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This paper computes the optimal progressivity of the income tax code in a dynamic general equilibrium model with household heterogeneity in which uninsurable labor productivity risk gives rise to a nontrivial income and wealth distribution. A progressive tax system serves as a partial substitute for missing insurance markets and enhances an equal distribution of economic welfare. These beneficial effects of a progressive tax system have to be traded off against the efficiency loss arising from distorting endogenous labor supply and capital accumulation decisions. A determination of the optimal progressivity of the income tax code therefore calls for a quantitative exploration. Using a utilitarian steady state social welfare criterion we find that the optimal US income tax is well approximated by a flat tax rate of 19.5% and a fixed deduction of about $3,700. The steady state welfare gains from a fundamental tax reform towards this tax system are equivalent to 0.4% higher consumption in each state of the world. An explicit computation of the transition path induced by a reform of the current towards the optimal tax system indicates, however, that a majority of the population currently alive would suffer welfare losses, calling into question the political feasibility of such fundamental income tax reform.