Productivity Growth: Theory and Measurement

Transcription

1 Productvty Growth: Theory and Measurement Davd T. Owyong INTRODUCTION Productvty growth has receved greater attenton from economsts and polcy makers n Asa n the 1990s. Ths s partly due to the work of Alwyn Young (1992, 1995) and Paul Krugman (1994), who argued that economc growth n Asa s drven by the accumulaton of the nputs n the producton process rather than by ncreases n productvty. In other words they, n partcular Krugman, beleve that the Asan economc mracle s largely attrbutable to an ncrease n the quantty and not the qualty of the factors of producton. Further analyss and evdence showed that as countres become more developed and move closer to the lmts of factor accumulaton, they rely more and more on ncreasng productvty to sustan the economc growth process. In fact, some studes (such as Rao and Owyong 1997) ndcate that ths process of productvty growth s already occurrng n the more developed economes n the regon. Nonetheless, polcy makers and economsts alke have begun to recognze more fully the mportance of technology and productvty n economc growth. In order to facltate ths process of understandng the movements n productvty and desgnng the rght polces to enhance t, t s crtcal to be able to frst get a handle on what exactly productvty s and how to measure t. Ths wll be the focus of ths artcle. SOME BASIC DEFINITIONS OF PRODUCTIVITY At ts most basc level, productvty s based on the economcs of the frm. It s measured as the rato of output to nput. Hstorcally, productvty s often expressed as the rato of output to the most lmted or crtcal nput, wth all the other nputs held constant. Agrcultural productvty s usually measured n bushels of wheat or corn per 19

2 20 APO Productvty Journal acre. In ndustres that requre sklled labor, whch s often n relatve shortage, output per worker s consdered as the most approprate measure of productvty. However, such sngle-factor-based measures of productvty suffer from obvous lmtatons. Frst, n most ndustres or sectors there may be several factors of producton that are of almost equal mportance, n whch case t mght be dffcult to choose among them. Second, the relatve mportance of nputs may change over tme. For nstance, the relatve mportance of labor may be low n the ntal stages of development when unemployment s hgh, but may become crtcal as the country becomes more developed because of declnng brth rates and an agng labor force. Total factor productvty s the the combned productvty of all nputs, and hence avods the problems faced by measures based on ust one factor. One does not have to choose any factor on whch to base productvty growth, snce all factors are ncluded. Furthermore, as wll be made clearer later, the mpact of each nput on total factor productvty s allowed to vary, hence takng nto account the possblty that the relatve mportance of factors may change over tme. As a result of these advantages, total factor productvty s the most commonly known and wdely used method of productvty measurement. TOTAL FACTOR PRODUCTIVITY: SOME MEASUREMENT CONCEPTS Smply defned, total factor productvty s the weghted average productvty of all nputs, where the weghts to these nputs are ther shares n the total cost of producton. Suppose for the moment that output s measured n some physcal unt, say tons. Then TFP s measured as the rato of output Y to aggregated nput X: TFP = Y X (1) Snce there are multple nputs, X has to be computed by aggregaton. Usng the defnton of Dvsa ndexes, the growth rate of the aggregated nput s equal to the weghted sum of the ndvdual nputs growth rates: dx X = I =1 v dx x (2) where x s quantty of nput and v s the weght assgned to nput.

3 Productvty Growth: Theory and Measurement 21 v = Unt cost of nput Unts of nput employed Total expendtures for all nputs (3) Consder that nstead of havng ust a sngle type of output, there are multple outputs. Usng Dvsa ndexes agan, t therefore follows that dy Y = J =1 w dy y (4) where y s the quantty of the th output produced, wth the weght w beng the share of total revenue contrbuted by the th output. Combnng (2) and (4) leads to the followng expresson for TFP growth: TFˆ P = w ŷ v xˆ (5) where the hats represent growth rates and the weghts are functons of the relevant prces and quanttes: w = q y Σ q y and v = p x Σ p x (6) where q and p are the prces of the th output and th nput, respectvely. The frm s assumed to maxmze profts subect to the constrant of the producton technology, whch s gven by where profts are gven as follows. (y 1,...,y J ) = F(x 1,...,x I ) (7) π= q y p x (8) If the producton technology follows constant returns to scale, then q y = p x (9) Totally dfferentatng the last equaton wth respect to tme and dvdng both sdes by the correspondng total value yelds w [qˆ + ŷ ] = v [pˆ + xˆ ] (10)

4 22 APO Productvty Journal Equaton (5) mples that the rate of growth of TFP s equal to the aggregate growth rate of output mnus the aggregate growth rate of nputs. In addton when (10) s used, t may be shown that TFˆ P = v.pˆ w.qˆ (11) whch s to say that the rate of TFP growth s equal to the average rate of growth of nput prces less the average rate of growth of output prces. Relatonshp among TFP Growth, Output Growth, and Labor Productvty Growth For small changes n a varable the rate of change from one tme perod t to t + 1 s closely approxmated by the correspondng dfference n logarthms. Thus, for any varable Z, Ẑ = Z t+1 Z t Z t ln Z t+1 ln Z t (12) Gven ths useful result, we can reformulate our measure of TFP n (5) by replacng all growth rates by the correspondng log dfferences. It follows that the growth rate of TFP s TFˆ P t = ln TFP t ln TFP t 1 = v,t (ln(y t /x,t ) ln(y t 1 /x,t 1 )) (13) where average expendture share v,t = 0.5(v,t + v,t 1 ). In ths form t becomes clear that the growth of total factor productvty s the weghted sum of the growth rates of all sngle factor productvtes. Put another way, output growth s equal to the sum of the TFP growth rate and the growth rate of the average nput. Ŷ t = v,t xˆ,t + TFˆ P t (14) Fnally, the growth rate of the productvty of any nput can be expressed n terms of the rates of growth of the ratos of all other nputs to that nput, and the growth of TFP. For the case of labor productvty, ths mples that

5 Productvty Growth: Theory and Measurement 23 Growth rate of labor productvty = (ln Y t ln x l,t ) (ln Y t 1 ln x l,t 1 ) = (ln Y t ln Y t 1 ) (ln x l,t ln x l,t 1 ) = v,t xˆ,t (1 v l )xˆ l,t + TFˆ P t l = v,t (xˆ,t xˆ l,t ) + TFˆ P t (15) l where the second last equalty follows from (14) and the last equalty uses 1 v l,t =Σ l v,t. Presence of a Nonvarable Factor We now relax the assumptons of nstantaneous adustment of all nputs. It s assumed that there s one quas-fxed factor, whose adustment s hndered by adustment costs or nsttutonal factors. Ths quas-fxed factor s usually taken to be captal. Hence the frm s problem s to mnmze the varable cost of producng output (y 1,...,y J ) subect to the fxed prces of the varable nputs as well as the fxed captal nput k. Mathematcally ths s expressed as mn VC = f(x 1,...,x I, k; y 1,...,y J ) subect to q y p x and (VC)/ y q where the frst constrant mples that total varable cost must be met or exceeded by total revenues, wth the second requrng that prce must exceed margnal cost. The shadow prce to the quas-fxed nput s p k = Σ q y Σ p x k 0 (16) Hence ths mples that n the short run when the captal nput s often fxed, TFP s measured by TFˆ P = w ŷ v xˆ v k k where v k = p k k/ p x + p k k Extenson to the Economy Level (17) At the level of the economy, we can thnk of an aggregate producton functon, whch for the moment s assumed to follow the Cobb- Douglas form:

6 24 APO Productvty Journal log Y t =α+βlog K t + (1 β) log L t + log u t (18) where the Cobb-Douglas assumpton s taken to be a frst approxmaton to a potentally much more complex relatonshp. Dfferentatng the above expresson wth respect to tme yelds Ŷ t =βkˆ t +γlˆ t + TFˆ P t (19) The parameters β and γ represent the share of total nput cost n the Cobb-Douglas formulaton, whch s n accordance wth the weghts that were used earler n the artcle. Measures of total factor productvty may then be obtaned by deductng the nput growth rates from output growth. Ths approach to decompose the total growth of output n the economy nto ts dfferent potental factors s called growth accountng. These factors are to explan output movements; what s left unexplaned (often called the resdual) s consdered as total factor productvty. To see more clearly the mplct restrctons of the smple Cobb- Douglas producton functon, consder a producton functon of the general form Y t = f(x 1, t,...,x I,t,t), where the tme trend varable t may be thought of as a proxy for total factor productvty growth. We can then wrte dy dt = I =1 f x dx dt + f t (20) Dvdng both sdes by Y, and recognzng that f/ x s by defnton the margnal product of factor (MP ), we get TFˆ P = Y/ t Y = dy/dt Y = Ŷ MP.x Y dx /dt x MP.x Y xˆ (21) Assumng perfect competton, the values of the margnal products equal factor prces,.e., P.MP = p, where P s the output prce and p s the prce of nput. The last expresson can therefore be wrtten as TFˆ P = Ŷ p x PY xˆ (22)

7 Productvty Growth: Theory and Measurement 25 where the weghts of the nput growth rates are the ratos of nput payments to total revenues. These weghts wll be equal to the share of total nput costs f we assume constant returns to scale, n whch case P.Y =Σ p x. Gven ths and by restrctng the number of nputs to captal and labor, the last equaton (22) reduces to the Cobb-Douglas form n equaton (19). Measurement of Captal and Other Inputs The data for generatng measures of productvty growth should nclude as many outputs and nputs of the frms as possble n order to reflect all producton and costs. Output s usually measured as an aggregate of all types of producton actvtes. The categores of nputs generally dentfed are captal, labor, energy, nonenergy ntermedate materals, and sometmes purchased servces. Inputs such as land and nventores are often ncluded n the measure of captal. The two potentally most problematc ssues that arse n data constructon nvolve the measurement of captal and aggregaton. Aggregaton s a problem because captal s clearly not homogeneous. As regards ts measurement and the constructon of a captal seres, t s also problematc snce t requres rethnkng the dea of current nput use. As a durable nput the servces from the avalable stock of captal, and the rental or user prces of these servces, are relevant values for the constructon of productvty growth measures, and nether of these s readly observable. Developng captal measures also requres consderaton of what types of nputs should be ncluded as components of the captal stock, whch s sometmes unclear. Measurement of Output Output measurement for a sngle-output frm s farly straghtforward, snce for a sngle output there s only one type of unt nvolved, say the number of pars of shoes or tons of steel. In ths case, therefore, an average prce per par or ton can generally be specfed n dollars as total sales dvded by the quantty of the output, and thus quantty and prce ndexes can drectly be computed. Even for ths smple case there are problems nvolved. For example, t s not mmedately clear how changes n qualty can be handled. In a few cases (tons of steel mght be an example) ths s not a crtcal ssue snce the product s qute homogenous. However n most other cases, such as the number of computers produced, the qualty of a partcular unt mght change over tme or across companes (as n dfferent brands). Another problem that may complcate the measurement of output

8 26 APO Productvty Journal s the exstence of nventores. Data are generally reported n terms of sales, whereas actual producton s the relevant output for the measurement of productvty. Inventores consttute the dfference between these two fgures. For the measurement of output, therefore, sales data should deally be adusted by net nventory change. In other words, the correct output seres to use s obtaned from addng sales to nventory change. For a frm that produces multple outputs, there are further dffcultes: how to add together goods that are measured n dfferent unts s a standard ndex-number problem? It s not an easy problem to deal wth. Whle determnng the total value of producton s relatvely straghtforward, dvdng ths value nto ts aggregate quantty and prce components s not. Ths aggregaton ssues wll be dealt wth n greater detal below. Measurement of Labor Labor nput s relatvely easy to measure compared to other nputs, snce labor statstcs are generally presented n terms of wage bll pad and the number of workers or person-hours. By dvdng the wage bll by the number of workers or person-hours, we obtan an estmate of the average wage rate. The number of person-hours s generally a better measure of true labor nput than number of workers, snce the latter does not reflect changes n the hours worked per worker. Measurement of Captal The most problematc nput to measure s probably captal. Frst, the categores are often not clearly defned. Although buldngs and structures, machnery and equpment, etc. are often accounted for, other categores that are potentally mportant are gnored. One such example s research and development, whch mght be consdered a long-term nvestment, and therefore a component of the captal stock. The man dffculty of measurng captal, however, s how to deal wth an nput that provdes a stream of servces over tme, and s often not consdered as part of the explct costs of the frm. A relevant measure of the avalable captal stock s computed as what s left of the captal nvestment n past tme perods for the frm. Ths s generally wrtten for each captal asset x k as x k,t = T x k,t,t τ = T s k,t,τ z k,t τ (23) τ=0 τ=0

9 Productvty Growth: Theory and Measurement 27 where T s the lfe of the durable good, x k,t,t τ s the stock of x k n tme perod t stll remanng from nvestment n perod t τ, s k,t,τ s defned as the physcal survval rate for age τ nvestment n tme perod t for asset k, and z k,t τ s gross nvestment n asset k at tme t τ. Ths summaton must be done for each asset ndvdually, and then the assets must be aggregated based on ther user costs, as wll be dscussed below. Determnng the level of x k,t for each asset therefore requres fndng a benchmark level of the stock n perod 0, deflatng the value of nvestment by relevant deflators (to convert to constant dollars) and cumulatng the nvestment from that pont on based on some assumpton about survval rates. Fndng a benchmark s sometmes dffcult, often requrng some udgment together wth past data and numbers from other studes. As for the deflators to be used n the second step, they may be obtaned from the output prce seres for the supplyng ndustres, such as offce equpment. The most dffcult s step three, whch requres us to characterze s k,t,τ, the physcal survval rate. There are a number of possble assumptons for ths: (a) one-hoss shay (the machne runs at full tlt untl t des), (b) constant exponental decay (.e., the decay per tme perod s a constant percentage, say δ%, whch mples that s τ = (1 δ) τ ), (c) straght lne or lnear deprecaton (e.g., 5% of the ntal captal stock n ts tme perod), (d) decelerated deprecaton (any method where the age-prce profle declnes slower than concave). The most common method s a form of exponental decay called the perpetual nventory method, based on geometrc deteroraton. Ths assumpton mples that captal servces never actually reach zero so every unt of nvestment s perpetually a part of the stock of captal. The perpetual nventory method essentally requres that K t = (1 δ t )K t 1 + I t 1 (24) where K t s the captal stock at the begnnng of tme t and I t 1 s the nvestment n perod t 1. Often a constant exponental rate s assumed for δ t, whch makes t fall under the category of constant exponental decay. Next a prce for the captal good needs to be obtaned. Snce the underlyng theory specfes the servce flow from captal as the relevant nput to measure, t s necessary to construct correspondng data seres measurng the servce flow prce. Ths concept leads to noton of the user cost of captal, whch not only ncludes the nvestment prce, but also adusts t by the nterest rate, the deprecaton rates, and government taxes and ncentves. Mathematcally ths s represented by the followng equaton:

10 28 APO Productvty Journal c t = TX t [r t J t 1 +δj t J t ] + b t (25) where b t represents the effectve property tax rate, J t s the asset prce at tme t, J t = J t J t 1 denotes the captal gans, r t s the rate of nterest, δ s the deprecaton rate, and TX t s the effectve rate of taxaton on captal ncome gven by TX t = (1 T t Θ t κ t )/(1 T t ) (26) where T t s the effectve corporate ncome tax rate, Θ t s the present value of deprecaton deductons for tax purposes on a dollar s nvestment over the lfetme of the good, and κ t s the effectve rate of the nvestment tax credt. Econometrc Issues Many dfferent functon forms have been used for the econometrc estmaton of productvty growth. The choce among dfferent functonal forms s generally based on the type of analyss to be carred out. Some functons smplfy computaton of elastcty formulas and specfcaton of constrants such as constant returns to scale, some facltate consderaton of dynamc nteractons, some allow curvature condtons to be drectly mposed, and some enhance the ablty to dentfy the dfference between short-run and long-run behavor. Most modern studes of producton technology, however, do rely on some type of flexble functonal form, whch allows generalty n terms of nteractons among arguments of the functon, such as substtuton among nputs. One example of a flexble functonal form whch has been used extensvely for the analyss of producton s the translog functon. The translog producton functon, assumng nstantaneous adustment of all nputs s of the form: ln Y t =α 0 +α K ln K t +α L ln L t +α t t B KK (ln K t ) 2 + B KL (ln K t )(ln L t ) + B Kt (ln K t ).t + 0.5B LL (ln L t ) 2 + B Lt (ln L t ).t + 0.5B tt t 2 (27) where the assumpton of constant returns to scale mples that α K +α L = 1, B KK + B KL = B LL + B KL = B Kt + B Lt = 0 (28)

11 Productvty Growth: Theory and Measurement 29 It s clear from observaton that the translog functon s a generalzaton of the Cobb-Douglas functonal form. The Cobb-Douglas form s restrctve n terms of the mplct substtuton assumptons: elastctes of substtuton between all nputs are one and shares of the nputs are constant. Extendng the Cobb-Douglas to the translog functon enables these constrants to be relaxed because cross-effects between nputs are recognzed and therefore more complex substtuton patterns can then be captured. CONCLUSION Wth the ncreasng recognton that productvty growth s the key to sustaned economc expanson, measurng productvty s becomng mportant to economsts and polcy makers alke. The accurate measurement of productvty growth plays an mportant role n provdng the nformaton economsts need to put forth better polcy recommendatons and for polcy makers to make the rght decsons. In ths artcle we have consdered some of the ways to capture ths elusve concept of productvty. Although much further research remans to be done n ths area, t s hoped that ths artcle wll prove helpful by clarfyng some of the concepts on productvty and by documentng some of the measurement methods employed by economsts. REFERENCES Krugman, Paul The Myth of Asa s Mracle, Foregn Affars, 73: Rao, Bhano and Davd Owyong Sources of Growth n the Sngapore Economy: Some New Results, paper presented at the Tape Internatonal Conference on Effcency and Productvty Growth, Academa Snca. Young, Alwyn A Tale of Two Ctes, NBER Macroeconomcs Annual 1992, Olver J. Blanchard and Stanley Fscher, eds. Cambrdge, MA: MIT Press, pp The Tyranny of Numbers: Confrontng the Statstcal Realtes of the East Asan Growth Experence, Quarterly Journal of Economcs, 110: