Appendix 1: Explaining Real Income Growth: The Translog Approach

Transcription

1 1 Appendix 1: Explaining Real Income Growh: The Translog Approach The Producion Theory Framework In his Appendix, we presen he producion heory framework used in he aricle New Esimaes of Real Income and Mulifacor Produciviy Growh for he Canadian Business Secor, published in he Fall 2012 issue of he Inernaional Produciviy Monior. 1 The main reference is Diewer and Morrison (1986), bu we also draw on he heory of he oupu price index, which was developed by Fisher and Shell (1972) and Archibald (1977). 2 This heory is he producer heory counerpar o he heory of he cos of living index for a single consumer (or household) ha was firs developed by he Russian economis, A. A. Konüs (1924). These economic approaches o price indexes rely on he assumpion of (compeiive) opimizing behavior on he par of economic agens (consumers or producers). Thus we consider only he marke secor of he economy in wha follows; i.e. ha par of he economy ha is moivaed by profi maximizing behavior. In our empirical work, we define he marke secor o be he Canadian business secor of he economy less he renal and owner occupied housing secors. 3 Iniially, we assume ha he marke secor of he economy produces quaniies of M (ne) 4 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 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. 5 1 This maerial is drawn from Diewer, Mizobuchi and Nomura (2005) and Diewer and Lawrence (2006). 2 The heory also draws on Diewer (1983: ), Kohli (1990, 2003, 2004, and 2006), Morrison and Diewer (1990), Fox and Kohli (1998) and Chaper 24 in he IMF, ILO, OECD, Eurosa, UNECE and he World Bank (2009). 3 The Canadian business secor excludes all of he general governmen secors such as schools, hospials, universiies, defense and public adminisraion where no independen measures of oupu can be obained. For owner occupied housing, oupu is equal o inpu and hence no produciviy improvemens can be generaed by his secor according o SNA convenions. Due o he difficulies involved in spliing up he residenial housing sock ino he renal and owner occupied porions, we omi he enire residenial housing sock and he consumpion of residenial housing services in our daa. However, we do include invesmen in residenial housing, since ha invesmen is par of he oupu of he marke producion secor. 4 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 and 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. 5 For a more explanaion for he meaning of hese properies, see Diewer (1974: 134). The assumpion ha S is a cone means ha he echnology is subjec o 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

2 2 Given a vecor of oupu prices P and a vecor of available primary inpus x, we define he period marke secor GDP funcion, g (P,x), as follows: 6 (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 primary 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 P,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 : 7 (2) y = P g (P,x ) ; = 0,1,2,.... Thus he period marke secor supply vecor y can be obained by differeniaing he period marke secor GDP funcion wih respec o he componens of he period oupu price vecor P. If he GDP funcion is differeniable wih respec o he componens of x a he poin P,x, 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 : 8 (3) W = x g (P,x ) ; = 0,1,2,.... Thus he period marke secor inpu prices W paid o primary inpus can be obained by differeniaing he period marke secor GDP funcion wih respec o he componens of he period inpu quaniy vecor x. The consan reurns 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 bu of course, 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 lieraure; see Kohli (1978 and 1991), Woodland (1982) and Feensra (2004:76). I was inroduced ino he economics lieraure by Samuelson (1953). 6 The funcion g (P,x) will be linearly homogeneous and convex in he componens of P and linearly homogeneous and concave in he componens of x; see Diewer (1974: 136). Noaion: P y m=1 M P m y m. 7 These relaionships are due o Hoelling (1932: 594). Noe ha P g (P,x ) [ g (P,x )/ P 1,..., g (P,x )/ P M ]. 8 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 ].

3 3 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 marke secor income; i.e. from (4), we have g (P,x ) = P y = W x. As an approximae welfare measure ha can be associaed wih marke secor producion, 9 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 ; i.e., define as follows: (5) 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: 10 (6) p P /P C ; w W /P C ; = 0,1,2,.... The firs and las equaliy in (5) imply ha period real income,, 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 facors: (Technical Progress or Toal 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: 11 (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 or echnical progress shif facors (p,x,): 12 9 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 producion 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. 10 Our approach is similar o he approach advocaed by Kohli (2004: 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. Our reamen of he balance of rade surplus or defici is also differen. 11 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 will also solve he profi maximizaion problem ha uses he normalized oupu prices p P/P C ; i.e., hey will also solve he problem defined by g (p,x).

4 4 (9) (p,x,) g (p,x)/g 1 (p,x) ; = 1,2,.... Thus (p,x,) measures he proporional change in he real income produced by he marke secor a he reference real oupu prices p and reference inpu quaniies used by he marke secor x where he numeraor in (9) uses he period echnology and he denominaor in (9) uses he period 1 echnology. Thus each choice of reference vecors p and x will generae a possibly differen measure of he shif in echnology going from period 1 o period. I is naural o choose special reference vecors for he measure of echnical progress defined by (9): a Laspeyres ype measure L ha chooses he period 1 reference vecors p 1 and x 1 and a Paasche ype measure P ha chooses he period reference vecors p and x : (10) L (p 1,x 1,) = g (p 1,x 1 )/g 1 (p 1,x 1 ) ; = 1,2,... ; (11) P (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 echnical change. 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 13 (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. 14 Thus we define he geomeric mean of (10) and (11) as follows: 15 (12) [ L 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. 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 Appendix ha are based on exac index number echniques. 12 This measure of echnical progress is due o Diewer and Morrison (1986; 662). A special case of i was defined earlier by Diewer (1983: 1063). 13 See Fisher (1922: 64). 14 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. 15 The heoreical produciviy change indexes defined by (10)-(12) were firs defined by Diewer and Morrison (1986: ) in he nominal GDP conex.

5 5 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 (p 1,p,x,s): 16 (13) (p 1,p,x,s) g s (p,x)/g s (p 1,x) ; s = 1,2,.... Thus (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 echnology ha is available during period s and using he reference inpu quaniies x. Each choice of he reference echnology s and he reference inpu vecor x will generae a possibly differen measure of he effec on real income of a change in real oupu prices going from period 1 o period. Again, i is naural o choose special reference vecors for he measures defined by (13): a Laspeyres ype measure L ha chooses he period 1 reference echnology and reference inpu vecor x 1 and a Paasche ype measure P ha chooses he period reference echnology and reference inpu vecor x : (14) L (p 1,p,x 1, 1) = g 1 (p,x 1 )/g 1 (p 1,x 1 ) ; = 1,2,... ; (15) P (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: 17 (16) [ L 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 inpu quaniies. Define a family of period real inpu quaniy growh facors (x 1,x,p,s): 18 (17) (x 1,x,p,s) g s (p,x )/g s (p,x 1 ) ; s = 1,2,.... Thus (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 used by he marke secor going from period 1 o, using he echnology ha is available during period s and using he reference real oupu prices p. Each choice of he reference echnology s and he reference real oupu price 16 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: and 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. 17 The indexes defined by (13)-(16) were defined by Diewer and Morrison (1986: 664) in he nominal GDP funcion conex. 18 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).

6 6 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. As usual, i is naural o choose special reference vecors for he measures defined by (17): a Laspeyres ype measure L ha chooses he period 1 reference echnology and reference real oupu price vecor p 1 and a Paasche ype measure P ha chooses he period reference echnology and reference real oupu price vecor p : (18) L (x 1,x,p 1, 1) = g 1 (p 1,x )/g 1 (p 1,x 1 ) ; = 1,2,... ; (19) P (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: 19 (20) [ L P ] 1/2 ; = 1,2,.... Recall ha marke secor real income for period was defined by (5) as equal o nominal period facor paymens W x deflaed by he household consumpion price deflaor P C. I is convenien o define as he period chain rae of growh facor for real income: (21) / 1 ; = 1,2,.... The Translog GDP Funcion Approach We now follow he example of Diewer and Morrison (1986: 663) and assume ha he log of he period (deflaed) GDP funcion, g (p,x), has he following ranslog funcional form: 20 (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 ; = 0,1,2,.... 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: 21 (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 ; 19 The heoreical indexes defined by (17)-(20) were defined in Diewer and Morrison (1986: 665) in he nominal GDP conex. 20 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. 21 There are addiional resricions on he parameers which are necessary o ensure ha g (p,x) is convex in p and concave in x. Noe ha when we divide he original prices by one of he prices, hen one of he scaled prices will be idenically equal o one and hence is logarihm will be idenically equal o zero.

7 7 (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. Diewer and Morrison (1986: 663) showed ha 22 if g 1 and g are defined by (22)-(30) above and here is compeiive profi maximizing behavior on he par of all marke secor producers for all periods, hen (31) = ; = 1,2,... where,, and are defined above by (21), (12), (16) and (20) respecively. In addiion, Diewer and Morrison (1986: ) showed ha, and could be calculaed using empirically observable price and quaniy daa for periods 1 and as follows: (32) ln = m=1 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 ) = ln P T (p 1,p,y 1,y ); (33) ln = n=1 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 ) = ln Q T (w 1,w,x 1,x ); (34) = / where P T (p 1,p,y 1,y ) is he Törnqvis (1936) and Törnqvis and Törnqvis (1937) oupu price index and Q T (w 1,w,x 1,x ) is he Törnqvis inpu quaniy index. Equaions (31) are in raes of growh. I is possible o obain counerpars o hese equaions in a levels form as follows. Thus we can express he level of real income in period in erms of an index of he echnology level or of Toal Facor Produciviy in period T, of he level of real oupu prices in period A, and of he level of primary inpu quaniies in period, B. 23 Thus we use he growh facors, and as follows o define he levels T, A and B : (35) T 0 1 ; T T 1 ; = 1,2,... ; (36) A 0 1 ; A A 1 ; = 1,2,... ; (37) B 0 1 ; B B 1 ; = 1,2,.... Using he above definiions and he exac equaions (31), we can esablish he following exac relaionship for he level of real income in period,, and he period levels for echnology, real oupu prices and inpu quaniies: (38) / 0 = T A B ; = 1,2, Diewer and Morrison 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. 23 This ype of levels presenaion of he daa is quie insrucive when presened in graphical form. I was suggesed by Kohli (1990) and used exensively by him; see Kohli (2003 and 2004) and Fox and Kohli (1998).

8 8 The Translog GDP Funcion Approach and Changes in he Terms of Trade For some purposes, i is convenien o decompose he aggregae period conribuion facor due o changes in all deflaed oupu prices ino separae effecs for each change in each oupu price. Similarly, i can someimes be useful o decompose he aggregae period conribuion facor due o changes in all marke secor primary inpu quaniies ino separae effecs for each change in each inpu quaniy. In his secion, we indicae how his can be done, making he same assumpions on he echnology ha were made in he previous secion. We firs model he effecs of a change in a single (deflaed) oupu price, 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 changes in all (deflaed) oupu prices are he following Laspeyres ype measure Lm ha chooses he period 1 reference echnology and holds consan oher oupu prices a heir period 1 levels and holds inpus consan a heir period 1 levels x 1 and a Paasche ype measure Pm ha chooses he period reference echnology and reference inpu vecor x and holds consan oher oupu prices a heir period levels: (39) 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,...; (40) 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: 24 (41) m [ Lm 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) in he previous secion, he argumens of Diewer and Morrison (1986: 666) can be adaped o give us he following resul: (42) ln 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 ; = 1,2,.... Noe ha ln m is equal o he mh erm in he summaion of he erms on he righ hand side of (32). This observaion means ha we have he following exac decomposiion of he period aggregae real oupu price conribuion facor ino a produc of separae price conribuion facors; i.e., we have under presen assumpions: (43) = M ; = 1,2,.... The above decomposiion is useful for analyzing how real changes in he price of expors (i.e. a change in he price of expors relaive o he price of domesic consumpion) and in he price of 24 The indexes defined by (39)-(41) were defined by Diewer and Morrison (1986: 666) in he nominal GDP funcion conex.

9 9 impors impac on he real income generaed by he marke secor. In he empirical illusraion which follows laer, we le M equal hree. The hree ne oupus are: Domesic sales (C+I+G); Expors (X) and Impors (M). Since commodiies 1 and 2 are oupus, y 1 and y 2 will be posiive bu since commodiy 3 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 (42) for m = 2 (where y m > 0) and for m = 3 (where y m < 0). As menioned above, i is also useful o have a decomposiion of he aggregae conribuion of inpu growh o he growh of real income ino separae conribuions for each imporan class of primary inpu ha is used by he marke secor. We now model he effecs of a change in a single inpu quaniy, say x n, going from period 1 o. Counerpars o he heoreical Laspeyres and Paasche ype quaniy indexes defined by (18) and (19) above for changes in inpu n are he following Laspeyres ype measure Ln ha chooses he period 1 reference echnology and holds consan oher inpu quaniies a heir period 1 levels and holds real oupu prices a heir period 1 levels p 1 and a Paasche ype measure Pn ha chooses he period reference echnology and reference real oupu price vecor p and holds consan oher inpu quaniies a heir period levels: (44) 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,... ; (45) Pn g (p,x )/g (p,x 1,...,x n 1,x n 1,x n+1,..., p 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: 25 (46) n [ Pn Pn ] 1/2 ; n = 1,...,N ; = 1,2,.... Under he assumpion ha he deflaed GDP funcions g (p,x) have he ranslog funcional forms as defined by (22)-(30) in he previous secion, he argumens of Diewer and Morrison (1986; 667) can be adaped o give us he following resul: (47) ln 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 ; = 1,2,.... Noe ha ln n is equal o he nh erm in he summaion of he erms on he righ hand side of (33). This observaion means ha we have he following exac decomposiion of he period aggregae inpu growh conribuion facor ino a produc of separae inpu quaniy conribuion facors; i.e., we have under presen assumpions: 25 The indexes defined by (52)-(54) were defined by Diewer and Morrison (1986: 667) in he nominal GDP funcion conex.

10 10 (48) = N ; = 1,2,.... References Archibald, R.B. (1977) On he Theory of Indusrial Price Measuremen: Oupu Price Indexes, Annals of Economic and Social Measuremen 6, Balk, B.M. (1998) Indusrial Price, Quaniy and Produciviy Indices, Boson: Kluwer Academic Publishers. Chrisensen, L.R., D.W. Jorgenson and L.J. Lau (1971) Conjugae Dualiy and he Transcendenal Logarihmic Producion Funcion, Economerica 39, Diewer, W.E., (1974) Applicaions of Dualiy Theory, pp in M.D. Inriligaor and D.A. Kendrick (ed.) Froniers of Quaniaive Economics, Vol. II, Amserdam: Norh- Holland. Diewer, W.E. (1980) Aggregaion Problems in he Measuremen of Capial, pp in The Measuremen of Capial, D. Usher (ed.) Chicago: The Universiy of Chicago Press. Diewer, W.E. (1983) The Theory of he Oupu Price Index and he Measuremen of Real Oupu Change, pp in Price Level Measuremen, W.E. Diewer and C. Monmarquee (eds.) Oawa: Saisics Canada. Diewer, W.E. (1997) Commenary, in Mahew D. Shapiro and David W. Wilcox (eds.) Alernaive Sraegies for Aggregaing Price in he CPI, The Federal Reserve Bank of S. Louis Review, 79:3, Diewer, W.E. and D. Lawrence (2006) Measuring he Conribuions of Produciviy and Terms of Trade o Ausralia s Economic Welfare, Repor by Meyrick and Associaes o he Produciviy Commission, Canberra, Ausralia. Diewer, W.E., H. Mizobuchi and K. Nomura (2005) On Measuring Japan s Produciviy, , Discussion Paper 05-22, Deparmen of Economics, Universiy of Briish Columbia, Vancouver, B.C., Canada, V6T 1Z1, December. Diewer, W.E. and C.J. Morrison (1986) Adjusing Oupu and Produciviy Indexes for Changes in he Terms of Trade, The Economic Journal 96, Eurosa, IMF, OECD, UN and he World Bank (1993) Sysem of Naional Accouns 1993, New York: The Unied Naions. Feensra, R.C. (2004) Advanced Inernaional Trade: Theory and Evidence, Princeon N.J.: Princeon Universiy Press.

11 11 Fisher, F.M. and K. Shell (1972) The Pure Theory of he Naional Oupu Deflaor, pp in The Economic Theory of Price Indexes, New York: Academic Press. Fisher, I. (1922) The Making of Index Numbers, Houghon-Mifflin, Boson. Fox, K.J. and U. Kohli (1998) GDP Growh, Terms of Trade Effecs and Toal Facor Produciviy, Journal of Inernaional Trade and Economic Developmen 7, Hoelling, H. (1932) Edgeworh s Taxaion Paradox and he Naure of Demand and Supply Funcions, Journal of Poliical Economy 40, IMF, ILO, OECD, Eurosa, UNECE and he World Bank (2009) Expor and Impor Price Index Manual, Washingon DC: Inernaional Moneary Fund. Kohli, U. (1978) A Gross Naional Produc Funcion and he Derived Demand for Impors and Supply of Expors, Canadian Journal of Economics 11, Kohli, U. (1990) Growh Accouning in he Open Economy: Parameric and Nonparameric Esimaes, Journal of Economic and Social Measuremen 16, Kohli, U. (2003) Growh Accouning in he Open Economy: Inernaional Comparisons, Inernaional Review of Economics and Finance 12, Kohli, U. (2004) Real GDP, Real Domesic Income and Terms of Trade Changes, Journal of Inernaional Economics 62, Kohli, U. (2006) Real GDP, Real GDI and Trading Gains: Canada, , Inernaional Produciviy Monior, Number 13, Fall, Konüs, A.A. (1924) The Problem of he True Index of he Cos of Living, ranslaed in Economerica 7, (1939), Morrison, C.J. and W.E. Diewer (1990) Produciviy Growh and Changes in he Terms of Trade in Japan and he Unied Saes, pp in Produciviy Growh in Japan and he Unied Saes, Chicago: Universiy of Chicago Press. Samuelson, P.A. (1953) Prices of Facors and Goods in General Equilibrium, Review of Economic Sudies 21, Samuelson, P.A. and S. Swamy (1974) Invarian Economic Index Numbers and Canonical Dualiy: Survey and Synhesis, American Economic Review 64, Sao, K. (1976) The Meaning and Measuremen of he Real Value Added Index, Review of Economics and Saisics 58,

12 12 Törnqvis, L. (1936) The Bank of Finland s Consumpion Price Index, Bank of Finland Monhly Bullein 10: 1-8. Törnqvis, L. and E. Törnqvis (1937) Vilke är förhällande mellan finska markens och svenska kronans köpkraf? Ekonomiska Samfundes Tidskrif 39, 1-39 reprined as pp in Colleced Scienific Papers of Leo Törnqvis, Helsinki: The Research Insiue of he Finnish Economy, Woodland, A.D. (1982) Inernaional Trade and Resource Allocaion, Amserdam: Norh- Holland.