An empirical test of marginal productivity theory · An empirical test of marginal productivity theory Martin Biewena,b,c,* and Constantin Weiserd aDepartment of Economics, University

An empirical test of marginal

productivity theory

Martin Biewena,b,c,* and Constantin Weiserd

aDepartment of Economics, University of Tübingen, Mohlstr. 36, Tübingen72076, GermanybIZA, Bonn, GermanycDIW, Berlin, GermanydDepartment of Economics, University of Mainz, Mainz, Germany

We explore an hitherto unused approach to testing marginal productivity theory.Our method rests on the simple idea that, under the assumption of a linearhomogeneous production function, residual profits are informative about thediscrepancies between factor payments and marginal products. Our empiricalapplication using data on manufacturing plants in Chile suggest moderate devia-tions from marginal productivity theory which depend on firm size.

Keywords: marginal productivity theory; distribution of income; robuststatistics

JEL Classification: D33; D22; D40

I. Introduction

Derived from the assumption of individual maximizationbehaviour and competitive markets, the hypothesis thatproduction factors are paid their marginal products is oneof the main ingredients of neo-classical economics. Ifone were to determine the share of contributions to the-oretical or applied economics making these assumptions,one would come to the conclusion that there are fewassumptions that are more widespread than the one thatproduction factors are employed up to the point wheretheir remuneration equals their marginal product. This is,of course, not to say that a large number of alternativetheories about the determination of factor remunerationsexist. Some of these theories will be discussed below.Given the great importance of marginal productivitytheory however, it is puzzling that the task of testingthis theory has received so little attention in the literature.

In this article, we take a fresh look at testing marginalproductivity theory empirically. We use a simple buthitherto unused approach that circumvents the need to

specify a production function and to compare the impliedmarginal products to factor payments. Our method rests onthe simple idea that, under the assumption that the produc-tion technology is linearly homogeneous, residual profits(or losses) are informative about whether each productionfactor was paid more or less than its marginal product. Thefact that we do not need to explicitly specify a productionfunction represents an advantage over previous attemptsto test marginal productivity theory as misspecification ofthe production technology is likely to yield biased testsabout whether or not factor payments equal marginalproducts. Our set-up is in this respect also less restrictivethan large parts of the literature which usually make theassumption that the production function is of the Cobb–Douglas form, the constant elasticity of substitution (CES)form, the Translog form, or similar parametric forms. Wealso do not make any behavioural assumptions such asprofit maximization, or assumptions on price taking orprice setting behaviour. As a consequence, our resultsrepresent powerful tests of whether such assumptions areconsistent with the data.

*Corresponding author. E-mail: [email protected]

Applied Economics, 2014Vol. 46, No. 9, 996–1020, http://dx.doi.org/10.1080/00036846.2013.864042

996 © 2013 Taylor & Francis

mailto:[email protected]

The validity or invalidity of marginal productivity the-ory has important implications for different areas. Asindicated above, marginal productivity theory rests onthe assumption that firms maximize profits (or minimizecosts), and that markets are competitive. Any deviationfrommarginal productivity theory will therefore indicate aviolation of either of these assumptions. Equality of factorpayments and marginal products also plays an importantrole in considerations of allocative efficiency. For exam-ple, in the general set-up of the first theorem of welfareeconomics, a necessary condition for allocative efficiencyis that production factors are paid their marginal products.Another area which is directly connected to the remunera-tion of production factors is the distribution of income.Indeed, this was one of the earliest questions economistsdealt with. Marginal productivity theory implies that eachproduction factor receives the share it marginally contri-butes to the overall product. This has both positive andnormative implications.1

We apply our method to a rich panel data set of manu-facturing plants in Chile. Our empirical results suggests thatthere are moderate deviations from marginal productivitytheory which depend on firm size. In large firms, capitalreceives more than its marginal contribution to output onaverage, intermediate inputs less, and labour about its mar-ginal product. In medium-sized firms, capital and labourappear to be engaged in sharing the rents from underpayingintermediate inputs with respect to their marginal contribu-tion. In small firms – which are often owner-managed –direct payments to capital are below its marginal product,but owners seem to extract part of the remuneration of theircapital by overpricing their own labour input in the firm.Wealso provide results in which we disaggregate the over-/underpayment of different forms of labour (managers, high-skilled, low-skilled etc.). As our literature review belowshows, our article also seems to be the first one that con-siders deviations from marginal productivity theory for thedifferent production factors capital, labour, and intermedi-ate inputs at the same time.

The rest of this article is structured as follows. In SectionII, we give an overview over related literature. Section IIIpresents the basic relationship between profits, factor pay-ments and marginal products, which we use to test marginalproductivity theory empirically. In Section IV, we discussour econometric methods. Section V presents our empiricalresults. In Section VI, we provide some interpretations ofour findings, and Section VII concludes.

II. Related Literature

In this section, we review some related literature. One ofthe earliest attempts to test marginal productivity theory

can be found in Bronfenbrenner and Douglas (1939) andGunn and Douglas (1940, 1942). In these studies, theCobb–Douglas production function is fitted to manufac-turing data at the industry level, and implied marginalproducts are compared to industry wage averages. Theauthors conclude that deviations from marginal productiv-ity theory are small. While the contribution of these arti-cles to economics is undisputed (especially given thepossibilities of data collection and analysis at the time),it is clear that the assumption of a Cobb–Douglas produc-tion function is very restrictive. Further limitations are theuse of value-added output and data at the industry levelwhich both imply difficult aggregation issues.

A slightly more sophisticated approach to testingmarginal productivity theory is adopted by Stein(1958), who derived two implications from this theoryand tested them using industry-level data on averagewages. Stein’s approach is also subject to the limitationsdescribed in the previous paragraph. Some of the pro-blems that arise in studies that use Stein’s methodologyor the methodology of the very early contributions arediscussed in a critique of Stein’s article given inArchibald (1960).

Another attempt to test whether capital and labour arepaid their marginal products is by Thurow (1968).Estimating different variants of aggregate Cobb–Douglasand CES production functions for the United States basedon time series data on output, capital and labour for theperiod 1929 to 1965, the article contrasts the impliedmarginal productivities with observed aggregate returnsto capital and labour over the same period. The authorfinds substantial discrepancies between implied marginalproducts and observed aggregate returns. He concludesthat, as a consequence, the economy is in disequilibrium,and discusses possible reasons. Apart from the possiblerestrictiveness of the production function used, theapproach used in Thurow’s (1968) study clearly faceseven more severe aggregation problems than the studiescited before. Thurow’s result that capital receives moreand labour receives less than its marginal product was alsoobtained earlier by Hildenbrand and Liu (1957) but basedon industry data.

A general discussion of methodological problemsinvolved in testing marginal productivity theory is pro-vided by Gottschalk and Tinbergen (1982). They listed anumber of possible issues such as different forms ofaggregation problems or the difficulty to find the correctspecification for the production function. However, as wewill show below, their claim that all tests of the marginalproductivity theory need to first specify a productionfunction and then compare implied marginal productiv-ities to observed factor payments does not seem to becorrect.

1 For a discussion, see e.g. Friedman (1976).

An empirical test of marginal productivity theory 997

The first study that used a more disaggregated schemeof production factors in a test of the marginal productivitytheory appears to be Gottschalk (1978). Using industrydata, Gottschalk (1978) distinguishes between a numberof different occupations in both production and adminis-tration. He also finds that capital receives more than itmarginal product, while labour receives less. Moreover,his results suggest that there are a number of discrepanciesbetween marginal productivity and wages paid acrossoccupations. Apart from the use of industry level data,Gottschalk’s results are also subject to the criticism that avery specific production function is assumed, and thatvalue-added data are used. In addition to theoretical pro-blems that may arise when working with value-addeddata,2 the implied omission of intermediate inputs as pro-duction factors does not allow one to address whetherthese factors receive more or less than their marginalproducts and may lead to omitted variable bias in theestimated deviations from marginal productivity theoryfor the input factors explicitly included in the analysis.

One of the first studies that uses microeconomic data ina test of the marginal productivity hypothesis is by Frank(1984). Recognizing the fact that it is hard to estimatemarginal productivity in production processes that involvemore than one factor, he focuses on a number of occupa-tions in which marginal productivity is relatively easilyobserved. The special focus of the article is on wage andproductivity differences within a given institution ratherthan on deviations from marginal productivity theory ingeneral. Using data on automobile and real estate salespersons as well as on university professors, and making anumber of assumptions on the production technology andfirm behaviour, he finds that wages are more compressedwithin institutions than productivities. This is an interest-ing result, although it is unclear to what extent it can begeneralized to occupations or sectors other than the onesconsidered.

Using a highly informative employer–employee dataset, Hellerstein et al. (1999) also examine whether wagedifferences between different types of workers are consis-tent with differences in estimated marginal productivities.Hellerstein et al. (1999) used micro-data, they distinguishbetween different types of labour, and they include mate-rials as a production factor in their estimated productionfunction. In our view, their study is one of the very fewgenuine empirical studies of the marginal productivityhypothesis. A limitation of their contribution is that theymake explicit functional form assumptions when specify-ing the production function. Moreover, as some of theother studies cited above, Hellerstein et al. (1999) onlyexamine deviations from the marginal productivity theoryfor the factor labour, but not for capital or intermediateinputs. Their results suggest that relative wages indeed

reflect relative marginal productivities, except for thecase of gender wage differentials. Using a very similarapproach, Hellerstein and Neumark (1999) investigategender wage and productivity differentials in firm datafrom Israel, concluding that relative wage differenceswith respect to gender are consistent with relative mar-ginal productivities.

A recent study that addresses a number of issues weconsider below is by Dobbelaere and Mairesse (2013).Using panel data on French firms, Dobbelaere andMairesse (2013) develop a comprehensive approach tothe joint estimation of production functions and imperfec-tions in product and labour markets. They theoreticallydistinguished between six different regimes characterizedby the perfectness or imperfectness of the product marketand different bargaining regimes for the labour market.They then show how to nest these six regimes in a regres-sion analysis that identifies the different parameters ofmarket imperfections and bargaining regimes. Theirempirical results suggest that the predominant regime isone of imperfect competition in the product market andefficient bargaining in the labour market (which impliesthat labour receives more than its marginal product). Alimitation of Dobbelaere and Mairesse’s study is that theyassume a Cobb–Douglas production function, and thatthey do not allow for the possibility of imperfect inputmarkets. An advantage of their study is that they canconsider in great detail heterogeneity across firms andindustries.

Taken together, when compared to previous contribu-tions to analyzing the empirical validity of marginal pro-ductivity theory, our study has the advantage that we useno parametric assumptions for modelling the productiontechnology that we do not make assumptions about inputand output market imperfections (in particular, we alsoallow for market power in input markets), that we canmake explicit statements about absolute (rather than rela-tive) deviations from marginal productivity remunerationand that we provide results for all three categories ofproduction inputs (capital, labour and intermediateinputs). A limitation of our approach is that it dependson the availability of information on firms’ profits (as wellas on factor inputs) which is sometimes hard to come byand that we can address heterogeneity across firms only toa limited extent (we consider heterogeneity with respect tofirm size but not with respect to other dimensions).

III. Profits and Marginal Productivity

In this section, we derive a simple equation that relatesfirms’ profits to discrepancies between factor paymentsand marginal products. To arrive at this equation, we only

2A general critique of using value-added data in the studies involving production functions is given in Basu and Fernald (1995).

998 M. Biewen and C. Weiser

make the assumption that the firms’ production function ishomogeneous, and that production at the firm level exhi-bits constant returns to scale. We acknowledge that bothassumptions may not be uncontroversial. However, bothof them are commonly made in both theory and empiricalapplications, suggesting that our approach is not morerestrictive than other approaches in the literature.Moreover, we make no further assumptions on the para-metric form of the production function, which makes ourapproach less restrictive than many contributions in theliterature. As to the assumption of constant returns toscale, there appears to be a consensus based on a largenumber of studies using a variety of methodologies anddata sets for different countries, suggesting that returns toscale are constant at the firm level. Examples includeGriliches and Ringstad (1971), Nguyen and Reznek(1991), Baily et al. (1992), Westbrook and Tybout(1993), Bregman et al. (1995), Levinsohn and Petrin(2003), Abraham and White (2006) and Basu et al.(2009). Evidence based on industry level data is moremixed, but findings presented, for example, in Basu andFernald (1995, 1997), Burnside et al. (1995), Oulton(1996), and Inklaar (2007) also suggest that there is littleevidence against the hypothesis that returns to scale areconstant. Below, we will test to what extent the assump-tion of constant returns to scale is consistent with our dataand assess possible biases that would follow if the assump-tion of constant returns to scale is not met.

We start with the accounting identity stating that reven-ues are equal to the sum of costs for the different produc-tion factors, profits and taxes

PY ¼ capital costs þ labour costsþ intermediate inputs

þ profitsþ taxes ¼ rK þ wLþ qI

þ profitsþ taxes;

(1)

where P denotes product price, Y denotes real output,K; L and I denote the input quantities of capital, labourand intermediate inputs and r;w and q denote their per-unit prices. Capital costs are meant to include all explicitpayments to capital holders, i.e. interest payments to cred-itors aswell as dividends and otherwithdrawals of equity byequity holders. It is important to note that residual profits (orlosses) retained in the firm automatically accrue to the firm’sowners and will therefore have to be taken into accountwhen assessing whether or not capital is over-/underpaidwith respect to its marginal productivity (see below).

Assuming homogeneity and constant returns to scale,real output Y can be written as

Y ¼ fKK þ fLLþ fI I ; (2)

where fK ; fL and fI denote the marginal products of capital,labour and intermediate inputs, respectively.

Combining (1) and (2) yields the basic equation

π ¼ profit þ taxes

P

� �

¼ fK � r

P

� �K þ fL � w

P

� �Lþ fI � q

P

� �I ;

(3)

i.e. real profits before taxes are the sum of deviationsbetween factor payments and marginal products, weightedby input quantities. Equation 3 will be our basic vehiclefor testing whether factor payments equal marginalproducts.

In the following, we will use the notation

π ¼ βKK þ βLLþ βI I (4)

with βK ¼ fK � rP

� �; βL ¼ fL � w

P

� �and βI ¼ fI � q

P

� �.

The case in which a particular production factor, e.g.labour, can be subdivided into different categoriesL1; L2; . . . ; Lk yields the analogous equation

π ¼ βKK þ βL1L1 þ βL2L2 þ . . .þ βLkLk þ βI I : (5)

It can be seen that in this case, the coefficient βL inEquation 4 identifies the (weighted) average over-/underpayment of the different labour categories overtheir marginal products because

βLL ¼ βL1L1 þ βL2L2 þ . . .þ βLkLk )βL ¼ βL1

L1Lþ βL2

L2Lþ . . .þ βLk

LkL:

(6)

The amount of over-/underpayment of production factorK; L1; L2; . . . ; Lk ; I can directly be read off Equation 5.Coefficients β that are greater than zero indicate under-payment and those that are less than zero indicate over-payment of a factor when compared to its marginalproduct. The magnitude of β directly represents the abso-lute amount of over-/underpayment per unit of the produc-tion factor in question, measured in real money units.3

It should be noted that, under our assumptions,Equation 4 or 5 hold for a given firm in a given period.It is not implausible that there is heterogeneity in thecoefficients βK ; βL and βI over time and/or across firms.We will investigate the variability of βK ; βL and βI overtime by carrying out yearly estimations. We will alsoconsider heterogeneity of βK ; βL and βI with respect tofirm size. For reasons explained below, we are not inthe position to address the potentially important ques-tion of further heterogeneity in βK ; βL and βI across

3 Focussing on real money units seems necessary as in some of our estimations we pool years with possibly very different price levels.


firms, i.e. all our results have to be interpreted asaverage deviations from marginal productivity remu-neration for a given subpopulation of firms.

IV. Data

For our empirical application, we use a rich panel data seton Chilean manufacturing plants earlier versions of whichhave been used in a number of well-known studies.4 Thedata set is called ENIA (‘Encuesta Nacional de la IndustriaManufactuerera'), and was generously provided to us bythe Instituto Nacional de Estadisticas de Chile (INE). Ourunbalanced panel covers 6634 manufacturing plants (mostof which are single-plant firms) over the period 2001 to2006.5 A particular advantage of our data set is that itincludes – besides general information on revenues, costs,intermediate inputs and capital – detailed information ondifferent types of labour inputs. This makes it possible tostudy whether and to what extent different types of labour(managers, high skilled, low skilled, etc.) are under-/overpaid with respect to their marginal product.

In the following, we give a brief description of the mainvariables used in our analysis. Our main dependent vari-able is real profit before taxes (see Equation 3), which wasconstructed as revenues minus all costs (excluding taxes).Total revenues and costs for all firms were computed bysumming up the revenues and costs from many differentsubcategories. As indicated above, we include, in thecapital costs, all explicit payments to either creditors orequity holders, i.e. we also include dividend payments andreported withdrawals of equity in the case of nonincorpo-rated firms. We deflate profits with a series of low-leveldeflators (base year = 2003) at the four-digit industry level(‘Clasification Industrial Internacional Uniforme de todaslas actividades economicas’, CIIU Rev.3) which is avail-able from the INE. Firms’ capital is measured in bookvalues and was deflated by a deflator for capital goods alsoprovided by the INE. Intermediate inputs comprise allnonlabour and noncapital inputs used by the plant toproduce output. We deflated intermediate inputs using aseries of low-level deflators for different types of inter-mediate inputs at the four-digit industry level. Labourinput is measured in hours per year for different labourcategories. We can distinguish between labour input byowners, managers, high-skilled workers that carry out

tasks directly related to production, low-skilled workersthat carry out tasks directly related to production, low-skilled workers that carry out tasks indirectly related toproduction, administrative personnel (accounting, electro-nic data processing, excluding secretarial staff), other ser-vices (secretarial staff, catering, drivers, security, cleaningpersonnel) and sales personnel.

Our data set is large enough to allow us to split oursample into several subsamples according to plant size.This ensures that the plants under consideration are suffi-ciently homogeneous, and it allows us to consider hetero-geneity between plants of different size. We define threegroups (i) plants with 10–25 workers, (ii) plants with 26–100 workers and (iii) plants with 101–1000 workers.6

Summary statistics for our main variables in all subsam-ples are given in Table A1. Figures 1–3 present someadditional information for the total sample along withinformation on the macroeconomic environment duringthe period under consideration. The figures show that theperiod 2001 to 2006 was one of stable growth both for themanufacturing sector and the rest of the economy withalmost constant and moderate inflation rates. Figure 2shows in addition that low-skilled production workersprovided by far the highest share of labour input, followedby high-skilled production workers. Labour input by othertypes of labour was less substantial.

V. Econometric Implementation

Our goal is to estimate variants of Equation 3 on our data.However, our initial attempts using OLS methods yieldedunrobust and implausible results. The reason is that mostof our variables, in particular our dependent variable(profit before taxes), turn out to be highly leptokurtic, i.e.they are characterized by a large share of extreme obser-vations (see Table A1). A common way to deal with non-normal, highly leptokurtic variables in regression analysisis to trim the sample, e.g., to discard the top and the bottom1% of the data. Figure 4 compares the distribution of ourdependent variable and that of important covariates withthe distribution after trimming for the subsample of firmswith 10–25 workers. It is clear that discarding the top andthe bottom 1% of observations reduces the problem butstill results in an extremely leptokurtic distribution. This is

4 See, for example, Lui (1993), Westbrook and Tybout (1993), Pavcnik (2002) and Levinson and Petrin (2003, 2008). Unfortunately,similar detailed data including information on the variables described above are hard to obtain for more prominent economies such as theUnited States or European countries. The need to make such data more generally accessible is also felt in other fields of economics, seeCard (2011).5We do not use earlier years because (i) the questionnaires were changed in 2001 rendering many variables incomparable over time, (ii)time-consistent, low-level price deflators for inputs and output are only available for the period 2001 to 2006, and (ii) the period 2001 to2006 was one of the stable business cycle conditions, see below.6 The subsamples were defined so that each subsample comprised sufficiently many observations. We excluded the small fraction of evenlarger plants as this would have added an extremely heterogeneous group of plants and would have exacerbated problems with extremeobservations, see the discussion below.


all the more remarkable as, by construction, the subsampleof firms with 10–25 workers should already exhibit arelatively high degree of homogeneity. We graduallyextended the share of discarded observations but this

only led to a sequence of varying and unrobust estimationresults which were also erratic across alternative specifica-tions.7 Even if one accepts the method of graduallyexcluding extreme observations, one faces the problem

050

0 00

01

000

000

Tho

usan

ds o

f pes

os/h

ours

per

yea

r

2001 2002 2003 2004 2005 2006

Year

Real profit before taxes Real output

Real capital Real intermediate inputs

Labor (hours per year)

Fig. 1. Input and output measures over time, firms with 10–1000 workersSource: ENIA, own calculations.

040

000

80 0

0012

0 00

0

Hou

rs p

er y

ear

2001 2002 2003 2004 2005 2006

Year

Owners Managers

High skilled (production) Low skilled (production)

Low skilled (non−production) Administration

Other services Sales personnel

Fig. 2. Labour input by category, firms with 10–1000 workersSource: ENIA, own calculations.

7 Similar experiences when using similar firm data are reported in Verardi and Wagner (2010).


that there is no natural way to decide how large the shareof excluded observations should be.

It is an interesting question whether the considerablenumber of extreme observations in our data are the resultof measurement error or whether they represent genuineheterogeneity among firms. In order to get an idea aboutwhich case is more likely, it is important to know that ourrevenue and cost variables (and therefore our profit vari-able) are the result of summing up a very detailed set ofdifferent categories (e.g. revenues from selling self-manufactured products, revenues from selling patentsor licences, revenues from transportation services,...,costs of different forms of labour input, costs of differentforms of capital payments, costs of different forms ofintermediate inputs such as electricity, fuel, materials,…). To the extent that there is measurement error ineach category, this may lead to an accumulation of mea-surement error for the aggregate revenues and cost mea-sures. On the other hand, we found that, for a given firm,a number of different categories are usually empty (e.g.because the given revenues or cost categories did notapply to this particular firm) suggesting that the largevariability in overall costs and revenues represents gen-uine heterogeneity rather than accumulated measurementerror. In any case, we will use statistical methods that areby construction robust to a substantial share of extremeobservations in our data. We emphasize that these meth-ods apply independently of whether these extreme

observations are the result of measurement error or ofgenuine heterogeneity. We also want to point out that themethods used by us explicitly avoid simply discarding anarbitrary number of extreme observations from the data.

Robust methods for data characterized by a potentiallylarge share of extreme observations have a long traditionin statistics, but have received relatively little attention ineconometrics. A good reference for robust statisticalmethods is Maronna et al. (2006). Up-to-date robustregression routines have recently been made availablefor Stata by Verardi and Croux (2009) and Jann (2010).8

The goal of robust methods is to provide estimationprocedures that are robust in the sense that estimationresults are not overly influenced by the presence ofextreme observations, while retaining efficiency proper-ties in the case when the data are not excessively char-acterized by extreme observations (e.g. in the case of aGaussian error distribution). The overall robustness of anestimator can be characterized by the so-called break-down point which is defined as the share of extremeobservations up to which the influence of such observa-tions on the estimates remains bounded.

In the following, we use the so-called MM estimatordeveloped by Yohai (1987) which combines a high break-down point and a high Gaussian efficiency.9 For theregression

yi ¼ θ0 þ θ1x1i þ . . .þ θpxpi þ εi ¼ θxi þ εi; (7)

500

550

600

650

700

Exc

hang

e ra

te (

peso

/US

dol

lar)

01

23

45

67

8

Per

cent

2001 2002 2003 2004 2005 2006Year

Real GDP growth InflationExchange rate (peso/US dollar)

Fig. 3. Macroeconomic variables for ChileSource: OECD.

8We use these routines (mmregress and robreg) for our estimations.9A good description of this estimator is given in Verardi and Croux (2009).


the MM estimator is defined as

θ̂MM ¼ argminθ

Xni¼1

ρriðθÞσ̂S

� �; (8)

where riðθÞ ¼ yi � θ0 � θ1x1i � . . .� θpxpi and σ̂S is arobust estimator of scale which is determined in a previousstep, see below. The function ρð�Þ is usually chosen as theTukey Biweight function

ρðuÞ ¼ 1� 1� uk

� �2h i3if juj � k

1 if juj > k:

((9)

The intuition behind the estimator defined in Equations 8and 9 is clear: instead of considering squared residualswhen minimizing the sum of deviations from the regres-sion plane, each residual undergoes a transformation ρð�Þsuch that the influence of large residuals is dampened(depending on the constant k).

02.

0e−

064.

0e−

066.

0e−

068.

0e−

061.

0e−

05

Den

sity

0 500 000 1 000 000 1 500 000 2 000 000 2 500 000

Real capital (in thousands of pesos)

02.

0e−

074.

0e−

076.

0e−

078.

0e−

07D

ensi

ty

−2.00e+07 0 2.00e+07 4.00e+07Real profit before taxes (in thousands of pesos)

02.

0e−

064.

0e−

066.

0e−

068.

0e−

06D

ensi

ty

−500 000 0 500 000 1 000 000 1 500 000 2 000 000Real profit before taxes (in thousands of pesos)

05.

0e−

071.

0e−

061.

5e−

062.

0e−

06D

ensi

ty

0 5 000 000 1.00e+07 1.50e+07 2.00e+07 2.50e+07Real intermediate inputs (in thousands of pesos)

01.

0e−

062.

0e−

063.

0e−

064.

0e−

06

Den

sity

0 1 000 000 2 000 000 3 000 000 4 000 000Real intermediate inputs (in thousands of pesos)

05.

0e−

081.

0e−

071.

5e−

072.

0e−

072.

5e−

07D

ensi

ty

0 5.00e+07 1.00e+08 1.50e+08 2.00e+08Real capital (in thousands of pesos)

Fig. 4. Illustration of extreme observations, firms with 10–25 workers. Left-hand panel: original sample. Right-hand panel:sample after discarding top and bottom 1%Source: ENIA, own calculations.


For the MM estimator, the normalizing scale σ̂S isrobustly determined in a first step. For this, defineσS ¼ σSðθÞ as the parameter that satisfies

1

n

Xni¼1

ρriðθÞσS

� �¼ E ρðZÞð Þ with Z,Nð0; 1Þ:

(10)

The robust estimator of scale σ̂S is then defined as thevalue that minimizes σSðθÞ with respect to θ. The mini-mizing parameter value θ̂S ¼ argminθ σSðθÞ is also calledS-estimator. An S-estimator has a high breakdown pointbut is rather inefficient. Yohai (1987) has shown thatcombining the first-step S-estimator in order to estimatethe scale parameter σ̂S with the second-step estimatordefined in Equation 8 leads to an estimator θ̂MM with ahigh breakdown point and a high efficiency in theGaussian case. Following Maronna et al. (2006), wechose a breakdown point of 50% and an efficiency relativeto the Gaussian case of 85% (which implies settingk ¼ 1:55 for the first-step S-estimator and k ¼ 3:44 for

the second-step estimator). Our results are practicallyidentical if we use lower levels of Gaussian efficiency,indicating that the choice of 85% does not come at the costof bias (setting efficiency at a too high level would lead tobias, in which case the low and the high efficiency esti-mator would significantly differ from each other). We alsoexperimented with lower breakdown points. This did notchange our results in any important way.

VI. Empirical Results

Yearly regressions

We start by estimating the robust regression describedpreviously year by year. The results presented inTables 1–3 suggest that the estimated coefficients areeither quite stable over time (capital and intermediateinputs) or statistically insignificant and unstable (labour).Given the relative stability of the observed patterns overtime, we defer a more detailed interpretation of the esti-mated coefficients to the next subsection where we carry

Table 1. Yearly regressions: firms with 10–25 workersa

Year 2001 (1101 Obs.) 2002 (1251 Obs.) 2003 (1286 Obs.)

Real capital .0786 (.0163) .0796 (.0023) .0253 (.0112)Real intermediate inputs .0371 (.0034) .0470 (.0081) .0394 (.0031)Labour (all categories) .0581 (.1080) –.2328 (.1051) –.1262 (.0948)Constant –16327.15 (4308.65) –3182.94 (3820.38) –2915.61 (3679.37)

Year 2004 (1375 Obs.) 2005 (1179 Obs.) 2006 (1120 Obs.)

Real capital .0506 (.0182) .0867 (.0059) .0743 (.0010)Real intermediate inputs .0293 (.0075) .0666 (.0099) .0876 (.0061)Labour (all categories) –.0965 (.0851) –.1474 (.0887) –.4100 (.1085)Constant –1666.94 (3309.84) –7181.17 (3337.97) –3338.86 (3849.97)

Source: ENIA, 2001–2006, own computations.Note: aRobust SEs in parentheses.


Year 2001 (1246 Obs.) 2002 (1372 Obs.) 2003 (1419 Obs.)

Real capital .0697 (.0006) .0809 (.0140) .0448 (.0024)Real intermediate inputs .0743 (.0024) .0075 (.0000) .0317 (.0045)Labour (all categories) .0487 (.1311) –.0437 (.1071) –.1053 (.0721)Constant –38469.04 (11890.36) 8303.077 (10286.21) 510.22 (7038.15)

Year 2004 (1522 Obs.) 2005 (1453 Obs.) 2006 (1434 Obs.)

Real capital .0537 (.0098) .0736 (.0011) .0619 (.0067)Real intermediate inputs .0640 (.0052) .0551 (.0047) .0316 (.0157)Labour (all categories) –.2588 (.0707) –.2622 (.0765) –.1427 (.0901)Constant 5464.08 (6973.95) –484.35 (7122.89) 1020.69 (9557.3)



out the regression on the pooled sample for the period2001 to 2006.10 Using the pooled sample with its muchlarger number of observations will also allow us to obtainstatistically significant results for the factor labour and, inmany cases, for its disaggregated subfactors (manageriallabour, high-skilled, low-skilled, etc.).

Pooled regression

Before we turn to a detailed discussion of the results forthe pooled sample, we return to the point that residualprofits (or losses) will eventually be owned by capitalholders and are therefore a part of the payments to theinput factor capital. Profits (after taxes) have therefore alsoto be subtracted when calculating the amount of over-/underpayment for the factor capital when compared to itsmarginal product. The total amount of over-/underpay-ment per unit of capital is therefore given by

fK � r

P

� ��

profitP

K¼ βK �

profitP

K¼ ~βK : (11)

In order to provide a full description of the over-/underpayment of capital, we also report in our results themean of ~βK over all firms in the sample.

In our initial estimations we noticed that for a number offirms, the ratio of profits/losses to capital appearing inEquation 11 was implausibly high or implausibly nega-tive. In order to avoid that the estimates of the average of~βK across firms were overly influenced by such extremevalues, we dropped firms in the bottom 1% and those intop 1% of the distribution of the profit to capital ratio.

Similarly, we detected a number of implausibly low orimplausibly high values of the hourly wage paid in a firm(which was computed as the firm’s cost of labour dividedby the number of hours employed by the firm), possibly aresult of coding or measurement error. In some cases, thisled to an implausible ranking of hourly wages acrossqualifications which was corrected when we droppedvery extreme observations. We, therefore, also droppedthe bottom 1% and the top 1% of hourly wage observa-tions. Apart from these two adjustments, we did not alterour data in any way. We emphasize that omitting thebottom and top 1% of profit-to-capital ratio and hourlywage observations did not change our robust regressionresults of our central Equation 3 in any important way. Theadjustments described were solely made in order to avoidimplausible summary statistics for hourly wages and thequantity in Equation 11.

Our results from applying the robust MM estimator tothe three subsamples (firms with 10–25 workers, 26–100workers and 101–1000 workers) are shown in Tables 4, 6and 7, respectively. Table 4 shows the results for the groupof firms with 10–25 workers. The first entry in column 1suggests that capital’s direct remuneration falls short of itsmarginal product by 6.15%.11 If profits and losses areincluded, capital is even more underpaid when comparedto its marginal product, namely by almost 13% on average.It is surprising that including profits decreases capital’sremuneration but this is easily explained by the fact thatover 50% of the firms with 10–25 workers actually operatewith losses (see Fig. 4). The cost of intermediate inputsalso falls short of their marginal contribution to output.More precisely, the underpayment of intermediate inputs


Year 2001 (698 Obs.) 2002 (735 Obs.) 2003 (738 Obs.)

Real capital .1913 (.0008) .0621 (.0123) .0610 (.0143)Real intermediate inputs .0922 (.0063) .1516 (.0112) .0863 (.0218)Labour (all categories) –.2988 (.2648) .3019 (.2450) .0557 (.3089)Constant 2.0e + 04 (1.1e + 05) –1.2e + 05 (1.0e + 05) 6.1e + 05 (1.4e + 05)

Year 2004 (824 Obs.) 2005 (866 Obs.) 2006 (829 Obs.)

Real capital .1430 (.0325) .1188 (.0072) .0919 (.0187)Real intermediate inputs .1158 (.0143) .0420 (.0128) .0845 (.0123)Labour (all categories) –.0373 (.2348) .0421 (.1638) –.0937 (.2027)Constant –2.3e + 05 (9.4e + 04) 5.5e + 04 (7.7e + 04) –6.5e + 04 (9.0e + 04)


10 The interpretation given there can easily be applied to the individual results in Tables 1–3.11According to Equation (4), capital is underpaid by an amount of .0615 times 1000 Chilean pesos per 1000 Chilean pesos capitalinvested (all variables except labour are expressed in thousands of Chilean pesos), or equivalently, by .0615 pesos per peso invested. Theover-/underpayment of intermediate inputs has a similar interpretation. It is the peso amount of over-/underpayment per peso spent onintermediate inputs. The interpretation of the labour coefficient is the over-/underpayment in thousands of pesos per unit of labour (i.e. perhour).


amounts to 0.042 pesos per peso spent on intermediateinputs (= 4.2%). By contrast, the factor labour seems to beslightly overpaid when compared to its marginal product.The overpayment amounts to .1259 thousand pesos perhour. In order to assess whether this is a large or a smallamount, one can compare this figure to the average hourlywage in firms with 10–25 workers, which is 2.034 thou-sand pesos (see first row of Table 5). Taken together, thissuggests that labour (including the labour input by own-ers) is overpaid by an amount of some 5% on average.

The other columns of Table 4 show some alternativespecifications at different aggregation levels of the factorlabour. The second but last column reveals that theobserved direct remuneration of owners’ labour substan-tially exceeded the corresponding marginal product.According to the estimates, owners’ labour was overpaidby 1.0501 thousand pesos per hour worked, whichamounts to an overpayment of some 22% (the averagehourly wage for labour supplied by owners was 4.655thousand pesos, see Table 5). This finding helps to resolvethe puzzle that firms in this size category systematicallyseem to make losses. It appears that owners extract part ofthe remuneration that should actually be attributed to theircapital through the remuneration of their labour (i.e. theyreceive a salary that is higher than their marginal product).This is plausible given the small size and the probablymostly unincorporated nature of these firms. The point

that owners’ income from capital and the remunerationof their labour are likely to mix up in practice is alsoconsidered in detail in Gollin (2002). The second but lastcolumn of Table 4 suggests that managers in firms with10–25 workers were also slightly overpaid (by some 7%),similarly low-skilled production workers (by 10%) andsales personnel (by 6%), although some of these effectsare only marginally significant. The intermediate results inTable 4 demonstrate that the estimates are consistentacross different levels of aggregation of the factor labour.

Turning to the results for firms with 26–100 workers,the first entry in Table 6 suggests that, not taking accountof profits, capital is underpaid by some 6.55%. If profitsare taken into account, capital is actually overpaid byabout by 13.65% when compared to its marginal product.Labour as a whole is also slightly overpaid when com-pared to its marginal product. The overpayment amountsto 198.5 pesos per hour, which is about 7.8% above themarginal product of an additional hour of labour (thehourly wage of labour employed in firms with 26–100workers is 2.530 thousand pesos, see Table 5). Again, theamount paid by the firms for intermediate inputs falls shortof their marginal contribution to output. The underpay-ment of intermediate inputs amounts to some 5%. Overall,it seems that firms with 26–100 workers generate rentsfrom underpaying intermediate products which are sharedby capital and labour. Again, the second but last column

Table 4. Pooled regression: firms with 10–25 workers (7312 Obs.)a

Real profit before taxes Specification 1 Specification 2 Specification 3

Real capital .0615 (.0137) .0807 (.0134) .0806 (.0134)Real capital (including profits) .1299 (.0575) .1491 (.0575) .1489 (.0574)Real intermediate inputs .0420 (.0069) .0417 (.0071) .0414 (.0074)Labour (all categories) –.1259 (.0513)

Labour category 1b –.7542 (.1886)Labour category 2 –.0763 (.0580)Labour category 3 –.1506 (.0741)Labour category 4 –.1216 (.0611)

Owners –1.0501 (.2601)Managers –.4319 (.2396)High skilled (production) –.0563 (.0622)Low skilled (production) –.1268 (.0623)Low skilled (nonproduction) –.0868 (.1679)Administration –.2415 (.1316)Other services .5378 (.2934)Sales personnel –.1655 (.0759)

Constantc –8479.72 (2286.48) –8803.16 (2300.33) –8565.38 (2356.76)

Source: ENIA, 2001–2006, own computations.Notes: aBootstrap SEs accounting for panel structure of the data in parentheses.bLabour category 1 = Owners, ManagersLabour category 2 = High skilled (production), AdministrationLabour category 3 = Sales personnelLabour category 4 = Low skilled (production and nonproduction), Other servicescAlso includes time dummies. Omitting constant and time dummies changes results only marginally.


Table 5. Real hourly wages, in thousands of pesosa

Firm size

10–25 workers 26–100 workers 101–1000 workers

Mean SD Mean SD Mean SD

All workers 2.034 1.220 2.530 1.560 3.227 2.472Labour category 1b 5.314 5.636 9.263 7.825 17.115 13.264Labour category 2 2.253 3.681 2.979 1.977 4.123 3.228Labour category 3 2.408 2.600 4.105 3.708 6.474 9.716Labour category 4 1.606 1.207 1.850 1.308 2.419 4.787

Owners 4.655 5.790 7.125 5.907 14.102 14.935Managers 6.017 4.933 10.368 8.476 17.229 13.290High skilled (production) 2.571 4.315 3.594 2.841 4.875 5.206Low skilled (production) 1.656 1.372 1.886 1.956 2.562 8.277Low skilled (nonproduction) 1.772 2.363 2.247 1.735 3.119 13.707Administration 2.100 1.524 2.793 1.886 4.002 5.141Other services 1.449 1.218 1.890 10.006 2.211 3.019Sales personnel 2.408 2.600 4.105 3.708 6.474 9.716

Source: ENIA, 2001–2006, own computations.Notes: aNominal wages deflated by CPI (base year 2003).bLabour category 1 = Owners, ManagersLabour category 2 = High skilled (production), AdministrationLabour category 3 = Sales personnelLabour category 4 = Low skilled (production and nonproduction), Other services

Table 6. Pooled regression: firms with 26–100 workers (8446 observations)a


Real capital .0655 (.0154) .0653 (.0144) .0651 (.0149)Real capital (including profits) –.1365 (.0352) –.1366 (.0354) –.1368 (.0354)Real intermediate inputs .0522 (.0170) .0525 (.0158) .0530 (.0152)Labour (all categories) –.1985 (.0822)


Owners –.9555 (.7229)Managers –.0150 (.6008)High skilled (production) –.1451 (.0826)Low skilled (production) –.1577 (.0718)Low skilled (nonproduction) –.4181 (.1437)Administration –.2687 (.2574)Other services –.3264 (.3459)Sales personnel –.2618 (.1534)

Constantc –2677.86 (5044.95) –2346.24 (5043.4) –1768.59 (5156.72)



gives more detailed results as to which types of labour areoverpaid. It seems that in particular, high-skilled produc-tion workers are slightly overpaid on average (by some4%), similarly low-skilled production workers (by some9%) and low-skilled workers whose work is only indir-ectly related to production (by some 18%), whichmight berelated to minimum wage laws. Note however, that inmost cases these effects are only marginally significant.For all other labour categories, one cannot reject thehypothesis that their remuneration equals their marginalcontribution to producing output at any conventional sig-nificance level.

The results for firms with 101–1000 workers are givenin Table 7. Again, capital is underpaid compared to itsmarginal product when profits are not taken into account.However, if profits are included, the remuneration ofcapital exceeds its marginal product by 358.4 pesos per1000 pesos invested. Again, intermediate inputs areunderpaid compared to their marginal contribution. Theamount of underpayment is higher for these larger firmsthan for the smaller firms discussed above (some 9%compared to 4% or 5%), which is plausible as largerfirms probably have more market power on input markets(see Section VII). The factor labour receives about itsmarginal product in firms with 101–1000 workers (seefifth row of Table 7). In these firms, workers may haveless bargaining power compared to smaller firms in whichthe factor labour appeared to engage in rent sharing with

the factor capital. A closer look at different types of labour(see second but last column of Table 7) reveals that noneof the different labour groups manages to earn more thantheir marginal products (except maybe owners whichhowever represent only a tiny fraction of workers in thegiven size class of firms).

Nonconstant returns to scale

In this section, we assess to what extent our results mightbe biased if returns to scale are not constant. As a basiccheck of the hypothesis of constant returns to scale, weestimate a CES production function

logðY Þ ¼ b0 � αρlog b1K

�ρ þ b2L�ρ þ b3I

�ρð Þ (12)

with b1 þ b2 þ b3 ¼ 1. The CES function is more flexiblethan, e.g., a Cobb–Douglas function and it has got theadvantage that the degree of homogeneity can be directlyread off the parameter α. The estimation results are shownin the upper panel of Table 8 for the case with the factorlabour in aggregated form, and in the lower panel for thecase with disaggregated labour categories. The estimatesfor α are very close to one in substantive terms, althoughthey are statistically different from one in some cases.

Table 7. Pooled regression: firms with 101–1000 workers (4690 Obs.)a


Real capital .1093 (.0341) .1073 (.0325) .1012 (.0321)Real capital (including profits) –.3584 (.0454) –.3603 (.0437) –.3665 (.0433)Real intermediate inputs .0892 (.0180) .0897 (.0172) .1016 (.0177)Labour (all categories) .1101 (.1551)

Labour category 1b –3.1273 (3.3483)Labour category 2 .2041 (.2760)Labour category 3 –.5343 (.4550)Labour category 4 .1951 (.1774)

Owners –11.6090 (6.7965)Managers –1.1584 (3.5937)High skilled (production) .2467 (.2865)Low skilled (production) .1266 (.1638)Low skilled (nonproduction) .7245 (.6058)Administration –1.6608 (1.2318)Other services 1.1175 (2.1841)Sales personnel –.5502 (.4538)

Constantc 22475.98 (77531.82) 39548.25 (79342.13) 72964.82 (81931.40)



In order to assess how this might bias our results, notethat in the case of a production function that is homoge-neous of degree α � 1, Equation 2 is replaced by

αY ¼ fKK þ fLLþ fI I ; (13)

where α is the scale elasticity (α < 1 for decreasing, andα > 1 for increasing returns to scale). An immediate con-sequence of Equation 13 is that in the case of decreasingreturns, the sum of marginal products does not exhaustthe overall product, while in the case of increasingreturns, it more than exhausts the overall product. Thismeans that in the case of decreasing returns, at least onefactor has to receive more than its marginal product,while in the case of increasing returns at least one factorhas to receive less than its marginal product.

The assumption of decreasing or increasing returns toscale modifies our main equation to

π ¼ profit þ taxes

P

� �

¼ fKα� r

P

� �K þ fL

α� w

P

� �Lþ fI

α� q

P

� �I :

(14)

As a consequence, the estimates β̂K , β̂L and β̂I will bebiased towards underpayment in the case of increasingreturns α > 1 and towards overpayment in the case ofdecreasing returns α < 1. In order to assess to what

extent our estimates might biased if the degree ofhomogeneity is, for example, α ¼ 1:04 (this is themost extreme value in Table 8), we consider the fol-lowing realistic example values for fK, fL and fI andr=P, w=P and q=P.

For the case of capital, consider

fKα� r

P

� �¼ 0:17

1:04� 0:10

� �¼ 0:063 )

fK � r

P

� �¼ 0:17� 0:10ð Þ ¼ 0:07;

(15)

where the left-hand side shows the estimated amount ofunderpayment, while the right-hand side displays the trueamount. A similar example could be given for intermedi-ate inputs. An example for labour is

fKα� r

P

� �¼ 2:43

1:04� 2:53

� �¼ �0:19 )

fL � w

P

� �¼ 2:43� 2:53ð Þ ¼ �0:10:

(16)

(These examples mimic the case of firms with 26–100workers given in Tables 5 and 6 with α ¼ 1:04 fromTable 8). We conclude that there may be some difference

Table 8. CES production function estimatesa

Firm size

10–25 workers 26–100 workers 101–1000 workers

With aggregated labour factorb0 .6857 (.1613) .4228 (.1617) .6366 (.1356)α 1.0162 (.0155) 1.0402 (.0139) 1.0230 (.0104)ρ –.8398 (.0670) –.5980 (.0927) –.6840 (.0979)Real capital .0720 (.0065) .0717 (.0073) .1099 (.0083)Real intermediate inputs .5250 (.0209) .5584 (.0244) .5405 (.0314)Labour .4028 (.0239) .3697 (.0237) .3495 (.0339)

With disaggregated labour factorb0 1.6907 (.2204) 1.5015 (.1525) 1.7815 (.2666)α .9904 (.0225) 1.0325 (.0124) 1.0031 (.0130)ρ –.6092 (.1122) –.6511 (.0640) –.4790 (.1200)Real capital .0558 (.0104) .0413 (.0067) .0692 (.0135)Real intermediate inputs .3821 (.0795) .2873 (.0448) .3935 (.0998)Labour category 1b .2788 (.0798) .2396 (.0642) .1960 (.1018)Labour category 2 .1426 (.0286) .2236 (.2236) .1727 (.0368)Labour category 3 –.0332 (.0373) .0380 (.0258) .0779 (.0304)Labour category 4 .1738 (.0273) .1699 (.0180) .0903 (.0203)

Source: ENIA, 2001–2006, own computations.Notes: aNonlinear least squares with clustered SEs in parenthesesbLabour category 1 = Owners, ManagersLabour category 2 = High skilled (production), AdministrationLabour category 3 = Sales personnelLabour category 4 = Low skilled (production and nonproduction), Other services


for the factor labour in some cases, but in general it seemsunlikely that our conclusions are reversed if α is notexactly equal to 1.

Apart from the above evidence in favour of approxi-mately constant returns to scale, it should be noted that ofthe two cases ‘strictly increasing returns’ and ‘strictlydecreasing returns', at least the case ‘strictly decreasingreturns’ seems unlikely even without reference to empiri-cal evidence (Basu and Fernald, 1995, 1997). First, thepossibility of replication suggests that returns to scaleshould in general not be falling. Second, extending theargument in Basu and Fernald (1995, 1997) to the case ofimperfect input markets (see Appendix), one can showthat under cost minimization

μ ¼ α � revenuescostsþ d

(17)

where μ is the markup-ratio, i.e. the ratio of price tomarginal cost, and d > 0 if firms have market power ininput markets. Basu and Fernald (1995, 1997) argue thatsharply decreasing returns to scale are implausiblebecause they tend to imply markup-ratios below one(firms price their products below marginal costs, whichdoes not make sense). Equation 17 shows that this is evenmore severe if firms have market power in input markets(for which we have evidence), because then implied mark-ups are further depressed (via d > 0).

Further robustness checks

As a further robustness check, we also carried out OLSestimations on a subsample which was ‘cleaned’ ofextreme values. A major virtue of robust statisticalmethods is that they are not only robust to (potentiallysubstantial amounts of) extreme observations but thatthey also provide ways to identify such extreme obser-vations. In the following, we exclude (in each subsam-ple) observations that are extreme values in the sensethat they are too far from the (robustly estimated)regression hyperplane. Our precise definition of anextreme value is an observation for whichjriðθ̂MM Þ=σ̂S j> c0:99, where c0:99 is the 99th percentileof the standard normal distribution (this is motivatedby the fact that the standardized MM residualsasymptotically follow a standard normal distribution).12

Tables A2–A4 show the results for OLS on the robustsubsamples. The estimates are almost identical to thoseof the MM regressions but SEs are in many cases much

lower (which is to be expected given that the elimina-tion of outliers also reduces sample variability).

We can also use the subsample cleaned of extremevalues in order to conduct further robustness checks. Inparticular, we apply a fixed-effects within-estimator onthis sample. This is a basic check of whether our resultsmay be biased by (time-invariant) unobserved firm dif-ferences that may be arbitrarily correlated with factorinputs. As Wooldridge (2005) has shown, this even cov-ers the case in which the parameters βK , βL and βI differacross firms in ways that may be correlated with (time-invariant) unobserved differences between firms (corre-lated random-coefficients model). The results of thewithin-estimator on the subsample cleaned of extremevalues are reported in Tables 9–11. In principle, theestimates are similar to OLS on this subsample or MMestimation on the full sample, but the coefficients gen-erally appear to be attenuated towards zero. For example,in firms with 10–25 workers, the underpayment of capital(in the case where profits are ignored) is about 3% com-pared to 6% in the robust regression case (see Table 4).Also, the underpayment of intermediate inputs and theoverpayment of labour vanishes, if the fixed-effects esti-mator is applied to the robust subsample of firms with10–25 workers. Similarly, in the case of firms with 26–100 workers, the underpayment of intermediate inputsand the overpayment of labour is also reduced towardszero (Table 10). In the case of firms with 101–1000workers, fixed-effects estimation on the robust subsam-ple suggests that real intermediate inputs and labourreceive about their marginal product, whereas capital isoverpaid (if profits are taken into account, see Table 11;the latter result is economically inconsistent as under theassumption of constant returns to scale, one productionfactor can only be overpaid if another one is underpaid).

It is well known that the within-estimator is much moresusceptible to measurement error than OLSmethods as thesignal to noise ratio may be drastically reduced by apply-ing the within-transformation. In this case, the within-estimator may even be more biased than the OLS estima-tor (Griliches and Hausman, 1986). Our evidence points inthis direction as the within-estimates are uniformly atte-nuated towards zero when compared to the OLS and MMregression results. In any case, given that OLS and within-estimates are generally quite close, we conclude that ourprevious estimates are not systematically biased due to theomission of time-invariant differences between firms. Ifanything, the fixed-effects results move our evidencefurther in the direction of the full validity of the marginal

12 In this way, we exclude so-called ‘vertical outliers’ as well as ‘bad leverage points’, see e.g. Verardi and Croux (2009) or Dehon et al.(2009). We do not exclude so-called ‘good leverage points’ which lie near the regression hyperplane but are characterized by extremevalues of the covariates. Dehon et al. (2009) argue that not excluding good leverage points may in some cases lead to an overestimation ofstatistical significance. As we are interested in detecting (possibly small) deviations from marginal productivity theory, it seems moreappropriate not to exclude good leverage points as they are likely to add statistical power.


Table 10. Fixed effects on robust subsample: firms with 26–100 workersa


Real capital .0386 (.0097) .0387 (.0097) .0399 (.0098)Real capital (including profits) –.1633 (.0293) –.1632 (.0293) –.1620 (.0293)Real intermediate inputs .0198 (.0099) .0204 (.0099) .0202 (.0100)Labour (all categories) –.1366 (.0655)Labour category 1b –.2139 (.4835)Labour category 2 –.1880 (.0958)Labour category 3 –.6575 (.2205)Labour category 4 –.0699 (.0735)

Owners .0278 (1.0873)Managers –.1860 (.5229)High skilled (production) –.2197 (.1083)Low skilled (production) –.0495 (.0827)Low skilled (nonproduction) –.0165 (.1864)Administration .0210 (.2762)Other services –.6428 (.4390)Sales personnel –.6725 (.2166)

Constantc 37244.10 (10066.03) 37905.82 (10219.55) 35378.55 (10575.29)

Size robust subsample 6759 6759 6753




Real capital .0318 (.0137) .0521 (.0137) .0534 (.0136)Real capital (including profits) .1002 (.0532) .1205 (.0532) .1218 (.0531)Real intermediate inputs .0091 (.0118) .0153 (.0118) .0138 (.0118)Labour (all categories) –.0928 (.0728)


Owners –.8744 (.3545)Managers –.2096 (.2801)High skilled (production) –.0074 (.0989)Low skilled (production) –.0693 (.0864)Low skilled (nonproduction) –.0110 (.1541)Administration –.2390 (.2030)Other services .5675 (.2756)Sales personnel –.1213 (.1312)

Constantc 3261.61 (3924.11) 716.92 (3984.59) 816.19 (3975.64)

Size robust subsample 6,112 6,114 6,113



productivity theory. Note that this robustness check doesnot address the potentially relevant but difficult to copewith endogeneity problem that arises if factor inputs varywith time-variant unobserved firm differences, e.g. pro-ductivity shocks. We believe however, that deviationsfrom marginal productivity remunerations should bemore closely related to firm characteristics which definethe market power of firms and which are therefore morelikely to be of a time-invariant or slowly changing nature.

We have also experimented with the robust fixed-effectsestimators described in Bramati and Croux (2007), whichare based on carrying out a ‘robust’ within-transformation(using the median instead of the mean over time) and thenapplying robust estimators to the transformed data. Thisbasically led to the situation that all regression coefficientswere identical to 0, both in economic and statistical terms(as these results are so easily summarized, we do not reportthem here). As indicated above, such a result is economic-ally inconsistent under the assumptions made if nonzeroprofits are present, as it would imply that capital is over-paid, but no other factor is underpaid. If anything, thiswould only be consistent under decreasing returns to scalein which marginal productivity remunerations do notexhaust the overall product so that one factor necessarilyreceives more than its marginal product. As explainedabove however, decreasing returns at the firm level arean implausible case. We find it more likely that the result

that all regression coefficients are equal to zero is due tothe fact that the fat-tailedness of our data reduces the signalto noise ratio of our variables after the within-transforma-tion so much that no significant relationships can beidentified.

Finally, we have also tried to address the question towhat extent the over- or underpayment of different factorsmight be heterogenous across firms by using the random-coefficient estimators proposed by Swamy (1970) (alsosee Dobbelaere and Mairesse, 2013). These methods alsoled to results that were economically implausible (or evennonsensical), or inconsistent over specifications (forexample results contradicted each other when differentaggregation levels of the factor labour were used). Apartfrom the fact that our sample sizes seem too small toestimate heterogeneity distributions, these methods arealso based on least squares and therefore seem ill-suitedfor our data.

VII. Interpretation of Results

Explanations for deviations from marginal productivitytheory

Our results suggest moderate deviations from marginalproductivity theory in some cases. How can such



Real capital .0755 (.0129) .0774 (.0135) .0826 (.0138)Real capital (including profits) –.3921 (.0290) –.3902 (.0293) –.3850 (.0295)Real intermediate inputs –.0071 (.0142) –.0047 (.0147) –.0006 (.0145)Labour (all categories) .1494 (.1491)

Labour category 1b –4.1866 (3.8176)Labour category 2 .2343 (.2773)Labour category 3 .1207 (.6370)Labour category 4 .1827 (.1797)

Owners –28.1480 (17.7654)Managers –.3637 (3.5978)High skilled (production) .2763 (.3115)Low skilled (production) .1396 (.2006)Low skilled (nonproduction) .2318 (.5428)Administration –1.5028 (.8011)Other services 1.0700 (1.4390)Sales personnel .1914 (.6691)

Constantc 8.7e + 05 (1.4e + 05) 8.7e + 05 (1.4e + 05) 8.5e + 05 (1.5e + 05)


Source: ENIA, 2001–2006, own computations.Notes: aBootstrap SEs accounting for panel structure of the data in parentheses.bLabour category 1 = Owners, ManagersLabour category 2 = High-skilled (production), AdministrationLabour category 3 = Sales personnelLabour category 4 = Low-skilled (production and nonproduction), Other servicescAlso includes time dummies. Omitting constant and time dummies changes results only marginally.


deviations be explained? We start with a basic model ofprofit maximization in possibly imperfect input and outputmarkets. The firm maximizes profit

π ¼ PðY ÞY � rðKÞK � wðLÞL� qðIÞIwith Y ¼ f ðK; L; IÞ; (18)

where PðY Þ denotes the inverse demand function foroutput, and rðKÞ;wðLÞ and qðIÞ denote the inverse supplyfunctions for inputs (with P

0 ðY Þ � 0, r0ðKÞ � 0,w

0 ðLÞ � 0 and q0 ðIÞ � 0). Profit maximization then

implies

fK ¼ r

P� 1þ εr;K� �1þ 1

εY ;P

� � ; fL ¼ w

P� 1þ εw;L� �1þ 1

εY ;P

� � ;

fI ¼ q

P� 1þ εq;I� �1þ 1

εY ;P

� � ;

(19)

where εY ;P; εr;K ; εw;L and εq;I are the elasticities of theinverse demand and the inverse supply functions,respectively. The first-order conditions show that profitmaximization requires that production factors are alwayspaid their marginal product or less, equality holding inthe case of perfect competition in the outputmarket (εY ;P ¼ �1) and in the input markets(εr;K ¼ εw;L ¼ εq;I ¼ 0). This means that under profitmaximization, a production factor will never receivemore than its marginal product, and underpayment of afactor is consistent with either monopolistic power in theoutput market (εY ;P ¼ �c > 0), or market power in theinput markets (for example, εw;L > 0 or εq;I > 0).

We found moderate but significant overpayment ofcapital, and, in some cases, of labour. While overpaymentof capital can be explained by the fact that residual profitsautomatically accrue to capital owners (see below), adirect implication of an overpayment of (nonowners’)labour would be that firms’ behaviour is not entirely con-sistent with pure profit maximization. Moreover, the factthat labour is overpaid rather than underpaid in some casesimplies that our data provides little evidence for theories ofmonopsonistic labour demand stating that employersexploit immobility, search frictions or preference hetero-geneity on the side of the employees in order to capturepart of their marginal product (Bhaskar et al., 2002;Manning, 2003; Cahuc and Zylberberg, 2004). On theother hand, our result that intermediate inputs are to acertain extent underpaid is consistent with the hypothesisthat firms exert some market power in markets for inter-mediate inputs. This is plausible given that long-termrelationships between firms and their suppliers may enable

the firms to enforce lower prices. A plausible predictionarising from the hypothesis that firms exert marketpower in intermediate input markets is that their mono-psonistic power is the greater, the larger the firm is. Thisprediction is supported by our results, i.e. the amount ofunderpayment increases with firm size (see third row ofTables 4, 6 and 7).

Another general explanation for factor payments belowmarginal products is uncertainty (output price uncertaintyand demand uncertainty). In the case of uncertainty, risk-averse firms will underpay (and underuse) inputs (Leland,1972; Batra and Ullah, 1974). However, as we do notobserve underpayment of all production factors (we onlyobserve underpayment of intermediate inputs), we con-clude that uncertainty plays no leading role in explainingdifferences between factor payments and marginal pro-ducts (or at least, it is dominated by other factors).Similarly, even without monopsonistic power in inputmarkets (εr;K ¼ εw;L ¼ εq;I ¼ 0), factor remunerationsmay fall short of marginal products if the firm has mono-polistic power in the output market (εY ;P ¼ �c < 0).Again, as we do not observe uniform underpayment ofall production factors (we only observe underpayment ofintermediate inputs), we conclude that monopolisticpower in the output market is unlikely to be the onlyexplanation for differences between factor remunerationsand marginal products, At least in the case of capital andlabour, there must be other aspects that dominate a possi-bly depressing effect of monopolistic power in outputmarkets on factor remunerations.

What are other aspects that possibly explain the (insome cases) existing overpayment of capital and labourover their marginal product? In the case of capital, anexplanation is easy as the firms’ owners may capture anyresidual income resulting after all other production factorshave been remunerated. They may even extract parts ofthe firm’s equity as a payment, although this would rundown the firm’s capital stock and therefore the firm own-ers’ assets. However, it is interesting that in our datacapital is only overpaid (compared to its marginal product)if residual profits are taken into account. Our results sug-gest that even if all explicitly reported payments to capitalholders are counted (interest payments, dividends andother payments to equity holders), the payments to capitalholders fall short of their marginal contribution to output(see first row of the tables showing the regression results).It is only after the residual income retained in the firms(this is our variable profit) are attributed to capital holdersthat capital’s remuneration on average exceeds its mar-ginal contribution to output.13

What are possible explanations for an overpayment ofthe factor labour? A leading model that potentially

13An exception is small firms where we suspect that remunerations of owners’ capital mix-up with remunerations of their labour inputs(see discussion of the empirical results).


explains how wages can exceed marginal products isMcDonald and Solow (1981). In this model, employeeshave bargaining power allowing them to bargain withemployers at the firm level over both wages and employ-ment. The intuition of the model lies in the insight that thesame level of profits may be reached with a higheremployment than the firm would choose if bargainingwas only over wages, or if firms were price-takers withrespect to wages. As employees prefer more employmentto less employment, efficient bargaining then implies thatemployment is extended to a point where the marginalproduct of labour is already below the paid wage. Themodel requires that employees have bargaining power.Possible sources of bargaining power are worker union-ization on the one hand, and the costs of firing, hiring andtraining workers on the other. Unionization in Chile isrelatively low (see Lawrence and Ishikawa, 2005) butemployees may be in the position to exploit the fact thatit is costly to replace them.

A similar argument is used in insider-outsider models,where wages may rise up to the point where they equalmarginal products plus turnover costs (Solow, 1985;Lindbeck and Snower, 1987, 1988). Turnover costs con-sist of the direct costs of hiring and firing but may alsoinclude the potential loss of specific human capital causedby the exit of workers who previously received training inthe firm. Cahuc and Zylberberg (2004, p. 264) describe thehold-up problem that arises when firms have the opportu-nity to invest in workers’ specific human capital, leadingto the situation that they have to share the returns to thisinvestment (and associated investments in productiontechnologies) with the workers. Note that in our data, weobserve some (not very substantial) overpayment of thefactor labour only in small and medium-sized firms. Thismay indicate that, in the absence of wide-spread officialcollective bargaining by trade unions, there may be impli-cit bargaining between employers and workers at the firmlevel, which is easier in small and medium-sized firmswhere the work force is smaller. Also, the personal rela-tionship between employees and firm owners tends to bemuch closer in smaller firms, which may lead to the situa-tion that firm owners care more about their employees’employment security than in large firms. This may beexploited by employees when bargaining about wagesand employment at the firm level.

VIII. Conclusion

In this article, we have explored a hitherto unusedapproach to testing the empirical validity of marginalproductivity theory. Our test is based on the fact thatunder the assumption of a linear homogeneous production

function, firms profits are informative about whether indi-vidual production factors are over- or underpaid withrespect to their marginal products. For our analysis, wedo not need to assume any specific parametric form for theproduction technology. This appears to be an advantageover previous attempts to test marginal productivity theoryas any kind of misspecification of the production functionmay lead to the spurious finding that factor payments do ordo not equal marginal products. We also do not makebehavioural assumptions such as profit maximization, orassumptions about price taking or price setting behaviour.As a consequence, we are able to carry out powerful directtests of whether such assumptions are consistent with thedata. Our analysis has also demonstrated that the simpli-city of our approach comes at the cost of having to dealwith typically highly fat-tailed profit data. We show thatthis requires the use of robust statistical methods. A lim-itation of our approach is that we can consider heteroge-neity of under- or overpayment of production factorsacross firms only to a limited extent (we classify firmsinto size classes). It would be interesting to investigatewhether such heterogeneity exists by using larger samplessizes and by looking at smaller subgroups of firms. Analternative would be to extent existing robust statisticalmethods to address parameter heterogeneity across firms,which appears quite challenging.

Our empirical results using data on manufacturing firmsin Chile suggest moderate deviations from marginal pro-ductivity theory that also depend on firm size. For smallfirms, we find that labour input by nonowners is overpaidin some cases, while capital and intermediate inputs tendto be slightly underpaid. The fact that the explicit pay-ments to capital fall short of its marginal product seems tobe related to an overpayment of firm owners’ labour input,indicating that in small firms the remuneration of firmowners’ labour input cannot separated from the remunera-tion of their capital input. For medium-sized firms, we alsofind that nonowners’ labour input is slightly overpaid insome cases, indicating that employees may have a certaindegree of bargaining power. Capital is also slightly over-paid in medium-sized firms but only if the residual profitsretained in firms are taken into account as a part of capi-tal’s remuneration. As in small firms, the contribution ofintermediate inputs to output is slightly larger than theircosts. The latter is also true for large firms, where theamount of underpayment of intermediate inputs is evenlarger than in small- and medium-sized firms, consistentwith the expectation that market power in input marketsincreases with firm size. In large firms, the profits obtainedfrom the underpayment of intermediate inputs solelyaccrue to capital holders, while the factor labour onlyreceives its marginal product. This indicates that in largefirms, employees do not have additional bargaining power.


We believe that our results provide interesting possibi-lities for future research. Our empirical application useddata for Chile whose economic system is that of a liberalmarket economy comparable to that of the United States.While the results presented here are probably also repre-sentative for other countries with similar economic condi-tions, this is certainly not the case for countries such asthose of continental Europe in which trade unions play amuch more important role, and market freedom isrestricted by a variety of institutional settings. We expectthis to have a severe impact on possible deviations offactor payments from marginal products. In any way, itwill be interesting to apply the approach described aboveto other sources on firm data such as those for the UnitedStates or major European countries. Such data bases areincreasingly opened up to scientific research (see, e.g.,Verardi and Wagner, 2010; Malgarini et al., 2013), andtherefore provide promising future applications of ourapproach. Depending on the richness of the data available,it will also be interesting to look at higher levels of dis-aggregation, for example to look at the amount of over- orunderpayment of different forms of capital such as equityand external capital, or that of different categories ofintermediate inputs.

Acknowledgements

We would like to thank an anonymous referee for helpfulcomments, Ben Jann for helpful discussions and JaimeEspina Ampuero (INE Chile), Luis Huergo, MarkusNiedergesäss and Nina Pavcnik for help with the data.All errors are our own.

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Appendix

In this appendix, we extend the relationship betweenreturns to scale α and the markup μ described in Basuand Fernald (1995) to the case of imperfect input markets.

Assume that firms are cost minimizers, i.e. they solvethe minimization problem

minK;L;I

rðKÞK þ wðLÞLþ qðIÞI s:t: Y ¼ f ðK; L; IÞ:(20)

The Lagrangian is

L ¼ rðKÞK þ wðLÞLþ qðIÞI � λðf ðK; L; IÞ � Y Þ:(21)

The markup is defined by μ ¼ P=MC (with marginal costMC). Minimization implies

MC ¼ d rðK�ÞK� þ wðL�ÞL� þ qðI�ÞI�f gdY

¼ λ (22)

@L@K

¼ r0ðKÞK þ rðKÞ � λfK ¼ 0

, fK ¼ r0ðKÞK þ rðKÞP

μ

(23)

@L@L

¼ w0ðLÞLþ wðLÞ � λfL ¼ 0

, fL ¼ w0ðLÞLþ wðLÞP

μ

(24)

@L@I

¼ q0ðIÞI þ qðIÞ � λfI ¼ 0

, fI ¼ q0ðIÞI þ qðIÞP

μ;

(25)

where stars indicate optimal values.


Under nonconstant returns to scale and using the first-order conditions,

α ¼ fKK

Yþ fLL

Yþ fI I

Y(26)

¼ ðr0ðKÞK þ rÞKPY

μþ ðw0ðLÞLþ wÞLPY

μþ ðq0ðIÞI þ qÞIPY

μ

(27)

, μ ¼ α � PY

ðrK þ wLþ qIÞ þ d¼ α � revenues

costsþ d;

(28)

where d ¼ r0ðKÞK2 þ w0ðLÞL2 þ q0ðIÞI2 � 0.


Tab

leA1.

Summarystatistics

(inthou

sandsof

pesos

orhou

rsper

year)

Firm

size

10–2

5workers(731

2Obs.)

26–1

00workers(844

6Obs.)

101–

1000

workers(469

0Obs.)

Mean

SD

Kurtosis

Skewness

Mean

SD

Kurt.

Skewn.

Mean

SD

Kurt.

Skewn.

Realp

rofitb

eforetaxes

1010

32.0

1.1e

+06

584.4

18.0

46559

1.0

3.5e

+06

1031

.426

.05.6e

+06

2.5e

+07

1031

.426

.0

Realp

rofit

9434

8.1

1.1e

+06

585.1

18.0

43610

8.0

3.4e

+06

1080

.126

.45.3e

+06

2.4e

+07

1080

.126

.4

Realcapital

2448

25.0

2.7e

+06

4904

.564

.71.1e

+06

5.0e

+06

197.6

12.9

1.1e

+07

4.3e

+07

197.6

12.9

Realintermediateinpu

ts38

2911.0

91587

2.0

144.8

9.6

1.8e

+06

1.1e

+07

1621

.537

.31.6e

+07

5.2e

+07

1621

.537

.3

Labou

r(allcatego

ries)

3967

9.7

1199

2.5

19.7

1.65

11619

4.0

5437

9.5

16.9

2.0

62329

4.0

49034

9.0

16.9

2.0

Labou

rcatego

ry1

3100

.429

64.5

17.7

2.1

5401

.752

56.0

51.9

4.0

1475

2.8

2173

1.0

51.9

4.0

Labou

rcatego

ry2

1417

4.2

1373

8.1

5.8

1.3

3558

4.9

3803

5.3

9.97

2.24

19285

5.0

32519

7.0

9.9

2.2

Labou

rcatego

ry3

3299

.86116

.712

.72.7

7214

.372

14.3

22.6

3.6

3047

4.8

8883

8.8

22.6

3.6

Labou

rcatego

ry4

1910

5.3

1498

1.1

3.6

.567

992.7

6799

2.7

12.7

1.6

385211.0

36327

4.0

12.7

1.6

Owners

1763

.923

40.7

8.0

1.6

1885

.628

27.7

7.3

1.8

1064

.527

12.5

7.3

1.8

Managers

1336

.423

09.4

34.6

3.7

3516

.149

06.6

63.4

4.7

1368

8.3

2142

6.7

63.4

4.7

High-skilled

(produ

ction)

1042

2.7

1326

3.5

5.5

1.4

2354

0.8

3531

9.1

11.9

2.6

13774

3.0

30599

3.0

11.9

2.6

Low

skilled

(produ

ction)

1601

3.5

1446

2.7

3.2

0.6

5456

4.1

4785

4.0

12.8

1.6

29631

2.0

31302

6.0

12.8

1.6

Low

-skilled

(non

prod

uctio

n)19

96.7

5156

.640

.85.0

8970

.617

570.8

81.9

5.8

6948

2.5

12973

6.0

81.9

5.8

Adm

inistration

3751

.543

56.5

23.6

3.1

1204

4.1

1319

0.6

31.5

3.8

55112.3

7150

8.7

31.5

3.8

Other

services

938.0

2403

.243

.74.9

3887

.677

66.2

65.4

5.7

1870

2.9

4218

4.3

65.4

5.7

Sales

person

nel

3299

.86116

.77

12.7

2.7

7214

.313

193.8

22.6

3.6

3047

4.8

8883

8.8

22.6

3.6

Source:E

NIA

,200

1–20

06,o

wncompu

tatio

ns.

Note:For

thedefinitio

nof

Labou

rcatego

ries,see

Table4.


Table A3. OLS on robust subsample: firms with 26–100 workersa


Real capital .0651 (.0026) .0652 (.0026) .0650 (.0026)

Real capital (including profits) –.1368 (.0279) –.1367 (.0279) –.1369 (.0279)

Real intermediate inputs .0516 (.0024) .0517 (.0024) .0525 (.0023)

Labour (all categories) –.2171 (.0368)

Labour –.4250 (.3378)

Labour category 2 –.2363 (.0552)



(continued )



Real capital .0594 (.0044) .0806 (.0026) .0802 (.0026)

Real capital (including profits) .1278 (.0500) .1490 (.0500) .1486 (.0500)


Labour (all categories) –.1103 (.0493)

Labour category 1b –.7464 (.1847)




Owners –1.0856 (.2285)

Managers –.3702 (.2456)

High skilled (production) –.0360 (.0563)

Low skilled (production) –.1350 (.0513)

Low skilled (nonproduction) –.0161 (.1622)

Administration –.2012 (.1242)

Other services .6027 (.2166)

Sales personnel –.1595 (.0767)

Constantc –8554.00 (2205.86) –9133.43 (2252.83) –8506.14 (2154.45)




Table A3. Continued


Owners –.9863 (.6024)

Managers –.0854 (.3928)

High skilled (production) –.1978 (.0642)

Low skilled (production) –.1600 (.0445)

Low skilled (nonproduction) –.4082 (.1242)

Administration –.3201 (.1624)

Other services –.3725 (.2516)


Constantc 2629.47 (4848.45) 3497.51 (4936.93) 3547.66 (5068.14)





Real capital .1094 (.0045) .1084 (.0046) .1036 (.0049)

Real capital (including profits) –.3583 (.0276) –.3593 (.0276) –.3640 (.0276)


Labour (all categories) .1303 (.0804)

Labour category 1b –4.937 (2.4052)

Labour category 2 .3600 (.1615)


Labour category 4 .1793 (.1008)

Owners –14.9068 (7.3548)

Managers –1.7233 (2.4516)

High skilled (production) .3551 (.1796)

Low skilled (production) .1063 (.1117)

Low skilled (nonproduction) .6729 (.3934)

Administration –1.3032 (.7065)

Other services .7319 (1.0783)


Constantc 7.9e + 04 (6.6e + 04) 1.1e + 05 (6.0e + 04) 1.2e + 05 (7.0e + 04)




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An empirical test of marginal productivity theory · An empirical test of marginal productivity theory Martin Biewena,b,c,* and Constantin Weiserd aDepartment of Economics, University