Private Money and Reserve Management in a Random Matching Model

We introduce an element of centralization in a random matching model of money that allows for private liabilities to circulate as media of exchange. Some agents, which we identify as banks, are endowed with the technology to issue notes and to record-keep reserves with a central clearinghouse, which we call the treasury. The liabilities are redeemed according to a stochastic process that depends on the endogenous trades. The treasury removes the banking technology from banks that are not able to meet the redemptions in a given period. This, together with the market incompleteness, gives rise to a reserve management problem for the issuing banks. We demonstrate that ``sufficiently patient'' banks will concentrate on improving their reserve position instead of pursuing additional issue. The model provides a first attempt to reconcile limited note issue with optimizing behavior by banks during the National Banking Era.