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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Incomplete Informatioity Higher-Order Beliefs
and Price Inertia
George-Marios Angeletos
Jennifer La'O
Working Paper 09- 18
March 31, 2009
RoomE52-251
50 Memorial Drive
Cambridge,
02142
MA
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http://www.archive.org/details/incompleteinformOOange
Incomplete Information, Higher-Order Beliefs
and Price
Inertia*
Jennifer La'O
George-Marios Angeletos
MIT
and
NBER
MIT
March
31, 2009
Abstract
This paper investigates how incomplete information impacts the response of prices to nominal
shocks.
Our
baseline model
nominal shocks with
is
noise.
a variant of the Calvo model in which firms observe the underlying
In this model, the response of prices
is
pinned down by three
parameters: the precision of available information about the nominal shock; the frequency of
price adjustment;
and the degree of strategic complementarity
synthesizes the broader lessons of the pertinent literature.
We
in pricing decisions.
next highlight that this synthesis
provides only a partial view of the role of incomplete information.
of information does not pin
down
This result
In general, the precision
the response of higher-order beliefs.
Therefore, one cannot
quantify the degree of price inertia without additional information about the dynamics of higherorder beliefs, or the agents' forecasts of inflation.
beliefs
We
with three extensions of our baseline model,
between the precision of information and higher-order
"Thi.s
paper was prepared
lor the
2008
SNIV-'MK
hosted by the Gerzensee Study Center, Switzerland.
pants.
We
highlight the distinct role of higher-order
all
of which break the tight connection
beliefs featured in previous work.
conference on "Incomplete Information and Mt)netary Policy,"
We
received useful
are particularly grateful to our discussants, Stephen Morris
the conference, Sergio Rebelo and Robert King.
comments from multiple conference
partici-
and Elmar Mertens, and to the organizers of
Introduction
1
How much, and how
mental questions
the business
in
cj'cle,
To address
quickly,
do prices respond to nominal shocks? This
macroeconomics:
as well as the
this question,
it is
power
of
monetary policy
economic
to control real
one strand of the literature has focused on
rigidities are
one of the most funda-
key to understanding the sources and the propagation of
tions in adjusting prices; this includes both convenient time-dependent
adjustment models. Price
is
menu
costs
activity.
and other
models and state-dependent
then identified as the key force behind price inertia. Another
strand of the literature has focused on informational frictions; this strand highlights that firnrs
fail
fric-
to adjust their price to nominal shocks, not because
it
is
may
costly or impossible to do so, but
rather because they have imperfect information about these shocks.
The
older literature formal-
ized this imperfection as a geographical dispersion of the available information (Lucas, 1972, Barro,
1976);
more recent contributions
ha\'e
proposed infrequent updating of information (Mankiw and
Reis, 2002; Reis, 2006) or rational inattention (Sims, 2003;
Wiederholt, 2008).
One way
Woodford, 2003, 2008; Machowiak and
or another, though, the key driving force behind price inertia
is
that
firms happen, or choose, to have imperfect knowledge of the underlying nominal shocks.
The
starting point of this paper
basehne model
is
is
a bridge between these two approaches.
In particular, our
a hybrid of Calvo (1983), Morris and Shin (2002), and Woodford (2003): on the
one hand, firms can adjust prices only infrequenth',
as in Calvo;
on the other hand, firms observe
the underl)'ing nominal shocks only with noise, similarly to Morris-Shin and Woodford.
Within
this baseline
impact of these shocks
model, the response of prices to nominal shocks
—
of a\'ailable information
is
— and
hence also the real
characterized by the interaction of three key parameters; the precision
about the underlying nominal shocks (equivalently, the
level of noise in
the firms' signals of these shocks); the frequency of price adjustment; and the degree of strategic
complementarity
their interaction
in pricing decisions.
is
also interesting.
That
all
three parameters should matter
The combination
of sticky prices
is
obvious; but
and strategic complementarity
implies that the incompleteness of information can have lasting effects on inflation and real output
even
if
the shocks
become commonly known very
<iuickly.
information about the shock at the time they set prices
to the extent that other firms
prices.
This
will find
had only incomplete information
is
it
because firms that
full
optimal to adjust only partly
at the
time they had set their
Moreover, incomplete information can help make inflation peak after
seems consistent with available evidence based on structural VARs.
lia\-e
real output,
which
These findings
and marginally extend, various lessons from the pertinent literature
sj'nthesize,
with regard to how frictions
in either price
adjustment or information about the underlying nominal
shocks impact the response of prices to these shocks. This synthesis has
its
own
value, as
it
provides
a simple and tractable incomplete-information version of the Calvo model that could readily be
taken to the data. Nevertheless, this synthesis
the main contribution of the paper
distinct role that higher-order beliefs play in the
how
this distinct role
The
not the main contribution of the paper. Rather,
to highlight that the aforementioned lessons miss the
first,
is,
is
dynamics of price adjustment and, second, to show
can be parsimoniously accommodated within our Calvo-like framework.
basic idea behind our contribution
the underlying nominal shock identifies
simple.
is
how
The
precision of the firms' information about
fast the firms' forecasts of the
shock respond to the true
shock; the more precise their information, the faster their forecasts converge to the truth. However,
need not also identify how
this
may
fast the forecasts of the forecasts of others
words, the prec:ision of available information pins
necessarily the response of higher-order beliefs.
down
adjust.
In other
the response of first-order beliefs, but not
But when
prices are strategic
complements, the
response of the price level to the underlying shock depends heavily on the response of higher-order
behefs.
It
follows that neither the precision of available information nor the degree of price rigidity
suffice for calibrating the degree of price inertia at the
To
better understand this point,
it is
macro
level.
useful to abstract for a
in particular, that all firms can adjust their prices in
moment from
sticky prices.
Assume,
any given period and that the prices they
set
are given by the following simple pricing rule;
...
where
pj is
,:
",
,
p,:
the price set by firm
i,
6
is
degree of strategic complementarity
on the information
price level,
where
level
p
£'''"
we
of firm
i.
infer that the
=
(1
- Q')E,(9-KaE,p
nominal demand, p
in pricing decisions,
-.,,/
the aggregate price
,.
^
level,
-
-
,
a £
(0,1)
is
the
and E, denotes the expectation conditional
Aggregating this condition and iterating over the expectations of the
aggregate price level must satisfy the following condition;
denotes the /c"*-order average forecast of
to an innovation in 6
is
.
^.'
It
then follows that the response of price
depends on the response of the entire sequence
of different orders of
beliefs, {£''^}^j, to that shock.
'The
fc"'-order average forecasts are defined recursively as follows:
forecasts of the underlying nominal shock;
£^
is
the cross-sectional
E^
mean
is
the cross-sectional
mean
of the firms' forecasts of £';
of the firms'
and so on.
When
information
is
perfect or at least
commonly
shared, then
E^ = E^
for all k.
It
then follows
that the response of prices to an innovation in 9 depends merely on the response of first-order beliefs,
which
in
turn
information
The
is
is
pinned down by the precision of the available information about
When,
6.
dispersed, higher-order beliefs need not coincide with first-order beliefs.
potential role of higher-order beliefs has been noted before by Morris and Shin (2002),
Woodford
(2003),
and
others.
However,
standard Gaussian example used
in the
literature, the sensitivity of higher-order beliefs to the
order beliefs: higher-order beliefs can be
of information about the shock
is
shock
less sensitive to
lower, in
follows that in the standard Gaussian
which case
is
in the
tightly connected to that of first-
the underlying shock only
if
the precision
first-order beliefs are also less sensitive.
It
However,
of the response of the price level to the shock.
once one goes away from the standard Gaussian example, this tight connection between
higher-order beliefs can break
pertinent
example the precision of information about the underlying
nominal shock remains the key determinant
We
instead,
— and the break can be quite
first-
and
significant.
highlight the crucial and distinct role of higher-order beliefs with three variants of our
baseline model.
All three variants retain the combination of infrequent price
Calvo) and noisy information about the underlying shocks (as
in
adjustment (as
in
Morris-Shin and Woodford), but
differentiate in the specification of higher-order beliefs.
In the
first
variant, firms face uncertainty, not only about the size of the aggregate
but also about the precision of the signals that other firms receive about
this shock.
nominal shock,
This extension
helps isolate the role of higher-order beliefs or, equivalently, the role of strategic uncertainty:
show how
this additional source of uncertamtj'
about the distribution
of precisions
we
can impact the
response of higher-order beliefs to the underlying shocks, and thereby the response of prices, without
necessarily affecting the response of first-order beliefs.
In the second variant,
we
the signals that other firms
more slowly
as
if
let
firms hold heterogeneous priors about the stochastic properties of
In this economy, firms expect the beliefs of others to adjust
recei\'e.
to the underlying shocks than their
they lived
in
a economy where
all
if
beliefs.
other firms had
themselves have. This in turn helps rationalize
the underlying nominal shocks even
own
why
They thus behave
less precise
equilibrium prices
the frequency of price adjustment
the inertia of higher-order beliefs; heterogeneous priors
is
equilibrium
information than what they
may
is
adjust very slowly to
arbitrarily high
firm has arbitrarily precise information about the underlying nominal shock.
is
in
Once
and each
again, the key
a convenient modeling device.
The aforementioned two
nominal shocks
how
variants focus on
economy. In the third and
in the
be the source of fluctuations
in
the economy
we show how
shocks. In particular,
higher-order beUefs impact the propagation of
final variant,
we show how higher-order
beliefs
can
— how they can themselves be one of the "structural"
variation in higher-order beliefs that
is
orthogonal to either the
underlying nominal shocks or the firms information about these shocks can generate fluctuations in
inflation
the
and
real
output that resemble those generated by "cost-push" shocks.
Combined, these flndings point out that a macroeconomist who wishes
to quantify the response of
economy
are these structural shocks
to its underlying structural shocks, or even to identify
in the first place,
may
what
need appropriate information, not only about the degree of price rigidity and
the firms' information (beliefs) about these shocks, but also about the stochastic properties of their
forecasts of the forecasts of others (higher-order beliefs).
Because the hierarchy of
beliefs
role of higher-order beliefs in
the contribution of the paper
is
an
infinitely
dimensional object, incorporating the distinct
macroeconomic models may appear
is
to
show that
this
is
not the case. All the models
are highly parsimonious and nevertheless allow for rich
The
rest of the
paper
is
to be a challenging task. Part of
dynamics
we present here
in higher-order beliefs.
organized as follows. Section 2 discusses the relation of our paper to
the literature. Section 3 studies our baseline model, which introduces incomplete information in the
Calvo model. Section 4 studies the variant with uncertainty about the precisions of one another.
Section 5 studies the variant that with heterogeneous priors. Section 6 turns to cost-push shocks.
Section 7 concludes with suggestions for future research. All proofs are in the Appendix,
2
,;
,,,
Related literature
The macroeconomics
literature
on informational
(1970), Lucas (1972, 1975), Barro (1976),
ture has been revived by
(2003),
Mankiw and
and subsequent work.^
Kmg
frictions has a long history, going
(1983) and
Townsend
back to Phelps
(1983). Recently, this litera-
Reis (2002), Morris and Shin (2002), Sims (2003),
This paper contributes to this literature in two ways:
Woodford
first,
by
studying the interaction of incomplete information with price rigidities within the Calvo model;
second, and most importantly, by furthering our understanding of the distinct role of higher-order
^See, e.g.,
Amato and
Shin (2006), Angeletos and La'O (2008, 2009a, 2009b), Angeletos and Pavan (2007),
Bacchetta and Wincoop (2005), Collard and Delias (2005), Hellwig (2005), Lorenzoni (2008, 2009), Mackowiak and
Wiederholt (2008, 2009), Nimark (2007, 2008), and Reis (2006).
beliefs.
Closely related in this respect
persed information has
verj' distinct
Angeletos and La'O (2009a), which emphasizes
is
how
dis-
implications for the business cycle than uncertainty about the
fundamentals.
Our paper
is
highly complementary to the papers by
2006). These papers
document how higher-order
underlying shocks than first-order
prior. "^
beliefs,
Woodford (2003) and Morris and Shin (2002,
beliefs
may respond
simply because they
a.re
information about the
less to
more a.nchored
to the
common
However, by adopting the convenience of a popular but very specific Gaussian information
structure, they have also restricted attention to settings where the response of higher-order beliefs
is
tightly tied to the response of first-order beliefs: in their settings, the response of higher-order beliefs
is
a
monotone transformation
of the response of first-order beliefs, thus precluding
any independent
role for higher-order beliefs.
Our
paper, instead, highlights that, whereas the response of first-order beliefs to the underlying
nominals shocks
is
pinned down solely by the
of higher-order beliefs
depends
about these shocks, the response
also on other sources of uncertainty, such as uncertainty
precision of others' information. Furthermore,
that higher-order beliefs respond very
all
level of uncertainty
little,
it
shows how with heterogeneous priors
or even not at
about the
it is
possible
to the underlying shocks even
all,
if
firms are nearly perfectly informed about these shocks (in which case first-order beliefs respond
nearly one-to-one with the shock).
Closely related are also the papers
bj'
Nimark (2008) and Klenow and
Willis (2007).
The former
paper studies a similar framework as ours, namely a Calvo model with incomplete information, along
with a more complex learning dynamics: unlike our paper and rather as
underlying shocks do not become
ies
common knowledge
a menu-cost model with sticky information.
after one-period delay.
Much
alike our baseline
Woodford (2003), the
in
The
latter
paper stud-
model, both papers study
the interaction of price rigidities and informational frictions. However, these papers do disentangle
the role of higher-order beliefs as
we do
incomplete information in these papers
shocks at the time of price changes.
dynamics of higher-oder
price inertia at the
level
^A
and
beliefs
macro
fast learning
level
in this paper. In particular,
is
the quantitative importance of
tied to the precision of information
In contrast, our paper shows
about the underlying
how one can
disentangle the
from the speed of learning, and uses this to argue that significant
can be consistent with both significant price
flexibility at the
micro
about the underlying nominal shocks.
similar role of higher-order beliefs has been highlighted by Allen, Morris
and Pavan (2007) and Bacchetta and Wincoop (2005) within the context of
and Shin (2005), Angeletos, Lorenzoni
financial markets.
Finally, the heterogeneous-priors variant of this
that paper
we consider
a real-business-cycle
the underlying productivity shocks.
We
certain type of sunspot-like fluctuations
model
paper builds on Angeletos and La'O (2009b), In
in
then show
—
i.e.,
how
fluctuations obtain also under a
common
dispersed information opens the door to a
fluctuations that cannot be explained by variation in
economic fundamentals or the
either the underlying
which firms have dispersed information about
prior,
firms' beliefs
but are easier to model with heterogeneous priors.
present paper complements this other work by illustrating
The
about these fundamentals. These
take the form of cost-push shocks in a new-keynesian model, and
how
these type of fluctuations can
how heterogeneous
priors
can
also
help rationalize significant inertia in the response of prices to nominal shocks.
The Calvo Model with Incomplete Information
3
In this section
we consider a
variant of the Calvo
model that allows firms
'
to have dispersed private
information about aggregate nominal demand.
Households and firms.
uum
indexed by
Along with the
The
is
populated by a representative household and a contin-
produce differentiated commodities.
of firms that
discrete,
The economy
i
£ {0, 1,2,
fact that there
is
...}.
There
is
Firms are indexed by
no capital, so that there
is
i
G
no saving
[0,1].
Time
is
in equilibrium.
a representative household, we can also abstract from asset trading.
preferences of the household are given by '^^p^U{Ct,Nt), with
'
..;v-,;
/..
where
/3
6
(0, 1) is
U{Ct,Nt)
the discount rate, Nt
is
the familiar
r]
>
the
is
CES
aggregator, Ci,t
is
'7-1
CJ
di
the consumption of the connnodity produced by firm
the elasticity of substitution across commodities.
demand
for the
commodity
of firm
i
is
given by
1
where
Pt=\j P^'^
di
^
^
is
,
the labor supplied by the household,
ais
= \ogCt~Nt,
the aggregate price index.
As
i,
and
usual, this specification implies that
The output
where
A,.;
is
of firm
i,
on the other hand,
is
'
given by
the idiosyncratic productivity shock and
e
e
(0, 1)
parameterizes the degree of dimin-
ishing returns.
By
the resource constraint for each commodity,
aggregate output
given by Yt
is
—
we have that
F,,t
—
Ci,i for all
nominal demand
Ct. Finally, aggregate
i
and therefore
given by the following
is
quantity-theor}' or cash-in-advance constraint:
= Qf
PtCt
Here, 6( denotes the level of aggregate nominal
demand
(aggregate nominal
GDP),
is
assumed
to
be exogenous, and defines the "monetary shock" of our model.
In
what
variables:
we use lower-case
follows,
dt
=
log6(,
yt
=
variables for the logarithms of the corresponding upper-case
logF^, pt
=
We
logPt, and so on.
also
assume that
all
exogenous
shocks are log-normally distributed, which guarantees that the equilibrium admits an exact loglinear solution.
Shocks and information.
Aggregate nominal demand
et^Ot-i
where
and
~
Vt
a^)
is
t.
summarized
in
i
~ A/'(0, at^)
is
employment and .consumption choices
are
we assume
known
=
imperfect during the morning
that a firm has about
6t
during the morning
is
Ot -fc,,,,
purely idiosyncratic noise
made
is
is
a Gaussian private signal of the following form:
that have the option to set prices) are
Finally,
random walk:
during, respectively, the morning and the evening of
The information
Xl,t
Six
to follow a
+ vt
Information about the current level of nominal demand
but perfect during the evening.
where
assumed
white noise. Each period has two stages, a morning and an evenmg. Let ij^
denote the information set of firm
Jj"j
period
A/'(0,
is
in
(i,i,d.
across firms). Pricing choices (for the firms
the morning, while information about
made
in
the evening, once
that the idiosyncratic productivity shock
to the firm from the beginning of the period.
a,,(
9t
6t is
imperfect;
has been publicly revealed.
follows a
random walk and that
it
The information
of firm
in the
i
morning of period
while her information in the evening of the same period
a
moment
that, in equilibrium, pt-i
no difference
if
we had included
of the firm. Similarly,
y,,t
the realized level of a firm's
The assumption that
nor inconsequential.
If
9t
demand
Townsend
evening information
its
(1983),
9t
at the
X/(_j U{9t}.
GDP
We
a
in the
will see in
if
information set
we had included
set.
end
of each period
is
after the
end of period
to
more decentralized trading structure and would complicate the
(2008),
and
less tractable:
would now have to keep tract of the entire
others, firms
have any simple recursive structure.
fast
However,
after a short delay.
it is
we avoid
Here,
keep the analysis highly tractable, only by assuming, as
knowledge
t,
commodity
history of their information in order to forecast the forecasts of others, and the equilibrium
would cease
random
neither
Furthermore, the dynamics would now become much
Nimark
o,i.t},
would thus make
}', it
remain dispersed
to
If^_^U{x,^t,
to limit the aggregation of information that takes place through
necessary micro- foundations.
as in
into
=
would thus make no difference
1;,;; it
becomes common knowledge
That would require
markets.
past values of the price level and real
we wished information about
we would need somehow
=
given by Ifj
is
and yt-i are functions of {9t-i,0t-2,
a function of
is
therefore given by l/^
t is
in
all
these complications, and
Lucas (1972), that
important to recognize that,
9t
becomes
in so doing,
convergence of beliefs about the past shocks and also rule out any heterogeneity
expectations of future shocks beyond the one
in their beliefs
dynamics
common
we impose a
in the agents'
about the current shock.
One may
expect that relaxing these properties would add to even more inertia, both because firms would learn
more slowly (Woodford, 2003) and because expectations
to public information (Morris
Price-setting behavior.
nominal shock
i
in
period
t is
is
= -^
more anchored
and Shin, 2006).
Consider
common knowledge
for a
moment
at the
the case where prices are flexible and the current
moment
firms set prices.
The optimal
price set
by firm
then given by
Pit
where ^
of future shocks could be
is
= M + rnc^.t
the monopolistic mark-up and mcj^t
in the evening of period
t.
The
latter
is
mc,j
is
the nominal marginal cost the firm faces
given by
=
1
wt
-
-\
1
e
a,,(
yi.t
e
£
where Wt
is
the nominal
wage
rate in period
From the
t.
representative household's optimality
condition for work,
-
Wt
From
Ct.
the consumer's demand,
-C( =
Ci,f
Fi-om market clearing,
real
=
pt
output
is
c,,;
=
y,.;
a =
and
yt-
~v{Pi,t -Pt)-
from the cash-in-advance constraint, aggregate
Finally,
given by
^
yt
-
Ot
Pt-
Combining the aforementioned conditions, we conclude that the
price full-information optimal price) of firm
Pit
=
i
in period
-
{I
+
a)et
apt
(i.e.,
the flexible-
given by
is
t
"target" price
+ ^r,t
(1)
where
e
+
{l
-
i)r]
and where
defines the degree of strategic complementarity in pricing decisions,
e
+
e
is
1
-
[I
e
it}ri
+
(I
-
e)i]
simply a linear transformation of the idiosyncratic productivity shock.
the
mean
If
of the idiosyncratic productivity shock so that the cross-sectional
prices
had been
firm would set
p,,(
=
and
flexible
p*^ in
imperfect information.
all
9t
had been common knowledge
1
its price,
the price
it
period, where A G
chooses
expectation of a weighted average of the current and
1E,,(
adjust
its
price (a
mean
of
the discount factor, A
measure
of
^,,(
is
zero.
begging of period
t,
the
periods and states. However, we have assumed that firms have only
— A during any given
the event that a firm changes
(0, 1) is
in the
henceforth normalize
Moreover, following Calvo (1983), we assume that a firm
price only with probability
where p G
We
how
{\
is
all
is
(0, 1).
It
may change
its
then follows that, in
equal (up to a constant) to
its
current
future target prices:
- 0\)Y^{0xy pU^
(2)
the probability that the firm won't have the option to
sticky prices are),
and
Et_f is
the expectation conditional on the
information set of firm
i
Combining conditions
chance to adjust
t^
in period
(1)
and
(2),
price in period
its
t is
we conclude
the price set by any firm that gets the
tliat
given by
oo
p,,t
=
(1
- 0X)
In the remainder of the paper,
Y, iP^y
we
Cl')
^^A+J +
treat condition (3) as
with the understanding though that
this rule
The economy
Equilibrium dynamics.
-
[(1
is
QE^,tP(+J
+ ^iAi,t+j]
(3)
were an exogenous behavioral
if it
actually fully microfounded in equilibrium.
dynamic game
effectively reduces to a
of
formation, with condition (3) representing the best response of the typical firm,
notion we adopt
is
response condition
standard Perfect-Bayesian Equilibrium.^
We
that the equilibrium strategy will have a linear form.
in
which the price
set
by a firm
Pi,t
for
some
=
in
period
i is
coefficients 61,62,^3-
=
bjpt-i+b2X,^t
This particular guess
is
The
eqiiilii)riinn
it is
a safe guess
+
justified
^t-i,a;t,(,<^i,t):
b3et-}+^i,t
(4)
by the following reasoning:
we
expect pt-i to matter because of a fraction of firms cannot adjust prices; Xij because
information about the current nominal shock
Ot;
in-
thus conjecture the existence of equilibria
a linear function of (pt_i,
P{Pt-i,9t-i,Xi,f^i,t)
incomplete
Because of the linearity of the best-
and the Gaussian specification of the information structure,
(3)
rule,
Ot-\ because
it is
the prior about
Of,
conveys
it
and
S,i^t
for
obvious reasons.
Given
and given the
this guess,
adjust prices in any given period,
Pt
=
Xpt-i
fact that only a
we
infer that the
+ il-X)J
J
''To
const
is
9t)
(2)
- A
of firms can
aggregate price level must satisfy
of the private signals (conditional
and Gt denotes the cross-sectional distribution
be precise, condition
1
PiPt-i.0t-i;x,OdFt{x)dGtiO
where Ft denotes the cross-sectional distribution
shock
randomly selected fraction
should have been written as
p,.,.
of the idiosyncratic shocks.
=
const
+
E,,,
(1
on the current
Given that
- 0X) YlTLo {P-^V P'+j
an endogenous quantity that involves second-order moments and that emerges due to
risk aversion.
>
P
where
However,
under the log-normal structure of shocks and signals that we have assumed, these second-order moments are invariant
with either the shock or the information of the firms, and
it
is
thus without any loss of generality to ignore the
aforementioned constant.
We
can safely ignore out-of-equilibrium paths by assuming that firms observe (at most) the cross-section
bution of prices, which guarantees that no individual deviation
10
is
detectable,
distri-
is
linear, that the cross-sectional
can re-write the above as
=
pt
average of X(
+
^Pt-i
=
Pt
—
(1
is 9t,
and that cross-sectional average
P[Pt-i^ ^t-v,
-^)
ClPt-l
+
C20t
+
of ^t is 0,
we
or equivalently as
9t, 0),
C39t-l
(5)
where
ci
=
+
A
Next, note that we can rewrite
p,,t
Using
p,,t
=
(4)
and
(5) into
(1-/3A)
[(1
=
(1
-
(1-A)6i,
-
=
(l-A)63.
i,.,t]
+
iPX)E^,tP^,t+l
(6)
form as
(3) in recursive
/3A) [(1
C3
= (l-A)62
C2
a) E,,tOt
+
aE,,tPt
+
we
the right-hand side of the above condition,
- q)
E,,t0f
+ aE^^tPt + ^^,t] + {P^)
=
Ei,t9t
=
+
[^iE„tPt
infer that the price
b2E^,tet
must
satisfy
+ b^E^^A+i + E,.,s^,,t+i]
.
(7)
Next, note that
Er^t9t+i
where k^
=
a^"^
is
—
—
+
Xit
and
9t-i
s a^"
the precision of the firms' signals and k^
prior about the innovation in
9.
Using these
rewrite the left-hand side of (7) as a linear function of p(_i,
with our conjecture
in (4),
is
it
is
and substituting
facts,
Jt,(,
=
E,.,^,,,+i
^(-i,
4,,t,
the precision of the
pt
from
and
common
(5) into (7),
4,^(.
we can
For this to coincide
necessary and sufficient that the coefficients (61,62,^3) solve the
following system:
61
=
(l-/3A)aci
62
=
[(1
-
-Q+
QC2)
+
(/?A)(6iC2
+
62
+
63)]
^3
=
1(1
-;3\){\-a +
ac2)
+
(/3A)(6iC2
+
bz
+
^3)]
/3A)(1
+
+(1 - ,3A)ac3
We
conclude that an equilibrium
Combining the conditions
is
for c\
(/3A)6iCi
+
it
"".^
pinned down by the joint solution of
and
(8)
(/3A)6iC3
61,
we
This equation admits two solutions: one with
the former solution because
^^
get that C\
ci
>
1
and
(8).
must solve the following equation:
and another with
leads to explosive price paths
11
(6)
ci
and henceforth
€
(A, 1).
We
ignore
limit attention to the
Note that
latter solution.
coefficient ci,
in the
which
this solution
independent of the information structure; indeed, the
is
endogenous persistence
identifies the
in the price level, coincides
with the one
standard (complete-information) Calvo model.
Given
remaining conditions define a linear system that admits a unique
this solution for ci, the
solution for the remainder of the coefficients.
+
Ci
straightforward to check that the solution satisfies
It is
which simply means that the price process
+
C2
=
C3
1,
homogenous
is
nominal
of degree one in the level of
demand. Furthermore,
_ A(l-ci)
which
nominal demand as
identifies the sensitivity of the price level to the current innovation in
an increasing function of the precision of available information.
We
therefore reach the following
characterization of the equilibrium.
Proposition
and
1.
(i)
There exists
an.
which the pricmg strategy
some
(ii)
Pt "= CiPt-l
+
C29t
+
The equilibrium values of
in Kx/kq; C3
The comparative
as follows.
long
^A
it
We
fast.
:
is
non-monotone
due
macro data
here.
The
This
to
fast,
(ii)
non-monotone
and decreasing
fact that these
in
cj
is
in A, decreasing in a,
in Kx/kq.
are illustrated in Figures
become widely
1,
2
and
.3.
The
baseline
impulse responses that wc report later
available.^
We
seems a good benchmark
.95
(which
most agents are
likely to
accordingly set
data are widely available .suggests
tliat
=
for
turn implies that their first-order beliefs are likely to converge to the truth
However, to the extent that this fact
not converge as
in a,
is
identify the length of a period with one year; this
takes for
qualification
is
for these figures, as well as for the
be well informed about them.
very
given by
the coefficients (C],C2,C3) satisfy the folloiuing properties:
statics described in part
parameterization we use
how
is
positive coefficients [hi, b2,b2,) anrf (cj, C2, C3).
and increasing
is
firm
C-iOt-l
increasing in X, increasing in a, and invariant to Kx/kq; €2
on,
of a
-
the aggregate price level is given by
"
for
m
equililmum
is
not
common
which could contribute to further inertia
12
knowledge,
in
it is
possible that higher-order beliefs
the response of prices.
do
means a discount
rate of
1%
about
per quarter), A
a =
equal to 1/3 per quarter), and
.85
—
.20
(which means a probabiUty of price change
(which means a quite strong complementarity
decisions); these values are consistent with standard calibrations of the Calvo model.
pricing
in
Lacking any
obvious estimate of the precision of information about the underlj'ing shocks, we set Kx/kq
means that the variance
this
in
nommal demand
Figure
1
is
of the forecast error of the typical firm
one half the variance of the innovation
plots the coefficients ci, C2,
the firm does not
is
non-monotonic
in
A
(it
We
about the current innovation
Calvo parameter
observe that
in a,
and
C3 is
We
observe that
non-monotonic
ci is
The comparative
models and
price level
coefficient
in the
is
an increasing function
decreasing and
is
statics described
some
in the
we
Woodford (2003)
point,
is
complementarity
see that C2
is
increasing
model
is
the
same
as in the
standard Calvo model, the aggregate
In our model, the
is
ci.
We
increase the length of this delay, then
or
Nimark
(2008), but only
is
till
common knowledge
we can obtain more
the shock becomes
optimal to stay closer to the past price
level
it is
only with a delay of
persistence, similarlj' to
common
driven solely by the Calvo rigidity.
under complete information, firms who can adjust
The
knowledge; after that
property, then, that Ci
almost the mechanical implication of
The impact
of q
on
their price following a
ci is also familiar:
monetary shock
'Note that only the ratio Ki/ks, and not the absolute values of Ki and Ke, matter
economy
C3.
In other words, once
we
fix Ki/fca,
Kg
is
even
will find
the higher the degree of strategic complementarity
between them and the firms that cannot adjust (and that are thus stuck to the past price
and
find that
standard Calvo model. This property, however,
increasing in the Calvo parameter A should be familiar:
of the
Calvo
unaffected by the incompleteness of information. In this sense, the persistence of
any subsequence persistence
ci,C2,
Hx/ 1^0,
in sticky-price
of the aggregate price process
the fact that a fraction A of firms do not adjust prices.
it
We
firms cannot adjust prices.
hinges on our assumption that the nominal shock becomes
If
is
a decreasing function
in o, C2 is
above are a hybrid of the results found
incomplete-information literature. As
prices in our baseline
one period.
A, C3
c\ is invariant.
which characterizes the persistence
this coefficient
m
Finally, Figure 3 plots these coefficients as functions of
in a.
persistent due to the fact that
is
the probability
increases for low values but decreases for high
the ratio of the precision of private signals to the precision of the prior.
in this ration, while C3
A,
increasing
is
ci
values). Figure 2 plots these coefficients as functions of q, the degree of strategic
in pricing decisions.
1;
itself.^
C3 as functions of the
a given period.
re^'ise its price in
decreasing in A, and co
and
=
only a scaling parameter.
to a one-standard-deviation change in v are invariant to
13
K.g.
By
for the
level).
equilibrium coefficients
implication, the impulse responses
Figure
Figure 2
I
C|.'!-c,
1.0
£']
OR
—
—
on
06
06
\
':
^1
Cl
—
—
\
04
^
'1
04
________^
<^J
--
02
0.2
::::==
06
04
0,2
OS
02
1,0
04
06
8
1
Figure 3
Where
C2
and
C3,
the incompleteness of information has a bite in our baseline model
level.
To understand how the
consider the choice of the price-setting firm.
The
past prices and past nominal shocks (which are
we
shock
by a firm
price chosen
is
common knowledge among
a linear combination of
all
firms)
and the firm's
is
unknown
find that the aggregate price level
is
a linear combination of past price levels
model with Gaussian
signals, the firm's
his private signal
less precise private
6t
own expectation
and the common
prior,
information relative to the prior,
As
in the current period).
in
of the
fundamental
is
Ot-i- This explains
is less
why
sensitive to the current shock Ot-i
less precise
6t-\.
lower k^/ho, they place
9t.
As a
less
If
firms
weight
result, the
and more anchored to the past shock
information (a lower Kx/ko) implies a lower
14
and
merely a weighted
which here coincides with
i.e.,
Aggregating
any static incomplete information
on their private signals than on their prior when forming their expectations of
average expectation
for
precision of information affects these coefficients,
past nominal shocks and of the average expectation of
combination of
peist
demand (which
expectation of current nominal
a.cross firms,
have
on the coefficients
which characterize, respectively, the price nnpact of the current and the
any given past price
own
is
C2
and a higher
C3.
Figure 4
The above
Impulse responses.
into the Calvo niodoi
of information
Figure 5
dampens
analysis highlights
how introducing incompleteness
the response of prices to the underlying nonunal shocks
becomes a key parameter
of information
— the precision
the djmamics of inflation along with the Calvo parameter
for
and the degree of strategic complementarity. To further appreciate
this,
we now study how the
precision of information affects the impulse responses of the inflation rate and real output to an
innovation
in
nominal demand.
Figures 4 and
real
output
—
is yt
these impulse responses. (Inflation in period
5 plot
dt
~
As
Pt-)
and a
—
.85.
which
is
our baseline; Kx/ng
We
before,
we
identify the period with a year
of a
given by pt
and
set
3
-
=
Pt-i, while
.95,
=
oo,
4,
we
A
=
.20
=
1,
which corresponds to the extreme of perfect information (as
in
—
0,
n^-fK-g
which corresponds to the alternative extreme, that of
no information about the current shock other than the prior
dynamics
is
then consider three alternative values for the precision of information:
the standard Calvo model); and Kx/ng
From Figure
t
(i.e.,
the past shock).
see that the incompleteness of information has important effects on inflation
relative to the
monetary shock on
complete information Calvo model.
inflation
is
First, the
As the
increasing in Kx/ng.
instantaneous impact effect
noise in private information increases,
prices react less initially to a nominal disturbance. Secondly, as the precision of private information
decreases,
we observe
that second period inflation becomes higher and highc-r.
demand now becomes common knowledge,
prices with low sensitivity to the
period greatly increase in the second period to reflect this
high values of Kj/Kg, this
inflation
is
is
where
As the jmst nominal
monetary shock
new information. Except
inflation reaches its peak.
last
for sufficiently
Lastly, although the decay rate of
constant after this date, because of the high inflation experienced in the second period,
lower Kxl ng leads to a higher level of inflation for
From Figure
5,
we then observe
all
subsequent periods.
that the impact effect of a
in the precision of private information.
Of
course, this
15
is
monetary shock on output
is
decreasing
simply the mirror image of what happens
to prices. Furthermore, like the impulse responses for inflation, real output
of
Kx/ng
It
is
higher for lower levels
subsequent periods.
interesting here to note that incomplete information has lasting effects on the levels of
is
inflation
This
for all
is
and
real
GDP
common knowledge
even though the shocks become
after just
one period.
precisely because of the interaction of incomplete information with price staggering
with strategic complementarity: by the time the shock becomes
common
and
knowledge, some firms
have already set their price on the basis of incomplete information about the shock; strategic
complementarity then guarantees that the firms that now have access to
find
it
optimal to respond to the shock as
Note
if
is
in contrast to the
we observe that the peak
A
still
of output occurs before
standard Calvo model which predicts strong price
increases during the period in which the shock
peaking before output.
information will
themselves had incomplete information.
that, except for high values of K^/Ke.
the peak in inflation. This
full
and therefore typically has
realized
is
similar observation has been
made
in
Woodford
important differences: Woodford (2003) abstracts from price staggering;
it
inflation
(2003), but with
also
two
assumes that past
shocks and past outcomes never become known, thus appearing to require an implausibly slow degree
of learning
about the underlying shocks.
Here,
we show how the empirically appealing property
that inflation peaks after real output can be obtained even with quite fast learning, provided one
interacts incomplete information with price staggering.
In the present model, the peak of inflation happens at
in 6.
This particular property
knowledge exactly one period
shock becomes
little
after the innovation takes place.
a.fter, sa.y,
The more
If
we extend the model
common
so that the
4 periods, then the peak in inflation can occur as late
general insight
information about the innovation and can
firms have accumulated
after the innovation
an artifact of the assumption that the shock becomes
common knowledge
as 4 periods after the shock.
have
is
most one period
is
that inflation can start low
rise in
if
firms initially
the early phases of learnmg, but once
enough information about the shock then
infiation will begin to
fall.
In
other words, the dynamics of learning are essential for the dynamics of inflation only as long as the
firms remain sufficiently uncertain about the shock; but once
tlie
firms have learned enoiigh about
the shock, the subsequent dynamics of inflation are determined primarily by the Calvo mec;hanics.
16
.
4
Uncertainty about precisions
The
analysis so far has focused on a Gaussian specification for the information structure that
quite standard in the pertinent Uterature.
Under
nominal shocks was determined by three parameters:
prices to
the degree of strategic complementarity; and
nominal shock. In
this section
we show
is
(i)
we show
that the response of
the degree of price rigidity;
(ii)
the precision of information about the underlying
(iii)
under a plausible variation of the information structure,
that,
knowledge of these parameters need not
insight
this specification,
is
suffice for calibrating the
The key
degree of price inertia.
down the response
that the precision of information about the underlying shock pins
of
the firms' forecasts of this shock, but not necessarily the response of their forecasts of the forecasts
of other firms
prices to the shock
its
own appeal
for the
in that
it
more general
for this
introduces
source of uncertainty:
a plausible
particular,
we introduce a second aggregate
we consider here
insight, the variant
it
allows firms to face
may have
uncertainty regarding the precision of information that other firms
demand. In
response of equilibrium
not only the former but also the latter.
is
Apart from serving as an example
has
and what matters
their higher-order beliefs);
(i.e.,
regarding nominal
state variable, which permits us to capture
uncertainty about the average precision of available information in the cross-section of the economy.
This new state variable
is
modeled
time and independent of the nominal shock
and
/
with probability 1/2. Let 7, k^,
<
and
K;
<
Kh-
The
ki
"type" of a firm
random
as a binary
variable,
and which takes each
df,
G {h,l}, which
St
of the
be scalars, commonly known to
is
now
given by the pair
all
(^i.t, Kj,*),
the particular signal the firm receives about the current nominal shock,
noise in this signal, and k^j
the
new
state variable
with probability 7 and
the precision of firm
i
precision.
when
as follows:
St
k,,(
is
is its
—
Kj,(
k.(
=
The
this state
with probability
ki
latter
1
-
is
—
St
with probability 7 and
«.,,(
signals have precision Ks
Note that a firm knows
a double
9t
role:
and a noisy
it
is
his
when
own
the state
«,(,
is s,
for s
€
—
is
~
=
Xj^t
and
k,/,
when
^t
< 7 <
+
A^(0, l/^i.t)
is
is
i
the state
with probability
1
is
Si,t
the
is
contingent on
the precision of firm
7; and, symmetrically,
because these realizations are independent across the firms, 7
whose
h.
with 1/2
where
specific to the firm
is
two possible values h
firms,
£,,f
over
is i.i.d.
1
-
k,,^
=
is S(
k^
=
/,
7. Finally,
also the fraction of the population
{/), /}.
but not the underlying state
Sf.
A
firm's
Kj^j
thus serves
both the precision of the firm's own information about the nominal shock
signal of the average precision in the cross-section of the economy.
the key difference from the baseline model
is
Therefore,
the property that firms face an additional source of
17
they face uncertainty regarding how informed other firms might be
informational heterogeneity:
about the nominal demand shock. Note then that the
heterogeneity:
when
when 7 =
instead 7 G (1/2,
1,
all
firms have the
same
coefficient
7 parameterizes the
and
precisions,
this fact
is
of this
common knowledge;
firms have different precisions, and each firm
1), different
level
is
uncertain about
the distribution of precisions in the rest of the population.
Furthermore, note that knowing the state variable
of the
nominal shock
only on
its
own
This
9t.
is
s>
would not help any firm improve
simply because belief of a firm about the nominal shock depends
precision (which the firm knows), not on the precisions of other firms (which the firm
does not know). Nevertheless, the firm would love to know
his forecast of the forecasts
expectation of
and actions
depends on both
Ot
of the firm also depe.nds on both
both on
9t
and on
Kf.
It
6t
and
his Xit
xu and
his
k.j^,
it
is
know the underlying innovation
own
a safe guess that the equilibrium choice
level
in &t>
depends
level,
but also because they don't
additional source of uncertainty that
better appreciate this point,
about
9.
know
Finally, note that, while the uncertainty
—
/
the uncertainty about
we have introduced
it
is
Sf.
Sf
matters only
when
This highlights the distinctive nature of the
0).
in this section.
useful to study the stochastic properties of the hierarchy
Next, for any
>
fc
of Eif[Ef''^]; that's the k"^ -order average beliefs.
of the current
0),
Let E^ denote the cross-sectional average of
state for the precisions being
2,
let
Ejt[^f]
conditional on the current
E^ denote the
cross-sectional average
average beliefs are functions
Clearl}-, all these
and past nominal shocks and the current precision state
s^.
Finally, let
E^ denote
the expectation of the fc"'-order average belief conditional on the nominal shocks alone (that
averaging across the two possible
st states).
E^ =
for
some constant
77^.
The constant
rji^
It is
r]k9t
+
{l-Tik)9t-u
= dEf/d9t
the price level to the underlying nominal shock
is
is,
easy to check that
thus identifies the sensitivity of the A'"'-order
average belief to the underlying nominal shock. As anticipated
The
not
matters for individual pricing behavior, and hence for aggregate prices, even when firms'
their pricing decisions are interdependent (a
of beliefs
Indeed, since an firm's
and therefore that the aggregate price
Kit
pricing decisions are strategically independent (a
To
because this could help him improve
of other firms in equilibrium.
precisely other firms are informed about this shock.
about
st
then follows that firms face imcertainty about the aggregate price
only because they do not
how
his forecast
in
the Introduction, the response of
determined by the
discussion in the Introduction had abstracted from price rigidities
18
sensitivities {rjk}-^
(i.e.,
it
had imposed A =
0),
but the
^
Figure 6
1
Figure 7
1)
_^
.
(18
;//^
(J,6
^
—
/
04
\
dtim
,ltlM
/
dtim
^
0.2
1
(tp/m
10
5
We
20
15
30
25
behavior of the hierarchy of beliefs
illustrate the
on the impact of the precision of information when
restrict k^
=
=
k-i
k,
this precision
is
In Figure
7.
common
identifies the
common
common
K,
-^
prior expectation, ^t-i, as the signals
for all
fc,
7?fc
—
1
>
as K -4 00. Finally,
Ot,
>
>
7/1
7/2
>
In Figure
information.
...
7,
>
we turn our focus
W'e then observe that
we
let
k^
rji
is
>
ki
with
it
fixed
pomt
is
??2
we can
more
and
773
independent of
depends not
affect the
as
perfect;
perfect, higher-
any k e
(0,cx)),
beliefs vary
with the coefficient
increase with 7.
That
is,
the sensitivity
7, while the sensitivities of higher-order
onl}'
on
first-order but also
on higher-order
response of the price level to the nominal shock even
does not affect the response of first-order
clearly
^
of information regarding the underlying precision state).
fact that the price level
though
^As mentioned
r]k
7.
we can expect that 7 should
is
and consider how the
invariant to 7, while
beliefs,
insight
for
We
to the impact of the uncertainty regarding the precision of others'
of first-order beliefs to the nominal shock
baseline model,
thus
converges
become
whenever the signals are informative but not
7 (which parameterizes the heterogeneity
Along with the
We
0.
In particular,
beliefs increase
9t
for all k,
as the signals
order beliefs are more anchored towards the prior than lower-order beliefs:
1
focus
precision of information).
become uninformative:
Second, the beliefs converge to the true underlying state,
0.
we
sensitivities of the
then observe the following qualitative properties. First, the hierarchy of behefs about
to the
6,
knowledge.
which case 7 becomes irrelevant) and consider how the
(in
shock vary with k (which now
beliefs to the
Figures 6 and
in
beliefs.
Indeed, following similar steps as in the
solve for the equilibrium as follows.^
general.
earlier, in
both our baseline model and
in all three variants of
of the best-response condition (3). For the current variant, this
nature of the information structure implies that this condition
However, the properties that
first-order beliefs
do not depend on
19
is
7,
is
it,
we
identify the equilibrium as the
with some abuse, since the non-Gaussian
not exact:
it
is
only a log-linear approximation.
while higher-order beliefs and hence equilibrium
Proposition
2.
(i)
There exists an equilibrium
{biPt-\ +
b2.hXi,t
+ bs^hdt-i
if
K'l.t
=
K,h
+
h.hXix
+
if K^^t
=
Ki
biPt-i
while the aggregate price level
ClPt-l
+ cs,kdt-i
if St
=
h
+ C2,h0t + C3M0t-l
if St
=
I
for some coefficients [bi.bo.h^h^hh.h^hd) "'^^
Let C2
=
^(c2,/i
+ C2,;)
b3j,,6t--[
C2
and
cs
depend on 7
This result
of 7.
We
is
is
=
cind C3
^(cs,/,
+ C3,;)
be the
mean
sensitivity of the price level to the
a
if
^
c\
'
0.
also illustrated in Figure 7,
which plots the
coeffirient C2
=
E[dpt/df)t] as a function
see that as 7 decreases, the sensitivity of the first-order beliefs to the current nominal shock
constant, while the sensitivity of the price level decreases.
staj's
The equilibrium value of
identical to that in the baseline model, while the equilibrium values of
and only
if
given by
(ci, C2,/i, C2,/, C3,/i, C3,;).
current and past nominal shock, averaging across the precision states.
does not depend on 7 and
is
given by
is
{cipt-i +C2^h0t
(ii)
which the pricing strategy of a firm
in.
Ss anticipated, this
because a
is
lower 7 decreases the sensitivity of second- and higher-order beliefs.
To
recap, the example of this section has highlighted how, even
of the firms forecasts to the underlying nominal shock, one could
how
for
if
one were to
still
fix
the sensitivity
have significant freedom
higher-order beliefs, and thereby equilibrium prices, respond to the shock. This
is
in
important
understanding the quantitative implications of incomplete information: to estimate the degree
of price inertia caused by incomplete information, one
may need
direct or indirect information,
not only about the firm's expectations about the underlying nominal shocks, but also about their
higher-order expectations.
Finally,
it
generate the possibility that
adjust their prices
when
informative); but
is
when
all
is
s
a variant of the model we have introduced here could
firms are perfectly informed about the nominal shock, are free to
and yet
fully,
the precision state
the signal
how
interesting to note
is
find
s
—
h
—
I,
some
all
it
optimal to adjust only partly.
firms get Ki
=
firms get k,
To
see this, suppose that
co (which means that their signals are perfectly
=
00 and others get k^
=
(which means that
completely uninformative). Under this scenario, when the precision state
outcomes do depend on
either the baseline
7,
do not hinge on
this approximation.
Finally, keep in
mind that condition
model or the two other variants that we consider subsequently.
20
is
(3)
s
is
=
h, all
exact in
However, this fact
firms are perfectly informed.
firm cannot
s
~
whether she
tell
is
is
not
knowledge,
perfectly informed because the state
is
s
=
/i
Tliis
is
because each
or because the state
is
but she was among the lucky ones to receive the perfectly precise signals. As a result, each
I
some other firms might not be informed and
firm must cissign positive probability to the event that
hence might not adjust their
find
common
it
optimal not to adjust fully to the shock.
perfectly
Of
But then because
prices.
mformed about the shock and
course, in this last
example the
It
complementarity every firm
of strategic
follows that there exist events
possibility that firms are perfectly informed
strictly less
events where some firms are relative uninformed and nevertheless find
shock because they expect that other firms
cannot generate situations where
other firms to be
less
all
firms are
nevertheless do not fully adjust their prices.
respond perfectly to the shock can occur only with probability
bit to the
where
will
will
it
than one: the
will also
be
optimal to respond quite a
be more informed. That
in all events firms are perfectly
and yet do not
this
is,
example
informed and nevertheless expect
informed. Indeed, based on the results of Kajii and Morris (1997) regarding
the robustness of complete-information equilibria to the introduction of incomplete information, one
can safely guess that
if
the
common
prior assigns probabihty near
1
where the
to the set of events
firms are nearly perfectly informed about the underlying nominal shock, then with probability near
1
the response of prices to the underlying nominal shock will be nearly the same as in the
knowledge benchmark.
Nevertheless, the results of this section do highhght
response of higher-order beliefs
5
is
common-
how quantifying the
essential for quantifying the response of prices to
nominal shocks.
Heterogeneous priors
In this section
we study how heterogeneous
priors regarding the signals firms receive can affect
the behavior of higher-order beliefs and thereby the response of prices to the underlymg nominal
shocks. In particular,
equilibrium as
if
we consider a system
of heterogeneous priors that induces firms to behave in
they lived in a world where other firms were
nominal shocks. This sustains a partially
underlying shock, even
if
of Section 3.
a firm
may change
We
its
equilibrium where firms react
they have nearly perfect information about
Apart from the mtroduction
model
self-fulfilling
informed about the underlying
less
again
of heterogeneous priors, the setup
let Of
follow an exogenous
price with probability
1
-
A, in
21
is
little
to the
it.
identical to our baseline
random walk
Calvo
process. In any given period,
which case the price
it
chooses
is
a weighted
average of
all
future target prices;
oo
p,,i
As
in the baseline
=
(1
- 8X)
^
{0Xy
[(1
- a)
+
E,,tO,+j
aE^,tPt+J]
model, each period firms learn perfectly the nominal demand of the previous
period, 9i-\, and receive a private signal of the current period's nominal
=
Xi,t
Ot
In particular, each firm believes that
believes the error in
its
own
its
private signal
own
is
demand:
+ ei,t.
However, firms disagree on the stochastic properties of the noise
i
(9)
signal
is
in their signals.
an unbiased signal of
drawn from the following
Of.
Specifically, firm
distribution:
e,,t~A^(0,l/K^).
At the same time, each firm believes that the private signals
firm
i
believes that the errors in the private signals of
all
of
all
other firms are biased. Specifically,
other firms are drawn independently from
the following distribution:
where
5i^t is
the bias that firm
i
believes to be present in the private signals of other firms. Finally,
assume that the perceived biases are negatively correlated with the innovation
(the nominal shock). Specifically, wc^
where x S
in the
[0, 1] is
assume
that, for all
i
and
in
we
the fundamental
all t,
a parameter that controls the correlation of the perceived bias with the innovation
nominal shock. Finally, these perceptions are commonly understood and mutually accepted:
^°
the firms have agreed to disagree.
To understand the
difference
between the baseline model (which had assumed a
and the current model (with allows
'"We have used a
for
heterogeneous priors),
related heterogeneous-priors specification in Angeletos
context and for different purposes.
role of heterogeneous priors:
document. For example,
in
We
refer the reader to that
paper
for a
it
is
either heterogeneous priors or a
we document how a
common
prior,
and La'O {2009b).
albeit within a different
more thorough discussion on the modeling
for
the type of
efl'ccts
we
certain type of sunspot-like fluctuations can obtain with
but the former maintain a higher
22
prior)
useful to consider the beliefs
they are convenient, but need not be strictly necessary
that paper
common
level of tractability.
of each firm about the average expectation of
expectation of the fundamental
In either model, each firm's
Of.
own
(first-order)
is
E^,tOt
'"'^
=
+
Xt
+
Kx
—^—
Kg
Kx
+
Kg
9t-l
In the baseline model, this implied that each firm believed that the average first-order expectation
in the rest of the popiilation satisfied
.^
Kj
That,
-
Kx
Kg
-\-
In contrast,
expectation
now
+
Kg
Kx
+
Kg
6t-\.
that firms have heterogeneous priors, each firm believes the average first-order
satisfies
^t
is,
Ei_,6'( H
;
Kx
—
the firm's second-order expectation
is
+
Kg
Therefore, the heterogeneous priors that
^
''(
;
Kx
beliefs,
Kg
the firm's second-order expectation was given by
is,
^i.tEi
That
+
now
^t-i'
;
given by
Kx
+
we have introduced
but they do affect second- and higher-order
beliefs:
Kg
in this section
the higher
\'
do not
is,
the
affect first-order
more each firm
believes that the beliefs of others will be less sensitive to innovations 0, even though
is
its
own
belief
not affected.
We now examine how
in
which the price
set
this affects equilibrium behavior.
by a firm
jt-t.t
for
some
Ft
( is
is
=
-P(P(-i>^i-i>2-,,()
Xpt-i
again an equilibrium
a linear function of (p;_i, 6'(_i,X(,
coefficients h\, 60,63. Accordingly, firm
Pt
where
=
in period
We conjecture once
+ (l-\)
=
i
J
6ip,_i
-K 62a-,,f
):
-h63^(-i
(10)
expects that the price level will satisfy
P(pt-uOt-i^x.)dFt{x)
the cross-sectional distribution of signals as perceived by the typical firm.
(11)
Our assump-
tion regarding the heterogeneous priors implies that each firm thinks that the cross-sectional
23
mean
of the signals in the rest of the population
is
—
(1
+
x)^t
It
X^t-i-
follows that each firm expects
the price level to satisfy
Pt
=
+ {l-X)P{pt-i,Ot-i.{l-x)Ot +
\pt-i
xOt-i)
or equivalently
Pt
=
+
CiPt-i
+
€20 1
0:^61-1
(12)
where
=A +
ci
But now
recall
(1-A)6i,
=
by the solution to
(8).
now
We
is
= (1-A)(63 +
coefficients (ci,C2,C3), the
given by the joint solution of (8) and (13).
(or, equivalently, as in
c\
and
for C2
C3 in (13),
(and hence also that of
61)
not the condition for
remains the same as
in
c\.
It
follows
our baseline
the standard Calvo model). Moreover, the price process continues to
be homogeneous of degree one, so that
ci
+
C2
+
C3
=
1.
C2
_
.
A(l-ci)
=
A
this last condition
now
Finally, the equilibrium value of ct
(..:_.,_,,
--
satisfies
(13)
62X).
conclude that the equilibrium values of the coefficients (61,62,63) and
that the equilibrium value of
Comparing
C3
x)
to set a price as in (10), with the coefficients (61,62.^3) defined
Note that x enters only the conditions
model
- X)h{l -
[l
from the baseline model that, no matter what are the
best response of a firm to (12)
(ci,C2,C3) are
C2
+
ci^^fi±^'
with the corresponding condition
for the baseline
model, we
observe that the equilibrium values of (ci,C2,C3) for the present model coincide with those of the
baseline
model
if
the precision of information in that model
^X
which
is
_
(I
~
\")
is
adjusted to the value k^ defined by
^I
,,
clearly decreasing in x- This observation in turn establishes a certain
isomorphism between
the present model and the baseline one: in the heterogeneous-prior economy that
we have introduced
same way
as in
for the fact
that
here, firms expect the price level to respond to the underlying nominal shock in the
a common-prior economy that
is
identical to the oaie in our baseline
the precision of available information
Along with the
to respond to
its
fact that,
is
model except
decreased from n^ to kx-
because of strategic complementarity (a >
own information
is
.s
0),
the incentive of a firm
lower the lower the expected response of the price level,
conclude that heterogeneous priors reduce the response of each firm to
the underlying nominal shock.
24
its
we
own information about
Proposition
In the equilibrium of the heterogeneous-priors economy, firms respond to their
3.
information in the same way as
that other firms have
to its
To
own
is a
m an a common-pnor economy m which the precision of information
decreasing function of x-
signal about the underlying
nominal shock
further appreciate this result, for the
and consider the
shock.
limit as k^ -^ 00,
moment
implication, the sensitivity 62 of a firm's price
is
also decreasing in x for given precision.
allow us to abstract from price rigidities (A
meaning that each firm
common-prior world,
In the
By
this
is
—
this limit,
if
»
would have guaranteed that prices move one-to-one with
now
let
each firm
it
optimal to adjust their prices
though they are perfectly informed about the shock and there
The preceding
analysis haii focused on
of prices to the underlying shock; in the
shock
is
how heterogeneous
is
no price
some p G
Proposition
!) One can the
[0,
4.
There
this point,
it is
i
=
is
much
even
alike the baseline
of prices after the
However,
model.
is
persistent
5t
=
-xvt
+ p5t,
extend the preceding analysis to show the following.
which depend on {\,p), such that the equi-
given by
P{pt-i,x,,t,0t-u5t-i)
^
6ip(-i
important to recognize that so
—we
effect,
follows an autoregressive process of the following form:
far
+b2Xu +
bjOt-i +b4St^i
we have used the model
only about what the firms expect the price level to do and
about this shock
than
St
exist coefficients (61,62,63,64),
librium strategy of firm
Pij
easily
less
priors affect the instantaneous response
model considered above, the dynamics
driven solely by the Calvo mechanics,
over time. To see this, suppose that bias
how they respond
to
make
predictions
to their information
have not used the model to make predictions about what we, as outside
observers or "econometricians", expect the price level to do.
make
along
rigidity.
heterogeneous priors can also affect these dynamics, to the extent that the perceived bias
At
1
perfectly informed about the shock but expects other firms to respond as
is
one-to-one in response to the nominal shock, thereby causing the shock to have a real
for
^
x
00 in such a way that the quantity k^ defined in (14) stays bounded away from oc. In
they were imperfectly informed; firms therefore find
initial
0)
about the
(nearly) perfectly informed
the nominal shock and hence that the nominal shock has no real effect. But
with Kx
=
This
is
where heterogeneous priors
things delicate. To analyze the equilibrium strategy of the firm as a function of their signals
(their "types"),
we do not need
to take a stand
on whether the firms signals are "truly" biased or
25
not;
we only need
predictions about
how
these beliefs will
response of the aggregate price
mapping from
and then we can use the model to make
to postulate the system of their beliefs
level to
beliefs to behavior;
we
map
In contrast, to analyze the impulse
into behavior.
the underlying shock,
know on what
also need to
nominal shock to the cross-sectional distribution
no more
it
is
the
suffices to characterize the
mapping from the underlying
of beliefs. In particular,
we must now take a stand
on how the cross-sectional average of signals relates to the underlying nominal shock.
Here,
we could assume
either that "econometrician" believes that the signals are biased, so that
the cross-sectional average of
cross-sectional average of
of
motion of the price
will
xu
+
x,( is 9i
is 9t. If
5t,
or that she believes that they are unbiased, so that the
we assume the former
scenario, then the econometrician's law
with the one in the minds of the firms.
level will coincide
In particular,
it
be given by
pt
with the function
P
= il-
X)pt-i
+
given by Proposition
4.
instead
If
econometrician's law of motion for the price level will
In particular,
it
will
be given by
ft
with the function
motions
P
=
-
X)pt-i
+
average of
we assume the
latter scenario,
from the one
::,.„,
,.,
mind
in the
,,.:
.
then the
of the firms.
„,;:;
;,,,
XPipt-uet,et-uSt-i),
given once again by Proposition
in that the cross-sectional
^t,0t-u5t-i),
differ
•
,-
(1
+
XP{pt-i,dt
Xtt is
4.
The only
assumed
to
be
(9(
difference in these
+ 6t
in
the
two law of
case
first
and
in
^
the latter case.
For the remainder of the analysis, we will assume the latter scenario, keeping though in
three properties.
First,
under
this scenario the
econometrician predicts that firms react to
because their signals are biased, but only because they believe that other firms
the two scenarios deliver similar qualitative properties as long as
tive difference
(9^
between the two scenarios vanishes as a
—
1.
>
q >
0.
And
will
do
so.
mind
Sf
not
Second,
third, the quantita-
Both of these properties are direct
implications of strategic complementarity.
Assuming the second scenario and combining the stochastic process
of
motion
for
the price
level,
we conclude that the dynamics
of 9t
and
r
'
e,-i
/J
6t
Pi
A/)2
+
A^3
Xb4
<5,-i
1
- A
+
A^i
P'-^
.
26
with the law
of the economy, as seen from the
perspective of the econometrician, are given by the following:
1
6t
"1
1
+
-\
C2
Vt
Figure
with the coefficients
8:
Figure
inflation
(61, 62! ^3
>
being determined as
^4)
system, along with the fact that real output
inflation
and output to
a positive
is
innovation in
j/(
=
-
6t
Vf (of
in
Proposition
pt- to
9:
output
We
4.
can then use this
simulate the impulse responses of
a size equal to the standard deviation of Vt).
Figures 8 and 9 illustrate these impulse responses for different values of \ and
(common-prior) model corresponds to x
keeping p
=
affects the
impact
=^
effect of the
in the period that the the innovation in
is
dampened by
But
as long as p
letting x
=
>
0,
—
p
The baseline
p.
anticipated,
we
see that letting
innovation but not
its
persistence.
0.
As
x >
but
In particular,
nominal demand materializes, the response of inflation
and by implication the positive
effect
on
real
output
is
amplified.
the dynamics following this initial period are determined solely by the Calvo
propagation mechanism and hence the persistence
is
the same as in the baseline model.
because any discrepancy between either first-order or higher-order
the shock vanishes after the
In contrast, letting p
initial period.
persist in higher order beliefs even after
it
has vanished
>
beliefs
This
is
and the true value of
permits the discrepancy to
in first-order beliids,
thereby contributing
to additional persistence in the real effects of the nominal shock.
To sum
changes
This
is
in
up, heterogeneous priors can help rationalize significant inertia the response of prices to
nommal demand simply
true no matter
how high
is
b}'
inducing inertia in the response of higher-order expectations.
the firms' precision of information, that
order beliefs to the nominal shock.
Moreover, this insight
is
is,
the sensitivity of
not specific to the contemporaneous
response of beliefs to the shock, but also to the entire dynamic adjustment of the
in
work that we do not report here because
in a heterogeneous-priors variant of
knowledge, thus allowing both
time.
But whereas
in
first-
beliefs,
beliefs
beliefs.
we have obtained
(2003), in which the shock never
and high-order
Woodford (2003) the
connected to that of first-order
of space limitations,
Woodford
first-
Indeed,
similar results
becomes common
converge to the truth only slowly over
rate of convergence of higher-order beliefs
is
tightly
heterogeneous priors permits us to break this connection, so
27
that higher-order behefs and prices
even
converge very slowly to their complete-information values
first-order beliefs converge very fast.
if
Finally, note that the present
precisions that
may
may
we considered
in the
model shares a
model with uncertain
bit of the flavor of the
previous section: there
we focused on
the possibility that firms
face uncertainty about the precision of other firms' information about the shock; here
showed how heterogeneous priors can induce firms
less precise
behave as
if
they expected other firms to have
In either case, the key
information than themselves.
behefs as opposed to first-order
to
we
the behavior of higher-order
is
beliefs.-'^
Heterogenous priors and cost-push shocks
6
In the preceding two sections
we highlighted how higher-order
beliefs
can induce inertia in the
response of prices to nominal shocks in the economy. In so doing, we focused on the role of higherorder beliefs for the propagation of certain structural shocks, namely nominal shocks. In this section,
building on Angeletos and La'O (2009b), we highlight
how
higher-order beliefs can themselves be
the source of a certain type of fluctuations in the price level and real output
resemble the ones generated by cost-push, or mark-up, shocks.
— fluctuations
that
.^
For this purpose, we modify the model of the previous section as follows.
have heterogeneous priors about their signals, but the perceived bias
is
Firms continue to
no more correlated with the
underlying nominal shock. Rather, the bias follows an independent stochastic process, given by
''.'
where
is
u>t
St
shows
similarity
that,
if
p5t-i +uJt
a Normally distributed shock that
Following similar steps as
"The
=
one
in
is i.i.d.
'
across time
(15)
'
'
^
and independent of
for all r.
0.^
the previous section, one can then show the following.
we obtain between our two models
is
fixes a specific hierarchy of beliefs,
priors from properties of any finite order of beliefs.
reminiscent of a point
one cannot
This
in
tell
made
apart a
in
Lipman
common
(2003).
That paper
prior from heterogeneous
turn suggests that in certain cases
it
may
be possible
to replicate, or approximate, the equilibrium behavior that obtains for any particular hierarchy of beliefs with either
a
common
prior or heterogeneous priors.
general equilibrium of our
At the same time, because the reduced-form game that characterizes the
economy admits a unique
rationalizable outcome,
and hence
also a unique correlated
equilibrium, the results of Kajii and Morris (1997) suggest that the complete-information equilibrium
the introduction of incomplete information.
to
make the
It tlien
follows that, witli a
common
prior,
predictions of the model under incomplete information sufficiently different.
28
one needs
is
robust to
sufficient "noise"
Figure
Figure 10 inflation
04
03
"'^
\
\
/i=0
'
'
2
10
--..0
/
*
\
output
1:
s
— —
/
02 [\
^
1
\V
'
—r
'
10
«
I
02
0.3
Proposition
i
is
5.
There exist coefficients (61,62,^3,^4,^5) such that the equilibrium strategy of firm
given by
Pi,t
=
+
biPt-i
b2X,j +b30t^i +b46t^i +b5UJt
Taking once again the perspective of an econometrician who iDeheves that the signals are unbiased,
we obtaui the unpulse responses
from the following dynamics:
of the price level
'
e,.:
1
dt-i
p
A62
Pt
In Figures 10 and 11,
innovation in
demand
we
+
A63
A64
illustrate the
1
- A +
1
+
A62
Pt-i
,\6i
Vt
+
t^'t
1
A65
impulse responses of inflation and real output to a positive
a shock causes firms to raise their prices even though aggregate nominal
Uf Such
As a
hasn't change.
result, inflation increases
and output contracts. The resulting fluctu-
ations thus resembled to those often identified as the impact of "cost-push" shocks.
this cost-push-like
shock
is
When
p
transitory; the moderately persistent effects on real output are then
merely to the Calvo mechanics.
When
instead p
>
0,
the bias
is itself
persistent,
=
0,
due
which contributes
to additional persistence in the real effects of the shock.
7
Conclusion
In this paper
justment
we studied how the combination
may dampen
of incomplete information
the response of prices to nominal shocks.
We
and infrequent price ad-
did so through a series of
variant models which progressively shifted focus from the stickiness of prices and the precision
We
thus
sought to highlight that quantifying the degree of price stickiness and the speed of learning
(i.e.,
of available information
about the shocks to the dynamics of higher-order expectations.
the rate at which first-order beliefs adjust to the shock) does not suffice for quantifying the rate
29
at
which higher-order behefs adjust, and therefore also does not
which prices adjust.
We
also illustrated
how
turn
We
may
sacrifice in analytical tractability
thus hope that more effort will be devoted to quantifying the behavior of expectations and
possibility, at least in principle,
economic agents.
We
and La'O (2009a),
in
find this
would be to
macro
How
level.
can
alike,
our eyes dispersed information
— and higher-order
may
beliefs
face
Angeletos
— are merely modeling
in
macro models
higher-order beliefs matter only to the extent that they impact forecasts of the
few macroeconomic variables, such as the price
employment. Therefore, we do not think that
beliefs.
in
about aggregate economic
equilibrium actions of other agents. Furthermore, in most macro models this
in forecasts of
One
be done?
both impractical and unnecessary. As we further argue
beyond the one that they face about the underlying fundamentals. Indeed,
and games
order
this
start conducting surveys of the higher-order beliefs of
devices for capturing the uncertainty that economic agents
or
—which
pave the way to bringing the ideas of this paper closer to the data.
their implications for the degree of price inertia at the
activity
rate at
the distinct role of higher-order beliefs could be readily
accommodated within the Calvo framework without any
in
suffice for quantifying the
Rather,
we
it is
level
and the
is
level of
essential to collect data
summarized
typically
aggregate output
on the details of higher-
believe that data on forecasts of economic activity can provide
guidance for quantifying the type of
effects
Turning to potential implications
one interprets the exogenous shock
for
in
we document
monetary
policy,
here.
more
direct
•
we wish
to
make
the following points.
our anal^'sis as an innovation in monetary policy, one
conclude that our results justify strong real effects
for
exogenous changes to monetary policy.
If
may
We
would not necessarily favor such an interpretation. The theory we have presented here does not
imply the same price inertia with respect to
that there
is
non-trivial lack of
all
shocks in the economy.
common knowledge about
note that information about changes in monetary policy
Rather,
in
it is
commonly understood
that this
the market regarding monetary policy, and financial
monetary
policy.
is
is
readily available
the case; there
pric(;s
In our eyes, these are indications that
significant lack of
crucial to
and
is
a
closely followed,
when
lot of
it
seems to
communication
adjust within seconds to changes in
assuming
common knowledge about
innovations to monetary policy might not be a terrible benchmark after
At the same time, we suspect a
is
the shock under consideration. But then
not only from participants in financial markets but also from the general public
matter. Moreover,
it
the
all.
common knowledge
for a variety of
other
"structural" shocks hitting the economy, such aggregate productivity shocks, financial shocks, labor-
30
market shocks, and so on.
We
then expect incomplete information to have more bite on monetary
policy in a different dimension: the interest-rate rule followed by the central bank can affect
much
the incompleteness of information about these shocks impacts equilibrium outcome. In par-
ticular, the central
designing
effects
how
it
bank can control the degree
of strategic complementarity in pricing decision by
response to realized inflation and output;
we have documented
in so doing,
here. Further exploring this possibility
it
can also mitigate the inertia
is left
for future
work.^^
^^The insight that the response of policy to macroeconomic outcomes can impact the decentraUzed use of information,
and thereby the response of the economy to both the fundamentals, and noise draws from Angeletos and Pavan
(2009).
See also Angeletos and La'O (2008) and Lorenzoni (2009) for some recent work on the design of optimal
monetary policy when information
is
dispersed.
31
Appendix
Proof of Proposition
The condition
1.
that determines
/(ci,A,/?)
=
cj
can be restated as
a,
(16)
where
_ (l-/3Aci)(ci-A)
^^'^'^'^)=ci(l-A)(l-/?A)As mentioned
Ci
G
(A, 1)
in the
main
two sohitions to
text, there are
this equation
—one
with
ci
>
1
and
—and we focus on the non-explosive one.
Clearly, that the aforementioned equation
independent of k^ and
is
Kg,
implying that
c\
is
independent of the information structure. Moreover,
dci
^
dX
df/dX ^
df/dci
dci
^
da
It
follows that ci
is
ci(l
X{1
1
—
ci
for C2
it
cj is
—
is
immediate that
non-monotone
in
is
.,,.--:
X[l-pc\)
o.
C3
it
is
for given c\
is
= A(l-ci)
A + ci^
'
'
'
,
immediate that
a can be
First,
(ci, C2,fi,C2,(, C3,/i, C3,i)
2.
,
main text that
/
C2 satisfies
_
C2 is increasing in
Kx/kB) and since cs
—
decreasing in KxJkq. Moreover, since the above expression
and
is
decreasing in
ci,
and since
is
c\ is itself
non-monotonic
in
increasing in q,
A and that C3
is
establish by numerical example.
The equilibrium can be
characterized in a similar fashion as in the
by aggregating the strategy of the firms, we
must
:
,
'
decreasing in q. Finally, the fact that C2
Proof of Proposition
baseline model.
- /3ci)(l - p\^)
- pX){l - Pcj)
increasing in both A and a. Next, recall from the
independent of
follows that C2
X){1
c2(l- A)(1-/?A)
df/dci
independent of [K^.Kg),
C2, it is
ci)(l
-
^
1
Co
Since
-
infer that the coefficients
solve the following system:
ci
= A+(l-
A)5i
C2,s
=
(1-A)[762,.
C3,,
=
(1
-
A) [763,,
32
.
.
+
(l-7)&2,-5]
+
(1
-
7)^3,-5]
.
= when
where we use the convention that -s
firms' best response,
we
=
s
/
=
h and —s
infer that the coefficients (&i,62,/ii^2,/.
h when
=
s
Next, by taking the
I.
solve the following system:
^3,/ii ^3,i)
""
61
=
((1-/?A)q
^^2,.
=
[
b3,s
=
[{l-pX){l-a) + {{l-3X)a+iPX)bi){^C2,s +
(1
-
+
/?A) (1
'
(/3A)6i)ci
- a) +
+(/3A) (i(62,,
+
Clearly, ci
again,
and
((1
+
- pX)a +
-
((1
+
62,,,)
+
flX)o^
(63,/
+
{pX)b,) (7C3,,
63,,))
+
+
(/3A)6i) (7C2,.
(1
{l~l)c2,-s)
7)c3,-s)
solution with ci S
Given
(0, 1).
Proof of Proposition
3.
In the
main
(62,5, 63,5. C2,5, C2^s)se{h.i}-
we showed that the equilibrium values
text
and (61,62,63) are determined by the solution
to (8)
and
(13);
Once
remainder of the conditions
this solution, the
consist a linear system, which admits a unique solution for the coefficients
in
7)c2,-.)
continue to be determined by the same equations as in the baseline model.
fej
we focus on the
determined as
-
-^
]
-
(1
that
ci
and
the baseline model and are thus independent of x; and that
of (ci, C2, C3)
continue to be
61
co satisfies
A(l-ci)
C2
A
which
is
+
ci
HO+\Ki
decreasing in \. Along with the fact that C2
=
(1
-
A)62(l
-
x)i
we
get that
A(l-ci)
(1-A)
which
is
also decreasing in x> since cj
Proof of Propositions 4 and
let
5.
G
To
+ ci^ + (ci-A)x
(A, 1).
nest both the
model
of Section 5
and that
of Section 6,
we
the bias be given by
St
We
A
then conjecture an equilibrium
in
=
-X'l'f
+^t + pSt-v
which the price
set
by
a
firm in period
t
is
b45t-i
+
a.
linear function
Oi {pt-^l,Xt,^t-lSt-l,uJt)^
P^,i
for
some
=
P{pt-i,Xi^t,dt-i,St^i,uJt)
=
bipt-i +b2Xij, +bsOt-i
coefficients 61,62,63,64,65.
33
+
bsojt
(17)
Our
mean
specification of the heterogeneous priors implies that each firm thinks that the cross-sectional
of the signals in the rest of the population
is
=
xt
+ 6t =
9t
-
{1
x)9t
+
Y^t-i
+
P^t-i
+ Wf
.
It
follows that each firm expects the price level to satisfy
Pt
=
Ap;-i
+
-
(1
X) P{pt-i,xt,6t-},5t-i,i^t)
or equivalently
Pt
=
ciPt-\
+
+ CiOt-i +
C29t
+ csuJt
c^St-i
where
ci
=
A
CA
=
{I
+ (1-A)6i,
-
we may
Next, note that
.-
p^,t
+
X) {b2P
C2
bi)
= (l-A)62(l-x),
,
C3
=
(1
-
A) (62\
+
63)
(19)
= (l-A)(62+65)
C5
'
write the firm
best response (9) as
i's
= il-PX)[{l-a)E,^tOt +
Given that the firm expects the price
aE,^tPt]
must be that the
it
"
i0X)E^,tm,t+i.
according to (18), for the firm's best response
level to evolve
to be consistent with our conjecture (17),
+
':
coefficients (61,62,63,64,65) solve the
following system:
61
=
[(l-/3A)Q
+
,..-•-'
(/?A)6i]ci
(20)
,
""^
62
=
[(1
-
+
ipX)
+
(pX) [bico
+
62
+
63
-
,y64)]
63
=
[(1
-PX){l-a + ac2) +
iPX)
+
(/?A) (61C2
+
60
+
63
-
,164)]
/JA) (1
- a+
ac2)
''''^
Kx
+ {l-6X)ac3 +
+
=
(1
- /?A)ac4 +
(/3A)6iC4
+
(/?A)64/5
65
=
(1-^A)qc5 +
(/?A)6iC5
+
(/3A)64
rium values
(19)
and
'
^
'
(20) gives us a
system of equations which characterizes the equilib-
for (61,62,63,64, 65, ci, C2, C3, C4,C5).
immediate that the conditions that determine
model. As
Kf)
ipX)biX
64
Combining conditions
It is
ipX)biC3
+
in
that has ci S
the baseline model,
(0, 1).
Given
we
cj
and
61
are identical to those in the baseline
ignore the solution that has ci
this solution, the
>
1
and focus on the solution
remaining conditions define a linear system, which has
a unique solution for the remaining coefficients.
34
.
References
[1]
Amato,
Jeffrey,
and Hyun Song Shin (2006), "Imperfect
tion Value of Prices,"
|2]
and the Informal
Economic Theory 27 213-241
,
Angeletos, George-Marios, and Jennifer La'O (2008), "Dispersed Information over the Business
Cycle: Optimal Fiscal and
[3]
Common Knowledge
Monetary
MIT
Policy,"
mimeo.
'
.
Angeletos, George-Marios, and Jennifer La'O (2009a), "Noisy Business Cycles," paper prepared
for the
2009
NBER
Macroeconomics Annual.
•
,
MIT
[4]
Angeletos, George-Marios, and Jennifer La'O (2009b), "Sentiments,"
|5]
Angeletos, George-Marios, Guido Lorenzoni, and Alessandro Pavan (2008), "Wall Street and
Silicon Valley: a Delicate Interaction,"
[6]
mimeo.
MIT/Northwestern University mimeo.
Angeletos, George-Marios, and Alessandro Pavan (2007), "Efficient Use of Information and
Social Value of Information," Econometrica 75:4, 1103-1142.
|7]
Angeletos, George-Marios, and Alessandro Pavan (2009), "Policy with Dispersed Information,"
Journal of the European Economic Association
[8]
Wincoop
Bacchetta, Philippe, and Eric van
7,
1-50
(2005),
"Can Information Heterogeneity Explain
the Exchange Rate Determination Puzzle?," American Economic Review 95.
|9]
Barro. Robert (1976), "Rational Expectations and the Role of Monetary Policy," Journal of
Monetary Economics
[10]
Bils,
Mark, and Pete Klenow (2004), "Some Evidence on the Importance of Sticky Prices,"
Journal of Political
|11]
Economy
12, 383-398.
Collard. Fabrice. and Harris Delias (2005), "Misperceived vs Unanticipated
sis,"
[13]
112, 947-985.
Calvo, Guillermo (1983), "Staggered Pric:es in a Utility Maximizing Framework," Journal of
Monetary Economics
|12]
2, 1-32.
CNRS-GREMAQ/University
of
Money:
A
Synthe-
Bern mimeo.
Hellwig, Christian (2005). "Heterogeneous Information and the Welfare Effects of Public Infor-
mation Disclosures,"
UCLA
mimeo.
35
[14]
Kajii, Atsushi,
and Stephen Morris (1997), "The Robustness of Equihbria to Incomplete Infor-
mation," Econometrica 65, 1283-1309.
|15|
Klenow, Pete, and Jonathan Willis (2007), "Sticky Information and Sticky Prices," Journal of
Monetary Economics
[16]
54, 79-99.
Lipman, Barton (2003), "Finite Order Implications
of
Common
Priors,"
Econometrica 71, 1255-
1267.
]17]
Lorenzoni, Guido (2008), "A Theory of
Demand
Shocks," American Economic Review, forth-
coming.
]18]
Lorenzoni, Guido (2009), "Optimal Monetary Policy with Uncertain Fundamentals and Dis-
persed Information," Review of Economic Studies, forthcoming.
]19]
Lucas, Robert E.,
Theory
]20]
[21]
Lucas, Robert E.,
of
Money," Journal of Economic
Jr.
(1975),
"An Equilibrium Model
of the Business Cycle," Journal of Political
Mackowiak, Bartosz, and Mirko Wiederholt (2008), "Optimal Sticky Prices under Rational
^mencan
jE^conomJc flewew, forthcoming.
'
-
Mackowiak, Bartosz, and Mirko Wiederholt (2009), "Business Cycle Dynamics under Rational
Inattention,"
[23[
and the Neutrality
83, 1113-1144.
Inattention,"
[22|
(1972), "Expectations
103-124.
4,
Economy
Jr.
European Central Bank/Nothwestern University mimeo.
Mankiw, N. Gregory, and Ricardo Reis
Proposal to Replace the
New Keynesian
(2002), "Sticky Information Versus Sticky Prices:
Phillips Curve," Quarterly Journal of
Economics
Morris, Stephen, and
ican
J25]
Hyun Song
Economic Review
Morris, Stephen, and
Shin (2002), "The Social Value of Public Information," Amer-
92, 1521-1534.
Hyun Song Shin
American Economic Review
[26]
117,
'
1295-1328.
[24]
A
',
,
(2006),
,
"The Inertia of Forward-Looking Expectations,"
96, 1521-1534.
Nimark, Kristoffer (2007), "Dynamic Higher Order Expectations," Reserve Bank of Australia
mimeo.
.
36
[27]
Nimark, Kristoffer (2008), "Dynamic Pricing and Imperfect
Monetary Economics
[28]
Phelps,
Edmund
55,
(1970), "Introduction:
New
Knowledge," Journal of
365D382.
The New Microeconomics
Theory," in A, A. Alchian, C,C. Holt et
Infl.aMon Theory,
Common
al.,
in
Employment and
Inflation
Microeconomic Foundations of Employment and
York: Norton.
Review of Economic Studies
129]
Reis, Ricardo (2006), "Inattentive Producers,"
[30]
Sims, Christopher (2003), "Implications of Rational Inattention," Journal of Monetary Eco-
nomics
[31]
50, 665-690.
Townsend, Robert M. (1983), "Forecasting the Forecasts of Others," Journal of
omy
[32]
73, 793-821.
Political
Econ-
91, 546-588.
Woodford, Michael (2003), "Imperfect
in P.
Aghion, R. Frydman,
Expectations in
J.
Common Knowledge
Stiglitz,
and M. Woodford,
Modern Macroeconomics:
In
Honor
of
and the Effects of Monetary Policy,"
eds..
Edmund
Knowledge, Information, and
S.
Phelps, Princeton University
Press.
[33]
Woodford, Michael (2008), "Information-Constrained State-Dependent Pricing," Columbia University
mimeo.
37