UNSW Business School Working Paper

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1 UNSW Business School Working Paper UNSW Business School Research Paper No ECON 4 Oupu Growh and Inflaion across Space and Time W.Erwin Diewer Kevin J. Fox This paper can be downloaded wihou charge from The Social Science Research Nework Elecronic Paper Collecion: hp://ssrn.com/absrac= business.unsw.edu.au Las Updaed 29 July 2014 CRICOS Code 00098G

2 Oupu Growh and Inflaion across Space and Time W. Erwin Diewer Universiy of Briish Columbia and UNSW and Kevin J. Fox* UNSW 22 February, 2015 Absrac I is common for comparisons o be made of oupu growh and inflaion across groups of counries, ye such comparisons can resul in inconsisencies. We address wo problems: (i) how o measure aggregae real oupu and inflaion for groups of counries and (ii) how o consruc measures of real GDP for a group of counries where he counry measures of real GDP are consisen across ime and space. A mehod is proposed for harmonizing conflicing esimaes of OECD member-counry real GDP, ensuring consisency over space and overall group consisency over ime. A new measure of OECD inflaion is also proposed. JEL Classificaion Numbers: C43, C82, E01. Key Words: Purchasing Power Pariies, PPPs, ICP, OECD counry saisics, inflaion, price and volume indexes, Fisher indexes, counry compeiiveness. *Conac auhor: School of Economics & Cenre for Applied Economic Research, Universiy of New Souh Wales, Sydney 2052 Ausralia, K.Fox@unsw.edu.au, Tel: Acknowledgemens: The auhors hank Ber Balk, Yuri Dikhanov, Dennis Fixler, Anne-Sophie Fraisse, Rober Inklaar, Paul McCarhy, Prasada Rao, Marshall Reinsdorf and Paul Schreyer for helpful commens on an earlier draf. Financial suppor from he Social Sciences and Humaniies Research Council of Canada and he Ausralian Research Council s Discovery Program (DP ) is graefully acknowledged. 1

3 Economericians have an ambivalen aiude owards economic daa. A one level, he daa are he world ha we wan o explain, he basic facs ha economiss purpor o elucidae. A he oher level, hey are he source of all our roubles. Zvi Griliches (1985; 196) 1. Inroducion Providing consisen esimaes of real GDP across counries and ime is imporan for many policy-relevan purposes, such as assessing convergence of living sandards; see Eurosa (2012) and he World Bank (2013). The OECD publishes esimaes of Purchasing Power Pariies (PPPs) on an annual basis and hese PPPs can be used o generae esimaes of real GDP for member counries ha are comparable across counries for he given year. However, he resuling esimaes of relaive real GDP are inconsisen wih naional esimaes of real GDP growh for he member counries. We use OECD daa for he years in order o sudy wo problems. Firs, how can esimaes be consruced of OECD aggregae real GDP and associaed measures of aggregae OECD inflaion? Index number heory is used o decompose naional nominal GDP ino price and quaniy (or volume) componens, bu consrucing esimaes of inflaion and real GDP growh for a group of counries ha use differen currencies is a more complicaed operaion. 1 Second, how can PPP informaion be used in conjuncion wih counry daa on real GDP growh o consruc esimaes of OECD member counry real GDP ha are in principle comparable across space and ime? Using our proposed soluion as a benchmark, we show ha if PPP daa are available only infrequenly, as is he case for he World Bank provided PPP daa used in he Penn World Tables, hen esimaes can differ considerably as new PPP informaion becomes available. This is raher inconvenien: sudies of compeiiveness and living sandards convergence across counries will wan o use real GDP series ha are no subjec o violen revision. In secion 2, we sudy he firs measuremen problem using jus naional daa and exchange rae informaion. Secions 3-6 use he OECD PPP daa and sudy he second measuremen problem (and revisi he firs problem). In secion 3, we propose a harmonized mehod for consrucing esimaes of member counry GDP volumes ha are comparable across ime and space. In secions 4 and 5, we compare our harmonized esimaes wih esimaes of member counry real GDP ha are generaed by using PPP 1 A he naional level, curren price (value) daa can ypically be decomposed ino a volume (or quaniy) series and price series. A he inernaional level, a second price componen eners he picure in he form of a conversion rae from he domesic o a common currency. The implicaion is ha values can be expressed a curren marke exchange raes (or curren inernaional prices, if purchasing power pariies - PPPs - are used); and a consan exchange raes (or consan inernaional prices). OECD (2001; 6). 2

4 daa for only one year. These base year esimaes are hen projeced o all oher years using naional growh raes of GDP. Secion 4 uses he PPPs for 2000 and secion 5 uses he PPPs for We find ha he resuling wo panel ses of real GDP esimaes are very differen from each oher and hey also are very differen from our harmonized esimaes developed in secion 3. Secion 6 considers he conex where PPPs are only available infrequenly, as is he case wih World Bank provided PPPs. 2 We use PPP informaion for 2000 and 2012 o provide inerpolaed esimaes of counry volumes, finding ha he curren inerpolaion mehod implemened in he widely used Penn World Tables did no work well wih our OECD daa base. In conras, we find ha our proposed mehod produces esimaes ha are much closer o our preferred harmonized esimaes of secion 3. Secion 7 concludes. 2. OECD Growh and Inflaion Using Marke Exchange Raes Our firs measure of aggregae GDP growh over he member counries in he OECD during he years uses naional growh raes of GDP and domesic prices convered ino US dollars a marke exchange raes. The aggregaion principle used o form OECD aggregae GDP volumes and prices in his secion is he same ha is used o aggregae prices and quaniies across differen regions in a counry: each commodiy in each region is regarded as a separae commodiy in he index number formula. In wha follows, we use he OECD ordering of counries, which is as in Table 1. Table 1: OECD Counry Codes 1= Ausralia 10= France 19= Korea 28= Slovenia 2= Ausria 11= Germany 20= Luxembourg 29= Spain 3= Belgium 12= Greece 21= Mexico 30= Sweden 4= Canada 13= Hungary 22= Neherlands 31= Swizerland 5= Chile 14= Iceland 23= New Zealand 32= Turkey 6= Czech Republic 15= Ireland 24= Norway 33=U.K. 7= Denmark 16= Israel 25= Poland 34= Unied Saes 8= Esonia 17= Ialy 26= Porugal 9= Finland 18= Japan 27=Slovak Republic The counry values for nominal GDP in he naional currencies for he years can be obained from he OECD elecronic daa base, OECD.Sa. 3 We conver hese 2 For example, he World Bank provided PPPs for 155 counries for he year 2005 and has jus provided a new se of PPPs for How can hese wo benchmark ses of PPPs be used in conjuncion wih naional daa in order o provide esimaes of counry real GDP ha are comparable across all years from 2005 o 2012? The inerpolaion mehod explained in secion 6 could be used in his conex. 3 OECD.Sa Table B1-GE: Gross domesic produc (expendiure approach); Naional currency, curren prices, millions, annual daa. 3

5 esimaes ino billions and denoe he esimae for counry n in year by V n. The corresponding volume esimaes can be obained from he same source, 4 and we similarly conver hese esimaes ino billions and denoe hese volumes (or quaniies) by Q n for n = 1,...,34 and = 2000,...,2012. The corresponding counry price level for counry n in year is defined as P n V n /Q n for n = 1,...,34 and = 2000,...,2012. These naional price levels and volumes are lised in he Appendix; see Tables A1 and A2. Since he counry volumes Q n are measured in domesic currency unis (which are no comparable across counries), we need o conver he domesic nominal values of GDP ino common currency unis using he average exchange raes for each year. In principle, he numeraire counry could be any of he 34 OECD counries bu i seems reasonable o choose he larges counry as he numeraire counry. The OECD has convenienly done his for us, convering each counry s nominal GDP ino US dollars a he average marke exchange raes for he given year. 5 We conver hese esimaes ino billions and denoe he US dollar esimae for nominal GDP for counry n in year by v n. The year, counry n US dollar price level for GDP, p n, is iniially defined as v n /Q n where he counry volumes or real oupus Q n have already been defined using naional daa. The resuling p n were normalized so ha p 2000 n = 1 for n = 1,...,34. The Q n were hen normalized in he opposie direcion so ha US dollar values were preserved. Denoe he resuling normalized Q n as q n for n = 1,...,34 and = 2000,..., These US dollar price levels p n and he corresponding volumes q n are lised in Tables A3 and A4 in he Appendix. We are now in a posiion o calculae aggregae OECD real oupu and he corresponding price level for he years using he price and volume daa, p n and q n, as inpus ino an index number formula. I will be useful o discuss he choice of index number formula in he conex of providing index levels for wo periods, say periods 0 and 1. 7 Suppose here are N commodiies o be aggregaed. Denoe he price and quaniy vecors for period by p [p 1,...,p N ] and q [q 1,...,q N ] for = 0,1. The value of ransacions in he N commodiies during period is defined as v N n=1 p n q n p q. 8 The problem of choosing funcional forms for he price and quaniy indexes is usually phrased as follows: find wo suiable funcions of 4N variables, a price index funcion P(p 0,p 1,q 0,q 1 ) and a quaniy index funcion Q(p 0,p 1,q 0,q 1 ), such ha he produc 4 OECD.Sa TableB1-GE: Gross domesic produc (GDP); Naional currency, consan prices, naional base year, millions, annual daa. 5 OECD.Sa Table B1-GE: Gross domesic produc (expendiure approach); US dollars, curren prices, curren exchange raes, millions, annual daa. 6 Noe ha v n = p n q n for n = 1,...,34 and = 2000,...,2012. For each n, he US dollar volumes q n are proporional o he naional volumes Q n ; i.e., we have q n = λ n Q n for = 2000,...,2012 for each counry n where λ n is he facor of proporionaliy for counry n. 7 For maerials on he hisorical developmen of index number heory, see Diewer (1993) and Balk (2008). 8 The inner produc of wo vecors x [x 1,...,x N ] and y [y 1,...,y N ] of he same dimension N is defined as x y N n=1 x n y n. 4

6 of hese wo funcions is equal o he value raio, v 1 /v 0. Thus he funcions P and Q are o saisfy he following equaion: (1) p 1 q 1 /p 0 q 0 = P(p 0,p 1,q 0,q 1 )Q(p 0,p 1,q 0,q 1 ). I can be seen ha if he funcional form for eiher he price or quaniy index is deermined hen he funcional form of he corresponding quaniy or price index is also deermined using equaion (1). 9 Two naural choices for he funcional form for he price index are he well-known Laspeyres and Paasche price indexes, P L and P P, defined as follows: 10 (2) P L (p 0,p 1,q 0,q 1 ) p 1 q 0 /p 0 q 0 ; (3) P P (p 0,p 1,q 0,q 1 ) p 1 q 1 /p 0 q 1. Using (1), i can be seen ha quaniy indexes ha mach up wih P L and P P are Q P and Q L defined as follows: (4) Q P (p 0,p 1,q 0,q 1 ) p 1 q 1 /p 1 q 0 ; (5) Q L (p 0,p 1,q 0,q 1 ) p 0 q 1 /p 0 q 0. The Paasche and Laspeyres price and quaniy indexes are equally plausible. The problem is ha hey can generae quie differen esimaes of growh and inflaion. A naural soluion o his problem is o ake a symmeric average of hese wo equally plausible esimaes; aking he geomeric mean of hese wo price indexes (and of he wo corresponding quaniy indexes) leads o indexes ha have very good axiomaic properies. 11 This leads o he Fisher (1922) ideal price and quaniy indexes, P F and Q F, defined as follows: 12 (6) P F (p 0,p 1,q 0,q 1 ) [P L (p 0,p 1,q 0,q 1 )P P (p 0,p 1,q 0,q 1 )] 1/2 ; (7) Q F (p 0,p 1,q 0,q 1 ) [Q L (p 0,p 1,q 0,q 1 )Q P (p 0,p 1,q 0,q 1 )] 1/2. 9 Once P and Q saisfying (1) have been chosen, he corresponding price levels for periods 0 and 1, say P 0 and P 1, and he corresponding quaniy (or volume) levels for periods 0 and 1, say Q 0 and Q 1, are generally deermined as follows: P 0 1; P 1 P(p 0,p 1,q 0,q 1 ); Q 0 v 0 = p 0 q 0 and Q 1 v 0 Q(p 0,p 1,q 0,q 1 ) = v 1 / P(p 0,p 1,q 0,q 1 ). Noe ha he price and quaniy indexes can be inerpreed as raios of aggregae price and quaniy levels; i.e., we have P(p 0,p 1,q 0,q 1 ) = P 1 /P 0 and Q(p 0,p 1,q 0,q 1 ) = Q 1 /Q I can be seen ha he Laspeyres price index uses he baske of period 0 quaniies, q 0, and prices ou his baske a he prices of period 0 (in he denominaor) and prices ou he same baske a he prices of period 1 (in he numeraor) and akes he raio of hese coss as he price index. The Paasche index is similar bu uses he baske of period 1 quaniies, q 1, as he common quaniy vecor in he numeraor and denominaor. 11 See Fisher (1922) and Diewer (1992) (1997). 12 I can be verified ha P F Q F = v 1 /v 0 ; i.e., he Fisher price and quaniy indexes saisfy equaion (1). 5

7 There is one more choice ha needs some discussion: namely, should fixed base or chained Fisher indexes be used when aggregaing over many periods? The chain sysem measures he change in prices going from one period o a subsequen period using a bilaeral index number formula involving he prices and quaniies peraining o he wo adjacen periods. 13 These one period raes of change (he links in he chain) are hen cumulaed o yield he relaive levels of prices over he enire period under consideraion. If he bilaeral price index is P, he chain sysem generaes he following sequence of price levels for he firs hree periods: (8) 1, P(p 0,p 1,q 0,q 1 ), P(p 0,p 1,q 0,q 1 ) P(p 1,p 2,q 1,q 2 ). The fixed base sysem of price levels using he same bilaeral index number formula P simply compues he level of prices in period relaive o he base period 0 as P(p 0,p,q 0,q ). The fixed base sequence of price levels for periods 0, 1 and 2 is: (9) 1, P(p 0,p 1,q 0,q 1 ), P(p 0,p 2,q 0,q 2 ). There are wo major problems associaed wih he use of fixed base indexes in he conex of annual ime series daa: (i) over longer periods of ime, i becomes more difficul o mach up producs in he curren period wih he corresponding producs in a disan base period, leading o less accurae index numbers; and (ii) fixed base indexes are subjec o revisions (ha can be subsanial) when he base period is finally changed. When using fixed base Paasche or Laspeyres indexes, he revision problem can become massive. 14 Thus a major advanage of he chain sysem in he conex of aggregaing annual daa is ha chaining will reduce he spread beween he Paasche and Laspeyres indexes. 15 These wo indexes each provide an asymmeric perspecive on he amoun of price change ha has occurred beween he wo periods under consideraion and i could be expeced ha a single poin esimae of he aggregae price change should lie beween hese wo esimaes. Chaining will usually lead o a smaller difference beween he wo and hence o esimaes ha are closer o he ruh The chain principle was inroduced independenly ino he economics lieraure by Lehr (1885; 45-46) and Marshall (1887; 373). Boh auhors observed ha he chain sysem would miigae he difficulies due o he inroducion of new commodiies ino he economy, a poin also menioned by T.P. Hill (1993; 388). Fisher (1911; 203) inroduced he erm chain sysem. 14 The US Bureau of Economic Analysis used o provide long erm esimaes of US GDP back o 1926 using fixed base Laspeyres volume indexes. When he base year was changed, he resuling Laspeyres esimaes of real GDP growh changed massively and his fac led he BEA o swich o chained Fisher indexes in he early 1990s. 15 See Diewer (1978; 895) and T.P. Hill (1988) (1993; ). 16 There is a more elaborae jusificaion for chaining annual daa ha is based on aggregaing over observaions ha have he mos similar price srucures; see R.J. Hill (2001), (2004) (2009) and Diewer (2009). Typically, adjacen annual observaions will have more similar price srucures han a pair of 6

8 For year = 2001,...,2012, denoe he chained Fisher aggregae OECD volume level for by Q and he corresponding US dollar price level by P, and define he OECD volume growh rae γ and he corresponding OECD US dollar inflaion rae ρ in percenage poins as follows: (10) γ 100[(Q /Q 1 ) 1] ; (11) ρ 100[(P /P 1 ) 1]. The chained Fisher OECD aggregae price and volume levels, P and Q, for he years are lised in Table 2 along wih he corresponding percenage poin annual growh raes, ρ and γ, for he years For comparison purposes, we also provide he aggregae OECD chained Laspeyres and Paasche indexes, P L and P P, over he same period. I can be seen ha he chained Fisher, Laspeyres and Paasche price levels are all very close o each oher so ha for his paricular applicaion, he choice of index number formula does no maer very much. Table 2: OECD Annual Aggregae Volumes Q and Price Levels in US Dollars P, P L and P P, Price Levels in Euros P EU, PPP Price Levels P PPP and Percenage Poin Changes, Year Q P P L P P γ ρ ρ EU P EU P ICP observaions chosen from differen decades. However, i is no always bes o use chained indexes. T.P. Hill (1993; 388), drawing on he earlier research of Szulc (1983) and T.P. Hill (1988; ), noed ha i is no appropriae o use he chain sysem when prices oscillae or bounce o use Szulc s (1983; 548) erm. This bouncing phenomenon can occur when aggregaing subannual daa when here are seasonal flucuaions or periodic sales (deeply discouned prices). However, in he conex of more or less smoohly rending prices and quaniies as is he usual case using annual daa, T.P. Hill (1993; 389) recommended he use of chained symmerically weighed indexes such as he Fisher ideal index. Thus in his paper, we will use chained Fisher indexes when aggregaing over counries. 7

9 The sample average of he year o year growh raes for OECD real GDP using US dollar weighs, he γ, was 1.58% per year. I can be seen ha here was only one year where OECD real growh was negaive: 2009 ( 3.64%). The sample average of he OECD inflaion raes ρ (measured in US dollars a marke exchange raes) was 3.24% per year. However, wha is sriking is he variabiliy in hese US dollar inflaion raes. The principles used o consruc our OECD aggregae measures of real GDP, Q, are he same principles used o consruc counry wide esimaes of real GDP wihin a counry. Esimaes of real GDP aggregae oupu growh over regions wihin he counry use regional price levels as weighs for he regional volumes. In consrucing naional esimaes of real oupu, he naional saisician does no assume ha he quaniies or volumes in each region are comparable across regions; all ha is assumed is ha whaever is being measured a he regional level is comparable over ime. This is he same principle ha is being used o consruc he above OECD real oupu measures Q ; here is no assumpion ha he counry unis of measuremen are comparable across counries. The one difference in our suggesed mehod for consrucing OECD real GDP as opposed o mehods used o consruc naional esimaes of real GDP is ha we needed o conver naional values of GDP ino a common currency using annual average marke exchange raes. We chose o make his conversion using US dollars as he numeraire currency. If we chose anoher currency o be he numeraire currency, he uni of measuremen would change, bu he overall OECD growh raes for real GDP would remain he same; i.e., he γ lised in Table 2 do no change if we convered all counry nominal GDP esimaes ino a differen common currency a annual average marke exchange raes and hen applied he same mehodology o consruc he overall OECD volume esimaes. 17 On he oher hand, swiching o a differen numeraire currency dramaically affecs he inflaion raes ρ ; he OECD aggregae price level esimaes P and he resuling inflaion raes ρ defined by (11) change wih each choice of a numeraire currency. In order o illusrae he dependence of he above OECD GDP inflaion raes on he choice of he numeraire counry, we compued he aggregae OECD price and volume levels, P EU and Q EU, using Germany as he numeraire counry. Thus insead of using he US dollar esimaes for nominal GDP for counry n in year defined earlier by v n, for Euro zone counries we use he repored naional value esimaes of GDP. For non-euro zone counries, we convered he v n ino Euros using he implied OECD exchange rae ha can be obained by dividing he naional value esimae of GDP for Germany (or any oher Euro zone counry) by he corresponding US dollar measure. The same Fisher index 17 In order for his saemen o be rue, we need our chosen bilaeral index number formula o saisfy he following wo ess: Q(λp 0,p 1,q 0,q 1 ) = Q(p 0,p 1,q 0,q 1 ) for all scalar λ > 0 and Q(p 0,λp 1,q 0,q 1 ) = Q(p 0,p 1,q 0,q 1 ) for all scalar λ > 0. The Fisher, Laspeyres and Paasche bilaeral quaniy indexes all saisfy hese homogeneiy-in-prices properies. 8

10 number mehodology was hen used o consruc P EU and Q EU. The resuling Euro based price index P EU and inflaion growh raes ρ EU 100[(P EU /P EU 1 ) 1] are lised in he las wo columns of Table 2. Comparing he inflaion measures using he US and hen Germany as he numeraire counries shows ha he resuling price levels, P and P EU, and inflaion raes, ρ and ρ EU, are compleely differen. P rends upward from 1.00 in 2000 o end up a 1.45 in 2012 whereas he Euro based OECD price level rends downward o 0.89 in 2008 and hen rend upward o end up a 1.04 in The explanaion for hese diverging resuls is simple: hey are driven by large exchange rae movemens over he sample period. 18 Our conclusion a his poin is ha our firs approach o measuring OECD real oupu and inflaion using naional GDP daa and marke exchange raes beween counries is (perhaps) saisfacory for measuring real oupu bu ha i is no saisfacory for measuring inflaion. A saisfacory inflaion measure will be inroduced in he following secion when we inroduce our second approach o measuring aggregae OECD inflaion. The analysis presened in his secion made no assumpion ha he goods and services produced in any counry were comparable o he goods and services produced in any oher counry. In he following secion, i will be assumed ha he goods and services produced in each counry are comparable across counries and differen measures of OECD growh and inflaion will be derived. 3. OECD Growh and Inflaion Measuremen Using Annual PPP Informaion The OECD (in close cooperaion wih Eurosa) produces an annual series of Purchasing Power Pariies (PPPs) ha enable he comparison of real GDP of member counries wih each oher. 19 For each OECD counry n and each year, PPP n is an esimae of he number of unis of he naional currency of counry n ha is required o purchase one dollar of US GDP in year. 20 We divide he counry n nominal value of GDP in year in domesic currency, V n, by he corresponding PPP n in order o obain an esimae, r n, of counry n s real GDP in year in unis ha are comparable across counries for year : 21 (12) r n V n /PPP n ; n = 1,...,34; = 2000,..., US prices in erms of Euros declined markedly from 2000 o 2008 and his explains he large number of negaive ρ EU over his period; he number of Euros i ook o buy one US dollar in 2000, 2008 and 2012 was 1.085, and 0.778, respecively. 19 The consrucion of hese PPPs is explained in he Eurosa and OECD PPP Manual; see Eurosa (2012). The Inernaional Comparison Program (ICP) of he World Bank consruced PPPs for over 150 counries for 2005 and The ICP mehodology is explained in World Bank (2013). 20 OECD.Sa, Table 4: PPPs and Exchange Raes; PPPGDP; Purchasing Power Pariies for GDP; Naional currency per US dollar; Annual; This able is reproduced in he Appendix as Table A5. 21 These relaive GDP volume measures for year are no comparable across years. 9

11 Once he r n have been calculaed, hey can be summed so ha r n=1 34 r n and hen he year counry n share of OECD real oupu can be defined as follows: 22 (13) s n r n /r ; n = 1,...,34; = 2000,...,2012. These counry shares of OECD real GDP are lised in Table 3, which enables he comparison of GDP volumes across all OECD counries wihin each year. 23 Noe ha counry 34, he US, has he larges share (around 35-37%), followed by counry 18, Japan, (10-11%) and counry 11, Germany (7%). We can use his informaion o consruc esimaes of overall real GDP growh and inflaion across OECD counries. A naural mehod is o use he counry shares in Table 3 as weighs for naional raes of growh of real GDP. The year growh facor for counry n can be defined as Q n /Q n 1 where Q n is counry n s GDP volume in year, and he OECD Laspeyres ype growh facor (or chain link) for year, Γ L, as he following weighed average of he naional growh facors: (14) Γ L n=1 34 s n 1 (Q n /Q n 1 ) ; = 2001,...,2012. The measure of OECD GDP volume growh defined by (14) is he mehod used by he OECD o calculae heir official measure of OECD volume growh. I cerainly is a sensible measure, using counry (one plus) growh raes going from year 1 o year, Q n /Q n 1, weighed by he counry real volume shares s n 1 for year 1, which were derived using PPPs. However, he above formula suffers from he same problem ha he sandard Laspeyres formula has: namely, i does no rea he periods in a symmeric fashion. The counerpar o he Laspeyres-ype formula defined by (14) is he following Paasche-ype formula: 24 (15) Γ P [ n=1 34 s n (Q n /Q n 1 ) 1 ] 1 ; = 2001,..., Noe ha he counry shares s n can be consruced wihou using counry exchange raes (in principle). Using definiions (12) and (13), i can be seen ha he s n can be wrien in he following form: s n = [V n /PPP n ]/[ N i=1 (V i /PPP i )] for all n and. Compare hese real shares s n wih he corresponding counry US dollar shares S n = [V n /e n ]/[ N i=1 (V i /e i )] defined in he Appendix. All of he measures derived in his secion are independen of counry exchange raes. 23 Row n+1 in he Table gives he shares for counry n where we use he sandard ordering of OECD counries lised in he previous secion. Since he PPPs used by he OECD are invarian o he choice of he numeraire counry (up o a scalar facor), i can be verified ha he counry shares lised in Table 3 are also invarian o he choice of numeraire counry. 24 Suppose ha here is only one homogeneous commodiy in each counry s GDP. Then he volume for counry n in year, Q n, should be equal o he number of unis of his homogeneous commodiy. Under hese condiions, i can be seen ha boh Γ L and Γ P equal 34 n=1 Q n / 34 n=1 Q 1 n. 10

12 Table 3: Counry Shares of OECD Real GDP n 2000 s n 2001 s n 2002 s n 2003 s n 2004 s n 2005 s n 2006 s n 2007 s n 2008 s n 2009 s n 2010 s n 2011 s n 2012 s n Noe: n denoes he counry code, given in Table 1. 11

13 The corresponding Fisher-ype formula for OECD volume growh for year is defined as follows: 25 (16) Γ F [Γ L Γ P ] 1/2 ; = 2001,...,2012. The growh facors (or chain link indexes) defined by (14)-(16) can be muliplied ogeher o generae OECD aggregae volume levels. The growh facors can also be ransformed ino growh raes, γ L, γ P and γ F (in percenage poins), by using he following definiions for = 2001,...,2012: (17) γ L 100[Γ L 1] ; γ P 100[Γ P 1] ; γ F 100[Γ F 1]. The annual OECD volume growh measures defined by (17) as well as our earlier US dollar weighed measures γ are lised in Table 4. I can be seen ha he Laspeyres, Paasche and Fisher measures of OECD growh explained in his secion are virually idenical so ha moving from he OECD Laspeyres-ype measure of overall volume growh o he Fisher measure did no make much difference for his daa se. 26 I can also be seen ha our preferred Fisher measure of OECD growh in comparable unis across counries, γ F, grew on average abou 1/10 of a percenage poin more rapidly per year han our preferred measure of OECD GDP growh using US dollar weighs, γ. Alhough his is no a large difference in growh raes, i is significan and so users need o decide which measure, γ F or γ, bes suis heir needs. The measure γ can be defined using jus naional informaion on domesic price and quaniy (or volume) indexes and exchange raes while he measure γ F requires informaion on domesic values, domesic volume indexes and PPPs. PPPs are no likely o be nearly as accurae as naional measures of price and volume change due o he difficulies in maching producs across counries. There are addiional difficulies wih he reamen of inernaional rade in he consrucion of PPPs. The γ measure has he problem ha large flucuaions in exchange raes can lead o flucuaions in he γ while he PPP based γ F measures are heoreically independen from exchange rae movemens. 27 Thus one has o weigh he disadvanage of possibly less 25 If he PPPs are independen of he choice of he numeraire counry (up o a scalar facor), hen he growh facors, Γ L Γ P and Γ F will no depend on he choice of he numeraire counry. 26 Recall ha he official OECD measure of real GDP growh is he Laspeyres measure, γ L. Our esimaes differ slighly from he official measures due o rounding. The exchange-rae-weighed growh raes γ should be somewha lower han he PPP-weighed growh raes γ F due o he Balassa-Samuelson effec and his expecaion is realized for he OECD daa. We would expec he divergence o grow as less rich counries are added o he lis of counries. 27 Exchange rae movemens do no direcly affec he domesic raes of growh (he Q n /Q n 1 ) bu as we have seen, hey do affec he weighs used o aggregae he counry real growh raes ino he overall OECD 12

14 reliable PPPs agains he advanage of having aggregae growh measures ha are independen from exchange rae movemens. 28 Table 4: Annual Percenage Poin Changes in OECD PPP Based Laspeyres, Paasche and Fisher Volume Measures γ L, γ P and γ F, US Dollar Weighed Volume Measures γ and Laspeyres, Paasche and Fisher PPP Based Inflaion Measures, ρ L, ρ P and ρ F : Year Average γ L γ P γ F The OECD real oupu shares, s n defined by (13), can also be used as weighs for naional GDP inflaion raes. We define he OECD Laspeyres, Paasche and Fisher PPP based chain link price indexes, Π L, Π P and Π F for = 2001,...,2012, as follows: 29 (18) Π L n=1 34 s n 1 (P n /P n 1 ) ; Π P [ n=1 34 s n (P n /P n 1 ) 1 ] 1 ; Π F [Π L Π P ] 1/2. These chain link indexes can be muliplied ogeher o generae he corresponding OECD aggregae price levels. The resuling Fisher OECD price level index for year is denoed by P PPP and i is lised in he las column of Table The inflaion growh facors can Laspeyres, Paasche and Fisher growh raes. Exchange rae flucuaions are large enough o maerially affec he weighs, which in urn lead o maerial flucuaions in he overall OECD volume growh raes. 28 I will generally be he case ha he s n will be greaer han he corresponding S n for counries n ha are relaively poor and hus he index of OECD aggregae real GDP growh defined in secion 2 will end o be a more pluocraic index (since rich counries ge larger share weighs in his index) compared o he more democraic index of OECD aggregae real GDP growh defined in secion 3. Thus one could choose beween he wo indexes based on one s preferences over weighs. We owe his poin o Marshall Reinsdorf. 29 The official OECD measure of household inflaion over member counries is he Laspeyres measure defined in (18) where household consumpion replaces GDP; see he OECD (2014). 30 This price index saisfies he ime reversal es whereas is Laspeyres and Paasche counerpars do no saisfy his imporan es. Hence he Fisher PPP based inflaion index P PPP is our preferred measure of OECD aggregae inflaion. γ ρ L ρ P ρ F 13

15 also be ransformed ino growh raes, ρ L, ρ P and ρ F in percenage poins, by using he following definiions for = 2001,...,2012: (19) ρ L 100[Π L 1] ; ρ P 100[Π P 1] ; ρ F 100[Π F 1]. These PPP based inflaion raes (in percenage poins) are lised in he las 3 columns of Table 4. The sample averages of he ρ L, ρ P and ρ F are 2.17, 2.10 and 2.14 percenage poins. Viewing Table 4, i can be seen ha here are some significan differences beween he hree measures of OECD inflaion ha are PPP based. 31 Comparing he numbers in ables 2 and 4, i can be seen ha he PPP based esimaes of OECD inflaion are much more reasonable han he esimaes ha were based on counry exchange raes ha were lised in Table 2, he ρ and ρ EU. Our conclusion is ha he OECD Fisher price index P PPP is a much beer measure of OECD inflaion han he indexes ha used exchange raes insead of PPPs. Now we come o he mos difficul problem: how can we use PPP informaion and naional growh raes o obain esimaes of member counry GDP volumes ha are comparable across ime and space? The Eurosa (2012) Manual offers he following advice: To race he evoluion of relaive GDP volume levels beween counries over ime, i is necessary o selec one of he reference years as a base year and o exrapolae is relaive GDP volume levels over he oher years. Exrapolaion is done by applying he relaive raes of GDP volume growh observed in he differen counries. This provides a ime series of volume indices a a consan uniform price level ha replicaes exacly he relaive movemens of GDP volume growh of each counry. Eurosa (2012; 18). We implemen his sraegy in secions 4 and 5 below, where we choose he relaive counry GDP volumes given by he counry shares of OECD aggregae GDP for 2000 (secion 4) and for 2012 (secion 5) and we use naional growh raes for counry GDP volumes o exrapolae hese base shares o all ime periods. However, i will be seen ha he resuling comparable counry volumes over ime and space differ considerably, depending upon which base year is chosen. This is raher inconvenien: sudies of compeiiveness of OECD counries and living sandards convergence across counries will wan o use counry volume series ha are no subjec o violen revision In view of hese differences in he hree indexes of OECD GDP inflaion, i may be preferable for he OECD o replace heir Laspeyres ype indexes of OECD household inflaion by heir Fisher counerpars. 32 McCarhy (2013; ) explains in some deail why esimaes of real GDP based on naional growh informaion do no mach up exacly wih relaive GDP esimaes based on PPP benchmark informaion. The PPP informaion is generally no as accurae as naional price index informaion due o he difficuly of maching represenaive producs across counries. However, counry mehodology for consrucing 14

16 Our suggesed soluion o he problem of harmonizing naional growh raes of GDP wih he counry shares of OECD aggregae real GDP ress on wo principles. Firs, he resuling harmonized esimaes of counry volumes mus be consisen wih he real annual cross counry volume shares s n lised in Table 3 above. Second, OECD aggregae real GDP growh mus be equal o he raes of aggregae growh generaed by our recommended Fisher indexes Γ F defined by (16). Using hese principles, he counry GDP volumes are uniquely deermined (up o a scalar unis-of-measuremen facor). To see his, firs define he OECD volume index ha chains he Γ F defined by (16) ino a ime series index, Q H. Define Q H as follows: (20) Q H ; Q H Q H 1 Γ F ; = 2001,...,2012. Now use he counry shares of OECD real GDP s n lised in Table 3 and he aggregae index Q H o define he following preliminary harmonized counry volumes for counry n in year, q Hn, as follows: (21) q Hn Q H s n ; n = 1,...,34; = 2000,...,2012. Noe ha for each year, n=1 34 q Hn = n=1 34 Q H s n = Q H ( n=1 34 s n ) = Q H and so he harmonized volumes saisfy he wo principles lised above. In principle, he counry volumes defined by (21) are independen of counry prices and exchange raes. 33 I is of ineres o define US dollar prices for real GDP for each counry. Recall ha he value of counry n s nominal GDP convered ino US dollars a marke exchange raes for year was defined as v n. The corresponding harmonized US dollar price of a uni of (comparable across counries) real GDP for counry n in year is defined as follows: (22) p Hn v n /q Hn ; n = 1,...,34; = 2000,...,2012. In order o make he harmonized volumes and prices defined by (21) and (22) comparable o he counry prices and volumes expressed in US dollars ha are lised in he Appendix naional price indexes differs considerably across counries; e.g., some counries may use ou of dae reference expendiure baskes, some counries use Carli indexes a he elemenary level while ohers use he Jevons or Duo indexes which generae lower esimaes of inflaion a he elemenary level and some counries may use qualiy adjusmen mehods more exensively han ohers. All of hese mehodological differences lead o inconsisencies beween he ime series and cross secional esimaes. Finally, he index number formulae used a he naional levels and in he consrucion of he benchmark PPPs are in general no ransiive and so i is impossible o achieve perfec consisency. 33 However, in pracice, he PPPs do no do a perfec job in eliminaing exchange rae effecs (since adjused exchange raes are used in place of rue PPPs o deflae inernaional rade flows). If he relaive PPPs are independen of he choice of he numeraire counry, hen he relaive volumes defined by (20) will also be independen of his choice. 15

17 in ables A3 and A4, we impose a normalizaion on he prices defined by (22) ha makes he price level for he US in 2000 equal o uniy; i.e., we divide all prices defined by (22) by a consan ha ses he resuling p H equal o 1 and he quaniies or volumes defined by (21) are all muliplied by his consan. The resuling normalized q Hn and p Hn are lised in ables 5 and 6. Noe ha q H = q and p H = p = 1 so ha counry GDP volumes are measured as muliples of a bundle of US GDP in he year Thus he price levels in Table 6 measure he US dollar value of consan bundle of GDP ha is (in heory) comparable across counries. The price levels in Table 6 are comparable across space and ime, whereas he price levels p n lised in Table A3 of he Appendix are only comparable across ime for each counry. From Table 6, i can be seen ha he counries wih he lowes price levels (in US dollars) in 2012 are counries 13, 21, 25 and 32 (Hungary, Mexico, Poland and Turkey) wih price levels in he 0.76 o 0.77 range. Counries wih relaively high price levels in 2012 are counries 1 (Ausralia, p H = 2.00), 4 (Canada, p H = 1.62), 7 (Denmark, 1.77), 9 (Finland, 1.58), 18 (Japan, 1.76), 20 (Luxembourg, 1.56), 23 (New Zealand, 1.55), 24 (Norway, 2.01), 30 (Sweden, 1.69) and 31 (Swizerland, 1.96). These price level esimaes are (imperfec) 34 indicaors of he compeiveness of he counry on inernaional markes, wih lower price levels indicaing greaer compeiveness. A problem wih he volume esimaes lised in Table 5 is ha hey do no respec naional growh raes of GDP by counry; only he aggregae OECD growh rae is respeced. In he following wo secions, we will derive alernaive counry volume esimaes ha are comparable over ime and space. These alernaive esimaes will respec counry growh raes bu hey will no reproduce he real OECD counry expendiure shares lised in Table 3 for all ime periods. 34 The price levels p Hn are imperfec indicaors of compeiveness because no all componens of GDP are inernaionally raded. Moreover, hese price levels are no independen of he choice of he numeraire currency (US dollars in his case). They are also imperfec because hey depend heavily on he accuracy of he underlying PPPs and hese PPPs are subjec o considerable error variances due o he difficulies involved in maching produc prices (and quaniies) across counries. 16

18 Table 5: Harmonized OECD Counry GDP Volumes in Comparable US Dollar Unis of Measuremen q Hn n 2000 q Hn 2001 q Hn 2002 q Hn 2003 q Hn 2004 q Hn 2005 q Hn 2006 q Hn 2007 q Hn 2008 q Hn 2009 q Hn 2010 q Hn 2011 q Hn 2012 q Hn

19 Table 6: Harmonized OECD Counry GDP Price Levels in Comparable US Dollar Unis of Measuremen p Hn n 2000 p Hn 2001 p Hn 2002 p Hn 2003 p Hn 2004 p Hn 2005 p Hn 2006 p Hn 2007 p Hn 2008 p Hn 2009 p Hn 2010 p Hn 2011 p Hn 2012 p Hn

20 4. OECD Growh and Inflaion Using Counry Annual GDP Volume Growh Raes and Base Period Shares of OECD Real GDP We generae comparable counry GDP volume esimaes for OECD counries covering he period by using he real GDP counry volume shares for 2000, he s n 2000 lised in Table 3 above, along wih he naional growh raes of counry real GDP relaive o 2000, he Q n /Q n 2000 lised in Table A2 of he Appendix. This is a ypical sraegy in forming esimaes of real GDP ha rely on PPPs ha are only produced infrequenly. Our purpose in lising hese esimaes is o evaluae how differen he resuling esimaes are from our preferred harmonized volume esimaes, q Hn, lised in Table 5 above. Define preliminary base period esimaes of counry GDP volumes for year and counry n, q Bn, as follows: (23) q Bn s n 2000 (Q n /Q n 2000 ) ; n = 1,...,34 ; = 2000,...,2012. The above esimaes are obviously based on he counry shares of real OECD GDP ha prevailed in 2000 (he s n 2000 ) and he long erm counry growh raes of real GDP (he Q n /Q n 2000 ). The companion counry US dollar price levels for counry n and year, p Bn, are defined as follows: (24) p Bn v n /q Bn ; n = 1,...,34 ; = 2000,...,2012 where v n is he nominal value of GDP for counry n in year convered ino US dollars a marke exchange raes for ha year. In order o make he volumes and prices defined by (23) and (24) comparable o he counry prices and volumes expressed in US dollars ha are lised ables 5 and 6 in he previous secion, we impose a normalizaion on he prices defined by (24) ha makes he price level for he US in 2000 equal o uniy; i.e., we divide all prices defined by (24) by a consan ha ses he resuling p B equal o 1 and he volumes defined by (23) are all muliplied by his consan. The resuling normalized p Bn are lised in Table The enries in ables 5 and 6 enable one o recover he US dollar values of GDP, equal o v n = p Hn q Hn for n = 1,...,34 and = 2000,...,2012. Then he q Bn can be recovered as v n /p Bn. 19