metadata: authors: 0: author: Basu, Susanto
1: author: Pascali, Luigi
2: author: Schiantarelli, Fabio
3: author: Serven, Luis
count: World
docna: 0: docna: Productivity, welfare and reallocation :
            theory and firm-level evidence
docty: Policy Research Working Paper
theme: Analysis of economic growth
majtheme: Economic management
subtopic: Macro-Fiscal Policy,Economic Adjustment and Lending,Construction Industry,Common Carriers Industry,Food & Beverage Industry,General Manufacturing,Pulp & Paper Industry,Plastics & Rubber Industry,Labor Markets,Disability,Public Sector Economics,Services & Transfers to Poor,International Trade and Trade Rules,Economic Assistance,Access of Poor to Social Services,Business Cycles and Stabilization Policies,Public Finance Decentralization and
            Poverty Reduction
teratopic: Private Sector Development,Macroeconomics and Economic Growth,Industry,Social Protections and Labor
abstracts: cdata!: A considerable literature has focused on
            the determinants of total factor productivity (TFP),
            prompted by the empirical finding that TFP accounts for the
            bulk of long-term growth. This paper offers a deeper reason
            for such focus: the welfare of a representative consumer is
            summarized by current and anticipated future Solow
            productivity residuals. The equivalence holds for any
            specification of technology and market structure, as long as
            the representative household maximizes utility while taking
            prices parametrically. This result justifies total factor
            productivity as the right summary measure of welfare, even
            in situations where it does not properly measure technology,
            and makes it possible to calculate the contributions of
            disaggregated units (industries or firms) to aggregate
            welfare using readily available data. Based on this finding,
            the authors compute firm and industry contributions to
            welfare for a set of European countries (Belgium, France,
            Great Britain, Italy, Spain) using industry-level and
            firm-level data. With additional assumptions about
            technology and market structure (specifically, that firms
            minimize costs and face common factor prices), the authors
            show that welfare change can be further decomposed into
            three components that reflect, respectively, technical
            change, aggregate distortions, and allocative efficiency.
            Then, using the appropriate firm-level data, they assess the
            importance of each of these components as sources of welfare
            improvement in the same set of European countries.
repnme: repnme: Productivity, welfare and reallocation :
            theory and firm-level evidence
display_title: Productivity, welfare and
            reallocation : theory and firm-level evidence
listing_relative_url: /research/2010/03/11882126/productivity-welfare-reallocation-theory-firm-level-evidence
topicv3: Manufacturing Sectors m1326521 1281,Government Expenditure m1326649 1112,Trade m1326924 698,Social Assistance m1327201 1466,Labor Productivity m1504060 2756
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Private Sector Development,Macroeconomics and Economic Growth,Industry,Social Protections and Labor
no_of_pages: 48
model_extractions: WPS5226 Policy Research Working Paper 5226 Productivity, Welfare and Reallocation Theory and Firm-Level Evidence _Susanto Basu Luigi Pascali Fabio Schiantarelli Luis Serven_ The World Bank Development Research Group Macroeconomics and Growth Team March 2010
Policy Research Working Paper 5226

# **Abstract**

A considerable literature has focused on the determinants of total factor productivity (TFP), prompted by the empirical finding that TFP accounts for the bulk of long-term growth. This paper offers a deeper reason for such focus: the welfare of a representative consumer is summarized by current and anticipated future Solow productivity residuals. The equivalence holds for any specification of technology and market structure, as long as the representative household maximizes utility while taking prices parametrically. This result justifies total factor productivity as the right summary measure of welfare, even in situations where it does not properly measure technology, and makes it possible to calculate the contributions of disaggregated units (industries or firms) to aggregate welfare using readily available data. Based on this finding, the authors compute firm and industry contributions to welfare for a set of European countries (Belgium, France, Great Britain, Italy, Spain) using industry-level and firm-level data. With additional assumptions about technology and market structure (specifically, that firms minimize costs and face common factor prices), the authors show that welfare change can be further decomposed into three components that reflect, respectively, technical change, aggregate distortions, and allocative efficiency. Then, using the appropriate firm-level data, they assess the importance of each of these components as sources of welfare improvement in the same set of European countries.

This paper—a product of the Macroeconomics and Growth Team, Development Research Group—is part of a larger effort in the department to assess the determinants of productivity and growth. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at lserven@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Productivity, Welfare and Reallocation: Theory and Firm-Level Evidence<sup>�</sup>

Susanto Basu Luigi Pascali Fabio Schiantarelli Boston College and NBER Boston College Boston College and IZA Luis Serven<sup>y</sup> World Bank March 1, 2010 JEL: D24, D90, E20, O47 Keywords: Productivity, Welfare, Reallocation, Technology, TFP

> �We would like to thank John Haltiwanger, Chad Syverson and the participants at the NBER Summer Institute, NBER Productivity Lunch, Ottawa Productivity Workshop and Bocconi University seminar for helpful discussions and suggestions.

> yBasu: Susanto.Basu@bc.edu; Pascali: Luigi.Pascali@bc.edu; Schiantarelli: Fabio.Schiantarelli@bc.edu; Serven: Lserven@worldbank.org

1
# 1 Introduction

How much of growth comes from innovation and technical advances, and how much from changes in allocative e¢ ciency? This question arises in a variety of contexts, in ...elds as diverse as growth and development, trade, and industrial organization. Yet, despite the importance of the question, there is no consensus regarding the answer. A large number of papers have proposed a bewildering variety of methods to measure the importance of allocative e¢ ciency, leading to a wide range of numerical estimates. Much of the confusion is due to the lack of an organizing conceptual framework for studying this issue. We propose such a framework, and then provide a quantitative answer using one particular set of data.

In starting such a project, one immediately faces the question: What do we mean by allocative e¢ ciency? Indeed, what do we mean by growth? We take the view that growth is an improvement in social well-being. While growth is commonly described in terms of GDP per worker or consumption per capita, these statistics are usually viewed as indicators of some deeper target. Their virtue, a considerable one, is that they can be generated from aggregate data, which are usually readily available. We ask if we can produce a more complete description of economic welfare and its change, while also restricting ourselves to aggregate data. Given an empirical method for characterizing aggregate welfare, allocative e¢ ciency is naturally de...ned as the increment to welfare achieved by reallocating productive resources to more e¢ cient uses, holding constant the aggregate quantities of resources used in production.

We undertake three tasks. First, we begin from a utility-maximization problem that is standard in the economics of growth and business cycles: We assume that a representative household with an in...nite horizon values both consumption and leisure, and maximizes utility subject to a standard intertemporal budget constraint.<sup>1</sup> We show that this standard speci...cation of the objective function implies, to a ...rst order, that welfare depends on the present discounted value of total factor productivity (TFP) for the aggregate economy and on the initial level of the capital stock. This result is “TFP without ...rms”— it is derived purely from the standard model of a price-taking, competitive household. Thus, our result holds for all speci...cations of technology and market structure, including ones where TFP does not measure technology, as long as consumers are free to choose the quantities of goods they purchase at prices they take as being outside their control. Here we follow the intuition of Basu and Fernald (2002), and supply a general proof of their basic proposition that TFP is relevant for welfare.

Second, we use this result to show that we can calculate the welfare contributions of particular sectors of the economy— which can be as large as industries and as small as individual ...rms. We present industry and ...rm contributions to welfare for a set of European OECD countries (Belgium,

> 1 While the valuation of leisure is not common in a growth context, it is quantitatively very important. Reviewing a large number of social goods that are valued by consumers but not counted in GDP, Nordhaus and Tobin (1973) found the omission of leisure the most signi...cant (with another imputation for the use of non-market time in home production the second most important). Our household maximization framework also corrects automatically for two other gaps that Nordhaus and Tobin ...nd are signi...cant: The need to subtract depreciation (moving to a NDP rather than a GDP framework), and the need to adjust for a growing population.

2
France, Great Britain, Italy, Spain), using industry data from EU-KLEMS and the Amadeus ...rmlevel data set. Among other things, we use these data to compare the distributions of ...rm-level productivities relative to the country means across the countries in our sample, and ask how much welfare would increase if, for example, Italian ...rms had the same relative productivity distribution as those in Great Britain. This analysis is akin to that of Hsieh and Klenow (2009), but it has a direct welfare interpretation and is more general because it does not require assumptions about the production technology.

Third, we show how to decompose welfare— aggregate TFP— into components due to technology, aggregate distortions, and allocative ine¢ ciency. Any such decomposition does depend on assumptions about production technology, adjustment costs, and industrial organization. Di¤erent assumptions will lead to di¤erent decompositions, but within the same overarching social-welfare framework. Finally, we implement one speci...c decomposition, again based on Basu and Fernald (2002), using ...rm-level data from a number of European countries represented in the Amadeus data set. We ...nd that welfare grows signi...cantly faster than technology changes, but improvements in allocative e¢ ciency usually account for a modest fraction of the gap between the two.

Our ...rst result clari...es the nature of the important link between welfare on the one hand and aggregate productivity and national income measurement on the other.<sup>2</sup> Our main goal in this section is to provide a clear objective for any decomposition of productivity. To have an economic interpretation, any such decomposition should indicate how productivity contributes to the ultimate target, which is social welfare. Under the usual assumptions and to a ...rst-order approximation, that target is a measure of productivity, aggregate TFP. But the method is more important than the speci...c result. A di¤erent speci...cation of the consumer’s problem may deliver a di¤erent result about the relationship between welfare and productivity. (In fact, we derive results in the paper showing that under certain conditions— for example, if there are distortionary taxes— the correct welfare measure may di¤er substantially from the usual Solow residual.) But it is still important for researchers interested in decomposing productivity or studying allocative e¢ ciency to relate their empirical method to the solution to some well-speci...ed maximization problem so that the implications of their decompositions for some ultimate welfare objective, which are usually left implicit in any such study, can be made explicit, and the necessary assumptions can be examined closely.

One bene...t of starting from a well-de...ned objective function is that it enables the researcher to take consistent, model-based positions on a variety of issues that bedevil the measurement of productivity and allocative e¢ ciency. For example, Baker and Rosnick (2007), reasoning that the ultimate object of growth is consumption, make the reasonable conjecture that one should de‡ate nominal productivity gains by a consumption price index to create a measure they call “usable productivity.”We begin from the assumption that consumption (and leisure) at di¤erent dates are the only inputs to economic wellbeing, but nevertheless show that output should be calculated in the

> 2 Earlier works also make a connection between the two. Some of the most important are Nordhaus and Tobin (1973), Weitzman (1976, 2003) and Hulten (1978). Our approach closely follows that of Basu and Fernald (2002).

3
conventional way, rather than being de‡ated by consumer prices.<sup>3</sup> To take another example, there is no consensus in the literature about the proper treatment of scale economies. Most researchers examine allocative e¢ ciency by asking whether ...rms with higher levels of Hicks-neutral technology produce more output. Others pose the same question in terms of labor productivity, which includes scale economies but does not subtract capital’s contribution to output. Using our framework, it is easy to show that only the Solow TFP index gives the correct welfare accounting. Unlike a pure technical change measure, the Solow residual includes scale e¤ects, which do contribute to welfare by producing more output for given inputs. Unlike labor productivity, the TFP residual subtracts the change in capital input valued at its opportunity cost to the consumer.

Our analytical results create several links between productivity and welfare. One important message is that welfare depends on the entire expected future path of TFP. Not surprisingly, the same size change in current TFP has very di¤erent e¤ects on welfare if it is expected to be persistent than if it is expected to be transitory. This result suggests new ways of assessing the importance of reallocation. To our knowledge, the literature does not examine the time-series properties, especially the persistence, of measures of allocative e¢ ciency. But our derivation shows that to understand the contribution that reallocation makes to growth, it is important to know the persistence as well as the mean. In principle, the allocative e¢ ciency component of TFP might be either more or less persistent than total TFP, making reallocation either more or less important than its average share would suggest.

So far we have been vague about whether our results relate to TFP in levels or in growth rates. In fact, our results apply to both. We show that the level of welfare for a representative consumer is, up to a ...rst-order approximation, proportional to the present discounted value of expected log levels of TFP. Welfare change for the consumer, on the other hand, is proportional to the change in log levels, i.e., to the present discounted value of TFP growth as we de...ne it (equal to the standard Solow productivity residual if there are zero economic pro...ts), plus an “expectation revision”term that depends on the di¤erence in expectations of future log levels of TFP between time t-1 and time t. Under perfect foresight, the expectation revision term is identically zero, and the change in welfare is proportional to the present discounted value of current and future Solow residuals alone. Starting from a well-posed optimization problem also forces us to confront two issues in national income and welfare measurement. First, our derivation shows that “consumption”should be de...ned as any good or service that consumers value, whether or not it is included in GDP. Similarly, "capital" should include all consumption that is foregone now in order to raise consumption possibilities for the future. These items include, for example, environmental quality and intangible capital. Of course, both are hard to measure and even harder to value, since there is usually no explicit market price for either good. But our derivation is quite clear on the principle that the environment, intangibles and other non-market goods should be included in our measure of “welfare TFP.”We follow conventional practice in restricting the output measure for our TFP variable

> 3 The other main adjustment by Baker and Rosnick, moving to a net measure of output as a starting point for productivity measurement, follows a long tradition of research on this topic, and is fully supported by our derivation.

4
to market output (and the inputs to measured physical capital and labor), but in so doing we, and almost everyone else, are mismeasuring real GDP and TFP. Second, our starting point of a representative-consumer framework implies that we automatically ignore issues of distribution that intuition says should matter for social welfare. We believe that distributional issues are very important. However, our objective of constructing a better welfare measure from aggregate data alone implies that we cannot incorporate measures of distribution into our framework. Thus, we maintain the representative-consumer framework, but without in any way minimizing the importance of issues that cannot be handled within that framework.

Having established that aggregate TFP is the natural measuring stick for aggregate welfare, we then ask the next natural question: Can one show what contribution a subset of the economy (which may be as small as a single ...rm) makes to the aggregate welfare index? The answer is yes, as shown by Domar (1961). Domar established that a correctly-weighted average of sectoral TFP residuals sums to Solow’s familiar aggregate index. We use a variant of his result to present the welfare contributions made by large sectors of the economy using the EU-KLEMS dataset. We compare the sources of welfare di¤erences across countries, asking what fraction of cross-country di¤erences are due to di¤erences in industrial structure as opposed to di¤erences in the welfare contributions of the same sector across countries. We then do a similar exercise using our ...rmlevel data over the period 1998-2004, and investigate the extent to which di¤erences in the relative productivity distribution of ...rms across countries contributes to di¤erences in welfare.

Finally, we decompose aggregate TFP into components. As we noted, while TFP is itself meaningful in welfare terms without any additional assumptions, we need to make assumptions about ...rm technology and behavior in order to decompose TFP in a meaningful way. We use a variant of the decomposition of Basu and Fernald (2002), which is derived by assuming that ...rms minimize costs and are price-takers in factor markets, but may have market power for the goods they sell and might produce with increasing returns to scale. As we also noted, di¤erent assumptions about technology would give di¤erent decompositions, without changing the essential features of the results. For example, Basu and Fernald (2002) assume that factors are freely mobile across ...rms, without adjustment costs, while Basu, Fernald and Shapiro (2001) extend the framework to include costly factor adjustment. Abel (2003) and Basu et al. (2001) show that adjustment costs are a special type of intangible capital, of a sort that needs to be accumulated in ...xed proportions with physical capital. Thus, accounting for adjustment costs in the empirical results would require us to impute an addition to measured output, which is conceptually the same issue as accounting for non-market consumption goods or for more general forms of intangible capital accumulation.

Some of the components in the decomposition we use can be clearly identi...ed as being due to reallocation, since they depend on marginal products of identical inputs not being equalized across ...rms. Other components depend on aggregate distortions, such as the average degree of market power and various tax rates. In order to estimate the reallocation terms, we need to estimate ...rmlevel marginal products. We do so using ...rm-level data for a number of manufacturing industries 5
across six European countries, as represented in the Amadeus data base.<sup>4</sup> We extend the existing decomposition to study reallocation both within and between industries, since the two kinds of reallocation may have di¤erent policy implications.

We use the Amadeus data to estimate production functions for ...rms within a number of manufacturing industries across six countries. We experimented with a variety of estimation methods to ensure that our main results were robust. We found that there is usually a substantial gap between our estimates of technical change for each manufacturing industry and that industry’s contribution to aggregate TFP growth (and hence welfare). However, for most countries, the majority of this gap is due to the aggregate distortions (especially when taxes are included in the decomposition). Reallocation strictly de...ned usually accounts for a small fraction of the gap.

The paper is organized as follows. We present the key equations linking productivity and welfare in Section 2, with the full derivation presented in an appendix. While our derivations link welfare to both TFP levels and growth rates, we choose to work mostly in growth rates, since there are wellknown di¢ culties in comparing TFP levels across industries and countries. In Section 3, we show how to identify sector and ...rm-level contributions to the productivity residual. Section 4 presents our data and discusses measurement issues. Section 5 assesses the contribution of di¤erent sectors and groups of ...rms to the productivity residual in ...ve European countries. Section 6 shows how the productivity residual can be decomposed into the respective contributions of reallocation and technology. We then switch to an econometric framework for decomposing the sources of welfare change in Section 7, and present the results in Section 8. We conclude in Section 9 with some re‡ections, and suggestions for future research.

# 2 The Productivity Residual and Welfare

It is intuitive that technology growth matters for welfare purposes, since our intuition suggests that technological progress is responsible for the secular increase in the standard of living. However, should we care about the Solow residual in an economy with non-competitive output markets, nonconstant returns to scale, and possibly other distortions? Here we build on the intuition of Basu and Fernald (2002) that a slightly modi...ed form of the Solow residual is welfare relevant even in those circumstances and derive rigorously the relationship between a modi...ed version of the productivity residual (in growth rates or log levels) and the intertemporal utility of the representative household. The fundamental result we obtain is that, to a ...rst-order approximation, utility re‡ects the present discounted values of productivity residuals.

Our results are complementary to those in Solow’s classic (1957) paper. Solow established that if there was an aggregate production function then his index measured its technical change. We now show that under a very di¤erent set of assumptions, which are disjoint from Solow’s, the familiar TFP index is also the correct welfare measure. The results are parallel to one another.

> 4 Petrin, White and Reiter (2009) also use ...rm-level data to implement a variant of the Basu-Fernald (2002) decomposition. They use U.S. Census data for manufacturing industries. We compare our results to theirs in Section 6.

6
Solow did not need to assume anything about the consumer side of the economy to give a technical interpretation to his index, but he had to make assumptions about technology and ...rm behavior. We do not need to assume anything about the ...rm side (which includes technology, but also ...rm behavior and industrial organization) in order to give a welfare interpretation, but we do need to assume the existence of a representative consumer. Both results assume the existence of a potential function (Hulten, 1973), and show that TFP is the rate of change of that function. Which result is more useful depends on the application, and the trade-o¤ that one is willing to make between having a result that is very general on the consumer side but requires very precise assumptions on technology and ...rm behavior, and a result that is just the opposite.

# 2.1 Approximating around the steady state

More precisely, assume that the representative household maximizes intertemporal utility: where Ci;t is the capita consumption of good i at time t, Lt are hours of work per capita, L is the time endowment, and Nt population. H is the number of households, assumed to be ...xed and normalized to one from now on. Xt denotes Harrod neutral technological progress, assumed to be common across all sectors. Population grows at constant rate n and Xt at rate g. For a well de...ned state state in which hours of work are constant we assume that the utility function has the King Plosser and Rebelo form(1988): with 0 < �< 1 or �> 1:<sup>5</sup> We assume that C() has constant returns to scale. De...ne ci;t+s =<sup>C</sup> X<sup>i;</sup> t<sup>t</sup> +<sup><u>+</u></sup> s<sup>s</sup> We can rewrite the utility function in a normalized form as follow: where � =<sup><u>(1+n</u></sup> (1+<sup><u>)(1+</u></sup> �<sup><u>g</u></sup> )<sup><u>)1��</u></sup> is assumed to be less than one. The budget constraint (with variables scaled by NtXt) is: New capital goods are the numeraire, kt = XKtNt t<sup>denotescapitalperunitofe¤ectivelabor,,</sup>

> 5 If � = 1, then the utility function must be U (C1; ::; CG; L � L) = log(C) � �(L � L): See King, Plosser and Rebelo (1988).

7
bt = Pt<sup>I</sup> <u>B</u><sup>X</sup> <u>t</u><sup>tNtarerealbonds:pK</sup> t<sup>=P</sup> Pt<sup>K</sup> t<sup>I,pL</sup> t<sup>=</sup> PPt<sup>I</sup> <u>t</u><sup>XLt,pC</sup> i;t<sup>=</sup> PPi<sup>C</sup> t<sup>I</sup> <u>;t</u> denote, respectively, the user cost of capital, the wage per hour of e¤ective labor, and the price of consumption goods. (1 + rt) is the real interest rate (again in terms of new capital goods) and �t = Pt<sup>I</sup> <u>�</u><sup>X</sup> <u>t</u><sup>tNtdenotespro...ts.</sup> Log linearizing around the non stochastic steady state, intertemporal household utility can be written (to a ...rst order approximation) as: where v is the steady state value of utility, xb = log xt � log x denote log deviation from the steady state. In obtaining this result we have used the FOC of the household maximization problem: and the log linear approximation of the budget constraint around the steady state: Equation (4) says that intertemporal utility (in log deviation from the steady state) equals the expected present discounted value of terms that represent the sum of the components of ...nal demand (in log deviation from the steady state), weighted by their steady state contribution to demand, minus primary inputs (in log deviation from the steady state) times their respective steady state factor prices.

# 2.2 Connecting with the productivity residual

We are now close to relating utility to a modi...ed version of the Solow residual. There are two options here. The ...rst one is to obtain a ...rst order approximation for the log level of utility in terms of the log level productivity residual. Simple manipulations allow us to rewrite log level utility as a function of expected future Solow residuals plus an initial (log) level productivity residual. The second one focuses instead on approximating the log change in utility over time.

8
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capital stock,<sup>b</sup> kt�1; since for any sequence of productivity, welfare is higher if the consumer starts with an higher initial endowment of capital.

To establish the relationship with the Solow residual (a growth rate concept) there are two options. One option is to use the fact that, for any variable x : In the appendix we show that log level utility,(4), implies that per capita (log) intertemporal utility can also be written as: where �log prt denotes the "modi...ed" Solow productivity residual: We use the word "modi...ed," for two reasons. First, we do not assume that the distributional shares of capital and labor add to one, as they would if there were zero economic pro...ts. Zero pro...ts are guaranteed in the benchmark case with perfect competition and constant returns to scale, but can also arise with imperfect competition and increasing returns to scale, as long as there is free entry, as in the standard Chamberlinian model of imperfect competition. Second, the distributional shares are calculated at their steady state values and, hence, are not time varying. Rotemberg and Woodford (1991) argue that in a consistent ...rst-order log-linearization of the production function the shares of capital and labor should be taken to be constant, and Solow’s (1957) use of timevarying shares amounts to keeping some second-order terms while ignoring others. Now log level productivity has been written as a combination of expected future Solow residuals and one initial productivity term in levels. Assume one is willing to make the assumption that an economy at time t-1 was at the steady state, so that log yt�1 � sL log Lt�1 � sK log kt�1 = log y � sL log L � sK log k In this special case simple algebra shows that vt depends upon the expected present discounted value of Solow residuals (from the present to in...nity).

where: An alternative and more satisfactory way to illustrate the relationship between welfare and the Solow residual (with no level term) is to return to (4) and take its di¤erence through time 10
(�vt = vt � vt�1): Using only the de...nition of value added in growth terms, equation (9), the growth rate of per capita utility can be written as follows: where Et�log prt+s represents the expected Solow residual while Et log prt+s � Et�1 log prt+s represents the revision in expectations of the log level of the productivity residual, based on the new information received between t-1 and t. In addition, �log Kt�1 captures the change in the initial endowment of capital. Finally, the constant f1 is: Note that the revision term in the second summation will reduce to a linear combination of the innovations in the stochastic shocks a¤ecting the economy at time t. Moreover, if we assume that the modi...ed Solow residual follows a simple stable ...rst order autoregressive process, then the current Solow residual, �log prt; is a su¢ cient statistic for all the terms in the ...rst summation. In this case, the growth in expected per capita utility is a linear function of today’s actual Solow residual, of innovations at time t in the stochastic processes driving the economy and of the change in initial endowment of capital.

# 2.3 Extensions

We now show that our method of using TFP to measure welfare can be extended to cover multiple types of capital and labor, taxes, and government expenditure. The ...rst extension modi...es our baseline results in only a trivial way, but the others all require more substantial changes to the formulas above. These results show that the basic idea of using TFP to measure welfare holds in a variety of economic environments, but also demonstrate the advantage of deriving the welfare measure from an explicit dynamic model of the household. The model shows exactly what modi...cations to the basic framework are required in each case, and demonstrates that some of these modi...cations are quantitatively signi...cant.

## 2.3.1 Multiple Types of Capital and Labor

The extension to the case of multiple types of labor and capital is immediate. For simplicity, we could assume that each individual is endowed with the ability to provide di¤erent types of labor 11
services, Lh;t and that the utility function can be written as: where L(:) is an homogenous function of degree 1, HL is the number of types of labor and Pt<sup>Lh</sup> denotes the payment to a unit of Lh;t:<sup>8</sup> Similarly consumers can accumulate di¤erent types of capitals Kh;t and rent them out at Pt<sup>Kh</sup> . Proceeding exactly as before, the same equations will characterize the relationship between utility and the Solow residual, with the only di¤erence that the latter is de...ned now as:

# 2.3.2 Taxes

Our derivation of section 2.2 requires only reinterpretation to apply exactly to an environment with either distortionary and/or lump-sum taxes. The reason is that all prices in the budget constraint, equation (4), are from the point of view of the consumer. Thus, if there are taxes, the prices should all be interpreted as after-tax prices. Therefore our derivation implicitly allows for proportional taxes on capital and labor income as well as sales or value-added taxes levied on consumption and/or investment goods. The variable that we have been calling “pro...ts,”�, is really any transfer of income that the consumer takes as exogenous. Thus, it can be interpreted to include lump-sum taxes or rebates.

However, for the sake of exposition, we shall interpret all prices in equation (4) as being from the point of view of a ...rm, and thus before all taxes. To modify (4) to allow for taxes, we de...ne some notation. Let �<sup>K</sup> be the tax rate on capital income, �<sup>L</sup> be the tax rate on labor income, �<sup>C</sup> i<sup>betheadvaloremtaxonconsumptiongoodsoftypei,and�Ibethecorrespondingtaxon</sup> investment goods. We assume that the revenue so raised is distributed back to individuals using lump-sum transfers. (We consider government expenditures in the next sub-section.) Then it is apparent that we arrive at the following modi...ed version of equation (4): To make contact with the data, note that the national accounts de...ne nominal expenditure using prices as perceived from the demand side. Thus, equation (11) can be written exactly as before and still be consistent with standard national accounts data:

> 8 We assume that the nature of the utility function is such that positive quantities of all types of labors are supplied.

12
On the other hand, the national accounts de...ne factor prices as perceived by ...rms, before income taxes. Thus, the data-consistent de...nition of the welfare residual with taxes needs to be based on a new de...nition of log prt. Rewrite equation (14) as: This new de...nition of log prt then needs to be applied to equations such as (12) and (13) in section 2.2.

While it is easy to incorporate taxes into the analysis— as noted above, they are present implicitly in the basic expressions derived in section 2.2— the quantitative impact of modeling taxes explicitly can be large. Suppose that output is produced using an aggregate, constant-returns-toscale production function of capital, labor and technology, as in Solow’s classic (1957) paper. Then, without distortionary taxes, only changes in technology change welfare.

Now suppose the average marginal tax rate on both capital and labor income is 30 percent, and the share of consumption in output is 0.60. Suppose the government manages to raise aggregate capital and labor inputs by 1 percent permanently without a change in technology (perhaps via a small cut in tax rates). Then the ‡ow increase in utility is equivalent to an increase in steadystate consumption of 0.5 percent. If the discount factor is 0.95 on an annual basis, the present value of this policy change is equivalent to a one-year increase in consumption of 10 percent of the steady-state level!

# 2.3.3 Government expenditure

With some minor modi...cation, our framework can be extended to allow for the provision of public goods and services. We illustrate this under the assumption that government activity is ...nanced with lump-sum taxes. Using the results from the previous subsection, it is straightforward to extend the argument to the case of distortionary taxes.

Assume that government spending takes the form of public consumption valued by consumers. We rewrite the instantaneous utility function as where G denotes per-capita public consumption, and we continue to assume that C(:) is homogenous of degree one in its arguments. The relevant welfare residual in equation (4) now becomes: 13
where gt = X<sup><u>Gt</u></sup> t<sup>.Thede...nitionofGDPindeviationfromsteadystateisnow:</sup> where sg = P<sup><u>P GYG</u></sup> Y<sup>andP Gisthepublicconsumptionde‡ator.Lets�</sup> g<sup>=Ugg</sup> �<sup><u>gbt+s</u></sup> . Then we can write: Hence in the presence of public consumption the Solow residual needs to be adjusted up or down depending on whether public consumption is under- or over-provided (i.e., s<sup>�</sup> g<sup>> sgor s�</sup> g<sup>< sg</sup> respectively). If the government sets public consumption exactly at the utility-maximizing level, s<sup>�</sup> g<sup>=sgandnocorrectionisnecessary.Inturn,inthestandardneoclassicalcaseinwhichpublic</sup> consumption is pure waste s<sup>�</sup> g<sup>=0,thewelfareresidualiscomputedonthebasisofprivate...nal</sup> demand –i.e., GDP minus government purchases.

What if government purchases also yield productive services to private agents? This could be the case if, for example, the government provides education or health services, or public infrastructure, which may be directly valued by consumers and may also raise private-sector productivity. In such case, the above expression remains valid, but it is important to note that the net contribution of public expenditure to welfare would not be fully by captured by (s<sup>�</sup> g<sup>�sg)gbt+s.Tothistermwe</sup> would need to add a measure of the productivity of public services, which in the expression is implicitly included in the productivity residual ybt+s � sLL<sup>b</sup> t+s � sK<sup>b</sup> kt+s.

# 3 Decomposing the Productivity Residual: Firm and Sector Level Contributions

The fundamental result from the previous section is that the growth in welfare is related to the expected present discounted value of the aggregate (modi...ed) Solow productivity residual. In this section we will argue that this aggregate e¤ect can be decomposed into the contribution of individual ...rms (or subset of ...rms). In order to do this we will look at aggregate value added, not from the expenditure side as we have done so far, but from the product side. More speci...cally, de...ne aggregate value added as the following Tornqvist/Divisia index of ...rm-level value added: 14
The corresponding index for producer prices is: Moreover, one can easily show that the following is true as an approximation: and As a result the aggregate Solow residual can be written as the weighted sum (with value-added weights) of the ...rm-level Solow residuals. More speci...cally: where �log prit is de...ned as: We can use this result to examine the sectoral sources of productivity growth, which is the key to welfare change, within a country. We can ask a variant of the same question for ...rms, as we explain in the results sub-section. Finally, we can compare cross-sectional summary statistics. For example, we can ask whether small or large ...rms contribute more to national welfare improvement.

# 4 Data and Measurement

Our main source of information is Amadeus, a comprehensive ...rm-level pan-European database developed by Bureau Van Dijk. For every ...rm it provides data on the industry where it operates (at the 4-digit NACE level), its location, the year of incorporation, the ownership structure and the number of employees, in addition to the complete balance sheets and the pro...t and loss accounts. The data set includes both publicly traded and non traded companies. We limit our analysis to a subset of countries: Belgium, France, Great Britain, Italy, and Spain. We focus on manufacturing companies with operating revenues greater than or equal to 2 million Euros and continuous observations within the period of analysis. (We restrict ourselves to the balanced panel because Amadeus does not supply census data; there is no way to distinguish between entry into the sample and actual entry into the economy.) We also use industry-level yearly data from the EU-KLEMS project, which provides output, input and price data for industries at roughly the 2-digit level of aggregation across a large number 15
of countries up to 2005. These countries are mostly, but not exclusively, European; the project also gives data for non-EU countries like Australia, Japan, Korea and the United States. The EU-KLEMS data are extensively documented by O’Mahony and Timmer (2009).

In addition to the non-parametric welfare-relevant index numbers presented in the next section, we will also estimate production functions using ...rm level data, allowing the coe¢ cients to vary across 2-digit industries for the period 1998-2005.<sup>9</sup> Before 1998 the number of ...rms in the survey is signi...cantly smaller in most countries. Between 1998 and 2000 many ...rms enter in the data set. The coverage provided by the dataset varies across these countries. In 2005 the aggregated sales of the ...rms represented in Amadeus represent between 20 percent and 45 percent of the manufacturing sector’s total production value, as documented in EU-KLEMS.

Our gross output proxy is (...rm level) revenues de‡ated by the sectoral value added de‡ator obtained from the EU-KLEMS data set, at the 2 digit level. All de‡ators used here will be at the 2 digit level and are obtained from EU-KLEMS. We are aware that using industry de‡ators in place of ...rm-level prices can cause problems (Klette and Grilliches (1996)), but ...rm-level price data for output are not available in Amadeus. Our proxy for labor input is manpower costs de‡ated by the labor services de‡ator. (For some countries, such as Italy, the number of employees ...gure is not reliable, since there is not a reporting requirement for the number of employees in the main section of the balance sheet.) Capital is the historical value of tangible ...xed assets divided by the price index for investment. We have also experimented with the perpetual inventory method, obtaining similar results. A measure for materials, intermediates and other services used in production has been computed using the following formula: materials = Operating Revenues - (Operating Pro...ts+Manpower costs+Depreciation). The ...gure obtained in this way is then de‡ated by the materials and services de‡ator. Given gross output and materials input, value added is constructed as a Tornqvist/Divisia index.

# 5 Sources of Welfare Di¤erences

Welfare change depends on the expected present discounted value of expected TFP growth as shown by equations (18) and (20). It is therefore important to investigate the time-series property of TFP growth. We do so in Table 1, using annual data from EU-KLEMS up to 2005 for the entire private economy for Belgium, France, Great Britain, Italy, and Spain. We use the measure of TFP developed in EU-KLEMS, based on the assumption of zero pro...ts and time varying distributional shares and present both country by country and pooled results. The persistence of TFP growth is a key statistic, since it shows how the entire summation of expected productivity residuals changes as a function of the new information about the TFP growth rate. For most countries the log level of the TFP index is well described by an AR(1) stationary process around a country-speci...c linear trend. Additional lags of log TFP are not signi...cant and the residual is white noise, as suggested by the Lagrange Multiplier test for residual serial correlation. The only exception is Spain, where

> 9 The use of a ...ner sectoral disaggregation is questionable if one wants to have enough ...rms in each sector for estimation purposes.

16
the coe¢ cient of log TFP (t-3) is signi...cantly di¤erent from zero at the 5% level and the LM test rejects the hypothesis of no serial correlation (up to the third order). Thus, for most countries the growth rate of TFP is well described by an ARMA(1,1) model. We henceforth focus on the current TFP growth rate, since for most countries the data do not reject the proposition that the current growth rate (or its innovation) gives all necessary information about the entire future path of TFP, and hence welfare.

We ...rst ask which sectors contributed the most to welfare change in these countries over the period of our study through their contribution to aggregate TFP growth. The results are in Table 2. We look at the contributions of ...ve major industry groups: Manufacturing, Utilities, Construction, Wholesale and Retail Trade and FIRE. For each country, we present in line 1 the mean of the Tornqvist index of TFP growth for these industries, which represent the overwhelming majority of private output. Interestingly, average TFP growth over this period is less than 1 percent per annum, even for the leading economies, France and Britain. The sectoral decompositions are also interesting. The next ...ve lines give average sectoral TFP growth rates (not growth contributions, which would multiply the growth rates by the respective sectoral weights, and give a mechanical advantage to large sectors). Manufacturing makes a positive contribution for each country. The contribution of FIRE (Finance, Insurance, Real Estate), on the other hand, is often negative, especially in Britain, which has become a ...nancial hub for the world.<sup>10</sup> But the humble utility sectors are the largest source of productivity growth on average (in every country other than Italy). Alesina et al (2005) suggest an explanation for this pattern based on deregulation of the utilities sectors in many European countries (with Italy a laggard in terms of the timing and pace of deregulation).

In Table 3, we look at the contributions of di¤erent groups of ...rms to welfare growth, now using our ...rm-level data from Amadeus for these countries. We now look at the average TFP growth rates of small and large ...rms, from 1998-2004. No very clear pattern emerges. Large ...rms have higher TFP growth rates in two countries (Belgium and Spain); small ...rms have higher growth rates in two others (Italy and Great Britain), and the contributions are basically identical in the remaining country, France.

We can further decompose productivity di¤erences across countries by applying the following decomposition, based on Griliches and Regev (1995). We wish to ask whether the di¤erence in the productivity growth rate of any pair of countries is due to di¤erences in their sectoral compositions or to di¤erences in the growth rates for each sector. Let i now index sectors (not ...rms) and C be one of the countries in our sample other than the UK.

> 10 However, measures of both nominal and real ...nancial sector output are often unreliable. See Wang, Basu and Fernald (forthcoming) for a model-based method for constructing ...nancial sector output. Basu, Inklaar and Wang (forthcoming) apply this theory to construct nominal bank output measures, and Inklaar and Wang (2007) provide a theory-consistent measure of real bank output in the United States.

17
In Table 4, we examine the results of the Griliches-Regev (1995) decomposition, investigating the sources of growth of each country’s TFP relative to that of Britain, which is the TFP growth leader over our period. The ...rst column describes the di¤erence in productivity change between Great Britain and the other economies in our sample (and is of course negative in all cases). The second column gives the amount of the di¤erence accounted for by cross country di¤erences in TFP growth for each sector, while the third column gives the amount of the di¤erence due to di¤erences in industrial structure (the share of each industry in the aggregate for that country). In most cases, cross country di¤erences in the growth rate of the same sector account for the great majority of the gap with the UK. The exception is France, which actually grows faster than Britain comparing the same sector in the two countries, but loses nearly two-tenths of a percentage point of TFP growth per year due to di¤erences in industrial structure.

In Table 5, we do an exercise designed to show whether the productivity patterns in each country are related to cross-country di¤erences in the shape of the distribution of productivity growth rates across ...rms. This is an exercise in the spirit of Hsieh and Klenow (2009). However, Hsieh and Klenow expended considerable e¤ort (and had to make a number of strong assumptions) in order to isolate ...rm-level technology within each country-sector. Our results show that if the object is to investigate the reasons for di¤erences in welfare change across countries, it is not necessary (and indeed not su¢ cient) to understand how technology di¤ers across ...rms; we should concentrate on di¤erences in the Solow residual instead. We do the following exercise. For our full sample of ...rms within each country, we calculate TFP, and then divide ...rm-level TFP growth by TFP growth for the aggregate of the ...rms in that country. We then divide the range of productivity growth rates into 10 bins, and ask what percentage of ...rm value-added is produced by ...rms in each standardized productivity decile. (We experimented with dividing the range of growth rates more ...nely, into 20 bins, with qualitatively similar results.) Finally, we ask how much faster or slower aggregate TFP would have grown if the standardized distribution for the country had been replaced by the standardized distribution for Great Britain.

The results are in Table 5. For ease of viewing the results, we also plot the distributions for each country and the distribution for Britain in Figure 1. We ...nd that replacing the distributions in Belgium and Spain with the British distribution would actually have caused those two countries to grow slightly more slowly. However, the same exercise for France and Italy shows that those two countries would each have had half a percentage point higher TFP growth per year over the full six years. This is a signi...cant di¤erence, especially for Italy where it approximately doubles the annual TFP growth for our aggregate of ...rms. Thus, there is some evidence that a portion of 18
the TFP growth di¤erences relative to Britain, which is the probably the least regulated and most "US-like" of the countries in our sample, is driven by di¤erences in institutions that allow weak ...rms to linger or prevent strong ...rms from expanding. The evidence is particularly strong in the case of Italy, which has been a conspicuous laggard in its rate of productivity growth over the last decade.

# 6 Decomposing the Productivity Residual: The Role of Reallocation and Technology

The great bene...t of an index-number approach, such as the one we take in the previous section, is that it provides interesting results without requiring formal econometrics. The cost is that we cannot then identify the components of productivity growth, such as technical change or scale economies. Having established that aggregate TFP is the natural measuring stick for aggregate welfare, we now proceed to decompose aggregate TFP into components. We choose to work in growth rates, since there are well-known di¢ culties in comparing TFP levels across industries and countries. As we noted, while TFP growth is itself meaningful in welfare terms without any additional assumptions, we need to make assumptions about ...rm technology and behavior in order to decompose it in a meaningful way. We use the decomposition of Basu and Fernald (2002), which is derived by assuming that ...rms minimize costs and are price-takers in factor markets, but may have market power for the goods they sell and might produce with increasing returns to scale. Some of the components in the decomposition we use can be clearly identi...ed as being due to reallocation, since they depend on marginal products of identical inputs not being equalized across ...rms. Other components depend on aggregate distortions, such as the average degree of market power and various tax rates.

## 6.1 Summary of the Basu and Fernald decomposition

Following Basu and Fernald (2002), in this paragraph we decompose changes in aggregate productivity into changes in aggregate technologies and changes in three non-technological components re‡ecting imperfections and frictions in output and factor markets. Suppose that each ...rm i has the following production function: where Qi is the gross output, Ki; Li and Mi are inputs of capital, labor and materials, Ti<sup>Q</sup> is a technology index and F<sup>i</sup> is an homogenous function. Assume that ...rms are price takers in factor markets but have market power in the output markets. Call PJi the price for factor J faced by ...rm i and �<sup>Q</sup> i<sup>themarkupthat...rmiimposedovermarginalcosts.ForanyinputJ,letF i</sup> J<sup>bethe</sup> marginal product. Firm i’s ...rst order condition implies: 19
Output growth, d log Qi, can be written as: where s<sup>Q</sup> J;i<sup>istherevenueshareofinputJoutofgrossoutput,d log T Q</sup> i denotes technology growth and d log Xi<sup>Q</sup> is revenue share weighted total input growth. Remember that our ultimate goal is decomposing the aggregate Solow residual. In the national account identity in closed economy, total expenditure equals the sum of ...rms’value added. Consider the standard Divisia index of ...rm level value added: and de...ne the change in aggregate primary inputs, d log Xi, as the share-weighted sum of the growth rates of capital and labor: After some algebra, taking into account that the ...rms’value added productivity residual d log pri equals d log Yi � d log Xi, we obtain: where: Let us move now to aggregate quantities. De...ne aggregate inputs as the simple sums of ...rmlevel quantities: K =<sup>PI</sup> i=1<sup>KiandL = PI</sup> i=1<sup>Li.</sup> Now de...ne aggregate output growth as a Divisa index of ...rm level value added: where wi is ...rm i’s share of nominal value added: wi = Pi<sup>YYi=P Y Yandde...neaggregateprimary</sup> input growth as: 20
where sJ is the share of input J out of total value added. After some algebraic manipulation, d log X can be written in terms of the weighted average of ...rm level primary input growth: d log X = PIi=1<sup>wid log Xi. Aggregate productivity growth, d log pr, is the di¤erence between aggregate output</sup> growth d log Y and aggregate inputs growth d log X. Basu and Fernald shows that after some manipulations, d log pr can be decomposed in the following way:<sup>11</sup> where: It is easy to provide an intuition for the welfare relevance of each term in which we have decomposed aggregate productivity. The ...rst term, <u>(�</u> � 1)d log X; is a direct consequence of imperfect competition. Consumers would prefer to provide more labor and capital and consume the extra goods produced, since their utility value exceeds the disutility of producing them. Hence aggregate productivity and welfare increases with aggregate primary input growth, and this is true even if ...rms have the same markup. In this sense, <u>(��1)d log X</u> re‡ects an aggregate distortion and should not be counted as part of "reallocation," which we use as shorthand for allocative e¢ ciency.

The third term, R�; represents the increase in productivity and welfare coming from the fact that primary inputs are directed towards ...rms with higher-than-average markups, since higher prices and markups express higher social valuation.

The terms <u>(��1)d log M=Q and RM</u> re‡ect the fact that a markup greater than one reduces the use of materials as well as primary inputs below the socially optimal level. This distortion is greater the greater is the markup. Note that if materials had to be used in ...xed proportion to output, d log Mi � d log Qi would equal zero and so would both <u>(�</u> � 1)d log M=Q and RM . (In other words, the distortions regarding primary inputs would summarize fully the distortions in input use due to markups that exceed one.) More speci...cally, <u>(�</u> � 1)d log M=Q re‡ects the distortion generated by an average markup above unity and RM re‡ects reallocation across ...rms with di¤erent markups (relative to <u>�).</u> Only the latter should be counted as part of reallocation. Finally the term d log T

> 11 We are assuming here that the price paid by each ...rm for capital and labor is the same. If it is allowed to di¤er, Basu and Fernald (2002) show that two additional terms should be added to the right hand side of (34): RK � <u>�</u><sup>PI</sup> i=1<sup>wisK;i</sup> h PKiPKi�PK i d log Ki and RL � <u>�</u><sup>PI</sup> i=1<sup>wisL;i</sup> h PLiPLi�PL i d log Li. These input reallocation terms represent gains from directing primary inputs towards ...rms where they have higher social valuation.

21
represents the contribution to productivity and welfare of changes in aggregate technology.

The Basu and Fernald decomposition can be extended by disaggregating R� into a within sectors and between sectors component. This is useful in assessing whether the gain from reallocation (if any) occur because resources are reallocated across industries or within industries across ...rms. Basu and Fernald used industry level data in their empirical exercise so they could at best evaluate the between component. (We say "at best" because if there are within-industry reallocation terms, then Basu and Fernald’s estimation using industry-level data would not give a consistent estimate of even the average industry markup, <u>�.</u> In general, one can estimate <u>�</u> correctly only by taking the average of ...rm-level markups, estimated using ...rm-level data.) If one uses ...rm-level data, one can discuss the relative importance of the within and between components. RM can also be decomposed into a within and between component, but there is a residual term.

Let P<sup>Y J</sup> Y<sup>J</sup> =<sup>P</sup> i�J<sup>P Y</sup> i<sup>Y</sup> i<sup>Jbe the total value added produced in industry J, wJ= P Y JYJ=P Y Y</sup> the share of industry J out of aggregate output and wi<sup>J</sup> = Pi<sup>YY</sup> i<sup>J=P Y JYJtheshareofvalue</sup> added of ...rm i in industry J. Denote with Qi a ...rm gross output and with Pi<sup>Q</sup> its price. Then wi<sup>QJ</sup> = Pi<sup>QQ</sup> i<sup>J=P QJQJ,whereP QJQJ= P</sup> i�J<sup>P Q</sup> ii<sup>QJ;representsthe...rmshareofindustrygross</sup> output. Finally, the primary inputs growth in industry J is d log X<sup>J</sup> = s<sup>J</sup> K<sup>d log KJ+ sJ</sup> L<sup>d log LJ,</sup> <u>Pi�J</u><sup>PK;iKi</sup> <u>Pi�J</u><sup>PL;iLi</sup> where s<sup>J</sup> K<sup>=</sup> P<sup>Y J</sup> Y<sup>J</sup> and s<sup>J</sup> L<sup>=</sup> P<sup>Y J</sup> Y<sup>J</sup> . De...ne R�<sup>JandR</sup> �<sup>Jastheindustryequivalentofthe</sup> reallocation terms R� and RM when aggregating over industry J rather than the entire economy, i.e. R�<sup>J=P</sup> i�J<sup>w</sup> i<sup>J</sup> ��i � <u>�</u><sup>~~J~~�</sup> d log Xi and RM<sup>J=P</sup> i�J<sup>w</sup> i<sup>J</sup> ��i � <u>�</u><sup>~~J~~�</sup> 1�s<sup>Q</sup> Ms<sup>Q</sup> M;i;i<sup>(d log Mi �d log Qi),where</sup> <u>�</u><sup>~~J~~</sup> =<sup>P</sup> i�J<sup>w</sup> i<sup>J�</sup> i<sup>.Wecandecomposethereallocationtermforprimaryinputs,R�;intoawithin</sup> and a between component (denoted by superscripts W and B, respectively) as follows: where R�<sup>W= P</sup> J<sup>K</sup> =1<sup>wJR</sup> �<sup>Jand R</sup> �<sup>B= P</sup> J<sup>K</sup> =1<sup>wJ �</sup> <u>�</u><sup>~~J~~</sup> � <u>��</u> d log X<sup>J</sup> :Note that the between component can be calculated on the basis of industry data only.

The decomposition for the reallocation term for materials, RM , is instead: where RM<sup>W= PK</sup> J=1<sup>wJR</sup> M<sup>J; R</sup> M<sup>B= PK</sup> J=1<sup>wJ �</sup> <u>�</u><sup>~~J~~</sup> � <u>��</u> (d log Q<sup>J</sup> �d log Y<sup>J</sup> ) and Rw�wQ =<sup>PK</sup> J=1<sup>wJ �</sup> <u>�</u><sup>~~J~~</sup> � <u>��</u> ( wi<sup>J�w</sup> i<sup>QJ</sup> )d log Qi: In the between component, d log Y<sup>J</sup> =<sup>P</sup> i�J<sup>w</sup> i<sup>Jd log YiistheDivisiaindex</sup> of industry value added,. d log Q<sup>J</sup> =<sup>P</sup> i�J<sup>w</sup> i<sup>QJ</sup> d log Qi is the Divisia index of gross output (using wi<sup>QJ</sup> as weights). The residual term, Rw�wQ, re‡ects the di¤erence between value added weights and gross output weights in aggregating ...rm level gross output within an industry.

# 7 Econometric Framework

The modi...ed Solow productivity residual can be essentially calculated from the data and requires no estimation if the distributional shares are observable (or if we observe the labor share and 22
assume approximately zero pro...ts). However, in order to break down the productivity residual into components that re‡ect aggregate distortions, reallocation and technology growth we must obtain estimates of the markups and of technology growth. We will do that by assuming that the (gross) production function in sector j is Cobb Douglas: where i denotes ...rms (i = 1; ::; Ij), t time (t = 1; ::; Tj), and small case variables logs. �jt is an industry speci...c common component of productivity, �i a time invariant ...rm level component and !it an idiosyncratic component. In our application using the Amadeus data set, Tj is small and Nj large.

We will experiment with di¤erent estimation methods: OLS, LSDV, Olley and Pakes, Di¤erence and System GMM (assuming that !it is either serially uncorrelated, or that it follows an AR(1) process). The advantages and disadvantages of each choice are well known, although there is no agreement on which estimator one should ultimately choose. One fundamental estimation problem is the endogeneity of the input variables, which are likely to be correlated both with �i and !it. Correlation with !it may re‡ect both simultaneity of input choices or measurement errors. Given the shortness of the panel, elimination of �i through a within transformation is not the appropriate strategy. Di¤erencing of (35) and application of the di¤erence GMM estimator (Arellano and Bond (1991)) is a possibility, but appropriately lagged values of the regressors may be poor instruments if inputs are very persistent. Application of the GMM System estimator (Blundell and Bond (1998) and Blundell and Bond ( 2000) is probably a better option. An alternative approach is the one proposed by Olley and Pakes (1996). This estimator addresses the simultaneity (and selection) problem by using ...rm investment as a proxy for unobserved productivity and requires the presence of only one unobserved state variable at the ...rm level and monotonicity of the investment function. We are not interested to take a stand in this paper on which one is the preferable estimation strategy. Fortunately for us, the results of the decomposition are insensitive to the choice of a particular estimator.

Having obtained estimates of the output elasticity for each factor we will recover the ...rm speci...c markup from the ...rst order conditions for materials, equation (33). In the Cobb Douglas case, this can be expressed as: where s<sup>Qthetimeaverageofthe...rmspeci...crevenueshareofmaterialsfor...rmi:Ahat</sup> M;i<sup>is</sup> denotes estimated values. We have chosen to focus on the FOC for materials because they are likely to be the most a ‡exible input. Whereas the labor share, s<sup>Q</sup> L;i<sup>;canbeeasilyrecoveredfrom</sup> the data, the same is not true for the capital share, s<sup>Q</sup> K;i<sup>,unlessoneiswillingtomakeassumptions</sup> about the user cost of capital, which is problematic in the presence of ...rm heterogeneity in the cost of ...nance. We have recovered the capital share from estimates of the markup described above and 23
of the elasticity of output with respect to capital, using: Alternatively we have obtained s<sup>Q</sup> K;i<sup>from:</sup> where<sup>b</sup> �j = b"jK<sup>+b"j</sup> L<sup>+b"j</sup> M<sup>isthedegreeofreturnstoscaleinsectorj.Theresultarerobusttothis</sup> choice.

# 8 Results

We will discuss now the empirical results obtained when the production function is estimated on the ...rm level data contained in Amadeus for Belgium, France, Great Britain, Italy, Spain over the period 1998-2005. To avoid overburdening the reader, we report results for selected estimators (OLS, System GMM, and Olley and Pakes) for only one of our countries, Belgium.

The estimation results for the elasticity of output with respect to each factor, for constant returns to scale and for average markups are reported in the tables 6, 7, and 9: Estimates are pretty standard and vary somewhat across estimators. Recall that materials include services together with materials and intermediates. The degree of returns to scale is very close to one in most sectors using OLS and System GMM, while it is slightly smaller, but still close to one, with the Olley and Pakes estimator. The estimate of "<sup>j</sup> K<sup>is greater for the OLS estimator and the smallest for the Olley</sup> and Pakes estimator. For ...ve sectors it is negative using the GMM System estimator with serially uncorrelated errors, although not signi...cantly so. The test of overidentifying restrictions and the test of second order serial correlation for the GMM System do not suggest major misspeci...cation issues for most sectors, which leads us to focus on this version of the GMM estimator, instead of the one allowing for an AR(1) error component in the level equation. The average estimated markup, obtained using (36), exceeds one in all sectors, whatever the estimator used. Moreover it is strictly greater than one for 64% of ...rms, using OLS, 70% using System GMM, and 63% using Olley and Pakes.

We ...nd markup estimates that are quite reasonable compared to existing estimates in the microeconometric literature<sup>12</sup> , albeit somewhat high relative to the macro literature. The numerical estimates in Tables 6 through 9, usually in the range of 1.10 to 1.25, seem quite small, but one needs to remember that these are markups on gross output. Converting to markups on value added using a representative materials share of 0.7, the markups are in the range of 1.43 to 3. Similarly, the implied pro...t rates are a bit on the high side. Using equation 38, the pro...t rate can

> 12 For example, Dobbelaure and Mairesse (2008) ...nd very similar markups using panel data for French ...rms.

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bj <u>�</u> be calculated as (1 � b<sup>Takingconstantreturnsasourmodalestimate,themarkuprangejust</sup> �<sup>Q</sup> i<sup>).</sup> discussed corresponds to pro...t rates in the range of 9 to 20 percent, expressed as a percentage of gross output.

Our estimates of the markup and thus of the pro...t rate are probably upper bounds. We do not control for variations in ...rm-level input utilization (changes in the number of shifts or variations in labor e¤ort), except through our use of time ...xed e¤ects. Thus, we remove variations in utilization due to common industry e¤ects but not due to ...rm-speci...c demand variation over time. Basu (1996) suggests that variable utilization is likely to bias upward the output elasticity of materials in particular, which is the parameter that has the largest impact on our estimates of markups and pro...t rates. Unfortunately, we do not have the ...rm-level data on hours worked per employee that would be necessary to implement the utilization control derived from the optimizing model of Basu, Fernald and Kimball (2006). Thus, our estimates of the average distortions coming from markup pricing, as summarized by the ...rst two terms in equation (34) are likely to be on the high side. But the fact that our estimated average markups are large does not create any particular bias in our estimates of the reallocation terms, which are our particular focus, since the reallocation terms involving markups depend on the gaps between ...rm-level and average markups.

In light of this discussion, it is interesting to look at the estimates of the various reallocation terms for our sample of six countries, which are presented in Tables 10 and 11. In Table 10 we report for each country in our sample, average productivity growth, d log pr, the sum of aggregate distortions, <u>(�</u> � 1)d log X + <u>(�</u> � 1)d log M=Q, the sum of the reallocation terms for primary factors and materials, R� + RM , and technology growth, d log T . The last column reports as residual the di¤erence between productivity, on the one hand, and the sum of aggregate distortions, of the reallocation terms, and of technological progress, on the other, i.e. the di¤erence between the left hand side and the right hand side of (34). This equation may not hold as an equality for three reasons: ...rst we do not observe the true value of the markup, but only its estimated value; (ii) whereas the labor share is observed in the data, calculations of the capital share depends upon a zero pro...t assumptions or an estimate of the markup and of the degree of returns to scale; (iii) as Basu and Fernald (2002) show, if the price paid for capital and labor di¤ers across ...rms, additional terms involving the di¤erence of factor prices for each ...rm from the average, multiplied by each factor growth rate will appear on the right hand side of (34).<sup>13</sup> First of all, we see from Table 10 robust average annual productivity growth for all countries in our sample of large ...rms. The case of Italy is particularly striking, since our sample of ...rms has an average productivity growth rate, d log pr, of 2.8 percent, while the EU-KLEMS database shows that for all of Italian manufacturing average TFP declined at a rate of 1.2 percent per year over our sample period. Second, we see that technical change was also positive for all countries, and

> 13 See footnote 11. Petrin, Reiter and White (2009) argue that changes in ...xed costs create yet another gap between the two sides of equation (34). However, changes in ...xed costs are equivalent to an additive technology shock, and to a ...rst-order approximation both additive and multiplicative technology shocks are already incorporated into the estimate of dlogT . Thus, changes in ...xed costs are not an additional gap between productivity growth and technological change.

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over 1 percent per year in all countries except Spain, where it averaged 0.5 percent. The strongest rates of technical change, in excess of 4 percent per year, were registered in France, which is usually found to be a high-productivity country in most cross-country studies, and in the United Kingdom, which had 2.2 percent average TFP growth in manufacturing over this time period.

Before discussing the results on reallocation, note that the residual is sizeable and we decide to allocate it to the aggregate distortion, reallocation, and technology growth component in proportion to their relative size. In Table 11 we report the proportion of aggregate productivity accounted for by each component, after this adjustment. The results suggest, ...rst, that in most countries most of productivity growth is accounted by technology growth. More speci...cally, technological progress accounts for the totality of productivity growth in Great Britain and in France, for a large fraction in Italy (.66%) and for a sizeable, but smaller fraction in Belgium and Spain (43% and 21% respectively). Second, aggregate distortions are quite important in Spain Belgium, and Italy, where they account for 85%, 55%, and 33% of productivity growth respectively. They are, instead rather small in Great Britain and in France. The reallocation terms for primary factors or materials accounts for a small proportion of productivity growth in all countries.<sup>14</sup> It follows that, unless one is willing to treat the entire residual as part of reallocation term, factor reallocation does not appear to be an important component of productivity growth.<sup>15</sup> Here the nature of the sample may work against ...nding strong results, since most of the ...rms are quite large in all the years they are observed. Reallocation e¤ects are most clearly apparent when ...rms that are small initially grow to a large size due to their superior productivity. There are probably fewer such ...rms in our sample than in the population, thus reducing the quantitative impact of reallocation. Petrin, White and Reiter (2009) come to the di¤erent conclusion that reallocation represents a large fraction of productivity growth, using manufacturing plant level data for the US. They calculate their reallocation term as the di¤erence between a Divisia index of ...rm level productivity growth and a Divisia index of technology growth. Thus, they include aggregate distortions as part of reallocation, which should not be the case if one wants to estimate and index of allocative e¢ ciency strictly de...ned. We also ...nd that aggregate distortions can be substantial for some countries.<sup>16</sup> Finally, although reallocation of factors towards uses where they have a higher social valuation has not been a large part of the improvement in productivity and welfare for the sample period we have analyzed, does not mean that a benevolent central planner could not achieve large welfare improvement from factor reallocation. This distinction between the historical decomposition we have presented and what could be potentially obtained should be kept in mind when drawing inferences from these results.

> 14 Because the reallocation term is so small, not much is learned from presenting its the decomposition in a within and between component.

> 15 If we treat the residual as re‡ecting the di¤erence in primary factor prices faced by ...rms and treat it entirely as part of the reallocation term, as in Basu and Fernald (2002), reallocation would account for approximately a third of productivity growth in Great Britain.

> 16 Petrin, White and Reiter (2009) implement the decomposition proposed by Petrin and Levinsohn (2008) which is a variant of the Basu and Fernald (2002) decomposition.

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# 9 Conclusions

We show that the present value of aggregate TFP growth, for a given initial endowment of capital, is a complete welfare measure for a representative consumer, up to a ...rst-order approximation. This result rigorously justi...es TFP, rather than technical change or labor productivity, as the central statistic of interest in any exploration of productivity, at all levels of aggregation. Importantly, the result holds even when TFP is not a correct measure of technical change, for example due to increasing returns, externalities, or imperfect competition. It also suggests that productivity decompositions should be oriented towards showing how particular features or frictions in an economy either promote or hinder aggregate TFP growth, which is the key to economic welfare. Our theoretical results point to a key role for the persistence of aggregate TFP growth, since welfare change is related to the entire expected time path of productivity growth in addition to the current growth rate. Finally, our derivation shows that in order to create a proper welfare measure, TFP has to be calculated using prices faced by households rather than prices facing ...rms. In modern, developed economies with high rates of income and indirect taxation, the gap between household and ...rm TFP can be considerable.

We use these central results to show that one can explore the sources of welfare change using both non-parametric index numbers and formal econometrics. The non-parametric approach has the great advantage of simplicity, and avoids the need to address issues of econometric identi...cation. Many interesting cross-country comparisons can be performed using the index-number approach, including calculating summary statistics of allocative e¢ ciency for each country based on ...rmlevel data. However, if one wants to ask how much of aggregate TFP growth is due to technical change, as opposed to scale economies or allocative ine¢ ciency, one does need to make additional assumptions and estimate production functions at the ...rm level. We show how one can decompose aggregate TFP growth in such a manner using ...rm-level data.

The results suggest that in the majority of OECD countries we have analyzed (Belgium, France, Great Britain, Italy, and Spain) most of productivity growth in manufacturing is accounted for by technology growth. This is particularly true for Great Britain and France. Moreover, aggregate distortions are quite important in many countries, such as Spain, Belgium, and Italy. Finally, the reallocation terms for primary factors or materials account for a small proportion of productivity growth in all countries over the period 1995-2005. We will explore in future research whether this result extends to other countries or time periods, or to other data sets less biased towards larger ...rms.

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