# INTEREST-RATE RULES: Rules that Involve a Lagged Inter est-Rate 3

Even higher values of с imply that initial increases in inflation are followed by such high real rates that the expected long run price level is lower than the initial price level so that j3°° is negative.

For a given initial increase in inflation and interest rates, higher values of с imply that the long run real rate rises more both because future short rates are expected to be higher and because future inflation is expected to be lower. Since unexpected increases in the long term real rate prevent prices from rising this means that, for given a, increases in G (and reductions in Vs) lead to smaller immediate price and interest rate increases the higher is c. This is reflected in Figure 12 which shows that, for each a, the interest rate is less variable the higher is c. It also shows that, not surprisingly, the variance of interest rates increases with a.
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While we have focused on stabilizing the variability of interest rates because of their implication for average inflation, the Fed also seems to be concerned with stabilizing the change in the funds rate from one week or month to the next. This would explain Rudebusch’s (1995) finding that changes in the target rate are followed by further changes in the same direction. Figure 13 thus displays the variance of the change in interest rates in the {a, c} plane. Interestingly, this figure is nearly identical to the figure for the variance of the interest rate itself. Thus, in our model, stabilization of the short term nominal rate is achieved in the same way as stabilization of the quarterly change in this rate.

As we saw in Figure 9, setting с to a very high value destabilizes the inflation rate. In part this is because sufficiently high values of с imply that increases in inflation at t must be matched by reductions in inflation in the future. These predictable movements in inflation both raise the variance of 7r and increase the loss L. For that reason, Figure 14 shows that L-Ь7г*2 reaches its lowest value for a low value of a and a moderate value of c. This minimum is very close to the point which minimizes L + 7T*2 within the family (2.2) since this minimum obtains when a, b and с equal 1.22, .06 and 1.28 respectively. This is the rule labeled Eq in Table 1.