The real interest rate would then fall (because the nominal interest rate responds little) and the resulting increase in output means that expected future inflation is lower than current inflation. Thus the change in the expected future path of inflation that is required to justify the initial change in inflation is consistent with expected future inflation converging back to the target inflation rate ix*. In this case, a stationary rational expectations equilibrium is possible in which such fluctuations occur simply because they are expected to. there
If, instead, a is large and positive, no such equilibrium is possible. Any increase in inflation above its unique saddle-path value is matched by increases in real interest rates which imply that output must fall. This, in turn, implies that expected future inflation rates must be higher than current inflation, given the nature of our AS curve. Thus, inflation must be expected to explode and, since this is not consistent with stationarity, inflation must equal its saddle-path value in the unique stationary equilibrium. Similarly, as mentioned above, the equilibrium is determinate when a and b are both negative.
Figure 4 presents contour lines for the value of our loss measure L + 7r*2 in the regions where equilibrium is determinate. Policy F0 appears as a star on this figure, at the point of a local minimum of the loss measure. However, the region of determinate equilibria with negative a and b also contains a local minimum. This point, which is shown with a star inside a circle, is actually the global minimum value. Nonetheless, we have chosen to present the local minimum F0 in Table 1, on the ground that restricting attention to values a > 0 corresponds to rules that are more similar to the Taylor and Henderson-McKibbin proposals. In addition, once we consider more general families of rules, we do find that the best rules involve tightening monetary policy (i.e., raising the funds rate) in response to inflation increases, as conventional wisdom (at least since the work of Wicksell (1907)) would indicate.
Similar contour plots for other statistics reported in Table 1 provide further insight into why our loss measure varies with a and b as it does. Figure 5 shows the contour plots of the variance of inflation, while Figure 6 shows the contour plots for the variance of (Y — Ys). These figures are essentially identical to each other, and they are both similar to the contour plot for L itself. There is thus no trade-off between stabilizing inflation and stabilizing (Y — У5); the same parameters stabilize both. This follows immediately from our AS curve which relates inflation to departures of Y from Ys. For the ranges considered in our figures, a wheel marks the global optimum for the performance criterion being considered. Thus, the figure shows that these variances become as small as possible when a is at its maximum possible value of 20 while b is set to a small negative number.