# EVOLUTION OF BELIEFS AND THE OPTIMALITY OF MONETARY POLICY RULES

*Fabio Scacciavillani* and
*Jasmina Arifovic* ()

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Fabio Scacciavillani: European Central Bank

No 300, Computing in Economics and Finance 2000 from Society for Computational Economics

**Abstract:**
Defining the rules for monetary policy has been the subject of an extensive literature in monetary economics since the 1968 AMA presidential address by Milton Friedman. Starting from the contribution by Barro and Gordon (1983) the prevalent approach has hinged on optimal control techniques to devise feedback policy reactions by the monetary authority, which typically faces a trade off between inflation and output growth.More recently the attention has focused on the robustness of monetary rules in the face of model uncertainty and lack of accurate information on the state of the economy. One branch of the literature has investigated this issue using the robust control techniques (e.g. Stock and Onatski (1998)). The other has concentrated on the search for monetary rules that result in lower inflation and output variability when applied to a set of different structural models (Levin, Wieland and Williams (1999), Rudebusch and Svensson (1999), Ball (1999)).This paper takes a novel approach by examining the properties of monetary rules in an economy populated by boundedly rational agents who have heterogeneous expectations about future inflation rates. In order to model the evolution of agents' behavior we use the genetic algorithm, which provides a natural framework to calculate explicit expectations. The evolution takes place through two distinct processes: a) The promotion of those expectations whose values turn out to be closer to the actual realizations of the inflation rate; b) The agentsÆ experiments with new values of expectations through random changes and recombination of those values that already exist in the population. Genetic algorithms impose low requirement on the computational ability of economic agents and provide a convenient framework for modeling decentralized learning with heterogeneity of agents' beliefs. They have been increasingly employed to model the behavior of economic agents in macroeconomic models, both as an equilibrium selection device and as a model of transitional, out-of- equilibrium dynamics (e.g. Arifovic (1995, 1996), Bullard and Duffy (1998a, 1998b), Dawid (1996)). They have also been successful in capturing the main features of the behavior of human subjects in controlled laboratory settings (e.g. Arifovic (1995, 1996)).We will start from a basic underlying model of the economy (as in many studies employing optimal control techniques) consisting of a linear aggregate demand equation (1) y(t+1) = a1*y(t) - a2*y(t-1) - a3*(i(t)-E[pi(t+1)|I(t)]) + u(t) and an equation governing the dynamics of inflation (2) pi(t+1) = pi(t) + b*y(t+1) + v(t), where y(t) represents the output gap at time t, pi(t) inflation and the errors term u(t) and v(t) are serially uncorrelated with mean zero. The E[x(t+1)|(I)t] operator in standard models represents the mathematical conditional expectations based on the set of information I(t).The optimal rule that a monetary authority minimizing a quadratic loss function defined over inflation and output gap would adopt in such an environment (and in more complex ones) has been studied extensively. Taylor (1999) and Clarida, Gali and Gertler (1999) provide extensive overviews of the existing literature. Clearly the value of the parameters in the (linear) feedback rule depends on the parameters of the objective function, however specific rules such as the 'Taylor rule' (see Taylor (1993)), which have provided a good fit to the data over an extended period, have gained prominence in the policy debate.Our contribution would be to investigate the dynamics of inflation and output gap in the economy described by equations (1) and (2) when the conditional expectation in (1) is replaced by the expectations formed by agents who have only an imperfect knowledge of the true parameters of the model and the structure of the economy. Specifically we devise an algorithm whereby we postulate that the artificial agents in the economy are characterized by their inflation expectations, which are initially uniformly distributed in the population. The conditional expectation in (1) is therefore replaced by the median of the agentsÆ expectations. The algorithm calculates the output gap and the inflation rate (for a given monetary policy rule) and eliminates those expectations that turn out to be relatively far away from the actual realization of the inflation rate. As a consequence the next generation of agents will have better forecasting abilities. On the other side, experimentation will generate new agents with 'unconventional' beliefs, which offset the tendency towards uniformity induced by imitation.In carrying out this exercise we will evaluate different rules such as those proposed by Taylor (1993) (which will be used as a benchmark), Ball (1999), Rudebusch and Svensson (1999), combined with different values of the genetic algorithm parameters. In addition, we will investigate the dynamics of the relevant variables in environments that are more complex both in terms of the features of the underlying economic model as well as in terms of the representation of agents' expectations. Finally we will try to examine how the uncertainty over the parameters of the model might affect the policy rule. This topic pioneered by Brainard (1967) has been recently examined among others by Cecchetti (1997), Wieland (1997), and Peersman and Smets (1999)).

**Date:** 2000-07-05

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