Temporary acceleration of inflation: what can a central bank learn from it?

• Author: Patron, Hilde
• Position: Setting monetary policy

1. Introduction

   High or even moderate inflation rates are a costly phenomenon. However, some central banks may be willing to trade oft" some inflation for lower unemployment, for higher revenue from money creation (seigniorage), or for both. The effectiveness with which they achieve these goals depends on the information and the expectations of the private sector and of the monetary authority.

   The importance of the information set and the learning behavior of the private sector have been analyzed extensively in the literature, but uncertainty and learning by the central bankers have not. (1) One of the most notable contributions in the field of learning in central banking is "Conquest of American Inflation" by Sargent (1999). In it, he studies the behavior of a central bank that is uncertain about the relationship between inflation and unemployment. At first, the central bank estimates the Phillips curve using past inflation and unemployment observations, then it econometrically reestimates the curve whenever new data becomes available. Sargent's book is aimed at studying the behavior of monetary policy and inflation in the United States during the last 40 years or so, and has proved to be very influential among academics.

   Another strand of the literature looks at central bankers that learn using Bayes' rule. Models of Bayesian central bankers include numerical studies, such as Wieland (1998, 2000), and theoretical studies, such as Bertocchi and Spagat (1993). (2) Both types of models find that truly optimal monetary policy involves some degree of experimentation, or some degree of deviation from static optimal policies with the purpose of increasing the central bank's information about the economic environment.

   This article is also a study of Bayesian central bankers. I use a theoretical two-period model to study the short-run dynamics of monetary policy when the central bank is uncertain about the economic environment but learns about it over time using Bayes' rule. As in most papers of learning in monetary economics, I do not study from the interactions between the learning processes of the central bank and the private sector. (3) I use the model to identify the conditions under which a temporary acceleration (or alternatively a deceleration) of inflation can increase the central bank's overall expected utility.

   Like Bertocchi and Spagat (1993), I look at the short-run dynamics in monetary policy, but unlike them I allow for a larger variety of monetary strategies (i.e., I use a very general central bank utility function) and a more general characterization of the uncertainty generation process. Also, unlike them, I work with a finite time horizon, which presumes that central bankers are appointed to a finite nonrenewable tenure, (4) that the government is pursuant of short-tern1 goals, or that it discounts the distant future very heavily. Working with a finite time horizon allows me to characterize the direction in which monetary policy is adjusted to optimally generate and to adjust to new information. Finally, unlike Bertocchi and Spagat, I introduce intrinsically dynamic elements into the model by allowing current policy actions to affect current and future economic variables. (5-6)

   I find that under very general conditions the central bank should experiment and that it should adjust monetary policy to new information. The nature of experimentation (its direction and magnitude) will depend on the particular uncertainty laced by the bankers. For example, if the central banker knows how the economy reacts to low money growth rates but is uncertain about it at high rates, then it is optimal to initially inflate "a lot" in order to learn about the economy. Alternatively, if the central banker knows how the economy reacts to large money growth rates but is unsure about it at low rates, then it is optimal for the banker to inflate "too little" in order to learn about the economy. (7)

   The rest of the article is organized as follows. In section 2, I present the basic model, and in section 3, I find the solution. In section 4, I discuss the informational aspects of the model, and in section 5, I present the main results of the article. In section 6, I extend the generic model of section 2 to incorporate intrinsically dynamic features. More particularly, I allow monetary policy to affect economic variables with a lag. In section 7, I conclude.

2. The Model

   Consider a two-period model with two agents: the central bank and the private sector. In each period the central bank, whose preferences are given by [U.sub.t], chooses the rate of growth of the money supply [g.sub.t], t = 1,2. The central bank is uncertain about the impact of g: on a target variable [m.sub.t], which is determined in part by the behavior of the private sector and in part by unobservable random shocks. The central bank initially makes decisions given some prior (exogenous) beliefs, and at the end of the first period, given the observations of [g.sub.t] and [m.sub.t], it updates beliefs using Bayes' rule.

   Before formalizing this generic setup further it is instructive to describe the role of the central bank and the private sector in more detail with some examples.

   Example 1: The Revenue Motive for Inflation

   Assume a central bank (or a government) that is traditional in the sense that it dislikes inflation for itself, but is unconventional in the sense that it is willing to accept some inflation for the sake of the revenue it receives from printing money. (8) In this example, the central bank's utility function can be represented by [U.sub.t]([g.sub.t],[m.sub.t]) - ([g.sup.2.sub.t]/2), where [g.sub.t] is used to denote the costs of inflation, and where x > 0 quantifies the central bank's wish for seigniorage, [S.sub.t], relative to its dislike for inflation. (9) Seigniorage in period t is given by the amount of goods and services that the government buys from printing money, or equivalently by the money growth rate times the demand for real balances [m.sub.t], [S.sub.t] - [g.sub.t][m.sub.t]. (10)

   Assume that the central bank is uncertain about the impact of money growth on the quantity of money demanded by the public. That is, assume that the central bank is uncertain about the parameters of the money demand function. Assuming that the demand for money is driven mostly by the public's expectations of inflation, which are increasing in the money growth rate, money demand can be written in reduced form as [m.sub.t] = l([g.sub.t],[OMEGA]) + [[epsilon].sub.t], where dl([g.sub.t],[OMEGA])/d[g.sub.t] = l([g.sub.t],OMEGA])

   In this application the central bank is uncertain about the parameters of the demand function. At the end of the first period, having observed the amount of money demanded (but not the random shock), the central bank updates beliefs about them using Bayes' rule.

   Example 2. The Output Motive for Inflation

   The general model considered in this article can also be applied to a more traditional central bank that dislikes inflation but is willing to tolerate some of it for the sake of output. Assuming that [g.sub.t] denotes the costs of inflation, and that [y.sub.t] denotes output, then [U.sub.t] can be represented by [U.sub.t]([g.sub.t], [y.sub.t]) = [y.sub.t] - ([g.sup.2.sub.t]/2) or possibly by [U.sub.t]([g.sub.t], [y.sub.t]) = [([y.sub.t] - [bar.y]).sup.2]/2 - ([g.sup.2.sub.t]/2) where [bar.y] is the bank's desired output level. Assume that the behavior of the economy can be summarized with a Lucas type supply function, [y.sub.t] = l([g.sub.t], [OMEGA]) + [[epsilon].sub.t] where dl([g.sub.t], [OMEGA])/d[g.sub.t] > 0, [OMEGA] is a vector of unknown parameters, and [[epsilon].sub.t] is a random shock to supply. (12) The function l([g.sub.t], [OMEGA]) shows how when the money growth rate increases, which increases the price level, firms hire more workers and increase production.

   In this example the central bank is assumed to be uncertain about the parameters of the supply equation. At the end of the first period the bank observes [g.sub.t] and [y.sub.t] and updates beliefs about the parameters of supply using Bayes' rule.

   There are of course several other possible strategies that bankers may pursue. They may, for example, be willing to tolerate inflation for lower unemployment and higher seigniorage; they may be trying to minimize inflation variability, around a target rate; or, their strategy may even include exchange rate considerations. Instead of restricting the model to a particular strategy, I assume that, in general, the central bank's period t utility function is given by [U.sub.t]([g.sub.t], [m.sub.t]), where [m.sub.t] is a target variable that is a function of [g.sub.t], of a parameter set [OMEGA], and of the realization of a real-valued random noise term, [[epsilon].sub.t], [m.sub.t] = l([g.sub.t], [OMEGA]) + [[epsilon].sub.t]. I assume that the random shock [[epsilon].sub.t] is distributed with a continuous, differentiable density function f([[epsilon].sub.t]) in -[infinity]

   I assume that the central bank is uncertain about the parameter set [OMEGA], and, thus, about the function l([g.sub.t], [OMEGA]). The nature of the uncertainty is as follows. First, I assume that the central bank can a priori recognize a small portion of l([f.sub.t], [OMEGA])) and, more particularly, the point (g, m) This can result from the assumption that at the beginning of period 1 the monetary authority has a large (time series) data set that exhibits a "reasonable" concentration of observations in a close neighborhood of or at the point (g,m). (15) Although the central bank knows the point (g, m), it is uncertain about the intercepts and the slope of the curve, which I assume (for tractability purposes), can take on two different equations. A very general depiction of the two possible mean curves is given in Figure 1. Following the notation of Figure 1 the uncertainty in the model can be formally characterized by Assumption 1. This is...