Countercyclical unemployment benefit extensions in the United States act as a propagation mechanism, contributing to the high persistence of unemployment following recent recessions, as well as the weak correlation between unemployment and productivity. We show this by modifying an otherwise standard frictional model of the labor market to incorporate a stochastic and state-dependent process for unemployment insurance estimated on US data. Accounting for movements in both productivity and unemployment insurance, our calibrated model is consistent with post-war labor-market dynamics. It explains the emergence of jobless recoveries in the 1990s, the low correlation between unemployment and productivity, and the apparent shifts in the Beveridge curve following recessions.