A duality theory of payment systems

We model the Central Bank's management of intraday liquidity in modern real-time gross settlement systems as a linear programming problem parameterized by different intraday monetary policies, such as reserve requirements, net debit caps and Lombard loans. We then use duality theory to determine the shadow-prices of constraints of each bank. These shadow-prices can be used by the Central Bank to set personalized intraday monetary policies in order to reduce idleness of money and to give a microfoundation of the too-big-to-fail policy.