unweighed Price Indexes Page 68 A.3 Weighed Price Indexes Page 69 A.4 Indexes which Use Annual Weighs Page 71 A.5 Mulilaeral Indexes Page 73 A.6 Mulilaeral Indexes Using Inflaion Adjused Carry Forward and Backward Prices Page 75 References Page 79

3 3 1. Inroducion Saisical agencies increasingly are approaching reail chains and asking hem o provide informaion on weekly values and quaniies sold of scanner coded producs and in many counries, his informaion is being provided o he relevan agency. This deailed weekly informaion on he value and quaniy of sales by produc means ha i is possible o consruc index numbers in real ime ha require produc informaion on prices and quaniies, such as he Fisher (1922) ideal index. The Consumer Price Index Manual 2 ha was published in 2004 recommended ha chained superlaive index numbers, 3 such as he Fisher index, be used in his siuaion when deailed price and quaniy informaion is available in real ime. However, experience wih scanner daa has shown ha he use of chained superlaive indexes leads o chain drif; i.e., a growing divergence beween he chained index and is counerpar fixed base index. 4 There are a leas hree possible real ime soluions o he chain drif problem: Use a fixed base index; Use a mulilaeral index; 5 Use annual weighs for a pas year. There are wo problems wih he firs soluion: (i) he resuls depend asymmerically on he choice of he base period and (ii) wih new and disappearing producs, he base period prices and quaniies may lose heir represenaiveness; i.e., over long periods of ime, maching producs becomes very difficul. 6 A problem wih he second soluion is ha as an exra period of daa becomes available, he indexes have o be recompued. 7 The 2 See ILO, IMF, OECD, Eurosa, UN and he World Bank (2004). 3 See Diewer (1976) on he concep of a superlaive index. Basically, a superlaive price index allows for an arbirary paern of subsiuion effecs beween producs. 4 Fisher (1922; 293) realized ha he chained Carli, Laspeyres and Young indexes were subjec o upward chain drif bu for his empirical example, here was no evidence of chain drif for he Fisher formula. However, Persons (1921) came up wih an acual empirical example where he Fisher index exhibied subsanial downward chain drif. Frisch (1936; 9) seems o have been he firs o use he erm chain drif. Boh Frisch (1936; 8-9) and Persons (1928; ) discussed and analyzed he chain drif problem. 5 The use of mulilaeral indexes in he ime series conex daes back o Persons (1921) and Fisher (1922; ), Gini (1931) and Balk (1980) (1981). Fisher (1922; 305) suggesed aking he arihmeic average of he Fisher sar indexes whereas Gini suggesed aking he geomeric mean of he sar indexes. 6 Persons (1928; ) has an excellen discussion on he difficulies of maching producs over ime. 7 This is no a major problem. A soluion o his problem is o use a rolling window of observaions and use he resuls of he curren window o updae he index o he curren period. This mehodology was suggesed by Ivancic, Diewer and Fox (2009) (2011) and is being used by he Ausralian Bureau of Saisics (2016). Ivancic, Diewer and Fox (2011) suggesed ha he movemen of he indexes for he las wo periods in he new window be linked o he las index value generaed by he previous window. However Krsinich (2016) in a slighly differen conex suggesed ha he movemen of he indexes generaed by he new window over he enire new window period be linked o he previous window index value for he second period in he previous window. Krsinich called his a window splice as opposed o he IDF movemen splice. De Haan (2015; 27) suggesed ha perhaps he linking period should be in he middle of he old window which he Ausralian Bureau of Saisics (2016; 12) erms a half splice. Ivancic, Diewer and Fox (2010) also suggesed ha he average of all links for he las period in he new window o

4 4 problem wih he hird possible soluion is ha he use of annual weighs will ineviably resul in some subsiuion bias. In any case, in his sudy, we will look a many of he commonly used fixed base indexes as well as six mulilaeral indexes and wo indexes ha use annual weighs. We will develop approximae and exac relaionships beween hese indexes and indicae likely differences (or biases ) beween he indexes. Saisical agencies are also using web-scraping o collec large number of prices as a subsiue for selecive sampling of prices a he firs sage of aggregaion. Thus i is of ineres o look a elemenary indexes ha depend only on prices, such as he Carli (1804), Duo (1838) and Jevons (1865) indexes, and compare hese indexes o superlaive indexes; i.e., under wha condiions will hese indexes adequaely approximae a superlaive index. 8 The wo superlaive indexes ha we will consider in his sudy are he Fisher (1922) and he Törnqvis 9 indexes. The reasons for singling ou hese wo indexes as preferred bilaeral index number formulae are as follows: (i) boh indexes can be given a srong jusificaion from he viewpoin of he economic approach o index number heory; 10 (ii) he Fisher index emerges as probably being he bes index from he viewpoin of he axiomaic or es approach o index number heory; 11 (iii) he Törnqvis index has a srong jusificaion from he viewpoin of he sochasic approach o index number heory. 12 Thus here are srong cases for he use of hese wo indexes when making comparisons of prices beween wo periods when deailed price and quaniy daa are available. When comparing wo indexes, wo mehods for making he comparisons will be used: (i) use second order Taylor series approximaions o he index differences; (ii) he difference beween wo indexes can frequenly be wrien as a covariance and i is possible in many cases o deermine he likely sign of he covariance. 13 When looking a scanner daa from a reail oule (or price and quaniy daa from a firm ha uses dynamic pricing o price is producs or services 14 ), a fac emerges: if a produc or a service is offered a a highly discouned price (i.e., i goes on sale), hen he quaniy sold of he produc can increase by a very large amoun. This empirical observaion will allow us o make reasonable guesses abou he signs of various covariances ha express he difference beween wo indexes. If we are aggregaing producs ha are close he observaions in he old window could be used as he linking facor. Diewer and Fox (2017) look a he alernaive mehods for linking. 8 We will also look a he properies of he CES price index wih equal weighs. 9 The usual reference is Törnqvis (1936) bu he index formula did no acually appear in his paper. I did appear explicily in Törnqvis and Törnqvis (1937). I was lised as one of Fisher s (1922) many indexes: namely number 123. I was explicily recommended as one of his op five ideal indexes by Warren Persons (1928; 86) so i probably should be called he Persons index. 10 The economic approach o index number heory is due o Konüs (1924). See Diewer (1976) for jusificaions for he use of hese wo indexes from he viewpoin of he economic approach o index number heory. 11 See Diewer (1992) or Chaper 16 of he Consumer Price Index Manual. 12 See Theil (1967; ) or Chaper 16 of he Consumer Price Index Manual. 13 This second mehod for making comparisons can be raced back o Borkiewicz (1923). 14 Airlines and hoels are increasingly using dynamic pricing; i.e., hey change prices frequenly.

5 5 subsiues for each oher, hen a heavily discouned price may no only increase he sales of he produc bu i may also increase he share of he sales in he lis of producs or services ha are in scope for he index. 15 I urns ou ha he behavior of shares in response o discouned prices does make a difference in analyzing he differences beween various indexes: in he conex of highly subsiuable producs, a heavily discouned price will probably increase he marke share of he produc bu if he producs are weak subsiues (which is ypically he case a higher levels of aggregaion), hen a discouned price will ypically increase sales of he produc bu no increase is marke share. These wo cases (srong or weak subsiues) will play an imporan role in our analysis. Secions 2 and 3 look a relaionships beween he fixed base and chained Carli, Duo, Jevons and CES elemenary indexes ha do no use share or quaniy informaion. Secion 4 looks a he relaionships beween he Laspeyres, Paasche, Geomeric Laspeyres, Geomeric Paasche, Fisher and Törnqvis price indexes. Secion 5 invesigaes how close he unweighed Jevons index is o he Geomeric Laspeyres, Geomeric Paasche P GP and Törnqvis P T price indexes. Secion 6 develops some relaionships beween he Törnqvis index and geomeric indexes ha use average annual shares as weighs. Secion 7 looks a he differences beween fixed base and chained Törnqvis indexes. Mulilaeral indexes make an appearance in secion 8: he fixed base Törnqvis index is compared o he GEKS and GEKS-Törnqvis (or CCDI) mulilaeral indexes. Secions 9 and 10 compare Uni Value and Qualiy Adjused Uni Value indexes o he Fisher index while secion 11 compares he Lowe index o he Fisher index. Secions look a various addiional mulilaeral indexes: he Geary Khamis in secion 12, he Weighed Time Produc Dummy index in secion 13 and Similariy Linked price indexes in secion 14. The Appendix evaluaes all of he above indexes for a grocery sore scanner daa se ha is publically available. However, he daa se had a number of missing prices and quaniies. Some of hese missing prices may be due o lack of sales or shorages of invenory. In addiion, how should he inroducion of new producs and he disappearance of (possibly) obsolee producs be reaed in he conex of forming a consumer price index? Hicks (1940; 140) suggesed a general approach o his measuremen problem in he conex of he economic approach o index number heory. His approach was o apply normal index number heory bu esimae (or guess a) hypoheical prices ha would induce uiliy maximizing purchasers of a relaed group of producs o demand 0 unis of unavailable producs. Wih hese virual (or reservaion or impued) prices in hand, one can jus apply normal index number heory using he 15 In he remainder of his sudy, we will speak of producs bu he same analysis applies o services.

6 6 augmened price daa and he observed quaniy daa. In our empirical example in he Appendix, we will use he scanner daa ha was used in Diewer and Feensra (2017) for frozen juice producs for a Dominick s sore in Chicago for 3 years. This daa se had 20 observaions where q n = 0. For hese 0 quaniy observaions, Diewer and Feensra esimaed posiive Hicksian reservaion prices for hese missing price observaions and hese impued prices are used in our empirical example in he Appendix. However, i is possible o use alernaive posiive reservaion prices ha do no rely on economerics. In secion 15 below, we will discuss one of hese alernaive mehods for consrucing reservaion prices for observaions when sales of a produc are zero. We used an inflaion adjused carry forward and backward mehodology o consruc alernaive reservaion prices for he missing prices and compared seleced indexes using he new reservaion prices o he corresponding indexes using he economerically esimaed reservaion prices. The resuling differences in our bes index numbers for our example daa se are lised in secion A.6 of he Appendix. The Appendix liss he Dominick s daa along wih he esimaed reservaion prices. The Appendix also has ables and chars of he various index number formulae ha are discussed in he main ex of he sudy. Secion 16 concludes. 2. Comparing CES Price Levels and Indexes In his secion, we will begin our analysis by considering alernaive mehods by which he prices for N relaed producs could be aggregaed ino a represenaive or aggregae price for he producs for a given period. We inroduce some noaion ha will be used in he res of he paper. We suppose ha we have colleced price and quaniy daa from a reail oule on N closely relaed producs for T ime periods. 16 Typically, a ime period is a monh. Denoe he price of produc n in period as p n and he corresponding quaniy during period as q n for n = 1,...,N and = 1,...,T. Usually, p n will be he period uni value price for produc n in period ; i.e., p n v n /q n where v n is he oal value of produc n ha is sold during period and q n is he oal quaniy of produc n ha is sold during period. We assume ha q n 0 and p n > 0 for all and n. 17 The resricion ha all producs have posiive prices associaed wih hem is a necessary one for much of our analysis since many popular index numbers are consruced using logarihms of prices and he logarihm of a zero price is no well defined. However, our analysis does allow for possible 0 quaniies and values sold during periods in he sample. Denoe he period sricly posiive price vecors and nonnegaive and non zero quaniy vecors as p [p 1,...,p N ] >> 0 N and q [q 1,...,q N ] > 16 The T periods can be regarded as a window of observaions, followed by anoher window of lengh T which has dropped he firs period from he window and added he daa of period T+1 o he window. The lieraure on how o link he resuls of one window o he nex window is discussed in Diewer and Fox (2017). We will no discuss his linking problem in he presen sudy. 17 In he case where q n = 0, hen v n = 0 as well and hence p n v n /q n is no well defined in his case. In he case where q n = 0, we will assume ha p n is a posiive impued price.

7 7 0 N respecively for = 1,...,T where 0 N is an N dimensional vecor of zeros. The inner produc of he vecors p and q is denoed by p q n=1 N p n q n > 0. Define he period sales (or expendiure) share for produc n as s n p n q n /p q for n = 1,...,N and = 1,...,T. The period sales share vecor is defined as s [s 1,...,s N ] > 0 N for = 1,...,T. In mos applicaions, he N producs are closely relaed and hey have common unis of measuremen (by weigh, or by volume or by sandard package size). In his conex, i is useful o define he period real share for produc n of oal produc sales, S n q n /1 N q for n = 1,...,N and = 1,...,T where 1 N is an N dimensional vecor of ones. Denoe he period real share vecor as S [S 1,...,S N ] for = 1,...,T. Define he generic produc weighing vecor as [ 1,..., N ]. We assume ha has sricly posiive componens which sum o one; i.e., we assume ha saisfies: (1) 1 N = 1 ; >> 0 N. Le p [p 1,...,p N ] >> 0 N be a posiive price vecor. The corresponding mean of order r of he prices p (wih weighs ) or CES price level, m r, (p) is defined as follows: 18 (2) m r, (p) [ N n=1 n p r n ] 1/r ; r 0; N n=1 (p n ) n ; r = 0. I is useful o have a special noaion for m r, (p) when r = 1: (3) p n=1 N n p n = p. Thus p is an weighed arihmeic mean of he prices p 1,p 2,...,p N and i can be inerpreed as a weighed Duo price level. 19 From Schlömilch s (1858) Inequaliy, 20 we know ha m r, (p) m s, (p) if r s and m r, (p) m s, (p) if r s. However, we do no know how big he gaps are beween hese price levels for differen r and s. When r = 0, m 0, (p) becomes a weighed geomeric mean or a weighed Jevons (1865) or Cobb-Douglas price level and i is of ineres o know how much higher he weighed Duo price level is han he corresponding weighed Jevons price level. Proposiion 1 below provides an approximaion o he gap beween m r, (p) and m 1, (p) for any r, including r = Hardy, Lilewood and Polya (1934; 12-13) refer o his family of means or averages as elemenary weighed mean values and sudy heir properies in grea deail. The funcion m r, (p) can also be inerpreed as a Consan Elasiciy of Subsiuion (CES) uni cos funcion if r 1. The corresponding uiliy or producion funcion was inroduced ino he economics lieraure by Arrow, Chenery, Minhas and Solow (1961). For addiional maerial on CES funcions, see Feensra (1994) and Diewer and Feensra (2017). 19 The ordinary Duo (1738) price level for he period prices p is defined as p D (1/N) N n=1 p n. Thus i is equal o m 1, (p ) where = (1/N)1 N. 20 See Hardy, Lilewood and Polya (1934; 26) for a proof of his resul.

8 8 Define he weighed variance of p/p [p 1 /p,...,p N /p ] where p is defined by (3) as follows: 21 (4) Var (p/p ) n=1 N n [(p n /p ) 1] 2. Proposiion 1: Le p >> 0 N, >> 0 N and 1 N = 1. Then m r, (p)/m 1, (p) is approximaely equal o he following expression for any r: (5) m r, (p)/m 1, (p) 1 + (½)(r 1)Var (p/p ) where Var (p/p ) is defined by (4). The expression on he righ hand side of (5) uses a second order Taylor series approximaion o m r, (p) around he equal price poin p = p 1 N where p is defined by (3). 22 Proof: Sraighforward calculaions show ha he level, vecor of firs order parial derivaives and marix of second order parial derivaives of m r, (p) evaluaed a he equal price poin p = p 1 N are equal o he following expressions: m r, (p 1 N ) = p p; p m r, (p 1 N ) = ; 2 ppm r, (p 1 N ) = (p ) 1 (r 1)( T ) where is a diagonal N by N marix wih he elemens of he column vecor running down he main diagonal and T is he ranspose of he column vecor. Thus T is a rank one N by N marix. Thus he second order Taylor series approximaion o m r, (p) around he poin p = p 1 N is given by he following expression: (6) m r, (p) p + (p p 1 N ) + (½)(p p 1 N ) T (p ) 1 (r 1)( T )(p p 1 N ) = p + (½)(p ) 1 (r 1)(p p 1 N ) T (p ) 1 ( T )(p p 1 N ) using (1) and (3) = p [1 + (½)(r 1)(p ) 2 (p p 1 N ) T ( T )(p p 1 N )] = m 1, (p)[1 + (½)(r 1)Var (p/p )] using (2), (3) and (4). Q.E.D. The approximaion (6) also holds if r = 0. In his case, (6) becomes he following approximaion: 23 (7) m 0, (p) N n=1 (p n ) n m 1, (p)[1 (½)Var (p/p )] = m 1, (p){1 (½) N n=1 n [(p n /p ) 1] 2 } = [ N n=1 n p n ]{1 (½) N n=1 n [(p n /p ) 1] 2 } 21 Noe ha he weighed mean of p/p is equal o N n=1 n p n /p = 1. Thus (4) defines he corresponding weighed variance. 22 For alernaive approximaions for he differences beween mean of order r averages, see Varia (1978; ). Varia s approximaions involve variances of logarihms of prices whereas our approximaions involve variances of deflaed prices. Our analysis is a variaion on his pioneering analysis. 23 Noe ha m 0, (p) can be regarded as a weighed Jevons (1865) price level or a Cobb Douglas (1928) price level. Similarly, p m 1, (p) can be regarded as a weighed Duo (1738) price level or a Leonief (1936) price level.

9 9 n=1 N n p n. Thus he bigger is he variaion in he N prices p 1,...,p N, he bigger will be Var (p/p ) and he more he weighed arihmeic mean of he prices, N n=1 n p n, will be greaer han he corresponding weighed geomeric mean of he prices, N n=1 (p n ) n. Noe ha if all of he p n are equal, hen Var (p/p ) will be equal o 0 and he approximaions in (6) and (7) become exac equaliies. Recall ha he uni value price and quaniy sold for produc n during period was defined as p n and q n for n = 1,...,N and = 1,...,T. A his poin, i is useful o define he Jevons (1865) and Duo (1738) period price levels for he prices in our window of observaions, p J and p D, and he corresponding Jevons and Duo price indexes, P J and P D, for = 1,...,T: (8) p D n=1 N (1/N)p n ; (9) p J n=1 N p n 1/N ; (10) P D p D /p D 1 ; (11) P J p J /p J 1 = n=1 N (p n /p 1n ) 1/N. Thus he period price index is simply he period price level divided by he corresponding period 1 price level. Noe ha he Jevons price index can also be wrien as he geomeric mean of he long erm price raios (p n /p 1n ) beween he period prices relaive o he corresponding period 1 prices. The weighed Duo and Jevons period price levels using a weigh vecor which saisfies he resricions (1), p D and p J, are defined by (12) and (13) and he corresponding weighed Duo and Jevons period price indexes, P D and P J, 24 are defined by (14) and (15) for = 1,...,T: (12) p D N n=1 n p n = m 1, (p ) ; (13) p J N n=1 (p n ) n = m 0, (p ) ; (14) P D p D /p 1 D = p / p 1 ; (15) P J p J /p 1 J = N n=1 (p n /p 1n ) n. Obviously, (12)-(15) reduce o definiions (8)-(11) if = (1/N)1 N. We can use he approximaion (7) for p = p 1 and p = p in order o obain he following approximae relaionship beween he weighed Duo price index for period, P D, and he corresponding weighed Jevons index, P J : (16) P J p J /p J 1 ; = 1,...,T = m 0, (p )/m 0, (p 1 ) using (2) and (13) m 1, (p ){1 (½) n=1 N n [(p n /p ) 1] 2 }/m 1, (p 1 ){1 (½) n=1 N n [(p 1n /p 1 ) 1] 2 } using (7) for p = p and p = p 1 where p p and p 1 p 1 24 This ype of index is frequenly called a Geomeric Young index; see Armknech and Silver (2014; 4-5).

10 10 = P D {1 (½) n=1 N n [(p n /p ) 1] 2 }/{1 (½) n=1 N n [(p 1n /p 1 ) 1] 2 } = P D {1 (½)Var (p /p )}/{1 (½)Var (p 1 /p 1 )}. In he elemenary index conex where here are no rends in prices in diverging direcions, i is likely ha Var (p /p ) is approximaely equal o Var (p 1 /p 1 ). 25 Under hese condiions, he weighed Jevons price index P J is likely o be approximaely equal o he corresponding weighed Duo price index, P D. Of course, his approximae equaliy resul exends o he case where = (1/N)1 N and so i is likely ha he Duo price indexes P D are approximaely equal o heir Jevons price index counerpars, P J. 26 However, if he variance of he deflaed period 1 prices is unusually large (small), hen here will be a endency for P J o exceed (o be less han) P D for > A higher levels of aggregaion where he producs may no be very similar, i is likely ha here will be divergen rends in prices over ime. In his case, we can expec Var (p /p ) o exceed Var (p 1 /p 1 ). Thus using (16) under hese circumsances leads o he likelihood ha he weighed index P J will be significanly lower han P D. Similarly, under he diverging rends in prices hypohesis, we can expec he ordinary Jevons index P J o be lower han he ordinary Duo index P D. 28 We conclude his secion by finding an approximae relaionship beween a CES price index and he corresponding weighed Duo price index P D. This approximaion resul assumes ha economeric esimaes for he parameers of he CES uni cos funcion m r, (p) defined by (2) are available so ha we have esimaes for he weighing vecor (which we assume saisfies he resricions (1)) and he parameer r which we assume saisfies r The CES period price levels using a weigh vecor which saisfies he resricions (1) and an r 1, p CES, and he corresponding CES period price indexes, P CES,r, are defined as follows for = 1,...,T: 25 Noe ha he vecors p /p and p 1 /p 1 are price vecors ha are divided by heir weighed arihmeic means. Thus hese vecors have eliminaed general inflaion beween periods 1 and. 26 The same approximae inequaliies hold for he weighed case. An approximaion resul similar o (16) for he equal weighs case where = (1/N)1 N was firs obained by Carruhers, Sellwood and Ward (1980; 25). 27 For our empirical example considered in Appendix 1, we found ha he sample mean of he P D was while he sample mean of he P J was so ha in general, he Duo price index was below he corresponding Jevons index. However, he period 1 variance, Var (p 1 /p 1 ) (for = (1/N)1 N ) was unusually large (equal o 0.072) as compared o he sample average of he Var (p /p ) (equal o 0.063) and his explains why P D was generally below P J for our paricular daa se. In general, for highly subsiuable producs, we expec he Jevons price index o lie below is Duo counerpar. When we dropped he daa for he firs year, we found ha he resuling sample mean of he P D was while he sample mean of he P J was so ha in his case, he Duo price index was above he corresponding Jevons index (as expeced). The daa for he firs year in our sample was unusual due o he absence of producs 2 and 4 for mos of he year. 28 Furhermore, as we shall see laer, he Duo index can be viewed as a fixed baske index where he baske is a vecor of ones. Thus i is subjec o subsiuion bias which will show up under he divergen price rends hypohesis. 29 These resricions imply ha m r, (p) is a linearly homogeneous, nondecreasing and concave funcion of he price vecor p. These resricions mus be saisfied if we apply he economic approach o price index heory.

11 11 (17) p CES,r [ n=1 N n p n r ] 1/r = m r, (p ) ; (18) P CES,r p CES,r /p CES,r 1 = m r, (p )/m r, (p 1 ). Now use he approximaion (6) for p = p 1 and p = p in order o obain he following approximae relaionship beween he weighed Duo price index for period, P D, and he corresponding period CES index, P CES,r for = 1,...,T: (19) P CES,r p CES,r /p CES,r 1 ; = m r, (p )/m r, (p 1 ) using (17) and (18) [m 1, (p )/m 1, (p 1 )][1 + (½)(r 1)Var (p /p )]/[1 + (½)(r 1)Var (p 1 /p 1 )] = P D {1 + (½)(r 1) n=1 N n [(p n /p ) 1] 2 }/{1 + (½)(r 1) n=1 N n [(p 1n /p 1 ) 1] 2 } where we used definiions (4), (12) and (14) o esablish he las equaliy in (19). Again, in he elemenary index conex wih no diverging rends in prices, we could expec Var (p /p ) Var (p 1 /p 1 ) for = 2,...,T. Using his assumpion abou he approximae consancy of he (weighed) variance of he deflaed prices over ime, and using (16) and (19), we obain he following approximaions for = 2,3,...,T: (20) P CES,r P J P D. Thus under he assumpion of approximaely consan variances for deflaed prices, he CES, weighed Jevons and weighed Duo price indexes should approximae each oher fairly closely, provided ha he same weighing vecor is used in he consrucion of hese indexes. 30 The parameer r which appears in he definiion of he CES uni cos funcion is relaed o he elasiciy of subsiuion ; i.e., i urns ou ha = 1 r. 31 Thus as r akes on values from 1 o, will ake on values from 0 o +. In he case where he producs are closely relaed, ypical esimaes for range from 1 o 10 when CES preferences are esimaed. Thus if we subsiue = 1 r ino he approximaion (19), we obain he following approximaions for = 1,...,T: (21) P CES,r P D [1 (½) Var (p /p )]/[1 (½) Var (p 1 /p 1 )]. The approximaions in (21) break down for large and posiive (or equivalenly, for very negaive r); i.e., he expressions in square brackes on he righ hand sides of (21) will pass hrough 0 and become meaningless as becomes very large. The approximaions become increasingly accurae as approaches 0 (or as r approaches 1). Of course, he approximaions also become more accurae as he dispersion of prices wihin a period becomes smaller. For beween 0 and 1 and wih normal dispersion of prices, he 30 Again, he approximae relaionship P CES,r P D may no hold if he variance of he prices in he base period, Var (p 1 /p 1 ), is unusually large or small. 31 See for example, Feensra (1994; 158) or Diewer and Feensra (2017).

12 12 approximaions in (21) should be reasonably good. However, as becomes larger, he expressions in square brackes will become closer o 0 and he approximaions in (21) will become more volaile and less accurae as increases from an iniial 0 value. A higher levels of aggregaion where he producs are no similar, i is likely ha here will be divergen rends in prices over ime and in his case, we can expec Var (p /p ) o exceed Var (p 1 /p 1 ). In his case, he approximae equaliies (20) will no longer hold. In he case where he elasiciy of subsiuion is greaer han 1 (so r < 0) and Var (p /p ) > Var (p 1 /p 1 ), we can expec ha P CES,r < P D and he gaps beween hese wo indexes will grow bigger over ime as Var (p /p ) grows larger han Var (p 1 /p 1 ). In he following secion, we will use he mean of order r funcion o aggregae he price raios p n /p 1n ino an aggregae price index for period direcly; i.e., we will no consruc price levels as a preliminary sep in he consrucion of a price index. 3. Using Means of Order r o Aggregae Price Raios In he previous secion, we compared various elemenary indexes using approximae relaionships beween price levels consruced by using means of order r o aggregae prices. In his secion, we will develop approximae relaionships beween price indexes consruced by using means of order r defined over price raios. In wha follows, i is assumed ha he weigh vecor saisfies condiions (1); i.e., >> 0 N and 1 N = 1. Define he mean of order r price index for period (relaive o period 1), P r,, as follows for = 1,...,T: (22) P r, [ N n=1 n (p n /p 1n ) r ] 1/r ; r 0; N n=1 (p n /p 1n ) n ; r = 0. When r = 1 and = (1/N)1 N, hen P r, becomes he period fixed base Carli (1804) price index, P C, defined as follows for = 1,...,T: (23) P C n=1 N (1/N)(p n /p 1n ). Wih a general and r = 1, P r, becomes he fixed base weighed Carli price index, P C, 32 defined as follows for = 1,...,T: (24) P C n=1 N n (p n /p 1n ). Using (24), i can be seen ha he weighed mean of he period long erm price raios p n /p 1n divided by P C is equal o 1; i.e., we have for = 1,...,T: 32 This ype of index is due o Arhur Young (1812; 72) and so we could call his index he Young index, P Y.

13 13 (25) n=1 N n (p n /p 1n P C ) = 1. Denoe he weighed variance of he deflaed period price raios p n /p 1n P C Var (p /p 1 P C ) and define i as follows for = 1,...,T: as (26) Var (p /p 1 P C ) N n=1 n [(p n /p 1n P C ) 1] 2. Proposiion 2: Le p >> 0 N, >> 0 N and 1 N = 1. Then P r, /P 1, approximaely equal o he following expression for any r for = 1,...,T: (27) P r, /P C 1 + (½)(r 1)Var (p /p 1 P C ) = P r, /P C is where P r, is he mean of order r price index (wih weighs ) defined by (22), P C is he weighed Carli index defined by (24) and Var (p /p 1 P C ) is he weighed variance of he deflaed long erm price raios (p n /p 1n )/P C defined by (26). Proof: Replace he vecor p in Proposiion 1 by he vecor [p 1 /p 11,p 2 /p 12,...,p N /p 1N ]. 33 Then he raio m r, (p)/m 1, (p) which appears on he lef hand side of (5) becomes he raio P r, /P 1, = P r, /P C using definiions (22) and (24). The erms p and Var (p/p ) which appear on he righ hand side of (5) become P C and Var (p /p 1 P C ) respecively. Wih hese subsiuions, (5) becomes (27) and we have esablished Proposiion 2. Q.E.D. I is useful o look a he special case of (27) when r = 0. In his case, using definiions (22) and (15), we can esablish he following equaliies for = 1,...,T: (28) P 0, N n=1 (p n /p 1n ) n = P J where P J is he period weighed Jevons or Cobb Douglas price index defined by (15) in he previous secion. 34 Thus when r = 0, he approximaions defined by (27) become he following approximaions for = 1,...,T: (29) P J /P C 1 (½)Var (p /p 1 P C ). Thus he bigger is he weighed variance of he deflaed period long erm price raios, (p 1 /p 11 )/P C,..., (p N /p 1N )/P C, he more he period weighed Carli index P C will exceed he corresponding period weighed Jevons index P J. When = (1/N)1 N, he approximaions (29) become he following approximae relaionships beween he period Carli index P C defined by (23) and he period Jevons index P J defined by (11) for = 1,...,T: In Proposiion 1, some prices in eiher period could be 0. However, Proposiion 2 requires ha all period 1 prices be posiive. 34 Again, recall ha Armknech and Silver (2014; 4) call his index he Geomeric Young index. 35 Resuls ha are essenially equivalen o (30) were firs obained by Dalén (1992) and Diewer (1995). The approximaions in (27) and (29) for weighed indexes are new. Varia and Suoperä (2018; 5) derived

14 14 (30) P J /P C 1 (½)Var (1/N)1 (p /p 1 P C ) = 1 (½) n=1 N (1/N)[(p n /p 1n P C ) 1] 2. Thus he Carli price indexes P C will exceed heir Jevons counerpars P J (unless p = p 1 in which case prices in period are proporional o prices in period 1 and in his case, P C = P J ). This is an imporan resul since from an axiomaic perspecive, he Jevons price index has much beer properies han he corresponding Carli indexes 36 and in paricular, chaining Carli indexes will lead o large upward biases as compared o heir Jevons counerpars. 37 The resuls in his secion can be summarized as follows: holding he weigh vecor consan, he weighed Jevons price index for period, P J is likely o lie below he corresponding weighed Carli index, P C, wih he gap growing as he weighed variance of he deflaed price raios, (p 1 /p 11 )/P C,..., (p N /p 1N )/P C, increases. 38 In he following secion, we urn our aenion o weighed price indexes where he weighs are no exogenous consans bu depend on observed sales shares. 4. Relaionships beween Some Share Weighed Price Indexes In his secion (and in subsequen secions), we will look a comparisons beween price indexes ha use informaion on he observed expendiure or sales shares of producs in addiion o price informaion. Recall ha s n p n q n /p q for n = 1,...,N and = 1,...,T. The fixed base Laspeyres (1871) price index for period, P L, is defined as he following base period share weighed arihmeic average of he price raios, p n /p 1n, for = 1,...,T: (31) P L n=1 N s 1n (p n /p 1n ). I can be seen ha P L is a weighed Carli index P C of he ype defined by (34) in he previous secion where s 1 [s 11,s 12,...,s 1N ]. We will compare P L wih is weighed geomeric mean counerpar, P GL, which is a weighed Jevons index P J where he alernaive approximaions. The analysis in his secion is similar o Varia s (1978; ) analysis of Fisher s (1922) five-ined fork. 36 See Diewer (1995) on his opic. 37 The same upward bias holds for weighed Carli indexes relaive o heir weighed Jevons counerpars. For our frozen juice daa lised in Appendix A, he 3 year sample average of he Carli indexes was as compared o he corresponding sample average of he Jevons indexes which was Since he Jevons price index has he bes axiomaic properies, his resul implies ha CPI compilers should avoid he use of he Carli index in he consrucion of a CPI. This advice goes back o Fisher (1922; 29-30). Since he Duo index will approximae he corresponding Jevons index provided ha he producs are similar and here are no sysemaic divergen rends in prices, Duo indexes can be saisfacory a he elemenary level. If he producs are no closely relaed, Duo indexes become problemaic since hey are no invarian o changes in he unis of measuremen. Moreover, in he case of nonsimilar producs, divergen rends in prices become more probable and hus he Duo index will end o be above he corresponding Jevons index due o subsiuion bias.

15 15 weigh vecor is = s 1. Thus he logarihm of he fixed base Geomeric Laspeyres price index is defined as follows for = 1,...,T: 39 (32) ln P GL n=1 N s 1n ln(p n /p 1n ). Since P GL and P L are weighed geomeric and arihmeic means of he price raios p n /p 1n (using he weighs in he period 1 share vecor s 1 ), Schlömilch s inequaliy implies ha P GL P L for = 1,...,T. The inequaliies (29), wih = s 1, give us approximaions o he gaps beween he P GL = P J and he P C = P L. Thus we have he following approximae equaliies for = s 1 and = 1,...,T: (33) P GL /P L 1 (½)Var (p /p 1 P L ) = 1 (½) n=1 N s 1n [(p n /p 1n P L ) 1] 2. Using our scanner daa se on 19 frozen frui juice producs lised in he Appendix for 39 monhs, we found ha Var (p /p 1 P C ) for 2 varied beween and wih a mean of Thus here was a considerable amoun of variaion in hese variance erms. The sample mean of he raios P GL /P L was so ha P GL was below P L by 1.27 percenage poins on average. The sample mean of he error erms on he righ hand sides of he approximaions defined by (33) was which is only below he sample mean of he raios P GL /P L by 0.1 percenage poins. The correlaion coefficien beween he raios P GL /P L and he corresponding error erms on he righ hand sides of (33) was Thus he approximae equaliies in (33) were quie close o being equaliies. The fixed base Paasche (1874) price index for period, P P, is defined as he following period share weighed harmonic average of he price raios, p n /p 1n, for = 1,...,T: (34) P P [ n=1 N s n (p n /p 1n ) 1 ] 1. We will compare P P wih is weighed geomeric mean counerpar, P GP, which is a weighed Jevons index P J where he weigh vecor is = s. Thus he logarihm of he fixed base Geomeric Paasche price index is defined as follows for = 1,...,T: (35) ln P GP n=1 N s n ln(p n /p 1n ). Since P GP and P P are weighed geomeric and harmonic means of he price raios p n /p 1n (using he weighs in he period share vecor s ), Schlömilch s inequaliy implies ha P P P GP for = 1,...,T. However, we canno apply he inequaliies (29) direcly o give us an approximaion o he size of he gap beween P GP and P P. Viewing definiion (34), i can be seen ha he reciprocal of P P is a period share weighed average of he reciprocals of he long erm price raios, p 11 /p 1, p 12 /p 2,..., p 1N /p N. Thus using definiion (34), we have he following equaions and inequaliies for = s and = 1,...,T: (36) [P P ] 1 = n=1 N s n (p 1n /p n ) 39 Varia (1978; 272) used he erms geomeric Laspeyres and geomeric Paasche o describe he indexes defined by (32) and (35).

16 16 N s n=1 (p 1n /p n ) n = [P GP ] 1 using definiions (35) where he inequaliies in (36) follow from Schlömilch s inequaliy; i.e., a weighed arihmeic mean is always equal o or greaer han he corresponding weighed geomeric mean. Noe ha he firs equaion in (36) implies ha he period share weighed mean of he reciprocal price raios, p 1n /p n, is equal o he reciprocal of P P. Now adap he approximae equaliies (29) in order o esablish he following approximae equaliies for = 1,...,T: (37) [P GP ] 1 /[P P ] 1 1 (½) n=1 N s n [(p 1n /p n [P P ] 1 ) 1] 2. The approximae equaliies (37) may be rewrien as follows for = 1,...,T: (38) P GP P P /{1 (½) n=1 N s n [(p 1n P P /p n ) 1] 2 }. Thus for = 1,...,T, we have P GP P P (and he approximae equaliies (38) measure he gaps beween hese indexes) and P GL P L (and he approximae equaliies (33) measure he gaps beween hese indexes). Laer we will show ha he inequaliies P GP P GL are likely if he N producs are close subsiues for each oher. Using our scanner daa se lised in he Appendix, we found ha he variance erms on he righ hand sides of (38), N n=1 s n [(p 1n P P /p n ) 1] 2, for 2 varied beween and wih a mean of Thus hese variances were bigger han he corresponding variances in (33) by one percenage poin on average. This means ha on average, P GL /P L > P P /P GP or on average for our sample, P GL P GP > P L P P, which in urn implies ha on average, P T > P F for our sample. 40 The sample mean of he raios P GP /P P was so ha P GP was above P P by 1.69 percenage poins on average. The sample mean of he error erms 1/{1 (½) N n=1 s n [(p 1n P P /p n ) 1] 2 } on he righ hand sides of he approximaions defined by (38) was which is idenical o he sample mean of he raios P GP /P P. The correlaion coefficien beween he raios P GP /P P and he corresponding error erms on he righ hand sides of (38) was Thus he approximae equaliies in (38) were quie close o being equaliies. Suppose ha prices in period are proporional o he corresponding prices in period 1 so ha p = p 1 where is a posiive scalar. Then i is sraighforward o show ha P P = P GP = P GL = P L = and he error erms for equaion in (34) and (39) are equal o 0. Define he period fixed base Fisher (1922) and Törnqvis 41 price indexes, P F and P T, as he following geomeric means for = 1,...,T: (39) P F [P L P P ] 1/2 ; (40) P T [P GL P GP ] 1/2. 40 P F and P T are defined by (39) and (40) below. 41 See Törnqvis (1936) and Törnqvis and Törnqvis (1937) and Theil (1967; ).

17 17 Thus P F is he geomeric mean of he period fixed base Laspeyres and Paasche price indexes while P T is he geomeric mean of he period fixed base geomeric Laspeyres and geomeric Paasche price indexes. Now use he approximae equaliies in (34) and (38) and subsiue hese equaliies ino (40) in order o obain he following approximae equaliies beween P T and P F for f = 1,...,T: (41) P T [P GL P GP ] 1/2 [P L P P ] 1/2 (p 1,p,s 1,s ) = P F (p 1,p,s 1,s ) where he approximaion error funcion (p 1,p,s 1,s ) is defined as follows for = 1,...,T: (42) (p 1,p,s 1,s ) {1 (½) n=1 N s 1n [(p n /p 1n P L ) 1] 2 } 1/2 /{1 (½) n=1 N s n [(p 1n P P /p n ) 1] 2 } 1/2. Thus P T is approximaely equal o P F for = 1,...,T. Bu how good are hese approximaions? We know from Diewer (1978) ha P T = P T (p 1,p,s 1,s ) approximaes P F = P F (p 1,p,s 1,s ) o he second order around any poin where p = p 1 and s = s We also know ha he approximaions in (33) and (38) are fairly good, a leas for our scanner daa se. Thus i is likely ha he error erms (p 1,p,s 1,s ) are close o Using our scanner daa se lised in he Appendix, we found ha he sample mean of he raios P T /P F was so ha P T was above P F by 0.19 percenage poins on average. The sample mean of he error erms (p 1,p,s 1,s ) defined by (42) was The correlaion coefficien beween he raios P T /P F and he corresponding error erms (p 1,p,s 1,s ) on he righ hand sides of (41) was Thus he approximae equaliies in (41) were quie close o being equaliies. However, if he producs were highly subsiuable and if prices and shares rended in opposie direcions, hen we expec ha he base period share weighed variance N n=1 s 1n [(p n /p 1n P L ) 1] 2 will increase as increases and we expec he period share weighed variance N n=1 s n [(p 1n P P /p n ) 1] 2 o increase even more as increases because as p n becomes smaller, [(p 1n P P /p n ) 1] 2 becomes bigger and he share weigh s n will also increase. Thus P T will end o increase relaive o P F as ime increases under hese condiions. The more subsiuable he producs are, he greaer will be his endency. Our enaive conclusion a his poin is ha he approximaions defined by (33), (38) and (41) are good enough o provide rough esimaes of he differences in he six price indexes involved in hese approximae equaliies. Empirically, we found ha he variance 42 This resul can be generalized o he case where p = p 1 and s = s However, he Diewer (1978) second order approximaion is differen from he presen second order approximaions ha are derived from Proposiion 2. Thus he closeness of (p 1,p,s 1,s ) o 1 depends on he closeness of he Diewer second order approximaion of P T o P F and he closeness of he second order approximaions ha were used in (33) and (38), which use differen Taylor series approximaions. Varia and Suoperä (2018) used alernaive Taylor series approximaions o obain relaionships beween various indexes.

18 18 erms on he righ hand sides of (38) ended o be larger han he corresponding variances on he righ hand sides of (33) and hese differences led o a endency for he fixed base Fisher price indexes P F o be slighly smaller han he corresponding fixed base Törnqvis Theil price indexes P T. 44 We conclude his secion by developing an exac relaionship beween he geomeric Laspeyres and Paasche price indexes. Using definiions (32) and (35) for he logarihms of hese indexes, we have he following exac decomposiion for he logarihmic difference beween hese indexes for = 1,...,T: 45 (43) lnp GP lnp GL = n=1 N s n ln(p n /p 1n ) n=1 N s 1n ln(p n /p 1n ) = n=1 N [s n s 1n ][lnp n lnp 1n ]. Define he vecors lnp [lnp 1,lnp 2,...,lnp N ] for = 1,...,T. I can be seen ha he righ hand side of equaion equaions (43) is equal o [s s 1 ] [lnp lnp 1 ], he inner produc of he vecors x s s 1 and y lnp lnp 1. Le x * and y * denoe he arihmeic means of he componens of he vecors x and y. Noe ha x * (1/N)1 N x = (1/N)1 N [s s 1 ] = (1/N)[1 1] = 0. The covariance beween x and y is defined as Cov(x,y) (1/N)[x x * 1 N ] [y y * 1 N ] = (1/N) x y x * y * = (1/N) x y 46 since x * is equal o 0. Thus he righ hand side of (43) is equal o N Cov(x,y) = N Cov(s s 1,lnp lnp 1 ); i.e., he righ hand side of (43) is equal o N imes he covariance of he long erm share difference vecor, s s 1, wih he long erm log price difference vecor, lnp lnp 1. Hence if his covariance is posiive, hen lnp GP lnp GL > 0 and P GP > P GL. If his covariance is negaive, hen P GP < P GL. We argue below ha for he case where he N producs are close subsiues, i is likely ha he covariances on he righ hand side of equaions (43) are negaive for > 1. Suppose ha he observed price and quaniy daa are approximaely consisen wih purchasers having idenical Consan Elasiciy of Subsiuion preferences. CES preferences are dual o he CES uni cos funcion m r, (p) which is defined by (2) above where saisfies (1) and r 1. I can be shown 47 ha he sales share for produc n in a period where purchasers face he sricly posiive price vecor p [p 1,...,p N ] is he following share: (44) s n (p) n p n r / i=1 N i p i r ; n = 1,...,N. Upon differeniaing s n (p) wih respec o p n, we find ha he following relaions hold: (45) lns n (p)/ lnp n = r[1 s n (p)] ; n = 1,...,N. 44 Varia and Suoperä (2018) also found a endency for he Fisher price index o lie slighly below heir Törnqvis counerpars in heir empirical work. 45 Varia and Suoperä (2018; 26) derived his resul and noiced ha he righ hand side of (43) could be inerpreed as a covariance. They also developed several alernaive exac decomposiions for he difference lnp GP lnp GL. Their paper also develops a new heory of excellen index numbers. 46 This equaion is he covariance ideniy which was firs used by Borkiewicz (1923) o show ha normally he Paasche price index is less han he corresponding Laspeyres index. 47 See Diewer and Feensra (2017) for example.

19 19 Thus lns n (p)/ lnp n < 0 if r < 0 (or equivalenly, if he elasiciy of subsiuion 1 r is greaer han 1) and lns n (p)/ lnp n > 0 if r saisfies 0 < r < 1 (or equivalenly, if he elasiciy of subsiuion saisfies 0 < < 1). If we are aggregaing prices a he firs sage of aggregaion where he producs are close subsiues and purchasers have common CES preferences, hen i is likely ha he elasiciy of subsiuion is greaer han 1 and hence as he price of produc n decreases, i is likely ha he share of ha produc will increase. Hence we expec he erms [s n s 1n ][lnp n lnp 1n ] o be predominanly negaive; i.e., if p 1n is unusually low, hen lnp n lnp 1n is likely o be posiive and s n s 1n is likely o be negaive. On he oher hand, if p n is unusually low, hen lnp n lnp 1n is likely o be negaive and s n s 1n is likely o be posiive. Thus for closely relaed producs, we expec he covariances on he righ hand sides of (43) o be negaive and for P GP o be less han P GL. We can combine his inequaliy wih our previously esablished inequaliies o conclude ha for closely relaed producs, i is likely ha P P < P GP < P T < P GL < P L. On he oher hand, if we are aggregaing a higher levels of aggregaion, hen i is likely ha he elasiciy of subsiuion is in he range 0 < < 1, 48 and in his case, he covariances on he righ hand sides of (43) will end o be posiive and hence in his case, i is likely ha P GP > P GL. We also have he inequaliies P P < P GP and P GL < P L in his case. 49 We urn now o some relaionships beween weighed and unweighed (i.e., equally weighed) geomeric price indexes. 5. Relaionships beween he Jevons, Geomeric Laspeyres, Geomeric Paasche and Törnqvis Price Indexes In his secion, we will invesigae how close he unweighed Jevons index P J is o he geomeric Laspeyres P GL, geomeric Paasche P GP and Törnqvis P T price indexes. We firs invesigae he difference beween he logarihms of P GL and P J. Using he definiions for hese indexes, we have he following log differences for = 1,...,T: (46) lnp GL lnp J = n=1 N [s 1n (1/N)][lnp n lnp 1n ] = NCov(s 1 (1/N)1 N, lnp lnp 1 ). In he elemenary index conex where he N producs are close subsiues and produc shares in period 1 are close o being equal, i is likely ha is posiive; i.e., if ln p 1n is unusually low, hen s 1n is likely o be unusually high and hus i is likely ha s 1n (1/N) > 48 See Shapiro and Wilcox (1997) who found ha = 0.7 fi he US daa well a higher levels of aggregaion. See also Armknech and Silver (2014; 9) who noed ha esimaes for end o be greaer han 1 a he lowes level of aggregaion and less han 1 a higher levels of aggregaion. 49 See Varia (1978; ) for a similar discussion abou he relaionships beween P L, P P, P F, P GL, P GP and P T. Varia exended he discussion o include period 1 and period share weighed harmonic averages of he price raios, p n /p 1n. See also Armknech and Silver (2014; 10) for a discussion on how weighed averages of he above indexes could approximae a superlaive index a higher levels of aggregaion.