Inflation Targeting with Imperfect Information∗
Aloisio Araujo†
Tiago Berriel
Rafael Santos
EPGE/FGV and IMPA
PUC-Rio
Central Bank of Brazil
July 21, 2014
Abstract
In a global games setup with imperfect information we show that low targets - the ones close to the
optimal inflation under perfect commitment - may be unattainable, leading to a trade-off between low
and credible targets. Moreover, as noisy public information helps to coordinate expectations around
the announced target, our paper supports unconventional policy prescriptions: higher targets and low
transparency should be considered by weaker economies. Results are based on a general central bank
loss function encompassing models traditionally used to discuss central bank decisions.
JEL Classification: E50; E59; E60;
Keywords: Inflation Targeting; Global Games; Imperfect Information
∗
We are grateful to Manuel Amador, Ricardo Cavalcanti, Guilhermo Calvo, Harold Cole, Luciana Fiorini, Stephen Morris,
Ricardo Reis, Nelson Souza. The views expressed in this paper do not necessarily reflect those of the Central Bank of Brazil.
†
Corresponding author: [email protected], Phone: 55-21-3799-5833, Fax: 55-21-2529-5129, Address: Estrada Dona
Castorina, 110 22460-320 Rio de Janeiro, Brazil
1
Introduction
The announcement of inflation targets by the monetary authority has been a trend in central banking
over the last 20 years. In 1990, New Zealand was the only country under a formal inflation targeting
regime. In 2006, the list of inflation targeters was long and diverse and included countries such as Australia, Brazil, Canada, Czech Republic, Iceland, Hungary, Mexico, Norway, Korea, Poland, Sweden,
Switzerland, UK and Turkey.1 Proponents of inflation targeting argue that it anchors expectations,
leads to less costly disinflations, and is less subject to confidence crises than alternative regimes, such
as exchange rate pegs.2
In this paper, we study the limits of target coordination. More specifically, we analyze the conditions for a low target to be preferred over a higher one. We define high targets as those close to
the discretionary inflation equilibrium, opposed to a low target which is close to the inflation level
under perfect commitment equilibrium. Results are based on a general central bank loss function
and are presented in two information setups: (i) a full-information model, a standard benchmark in
the literature and (ii) a model where private agents have imperfect information about central bank
commitment technology. An information disadvantage by the agents with respect to Central Banks
preferences or policy parameters has been viewed as a step towards a more realistic model.3
In a full-information model, we show that a target may be sufficiently high in order to lead to a
unique equilibrium, while a low target may lead to multiple equilibria, depending on the central bank
loss function. Thus, we highlight the restrictions to low targets to be delivered, limiting the arguments
of the proponents of always very low inflation target.
In a global games setup, where each agent receives one private noisy signal and one public noisy
signal about the central bank commitment technology, the above results remain: lower targets require
stronger conditions over the loss function in order to assure a unique equilibrium. Regarding the
imperfect information setup, we show that for a given central bank loss function, a noisier public
signal improves coordination and allows for a lower target to be implemented. Accordingly, too much
transparency represented by precise public signal increases the expected losses from ambitious low
1
2
3
For more details, Carvalho and Goncalves (2008)
See Kumhof et al. (2007).
See Orphanides and Williams (2007), Milani (2007), and Cogley et al. (2014).
2
targets.
4
The intuition for limited ability to coordinate expectations around aggressive low targets goes
as follow: low targets for inflation are far from discretionary inflation level, fueling central bank’s
temptation to renegade it later, even when reputational costs are considered. Agents take this into
consideration when forming their expectations about the subsequent behavior of the central bank.
Agents doubts about the target reinforce the central bank’s temptation to renegade it, which in turn
reinforces doubts, and so on. This reinforcement channel reduces the coordination of the expectations around the target. It also becomes more important when agents tend to agree to each other,
which is the case with a precise public signal. Consequently, low targets may result in worse equilibrium outcomes and less credible central bank policy announcements, especially under precise public
knowledge.
Positively, while making explicit the limitations of inflation targeting regimes, our model addresses
the inflation target heterogeneity across countries and years. Considering perfectly credible policy
makers, common knowledge and homogeneous inflation costs, targets should always and everywhere
be very low. In our model, imperfect commitment technology explains the presence of high targets
for inflation. Moreover, heterogeneity in institution transparency, which leads to heterogeneity in the
precision of the available public information, would generate endogenous reasons for heterogeneity
not only in target level. Figure 1 suggests that highly committed central banks tend to announce
lower targets and to be more transparent.5 Normatively, our results are a word of caution against low
targets , where policy makers neglect the trade-off between credible and low targets.
We illustrate our general results with two examples of loss functions in this paper. First, we present
the Barro-Gordon model with quadratic loss function and a neoclassical Phillips curve. As expected,
coordination is improved with noisier public information. This result is explained as follows: more
disperse beliefs helps the central bank to influence expectations as it limits speculation from private
agents. In addition, more aggressive targets are easily achieved with (i) central banks more concerned
with inflation, i.e. larger weights of inflation in the loss function; and (ii) with lower sensitivity of
output to changes in inflation, implied by the Phillips curve. The intuition is that it is hard to influence
4
5
See Morris and Shin (2002) and Svensson (2006) for a discussion of the role of transparency in global games setup.
Data is available in Eichengreen and Dincer (2009) and IMF (2005).
3
14
New Zeland
Canada
10
Dincer, Eichengreen, NBER (2007)
Transparency Index as of 2005
Sweden
UK
12
Switzerland
Australia
Israel
Norway
Iceland
8
Korea
Peru
Chile
6
Mexico
Brazil
Colombia
4
2
0
0
2
4
6
Inflation Target Upper Limit as of 2005
World Economic Outlook 4, IMF
Figure 1: Transparency vs Announced Target
4
8
agents’ expectations when the gains for the central bank of deviating are large.
Our second model considers the trade-off between inflating away debt, which leads to costs of high
inflation, and increasing taxes to service debt, which is costly due to distortionary taxation. With a
completely different loss-function, we obtain the same results of the Barro-Gordon model with respect
to effects of institutional transparency. In addition, we show that the larger the marginal cost of
inflation, the easier it is to coordinate expectations and deliver a more aggressive target.
Regarding the related literature, in a seminal contribution, Morris and Shin (1999) established
that the well-known result of multiplicity of equilibrium due to speculative attacks reverts to the
unique equilibrium result if the common knowledge is broken. Even if one departs from the common
knowledge benchmark by introducing an infinitesimal amount of private uncertainty. Morris and
Shin (2001) also introduced a new framework with exogenous public information and showed that the
previous multiplicity result with lack of common knowledge may survive when public information is
precise enough when compared to private information. The intuition for this result is that precise
public information minimizes the role of private information. Angeletos and Werning (2006) add
endogenous public information to this class of models. Our framework was built based on Morris and
Shin (2001) and it is similar to the model declared in the first part of Angeletos and Werning (2006),
where the game of status-quo defense has two stages and exogenous information structure. To this
previous literature (i) we add a third initial stage where the status-quo - defined as the inflation target
- must be selected; (ii) we add a non-linear payoff function of the policy maker; and (iii) we study the
information structure exogenous and dependent on the variance of signals, following Morris and Shin
(2001), to the inflation targeting problem.
The rest of the paper is organized as follows. Section 2 develops the model based on a general loss
function and results are explored. In section 3 two examples of loss functions are presented. Section
4 concludes.
2
Model
We consider a three stage game played by the central bank and a continuum of private agents The
sequence of actions are taken as follows: in the first stage, the central bank announces the target for
5
inflation πa . Given the announced target and their information set, in the second stage agents form
their expectations, πe . In the last stage, the central bank chooses the actual inflation π. By setting an
inflation target, central bank aims to coordinate second stage expectations on a credible inflation, that
dominates the discretionary level of inflation. In the third stage, the actual inflation is selected by the
central bank. We also assume that, given inflation expectations, there is a trade-off between aggregate
output and inflation. The neoclassical Phillips curve is an example that fits this last assumption.
More specifically, the problem of central banker consists in minimizing a generic loss function,
L(π, π e ). Note that, in this loss function we have already substituted the output variable, using the
relation between output, inflation and inflation expectations discussed above. We assume that L(π, π e )
is a function of class C3 mapping R2 into R. We also assume that the loss-function satisfies:
∂2L
∂2L
∂2L
∂2L
>
0;
<
0;
>
−
∂π 2
∂π∂π e
∂π 2
∂π∂π e
(1)
The first inequality states that the loss function is convex on inflation. The second shows that
higher inflation expectations decrease the marginal losses of inflation. This assumption makes clear
the relevance of inflation expectations in the willingness of the policy makers to deliver low inflation
rates. It also makes the central bank best response on the third stage be positevely correlated with
inflation expectations. The last inequality assumes that the effect of increasing inflation on marginal
loss of inflation is larger than that of increasing inflation expectations. These assumptions are trivially
satisfied in the linear-quadratic framework of Barro-Gordon, for example.
The objective of the central bank in the third stage of the game is to set the inflation rate π of the
economy by minimizing its loss function:
L (π, πe ) + Iκ,π
Iκ,π =


 0 if π = πa

 κ if π 6= πa
The indicator function Iκ,π is equal either to zero whenever the actual inflation π matches the pre
committed inflation πa , or to the scalar κ otherwise. Therefore, there are costs from the central bank
6
perspective when the actual inflation is not the optimal one, and also when the central bank does not
delivery its inflation announcement. This way, we assume there is a commitment technology, κ ∈ R,
which is added to the loss function in case the central bank does not fulfill the announced target.
Moreover, the central bank can set the inflation target in the first stage of the game to try to drive
expectations and its own third stage decision, minimizing, not the value of L, but the expected value
of L, since we can make πe subject to uncertainty when the target is announced (subsection (2.1)).
The central bank decides in the third stage and also in the first stage, according to:
Third stage: π = arg min L (π, πe ) + Ik,π
h
i
First stage: πa = arg min E min {L (π, πe (πa )) + Ik,π }
π
Assuming the central bank finds itself in the last stage and takes agents expectation as given, one
can define π ∗ = arg min L(π|πe ), and π ∗ (πe ) is the central bank best response function on this stage.
As
∂2L
∂π 2
> 0 for all πe , there is only one π ∗ (πe ) well defined for each πe . Also, π ∗ is increasing in πe .
We collect these results in proposition (1) below. All proofs are in the appendix.
Proposition 1. Let the function L(π, πe ) be of class C 3 , mapping from R2 (actual and expected
inflation) into R (loss) and satisfying (1). For each πe , there is only one π ∗ (πe ) ∈ [π, π] , where π and
π are constants. Moreover, π ∗ (.) is increasing in πe and there is only one πe satisfying π ∗ (πe ) = πe .
This ensures the existence of an equilibrium of rational expectations, i.e. π ∗ (πe ) = πe for only
one πe ∈ R. We call this fixed point as πD , D for the discretionary equilibrium inflation, since the
central bank ignores its effect on agents expectations. Now, let (πG , πG ) = arg min L(π, πe ) s.t. π = πe .
(πG , πG ) is the solution of the problem under perfect commitment where the central bank incorporates
a rational expectations response of the agents before hand. It is straightforward to see that LG ≤ LD ,
where Ln ≡ L (πn , πn ), n ∈ {G, D}.
We ignore the knife edge case where LG = LD . In all other cases, we have a typical problem of the
temporal inconsistency: only under full commitment the best equilibrium LG can be implemented.
In the first stage, whenever there is a commitment technology available, the central bank tries to
7
coordinate private agents expectations around a target to reduce the loss from the discretionary
outcome, LD , into the direction of the full commitment outcome, LG . Formally, we assume that
πa ∈ Πa and Πa ≡ π ∈ R|L (π, π) < LD .
Next, we study two cases that differs from each other on how agent form their expectations. In
the first, all agents have the same information set, since they know with certainty the value of κ. In
the second case, κ is not directly observed.
2.1
Full information
In this section, we assume agents directly observe κ, i.e. the commitment technology of the central
bank. We can also define a minimum value for κ for which, given an announced target and an expected
inflation, the central bank does not have incentives to deviate from the target.
k(πa , πe ) ≥ L (πa , πe ) − L (π ∗ (πe ) , πe )
(2)
Proposition 2. For any κ > 0, it is always possible to announce some target πa satisfying both
π ∗ (πa ) = πa and π ∗ (πD ) = πa . For a sufficient high κ, it is possible to announce πa = πG satisfying
both π ∗ (πG ) = πG and π ∗ (πD ) = πG .
Proposition (2) makes clear that given a commitment technology, there is a role under perfect
information for inflation targeting. The central bank can move the equilibrium away from the discretionary outcome. It also shows that for a good enough commitment technology, the central bank can
always deliver the good outcome of full commitment.
By assuming that as expected inflation approaches the discretonary inflation level, the loss function
2
become more concave ( ∂∂πL2 becomes greater), we can also see that for targets closer to πG , a higher
technology commitment is required to implement the target as the unique equilibrium. Conversely,
as πa gets close to πD , a lower κ is required to implement the target and the multiplicity equilibrium
region shrinks. Proposition (3) makes these points clear.
3
L(π,πe )
Proposition 3. Let the sign (πD − πG ) ∂ ∂π
be non negative. Then k (πa , πD ) − k (πa , πa ) > 0
2 ∂π
e
and such difference increases as πa becomes closer to πG and more distant from πD .
8
Figure 2: Multiplicity and Uniqueness of Equilibrium
As a direct consequence of proposition (3), whenever κ ∈ (0, k (πG , πD )) , there is a trade-off for
the target selection: aggressive targets that are closer to πG but subject to multiple equilibria versus
conservative target that is the unique equilibrium but closer to πD . If κ > k (πG , πD ) , then πa = πG
is the best target to be selected and assures π = πG .
Figure 2 shows these points graphically. As πa moves from πG in the direction of πD , the multiplicity region between the two curves shrinks. Moreover, targets will be implemented as unique
equilibrium only by highly committed central banks, i.e. central banks with κ above the upper curve
in the graph. For κ below the lower curve, the central bank never delivers the announced target.
2.2
Imperfect Information
Now suppose that πe is formed by agents that do not directly observe κ and therefore have imperfect
information about the commitment technology. This imperfect information may lead to disagreement
9
between private agents. Still, a very high κ ensures πe = πa . Also, πe = πD is the unique equilibrium
when κ ≤ 0,. Nonetheless, for some κ in between these two extreme cases, agents may disagree
about what should be the next π. In this sense, by aggregating expectations we may have πe ∈
[min(πD , πG ), max(πD , πG )] .
In this setup, we assume a continuum of private agents normalized to one. They receive their
public and private signals about the reputational costs of the central bank, compute their expected
inflation and then decide to trust or not the announced target. Agent j decision is represented by
αj ∈ {0, 1}. The expected j0s payoff is equal to 1 − αj (gs − c) .When αj equals to zero means that
agent j does not trust the announced target. This way, he acts based on his beliefs, pushing inflation
expectations up. If one considers the agents of this economy as firms, α = 0 implies a decision to
readjust its price. We are assuming that the gain for not trusting the announced target, gs , depends
on the central bank’s response. If the target is sustained, then gs is equal to 0, otherwise it is equal to
1. By assumption, the parameter c (0, 1) . With this payoff structure, betting against the target is ex
post a good deal only when the target is abandoned. It is straightforward that agents have incentives
to predict central bank’s behavior correctly.
Agents have heterogeneous information sets. We assume that reputational cost κ is known by the
central bank but not by private agents. Instead of observing κ directly, each agent j observes a public
signal sp and a private signal sj , both informative for predicting κ.
sj = κ + σεj ; σ > 0 and εj ∼ N (0, 1)
(3)
sp = κ + σp εp ; σp > 0 and εp ∼ N (0, 1)
(4)
where εj is assumed to be independent of k and εj0 for all j0 6= j and εp is also assumed to be
independent of k and εj .
The agents’ asymmetric information gives rise to coordination motive from strategic complementarity in their actions. More specifically, since it is more likely for the central bank to abandon the
target during confidence crises, agents’ incentives not to trust the central bank are increasing not only
in their private pessimism, but also in the public perception of a low reputational cost. Formally, each
10
agent j must decide αj in the second stage of the game as follow:
αj = arg max E
αj ∈{0,1}
gs =
1 − αj (gs − c)
(5)


 0 if π = πa

 1 if π 6= πa
Note that the expected inflation in the Phillips curve does not necessarily match the expected
inflation of an agent. The diversity of the primitives behind the decision making process that leads
to a Phillips Curve may include fixed cost associated to price adjustments or information frictions
at an individual level. Instead of laying out one of these primitives and the resulting aggregation
process, we opt for a shortcut to define expected inflation. We set a payoff structure for each private
agent that produces incentives for them to expect the very same rates of inflation observed in one
of the possible equilibria obtained in the full information model with rational expectations. At the
same time, as assumed information friction generates disagreement, the aggregated expected inflation
under imperfect information can rely at any point inside the continuum interval limited by the inflation
rates obtained in the full information model with rational expectations. Our qualitative result, i.e.
disperse beliefs due to imperfect information shrinks the multiplicity region, does not depend on how
an individual agent compute the expected inflation but on the fact that they may disagree, pushing
aggregated expected inflation far from extreme values.
We turn next to the definition of equilibrium.
2.2.1
Equilibrium
Results are based on monotone equilibria defined as perfect Bayesian. For each public signal sp , an
agent j decides to not be aligned with the target if and only if his private signal sj is less than some
threshold s∗ (sp , πa ) . Hence, an equilibrium is defined as a collection of πa∗ , αj (s∗ (sp , πa )), π ∗ (sp , πa )
such that πa∗ solves the first stage problem and, given πa∗ , αj (s∗ (sp , πa∗ )) solves the second stage problem
for all agent j ∈ [0, 1] and for all sp ∈ R, and π ∗ (sp , πa ) is the optimal central bank reaction function
that solves the third stage problem for all sp ∈ R, given πa . Next, we provide some definitions and
11
key results to clarify the equilibrium computation.
The mass of agents that ends up not aligned with the target is naturally given by:
prob(sj < s∗ (sp , πa )) = Φ(
s∗ (sp , πa ) − k
)
σ
where Φ (.) denotes the cumulative distribution function for the standard normal. We define the
target-coordination efficiency α as how likely an arbitrary agent is to make decisions aligned with the
target achievement:
∗
Z1
α (s ) ≡
αj dj = 1 − Φ(
s∗ (sp , πa ) − k
)
σ
0
From the central bank perspective, α∗ summarizes the relevant information contained in the private
actions. Note that each private agent solves his problem in the second stage, contributing to increase
α if and only if αj equals to one. Additionally, the expected payoff from any j who decides to not
be aligned with the target must be equal to zero whenever sj = s∗ (sp , πa ) . It implies the following
indifference condition:
Φ−1 (c) =
k ∗ (sp , πa ) −
σp2 σ 2
σp2 +σ 2
r
s∗ (sp ,πa )
σ2
+
sp
σp2
σp2 σ 2
σp2 +σ 2
The central bank solves its problem in the third stage and it sustains the target if and only if κ is
greater than the critical reputational cost k ∗ , which is given by the following indifference condition:
k ∗ = k (πa , πe∗ )
πe∗ ≡ Agregated Expectation under s∗ (sp , πa )
Then, to compute k ∗ and s∗ , we need first to compute πe , which must be an aggregation of each
private agent outcome. We define the expected inflation based on private agent decisions:
πe∗ ≡ α (s∗ ) πa + (1 − α (s∗ )) πD
where the first term represents the inflation target weighted by the mass of agents deciding aligned
with the target, and the second term represents the discretionary inflation weighted by the mass of
12
agents deciding to be not aligned with the target.
Given any πa , k ∗ (sp ) and s∗ (sp ) can be computed by solving the indifference conditions just
presented. Alternatively, in the third stage, π ∗ (sp , πa ) can be easily computed for each given pair
(s(sp ), sp ) . After computing π ∗ ((s(sp ), sp )) , the function s∗ (sp ) can also be computed by each private
agent in the second stage problem. Given these both stage decisions, central bank selects the optimal
target for inflation to be announced in the first stage, knowing that the target affects both the second
and the third stage decisions.
Next, we show a sufficient condition for unique equilibrium in the context of imperfect information.
2
Proposition 4. Given an announced target, if − ∂∂πL(.,.)
<
e ∂π
√
2π
(πa −πD )2
σp2
σ
, then the equilibrium is
unique for every public signal.
As it is clear, precision of public signal relative to the private signal makes it harder to achieve
the conditions for uniqueness of equilibrium. The intuition is again clear: a lot of information on
the public signal makes coordenation easy, leading to more than one possible outcomes. Therefore,
transparency on reputational costs leads to multipicity. If πa is very close to πD uniqueness is achieved,
since the agents will not doubt the central bank when gains in deviating are insignificant. Finally, if
the marginal loss of inflation is higly affected by expectations, coordination becomes more important
and multiplicity arises.
Proposition 5. Let the variance of the public signal be high enough to ensure the uniqueness for all
i
h 2
public signals and for all target-candidates: σp2 > √σ2π − ∂∂πL(.,.)
(πG − πD )2 . In such a case, more
e ∂π
conservative targets improves the target coordination as it increases α∗ .
As a result of proposition (5), we have that within the unique equilibrium region, announced
targets closer to πD tends to increase agents belief that the target will be implemented. This way, our
framework implies a trade-off between credible and more aggressive targets.
13
3
Examples
3.1
Barro-Gordon model
We focus in choosing the inflation targeting in a static optimal monetary policy problem of Barro
and Gordon (1983) and Cukierman and Liviatan (1991). We consider the same three games stage
discussed in section (2).
We also assume that, given inflation expectations, there is a trade-off between aggregate output
and inflation. We chose the simplest workhorse framework that delivers this trade-off and keeps
tractability in a model that lacks common knowledge. This way, we assume a neoclassical Phillips
Curve. More specifically, there is an exogenous variable yn that will equal the output y whenever
the inflation is equal to the private agents expectations. If inflation turns out to be larger than the
expected inflation, output is higher than yn .
6
π = πe + ϕ (y − yn )
(6)
where ϕ is the trade-off between inflation surprises and output deviations, i.e., the slope of the Phillips
curve.
We restrict the central bank to minimize the loss function below:
L = π 2 + λ (y − y ∗ )2 + Iκ,π
(7)
where y ∗ is the central bank target for output and λ is the relative cost of output deviations with
respect to inflation deviations. This loss function is clearly a special case of the one discussed in
section (2) that satisfies conditions (1).
One could microfounded a quadratic loss-function as the one showed above. In a model with (i)a
continuun of products, that are subject to Dixit-Stiglitz demand functions, (ii) production function
that only has labor as an input, one can get that social welfare for the representative agent is proportional to the cross sectional variance of products prices and squared output. Some price frictions, such
as rigidity a la Calvo, would allow us to write the cross section variance of product prices as squared
6
Woodford (1999), chapter 2 for a in depth discussion and derivation of this Phillips Curve.
14
inflation.
In the third stage the central bank minimizes eqn. (7), subject to eqn.(6), taking agents expectation
as given. In the case where k = 0, the central bank always chooses the discretionary equilibrium rate
of inflation as below:
πD =
λ ∗
(y − yn )
ϕ
(8)
In the case of k > 0, the inflation choice in the third stage is no longer straightforward, since it
depends on the expected inflation and on the announced target. In this case, the the central bank can
set the inflation target in the first stage of the game to drive expectations and to drive its own third
stage decision, minimizing the expected value for (7). In particular, if k were extremely large, then a
zero target would be announced, leading always to a zero inflation economy. Formally, the announced
target for inflation solves the first stage problem if the condition below is met:
(
"
2
πa = arg min E min π + λ
π
π − πe (πa )
+ (yn − y ∗ )
ϕ
)#
2
+ Iκ,π
(9)
The description of agents information structure and equilibrium definition is completely analogous
to section (2).
3 L(π,π )
e
λ
∗ (π ) =
We verify conditions (1) and also that sign (πD − πG ) ∂ ∂π
=
0
≥
0.
Since
π
πe −
e
2 ∂π
2
λ+ϕ
e
h
i
ϕλ
(yn − y ∗ ) , we have π ∗0 (πe ) > 0 and only one πe such that πe = π ∗ (πe ) . From proposition 4, a
λ+ϕ2
sufficient condition for the target πa to assure a unique equilibrium is:
πa
∈
πa , πD
√
πa ≡ πD − ϕσp
2π
2σλ
!1/2
It is also straightfoward to show that the loss function with lower weight on the activity, steeper
Phillips Curve, noisier public information, and more precise private information contribute to efficient
15
coordination under more aggressive targets.
7
Finally, we provide a numerical example, restricting
0.
σp
σ
to ensure uniqueness even when πa = πG =
8
Figure (3) shows the equivalente of Figure (2) for this numerical example. In between the dotted
lines we have the region of multiple equilibria in the case of full-information. The full line shows what
is the minimum deviation cost k(πa , πe ) required for each target to be delivered in stage 3. Note that
the full line considers a single realization of public signal and, therefore, the expected inflation and
the resulting minimum deviation cost plotted depends only on the target announced. If that central
bank could know the specific public signal before the target announcement, the target announcement
would be around 1.25%, and this target would be actually delivered.
In pratice, the central bank does not observe the private signal before annoucing and implementing
the target. For this reason, we plot for the same values of parameters the expected loss of the central
bank for different announced targets. Figure (4) shows that, for the same k = 8, the announced target
before observing the public signal that minimizes expected loss is around 2.5%. Moreover, expected
loss is decreasing in the target when the target is too aggressive. Finally, figure (4) shows that it may
be optimal to have a higher expected inflation due to higher target in order to avoid states where
actual inflation is above announced target.
7
dπa
dλ
=
dπa
dϕ
=
dπa
dσp
=
dπa
dσ
=
1/2
For this we need σp > ϕλ2 (y ∗ − yn )
σp = 10, k = 8, and c = 0.35
8
2σλ
√
2π
−3/2
2σλ
√
>0
2π
√ !1/2
λ ∗
2π
− 2 (y − yn ) − σp
<0
ϕ
2σλ
√ !1/2
2π
−ϕ
<0
2σλ
−3/2
λϕσp 2σλ
√
√
>0
2π
2π
(y ∗ − yn ) ϕσp σ
+ √
ϕ
2π
. We set the following parameters: (y ∗ − yn ) = 20, λ = 0.2, ϕ = 1, σ = 5,
16
k ( πa , π*e(sp = k) )
20
15
10
πe = πD
πe = πa
5
0
0
1
πa 2
3
Figure 3: Target given Public Signal
17
4
4
98
Expected Inflation (LHS)
Expected Loss Function (RHS)
3.5
96
3
94
2.5
0
1
πa2
3
92
4
Figure 4: Expected Loss and Expected Inflation as a function of Announced Target
18
3.2
Fiscal Model
In this section, we present an alternative with a different loss function. Here, the policy maker (i)
announces the inflation target in the first stage and (ii) may implement a high inflation rate to avoid
costly distortionary taxes, τ , in stage three. Private agents have again the same role: given the
announced target and their private information, they form their expectations, πe , in stage two.
As in the previous sections, there are reputation costs associated with not delivering the promised
target, represented by Iκ,πe .
L (π, πe ) ≡
π
η
η
+ λ exp (d − (π − πe )) + Iκ,πe
(10)
Here, λ is the relative weight of distortive taxes in the policy maker’s loss function and τ is the tax
rate. The policy maker starts with an exogenous amount of nominal debt, D, in the first stage, that
has to be paid in the third stage, in order to satisfy the terminal condition in this finite time model.9
The nominal rate servicing the debt depends on an exogenous and fixed real interest rate, r and on the
inflation expectations πe . The government must pay its obligation by imposing distortionary taxes.
However, inflation reduces its real obligations and, thus, the required taxes adjustments. This way,
the policy maker budget constraint in real terms is given by:
(r + Πe )
D̄ = (1 + τ )
Π
(11)
where Πe is the expected inflation, Π is the actual inflation and D̄ is D divided by today’s price
level.10 This way, defining d = log(D̄) as the log of real government debt, a log-linear approximation
to equation above leads to.
d = τ + (π − πe )
(12)
By assuming η > 1, λ > 0, d > 0, and (y ∗ > yn ), the conditions (1) are satisfied. Note that π ∗
1 d
solves: π η−1 = λ exp (d − π + πe ), πD = λ η−1 exp η−1
> 0, and πG = 0 < πD . One can also show
3
L(π,πe )
that sign (πD − πG ) ∂ ∂π
= λ exp (d − (π − πe )) ≥ 0.
2 ∂π
e
9
10
It is straightforward to add government expenditure in the third stage.
Naturally, log(Πe ) and log(Π) are approximately πe and π.
19
From proposition 4, a sufficient condition for the target πa be unique is:
πa ∈ πa , πD
σp λ exp (d + πD ) −1/2
√
πa = πD − √
σ
2π
It is also straightfoward to show that the loss function with lower weight on inflating debt temptation, a larger power on the marginal cost of inflation, noisier public information (disagreement), and
more precise private information contribute to efficient coordination under more aggressive targets.11
Despite a different loss functions, the lessons from figures (3) and (4) remain qualitatively the
same, as one can observe from figures (5) and (6).
4
Conclusion
In this paper, we show the limits of the inflation target coordination and how such limits depend
on the target level and on the information structure of the economy. Accordingly, the target region
where the equilibrium target-announced followed by target-delivered is the unique possible equilibrium
11
Notice that
−3/2 dπa
λ exp (d + πD ) λ exp (d + πD )
1
1
σp λ exp (d + πD )
√
√
√
=
πD + √
+
πD
dλ
λ (η − 1)
(η − 1)
2 σ
2π
λ 2π
λ 2π
−1/2 dπa
1
σp
λ exp (d + πD )
1
√
=
πD + √
1+
πD > 0
dλ
λ (η − 1)
(η − 1)
2λ σ
2π
1 d
πD = λ η−1 exp
η−1
1 − λ η−1
dπD
d
=
2 exp η − 1 [1 + ln λ] < 0
dη
(η − 1)
1 "
#
−3/2
− λ η−1
dπa
d
σp λ exp (d + πD )
√
√
=
exp (d + πD ) < 0
2 exp η − 1 [1 + ln λ] 1 + 2 σ
dη
2π
(η − 1)
−1/2
dπa
1 λ exp (d + πD )
√
√
=−
<0
dσp
σ
2π
−1/2
dπa
σp λ exp (d + πD )
√
= (1/2) 3
>0
dσ
2π
σ2
20
k ( πa , π*e(sp = k) )
3
2
πe = πD
1
0
0
πe = πa
2
πa
4
6
Figure 5: Expected Loss and Expected Inflation as a function of Announced Target
21
6
12
Expected Inflation (LHS)
Expected Loss Function (RHS)
5
11
4
10
3
0
1
2
πa 3
4
5
9
Figure 6: Expected Loss and Expected Inflation as a function of Announced Target
22
shrinks as the precision of the public information increases. Moreover, by considering an information
structure that assures uniqueness, a non-aggressive target is easier to be delivered than aggressive
targets, i.e. those close to the optimal inflation level under perfect commitment technology. Hence,
we showed that there is a trade-off between low and credible inflation targets.
The general message of this paper can provide guidance to current real-world conduction of monetary policy. One example is the interaction between the fiscal solvency in advanced economies and
the limits to fight inflation. Such limits are faced by policy makers when the interest rate must be
increased to curb accelerating prices, but the fiscal budget constraint brings incentives to avoid higher
rates. According to our model, the fiscal limits of the coordination around policy announcements may
become tighter under too much transparency. This facts lead to unconventional policy prescriptions
to weaker economies: higher inflation targets and less transparency, precisely what is suggested in the
Figure 1 of the paper.
23
References
Angeletos, George-Marios and Ivan Werning, “Crises and Prices: Information Aggregation,
Multiplicity and Volatility,” American Economic Review, 2006, 96, 1720–1736.
Barro, Robert and David Gordon, “A Positive Theory of Monetary Policy in a Natural Rate
Model,” Journal of Political Economy, 1983, 91, 589–610.
Carvalho, Alexandre and Carlos Eduardo Soares Goncalves, “Who Chooses Inflation
Targeting,” Economic Letters, 2008, 99.
Cogley, Timothy, Christian Matthes, and Argia M. Sbordone, “Optimized Taylor Rules
for Disinflation When Agents are Learning,” Working Paper 14-7, Federal Reserve Bank of Richmond 2014.
Cukierman, Alex and Nissan Liviatan, “Optimal accommodation by strong policymakers
under incomplete information,” Journal of Monetary Economics, 1991, 27, 99–127.
Eichengreen, Barry and Nergiz Dincer, “Central Bank Transparency: Causes, Consequences,
and Updates,” NBER Working Papers, 2009, 14791.
IMF, “Chapter 4 Does inflation Targeting Work in Emerging Markets?,” World Economic Outlook,
2005, September.
Kumhof, Michael, Shujing Li, and Isabel Yan, “Balance of Payments and Inflation Targeting,” Journal of International Economics, 2007, 72, 242–264.
Milani, Fabio, “Expectations, learning and macroeconomic persistence,” Journal of Monetary
Economics, October 2007, 54 (7), 2065–2082.
Morris, Stephen and Hyun Song Shin, “Unique Equilibrium in a Model of Self-Fulfilling
Currency Attacks,” American Economic Review, 1999, 88.
and
, “Rethinking Multiple Equilibria in Macroeconomic Modeling,” in “NBER Macroe-
conomics Annual 2000, Volume 15” NBER Chapters, National Bureau of Economic Research,
Inc, October 2001, pp. 139–182.
and
, “The Social Value of Public Information,” American Economic Review, 2002, 92,
1521–1534.
24
Orphanides, Athanasios and John C. Williams, “Inflation targeting under imperfect knowledge,” Economic Review, 2007, pp. 1–23.
Svensson, Lars, “Social Value of Public Information: Morris and Shin (2002) is actually Pro
Transparency, Not Con,” American Economic Review, 2006, 96, 448–451.
Woodford, Michael, “Optimal monetary policy inertia,” The Manchester School, 1999, (67),
31–35.
25
Appendix
Proof of Proposition 1 arg min L(., πe ) always exists as the domain is limited for the closed
interval [π, π] and it is unique for every πe since
∂2L
∂π 2
in πe . To see it, note that, if πe increases to πe 0, then
interior solution, then
∂L(π ∗ (πe ),πe 0)
∂π
> 0 ∀πe . As
∂L(π ∗ (πe ),πe )
∂π
∂2L
∂πe ∂π
>
< 0, π ∗ (.) is increasing
∂L(π ∗ (πe ),πe 0)
.
∂π
If π ∗ (πe ) is an
< 0 and π ∗ (πe 0) must be higher than π ∗ (πe ) . If π ∗ (πe ) = π, then
π ∗ (πe 0) = π ∗ (πe ) . If π ∗ (πe ) = π, then π ∗ (πe 0) ≥ π ∗ (πe ). From
the implicit function theorem, we have
dπ ∗ (πe )
dπe
=
−∂ 2 L(π ∗ (πe ),πe )
∂πe ∂π
∂ 2 L(π ∗ (πe ),πe )
∂π 2
∂L(π ∗ (πe ),πe )
∂π
= 0, after applying
< 1. As a result, there is only one πe
satisfying π ∗ (πe ) = πe . We name the fixed point π ∗ (πe ) = πe as πD , D for the discretionary
equilibrium.
Proof of Proposition 2 As (πD , πD ) is restricted to be the best response function given
πe = πD , it is straightforward that L (πG , πG ) ≤ L (πD , πD ) . For simplicity, we ruled out the
very particular and not interesting case of L (πG , πG ) = L (πD , πD ) . First, consider a point placed
on the plan (πa , πe ) . If we consider πa = πD , such a point would be equal to (πD , πD ) as, for this
particular target, expectations different of πD cannot be confirmed by the actual inflation. Then, consider πa = πD ± ε, for some small ε > 0. Since
∂2L
∂π 2
2
> − ∂π∂ eL∂π , we know that π ∗ (πD ± ε) = πD ± n.ε,
for some n ∈ (0, 1) . Therefore, k (πa , πe = πa ) ≡ L (πD ± ε, πD ± ε) − L (πD ± n.ε, πD ± ε) and
k (πa , πe = πD ) ≡ L (πD ± ε, πD ) − L (πD , πD ) . The distance between (πD , πD ) and (πD ± ε, πD )
is greater than the distance between (πD ± ε, πD ± ε) and (πD ± n.ε, πD ± ε) . As (πD , πD ) is a
local minimum, we know that
∂L(πD ,πD )
∂π
= 0. Therefore, for some ε close enough to zero, we have
k (πa , πe = πD ) > k (πa , πe = πa ) . In other words, if πa is close enough to πD , there is strategic complementarity between public expectations and central bank actions: expectations that central bank will
not deliver the target increases the incentive to central bank to not deliver the target. Let κ > 0 be the
cost of to not delivery the target. There is a small enough ε such that L (πD ± ε, πD ) − L (πD , πD ) <
κ, making both π ∗ (πa ) = πa and π ∗ (πD ) = πa . By starting at πD and moving a little from πD into
the direction of πG , it is possible to find πa ∈ {π ∈ R|L (π, π) < L (πD , πD )} such that π ∗ (πa ) = πa
and π ∗ (πD ) = πa . Now let κ be higher than max {k (πa = πG , πe = πD ) , k (πa = πG , πe = πa )} .
Then π ∗ (πG ) = πG and π ∗ (πD ) = πG .
Proof of Proposition 3 Consider a point on the plan (πa , πe ) , and let k (πa ) = L (πa , πa ) −
26
L (π ∗ (πa ) , πa ) and k (πa ) = L (πa , πD )−L (πD , πD ) . As
∂L
∗
|
∂π (π,πe )=(π (πa ),πa )
= 0 and
∂L
|
∂π (π,πe )=(πD ,πD )
= 0, and knowing that π ∗ (πa ) ∈ [min {πa , πD } , max {πa , πD }] , by having sign (πD − πG )
∂ 3 L(π,πe )
∂π 2 ∂πe
≥ 0, we have k (πa ) > k (πa ) . To see it, note that the distance between πa and πD is higher than the
2
dπ ∗ (π )
∂ L(π,πe )
distance between πa and π ∗ (πa ) , since dπe e < 1. Suppose πD −πG > 0. Then, ∂π∂ e
≥0
∂π 2
and πD > πa , making k (πa , πd ) > k (πa , πa ) . As πa becomes lower and closer to πG , the distance
k (πa , πd ) − k (πa , πa ) becomes higher. Now suppose πD − πG < 0. Then,
∂
∂πe
∂ 2 L(π,πe )
∂π 2
≤ 0 and
πD < πa , making again k (πa , πd ) > k (πa , πa ) . As πa becomes higher and closer to πG , the distance
k (πa , πd ) − k (πa , πa ) becomes higher. Knowing that the difference between k (πa , πd ) and k (πa , πa )
is given by:
L (πa , πD ) − L (πD , πD ) − [L (πa , πa ) − L (π ∗ (πa ) , πa )]
by replacing π ∗ (πD ) with πD , it is straightforward that:
lim k (πa ) = 0 = lim k (πa ) = 0
πa →πD
πa →πD
Summarizing, if πa moves from πD into direction of πG , π ∗ (πa ) moves into the same direction. But
since
∂2L
∂π 2
2
> − ∂π∂ eL∂π , π ∗ (πa ) does not move as much as πa . As a result, the distance |π ∗ (πa ) − πa |
increases as πa moves from πD into direction of πG . The same is true for the distance |πD − πa | .
Moreover, |πD − πa | increases more than |π ∗ (πa ) − πa | as πa moves from πD into direction of πG .
And as sign (πD − πG )
∂ 3 L(π,πe )
∂π 2 ∂πe
≥ 0, we conclude that k (πa ) increases more than k (πa ) as πa moves
from πD into direction of πG .
Proof of Proposition 4 For any given πa , α∗ (sp ) summarizes the equilibrium for the expectations:
L (πa , πe∗ )
− L (π
∗
(πe∗ ) , πe∗ )
πe∗
σp2
σp2 −1
p
= √ Φ (c) + s − Φ−1 (α∗ )
σ
τ
∗
∗
= α πa + (1 − α ) πD
where
τ≡
σp2 σ 2
σp2 + σ 2
Once α∗ is computed, both πe (α∗ ) and π ∗ (πe ) are uniquely determined. The existence of at
27
least one α∗ is straightforward since the right hand side of the above equation may vary from minus
infinite to infinite when one changes α ∈ [0, 1] , while the left hand side remains limited to the
set k (πa ) , k (πa ) . Considering α ∈ [0, 1] , the lowest absolute slope for the RHS is given by
σp2
σ
−1 min dΦ dα(α) . Taking the LHS, the slope on α is given by
∂L (πa , πe∗ ) ∂L (π ∗ (πe∗ ) , πe∗ ) ∂L (π ∗ (πe∗ ) , πe∗ ) dπ ∗ (πe )
−
−
∂πe
∂πe
∂π ∗
dπe
∗
∗
∗
∗
∂L (πa , πe ) ∂L (π (πe ) , πe ) dπe
=
−
≤0
∂πe
∂πe
dα
dπe
dα
And then, the highest absolute slope for the LHS is limited by:
2
∂
L
(.,
.)
(πa − πD )
(πa − π ∗ (πe∗ )) max ∂πe ∂π or limited by
2
∂ L (., .) (πa − πD ) max ∂πe ∂π 2
Therefore, a sufficient requirement for uniqueness to each sp is:
2
−1
∂ L (., .) σp2
dΦ (α) (πa − πD ) max <
min ∂πe ∂π σ
dα 2
or:
√
2
σp
∂ 2 L (., .)
2π
−
<
∂πe ∂π
(πa − πD )2 σ
Proof of Proposition 5 Under uniqueness assumption (sufficient noisy on public information)
more conservative targets improves the target coordination (increases α∗ ). We know that α∗ (sp , πa )
28
summarizes the equilibrium as follow:
α∗ solves ΨLHS (α) = ΨRHS (α)
ΨLHS (α) ≡ L (πa , πe ) − L (π ∗ (πe ) , πe )
πe = απa + (1 − α) πD
2
σp2 −1 ∗
σp −1
p
ΨRHS (α) ≡ √ Φ (c) + s − Φ (α )
σ
τ
2 2
σp σ
τ≡ 2
σp + σ 2
and because we have
∂2L
∂πe ∂π
<0:
dΨLHS (α)
=
dα
∂L (πa , πe∗ ) ∂L (π ∗ (πe∗ ) , πe∗ )
−
∂πe
∂πe
(πa − πD ) < 0
Evaluating ΨLHS at α = 0 :
ΨLHS (0) ≡ L (πa , πD ) − L (πD , πD )
dΨLHS (0)
∂L (πa , πD )
=
dπa
∂πa
∂L(πa ,πD )
∂πa
In case πG < πD , i.e. πa ∈ (πG , πD ) , then
< 0. In case πD < πG , then
∂L(πa ,πD )
∂πa
> 0.
Therefore, as πa moves closer to πD , ΨLHS (0) reduces.
Evaluating ΨLHS at α = 1 :
ΨLHS (1) ≡ L (πa , πa ) − L (π ∗ (πa ) , πa )
dΨLHS (1)
dL (πa , πa ) dL (π ∗ (πa ) , πa )
=
−
dπa
dπa
dπa
In case πG < πD , i.e. πa ∈ (πG , πD ) , then
Also,
dL(π ∗ (πa ),πa )
dπa
dL(πa ,πa )
dπa
> 0, since πa ∈ Πa ≡ π ∈ R|L (π, π) < LD .
> 0. Take for example πa = πD − ε for some ε > 0 and increases πa up to πD .
Alternatively, note that
dL(π ∗ (πa ),πa )
dπa
L (πa , πa ) − k (πa ) , therefore,
=
∂L(π ∗ (.),πa )
∂πe
∂L(π ∗ (.),πa )
∂πe
πD < πG , i.e. πa ∈ (πD , πG ) , then
=
+
dL(πa ,πa )
dπa
dL(πa ,πa )
dπa
∂L(π ∗ (.),πa ) dπ ∗
∂π
dπa
−
=
∂L(π ∗ (.),πa )
.
∂πe
dk(πa )
dΨLHS (1)
dπa .Then,
dπa
dk(πa )
dπa
< 0. In case
< 0, since πa ∈ Πa ≡ π ∈ R|L (π, π) < LD . Also,
29
=
As L (π ∗ (.) , πa ) =
dL(π ∗ (πa ),πa )
dπa
that
< 0. Take for example πa = πD and increases πa up to πD + ε. Alternatively, note
dL(π ∗ (πa ),πa )
dπa
therefore,
=
∂L(π ∗ (.),πa )
∂πe
∂L(π ∗ (.),πa )
∂πe
=
+
dL(πa ,πa )
dπa
∂L(π ∗ (.),πa ) dπ ∗
∂π
dπa
−
=
∂L(π ∗ (.),πa )
.
∂πe
dk(πa )
dΨLHS (1)
dπa .Then,
dπa
=
As L (π ∗ (.) , πa ) = L (πa , πa ) − k (πa ) ,
dk(πa )
dπa
> 0. Therefore, as πa moves closer
to πD , ΨLHS (1) reduces. As a result α∗ increases as πa approaches to πD .
30