The Friedman rule and inflation targeting

We use concepts from the financial economics discipline – and in particular the methods of continuous time finance – to develop a monetarist model under which the rate of inflation evolves in terms of a first-order mean reversion process based on a ‘white noise’ error structure. The Fokker–Planck (i.e. the Chapman–Kolmogorov) equation is then invoked to retrieve the steady-state (i.e. unconditional) probability distribution for the rate of inflation. Monthly data for the UK Consumer Price Index (CPI) covering the period from 1988 until 2012 are then used to estimate the parameters of the probability distribution for the UK inflation rate. The parameter estimates are compatible with the hypothesis that the UK inflation rate evolves in terms of a slightly skewed and highly leptokurtic probability distribution that encompasses non-convergent higher moments. We then determine the Hamilton–Jacobi–Bellman fundamental equation of optimality corresponding to a monetary policy loss function defined in terms of the squared difference between the targeted rate of inflation and the actual inflation rate. Optimising and then solving the Hamilton–Jacobi–Bellman equation shows that the optimal control for the rate of increase in the money supply will be a linear function of the difference between the current rate of inflation and the targeted inflation rate. The conditions under which the optimal control will lead to the Friedman rule are then determined. These conditions are used in conjunction with the Fokker–Planck equation and the mean reversion process describing the evolution of the inflation rate to determine the probability distribution for the inflation rate under the Friedman rule. This shows that whilst the empirically determined probability distribution for the UK inflation rate meets some of the conditions required for the application of the Friedman rule, it does not meet them all.