The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Federal Reserve Bank of Chicago
Constrained Discretion and Central Bank
Transparency
Francesco Bianchi and Leonardo Melosi
July 2014
WP 2014-16
Constrained Discretion and Central Bank Transparencyy
Francesco Bianchi
Leonardo Melosi
Duke University
Cornell University
CEPR and NBER
Federal Reserve Bank of Chicago
July 2014
Abstract
We develop and estimate a general equilibrium model in which monetary policy can deviate
from active in‡ation stabilization and agents face uncertainty about the nature of these deviations.
When observing a deviation, agents conduct Bayesian learning to infer its likely duration. Under
constrained discretion, only short deviations occur: Agents are con…dent about a prompt return
to the active regime, macroeconomic uncertainty is low, welfare is high. However, if a deviation
persists, agents’beliefs start drifting, uncertainty accelerates, and welfare declines. If the duration
of the deviations is announced, uncertainty follows a reverse path. For the U.S. transparency lowers
uncertainty and increases welfare.
JEL classi…cation: E52, D83, C11.
y Correspondence
to: Francesco Bianchi, Department of Economics, Duke University, 213 Social Sciences
Building, Durham, NC, 27708-0089, USA. Email: francesco.bianchi@duke.edu. Leonardo Melosi, Federal Reserve Bank of Chicago, 230 South LaSalle street, Chicago, IL 60604-1413, USA. Email: lmelosi@frbchi.org.
We wish to thank Martin Eichenbaum, Cristina Fuentes-Albero, Jordi Gali, Pablo Guerron-Quintana,
Yuriy Gorodnichenko, Narayana Kocherlakota, Frederick Mishkin, Matthias Paustian, Jon Steinsson, Mirko
Wiederholt, Michael Woodford, and Tony Yates for very helpful comments and discussions. We also thank
seminar participants at the NBER Summer Institute, Columbia University, UC Berkeley, Duke University,
the SED conference in Cyprus, the Bank of England, the ECB Conference “Information, Beliefs and Economic Policy,”the Midwest Macro Meetings 2014, and the Philadelphia Fed. Part of this paper was written
while Leonardo Melosi was visiting the Bank of England, whose hospitality is gratefully acknowledged. The
views expressed in this paper are those of the authors, and not those of the Bank of England. The views in
this paper are solely the responsibility of the authors and should not be interpreted as re‡ecting the views
of the Federal Reserve Bank of Chicago or any other person associated with the Federal Reserve System.
Francesco Bianchi gratefully acknowledges …nancial support from the National Science Foundation through
grant SES-1227397.
1
Introduction
The last two decades have witnessed two major breakthroughs in the practice of central
banking worldwide. First, most central banks have adopted a monetary policy framework
that Bernanke and Mishkin (1997) have termed constrained discretion. Bernanke (2003)
explains that under constrained discretion, the central bank retains some ‡exibility in deemphasizing in‡ation stabilization so as to pursue alternative short-run objectives such as
unemployment stabilization. However, such ‡exibility is constrained to the extent that
the central bank should maintain a strong reputation for keeping in‡ation and in‡ation
expectations …rmly under control. Second, many countries have taken remarkable steps to
make their central bank more transparent (Bernanke et al., 1999 and Mishkin 2001).1
As a result of these changes, some key questions lie at the heart of modern monetary policy
making. First, for how long can a central bank de-emphasize in‡ation stabilization before
the private sector starts fearing a return to a period of high and volatile in‡ation as in ’70s?
Second, does transparency play an essential role for e¤ective monetary policy making? In
other words, should a central bank be explicit about the future course of monetary policy?
The recent …nancial crisis has triggered a prolonged period of accommodative monetary
policy that some members of the Federal Open Market Committee fear could lead to a
disanchoring of in‡ation expectations.2 As a result, the research questions outlined above
are at the center of the policy debate.
In order to address them, we develop and estimate a model in which the anti-in‡ationary
stance of the central bank can change over time and agents face uncertainty about the nature
of deviations from active in‡ation stabilization. When monetary policy alternates between
prolonged periods of active in‡ation stabilization, active regime, and short periods during
which the emphasis on in‡ation stabilization is reduced, short-lasting passive regime, the
model captures the monetary approach described as constrained discretion. However, the
central bank can also engage in a prolonged deviation from the active regime and move to
a long-lasting passive regime. Agents in the model are fully rational and able to infer if
monetary policy is active or not. However, when the passive rule prevails, they are uncertain
about the nature of the observed deviation. In other words, agents are not sure if the central
bank is engaging in a short or long-lasting deviation from the active regime. The central bank
1
Since May 1999, the Federal Open Market Committee (FOMC) has included explicit language about
the likely future policy stance in its o¢ cial statements, as documented in Rudebusch and Williams (2008).
Industrialized countries such Canada, Spain, Sweden, and the United Kingdom have publicly announced
a target range for in‡ation and also introduced a wide variety of instruments for communicating with the
public. These include regular release of macroeconomic forecasts, discussions of the policy responses to keep
in‡ation on target, and prompt releases of minutes.
2
As an example see Plosser (2012).
1
can then follow two possible communication strategies: Transparency and no transparency.
Under no transparency, the nature of the deviation is not revealed. Under transparency, the
duration of every deviation is announced.
Under no transparency, when passive monetary policy prevails, agents conduct Bayesian
learning in order to infer the likely duration of the deviation from active monetary policy.
Given that the behavior of the monetary authority is unchanged across the two passive
regimes, the only way for rational agents to learn about the nature of the deviation consists
of keeping track of the number of consecutive deviations. As agents observe more and more
realizations of the passive rule, they become increasingly convinced that the long-lasting
passive regime is occurring. As a result, the more the central bank deviates from active
in‡ation stabilization, the more agents become discouraged about a quick return to the
active regime. We then solve the model keeping track of the joint evolution of policy makers’
behavior and agents’ beliefs using the methods developed in Bianchi and Melosi (2014b).
The latter methods are based on the idea of expanding the number of regimes to take into
account the learning mechanism. Once a regime is de…ned in terms of both policy makers’
behavior and agents’beliefs, the model can be solved using any of the methods developed
for perfect information Markov-switching models. The resulting solution implies that the
model dynamics evolve over time in response to the evolution of policy makers’behavior and
agents’beliefs.
The ability of generating smooth changes in agents’beliefs in response to central bank
actions makes the model an ideal laboratory to study the macroeconomic and welfare implications of constrained discretion. In the model, social welfare is shown to be a function
of agents uncertainty about future in‡ation and future output gaps. Both of these measures
of uncertainty keep increasing as agents observe more and more deviations from the active
policy and update their beliefs about the duration of the passive policy. In standard models, monetary policy a¤ects agents’welfare by in‡uencing the unconditional variances of the
endogenous variables. In our nonlinear setting, policy actions exert dynamic e¤ects on uncertainty. Therefore, welfare evolves over time in response to the short-run ‡uctuations of
uncertainty. To our knowledge, this feature is new in the literature and allows us to study
changes in the macroeconomic risk due to policy actions and communication strategies and
the associated welfare implications.
We measure uncertainty taking into account agents’beliefs about the evolution of monetary policy. As long as the number of deviations from the active regime is low, the increase
in uncertainty is very modest and in line with the levels implied by the active regime. This
is because agents regard the early deviations as temporary. However, as the number of deviations increases and fairly optimistic agents become fairly pessimistic, uncertainty starts
2
increasing and eventually converges to the values implied by the long-lasting passive regime.
As a result, for each horizon, our measure of uncertainty is now higher than its long run value.
This is because agents take into account that while in the short run a prolonged period of
passive monetary policy will prevail, in the long run the economy will surely visit the active
regime again. Therefore, an important result arises: Deviations from the active regime that
last only a few periods have no disruptive consequences on welfare because they do not have
a large impact on agents’uncertainty regarding future monetary policy. Instead, if a central
bank deviates for a prolonged period of time, the disanchoring of agents’uncertainty occurs,
causing sizeable welfare losses.
The model under the assumption of no transparency is …tted to U.S. data. In line with
previous contributions, we identify prolonged deviations from active monetary policy in the
’60s and the ’70s. However, we also …nd that the Federal Reserve has recurrently engaged
in short-lasting passive policies since the early ’80s, supporting the view that constrained
discretion has been the predominant approach to U.S. monetary policy in the last three
decades. In the analysis, we abstract from the reasons why the Federal Reserve has engaged
in such deviations. In fact, we consider such recurrent deviations as a given of our analysis.
The objective of this paper is to use the estimated model to evaluate how quickly agents’
beliefs respond to policy makers’ behavior and announcements, what this implies for the
evolution of uncertainty and welfare, and what the potential gains are from reducing the
uncertainty about the conduct of monetary policy.
The paper introduces a practical de…nition of reputation: a central bank has strong
reputation if it is less likely to engage in long-lasting deviations from active policies. It
is worth pointing out that the proposed de…nition of central bank reputation is not only
re‡ected in the in sample frequency of regime changes, but it also manifests itself a¤ecting
agents’beliefs and, consequently, the general equilibrium properties of the macroeconomy.
Therefore, the proposed de…nition of central bank reputation has the advantage of being
measurable in the data, while at the same time being in line with the seminal contributions
of Kydland and Prescott (1977), Barro and Gordon (1983), and Gali and Gertler (2007).
The Federal Reserve is found to bene…t from strong reputation. Based on the estimates,
pessimism and hence agents’ uncertainty about future in‡ation change very sluggishly in
response to deviations from active monetary policy. In fact, uncertainty is found to stay anchored and move only very gradually as the Federal Reserve deviates from active monetary
policy. This …nding has the important implication that the Federal Reserve can conduct
passive policies for up fairly large number of years before the disanchoring of in‡ation expectations and an overall increase in macroeconomic uncertainty occur.
While this result implies that the Federal Reserve can successfully implement constrained
3
discretion even without transparency, our …ndings suggest that increasing transparency
would improve welfare. The estimated model suggests that the welfare gains from transparency range between 0.67% to 6.63%. A transparent central bank systematically announces
the duration of any deviation from the active regime beforehand. The implications of such a
communication strategy vary based on the nature of the deviation. When the central bank
engages in a short lasting deviation, announcing its duration immediately removes the fear
of the ’70s: When agents are not informed about the exact nature of an observed deviation, whenever a short deviation occurs, ex-ante agents cannot rule out the possibility of
a long-lasting deviation of the kind that characterized the early part of the sample. As a
result, ex-post, agents turn out to have overstated the persistence of the observed deviation.
How large this e¤ect is depends on the central bank reputation, captured by the conditional
probability of engaging in a long lasting deviation.
When instead a deviation is in fact long lasting, the model allows us to highlight an
important trade-o¤ associated with transparency. First, in the short run being transparent
reduces welfare because agents are told that passive monetary policy will prevail for a while
and thereby future shocks are expected to have more dramatic in‡ationary/de‡ationary
consequences. It follows that, if the duration of the announced deviation is long enough,
over the early periods uncertainty is higher than when no announcement is made. This
short-run e¤ect on welfare arises because the central bank publicly commits to a policy that
de-emphasizes in‡ation stabilization for the announced number of future periods. Agents
understand that such a commitment to follow the announced policy course limits the central
bank’s ability of countering the in‡ationary consequences of future shocks that might occur
during the implementation of the announced policy. Therefore, the announcement leads to
a higher macroeconomic risk and associated detrimental e¤ects on welfare. Second, as time
goes by, agents know that the prolonged period of passive monetary policy is coming to an
end. This leads to a reduction in the level of uncertainty at every horizon with an associated
improvement in welfare. Notice, that this is exactly the opposite of what occurs when no
announcement is made: Agents, in this case, become more and more discouraged about the
possibility of moving to the active regime and uncertainty increases. To our knowledge, this
is the …rst paper that studies this critical trade-o¤ and its welfare implications through the
lens of an estimated DSGE model.
This paper makes three main methodological contributions to the existing literature.
First, we estimate a microfounded general equilibrium model with changes in policy makers’
behavior and Bayesian learning. To the best of our knowledge, this is the …rst paper that
estimates a DSGE model with Markov-switching parameters and Bayesian learning.3 Second,
3
The learning mechanism implies that agents’beliefs are not invariant to the duration of a certain policy.
4
we show how to model systematic and recurrent policy makers’announcements in a general
equilibrium framework. In light of the recent development of forward guidance, we believe
that this contribution should be of independent interest. Finally, we show how to characterize
and compute social welfare in a Markov switching DSGE model with Bayesian learning and
announcements. In doing so, we combine the methods developed by Bianchi (2013a) to
measure uncertainty in MS-DSGE models with the solution methods for MS-DSGE models
with learning developed by Bianchi and Melosi (2014b) and the solution methods for MSDSGE models with announcements developed in this paper.
Our modeling framework goes beyond the assumption of anticipated utility that is often
used in the learning literature.4 Such an assumption implies that agents forecast future
events assuming that their beliefs will never change in the future. Instead, agents in our
models know that they do not know. Therefore, when forming expectations, they take into
account that their beliefs will evolve according to what they will observe in the future. In our
context, it is possible to go beyond the anticipated utility assumption because we can keep
track of the joint evolution of policymakers’behavior and agents’beliefs. Using anticipated
utility would break the link between the observed policy path and the future policy course.
This link is key for the dynamics of uncertainty. To understand why, consider a prolonged
sequence of deviations from the active regime. This would have two e¤ects. First, monetary
policy is less active in stabilizing in‡ation. Second, agents become more pessimistic about a
return to the active regime. Both e¤ects are re‡ected in the model solution with important
consequences for the expected impact of future shocks and, consequently, the evolution of
uncertainty and welfare.
This paper is part of a broader research agenda that aims to model the evolution of
agents’ beliefs in general equilibrium models. In Bianchi and Melosi (2014a), we study a
model in which the current policy makers’ behavior in‡uences agents’ beliefs about the
way debt will be stabilized. In Bianchi and Melosi (2013), we develop methods to study
the evolution of agents’beliefs in general equilibrium models. Unlike those two papers, in
this paper we conduct a full estimation of a DSGE model with parameter instability and
information frictions. We use the results to assess how anchored in‡ation expectations and
uncertainty are in the U.S. economy and to investigate the welfare implications of forwardlooking communication by the Federal Reserve.
Eusepi and Preston (2010) study monetary policy communication in a model where
Therefore, the model captures a very intuitive idea: Agents in the late ’70s were arguably more pessimistic
about a return to the active regime with respect to the early ’70s. This feature was not present in previous
contributions such as Bianchi (2013b) and Davig and Doh (2008).
4
For some prominent examples see Marcet and Sargent (1989b,a) Cho, Williams, and Sargent (2002), and
Evans and Honkapohja (2001, 2003).
5
agents face uncertainty about the value of model parameters. Unlike Eusepi and Preston
(2010), agents in our model are not bounded rational, they only have incomplete information.
Cogley, Matthes, and Sbordone (2011) address the problem of a newly-appointed central
bank governor who inherits a high average in‡ation rate from the past and wants to disin‡ate.
In their model, agents conduct Bayesian learning over the coe¢ cients that characterize the
conduct of monetary policy, but they are bounded rational to the extent that use anticipated
utility to form expectations. In our model, regime changes are recurrent, agents learn about
the regime in place as opposed to the Taylor rule parameters, and expectations re‡ect the
possibility of changes in beliefs and policy makers’behavior. Finally, the tractability of our
approach allows us to conduct a full estimation.
Schorfheide (2005) considers an economy in which agents use Bayesian learning to infer
changes in a Markov-switching in‡ation target. In that paper agents solve a …ltering problem
to disentangle a persistent component from a transitory component. The learning mechanism
is treated as external to the model, implying that the model needs to be solved in every
period in order to re‡ect the change in agents’beliefs regarding the persistent and transitory
components. Consequently, when agents form expectations they do not take into account
how their beliefs will respond to future observations. On the contrary, in this paper agents
form expectations by knowing that they do not know. Furthermore, the method developed
in Schorfheide (2005) cannot be immediately extended to models in which agents learn about
changes in the stochastic properties of the model parameters.
The paper is then related to a growing literature that models parameter instability to
capture changes in the evolution of the macroeconomy. This consists of three branches: Davig
and Leeper (2007), Farmer, Waggoner, and Zha (2009), and Foerster, Rubio-Ramirez, Waggoner, and Zha (2011) develop solution methods for Markov-switching rational expectations
models, Justiniano and Primiceri (2008), Benati and Surico (2009), Bianchi (2013b), Bianchi
and Ilut (2013), Davig and Doh (2008), and Fernandez-Villaverde and Rubio-Ramirez (2008)
introduce parameter instability in estimated dynamic equilibrium models, while Sims and
Zha (2006), Primiceri (2005), Cogley and Sargent (2005), and Boivin and Giannoni (2006)
work with structural VARs. Finally, our work is also linked to papers that study the impact
of monetary policy decisions on in‡ation expectations, such as Nimark (2008), Mankiw, Reis,
and Wolfers (2004), Del Negro and Eusepi (2010), and Melosi (2014a and 2014b).
This paper is organized as follows. Section 2 introduces the baseline model. In Section
3, we show how to solve the model under no transparency and transparency. In Section
4, the model under the assumption of no transparency is …tted to U.S. data. In Section 5
we use the estimated model to assess the welfare implications of introducing transparency.
In Section 6 we study some extensions and assess the robustness of our results. Section 7
6
concludes.
2
The Model
The model is a prototypical three-equation New-Keynesian model (Clarida, Gali, and Gertler,
2000 and Woodford, 2003), which has been used for empirical studies (Lubik and Schorfheide,
2004). We make two main departures from this standard framework. First, we assume that
households and …rms have incomplete information, in a sense to be made clear shortly.
Second, we assume parameter instability in the monetary policy rule.
Households: The representative household maximizes
E
hP
1
t=0
t
Gt (1
1
)
Ct1
(1 + )
1
Nt1+
i
jF0 ;
where Ct is composite consumption and Nt are hours worked in period t. The parameter
2 (0; 1) is the discount factor, the parameter
0 is the inverse of the Frisch elasticity
of labor supply. E [ jF0 ] is the expectation operator conditioned on information of private
agents available at time 0. The information set Ft contains the history of all model variables
and volatility regimes vt but not the history of policy regimes pt that, as we shall show,
determine the parameter value of the central bank’s reaction function. Gt is an exogenous
process a¤ecting the discount factor of households and is assumed to follow a stationary
…rst-order autoregressive process:
ln Gt = 1
g
ln G +
g
ln Gt
1
+
g;
gt ;
v
t
N (0; 1) :
gt
(1)
where gt is an i.i.d. Gaussian shock and g; vt is determined by the exogenous variable vt ,
which is assumed to follow a discrete Markov-switching process. As it is common in the
literature we assume G = 1 implying that the discount factor in steady state is given by .
We refer to gt as preference shock.
Composite consumption in period t is given by the Dixit-Stiglitz aggregator
Ct =
R1
1 1="t
C
di
0 it
"t
"t 1
;
where Cit is consumption of a di¤erentiated good i in period t and "t > 1 determines
the elasticity of substitution between consumption goods. The elasticity of substitution is
determined by the following exogenous process:
ln Mt = (1
m ) ln M
+
m
ln Mt
7
1
+
m;
v
t
mt ;
mt
N (0; 1)
(2)
where Mt = ("t 1) 1 and mt is referred to as price markup shock. Analogously to the
preference shocks, the standard deviation of the markup shock m;t is determined by the
discrete Markov-switching process vt :
The ‡ow budget constraint of the representative household in period t reads
Pt Ct + Bt = Rt 1 Bt
1
+ Wt Nt + Dt
Tt ;
where Pt is the price level in period t, Bt 1 is the stock of one-period nominal government
bonds held by the household between period t 1 and period t, Rt 1 is the gross nominal
interest rate on those bonds, Wt is the nominal wage rate, Dt are nominal aggregate pro…ts,
and Tt are nominal lump-sum taxes in period t. The price level is given by
R1
Pt =
0
Pit1
"t
di
1=(1 "t )
(3)
:
In every period t, the representative household chooses a consumption vector, labor
supply, and bond holdings subject to the sequence of the ‡ow budget constraints and a noPonzi-scheme condition. The representative household takes as given the nominal interest
rate, the nominal wage rate, nominal aggregate pro…ts, nominal lump-sum taxes, and the
prices of all consumption goods.
Firms: There is a continuum of monopolistically competitive …rms of mass one. Firms
are indexed by i. Firm i supplies a di¤erentiated good i. Firms face Calvo-type nominal
rigidities and the probability of re-optimizing prices in any given period is given by 1
independent across …rms. Those …rms that are not allowed to re-optimize index their prices
to the steady-state in‡ation rate . Those …rms that are allowed to re-optimize their price
choose their price Pt (i) so as to maximize:
P1
k=0
k
Et Qt;t+k
k
Wt+k Nt+k (i) jFt
Pt (i) Yt+k (i)
where Qt;t+k is the stochastic discount factor measuring the time t utility of one unit of
consumption good available at time t + k, Nt (i) is amount of labor hired, and Yt (i) is the
amount of di¤erentiated good produced by …rm i. Firms are endowed with an identical
technology of production:
Yt (i) = Zt Nt (i)1 :
The variable Zt captures exogenous shifts of the marginal costs of production and is assumed
to follow a stationary …rst-order autoregressive process:
ln Zt = (1
z ) ln Z
+
z
ln Zt
8
1
+
z;
v
t
zt ;
zt
N (0; 1) :
(4)
We refer to the innovations zt as technology shocks. Again, the Markov-switching process
v
t determines the volatility regime for the technology shock. Re-optimizing …rms face a
sequence of demand constraints:
k
Yt+k (i) =
Pt (i) =Pt+k
"t
Yt+k
Policy Makers: There is a monetary authority and a …scal authority. The ‡ow budget
constraint of the …scal authority in period t reads
Tt + Bt = Rt 1 Bt 1 :
The …scal authority has to …nance maturing government bonds. The …scal authority can
collect lump-sum taxes or issue new government bonds. We assume that the …scal authority
follows a Ricardian …scal policy. The monetary authority sets the nominal interest rate Rt
according to the Taylor rule
Rt = Rt
p
r; t
1
"
t
p
; t
p
y; t
Yt
Yt
#1
p
r; t
exp
r;
v
t
rt
;
rt
N (0; 1)
(5)
where t = (Pt =Pt 1 ) is in‡ation and Yt is aggregate output in period t, and Yt is the
potential output. The variable rt captures non-systematic exogenous deviations of the
nominal interest rate Rt from the rule. The standard deviation of the monetary shock is
assumed to depend on the volatility regime vt that follows a discrete Markov process. The
variable pt is the policy regime that determines the policy coe¢ cients of the rule re‡ecting
the emphasis of the central bank on in‡ation stabilization relative to output gap stabilization
in any period t.
2.1
Volatility and Policy Regimes
The standard deviations of the preference shocks, the markup shocks, the technology shocks,
and the monetary shocks are determined by the volatility of regime vt . The volatility regime
follows a discrete Markov process and can assume two values: High and Low. The low
volatility regime is characterized by standard deviations that are strictly smaller than those
associated with the high volatility regime. Transition matrix that governs the evolution of
the two volatility regimes vt is the following:
Pv =
"
pH
1 pH
1 pL
pL
9
#
where pH (pL ) captures the probability of staying in the high (low) volatility regime. Unlike
the policy regimes pt , the realizations of the volatilities regimes vt are perfectly observed by
the agents (i.e., vt 2 Ft , any t).
We model changes in the central bank’s emphasis on in‡ation and output gap stabilization
by introducing a Markov-switching process pt with three regimes that evolve according to
the matrix:
2
3
p11
p12 p13
6
7
Pp = 4 1 p22 p22 0 5
(6)
1
p33
0
p33
The realized regime determines the monetary policy parameters of the central bank’s reaction
function. In symbols, for j 2 f1; 2; 3g:
R
(
p
t
= j) ;
(
p
t
= j) ;
y
(
p
t
2
6
= j) = 4
A
R;
P
R;
P
R;
A
;
P
;
P
;
A
y
P
y
P
y
3
, if j = 1
7
, if j = 2 5
, if j = 3
(7)
Under Regime 1, the active regime, the central bank’s main goal is to stabilize in‡ation
and the Taylor principle is satis…ed
( pt = 1) = A
1. Under Regime 2, the shortlasting passive regime, the central bank de-emphasizes in‡ation stabilization by deviating
from the Taylor principle P < 1, but only for short periods of time (on average). The
same parameter combination also characterizes Regime 3, the long-lasting passive regime.
However, under Regime 3 deviations are generally more prolonged. In other words, Regime
2 is less persistent than Regime 3: p22 < p33 . Summarizing, the two passive regimes do not
di¤er in terms of the response to in‡ation P and the output gap Py , but only in terms of
their relative persistence.
The three policy regimes are meant to capture the recurrent changes in the Federal
Reserve’s attitude toward in‡ation and output stabilization in the postwar period. A number
of empirical works (Clarida, Gali, and Gertler, 2000, Lubik and Schorfheide, 2004, Bianchi,
2013) have documented that the Federal Reserve de-emphasized in‡ation stabilization for
prolonged periods of time in the 1970s. Furthermore, as argued by Bernanke (2003), while
the Federal Reserve has been mostly focused on actively stabilizing in‡ation and in‡ation
expectations starting from the early 1980, it has also occasionally engaged in short-lasting
policies whose objective was not stabilizing in‡ation in the short run. This monetary policy
approach has been dubbed constrained discretion. We introduce this three-regime structure
so as to give the model enough ‡exibility to explain both the long-lasting passive monetary
policy of the 1970s as well as the recurrent and short-lasting passive policies of post-1970s.
The probabilities p11 , p12 , p22 govern the evolution of monetary policy when the central
10
bank follows constrained discretion. The larger p12 is vis-a-vis to p11 , the more frequent the
short-lasting deviations are. The larger p22 is, the more persistent the short-lasting deviations
are. The probability p13 controls how likely it is that constrained discretion is abandoned
in favor of a prolonged deviation from the active regime. The ratio p12 = (1 p11 ) captures
the relative probability of a short-lasting deviation conditional on having deviated to passive
regimes and can be interpreted as a measure of central bank’s reputation. This is because this
composite parameter controls how likely it is that the central bank will abandon constrained
discretion the moment it starts deviating from the active regime. When p12 = (1 p11 ) is close
to unity, agents expect that the central bank will refrain from engaging in 1970s-style longlasting passive policies that - as we shall show - are invariably associated with heightened
in‡ation instability. As it will become clear later on, central bank reputation has deep
implications for the general equilibrium properties of the macroeconomy. Therefore, the
parameters of the transition matrix do not only a¤ect the frequency with which the di¤erent
regimes are observed, but also the law of motion of the economy across the di¤erent regimes.
This is because agents are fully rational and form expectations taking into account the
possibility of regime changes, implying that their beliefs matter for the way shocks propagate
through the economy. Therefore, the proposed de…nition of central bank reputation has the
important advantage of being measurable, even over a relatively short period of time.
2.2
Communication Strategies
In the model, regime changes do not a¤ect the steady state, but only the way the economy
propagates around it. We then log-linearize the model around the steady-state equilibrium.
We then obtain:5
1
yt = E (yt+1 jFt )
t
=
Et (
t+1 jFt )
R;
rt
1
gt =
g gt 1
+
g;
v
t
gt
zt =
z zt 1
+
z;
v
t
zt
m;
v
t
mt =
m mt 1
+ 1
+
R;
t+1 jFt )]
Et (
+ (yt
rt =
p
t
[it
p
t
z ) + mt
ht
;
p
t
t
+
(8)
+ gt
y;
p
t
(yt
i
zt ) +
(9)
r;
v
t
r;t
(10)
(11)
(12)
(13)
m;t
where lowercase variables denote log-deviations of uppercase variables from their steady
(1
)+ +
is the slope of the Phillips curve. The model
state equilibrium and
(1
)(1 )
1
+ "
can then be solved under di¤erent assumptions on what the central bank communicates to
5
Following Lubik and Schorfheide (2004) we rescale the preference process Gt .
11
agents about the future monetary policy course. Central bank communication a¤ects agents’
information set Ft . We consider two cases: no transparency and transparency.
If the central bank is not transparent, it never announces the duration of passive policies.
We call this approach no transparency. We make a minimal departure from the assumption
of perfect information assuming that agents can observe the history of all the endogenous
variables, the history of the structural shocks as well as the history of the volatility regimes
p
v
t but not the policy regimes t . It should be noted that agents are always able to infer if
monetary policy is currently active or passive. However, when monetary policy is passive,
agents cannot immediately …gure out whether the short-lasting Regime 2 or the long-lasting
Regime 3 is in place. To see why, recall that the two passive regimes are observationally
equivalent to agents, given that p and py are the same across the two regimes. Therefore,
agents conduct Bayesian learning in order to infer which one of the two regimes is in place.
In the next section we will discuss how agents’ beliefs evolve as agents observe more and
more deviations from the active regime.6
Under transparency all the information held by the central bank is communicated to
agents. We assume that the central bank knows for how long it will be deviating from active
monetary policy. Therefore, a transparent central bank announces the duration of passive
policies, revealing to agents exactly when monetary policy will switch back to the active
regime. It is important to emphasize that agents form their beliefs by taking into account
that the central bank will systematically announce the duration of passive policy. We assume
that central bank’s announcements are truthful and are believed as such by rational agents.
In Section 7, we will consider the case in which the central bank has much less information
about the duration of its policy course and can only announce the likely duration of the
passive policies; that is, the type of passive regime (i.e., pt 2 f2; 3g) that the central bank
will carry out. This case corresponds to a form of transparency in which the central bank
communicates only the likely duration rather than the actual duration of the passive policy.
3
Beliefs Dynamics and Model Solution
Di¤erent communication strategies imply di¤erent dynamics of beliefs and hence di¤erent
solution methods. Let us …rst discuss how to solve the model in which the central bank is
6
It might be argued that the central bank could try to signal the kind of deviation perturbing the Taylor
rule parameters across the two rules. For example,
(st = 3) =
(st = 2) + for 6= 0 and small.
However, the point of the paper is exactly to capture agents’ uncertainty about the duration of passive
policies. Therefore, the model would be extended to allow for a total of four passive regimes: a long-lasting
Regime 4 in which
=
(st = 2) and p44 > p22 and a short-lasting Regime 5 in which
=
(st = 3)
and p55 < p33 .
12
not transparent. Since agents know the history of endogenous variables and shocks, they can
exactly infer the policy mix that is in place at each point in time. However, while the active
regime is fully revealing, when the passive regime is prevailing, agents do not know whether
the central bank is engaging in a short-lasting deviation or a long-lasting one. Agents have to
learn the nature of the deviation in order to form expectations over the endogenous variables
of the economy.
To solve the model under no transparency we use the methods developed in Bianchi
and Melosi (2014b). We brie‡y report the main features of this solution method so as to
make this paper self-contained. Denote the number of consecutive deviations from the active
regime at time t as t 2 f0; 1; :::g, where t = 0 means that monetary policy is active at time
t. Conditional on having observed t 1 consecutive deviations from the active regime at
time t, agents believe that the central bank will keep deviating in the next period t + 1 with
the following probability:7
prob f
t+1
6= 0j
t
p22 (p12 =p13 ) (p22 =p33 ) t 1 + p33
:
6 0g =
=
(p12 =p13 ) (p22 =p33 ) t 1 + 1
(14)
Equation (14) makes it clear that prob f t+1 6= 0j t 6= 0g = prob f t+1 6= 0jFt g as t is a
su¢ cient statistic for the probability of being in the passive regime next period. Furthermore,
this equation captures the dynamics of agents’beliefs about observing yet another period of
passive policy in the next period, which is the key state variable we use to solve the model
under no transparency. It should be observed that this equation has a number of properties
that are quite insightful to the key mechanism of the model at hand. It is useful to observe
that the probability of observing yet another period of passive policy in the next period is
a weighted average of the probabilities p22 and p33 with weights that vary with the number
of consecutive periods of passive policy t . When agents observe the central bank deviating
from the active regime for the …rst time ( t = 1), the weights for the probabilities p22 and
p33 are p12 = (1 p11 ) and p13 = (1 p11 ), respectively. These weights can be interpreted as
agents’priors about which passive regime is in place when the …rst deviation is observed.
As more and more periods of passive policy are observed ( t "), the weight assigned to
the short-lasting passive Regime 2 monotonically decreases due to the fact that p33 > p22 .
Consequently, as the …rst period of passive policy is observed, agents’beliefs about observing
a passive policy in the next period are at their lower bound. Furthermore, as the central
bank keeps on deviating, agents get increasingly convinced that the economy has entered a
long-lasting deviation, given that under this policy regime long deviations are more likely.
7
This result can be derived by applying the Bayes’theorem and then combining the resulting probabilities
with the transition matrix H. The proof is straightforward and is shown in Bianchi and Melosi (2014b).
13
Importantly, how low is the lower bound for the probability of observing yet another
period of passive policy will depend on the level of the central bank’s reputation. High
reputation makes the weight p12 = (1 p11 ) close to one, implying that the probability of
observing a second consecutive period of passive policy will be very close to p22 , the value
associated with a short lasting deviation. When reputation is high, it is very unlikely that the
central bank engages in a long-lasting passive policy. Therefore, as the …rst period of passive
policy is observed, agents are quite con…dent to have entered the short-lasting passive regime
(Regime 2). If the central bank keeps deviating from the active regime, agents will eventually
become convinced of being in the long-lasting passive regime (Regime 3) regardless of the
level of the central bank’s reputation, p12 = (1 p11 ).8 After a su¢ ciently long-lasting passive
policy, the probability of observing an additional deviation in the next period degenerates to
the persistence of the long-lasting Regime 3. Formally: lim prob f t+1 6= 0j t 6= 0g = p33 .
t !1
Hence, p33 is the upper bound for the probability that agents attach to staying in the passive
regime next period. It follows that rational agents cannot get more convinced to observe yet
another passive policy in the next period than when they are sure to be in the long-lasting
Regime 3. More formally, for each e > 0, there exists an integer
such that:
p33
prob f
t+1
6= 0j
t
g < e;
=
(15)
with the important result that for any t > , agents’beliefs can be e¤ectively approximated
using the properties of the long-lasting passive regime (Regime 3).
Endowed with these results, we can solve the model under no transparency by expanding
the number of regimes in order to take into account the evolution of agents’beliefs. Now each
regime is characterized by central bank’s behavior and the number of observed consecutive
deviations from the active policy at any time t t 2 f0; 1; :::; g : The mapping to the
parameter values of the policy rule is as follows:
r
(
t
= j) ;
(
t
= j) ;
y
(
t
= j) =
"
P
r ;
A
r;
P
A
;
;
P
y
A
y
, if j = 0
, if 1 j <
#
(16)
The transition matrix for this new set of regimes t 2 f0; 1; :::; g can be derived by equation
(14) as shown in Appendix A.
Endowed with these results regarding the dynamics of agents’beliefs, one can recast the
8
We abstract from the extreme case in which the central bank’s reputation is such that p12 = (1 p11 ) = 1.
In this case, agents’ beliefs will not evolve at all as the central bank deviates Another limit case is when
the central bank’s reputation is at its lowest; that is, p12 = (1 p11 ) = 0. In this case, agents know that any
passive policy is surely of the long-lasting type and do not update their beliefs during the implementation
of the passive policy. We do not consider these two extreme cases in this paper.
14
Markov-switching DSGE model under no transparency and learning as a Markov-switching
Rational Expectations model. Now regimes are de…ned in terms of the observed consecutive
duration of the passive regimes, t , which, unlike the primitive set of policy regime pt 2
f1; 2; 3g, belongs to the agents’information set Ft . This result allows us to solve this model
by applying any of the methods developed to solve Markov-switching rational expectations
models, such as Davig and Leeper (2007), Farmer, Waggoner, and Zha (2009), and Foerster,
Rubio-Ramirez, Waggoner, and Zha (2011). We use Farmer, Waggoner, and Zha (2009) to
solve the model with learning once the policy regimes are rede…ned as described above.
It is worth emphasizing that this way of recasting the learning process allows us to
tractably model the behavior of agents that know that they do not know. In other words,
agents are aware of the fact that their beliefs will change in the future according to what they
observe in the economy. This represents a substantial di¤erence with the anticipated utility
approach, in which agents form expectations without taking into account that their beliefs
about the economy will change over time. Furthermore, our approach di¤ers from the one
traditionally used in the learning literature in which agents form expectations according to a
reduced form law of motion that is updated recursively (for example, using discounted least
squares regressions). The advantage of adaptive learning is the extreme ‡exibility given that,
at least in principle, no restrictions need to be imposed on the type of parameter instability
characterizing the model. However, such ‡exibility does not come without a cost, given that
agents are not really aware of the model they live in, but only of the implied law of motion.
Instead, in this paper, agents fully understand the model and they are aware of the trade-o¤s
that characterize it. However, they are uncertain about the central bank’future behavior,
and this uncertainty has important consequences for the law of motion of the economy.
When the central bank is transparent, the exact duration of every deviation from active
policy is truthfully announced. In this model the number of announced deviations from
the active policy yet to be carried out at is a su¢ cient statistic that captures the dynamics
of beliefs. Hence, we rede…ne the set of policy regimes in terms of this variable with the
following mapping to the parameter values of the policy rule:
r
(
a
t
= j) ;
(
a
t
= j) ;
y
(
a
t
= j) =
"
P
r ;
A
r;
P
A
;
;
P
y
A
y
, if j = 0
, if 1 j <
a
#
(17)
where a is a large number at which we truncate the rede…ned set of regimes.9 The regimes
0
a
a
eA = [e
g governed by the ( a + 1) ( a + 1) transition matrix P
p10
e20
t 2 f0; 1; :::;
A; p
A ] , where
pe1A is a 1 ( a + 1) vector whose j th element peA (j) is p11 if j = 1 and p12 pj22 2 p21 +p13 pj33 2 p31
9
Since p33 < 1, it can be easily show that the higher the truncation regime a , the lower the probability
that the realized duration is larger than a , the lower the approximation error.
15
(the probability that the realized passive policy will last exactly j 1 consecutive periods)
a
for any 2 j
+ 1. The a ( a + 1) matrix pe2A is de…ned as I a ; 0 a 1 , where I a is
a
identity matrix and 0 a 1 is a 1 column vector of zeros. Note that regimes are
a a
ordered from the smallest number of deviations (zero, the active policy) to the largest one
( a ).
Similarly to the case of no transparency, we have recasted the Markov-switching DSGE
model under transparency as a Markov-switching Rational Expectations model, in which
the regimes are rede…ned in terms of the number of announced deviations from the active
regimes yet to be carried out, at , which, unlike the policy regime pt , belongs to the agents’
information set Ft . This result allows us to solve the model under transparency by applying
any of the methods developed to solve Markov-switching rational expectations models.
4
Empirical Analysis
In order to put discipline on the parameter values, the model under the assumption of no
transparency is …tted to US data. We believe that the model with a non-transparent central
bank is the better suited to capture the Federal Reserve communication strategy in our
sample that ranges from mid-1950s to prior the great recession. We then use the results to
quantify the Federal Reserve reputation and the potential gains from making the Federal
Reserve more transparent.
This section is organized as follows. Section 4.1 brie‡y deals with the Bayesian estimation
of the model. In Section 4.2 we show the evolution of agents’beliefs about future monetary
policy, which is key to understand the welfare implications of transparency.
4.1
Data and Estimation
For observables, we use three series of U.S. quarterly data: the (HP …ltered) real GDP
per capita, the annualized quarterly in‡ation (GDP de‡ator), and the Federal Funds Rate
(FFR). The sample spans from 1954:III to 2008:I. Table 1 reports the prior and the posterior
distribution of model parameters.10 To keep the dimensionality of the state space tractable,
we measure the output gap using the HP-…ltered GDP. For a detailed discussion of the
estimation strategy see Bianchi (2013b). We shut down the process for the technology zt as
its parameters cannot be identi…ed. The parameter rr denotes the steady-state equilibrium
real interest rate. The parameter
is the standard deviation of the measurement error
associated with in‡ation.
10
The convergence statistics of the Gibbs sampler are reported in Appendix B.
16
Name
A
A
y
A
R
P
P
y
P
R
p11
p22 =p33
p33
p12 = (1 p11 )
1
g
100 rr
100
100 H
R
100 H
g
100 H
100 L
R
100 L
g
100 L
pH
pL
100
Mode
Posterior
Mean
5%
1:6033
0:2450
0:6253
0:7456
0:3706
0:8725
0:7887
0:9132
0:9538
0:9186
4:5927
0:0185
0:8335
0:9462
0:4230
0:5151
0:3029
0:2942
1:3700
0:0714
0:1349
0:4798
0:9086
0:9646
0:2772
1:6332
0:2763
0:6899
0:7587
0:4194
0:8549
0:7825
0:8326
0:9401
0:8773
4:6988
0:0173
0:8312
0:9288
0:4359
0:5142
0:3228
0:3128
2:2249
0:0771
0:1420
0:7567
0:8856
0:9545
0:2741
1:2896
0:0934
0:5384
0:4926
0:2056
0:7269
0:6254
0:6529
0:8862
0:7738
3:0468
0:0079
0:7862
0:8909
0:3404
0:4693
0:2613
0:2364
1:1936
0:0654
0:1127
0:4383
0:7951
0:9198
0:2369
95%
Type
Prior
Mean
Std.
2:0583
0:6042
0:8916
1:0537
0:6867
0:9107
0:9251
0:9438
0:9779
0:9595
6:7451
0:0290
0:8755
0:9617
0:5330
0:5582
0:3973
0:3971
3:7360
0:0908
0:1757
1:2573
0:9515
0:9796
0:3131
N
G
B
N
G
B
B
B
B
B
G
G
B
B
G
G
IG
IG
IG
IG
IG
IG
Dir
Dir
IG
1:8
0:25
0:7
0:8
0:25
0:7
0:85
0:75
0:9
0:9
3
0:3
0:8
0:7
0:6
0:5
0:31
0:38
1
0:31
0:38
1
0:8333
0:8333
0:15
0:3
0:15
0:15
0:3
0:15
0:15
0:1
0:1
0:05
0:05
1
0:2
0:1
0:15
0:3
0:025
0:4
0:4
0:8
0:4
0:4
0:8
0:1034
0:1034
0:3
Table 1: Posterior modes, means, and 90% error bands of the model parameters. Type N, G,
B, IG stand for Normal, Gamma, Beta, Inversed Gamma density, respectively. Dir stands
for the Dirichelet distribution
At the posterior mode, the passive policy implies a higher output-gap coe¢ cient y than
that implied by the active policy. The probability of being in the short-lasting passive regime
conditional on having switched to passive policies, p12 = (1 p11 ), plays a critical rule in the
model. As noticed in Section 2, this parameter value relates to the strength of the Federal
Reserve reputation to refrain from long-lasting deviations. This parameter is found to be
fairly close to one, con…rming that the Federal Reserve has strong reputation. This number
means that as agents observe a deviation from the active regime, they expect that the Federal
Reserve is conducting a short-lasting passive policy with probability 0:92.
Figure 1 shows the estimated probabilities of the active policy regime (upper panel)11 and
the estimated probabilities of the high volatility regimes (lower panel). In line with previous
studies, it emerges that the 1970s and the early 1980s were periods of high volatility. While
11
As discussed in Section 3, we estimate the model after we have rede…ned the set of regimes as the
number of consecutive deviations from the active policy t . Therefore, we cannot tease out the evolution of
the probability of the short-lasting passive regime and the long-lasting passive regime.
17
Probability Active Monetary Policy
0.8
0.6
0.4
0.2
0
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
1995
2000
2005
Probability High Volatility Regime
1
0.8
0.6
0.4
0.2
0
1955
1960
1965
1970
1975
1980
1985
1990
Figure 1: Estimated probability of the active monetary policy regime and the high volatility
regime.
the 1970s were characterized by a fairly long-lasting passive policy, the Federal Reserve has
alternated periods of active policy to periods of rather short-lasting passive policy in the
post-1970s. The most recent approach to monetary policy closely resembles the idea of
constrained discretion discussed by Bernanke (2003); that is, a monetary policy approach
whose main objective is to stabilize in‡ation through active policies but the central bank
may sometimes de-emphasize in‡ation stabilization for rather short periods of time.
4.2
Beliefs Dynamics
Monetary policy decisions on stabilizing in‡ation and communication strategies critically
a¤ect the social welfare and the macroeconomic equilibrium by in‡uencing agents’pessimism
about future monetary policy. In this paper, we will use the word pessimism to precisely
mean agents’ expectations about the duration of an observed passive policy. A high level
of pessimism means that agents expect an observed passive policy to last for relatively
long; that is, close to the expected duration of the long-lasting passive Regime (1 p33 ) 1 .
While expecting a longer lasting deviation from the active regime is not necessarily welfare
decreasing, we will show that expecting a prolonged period of passive policy impairs social
welfare in the estimated model.
We measure pessimism by computing the expected number of consecutive periods of
passive monetary policy. Let us assume that the central bank decides to engage in passive
18
No Transparency
Transparency
22
100
90
20
80
18
70
Pessimism
16
60
14
50
40
12
30
10
20
8
10
6
0
20
40
60
80
100
Consecutive Periods of Passive Policy
20
40
60
80
100
Consecutive Period of Passive Policy
Figure 2: Pessimism on the vertical axis is measured as the number of expected consecutive
deviations ahead. On the left plot the two horizontal lines denote the smallest lower bound
(1 p22 ) 1 and upper bound of pessimism (1 p33 ) 1 . These statistics are computed at the
posterior mode.
policies lasting one hundred consecutive periods. While a policy of such a long duration is
clearly quite implausible for the U.S., this example is illustrative of how transparency a¤ects
pessimism relative to no transparency. Figure 2 reports the evolution of pessimism under no
transparency (left graph) and under transparency (right graph) at the posterior mode.12 The
two horizontal lines mark the smallest lower bound and upper bound for pessimism. The
former is given by the expected duration of the short-lasting passive Regime (1 p22 ) 1 . The
smallest lower bound is attained at the …rst period of passive policy only if the conditional
probability of a long lasting deviation is zero: p12 = (1 p11 ) = 1. The left graph shows that
the intercept of the solid line is quite close to the bottom dashed line, implying that agents’s
mostly expect that the Federal Reserve is engaging in a short-lasting deviation as the …rst
period of passive policy is observed. This result is due to the fact that the Federal Reserve’s
12
Under no transparency, the pessimism after having observed
policy is computed as follows:
prob f
p
t
= 2j t g
1
1
+ [1
p22
prob f
p
t
t
consecutive deviations from the active
= 2j t g]
1
1
p33
where prob f pt = 2j t g = p12 p22t 1 = p12 p22t 1 + p13 p33t 1 is the probability of being in Regime 2 conditional
on having observed t consecutive periods of passive regime.
19
reputation is estimated to be fairly high (p12 = (1 p11 ) = 0:92).
The upper bound for pessimism is given by the expected duration of the long-lasting
passive policy (1 p33 ) 1 and is attained only after a very large number of consecutive
deviations from the active regime. Such a gradual increase in pessimism suggests that the
Federal Reserve can enjoy a great deal of leeway in deviating from active monetary policy in
order to stabilize alternative short-lasting objectives. This result is again due to the strong
reputation of the Federal Reserve. If the reputation coe¢ cient p12 = (1 p11 ) were close to
zero, then the expected number of consecutive deviations would experience a larger jump
and, hence, the convergence to the upper bound would be much faster than what shown in
Figure 2.
As shown in the right graph, pessimism follows an inverse path under transparency.
Unlike the case of no transparency, agents’pessimism is very high at the initial stage of the
deviation from the passive policy but it decreases as the time goes by. This result comes
from assuming that agents are fully rational and the announcement is truthful. As the
one hundred periods of passive monetary policy are announced (t = 0), an immediate rise in
pessimism occurs. As the number of periods of passive policy yet to be carried out decreases,
agents’pessimism declines accordingly. At the end of the policy (t = 100), pessimism reaches
its lowest level, with agents expecting to return to the active regime with probability one
in the following period. It should be noted that at the end of the announced deviation
transparency allows the central bank to lower agents’ pessimism below the smallest lower
bound attainable under no transparency: This result emerges because the central bank is
able to inform agents about the exact period in which the passive policy will be terminated.
This assumption will be relaxed in Section 6.2.
To sum up, Figure 2 allows us to isolate two important e¤ects of transparency on agents’
pessimism about future monetary policy: (i) transparency raises pessimism at the beginning
of the policy; (ii) transparency anchors down pessimism at the end of the policy. As we shall
show, these two e¤ects play a critical role for the welfare implications of transparency.
5
Welfare Implications of Transparency
In this section, we use the model to assess the welfare implications of introducing transparency. Before proceeding, it is worth emphasizing that the regime changes considered in
this paper do not a¤ect the steady state, but only the way the economy ‡uctuates around the
steady state. The period welfare function can then be obtained by taking a log-quadratic
approximation of the representative household’s utility function around the deterministic
20
steady state:13
Wi (st (i) ;
v
t;
)=
P1
h=1
h
("= ) vari (
t+h jst
(i) ;
v
t)
+
+
+
1
vari (yt+h jst (i) ;
v
t)
(18)
, vari ( ) stands for the stochastic variance associated
where i 2 fN; T g,
+1
with agents’forecasts of in‡ation, and the output gap at horizon h, and , is the vector of
model parameter. The subscript i refers to the communication strategy: i = N stands for
the case of no transparency, while i = T denotes transparency. Finally, st (i) denotes the
policy regime: st (i = N ) = t and st (i = T ) = at . Recall from Section 3, the policy regime
st is the observed duration of passive policy for the case of no transparency and the number
of periods of announced passive policy yet to be carried out in the case of transparency. The
steady-state demand elasticity " = (1 + ) = , the Frisch labor supply elasticity parameter
, and the capital labor share are not identi…able. We set this parameter equal to 6, which
Rotemberg and Woodford (1999) argue to deliver plausible markups for the U.S. economy.
Following Rios-Rull, Schorfheide, Fuentes-Albero, Kryshko, and Santaeulalia-Llopis (2012)
we calibrate the (inverse) Frisch labor supply elasticity parameter
to 0:5. The capital
income share is set equal to 0:3.
The output gap enters the welfare function because it re‡ects the di¤erence between
the marginal rate of substitution and the marginal product of labor, which is a measure
of the economy’s aggregate ine¢ ciency (Woodford, 2003, Steinsson (2003), and Gali, 2008).
In‡ation deviations from its steady-state level reduce welfare by raising price dispersion. The
elasticity of substitution between two di¤erentiated goods " raises the weight of in‡ation
‡uctuations relative to the output gap because it ampli…es the welfare losses associated with
any given price dispersion. Nominal rigidities, whose size is inversely related to the slope of
the New Keynesian Phillips curve , raise the degree of price dispersion resulting from any
given deviation from the steady-state in‡ation rate.
Equation (18) makes it explicit that social welfare depends on agents’uncertainty about
future in‡ation and the future output gap. It should be noted that agents’ uncertainty
in any given period captures the macroeconomic risk associated with the observed policy
regime and communication strategy, st (i). Unlike standard New Keynesian models with
…xed parameters where welfare is merely a function of the unconditional variance of in‡ation
and the output gap, our model allows to study the dynamic e¤ects of policy actions and
forward-looking communication on welfare. Furthermore, the learning mechanism plays an
+
1
13
The derivation closely follows Woodford (2003), Gali (2008), Coibion, Gorodnichenko, and Wieland
(2012). Furthermore, we assume that the ine¢ ciency generated by the market power are removed by the
suitable choice of subsidies so that the steady-state equilibrium is e¢ cient. A detailed derivation of the
welfare function is in Appendix C.
21
;
important role in our welfare analysis by linking the concept of reputation, which can be
directly measured in the data, to the central bank’s ability of controlling the dynamics of
the macroeconomic risk associated with policy actions. This last point will be the focus of
the next session.
To assess the desirability of transparency, we compute the model predicted welfare
gains/losses from transparency as follows:
We ( ) =pT ( at ;
v 0
t)
WT ( at ;
v
t;
)
pN ( t ;
v 0
t)
WN ( t ;
v
t;
)
(19)
where pT ( a ; vt ) stands for the vector of the ergodic joint probabilities of a passive policy of announced duration at and volatility regime vt . pN ( t ; vt ) stands for the vector of
ergodic joint probabilities of a passive policy of observed duration t and volatility regime
v
t are realized. It is important to emphasize that welfare gains from transparency are not
conditioned on a particular shock or policy path. Instead, the welfare gain is measured by
the unconditional long-run change in welfare that arises if the central bank systematically
announces the duration of any deviation from active monetary policy.
Uncertainty about the future output gap plays only a minor role for social welfare since
the estimated value of the slope of the Phillips curve is very small (see Table 1) and
standard calibrations for the elasticity of substitution " range from 6 to 10. Such a ‡at
Phillips curve is a standard …nding when DSGE models are estimated using U.S. data.
Therefore, welfare turns out to be tightly related to agents’uncertainty about future in‡ation,
which, as we shall show, depends on the time-varying level of pessimism about observing
a future switch to active monetary policy. If the central bank has lower reputation, agents
take into account that long-lasting deviations from the active regime are more frequent
and potentially more persistent. Consequently, agents expect more drastic in‡ationary or
de‡ationary consequences from future shocks and thereby they become more uncertain about
future in‡ation as the central bank engages in passive policies. As a result, social welfare
deteriorates faster than in the case of strong reputation. As shown in Section 4.2, the
level of pessimism responds to central bank behavior, namely the frequency and duration of
deviations from active policy and the communication strategy.
Section 5.1 outlines how uncertainty evolves as the central bank conducts passive policies
of di¤erent duration and under di¤erent communication strategies. In Section 5.2, we use
the model to assess the welfare implications of increasing central bank transparency.
22
5.1
Evolution of Uncertainty
We have shown that agents’uncertainty about future in‡ation crucially a¤ects social welfare
in the estimated model. In this section, we will show how uncertainty is tightly linked to
agents’pessimism about observing active monetary policy in the future. As shown in Section
4.2, transparency has two e¤ects on pessimism: (i) pessimism rises at the beginning of the
policy (henceforth, the short-run e¤ect of transparency on pessimism); (ii) pessimism is
anchored down at the end of the policy (henceforth, the anchoring e¤ect of transparency on
pessimism). As we shall show, these two e¤ects play a critical role for the welfare implications
of enhancing the transparency of the central bank.
To illustrate how uncertainty responds to pessimism, we consider the case in which the
Federal Reserve conducts a forty-quarter-long deviation from active monetary policy.14 Figure 3 illustrates the evolution of uncertainty about in‡ation at di¤erent horizons under no
transparency, left panel, and under transparency, right panel. At each point in time, the
evolution of agents’ uncertainty is measured by the h-period ahead standard deviation of
in‡ation at di¤erent horizons re-scaled by the standard deviation conditional on monetary
policy being currently active and given the communication strategy:
sdi (
t+h j t
;
v
t)
= 100
hp
vari (
t+h jst
(i) ;
v
t)
p
vari (
t+h jst
(i) = 0;
i
v
t)
where i 2 fN; T g and vt 2 fL; Hg.15 We analytically compute the conditional standard
deviations taking into account regime uncertainty using the methods described in Bianchi
(2013a).
It should be noted that the normalizing factor vari ( t+h jst (i) = 0; vt ) is not the same
across communication strategies i 2 fN; T g, implying that the left and right panels are not
directly comparable. This is because transparency determines an overall reduction in uncertainty that manifests itself also under the active regime, even if under the active regime
no announcement is made. A transparent central bank enjoys lower uncertainty even when
monetary policy is active because agents understand that should a passive policy of any
durations be implemented in the future, the central bank will announce its duration before14
The analysis is conducted for an economy at the steady-state and hence without conditioning on a
particular shock. The exercise is only conditioned on the policy path and intends to facilitate the exposition
of the welfare implications of transparency in the next section.
15
The graphs plot the results for h from 1 to 60: At horizon h = 0, uncertainty is zero as agents observe
current in‡ation. We rescale the uncertainty using the volatility conditional on being in the active regime
at horizon h = 0 so as to purge the e¤ect of heteroskedasticity on agents’ uncertainty from the graph. In
the interest of space, we report only the dynamics of uncertainty conditional on being in the low volatility
regime while the passive policy is carried out. The dynamics of uncertainty conditional on being in the high
volatility regime follows a very similar pattern and is available upon request.
23
Uncertainty about Inflation
No Transparency
Transparency
0.06
0.06
0.04
0.04
0.02
0.02
0
0
60
60
40
40
30
20
20
Horizon
40
40
30
20
20
10
Horizon
Time
10
Time
Figure 3: Evolution of uncertainty about in‡ation at di¤erent horizons (h) over forty periods
of passive policy (time) under low volatility (i.e., vt = L). The vertical axis reports the
standard deviations (rescaled for the standard deviation associated with currently being in
the active regime) in percentage points at the posterior mode.
hand. As it will soon become clear, such a communication strategy is e¤ective in reducing
uncertainty by removing the fear of a long lasting deviation for the frequent short lasting
deviations and creating an anchoring e¤ect for the sporadic long lasting deviations. However,
Figure 3 is illustrative of the dynamics of uncertainty about future in‡ation as a prolonged
passive policy is carried out. When we will compute the welfare implications of transparency,
we will not normalize our measure of uncertainty.
As shown in the left graph, when the central bank does not announce its policy course
beforehand, uncertainty about future in‡ation is relatively low at the beginning of the policy
because agents interpret the …rst deviations from active policy as short lasting. This result
is driven by the high reputation of the Federal Reserve, implying that agents attach 92%
probability of being in the short-lasting passive regime as the …rst period deviation from
active policy is observed. As more and more periods of passive policy are observed, agents
get progressively more persuaded that the observed deviation may have a long-lasting nature and uncertainty about future in‡ation gradually takes o¤. Uncertainty rises because
expecting a longer spell of passive policies raises concerns about the central bank’s ability of
24
controlling the in‡ationary consequences of future unanticipated shocks. Note that the increase in uncertainty occurs at every horizon because agents expect passive monetary policy
to prevail for many periods ahead and thereby anticipate that the in‡ationary/de‡ationary
consequences of future shocks will be more severe. It is worth emphasizing that the pattern of agents’uncertainty over time mimics the evolution of pessimism depicted in Figure
2. Summarizing, under no transparency, following a prolonged deviation from the active
regime uncertainty starts low and then gradually accelerates. Since higher uncertainty maps
into higher welfare loss, the progressive disanchoring of uncertainty about future in‡ation is
a reason of concern for a non-transparent central bank.
The right graph illustrates the dynamics of uncertainty about future in‡ation in the case
of transparency. Upon announcement agents become suddenly more uncertain about future
in‡ation because of the short-run e¤ect of transparency on pessimism. This is captured
by the pronounced hump-shaped dynamics of short- and medium-horizon uncertainty. The
announcement commits the central bank to follow a passive policy for the next forty periods,
causing agents to expect more dramatic in‡ationary/de‡ationary consequences from all those
disturbances that will materialize during the implementation of the announced policy path.
The rise in uncertainty upon the announcement is clearly welfare decreasing and captures the
main reason why the central bank may be reluctant to explicitly communicate their future
policy course.
Compared to the case of no transparency, short-horizon uncertainty is larger at the
beginning of the policy. However, at this early stage of the passive policy, uncertainty about
forty-quarter-ahead in‡ation appears to be smaller in the case of transparency. This result
is due to the anchoring e¤ect of transparency on pessimism. While agents know monetary
policy will be passive for forty quarters, they also know there will be a switch to the active
regime thereafter. Announcing the timing of the return to active monetary policy determines
a fall in uncertainty in correspondence of the horizons that coincide with announced date
(40 quarters in this numerical example). In the graph, such a decline in uncertainty shows
up as a valley in the surface representing the level of uncertainty. As we shall show, this
feature of transparency has the e¤ect of raising social welfare by systematically anchoring
agents’uncertainty at the end of prolonged deviations from the active regime.
While under no transparency uncertainty increases across all horizons as the policy is
implemented, under transparency uncertainty decreases across all horizons because agents
are aware that the end of the prolonged period of passive monetary policy is approaching.
This depends on the anchoring e¤ect of transparency on pessimism. Furthermore, announcing the duration of passive policies triggers a fall in uncertainty at every horizon because it
eliminates policy uncertainty for the duration of the announced policy. Even though this
25
e¤ect is not easy to observe in Figure 3, this is certainly an additional welfare-increasing
e¤ect. Finally, under the active regime uncertainty about future in‡ation is found to be
lower at every horizon under transparency. This result tends to raise the social welfare associated with transparency. Note that this …nding does not hold for all parameter values and
hence is due to the estimated parameters for the U.S. In the next section, we will show that
this last fact plays a critical role for the welfare comparison between transparency and no
transparency.
For the sake of brevity, we do not discuss the evolution of uncertainty about the output
gap. As mentioned before, since the estimated value of the slope of the Phillips curve
is very small when compared to the elasticity of substitution between goods ", uncertainty
about future output plays a negligible role in our welfare analysis.
It is also important to point out that the variance about future in‡ation conditional on
monetary policy to be current active is lower under transparency than that under no transparency at all horizons. In symbols, varN ( t+h j t = 0; vt ) > varT ( t+h j at = 0; vt ) :This
point is important and will be deserve more attention in the next sections.
5.2
Welfare Gains from Transparency
This section derives the welfare gains from enhancing central bank transparency in our model
estimated to the U.S. economy. In Section 5.2.1, we conduct a numerical exercise to illustrate
the dynamics of the welfare gains/losses from transparency during the implementation of a
passive policy. In Section 5.2.2, we compute and discuss the model predicted welfare gains
from transparency.
5.2.1
A Numerical Example
For the sake of illustrating the dynamics of welfare, let us consider a passive policy of duration
40 quarters.16 Figure 4 shows the dynamics of welfare Wi (st (i) ; vt ; ), de…ned in equation
(18), over time as this policy is implemented under the two communication schemes: no
transparency i = N and transparency i = T . We make this computation conditional on
being in the high volatility regime (left graph) and in the low volatility regime (right graph)
at time t. It should be observed that under the high volatility regime the welfare under
transparency (red solid line) is lower than the welfare under no transparency (blue dashed
line) at an early stage of the passive policy. Under the low volatility regime the welfare
associated with transparency is always higher than that associated with no transparency
16
This is a numerical example and is made for the sake of illustrating the evolution of welfare. We pick a
fairly prolonged deviation from the active regime so as to make these dynamics more visible in the graphs.
Such a long-lasting passive policy has a low probability of occuring based on our estimates.
26
High Volatility
-474
Low Volatility
No Transparency
Transparency
-441
-442
-476
-443
-444
Welfare
Welfare
-478
-480
-445
-446
-447
-482
-448
-449
-484
-450
-486
-451
10
20
Time
30
40
10
20
Time
30
40
Figure 4: Evolution of welfare Wi (st (i) ; vt ; ) de…ned in equation (18) as a passive policy
of duration 40 quarters is implemented under no transparency (i = N ), the blue dashed
line, and under transparency (i = T ), the red solid line. Parameter values are set at their
posterior mode.
at any time. However, larger gains from transparency, measured by the vertical distance
between the two lines, are reaped at the end of the passive policy. As discussed earlier,
when the announcement is made, agents become suddenly more pessimistic and hence being
transparent lowers welfare compared to no transparency at the beginning of the policy.
However, transparency lowers pessimism as the passive policy is implemented because
agents expect less and less periods of passive policy ahead. Therefore, welfare generally
increases as the passive policy is implemented. In contrast welfare is downward sloping
under no transparency. When the central bank does not communicate the duration of passive
policies, agents’pessimism gradually unfold, progressively lowering welfare.
5.2.2
Model Predicted Welfare Gains from Transparency
To assess the welfare gains from transparency we use formula (19), which combines the
welfare associated with the augmented policy regimes ( t for the case of no transparency and
a
t for the case of transparency) and their ergodic probabilities. To facilitate the comparison,
we rede…ne the regimes under transparency at in terms of observed periods of passive policy
t and recompute welfare under transparency associated with these new set of regimes as
27
Low Volatility
Welfare gains
from transparency
perc of ss consumption
Welfare gains
from transparency
perc of ss consumption
High Volatility
10
9
8
7
10
9
8
7
0
20
40
60
Observed Periods of Passive Policy
0
20
40
60
Observed Periods of Passive Policy
0.08
Ergodic Prob.
Ergodic Prob.
0.25
0.06
0.04
0.02
0
0.2
0.15
0.1
0.05
0
20
40
60
Observed Periods of Passive Policy
20
40
60
Observed Periods of Passive Policy
Figure 5: The upper graphs report the dynamics of the welfare gains from transparency as a
function of the observed periods of passive policy ( t ). The lower graph reports the ergodic
probability of observing the periods of passive policy on the x-axis ( t ). Parameter values
are set at their posterior mode.
shown in Appendix D.
The upper panels of Figure 5 shows the welfare gains from being transparent associated
with having observed passive policy for t periods. Once again, we consider the case in which
the economy is initially under the high and low volatility regimes. The lower panels report
the ergodic probabilities associated with these events. It should be observed that the welfare
gains from transparency grow fast for passive policies of short duration and then slow down as
their duration increases. This negative second-order derivative is explained by the fact that
announcing deviations of longer and longer durations progressively strengthens the short-run
e¤ect of transparency on pessimism which raises the risk of macroeconomic instability, as
shown in Figure 3. In fact, for extremely long-lasting passive policy welfare gains from transparency start declining. However, the lower graphs show that such long-lasting deviations
have virtually zero probability to occur in the US economy and hence do not signi…cantly
in‡uence the computation of the model predicted welfare gains from transparency based on
equation (19). It should be observed that welfare gains from transparency are positive for
all the durations of passive policies reported on the x-axis under both low volatility and
high volatility. This result implies that the anchoring e¤ects due to transparency dominate
28
its short-run e¤ects. In other words, transparency is welfare improving because it allows
the central bank to e¤ectively sweep away the fear of a return to the 1970s-type of passive
policies.
Furthermore, it is worth emphasizing that the upper plots capture the welfare gains from
systematically announcing the duration of passive policies. This explains why when the
central bank conducts an active policy ( t = 0), the welfare gains from transparency are not
zero. They are, in fact, positive capturing the welfare gains from expecting that the central
bank will systematically and truthfully announce the duration of any future passive policy.
This result is the mirror image of what discussed in the previous section: Transparency
implies that under the active regime uncertainty is lower at every horizon because agents
anticipate the future conduct of monetary policy.
The overall welfare gain is obtained combining the welfare gains conditional on having
observed a speci…c number of deviations, with their corresponding ergodic probabilities.
The plausible durations of passive policies in the U.S. are shown in the lower graphs, which
report the ergodic probabilities of observing a policy regime of a given duration conditional
on currently being in the high volatility regime and in the low volatility regime. Quite
interestingly, in the case of low volatility the ergodic probability appears to be less skewed
to the right than when the macroeconomic volatility is high. In both cases, the distribution
attributes large probability mass to active policies and fairly small probability mass of passive
policies of duration longer than twenty quarters.
To sum up, Figure 5 shows that welfare gains from transparency are positive for passive
policies of plausible durations for the U.S. under both volatility regimes, implying that the
model predicted welfare gains from transparency computed in equation (19) are positive based
on our estimates. More precisely, the model’s predicted welfare gains from transparency
amount to roughly 6.63% of steady-state consumption for the U.S. economy.
6
Robustness and Extensions
In this section we conduct a robustness exercise and consider an extension of the benchmark communication strategy. In Section 6.1 we investigate whether transparency is welfare
increasing for passive policies of every plausible duration. In Section 6.2, we relax the assumption that the central bank knows exactly the realized duration of the ongoing passive
policy and investigate whether transparency would still deliver welfare improvements. Specifically, we study the e¤ects of a central bank that announces the likely duration of passive
policies revealing which type of passive regime is in place as opposed to announcing the exact
duration of the deviation.
29
Welfare gains
from transparency
perc of ss consumption
7.5
High Volatility
Low Volatility
7
6.5
0
5
10
15
20
25
Realized Duration of Passive Policy
High Volatility
30
35
40
Low Volatility
0.25
Ergodic Prob.
Ergodic Prob.
0.08
0.06
0.04
0.02
0
10
20
30
40
Duration of the Passive Policies
0.2
0.15
0.1
0.05
0
10
20
30
40
Duration of the Passive Policies
Figure 6: Upper graph: Average per-period welfare gains from transparency associated with
policy of a given duration. Lower graphs: the ergodic distribution of the duration of passive
policies. A passive policy with duration of zero period on the x-axis corresponds to an active
policy. Parameter values are set at their posterior mode.
6.1
Short-Run Bene…ts from Transparency
In the previous section we showed that embracing transparency would raise the social welfare
compared to no transparency. The computation of expected welfare gains from transparency
is obtained using the ergodic distribution of the policy regimes and hence captures the
long-run gains. Furthermore, it should be noted that these long-run welfare gains have
been computed under the assumption that agents understand that the central bank will
systematically announce the duration of every passive policy. The fact that welfare gains
from transparency are positive in the active regime (when no announcement is actually
made) suggests that this systematic feature of the central bank’s communication policy
contributes to raise the welfare gains from transparency. A transparent central bank enjoys
higher welfare when monetary policy is active because agents understand that should a
passive policy of any durations be implemented in the future, the central bank will announce
its duration beforehand. However, it remains to be seen if the central bank is better o¤
following transparency for any possible duration of the deviation. In other words, are there
deviations for which the central bank would rather be not transparent?
We …nd that the gains from transparency occur for every plausible duration of the passive
30
policy. To see this, the upper plot of Figure 6 shows the dynamics of the average per-period
welfare gains from transparency associated with passive policy of various durations, while
the lower plots in Figure 6 report the corresponding ergodic probabilities. The important
result is that welfare gains are positive for all plausible durations of the passive policies. This
…nding suggests that the central bank is better o¤ by announcing passive policies of every
plausible duration. Quite interestingly, the upper graph suggests that the central bank is
better o¤ even if it has to announce passive of policy of fairly long duration. This is an
important result that implies that the overall reduction of uncertainty that occurs thanks to
transparency overcomes any short run loss associated with announcing a prolonged deviation
fro active in‡ation stabilization.
It should be noted the di¤erence between the welfare gains from transparency in Figure 5
and those of Figure 6. Figure 5 reports the welfare gains from having announced the duration
of the ongoing passive policy when t deviations out of the announced at
t have been
observed. Figure 6 shows the average per-period welfare gains, should the Federal Reserve
decide to announce a passive policy of a certain duration. The latter measure evaluates the
average of welfare gains from transparency across periods of policy implementation whereas
the former measure coupled with the ergodic probability distribution of the observed durations of policies ( t ) captures the expected bene…t from being transparent over the long
run.
6.2
Limited Information
We have modeled transparency as a communication strategy in which the central bank shares
all the information about the policy regime to households and …rms. Since we assume that
the central bank knows the exact duration of its passive policies, transparency implies that
such information is shared with the public. In this section, we relax the assumption that
the central bank knows the exact duration of passive policies. Rather, we assume that the
central bank knows only the expected duration of the deviations from the active regime;
that is, the bank perfectly knows only if the passive policy is short-lasting, Regime 2, or
long-lasting, Regime 3. Now, under transparency the central bank truthfully announces the
expected duration of passive regimes to households and …rms. It should be noted that since
the central bank truthfully tells the type of passive regime that is realized to agents, the
model boils down to a MS DSGE model with perfect information given that now the history
of policy regimes pt 2 f1; 2; 3g belongs to the agents’information set, Ft .
The upper graphs of Figure 7 show the welfare gains from transparency associated with
observing di¤erent durations of passive policies under the two volatility regimes. The lower
31
Low Volatility
Welfare gains
from transparency
perc of ss consumption
High Volatility
2.5
2
1.5
1
2.5
2
1.5
1
0
20
40
60
Observed Period of Passive Policy
0
20
40
60
Observed Period of Passive Policy
0.08
Ergodic Prob.
Ergodic Prob.
0.25
0.06
0.04
0.02
0
0.2
0.15
0.1
0.05
0
20
40
60
Observed Period of Passive Policy
20
40
60
Observed Period of Passive Policy
Figure 7: The upper graphs report the dynamics of the welfare gains from transparency as a
function of the observed periods of passive policy ( t ). The lower graph reports the ergodic
probability of observing the periods of passive policy on the x-axis ( t ). Parameter values
are set at their posterior mode.
graphs report the ergodic probability of observing passive policies of di¤erent durations where
the one with zero duration corresponds to active policy.17 The important result that emerges
from this graph is that welfare gains from transparency are always positive for policies of
any plausible duration and any volatility regime.
Quite interestingly, the pattern of the welfare gains from transparency is qualitatively
similar to the one depicted in Figure 5 reporting the welfare gains from transparency when
the central bank has full information. The main di¤erence is the duration of passive policy
above which the welfare gains from transparency starts declining. The turning point occurs
at shorter durations in the case of limited information of the central bank. The welfare gains
from transparency are reduced because while the central bank is still able to remove the fear
of a long lasting deviation from the active regime whenever a short lasting deviation occurs,
the anchoring e¤ect deriving from exactly announcing when the active regime will be again
in place is lost.
Computing the upper graphs requires to transform the primitive regimes, pt 2 f1; 2; 3g ; into the set of
regimes used for the case of no transparency that are de…ned in terms of the observed durations of passive
policies t . The details of this transformation are provided in Appendix D.
17
32
Welfare gains from transparency are smaller than in the case of full information by the
central bank. In fact, the model predicted welfare gains from transparency amounts to
0.67% of steady-state consumption. Thus, our analysis suggests that the welfare gains from
transparency are positive and are quanti…ed to range from 0.67% to 6.63% depending on the
degree of information of the central bank.
7
Concluding Remarks
In the model, the central bank alternates active policies aimed to stabilize in‡ation and
passive policies that de-emphasize in‡ation stabilization. Agents observe when monetary
policy becomes passive but they face uncertainty regarding its nature. Importantly, when
passive policies are observed, they cannot rule out the possibility that a persistent sequence
of deviations is in fact a return to the kind of monetary policy that characterized the 1970s.
Instead, they have to keep track of the number of deviations to learn if monetary policy
entered a short-lasting or a long-lasting period of passive monetary policy. The longer
the deviation from the active policy is, the more pessimistic about the evolution of future
monetary policy agents become. This implies that as the central bank keeps deviating,
uncertainty increases and welfare deteriorates.
We develop a general equilibrium model in which the central bank can deviate from active
in‡ation stabilization. Agents observe when monetary policy becomes passive but they face
uncertainty regarding its nature. Importantly, when passive policies are observed, they
cannot rule out the possibility that a persistent sequence of deviations is in fact a return to
the kind of monetary policy that characterized the 1970s. Instead, they have to keep track of
the number of deviations to learn if monetary policy entered a short-lasting or a long-lasting
period of passive monetary policy. The longer the deviation from the active policy is, the
more pessimistic about the evolution of future monetary policy agents become. This implies
that as the central bank keeps deviating, uncertainty increases and welfare deteriorates.
When the model is …tted to U.S. data, we …nd that the Federal Reserve bene…ts from
strong reputation. As a result, the Federal Reserve can deviate for a fairly prolonged period
of time from active monetary policy before losing control over agents’ uncertainty about
future in‡ation. Nevertheless, increasing the transparency of the Federal Reserve would
improve welfare by anchoring agents’pessimism when facing exceptionally prolonged periods
of passive monetary policy and removing the fear of the ’70s for the frequent short lasting
deviations.
In the model, agents learn only the persistence of passive policies, while the active regime
is fully revealing. This implies that agents’expectations are completely revised as soon as
33
the central bank returns to the active regime. In Bianchi and Melosi (2014b) we develop a
more general methodology that could be used to study a model in which agents have to learn
about the likely duration of both passive and active policies. This extension implies that
central bank reputation varies over time. While this feature is very interesting, it would make
the task of solving the model computationally challenging, preventing us from estimating the
model. We regard estimation as an important ingredient of the paper because the proposed
framework is new in the literature, with the result that the parameters controlling central
bank reputation cannot be borrowed from previous contributions. Furthermore, we believe
that this extension is unlikely to a¤ect the main conclusions of the paper. This is because
announcing the return to a long lasting period of active monetary policy would still have the
e¤ect of anchoring agents’pessimism and uncertainty.
A nice feature of the paper is to introduce a convenient way to model gradual changes
in beliefs about future policy decisions and macroeconomic outcomes. In the parsimonious
setting studied in this paper, we have shown that waves of agents’pessimism or optimism
about future policy actions play a central role in shaping the response of macroeconomic
variables and households’welfare to macroeconomic shocks in forward-looking rational expectations models. Expanding the analysis to state-of-the-art monetary DSGE models such
as Christiano, Eichenbaum, and Evans, 2005 and Smets and Wouters, 2007 would be of great
interest, but quite challenging from a computational point of view.
34
References
Barro, R. J., and D. B. Gordon (1983): “Rules, discretion and reputation in a model
of monetary policy,”Journal of Monetary Economics, 12(1), 101–121.
Benati, L., and P. Surico (2009): “VAR Analysis and the Great Moderation,”American
Economic Review, 99(4), 1636–52.
Bernanke, B. S. (2003): “Constrained Discretion" and Monetary Policy,” Discussion paper, Remarks by Governor Ben S. Bernanke.
Bernanke, B. S., T. Laubach, F. S. Mishkin, and A. S. Posen (1999): In‡ation
Targeting: Lessons from the International Experience. Princeton University Press.
Bernanke, B. S., and F. S. Mishkin (1997): “In‡ation Targeting: A New Framework
for Monetary Policy?,”Journal of Economic Perspectives, 11(2), 97–116.
Bianchi, F. (2013a): “Methods for Markov-Switching Models,”Manuscript.
(2013b): “Regime Switches, Agents’Beliefs, and Post-World War II U.S. Macroeconomic Dynamics,”Review of Economic Studies, 80(2), 463–490.
Bianchi, F., and C. Ilut (2013): “Monetary/Fiscal Policy Mix and Agents’ Beliefs,”
CEPR Discussion Papers 9645, C.E.P.R. Discussion Papers.
Bianchi, F., and L. Melosi (2013): “Escaping the Great Recession,” Duke University
mimeo.
(2014a): “Dormant Shocks and Fiscal Virtue,” in NBER Macroeconomics Annual
2013, ed. by J. Parker, and M. Woodford, vol. 28 of NBER Chapters, pp. 1–46. National
Bureau of Economic Research, Inc.
(2014b): “Modeling the Evolution of Expectations and Uncertainty in General
Equilibriumy,”Duke University mimeo.
Boivin, J., and M. P. Giannoni (2006): “Has Monetary Policy Become More E¤ective?,”
The Review of Economics and Statistics, 88(3), 445–462.
Cho, I.-K., N. Williams, and T. J. Sargent (2002): “Escaping Nash In‡ation,”Review
of Economic Studies, 69(1), 1–40.
35
Christiano, L. J., M. Eichenbaum, and C. L. Evans (2005): “Nominal Rigidities and
the Dynamic E¤ects of a Shock to Monetary Policy,”Journal of Political Economy, 113(1),
1–45.
Clarida, R., J. Gali, and M. Gertler (2000): “Monetary Policy Rules And Macroeconomic Stability: Evidence And Some Theory,” The Quarterly Journal of Economics,
115(1), 147–180.
Cogley, T., C. Matthes, and A. M. Sbordone (2011): “Optimal disin‡ation under
learning,”Sta¤ Reports 524, Federal Reserve Bank of New York.
Cogley, T., and T. J. Sargent (2005): “Drift and Volatilities: Monetary Policies and
Outcomes in the Post WWII U.S,”Review of Economic Dynamics, 8(2), 262–302.
Coibion, O., Y. Gorodnichenko, and J. Wieland (2012): “The Optimal In‡ation
Rate in New Keynesian Models: Should Central Banks Raise Their In‡ation Targets in
Light of the Zero Lower Bound?,”Review of Economic Studies, 79(4), 1371–1406.
Davig, T., and T. Doh (2008): “Monetary policy regime shifts and in‡ation persistence,”
Research Working Paper RWP 08-16, Federal Reserve Bank of Kansas City.
Davig, T., and E. M. Leeper (2007): “Generalizing the Taylor Principle,” American
Economic Review, 97(3), 607–635.
Del Negro, M., and S. Eusepi (2010): “Fitting observed in‡ation expectations,” Sta¤
Reports 476, Federal Reserve Bank of New York.
Eusepi, S., and B. Preston (2010): “Central Bank Communication and Expectations
Stabilization,”American Economic Journal: Macroeconomics, 2(3), 235–71.
Evans, G., and S. Honkapohja (2001): Learning and Expectations In Macroeconomics.
Princeton University Press, Princeton.
Evans, G. W., and S. Honkapohja (2003): “Expectations and the Stability Problem for
Optimal Monetary Policies,”Review of Economic Studies, 70(4), 807–824.
Farmer, R. E., D. F. Waggoner, and T. Zha (2009): “Understanding Markov-switching
rational expectations models,”Journal of Economic Theory, 144(5), 1849–1867.
Fernandez-Villaverde, J., and J. F. Rubio-Ramirez (2008): “How Structural Are
Structural Parameters?,” in NBER Macroeconomics Annual 2007, Volume 22, NBER
Chapters, pp. 83–137. National Bureau of Economic Research, Inc.
36
Foerster, A. T., J. Rubio-Ramirez, D. Waggoner, and T. Zha (2011): “Perturbation Methods for Markov-Switching Models,”Mimeo Duke University.
Gali, J. (2008): Monetary Policy, In‡ation, and the Business Cycle: An Introduction to
the New Keynesian Framework. Princeton University Press.
Gali, J., and M. Gertler (2007): “Macroeconomic Modeling for Monetary Policy Evaluation,”Journal of Economic Perspectives, 21(4), 25–46.
Justiniano, A., and G. E. Primiceri (2008): “The Time-Varying Volatility of Macroeconomic Fluctuations,”American Economic Review, 98(3), 604–41.
Kydland, F. E., and E. C. Prescott (1977): “Rules Rather Than Discretion: The
Inconsistency of Optimal Plans,”Journal of Political Economy, 85(3), 473–91.
Lubik, T. A., and F. Schorfheide (2004): “Testing for Indeterminacy: An Application
to U.S. Monetary Policy,”American Economic Review, 94(1), 190–217.
Mankiw, N. G., R. Reis, and J. Wolfers (2004): “Disagreement about In‡ation Expectations,” in NBER Macroeconomics Annual 2003, Volume 18, NBER Chapters, pp.
209–270. National Bureau of Economic Research, Inc.
Marcet, A., and T. J. Sargent (1989a): “Convergence of Least-Squares Learning in
Environments with Hidden State Variables and Private Information,”Journal of Political
Economy, 97(6), 1306–22.
Marcet, A., and T. J. Sargent (1989b): “Convergence of least squares learning mechanisms in self-referential linear stochastic models,” Journal of Economic Theory, 48(2),
337–368.
Melosi, L. (2014a): “Estimating Models with Dispersed Information,”American Economic
Journal: Macroeconomics, 6(1), 1–31.
(2014b): “Signaling E¤ects of Monetary Policy,” Mimeo Federal Reserve Bank of
Chicago.
Mishkin, F. S. (2002): “In‡ation Targeting,” in An encyclopedia of macroeconomics, ed.
by B. Snowdon, and H. Vane, Elgar Original Reference Series. Edward Elgar.
Nimark, K. (2008): “Dynamic Pricing and Imperfect Common Knowledge,” Journal of
Monetary Economics, 55(8), 365–382.
37
Plosser, C. I. (2012): “Economic Outlook and Monetary Policy,” Policy Speech at the
CFA Society of Philadelphia/The Bond Club of Philadelphia, September 25.
Primiceri, G. E. (2005): “Time Varying Structural Vector Autoregressions and Monetary
Policy,”Review of Economic Studies, 72(3), 821–852.
Rios-Rull, J.-V., F. Schorfheide, C. Fuentes-Albero, M. Kryshko, and
R. Santaeulalia-Llopis (2012): “Methods versus substance: Measuring the e¤ects
of technology shocks,”Journal of Monetary Economics, 59(8), 826–846.
Rotemberg, J. J., and M. Woodford (1999): “The Cyclical Behavior of Prices and
Costs,” in Handbook of Macroeconomics, ed. by J. B. Taylor, and M. Woodford, vol. 1 of
Handbook of Macroeconomics, chap. 16, pp. 1051–1135. Elsevier.
Rudebusch, G. D., and J. C. Williams (2008): “Revealing the Secrets of the Temple:
The Value of Publishing Central Bank Interest Rate Projections,” in Asset Prices and
Monetary Policy, NBER Chapters, pp. 247–289. National Bureau of Economic Research,
Inc.
Schorfheide, F. (2005): “Learning and Monetary Policy Shifts,” Review of Economic
Dynamics, 8(2), 392–419.
Sims, C. A., and T. Zha (2006): “Were There Regime Switches in U.S. Monetary Policy?,”
American Economic Review, 96(1), 54–81.
Smets, F., and R. Wouters (2007): “Shocks and Frictions in US Business Cycles: A
Bayeasian DSGE Approach,”American Economic Review, 97(3), 586–606.
Steinsson, J. (2003): “Optimal monetary policy in an economy with in‡ation persistence,”
Journal of Monetary Economics, 50(7), 1425–1456.
Woodford, M. (2003): Interest and Prices. Princeton University Press, Princeton.
38
A
Solving the Model with No Transparency
It is very important to emphasize that the evolution of agents’beliefs about the future conduct of monetary policies plays a critical role in the Markov-switching model with learning.
In fact, three policy regimes pt are not a su¢ cient statistic for the dynamics of the endogenous variables in the model with learning. Instead, agents expect di¤erent dynamics for next
period’s endogenous variables depending on their beliefs about a return to the active regime.
To account for agents learning we expand the number of regimes and rede…ne them as a
combination between central bank’behaviors and agents’beliefs. Bianchi and Melosi (2014b)
show that the Markov-switching model with learning described previously can be recast in
terms of an expanded set of ( t + 1) > 3 new regimes, where t > 0 is de…ned by the
condition (15). These new set of regimes constitute a su¢ cient statistics for the endogenous
variables in the model as they capture the evolution of agents’ beliefs about observing a
switch to the active regime in the next period. The + 1 regimes are given by
[(
p
t
= 1;
t
= 0) ; (
p
t
6= 1;
t
= 1) ; (
p
t
6= 1;
t
= 2) ; :::; (
p
t
6= 1;
and the transition matrix Pep is de…ned using equation (14); that is,
2
6
6
6
6
6
e
Pp = 6
6
6
6
6 1
4
1
B
p11
1
1
p12 p22 +p13 p33
p12 +p13
p12 p222 +p13 p233
p12 p22 +p13 p33
..
.
p22 (p12 =p13 )(p22 =p33 )
(p12 =p13 )(p22 =p33 )
p22 (p12 =p13 )(p22 =p33 )
(p12 =p13 )(p22 =p33 )
2
+p33
+1
1
+p33
1
+1
2
p12 + p13
0
0
t
=
)] ;
p12 p22 +p13 p33
p12 +p13
::: 0
::: 0
0
0
0
..
.
0
..
.
::: 0
. . . ..
.
0
..
.
0
0
0
0
0
0
0
0
p22 (p12 =p13 )(p22 =p33 )
(p12 =p13 )(p22 =p33 )
p22 (p12 =p13 )(p22 =p33 )
(p12 =p13 )(p22 =p33 )
3
2
+p33
+1
1
+p33
1
+1
2
Convergence
Table 2 reports results based on the Brooks-Gelman-Rubin potential reduction scale factor
using within and between variances based on the four multiple chains used in the paper.
The …ve chains consist of 540; 000 draws each, the …rst 40; 000 draws are dropped, and of
the remaining draws 1 every 1; 000 draws is saved. The numbers are very close to 1 and
therefore well below the 1:2 benchmark value used as an upper bound for convergence.
39
7
7
7
7
7
7:
7
7
7
7
5
Potential Scale Reduction Factor
Parameter PSRF
Parameter
PSRF Parameter PSRF Parameter PSRF
mp
A
H
1:00
H11
1:01
1:01
1:00
R
mp
mp
A
H
1:00
H22 =H33
1:00
1:00
1:00
y
g
mp
A
H
1:00
H
1:00
1:00
1:00
g
33
m
R
mp
mp
P
L
1:00 H12 = (1 H11 ) 1:00
1:02
1:00
m
R
P
L
1:00
1:00
r
1:00
1:05
H11
g
y
P
L
1:00
H22
1:00
1:00
1:01
m
R
1:00
Table 2: The table reports the Gelman-Rubin Potential Scale Reduction Factor (PSRF) for
four chains of 540,000 draws each (1 every 1000 is stored). Values below 1.2 are regarded as
indicative of convergence.
C
Welfare Function
It is worth emphasizing that the regime changes considered in this paper do not a¤ect the
steady state, but only the way the economy ‡uctuates around the steady state. Therefore, we are going to follow the literature and we derive a second order approximation of
the representative household’s utility function around the steady state. The representative
household’s utility function are described by the following function:
Nt1+
1+
C1
U (Ct ; Nt ) = t
1
De…ne the x
bt = ln XXt as the log-deviation of the generic variable Xt from its own steady-state
value. The utility function above can be equivalently written as
U (Ct ; Nt ) =
C1
)^
ct
e(1
1
N 1+ e(1+
1+
)^
nt
The second order Taylor expansion around the steady-state equilibrium yields
U (Yt ; Nt ) ' C 1
' Y1
c^t
1
N 1+ n
^t + C 1
2
1
y^t + (1
) y^t2
2
(1
N 1+
1 1+
N
(1 + ) n
^ 2t
2
1
n
^ t + (1 + ) n
^ 2t
2
) c^2t
where in the second line we use the market clearing condition c^t = y^t .
Using the production function we can write
Nt (i)1
=
40
Yt (i)
Zt
(20)
Using the fact that the demand for the variety produced by …rm i is Yt (i) =
obtain
"t
1
Yt 1
Pt (i) 1
Nt (i) =
Zt
Pt
Pt (i)
Pt
"t
Yt we
Integrating both sides of the above equation across …rms i, we obtain
Nt =
Z
1
Yt
Zt
1
"t
Pt (i)
Pt
1
di
R
where we use the fact that the labor market clearing requires that Nt = Nt (i) di. The
above equation can be log-linearized around a symmetric steady state to get the following:
(1
where we de…ne dt
(1
) ln
R
)n
^ t = (^
yt
(21)
zt ) + dt
"t
1
Pt (i)
Pt
di.
Lemma 1 In a neighborhood of a symmetric hsteady state,
i 1 and up to a second-order ap1+
("
1)
proximation, dt ' 2" vari ln Pt (i) ; where
is the strategic complementarity
(1
)
parameter.
Proof. Let p^t (i) ln Pt (i) ln Pt . Note that p^t (i) is a stationary variable because of the
assumption of perfect indexation of price to steady-state in‡ation. Notice that
"t
Pt (i)
Pt
"t
1
= exp
p^t (i)
1
Taking the second-order approximation of this object around the symmetric steady state:18
Pt (i)
Pt
"t
"
1
'1
1
p^t (i) +
"2
2 (1
^2t
2p
)
(i)
Integrating both sides of this equation across …rms leads to
Z
Pt (i)
Pt
"t
"
1
di ' 1
1
Z
p^t (i) di +
"2
2 (1
Note that from the de…nition of the price level we obtain 1 =
2
)
R
Z
p^2t (i) di
ep^t (i)(1
"t )
(22)
di. Taking the
18
Note that since pbt (i) is equal to zero in steady state, taking the Taylor expansion with respect to the
elasticity of substitution ^"t ln ("t =") would be immaterial. To keep the derivation notationally tractable,
we do not take the Taylor expansion with respect to ^"t .
41
second-order approximation of this expression yields
0 ' (1
and after rearranging
Z
")
Z
p^t (i) di +
p^t (i) di =
"
")2
(1
2
1
2
Z
Z
p^t (i)2 di
p^t (i)2 di
(23)
Substituting equation (23) in equation (22) allows us to write
Z
Pt (i)
Pt
"t
"
1
di = 1
1
"
1
2
Z
"2
2
p^t (i) di +
2 (1
2
)
Z
p^t (i)2 di
After same straightforward manipulation, we obtain
Z
"t
Pt (i)
Pt
1
1 "
di = 1 +
21
1
Z
p^t (i)2 di:
(24)
R
R
Recalling the de…nition of p^t (i) we can write Ei p^t (i)2
p^t (i)2 di = [ln Pt (i) ln Pt ]2 di.
Note also that up to a …rst-order approximation, the price index equation implies that ln Pt =
R
Ei ln Pt (i).19 Hence, we can write Ei p^t (i)2 ' (ln Pt (i) Ei ln Pt (i))2 di = vari (ln Pt (i) =Pt ) :
Then we can use this result to write
Z
"t
Pt (i)
Pt
1
di = 1 +
1 "
21
1
vari (ln Pt (i))
(25)
Finally, combining the de…nition of dt with the equation above leads to the following:
dt
(1
'
) ln
Z
Pt (i)
Pt
"t
1
di
"
vari (ln Pt (i))
2
where in the last line we use the fact that ln (1 + n) ' n for n su¢ ciently small. QED.
Using equation (21) we can rewrite the second-order approximation of the utility function
R
Write the price index as 1 = ep^t (i)(1 ") di: Taking the loglinearization yields 0 = Ei p^t (i). Recall that
p^t (i) ln Pt (i) ln Pt , implying ln Pt = Ei ln Pt (i).
19
42
(20) as follows:
U (Ct ; Nt ) ' y 1
y^t +
1
(1
2
) y^t2
(26)
N 1+
"
11+
y^t +
vari fln Pt (i)g +
1
2
21
+t:i:p: + h:o:t:
zt )2
(^
yt
where t:i:p: collects all terms independent of policies (e.g., zt ) and h:o:t: stands for higherorder term. We can write
U (Yt ; Nt )
'
Y1
1
(1
) y^t2
2
N N
1
11+
"
vari fln Pt (i)g +
y^t +
Y Y 1
2
21
+t:i:p: + h:o:t:
(27)
y^t +
(^
yt
zt )2
E¢ ciency of the steady state implies that20
N
C
= MP N
= (1
)
Y
N
Substituting the last equation into equation (27) and re-arranging yield:
U (Yt ; Nt )
'
Y1
1
+
+
2
1
+t:i:p: + h:o:t:
y^t2
"
1+
vari fln Pt (i)g +
y^t zt
2
1
(28)
Note that the log-deviation of the e¢ cient level of output from its steady-state level is given
by y^te = (1 1+)+ + zt . Hence, we can substitute this expression for zt and write
U (Yt ; Nt )
'
Y1
"
vari fln Pt (i)g
2
+t:i:p: + h:o:t:
20
1
2
+
+
1
y^t2
2^
yt y^te
(29)
As standard, we assume that the ine¢ ciency generated by the market power are removed by the suitable
choice of subsidies so that the steady-state equilibrium can be regarded e¢ cient.
43
We can complete the square and write
U (Ct ; Nt )
'
Y1
1
2
"
vari fln Pt (i)g
2
+t:i:p: + h:o:t:
+
+
(^
yt
1
y^te )2
(30)
Accordingly, a second-order approximation to the consumer’s welfare function can be expressed as a fraction of steady-state consumption (and up to additive terms independent of
policy) as follows21
1 X
E0
2
t=0
1
W=
t
"
vari fln Pt (i)g +
+
+
1
(^
yt
y^te )2 ;
(31)
where E0 is the expectation operator conditional on the information set that agents has at
time 0, F0 .
Lemma 2 Under Calvo pricing with non-zero in‡ation steady state and perfect indexation
P
P
t
t 2
vari fln Pt (i)g = (1 )(1 ) 1
to steady state in‡ation 1
^t .
t=0
t=0
Proof. In each period the distribution of prices is given by times the price distribution in
the previous period times the gross steady-state in‡ation rate plus an atom of height (1
)
at the optimal reset price. Let us denote
Pt
(32)
Ei ln Pt (i) :
Observe that Calvo pricing implies
Pt
Pt
1
ln
= Ei ln Pt (i)
=
Ei (ln Pt
=
Ei ln Pt
Pt
1
1
ln
1
(i) + ln
(i)
Pt
)
1
Pt
+ (1
ln
+ (1
) ln Pt
Pt
1
) ln Pt
1
ln
Pt
1
ln
(33)
where ln Pt is the optimal resetting price for those …rms that are allowed to re-optimize their
price and
is the steady-state gross in‡ation rate. Notice that Ei ln Pt 1 (i)
Pt 1 and
hence Ei ln Pt 1 (i) Pt 1 = 0. Therefore,
Pt
Pt
1
ln
= (1
) ln Pt
21
Pt
1
ln
(34)
Since we analyze welfare for an economy that is currently at the steady state, we set the value of the
exogenous process Gt = G = 1:
44
Analogously, denote the cross-sectional variance of prices vari fln Pt (i)g as
equivalently expressed as
= vari ln Pt (i) Pt
h
= Ei ln Pt (i) Pt
t
ln
1
2
ln
1
i
Ei ln Pt (i)
Pt
This can be
t:
2
ln
1
(35)
Observe that the property of the cross-sectional distribution of prices under Calvo pricing
allows us to write
Ei
h
ln Pt (i)
Pt
1
2
ln
Notice that since Ei ln Pt
Ei
h
1
i
= Ei
(i)
Pt
ln Pt
1
1
(i)
h
ln Pt
1
(i)
2
Pt
1
i
+(1
) ln Pt
Pt
1
ln
2
(36)
= 0, then
2
Pt
1
i
= vari [ln Pt
1
(i)]
(37)
t 1
Also taking the square of both sided of equation (34) implies that
(1
) ln Pt
Pt
2
ln
1
=
1
Pt
1
Pt
2
ln
1
(38)
Using the results in equations (37) and (38) in combination with equation (36) yields
Ei
h
ln Pt (i)
Pt
2
ln
1
i
=
t 1
+
1
1
Pt
Pt
1
2
ln
(39)
Plugging equation (39) into equation (35) and also observing that the de…nition (32) implies
2
that the second term of equation (35) is equal to Pt Pt 1 ln
allow us to leads to
t
=
t 1
+
=
t 1
+
1
1
1
Pt
Pt
1
ln
Pt
Pt
1
ln
2
2
Pt
Pt
1
ln
2
:
Now note that up to a …rst-order approximation, the price index equation implies that
ln Pt = Pt and hence the last term of the recursive equation for price dispersion is the
log-deviation of in‡ation from its steady state value , which we denote with ^ t . Finally,
t
=
t 1
+
45
1
^ 2t :
Integrating forward for any given (small) initial degree of price dispersion
t
t+1
=
0+
t
X
t s
Since the initial price dispersion
t 0, one obtains
1
X
is independent of any policy implemented in periods
0
t
t
we obtain:
^ 2s :
1
s=0
0
=
t=0
(1
) (1
)
1
X
t 2
^t
t=0
QED.
Hence the welfare function can be rewritten as follows
W=
1 X
E0
2
t=0
where we use the de…nition
gap.
D
1
(1
t
)(1
"
)
^ 2t +
+
and b
y~t
+
y^te )2
(^
yt
1
(40)
y^te ) is the welfare-relevant output
(^
yt
Transformation of Regimes under Transparency
In Figure 5 we express welfare under transparency in terms of number of observed deviations from the active regime. This corresponds to the de…nition of policy regime under no
transparency. This is done in order to facilitate the analysis of how the welfare gains from
transparency varies with passive policies of duration :
Let us compute the probability that i consecutive periods of passive policy has been
announced conditional on having observed period of passive policy:
i 1
p12 pi22 1 p21 + p13 p33
p31
for any
a+
1
j 1
p12 p22 p21 + p13 pj33 1 p31
j=
(i) = P
i
+
a
Note that the numerator captures the probability that a deviation of duration i is realized and
hence announced (recall all announcements are truthful). The denominator is the probability
of (announcing) a passive policy lasting periods or longer (up to the truncation ).
The welfare associated with a policy that has been deviating for consecutive periods
under transparency is given by
f
WT ( ;
v
; )=
a
X
(j + ) WT (
j=0
46
a
= j;
v
; )
(41)
Note the di¤erence from WT ( a ; v ; ) in equation (18), which is the welfare function de…ned
in terms of policy regimes for the case of transparency (i.e., a the number of announced
deviations yet to be carried out). WT ( ; ) is the welfare under transparency associated with
a announcing a periods of passive policy. f
WT ( ; v ; ) is the welfare under transparency
associated with having observed consecutive periods of passive policy. We can show that
this recasting of policy regimes leads to a negligible approximation error as
pN ( ;
v 0
t)
f
WT ( )
pT ( a ;
v 0
t)
WT ( a ;
v
; )
When we compute the welfare gains from transparence under limited information by the
central bank in Section 6.1, we compute WT ( ; v ; ) = Wp (st = 2) + (1
) Wp (st = 3)
where Wp denotes the welfare under perfect information and st 2 f1; 2; 3g the primitive set
of policy regimes and the weight is de…ned as follows:
( )=
p12 p22 1 p21
;
p12 p22 1 p21 + p13 p33 1 p31
which capture the probability of being in the short-lasting passive regime conditional on
having observed consecutive deviations from the active policy.
47
Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.
Corporate Average Fuel Economy Standards and the Market for New Vehicles
Thomas Klier and Joshua Linn
WP-11-01
The Role of Securitization in Mortgage Renegotiation
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff
WP-11-02
Market-Based Loss Mitigation Practices for Troubled Mortgages
Following the Financial Crisis
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff
WP-11-03
Federal Reserve Policies and Financial Market Conditions During the Crisis
Scott A. Brave and Hesna Genay
WP-11-04
The Financial Labor Supply Accelerator
Jeffrey R. Campbell and Zvi Hercowitz
WP-11-05
Survival and long-run dynamics with heterogeneous beliefs under recursive preferences
Jaroslav Borovička
WP-11-06
A Leverage-based Model of Speculative Bubbles (Revised)
Gadi Barlevy
WP-11-07
Estimation of Panel Data Regression Models with Two-Sided Censoring or Truncation
Sule Alan, Bo E. Honoré, Luojia Hu, and Søren Leth–Petersen
WP-11-08
Fertility Transitions Along the Extensive and Intensive Margins
Daniel Aaronson, Fabian Lange, and Bhashkar Mazumder
WP-11-09
Black-White Differences in Intergenerational Economic Mobility in the US
Bhashkar Mazumder
WP-11-10
Can Standard Preferences Explain the Prices of Out-of-the-Money S&P 500 Put Options?
Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein
WP-11-11
Business Networks, Production Chains, and Productivity:
A Theory of Input-Output Architecture
Ezra Oberfield
WP-11-12
Equilibrium Bank Runs Revisited
Ed Nosal
WP-11-13
Are Covered Bonds a Substitute for Mortgage-Backed Securities?
Santiago Carbó-Valverde, Richard J. Rosen, and Francisco Rodríguez-Fernández
WP-11-14
The Cost of Banking Panics in an Age before “Too Big to Fail”
Benjamin Chabot
WP-11-15
1
Working Paper Series (continued)
Import Protection, Business Cycles, and Exchange Rates:
Evidence from the Great Recession
Chad P. Bown and Meredith A. Crowley
WP-11-16
Examining Macroeconomic Models through the Lens of Asset Pricing
Jaroslav Borovička and Lars Peter Hansen
WP-12-01
The Chicago Fed DSGE Model
Scott A. Brave, Jeffrey R. Campbell, Jonas D.M. Fisher, and Alejandro Justiniano
WP-12-02
Macroeconomic Effects of Federal Reserve Forward Guidance
Jeffrey R. Campbell, Charles L. Evans, Jonas D.M. Fisher, and Alejandro Justiniano
WP-12-03
Modeling Credit Contagion via the Updating of Fragile Beliefs
Luca Benzoni, Pierre Collin-Dufresne, Robert S. Goldstein, and Jean Helwege
WP-12-04
Signaling Effects of Monetary Policy
Leonardo Melosi
WP-12-05
Empirical Research on Sovereign Debt and Default
Michael Tomz and Mark L. J. Wright
WP-12-06
Credit Risk and Disaster Risk
François Gourio
WP-12-07
From the Horse’s Mouth: How do Investor Expectations of Risk and Return
Vary with Economic Conditions?
Gene Amromin and Steven A. Sharpe
WP-12-08
Using Vehicle Taxes To Reduce Carbon Dioxide Emissions Rates of
New Passenger Vehicles: Evidence from France, Germany, and Sweden
Thomas Klier and Joshua Linn
WP-12-09
Spending Responses to State Sales Tax Holidays
Sumit Agarwal and Leslie McGranahan
WP-12-10
Micro Data and Macro Technology
Ezra Oberfield and Devesh Raval
WP-12-11
The Effect of Disability Insurance Receipt on Labor Supply: A Dynamic Analysis
Eric French and Jae Song
WP-12-12
Medicaid Insurance in Old Age
Mariacristina De Nardi, Eric French, and John Bailey Jones
WP-12-13
Fetal Origins and Parental Responses
Douglas Almond and Bhashkar Mazumder
WP-12-14
2
Working Paper Series (continued)
Repos, Fire Sales, and Bankruptcy Policy
Gaetano Antinolfi, Francesca Carapella, Charles Kahn, Antoine Martin,
David Mills, and Ed Nosal
WP-12-15
Speculative Runs on Interest Rate Pegs
The Frictionless Case
Marco Bassetto and Christopher Phelan
WP-12-16
Institutions, the Cost of Capital, and Long-Run Economic Growth:
Evidence from the 19th Century Capital Market
Ron Alquist and Ben Chabot
WP-12-17
Emerging Economies, Trade Policy, and Macroeconomic Shocks
Chad P. Bown and Meredith A. Crowley
WP-12-18
The Urban Density Premium across Establishments
R. Jason Faberman and Matthew Freedman
WP-13-01
Why Do Borrowers Make Mortgage Refinancing Mistakes?
Sumit Agarwal, Richard J. Rosen, and Vincent Yao
WP-13-02
Bank Panics, Government Guarantees, and the Long-Run Size of the Financial Sector:
Evidence from Free-Banking America
Benjamin Chabot and Charles C. Moul
WP-13-03
Fiscal Consequences of Paying Interest on Reserves
Marco Bassetto and Todd Messer
WP-13-04
Properties of the Vacancy Statistic in the Discrete Circle Covering Problem
Gadi Barlevy and H. N. Nagaraja
WP-13-05
Credit Crunches and Credit Allocation in a Model of Entrepreneurship
Marco Bassetto, Marco Cagetti, and Mariacristina De Nardi
WP-13-06
Financial Incentives and Educational Investment:
The Impact of Performance-Based Scholarships on Student Time Use
Lisa Barrow and Cecilia Elena Rouse
WP-13-07
The Global Welfare Impact of China: Trade Integration and Technological Change
Julian di Giovanni, Andrei A. Levchenko, and Jing Zhang
WP-13-08
Structural Change in an Open Economy
Timothy Uy, Kei-Mu Yi, and Jing Zhang
WP-13-09
The Global Labor Market Impact of Emerging Giants: a Quantitative Assessment
Andrei A. Levchenko and Jing Zhang
WP-13-10
3
Working Paper Series (continued)
Size-Dependent Regulations, Firm Size Distribution, and Reallocation
François Gourio and Nicolas Roys
WP-13-11
Modeling the Evolution of Expectations and Uncertainty in General Equilibrium
Francesco Bianchi and Leonardo Melosi
WP-13-12
Rushing into American Dream? House Prices, Timing of Homeownership,
and Adjustment of Consumer Credit
Sumit Agarwal, Luojia Hu, and Xing Huang
WP-13-13
The Earned Income Tax Credit and Food Consumption Patterns
Leslie McGranahan and Diane W. Schanzenbach
WP-13-14
Agglomeration in the European automobile supplier industry
Thomas Klier and Dan McMillen
WP-13-15
Human Capital and Long-Run Labor Income Risk
Luca Benzoni and Olena Chyruk
WP-13-16
The Effects of the Saving and Banking Glut on the U.S. Economy
Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti
WP-13-17
A Portfolio-Balance Approach to the Nominal Term Structure
Thomas B. King
WP-13-18
Gross Migration, Housing and Urban Population Dynamics
Morris A. Davis, Jonas D.M. Fisher, and Marcelo Veracierto
WP-13-19
Very Simple Markov-Perfect Industry Dynamics
Jaap H. Abbring, Jeffrey R. Campbell, Jan Tilly, and Nan Yang
WP-13-20
Bubbles and Leverage: A Simple and Unified Approach
Robert Barsky and Theodore Bogusz
WP-13-21
The scarcity value of Treasury collateral:
Repo market effects of security-specific supply and demand factors
Stefania D'Amico, Roger Fan, and Yuriy Kitsul
Gambling for Dollars: Strategic Hedge Fund Manager Investment
Dan Bernhardt and Ed Nosal
Cash-in-the-Market Pricing in a Model with Money and
Over-the-Counter Financial Markets
Fabrizio Mattesini and Ed Nosal
An Interview with Neil Wallace
David Altig and Ed Nosal
WP-13-22
WP-13-23
WP-13-24
WP-13-25
4
Working Paper Series (continued)
Firm Dynamics and the Minimum Wage: A Putty-Clay Approach
Daniel Aaronson, Eric French, and Isaac Sorkin
Policy Intervention in Debt Renegotiation:
Evidence from the Home Affordable Modification Program
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
Tomasz Piskorski, and Amit Seru
WP-13-26
WP-13-27
The Effects of the Massachusetts Health Reform on Financial Distress
Bhashkar Mazumder and Sarah Miller
WP-14-01
Can Intangible Capital Explain Cyclical Movements in the Labor Wedge?
François Gourio and Leena Rudanko
WP-14-02
Early Public Banks
William Roberds and François R. Velde
WP-14-03
Mandatory Disclosure and Financial Contagion
Fernando Alvarez and Gadi Barlevy
WP-14-04
The Stock of External Sovereign Debt: Can We Take the Data at ‘Face Value’?
Daniel A. Dias, Christine Richmond, and Mark L. J. Wright
WP-14-05
Interpreting the Pari Passu Clause in Sovereign Bond Contracts:
It’s All Hebrew (and Aramaic) to Me
Mark L. J. Wright
WP-14-06
AIG in Hindsight
Robert McDonald and Anna Paulson
WP-14-07
On the Structural Interpretation of the Smets-Wouters “Risk Premium” Shock
Jonas D.M. Fisher
WP-14-08
Human Capital Risk, Contract Enforcement, and the Macroeconomy
Tom Krebs, Moritz Kuhn, and Mark L. J. Wright
WP-14-09
Adverse Selection, Risk Sharing and Business Cycles
Marcelo Veracierto
WP-14-10
Core and ‘Crust’: Consumer Prices and the Term Structure of Interest Rates
Andrea Ajello, Luca Benzoni, and Olena Chyruk
WP-14-11
The Evolution of Comparative Advantage: Measurement and Implications
Andrei A. Levchenko and Jing Zhang
WP-14-12
5
Working Paper Series (continued)
Saving Europe?: The Unpleasant Arithmetic of Fiscal Austerity in Integrated Economies
Enrique G. Mendoza, Linda L. Tesar, and Jing Zhang
WP-14-13
Liquidity Traps and Monetary Policy: Managing a Credit Crunch
Francisco Buera and Juan Pablo Nicolini
WP-14-14
Quantitative Easing in Joseph’s Egypt with Keynesian Producers
Jeffrey R. Campbell
WP-14-15
Constrained Discretion and Central Bank Transparency
Francesco Bianchi and Leonardo Melosi
WP-14-16
6
@FedFRASER