Decomposing Value Added Growth Over Sectors into Explanatory Factors

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1 Business School Decomposing Value Added Growh Over Secors ino Explanaory Facors W. Erwin Diewer (UBC and UNSW Ausralia) and Kevin J. Fox (UNSW Ausralia) EMG Workshop UNSW 2 December 2016

2 Summary Decompose nominal value added growh over muliple secors ino explanaory facors. For a single secor, explanaory facors are efficiency changes, changes in oupu prices, changes in primary inpus, changes in inpu prices, echnical progress, and reurns o scale.

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4 Summary Need secor s bes pracice echnology for he wo periods under consideraion. Could use economeric or nonparameric (DEA) echniques We use Free Disposal Hull approach no convexiy assumpions Our approach has he advanage ha i does no involve economeric esimaion, and involves only observable daa. Simple enough o be implemened by saisical agencies If efficien in boh periods, can use he index number echniques of Diewer-Morrison (1986)/Kohli (1990). Address he problem of aggregaing over secors.

5 Cos Consrained Value Added Funcion for a Secor A secor produces M ne oupus, y [y 1,...,y M ], using N primary inpus x [x 1,...,x N ] 0 N. If y m > 0, hen he secor produces he mh ne oupu during period while if y m < 0, hen he secor uses he mh ne oupu as an inermediae inpu. Sricly posiive vecor of ne oupu prices p [p 1,...,p M ] >> 0 M and sricly posiive vecor of inpu prices w [w 1,...,w N ] >> 0 N Period producion possibiliies se for he secor S

6 Cos Consrained Value Added Funcion for a Secor S saisfies he following regulariy condiions: (i) S is a closed se; (ii) for every x 0 N, (0 M,x) S ; (iii) if (y,x) S and y * y, hen (y *,x) S (free disposabiliy of ne oupus); (iv) if (y,x) S and x * x, hen (y,x * ) S (free disposabiliy of primary inpus); (v) if x 0 N and (y,x) S, hen y b(x) where he upper bounding vecor b can depend on x (bounded primary inpus implies bounded from above ne oupus).

7 Cos Consrained Value Added Funcion for a Secor Period cos consrained value added funcion: R (p,w,x) max y,z {p y : (y,z) S ; w z w x} R (p,w,x) is well defined even if here are increasing reurns o scale in producion; i.e., he consrain w z w x leads o a finie value for R (p,w,x). If (y *,z * ) solves his consrained maximizaion problem, hen secoral value added p y is maximized subjec o he consrains ha (y,z) is a feasible producion vecor and primary inpu expendiure w z is equal o or less han observed primary inpu expendiure w x.

8 Cos Consrained Value Added Funcion for a Secor Observed value added, p y, may no equal he opimal value added. Value added efficiency of he secor during period : e p y /R (p,w,x ) 1 The cos consrained valued added funcion has some ineresing properies. If S is a cone, so ha producion is subjec o consan reurns o scale, can show ha R (p,w,x) w x/c (w,p) where c (w,p) is he uni cos funcion for producing a uni of value added.

9 Decomposing Value Added Growh for a Secor ino Explanaory Facors Change in value added efficiency ε e /e 1 = [p y /R (p,w,x )]/[p 1 y 1 /R 1 (p 1,w 1,x 1 )] If ε > 1, hen value added efficiency has improved going from period 1 o whereas i has fallen if ε < 1.

10 Decomposing Value Added Growh for a Secor ino Explanaory Facors Follow mehod of Konüs (1939) and Allen (1949) o define various families of indexes ha vary only one of he four ses of variables,, p, w and x, beween he wo periods under consideraion and hold consan he oher ses of variables. Family of oupu price indexes: α(p 1,p,w,x,s) R s (p,w,x)/r s (p 1,w,x). Two alernaives: α L α(p 1,p,w 1,x 1, 1) R 1 (p,w 1,x 1 )/R 1 (p 1,w 1,x 1 ) ; α P α(p 1,p,w,x,) R (p,w,x )/R (p 1,w,x ). Preferred overall measure of oupu price growh: α [α L α P ] 1/2

11 Decomposing Value Added Growh for a Secor ino Explanaory Facors Family of inpu quaniy indexes: β(x 1,x,w) w x /w x 1. β L w 1 x /w 1 x 1 ; β P w x /w x 1. Preferred overall measure of inpu quaniy growh: β [β L β P ] 1/2.

12 Decomposing Value Added Growh for a Secor ino Explanaory Facors Family of inpu mix indexes: γ(w 1,w,p,x,s) R s (p,w,x)/r s (p,w 1,x) More accurae o say ha γ(w 1,w,p,x,s) represens he hypoheical proporional change in cos consrained value added for he period s reference echnology due o he effecs of a change in he inpu price vecor from w 1 o w when facing he reference ne oupu prices p and he reference vecor of inpus x. γ LPP γ(w 1,w,p 1,x,) R (p 1,w,x )/R (p 1,w 1,x ); γ PLL γ(w 1,w,p,x 1, 1) R 1 (p,w,x 1 )/R 1 (p,w 1,x 1 ). γ [γ LPP γ PLL ] 1/2.

13 Decomposing Value Added Growh for a Secor ino Explanaory Facors Family of echnical progress indexes: τ( 1,,p,w,x) R (p,w,x)/r 1 (p,w,x) τ L τ( 1,,p 1,w 1,x ) R (p 1,w 1,x )/R 1 (p 1,w 1,x ). τ P τ( 1,,p,w,x 1 ) R (p,w,x 1 )/R 1 (p,w,x 1 ). Recall, if he reference echnologies in periods and 1 are cones, R (p,w,x) = w x/c (w,p) and R 1 (p,w,x) = w x/c 1 (w,p). Thus in he case where he reference echnology is subjec o CRS, hese mixed indexes of echnical progress are independen of x and hen rue Laspeyres and Paasche ype indexes.

14 Decomposing Value Added Growh for a Secor ino Explanaory Facors Family of (global) reurns o scale measures: δ(x 1,x,p,w,s) [R s (p,w,x )/R s (p,w,x 1 )]/[w x /w x 1 ]. δ L δ(x 1,x,p 1,w 1, 1) R 1 (p 1,w 1,x )/R 1 (p 1,w 1,x 1 )]/[w 1 x /w 1 x 1 ]; δ P δ(x 1,x,p,w,) [R (p,w,x )/R (p,w,x 1 )]/[w x /w x 1 ]. δ [δ L δ P ] 1/2

15 Decomposing Value Added Growh for a Secor ino Explanaory Facors Six explanaory growh facors: 1. Change in cos consrained value added efficiency: ε e /e 1 2. Change in oupu prices: α(p 1,p,w,x,s) 3. Change in inpu quaniies: β(x 1,x,w) 4. Change in inpu prices: γ(w 1,w,p,x,s) 5. Changes due o echnical progress: τ( 1,,p,w,x) 6. Reurns o scale measure: δ(x 1,x,p,w,s)

16 Decomposing Value Added Growh for a Secor ino Explanaory Facors Exac decomposiions of observed value added: p y /p 1 y 1 = ε α P β L γ LPP δ L τ L ; p y /p 1 y 1 = ε α L β P γ PLL δ P τ P. Take he geomeric mean of boh sides of he above equaions o ge our preferred decomposiion: p y /p 1 y 1 = ε α β γ δ τ. Can re-organise o ge: TFPG {[p y /p 1 y 1 ]/α }/β = ε γ δ τ

17 A Nonparameric Approximaion o he Cos Consrained Value Added Funcion Assume ha he producion uni s period producion possibiliies se S is he conical free disposal hull of he period acual producion vecor and pas producion vecors. LP problem: R (p,w,x) max λ {p (Σ s=1 y s λ s ) ; w (Σ s=1 x s λ s ) w x ; λ 1 0,..., λ 0} = max s {p y s w x/w x s : s = 1,2,...,} = w x max s {p y s /w x s : s = 1,2,...,} = w x/min s {w x s /p y s : s = 1,2,...,} = w x/c (w,p) where c (w,p) is he period nonparameric uni cos funcion

18 Naional Value Added Growh Decomposiions: The Secoral Weighed Average Approach Secoral value added decomposiion, for each secor k: v k /v k, 1 = α k β k γ k δ k ε k τ k Period share of naional value added for secor k: s k v k /v Can use period -1 or period shares o aggregae: v /v 1 = Σ k=1 K s k, 1 α k β k γ k δ k ε k τ k v /v 1 = [Σ k=1 K s k (α k β k γ k δ k ε k τ k ) 1 ] 1

19 Naional Value Added Growh Decomposiions: The Secoral Weighed Average Approach Nice! Bu hese exac decomposiions don lead o simple decomposiions ino naional explanaory facors. Define (logarihms of) weighed naional explanaory facors: ln α Σ K k=1 (1/2)(s k + s k, 1 )ln α k ; ln β Σ K k=1 (1/2)(s k + s k, 1 )ln β k ; ln γ Σ K k=1 (1/2)(s k + s k, 1 )ln γ k ; ln δ Σ K k=1 (1/2)(s k + s k, 1 )ln δ k ; ln ε Σ K k=1 (1/2)(s k + s k, 1 )ln ε k ; ln τ Σ K k=1 (1/2)(s k + s k, 1 )ln τ k.

20 Naional Value Added Growh Decomposiions: The Secoral Weighed Average Approach Use some approximaions (drawing on Schlömilch s inequaliy) o wrie: ln v /v 1 Σ k=1 K (1/2)(s k + s k, 1 )ln(v k /v k, 1 ) = Σ k=1 K (1/2)(s k + s k, 1 )ln(α k β k γ k δ k ε k τ k ) = ln α + ln β + ln γ + ln δ + ln ε + ln τ Naional Toal Facor Produciviy Growh: TFPG [v /v 1 ]/α β γ δ ε τ

21 Naional Value Added Growh Decomposiions: The Secoral Weighed Average Approach Assume ha he echnology of each secor can be represened by a ranslog value added funcion wih he resricions on echnical progress ha are described in Diewer and Morrison (1986) and Kohli (1990). These papers also assumed consan reurns o scale and compeiive profi maximizing behavior. Under hese assumpions: v k /v k, 1 = α k β k τ k where α k urns ou o be he period Törnqvis value added oupu price index for secor k and β k is he period Törnqvis primary inpu quaniy index for secor k. v /v 1 α β τ ; Can be implemened using index numbers; i.e. no necessary o have esimaes for secoral bes pracice funcions.

22 Naional Value Added Growh Decomposiions: The Secoral Weighed Average Approach This is a boom up approach; sar a he secor level and aggregae up o he naional level. No clear ha he correc definiion of naional TFPG [v /v 1 ]/α β is correc. Now look a a op down approach.

23 Naional Value Added Growh Decomposiions: The Naional Cos Consrained Value Added Funcion Approach Secor k share of naional bes pracice value added in period : σ k R k (p k,w k,x k )/R (p,w,x ) Naional efficiency Level: e v /R (p,w,x ) = Σ K k=1 σ k e k Naional efficiency change: ε e /e 1 = [Σ K k=1 σ k e k ]/[Σ K k=1 σ k, 1 e k, 1 ]

24 Naional Value Added Growh Decomposiions: The Naional Cos Consrained Value Added Funcion Approach Using a similar approach for oher componens, and similar definiions as for he explanaory componens as before, we ge he following exac decomposiion of naional value added growh: v /v 1 = α β γ δ ε τ Can derive approximaions o all six naional explanaory facors, so ha we ge: v /v 1 = α β γ δ ε τ α β γ δ ε τ Which is he same decomposiion ha we had for he boom up approach.

25 TFP Growh for he U.S. Corporae Nonfinancial Secor, Use he (BEA, BLS, Fed Reserve) Inegraed Macroeconomic Accouns o consruc a daa se for wo major secors of he U.S. economy in Diewer and Fox (2016) : Alernaive User Coss, Raes of Reurn and TFP Growh Raes for he US Nonfinancial Corporae and Noncorporae Business Secors: Secor 1: US Corporae Nonfinancial Secor Secor 2: US Noncorporae Nonfinancial Secor

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27 TFP Growh for he U.S. Corporae Nonfinancial Secor, There was a subsanial decline in value added efficiency over he years TFP has grown a a slower han average rae since The level of TFP also fell in he 1974, 1979, 1982, 1989 and 2001 recessions when efficiency growh dipped below one. On he whole, TFP growh in he U.S. Corporae Nonfinancial Secor has been saisfacory.

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29 TFP Growh for he U.S. Noncorporae Nonfinancial Secor, The loss of value added efficiency in Secor 2 was massive over he 20 years This loss of efficiency dragged down he level of Secor 2 TFP over hese years. TFP growh resumed in 1994 and was excellen unil 2006 when TFP growh again salled wih he excepion of wo good years of growh in 2011 and Illusraes he adverse influence of recessions when oupu falls bu inpus canno be adjused opimally due o he fixiy of many capial sock (and labour) componens of aggregae inpu. Under hese circumsances, producion akes place in he inerior of he producion possibiliies se and for Secor 2, he resuling wase of resources was subsanial.

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32 Summary Derived decomposiions of nominal value added growh (and TFP growh) for a single secor ino explanaory facors. We also used wo alernaive approaches o relaing he secoral decomposiions o a naional growh decomposiion: a weighed average secoral approach and a naional value added funcion approach. A main advanage of our new approach is ha our new nonparameric measure of echnical progress never indicaes echnical regress. During recessions, value added efficiency drops below uniy and depresses TFP growh.

33 Summary For our U.S. daa se, TFP growh is well explained as he produc of value added efficiency growh imes he rae of echnical progress. For he U.S. Noncorporae Nonfinancial Secor, we found ha he cos of recessions was paricularly high. Implemenaion of he decomposiions can provide key insighs ino he drivers of economic growh a a deailed secoral level. Hence, we believe ha hey will provide new insighs ino he sources of economic growh. Our decomposiions may also indicae daa mismeasuremen problems ha can hen be addressed by saisical agencies.

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