technological change, technological catch-up, and capital ... · technological change,...

Technological Change Technological Catch-up and Capital

Deepening Relative Contributions to Growth and Convergence

By SUBODH KUMAR AND R ROBERT RUSSELL

We decompose labor-productivity growth into components attributable to (1) tech-nological change (shifts in the world production frontier) (2) technologicalcatch-up (movements toward or away from the frontier) and (3) capital accumu-lation (movement along the frontier) The world production frontier is constructedusing deterministic methods requiring no speci cation of functional form for thetechnology nor any assumption about market structure or the absence of marketimperfections We analyze the evolution of the cross-country distribution of laborproductivity in terms of the tripartite decomposition nding that technologicalchange is decidedly nonneutral and that both growth and bipolar internationaldivergence are driven primarily by capital deepening (JEL O30 O47 D24)

Renewed recognition of the enormous poten-tial welfare gains from faster economic growthhas led to a resurgence of interest in the theoryof the growth process and in the empirical pat-terns of growth Unlike the studies of the1960rsquos however the recent empirical literaturetakes a distinctly international perspective fo-cusing especially on the question of whetherthere is a tendency for growth ratesmdashor moreimportantly growth pathsmdashof the worldrsquoseconomies to converge narrowing the gap be-tween the poor and the rich

While research over the last 10 to 15 yearshas considerably improved our understandingof these issues it has also generated a certainamount of controversy Exogenous growth the-

ory building on the pioneering model of RobertW Solow (1956) points to technologicalprogress as the source of persistent growthwhile endogenous growth theory building onthe more recent research of Paul M Romer(1986) and Robert E Lucas Jr (1988) empha-sizes a broad measure of physical and humancapital as the principal engine of growth At thesame time exogenous growth theory focuses oncapital accumulation as the source of (condi-tional) convergence while endogenous growththeory emphasizes differences in technologyacross countries and over time as the source ofthe presence or lack of convergence Beginningwith William J Baumol (1986) these hypotheseshave been subjected to extensiveempirical testing

Much of the theoretical and empirical litera-ture is summarized in Robert Barro and XavierSala-i-Martin (1995) Jonathan Temple (1999)and the 1996 Economic Journal symposium(Andrew B Bernard and Charles I Jones1996b Steven N Durlauf 1996 Oded Galor1996 Danny Quah 1996b and Sala-i-Martin1996) The concluding paragraph of the cogentsummary of these con icting views by Bernardand Jones (1996b p 1043) suggests the direc-tion of future research in this area

For the theoretical and empirical reasonsoutlined here we think that future workon convergence should focus much morecarefully on technology Why do coun-

Kumar Providian Financial 123 Mission Street SanFrancisco CA 94105 (e-mail Subodh_KumarProvidiancom) Russell Department of Economics University ofCalifornia Riverside CA 92521 (e-mail rcubedmailucredu) Frequent and extensive discussions with our UCRcolleagues Xu Chang Jang-Ting Guo Daniel Hendersonand Aman Ullah have been very helpful in our research onthis paper We have also bene ted from the comments andcritiques of two referees and of Paul Beaudry MarcelleChauvet Denise Doiron Steve Dowrick Prasanta Pattan-aik Dan Primont Bill Schworm Alan Woodland and otherparticipants in seminars at the Auckland University Aus-tralian National University Oregon State University Syd-ney University the University of British Columbia theUniversity of California-Riverside and Waikato UniversityEconomics Departments and at the 1998 Georgia Produc-tivity Conference

527

tries have different levels of technologyHow do technologies change over timeHow do we measure technologymdashis itsuf cient to simply consider a labor-augmenting technology factor or are otherdifferences in the production function im-portant How much of convergence thatwe observe is due to convergence in tech-nology versus convergence in capital-labor ratios

This paper addresses each of these questions(in varying degrees) In particular we decom-pose the growth of labor productivity a crudemeasure of welfare1 into three componentsthose attributable to (1) technological change(2) technological catch-up and (3) capital ac-cumulation The rst component re ects shiftsin the world production frontier determinedconceptually by the state-of-the-art potentiallytransferable technology the second re ectsmovements toward (or away from) the frontieras countries adopt ldquobest practicerdquo technologiesand reduce (or exacerbate) technical and alloca-tive inef ciencies and the third re ects move-ments along the frontier The world productionfrontier at each point in time is constructedusing deterministic nonparametric (mathemati-cal programming) methods (essentially ndingthe smallest convex cone enveloping the data)and ef ciency is measured as the (output-based)distance from the frontier These data-drivenmethods do not require speci cation of anyparticular functional form for the technologynor do they require any assumption about mar-ket structure or about the absence of marketimperfections indeed market imperfections aswell as technical inef ciencies are possible rea-sons for countries falling below the worldwideproduction frontier We calculate each of theabove three components of labor-productivitychanges for 57 countries over the 1965ndash1990period These methods serve one important ob-jective of this paper to develop a link alreadyestablished by Rolf Fare et al (1994) betweentwo voluminous literatures the macroeconomic

convergence literature alluded to above and the(deterministic) frontier production function lit-erature based on the pioneering work of Mi-chael J Farrell (1957) and Sydney Afriat (1972)and nicely exposited in Fare et al (1995)

Very much in the spirit of Quahrsquos (19931996b 1997) suggested approach (also adoptedby Galor [1996] and Jones [1997]) we analyzethe evolution of the entire distribution of thethree growth factors technological changetechnological catch-up and capital accumula-tion Quah has argued compellingly that analy-ses based on standard regression methodsfocusing on rst moments of the distributioncannot adequately address the convergence is-sue These arguments are buttressed by the em-pirical analyses of Quah and others posing arobust stylized fact about the internationalgrowth pattern that begs for explanation A plotof the distribution of output per worker across57 countries in 1965 and 1990 appears in Figure1 (The data and the kernel-based method ofsmoothing the distribution are described be-low) Over this 25-year period the distributionof labor productivity was transformed from aunimodal into a bimodal distribution with ahigher mean This transformation in turn meansthat while in 1965 there were many countries inthe middle income group in 1990 the world hadbecome divided as a stylized fact into twocategories the rich and the poor Quah(1996a b 1997) refers to this phenomenon asldquotwo-clubrdquo or ldquotwin-peakrdquo convergence aphenomenon that renders suspect analysesbased on the rst moment (or even higher mo-ments) of this distribution Our analysis isaimed at explaining this bipolarization of thedistribution of output per worker as well as itsgrowth pattern in terms of the tripartite decom-position described above As such it buildsupon Quahrsquos insights about the need to examinethe ldquodynamics of the entire cross-section distri-butionrdquo (Quah 1997 p 29) In addition weexploit recent developments in nonparametricmethods (Yanqin Fan and Aman Ullah 1999) totest formally for the statistical signi cance ofthe relative contributions of the three compo-nents of the decomposition of productivitychanges to changes in the distribution of laborproductivity

Although the analysis is quite simple ityields somewhat striking results (1) While

1 As pointed out by Jones (1997) labor productivitymight well be a better measure of welfare than measuredoutput per capita especially in countries with substantialnonmarket production activity since both the numeratorand the denominator of labor productivity correspond to themarket sector

528 THE AMERICAN ECONOMIC REVIEW JUNE 2002

there is substantial evidence of technologicalcatch-up (movements toward the productionfrontier) with the degree of catch-up directlyrelated to initial distance from the frontier thisfactor apparently has not contributed to conver-gence since the degree of catch-up appearsnot to be related to initial productivity (2)Technological change is decidedly nonneutralwith no improvementmdashindeed possibly someimplosionmdashat very low capitalndashlabor ratiosmodest expansion at relatively low capitalndashlaborratios and rapid expansion at high capitalndashlaborratios (3) Both growth and bipolar interna-tional divergence are driven primarily by capitaldeepening

We should emphasize at the outset howeverthat the analysis in the tradition of measure-ment and index number theory does not purportto provide fundamental reasons for the phenom-ena that are measured it is basically a growth-accounting exercise with a new twist The ndings are potentially consistent with differentmodels of economic growth and different fun-damental causes of the growth process Never-theless by developing a link between themacroeconomic convergence literature and the(deterministic) frontier production function lit-erature our approach has certain advantagesover the standard (regression-based) growth-

accounting literature It distinguishes betweentechnological catch-up (ef ciency improve-ments) and technological change (shifts in theproduction frontier) requires no speci cation offunctional form presumes no particular institu-tional or market structure and does not requireneutrality of technological change Our ap-proach also goes beyond the standard growth-accounting literature by attempting to accountfor the shift over time in the entire (ldquoworld-widerdquo) distribution of productivity Again itdoes not purport to provide fundamental rea-sons for the shifts Thus our approach to ac-counting for the evolution of the distribution ofproductivity is complementary to and consistentwith Quahrsquos (1997) analysis based on the con-ditioning of the productivity distribution onfundamental factors (speci cally geographicalproximity to rich countries and openness totrade)

Section I jointly constructs the world produc-tion frontier at the end points of our sampleperiod 1965 and 1990 and calculates and ana-lyzes ef ciency levelsmdashdistances from thefrontiermdashof individual economies using datafrom the Penn World Tables (Robert Summersand Alan Heston 1991) Section II constructsand analyzes the tripartite decomposition ofchanges in labor productivity over the sample

FIGURE 1 DISTRIBUTIONS OF OUTPUT PER WORKER 1965 AND 1990

529VOL 92 NO 3 KUMAR AND RUSSELL TECHNOLOGY AND CAPITAL DEEPENING

period Section III analyzes the shifts in theoverall distribution of productivity in termsof this tripartite decomposition Section IVconcludes

I Nonparametric Construction of Technologiesand Ef ciency Measurement

(Technological Catch-up)

A Data Envelopment Analysis

Our approach to constructing the worldwideproduction frontier and associated ef ciencylevels of individual economies (distances fromthe frontier) is nonparametric The basic idea isto envelop the data in the ldquosmallestrdquo or ldquotight-est ttingrdquo convex cone and the (upper) bound-ary of this set then represents the ldquobest practicerdquoproduction frontier This data-driven approachwhich is implemented with standard mathemat-ical programming algorithms requires no spec-i cation of functional form though it doesrequire an assumption about returns to scale ofthe technology (as well as free input and outputdisposability)2 In our simple case we deal withonly three macroeconomic variables aggregateoutput and two aggregate inputs labor and cap-ital Let Yt

j Ltj Kt

j t 5 1 T j 5 1 Jrepresent T observations on these three vari-ables for each of the J countries The constant-returns-to-scale reference technology (the ldquoFarrellconerdquo) for the world in period t is de ned by

(1) Tt 5 5 Y L K [ R13 Y O

j

zjY tj L

$Oj

zjL tj K $ O

j

zjKtj zj $ 0 j6

In this construction each observation is inter-preted as a unit operation of a linear process z j

represents the level of operation of that processand every point in the technology set is a linearcombination of observed outputinput vectors ora point dominated by a linear combination ofobserved points The constructed technology isa polyhedral cone and isoquants are piecewiselinear3

Of course typically some observed inputndashoutput combinations will be redundant in con-structing the technologies in that the observedoutput can be produced by some other process(generated by an alternative observed inputndashoutput vector) or by some linear combination ofother processes using less of one input and nomore of the other These dominated processesare technologically inef cient The Farrell(output-based) ef ciency index for country j attime t is de ned by

(2) E~Y tj L t

j K tj

5 min$lz Y tjl L t

j K tj [ Tt

This index is the inverse of the maximalproportional amount that output Y t

j can beexpanded while remaining technologically fea-sible given the technology Tt and the inputquantities Lt

j and Ktj it is less than or equal to 1

and takes the value of 1 if and only if the jtobservation is on the period t production fron-tier In this case of a scalar output the output-based ef ciency index is simply the ratio ofactual to potential output evaluated at the actualinput quantities but in multiple-output technol-ogies the index is a radial measure of the (pro-portional) distance of the actual output vectorfrom the production frontier

The Farrell ef ciency index can be calculatedby solving the following linear program foreach observation

2 As noted in the introduction a fully general expositionof this approach based on the early work of Farrell (1957)and Afriat (1972) can be found in Fare et al (1995) Thesereferences are aimed primarily at economists the manage-ment-science approach to essentially the same methods be-gan with the paper by Abraham Charnes et al (1978) whocoined the evocative term ldquodata envelopment analysisrdquo(DEA) and is comprehensively treated in Charnes et al(1994)

3 The nonincreasing-returns-to-scale (NIRS) technologyis constructed by restricting the process operation levels tosatisfy 0 z j 1 for all j so that observed processes canbe radially contracted but not expanded The variable-returns-to-scale (VRS) technology is constructed by addingthe restriction yenj z j 1 resulting in increasing returns toscale at low levels of inputs By construction ef ciencyindexes calculated under the assumption of constant returnsto scale are no higher than those calculated under theassumption of NIRS which in turn are no greater than thoseconstructed under the assumption of VRS (see Fare et al[1995] for details)


minl z1 zJ

l subject to

Y jl Ok

zkYtk

L j $ Ok

zkL tk

K j $ Ok

zkK tk

zk $ 0 k

The solution value of l in this problem is thevalue of the ef ciency index for country j attime t

The world production frontier in this con-struction and the associated ef ciency in-dexes should be interpreted quite broadly toencompass institutions and policies as wellas purely technological phenomena Thus acountry can fall short of the frontier becauseof for example inadequate nancial institu-tions or inapposite regulatory interventionAlso the frontier is de ned relative to theldquobest practicerdquo of those countries in the sam-ple and of course the ldquotruerdquo frontier could beabove the constructed frontier Neverthelessin our view this approach based on the con-struction of a global production frontier forthe world has advantages over the standardapproach of measuring productivity shortfalland catch-up relative to the US economywhich reduces the best-practice frontier to apoint and confounds the consequences of (rel-ative) undercapitalization on the one hand andinef cient utilization of factor supplies on theother hand Moreover even studies of tech-nological change based on total factor pro-ductivity while taking account of capitaldeepening require Hicks neutrality of tech-nological change in order to represent thestate of technology by a scalar as in theclassic study of technological change by Solow(1957)

B Data

We consider a sample of 57 countries overthe period 1965ndash1990 using data from the Penn

World Tables (version 56)4 This data set in-cludes developing countries and newly industri-alized countries (NICs) as well as the originalOECD countries The included countries areidenti ed in Table 1 Our measure of aggregateoutput is real gross domestic product (RGDPCHmultiplied by POP in the Penn Tables) Ouraggregate inputs capital stock and employmentare retrieved from capital stock per worker andreal GDP per worker (KAPW and RGDPW)Real GDP and the capital stock are measured in1985 international prices

C Technological Catch-up

Table 1 lists the ef ciency levels of each of the57 countries for the beginning and end years ofour sample 1965 and 19905 Note that not onlythe United States but also Sierra Leone andParaguay have ef ciency scores of 10 in bothyears so that all three countries are on thefrontier in both years Also on the frontier in1965 is Argentina while Hong Kong and Lux-embourg have ef ciency scores of 10 in 19906

It might seem peculiar that Paraguay andespecially Sierra Leone are on the frontier inboth years The literal interpretation of this nd-ing is that Sierra Leone one of the poorestcountries in our sample is poor because it is soterribly undercapitalized not because it makesinef cient use of the meager capital inputs thatit has Another (perhaps more plausible) inter-pretation is that the DEA method of construct-ing the best-practice frontiermdasha lower bound onthe frontier under the assumption of constant

4 These are the countries for which complete data setsare available though as is common in the convergenceliterature we report our results with two major oil-producingcountries Iran and Venezuela excluded Including thesetwo countries has no signi cant effect on the results thoughit is worth noting that when they are included both are onthe production frontier in 1965 All calculations in thispaper including these two countries are available (as Ap-pendix B) from the authors upon request

5 Our ef ciency calculations were carried out using thesoftware OnFront available from Economic Measurementand Quality i Lund AB (Box 2134 S-220 02 Lund Sweden[wwwemqse])

6 These results contrast with those of Fare et al (1994)who found that the United States was the only countrydetermining the frontier in each year The difference isattributable to the fact that they consider only OECDcountries


returnsmdashfails to identify the ldquotruerdquo but un-known frontier especially at low capitalndashlaborratios We discuss this issue again below but atthis point we note that the DEA method ispowerful enough to exclude more than 50 of the57 countries from the frontier in each of the twoyears 1965 and 1990 yet we cannot excludeSierra Leone from the frontier in either yearMoreover the levels of operation of moreadvanced countries cannot be scaled back (ra-dially) to a production point that dominates SierraLeone by producing its output with less of oneinput and no more of the other Finally otherpoorly capitalized countries (ie other observedpoints in capitalndashlabor space near Sierra Leone)like Ivory Coast Madagascar Malawi and SriLanka have very low ef ciency scores in 1965they appear to be overwhelmingly dominated interms of ef ciency by Sierra Leone7

It has often been argued that nonconvergenceor slow convergence in the level of output perworker is primarily caused by slow technolog-ical catch-up For example Quah (1997) sug-gests that it is the pattern of technologicaldiffusion that is responsible for the emergingbimodal distribution of productivity while NGregory Mankiw et al (1992) and Barro andSali-i-Martin (1995) highlight slow diffusion oftechnology as the reason for the slow (approx-imately 2 percent per year) speed of conver-gence In the context of our structure the stateof (worldwide) technology is represented by aproduction surface in outputinput space andtechnological catch-up is represented by move-ments toward the frontier re ected by increasesin ef ciency In Figure 2 the dotted line andsolid line respectively show the distributionsof the level of ef ciency across countries in1965 and 1990 obtained under the assumption

7 Having said this we should note that these mathemat-ical programming methods take no account of measurementerror sampling error and other stochastic phenomena Re-cent research (Leopold Simar 1996 Alois Kneip et al1998 Irene Gijbels 1999 Simar and Paul W Wilson 2000)has made substantial progress on the use of bootstrappingmethods to construct con dence intervals around ef ciencyindexes In this paper we are less concerned about thestatistical signi cance of inef ciency of individual countriesthan about the statistical signi cance of changes in thedistributions of ef ciency indexes and the components ofthe tripartite decomposition of productivity changes Withrespect to the latter we do employ bootstrappingmethods tocalculate signi cance levels for distribution shifts in SectionIII below

TABLE 1mdashEFFICIENCY INDEXES FOR 57 COUNTRIES1965 AND 1990

Country 1965 1990

Argentina 100 065Australia 076 082Austria 085 073Belgium 070 086Bolivia 050 041Canada 079 093Chile 085 065Colombia 041 045Denmark 076 070Dominican Republic 072 051Ecuador 038 036Finland 051 073France 079 083Germany West 069 078Greece 055 060Guatemala 081 073Honduras 045 041Hong Kong 045 100Iceland 096 087India 037 041Ireland 071 085Israel 060 084Italy 067 088Ivory Coast 066 047Jamaica 056 052Japan 060 061Kenya 026 029Korea South 043 061Luxembourg 076 100Madagascar 037 021Malawi 028 033Mauritius 094 097Mexico 085 074Morocco 074 086Netherlands 084 088New Zealand 084 071Nigeria 037 040Norway 061 078Panama 044 033Paraguay 100 100Peru 058 040Philippines 042 047Portugal 067 078Sierra Leone 100 100Spain 093 082Sri Lanka 032 033Sweden 081 076Switzerland 084 086Syria 042 065Taiwan 052 059Thailand 044 056Turkey 050 055United Kingdom 099 095United States 100 100Yugoslavia 069 059Zambia 042 029Zimbabwe 017 023Mean 0642 0658


of constant returns to scale The prominent shiftin the probability mass towards 10 between1965 and 1990 indicates a predominant move ofeconomies towards the frontier over time8

Moreover a generalized least-squares (GLS)regression of the change in ef ciency on thelevel of ef ciency in 1965 has a coef cient of2035 with a t-statistic of 2257 indicatingthat the less ef cient countries in 1965 haveon balance bene ted more from ef ciency

improvements than have the more ef cientcountries These two facts however do notnecessarily imply that there is a tendency fortechnology transfer to reduce the gap betweenthe rich and the poor since it is possible thatrelatively rich countries which can also fallshort of the frontier bene t from ef ciencyimprovements as much as or more than poorercountries We will see in the next section thatthis appears to be the case

II Tripartite Decomposition of the FactorsAffecting Labor Productivity

A Conceptual Decomposition

Our decomposition of the factors affectingproductivity growth exploits the assumption ofconstant returns to scale in which case thebenchmark technology sets can be drawn in ky space where k 5 KL and y 5 YL Verysimple hypothetical (polyhedral) technologiesfor two periods say a base period b and acurrent period c are drawn in Figure 3 In thissimple example the single kink in each of thepolyhedral technologies would indicate that asingle economymdashthe only ef cient economymdashde nes the frontier The two points kb yb

and kc yc represent observed values of the

8 Additional calculations indicate that this convergenceis robust with respect to the assumption about returns toscale That is Figure 2 is not perceptibly affected by recal-culation of the ef ciency indexes under the alternative as-sumptions of nonincreasing returns to scale or variablereturns to scale (de ned in footnote 3 above) Of course afew individual-countryef ciency indexes are quite sensitiveto the assumption about returns to scale As would beexpected the signi cant changes occur at very low levels ofproductivity Most notably under the NIRS assumptionIndia and Nigeria join Sierra Leone on the low-productivityend of the frontier in both 1965 and 1990 Ivory Coastbecomes part of the frontier in 1965 and Morocco andMexico (both a little higher up the productivity scale) moveto the frontier in 1990 Interestingly relaxation to VRS hasno appreciable effect on any index All of the ef ciencyindexes calculated under these alternative returns-to-scaleassumptions as well as the associated distributions compa-rable to Figure 2 are available (as Appendix C) from theauthors upon request

FIGURE 2 DISTRIBUTIONS OF EFFICIENCY INDEX 1965 AND 1990


two ratios in the two periods for some hypo-thetical economy By construction potentialoutputs for this economy in the two periodsare y b(kb) 5 ybeb and y c(kc) 5 ycec whereeb and ec are the values of the ef ciencyindexes in the two periods calculated as in(2) above Therefore

(3)yc

yb5

e c 3 y c ~kc

eb 3 b ~kb

Multiplying top and bottom by yb(kc) the po-tential outputndashlabor ratio at current-period cap-ital intensity using the base-period technologywe obtain

(4)yc

yb5

ec

eb3

y c ~kc

y b ~k c 3

y b ~k c

y b ~kb

This identity decomposes the relative change inthe outputndashlabor ratio in the two periods into (i)the change in ef ciencymdashie the change in thedistance from the frontier (the rst term on theright) (ii) the technology changemdashie the shiftin the frontier (the second term) and (iii) theeffect of the change in the capitalndashlabor ratiomdashie movement along the frontier (the thirdterm)

The decomposition in (4) measures techno-logical change by the shift in the frontier in

the output direction at the current-periodcapitalndashlabor ratiomdashfrom point C to point Din Figure 3mdashand it measures the effect ofcapital accumulation along the base-periodfrontiermdashfrom point B to point C We canalternatively measure technological changeat the base-period capitalndashlabor ratiomdashfrompoint B to point E in Figure 3mdashand capi-tal accumulation by movements along thecurrent-period frontiermdashfrom point E topoint Dmdash by multiplying the top and bottomof (3) by the potential outputndashlabor ratioat base-period capital intensity using thecurrent-period technology y c(kb) yieldingthe decomposition

(5)yc

yb5

ec

eb3

y c ~kb

y b ~k b 3

y c ~k c

y c ~kb

Thus the decomposition of (discrete) produc-tivity changes (not attributable to ef ciencychanges) into the technological-change andcapital-deepening components is path depen-dent and the choice between (4) and (5) isarbitrary There is no avoiding this arbitrarinessunless technological change is Hicks neutral inwhich case the proportional vertical shift in thefrontier is independent of the value of thecapitalndashlabor ratio It is this assumption (alongwith constant returns to scale) that enabledSolow (1957) and the legions of growth ac-countants who have followed his lead to unam-biguously decompose productivity growth intocomponents attributable to technological changeand capital deepening But without constrainingtechnological change to be Hicks neutral theproportional (vertical) shift in the frontier variesin unspeci ed ways In fact this arbitrariness isnot per se attributable to the decompositionitself It is endemic to the basic task of measur-ing technological change as is evident in thenecessity of normalizing on one of two technol-ogies in the (Malmquist) productivity indexproposed in the pioneering paper of Douglas WCaves et al (1982a)9

9 In fact it is a geometric average of two indexes oftechnological change normalized on different tech-nologies that is ldquoexactrdquo for a translog production tech-nology (with certain intertemporal restrictions on thetranslog second-order coef cients) For subsequent ldquoidealrdquo

FIGURE 3 ILLUSTRATION OF TRIPARTITE DECOMPOSITION


We have carried out the subsequent analysisfollowing both paths and while the results dif-fer signi cantly for many countries the overalldistribution of changes and hence our basicresults are not sensitive to the path chosen10

For the purpose of reporting our results wefollow the approach of Caves et al (1982a) andFare et al (1994b) by adopting the ldquoFisheridealrdquo decomposition based on geometric aver-ages of the two measures of the effects of tech-nological change and capital accumulationobtained by multiplying top and bottom of (3)by ( yb(kc) y c(kb))12

(6)yc

yb5

ec

ebX y c ~kc

y b ~kc 3

y c ~kb

y b ~kb D12

3 X y b ~kc

y b ~kb 3

y c ~kc

y c ~k b D12

5 EFF 3 TECH 3 KACCUM

B Empirical Results

We have carried out the above calculationsfor each ve-year interval in our sample butconcentrate here on an analysis of the changefrom the beginning to the end of our sampleperiod 1965ndash199011 Table 2 lists the per-centage changes from 1965 to 1990 in laborproductivity and each of the three compo-nents (i) change in ef ciency (ii) technolog-ical change and (iii) capital deepening for all57 countries along with the sample meanpercentage changes The overall averagesprovide striking evidence that most of theworldwide productivity improvement overthis period was attributable to capital accu-mulation with technological progress and ef- ciency changes (technological catch-up)

accounting for less than 15 percent and 10percent respectively of the growth12

A few observations about individual econo-mies are worth making Note rst that our cal-culations suggest that the four Asian ldquogrowthmiraclesrdquo with output per worker more thantripling in Hong Kong and Japan quadruplingin Taiwan and more than quintupling in SouthKorea over this 25-year span (Singapore is notin our data set) have substantially differentexplanations in terms of our tripartite decompo-sition the Japan South Korea and Taiwangrowth spurts were driven primarily by capitalaccumulation whereas that of Hong Kong re-sulted primarily from ef ciency improvementsalthough capital accumulation was important aswell13 On the other side of the ledger theArgentina stagnation (with a mere 5-percentincrease in productivity over this period) is pri-marily attributablemdashperhaps surprisinglymdashtoa collapse in ef ciency with a relatively slowrate of technological change also a factorwhereas growth attributable to capital accumu-lation in Argentina was above average Finally

productivity index number developments based on Caves etal (1982a) see Caves et al (1982b) W E Diewert (1992)Fare et al (1994) and Bert M Balk (1998)

10 Calculations carried out for each of these alternativepaths are available (as Appendix D) from the authors uponrequest

11 All calculations for the ve-year intervals are avail-able as Appendix E from the authors

12 The nding that capital deepening has been a majorcontributor to growth while consistent with some of thestandard regression-based growth-accounting studies [seeeg Temple (1998) for citations] is at odds with othersMost notably Robert E Hall and Jones (1999) using verydifferent growth-accounting methods nd that physicalcapital along with human capital (educational attainment)do not account for a large proportion of the difference inproductivity across countries Their (model-based) methoddiffers not only from ours but also from standard regression-based growth-accounting methods in that they decomposedifferences in output per worker into that attributable todifferences in the capitalndashoutput ratio educational attain-ment and differences in (Hicks-neutral) technology Themotivation for attributing to capital only the productivitychange generated by changes in the capitalndashoutput ratiorather than by changes in the capitalndashlabor ratio is thatalong a neoclassical steady-state growth path driven en-tirely by technological progress the capitalndashoutput ratiowould be constant while the capitalndashlabor ratio would riseso that standard growth-accounting methods would (errone-ously) attribute much of the growth of labor productivity tocapital deepening It would seem that a similar type ofdecomposition could be carried out using the nonparametricmethods of this paper Such a study would probably at-tribute less of the productivity change to capital and more totechnological change than does our approach But the re-sults would be model dependent

13 These results appear to be roughly consistent with theconclusions of the thorough analysis of these growth phe-nomena by Alwyn Young (1995)


TABLE 2mdashPERCENTAGE CHANGE OF TRIPARTITE DECOMPOSITION INDEXES 1965ndash1990

CountryOutput per

worker 1965Output per

worker 1990

Percentage changein output per

worker

Contribution to percentage change inoutput per worker of

Change inef ciency

Change intechnology

Capitaldeepening

Argentina 12818 13406 46 23548 179 5926Australia 21246 30312 427 820 1387 1580Austria 13682 26700 951 21460 1542 9798Belgium 17790 31730 784 2241 1268 2931Bolivia 4005 5315 327 21870 515 5524Canada 22245 34380 546 1667 1172 1858Chile 10169 11854 166 22387 192 5024Colombia 5989 10108 688 759 241 5317Denmark 17955 24971 391 2769 1284 3352Dominican Republic 4544 6898 518 22923 864 9744Ecuador 4993 9032 809 2362 2211 9173Finland 13938 27350 962 4307 1165 2285France 17027 30357 783 413 1633 4718Germany West 17282 29509 707 1328 1438 3178Greece 7721 17717 1295 958 308 10314Guatemala 5784 7435 285 21022 942 3085Honduras 3633 4464 229 2864 688 2584Hong Kong 6502 22827 2511 12000 239 5585Iceland 15010 24978 664 2957 208 8026India 1792 3235 805 1240 1565 3888Ireland 10322 24058 1331 1949 120 9275Israel 12776 23780 861 3950 234 3038Italy 14163 30797 1174 3186 1332 4552Ivory Coast 2674 3075 150 22911 2703 7449Jamaica 5336 5146 236 2829 622 2100Japan 7333 22624 2085 307 1519 15987Kenya 1377 1863 353 1437 2416 2472Korea Republic of 3055 16022 4245 4172 287 25973Luxembourg 21238 37903 785 3200 2440 868Madagascar 2220 1561 2297 24419 1786 690Malawi 846 1217 439 1733 24266 11380Mauritius 6496 10198 570 291 988 3883Mexico 11536 17012 475 21333 207 6671Morocco 4428 6770 529 1724 1657 1187Netherlands 20628 31242 515 531 1116 2938New Zealand 23658 25413 74 21560 927 1648Nigeria 1481 2082 406 800 21385 5110Norway 17233 29248 697 2636 3304 096Panama 6020 7999 329 22483 2086 7832Paraguay 3910 6383 632 000 21457 9108Peru 8162 6847 2161 23202 141 2168Philippines 3326 4784 438 1028 788 2090Portugal 6189 16637 1688 1550 480 12206Sierra Leone 2640 2487 258 000 25774 12292Spain 12451 26364 1117 21230 708 12547Sri Lanka 3337 5742 721 328 298 6178Sweden 20870 28389 360 2534 1263 2759Switzerland 23660 32812 387 259 2844 525Syria 7634 15871 1079 5490 019 3396Taiwan 4394 18409 3190 1479 960 23301Thailand 2292 6754 1947 2825 1261 10405Turkey 3765 8632 1293 994 661 9560United Kingdom 16645 26755 607 2381 137 6485United States 28051 36771 311 000 989 1929Yugoslavia 5320 10007 881 21529 660 10832Zambia 3116 2061 2339 22950 1613 21921Zimbabwe 2188 2437 114 3715 250 22077Mean 7506 523 614 5854


the disastrous 30-percent collapses of produc-tivity in two African countriesmdashMadagascarand Zambiamdashwere generated primarily by thedeterioration in ef ciency with slow or nega-tive capital accumulation also a factor

Figure 4 summarizes these calculations byplotting the four growth rates (labor productiv-ity and its three components) against output perworker in 1965 GLS regression lines are alsoplotted Figure 4(a) indicates that the increase inaverage productivity re ects positive growthover this period for almost all countries Theprominent spikes at the lower incomes re ectthe economic emergence of the Asian ldquomiraclerdquocountries and is consistent with our earlier ob-servation about the movement of probabilitymass from the lower-middle-income group tothe higher-income group in the cross-countryincome distribution The negative slope coef -cient of the regression line while not statisti-

cally signi cant without inclusion of criticalconditioning variables is essentially the empir-ical result that has led many to argue that world-wide productivity growth patterns supportconvergence an argument that has met withcogent criticism from Quah (1993 1996a b1997) and his suggestion that we need to studythe entire distribution of growth patterns to un-derstand these complex issues

Figure 4(b) showing the relationship betweenthe contribution of ef ciency to productivitygrowth and the initial level of productivityevinces no clear pattern with many negative aswell as positive changes The regression slopecoef cient is not statistically signi cant sug-gesting that technological catch-up illustratedby the substantial movement of probabilitymass toward full ef ciency in Figure 2 hasdone little if anything to lower income inequal-ity across countries Apparently technology

FIGURE 4 PERCENTAGE CHANGES BETWEEN 1965 AND 1990 IN OUTPUT PER WORKER AND THREE DECOMPOSITION INDEXESPLOTTED AGAINST 1965 OUTPUT PER WORKER

Note Each panel contains a GLS regression line


transfer has bene ted relatively rich countries asmuch as relatively poor countries

Figure 4(c) indicates that while technologi-cal change has contributed positively to growthfor most countries the pattern is very dissimilarto that of overall productivity growth withsome striking examples of technological regressfor low-income countries and larger-than-averagecontributions to growth for most high-incomecountries The positive regression slope coef -cient is statistically signi cant suggesting thatrelatively wealthy countries have bene ted morefrom technological progress than have less de-veloped countries

Figure 4(d) on the other hand suggests thatthe pattern of productivity growth attributableto capital accumulation is remarkably similar tothe overall pattern of changes in labor produc-tivity Indeed the regression slope coef cientsare almost identical in panels (a) and (d) Thusit appears that the growth pattern may have beendriven primarily by the pattern of capitalaccumulation

Figures 5 and 6 contain the empirically con-structed production frontiers in 1965 and 1990along with scatterplots of labor productivity andthe capitalndashlabor ratio Each kink is an actualobservation on these ratios for an (indicated)economy with a 10-ef ciency index for thatyear Figure 7 which superimposes these two

production frontiers provides strong evidencethat technological change over this period hasbeen decidedly nonneutral In particular Hicks-neutral technological change would shift thefrontier in k-y space vertically by the sameproportional amount at all capitalndashlabor ratios ashift that is inconsistent with the results in Fig-ure 714 Although owing to the scale there isnot much visual resolution in this gure at verylow capitalndashlabor ratios it is nevertheless evi-dent that there appears to be some substantialimplosion of the frontier at the very lowestcapitalndashlabor ratios (note the large negativetechnological-change indexes in Table 2 for verypoormdashand very poorly capitalizedmdashcountrieslike Malawi Nigeria and Sierra Leone) anoutward shift of about 20 percent in the frontierfor higher but still quite low capitalndashlabor ra-tios very little change in the frontier for themiddle of the distribution of capitalndashlabor ra-tios and a sizable 30-percent expansion of po-tential output at very high capitalndashlabor ratios(note the large technological-change indexes forNorway and Switzerland in Table 2)

14 The shift in the technologies is also inconsistent withHarrod-neutral (labor-augmenting) technological changewhich would shift the frontier radially (ie by equal pro-portional factors along rays from the origin)

FIGURE 5 1965 WORLD PRODUCTION FRONTIER


As highly capitalized countries also tend tobe countries with high per capita incomes thesubstantial outward shift in the frontier at highcapitalndashlabor ratios in Figure 7 means thattechnological change has primarily bene tedhigh-income countries an observation that is

consistent with Figure 4(c) above Perhaps thisis not surprising assuming that technologicalchange takes place primarily in highly capital-ized economies

The technological degradation apparent atvery low capitalndashlabor ratios should be taken


FIGURE 7 1965 AND 1990 WORLD PRODUCTION FRONTIERS


with a grain of salt For one thing it is notclear how the world frontier could implode atsome capitalndashlabor ratios Does knowledgedecay Were ldquoblueprintsrdquo lost It is perhapsmore likely that the ldquobest-practicerdquo frontierconstructed by the DEA technique is wellbelow the ldquotruerdquo but unobservable frontier atvery low capitalndashlabor ratios and thereforethat the apparent technological degradation atthese low levels of capitalization are in factef ciency declines

The nature of technological change between1965 and 1990 can be further explicated byreferring to ldquounit isoquantsrdquo for the two yearsdrawn in Figure 8 (Under the maintained as-sumption of constant returns isoquants in eachyear are simply radial expansions or contrac-tions of these ldquounitrdquo isoquants) The shift inthe unit isoquant re ects technological progressin those regions of capitalndashlabor space wherethe unit isoquant has shifted inward and tech-nological regress in those regions where theunit isoquant has shifted outward Again thelargest salutary shifts in the unit isoquant occurat very high capitalndashlabor ratios where the ra-dial contraction is on the order of 25 percentindicating that the 1990 technology could pro-duce a given output with about 25 percent less

capital and labor than could the 1965 tech-nology15 The implosion of the frontier at verylow capitalndashlabor ratios is re ected by the out-ward movement of the unit isoquant near thelabor axis indicating that producing the sameoutput in 1990 as in 1965 at these low levels ofcapitalization require very large proportionalincreases in the two factors of production (onthe order of 30 percent)

Technological change is labor saving (capitalusing) when the slope of the isoquant becomesless steep and capital saving (labor using) whenit becomes steeper Because of homogeneity ofthe technology these shifts are constant alongrays hence the regions of labor-saving andcapital-saving technological change are conesThe isoquant map in Figure 8 is partitioned into ve cones technological change is labor savingin the cones labeled A C and E and capitalsaving in those labeled B and D16

15 This construction is an informal rendition of the input-based measure of technology change see Fare et al (1995)for details

16 Formally the ldquosloperdquo at cusps of the polyhedral tech-nology must be de ned in terms of subdifferentials in anyevent in regions of labor-saving technological change thecapitalndashlabor ratio would be either unchanged or increased

FIGURE 8 1965 AND 1990 ISOQUANTS


III Analysis of Productivity Distributions

We now turn to an analysis of the distributiondynamics of labor productivity As emphasizedby Quah (1996a b 1997) this approach islikely to be more informative than summarymeasures like the conditional mean or varianceas is implicit in regression analysis First andsecond moments of a distribution are of courseespecially uninformative in the case of bimodaldistributions like that in Figure 1 In particularour objective is to assess the degree to whicheach of the three components of productivitychange account for the deformation of the dis-tribution of labor productivity from a unimodalto a bimodal distribution with a higher mean

between 1965 and 1990 illustrated in Figure 1and repeated here in Figure 9(a) The distribu-tions we employ are nonparametric kernel-baseddensity estimates essentially ldquosmoothedrdquo histo-grams of productivity levels17

Rewrite the tripartite decomposition of labor-productivity changes in equation (6) as follows

(7) yc 5 ~EFF 3 TECH 3 KACCUMyb

Thus the labor-productivity distribution in1990 can be constructed by successively multi-plying labor productivity in 1965 by each of thethree factors This in turn allows us to constructcounterfactual distributions by sequential intro-duction of each of these factors (where b 51965 and c 5 1990)

at given factor prices under the assumption of output-constrained cost minimization 17 See the Appendix for the particulars

FIGURE 9 COUNTERFACTUAL DISTRIBUTIONS OF OUTPUT PER WORKER

Notes In each panel the solid curve is the actual 1990 distribution and the solid vertical line represents the 1990 mean valueThe dotted curve in panel (a) is the actual 1965 distribution and the dotted vertical line represents the 1965 mean value Thedotted curves in panels (b) and (c) are counterfactual distributions isolating sequentially the effects of ef ciency change andtechnological change on the 1965 distribution and the dotted vertical lines represent the respective counterfactual means


The counterfactual 1990 labor-productivitydistribution of the variable

(8) yE 5 EFF 3 y b

isolates the effect on the distribution of changesin ef ciency only assuming a stationary worldproduction frontier and no capital deepeningthis distribution is illustrated in Figure 9(b) alongwith the (actual) 1990 distribution To facilitateinterpretation of this counterfactual distributionnote that if ef ciency did not change for anycountry the ldquocounterfactualrdquo distribution inFigure 9(b) would be identical to the actual1965 distribution in Figure 9(a) The mean ofthe counterfactual distribution of yE is indi-cated along with the actual 1965 and 1990means by a vertical line The small shift inmean labor productivity from 1965 re ects thecontribution of the gain in ef ciency to theworld average output per worker The gain inprobability mass at the upper-middle-incomelevel and a loss at the lower-middle-incomezone is consistent with conventional wisdomcon rmed in Figure 1 that many newly devel-oped countries have moved their economiessigni cantly towards the production frontierNevertheless the single-peaked character of thedistribution of per capita income is more or lessmaintained

The counterfactual distribution of the vari-able

(9) yET 5 ~EFF 3 TECHyb 5 TECH 3 yE

drawn in Figure 9(c) then isolates the effect oftechnological change as compared to the distri-bution in Figure 9(b) and the joint effect ofef ciency change and technological change ascompared to the distribution in Figure 9(a) Thesmall change in average output per worker fromthat in Figure 9(b) re ects the fact that techno-logical change has not contributed much to theincrease in worldwide per capita income Theshift of probability mass toward both tails of thedistribution indicates that technological changehas contributed to the welfare of relatively richcountries more than to that of poorer countriesThis result is of course consistent with the shiftin the production frontier during this period

The world distribution of income per workerremains unimodal after adjusting for ef ciencychanges and technologicalprogress (or regress)

For completeness the distribution in Figure9(d) re ects all three adjustments of the labor-productivity distribution and therefore is coin-cident with the actual distribution in 1990 Ascompared to Figure 9(c) this distribution iso-lates the effect of capital accumulation Com-parison of these two distributions includingtheir means provides strong evidence that cap-ital accumulation is the primary driving force inincreasing labor productivity and differencesacross countries in capital-accumulation histo-ries are primarily responsible for the emergenceof the bimodal structure of the distribution oflabor productivity

Of course the order in which the three ad-justments of labor productivity are introduced isarbitrary Nevertheless essentially the samestory emerges in every case In Figure 10 forexample we rst introduce the effects of tech-nological change then ef ciency change and nally capital accumulation again the bimodaldistribution emerges only at the last stage

Figure 11 which rst introduces the effect ofcapital accumulation provides perhaps the mostdramatic evidence that it is this factor that pri-marily accounts for bipolarization since thebimodal distribution emerges immediately inFigure 11(b) and persists with the other adjust-ments rst for technological change and thenfor ef ciency change

Figure 12 however provides some evidencethat ef ciency changes have also contributed tothe bipolarization of productivities The intro-duction of the effect of ef ciency changes inpanel (c) accentuates the bipolarization gener-ated by capital accumulation in panel (b) asmass is shifted from the middle of the distribu-tion to the upper end Technological changethen ameliorates this effect somewhat in panel(d)

We can exploit recent developments in non-parametric methods to test formally for the sta-tistical signi cance of differences betweendistributions in Figures 9ndash12mdashto test indi-rectly that is for the statistical signi cance ofthe relative contributions of the three compo-nents of the decomposition of productivitychanges to changes in the distribution of laborproductivity In particular Fan and Ullah


(1999) have proposed a nonparametric test forthe comparison of two unknown distributionssay f and gmdashthat is a test of the null hypothe-sis H0 f( x) 5 g( x) for all x against thealternative H1 f( x) THORN g( x) for some x18

Some test statistics along with critical val-ues for various signi cance levels are shownin Table 3 The rst row indicates that thebaseline distributions in panel (a) of eachof the diagramsmdashthe actual 1965 and 1990distributionsmdashare signi cantly different evenat the 1-percent signi cance level The null hy-pothesis of no difference between the actual1990 distribution and the counterfactual 1990distribution assuming changes in ef ciency only(with no technological change and no capital

deepening) is similarly rejected at the 1-percentlevel (row 2) At the 1-percent level it is notpossible to reject the equivalence of the 1990distribution and the counterfactual distributionassuming only technological change but thishypothesis is rejected at the 5-percent levelBecause of the small size of our sample theT-test is likely to have very low power hencea 1-percent signi cance level is likely to be fartoo stringent In row 4 we are unable to rejectthe hypothesis that the 1990 distribution and thecounterfactual distribution incorporating onlycapital deepening are the same even at the10-percent level Finally in the last two rows ofTable 3 we test for differences between the1990 distribution and the counterfactual distri-butions incorporating the effects of ef ciencychanges and capital accumulation (row 5) andof technological change and capital accumula-tion (row 6) it is not possible to reject the

18 See the Appendix for an exact description of the teststatistic


Notes In each panel the solid curve is the actual 1990 distribution and the solid vertical line represents the 1990 mean valueThe dotted curve in panel (a) is the actual 1965 distribution and the dotted vertical line represents the 1965 mean value Thedotted curves in panels (b) and (c) are counterfactual distributions isolating sequentially the effects of technological changeand ef ciency change on the 1965 distribution and the dotted vertical lines represent the respective counterfactual means


hypothesis of identical distributions in eithercase at the 10-percent level

Thus at both the 5-percent and the 10-percentsigni cance levels the gestalt conclusions drawnfrom examination of Figures 9ndash12mdashnamelythat capital deepening is the driving force be-hind the bipolar divergence of the productivitydistribution over our sample periodmdashare (ro-bustly) con rmed by these statistical tests

IV Concluding Remarks

Several caveats need to be emphasized Firstas noted at the outset the analysis in the tradi-tion of measurement and index number theorydoes not purport to provide reasons for thephenomena that are measured it is basically agrowth-accounting exercise with a new twistAs in the original growth-accounting paper bySolow (1957) we provide only proximatemdashasopposed to fundamentalmdashcontributions of the

three identi ed factors to growth and conver-gence the resultant allocations to the three fac-tors could be consistent with more than onemodel of economic growth On the otherhand our approach unlike standard growth-accounting exercises does not require neu-trality of technological change or speci cationof a functional form for the technology

Second we have focused in this paper on thethree macroeconomic variables commonly usedin empirical studies of convergence potentiallyimportant variables (eg human capital andnatural resources) are omitted Third as is wellknown the capital stock data in the Penn WorldTable are measured with considerable error andthis should be taken into account in assessingour results Fourth the level of aggregation ishighly macroeconomic and some recent con-vergence papers (see eg Bernard and Jones1996a) have suggested that industry-speci canalyses might be more appropriate for thestudy of convergence especially that attribut-


Notes In each panel the solid curve is the actual 1990 distribution and the solid vertical line represents the 1990 mean valueThe dotted curve in panel (a) is the actual 1965 distribution and the dotted vertical line represents the 1965 mean value Thedotted curves in panels (b) and (c) are counterfactual distributions isolating sequentially the effects of capital deepening andef ciency change on the 1965 distribution and the dotted vertical lines represent the respective counterfactual means


able to technology transfer Fifth our long-runanalysis has not taken short-run economic uc-tuations into account Finally our sample of 57countries is arbitrary determined by the crite-

rion that consistent data were available for all ofthem and not for others in the Penn WorldTables Each of these caveats suggests topicsfor additional research


Notes In each panel the solid curve is the actual 1990 distribution and the solid vertical line represents the 1990 mean valueThe dotted curve in panel (a) is the actual 1965 distribution and the dotted vertical line represents the 1965 mean value Thedotted curves in panels (b) and (c) are counterfactual distributions isolating sequentially the effects of capital deepening andtechnological change on the 1965 distribution and the dotted vertical lines represent the respective counterfactual means

TABLE 3mdashDISTRIBUTION HYPOTHESIS TESTS

Null hypothesis (H0)T-test

statistics

10-Percentsigni cance level

(critical value 128)





f(y90) 5 g(y65) 354 H0 rejected H0 rejected H0 rejectedf(y90) 5 gE(y65 3 EFF) 280 H0 rejected H0 rejected H0 rejectedf(y90) 5 gT(y65 3 TECH) 206 H0 rejected H0 rejected H0 not rejectedf(y90) 5 gK(y65 3 KACCUM) 077 H0 not rejected H0 not rejected H0 not rejectedf(y90) 5 gEK(y65 3 EFF 3

KACCUM) 068 H0 not rejected H0 not rejected H0 not rejectedf(y90) 5 gEK(y65 3 TECH 3

KACCUM) 2008 H0 not rejected H0 not rejected H0 not rejected

Notes The functions f and g are (kernel) distribution functions for the actual data in 1990 and 1965 respectively gE gT gK gEK and gTK are counterfactual distributions obtained by adjusting the 1965 data for the effects ofrespectively ef ciency changes technological change capital deepening both ef ciency changes and capital deepening andboth technological change and capital deepening


Nevertheless we believe that our simple ap-proach to decomposing labor productivity shiftsinto three factors and analyzing the shift in thedistribution of productivity as well as the shiftin the production frontier and distances of econ-omies from the frontier is quite responsive tothe questions posed by Bernard and Jones(1996b) and quoted in our introductory remarksIt provides a method of measuring separatelythe effects of technological catch-up technicalchange and capital accumulation on labor pro-ductivity without imposing any assumptionabout the functional form of the productionfunction or about allocative or technical ef -ciency as required by estimation methods usingstandard econometric techniques This methodsuggests the following answers to the BernardJones questions

(1) There is substantial evidence of technolog-ical catch-up as countries have on averagemoved toward the worldwide productionfrontier even as the frontier itself hasmoved outward at most capitalndashlabor ratiosbut this catch-up does not seem to havebeen a force for convergence as relativelyrich as well as relatively poor countrieshave bene ted from catch-up

(2) Technological change has been decidedlynonneutral apparently bene ting rich coun-tries more than the poor

(3) It is primarily capital deepening as opposedto technological change or technologicalcatch-up that has contributed the most toboth growth and bipolar international diver-gence of economies

APPENDIX

Each distribution in the paper is a kernel-based estimate of a density function f( x) of arandom variable x based on the standard nor-mal kernel function and optimal bandwidth

f~x 51

nh Oj 5 1

J

kX x j 2 x

h D

where 2 `` k(c)dc 5 1 and c 5 ( xj 2 x)h

In this construction h is the optimal windowwidth which is a function of the sample size nand goes to zero as n ` We assume that k

is a symmetric standard normal density func-tion with nonnegative images See AdrianPagan and Ullah (1999) for details

The statistic used to test for the differencebetween two distributions predicated on theintegrated-square-error metric on a space ofdensity functions I( f g) 5 x ( f( x) 2g( x))2dx is

T 5nh12I

sN~0 1

where

I 51

n2h Oi 5 1

n Oj 5 1j THORN i

n

kX x i 2 x j

h D 1 kX y i 2 y j

h D2 kX y i 2 x j

h D 2 kX x i 2 y i

h Dand

s2 51

n2hp12 Oi 5 1

n Oj 5 1

n

kX x i 2 x j

h D1 kX y i 2 y j

h D 1 2kX x i 2 y j

h D

Qi Li (1996) has established that this test sta-tistic is valid for dependent as well as indepen-dent variables As shown by Fan and Ullah(1999) the test statistic asymptotically goes tothe standard normal but the sample in our studycontains only 51 observations Thus we do abootstrap approximation with 500 replicationsto nd the critical values for the statistic at the5-percent and 1-percent levels of signi cance

REFERENCES

Afriat Sydney ldquoEf ciency Estimation of Pro-duction Functionsrdquo International EconomicReview October 1972 13(3) pp 568ndash98

Balk Bert M Industrial price quantity andproductivity indices The micro-economictheory and an application Boston MA Klu-wer 1998

Barro Robert J and Sala-i-Martin Xavier Eco-


nomic growth New York McGraw-Hill1995

Baumol William J ldquoProductivity Growth Con-vergence and Welfare What the Long-runData Showrdquo American Economic ReviewDecember 1986 76(5) pp 1072ndash85

Bernard Andrew B and Jones Charles I ldquoPro-ductivity across Industries and CountriesTime Series Theory and Evidencerdquo Reviewof Economics and Statistics February 1996a78(1) pp 135ndash46

ldquoTechnology and Convergencerdquo Eco-nomic Journal July 1996b 106(437) pp1037ndash44

Caves Douglas W Christensen Laurits R andDiewert W Erwin ldquoMultilateral Compari-sons of Outputs Inputs and Productivity Us-ing Superlative Index Numberrdquo EconomicJournal March 1982a 92(365) pp 73ndash86

ldquoThe Economic Theory of Index Num-bers and the Measurement of Input Outputand Productivityrdquo Econometrica November1982b 50(6) pp 1393ndash414

Charnes Abraham Cooper William W LewinArie Y and Seiford Lawrence M Data envel-opment analysis Theory methodology andapplication Boston MA Kluwer 1994

Charnes Abraham Cooper William W andRhodes E ldquoMeasuring the Ef ciency of De-cision-Making Unitsrdquo European Journal ofOperational Research 1978 3 pp 429ndash44

Diewert W E ldquoFisher Ideal Output Input andProductivity Indexes Revisitedrdquo Journal ofProductivity Analysis September 1992 3(3)pp 211ndash48

Durlauf Steven N ldquoOn Convergence and Diver-gence of Growth Rates An IntroductionrdquoEconomic Journal June 1996 106(6) pp1016ndash18

Fan Yanqin and Ullah Aman ldquoOn Goodness-of- t Tests for Weakly Dependent ProcessesUsing Kernel Methodrdquo Journal of Nonpara-metric Statistics 1999 11(1ndash3) pp 337ndash60

Fare Rolf Grosskopf Shawna and Lovell C AKnox Production frontiers Cambridge UKCambridge University Press 1995

Fare Rolf Grosskopf Shawna Norris M andZhang Z ldquoProductivity Growth TechnicalProgress and Ef ciency Change in Industri-alized Countriesrdquo American Economic Re-view March 1994 84(1) pp 66ndash83

Farrell Michael J ldquoThe Measurement of Pro-

ductive Ef ciencyrdquo Journal of the Royal Sta-tistical Society Series A General 1957120(3) pp 253ndash82

Galor Oded ldquoConvergence Inferences fromTheoretical Modelsrdquo Economic Journal July1996 106(437) pp 1056ndash69

Gijbels Irene Mammen Enno Park Byeong Uand Simar Leopold ldquoOn Estimation ofMonotone and Concave Frontier FunctionsrdquoJournal of the American Statistical Associa-tion March 1999 94(445) pp 220ndash28

Hall Robert E and Jones Charles I ldquoWhy DoSome Countries Produce So Much More Out-put Per Worker Than Othersrdquo QuarterlyJournal of Economics February 1999 114(1)pp 83ndash116

Jones Charles I ldquoOn the Evolution of the WorldIncome Distributionrdquo Journal of EconomicPerspectives Summer 1997 11(3) pp 19ndash36

Kneip Alois Park Byeong U and SimarLeopold ldquoA Note on the Convergence ofNonparametric DEA Estimators for Produc-tion Ef ciency Scoresrdquo Econometric TheoryDecember 1998 14(6) pp 783ndash93

Li Qi ldquoNonparametric Testing of Closenessbetween Two Unknown Distribution Func-tionsrdquo Econometric Reviews August 199615(3) pp 261ndash74

Lucas Robert E Jr ldquoOn the Mechanics of Eco-nomic Developmentrdquo Journal of MonetaryEconomics July 1988 22(1) pp 3ndash42

Mankiw N Gregory Romer David and WeilDavid N ldquoA Contribution to the Empirics ofEconomic Growthrdquo Quarterly Journal ofEconomics May 1992 107(2) pp 407ndash37

Pagan Adrian and Ullah Aman Nonparametriceconometics Cambridge UK CambridgeUniversity Press 1999

Quah Danny ldquoGaltonrsquos Fallacy and Tests ofthe Convergence Hypothesisrdquo ScandinavianJournal of Economics December 199395(4) pp 427ndash43

ldquoConvergence Empirics across Econ-omies with (Some) Capital Mobilityrdquo Jour-nal of Economic Growth March 1996a 1(1)pp 95ndash124

ldquoTwin Peaks Growth and Conver-gence in Models of Distribution DynamicsrdquoEconomic Journal July 1996b 106(437) pp1045ndash55

ldquoEmpirics for Growth and Distribu-tion Strati cation Polarization and Conver-


gence Clubsrdquo Journal of Economic GrowthMarch 1997 2(1) pp 27ndash59

Romer Paul M ldquoIncreasing Returns and Long-run Growthrdquo Journal of Political EconomyOctober 1986 94(5) pp 1002ndash37

Sala-i-Martin Xavier ldquoThe Classical Approachto Convergence Analysisrdquo Economic Jour-nal June 1996 106(6) pp 1019ndash36

Simar Leopold ldquoAspects of Statistical Analysisin DEA-Type Frontier Modelsrdquo Journal ofProductivity Analysis July 1996 7(2ndash3) pp177ndash85

Simar Leopold and Wilson Paul W ldquoA GeneralMethodology for Bootstrapping in Non-parametric Frontier Modelsrdquo Journal of Ap-plied Statistics August 2000 27(6) pp779ndash802

Solow Robert M ldquoA Contribution to the Theoryof Economic Growthrdquo Quarterly Journal ofEconomics February 1956 70(1) pp 65ndash94

ldquoTechnical Change and the AggregateProduction Functionrdquo Review of Economicsand Statistics August 1957 39(3) pp 312ndash20

Summers Robert and Heston Alan ldquoThe PennWorld Table (Mark 5) An Expanded Setof International Comparisons 1950ndash1988rdquoQuarterly Journal of Economics May 1991106(2) pp 327ndash68

Temple Jonathan ldquoEquipment Investment andthe Solow Modelrdquo Oxford Economic PapersJanuary 1998 50(1) pp 39ndash62

ldquoThe New Growth Evidencerdquo Journalof Economic Literature March 1999 37(1)pp 112ndash56

Young Alwyn ldquoThe Tyranny of Numbers Con-fronting the Statistical Realities of the EastAsian Growth Experiencerdquo Quarterly Jour-nal of Economics August 1995 110(3) pp641ndash80


tries have different levels of technologyHow do technologies change over timeHow do we measure technologymdashis itsuf cient to simply consider a labor-augmenting technology factor or are otherdifferences in the production function im-portant How much of convergence thatwe observe is due to convergence in tech-nology versus convergence in capital-labor ratios

This paper addresses each of these questions(in varying degrees) In particular we decom-pose the growth of labor productivity a crudemeasure of welfare1 into three componentsthose attributable to (1) technological change(2) technological catch-up and (3) capital ac-cumulation The rst component re ects shiftsin the world production frontier determinedconceptually by the state-of-the-art potentiallytransferable technology the second re ectsmovements toward (or away from) the frontieras countries adopt ldquobest practicerdquo technologiesand reduce (or exacerbate) technical and alloca-tive inef ciencies and the third re ects move-ments along the frontier The world productionfrontier at each point in time is constructedusing deterministic nonparametric (mathemati-cal programming) methods (essentially ndingthe smallest convex cone enveloping the data)and ef ciency is measured as the (output-based)distance from the frontier These data-drivenmethods do not require speci cation of anyparticular functional form for the technologynor do they require any assumption about mar-ket structure or about the absence of marketimperfections indeed market imperfections aswell as technical inef ciencies are possible rea-sons for countries falling below the worldwideproduction frontier We calculate each of theabove three components of labor-productivitychanges for 57 countries over the 1965ndash1990period These methods serve one important ob-jective of this paper to develop a link alreadyestablished by Rolf Fare et al (1994) betweentwo voluminous literatures the macroeconomic

convergence literature alluded to above and the(deterministic) frontier production function lit-erature based on the pioneering work of Mi-chael J Farrell (1957) and Sydney Afriat (1972)and nicely exposited in Fare et al (1995)

Very much in the spirit of Quahrsquos (19931996b 1997) suggested approach (also adoptedby Galor [1996] and Jones [1997]) we analyzethe evolution of the entire distribution of thethree growth factors technological changetechnological catch-up and capital accumula-tion Quah has argued compellingly that analy-ses based on standard regression methodsfocusing on rst moments of the distributioncannot adequately address the convergence is-sue These arguments are buttressed by the em-pirical analyses of Quah and others posing arobust stylized fact about the internationalgrowth pattern that begs for explanation A plotof the distribution of output per worker across57 countries in 1965 and 1990 appears in Figure1 (The data and the kernel-based method ofsmoothing the distribution are described be-low) Over this 25-year period the distributionof labor productivity was transformed from aunimodal into a bimodal distribution with ahigher mean This transformation in turn meansthat while in 1965 there were many countries inthe middle income group in 1990 the world hadbecome divided as a stylized fact into twocategories the rich and the poor Quah(1996a b 1997) refers to this phenomenon asldquotwo-clubrdquo or ldquotwin-peakrdquo convergence aphenomenon that renders suspect analysesbased on the rst moment (or even higher mo-ments) of this distribution Our analysis isaimed at explaining this bipolarization of thedistribution of output per worker as well as itsgrowth pattern in terms of the tripartite decom-position described above As such it buildsupon Quahrsquos insights about the need to examinethe ldquodynamics of the entire cross-section distri-butionrdquo (Quah 1997 p 29) In addition weexploit recent developments in nonparametricmethods (Yanqin Fan and Aman Ullah 1999) totest formally for the statistical signi cance ofthe relative contributions of the three compo-nents of the decomposition of productivitychanges to changes in the distribution of laborproductivity

Although the analysis is quite simple ityields somewhat striking results (1) While

1 As pointed out by Jones (1997) labor productivitymight well be a better measure of welfare than measuredoutput per capita especially in countries with substantialnonmarket production activity since both the numeratorand the denominator of labor productivity correspond to themarket sector













j Ltj Kt


(1) Tt 5 5 Y L K [ R13 Y O

j

zjY tj L

$Oj

zjL tj K $ O

j

zjKtj zj $ 0 j6




(2) E~Y tj L t

j K tj

5 min$lz Y tjl L t

j K tj [ Tt









minl z1 zJ

l subject to

Y jl Ok

zkYtk

L j $ Ok

zkL tk

K j $ Ok

zkK tk

zk $ 0 k



B Data














Country 1965 1990














(3)yc

yb5

e c 3 y c ~kc

eb 3 b ~kb


(4)yc

yb5

ec

eb3

y c ~kc

y b ~k c 3

y b ~k c

y b ~kb




(5)yc

yb5

ec

eb3

y c ~kb

y b ~k b 3

y c ~k c

y c ~kb







(6)yc

yb5

ec

ebX y c ~kc

y b ~kc 3

y c ~kb

y b ~kb D12

3 X y b ~kc

y b ~kb 3

y c ~kc

y c ~k b D12


B Empirical Results











CountryOutput per


worker 1990


worker


Change inef ciency

Change intechnology

Capitaldeepening











































(8) yE 5 EFF 3 y b


































statistics
















APPENDIX


f~x 51

nh Oj 5 1

J

kX x j 2 x

h D





T 5nh12I

sN~0 1

where

I 51

n2h Oi 5 1

n Oj 5 1j THORN i

n

kX x i 2 x j

h D 1 kX y i 2 y j

h D2 kX y i 2 x j

h D 2 kX x i 2 y i

h Dand

s2 51

n2hp12 Oi 5 1

n Oj 5 1

n

kX x i 2 x j

h D1 kX y i 2 y j

h D 1 2kX x i 2 y j

h D


REFERENCES

























































j Ltj Kt


(1) Tt 5 5 Y L K [ R13 Y O

j

zjY tj L

$Oj

zjL tj K $ O

j

zjKtj zj $ 0 j6




(2) E~Y tj L t

j K tj

5 min$lz Y tjl L t

j K tj [ Tt









minl z1 zJ

l subject to

Y jl Ok

zkYtk

L j $ Ok

zkL tk

K j $ Ok

zkK tk

zk $ 0 k



B Data














Country 1965 1990














(3)yc

yb5

e c 3 y c ~kc

eb 3 b ~kb


(4)yc

yb5

ec

eb3

y c ~kc

y b ~k c 3

y b ~k c

y b ~kb




(5)yc

yb5

ec

eb3

y c ~kb

y b ~k b 3

y c ~k c

y c ~kb







(6)yc

yb5

ec

ebX y c ~kc

y b ~kc 3

y c ~kb

y b ~kb D12

3 X y b ~kc

y b ~kb 3

y c ~kc

y c ~k b D12


B Empirical Results











CountryOutput per


worker 1990


worker


Change inef ciency

Change intechnology

Capitaldeepening











































(8) yE 5 EFF 3 y b


































statistics
















APPENDIX


f~x 51

nh Oj 5 1

J

kX x j 2 x

h D





T 5nh12I

sN~0 1

where

I 51

n2h Oi 5 1

n Oj 5 1j THORN i

n

kX x i 2 x j

h D 1 kX y i 2 y j

h D2 kX y i 2 x j

h D 2 kX x i 2 y i

h Dand

s2 51

n2hp12 Oi 5 1

n Oj 5 1

n

kX x i 2 x j

h D1 kX y i 2 y j

h D 1 2kX x i 2 y j

h D


REFERENCES














































minl z1 zJ

l subject to

Y jl Ok

zkYtk

L j $ Ok

zkL tk

K j $ Ok

zkK tk

zk $ 0 k



B Data














Country 1965 1990














(3)yc

yb5

e c 3 y c ~kc

eb 3 b ~kb


(4)yc

yb5

ec

eb3

y c ~kc

y b ~k c 3

y b ~k c

y b ~kb




(5)yc

yb5

ec

eb3

y c ~kb

y b ~k b 3

y c ~k c

y c ~kb







(6)yc

yb5

ec

ebX y c ~kc

y b ~kc 3

y c ~kb

y b ~kb D12

3 X y b ~kc

y b ~kb 3

y c ~kc

y c ~k b D12


B Empirical Results











CountryOutput per


worker 1990


worker


Change inef ciency

Change intechnology

Capitaldeepening











































(8) yE 5 EFF 3 y b


































statistics
















APPENDIX


f~x 51

nh Oj 5 1

J

kX x j 2 x

h D





T 5nh12I

sN~0 1

where

I 51

n2h Oi 5 1

n Oj 5 1j THORN i

n

kX x i 2 x j

h D 1 kX y i 2 y j

h D2 kX y i 2 x j

h D 2 kX x i 2 y i

h Dand

s2 51

n2hp12 Oi 5 1

n Oj 5 1

n

kX x i 2 x j

h D1 kX y i 2 y j

h D 1 2kX x i 2 y j

h D


REFERENCES


















































Country 1965 1990














(3)yc

yb5

e c 3 y c ~kc

eb 3 b ~kb


(4)yc

yb5

ec

eb3

y c ~kc

y b ~k c 3

y b ~k c

y b ~kb




(5)yc

yb5

ec

eb3

y c ~kb

y b ~k b 3

y c ~k c

y c ~kb







(6)yc

yb5

ec

ebX y c ~kc

y b ~kc 3

y c ~kb

y b ~kb D12

3 X y b ~kc

y b ~kb 3

y c ~kc

y c ~k b D12


B Empirical Results











CountryOutput per


worker 1990


worker


Change inef ciency

Change intechnology

Capitaldeepening











































(8) yE 5 EFF 3 y b


































statistics
















APPENDIX


f~x 51

nh Oj 5 1

J

kX x j 2 x

h D





T 5nh12I

sN~0 1

where

I 51

n2h Oi 5 1

n Oj 5 1j THORN i

n

kX x i 2 x j

h D 1 kX y i 2 y j

h D2 kX y i 2 x j

h D 2 kX x i 2 y i

h Dand

s2 51

n2hp12 Oi 5 1

n Oj 5 1

n

kX x i 2 x j

h D1 kX y i 2 y j

h D 1 2kX x i 2 y j

h D


REFERENCES

























































(3)yc

yb5

e c 3 y c ~kc

eb 3 b ~kb


(4)yc

yb5

ec

eb3

y c ~kc

y b ~k c 3

y b ~k c

y b ~kb




(5)yc

yb5

ec

eb3

y c ~kb

y b ~k b 3

y c ~k c

y c ~kb







(6)yc

yb5

ec

ebX y c ~kc

y b ~kc 3

y c ~kb

y b ~kb D12

3 X y b ~kc

y b ~kb 3

y c ~kc

y c ~k b D12


B Empirical Results











CountryOutput per


worker 1990


worker


Change inef ciency

Change intechnology

Capitaldeepening











































(8) yE 5 EFF 3 y b


































statistics
















APPENDIX


f~x 51

nh Oj 5 1

J

kX x j 2 x

h D





T 5nh12I

sN~0 1

where

I 51

n2h Oi 5 1

n Oj 5 1j THORN i

n

kX x i 2 x j

h D 1 kX y i 2 y j

h D2 kX y i 2 x j

h D 2 kX x i 2 y i

h Dand

s2 51

n2hp12 Oi 5 1

n Oj 5 1

n

kX x i 2 x j

h D1 kX y i 2 y j

h D 1 2kX x i 2 y j

h D


REFERENCES
















































(6)yc

yb5

ec

ebX y c ~kc

y b ~kc 3

y c ~kb

y b ~kb D12

3 X y b ~kc

y b ~kb 3

y c ~kc

y c ~k b D12


B Empirical Results











CountryOutput per


worker 1990


worker


Change inef ciency

Change intechnology

Capitaldeepening











































(8) yE 5 EFF 3 y b


































statistics
















APPENDIX


f~x 51

nh Oj 5 1

J

kX x j 2 x

h D





T 5nh12I

sN~0 1

where

I 51

n2h Oi 5 1

n Oj 5 1j THORN i

n

kX x i 2 x j

h D 1 kX y i 2 y j

h D2 kX y i 2 x j

h D 2 kX x i 2 y i

h Dand

s2 51

n2hp12 Oi 5 1

n Oj 5 1

n

kX x i 2 x j

h D1 kX y i 2 y j

h D 1 2kX x i 2 y j

h D


REFERENCES















































CountryOutput per


worker 1990


worker


Change inef ciency

Change intechnology

Capitaldeepening











































(8) yE 5 EFF 3 y b


































statistics
















APPENDIX


f~x 51

nh Oj 5 1

J

kX x j 2 x

h D





T 5nh12I

sN~0 1

where

I 51

n2h Oi 5 1

n Oj 5 1j THORN i

n

kX x i 2 x j

h D 1 kX y i 2 y j

h D2 kX y i 2 x j

h D 2 kX x i 2 y i

h Dand

s2 51

n2hp12 Oi 5 1

n Oj 5 1

n

kX x i 2 x j

h D1 kX y i 2 y j

h D 1 2kX x i 2 y j

h D


REFERENCES






















































































(8) yE 5 EFF 3 y b


































statistics
















APPENDIX


f~x 51

nh Oj 5 1

J

kX x j 2 x

h D





T 5nh12I

sN~0 1

where

I 51

n2h Oi 5 1

n Oj 5 1j THORN i

n

kX x i 2 x j

h D 1 kX y i 2 y j

h D2 kX y i 2 x j

h D 2 kX x i 2 y i

h Dand

s2 51

n2hp12 Oi 5 1

n Oj 5 1

n

kX x i 2 x j

h D1 kX y i 2 y j

h D 1 2kX x i 2 y j

h D


REFERENCES




































































statistics
















APPENDIX


f~x 51

nh Oj 5 1

J

kX x j 2 x

h D





T 5nh12I

sN~0 1

where

I 51

n2h Oi 5 1

n Oj 5 1j THORN i

n

kX x i 2 x j

h D 1 kX y i 2 y j

h D2 kX y i 2 x j

h D 2 kX x i 2 y i

h Dand

s2 51

n2hp12 Oi 5 1

n Oj 5 1

n

kX x i 2 x j

h D1 kX y i 2 y j

h D 1 2kX x i 2 y j

h D


REFERENCES


















































APPENDIX


f~x 51

nh Oj 5 1

J

kX x j 2 x

h D





T 5nh12I

sN~0 1

where

I 51

n2h Oi 5 1

n Oj 5 1j THORN i

n

kX x i 2 x j

h D 1 kX y i 2 y j

h D2 kX y i 2 x j

h D 2 kX x i 2 y i

h Dand

s2 51

n2hp12 Oi 5 1

n Oj 5 1

n

kX x i 2 x j

h D1 kX y i 2 y j

h D 1 2kX x i 2 y j

h D


REFERENCES














































technological change, technological catch-up, and capital ... · technological change,...

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