The Optimal Taylor Rule
We have shown that in a model with endogenous unemployment persistence, as in the model in this chapter, the Taylor rule results in inflation persistence. One can show that inflation persistence would also be associated with optimal monetary policy if the loss function of the central bank depends on both deviations of inflation from target and deviations of unemployment from its natural rate.24
17.8.1 Optimal Inflation Policy
To derive optimal monetary policy, we have to specify an appropriate social welfare function.
Assuming that the optimal steady state inflation rate is equal to π*, the only other distortion in this model is the deviation of unemployment from its natural rate. Thus, we can assume that the central bank would seek to minimize an intertemporal loss function that depends on deviations of inflation from its steady state optimal rate π*, and deviations of unemployment from its natural rate
. This can be written as 
where β is the discount factor, β = 1/(1 + ρ); ρ is the pure rate of time preference; and ζ is the relative weight attached by the central bank to deviations of unemployment from its natural rate, relative to deviations of inflation from target.25
The optimal policy is the one that minimizes (17.94) subject to the dynamic stochastic expectational Phillips curve (17.38). This policy is called the optimal time-consistent contingent policy, because the central bank has no short-run incentive to deviate from it, and the choice of inflation will depend on the current state of the economy, summarized by the deviations of the current unemployment rate from its natural rate.26
Under this policy, there is a clash between the objectives of the monetary authorities and the objectives of wage-setting insiders with regard to unemployment. The central bank seeks to minimize deviations of unemployment from its natural rate, whereas wage setters seek to minimize deviations of unemployment from a weighted average of the natural rate and past unemployment.
This clash is what accounts for the persistence of inflation under the contingent optimal policy.From the first-order conditions for a minimum of (17.94) subject to (17.38), we get
Using (17.38), Okun’s Law (17.41), to substitute for current and expected future deviations of the unemployment rate from its natural rate and after rearranging, we get
The rational expectations solution of (17.96) is given by
From (17.97), deviations of the optimal, time consistent, contingent inflation rate from the inflation target π* display the same degree of persistence as the persistence of deviations of unemployment from its natural rate. The reason is that the central bank allows inflation to fluctuate so as to minimize deviations of unemployment from its natural rate. Because these deviations in unemployment display persistence, deviations of inflation from target will also display persistence under the optimal contingent policy.27
Note that the persistence of inflation under the optimal time-consistent contingent monetary policy does not affect the persistence of unemployment. The reason is that wage setters can anticipate the persistent part of the inflation process, incorporate it in their expectations when they set nominal wages, and neutralize the effects of persistent inflation on unemployment. Thus, the only element of monetary policy that matters for unemployment is the unanticipated part, which is a function of the current productivity shock.
17.9
More on the topic The Optimal Taylor Rule:
- Monetary Policy Shocks and the Optimal Policy Rule
- Financial Frictions in a Model with Unemployment Persistence and Nominal Wage Contracts
- Conclusion
- The perfectly competitive new classical models that we analyzed in chapters 13 and 14 are examples of DSGE models in which wages and prices are perfectly flexible and equilibrate both the product and labor markets.
- Relevant Works