Using the Two-Stage Approach to Price Index Aggregation

Transcription

1 Oaa Grou Meeng, 3 Ung he To-Sage Aroach o Prce Inde Aggregaon Toc: Samlng and Elemenary Aggregae; Aggregaon Aravndan Jayanha and Le Conn Abrac Th aer aee he raccal mlcaon for Naonal Sacal Offce (NSO) of mlemenng he o-age aroach o rce nde aggregaon reened n he Inernaonal Moneary Fund (IMF) Prce Inde Proceor ofare uer gude (UNECE 9). The o-age aroach rereened by a hor erm rce nde, hch udae a long erm rce nde. Th aer eamne amle change and qualy adumen hn he o-age aroach. The Auralan Bureau of Sac (ABS) conderng he mlemenaon of he o-age aroach o rce nde aggregaon a ar of a roec o udae bune rocee, yem and mehodologe ued o roduce rce ndee.

2 . Inroducon The Auralan Bureau of Sac (ABS) roduce a de range of rce ndee ncludng conumer, roducer, labour, nernaonal rade and houng. Varou bune rocee, yem and mehodologe are ued o comle each of hee ndee. The ABS ha commenced a roec o udae bune rocee, echnology and mehodologe ued o roduce rce ndee and harmone hee here oble. The ABS conderng he mlemenaon of he To-Sage aroach o rce nde aggregaon a ar of h broader roec. Th aer ummare he curren aroach o rce nde aggregaon n Aurala, defne he o-age aggregaon mehodology and eamne amle change and qualy adumen. Fuure challenge and lkely mlcaon for a Naonal Sacal Offce are alo eamned.. Curren aroach o rce nde roducon n Aurala The maory of ABS rce ndee are aggregaed: a. a he elemenary aggregae level ung he Jevon or Duo aggregaon aroache; and b. a he uer level ung he arhmec Loe or Young aroache. The ABS ue a drec nde aroach hn he elemenary aggregae of mo rce ndee (e.g. CPI and PPI). Th mean ha curren erod rce are comared o bae (rce reference) erod rce. Th can reen challenge for comlng rce ndee over long erod hen he em beng rced are relaced becaue hey re ermanenly mng or no longer rereenave, and no rce e n he rce reference erod for ne em. Mo ABS rce ndee ue he eendure hare form of he Loe or Young nde ece for he Wage Prce Inde hch ue he quany form of he Laeyre formulae a he loe level, h quany ereed a he number of ob or hour. The ABS CPI an eamle of h a he bae erod (rce reference erod) udaed every 6 year.

3 The frequency of nde egh udae and amle change vare from nde o nde. The nernaonal rade rce ndee and age rce ndee eendure egh are udaed annually, herea he Auralan conumer rce nde currenly udae he ublhed uer level eendure egh every year. Belo he ublhed level he ABS underake regular nvegaon no amle for reve and manenance, udang rucure and egh erodcally o reflec curren eendure aern. The ABS currenly ue an nde change facor o adu drec rce ndee a amle change from he revou erod o he curren erod. The facor calculaed a he rao of o drec ndee n he revou erod, h each nde rereenng he dfferen amle beeen he revou and curren erod. Each facor unque o he rce nde calculaed, comoundng from erod o erod. 3. To-age aroach o aggregaon The mehodology for roducng rce ndee by Naonal Sacal Offce ell defned h nernaonal manual, be racce gude and an acve roducer communy. Counre roduce a range of rce ndee ncludng conumer, roducer, houng, nernaonal rade and labour co. Prce ndee are generally roduced ung arhmec Laeyre-ye ndee a he uer level and arhmec or geomerc ndee of rce or rce relave a he elemenary aggregae level. In addon, here a range of oher rce nde formulae ha ha been develoed, ncludng recen develomen o comle ndee from admnrave daa (e.g. canner daa). The follong econ oulne he dervaon of he o-age aggregaon formulae from andard nde formulae. Furher nformaon of he dervaon can be found n Chaer 3 of he Prce Inde Proceor Sofare CPI manual, IMF (9). Whle he dervaon are generally for he Laeyre nde, n racce he quane ued n he nde generally come from a erod ror o he rce reference erod, reulng n a Young or Loe nde beng calculaed. Therefore, he erm Laeyre or Laeyre-ye ued o generale hee cae. The dervaon reve an arhmec aggregaon aroach 3. 3 Aachmen ho he Geomerc alcaon of he o-age aroach. 3

4 4. Facorng he Laeyre formulae The dervaon of he o-age aroach o aggregaon can be hon by he facoraon of he clac Laeyre Inde rereenaon: I q M q Laeyre q M q M M () Where M he amle of em a me =. Furher deal on noaon ued hn he reor rovded n Aend. The clacal form can be ereed a he nde for a comonen of an nde, called an Elemenary Aggregae () and decomoed no he follong facor: I q q ( Laeyre )... M q M q M M () Le: Eendure hare: q q M M Long Term Prce Relave (LTPR): LTPR... Shor Term Prce Relave (STPR): STPR Th lead o an nermedae, facored form: I LTPR STPR (3) ( Laeyre ) M Follong he alcaon of he aggregae relaonh (ee Aend ): LTPR LTPR (4) Where, eendure hare are rce udaed and recaled h he follong equaon: q LTPR LTPR q LTPR LTPR M M M (5) 4

5 By rce udang he eendure egh, h eacly equal he drec rce nde aroach. The LTPR equvalen o he nde he long erm rce relave of he elemenary aggregae, hch I of he rce amle. Subung (4) no (3), he follong raccal and raccal form of he o-age aroach o aggregaon follo: I LTPR STPR ( Laeyre ) M I LTPR STPR ( Laeyre ) M I LTPR STPR (6) ( Laeyre ) Equaon 6 ho ha he curren erod movemen from erod - o ung a fed bake from erod can be decomoed no an aggregae Long Term Prce Relave from erod o - LTPR, a rce udaed eendure hare and curren erod hor erm rce relave movemen STPR. In order o ncororae he relnkng of rce ndee a arcular on n me ncororang he lnk erod rce nde ( I ), he fnal form of he o-age aroach o aggregaon formulae oulned by he defnon decrbed n able : Table : To-age aroach o aggregaon: (Arhmec formulae) I I LTPR STPR ( Laeyre ) STPR STPR M Where, LTPR LTPR LTPR STPR LTPR STPR ; ; ; LTPR M For all M ;,,3,... 5

6 5. Imlemenaon of he o-age aroach o aggregaon The mlemenaon of he o-age aroach o aggregaon ue a o erod ndo 4. By ung he aggregae relaonh - he rce udaed eendure egh relaonh decrbed n equaon (4) - he aggregaon rocedure able o focu urely on a o erod ndo. One advanage of h mehod here no need o aler or ue bae erod value hen amle change occur. In urn, he reulng nde ranve ung amle change rule dcued n econ 6 o rean he fed bake aroach. 6. Samle change under he o-age aggregaon aroach The curren erod movemen can be ereed n a form hch only requre daa from he revou and curren erod (a hon n equaon 6). Th mlfe he handlng of amle change each erod. Hoever, n order o manan he fed bake conce and no nroduce ba n he nde, rule urroundng he changng of amled em mu be adhered o. A formal roof of amle change mac hn a Laeyre o-age nde are rovded n aend 3 & Addng ne em For he alcaon of amle change for a Laeyre Inde (of he Loe and Young form), he addon of em o a rce amle omeme afer he lnk erod mu be nered n a manner hch equvalen o mung all revou erod rce off he rce amle movemen back unl he lnk erod (=). The rule ha govern h roce are a follo: If a me=b, a ne em added o he rce amle, here b, In order o add an em o he amle, he follong mu be knon:. Prevou erod rce: b. Curren erod rce: b. The quany or eendure egh of em a a on n me, here: o If quany q knon, for any me erod, he key aumon of a Laeyre nde ha quane have no changed beeen he lnk erod and erod, hu q q. 4 Th aer focue on he Laeyre ye (arhmec) aroach o o-age aggregaon. Hoever, ee aachmen for he mang of clac nde formulae (boh arhmec and geormerc) o he o-age aroach o aggregaon. 6

7 o For arhmec alcaon: If em egh, he lnk erod em egh ll be derved by: knon, for any me erod LTPR, here LTPR he elemenary aggregae long erm rce relave equvalen o he nde: aled here. I.The above aumon regardng quany erm alo o If eendure hare knon n relaon o he oher amled roduc, for any me erod, he erod egh of em can be found by: ; and hen aly he ame rocedure led above for M eendure egh. The alcaon of hee rule hen a ne em nered no a amle ll mmc he effec of revere mung a ne rce obervaon off he rce amle movemen and re-calculang he rce nde. 6. Removng em from he bake The removal of em mly he roce of recalng he eendure hare elemen o hey um o. Th occur hn he aggregaon formulae elf, a he eendure egh are caled hn he rce amle M, uch ha he follong eendure hare conran uheld each erod: M 6.3 Samle change h o-age aroach o aggregaon In racce The follong o able llurae he mechanm reen hn he o-age aroach o aggregaon roce. The eamle ue he andard eghed Laeyre Inde, h each lnk erod eendure egh rereened a: q Table : Eamle of an Elemenary Aggregae rce amle n he lnk erod Eendure Wegh: Eendure Share: Prce: STPR Tme = Tme = $ $ $ $ STPR STPR STPR STPR.5 M LTPR LTPR STPR.5.5 In he follong erod, he o-age aroach o aggregaon roce ll remove all mac of addng a ne em ( 3) o a rce amle adherng o he fed bake naure of a rce nde, hle allong he amled em o change. The ne em ha a knon eendure egh and rce a me erod. 7

8 Table 3: Elemenary Aggregae rce amle n he ne erod Samle Change: Removal of em and he addon of em 3, h a knon eendure egh. Eendure Wegh: Eendure Share: Prce: STPR.5 4 / 7 Tme = Tme = $ $ STPR Removed Removed $ Removed Removed / 7 (Ne Iem) 3 $3 3 $ STPR3 3 To-Sage Aggregaon: 4 3 STPR STPR M LTPR LTPR LTPR STPR Drec Inde: (hou nde change facor) I Chaned Inde: I Wh he rce udang and recalng of eendure egh o form eendure hare, he effec of amle change doe no mac he meaure of rce change hch occur n he chaned nde hon n able 3. Th on furher lluraed n grah, hong ha he drec and chaned rce nde ould have been had he amle change no be accouned for. 8

9 7. Removng qualy change mac A key requremen of rce ndee o rce o conan qualy. The nernaonal manual dcu a range of drec and ndrec qualy adumen echnque ha can be aled. A key requremen o enure ha he mac of he qualy change doe no affec he fed bake aroach o nde calculaon. Whn he nde facoraon hon n econ 4 equaon () and (5), he udae of eendure hare a hon o be oble by ung eher he mo recen rce relave or he long erm rce relave. Hoever, f here a change n qualy over me, h ll mean he ra rce obervaon change ll no longer equal he qualy adued nde movemen. In order o caure h change, a qualy adued hor erm rce relave ued: Qualy Adued STPR: STPR Qfacor Where, Qfacor he rooron of qualy change recorded for he em; f no qualy change occur, he Qfacor equal o. In order o enure ha change n qualy do no caue an nde o loe ranvy, he choen aroach hen udang eendure hare egh o ue rce relave. In arcular, he long erm rce relave for each obervaon a choen, a each rce relave meaured on he qualy adued rce movemen of each em. Th conce can be een n he follong eamle of he o aroache o egh udae, ung a large qualy change for llurave uroe: Prce udae aroach: Prce relave udae aroach: q M M q LTPR M M q q LTPR If no qualy change occur, hen he rce relave udaed eendure hare ll equal he ra rce udaed value, a: daa: LTPR Table 4a: Eamle of qualy change on rce amle value Quany Wegh: q q q Prce:. When aled o numercal Tme = Tme = Tme = STPR $ $ $ * [Qual. Adu =] % qualy ncreae = No rce change $ $ Qualy Ad. STPR: STPR $.5 9

10 The o roduc rce amle reurn o he lnk erod rce and he reulng rce nde mu reurn o lnk erod value o be ranve. The Calculaon belo llurae he oucome of each rce udae aroach and he benef of he rce relave udae aroach. Table 4b: Aroache o nde calculaon h qualy change Tme erod : Inde Calculaon: Aroach : Prce Udae Aroach : Prce Relave Udae Eendure Share: ( ) ( ) ( ) ( ) Aggregae STPR & LTPR: STPR.5 (Correc) LTPR.5.5 Tme erod : Inde Calculaon: Eendure Share: A, LTPR Aggregae STPR & LTPR: STPR.5 (Correc) LTPR.5.5 {,} Aroach : Prce Udae Eendure Share: * * * ( ) () ( ) () Aggregae STPR & LTPR: STPR.75 (Incorrec) LTPR Aroach : Prce Relave Udae Eendure Share: 3 3 Aggregae STPR & LTPR: STPR (Correc) LTPR A he eamle ho, he ue of rce o udae eendure egh mean ha he ne em qualy ued o calculae he eendure egh. Alernavely, by ung he qualy adued STPR o udae he eendure egh each erod, he fed bake aroach uheld conen h calculang he rce relave and bae erod eendure hare. Therefore, he aroach recommended here o ue he qualy adued rce relave mehod o udae eendure egh and rean he fed bake eendure hare.

11 8. Analycal meaure on conrbuon Pon conrbuon a meaure of ho much each comonen conrbue o he allgrou rce nde for he curren erod, regardle of ha comonen level n he nde. Pon conrbuon allo for he nde o be decomoed no addve comonen. For eamle, he on conrbuon of a COICOP ub-dvon made u of he um of he on conrbuon from comonen COICOP grou, hch n urn are made u of he um of he comonen clae, hch n urn are made u from he on conrbuon of he elemenary aggregae. Pon conrbuon a combnaon of o hng: he egh of he comonen n he lnk erod, and he rooron by hch ha comonen rce ha changed nce he lnk erod. 8. Calculaon of on conrbuon and he o-age aroach Pon conrbuon are calculaed by akng he rao of he curren erod egh of he comonen, o he egh of he roo nde (n general), and mullyng h rao by he roo nde number. The roo nde he uer level nde ha he on conrbuon meaured agan. I can be calculaed ung eher curren erod value or value from he lnk erod k. The advanage of ung he lnk erod k ha on conrbuon can be calculaed before he roo level nde aggregaed. PC PC PC I P. ROOT ROOT k ROOT IP. ROOT k ROOT ROOT k I P. ROOT k ROOT An Eamle of h een h he calculaon of he on Conrbuon December quarer for he Fru EC n he CPI: Lnk Perod Aroach: PC Dec Dec June FRUIT FRUIT ICPI June CPI Curren Perod Aroach: PC Dec Dec Dec FRUIT FRUIT ICPI Dec CPI

12 Converng h rocedure o he o-age aroach, PC PC PC k I P. ROOT k ROOT k k IP. ROOT k k ROOT I I k k P. P. I P. ROOT k k ROOT I P. I P. PC I LTPR STPR k k P. ROOT For eamle, CPI Pon Conrbuon December quarer for he Fru EC n he CPI: To-age aroach on conrbuon: PC I LTPR STPR Dec June June Se Dec FRUIT CPI FRUIT FRUIT FRUIT Where, June FRUIT he eendure hare of fru n relaon o he roo nde hch he CPI all-grou level n h eamle. 9. Praccal Imlcaon for Naonal Sacal Offce The o-age aroach o aggregaon a robu mehod of aggregang rce value and handlng amle change and qualy adumen. The mehod doe rovde ome raccal challenge, ncludng he develomen of ne bune rocee and yem. One eamle of requrng udaed bune rocee relae o he ably o oberve udaed egh. Th may caue analy o udae or change egh baed on hor run hock n urn ncreang he rk of chan drf and lo of ranvy. In order o accoun for h, NSO mu manan roer amle manenance rocedure o mgae h rk. The ably o make revon o rce ndee ung he o-age aroach alo of nere o he ABS 5. I clear ha he o-age aroach o aggregaon caer for a o erod ndo, o all revon mu be aled o he erod n queon and rogrevely udaed n ubequen erod. Wh he ue of clearly defned bune 5 The ABS announced n, a ar of he PPI reve, ha from he Seember quarer 4, he PPI and ITPI ll be reved o accommodae mroved daa n ubequen quarer. See ABS Ca.no Informaon Paer: Oucome of he Reve of he Producer and Inernaonal Trade Prce Indee,

13 rocee and yem alcaon, NSO ll be able o ncororae revon no rce ndee.. Fuure ork Furher nvegaon by he ABS no he o-age aroach o aggregaon ancaed n fuure. The ABS arcularly keen o underake an economc aemen of he geomerc aggregaon a he uer level of he rce ndee. Deal of ho h mgh be aled hn he o-age aggregaon mehod are gven n Aachmen. The able n Aachmen ho he dervaon of eendure hare and eendure egh from a number of rce nde formulae hn he o-age aggregaon cone. Concluon Th aer ha rogreed he nermedae form of he o-age aroach o aggregaon o he fnal raccal form. The elemenary aggregae long erm rce relave can mly be udaed by he rce udaed eendure hare and hor erm rce relave of each rce obervaon. In urn, h enable a o erod ve of rce obervaon for hor erm aggregaon calculaon, hch hen udae a chaned longer erm elemenary aggregae rce relave and nde allong for amle o be changed n a mle manner each erod. Alo hon he ue of qualy adued rce relave o udae he eendure egh and roduce rce ndee ha rean ranvy hen em quale change. Fnally, ancaed ha he o-age aroach o nde aggregaon ll lead o enablng amle flebly and more ranaren rce nde aggregaon yem alcaon o be develoed. 3

14 Aachmen : Geomerc alcaon of o-age aroach The Geomerc Laeyre rce nde formulae (alo he Geomerc Loe and Geomerc Young nde), can alo ue he o-age aroach o aggregaon. The key dfference beeen he arhmec and geomerc aggregaon ha he eendure egh are no rce udaed from he lnk erod (=) onard. The Geomerc Young nde ued a conen h unary elacy of ubuon (ILO Conumer Prce Inde Manual, chaer,.35, g. 5). A a reul, he eendure egh and eendure hare for geomerc alcaon ll be a follo: ; M M ; for all,,3,... The geomerc formulae ll requre he calculaon of he hor erm rce relave (STPR) o be calculaed geomercally. Follong he calculaon of he geomerc STPR, he nde calculaon done a er he arhmec formulaon. The comlee formulae are decrbed a follo: Table 5: To-age aroach o aggregaon: (Geomerc formulae) I I LTPR STPR ( Geo Laeyre ) ( Geo) ( ) e log STPR Geo STPR e STPR M M Where, LTPR LTPR STPR LTPR STPR ; ; ; M M For all M ;,,3,... A able 6 ho, by akng logarhm of he rce relave value, a geomerc aggregaon can be ereed n an addve form. Th form can lead o greaer flebly hn yem baed on h roce. 4

15 Aachmen : Inde formulae rereened under he o-age aroach o aggregaon: In addon o he amle change able of he o-age aroach o aggregaon, a core arbue of he mehod he ably o harmone he alcaon of a ue of rce ndee ued n racce under o dncve funcon Arhmec and Geomerc. Th done by defnng each rce nde varan ued n racce by he eendure egh allocaed o hem. Table 6 llurae he eendure egh of varou rce nde and elemenary aggregae ndee hch can be aled hn he o-age aggregaon frameork. 5

16 Table 6: Mang clacal Prce nde formulae o o-age aroach o aggregaon Prce Inde Tye: Eendure Wegh & Eendure Share for To-Sage Aggregaon: Arhmec o-age aggregaon: I LTPR STPR ( Laeyre ) M Arhmec Laeyre-ye Inde: Laeyre: (n eendure hare and quany form): I q M q Laeyre q M q M M q q LTPR q LTPR M Includng he Generc Cae of he: I Loe Inde: q b b M q Loe b b q M q M M Young Inde: I q b b Young b b M q M Carl Inde (Average of Prce Relave): Loe: q b q LTPR b b q LTPR M Young: q b b q LTPR b b b b q LTPR M Carl: I Carl M M LTPR LTPR M Duo Inde (Rao of Average Prce): Duo: I Duo M M M M M M LTPR LTPR M 6

17 Geomerc o-age aggregaon: I I LTPR STPR ( GeoLaeyre ) M Where, ( ) e log STPR Geo STPR e STPR M M Geomerc Laeyre (alo Young and Loe) Inde: I Laeyre Jevon Inde: q M q M (Equally Weghed Geomerc average of rce relave) I Jevon M M Geomerc Laeyre: q q q M For all,,,... Jevon: M For all,,,... 7

18 Reference: Prce Inde Proceor ofare uer gude. h://.unece.org/fleadmn/dam/a/sw_c_/cpi_uer_manual_augu_ 9.df Armknech, P and Slver, M (), Po-Laeyre: The Cae for a Ne Formula for Comlng Conumer Prce Indee. Avalable a h://.mf.org/eernal/ub/f///5.df Auralan Bureau of Sac (), Generc Sacal Bune Proce Model (GSBPM) and conrbuon o modellng bune rocee- eerence from he Auralan Bureau of Sac. Preened a he Tenh Managemen Semnar for he Head of Naonal Sacal Offce n Aa and he Pacfc ILO, IMF, OECD, Euroa, UNECE and he World Bank (4), Conumer Prce Inde Manual: Theory and Pracce. Geneva. Avalable a: h://.lo.org/ublc/englh/bureau/a/gude/c/nde.hm ILO, IMF, OECD, Euroa, UNECE and he World Bank (4), Producer Prce Inde Manual. Geneva. Avalable a:.mf.org/eernal/n/a/eg/nde.hm Inernaonal Moneary Fund (IMF), Prce Inde Proceor Sofare CPI manual, IMF (9) Chaer 3. Avalable a h://.unece.org/fleadmn/dam/a/sw_c_/cpi_uer_manual_augu_ 9.df UNECE (9), Generc Sacal Bune Proce Model, Veron 4.. UNECE Secrearae. Avalable a h://.unece.org/a/laform/dlay/me/generc+sacal+bune+pr oce+model+paer UNECE (3), Generc Sacal Informaon Model. Acceed on February 3. Avalable a: h://.unece.org/a/laform/age/veage.acon?ageid=

19 Aend. Noaon Ued Whn Reor: I X Aggregae Prce nde a me, here X can be an Elemenary Aggregae () or Uer level nde. q M Prce of roduc a me Quany of roduc a me Se of roduc hn he rce amle a me STPR Shor erm rce relave of roduc (or ) a me LTPR Long erm rce relave of roduc (or ) a me PC Lnk erod eendure egh of roduc (or ) a me Prce udaed eendure egh of roduc (or ) a me Lnk erod eendure hare of roduc (or ) a me Prce udaed eendure hare of roduc (or ) a me (alo knon a a hybrd eendure hare) Pon Conrbuon of (or Roo level) a me. Aggregae Relaonh Dervaon: In order o derve he aggregae eendure relaonh hon n equaon (4) hn he aer: LTPR LTPR Le:, he follong roere mu aly: The lnk erod eendure hare: ; M The Elemenary Aggregae long erm rce relave: LTPR STPR STPR... STPR ; LTPR LTPR M ; here Prce udaed and recaled eendure hare (alo knon a hybrd eendure hare): LTPR LTPR M ; and The Conran a any erod ha he um of all a rce amle eendure hare mu equal, e. M 9

20 Therefore he follong mu hold: LTPR LTPR LTPR LTPR M M M LTPR LTPR LTPR LTPR M M M LTPR LTPR M LTPR LTPR M LTPR LTPR M Inuvely, h oucome decrbe he mechanm of rce udang and recalng (normalaon), h each h roduc rce movemen dvergence from he aggregae meaure beng caured by he rce udaed eendure hare. 3. Samle change Provng Samle change manan a Laeyre Inde ere: A hon hn he aer, he o age aroach o aggregaon enable he Laeyre nde roduced o rean rce nde ranvy hen rce amle change. In order o formally ho he rce nde back ere unchanged h he addon of ne em hn he rce amle under he condon reened n econ 6 of he reor, he follong can be een: If e have an elemenary aggregae rce nde: I LTPR STPR STPR STPR STPR... If a me b, a ne e of em E b ha been added o he rce amle M b uch ha M M \ E b b b Then, n order o ho ha he back ere nde, I STPR STPR... STPR b b ha no changed, e mu ho ha for he arhmec Laeyre rce nde: b b b STPR STPR : {,.., b } (a) Mb Ung amle change rule (ee econ 6), e kno ha for all E : b STPR b STPR ; b LTPR b LTPR ; b b b LTPR

21 If e eand he um of equaon (a): Mb STPR b b STPR STPR b b b b Mb Eb STPR STPR b b b b Mb Eb b b b b STPR STPR b b b b M b Mb E b Eb Mb Eb Mb STPR STPR b b b b b b Mb Eb Eb If e hen mully equaon (b) by: Mb STPR b b Mb Mb b b ; e ll ge: b b b b b Mb b b STPR STPR b Mb E b Mb Eb Mb b b b M b b b Eb STPR b b b STPR b M b Mb Eb Mb Mb b b M b b b Eb STPR b b STPR b Mb Eb Mb STPR STPR b b Mb b b Mb Eb b b b Mb Eb b Mb STPR STPR : {,,..., b } Mb b b b Therefore he addon of rce obervaon o he rce amle ung he rule decrbed n econ 6 reul n no change o he rce nde back ere, hu uholdng he Laeyre rce nde ranvy roery for he o age aroach o rce nde aggregaon. (b)

22 4. Samle change Provng Samle change manan a Geomerc Laeyre Inde ere: Smlar o aend 3, he follong ll ho ha he o age aroach o aggregaon alo manan geomerc Laeyre rce nde ranvy. Th can be hon a follo: If e have an elemenary aggregae rce nde: I LTPR STPR STPR STPR STPR... If a me b, a ne e of em E b ha been added o he rce amle M b uch ha M M \ E b b b Then, n order o ho ha he back ere nde, I STPR STPR... STPR b b ha no changed, e mu ho ha for he geomerc Laeyre rce nde: b b b STPR : {,.., } b b M (c) b Ung amle change rule (ee econ 6), e kno ha for all geomerc cae: E, n he b STPR b STPR ; b LTPR b LTPR b If e eand he roduc of equaon (c): Mb b b b b b b b e loge log b e b Mb \ E b E b e log log b b b b e b e STPR Mb \ E b Eb e b b b b log E e STPR b b Mb \ E b (d) If e aly he follong fac o equaon (d): b b b Mb b b b b b b Mb M M \ E ;

23 We ge he follong: Mb b b b b b b b M b b e log E b b e STPR b b M b Mb b M b e b b b b M b b log b e STPR b b M b Mb b M b e log e e b b b b Mb b e b b M b Mb b Mb b b Mb b b log b e b M b STPR STPR b E b STPR b b E b b b E b e b b b M b b log b b e STPR E b b Mb Mb (e) Therefore, n ho (c) rue, e mu ho: b b b M b b b E b b Mb Mb (f) Thu, eandng he erm of (f) e fnd ha e fnd he follong: b b b b b b b b E b Mb Mb Mb Mb 3

24 b b b b b b b b E b Mb Mb Mb Mb b b b b b b E b Mb Mb Mb b b Mb b b Mb Eb b b b Mb Mb b b b b b Mb Eb Mb (g) If e kno ha a key fac for all em erod (=) eendure egh reman fed, Subung h relaonh no (g),e ge: Mb b b b b Mb Mb Eb b b Mb Mb b b Mb Eb M of Laeyre ndee ha he lnk b : {,,..., } Thu, a e ve hon (f) rue for all, equaon (e) become: Mb b b e log b STPR STPR b b e Thu he addon of ne roduc o a geomerc Laeyre rce nde accordng o he rule ecfed n econ 6 of he reor ll no mac he rce nde back ere, for all value of {,,..., b }. 4