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Interest rate rules, inflation and the Taylor principle: an analytical exploration.

Article Excerpt
Summary. The purpose of this article is to characterize optimal interest rate rules in the framework of a dynamic stochastic general equilibrium model, and notably to scrutinize the "Taylor principle", according to which the nominal interest rate should respond more than one for one to inflation. This model yields explicit solutions for the optimal rule. We find that the elasticity of response depends on numerous factors, such as the degree of price rigidity, the autocorrelation of the underlying shocks, or which measure of inflation is used. In general the optimal elasticity of the interest rate with respect to inflation needs not be greater than one.

Keywords and Phrases: Interest rate rules, Taylor rules, Inflation, Optimal monetary policy.

JEL Classification Numbers: E5, E52, E58.

1 Introduction

The purpose of this article is to characterize optimal interest rate rules in the framework of a dynamic stochastic general equilibrium model, and notably to examine the much debated subject of how nominal interest rates should react to inflation.

Indeed following Taylor's (1993) influential article, many authors have studied interest rate rules where central bank policy is a function of endogenous variables such as inflation and output. A particularly scrutinized issue has been the response to inflation (1), and notably the "Taylor principle", according to which the nominal interest rate should respond more than one for one to inflation (2). In a nutshell, if the interest rate rule is of the form (omitting constants) [i.sub.t] = [gamma][[pi].sub.t], then the Taylor principle says that [gamma] should be greater than 1.

The basic framework we shall use for this investigation is that of a dynamic monetary economy subject to stochastic shocks, like productivity shocks. We shall develop a simple model for which we will be able to compute explicit solutions for the optimal interest rate rules (3). The optimal interest rate policy is obtained through maximization of households' utility, subject to the laws of motion of the economy. In such a framework, and under rational expectations, the natural expression of optimal interest rules would be a function linking the interest rate to all observable shocks.

But, as indicated above, our main interest will be in rules expressed as a function of inflation, and in examining the size of their elasticity of response. We can identify two channels through which inflation enters the response function, which we will call "intrinsic" and "surrogate".

(a) The "intrinsic" channel actually concerns not current inflation but expected inflation, for the following natural reason: expected inflation appears in the households's demand functions, and therefore in the dynamic equations of the economy. Consequently it will also generally appear in the optimal interest rate rules.

(b) The second way inflation appears in the interest rate rules is as a "surrogate" for the underlying shocks when these are omitted from the policy function. In that respect, the argument of the function can be either expected inflation or current inflation (it could even be some past inflation).

So, in order to investigate the optimal response of interest rates to inflation, we shall derive a number of optimal interest rate rules taking into account the two above channels.

In order to disentangle them, we shall first derive an optimal interest rate rule when both expected inflation and shocks appear independently as arguments.

We shall then derive, in the tradition of dynamic stochastic general equilibrium (DSGE) models, interest rate rules that respond to shocks only. This will show that the optimal response to shocks depends on the nature of the underlying nominal rigidities (4).

Finally we shall investigate, for various types of rigidities, a number of interest rate rules where current or expected inflation act as surrogates for the underlying shocks. We shall see that the optimal degree of response to inflation depends on numerous factors, like which measure of inflation (current or expected) is used, the nature and degree of price rigidities, and the autocorrelation of shocks. But we will find that the corresponding elasticity can be smaller or greater than one, depending on the values of the relevant parameters, and thus not systematically greater than one.

2 The model

2.1 The agents

We shall consider a monetary overlapping generations model (Samuelson, 1958) with production. The economy includes representative firms and households, and the government.

Households of generation t live for two periods, work [N.sub.t] and consume [C.sub.1t] in period t, consume [C.sub.2t+1] in period t + 1. They maximize the expected value of the following two period utility:

[U.sub.t] = [alpha][[C.sub.1t.sup.1-[sigma]]/[1-[sigma]]] + [[C.sub.2t+1.sup.1-[sigma]]/[1-[sigma]]] - [N.sub.t] [sigma] [greater than or equal to] (1)

Households are submitted in each period of their life to a "cash in advance" constraint:

[M.sub.1t] [greater than or equal to] [P.sub.t][C.sub.1t] [M.sub.2t+1] [greater than or equal to] [P.sub.t+1][C.sub.2t+1] (2)

The total quantity of money is [M.sub.t] = [M.sub.1t] + [M.sub.2t]. Since the young household starts his life without any asset, he has to borrow [P.sub.t][C.sub.1t] from the bank at the interest rate [i.sub.t] in order to satisfy the cash in advance constraint. Consequently the bank makes profits [[LAMBDA].sub.t], equal to:

[[LAMBDA].sub.t] = [i.sub.t][P.sub.t][C.sub.1t] (3)

To simplify calculations we assume that these profits [[LAMBDA].sub.t] are redistributed lump-sum to the young households (5).

The representative firm in period t produces output [Y.sub.t] with labor [N.sub.t] via the production function:

[Y.sub.t] = [Z.sub.t][N.sub.t] (4)

where [Z.sub.t] is a technological shock common to all firms. We assume that the firms belong to the young households, to which they distribute their profits, if any.

2.2 Government policy and the optimality criterion

The government has essentially one policy instrument, the nominal interest rate [i.sub.t].

In order to evaluate the optimality properties of potential interest rate policies, we shall use the criterion...