MEASURING AND DECOMPOSING CAPITAL INPUT COST

Transcription

1 row_ Revew of ncome and Wealh Seres 57 Number 3 Sepember 20 DO: 0/j x MEASURNG AND DECOMOSNG CATAL NUT COST by Ber M Balk* Erasmus Unversy and Sascs Neherlands The measuremen of oal facor producvy change (or dfference) vs-à-vs labor producvy change crucally depends on he measuremen and decomposon of capal npu cos Ths paper dscusses he bascs of s measuremen and shows ha one can dspense wh he usual neoclasscal assumpons By vrue of s srucural feaures he measuremen model s applcable o ndvdual esablshmens and aggregaes such as ndusres secors or economes nroducon Measurng he value of capal as sock or servce flow and he decomposon n prce and quany componens s a horny ssue wh deep hsorcal roos and a large leraure recenly summarzed by Dewer and Schreyer (2008) A classc paper s Hulen (990) SNA (2008 chaper 20) provdes a non-echncal nroducon buldng on he mporan OECD (2009) manual The focus of hs paper s on capal as npu of a producon process The background s measuremen of oal (or mul-) facor producvy change for whch he measuremen and decomposon of capal npu cos s crucal Ths paper develops he heory wh a vew o praccal mplemenaon Along he way s shown ha here s no need for he usual neoclasscal assumpons The conrbuons of hs paper can be summarzed n he followng pons: Though he leraure acknowledges he fac ha for he reamen of capal npu s necessary o dsngush beween me perods as pons of me (ha s nsances of a real varable) and nervals wh a defne lengh n he subsequen mahemacs hs dsncon ofen s no mananed Ths paper suggess a noaon o deal wh hs problem 2 The leraure by and large seems o assume ha nvesmens ha have maeralzed durng a ceran me perod already have become operaonal a he begnnng of he same perod n hs paper s explcly assumed ha nvesmens become operaonal a each perod s mdpon 3 The leraure grosso modo neglecs he fac ha nvesmens concern no only new asses bu also used asses and ha here s a subsanal rade n used asses n hs paper s explcly assumed ha nvesmens can be of varous ages Noe: The vews expressed n hs paper are hose of he auhor and do no necessarly reflec any polcy of Sascs Neherlands Two referees are hanked for useful commens on a prevous verson *Correspondence o: Ber M Balk Roerdam School of Managemen Erasmus Unversy O Box DR Roerdam The Neherlands (bbalk@rsmnl) Revew of ncome and Wealh 20 nernaonal Assocaon for Research n ncome and Wealh ublshed by Blackwell ublshng 9600 Garsngon Road Oxford OX4 2DQ UK and 350 Man S Malden MA 0248 USA 490

2 4 The dealed dscusson of mplemenaon ssues hghlghs he role of models assumpons and approxmaons and s nsrumenal for he desgn of varous ypes of sensvy analyss n hs paper he ex pos accounng pon of vew s used Ths s conssen wh he sascan s pon of vew whch s by and large acceped for he componens of value added and labor npu cos For revenue nermedae npus cos and labor npu cos one smply uses he observed (money) values whereby a he lowes level of aggregaon prces are compued as raos of observed values and observed quanes; ha s prces are un values ex pos measured Of course a dsncve feaure of capal npu cos s ha hs cos as such canno be observed mpuaons mus be made no only o arrve a (an esmae of ) he cos bu also o enable one o spl he cos n prce and quany componens mpuaons are always more or less arbrary and depend on he purpose of he accounng exercse One has o be clear abou hs The layou of hs paper s as follows n Secon 2 he fundamenal KLEMS-Y npu oupu model of a producon un s skeched Ths provdes he framework for wha follows Secon 3 nroduces our noaon derves he un user cos formulas for asses avalable a he begnnng of an accounng perod as well as for asses nvesed durng hs perod and dscusses he decomposon of capal npu cos change n prce and quany ndces Secon 4 dscusses he relaon wh capal sock measures Un user cos depends on prces (or valuaons) of he asses bu hese prces are no observable Hence Secon 5 dscusses he defnon of such prces from expeced values of he varables nvolved Ths gves rse o a decomposon of oal user cos no four componens namely he cos of wang he cos of ancpaed me-seres deprecaon he cos of unancpaed revaluaon and he cos of ax An mporan role n he cos of wang s played by he neres rae whch s also called he rae of reurn There appear o be several conceps of hs rae; hey are dscussed n Secon 6 n Secon 7 he ssue of aggregaon s consdered Secon 8 dscusses some ssues of mplemenaon and Secon 9 concludes 2 The Basc KLEMS-Y Model Le us consder a sngle producon un Ths could be an esablshmen or plan a frm an ndusry a secor or even an enre economy For he oupu sde as well as for he npu sde of he un here s some ls of commodes (accordng o some classfcaon scheme) A commody s hereby defned as a se of closely relaed ems whch for he purpose of analyss can be consdered as equvalen eher n he sac sense of her quanes beng addve or n he dynamc sense of dsplayng equal relave prce or quany changes deally hen for any accounng perod consdered (ex pos) say a year each commody comes wh a value (n moneary erms) and a prce and/or a quany A he oupu sde he prces mus be hose receved by he un whereas a he npu sde he prces mus be hose pad s assumed ha he un does no delver o self u oherwse all he nra-un delveres are need ou The npus are cusomarly classfed accordng o he KLEMS forma The leer K denoes he class of owned reproducble capal asses The commodes 49

3 here are he asse-ypes sub-classfed by age caegory Cohors of asses are assumed o be avalable a he begnnng of he accounng perod and n deeroraed form (due o ageng wear and ear) sll avalable a he end of he perod nvesmen durng he perod adds enes o hese cohors whle desnvesmen breakdown or reremen remove enes Examples nclude buldngs and oher srucures land machnery ranspor and CT equpmen all sors of ools As wll be dscussed laer n deal accounng rules mply ha quanes sough are jus he quanes of all hese cohors of asses (ogeher represenng he producve capal sock) whereas he relevan prces are her un user coss (per ype age combnaon) consruced from mpued neres raes deprecaon profles (ancpaed) revaluaons and ax raes The sum of quanes mes prces hen provdes he capal npu cos of a producon un The leer L denoes he class of labor npus; ha s all he ypes of work ha are mporan o dsngush cross-classfed for nsance accordng o educaonal aanmen gender and experence (whch s usually proxed by age caegores) Quanes are measured as hours worked (or pad) and prces are he correspondng wage raes per hour Where applcable mpuaons mus be made for he work execued by self-employed persons The sum of quanes mes prces provdes he labor npu cos (or he labor bll or labor compensaon as s somemes called) The classes K and L concern so-called prmary npus The leers E M and S denoe hree dsjunc classes of so-called nermedae npus Frs E s he class of energy commodes consumed by a producon un: ol gas elecrcy and waer Second M s he class of all he (physcal) maerals consumed n he producon process whch could be sub-classfed no raw maerals semfabrcaes and auxlary producs Thrd S s he class of all he busness servces whch are consumed for mananng he producon process Ths class ncludes he servces of leased capal asses as well as all he ousourced acves Though s no a all a rval ask o defne precsely all he nermedae npus and o classfy hem can safely be assumed ha a he end of each accounng perod here s a quany and a prce assocaed wh each of hose npus Then for each accounng perod producon cos s defned as he sum of prmary and nermedae npu cos A he oupu sde he leer Y denoes he class of commodes goods and/or servces whch are produced by he un Though n some ndusres such as servces ndusres or ndusres producng manly unque goods defnonal problems are formdable can safely be assumed ha for each accounng perod here are daa on quanes produced For uns operang on he marke here are also prces The sum of quanes mes prces hen provdes he producon revenue and apar from axes on producon revenue mnus cos yelds prof The suaon as pcured n he precedng paragraphs s ypcal for a un operang on a (oupu) marke Because a non-marke un has no oupu prces here s no revenue Though here s cos lke for marke uns here s no prof Naonal accounans usually resolve he problem here by defnng he revenue of a non-marke un o be equal o s cos hereby seng prof equal o 0 492

4 Cash flow s defned as revenue mnus nermedae npus cos and labor npu cos Thus cash flow mnus capal npu cos equals prof Cash flow when posve can be seen as he (gross) reurn o capal npu Balk (200) called hs he K-CF model Of course hs model only makes sense when he producon un acually owns capal asses Bu le us assume ha hs s ndeed he case 3 Capal npu Cos The K-CF model provdes a good pon of deparure for a dscusson of he measuremen of capal npu cos 2 Cash flow as defned n he foregong s he (ex pos measured) moneary balance of all he flow varables Capal npu cos s dfferen snce capal s a sock varable Bascally capal npu cos s measured as he dfference beween he book values of he producon un s owned capal sock a begnnng and end of he accounng perod consdered Our noaon mus reflec hs The begnnng of perod s denoed by - and s end by Thus a perod s an nerval of me = [ - ] where - = ( - ) and = ( ) - Occasonally he varable wll also be used o denoe he mdpon of he perod All he asses are supposed o be economcally born a mdpons of perods wheher hs has occurred nsde or ousde he producon un under consderaon Thus he age of an asse of ype a (he mdpon of ) perod s a non-negave neger number j = 0J The age of hs asse a he begnnng of he perod s j - 05 and a he end j 05 The economcally maxmal servce lfe of asse ype s denoed by J 3 The openng sock of capal asses s he nherance of pas nvesmens and desnvesmens; hence he openng sock consss of cohors of asses of varous ypes each cohor comprsng a number of asses of he same age By convenon asses ha are dscarded (normally rered or premaurely scrapped) or sold durng a ceran perod are supposed o be dscarded or sold a he end of ha perod; ha s a Second-hand asses ha are acqured durng perod from oher producon uns are supposed o be acqured a he begnnng of he nex perod ( ) - However all oher acqusons of second-hand asses and hose of new asses are supposed o happen a he mdpon of he perod and such asses are supposed o be mmedaely operaonal Hence all he asses ha are par of he openng sock reman acve hrough he enre perod [ - ] The perod nvesmens are supposed o be acve hrough he second half of perod ha s [ ] u oherwse he sock of capal asses a he mdpon of he perod s he same as he sock a - he begnnng of he perod bu 05 perod older A he mdpon of he perod he nvesmens of varous age are added o he sock Noce however ha he closng sock a Cash flow s also called gross or varable prof The Naonal Accouns erm s gross operang surplus 2 Ths and he nex secon lean heavly on Balk and van den Bergen (2006) 3 s of course a smplfcaon o assume ha he economcally maxmal servce lfe of an asse ype s some gven consan Acually a me superscrp should be added See Dewer (2009) for some heorecal consderaons 493

5 he end of he perod s no necessarly dencal o he openng sock a ( ) - because of he convenon on sale acquson and dscard of asses Le K denoe he quany (number) of asse ype ( = ) and age j (j = J ) a he mdpon of perod These quanes are non-negave; some of hem mgh be equal o 0 Furher le denoe he (non-negave) quany (number) of asse ype ( = ) and age j ( j = 0J ) ha s added o he sock a he mdpon of perod The followng relaons are useful o keep n mnd: () K j = K ( j = ) (2) 0 = K 0 5 (3) K = K j ( j = ) (4) ( ) K ( j ) 0 5 = K j B j ( j = 0 ) (5) ( ) K ( J ) 0 5 = 0 where B j denoes he balance of sale acquson and dscard a n general esmaes of he K and B varables are generaed from dealed nvesmen and desnvesmen surveys combned wh he erpeual nvenory Mehod The level of deal can very beween uns We are now ready o defne he concep of user cos for asses ha are owned by he producon un The frs dsncon ha mus be made s beween asses ha are par of he openng sock of a perod and nvesmens whch are made durng hs perod Consder an asse of ype ha has age j a he mdpon of perod s prce (or valuaon) a he begnnng of he perod s denoed by j and s prce (or valuaon) a he end of he perod by j For he me beng we consder such prces as beng gven and pospone her precse defnon o a nex secon The prces are assumed o be non-negave; some mgh be equal o 0 n any case J = 0; ha s an asse ha has reached s economcally maxmal age n perod s valued wh a zero prce a he end of hs perod The (ex pos) un user cos over perod of an openng sock asse of ype ha has age j a he mdpon of he perod s hen defned as (6) u r τ ( j J ) ( ) = j j j There are hree componens here Le us sar wh he second mos mporan one Ths par of expresson (6) j j 05 s he value change of he asse beween he begnnng and end of he accounng perod Among naonal accounans hs 494

6 value change s called (nomnal) me-seres deprecaon combnes he effec of he progress of me from - o wh he effec of ageng from j - 05 o j 05 n general he dfference beween he wo prces (valuaons) comprses he effec of exhauson deeroraon and obsolescence The hrd componen τ denoes he specfc ax(es) ha s (are) leved on he use of an asse of ype and age j durng perod j Fnally he frs componen r s he prce (or valuaon) of hs asse a he begnnng of he perod when s age s j - 05 mes an neres rae r Ths componen reflecs he premum ha mus be pad by he user of he asse o s owner o preven beng sold rgh a he begnnng of he perod and he revenue used for mmedae consumpon; s herefore also called he prce of wang 4 Anoher nerpreaon s o see hs componen as he acual or mpued neres cos o fnance he moneary capal ha s ed up n he asse; s hen called opporuny cos Anyway s a sor of remuneraon whch because here mgh be a rsk componen nvolved s specfc for he producon un and he asse hough he las complcaon s usually no aken no accoun 5 Un user cos as defned by expresson (6) s also called renal prce because can be consdered as he renal prce ha he owner of he asse as owner would charge o he owner as user u oherwse un user cos s lke a lease prce 6 Though (6) has been jusfed from varous vewpons he expresson bascally goes back o Walras (874 p 269) One parcular jusfcaon s worh recallng Rearrangng expresson (6) delvers (7) r = ( u τ ) ( j = J ) j j j Sellng he asse a he begnnng of he perod and generang a reurn from he proceeds of he sale should cover he renal prce mnus ax and he money necessary for buyng back he asse a he end of he perod when s a full perod older Thus equaon (7) could be seen as he oucome of an arbrage process There s however no behavoral assumpon nvolved because he renal prce s no an ndependen varable bu defned by he same equaon The renal prce s precsely equal o he cos of usng hs asse durng one perod whch s he sum of value change ax and opporuny cos 7 Le us now urn o he un user cos of an asse of ype and age j ha s acqured a he mdpon of perod To keep hngs smple hs user cos s analogous o expresson (6) defned as 4 Accordng o Rymes (983) hs namng goes back o gou 5 SNA (2008 par 630) mplcly prescrbes ha for non-marke uns he neres rae r mus be se equal o 0 6 should be noed ha all he operaonal coss assocaed wh he use of a parcular asse are accouned for as nermedae or labor npus cos For nvesmen decsons expeced values of such cos componens mus of course be consdered ogeher wh expeced user cos 7 n he case of rreversble nvesmens such as elecrcy neworks bascally a slgh modfcaon of expresson (6) mus be used as argued by Dewer e al (2009 chaper 0) The frs par remans as s; n he second par now called amorzaon amoun nsead of he acual end-of-perod prce he expeced end-of-perod prce mus be used; and he ax componen dsappears 495

7 (8) ( ) ( ) = v ( 2) r j j j 2 τ ( j 0 ) The dfference from he prevous formula s ha here he second half of he perod nsead of he enre perod s aken no accoun 8 Toal user cos over all asse ypes and ages for perod s hen naurally defned by (9) C u K v K = j = = We see ha he se of commodes K consss of wo subses correspondng respecvely o he ype age classes of asses ha are par of he openng sock and he ype age classes of asses ha are acqured laer The dmenson of he frs se s = J and he dmenson of he second se s = ( J ) The npu prces are gven by expressons (6) and (8) respecvely whle he quanes are gven by K and respecvely 0 The nex ask s o spl for any wo perods = 0 he cos rao CK CK 0 no a prce ndex and a quany ndex or he cos dfference CK CK no a prce ndcaor and a quany ndcaor There s some choce here and gudance can be obaned from Balk (2008) For example he Laspeyres quany ndex reads (0) Q L K ( 0 ) 0 0 uk v = j = = uk j = v = = When all or a number of he maxmal lfe mes J are me-dependen a deour mgh be necessary Frs one compues a prce ndex K( 0) on he common 0 vnages and nex an mplc quany ndex as ( CK CK) K( 0 ) f all he varables occurrng n expresson (9) were observable hen our sory would almos end here However hs s no he case Though he quany varables are n prncple observable he prce varables are no To sar wh he expressons (6) and (8) conan prces (valuaons) for all asse ypes and ages bu excep for new asses and where markes for second-hand asses exs such prces are no observable Thus we need models Bu frs we urn o he relaon beween measures of capal npu cos and capal sock 8 The facor (/2)r s mean as an approxmaon o ( r ) /2 - and he facor ( 2)τ 2 approxmaon o (( τ j05) ) j as an 496

8 4 The Relaon wh Capal Sock Measures The se of quanes { K ; = ; j = 0 J } represens he so-called producve capal sock of he producon un Ths s an enumeraon of he asses ha make producon durng perod possble The oal value of hese asses a he mdpon of perod can be calculaed as () NCS = K = j = = Ths value s called he ne (or wealh) capal sock Noce ha hough he quanes n expresson () are he same as hose occurrng n (9) he prces are dfferen For any wo perods = 0 he rao NCS /NCS 0 can also be spl no a prce ndex and a quany ndex For example he Laspeyres quany ndex reads (2) Q L NCS ( 0 ) 0 K j = = = 0 0 K = j = = L Boh quany ndces Q NCS ( 0 ) and Q L K ( 0 ) measure he volume change of he producve capal sock The prces used o wegh he quanes are dfferen L however The quany ndexq NCS ( 0 )s called a volume ndex of he capal sock whereas Q L K ( 0 ) s called a volume ndex of capal servces Ther numercal dfference can be apprecable (see for example Coremberg 2008) f here are no ransacons n second-hand asses hen he number of asses K s equal o he number of new nvesmens of j perods earler j adjused for 0 he probably of survval Then expresson (9) reduces o (3) J j CK = u v 0 = j = 0 0 f n addon new nvesmens are supposed o sar her economc lfe a he begnnng of he frs-nex perod hen he las expresson reduces furher o (4) C K = J u j = 0 j = whch s he classcal formula for he value of capal servces Ths formula can be rewren as (5) J j 0 = j = CK = u ( u u ) 497

9 provded ha u 0 f could be assumed ha for each asse ype he un user cos raos are ndependen of me 9 ha s (6) u u = φ ( = ; j = ) hen expresson (5) reduces o (7) CK u j = φ 0 = J j = Noce ha f = ( = ) Moreover common sense suggess ha 0 < f ( = ; j = J ) For each asse = he coeffcens f serve o ransform (lnearly) asses of age 2 o J no asses of age The se {f φ } s called he age-effcency profle of asse ype The par beween brackes n expresson (7) s called he producve capal sock of asse ype measured n effcency uns u oherwse gven he age effcency profle he uns of measuremen become uns of age and he only un user cos ha s needed for he compuaon of (7) s he user cos of a one year old asse Under our wo assumpons no ransacons n second-hand asses and new nvesmens sar her economc lfe a he begnnng of he frs-nex perod expresson () reduces o (8) NCS = J j = 0 j = whch s he classcal formula for he ne capal sock value Ths formula can be rewren as (9) 0 J j 0 0 = j = NCS = ( ) provded ha 0 0 f could be assumed ha for each asse ype he prce (valuaon) raos are ndependen of me ha s (20) 0 = ϕ ( = ; k = ) hen expresson (9) reduces o (2) NCS j = 0 ϕ 0 = J j = 9 Auhors such as Hulen (990) sugges ha n equlbrum hs wll be he case Hulen (2009) calls a srong assumpon 498

10 Snce ageng n general reduces he value of an asse s o be expeced ha he coeffcens j are smaller han and her successve magnudes declnng The se {j ϕ } s called he age prce profle of asse ype Accordng o he leraure he producve capal sock a curren prces s defned as (22) CS j = 0 φ 0 = J j = where s he prce (valuaon) a of an asse of ype and age 0; ha s a new 0 asse s neresng o compare expressons (2) and (22) Whle he value of he producve capal sock s obaned by aggregang he varous vnages by he age effcency profle he ne value of he capal sock s obaned by aggregang he vnages by he age prce profle These wo profles are n general dfferen 0 As s clear from expresson (7) s he producve capal sock ha plays a role n he deermnaon of he value of capal servces However he more prmve expresson for he value of capal servces s expresson (4) where he vnages are aggregaed by her un user coss We now reurn o where we lef off a he end of he prevous secon 5 The Relaon Beween Asse rce and Un User Cos Consder expresson (6) and rewre n he form (23) ( ) = u τ = r j j ( j ) For any asse ha s no premaurely dscarded wll be he case ha s value a he end of perod s equal o s value a he begnnng of perod ; formally ( ) j = ( j ) Subsung hs no expresson (23) and rewrng agan one obans (24) ( ) = ( ) j = u j J ( j ) τ ( ) r Ths expresson lnks he prce of an asse a he begnnng of perod wh s prce a he begnnng of perod beng hen perod older Bu a smlar relaon lnks s prce a he begnnng of perod wh s prce a he begnnng of perod 2 beng hen agan perod older (25) ( ) = ( 2) = u ( j ) j r ( ) ( j ) 0 5 τ j ( j J ) 0 Under mld regulary condons can be shown ha a geomerc age effcency profle f = ( - d ) j- mples a geomerc age prce profle j = ( - d ) j and vce versa See SNA (2008 par ) for a smple proof Usng expressons (4) and (8) as pons of deparure for he consrucon of prce and quany ndces was frs suggesed by Dewer and Lawrence (2000) 499

11 Ths can be connued unl (26) ( ) = j = u j r ( j) ( ) j j J J τ ( j J ) j snce we know ha J 05 (24) ec one fnally obans ( ) ( ) j = J 05 = 0 Subsung expresson (25) no (27) j05 u τ u j τ j = r r r ( ) ( ) J j j uj τ J ( r ) r ( ) J j Here maeralzes wha s known as he fundamenal asse prce equlbrum equaon Noce however ha here was no equlbrum whaever ha may mean assumed here and ha here are no oher economc behavoral assumpons nvolved; s jus a mahemacal resul Expressons (23) and (27) are dual The frs derves he (ex ax) un user cos from dscouned asse prces whle he second derves he asse prce as he sum of dscouned fuure (ex ax) un user coss; he dscounng s execued by means of fuure neres raes A mahemacal ruh lke expresson (27) however s no mmedaely helpful n he real world A he begnnng or even a he end of perod mos f no all of he daa ha are needed for he compuaon of he asse prces j 05 and j are no avalable Thus n pracce expresson (27) mus be flled n wh expecaons and hese depend on he pon of me from whch one looks a he fuure A raher naural vanage pon s he begnnng of perod ; hus he operaor E placed before a varable means ha he expeced value of he varable a - s aken Modfyng expresson (27) he prce a he begnnng of perod of an asse of ype and age j - 05 s gven by (28) j05 E ( u τ ) E ( u j τ j ) E r E r E r E j E E u τ j ( E ) E E j E r E r ( ) ( )( ) ( ) Noce n parcular ha n hs expresson he economcally maxmal age as expeced a he begnnng of perod E J occurs u oherwse a he begnnng of perod he remanng economc lfeme of he asse s expeced o be E j05 perods 2 For each of he comng perods here s an expeced (ex ax) renal and he (wh expeced neres raes) dscouned renals are summed Ths sum consues he prce (value) of he asse 2 See Erumban (2008) on he esmaon of expeced lfemes for hree ypes of asses n a number of ndusres 500

12 Smlarly he prce a he end of perod of an asse of ype and age j 05 s gven by (29) j ( j ) ( ) ( ) 2 2 E u ( j j ( ) τ ) E ( u j 2 τ j 2 ) = ( ) ( ( ) E r E r ) E r ( ) ( ) ( ) E j E E u ( ) E τ j ( ( ) E J ) ( ) ( ) ( ) E ( E r ) E r j ( ) ( ) ( ) 2 Noce ha hs prce depends on he economcally maxmal age as expeced a he begnnng of perod (whch s he end of perod ) ( E ) J whch may or may no dffer from he economcally maxmal age as expeced one perod earler E J The las menoned expeced age plays a role n he prce a he end of perod of an asse of ype and age j 05 as expeced a he begnnng of hs perod (30) E j E ( u ) E r 2 ( τ j 2 ) ( E r ) E r 2 j τ j E u j 2 ( ) E j E j u E τ E E j E ( E r ) ( E r ) ( ) 2 Expresson (30) was obaned from expresson (28) by deleng s frs erm as well as he frs perod dscoun facor E r Ths reflecs he fac ha a he end of perod he asse s remanng lfeme has become shorer by one perod Generally one may expec ha E j j Expresson (29) dffers from expresson (30) n ha expecaons are a ( ) - nsead of - Snce one may expec ha due o echnologcal progress he remanng economc lfeme of any asse shorens ha s E < E J ( ) expresson (29) conans fewer erms han expresson (30) Generally one may expec ha j < E j ; ha s he acual prce of an asse a he end of a perod s less han or equal o he prce as expeced a he begnnng Armed wh hese nsghs we reurn o he un user cos expressons (6) and (8) Naural decomposons of hese wo expressons are (3) ( ) ( ) = u r j j = E j E j j τ ( j ) and ( ) ( ) ( ) = v r j j j (32) = ( 2) E E j j τ ( j 0 J ) 50

13 As before he frs erm a eher rgh-hand sde represens he prce of wang The second erm beween brackes s called ancpaed me-seres deprecaon and could be decomposed furher no he ancpaed effec of me (or ancpaed revaluaon) and he ancpaed effec of ageng (or ancpaed crossseconal deprecaon) The hrd erm also beween brackes s called unancpaed revaluaon We wll come back o hese erms laer The underlyng dea s ha a he begnnng of each perod or n he case of nvesmen a he mdpon economc decsons are based on ancpaed raher han realzed prces The fourh erm n he wo decomposons s agan he ax erm s here assumed ha wh respec o wang and ax ancpaed and realzed prces concde Subsung expressons (3) and (32) no expresson (9) one obans he followng aggregae decomposon: (33) C = r K ( 2 ) r K j05 j = j = = J ( j ) K j E je j 0 5 = j = = ( E j j ) K E j j = j = = τk ( ) 2 τ = j = = ( ) ( ) On he frs lne afer he equaly sgn we have he aggregae cos of wang (34) J CKw r j05 K ( 2) j = j = = J Noce ha he par beween brackes dffers slghly from he ne capal sock as defned by expresson () can be nerpreed as he value of he producon un s producve capal sock as used durng perod On he second lne afer he equaly sgn n expresson (33) we have he aggregae cos of ancpaed me-seres deprecaon (35) ( ) ( ) j = j = = CKe j E j K E j On he hrd lne we have he aggregae cos of unancpaed revaluaon (36) CKu E j j K E j j = j = ( ) = ( ) Fnally on he fourh lne we have he aggregae cos of ax 502

14 (37) Kax = j = = C τ K ( 2) τ Usng all hese defnons expresson (33) reduces o (38) C = C C C C K Kw Ke Ku Kax Thus capal npu cos can raher naurally be spl no four meanngful componens 6 Raes of Reurn The K-CF model s governed by he followng accounng deny where npu caegores are placed lef and oupu caegores are placed rgh of he equaly sgn: (39) C C C C Π = CF Kw Ke Ku Kax where CF denoes ex pos cash flow generaed by he operaons of he producon un durng perod and prof s defned by hs deny The nex model s based on he dea ha he (ex pos) cos of me-seres deprecaon plus ax should be reaed n he same way as he cos of nermedae npus and hus subraced from cash flow Hence he oupu concep s called ne cash flow and he remanng npu cos s he wang cos of capal Ths s called he K-NCF model whch s governed by (40) C Π = CF ( C C C ) NCF Kw Ke Ku Kax Ths accounng relaon provdes an excellen pon of deparure for a dscusson of he neres rae r whch deermnes he aggregae cos of wang or opporuny cos C accordng o expresson (34) Usng expresson (34) equaon (40) Kw can be rewren as (4) rucs Π = NCF where (42) j05 j = j = = j = UCS K ( 2) whch can be nerpreed as he (value of he) producon un s capal sock as used durng perod rovded ha NCF 0 equaon (4) hen says ha apar from prof ne cash flow provdes he reurn o he (owner of he) capal sock Ths s he reason why r s also called he rae of reurn 503

15 n prncple he value of he capal sock as well as he ne cash flow are emprcally deermned Tha leaves an equaon wh wo unknowns namely he rae of reurn r and prof Seng = 0 and solvng equaon (4) for r delvers he so-called endogenous or nernal or balancng rae of reurn Ths soluon s of course specfc for he producon un Ne cash flow s calculaed ex pos snce conans oal me-seres deprecaon Thus he endogenous rae of reurn as calculaed from expresson (4) s also an ex pos concep The alernave s o specfy some reasonable exogenous value for he rae of reurn say he annual percenage of headlne C change plus somehng Then of course prof follows from equaon (4) and wll n general be unequal o 0 The endogenous rae of reurn s defned by he equaon (43) r UCS = NCF endo Combnng hs wh expresson (4) delvers he followng relaon beween he endogenous and some exogenous rae of reurn: (44) r = r Π UCS endo Hence prof s posve f and only f rendo > r Then relaon (44) can be nerpreed as sayng ha r endo absorbs prof A varan of he K-NCF model s obaned by consderng unancpaed revaluaon whch s he unancpaed par of me-seres deprecaon as a componen ha mus be added o prof: 3 (45) r Π* = CF ( C C ) NNCF Kw Ke Kax Ths deny a he same me serves as he defnon of prof from normal operaons * The relaon beween he wo prof conceps s (46) Π* Π = CKu The accounng deny of he K-NNCF model gven by expresson (45) can be rewren as (47) r UCS Π* = NNCF Now provded ha NNCF * 0 normal ne cash flow s seen as he reurn o he (owner of he) capal sock Seng * = 0 and solvng equaon (47) for r delvers wha can be called he normal endogenous rae of reurn defned by (48) r* UCS = NNCF endo 3 n he model of Hulen and Schreyer (2006) oal (= unancpaed plus ancpaed) revaluaon s added o prof Ths s conssen wh SNA (2008) s prescrpon for non-marke uns 504

16 Usng expressons (47) (46) and (44) appears ha (49) r* = r C UCS endo endo Ku Thus he normal endogenous rae of reurn absorbs no only prof bu also he moneary value of all unancpaed asse revaluaons Alernavely as n he prevous model one can specfy some reasonable exogenous value for he rae of reurn Then of course * follows from equaon (47) and by subracng he value of all unancpaed asse revaluaons C Ku one obans prof The wo expressons (4) and (47) and her underlyng models are o be consdered as polar cases n he frs all he unancpaed revaluaons (ha s he whole of C Ku ) are consdered as nermedae cos whereas n he second hey are consdered as belongng o prof Clearly posons n beween hese wo exremes are hnkable For some asse ypes unancpaed revaluaons mgh be consdered as nermedae cos and for oher ypes hese revaluaons mgh be consdered as belongng o prof A number of conclusons can be drawn Frs here s no sngle concep of he endogenous rae of reurn There s raher a connuum of possbles dependng on he way one wans o deal wh unancpaed revaluaons Second an endogenous rae of reurn of whaever varey can only be calculaed ex pos Ne cash flow as well as normal ne cash flow requre for her compuaon ha he accounng perod has expred Thrd we are assumng ha all he npus and oupus are correcly observed Unobserved npus and oupus and measuremen errors lead o a dsored prof fgure Snce an endogenous rae of reurn can be sad o absorb prof see expressons (44) and (49) he exen of undercoverage also has mplcaons for he nerpreaon of he rae of reurn (see also Schreyer 200) u oherwse snce an endogenous rae of reurn closes he gap beween he npu and he oupu sde of he producon un s nfluenced by all sors of measuremen errors Fnally he concep of an endogenous rae of reurn does no make sense for non-marke uns snce here s no accounng deny based on ndependen measures a he npu and he oupu sde Ye non-marke uns use capal lke marke uns The accounng pon of vew as used n hs paper mples an ex pos user cos concep However economc decson processes are usually based on ancpaed magnudes of ceran varables n parcular nvesmen and oher producon decsons are based on ex ane user coss How do he wo conceps relae? Ths s one of he quesons consdered n an nrgung arcle by Oulon (2007) He proposed a hybrd approach whch n our seup can be summarzed as follows Ex ane capal npu cos s calculaed as (50) ˆ C C C C K Kw Ke Kax where C Kw s based on some requred rae of reurn r (perhaps derved from pas endogenous raes of reurn) and C s based on expeced end-of-perod prces Ke 505

17 The ex ane capal npu cos rao for perod relave o perod 0 ˆ C Cˆ 0 K K can be decomposed no a prce ndex ˆ K ( 0 ) and a quany ndex Q ˆ K ( 0 ) and s hs quany ndex ha s supposed o ac as he capal npu quany ndex Ex pos capal npu cos plus prof appears o be (5) C C C C Π Kw Ke Ku Kax and hs s by defnon equal o CF The cash-flow based oal facor producvy (TF) ndex for perod relave o perod 0 s genercally defned as Q CF ( 0)/Q K( 0) where Q CF ( 0) s a cash-flow based oupu quany ndex and Q K( 0) s he quany ndex componen of he ex pos capal npu cos rao (see Balk 200 p S243) f nerpre Oulon (2007) correcly hs suggeson s o calculae he TF ndex nsead by he followng formula: (52) QCF ( 0) CK CF QK ( ˆ ) ˆ ( ) We see here ha n he denomnaor he ex ane capal npu quany ndex s mulpled by he share of ex ane capal npu cos n ex pos cash flow (whch s equal o ex pos capal npu cos under an endogenous rae of reurn) Usng he famlar produc relaons lnkng prce ndex quany ndex and value rao Oulon s TF ndex can be wren as (53) QCF ( 0) CF CF 0 CK CF QK 0 CK K ( ) ( ) = ( ) ˆ ˆ ˆ ˆ ( ) Ths s deflaed ex pos cash flow dvded by deflaed ex ane capal npu cos s no compleely clear wha hs rao s supposed o measure 7 Aggregaon Le us now consder an ensemble of producon uns K Le here be a common classfcaon of asse ypes and ages All he prce quany and value varables dscussed n he foregong should be adorned wh he superscrp k Then for each producon un he K-NCF accounng relaon reads (54) k k k k r UCS Π = NCF ( k K ) f here are no ax wedges so ha he values of ougong and ncomng rade flows beween he producon uns cancel hen aggregaon reduces o smple addon and he K-NCF accounng relaon for he aggregae consdered as a bg producon un reads (55) K k K k r UCS Π = NCF k K k K 506

18 Afer havng emprcally flled n he capal sock and cash flow varables we are lef wh a large number of nerrelaed unknowns For reachng conssency here are wo approaches a boom-up approach and a op-down approach respecvely The boom-up approach sars no unexpecedly wh relaons (54) Aggregae prof s hen se equal o he sum of ndvdual profs (56) Π K k = Π k K and he aggregae rae of reurn s se such ha equaon (55) holds Bu hs means ha (57) K k k k r UCS = r UCS k K k K or (58) r K k K = k r UCS k K UCS k k Thus he rae of reurn for he aggregae s a weghed mean of he raes of reurn for he ndvdual uns he weghs beng shares of he value of he producve capal sock as used durng perod The op-down approach sars wh relaon (55) For each ndvdual producon un we hen se (59) k K r = r ( k K ) One checks mmedaely ha hs mples ha equaon (56) holds; ha s aggregae prof s he sum of all he ndvdual profs When r K s he endogenous rae of reurn hen S k K k = 0 bu noce ha he ndvdual k need no be equal o zero; pu oherwse he endogenous rae of reurn for he aggregae does no necessarly concde wh he endogenous raes of reurn for he ndvdual producon uns 8 Some mplemenaon ssues There reman a number of mplemenaon ssues o dscuss For hs he reader s nved o reurn o expresson (33) To ease he presenaon a perod s now se equal o a year The quanes {K ; = ; j = J } and { ; = ; j = 0J } are usually no avalable nsead as s usually he case he erpeual nvenory Mehod generaes esmaes of he openng sock of asses a perod - 507

19 prces { j K = j K ; = ; j = } and he nvesmen j Survey generaes esmaes of md-perod values { ; = ; j = 0J } Models for me-seres deprecaon are brefly dscussed n he Appendx The me-seres deprecaon of an asse of ype and age j ha s avalable a he begnnng of perod s n pracce frequenly modeled as (60) j j05 = ( δ ) where denoes he roducer rce ndex (or a kndred prce ndex) ha s applcable o new asses of ype and d s he annual cross-seconal deprecaon rae ha s applcable o an asse of ype and age j Ths deprecaon rae deally comes from an emprcally esmaed age prce profle Thus me-seres deprecaon s modeled as a smple mulplcave funcon of wo ndependen facors The frs whch s plus he annual rae of prce change of new asses of ype concerns he effec of he progress of me on he value of an asse of ype and age j The second - d > 0 concerns he effec of ageng by one year on he value of an asse of ype and age j Ageng by one year causes he value o declne by d 00 percen Smlarly ancpaed me-seres deprecaon s modeled as (6) E j j05 = E ( δ ) n hs expresson nsead of he annual rae of prce change of new asses as observed ex pos he annual rae as expeced a he begnnng of perod s aken Bu wha o expec? There are of course several opons here The frs ha comes o mnd s o use some pas observed rae of change of or a more general Second one could assume ha expecedly he rae of prce change of new asses s equal o he rae of change of a (headlne) C and use he realzed expecaon : 4 (62) E C = C Under he las assumpon ancpaed me-seres deprecaon s measured as (63) E j j05 C = ( δ ) C 4 Ths corresponds o he SNA (2008 par 287) advce on calculang neural holdng gans 508

20 and combnng expressons (60) and (63) unancpaed revaluaon s measured by (64) E j j05 j C = ( δ ) C j05 Smlar expressons hold for asses ha are acqured a he mdpon of perod excep ha we mus make a dsncon beween new and used asses The meseres deprecaon for an asse of ype and age j s modeled as (65) = ( δ 0 ) 0 j = ( 2) j = j δ ( ) The ancpaed me-seres deprecaon s measured by (66) E E C = ( δ 0 ) 0 C j C = ( 2) j = j C δ ( ) and unancpaed revaluaon s measured by (67) E C = 0 0 C ( δ 0 ) E j j C = j j J j C ( 2) = δ ( ) An mporan queson s n wha crcumsances do he un user coss u and v become non-posve? Consder for nsance expresson (3) and subsue expressons (63) and (64) Ths yelds (68) u j05 C C = r ( δ ) C C = r τ ( δ ) j05 τ ( δ ) j05 Hence u 0 f and only f 509

21 (69) r τ δ j05 n ceran exreme cases hs could ndeed happen Consder asses wh a very low cross-seconal deprecaon rae (such as ceran buldngs or land) and a very hgh revaluaon rae (or rae of prce ncrease) A low neres-plus-ax rae hen could lead o negave un user coss u oherwse when he ex pos revaluaon (as measured by a ) more han offses neres plus ax plus deprecaon hen he un user cos of such an asse becomes negave f he unancpaed revaluaon s excluded from he user cos ha s un user cos s measured by (70) u C τ = r ( δ ) C j j05 hen u 0 f and only f (7) C C r τ δ j05 The lkelhood ha such a suaon wll occur s small For hs o happen expeced revaluaon (as measured by a C) mus more han offse neres plus ax plus deprecaon 9 Concluson The approach followed n hs paper s ha capal npu cos s he cos of usng capal asses much lke labor npu cos s he cos of employng workers However unlke he buldng blocks for labor npu cos quanes of asses and un user coss canno smply be observed The usual neoclasscally nspred approach o measurng and decomposng capal npu cos seems o res on a number of darng behavoral assumpons We are followng here he accounng approach whch on he one hand avods such assumpons and on he oher hand shows he degrees of freedom one has n mplemenng measuremen Some would say ha freedom means arbrarness or even anarchy hope o have made clear ha hs s no he case however Surely here s a lo of freedom bu ha forces us o hnk abou he choces ha mus be made f choces are made o accommodae neoclasscal behavoral assumpons fne Bu whn he framework explcaed n hs paper alernave choces can be argued for u oherwse he accounng framework s so general ha he consequences of varan assumpons can be explored and he assumpons hemselves be esed a her relevance To wrap up s useful o revs our man seps As sad he pon of deparure s ha capal npu cos s he cos of usng (owned) asses The quanes of hose asses accordng o ype age classes a some level of deal a he begnnng and end 50

22 of each accounng perod are esmaed from nvesmen surveys and oher daa sources combned wh some varan of he erpeual nvenory Mehod These quanes consue he producve capal sock of he producon un under consderaon There appears o be no need for he concep of capal servce or he assumpon ha he servce rendered by a ceran asse s proporonal o s quany Ths concep s as were replaced by he user cos concep The un user cos of an asse of ceran ype and age s bascally deermned by he dfference beween s begnnng- and end-of-perod prce plus an opporuny cos deermned by a ceran neres rae and an amoun of ax There s no behavoral assumpon underlyng hs specfcaon bu here are a number of as ye undeermned varables rces and un user coss appear o be muually deermned by he fundamenal asse prce equaon whch here maeralzes as a purely mahemacal resul rces appear o depend on fuure un user coss bu he fuure s unknown; hus hs s where expecaons ener he pcure appears useful o dsngush beween acual and expeced prces and o decompose he dfference beween begnnng- and end-of-perod acual prce no an expeced and an unexpeced par where s assumed ha expecaons are formed a he begnnng of he perod The dfference beween begnnngand (expeced) end-of-perod prces s hen convenonally modelled by a smple mulplcave relaon n whch roducer and Consumer rce ndces fgure as well as emprcally deermned asse-specfc deprecaon raes The mporan remander s he deermnaon of he opporuny cos componen whch urns ou o be equal o he value of he producve capal sock mes an neres rae The key assumpon of he neoclasscal seup s o deermne hs neres rae called rae of reurn endogenously by seng capal npu cos equal o cash flow (or gross operang surplus); pu oherwse by seng prof as he dfference beween cash flow and capal npu cos equal o zero Ths of course accommodaes he assumpon of compeve prof maxmzaon under a consan-reurns-o-scale echnology bu here s nohng n he accounng sysem ha necessaes such an assumpon And he assumpon self s a varance wh much emprcal evdence Moreover as demonsraed n Secon 6 here s no a unque endogenous rae of reurn bu a connuum of possbles dependng on he way one wans o deal wh unexpeced revaluaons The boom lne could be he advce o sascal agences o use he degrees of freedom avalable n he measuremen sysem o accommodae dfferen sors of users References Balk B M rce and Quany ndex Numbers: Models for Measurng Aggregae Change and Dfference Cambrdge Unversy ress New York 2008 An Assumpon-Free Framework for Measurng roducvy Change Revew of ncome and Wealh 56 Specal ssue S Balk B M and D A van den Bergen The Cos of Capal npu: Calculaon Mehods Revsed verson of a paper presened a he Capal Measuremen Workshop Oawa May Coremberg A The Measuremen of TF n Argenna : A Case of he Tyranny of Numbers Economc Cycles and Mehodology nernaonal roducvy Monor

23 Dewer W E The Aggregaon of Capal over Vnages n a Model of Emboded Techncal rogress Journal of roducvy Analyss Dewer W E and D Lawrence rogress n Measurng he rce and Quany of Capal n L Lau (ed) Economercs and he Cos of Capal: Essays n Honor of Dale W Jorgenson MT ress Cambrdge MA 2000 Dewer W E and Schreyer The Measuremen of Capal n S N Durlauf and L E Blume (eds) The New algrave Dconary of Economcs 2nd edon algrave Macmllan 2008 Dewer W E D Lawrence and J Fallon The Theory of Nework Regulaon n he resence of Sunk Coss Economc nsghs Hawker ACT Ausrala 2009 Erumban A A Lfemes of Machnery and Equpmen: Evdence from Duch Manufacurng Revew of ncome and Wealh Hulen C R The Measuremen of Capal n E R Bernd and J E Trple (eds) Ffy Years of Economc Measuremen Sudes n ncome and Wealh Volume 54 Unversy of Chcago ress Chcago and London 990 Growh Accounng Workng aper 534 Naonal Bureau of Economc Research Cambrdge MA 2009 Hulen C R and Schreyer ncome Deprecaon and Capal Gans n an neremporal Economc Model Ffh Oawa roducvy Workshop May OECD Measurng Capal; OECD Manual 2nd edon Organsaon for Economc Co-operaon and Developmen ars 2009 Oulon N Ex os versus Ex Ane Measures of he User Cos of Capal Revew of ncome and Wealh Rymes T K More on he Measuremen of Toal Facor roducvy Revew of ncome and Wealh Schreyer Measurng Mul-Facor roducvy when Raes of Reurn are Exogenous n W E Dewer B M Balk D Fxler K J Fox and A O Nakamura (eds) rce and roducvy Measuremen Volume 6 Trafford ress 200 (avalable a wwwvancouvervolumescom) SNA Sysem of Naonal Accouns European Commsson nernaonal Moneary Fund Organsaon for Economc Co-operaon and Developmen Uned Naons World Bank 2008 Walras L Elemens of ure Economcs (ranslaed by W Jaffé Rchard D rwn Homewood L 954) 874 Supporng nformaon Addonal Supporng nformaon may be found n he onlne verson of hs arcle: Appendx: Decomposons of Tme-Seres Deprecaon lease noe: Wley-Blackwell are no responsble for he conen or funconaly of any supporng maerals suppled by he auhors Any queres (oher han mssng maeral) should be dreced o he correspondng auhor for he arcle 52