This relationship can easily be derived from the equilibrium conditions (1.17), (1.20), and (1.21), together with the requirement that
Note that pt is an exogenous stochastic process, that can be expressed as a function of the history of the shocks eit. this
Equation (1.37) represents the only restriction implied by our model on the behavior of Rt given the evolution of Xt. For any given process for Xu the variance of Rt is obviously minimized by setting Rt+\ equal to the right-hand side of (1.37). In the case where one wishes to stabilize prices completely, this means that Rt+\ is given by pt, as discussed in Rotemberg and Woodford (1997).
This means that the interest rate at t + 1 must rise whenever (Gt+1 — VJ+j) increases unexpectedly at t. If, instead, upwards revisions in (Gf+i — y^i) are matched by upwards revisions in Xt+i, Rt+i need not rise as much. In other words, if inflation is allowed to respond to these shocks, the interest rate does not have to respond as much to them.
We propose a simple representation of the quantitative connection between average inflation and the variability of interest rates as in Rotemberg and Woodford (1997). In particular, we suppose that, along any equilibrium path, the lowest possible value of r* (and 7Г*) consistent with a given degree of interest-rate variability is given by
where cr(R) refers to the standard deviation of the unconditional distribution for Rt in the stationary equilibrium associated with a given policy rule. We let the factor к equal 2.26, which is the ratio of the mean funds rate to its standard deviation under the historical regime so that, in effect, we are assuming that this is the minimum possible value for this ratio.22 For any monetary policy rule we consider, we thus compute the variance of the nominal funds rate, and then use (1.39) to determine the associated value of 7r*. We then compare policy rules according to how low a value they imply for the overall deadweight loss measure L + tt*2.