# INTEREST-RATE RULES: Rules that Involve a Lagged Inter est-Rate 2

The result is that the economy stays on a non-explosive path in which increases in inflation are matched by subsequent reductions in inflation which ensure that the interest rate does not explode. In fact, higher values of с actually increase the range of values of a for which a determinate equilibrium exists, by helping to solve the problem of indeterminacy discussed above.

The figure also shows that, within the range being considered, the goal of inflation stabilization is furthered by setting a to as large as possible. The variance of inflation reaches its minimum value (over the range of rules shown in the figure) when a equals 20 and с takes a positive value less than one. If the range of the figure were extended, the optimum would involve even higher values of a. Thus, the key to inflation stabilization remains making sure that the interest rate reacts vigorously to inflation.

Interestingly, a higher value of с turns out to be better if one seeks to stabilize the long run price level. This can be seen in Figure 10 which shows that, for any given value of a, the variance of Ap°° reaches a minimum of zero for с equal to one. Further insight into this behavior of the variance of Ap°° can be obtained from Figure 11 which shows (3°° as a function of a and c. This figure shows that, when с is zero. j3°° is greater than one so that initial increases in inflation are followed by further inflation.
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The reason for this is that an increase in Gt raises the price level at t somewhat in spite of the increase in interest rates that takes place at t. But, unless the price level continues rising, interest rates would immediately return back to their steady state level. The result is that, in equilibrium, prices do keep rising because the initial increase in prices means that marginal cost has gone up for the firms which did not raise their price at t. Consequently, increases in the price level at t are followed by further increases in prices which, admittedly, are kept somewhat in check by the fact that the interest rate remains somewhat above the steady state for some time.

If, instead, с is made higher, the interest rate tends to stay high after an increase in G even if the price level ceases to rise. This means that firms can be induced not to change their prices in the aftermath of an increase in G. The result is that initial increases in prices are followed by smaller increases so that is smaller and the variance of Др°° falls. Setting с equal to one as suggested by Fuhrer and Moore (1995) makes p°° equal to zero so that the shocks have no effect on the long run price level.