Some results on the optimality and implementation of the Friedman rule in the Search Theory of Money

I characterize a large family of monetary policies that implement Milton Friedman's prescription of zero nominal interest rates in a monetary search economy with multiple assets and aggregate uncertainty. This family of optimal policies is defined by two properties: (i) the money supply must be arbitrarily close to zero for an infinite number of dates, and (ii) asymptotically, on average (over the dates when fiat money plays an essential role), the growth rate of the money supply must be at least as large as the rate of time preference.