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.

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