15. Basic Index Number Theory

5. Basc Idex Numer Theory A. Iroduco The aswer o he queso wha s he Mea of a gve se of magudes cao geeral e foud, uless here s gve also he ojec for he sake of whch a mea value s requred. There are as may kds of average as here are purposes; ad we may almos say, he maer of prces as may purposes as wrers. Hece much va coroversy ewee persos who are lerally a cross purposes. (F.Y. Edgeworh, 888, p. 347) 5. The umer of physcally dsc goods ad uque ypes of servces ha cosumers ca purchase s he mllos. O he usess or produco sde of he ecoomy, here are eve more producs ha are acvely raded. The reaso s ha frms o oly produce producs for fal cosumpo, hey also produce expors ad ermedae producs ha are demaded y oher producers. Frms collecvely also use mllos of mpored goods ad servces, housads of dffere ypes of laor servces, ad hudreds of housads of specfc ypes of capal. If we furher dsgush physcal producs y her geographc locao or y he seaso or me of day ha hey are produced or cosumed, he here are llos of producs ha are raded wh each year ay advaced ecoomy. For may purposes, s ecessary o summarze hs vas amou of prce ad quay formao o a much smaller se of umers. The queso ha hs chaper addresses s he followg: How exacly should he mcroecoomc formao volvg possly mllos of prces ad quaes e aggregaed o a smaller umer of prce ad quay varales? Ths s he asc dex umer prolem. 5.2 I s possle o pose he dex umer prolem he coex of mcroecoomc heory; ha s, gve ha we wsh o mpleme some ecoomc model ased o producer or cosumer heory, wha s he es mehod for cosrucg a se of aggregaes for he model? However, whe cosrucg aggregae prces or quaes, oher pos of vew (ha do o rely o ecoomcs) are possle. Some of hese alerave pos of vew wll e cosdered hs chaper ad he ex chaper. Ecoomc approaches wll e pursued Chapers 7 ad 8. 5.3 The dex umer prolem ca e framed as he prolem of decomposg he value of a welldefed se of rasacos a perod of me o a aggregae prce mulpled y a aggregae quay erm. I urs ou ha hs approach o he dex umer prolem does o lead o ay useful soluos. Therefore, Seco B, he prolem of decomposg a value rao perag o wo perods of me o a compoe ha measures he overall chage prces ewee he wo perods (hs s he prce dex) mulpled y a erm ha measures he overall chage quaes ewee he wo perods (hs s he quay dex) s cosdered. The smples prce dex s a fxed-aske dex. I hs dex, fxed amous of he quaes he value aggregae are chose, ad he hs fxed aske of quaes a he prces of perod 0 ad perod are calculaed. The fxed-aske prce dex s smply he rao of hese wo values, where he prces vary u he quaes are held fxed. Two aural choces for he fxed aske are he quaes rasaced he ase perod, perod 0, or he quaes rasaced he curre perod, perod. These wo choces lead o he Laspeyres (87) ad Paasche (874) prce dces, respecvely. 5.4 Uforuaely, he Paasche ad Laspeyres measures of aggregae prce chage ca dffer, somemes susaally. Thus, Seco C cosders akg a average of hese wo dces o come up wh a sgle measure of prce chage. Seco C. argues ha he es average o ake s he geomerc mea, whch s Irvg Fsher s (922) deal prce dex. I Seco C.2, sead of averagg he Paasche ad Laspeyres measures of prce chage, akg a average of he wo askes s cosdered. Ths fxed-aske approach o dex umer heory leads o a prce dex advocaed y Walsh (90, 92a). However, oher fxed-aske 370

5. Basc Idex Numer Theory approaches are also possle. Isead of choosg he aske of perod 0 or (or a average of hese wo askes), s possle o choose a aske ha peras o a erely dffere perod, say, perod. I fac, s ypcal sascal agecy pracce o pck a aske ha peras o a ere year (or eve wo years) of rasacos a year efore perod 0, whch s usually a moh. Idces of hs ype, where he wegh referece perod dffers from he prce referece perod, were orgally proposed y Joseph Lowe (823), ad Seco D dces of hs ype wll e suded. They wll also e evaluaed from he axomac perspecve Chaper 6 ad from he ecoomc perspecve Chaper 7. 5.5 I Seco E, aoher approach o he deermao of he fucoal form or he formula for he prce dex s cosdered. Ths approach, devsed y he Frech ecooms Dvsa (926), s ased o he assumpo ha prce ad quay daa are avalale as couous fucos of me. The heory of dffereao s used o decompose he rae of chage of a couous me value aggregae o wo compoes ha reflec aggregae prce ad quay chage. Alhough Dvsa s approach offers some sghs, 2 does o offer much gudace o sascal ageces erms of leadg o a defe choce of dex umer formula. 5.6 I Seco F, he advaages ad dsadvaages of usg a fxed-ase perod he laeral dex umer comparso are cosdered versus always comparg he curre perod wh he prevous perod, whch s called he cha sysem. I he cha sysem, a lk s a dex umer comparso of oe perod wh he prevous perod. These lks are mulpled o make comparsos over may perods. Idces of hs ype wll o appear Chaper 9, where mos of he dex umer formulas exhed Chapers 5 8 wll e llusraed usg a arfcal daa se. However, dces where he wegh referece perod dffers from he prce referece perod wll e llusraed umercally Chaper 22, where he prolem of seasoal producs wll e dscussed. 2 I parcular, ca e used o jusfy he cha sysem of dex umers, whch wll e dscussed Seco E.2. B. Decomposo of Value Aggregaes o Prce ad Quay Compoes B. Decomposo of value aggregaes ad he produc es 5.7 A prce dex s a measure or fuco ha summarzes he chage he prces of may producs from oe suao 0 (a me perod or place) o aoher suao. More specfcally, for mos praccal purposes, a prce dex ca e regarded as a weghed mea of he chage he relave prces of he producs uder cosderao he wo suaos. To deerme a prce dex, s ecessary o kow () Whch producs or ems o clude he dex, () How o deerme he em prces, () Whch rasacos ha volve hese ems o clude he dex, (v) How o deerme he weghs ad from whch sources hese weghs should e draw, ad (v) Whch formula or mea should e used o average he seleced em relave prces. All he aove prce dex defo quesos excep he las ca e aswered y appealg o he defo of he value aggregae o whch he prce dex refers. A value aggregae V for a gve colleco of ems ad rasacos s compued as (5.) V pq, where p represes he prce of he h em aoal currecy us, q represes he correspodg quay rasaced he me perod uder cosderao, ad he suscrp defes he h elemeary em he group of ems ha make up he chose value aggregae V. Icluded hs defo of a value aggregae s he specfcao of he group of cluded producs (whch ems o clude) ad of he ecoomc ages egagg rasacos volvg hose producs (whch rasacos o clude), as well as he valuao ad me of recordg prcples movag he ehavor of he ecoomc ages uderakg he rasacos (deermao of prces). The cluded elemeary ems, her valuao (he p ), 37

Producer Prce Idex Maual he elgly of he rasacos, ad he em weghs (he q ) are all wh he doma of defo of he value aggregae. The precse deermao of he p ad q was dscussed more deal Chaper 5 ad oher chapers. 3 5.8 The value aggregae V defed y equao (5.) referred o a cera se of rasacos perag o a sgle (uspecfed) me perod. Now, cosder he same value aggregae for wo places or me perods, perods 0 ad. For he sake of defeess, perod 0 s called he ase perod ad perod s called he curre perod. Assume ha oservaos o he ase-perod prce ad quay vecors, p 0 [p 0,,p 0 ] ad q 0 [q 0,,q 0 ], respecvely, have ee colleced. 4 The value aggregaes he wo perods are defed he ovous way as (5.2) 0 0 0 ;. V V 5.9 I he prevous paragraph, a prce dex was defed as a fuco or measure ha summarzes he chage he prces of he producs he value aggregae from suao 0 o suao. I hs paragraph, a prce dex P(p 0,p,q 0,q ) alog wh he correspodg quay dex (or volume dex) Q(p 0,p,q 0,q ) s defed as wo fucos of he 4 varales p 0,p,q 0,q (hese varales descre he prces ad quaes perag o he value aggregae for perods 0 ad ), where hese wo fucos sasfy he followg equao: 5 (5.3) V/V Pp,p,q,q ( ) 0 0 0 0 0 Q( p,p,q,q ). 3 Ralph Turvey ad ohers (989) have oed ha some values may e dffcul o decompose o uamguous prce ad quay compoes. Some examples of values dffcul o decompose are ak charges, gamlg expedures, ad lfe surace paymes. 4 Noe ha s assumed ha here are o ew or dsappearg producs he value aggregaes. Approaches o he ew goods prolem ad he prolem of accoug for qualy chage are dscussed Chapers 7, 8, ad 2. 5 The frs perso o sugges ha he prce ad quay dces should e joly deermed o sasfy equao (5.3) was Irvg Fsher (9, p. 48). Frsch (930, p. 399) called equao (5.3) he produc es. If here s oly oe em he value aggregae, he he prce dex P should collapse o he sgle-prce rao p /p 0, ad he quay dex Q should collapse o he sgle-quay rao q /q 0. I he case of may ems, he prce dex P s o e erpreed as some sor of weghed average of he dvdual prce raos, p /p 0,, p /p 0. 5.0 Thus, he frs approach o dex umer heory ca e regarded as he prolem of decomposg he chage a value aggregae, V /V 0, o he produc of a par ha s due o prce chage, P(p 0,p,q 0,q ), ad a par ha s due o quay chage, Q(p 0,p,q 0,q ). Ths approach o he deermao of he prce dex s he approach ake he aoal accous, where a prce dex s used o deflae a value rao o oa a esmae of quay chage. Thus, hs approach o dex umer heory, he prmary use for he prce dex s as a deflaor. Noe ha oce he fucoal form for he prce dex P(p 0,p,q 0,q ) s kow, he he correspodg quay or volume dex Q(p 0,p,q 0,q ) s compleely deermed y P; ha s, y rearragg equao (5.3): (5.4) Qp,p,q,q ( 0 0 ) ( V/V 0 ) 0 0 /Pp,p,q,q ( ). Coversely, f he fucoal form for he quay dex Q(p 0,p,q 0,q ) s kow, he he correspodg prce dex P(p 0,p,q 0,q ) s compleely deermed y Q. Thus, usg hs deflao approach o dex umer heory, separae heores for he deermao of he prce ad quay dces are o requred: f eher P or Q s deermed, he he oher fuco s mplcly deermed y he produc es, equao (5.4). 5. I he ex suseco, wo cocree choces for he prce dex P(p 0,p,q 0,q ) are cosdered, ad he correspodg quay dces Q(p 0,p,q 0,q ) ha resul from usg equao (5.4) are also calculaed. These are he wo choces used mos frequely y aoal come accouas. B.2 Laspeyres ad Paasche dces 5.2 Oe of he smples approaches deermg he prce dex formula was descred grea deal y Joseph Lowe (823). Hs approach o 372

5. Basc Idex Numer Theory measurg he prce chage ewee perods 0 ad was o specfy a approxmae represeave produc aske, 6 whch s a quay vecor q [q,,q ] ha s represeave of purchases made durg he wo perods uder cosderao, ad he o calculae he level of prces perod relave o perod 0 as he rao of he perod cos of he aske, aske, 0, o he perod 0 cos of he. Ths fxed-aske approach o he deermao of he prce dex leaves ope he followg queso: How exacly s he fxedaske vecor q o e chose? 5.3 As me passed, ecoomss ad prce sascas demaded a more precso wh respec o he specfcao of he aske vecor q. There are wo aural choces for he referece aske: he ase perod 0 produc vecor q 0 or he curre perod produc vecor q. These wo choces led o he Laspeyres (87) prce dex 7 P L defed y equao (5.5) ad he Paasche (874) prce dex 8 P P defed y equao (5.6): 9 6 Joseph Lowe (823, Appedx, p. 95) suggesed ha he produc aske vecor q should e updaed every fve years. Lowe dces wll e suded more deal Seco D. 7 Ths dex was acually roduced ad jusfed y Drosch (87a, p. 47) slghly earler ha Laspeyres. Laspeyres (87, p. 305) fac explcly ackowledged ha Drosch showed hm he way forward. However, he coruos of Drosch have ee forgoe for he mos par y laer wrers ecause Drosch aggressvely pushed for he rao of wo u values as eg he es dex umer formula. Whle hs formula has some excelle properes, f all he producs eg compared have he same u of measureme, he formula s useless whe, say, oh goods ad servces are he dex aske. 8 Aga, Drosch (87, p. 424) appears o have ee he frs o explcly defe ad jusfy hs formula. However, he rejeced hs formula favor of hs preferred formula, he rao of u values, ad so aga he dd o ge ay cred for hs early suggeso of he Paasche formula. 9 Noe ha P L (p 0,p,q 0,q ) does o acually deped o q, ad P P (p 0,p,q 0,q ) does o acually deped o q 0. However, does o harm o clude hese vecors, ad he oao dcaes ha he reader s he realm of laeral dex umer heory; ha s, he prces ad quaes for a value aggregae perag o wo perods are eg compared. (5.5) (5.6) 0 pq 0 0 PL ( p, p,q,q ) ; 0 0 pq 0 0 P ( p, p,q,q ). 0 5.4 The aove formulas ca e rewre a maer ha s more useful for sascal ageces. Defe he perod reveue share o produc as follows: (5.7) ad 0,. / j j j s pq pq for,..., The, he Laspeyres dex, equao (5.5), ca e rewre as follows: 0 (5.8) 0 0 0 0 0 L(,,, ) / j j j P p q usg defos equao (5.7). 0 0 0 0 0 ( p / p ) / pjqj j 0 0 ( p / p ) s, Thus, he Laspeyres prce dex, P L ca e wre as a ase-perod reveue share-weghed arhmec average of he prce raos, p /p 0. The Laspeyres formula (ul he very rece pas) has ee wdely used as he ellecual ase for PPIs aroud he world. To mpleme, a sascal agecy eeds oly o collec formao o reveue shares s 0 for he dex doma of defo for he ase perod 0 ad he collec formao o em prces aloe o a ogog ass. Thus, he Laspeyres PPI ca e produced o a mely ass whou curre-perod quay formao. 0 Ths mehod of rewrg he Laspeyres dex (or ay fxed-aske dex) as a share-weghed arhmec average of prce raos s due o Irvg Fsher (897, p. 57; 9, p. 397; 922, p. 5) ad Walsh (90, p. 506; 92a, p. 92). 373

Producer Prce Idex Maual 5.5 The Paasche dex ca also e wre reveue share ad prce rao form as follows: 0 0 0 (5.9) P ( p, p, q, q ) pjqj j 0 ( p p ) pq pjqj j 0 ( p p ) s 0 ( p p ) s, usg defos equao (5.7). Thus, he Paasche prce dex P P ca e wre as a perod (or curre-perod) reveue shareweghed harmoc average of he em prce raos p /p 0. 2 The lack of formao o curreperod quaes preves sascal ageces from producg Paasche dces o a mely ass. 5.6 The quay dex ha correspods o he Laspeyres prce dex usg he produc es, equao (5.3), s he Paasche quay dex; ha s, f P equao (5.4) s replaced y P L defed y equao (5.5), he he followg quay dex s oaed: (5.0) pq 0 0 P ( ). 0 pq Q p, p,q,q Noe ha Q P s he value of he perod quay vecor valued a he perod prces,, dvded y he (hypohecal) value of he perod 0 quay vecor valued a he perod prces, 0. Thus, he perod 0 ad quay vecors are valued a he same se of prces, he curreperod prces, p. 5.7 The quay dex ha correspods o he Paasche prce dex usg he produc es, equao (5.3), s he Laspeyres quay dex; ha s, f P equao (5.4) s replaced y P P defed y equao (5.6), he he followg quay dex s oaed: (5.) 0 p q 0 0 L ( ). 0 0 p q Q p,p,q,q Noe ha Q L s he (hypohecal) value of he perod quay vecor valued a he perod 0 prces, 0 p q, dvded y he value of he perod 0 quay vecor valued a he perod 0 prces, 0 0. Thus, he perod 0 ad quay vecors are valued a he same se of prces, he aseperod prces, p 0. 5.8 The prolem wh he Laspeyres ad Paasche dex umer formulas s ha hey are equally plausle, u, geeral, hey wll gve dffere aswers. For mos purposes, s o sasfacory for he sascal agecy o provde wo aswers o hs queso: 3 wha s he es overall summary measure of prce chage for he value aggregae over he wo perods queso? Thus, he followg seco, s cosdered how es averages of hese wo esmaes of prce chage ca e cosruced. Before dog hs, we ask wha s he ormal relaoshp ewee he Paasche ad Laspeyres dces? Uder ormal ecoomc codos, whe he prce raos perag o he wo suaos uder cosderao are egavely correlaed wh he correspodg quay raos, ca e show ha he Laspeyres prce dex wll e Ths mehod of rewrg he Paasche dex (or ay fxed-aske dex) as a share-weghed harmoc average of he prce raos s due o Walsh (90, p. 5; 92a, p. 93) ad Irvg Fsher (9, pp. 397 98). 2 Noe ha he dervao equao (5.9) shows how harmoc averages arse dex umer heory a very aural way. 3 I prcple, sead of averagg he Paasche ad Laspeyres dces, he sascal agecy could hk of provdg oh (he Paasche dex o a delayed ass). Ths suggeso would lead o a marx of prce comparsos ewee every par of perods sead of a me seres of comparsos. Walsh (90, p. 425) oed hs possly: I fac, f we use such drec comparsos a all, we ough o use all possle oes. 374

5. Basc Idex Numer Theory larger ha he correspodg Paasche dex. 4 I Appedx 5., a precse saeme of hs resul s preseed. 5 Ths dvergece ewee P L ad P P suggess ha f a sgle esmae for he prce chage ewee he wo perods s requred, he some sor of evely weghed average of he wo dces should e ake as he fal esmae of prce chage ewee perods 0 ad. Ths sraegy wll e pursued he followg seco. However, should e kep md ha, usually, sascal ageces wll o have formao o curre reveue weghs ad, hece, averages of Paasche ad Laspeyres dces ca e produced oly o a delayed ass (perhaps usg aoal accous formao) or o a all. C. Symmerc Averages of Fxed-Baske Prce Idces C. Fsher dex as a average of he Paasche ad Laspeyres dces 5.9 As was meoed he prevous paragraph, sce he Paasche ad Laspeyres prce dces are equally plausle u ofe gve dffere esmaes of he amou of aggregae prce chage ewee perods 0 ad, s useful o cosder akg a evely weghed average of hese fxedaske prce dces as a sgle esmaor of prce chage ewee he wo perods. Examples of such 4 Peer Hll (993, p. 383) summarzed hs equaly as follows: I ca e show ha relaoshp (3) [ha s, ha P L s greaer ha P P ] holds wheever he prce ad quay relaves (weghed y values) are egavely correlaed. Such egave correlao s o e expeced for prce akers who reac o chages relave prces y susug goods ad servces ha have ecome relavely less expesve for hose ha have ecome relavely more expesve. I he vas majory of suaos covered y dex umers, he prce ad quay relaves ur ou o e egavely correlaed so ha Laspeyres dces ed sysemacally o record greaer creases ha Paasche wh he gap ewee hem edg o wde wh me. 5 There s aoher way o see why P P wll ofe e less ha P L. If he perod 0 reveue shares s 0 are exacly equal o he correspodg perod reveue shares s, he y Schlömlch's (858) Iequaly (see Hardy, Llewood, ad Polyá, 934, p. 26), ca e show ha a weghed harmoc mea of umers s equal o or less ha he correspodg arhmec mea of he umers ad he equaly s src f he umers are o all equal. If reveue shares are approxmaely cosa across perods, he follows ha P P wll usually e less ha P L uder hese codos; see Seco D.3. symmerc averages 6 are he arhmec mea, whch leads o he Drosch (87, p. 425) Sdgwck (883, p. 68) Bowley (90, p. 227) 7 dex, P DR (/2)P L + (/2)P P, ad he geomerc mea, whch leads o he Irvg Fsher 8 (922) deal dex, P F, defed as 0 0 0 0 (5.2) PF( p, p, q, q ) PL( p, p, q, q ) 2 ( 0,, 0, ) 2. P P p q A hs po, he fxed-aske approach o dex umer heory s rasformed o he es approach o dex umer heory; ha s, o deerme whch of hese fxed-aske dces or whch averages of hem mgh e es, desrale crera or ess or properes are eeded for he prce dex. Ths opc wll e pursued more deal he ex chaper, u a roduco o he es approach s provded he prese seco ecause a es s used o deerme whch average of he Paasche ad Laspeyres dces mgh e es. 5.20 Wha s he es symmerc average of P L ad P P o use as a po esmae for he heorecal cos-of-lvg dex? I s very desrale for a prce dex formula ha depeds o he prce ad quay vecors perag o he wo perods uder cosderao o sasfy he me reversal es. 9 A 6 For a dscusso of he properes of symmerc averages, see Dewer (993c). Formally, a average m(a,) of wo umers a ad s symmerc f m(a,) m(,a). I oher words, he umers a ad are reaed he same maer he average. A example of a osymmerc average of a ad s (/4)a + (3/4). I geeral, Walsh (90, p. 05) argued for a symmerc reame f he wo perods (or coures) uder cosderao were o e gve equal mporace. 7 Walsh (90, p. 99) also suggesed hs dex. See Dewer (993a, p. 36) for addoal refereces o he early hsory of dex umer heory. 8 Bowley (899, p. 64) appears o have ee he frs o sugges he use of hs dex. Walsh (90, pp. 428 29) also suggesed hs dex whle commeg o he g dffereces ewee he Laspeyres ad Paasche dces oe of hs umercal examples: The fgures colums (2) [Laspeyres] ad (3) [Paasche] are, sgly, exravaga ad asurd. Bu here s order her exravagace; for he earess of her meas o he more ruhful resuls shows ha hey sraddle he rue course, he oe varyg o he oe sde aou as he oher does o he oher. 9 See Dewer (992a, p. 28) for early refereces o hs es. If we wa he prce dex o have he same propery as a sgle-prce rao, he s mpora o sasfy he me reversal es. However, oher pos of vew are poss- (coued) 375

Producer Prce Idex Maual dex umer formula P(p 0,p,q 0,q ) sasfes hs es f (5.3) P p,p,q,q / Pp,p,q,q 0 0 0 0 ( ) ( ) ; ha s, f he perod 0 ad perod prce ad quay daa are erchaged ad he dex umer formula s evaluaed, he hs ew dex P(p,p 0,q,q 0 ) s equal o he recprocal of he orgal dex P(p 0,p,q 0,q ). Ths s a propery ha s sasfed y a sgle prce rao, ad seems desrale ha he measure of aggregae prce chage should also sasfy hs propery so ha does o maer whch perod s chose as he ase perod. Pu aoher way, he dex umer comparso ewee ay wo pos of me should o deped o he choce of whch perod we regard as he ase perod: f he oher perod s chose as he ase perod, he he ew dex umer should smply equal he recprocal of he orgal dex. I should e oed ha he Laspeyres ad Paasche prce dces do o sasfy hs me reversal propery. 5.2 Havg defed wha meas for a prce dex P o sasfy he me reversal es, he s possle o esalsh he followg resul: 20 he Fsher deal prce dex defed y equao (5.2) aove s he oly dex ha s a homogeeous 2 symmerc average of he Laspeyres ad Paasche prce dces, P L ad P P, ad sasfes he me reversal es equao (5.3) aove. Thus, he Fsher deal prce dex emerges as perhaps he es evely weghed average of he Paasche ad Laspeyres prce dces. 5.22 I s eresg o oe ha hs symmerc aske approach o dex umer heory daes ack o oe of he early poeers of dex umer heory, Arhur L. Bowley, as he followg quoaos dcae: If [he Paasche dex] ad [he Laspeyres dex] le close ogeher here s o furher dffculy; f hey dffer y much hey may e regarded as feror ad superor lms of he dex umer, whch may e esmaed as her arhmec mea as a frs approxmao. (Arhur L. Bowley, 90, p. 227) Whe esmag he facor ecessary for he correco of a chage foud moey wages o oa he chage real wages, sascas have o ee coe o follow Mehod II oly [o calculae a Laspeyres prce dex], u have worked he prolem ackwards [o calculae a Paasche prce dex] as well as forwards. They have he ake he arhmec, geomerc or harmoc mea of he wo umers so foud. (Arhur L. Bowley, 99, p. 348) 22 5.23 The quay dex ha correspods o he Fsher prce dex usg he produc es, equao (5.3), s he Fsher quay dex; ha s, f P equao (5.4) s replaced y P F defed y equao (5.2), he followg quay dex s oaed: 0 0 0 0 (5.4) QF( p, p, q, q ) QL( p, p, q, q ) 2 ( 0,, 0, ) 2. P Q p q Thus, he Fsher quay dex s equal o he square roo of he produc of he Laspeyres ad Paasche quay dces. I should also e oed ha Q F (p 0,p,q 0,q ) P F (q 0,q,p 0,p ); ha s, f he role of prces ad quaes s erchaged he Fsher prce dex formula, he he Fsher quay dex s oaed. 23 5.24 Raher ha ake a symmerc average of he wo asc fxed-aske prce dces perag o wo suaos, P L ad P P, s also possle o reur o Lowe s asc formulao ad choose he aske vecor q o e a symmerc average of he ase- ad curre-perod aske vecors, q 0 ad q. The followg suseco pursues hs approach o dex umer heory. le. For example, we may wa o use our prce dex for compesao purposes, whch case sasfaco of he me reversal es may o e so mpora. 20 See Dewer (997, p. 38). 2 A average or mea of wo umers a ad, m(a,), s homogeeous f whe oh umers a ad are mulpled y a posve umer λ, he he mea s also mulpled y λ; ha s, m sasfes he followg propery: m(λa,λ) λm(a,). 22 Irvg Fsher (9, pp. 47 8; 922) also cosdered he arhmec, geomerc, ad harmoc averages of he Paasche ad Laspeyres dces. 23 Irvg Fsher (922, p. 72) sad ha P ad Q sasfed he facor reversal es f Q(p 0,p,q 0,q ) P(q 0,q,p 0,p ) ad P ad Q sasfed he produc es equao (5.3) as well. 376

5. Basc Idex Numer Theory C.2 Walsh dex ad heory of pure prce dex (5.6) 0 0 j j j s pq/ pq for,2,...,. 5.25 Prce sascas ed o e very comforale wh a cocep of he prce dex ased o prcg ou a cosa represeave aske of producs, q (q,q 2,,q ), a he prces of perod 0 ad, p 0 (p 0,p 0 2,,p 0 ) ad p (p,p 2,,p ), respecvely. Prce sascas refer o hs ype of dex as a fxed-aske dex or a pure prce dex, 24 ad correspods o Ks s (924, p. 43) uequvocal prce dex. 25 Sce Joseph Lowe (823) was he frs perso o descre sysemacally hs ype of dex, s referred o as a Lowe dex. Thus, he geeral fucoal form for he Lowe prce dex s (5.5) 0 0 Lo(,, ) / 0 s( p / p ), P p where he (hypohecal) hyrd reveue shares s 26 correspodg o he quay weghs vecor q are defed y 24 See Seco 7 Dewer (200). 25 Suppose, however, ha for each commody, Q Q, he fraco, (P Q) / (PQ), vz., he rao of aggregae value for he secod u-perod o he aggregae value for he frs u-perod s o loger merely a rao of oals, also shows uequvocally he effec of he chage prce. Thus, s a uequvocal prce dex for he quaavely uchaged complex of commodes, A, B, C, ec. I s ovous ha f he quaes were dffere o he wo occasos, ad f a he same me he prces had ee uchaged, he precedg formula would ecome (PQ ) / (PQ). I would sll e he rao of he aggregae value for he secod u-perod o he aggregae value for he frs u-perod. Bu would e also more ha hs. I would show a geeralzed way he rao of he quaes o he wo occasos. Thus s a uequvocal quay dex for he complex of commodes, uchaged as o prce ad dfferg oly as o quay. Le e oed ha he mere algerac form of hese expressos shows a oce he logc of he prolem of fdg hese wo dces s decal (Sr George H. Ks, 924, pp. 43 44). 26 Irvg Fsher (922, p. 53) used he ermology weghed y a hyrd value, whle Walsh (932, p. 657) used he erm hyrd weghs. 5.26 The ma reaso why prce sascas mgh prefer a memer of he famly of Lowe or fxed-aske prce dces defed y equao (5.5) s ha he fxed-aske cocep s easy o expla o he pulc. Noe ha he Laspeyres ad Paasche dces are specal cases of he pure prce cocep f we choose q q 0 (whch leads o he Laspeyres dex) or f we choose q q (whch leads o he Paasche dex). 27 The praccal prolem of pckg q remas o e resolved, ad ha s he prolem addressed hs seco. 5.27 I should e oed ha Walsh (90, p. 05; 92a) also saw he prce dex umer prolem he aove framework: Commodes are o e weghed accordg o her mporace, or her full values. Bu he prolem of axomery always volves a leas wo perods. There s a frs perod, ad here s a secod perod whch s compared wh. Prce varaos have ake place ewee he wo, ad hese are o e averaged o ge he amou of her varao as a whole. Bu he weghs of he commodes a he secod perod are ap o e dffere from her weghs a he frs perod. Whch weghs, he, are he rgh oes hose of he frs perod? Or hose of he secod? Or should here e a comao of he wo ses? There s o reaso for preferrg eher he frs or he secod. The he comao of oh would seem o e he proper aswer. Ad hs comao self volves a averagg of he weghs of he wo perods. (Correa Moyla Walsh, 92a, p. 90) Walsh s suggeso wll e followed, ad hus he h quay wegh, q, s resrced o e a average or mea of he ase-perod quay q 0 ad he curre-perod quay for produc q, say, m(q 0,q ), for,2,,. 28 Uder hs assump- 27 Noe ha he h share defed y equao (5.6) hs case s he hyrd share s pq Σ pq, whch uses 0 0 he prces of perod 0 ad he quaes of perod. 28 Noe ha we have chose he mea fuco m(q 0,q ) o e he same for each em. We assume ha m(a,) has he followg wo properes: m(a,) s a posve ad couous fuco, defed for all posve umers a ad, ad m(a,a) a for all a > 0. 377

Producer Prce Idex Maual o, he Lowe prce dex equao (5.5) ecomes (5.7) 0 pmq (, q) 0 0 Lo (,,, ). 0 0 p jmq ( j, qj) j P p q 5.28 To deerme he fucoal form for he mea fuco m, s ecessary o mpose some ess or axoms o he pure prce dex defed y equao (5.7). As Seco C., we ask ha P Lo sasfy he me reversal es, equao (5.3) aove. Uder hs hypohess, s mmedaely ovous ha he mea fuco m mus e a symmerc mea; 29 ha s, m mus sasfy he followg propery: m(a,) m(,a) for all a > 0 ad > 0. Ths assumpo sll does o p dow he fucoal form for he pure prce dex defed y equao (5.7) aove. For example, he fuco m(a,) could e he arhmec mea, (/2)a + (/2), whch case equao (5.7) reduces o he Marshall (887) Edgeworh (925) prce dex P ME, whch was he pure prce dex preferred y Ks (924, p. 56): (5.8) {( + )/2} 0 q 0 0 ME (,,, ). 0 0 pj{ ( qj + qj) /2} j P p q 5.29 O he oher had, he fuco m(a,) could e he geomerc mea, (a) /2, whch case equao (5.7) reduces o he Walsh (90, p. 398; 92a, p. 97) prce dex, P W : 30 29 For more o symmerc meas, see Dewer (993c, p. 36). 30 Walsh edorsed P W as eg he es dex umer formula: We have see reaso o eleve formula 6 eer ha formula 7. Perhaps formula 9 s he es of he res, u ewee ad Nos. 6 ad 8 would e dffcul o decde wh assurace (C.M. Walsh, 92a, p. 03). Hs formula 6 s P W defed y equao (5.9), ad hs 9 s he Fsher deal defed y equao (5.2) aove. The Walsh quay dex, Q W (p 0,p,q 0,q ), s defed as P W (q 0,q,p 0,p ); ha s, prces ad quaes equao (5.9) are erchaged. If he Walsh quay dex s used o deflae he value rao, a mplc prce dex s oaed, whch s Walsh s formula 8. (5.9) 0 q 0 0 W (,,, ). 0 0 p j qq j j j P p q 5.30 There are may oher possles for he mea fuco m, cludg he mea of order r, [(/2)a r + (/2) r ] /r for r 0. To compleely deerme he fucoal form for he pure prce dex P Lo, s ecessary o mpose a leas oe addoal es or axom o P Lo (p 0,p,q 0,q ). 5.3 There s a poeal prolem wh he use of he Marshall-Edgeworh prce dex, equao (5.8), ha has ee oced he coex of usg he formula o make eraoal comparsos of prces. If he prce levels of a very large coury are compared wh he prce levels of a small coury usg equao (5.8), he he quay vecor of he large coury may oally overwhelm he fluece of he quay vecor correspodg o he small coury. 3 I echcal erms, he Marshall- Edgeworh formula s o homogeeous of degree 0 he compoes of oh q 0 ad q. To preve hs prolem from occurrg he use of he pure prce dex P K (p 0,p,q 0,q ) defed y equao (5.7), s asked ha P Lo sasfy he followg varace o proporoal chages curre quaes es: 32 (5.20) P p q P p q 0 0 0 0 Lo(,,, λ ) Lo(,,, ) 0 0 for all p, p, q, q ad all λ> 0. The wo ess, he me reversal es equao (5.3) ad he varace es equao (5.20), eale oe o deerme he precse fucoal form for he pure prce dex P Lo defed y equao (5.7) aove: he pure prce dex P K mus e he Walsh dex P W defed y equao (5.9). 33 5.32 To e of praccal use y sascal ageces, a dex umer formula mus e ale o e expressed as a fuco of he ase-perod reveue shares, s 0 ; he curre-perod reveue shares, s ; 3 Ths s o lkely o e a severe prolem he meseres coex where he chage quay vecors gog from oe perod o he ex s small. 32 Ths s he ermology used y Dewer (992a, p. 26). Vog (980) was he frs o propose hs es. 33 See Seco 7 Dewer (200). 378

5. Basc Idex Numer Theory ad he prce raos, p /p 0. The Walsh prce dex defed y equao (5.9) aove ca e rewre hs forma: (5.2) P ( p, p, q, q ) W 0 0 j j C.3 Coclusos j p p qq 0 qq 0 0 j j j ( p / p p ) s s 0 0 ( p / p p ) s s 0 0 0 j j j j j s s p p 0 0 s s p p 0 0 j j j j 5.33 The approach ake o dex umer heory hs seco was o cosder averages of varous fxed-aske prce dces. The frs approach was o ake a evehaded average of he wo prmary fxed-aske dces: he Laspeyres ad Paasche prce dces. These wo prmary dces are ased o prcg ou he askes ha pera o he wo perods (or locaos) uder cosderao. Takg a average of hem led o he Fsher deal prce dex P F defed y equao (5.2) aove. The secod approach was o average he aske quay weghs ad he prce ou hs average aske a he prces perag o he wo suaos uder cosderao. Ths approach led o he Walsh prce dex P W defed y equao (5.9) aove. Boh hese dces ca e wre as a fuco of he ase-perod reveue shares, s 0 ; he curre-perod reveue shares, s ; ad he prce raos, p /p 0. Assumg ha he sascal agecy has formao o hese hree ses of varales, whch dex should e used? Experece wh ormal me-seres daa has show ha hese wo dces wll o dffer susaally, ad hus s a maer of choce whch of hese dces s used pracce. 34 Boh hese dces are examples of 34 Dewer (978, pp. 887 89) showed ha hese wo dces wll approxmae each oher o he secod order aroud a equal prce ad quay po. Thus, for ormal meseres daa where prces ad quaes do o chage much (coued). superlave dces, whch wll e defed Chaper 7. However, oe ha oh hese dces rea he daa perag o he wo suaos a symmerc maer. Hll commeed o superlave prce dces ad he mporace of a symmerc reame of he daa as follows: Thus ecoomc heory suggess ha, geeral, a symmerc dex ha assgs equal wegh o he wo suaos eg compared s o e preferred o eher he Laspeyres or Paasche dces o her ow. The precse choce of superlave dex wheher Fsher, Törqvs or oher superlave dex may e of oly secodary mporace as all he symmerc dces are lkely o approxmae each oher, ad he uderlyg heorec dex farly closely, a leas whe he dex umer spread ewee he Laspeyres ad Paasche s o very grea. (Peer Hll, 993, p. 384) 35 D. Aual Weghs ad Mohly Prce Idces D. Lowe dex wh mohly prces ad aual ase-year quaes 5.34 I s ow ecessary o dscuss a major praccal prolem wh he heory of aske-ype dces. Up o ow, has ee assumed ha he quay vecor q (q,q 2,,q ) ha appeared he defo of he Lowe dex, P Lo (p 0,p,q) defed y equao (5.5), s eher he ase-perod quay vecor q 0 or he curre-perod quay vecor q or a average of he wo. I fac, erms of acual sascal agecy pracce, he quay vecor q s usually ake o e a aual quay vecor ha refers o a ase year, say, ha s efore he ase perod for he prces, perod 0. Typcally, a sascal agecy wll produce a PPI a a mohly or quarerly frequecy, u, for he sake of defeess, a mohly frequecy wll e assumed wha follows. Thus, a ypcal prce dex wll have he form P Lo (p 0,p,q ), where p 0 s he prce vecor perag o he ase-perod moh for prces, moh 0; p s he prce vecor perag o he curre-perod moh for prces, moh, say; gog from he ase perod o he curre perod, he dces wll approxmae each oher que closely. 35 See also Peer Hll (988). 379

Producer Prce Idex Maual ad q s a referece aske quay vecor ha refers o he ase year, whch s equal o or efore moh 0. 36 Noe ha hs Lowe dex P Lo (p 0,p,q ) s o a rue Laspeyres dex (ecause he aual quay vecor q s o equal o he mohly quay vecor q 0 geeral). 37 5.35 The queso s hs: why do sascal ageces o pck he referece quay vecor q he Lowe formula o e he mohly quay vecor q 0 ha peras o rasacos moh 0 (so ha he dex would reduce o a ordary Laspeyres prce dex)? There are wo ma reasos: Mos ecoomes are sujec o seasoal flucuaos, ad so pckg he quay vecor of moh 0 as he referece quay vecor for all mohs of he year would o e represeave of rasacos made hroughou he year. Mohly household quay or reveue weghs are usually colleced y he sascal agecy usg a esalshme survey wh a relavely small sample. Hece, he resulg weghs are usually sujec o very large samplg errors, ad so sadard pracce s o average hese mohly reveue or quay weghs over a ere year (or some cases, over several years), a aemp o reduce hese samplg errors. I oher saces, where a esalshme cesus s used, he repored reveue weghs are for a aual perod. The dex umer prolems ha are caused y seasoal mohly weghs wll e suded more deal Chaper 22. For ow, ca e argued ha he use of aual weghs a mohly dex umer formula s smply a mehod for dealg wh he seasoaly prolem. 38 36 Moh 0 s called he prce referece perod, ad year s called he wegh referece perod. 37 Trple (98, p. 2) defed he Lowe dex, callg a Laspeyres dex, ad callg he dex ha has he wegh referece perod equal o he prce referece perod a pure Laspeyres dex. Trple also oed he hyrd share represeao for he Lowe dex defed y equao (5.5) ad equao (5.6). Trple oed ha he rao of wo Lowe dces usg he same quay weghs was also a Lowe dex. 38 I fac, usg he Lowe dex P Lo (p 0,p,q ) he coex of seasoal producs correspods o Bea ad Se s (924, p. 3) Type A dex umer formula. Bea ad (coued) 5.36 Oe prolem wh usg aual weghs correspodg o a perhaps dsa year he coex of a mohly PPI mus e oed a hs po. If here are sysemac (u dverge) reds produc prces, ad cosumers or usesses crease her purchases of producs ha decle (relavely) prce ad decrease her purchases of producs ha crease (relavely) prce, he he use of dsa quay weghs wll ed o lead o a upward as hs Lowe dex compared wh oe ha used more curre weghs, as wll e show elow. Ths oservao suggess ha sascal ageces should ge up-o-dae weghs o a ogog ass. 5.37 I s useful o expla how he aual quay vecor q could e oaed from mohly reveues o each produc durg he chose ase year. Le he moh m reveue of he referece populao he ase year for produc e v,m, ad le he correspodg prce ad quay e p,m ad q,m, respecvely. Value, prce, ad quay for each produc are relaed y he followg equaos: m, m, m, (5.22) v ;,...,; m,...,2. For each produc, he aual oal q ca e oaed y prce-deflag mohly values ad summg over mohs he ase year as follows: (5.23) q v 2 m, 2, m ; m, q,...,, m p m where equao (5.22) was used o derve equao (5.23). I pracce, he aove equaos wll e evaluaed usg aggregae reveues over closely relaed producs, ad he prce p,m wll e he moh m prce dex for hs elemeary produc group year relave o he frs moh of year. 5.38 For some purposes, s also useful o have aual prces y produc o mach he aual quaes defed y equao (5.23). Followg aoal come accoug coveos, a reaso- Se made hree addoal suggesos for prce dces he coex of seasoal producs. Ther coruos wll e evaluaed Chaper 22. 380

5. Basc Idex Numer Theory ale 39 prce p o mach he aual quay q s he value of oal reveue for produc year dvded y q. Thus, we have 2, m (5.24) p v q ;,..., m 2 m, v m 2 m, m, v p m 2 m, m, s ( p ), m ; usg equao (5.23) where he share of aual reveue o produc moh m of he ase year s (5.25) s m, v m, 2 k, v k ;,...,. Thus, he aual ase-year prce for produc, p, urs ou o e a mohly reveue-weghed harmoc mea of he mohly prces for produc he ase year, p,, p,2,, p,2. 5.39 Usg he aual produc prces for he ase year defed y equao (5.24), a vecor of hese prces ca e defed as p [p,,p ]. Usg hs defo, he Lowe dex ca e expressed as a rao of wo Laspeyres dces where he prce vecor p plays he role of ase-perod prces each of he wo Laspeyres dces: (5.26) P ( p, p, q ) Lo 0 0 39 Hece, hese aual produc prces are esseally uvalue prces. Uder codos of hgh flao, he aual prces defed y equao (5.24) may o loger e reasoale or represeave of prces durg he ere ase year ecause he reveues he fal mohs of he hghflao year wll e somewha arfcally low up y geeral flao. Uder hese codos, he aual prces ad aual produc reveue shares should e erpreed wh cauo. For more o dealg wh suaos where here s hgh flao wh a year, see Peer Hll (996). pq s p p 0 0 s p p 0 PL( p, p, q )/ PL( p, p, q ) / ( / ) / ( / ), where he Laspeyres formula P L was defed y equao (5.5) aove. Thus, he aove equao shows ha he Lowe mohly prce dex comparg he prces of moh 0 wh hose of moh usg he quaes of ase year as weghs, P Lo (p 0,p,q ), s equal o he Laspeyres dex ha compares he prces of moh wh hose of year, P L (p,p,q ), dvded y he Laspeyres dex ha compares he prces of moh 0 wh hose of year, P L (p,p 0,q ). Noe ha he Laspeyres dex he umeraor ca e calculaed f he ase-year produc reveue shares, s, are kow alog wh he prce raos ha compare he prces of produc moh, p, wh he correspodg aual average prces he ase year, p. The Laspeyres dex he deomaor ca e calculaed f he ase-year produc reveue shares, s, are kow alog wh he prce raos ha compare he prces of produc moh 0, p 0, wh he correspodg aual average prces he ase year, p. 5.40 Aoher covee formula for evaluag he Lowe dex, P Lo (p 0,p,q ), uses he hyrd weghs formula, equao (5.5). I he prese coex, he formula ecomes (5.27) P ( p, p, q ) Lo 0 0 ( p / p ) 0 0 p pq 0 0 0 s p, where he hyrd weghs s 0 usg he prces of moh 0 ad he quaes of year are defed y (5.28) pq 0 0 0 p jq j j s ;,..., 38

Producer Prce Idex Maual 0 pq ( p / p ) 0 pq j j( pj / pj) j Equao (5.28) shows how he ase-year reveues, p q, ca e mulpled y he produc prce dces, p 0 /p, o calculae he hyrd shares. 5.4 Oe addoal formula for he Lowe dex, P Lo (p 0,p,q ), wll e exhed. Noe ha he Laspeyres decomposo of he Lowe dex defed y he hrd le equao (5.26) volves he very log-erm prce relaves, p /p, ha compare he prces moh, p, wh he possly dsa ase-year prces, p. Furher, he hyrd share decomposo of he Lowe dex defed y he hrd le equao (5.27) volves he logerm mohly prce relaves, p /p 0, whch compare he prces moh, p, wh he ase moh prces, p 0. Boh hese formulas are o sasfacory pracce ecause of he prolem of sample aro: each moh, a susaal fraco of producs dsappears from he markeplace, ad hus s useful o have a formula for updag he prevous moh s prce dex usg jus moh-over-moh prce relaves. I oher words, log-erm prce relaves dsappear a a rae ha s oo large pracce o ase a dex umer formula o her use. The Lowe dex for moh +, P Lo (p 0,p +,q ), ca e wre erms of he Lowe dex for moh, P Lo (p 0,p,q ), ad a updag facor as follows: (5.29) P ( p, p, q ) Lo + p q 0 + 0 p q + pq 0 pq. + p q 0 PLo ( p, p, q ) pq P p 0 Lo (,, ) p + p pq + 0 p PLo ( p, p, q ) s, p where he hyrd weghs s are defed y (5.30) pq p jq j j s ;,...,. Thus, he requred updag facor, gog from moh o moh +, s he cha-lked dex + ( ) s p p, whch uses he hyrd share weghs s correspodg o moh ad ase year. 5.42 The Lowe dex P Lo (p 0,p,q ) ca e regarded as a approxmao o he ordary Laspeyres dex, P L (p 0,p,q 0 ), ha compares he prces of he ase moh 0, p 0, wh hose of moh, p, usg he quay vecor of moh 0, q 0, as weghs. There s a relavely smple formula ha relaes hese wo dces. To expla hs formula, s frs ecessary o make a few defos. Defe he h prce relave ewee moh 0 ad moh as 0 (5.3) r p / p ;,...,. The ordary Laspeyres prce dex, gog from moh 0 o, ca e defed erms of hese prce relaves as follows: (5.32) P ( p, p, q ) L 0 0 p 0 0 0 0 0 0 pq p p 0 s 0 0 0 p pq 382

5. Basc Idex Numer Theory 0 s r r, where he moh 0 reveue shares s 0 are defed as follows: (5.33) s 0 pq 0 0 0 0 p jq j j ;,...,. 5.43 Defe he h quay relave as he rao of he quay of produc used he ase year, q, o he quay used moh 0, q 0, as follows: 0 (5.34) q / q ;,...,. The Laspeyres quay dex, Q L (q 0,q,p 0 ), ha compares quaes year, q, wh he correspodg quaes moh 0, q 0, usg he prces of moh 0, p 0, as weghs ca e defed as a weghed average of he quay raos as follows: 0 p q 0 0 L 0 0 p q q 0 0 0 p q q 0 0 pq q 0 s 0 q 0 s * (5.35) Q ( q, q, p ) 5.44 Usg equao (A5.2.4) Appedx 5.2, he relaoshp ewee he Lowe dex P Lo (p 0,p,q ) ha uses he quaes of year as weghs o compare he prces of moh wh moh 0 ad he correspodg ordary Laspeyres dex P L (p 0,p,q 0 ) ha uses he quaes of moh 0 as weghs s defed as (5.36) P ( p, p, q ) Lo 0 P ( p, p, q ) + L 0 0 0 ( r r )( ) s 0 Q q q p 0 0 L (,, ) Thus, he Lowe prce dex usg he quaes of year as weghs, P Lo (p 0,p,q ), s equal o he usual Laspeyres dex usg he quaes of moh 0 as weghs, P L (p 0,p,q 0 ), plus a covarace erm ( r r )( ) s 0 ewee he prce relaves r p / p 0 ad he quay relaves q /q 0, dvded y he Laspeyres quay dex Q L (q 0,q,p 0 ) ewee moh 0 ad ase year. 5.45 Equao (5.36) shows ha he Lowe prce dex wll cocde wh he Laspeyres prce dex f he covarace or correlao ewee he moh 0 o prce relaves r p /p 0 ad he moh 0 o year quay relaves q /q 0 s zero. Noe ha hs covarace wll e zero uder hree dffere ses of codos: If he moh prces are proporoal o he moh 0 prces so ha all r r*, If he ase year quaes are proporoal o he moh 0 quaes so ha all *, ad If he dsruo of he relave prces r s depede of he dsruo of he relave quaes. The frs wo codos are ulkely o hold emprcally, u he hrd s possle, a leas approxmaely, f purchasers do o sysemacally chage her purchasg has respose o chages relave prces. 5.46 If hs covarace equao (5.36) s egave, he he Lowe dex wll e less ha he Laspeyres, ad, fally, f he covarace s posve, he he Lowe dex wll e greaer ha he Laspeyres dex. Alhough he sg ad magude of he covarace erm s ulmaely a emprcal maer, s possle o make some reasoale cojecures aou s lkely sg. If he ase year precedes he prce referece moh 0 ad here are log-erm reds prces, he s lkely ha hs covarace s posve, ad hece ha he Lowe -. 383

Producer Prce Idex Maual dex wll exceed he correspodg Laspeyres prce dex; 40 ha s, (5.37) P p P p 0 0 0 Lo(,, ) > L (,, ). To see why hs covarace s lkely o e posve, suppose ha here s a log-erm upward red he prce of produc so ha r r* (p / p 0 ) r* s posve. Wh ormal susuo resposes, 4 q / q 0 less a average quay chage of hs ype (*) s lkely o e egave, or, upo akg recprocals, q / q less a average quay chage of 0 hs (recprocal) ype s lkely o e posve. Bu f he log-erm upward red prces has perssed ack o he ase year, he * (q / q 0 ) * s also lkely o e posve. Hece, he covarace wll e posve uder hese crcumsaces. Moreover, he more dsa s he wegh referece year from he prce referece moh 0, he gger he resduals * wll lkely e ad he gger wll e he posve covarace. Smlarly, he more dsa s he curre-perod moh from he aseperod moh 0, he gger he resduals r r* wll lkely e ad he gger wll e he posve covarace. Thus, uder he assumpos ha here are log-erm reds prces ad ormal susuo resposes, he Lowe dex wll ormally e greaer ha he correspodg Laspeyres dex. (5.38) pq 0 P (,, ). 0 p q P p As was dscussed Seco C., a reasoale arge dex o measure he prce chage gog from moh 0 o s some sor of symmerc average of he Paasche dex P P (p 0,p,q ) defed y equao (5.38) ad he correspodg Laspeyres dex P L (p 0,p,q 0 ) defed y equao (5.32). Adapg equao (A5..5) Appedx 5., he relaoshp ewee he Paasche ad Laspeyres dces ca e wre as follows: (5.39) P p P p 0 0 0 P(,, ) L(,, ) 0 ( r r )( u u ) s + 0 0 QL ( q, q, p ) where he prce relaves r p / p 0 are defed y equao (5.3) ad her share-weghed average r* y equao (5.32), ad he u, u* ad Q L are defed as follows: 0 (5.40) u q / q ;,...,,, 5.47 Defe he Paasche dex ewee mohs 0 ad as follows: (5.4) 0 L( 0,, 0 ), u s u Q q q p 40 I s also ecessary o assume ha purchasers have ormal susuo effecs respose o hese log-erm reds prces; ha s, f a produc creases (relavely) prce, s quay purchased wll decle (relavely), ad f a produc decreases relavely prce, s quay purchased wll crease relavely. Ths reflecs he ormal marke equlrum respose o chages supply. 4 Walsh (90, pp. 28 82) was well aware of susuo effecs, as ca e see he followg comme ha oed he asc prolem wh a fxed-aske dex ha uses he quay weghs of a sgle perod: The argume made y he arhmec averags supposes ha we uy he same quaes of every class a oh perods spe of he varao her prces, whch we rarely, f ever, do. As a rough proposo, we a commuy geerally sped more o arcles ha have rse prce ad ge less of hem, ad sped less o arcles ha have falle prce ad ge more of hem. ad he moh 0 reveue shares s 0 are defed y equao (5.33). Thus, u* s equal o he Laspeyres quay dex ewee mohs 0 ad. Ths meas ha he Paasche prce dex ha uses he quaes of moh as weghs, P P (p 0,p,q ), s equal o he usual Laspeyres dex usg he quaes of moh 0 as weghs, P L (p 0,p,q 0 ), plus a 0 covarace erm ( r r )( u u ) s ewee he prce relaves r p / p 0 ad he quay relaves u q / q 0, dvded y he Laspeyres quay dex Q L (q 0,q,p 0 ) ewee moh 0 ad moh. 5.48 Alhough he sg ad magude of he covarace erm s aga a emprcal maer, s possle o make a reasoale cojecure aou s lkely sg. If here are log-erm reds prces, ad purchasers respod ormally o prce chages her purchases, he s lkely ha hs covar- 384

5. Basc Idex Numer Theory ace s egave, ad hece he Paasche dex wll e less ha he correspodg Laspeyres prce dex; ha s, (5.42) P p < P p. 0 0 0 P(,, ) L(,, ) To see why hs covarace s lkely o e egave, suppose ha here s a log-erm upward red he prce of produc 42 so ha r r* (p / p 0 ) r* s posve. Wh ormal susuo resposes, q / q 0 less a average quay chage of hs ype (u*) s lkely o e egave. Hece, u u* (q / q 0 ) u* s lkely o e egave. Thus, he covarace wll e egave uder hese crcumsaces. Moreover, he more dsa s he ase moh 0 from he curre-moh, he gger magude he resduals u u* wll lkely e ad he gger magude wll e he egave covarace. 43 Smlarly, he more dsa s he curre-perod moh from he ase-perod moh 0, he gger he resduals r r* wll lkely e ad he gger magude wll e he covarace. Thus, uder he assumpos ha here are log-erm reds prces ad ormal susuo resposes, he Laspeyres dex wll e greaer ha he correspodg Paasche dex, wh he dvergece lkely growg as moh ecomes more dsa from moh 0. 5.49 Pug he argumes he hree prevous paragraphs ogeher, ca e see ha uder he assumpos ha here are log-erm reds prces ad ormal susuo resposes, he Lowe prce dex ewee mohs 0 ad wll exceed he correspodg Laspeyres prce dex, whch ur wll exceed he correspodg Paasche prce dex; ha s, uder hese hypoheses, (5.43) 0 0 0 0 P ( p, p, q ) > P ( p, p, q ) > P ( p, p, q ). Lo L P Thus, f he log-ru arge prce dex s a average of he Laspeyres ad Paasche dces, ca e 42 The reader ca carry hrough he argume f here s a log-erm relave decle he prce of he h produc. The argume requred o oa a egave covarace requres ha here e some dffereces he log-erm reds prces; ha s, f all prces grow (or fall) a he same rae, we have prce proporoaly, ad he covarace wll e zero. 43 However, Q L u* may also e growg magude, so he e effec o he dvergece ewee P L ad P P s amguous. see ha he Laspeyres dex wll have a upward as relave o hs arge dex, ad he Paasche dex wll have a dowward as. I addo, f he ase year s pror o he prce referece moh, moh 0, he he Lowe dex wll also have a upward as relave o he Laspeyres dex ad hece also o he arge dex. D.2 Lowe dex ad mdyear dces 5.50 The dscusso he prevous paragraph assumed ha he ase year for quaes preceded he ase moh for prces, moh 0. However, f he curre-perod moh s que dsa from he ase moh 0, he s possle o hk of he ase year as referrg o a year ha les ewee mohs 0 ad. If he year does fall ewee mohs 0 ad, he he Lowe dex ecomes a mdyear dex. 44 The Lowe mdyear dex o loger has he upward ases dcaed y he equales equao (5.43) uder he assumpo of log-erm reds prces ad ormal susuo resposes y quaes. 5.5 I s ow assumed ha he ase-year quay vecor q correspods o a year ha les ewee mohs 0 ad. Uder he assumpo of log-erm reds prces ad ormal susuo effecs so ha here are also log-erm reds quaes ( he oppose dreco o he reds prces so ha f he h produc prce s redg up, he he correspodg h quay s redg dow), s lkely ha he ermedae-year qua- 44 Ths cocep ca e raced o Peer Hll (998, p. 46): Whe flao has o e measured over a specfed sequece of years, such as a decade, a pragmac soluo o he prolems rased aove would e o ake he mddle year as he ase year. Ths ca e jusfed o he grouds ha he aske of goods ad servces purchased he mddle year s lkely o e much more represeave of he paer of cosumpo over he decade as a whole ha askes purchased eher he frs or he las years. Moreover, choosg a more represeave aske wll also ed o reduce, or eve elmae, ay as he rae of flao over he decade as a whole as compared wh he crease he CoL dex. Thus, addo o roducg he cocep of a mdyear dex, Hll also roduced he dea of represeavy as. For addoal maeral o mdyear dces, see Schulz (999) ad Okamoo (200). Noe ha he mdyear dex cocep could e vewed as a close compeor o Walsh s (90, p. 43) mulyear fxed-aske dex, where he quay vecor was chose o e a arhmec or geomerc average of he quay vecors he perod. 385

Producer Prce Idex Maual y vecor wll le ewee he mohly quay vecors q 0 ad q. The mdyear Lowe dex, P Lo (p 0,p,q ), ad he Laspeyres dex gog from moh 0 o, P L (p 0,p,q 0 ), wll sll sasfy he exac relaoshp gve y equao (5.36). Thus, P Lo (p 0,p,q ) wll equal P L (p 0,p,q 0 ) plus he covarace erm ( r r )( ) s QL( q, q, p ), 0 0 0 where Q L (q 0,q,p 0 ) s he Laspeyres quay dex gog from moh 0 o. Ths covarace erm s lkely o e egave, so ha (5.44) P p P p 0 0 0 L(,, ) > Lo(,, ). To see why hs covarace s lkely o e egave, suppose ha here s a log-erm upward red he prce of produc so ha r r* (p / p 0 ) r* s posve. Wh ormal susuo resposes, q wll ed o decrease relavely over me, ad sce q s assumed o e ewee q 0 ad q, q /q 0 less a average quay chage of hs ype, r* s lkely o e egave. Hece u u* (q / q 0 ) * s lkely o e egave. Thus, he covarace s lkely o e egave uder hese crcumsaces. Uder he assumpos ha he quay ase year falls ewee mohs 0 ad ad ha here are log-erm reds prces ad ormal susuo resposes, he Laspeyres dex wll ormally e larger ha he correspodg Lowe mdyear dex, wh he dvergece lkely growg as moh ecomes more dsa from moh 0. 5.52 I ca also e see ha uder he aove assumpos, he mdyear Lowe dex s lkely o e greaer ha he Paasche dex ewee mohs 0 ad ; ha s, (5.45) P p P p 0 0 Lo(,, ) > P (,, ). To see why he aove equaly s lkely o hold, hk of q sarg a he moh 0 quay vecor q 0 ad he redg smoohly o he moh quay vecor q. Whe q q 0, he Lowe dex ecomes he Laspeyres dex P L (p 0,p,q 0 ). Whe q q, he Lowe dex ecomes he Paasche dex P P (p 0,p,q ). Uder he assumpo of redg prces ad ormal susuo resposes o hese redg prces, was show earler ha he Paasche dex wll e less ha he correspodg Laspeyres prce dex; ha s, ha P P (p 0,p,q ) was less ha P L (p 0,p,q 0 ); recall equao (5.42). Thus, uder he assumpo of smoohly redg prces ad quaes ewee mohs 0 ad, ad assumg ha q s ewee q 0 ad q, we wll have (5.46) P p P p < (,, ). 0 0 P(,, ) < Lo(,, ) 0 0 PL p Thus, f he ase year for he Lowe dex s chose o e ewee he ase moh for he prces, moh 0, ad he curre moh for prces, moh, ad here are reds prces wh correspodg reds quaes ha correspod o ormal susuo effecs, he he resulg Lowe dex s lkely o le ewee he Paasche ad Laspeyres dces gog from mohs 0 o. If he reds prces ad quaes are smooh, he choosg he ase year halfway ewee perods 0 ad should gve a Lowe dex ha s approxmaely halfway ewee he Paasche ad Laspeyres dces ad hece wll e very close o a deal arge dex ewee mohs 0 ad. Ths asc dea has ee mplemeed y Okamoo (200) usg Japaese cosumer daa, ad he foud ha he resulg mdyear dces approxmaed he correspodg Fsher deal dces very closely. 5.53 I should e oed ha hese mdyear dces ca e compued oly o a rerospecve ass; ha s, hey cao e calculaed a mely fasho as ca Lowe dces ha use a ase year efore moh 0. Thus, mdyear dces cao e used o replace he more mely Lowe dces. However, hese mely Lowe dces are lkely o have a upward as eve gger ha he usual Laspeyres upward as compared wh a deal arge dex, whch was ake o e a average of he Paasche ad Laspeyres dces. 5.54 All of he equales derved hs seco res o he assumpo of log-erm reds prces (ad correspodg ecoomc resposes quaes). If here are o sysemac log-ru reds prces ad oly radom flucuaos aroud a commo red all prces, he he aove equales are o vald, ad he Lowe dex usg a pror ase year wll proaly provde a perfecly adequae approxmao o oh he Paasche ad Laspeyres dces. However, here are some reasos for elevg ha some log-ru reds prces exs: 386

5. Basc Idex Numer Theory () The compuer chp revoluo of he pas 40 years has led o srog dowward reds he prces of producs ha use hese chps esvely. As ew uses for chps are developed, he share of producs ha are chpesve has grow, whch mples ha wha used o e a relavely mor prolem has ecome a major prolem. () Oher major scefc advaces have had smlar effecs. For example, he veo of fer-opc cale (ad lasers) has led o a dowward red elecommucaos prces as osolee echologes ased o copper wre are gradually replaced. () Sce he ed of World War II, a seres of eraoal rade agreemes have dramacally reduced arffs aroud he world. These reducos, comed wh mprovemes rasporao echologes, have led o a rapd growh of eraoal rade ad remarkale mprovemes eraoal specalzao. Maufacurg acves he more developed ecoomes have gradually ee ousourced o lower-wage coures, leadg o deflao goods prces mos coures. However, may servces cao e readly ousourced, 45 ad so o average he prce of servces reds upward whle he prce of goods reds dowward. (v) A he mcroecoomc level, here are remedous dffereces growh raes of frms. Successful frms expad her scale, lower her coss, ad cause less successful compeors o wher away wh her hgher prces ad lower volumes. Ths leads o a sysemac egave correlao ewee chages em prces ad he correspodg chages em volumes ha ca e very large. Thus, here s some a pror ass for assumg log-ru dverge reds prces ad hece some ass for cocer ha a Lowe dex ha uses a ase year for quay weghs ha s pror o he ase moh for prces may e upward ased, compared wh a more deal arge dex. 45 However some servces ca e eraoally ousourced; for example, call ceers, compuer programmg, ad arle maeace. D.3 Youg dex 5.55 Recall he defos for he ase-year quaes, q, ad he ase-year prces, p, gve y equao (5.23) ad equao (5.24). The ase-year reveue shares ca e defed he usual way as follows: (5.47) pq pkqk k s ;,...,. Defe he vecor of ase-year reveue shares he usual way as s [s,,s ]. These ase-year reveue shares were used o provde a alerave formula for he ase year Lowe prce dex gog from moh 0 o defed equao (5.26) as P Lo (p 0,p,q ) 0 s ( p / p ) s ( p / p ). Raher ha usg hs dex as her shor-ru arge dex, may sascal ageces use he followg closely relaed dex: 0 0 (5.48) PY p p s s ( p p ) (,, ). Ths ype of dex was frs defed y he Eglsh ecooms Arhur Youg (82). 46 Noe ha here s a chage focus whe he Youg dex s used compared wh he dces proposed earler hs chaper. Up o hs po, he dces proposed have ee of he fxed-aske ype (or averages of such dces), where a produc aske ha s somehow represeave for he wo perods eg compared s chose ad he purchased a he prces of he wo perods, ad he dex s ake o e he rao of hese wo coss. O he oher had, for he Youg dex, oe sead chooses represeave reveue shares ha pera o he wo perods uder cosderao ad he uses hese shares o calculae he overall dex as a share-weghed average of he dvdual prce raos, p / p 0. Noe ha hs share-weghed average of prce raos vew of dex umer heory s a dffere from he vew ake a he egg of hs chaper, whch vewed he dex umer prolem as he prolem of decomposg a value rao o he produc of wo erms, oe of whch expresses he amou of 46 Walsh (90, p. 536; 932, p. 657) arues hs formula o Youg. 387

Producer Prce Idex Maual prce chage ewee he wo perods ad he oher ha expresses he amou of quay chage. 47 5.56 Sascal ageces somemes regard he Youg dex defed aove as a approxmao o he Laspeyres prce dex P L (p 0,p,q 0 ). Hece, s of eres o see how he wo dces compare. Defg he log-erm mohly prce relaves gog from moh 0 o as r p /p 0 ad usg equaos (5.32) ad (5.48), (5.49) sce P p p s P p 0 0 0 Y(,, ) L(,, ) p 0 p s 0 s 0 p p 0 p 0 s s 0 s s r p 0 0 s s r r + r s s 0 s s r r, 0 s s ad defg 47 Irvg Fsher s 922 ook s famous for developg he value rao decomposo approach o dex umer heory, u hs roducory chapers ook he share-weghed average po of vew: A dex umer of prces, he, shows he average perceage chage of prces from oe po of me o aoher (922, p. 3). Fsher we o o oe he mporace of ecoomc weghg: The precedg calculao reas all he commodes as equally mpora; cosequely, he average was called smple. If oe commody s more mpora ha aoher, we may rea he more mpora as hough were wo or hree commodes, hus gvg wo or hree mes as much wegh as he oher commody (922, p. 6). Walsh (90, pp. 430 3) cosdered oh approaches: We ca eher () draw some average of he oal moey values of he classes durg a epoch of years, ad wh weghg so deermed employ he geomerc average of he prce varaos [raos]; or (2) draw some average of he mass quaes of he classes durg he epoch, ad apply o hem Scrope s mehod. Scrope s mehod s he same as usg he Lowe dex. Walsh (90, pp. 88 90) cossely sressed he mporace of weghg prce raos y her ecoomc mporace (raher ha usg equally weghed averages of prce relaves). Boh he value rao decomposo approach ad he share-weghed average approach o dex umer heory wll e suded from he axomac perspecve he followg chaper; see also Secos C ad E Chaper 6. ( ) 0 L 0 0. r* s r P p, p, q Thus, he Youg dex P Y (p 0,p,s ) s equal o he Laspeyres dex P L (p 0,p,q 0 ) plus he covarace ewee he dfferece he aual shares perag o year ad he moh 0 shares, s s 0, ad he devaos of he relave prces from her mea, r r*. 5.57 I s o loger possle o guess he lkely sg of he covarace erm. The queso s o loger wheher he quay demaded goes dow as he prce of produc goes up (he aswer o hs queso s usually yes) u does he share of reveue go dow as he prce of produc goes up? The aswer depeds o he elascy of demad for he produc. However, le us provsoally assume ha here are log-ru reds produc prces, ad f he red prces for produc s aove he mea, he he reveue share for he produc reds dow (ad vce versa). Thus, we are assumg hgh elasces or very srog susuo effecs. Assumg also ha he ase year s efore moh 0, he uder hese codos, suppose ha here s a log-erm upward red he prce of produc so ha r r* (p / p 0 ) r* s posve. Wh he assumed very elasc purchaser susuo resposes, s wll ed o decrease relavely over me. Sce s s assumed o e efore s 0, s 0 s expeced o e less ha s, or s s 0 wll lkely e posve. Thus, he covarace s lkely o e posve uder hese crcumsaces. Hece wh logru reds prces ad very elasc resposes of purchasers o prce chages, he Youg dex s lkely o e greaer ha he correspodg Laspeyres dex. 5.58 Assume ha here are log-ru reds produc prces. If he red prces for produc s aove he mea, he suppose ha he reveue share for he produc reds up (ad vce versa). Thus, we are assumg low elasces or very weak susuo effecs. Assume also ha he ase year s efore moh 0, ad suppose ha here s a log-erm upward red he prce of produc so ha r r* (p / p 0 ) r* s posve. Wh he assumed very elasc susuo resposes, s wll ed o crease relavely over me, ad, sce s s assumed o e efore s 0, we wll have s 0 greaer ha s, or s s 0 s egave. Thus, he covarace s lkely o e egave uder 388

5. Basc Idex Numer Theory hese crcumsaces. Hece wh log-ru reds prces ad very elasc resposes of purchasers o prce chages, he Youg dex s lkely o e less ha he correspodg Laspeyres dex. 5.59 The prevous wo paragraphs dcae ha, a pror, s o kow wha he lkely dfferece ewee he Youg dex ad he correspodg Laspeyres dex wll e. If elasces of susuo are close o, he he wo ses of reveue shares, s ad s 0, wll e close o each oher ad he dfferece ewee he wo dces wll e close o zero. However, f mohly reveue shares have srog seasoal compoes, he he aual shares s could dffer susaally from he mohly shares s 0. 5.60 I s useful o have a formula for updag he prevous moh s Youg prce dex usg oly moh-over-moh prce relaves. The Youg dex for moh +, P Y (p 0,p +,s ), ca e preseed erms of he Lowe dex for moh, P Y (p 0,p,s ), ad a updag facor as follows: (5.50) + 0 + 0 p + 0 s ( p / p ) 0 PY ( p, p, s ) 0 s ( p / p ) + 0 p p 0 PY ( p, p, s ) 0 ( p / p ) P ( p, p, s ) s Y usg equao (5.47) p ( / ) ; p p + pq 0 0 p p Y (,, ) 0 pq ( p / p ) 0 0 + Y(,, ) ( / ), P p p s P p p s s p p where he hyrd weghs s 0 are defed y 0 s ( p / p ) 0 sk pk pk k ;,...,. ( / ) Thus, he hyrd weghs s 0 ca e oaed from he ase-year weghs s y updag hem; ha s, y mulplyg hem y he prce relaves (or dces a hgher levels of aggregao), p / p 0. Thus, he requred updag facor, gog from moh o moh +, s he cha-lked dex, 0 + s ( p / p ), whch uses he hyrd reveueshare weghs s 0 defed y equao (5.5). 5.6 Eve f he Youg dex provdes a close approxmao o he correspodg Laspeyres dex, s dffcul o recommed he use of he Youg dex as a fal esmae of he chage prces gog from perod 0 o, jus as was dffcul o recommed he use of he Laspeyres dex as he fal esmae of flao gog from perod 0 o. Recall ha he prolem wh he Laspeyres dex was s lack of symmery he reame of he wo perods uder cosderao. Tha s, usg he jusfcao for he Laspeyres dex as a good fxed-aske dex, here was a decal jusfcao for he use of he Paasche dex as a equally good fxed-aske dex o compare perods 0 ad. The Youg dex suffers from a smlar lack of symmery wh respec o he reame of he ase perod. The prolem ca e explaed as follows. The Youg dex, P Y (p 0,p,s ), defed y equao (5.48), calculaes he prce chage ewee mohs 0 ad, reag moh 0 as he ase. Bu here s o parcular reaso o rea moh 0 as he ase moh oher ha coveo. Hece, f we rea moh as he ase ad use he same formula o measure he prce chage from moh ack o moh 0, he dex P Y (p 0,p,s ) s p p 0 ( / ) would e approprae. Ths esmae of prce chage ca he e made comparale o he orgal Youg dex y akg s recprocal, leadg o he followg reased Youg dex, 48 P Y *(p 0,p,s ), defed as (5.5) 0 0 pq ( p / p ) 0 pkqk pk pk k s ( / ) 48 Usg Irvg Fsher s (922, p. 8) ermology, P Y *(p 0,p,s ) /[P Y (p,p 0,s )] s he me ahess of he orgal Youg dex, P Y (p 0,p,s ). 389

Producer Prce Idex Maual (5.52) 0 0 Y (,, ) ( / ) 0 s p p P p p s s p p ( / ). Thus, he reased Youg dex, P Y *(p 0,p,s ), ha uses he curre moh as he al ase perod s a share-weghed harmoc mea of he prce relaves gog from moh 0 o moh, whereas he orgal Youg dex, P Y (p 0,p,s ), s a shareweghed arhmec mea of he same prce relaves. 5.62 Fsher argued ha a dex umer formula should gve he same aswer o maer whch perod was chose as he ase: Eher oe of he wo mes may e ake as he ase. Wll make a dfferece whch s chose? Ceraly, ough o ad our Tes demads ha shall o. More fully expressed, he es s ha he formula for calculag a dex umer should e such ha wll gve he same rao ewee oe po of comparso ad he oher po, o maer whch of he wo s ake as he ase. (Irvg Fsher, 922, p. 64) 5.63 The prolem wh he Youg dex s ha o oly does o cocde wh s reased couerpar, u here s a defe equaly ewee he wo dces, amely (5.53) P p p s P p p s 0 0 Y (,, ) Y(,, ), wh a src equaly provded ha he perod prce vecor p s o proporoal o he perod 0 prce vecor p 0. 49 Thus, a sascal agecy ha 49 These equales follow from he fac ha a harmoc mea of M posve umers s always equal o or less ha he correspodg arhmec mea; see Walsh (90, p. 57) or Irvg Fsher (922, pp. 383 84). Ths equaly s a specal case of Schlömlch s (858) Iequaly; see Hardy, Llewood ad Polyá (934, p. 26). Walsh (90, pp. 330 32) explcly oed he equaly equao (5.53) ad also oed ha he correspodg geomerc average would fall ewee he harmoc ad arhmec averages. Walsh (90, p. 432) compued some umercal examples of he Youg dex ad foud g dffereces ewee ad hs es dces, eve usg weghs ha were represeave for he perods eg compared. Recall ha he Lowe dex ecomes he Walsh dex whe geomerc (coued) uses he drec Youg dex P Y (p 0,p,s ) wll geerally show a hgher flao rae ha a sascal agecy ha uses he same raw daa u uses he reased Youg dex, P Y *(p 0,p,s ). 5.64 The equaly equao (5.53) does o ell us y how much he Youg dex wll exceed s reased me ahess. However, Appedx 5.3, s show ha o he accuracy of a cera secod-order Taylor seres approxmao, he followg relaoshp holds ewee he drec Youg dex ad s me ahess: 0 0 (5.54) PY( p, p, s ) PY ( p, p, s ) 0 + PY ( p, p, s ) Var e, where Var e s defed as (5.55) Var e s e e. The devaos e are defed y + e r / r* for,, where he r ad her weghed mea r* are defed y 0 (5.56) r p / p ;,,, (5.57) r s r, whch urs ou o equal he drec Youg dex, P Y (p 0,p,s ). The weghed mea of he e s defed as (5.58) e s e, whch urs ou o equal 0. Hece, he more dsperso here s he prce relaves p / p 0, o he accuracy of a secod-order approxmao, he more he drec Youg dex wll exceed s couerpar ha uses moh as he al ase perod raher ha moh 0. mea quay weghs are chose, ad so he Lowe dex ca perform well whe represeave weghs are used. Ths s o ecessarly he case for he Youg dex, eve usg represeave weghs. Walsh (90, p. 433) summed up hs umercal expermes wh he Youg dex as follows: I fac, Youg s mehod, every form, has ee foud o e ad. 2 390

5. Basc Idex Numer Theory 5.65 Gve wo a pror equally plausle dex umer formulas ha gve dffere aswers, such as he Youg dex ad s me ahess, Irvg Fsher (922, p. 36) geerally suggesed akg he geomerc average of he wo dces. 50 A eef of hs averagg s ha he resulg formula wll sasfy he me reversal es. Thus, raher ha usg eher he ase perod 0 Youg dex, P Y (p 0,p,s ), or he curre perod Youg dex, P Y *(p 0,p,s ), whch s always elow he ase perod 0 Youg dex f here s ay dsperso relave prces, seems preferale o use he followg dex, whch s he geomerc average of he wo aleravely ased Youg dces: 5 0 (5.59) P ( p, p, s ) Y 0 0 /2 (,, ) Y Y(,, ). P p p s P p p s If he ase-year shares s happe o cocde wh oh he moh 0 ad moh shares, s 0 ad s, respecvely, he me-recfed Youg dex P Y **(p 0,p,s ) defed y equao (5.59) wll cocde wh he Fsher deal prce dex ewee mohs 0 ad, P F (p 0,p,q 0,q ) (whch wll also equal he Laspeyres ad Paasche dces uder hese codos). Noe also ha he dex P Y ** defed y equao (5.59) ca e produced o a mely ass y a sascal agecy. 50 We ow come o a hrd use of hese ess, amely, o recfy formulae,.e., o derve from ay gve formula whch does o sasfy a es aoher formula whch does sasfy ;. Ths s easly doe y crossg, ha s, y averagg aheses. If a gve formula fals o sasfy Tes [he me reversal es], s me ahess wll also fal o sasfy ; u he wo wll fal, as were, oppose ways, so ha a cross ewee hem (oaed y geomercal averagg) wll gve he golde mea whch does sasfy (Irvg Fsher, 922, p. 36). Acually, he asc dea ehd Fsher s recfcao procedure was suggesed y Walsh, who was a dscussa for Fsher (92), where Fsher gave a prevew of hs 922 ook: We merely have o ake ay dex umer, fd s ahess he way prescred y Professor Fsher, ad he draw he geomerc mea ewee he wo (Correa Moyla Walsh, 92, p. 542). 5 Ths dex s a ase-year weghed couerpar o a equally weghed dex proposed y Carruhers, Sellwood, ad Ward (980, p. 25) ad Dalé (992a, p. 40) he coex of elemeary dex umer formulas. See Chaper 20 for furher dscusso of hs uweghed dex. E. Dvsa Idex ad Dscree Approxmaos E. Dvsa prce ad quay dces 5.66 The secod road approach o dex umer heory reles o he assumpo ha prce ad quay daa chage a more or less couous way. 5.67 Suppose ha he prce ad quay daa o he producs he chose doma of defo ca e regarded as couous fucos of (couous) me, say, p () ad q () for,,. The value of producer reveue a me s V() defed he ovous way as (5.60) V () p() q(). 5.68 Now suppose ha he fucos p () ad q () are dffereale. The oh sdes of equao (5.60) ca e dffereaed wh respec o me o oa (5.6) V () p () q () + p () q (). Dvde oh sdes of equao (5.6) hrough y V() ad, usg equao (5.60), he followg equao s oaed: p q() + p() q () (5.62) V ()/ V() p () q () j j p () q () () + s() () q () s p, where he me reveue share o produc, s (), s defed as (5.63) s () m p () q () p () q () m m j for,,. 5.69 Fraços Dvsa (926, p. 39) argued as follows: suppose he aggregae value a me, 39

Producer Prce Idex Maual V(), ca e wre as he produc of a me prcelevel fuco, P(), say, mulpled y a me quay-level fuco, Q(), say; ha s, we have (5.64) V () PQ () (). Suppose, furher, ha he fucos P() ad Q() are dffereale. The, dffereag equao (5.64) yelds (5.65) V () P () Q() + P() Q (). Dvdg oh sdes of equao (5.65) y V() ad usg equao (5.64) leads o he followg equao: (5.66) V () P () Q () +. V () P () Q () 5.70 Dvsa compared he wo expressos for he logarhmc value dervave, V ()/V(), gve y equao (5.62) ad equao (5.66). He smply defed he logarhmc rae of chage of he aggregae prce level, P ()/P(), as he frs se of erms o he rgh-had sde of equao (5.62), ad he smply defed he logarhmc rae of chage of he aggregae quay level, Q ()/Q(), as he secod se of erms o he rgh-had sde of equao (5.62). Tha s, he made he followg defos: (5.67) (5.68) P () p () s () ; P() p () Q () q () s (). Q () q() 5.7 Equaos (5.67) ad (5.68) are reasoale defos for he proporoal chages he aggregae prce ad quay (or quay) levels, P() ad Q(). 52 The prolem wh hese defos s ha ecoomc daa are o colleced couous me; hey are colleced dscree me. I oher words, eve hough rasacos ca e 52 If hese defos are appled (approxmaely) o he Youg dex suded he prevous seco, he ca e see ha for he Youg prce dex o e cosse wh he Dvsa prce dex, he ase-year shares should e chose o e average shares ha apply o he ere me perod ewee mohs 0 ad. hough of as occurrg couous me, o producer records hs or her purchases as hey occur couous me; raher, purchases over a fe me perod are cumulaed ad he recorded. A smlar suao occurs for producers or sellers of producs; frms cumulae her sales over dscree perods of me for accoug or aalycal purposes. If s aemped o approxmae couous me y shorer ad shorer dscree me ervals, emprcal prce ad quay daa ca e expeced o ecome creasgly errac, sce cosumers make purchases oly a dscree pos of me (ad producers or sellers of producs make sales oly a dscree pos of me). However, s sll of some eres o approxmae he couous me prce ad quay levels, P() ad Q() defed mplcly y equaos (5.67) ad (5.68), y dscree me approxmaos. Ths ca e doe wo ways. Eher mehods of umercal approxmao ca e used or assumpos aou he pah ake y he fucos p () ad q () (,,) hrough me ca e made. The frs sraegy s used he followg seco. For dscussos of he secod sraegy, see Vog (977; 978), Va Ijzere (987, pp. 8 2), Vog ad Bara (997), ad Balk (2000). 5.72 There s a coeco ewee he Dvsa prce ad quay levels, P() ad Q(), ad he ecoomc approach o dex umer heory. However, hs coeco s es made afer oe has suded he ecoomc approach o dex umer heory Chaper 7. Sce hs maeral s raher echcal, appears Appedx 7.. E.2 Dscree approxmaos o couous-me Dvsa dex 5.73 To make operaoal he couous me Dvsa prce ad quay levels, P() ad Q() defed y he dffereal equaos (5.67) ad (5.68), s ecessary o cover o dscree me. Dvsa (926, p. 40) suggesed a sraghforward mehod for dog hs coverso, whch we ow oule. 5.74 Defe he followg prce ad quay (forward) dffereces (5.69) P P() P(0); (5.70) p p () p (0);,...,. Usg he aove defos 392

5. Basc Idex Numer Theory (5.7) P() P(0) + P P(0) P(0) P + + P(0) m pq (0), p (0) q (0) usg equao (5.67) whe 0 ad approxmag p (0) y he dfferece p { + } m m m p() q(0) pm(0) qm(0) m 0 0 PL p q m p (0) (0) p (0) q (0) (,,, ), where p [p (),,p ()] ad q [q (),,q ()] for 0,. Thus, ca e see ha Dvsa s dscree approxmao o hs couous-me prce dex s jus he Laspeyres prce dex, P L, defed y equao (5.5). 5.75 Bu ow a prolem oed y Frsch (936, p. 8) occurs: sead of approxmag he dervaves y he dscree (forward) dffereces defed y equaos (5.69) ad (5.70), oher approxmaos could e used ad a wde varey of dscree me approxmaos ca e oaed. For example, sead of usg forward dffereces ad evaluag he dex a me 0, oe could use ackward dffereces ad evaluae he dex a me. These ackward dffereces are defed as (5.72) p p (0) p ();,...,. Ths use of ackward dffereces leads o he followg approxmao for P(0) / P(): (5.73) P(0) P() + P P() P() P + + P() m m m pq() p () q () m usg equao (5.67) whe ad approxmag p () y he dfferece p : { + } m p () () m p () q () m p (0) q () m m m p () q (), P p q ( 0,, 0, ) P where P P s he Paasche dex defed y equao (5.6). Takg recprocals of oh sdes of equao (5.73) leads o he followg dscree approxmao o P() / P(0): (5.74) P() P. P(0) 5.76 Thus, as Frsch 53 oed, oh he Paasche ad Laspeyres dces ca e regarded as (equally vald) approxmaos o he couous-me Dvsa prce dex. 54 Sce he Paasche ad Laspeyres dces ca dffer cosderaly some emprcal applcaos, ca e see ha Dvsa s dea s o all ha helpful deermg a uque dscree me dex umer formula. 55 Wha s useful aou he Dvsa dces s he dea ha as he dscree u of me ges smaller, dscree ap- 53 As he elemeary formula of he chag, we may ge Laspeyres or Paasche s or Edgeworh s or early ay oher formula, accordg as we choose he approxmao prcple for he seps of he umercal egrao (Ragar Frsch, 936, p. 8). 54 Dewer (980, p. 444) also oaed he Paasche ad Laspeyres approxmaos o he Dvsa dex usg a somewha dffere approxmao argume. He also showed how several oher popular dscree me dex umer formulas could e regarded as approxmaos o he couous-me Dvsa dex. 55 Trved (98) sysemacally examed he prolems volved fdg a es dscree me approxmao o he Dvsa dces usg he echques of umercal aalyss. However, hese umercal aalyss echques deped o he assumpo ha he rue couous-me mcro prce fucos, p (), ca e adequaely represeed y a polyomal approxmao. Thus, we are led o he cocluso ha he es dscree me approxmao o he Dvsa dex depeds o assumpos ha are dffcul o verfy. 393

Producer Prce Idex Maual proxmaos o he Dvsa dces ca approach meagful ecoomc dces uder cera codos. Moreover, f he Dvsa cocep s acceped as he correc oe for dex umer heory, he he correspodg correc dscree me couerpar mgh e ake as a weghed average of he cha prce relaves perag o he adjace perods uder cosderao, where he weghs are somehow represeave for he wo perods uder cosderao. F. Fxed-Base versus Cha Idces 5.77 Ths seco 56 dscusses he mers of usg he cha sysem for cosrucg prce dces he me seres coex versus usg he fxedase sysem. 57 5.78 The cha sysem 58 measures he chage prces gog from oe perod o aoher usg a laeral dex umer formula volvg he prces ad quaes perag o he wo adjace perods. These oe-perod raes of chage (he lks he cha) are he cumulaed o yeld he relave levels of prces over he ere perod uder cosderao. Thus, f he laeral prce dex s P, he cha sysem geeraes he followg paer of prce levels for he frs hree perods: (5.75) 0 0, P( p, p, q, q ), 0 0 2 2 P( p, p, q, q ) P( p, p, q, q ). 5.79 O he oher had, he fxed-ase sysem of prce levels usg he same laeral dex umer formula P smply compues he level of prces 56 Ths seco s ased largely o he work of Peer Hll (988; 993, pp. 385 90). 57 The resuls Appedx 7. provde some heorecal suppor for he use of cha dces ha s show ha uder cera codos, he Dvsa dex wll equal a ecoomc dex. Hece, ay dscree approxmao o he Dvsa dex wll approach he ecoomc dex as he me perod ges shorer. Thus, uder cera codos, cha dces wll approach a uderlyg ecoomc dex. 58 The cha prcple was roduced depedely o he ecoomcs leraure y Lehr (885, pp. 45 6) ad Marshall (887, p. 373). Boh auhors oserved ha he cha sysem would mgae he dffcules ecause of he roduco of ew producs o he ecoomy, a po also meoed y Peer Hll (993, p. 388). Irvg Fsher (9, p. 203) roduced he erm cha sysem. perod relave o he ase perod 0 as P(p 0,p,q 0,q ). Thus, he fxed-ase paer of prce levels for perods 0,, ad 2 s (5.76) 0 0 0 2 0 2, P( p, p, q, q ), P( p, p, q, q ). 5.80 Noe ha oh he cha sysem ad he fxed-ase sysem of prce levels defed y equaos (5.75) ad (5.76), he ase-perod prce level s equal o. The usual pracce sascal ageces s o se he ase-perod prce level equal o 00. If hs s doe, he s ecessary o mulply each of he umers equaos (5.75) ad (5.76) y 00. 5.8 Because of he dffcules volved oag curre-perod formao o quaes (or, equvalely, o reveues), may sascal ageces loosely ase her PPI o he use of he Laspeyres formula equao (5.5) ad he fxed-ase sysem. Therefore, s of some eres o look a he possle prolems assocaed wh he use of fxed-ase Laspeyres dces. 5.82 The ma prolem wh he use of fxedase Laspeyres dces s ha he perod 0 fxed aske of producs ha s eg prced ou perod ofe ca e que dffere from he perod aske. Thus, f here are sysemac reds a leas some of he prces ad quaes 59 he dex aske, he fxed-ase Laspeyres prce dex, P L (p 0,p,q 0,q ), ca e que dffere from he correspodg fxed-ase Paasche prce dex, P P (p 0,p,q 0,q ). 60 Ths meas ha oh dces are lkely o e a adequae represeao of he moveme average prces over he me perod uder cosderao. 5.83 The fxed-ase Laspeyres quay dex cao e used forever; eveually, he ase-perod quaes q 0 are so far removed from he curreperod quaes q ha he ase mus e chaged. 59 Examples of rapdly dowward redg prces ad upward redg quaes are compuers, elecroc equpme of all ypes, Iere access, ad elecommucao charges. 60 Noe ha P L (p 0,p,q 0,q ) wll equal P P (p 0,p,q 0,q ) f eher he wo quay vecors q 0 ad q are proporoal or he wo prce vecors p 0 ad p are proporoal. Thus, o oa a dfferece ewee he Paasche ad Laspeyres dces, oproporoaly oh prces ad quaes s requred. 394

5. Basc Idex Numer Theory Chag s merely he lmg case where he ase s chaged each perod. 6 5.84 The ma advaage of he cha sysem s ha uder ormal codos, chag wll reduce he spread ewee he Paasche ad Laspeyres dces. 62 These dces provde a asymmerc perspecve o he amou of prce chage ha has occurred ewee he wo perods uder cosderao, ad could e expeced ha a sgle po esmae of he aggregae prce chage should le ewee hese wo esmaes. Thus, he use of eher a chaed Paasche or Laspeyres dex wll usually lead o a smaller dfferece ewee he wo ad hece o esmaes ha are closer o he ruh. 63 5.85 Peer Hll (993, p. 388), drawg o hs earler research (988, pp. 36 37) ad ha of Szulc (983), oed ha s o approprae o use he cha sysem whe prces oscllae, or ouce, o use Szulc s (983, p. 548) erm. Ths pheomeo ca occur he coex of regular seasoal flucuaos or he coex of prce wars. However, he coex of roughly mooocally chagg prces ad quaes, Peer Hll (993, p. 389) recommeded he use of chaed symmercally weghed dces (see Seco C). The Fsher ad Walsh dces are examples of symmercally weghed dces. 5.86 I s possle o e a more precse regardg uder whch codos oe should or should o cha. Bascally, oe should cha f he prces ad quaes perag o adjace perods are more smlar ha he prces ad quaes of more dsa perods, sce hs sraegy wll lead o a arrowg of he spread ewee he Paasche ad Laspeyres dces a each lk. 64 Oe eeds a 6 Regular seasoal flucuaos ca cause mohly or quarerly daa o ouce, usg Szulc s (983) erm, ad chag oucg daa ca lead o a cosderale amou of dex drf. Tha s, f afer 2 mohs, prces ad quaes reur o her levels of a year earler, he a chaed mohly dex wll usually o reur o uy. Hece, he use of chaed dces for osy mohly or quarerly daa s o recommeded whou careful cosderao. 62 See Dewer (978, p. 895) ad Peer Hll (988; 993, pp. 387 88). 63 Ths oservao wll e llusraed wh a arfcal daa se Chaper 9. 64 Walsh, dscussg wheher fxed-ase or chaed dex umers should e cosruced, ook for graed ha he precso of all reasoale laeral dex umer for- (coued) measure of how smlar are he prces ad quaes perag o wo perods. The smlary measures could e relave or asolue. I he case of asolue comparsos, wo vecors of he same dmeso are smlar f hey are decal ad dssmlar oherwse. I he case of relave comparsos, wo vecors are smlar f hey are proporoal ad dssmlar f hey are oproporoal. 65 Oce a smlary measure has ee defed, he prces ad quaes of each perod ca e compared usg hs measure, ad a ree or pah ha lks all of he oservaos ca e cosruced where he mos smlar oservaos are compared usg a laeral dex umer formula. 66 R. J. Hll (995) defed he prce srucures ewee he wo coures o e more dssmlar he gger s mulas would mprove, provded ha he wo perods or suaos eg compared were more smlar ad, for hs reaso, favored he use of chaed dces: The queso s really, whch of he wo courses [fxed-ase or chaed dex umers] are we lkely o ga greaer exacess he comparsos acually made? Here he proaly seems o cle favor of he secod course; for he codos are lkely o e less dverse ewee wo coguous perods ha ewee wo perods say, ffy years apar (Correa Moyla Walsh, 90, p. 206). Walsh (92a, pp. 84 85) laer reeraed hs preferece for chaed dex umers. Fsher also made use of he dea ha he cha sysem would usually make laeral comparsos ewee prce ad quay daa ha were more smlar ad hece he resulg comparsos would e more accurae: The dex umers for 909 ad 90 (each calculaed erms of 867 877) are compared wh each oher. Bu drec comparso ewee 909 ad 90 would gve a dffere ad more valuale resul. To use a commo ase s lke comparg he relave heghs of wo me y measurg he hegh of each aove he floor, sead of pug hem ack o ack ad drecly measurg he dfferece of level ewee he ops of her heads (Irvg Fsher, 9, p. 204). I seems, herefore, advsale o compare each year wh he ex, or, oher words, o make each year he ase year for he ex. Such a procedure has ee recommeded y Marshall, Edgeworh ad Flux. I largely mees he dffculy of o-uform chages he Q s, for ay equales for successve years are relavely small (Irvg Fsher, 9, pp. 423 24). 65 Dewer (2002) akes a axomac approach o defg varous dces of asolue ad relave dssmlary. 66 Irvg Fsher (922, pp. 27 76) hed a he possly of usg spaal lkg; ha s, of lkg coures smlar srucure. However, he moder leraure has grow due o he poeerg effors of R.J. Hll (995; 999a; 999; 200). R.J. Hll (995) used he spread ewee he Paasche ad Laspeyres prce dces as a dcaor of smlary ad showed ha hs crero gves he same resuls as a crero ha looks a he spread ewee he Paasche ad Laspeyres quay dces. 395

Producer Prce Idex Maual he spread ewee P L ad P P ; ha s, he gger s max {P L /P P, P P /P L }. The prolem wh hs measure of dssmlary he prce srucures of he wo coures s ha could e he case ha P L P P (so ha he R. J. Hll measure would regser a maxmal degree of smlary), u p 0 could e very dffere from p. Thus, here s a eed for a more sysemac sudy of smlary (or dssmlary) measures o pck he es oe ha could e used as a pu o R. J. Hll s (999a; 999; 200) spag ree algorhm for lkg oservaos. 5.87 The mehod of lkg oservaos explaed he prevous paragraph ased o he smlary of he prce ad quay srucures of ay wo oservaos may o e praccal a sascal agecy coex, sce he addo of a ew perod may lead o a reorderg of he prevous lks. However, he aove scefc mehod for lkg oservaos may e useful decdg wheher chag s preferale or wheher fxedase dces should e used whle makg moho-moh comparsos wh a year. 5.88 Some dex umer heorss have ojeced o he cha prcple o he grouds ha has o couerpar he spaal coex: They [cha dces] oly apply o eremporal comparsos, ad coras o drec dces hey are o applcale o cases whch o aural order or sequece exss. Thus, he dea of a cha dex, for example, has o couerpar erregoal or eraoal prce comparsos, ecause coures cao e sequeced a logcal or aural way (here s o k + or k coury o e compared wh coury k). (Peer vo der Lppe, 200, p. 2) 67 Ths s correc, u R.J. Hll s approach does lead o a aural se of spaal lks. Applyg he same approach o he me-seres coex wll lead o a se of lks ewee perods ha may o e moh o moh, u wll may cases jusfy yearover-year lkg of he daa perag o he same 67 I should e oed ha vo der Lppe (200, pp. 56 8) s a vgorous crc of all dex umer ess ased o symmery he me seres coex, alhough he s wllg o accep symmery he coex of makg eraoal comparsos. Bu here are good reasos o o ss o such crera he eremporal case. Whe o symmery exss ewee 0 ad, here s o po erchagg 0 ad (Peer vo der Lppe, 200, p. 58). moh. Ths prolem wll e recosdered Chaper 22. 5.89 I s of some eres o deerme f here are dex umer formulas ha gve he same aswer whe eher he fxed-ase or cha sysem s used. Comparg he sequece of cha dces defed y equao (5.75) aove o he correspodg fxed-ase dces, ca e see ha we wll oa he same aswer all hree perods f he dex umer formula P sasfes he followg fucoal equao for all prce ad quay vecors: (5.77) 0 2 0 2 0 0 P( p, p, q, q ) P( p, p, q, q ) 2 2 P( p, p, q, q ). If a dex umer formula P sasfes equao (5.77), he P sasfes he crculary es. 68 5.90 If s assumed ha he dex umer formula P sasfes cera properes or ess addo o he crculary es aove, 69 he Fuke, Hacker, ad Voeller (979) showed ha P mus have he followg fucoal form creded orgally o Koüs ad Byushges 70 (926, pp. 63 66): 7 68 The es ame s creded o Irvg Fsher (922, p. 43), ad he cocep was orgally creded o Wesergaard (890, pp. 28 9). 69 The addoal ess are () posvy ad couy of P(p 0,p,q 0,q ) for all srcly posve prce ad quay vecors p 0,p,q 0,q ; () he dey es; () he commesuraly es; (v) P(p 0,p,q 0,q ) s posvely homogeeous of degree he compoes of p ; ad (v) P(p 0,p,q 0,q ) s posvely homogeeous of degree zero he compoes of q. 70 Koüs ad Byushges show ha he dex defed y equao (5.78) s exac for Co-Douglas (928) prefereces; see also Pollak (983a, pp. 9 20). The cocep of a exac dex umer formula wll e explaed Chaper 7. 7 Ths resul ca e derved usg resuls Echhor (978, pp. 67 68) ad Vog ad Bara (997, p. 47). A smple proof ca e foud Balk (995). Ths resul vdcaes Irvg Fsher s (922, p. 274) uo. He assered ha he oly formulae whch coform perfecly o he crcular es are dex umers whch have cosa weghs Irvg Fsher (922, p. 275) we o o asser, Bu, clearly, cosa weghg s o heorecally correc. If we compare 93 wh 94, we eed oe se of weghs; f we compare 93 wh 95, we eed, heorecally a leas, aoher se of weghs. Smlarly, urg (coued) 396

5. Basc Idex Numer Theory (5.78) 0 0 KB (,,, ) P p q α p 0 p, where he cosas α sasfy he followg resrcos: (5.79) α ad α > 0 for,...,. Thus, uder very weak regulary codos, he oly prce dex sasfyg he crculary es s a weghed geomerc average of all he dvdual prce raos, he weghs eg cosa hrough me. 5.9 A eresg specal case of he famly of dces defed y equao (5.78) occurs whe he weghs α are all equal. I hs case, P KB reduces o he Jevos (865) dex: (5.80) 0 0 p J (,,, ). 0 p P p q 5.92 The prolem wh he dces defed y Koüs ad Byushges ad Jevos s ha he dvdual prce raos, p / p 0, have weghs (eher α or / ) ha are depede of he ecoomc mporace of produc he wo perods uder cosderao. Pu aoher way, hese prce weghs are depede of he quaes of produc cosumed or he reveues o produc durg he wo perods. Hece, hese dces are o really suale for use y sascal ageces a hgher levels of aggregao whe reveue share formao s avalale. 5.93 The aove resuls dcae ha s o useful o ask ha he prce dex P sasfy he crculary es exacly. However, s of some eres o fd dex umer formulas ha sasfy he crculary es o some degree of approxmao, sce he use of such a dex umer formula wll lead o measures of aggregae prce chage ha are more or less he same wheher we use he cha or fxed-ase sysems. Irvg Fsher (922, p. 284) from me o space, a dex umer for comparg he Ued Saes ad Eglad requres oe se of weghs, ad a dex umer for comparg he Ued Saes ad Frace requres, heorecally a leas, aoher. foud ha devaos from crculary usg hs daa se ad he Fsher deal prce dex P F defed y equao (5.2) aove were que small. Ths relavely hgh degree of correspodece ewee fxed-ase ad cha dces has ee foud o hold for oher symmercally weghed formulas lke he Walsh dex P W defed y equao (5.9). 72 Thus, mos me-seres applcaos of dex umer heory where he ase year fxedase dces s chaged every fve years or so, wll o maer very much wheher he sascal agecy uses a fxed-ase prce dex or a cha dex, provded ha a symmercally weghed formula s used. 73 Ths, of course, depeds o he legh of he me seres cosdered ad he degree of varao he prces ad quaes as we go from perod o perod. The more prces ad quaes are sujec o large flucuaos (raher ha smooh reds), he less he correspodece. 74 5.94 I s possle o gve a heorecal explaao for he approxmae sasfaco of he crculary es for symmercally weghed dex umer formulas. Aoher symmercally weghed formula s he Törqvs dex P T. 75 The aural logarhm of hs dex s defed as follows: (5.8) 0 0 l PT ( p, p, q, q ) 0 p ( s + s ) l 0 2 p, where he perod reveue shares s are defed y equao (5.7) aove. Alerma, Dewer, ad Feesra (999, p. 6) show ha f he logarhmc 72 See, for example, Dewer (978, p. 894). Walsh (90, pp. 424 ad 429) foud ha hs hree preferred formulas all approxmaed each oher very well, as dd he Fsher deal for hs arfcal daa se. 73 More specfcally, mos superlave dces (whch are symmercally weghed) wll sasfy he crculary es o a hgh degree of approxmao he me seres coex. See Chaper 7 for he defo of a superlave dex. I s worh sressg ha fxed-ase Paasche ad Laspeyres dces are very lkely o dverge cosderaly over a fveyear perod f compuers (or ay oher produc ha has prce ad quay reds dffere from he reds he oher producs) are cluded he value aggregae uder cosderao. See Chaper 9 for some emprcal evdece o hs opc. 74 Aga, see Szulc (983) ad Peer Hll (988). 75 Ths formula was mplcly roduced Törqvs (936) ad explcly defed Törqvs ad Törqvs (937). 397

Producer Prce Idex Maual prce raos l (p / p ) red learly wh me, ad he reveue shares s also red learly wh me, he he Törqvs dex P T wll sasfy he crculary es exacly. 76 Sce may ecoomc me seres o prces ad quaes sasfy hese assumpos approxmaely, he he Törqvs dex wll sasfy he crculary es approxmaely. As wll e see Chaper 9, geerally he Törqvs dex closely approxmaes he symmercally weghed Fsher ad Walsh dces, so ha for may ecoomc me seres (wh smooh reds), all hree of hese symmercally weghed dces wll sasfy he crculary es o a hgh eough degree of approxmao so ha wll o maer wheher we use he fxed-ase or cha prcple. 5.95 Walsh (90, p. 40; 92a, p. 98; 92, p. 540) roduced he followg useful vara of he crculary es: (5.82) 0 0 2 2 P( p, p, q, q ) P( p, p, q, q ) T 0 T 0... (,,, ) P p q. The movao for hs es s he followg. Use he laeral dex formula P(p 0,p,q 0,q ) o calculae he chage prces gog from perod 0 o, use he same formula evaluaed a he daa correspodg o perods ad 2, P(p,p 2,q,q 2 ), o calculae he chage prces gog from perod o 2,. Use P(p T,p T,q T,q T ) o calculae he chage prces gog from perod T o T. Iroduce a arfcal perod T + ha has exacly he prce ad quay of he al perod 0 ad use P(p T,p 0,q T,q 0 ) o calculae he chage prces gog from perod T o 0. Fally, mulply all hese dces, ad, sce we ed up where we sared, he he produc of all of hese dces should deally e. Dewer (993a, p. 40) called hs es a mulperod dey es. 77 Noe ha f T 2 (so ha he umer of perods s 3 oal), he Walsh s es reduces o Fsher s (92, p. 534; 922, p. 64) me reversal es. 78 5.96 Walsh (90, pp. 423 33) showed how hs crculary es could e used o evaluae he worh of a laeral dex umer formula. He veed arfcal prce ad quay daa for fve perods ad added a sxh perod ha had he daa of he frs perod. He he evaluaed he rgh-had sde of equao (5.82) for varous formulas, P(p 0,p,q 0,q ), ad deermed how far from uy he resuls were. Hs es formulas had producs ha were close o. 79 5.97 Ths same framework s ofe used o evaluae he effcacy of chaed dces versus her drec couerpars. Thus, f he rgh-had sde of equao (5.82) urs ou o e dffere from uy, he chaed dces are sad o suffer from cha drf. If a formula does suffer from cha drf, s somemes recommeded ha fxed-ase dces e used place of chaed oes. However, hs advce, f acceped, would always lead o he adopo of fxed-ase dces, provded ha he laeral dex formula sasfes he dey es, P(p 0,p 0,q 0,q 0 ). Thus, s o recommeded ha Walsh s crculary es e used o decde wheher fxed-ase or chaed dces should e calculaed. However, s far o use Walsh s crculary es as he orgally used ; ha s, as a approxmae mehod for decdg he force of a parcular dex umer formula. To decde wheher o cha or use fxed-ase dces, oe should decde o he ass of how smlar are he oservaos eg compared ad choose he mehod ha wll es lk he mos smlar oservaos. 5.98 Varous properes, axoms, or ess ha a dex umer formula could sasfy have already ee roduced hs chaper. I he followg chaper, he es approach o dex umer heory wll e suded a more sysemac maer. 76 Ths exacess resul ca e exeded o cover he case whe here are mohly proporoal varaos prces ad he reveue shares have cosa seasoal effecs addo o lear reds; see Alerma, Dewer, ad Feesra (999, p. 65). 77 Walsh (92a, p. 98) called hs es he crcular es, u sce Irvg Fsher also used hs erm o descre hs rasvy es defed earler y equao (5.77), seems es o sck o Irvg Fsher s ermology sce s well esalshed he leraure. 78 Walsh (92, pp. 540 4) oed ha he me reversal es was a specal case of hs crculary es. 79 Ths s esseally a vara of he mehodology ha Irvg Fsher (922, p. 284) used o check how well varous formulas correspoded o hs verso of he crculary es. 398

5. Basc Idex Numer Theory Appedx 5.: Relaoshp Bewee Paasche ad Laspeyres Idces 5.99 Recall he oao used Seco B.2. Defe he h relave prce or prce relave r ad he h quay relave as follows: (A5..) r ; ;,...,. p 0 0 q Usg equao (5.8) aove for he Laspeyres prce dex P L ad equao (A5..), we have (A5..2) 0 * L P rs r ; ha s, we defe he average prce relave r* as he ase-perod reveue share-weghed average of he dvdual prce relaves, r. 5.00 Usg equao (5.6) for he Paasche prce dex P P, we have (A5..3) pq P 0 pmqm m 0 0 r pq 0 0 mpmqm m 0 r s 0 msm m usg equao (A5..) * * 0 * ( r r )( ) s + r, 0 ms m m usg equao (A5..2) ad 0 s ad where he average quay relave * s defed as (A5..4) * 0 L s Q, where he las equaly follows usg equao (5.), he defo of he Laspeyres quay dex Q L. 5.0 Takg he dfferece ewee P P ad P L ad usg equao (A5..2) equao (A5..4) yelds (A5..5) P P r r s * * 0 P L ( )( ). Q L Now le r ad e dscree radom varales ha 0 ake o he values r ad, respecvely. Le s e he jo proaly ha r r ad for,,, ad le he jo proaly e 0 f r r ad j where j. I ca e verfed ha he * * 0 summao ( r r )( ) s o he rgh-had sde of equao (A5..5) s he covarace ewee he prce relaves r ad he correspodg quay relaves. Ths covarace ca e covered o a correlao coeffce. 80 If hs covarace s egave, whch s he usual case he cosumer coex, he P P wll e less ha P L. If s posve, whch wll occur he suaos where supply codos are fxed (as he fxedpu oupu prce dex), u demad s chagg, he P P wll e greaer ha P L. Appedx 5.2: Relaoshp Bewee Lowe ad Laspeyres Idces 5.02 Recall he oao used Seco D.. Defe he h relave prce relag he prce of produc of moh o moh 0, r, ad he h quay relave,, relag quay of produc ase year o moh 0,, as follows: p q (A5.2.) r ; 0 0 p q,,. 80 See Borkewcz (923, pp. 374 75) for he frs applcao of hs correlao coeffce decomposo echque. 399

Producer Prce Idex Maual As Appedx 5., he Laspeyres prce dex P L (p 0,p,q 0 ) ca e defed as r*, he moh 0 reveue share-weghed average of he dvdual prce relaves r defed equao (A5.2.), excep ha he moh prce, p, ow replaces perod prce, p, he defo of he h prce relave r : L 0 (A5.2.2) r rs P. 5.03 The average quay relave * relag he quaes of ase year o hose of moh 0 s defed as he moh 0 reveue share-weghed average of he dvdual quay relaves defed equao (A5.2.): (A5.2.3) * 0 L s Q, where Q L Q L (q 0,q,p 0 ) s he Laspeyres quay dex relag he quaes of moh 0, q 0, o hose of he year, q, usg he prces of moh 0, p 0, as weghs. 5.04 Usg equao (5.26), he Lowe dex comparg he prces moh wh hose of moh 0, usg he quay weghs of he ase year, s equal o (A5.2.4) P ( p, p, q ) Lo 0 0 0 0 0 usg equao (A5.2.) 0 0 0 pq pq 0 0 0 0 pq pq p 0 0 0 pq p 0 0 pq usg equao (A5.2.3) 0 0 r pq 0 0 pq 0 0 0 r s ( r r ) s r s usg equao (A5.2.) + 0 0 ( r r ) r s s + 0 ( r r ) s + r usg equao (A5.2.3) 0 0 ( r r )( ) s ( r r ) s + + r 0 0 ( r r )( ) s rs r + + r 0 ( r r )( ) s + P ( p, p, q ) + L 0 0 r sce ( r r )( ) s rs 0 Q q q p 0 0 L (,, ) 0 sce usg equao (A5.2.2), r* equals he Laspeyres prce dex, P L (p 0,p,q 0 ), ad usg equao (A5.2.3), * equals he Laspeyres quay dex, Q L (q 0,q,p 0 ). Thus, equao (A5.2.4) ells us ha he Lowe prce dex usg he quaes of year as weghs, P Lo (p 0,p,q ), s equal o he usual Laspeyres dex usg he quaes of moh 0 as weghs, P L (p 0,p,q 0 ), plus a covarace erm ( r r )( ) s 0, r ewee he prce relaves r p / p 0 ad he quay relaves q / q 0, dvded y he Laspeyres quay dex Q L (q 0,q,p 0 ) ewee moh 0 ad ase year. 400

5. Basc Idex Numer Theory Appedx 5.3: Relaoshp Bewee Youg Idex ad Is Tme Ahess 5.05 Recall ha he drec Youg dex, P Y (p 0,p,s ), was defed y equao (5.48) ad s me ahess, P Y *(p 0,p,s ), was defed y equao (5.52). Defe he h relave prce ewee mohs 0 ad as 0 (A5.3.) r p / p ;,...,, ad defe he weghed average (usg he aseyear weghs s ) of he r as (A5.3.2) r s r, whch urs ou o equal he drec Youg dex, P Y (p 0,p,s ). Defe he devao e of r from her weghed average r* usg he followg equao: (A5.3.3) r r ( + e );,...,. If equao (A5.3.3) s susued o equao (A5.3.2), he followg equaos are oaed: (A5.3.4) r s r ( + e ) (A5.3.5) e r + r se, sce s. s e 0. Thus, he weghed mea e* of he devaos e equals 0. 5.06 The drec Youg dex, P Y (p 0,p,s ), ad s me ahess, P Y *(p 0,p,s ), ca e wre as fucos of r*, he weghs s ad he devaos of he prce relaves e as follows: (A5.3.6) P p p s r 0 Y (,, ) ; 0 Y (,, ) ( + ) r s ( + e). (A5.3.7) P p p s s { r e } 5.07 Now, regard P Y *(p 0,p,s ) as a fuco of he vecor of devaos, e [e,,e ], say, P Y *(e). The secod-order Taylor seres approxmao o P Y *(e) aroud he po e 0 s gve y he followg expresso: 8 (A5.3.8) P () e Y j j j [ ] 2 r + r se+ r ssee r s e r + r 0 + r s sjej e r* s e e j usg equao (A5.3.5) [ 0] r + r s e r s e e usg equao (A5.3.5) 0 0 Y(,, ) Y(,, ) P p p s P p p s s e e usg equao (A5.3.6) 0 0 P ( p, p, s ) P ( p, p, s )Var e, Y Y where he weghed sample varace of he vecor e of prce devaos s defed as (A5.3.9) Var e s e e. 5.08 Rearragg equao (A5.3.8) gves he followg approxmae relaoshp ewee he drec Youg dex P Y (p 0,p,s ) ad s me ahess P Y *(p 0,p,s ), o he accuracy of a secod-order Taylor seres approxmao aou a prce po where he moh prce vecor s proporoal o he moh 0 prce vecor: (A5.3.0) P p p s P ( p, p, s ) + P ( p, p, s ) Var e. 0 Y (,, ) 0 0 Y Y Thus, o he accuracy of a secod-order approxmao, he drec Youg dex wll exceed s me 8 Ths ype of secod-order approxmao s creded o Dalé (992a, p. 43) for he case r* ad o Dewer (995a, p. 29) for he case of a geeral r*. 2 2 2 2 40

Producer Prce Idex Maual ahess y a erm equal o he drec Youg dex mes he weghed varace of he devaos of he prce relaves from her weghed mea. Thus, he gger he dsperso relave prces, he more he drec Youg dex wll exceed s me ahess. 402