A Decomposition of U.S. Business Sector TFP Growth into Technical Progress and Cost Efficiency Components

Transcription

1 A Decomposiion of U.S. Business Secor TFP Growh ino Technical Progress and Cos Efficiency Componens Erwin Diewer (Universiy of Briish Columbia and Universiy of New Souh Wales, Ausralia) Kevin Fox (Universiy of New Souh Wales, Ausralia) Paper prepared for he 34 h IARIW General Conference Dresden, Germany, Augus 21-27, 2016 Session 7C: Explaining Produciviy Trends II Time: Friday, Augus 26, 2016 [Morning]

2 A Decomposiion of U.S. Business Secor TFP Growh ino Technical Progress and Cos Efficiency Componens W. Erwin Diewer and Kevin J. Fox 1 June 3, 2016 Discussion Paper 16-04, Vancouver School of Economics, The Universiy of Briish Columbia, Vancouver, Canada, V6T 1L4. Absrac One of he problems wih index number mehods for compuing TFP growh is ha during recessions, hese mehods show declines in TFP and his seems o imply ha echnical progress is negaive during hese periods. This is raher implausible since i implies echnological regress; i.e., ha ha he producion fronier has conraced. The paper works ou a nonparameric mehod where one can decompose TFP growh ino wo componens: a echnical progress componen (i.e., a shif in he producion fronier over ime) and an inefficiency componen ha is due o he fixiy of capial and labour in he shor run. The new decomposiion is illusraed using he new Bureau of Economic Analysis (BEA) Inegraed Macroeconomic Accouns which faciliaed he consrucion of a se of produciviy accouns for wo key secors of he US privae business secor: he Corporae Nonfinancial Secor and he Noncorporae Nonfinancial Secor. The analysis sheds ligh on produciviy growh slowdowns over he period 1960 o Key Words Toal Facor Produciviy, user coss, measures of echnical progress, measures of echnical and allocaive inefficiency, nonparameric cos funcions, nonparameric producion heory, Fisher ideal indexes Journal of Economic Lieraure Classificaion Numbers C43, C61, C67, C82, D24, E22. 1 W. Erwin Diewer: School of Economics, Universiy of Briish Columbia, Vancouver B.C., Canada, V6T 1Z1 and he School of Economics, UNSW Ausralia, Sydney 2052 ( erwin.diewer@ubc.ca) and Kevin J. Fox: School of Economics, UNSW Ausralia, Sydney 2052 ( K.Fox@unsw.edu.au). The firs auhor graefully acknowledges he financial suppor of he SSHRC of Canada, and boh auhors graefully acknowledge he financial suppor of he Ausralian Research Council (DP ).

3 1. Inroducion 2 Toal Facor Produciviy (TFP) Growh is usually defined as aggregae oupu growh divided by aggregae inpu growh. 2 Oupu growh and inpu growh are usually measured using (bilaeral) index numbers. During recessions when oupus fall more rapidly han inpus, hese index number measures of TFP growh usually show ha produciviy has declined. Since index number measures of TFP growh are ofen inerpreed as measures of echnical progress, such declines in measured produciviy are somewha embarrassing since hey seem o imply echnological regress; i.e., he producion fronier has somehow managed o shrink. Aiyar, Dalgaard and Moav (2008; 127) noe ha here are insances of los knowledge and echnical regress in pre-indusrial imes as, before he prining press and widespread lieracy, echnological knowledge was ypically embodied in humans. Negaive populaion and land produciviy shocks could induce he neglec of echniques rendered emporarily unprofiable. 3 The ransmission of hese echniques o he following generaions would hen be los. However, hey also noe (p. 126) he following in relaion o he relevance of such ancien hisory o our modern economies. The phenomenon of echnological regress canno be reconciled wih sandard macroeconomic models of innovaion (Romer 1990; Grossman and Helpman 1991; Aghion and Howi 1992; Korum 1997; Weizman 1998; Olsson 2000) or adopion (Nelson and Phelps 1966; Eaon and Korum 1999). Knowledge in hese heories is conceived as a sock which may sagnae bu never shrinks. Nor is his surprising; he heories are buil o explain he facs of he conemporary world, which is mosly characerized by purposive R&D in rich counries, and he adopion of foreign echnology by low-income counries. Hence, in modern imes, any inerpreaion of declines in produciviy as represening echnological regress is problemaic for economic modelling and policy formulaion. Of course, serious sudens of produciviy measuremen have long realized ha facors oher han echnical change can explain changes in TFP; see e.g. Färe, Grosskopf, Norris and Zhang (1994), Syverson (2011) and references herein. Two of hese explanaory facors are: Changes in echnical and allocaive inefficiency and Nonconsan reurns o scale in producion. We will no be able o deal wih he second explanaory facor in his paper (we will assume consan reurns o scale echnologies) bu we will be able o decompose TFP growh ino he produc of an efficiency facor (a movemen owards he producion fronier) imes a echnical progress facor (an expansion of he producion possibiliies se for he secor of he economy under consideraion). The novely in our decomposiion is ha (i) i is a nonparameric mehod; (ii) i can be readily compued and (iii) our measure of TFP growh is quie closely relaed o he sandard Fisher index of TFP growh. However, our mehod does have a drawback (in addiion o assuming consan reurns o scale): our mehodology requires ha 2 See Jorgenson and Griliches (1967) or Balk (1998). 3 Examples of echnical regress in hisory include he loss of hydraulic mining echnics following he decline of he Roman Empire (Wilson 2002) and he loss of boa building echnologies following socieal collapse in Easer Island (Diamond 2005).

4 here be only one oupu. In many empirical conexs of ineres his may no be a major issue, such as he sandard macroeconomic conex of considering he TFP growh of a counry using naional accouns daa. Given he recen heighened ineres in U.S. produciviy growh due o he measured produciviy growh slowdown since 2004, 4 he new decomposiion is applied o he new U.S. Bureau of Economic Analysis (BEA) Inegraed Macroeconomic Accouns. These new accouns enabled he consrucion of a se of produciviy accouns for wo key secors of he US privae business secor: he Corporae Nonfinancial Secor and he Noncorporae Nonfinancial Secor. Our mehodology is explained in he following secion, and is fairly closely relaed o he nonparameric producion mehodology developed by Farrell (1957). Secions 3 and 4 apply he mehodology o our wo secors of he U.S. economy: he Corporae Financial Secor and he Noncorporae Nonfinancial Secor. Secion 4 concludes. Appendix A briefly describes and liss our daa se, which covers he years Appendix B looks a how our mehodology could be exended o he many oupu case The Decomposiion of TFP Growh ino Technical Progress and Efficiency Componens Our analyic framework broadly follows he cos decomposiion framework explained in Secion 2 of Diewer (2014) excep ha in he presen paper, we assume a consan reurns o scale bes pracice echnology wih one oupu. A significan innovaion is ha we are explici abou he deerminaion of he relevan bes pracice echnology, by assuming ha he bes pracice echnology producion possibiliies se is he convex, conical hull of pas observaions on he producion uni up o and including he curren period. 5 Our approach can also be regarded as a modificaion of Farrell s (1957) one oupu, consan reurns o scale nonparameric model of efficien producion where we adap his mehod of analysis o he ime series conex. We assume ha we have informaion on he period oupu produced by a producion uni, y > 0, on he period inpus used, x [x 1,...,x N ] > 0 N, and on he inpu prices faced during period by he producion uni, w [w 1,...,w N ] > 0 N for periods = 1,...,T. Define he period observed oal inpu cos as C w x n=1 N w n x n for = 1,...,T. 6 If we divide each componen of x by he period oupu level y, we obain he vecor of observed inpu-oupu coefficiens for period, z x /y for = 1,...,T. If we weigh he componens of z by he componens of w and sum he resuling erms, we obain he period observed uni cos c for he producion uni; i.e., we have: (1) c w z = w x /y = C /y ; = 1,...,T. 4 See Gordon (2012), Mokyr, Vickers and Ziebarh (2015), Byrne, Fernald and Reinsdorf (2016), and Syverson (2016). 5 The use of nonparameric mehods for esimaing echnical progress can be raced back o Diewer (1980b; 264) (1981), Diewer and Parkan (1983; ) and Tulkens (1993; ). 6 We assume C > 0 for each.

5 We assume ha he producion uni s period producion possibiliies se can be adequaely approximaed by he convex 7 conical free disposal hull of he period acual producion vecor and pas producion vecors ha are in our sample of ime series observaions for he uni. 8 Using his assumpion, we define he period bes pracice uni cos funcion if he producer faces he nonnegaive, nonzero vecor of inpu prices w > 0 N, c(w,), as follows: (2) c(w,) min {w ( s=1 x s s ) ; s=1 y s s 1 ; 1 0,..., 0},..., 1 = min {w x s /y s : s = 1,2,...,} = min {w z s : s = 1,2,...,} where he second line in (2) follows from he firs line since here is only one consrain in he linear programming problem defined by he firs line and so he soluion o he LP problem is a simple minimum of numbers. The bes pracice uni cos funcion, c(w,), can be used o define he period cos efficiency for he producion uni. I will no necessarily be he case ha he producion uni being sudied achieves he bes pracice level of coss in period ; i.e., he following inequaliies will be saisfied: (3) c = w z c(w,) ; = 1,...,T. Thus he observed period uni cos, c, will be equal o or greaer han he bes pracice minimum uni cos, c(w,). Define he cos efficiency of he producion uni during period, e, as follows: 9 (4) e c(w,)/w x 1 ; = 1,...,T where he inequaliies in (4) follow from he inequaliies in (3). We noe ha his definiion of cos efficiency is equivalen o Farrell s (1957; 255) measure of overall efficiency, which combined his measures of echnical and allocaive inefficiency. Given he above definiion of cos efficiency in period, we can define an index of he change in he producion uni s cos efficiency going from period 1 o as follows: (5) e /e 1 = [c(w,)/c ]/[c(w 1, 1)/c 1 ]. Thus if > 1, hen he cos efficiency of he producion uni has improved going from period 1 o whereas i has fallen if < The convexiy assumpion is no required in he one oupu case. If we assume ha he bes pracice echnology can be represened by he conical Free Disposal Hull (FDH) of pas observaions, hen he corresponding bes pracice uni cos funcion is sill defined by (2). For references o he FDH approach o nonparameric producion heory, see he pioneering conribuions by Tulkens (1986) (1993) and his coauhors and he subsequen papers by Diewer and Fox (2014) (2016a). Our approach can be viewed as applying cos daa o Tulkens (1993; ) Sequenial FDH measuremen of efficiency and local echnical progress approach. 8 This approximaion will become more adequae for he laer observaions in our sample. 9 This definiion follows Balk (1998; 28). Noe ha e 1 is equal o uniy.

6 We now use he bes pracice uni cos funcion o define a family of measures of echnical progress, (w,); i.e., measures of expansion in he producion possibiliies se going from period 1 o. For a reference vecor of inpu prices w >> 0 N, define (w,) as follows: 10 (6) (w,) c(w, 1)/c(w,). In he regulaory lieraure, i is quie common o specify echnical progress in erms of downward shifs in he cos funcion over ime. Thus in definiion (6), we pick a reference vecor w and use he bes pracice echnology of period 0 o calculae he minimum cos of producing one uni of oupu a he inpu prices w using he period 1 and bes pracice echnologies a hose imes. This gives rise o he uni coss, c(w, 1) and c(w,), respecively. If here is posiive echnical progress going from period 1 o, hen c(w,), will generally be less han c(w, 1) and hence (w,) = c(w, 1)/c(w,) will be greaer han one and his measure of echnical progress is he reciprocal of he degree of proporional cos reducion ha resuls from he expansion of he underlying bes pracice echnology ses due o he passage of ime. For each choice of a reference vecor of inpu prices w, we obain a (possibly differen) measure of exogenous cos reducion and hence of echnical progress. For he period measures, i is naural o choose w o be w 1 or w, leading o he following Laspeyres and Paasche ype measures of period echnical progress: (7) L (w 1,) = c(w 1, 1)/c(w 1,) ; = 2,3,...,T; (8) P (w,) = c(w, 1)/c(w,) ; = 2,3,...,T. Since boh of he above measures of echnical progress, L and P, are equally represenaive, a single esimae of echnical progress should be se equal o a symmeric average of hese wo esimaes. We choose he geomeric mean as our preferred symmeric average 11 and hus our preferred summary measure of echnical progress going from period 1 o is he following Fisher ype index of echnical progress, ; for = 2,...,T: 12 (9) [ L P ] 1/2. We now urn our aenion o measures of he effecs on bes pracice uni cos of inpu price change. We use he period bes pracice uni cos funcion c(w,) in order o define a family of inpu price indexes going from period 1 o period, (w 1,w,s), as follows: This ype of cos based measure of echnical progress can be raced back o Saler (1960). Balk (1998; 58) defined a family of indexes similar o ha defined by (6) by using he bes pracice oal cos funcion in place of he bes pracice uni cos funcion. The definiions (6)-(13) are specializaions (o he case of one oupu) of he cos funcion based definiions used by Diewer (2011; ) (2012; ) o decompose oal cos growh ino explanaory facors using a reference bes pracice cos funcion. Wha is missing in Diewer s decomposiions is he change in cos efficiency erm defined by (5). 11 This will ensure ha he resuling measure of echnical progress saisfies he ime reversal propery; i.e., if we reverse he role of ime and recalculae he measure of echnical progress, we obain he reciprocal of he original measure when we ake he geomeric average. 12 Fisher (1922) defined his ideal index as he geomeric mean of he corresponding Laspeyres and Paasche indexes and noed ha he resuling index saisfied he ime reversal es. 13 If he number of oupus is equal o one, hen he family of indexes defined by (10) reduces o a Konüs (1939) rue cos of living index family in he case of homoheic preferences where c(w,) is he consumer s uni uiliy expendiure funcion using period preferences and oupu is inerpreed as a uiliy level. Shephard (1953) developed his heory of inpu price indexes for he case of a consan reurns o scale producion funcion.

7 6 (10) (w 1,w,s) c(w,s)/c(w 1,s) ; = 2,3,...,T. Thus he inpu price index (w 1,w,s) defined by (10) is equal o he (hypoheical) uni cos c(w,s) of producing one uni of oupu when he producion uni faces he period observed vecor of inpu prices w, divided by he uni cos c(w,s) of producing one uni of oupu when he producion uni faces he period 1 observed vecor of inpu prices w 1, where in boh cases, he producion uni has access o he bes pracice echnology of period s. Hence for each choice of echnology, we obain a (possibly differen) inpu price index. I is naural o choose s o be equal o eiher 1 or. Thus for hese choices of s, we obain he following Laspeyres and Paasche ype inpu price indexes, L and P, ha measure inpu price change going from period 1 o : (11) L (w 1,w, 1) = c(w, 1)/c(w 1, 1) ; = 2,3,...,T; (12) P (w 1,w,) = c(w,)/c(w 1,) ; = 2,3,...,T. Since boh inpu price indexes, L and P, are equally represenaive, a single esimae of inpu price change should be se equal o a symmeric average of hese wo esimaes. We again choose he geomeric mean as our preferred symmeric average and hus our preferred overall measure of inpu price growh going from period 1 o is he following Fisher ype heoreical inpu price index, : (13) [ L P ] 1/2 ; = 2,3,...,T. The inpu price index can be used o deflae he (oal) cos raio going from period o 1, C /C 1, in order o obain he following implici inpu quaniy index going from period 1 o, : (14) [C /C 1 ]/ ; = 2,3,...,T. Following Jorgenson and Griliches (1967), Toal Facor Produciviy Growh going from period 1 o, TFP G, can be defined as (one plus) oupu growh, y /y 1, divided by (one plus) inpu growh, : (15) TFPG [y /y 1 ]/ ; = 2,3,...,T. We can use he above definiions o show ha TFP growh is equal o he produc of cos efficiency growh imes echnical progress; i.e., we can show ha TFPG = where is defined by (5) and is defined by (9). To show his, sar off wih definiion (15): (16) TFPG [y /y 1 ]/ ; = 2,3,...,T. = [y /y 1 ] /[C /C 1 ] using (14) = [C 1 /C ][c(w,)/c(w 1, 1)][c(w 1, 1)/c(w,)] rearranging erms = [c(w 1, 1)/c(w,)] using (5) = [c(w 1, 1)/c(w,)][ L P ] 1/2 using (13) = [c(w 1, 1)/c(w,)][c(w, 1)/c(w 1, 1)] 1/2 [c(w,)/c(w 1,)] 1/2

8 using (11) and (12) = [c(w, 1)/c(w,)] 1/2 [c(w 1, 1)/c(w 1,)] 1/2 rearranging erms = [ P ] 1/2 [ L ] 1/2 using (7) and (8) = using (9). 7 As will be seen in subsequen secions of he paper, he decomposiion of TFP growh defined by (16) seems o work well in pracice; i.e., he ime echnical progress measure never falls below 1 (so here is never any echnological regress using our mehodology) and he ime measure of cos efficiency change falls below 1 during recession years as anicipaed. 14 Finally, i is convenien o conver he above growh decomposiion ino a levels decomposiion. Thus we sar off by seing he period 1 inpu level, X 1, and he period 1 oupu level, y 1, 15 equal o period 1 oal cos, C 1 : (17) X 1 = y 1 C 1. The period 1 TFP level TFP 1, cos efficiency level E 1 and echnology level T 1 are all se equal o 1: (18) TFP 1 = E 1 = T 1 1. The period inpu level X, TFP level TFP, cos efficiency level E and echnology level T for greaer han one are defined recursively as follows: (19) X X 1 ; TFP y /X ; E E 1 ; T T 1 ; = 2,3...,T. Using he above definiions, i can be shown ha we have he following decomposiion of he produciviy level a ime ino he produc of he ime level of cos efficiency imes he ime level of echnology; i.e., we have: 16 (20) TFP = y /X = E T ; = 1,...,T. We will illusrae he decomposiions defined by (16) and (20) using U.S. daa in he following secions. 3. TFP Decomposiions for he U.S. Corporae Nonfinancial Secor The US Bureau of Economic Analysis (BEA), in conjuncion wih he Bureau of Labor Saisics (BLS) and he Board of Governors of he Federal Reserve, have developed a new se of producion accouns (he Inegraed Macroeconomic Accouns or IMA) for wo major 14 Our measure of echnical progress is no perfec: in a given period, cos inefficiency could be large enough o negae he effecs of acual echnical progress so ha our nonparameric indicaor of echnical progress will indicae ha here is no echnical progress when in fac here was echnical progress. 15 This implies ha we are choosing a paricular uni of measuremen for aggregae oupu. 16 Of course, equaion (20) can be rearranged o give a decomposiion of real oupu of he producion uni in period, y, ino he produc of he period inpu level X imes he period level of cos efficiency E imes he period level of echnology T. This is similar o Kohli s (1990) decomposiion of nominal GDP ino he produc of explanaory facors. See also Fox and Kohli (1998) for a similar decomposiion.

9 privae secors of he US economy: he Corporae Nonfinancial Secor and he Noncorporae Nonfinancial Secor. The Balance Shee Accouns in he IMA cover he years bu do no provide a decomposiion of oupu, inpu and asse values ino price and quaniy componens. Diewer and Fox (2016b) provided such a decomposiion and we will use heir daa in his sudy. The daa for he wo subsecors are described and lised in Appendix A. In his secion, we will use heir oupu and inpu daa for he U.S. Corporae Nonfinancial Secor for he 55 years The year oupu y is real value added 17 and he corresponding year value added deflaor is denoed as p. The en inpus used by his secor are labour and he services of nine ypes of asse, which are lised in Appendix A. The year inpu vecor is x [x 1,x 2,...,x 10 ] where x 1 is year labour inpu measured in billions of 1960 dollars and x 2,...,x 10 are capial service inpus measured in billions of 1960 capial sock dollars. The corresponding year inpu price vecor is w [w 1,w 2,...,w 10 ] for = 1960,...,2014. I is sraighforward o do he compuaions using he above daa o compue he year cos efficiency facors e defined by (4), he year change in cos efficiency facors defined by (5), he year measures of echnical progress defined by (9), he nonparameric inpu price index defined by (13), he nonparameric inpu quaniy (or volume) index defined by (14) and he nonparameric Toal Facor Produciviy Growh facor for year, TFPG [y /y 1 ]/ defined by (15). These growh facors are lised in Table 1 below along wih he oupu growh facors, y /y 1. For comparison purposes, we also compued chained Fisher (1922) inpu indexes where * denoes he year Fisher chain link. 18 These Fisher inpu indexes were hen used o form a convenional Fisher TFP growh index, TFPG *, which is defined as follows: (21) TFPG * [y /y 1 ]/ * ; = 1961,...,2014. The growh facors y /y 1, * and he Fisher Toal Facor Produciviy Growh index, TFPG * are also lised in Table 1. I can be verified ha he TFP growh decomposiion defined by (16) in he previous secion holds; i.e., for each year, nonparameric TFP growh TFPG equals he produc of cos efficiency growh imes he year cos based echnical progress measure. I can also be seen when cos efficiency in year, e, is less han one, hen he year echnical progress measure always equals one so ha here is no echnical progress in years where he cos efficiency is less han one. Our nonparameric measure of echnical progress is always equal o or greaer han one; i.e., our measure never indicaes echnological regress. Anoher imporan empirical regulariy emerges from Table 1: for mos years, when he year cos efficiency growh facor is equal o one, he Fisher index measure of TFP growh, TFPG *, is equal o our nonparameric measure of TFP growh, TFPG. This follows from he fac, ha for mos years, he Fisher index of inpu growh, *, is equal o he corresponding nonparameric index of inpu growh,. Thus empirically, our nonparameric indexes of TFP growh are no all ha differen from ordinary Fisher indexes of TFP growh. Over he 55 years in our sample, he geomeric average nonparameric rae of TFP growh TFPG was 1.688% per year while he corresponding Fisher index rae of TFP 8 17 The published daa for his secor did no allow Diewer and Fox o decompose real value added ino gross oupu and inermediae inpu componens. 18 The year Fisher chain link inpu index is defined as * [(w 1 x w x )/(w 1 x 1 w x 1 )] 1/2.

10 growh TFPG * was 1.693% per year. 19 These average raes of TFP growh are very large indeed! Table 1: U.S. Corporae Nonfinancial Fisher TFP Growh TFPG *, Nonparameric TFP Growh TFPG, Cos Efficiency Growh Facors, Technical Progress Growh Facors, Nonparameric Inpu Growh Facors, Fisher Inpu Growh Indexes *, Oupu Growh Facors y /y 1 and Cos Efficiency Facors e 9 Year TFPG * TFPG * y /y 1 e These percenages are acually percenage poins. The corresponding geomeric average rae of growh for our nonparameric rae of inpu growh was 1.881% per year while he corresponding Fisher index rae of inpu growh * was 1.886% per year.

11 Table 2: Corporae Nonfinancial Fisher TFP Levels TFP *, Nonparameric TFP Levels TFP, Cumulaed Cos Efficiency Levels E, Technology Levels T, Real Value Added y, Nonparameric Aggregae Inpu X and Fisher Year Inpu Levels X * Year TFP * TFP E T y X X *

12 11, From Table 1, i can be seen ha here were cos efficiencies e below 1 for he following eleven years: 1969, 1970, 1974, 1979, 1980, 1982, 1989, 2001, 2007, 2008 and For he mos par, hese years were recession years for he U.S. economy. These efficiency declines were less han one percen in mos years excep here was a 3.28% efficiency decline for his secor in Thus in general, cos inefficiencies are a ransiory phenomenon which show up during recessions bu which vanish when he economy recovers from he recession. Neverheless, he cos inefficiency facors e provide a rough and ready numerical esimae of he coss of a recession. During he Grea Recession, his cos was significan and i illusraes he imporance of macroeconomic sabiliy in prevening recessions. Recall ha equaions (17)-(19) in he previous secion convered he growh decomposiion defined by (16) ino he levels decomposiion TFP y /X = E T defined by (20) where TFP is he level of nonparameric TFP in year, y is oupu in year, X is he level of nonparameric inpu in year, E is he cumulaed level of cos efficiency in year (which is equal o e, he cos efficiency facor defined by (4) above) and T is he cumulaed level of echnical progress for year. Define X * as he Fisher chained index level of inpu for year and define he Fisher index level of TFP in year as TFP * y /X *. The Fisher inpu and produciviy level series, X * and TFP *, are also lised in Table 2 for comparison purposes. I can be seen ha hese indexes are very close o heir nonparameric counerpars, X and TFP. I can also be seen ha he echnology level series, T, is also very close o TFP and TFP *. The nonparameric TFP levels TFP, cumulaed cos efficiency facors E (equal o he year cos efficiency facor e ) and he year level of echnology T are ploed in Figure 1. I can be seen ha excep for recession years when e is less han 1, he year nonparameric TFP level TFP is equal o he corresponding level of echnology index T. During he years when e is less han 1, TFP falls below he echnology index, T, since TFP /T is equal o e. I can be seen ha he rae of echnical change in he Corporae Nonfinancial secor slowed down beween he years , grew rapidly during , slowed down during , grew rapidly during and hen experienced a major slowdown and hen grew modesly during Figure 1: Corporae Nonfinancial Secor TFP Levels TFP, Cumulaed Cos Efficiency Facors E and Levels of Technology T 20 Noe ha he oupu decline in 2009 was 7.10% while he nonparameric inpu decline was 4.83%. Thus during recessions when demand drops unexpecedly, many inpus are quasi-fixed and canno drop a he same rae, leading o a TFP decline. Thus during recession years, he observed oupu and inpu combinaion generally ends up in he inerior of he producion possibiliies se.

13 12 4. TFP Decomposiions for he U.S. Noncorporae Nonfinancial Secor In his secion, we will use he Diewer and Fox (2016b) oupu and inpu daa for he U.S. Noncorporae Nonfinancial Secor for he 55 years We use he same noaion ha was used in he previous secion. Now he year oupu y is Noncorporae Nonfinancial real value added and he corresponding year value added deflaor is denoed as p. The fifeen inpus used by his secor are labour and he services of foureen ypes of asse, which are lised in Appendix A. The year inpu vecor is x [x 1,x 2,...,x 15 ] where x 1 is year labour inpu measured in billions of 1960 dollars and x 2,...,x 15 are capial service inpus measured in billions of 1960 capial sock dollars. The corresponding year inpu price vecor is w [w 1,w 2,...,w 15 ] for = 1960,...,2014. Again, i is sraighforward using he above daa o compue he year cos efficiency facors e defined by (4), he year change in cos efficiency facors defined by (5), he year measures of echnical progress defined by (9), he nonparameric inpu price index defined by (13), he nonparameric inpu quaniy (or volume) index defined by (14) and he nonparameric Toal Facor Produciviy Growh facor for year, TFPG [y /y 1 ]/ defined by (15). These growh facors are lised in Table 3 along wih he oupu growh facors, y /y 1. For comparison purposes, we also compued chained Fisher inpu indexes where * denoes he year Fisher chain link. These Fisher inpu indexes were hen used o form he Fisher TFP growh index, TFPG *, which was defined by (21) in he previous secion. The oupu and inpu growh facors y /y 1, * and he resuling Fisher Toal Facor Produciviy Growh index, TFPG * are also lised in Table 3. Table 3: U.S. Noncorporae Nonfinancial Fisher TFP Growh TFPG *, Nonparameric TFP Growh TFPG, Cos Efficiency Growh Facors, Technical Progress Growh Facors,

14 Nonparameric Inpu Growh Facors, Fisher Inpu Growh Indexes *, Oupu Growh Facors y /y 1 and Cos Efficiency Facors e Year TFPG * TFPG * y /y 1 e I can be verified ha he TFP growh decomposiion defined by (16) holds; i.e., for each year, nonparameric TFP growh TFPG equals he produc of cos efficiency growh imes he

15 year cos based echnical progress measure. As in he previous secion, i can also be seen when cos efficiency in year, e, is less han one, hen he year echnical progress measure always equals one so ha here is no echnical progress in years where he cos efficiency is less han one. Over he 55 years in our sample, he geomeric average nonparameric rae of TFP growh TFPG was 1.210% per year while he corresponding Fisher index rae of TFP growh TFPG * was 1.240% per year. 21 These average raes of TFP growh are subsanial bu considerably smaller han he corresponding average raes of TFP growh for he Corporae Nonfinancial Secor (which were 1.688% and 1.693% per year). From Table 3, i can be seen ha here were cos efficiencies e below 1 for he 20 consecuive years and in addiion, for he years 1995, 2007, 2009 and Some of he cos efficiency declines were subsanial; e.g., for 1983, e 1983 was equal o , indicaing ha opimal inpu cos was only 82.32% of observed inpu cos for ha year. Thus he behavior and echnology of he Noncorporae Nonfinancial Secor is very differen from he behavior and echnology of he Corporae Financial Secor. 23 Again recall ha equaions (17)-(19) in secion 2 above convered he growh decomposiion for TFP ino he levels decomposiion TFP y /X = E T defined by (20). Again define X * as he Fisher chained index level of inpu for year and again define he Fisher index level of TFP in year as TFP * y /X *. The Fisher inpu and produciviy level series for he Noncorporae Nonfinancial Secor, X * and TFP *, are also lised in Table 4 for comparison purposes. I can be seen ha hese indexes are very close o heir nonparameric counerpars, X and TFP. However, i is no longer he case ha he echnology level series, T, is always close o TFP and TFP * : when e is subsanially less han 1, TFP and TFP * are subsanially below T. However, we noe ha a he end of he sample period, TFP *, TFP and T end up a , and respecively. Noe ha y, X and X * are measured in billions of consan 1960 dollars. Table 4: Noncorporae Nonfinancial Fisher TFP Levels TFP *, Nonparameric TFP Levels TFP, Cumulaed Cos Efficiency Levels E, Technology Levels T, Real Value Added y, Nonparameric Aggregae Inpu X and Fisher Year Inpu Levels X * Year TFP * TFP E T y X X * The corresponding geomeric average raes of growh for our nonparameric rae of inpu growh was 1.137% per year while he corresponding geomeric average Fisher index rae of inpu growh * was 1.166% per year. 22 Evidenly i ook he Noncorporae Secor some 20 years o fully recover from he effecs of he firs oil shock recession in Anoher possibiliy is ha here is a considerable amoun of measuremen error in our daa for his secor. 23 The Noncorporae Secor uses land and srucure services more inensively while he Corporae Secor uses machinery and equipmen services more inensively. Thus during recessions, i is much more difficul for he Noncorporae Secor o reduce is inpu usage due o he fixiy of is land and srucure inpus.

16 The nonparameric TFP levels TFP, cumulaed cos efficiency facors E (equal o he year cos efficiency facor e ) and he year level of echnology T for he Noncorporae Secor are graphed in Figure 2. I can be seen ha excep for recession years when e is less han 1, he year nonparameric TFP level TFP is equal o he corresponding level of echnology index T. During he years when e is less han 1, TFP falls below he echnology index, T, since TFP /T is equal o e. Noe ha he level of echnology index T is consan for he years and here is anoher fla spo on he plo for T for he years Finally, we noe ha here were wo periods of rapid echnological progress for his secor: he years and he years

17 Figure 2: Noncorporae Nonfinancial Secor TFP Levels TFP, Cumulaed Cos Efficiency Facors E and Levels of Technology T Conclusion We have provided a fairly simple decomposiion of TFP growh ino echnical progress and echnical efficiency componens. These componens are relaively simple o compue and could be used by saisical agencies o provide more insigh ino he sources of Toal Facor Produciviy Growh. In paricular, he efficiency componen, when less han one shows he cos o he economy of recessions and underlines he imporance of achieving macroeconomic sabiliy. For he U.S. Corporae Nonfinancial Secor here were cos inefficiencies in he following eleven years: 1969, 1970, 1974, 1979, 1980, 1982, 1989, 2001, 2007, 2008 and For 2009 here was a 3.28% efficiency decline, providing a numerical esimae of he cos of he Grea Recession. The resuls for he Noncorporae Nonfinancial Secor were quie differen, wih cos inefficiencies for weny consecuive years , and also for he years 1995, 2007, 2009 and For 1983, cos inefficiency was equal o , meaning ha opimal inpu cos was only 82.32% of observed inpu cos for ha year. Hence he behavior and echnology of he Noncorporae Nonfinancial Secor is very differen from he behavior and echnology of he Corporae Financial Secor. 24 Such differences beween major secors of he economy are informaive as o he sources of produciviy slowdowns and possible policy responses. 24 The finding ha cos inefficiency is so much bigger in he Noncorporae Secor can be a leas parially explained by he fac ha his secor uses land and srucures much more inensively han he Corporae Secor, which uses machinery and equipmen more inensively. Thus when here is a recession, due o he quasi-fixed naure of he land and srucure inpus, he Noncorporae Secor canno reduce heir use of hese inpus.

18 We have also shown ha sandard Fisher index measure of TFP growh will provide an adequae long run measure of growh in TFP. Thus our mehodology should be viewed as a complemen o sandard saisical agency measures of TFP growh. Our decomposiion is differen from sandard growh accouning decomposiion approaches and gives policy makers informaion on wo imporan componens of TFP growh, while avoiding he inerpreaion of produciviy declines implying disappearing echnologies. In Appendix B, we show how our one oupu mehodology can be exended o he case of many oupus. However, i should be noed ha he muliple oupu join cos funcion C(y,w,) defined by equaion (B1) in Appendix B may no be well defined; i.e., he linear program defined in (B1) may no have a feasible soluion if he oupu vecor y has some negaive componens in i ha represen amouns used of inermediae inpus. Even if all componens of he y vecor are posiive, he opimal coss defined by C(y,w,) can be quie volaile and his will end o make he mehodology based on his nonparameric cos funcion impracical for saisical agencies o use. Thus he one oupu mehodology ha we have illusraed above may be he mos suiable applicaion for saisical agencies ha provide TFP measures. Appendix A: U.S. Corporae and Noncorporae Nonfinancial Secor Daa The oupu and inpu price daa ha we use in his paper are drawn from Diewer and Fox (2016b). These daa were consruced from Bureau of Economic Analysis (BEA) Naional Income and Produc Accouns Tables, BEA Capial Sock Tables, he Inegraed Macroeconomic Accouns Tables consruced by he BEA, Bureau of Labor Saisics and he Board of Governors of he Federal Reserve and published by he BEA. Diewer and Fox also used oher informaion on he price and quaniy of farm land and on commercial propery in he U.S. Diewer and Fox used he daa from hese sources o consruc produciviy accouns for wo subsecors of he U.S. privae secor: he Corporae Nonfinancial Secor and he Noncorporae Nonfinancial Secor. For convenience, we lis he daa used for hese wo subsecors in he presen sudy in his Appendix. 25 The user coss for he various capial socks ha we lis here are he user coss ha use prediced end of period asse prices raher han acual end of period asse prices in he user cos formula. Our use of prediced prices dramaically smoohed he resuling user coss and in paricular, eliminaed he negaive user coss ha occurred when acual end of period asse prices were used in he user cos formula. We used balancing raes of reurn in our user coss so ha he value of inpus was equal o he corresponding value added for each year and for each of our wo secors. The oupu and inpu price informaion for he U.S. Corporae Nonfinancial Secor are lised in Table A1. The firs inpu is labour inpu and he remaining inpu prices are user coss 26 for Diewer and Fox (2016b) did no consruc daa for he hird subsecor of he U.S. Privae Secor economy which is he Financial Secor. There is no general consensus on how o measure oupus and inpus for he Financial Secor. 26 Diewer and Fox used end of period user coss as he price for he services of a uni of capial. The period user cos u for an asse wih beginning of year asse price P is defined as u [1+r (1+i )(1 )]P = [r i + (1+i ) ]P where is he geomeric depreciaion rae, r is he year cos of capial and i is eiher he ex pos asse inflaion rae for year or an anicipaed asse inflaion rae. This formula for he user cos of capial was obained by Chrisensen and Jorgenson (1969; 302) using ex pos asse inflaion raes. Diewer and Fox used anicipaed asse inflaion raes where he anicipaed rae in year was essenially he geomeric average growh rae of he asse price over he 25 years prior o year.