As just noted, the finding that the federal funds rate should be very aggressive under the optimal policy depends crucially on the manner in which private agents form their expectations, the subject of exhibit 3. The previous exercise assumed that those expectations are formed under the belief that the FOMC is following its historical policy rule, even if that rule is changed going forward. In practice, private agents are likely to take into account changes in the way policy is implemented.
Consider what happens when the private sector understands the policy rule that the central bank is actually using. As summarized in the top panel, if policymakers are following an inertial policy rule, private agents will expect the initial response of the federal funds rate to a macroeconomic shock to be followed by additional policy changes in the same direction. Moreover, those actions will be expected to unwind only gradually as the shock dissipates. Expectations of this persistent response of policy will be incorporated into current asset prices and economic decisions, thus bringing forward the effects of those future policy actions. As a result, an inertial policy response can have an immediate and sizable impact on economic variables, even with relatively small movements in the federal funds rate in each period. By contrast, the large but transitory movements in the federal funds rate that were found to be optimal in the previous exhibit will be less effective, because private agents will recognize the change in the policy rule and look through the near-term swings in the interest rate.
To illustrate the importance of this consideration, we conduct an experiment in which the public forms its expectations as a weighted average of forward- and backward-looking terms. As the middle panel notes, the degree of forward-looking behavior is governed by a single parameter, φ. When φ equals zero, the model corresponds to the version of FRB/US used in exhibit 2, in which expectations are formed using a “backward-looking” vector autoregression (VAR) model. When φ equals unity, expectations are rational—meaning that households and firms fully understand the structure of the model and the policy rule in forming their expectations.
With this set-up, we can calculate the optimal policy rule for different degrees of forward-looking behavior. As shown in the bottom left panel, the optimal coefficient on the lagged federal funds rate moves higher as the parameter φ increases. That is, the optimal monetary policy rule becomes more inertial when the public is forward- looking, since expectations of future policy actions lead to larger and more-persistent movements in bond rates and other asset prices that effectively counterbalance the persistent effects of macroeconomic disturbances. Note, however, that the historical estimate of the coefficient on the lagged federal funds rate (0.76) is reached only when expectations are formed in an almost completely forward-looking manner. The bottom right panel shows the complete optimal policy rule for three choices of the degree of forward-looking behavior. As just noted, in the case in which φ equals unity, the coefficient on the lagged policy rate is quite high—even higher than that from the estimated rule. Nevertheless, the coefficients on the output gap and inflation are about three times larger than their estimated values. Thus, the optimal rule still calls for much more volatile movements in the federal funds rate than are observed.
Another possible source of gradualism in policy setting is parameter uncertainty, the topic of exhibit 4. In the analysis so far, policymakers have been assumed to know the exact structure of the economy; uncertainty has entered only through additive error terms. To see the effects of this assumption, consider the situation in the top left panel, which shows the relationship between the output gap (plotted on the horizontal axis), and the real interest rate, r, relative to its equilibrium level, r* (plotted on the vertical axis). The solid line represents policymakers’ best estimate of the level of the output gap that would be realized at any given setting of the real interest rate. But because of an additive error term, the realized outcome could lie above or below the central estimate, as indicated by the shaded region. As noted in the top right panel, the amount of uncertainty is not affected by the policy decision. For this reason, it turns out that additive uncertainty has no effect on the optimal policy setting. That is, policymakers should ignore the uncertainty and set policy based on their best estimate of the likely outcome for macroeconomic variables.
However, this framework neglects the obvious fact that policymakers face considerable uncertainty about the values of key parameters in their models of the economy. To see the implications of this type of uncertainty, suppose policymakers are unsure of the value of the policy multiplier, as shown in the middle left panel. The policymakers’ best estimate of the relationship again is represented by the solid line, but the actual slope of the relationship could be higher or lower. In those circumstances, policymakers face more uncertainty about the outcome for the output gap the further the real interest rate deviates from its equilibrium level, as indicated by the shaded region. As we note in the middle right panel, this example highlights the key implication of parameter uncertainty—that uncertainty about future economic conditions will be importantly affected by current monetary policy decisions—a factor that policymakers should take into account in formulating those decisions. Under the objectives assumed, policymakers will tend to shade their policy actions toward choices that reduce uncertainty about future levels of unemployment and inflation. Of course, the middle left panel presents just one specific example of parameter uncertainty. Many other parameters are also unknown, in which case the variance-minimizing policy will not be to hold the real interest rate at its equilibrium level. In general, the effects of parameter uncertainty depend crucially on which parameters are unknown and on the variances and covariances of those parameters.
Unfortunately, the complexity of FRB/US makes it difficult to incorporate parameter uncertainty directly into simulations of that model. But we can quantify the effects of parameter uncertainty using the simpler VAR model utilized to characterize expectations in FRB/US. As summarized in the bottom left panel, the VAR captures the dynamics of key macroeconomic variables, including inflation and the output gap. In addition, the VAR provides a convenient measure of parameter uncertainty—namely, the variance-covariance matrix of the estimated coefficients. We can use this measure to assess the effect of parameter uncertainty on the optimal monetary policy rule.
The results are shown in the bottom right panel in two steps. We first compute the optimal policy rule that ignores parameter uncertainty by assuming that the estimated coefficients from the VAR are known with certainty. As shown in the first line of the table, the optimal policy under those assumptions is much more aggressive than the estimated policy rule, as reflected in the larger coefficients on the output gap and inflation and the smaller coefficient on the lagged policy rate. These findings are qualitatively similar to those found earlier using the FRB/US model. However, when we allow policymakers to recognize that the coefficients of the VAR are uncertain, they choose a policy with smaller coefficients on inflation and the output gap and a larger coefficient on the lagged interest rate. Parameter uncertainty therefore moves the optimal rule in the direction of the estimated rule, but it seems to fall well short of explaining the observed degree of smoothness in the federal funds rate.