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).

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