Unconventional optimal open market purchases

We build a model in which verifiability of private debt and a timing mismatch in debt settlements can lead to a liquidity problem in the financial market. The central bank can respond to the liquidity problem by adopting an unconventional monetary policy that purchases private debts in the open market. This policy is effective if the timing mismatch is nominal (i.e., a settlement participation risk). It is ineffective if the limited participation is driven by a real shock (i.e., preference shock).