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This paper studies optimal monetary policy in a model where inflation is persistent. Two types of price setters are assumed to exist. One acts rationally given Calvo-type constraints on price setting. The other type sets prices according to a rule-of-thumb. This results in a Phillips curve with both a forward-looking term and a backward-looking term. The Phillips curve nests a standard purely forward-looking Phillips curve as well as a standard purely backward-looking Phillips curve as special cases. A cost push supply shock is derived from microfoundations by adding a time varying income tax and by making the elasticity of substitution between goods stochastic. A central bank loss function for this model is derived from a second-order Taylor approximation of the household's welfare function. Optimal monetary policy for different relative values of the forward- and backward-looking terms is then analyzed for both the commitment case and the case of discretion.

Ever since the publication of Phillips’ (1958) famous paper documenting the apparent tradeoff between inflation and unemployment, the Phillips curve has been a central piece of macroeconomics. Few ideas in economics have been as controversial, as influential, and undergone as many fundamental revisions. Since Friedman (1968) and Phelps (1967) it has been widely appreciated that inflationary expectations are an important element of the Phillips curve. Two very different approaches to modeling how inflationary expectations enter the Phillips curve have been most popular in the literature. One approach uses lagged values of inflation as a proxy for current inflationary expectations. According to this approach, the Phillips curve takes the following form: where πt is inflation in period is the output gap in period t, while A(L) and B(L) are polynomials in the lag operator. We will refer to this as the “acceleration” Phillips curve. 1 Alternatively, it is often assumed that inflationary expectations are formed rationally in an environment of staggered price and wage adjustments. These assumptions result in a Phillips curve of the following form: where Etπt+1 is the conditional expectation of πt+1 at date t. We will refer to this as the “new Keynesian” Phillips curve. 2 Neither of these two specifications, however, seems adequate to capture the behavior of inflation in actual economies. The acceleration Phillips curve fails to capture the fact that individuals and firms do not form their expectations about inflation in a rigid and mechanical manner. For instance, it is well documented that inflationary expectations can be drastically altered by a sharp change in macroeconomic policy.3 On the other hand, the new Keynesian Phillips curve fails to capture the fact that inflation is highly persistent. According to it firms completely front load changes in prices in response to “news” about future profits. Empirical studies do not validate this prediction. Several recent studies which seek to estimate Phillips curves of this type find that they fit the data poorly (see e.g., Fuhrer and Moore, 1995; Fuhrer, 1997; Gali and Gertler, 1999; Roberts, 2000). Evidence from VAR studies also show that the response of inflation to shocks is “hump-shaped” rather than front loaded. In recent years increasing attention has been given to the following hybrid specification of the Phillips curve: equation(1) Fuhrer and Moore (1995) derive a Phillips curve of this type with χ1=χ2=0.5 from a model with two period overlapping wage contracts. They estimate this equation and conclude that it fits recent U.S. data better than either a purely forward-looking or purely backward-looking Phillips curve. Gali and Gertler (1999) derive a Phillips curve of this type from a model with staggered price setting with the additional assumption that a fraction of the producers set their prices according to a rule of thumb. They then estimate this model and report values for χ1 and χ2 close to 0.8 and 0.2, respectively. They are able to reject both the purely forward-looking Phillips curve and the purely backward-looking Phillips curve. Other recent papers, discussed below, come to similar conclusions, although the estimated relative values of χ1 and χ2 vary greatly between studies. In light of these facts, and the importance of the Phillips curve for the conduct of monetary policy, it is surprising how little work has sought to analyze and compare optimal monetary policy for different relative weights of χ1 and χ2. This is especially surprising given the current emphasis on the analysis of robustness of different types of monetary policy. Surely, variation in the relative weights on the forward- and backward-looking terms in the Phillips curve is an important dimension of such robustness analysis. The principal goal of this paper is to partially fill this hole in the literature. Another goal of the paper is the derivation of microfoundations for several popular deviations from the benchmark new Keynesian model. We derive a hybrid Phillips curve by assuming that a fraction of the producers set their prices according to a rule of thumb. This approach to deriving a hybrid Phillips curve has recently been used by Gali and Gertler (1999). The rule of thumb we choose is however a generalization of the rule of thumb chosen my Gali and Gertler. It has the theoretically appealing property that it nests the standard new Keynesian Phillips curve and the standard acceleration Phillips curve as special (limit) cases. A second theoretical innovation of the paper is a derivation of a “cost push” shock to the Phillips curve. Actually, we model two potential sources of such shocks: time varying income taxes and time varying monopoly power of producers. It turns out that for reasonable calibrations of our model variation in taxes results in very small shocks while reasonable variation of the monopoly power of producers is capable of creating large disturbances to the Phillips curve. Our model also includes other “supply” disturbances which do not constitute cost push shocks.4 The reason is that they represent movements in the efficient level of output which monetary policy does not optimally react to. In Section 2 we present the derivation of our model. In Section 3 we analyze optimal responses of the economy to cost push supply shocks. In Section 4 we conclude with a brief discuss of the main insights that can be drawn from the analysis presented in this paper as well as a discussion of some future extensions of this research.

In this paper, we have extended the benchmark new Keynesian macro-model by making the Phillips curve a convex combination of a forward-looking term and a backward-looking term. This was motivated by our assumption that a fraction ω of the producers in the economy set their prices according to a rule of thumb. We have seen that the main features of optimal policy in the purely forward looking case, such as the importance of commitment, carry over to this hybrid case. However, we have also seen that some features of the solution change in important ways. In our primarily backward looking cases it is optimal to bring inflation back down to zero in a gradual manner instead of the immediate overshooting that characterizes the purely forward-looking case. Also, in the backward-looking cases it is optimal to endure a much larger contraction of output in order to avoid getting too much inflation into the system. These features of our hybrid cases seem to correspond quite well with actual central bank policy. The sharp overshooting of inflation in the period immediately following a supply shock which is optimal in the purely forward looking case does not seem to correspond to the way actual central banks react to supply shocks. Quite to the contrary, actual central banks often seek to gradually bring inflation back in line with their target. The policies of both the Bundesbank and the Federal Reserve in the early 1990s are a good example of this type of behavior. Woodford (1999, Chapter 7) notes that an important feature of recent monetary policy by central banks such as the Federal Reserve is a high degree of interest rate inertia. Woodford argues that this type of behavior can be explained as being a feature of the optimal response of a central bank optimizing under commitment in a purely forward-looking economy. This explanation is however not consistent with the behavior of inflation in actual economies. Such behavior within a purely forward-looking model should produce sharp overshooting of inflation. The actual behavior of inflation is more in line with the responses produced by a central bank optimizing with commitment in our hybrid cases. The theoretical discussion in Section 2 also brought some interesting points to light. We added a time varying income tax to the model and made the monopoly power of the producers in the economy stochastic. We then showed that both of these extensions of the benchmark model resulted in a cost push supply shocks being added to the Phillips curve. However, we also found that reasonably sized tax shocks resulted in only miniscule cost push shocks while reasonably sized shocks to the market power of producers resulted in large shocks. We were actually quite surprised to see how much instability such shocks could create. The model studied in this paper is a closed economy model. In future research, we are particularly interested in extending the model to a small open economy setting. Surprisingly little work to date has sought to analyze the optimal role of the exchange rate in the monetary policy of small open economies. A particularly interesting feature of this extension is the derivation of an appropriate central bank loss function in such a setting. Another important extension would be to derive a fully general second-order approximation to welfare and analyze optimal policy given this loss function and a second-order approximation to the structural equations of our model. A feature of the our model that we are particularly uneasy about is the extremely low weight which the central bank loss function we derive puts on deviations of output from potential. As a result of this low value the optimal tradeoff between stabilizing output and stabilizing inflation is seriously skewed towards the stabilization of inflation, much more so than we think is reasonable. It is possible that this is due to the fact that all the frictions that we have introduced in our model are frictions to price adjustments. If frictions were introduced evenhandedly to every part of the model this would probably raise the relative weight on the output gap in the central bank loss function. The following two types of frictions, for instance, seem likely to become important pieces of more realistic models in this genre: rule-of-thumb consumers such as the ones introduced by Campbell and Mankiw (1989) and labor market frictions which result in persistence in the level of unemployment. Both of these types of frictions would most likely raise the relative weight on the output gap in the central bank loss function. Hopefully, we will be able to shed some light on these issues in future research.