The classical Fisher equation asserts that in a nonstochastic economy, the inflation rate must equal the diﬀerence between the nominal and real interest rates. We extend this equation to a representative agent economy with real uncertainty in which the central bank sets the nominal rate of interest. The Fisher equation still holds, but with the rate of inflation replaced by the harmonic mean of the growth rate of money. Except for logarithmic utility, we show that on almost every path the long-run rate of inflation is strictly higher than it would be in the nonstochastic world obtained by replacing output with expected output in every period. If the central bank sets the nominal interest rate equal to the discount rate of the representative agent, then the long-run rate of inflation is positive (and the same) on almost every path. By contrast, the classical Fisher equation asserts that inflation should then be zero. In fact, no constant interest rate will stabilize prices, even if the economy is stationary with bounded i.d.d. shocks. The central bank must actively manage interest rates if it wants to keep prices bounded forever. However, not even an active central bank can keep prices exactly constant.
Karatzas, Ioannis; Shubik, Martin; Sudderth, William D.; and Geanakoplos, John, "The Harmonic Fisher Equation and the Inflationary Bias of Real Uncertainty" (2003). Cowles Foundation Discussion Papers. 1694.