Inflationary shocks now lead to delayed increases in interest rates which imply delayed reductions in inflation. The rule then requires that subsequent interest rates fall so that inflation rises once again. For a sufficiently strong reaction of interest rates to lagged inflation, i.e., a high value of a, the resulting oscillations are explosive. Thus, the parameters that minimize the variance of тг in the case of a contemporaneous rule no longer do so when the government can only react with a delay. In particular, this minimization now requires that a be equal to about 15.
Figure 16 which gives the contour plot for the variance of тг when b is set to zero while a and с are allowed to vary tells a similar story. Again, high values of a lead to explosive equilibria. By contrast, high values of с with low values of a, do not. Note that high values of с coupled with moderate values of a mean that the eventual reaction of interest rates to increased inflation is extremely large. Nonetheless, these rules are less destabilizing than having the interest rate respond strongly to inflation after a delay of one quarter.
Even in the case of rules that react with a lag, the stabilization of interest rates continues to require high values of с together with small values of a. The result is that Figure 17 shows that L + тг*2 achieves a minimum for a combination of a and с that is quite similar to the combination that was optimal in the case where the interest rate reacted contemporaneously. Moreover, the minimum value of L 4- 7r*2 within the family (2.3) is obtained for very similar parameters. In particular, it requires that a, b and с be equal to 1.27, .08 and 1.13 respectively.
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This is the rule called E\ in Table 1. Clearly, these parameters are very similar to those (the rule E0) that minimize L + n*2 when contemporaneous data are used. What is more surprising, however, is that Table 1 indicates that the minimized value of L + 7r*2 is very similar in the two cases. In other words, this welfare criterion equals 1.10 when the best contemporaneous rule is used while it equals 1.13 when the best of the rules that respond to lagged values is used. Recall that the units of this welfare criterion are squares of percentage yearly inflation rates. Thus, the difference in loss is equivalent to the difference between having a completely stable annual inflation rate of 1.06 percent per year and having a completely stable annual inflation rate of 1.05 percent per year. This difference is trivial.
This similarity is not surprising once one recognizes that the optimal contemporaneous rules involves a high value of c. This means that, even in the case of contemporaneous rules, most of the reaction of interest rates to an inflationary shock such as an increase in G or a reduction in Ys takes place with a delay. Given this, it is not surprising that the further delay that comes about from responding to inflation and output with a lag has trivial welfare consequences. Prom an economic perspective, what is important is that delayed responses still allow for substantial revisions in long term real interest rates, and it is these which help stabilize inflation.