Optimal information acquisition and monetary policy

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Journal of Macroeconomics 30 (2008) 1370–1389 www.elsevier.com/locate/jmacro

Optimal information acquisition and monetary policy q Thomas E. Cone

*

Department of Business Administration and Economics, SUNY College at Brockport, 119 Hartwell Hall, Brockport, NY 14420, United States Received 23 January 2007; accepted 25 September 2007 Available online 12 October 2007

Abstract I study optimal monetary policy with an expectational AS curve and private agents who optimally choose their amount of information pertinent to predicting policy. Shocks with time-varying variance (ARCH) induce interesting information acquisition (IA) dynamics; optimal IA affects optimal policy and vice versa. Under discretion, IA dynamics cause time-varying effectiveness of policy because of the expectational AS curve; policy may be rendered completely ineffective. In policy game equilibrium, a fall in the shock’s variance typically induces less IA and raises welfare. In one exceptional case the opposite occurs, a result which does not require implausible unstable equilibria. An agent becoming informed increases the endogenous component of economic volatility; IA therefore has a negative externality. Under commitment policy’s effectiveness is again time-varying, but policy is never completely ineffective: commitment enables the central bank to credibly limit policy’s volatility; this limits private agents’ incentive to become informed, so limits expectation-induced policy neutrality. Ó 2007 Elsevier Inc. All rights reserved. JEL classification: E52; E58; D83; D84 Keywords: Information; Expectations; Monetary policy

q

I am indebted to participants of the July 2006 Conference on Learning and Monetary Policy hosted by the Federal Reserve Bank of Saint Louis and three anonymous referees for helpful comments. * Tel.: +1 585 395 5524; fax: +1 585 395 2542. E-mail address: [email protected] 0164-0704/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2007.09.008

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1. Introduction This paper presents optimal monetary policy for a model in which private agents can choose whether to acquire information that helps them predict policy actions. Private sector (PS) information acquisition is important in the model because a monetary policy action has real effects only to the extent that it surprises the PS. This means that as the PS’s information acquisition changes over time, the effectiveness of monetary policy also changes, a phenomenon that would appear to an econometrician as an output-inflation tradeoff with a time-varying slope. This topic is important for monetary policy in empirical economies; to the extent that policy neutralization occurs in such economies when private agents become informed, their information acquisition affects the results of monetary policy.1 In the model, interesting dynamics of information acquisition are produced by output shocks that have time-varying variance—specifically autoregressive conditional heteroskedasticity, or ARCH. The monetary authority’s offsetting of ARCH shocks causes inflation to also exhibit ARCH; furthermore, inflation is the variable the PS might become informed about, but its ARCH behavior implies that becoming informed is worth the cost only in some periods. Recently, interest in the effects of the PS’s information on monetary policy has increased; in particular see Woodford (2002), Ball et al. (2005), and Branch et al. (2006a). This paper advances the literature on this topic in several ways. One way is that PS agents are allowed to have different costs of acquiring information, which seems to be unique in the information strand of the monetary policy literature. Another way is the incorporation of ARCH shocks. The third and most important way is the inclusion of both private agents who optimally choose their information and a monetary authority (MA) that optimally conducts monetary policy. The only other paper with both optimal information acquisition (IA) and an optimizing MA is that of Branch et al. (2006a) (discussed below); the rest of the literature has only one of these features. There is of course a vast literature on optimal monetary policy under exogenous PS information sets. Ball et al. (2005) is part of this literature in that each private agent in their setting has an exogenous probability of information updating. Regarding optimal IA, Evans and Ramey (1992) study a model in which private agents endogenously (optimally in some cases) acquire information under ad hoc monetary policy. Hahm (1987) studies the PS’s optimal IA but does not describe optimal monetary policy response to output shocks. Neither does Hahm consider private agents with different costs of becoming informed, ARCH shocks, or the differences between discretionary and committed policy, which I do below. Branch et al. (2006a) study optimal monetary policy in a model with endogenous IA but assume all PS agents have the same cost function for IA. They also assume the shocks to which the MA responds are homoskedastic, so that the IA problem is not intrinsically time-varying. 1 Since Lucas (1972, 1973) formalized the view that only surprise changes in the money supply have real effects, some work has disputed that view. Woodford (2002) rescues the ‘‘surprise’’ strand of the transmission literature from the principal objections which have been raised against it. Those objections are (1) the theoretical objection that monetary aggregates are almost immediately and freely available, so that private agents can not plausibly be assumed to be surprised, and (2) the empirically based objection that the time profile of output’s response to a monetary shock is much more spread out than the very short lag with which monetary data are published. Woodford models agents as not observing monetary data in real time with perfect precision because they have finite bandwidth in the information-theoretic sense. For reasonable parameterizations his model implies a time profile of output’s response to a monetary shock that agrees well with the data.

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In this paper the following results emerge. First, under discretion, changing IA causes policy’s effectiveness to change over time. This provides a possible explanation for timevarying output-inflation tradeoffs, e.g., the weakening of the tradeoff in the US in the 1970s, when the economy became more volatile.2 Also, the effects of a change in the shock’s variance can be path-dependent in that a switch from the low-variance regime to the high-variance regime does not necessarily induce a mirror image result of a switch from the high-variance regime to the low-variance regime. Second, it is possible, although only in one exceptional case, that a decreased variance of the shock induces more private agents to become informed and lowers welfare, and though such comparative statics are the exception, they do not require equilibria which are implausible on stability grounds. Third, aside from responding to economic volatility, the fraction of agents who are informed also affects the volatility. The higher is the fraction of agents who are informed, the higher is inflation’s volatility; informed agents therefore impose a negative externality on uninformed ones. This result is novel. In previous work IA has positive external effects on the uninformed, because of the information-revealing behavior of the informed, cf. the free rider problem in financial markets.3 Fourth, under commitment some private agents may acquire information in equilibrium, though the MA never chooses policy that would induce all private agents to be informed (as can happen under discretion). Finally, for both discretion (with one exception) and commitment (always), policy is less effective precisely when it would be most useful, when the shock’s variance is high. With discretion this is because the PS becomes informed when output volatility, which the MA would like to offset, is high; this IA behavior tends to neutralize policy in high-variance periods and in fact can completely neutralize it. With commitment it is because the MA is optimally less responsive to shocks in high-variance periods to limit the PS’s information acquisition, and preserve some policy effectiveness. Thus policy is of limited help in high-volatility periods, partly because the policy action has a weak effect on output and partly because the policy action itself is restrained. As explained more thoroughly in Section 3, the above results on discretion are consistent with the stylized fact that US inflation and GDP exhibit correlated regime shifts in their volatilities. Branch and Evans (2007) provide references to empirical work documenting this stylized fact and note, ‘‘[T]he volatility of GDP and inflation vary over time. In particular, each series appears to move in tandem and alternate between high and low variance regimes.’’ The main interest of Branch et al. (2006a) is conditions under which the usual policy tradeoff between output volatility and price level volatility disappears, so that both volatilities could be lowered simultaneously. Their focus differs from the present one in that they are mainly concerned with shifts between equilibria for static environments, i.e., they do not rely on ARCH shocks to motivate discussion of these volatility changes. However, for this very reason their model cannot account for such shifts since it requires going beyond them to explain why the PS and the MA would suddenly change from one Nash 2

In empirical economies there is a similar relationship across countries; price level shocks have less effect on output in countries with higher variances of the price level; see, e.g., Lucas’s classic (1973) paper, Apergis and Miller (2004). 3 The only other paper I know of in which a negative externality from information processing can arise is Evans and Ramey (1992). In that paper, though, the reason is different; it is not caused by the MA’s optimal response to the fraction of informed agents, which is what drives the effect here.

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equilibrium to another. The present paper motivates such equilibrium switching as occurring when the shock’s variance changes. However, in a companion paper, Branch et al. (2006b) provide an analysis of a possible mechanism for a one-way shift, MA learning. While the motivation of this paper is theoretical, it can also be viewed as a contribution to the recent literature on the high macroeconomic volatility of the 1970s and the decline in volatility since 1983, the Great Moderation. One strand of this literature has sought to explain this episode in terms of the Federal Reserve System entertaining overly optimistic beliefs about what monetary policy could accomplish in the 1970s, leading to excessive policy activism, and then learning its way to more resigned beliefs that led to the Great Moderation. Examples of this literature include Sargent (1999), Sargent et al. (2006), and Branch et al. (2006b). Another strand has emphasized the role of changing variances of exogenous shocks that have buffeted the economy; see, e.g., Sims and Zha (2006). This paper strengthens the second kind of argument because the endogenous IA is an amplification mechanism by which a small amount of exogenous ARCH can help explain a large amount of overall macro ARCH (overall in the sense of including both the exogenous and endogenous components of macro volatility). The endogenous IA’s other contribution is to help explain the policy neutralization that was associated with the higher volatility in the 1970s, which the exogenous ARCH by itself does not address. The ‘‘learning’’ strand of the literature is more optimistic because it can imply that a recurrence of the 1970s experience is unlikely, depending on what exactly the Fed has learned (but see the ‘‘Reservations’’ section in Sargent, 1999). The ‘‘shocks’’ argument, at least for discretionary policy, is pessimistic regarding macro volatility; it implies a 1970s recurrence is inevitable because the high-variance regime eventually will return. 2. Benchmark results 2.1. Uninformed private sector Before studying endogenous IA, I review some benchmark results from the extant literature with exogenous information and iid shocks. The basic model is a standard linear-quadratic framework; it has a quadratic MA loss minimized subject to an expectational AS curve as in, e.g. Rogoff (1985), Sargent (1999), Arifovic and Sargent (2003). Time is discrete and infinite. In period t the MA chooses inflation to minimize 2

lt ¼ ap2t þ ðy t  ky Þ ;

ð1Þ

where a > 0, k P 1, pt is the rate of inflation at time t, yt is output at time t, and y is the natural level of output. If k > 1, given the aggregate supply curve below, the standard positive inflation bias arises.4 The MA minimizes this loss subject to the constraint imposed by the aggregate supply curve

4

Several possible reasons the MA wants output to be higher than y appear in the literature. One is that there are distortionary non-monetary policies that make y smaller than the socially optimal level of output. Another possibility is that y is suboptimally low because the private agents are monopolitic competitors. Another possibility is that there is a positive externality created by output. Each private agent takes the external benefit as exogenous; the MA takes the external benefit into account. Yet another possibility is that the MA simply has different preferences from the PS; perhaps there are political economy reasons it wants higher production.

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y t ¼ y þ bðpt  E½pt jI t Þ þ ut ;

ð2Þ 2

where b > 0 and the shock ut is an iid random variable with mean zero, variance r , and pdf f(ut), which the MA observes contemporaneously and before setting inflation. ut is a distortionary shock, e.g., associated with time-varying distortionary taxes, such that the MA would like to offset it to the extent that it moves output away from ky . E[ptjIt] is the rational expectation of date t inflation conditional on the PS’s start-of-date-t information set It, the set of all variables dated t  1 or earlier.5 PS agents are for the time being assumed to be unable to improve their contemporaneous information. One rationale for the MA’s superior information is that it has first access to economic data. Another is that the MA generates a private forecast of output that is superior to the PS’s forecast, as in Canzoneri (1985), Walsh (1995), and Romer and Romer (2000), although in this interpretation the MA would base policy not on ut but on its forecast of ut. Thus, the timing of the game is as follows: the PS sets its inflation expectation, the MA observes the shock ut, the MA sets inflation, yt is determined according to (2). Under discretionary policy with a myopic MA the game between the PS and the MA has a unique rational expectations equilibrium (REE) level of inflation, b b ut ; pt ¼ ðk  1Þy  a a þ b2

ð3Þ

with the PS’s expectation of inflation being ba ðk  1Þy , and REE output is y t ¼ y þ

a ut : a þ b2

ð4Þ

The MA partially offsets shocks’ effect on output; monetary policy cannot be used to systematically raise output. Inflation’s variance is [b/(a + b2)]2r2; note this is also the PS’s expected squared inflation forecast error. With commitment the MA commits to a policy p(ut) to minimize the expected loss in a given period (since there are no connections between periods the one-period optimum is also the dynamic optimum) subject to the constraint that E[ptjIt] is the rational expectation of p(ut). The optimal committed policy is pðut Þ ¼ 

b ut ; a þ b2

ð5Þ

with the PS’s expectation of inflation being zero, and output is y t ¼ y þ

a ut ; a þ b2

ð6Þ

the same as output in the discretionary case. Under commitment the MA responds to shocks just as strongly as under discretion but average inflation is zero instead of positive. 5

As is well known, aggregate supply curves of this form arise from microfoundations in many models, e.g., nominal wage contracts models as in Fischer (1977). In Fischer models firms produce with a Cobb-Douglas production technology; firms and workers negotiate the nominal wage for date t at the end of t  1, and workers agree to supply whatever quantity of labor firms demand at that wage at date t.

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2.2. Informed private sector In this section I derive results for PS agents who are all exogenously endowed with as much information as the MA; these results are useful for the endogenous IA model below. Thus, the timing of the game is as follows: the PS and the MA observe ut, the PS sets its inflation expectation, the MA sets inflation, yt is determined according to (2). Since the PS observes ut contemporaneously and knows the MA’s preferences, it can then predict period t inflation. Under discretion REE inflation, which is also the PS’s inflation expectation, is b b pt ¼ ðk  1Þy  ut ; a a

ð7Þ

and output is y t ¼ y þ ut . REE inflation is more responsive to shocks when the PS is contemporaneously informed about the shock. Since actual inflation equals expected inflation monetary policy has no effect on output and so the output shock is felt fully on output. In the canonical time inconsistency model inflation is positive on average without raising average output; here inflation also varies without stabilizing output. The optimal committed monetary policy with an always-informed PS is simply pt = 0 and output is again y t ¼ y þ ut . If the PS is informed monetary policy cannot affect output, so the best policy is simply to commit to zero inflation. 3. Information acquisition with ARCH shocks This section introduces autoregressive variance of the output shock and endogenous PS choice of whether to observe the shock contemporaneously. Some recent empirical studies finding autoregressivity in output shocks’ variances are Altissimo and Violante (2001), Ho and Tsui (2002), and Sims and Zha (2006). ARCH output shocks generate non-trivial IA dynamics because they cause the MA’s optimal inflation to also be ARCH, so the PS’s optimal choice of whether to become informed varies over time. Below I specify an ARCH process such that the period t  1 squared shock tells agents the variance of the period t shock, thus influencing their date-t IA choice. Endogenous IA choice requires that more be said about the private sector agents. Suppose there is a [0, 1] continuum of them, with agent n having IA cost zn = c + dn with c P 0, d P 0.6 Here zn is the time, effort and resource cost necessary for n to become informed, i.e., to observe ut contemporaneously or to infer it from other data. Let it denote the fraction of agents who are informed in period t. I assume all PS agents affect aggregate outcomes equally and that all agents know the current value of it as they form their inflation forecasts,7 so the AS curve is y t ¼ y þ bpt  bfit E½pt jI t [ it [ ut  þ ð1  it ÞE½pt jI t [ it g þ ut :

ð8Þ

This is the AS curve appropriate for the following timing: having observed ut1 at the start of period t, agent n knows r2t and has the opportunity to pay zn and observe ut. After the PS converges to an equilibrium value of it and the informed observe ut, the MA sets infla6

In what follows I assume d > 0; the case of d = 0 is taken up in a separate section below. This is appropriate if the PS agents play a Nash equilibrium in IA choice. Equilibrium is non-trivial because, as will become apparent below, an agent’s expected payoff of becoming informed depends on the proportion of other agents who are informed. 7

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tion according to the policy regime: myopically taking ut and it as given for discretion, and according to its committed policy for commitment. Then output is realized. Private agent n’s loss is a function of the IA cost n pays and the squared inflation forecast error, a standard form for forecast error losses; see, e.g., Evans and Ramey (1992), Branch et al. (2006a): 2

lnt ¼ Z n;t þ ðE½pt jXn;t   pt Þ ;

ð9Þ

where Zn,t 2 {0, zn} is the cost n pays for IA, Zn,t = 0 meaning n remains ignorant at t, and Xn,t is n’s endogenous date t information set; Xn,t = It [ it if n remains ignorant and Xn,t = It [ it [ ut if n becomes informed. Agent n minimizes the expectation of (9) conditional on the default information set It [ it, acquiring information only if the IA cost zn is strictly less than the expected loss under ignorance.8 The expected loss under ignorance is the mean-centered variance of pt conditional on It; this variance changes over time due to the MA’s response to the ARCH shock. Note that while their IA costs differ, the expected cost of remaining ignorant (given some value of it) is the same for all private agents. In a given period one of the terms in (9) is zero; if n remains ignorant the first term is zero and if n becomes informed the second term is zero. The above loss of course suppresses losses from factors other than forecast errors; as Branch et al. (2006a) remark regarding their similar approach, ‘‘Agents are in effect splitting their decision problem into separate optimization and forecasting problems, a procedure that is often followed, for example, in the least squares learning literature.’’ For the shock’s autoregressive variance I assume ut has pdf ft(ut) 2 {flow(Æ), fhigh(Æ)}, where ft(Æ) has mean zero and variance r2t 2 fr2low ; r2high g, r2low being the variance of the pdf flow(Æ) and r2high ð> r2low Þ being the variance of the pdf fhigh(Æ). Because the variance is the object of interest for the PS, I express the following assumptions in terms of the variance: for some constant x > 0, if u2t1 2 ½0; x then r2t ¼ r2low and if u2t1 > x then r2t ¼ r2high .9 The variance is a two-state Markov process since—given flow(Æ) and fhigh(Æ)—r2t implies the probability of u2t 6 x, and so implies the probabilities of r2tþ1 ¼ r2low and r2tþ1 ¼ r2high . This specification is quite general. It is not necessary to assume, for example, that flow(Æ) and fhigh(Æ) are continuous or symmetric around zero or that they come from the same family of functions parameterized by r2t . 3.1. Discretionary policy In this section I first describe the Nash equilibria of discretionary policy, given the timing as discussed in the previous section. In REE, E[ptjIt [ it [ ut] = pt,10 so b b pt ¼ ðk  1Þy  ut a a þ ð1  it Þb2 8

ð10Þ

Since each PS agent is a measure-zero fraction of the population, his individual IA decision has zero effect on policy’s effectiveness. Thus, agent n faces no tradeoff of thwarting stabilization policy, if n becomes informed, versus helping stabilization policy, if n remains ignorant. So n need not consider any effects of his IA choice on monetary policy or other agents’ behavior; n simply minimizes the expectation of lnt . 9 While it is not required for the results here, it is natural to assume flow(Æ) and fhigh(Æ) are such that 0 < Pr½u2t > xjr2t ¼ r2low  < Pr½u2t > xjr2t ¼ r2high  a n d 0 < Pr½u2t 6 xjr2t ¼ r2high  < Pr½u2t 6 xjr2t ¼ r2low . T h i s ensures the ‘‘right’’ kind of autocorrelation in the variance; it is positively, not negatively, autocorrelated. 10 In the solution procedure the MA optimizes for general expectations first, then REE is imposed.

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and y t ¼ y þ

a ut : a þ ð1  it Þb2

ð11Þ

Note that if it = 0 then (10) reduces to (3) and if it = 1 then (10) reduces to (7). Inflation’s variance conditional on It [ it is  2 b 2 r2t : ð12Þ varðit ; rt Þ ¼ p a þ ð1  it Þb2 From (12) it is apparent that the PS’s IA choice does not only respond to inflation’s variance; it also affects it. Because inflation’s variance is an increasing function of it, informed agents impose a negative externality on uninformed ones. I return to this point in the section on expected losses below. A Nash equilibrium for date t has two components. One is the PS agents, who move first, coordinating on an equilibrium value of it. The second part is the MA’s optimal behavior given it, Eq. (10). Definition 1. A Nash equilibrium for date t consists of MA policy (10), a value of it, and an IA choice for each private agent n such that n is informed iff zn < varðit ; r2t Þ: p

ð13Þ

The number and nature of date t Nash equilibria vary with parameters and whether r2t ¼ r2low or r2t ¼ r2high and I now turn to describing the various cases. 3.1.1. Nash Equilibrium values of it Plainly an equilibrium it has all agents n < it being informed (e.g., if it = 0.75 then it is agents 0–0.75 who are informed, not some other set of agents with measure 0.75). If in period t there is one it 2 (0, 1) such that zit ¼ varp ðit ; r2t Þ I denote it by it ; if there are two such it’s I denote them by it;1 and it;2 with it;1 < it;2 . 11 Then the set of Nash equilibrium values of it has a simple description: if there is one such intersection it it is an equilibrium value of it, and if there are two intersections then both, it;1 and it;2 , are equilibrium values of it. Also, if varp ð0; r2t Þ 6 z0 then it = 0 is an equilibrium and if varp ð1; r2t Þ P z1 then it = 1 is an equilibrium. The positive-measure cases12 and their equilibria are: 1. varp ðit ; r2t Þ may lie above zit for all it (see Fig. 1). In this case the equilibrium has it = 1. 2. varp ðit ; r2t Þ may lie below zit for all it 2 [0, 1]. In this case it = 0. 3. There may be one intersection it , with varp ðit ; r2t Þ lying above zit for it 2 ½0; it Þ and below zit for it 2 ðit ; 1. In this case it ¼ it . 4. There may be one intersection at it , with varp ðit ; r2t Þ lying below zit for it 2 ½0; it Þ and above zit for it > it . In this case the equilibrium values of it are 0, it , and 1. 5. There may be two intersections, so that varp ðit ; r2t Þ lies above zit for it 2 ½0; it;1 Þ, below zit for it 2 ðit;1 ; it;2 Þ, and above zit for it > it;2 . The equilibrium values of it are it;1 , it;2 , and 1. 11 The fact that zn is linear in n together with the shape of varp ðit ; r2t Þ (see (12)) imply that there are at most two such intersections. 12 I ignore knife-edge cases, e.g., varp ðit ; r2t Þ being tangent to the line c + dit.

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Fig. 1. Some possible cases for discretionary policy. The horizontal axis is n in [0, 1]; the units on the vertical axis are utils. The upward-sloping line is the IA cost as a function of n. The three convex curves are three possible cases arising from the MA’s best-response behavior, giving inflation’s variance as a function of n = i. From the top down these depict cases 1, 4, and 2.

Fig. 1 illustrates some possible cases. The best-response dynamics in IA choice are simple; from the starting value of it they move to the right wherever varp ðit ; r2t Þ lies above zit and move to the left wherever varp ðit ; r2t Þ lies below zit . If the best-response dynamics move toward an equilibrium from nearby points that equilibrium is stable and if the dynamics move away it is unstable. I use stability as an equilibrium selection criterion and mostly ignore unstable equilibria; in case 4, it is unstable and in case 5, it;2 is unstable. There is an infinity of overall Nash equilibria in the full intertemporal game since the PS could coordinate on switching between date t equilibria as a function of t (if there are multiple date t equilibria). I assume they do not switch unless a change between r2low and r2high compels it. This seems reasonable for an economy with a large number of agents, because coordinating on an uncompelled change in equilibrium would be difficult. 3.1.2. Comparative statics For the topic of information acquisition the main parameters of interest are r2t , c and d. Equilibrium it in the interior of [0, 1] are given by solutions to

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2 b r2t ¼ c þ dit : a þ ð1  it Þb2

ð14Þ

To derive the comparative statics effects of r2low or r2high , given that the equilibrium remains an interior one, first find the effect on the equilibrium fraction of informed agents.13 Totally differentiating (14) gives oit b2 ½a þ ð1  it Þb2  ¼ :  3 or2t d a þ ð1  it Þb2  2b4 r2t

ð15Þ

This can be signed for any particular interior equilibrium using the equilibrium’s stability or instability. For stable interior equilibria, varp(Æ) intersects z(Æ) from above and it can be shown that this implies oit =or2t is positive. In this kind of equilibrium, then, the intrinsic noisiness of the economy, r2t , affects IA in an intuitively sensible way; the worse the noise, the greater the fraction of agents who acquire information. Also, that oit =or2t is positive implies ovarp ðit ; r2t Þ oit ¼d 2 ort or2t

ð16Þ

is positive. In such cases, e.g., case 3, inflation’s variance is increasing in r2t for two reasons. One is the direct effect, due to the fact that the MA sets inflation to offset the shock; the other reason is the indirect effect due to the effect of r2t on the equilibrium it. Both of these effects can be seen by inspection of (12). Applying similar arguments to vary ðit ; r2t Þ, output’s variance conditional on It [ it, we have the following: Proposition 1. For stable interior equilibria

oit or2t

> 0,

ovarp ðit ;r2t Þ or2t

> 0, and

ovary ðit ;r2t Þ or2t

> 0.

In stable equilibria, the higher is the shock’s variance, the higher is the equilibrium measure of informed agents, and thus the smaller is monetary policy’s effect on output. Because of this, the MA sets a higher responsiveness of inflation to the output shock, so inflation’s variance is higher. Output’s variance is also higher though, because the higher it reduces policy’s effectiveness via adjustments to expectations. In fact, these comparative statics can also be shown to hold without assuming that equilibria remain interior, unless the constraint i 6 1 binds or the exception in the next proposition holds. Opposite comparative statics can be shown to prevail in the implausible unstable interior equilibria. One can consider the effects of a discrete difference in r2low or in r2high , as well as the infinitesimal differences of derivatives, thus bringing out the effects of a change in the cases. (As explained below, interpreting these effects as comparative statics is less appropriate than interpreting them as changes over time.) In particular, it may not be in an interior equilibrium for two different values of r2t , e.g., if for a small value of r2t case 2 pertains and so it = 0, while for a high value of r2t case 1 pertains and so it = 1. Then (14)–(16) do not apply, but it is still possible to show the following. 13 Due to the assumption that PS agents do not switch between equilibria if it is avoidable, the comparative for some endogenous variable x are also the effects on x of a real-time switch between r2low statics orox2 and orox 2 high 2 low and rhigh , with one exception mentioned below.

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Proposition 2. Suppose r2t switches from r2high to r2low and this induces a switch from case 5 Dit with it ¼ it;1 < 1 to case 4 with it = 1. The observed effects, in the dynamics sense, are Dr 2 < 0 and

Dvarp ðit ;r2t Þ Dr2t

< 0, and

Dvary ðit ;r2t Þ Dr2t

t

cannot be signed.

Proof. The sign of Dit =Dr2t is obvious. varp ðit ; r2t Þ ¼ c þ dit;1 initially and varp ðit ; r2t Þ > c þ d after the switch to it = 1, so Dvarp ðit ; r2t Þ=Dr2t is negative. For the last part, note from (11) that " #2 a 2 2 r2high ; D varðit ; rt Þ ¼ rlow  y a þ ð1  it;1 Þb2 the sign of which cannot be determined.

h

The effect on output’s variance is ambiguous because the direct effect of a drop in the shock’s variance lowers output’s variance, but the rise in IA makes stabilization policy less effective, which raises output’s variance. The change in interpretation, from comparative statics to a change over time, is compelled by the implicit time direction of the assumption that agents only switch equilibria if they must. If agents start playing case 5 with it ¼ it;1 < 1 and the equilibrium they are playing is destroyed by a fall in variance, going to case 4 with it = 1 is one of the things that can happen (it = 0 is the other). On the other hand, if they were to start at case 4 with it = 1 and the variance were to rise, staying at it = 1 with case 5 is the only thing that can happen, by the assumption that they only switch between equilibria if they must. So, going from case 4 to case 5 is not the simple mirror image of the opposite change, hence no conclusions can be drawn without specifying the direction of the change. Proposition 1 can be summarized as follows: it changes weakly, and varp ðit ; r2t Þ, and vary ðit ; r2t Þ change strictly, in the same direction as r2t . This supports the ‘‘change in exogenous shocks’’ explanation for the volatility of the 1970s, followed by the Great Moderation, because a change in r2t induces a change in IA which magnifies the effect on overall economic volatility, e.g., an increase in the shock’s variance not only worsens economic volatility directly, but also makes more PS agents become informed. This reduces monetary policy’s ability to stabilize away the shock’s effect on output, thus raising output’s variance. It also induces an endogenous increase in monetary activism as the MA responds to policy’s reduced effectiveness, thus raising inflation’s variance. This looks something like the 1970s. It also is consistent with the stylized fact, noted by Branch and Evans (2007) and the empirical work cited therein, that US inflation and GDP display regime shifts between high and low volatilities, and that these regime shifts are contemporaneously positively correlated. In the conditions covered by Proposition 1 the effect of IA will always be to amplify the effect of r2t on the economy. If r2t switches from r2high to r2low the IA behavior helps to reduce overall volatility, thus buttressing the ‘‘change in exogenous shocks’’ explanation (e.g., Sims and Zha (2006)) for the Great Moderation. It is particularly unfortunate that high-volatility periods, in which stabilization policy would be most useful, are precisely those in which it is rendered ineffective. In data sets of such an economy one observed relationship would be that, on average, inflation has a greater effect on output in periods when inflation’s variance is low, and is less able to affect output in periods when inflation’s variance is high. Again, a relevant empirical

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episode is the 1970s in the US, when inflation’s variance rose and it lost its ability to affect real variables. However, there is also the exceptional case documented in Proposition 2. This ‘‘backwards’’ result does not occur due to implausible unstable equilibria. In fact it occurs under assumptions designed to be as plausible as possible, i.e., unstable equilibria are ignored and agents are assumed not to switch between equilibria if it is avoidable. The result emerges because the initial equilibrium is destroyed by a change in the exogenous variance, so agents must switch to another equilibrium. Proposition 2 assumes they switch to an equilibrium with it = 1, but they could also switch to another equilibrium for case 4, it = 0. If they switch to it = 0, Proposition 1 is exhaustive. Both case 4 equilibria are stable, so there is no basis for deeming either more plausible than the other. Branch et al. (2006a) study optimal monetary policy in the presence of endogenous IA and reach similar conclusions regarding stable interior equilibria. In the symmetric equilibria in IA they study, the extent of IA undertaken by each agent (IA choice is not binary in their model) is increasing in the variance of each of the model’s two shocks. Their model also allows for a change in either variance to force a jump to a different equilibrium, but they do not discuss that possibility. Branch et al.’s analysis differs from the current one in that a change in economic volatility is likely to be permanent in their analysis (especially in their companion paper which explicitly takes up the idea of MA learning). Here, the ARCH shock implies that sooner or later the good effects will be undone as r2t reverts to r2high . I turn next to comparative statics for the IA cost parameters c and d. I ignore equilibrium switching for these parameters and list the following proposition: ovar ði ;r2 Þ

oit p t t t Proposition 3. For stable interior equilibria oi , oc oc < 0 and od < 0. It follows that ovary ðit ;r2t Þ ovarp ðit ;r2t Þ ovary ðit ;r2t Þ , , and are negative. For i = 0 or i = 1, all six derivatives are zero. t t oc od od

The proof is in Appendix 1. Sensible comparative statics—higher IA costs leading to fewer agents being informed— obtain in stable interior equilibria. Finally, there is the possibility d = 0, i.e., all agents have the same IA cost. The set of equilibrium it is simple; there are only three positive-measure cases. One case has varp ðit ; r2t Þ lying above c, in which case everyone will be informed, another has varp ðit ; r2t Þ lying below c, in which case no one will be informed, and there is a case in which varp ðit ; r2t Þ intersects zn once, from below. This case has three equilibria, it equal to 0, it , or 1; only the corner equilibria are stable and their comparative statics are as above. 3.1.3. The effectiveness of monetary policy The effectiveness of monetary policy, oy/op, is given by oy ¼ bð1  it Þ; op which varies between 0 and b. In stable interior equilibria higher r2t induces higher it and therefore less effective policy. The effect of a change in r2t on policy’s effectiveness is discrete if it leads to a change between equilibria. For example, a small value of r2t may imply

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case 5 with it ¼ it;1 , while a larger value of r2t may imply case 1 with it = 1, so the value of r2t makes the difference between policy being somewhat effective and completely ineffective. The dynamic analog of this example would be that a switch from r2low to r2high makes policy suddenly change from somewhat effective to completely ineffective. I next turn to the econometric implications of such switching. u t would An econometrician estimating a 1960s-era tradeoff of the form y t ¼ a þ ^bpt þ e seem to detect regime changes in the value of ^ b; these would be especially noticeable if a change in r2t induced a jump between different equilibria. Also, the variance of the residuals would be autoregressive, and negatively correlated with the estimated value of ^b due to private agents’ IA behavior (ignoring unstable interior equilibria). An econometrician incorporating survey data on expectations, estimating y t ¼ aþ ^ u t where pet is the PS’s average expectation, would generate a ^b that bðpt  pet Þ þ e would be an unbiased estimate of b. However, the econometrician would observe the following: (1) the variance of the residuals would not be constant but autoregressive, (2) the expectations data would sometimes be highly correlated with the residuals (i.e., in sequences of periods in which it was high) and sometimes not, (3) the degree of correlation would itself be positively correlated with the residuals’ variance (in stable equilibria), (4) there would be a perfect multicolinearity problem between expectations and inflation in sequences of periods in which it = 1.

3.1.4. Expected losses I do not take a position on whether the MA’s loss is an approximation of PS agents’ losses net of forecast errors. As is well known, that is the case for quadratic losses like (1) in some environments; in others the best interpretation is that the loss represents policymaker preferences that differ from the PS’s preferences. Remaining agnostic on this matter, I analyze the MA and PS losses separately. In some cases the comparative statics of E(lt) and Eðlnt Þ are the same, so the interpretation does not matter anyway. For PS agents the interest is agents’ welfare during their entire lives, not just from living in any particular variance regime. For this reason, it is not necessary to study the expected losses conditional on r2t ; the welfare analysis uses the unconditional expectations of the losses.14 The next two propositions take up the effect of r2t and state that higher values of r2t are worse. Proposition 4. Suppose d = 0. In the stable (i.e., corner) equilibria oEðlnt Þ or2t

> 0 for all n. If it = 1 then

Proof. See Appendix 1.

14

oEðlnt Þ or2t

oEðlt Þ or2t

> 0. If it = 0 then

¼ 0 for all n.

h

This distinction is of little consequence anyway, because for a given r2high (respectively, r2low ), a change in r2low (respectively, r2high ) affects the conditional and unconditional expected losses in the same direction.

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Proposition 5. If d > 0, higher r2t is Pareto inferior to lower r2t within a given stable interior equilibrium or corner equilibria. Proof. See Appendix 1.

h

For stable interior equilibria in Proposition 5, an increase in r2t hurts the MA because it induces an increase in the variances of inflation and output. For the PS, an agent n who was informed at the lower value of it is not affected by the change; n’s loss is still zn. Agents who are uninformed in both situations are ex ante worse off, since their expected forecast error is higher with a higher r2t . Note the deterioration in their expected welfare is not only due to the direct effect of higher shock’s variance on inflation’s variance, but also due to the indirect effect, the externality caused by other agents’ IA behavior. PS agents who are uninformed for low r2t and informed for high r2t have a change from an expected loss of varp ðit ; r2t Þ to an expected loss of zn. Plainly they are worse off; the fact that n was uninformed previously means n’s expected loss under ignorance was less than zn. Finally, consider the possibility of a switch from case 5 with it ¼ it;1 < 1 to case 4 with it = 1. In this scenario lower r2t is worse for the PS, leaving some agents’ losses unchanged—i.e., those who are informed in both equilibria—and raising other agent’s expected losses, i.e., those who switch from uninformed to informed; the fact that they were uninformed initially proves that they were better off initially. I turn next to the welfare effects of c and d. For it = 0 a small increase in c or d does not affect either the MA’s or the PS’s expected loss. For it = 1 an increase in c or d does not affect the MA’s expected loss, but it does affect the PS’s loss since the cost of becoming informed rises for all agents. For stable interior equilibria, recall that oit/oc < 0 and oit/od < 0. It follows that the MA’s expected loss is decreasing in d and in c since   2ab2 a þ b2 r2t oit oEðlt Þ ; ð17Þ ¼ 3 oc a þ ð1  it Þb2 oc   2ab2 a þ b2 r2t oit oEðlt Þ : ¼ 3 od a þ ð1  it Þb2 od

ð18Þ

That is, higher IA costs induce fewer informed agents, which improves the MA’s ability to stabilize the economy. For the PS welfare results are more complicated. In stable interior equilibria oit/oc and oit/od are negative. Those who are uninformed for both values of the changing parameter, and those who switch from informed to uninformed, have lower expected losses, because inflation’s variance is lower for lower it. However, those who are informed for both values of the parameter pay higher IA costs. 3.2. Committed policy The policy problem under commitment is generically intractable, but we can draw the following conclusions (see Appendix 2 for details). The MA never allows all PS agents to be informed, and for some parameters no agents are. More precise results are possible for special cases.

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3.3. Special case: c = 0 Suppose c = 0, so that agent n = 0 incurs no cost of becoming informed and agents n  0 incur arbitrarily small costs. For example, these agents may be at the heart of the financial system and so have easy access to economic data. In this case the MA could induce complete ignorance only by having inflation be completely unresponsive to shocks; it is easy to show then that complete ignorance is not optimal. Proposition 6. If c = 0 and d > 0 the MA’s optimum has i 2 ð0; 1Þ in every period. Proof. See Appendix 2. 3.4. Special case: d = 0 If d = 0, that is, all PS agents have the same cost c of acquiring information, the MA has two options. By committing to set inflation’s variance equal to or less than c, it can induce i ¼ 0. If inflation’s variance is greater than c, the equilibrium value of i is one. Plainly the MA would never choose to induce i ¼ 1, so it chooses the optimal policy that induces var½pðit ; r2t ; ut ÞjI t  6 c.15 Therefore the MA commits to (5) if the variance associ2 ated with that policy is no greater than c, i.e., ½b=ða þ b2 Þ r2t 6 c. Otherwise the MA must restrain inflation’s response to the shocks so that its variance is c, using (22) with d = 0 (see Appendix 2): rffiffiffiffiffi c 2  ut : ð19Þ pði; rt ; ut Þ ¼  r2t That this induces i ¼ 0 is implied by the assumption that an agent indifferent about becoming informed does not do so. If d = c = 0, all PS agents will be informed unless inflation’s variance is zero. In this case the MA can never achieve any stabilization effects, so it simply commits to setting inflation to zero in all periods. 3.5. Expected losses I consider the effects of r2t , d, and c . While the comparative statics can be counterintuitive with discretion, that happens only in unstable equilibria or equilibrium switching, which the MA can rule out with commitment. Plainly if the constraint imposed by the PS’s possible IA binds, higher c makes every PS agent worse off and makes the MA better off, because higher c allows the MA to increase inflation’s variance. For d the same analysis applies. Higher r2t makes everyone, PS and MA, worse off, whether or not the constraint binds. This is because even if it does not bind, there is still the direct effect of the shock’s variance on economic volatility, which all players dislike. If c is large enough that the constraint does not bind, a small change in c or d do not affect either MA’s or the PS’s expected losses. A small increase in r2t increases both the MA’s and the PS’s expected losses. For d = 0, if the constraint binds an increase in 15

Here var½pðit ; r2t ; ut ÞjI t  is used to denote the conditional variance of inflation to emphasize that under commitment the variance is not necessarily varp ðit ; r2t Þ, the variance under discretionary policy.

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r2t does not affect the PS’s loss, because the MA’s optimum is to keep inflation’s variance equal to c (see (19)). 4. Conclusion This paper has presented results for optimal information acquisition in a setting in which information acquisition affects the effectiveness of monetary policy. Under discretionary policy, the endogenous IA causes a time-varying inflation-output tradeoff, and in fact it tends to weaken monetary policy exactly when monetary policy would be most useful, in high-volatility periods. Under commitment the MA can manage policy’s effectiveness by committing to low-variance policy rules that limit the PS’s incentive to become informed. That is, if the MA ‘‘doesn’t abuse it, then it won’t lose it.’’ However, the required restraint also limits the MA’s ability to stabilize the economy. The model also brings out a novel externality: informed agents, by increasing the MA’s optimal inflation variance, impose a negative externality on uninformed ones. The empirical episode of most interest in light of these theoretical results is the change in US macroeconomic volatility over the last few decades. The paper’s endogenous IA choice provides a mechanism by which regime changes in an exogenous shock’s variance can have amplified effects on overall macro volatility; thus it buttresses the ‘‘changes in shocks’’ explanation of the high-volatility of the 1970s and the subsequent Great Moderation. Appendix 1. Proofs A.1. Proof of Proposition 3 For interior equilibria, differentiate (14) with respect to c:  3 a þ ð1  it Þb2 oit ¼   oc 2b4 r2  d a þ ð1  it Þb2 3 t and with respect to d: oit oit ¼ it : od oc working with these two derivatives, one can easily show the part of Proposition 3 that applies to stable interior equilibria. The part for boundary equilibria is obvious. h A.2. Proof of Proposition 4 First consider the effect on the MA’s expected loss, for corner equilibria,  2 ovary a ¼ >0 or2t a þ ð1  it Þb2

oEðlt Þ . or2t

Using (11) and (12), we have

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and  2 ovarp b ¼ > 0: or2t a þ ð1  it Þb2 oEðlt Þ > 0 in corner equilibria. For or2t ovarp ðit ;r2t Þ > 0 in that case. The result or2t

Therefore

the PS, the result for it = 0 follows from the

fact that is always zn.

for it = 1 follows because n’s loss in that case

A.3. Proof of Proposition 5 An uninformed agent n’s expected loss is Eðlnt Þ ¼ varp ðit ; r2t Þ and the MA’s loss in REE is

     2 a þ b2 y ðk  1Þ a a þ b2 a þ b2 2 2 2 y ðk  1Þ  ut þ  lt ¼  u: 2 2 t a a þ ð1  it Þb2 a þ ð1  it Þb 

The MA’s unconditional expected loss is   a þ b2 2 y ðk  1Þ2 Eðlt Þ ¼ a ( " # " #)   r2high r2low 2 þa aþb P low  2 þ ð1  P low Þ  2 ; a þ ð1  it Þb2 a þ ð1  it Þb2 where Plow is the unconditional proportion of the time that r2t ¼ r2low . So ( )   oEðlt Þ 2b2 r2low oit ðr2low Þ 1 2 ¼ a a þ b P low  þ 3 2 ; or2low or2low a þ ð1  it ðr2low ÞÞb2 a þ ð1  it ðr2low ÞÞb2 ð20Þ   oEðlt Þ ¼ a a þ b2 ð1  P low Þ 2 orhigh 9 8 > > 2 2 2 = < 2b rhigh oit ðrhigh Þ 1 :  h þ i3 h i 2> > or2high : a þ ð1  it ðr2 ÞÞb2 a þ ð1  it ðr2high ÞÞb2 ; high

ð21Þ

There are several cases if d > 0: tÞ (1) If it = 20 initially and a change nin r2t does not affect it, then oEðl > 0, and because or2t Dvarp ðit ;rt Þ oEðlt Þ > 0, it follows that > 0 for all n. 2 2 Drt ort oEðln Þ tÞ (2) If it = 1 initially and a change in r2t does not affect it, then oEðl > 0 and or2t ¼ 0 for or2t t all n. oit (3) For stable interior equilibrium values of it, recall or 2 > 0 (Proposition 1), and this t

tÞ tÞ > 0 and oEðl > 0. For the PS: an agent n who was informed at the lower implies oEðl or2 or2 low

high

value of i is not affected one way or another by the change; n’s loss is still zn. Agents who are uninformed in both situations are ex ante worse off, since their expected

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forecast error is higher with a higher r2t (Proposition 1). PS agents who were uninformed before and are now informed have a change from an expected loss of varp ðit ; r2t Þ to an expected loss of zn. They are worse off; that agent n was uninformed previously means n’s loss under ignorance was less than zn and now it is zn. A similar analysis can be shown without the assumption that equilibria remain interior, except that the effect of r2t on it is only weakly positive.

Appendix 2 For commitment I assume the MA can observe only the aggregate measure of agents who are informed, not individual agents (under discretion it does not matter whether the MA observes individuals). This prevents the MA from using threats or promises about offequilibrium events because an agent will defect from any value of it the MA tries to induce if, in that period, the expected loss of remaining ignorant is greater than that agent’s IA cost. This is because each agent, having measure zero, knows his behavior will not affect the aggregate behavior the MA observes. I also restrict the MA to pure strategies as functions of it and ut; this is in fact a significant restriction; see Cone (2006). With commitment the MA faces a tradeoff; making inflation more responsive to shocks stabilizes output more through the effect on uninformed agents, but also may induce fewer uninformed agents. This tradeoff also exists under discretion but the MA ignores it because it optimizes myopically given it. The tradeoff makes it necessary for the MA to consider how its inflation rule affects the equilibrium it. This implies a three-step procedure for solving the MA’s problem. The first step is to check whether the constraint imposed by the PS’s possible IA binds. This involves checking, for each value of r2t , whether the discretionary variance of inflation for it = 0 is small 2 enough to prevent all IA, ½b=ða þ b2 Þ r2t 6 c. If this inequality holds for a given value of r2t the MA’s optimum is simple; it commits, for periods with that value of r2t , to (10) with it = 0 but with mean inflation of zero. 2 If ½b=ða þ b2 Þ r2t > c for one or both values of r2t the MA proceeds to the second step. For this step note the MA can induce any it 2 [0, 1] as an equilibrium by committing to a policy with appropriate variance (unless d = 0, a possibility taken up below). If the MA desires it ¼ i it simply commits to a policy rule pðit ; r2t ; ut Þ such that var½pðit ; r2t ; ut ÞjI t  ¼ c þ di, no matter what it the PS coordinates on at t. (The notation var½pðit ; r2t ; ut ÞjI t  emphasizes that under commitment the variance is not necessarily the discretionary variance varp ðit ; r2t Þ.) That commitment in fact induces the PS to coordinate on it ¼ i. Therefore, the MA’s second step is to find, for each i 2 ½0; 1, the loss-minimizing policy rule such that var½pðit ; r2t ; ut ÞjI t  ¼ c þ di. The third step is to compare the implied expected losses for all i; the i implying the lowest expected loss is the optimum. It can be shown by variational arguments that the solution to the second-step problem, among all functions p(Æ) that are C1 in ut, is the policy function16,17 16

Proof available upon request. Note that if i ¼ 0 this policy implies an inflation variance of c, which also the variance implied by the b 2 2 inequality ½aþb 2  rt 6 c when it holds with equality. That is, the MA’s two subproblems overlap where PS agent 0 is just indifferent about becoming informed. 17

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sffiffiffiffiffiffiffiffiffiffiffiffi c þ di ut : pðit ; r2t ; ut Þ ¼  r2t

ð22Þ

In the third step the MA substitutes (22) into its expected loss function and minimizes over i subject to 0 6 i 6 1. Suppressing the term r2t þ ð1  kÞ2 y 2 , maximizing the negative of the loss gives an optimization problem that has at least one solution, since the objective function is a continuous function of the choice variable and is defined over a compact feasible set of that choice variable. The Lagrangian is max L ¼ 2b i

qffiffiffiffiffi i pffiffiffiffiffiffiffiffiffiffiffiffi h r2t ð1  iÞ c þ di  a þ b2 ð1  iÞ2 ½c þ di þ l1i  l2i;

ð23Þ

where l1 and l2 are the Lagrangian multipliers on the constraints i P 0 and i 6 1, respectively. It is easy to show i ¼ 1 is not a solution, so the relevant necessary conditions for a solution are 0

6

i < 1;

ð24Þ

l1i ¼ 0; pffiffiffiffiffi r2t oL ¼ 0 ¼ bdð1  iÞ þ 2b2 ð1  iÞ½c þ di 1=2  oi ½c þ di qffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi 2  2b r2t c þ di  db2 ð1  iÞ  da þ l1 :

ð25Þ

It is evident from (25) that the second-order condition for a global optimum will not be satisfied. Furthermore, (25) is analytically solvable because it is isomorphic to a quinpnot ffiffiffiffiffiffiffiffiffiffiffiffi tic equation in the variable c þ di. However, we can draw the following conclusions. The MA never allows all PS agents to be informed; this is intuitively sensible because in that case monetary policy cannot stabilize output at all. In some regions of the parameter space the MA will conduct policy in a way consistent with some PS agents engaging in IA. In other regions the MA will conduct policy such that there is no IA. In particular, if c > 0 and r2t is small enough, the MA can set inflation as in (5), pt = [b/(a + b2)]ut, without triggering any IA. B.1. Proof of Proposition 6 For c = 0 the step 1 inequality will obviously not be satisfied and the constraint for achieving i ¼ 0 is var½pðit ; r2t ; ut ÞjI t  6 0, which is identical to var½pðit ; r2t ; ut ÞjI t  ¼ c þ di, the constraint that is used in step two above. Therefore the appropriate first-order condition is (25) with c = 0:   pffiffiffi 1   oL 1=2  þ db2 4i  3i2  da  db2 þ l1 : ¼ brt d 1=2  3i  oi i limi#0 oL ¼ 1, so i ¼ 0 is not a solution. As noted above, i ¼ 1 is not a solution, and there is oi at least one solution. It follows that the solution is interior. h

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