Efficiency Estimation from Cobb-Douglas Production Functions with ComposedError

Wim Meeusen; Julien van Den Broeck

International Economic Review, Vol. 18, No. 2. (Jun., 1977), pp. 435-444.

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INTERNATIONAL ECONOMIC REVIEW Vol. 18, No. 2,June, 1977

EFFICIENCY ESTIMATION FROM COBB-DOUGLAS PRODUCTION FUNCTIONS WITH COMPOSED

ERROR*

R Y Wlbl MEEUSFN AND JUI,IENVAU DEN I ~ R O E C K '

1 . INTRODUCTION

In the recent economic literature (Aigner and Chu r21, Fgrsund and Hjalmarsson [5], Seitz [ I l l , and Timnier [12]), in continuation of the research initiated by Farrell [4], much attention has been paid to the estiination of productive efficien- cy by means of frontier production functions. In this journal, Richmond [lo], developing a suggestion made by Afriat [I], carried out the estimation of the Cobb-Douglas production frontier by letting

and applying ordinary least squares to tlie logarithmic transforination of (I), after putting k,=e c Z t ,assigning a Gamma distribution G,(z ; n) to the random variable Z, and introducing the statistical error c,= n -2,. y, is the actual output, x, is the corresponding (M x 1 ) vector of inputs, and f(s,) is the Cobb-Douglas frontier production function.

It follows that the random variable K, with observed values ic,(t= 1,..., T ) , represents the technical level of efficiency at the firin level, i.e., the proportion between actual and potential output.

t,=2-1' is a consistent although upward biased estimator of the average Richmond-efficiency level of the econonly (or industry) given by

h=(Co,2)/(T-IW- I ) is the OLS estimator of the variance 11 of the random vari- t

able V. Of course, the estimates of the production elasticities are not affected in the

foregoing specification: the Afriat-Richmond production frontier comes about by a inere upward shift on the logarithmic scale of the corresponding "average" production function.

* Manuscript received January 16, 1976; revised May 21, 1976. We should like to express our gratitude to Jacques Mairesse of INSEE at Paris for perrriitt-

ing us to use the data of the 1962 French Census of Manufacttiring Industries. We are also grateful to Dr. F1. Deckers and G. Peperstraete of the Nuclear Rcsearch Centre at Mol (Belgi~~m) for their asistance with the calculations.

436 MEEUSEN AND VAN DEN Bf<OECK

It is our purpose to propose an alternative approach to the estimation of the frontier production functio~?, since the Afriat-Richn~ond measure yields, as we shall sl?ow, systeinatically underestimated values for productive eficieiicy.

Contrary to Afriat and Rich~noiid, who proceed as if every disturbance from the theoretical value 011the frontier results solely from Iluman errors (iiiformation deficiencies, adjustment costs, managerial errors on tlie level of the techllical operatio11 of the planl, ctc.), we shall introduce in our inodel oE the Croritier production fi~ilction, next to a dist~1rb:lnce due to inefficiency. a statistical dis- turbance due to randomness i l l the real sense and to specification and measure-ment errors.

2. THE EFFICIENCY MODEL WITH COMPOSED ERROR

We propose an efficie~lcy model nith a composed multiplicative disturbance term, i.e., the product of a "true" error term and an elfciency measure:

lit is an eficiency ineasure and is distributed in the (0, 1) interval; u, is a dis-turba~ice in the proper sense and is distributed in the (0, co) interval. u,,k, a i ~ dstare niiitually independent.

Considering the Cobb-Douglas production model and putting l;,=exp(-r,) and u, =exp ( - c,), we get

We assume that tlie z , as \lie11 as the c, lia\ie a specified theoretical distribution. Tlie convolution of those distributions and a subseq~rent straightforward applica- tion c f the inaximuui likelihood ~ilethod yields estimates of tlie relevant parameters.

It is clear that the two types of errors and not separated with regard to each piece of availeble infi>rniation. Once the parameters of the digtribution obtained througti convolution are estimated, however certain characteristics, such as mo- ments, of tlie pd f ' s of respectively K and U can be derived. The two kinds of errors are separated a posteriori in this sense.

It is assumed that the L., are a random sample of a Gaussian distribution, with zero mean and variance a2.

The 2, are of course distributed in the interval (0, co). It turns out that assigning an exponential distribution to 2 leads to acceptable

properties of the moclel, borli from a theoretical and compiitatior~al point of view.

Other distributions, which from an eco~ io~n ic point of view seemed to be more adecl~late (Illat is, distributions the p i l f of which have a modal value between 0 and 1, such as the beta distribution or its 1.ig11t-truncated version) led to ~IISL!I.-

mouiitab!e dillici~ities in finding a neat expression for the ydJ' of the composed error.

EFFICIENCY ESTIMATION

For the pdf of Z we put

We get for the pdf of K = e-=

= 0 , elsewhere.

The p:l,f' of k is monolonc decre~lsing or increasing for 0< 1.< 1 , resp. 3. > I . If ;.== 1, li is illlifor~~ily i i id istr ib~~ted tile interval.

FIG. 1 a FIG. 1 b

There are three possible shapes for the pdf corresponding to the situation where higher efficiencies have higher probability densities. From an eco~loillic point of view o ~ l y these three cases can be considered as meaningful (Figure 1 a).

This is it1 contrast with the pdf for I<, g,(k; n ) , generated by a Gamma proba- bility density function (the procedure followed by Afriat and Richmond) where the above n~entioned situation only corresponds to one specific intervzl for the rele- vant parameter (0<17<1 ; see Figure lb). The exponential pdf for Z clearly i~iiposes fewer restrictions on tlle distribution of efficiency than the Gamma pdf.

An additional argumetit in favor of the exponential pdf consists in the finite values of the probability density corresponding to a maximal level of efficiency. This is not the case with the pdf for K = e-' resulting from a Gamma-distribution for 2.

Most of all, the choice of an expoilelltial distribution for 2,leads to neat ex- pressions for the average efficiency &,= E(K), and for the pdf of the cornposed error term W =Z + V and its moments.

For the avcrage efficiency level E,, we ~ b t i i i ~ l

43 8 MEEUSEN AND VAN DEN HROECK

Using the general expression for the pdf of a sun1 of two independent random variables with respective pdf's f,(z) and f,,(v), we get after some transforii~ations

The following i?loments can easily be computed :

The above expression for 11, is useful in determining which part of the total regres- sion error is attributable to respectively inetficiellt operating and "true" statistical disturbance. The skewness measure is positive and therefore indicates asym- metry to the right. From p , we conclude that the distribution of the composed error is leptocurtic, i.e., the corresponding pdf is sharper around the mode than normal. If /? tends to infiliity, i.e., efficiency approaches the 1 0 0 x level, f,(,v) obviously tends to the ~ l o r n ~ a l distribution.

It is clear that the average efficiency like defined in expression ( 3 ) can now no longer be interpreted, as do Afriat and XichmonJ, as the average proportioll p betheen the actual and potential production level, since it holds that

It follows immediately that tlie average value of the proportio~l p is a thoroughly misleading measure of efficiency.

3. DATA AND ESTIhfATION PROCEDURE

The data from the 1962 French Census of Manufacturing Industries have beeti used for a first verification of the foregoing model. The same data have already bee11 handled by Mairesse [t?] to estimate production functions at the micro- level for France. I-Ie compared them with similar production function results for Norway, estiinated by Griliches and Ringstad [ 6 ] . The results of the latter enabled Richmond to implement his nletliod to measure productive efficien- cy. Ten industries have been selected; in decreasing order of the number of firms at issue: nlachine construction and mechanical tools (953) , textiles (824) , electrical engineerislg (679) , paper (523) , industrial chemicals (415) , vellicles and

EFFICIENCY ESTIMATION 439

cycles (356), footwear (341), milk products (320), sugar-works, distilleries and beverages (292), and glass products (135). In each industry, the sample consists of all corporate firms einployi~lg at least 20 workers and other salaried enlployees. The production variable is measured by the value added at factor prices. Tlie labor variable is the ~unweighted sum of workers.

For the capital variable only the book value of gross fixed assets was available. This seeins justifiable: Griliches and Ringstad used both a capital service flow measure based on fire insurance values of machinery and buildings and given depreciation rates, and capital stock measures; their results were practically uii- affected.

We define the loglikeliliood function as

log L(1vlO) = C log f,,(\l~,/O) . 1

0 is the shorthand cxpression for tlie parameter vector with elements 2 , o, log A , Dl, and /?,. A is the multiplicative constant. /?, aiid /?,are respectively tlie pro- duction elasticities of capital and labor.

The maximum likelihood estiinators 0of the parameters of the new model have bee11 estimated by applying a co~nbination of three minimization methods. Tlie first method consists of a Monte Carlo searching technique finding the neighbor- hood of the global minimum. It leads to reasonable starting values for the next minimization stage: simplices are formed which adapt themselves to tlie local func- tion landscape, aiid contract in tlie neighborhood of a minimum [9]. The third and last minimization technique is based on a variable metric method by. Davidon [3]. The sequential application of these techniques provides for reliable maximum likelihood estimators and in most cases for their standard errors. The computer program supplies in addition the inverse of the Hessian of the log- likelihood function in 8.

Since the sample sizes are large and tlie Cramer-Rao bound therefore is a close approxiinatioii of the real covariance matrix, and since -as it will appear -

~ ( M s )is nearly normal i11 most instances and Q should then be "nearly sufiicient", the inverse of tlie Hessian in 0can reasonably be looked upon as a good estimatioii of the actual covariance matrix of the parameter Q. In those cases when the loglikelihood functio~i is not approximately parabolic in the neighborhood of the maximum, the program supplies an estimate of the standard error (in tlie shape of a confidence interval which is obviously no longer symmetrical around tlie cstirnated value of the parameter) obtained by means of a scanning p r o c e d ~ r e . ~

4. RESULTS OF THE EMPIRICAL ANALYSIS

Comparing the elasticities of production of both the average and the frontier Cobb-Douglas model (Table I), we see that the elasticities of production remain

These methods have been incorporated in a package of programs called MINUIT, by .[. ia i i i cs and M. Rooi of CERN at Gcncva.

440 MEEUSEN AND VAN DEN BROECK

practically the same. In spite of the nonnormality of the distribution of the composed error the regression coefficients have been very little affected. The re- turns to scale remain constant for the industries under consideration. Only the intercept changes in an ~lpward direction. This is as expected. Although in this case the production frontier turns out to be a neutral shift of the average psoduc- tion function, this shall not necessarily be so in other applications.

From the results in Table 2 it follows that the most important part (at least 75% and in most cases much more) of tlie variance of the convoluted distrib~ltion is due to statistical errors and therefore cannot be attributed to inefficiency. Neglecting this fact may be fatal for eficiency measureme!?t. The efficiency parameter 3, ranges from 16.684 (Glass products) to 2.41 80 (Sugar-works, distili- ery, beverages) and is highly significant (at the level of 9573 for all the industries.

When we calculate the average efficiency E , of the composed error model and compare it with the efficiencies computed according to Richmond's method, (see Table 3) it is easy to see that the latter are improbably low. This is not the case for the composed error efficiency, which looks economically more plausible. As E , is a monotone transformation of 2, the industry with the highest i, has also the highest average efficiency, i.e., glass products with efficiency equal to 0.9435. The sector sugar-works, distillery and beverages has the smallest average efliciency namely 0.7074. Although for sugar-works, distillery and beverages the lower extreme for E, corresponds to the lower extreme for tlie Richtnond measure c, this does not seem to imply a correspondence of a ]nore general nature. 0 1 2 the contrary, the low correlation coefficient (0.254) between the "co~i~posed" efficien-cy measure and tlie Rich~nond measure leads to an opposite conclusion as i t is not significantly different fro111 zero at the 907; level.

It is of interest to investigate the relationship between tlie structural para~neters and the efficiency in terms of their correlation (see Table 4). There is a slight negative relation between the labor elasticity of production and the eEiciency parameter (-0.031). The correlation between the capital elasticity of production and the efficiency parameter is greater and positive (0.430).

Their difference is not significant at an appreciable level. Consequently there is no evidence on the existence of a relationship between the structural parameters and efficiency. It is however evident that the mere consideration of ten industries is insufficient and that further research on this question is required.

As to tlie sources of the established deviations from the 100% level of productive efficiency, they might partially overlap with the reasons for Leibenstein's X-inefficiency: the contracts for labor are frequently incomplete, and not all inputs are marketed or, if marketed, are not available on equal terms to all buyers (pa- tents) [9, (412)l. Only a conlplete investigation of the market structure and the internal organization and operation of the firms could give a more conclusive answer on this issue.

It may also be interesting to calculate the proportion of firms with productive efficiency at least equal to some given value K, where rc is some given constant (0 <K <1). We have

EFFICIENCY ESTIMATIOI\'

-- - - -

MEkUSEI\I AND VAN DEN BROLCK

TABLE 2

COlMPARISOS BETCVI-FN THE VARlAXCES OF THE ERRORS OF THE STANDARD AX11

THE COhlPOSED ERROR CODE-DOU(;LASMODEL -~ --- ---

I Standard i 1 Cobb-Douglas / Coniposed el-)or

Industry

~ ~.

1) Glass products 2) Milk products 3) Textiles 4) h/iachine construction

Mechnical tools 5) Fleclrical inacliinc~ y 6 ) Vclriclcs and cyclcs 7) Int1~1stl~i:tlcl1emicals 8) Paper 9) Foot\veer

10) Sugar works, tlistillcry and beverages j 1.0935 ! 0.91 15 O , l 7 l ~ ~

-- -.- --- - ~

TABLE 3

C'OhlP4RISON BET\VI,LN TI-IF AFRIAT-KICFIWONI>AN11 COMI'OSLD Ll<ROR

PRODUCTIVE 1tFICItNCY -- ~~ ~ -- ~ - -: Afriat-liichniond Composed

~ - ,

Industry Ga~nma errorI

Ec -- . ! E -I: .

1 ) -~ Glass products 1

.

0.5697 0.9435 2) Milk products 0.6333 0.9265I3) 'Textiles ~ 0.5iS3

I 0.9089

4) Machine constri~ction I , i

and Mechanical tools 0.606:1 0.9036 5) Electl.ical X.1aciiinel.y 0.5565 0.8595 6) Vchicles and cycles 0.5789 0.8249 7) Industrial chemicals 0.4987 0.7901i I8) Paper 0.5864 0.7888 9) Foot~vear 0.7173 I 0.7560

10) Sugar ~vorks, distillery I I and beverages 0.4686 0.7069

('OKI<I.LA1 ION hlA'I R I Y 01: Tllt: I'ARAML K K S 01:'I'FIE C:OhlI'OSkI> 1:RIIUlI hlOl)l 1, . - - -- ..--- --

p g 1 2 ii n p~

I Ii1.000 j i-0.031 I 1.000 0.167 I - 0 . 1 5 0 , 1.000 . ..- - . . . .

~

EFFICIENCY ESTIMATION 443

These proportions are given by way of illustration in Table 5 for various proba- bility and efficiency levels.

A reinarkable feature of the results which should finally not be left ~undiscussed is the often slight, and occasio~lally important, difference between the standard errors on the productioll elasticities in the standard Cobb-Douglas inodel on the one hand, and in the composed error Cobb-Douglas model on the other. The latter are systematically smaller than the former. These differences in our opinion shed some light upon the extent of inisspecification involved in the es t i~nat io~l of production elasticities by iiieans of a standard Cobb-Douglas model, that is, a rnodel i n whicli 110 cxplicit account has hccn takcn of the p l icnoi i~cno~~ of proil~lc- tivc cficiency.

Taking into account the purely statical definition of efficiency :is adopted in the co~nposed error model, we arrive at an average sectoral efficiency for the in- dustries at issue which lies between .70 and .94. There is no significailt sta.tistica1 relationship between our eficiency lneasure a11d Iiichmond's. The latter arrives at results which are systematically lower than ours, which is only natural con-sidering his negligence of the statistical error. The evidence from the limited nuinber of industries which were examined was not conclusive iil finding a relation- ship between the efficiency phenomenon and the other stl.~~ctural characteristics of the production process. The aprioristic choice of the exponential d i s t r ib~~ t io i~ for the efficiency variable and, for that matter, of the Cobb-Douglas specificatiorl itself, is, of course, debatable. I t could, therefore, be ~ ~ s e f u l to conlpute the sen- sitivity of our results with respect to this particular choice.

REFERENCES

[ 1 ] AFRIAT,S., "Efficiency Estimation of Production Fiinctions," Itiicri~afiotinl Ec.oiroiiric. Re~jien,,XI11 (October, 1972), 568-598.

[ 21 AIGNER,D. AND S. CI%U,''On Estimating the 1ndusit.y P~.oduction Function," /liirc~ric,tri: Erorrorriic R e i ~ i r ~ ~ . , LVlIl (September, 1968), 826--839.

1 3 ] D a v r o o ~ ,W., "Variable Metric Method for Minimization," C'oiiil~i~/r/.Jor~r.titrl,X ( I 9hX), 406.

[ 4 ] FARRELL,M., "The Measusenlent of Productive Efficiency", Ju~i~.r~nluf file Kuj~nl Sintisti- cal Society-Series A , CXX (Part 111, 1957), 253-290.

[ S ] FBRSUND, L. I-IJALMARSSON, the Measurement of Productive Efficiency,"F. AND "On The Slvedi~h Jourr~nl oj' Ecot~owzics, LXXVI (June, 1974), 141-154.

[ 6 ] GRILICHES,Z. AND V. KINGSTAD,Ecot~otnics of Scale und the For,trr of ' the Pro(l~lctioiz Frlriction (Amsterdam: North Holland Publishing Company, 1971).

171 I.FIRI NSTLIN,H., "Allvcativ~ Eficicncy vs. 'X-Eflicicncy'," The ,4iii~ri(,irtr E(,utwiiii( Rci~ioio,

444 MEEUSEN AND VAN DEN BROECK

LVI (June, 1966), 392-41 5. [ 8 ] MAIRESSE, J., "Colnparisonof Production Function Estimates on the French and

Norwegian Census of Man~lfacturing Industries," in F. L. Altmann, 0.Kyn and H. I. Wagener, eds, On the Mc~sriremcntof' Fuctor. Producii~.itir.s. Tlieorc~tical Problenrs orld Etl>pirical Rr.~irlts, Papers and Proceedings of tlic 2nd Reircnburg Symposium (GOttingen: Vandenbroeck and Ruprecht, 1976).

[ 9 ] N E L U ~ R ,J . A ~ DR. M~ALI, "A Sin~picx hlelhod Sol. Function Minimi/c~rion", Corli/)ufcr Jo~o.nol,VII (1967), 308-313.

[lo] RICH~IOND, Ecoi7oitiic Rrvi~.,J., "Estimating the Eficiency of Prod~~ction," Ii~tc~rriotiotrrr! XV (June, 1974), 5 15-521.

[l l] SEITZ,W., "The Measurement of Efficiency Relative to a Frontier Production F~~nction," Americar2 Journal of Agricultural Econo171ics, LII (November, 1970), 505-5 1 1.

[12] TIMMER,C., "Using a Probabilistic: Frontier Production Function to Measure Technical Efficiency," Joilrtml of Political Ecorloitij~, LXXlX (July-August, 1971), 776-794.

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Efficiency Estimation from Cobb-Douglas Production Functions with Composed ErrorWim Meeusen; Julien van Den BroeckInternational Economic Review, Vol. 18, No. 2. (Jun., 1977), pp. 435-444.Stable URL:

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References

1 Efficiency Estimation of Production FunctionsS. N. AfriatInternational Economic Review, Vol. 13, No. 3. (Oct., 1972), pp. 568-598.Stable URL:

http://links.jstor.org/sici?sici=0020-6598%28197210%2913%3A3%3C568%3AEEOPF%3E2.0.CO%3B2-Z

2 On Estimating the Industry Production FunctionD. J. Aigner; S. F. ChuThe American Economic Review, Vol. 58, No. 4. (Sep., 1968), pp. 826-839.Stable URL:

http://links.jstor.org/sici?sici=0002-8282%28196809%2958%3A4%3C826%3AOETIPF%3E2.0.CO%3B2-R

5 On the Measurement of Productive EfficiencyFinn R. Førsund; Lennart HjalmarssonThe Swedish Journal of Economics, Vol. 76, No. 2. (Jun., 1974), pp. 141-154.Stable URL:

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7 Allocative Efficiency vs. "X-Efficiency"Harvey LeibensteinThe American Economic Review, Vol. 56, No. 3. (Jun., 1966), pp. 392-415.Stable URL:

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11 The Measurement of Efficiency Relative to a Frontier Production FunctionWesley D. SeitzAmerican Journal of Agricultural Economics, Vol. 52, No. 4. (Nov., 1970), pp. 505-511.Stable URL:

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12 Using a Probabilistic Frontier Production Function to Measure Technical EfficiencyC. P. TimmerThe Journal of Political Economy, Vol. 79, No. 4. (Jul. - Aug., 1971), pp. 776-794.Stable URL:

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