10. XMPI Calculation in Practice

Transcription

1 10. XMPI Calculaton n Practce A. Introducton 10.1 Ths chapter provdes a general descrpton wth examples of the ways n whch XMPIs are calculated n practce. The methods used n dfferent countres are not exactly the same, but they have much n common As a result of the greater nsghts nto the propertes and behavor of prce ndces that have been acheved n recent years, t s now recognzed that some tradtonal methods may not necessarly be optmal from a conceptual and theoretcal vewpont. Concerns have also been voced n a number of countres about possble bases that may be affectng XMPIs. These ssues and concerns need to be addressed n the Manual. Of course, the methods used to comple XMPIs are nevtably constraned by the resources avalable, manly for collectng and processng prces. In some countres, the methods used may be severely constraned by a lack of resources The calculaton of XMPIs usually proceeds n two stages. Frst, prce ndces are estmated for the unweghted elementary aggregates, and then these elementary prce ndces are averaged to obtan hgher-level ndces usng the relatve values of the trade for the elementary aggregates as weghts. Secton B starts by explanng how the elementary aggregates are constructed and whch economc and statstcal crtera need to be taken nto consderaton n defnng the aggregates. The ndex number formulas most commonly used to calculate the elementary ndces are then presented and ther propertes and behavor llustrated usng numercal examples. The pros and cons of the varous formulas are consdered together wth some alternatve formulas that mght be used. Much more detal on the propertes of elementary aggregate ndex number formulas s provded n Chapter 21. The problems created by dsappearng and new commodtes are also explaned as are the dfferent ways of mputng for mssng prces, consdered n further detal n Chapter The data for the measurement of prce changes at the elementary aggregate level may be compled usng unt value ndces from customs documentaton as apposed to prce relatves of detaled descrptons of commodtes from establshment surveys. Unt value ndces ratos of the value of exports/mports dvded by the quantty n one perod compared wth that n a reference perod may be used to represent the prce changes of a commodty classfcaton at the elementary level, pror to aggregaton by weghts, only f the tems wthn the classfcaton are homogeneous. Ths s unlkely for unt values from customs data. The reader s referred to Chapter 2 for ssues concernng such a use of a unt value ndces as plug ns or proxes for a prce relatves at the elementary level. If more than one unt value ndex from customs data or prce relatve from establshment surveys are to be aggregated as unweghted averages at the elementary level, then the dscusson n Secton B of ths chapter apples. A detaled commodty classfcaton may have more than one seres of prce relatves representng the prce change of more than one traded tem. Secton B consders and llustrates the formulas that may be used to aggregate there prce relatves gven no nformaton on weghts s avalable. Ths defnes the use of an elementary ndex number formula. The frst order soluton to the problem s of course to attempt ot

2 obtan weghtng nformaton on the relatve shares of mports/exports the prce relatves represent. Secton B, and Chapter 21, addresses the ndex number problem when no such nformaton s avalable Secton C of the chapter s concerned wth the calculaton of hgher-level ndces, when nformaton on weghts s avalable. If elementary aggregate ndces, be they based on unt value ndces from customs data or prce relatves from establshment surveys, are to be aggregated usng weghts that s, be aggregated at the hgher level then the concern s wth the choce of weghted formulas as dscussed n Secton C of ths chapter. The focus s on the ongong producton of a monthly prce ndex n whch the elementary prce ndces are averaged, or aggregated, to obtan hgher-level ndces. Prce-updatng of weghts, chan lnkng, and reweghtng are dscussed, wth examples provded. The problems assocated wth ntroducton of new elementary prce ndces and new hgher-level ndces nto the XMPI are also covered. The secton explans how t s possble to decompose the change n the overall ndex nto ts component parts. Fnally, the possblty of usng some alternatve and rather more complex ndex formulas s consdered Secton D concludes wth data edtng procedures, snce these are an ntegral part of the process of complng XMPIs. It s essental to ensure that the rght data are entered nto the varous formulas. There may be errors resultng from the ncluson of ncorrect data or from enterng correct data napproprately and errors resultng from the excluson of correct data that are mstakenly beleved to be wrong. The secton examnes data edtng procedures that try to mnmze both types of errors. B. Calculaton of Prce Indces for Elementary Aggregates 10.7 XMPIs typcally are calculated n two steps. In the frst step, the elementary prce ndces for the elementary aggregates are calculated. In the second step, hgher-level ndces are calculated by averagng the elementary prce ndces. The elementary aggregates and ther prce ndces are the basc buldng blocks of the XMPI. B.1 Constructon of elementary aggregates 10.8 Elementary aggregates are constructed by groupng ndvdual commodtes and ndvdual servces nto groups of relatvely homogeneous commodtes or servces. They may be formed for groups of commodtes or servces rrespectve of the country of destnaton (export) or the country of orgn (mport), but t s also possble to form elementary aggregates accordng to country of destnaton or orgn, or for dfferent types of establshments. The actual formaton of elementary aggregates thus depends on the crcumstances and the avalablty of nformaton, and they may therefore be defned dfferently n dfferent countres. However, some key ponts should be observed: Elementary aggregates should consst of groups of commodtes or servces that are as smlar as possble, and preferably farly homogeneous. They should also consst of commodtes that may be expected to have smlar prce movements. The objectve should be to try to mnmze the dsperson of prce

3 movements wthn the aggregate. The elementary aggregates should be approprate to serve as strata for samplng purposes n lght of the samplng regme planned for the data collecton Each elementary aggregate, whether relatng to the whole export or mport, the country of destnaton or orgn, or a group of establshments, wll typcally contan a very large number of ndvdual commodtes or servces. Unt value ndces from customs data beneft from coverng the vast majorty of transactons for merchandse goods. 1 However, for establshment surveys, n practce only a small number can be selected for prcng. When selectng the commodtes, the followng consderatons need to be taken nto account: The transactons selected should be ones wth prce movements that are beleved to be representatve of all the commodtes wthn the elementary aggregate. The number of transactons wthn each elementary aggregate for whch prces are collected should be large enough for the estmated prce ndex to be statstcally relable. The mnmum number requred wll vary between elementary aggregates, dependng on the nature of the commodtes and ther prce behavor. The object s to try to track the prce of the same product over tme for as long as the product contnues to be representatve. The commodtes selected should therefore be ones that are expected to reman on the market for some tme so that lke can be compared wth lke and problems assocated wth dsappearng commodtes and selecton of replacements be reduced. The need for montorng prces of commodtes of matched qualty s because the am of the measure s to be one of pure prce changes unaffected by changes n the qualty composton over tme, as may be the case wth unt values The ndvdual commodtes should be grouped nto elementary aggregates by use of a product (commodty) or actvty (ndustry) classfcaton. For example the Harmonzed Commodty Descrpton and Codng System (HS), or the Internatonal Standard Industral Classfcaton of Economc Actvtes (ISIC). It s useful to assgn a detaled product or actvty code to each sampled commodty n order to facltate the groupng of ndvdual observatons nto elementary aggregates and the calculaton of elementary ndces. Smlarly, the elementary aggregates should be approprately coded to allow further aggregaton nto hgher-level ndces. Ths s dealt wth below n secton C.1.1. The classfcatons are presented n more detals n Chapter 4. B.2 Calculaton of elementary prce ndces An elementary prce ndex s the prce ndex for an elementary aggregate. Varous methods and formulas may be used to calculate elementary prce ndces. Ths secton provdes a summary of pros and cons that statstcal offces must evaluate when choosng a 1 In practce the coverage s less so snce large proportons of trade may be deleted f no quantty nformaton s avalable, the unt value changes are consdered to be outlers, and customs documentaton does not cover the commodtes concerned.

4 formula at the elementary level; Chapter 21 provdes a more detaled dscusson Often t s not possble to obtan nformaton about the relatve mportance of the ndvdual commodtes that enters nto the elementary aggregates. Or t may be consdered too tme consumng and resource demandng to obtan and mantan ndvdual weghts, compared wth the possble mprovements the use of such weghts would add to the ndex. If such nformaton has to be collected from the respondents t wll also add to the establshments response burden. In many countres t s thus practce that much aggregaton s wthout the use of weghtng data. Ths secton, therefore, focuses on the calculaton of unweghted elementary prce ndces. The calculaton of weghted elementary ndces s dealt wth n secton C The methods statstcal offces most commonly use are llustrated by means of a numercal example n Table It s assumed that prces are collected for four representatve commodtes wthn an elementary aggregate. The qualty of each commodty remans unchanged over tme so that the month-to-month changes compare lke wth lke. No weghts are appled. Assume ntally that prces are collected for all four commodtes n every month covered so that there s a complete set of prces. There are no dsappearng commodtes, no mssng prces, and no replacement commodtes. These are qute strong assumptons because many of the problems encountered n practce are attrbutable to breaks n the contnuty of the prce seres for the ndvdual transactons for one reason or another. The treatment of dsappearng and replacement commodtes s taken up n secton B Three wdely used formulas that have been, or stll are, n use by statstcal offces to calculate elementary prce ndces are llustrated n Table It should be noted, however, that these are not the only possbltes and some alternatve formulas are consdered later. The frst s the Carl ndex for = 1,., n commodtes. It s defned as the smple, or unweghted, arthmetc mean of the prce ratos, or prce relatves, for the two perods, 0 and t, to be compared. 0: t (10.1) P C t 1 p = 0 n p The second s the Dutot ndex, whch s defned as the rato of the unweghted arthmetc mean prces. 0: t (10.2) P D = 1 n 1 n p p t 0 The thrd s the Jevons ndex, whch s defned as the unweghted geometrc mean of the prce ratos, whch s dentcal wth the rato of the unweghted geometrc mean prces.

5 (10.3) P 0: t J 1 n t p = = 0 p t ( p ) 0 ( p ) 1 n 1 n Each month-to-month ndex shows the change n the ndex from one month to the next. The chaned month-to-month ndex lnks together these month-to-month changes by successve multplcaton. The drect ndex compares the prces n each successve month drectly wth those of the reference month, January. By smple nspecton of the varous ndces, t s clear that the choce of formula and method can make a substantal dfference n the results obtaned. Some results are strkng n partcular, the large dfference between the chaned Carl ndex for July and each of the drect ndces for July, ncludng the drect Carl The propertes and behavor of the dfferent ndces are summarzed n the followng paragraphs and explaned n more detal n Chapter 21. Frst, the dfferences between the results obtaned by usng the dfferent formulas tend to ncrease as the varance of the prce relatves, or ratos, ncreases. The greater the change n the dsperson of the prce movements, the more crtcal the choce of ndex formula and method becomes. If the elementary aggregates are defned so that the prce movements wthn the aggregate are mnmzed, the results obtaned become less senstve to the choce of formula and method Certan features dsplayed by the data n Table 10.1 are systematc and predctable and follow from the mathematcal propertes of the ndces. For example, t s well known that an arthmetc mean s always greater than, or equal to, the correspondng geometrc mean the equalty holdng only n the trval case n whch the numbers beng averaged are all the same. The drect Carl ndces are therefore all greater than the Jevons ndces, except n May and July when the four prce relatves based on January are all equal. In general, the Dutot ndex may be greater or less than the Jevons ndex, but tends to be less than the Carl ndex One general property of geometrc means should be noted when usng the Jevons formula. If any one observaton out of a set of observatons s zero, ts geometrc mean s zero, whatever the values of the other observatons. The Jevons ndex s senstve to extreme changes n prces, and t may be necessary to mpose upper and lower bounds on the ndvdual prce relatves of, say, 10 and 0.1, respectvely, when usng the Jevons. Of course, extreme observatons are often the results of errors of one knd or another, and so extreme prce movements should be carefully checked n any case. More detals on data edtng can be found n secton D Another mportant property of the ndces llustrated n Table 10.1 s that the Dutot and the Jevons ndces are transtve, whereas the Carl ndex s not. Transtvty means that the chaned monthly ndces are dentcal wth the correspondng drect ndces. Ths property s mportant n practce, because many elementary prce ndces are n fact calculated as chan ndces that lnk together the month-to-month-ndces. The ntranstvty of the Carl ndex s llustrated dramatcally n Table 10.1, n whch each of the four ndvdual prces n May returns to the same level as t was n January, but the chaned Carl ndex regsters an ncrease of almost 14 percent over January. Smlarly, n July, although each ndvdual prce s exactly 10 percent hgher than n January, the chaned Carl ndex regsters an ncrease of

6 29 percent. These results would be regarded as perverse and unacceptable n the case of a drect ndex, but even n the case of the chaned ndex, the results seems so ntutvely unreasonable as to undermne the credblty of the chaned Carl ndex. The prce changes between March and Aprl llustrate the effects of prce bouncng, n whch the same four prces are observed n both perods, but they are swtched between the dfferent commodtes. The monthly Carl ndex from March to Aprl ncreases, whereas both the Dutot and the Jevons ndces are unchanged The message emergng from ths bref llustraton of the behavor of just three possble formulas s that dfferent ndex numbers and methods can delver very dfferent results. Index complers have to famlarze themselves wth the nterrelatonshps between the varous formulas at ther dsposal for the calculaton of the elementary prce ndces so that they are aware of the mplcatons of choosng one formula rather than another. However, knowledge of these nterrelatonshps s not suffcent to determne whch formula should be used, even though t makes t possble to make a more nformed and reasoned choce. It s necessary to appeal to addtonal crtera to settle the choce of formula. Two man approaches may be used, the axomatc and the economc approaches. B.2.1 Samplng propertes of elementary prce ndces The nterpretaton of the elementary aggregate ndces s related to the way n whch the sample of commodtes s drawn. Hence, f the commodtes n the sample are selected wth probabltes proportonal to the populaton value shares n the prce reference perod, The sample (unweghted) Carl ndex provdes an unbased estmate of the populaton Laspeyres prce ndex. The sample (unweghted) Jevons ndex provdes an unbased estmate of the populaton Geometrc Laspeyres prce ndex (see equaton (10.5)) If the commodtes are sampled wth probabltes proportonal to populaton quantty shares n the prce reference perod, the sample (unweghted) Dutot ndex would provde an estmate of the populaton Laspeyres prce ndex. However, f the basket for the Laspeyres ndex contans dfferent knds of products whose quanttes are not addtve, the quantty shares, and hence the probabltes, are undefned. B.2.2 Axomatc approach to elementary prce ndces As explaned n Chapters 17 and 21, one way to decde on an approprate ndex formula s to requre t to satsfy certan specfed axoms or tests. The tests throw lght on the propertes possessed by dfferent knds of ndces, some of whch may not be ntutvely obvous. Four basc tests llustrate the axomatc approach. Proportonalty Test: If all prces are λ tmes the prces n the prce reference perod (January n the example), the ndex should equal λ. The data for July, when every prce s 10 percent hgher than n January, show that all three drect ndces satsfy ths test. A specal case of ths test s the dentty test, whch requres that f the prce of every commodty s the same as n the reference perod, the ndex should be equal to unty (as n May n the example).

7 Changes n the Unts of Measurement Test (or Commensurablty Test): The prce ndex should not change f the quantty unts n whch the commodtes are measured are changed for example, f the prces are expressed per lter rather than per pnt. The Dutot ndex fals ths test, as explaned below, but the Carl and Jevons ndces satsfy the test. Tme Reversal Test: If all the data for the two perods are nterchanged, then the resultng prce ndex should equal the recprocal of the orgnal prce ndex. The Carl ndex fals ths test, but the Dutot and the Jevons both satsfy the test. The falure of the Carl ndex to satsfy the test s not mmedately obvous from the example but can easly be verfed by nterchangng the prces n January and Aprl, for example, n whch case the backward Carl for January based on Aprl s equal to 91.3 whereas the recprocal of the forward Carl ndex s 1/132.5, or Transtvty Test: The chaned ndex between two perods should equal the drect ndex between the same two perods. The example shows that the Jevons and the Dutot ndces both satsfy ths test, whereas the Carl ndex does not. For example, although the prces n May have returned to the same levels as n January, the chaned Carl ndex regsters Ths llustrates the fact that the Carl ndex may have a sgnfcant bult-n upward bas Many other axoms or tests can be devsed, as presented n Chapter 17, but the above (summarzed n Table 10.2) are suffcent to llustrate the approach and also to throw lght on some mportant features of the elementary ndces under consderaton here The sets of commodtes covered by elementary aggregates are meant to be as homogeneous as possble. If they are not farly homogeneous, the falure of the Dutot ndex to satsfy the unts of measurement, or commensurablty, test can be a serous dsadvantage. Although defned as the rato of the unweghted arthmetc average prces, the Dutot ndex may also be nterpreted as a weghted arthmetc average of the prce ratos n whch each rato s weghted by ts prce n the base perod. 2 However, f the commodtes are not homogeneous, the relatve prces of the dfferent commodtes may depend qute arbtrarly on the quantty unts n whch they are measured Consder, for example, salt and pepper, whch are found wthn the same Central Product Classfcaton subclass. Suppose the unt of measurement for pepper s changed from grams to ounces whle leavng the unts n whch salt s measured (say klos) unchanged. Because an ounce of pepper s equal to grams, the prce of pepper ncreases by more than 28 tmes, whch effectvely ncreases the weght gven to pepper n the Dutot ndex by more than 28 tmes. The prce of pepper relatve to salt s nherently arbtrary, dependng 2 Ths can be seen by rewrtng equaton (10.2) above as P 0:t D 1 = n 1 n t ( ) p p p 0 0 p 0

8 entrely on the choce of unts n whch to measure the two goods. In general, when there are dfferent knds of commodtes wthn the elementary aggregate, the Dutot ndex s not acceptable The Dutot ndex s acceptable only when the set of commodtes covered s homogeneous, or at least nearly homogeneous. For example, the Dutot ndex may be acceptable for a set of apple prces, even though the apples may be of dfferent varetes, but not for the prces of dfferent knds of fruts, such as apples, pneapples, and bananas, some of whch may be much more expensve per tem or per klo than others. Even when the commodtes are farly homogeneous and measured n the same unts, the Dutot ndex s mplct weghts may stll not be satsfactory. More weght s gven to the prce changes for the more expensve commodtes, but more expensve tems may not account for the hghest revenue shares. For example, f one type of apple ncreased n prce much more than the others, there wll be ncreased dsperson n prce relatves and the Dutot and Jevons ndces wll become much further apart. The mplct weghtng of the base perod prce for the Dutot ndex wll be open to queston gven ts relatvely larger prce n the current perod It may be concluded that from an axomatc vewpont, both the Carl and the Dutot ndces, although they have been and stll are wdely used by statstcal offces, have serous dsadvantages. The Carl ndex fals the tme reversal and transtvty tests. In prncple, t should not matter whether we choose to measure prce changes forward or backward n tme. We would expect the same answer, but ths s not the case for the Carl ndex. Chaned Carl ndces may be subject to a sgnfcant upward bas and should not be appled. The Dutot ndex s meanngful for a set of homogeneous commodtes but becomes ncreasngly arbtrary as the set of commodtes becomes more dverse. On the other hand, the Jevons ndex satsfes all the tests lsted above and also emerges as the preferred ndex when the set of test s enlarged, as shown n Chapter 21. From an axomatc pont of vew, the Jevons ndex s clearly the ndex wth the best propertes, even though t may not have been used much untl recently. B.2.3 Economc approach to elementary prce ndces The objectve of the economc approach s to estmate an deal or true economc ndex for the elementary aggregates. For an output prce ndex from a resdent s perspectve, the export prce ndex, the economc ndex s one consstent wth the economc theory of revenue-maxmzng establshments (exporters). For an nput prce ndex from a resdent s perspectve, such as the mport prce ndex, the economc ndex s one consstent wth the economc theory of cost-mnmzng purchasers (mporters). As explaned n Chapter 18, the behavoral assumpton are reversed from the nonresdent s perspectve: mports are from nonresdent exporters (revenue-maxmzng establshments) and exports are to nonresdent mporters (cost-mnmzng purchasers). The theoretcal background of nput and output prce ndces s explaned n detal n Chapters 18 and 21. In ths chapter for smplcty of exposton a resdent s perspectve s taken and the reader can dentfy the mplcatons for XMPIs the nonresdents perspectve by reversng the behavoral assumptons or consderng the matter further n Chapter 18.

9 B Output export prce ndces, XPI For the XPI from the resdent producer s perspectve, the commodtes for whch respondents provde prces are treated as a basket of goods and servces sold by establshments to provde revenue. The resdent producng establshments are assumed to arrve at ther decson about the quanttes of outputs to supply on the bass of revenuemaxmzng behavor. As explaned n Chapters 1, 16, and 18, an deal theoretcal economc output prce ndex measures the rato of revenues between two perods that an establshment can attan when faced wth fxed technologes and nputs. Changes n the ndex, therefore, arse only from changes n prces. The technology s assumed to be held fxed, although the revenue-maxmzng producer can make substtutons between the commodtes suppled n response to changes n ther relatve prces In the absence of nformaton about quanttes or trade values wthn an elementary aggregate, an deal ndex can be estmated only when certan condtons are assumed to preval There are two specal cases of some nterest. The frst case s when producers contnue to produce the same relatve quanttes rrespectve of the relatve output prces on the market. When producers contnue to produce the same quanttes even f relatve prces change, the cross elastctes of supply are zero. In ths case a Laspeyres ndex would provde an exact measure of the deal economc output prce ndex and a Carl ndex 3 calculated for a random sample of products would provde an estmate of the deal economc ndex, provded that the products are selected wth probabltes proportonal to the populaton revenue shares The second case occurs when producers are assumed to vary the quanttes they produce n nverse proporton to the changes n relatve output prces. The cross-elastctes of supply are all unty, the revenue shares remanng the same n both perods. In ths case the Geometrc Laspeyres ndex would provde an exact measure of the deal ndex. The Jevons ndex calculated for a random sample of products would provde an unbased estmate of the deal economc ndex provded that the products are selected wth probabltes proportonal to the populaton expendture shares In the economc approach, the choce of ndex formula rests on whch s lkely to approxmate more closely the underlyng deal economc ndex n other words, whether the (unknown) cross-elastctes are lkely to be closer to unty or zero, on average. In practce, the cross-elastctes could take on any value rangng up to + for an elementary aggregate consstng of a set of strctly homogeneous products that s, perfect substtutes. In some ndustres supply s relatvely unresponsve to changes n demand, and producers tend to produce the same relatve quanttes rrespectve of changes n relatve prces. In such cases the Carl ndex s lkely to provde a closer approxmaton to the deal economc ndex than 3 The case for Carl n ths context s outlned for fxed-base and not chaned-base comparsons. A chaned Carl when used for commodtes whose prce and quantty changes are not smooth, such as seasonal goods, s subject to severe bas.

10 the Jevons ndex, but only f the samplng of prces collected s wth probablty proportonate to revenue shares n the reference perod In compettve, demand-led ndustres producers wll tend to produce more of those commodtes whose relatve prce has ncreased. Under such condtons none of the three ndces are approprate for provdng a close estmate of the economc ndex. However, the Carl ndex s more lkely to provde a closer approxmaton to the deal economc ndex than the Jevons ndex, whch must be vewed as havng a downward bas compared to the two other ndces In the specal case where producers substtutes to keep constant revenue shares the Jevons ndex would be an exact measure of the economc ndex. Thus, only f producers tend to produce more of those goods whose prces have fallen and less of those whose prces have ncreased the Jevons ndex s lkely to provde the better estmate of the economc ndex. Ths stuaton, however, s not very lkely. It mght be the case that producers when makng ther producton decson also take nto account other factors such as ther market poston and expected demand. In such cases producers may focus on growng markets even f the relatve prces of products on such markets are fallng. Lkewse, some producers that supply only one specfc output may not be able to change the producton, and wll contnue to supply the same product even f ts relatve prce s fallng. Such condtons may, to some extent, justfy the use of the Jevons ndex All three ndces can be justfed from the economc approach dependng on the market condtons, the degree of substtuton, the homogenety of the elementary aggregates and the desgn of the sample. It should be noted that the elementary aggregates should be delberately constructed to group together smlar products that are close substtutes for each other It may be concluded that on bass of the axomatc approach, the Jevons emerges as the preferred ndex. On the bass of the economc approach, arguments can be found for all three ndces. Whch ndex wll provde the better estmate of the economc ndex wll depend on the type of producton and the market condtons under consderaton. Thus, the ndex compler must make a judgment on the bass of the nature of the products ncluded n the elementary aggregates. However, samplng wth probablty proportonate to value shares and judgments of Leontef producton functons are restrctve condtons to argue for the Carl and general rule, gven ts poor axomatc propertes, s not to use t B Input prce ndces, MPI For the MPI, the commodtes for whch prces are collected are treated as a basket of goods and servces purchased by the mportng establshment An deal economc nput prce ndex measures the rato of hypothetcal nput costs that the establshment must pay to produce the same vector of outputs, wth fxed producton technology. Changes n the ndex arse only from changes n prces. The establshment (mporter) s assumed to make substtutons between nputs n response to changes n the relatve prces on nputs n order to mnmze costs.

11 10.40 It s worth notcng the parallel to the cost-of-lvng consumer prce ndex (COLI). The COLI measures the rato of the mnmum costs of purchasng goods and servces n two perods, whch allow the consumer to keep constant utlty or welfare. The basket of goods and servces purchased n the two perods s not necessarly dentcal as the consumer s supposed to substtute n response to changes n relatve prces of consumer goods and servces. In a smlar way the mporter s assumed to substtute among varous nputs n response to changes n relatve prces of nputs. Further theoretcal explanatons of nput prce ndces are provded n chapters 18 and As for the export prce ndex, there are two specal cases of some nterest. The frst case s when the establshment contnues to mport the same relatve quanttes of nputs whatever ther relatve prces. The technology by whch nputs are translated nto outputs n economc theory s descrbed by a producton functon, and a producton functon wth such a restrctve reacton to relatve nput prce changes s descrbed as a Leontef producton functon. Wth a Leontef producton functon a Laspeyres ndex would provde an exact measure of the deal economc ndex. In ths case, the Carl ndex calculated for a random sample of commodtes would provde an estmate of the deal economc ndex provded that the commodtes are selected wth probabltes proportonal to the populaton trade value shares. If the products were selected wth probabltes proportonal to the populaton quantty shares, the sample Dutot would provde an estmate of the populaton Laspeyres The second case occurs when establshments are assumed to vary the quanttes they mport n nverse proporton to the changes n relatve nput prces. The cross-elastctes are all unty and the mport value shares reman constant. In ths case the Geometrc Laspeyres ndex would provde an exact measure of the deal ndex. The Jevons ndex calculated for a random sample of products would provde an unbased estmate of the deal economc ndex provded that the products are selected wth probabltes proportonal to the populaton expendture shares The choce of ndex formula rests on whch s lkely to approxmate more closely the underlyng deal economc ndex,.e. whether the (unknown) cross-elastctes are lkely to be closer to unty or zero, on average. In some ndustres demand for nputs s relatvely unresponsve to changes n prces and establshments tend to mport the same relatve quanttes rrespectve of changes n ther prces. In such cases the Carl ndex, or the Dutot ndex, are lkely to provde a closer approxmaton to the deal economc ndex than the Jevons ndex, whch may be vewed as havng a downward bas. If establshments tend to substtute between nputs as a response to change n relatve prces, the Jevons ndex may provde the better estmate of the economc ndex, and the Carl may be vewed as havng an upward bas All three ndces can be justfed from the economc approach dependng on the degree of substtuton between nputs, the homogenety of the elementary aggregates and the desgn of the sample. As the elementary aggregates should be constructed to nclude products that are close substtutes ths may lend some support to assume n general some degree of substtuton It may be concluded that on bass of the axomatc approach, the Jevons emerges as

12 the preferred ndex. On the bass of the economc approach, Jevons n general also appears as the better choce of an nput prce ndex as t allows for some substtuton. However, n some ndustres there may be very low or even no substtuton between nputs. Hence, whch ndex wll provde the better estmate of the economc ndex wll depend on the type of producton and the market condtons under consderaton. Thus, the ndex compler must make a judgment takng these factors nto account In summary t should be noted that the Jevons ndex does not mply, or assume, that the trade value shares reman constant. Obvously, the Jevons can be calculated whether changes occur n the value shares n practce. What the economc approach shows s that f the value shares reman constant, or roughly constant, then the Jevons ndex can be expected to provde a good estmate of the underlyng deal economc ndex. Smlarly, f the relatve quanttes reman constant, then the Carl ndex can be expected to provde a good estmate, but they do not actually mply that quanttes reman fxed. Reference should be made to Secton F of 21 for a more rgorous statement of the economc approach. B.3 Chaned versus drect ndces for elementary aggregates In a drect elementary ndex, the prces of the current perod are compared drectly wth those of the prce reference perod. In a chaned ndex, prces n each perod are compared wth those n the prevous perod, and the resultng short-term ndces are then multpled, or chaned, to obtan the long-term ndex, as llustrated n Table Provded that prces are recorded for the same set of commodtes n every perod, as n Table 10.1, any ndex formula defned as the rato of the average prces wll be transtve that s, the same result s obtaned whether the ndex s calculated as a drect ndex or as a chaned ndex. In a chaned ndex, successve numerators and denomnators wll cancel out, leavng only the average prce n the last perod dvded by the average prce n the prce reference perod, whch s the same as the drect ndex. Both the Dutot and the Jevons ndces are therefore transtve. As already noted, however, a chan Carl ndex s not transtve and should not be used because of ts upward bas. The drect Carl fals, as noted above, the tme reversal test and s not generally advsed. Nevertheless, the drect Carl remans an opton Although the chaned and drect versons of the Dutot and Jevons ndces are dentcal when there are no breaks n the seres for the ndvdual commodtes, they offer dfferent ways of dealng wth new and dsappearng commodtes, mssng prces, and qualty adjustments. In practce, commodtes contnually have to be dropped from the ndex and new ones ncluded, n whch case the drect and the chan ndces may dffer f the mputatons for mssng prces are made dfferently When a replacement commodty has to be ncluded n a drect ndex, t often wll be necessary to estmate the prce of the new commodty n the prce reference perod, whch may be some tme n the past. The same happens f, as a result of an update of the sample, new commodtes have to be lnked nto the ndex. Assumng that no nformaton exsts on the prce of the replacement commodty n the prce reference perod, t wll be necessary to estmate t usng prce relatves calculated for the commodtes that reman n the elementary

13 aggregate, a subset of these commodtes, or some other ndcator. However, the drect approach should be used only for a lmted perod. Otherwse, most of the reference prces would end up beng mputed, whch would be an undesrable outcome. Ths effectvely rules out the use of the Carl ndex over a long perod, as the Carl ndex can only be used n ts drect form anyway, and even then beng subject to bas due to ts falure of the tme reversal test, beng unacceptable when chaned. Ths mples that, n practce, the drect Carl ndex may be used only f the overall ndex s chan lnked annually, or at ntervals of two or three years In a chaned ndex, f a commodty becomes permanently mssng, a replacement commodty can be lnked nto the ndex as part of the ongong ndex calculaton by ncludng the commodty n the monthly ndex as soon as prces for two successve months are obtaned. Smlarly, f the sample s updated and new commodtes have to be lnked nto the ndex, ths wll requre successve old and new prces for the present and the precedng month. However, for a chan ndex, the mssng observaton wll affect the ndex for two months, snce the mssng observaton s part of two lnks n the chan. Ths s not the case for a drect ndex where a sngle, non-estmated mssng observaton wll affect only the ndex n the current perod. For example, when comparng perods 0 and 3, a mssng prce of a commodty n perod 2 means that the chaned ndex excludes the commodty for the last lnk of the ndex n perods 2 and 3, whle the drect ndex ncludes t n perod 3 (snce a drect ndex wll be based on commodtes wth prces avalable n perods 0 and 3). However, n general, the use of a chaned ndex can make the estmaton of mssng prces and the ntroducton of replacements easer from a computatonal pont of vew, whereas t may be nferred that a drect ndex wll lmt the usefulness of overlap methods for dealng wth mssng observatons. Mssng prce observatons are dscussed further n Secton B The drect and the chaned approaches also produce dfferent by-products that may be used for montorng prce data. For each elementary aggregate, a chaned ndex approach gves the latest monthly prce change, whch can be useful for both edtng data and mputng mssng prces. By the same token, however, a drect ndex derves average prce levels for each elementary aggregate n each perod, and ths nformaton may be a useful by-product. However, the avalablty of cheap computng power and spreadsheets allows such byproducts to be calculated whether a drect or a chaned approach s appled, so that the choce of formula should not be dctated by consderatons regardng by-products. B.4 Consstency n aggregaton Consstency n aggregaton means that f an ndex s calculated stepwse by aggregatng lower-level ndces to obtan ndces at progressvely hgher levels of aggregaton, the same overall result should be obtaned as f the calculaton had been made n one step. If the elementary ndces are calculated usng one formula, and then averaged to obtan the hgher-level ndces usng another formula, the resultng XMPI s not consstent n aggregaton. However, t may be argued that consstency n aggregaton s not necessarly an mportant or even approprate crteron. There may be dfferent elastctes of substtuton wthn elementary aggregates compared to the elastctes between elementary aggregates. Ths may be an argument for usng a dfferent ndex formula at dfferent level of aggregaton. Also t may be unachevable, partcularly when the amount of nformaton avalable on

14 quanttes and trade values s not the same at the dfferent levels of aggregaton The Carl ndex s consstent n aggregaton wth a hgher-level Laspeyres ndex f the commodtes are selected wth probabltes proportonal to trade values n the prce reference perod. The Dutot and the Jevons ndces are not consstent n aggregaton wth a hgher-level Laspeyres ndex. However, as explaned below, the XMPIs actually calculated by statstcal offces are usually not true Laspeyres ndces anyway, even though they may be based on fxed baskets of goods and servces. If the hgher-level ndex were to be defned as a Geometrc Laspeyres ndex, consstency n aggregaton could be acheved by usng the Jevons ndex for the elementary ndces at the lower level, provded that the ndvdual commodtes are sampled wth probabltes proportonal to trade values. Although unfamlar, a Geometrc Laspeyres ndex has desrable propertes from an economc pont of vew and s consdered agan n secton B.6. B.5 Mssng prce observatons The prce of a commodty may not be collected n a partcular perod ether because the commodty s mssng temporarly or because t has permanently dsappeared. The two classes of mssng prces requre dfferent treatments. Temporary unavalablty may occur for seasonal commodtes (partcularly for frut, vegetables, and clothng) because of supply shortages or possbly because of some collecton dffculty (for example, an establshment was closed or a respondent was on vacaton). The treatment of seasonal commodtes rases a number of partcular problems. These are dealt wth n Chapter 23 and wll not be dscussed here. B.5.1 Treatment of temporarly mssng prces In the case of temporarly mssng observatons for commodtes one of four actons may be taken: Omt the commodty for whch the prce s mssng so that a matched sample s mantaned (lke s compared wth lke), even though the sample s depleted. Carry forward the last observed prce. Impute the mssng prce by the average prce change of the prces that are avalable n the elementary aggregate. Impute the mssng prce by the prce change of a comparable commodty from a smlar establshment The prce development for a gven commodty may be dfferent accordng to the country to where t s exported or from where t s mported. Ths may be due to dfferent prce trends n the countres or exchange rate changes. An elementary ndex thus may contan commodtes from several countres, and the prce development of a mssng commodty for a specfc country may be unusual compared to the average prce development of the remanng ones. When mputng a prce by the prce development of another commodty or a group of commodtes consderatons should, therefore, be gven to the country of orgn (mports) or destnaton (exports).

15 10.58 Omttng an observaton from the calculaton of an elementary ndex s equvalent to assumng that the prce would have moved n the same way as the average of the prces of the commodtes that reman ncluded n the ndex. Omttng an observaton changes the mplct weghts attached to the other prces n the elementary aggregate Carryng forward the last observed prce should be avoded wherever possble and s acceptable only for a very lmted number of perods. Specal care needs to be taken n perods of hgh nflaton or when markets are changng rapdly as a result of a hgh rate of nnovaton and commodty turnover. Whle smple to apply, carryng forward the last observed prce bases the resultng ndex toward zero change. In addton, there s lkely to be a compensatng step-change n the ndex when the prce of the mssng commodty s recorded agan. The adverse effect on the ndex wll be ncreasngly severe f the commodty remans unprced for some length of tme. In general, carry forward s not an acceptable procedure or soluton to the problem unless t s certan the prce has not changed Imputaton of the mssng prce by the average change of the avalable prces may be appled for elementary aggregates when the prces can be expected to move n the same drecton. The mputaton can be made usng all the remanng prces n the elementary aggregate. As already noted, ths s numercally equvalent to omttng the commodty for the mmedate perod, but t s useful to make the mputaton so that f the prce becomes avalable agan n a later perod, the sample sze s not reduced n that perod. In some cases, dependng on the homogenety of the elementary aggregate, t may be preferable to use only a subset of commodtes from the elementary aggregate to estmate the mssng prce. In some nstances, ths may even be a sngle comparable commodty from a smlar type of establshment whose prce change can be expected to be smlar to the mssng one Table 10.3 llustrates the calculaton of the prce ndex for an elementary aggregate consstng of three commodtes, where one of the prces s mssng n March. The upper part of Table 10.3 shows the ndces where the mssng prce has been omtted from the calculaton. The drect ndces are therefore calculated on the bass of A, B, and C for all months except March, where they are calculated on bass of B and C only. The chaned ndces are calculated on the bass of all three prces from January to February and from Aprl to May. From February to March and from March to Aprl, the monthly ndces are calculated on the bass of B and C only For both the Dutot and the Jevons, the drect and chan ndces now dffer from March onward. The frst lnk n the chaned ndex (January to February) s the same as the drect ndex, so that the two ndces are dentcal numercally. The drect ndex for March gnores the prce decrease of commodty A between January and February, whle ths s taken nto account n the chaned ndex. As a result, the drect ndex s hgher than the chaned ndex for March. On the other hand, n Aprl and May, where all prces agan are avalable, the drect ndex catches the prce development, whereas the chaned ndex fals to track the development n the prces In the lower half of Table 10.3, the mssng prce for commodty A n March s mputed by the average prce change of the remanng commodtes from February to March. Whle the ndex may be calculated as a drect ndex comparng the prces of the present

16 perod wth the reference perod prces, the mputaton of mssng prces should be made on bass of the average prce change from the precedng to the present perod, as shown n the table. Imputaton on the bass of the average prce change from the prce reference perod to the present perod should not be used snce t gnores the nformaton about the prce change of the mssng commodty that has already been ncluded n the ndex. The treatment of mputatons s dscussed n more detal n Chapter A specal case of mssng prces occurs when prces are recorded wth dfferent frequency, for example f some prces are recorded monthly whle others only quarterly or every half-year. If the ndex s compled on a monthly bass there wll then be a need to temporarly update the prces recorded wth a lower frequency. The optons for updatng the lower frequency prces wll be the same as those descrbed above. B.5.2 Treatment of commodtes that have permanently dsappeared and ther replacements Commodtes may dsappear permanently for varous reasons. The commodty may dsappear from the market because new commodtes have been ntroduced or an establshment from whch prces have been collected leave the market. When commodtes dsappear permanently, a replacement commodty has to be sampled and ncluded n the ndex. The replacement commodty should deally be one that accounts for a sgnfcant proporton of sales, s lkely to contnue to be sold for some tme, and s lkely to be representatve of the prce changes of the market that the old commodty covered The tmng of the ntroducton of replacement commodtes s mportant. Many new commodtes are ntally sold at hgh prces that then gradually drop over tme, especally as the value of sales ncreases. Alternatvely, some commodtes may be ntroduced at artfcally low prces to stmulate demand. In such cases, delayng the ntroducton of a new or replacement commodty untl a large volume of sales s acheved may mss some systematc prce changes that ought to be captured by XMPIs. It may be desrable to try to avod forced replacements caused when commodtes dsappear completely from the market and to try to ntroduce replacements when sales of the commodtes they replace are decreasng and before they cease altogether Table 10.4 shows an example where commodty A dsappears after March and commodty D s ncluded as a replacement from Aprl onward. Commodtes A and D are not avalable on the market at the same tme, and ther prce seres do not overlap. To nclude the new commodty n the ndex from Aprl onward an mputed prce needs to be calculated ether for the base perod (January) f a drect ndex s beng calculated, or for the precedng perod (March) f a chaned ndex s calculated. In both cases, the mputaton method ensures that the ncluson of the new commodty does not, n tself, affect the ndex In the case of a chaned ndex, mputng the mssng prce by the average change of the avalable prces gves the same result as f the commodty s smply omtted from the ndex calculaton untl t has been prced n two successve perods. Ths allows the chaned ndex to be compled by smply channg the month-to-month ndex between perods t 1 and t, based on the matched set of prces n those two perods, on to the value of the chaned

17 ndex for perod t 1. In the example, no further mputaton s requred after Aprl, and the subsequent movement of the ndex s unaffected by the mputed prce change between March and Aprl In the case of a drect ndex, however, an mputed prce s always requred for the reference perod to nclude a new commodty. In the example, the prce of the new commodty n each month after Aprl stll has to be compared wth the mputed prce for January. As already noted, to prevent a stuaton n whch most of the reference perod prces end up beng mputed, the drect approach should be used only for a lmted perod of tme The stuaton s somewhat smpler when there s an overlap month n whch prces are collected for both the dsappearng and the replacement commodty. In ths case, t s possble to lnk the prce seres for the new commodty to the prce seres for the old commodty that t replaces. Lnkng wth overlappng prces nvolves makng an mplct adjustment for the dfference n qualty between the two commodtes, snce t assumes that the relatve prces of the new and old commodty reflect ther relatve qualtes. For perfect or nearly perfect markets, ths may be a vald assumpton, but for certan markets and commodtes t may not be so reasonable. The queston of when to use overlappng prces s dealt wth n detal n Chapter 8. The overlap method s llustrated n Table In the example, overlappng prces are obtaned for commodtes A and D n March. Ther relatve prces suggest that one unt of commodty A s worth two unts of commodty D. If the ndex s calculated as a drect Carl ndex, the January base perod prce for commodty D can be mputed by dvdng the prce of commodty A n January by the prce rato of A and D n March A monthly chan ndex of arthmetc mean prces wll be based on the prces of commodtes A, B, and C untl March, and from Aprl onward by B, C, and D. The replacement commodty s not ncluded untl prces for two successve perods are obtaned. Thus, the monthly chaned ndex has the advantage that t s not necessary to carry out any explct mputaton of a reference prce for the new commodty If a drect ndex s calculated as the rato of the arthmetc mean prces, the prce of the new commodty needs to be adjusted by the prce rato of A and D n March n every subsequent month, whch complcates computaton. Alternatvely, a reference perod prce of commodty D for January may be mputed. However, ths results n a dfferent ndex because the prce relatves are mplctly weghted by the relatve reference perod prces n the Dutot ndex, whch s not the case for the Carl or the Jevons ndex. For the Jevons ndex, all three methods gve the same result, whch s an addtonal advantage of ths approach Problems wth mssng prces may be partcular n smaller countres or for commodty groups, even n larger countres, where the number of reportng establshments s very lmted. From tme to tme establshments wll leave the market and no replacement can be found. If there are stll prces recorded the elementary ndex can be contnued on the bass of the remanng prces. However, n some nstances all prces for an elementary aggregate may dsappear. In ths case, t wll be necessary to assgn the weght to another elementary aggregate or to mpute or carry forward the elementary ndex untl the next revson of the

18 ndex The statstcal offce may try to reduce the problems assocated wth mssng prces by defnng the elementary aggregates not too narrow. Broader defned elementary aggregates wll reduce problems wth mssng prces and help facltate a smooth, regular complaton of the ndex. However, markets change over tme and the ndex should reflect ths. Ths s dealt wth n more detals n sectons C.6.4 and C.6.5 on the ntroducton of new elementary and hgher-level ndces n the overall prce ndex. B.6 Calculaton of elementary prce ndces usng weghts Whenever possble weghts that reflect the relatve mportance of the sampled commodtes may be ntroduced n the calculaton of the elementary ndces. For certan elementary aggregates, nformaton about the export or mport of partcular commodtes may be obtaned from exstng trade and ndustry sources, or the statstcal offce can work wth establshment respondents to obtan weghtng data, as dscussed n Chapter 5. In addton the growng use of electronc recordng of transactons n many countres, n whch both prces and quanttes are mantaned, means that valuable new sources of nformaton may become ncreasngly avalable to statstcal offces For example, assume that the number of mporters of a certan commodty such as gasolne s lmted. The market shares of the mporters may be known from busness survey statstcs and can be used as weghts n the calculaton of an elementary aggregate prce ndex for gasolne A specal stuaton occurs n the case of tarff prces. A tarff s a lst of prces for the provson of a partcular knd of good or servce under dfferent terms and condtons. One example s electrcty for whch one prce s charged durng daytme and a lower prce s charged at nght. Another example may be arlne passenger fares sold at one prce to some passengers and at lower prces to others. In such cases, t s approprate to assgn weghts to the dfferent tarffs or prces to calculate the prce ndex for the elementary aggregate Weghts wthn elementary aggregates may be updated ndependently and possbly more often than the elementary aggregate weghts themselves If weghtng data s avalable for all the ndvdual commodtes wthn an elementary aggregate, the elementary prce ndex can be calculated as a Laspeyres prce ndex, or as a Geometrc Laspeyres ndex; both are dscussed further n Chapter 21. The Laspeyres prce ndex s defned as (10.4) P p q w p w p q t 0 t 0 0 0: t 0 0 L = =, = 0 0 p q p p q, As the quanttes are often unknown, the ndex usually wll have to be calculated by weghtng together the ndvdual prce ratos by ther trade value shares n the prce reference perod, w 0. The avalable weghtng data may refer to an earler perod than the prce reference perod, but may stll provde a good estmate. A more general verson of (10.4)

19 would be that of a Lowe or a Young ndex, where the weghts are not necessarly those of the prce reference perod. These two ndces are dscussed n more detals n secton C.3. Note that f all the weghts were equal, equaton (10.4) would reduce to the Carl ndex. If the weghts were proportonal to the prces n the reference perod, t would reduce to the Dutot ndex The Geometrc Laspeyres ndex s defned as (10.5) P 0:t GL W0 t ( p ) 0 ( p ) 0 W t p 0 =, w 1 0 = 0 W =, p where the weghts, w 0 are agan the trade value shares n the reference perod. When the weghts are all equal, equaton (10.5) reduces to the Jevons ndex. If the trade value shares do not change much between the weght reference perod and the current perod, then the geometrc Laspeyres ndex approxmates a Törnqvst ndex. A more general verson of (10.5) would be that of a Geometrc Young ndex, where the weghts are not necessarly those of the prce reference perod The weghts may be attached to the ndvdual prce observatons or to groups of prce observatons. For example two establshments may both report, say, 5 prces that enter nto the calculaton of an elementary aggregate prce ndex. However, the only weghtng nformaton may refer to the overall relatve market share of the two establshments rather than to the ndvdual commodtes. Thus, f the relatve market shares are 40/60, the two groups of prces may be weghted accordng to the 40/60 shares of the establshments Table 10.6 provdes an example of calculaton of an elementary ndex usng weghts. The elementary aggregate conssts of three commodtes for whch prces are collected monthly. The trade value shares are estmated to 0.80, 0.17 and 0, One opton s to calculate the ndex as the weghted arthmetc mean of the prce ratos, whch gves an ndex of The ndvdual prce changes are weghted accordng to ther explct weghts, rrespectve of the prce levels. Ths corresponds to the calculaton of a Laspeyres prce ndex, where the prce ratos and the weghts refer to the same reference month. The ndex may also be calculated as the weghted geometrc mean of the prce ratos, the so-called geometrc Laspeyres ndex, whch gves an ndex of A thrd opton could be to calculate the ndex as the rato of the weghted arthmetc mean prces. As already noted, an elementary ndex should only be based on arthmetc mean prces f t ncludes homogenous products measured n the same unt; otherwse t s not meanngful to calculate an average prce. In practce, ths wll also mean that the prce level of the products should be more or less the same. Secondly, ths approach weghts the prce changes accordng to the relatve prce level n the reference perod. Hence, the ncrease of 28,6% on commodty A that accounts for 80% of the market s weghted down because of ts relatve low prce, resultng n an ndex of Ths calculaton s msleadng, however The dfference between the two arthmetc methods can be llustrated by an example:

20 Assume an elementary aggregate wth two commodtes, X and Y of equal weghts (50/50). The prce of X s constant 90, the prce of Y ncreases from 10 to 12. The weghted arthmetc mean of the prce ratos gves 90/ /10 = The rato of arthmetc weghted prces gves (90+12)/(90+10) = In the frst approach, the prce ncreases of the two commodtes are equally weghted, whch gves an ncrease of 10%. The problem n the second approach s that t weghts the 0% prce ncrease on X by 90/100, and the 20% ncrease of Y by only 10/100, whch gves an overall ncrease of 2%. Ths can only be justfed, f the weghts are proportonal to the relatve prce level n the reference perod. That s, f the weght of X s 90 and that of Y s 10, whch, however, contradcts the assumpton of 50/50 weghts. Because of the calculaton method the weghts are twsted accordng to the relatve prce levels resultng n a msleadng ndex Weghtng nformaton at the very detaled level s resource demandng to obtan and update. Ths has to be balanced aganst the possble gans n terms of a more accurate prce ndex, and n some cases, t may be the better opton to use an unweghted approach. B.7 Some alternatve ndex formulas Another type of average s the harmonc mean. In the present context, there are two possble versons: ether the harmonc mean of prce ratos or the rato of harmonc mean of prces. The harmonc mean of prce ratos s defned as 0: t (10.6) P HR = 1 n 1 p p 0 t The rato of harmonc mean prces s defned as (10.7) n p 0 0: t RH = t n p P Nether formula appears to be used much n practce, perhaps because the harmonc mean s not a famlar concept and would not be easy to explan to users. However, at an aggregate level, the wdely used Paasche ndex s a weghted harmonc average The rankng of the three common types of mean s always arthmetc mean geometrc mean harmonc mean. It s shown n Chapter 21 that, n practce, the Carl ndex, the arthmetc mean of the prce ratos, s lkely to exceed the Jevons ndex, the geometrc mean, by roughly the same amount that the Jevons exceeds the harmonc mean, equaton (10.6). The harmonc mean of the prce ratos has the same knds of axomatc propertes as the Carl but wth opposte tendences and bases. It fals the transtvty and tme reversal tests dscussed earler. Equaton (10.7), lke the Dutot ndex, fals the commensurablty test and would be an acceptable possblty