Capital Services and Total Factor Productivity Measurements: Impact of Various Methodologies for Belgium

Transcription

1 Capal Servces and Toal Facor Producvy Measuremens: Impac of Varous Mehodologes for Belgum Bernadee Baour, Geer Bryon, Chanal Kegels, 1 Federaal Planbureau March 2007 Absrac Ths Workng Paper presens he dfferen mehodologes currenly used o consruc a volume ndex of capal servces and analyzes he effecs of mehodologcal changes on capal servces and oal facor producvy esmaes for Belgum over he perod The measuremen of capal servces s realzed n wo seps. Frs, producve capal socks have o be esmaed for each ype of asse. Two mehodologes are generally used: he geomerc and he hyperbolc profle. Secondly, hese socks are aggregaed, usng he user coss of capal (ex-ane or ex-pos approach) as weghs o derve an overall ndex. For he economy as a whole and he enre perod, under an ex-pos approach, he volume ndces of capal servces esmaed wh a hyperbolc age-effcency profle grow a a hgher rae han he ndces esmaed wh a geomerc profle. Ths general concluson s, however, no observed n every secor. Under an ex-ane approach, he dfferen volume ndces are que smlar for he whole economy, even f he ndces grow generally a a slghly hgher rae n he case of a geomerc paern. A hgher growh rae of he volume ndces generaes a hgher capal conrbuon and, consequenly, a lower fp conrbuon. Over long perods of me, he dfferen fp esmaes are relavely smlar. Over shorer perods, he dfferen mehodologes generae more sgnfcan varaons n he fp conrbuon. Jel Classfcaon C81, D24, O40. Keywords - Producve capal socks, User coss, Capal servces, TFP. 1 Acknowledgemens: The auhors are graeful o Chanal Vandevoorde from he ranslaon eam for her careful readng. The daa analysed are comng from he Belgan par of he euklems projec funded by he European Commsson, Research Drecorae General as par of he 6h Framework Programme, Prory 8, Polcy Suppor and Ancpang Scenfc and Technologcal Needs.

2 Execuve Summary Ths workng paper presens he dfferen mehodologes currenly used o consruc a volume ndex of capal servces and analyzes he effecs of mehodologcal changes on capal servces and oal facor producvy esmaes for Belgum over he perod Ths workng paper s realzed whn he framework of he European EUKLEMS projec, n whch he Federal Plannng Bureau parcpaes. TFP measuremens are essenal o sudy economc growh deermnans. They depend on he avalably and qualy of daa concernng he oher sources of growh: he conrbuon of capal and labour. Accordng o he OECD recommendaons, he approprae measuremen of capal npu for producvy sudes s he flow of servces produced by capal asses, raher han capal socks. In Belgum, as n oher many counres, no offcal daa ex on capal servces. The measuremen of he capal servces s realzed n wo seps. Frs, producve capal socks have o be esmaed for each ype of nvesmen goods called asses. Then, hese socks are aggregaed wh he user coss of capal as weghs o derve a global ndex measurng he producve conrbuon of all ypes of capal asses o oupu growh. Two mehodologes are generally used o consruc producve capal socks: he geomerc profle and he hyperbolc profle. For each profle, several assumpons have o be made such as he choce of he funconal form of he reremen funcon or of he maxmum servce lfe. For he compuaon of he user coss, we have he choce beween an ex-ane and an ex-pos rae of reurn of capal. Afer a heorecal presenaon of he dfferen mehodologes currenly used o consruc a volume ndex of capal servces, he resuls show ha for he whole economy and for he majory of capal asses, he esmaon of producve capal socks s hgher n level and growh rae, when s compued wh a hyperbolc age-effcency profle han when a geomerc profle s assumed. The dfferen mehodologcal opons whn each age-effcency profle also have consequences on producve capal socks. Concernng he user coss of capal, he evoluon of he esmaon seems o be more nfluenced by he choce beween an ex-pos and an ex-ane rae of reurn han by he assumed deprecaon profle (geomerc or hyperbolc). For he economy as a whole and he enre perod , under an ex-pos approach, he volume ndces of capal servces esmaed wh a hyperbolc age-effcency profle grow a a hgher rae han he ndces esmaed wh a geomerc profle. Ths general concluson s, however, no observed n every secor. Under an ex-ane approach, he dfferen volume ndces are que smlar for he whole economy, even f he ndces grow generally a a slghly hgher rae n he case of a geomerc paern. In he case of a hyperbolc profle, a longer maxmum servce lfe and/or a hgher β parameer (slower loss of effcency) ncrease he volume ndex of capal servces n level and n growh rae. A hgher growh rae of he volume ndces generaes a 2

3 hgher capal conrbuon o oupu growh and, consequenly, a lower fp conrbuon. Concernng he reremen funcon, he growh raes of ndces are slghly hgher wh a delayedlnear funcon, han wh a lognormal and are he lowes wh a lnear funcon. In he case of a geomerc profle, he choce beween he wo esed deprecaon raes has no nfluence on he volume ndex of capal servces for he oal economy. However, dfferences appear n he volume ndces of ndusres. Over long perods of me, he dfferen fp esmaes are relavely smlar. Over shorer perods, he dfferen mehodologes generae more sgnfcan varaons n he TFP conrbuon. Over he recen perod , he average annual TFP conrbuon o oupu growh for he whole economy s esmaed beween 0.04% and 0.27%, accordng o he made assumpons. I also appears ha, for he whole economy, he varaons n TFP conrbuon are larger when an expos user cos s used. In summary, hs sensvy analyss shows ha mehodologcal choces have ceran consequences on he volume ndex of capal servces and consequenly on he esmaon of he TFP conrbuon o growh. These consequences are more mporan when he analyss s realzed a he ndusry level and when he suded perod s shor, underlyng he mporance of long-erm perspecve o analyze producvy evoluon. I seems credble ha he approprae ageeffcency profle dffers accordng o he knd of asses and he ndusry n whch hese asses are used. Emprcal sudes on age-effcency and reremen profles of asses are necessary o deermne he bes appropraed profles. 3

4 Table of Conens Execuve Summary... 2 Table of Conens Inroducon The growh accounng framework Capal servces esmaes The producve capal sock The volume ndex of capal servces The ne and he gross capal sock 18 4 Resuls The daa Resuls Conclusons References Annex A.1 Descrpon of he secors A.2 Descrpon of he asses A.3 Average servce lfe of asses by ndusry A.4 Daa consrucon and assumpons A.4.1 Consrucon of long nvesmen seres crossng secors and producs 40 A.4.2 Consrucon of nvesmen seres n IT and communcaons equpmen 40 A.4.3 Consrucon of sofware nvesmen seres 42 A.4.4 Harmonzed prce daa 42 Papers ssued n he seres of he EU KLEMS projec

5 1. Inroducon One of he man sources of economc growh s he ncrease n he oal producvy of npus used n he producon process, whch reflecs greaer overall effcency n he ulsaon of npus due o nnovaon. Therefore, measurng oal facor producvy (TFP) s essenal o sudy economc growh deermnans. TFP measuremens depend on he avalably and qualy of daa concernng he oher sources of growh: he conrbuon of capal and labour. TFP growh s ndeed esmaed as a resdual n he decomposon of economc growh and ncludes, herefore, measuremen errors n he absence of perfec sascal measuremens of capal and labour, especally n erms of qualy mprovemens. In many producvy sudes, he conrbuon of capal s measured by usng socks of asses. However, accordng o he OECD Manuals Measurng producvy and Measurng capal, he approprae measuremen of capal npu s he flow of servces produced by capal asses, raher han capal socks for he followng reasons: The oher varables n he growh accounng model are all flows. The use of ne or gross socks s herefore no conssen. Gross or ne socks do no reflec he producve effcency of capal asses declnng wh he age. The gross capal sock values all asses n use as f hey were sll new and he ne capal sock measures he marke value of capal asses. The reducon n he value of fxed asses (deprecaon/consumpon of fxed capal) s no necessarly dencal o he loss of producve effcency. In calculang capal socks, each asse n he socks s weghed by s marke value, ndependenly of s servce lfe. Consequenly, o have a good measuremen of he TFP resdual, s mporan o evaluae flows of capal servces, even f no offcal daa exs, as s he case n Belgum. The measuremen of capal servces s realzed n wo seps. The frs sep consss n he calculaon of producve capal socks, whch are equvalen o he quany of servces produced by each ype of asse. The second sep s he consrucon of an aggregae measuremen of he producve conrbuon of he dfferen ypes of capal asses (he volume ndex of capal servces). The weghs used o derve an overall ndex are he user coss of capal or renal prces, whch correspond o he prce of he servces. To compue he producve capal socks and he user coss of capal, alernave mehodologes exs. The producve capal sock can be esmaed, for example, by usng a geomerc or hyperbolc age-effcency funcon. These funcons can be combned wh a normal, a lnear or a smulaneous ex reremen funcon. For he compuaon of he user coss, we have he choce beween an ex-ane and an ex-pos rae of reurn. The objecve of hs paper s o presen he dfferen mehodologes currenly used o consruc a volume ndex of capal servces and o es how sensve he measuremen of capal servces s o mehodologcal changes. We also evaluae he mpac of he dfferen capal servces esmaes on he measuremen of he oal facor producvy. We provde resuls for Belgum for 5

6 he perod Ths paper s realzed whn he framework of he European EUKLEMS projec, n whch he Federal Plannng Bureau parcpaes. Ths projec ams o analyze he evoluon of producvy a a dealed ndusry level by creang a daabase on measures of economc growh, producvy, employmen creaon, capal formaon and echnologcal change a he ndusry level for all European Unon member saes from 1970 onwards. In hs daabase, capal servces are measured accordng o a common mehodology ha s compared o he oher mehodologes developed n hs paper. The srucure of he paper s as follows. Chaper 2 places he analyss n he growh accounng framework, chaper 3 presens he man mehodologes o esmae capal servces, chaper 4 shows he resuls for Belgum and chaper 5 conans a concluson. 6

7 2. The growh accounng framework Capal and labour servces measuremens allow o assess he conrbuons of capal and labour npus o economc growh and o esmae he conrbuon of oal facor producvy (TFP), by usng he general growh accounng framework. Ths model, poneered by Solow (1957), sars wh a neoclasscal producon funcon 2, Y = Af ( K, L) where Y s he GDP level, A he echnology level, K he capal npu ha ncludes he dfferen ypes of capal npus (K 1, K 2,, K n ), L he labour npu. f ( K, L) f ( K, L) Y = Af ( K, L) + A K + A L K L Y A F K K K F L L L g = = + + Y A f ( K, L) K f ( K, L) L Under he assumpons of compeve produc and facor markes and consan reurns o scale, he producon facors are remuneraed a her margnal producvy, whch means: w = AF r = AF L K We can hen rewre he growh rae of he economy as: A K L g = + α + ( 1 α) A K L where α s he share of capal n he GDP and ( 1 α) he share of labour n he GDP. The par of GDP growh ha s no explaned by he capal and labour raes of growh s assumed o be he TFP growh and s called he Solow resdual. Dfferen esmaes of capal growh raes herefore have an mpac on he TFP resdual. 2 In hs producon funcon, he augmenaon of A s Hcks-neural. The echncal progress allows o produce more wh a same proporon of decrease n npus use. 7

8 3. Capal servces esmaes 3.1 The producve capal sock To quanfy he conrbuon of capal o he producon process, s recommended o esmae he flow of servces delvered by each ype of asse, durng a perod of me. These flows of servces called capal servces, correspond o a quany concep and no o a prce or value concep of capal. For example, he servce flows generaed by a ruck are onneklomeres, or cubc meres of sorage space for a warehouse (OECD (2001a)). The prce of capal servces s he user cos of capal or he renal prce. Due o wear and ear ha reduce he asse effcency, he quany of servces produced by an asse declnes durng s servce lfe (oal perod durng whch he asse remans n use n he producve process). These flows of servces produced by an asse durng he accounng perod are usually no observable and have o be measured by a proxy (Iomm, Jona-Lasno (2005)). Capal servces are radonally approxmaed by assumng ha he servce flows from an asse are a consan proporon of he producve capal sock of he asse. For an asse wh a servce lfe of T years, he producve capal sock s defned as a sum of he pas nvesmen weghed by he age-effcency profle whch capures he loss of producve effcency of he asse as ages. Insead of usng he concep of producve capal sock, he OECD Manual Measurng Capal says ha he asses of a parcular ype are convered no sandard effcency uns, because producve capal sock s a very dfferen concep wh regards o he usual ne or gross sock. There s very lle emprcal evdence on he way n whch asses lose her effcency wh age (OECD (2001a)). Two age-effcency profles are radonally used o descrbe he evoluon of he producve effcency of he asse wh s age: he geomerc profle and he hyperbolc profle. The measuremen of producve capal sock of he dfferen knds of asses s he frs sep o esmae he volume ndex of capal servces,.e. an aggregae measuremen of he producve conrbuon of he dfferen knds of asses Geomerc age-effcency profle The geomerc age-effcency profle assumes ha he producve effcency of an asse falls a a consan rae each year. I means ha effcency declnes by he larges amoun n he frs years of servce and hen by decreasng amouns each subsequen year. The geomerc profle s emprcally frequenly used, because of s smplcy. The geomerc profle s used by Sascs Canada, by many researchers lke Jorgenson (for example: Jorgenson and Sroh (2000)), Fraumen (1997), O Mahony (2002), Hulen and Wyckoff (1996) 3, Oulon (2001). The European EUKLEMS projec also apples a geomerc paern o esmae producve capal socks for all counres. 3 Ced n Oulon (2001). 8

9 The producve capal sock (S) for an asse a me s gven by: τ = 0 τ I τ S = (1 δ ) Schreyer (2003) I τ δ he geomerc depre- wh nvesmen a consan prce n asse of ype a me -τ and caon rae of he asse of ype. I s hen assumed ha he flow of capal servces K s proporonal o he producve sock S: K = λ S The proporonaly facor λ measures he ulsaon of capacy of he capal sock whch vares wh he busness cycle. For smplcy, hs facor s frequenly se o uny for all ypes of asses 4. The volume ndex of capal servces esmaed n hs case measures hen he poenal flow of capal servces ha could be provded by he capal sock and no he flow of servces ha were acually produced (OECD (2001a)). These flucuaons of he rae of ulsaon of he capal sock ha are no aken no accoun are pcked up by he resdual TFP. In he geomerc age-effcency funcon, maxmum servce lfe s assumed o go owards nfny. By smplfcaon, we have lmed n our calculaons he nfny o 2 mes he servce lfe of he asse. In he case of he geomerc profle, no reremen funcon s explcly formulaed (Schreyer, Dewer and Harrson (2005)). I s assumed ha he geomerc declne capures boh he effecs of wear and ear and reremen (Schreyer (2003)). So, no specfc reremen funcon descrbng he share of asses sll n servce durng he servce lfe s necessary n he formula. The deprecaon rae (δ) depends on he declnng balance rae (R) and he servce lfe of each asse n (δ=r/n). To esmae he deprecaon rae, we use, lke many auhors, he BEA declnngbalance raes gven by Fraumen (1997) and he servce lfe of he asses gven by he Belgan naonal accouns. Some auhors use he double declnng balance mehod (R=2). Oher auhors use drecly he deprecaon rae gven by Fraumen (1997). Fnally, for ICT asses, deprecaon raes are ofen se a he common raes used by Jorgenson and Sroh (2000) and Oulon (2001): for compung equpmen, for sofware and for communcaons equpmen Hyperbolc age-effcency profle Wh a hyperbolc age-effcency profle, he effcency declnes slowly n he frs perod and a an ncreasng rae owards he end of he asse s lfe. 4 See Schreyer e al.(2003) for anoher assumpon, ced n Schreyer (2003). 9

10 For each of he homogenous asse ypes, he producve sock a me s consruced as follows 5 : K = T τ = 0 τ τ h. F. I τ In hs expresson, I τ s he real nvesmen expendure on asse ype a me -τ. The facor h τ s he effcency profle. Ths funcon s declnng and akes values beween uny (when an asse s new) and zero (when he asse has los s oal producve capacy). F τ s he reremen funcon ha ndcaes he share of asses of age τ ha are sll n servce a me. F τ s also declnng and akes values beween uny (when all asses are n servce) and zero (when all asses of a parcular vnage have been rered) (OECD (2001a)). T s he maxmum servce lve. Indeed, s unlkely ha all asses of a parcular ype dsappear exacly a he average servce lfe. Afer he maxmum servce lfe, all capal goods are rered. In Schreyer (2003), he maxmum servce lfe s equal o 1.5 mes he average servce lfe (n). The Insue of Naonal Accouns (2002:19) assumes ha T=2n for s calculaon of he sock of non-ict asses. Several oher relaons beween average and maxmum servce lfe are found n he leraure. For nsance, van der Wel (2001) and Vjselaar and Albers (2002) ake T=n n he above formula, Menem e al. (1998) assume ha T = nβ 2 /[β + (1-β) log(1-β)] and Mohr and Glber (1996) ake he expeced value of he hyperbolc funcon for all possble servce lves beween 0.5n and 1.5n. 6 Accordng o Pamukçu and Van Zandweghe (2002), he resuls are sensve o he choce of T. The hyperbolc age-effcency profle s used by he Ausralan Bureau of Sascs and he US Bureau of Labor Sascs, and by oher researchers such as Wolfl and Hajkova (2006), Mas, Pérez and Urel (2006), and Schreyer and Dupon (2006). Accordng o he OECD (2001b), hs profle s also a plausble paern o many knds of asses. a. The hyperbolc effcency funcon The hyperbolc effcency profle of a τ-year old asse s calculaed by he followng funcon: ( T τ ) = ( T βτ ) h τ (Schreyer (2003)) wh T he maxmum servce lfe and β he slope coeffcen. Pamukçu and Van Zandweghe (2002) have observed ha varaons n he value of β have a neglgble mpac on he resuls. Schreyer (2003) ses he parameer a 0.8. The US Bureau of Labour Sascs and he Ausralan 5 I s assumed ha he flow of capal servces s equal o he producve capal sock. 6 Ced n Pamukçu and Van Zandweghe (2002). 10

11 Bureau of Sascs se he parameer a 0.5 for machnery and equpmen and a 0.75 for buldngs and srucures. A hgher value of β mples ha he asse loses s effcency more slowly han n he case of a smaller value. Wh β se a 1, he hyperbolc funcon gves a consan or one-hoss shay age-effcency profle ha wll be explaned n he nex secon. Wh β se a 0, he funcon h s equvalen o a lnear funcon. The Ausralan Bureau of Sascs uses hs value of β for mneral exploraon because beleves ha here s no effcency declne n he knowledge acqured hrough explorng for mnerals (OECD (2001a)). Fgure 1 shows he dfference beween wo hyperbolc and wo geomerc age-effcency profles for an asse wh an average servce lfe of 11 years (communcaons equpmen). For he hyperbolc profles, he parameer β s fxed a 0.8 n he frs case and a 0.5 n he second case. The maxmum servce lfe s equal o 1.5 mes he average servce lfe. For he geomerc profles, he deprecaon rae s calculaed wh a declnng balance rae se a 2 n he frs case and s fxed o he usual rae (0.115) gven by Jorgenson and Sroh (2000) n he second case. Fgure 1 shows ha he hyperbolc paern generaes a slower loss n producve effcency han he geomerc paern over he frs years of he asse s servce lfe and a faser fall a he end of he servce lfe. In no case, he asse s scrapped afer 11 years. Fgure 1: Hyperbolc and geomerc age-effcency paerns Average servce lfe 11 years Hyperbolc_bea0.8 Hyperbolc_bea0.5 Geomerc_R/n Geomerc_0.115 b. The reremen funcon Four ypes of reremen funcons are radonally used: smulaneous ex, lnear, delayed lnear and bell-shaped. The probably densy funcons gven below ndcae he rae of reremen (R) over he lfemes of he longes-lved member of a group of asses of a parcular ype (= moraly funcons) and he cumulave funcons (= he survval funcons) represen he proporon of he members of a group of asses ha are sll n servce a each pon durng he lfeme of he longes-lved member of he group (OECD (2001a)). F τ s declnng wh me and 11

12 akes values beween uny (when all asses are n servce) and zero (when all asses of a parcular vnage have been rered). The lnear funcon The lnear funcon assumes ha he asses are dscarded a he same rae each year. Fgure 2: The lnear reremen funcon R N - Average servce lfe 100% N - Average servce lfe The delayed lnear funcon Wh he lnear paern, he reremens sar mmedaely afer he asses are nsalled and hs s generally consdered as an unrealsc assumpon. Wh a delayed lnear reremen paern, reremens sar laer and fnsh sooner. Fgure 3: The delayed lnear funcon R N - Average servce lfe 12

13 100% N - Average servce lfe The smulaneous ex The smulaneous ex moraly funcon assumes ha all asses are rered from he capal sock a he momen when hey reach her average servce lfe. Fgure 4: The smulaneous ex R N - Average servce lfe 100% N - Average servce lfe The bell-shaped funcon Wh a bell-shaped moraly paern, reremens sar gradually afer he nsallaon of he asses, reach a peak around he average servce lfe and hen slow down n laer years. Several mahemacal funcons are avalable o produce bell-shaped reremen paerns: gamma, quadrac, Webull, Wnfrey, normal and lognormal funcons. The las wo are he mos wdely used funcons and are also hese ha we have esed. 13

14 Fgure 5: The bell-shaped funcon R N - Average servce lfe 100% N - Average servce lfe Accordng o he OECD Manual Measurng Capal, smulaneous ex and lnear declne are clearly unrealsc. Among he oher wo dscard paerns delayed lnear and bell-shaped s he bell-shaped dsrbuons ha are he mos plausble Oher age-effcency profles There exs wo oher age-effcency profles, more rarely used n he emprcal analyses: he consan and he lnear profle. The consan age-effcency profle or he one-hoss shay paern corresponds o a suaon where he asses do no lose effcency and produce consequenly he same quany of servces durng her servce lfe. Accordng o he OECD (2001a), here are probably raher few asses ha manan consan effcency hroughou her workng lves. The consan profle s a parcular case of he hyperbolc profle, wh he parameer β se a 1. Wh he lnear profle, he effcency falls by a consan amoun each year, whch means ha he effcency falls a a faser rae as he asse ages. Ths profle can be seen as a compromse beween he geomerc and he hyperbolc profle. 14

15 3.2 The volume ndex of capal servces Because many dfferen knds of asses nervene n he producon process, an aggregae measuremen of capal servces has o be calculaed. Jorgenson (1963) and Jorgenson and Grlches (1967) were he frs o develop aggregae capal servces measuremens ha ake he heerogeney of asses no accoun. They defned he flow of quanes of capal servces ndvdually for each ype of asse, and hen appled asse-specfc user coss as weghs o aggregae across servces from he dfferen ypes of asses The user cos The prce of capal servces s gven by he user cos of capal or he renal prce of servces. These coss reflec he amoun ha would be blled on a funconng marke for he renng of an asse for one perod. However, here are ofen no complee markes for capal servces because many capal goods are owned and renal prces have o be esmaed. The mplc ren ha capal goods owners pay hemselves gves rse o he ermnology user coss of capal (Schreyer (2003)). The produc of he prce of capal servces (user cos of capal) and he quany of servces (he producve capal sock) gves he value of capal servces. The compuaon of he user cos/renal prce s derved from he assumpon ha n a wellfunconng asse marke, he purchase prce of a capal good equals he dscouned value of he renals/value of servces ha he asse s expeced o generae n he fuure. Ths equlbrum condon corresponds o he equaon (Schreyer (2003)): q Wh, ( τ + 1), o = uc+ τ + 1, τ (1 + r) τ = 0 q, o he purchase prce n year of a new (zero year old) asse of ype and τ +1, τ (1) uc + he renal prce ha hs asse s expeced o fech τ perods laer (+τ), when he age of he asse wll be τ. The formulaon of equaon (1) mples ha he renals are receved a he end of each perod and ha he asse s acqured a he begnnng. Then he prce for a one year old asse n he perod +1 s equal o: q ( τ + 1) + 1,1 = uc + τ + 2, τ + 1(1 + r) τ = 0 (2) By subracng (2) from (1) mulpled by (1+r), we oban: q+ 1,1 q+ 1,0 uc + 1,0 = q, o( 1+ r) q+ 1, 1, whch s equvalen o uc + 1,0 = q, o(1 + r ) q q + 1,0, o 15

16 Fnally, hs expresson can be ransformed no uc + 1,0 = q,0 ( r + δ + 1 ζ δ ζ + 1) or ( uc,0 = q 1,0 r + δ ζ + δ ζ ) 7 wh q, s+ 1 δ, he rae of deprecaon ( 1 ), whch corresponds o he loss n marke value of a q, s capal good due o ageng. Ths rae s relaed o he age-effcency profle chosen. In he case of geomercally declnng effcency, he deprecaon rae s usually he rae used n he consrucon of he producve capal sock, assumng ha he age-prce 8 and he ageeffcency profle follow he same geomerc paern (wh a consan rae). However, n he case of a hyperbolc paern, he deprecaon rae has o be calculaed. We followed he way recommended by he OECD Manual (Annex 5 of Measurng producvy ), whch consss n he calculaon of he age-prce profle and hen he ne wealh sock 9. ζ, a revaluaon or capal gans erm defned as he expeced asse prce change beween q,0 he begnnng and he end of he perod (of a new asse): ζ 1. Ths erm s nroduced n he equaon wh a negave sgn, whch means ha a fall n asse prces rases user q 1,0 coss. r, he ne rae of reurn, whch corresponds o he expeced remanng remuneraon for he capal owner once deprecaon and asse prce changes have been aken no accoun. There are manly wo approaches o he measuremen of he rae of reurn: an endogenous rae and an exogenous rae. 7 I s a pre-ax measuremen of user cos of capal, because here are no ax varables. Some auhors leave ou he neracon erm δ ζ by smplfcaon. We added a consran o hs equaon: he user cos canno be negave and s se o zero n case of negave value. A negave value can ndeed be observed n some cases. 8 The age-prce profle of an asse descrbes he declne n he prce of an asse as ages. Ths profle, relaed o he age-effcency profle, wll be explaned n more deals n pon C The ne and he gross capal sock of hs chaper. 9 The deprecaon rae s obaned by calculang he followng equaons : - Calculaon of he age-prce profle: T s (1+ τ ) hτ + s. Fτ + s.(1.04) τ z s = T (1+ τ ) hτ. Fτ.(1.04) τ wh s : age of asse - Calculaon of he ne wealh capal sock a consan prces: = T K zτ. Fτ. I τ τ = 0 N, N, - Calculaon of he real deprecaon: D = K + - Calculaon of he deprecaon rae: 1 D δ = N, K K I N, 16

17 c. The endogenous ne rae (ex-pos rae) The endogenous rae of reurn s calculaed as he ex-pos rae ha exhauss all non-labour ncomes compued n he Naonal Accouns. Gross operang surplus plus he capal par of mxed ncome of self-employed are consdered as compensaon for capal. The ne rae of reurn r s hen compued by solvng he followng equaon: P Q w L = Σuc K = r Σ q 1 K + Σ( δ ζ + δ ζ ) q 1 So he rae of reurn s equal o: K r = CAP ( δ ζ + δ ζ ) q q 1 K 1 K, wh CAP: he capal compensaon (gross operang surplus plus he capal par of mxed ncome) Ths mehod s used by he Ausralan Bureau of Sascs and he US Bureau of Labour Sascs and requres ha he underlyng producon funcon exhbs consan reurns o scale, ha markes are compeve and he expeced (ex-ane) rae of reurn equals he realzed (ex-pos) rae of reurn (OECD (2001a)). An assumpon has also o be made o decompose he mxed ncome of self-employed no labour and capal componens. In Belgum, we do no have daa on labour compensaon for selfemployed. In hs paper, we esmaed hs compensaon by assumng ha he compensaon per self-employed person s equal o he compensaon per employee. A lm s, however, mposed by secor: he esmaed labour compensaon for self-employed canno exceed he mxed ncome. d. The exogenous ne rae (ex-ane rae) An alernave s o esmae an exogenous rae from marke neres raes ha play a key role n deermnng raes of reurn. Indeed, he decson o purchase a capal asse s parally based on he comparson beween he expeced reurn of he asse and he neres raes from markes. Furhermore, nvesmen n capal asses s ofen a leas parly fnanced by loans. To compue an exogenous ne rae, we followed he mehodology used by Schreyer (2003). The expeced nomnal neres rae for every year s hen compued as: r = [(1 + rr)(1 + p)] 1 where p s he expeced value of he consumer prce ndex and rr a consan value represenng he expeced real neres rae. 17

18 The expeced real neres rae rr s compued by akng a seres of annual observed nomnal raes (un-weghed average of neres rae wh dfferen maures) and deflang hem by he consumer prce ndex. The resulng seres of real neres raes s averaged over he concerned perod o yeld a consan value for rr. To oban a measuremen for p, he expeced overall nflaon, we consruc a 5-year movng average of he rae of change of he consumer prce ndex: MACPI 5 = CPI 1 = s s 5 where CPI s he rae of change of he consumer prce ndex. Wh an ex-ane rae of reurn, he value of capal servces s no necessarly equal o he capal compensaon derved from he Naonal Accouns, whch makes he growh accounng exercses more dffcul The volume ndex of capal servces The aggregaon of he dfferen producve socks of capal asses s made by usng he user coss of capal as weghs. User coss are prces for capal servces and, under compeve markes and equlbrum condons, hese prces reflec he margnal producvy of he dfferen asses. User cos weghs hus provde a means o ncorporae dfferences n he producve conrbuon of heerogeneous nvesmens no he overall measuremen of capal npu (OECD (2001a)). A usual and heorecally recommended ndex formula o consruc a volume ndex of capal servces s he Törnqvs ndex: + K ) = 0.5( v+ 1 + v )ln( K + ln( K 1 K ) 1 where 3.3 The ne and he gross capal sock uc K K The gross capal sock and he ne capal sock are measuremens of capal ha are frequenly avalable, bu ha do no measure he producve effcency of he asses. The gross capal sock corresponds o he sum of pas nvesmens, correced only for a reremen paern. I gnores deeroraon of asses wh age and values pas nvesmens a as new prces. Consequenly, he gross capal sock can be consdered as a parcular case of producve sock, where he effcency of he asses remans fully unalered unl he end of he servce lfe (whch s called a one-hoss shay paern). v uc The ne capal sock, somemes called he wealh sock, measures he marke value of capal asses. I corresponds o he sum of pas nvesmens, correced for reremen and for loss n 18

19 value due o ageng. Ths measuremen of wealh s publshed n he balance shees of he Naonal Accouns. The ne sock s usually derved from he gross sock by deducng he consumpon of fxed ncome or deprecaon measurng he loss n value of capal goods as ages. Deprecaon has o be dsngushed from effcency declne ha reflecs he loss of producve effcency as asses age and ha s assocaed wh he producve capal sock. The ne sock can also be derved by frs assumng an age-effcency profle and hen by esmang he correspondng age-prce profle whch descrbes he change (usually he declne) n he prce of an asse as ages and whch s lnked o he deprecaon concep. Indeed, he ne capal sock a consan prces can be obaned by usng an age-prce profle (z): K N, = T τ = 0 τ τ z. F. I τ wh F τ he reremen funcon and I τ he real nvesmen. The age-prce profle and he age-effcency profle of an asse are relaed, bu are no necessarly dencal. I wll only be dencal f he wo profles concde. The age-prce profle can be derved from he ageeffcency profle accordng o he formula explaned n foonoe 8. In he case of a geomerc age-effcency profle, s usually assumed ha he effcency and he marke value of he asse declne a a consan rae ndependen of me, whch consderably smplfes he calculaons. The age-prce profle can also be obaned wh oher mehods such as economerc sudes of used asse markes (Hulen and Wykoff (1980) 10 ) or by assumng a paern (OECD (2001b)). The oal value of he ne sock s obaned as follows: q W = Σ q W (Schreyer (2003)) Wh q : he marke prce of he asse. The rae of change of he ne sock a consan prces depends on he ndex number formula chosen for he aggregaon of asses. One soluon s o use a Törnqvs ndex: + W ) = 0.5( w w )ln( W + ln( W 1 W ) 1 where w q,0w q,0 W However, he majory of counres use a Laspeyres-ype volume ndex (Schreyer (2003)). To llusrae he dfference beween he ne capal sock and he capal servces, Fgure 6 shows he wo measuremens for he enre economy. Boh measuremens are compued wh a Törnqvs ndex. The socks are based on a geomerc paern, assumng ha he effcency and he marke value of he asses declne a he same rae. Schreyer and Pla (2001) observed ha n he emprcal applcaons, he growh rae of capal servces s generally hgher han he growh 19

20 rae of he ne capal sock. So, f he ne capal sock s used as a measuremen of capal npu n he producvy analyses, he growh of TFP could be overesmaed, compared wh TFP growh lnked o capal servces. Fgure 6: Capal servces and ne capal sock, Toal economy, 1970= Ne capal sock Capal servces 10 Ced n OECD (2001b). 20

21 4 Resuls 4.1 The daa The dfferen measuremens of capal servces are realzed by usng nvesmen daa consruced on he bass of he Belgan Naonal Accouns publshed n November 2006 whn he framework of he European EUKLEMS projec n whch he Federal Plannng Bureau s nvolved. We used offcal daa on nvesmen publshed eher by produc (7), or by secor (29) from 1995 o 2004 (a curren and consan prces). Then, we used unpublshed offcal daa and we made several assumpons, coheren wh he Naonal accouns o consruc 3 ICT asses, daa from 1853 o 2004 and crossng 29 secors and 9 producs and hedonc prce ndces. The mehodology followed for consrucng he daa, he descrpons of he 29 secors covered and he descrpon and he servce lves of he 9 producs covered are dealed n he annex. The oher daa used o esmae he oal facor producvy come from he Naonal Bank of Belgum for he perod For he perod before 1995, daa are esmaons made whn he framework of he EUKLEMS projec, whch are compable wh he Naonal Accouns. 4.2 Resuls Ths chaper presens measuremens of producve capal socks, user coss and capal servces esmaed by means of dfferen mehodologes o es he sensvy of hese measuremens o mehodologcal changes. Then, by usng he resuls on capal servces, he daa on hours worked by employees and self-employed and he esmaon of he coss shares of npus, we calculae he conrbuon of TFP o GDP growh, n order o evaluae he mpac of mehodologcal changes on TFP conrbuon The producve capal sock e. The hyperbolc profle The hyperbolc producve capal sock s calculaed by usng a hyperbolc age-effcency funcon and a reremen funcon. As sad before, s possble o choose dfferen opons n each of hese wo funcons. Varaon of he parameers of he age-effcency funcon Table 1 presens he producve capal sock n IT equpmen (5 years of average servce lfe) for he elecrcal and opcal equpmen secor (DL) a dfferen years. The second par of he able provdes he average annual growh rae of hs sock beween hese years. Dfferen opons are chosen for he maxmum servce lfe T and for he β parameer accordng o pracces observed n oher papers or leraure. The maxmum servce lfe T can be equal o he average servce lfe n, o 1.5n and o 2n. The β parameer can be se a 0.5, 0.8, 0.75 and 1. Wh he β parameer se a 1, he effcency funcon s n fac a consan funcon and wh he parameer se a 0, s a lnear funcon. The reremen funcon chosen s a normal funcon for all esmaons. 21

22 Table 1 shows ha he producve capal sock n level ncreases when he maxmum servce lfe and he β parameer ncrease. A hgher value for β means ha he asse loses s effcency more slowly. Varaons n producve socks are hgher when he maxmum servce lfe changes han when he β parameer changes. Indeed, a maxmum servce lfe beween n and 2n generaes a producve capal sock beween and 543.2; whle, he varaon of he β parameer beween 0.5 and 0.8 generaes a producve capal sock beween and The producve sock obaned wh a lnear age-effcency profle s he lowes. The producve sock obaned by usng a consan funcon (β=1) s logcally very hgh compared o he oher opons. Ths exreme case, n whch he asse does no lose effcency, s n fac never used, because does no seem realsc. A smlar concluson can be drawn from he growh raes. Table 1: Producve capal sock n IT equpmen (of he elecrcal and opcal equpmen secor) Levels Hyperbolc, T=2n, β= Hyperbolc, T=1.5n, β= Hyperbolc, T=n, β= Hyperbolc, T=1.5n, β= Hyperbolc, T=1.5n, β= Lnear, T=1.5n, β= Consan, T=1.5n, β= Average annual growh raes Hyperbolc, T=2n, β= % 32.9% 32.9% 8.1% 23.5% Hyperbolc, T=1.5n, β= % 32.5% 32.5% 7.6% 23.4% Hyperbolc, T=n, β= % 31.4% 31.4% 6.3% 23.1% Hyperbolc, T=1.5n, β= % 32.4% 32.4% 7.5% 23.4% Hyperbolc, T=1.5n, β= % 31.9% 31.9% 6.9% 23.3% Lnear, T=1.5n, β=0 30.6% 31.1% 31.1% 6.2% 23.1% Consan, T=1.5n, β=1 30.0% 33.2% 33.2% 8.5% 23.6% Table 2 presens he producve capal sock n oher consrucons for he same secor. Ths asse has an average servce lfe of 35 years n hs secor. The same observaons appear for hs asse wh a much hgher servce lfe. 22

23 Table 2: Producve capal sock n oher consrucons (of he elecrcal and opcal equpmen secor) Levels Hyperbolc, T=2n, β= Hyperbolc, T=1.5n, β= Hyperbolc, T=n, β= Hyperbolc, T=1.5n, β= Hyperbolc, T=1.5n, β= Lnear, T=1.5n, β= Consan, T=1.5n, β= Average annual growh raes Hyperbolc, T=2n, β= % 3.1% 3.1% 0.0% 3.31% Hyperbolc, T=1.5n, β= % 3.1% 3.1% -0.1% 3.27% Hyperbolc, T=n, β= % 2.8% 2.8% -0.8% 3.04% Hyperbolc, T=1.5n, β= % 3.0% 3.0% -0.2% 3.25% Hyperbolc, T=1.5n, β= % 2.9% 2.9% -0.4% 3.13% Lnear, T=1.5n, β=0 4.7% 2.7% 2.7% -0.8% 2.95% Consan, T=1.5n, β=1 5.0% 3.3% 3.3% 0.2% 3.39% Varaon of he profle of he reremen funcon Table 3 presens he producve capal sock n IT equpmen for he secor DL esmaed wh dfferen reremen funcons. For he parameers of he age-effcency funcon, we ook usual values for T (=1.5n) and β (=0.8). The wo bell-shaped funcons (normal and lognormal) generae smlar socks and average growh raes for he consdered perods. On he oher hand, lower socks and growh raes are obaned when usng he lnear reremen funcon. Gven he shor average servce lfe of IT equpmen, smulaneous ex and delayed lnear funcons generae dencal resuls, whch are hgher n level and n growh rae compared o normal reremen. Table 3: Producve capal sock n IT equpmen (of he elecrcal and opcal equpmen secor) Levels Hyperbolc, T=1.5n, β=0.8, Normal reremen Hyperbolc, T=1.5n, β=0.8, Log normal reremen Hyperbolc, T=1.5n, β=0.8, Smulaneous ex reremen Hyperbolc, T=1.5n, β=0.8, Lnear Hyperbolc, T=1.5n, β=0.8, Delayed Lnear Average annual growh raes Hyperbolc, T=1.5n, β=0.8, Normal reremen 30.1% 32.5% 32.5% 7.6% 23.4% Hyperbolc, T=1.5n, β=0.8, Log normal reremen 30.1% 32.5% 32.5% 7.5% 23.4% Hyperbolc, T=1.5n, β=0.8, Smulaneous ex reremen 23.6% 30.0% 33.3% 33.3% 8.4% Hyperbolc, T=1.5n, β=0.8, Lnear 30.8% 31.2% 31.2% 6.4% 23.1% Hyperbolc, T=1.5n, β=0.8, Delayed Lnear 30.0% 33.3% 33.3% 8.4% 23.6% Table 4 presens he producve capal sock n oher consrucons for he same secor. The wo-bell shaped funcons have smlar resuls. For hs asse wh a hgher servce lfe, he s- 23

24 mulaneous ex and he delayed lnear reremen funcons are dfferen, bu que close. Once agan, he resuls are very dfferen when usng a lnear reremen funcon. Table 4: Producve capal sock n oher consrucons (of he elecrcal and opcal equpmen secor) Levels Hyperbolc, T=1.5n, β=0.8, Normal reremen Hyperbolc, T=1.5n, β=0.8, Log normal reremen Hyperbolc, T=1.5n, β=0.8, Smulaneous ex reremen Hyperbolc, T=1.5n, β=0.8, Lnear Hyperbolc, T=1.5n, β=0.8, Delayed Lnear Average annual growh raes Hyperbolc, T=1.5n, β=0.8, Normal reremen 5.0% 3.1% 3.1% -0.1% 3.27% Hyperbolc, T=1.5n, β=0.8, Log normal reremen 5.0% 3.1% 3.1% -0.2% 3.27% Hyperbolc, T=1.5n, β=0.8, Smulaneous ex reremen 3.44% 5.2% 3.2% 3.2% -0.1% Hyperbolc, T=1.5n, β=0.8, Lnear 4.5% 2.7% 2.7% -0.7% 2.95% Hyperbolc, T=1.5n, β=0.8, Delayed Lnear 5.1% 3.2% 3.2% -0.3% 3.37% f. The geomerc profle Wh he geomerc age-effcency profle, he value of he producve capal sock depends on he esmaon of he deprecaon rae. The deprecaon rae s usually compued by dvdng he declnng-balance raes of Fraumen (1997) by he average servce lfe of he asses. The deprecaon raes, ha we called gamma 1, correspond o hs compuaon and use he average servce lves gven by he Naonal Bank of Belgum for non-ict asses and sofware. For he oher wo ICT asses -IT equpmen and communcaons equpmen- we followed he assumpons of he US Bureau of Economc Analyss (see Annex C). The deprecaon raes gamma 2 are raes used n he European EUKLEMS projec for all counres. For ICT asses, hese raes are hose gven by Jorgenson and Sroh (2000) and Oulon (2001) and are ofen used by many auhors for ICT asses. Table 5 shows producve capal sock n he ICT asse IT equpmen for he secor elecrcal and opcal equpmen (DL). Gamma 1 s equal o 0.4 and gamma 2 o The producve capal socks and average growh raes n he consdered perods are hgher wh he lower deprecaon rae. The same concluson appears from Table 6 showng he producve capal socks for oher consrucons n secor DL. In hs secor, gamma 1 s equal o and gamma 2 o

25 Table 5: Producve capal sock n IT equpmen (of he elecrcal and opcal equpmen secor) Levels Geomerc, gamma Geomerc, gamma Average annual growh raes Geomerc, gamma % 29.5% 29.5% 4.8% 22.7% Geomerc, gamma % 30.4% 30.4% 6.1% 23.1% Table 6: Producve capal sock n oher consrucons (of he elecrcal and opcal equpmen secor) Levels Geomerc, gamma Geomerc, gamma Average annual growh raes Geomerc, gamma 1 3.8% 2.6% 2.6% -0.1% 2.81% Geomerc, gamma 2 3.7% 2.5% 2.5% -0.5% 2.70% g. Geomerc versus hyperbolc Fgures 7 and 8 compare some resuls obaned above wh hyperbolc, geomerc and consan funcons for he wo asses: IT equpmen and oher consrucons n he elecrcal and opcal equpmen secor. In boh cases, he producve capal sock esmaed by usng a hyperbolc age-effcency profle, combned wh a log-normal reremen funcon (curren funcon), s hgher n level han he sock esmaed wh a geomerc funcon. In he case of he hyperbolc profle, ncreasng he maxmum servce lfe from 1.5n o 2n wll generae an even hgher producve sock. In boh cases, he geomerc profle esmaed wh he EUKLEMS deprecaon raes (gamma 2) seems smlar o he hyperbolc profle, combned wh a lnear reremen funcon. Fnally, he consan funcon generaes logcally he hghes producve capal sock. For he whole economy and for he majory of asses, he esmaon of producve capal socks s hgher n level, when s realzed wh a hyperbolc profle (wh mean parameers: T=1.5n and β=0.75) han when a geomerc profle s assumed. If a maxmum servce lfe of 2n and a β parameer of 0.8 are chosen, he hyperbolc esmaon s even hgher for all asses, excep for producs of agrculure. 25

26 Fgure 7: Hyperbolc versus geomerc: Producve capal sock n IT equpmen (of he elecrcal and opcal equpmen secor) Hyperbolc_1.5n_0.8_lognormal Geomerc_gamma1 Hyperbolc_1.5n_0.8_lnear Consan_1.5n_1_normal Geomerc_gamma2 Fgure 8: Hyperbolc versus geomerc: Producve capal sock n oher consrucons (of he elecrcal and opcal equpmen secor) Hyperbolc_1.5n_0.8_lognormal Geomerc_gamma1 Hyperbolc_1.5n_0.8_lnear Consan_1.5n_1_normal Geomerc_gamma User coss The esmaon of he user coss of capal depends on several varables: he rae of reurn whch can be compued ex-ane or ex-pos, he rae of deprecaon whch depends on he age-effcency profle chosen and he revaluaon erm. So s possble o use ex-pos or ex-ane user coss esmaed eher wh a geomerc deprecaon rae or a hyperbolc deprecaon rae. An ex-pos approach s used n he EUKLEMS daabase. Fgure 9 presens for example, dfferen compuaons of he rae of reurn of capal for he whole economy: an ex-ane rae of reurn (REA) based on he mehodology used by Schreyer (2003) and explaned before, an ex-pos rae of reurn based on a geomerc age-effcency pro- 26

27 fle 11 and an ex-pos rae based on a hyperbolc age-effcency profle. I should be noed ha hese raes do no ake no accoun he fscal effec creaed by changes n ax raes on busness ncome. Fgure 9 shows ha he wo ex-pos raes of reurn follow he same evoluon, bu he hyperbolc rae s sysemacally below he geomerc rae. The hree raes of reurn have declned over he whole perod. However, he geomerc ex-pos rae s frequenly above he exane rae and he hyperbolc ex-pos rae s frequenly below. Fgure 9: Rae of reurn (for oal economy) REA REP_Geomerc REP_hyperbolc Fgure 10 presens he user coss of he asse oher consrucons n he secor DL. These user coss are calculaed accordng o he four assumpons ced above 12. The evoluon of he esmaon of he user cos seems o be more nfluenced by he choce beween an ex-pos or ex-ane rae of reurn han by he chosen deprecaon profle. Indeed, he wo ex-ane esmaons and he wo ex-pos esmaons seem o follow a smlar evoluon. Usng he ex-ane mehod or he ex-pos mehod seems o generae que dfferen user coss. The wo ex-ane esmaons, consruced wh he same ex-ane rae of reurn (REA), are smlar. Ths means ha he hyperbolc and he geomerc deprecaon raes are smlar for hs asse. I s no always he case for he oher asses. Fgures 10 and 11 show ha, on he oher hand, he geomerc ex-pos user cos s always hgher han he hyperbolc ex-pos user cos. Ths gap beween he user coss s explaned n hs case by he rae of reurn. The dfference beween he ex-ane and ex-pos esmaons s less pronounced when he user coss are calculaed for he whole economy (see fgure 11). 11 The choce beween gamma 1 and gamma 2 s of no mporance because hey gve praccally dencal ex-pos raes of reurn. 12 The hyperbolc profle s combned wh a lognormal reremen funcon (oher assumpons: T=1.5n, β=0.8) and he geomerc profle s esmaed wh he euklems deprecaon raes (gamma 2). 27

28 Fgure 10: User cos of he asse oher consrucons (n he elecrcal and opcal equpmen secor) UCEA_hyperbolc UCEP_hyperbolc UCEA_geomerc UCEP_geomerc Fgure 11: User cos of he asse oher consrucons (for oal economy) UCEA_hyperbolc UCEP_hyperbolc UCEA_geomerc UCEP_geomerc Volume ndex of capal servces Table 7 gves he volume ndex of capal servces (1980=100) of he elecrcal and opcal equpmen secor (DL) for several years. Table 8 gves he same resuls for he oal economy. For he hyperbolc esmaon, he wo more realsc reremen funcons have been seleced: a bell-shaped funcon (lognormal), and a delayed-lnear funcon. The base s se o 1980 because mporan dfferences beween he ex-ane and he ex-pos esmaon appear n he begnnng of he sevenes, whch nfluence he whole perod f he base s se o Wh he base n 1980, s possble o observe wha happens before and afer However, growh raes on he whole perod are also gven. 28

29 A he end of he consdered perod, he volume ndex of capal servces akes he lowes value when he lnear age-effcency profle s chosen (wh ex-pos rae of reurn) n Table 7 and he geomerc paern (gamma 2) wh an ex-pos rae of reurn s chosen n Table 8. I akes he hghes value when he geomerc profle s seleced (ex-ane, gamma 2) n Table 7 and when he hyperbolc profle s seleced (ex-pos, lognormal) n Table 8. The choce of he ageeffcency profle seems o have an nfluence on he evoluon of he volume ndex of capal servces, bu he choce beween an ex-ane and an ex-pos rae of reurn seems also mporan, n parcular n he case of a geomerc profle for he secor DL. For he whole economy and he enre perod, f an ex-pos approach s used, he volume ndces of capal servces esmaed wh a hyperbolc age-effcency profle grow a a hgher rae han he ndces esmaed wh a geomerc profle, whaever he reremen funcon (lognormal or delayed-lnear) or he geomerc deprecaon raes (gamma 1 or gamma 2 of he euklems projec) chosen. Ths general concluson s observed n almos all secors over he whole perod I s, however, less frequen over he perod (see DL secor n Table 7). I s he lnear age-effcency profle whch provdes he mos smlar resuls o he geomerc resuls. The oppose of hs concluson s observed n boh ables wh an ex-ane approach: he volume ndces grow a a somewha hgher rae when a geomerc profle s adoped, bu o a lesser exen. Indeed, hs s only observed for he whole economy f he consdered perod s reduced o The dfferences observed n ndces n he case of an ex-ane approach are more apparen n he DL secor han n he whole economy, where he ndces are que smlar. In he case of a hyperbolc profle, akng a longer maxmum servce lfe and/or a hgher β parameer ncrease he value and he growh rae of he volume ndex of capal servces. Indeed, a longer maxmum servce lfe and/or a hgher β parameer (slower loss of effcency) generae a hgher producve capal sock and growh rae of he sock over he consdered perod. Concernng he reremen funcon, he same concluson appears from he wo ables: n 2004, he ndex s slghly hgher n level and n growh rae wh a delayed-lnear funcon, hen wh a lognormal and s he lowes wh a lnear funcon (wh T=1.5n n each case). In he case of a geomerc profle, he choce beween he gamma 1 and gamma 2 deprecaon raes has no nfluence on he volume ndex of capal servces for he whole economy. However, dfferences appear n he ndces of secors (see Table 7). 29