This paper derives a central bank's optimal liquidity supply towards a money market with an unrestricted lending facility. We show that when the effect of liquidity on market rates is not too small, and the monetary authority cares for both interest rates and liquidity conditions, then the optimal allotment policy may entail a discontinuous
reaction to initial conditions. In particular, the model predicts a threshold level of liquidity below which the central bank will not bail out the banking system. An
estimation of the liquidity effect for the euro area suggests that the discontinuity might have contributed to the Eurosystem's tight response to occurrences of underbidding during the period June 2000 through March 2004.