Making a big contributes to stabilization because it ensures that interest rates rise a lot when either G rises or Ys falls. This ensures that inflation does not rise much in either case and that, at least after the demand for output adjusts to changes in real rates, output does not rise in the former case while it declines substantially in the latter.
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As Figure 7 indicates, the rule that minimizes L by setting a equal to 20 leads to very variable interest rates. This is in part due to the delays in the response of output to interest rates. These delays imply that changes in Gt that become known at t — 1 inevitably change output at t since Ct is predetermined. This leads firms to raise their prices at t unless long term real interest rates rise unexpectedly. With с equal to zero, this means that prices can only be stabilized if the nominal interest rate at t rises a great deal.
The resulting variability of interest rates then requires a high average inflation rate for interest rates never to be negative. This high inflation is so costly, at least relative to the benefits of the additional stabilization that is possible with a high value of a, that the contour plots for the variance of the interest rate are essentially identical to the contour plots for L + 7Г*2. The point that minimizes the variance of interest rates has a sufficiently stable inflation to be quite desirable as far as total welfare is concerned.
It is interesting to note that the stabilization of output requires a quite different set of parameters. This is demonstrated in Figure 8 which gives the contour plots for the variance of output. This variance is reduced by keeping a small and positive while making b very large. Not surprisingly, output is stabilized if the real interest rate is raised significantly by the central bank whenever output rises while it is lowered when output declines. What is interesting here is that the effect of the policy parameters on the variance of (Y — Ys), which are essentially the same as the effects on L, are very different from the effects on the variance of Y. The reason is that the VAR of Rotemberg and Woodford (1997) identifies large short run fluctuations in Ys. As long as these are treated as variations in the welfare maximizing level of output, setting b large is not desirable and, indeed, stabilization of (Y — Ys) requires that b be negative at least when a is 20.
Even higher values of a reduce the variance of (Y — Ys) still further. Obviously, the result that the stabilization of У relative to Ys requires very different policies from those that stabilize output relative to trend is very sensitive to the assumption that our estimate of Ys is indeed the welfare maximizing level of output. This conclusion would presumably change dramatically if movements in Ys were viewed as resulting from changes in distortions such as changes in desired markups. From an empirical point of view these two interpretations may be difficult to disentangle because we identify Ys by measuring shifts in the empirically estimated aggregate supply equation given by (1.22). Unfortunately, changes in desired markups will shift this equation just as much as changes in technology or other changes in the welfare maximizing level of output.