Alternative Methods for Measuring Productivity Growth Including Approaches When Output is Measured With Chain Indexes

Transcription

1 Alternatve Methods for Measurng Productvty Growth Includng Approaches When Output s Measured Wth Chan Indexes Wllam D. Nordhaus 1 June 24, 2002 Abstract The present study s a contrbuton to the theory of the measurement of productvty growth. It frst examnes the welfare-theoretc bass for measurng productvty growth and shows that the deal welfaretheoretc measure s a chan ndex of the productvty growth rates of dfferent sectors whch uses current nomnal output weghts. Second, t lays out a technque for decomposng productvty growth whch separates aggregate productvty growth nto four factors the pure productvty effect, the effect of changng shares or Baumol effect, the effect of dfferent productvty levels or Denson effect, and a fxedweght drft term. Fnally, t shows how to apply the theoretcally correct measure of productvty growth and ndcates whch of the four dfferent components should be ncluded n a welfare-orented measure of productvty growth. The study concludes that none of the measures generally used to measure productvty growth s consstent wth the theoretcally correct measure. 1 I am grateful for helpful comments by Erwn Dewert, Paul Samuelson, and Thjs ten Raa. Ths verson s an augmented verson of a smlar paper dated November 20, The major change are a correcton of an erroneous approxmaton formula, an addton of a secton dealng wth chan-weghted output ndexes, and updatng of the emprcal estmates. Verson s welf_ doc -1-

2 Measurng productvty growth has been a growth ndustry wthn economcs for half a century. Over ths perod, there have been substantal changes and mprovements n the constructon of the underlyng data and methods. Partcularly notable are mprovements n measurng output and prces and n mplementng mproved ndexes, notably the use of superlatve prce and output measures by government statstcal agences. 2 Productvty growth s usually taken to be an obvous ndex of welfare. Paul Krugman put t succnctly, Productvty sn't everythng, but n the long run t s almost everythng. 3 The lnk between productvty growth and economc welfare s actually not obvous. There has, however, been surprsngly lttle attenton to the constructon of productvty measures. The present paper s part of a larger study whch s devoted to analytcal and emprcal questons n productvty measurement. 4 5 It makes three contrbutons to understandng the measurement of productvty. Frst, t examnes the welfare-theoretc bass for measurng productvty growth. Second, t lays out a technque for decomposng productvty growth whch dvdes aggregate productvty trends nto four factors that contrbute to the growth n economy-wde productvty. Fnally, t dscusses the approprate way to apply the deal welfare-theoretc measure n practce. The major practcal result of ths study s that current measures of productvty growth are generally napproprate from the pont of vew of reflectng economc welfare. We propose an alternatve measure of productvty 2 A dscusson of the use of Fsher ndexes n the natonal ncome and product accounts s found at Survey of Current Busness, vol. 72, Aprl 1992, pp and J. Steven Landefeld and Robert P. Parker, BEA's Chan Indexes, Tme Seres, and Measures of Long-Term Economc Growth, Survey of Current Busness, vol. 77, May 1997, p p Paul Krugman, The Age of Dmnshed Expectatons, MIT Press, Cambrdge, Mass., 1990, 4 See Wllam D. Nordhaus, Alternatve Methods for Measurng Productvty Growth, November 6, 2000, avalable at 5 Wllam D. Nordhaus, Productvty Growth and the New Economy, November 13, 2000, avalable at Ths study has been revsed as of June 2002 and s currently scheduled to appear n Brookngs Papers on Economc Actvty, 2002:2. -2-

3 growth rate, the chan ndex productvty growth rates, whch better approxmates the deal ndex. I. Welfare Aspects of Productvty Measures We begn wth the queston of the deal approach to measurng productvty growth. We approach ths ssue usng the tools of ndex number theory. 6 For smplcty, we assume that all output s devoted to consumpton goods and that consumpton goods are mmedately used up (.e., there are no durable goods). 7 We further assume that the approprate measure of real ncome s a smooth utlty functon of the form (1) U t = U(C 1t, C 2t,..., C nt ) where C t s the flow of servces from consumpton goods at tme t, and there are n goods, = 1,... n. We do not assume any partcular form for U, but we do assume that the utlty functon s smooth and homothetc. Under ths assumpton, we can construct ndexes of real ncome changes by takng the weghted average growth of ndvdual components. It wll be convenent to smplfy by assumng that each good s produced by prmary factors alone, so C t = F (S t ), where F s a constant returns to scale producton functon for ndustry and S t s a scalar ndex of nputs nto the ndustry (for example, S mght be a Cobb-Douglas functon of the relevant nputs). If A t s total factor productvty n sector, we can then wrte the producton functon as C t = A t S t. For the dscusson n ths secton, we assume that the economy s characterzed by perfect competton, that all factors are prced at ther margnal products, and that all goods are prced at ther margnal costs. Ths assumpton removes nfluences of mperfect competton and the dstortons that may arse from ndrect taxaton. Fnally, we assume that households have dentcal utlty functons and endowments. In addton, we make three smplfyng normalzatons. Frst, each household supples one unt of the composte nput, S. Second, we normalze the prce of the 6 There are many excellent references to the theory of ndex numbers. A succnct formal statement s W. Erwn Dewert, Index Numbers, n J. Eatwell, M. Mlgate, and P. Newman, eds., The New Palgrave: A Dctonary of Economcs, vol.1, London: The Macmllan Press, 1987, pp Durable goods and nvestment can be added by usng the approach ntroduced by Martn Wetzman, On the Welfare Sgnfcance of Natonal Product n a Dynamc Economy, Quarterly Journal of Economcs, vol. 90, 1976, pp

4 composte nput S to be unty. These normalzatons mply that each household has 1 unt of ncome. Thrd, we assume that ntal prce and level of productvty are equal to 1. Under these assumptons, the prce of each good s gven by: (2) q t = 1/A t. We now consder the expendture functon, V, that comes from maxmzng the utlty functon n (1) subject to the budget constrant: (3) E t = V (q 1t, q 2t,..., q nt, U t ) where E t s expendture. Note that the ncome term has been suppressed because we normalze ncome to be unty. Dfferentatng (3) wth respect to tme yelds: E t = (MV/Mq 1t ) 1t q + (MV/Mq 2t ) q 2t + þ + (MV/Mq nt ) nt q where a rased dot over a varable ndcates a tme dervatve. Usng the propertes of the expendture functon, we have E t = C 1t 1t q + C 2t q 2t + þ + C nt nt q Dvdng by E t and multplyng and dvdng each term on the rght hand sde by the relevant q t yelds (4) E t /E t = C 1t q 1t [ q 1t /q 1t ]/E t+ C 2t q 2t [ q 2t /q 2t ]/E t + þ + C 2t q nt [ q nt /q nt ]/E t Takng the logarthmc dervatve of (2) and substtutng nto (4) yelds: (5) g(e t) = - [F 1t g(a 1t ) + F 2t g(a 2t ) + þ + F nt g(a nt ) ] where F t = C t q t /E t = the share of good n total nomnal spendng at tme t. In what follows, we use the notaton that g(x t ) = x t /x t = the rate of growth of x t for ether contnuous or dscrete varables as approprate. We now proceed to determne the growth n real ncome due to changes n the total factor productvtes n dfferent ndustres. Defnng real ncome as R t, the growth n real ncome can be calculated as the growth n R t over tme. Snce (5) represents the -4-

5 declne n total expendture or ncome necessary to attan a constant utlty, by homothetcty the growth n real ncome that can be attaned wth the actual consumpton shares, productvty levels, and prces s therefore: (6) g(r t) = F 1t g(a 1t ) + F 2t g(a 2t ) + þ + F nt g(a nt ) In words, the growth rate of real ncome or real output s the chan-weghted ndex of sector-level productvty growths. The weghts n the ndex are the current nomnal shares of each good n total nomnal consumpton. Wth dscrete tme, equaton (6) should be calculated as an equaton n growth rates usng Fsher or other superlatve weghts. We now see how equaton (6) apples to the queston of the deal welfaretheoretc measure of productvty n an economy wth many sectors experencng varyng rates of productvty growth. The major result s that the deal measure of productvty growth s a weghted average of the productvty growth rates of dfferent sectors. Ths formula s very smlar to that currently used n constructng superlatve ndexes of prces and output. The mportant pont s that the ndexes used n the approprate measure are chan ndexes of productvty growth rather than dfferences n the growth rates or ndexes of output and nputs. II. Decomposng Actual Productvty Growth nto ts Components In ths secton, we turn to the queston of how productvty growth s actually measured. It wll be convenent to begn wth aggregate measures of productvty growth and break them nto ther major components. We wll see that the welfare analyss of the prevous secton fts very neatly nto ths decomposton. Productvty Accountng Movements n aggregate productvty are composed of both productvty changes n ndvdual ndustres and changes n the composton of ndustres. Research on decomposng the aggregate between the two components goes back many years, but t has not been updated to reflect changes n measurement of output. Ths secton revews the ssues and updates pror work to reflect the shft from fxed-weghted quantty ndexes to chan ndexes. Productvty wth fxed weghts ( old-style output ) We begn wth a revew of the decomposton of output growth usng fxedweghted quantty ndexes (references to the lterature wll be provded n the -5-

6 dscusson secton below). To begn wth, we consder productvty accountng where output s measured n the old style -- that s, wth fxed weghts. Consder aggregates of output (X t ), composte nputs (S t ), and total or partal factor productvty (A t =X t /S t ). The aggregates are the weghted sums of ndustry output and nputs (X t and S t ), where for smplcty we measure output and nputs n base-year prces so that they can be summed to form the aggregates. We can rewrte these as bult up from ndustry values ( = 1,..., N) as follows: X t = 3 X t S t = 3 S jt j and aggregate productvty s: (6) A t = X t /S t = (3X t )/(3 j S jt ) For smplcty, I wll begn by analyzng productvty growth n contnuous tme under the assumpton of smooth seres; the dscrete analyss s extremely messy. 8 Takng the logarthmc dervatve of (6) yelds: g(a t ) = g(x t ) - g(s t ) = ' ( X t / X t )(X t /X t ) - ' ( S t / S t )(S t /S t ) (7) g(a t ) = ' g(x t )z t - ' g(s t ) w t where z t = X t /X t = the share of real output of ndustry n total real output and w t = S t /S t = the share of nputs of ndustry n total nputs. Next add and subtract ' g(x t )σ t and ' g(s t )σ t from the rght-hand sde of (7): 8 So messy that, n the earler draft dervng an exact dscrete formula for the decomposton, I employed an nexact approxmaton whch led to an error when appled to fxed-weght quantty ndexes. I am grateful to Thjs ten Raa for pontng out the error. -6-

7 (8) g(a t ) = ' g(a t )σ t + ' g(s t )[σ t - w t ]+ ' g(x t )[z t - σ t ] Fnally, add and subtract ' productvty calculatons: g(a t )σ,0, where t = 0 s the base perod for (9) g(a t ) = ' g(a t )σ,0 + ' g(a t )[σ t - σ,0 ] + ' g(s t )[σ t - w t ] + ' g(x t )[z t - σ t ] The four terms n (9) are the fxed-weght average productvty growth, the Baumol effect, the Denson effect, and the fxed-weght drft term. I wll dscuss each of the terms n a later secton. Productvty wth chan weghts ( new-style output ) Decompostons for fxed-weght quantty ndexes lke those n the pror secton have been known for at least a quarter century. Snce that tme, the Unted States and some other countres have ntroduced superlatve ndexes that remove some of the problems that arse wth fxed-weght quantty ndexes. The U.N. System of Natonal Accounts (SNA) recommends usng annual chan ndexes, 9 and the U.S. and Canada have adopted chan Fsher ndexes for ther natonal accounts. 10 Consder next the decomposton of productvty growth where output s measured usng chan ndexes, focusng prmarly on the Fsher deal ndex. All varables are as n the last secton except that output s measured as a chan ndex. Consder the growth of X t, dealng now for the moment n dscrete tme. The exact defnton of the aggregate ndex, X t, wll dffer dependng upon how the chan ndex s calculated. If the ndex s calculated usng Fsher s deal ndex, then the growth of output s calculated as a Fsher ndex (F t ), whch s a geometrc average of the growth 9 The SNA summary states, The System also provdes specfc gudance about the methodology to be used to comple an ntegrated set of prce and volume ndces for flows of goods and servces, gross value added and GDP that are consstent wth the concepts and accountng prncples of the System. It s recommended that annual chan ndces should be used where possble, although fxed base ndces may also be used when the volume measures for components and aggregates have to be addtvely consstent for purposes of economc analyss and modellng. (SNA, 1993, paragraph 1.17). The reference s avalable at 10 A full dscusson of the ntroducton of Fsher ndexes s provded on the web ste of the Bureau of Economc Analyss at -7-

8 rates usng Laspeyres (L t ) and the Paasche (P t ) ndexes. More precsely, X t = X t-1 F t = X t-1 (L t P t ) ½, where L t = 3(X t q,t-1 )/3(X,t-1 q,t-1 ) and P t = 3(X t q t )/3(X,t-1 q t ). Takng logarthms and combnng the expressons, we have )ln(x t ) = ½ { ln[3(x t q,t-1 )/3(X,t-1 q,t-1 )] + ln [3(X t q t )/3(X,t-1 q t )] } A lttle manpulaton shows that L t = 3[(1+g t ) F,t-1 )], whle P t = { 3[(1+g t ) -1 F,t )] } -1, where F,t s agan the share of ndustry n nomnal output for perod t. For smooth seres and small tme ntervals, we can take the frst-order approxmaton of these equatons, whch yelds the approxmaton to the Fsher ndex as follows: (10) )ln(x t ) = ln(f t ) = 3 g(x t) F,tav where F,tav s the average share of F for the current and prevous perod. It s useful to note that the approxmaton formula n (10) s actually the Tornqvst ndex. Applyng the Fsher formula n (10), we can derve the growth of productvty as: )ln(a t ) = 3 g(x t ) F,tav - 3 g(s t ) w,t-1 Movng to contnuous tme and then addng and subtractng 3 (11) g(a t ) = 3 g(a t) F,t + 3 g(s t) [F,t - w,t ] g(s t) F,t yelds: The nterpretaton of (11) s that the aggregate rate of productvty growth s equal to the weghted average productvty growth of the ndvdual sectors plus the dfference-weghted average of the growth of nputs. The weghts on productvty growth are the shares of nomnal outputs whle the dfference-weghts on nput growth are the dfferences between output and nput shares. (A symmetrcal formula could be derved where the roles of nput and output shares are reversed.) One fnal formula ncludes the role of changng shares of output. Add and -8-

9 subtract 3 g(a,0) F,0 from equaton (11 ) and rearrange terms. Ths yelds the decomposton of productvty growth for chan quantty ndexes: (12) g(a t ) = 3 g(a,t )F,0 + 3 g(a t) [F,t - F,0 ] + 3 g(s t)[f,t - w t ] For easy reference, I repeat equaton (9) for fxed-weght quantty ndexes: (9) g(a t ) = ' g(a t )σ,0 + ' g(a t ) [σ t - σ,0 ] + ' g(s t )[σ t - w t ] + ' g(x t )[z t - σ t ] Interpretaton Equatons (9) and (12) are dentcal except for the fourth term of (9), whch s the fxed-weght drft term. The other three terms are, n order: the pure (fxed-weght) productvty term usng fxed nomnal output weghts for a gven year (t = 0, whch mght be dfferent from the base year for a fxed-prce-base quantty ndex); the Baumol effect whch reflects the dfference between current nomnal output weghts and baseyear nomnal output weghts; and the Denson effect whch reflects the nteracton between the growth of nputs and the dfference between output and nput weghts. We now dscuss each term n detal. Fxed-weght drft term. The fourth term n (9) represents the fxed-weght drft term The nterestng part of the fourth term s (z t - F t ). Ths term s the dfference between the share of ndustry n total real output and ts share n total nomnal output. Its propertes are well-known: When the relatve prces of year t are very close to those of the prce-base perod, ths term wll be close to zero. On the other hand, as we move further from the prce-base perod, and as relatve prces dverge from the prce-baseyear prces, relatve real outputs dverge from relatve nomnal outputs, and the fourth term wll tend to be nonzero. Real output wth a Laspeyres fxed-base quantty ndex tends to grow more slowly than a chan ndex n perods before the base year and more rapdly n perods after the base year. The dvergence of relatve real outputs from relatve nomnal outputs wth old-style fxed-weght quantty ndexes motvates the name fxed-weght drft term. Ths term vanshes wth the ntroducton of chan ndexes (or more precsely, well-constructed superlatve ndex numbers) because real output shares used n calculatng the growth rates are to a frst approxmaton equal to nomnal output shares. Pure Productvty Effect. The frst term on the rght hand sde of equatons (9) and (12) s a fxed-weghted average of the productvty growth rates of dfferent sectors. -9-

10 More precsely, ths measures the sum of the growth rates of dfferent ndustres weghted by base-year nomnal output shares of each ndustry. Another way of nterpretng the pure productvty effect s as the productvty effect f there were no change n output composton among ndustres. Ths s a pure measure of the underlyng trends n technology n dfferent ndustres that omts the effect of changes n spendng patterns. The Baumol effect. The second term captures the nteracton between the dfferences n productvty growth and the changng shares of dfferent ndustres over tme. Ths effect has been emphaszed by Wllam Baumol n hs work on unbalanced growth. 11 The Baumol effect occurs when those sectors wth relatvely slow productvty growth also have rsng nomnal output shares. Ths syndrome s often attrbuted to the servce ndustres, wth lve performances of Mozart strng quartettes beng a muchcted example. The Baumol effect, also sometmes called the cost dsease, s often msunderstood as sayng that low-productvty-growth sectors have hgher than average sectoral nflaton rates. Ths proposton s vrtually unversal and completely unremarkable. The true Baumol effect arses when ndustres wth low productvty growth and rsng relatve prces also have rsng shares of nomnal output n effect, beng ndustres wth relatvely prce-nelastc demand. Whether such cases are the norm or not s n fact a major emprcal queston. The usual example of health care probably does not prove the case because the prce ndexes used to estmate the prces of medcal care are hghly defectve. Estmates from a companon paper ndcate that the Baumol effect for the Unted States n the perod was actually slghtly postve rather than negatve as generally supposed. 12 Denson Effect. The thrd terms n (9) and (12) capture level effects due to ether changng shares of nputs on aggregate productvty. I label these the Denson effect, after Edward Denson who ponted out that the movement from low-productvty-level agrculture to hgh-productvty-level ndustry would rase productvty even f the productvty growth n the two ndustres were zero. Denson showed that ths effect was an mportant component of overall productvty growth n much of the twenteth century See Wllam J. Baumol, Macroeconomcs of Unbalanced Growth: The Anatomy of Urban Crss, The Amercan Economc Revew, vol. 57, no. 3, June 1967, pp Ths was updated and revsed n Wllam J. Baumol, Sue Anne Batey Blackman, and Edward N. Wolff, Unbalanced Growth Revsted: Asymptotc Stagnancy and New Evdence, The Amercan Economc Revew, vol. 75, no. 4, September 1985, pp See the references n footnote A number of studes found ths syndrome. See partcularly hs studes of postwar Europe n Why Growth Rates Dffer, Brookng, Washngton, D.C., Ths fndng was also dentfed n Nordhaus, -10-

11 We can get a more nterestng nterpretaton of the Denson effect by manpulatng t as follows. Defne R t = A t /A t = productvty n sector relatve to aggregate productvty. Then, note that R t w t = σ t and that w t [g(s t )- g(s t )] }= approprate nto equaton (12), we have: S t. Substtutng all these where Denson effect = ' g(s t )[σ t - w t ] = ' σ t [ g(s t )- g(s t ) ] = ' R t {w t [g(s t )- g(s t )] } or (13) Denson effect = ' R t w t The Denson effect s therefore equal to the sum of the changes n nput shares of dfferent ndustres weghted by ther relatve productvty levels. The classcal Denson effect occurred when the share of nputs n agrculture fell and agrculture was a lowproductvty-level ndustry. In ths case, the Denson effect was be postve, ndcatng that the aggregate productvty ncrease was hgher than the weghted average productvty ncreases n all ndustres. More generally, when the nput share of nputs of hgh-productvty-level ndustres rses at the expense of low-productvty-level ndustres, ths wll tend to rase aggregate productvty. Approprate Treatment of the Dfferent Effects A major queston n measurng productvty growth concerns the approprate constructon of ndexes. Whch of the four components of equaton (9) or three n (12) should be ncluded f our productvty measures are to be a useful measure of welfare? For ths dscusson, we turn as an applcaton to changes n labor productvty that s, we nterpret the varable S as labor hours worked. The queston then becomes what s the deal measure of labor productvty? Ths queston can be answered by comparng measured productvty growth n equaton (9) or (12) wth the deal productvty growth measure shown n equaton (6). A comparson of the three equatons shows that the deal ndex of productvty growth from a welfare-theoretc perspectve ncludes the frst two terms n (9) or (12) but excludes the other two terms. op. ct., 1972, whch used a slghtly dfferent verson of the decomposton for fxed-weght ndexes. -11-

12 Ths mples that the pure productvty effect and Baumol effect should be ncluded n a welfare-orented measure of productvty growth. The reason for the pure productvty effect s ntutve. Addtonally, the Baumol effect reflects the mpact of changng expendture shares on the overall productvty measure. If spendng s ncreasngly devoted to sectors that have low productvty growth, then ths mples that our economc welfare wll ndeed be growng relatvely slowly. Ths also mples that the fxed-weght drft term should be excluded from a welfare-theoretc measure of productvty growth. Ths s fortunate, gven the unversal movement away from fxed-weght ndexes of output growth noted above. The reason that t s approprate to exclude the fxed-weght drft term s that the welfare weghts on growth n outputs are current nomnal output, whch are the weghts used by chan ndexes. In economc terms, snce dealzed consumers set relatve margnal utltes each perod equal to current relatve market prces, the welfare weghts should be current relatve market prces rather than relatve prces of some hstorcal base year. Less obvous, but equally true, s that the Denson effect should normally be excluded from an deal productvty ndex. To understand the reason for ts excluson requres some dscusson of the potental sources of the Denson effect. Ths pont s most easly understood usng the Denson effect n equaton (13). The shows that the Denson effect s equal to the weghted average of the productvty relatves and the change n the shares of nputs. In other words, the Denson effect arses because of dfferences n the levels of productvty by ndustry. We can dentfy three major reasons for dfferences. The frst s that dfferences n productvty levels reflect dfferences n nputs whch are not captured by our productvty measures. For ths frst case, the Denson effect should be excluded from a welfare-orented measure of productvty because nterndustry shfts produce spurous changes n productvty growth. If n fact the levels of total factor productvty are equal n all ndustres, then the Denson effect would by constructon be zero. 14 A second reason for dfferences n productvty levels would arse because of dfferences n ndrect taxaton. A thrd reason would arse from dsequlbrum n nput or output markets, because of slow mgraton of labor from farmng to ndustry, or because of market power. If the second and thrd reasons were the major source of dfferences n productvty levels, then the treatment would be more complex and some or all of the Denson effect mght be approprately ncluded n a welfare-theoretc measure of productvty growth. As n other cases where tax wedges or other 14 Ths ponts to an mportant reason for constructng complete measures of nputs. In prncple, f all dfferences n productvty levels are due solely to dfferences n qualty and quantty of all nputs, a complete and accurate accountng system would show that total factor productvtes were equal n all ndustres. At ths pont, the Denson effect would be calculated to be zero. -12-

13 dstortons apply, the approprate treatment wll usually be somewhere between ncluson and excluson dependng upon the relevant elastctes. An examnaton of the actual patterns of labor productvty across sectors suggests that the dfferences n productvty arse prmarly because of the frst reason, dfferences n nputs whch are not captured by our productvty measures. In 1998, the level of labor productvty (gross output per hour worked) dffered by more than a factor of 100 across major ndustres. One major source of dfference comes from dfferences n captal ntensty or n labor sklls across ndustres. For example, the hghest gross output per person employed n 1998 at the two-dgt level was n nonfarm housng servces, wth a productvty level 34 tmes that of the overall economy. The hgh productvty arose because ths sector s essentally entrely mputed rents. Other hgh productvty ratos are found n captal-ntensve sectors such as ppelnes, ol and gas extracton, and telephone servces. Smlarly, hgh productvty levels are found n sectors wth specalzed and hghly compensated sklls such as securty and commodty brokers. At the other end of the spectrum are ndustres wth low-sklled workers, such as prvate households, personal servces, and apparel. The second and thrd sources of productvty dfferences appear less sgnfcant today. The ndustry whch gave rse to the Denson effect, farmng, had a productvty rato of 90 percent of the total economy n Ths suggests that dsequlbrum n labor mgraton patterns s a relatvely unmportant source of productvty dfferences today. Moreover, there are few two-dgt ndustres where ndrect taxes are a large share of gross output. The major case s tobacco products, where ndrect taxes were 60 percent of gross output n Among one-dgt ndustres, the rato of ndrect busness taxes to gross output n 1998 ranged from a low of 2 percent n constructon to a hgh of 21 percent n wholesale trade. Whle these dfferences are not trval, they are much smaller than the dfferences n productvty due to dfferng captal ntenstes or labor qualtes. Numercal examples We can llustrate the problems dscussed here usng a numercal example. For ths example, we use a Fsher chan ndex of the growth of output (so the fxed-weght drft term does not apply). Unfortunately, the calculatons are tedous, so I have omtted the underlyng detals. Table 1 llustrates how the Denson effect can provde msleadng estmates of productvty growth f not properly calculated. It shows an economy wth two ndustres wth dfferng levels of productvty. Industry 1 s a hgh productvty level sector whle ndustry 2 s a low productvty level sector. The bottom of the table shows dfferent ways of calculatng productvty growth. Lne 1 shows the welfare-theoretc measure of productvty growth from equaton (6), whch ncludes the pure productvty effect and the Baumol effect. Lnes 5 shows the standard calculaton -13-

14 (smply output per hour) and lne 6 shows the sum of the three effects. Lne 7 shows the error, whch s due to omtted second order effects. For chan ndexes, the drect calculaton s also the dfference of growth rates methodology, whch s the approach used by the BLS and many scholars. For the example n Table 1, both the aggregate on lne 6 and the dfference of growth rates methodologes gve msleadng results because they nclude the Denson effect. (If they use chan ndexes, happly, they do not ncluded the fxed-weght drft term.) Aggregate productvty grows at percent. The Denson effect s negatve (see lne 4), so the aggregate estmate of productvty growth understates the approprate method whch ncludes only the pure productvty effect and the Baumol effect (shown on lne 3). Hence the welfare measure of productvty growth s understated. Table 2 shows an example wth a strong negatve Baumol effect (llustratng hs sck-servces syndrome). Here, the ntal levels of productvty n the two sectors are equal, but ndustry 1 shows a declnng share of hours along wth strong productvty growth whle ndustry 2 s technologcally stagnant wth a relatvely large growth n hours. For ths case, the Baumol effect s strongly negatve (-1.33 percent n lne 3). The postve pure productvty effect s pulled down by the large negatve Baumol effect. In ths case, the standard calculaton whch ncludes the Baumol effect -- provdes a reasonably accurate estmate of the welfare-theoretc estmate because the Denson effect s small. Applcaton to aggregate U.S. data We can llustrate the procedures usng actual data on labor productvty for the Unted States. These data are derved from two companon papers on the subject. The frst companon paper presents a new data set on aggregate and ndustral productvty derved from ncome-sde data. 15 The second companon paper apples the concepts n ths paper and the data n the frst paper to estmatng productvty growth and the role of the new economy n the recent productvty upsurge. 16 Fgure 1 and Table 3, whch are drawn from the second paper, show a comparson of two measures of labor productvty for the overall economy over the perod The seres called deal measure s the welfare-theoretc ndex derved n a manner defned n equaton (6) above. Ths shows the rate of productvty growth that best measures the growth n average lvng standards. The measure labeled 15 See footnote See footnote

15 GDI productvty s the growth of total labor productvty, measured as ncome-sde GDP per hour worked. The results show a sgnfcant dfference between the two concepts. The deal or welfare-theoretc measure s hgher than standard labor productvty n two of the three subperods. The dfferences are substantal n the most recent perod, where the deal ndex exceeds the standard measure by 0.58 percentage ponts per year. On average, the deal or welfare-theoretc measure over the entre perod was 0.13 percentage ponts per year hgher than total ncome-sde productvty growth. Current Approaches to Constructng Productvty Measures Gven the vast lterature on productvty growth, there s surprsngly lttle dscusson of the welfare-theoretc nterpretaton of alternatve measures. There s of course a vast lterature on the constructon of deal ndexes of prces and output. One of the few studes to apply these to productvty s by Caves, Chrstensen, and Dewert. 17 Ths study does not address the ssue of dfferng levels of productvty n dfferent ndustres, however. One mportant study of the relatonshp between alternatve measures of productvty and welfare theory was by Baumol and Wolff, whch recommended the use of what they called a deflated ndex of total factor productvty. 18 Ths ndex s constructed by deflatng nomnal labor productvty by the an economy-wde average of the real wage. Whle ths formula does correct for the Denson effect, t does not appear to dentfy the need for a chan ndex to measure welfare mprovements of productvty growth. Early appled analyses of total factor productvty and labor productvty dd not analyze the approprate ndex from a welfare-theoretc pont of vew. 19 Alternatve approaches were used n these studes, but the central approaches were generally ones whch weghted productvty growth by fxed output weghts. Ths approach s clearly napproprate and can gve msleadng results f productvty growth dffers by sector. 17 Douglas W. Caves, Laurts R. Chrstensen, and W. Erwn Dewert, The Economc Theory of Index Numbers and the Measurement of Input, Output, and Productvty, Econometrca, Vol. 50, no. 6, Nov. 1982, pp Wllam J. Baumol and Edward N. Wolff, On Interndustry Dfferences n Absolute Productvty, The Journal of Poltcal Economy, Vol. 92, No. 6, Dec., 1984), pp See Wllam D. Nordhaus, The Recent Productvty Slowdown, Brookngs Papers on Economc Actvty, 1972, no. 3, pp ; Martn N. Baly, The Productvty Growth Slowdown by Industry, Brookngs Papers on Economc Actvty, 1982, no. 2, pp ; and Edward F. Denson, Accountng for Slower Growth: the Unted States n the 1970s, Washngton, Brookngs,

16 The work of Jorgenson and Grlches and later work by them and co-authors derve measures of productvty growth from general transformaton functons. 20 Ths approach led to the suggeston, poneered by Grlches and Jorgenson, that productvty growth be estmated as the dfference between Dvsa ndexes of output growth and nput growth (these are essentally Tornqvst ndexes). Ths approach, whch we call the dfference of growth rates approach, wll be close to the deal approach f productvty levels n dfferent ndustres are close, but t wll not n general provde the correct result from a welfare theoretc pont of vew f the Denson effect s present. (See Tables 1 and 2.) The Bureau of Labor Statstcs currently uses the dfference of growth rates approach n ts productvty measures. 21 The output measures are currently chan ndexes of output usng Fsher or Tornqvst weghts. Labor nputs are ether hours or weghted hours at work. Measures of productvty are calculated as the dfferences between the growth n output and the growth of nputs. The BLS measures therefore suffer from the defcency that t ncludes the Denson effect. More generally, t s not a chan ndex of productvty growth, whch s the preferred measure. In summary, none of the current approaches to estmatng productvty growth appear to be well-grounded n welfare economcs. Rather, assumng that dfferences n productvty growth are due to dfferences n unmeasured nputs, the approprate measure would be a chan ndex of productvty growth of dfferent sectors weghted by current expendture or current-value nputs shares. In terms of the decomposton derved above, the approprate measure would be total productvty growth after removng both the fxed-weght drft term and the Denson effect. Alternatvely, the approprate measure would be the pure productvty effect plus the Baumol effect. It s useful to note, that none of the current measures of productvty follow the approprate procedure for measurng productvty growth. 20 The major studes are usefully summarzed n Dale W. Jorgenson, Productvty: Volume I, Postwar U.S. Economc Growth, MIT Press, Cambrdge, MA., See especally Chapter 3, The Explanaton of Productvty Change, wth Zv Grlches. 21 See Kent Kunze, Mary Jablonsk, and Vrgna Klarqust, BLS Modernzes Industry Labor Productvty Program, Monthly Labor Revew, vol. 118, no. 7, July 1995, pp and John Duke and Lsa Usher, BLS Completes Major Expanson of Industry Productvty Seres, Monthly Labor Revew, vol. 121, no. 9, September 1998, pp

17 Table 1 Example of Alternatve Measures of Productvty Growth Showng Strong Denson Effect Illustraton of Denson Effect Perod Industry 1 Industry 2 Sum Fsher Index Real output (perod 1 prces) Hours Components of productvty growth 1 Welfare productvty growth 1.773% 0.000% 1.773% 2 Pure productvty effect (early weghts) 1.784% 0.000% 1.784% 3 Baumol effect % 0.000% % 4 Denson effect % % % 5 Sum ( ) 1.283% % 0.705% 6 Drect calculaton 0.704% 7 Error (5-6) 0.001% -17-

18 Table 2 Example of Alternatve Measures of Productvty Growth Showng Strong Baumol Effect Illustraton of Baumol Effect Perod Industry 1 Industry 2 Sum Fsher Index Real output (perod 1 prces) Hours Components of productvty growth 1 Welfare productvty growth 9.741% % 2.247% 2 Pure productvty effect (early weghts) % % 3.577% 3 Baumol effect % % % 4 Denson effect 0.000% 0.000% 0.000% 5 Sum ( ) 9.741% % 2.247% 6 Drect calculaton 2.257% 7 Error (5-6) % -18-

19 Fgure 1 Alternatve Measures of Labor Productvty for Overall Economy, % Actual Ideal Productvty growth (percent per year) 3% 2% 1% 0% -1% Source: Source: Output_Create_ xls -19-

20 Table 3 Alternatve Measures of Labor Productvty for Overall Economy, (1) (2) (3) (2) - (1) (3) - (1) Ideal measure 1.44% 0.72% 2.37% -0.72% 0.92% GDI measure 1.20% 1.11% 1.79% -0.09% 0.59% Note: The deal measure s an ndex of productvty growth that ncludes the fxedweght and Baumol effects but excludes the Denson effect. The GDI measure s productvty growth calculated as the growth n output less the growth n nputs. Source: Output_Create_ xls -20-