The Optimal Distribution of Income Revisited

This paper revisits the optimal distribution of income model in Fair (1971). This model is the same as in Mirrlees (1971) except that education is also a decision variable and tax rates are restricted to lie on a tax function. In the current paper the tax-rate restriction is relaxed. As in Fair (1971), a numerical method is used. The current method uses the DFP algorithm with numeric derivatives. Because no analytic derivatives have to be taken,it is easy to change assumptions and functional forms and run alternative experiments. Gini coefficients are computed, which provides a metric for comparing the redistributive effects under different assumptions. Ten optimal marginal tax rates are computed per experiment corresponding to ten tax brackets. The sensitivity of the results to the four main assumptions of the model are examined: 1) the form of the social welfare function that the government maximizes, 2) the form of the utility function that each individual maximizes, 3) the distribution of ability across individuals, and 4) the rate of return to education. The changes in the Gini coefficient from before-tax income to after-tax income for the experiments are compared to actual changes from various countries. Experiments using a lognormal distribution of ability match the data better than those using a lognormal distribution with a Pareto tail---there is less actual redistribution than a Pareto tail implies. The numerical approach in this paper has advantages over the use of analytic expressions. When functional forms are changed, it may be easier to run a new numerical experiment then use an analytic expression, which can be complicated. Also, although not done in this paper, individual heterogeneity is straightforward to handle. The coding can have a different utility function for each individual. And different assumptions about education can be easily incorporated. The approach also shows the problematic nature of assuming a quasi-linear utility function---a utility function with no income effects.