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

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]) < 0, [OMEGA] is a vector of unknown parameters, and [[epsilon].sub.t] a random shock to demand with known density f([[epsilon].sub.t]). (11)

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] < [[epsilon].sub.t] < [infinity], with zero expectation, and satisfying the monotone likelihood ratio property (MLRP), that is. f'([[epsilon].sub.t])/f([[epsilon].sub.t]) is strictly decreasing in [[epsilon].sub.t]. (13) The shock [[epsilon].sub.t] is meant to capture all the variables excluded from the analysis, as well as the stochastic nature of the economy. (14)

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 an analogous information structure to that in Mirman, Samuelson, and Urbano (1993) adapted to central banking.

ASSUMPTION 1. Assume that l([g.sub.t], [OMEGA]) can take on one of the following two representations,

[m.sub.t] = l([g.sub.t], [bar.[OMEGA]]) + [[epsilon].sub.t], [m.sub.t] = l([g.sub.t], [[OMEGA].bar]) + [[epsilon].sub.t],

and assume that there exists a value g [member of] R, such that l(g, [OMEGA]) = l(g, [[OMEGA].bar]). Without loss of generality, assume that [for all] [g.sub.t] < g l([g.sub.t], [OMEGA]) - l([g.sub.t], [[OMEGA].bar]) < 0, and [for all] [g.sub.t] > g l([g.sub.t], [OMEGA]) - l([g.sub.t], [[OMEGA].bar]) > 0. Hence, [for all] [g.sub.t] l'([g.sub.t], [for all]) - l'([g.sub.t], [[OMEGA].bar]) > 0, where l'([g.sub.t], [OMEGA]) = dl([g.sub.t], [OMEGA])/d[g.sub.t].

Since the central bank does not fully know the entire function l([g.sub.t], [[OMEGA].bar]), it initially makes decisions based on prior beliefs. Let [[mu].sub.1] be the central bank's prior belief that [OMEGA] = [OMEGA].

In period 1, the central bank chooses a growth rate for the money supply [g.sub.t], and values of [[epsilon].sub.t] and, hence, [m.sub.1] are realized. The value of [m.sub.1] is observed but not the value of [[epsilon].sub.t], which prevents the central bank from learning the parameter set [OMEGA]. However, given [m.sub.1], [g.sub.1] and given the known distribution of [[epsilon].sub.1], the central bank updates beliefs using Bayes' rule. Letting [[mu].sub.2] denote the posterior belief that [OMEGA] = [OMEGA] then, by Bayes' rule,

[[mu].sub.2] = [[mu].sub.1]f([m.sub.1] - l[[g.sub.1], [bar.[OMEGA]])/(1 - [[mu].sub.1])f([[mu].sub.1] - l[[g.sub.1], [[OMEGA].bar]) + [[mu].sub.1]f([m.sub.1] - l[[g.sub.1], [bar.[OMEGA]]]) = [[mu].sub.1][bar.f]/(1 - [[mu].sub.1])[f.bar] + [[mu].sub.1][bar.f].

Given posterior beliefs, the central bank chooses the second period money growth rate [g.sub.2].

3. The Solution

Within each period the money growth rate is chosen by the central bank to maximize the discounted expected value of [U.sub.t]([g.sub.t], [m.sub.t]). Given the nature of the uncertainty in the model, the expected utility in period t, [E.sub.t][U.sub.t]([g.sub.t], [m.sub.t]), is given by

[E.sub.t][U.sub.t]([g.sub.t], l[[g.sub.t], [OMEGA]] + [[epsilon].sub.t]) = [[mu].sub.t][U.sub.t]([g.sub.t], [bar.[OMEGA]]) + (1 - [[mu].sub.t]) [U.sub.t]([g.sub.t], [[OMEGA].bar]),

where [E.sub.t] denotes the expectations operator given the beliefs and information at the beginning of period t and where

[U.sub.t]([g.sub.t], [OMEGA]) = [[integral].sup.[infinity].sub.-[infinity]] [U.sub.t]([g.sub.t], l[[g.sub.t], [OMEGA]] + [[epsilon].sub.t])f([[epsilon].sub.t])d[[epsilon].sub.t].

I assume that [U.sub.t] and l are twice continuously differentiable, and that [[differential].sup.2][E.sub.t][U.sub.t]/[differential][g.sup.2.sub.t] < 0. This latter assumption ensures the existence of a unique solution in the static utility maximization problem.

Since posterior beliefs are a function of prior beliefs and of the first period money growth rate, I initially solve for the second period solution. In period 2, the monetary authority maximizes,

[E.sub.2][U.sub.2]([g.sub.2], l[[g.sub.2], [OMEGA]] + [[epsilon].sub.2]),

with respect to [g.sub.2]. Let [g.sup.*.sub.2]([[mu].sub.2]) denote the second period optimal money growth rate, (16) and let V([[mu].sub.2]) denote the second period value function, that is,

V([[mu].sub.2]) = [[mu].sub.2][U.sub.2]([g.sup.*.sub.2][[[mu].sub.2], [bar.[OMEGA]]) + (1 - [[mu].sub.2])[U.sub.2]([g.sup.*.sub.2][[[mu].sub.2]], [[OMEGA].bar]).

In the first period the government chooses a rate of growth of money by maximizing expected utility over the two periods. Since in period 1 the posterior beliefs [[mu].sub.2] are a random variable whose distribution function depends upon first period money growth rate and upon the distribution of [m.sub.1] implied by [g.sub.1], I have to take the expectation of the second period value function over all possible values of [m.sub.1]. Assuming that the time discount parameter is given by [delta], then the first period problem is given by

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

W([g.sub.1]) = [[integral].sup.[infinity].sub.-[infinity]] V([[mu].sub.2])h([m.sub.1], [g.sub.1])d[m.sub.1],

and where h([m.sub.1], [g.sub.1]) is the density' of [m.sub.1] implied by [g.sub.1], which is given by

h([m.sub.1], [g.sub.1]) = [[mu].sub.1]f([m.sub.1] - l[[g.sub.1], [bar.[OMEGA]]) + (1 - [[mu].sub.1])f([m.sub.1] - l[[g.sub.1], [[OMEGA].bar]).

Let [G.sup.*.sub.1] denote the set of maximizers for Equation 1 with [g.sup.*.sub.1]([[mu].sub.1]) an element of [G.sup.*.sub.1]. (17)

4. The Government's Learning and the Informativeness of [m.sub.1]

In the central bank's optimization problem (Eqn. 1) periods 1 and 2 are connected by posterior beliefs. Thus, when the central bank chooses the first period money growth rate [g.sup.*.sub.1]([[mu].sub.1]), it accounts for the effect of period 1 money growth on posterior beliefs and on future utility. A central bank that does not account for the effect of first period money growth on posterior beliefs or their distribution and that does not update beliefs once new information becomes available is called a "nonlearner." Since posterior beliefs are the only connection between periods, (18) a nonlearner is also a myopic central bank.

DEFINITION. A central bank is said to be myopic if it maximizes first period utility without concern for the effect of first period money growth on the future. In period 1 the myopic central bank chooses [g.sub.myopic]([[mu].sub.1]) that maximizes static expected utility given prior beliefs, that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

If in period 1 the central bank sets a money growth rate [g.sup.*.sub.1]([[mu].sub.1]) different from the myopic rate [g.sub.myopic]([[mu].sub.1]), then it is said to experiment. The formal definitions of experiments and information given below are based on Blackwell (1951, 1953). (20) In this model, an experiment is a rate of growth of money [g.sub.1]. For each experiment there are two density functions, (f[[m.sub.1] - l([g.sub.1],[OMEGA]], f[[m.sub.1] - l([g.sub.1],[[OMEGA].bar]]), which indicate the conditional probability (or "likelihood") of observing [m.sub.1], given each possible state of nature ([OMEGA], [[OMEGA].bar]). For each experiment there is also a density of [m.sub.1], h([m.sub.1], [g.sub.1]), which shows the overall prior probability of observing [m.sub.1]. Using Blackwell's (1953) sufficiency criterion, given two experiments [g'.sub.1] and [g''.sub.1], the experiment [g'.sub.1] is said to be more informative than [g''.sub.1] if [g''.sub.1] is a "garbling" of [g'.sub.1]. Equivalently, as shown by Kihlstrom (1984), more informative experiments can be defined as follows.

LEMMA 1. Consider two experiments [g'.sub.1] and [g''.sub.1]. Then, [g'.sub.1] is said to be more informative than [g''.sub.1] if every utility maximizer prefers [g'.sub.1] over [g''.sub.1], that is, if

[integral] [chi]([[mu].sub.2][[m.sub.1], [g'.sub.1]])h([m.sub.1], [g'.sub.1])d[m.sub.1] [greater than or equal to] [integral] [chi]([[mu].sub.2][[m.sub.1], [g''.sub.1]])h([m.sub.1], [g''.sub.1]) d[m.sub.1],

for every continuous, convex function [chi]([[mu].sub.2][[m.sub.1], *]) : [0,1] [right arrow] R.

More informative money growth rates are thus preferred to less informative rates, if and only if the central bank's value function is convex in posterior beliefs.

5. Implications of Learning on the Optimal Monetary Policy

In this section, I show that the monetary authority values information and thus that it should deviate from the myopic money growth rate in order to increase information. Furthermore I establish sufficient conditions under which experimentation means increasing the money growth rate above the myopic rate. (For continuity of exposition, most of the proofs in the article are presented in the Appendix).

PROPOSITION 1. The central bank's second period value function is strictly convex in posterior beliefs, [d.sup.2]V([[mu].sub.2])/d[[mu].sup.2.sub.2] > 0. Information is thus said to be valuable.

PROOF: See Appendix A.

An immediate consequence of Proposition 1 is that if the central bank can affect the informativeness of [m.sub.1], it should deviate from the myopic rate in order to increase information (Proposition 2).

PROPOSITION 2. Assume that different money growth rates yield different information. Then, the government experiments in the direction that increases information, that is, if dW([g.sub.1])/d[g.sub.1] > (<)0 then [g.sup.*.sub.1]([[mu].sub.1]) > (<) [g.sub.myopic] ([[mu].sub.1]).

PROOF. The proof follows from concavity of [E.sub.1][U.sub.1]([g.sub.1], l[[g.sub.1], [OMEGA]] + [[epsilon].sub.1]) on [g.sub.1]. QED.

To establish the conditions under which the central bank experiments, and the direction of experimentation, note that after some algebra (in Appendix B) the term dW([g.sub.1])/d[g.sub.1] can be written as follows,

(2) dW([g.sub.1])/d[g.sub.1] = (l'[[g.sub.1], [bar.[OMEGA]]] - (l'[[g.sub.1], [[OMEGA].bar]) [[integral].sup.[infinity].sub.-[infinity]] [d.sup.2]V([[mu].sub.2])/d[[mu].sup.2.sub.2] d[[mu].sub.2]/ d[m.sub.1] [[mu].sub.2](1 - [[mu].sub.1])[f.bar]d[m.sub.1].

By Assumption 1, l'([g.sub.1], [bar.[OMEGA]]) - l'([g.sub.1], [[OMEGA].bar]) > 0. Hence, to sign and interpret Equation 2, I need to determine the sign of d[[mu].sub.2]/d[m.sub.1].

PROPOSITION 3. Given the MLRP, if [g.sub.1] < g, then d[[mu].sub.2]/d[m.sub.1] < 0, and thus information decreases with the money growth rate; if [g.sub.1] > g, then d[[mu].sub.2]/d[m.sub.1] > 0, and thus infonnation increases with the money growth rate.

PROOF. See Appendix C.

The intuition behind Proposition 3 can be explained using either of the two panels of Figure 1. To the right of point g, for example, the possible density functions of [m.sub.1] implied by [g.sub.1] spread apart and superpose less as [g.sub.1] increases. Thus a greater [g.sub.1] spreads the means of the two possible curves apart making them more distinguishable (Creane, 1996).

Given Proposition 3, I know that, in fact, the central bank should deviate from the myopic money growth rate in order to learn. However, to determine the direction of experimentation I must be able to order the two possible mean functions l([g.sub.1], [bar.[OMEGA]]) and I([g.sub.t], [[OMEGA].bar]). I therefore restrict the model to the two possible alternatives stated in Assumptions 2 and 3, both of which are particular cases of Assumption 1. Either of these two assumptions ensures that the central bank's beliefs are monotonic in [m.sub.1] and thus that information is monotonic in the first period money growth rate.

ASSUMPTION 2. Assume that the value of g is "sufficiently small" and thus that (for the relevant range of money growth rates), l([g.sub.1], [bar.[OMEGA]]) - I([g.sub.t], [[OMEGA].bar]) > 0.

ASSUMPTION 3. Assume that the value of g is "sufficiently high" and thus that (for the relevant range of money growth rates), l([g.sub.1], [bar.[OMEGA]]) - I([g.sub.t], [[OMEGA].bar]) < 0.

PROPOSmON 4. (i) Let Assumption 2 hold. Then a rational central bank should experiment in period 1 in order to increase information about l([g.sub.1], [OMEGA]) by setting a money growth rate that is higher than the myopic rate, that is, [g.sup.*.sub.1]([[mu].sub.1]) > [g.sub.myopic]([[mu].sub.1]).

(ii) Let Assumption 3 hold. Then a rational central bank should experiment in period 1 in order to increase information about l([g.sub.1], [OMEGA]) by setting a money growth rate that is lower than the myopic rate, that is, [g.sup.*.sub.1]([[mu].sub.1]) > [g.sub.myopic]([[mu].sub.1]).

PROOF. The proof follows from concavity of [E.sub.1][U.sub.1]([g.sub.1],l[[g.sub.1], [OMEGA]] + [[epsilon].sub.1]) on [g.sub.1]. QED.

According to Proposition 4, the central bank's optimal amount of experimentation is a function of the bank's prior information. For example, if Assumption 2 holds, that is, if the central bank's prior (and exogenous) knowledge about l([g.sub.1],[OMEGA]]) corresponds to a "low" enough money growth rate, then increasing the money growth rate above the myopic rate increases overall expected utility, even if it reduces it in the first period. Assumption 2 is consistent, for example, with the idea that the economic environment has been stable and monetary policy disciplined for some time. Whether the initially expansionary policy is optimal on a transitory or on a more permanent basis depends on the information that the central bank receives at the end of the first period and, thus, depends on posterior beliefs. For example, if posterior beliefs are equal to prior beliefs, the central bank lowers the money growth rate in period 2 to equal the myopic rate.

PROPOSITION 5. Assume that d[bar.U]([g.sub.t],[bar.[OMEGA]])/d[g.sub.t] > (<)d[bar.U]([g.sub.t],[[OMEGA].bar])/ d[g.sub.t]. Then, the optimal money growth rate in period 2 is increasing (decreasing) in posterior beliefs. That is, for any two beliefs [[mu].sub.2] > [[mu]'.sub.2] [member of] (0,1) if [[mu].sub.2] > [[mu]'.sub.2] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

PROOF. See Appendix D.

Uncertainty and learning by central bankers thus leads to active monetary policy in two ways. On the one hand, bankers should deviate from short-run optimal policies in order to learn. Under Assumption 2 this implies an initially "high" inflation rate; under Assumption 3 it implies an initially "low" inflation rate. (21) On the other hand, bankers should actively adjust to new information. New information determines whether inflation hikes (or dips) are quickly reversed or whether they are maintained over time.

The volatile character of monetary policy implied by uncertainty and learning can have serious implications in the economy. For example, in an already highly inflationary environment (one in which the government relies heavily on seigniorage), experimentation and learning can either aggravate the inflationary situation (provided Assumption 2 is valid) or it can ameliorate it somewhat (provided Assumption 3 is correct).

6. An Intrinsically Dynamic Model

In the basic model of section 2, the outcome variable [m.sub.t] is assumed to be a function of only current monetary policy [g.sub.t] and not a function of past money growth rates. In some cases it would be more realistic to assume that [m.sub.t] is a function of current and past money growth rates (or even possibly past values of m). For example, if [m.sub.t] denotes the demand for money, it is sensible to assume that it takes time for people to adjust to desired money balances, and thus that [m.sub.t] is a function of current and past money growth rates. If [m.sub.t] denotes a Phillips curve, then under some circumstances it is sensible to assume that there exists some degree of unemployment persistence.

Assume then that at any time, the central bank maximizes the discounted expected value of [U.sub.t]([g.sub.t],[m.sub.t]), where [m.sub.t] is now given by [m.sub.t] = l([g.sub.t],[g.sub.t-1],[OMEGA]) + [[epsilon].sub.t]. (22)

Assume, as before, that [OMEGA][member of]{[bar.[OMEGA]],[[OMEGA].bar]} is unknown, and that [[mu].sub.1] is the central bank's prior belief that [OMEGA] = [bar.[OMEGA]]. Also, assume that the MLRP holds, and to simplify the exposition of results, assume that the relevant range of money growth rates is given by region II in Figure 1.

ASSUMPTION 4. Assume that for the relevant range of money growth rates l([g.sub.t],[g.sub.t-1],[bar.[OMEGA]]) > l([g.sub.t],[g.sub.t-1],[[OMEGA].bar]) and d/([g.sub.t],[g.sub.t-1],[bar.[OMEGA]])/d[g.sub.t] > dl([g.sub.t],[g.sub.t-1] [[OMEGA].bar])/d[g.sub.t].

Given that the period t expected utility function can be written as

[E.sub.t][U.sub.t]([g.sub.t],l[[g.sub.t], [g.sub.t-1], [OMEGA]] + [[epsilon].sub.t]) = [[mu].sub.t][U.sub.t]([g.sub.t], [g.sub.t-1], [bar.[OMEGA]]) + (1 - [[mu].sub.t])[[bar.U].sub.t]([g.sub.t], [g.sub.t-1], [[OMEGA].bar]),

where

[U.sub.t]([g.sub.t], [g.sub.t-1], [OMEGA]) = [[integral].sup.[infinity].sub.-[infinity]] [U.sub.t]([g.sub.t], l[[g.sub.t], [g.sub.t-1], [OMEGA]] + [[epsilon].sub.t])f ([[epsilon].sub.t])d[[epsilon].sub.t],

then the period 2 optimal money growth rate [g.sup.*.sub.2]([g.sub.1],[[mu].sub.2]) is found according to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This equation yields the second period value function

V([g.sub.1], [[mu].sub.2] = [E.sub.2][U.sub.2]([g.sup.*.sub.2] [[g.sub.1], [[mu].sub.2], l[[g.sup.*.sub.2]([g.sub.1], [[mu].sub.2]), [g.sub.1], [OMEGA]] + [[epsilon].sub.2].

Similarly, for a given initial value [g.sub.0], the period 1 money growth rate is found by choosing [g.sub.1] to maximize

[E.sub.1][U.sub.1]([g.sub.1], l[[g.sub.1], [g.sub.0], [OMEGA]] + [[epsilon].sub.1]) + [delta][[integral].sup.[infinity].sub.-[infinity]] V([g.sub.1], [[mu].sub.2])h([m.sub.1], [g.sub.1], [g.sub.0])d[[m.sub.1],

where

h([m.sub.1], [g.sub.1], [g.sub.0] = [[mu].sub.1]f([m.sub.1] - l[[g.sub.1], [g.sub.0], [bar.[OMEGA]]]) + (1 - [[mu].sub.1])f ([m.sub.1] - l[[g.sub.1], [g.sub.0], [[OMEGA].bar]])

and

[[mu].sub.2] = [[mu].sub.1]f([m.sub.1] - l[[g.sub.1], [g.sub.0], [bar.[OMEGA]]]) / h([m.sub.1], [g.sub.1], [g.sub.0]).

Let [G.sup.*.sub.1] denote the set of period 1 maximizers and let [g.sup.*.sub.1]([g.sub.0],[[mu].sub.1]) be an element of [G.sup.*.sub.1]. (23)

To study the effect of updating beliefs on the optimal money growth rate, I compare [g.sup.*.sub.1]([g.sub.0], [[mu].sub.1]) with the optimal money growth rate set by a central bank that does not consider the effect that current policy has on posterior beliefs and that does not update beliefs or their distribution once new observations become available, that is, the nonlearner central bank. A nonlearner always makes decisions based on prior beliefs.

DEFINITION. A central bank is said to be a nonlearner if it does not account for the effect of the first period money growth rate on posterior beliefs or their distribution and does not update beliefs once new observations become available. Let [g.sub.nonlearner] ([g.sub.0],[[mu].sub.1]) denote the nonlearner's period 1 money growth rate, then

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assume that the nonlearner's utility function is strictly concave in [g.sub.1] and thus that [g.sub.nonlearner]([g.sub.0],[[mu].sub.1]) denotes a unique maximum.

To establish the conditions under which the optimal money growth rate in this intrinsically dynamic model [g.sup.*.sub.1]([g.sub.0],[[mu].sub.1]) differs from the nonlearner's rate [g.sub.nonlearner]([g.sub.0], [[mu].sub.1]), I compare the first-order conditions of the nonlearner with the fully rational central bankers.

PROPOSITION 6. If (d/d[g.sub.1]) ([[integral].sup.[infinity].sub.-[infinity]])V[g.sub.1],[[mu].sub.2], then the fully optimal first period money growth rate is above (below) the nonlearner's rate, that is, [g.sup.*.sub.1]([g.sub.0],[[mu].sub.1]) > (<)[g.sub.nonlearner] ([g.sub.0],[[mu].sub.1]).

PROOF. The proof follows from concavity of [E.sub.1][U.sub.1] ([g.sub.1],l[[g.sub.1],[g.sub.0], [OMEGA]] + [[epsilon].sub.1]) + [delta]V([g.sub.1],[[mu].sub.1]) on [g.sub.1]. QED.

Differentiating ([[integral].sup.[infinity].sub.-[infinity]] V[[g.sub.1],[[mu].sub.2]]h[[m.sub.1],[g.sub.1],[g.sub.0]]d[m.sub.1] - V[[g.sub.1],[[mu].sub.1]]) with respect to [g.sub.1] yields

[[integral].sup.[infinity].sub.-[infinity]][differential]V ([g.sub.1], [[mu].sub.2])/[differential][g.sub.1]h([m.sub.1], [g.sub.1], [g.sub.0])d[m.sub.1] - [differential]V([g.sub.1], [[mu].sub.1])/ [differential][g.sub.1] + [[integral].sup.[infinity].sub.-[infinity]] dV([g.sub.1], [[mu].sub.2])/d[[mu].sub.2]d[[mu].sub.2]/d[g.sub.1] h([m.sub.1], [g.sub.1], [g.sub.0])d[m.sub.1]+ [[integral].sup.[infinity].sub.-[infinity]] V([g.sub.1], [[mu].sub.2]) dh([m.sub.1], [g.sub.1], [g.sub.0])/d[g.sub.1]d[m.sub.1].

The term [[integral].sup.[infinity].sub.-[infinity]](dV[[g.sub.1], [[mu].sub.2]]/d[[mu].sub.2])(d[[mu].sub.2]/d[g.sub.1])h([m.sub.1], [g.sub.1],[g.sub.0])d[m.sub.1] + [[integral].sup.[infinity].sub.-[infinity]] V([g.sub.1],[[mu].sub.2])(dh[[m.sub.1],[g.sub.1],[g.sub.0]]/d[g.sub.1]) d[m.sub.1] shows the effect that first period money growth has on the second period utility via posterior beliefs ([[mu].sub.2]) and their distribution h([m.sub.1],[g.sub.1],[g.sub.0]). Call this, the learning effect. Note that the learning effect appears in both the basic model of section 2 (in which dl[[g.sub.t],[g.sub.t-1],[OMEGA]]/d[g.sub.t-1] = 0) and in this extended model.

After some algebra the learning effect can be written as follows (25)

(dl[[g.sub.1], [g.sub.0], [bar.[OMEGA]]/d[g.sub.1] - dl[[g.sub.1], [g.sub.0], [[OMEGA].bar]/d[g.sub.1]) ([[integral].sup.[infinity].sub.-[infinity]] [d.sup.2]V[[g.sub.1], [[mu].sub.2]]/d[[mu].sup.2.sub.2] d[[mu].sub.2]/d[[mu].sub.1] [[mu].sub.2][1 - [[mu].sub.1][f.bar]d[[mu].sub.1]).

PROPOSITION 7. Assume that the MLRP and Assumption 4 hold. Then, the learning effect is positive.

PROOF. See Appendix E.

The learning effect therefore determines that the fully rational central bank should increase the money growth rate to learn about its environment. However, in this dynamic model there is a new effect, [[integral].sup.[infinity].sub.-[infinity]]([differential] V[[g.sub.1], [[mu].sub.2]]/[differential][g.sub.1])h([[mu].sub.1], [g.sub.1],[g.sub.0])d[m.sub.1] - [differential]V([g.sub.1], [[mu].sub.1])/ [differential][g.sub.1] call it the lagged-policy effect, which may work in the opposite direction. The lagged-policy effect shows the effect that first period monetary policy has on the future economic environment, [m.sub.2], and through it on the central bank's expected second period utility.

PROPOSITION 8. If [differential]V([g.sub.1], [[mu].sub.2])/ [differential][g.sub.1] is convex (concave) in [m.sub.1], then the lagged-policy effect is positive (negative).

PROOF. See Appendix F.

If [differential]V([g.sub.1], [[mu].sub.2])/[differential] [g.sub.1] is convex in [m.sub.1], it is clear that the fully optimal money growth rate is larger than the nonlearner's rate. However, if [differential]V([g.sub.1], [[mu].sub.2])/[differential][g.sub.1] is concave in [m.sub.1], the learning effect and the lagged-policy effect work in opposite directions. The relative size of the two effects will then determine whether the fully optimal money growth rate is larger or smaller than the nonlearner's rate.

To summarize, when current monetary policy actions are allowed to affect future economic outcomes, a rational central bank that updates beliefs and their distribution once new observations become available has to deal with two effects: the effect that its current actions have on the amount of information (on posterior beliefs and their distribution) and the effect that they have on future economic variables. The interaction between these two effects determines whether or not a temporary acceleration of money growth increases expected utility.

7. Conclusions

In this article I assume that the monetary authority makes decisions without fully knowing the effect of monetary policy on the economic environment but learns about it over time using Bayes' rule. I find that the central bank's process of updating beliefs leads to volatility of monetary policy. Consider first the basic model in which current policy actions only affect current economic variables. Then, assuming that higher money growth rates are more (less) informative, the central bank initially increases (decreases) the money growth rate above (below) the myopic rate to learn about its environment. Depending on the information that the central bank receives in the first period, it adjusts the rate of growth of money in the second period.

When monetary policy affects the present and future economic environments, a rational central bank that updates beliefs and their distribution once new observations become available has to deal with two effects: the effect that its current actions have on the amount of information (the learning effect), and the effect that they have on the future economic variables (the lagged-policy effect). The interaction between these two effects determines whether or not the central bank manipulates first period policy to increase its information.

It is important to note that throughout the article the central bank was assumed to choose money supply. An alternative specification in which the bank chooses the interest rate is also possible. However, behavioral equations must be modified to appropriately reflect the relationship between interest rates and other variables. It is also important to note that when monetary decisions are made within certain, deterministic environments, the choice of the instrument of monetary policy is innocuous. However, when faced with uncertainty the choice of the instrument of monetary policy becomes relevant. (26) Hence, within the context of this model, it is possible that the central bank can do better with an interest rate rule rather than with a money growth rate rule. (27) The model in this article could therefore be extended to include a specific functional relationship between the nominal interest rate and the rate of growth of money supply. Doing so would allow a study of the optimal way to conduct monetary policy and the optimal way to generate information.

Appendix A: Proof of Proposition 1

PROPOSITION 1. The central bank's second period value function is strictly convex in posterior beliefs, [d.sup.2]V([[mu].sub.2])/d[[mu].sup.2.sub.2] > 0. Information is thus said to be valuable.

PROOF. Since [g.sup.*.sub.1]([[mu].sub.2]) yields a (unique) maximum in period 2 for posterior [[mu].sub.2], then V([[mu].sub.2]) > [[mu].sub.2][U.sub.2]([g.sup.*.sub.2][[[mu].sub.2], [bar.OMEGA]) + (1 - [[mu].sub.2][[bar.U].sub.2]([g.sup.*.sub.2] [[[mu].sub.2]],[[OMEGA].bar]), for all [[mu].sub.2] [not equal to] [[mu].sub.2]. Then, [theta]V([[mu].sub.2]) + (1 - [theta])V ([[mu]'.sub.2]) > V ([[mu].sub.2]), for [theta] [member of] (0, 1) and [[mu].sub.2] = [theta][[mu].sub.2] + (1 - [theta])[[mu]'.sub.2]. Thus, V([[mu].sub.2]) is strictly convex in it [[mu].sub.2]. QED.

Appendix B: Derivation of Equation 2

In this Appendix I show, following Mirman, Samuelson, and Schlee (1994, pp. 382-3), that

dW([g.sub.1])/d[g.sub.1] = [bar.l'] - [l'.bar]) [[integral].sup.[infinity].sub.-[infinity]] [d.sup.2]V([[mu].sub.2]) / d[[mu].sub.2] d[[mu].sub.2]/d[m.sub.1] [[mu].sub.2](1 - [[mu].sub.1]) [f.bar]d[[mu].sub.1],

where [bar.f] = f([m.sub.1] - l[[g.sub.1], [bar.[OMEGA]], [f.bar] = f([m.sub.1] - l[[g.sub.1],[[OMEGA].bar]), [bar.l] = l([g.sub.1],[bar.[OMEGA]]) and [l.bar] = l([g.sub.1],[[OMEGA].bar]).

Differentiating W([g.sub.1]) reside the integral,

(A1) dW([g.sub.1])/d[g.sub.1] = [[integral].sup.[infinity].sub.-[infinity]] (dV[[[mu].sub.2]/d[[mu].sub.2] d[[mu].sub.2] /d[g.sub.1]h[[m.sub.1], [g.sub.1]] + V[[[mu].sub.2] dh[[m.sub.1], [g.sub.1]]/d[g.sub.1])d[m.sub.1].

Integrating the second term of Equation A1 by parts,

(A2) dW([g.sub.1])/d[g.sub.1] = [[integral].sup.[infinity].sub.-[infinity]] (dV[[[mu].sub.2]/d[[mu].sub.2] d[[mu].sub.2]/d[g.sub.1]h[[m.sub.1], [g.sub.1]] + dV[[[mu].sub.2] d[[mu].sub.2]/d[m.sub.1] [[[mu].sub.1][bar.f][bar.l'] + (1 - [[mu].sub.1])[f.bar][l'.bar]]) d[m.sub.1].

Now note, using Bayes' rule, that d[[mu].sub.2]/d[m.sub.1] = ([[mu].sub.1][bar.f] - [[mu].sub.2][dh([m.sub.1],[g.sub.1]) /h([m.sub.1],[g.sub.1]) and

(A3) d[[mu].sub.2]/d[g.sub.1] = -[[mu].sub.1][bar.f'][bar.l'] - [[mu].sub.2] (dh[[m.sub.1], [g.sub.1]]/d[g.sub.1])/h([m.sub.1], [g.sub.1]) = -[bar.l']d[[mu].sub.2]/d[[mu].sub.1] + [[mu].sub.2] (1 - [[mu].sub.1])[f'.bar](l'.bar] - [bar.l'])/h([m.sub.1], [g.sub.1])

Plugging Equation A3 into Equation A2 and canceling terms,

(A4) dW([g.sub.1])/d[g.sub.1] = ([l'.bar] - [bar.l']) ([[integral].sup.[infinity].sub.-[infinity]]dV[[[mu].sub.2]/ d[[mu].sub.2] d[[mu].sub.2]/d[m.sub.1] [1 - [[mu].sub.1]] [f.bar]d[m.sub.1] + [[integral].sup.[infinity].sub.-[infinity]] dV[[[mu].sub.2]]/d[[mu].sub.2] [[mu].sub.2][1 - [[mu].sub.1]] [f'.bar]d[m.sub.1]).

Integrating by parts the second term in the right-hand side of Equation A4 yields Equation 2. QED.

Appendix C: Proof of Proposition 3

PROPOSITION 3. Given the MLRP, if [g.sub.1] < g, then d[[mu].sub.2] d[m.sub.1] < 0, and thus information decreases with the money growth rate; if [g.sub.1] > g, then d[[mu].sub.2]/d[m.sub.1] > 0, and thus information increases with the money growth rate.

PROOF: From Bayes' rule, d[[mu].sub.2]/d[m.sub.1] = [[mu].sub.1] (1 - [[mu].sub.1])([bar.f][f'.bar] - [f'.bar][bar.f])/ [([[mu].sub.1][bar.f] + [1 - [[mu].sub.1]][f.bar]).sup.2], then, given the MLRP, if [g.sub.1] > (<), g then d[[mu].sub.2]/d[m.sub.1] > (<)0. QED.

Appendix D: Proof of Proposition 5

PROPOSITION 5. Assume that dU([g.sub.t],[bar.OMEGA]])/ d[g.sub.t] > (<)dU([g.sub.t],[[OMEGA].bar]/d[g.sub.t]. Then, the optimal money growth rate in period 2 is increasing (decreasing) in posterior beliefs. That is, for any two beliefs [[mu].sub.2], [[mu]'.sub.2] [member of] (0,1) if [[mu].sub.2] > [[mu]'.sub.2] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

PROOF. Let [E.sub.2]U([[mu].sub.2]) = [[mu].sub.2]U ([g.sub.2],[bar.[OMEGA]]) + (1 - [[mu].sub.2])U([g.sub.2], [[OMEGA].bar]), then, [E.sub.2]U([[mu]'.sub.2]) = [E.sub.2] U[[mu].sub.2] + ([[mu]'.sub.2]. - [[mu].sub.2])(U[[g.sub.2], [bar.OMEGA]] - U[[g.sub.2],[[OMEGA].bar]]). Differentiating [E.sub.2]U([[mu]'.sub.2]) with respect to [g.sub.2] and setting it equal to zero yields

D[E.sub.2]U([[mu]'.sub.2])/d[g.sub.2] = d[E.sub.2]U([[mu].sub.2])/ d[g.sub.2] + ([[mu]'.sub.2] - [[mu].sub.2])(dU[[g.sub.2], [bar.[OMEGA]]/d[g.sub.2] - U[[g.sub.2], [[OMEGA].bar]/d[g.sub.2]]) = 0.

The proof thus follows from concavity of [E.sub.2]U([[mu].sub.2]). QED.

Appendix E: Proof of Proposition 7

PROPOSITION 7. Assume that the MLRP and Assumption 4 hold. Then, the learning effect is positive.

PROOF. The learning effect is given by ([dl([g.sub.1],[g.sub.0], [bar.OMEGA]])/d[g.sub.1] - dl([g.sub.1],[g.sub.0],[[OMEGA].bar])/ d[g.sub.1]])([[integral].sup.[infinity].sub.-[infinity]][d.sup.2] V([g.sub.1], [[mu].sub.2])/d[[mu].sup.2.sub.2]][d[[mu].sub.2]/ d[m.sub.1]][[mu].sub.2][1 - [[mu].sub.1][f.bar]d[m.sub.1]). By Assumption 4, ([dl([g.sub.1],[g.sub.0],[bar.[OMEGA]]/d[g.sub.1]] - [dl([g.sub.1],[g.sub.0],[bar.[OMEGA]])/d[g.sub.1]] - [dl([g.sub.1], [g.sub.0], [[OMEGA].bar])/d[g.sub.1]]) > 0; Assumption 4 combined with the MLRP and Bayes' rule implies that d[mu].sub.2]/[dm.sub.1] > 0. Finally. since V([g.sub.1], [mu].sub.2]) is the supremum of functions linear in [mu].sub.2]. then [d.sup.2]V([g.sub.1], [mu].sub.2])/ d[mu].sup.2.sub.2] > 0. Then the learning effect is positive. QED.

Appendix F: Proof of Proposition 8

PROPOSITION 8. If [differential]V([g.sub.1], [mu].sub.2])/[differential] [g.sub.1] is convex (concave) in [m.sub.1], then the lagged-policy effect is positive (negative).

PROOF. If [differential]V([g.sub.1], [mu].sub.2])/[differential] [g.sub.1] is convex (concave) in [m.sub.1], then by Jensen's inequality.

[[integral].sup.[infinity].sub.-[infinity]] [differential]V ([g.sub.1], [mu].sub.2])/[differential][g.sub.1] h([m.sub.1], [g.sub.1], [g.sub.0]) d[m.sub.1] > (<) [differential]V([g.sub.1], [[integral].sup.[infinity].sub.-[infinity]][mu].sub.2]h[[m.sub.1], [g.sub.1], [g.sub.0]]d[m.sub.1]/[differential][g.sub.1] = [differential] V([g.sub.1], [m.sub.1])/[differential][g.sub.1].

Hence, if [differential]V([g.sub.1], [mu].sub.2])/[differntial] [g.sub.1] is convex (concave) in [m.sub.1]. the lagged policy effect is positive (negative). QED.

Appendix G: Proof of Footnote 27

In this appendix I show with a very simple one period example that when the central bank makes decisions under uncertainty, then interest rate and money growth rate rules are not equivalent, and, furthermore, that interest rate roles may be preferable.

Assume that the central bank's utility function is given by S - ([g.sup.2]/2), and assume a linear money demand function, m = a + bi + [epsilon], where a > 0, b < 0. and where the expected value of [epsilon] is zero. Assume that the nominal interest rate is a linear function of the money growth rate, i = A + Bg. where A [greater than or equal to] 0 and B > 0. Finally. assume that the central bank is uncertain about the parameter B, which can take on two possible values. B [member of] {[B.bar], B}. Let [mu] equal the probability that B = B.

If the central bank chooses the rate of growth of money, it maximizes [U.sub.g] = (a + bA + bE[B]g)g - ([g.sup.2]/2), with respect to g, where E(B) = [mu]B + (1 - [mu])[B.bar]. As a result, the optimal money growth rate is given by [g.sup.*] = (a + bA)/[1 - 2bE(B)], which yields expected utility [U.sub.g] = [(a + bA).sup.2]/(2[1 - 2bE(B)]).

If the central bank chooses an interest rate rule, it maximizes [U.sub.i] = (a + bi)(i - A)E(1/B) - ([[i - A].sup.2]/2)E(1/[B.sup.2]) with respect to i, where E(1/B) = [mu](1/B) + (1 - [mu])(1/[B.bar]), and E(1/ [B.sup.2]) = [mu](1/[B.sup.2]) + (1 - [mu])(1/[[B.bar].sup.2]). As a result, the optimal interest rate is given by [i.sup.*] = [AE(1/[B.sup.2]) + (a - bA)E(1/B)]/[E(1/[B.sup.2]) - 2bE(1/B)]. and the expected utility is given by [U.sub.i] = [[(a + bA).sup.2]E[(1/B).sup.2]]/(2[E(1/[B.sup.2]) - 2bE(1/B)]).

In general, [U.sub.g] [not equal to] [U.sub.i], but some numbers can help make this point more obvious. For example, if [mu] = 0.5. b = -1, a - 10, A = 0, B = 2, and [B.bar] = 1, then [U.sub.g] = 12.5 and [U.sub.i] = 13.24. interest rate roles might therefore be superior to money growth rate rules. QED.

(1) The role that the expectations of the private sector play in effective monetary policy making is evident in both the time inconsistency literature (see, e.g., Kydland and Prescott 1977 and Barro and Gordon 1983) and in the hyperinflations literature (as can be seen in Cagan 1956; Sargent and Wallace 1981; and Marcet and Nicolini 2003).

(2) Other theoretical models of Bayesian central bankers include Balvers and Cosimano (1994). Caplin and Leahy (1996), and Kasa (1999).

(3) Caplin and Leahy (1996) are an important exception since they explicitly model the strategic interactions between the central bank and the private sector learning processes. For more on this topic, see Wieland (1998).

(4) As an example of nonrenewable fixed appointments consider the European Central Bank (ECB). The term of office of the executive board of the ECB (the president, vice-president, and the other four members) is eight years and is nonrenewable.

(5) This article is more similar to other models of information and learning that lie outside the context of monetary economics, such as the models of monopoly experimentation by Mirman, Samuelson, and Urbano (1993) and the model of experimentation within signal dependent problems by Datta, Mirman, and Schlee (2002). In fact, a lot of the notation in this model follows Datta, Mirman, and Schlee (2002).

(6) Bertocchi and Spagat (1998) is an example of a macroeconomic model that incorporates signal dependency features.

(7) Although the results of the article show that in theory it is optimal to experiment under very general conditions, it is not clear that in reality central bankers behave in this way. Some authors, however, have wondered whether certain policies observed in the past were in fact a process of search. For example, Bertocchi and Spagat (1993, p. 170), when referring to the disinflationary episode in the U.S. economy of the late 1970s and early 1980s, said that "one could argue that the 1979-82 'Volcker experiment' is an example of precisely such an experimental policy." Caplin and Leahy (1996, p. 689) were also motivated to model monetary policy as a process of search by the behavior of central bankers, and more particularly of the Fed and the Bundesbank during the 1990-1992 recession when they "appeared to be searching for the optimal stimulus. They would lower interest rates somewhat, then wait to see how the economy responded. If the recession continued, they lowered rates again. If signs of recovery became apparent, they held steady and turned their attention towards inflation." Finally, Joseph Stiglitz (1997, p. 10), when Chairman of the Council of Economic Advisers, asked "are there policies that can affect the degree of uncertainty about the value of NAIRU or of policy tradeoffs?"

(8) This type of situation can arise within economies with weak governments and civil disorder (Capie 1986), within governments involved in wars of attrition over fiscal reforms (Alesina and Drazen 1991), or within countries with inefficient or unsophisticated tax systems, common in countries with high political instability and polarization (Cukierman, Edwards, and Tabellini 1992).

(9) This example is based on Cukierman (1988). He, however, uses a more complex specification of this utility function by assuming that the parameter x varies stochastically over time.

(10) Alternatively, a central bank that needs seigniorage can have a utility function such as [U.sub.t]([g.sub.t], [m.sub.t]) = -[([S.sub.t] - [bar.S]).sup.2], where [bar.S] represents the bank's seigniorage target.

(11) Note that the expectations of inflation of the private sector are not explicitly modeled in the example.

(12) Note once more that expectations of inflation are not explicitly modeled. In this particular example this implies that the shock [[epsilon].sub.t] must, at the very least, contain the unexpected component of monetary policy.

(13) The MLRP derives its name from the facts that f'([epsilon])/f([epsilon]) = d log f([epsilon]/d[epsilon], and that the function log f([[epsilon].sub.t]) = log f([m.sub.1] - l[[g.sub.t], [OMEGA] is the likelihood function lot this model in which [m.sub.1] is the endogenous variable and [OMEGA] is the parameter to be estimated. For more on the MLRP, I refer the reader to Laffont (1989. pp. 185-6). The MLRP can also be found in several applications, some of which are Mirman, Samuelson, and Urhano (1993) and Jeitschko and Mirman (2002), among others.

(14) Since [[epsilon].sub.t] is distributed independently from [g.sub.t], and since by assumption -[infinity] < [[epsilon].sub.t] < [infinity], it is possible to have negative values of [m.sub.t]. This assumption significantly simplifies the analysis, and it is traditionally used in the learning literature given the transparency of the results (see, for example, Mirman, Samuelson, and Urbano 1993 and Mimlan, Samuelson, and Schlee 1994). However, a more "realistic" assumption does not affect the qualitative results of the model, and thus little or nothing is gained by, a more realistic setup. Moreover, one can assume that the probability of negative observations is small enough that it can be neglected.

(15) This model fails to provide a rationale of how the monetary authority comes to know a point in the curve. Technically speaking, since I assume that there are two possible functions, the monetary authority must know a priori either a point in the curve or the slope of the curve at each money growth rate, The latter, however, seems to be a much stronger presumption.

(16) Given curvature assumptions, the second order condition is satisfied with strict inequality, and thus [g.sup.*.sub.2]([[mu].sub.2]) denotes a unique maximum.

(17) Note that the solution to Equation 1 is not necessarily unique since W([g.sub.1]) is not necessarily concave.

(18) In Section 7, I introduce a variation of the model in Milch there are additional dynamic features.

(19) Note that [g.sub.myopic]([[mu].sub.1]) and [g.sub.2]([[mu].sub.2]) only differ in the value of [mu]. Given curvature assumptions, [g.sub.myopic]([[mu].sub.1]) exists and is a unique maximum to the myopic problem.

(20) Other expositions of Blackwell's results can be found in Kihlstrom (1984), Laffont (1989), and Hirshleifer and Riley (1992).

(21) Which of the two assumptions is more appropriate is an empirical question that is not addressed in this article.

(22) Datta, Mirman, and Schlee (2002) present a similar model of experimentation in signal dependent problems, in which, following my notation, the outcome variable can be written as [m.sub.t] = l([g.sub.t], [m.sub.t-1], [OMEGA]) + [[epsilon].sub.t].

(23) Assuming that [[differential].sup.2][E.sub.t][U.sub.t]([g.sub.t],[g.sub.t-1],[OMEGA]] + [[epsilon].sub.t])/[differential][g.sup.2.sub.t] < 0, then [g.sup.*.sub.2]([g.sub.1],[[mu].sub.2]) exists and is a unique maximum. The first period maximum [g.sup.*.sub.1]([g.sub.0],[[mu].sub.1]), however, is not necessarily unique given possible nonconcavities in [[integral].sup.[infinity].sub.-[infinity]] V([g.sub.1],[[mu].sub.2])h([m.sub.1],[g.sub.1],[g.sub.0])d[m.sub.1].

(24) Note that if dl([g.sub.t],[g.sub.t-1], [OMEGA])/d[g.sub.t-1] = 0, then the definitions of a nonlearner and a myopic central bank are equivalent.

(25) The algebra involved in this transformation is virtually identical to that followed in Appendix B and it is thus omitted from the article.

(26) This point was first brought forth in a seminal paper by Poole (1970).

(27) A very simple example that proves this point is presented in Appendix G.

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Hilde Patron, Department of Economics and Finance, Louisiana Tech University, Ruston, LA 71272, USA; Email [e-mail].

I wish to thank Ronald Balvers; Bob Chirinko; Carl Davidson; Arturo Galindo: Mike Highfield; Thoinas Jeitschko; Leonard Minnan; Rowena Pecchenino: Ken Roskelley; the seminar participants in Butler University, California State University at Los Angeles, Drake University, Emory University, Louisiana Tech University, Michigan State University, and the Midwest Theory Meetings; and two anonymous referees for helpful comments and suggestions. All remaining errors are the sole responsibility of the author.

Received August 2002; accepted August 2004.