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This PDF is a selection from an out-of-print volume from the National Bureau of Economic Research

Volume Title: Annals of Economic and Social Measurement, Volume 4, number 4

Volume Author/Editor: NBER

Volume Publisher: NBER

Volume URL: http://www.nber.org/books/aesm75-4

Publication Date: October 1975

Chapter Title: Production Functions for Industrial Establishments of Different Sizes: The Chilean Case

Chapter Author: Patricio Meller

Chapter URL: http://www.nber.org/chapters/c10420

Chapter pages in book: (p. 595 - 634)

.InnaIs / Ecu,mni (itu! .Soeja/ ,Ieasicri,iu'nt 44. 975

PRODUCTION FUNCTIONS FOR

INDUSTRIAL ESTABlISHMENTS OF DIFFERENT SIZES:THE CHftEAN CASE*

wr PATRIUJO MaLER

Lsing single eq uaiio'i ,nodel.s ICES and Cobb --Doughis), prodwi niH jiI1t( t(ijfls were t'',fini,ijed br ('/iih'anindustrial estahhshmenis : these estahli.thments were c/used according to their sizes. The purpose olthe paper is to examine if all esfahli.shnienss vi the same industr hare lilt' came or difierecii productionfunctions Chow tests and I/U' Iran.c!og production function were used to test this hi-pothesis .%hcrc'uit'r,the trend wit/i establishment SI:r of tile zec-hnohgicul parameters is examined.

I. INTROIuCTION

The main purpose of this paper is to study the technological characteristics ofindustrial establishments of different sizes. For each one of 21 Chilean industriesthe study analyzes the relationship between the size of establishments and thetechnology they utilize.

The main result can be summarized as follows: the empirical evidenceobtained supports the assumption of the existence of structural heterogeneitywithin the industrial sector--i.e., technology varies at the industry level as thesize of establishment increases. in other words, the industry is not a collection of"representative firms," and the size of establishment seems to be an importantelement in determining the technological characteristics of an industrial establish-ment.

The relation between establishment size and type of technology is examinedthrough the econometric estimation of production functions. The establishmentswithin each industry are first classified by size, and then production functionsare estimated for each size grouping of establishment. The breakdown of establish-ments by size groupings permits an examination of variations in technologicalparameters (product-productive factor elasticities, scale economies, elasticity ofsubstitution between factors) as establishment size varies. Such an examinationcan assist in answering several questions which have important economic policyimplications. How does the product-capital elasticity vary as the size of establish-ment increases? To what extent do economies of scale exist? What happens tothe elasticity of substitution between factors as the size of establishment increases(e.g., do isoquants become right angles or straight lines as we move farther awayfrom the origin)?

The validity of the methodology used, i.e., estimating production functionsfor establishments of different sizes within an industry, is verified by the Chowmethod and the translogarithmic production function method.

* This paper is a revised version of Chapter III of the author's unpublished Ph.D. Thesis "Pro-duction Functions and Efficiency Frontiers for Industrial Establishments of Different Sues. TheChilean Case. Year 1967," University of California. Berkeley. Jan. 1975.

595

The following paragraphs discuss the implicit theoretical framework employedto determine the technological characteristics ofdilferent firms within an industry.

Numerous economists have pointed out that a frequent phenomenon inunder-developed economies is the prolonged existence in a given economic sectorof firms employing substantially diverse production techniques. This phenoniençnis commonly called economic dualism.' Spaventa and others (see footnote I)have explained dualism in the following way. Some economies have had a non-homogeneous growth process in which an important segment of the system hasstagnated while the remainder has undergone sustained development. The conse-quence is the coexistence of two groups of firms with sharply different (l%'flamjccharacteristics which, in turn, have important implications for income distributionand labor absorption.2 The existence ofdiffercnt growth rates for different segmentsof an economy is, in itself, not a particularly striking fact. What is striking is theprolonged existence of dualism in underdeveloped countries.3 A wide range ofhypotheses have been formulated to explain the causes, consequences, and pro-longed survival of dualism.4 This study examines in some detail the phenomenoncalled intra-dualism, i.e., dualism within a given industrial sector as opposed todualism among different branches of industry or among diffesent economicsectors (e.g., agriculture versus industry).

The literature on economic dualism generally assumes an equivalenceamong size, modern technology, and efficiency. The prevalent premise is thatlarger establishments use more modern techniques and are more efficient thansmaller firms, but the equivalence among those three concepts is still an empiricallyopen question.

A. Pinto has generalized dualism by examining the feasibility and the conse-quences of more than two types of firms existing within an industrya situationknown as structural heterogeneity. Both Pinto and di Fillipo suggest that thecoexistence of different types of firms within a given industry results from theutilization of different technologies (where 18th and 19th century technologiescoexist with those of the 20th century). The technological differences amongfirms becomes a fundamental factor in explaining the labor productivity differ-entials among industrial establishments and the low rate of labor absorption by

C. Lutz, "The Growth Process in a Dual Economic System' (Quarterli Rtm'a. Sept. 958):L. Spaventa, "Dualism in Economic Growth" (Quarterlt' Rt't'iew. Dcc, 1959j S. H. Wellisi. TheCoexistence of Large and Small Firms: A Study of the Italian Mechanical Industries" (Quarti'rlvJournal of Eco,i.. Feb. 1957); T. Watanabe, "Economic Aspects of Dualism in the industrial Develop.inent of Japan' (Econ. Dc,. and Cult. Ch.. April 1965); A. Pinto. "Natuialeia c Implicaciones de a'heterogeneidad estructural' de Ia America Lalina" tTrirneszre Econ.. No. 145, Enero 1970): R R.Nelson, T. P. Schulti, and R. L. Slighton, Structural Change in a Det'elopnig Economy (Princeton UnivPress, 1971).

2 D. Turnham and I. Jaeger, The' Emp!orment Problem in Less Dcie'lope'd (4,i,nlrics (OECD.Paris l97lj; C.1.E.S.. "El Emplco y el Crecimiento en Ia Esirategia del Desarrollo de America Latinalmplicaciones para Ia Decada de los Setenia" (VII Reunion Anual dcl C.I.E.S.. Sept 1971)Leibenstejn, "Technical Progress, the Production Function and Dualism' (Quarte'rliRenew. Dec. 1960); T. Watanabe. op. cii.In addition to the bibliography of footnote I, additional bibliography with a survey on thesubject is found in H. Ellis, "Las Economias Duales y el Progreso" (Re'iista il" Eo,iomiu Ltino'americana No.3. 1961).

596

the industrial sector.S Structural heterogeneity as described by A. Pinto corres-ponds to R. Nelson's arguments that the use of a single production function tocharacterizerize an entire industry in an underdeveloped country conceals morethan it reveals, and that labor productivity differentials among industrial establish-ments cannot be simply explained by varying capital-labor ratios.ô

In operationalizing the structural heterogeneity hypothesis, the sue ofestablishment has been used as the fundamental variable in determining the typeof technology used by a fIrm. Furthermore, as previously mentioned, establish-ments of each industry are classified by size variable, and production functionsare estimated separately for each size grouping of establishment.

The study, finally, calls into question the utility of production functions as amicroecOnOrnic tool. l'he economic literature Contains many articles concerningthe variety of results obtained from econometric estimations of production func-tions. The sizable variations obtained in the values of the difl'erent technologicalelasticities (product-productive factors, economies of scale, factor substitution).should lead one to question their economic meaning. Why do the estimatedparameters have such great instability? Are there any economic hypotheses whichare not fulfilled'! If so, which are they?

The econometric estimation of production functions in this study permitscloser examination of the validity of economic policies based on the results ofproduction function analyses, particularly results concerning economies of scaleand elasticity of substitution.

Given the data available (i.e.. cross-section series), only a static analysis ispossible. Then when variations of different technological and economic variablesin the size groupings of establishment are examined, the consequences of thedualistic phenomenon at a given moment of time will be determined.

The data used in this study come from the Chilean Industrial ManufacturingCensus of 1967 disaggregated at the establishment level (11,468 establishmentsemploying 5 or more persons). For the purposes of this study, 21 industries willbe selected at ISIC International Standard Industrial Classification), four digitclassification level, and a separate analysis will be made for each of the 21 industries.

2. MErnouowGY

Different productive techniques are used by a set of firms of the same industryat a given point in time. 'rhe implicit assumption that firms in the same industryuse the same production function does not help to explain what happens inreality: on the contrary, it probably obscures a lot. Once the assumption ismade that all firms in the same industry use the same production function, theproductive techniques selected are a function of relative prices of the productive

A. Pinto y A. di Fillipo. "Notas sobre a Estralegia de Ia Distribuoo y Ia Redisribucion delIngreso en America Latina" (Semin. Internac de Distr. dcl Ingreso, Ceplan. mimec, Chile, Marzo1973).

6 R. R. Nelson et al.. op. cit.

597

factors, and the combination of these productive factors explains the (liflerencesin the productivity of labor.7

The working hypothesis utilized here is that the production functjoji variesfor establishments of different size.

The existence of different production functions within the same industrywill allow us to examine more closely some of the so-called "dual" characteristicsobserved in the industrial sector of an underdeveloped country. The term dualismis used here in the sense of intra-industrial, that is, to refer to the coexistence offirms using modern and old techniques in the same industry. The usual assumptionis that the largest firms are the most modern while the smallest are the mostbackward.8 With the existence of different production functions it is possible toexamine the empirical validity of the dualistic Concept.

The production function will be used as a tool to permit us to present in acompact form the most important technological characteristics of an industry.We will also try to identify the degree and types of variations of those technologicalcharacteristics for different size establishments within the same industry.

The method of estimating production functions used will be similar to thatused by Griliches and Ringstad,9 with some small variations. Each of the 21industries will be divided into 5 categories of different size-establishments: 5 to 9persons employed, 10 tc 19,20 to 49, 50 to 99, and 100 or more people employed,'0Then we will proceed to estimate production functions (Cobb-Douglas and CES)in each four-digit industry for each individual establishment grouping and for theindustry as a whole.

The criterion followed to estimate the different production function elasticitieshas been to employ a model which allows estimating each elasticity as a firstorder parameter. Even though every estimation model is different, all modelsused are very similar and it is assumed they are close approximations of the truemodel. Using that functional form which has a comparative advantage in theestimation of each elasticity is, in our judgment, the approach that will minimizethe unstable fluctuations of the estimators.

The Cobb-Douglas function was used to obtain estimates of the product-capital and product-labor elasticities, Kmenta's linearization of the CES was

1R. R. Nelson etal., op. ci:., pp. 91 92; B. S. Minhas. .4,, Inter,,,tjmu,j ('onrparLo,, ol hutor ("oiand Fu,'tor Use(North44olland 1963). pp. 30--3lTraditionally, the relationship between modern firm and large sue and backward lIrni andsmall size !s based on the following. New production techniques require a large solume of productionin order to take advantage of economies of scale they are very capital-intensive which means a largeinvestment, and are developed in the more industrialized countries in order to operate in larger markets.etc. Small firms use production techniques that are more labor-intensive, which is the factor moderntechnology tries to save, given the factor endowment prevalent in industrialized Countries which arethe ones in the vanguard of technological progress, J. Governeur. Productiri:v and Factor Proportionsin Less Dete!oped Counlrie.s(Clarendon Press, 1971); B. Singh. The &'oiumies of Small.S calt' !,,,1us,ri's(Asia Publishing House, 1961): NI. C. Shetty. Small-Scale and Household !sslusiries in a DecelopingEconomy (Asia Publishing House. 1963)Z. Griliches and V. Ringstad, Ecc,ioni it's of Scale and the Form f the Production Function(North-Holland, 1971).

'° See P. Meller, Ph.D. Thesis, op. ci!., a discussion of different variables that could be used forclassificatory establishment size purposes. The main empirical conclusion is that "when an establish-ment is large, ii is generally large in all its dimensions"_i,e. size groupings will not change very muchby the use of different size variables,

598

used for the economics of scale estimator. The traditional CES of Arrow et al.,''was used to obtain an estimator of the elasticity of substitution.'2

It is interesting to note that if either of the two traditional production functionsis used, i.e.. CES and/or CobbDouglas, the hypothesis that different-size firmspossess different production functions is compatible with the usual assumptionin economiC theory of the existence of a U-shaped average cost cur'e. The returnsto scale revealed by production functions of different-size establishment will bea test of the validity of the existence of a U-shaped average cost curve for anindustry.

The ordinary least-squares single-equation method is used to estimate theparameters of the production function. '"

The industrial establishments have been subjected to a rigorous selectionprocess. (See Appendix.) The criterion of selection has been the quality ratherthan the quantity of observations. The use of this criterion results in the exclusionof establishments in which some of the variables that are used in the econometricestimates are doubtful or whose data should be omitted.

In relation to the problem of measurement of variables after a series ofexperiments with alternative measurements, we have adopted a procedure similarto the one used by Griliches and Ringstad.'6 Labor requirements were measuredby the number of "equivalent" man-days employed by the establishment, andcapital requirements were measured by a flow of capital services obtained byusing book values (see footnote 16).

K. J. Arrow et al., "Capital-Labor Substitution and Economic Efficiency" (Re,. of Econ. andStat., August 1961).

12 Several other functional forms were estimated: Generalized CES form (J. Katz. ProductionFunctions. Foreign Inuestnient and Growth. North-Holland, 1969): I-lildebrand-Liu and Nerlove IM.Nerlove. "Recent Empirical Studies ol the CES and Related Production Functions." ed. M. Brown.The Theory and Empir. Anal, of Prod., N.B.E.R., l967) V. Mukerji("A Generalized SMAC Functionwith Constant Ratios of Elasticities of Substitution," Rev, of Ec. Stud.. 1963).

IS A. Walters, An Introduction to Econometrics (Macmillan, 1968), p.290.14 The estimators obtained by least-squares single-equation have many advantages: simplicity

of computation, small standard errors of the coefficients, and a high level of efficiency for prediction.But they also have many disadvantages, and the main one is that if the true model corresponds to asimultaneous equation model, then the least-squares single-equation estimators will be both biased and

not consistent. See A. A. Walters. "Production and Cost Functions: An Econometric Survey" (Econo-inetrica, Jan. 1963). pp. 18-22.

iS See P. Meiler, "Efficiency Frontiers for Industrial Establishments of Different Sizes" (Explora-lions in Economic Research, forthcoming), for an extensive discussion of the possibility ofestimating

production functions from cross-section data: i.e.. the factors explaining the existence of diffientproductive techniques in a particular industry are discussed.

16 Z. Grilches and V. Ringslad. op. cit., pp. 22-29. The variables are measured in the following

way:1: value added measuied in E-1967 (escudos of year 1967).L: labor factor measured in number of equivalent man-days, where the number of equivalent workers

is: L, = m1 (w2/w1)m2 + (2w2/w1)m3 in,. in2 and m3 are the number of blue collar, white collar

workers, and entrepreneurs. and w1 and w2 are the average wages received by blue and white

collar workers.K: capital factor measured in E°-l967, computed as a flow according to the following expression:

K = O.IOKM + 0.03K1 0.2OKr f 0.lO(Ku + K1 + K K,), where KM K1, K and K, are

the book values of machinery, buildings, vehicles and inventory goods. Linear depreciation rates

of 0.10, 0.03 and 0.20 have been used for machinery, buildings and vehicles, and a 10 percent real

interest rate is used as an alternative cost for immobilized capital.

599

LThe labor factor could have been measured through the number of man-days(designated by N1) without using that transformation to equivalent worker whichwould consider [he factor of difFerences in quality of labor. The simple co, relationcoefficients between L and N1 for the establishments within the 21 industrieshave values close to 1.0 (see Table I). This indicates it does not make any difFerencewhether the labor variable is measured by the number of man-days or by thenumber of equivalent worker-days.

TABLESIsu'I.E ('ORRELATION COFFIi('IENlS Bi rwi:i N nil ClrosN MFASURI:..MENTS ANI) AlTERNATIVE MEASUEEMFNTS OF Till LAIIoR ANt)

CAPITAL FAC1ORS SEPARA tEl) BY I NOUSI RY

Note L: number of equivaleni man-days: N1 number of man-days K flow of capital serices(see footnote 16); Ks,: book value ofniachinery KKW: number of KWh; K,,: number ofinslalled lip;K5: sum of the book values of machinery. buildings, vehicles and inventory goods

Different alternative measurements (proxy variables) could be used for thecapital variable. Among these are two flow variables: the previously definedcapital services, called K, and the number of KWh of consumed electricity, KK:also, the following stock variables could be used: the number of lIP installedcorresponding to the machinery related to the production process, K11, the totalbook value of the fixed assets plus the stocks of goods and inputs, measured inE° 1967, K, and lastly, KM, the book value of the machinery, measured in E°estimated to be the most reliable book value provided by the establishments.Above we present a table of simple correlation coefficients between K and thedifferent measurements of the capital factor for each one of the 21 industries600

Type ofIndustry

'SICCode L.N1 K:KM K;K K:K,, k:iç3111 0.969 0.959 0.806 0.597 0 9963112 0.992 0.983 0.892 0.874 0.9993116 0.981 0.972 0.622 0.680 0.9973117 0.984 0.970 0.866 0.837 09%3121 0.982 0.949 0.560 0.806 09973131 0.960 0.824 0.152 0.488 0.9993211 0.996 0.996 0.925 0.865 09983213 0.992 0.994 0.839 0.888 0.9953220 0.984 0.971 0.864 0.326 0.9973231 0.981 0.961 0.904 0887 09973240 0.989 1)985 0.575 0.641 09923311 0.988 0.913 0.675 0.707 0.9953320 0.985 0.942 0.875 0.875 0.9993420 0.884 0.992 0.949 0.672 0.9993560 0.990 0.994 0.927 0.880 0.9943693 0.984 0.936 0.558 0.926 09943710 0.989 0.989 0.697 0.840 09983813 0.989 0.986 0.887 0.536 0.9983819 0.983 0.989 0.810 0.867 09993829 0.976 0.992 0.728 0.790 09Q93843 0.973 0.832 0.864 0.389 0.995

(Table I). Most of the correlation coefficients are significant at the I percent level(see appendix for number of observations per industry).

Another variable needed for econometric estimation is w, average wages.In this case, w includes average wages received by blue and white collar workersworking in a given establishment, plus social legislation payments (employer'scontributions, bonuses, and child allowances) i.e., w represents the establishnient'scost of labor.

3. Esiistrtor'J OF 11Th VALUE ADDED-PR0I)ucTIvF FACTORS Ei ASTICITIES

To obtain estimators of elasticities of value added-producti.c factors onlythe CobbDouglas function was used because certain of its properties seem moresuitable in this case. Specifically, this means :1. ProductproductjvefactOrelas(icjtjescan be obtained directly as parameters of the first order: ii. Those elasticities areconstant for any production level (not the case for the CES) and iii. It is notnecessary to make assumptions regarding the type of market structure nor aboutthe firms decision-making policy.

We use the following notation for the Cobb-Douglas function:

Y = ALKwhere A, a, and are parameters of the functions and Y, L, and K corrcspondto aggregate value, labor, and capital respectively. Dividing this expression by Land denoting the parameter of economies of scale Iz a + fi I, we arrive atthe form traditionally used by Grilichcs :17

log = log A + b log + It log L.

The advantage of this form is that, in addition to obtaining the product-capital elasticity's directly, we also obtain directly the parameter It, i.e., theeconomies of scale. It is possible to verify directly whether the function has ordoes not have constant returns by testing the null hypothesis: if : h = 0.

The labor-capital elasticity, a, is found using the estimators of fi and !i,and the equation of definition, h = a ± fJ - 1. The estimator a found by thisindirect method is not biased if 1 and are unbiased estimators.'

The division of establishments into different size groupings was done todetermine whether variations occurred in different parameters of the productionfunction across different-size establishments. Let us see how fi varies as the sizeof the establishment is increased.

'Z. Griliches, "Production Functions in Manufacturing: Some Preliminary Results." TheTheory and Emp,rical Ana/v,js ofProducilon, op. cii., pp. 275-340; (iriliches and Ringstad, op. cit.,p.63.

IS Jf Y = .4LK0, Y/K = .4pL'K l

= (K/Y)(PYPK) = .4flLK'/Y ='

Let ft and 1 be the estimators for fi and I,. which are found by using the method ofteast squaresThen, if I + & - ft

E-E(l + -ft)= I +E1,-.Ejl= I + h ... =

601

(I)

(2)

The following table contains estimators for /3 calculated as a simple arithmeticaverage20 of the individual values of each of the 21 industries, for each sue groupof establishments

TAH[.l 2AVERA6I VAJUIS lop mr AooRii;ATr VAI.t:i:-CAi'iFAI 1tASJi(I I Y liv Si,'i 4)1

ES IAIIIISIIM NT

5(09 10 to 19 20 to 49 50(099 ll)orfllorepeople people people people people

As can be seen, the two largest size grouping of establishments have averagevalues far greater than the sma1er groups. This suggests that if the marginalproductivity of capital were t'ie same across firms, the smaller establishmentspossess a larger average Y/K (value added per unit of capital) than would thelarger establishments.2'It is difficult to find a definite trend of stable behavior for the large majorityof industries. However, in 16 of the 21 industries, the largest values fort? lie in thetwo classes containing the largest establishments, while in 15 ofthe2l industriesthe lowest values for /3 lie in the two classes with the smallest establishments.In the majority of the industries the largest establishments have larger valueadded-capital elasticity than the smallest establishments: however, there are caseswhere the situation is reversed. Even within the same industry the coefficient

varies considerably (see Table A-4): this behavior seems to suggest that the useof the same CobbDouglas function for a set of establishments within the sameindustry is not valid.Frequently, relative participation of the productive factors in the aggregatevalue, SK and SL are employed as estimators of the respective elasticities of outputwith respect to factor inputs, i.e. and This procedure is theoreticallyjustifIed assuming a firm has a CobbDouglas production function, enjoysconstant returns, faces competitive markets in goods as well as factors, and usesthe criterion of profit maximization. In this case:

= = /3 and '}L = =I will try different types of tests to check the hypothesis that the /3 estimatoris equal to the share of capital SK. Even though the different tests give contradictoryresults, my own conclusion is that there is no relationship between I? and SK.

20 have calculated the eIasticitj for each size grouping of establishments by using a simplearithmetic average of (he values obtained separately for each size group in each industry. Alternatemethods for obtaining this value would be (1) to find an estimator by using a regression for the set ofestablishments of a determined size group; (2) to calculate a weighted average of the values obtainedfor each industry separately. Either of these two alternative methods implies a different weight givento different industries. In this study I try to examine the characteristi of establishments of differentS1Zes in general and wish to avoid the influence of what might happen in one particular industry on thegeneral conclusions Thus, I gave the same weight to each industry, no matter what size it has.Negative values of f3were excluded, but their inclusion would not have changed the ranking of ft21 This would imply that if capital is the only scarce resource, small firms should be preferredbecause they maximize the output by unit of capital.

602

Average values of/ 0.401 0.34! 0.40k 0515 0

Table A-4 of the Appendix denotes values furnished by SK and fi. A secondtable where c is obtained indirectly was omitted because its values were veryunstable, even occasionally negative, while other values were much greater than1.0. Both of these situations are difficult to ilccept I[Oifl an economic point of

view.22In Table A-4 the values for S and fi are separated by industry for each one

of the five establishment size groupings and for the whole industry. It is difficultto determine an stable relation between values S and /3. In the following twoTables, 3 and 4. we calculated the coefficients of correlation between SK and j,separated by industry and by establishment size.

TABLE 3SistLL CORRELATION COEFFICIENTS BETWEEN

ANt) S SIPARA1FU IIY INDUSTRY

Type of Industry Correlation Coefficients

3111 0.600*

3112 -0.7903116 0.328*

3117 0704*3121 0353'3132 0.351'3211 0.291'3213 0.463'3220 0.071'3231 0.163'3240 0.184'3311 0.007'3320 0.340'3420 0.684'3560 0.475'3693 0.041*

3710 0.118'3813 0.677'3819 0.008'3829 0.091'3843 0.415'

* Correlation coefficient values not signi-ficant at 5 percent.

Finally, the correlation coefficient between (3 and SK taken at the industry

level for the set of 21 industries was 0.278.From Tables 3 and 4 and the previously given correlation coefficient. it is

indicated that there is no linear association between fi and SK. In other words. it

appears that (3 and SK measure different things, a result also encountered by

Griliches and Ringstad.23 However, in order to render a conclusive judgment.

I would test the null hypothesis: H0:fl SK for each one of the cases analyzed,

22 A negative value added-productive factor elasticity would stem from the fact that the marginalproductivity of that factor is negative. i.e.. a greater use of that factor implies a decrease in the value

added and would result in an irrational employment of that factor.Griliches and Ringstad, op. cit.. pp. 73-75. Griliches and Ringstad arrive at this conclusion

using rank correlation coefficients between a and S,.

603

J

5 to 9lOto 1920 to 4950 to 99100 and more

\Vhote lndustry

101Al.

tAitli 4SIMI'tt ('oRRtistt (IIIEII(II\I5 III Wi! N

AN!) S5 Sip.AitAlfI) BY ESIA!!! INJIMIN t Si,tSue of Establushnieni Correla to!) ( oellcjcn k

S to 9 person -- 0.055k10 to 19 persons OI)5220 to 49 peisons 0 299'SOlo99persons --0.121'tOO or more persons 0.295'

Correlation coefficient values not signilicantat 5 percent

NUMBER ot TIMES IIIATts A(cEt'TEI) OR RFJF(-j

Sue of Estabhshinent

TABLE 5nut: Nuti. FivIw titsis !f: /1

It) AT A Euvtj ot S!(;N:ti(AN(, o.5 PERCENu

/15 Rejected lmpos1ble to Reject !I

S

9644

IS1414

13

44 76

using the t values given in Table A-4.24 The results of this test are given in Table 5.This table suggests that it is not possible to reject the null hypothesis. Therelative share of the capital factor. S, is a good estimator of capital-aggregatevalue elasticity, at least for 63.3 percent of the cases. However, this hypothesis wasrejected in those industries or size groupings containing the largest number ofobservations Therefore, an increase in the number of observations in each casecould make possible the rejection of the null hypothesis H0 fl =r SK in a greatnumber of cases. From an empirical standpoint, the estimator of product-capitalelasticity S is inadequate for the industry as a whole, due to the rejection of thishypothesis in 12 0121 cases examined and at a significance level of 5 percent.is greater than estimator fi in 74 percent of the cases. Then, when SK is used asan estimator of product-capital elasticity, this elasticity is Overestimated in 74percent of the cases and underestimated in the remaining 26 percent of the cases.The magnitude of overestjnlatjon (and/or underestimation) fluctuates considerablyand no regular pattern of behavior emerged.But, even when the hypothesis H:f3 = S, is rejected for a large number ofcases, it does not imply the rejection of the basic assumptions such as perfectcompetition, constant returns, and profit maximization s, is empirically biased- Thj I test faces a serious objection due to the fact that S ts also a random ariahJe therefore

the i values obtained wilt be over-estimated making it easier to reject null hypothesis

604

I

upward because profits earned by the industry are iiiclttki When determiningthe relative shares of the capital factor in the aggregate tltii'. This is a partialexplanatiOr of why SK is greater than fin the maJorit of cases observed, I loweimperfections in the commodities and factors market also contribute a priorito make SK greater than II (in a short run model),25

4. EsriMvrio OF liii: E('oNo1fls 01: S'\1.E

The jaranieter for the economies of scale was estimated using tWO difl'ercntmethods: the Cobb-Douglas function and the linearization of Kmcnta of thegeneralized CES function.2

The notation for the generalized CIiS function is as follows

where ', . p. p are the CES function traditional paranletcrs.2Using the linearization of Knenta (expansion of(3) by a Taylor series around

the value of p = 0), the regression equation becomes:

log Y/L. = log ; + p log K/I. + It log L -F a0[iog KL]2

in which It p - I, being the scale elasticity. The expression (4) is equivalent to(2) plus the term a0[log K/LI2. Thus by testing the null hypothesis H0 a, - 0,we can determine whether the Cobb-Douglas function is or is not acceptable asan estimation model.

The economies of scale parameters estimated by the Cobb--Douglas andKrnenta functions are presented in Table A-5. In most cases the values obtainedwith the two indicators are highly similar. For this reason I need only examinethe type of variation evidenced by only one of them. I chose the Kmenta approx-imation because according to Monte Carlo's studies, the biases found in theestimator of the parameter for economies of scale arc insignificant when thisspecification is used.28

25 A model with imperfect competit ion in the market of goods and profit ma \Irnh/at ion h the

firm glcs: a/S, = qI = -- ,i sthere ; is the price eIasticit of the products demand. As and N, are

positive, for i to be negative t is necessary > N. We reach the same condition through a model of

imperfect competition in the labor market asailable to the firm. If we start from the ret9rns to scaleh and a and we calculated /1 (in this study the estimated elasticity), we woald obtain it 5 Fhis

explanation is not at all satisfactory from an empirical point of view, because 45. percent of the 't

elasticities would be "positise." (a has been indirectly obtained in this study IThe economies of scale parameter is obtained as a first order parameter. I his parameter is

constant throughout the whole estimation range and It S not necessary to make an h PC of issiiiiip-

tions about the market structure and or the firm's behavior.ix the efficiency parameter, ) is the distribution paranicter. p is he stihstituiioil parameter.

and p is the economies of scale parameter.1 G. S. Maddala and J. B. Kadana. "Fsfintion of Returns to Scale and the Elasticity of Sub-

stitution" (Econometrics., July 1967). pp. 421 422.

60

Y = '[( I -- O)L -I- oK

Excluding those values for economies of scale where IIj > 1.0. the figures inTable A-5 reveal the following information

TABLE 6NUMDFR OF EST:MA[ORS WHICH INI)i('ATi E(ONOSI!IS OR DISF(ONO,,1Ils (0 .Scr in

Sizr ot ESTARIISIMI Ni 1Ok i iw Sir OF 2! lNmsrkli s

Sue of t:siabi i.hnicniIn(luslr\

5 9 0 19 20 49 51) 99 1(X) Totalor more

Number of estimatorswhich mdicate economiesof scale

Number of estimatorswhich indicatedisccononiies of scale

2 3 5

16 5 IS S

11 we had estimated production functions for the industrial sector as awhole and for each industry separately, the conclusion inferred from the empiricalresults would be that nearly all Chilean industries are found to have economiesof scale. In some cases, these economies of scale reach 80 percent and their average(simp'e arithmetic mean) is 26.4 percent. These results might suggest that goern-mental policy should favor the appearance and survival of the largest establish-ments within the industrial sector in order to take advantage of these economiesof scale. However, this type of result is highly questionable, and below we willprovide some reasons to support our doubts. If we examine each industry separately,divided into different size groupings, and observe the Ti estimators obtained foreach one of those classes, the results are, in general, quite strange:

First, it is surprising to observe that even in the establishments of smallestsize, 16 estimators out of a total of 18 indicate the presence of diseconomies ofscale. A similar result was obtained in the two next larger classes: as a result, forthe three smallest size establishments (5 to 49 persons), a 10 percent increase inthe productive factors L and K, increases the aggregate value by less than 10 per-cent (for 15 of the 21 industries).

In a large number of industries each one of the size groupings evidencesa negative Ti estimator, although positive economies of scale are obtained for theindustry as a whole. One possible explanation for this phenomenon could be theone pictured in Figure 1, where we observe increasing average cost curves foreach size group of establishments and a decreasing average cost curve (economiesof scale) for the whole industry.

In various industries the Ti estimator increases with the size of establish-ment, but goes from negative values to positive values. This could be an excellenttest for proving that the average cost curve in various industries "takes the formof an inverted U."

An easy way out of all these strange results is to verify the null hypothesis:As shown in Table A-5, the null hypothesis H0:h = 0 is accepted in 77.8 percentof the cases considering only Ti estimators obtained by the approximation of

606

() 20

[

I

(5)

Figure I

Kmenta for different size groups. This suggests that the hypothesis of constantreturns to scale for each size class of establishments is rejected in only 22.2 percentof the cases (with a significance level of 5 percent). Meanwhile, at the level of thewhole industry, the hypothesis of constant returns to scale is relected in 14 of21 industries, with a significance level at 5 percent. Finally, as can be observedin Table A-5, in each size grouping almost 70 percent of the estimators indicatethe presence of diseconomies of scale. Diseconomies of scale prevail in Chileanindustry at the level of each size grouping of establishments.

In summary, not much confidence should be given to the results obtained forthe economies of scale parameter.

5. ESTIMATION OF THE ELASTICITY OF SUBSTITUTION

The traditional Arrow ci al. method (ACMS)29 was used to estimate theelasticity of substitution a.°

The estimation model is:

Ylog = a ± a log w.

The implicit economic assumptions in this model are that the establishmentoperates in a competitive market (goods and factors), maximizes profits, andthat there exist constant returns to scale.

Values of a will be provided at the total industry level and at the estabJishment-size level. At the end of this section, the trend that a has across an industry isdiscussed,

(i) Values ala at total industry level. For 14 o121 industries values of elasticityof substitution a fluctuate between (18 and 1.2. Except for 2 cases, the total of in-

Arrow et at., op. cii.° Two other methods were used to obtain estimators for o as pointed out previously the Katz

and Nerlove methods (see footnote 12). but they were rejected because of their theoretical limitationsand empirical inconsistencies. See P. MeIler's Ph.D. Thesis, o,. cit.. for a lull discussion of these issues.

607

V

\ (ILiStries has a values over 0.65. Only one negative value is obtained at the industrylevel and only 2 of the 20 positive values are not significant at 5 percent. (Table A-fl.)

(ii) Value.s oja at esiabjishpnent-s,ze let'el. Taking a simple arithmetic averageof the elasticity of substitution3' for each size grouping of establishment I obtainthe following results:

TABLE 7Anmisicric AVFRAGFS OF SuBsTITuTIoN ELASTICITIES FOR SI/f CiROtJi'IN(;S

OF ESTABLIShMENT

10059 10-19 20-49 50 99 and more

Values of a 0857 0839 0.764 0.670 I 266

There is a clearly decreasing tendency in the a values in establishments of5 to 99 persons (momentarily excluding the largest Ones). Considering these fivesize groupings, we can think of a U -shape relation between the a values and thesize groupings of establishments. This coincides with the results obtained by Abein the study of Japanese firms.32 Establishments of 50 to 99 persons probablyhave a more inflexible technology, while larger establishments probably use moreflexible techniques.

This empirical result also supports Leibenstein's thesis33 that for any giventype of technology, the instrument of relative prices will be more likely to affectthe choice of techniques in larger establishments than in smaller ones, becausesmall variations in the relative prices of productive factors will produce a relativdygreater impact in those establishments using more elastic techniques from thesubstitution point of view.

This conclusion requires the following qualification: the first four sizegroupings of establishments from 5 to 99 persons could be thought of as corres-ponding to size groupings of establishments approximately comparable to eachother in terms of range of technology. Meanwhile, the firms employing 100 ormore persons are an open grouping of fairlyextensive range, clearly different fromthe range covered by the other four. Therefore, it was possibl'e to predict a priorihigh values obtained for this large grouping.

(iii) Values of a in each industry. It is interesting to know the tendency theelasticity of substitution has in each particular industry. Six possible tendenciesof a related to the increase of establishment size groupings were chosen : increasing,decreasing, constant, (all values no different in more than 0.2), fl-form, U-formand indefinite. Those industries with less than 4 values of a were excluded (theyeither had negative values or had a small number of observations).

Negative values of a have been excluded, hut if they were considered, the ranking of a valueswould not change.- M. A. Abe, "The Growth Pati, of Firms and the Development Process of the Economy: TheCase of Japan" (The Developing Economies, June 1972), p.201.H. Leibensje,n "Technical Progress, the Production Function and Dualism" (Bunco Nu:ionh'del Lacoro Quarter!;: Review, March 1960), pp. 348-351.

608

TABLE 8Ntllnik OF INI)USLkIIs HAVINGA DF1IBM1NEI) TFNDIN(y Br-I WiEN Ci AM) Stir GROUPIN(;S

(JJ ESFAIII!SIIMrNT

Excluding largeTendencies estahlishrnem

of groupings

IncreasingDecreasingConstantfl-form 3U-form 4indefInite 6

There is no definite tendency for a to vary uniformly across the different sizegroupings. Furthermore, the elasticity of substitution does not remain constantas the size of establishment varies, suggesting that the isoquant map is not honio-thetic.

Considering the magnitudes taken by the elasticity of substitution, Table 9provides the number of a values obtained by size groupings and range of values.

TABLE 9NIJMnrR OF ELASTICITY OF SUIISTITUJION VAluEs sy SizrGouptNGs oi Esr.misntrns ANn RANUF OF VAlUES

The values in this table agree with the condensed and analyzed data ofTable 7.

(iv) Tests ofa for values 0 and I. The traditional tests made with the elasticityof substitution correspond to the hypothesis a = 0 (fixed proportions function)and a = I (CobbDouglas function). In the case H0 :a = 0, Table A-6 shows thatthis hypothesis is rejected in 83 of 120 cases (69.2 percent) at a significance levelof 5 percent. These results indicate that Chilean industry, at a level of differentestablishment size, does not present a rigid technological structure of fixedproportions. I could also calculate the value of the statistics t in Table A-6 forthe null hypothesis H0:a = I. However, taking advantage of the Kmenta ap-proximation, it is simpler to use a direct test where the quadratic term parameterpermits immediate verification of the null hypothesis by showing whether ornot the production function is Cobb-Douglas. This test for the different sizegroupings of establishments shows that the null hypothesis (the production func-tion is not CobbDouglas) is rejected in 7 out of 120 cases (58) percent) at a sig-

609

Sizegroupings

Range of Values

0.00-0.50 0510.80 0.81 -1.20 I-1.21 and more

5-9 3 5 8 310-13 2 8 5 320-49 6 7 4 350-99 8 2 5

l00andrnore I I 6 6

;------

(6)

nificance level of 5 percent. As a result. I do not have to discard the conclusions andmagnitudes previously obtained for the product-capital elasticity and thceconomies of scale, which critically depend on the condition that the Cobb-Douglas function be consistent with the data.

6. HoMorneTiciTy TESTS OF THE PRootrcTIoN FUNC-nON

The central point to be examined in this section will be to find out whetherthere is a structural change in the technological parameters of the productionfunction when I go from one size grouping of establishments to the next.

In the previous sections, where I examined the estimated values for theproduct-capital, economies of scale, and substitution elasticities, I affirmed theseparameters were subject to change for different size groupings. This alreadysuggests that a production function with constant elasticity is not an adequatetool for synthesizing the technological characteristics of the establishments ofdiffering sizes in the same industry.

The traditional econometric procedure for examining the hypothesis ofstructural change in the parameters is the Chow test. An alternative way toexamine the null hypothesis as to whether the production function is homotheticis using a heterothetic function, the trans-logarithm production function.34 Inthis study I will use these two procedures to verify the homothetiticity of theproduction function.

(a) Chow Test

To examine the hypothesis of a structural change in the parameters, a generalChow Test will be used. This test attempts to verify whether the breakdown ofthe establishments into 5 size groupings is significant or noti.e., the null hypo-thesis showing no difference between the five size groupings. If I assume thatvector /3. represents the vector of technological characteristics of size grouping 1,the H0 could be written: Ho:fl = /32 = /33 = [34 =

The alternative hypothesis is that the five size groupings are different fromeach other. This implies that for each size grouping of establishments thereprevails a different production function, or that the production function for theindustry does not possess constant elasticities (product-inputs, economies ofscale, and substitution).Let K be the number of explanatory variables for the regression and N thenumber of observations

Q1 will be the sum of the squared residuals of the regressionfor size grouping i, and QT the sum of the squared residuals for the whole industry(the five size groupings Simultaneously).The Chow Test for testing the hypothesis H0 takes the following expressionfor the case of five subsamples

QT-1Q N-5(K+ 1)FG = -= K + 1) -'

L. R. Christensen, D. W. Jorgenson, and L. J. Lie, "Transcendental Logarithmic ProduclionFunctions" (Rev, of Econ, and Stat, Feb. 1973).D. S. Huang, Regression and Econometric Methods (ViIey and Sons, 1970),

610

-

- .1- -.: -_._-

*

whereFGhaSadistribution Fwith4(K ± 1)and N - 5(K + l)degreesoffreedom.Table 10 provides the F4 values for the three different functional forms used inthis study.

TABLE 10F STAJISTICS OF CHOW TEST FOR EACH INI)USTRV AND

FOR IIREE DIFFERENT FUNCTONA1. FORMS

Values of F not significani at 5 percent.

Table 10 indicates that the null hypothesis according to which no differencesexist between technological parameters of establishments of different sizes isiejected in over 60 percent of the industries considered.

(b) The Trans-logarithmic Production Function

The trans-logarithmic production function is obtained by expanding thequadratic term of the linear approximation of Kmenta:

log = a0 4 a, log + a2 log L ± a3(log K - log L)2

Y Ka0 + a1 log + a2IogL + a31(IogK)2 + a32(logL)2

(7) -2a33(log K)(log L)

The trans-logarithmic function is heterothetic, while the Kmenta approxima-tion is homothetic. To verify the homothetic hypothesis for the productionfunction, Griliches and Ringstad suggested three alternative forms,36 one of

Grilichcs and Ringstad, op. ci:., pp. 10 and 88.

611

Type olIndustry

ACMSFunction

Cobb Douglas Kmenta'sFunction Linearization

311 I 1.668' 5.403 3.9823112 0795' 3.743 2.7543116 3.158 2.032 1.901'3117 7.775 10.100 14.5703121 3.612 2.890' 3.7823132 14.375 13.240 10.3633211 1.527* 4.057 3.5823213 4.467 2.833 2.1503220 5.657 2.519 2.6093231 0.484* 1.952* 1.441*3240 2.522 5.469 4.0853311 ó.164 3.747 3.5023320 1.833' 1.167' 0.820'3420 2.981 2.232 1.676'3560 1.118' 2.837 2.20!3693 4.870 3.243 2.4583710 1.958* 1.995' 1.771*3813 4.559 1.175* 1.396*3819 3.769 1.985 1.476'3829 3.818 1.830' 1.572*3843 1.236* 1.987 1.723

which, the simpler one, will he used here. This method used the Kmenta functiotias the null hypothesis and the trans-logarithmic function as the alternativehypothesis. We have:

H0: Knienta function is the adequate modelH1: Trans-logarithmic function is the adequate model

Let Q be the sum of the squared residuals of the regression for the Kmentafunction, Q the sum of the squared residuals for the regression in the trans-logarithmic function. The statistics for examining the null hypothesis is.

(8) F - Q)/2Q1,/IN (K +- I)]

which has an F distribution with 2 and N (K + 1) degrees of freedom. Thedegrees of freedom in the numerator correspond to the difference in the numberof independent variables found between the Kmenta function and the traiis-logarithmic function. The degrees of freedom in the denominator correspond tothe trans-logarithmic function.

An increase in the statistic F in expression (8) indicates a considerable increasein the sum of the squared residuals when the heterothetic function becomes thehomothetic function. In other words, there is a resulting unexplained increasein the variation of the observations when the hoinothetjc function is used. Whenthe statistic F in expression (8) has a significantly small value, it indicates thatthere is no visible change in the explanation given for the behavior ofestablishmeritsin one industry when a heterothetic function replaces a homothetic function.The statistic F in expression (8) has been calculated for each Industry: the valuesappear in Table I I. Negative F values were obtained for five industries; this in-dicates the homothetic function produces better fits than those of the heterotheticfunction, once the respective coefficients R2 have been adjusted by the correspond-ing degrees of freedom. I believe the basic reason for the number of industriesrejecting the homothetic function not being larger is that the trans-logarithmicfunction does not have a good fit for the majority of the industries, as can beseen in Table 11.

Table 11 shows the homothetjc test at the industry level rejected for halfof the industries with a significant level at 5 percent.37The above results (Chow test and trans-logarithmic production function test)suggest that for a majority of Chilean industries, there is no one single productionfunction with constant elasticities which reflects the technological characteristicsof the firms in that industry, in other words, to determine the magnitudes of thetechnological elasticities in industrial establishments each size grouping mustbe studied separately (or, a production function with variable elasticities shouldbe used).

7. SOME REFLECTIONS ON THE METHODOLOGY OF PRODUCTION FUNCTIONSEC0NONIEt-Ric ESTIMATION

This research was started in the belief that the available disaggregated dataon industrial establishments would permit the estimation of production functionsis important to denote that in Griliches' and Ringstads study there is only one industryfor which the honiothetic hypouiesis is rejected In this paper, the number of industries for which thealternative hypothesis is accepted is considerable

612

TABLE 11HoMonunc TFST FOR THE PRouucTloN Fur4(-1 ION Usi i m IRANS-

L(X,AR!i}IMIC lUNCTION

Type of Values for F Degrees of Freedom (Trans-logarithmic

613

* Values of F not significant at 5 percent.Negative values for F which imply that the Kmen;a function has a

better fit than the trans-logarithmic function.

and give "excellent" results. By "excellent" results it was understood that theestimators of different elasticities would have the signs suggested by economictheory and magnitudes corresponding to those obtained in other empiricalstudies: that the values of the estimators would be quite stable for the differentfunctions that were to be estimated; that the statistics t and F would be highenough to reject any null hypothesis which might contradict economic theory;and that the values obtained for R2 would be close to 1.0.

The first econometric estimations contradicted these expectations. When Iexamined the literature on the econometric estimations of production functions,the results obtained generally agreed with the ideal conditions described aboveand drew my attention to the excellent fits shown by the R2 coefficients (betterthan 0.90, generally). The findings of Griliches and Ringstad are one of the fewexceptions to that rule. This study, just like that of Griliches and Ringstad, usesprimary data, i.e., at the establishment level. Why is it then that studies using anideal source of information (from the production function point of vtew) furnishresults worse than those studies which have no possibility of selecting and refiningthe data? The values of R2 obtained in this study are provided lfl the followingtable:

Industry (see expression K) for F function)

3111 7.495 2-169 (13973112 6.492 2- 39 0.6813116 3.693 2-125 0.3523117 12.770 2-623 0.1973121 2.333* 2- 41 0.4 Il3132 15.846 2-548 0.1993211 --0.789 2-220 0.3203213 7.810 2-138 0.4883220 -3.346 2-187 0.1723231 1.046* 2- 55 0.3593240 1.641* 2-136 0.4933311 - I3.358 2-414 0.1583320 - I2.352 2-174 0.1943420 7.914 2 -144 0.5853560 3.424 2-- 45 0.4023693 --0.315 2- 58 0.1883710 0.050* 2-- 42 0.5743813 12.857 2- 80 (1.5643819 -1.446k 2-115 0.4333829 0.528* 2- 91 0.4713843 3.877 2- 76 0.548

LiTABLE 12

NUMBER OF VAlUES FUR R2 FOR EA(iI FORMUlA I ION AM) DyUI ALW-.

Disaggregated data at the establishment level allow the elimination of thoseobservations which provide unreliable information. In this study, where the givendata were supposed to contain a sufficient number of observations at the 4-digitsindustry level, the establishments were subjected to a rigid selection process(see appendix) in which the quality of observations was emphasized over theirquantity.

After this selection process, I proceeded to examine the effects of measuringthe labor and capital variables by different alternatives. The alternative measure-ments for the different variables showed no significant variation among themselves.This had already been predicted by the high correlation coefficients found foralternative forms of measurement for different proxy variables employed. Thisresult indicates that biases introduced by the type of measurement used do notalter significantly the results already obtained. Therefore, I felt that it was irrelevantto calculate the magnitudes of those biases.

Finally, I have estimated and tested several different functions (others arenot discussed explicitly in this study) and all had equally low fits, some with oddsigns for some of the parameters.

In short, the basic information used is taken at the establishment level:only establishments with reliable data are selected; the economic variables aremeasured in several different ways: a variety of functions are estimated: thenumber of observations is sufficiently large: and the number and type of industriesexamined is large and varied. In spite of all these "precautions," the degree ofvariance in the behavior of establishments is considerably greater than that whicheconomic theory can explain.

In light of the results obtained, re-examining the economic assumptionsinvolved in some of the functions (perfect competition and maximization ofprofits) shows that the norm of bad results is also indifferent to the economicassumptions made. It is necessary to take a closer look at the production functionconcept. The series of bad results would indicate that the concept of the productionfunction is not useful for studying the behavior of the firm; but, if the productionfunction concept is irrelevant at the microeconomic level, having already beencriticized at the macroeconomic level, what is its use?

Re-examining the notion of the production function, I see that the essentialcondition of the concept is including only productively efficient techniques. Thus.a process of selection is assumed to have taken place eliminating the inefficienttechniques and leading to a uni-valued function. This basic premise of the produc-

614

Range of Values of R2Functional

models 0.00 0.24 0.25 049 0.50 0.74 0.75 I 00

ACMS 66 35 13 3

Cobb-Douglas 43 44 26 7Kmenta 50 42 21 7

r

tion function concept was not applied in the selection of observations includedto estimate the production functions in each industry.

There are two alternatives to the traditional method of production futjns'econometric estimation for obtaining the production function that would onlyinclude the efficient techniques:38 (1) The engineering approach, where inefficienttechniques are eliminated by monetary cost considerations, and (2) the linear (Orquadratic) programming method where the function is fitted to the data byminimizing the sum of deviations on one side of the curve or on it. A simplervariation of this second approach is Farrell's method, which provides the efficiencyfrontier for all observations.39

8. CONCLUSIONS

Diftèrent production functions exist Jr an industry's establishments

Through the application of the general Chow test I concluded that fordifferent functional specifications it is possible to reject in at least 16 of2l industriesthe hypothesis of no technological differences existing among the different sizegroupings of establishments. There is an increase in the number of industries forwhich it is not possible to employ only one homothetic production function, sincethe fit of the different functional specifications improves.

The use of the trans-logarithmic production function does not give a con-clusive result. The hypothesis that the production function is homothetic can berejected in only 10 of 21 industries.

Finally, the technological parameter elasticities (product-capital, economiesof scale, and factor substitution) obtained for each size grouping are usuallydifferent at each industry level. Therefore, the need to discriminate between thedifferent size groupings of industrial establishments is clearly suggested from apractical point of view.

These results could be used as empirical evidence to support the assumptionof structural heterogeneity within the industrial sector, i.e., technology varies atthe industry level as the size of establishment increases.

The former discussion suggests that caution must be exercised when usingindustry level data. The industry is not a collection of "representative firms," andthe degree of heterogeneity is fairly high. The size of establishment seems to bean important element in determining the economic and technological character-istics of an industrial establishment.

Principal results of production JI4nctions estimation

(a) Value added-cpital elasticity(i) There is no (.imp1e) correlation between the value added-capital elasticity,

fi, and the relative si are of capital in value added, SK. In other words, if S, isused as a value added-capital elasticity estimator, the elasticity would be over-estimated in 75 percent of the cases. The magnitude of such overestimationfluctuates considerably.

3S L. Johansen. Production Functions (North-Holland. 1912). ChapterS.M. J. Farrell, "The Measurement of Productive Efficiency" jfournal of the Roa1 5ta. SOC.,

Series A, Vol. 120, Part 3, 1957) pp. 253-281, This is done in P. MetIer's Ph.D. Thesis, op. Cit.

615

I

/In the majority of industries. 16 of 21, the largest establishments have a

greater value added-capital elasticity than the smallest ones.Size groupings 5-9. 10-19. and 20-49 persons have / elasticities of

approximately 0.40: size grouping 50 99 persons has elasticity higher than 0.50;and establishments of 100 or more persons have fi values close to 0.60.

(b) Economies of scaleEconomies of scale can he observed in 20 of 21 industries, at the industry

level. These economics of scale fluctuate between 10 and 35 percent in 14 industries;26.4 percent (simple arithmetic mean) of economies of scale is observed for the21 industries.

The results obtained for economies of scale are highly questionable,given the odd results obtained at the size of establishment level. In the latter case,there are diseconomies of scale for all size groupings of establishment, even forthe smallest ones employing 5-9 and 10-19 persons. Moreover, when differentsize groupings reveal the variation pattern of scale economies across size classes,some industries present an average cost curve with an inverted U form.

The test for constant returns to scale is accepted for almost 80 percentof the size groupings when estimated separately. But this test is rejected at theindustry level in 14 ot'21 industries.(c) Elasticity of substitution

(1) The ACMS method (Arrow etal.) produces adequate values for the elasticityof substitution a. These values arc consistent with the expectations provided byeconomic theory. Furthermore, these values fluctuate only within small rangesof magnitude.

The elasticity of substitution varies among industries and also among thedifferent size groupings in an industry. At the industry level, a oscillates between0.8 and 1.2 in 14 of 21 industries, and save two exceptions, the industries as awhole have a values above 0.65. No regular pattern of behavior is observed in thevariation pattern of a in the size groupings of an industry. Considering the fivesize groupings, a would show a U shaped pattern as the establishment size increases.The fourth group (50-99 persons) has the lowest value of a, and the largest group(100 and more persons) has the highest a value. However, the largest groupcannot really be compared to the rest because of its much greater size range.As the latter is an open group it is not strange to find there the highest estimatedvalues of a.

The traditional tests of values that a may take are related to values of 0(fixed proportions function) and I (Cobb-Douglas function). The hypothesis a = 0is rejected in almost 70 percent of the cases. The Chilean industrial structure, ingeneral, does not show technological inflexibility. The hypothesis a = I is onlyrejected in about 6 percent of the cases.

3. Firs of different functional specifications, measured by R2 values, are generallypoor for ailfunctional forms

The traditional ACMS specification produces the worst fits, with 87 percentof the cases having an R2 below 0.50. The Cobb-Douglas and Kmcnta functionshave 75 percent of the cases with R2 below 0.50.

616

I

These poor fits, the odd results obtained for the economies of scale, and thegreat instability and fluctuation of different estimated parameters introducedoubts regarding the utility of the production tunction concept at the micro-economic level. However, to defend this concept at micro level and to explainthe problems iiieiitioacd in the previous paragraph, we would have to indicatethat in the econometric estimation there is one requirement which has not beenconsidered, namely the technical efficiency condition that must fulfill the differentproductive techniques.

The fundamental question to be asked is: How reliable are the results andwhat is their use? These results are merely of a descriptive nature and, given theirhigh variability, are not appropriate for suggesting any kind of economic policymeasure. The purpose of the estimators obtained is to show average valuesindicating an existing empirical situation which in some cases is very differentfrom the one predicted by the theoretical model.

APPENDIX

Data used in this study correspond to a four-digit disaggregation industrylevel, ISIC classification. Basic data consists of primary information at industrylevel for the Chilean Industrial Sector Manufacturing Census of 1967.

The 21 industries shown in Table A-i were selected according to a flexibleapplication of the following criteria: (1) Each chosen industry should countwith a "sufficient" number of observations to enable a meaningful econometricestimation in the different size groupings of establishments (2) industries chosen

TABLE A-IISIC Cooa AND NAME OF THE 21 INDUSTRIES SELEC-rED

ISIC Code Name of Industry

3111 Cattle slaughtering preparation and storing of meat3112 Manufacturing of dairy products3116 Mill products3117 Manufacturing of bakery products312! Processing of various food products3132 Wine industries3211 Spinning, weaving, and textile finishing3213 Knitting factories3220 Clothing factories, except shoes3231 Tannery and finishing workshops3240 Shoe factory, except plastic or rubber3311 Sawmills, barracks, and wood workshops3320 Furniture and accessory factories3420 Printing presses and publishing companIes3560 Plastic products factories3693 Cement products factories3710 Iron and steel basic industries3l3 Structural metal products factories3819 Nonspecific metal pioducts factories3829 Machinery and equipment manufacturing3843 Spare parts and accessories for motorized vehicles

617

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ual t

o ze

roV

alue

add

ed le

ssor

equ

al to

zer

o

Pay

men

ts to

Cap

ia1

Fac

tor

less

or

equa

l to

zero

Mac

hine

ry B

ook

Val

ue e

qual

toze

ro

Tot

al N

umbe

r of

Elim

nate

dE

stab

liThm

ents

3111

30

132

30

216

324

3112

I0

141

00

04!

3116

30

127

00

129

3117

100

253

40

23

526

3121

10

024

00

025

3132

430

1334

2()

10

3032

110

00

125

00

112

532

133

00

199

00

020

032

205

00

483

00

04!

532

310

00

310

01

2!32

401

01

200

02

02(

X)

3311

244

022

096

02

195

3320

50

l28

70

I3

2834

202

01

233

01

122

335

601

II

6I

00

6636

932

00

620

00

6337

100

00

120

00

238

13I

00

III0

00

fl238

193

02

117

00

I11

738

290

00

660

00

6638

430

00

630

00

63

TO

TA

L32

824

443

06II

2843

7!

TABLE A-3NUMBER OF OBSERVATIONS tJsFfl FOR PRtMJ(TlON FtN( iiC1 r MAt

Establishment Sue Cla5

should produce more or less homogeneous products and (3) there should beat least one industry for each two-digit ISIC classification.

These 21 industries have 8021 establishments. These establishments weresubmitted to the following selection criteria:

I. Number of persons employed per establishment is less than 5)Number of days worked per establishment is equal to 0.Total number of workers and employees is equal to 0.Book value of machinery is equal toO.Book value of buildings is equal to 0.Added value is less than or equal to 0.Payment to capital factor, obtained as the difference between value addedand total labor cost, is less than or equal to 0.

In most cases 0 does not literally mean iero but reflects the omission ofinformation.

The establishnents that did not meet any one of the previous criteria wereexcluded from the sample, The number of establishments was drastically reducedfrom 8021 to 3650 (see Table A-2).

t should be pointed out that over 80 percent of the eliminated establishmentsbelong to the two smallest size groupings (5-9 and 10-19 people employed).

l In spite of the fact that the Industrial Census should include establishments employing at leastfive persons, there are 328 establishments violating this rule.

619

Type ofIndustry

5 9 10 19 2049 5099persons persons persons persons

1(X) and morepersons

Totallndustr

3W 78 36 39 16 63112 II II 12 8 II 533ll6 29 30 50 20 2 313117 210 249 143 15 12 6293121 9 16 12 4 6 473132 357 131 56 6 4 5543211 22 37 73 36 58 2263213 36 29 44 20 16 1453220 60 4l 43 20 29 1933231 9 13 20 II 8 613240 30 37 31 20 14 4'3311 124 129 97 33 31 4203320 79 47 37 0 7 1803420 48 25 39 4 9 453560 6 ) 19 7 JO SI3693 21 20 17 5 643710 7 6 17 4 12 483813 l9 27 20 8 2 863819 45 30 32 8 6 1213829 22 4 27 21 13 973843 IS 21 19 12 IS 82

TOTAF 1.237 960 847 296 308 3.650

TA

BLE

A-4

CA

PIT

ALS

SH

AR

E A

ND

PR

OD

UC

T-C

AP

ITA

L E

LAS

TIC

ITY

BY

ES

TA

BLI

SH

ME

NT

SIZ

EA

ND

IND

US

TR

Y

Val

ue A

dded

- P

aym

ents

to L

abor

Fac

tor

Cap

ital S

hare

Val

ue A

dded

9: R

egre

ssor

Coe

ffici

ent o

f Equ

ition

: log

1/ L

= lo

g A

+ /3

log

K/L

+ 6

log

L (S

ee T

able

A-7

)

r: S

tatis

tic fo

r nu

ll hy

poth

esis

H0:

/3 =

Typ

e of

Indu

stry

Cilu

Cod

e

Est

ablis

hmen

t Siz

e C

lass

5--9

per

sons

10-1

9 pe

rson

s20

-49

pers

ons

S/9

/9z

S

3111

0.35

!0.

269

-0.6

300.

58!

0.58

50.

020'

0.66

10.

436

- :3

01'

3112

0.48

41.

547

8.24

40,

711

0.11

8l.7

19'

0.65

61.

411

2.18

8'31

160.

685

0.43

7-

1.28

8'0.

716

0.49

7--

0.9

48*

0.71

40.

372

- 3.

455

3117

0.51

20.

227

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960.

4.68

0.22

6-6

.050

0.45

30.

101

-'.18

431

210.

392

0.34

0-0

.l75'

0.62

8-0

.069

-3.7

070.

698

0.67

90.

089'

3132

-0.6

110.

172

-9.9

770.

741

0.13

5-8

.684

0.79

20.

367

- 3.

455

3211

0.57

50.

915

2.65

10.

689

0.33

5-3

.222

0.59

60.

513

- :1

23'

3213

0.67

50.

559

-0.9

92'

0.68

00.

193

-4.1

490.

684

0.38

8-2

.696

3220

0.38

70.

339

-0.3

74'

0.62

60.

029

-4.7

550.

618

0.16

0-2

.802

3231

0.72

10.

264

-5.1

350.

733

0.20

2'-2

.313

0.62

50.

459

-0.9

27'

3240

0.53

00.

481

-0.8

18'

0.48

80.

236

-2.0

660.

511

0.36

3-

:244

'33

110.

529

0.40

3-

1.82

0'0.

588

0.35

3-3

.35!

0.37

80.

368

- .0

15'

3320

0.38

80.

352

-0.3

58'

0.41

30.

420

0.59

4'0.

545

0.46

8-0

.849

'34

200.

515

0.38

4-

1.31

0'0.

443

0.43

20.

666'

0.48

30.

451

-0.2

17'

3560

0.52

80.

259

-0.8

39'

0.75

60.

755

-0.0

05'

0.62

70.

433

- 1.

576'

3693

0.52

90.

169

-3.7

190.

555

0.40

8-0

.477

'0.

227

0.14

6-0

.500

'37

100.

414

0.51

10.

558'

0.54

00.

333

-0.9

08'

0.60

80.

346

-- 1

.664

'38

130.

361

0.08

3-2

.248

0.50

30.

366

- 1:

458'

0481

0.16

3-

1.60

0'38

190,

531

0.22

3-2

.885

0.57

20.

442

- 1.

284'

0.60

60.

399

-98

0'38

290.

318

0.42

00.

495'

0.35

90.

265

-0.7

39'

0.27

70.

417

1102

'38

430.

603

0.60

0-0

.017

'0.

645

0.39

4-

1.70

2'0.

635

0.12

4-3

498

TA

BLE

A-4

---(

cont

.)

-

': va

lues

non

-sig

nific

ant a

t 5 p

erce

nt.

(a)

Val

ues

not c

ompu

ted

beca

use

num

ber

of o

bser

vatio

ns w

as e

qual

or

less

than

five

.

Typ

e of

Est

ablis

hmen

t Siz

e C

lass

Tot

al In

dust

ryIn

dust

ryC

IJU

50-9

9 pe

rson

s10

0an

dm

ore

pers

ons

Cod

eS

S,

ftt

3111

0.68

00.

690

-0.0

53'

0.66

60.

598

-0.2

89'

0.50

80.

692

2.32

631

120.

774

0.36

9-0

.707

'0.

670

0.54

8-0

.667

'0.

652

1.01

925

6631

160.

704

-0.3

02-5

.427

0.35

4(a

)(a

)0.

701

0.46

5-

2.98

731

170.

461

0.61

60.

861*

0.59

50.

853

2.66

00.

482

0.29

9--

7.03

831

210.

783

(a)

(a)

0,70

81.

903

1.30

6'0.

624

0.28

0--

2.89

031

320.

817

0.58

0-

1.65

8*0.

675

(a)

(a)

0.66

303

63.

8.33

332

110.

563

0.33

71.

694'

0.54

50.

351

-3.9

950.

591

0.49

5--

2.06

532

130.

577

0.66

40.

475*

0.63

00.

274

-2.7

010.

660

0.50

9-2

.437

3220

0.61

50.

172

-2.1

500.

582

1321

1.97

7'0.

362

0.40

60.

593'

3231

0.53

10.

507

-0.0

47'

0.49

4(a

)(a

)0.

628

0.44

1--

2.3

4032

400.

481

0.35

0-0

.830

'0.

480

0.47

8-0

.174

'0.

500

0.52

10.

356*

3311

0.46

30.

018

-3.6

560.

350

0.38

60.

274'

0.49

10.

44)9

--2.

222

3320

0.36

00.

939

3.21

80.

482

0.57

20.

236'

0.42

90.

445

0.27

8'34

200.

400

0.64

91.

618'

0.43

80.

601

1.24

2'0.

473

0.62

53.

118

3560

0.63

40.

994

1,23

9*0.

634

0.08

4-2

.170

'0.

651

0.48

7--

1.68

9'36

930.

553

(a)

(a)

0.46

9(a

)(a

)0.

458

0.31

3_.

l.918

*37

100.

641

(a)

(a)

0.51

30.

366

-0.7

93'

3.31

70.

547

0.00

0'38

130.

529

0.56

60.

145'

0.34

00.

253

-0.4

09'

0.44

70.

338

-. 1

.697

'38

190,

559

0.24

8-

1.27

5'0.

509

0.50

4-0

.042

'0.

562

0.41

1-

2.64

238

290.

496

0.39

4-0

.390

'0.

419

0.55

31.

139*

0.36

50.

559

2.70

238

430.

556

0.56

40.

053'

0711

1.01

92.

217

0.63

40.

506

- 1.

803'

TA

BLE

A-5

EC

ON

OM

IES

OF

SC

ALE

AC

CO

RD

ING

TO

CO

BB

-DO

UG

LAS

AN

D K

ME

NT

A P

RO

DU

CT

ION

FU

NC

TIO

NS

BY

ES

TA

BLI

SH

ME

NT

SIz

E A

ND

IND

US

TE

Y

Hc_

0: C

obb-

Dou

glas

eco

nom

ics

of s

cale

. (T

able

A-7

)h:

Km

enta

eco

nom

ies o

f sca

le. (

Tabl

eA

-8)

Est

ablis

hmen

t Siz

e C

lass

'h v

alue

s w

hose

t st

atis

tics

in n

on-s

igni

fican

t at 5

per

cen

t. S

ee T

able

s A

-7 a

nd A

-8.

(a)

Val

ues

not c

ompu

ted

beca

use

num

ber

of o

bser

vatio

ns w

as e

qual

or

less

than

five

.

Typ

e of

lndu

stty

CII

UC

ode

Tot

al In

dust

ry5-

9 pe

rson

slO

--19

per

sons

20-4

9 pe

rson

s50

-99

pers

ons

100

and

mor

e pe

rson

s

h.0

h,,

h.0

/Ic.4

)h1

c_D

3111

-0.6

56-0

.660

-1.3

50-1

.354

-0.1

74'

O.l8

l'0l

53'

-0.2

47'

0.74

8*0.

363*

-0.1

19-0

146

3112

0.14

8*0.

040*

0.39

0'0.

469*

0.48

8*0.

194*

-0.9

92'

0.69

9'-0

.471

'00

49'

0.29

40.

308

3116

-0.2

93'

-0.3

450.

125

0.17

7*04

95'

0.53

01.

062

1.05

8(a

)(a

)0.

263

0.26

031

17-0

.707

-0.7

13-0

.857

-0.8

21-0

,140

'-0

.380

'0.

236'

0.32

40.

243'

0.24

1'0.

080

0.80

031

210.

787'

1.00

3'-0

.191

'0.

190w

0.21

80.

473'

(a)

(a)

1.22

8'1.

141'

0.45

40.

461

3132

-0,4

00-0

.396

-0.3

74-0

.374

0.17

7*0.

294*

0.15

2'0.

139*

(a)

(a)

0.06

200

65j

3211

-0.6

13'

-0.5

97'

-0.2

67'

-0.3

95'

--0.

575

-0.5

66-0

.142

'-0

.145

'--

0.02

3'-0

.022

'0.

070'

0007

'1'

.)32

13-0

.249

'-0

.104

'-0

.747

'-0

.147

'-0

.353

'--

0.35

0'-0

.668

'-0

.636

'-0

.007

'-0

.006

'0.

216

017

3220

-0.2

77'

-0.3

31'

-0.7

38-0

.779

-114

04'

-0.2

86'

-0.3

54'

0235

'0.

061*

-0.1

25'

0.25

502

3832

310.

233

0.28

9'1.

087

- 1.

044

-0.1

20-0

.155

'-0

.404

'-0

.499

'(a

)(a

)0.

128

0.13

832

40-

1.40

3-

1.36

5--

1.18

0'-0

.184

'-0

.513

-0.5

50-0

.289

-0.4

510.

265

0.25

70.

291

0291

3311

-0.0

15'

-0.0

23'

-0.1

58'

-0.0

17-0

.124

'-0

.119

'0.

082'

0.01

7*0.

184'

0.18

1'0.

060

0550

3320

-0.5

2'-0

.084

'-0

.279

'-0

.303

'-0

.363

'-0

.298

'0.

000

-0.0

16'

0.58

5'0.

261

0.29

903

1034

20-0

.791

-0.7

93-0

.680

'-0

.662

'-0

.311

'-0

.318

'-0

.967

0.02

7'0.

264'

-0.2

66'

0.02

1'0.

021'

3560

-0.9

72'

-0.9

80'

0.32

2'-0

.281

'-0

.434

'-0

.440

'-0

.436

'-0

.866

'0.

453'

0.45

7'0.

088'

0.09

536

93-0

.532

'-0

.551

'0,

864'

-0.8

55'

-0.8

71-

1.21

7(a

)(a

)(a

)(a

)0.

157'

0.16

2'37

10-0

.103

'-3

.621

'-0

,325

'-0

.880

'-0

.455

'-0

.533

'(a

)(a

)-.

1.05

0'-0

.185

'0.

118'

0115

'38

13-0

.421

'-0

.199

'--

0.24

9'-0

.425

'-0

.261

'-0

.257

0.22

4'0.

183'

0.20

4'-0

.009

'0.

343

0334

3819

-0.5

56'

-0.5

80'

-0.3

01'

-0.1

93'

-0.1

26'

-0.1

96'

-0.1

29'

-0.2

93'

0.16

1'-0

.047

'0.

318

0326

3829

-0.4

95'

-0.4

70'

-0,8

68'

- 1.

383

0.31

1'0.

3 12

'-0

.183

'0.

076'

0.41

70.

457

0.23

002

2438

43-0

.152

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.281

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623

-0.4

61'

0.52

2'-0

.549

'-0

.493

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.580

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035'

0.07

1'0

353

0353

Typ

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Indu

stry

CIIU

(T

odc

TA

BLE

A-6

AC

MS

PR

OD

UC

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N F

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CT

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EC

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OM

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RE

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BY

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TA

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n F

unct

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K+

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Reg

ress

ion

Eqt

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n: lo

g Y

,'L -

a 'I

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og

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abI,s

hrnc

nt S

ize

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ss

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ons

10-1

9 pe

rson

s

R2

0

70-4

9 pe

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s

NJ

3111

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9194

50.

250

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401.

598

0.33

4-3

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0.90

101

:

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59)

(516

2)(-

9.45

4)(4

308)

(-8

165)

(7.4

34)

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338

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178

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943

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0,07

3

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2407

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( -

35.7

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0)I

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5871

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(-63

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529

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366)

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7547

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3.77

604

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051

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460

775)

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(2.7

70)

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6)10

501)

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6)

-3'

70-4

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The distribution of the sample by establishment size groupings is shown in TableA-3. Nevertheless, in spite of the large number of eliminated observations, thesample comprises over 30 percent of thc total nuinbei of establishments for thetwo smallest size groupings and over 70 percent of the total number of establish-ments for the two largest sizegroupings (50-99 and 100 and more people employed)

Catholic tinh'ers it v, Chile

634

I

volume title: annals of economic and social measurement, … · internac de distr. dcl ingreso,...

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