arrow paper on micro theory

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Capital-Labor Substitution and Economic EfficiencyAuthor(s): K. J. Arrow, H. B. Chenery, B. S. Minhas, R. M. SolowSource: The Review of Economics and Statistics, Vol. 43, No. 3 (Aug., 1961), pp. 225-250Published by: The MIT PressStable URL: http://www.jstor.org/stable/1927286.

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h e ev i ew o conomics a n d StatisticsVOLUME XLIII AUGUST I96I NUMBER 3

CAPITAL-LABOR SUBSTITUTION ANDECONOMIC EFFICIENCY1K. J. Arrow,H. B. Chenery,B. S. Minhas,and R. M. Solow

IN manybranchesof economic heory,it isnecessary to make some assumption aboutthe extent to which capital and labor are sub-stitutable for each other. In the absence of em-pirical generalizationsabout this phenomenon,theorists have chosen simple hypotheses, whichhave become widely accepted through frequentrepetition. Two competing alternatives hold thefield at present: the Walras-Leontief-Har-rod-Domar assumption of constant input co-efficients; and the Cobb-Douglas function,which impliesa unitaryelasticity of substitutionbetween labor and capital. From a mathemati-cal point of view, zero and one are perhaps themost convenient alternatives for this elasticity.Economic analysis based on these assumptions,however, often leads to conclusions that are un-duly restrictive.The crucial nature of the substitution as-sumption can be illustrated in various fields ofeconomictheory:(i) The unstable balance of the Harrod-Domar model of growth depends in a criticalway on the asumption of zero substitution be-tween labor and capital, as Solow [I5], Swan[I8], and others have shown.(ii) The effects of varying factor endow-ments on international tradehinge on the shapeof particularproductionfunctions. In this case,either zero or unitary elasticities of substitutionin all sectors of the economylead to Samuelson's

strong assumption as to the invariability of theranking of factor proportions. Variations inelasticity among sectors imply reversals of fac-tor intensities at different factor prices withquite differentconsequences for trade and factorreturns.(iii) In analyzing the relative shares of in-come receivedby the factors of production,it istempting to assume unit elasticity of substitu-tion to agree with the supposed constancy ofthe labor share in the United States. Recentwork has called into question both the observedconstancy and the necessity of the assumption

[9, I7].Turning to empirical evidence, we find everyindication of varying degrees of substitutabilityin different types of production. Technologicalalternatives are numerous and flexible in somesectors, limited in others; and uniform substi-tutability is most unlikely. The difference inelasticities is confirmedby direct observationofcapital-labor proportions, which show muchmorevariation among countries in some sectorsthan in others.The starting point for the present study wasthe empirical observation that the value addedper unit of labor used within a given industryvaries across countries with the wage rate.Evidence of this relationship for 24 manufac-turing industries in a sample of I9 countries isgiven in section I. A regression of the laborproductivity on the wage rate shows a highlysignificantcorrelationin all industries and alsoa considerablevariation in the regression coeffi-cients.These empirical findings led to attempts toderivea mathematical functionhavingthe prop-erties of (i) homogeneity, (ii) constant elastici-ty of substitution betweencapital andlabor,and(iii) the possibility of different elasticities for

[ 2251

1 This study grows out of the research program of theStanford Project for Quantitative Research in Economic De-velopment. It is one in a series of analyses based on inter-national comparisons of the economic structure. HendrikHouthakker contributed substantially to the formulation ofboth the statistical and theoretical analyses. Arrow's par-ticipation was aided by Contract 25I(33), Task N4047-co4,Office of Naval Research.2 It is only fair to note that the general equilibrium the-ories of Walras and Leontief never assume fixed proportionsfor gross aggregates like capital and labor.


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2 26 THE REVIEW OF ECONOMICS AND STATISTICSdifferent industries. In section II it is shownthat there is one general production functionhaving these properties; it includes the Leon-tief and Cobb-Douglas functions as specialcases.3The function contains three parameters,which are identified as the substitution param-eter, the distribution parameter, and the effi-ciency parameter.To test the validity of this formulation, weexamine in section III the fragmentary infor-mationavailable on directuse of capitaland alsothe deviations from the regression analysis.These tests, while inconclusive, suggest theworking hypothesis that the efficiency param-eter varies fromcountry to country but that theother two are constants for each industry.In this form, the constant-elasticity-of-sub-stitution (CES) production function implies anumber of predictable differences in the struc-ture of production and trade between countrieshaving differentrelative factor costs. Some ofthese are investigated in section IV through acomparison between Japan and the UnitedStates of factor use and relativeprices. The re-sults indicate the extent of substitutability be-tween labor and capital in all sectors of theeconomy and also support the hypothesis ofvarying efficiencygiven in section III.

Finally, the CES production function is ap-plied in section V to a time-seriesanalysis of allnon-farmproductionin the United States. Theresults show an over-all elasticity of substitu-tion between capital and labor significantlylessthan unity and provide a further test of thevalidity of the productionfunction itself.I. Variation in LaborInputs with Labor CostInternational comparisons are probably thebest available source of information on the ef-

fect of varying factor costs on factor inputs.The observed range of variation in the relativecosts of laborandcapital is of the order of 30: I,which is much greater than that observed in asingle country over any period for which dataareavailable. The observationson factor inputsrefer to a specific industry or set of technologi-cal operationsrather than to the different ndus-tries conventionally employed in cross-sectionstudies within a single country. Finally, takingThis function and its properties were arrived at inde-pendently by Solow and Arrow.

observationsthat are close together in time, onecan assume access to approximately the samebody of technological knowledge; while notstrictly true, this hypothesis is more valid thanthe same assumption applied to time-seriesanalysis.A. DataThe substantialnumberof industrialcensusesin the postwar period that use comparable in-dustrial classifications makes it possible to ex-ploit some of these potential advantages of in-ter-countryanalysis. The sampleused, the datacollected, and the relationshipsexploredare de-terminedprimarily by the nature of the censusmaterials.

Countries n the sample. The sampleconsistsof countries having the requisite wage and out-put data in a reasonable number of industries.The countries, averagewage rates, and numberof industries available for each are shown inTable i. The data pertain to different yearsbetween I949 and I955.TABLE I. - COUNTRIES IN THE SAMPLE

Averagewagea Number ofYear of (Current industriesCountry Census dollars) usedbi. United States I954 384I 242. Canada 1954 3226 233. New Zealand I955/56 I980 224. Australia I955/56 I926 245. Denmark I954 I455 246. Norway I954 1393 227. Puerto Rico 1952 II82 178. United Kingdom I951 I059 249. Colombia I953 924 24i o. Ireland I 953 900 I 5I I. Mexico I95I 524 2 1

I2. Argentina 1950 5I9 2413. Japan 1953 476 23I4. El Salvador I951 445 i6I 5. Brazil 1949 436 I0i6. S. Rhodesia 1952 384 617. Ceylon 1952 26I I Ii8. India 1953 24I I 7I9. Iraq I954 2I3 2

a Unweightedaverage of wages in industries in sample.b Industry data are given in the appendix.

Industries. Data were collected for all indus-tries at the three-digit level of the United Na-tions International Standard Industrial Classi-fication having sufficientobservations (at leastio). The 24 industries analyzed are listed inTable 2 and defined in the ISIC. There is, of


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CAPITAL-LABOR SUBSTITUTION AND ECONOMIC EFFICIENCY 227course, considerable variation in the composi-tion of outputwithin a given industrialcategoryamong countries at different income levels,which cannot be allowed for here.Labor inputs and costs. Labor inputs aremeasured in man-years per $iooo of value add-ed. They include productionworkers, salariedemployees, and working proprietors. Laborcosts are measuredby the average annual wagepayment, computed as the total wage bill divid-ed by the number of employees. The data onwage payments for different countries includevarying proportions of non-wage benefits, andwe made no allowance for such variations. Thedata on employmentare not corrected for inter-country differences in the number of hoursworkedper year or the age and sex compositionof the labor force. The data for each industryare given in the appendix.EIxchangerates. All conversions from localcurrencyvalues into U.S. dollarswere at officialexchange rates or at free market rates wheremultiple exchange rates prevailed. No allow-ance was made for the variation in the purchas-ing power of the dollar between different cen-sus years.

Capital inputs. Data on capital inputs orrates of return are available only for a smallnumber of countries and industries. They aretherefore omitted from the initial statisticalanalysis and utilized in section III to test thevalidity of the production function that is pro-posed in section II.B. RegressionAnalysisThe variables available for statistical analy-sis are as follows:V : value addedin thousands of U.S. dol-larsL : labor input in man-yearsW : money wage rate (total laborcost di-vided by L) in dollarsper man-year.

As an aid in formulatingthe regressionanal-ysis, we make the followingpreliminaryassump-tions, the validity of which will be examined insection III.(i) Prices of products and material inputsdo not vary systematically with the wage level.(2) Overvaluation or undervaluationof ex-change rates is not related to the wage level.(3) Variation in average plant size does notaffect the factor inputs.

TABLE 2. - RESULTS OF REGRESSION ANALYSIS aRegression Test of Significanceon bequations Standard Coeff. ofisic error deter. Degreesof Confidence evel forNo. Industry Log a b Sb R2 freedom b differentfrom I

202 Dairy products .4I9 .72I .073 .92I I4 99%?203 Fruit and vegetable canning .355 .855 .075 .9IO I2 90205 Grain and mill products .429 .909 .o96 .855 I4 *206 Bakery products .304 g900 .o65 .927 I4 8o207 Sugar .43I .78I .JI5 .790 II 90220 Tobacco *564 .753 .I5I .629 I3 8o23I Textile - spinning and weaving .296 .809 .o68 .892 I6 98232 Knitting mills .270 .785 .o64 .9I5 I3 99250 Lumber and wood .279 .86o .o66 .9IO i6 95260 Furniture .226 .894 .042 .952 I4 9527I Pulp and paper .478 .965 .IOI .858 I4 *280 Printing and publishing .284 .868 .056 .940 I4 9529I Leather finishing .292 .857 .062 .92I I2 953II Basic chemicals .460 .83I .070 .898 I4 953I2 Fats and oils .5I5 .839 .090 .869 12 903I9 Miscellaneous chemicals .483 .895 .059 .938 14 9033I Clay products .273 .9I9 .o98 .878 II332 Glass .285 .999 .o84 .92I II *333 Ceramics .2IO .90I .044 .974 IO 95334 Cement .560 .920 .I49 .770 IO34I Iron and steel .363 .8II .05I .936 II 99342 Non-ferrous metals .370 I.OII .J20 .886 8 *350 Metal products .30I .902 .o88 .897 II *370 Electric machinery .344 .870 .ii8 .804 I2 *

a From data given in the Appendix.* Not significant at 8o% or higher levels of confidence.


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228 THE REVIEW OF ECONOMICS AND STATISTICS(4) The same technological alternatives areavailable to all countries.On these assumptions,we can treat $iooo ofvalue addedas a unit of physical output in eachindustry. We also assume a single production

function for all countries, which implies thattherewill be a determinaterelationbetween thelabor input per unit of value added and thewage rate. Before exploring the possible formsof this function in detail, we tested two simplerelations among the three variables statisti-cally:V = c + dW + (ia)log - = loga + blogW +,e. (ib)LBoth functions give good fits to the observa-tions, the logarithmicformbeing somewhatbet-ter. The results of the latter regression areshown in Table 2.4 It is apparent from thesmall standard errors of b and the high coeffi-cient of determination k2 that the fit is rela-tively good. In 20 out of 24 industries,over 85per cent of the variation in labor productivityis explained by variation in wage rates alone.5

C. Implied Properties of theProduction FunctionThe regression analysis provides an impor-tant basis for the derivation of a more generalproduction function: the finding that a linearlogarithmic function provides a good fit to theobservations of wages and labor inputs. Thetheoretical analysis of the next section willthereforestart from this assumption.It is shown in section II that under the as-sumptions made here the coefficient b is equalto the elasticity of substitution between labor

and capital. It is therefore of interest to deter-mine the number of industries in which theelasticity is significantly different from o or i,the values most commonlyassumed for it. Re-

sults of a t test of the second hypothesis aregiven in Table 2. In all cases, the value of b issignificantly differentfrom zero at a 90 per centlevel of confidence. In I4 out of 24 industriesit is significantlydifferentfrom i at go per centor higher levels of confidence. We thereforereject these hypotheses as inadequate descrip-tions of the possibilities for combining laborand capital, and we proceed to derive a pro-duction function that allows for a differentelasticity in each industry.

II. A New Classof ProductionFunctionsSection I presents observations on the rela-tion between V/L and w within each of severalindustries at a single point of time. It is a

natural first step to give an account of the re-sults in terms of profit-maximizingresponsesto given factor prices. Under the assumptionsof constant returns to scale and competitive la-bor markets, the standardtheory of productionshows how any particular production functionentails a particular relation between V/L andw. We shall show that the reverse implicationalso holds: that a particular relation betweenVIL and w determines the corresponding pro-duction function up to one arbitrary constant.A. Output per unmitf Labor, Real Wages,and the Production Function un-der Constanit ReturnsIf the production function in a particular in-dustry is written V = F(K,L), and assumed tobe homogeneous of degree one, then V/L= F(K/L, i); and if we put V/L = y, K/L = x, wecan say y = f (x). In these terms the marginalproducts of capital and labor are f' (x) andf (x) - xf' (x) respectively. Let w be the wagerate with output as nume'raire. If the labor andproductmarketsare competitive then

w = f(x) - xf'(x) (2)which can be inverted to give a functional rela-tion between x and w, and thence, since y= f(x),a monotone increasing relation between y andw. Conversely, suppose we begin (as we do)with such an observed relation between y andw, say y = +(w). Then from (i) we see that

y = 0(y - x- dy(3)dx

4 Independently of this study, J. B. Minasian has fittedequation (ib) to U. S. interstate data for a number of in-dustries in "Elasticities of Substitution and Constant-OutputDemand Curves for Labor," Journal of Political Economy,June I96I, 26I-270. (Note added in proof.)5 For the economy as a whole, the level of wages dependson the level of labor productivity, but for a given industrythe labor input per unit of output is adjusted to the prevail-ing wage level in the country with relatively small deviations

due to the relative profitability of the given industry.


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CAPITAL-LABOR SUBSTITUTION AND ECONOMIC EFFICIENCY 229which is a differential equation for y(x). Itwill have a solution

y=f(x;A) (4a)whereA is a constant of integration. Returningto the original variables we get the one-param-eter family of productionfunctions

V = Lf (K/L; A). (4b)Of course for (4) to do duty as a productionfunction it should have positive marginal pro-ductivities for both inputs and be subject to theusual diminishing returns when factor-propor-tions vary. An elementary calculation showsthat these conditions are equivalent to requir-ing that f' (x) > o and f"(x) < o. The lattercondition is also sufficient to permit the inver-sion of (2 ) . Geometrically these conditionsstate that outputper unit of labor is an increas-ing function of the input of capital per unit oflabor, convex from above, just as the curve isnormally drawn. In addition one would desirethat f (x) > o for x > o. All these requirementsshouldhold for at least somevalue of A.This way of generatingproduction functionsbrings to light a connection with the elasticityof substitution which does not seem to havebeen noticed in the literature, although closelyrelated results were obtained by Hicks andothers (see Allen [2], 373). The slight differ-ence has to do with the treatment of productprice. Let s stand for the marginalrate of sub-stitution between K and L (the ratio of themarginalproduct of L to that of K). Then theelasticity of substitution c- is defined simply asthe elasticity of K/L with respect to s, along anisoquant. For constant returns to scale it turnsout,6in our notation,

f'(f - Xf')xf" (5)Now consider the relation between y and was determined implicitly by (2). Differentiat-ing with respectto w we obtain

f. dxdy _ftt dxdy f, dxdydy dw dy dw dy dwdx Iand since d- =dy 7'dy_ fdw xf"

On all this see Allen [2], 340-43.

Thus the elasticity of y with respect to w is,from (2),wdy fJ'(f xf') (6)y dw xf" =

That is to say, if the relationbetween V/L andw arises from profit-maximizationalong a con-stant-returns-to-scaleproduction function, theelasticity of the resulting curve is simply theelasticity of substitution. Information about a-can be obtained,under these assumptions,fromobservationof the joint variation of output perunit of laborand the real wage.We may also observe another simple and in-teresting relation associated with productionfunctions homogeneous of degree one. As hasbeen seen the marginalproductivity of capitalis a decreasing function of x, the capital-laborratio, while the marginalproductivity of laboris of course an increasing function. Hence, forcompetitive markets, the gross rental, r, meas-ured with output as numefraire, is a decreasingfunction of the wage rate. More specifically,we may differentiate the relations, r = f' (x)and (2) to yield,dr dw

= f"(x); dw= - xf" -f = -xf"dx dxso thatdr (dr ) (dw I L

dw dx dx x K'whence the elasticity of the rate of returnwithrespect to the wage rate is,

w dr wLr dw rK '

i.e., the ratio of labor's share to capital's sharein value added.B. Rationalizingthe Data of Section IWe found in section I that in generala linearrelationship between the logarithmsof V/L andw, i.e.,logy = loga + blogw (8)gives a good fit. Along such a curve, the elas-ticity of y with respect to w is constant andequal to b. We are forewarnedthat the impliedproduction function will have a constant elas-ticity of substitution equal to b, so that in de-ducing it we provide a substantial generaliza-tion of the Cobb-Douglas function. Indeed the


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230 THE REVIEW OF ECONOMICS AND STATISTICSCobb-Douglas family is the special case b= Iin (8). Our empirical results imply that elas-ticities of substitution tend to be less than one,whichcontrastsstronglywith the Cobb-Douglasview of the world. We will return subsequentlyto the distributional and other implications ofthis conclusion.The differentialequation (3) becomes

log y = log a + b log (y-xdy). (9)dxdyTakingantilogarithmsnd solvingfor dyv we

finddy al/b y - yl/b y(i - ayP)dx al/b x x

where we have set a= a-'/b and P i forconvenience. The equation

dx dyx y(I - ayP)

has a partial-fractionsexpansion:dx dy ayP-1 dyX y I - ayP

which can be integratedto yieldI Ilog x = log y- -log (i - ayP) + -log 8p por

xP gypI - ayPwhich in turn can be solved for yP,and then y,to give

y = XQ(3 + aXP) -/P = (/3X-P + a)-1/P (IO)Writtenout in full the productionfunctionis:

V =L(,/K-PLP + a)-1/P= (3K P + aL-P)k1p (II)As for our requirementson the shape of theproduction function, it is clear that y > o forx > o as long as a > o and 3> o. Differentia-tion of (io) shows that the only requirementforpositive marginal productivities is , > o. Asecond differentiationyields one further condi-tion for diminishing returns, namely p + I > owhich is equivalent to b > o and in accordancewith our empiricalresults.The family of productionfunctionsdescribed

by (io) or (ii) comprisesall those which ex-

hibit a constant elasticity of substitution for allvalues of K/L. To be precise, the elasticity ofsubstitution a- = i / ( i + p) = b. For this reasonwe will call ( iO) or ( i i ) a constant-elasticity-of-substitution production function (abbrevi-ated to CES).7 Admissiblevalues of p run from--I to oo, which permits o- to range from + coto o. Since our empiricalvalues of b are almostall significantly less than one, they imply posi-tive values of p and elasticities of substitutionin differentindustriesgenerally less than unity.C. Properties of the CES Production FunctionWe can write (i o) and (i i ) more symmetri-cally by setting a+:1 = y-P and /3yP 8, inwhich notation they become

y y [8x P + (I_8)]-1/p (I2)V = y[8KP + (i -8)L-P] -'/P (I3)A change in the parametery changesthe out-put for any given set of inputs in the same pro-portion. It will therefore be referred to as the(neutral) efficiencyparameter. The parameterp, as has just been seen, is a transform of theelasticity of substitutionand will be termedthesubstitution parameter. It will be seen below(equation 23) that for any given value of C-

(equivalently, for any given value of p), thefunctional distributionof income is determinedby 8, the distributionparameter-Apart from the efficiency parameter (whichcan be made equal to one by appropriatechoiceof output units), (I3) is a class of functionknown in the mathematical literature as a"mean value of order -p.") 8The lowest admissiblevalue for p is - i; thisimplies an infinite elasticity of substitution andtherefore straight-line isoquants. One verifiesthis by putting p = -I in (I 3).For values of p between - I and o we haveelasticities of substitution greater than unity.From (I2) we see that y-*oc as x-*co, andy >y(i-8)-l/P as x-*o. That is to say, out-put per unit of labor becomes indefinitely largeas the ratio of capital to labor increases; but asthe capital/labor ratio approacheszero, the av-7 We note that Trevor Swan has independently deducedthe constant-elasticity-of-substitution property of (ii). Thefunction itself was used by Solow [I5], 77, as an illustration.8 See Hardy, Littlewood, and P6lya [71, I3. It may alsobe shown that the function (I3) is the most general func-

tion which can be computed on a suitable slide rule.


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CAPITAL-LABOR SUBSTITUTION AND ECONOMIC EFFICIENCY 21Ierage product of labor approaches a positivelower limit.The case p = o yields an elasticity of sub-stitution of unity and should, therefore, leadback to the Cobb-Douglasfunction. This is notobvious from (I3), since as p-*o the right-handside is an indeterminate form of the type i??.But in fact the limit is the Cobb-Douglas func-tion. This can be seen (a) by direct applicationof L'Hopital's Rule to (I3); (b) by integra-tion of (g) with b = i; or (c) by appealing tothe purely mathematicaltheoremthat the meanvalue of order zero is the geometric mean.9Thus the limiting form of (I3) at p = o is in-deed V = yKIL1-. 1OFor o < p < oo, which is the empirically in-teresting case, we have o- < i. The behavior isquite different from the case - i < p < o. AsX?co, y>y7(i 8) -1/P; as x-*o, y-*o. Thatis, as a fixed dose of labor is saturated withcapital, the output per unit of labor reaches anupper limit. And as a fixed dose of capital issaturated with labor, the productivity of labortends to zero.Whenever p> - i, the isoquants have theright curvature (p =- i is the case of straight-line isoquants, and p < - i is ruled out pre-cisely because the isoquants have the wrongcurvature). The cases p < o and p> o are dif-ferent; when p < o, the isoquants intersect theK and L axes, while when p> o, the isoquantsonly approach the axes asymptotically. Bothcases are illustrated in Chart i of section IV.Our survey of possible values of p concludeswith two final remarks. The case p = i, a- = '2is seen to be the ordinaryharmonicmean. Andas p-- oo, the elasticity of substitution tends tozero and we approachthe case of fixed propor-tions. We may prove this by making the ap-propriate limiting process on (I3). And onceagain the general theory of mean values assuresus that as a mean value of order - oo we have"

lim y[8K-P+(I-8)L-P]-1/Pp ->oo

T r min (K,L) = min f n (I4)This represents a svstem of rig)ht-ang)led iso-;0

quants with corners lying on a 450 line fromthe origin. But it is clearly more general thanthat, since the location of the corners can bechangedsimply measuringK and L in differentunits.So far we have simply provided one possiblerationalizationof the data of section I. We turnnext to some of the testable implicationsof themodel, and in so doing we consider the possi-bility of lifting or at least testing the hypothesisof constant returns to scale. Further economicimplications of the CES production functionare discussed in section IV below.

D. Testable Implications of the Modeli. Returns to scale. So far we have assumed

the existence of constant returns to scale. Thisis more than just convenience; it is at leastsuggested by the existence of a relationshipbetween V/L and w, independent of the stockof capital. Indeed, homogeneity of degree one(together with competition in the labor andproduct markets) entails the existence of sucha relationship. Clearly,not all productionfunc-tions admit of a relationshipbetween V/L andw = DV/IL; the class which does so, however,is somewhat broader than the homogeneousfunctions of degree one. We have the followingprecise result: if the labor and productmarketsare competitive, and if profit-maximizingbe-havior along a production function V= F(K,L)leads to a functional relationship between wand V/L, then F(K,L) = H(C(K),L) whereHis homogeneous of degree one in C and L, andC is an increasingfunctionof K.In proof, since w = DV/DL, we can writethis functional relationas:DV VDL =*LJ

Since this holds independently of K we mayhold K constant and proceed as with an ordi-nary differentialequation. Introducingy= V/L,we have L Dy/DL + y = DV/DL and thereforeDy/DL = k(y) - y . Since K is fixed we mayLwrite thisdy dL

k(y)-y Land integrateto getL = Cg(y) ('5)

'Hardy, Littlewood, and P6lya [7], I5, Theorem 3."0This special case reinforces our singling-out of a as adistribution parameter."Hardy, Littlewood, and P6lya [7], I5, Theorem 4.


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232 THE REVIEW OF ECONOMICS AND STATISTICSwhereg (y) = exp Y dy ; andC,thecon-h(y)-ystant of integration must be taken as a functionof K. (We also assume h(y) < y; that is, theaverage productivity of labor exceeds its mar-ginal productivity.) Upon inversion of (I 5) wehave y as a function of L/C (K) alone, sayy= G(L/C(K)) and therefore

V = LG LK H(C(K),L) (6)as asserted, where H is homogeneousof degreeone in its arguments. If K is to have positivemarginalproductivity, C must be an increasingfunctionof K, since g(y) is decreasing.Thus under our assumptions production ex-hibits constant returns to scale, not necessarilyin K and L, but in C(K) and L. We have con-stant returnsto scale if C is proportionalto K.But C(K) can be given an interpretation n anycase. Let P represent all non-labor income,whether returns to capital or not. Then byEuler's Theorem, P = CDH/DC. And v/aDK= (DH/DC) (dC/dK). Hence

C PDv (I7)DK

so that C/C' represents the "present value" ofthe stream of profits, discounted by the mar-ginal productivity of capital.The argument leading to (I6) provides uswith an empiricaltest of the hypothesis of con-stant returns to scale. As we have noted, thelatter is equivalent to C(K)/K being constant.But from (I5),C LI (8)K K g(y) xg(y)

So a stringent test of the hypothesis is thatxg(y) be constant within any industryand overall countries for which we have data on capital.The stringency of the test comes from the factthat it relies on data (namely K) which havenot been used in the previous analysis. If thetest is passed, then not only have we validatedthe assumption of constant returns, but also(I 5) and with it ourwhole approach to the pro-duction function.When h(y) is obtained by solving for w in(8), the integration needed to determine g(y)

is a repetition of the argumentleading to (io).Then,- (I - ayP -l/pxg(y) K

Since ,Bis a constant, a test of constancy of re-turns to scale is obtained by the condition thatC = (-k (I - ayP)-/P ('9)

is a constant.2. Capital and the rate of return. It shouldnot be overlookedthat up to the previous para-graph our production functions have appearedonly as rationalizationsof the observedrelationbetweeny andw underassumptions about com-

petition. We can not be sure that they do infact describe production relations (i.e., holdingamong V, L, and K), and it is indeed intrinsi-cally impossible to know this without data onK, or equivalentlyon the rate of return. Shouldsuch data be available, however, we can per-formsome furthervery strongtests of the wholeapproach.Suppose we have observations on K for aparticular industry across several countries.Then we know x as well as y and we can testdirectly whether our deduced production func-tion (3) or (4) does in fact hold for some valueof A. If it does, then this provides an estimateof A and a stringent external check on thevalidity of our approach.This is merely a rephrasing of our test forconstant returns, to emphasize that it reallygoes somewhat further; if the hypothesis ofconstant returns to scale is accepted, so is thevalidity of the impliedproductionfunction.3. Neutral variations in efficiency.From theargument leading to (io) and (ii), it is seenthat the parametersa and p are deriveddirectlyfrom our empiricalestimates of a and b in sec-tion I. But ,Bis a constant of integration andcan be determinedonly from observed data in-cluding measurements of K or x. Now the testquantity c in (i9) depends on a and p, butnot on ,3, on the assumption that /B s constantacross countries. Failure of data to pass thestringent test based on (I9) may be read assuggesting that ,8 varies across countries whilea and p are the same. From (i i), this is equiva-lent to the statement that the efficiency of use


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CAPITAL-LABOR SUBSTITUTION AND ECONOMIC EFFICIENCY 233of capital varies from country to country, butnot the efficiencyof use of labor.A more symmetrical (and more plausible)possibility is that international differences inefficiency affect both inputs equally. Thisamounts to assumingin (13) that the efficiencyparameter y varies from country to countrywhile 8 and p remain constant. Since //a= 8/(i-8), we can put this by saying that /8and a vary proportionately. We can provide atest of this hypothesis.From the definition of the elasticity of sub-stitution and its constancy and the competitiveequivalence of factor price ratios and marginalrates of substitution, it follows that w/r is pro-portional to (K/L)'10 = (K/L)(1+P). It is easyenough to calculate the constant of proportion-ality directly; we have

W I8 K Al+Pr 8 L (20)and2/a /=/(I 8) = (r) (K )l+P (2I)

Thus for countries from which we have data onr and K, and given our estimate of p for anindustry, we may compare the values of theright-hand side of (2I). If they are constantor nearly so, we conclude that there are neutralvariationsin efficiencyfromcountry to country,and we are able simultaneously to estimate 8.Then from 8 and p we can use (I2) to estimatethe efficiency parameter y in each country in-volved, for this particularindustry.4. Factor intensity and the CES productionfunction. From (20) we see that:K wX ~=(~Y.(2 oa)L I-S rJ 2a

Now imagine two industries each with a CESproduction function although with differentparameters,and buying labor and capital in thesame competitivemarket. ThenX1 81 Aal 82 A-2 W Aa-2X2 I-81 I -82 r

W(y- r2) (22)

If 01 =0 (i.e., Pi = P2), then this relative fac-tor-intensity ratio is independent of the factorprice ratio. That is, industry one, say, is morecapital-intensive than industry two, at all pos-

sible price ratios. This is the case both for theCobb-Douglas function (o-, = 0-2 = i) and thefixed-proportionscase (o-, = 0-2 = a). But onceO- z o-2, this factor-intensity property disap-pears and it is impossible to characterize oneindustry as more capital-intensive than theother independentlyof factor prices. For (22)says quite clearly that there is always a criticalvalue of w/r at which the factor-intensity ratiox1/x2 flips over from being greater than unityto being less. There is only one such criticalvalue at which the industrieschangeplaces withrespect to relative capital-intensity. The natureof the switch is in accord with common sense:as wages increase relative to capital costs, ulti-mately the industry with the greater elasticityof substitution becomes more capital-intensive.Such switches in relative factor intensity shouldbe observable if one compares countries withvery differentfactor-pricestructures,which wehave done for Japan and the United States insection IV.The relative factor-intensity ratio plays animportantrole in discussions of the tendency ofinternational trade in commodities to equalizefactorprices in differentcountries (and for thatmatter, in the moregeneralproblemof the rela-tion betweenfactor pricesand commoditypricesin any generalequilibriumsystem).5. Time series and technological change. TheCES production function is intrinsically diffi-cult to fit directly to observationson outputandinputs because of the non-linear way in whichthe parameterp enters. But, provided technicalchange is neutral or uniform, we may use theconvenient factor-priceproperties of the func-tion to analyze time series and to estimate themagnitudeof technicalprogress.A uniform technical change is a shift in theproduction function leaving invariant the mar-ginal rate of substitution at each K/L ratio.From (I3) and (20), uniform technical prog-ress affects only the efficiency parametery, andnot the substitution or distributionparameters,p or o-.One notes from (20) that

wL i-8 ( K PrK 8 L(23)

which is independent of y. Hence if historicalshifts in a CES functionare neutral, (23) should


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234 THE REVIEW OF ECONOMICS AND STATISTICShold over time, and its validity provides a testof the hypothesis of neutrality.Suppose we have observations on x and onw/r at two points on the production function,say two countries or two points of time in thesame country. Then, from (2oa),Xl [ (w/r)l]J (24)

x2 (wlr) 2Thus an estimate of o-may be made. Note fur-ther that, since y does not enter into (20a), theestimate is valid even if the efficiencyparameterhas changedbetween the two observations,pro-vided the distribution and substitution param-eters have not, that is, provided that tech-nological change is neutral.12If the hypothesis of neutrality is acceptable,we may try to trace the shifts in the efficiencyparameter over time. One way to do this is togo back to (8). From V/L = awb and the defi-nitions of the parametersa,o-,8, and y, one cal-culates first that

wL I'1L _ VJ)Wl-b = aaWl-of= I S)ay W . (25)Two possibilitiesnowpresentthemselves.Forgiven values of the parameters o- and 8, one

F ~~~~~~~~~~~1aWAV 1A1'aPA WA -1B +S- ~;-)-II __ -FA i28PB WB 'YA 2 i Y a-) (28

can use (25) to compute the implied time-pathof y. Or alternatively one may assume a con-stant geometric rate of technologicalchange, sothat y(t) = y, io\t, and fitlog (W) = [ log (I -8) + (a-I) logy0]

+ (I--a) logw + A(U-I)t (26)to estimate o- and X. We return to this subjectin section V.6. Variation in commodity prices amongcountries. The accepted explanation of the vari-ation in commodity prices among countries isbased on differences in capital intensity andfactor costs. In our production function the

capital intensity depends both on o- and 8 (in-stead of only on 8, as in the Cobb-Douglasfunc-tion), and we also allow for differencesin effi-ciency among sectors. The corresponding ex-planation of price differencesis therefore morecomplex.The price of a commodity in our model isdefined as the direct labor and capital cost perunit of value added:1 RP=Wl+Rk=Wl i+-( .x

where W and R are wages and returnon capitalin money terms (rather than using output asnumeraire), I = L/V = i/y, and k = K/V.

Substituting from (I 2) for the labor coeffi-cient gives:w ~~~~~~rP - [8x-P + (I 8) 1/P [- * x + I (27)7Y w

in which the price of a commodity depends onfactor costs and capital intensity. For a givenproduction function, the ratio of prices in coun-tries A and B can be stated as a function of thefactor prices only by using (2oa) to eliminate x:

The empirical significance of this result is dis-cussed in section IV-C.III. Tests of the CES ProductionFunction

The CES function may describe productionrelations in an industry with varying degreesofuniformity across countries. Two tests wereoutlined in section II-D that enable us to makea tentative choice among three hypotheses: (i)all three parameters the same in all countries,(ii) same o- and one other parameter the same,(iii) only o- the same. The evidence presentedin section A below rejects the first hypothesisbut supports the second. Furthermore, thereappears to be some uniformity in the efficiencylevels of different industries in the same coun-try; this possibility is analyzed in section B.12 This method of estimating the elasticity of substitu-tion has been used by Kravis [9], 940-4I.


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CAPITAL-LABOR SUBSTITUTION AND ECONOMIC EFFICIENCY 235In section C we investigate the possible sourcesof bias in our previous estimates of o- in thelight of these findings.A. Generalityof the ProductionFunctionThe two tests given in (i 9) and (2 i) requireestimates of either the capital stock or the rateof return on capital. Although such data arenotoriouslyscarce and unreliable, we have beenable to assemble comparable information on

rates of returnin four of the industriesin Table2 covering from three to five countries in eachindustry.13The capital stock can be estimatedfrom the rate of return, r, by the relation:K= (V-7m)rL/r.

The test of constancy of all parameters in-volves the computation of c in equation (I9).If there is no variation in efficiency, this num-ber should remainconstant. This calculation isshown in Table 3-A. The extent of variation inc is indicated by taking its ratio to the geomet-ric mean for each industry,J.It is clear from these results that the hypoth-esis of constancy in all three parameters mustbe rejected, since there is a large variation in c

in all four industries. We therefore abandonthe idea that efficiency is the same among coun-tries and look for constancy in either a, /3, or S.The first would imply that variations in effi-ciency apply entirely to capital (assumed inthe computation of c in Table 3-A); the secondthat they apply entirely to labor; and the lastthat they affect both factors equally. The logicof the test was indicated in section JI-D. Thethree possibilities are evaluated in Tables 3-B

TABLE 3.- TESTS OF THE CES PRODUCTION FUNCTION aA. Test of Constant Efficiencyb

23I. Spinning&Weaving 31I. Basic Chemicals 34I. Ironand Steel 350. Metal ProductsC c c c

w r c w r c w r c w r cUnited States 2920 .0526 I.72 4754 .2083 2.3I 4387 *I984 2.96 43I4 *I359 2.35Canada 2708 .0403 i.i6 4036 .2II5 2.0I 3769 *I740 2.13 3507 .I07I I.37United Kingdom 874 .2022 I.44 .. I224 *I5I3 0.62Japan 287 .I902 0.55 563 .2373 .70 664 .I9II 0.49 422 .2245 0.3IIndia 276 *I543 o.63 320 .2200 .5I 450 .2686 0.53B. Test of Constant 5'

fi/a, at 'Yi fi/a, at 'Yi fi/ai at, 'i fi/a, aj 'YiUnited States I.I55 .536 i.oi6 3.283 .767 I.02I I.564 .6io I-733 I.039 .509 I-736Canada I.352 .575 .77I 3.289 .767 .839 I.765 .638 I-378 i.i8o .541 I.292United Kingdom I-5I4 .602 .880 .. I.. .265 .559 .857Japan I.833 .647 .428 2.785 .734 .567 I.46I .584 .687 .968 .492 .6I9India .. .. . 2.967 .748 .44I 1495 .599 .65 7Mean .590 .754 .598 *5I4Coefficient of Variationd 5.85 % 1.72 % 3.55 % 3.50 %C. Test of Constancy of a and P

at et at pis ai 6 i at ,8United States .462 '534 .233 .765 .343 .536 .462 .480Canada .452 .6iI .242 .796 .336 .593 .448 .529United Kingdom 410 .62I .. .458 .579 ..Japan .43 I .790 .297 .826 .444 .648 .53 5 .5I8India .. .. .298 .883 .442 .66iMean .439 .639 .268 .8I8 .405 .603 .482 .509Coefficient of Variationd 4.I7%o ii.85%o II.I9% 8.04% I2.84% 6.77% 77-4I% 3-79%a Sourcesof data: See text.b Computed from equation (I9).c Computedfrom equations (2I) and (12), using country values of 3 in computing 'y.d Defined as -X _ where Xs is the country value, X is the industry mean, and N is the number of observations for the industry.NX

'3 The rates of return on capital were estimated frombalance sheets of different industries. Capital was measuredby net fixed assets (including land) plus cash and workingcapital. All financial investments were excluded. Total re-turns to capital were taken to be equal to gross profit fromoperations (excluding other income) minus depreciation.For further details see [I2].


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236 THE REVIEW OF ECONOMICS AND STATISTICSand 3-C by computing the coefficientof varia-tion foreachparameter n each industry.Of these three possibilities, the constancy of8, implying neutral variations in efficiency, ismuch the closest approximationwhile there islittle to choose between the other two. For thefour industries taken together, the coefficientof variation in 8 is only 3.6 per cent, while it ismore than twice as large for the other two pa-rameters.We thereforetentatively accept equa-tion (I2) or (I3) as the basic form of the CESproduction function. Constantc- and 8 charac-terize an industry in all countries, and differ-ences in efficiency are assumed to be concen-trated in y,.The conclusionthat observationson the sameindustryin differentcountriesdo not come fromthe same production function is so importantthat it should be tested in a way that does notdependon our particularchoice of a productionfunction. If in fact all countriesfell on the sameproduction function, homogeneous of degreeone, then a high wage rate must arise from ahigh capital-labor ratio, which must, in turn,imply a low rate of return on capital. FromTable 3 we see there is by and large an inversecorrelationbetween wages and rates of return,but the variation in the latter seems muchsmaller than is consistent with the wide varia-tions in wage rates. This impression can beconfirmed quantitatively with the aid of for-mula (7).It is there noted that, for points on the sameproduction function, homogeneous of degreeone, the rate of returnis a function of the wagerate, with an elasticity which is negative andequal in magnitudeto the ratio of labor's shareto capital's. We proceed as follows. Let v bethe smallest observedvalue of this ratio. Thenthe elasticity of r, the rate of return, with re-spect to the wage rate, w, cannot exceed -v,so that,

og rli

where the subscriptsrefer to any two countries.If we choose the countries so that wo> w1,multiply through by the negative quantity log(w1/wo), and take antilogarithms, we find r1

',?) = ri, say. Then, if the two coun-tries were on the same production function, therate of return in the low-wage country couldnot fall below the limit r1.For each of the industries in Table 3, a com-parison was made of the rates of return in thelowest-wage country, with the correspondinglower bound r1, computed from the highest-wage country (the United States in each case).The results of this computation follow:Industry 23I 3II 34I 350r I.g9o .220 .269 .225

ii .244 .667 I.2I3 I.749Thus in each case the actual rate of return inthe lowest-wage country falls below the theo-retical minimumconsistent with the assumptionof a uniform production function, and in mostcases very far below. The results of section Aare thus strongly confirmed.B. Effects of VaryingEfficiencySince we have revised our interpretation ofthe empirical evidence on the elasticity of sub-stitution, we can no longer take the regressioncoefficient b in section I as equal to o-. We nowpresent a formula for determining o- from bwhen efficiencyis known to vary with the wagerate and indicate the magnitude of the correc-tion involved. We then examine the residualsfrom the regression equations for further evi-dence of varying efficiency or other sources ofbias in estimation.

i. Estimation of o-. It is plausible to assumethat in each industry the efficiency parameter yvaries among countries with the wage rate.Since the wage rate increases with both y andx, a country with high y is also likely to havebeen more efficientin the past and to have hadhigh income and savings. Thus we expect x tobe positively correlated with y across countriesand y to increase with w. Assume for conven-ience that this variationtakes the form:

(A ) (WA )e (29)YB WB

where the subscriptsrefer to countriesA and B.The effect of variation in efficiency on the out-put per unit of labor (y) can be shown from(25) to be:

YB YA WA


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CAPITAL-LABOR SUBSTITUTION AND ECONOMIC EFFICIENCY 237Substituting from (29) for the efficiency ratioand taking logs we get a formula comparableto equation (8) from which our elasticity esti-mates were derived:

log Y = (a + e - ea) log (W(3AY'B WBComparing (8) and (3I), we see that the re-gressioncoefficient b is equal to (o-+e-eo-), or

b-e (32)b-ei - eTherefore it is only when efficiency does notvary with w - i.e. when e = o - that b isequal to o-. For e > o, o- must be still smallerthan b, and therefore, a fortiori, less than iwhenb < i.

To get a rough idea of the magnitude of thecorrection,we normalizedthe y's in Table 3 sOthat the United States value equals one in eachcase and then fitted log y to log w by leastsquares.14We obtain the following result fromthe combinedsampleof I4 observations:logy .323 log w -.039 R2 = .82.

(.043)The separate industriesvary somewhat,but thenumber of observationsin each is too small forreliable estimates. Another source of informa-tion on e is provided by the comparisons ofJapan with the United States in section IV,which cover io manufacturing ndustries. Herethe median value of (yJ/yu) is about .35, cor-responding o a value of e of about .5.For values of b less than i, equation (32)shows that variationof efficiencywith the wagerate will reduce the estimate of a-. Taking e= .3,values of b of .9, .8, and .7 yield values of .86,.7i, and .57. At the median value of b=.87observedin Table 2, the correspondingC-is .8i.

2. Residual variation by country. The ex-tent of the deviationof observedvalues of valueaddedper unit of labor in each country fromthevalues predicted by the regressionequations isshown in Table 4. Apart from errors of ob-servation, there are three main causes of thesedifferencesbetween the predicted value (9= a+ b log p ) and the observedvalue (y = V/L):

(a) Variations in efficiency, which affect Lonly.(b) Variations in commodity prices, whichaffect both V and W/P. The net effectdepends on the magnitude of the differ-ence (i - b).(c) Variations in the exchange rate, whichaffect V but not W/P.The following deviations in each factor areassociated with positive and negative values of(y - ):

Positive Residuals Negative ResidualsEfficiency Relatively efficient Relatively inefficientCommodity price High LowExchange rate Overvalued Undervalued

TABLE 4.- RESIDUAL VARIATION IN (V/L)BY COUNTRY aNumber of Industries

BetweenAverage Averageb Above +5% and BelowV (%) +5% -5% - 5s%io() (2) (3) (4) (5)United States 384I +5 I0 II 3Canada 3226 +5 9 II 3New Zealand i980 +4 7 7 8Australia I926 -I2 I 3 20Denmark I455 -8 2 6 i6Norway I393 -9 2 6 I5Puerto Rico II82 +22 9 I 8United Kingdom I059 -II I 4 I9Colombia 924 +I4 i6 2 6Ireland 900 -i8 0 2 I2Mexico 524 +3 2 I9 3 XArgentina 5I9 +I0 I2 5 7Japan 476 +7 9 5 9Salvador 445 + I2 Io I 5Brazil 436 +33 9 0 1Southern Rhodesia 384 -i8 0 2 4Ceylon 26I +7 5 a 6India 24I -23 0 2 i6Iraq 2I3 +I I 0 I

a Residual Ay = y derived from Table 2.b Arithmetic mean of (Ay/y).

Although we cannot separate these causes incountries for which we do not have observationsof relativeprices, the observedresidualshelp inthe interpretation of our previous results. Ofthe five countries analyzed in Table 3, theUnited States, Canada, and Japan have smallaveragedeviations, and hence little countrybiasis introduced into any conclusions based onthem. The United Kingdom and India havepredominantlynegative residuals in V/L, prob-ably due to undervalued exchange rates. Cor-rection of Table 3-B to allow for these possible

"This procedure is not strictly correct, since y is com-puted from an assumed value of a which is subsequently tobe corrected, but it is roughly valid since y is insensitive tovariations in a.


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238 THE REVIEW OF ECONOMICS AND STATISTICSbiases does not significantlyaffect the estimatesof relative efficiency,however.It seems plausible to interpret the systematiccountry deviations as due mainly to differencesin exchange rates or in the level of protection.The United States, Canada, and Latin Americahave predominantly positive residuals, probab-ly due to overvalued exchange rates and (inLatin America) high levels of protection.West-ern Europe and India have predominantlynegative deviations, probably because of rela-tively undervalued exchange rates. Variationsin exchange rates and prices introduce a biasin estimation only if they are systematicallyrelated to the wage rate, which does not seemto be the case.

Although another comparative study [5]strongly suggests the importance of economiesof scale, their effects are not apparent here inthe residual variation in V/L. Larger plantsize may account for part of the higher efficien-cy and positive deviations in the United States,but any such effect in other countries havinglarge markets is concealed by the other sourcesof variation.C. Effects of Price VariationOf the three sources of bias discussed in thepreceding section, the variation in commodityprices is probably the least important becauseit has a similar effect on both variables in theregression analysis. Since some data on rela-tive prices among countries are available, how-ever, it is desirable to test the magnitudeof theerror introducedby ignoring prices.When commodity prices are known, the re-gression equation of (8) should be restated asfollows, using the commodity price as the nu-merairefor both value added and wages:

log( a+ b log() (8a)If prices are uncorrelated with wages, theiromission affects the standard error but not themagnitude of the regression coefficient b. Ifprices are correlatedwith wages, the correctionin the estimate of C-would be given by an equa-tion similar to (32 ).15 For example, an inverserelation between wages and prices would raisethe estimateof C-for values of b less than i.

15If (PA/PB) = (WA/WB)t, then a = (b-f)/(i+f) if bis estimated from (8).

To test the quantitative significance of thiscorrection, we have been able to assemble datafor only two industries, neither of which cor-responds entirely either in coverage or time tothe original data.'6 Estimating b alternativelyfrom equations (8) and (8a) for the elevencountries available gives the following results:Eq. (8) Eq. (8a)

Furniture (260) .8i4 .780(.045) (.Io4)Knitting mill products (23 2) .692 .755(.035) (.039)

In neither case is there a significant differencebetween the two estimates. Although this testby itself is by no means conclusive, such otherevidence as is available on relative prices doesnot suggest that there aremany sectors in whichthe estimate of o- would be significantlyaffectedby this correction.

IV. FactorSubstitution ndtheEconomicStructureVariationsin productionfunctions among in-dustries have a substantial effect on the struc-tural features of economies at different levelsof income. In the present section, we shall in-vestigate the effects on factor proportions,com-

modity prices, and comparativeadvantage thatstem from differencesin the parameters of theCES productionfunction.16The sectors covered are both consumer goods, since wewere unable to find comparable data on intermediate prod-ucts for any substantial number of countries. The pricesused for sector 232 apply to all clothing rather than to 232only. The price indexes are as follows for the ii countries:

Price of Furniture Price of Knitted GoodsUnited States I00.00 100.00Canada 154-70 148.9IAustralia 8I.95 73-58New Zealand 94.82 I03.46United Kingdom 66.46 60.30Denmark 89.4I 77-I3Norway 94.3 7 89.90Argentina 223.00 139.0IBrazil 145.90 97.20Colombia I82.60 2 25.70Mexico I75.80 I24-58Data are taken from Internationaler Vergleich der Preisfir die Lebenshaltung, Ergainzungsheft Nr. 4 Zu Reiche 9,Einzelhandelspreise in Ausland, Verlag W. KohlhammerGMBH, Stuttgart und Mainz, Jahrgang, I959. The originaldata are in deutschmark purchasing power equivalents, fromwhich the implied prices indexes were derived by taking theUnited States as a base. The exchange rates used in convert-ing the prices to dollars were the ones that were used insection I.


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CAPITAL-LABOR SUBSTITUTION AND ECONOMIC EFFICIENCY 239To carry out this analysis, it is necessary tohave some indication of the values of the threeparameters n sectors of the economy other thanthose examined in section I, and hence to havesome direct observations on the use of capital.

For this purpose, we shall determine the pa-rameters in the production function from dataon comparable sectors in Japan and the UnitedStates. Although these two-point estimatesmayhave substantial errors in individual sectors,the over-all results of this second method ofestimation support the principal results of ourearlier analysis and lead to some more generalconclusions.A. Production Functions from

US..-Japanese ComparisonsThe United States and Japan were selectedfor this analysis because of the availability ofdata on factor use, factor prices, and commodi-ty prices in a large number of sectors.'7 Theyalso are convenient in having large differencesin relative factor prices and factor proportions.The errors involved in estimating the elasticityof substitution are therefore less than theywould be if there were less variation in the ob-served values. (For the data in section I, esti-mates based only on the United States andJapan differed by less than i O per cent on theaverage from the regressionestimates.)The elasticity of substitution can be esti-mated from these data by means of equation(24):

Xi (K/L) j rjXU (K/L) u Wu

ru Jwhere subscripts indicate the country. Thismethod of estimation has the advantage of uti-lizing direct observations of capital as well aslabor and of being independent of the varyingvalue of the efficiencyparameter y.The data for this calculation are taken frominput-output studies in the two countries andare summarized n Table 5. The main concep-

tual differencefrom section III is in the defini-tion of capital, which here includes only fixedcapital. The labor cost in Japan makes allow-ance for the varying proportions of unpaidfamily workers in each sector. The variation inrelative factor costs shown in column (4) isdue entirely to differencesin labor costs, sincethe relative cost of capital is assumed to be thesame for all sectors.The values of C-derived by this method varyconsiderably more than those derived fromwage and labor inputs alone in section I. How-ever, for the I2 manufacturingsectors in whichboth are available, there is a significantcorrela-tion of .55 between the two estimates.'8 Theweighted median of a- for these sectors is .93 ascomparedwith .87 by the earlieranalysis. Themedian c- is also .93 for all manufacturing. Theomission of working capital provides a plau-sible explanation of this difference, since thelittle evidence available indicates that stocksof materials and goods in process are generallyas high in low-wage as in high-wage countries.The elasticity of substitution between workingcapital and labor is therefore probably muchless than unity. This correctionis particularlyimportant n tradeand in manufacturingsectorshaving small amountsof fixedcapital.Since these two-country estimates are rea-sonably consistent with our earlier findings forthe manufacturing sectors, we will tentativelyaccept them as indicative of elasticities of sub-stitution in non-manufacturing sectors, withqualificationsfor the omission of working capi-tal. Here the most notable results are the rela-tively high elasticities in agriculture and min-ing, and the low elasticity in electric power.'9In trade, the omission of working capital prob-ably leads to a serious overestimate of the elas-ticity of substitution, while for other serviceswe have no comparabledata. The evidence ofrelative prices, however, suggests an elasticityfor personal services, at least, of substantiallyless than unity.

"7The compilation of these data on a comparable basishas been done by Gary Bickel, who is conducting an exten-sive analysis of the relation between factor proportions andrelative prices in the two countries. Further discussion ofthe data is given by Bickel [41.

18 In some sectors the correspondence between the in-dustries covered is very imperfect because the earlier esti-mates are on a 3-digit basis and cover only part of the 2-digit class.19 The transport sector involves a very large difference

in product mix, and the reliability of the estimate is doubtful.


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240 THE REVIEW OF ECONOMICS AND STATISTICSCHART I.-C.E.S. PRODUCTIONS FUNCTIONS

LLABOR

x-JAPANESE FACTORPROPORTIONS\D LU.S. FACTOR PROPORTIONS

\ / | PARAMETERS/ ~~~~~~~~~~~~A5 .25/ E C .8 .2

A / - x=2.0 D .8 .8A ~~~~~~~~~~~~~E .4 .05C

x~~~-A=S~~~~~~~~~~~~~~~~~~~~~~~~D

CAPITAL

On the basis of this comparison,we have con-structed the five isoquants in Chart i to illus-trate the range of variation in o- and 8. Sectors

in Table 5 correspondingapproximately o thesesets of values are indicated in Table 6. TheTABLE 6.- ILLUSTRATIVE COMBINATIONS OF cy AND 8

6 ExamplesA I.I5 .25 Agriculture, mining, paper, non-ferrous metalsB i.O .2 Steel, rubber, transport equipmentC .8 .2 Textiles, wood products, grain milllingD .8 .8 Electric powerE .4 .05 Apparel, personal serviceseffect of increasingo- in flattening the isoquantis shown by comparing E, C, B and A, whilethe effect of 8 on the capital intensity is shownby comparing C and D. The optimum factorproportions at average Japanese and UnitedStates factor prices are also indicated to illus-trate the discussionin the next section.

TABLE 5.- CALCULATION OF ELASTICITY OF SUBSTITUTION FROM FACTOR INPUTSAND FACTOR PRICES: JAPAN VS. UNITED STATESa

Capital Intensity ParametersEstimated Regression- Relative Factor Cost EstimateNo. Sector Xu XJ XJ/Xu (Wj/Wu X ru/rj) a0 of a(I) (2) (3) (4) (5) (6) (7)

I Primary ProductionOI, 02, 03 Agriculture I9.5I .367 .019 .036 I.20 .39604 Fishing 3.24 .490 .J52 *I33 .94 .20IIo Coal mining 4.87 .534 .IIO .093 .93 .i82I2 Metal mining I3.34 .566 .042 J107 I.4I .2I5I3 Petroleum & natural gas 40.57 .722 .oi8 .o96 I.7I .265I4, I9 Non-metallic minerals IO-37 .777 .075 .III i.i8 .260

II Manufacturing205 Grain mill. production 5.36 .549 .J03 .o6o .8i .286 .9I20, 22 Processed food 5-II .374 .073 .o6I .93 .327 .8223 Textiles 2.76 .340 .J23 .073 .80 .J59 .8I232, 243 Apparel .99 .329 .332 .07I .42 .055241, 242, 29 Leather products I.0I J.90 .I89 .o98 .72 .05I .8625, 26 Lumber and wood prod. 3.58 .3I0 .o87 .054 .84 .i98 .8727 Paper 7.3I .528 .072 .099 I.I4 .204 .9628 Printing and publishing 3.45 .I43 .042 .072 I.2I .092 .8730 Rubber 3.73 .332 .o89 .o84 .98 .I473I Chemicals 8.32 I.I25 .I35 *I57 .90 .325 .85321, 329 Petroleum products 38.I8 .360 .094 .J5I I.04 .550322,329 Coal 35.85 I.895 053 .I I3 I.35 .36533 Non-metal. min. prod. 5.95 *4I4 .070 .o84 I.08 .J97 .9534I, 35 Iron and steel 8.6o .986 .JI5 .II5 I.00 .273 .85342 Non-ferrous metals II.45 I.I5I .JOI .I23 I.I0 1.287 I.0I36, 37 Machinery 4.86 .469 .o97 .o83 .93 .i87 .8738I Shipbuilding 4.76 .477 .IOO .094 .97 I 74382 Transport equipment 5.oI .378 .075 .o83 I.04 .j69

III Utilities and Services5II Electric power 46.I3 IO.50 .228 .I64 .82 .8I96i Trade 5.93 .349 .o59 .079 I.I2 .I877I Transport I5.7I .3I6 .020 .io6 I.74 .I70a SOURCES:COlS. (I), (2), and (4) are taken from Bickel [4], based on U.S. and Japanese input-output materials; rjlru assumed to be 1.47 for all sectors.Col. (5) is calculated from equation (33).Col. (6) is calculated from equation (2oa).

Col. (7) aggregatedfrom Table 2 using the averageproportionsof value added in the two countriesas weights.


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CAPITAL-LABOR SUBSTITUTION AND ECONOMIC EFFICIENCY 24IB. Factor Costs and Factor ProportionsAs shown in equation (2oa), the variation infactor proportionsamong countries, and amongsectors in the same country, depends on o-, 8,and the relative factor costs. The U.S.-Japanesedata are used in Chart 2 to provide a graphical

CHART 2. - FACTOR COSTS AND OPTIMUM FACTORPROPORTIONS

(Logarithmic Scale)

20 -CHEMICALS LTEXTILESu100ON

APPAREL METAL FERRMININ METAL

w 1 < C / gPOWERAGIULTURE

2 - X/-U.S. FACTOR PROPORTIONS

-JAPNESE FACTOR PROPORTIONSI 0 2 5 10 20 K

CAPITAL-LABOR RATIO

illustration of both types of variation. Whenboth variables are expressedas logarithms, thecapital-labor ratio is a linear function of therelative factor cost:log x = a log + wog-.In the United States, wages vary relativelylittle among sectors of the economy and thevariation in capital intensity is due almost en-tirely to differences n 8 and o-. In Japan, how-ever, population pressure and underemploy-ment are reflected in large wage differentialsamong sectors. Chart 2 shows that low wagesrather than the production function account forthe low capital intensity in Japanese agricul-ture; if there were as small a wage differential

as in the United States, agriculture would be-come a relatively capital-intensive sector inJapan on this analysis. Similarly, high wagescontribute to the relatively high capital inten-sity observedin Japan in sectors like power andnon-ferrousmetals.As between countries, changes in the relativecapital intensity of sectors have great signifi-cance for international trade. Our results indi-cate that changes in the orderingof sectors byfactor proportions are normal rather than ex-ceptional occurrences. The only sectors that

are fairly immune to them are ones like powerand apparel that have extremely high or lowvalues of 8. A type that shifts its relative posi-tion a great deal is illustrated in Chart 2 bymetal mining, which is quite capital intensivein the United States and quite labor intensive inJapan because of its high elasticity of substitu-tion. For the less extremecases, wage differen-tials of the magnitude of that between Japanand the United States will cause factor reversalseven with relatively small differences n elastic-ity, but for smaller wage differencesthe rank-ing would be more constant.Despite the approximatenature of our find-ings, the evidence of quantitatively significantreversals in capital intensity is too strong to beignored.20The assumptionof invariance in theranking of commoditiesby factor intensity thatis used by Samuelson [ I3] and other tradetheorists appears to have very limited empiri-cal application.The varying possibilities for factor substitu-tion also have important consequences for theallocation of labor and capital at different in-come levels. If there were no such variation,the distribution of labor and capital by sectorwould correspondto the distribution of outputexcept for differences in efficiency.2' In actu-ality, there are significant departures. Risingincome leads to a declining share of primaryproductionin total output and to an even morerapid decline in primary employment becauseof the high elasticity of substitution.22On theother hand, the observed rise in the share oflabor in the service sectors as income rises isprobably due primarily to a low elasticity ofsubstitution, since the share of service outputdoes not appearto rise significantly [5]C. Variations in Efficiency and PricesAlthough they have received most attentionin trade theory, factor proportions are not theonly determinants of relative prices. A com-plete explanationmust also take account of dif-

'This aspect of our results and its implication for theproblem of factor price equalization will be discussed byMinhas in a separate paper.21 Sector growth models using the Cobb-Douglas functionare discussed by Johansen [8].22 In the case of underemployment, this tendency may beoffset by low wages and low efficiency in agriculture, as inJapan.


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242 THE REVIEW OF ECONOMICS AND STATISTICSferences in relative efficiency among countriesand industries, about which there is little sys-tematic knowledge. We first present measuresof relative efficiency n each sector derived fromthe Japan-United States comparison and thenexplore the combined effects of variation in allthree parameters on relative prices amongcountries.

i. Estimates of relative efficiency. Havingderived values of o- and 8 in section IV-A fromthe propertiesof the productionfunction alone,we can now estimate relative efficiency, yJ/yu,by introducing information on relative com-modity prices. The most direct method, whichwill be followedhere, is to use commodity pricesto determine isoquants for each country sepa-rately and then to derive the relative efficiencyfrom a comparisonof the isoquants. Othereffi-ciency implications of observed prices are con-sideredlater.

CHART 3. - THE METHOD OF CALCULATINGRELATIVE EFFICIENCY

LABOR

JAPANESEK RATIO

U K RATIOL

L- -2-L-2I

K2 K3K1 CAPITAL

This method of measuringrelative efficiencybetween two countries is illustrated in Chart 3.We observe factor proportionsand factorpricesfor the United States at point i and for Japanat point 3. The assumptionof neutralefficiencydifferencesenables us to derive o- and 8 fromthese two observations,as was done in Table 5,and thus to determine the isoquants througheach point. Given the labor input at point i(L,), we can derive the labor input on theUnited States isoquant at point 2 fromequation(I2) accordingto the followingformula:

v -1- L1 [8x P+ (I PYu -1- L2 [8Xj-P + (I -8) ] P (33)

where xu and XJ are the capital intensities inthe United States and Japan. Point 2 representsthe combinations of factor inputs required toproduce a unit of output with Japanese factorproportions and the United States efficiencylevel, yu. Point 3 represents the amount of la-bor and capital actually used in Japan to pro-duce the same output, i.e., the labor and capitalinputs per thousand dollars of output correctedfor the differences n relativeprices of the givencommodity. Assuming neutrality in the effectson the two inputs, relative efficiency can bemeasuredby the ratio of either labor or capitalused at points 2 and 3:Y.T L2 K2Yu L3 K3Given the properties of the CES productionfunction, the measureof relative efficiencydoesnot depend on the point chosen, and we obtainthe same result by taking L1jL4. The calcula-tion of relative efficiencies by this method isshown in Table 7 for all of the sectors from

TABLE 7. - CALCULATION OF RELATIVE EFFICIENCY:JAPAN VS. UNITED STATESaEfficien-Values for Chart3 cy Ratio

ISIC No. Sector LI L. L3 J/lyuPrimiary Production

OI-03 Agriculture .083 *468 3.740 .J3IO Coal mining .I66 .245 1.724 .I412 Metal mining .II7 .2 79 .876 .3 214, I5 Non-metallicmin. J143 .326 .583 .56

Manufcturing20-22 Processed foods .046 .083 .339 .2523 Textiles .104 .I45 .332 .4424I, 242, 29 Leather products .IIO .123 .291 .4227 Paper .o63 .112 .33 2 .3 23I Chemicals .05I .093 .269 .35321, 329 Petroleum prod. .025 .095 .077 1.23322, 329 Coal products .047 .203 .2I5 .9533 Non-metallic min.products .122 .150 .859 .17341, 35 Iron and steel .o83 .153 .17I .90342 Non-ferrous met. .058 .II6 .I69 .69a SOURCES:Li = U.S. labor input in man-yearsper $IOOOf output from BickelL4W.L2= Japanese labor input for the same output at U.S. efficiencylevel from equation (33).L3 = Actual Japanese labor input from Bickel [41.

yJl/8u = L2/L3.


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CAPITAL-LABOR SUBSTITUTION AND ECONOMIC EFFICIENCY 2 43Table 5 for which relative commodity pricesare also available.In manufacturing,the medianefficiency evelis .43 (or .35 weightedby value added) which isabout the same as the averageratio of Japaneseand American efficiency determined in sectionIII.23 In primaryproduction it is considerablylower, with agriculture and coal mining onlyone seventh of the American evel.Three factors may be suggested to explainthese differences n relative efficiency.(i) Limited natural resourcesdoubtless ex-plain a largepart of the lower efficiencyof capi-tal and labor in primary production in Japan.(ii) Competitive pressure in exports andimport substitutes was suggested in [6] to bea cause of more rapid productivity increases inthese sectors in Japan. Inefficientsectors (agri-culture, mining, food, non-metallic mineralproducts) produce for the home market inJapan and are protected by either transportcosts or tariffs from foreigncompetition.(iii) Relatively efficient sectors in Japan(petroleum products, coal products, steel, non-ferrous metals) are characterizedby high capi-tal intensity, large plants, and continuousproc-essing. There may be technologicalreasonswhyit is easier to achieve comparableefficiencylev-els under these conditions.Tests of these and other hypotheses regard-ing relative efficiencymust await similarstudiesfor othercountries.

2. Relative commodity prices. Since we nowhave estimates of all three parameters in theproduction unction,we can investigatetheirim-portance to the determinationof relative com-modity prices.To do this, we substitute representativeval-ues of wlr 24 for the United States, WesternEurope,andJapan in formula (28):

| ____) A 1_

P B W-8 YA ( A(WB A -

and use it to construct curves of constant rela-tive prices (isoprice curves). As indicated insection II-D-6, prices in our analysis refer onlyto the direct cost of labor and capital. Threesuch curves are shown in Chart 4. Curve ICHART 4. - EFFECTS OF C AND 8 ON RELATIVE PRICES

CIllrIE I1.8

1.6LOWER PRICES INHIGH-WAGE COUNTRY14 -

1.2- xApB rB

COMPARISONS P tA1.0 x B OF: A4I -JAPAN-U.S. .333\II-JAPAN-US. .17 7MI\EUROPE-US. .3 33.8 Cx xD

HIGHERPRICES INHIGH-WAGECOUNTRY

.2-

.1 .2 4 .6 .8 1.0

assumes the typical efficiency ratio of .33 be-tween Japan (B) and the United States (A),and equal prices. However, the average ratioof Pj/Pu actually calculated for the manufac-turing sectors in Table 7 is about .53, a valuewhich is illustrated by curve II. Since changesin efficiencyhave the same effect as changes inrelative prices, however, curve II can also beinterpretedas a line of equal prices and relativeefficiency of .i8. Curve III corresponds tocurve I for the Europe-United States compari-son, with an efficiency ratio of .33 and equalprices.The general principle illustrated by Chart 4is that high values of 8, a-, or the YA 71B ratiolead to lower prices in the high-wage country(A). A higher value of o- can offset a low value

23 None of the four examples in section III correspondsvery well to the larger industries used here; the main differ-ence in yJ/yu is in steel, which shows a much lower efficiencyin Table 3.24 Valuesof w/r assumedare: United States, 2I.3; Eu-rope, 5.o; Japan, I.42.


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244 THE REVIEW OF ECONOMICS AND STATISTICSof 8 to a considerable extent. Representativecombinationsof o- and 8 are also shown in Chart4, where the five illustrative sets of productionparameters of Chart i are plotted. For a con-siderable number of the industrial sectors inTable 5, illustrated by the range of pointsA-B-C, variation in the elasticity of substitu-tion seems to be more importantthan variationin 8 in determiningrelative prices.Actual price differences between Japan andthe United States are affected as much by thecost of purchasedinputs as by the value addedcomponent. Calculations for the ten manufac-turing sectors in Table 7 based on (28) give arange of direct costs in Japan of .3 to .95 of theUnited States value, but this element is onlyabout 35 per cent of total cost on the average.The average price of purchasedinputs in thesesectors ranges from .93 to I.70 of their cost inthe United States, which more than makes upfor the lower cost of the factors used directly.25An adequate explanation of the differences inrelative prices therefore requiresan analysis oftotal factor use rather than of the direct use byitself.26

V. SubstitutionndTechnologicalChangeA. Technological Change,Labor'sShare,and the Wage Rate

i. Historical changes in labor's share. Insection II-D-5, some implications of the CESproduction function for time series were de-rived. In particular (25), it was shown thatlabor'sshare was governedby the relation,=L (-) . (25a)V 7YUnder the assumption of neutral technologicalchange, the only parameter that varies is y.

For an elasticity of substitution less than one,

labor's share rises when wage rates increasemore rapidly than technologicalprogress. (Foro > I the relation is reversed; for o- = I, we arein the familiarCobb-Douglascase wherelabor'sshare is independent of both neutral techno-logical progress and wage-rate changes.) Thehistorically observed relative constancy of la-bor's share can be understood in these terms;labor'sshare is the resultant of offsettingtrends;and further, for o- not too far from I, it is arelatively insensitive function of them.27If in particular we add the assumption thattechnological change proceeds at a constantgeometricrate, we have [cf. (26)]wLlog - = ao+ a, logw+ a2t, (34)

whereao = crlog (I-8) + ( I-I) logyO, (35)a, = I-,a2 = -(I-). (35a)From estimates of a, and a2, it is easy, from(35a), to solve for estimates of X and of theelasticity of substitution, which is I-a,.Equation (34) was fitted by least squaresto thedata for the United States non-farmproduction,

I909-49, given in Solow; 28 it was found thata,= .43I and a2 =- .003. The correspondingestimate of o- is .569 and that of X= .oo8, whichcorrespondsto an annualrate of growth of pro-ductivity of i.83 per cent. This figure agreespretty well with most earlier estimates (Solow,[I4], 36, gives i.5 per cent; see also Abramo-vitz [ I ], I I).-We can test for the significanceof the differ-ence of the elasticity of substitution from itsCobb-Douglas hypothetical value of I, whichimplies that a, and a2 are both zero. The testthen is equivalent to that for the significanceof the multiplecorrelationcoefficient,which hasa value of .740; an F-test shows that, for 4I

25 Japan is somewhat exceptional among industrial coun-tries in its dependence on imported materials, and betweenanother pair of countries the variation in direct factor costmight be more indicative of the variation in total cost ofproduction.' The extent to which price differences between the twocountries can be explained by total factor use is analyzedby Bickel [4]. He shows that the average capital intensity(reflecting 3) combined with the ratio of total factor inten-sities, which indicates a weighted average elasticity of sub-stitution throughout the economy, gives a reasonably goodprediction of relative prices. This result can be derived fromequation (28) on the assumption that yJ/yu does not varygreatly.

27However, (25a) does not provide a true causal analysisof the changes in labor's share if the wage rate is deter-mined simultaneously with the other variables of the sys-tem. The wage rate may be treated as exogenous as in, forexample, Lewis's model of economic growth [ii], which isapplicable to those economies in which there is large ruraldisguised unemployment.28 The data were derived from the columns of Table iin Solow [I4] as follows: wL/V is obtained by subtract-ing column (4) (share of property in income) from i;w = (wL/V) (V/L), where V/L is column (5) (private non-farm GNP per man-hour); t is time measured in years

from I929.


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CAPITAL-LABOR SUBSTITUTION AND ECONOMIC EFFICIENCY 245observations, the value is highly significant.The alternative of capital-output and labor-output ratios which are fixed at any instant oftime correspondsto a, = i. Since the standarderror of estimate of a, is .042, this hypothesismust also be rejected.As in the cross-sectionstudies, there is strongevidence that the elasticity of substitution isbetween zero and one. There may be some in-consistency with the results of the cross-sectionstudy of manufacturingsectors in section I inthat the time series estimate of the elasticity ofsubstitution is considerably lower than thatfound in the international comparisons. How-ever, the time series data include services aboutwhose elasticity of substitution we know little.

Estimation of (34) by least squares must beregarded only as an approximation. There isan unknownsimultaneousequations bias in theprocedure. However, more accurate methodswould require a more detailed specification ofthe types of errors appearing in the marginalproductivity relation (34) and the productionfunction, a specification that we are not pre-pared to make. The results must therefore beregarded as tentative; the difficulties noted insection B below are undoubtedly related to thechoice of estimationmethods.2. A test of the productionfunction. As inthe cross-section study, we can try to test theproduction function implied by the precedingresults; this test, again, is a stringent one inthat it uses capital data which have not beenemployed in fitting the marginal productivityrelation. The fitting of (34) has yielded esti-mates of o- and X but only one relation (35)between y0 and 8. We now use capital data toobtain separate estimates of these parameters.Define,

Iq= antilog?from (35), since

P= (I-cr)po

(I-8c)If we let y = yo (io)t in the production func-

tion and expressyo in terms of q, we find, aftersome manipulation that,X- =8 ) (36)

where,XI = q (K )IPxt x2 K= (K) (36a)

As in the analysis of the internationalcompari-sons, the strictest test would be the constancyof the left-hand side of (36), which would im-ply an exact fit and simultaneously give an esti-mate of 8. In the absence of strict constancy,we can estimate 8/( i-8) as the average,X- X2; this yields the estimate, 8= .5I9.29If we measure time in years from I929, thevalue of ao was -.o8o, so that the value of yois .584. The production function for UnitedStates, non-farm output, I929-49, is given by,V = .584 (IOI83)t (.5I9 K- .756+ .48i L- .756)-1-322 (37)

We have tested this production functionagainst actual output; the fit appears satisfac-tory. Out of 4I years, the prediction error isnot more than 4 per cent of the predictedvaluein 22 years and not more than 8 per cent in 3I.The maximumerrors in prediction were - I3.3per cent (I933) and +IO.7 per cent (I909).It is further significant that all five of theyears in which actual output fell short of pre-dicted by more than 8 per cent were the depres-sion years, I930-34. This is reasonable ontheoretical grounds; the immobility created bysevere unemploymentof both capital and laborcauses inefficiency in the utilization of thoseresources that are employed (for a develop-ment of this argument,see Arrow [31). If thedepression years had been excluded we mighthave expected a still better fit.B. Relative Shares and theCapital-LaborRatioAnother test, with less satisfactory results,is that given by (23). In logarithmicform, theratio of labor's to capital's share is related tothe capital-laborratio by:

9 X1 and X2 are again calculated from Table i in [I4];K/L is given in column (6) (employed capital per man-hour); K/V = (K/L) (VIL), where V/L is column 5.


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246 THE REVIEW OF ECONOMICS AND STATISTICSlog( W ) = log(I-8) log (38)For p positive, the steady secular rise in thecapital-laborratiowould give rise to an increasein labor's share, but one which would be verymoderateunless p were large.Relation (38) was fittedby least squares; thedata were the same as those used in the preced-ing section.30 The estimate of p was - .095,whichimplies an elasticity of substitutionslight-ly greater than i. The standard error of esti-mate of p is .o98, so that the hypothesis of apositive value for p is not rejected, but thevalues are inconsistent with those found in sec-tion A.The reasons for the contradictionof the two

estimates are not clear. If (34) and the produc-tion function (37) both held exactly, then (38)wouldhave to hold exactly with the samevaluesof p and 8. Hence the discrepancymust be dueto the different assumptions about the errorsimplicit in the statistical estimation methods.This problem remains an open one for thepresent.Kravis ( [9], 940-4 ) has applied essentiallythe same method, in the form of (24), to datafor the entire economy (rather than only thenon-farm portions, as here). His estimate ofthe elasticity of substitution is .64, which ismuchcloser to the results of section A.

VI. ConclusionThis article has touched on a wide range ofsubjects: the pure theory of production, thefunctional distribution of income, technologicalprogress, internationaldifferencesin efficiency,the sources of comparativeadvantage. In partthis broad scope reflects, as our introduction

suggests, the fundamentaleconomicsignificanceof the degree of substitutability of capital andlabor. In part it points to a wide variety ofunsettled questions which are left for futureresearchand better data. (In part, no doubt, itis simply due to the large number of authors ofthe paper ) Since our work does not lend itselfto detailed summary,we content ourselves witha brief reprise of some of our findings, somespeculationabout others, and some suggestionsfor future research.30 wL/rK = (wL/V)/[i - (wL/V)].

A. FindingsWe have produced some evidence that theelasticity of substitution between capital andlabor in manufacturing may typically be lessthan unity. There are weaker indications thatin primary production this conclusion is re-versed. Although our original evidence comesfrom an analysis of the relationship betweenwages and value added per unit of labor, wehave interpreted it by introducing a new classof production functions, more flexible and (wethink) more realistic than the standard ones.Althoughwe beganourempiricalworkon thenaivehypothesisthat observationswithin a givenindustrybut for differentcountriesat about thesame time can be taken as coming from a com-mon production function, we find subsequentlythat this hypothesis cannot be maintained. Butwe get reasonablygood results when we replaceit by the weaker, but still meaningful,assump-tion that international differences in efficiencyare approximatelyneutral in their incidence oncapital and labor. A closer analysis of inter-national differences in efficiency leads us tosuggest that this factor may have much to dowith the pattern of comparative advantage ininternationaltrade.Finally, our formulation contributes some-thing to the much-discussedquestion of func-tional shares. If, on the average, elasticities ofsubstitution are less than unity, the share ofthe rapidly-growingfactor, capital, in nationalproduct should fall. This is what has actuallyoccurred. But in the CES production functionit is possible that increases in real wages be off-set by neutral technological progress in theireffect on relative shares.B. Speculation

In his original work on what has since cometo be known as the "Leontief scarce-factorpar-adox," Professor Leontief [iO] advanced tenta-tively the hypothesis that the United Statesexports relatively labor-intensivegoods not be-cause labor is relatively abundant when meas-ured conventionally, but because the efficiencyof American abor is somethinglike three timesthe efficiencyof overseas labor. In ournotation,this amounts to the suggestion that interna-tional differences in efficiencytake the form ofvariations in 8/a or, equivalently, of 8. We


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CAPITAL-LABOR SUBSTITUTION AND ECONOMIC EFFICIENCY 247have proposed instead the hypothesis that //ais constant across countries, while differencesin efficiencyare neutral. But we have also foundsome slight indications,in comparingJapan andthe United States, that the Americanadvantagein efficiencytends to be least in capital-intensiveindustries. This pattern, if it were verified,would seem to lead to an alternative interpreta-tion of the Leontief phenomenon. But it alsoopens wide the question of why this associationbetween differential efficiency and capital in-tensity should occur. Some possible explana-tions were mentioned in section IV-C, but thereadercan think of others. We may be missingsomethingimportantby excludingthird factors,or external effects, or the importance of grossinvestment itself as a carrierof advanced tech-nology into a sector.Another active area of economic researchwhere our results may have some interest is thetheory of economic growth. An as yet unpub-lished paper 31 by J. D. Pitchford of MelbourneUniversity considers the introduction of a CESproductionfunction into a macroeconomicmod-el of economic growth and concludes that atleast in some cases this am

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