Optimal pricing of intra-day liquidity

It is a puzzling fact that many central banks choose to lend intra-day funds at an interest rate of zero (or very close to zero), while the interest rate on overnight funds is much higher. I build a general equilibrium model where intra-day liquidity is needed because it is costly to make precise the time at which payments are received. If liquidity shocks are uninsurable, a necessary and sufficient condition for an equilibrium to be efficient is that the nominal intra-day interest rate be zero. This is true despite the fact that the overnight nominal rate is strictly positive (the reverse of the discount factor). Since a market intra-day rate will not always be zero, this creates a role for the central bank to supply intra-day liquidity. I allow for the possibility of moral hazard and study policies commonly used by central banks to reduce their exposure to risk. I show that collateralized lending achieves the efficient allocation, while, for certain parameters, caps on borrowing cannot prevent moral hazard.