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Suppose we want to estimate for many goods in the economy (or classifications thereof) the preferences of each household for these goods. We may want to restrict the analysis to specific utility classes, e.g. translog or CES utility. Naturally, we can not obtain the utility function but only the value function.

To my knowledge the Nielsen Datasets from U Chicago seem to provide such a possibility, but have highly restrictive access. What are good alternatives?

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As I understand it, you're trying to estimate a demand system for different goods.

Have a look at Aguiar and Bils (forthcoming AER), who use the consumption expenditure survey to do exactly that. The previous link also contains their data and code.

From their abstract:

We do so by constructing an alternative measure of consumption expenditure, using data from the Consumer Expenditure Survey (CE), that employs a demand system to correct for systematic measurement error. Specifically, we consider trends in the relative expenditure of high-income and low-income households for different goods with different expenditure elasticities.

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