The Definition and Measurement of Productivity* Mark Rogers

Transcription

1 The Defnon and Measuremen of Producvy* Mark Rogers Melbourne Insue of Appled Economc and Socal Research The Unversy of Melbourne Melbourne Insue Workng Paper No. 9/98 ISSN ISBN May 1998 *Ths paper s he resul of work beng underaken as par of a collaborave research program enled The Performance of Ausralan Enerprses: Innovaon, Producvy and Profably. The projec s generously suppored by he Ausralan Research Councl and he followng collaborave parners: Ausrala Tax Offce, Commonwealh Offce of Small Busness, IBIS Busness Informaon Py Ld, Indusry Commsson, and Vcoran Deparmen of Sae Developmen. The vews expressed n hs paper represen hose of he auhor and no necessarly he vews of he collaborave parners. Melbourne Insue of Appled Economc and Socal Research The Unversy of Melbourne Parkvlle, Vcora 3052 Ausrala Telephone (03) Fax (03) Emal [email protected] WWW Address hp://

2 Absrac Ths paper nroduces he varous mehods ha have been used o analyse producvy. Producvy s defned as he rao of oupu o npu for a specfc producon suaon. Producvy changes can be caused by eher movemens n he 'bes pracce' producon echnology, or a changes n he level of effcency. The paper dscusses he varous problems encounered n measurng producvy when here are mulple oupus and npus. Also, he problems concernng he measuremen of npus and oupus are dscussed. Mehods ha analyse he level of neffcency whn a sample of frms are revewed. These nclude daa envelopmen analyss, sochasc producon funcons and panel daa mehods. Lasly, a few Ausralan producvy sudes are revewed o llusrae he emprcal use of he varous defnons and echnques. Keywords: producvy, daa envelopmen analyss, sochasc producon froners, panel daa. 2

3 Curren workng papers from he 'Performance of Ausralan Enerprses' projec Tle Number Auhor(s) The Theory and Measuremen of Profably 7/98 Gow/Kells The Defnon and Measuremen of Producvy 9/98 Rogers The Defnon and Measuremen of Innovaon 10/98 Rogers 3

4 Conens 1 INTRODUCTION 5 2 THEORY OF PRODUCTIVITY MEASUREMENT DEFINING PRODUCTIVITY FOR A SINGLE FIRM INDEX NUMBERS 10 3 MEASURING INPUTS AND OUTPUTS OUTPUT LABOUR CAPITAL 12 4 EFFICIENCY AND PRODUCTIVITY 13 5 ANALYSIS OF EFFICIENCY DATA ENVELOPMENT ANALYSIS STOCHASTIC PRODUCTION FRONTIER APPROACH PANEL DATA METHODS 19 6 AUSTRALIAN STUDIES 20 7 CONCLUSION 23 4

5 1 Inroducon The ssues surroundng he defnon and measuremen of producvy have been he opc of research for a varey of dscplnes, ncludng accounancy, economcs, engneerng and operaons research. A a basc level, he concep of 'producvy' s relavely easy o defne. I s he rao of oupu o npu for a specfc producon suaon. Rsng producvy mples eher more oupu s produced wh he same amoun of npus, or ha less npus are requred o produce he same level of oupu. In eher case, s no dffcul o undersand he mporance of producvy changes for general welfare, ncludng envronmenal concerns. The concep of producvy s lnked closely wh he ssue of effcency. If a frm s effcen s sad o be operang on he producon froner (.e. s achevng 'bes pracce'), where he producon froner s defned a some pon n me wh reference o a parcular se of frms (.e Ausralan frms n a ceran ndusry). Rsng effcency would herefore mply rsng producvy. Equally, he shf ouwards of a producon froner also mples producvy growh. In hs paper, herefore, producvy growh encompasses boh changes n effcency and changes n bes pracce. The ssue of producvy and effcency s elaboraed on n secon 4. Alhough he basc concep of producvy s sraghforward, dffcules are soon encounered when one confrons varous measuremen problems, he presence of mulple npus and oupus, and uncerany over how o model he producon process. In addon, producvy measures can be made a he process, plan, frm, ndusry or economy level; each of whch nvolves some specfc ssues and conceps. Ths paper ams o nroduce he varous ssues and echnques nvolved n producvy analyss. The paper s wren a he sar of a projec whch wll analyse frm performance, hence he paper concenraes on ssues surroundng frm-level producvy. Issues specfc o macroeconomc producvy analyss are no sressed alhough, of course, macroeconomc daa orgnaes from measuremen of frm-level daa. The srucure of he paper s as follows. Secon 2 oulnes some basc heory of producvy measuremen for a sngle frm (or workplace). The cenral aspec of such 5

6 work s he producon funcon from whch varous producvy measures can be consruced on he bass of ceran assumpons abou frm behavour and marke condons. Secon 2 also dscusses he role of ndex number heory o consruc producvy measures. Secon 3 consders some he problems nvolved wh measurng oupus and npus. A a heorecal level hese are easy o defne bu n pracce daa are ofen no avalable o allow he heorecally correc measuremen of oupus and npus. Secon 4 reurns o he ssue of effcency and producvy and clarfes he pons made n he openng paragraph above. Secon 4 also provdes he heorecal framework for secon 5 whch consders quanave echnques o denfy relave levels of effcency when a daa se of many frms s avalable. The essenal concep n such work s he consrucon of a heorecal producon froner from a sample of frms. Ths can be done usng mahemacal programmng (a echnque from operaons research) or economerc mehods. These echnques have been developed no only o assess effcency a a pon n me bu also o assess how effcency changes over me, ncludng he shf n he producon froner. Secon 6 consders a few Ausralan sudes no producvy. Ths s done o provde some examples of how he mehods and ssues dscussed n prevous secons have been used. Secon 7 concludes. 2 Theory of producvy measuremen 2.1 Defnng producvy for a sngle frm The mos basc case o consder s ha of a sngle frm ha produces one oupu (y) usng a sngle npu (x), he rao of hese wo (y/x) yelds a measure of he level of producvy. Boh oupu and npu are measured n real uns (.e. her level reflecs he real quany of eher oupu or npu). Dewer (1992), n fac, consders hs rao an oupu-npu coeffcen and only arbues he word 'producvy' o he measuremen of changes n such coeffcens over me (),.e. ϑ1, [1] prod y ϑ1 x ϑ1. y x In such a smple sngle oupu, sngle npu case here are varous equvalen ways of 6

7 assessng producvy change beween wo perods. Dewer (1992) lss fve mehods ha yeld equvalen measures of producvy (bascally hese ulse daa on coss and prces o express [1] n dfferen ways). Dewer (1992) also refers o he Jorgenson and Grlches (1967) mehod of usng only npu prce (w) and oupu prce (p) daao yeld he rao ϑ1, [2] w p ϑ1 ϑ1. w p If a frm's revenues equals s coss hen [2] s an equvalen producvy measure o [1] (whch can be shown by usng he deny py=wx). Ths laer producvy measure s an ndcaon of how varous assumpons, n hs case ha of perfec compeon (revenue equal o coss), may be used o allow producvy measuremen when only ceran daa s avalable. These varous ways of calculang a well-defned producvy measure for he sngle npu, sngle oupu case are prmarly llusrave, snce n pracce almos all neresng cases nvolve mulple oupus or npus or boh. When we move o consder hese cases he analyss becomes much more complex. In Dewer's words "we are faced wh a bewlderng array of alernave concepual measures of producvy change" (Dewer, 1992, p.169). To ake a smple example, suppose ha a frm has one oupu (y) and hree npus: labour (l), capal (k) and maerals (m) and ha he relaonshp beween hem (a me )sgvenby [3] y Ζ f ( l, k, m). Even n such a suaon, s sll possble o specfy paral producvy measures, such as y/l or y/k (and consder eher he level or he growh raes of such raos). Indeed, here may be specfc reasons for calculang such paral producvy measures (e.g. wage negoaors may wsh o sudy he level/pah of y/l). However, he use of paral producvy measures can be msleadng snce dfferen npus can subsue for each oher (e.g. f k s ncreased whle all oher npus are held consan, hs rases y and also he paral producvy measure y/l, even hough here may have 7

8 been no ncrease n he producvy of labour per se). Ths ype of suaon has led people o adop mulfacor measures of producvy. A basc adjusmen s o consder value added (y a = y - m) raher han gross oupu as he lef hand sde varable n [3] (see Morrson, 1993, p.29). Ths s one mehod of removng he nfluence of maerals npu so as o consder y a /l or y a /k. To decompose he movemens of y we can dfferenae [3] wh respec o me, yeldng, [4] y f l lη f k kη f m mη df d, where he do noaon s used for a me dervave. Dvdng equaon [4] by y yelds [5] y f l l f k k f m m df Η Η Η 1 y l y l k y k k y m d y. In words, equaon [5] saes ha he growh of oupu s a weghed sum of he growh raes of he npus, and an addonal erm whch ncludes he shf n he producon funcon (f(.)) over me. The laer erm s referred o as he rae of mulfacor producvy (MFP) growh or, somemes, he rae of echncal advance. In eher case represens he oupu change no accouned for by changes n npus or changes n npu mx. For emprcal work, equaon [5] s no drecly useable as sands. The problem s assumng approprae (real) measures of y, k, l and m are avalable ha he margnal produc erms (e.g. f/ l= MP) are no drecly observable. Pas researchers have, herefore, resored o heorecal assumpons o yeld values for he margnal producs whch, n urn, allows he calculaon of MFP. To llusrae, assume ha he frm s prof maxmsng and subjec o consan reurns o scale (hese assumpons mply he frm operaes n a perfecly compeve ndusry). In such a suaon, he frm's maxmum profs are acheved when he value of he margnal produc of an npu equals he npu prce, e. p.mp = w,whereps he oupu un prce and w s he 8

9 npu prce. Ths equaon allows us o subsue ou he MP erms n [5] and wre he more famlar (see Morrson, 1993, p.45 for a more dealed dervaon) y y l l m m [6] Η Η Η mfp, S l S k k k S m where S represens he relave npu share of value added oupu. Ths expresson s he mehod used by he ABS o provde mulfacor producvy esmaes a he aggregae level (see Aspen, 1990). The dervaon of mulfacor producvy can also be acheved by usng he cos srucure of a frm. Mahemacally, he dervaon above s based on he producon funcon or 'prmal' measure; an equvalen mehod of expressng he consrans faced by a frm s o use he cos funcon or 'dual' measure. Usng he cos funcon, C(w, Y, ), and agan he assumpons of consan reurns and cos mnmsaon, mulfacor producvy can be shown o equal [7] mfp c c c w j x j w j ϑ, C w j j where C equals oal coss, c (= C/Y) s un coss, w s npu prces and x s npu quanes (for j npus). Under he assumpons of consan reurns o scale and cos mnmsaon can be shown ha he measure for mulfacor producvy n [7] s dencal o ha n [6] (see Morrson, 1993, p.48 for a full dervaon of [7] and proof of hs saemen). Noe ha he dual mehod would requre dealed daa on he prces of npus. Wheher he prmal or dual mehodologes are used n he measuremen of producvy s dependen on he assumpons made. As saed, he mehods assume ha frms experence consan reurns o scale and underake prof maxmsaon (cos mnmsaon). If, n realy, hese assumpons are no rue, he mulfacor producvy measure may be based. Moreover, he producon and cos funcon have 9

10 an mplc assumpon ha npus can be adjused nsananeously (.e. no adjusmen coss). In he real world he level of varous npus may be fxed over he shor or medum erm (capal beng he mos obvous example bu even labour and oher npus may exhb some adjusmen coss). The problem of 'npu fxy' agan mean ha he equaons for mulfacor producvy are lkely o be based. There are varous mehods ha can be used o adjus for non-consan reurns o scale and npu fxy. These nvolve a more general reamen of equaon [5] so as no o mpose hese assumpons. Morrson (1993, Chapers 4, 6 and 7) deals some of hese mehods. Hall (1986) provdes he nal paper n a leraure ha seeks o adjus MFP measures for he degree of marke power, as proxed by he mark-up rao (whch s defned as he rao of oupu prce o margnal coss). Agan, Morrson (1993, p.111 and Chaper 9) conans a full dscusson. 2.2 Index numbers An alernave way o consder he problem of assessng mulfacor producvy comes from ndex number heory. In shor, ndex number heory consders how o oban a sngle ndex of, say, he prces of goods over me, when here are many goods. Varous ndces are commonly used ncludng Laspeyres, Paasche, Fsher Ideal and he Tornqvs ndex. The Tornqvs ndex can be appled o he problem of creang an ndex for oupu from a number of npus o yeld, [8] ln Y Y ϑ1 where S S 1 2 ln w x x, ϑ1,, ϑ1 x, ϑ1 w, ϑ1 x, ϑ1 1 2 w, x, w, x, Ths, somewha nmdang equaon, smply saes ha oupu growh s a weghed sum of he npu growh raes 1, where he weghs are he average npu shares n oal 1 Noe ha ln(x /x -1 ) s a dscree approxmaon of dx/d/x. 10

11 coss. Ths, herefore, s almos dencal o equaon [5], whch s saed n connuous me. Thus, when a daa seres for oupu and npus are avalable he Tornqvs ndex can be used o yeld MFP, by subracng he lef hand sde of [8] from he rgh hand sde. Ths s he normal mehod used by he ABS (see Aspen, 1990). I s worhwhle elaborang a lle more of he Tornqvs ndex number approach o producvy. In parcular, should be noed ha he ndex s a dscree approxmaon o he connuous me equaon [5]. Ths rase he queson of how good an approxmaon s? To answer hs an exac funconal form for he producon funcon f( ) needs o be consdered. When he producon funcon s ranslog, Dewer (1976) has shown ha he Tornqvs ndex s exac (e. he ndex [8] s he approprae choce for emprcal work). Moreover, snce he ranslog funcon s 'flexble' (.e. provdes a second order approxmaon o an aggregaor funcon), Dewer calls he Tornqvs ndex superlave. 3 Measurng npus and oupus The varous mehods dscussed above have no menoned he dffcules ha are encounered when praccal measures of producvy are sough. Ths secon oulnes some of he basc measuremen ssues ha face researchers. More exensve dscussons are conaned n Morrson (1993) and Baly and Gordon (1988). 3.1 Oupu The dscusson above has referred o measures of real oupu, eher gross or value added. In a me seres sudy hs mples he need for prce deflaors whch may or may no be avalable a he, preferred, frm or ndusry level. A more dffcul ssue s ha oupu qualy may ncrease over me, even hough un prces are sac or declnng (a prme example s personal compuers). There s a leraure on mehods o adjus for qualy ssues (see Gandal, 1994, for a compuer sofware example), and ndeed some sascal agences adjus he oupu of ceran secors (Grlches, 1994, noes ha he US governmen nroduced a qualy adjused (hedonc) prce seres for compuers n 1986). Calculaon of real oupu n servce ndusres also presens problems. Lowe (1995) dscusses he problems of measurng he oupu of he real 11

12 secor when qualy changes, such as an ncrease n openng hours, have occurred. Rzman (1995), n a sudy of Ausrala bankng, assumes ha real oupu s proporonal o he level of deposs and loans (.e. assumng ha an oupu flow s proporonal o a sock). There are, herefore, varous dffcul measuremen ssues and adjusng for hese wll depend on he specfc daa avalable and varous assumpons. Oupu should be defned as he real oupu produced n a se me perod. The sales or revenue fgure normally repored n accouns wll no concde wh hs f nvenory levels have rsen or fallen over he perod. Hence, adjusmens for he level of nvenores should be made and also, f possble, he mpac of any oupu gven away for promoons, ec Labour Labour quany s normally measured n erms of he number of employees. Adjusmens should be made for he exen of par me work, hence he dea of 'full me equvalen' employees. Even f hese adjusmens are possble, here s also he ssue of adjusng for hours worked by, for example, he ncluson of overme. Adjusmens for he qualy of labour s also an ssue o be consdered. In heory, labour could be spl no varous separae npus dependng on skll, educaon or oher classfcaon, and each of hese could be enered as a separae npu n [5]. Such an approach would requre he necessary daa on he quanes of each and he wage bll of each caegory. 3.3 Capal The measuremen of capal s, perhaps, he mos problemac of npus o measure. Imporanly, he producvy measures dscussed above are concerned wh he real value of capal servces n a se me perod. Ths s no he same as he sock of capal held by a frm or ndusry; nsead we need o measure he flow of servces from such a sock. Morrson (1993, p.139) repors ha one mehod of acklng hs 2 See Dewer and Smh (1994) for a dealed case sudy of nvenores and producvy n a dsrbuon frm. 12

13 ssue s o: frs, creae a capal sock seres by addng up he nvesmen n dfferen asses over me, allowng for deprecaon, manenance, nflaon n asse prces, ec; second, assume he flow of capal servces s a consan fracon of hs sock; hrd, calculae a prce for each componen of capal servces by usng he prce of relevan capal goods. The vecor of servce flows and servce prces can hen be aggregaed no a sngle capal flow and prce (usng an approprae ndex). The daa requremens for such a procedure, however, mean ha such a mehod s ofen mpossble n pracce. Insead, some researchers have assumed ha he value flow of capal for a frm equals sales less all varable coss (effecvely a 'resdual' measure of he value of capal). A capal sock s also calculaed (usng bes avalable daa). Then he researcher assumes ha he flow value equals he prce of capal servces mulpled by he quany (sock of capal). Ths, n urn, allows a prce for capal servces o be calculaed. Ths mehod calculaes he prce of capal from he wo oher componens. An alernave mehod s o assume a prce of capal (say, o reference o bond prces) and hen calculae he value of capal servces. 4 Effcency and producvy The above dscusson focussed only on producvy measuremen wh no menon of he concep of 'effcency'. Effcency s normally defned as comprsng of wo componens: echncal and allocave effcency. Techncal neffcency occurs f a frm s no obanng maxmal oupu from a se of npus. Allocave neffcency occurs when a frm fals o choose he opmal balance of npus gven npu prces (even hough may be obanng maxmal oupu from he npus acually used). To llusrae, Fgure 1 s reproduced from Coell (1995, p.222), 13

14 Fgure 1 Techncal and allocave effcency L/Y F P A Q R E F O A K/Y The fgure has axes for he npu of labour and capal (per un of oupu). The (unknown) bes pracce un soquan s gven by FF (noe he axes are n npu per un of oupu). The npu combnaon o produce a sngle un of oupu for a sngle frm s shown by he pon P (a he end of he lne OP). Thus, he frm s producng ousde he effcen un soquan and can be ermed neffcen. Techncal neffcency s measured by he rao OQ/OP (.e. he hypohecal exen o whch npus can be proporonally reduced whou reducng oupu). Allocave effcency can only be deermned n relaon o he prces of npus. The socos lne AA llusraes he npu prce rao. Gven hs rao, producon a pon Q s no opmal and he frm should locae a pon E. A measure of he neffcency caused by hs non-opmal pon s OR/OQ and s ermed he exen of allocave neffcency. Thus, overall economc neffcency can be shown by he rao OR/OP. Overall neffcency can be hough of as he combnaon of echncal and allocave neffcency (represened by he rao OR/OQ). The measuremen of producvy dscussed n secon 3 assumes ha all frms were fully effcen (.e. all operae on he producon froner and selec opmal quanes 14

15 of each npu). Ths assumpon was mplc n he prof maxmsng/cos mnmsng assumpons made. In realy, we would no expec all frms o be fully effcen. If hs s he case, hen he measure of TFP arsng from, say, a Tornqvs ndex, may be parly due o a shf n he producon froner and parly due o frms movng closer o he froner over me (.e. becomng more effcen). Ths means ha TFP may be measurng boh 'effcency' mprovemens and 'shfs n he producon froner' and should no, herefore, be defned as 'echncal change' (unless varous adjusmens for effcency changes have been made). Ths dsncon can cause confuson as some researchers consder 'producvy growh' as only referrng o he shf n he froner. Here we follow Grosskopf (1993, p.160) who saes, "I defne producvy growh as he ne change n oupu due o change n effcency and echncal change, where he former s undersood o be he change n how far an observaon s from he froner of echnology and he laer s undersood o be shfs n he producon froner". 3 In hs paper, herefore, we consder he analyss of neffcency as an aspec of producvy. Such analyss reles on havng a daa se of varous frms (wh nformaon on npu and oupu levels) and usng hs nformaon o consruc a heorecal producon froner. 5 Analyss of effcency There are hree man caegores of echnques ha can be used: daa envelopmen analyss, sochasc producon froner, and panel daa echnques. Each of hese echnques has an exensve leraure and our am here s o provde an overvew wh assocaed references. 3 There s a furher possble confuson arsng from callng shfs n he producon froner 'echncal change' snce hs may creae he mpresson ha echnologcal change only accouns for a fracon of overall oupu growh. Such an mpresson s only correc f echnologcal change and npu accumulaon are ndependen (e.g. new echnology does no gve rse o nvesmen). See Fagerberg (1994) for furher dscusson of hese ssues. 15

16 5.1 Daa envelopmen analyss The daa envelopmen analyss (DEA) mehod s based on work by Farrell (1957) and Koopmans (1951), alhough s frs full mplemenaon was by Charnes, Cooper and Rhodes (1978). Useful recen summares of daa envelopmen analyss mehods are gven by Al and Seford (1993) and Coell (1995), boh of whch we draw on here. 4 Daa envelopmen analyss (DEA) s a mahemacal programmng approach o assessng he neffcency of a frm (plan) relave o a sample of frms (plans). In essence, he approach maps ou a hypohecal producon froner on he bass of all he oupu and npu nformaon from he sample. Ineffcency s hen assessed by he dsance a frm s specfc oupu/npu mx s from he froner. Such a dsance can be assessed n erms of he amoun of npus ha are wased n producng a gven oupu, or he level of oupu ha could be produced gven he level of npus. The mehod used o denfy he producon funcon nvolves he soluon o a mahemacal programmng problem for each of he frms n he sample. If here are n frms whch produce a sngle (dencal) oupu y wh wo npus (l, k), he mahemacal programmng problem for frm s max, l, k, y - l l k k [9] s.. y - ll kk l k for 1 1,..., n 4 See also Seford and Thrall (1990) for a revew. 16

17 where,, and are parameers. The soluon o hs problem defnes a 'hyperplane' (n hs case hs s smply a plane n y,l,k-space) whch can be nuvely hough of as he closes froner o he oupu/npu combnaons of frm j. Noe ha hs froner s defned usng he nformaon on all he oher frms n he sample (hence he n consrans n [9]). If frm j s locaed on hs hyperplane s defned as effcen. If he frm s no locaed on hs hyperplane s neffcen, and a measure of he level of neffcency can be derved from he dsance he acual oupu/npu vecor s from he froner. A smlar hyperplane can be defned for each of he frms n he sample and hese can hen be regarded as formng he varous faces of he overall producon froner. Some frms may share he same 'hyperplane. In Coell s (1995, p.231) words, he overall producon froner can be vsualsed as a number of nersecng planes formng a gh fng cover over a scaer plo of pons n hree-dmensonal space. The soluon o he n programmng problems s normally pursued by usng he dual represenaon of he above problems (see Al and Seford, 1993, p.124, or Coell, 1995, p.231). Equaon [9] has effecvely defned he producon process as a lnear combnaon of npus/oupus. The presence of n he maxmsaon equaon means ha each hyperplane does no have o pass hrough he orgn (.e. here s no requremen ha zero npus produce zero oupus). Alhough hs may seem odd, allows a more flexble overall producon froner, whch s normally called he 'varable reurns o scale' case. If s se o equal zero, all hyperplanes are forced o pass hrough he orgn, whch s known as he consan reurns o scale case. Noe also ha he hyperplanes need no be lnear combnaons of oupus and npus y, k and l, snce hey could be defned n logarhms, yeldng a Cobb-Douglas based froner. The DEA approach, accordng o Al and Seford (1993), has been used exensvely n sudes on effcency (hey reference a leraure revew n 1990 wh 400 DEA references) and s, herefore, a well esablshed mehod. Poenal drawbacks of he approach nclude he fac ha daa pons are aken as represenng "rue" values, wh no allowance for measuremen error and oher nose. Also, he process of creang a producon froner from a se of nersecng hyperplanes can somemes cause 17

18 oddes n measures of effcency (essenally neffcency s measured as he dsance o he neares plane, even f hs plane s parallel o an axs whch mples one npu could be reduced whou loss of oupu 5 ). Anoher poenal drawback s dscussed by Sengupa (1998) who shows ha DEA analyss can be msleadng f does no correcly ncorporae capal goods npus. 5.2 Sochasc producon froner approach The DEA mehod oulned above akes each daa pon as an accurae descrpon of he producon process. Ths may no be he case snce errors can occur n he measuremen of oupus or npus. The lkelhood of such errors has led o he sochasc producon froner (SPF) approach, whch s a mehod ha allows for such 'nose' n daa pons. Agan, he SPF approach s concerned wh he consrucon of a producon froner on he bass of daa on a sample of frms. For example, suppose ha frm s producon can be represened by [10] y Ζ f ( x ; ϒ ) v u Ζ1,..., n where refers o one of n frms. The v and u erms are sochasc. For example, v could be defned as havng a normal dsrbuon, whle u has a posve half normal dsrbuon. The reasonng s ha v capures possble measuremen errors, ec whle u represens he dsance observaon s from he froner, hence u eners as a negave n [10]. Equaon [10] can be solved by eher maxmum lkelhood or COLS (correced OLS) o yeld he coeffcens. 6 Furhermore, usng he avalable values for he resduals (v + u ) s also possble o calculae he mean or mode of he dsrbuon of u (see Bauer, 1990, p.42-3). An mporan ssue n he SPF approach s he choce of dsrbuon for u. Obvously, he dsrbuon needs o be one-sded, bu hey are a range of possble forms ncludng he exponenal, runcaed normal and wo-parameer gamma (see Greene, 1990). 5 Coell (1995, p.233) llusraes hs pon. He repors on some mehods ha ry o aver hs problem. 6 Coell (1995, p.225) saes ha he maxmum lkelhood mehod s o be preferred (followng a Mone Carlo expermen on he dfference beween COLS and ML). 18

19 Followng from hs s he fac ha he effcency measures from he SPF may be sensve o he dsrbuon used. The SPF approach may be augmened n varous famlar ways. Frs, he funconal form used for he producon funcon can be Cobb-Douglas, ranslog or oher form. Second, he cos dual represenaon can also be used f here s daa on npu prces. There are also oher, less famlar, modfcaons, for example, Kalrajan and Obwona (1994a, 1994b) modfy [10] so ha he coeffcens are random. As hey noe, equaon [10] assumes ha all frms share he same coeffcens, wh movemen n he producon froner beng a neural shf from he curren poson. One mehod of relaxng hs assumpon s he use of a random coeffcens model. The SPF approach s able o deal wh he fac ha measures of oupus and npus have an elemen of randomness, however, hs comes a he cos of mposng a funconal form for he producon funcon and assumng a dsrbuon for neffcency. Harrs (1992, p.203) noes ha use of dfferen funconal forms n SPF analyss can lead o dfferen effcency measures. In conras, he DEA mehod does no have hs drawback, n Bauer s words (1990, p.29), "The Chef advanage of he mahemacal programmng or DEA approach s ha no explc funconal form s mposed on he daa." 5.3 Panel daa mehods The esmang equaon shown by [10] can also be esmaed f panel daa s avalable. Gven T perods of daa he equaon s now [11] y Ζ f x ; ϒ ) v u Ζ 1,..., n,.., T., (,,, Ζ 1 Esmaon of [11] s only dfferen from he cross-seconal esmaon [10] n ha has greaer degrees of freedom (alhough noe he mplc assumpon ha he coeffcens,, are consan over me somehng ha can be esed for (see Balag, 19

20 1995, Chaper 4). 7 Dfferen esmaon mehods can be used f he effcency erm, u, s consdered consan over me (.e. u ). Ths hen allows he use of random or fxed effecs models (see Greene, 1993, or Balag, 1995, for basc explanaons of such esmaors). The assumpon ha he neffcency erms (u s) are consan overme s a major assumpon, whch may be vald only for a small number of years. Kumbhakar (1990) and Baese and Coell (1992) sugges mehods of allowng for sysemac varaon n he u 's overme, bu hs agan mgh be consrued as mposng major assumpons on effcency adjusmen. Balag, Grffn and Rch (1995) esmae a ranslog cos funcon whch conans boh an ndusry wde echncal change erm and ndvdual frm effcency erms. The former reflecs he overall shf n he producon funcon and s allowed o be a funcon no only of me, bu also oupu and prces. Use of he ndusry echncal change and he resdual from he regresson allows an ndex of frm specfc echncal change o be calculaed. They compare hs wh a radonal TFP measure (calculaed for he same daa se of 24 arlnes over 1971 o 1986) and, alhough he overall correlaon coeffcen beween he percenage changes of each s 0.61, hey fnd subsanal dfferences beween he measures. 6 Ausralan sudes Ths secon dscusses a number of Ausralan based sudes o llusrae he use of he echnques menoned above. Table 1 provdes a summary of some recen frm-level or specfc ndusry producvy sudes (for a more complee revew of producvy sudes see Dawkns and Rogers, 1998). 7 A panel daa se may also allow he calculaon of frm level neffcency measures, even when usng he SPF approach, see Baese and Coell (1988) for a dscusson and an example on hree years of dary farm daa n Vcora and New Souh Wales. 20

21 Table 1 Examples of Ausralan frm-level producvy sudes Auhor, Year Tle Deals Esho and Sharpe, 1996 Rzman, 1995 Harrs, 1992 Tasman Pacfc Bureau of Indusry Economcs, X-Effcency n Ausralan Permanen Buldng Socees, Producvy n Ausralan Bankng: Two Alernave Approaches Techncal Effcency n Ausrala: Phase I The Scope for Producvy Improvemen n Ausrala's Open Cu Black Coal Indusry, 1997 Inernaonal Performance Indcaors: Elecrcy Updae SPF approach (wh quadrac me rend for effcency) Tornqvs ndexes (TFP) and economerc analyss of cos funcons SPF approach usng manufacurng census (1977) daa a 4 dg level Mul-laeral producvy ndex (TFP) and paral producvy measures MFP and DEA analyss Lawrence e al, 1991 The Comparave Effcency of Sae Elecrcy Auhores Mul-laeral ndex (TFP) producvy Noe: he Table summarses some frm-level sudes ha have used smlar echnques o hose dscussed n hs paper. Sudes based on survey daa responses are no ncluded, alhough hey are brefly dscussed below. Lawrence, Swan and Zesch (1991) look a he producvy of Ausralan sae elecrcy supplers. 9 They employ a growh accounng approach based on he ranslog mullaeral npu ndex (see Caves, Chrsensen and Dewer, 1982). Ths ndex s preferable o he Tornqvs when he daa s cross-seconal or panel n naure. 10 Ths preference s movaed by he fac ha he ndex s ransve (.e. ha he comparson of producvy beween wo uns should be he same wheher 8 There are a number of BIE sudes n he Inernaonal Performance Indcaors projec, ncludng gas, elecommuncaons and coasal shppng. These repors use varous mehods of analysng producvy bu are no lsed for reasons of space. There are also sudes by he Indusry Commsson and relaed bodes whch nclude frm-level producvy analyss (e.g. Indusry Commsson, 1995). 9 In fac, he auhors use he erm effcency, bu accordng o our defnon above, we can nerpre her paper as beng concerned wh producvy. 10 Caves e al (1982, p.84) sugges may no be preferable for purely me seres sudes. 21

22 compared drecly or hrough a hrd observaon). The parcular form of he ndex used by Lawrence e al (1991) compares he TFP beween elecrcy provders m and n accordng o, [12] log( TFP m / TFP ) n j ( S ( R jm m Η R Η S * j * Y )log Y X )log X m * jm * j Η j ( R ( S n jn Η R Η S * * j Y )log Y n * X )log X jn * j where he * ndcaes he average over all observaons, R s revenue share, S s cos share, refers o oupus and j o npus. In erms of measurng he capal npu, Lawrence e al use a seres for he user cos of capal servce, whch akes accoun of "nvesmen sreams, asse lves, neres durng consrucon and capacy commssoned each year" (Lawrence e al, 1991, p.186). The sudy fnds que large dfferences n TFP across saes, alhough he dfferences have narrowed overme. Esho and Sharpe (1996) use a panel daa approach o nvesgae he effcences of Ausralan permanen buldng socees over he perod They use a varan of he fxed effecs panel daa model, whch allows frm effcences o be a quadrac funcon of me. Usng a ranslog cos funcon, Esho and Sharpe consder hree oupu measures (housng loans, curren asses and average deposors' balances) and wo npus prces (cos of funds and a wage ndex). They fnd an average level of frm neffcency of 25% usng he economerc approach. Ths hey compare o an accounng based approach (usng raos of operang expense o oal asses, and ncome o operang expenses), fndng ha f such raos are averaged over seven or more years hey are "remarkably smlar" (Esho and Sharpe, 1996, p.254) o he resuls of he economerc analyss. The Bureau of Indusry Economcs (1995) underook a sudy on he elecommuncaons ndusry ha ulsed varous producvy echnques. Varous paral producvy measures were calculaed (e.g. revenue per elephone lne, revenue per employee), bu he repor also recognsed he problems of measurng oupu (e.g. ssues concernng he qualy of servce). An oupu ndex comprsng of 22

23 he oal number of calls and he oal number of lnes was used, wh he weghs of 30:70 used respecvely (he weghs are based on he assessmen ha around 70% of employees were used n mananng neworks, and 30% were used n 'raffc' relaed funcons). To calculae an annual user cos of capal (.e. he value of capal servces) he repor used an esmae gven by [13] VC ( g d ϑ P) K, where, g s opporuny cos of holdng capal (aken as he 10 year bond rae), d s he declnng balance deprecaon rae (equal o 13% assumng asse lfe of 13 years), dp/d s he annual rae of change n capal prces, and K s he (esmaed) sock of capal. There are also sudes based on survey daa whch rely on he answers o quesons concernng producvy levels and change. For example, he Ausralan Workplace and Indusral Relaons Surveys (AWIRS) ask he general manager of a workplace how s producvy compares wh wo years ago and also how producvy compares wh s major compeors (n boh cases usng a fve pon scale). Alhough such a subjecve rankng of producvy may have s lmaons, should be noed ha all he varous quanave mehods dscussed above also have lmaons. Many of he sudes based on he AWIRS daa have concerned ndusral relaons and producvy (see Crocke e al, 1990, and Drago and Wooden, 1992), alhough some sudes consder oher aspecs (see Blanchflower and Machn, 1996, on produc marke compeon and producvy). These sudes end o use he responses o he producvy quesons as dependen varables n ordered prob regressons wh a hos of explanaory varables descrbng workplace and marke characerscs. 7 Concluson Ths paper has consdered a varey of mehods for analysng he level and growh of producvy. The nal secons dscussed how producvy measures can be derved from a producon funcon. Varous assumpons have o be made o allow emprcal applcaon, ncludng he assumpons of prof maxmsaon (cos mnmsaon) and 23

24 consan reurns o scale. Secon 2.2 also showed how he heorecal mehods are smlar o mehods based on ndex number heory. In parcular, he Tornqvs or ranslog mullaeral ndex are he mos common emprcal mehods of creang an ndex of MFP. Secon 3 dscussed he problems of measurng varous oupus and npus. These problems should no be underesmaed, as n any parcular sudy measuremen problems are lkely o rase he possbly of subsanal bas n producvy esmaes. As explaned n secon 4, hs paper vews effcency changes as beng one elemen of producvy changes, hence he paper also revews he leraure on effcency analyss. Ths analyss ncludes he echnques of daa envelopmen analyss (DEA), sochasc producon froner (SPF) and panel daa mehods. Each of hese mehods has been he opc of exensve research and has ceran benefs and drawbacks. The choce of echnque may also be deermned by he naure of he daa se. A few sudes compare some of he resuls of usng more han one echnque on he same daa se. These show ha he resuls from dfferen echnques can be sgnfcanly dfferen. Ths agan suggess ha producvy sudes are complex, wh he possbly ha varous measuremen problems and echncal assumpons wll nfluence he resuls. 24

25 Bblography Al, A. I. and L. M. Seford "The Mahemacal Programmng Approach o EffcencyAnalyss".nH.-O.Fred,C.A.K.LovellandS.-S.e.Schmd(ed.) The measuremen of producve effcency: Technques and applcaons. New York; Oxford; Torono and Melbourne, Oxford Unversy Press.: Aspen, C "Esmaes of Mulfacor Producvy, Ausrala", ABS Occassonal Paper , AGPS, Canberra. Baly, M. N. and R. J. Gordon "The Producvy Slowdown, Measuremen Issues, and he Exploson of Compuer Power", Brookngs Papers on Economc Acvy 0(2): Balag, B Economerc Analyss of Panel Daa. NewYork,JohnWley. Balag, B. H., J. M. Grffn, e al "The Measuremen of Frm-Specfc Indexes of Techncal Change", Revew of Economcs and Sascs 77(4): Baese, G. and Coell, T. (1992). "Froner Producon Funcons, Techncal Effcency and Panel Daa: Wh Applcaon o Paddy Farmers n Inda", Journal of Producvy Analyss 3, Bauer, P. W "Recen Developmens n he Economerc Esmaon of Froners", Journal of Economercs 46(1-2): Blanchflower, D. and S. Machn "Produc Marke Compeon, Wages and Producvy: Inernaonal Evdence from Esablshmen-level Daa", Cenre for Economc Performance, Dscusson Paper 286. Bureau of Indusry Economcs, "Inernaonal Performance Indcaors Elecrcy Updae", Research Repor 54, AGPS, Canberra Bureau of Indusry Economcs "Inernaonal Performance Indcaors Telecommuncaons", Research Repor 65, AGPS, Canberra. Caves, D. W., L. R. Chrsensen, e al "Mullaeral Comparsons of Oupu, Inpu, and Producvy Usng Superlave Index Numbers", Economc Journal 92(365): Charnes, A., W. Cooper, e al "Measurng he Effcency of Decson Makng Uns", European Journal of Operaons Research 2: Coell, T. J "Recen Developmens n Froner Modellng and Effcency Measuremen", Ausralan Journal of Agrculural Economcs 39(3): Crocke, G, Dawkns, P, Mller, P and Mulvey,C. 1992, "The Impac of Unons on Workplace Producvy n Ausrala", Ausralan Bullen of Labour, 18, 2,

26 Dawkns, P. and Rogers, M A General Revew of Producvy Analyses n Ausrala, paper presened a Indusry Commsson/ANU Conference on Producvy and Mcroeconomc Reform, Canberra, February Dewer, W. E "Exac and Superlave Index Numbers", Journal of Economercs 4(2): Dewer, W. E "The Measuremen of Producvy", Bullen of Economc Research 44(3): Dewer, W. E. and A. M. Smh "Producvy Measuremen for a Dsrbuon Frm", Naonal Bureau of Economc Research Workng Paper 4812: 14. Drago, R. and Wooden, M., 1992, "The Ausralan Indusral Relaons Survey and Workplace Performance", Ausralan Bullen of Labour, 18, 2, Esho, N. and I. Sharpe "X-Effcency of Ausralan Permanen Buldng Socees, ", The Economc Record 72(218): Fagerberg, J "Technology and Inernaonal Dfferences n Growh Raes", Journal of Economc Leraure, XXXII, Farrell, M "The Measuremen of Producve Effcency", Journal of he Royal Sascal Socey A CXX(3): Gandal, N "Hedonc Prce Indexes for Spreadshees and an Emprcal Tes for Nework Exernales", Rand Journal of Economcs 25(1): Greene, W. H "A Gamma-dsrbued Sochasc Froner Model", Journal of Economercs 46: Greene, W. H Economerc Analyss. New York, MacMllan. Grlches, Z "Producvy, R&D, and he Daa Consran", Amercan Economc Revew 84(1): Grosskopf, S "Effcency and Producvy". n H.-O. Fred, C. A. K. Lovell and S.-S. e. Schmd (ed.) The measuremen of producve effcency: Technques and applcaons. New York; Oxford; Torono and Melbourne, Oxford Unversy Press: Hall, R. E "Marke Srucure and Macroeconomc Flucuaons", Brookngs Papers on Economc Acvy 0(2): Harrs, C. M "Techncal Effcency n Ausrala: Phase 1". n R.-E. Caves (ed.) Indusral effcency n sx naons.. In assocaon wh Sheryl D. Baley e al. Cambrdge and London, MIT Press.: Indusry Commsson Governmen Tradng Enerprses Performance Indcaors o , AGPS, Canberra 26

27 Jorgenson, D. and Z. Grlches "The Explanaon of Producvy Change", Revew of Economc Sudes 34: Kalrajan, K. P. and M. B. Obwona 1994a. "Froner Producon Funcon: The Sochasc Coeffcens Approach", Oxford Bullen of Economcs and Sascs 56(1): Kalrajan, K. and M. Obwona 1994b. "A Measuremen of Frm- and Inpu-specfc Techncal and Allocave Effcences", Appled Economcs 26: Koopmans, T "An Analyss of Producon as an Effcen Combnaon of Acves". n T. Koopmans (ed.) Acvy Analyss of Producon and Allocaon. New York, Wley. Kumbhakar, S "Producon Froners, Panel Daa, and Tme-Varyng Techncal Effcency", Journal of Economercs 46: Lawrence, D., S. Peer, e al "The Comparave Effcency of Sae Elecrcy Auhores". n Johnson e. al. (ed.) Conemporary Issues n Ausralan Economcs. Melbourne, MacMllan. Lowe, P "Labour Producvy Growh and Relave Wages: ", Producvy and Growh RBA Conference. Morrson, C. J A mcroeconomc approach o he measuremen of economc performance: Producvy growh, capacy ulzaon, and relaed performance ndcaors. 3. New York; Hedelberg; London and Tokyo, Sprnger. Tasman Pacfc "The scope for producvy mprovemen n Ausrala's open cu black coal ndusry", Repor Ocober(Tasman Asa Pacfc Ld, Canberra). Rzman, D "Producvy n Ausralan Bankng: Two Alernave Approaches", Melbourne Unversy, Economcs, Occasonal Paper 14. Seford, L. M. and R. M. Thrall "Recen Developmens n DEA: The Mahemacal Programmng Approach o Froner Analyss", Journal of Economercs 46(1-2): Sengupa, J "Tesng for Techncal Change: a Nonparamerc Applcaon", Appled Economcs Leers 5: