Countercyclical unemployment benefit extensions in the United States act as a propagation mechanism, contributing to both the high persistence of unemployment and its weak correlation with 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 unemployment dynamics of the past 50 years. In particular, it explains the emergence of jobless recoveries in the 1990's as well as their absence in previous recessions, the low correlation between unemployment and labor productivity, and the apparent shifts in the Beveridge curve following recessions. Next, we embed this mechanism into a medium-scale DSGE model, which we estimate using standard Bayesian methods. Both shocks to unemployment benefits and their systematic component are shown to be important for the sluggish recovery of employment following recessions, in particular the Great Recession, despite the fact that shocks to unemployment benefits account for little of the overall variance decomposition. If we also incorporate other social safety nets, such as food stamps (SNAP), the estimated model assigns an even bigger role to policy in explaining sluggish labor market recovery. We also find that unemployment benefit extensions prevented deflation in the last three recessions, thus acting similarly to a wage markup shock.