Liquidity backstops have important implications for financial stability. In this paper, we provide a microfoundation for the important role of liquidity backstops in mitigating runs (or, conversely, the role of the lack of liquidity backstops in exacerbating runs) based on a dynamic model of debt runs. We focus on the municipal bond markets for variable rate demand obligations (VRDOs) and auction rate securities (ARS). The different experiences in these markets during the recent financial crisis of 2007-09 provide a natural experiment to identify the value of a liquidity backstop in mitigating runs. Through structural estimation of the model, we show that the value of a liquidity backstop is about 14.5 basis points per annum. The results in this paper shed light on one central difference between shadow banks and traditional banks in terms of their differential access to public liquidity backstops.