1 AN INTRODUCTION TO CONSUMER PRICE INDEX METHODOLOGY

Transcription

1 AN INTRODUCTION TO CONSUMER PRICE INDEX METHODOLOGY. A prce dex s a measure of the proportoate, or percetage, chages a set of prces over tme. A cosumer prce dex (CPI) measures chages the prces of goods ad servces that households cosume. Such chages affect the real purchasg power of cosumers comes ad ther welfare. As the prces of dfferet goods ad servces do ot all chage at the same rate, a prce dex ca oly reflect ther average movemet. A prce dex s typcally assged a value of uty, or 00, some referece perod ad the values of the dex for other perods of tme are teded to dcate the average proportoate, or percetage, chage prces from ths prce referece perod. Prce dces ca also be used to measure dffereces prce levels betwee dfferet ctes, regos or coutres at the same pot tme..2 Much of ths maual ad the assocated ecoomc lterature o prce dces s cocered wth two basc questos: Exactly what set of prces should be covered by the dex? What s the most approprate way whch to average ther movemets? These two questos are addressed the early sectos of ths troducto..3 Cosumer prce dces (CPIs) are dex umbers that measure chages the prces of goods ad servces purchased or otherwse acqured by households, whch households use drectly, or drectly, to satsfy ther ow eeds ad wats. Cosumer prce dces ca be teded to measure ether the rate of prce flato as perceved by households, or chages ther cost of lvg (that s, chages the amouts that the households eed to sped order to mata ther stadard of lvg). There eed be o coflct betwee these two objectves. I practce, most CPIs are calculated as weghted averages of the percetage prce chages for a specfed set, or basket, of cosumer products, the weghts reflectg ther relatve mportace household cosumpto some perod. Much depeds o how approprate ad tmely the weghts are..4 Ths chapter provdes a geeral troducto to, ad overvew of, the methodology for complg CPIs. It provdes a summary of the relevat theory ad practce of dex umber complato that s teded to facltate the readg ad uderstadg of the detaled chapters that follow, some of whch are evtably qute techcal. It descrbes all the varous steps volved CPI complato startg wth the basc cocept, defto ad purpose of a CPI, followed by the samplg procedures ad survey methods used to collect ad process the prce data, ad fshg wth a summary of the actual calculato of the dex ad ts dssemato..5 A troductory presetato of CPI methodology has to start wth the basc cocept of a CPI ad the uderlyg dex umber theory, cludg the propertes ad behavour of the varous kds of dex umber that are, or mght be, used for CPI purposes. I prcple, t s ecessary to settle what type of dex to calculate before gog o to cosder the best way whch to estmate t practce, takg accout of the resources avalable.

2 .6 The ma topcs covered ths chapter are as follows: the orgs ad uses of CPIs; basc dex umber theory, cludg the axomatc ad ecoomc approaches to CPIs; elemetary prce dces ad aggregate CPIs; the trasactos, actvtes ad households covered by CPIs; the collecto ad processg of the prces, cludg adjustg for qualty chage; the actual calculato of the CPI; potetal errors ad bas; orgazato, maagemet ad dssemato polcy. I cotrast, ths maual, the chapters dealg wth dex theory come later o; thus the presetato ths chapter does ot follow the same order as the correspodg chapters of the maual..7 It s ot the purpose of ths troducto to provde a complete summary of the cotets of the maual. The objectve s rather to provde a short presetato of the core methodologcal ssues wth whch readers eed to be acquated before tacklg the detaled chapters that follow. Some specal topcs, such as the treatmet of certa dvdual products whose prces caot be drectly observed, are ot cosdered here as they are ot cetral to CPI methodology. The orgs ad uses of cosumer prce dces.8 CPIs must serve a purpose. The precse way whch they are defed ad costructed depeds very much o what they are meat to be used for, ad by whom. As explaed Chapter 5, CPIs have a log hstory datg back to the eghteeth cetury. Laspeyres ad Paasche dces, whch are stll wdely used today, were frst proposed the 870s. They are explaed below. The cocept of the cost of lvg dex was troduced early the tweteth cetury..9 Tradtoally, oe of the ma reasos for complg a CPI was to compesate wageearers for flato by adjustg ther wage rates proporto to the percetage chage the CPI, a procedure kow as dexato. For ths reaso, offcal CPIs teded to become the resposblty of mstres of labour, but most are ow compled by atoal statstcal offces. A CPI that s specfcally teded to be used to dex wages s kow as a compesato dex..0 CPIs have three mportat characterstcs. They are publshed frequetly, usually every moth but sometmes every quarter. They are avalable quckly, usually about two weeks after the ed of the moth or quarter. They are also usually ot revsed. CPIs ted to be closely motored ad attract a lot of publcty.. As CPIs provde tmely formato about the rate of flato, they have also come to be used for a wde varety of purposes addto to dexg wages. For example: CPIs are wdely used to dex pesos ad socal securty beefts. CPIs are also used to dex other paymets, such as terest paymets or rets, or the prces of bods. 2

3 CPIs are also commoly used as a proxy for the geeral rate of flato, eve though they measure oly cosumer flato. They are used by some govermets or cetral baks to set flato targets for purposes of moetary polcy. The prce data collected for CPI purposes ca also be used to comple other dces, such as the prce dces used to deflate household cosumpto expedtures atoal accouts, or the purchasg power partes used to compare real levels of cosumpto dfferet coutres..2 These vared uses ca create coflcts of terest. For example, usg a CPI as a dcator of geeral flato may create pressure to exted ts coverage to clude elemets that are ot goods ad servces cosumed by households, thereby chagg the ature ad cocept of the CPI. It should also be oted that because of the wdespread use of CPIs to dex a wde varety of paymets ot just wages, but socal securty beefts, terest paymets, prvate cotracts, etc. extremely large sums of moey tur o ther movemets, eough to have a sgfcat mpact o the state of govermet faces. Thus, small dffereces the movemets of CPIs resultg from the use of slghtly dfferet formulae or methods ca have cosderable facal mplcatos. CPI methodology s mportat practce ad ot just theory. Choce of dex umber.3 The frst questo s to decde o the kd of dex umber to use. The extesve refereces dealg wth dex theory the bblography reflect the fact that there s a very large lterature o ths subject. May dfferet kds of mathematcal formulae have bee proposed over the past two cetures. Whle there may be o sgle formula that would be preferred all crcumstaces, most ecoomsts ad complers of CPIs seem to be agreed that, prcple, the dex formula should belog to a small class of dces called superlatve dces. A superlatve dex may be expected to provde a approxmato to a cost of lvg dex. A characterstc feature of a superlatve dex s that t treats the prces ad quattes both perods beg compared symmetrcally. Dfferet superlatve dces ted to have smlar propertes, yeld smlar results ad behave very smlar ways. Because of ther propertes of symmetry, some kd of superlatve dex s also lkely to be see as desrable, eve whe the CPI s ot meat to be a cost of lvg dex..4 Whe a mothly or quarterly CPI s frst publshed, however, t s varably the case that there s ot suffcet formato o the quattes ad expedtures the curret perod to make t possble to calculate a symmetrc, or superlatve, dex. Whle t s ecessary to resort to secod-best alteratves practce, beg able to make a ratoal choce betwee the varous possbltes meas havg a clear dea of the target dex that would be preferred prcple. The target dex ca have a cosderable fluece o practcal matters such as the frequecy wth whch the weghts used the dex should be updated..5 A comprehesve, detaled, rgorous ad up-to-date dscusso of the relevat dex umber theory s provded Chapters 5 to 23 of the maual. The followg sectos provde a summary of ths materal. Proofs of the varous propostos or theorems stated ths chapter are to be foud the later chapters, to whch the reader may refer for further explaato. Prce dces based o baskets of goods ad servces.6 The purpose of a dex umber may be explaed as comparg the values of households expedtures o cosumer goods ad servces two tme perods. Kowg that 3

4 expedtures have creased by 5 per cet s ot very formatve f we do ot kow how much of ths chage s attrbutable to chages the prces of the goods ad servces, ad how much to chages the quattes purchased. The purpose of a dex umber s to decompose proportoate or percetage chages value aggregates to ther overall compoets of prce ad quatty chage. A CPI s teded to measure the prce compoet of the chage households cosumpto expedtures. Oe way to do ths s to measure the chage the value of a aggregate, holdg the quattes costat. Lowe dces.7 Oe very wde, ad popular, class of prce dces s obtaed by defg the dex as the percetage chage, betwee the perods compared, the total cost of purchasg a gve set of quattes, geerally descrbed as a basket. The meag of such a dex s easy to grasp ad to expla to users. Ths class of dex s called a Lowe dex ths maual, after the dex umber poeer who frst proposed t 823 (see Chapter 5). Most statstcal offces make use of some kd of Lowe dex practce..8 Let there be products the basket wth prces p ad quattes q, ad let the two perods compared be 0 ad t. The Lowe dex, P Lo, s defed as follows: P = Lo = p q p t 0 q.9 I prcple, ay set of quattes could serve as the basket. The basket does ot have to be restrcted to the quattes purchased oe or other of the two perods compared, or deed ay actual perod of tme. The quattes could, for example, be arthmetc or geometrc averages of the quattes the two perods. For practcal reasos, the basket of quattes used for CPI purposes usually has to be based o a survey of household cosumpto expedtures coducted a earler perod tha ether of the two perods whose prces are compared. For example, a mothly CPI may ru from Jauary 2000 owards, wth Jauary 2000 = 00, but the quattes may be derved from a aual expedture survey made 997 or 998, or eve spag both those years. As t takes a log tme to collect ad process expedture data, there s usually a cosderable tme lag before such data ca be troduced to the calculato of CPIs. The basket may also refer to a year, whereas the dex may be compled mothly or quarterly..20 The perod whose quattes are actually used a CPI s descrbed as the weght referece perod ad t wll be deoted as perod b. Perod 0 s the prce referece perod. As just oted, b s lkely to precede 0, at least whe the dex s frst publshed, ad ths s assumed here, but b could be ay perod, cludg oe betwee 0 ad t, f the dex s calculated some tme after t. The Lowe dex usg the quattes of perod b ca be wrtte as follows: P Lo t b p q = t 0 0b 0b ( p p ) s s = 0 b = p q = where 0 b p q (.) p q The dex ca be wrtte, ad calculated, two ways: ether as the rato of two value aggregates, or as a arthmetc weghted average of the prce ratos, or prce relatves, p t / p 0, for the dvdual products usg the hybrd expedture shares s 0b as weghts. The = 0 b 4

5 expedtures are descrbed as hybrd because the prces ad quattes belog to two dfferet tme perods, 0 ad b respectvely. The hybrd weghts may be obtaed by updatg the actual expedture shares perod b, amely p b q b / p b q b, for the prce chages occurrg betwee perods b ad 0 by multplyg them by the prce relatves b ad 0, amely p 0 / p b. Lowe dces are wdely used for CPI purposes. Laspeyres ad Paasche dces.2 Ay set of quattes could be used a Lowe dex, but there are two specal cases whch fgure very prometly the lterature ad are of cosderable mportace from a theoretcal pot of vew. Whe the quattes are those of the prce referece perod, that s whe b = 0, the Laspeyres dex s obtaed. Whe quattes are those of the other perod, that s whe b = t, the Paasche dex s obtaed. It s ecessary to cosder the propertes of Laspeyres ad Paasche dces, ad also the relatoshps betwee them, more detal..22 The Laspeyres prce dex, P L, s defed as: P L = t 0 p q = 0 0 p q = t 0 ( p p ) = s 0 (.2) where s 0 deotes the share of the actual expedture o commodty perod 0: that s, p 0 q 0 / p 0 q The Paasche dex, P P, s defed as: t t p q = = 0 t = p q = t 0 ( p p ) t P P s (.3) where s t deotes the actual share of the expedture o commodty perod t; that s, p t q t / p t q t. Notce that the Paasche dex s a weghted harmoc average of the prce relatves that uses the actual expedture shares the later perod t as weghts. It follows from equato (.) that the Paasche dex ca also be expressed as a weghted arthmetc average of the prce relatves usg hybrd expedture weghts, whch the quattes of t are valued at the prces of 0. Decomposg curret value chages usg Laspeyres ad Paasche dces.24 Laspeyres ad Paasche quatty dces are defed a smlar way to the prce dces, smply by terchagg the p ad q values formulae (.2) ad (.3). They summarze chages over tme the flow of quattes of goods ad servces cosumed. A Laspeyres quatty dex values the quattes at the fxed prces of the earler perod, whle the Paasche quatty dex uses the prces of the later perod. The rato of the values of the expedtures two perods (V) reflects the combed effects of both prce ad quatty chages. Whe Laspeyres ad Paasche dces are used, the value chage ca be exactly decomposed to a prce dex tmes a quatty dex oly f the Laspeyres prce (quatty) dex s matched wth the Paasche quatty (prce) dex. Let P La ad Q La deote the Laspeyres prce ad quatty dces ad let P Pa ad Q Pa deote the Paasche prce ad quatty dces: the, P La Q Pa V ad P Pa Q La V. 5

6 .25 Suppose, for example, a tme seres of household cosumpto expedtures at curret prces the atoal accouts s to be deflated by a prce dex to show chages real cosumpto. To geerate a seres of cosumpto expedtures at costat base perod prces (whose movemets are detcal wth those of the Laspeyres volume dex), the cosumpto expedtures at curret prces must be deflated by a seres of Paasche prce dces. Ratos of Lowe ad Laspeyres dces.26 The Lowe dex s trastve. The rato of two Lowe dces usg the same set of q b values s also a Lowe dex. For example, the rato of the Lowe dex for perod t+ wth prce referece perod 0 dvded by that for perod t also wth prce referece perod 0 s: = = = Lo t, b 0 t = + = p q b t q p b q p = = P t (.4) = + t b 0 b + t b Ths s a Lowe dex for perod t+ wth perod t as the prce referece perod. Ths kd of dex s, fact, wdely used to measure short-term prce movemets, such as betwee t ad t+, eve though the quattes may date back to some much earler perod b. p q p q p q.27 A Lowe dex ca also be expressed as the rato of two Laspeyres dces. For example, the Lowe dex for perod t wth prce referece perod 0 s equal to the Laspeyres dex for perod t wth prce referece perod b dvded by the Laspeyres dex for perod 0 also wth prce referece perod b. Thus, = 0 b t b t b b b p q p q p q = = = P P Lo = = = P p q p q p q = 0 b = b b t La 0 La (.5) Updated Lowe dces.28 It s useful to have a formula that eables a Lowe dex to be calculated drectly as a cha dex, whch the dex for perod t+ s obtaed by updatg the dex for perod t. Because Lowe dces are trastve, the Lowe dex for perod t+ wth prce referece perod 0 ca be wrtte as the product of the Lowe dex for perod t wth prce referece perod 0 multpled by the Lowe dex for perod t+ wth prce referece perod t. Thus, t+ b t b t+ b p q pq p q = = = = 0 b 0 b t b p q p q pq = = = (.6) t b pq t+ = p tb = s t 0 b = p p q = tb where the expedture weghts s are hybrd weghts defed as: s tb p q t b = p q t b (.7) 6

7 .29 Hybrd weghts of the kd defed equato (.7) are ofte descrbed as prceupdated weghts. They ca be obtaed by adjustg the orgal expedture weghts p b q b / p b q b by the prce relatves p t / p b. By prce-updatg the expedture weghts from b to t ths way, the dex betwee t ad t+ ca be calculated drectly as a weghted average of the prce relatves p t+ / p t wthout referrg back to the prce referece perod 0. The dex ca the be lked o to the value of the dex the precedg perod t. Iterrelatoshps betwee fxed basket dces.30 Cosder frst the terrelatoshp betwee the Laspeyres ad the Paasche dces. A well-kow result dex umber theory s that f the prce ad quatty chages (weghted by values) are egatvely correlated, the the Laspeyres dex exceeds the Paasche dex. Coversely, f the weghted prce ad quatty chages are postvely correlated, the the Paasche dex exceeds the Laspeyres dex. The proof s gve Appedx 5. of Chapter 5..3 As cosumers are usually prce-takers, they typcally react to prce chages by substtutg goods or servces that have become relatvely cheaper for those that have become relatvely dearer. Ths s kow as the substtuto effect, a pheomeo that fgures prometly ths maual ad the wder lterature o dex umbers. Substtuto teds to geerate a egatve correlato betwee the prce ad quatty relatves, whch case the Laspeyres dex s greater tha the Paasche dex, the gap betwee them tedg to wde over tme..32 I practce, however, statstcal offces do ot calculate Laspeyres or Paasche dces but stead usually calculate Lowe dces as defed equato (.). The questo the arses of how the Lowe dex relates to the Laspeyres ad Paasche dces. It s show the text of Chapter 5, ad also Appedx 5.2, that f there are persstet log-term treds relatve prces ad f the substtuto effect s operatve, the Lowe dex wll ted to exceed the Laspeyres, ad therefore also the Fsher ad the Paasche dces. Assumg that perod b precedes perod 0, the rakg uder these codtos wll be: Lowe Laspeyres Fsher Paasche Moreover, the amout by whch the Lowe exceeds the other three dces wll ted to crease, the further back tme perod b s relato to perod The postog of perod b s crucal. Gve the assumptos about log-term prce treds ad substtuto, a Lowe dex wll ted to crease as perod b s moved backwards tme, or to decrease as perod b s moved forwards tme. Whle b may have to precede 0 whe the dex s frst publshed, there s o such restrcto o the postog of b as prce ad quatty data become avalable for later perods wth passage of tme. Perod b ca the be moved forwards. If b s postoed mdway betwee 0 ad t, the quattes are lkely to be equ-represetatve of both perods, assumg that there s a farly smooth trasto from the relatve quattes of 0 to those of t. I these crcumstaces, the Lowe dex s lkely to be close to the Fsher ad other superlatve dces, ad caot be presumed to have ether a upward or a dowward bas. These pots are elaborated further below, ad also Chapter It s mportat that statstcal offces take these relatoshps to cosderato decdg upo ther polces. There are obvously practcal advatages ad facal savgs from cotug to make repeated use over may years of the same fxed set of quattes to calculate a CPI. However, the amout by whch such a CPI exceeds some coceptually 7

8 preferred target dex, such as a cost of lvg dex (COLI), s lkely to get steadly larger the further back tme the perod b to whch the quattes refer. Most users are lkely to terpret the dfferece as upward bas. A large bas may uderme the credblty ad acceptablty of the dex. Youg dex.35 Istead of holdg costat the quattes of perod b, a statstcal offce may calculate a CPI as a weghted arthmetc average of the dvdual prce relatves, holdg costat the reveue shares of perod b. The resultg dex s called a Youg dex ths maual, aga after aother dex umber poeer. The Youg dex s defed as follows: t b b b p b pq PYo s 0 where s (.8) = p b b p q = I the correspodg Lowe dex, equato (.), the weghts are hybrd reveue shares that value the quattes of b at the prces of 0. As already explaed, the prce referece perod 0 s usually later tha the weght referece perod b because of the tme eeded to collect ad process ad reveue data. I that case, a statstcal offce has the choce of assumg that ether the quattes of perod b rema costat or the expedture shares perod b rema costat. Both caot rema costat f prces chage betwee b ad 0. If the expedture shares actually remaed costat betwee perods b ad 0, the quattes must have chaged versely respose to the prce chages, whch mples a elastcty of substtuto of uty..36 Whereas there s a presumpto that the Lowe dex wll ted to exceed the Laspeyres dex, t s more dffcult to geeralze about the relatoshp betwee the Youg dex ad the Laspeyres dex. The Youg could be greater or less tha the Laspeyres depedg o how sestve the quattes are to chages relatve prces. It s show Chapter 5 that wth hgh elastctes of substtuto (greater tha uty) the Youg wll ted to exceed the Laspeyres, whereas wth low elastctes the Youg wll ted to be less tha the Laspeyres..37 As explaed later ths chapter, the Lowe dex may be preferred to the Youg dex because the Youg dex has some udesrable propertes that cause t to fal some crtcal dex umber tests (see also Chapter 6). Geometrc Youg, Laspeyres ad Paasche dces.38 I the geometrc verso of the Youg dex, a weghted geometrc average s take of the prce relatves usg the expedture shares of perod b as weghts. It s defed as follows: b s t p P GYo 0 (.9) = p where s b s defed as above. The geometrc Laspeyres s the specal case whch b = 0; that s, the expedture shares are those of the prce referece perod 0. Smlarly, the geometrc Paasche uses the expedture shares of perod t. It should be oted that these geometrc dces caot be expressed as the ratos of value aggregates whch the quattes are fxed. They are ot basket dces ad there are o couterpart Lowe dces..39 It s worth recallg that for ay set of postve umbers the arthmetc average s greater tha, or equal to, the geometrc average, whch tur s greater tha, or equal to, the 8

9 harmoc average, the equaltes holdg oly whe the umbers are all equal. I the case of utary cross-elastctes of demad ad costat expedture shares, the geometrc Laspeyres ad Paasche dces cocde. I ths case, the rakg of the dces must be the ordary Laspeyres the geometrc Laspeyres ad Paasche the ordary Paasche, because the dces are, respectvely, arthmetc, geometrc ad harmoc averages of the same prce relatves whch all use the same set of weghts..40 The geometrc Youg ad Laspeyres dces have the same formato requremets as ther ordary arthmetc couterparts. They ca be produced o a tmely bass. Thus, these geometrc dces must be treated as serous practcal possbltes for purposes of CPI calculatos. As explaed later, the geometrc dces are lkely to be less subject tha ther arthmetc couterparts to the kds of dex umber bases dscussed later sectos. Ther ma dsadvatage may be that, because they are ot fxed basket dces, they are ot so easy to expla or justfy to users. Symmetrc dces.4 A symmetrc dex s oe that makes equal use of the prces ad quattes both the perods compared ad treats them a symmetrc maer. There are three partcular symmetrc dces that are wdely used ecoomc statstcs. It s coveet to troduce them at ths pot. As already oted, these three dces are also superlatve dces..42 The frst s the Fsher prce dex, P F, defed as the geometrc average of the Laspeyres ad Paasche dces; that s, P P P (.0) F L P.43 The secod s the Walsh prce dex, P W. Ths s a basket dex whose quattes cosst of geometrc averages of the quattes the two perods; that s, = W P = p p t 0 q q t t 0 q q 0 (.) By takg a geometrc rather tha a arthmetc average of the quattes, equal weght s gve to the relatve quattes both perods. The quattes the Walsh dex ca be regarded as beg equ-represetatve of both perods..44 The thrd dex s the Törqvst prce dex, P T, defed as a geometrc average of the prce relatves weghted by the average expedture shares the two perods. P T = t ( p p ) = 0 σ (.2) where σ s the arthmetc average of the share of expedture o product the two perods. t s + s = 2 0 σ (.3) where the s values are defed as equatos (.2) ad (.3) above..45 The theoretcal attractos of these dces become more apparet the followg sectos o the axomatc ad ecoomc approaches to dex umbers. 9

10 Fxed base versus cha dces.46 Ths topc s examed Chapter 5. Whe a tme seres of Lowe or Laspeyres dces s calculated usg a fxed set of quattes, the quattes become progressvely out of date ad creasgly rrelevat to the later perods for whch prces are beg compared. The base perod, whch quattes are set, has to be updated sooer or later ad the ew dex seres lked to the old. Lkg s evtable the log ru..47 I a cha dex, each lk cossts of a dex whch each perod s compared wth the precedg oe, the weght ad prce referece perods beg moved forward each perod. Ay dex umber formula ca be used for the dvdual lks a cha dex. For example, t s possble to have a cha dex whch the dex for t+ o t s a Lowe dex defed as p t+ q t-j / p t q t-j. The quattes refer to some perod that s j perods earler tha the prce referece perod t. The quattes move forward oe perod as the prce referece perod moves forward oe perod. If j = 0, the cha Lowe becomes a cha Laspeyres, whle f j =, t becomes a cha Paasche..48 The CPIs some coutres are, fact, aual cha Lowe dces of ths geeral type, the quattes referrg to some year or years that precede the prce referece perod 0 by a fxed perod. For example, the 2 mothly dces from Jauary 2000 to Jauary 200, wth Jauary 2000 as the prce referece perod, could be Lowe dces based o prceupdated expedtures for 998. The 2 dces from Jauary 200 to Jauary 2002 are the based o prce updated expedtures for 999, ad so o..49 The expedtures lag behd the Jauary prce referece perod by a fxed terval, movg forward a year each Jauary as the prce referece perod moves forward oe year. Although, for practcal reasos, there has to be a tme lag betwee the quattes ad the prces whe the dex s frst publshed, t s possble to recalculate the mothly dces for the curret year later, usg curret expedture data whe they evetually become avalable. I ths way, t s possble for the log-ru dex to be a aually chaed mothly dex, wth cotemporaeous aual weghts. Ths method s explaed more detal Chapter 9. It s used by oe statstcal offce..50 A cha dex has to be path depedet. It must deped o the prces ad quattes all the terveg perods betwee the frst ad last perods the dex seres. Path depedecy ca be advatageous or dsadvatageous. Whe there s a gradual ecoomc trasto from the frst to the last perod, wth smooth treds relatve prces ad quattes, chag wll ted to reduce the dex umber spread betwee the Lowe, Laspeyres ad Paasche dces, thereby makg the movemets the dex less depedet o the choce of dex umber formula..5 If there are fluctuatos the prces ad quattes the terveg perods, however, chag may ot oly crease the dex umber spread but also dstort the measure of the overall chage betwee the frst ad last perods. For example, suppose all the prces the last perod retur to ther tal levels perod 0, whch mples that they must have fluctuated betwee; a cha Laspeyres dex wll ot retur to 00. It wll ted to be greater tha 00. If the cycle s repeated wth all the prces perodcally returg to ther orgal levels, a cha Laspeyres dex wll ted to drft further ad further above 00 eve though there may be o log-term upward tred the prces. Chag s therefore ot advsed whe prces fluctuate. Whe mothly prces are subject to regular ad substatal 0

11 seasoal fluctuatos, for example, mothly chag caot be recommeded. Seasoal fluctuatos cause serous problems, whch are aalysed Chapter 22. Whle a umber of coutres update ther expedture weghts aually, the 2-mothly dces wth each year are ot cha dces but Lowe dces usg fxed aual quattes..52 The Dvsa dex. If the prces ad quattes are cotuous fuctos of tme, t s possble to partto the chage ther total value over tme to prce ad quatty compoets followg the method of Dvsa. As show Chapter 5, the Dvsa dex may be derved mathematcally by dfferetatg value (.e. prce multpled by quatty) wth respect to tme to obta two compoets: a relatve-value-weghted prce chage ad a relatve-value-weghted quatty chage. These two compoets are defed to be prce ad quatty dces, respectvely. The Dvsa s essetally a theoretcal dex. I practce, prces ca be recorded oly at dscrete tervals, eve f they vary cotuously wth tme. A cha dex may, however, be regarded as a dscrete approxmato to a Dvsa. The Dvsa dex tself offers lmted practcal gudace about the kd of dex umber formula to choose for the dvdual lks a cha dex. Axomatc ad stochastc approaches to dex umbers.53 Varous axomatc approaches to dex umbers are explaed Chapter 6. These approaches seek to determe the most approprate fuctoal form for a dex by specfyg a umber of axoms, or tests, that the dex ought to satsfy. They throw lght o the propertes possessed by dfferet kds of dces, some of whch are ot tutvely obvous. Idces that fal to satsfy certa basc or fudametal axoms, or tests, may be rejected completely because they are lable to behave uacceptable ways. A axomatc approach may also be used to rak dces o the bass of ther desrable, ad udesrable, propertes. Frst axomatc approach.54 The frst approach s the tradtoal test approach poeered by Irvg Fsher. The prce ad quatty dces are defed as fuctos of the two vectors of prces ad two vectors of quattes relatg to the two perods compared. The prces ad quattes are treated as depedet varables, whereas the ecoomc approach to dex umbers cosdered later ths chapter the quattes are assumed to be fuctos of the prces..55 Chapter 6 starts by cosderg a set of 20 axoms, but oly a selecto of them s gve here by way of llustrato. T: postvty the prce dex ad ts costtuet vectors of prces ad quattes should be postve. T3: detty test f the prce of every product s detcal both perods, the the prce dex should equal uty, o matter what the quatty vectors are. T5: proportoalty curret prces f all prces perod t are multpled by the postve umber λ, the the ew prce dex should be λ tmes the old prce dex;.e., the prce dex fucto s (postvely) homogeeous of degree oe the compoets of the perod t prce vector. T0: varace to chages the uts of measuremet (commesurablty test) the prce dex does ot chage f the uts whch the products are measured are chaged. T: tme reversal test f all the data for the two perods are terchaged, the the resultg prce dex should equal the recprocal of the orgal prce dex. T4: mea value test for prces the prce dex les betwee the hghest ad the lowest prce relatves.

12 T6: Paasche ad Laspeyres boudg test the prce dex les betwee the Laspeyres ad Paasche dces. T7: mootocty curret prces f ay perod t prce s creased, the the prce dex must crease..56 Some of the axoms or tests ca be regarded as more mportat tha others. Ideed, some of the axoms seem so heretly reasoable that t mght be assumed that ay dex umber actually use would satsfy them. For example, test T0, the commesurablty test, says that f the ut of quatty whch a product s measured s chaged, say, from a gallo to a ltre, the dex must be uchaged. Oe dex that does ot satsfy ths test s the Dutot dex, whch s defed as the rato of the arthmetc meas of the prces the two perods. As explaed later, ths s a type of elemetary dex that s fact wdely used the early stages of CPI calculato..57 Cosder, for example, the average prce of salt ad pepper. Suppose t s decded to chage the ut of measuremet for pepper from grams to ouces whle leavg the uts whch salt s measured (for example, klos) uchaged. As a ouce s equal to grams, the absolute value of the prce of pepper creases by over 28 tmes, whch effectvely creases the weght of pepper the Dutot dex by over 28 tmes..58 Whe the products covered by a dex are heterogeeous ad measured dfferet physcal uts, the value of ay dex that does ot satsfy the commesurablty test depeds o the purely arbtrary choce of uts. Such a dex must be uacceptable coceptually. If the prces refer to a strctly homogeeous set of products that all use the same ut of measuremet, the test becomes rrelevat..59 Aother mportat test s T, the tme reversal test. I prcple, t seems reasoable to requre that the same result should be obtaed whchever of the two perods s chose as the prce referece perod: other words, whether the chage s measured forwards tme,.e., from 0 to t, or backwards tme from t to 0. The Youg dex fals ths test because a arthmetc average of a set of prce relatves s ot equal to the recprocal of the arthmetc average of the recprocals of the prce relatves. The fact that the coceptually arbtrary decso to measure the chage prces forwards from 0 ad t gves a dfferet result from measurg backwards from t to 0 s see by may users as a serous dsadvatage. The falure of the Youg dex to satsfy the tme reversal test eeds to be take to accout by statstcal offces..60 Both the Laspeyres ad Paasche fal the tme reversal test for the same reasos as the Youg dex. For example, the formula for a Laspeyres calculated backwards from t to 0, P BL, s: P BL = = = p 0 p q t q t t P P (.4) Ths dex s detcal to the recprocal of the (forwards) Paasche, ot to the recprocal of the (forwards) Laspeyres. As already oted, the (forwards) Paasche teds to regster a smaller crease tha the (forwards) Laspeyres so that the Laspeyres dex caot satsfy the tme reversal test. The Paasche dex also fals the tme reversal test. 2

13 .6 I cotrast, the Lowe dex satsfes the tme reversal test provded that the quattes q b rema fxed whe the prce referece perod s chaged from 0 to t. The quattes of a Laspeyres dex are, however, those of the prce referece perod by defto, ad must chage wheever the prce referece perod s chaged. The basket for a forwards Laspeyres s dfferet from that for a backwards Laspeyres, ad the Laspeyres fals the tme reversal test cosequece..62 Smlarly, the Lowe dex s trastve whereas the Laspeyres ad Paasche dces are ot. Assumg that a Lowe dex uses a fxed set of quattes, q b, whatever the prce referece perod, t follows that Lo 0, t = Lo 0, t-k Lo t-k, t where Lo 0,t s the Lowe dex for perod t wth perod 0 as the prce referece perod. The Lowe dex that compares t drectly wth 0 s the same as that calculated drectly as a cha dex through perod t-k..63 If, o the other had, the Lowe dex s defed such a way that quattes vary wth the prce referece perod, as the dex p t+ q t j / p t q t j cosdered earler, the resultg cha dex s ot trastve. The cha Laspeyres ad cha Paasche dces are specal cases of ths dex..64 I the real world, the quattes do chage ad the whole pot of chag s to eable the quattes to be cotually updated to take accout of the chagg uverse of products. Achevg trastvty by arbtrarly holdg the quattes costat, especally over a very log perod of tme, does ot compesate for the potetal bases troduced by usg out-ofdate quattes. Rakg of dces usg the frst axomatc approach.65 I Chapter 6 t s show ot oly that the Fsher prce dex satsfes all the 20 axoms lsted but also, more remarkably, that t s the oly possble dex that ca satsfy all 20 axoms. Thus, o the bass of ths partcular set of axoms, the Fsher clearly domates other dces..66 I cotrast to Fsher, the other two symmetrc (ad superlatve) dces defed equatos (.) ad (.2) above do ot emerge so well from the 20 tests. I Chapter 6, t s show that the Walsh prce dex fals four tests whle the Törqvst dex fals e tests. Nevertheless, the Törqvst ad the Fsher may be expected to approxmate each other qute closely umercally whe the data follow relatvely smooth treds, as show Chapter Oe lmtato of the axomatc approach s that the lst of axoms s evtably somewhat arbtrary. Some axoms, such as the Paasche ad Laspeyres boudg test faled by both Törqvst ad Walsh, could be regarded as dspesable. Addtoal axoms or tests ca be evsaged, ad two further axoms are cosdered below. Aother problem wth a smple applcato of the axomatc approach s that t s ot suffcet to kow whch tests are faled. It s also ecessary to kow how badly a dex fals. Falg badly oe major test, such as the commesurablty test, mght be cosdered suffcet to rule out a dex, whereas falg several mor tests margally may ot be very dsadvatageous. Some further tests.68 Cosder a further symmetry test. Reversg the roles of prces ad quattes a prce dex yelds a quatty dex of the same fuctoal form as the prce dex. The factor 3

14 reversal test requres that the product of ths quatty dex ad the orgal prce dex should be detcal wth the chage the value of the aggregate questo. The test s mportat f, as stated earler, prce ad quatty dces are teded to eable chages the values of aggregates over tme to be factored to ther prce ad quatty compoets a ecoomcally meagful way. Aother terestg result gve Chapter 6 s that the Fsher dex s the oly prce dex to satsfy four mmal tests: T (postvty), T (tme reversal test), T2 (quatty reversal test) ad T2 (factor reversal test). As the factor reversal test mplctly assumes that the prces ad quattes must refer ether to perod 0 or to perod t, t s ot relevat to a Lowe dex whch three perods are volved, b, 0 ad t..69 As show earler, the product of the Laspeyres prce (quatty) dex ad the Paasche quatty (prce) dex s detcal wth the chage the total value of the aggregate questo. Thus, Laspeyres ad Paasche dces may be sad to satsfy a weak verso of the factor reversal test that dvdg the value chage by a Laspeyres (Paasche) prce dex does lead to a meagful quatty dex,.e., the Paasche (Laspeyres), eve though the fuctoal forms of the prce ad quatty dces are ot detcal..70 Aother test dscussed Chapter 6 s the addtvty test. Ths s more mportat from the perspectve of quatty tha prce dces. Prce dces may be used to deflate value chages to obta mplct quatty chages. The results may be preseted for sub-aggregates such as broad categores of household cosumpto. Just as expedture aggregates at curret prces are, by defto, obtaed smply by summg dvdual expedtures, t s reasoable to expect that the chages the sub-aggregates of a quatty dex should add up to the chages the totals the addtvty test. Quatty dces such as Laspeyres ad Paasche that use a commo set of prces to value quattes both perods must satsfy the addtvty test. Smlarly, the Lowe quatty dex defed as p j q t / p j q 0 s also addtve. The Geary Khams quatty dex (see Aex 4) used to make teratoal comparsos of real cosumpto ad gross domestc product (GDP) betwee coutres s a example of such a Lowe quatty dex. It uses a arthmetcally weghted average of the prces the dfferet coutres as the commo prce vector p j to compare the quattes dfferet coutres..7 Smlarly, a average of the prces two perods ca be used to value the quattes tertemporal dces. If the quatty dex s also to satsfy the tme reversal test, the average must be symmetrcal. The varace to proportoal chages curret prces test (whch correspods to test T7 lsted Chapter 6, except that the roles of prces ad quattes are reversed) requres that a quatty dex deped oly o the relatve, ot the absolute, level of the prces each perod. The Walsh quatty dex satsfes ths test, s addtve ad satsfes the tme reversal test as well. It emerges as a quatty dex wth some very desrable propertes..72 Although the Fsher dex tself s ot addtve, t s possble to decompose the overall percetage chage a Fsher prce, or quatty, dex to addtve compoets that reflect the percetage chage each prce or quatty. A smlar multplcatve decomposto s possble for a Törqvst prce or quatty dex. The stochastc approach ad a secod axomatc approach.73 Before cosderg a secod axomatc approach, t s coveet to take the stochastc approach to prce dces. The stochastc approach treats the observed prce chages or relatves as f they were a radom sample draw from a defed uverse whose 4

15 mea ca be terpreted as the geeral rate of flato. There ca, however, be o sgle uque rate of flato. May possble uverses ca be defed, depedg o whch partcular sets of expedtures or trasactos the user s terested. Clearly, the sample mea depeds o the choce of uverse from whch the sample s draw. Specfyg the uverse s smlar to specfyg the scope of a CPI. The stochastc approach addresses ssues such as the approprate form of average to take ad the most effcet way to estmate t from a sample of prce relatves, oce the uverse has bee defed..74 The stochastc approach s partcularly useful whe the uverse s reduced to a sgle type of product. Because of market mperfectos, there may be cosderable varato the prces at whch the same product s sold dfferet outlets ad also the prce chages observed. I practce, statstcal offces have to estmate the average prce chage for a sgle product from a sample of prce observatos. Importat methodologcal ssues are rased, whch are dscussed some detal Chapter 7 ad Chapter 20. The uweghted stochastc approach.75 I Chapter 6, the uweghted stochastc approach to dex umber theory s explaed. If smple radom samplg has bee used, equal weght may be gve to each sampled prce relatve. Suppose each prce relatve ca be treated as the sum of two compoets: a commo flato rate ad a radom dsturbace wth a zero mea. Usg least squares or maxmum lkelhood, the best estmate of the commo flato rate s the uweghted arthmetc mea of prce relatves, a dex formula kow as the Carl dex. Ths dex s the uweghted verso of the Youg dex ad s dscussed further below, the cotext of elemetary prce dces..76 If the radom compoet s multplcatve, ot addtve, the best estmate of the commo flato rate s gve by the uweghted geometrc mea of prce relatves, kow as the Jevos dex. The Jevos dex may be preferred to the Carl o the grouds that t satsfes the tme reversal test, whereas the Carl does ot. As explaed below, ths fact may be decsve whe determg the fuctoal form to be used to estmate the elemetary dces compled the early stages of CPI calculatos. The weghted stochastc approach.77 As explaed Chapter 6, a weghted stochastc approach ca be appled at a aggregatve level coverg sets of dfferet products. As the products may be of dfferg ecoomc mportace, equal weght should ot be gve to each type of product. The products may be weghted o the bass of ther share the total value of the expedtures, or other trasactos, some perod or perods. I ths case, the dex (or ts logarthm) s the expected value of a radom sample of prce relatves (or ther logarthms) whose probablty of selecto s proportoal to the expedture o that type of product some perod, or perods. Dfferet dces are obtaed depedg o whch expedture weghts are used ad o whether the prce relatves or ther logarthms are used..78 Suppose a sample of prce relatves s radomly selected wth the probablty of selecto proportoal to the expedture o that type of product the prce referece perod 0. The expected prce chage s the the Laspeyres prce dex for the uverse. Other dces may, however, also be obtaed usg the weghted stochastc approach. Suppose both perods are treated symmetrcally ad the probabltes of selecto are made proportoal to the arthmetc mea expedture shares both perods 0 ad t. Whe these weghts are appled to the logarthms of the prce relatves, the expected value of the logarthms s the 5

16 Törqvst dex, also kow as the Törqvst Thel dex. From a axomatc vewpot, the choce of a symmetrc average of the expedture shares esures that the tme reversal test s satsfed, whle the choce of the arthmetc mea, as dstct from some other symmetrc average, may be justfed o the grouds that the fudametal proportoalty curret prces test, T5, s thereby satsfed..79 By focusg o prce chages, the Törqvst dex emerges as a dex wth some very desrable propertes. Ths suggests a secod axomatc approach to dces, whch the focus s shfted from the dvdual prces ad quattes used the tradtoal axomatc approach, to prce chages ad values shares. A secod axomatc approach.80 A secod axomatc approach s examed Chapter 6 whch a prce dex s defed as a fucto of the two sets of prces, or ther ratos, ad two sets of values. Provded the dex s varat to chages uts of measuremet,.e., satsfes the commesurablty test, t makes o dfferece whether dvdual prces or ther ratos are specfed. A set of 7 axoms s postulated whch are smlar to the 20 axoms cosdered the frst axomatc approach..8 It s show Appedx 6. that the Törqvst, or Törqvst Thel, s the oly prce dex to satsfy all 7 axoms, just as the Fsher prce dex s the oly dex to satsfy all 20 tests the frst approach. However, the Törqvst dex does ot satsfy the factor reversal test, so that the mplct quatty dex obtaed by deflatg the chage value by the Törqvst prce dex s ot the Törqvst quatty dex. The mplct quatty dex s therefore ot best the sese of satsfyg the 7 axoms whe these are appled to the quatty, rather tha prce, dces..82 Zero prces may cause problems for dces based o prce ratos, ad especally for geometrc averages of prce ratos. I partcular, f ay prce teds to zero, oe test that may be appled s that the prce dex ought ot to ted to zero or plus fty. The Törqvst does ot meet ths test. It s therefore proposed Chapter 6 that whe usg the Törqvst dex, care should be take to boud the prces away from zero order to avod a meagless dex umber..83 Fally, Chapter 6 exames the axomatc propertes of the Lowe ad Youg dces. The Lowe dex emerges qute well from the axomatc approach, satsfyg both the tme reversal ad crcularty tests. O the other had, the Youg dex, lke the Laspeyres ad Paasche dces, fals both tests. As already explaed, however, the attractveess of the Lowe dex depeds more o how relevat the fxed quatty weghts are to the two perods beg compared, that s o the postog of perod b, tha ts axomatc propertes..84 Although the best dces emergg from the two axomatc approaches, amely Fsher ad Törqvst, are ot the same, they have much commo. As already oted, they are both symmetrc dces ad they are both superlatve dces. Although ther formulae are dfferet, they may be expected to behave smlar ways ad regster smlar prce movemets. The same type of dces keep emergg as havg desrable propertes whchever approach to dex theory s adopted, a cocluso that s reforced by the ecoomc approach to dex umbers, whch s explaed Chapter 7. 6

17 Cost of lvg dex.85 Approachg the cosumer prce dex from the stadpot of ecoomc theory has led to the developmet of the cocept of a cost of lvg dex (COLI). The theory of the COLI was frst developed by Kous (924). It rests o the assumpto of optmzg behavour o the part of a ratoal cosumer. The COLI for such a cosumer has bee defed succctly as the rato of the mmum expedtures eeded to atta the gve level of utlty, or welfare, uder two dfferet prce regmes. A more precse defto ad explaato are gve Chapter Whereas a Lowe dex measures the chage the cost of purchasg a fxed basket of goods ad servces resultg from chages ther prces, a COLI measures the chage the mmum cost of matag a gve level of utlty, or welfare, that results from chages the prces of the goods ad servces cosumed..87 A COLI s lable to possble msterpretato because households welfare depeds o a varety of physcal ad socal factors that have o coecto wth prces. Evets may occur that mpge drectly o welfare, such as atural or ma-made dsasters. Whe such evets occur, households may eed to crease ther cosumpto of goods ad servces order to compesate for the loss of welfare caused by those evets. Chages the costs of cosumpto trggered by evets other tha chages prces are rrelevat for a CPI that s ot merely teded to measure chages the prces of cosumer goods ad servces but s geerally terpreted by users as measurg prce chages, ad oly prce chages. I order to qualfy as a CPI, a COLI must therefore hold costat ot oly the cosumer s prefereces but all the o-prce factors that affect the cosumer s welfare ad stadard of lvg. If a CPI s teded to be a COLI t must be codtoal o: a partcular level of utlty or welfare; a partcular set of cosumer prefereces; a partcular state of the physcal ad socal evromet. Of course, Lowe dces are also codtoal as they deped o the partcular basket of goods ad servces selected..88 Lowe dces ad COLIs have commo the fact that they may both be defed as the ratos of expedtures two perods. However, whereas, by defto, the quattes are fxed Lowe dces, they vary respose to chages relatve prces COLIs. I cotrast to the fxed basket approach to dex theory, the ecoomc approach explctly recogzes that the quattes cosumed are actually depedet o the prces. I practce, ratoal cosumers may be expected to adjust the relatve quattes they cosume respose to chages relatve prces. A COLI assumes that a cosumer seekg to mmze the cost of matag a gve level of utlty wll make the ecessary adjustmets. The baskets of goods ad servces the umerator ad deomator of a COLI are ot therefore exactly the same..89 The observed expedture of a ratoal cosumer the selected base perod may be assumed to be the mmum expedture eeded to acheve the level of utlty ejoyed that perod. I order to calculate a COLI based o that perod, t s ecessary to kow what would be the mmum expedture eeded to atta precsely the same level of utlty f the prces prevalg were those of the secod perod, other thgs remag equal. The quattes purchased uder these assumed codtos are lkely to be hypothetcal. They wll ot be the 7

18 quattes actually cosumed the secod perod f other factors, cludg the resources avalable to the cosumer, have chaged..90 The quattes requred for the calculato of the COLI at least oe of the perods are ot lkely to be observable practce. The COLI s ot a operatoal dex that ca be calculated drectly. The challege s therefore to see whether t s possble to fd methods of estmatg a COLI drectly or at least to fd upper ad lower bouds for the dex. There s also cosderable terest establshg the relatoshp betwee a COLI ad Lowe dces, cludg Laspeyres ad Paasche, that ca be calculated. Upper ad lower bouds o a cost of lvg dex.9 It follows from the defto of a Laspeyres dex that, f the cosumer s come were to chage by the same proporto as the chage the Laspeyres dex, the cosumer must have the possblty of purchasg the same basket of products as the base perod. The cosumer caot be worse off. However, f relatve prces have chaged, a utltymaxmzg cosumer would ot cotue to purchase the same quattes as before. The cosumer would be able to acheve a hgher level of utlty by substtutg, at least margally, products that have become relatvely cheaper for those that have become dearer. As a COLI measures the chage the mmum expedtures eeded to mata a costat level of utlty, the COLI based o the frst perod wll crease by less tha the Laspeyres dex..92 By a smlar le of reasog, t follows that whe relatve prces chage, the COLI based o the secod perod must crease by more tha the Paasche dex. As explaed more detal Chapter 7, the Laspeyres dex provdes a upper boud to the COLI based o the frst perod ad the Paasche a lower boud to the COLI based o the secod perod. It should be oted that there are two dfferet COLIs volved here: oe based o the frst perod ad the other based o the secod perod. I geeral, however, the two COLIs are ulkely to dffer much..93 Suppose that the theoretcal target dex s a COLI, but that, for practcal reasos, the CPI s actually calculated as a Lowe dex whch the quattes refer to some perod b that precedes the prce referece perod 0. Oe mportat cocluso to be draw from ths prelmary aalyss s that as the Lowe may be expected to exceed the Laspeyres, assumg log-term prce treds ad substtuto, whle the Laspeyres may tur be expected to exceed the COLI, the wdely used Lowe dex may be expected to have a upward bas. Ths pot has had a profoud fluece o atttudes towards CPIs some coutres. The bas results from the fact that, by defto, fxed basket dces, cludg Laspeyres, do ot permt ay substtuto betwee products respose to chages relatve prces. It s therefore usually descrbed as substtuto bas. A Paasche dex would be expected to have a dowward substtuto bas. Some specal cases.94 The ext step s to establsh whether there are specal codtos uder whch t may be possble to measure a COLI exactly. I Chapter 7 t s show that f the cosumer s prefereces are homothetc that s, each dfferece curve has the same shape, each beg a uform elargemet, or cotracto, of each other the the COLI s depedet of the utlty level o whch t s based. The Laspeyres ad Paasche dces provde upper ad lower bouds to the same COLI. 8