holm lemseminar 2010

8/22/2019 Holm Lemseminar 2010

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Applying Prices Equationin empirical research

Perspectives on decomposition analyses

of productivity change

Jacob Rubk HolmVersion of September 28, 2010

Abstract

This paper discusses and demonstrates how Prices Equation may be

applied in productivity studies. It derives and discusses the interpretation

of the mathematics of the equation. Hereby it combines evolutionary theo-

rizing on generalized Darwinism with the vast literature on decomposition

studies of productivity change.

It is shown that the inter firm reallocation effect of decomposition

studies describes a special case of generative selection where fitness is alinear function of productivity.

It is discussed how the recursive formulation of Prices Equation allows

decomposition studies to explicitly take into account that not all firms are

competing with each other. They are part of different populations. And

it is demonstrated how the evolution of Danish productivity from 1992 to

1999 can be decomposed, taking into account that firms competing on the

basis of labour productivity are competing not with all other firms but

only with those employing similar production processes.

Introduction

The central part of George Prices general mathematical theory of selection

is Prices Equation.1 The equation specifies a decomposition of the change in aweighted mean characteristic of a population from one point in time to another.Within economics there are not many explicit references to Prices work. Thefew that refer to Price generally deal with theorizing on industrial dynamics. Inempirical work, however, there is a vast literature already implicitly employingPrices Equation for the decomposition of productivity change without explic-itly noticing the kinship of decomposition techniques and Prices Equation. It

IKE, Department of Business Studies, Aalborg University. [email protected] sometimes called the Price equation. As this term may be confusing in economics it

will not be used in this paper.

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is thus necessary to evaluate the additional insight into decomposition stud-

ies that is already available from the interpretation of Prices Equation. Thecontribution of the present paper is to highlight the close relationship betweenPrices Equation and decomposition studies and to evaluate the insight gainedfrom acknowledging that these studies in fact apply Prices equation.

In the following section Prices Equation will be formally presented andits relationship to the methodology used in productivity decomposition will bedemonstrated. Then, in section 2, the interpretation of the equations compo-nents as suggested by an evolutionary perspective will be discussed. Section 3provides some methodological considerations and presents the data used, and insection 4 the results of decomposing the evolution in Danish labour productivityfrom 1992 to 1999 are presented. Section 5 discusses the results and evaluatesthe additional insight to productivity decomposition generated by Prices Equa-tion.

1 Prices Equation and its relatives

Prices Equation was first developed as a tool in biology for modelling the evo-lution of gene frequencies but was also propagated as a contribution to a generalmathematical theory of selection with application in all forms of evolutionaryanalysis not just the biological sort (Frank, 1995; Price, 1995, 1970). In biologyit has since been applied not just in modelling but also in empirical research,mostly with the aim of supporting the results from modelling efforts (van Veelen,2005).

Within economics two different strands of literature employing Prices Equa-tion may be identified. Some researches include it in discussions related to gen-eralised Darwinism as a tool for studying the evolution of routine frequencies an approach very similar to the original application where it was envisaged as atool for studying the evolution of gene frequencies. Examples of such literatureare Hodgson and Knudsen (2006b) and Knudsen (2004).

Another strand of literature on Prices Equation in economics employs it inmodelling the evolution of a characteristic (which tends to be some measure ofproductivity) within a population of firms. This strand is related to the lit-erature on agent-based/simulation models of economic evolution in the sensethat it discusses Prices Equation as en extension to the replicator dynamics ofsuch models. The equation thereby becomes a tool for mapping the balance be-tween intra firm and inter firm forces in the evolution of productivity. Examplesin this vein include Metcalfe and Ramlogan (2006), Metcalfe (1998, 1994) and

Andersen (2004).There are so far no empirical applications within either of the two strands

identified above. There is however a very large third strand of literature onaccounting for productivity growth by decomposing productivity change intoa number of effects. There are several equations in use for decomposition inthe literature. The equations are all similar in that they strive to decomposeproductivity change into an inter firm component, an intra firm component andthe effects of entry and exit. The equations differ in the balance between ease ofinterpretation versus robustness to measurement error but there is one equationin particular, which is included in most research and even often employed as

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the only equation.2 This equation may readily be rewritten as Prices Equation

and thus any insight gained towards interpretation of Prices Equation can beused here as well. This specific equation was first suggested in Foster et al.(1998) in which a thorough study of the evolution of American productivity isundertaken.

1.1 Examples of decomposition studies

The study by Foster et al. (1998) decomposes both labour productivity and to-tal factor productivity growth (TFP) in a number of American manufacturingand service industries and the research is extended in Foster et al. (2002) tothe US retail industry. In the later study, however, only the change in labourproductivity is decomposed. Other studies of manufacturing industries includeDisney et al. (2003) for the UK and Cantner and Kruger (2008) for Germany,

while Andersson (2006) conducts a study of productivity growth in the entireSwedish economy. Other studies have sought to perform international compari-son, for example Scarpetta et al. (2002), which is a study of the manufacturingand service sectors of select OECD countries without disaggregating to specificindustries. A similar analysis based on OECD data is presented in Bartelsmanet al. (2004). A common justification of studying productivity growth by decom-position analysis is that it can contribute insight into the idiosyncratic industrydynamics underlying the well-documented, large and persistent differences inproductivity among firms and industries (see e.g. Syverson (2010) and Bar-telsman and Doms (2000) for discussions of the heterogeneity). As populationdynamics are precisely what George Price sought to quantify mathematically itis very reasonable to expect his general mathematical theory of selection will

contribute to the understanding of the decomposition literature.

1.2 Prices Equation

The classic reference for Prices Equation is Price (1970) but this preliminarycommunication is quite short and directed at biology, whereas the posthu-mously published Price (1995) elevates the discussion to a much more generallevel.3 It is not easy to say when Prices Equation was first applied in economics.Several applications of replicator dynamics in evolutionary game theory comevery close to Prices Equation without actually referring to it and much work inthe 1990s by J. S. Metcalfe explicitly employ Prices ideas in model building; e.g.Metcalfe (1998, 1994). For an example of Prices Equation in game theory seeBowles (2004) ch. 13. For a formal documentation of the relationship between

Prices Equation and, inter alia, replicator dynamics and the Lotka-Volterraequations of evolutionary game theory see Page and Nowak (2002).

The variety of contributions means that there is a wide array of notationsbeing used for Prices Equation. I will adhere to the following notation, whichcomes very close to the notations of Andersen (2004) and Frank (1995): Uppercase letters denote population level means at the highest level of aggregationand lower case letters denote firm level values and means of sub-populations.

2The formal presentation of this equation follows section 1.3.3It is not that Price did not publish more on his work during his lifetime. But the exposition

in Price (1995) with the accompanying paper by Steven Frank (Frank, 1995) is a very thoroughexposition of the selection mathematics developed by George Price.

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Table 1: Definitions

Formal Description

xi Firm size

zi Firm productivity

X =

i xi Population size

si = xi/X Firm share

wi = x

i/xi Firm fitness (growth rate plus one)

zi = z

i zi The evolution of productivity

Z =

i sizi = E(zi) Population (mean) productivity

Z = Z

Z The evolution of pop. productivityW = X/X Population fitness

Cov(wi, zi) =

i si(wi W)(zi Z) Covariance of fitness and productivity

Adding a prime to a variable denotes the end year, as opposed to the baseyear, and a prefixed denotes the difference between base and end years. Thesubscript i denotes firms and subscript j denotes sub-populations (i.e. indus-tries). Cov(ai, bi) is the population covariance between random variables a andb weighted by firm size and E(ai) is the population mean of a weighted by firmsize, i.e. it can also be written as capital A. Table 1 contains the descriptionsas well as the mathematical definitions of the components of Prices Equation.

Equation 1 is Prices Equation in the form where it can be used to decomposethe change in productivity into an inter and an intra firm component.

Z =Cov(wi, zi)

W+

E(wizi)

W(1)

The change in population level labour productivity between the base andend years is equal to the sum of two terms.4 The first term is based on thecovariance between growth and productivity and is referred to as the selectioneffect or the inter firm effect. It indicates to what degree the growth of aggregateproductivity can be attributed to relatively high productivity firms growingmore than relatively low productivity firms. The second term is sometimesreferred to as the innovation effect or the intra firm effect. It is the part of

productivity growth that may be attributed to processes internal to firms. Itwill be argued below that it is better to label this term as the learning effectbut during the below formal presentation of Prices Equation it will be referredto as the innovation effect.

At this stage an insight can already be added to the literature on productiv-ity decomposition: the inter firm effect is dependent on the covariance of firmgrowth and productivity in the base year. If the relationship between productiv-ity and growth is not linear this covariance will be weak even if growth is highly

4Proof that the change in a weighted mean may be decomposed in this manner and thatPrices Equation is thus an identity is given in the appendix to Andersen (2004) and willnot be repeated here.

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dependant on productivity. The decomposition will then ascribe too little of

the change in population mean productivity to the inter firm redistribution ofresources following from differences in productivity.An alternative specification of Prices Equation is reached by multiplying by

W and is given in equation 2.

WZ = Cov(wi, zi) + E(wizi) (2)

The interpretation of what exactly is being decomposed is a bit more trickyin equation 2 than in equation 1. It could be interpreted as the gross change inwelfare contribution of the population: the change in mean productivity mul-tiplied by the size growth. Notice that the expression within the expectationsoperator on the right is the firm level equivalent of the population level termon the left. This means that Prices Equation can be substituted, very ele-

gantly, into itself as many times as the researcher may desire and thus allowfor a multilevel study of evolution. It is thus possible to describe each agentof the population as a population in its own right. Most studies employingproductivity decomposition map how industry level productivity has changedthrough inter and intra firm processes. A few provide separate evidence on howeven more aggregate mean productivity has changed by inter and intra industryprocesses. But Prices Equation illustrates how both levels of dynamics andeven additional levels may be mapped in just one decomposition. Multilevelapplication will be explored later in this paper and for this analysis equation 2will be expanded and rewritten so that it once again becomes clear just what isbeing decomposed.

1.3 The road to decompositionPrices Equation as specified in the previous section cannot be applied directlyto firm data, as two crucial phenomena in the evolution of a population of firmsis not taken into account in this form. These are entry and exit. In order to takethese phenomena into account it is necessary to distinguish between three setsof firms: Those that exist in both the base and end years, those that exist inonly the end year and those that exist in only the base year. These sets will belabelled the C, N and X sets respectively (Continuing, eNtering and eXiting).

The two terms of Prices Equation refer to the contribution of the C-set toproductivity growth. in order to specify the contributions of the N and X-setsit helps to expand the covariance and expectation terms so as to make explicitwhich firms are included in the computations.

The covariance and expectation operators in equation 1 can be expanded to

Z =

i si(wi W)(zi Z)

W+

i siwizi

W

and soZ =

i

si(wi/W 1)(zi Z) +i

si(wi/W)zi

and as siwi/W =xiXxixi

/X

X= si

Z =i

(si si)(zi Z) +i

sizi

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in order to indicate that this only refers to firms present in both end and base

yearsZ =

iC

si(zi Z) +iC

sizi

and adding the contributions of the N- and X-sets

Z =iC

si(zi Z) +iC

sizi +iN

si(z

i Z) iX

si(zi Z) (3)

The four terms of equation 3 thus correspond to the selection and innovationeffects, and the effects of entry and exit. The entry effect will contribute pos-itively (negatively) to productivity growth when the productivity of new firmsis higher (lower) then population productivity in the base year. The exit effectwill contribute positively (negatively) when firms exiting the population haveproductivity lower (higher) then population productivity in the base year.

Equation 3 is almost identical to the decomposition technique introducedby Foster et al. (1998) and used in the studies mentioned earlier. The onlydifference is that the innovation effect is typically divided up into two differentterms, as in equation 4.

iC

sizi =iC

sizi +iC

sizi (4)

The first term on the right hand side of equation 4 is termed the intra firmeffect and the second term the cross level effect. The cross level effect is alsosometimes referred to as a covariance term but as it is the inter firm selectioneffect that may by rewritten as a covariance scaled by population fitness, thislabel for the cross level effect is misleading. The justification for separating the

cross level effect from the intra firm effect is robustness to measurement error.As the aim of the present paper is to evaluate the insight from Prices Equationfor productivity decomposition the innovation effect will not be split up in thedecompositions performed in the present paper.

Another problem with equation 3 is that there may be multiple forms ofentry and exit. It is not certain that a firm that appears for the first time inan industry is newly set up. It may very well have been operating in anotherindustry for some time before the relative importance of its various activitiesevolved so that it was reclassified in the statistical industry classification system.And a corresponding problem holds for exits. This problem could be taken intoaccount by defining two additional sets of firms so that there would be twosets of exiting firms and two sets of entering firms. These would be based onwhether the firms close down or not when exiting and whether the firms arenewly set up or not when entering. The data used in this paper allows for theidentifying the both types of entry and exit (the data will be formerly describedin section 3.2). The question is, however, whether this complication is justified.It does not seem warranted to add extra terms to the decomposition unless theserepresent effects that have particular interest. The focus here is on the extrainsight into productivity decomposition gained from Prices Equation and thusthe complication of multiple entry and exit effects is avoided.

Thus the equation used to decompose the evolution of productivity in singlelevel populations in the current paper will be equation 3. But before demon-strating the technique and presenting the results the interpretation of the termswill be discussed.

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2 The evolutionary interpretation

Prices Equation is for mapping how a characteristic changes over time in anevolving population. This does not mean that the interpretation of the decom-position needs to have analogies with biology but it means that the focus ofthe analysis necessarily is on populations and the dynamics within them ratherthan on the agents of the population. Employing such a population perspectiveentails recognizing that there are irreducible forces at the population level thatare lost if focus is only on the individual agents (Andersen, 2004; Zinovyeva,2004). In evolutionary economics these population level forces are conceptual-ized as the forces of competition and thus competition is conceptualized as aprocess (as opposed to competition as a state). The process of competition iswhat drives economics evolution.

2.1 Creating, sorting and consuming variety

Competition is an open ended process of novelty generation and reallocation ofproductive resources. Firms perform innovations in attempts to gain decisivecompetitive advantages over competitors. They change their strategies based onthe innovation of these same competitors and in anticipation of future actionsby competitors. And also in response to and in anticipation of changes inconditions external to the population. Firms prosper or decline as a result ofthe interaction between their own innovation activities, the innovation activitiesof competitors and the external factors setting the premises for the interaction.

Competition may thus be characterised by three mechanisms: the disequili-brating mechanism of novelty creation, the equilibrating reallocation mechanismof firms being evaluated in market competition and retention in the sense ofcharacteristics of successful firms being retained within the population. Thesethree elements of industrial evolution; innovation, selection and inertia, are notall captured equally by decomposition studies. The first element, innovation, isconsigned to the intra firm effect and treated more or less as a black box: de-composition aims to map the population level dynamics of productivity growth.Not the intra firm processes. Inertia is not considered explicitly either. Ifthere is to be selection based on productivity it is necessary that productivity isnot a wildly fluctuating characteristic of a firm but rather, productivity needsto be a parameter upon which firms can be credibly distinguished from eachother. Whereas decomposition thus tells us little about the innovation and in-ertia elements of industrial evolution it quantifies the selection element witha few caveats: decomposition quantifies only linear, single factor selection. I.e.

decomposition tells us nothing about non-linear effects of productivity upon dif-ferentiated growth rates and it tells nothing about other sources of differentiatedgrowth.

There are various approached in evolutionary economics to the application ofa population perspective and evolution through competition. These range fromthose working to generalise biological evolution into a set of principles that applyto all occurrences of evolution a generalised Darwinism (Aldrich et al. (2008);Hodgson and Knudsen (2006b)) while others argue for constructing a generaltheory of evolution of which the biological and economic sorts are particularoccurrences (Winter (1987); Witt (2003), ch. 7).5.

5Winter (1987) constructs a tongue-in-cheek evolutionary model of a library arguing that

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Although these approaches are not, on the face of them, radically different

or even mutually exclusive there is nevertheless ample room for controversy, ase.g. the discussions in Buenstorf (2006) or Hodgson and Knudsen (2006a) show.In this paper I will attempt to tread the middle ground and apply the followingdefinitions inspired by my reading of this literature.

Evolution is open ended, internally propelled change over time of a charac-teristic in a population. Thus Z is evolution. There are two types of selection:subset selection; when the units being selected upon are divided up into win-ners and losers and generative selection; when the distinction betweenwinners and losers is a matter of degree. The selection effect of equation 3 isthus an expression for generative selection (growth, w, determined by the rank-ing according to productivity, z) in the evolution of Z, whereas the exit effectis the more radical form of selection, subset selection.

The process of selection consumes variety. Subset selection obviously de-

creases variety while generative selection should, in the extreme case of firmswith fixed productivities, make the distribution of productivity collapse uponthe optimal (i.e maximum) value. That is, all resources are put to work in themost productive way. The real world is however far removed from the extremecase: novelty is constantly being generated and adds variety to the selectionprocess. Equation 3 allows for two sources of variety: change within existingfirms (the innovation effect) and the entry of new firms (the entry effect).

That variety is the fuel of evolution is intuitively straightforward but it mayalso be demonstrated mathematically, as is done e.g. in chapter 2 of Metcalfe(1998). Not only does Metcalfe show that growth in mean performance (whenthere is no novelty being introduced) can be mathematically linked to the vari-ance of performance, he also shows that the decrease in variance is proportional

to the skewness of the distribution.An important caveat applies to the interpretation of the innovation and en-

try effects as contributors of variety. In an economic sense innovation is anyactivity that leads to the creation of (possibly negative) quasi rents. The inno-vation effect in equation 3 thus captures more than just economic innovation.It captures any intra firm change, including adaptive changes to environmentalshifts.6 Thus, intra firm change is not just innovation but learning in a broadersense. Thus, I shall use the term intra firm learning effect. Similarly, the entryeffect captures the effect on productivity by new firms regardless of whetherthese bring novelty or are mere clones of existing firms.

This theoretical ambiguity means that the magnitude of the innovation (orlearning) effect will be sparsely interpreted. Focus will be on whether the selec-tion effect captures inter firm processes sufficiently and the insight gained frommultilevel application of Prices equation.

One could just as well, however, take evolutionary bibliography as the prototypical evolution-ary science ... (p. 615)

6This caveat is very general for Prices Equation though it is more obvious in its originalapplication in biology than in economics. When studying gene frequencies one must takeinto account that the selection environment is constantly changing and that the genes thatare beneficial for survival and/or reproduction in one selection environment may not be inother selection environments. Thus Prices Equation should be used to study evolution overrelatively short time spans. High productivity, on the contrary, is generally a competitiveadvantage for firms, as it provides a ceteris paribus cost advantage.

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3 Some considerations

zi is the productivity of firm i. But it matter a great deal just how this pro-ductivity is measured. Productivity as a parameter for competition generallyrefers to costs: the more output a firm can generate from given inputs the lowerits costs and thus the better its competitiveness. This is straightforward whenstudying the evolution of labour productivity by decomposition: the competi-tive advantage assumed to determine generative selection in the population isa cost advantage achieved by lower labour requirements. If, on the other hand,one is decomposing the evolution of total factor productivity (TFP) using de-composition the interpretation of economic selection is more open to theorizing.TFP advantages can be argued to stem from the firms ability to innovate andcompetition thus takes on a more dynamic character with more innovative firmsoutgrowing others.7 When decomposing changes in labour productivity the in-

ter firm selection effect should only be interpreted as the effect of generativeselection acting on firms with cost advantages while when decomposing changesin TFP it could arguably also be interpreted as the effect of generative selectionacting on more innovative firms.

It is very common in the literature to decompose both the evolution of labourproductivity and TFP but the differences in interpretation of the decompositionsare not commonly heeded. And, as a further complication, the way in whichlabour productivity or TFP is measured matters to the interpretation as well.Labour productivity is commonly measured as the log of output per unit oflabour input while in a few instances (e.g. Andersson (2006)) it is simply outputper unit of labour input. The interpretations differ in that in the former casecompetitive advantage is relative while in the latter case it is absolute i.e.

using logs means that the effect on generative selection of a 10 percent. costadvantage is independent of cost level, while not using logs means that the effectof a 1,000$ cost advantage is independent of cost level. Intuitively, using logslooks like the right approach. But this means that the worst performing firmsare automatically censored from the data as only firms with positive outputcan be used.8 In section 4.1 the correlations of labour productivity and firmgrowth rates with and without the logarithmic transformation will be compared,and it will compared to the Spearman rank correlation (), which is robustto monotonous transformations, such as the logarithmic. will thus measurewhether there is a tendency for more productive firms to grow more than lessproductive firms without any assumptions about the character of relationshipbetween productivity and growth.

When decomposing TFP, on the other hand, the robustness to variations in

the theory of production becomes important as the theory chosen will determinehow TFP is computed. The most common theory of production is a Cobb-Douglas form taking labour, capital and materials as input (e.g. Disney et al.(2003); Foster et al. (2002, 1998)) but there are also alternatives (e.g. the frontierfunction approach used by Cantner and Kruger (2008) or an index numbers

7A common interpretation of TFP is as technological change. As argued by Lipsey andCarlaw (2004) there is a large number of factors causing TFP to be a biased measure oftechnological change. The authors argue that a better interpretation of changes in TFP isas an imperfect measure of the quasi rents associated with innovative activities. That is, asphrased by Syverson (2010), as changes in output when holding inputs constant.

8Gross output is of course never negative but value added can be.

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method as used by e.g. Scarpetta et al. (2002)). The choice of theory will imply

a set of assumptions; often perfect substitutability of labour, capital and perhapseven inputs. The trick of measuring capital may be overcome by assuming thatfirms operate under perfect competition and with constant returns to scale. Inthis paper I will focus on labour productivity. The richness of the insight intothe decomposition of productivity gained from Prices Equation would exceedthe scope of a single paper were I to consider labour productivity and TFP inequal detail.

Labour productivity suffers from measurement problems as well. Studyingany single factor productivity, labour productivity included, entails ignoringother factors. This is justifiable when firms can be assumed to be similar alongthese other factors; in particular this means similar in their use of physical andhuman capital. Such an assumption has merit at a very disaggregate level when the firms being compared generate their principal shares of value added

from similar production processes and thus have same industry classificationcodes. It should thus be expected that the lower the level of aggregation thestronger the role of the selection effect in the evolution of labour productivity.

3.1 Weights and populations

The considerations in this section have so far focussed on the choice ofz, produc-tivity, in the decomposition. But an equally important consideration pertainsto the choice of x, the measure of firm size. In the earlier modelling efforts em-ploying Prices Equation (e.g. Metcalfe and Ramlogan (2006); Metcalfe (1998))a firms share (si) of a population is often conceptualised at its market shareand thus size is measured by output. In decomposition studies, however, it is

more common to measure firm size by labour input. In the empirical analyses ofthe present paper populations will be delimited by industry codes and thus byproduction processes rather than by the goods produced. Using output weightswould implicitly equate industries and product markets, which does not seemwarranted. Using labour input for size in connection with populations delimitedby industry codes has stronger merit the lower the level of aggregation. Themore similar firms are with regards to their production processes, the more theycan be expected to require qualitatively similar labour services.

The various possible combinations of z and x means that population meanproductivity, Z, will rarely be equal to aggregate productivity.9. For this reasonit is common to refer to the variable being decomposed as an index of produc-tivity. I will refer to Z as population (mean) productivity.

x will be measured by full-time equivalent employment (FTE), output as

value added (VA) and thus z will be the logarithmic transformation of VA perFTE. This means that firms with negative or zero VA will be excluded from theanalyses altogether. Furthermore, I will not consider populations of less than20 firms. The availability of the data (described further in section 3.2) meansthat the firms used to decompose at the aggregate level can be decomposedinto 39 two digit industries. These are listed in table A-1 page 23. The righthand column of the table lists short labels that will be used for reference to the

9The one exception is the study by Andersson (2006), where size is measured as labourinput and productivity is output per unit of labour input. I will decompose the evolutionof ln

Output

Labour input

and as ln

weightivaluei

= weighti ln(valuei) my decompositionscannot be said to refer to aggregate productivity.

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industries. At the two digit level 5 of the 39 industries have less than 20 firms

and these are indicated by asterisks in table A-1.10

Thus the more disaggregate the level of analysis the fewer firms are includedin the decompositions. Notice also how the meaning of entry and exit changeswhen moving to more disaggregate levels. Firms that are considered part ofthe C set at the most aggregate level may switch between industries and thusbe the victim of subset selection in one industry and contribute novelty in theother.

3.2 Data material

The Company and Industry Statistics database11 of Statistics Denmark is em-ployed in the empirical research of the following sections. The database containsvarious financial data from the balance sheets and annual accounts of all private

firms in Denmark as well as a number of estimated key indicators. The databaseis available for the years 1992 to 1999 (where the database was discontinued).Only three variables from the database are used here and only for the years 1992and 1999. The variables are: value added (VA), full-time equivalent employ-ment (FTE) and industry. VA in 1999 is deflated to 1992 prices using the priceindex for the domestic supply of goods, which is obtained from the web page ofStatistics Denmark (www.statistikbanken.dk).More disaggregate indexes forrevenue in parts of the manufacturing sector are also available but will not beemployed. What matters for competitive advantage is not the physical quantityof output but the real value of output. VA and FTE are not available for everyfirm in the database; in particular, there is no data for a number of serviceindustries (utilities and a few entertainment industries such as theatre and ra-

dio/television) and the entire primary sector. The firms included do, however,represent a significant share of economic activity in Denmark.The analyses of section 4.1 will compare correlations at various levels of

aggregation but the data will be almost equally representative at the variouslevels. When studying the most aggregate level and thus not excluding any firmfor belonging to small populations, the firms included in the analyses accountfor 89.2 percent of private sector employment in 1992 and 83.0 percent in 1999.The number of firms employed decreases whenever the investigation moves toa more disaggregate level. At the most disaggregate level used in section 4.1,the four digit level, the firms included account for 85.1 percent of private sectoremployment in 1992 and 78.9 percent in 1999. In section 4.2 all analyses willbe based on the group of firms that can be aggregates into two digit industriesof at least 20 firms. These firms account for 89.0 percent of total private sector

employment in 1992 and 82.8 percent in 1999. As will be seen, the analyses ofsection 4.2 could alternatively have been performed at others level of aggregationand the two digit level may seem somewhat arbitrary. However, the main aim ofthis paper is to demonstrate and discuss decomposition analyses based on PricesEquation and for this purpose the relative ease of presenting an overview of theresults by-industry at the two digit level outweighs theoretical considerations.

10At the three digit level 58 of 159 industries are excluded and at the four digit level 170 of348 are excluded.

11Original Danish name: Firma- og Ressourceomradestatistik.

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3.3 The equations

The decomposition technique used will be equation 3 and it is repeated here asequation 5 for reference and to make the labelling of the various terms explicit.

Z =iC

si(zi Z) (Selection effect)

+iC

sizi (Learning effect)

+iN

si(z

i Z) (Entry effect)

iX

si(zi Z) (Exit effect)

(5)

Whenever results are presented the order of the effects will be as above. Insection 4.2.3 I will exploit the recursive form of Prices Equation as expressed inequation 2 in order to perform a multilevel (here; two levels) decomposition ofthe evolution of labour productivity. Replacing the subscript is with subscriptjs for industry the equation becomes:

WZ = Cov(wj , zj) + E(wjzj) (6)

By adding subscript js to equation 5 and substituting for zj in equation 6the second term on the right may also be written as:

E(wjzj) =j sjwji

Cj

sij(zij zj) + i

Cj

sijzij

+iNj

sij(z

ij zj) iXj

sij(zij zj)

(7)

And thus, substituting equation 7 into equation 6 and multiplying by W1,the equation used for the two level decomposition becomes (cf. the derivationof equation 3 page 6):

Z =j

sj(zj Z) (Industry selection effect)

+

jsj

iCjsij(zij zj) (Firm selection effect)

+j

sjiCj

sijzij (Learning effect)

+j

sjiNj

sij(z

ij zj) (Entry effect)

j

sjiXj

sij(zij zj) (Exit effect)

(8)

The first term of this decomposition is the industry selection effect or theinter industry effect. It captures generative selection among industries and isnot expected to contribute much to overall evolution i.e. cost advantages are

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Table 2: Correlations of growth and productivity

Measure of correlation

Aggregation corr(wi, zi) corr( wi, ln zi) (wi, zi)

All firms 0.060 0.057 0.225

Av. 2-digit 0.119(0.133) 0.101(0.118) 0.236(0.077)

Av. 3-digit 0.151(0.205) 0.138(0.193) 0.243(0.111)

Av. 4-digit 0.164(0.248) 0.151(0.238) 0.250(0.133)

wi =Li,1999Li,1992

. zi =Yi,1992Li,1992

. =Spearman rank correlation. L is

FTE labour, Y is output (VA in 1992 DKK) and subscripts refer toyears. Weighted sample standard deviations in parentheses. Source:own calculations.

not expected to be a major force in the evolution of industry structure. Thesecond term is the firm selection effect or the inter firm intra industry effect. Aswith the selection effect in equation 5 it captures the generative selection amongfirms. The third effect is the learning or intra firm effect, which has the sameinterpretation and definition as what is elsewhere referred to as the innovationeffect, and which was discussed earlier.

The fourth and fifth terms are the entry and exit effects respectively andwhile their interpretations are the same as above (bringer of novelty and subsetselection) it must be kept in mind that some firms contribute dually to theseeffects: by leaving one industry and entering another they increase variety inthe latter industry and decrease variety in the former.

4 Example results for labour productivity

In this section it will first be studied whether selection is linear and thus cap-tured by a covariance term, as assumed in both Prices Equation and similardecomposition techniques. It will secondly be demonstrated, through a num-ber of examples based on Danish data, how Prices Equation can be used tocreate multilevel productivity analyses that takes into account redistribution ofresources at multiple levels.

4.1 Is selection linear?

It was discussed in section 1 how both the inter firm effect of the popular

technique for decomposing productivity growth and the selection effect of PricesEquation rely on there being a linear relationship between productivity andgrowth. The consequence of this assumption for studying the growth of labourproductivity was discussed in section 3. To get a picture of the relationshipbetween labour productivity and growth, various measures of correlation atvarious levels of aggregation have been computed and the results are presentedin table 2.

The table presents the correlations between productivity in 1992 and growthrate from 1992 to 1999 for all firms in the top row. The following three rowsshow the average correlations at the 2, 3 and 4 digit levels weighted by industryemployment in 1992. In column two productivity is output per unit of labour

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input and in column three this productivity is logged. Column four shows the

rank correlation which is independent of whether or not the log transformationis applied. As mentioned in the presentation of the data in section 3.2 industriesof less than 20 firms are excluded but this matters very little for the share ofeconomic activity in Denmark included in the computations.

The table clearly shows that the more disaggregated the level of computa-tion, the stronger the correlation between labour productivity and employmentgrowth from 1992 to 1999. This holds regardless of whether productivity is log-transformed or not and whether Pearsons or Spearmans correlation coefficientis interpreted. This was to be expected. The more disaggregate the level ofcomputation, the more meaningful it is to say that the firms making up a givenindustriy are in competition and that labour costs is a parameter for compet-ing. It is worth noticing that the Pearson correlation coefficient increases almostthree fold when disaggregating from the top level down to four digit industries,

while the Spearman correlation coefficient only increases slightly though it isconsistently a good deal higher than the Pearson coefficient. This could indicatethat there is a tendency for more productive firms to outgrow others but thatthe relationship is not linear.12

The Pearson correlation is slightly higher for the un-transformed produc-tivity measure than for the transformed measure. However, this difference isvery small when also considering the standard deviation and the evidence intable 2 does thus not allow inferences to be made regarding the role of costs incompetition. That is, whether competitive advantages should be measured byabsolute or relative difference in costs.

The mean and standard deviations in table 2 gives an initial idea of the dis-tributions of the correlations across industries but the full picture is obtained

from the histograms in figure A-1, page 22.13

The histograms show clearly thatwhile there are a number of industries with negative Pearson correlation thereare very few industries with negative Spearman correlation; notice the dashedvertical reference line at zero. This again indicates that firms with higher pro-ductivity outgrow others but also that the relationship is more complicated thanis captured by a linear relationship with or without the logarithmic transforma-tion. Notice also how the distribution of Spearman correlations tends towards aunimodal, left skewed shape, especially at the three and four digit level, whereasthe Pearson correlations seem to have fatter tails (higher kurtosis).

Even though the linear relationship assumed in the decomposition techniquesseems questionable, there still is, on average, a positive and non-negligible Pear-son correlation between productivity and growth. And it must also be kept in

12

In order to investigate this further I also tried doing a number of OLS regressions with wias the dependent variable and productivity and squared productivity as explanatory variables with and without the logarithmic-transformation. It is difficult to make any generalizationsabout the results. For some industries the slopes are not at all significant while in othersthey are highly significant. The slope of squared productivity is sometimes negative thoughmostly positive. The variation in the standard errors of the slope estimates across industriesmay have as much to do with the varying number of firms in the industries as with differencesin the relationship between productivity and growth. Even when both slopes are significantit is often the case that R2 < 0.1 and in several instances even R2 < 0.01. The resultsare generally the same whether or not the log transformation is applied. This indicates thatcost competition has very different roles across the industries and further development in thisdirection should probably start at studying the role of cost competition in specific industries.

13The data in table 2 is weighted and thus do not describe the distributions pictured infigure A-1.

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Table 3: Labour productivity in 1992 and 1999

Z Z Z X X W

12.854 13.004 0.150 910,243 984,237 1.081

Source: own calculations.

mind that the decompositions do not take the correlations as a point of de-parture. Rather, the equations may be rewritten as a covariance in order tofacilitate interpretation and the relatively low correlations do not imply thatdecomposition studies should be given up altogether. The following sections re-port the results of decomposition analyses at various levels of aggregation basedon the log of labour productivity.

4.2 Decomposition analyses

In this section the evolution from 1992 to 1999 of Danish logarithmically trans-formed labour productivity will be decomposed with reference to three levelsof aggregation. Z refers to population mean of productivity at the most aggre-gate level, zj is mean productivity at the two digit industry level and zij is theproductivity of firm i in industry j. The top level of aggregation includes allfirms for which data is available excluding those for which the correspondingtwo digit industry has less than 20 firms and those with negative value added,cf. section 3.

Table 3 present an overview of labour productivity in Denmark in 1992 and1999. The population mean productivity was 12.854 in 1992 corresponding to

382,314 DKK per FTE labour unit.14 Seven years later productivity had grownby 16 percent in real terms corresponding to an average growth rate of 2.2percent per year.15 At the same time FTE employment rose by 8.1 percentfrom 910,243 to 984,237.

In the following three subsection the evolution of the labour productivityindex (i.e. the 0.150) will be decomposed. At first aggregately; treating all firmsas part of just one large population. Secondly, the decomposition will be under-taken on a by-industry basis at the two digit level highlighting the discreprenciesacross industries. And lastly the information from the by-industry decompo-sition will be used to do a multilevel decomposition of top level productivityevolution.

4.2.1 The evolution of labour productivity

The decomposition of the evolution of labour productivity according to equation5 is shown in figure 1. The order of the effects is the same in figure 1 as inequation 5: the selection, learning, entry and lastly the exit effect.

The lions share of evolution is attributed to intra firm change; or what hashere been labelled the learning effect. This effect accounts for 51 percent oftotal evolution. The effect of firms entering the population is also quite large:the entry effect accounts for 23 percent of total change. Notice, however, that

14e12.854 = 382, 31415e0.150 = 1.1618 and 7

1.1618 1 = 0.022

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0.150100%

= 0.02315%

+ 0.07651%

+ 0.03523%

+ 0.01612%

Figure 1: Decomposition of the evolution of labour productivity

some survivor bias may be expected, as firms that start up after 1992 but areclosed down before 1999 cannot be included in the decomposition. The totalcontribution of the two effects argued to represent the creation of novelty is thus74 percent of total productivity growth.

The selection effect; the linear tendency for firms with higher levels of pro-ductivity to grow more than firms with low levels of productivity, accounts for15 percent of total evolution, while the exit effect accounts for the remaining 12percent. This makes the effects relatively minor in an accounting sense but the

total effect of selection, the sum of generative and subset selection, accounts forslightly more than a fourth of total evolution.

It is not often seen in the literature that the top level productivity is decom-posed, as the validity of comparing firms based on labour productivity dependson the similarity of the activities of the firms. The decomposition of top levelproductivity here is undertaken for comparison with the more disaggregate de-compositions that follow.

4.2.2 Two digit industries

In the following analysis industries will be denoted by j. The evolution of labourproductivity (zj) in 34 Danish two digit industries is reported in table A-2,which uses short labels to refer to the industries. The industries are defined in

table A-1. Both tables are in the appendix.In table A-2 the selection, learning, entry and exit effects are reported as

shares of numerical total evolution. The sum of the four effects will therefore beequal to 1 in industries with growing productivity and equal to 1 in industrieswith declining productivity.

The first thing to notice from table A-2 is that there are quite a numberof negative effects. A negative entry (exit) effect means that firms entering(exiting) the population have lower (higher) than mean productivity and theirentry (exit) thus contributes to lowering the mean. A negative learning effectmeans that intra firm change tends to decrease productivity and a negativeselection effect means that the covariance between productivity in the baseyear and growth over the period is negative. It must be kept in mind that

productivity is just one characteristic of the firms. In some industries theremay be other factors that are more important for competition and thus forselection.

It is also worth noticing that there is a tendency for the decomposition toreturn extreme results when the change being decomposed is very small. Themost extreme case is the industry for manufacturing of radio and communica-tions equipment (Comm.), where zj is equal to -0.018 corresponding to anaverage yearly percentage change in productivity of -0.26 percent. The resultof decomposing this small change is that the learning as well as the entry effecthas a magnitude of more than nine times the total change in productivity. Noconclusion can be inferred from this extreme result but neither should one ex-

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pect to learn much from decomposing productivity change when productivity

does not change. Similarly, the decomposition tends to give good meaning andreturn effects in the ]0 , 1[ range whenever productivity growth is strong. Twosuch examples are the manufacturing industries of wearing apparel (Clothes)and leather and footwear (Leather), which have seen changes of 0.445 and 0.472respectively but highly disparate underlying forces. In Clothes the change isto a large extent driven by the selection effect (43.2 percent) but notice alsothat the amount of labour input employed in the industry has been more thanhalved (wj = 0.46). This is consistent with low productivity firms downsizingand potentially off shoring production without closing down completely. Werethe firms to terminate all activities or just enough to be reclassified in the in-dustry classification system they would contribute to the exit effect instead. InLeather, on the other hand, productivity growth stems almost exclusively fromchange internal to the firms (the learning effect equals 80.2 percent of total

change) and employment has only decreased slightly over these seven years.There are only a few industries in which the selection effect accounts for a

large share of productivity growth relative to the top level decomposition of theprevious section.16 Interestingly, a number of the industries with strong selec-tion effects are service industries. The covariance of productivity and growthseems very strong in the human health and social services (Health) though thisindustry is arguably in the group of industries with low productivity change andthereby not a meaningful decomposition. A better example may be recreational,cultural and sporting activities (Recr.) where the selection effect accounts for25.2 percent of a 0.334 change in mean productivity the largest productivitygrowth among the service industries.

The decomposition of productivity growth at the two digit level has shown

varying results. That the forces driving the evolution of productivity variesacross industries is to be expected but it is most likely not possible to generalizemuch regarding the background of the variations in the primacy of the effects.What is needed is industry studies. Here, however, the analysis of evolutionat the two digit level serves to give a sense of the dynamics underlying toplevel evolution. And in the following section it will be demonstrated how thedynamics at the two digit level can be taken into account when studying theevolution of mean productivity at the top level.

4.2.3 A multilevel decomposition

In this section it will be demonstrated how multilevel decomposition of pro-ductivity change can add detail to the interpretation of the selection effect in

particular. It was argued that when decomposing top level mean productivitychange (Z = 0.150, cf. table 3) the inter firm selection effect was not reallymeaningful as comparison of labour productivity across the entire Danish pri-vate sector has very little merit. The merit of the comparison increases withdisaggregation and table A-2 showed how the decomposition of productivitychange across the industries reveals some very heterogeneous dynamics. Figure2 shows how the change en population mean productivity at the top level isdecomposed using equation 8 and the information from table A-2.

The effects in figure 2 are the industry selection, firm selection, learning,

16Where the selection effect accounted for 15 percent of total change.

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0.150100%

= 0.0010%

+ 0.01913%

+ 0.07449%

+ 0.03926%

+ 0.01711%

Figure 2: Decomposition of the evolution of labour productivity

entry and exit effects respectively. The inter industry selection effect is verysmall. Indeed, there was very little reason to expect a priori that the evolution ofindustry structure is governed by the mean labour productivity at the industrylevel. But separating out this effect means that the generative selection of firmsis quantified at the appropriate level of aggregation. Here, this inter firm intraindustry selection effect accounts for 13 percent of evolution, i.e. slightly lessthan was found when decomposing at the top level. The selection effect in figure1 is potentially biased by industry idiosyncrasies. That is, by some industries

expanding at the cost of others for reasons that are unrelated to productivitybut nevertheless show up as a covariance between growth and productivity. Byrestricting the selection effect to comparison of firms at the two digit level suchbias is controlled for.

It is not just the selection effect that is close to identical in figures 1 and2. So are the learning, entry and exit effects. Regarding the entry and exiteffects this is a bit surprising. At the two digit level there is a lot of entries andexits that are not captured at the top level, as the firms in question are not newstart-ups or close down but rather move between industries. This phenomenoncould be included in the decomposition by defining multiple sets of entering andexiting firms according to future and past existence of the firms. However, thisis left for future research.

The motivation for restricting the inter firm selection effect to specific sub-populations is determined by theory and closely intertwined with the choice ofproductivity measure and the choice of size measure/weighting scheme. Here,the focus has been on demonstrating the multilevel decomposition technique andthis lead to a definition of sub-populations at the two digit level. This resultsin a relatively low number of sub-populations and therefore has the appealingfeature that an exhaustive overview of the complete set of sub-populations canbe presented, as also mentioned in section 3.2. It was thus ignored that the cor-relation of productivity and growth was shown to increase with disaggregationin section 4.1.

5 Conclusions

Even though George Prices general mathematical theory of selection was orig-inally formulated for biology it also carries important lessons for the field ofeconomics. It highlights the limitations and caveats to keep in mind when in-terpreting the inter firm selection effect and it inspires to rewriting the mostcommon decomposition technique so that it becomes capable of taking intoaccount possible multilevel population structures. In hindsight the multilevelproperty could also have been inferred from the traditional decomposition equa-tions as long as the cross level effect is not separated from the intra firm in-novation/learning effect. This, however, is not the norm. Prices methodologybrings out the linearity assumption and the multilevel property for all to see.

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Combining Prices selection mathematics on the one side with theorizing in

evolutionary economics on generalized Darwinism and the population perspec-tive on the other adds insight to the meaning of the terms in decompositionstudies. It makes clear how the meaning of the selection effect is highly depen-dent on the chosen measure of productivity and the applied weighting scheme.Thus performing multiple decompositions employing different measure of pro-ductivity and different weighting schemes cannot be considered tests for robust-ness of the results. They should rather be seen as different analyses in their ownright and varying decomposition results should be expected.

The assumption of linear selection has been subjected to a preliminary anal-ysis in this paper. It was shown that for labour productivity weighted by em-ployment the linear relationship is simplistic. The result was found whether ornot labour productivity was logarithmically transformed and at various levels ofaggregation. But it cannot be surprising that the relationship between produc-

tivity and growth varies across industries and common decomposition techniquesmay yet be the best one-size-fits-all tools. This, however, also strongly impliesthat more could be learned about economics selection in industry specific stud-ies.

The demonstration of the application of multilevel decomposition served tohighlight the possibilities of the technique but did not add much to empiricalknowledge of the evolution of labour productivity. The inter industry selectioneffect was practically zero and the inter firm intra industry selection effect waslargely the same as when the extra level was not taken into account. In thisspecific case, it could be argued that the multilevel decomposition is an unnec-essary complication and that the simpler decomposition should be preferred ongrounds of parsimony. But it is important to stress that the application here

was chosen for ease of exposition: the data only allows for 34 industries at thetwo digit level and thus the results could be presented by industry as well. Themultilevel equations will find application in research projects with an explicittaxonomic side; e.g. studies that include taxonomies of labour quality and thusallow for grouping of firms according to their human capital profile. Other ex-amples are studies that seek to study the tendency (at least in Denmark) foreconomics activity to be increasingly centred around larger cities. By includinga spatial level in the decomposition it would be possible to include the effect oflocation on the reallocation of labour resources.

The paper suggests a number of paths for further research. Experimentingwith various delimitations of subpopulations is just one (grouping of firms byoutput rather than production processes would also be interesting). The demon-stration of the technique presented here also uncovered an interesting facet ofmultilevel decomposition studies: firms may jump from one subpopulation toanother. The role of this effect was not studied at all here. And most impor-tantly: this paper has focussed on labour productivity but research is also needon the consequences of Prices insight for decomposition of TFP change.

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Two digit industries

Three digit industries

Four digit industries

Left column: corr(wi, zi), centre column: corr(wi, ln zi) and right column: (wi, zi). The verticalaxis is 0 to 30 percent. and the horizontal axis is -1 to 1 with a dashed reference line at 0. Source:own calculations.

Figure A-1: The distribution of correlations across industries

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Table A-1: Two digit populations

Two digitclassification

Description Label

Manufacturing of . . .

15 Food and beverages Food.Bev.16 Tobacco* Tobacco17 Textiles Text.18 Wearing apparel Clothes19 Leather and footwear Leather20 Wood and wood products Wood21 Pulp, paper and paper products Paper

22 Printing and publishing Publ.23 Refined petroleum products* Oil24 Chemicals Chemicals25 Rubber and plastic products Rubber26 Other non-metallic mineral products Mineral27 Basic metals Metal28 Metal products M.Prod.29 Machinery and equipment Mach.30 Electronic components* Elect.31 Other electric machinery and equipment O.elec.32 Radio and communications equipment Comm.33 Medical and optical instruments Med.Opt.

34 Motor vehicles Motor35 Other transport equipment Transp.36 Furniture and manufacturing n.e.c. Furni.37 Recycling* Recyc.

Services

45 Construction Constr.50 Sales and services of motor vehicles Vehicl.51 Wholesale Wholes.52 Retail and repair shops Retail55 Hotels and restaurants Hotels60 Land transport L.trans.

61 Water transport W.trans.62 Air transport* A.trans.63 Supporting transport activities O.trans.70 Real estate activities R.estate71 Rental of transport equipment and machinery Rental72 Computer and related activities Computers74 Consultancy and cleaning activities Consul.85 Human health and social services Health92 Recreational, cultural and sporting activities Recr.93 Other service activities Others

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Table A-2: Evolution of labour productivity (zj), 1992-1999

Industry Selection Learning. Entry Exit zj wj

Manufacturing

Food.Bev. 0.029 0.670 0.121 0.180 0.175 0.81Text. 0.277 0.385 0.279 0.058 0.159 0.78Clothes 0.432 0.025 0.286 0.257 0.445 0.46Leather 0.090 0.802 0.013 0.095 0.472 0.82Wood 0.210 0.333 0.099 0.358 0.135 1.28Paper 2.339 2.498 0.977 0.181 0.015 0.86

Publ. 0.143 0.595 0.314 0.052 0.264 0.96Chemicals 0.027 0.562 0.612 0.202 0.259 1.11Rubber 0.300 0.581 0.183 0.064 0.148 1.07Mineral 0.187 0.924 0.017 0.281 0.141 1.03Metal 0.775 1.573 1.107 2.455 0.047 1.39M.Prod. 0.184 0.425 0.275 0.116 0.166 0.94Mach. 0.093 0.561 0.261 0.085 0.174 1.02O.elec. 2.043 1.307 0.312 0.576 0.036 1.29Comm. 1.602 9.439 9.350 2.513 0.018 1.20Med.Opt. 0.044 0.412 0.478 0.066 0.406 1.07Motor 0.176 0.549 0.266 0.008 0.226 1.24Transp. 0.102 1.389 0.358 0.133 0.128 0.69

Furni. 0.260 0.518 0.158 0.064 0.241 1.10Services

Constr. 0.159 0.619 0.337 0.116 0.187 1.19Vehicl. 0.166 0.527 0.240 0.067 0.273 1.15Wholes. 0.220 0.457 0.102 0.425 0.101 1.12Retail 0.051 0.443 0.033 0.576 0.077 1.00Hotels 0.060 0.207 0.608 0.126 0.270 1.01L.trans. 0.227 0.530 0.224 0.020 0.180 1.14W.trans. 0.512 1.060 0.572 0.121 0.258 0.29O.trans. 0.083 0.436 0.544 0.102 0.244 1.25R.estate 0.232 0.266 1.022 0.055 0.216 1.71Rental 0.115 0.507 0.584 0.809 0.070 1.43

Computers 0.266 0.044 0.342 0.880 0.051 2.05Consul. 0.060 0.306 0.518 0.236 0.043 1.37Health 1.084 1.093 0.303 0.707 0.067 1.46Recr. 0.252 0.364 0.185 0.199 0.334 1.52Others 0.371 0.807 0.094 0.084 0.103 1.16

Effects are reported as shares of zj . zj is population mean productivity in industry j.

Productivity is lnYi,1992Li,1992

Y is output (VA in 1992 DKK), L is FTE labour and subscripts

refer to years. Source: own calculations.

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holm lemseminar 2010

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