On Measuring the Productivity and. the Standard of Living in Japan,

Transcription

1 KEO Discussion Paper No. 115 On Measuring he Produciviy and he Sandard of Living in Japan, W. Erwin Diewer, Hideyuki Mizobuchi, and Koji Nomura February 2009 Absrac The paper looks a he conribuion of he marke secor o changes in Japan's living sandards over he years I considers real income as a measure for living sandards and decomposes is growh rae ino hree componens: componens due o changes in produciviy, in real oupu prices (including changes in he erms of rade) and in primary inpu growh. The exac index number approach developed by Diewer and Morrison (1986) and Kohli (1990) (2004) is adaped o his real income conex. Finally, he paper swiches from a gross oupu concep o a heoreically preferred ne oupu concep. In he ne oupu conex, i urns ou ha he role of capial deepening as a conribuor o higher living sandards diminishes and he role of produciviy and labour growh becomes more imporan. The firs auhor is a he Deparmen of Economics, Universiy of Briish Columbia, Vancouver, Canada and he las wo auhors are a Keio Economic Observaory (KEO), Keio Universiy, Tokyo, Japan.

2 Conens 1. Inroducion The Theoreical Framework Daa Consrucion The Definiion of he Marke Secor The Daabase for Inpus and Oupus User Coss and Real Raes of Reurn Resuls Japanese Produciviy Growh: A Convenional Approach The Decomposiion of Deflaed GDP Growh The Decomposiion of Deflaed NDP Growh Conclusion Appendix A. Daa Consrucion Appendix B. Tables of Prices, Quaniies, and Value for Inpus and Oupus References Tables and Figure Table 1: Lis of Capial Services, Invenory Services, and Land Services Table 2: Chained Fisher Indexes of Oupu, Inpu and Produciviy Growh and he Balancing Real Rae of Reurn in he Japanese Economy, Table 3: The Decomposiion of Marke Secor Real Income Growh ino Translog Produciviy, Real Oupu Price Change and Inpu Quaniy Conribuion Facors Table 4: The Decomposiion of Marke Secor Real Income Level ino Cumulaive Produciviy, Real Oupu Price Change and Inpu Quaniy Conribuion Facors Table 5: The Quaniy and Price of Gross Invesmen, Depreciaion, Ne Invesmen, Capial Services, Waiing Capial Services (Billion Yen) Table 6: The Decomposiion of Marke Secor Ne Real Income Growh ino Translog Produciviy, Real Oupu Price Change, and Inpu Quaniy Conribuion Facors Table 7: The Decomposiion of Marke Secor Ne Real Income Cumulaive Growh ino Produciviy, Real Oupu Price Change, and Inpu Quaniy Conribuion Facors using he Translog Ne Produc Approach Table B- 1: Marke Secor Oupu and Inpu Prices for Japan Table B- 2: Marke Secor Oupu and Inpu Quaniies for Japan (Billion Yen) Figure 1: The Real Rae of Reurn for he Years

3 On Measuring he Produciviy and he Sandard of Living in Japan, W. Erwin Diewer, Hideyuki Mizobuchi and Koji Nomura February Inroducion The Japanese economy has experienced remendous growh in he pas half cenury bu here has been a pronounced slowdown for more han 10 years afer he bubble economy collapsed. 1 This unique experience of he pos war Japanese economy has been aracing he ineres of a lo of researchers. However, mos of hese sudies have been concerned wih he growh of producion and produciviy of he Japanese economy. These sudies quanify he various sources of Japanese economic growh and measure he producive efficiency of he Japanese economy. 2 In his paper, we are primarily concerned wih increases in he sandard of living in he pos-war Japanese economy. In paricular, we focus on he conribuion of Japan s marke secor o improvemens in Japanese living sandards over he period We have excluded he general governmen secor from our analysis because his secor may provide is oupus wihou charging for hem or i may sell hem for prices ha are no economically significan. Our analysis will rely on he assumpions of revenue maximizing and cos minimizing behavior, assumpions ha are generally no applicable for he governmen secor. There are oher measuremen difficulies associaed wih he governmen secor. Because i is difficul o measure many oupus in he governmen secor, he Sysem of Naional Accouns (see he UN (1993)) recommends measuring he value of hese difficul o measure oupus of he general governmen as he cos of producing he oupus and 1993 SNA recommends seing he price of hese oupus equal o an inpu cos index. This is no a proper way of measuring oupu from a heoreical poin of view. 4 The household secor is also a producer in he SNA. In paricular, his secor produces impued ren from owner-occupied houses. However, his is a special ype of producion where produciviy improvemens canno ake place and so we have decided o exclude his secor from our concep of he marke producion secor. Therefore, we compleely ne ou he general governmen and he household secor from our framework in order o avoid he above menioned difficulies. Real Gross Domesic Produc (GDP) is ofen used as a proxy for a counry s living sandard. However, real GDP is a measure of he level of producion and i is well known ha real GDP can be a very misleading indicaor of a counry s welfare in he face of changing erms of rade. 5 Real (Gross) Income, which deflaes nominal GDP by he price of domesic consumpion goods is more appropriae as a measure of economic welfare. Economic welfare comes from consumpion. Real income capures how much consumpion people can purchase for heir income. For example, if he nominal oupu is consan The firs auhor is a he Deparmen of Economics, Universiy of Briish Columbia, Vancouver, Canada and he las wo auhors are a Keio Economic Observaory (KEO), Keio Universiy, Tokyo, Japan. 1 Alhough Japanese economy is in a period of economic recovery since 2002, is growh is sill modes. 2 See Hayashi and Nomura (2005), Hayashi and Presco (2002), Jorgenson and Nishimizu (1978), Jorgenson, Kuroda, and Nishimizu (1987), Jorgenson and Nomura (2005), Nishimizu and Hulen (1978) and Nomura (2004). 3 We call i 1993 SNA hereafer. 4 There is an addiional difficuly wih he SNA reamen of inpu cos in he governmen secor: he user cos of capial in he governmen secor is jus depreciaion wih no impuaion for he financial cos of capial. Thus if a governmen building is sold o he privae secor, GDP will increase! 5 Hamada and Iwaa (1984) show ha real GNP (equivalen o real GDP in our paper) may overesimae naional welfare when he erms of rade deeriorae for a counry. Kohli (2004) showed ha an improvemen in he erms of rade can acually lead o a fall in real GDP. 1

4 over wo consecuive periods, an improvemen in he erms of rade should increase economic welfare by enhancing he purchasing power of domesic households. Real Income always increases in his siuaion. However, he convenional Laspeyres ype real GDP may decrease in his case. We shed he new ligh on he measuremen of he sandard of living in Japan for he years by focusing on he real income generaed by he marke secor. We aemp o measure he deerminans of real income growh in Japan. We adap he analyic framework for produciviy measuremen ha was developed by Diewer and Morrison (1986) and Kohli (1990) (2004) o he real income growh conex. Adaping heir resuls, we find ha he main deerminans of growh in real income generaed by he marke secor of he economy are: Technical progress or improvemens in Toal Facor Produciviy (TFP); Growh in nonconsumpion real domesic oupu prices, growh in he real prices of expors and falls in he real prices of impors; and Growh in primary inpus. Using our adaped decomposiion formula, all of hese hree deerminans can be calculaed using only he observable price and quaniy daa over he years 1955 o A difficuly in applying his decomposiion formula is uncerainy as o wha is he correc user cos of capial. In paricular, should an exogenous rae of reurn be used in he formula or should an endogenous balancing rae of reurn ha makes he value of inpus equal o he value of oupus be used. Schreyer (2007) surveys several differen procedures for esimaing raes of reurn. We use ex pos balancing (real) raes of reurn. These balancing real raes of reurn seem o capure business condiions in he pos war Japanese economy. GDP in a closed economy wih no governmen is simply consumpion plus gross invesmen plus changes in invenory. Economiss have argued for a long ime ha Ne Domesic Produc (NDP), which equals consumpion and ne invesmen, is a more appropriae welfare measure han GDP. 6 The problem wih he gross concep is ha i gives us a measure of oupu ha is no susainable. In order o move o a more heoreically appropriae NDP concep using our framework for he deerminans of real income growh, i is necessary o rea depreciaion as an offse o gross invesmen. Thus we ake he depreciaion erm in he user cos formula ou of coss and rea i as a negaive oupu ha will ac as an offse o gross invesmen. Our heoreical resuls, explaining he growh of gross real income generaed by he marke secor, can sill be applied o he ne real income concep. Our daase of he marke secor has been consruced from differen daa sources. For ne oupu daa, we heavily depend on daa in Japanese Sysem of Naional Accouns. 7 For capial socks and invesmen, we also made exensive use of he invesmen and asse daa in he KEO daabase. For labour inpu, we consruced oal hours of works for hree differen workers; he self-employed, family workers, and employees from he Japanese Labour Force Survey and he Japanese naional accouns. One limiaion of our sudy mus be noed and ha is he fac ha we have no indusry deail in our daa base and hus we canno locae he conribuions of TFP growh in individual indusries o he aggregae marke secor TFP growh. 8 6 See Marshall (1890), Pigou (1924), Samuelson (1961), and Weizman (1976). More recen papers ha argue for he ne produc framework are Diewer and Fox (2005), Oulon (2004), and Weizman (2003). 7 In 1978, he Japanese sysem of naional accouns was revised o comply wih he guidelines proposed by he Unied Naions Sysem of Naional Accouns (1968 SNA). In 2000, i was revised o comply wih he guidelines newly proposed by he Unied Naions Sysem of Naional Accouns (1993 SNA). 8 Moreover, we canno measure he conribuions o aggregae TFP growh of shifs in labour resources from less producive secors o more producive secors. I comes from he aggregae naure of our labour daa. Since we consider hree ypes of labour such as employees, he self-employed, and family workers, we can 2

5 The remainder of his paper is organized as follows. In secion 2, we ouline a framework for measuring produciviy and decomposing real income growh ino several explanaory facors. This framework is an adapaion of a mehodology inroduced by Diewer and Morrison (1987). In secion 3, we explain our daa consrucion. The rae of reurn is endogenously deermined from observed daa. The procedure for deermining he rae of reurn is explained here. In secion 4 of he paper, we presen a convenional TFP growh accouning for he marke secor of he Japanese economy for he years There have been several recen sudies ha do more or less he same hing so one migh quesion he value of ye anoher sudy of Japanese TFP growh. 9 However, all of hese alernaive sudies cover a much shorer period (wih he excepion of Nomura (2004)) and here are oher significan differences. In he presen sudy, we consider only he marke secor of he Japanese economy where produciviy improvemens are possible under curren naional income accouning convenions. Moreover, our focus is no on TFP growh raes per se bu raher he conribuion of TFP growh o real income growh; i.e., he exising sudies do no follow our reamen of changes in he erms of rade. In secion 4.2, he gross income mehodology developed in secion 2 is implemened using our Japanese marke secor daa base for he years and he ne income mehodology is implemened in secion 4.3. In secion 5, we conclude he paper. 2. The Theoreical Framework In his secion, we presen he producion heory framework which will be used in he remainder of he paper. The main references are Diewer and Morrison (1986) and Kohli (1990). 10 Iniially, we assume ha he marke secor of he economy produces quaniies of M (ne) 11 oupus, y [y 1,...,y M ], which are sold a he posiive producer prices P [P 1,...,P M ]. We furher assume ha he marke secor of he economy uses posiive quaniies of N (primary) 12 inpus, x [x 1,...,x N ] which are purchased a he posiive primary inpu prices W [W 1,...,W N ]. In period, we assume ha here is a feasible se of oupu vecors y ha can be produced by he marke secor if he vecor of primary inpus x is uilized by he marke secor of he economy; denoe his period producion possibiliies se by S. We assume ha S is a closed convex cone ha exhibis a free disposal propery. 13 only capure he conribuion of shifs in labour resources wihin hese hree ypes. On he oher hand, indusry price and quaniy daa are nooriously unreliable due o he lack of deailed surveys on gross oupus and inermediae inpus, for service indusries in paricular. A final reason for our use of aggregae marke secor daa is ha reliable breakdowns of expors and impors by indusry are no available. 9 Some of he recen sudies are Hayashi and Nomura (2005), Hayashi and Presco (2002), Miyagawa, Io and Harada (2004) and Jorgenson and Nomura (2005). 10 The heory also draws on Samuelson (1953), Diewer (1974; ) (1980) (1983; ), Fox and Kohli (1998), Kohli (1978) (1991) (2003) (2004a) (2004b), Morrison and Diewer (1990), Samuelson (1953) and Sao (1976). This paper is essenially an exended version of Diewer, Mizobuchi and Nomura (2005) using newly available daa. The heoreical framework explained in his secion was recenly used by Diewer and Lawrence (2006). 11 If he mh commodiy is an impor (or oher produced inpu) ino he marke secor of he economy, hen he corresponding quaniy y m is indexed wih a negaive sign. We will follow Kohli (1978) (1991) and Woodland (1982) in assuming ha impors flow hrough he domesic producion secor and are ransformed (perhaps only by adding ransporaion, wholesaling and reailing margins) by he domesic producion secor. The recen exbook by Feensra (2004; 76) also uses his approach. 12 Primary inpus only include labour and capial services from reproducible asses, invenories and land. Inermediae inpus fall ino he caegory of (ne) oupus. 13 For a more explanaion for he meaning of hese properies, see Diewer (1973) (1974; 134), Woodland (1982) or Kohli (1978) (1991). The assumpion ha S is a cone means ha he echnology is subjec o 3

6 Given a vecor of oupu prices P and a vecor of available inpus x, we define he period marke secor GDP funcion, g (P,x), as follows: 14 (1) g (P,x) max y {P y : (y,x) belongs o S } ; = 0,1,2,.... Thus, marke secor GDP depends on (which represens he period echnology se S ), on he vecor of oupu prices P ha he marke secor faces and on x, he vecor of inpus ha is available o he marke secor. If P is he period oupu price vecor and x is he vecor of inpus used by he marke secor during period and if he GDP funcion is differeniable wih respec o he componens of P a he poin x, hen he period vecor of marke secor oupus y will be equal o he vecor of firs order parial derivaives of g (P,x ) wih respec o he componens of P; i.e., we will have he following equaions for each period : 15 (2) y = P g (P,x ) ; = 0,1,2,.... If he GDP funcion is differeniable wih respec o he componens of x a he poin P, hen he period vecor of inpu prices W will be equal o he vecor of firs order parial derivaives of g (P,x ) wih respec o he componens of x; i.e., we will have he following equaions for each period : 16 (3) W = x g (P,x ) ; = 0,1,2,.... The consan reurn o scale assumpion on he echnology ses S implies ha he value of oupus will equal he value of inpus in period ; i.e., we have he following relaionships: (4) g (P,x ) = P y = W x ; = 0,1,2,.... The above maerial will be useful in wha follows. However, our focus is no on GDP; insead, our focus is on he income generaed by he marke secor or more precisely, on he real income generaed by he marke secor. However, since marke secor GDP (he value of marke secor producion) is disribued o he facors of producion used by he marke secor, nominal marke secor GDP will be equal o nominal income of he marke secor. As an approximae welfare measure ha can be associaed wih consan reurns o scale. This is an imporan assumpion since i implies ha he value of oupus should equal he value of inpus in equilibrium. In our empirical work, we use an ex pos rae of reurn in our user coss of capial, which forces he value of inpus o equal he value of oupus for each period. The funcion g is known as he GDP funcion or he naional produc funcion in he inernaional rade lieraure (see Kohli (1978)(1991), Woodland (1982) and Feensra (2004; 76). I was inroduced ino he economics lieraure by Samuelson (1953). Alernaive erms for his funcion include: (i) he gross profi funcion; see Gorman (1968); (ii) he resriced profi funcion; see Lau (1976) and McFadden (1978); and (iii) he variable profi funcion; see Diewer (1973) (1974) (1993). 14 The funcion g (P,x) will be linearly homogeneous and convex in he componens of P and linearly M homogeneous and concave in he componens of x; see Diewer (1973) (1974; 136). Noaion: P y m=1 P m y m. 15 These relaionships are due o Hoelling (1932; 594). Noe ha P g (P,x ) [ g (P,x )/ P 1,..., g (P,x )/ P M ]. 16 These relaionships are due o Samuelson (1953) and Diewer (1974; 140). Noe ha x g (P,x ) [ g (P,x )/ x 1,..., g (P,x )/ x N ]. 4

7 marke secor, 17 we will choose o measure he real income generaed by he marke secor in period in erms of he number of consumpion bundles ha he nominal income could purchase in period. Therefore, we define ρ as follows: (5) ρ W x /P C ; = 0,1,2,... = w x = p y = g (p,x ) where P C > 0 is he period consumpion expendiures deflaor and he marke secor period real oupu price p and real inpu price w vecors are defined as he corresponding nominal price vecors deflaed by he consumpion expendiures price index; i.e., we have he following definiions: 18 (6) p P /P C ; w W /P C ; = 0,1,2,.... The firs and las equaliy in (5) imply ha period real income, ρ, is equal o he period GDP funcion, evaluaed a he period real oupu price vecor p and he period inpu vecor x, g (p,x ). Thus he growh in real income over ime can be explained by hree main facor; produciviy growh, growh in real oupu prices and he growh of primary inpus. We will shorly give formal definiions for hese hree growh facors. Using he linear homogeneiy properies of he GDP funcions g (P,x) in P and x separaely, we can show ha he following counerpars o he relaions (2) and (3) hold using he deflaed prices p and w: 19 (7) y = p g (p,x ) ; = 0,1,2,... (8) w = x g (p,x ) ; = 0,1,2,.... Now we are ready o define a family of period produciviy growh facors τ(p,x,): (9) τ(p,x,) g (p,x)/g 1 (p,x) ; = 1,2,.... Thus, τ(p,x,) measures he proporional change in he real income produced by he marke secor ha is induced by he echnical change going from period 1 o, facing he reference real oupu prices p and using reference inpu quaniies x. We can choose he reference vecors for he measure of echnical progress defined by (9) from he curren year or he previous year: a Laspeyres ype measure τ L ha 17 Since some of he primary inpus used by he marke secor can be owned by foreigners, our measure of domesic welfare generaed by he marke secor is only an approximae one. Moreover, our suggesed welfare measure is no sensiive o he disribuion of he income ha is generaed by he marke secor. 18 Our approach is similar o he approach advocaed by Kohli (2004b; 92), excep he essenially deflaes nominal GDP by he domesic expendiures deflaor raher han jus he domesic (household) expendiures deflaor; i.e., he deflaes by he deflaor for C+G+I, whereas we sugges deflaing by he deflaor for C. Anoher difference in his approach compared o he presen approach is ha we resric our analysis o he marke secor GDP, whereas Kohli deflaes all of GDP (probably due o daa limiaions). Our reamen of he balance of rade surplus or defici is also differen. 19 If producers in he marke secor of he economy are solving he profi maximizaion problem ha is associaed wih g (P,x), which uses he original oupu prices P, hen hey equivalenly solve he profi maximizaion problem ha is associaed wih g (p,x), which uses he normalized oupu prices p P/P C ; i.e., Therefore, heir behaviour can be described by using eiher g (P,x) or g (p,x). 5

8 chooses he period 1 reference vecors p 1 and x 1 and a Paasche ype measure τ P ha chooses he period reference vecors p and x : (10) τ L τ(p 1,x 1,) = g (p 1,x 1 )/g 1 (p 1,x 1 ) ; = 1,2,... ; (11) τ P τ(p,x,) = g (p,x )/g 1 (p,x ) ; = 1,2,.... Since boh measures of echnical progress are equally valid, i is naural o average hem o obain an overall measure of produciviy growh. If we wan o rea he wo measures in a symmeric manner and we wan he measure o saisfy he ime reversal propery from index number heory 20 (so ha he esimae going backwards is equal o he reciprocal of he esimae going forwards), hen he geomeric mean will be he bes simple average o ake. 21 Thus we define he geomeric mean of (10) and (11) as follows: 22 (12) τ [τ L τ P ] 1/2 ; = 1,2,.... A his poin, i is no clear how we will obain empirical esimaes for he heoreical produciviy growh indexes defined by (10)-(12). One obvious way would be o assume a funcional form for he GDP funcion g (p,x), collec daa on oupu and inpu prices and quaniies for he marke secor for a number of years (and for he consumpion expendiures deflaor), add error erms o equaions (7) and (8) and use economeric echniques o esimae he unknown parameers in he assumed funcional form. However, economeric echniques are generally no compleely sraighforward: differen economericians will make differen sochasic specificaions and will choose differen funcional forms. 23 Moreover, as he number of oupus and inpus grows, i will be impossible o esimae a flexible funcional form. Thus we will sugges mehods for implemening measures like (12) in his paper ha are based on exac index number echniques. We urn now o he problem of defining heoreical indexes for he effecs on real income due o changes in real oupu prices. Define a family of period real oupu price growh facors α(p 1,p,x,s): 24 (13) α(p 1,p,x,s) g s (p,x)/g s (p 1,x) ; s = 1,2,.... Thus α(p 1,p,x,s) measures he proporional change in he real income produced by he marke secor ha is induced by he change in real oupu prices going from period 1 o, using he reference echnology ha is available during period s and using he reference inpu quaniies x. Thus, each choice of he reference period for echnology s and he reference inpu vecor x will generae a possibly 20 See Fisher (1922; 64). 21 See he discussion in Diewer (1997) on choosing he bes symmeric average of Laspeyres and Paasche indexes ha will lead o he saisfacion of he ime reversal es by he resuling average index. 22 The heoreical produciviy change indexes defined by (10)-(12) were firs defined by Diewer and Morrison (1986; ). See Diewer (1993) for properies of symmeric means. 23 The esimaion of GDP funcions such as (19) can be conroversial, however, since i raises issues such as esimaion echnique and sochasic specificaion.... We herefore prefer o op for a more sraighforward index number approach. Ulrich Kohli (2004a; 344). 24 This measure of real oupu price change was essenially defined by Fisher and Shell (1972; 56-58), Samuelson and Swamy (1974; ), Archibald (1977; 60-61), Diewer (1980; ) (1983; 1055) and Balk (1998; 83-89). Readers who are familiar wih he heory of he rue cos of living index will noe ha he real oupu price index defined by (13) is analogous o he Konüs (1924) rue cos of living index which is a raio of cos funcions, say C(u,p )/C(u,p 1 ) where u is a reference uiliy level: g s replaces C and he reference uiliy level u is replaced by he vecor of reference variables x. 6

9 differen measure of he effec on real income of a change in real oupu prices going from period 1 o period. Again, we can choose he reference vecors for he measure of oupu price change defined by (13) from he curren year or he previous year: a Laspeyres ype measure α L ha chooses he period 1 reference echnology and reference inpu vecor x 1 and a Paasche ype measure α P ha chooses he period reference echnology and reference inpu vecor x : (14) α L α(p 1,p,x 1, 1) = g 1 (p,x 1 )/g 1 (p 1,x 1 ) ; = 1,2,... ; (15) α P α(p 1,p,x,) = g (p,x )/g (p 1,x ) ; = 1,2,.... Since boh measures of real oupu price change are equally valid, i is naural o average hem o obain an overall measure of he effecs on real income of he change in real oupu prices: 25 The simple geomeric mean is jusified from he same reason as ha for he echnical progress shif funcion. (16) α [α L α P ] 1/2 ; = 1,2,.... Finally, we look a he problem of defining heoreical indexes for he effecs on real income due o changes in real oupu prices. Define a family of period real inpu quaniy growh facors β(x 1,x,p,s): 26 (17) β(x 1,x,p,s) g s (p,x )/g s (p,x 1 ) ; s = 1,2,.... Thus β(x 1,x,p,s) measures he proporional change in he real income produced by he marke secor ha is induced by he change in inpu quaniies going from period 1 o, using he echnology ha is available during period s and facing he reference real oupu prices p. Thus, each choice of he reference period for echnology s and he reference real oupu price vecor p will generae a possibly differen measure of he effec on real income of a change in inpu quaniies going from period 1 o period. Again, We can choose he reference vecors for he measure of inpu quaniy change defined by (13) from he curren year or he previous year: a Laspeyres ype measure β L ha chooses he period 1 reference echnology and reference real oupu price vecor p 1 and a Paasche ype measure β P ha chooses he period reference echnology and reference real oupu price vecor p : (18) β L β(x 1,x,p 1, 1) = g 1 (p 1,x )/g 1 (p 1,x 1 ) ; = 1,2,... ; (19) β P β(x 1,x,p,) = g (p,x )/g (p,x 1 ) ; = 1,2,.... Since boh measures of real inpu growh are equally valid, i is naural o average hem o obain an overall measure of he effecs of inpu growh on real income: 27 The simple geomeric mean is jusified from he same reason as ha for he echnical progress shif funcion. (20) β [β L β P ] 1/2 ; = 1,2, The indexes defined by (13)-(16) were defined by Diewer and Morrison (1986; 664) in he nominal GDP funcion conex. 26 This ype of index was defined as a rue index of value added by Sao (1976; 438) and as a real inpu index by Diewer (1980; 456). 27 The heoreical indexes defined by (17)-(20) were defined in Diewer and Morrison (1986; 665) in he nominal GDP conex. 7

10 We provide all he heoreically moivaed measures such as τ L, α L, β L, τ P, α P, and β P. Now, we consider how o calculae hese measures from he observed daa. On he firs sigh, he effor is hopeless. Under he consan reurns o scale assumpion on he echnology se S, we have (21) p y = g (p,x ) I means ha we observe values for he denominaor of Laspeyres ype measure such as τ L, α L, β L and for he numeraor of Paasche ype measure such as τ P, α P, β P. However, since we only know he denominaor or he numeraor of indicaors, we canno calculae even any indicaors of τ L, α L, β L, τ P, α P, and β P. For example, he numeraor of Laspeyres ype measure echnical progress g (p 1,x 1 ) is he hypoheical revenue ha would resul from using he period echnology wih he period 1 inpu quaniies and real oupu prices. Forunaely, hese hypoheical revenues can be inferred from observed daa if he period revenue funcion follows he Translog funcional form. The following Translog funcional form is known as he flexible funcional form which approximaes he arbirary revenue funcion by he second order: 28 (22) lng (p,x) a 0 + m=1 M a m lnp m + (1/2) m=1 M k=1 M a mk lnp m lnp k + n=1 N b n lnx n + (1/2) n=1 N j=1 N b nj lnx n lnx j + m=1 M n=1 M c mn lnp m lnx n ; = 1,2,.... Noe ha he coefficiens for he quadraic erms are assumed o be consan over ime. The coefficiens mus saisfy he following resricions in order for g o saisfy he linear homogeneiy properies ha we have assumed in secion 2 above: 29 (23) m=1 M a m = 1 for = 0,1,2,...; (24) n=1 N b n = 1 for = 0,1,2,...; (25) a mk = a km for all k,m ; (26) b nj = b jn for all n,j ; (27) k=1 M a mk = 0 for m = 1,...,M ; (28) j=1 N b nj = 0 for n = 1,...,N ; (29) n=1 N c mn = 0 for m = 1,...,M ; (30) m=1 M c mn = 0 for n = 1,...,N. Theorem 1: Adapaion of Diewer and Morrison (1986) 30 : If g 1 and g are defined by (22)-(30) above and here is compeiive profi maximizing behaviour on he par of all marke secor producers for all period, we have 28 This funcional form was firs suggesed by Diewer (1974; 139) as a generalizaion of he ranslog funcional form inroduced by Chrisensen, Jorgenson and Lau (1971). Diewer (1974; 139) indicaed ha his funcional form was flexible. 29 There are addiional resricions on he parameers which are necessary o ensure ha g (p,x) is convex in p and concave in x. 30 Diewer and Morrison (1986) esablished heir proof using he nominal GDP funcion g (P,x). However, i is easy o rework heir proof using he deflaed GDP funcion g (p,x) using he fac ha g (p,x) = g (P/P C,x) = g (P,x)/P C using he linear homogeneiy propery of g (P,x) in P. This argumen is also rue for heorem 2 and 3. 8

11 (31) τ = p y /(α β p -1 y -1 ) ; = 1,2,... where (32) lnα = Σ m=1 (1/2)[(p m 1 y m 1 /p 1 y 1 ) + (p m y m /p y )] ln(p m /p m 1 ) ; (33) lnβ = Σ n=1 (1/2)[(w n 1 x n 1 /w 1 x 1 ) + (w n x n /w x )] ln(x n /x n 1 ); The above heorem shows how we can calculae he heoreically moivaed measures such as τ, α, and β from observed daa. Anoher conribuion of he heorem is o show ha he real income growh can be exacly decomposed ino he hree explanaory facors: produciviy growh, real oupu price change, and inpu quaniy change. For some purposes, i is convenien o decompose he aggregae period conribuion o real income growh due o changes in all real oupu prices α ino separae effecs for he change in each real oupu price. Similarly, i can someimes be useful o decompose he aggregae period conribuion o real income growh due o changes in all inpu quaniies β ino separae effecs for he change in each inpu quaniy. We indicae how his can be done, making he same assumpions on he echnology ha we have made so far. We firs model he effec of he change in he real oupu price of oupu m, say p m, going from period 1 o. Counerpars o he heoreical Laspeyres and Paasche ype price indexes defined by (14) and (15) above for all he oupus are he following Laspeyres ype measure α Lm ha chooses he period 1 reference echnology, holds consan oher real oupu prices a heir period 1 levels, and holds inpu quaniies consan a heir period 1 levels and a Paasche ype measure α Pm ha chooses he period reference echnology, holds consan oher real oupu prices a heir period levels, and holds inpu quaniies consan a heir period levels; (34) α Lm g 1 (p 1 1,...,p m 1 1,p m,p m+1 1,..., p M 1,x 1 )/g 1 (p 1,x 1 ) ; m = 1,...,M; = 1,2,... ; (35) α Pm g (p,x )/g (p 1,...,p m 1,p m 1,p m+1,..., p M,x ) ; m = 1,...,M; = 1,2,.... Since boh measures of real oupu price change are equally valid, i is naural o average hem o obain an overall measure of he effecs on real income of he change in he real price of oupu m: 31 (36) α m [α Lm α Pm ] 1/2 ; m = 1,...,M ; = 1,2,.... Under he assumpion ha he deflaed GDP funcions g (p,x) have he ranslog funcional forms as defined by (22)-(30), he argumens of Diewer and Morrison (1986; 666) provide he following exac decomposiion of he period aggregae real oupu price conribuion facor α ino a produc of separae price conribuion facors α m : Theorem 2: Adapaion of Diewer and Morrison (1986): If g 1 and g are defined by (22)-(33) above and here is compeiive profi maximizing behaviour on he par of all marke secor producers for all periods, hen we have (37) α = α 1 α 2... α M ; = 1,2,.... where 31 The indexes defined by (34)-(36) were defined by Diewer and Morrison (1986; 666) in he nominal GDP funcion conex. 9

12 (38) lnα m = (1/2)[(p m 1 y m 1 /p 1 y 1 ) + (p m y m /p y )] ln(p m /p m 1 ) ; m = 1,...,M. For example, we consider he case where here are four ne oupus: 1: Domesic sales; 2: Invesmen; 3: Expors and; 4: Impors. Since commodiies 1, 2, and 3 are oupus, y 1, y 2, and y 3 will be posiive bu since commodiy 4 is an inpu ino he marke secor, y 3 will be negaive. Hence an increase in he real price of expors will increase real income bu an increase in he real price of impors will decrease he real income generaed by he marke secor, as is eviden by looking a he conribuion erms defined by (38) for m = 3(where y m > 0) and for m = 4(where y m < 0). The above decomposiion (38) is useful for analyzing he impacs ha he changes in he real price of expors (i.e., a change in he price of expors relaive o he price of domesic consumpion) and in he real price of impors make on he real income generaed by he marke secor. We now model he effecs of he change in he quaniy of inpu n, say x n, going from period 1 o. Individual counerpars o he overall heoreical Laspeyres and Paasche ype quaniy indexes defined by (18) and (19) above for all he inpus are he following Laspeyres ype measures for inpu n β Ln ha chooses he period 1 reference echnology and holds consan real oupu prices a heir period 1 levels and holds oher inpu quaniies (oher han inpu n) consan a heir period 1 levels and he Paasche ype measures for inpu n β Pn ha chooses he period reference echnology and hold consan real oupu prices a heir period levels and hold oher inpu quaniies consan a heir period levels; (39) β Ln g 1 (p 1,x 1 1,...,x n 1 1,x n,x n+1 1,..., x N 1 )/g 1 (p 1,x 1 ) ; n = 1,...,N; = 1,2,... ; (40) β Pn g (p,x )/g (p,x 1,...,x n 1,x n 1,x n+1,..., x N ) ; n = 1,...,N; = 1,2,.... Since boh measures of inpu change are equally valid, as usual, we average hem o obain an overall measure of he effecs on real income of he change in he quaniy of inpu n: 32 (41) β n [β Ln β Pn ] 1/2 ; n = 1,...,N ; = 1,2,.... Under he assumpion ha he deflaed GDP funcions g (p,x) have he Translog funcional forms as defined by (22)-(30), he argumens of Diewer and Morrison (1986; 666) can be adaped o provide he following exac decomposiion of he period aggregae inpu growh conribuion facor β ino a produc of separae inpu conribuion facors β n : Theorem 3: Adapaion of Diewer and Morrison (1986): If g 1 and g are defined by (22)-(30) above and here is compeiive profi maximizing behaviour on he par of all marke secor producers for all periods, hen we have (42) β = β 1 β 2... β N ; = 1,2,.... where 32 The indexes defined by (33)-(41) were defined by Diewer and Morrison (1986; 667) in he nominal GDP funcion conex. 10

13 (43) lnβ n = (1/2)[(w n 1 x n 1 /w 1 x 1 ) + (w n x n /w x )] ln(x n /x n 1 ) ; n = 1,...,N. Subsiuing he resuls in Theorems 2 and 3 ino he decomposiion in Theorem 1 and rearranging he erms, we obain he following formula; (44) ρ /ρ -1 = τ m=1 M α m n=1 N β n ; = 1,2,.... Thus he growh in real income over ime can be explained by hree main facors: produciviy growh, he growh of real oupu prices, and he growh of inpu quaniies. Raher han look a explanaory facors for he growh in real income, i is someimes convenien o express he level of real income in period in erms of cumulaed period o period growh facors. Thus (45) below defines a period index of he echnology level or of Toal Facor Produciviy in period relaive o period 0, T ; (46) defines a period cumulaed index of he level of real oupu m price change since period 0, A m ; and (47) defines a period cumulaed index of he growh of inpu n since period 0: (45) T 0 1; T T -1 τ ; = 1,2,,; (46) A 0 m 1; A 1 m A m α m ; m = 1,2,,M and = 1,2,,; (47) B 0 n 1; B 1 n B 0 n β n ; n = 1,2,,N and = 1,2,,; Using he appropriae equaliies (44) for he chain links ha appear in (45)-(47), we can esablish he following exac relaionship for he level of real income in period, ρ, relaive o is counerpar in period 0, ρ 0, and he cumulaed growh facors defined by (45)-(47): (48) ρ /ρ 0 = T m=1 M A m n=1 N B n ; = 1,2, Daa Consrucion 3.1 The Definiion of he Marke Secor In his paper, we focus on he producion of he marke secor wihin he whole economy. This is he secor which plays he criical role for economic growh. Following he 1993 SNA, we classify he producers for he whole economy ino he following five muually exclusive insiuional secors; IU1; Non-financial corporaions; IU2; Financial corporaions; IU3; General governmen; IU4; Households 33 ; IU5; Non-profi insiuions serving households. In our definiion, he marke secor consiss of IU1, IU2, and IU5. Thus, we subrac he governmen s and he households oupus and inpus from aggregae oupu, capial services and labour inpu in order o perform he growh accouning exercise explained in he previous secion. There are wo ypes of ransacions beween he marke secor and general governmen secor; (1) he sales of goods and services of he general governmen secor o he marke secor and (2) he purchases of inermediae inpus by he general governmen secor from he marke secor. Noe ha he sales of goods and services by he general governmen secor are considered as inpus for he marke secor while he purchases of inermediae inpus by he general governmen secor are regarded as oupus by he marke 33 According o UN (1993), he unincorporaed enerprises are pars of he household secors. However, we added he unincorporaed enerprises o IU1 or IU2 based on he characerisics of he enerprise. Therefore, in our classificaion, households only includes household own-accoun producers. 11

14 secor. Thus, he difference beween (1) and (2) is he ne oupu by he marke secor which has been purchased by he general governmen secor. In he Japanese naional accouns, he impued ren of he owner-occupied dwellings is he major oupu by he household secor. 34 The households secor produces impued ren by making use of capial services from he sock of owner-occupied houses. Therefore, we need o subrac he impued ren from he consumpion oupu and he sock of owneroccupied houses from he oal sock of residenial 35 buildings. 3.2 The Daabase for Inpus and Oupus Our final daa se consiss of price and quaniy series for he variables lised below. They are he inpus and oupus for he marke secor. We also followed he convenions inroduced by Jorgenson and Griliches on he reamen of indirec axes; i.e., we adjused prices for ax wedges whenever possible so ha he adjused prices reflec he prices ha producers face. 36 We also included he services of invenories and land (uilized by he marke secor) as capial inpus. The ne oupus in our mos disaggregaed daa base are as follows: C; Domesic final consumpion expendiure of households (excluding he impued ren of owner-occupied dwellings); N; Final consumpion expendiure of privae non-profi insiuions serving households; G; Ne purchases of goods and services by he general governmen from he marke secor; X; Expors of goods and services (including direc purchases in he domesic marke by nonresiden households since his is a source of revenue o he marke secor); M; Impors of goods and services (excluding direc purchases abroad by residen households since his is no a source of revenue o he Japanese marke secor); I1 o I95; Gross invesmens for 95 asse caegories; IV1 o IV4; Change in invenories for 4 ypes of invenory asse. The primary inpus in our mos disaggregaed daa base are as follows: K1 o K95; Capial services from 95 classes of fixed assses; KIV1 o KIV4; Invenory services for 4 classes of invenory asse; LD1 o LD4; Marke secor land services and LB1 o LB3; Labour inpu for 3 ypes of labour. Dividing he prices of inpus and oupus by he price of domesic final consumpion expendiure of households P C, we consruced he corresponding real inpu and oupu prices. The ne oupus C, N, G, X, and M have been consruced as aggregaes of he ne oupu componens lised above and are essenially based on he daa of he Japanese naional accouns. Since he Japanese naional accouns experienced several revisions, here are differen daa series even for he same variable. We chose he reference daa series from he mos recen publicaion and exended hese series by using he growh 34 We ignore oher own-accoun producion by households. 35 The household secor purchases goods and services for he mainenance of owner-occupied houses. The expenses on hese goods and services are produced by he marke secor and hus are par of our marke secor oupu. 36 Thus our suggesed reamen of indirec commodiy axes in an accouning framework ha is suiable for produciviy analysis follows he example se by Jorgenson and Griliches who advocaed he following reamen of indirec axes: In our original esimaes, we used gross produc a marke prices; we now employ gross produc from he producers poin of view, which includes indirec axes levied on facor oulay, bu excludes indirec axes levied on oupu. Jorgenson and Griliches (1972; 85). 12

15 raes of oher daa series from earlier publicaions. 37 consrucion mehods is provided in appendix. A more deailed explanaion of our daa Table 1: Lis of Capial Services, Invenory Services, and Land Services Name K1 Trees (growh) K53 Oher ships K2 Livesock (growh) K54 Railway vehicles K3 Texile producs K55 Aircraf K4 Wooded producs K56 Bicycles K5 Wooden furniure and fixures K57 Transpor equipmen for indusrial use K6 Meallic furniure and fixures K58 Oher ranspor equipmen K7 Nuclear fuel rods K59 Camera K8 Meallic producs K60 Oher phoographic and opical insrumens K9 Boilers and urbines K61 Waches and clocks K0 Engines K62 Physics and chemisry insrumens K11 Conveyors K63 Analyical and measuring insrumens and esing machines K12 Refrigeraors and air condiioning apparaus K64 Medical insrumens K13 Pumps and compressors K65 Miscellaneous manufacuring producs K14 Sewing machines K66 Residenial consrucion (wooden) K15 Oher general indusrial machinery and equipmen K67 Residenial consrucion (non-wooden) K16 Mining, civil engineering and consrucion machinery K68 Non-residenial consrucion (wooden) K17 Chemical machinery K69 Non-residenial consrucion (non-wooden) K18 Indusrial robos K70 Road consrucion K19 Meal machine ools K71 Sree consrucion K20 Meal processing machinery K72 Bridge consrucion K21 Agriculural machinery K73 Toll road consrucion K22 Texile machinery K74 River improvemen K23 Food processing machinery K75 Erosion conrol K24 Sawmill, wood working, veneer and plywood machinery K76 Seashore improvemen K25 Pulp equipmen and paper machinery K77 Park consrucion K26 Prining, bookbinding and paper processing machinery K78 Sewer consrucion K27 Casing equipmen K79 Sewage disposal faciliies K28 Plasic processing machinery K80 Wase disposal faciliies K29 Oher special indusrial machinery, nec K81 Harbor consrucion K30 Oher general machines and pars K82 Fishing por consrucion K31 Office machines K83 Airpor consrucion K32 Vending, amusemen and oher service machinery K84 Agriculural consrucion K33 Elecric audio equipmen K85 Fores road consrucion K34 Radio and elevision ses K86 Foresry proecion K35 Video recording and playback equipmen K87 Railway consrucion K36 Household elecric appliance K88 Elecric power faciliies K37 Elecronic compuer and peripheral equipmen K89 Telecommunicaion faciliies K38 Wired communicaion equipmen K90 Oher civil engineering and consrucion K39 Radio communicaion equipmen K91 Plan engineering K40 Oher communicaion equipmen K92 Mineral exploraion K41 Applied elecronic equipmen K93 Cusom sofware K42 Elecric measuring insrumens K94 Pre-packaged sofware K43 Generaors K95 Own-accoun sofware K44 Elecric moors KIV1 Finished-goods invenory K45 Relay swiches and swichboards KIV2 Work-in-process invenory K46 Oher indusrial heavy elecrical equipmen KIV3 Work-in-process invenory for culivaed asses K47 Elecric lighing fixures and apparaus KIV4 Maerial invenory K48 Passenger moor vehicles LD1 Land for agriculural use K49 Trucks, buses and oher vehicles LD2 Land for indusrial use K50 Two-wheel moor vehicles LD3 Land for commercial use K51 Moor vehicle pars LD4 Land for residenial use K52 Seel ships 37 The daa from he Japanese naional accouns are aken from publicaions such as he Annual Repor on Naional Income Saisics 1975, Repor on Naional Accouns from 1955 o 1998, Annual Repor on Naional Accouns of 2000, and Annual Repor on Naional Accouns of See appendix for he furher explanaion. 13

16 Quaniies of capial services, invenory services, and land services are proporional o capial socks of fixed asses, invenories, and land a he beginning of he year. Capial socks have been consruced by applying he perpeual invenory mehod using he gross invesmen daa (I1 o I95) o he iniial socks of 1955, using asse specific depreciaion raes suggesed by Nomura (2004). Changes in Invenories IV1-IV4 are calculaed by using he difference beween he invenory sock in he curren period and he invenory sock in he previous period. The iniial socks for fixed asses, invenory socks and land and he price and quaniy series for invesmens have been aken from KEO daabase. This is a comprehensive produciviy daabase for he Japanese economy consruced by he Keio Economic Observaory (KEO) a Keio Universiy. 38 The deailed procedures used o consruc prices and quaniies for capial socks and services are explained in Nomura (2004). For residenial srucures and residenial land, we use oher daa sources in order o ne ou he socks held by he household secor, which corresponds o he socks of residenial land and srucures used by owner-occupier households. Our procedures will be explained in he following subsecion. Prices of capial services for he 95 ypes of reproducible capial services K1-K95, for he 4 ypes of invenory services KIV1-KIV4 and he four ypes of land services LD1-LD4 are calculaed by applying a user cos formula which we will explain in he nex secion. Esimaes of he quaniies of labour services LB1-LB4 are based on hours of work. There are hree differen ypes of workers; he self-employed, family workers, and employees. Afer we calculae he average labour income and he hours of works for each ype of worker, we aggregae hese hree ypes of labour ino a labour aggregae using a superlaive index number formula. We explain how we calculae he average labour income for hree ypes of workers in he daa appendix. Invesmens I1-I95 correspond o he capial services K1-K95. Changes in invenories IV1-IV4 correspond o he invenory socks KIV1-KIV4. The Table 1 liss he names of all capial socks, invenory socks and land socks. The following subsecion explains how we consruced he price, quaniy and value series for he capial inpus. The deails on how our labour inpus were consruced may be found in Appendix A. 3.3 User Coss and Real Raes of Reurn Prices of capial services, invenory services and land services are esimaed as user coss of capial. User coss are consruced for each of our capial sock componens. The general formula for a user cos for a capial sock componen in year, u, is as follows: (49) u = P K (1 δ)p K (1+i )/(1+r ) where P K is he beginning of year asse price for he capial sock componen, δ is he geomeric depreciaion rae ha applies o he asse, 39 i is he amoun of asse specific price change ha is expeced o occur over he course of year and r is he nominal rae of reurn (or opporuniy cos of capial) ha producers face a he beginning of year. Thus he user cos of a durable inpu is equal o is purchase cos or opporuniy cos a he beginning of he year, P K, less he discouned expeced value of he depreciaed asse a he end of he year. 40 The nominal rae of reurn r can be decomposed ino a real rae of reurn componen r * and an expeced general inflaion componen, i *, by using he following formula: 38 This daabase also includes labour inpus. Kuroda, Shimpo, Nomura, and Kobayashi (1997) explains he daa consrucion of labour inpus in deail. In his paper, we use only heir relaive wages for employees, he self-employed and family workers from KEO daabase. 39 We assume ha he depreciaion raes for he invenory socks and land componens are We have emporarily negleced ax facors for simpliciy. 14

17 (50) 1+r = (1+r * )(1+ i * ). The pracical problem for he economic saisician wih a user cos formulae of he ype defined by (49) and (50) is ha here is uncerainy abou how exacly o esimae he depreciaion rae δ, he relevan real rae of reurn r * and he wo anicipaed inflaion raes, ρ (general inflaion rae) and i (asse specific inflaion rae). If we esimae i * and i as acual ex pos inflaion raes, we will almos cerainly generae user coss u which are negaive for some years, which is no sensible in our conex since we wan our user coss o closely approximae marke renal raes for he asses and hese raes would no be negaive. Even if we esimae i * and i by smoohing he ex pos values for hese variables or using a forecasing model, wih Japanese daa, we will ineviably generae some negaive user coss for land componens, due o he very rapid land price inflaion ha occurred in Japan during he 1980 s. We decided o avoid hese negaive user cos problems by assuming ha producers expec he asse specific inflaion rae i o equal he general inflaion rae i * ; i.e., we make he following assumpion: (51) i = i *. If we subsiue (50) and (51) ino (49), we find ha our user cos formula simplifies as follows: 41 (52) u = P K (1 δ)p K /(1+r * ) = [r * + δ]p K /(1+r * ). We approximae he beginning of he year asse price P K by he corresponding year invesmen price PI. Thus our final user cos formula has he following generic form where τ is he business ax rae: 42 (53) u [r * + τ + δ]p I /(1+r * ). There remains he problem of finding a suiable real ineres rae series, r *, which will be discussed below. Once r * has been deermined, he user cos defined by (53) can be calculaed for each of our 95 ypes of capial services, 4 ypes of invenory services, and 4 ypes of land services. The corresponding beginning of year capial socks have already been described and abled. We normalize he resuling user coss so ha hey are 1 in 1955 and offseing normalizaions are made o our capial quaniy series. The resuling inpu prices are W K1 -W K95, W KIV1 -W KIV4, and W LD1 -W LD4 and he resuling quaniies are x K1 -x K95, x KIV1 -x KIV4, and x LD1 -x LD4. The price series and he quaniy series for 13 aggregaed inpus a nd oupus are lised in Appendix B. 43 Finally, we discuss how we obained a suiable real ineres rae series r *. The rae we chose is an economy wide ex pos real rae of reurn. If we use he user cos formula (53) in order o form prices for capial services, hen we can rearrange he value of oupus equals he value of primary inpus equaion ino a single (linear) equaion involving an unknown real rae of reurn. The soluions o hese equaions (one for each year) are he series of ex pos real ineres raes, r *, lised below in Figure 1. The average rae for he years is 3.25%. I is consisen wih he long run observed economy wide real raes of reurn for mos OECD counries which fall in he 3 o 5 per cen rage (Diewer (2004)). The performance of he Japanese economy over he years can be divided ino 5 periods: 41 For discussions abou alernaive assumpions for user cos formulae, see Schreyer (2001) (2007) (2008) and Diewer (1980) (2004) (2005b). 42 For business srucures and land, we need o add he appropriae specific propery ax raes o he general ax rae τ. 43 More deailed daa will be provided o he ineresed reader. 15