The signaling role of policy actions

Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party

websites are prohibited.

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information

regarding Elsevier’s archiving and manuscript policies are encouraged to visit:

http://www.elsevier.com/copyright

The signaling role of policy actions

Romain Baeriswyl^a,, Camille Cornand^b

aMunich Graduate School of Economics and Swiss National Bank, Boersenstrasse 15, P.O. Box, CH-8022 Zurich, Switzerland

bCNRS - BETA and University of Strasbourg, France

a r t i c l e i n f o

Article history:

Received 13 July 2007 Received in revised form 31 May 2010

Accepted 4 June 2010 Available online 12 June 2010

a b s t r a c t

In an economy affected by shocks that are imperfectly known, the monetary instrument takes on a dual stabilizing role: as a policy response that directly influences the economy and as a vehicle for information that reveals the central bank’s assessment to firms. Because mark-up shocks cannot be neutralized by monetary policy, providing firms with more information about these shocks exacerbates their reaction and creates a larger distortion. Recognizing the signaling role of its instrument, the central bank distorts its policy response in order to optimally shape firms’ beliefs. While providing firms with more information is always detrimental to the output gap, it has a more subtle effect on price dispersion depending on whether information is provided through the transparency channel or through the signaling channel. Although more transpar- ency is always detrimental to welfare, the information that is conveyed by the monetary instrument improves welfare when firms’ coordination is highly valuable.

&2010 Elsevier B.V. All rights reserved.

1. Introduction

In the ongoing debate on the social value of public information, most of the literature considers information as being disclosed by means of an explicit and official statement made by an institution such as a central bank.¹However, taking a policy action also conveys information as an implicit communication. For instance, the implementation of monetary policy reveals to the market the economic assessment of the central bank and discloses public information even in the absence of an explicit statement. Consequently, the response of a policy maker takes a dual role: as an action that directly influences economic outcomes and as a vehicle for information that influences the beliefs of market participants. A policy maker should naturally account for the signaling role of its policy action when determining the optimal action to take. Although there is a growing pool of literature on the desirability of central bank transparency, it has largely abstracted from the interaction between the choice to be transparent and the optimal design of monetary policy.²

To illustrate the signaling role of the policy action, we consider the optimal conduct of monetary policy in an economy where monetary frictions are caused by imperfect information. In this environment, communication turns out to be an essential component in designing an optimal monetary policy because it drives the degree of information imperfection and

Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/jme

Journal of Monetary Economics

0304-3932/$ - see front matter&2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.jmoneco.2010.06.001

$An earlier version of this paper was presented at the conference on ‘‘Monetary Policy, Transparency, and Credibility’’ at the Federal Reserve Bank of San Francisco, March 2007.

Corresponding author. Tel.: + 41 44 631 31 21.

E-mail address:Romain.Baeriswyl@snb.ch (R. Baeriswyl).

1SeeGeraats (2002)for an overview and the literature in the vein ofMorris and Shin (2002)seminal beauty-contest paper, for instanceHellwig (2005)andLorenzoni (2010).

2Some exceptions areAngeletos et al. (2006),Hellwig et al. (2006), andWalsh (2007)who analyze the signaling role of policy choices on market participants in different contexts.

the real effects of policy choices. The central bank may disclose more information to the private sector through two channels. First, the central bank explicitly discloses more information when it becomes more transparent with respect to its economic assessment or its monetary instrument; this is the form of communication with which the literature usually deals. Second, for a given level of transparency, the central bank may implicitly disclose more information about its economic assessment through the signaling role of its monetary policy whenever the market’s interpretation of it is ambiguous.³The central bank thus determines its monetary policy action by optimally balancing both its direct impact on the economy and the information it conveys to market participants. In particular, if the central bank wishes to withhold information from the markets, it should adjust its monetary policy so as not to release too much information through its policy action.

The economy is hit by two types of disturbances, namely, a stochastic labor supply shock that induces parallel variations in both the efficient and the equilibrium level of output, and a stochastic mark-up shock that induces variations in the equilibrium level of output but leaves the efficient level unaffected. Providing better information reduces frictions and helps economic agents reach the equilibrium level of output that would prevail in a frictionless economy. AsAngeletos and Pavan (2007)emphasize, providing more information improves welfare to the extent that the equilibrium and the efficient level of output are symmetrically affected by shocks. Consequently, withholding information about the mark-up shock reduces the output gap because it prevents the economy from moving too closely to the frictionless equilibrium level of output and, thereby, from deviating too much from the efficient level.⁴In cases where firms perfectly observe the monetary instrument but where the central bank does not disclose any explicit announcement about its economic assessment, firms cannot properly decipher the rationale behind the policy action. The central bank can exploit this ambiguity by distorting its response to labor supply and mark-up shocks in order to optimally shape firms’ beliefs.

Because the central bank seeks to maximize the welfare of the representative agent, the welfare loss increases with both the output gap and price dispersion. Providing firms with more information on mark-up shocks systematically exacerbates the output gap. As a result, when the output gap is relatively more costly, the central bank strengthens its response to labor supply shocks to reduce the amount of information about mark-up shocks that is conveyed by its chosen monetary instrument. However, providing firms with more information has a more subtle effect on price dispersion depending on whether information is provided through the transparency channel or through the signaling channel. On the one hand, providing firms with more information through the transparency channel is always detrimental to welfare because it impairs the trade-off between output gap and price dispersion. This suggests that the result ofCukierman (2001), which emphasizes the benefit of opacity for the trade-off between employment and inflation caused by mark-up shocks, carries over into a framework where price dispersion matters, even if one would expect transparency to favor firms’ coordination.

On the other hand, providing firms with more information through the signaling role of the monetary instrument, for a given level of transparency, enhances firms’ coordination and improves welfare when price dispersion is relatively more costly. In this case, the central bank weakens its response to labor supply shocks to make its instrument more informative about mark-up shocks and to reduce price dispersion. The fact that more public information enhances firms’ coordination has been stressed by Hellwig (2005) and Woodford (2005). Briefly, while more transparency is always detrimental to welfare, disclosing more information to firms through the signaling role of the monetary instrument improves welfare when coordination is socially highly valuable.⁵

The paper is structured as follows. Section 2 presents the economy. Section 3 derives the optimal monetary policy for benchmark cases with homogeneous information; it highlights the welfare effect of information with respect to labor supply and mark-up shocks. Section 4 presents the optimal monetary policy under heterogeneous information as a function of the communication strategy of the central bank. Finally, Section 5 concludes.

2. The economy

The economy is derived from a small-scale general equilibrium model with flexible prices, populated by a representative household, a continuum of monopolistic competitive firms, and a central bank. Two types of stochastic shocks hit the economy: a labor supply shock and a mark-up shock. The nominal aggregate demand is determined by the central bank, which maximizes the utility of the representative household. Apart from the informational structure, the analysis is based on the model developed byAdam (2007).

3Empirically, the signaling role of monetary policy has been well documented byRomer and Romer (2000). Using US data, they show that ‘‘the Federal Reserve’s actions signal its information’’ and that ‘‘commercial forecasters raise their expectations of inflation in response to contractionary Federal Reserve actions[y]’’ (p. 430).

4Angeletos and La’O (2010)also emphasize the welfare effect of information about mark-up shocks within a micro-founded business cycle model.

5Although the current analysis is based on the methodology developed byMorris and Shin (2002), the detrimental effect of transparency has little to do with the overreaction of firms to a noisy announcement by the central bank. Information about mark-up shocks is indeed detrimental to welfare even if it is perfect. In that respect, the current analysis departs fromWalsh (2007), who also accounts for the signaling role of the monetary instrument but highlights the detrimental effect of firms’ reaction to the central bank’s noisy information about mark-up shocks. In his framework, transparency improves welfare when the central bank obtains more accurate information on mark-up shocks.

2.1. Representative household

The representative household chooses its aggregate composite goodYand labor supplyLin order to maximize its utility subject to its budget constraint,

UðYÞnVðLÞ,

s:t: WLþP¼PYþT:

Wdenotes the competitive wage,Pthe profits the household gets from firms, andTthe nominal transfer from the central bank. The parametern is a stochastic labor supply shock with EðnÞ ¼1, that induces variations in the efficient level of output.Yis the composite good defined by the Dixit–Stiglitz aggregator

Y¼ Z 1

0

ðY_iÞ^ðy1Þ=ydi

" #y=ðy1Þ

,

wherey41 is the parameter of price elasticity of demand and whereYiis the good produced by firmi.yis stochastic with EðyÞ ¼y and induces variations in the desired mark-up of firms and thereby in the equilibrium level of prices and of the output.Pis the appropriate price index which solvesPY¼R1

0P_iY_idi.

2.2. Firms

Each firmiproduces a single differentiated goodYiwith one unit of laborLiaccording to the simple production function Li¼Yi:

The profit maximization problem of firmiis given by maxP_i E½ð1þiÞP_iY_iðP_iÞWY_iðP_iÞjG_i,

whereiis an output subsidy that offsets the efficiency detrimental effect of the mark-up andGiis the information set of firmi. Linearizing the first order condition of firmi’s problem around its steady state delivers

p_i¼E_i½pþxðyyÞ þu, ð1Þ

whereE_iis the expectation operator conditional on firmi’s informationG_i and where small letters indicate percentage deviation from the steady state. The pricing rule (1) states that firms set their price as a function of their expectations of the overall price level p, the real output gap yy*, and the mark-up shock u. The deviation of the efficient level of output y* from its steady state is determined by the stochastic labor supply shifter n. The parameter x¼ U⁰⁰ðYÞY=UuðYÞ þV⁰⁰ðYÞY=VuðYÞdetermines the sensitivity of the optimal price to the output gap and is increasing in the risk aversion of the household. The optimal pricing also depends on the expected value of the mark-up shockugiven by u¼ ðyyÞ=ðyðy1ÞÞ, wherey is the price elasticity of demand at its steady state level. Firms find it optimal to increase their price when the price elasticity of demandyfalls below its steady state valuey.

Using the fact that the nominal aggregate demandqcan be expressed asq=y+p, the pricing rule (1) can be rewritten as

p_i¼E_i½ð1xÞpþxqxyþu: ð2Þ

x determines whether prices are strategic complements or substitutes. Prices are realistically assumed to be strategic complements, i.e., 0oxr1: each firm tends to rise its own price when it expects other firms to do so.

The labor supply shock and the mark-up shock are assumed to be normally and independently distributed with the following properties:

yNð0,s²_yÞ,

uNð0,s²_uÞ:

2.3. Informational structure

Monetary frictions arise because of information imperfections. This is reminiscent of insight byPhelps (1970), which states that information imperfections play a crucial role in the monetary transmission mechanism.⁶

6The recent revival of interest inPhelps (1970)insight includes the work ofAdam (2007),Hellwig (2002),Mankiw and Reis (2002), andWoodford (2003). These authors emphasize the realistic dynamics of models relying on information imperfections when firms’ prices are strategic complements.

2.3.1. Information of the central bank

The central bank receives a signal in private on both the labor supply and the mark-up shock. Each signal deviates from the true value of the shock by an error term, which is normally distributed:

y_cb¼yþZ withZNð0,s²_ZÞ,

ucb¼uþm withmNð0,s²_mÞ,

whereZandmare independently distributed.

2.3.2. Information of firms

Two sources of information imperfections with respect to firms are introduced.

On the one hand, followingMankiw and Reis (2002), information spreads slowly through the economy. According to this assumption of information stickiness, only a fractionaof firms are informed about the current economic development, while the remaining fraction 1aof firms do not receive any contemporaneous information update at all. As emphasized in Section 3, this first source of frictions allows to solve the problem of price level indeterminacy and to derive the optimal monetary policy.

On the other hand, the information received by the fractionaof informed firms is noisy and heterogeneous, which entails fundamental and strategic uncertainty. The information received by the informedatype firms is threefold.

First, each informedatype firmireceives a private signal on the mark-up shocku_i, which may be interpreted as a private estimate. The private signal of each firm deviates from the true mark-up shock by an error term, which is normally distributed:

u_i¼uþr_i withr_iNð0,s²_rÞ,

wherer_iis identically and independently distributed acrossatype firms.

Second, theatype firms eventually observe the monetary instrument implemented by the central bank. The signal released by the central bank on its instrument can be generally expressed as

qi¼qþj_i withj_iNð0,s²_jÞ:

Whenever the central bank is transparent with respect to its monetary instrument (s²_j¼0), the nominal level of aggregate demandqis common knowledge among the informedatype firms. By contrast, whenever the central bank is opaque in this respect (s²_j-1), firms cannot observe the instrument. By making its instrument public, the central bank gives an indication to firms about its own beliefs on the state of the economy. However, firms are unable to properly understand the central bank’s assessment of the economy: because the central bank responds to two shocks, the monetary instrument does not allow firms to decipher the rationale behind the implemented policy unless the central bank discloses more information.

Third, whenever the central bank is transparent with respect to its monetary instrument, theatype firms eventually observe an additional public signal that completely eliminates the informational asymmetry between itself and theatype firms. A fully transparent central bank directly discloses its signal on the efficient level of outputy*cbso that theatype firms are able to properly interpret the rationale for the monetary instrument.⁷The signal released by the central bank on its economic assessment can be generally expressed as

y_cb,i¼y_cbþf_i, withf_iNð0,s²_fÞ:

The case of transparency with respect to its economic assessment is captured bys²_f¼0, and the case of opacity is captured bys²_f-1.

2.4. The central bank

The central bank seeks to maximize the expected utility of the representative household by adjusting its monetary instrument, the nominal demandq, conditional on its own information. With our informational setup, the second-order approximation of the welfare of the representative household implies that the central bank seeks to minimize the following unconditional expected loss

EðLÞ ¼min

q E ðyyÞ²þy x

1a a ^p

2þaðg²₁s²_jþg²₂s²_fþg²₃s²_rÞ

" #

, ð3Þ

subject to the pricing equation of firms (2).⁸The coefficientsg₁,g₂, andg₃are the weights assigned in equilibrium by firms to their signals on, respectively, the monetary instrument, the central bank’s disclosure, and the mark-up shock, as defined

7One may think of different types of announcement that would reveal the central bank’s signals to firms. In practice, the publication of inflation forecasts and/or targets appears to be the main form of announcement adopted by transparent central banks.

8For more details, see Appendix A available online, which derives the approximation of the welfare of the representative household according to the informational structure of the economy.

in Section 4.1. The cost of the price dispersion increases relative to that of the output gap with the price elasticity of demandy (with more competition).

Because both fundamental shocks and error terms are independently normally distributed, and because the welfare function is quadratic, the optimal instrument rule of the central bank that determines the nominal aggregate demandqis a linear combination of its signals and can be written as

q¼z₁ðyþZÞ þz₂ðuþmÞ:

z1andz2describe how the central bank sets the nominal aggregate demand in response to its signal on both shocks.

2.5. Timing of events

The sequence of events is as follows. First, the communication strategy of the central bank is determined and is common knowledge among firms.⁹Second, the nature draws the labor supply shock y*and the mark-up shocku. The central bank observes both shocks with an error term and sets its monetary instrumentq. According to its communication strategy, the central bank may reveal its instrument to the public and may make an explicit announcementy*cb. Based on their private signal on the mark-up shock ui and—when available—on the monetary instrument qi as well as on the announcement of the central banky*_cb,i, firms then simultaneously determine their price. Finally, the household demands products for consumption, and production takes place.

3. Homogeneous information

The mechanism of the model is illustrated within benchmark information settings under homogeneous information.

First, it is shown that in a frictionless economy, the optimal monetary policy is indeterminate. Second, information stickiness is introduced for resolving the indeterminacy problem and for illustrating the welfare effect of information about mark-up shocks.

3.1. Perfect information

The case of perfect information is captured when there is no information stickiness (a¼1) and when the error terms are zero:s²_Z¼0,s²_m¼0, ands²_r¼0. With perfect information, there is no price dispersion across firms because all firms set the same price, and the monetary policy problem becomes

EðLÞ ¼min

q E½ðyyÞ² ð4Þ

s:t: p¼qyþ1

x^{u, ð5Þ}

q¼yþp: ð6Þ

Plugging (6) into (5) shows that an output gap (and a loss) appears whenever there is a mark-up shock: yy¼ u=x. Because there is no friction on the price level (orpdoes not enter the loss function (4)), the optimal monetary policy is indeterminate:

q¼y1 x^uþp:

The nominal aggregate demandq can be arbitrarily chosen by the central bank, leading the price levelpto take on the corresponding value. The unconditional expected loss is given by

EðLÞ ¼s²_u

x^{2 ð7Þ}

and is independent from the policy implemented by the central bank.

3.2. Perfect sticky-information

In order to solve the indeterminacy problem of monetary policy that arises in the frictionless economy, sticky information as described in Section 2.3.2 is introduced. In this setup, only a fraction 0oao1 of firms are assumed to get an information update; the other 1afirms remain completely uninformed. However, the error terms of the central bank and of theatype firms remain zero:s²_Z¼0,s²_m¼0, ands²_r¼0. With perfect sticky-information the monetary policy problem

9The choice of the communication strategy occurs before the central bank observes its signals on the shocks. We abstract here from the discussion on whether it is optimal for the central bank to rely on its signals to choose its communication strategy and how firms would accordingly adjust their beliefs.

of the central bank becomes

EðLÞ ¼min

q E ðyyÞ²þy x

1a a ^p

2

" #

ð8Þ

s:t: p¼ a

1aþax^ðxqxyþuÞ,

q¼yþp:

Solving the problem (8) delivers the optimal monetary policy

q¼y aðy1Þ 1aþayx^u

and yields a price level and an output gap given byp¼au=ð1aþayxÞandyy¼ ayu=ð1aþayxÞ, which implies an unconditional expected loss equal to

EðLÞ ¼ ay

xð1aþayxÞs²_u:

The optimal monetary policy indicates that labor supply shocks are perfectly accommodated by the central bank. The monetary instrument simultaneously closes the output gap and eliminates price deviations induced by labor supply shocks independently of the shareaof informed firms.

By contrast, mark-up shocks cannot be neutralized by the central bank. Increasing the share a of informed firms strengthens the aggregate reaction to mark-up shocks. As a result, the response of the central bank to mark-up shocks strengthens (@q=@ao0), and the price level and the output gap deviations increase, all of which lead to a larger unconditional expected loss. Therefore, improving information among firms is detrimental to welfare. While information about mark-up shocks is privately desirable according to the optimal pricing rule of firms (2), it is socially undesirable because of the inefficiency wedges it creates.¹⁰

The loss associated with mark-up distortions increases with the price elasticity of demandy. Therefore, the central bank responds more aggressively to mark-up shocks when the price elasticity of demand increases (@q=@yo0) and perfectly stabilizes the price level for infinite price elasticity.

4. Heterogeneous information

Let us now consider the more realistic case where firms have heterogeneous information. First, the general equilibrium of the economy is established, and then the optimal monetary policy is derived according to three communication strategies of the central bank.

4.1. Equilibrium

This section solves the perfect Bayesian equilibrium and derives the optimal behavior of firms according to their information set on the monetary instrument and on the central bank’s assessment of the economy. The information set of atype firms is composed of a private signal on the mark-up shockui, a signal on the nominal aggregate demandqi, and a signal on the central bank‘s assessment of the labor supply shocky*cb,i.

For setting its optimal price according to (2), eachatype firm solves the inference problemE½q,y,y_cb,ujq_i,y_cb,i,u_ithat is defined by

E q y y_cb u

qi,y_cb,i,ui

0 BB B@

1 CC CA¼X

q_i y_cb,i

u_i 0 B@

1 CA¼

O11 O12 O13

O₂₁ O₂₂ O₂₃ O31 O32 O33

O41 O42 O43

0 BB BB

@

1 CC CC A

q_i y_cb,i

u_i 0 B@

1

CA, ð9Þ

with X¼VuoV¹_oo, where Vuo is the covariance matrix of the unobserved expected variables and the observed signals andVoo is the covariance matrix of the observed signals themselves. It is important to see from the covariance

10At the limit, whenaconverges to 1, the output gap and the unconditional expected loss are identical as under perfect information. However, the price level and the monetary policy are determinate at the limit.

matrix

Vuo¼

z²₁ðs²_yþs²_ZÞ þz²₂ðs²_uþs²_mÞ z₁ðs²_yþs²_ZÞ z₂s²_u z₁s²_y s²_y 0 z₁ðs²_yþs²_ZÞ s²_yþs²_Z 0 z₂s²_u 0 s²_u 0

BB BB B@

1 CC CC CA

that firms rationally use their signals to form their expectations. For instance, the covariance between the mark-up shock and the monetary instrumentz₂s²_u highlights the signaling role that the monetary instrument plays in the formation of firms’ rational expectations.¹¹

FollowingMorris and Shin (2002),atype firms set their price according to the following linear pricing rule:

p_i¼g₁q_iþg₂y_cb,iþg₃u_i:

The equilibrium response ofatype firms to their signals is given by the system of simultaneous equations¹²:

g₁¼að1xÞðg₂O₃₁þg₃O₄₁Þ þxðO₁₁O₂₁Þ þO₄₁

1að1xÞO₁₁ ^,

g₂¼að1xÞðg₁O₁₂þg₃O₄₂Þ þxðO₁₂O₂₂Þ þO₄₂ 1að1xÞO32

,

g₃¼að1xÞðg₁O₁₃þg₂O₃₃Þ þxðO₁₃O₂₃Þ þO₄₃

1að1xÞO₄₃ ^{: ð10Þ}

The central bank chooses its monetary instrument to minimize the expected loss (3) subject to (10) given the precision of its information.

In the following sections, we derive the optimal monetary policy for three communication strategies employed by a central bank. First,transparencyrefers to the case where there is no information asymmetry between the central bank and informedatype firms. Second,opacitydescribes the case where the central bank does not disclose any information with respect to its monetary instrument and its economic assessment. Third, intermediate transparency depicts the most interesting situation, where the central bank is transparent with respect to its monetary instrument but does not disclose any additional information about its economic assessment.

While the current section presents the equilibrium for any degree of information stickinessa, in the remainder of the paper, we concentrate on the limiting case, where the share of informed firmsagoes to one. This allows us to solve the indeterminacy problem that occurs in the absence of stickiness while focusing on the heterogeneous nature of information as frictions.

4.2. Optimal monetary policy under transparency

Under transparency, the informedatype firms perfectly observe the monetary instrumentqand the central bank‘s assessment of the labor supply shocky_cb* . Because there are two shocks affecting the economy, the combination of both observations removes the informational asymmetry between the central bank and the atype firms. Transparency is modeled with perfect firms’ signals on the monetary instrumentqand on the central bank‘s announcementy*_cb, that is, withs²_j¼s²_f¼0.¹³

Solving the monetary policy problem under transparency and taking the limit whenaconverges to one delivers the optimal coefficients of monetary policy¹⁴:

z_1,T¼ s²_y

s²_yþs²_Z^,

z2,T¼ y1 yx

s²_u s²_uþs²_m^:

11Since firms know how the central bank responds to mark-up shocks, i.e.,z2 is common knowledge, they use their signal on the monetary instrument to infer mark-up shocks as well as their private signal on mark-up shocks to infer the monetary instrument. However, when the monetary instrument is perfectly observable, signals on labor supply and mark-up shocks do not contain any information that is relevant to infer the instrument andO12¼O13¼0.

12See Appendix B available online for the derivation.

13We consider the case of a credible central bank and abstract from the discussion on whether it would be optimal for the central bank to cheat the private sector by disclosing a falsified economic assessment.

14Appendix C available online presents the optimal monetary policy under transparency for general values ofa.

As z_1,T and z_2,T indicate, the central bank accommodates variations in the efficient level of output but implements a restrictive policy in response to mark-up shocks. The strength of central bank’s response to mark-up shocks increases with the precision of its information s²_m, with the price elasticity of demand y, and with the degree of strategic complementarities 1x.

Implementing the optimal monetary policy under transparency yields an unconditional expected loss given by EðLTÞ ¼ATþBTþCTþDT,

where

AT¼s²_u x^2,

BT¼ s²_ys²_Z s²_yþs²_Z^,

CT¼ 2xs²_us²_ms²_rðs²_uþs²_mÞ

x²ðs²_rðs²_uþs²_mÞ þxs²_us²_mÞ²s²_us²_ms²_r, and

DT¼ yxs²_us²_m

x²ðs²_rðs²_uþs²_mÞ þxs²_us²_mÞ²s²_us²_ms²_r:

The unconditional expected loss can be split into the output gap component (AT,BT, andCT) and the price dispersion component (DT).ATstands for the loss under perfect information, as derived in Section 3.1.BTis the incremental loss due to the output gap that arises when the central bank is unable to perfectly accommodate the labor supply shock because of its imperfect information.C_Tcaptures the mitigation of the output gap that arises owing to uncertainty surrounding the mark- up shock.C_Tincreases in absolute value with the inaccuracy of firms’ private informations²_r, with the inaccuracy of central bank informations²_m, and with the degree of strategic complementarities 1x. When firms’ private information and central bank information is totally noisy,CTperfectly offsetsAT, that is, lim_s2

r,s²m-1CT¼ s²_u=x².

The loss associated with the price dispersionDTincreases when firms respond more strongly to their private signals because of more inaccurate central bank informations²_mor stronger strategic complementarities 1x. The price dispersion also becomes more costly when the economy is more competitive, that is, when the price elasticity of demand y increases. The precision of firms’ private information has, however, an ambiguous effect on price dispersion:

@DT=@s²_r403s²_roxs²_us²_m=ðs²_uþs²_mÞ. When firms’ private information is highly precise, reducing the precision of that information increases price dispersion because firms tend to assign a large weight to it. By contrast, when the precision of firms’ private information is sufficiently low, reducing its precision further reduces price dispersion because firms respond less strongly to it.

Overall, the uncertainty surrounding the mark-up shock improves welfareðCTþDTo0Þwhen the weight assigned to the output gap is relatively large, that is, when the price elasticity of demandy is low. The positive effect of the uncertainty surrounding the mark-up shocks on the output gap CT dominates the negative effect of the uncertainty on the price dispersion DT when yot, where t¼2þ ðs²_uþs²_mÞs²_r=xs²_us²_m. Similarly, we have C^T+D^T=0 when y¼t, and CTþDT40 wheny4t.

4.3. Optimal monetary policy under opacity

Under opacity, the informedatype firms do not observe the monetary instrumentqor the central bank‘s assessment of the labor supply shocky*_cb. They only get their private signal on the mark-up shocku_i. The case of opacity is modeled with infinite noise on firms’ signal on the monetary instrument and on the central bank‘s assessment, i.e.,s²_j,s²_f-1.

Solving the monetary policy problem under opacity and taking the limit whenaconverges to one delivers the optimal coefficients of monetary policy¹⁵:

z_1,O¼ s²_y

s²_yþs²_Z^,

z_2,O¼ ðy1Þs²_rs⁴_u

s²_mðs²_rþxs²_uÞ²þs²_rs²_uðxys²_uþs²_rÞ:

As under transparency,z_1,O indicates that the central bank tries to fully accommodate variations in the efficient level of output according to the precision of its signal. The central bank’s response to mark-up shocksz_2,Ois always restrictive and

15Appendix D available online presents the optimal monetary policy under opacity for general values ofa.

strengthens with the precision of its informations²_m, with the price elasticity of demandy, and with the degree of strategic complementarities 1x. The effect of the precision of firms’ private informations²_r is not monotone on the response to mark-up shocks:@jz_2,Oj=@s²_r403s⁴_rox²s⁴_us²_m=ðs²_uþs²_mÞ. As already mentioned, the strength of the central bank’s response to mark-up shocks increases with the weight assigned to the price dispersion (y). Similarly, the central bank responds more strongly when the price dispersion across firms is relatively high. This arises when the inaccuracy of firms’ private information is intermediate. When firms’ private information is either perfectly accurate (s²_r¼0) or perfectly noisy (s²_r-1), there is no price dispersion, and the central bank does not respond to mark-up shocks.

Implementing the optimal monetary policy under opacity yields an unconditional expected loss given by EðLOÞ ¼AOþBOþCOþDO,

where

AO¼s²_u x^2,

BO¼ s²_ys²_Z s²_yþs²_Z^,

CO¼s²_u x^{2 þ}

x²s⁴_ms⁴_uþ2xs²_ms²_rs²_uðs²_mþys²_uÞ þ ðs²_mþs²_uÞðs²_mþy²s²_uÞs⁴_r ðs²_mðxs²_uþs²_rÞ²þ ðxys²_uþs²_rÞs²_us²_rÞ² s⁶_u, and

DO¼ yxðs²_rðs²_uþs²_mÞ þxs²_us²_mÞ²s²_u x²ðs²_mðxs²_uþs²_rÞ²þ ðxys²_uþs²_rÞs²_us²_rÞ²

s²_us²_r:

Similarly to the case of transparency,AOstands for the loss under perfect information, andBOstands for the incremental output gap that arises when the central bank is unable to perfectly accommodate the labor supply shock because of its imperfect information. When firms’ private information is totally noisy, C_O, the welfare effect on the output gap of uncertainty on mark-up shocks, perfectly offsetsA_O, that is, lim_s2

r-1C_O¼ s²_u=x².

The analysis of the combined effect of the uncertainty surrounding the mark-up shock on the output gap and on the price dispersion under opacity shows that the positive welfare effect of uncertainty on the output gap dominates the negative effect on the price dispersion (COþDOo0) when yot. Both the positive and negative effects cancel out (C_O+D_O=0) when y¼t, and the negative effect on price dispersion dominates the positive effect on the output gap ðCOþDO40Þwheny4t. Note that this result is the same as under transparency.

4.4. Unconditional expected loss under transparency vs. opacity

The effect of transparencyversus opacity on the unconditional expected loss induced by labor supply and mark-up shocks is now analyzed.

4.4.1. Loss induced by labor supply shocks

The unconditional expected loss induced by labor supply shocks is independent of the disclosure strategy of the central bank; that is,B_T=B_O. This result maya priorilook surprising in regard to the analysis byAngeletos and Pavan (2007). The latter show that the accuracy of information with respect to labor supply shocks improves welfare while the commonality of information may decrease welfare if it destabilizes the weight assigned to competing signals. One would thus expect the disclosure regime to affect how the economy reacts to labor supply shocks. However, the current framework presents two particularities that rationalize the irrelevance of disclosure. First, the central bank accommodates labor supply shocks with its instrument up to the accuracy of its information. Second, firms do not get any private signal on labor supply shocks.

Therefore, the central bank’s announcement neither impairs the stabilizing role of its monetary instrument nor destabilizes firms’ responses. Expecting the central bank to accommodate labor supply shocks, firms indeed choose not to respond to them.

4.4.2. Loss induced by mark-up shocks

By contrast, the central bank‘s disclosure strategy matters in the case of losses associated with mark-up shocks because they cannot be neutralized by the central bank. The unconditional expected loss under opacity is always smaller than or equal to the loss under transparency:COþDOrCTþDT. In the particular case wherey¼t, the loss is equal under both disclosure regimes (CT+DT=CO+DO= 0).

Whenyot, opacity is superior to transparency in terms of welfare because it yields a lower output gap (COoCT). By reducing firms’ uncertainty, transparency enhances the reaction of firms to mark-up shocks, which exacerbates the output gap. Wheny4t, opacity is superior to transparency because it yields a lower price dispersion (D_OoDT). It is surprising that transparency leads to a larger price dispersion than opacity when the price elasticity of demand is high. One would