across countries would require us to adjust for di¤erences in the risk of country portfolios, which is a formidable undertaking. Fortunately, we are able to show that these di¢ culties disappear if we switch to using real absorption rather than GDP as the measure of output.<sup>3</sup> In this case, exactly the same three terms that summarize welfare in the closed economy are also su¢ cient statistics in the open economy. Thus, we can measure welfare change empirically in ways that are invariant to the degree of openness of the economy.

These results pertain to the evolution of welfare in individual economies over time. The indexes we obtain are not comparable across countries. Thus it is natural to ask whether our methods shed any light on a pressing and long-standing question, the measurement of relative welfare across countries using a method ...rmly grounded in economic theory. It turns out that they do. Perhaps our most striking ...nding is the result that we can use data on cross-country di¤erences in TFP and capital intensity, long the staples of discussion in the development and growth literatures, to measure di¤erences in welfare across countries. More precisely, we show that productivity and the capital stock su¢ ce to calculate di¤erences in welfare across countries, with both variables computed as log level deviations from a reference country.

To understand this result, it helps to deepen the intuition o¤ered above. Our analysis is based on a dynamic application of the envelope theorem, and it shows that the welfare of a representative agent depends to a ...rst order on the expected time paths of the variables that the agent takes as exogenous. In a dynamic growth context, these variables are the prices for factors the household supplies (labor and capital), the prices for the goods it purchases (consumption and investment), and beginning-of-period household assets, which are predetermined state variables and equal to the capital stock in a closed economy. Apart from this last term, the household’s welfare depends on the time paths of prices, which are exogenous to the household. Thus, the TFP that is directly relevant for household welfare is actually the dual Solow residual. We use the national income accounts identity to transform the dual residual into the familiar primal Solow residual.<sup>4</sup> Our cross-country welfare result comes from using the link between welfare and exogenous prices implied by economic theory to ask how much an individual’s welfare would di¤er if he faced the sequence of prices, not of his own country, but of some other country. We can perform the thought experiment of having a US consumer face the expected time paths of all goods and factor prices in, say, France, and also endow him with beginning-of-period French assets rather than US assets. The di¤erence between the resulting level of welfare and the welfare of remaining in the US measures the gain or loss to a US consumer of being moved to France. Note that our welfare comparisons are from a de...nite point of view–in this example, from the view of a US consumer. In principle, the result could be di¤erent if the USA-France comparison is made by a French consumer, with di¤erent preferences over consumption and leisure. Fortunately, our empirical results are qualitatively unchanged and quantitatively little a¤ected by the choice of the "reference country" used for these welfare comparisons.

> 3 We are indebted to Mikhail Dmitriev for pointing out this result.

> 4 See Barro and Sala-i-Martin (2004, section 10.2).

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