The public sector hires disproportionately more educated workers. Using US microdata, we show that the education bias also holds within industries and in two thirds of 3-digit occupations. To rationalize this finding, we propose a model of private and public employment based on two features. First, alongside a perfectly competitive private sector, a cost-minimizing government acts with a wage schedule that does not equate supply and demand. Second, our economy features heterogeneity across individuals and jobs, and a simple sorting mechanism that generates underemployment - educated workers performing unskilled jobs. The equilibrium model is parsimonious and is calibrated to match key moments of the US public and private sectors. We find that the public-sector wage differential and excess underemployment account for 15 percent of the education bias, with the remaining accounted for by technology. In a counterintuitive fashion, we find that more wage compression in the public sector raises inequality in the private sector. A 1 percent increase in unskilled public wages raises skilled private wages by 0.07 percent and lowers unskilled private wages by 0.06 percent.
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