سیاست های پولی فعال و غیر فعال در مدل نسل های متداخل

We consider an overlapping generations model in which the growth rate of money is determined either by inflation forecast targeting or by inflation targeting. New money is distributed via lump-sum transfers to old agents. We study how the responsiveness of the policy rule with respect to (expected) inflation affects determinacy and stability of the monetary steady state. A policy rule is called active (passive) if it responds strongly (weakly). Active inflation forecast targeting reinforces mechanisms that lead to indeterminacy. Active inflation targeting, on the other hand, makes indeterminacy less likely but can create instability of the monetary steady state.

In recent years, many prominent central banks have adopted inflation targeting or inflation forecast targeting as their preferred framework for the conduct of monetary policy. In a parallel development in the academic literature, instrument rules with varying degrees of responsiveness to inflation or inflation forecasts have been explored across a number ofdifferent models; see, e.g., Taylor (1999) or Benhabib et al. (2001a). Following this line of research, the present paper studies a standard overlapping generations model under the assumption that monetary policy is implemented through an instrument rule of the form lnμt − ln n = ln g − η(lnπt − ln g) (1) or lnμt − ln n = ln g − η(lnπt−1 −ln g). (2) Here μt and πt are the gross rates of nominal money growth and inflation from period t to period t + 1, and n is the (constant) gross rate of population growth. The target for the adjusted nominal money growth rate, μt/n, is given by g and the elasticity of the nominal money growth rate with respect to deviations of inflation from target is −η, where η is a real number greater than −1. We refer to (1) as a policy rule for inflation forecast targeting and to (2) as a policy rule for inflation targeting.1 The reason is that, at the time when the central bank sets the money growth rate μt , the most recent observation of inflation is πt−1 whereas πt has not yet been observed and is therefore to be interpreted as an inflation forecast. In other words, Eq. (1) describes a forward-looking policy rule while (2) is a backward-looking one. The main goal of the paper is to study how the policy parameter η affects the stability and determinacy of the monetary steady state under perfect foresight. If η = 0, then Eqs. (1) and (2) require the nominal money supply to grow at the constant gross rate μt = gn. In other words, η = 0 corresponds to a regime of strict money growth targeting in which the adjusted nominal money growth rate is fixed at g. For η >0, the adjusted nominal money growth rate is decreasing with respect to actual (or expected) inflation. This means that the nominal money supply is contracted if inflation is (expected to be) above target and it is expanded if inflation is (expected to be) below target. If η ∈ (−1, 0), on the other hand, the adjusted nominal money growth rate increases with (expected) inflation but less than onefor- one. We shall call a rule with η >0 an active rule and a rule with η ∈ (−1, 0) a passive rule.2 We conduct our analysis in the framework of a standard overlapping generations model. Determinacy and stability of the monetary steady state in this model have been thoroughly investigated under the assumption of a constant money supply or a constant growth rate of the money supply, see, e.g., Azariadis (1993, Chapter 24.1) Policy rules similar to those considered here have been studied in Grandmont (1986).3 The present paper generalizesor adapts some of these results to an economy in which the central bank uses either (1) or (2) and distributes newly created money in the form of lump-sum transfers to old agents. We focus our attention on the question of whether the monetary steady state is a locally unique equilibrium or not, and on how the answer to this question depends on the policy parameter η. The lack of local uniqueness, which is usually referred to as indeterminacy, makes it impossible to compute comparative static properties of the equilibrium, casts doubt on the applicability of the perfect foresight assumption, and reduces the predictive power of the model.4 Indeterminacy arises when the monetary steady state is a locally asymptotically stable fixed point of the forward perfect foresight dynamics. In this case, it is also well known that there exist stationary sunspot equilibria close to themonetary steady state. This has been proved in a very general setting by Woodford (1986). Conversely, if the monetary steady state is a repelling fixed point of the forward perfect foresight dynamics, then indeterminacy is ruled out and there do not exist stationary sunspot equilibria locally around the monetary steady state; see, e.g., Laitner (1989). These results show that there is a close link between indeterminacy of the monetary steady state and the susceptibility of the economy to extrinsic uncertainty, a link that underlines the importance of characterizing the conditions under which indeterminacy occurs. The existence of stationary sunspot equilibria is also related to the existence of deterministic periodic equilibria. This relation has been explored for example by Azariadis and Guesnerie (1986) and Grandmont (1985, 1986). These authors find that, in the overlapping generations models under consideration, both stationary sunspot equilibria and deterministic cycles exist whenever the monetary steady state is a stable fixed point of the perfect foresight dynamics. The results are based on a global analysis of the equilibrium dynamics and do not necessarily generalize to other models. In contrast, the link between indeterminacy and stationary sunspot equilibria and, in particular, the results by Laitner (1989) and Woodford (1986) mentioned above depend only on local properties of the perfect foresight dynamics and are not restricted to the overlapping generations model.5 The present paper applies these local results to the overlapping generations model with policy rules (1) and (2) but it does not attempt to replicate a global analysis in the spirit of Azariadis and Guesnerie (1986) and Grandmont (1985, 1986). We do believe, however, that a global analysis along the lines of Grandmont (1985, 1986) would be possible in the present model and would lead to essentially the same results. Substantiation for this conjecture will be provided at the end of Section 2. In Section 2 we present the model and derive the equilibrium conditions under the inflation forecast targeting rule (1). We prove that, by making the policy rule more active (i.e., by increasing η), the set of economies for which the monetary steady state is indeterminate increases. This shows that active inflation forecast targeting renders the economy more susceptible to endogenous business cycles than strict money growth targeting or passive inflation forecast targeting. The intuition for this result is quite simple. Suppose that πt , the forecast for inflation from period t to period t + 1, is high. Under active inflation fore-cast targeting, this implies that monetary policy will be tightened. Young households in period t therefore rationally expect to receive low transfers during their old age, which increases their incentive to transfer wealth from period t to period t + 1. This obviously increases their demand for money (the sole store of value) while it reduces aggregate demand for goods in period t . Consequently, the price level in period t goes down and this helps to validate the high expected inflation rate πt . We therefore conclude that active inflation forecast targeting reinforces the mechanism which generates indeterminacy of the monetary steady state in the overlapping generations model in the first place, and which results from the interplay between income and substitution effects. Section 3 studies equilibria under the instrument rule (2). We study this case because several authors have claimed that backward-looking instrument rules perform better than forward-looking ones; see Benhabib et al. (2001b, 2003) and references therein. We can confirm within our framework that, by switching from an active forward-looking rule to the corresponding backward-looking rule, the set of economies for which indeterminacy occurs is reduced. If we also allow for passive rules, however, this is not true. Furthermore, we find that the monetary steady state can become unstable, if an active backward-looking instrument rule is used. This is a consequence of the fact that the backward-looking rule (2) cannot be applied in the first period of the model but only from period 2 onwards. Thus, the money growth rate in period 1 becomes an additional policy parameter.We shall see in Section 3 that, under very active inflation targeting rules (i.e., η in (2) is sufficiently high), equilibria of the economy cannot converge to the monetary steady state unless the money growth rate in the first period coincides exactly with its steady state value. Instability in this sense indicates a lack of robustness of the monetary steady state under the inflation targeting scheme. The paper concludes with Section 4, where we summarize our findings and discuss possible caveats and extensions to our analysis.

The purpose of the present paper was to study how different monetary policy rules affect the determinacy and stability of the monetary steady state in the standard overlapping generations model used by Grandmont (1985). We have considered inflation forecast targeting rules, which prescribe the money growth rate as a function of the rational forecast of current inflation, and inflation targeting rules, for which the money growth rate depends on the actual past inflation rate. Our main findings are that active inflation forecast targeting reinforces those mechanisms that lead to indeterminacy of the monetary steady state, but that active inflation targeting weakens those mechanisms. We have also found that the application of an active backward-looking rule can make the monetary steady state unstable. There are several restrictions of our analysis that we want to point out. First and foremost, the model we have used is a very stylized one and it does not include several features that figure prominently in other studies of monetary policy rules. In particular, the present model does not include any form of imperfect competition or nominal rigidity.We have deliberately chosen this simple framework in order to emphasize the way in which monetary policy rules can reinforce some of the mechanisms that are known to generate endogenous business cycles in such an idealized world. We have also assumed that monetary policy operates through lump-sum transfers and not through proportional transfers or open market operations. This is admittedly an oversimplification but we believe that the main conclusions of our paper would remain valid under a more realistic description of monetary policy. Modeling open market operations would require a second asset (government debt) besides money, which would complicate the analysis considerably. Assuming proportional transfers (interest bearing money) would render monetary policy superneutral such that the responsiveness of the policy rules would not have any effect on the perfect foresight equilibrium dynamics.20 Second, we have restricted ourselves to the equilibrium dynamics locally around the monetary steady state. We are aware that a local analysis of this kind is not sufficient to fully evaluate the properties of monetary policy rules, but we think that it is a necessary and important first step towards a full understanding of the equilibrium dynamics under various rules. A global analysis would be particularly important for the case in which the monetary steady state is unstable (see, e.g., Theorem 2(a)) and could be carried out using arguments analogous to those in Grandmont et al. (1998). We leave a more complete investigation of the global dynamics for another paper. It is also worth mentioning that our restriction to a local analysis justifies considering only the restricted class of policy rules which are linear in logarithms. As a matter of fact, all that we need to know about these rules for theanalysis of stability and determinacy of the monetary steady state is their value and slope (or elasticity) at the steady state. Finally, we have focused entirely on equilibria under perfect foresight and have not considered any learning dynamics. It is well known (see, e.g., Grandmont, 1985) that the stability properties of equilibria are usually reversed if one considers learning dynamics instead of perfect foresight equilibria. Since the literature still has not come to a unanimous view on the question of which of the two approaches (learning or rational expectations) is the more relevant one, we feel that our restriction to one of them is justified. We do, however, caution the reader that the assumption of rational expectations is an important one for our analysis and that it should be born in mind when the results of the present paper are interpreted.