Reconciling Gross Output TFP Growth with Value Added TFP Growth

Transcription

1 Reconciling Gross Oupu TP Growh wih Value Added TP Growh Erwin Diewer Universiy of Briish Columbia and Universiy of New Souh Wales ABSTRACT This aricle obains relaively simple exac expressions ha relae value added oal facor produciviy growh (TP) in a value added framework o he corresponding measures of TP growh in a gross oupu framework when Laspeyres or Paasche indexes are used o aggregae oupus and inpus Basically, as he inpu base becomes smaller, he corresponding esimaes of TP growh become larger A fairly simple approximae relaionship beween isher indexes of gross oupu TP growh and he corresponding isher index of value added TP growh is also derived The mehodology developed in his aricle has a number of applicaions SCHREYER (2:26) DEVELOPED AN approximae formula o relae oal facor produciviy growh (or mulifacor produciviy growh) in a gross oupu model of producion o TP (or MP) growh in a value added seing In his aricle, we ake anoher look a his issue 2 and develop an exac relaionship beween he wo measures when Laspeyres (or Paasche) oupu and inpu indexes are used o compue aggregae growh raes of inpus and oupus We develop rules relaing he wo produciviy conceps ha are simpler han he exising rules ha have been developed in he lieraure Secions and 2 discuss he consrucion of he aggregae oupus and inpus using he Laspeyres and Paasche formulae respecively, while Secion 3 uses he isher (922) ideal index number formula o aggregae inpus and oupus Secion 4 concludes The Laspeyres Case We firs consider he siuaion where Laspeyres indexes are used o aggregae oupus and inpus or simpliciy, consider a siuaion where we wan o compue gross oupu and value added produciviy growh raes for a producion uni ha produces gross oupu q Y > a prices p Y >, uses inermediae inpu q M > a prices p M > and uses primary inpu q a prices for, 3 X > p X > The auhor is a Professor of Economics a he Universiy of Briish Columbia and he Universiy of New Souh Wales The auhor graefully acknowledges he financial suppor of an Ausralian Research Council Linkage Gran (projec number LP88495) and a SSHRC gran This aricle is a revised version of UBC Deparmen of Economics Discussion Paper 4-2 The auhor hanks Kevin ox and an anonymous referee for helpful commens erwindiewer@ubcca 2 See Balk (29) for a comprehensive discussion on his opic We obain approximaely he same resul as Balk obains bu our mehod of derivaion is much simpler 3 In he case where here are many inpus and oupus, he oupu and inpu quaniy raio aggregaes, q Y > / q Y >, q M > / q M > and q X > / q X > he oupu and inpu quaniy raio aggregaes can be inerpreed as Laspeyres indexes of he micro quaniies The corresponding aggregae price raios are of course Paasche price indexes 6 N UMBER 29, ALL 25

2 Define he period value of gross oupu as p Y q Y for,, while inermediae inpus and primary inpus are defined as v M p M q M, and v X p X q X Define (one plus) he growh raes of oupu, inermediae inpus and primary inpus as Γ Y, Γ M and Γ X as follows: 4 () Γ Y q Y qy ; Γ M q M qm and Γ X q X qx We now need o define value added TP growh We use he Laspeyres index number formula o consruc an aggregae of gross oupu less inermediae inpu We assume ha he quaniies of gross oupu and inermediae inpu are posiive bu we change he sign of he price of inermediae inpus from a posiive sign o a negaive sign in order o consruc a Laspeyres index of real value added Thus (one plus) real value added TP growh using a Laspeyres quaniy index o consruc real value added growh,, is defines as follows: (6) [ S Y ΓY + S M ΓM Γ X, We assume ha he value of inpus equals he value of oupus in period : where he period value added oupu cos shares 6 and are defined as follows: 7 S Y S M (2) v M (One plus) gross oupu TP growh using Laspeyres quaniy indexes, follows: 5 (3) Γ Y [ ΓM + s X ΓX, is defined as (7) (8) S Y [ v M [ ( v M vy ) [ S M v M [ v M S Y [ using (4); using (7); where he period inpu cos shares are defined as follows: s X (4) v M [ v M using (2); v M vy (5) s X v X [ v M using (4) and Now subsiue (7) and (8) ino (6) and we obain he following expression for (one plus) value added TP growh: (9) [ Γ Y ΓM ( ) Γ x We can also subsiue (5) ino (3) and obain he following expression for (one plus) gross oupu TP growh: () Γ Y [ ΓM + ( 4 The capial gamma is called a growh facor 5 Noe ha ΓM + s X ΓX is he Laspeyres quaniy index of all inpus 6 These shares sum o one bu of course S M is negaive and so he share will no be bounded from below by and above by 7 Noe ha we did no change he sign of v M > and > Thus, using (7), S M < We assume ha period nominal value added, v M, is greaer han I NTERNATIONAL PRODUCTIVITY MONITOR 6

3 Comparing (9) and (), i can be seen ha boh value added and gross oupu TP (one plus) growh raes can be expressed in erms of (one plus) he growh raes of oupu, inermediae inpu and primary inpu ( Γ Y, Γ M and Γ X ) and he share of inermediae inpu in oal inpu, Now define he rae of gross oupu TP growh as equal o less and he rae of real value added TP growh as equal o less We can obain he folowing alernaive expressions for and : () (2) using (9) When we consruc he raio of value added TP growh o gross oupu TP growh and we obain he following exac formula relaing he wo produciviy conceps: 8 (3) ( Γ Y [ ΓM + ( ) Γ Y ΓM [ ( [ ΓM + ( using () ([ Γ Y ΓM ( ) [ Γ Y ΓM ( ( ) Γ X ( ) ([ ΓM + ( ) Γ X ( s X ) [ ΓM + ( Γ X using (5) [ vx [ ΓM + ( Γ X using (2) and (5) Thus he larger is he share of inermediae inpus in oal period inpu,, he larger will be ( ) and hence, he greaer will be value added TP growh relaive o gross oupu TP growh 9 Similarly, he larger is (one plus) aggregae inpu growh ΓM + ( over he wo periods being compared relaive o (one plus) primary inpu growh Γ X, he larger will be value added TP growh relaive o gross oupu TP growh Typically, (one plus) aggregae inpu growh will no be all ha differen from (one plus) primary inpu growh (boh erms will be close o uniy) so he erm( ) s X vx will explain almos all of he difference in he wo TP growh raes Noe ha he las equaion in (3) shows ha is approximaely equal o vx, he value of gross oupu,, divided by he value of primary inpus, v X, which in urn is equal o value added, v M, using our assumpion (2) The raio of gross oupu o value added is frequenly called he Domar facor (Balk, 29:249) The above resuls can readily be generalized o many oupus and inpus due o he consisency in aggregaion properies of he Laspeyres formula 8 We need o assume ha 9 Noe ha is equal o s X, he share of primary inpus in oal inpus used in period The firs equaion in (2) is roughly equivalen o Domar s (96:725) equaion (46) while he hird equaion in (2) is roughly equivalen o Balk s (29:248) equaion (2) Thus if (Laspeyres) gross oupu TP growh is 5 per cen and he period primary inpu share of oal inpu s is X / 2, hen (Laspeyres) value added TP growh will be approximaely ( s if X ) ( 5% ) % s X / 3, hen ( 3) ( 5% ) 5% 62 N UMBER 29, ALL 25

4 The Paasche Case We now consider he siuaion where Paasche indexes are used o aggregae oupus and inpus Again, define (one plus) he growh raes of oupu, inermediae inpus and primary inpus as Γ Y, Γ M and Γ X by equaions () when we are dealing wih muliple oupus and inpus, hese raios are o be inerpreed as being equal o Paasche indexes of gross oupu, inermediae inpus and primary inpus We assume ha he value of inpus equals he value of oupus in period : (4) v M quaniies of gross oupu and inermediae inpu are posiive bu we change he sign of he price of inermediae inpus from a posiive sign o a negaive sign in order o form a Paasche index of real value added Thus (one plus) real value added TP growh using a Paasche quaniy index o consruc real value added growh,, is defined as follows: 2 (8) - - [ S Y ΓY + S M ΓM Γ X where he period value added oupu expendiure shares, 3 S Y and S M, are defined as follows: 4 (One plus) gross oupu TP growh using, is defined as fol- Paasche quaniy indexes, lows: (9) S Y [ [ ( v M vy ) [ v M using (6); (5) Γ Y [ ΓM + s X ΓX where he period inpu cos shares s X are defined as follows: and (2) S M v M [ v M S Y [ using (9) (6) (7) v M [ v M v M vy v X [ v M s X using (4); using (6) Now subsiue (9) and (2) ino (8) and we obain he following expression for (one plus) value added Paasche TP growh: We now need o define Paasche value added TP growh We use he Paasche index number formula o consruc an aggregae of gross oupu less inermediae inpu We assume ha he (2) [ S Y ΓY + S M ΓM Γ X Γ X [ S Y ΓY + S M ΓM Γ X Γ Y ΓM ( ) [ Noe ha [ ΓM + s X ΓX is he Paasche quaniy index of all inpus 2 Noe ha [ S Y ΓY + S M ΓM is he Paasche real value added index 3 These shares sum o one bu of course S M is negaive and so he shares will no be bounded from below by and above by We assume ha v M > 4 Noe ha we did no change he sign of v M > and > Thus, using (2), S M < I NTERNATIONAL PRODUCTIVITY MONITOR 63

5 We can also subsiue (7) ino (5) and obain he following expression for (one plus) gross oupu Paasche TP growh: (22) Comparing (2) and (22), i can be seen ha boh value added and gross oupu Paasche TP (one plus) growh raes can be expressed in erms of (one plus) he growh raes of (Paasche) oupu, inermediae inpu and primary inpu ( Γ Y, Γ M and Γ X ) and he period share of inermediae inpu in oal inpu, Now compue ( ) and ( ) using (2) and (22): (23) Γ Y [ ΓM + ( [ + ( Γ Y ΓM ( ) [ Γ Y ΓM ( ) Γ X Now ake he raio of (23) and (24) and we obain he following ideniy: 5 (25) ( ) [ [( ) Define he Paasche rae of gross oupu TP growh as equal o less and he Paasche rae of real value added TP growh as equal o less : (26) (27) Now muliply boh sides of (25) by and we obain he following equaion: (28) [ ΓM + ( [( [ [ ΓM + ( [ Γ Y ΓM [( using (25) ( [( ( ) [ S Y ΓY + S M ΓM Γ Y using (2) and (22); (24) ( ) Γ Y ( [ Γ Y ΓM [ ΓM + ( s X ) [ S Y ΓY + S M ΓM Γ Y using (7); ( vx )[ S Y ΓY + S M ΓM Γ Y using (7) and (4) ( [ ΓM + ( Thus he larger is he share of inermediae inpus in oal period inpu,, he larger will be ( ) and hence he larger will be value added Paasche TP growh relaive o 5 We need o assume ha ( ) which will imply ha is also no equal o since > 64 N UMBER 29, ALL 25

6 Paasche gross oupu TP growh Of course, ( ) is equal o s X which in urn is equal o he period Domar augmenaion facor, vx, which in urn is equal o he period value of gross oupu divided by he period value of primary inpu 6 Similarly, he larger is (one plus) real value added growh [ S Y ΓY + S M ΓM over he wo periods being compared relaive o (one plus) gross oupu growh Γ, 7 Y he larger will be Paasche value added TP growh relaive o Paasche gross oupu TP growh Noe he similariy of Paasche formula (28) o our earlier Laspeyres formula (3) Typically, (one plus) real value added growh will no be all ha differen from (one plus) gross oupu growh (boh erms will be close o uniy) so he Paasche Augmenaion acor ( ) s X vx will explain almos all of he difference in he wo Paasche TP growh raes The above resuls can readily be generalized o many oupus and inpus due o he consisency in aggregaion properies of he Paasche formula The isher Case (One plus) he isher (922) index of value added produciviy growh,, is defined as he geomeric mean of he corresponding Laspeyres and Paasche measures of (one plus) value added TP growh, and : (29) -- 2 [ Similarly, (one plus) he isher (922) index of gross oupu,, is defined as he geomeric mean of he corresponding Laspeyres and Paasche measures of (one plus) gross oupu TP, and : (3) inally, define he isher (922) indexes of value added and gross oupu produciviy growh, and, as follows: (3) (32) Equaions (9) and () give exac expressions for he Laspeyres indexes of and, while equaions (2) and (22) give exac expressions for he Paasche indexes of and Hence we could use hese expressions o calculae he isher variables (29)-(32) and we would end up wih an exac expression for he raio of isher index value added TP growh o isher gross oupu growh However, he resuling expression is difficul o inerpre and so we will resor o a differen sraegy ha makes use of equaions (3) and (28) bu also involves approximaing he geomeric means ha define and Π by corresponding arihmeic means 8 G The firs sep in our sraegy is o define he righ hand side equaions (3) and (28) as he consans γ and γ : 9 (33) γ [ -- 2 ( s X ) [ ΓM + ( Γ X 6 Assumpion (4) ensures ha vx > 7 Noe ha [ S Y ΓY + S M ΓM is he Paasche index of real value added growh and Γ Y is o be inerpreed as he Paasche index of gross oupu growh if here are many oupus and we are aggregaing oupus in wo sages using he Paasche formula 8 These approximaions will ypically be very close 9 These consans can be regarded as magnificaion facors; hey magnify he gross oupu TP growh raes ino he corresponding real value added TP growh raes I NTERNATIONAL PRODUCTIVITY MONITOR 65

7 (34) π VA Using definiions (), (2), (26) and (27), equaions (33) and (34) imply he following relaions: (35) ( ) γ( ) (36)( ) γ ( ) Using definiions (29) and (3), we have he following equaions: (37) where we have approximaed he geomeric mean by an arihmeic mean: where he las equaion follows using (35) and (36) Using definiions (3) and (32), we have he following equaions: (38) γ ( s X ) [ S Y ΓY + S M ΓM Γ Y where we have approximaed he geomeric mean by an arihmeic mean -- 2 [ ( 2) + ( 2) ( 2) [ + ( 2) [ ( 2)γ[ ( 2)γ + [ [ 2 ( 2) + ( 2) ( 2) [ + ( 2) [ If ( 2) [ + ( 2) [ is no equal o zero, we can ake he raio of o and using (37) and (38), we obain he following approximae relaionship of isher value added TP growh o isher gross oupu TP growh: (39) where w [ [ + [ (4) γ[ γ + [ [ + [ wγ + ( w)γ Thus is approximaely equal o a weighed average of he Laspeyres and Paasche magnificaion facors, γ and γ 2 If we are willing o make a furher approximaion ha he Laspeyres and Paasche indexes of gross oupu growh are approximaely equal so ha, hen we obain he following very simple approximae relaionship beween isher value added TP growh and isher gross oupu TP growh: ( 2)γ + ( 2)γ Conclusion We have obained relaively simple exac expressions for he relaionship beween he rae of gross oupu TP growh and he corresponding rae of real value added TP growh when he Laspeyres or Paasche index number formulae are used o aggregae inpus and oupus We also obained a simple approximae expression relaing value added TP growh o gross oupu TP growh 2 The weighs for γ and γ will sum o uniy bu hey need no be nonnegaive unless and have he same sign This will almos always be he case empirically 66 N UMBER 29, ALL 25

8 when he isher formula is used o aggregae inpus and oupus Generally speaking, gross oupu TP growh is magnified when we move o value added TP growh and he magnificaion facor is approximaely equal o he reciprocal of he share of primary inpus in oal inpu use The same mehodology can be used in oher siuaions or example, we may wan o compare value added produciviy growh o ne value added produciviy growh where he laer concep akes depreciaion ou of capial services and reas i as an inermediae inpu expense The resuling ne value added TP growh will be equal o a magnificaion facor imes he corresponding radiional value added TP growh and he magnificaion facor will be approximaely equal o he reciprocal of he share of labour and waiing services in radiional labour and capial services (which include depreciaion in he user coss of capial) 2 In some regulaory conexs, we may wan o compue TP growh wih labour added regarded as an inermediae inpu and only capial services in he primary inpu base and compare his TP growh wih more radiional measures Again a magnificaion facor will emerge 22 Generally speaking, he smaller we make he inpu base in he produciviy measure, he larger will be he rae of TP growh Calver (25) has provided an empirical illusraion of his relaionship for Ausralian indusries Oher examples of his magnificaion effec can be found in Schreyer (2) and Balk (29) References Balk, BM (29) On he Relaion beween Gross Oupu and Value Added Based Produciviy Measures: The Imporance of he Domar acor, Macroeconomic Dynamics, Vol 3 (Supplemen 2), pp Calver, M (25) On he Relaionship beween Gross Oupu Based TP Growh and Value Added Based TP Growh: An Illusraion Using Daa from Ausralian Indusries, Inernaional Produciviy Monior, No 29, all Diewer, WE (24) US TP Growh and he Conribuion of Changes in Expor and Impor Prices o Real Income Growh, Journal of Produciviy Analysis, Vol 4, pp 9-39 Diewer, WE and E Yu (22) New Esimaes of Real Income and Mulifacor Produciviy Growh for he Canadian Business Secor, 96-2, Inernaional Produciviy Monior, Number 24, all, pp Domar, ED (96) On he Measuremen of Technological Change, Economic Journal, Vol 7, pp isher, I (922) The Making of Index Numbers (Boson: Houghon-Mifflin) Lawrence, DA, WE Diewer and KJ ox (26) Who Benefis from Economic Reform: The Conribuion of Produciviy, Price Changes and irm Size o Profiabiliy, Journal of Produciviy Analysis, Vol 26, pp -3 Schreyer, P (2) OECD Produciviy Manual: A Guide o he Measuremen of Indusry-Level and Aggregae Produciviy Growh (Paris: OECD) 2 or applicaions of his ne value added approach and esimaes of he resuling magnificaion facors, see Diewer (24) and Diewer and Yu (22) 22 or an applicaion of his approach, see Lawrence, Diewer and ox (26) I NTERNATIONAL PRODUCTIVITY MONITOR 67