A random matching environment is constructed where banks mitigate a mismatch between the timing of investment payoffs and when agents wish to consume. Claims on banks may serve as media of exchange, i.e. private money. Two problems can emerge with private money. First, there may exist welfare-dominated equilibria where banks hold low-return assets. Second, private media of exchange may be subject to lemons problems. In spite of these problems, the introduction of fiat money can decrease welfare, as this displaces private money and results in a crowding out of productive intermediation.
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