Optimal Monetary Policy and the Taylor Rule

We can now analyze interest rate rules and compare them to the optimal monetary policy. Let us focus on the Taylor rule. As discussed in chapters 14–17, the principle of the Taylor interest rate rule requires the central bank to set its target for the nominal interest rate as a function of three variables:

1.

2. deviations of current inflation from the inflation target of the central bank, and

3. deviations of output from its natural rate.

Assume that this generalized Taylor rule takes the form

where ϕ_π, ϕ_y > 0.¹⁴

By substituting (20.31) in the new neoclassical synthesis IS curve (20.1) and then substituting in the expectations-augmented Phillips curve (20.2), we get the following process for inflation:

where

Note that the inflationary process depends on both parameters of the Taylor rule, because deviations of output from its natural rate cause unanticipated inflation. In addition, the effects of productivity shocks on inflation also depend on the parameters of the Taylor rule.

One can show that the process (20.32) is stable if γ₁ + γ₂ < 1. A sufficient condition for this is

Condition (20.33) is the Taylor principle mentioned in the analysis of the new Keynesian models in chapters 16 and 17.

From (20.34), the fluctuations of inflation around the target of the monetary authorities π^* only depend on the current innovation in productivity, as is the case for the optimal policy analyzed in section 20.3.

The Taylor rule (20.31) would be equivalent to the optimal policy, if γ₁ was equal to the optimal response of inflation to innovations in productivity ψ in equation (20.26). From the definition of ψ in (20.17) and the definition of γ₁ in (20.32), this requires that ϕ_π and ϕ_y must satisfy

As long as the policy parameters are chosen to satisfy (20.35), a Taylor rule can fully replicate the optimal policy in this model.

Solving (20.35) for the optimal ϕ_π, we get

From (20.36), ϕ_y must be chosen so that ϕ_π > 1, which requires

Thus, any pair of ϕ_π and ϕ_y that satisfies (20.36) results in an optimal Taylor rule.

Note that the Taylor rule specified in (20.31) is not the constant intercept rule proposed by Taylor [1993, 1999] but a rule that also requires the nominal interest rate to respond to changes in the natural real interest rate. The constant intercept Taylor rule is not an optimal rule if there are fluctuations in the natural real interest rate, due to productivity or other real shocks, as in our model.¹⁵

20.6