[영문]We analyze the insurance provided by the U.S. social security and income tax system within a model in which agents receive idiosyncratic, wage rate shocks that are privately observed. We consider two reforms: a piecemeal reform that optimally chooses the social security benefit function and a radical reform that eliminates the entire social insurance system and replaces it with an optimal tax on lifetime earnings. The radical reform outperforms the piecemeal reform and achieves nearly all of the maximum possible welfare gain when wages differ permanently over the lifetime. When wage shocks match properties in U.S. data, the piecemeal reform outperforms the radical reform.