Using the Mirrlees optimal income tax model with quasi-linear preferences, the paper examines conditions for marginal tax rates to be rising at high income levels and declining in an interval containing the modal skill. It examines conditions for the marginal tax rate to be higher at a low skill level than at the high skill level with the same density--an argument only holding for skill levels above a cutoff where resources of a worker are marginally of the same value as resources of the government. Data on earnings rates are presented.

Advancing annuity demand theory, we present sufficient conditions for the optimality of full annuitization under market completeness which are substantially less restrictive than those used by Menahem E. Yaari (1965). We examine demand with market incompleteness, finding that positive annuitization remains optimal widely, but complete annuitization does not. How uninsured medical expenses affect demand for illiquid annuities depends critically on the timing of the risk. A new set of calculations with optimal consumption trajectories very different from available annuity income streams still shows a preference for considerable annuitization, suggesting that limited annuity purchases are plausibly due to psychological or behavioral biases.