The sub-optimality of the Friedman rule and the optimum quantity of money

According to the logic of the Friedman rule, the opportunity cost of holding money faced by private agents should equal the social cost of creating additional fiat money. Thus nominal rates of interest should be zero. This logic has been shown to be correct in a number of contexts, with and without various distortions. In practice, however, economies that have confronted very low nominal rates of interest over extended periods have been viewed as performing very poorly rather than as performing very well. Examples include the U.S. during the Great Depression, or Japan during the last decade. Indeed economies experiencing low nominal interest rates have often suffered severe and long-lasting recessions. This observation suggests that the logic of the Friedman rule needs to be reassessed. We consider the possibility that low nominal rates of interest imply that fiat money is a good asset. As a result, agents are induced to hold an excessive amount of savings in the form of money, and a suboptimal amount of savings in other, more productive forms. Hence low nominal interest rates can lead to low rates of investment and, in an endogenous growth model, to low rates of real growth. This is a cost of following the Friedman rule. Benefits of following the Friedman rule include the possibility that banks will provide considerable liquidity, reducing the cost of transactions that require cash. With this trade-off, we describe conditions under which the Friedman rule is and is not optimal. Finally, our model predicts that excessively high rates of inflation, and nominal rates of interest, are detrimental to growth. This implication of the model, which is consistent with observation, in turn implies that there is a nominal rate of interest that maximizes an economy's real growth rate. We characterize this interest rate, and we describe when it is and is not optimal to drive the nominal rate of interest to its growth maximizing level.