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Following the crisis of 2008, several central banks engaged in a new experiment by setting negative policy rates. Using aggregate and bank level data, we document that deposit rates stopped responding to policy rates once they went negative and that bank lending rates in some cases increased rather than decreased in response to policy rate cuts. Based on the empirical evidence, we construct a macro-model with a banking sector that links together policy rates, deposit rates and lending rates. Once the policy rate turns negative, the usual transmission mechanism of monetary policy through the bank sector breaks down. Moreover, because a negative policy rate reduces bank profits, the total effect on aggregate output can be contractionary. A calibration which matches Swedish bank level data suggests that a policy rate of -0.50 percent increases borrowing rates by 15 basis points and reduces output by 7 basis points.