If optimal tax theory is to be the basis for calculating tax rates, a close understanding is required of the relationship between the structure of preferences and the configuration of optimal tax rates. Otherwise hypotheses chosen by the econometrician for practical convenience may completely determine the results, independently of measurement. This paper explores the relationship between various types of separability, particularly weak and implicit separability, and optimal tax rates in the various models discussed in the literature. The use of distance functions and the Antonelli matrix provides a significant unification of previously disparate results. IN THE FINAL ANALYSIS, optimal tax theory should be the basis for actual calculation of tax rates. Although recently there have been great advances in theoretical results and in our understanding of their meaning, we are still some way from a working knowledge of whether uniform commodity taxes are in practice optimal or, if not, which commodities should be discriminated against. Present theoretical formulae do not yield clear-cut results except in special cases and it has recently become clear that optimal rates depend crucially on the detailed structure of consumer preferences. For example, Atkinson and Stiglitz [3] show that with an optimal nonlinear income tax, discriminatory commodity taxes are only necessary to the extent that individual commodities are not weakly separable from leisure. More recently, Deaton [6] has shown that a similar result holds for what is perhaps the most interesting of the standard models, that where there are many consumers and only a linear income tax and proportional commodity taxes are allowed. In this case, separability between goods and leisure, together with linear Engel curves for goods, removes the need for differential commodity taxation. In consequence, nothing can be learned about commodity taxes from consumer demand studies in which commodity demands are explained conditionally on total expenditure and commodity prices and which assume linear Engel curves. All such studies require separability from leisure as a maintained hypothesis and so are consistent with uniform commodity taxation. These results suggest that the prospects for meaningful empirical calculations of tax rates are bleak. Econometricians estimating commodity demand and labor supply equations make generous use of separability assumptions to enable estimation at all. In consequence, it is likely that empirically calculated tax rates, based on econometric estimates of parameters, will be determined in structure, not by the measurements actually made, but by arbitrary, untested (and even unconscious) hypotheses chosen by the econometrician for practical convenience. To remedy this situation, and as a prelude to fruitful empirical work, it is necessary to have a more explicit understanding of how preference structure affects optimal tax rates. Such is the object of this paper. Three different
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