With the emergence of electronic money and prepaid cards, the debate about private money has reemerged. Stephen Williamson has long been and advocate of private issuance of money, even the complete privatization of this central bank activity. But he also acknowledges that there can be a lemons problem, as the quality of issued notes may vary. Think for example of gift certificates which may lose their value when the issuing store goes bankrupt or, more often, when they have an expiration date. This may call for some regulation.
Among proposed regulations is the requirement that private money be exchanged at par with "official" money. Clearly, if private money is perfectly credible, this would be an equilibrium outcome anyway. But if there are some doubts about private money issuers, their money should be redeemed at a discount in equilibrium. Imposing parity could clearly constrain the economy away from some optimum. This is the argument of David Mills in the lead article of the latest International Economic Review.
I like very much the money search approach, which is arguably still very crude, but has the immense merit to give true micro-foundations for money, instead of money-in-the-utility, ad hoc nominal rigidities or assumed Phillips curves. It allows to explicit the impact of institutions or policy on agents' behaviors. Search theory can also make statements about welfare.
In this case, the modeling allows to explicit the differences between a regime where market participants are free to determine the price of private money and free to accept it or not. It highlights how much stronger the gambles of accepting private money becomes when its price is fixed and the monitoring of private money issuers are imperfectly monitored. The essential issue is that with parity, price cannot function as signals anymore.