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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Noisy Business Cycles
George-Marios Angeletos
Jennifer La'O
Working Paper 09- 1
May 14, 2009
RoomE52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research
Network Paper Collection
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1
!
4302
at
Noisy Business Cycles*
George-Marios Angeletos
MIT
and
Jennifer La'O
NBER
May
MIT
14,
2009
Abstract
This paper investigates a real-business-cycle economy that features dispersed information
about the underlying aggregate productivity shocks, taste shocks, and
monopoly power.
We
show how the dispersion
inertia in the response of
of information can
macroeconomic outcomes
run response of employment to productivity shocks;
to such shocks;
(iii)
only a small fraction of high-frequency fluctuations;
business cycle; (v) formalize a certain type of
(vi)
generate cyclical variation
in
(i)
contribute to significant
(ii)
induce a negative short-
imply that productivity shocks explain
(iv)
contribute to significant noise in the
demand shocks
within an
Finally,
economy; and
about the underlying fundamentals: they
on the heterogeneity of information and the strength of trade linkages
the level of uncertainty.
RBC
observed Solow residuals and labor wedges. Importantly, none
of these properties requires significant uncertainty
rest
— potentially — shocks to
in
none of these properties are symptoms of
the economy, not
inefficiency:
apart
from undoing monopoly distortions or providing the agents with more information, no policy
intervention can improve
JEL
upon the equilibrium
allocations.
codes: C7, D6, D8,
Keywords:
Business cycles, fluctuations, heterogeneous information, informational frictions,
noise, strategic complementarity, higher-order beliefs.
'This paper was prepared
for
the 2009
NBER Macroeconomics Annual. We
to our discussants, Christian Hellwig and Robert King, and the organizers,
Michael Woodford.
Emmanuel
We
also received useful
Farhi, Mikhail Golosov,
are grateful for their detailed feedback
Daron Acemoglu, Kenneth Rogoff, and
comments from Philippe Bacchetta, Ricardo Caballero, V,V. Chari,
Guido Lorenzoni, Ellen McGrattan, Patrick Kehoe, Ricardo Lagos, Stephen
Morris, Alessandro Pavan, Ricardo Reis, Robert Shimer, Robert Townsend, and seminar participants at
University of Wisconsin at Madison, the Federal Reser-\'e
Bank, the 2009 conference
in
honor of Truman Bewley
Bank
at the University of Texas, Austin,
Macroeconomics Annual. Email: angeletOmit edu, jenlao9mit edu.
.
MIT, the
of Minneapolis, the Federal Reserve Board, the
.
and the 2009
World
NBER
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/noisybusinesscycOOange
Introduction
1
There
King (1982), and
(1976),
policy.
a long tradition in macroeconomics, going back to Phelps (1970), Lucas (1972, 1975), Barro
is
others, to use information frictions to break the neutrality of
This literature has recently been revived by
Mackowiak and Wiederholt
(2003a, 2008),
Mankiw and
and
(2008),
others.
formalizations of the origins of informational frictions, most of
theme
monetary
of breaking
but
frictions,
shift focus to
neutrality.
In this paper,
we
Reis (2002), Sims (2003),
While
it
this
Woodford
work has proposed new
has remained focused on the old
are also concerned with informational
we study how informational
a very different theme:
monetary
frictions
impact the
response of the economy to aggregate productivity shocks, and other real shocks, within the context
of a micro-founded real-business-cycle model.
This
shift is
motivated by two considerations.
some type
require significant lack of information, or
policy
is
debatable.
Indeed, the
first
the criticism that such information
is
First, the empirical relevance of theories that
of unawareness,
about the current monetary
generation of the aforementioned literature succumbed to
widely, readily,
and cheaply
available.^
that the dispersion of information about the real shocks hitting the
than the one about the conduct of monetary
policy.
economy
economic fundamentals matters
not only
how
this heterogeneity
for
more severe
far
is
far
factors such as the value of certain assets,
And
the health of the financial system, or the broader economic fundamentals.
how
is
In the ongoing crisis, for example, there
more uncertainty, and disagreement, about non-monetary
literature has little to say about
Second, we contend
yet, the pertinent
the heterogeneity of information about the real underlying
macroeconomic outcomes. FinaUy, we would
may impact
like to
understand
the positive properties of the business cycle but also
normative properties.
its
Motivated by these considerations,
RBC
canonical
first
the
paper introduces dispersed information
model, where nominal prices are
show that the dispersion
RBC
this
paradigm
— indeed
in
flexible
ways that might
that New-Keynesians have raised against the
'The new generation has attempted to escape
or to process
available,
it
may
still
plausible,
it
irrelevant.
We
impl}' that technology shocks explain only a small
RBC
this criticism
same time helping overcome
paradigm,
We
certain criticisms
next show that this significant
by postulating that, even
if
such information
is
readily
be hard to update one's information sufficiently frequently (Mankiw and Reis, 2002)
and absorb such information
While these ideas are
an otherwise
of information can significantly alter certain positive properties of
fraction of high-frequency business cycles, while at the
and cheaply
and monetary factors are
in
sufficiently well
(Woodford, 2003a, 2008; Mackowiak and Wiederholt, 2008).
seems hard either to gauge their quantitative importance or to reconcile them with
the fact that financial markets respond nearly instantaneously to any news about monetary policy, or that a variety
of economic agents appear to pay close attention to monetary policy.
change
in
the positive properties of the
RBC
paradigm happens without
affecting one
important
normative lesson: as long as there are no monopoly distortions, the equihbrium allocations coincide
with the solution to a certain planning problem, leaving no room
These
results should not
Our primary
paradigm.
be interpreted narrowly only as an attack against the New-Keynesian
goal
is
to provide a clean theoretical
models.
And
yet,
our framework
Our framework and
results
we
analysts ahke. In this regard,
enough
rich
is
may
benchmark
for the positive
and
Abstracting from nominal frictions best serves
normative implications of dispersed information.
this purpose.
for stabilization policies.
to nest the real
backbone
RBC
thus prove equally useful for
makes not only a
believe that our paper
New-Keynesian
of
and New-Keynesian
specific contribution into
business-cycle theory but also a broader methodological contribution.
Preview of model. The backbone
of our
from capital to simplify the analysis, but allow
model
for
is
a canonical
RBC
crucial role for aggregate fluctuations
whether each of the goods
is
Keynesian models,
facilitating
in a
it
introduces a
highlighted in Angeletos and La'O (2009b), play a
when, and only when, information
produced
combined with monopoly power,
first
abstract
a continuum of differentiated commodities. This
multi-good (or multi-sector) specification serves two purposes. First and foremost,
certain type of trade interactions that, as
We
economy.
is
dispersed; this
competitive or monopolistic fashion.
this specification
Second,
is
true
when
permits us to nest the real backbone of New-
a translation of our results to such models.^ Accordingly, while the
core of our analysis focuses on shocks to technology (TFP), in principle
we
also allow for
two other
types of shocks to the fundamentals of the economy: taste shocks (shocks to the disutility of labor),
and mark-up shocks (shocks
to the elasticity of
demand). However, none
of our results rests
on the
presence of either monopoly power or these additional shocks.
The
only friction featured in our model
is
that certain economic decisions have to be
made
under heterogeneous information about the aggregate shocks hitting the economy. The challenge
to incorporate this informational friction without an
or the tractability of the analysis.
Towards
undue
this goal,
sacrifice in either the
we formalize
is
micro-foundations
this friction
with a certain
geographical segmentation, following similar lines as Lucas (1972), Barro (1976), Townsend (1983),
and Angeletos and La'O (2008, 2009b). In particular, we assume that each period firms and workers
meet
in different "islands
"and have to make their employment and production decisions while facing
uncertainty about the shocks hitting other islands. At the same time, we assume that consumption
choices take place in a centralized market, where information
is
homogenous, and that households
are "big families", with fully diversified sources of income. This guarantees that our
"Indeed,
all
the results
we document
in this
economy admits
paper directly extend to a New-Keynesian variant as long as monetary
policy replicates flexible-price allocations, which in certain cases
is
the optimal thing to do (Angeletos and Lao, 2008),
a representative consumer and maintains high tractabihty in analysis despite the fact that
some key
economic decisions take place under heterogeneous information.
Positive results.
As mentioned, the core
on the special case where
of our analysis focuses
firms are competitive and the onlj' shocks hitting the fundamentals of the
(TFP) shocks which makes the
In standard
(i)
respond
fast
RBC
models
significant even
in
We show
Some
macroeconomic outcomes.
the agents face
if
little
that the dispersion of information induces
Perhaps paradoxically,
employment responds negatively
the data; have pointed out that that this fact
fact to
paradigm.
this inertia
is
Although
to productivity shocks
inconsistent with standard
argue in favor of New-Keynesian models
Kimball, 2006; Gali and Rabanal, 2004).
this fact
(e.g.,
RBC
(iii)
Many
In the
within the
it
RBC
RBC
remains debatable
(e.g.,
We
paradigm, technology shocks account
is
for the
What
of information can induce technology shocks to explain
its
is,
monetary
nature:
drives the residual variation
noise in available information, that
of information
bulk of short-run fluctuations.
only a small fraction of the high-frequency variation in the business cycle.
(iv)
Christiano,
empirically implausible and have favored New-Keynesian
show that the dispersion
business cycle remains neoclassical in
and
paradigm.
economists have argued that this
alternatives.
models; and
Gali, 1999; Basu, Fernald
Eichenbaum and Vigfusson, 2003; McGrattan, 2004), we show that the dispersion
can accommodate
can be
uncertainty about the underlying shocks.
researchers have argued that
have used this
RBC
Hansen, 1985; Prescott, 1986), macroeconomic outcomes
(e.g.,
and strongly to technology shocks.
inertia in the response of
(ii)
analysis directly comparable to the
economy are technology
m
And
yet, the entire
factors play no role whatsoever.
short-run fluctuations in our model
is
simply the
correlated errors in the agents' expectations of the underlying
technology shocks. Most interestingly, we show that the fraction of short-run volatility that
to such noise can be arbitrarily high even
is
due
the agents are nearly perfectly informed about the
if
underlying technology shocks.
(v) These noise-driven fluctuations help formalize a certain type of
RBC
in
setting.
The
associated errors in forecasting economic activity can be interpreted as variation
expectations of "aggregate demand".
while decreasing
"demand shocks "within an
its
They
correlation with output.
help increase the relative volatility of emplo\'ment
An
identification strategy as in
Blanchard and
Quah
(1989) or Gall (1999) would likely identify these shocks as ''demand "shocks.
(vi)
These noise-driven fluctuations involve countercyclical variation
and procyclical variation
in
in
Solow residuals, consistent with what observed
these cyclical variations can be significant even
the underlying technology shocks.
if
measured labor wedges,
in the data.
Once again,
the agents are nearly perfectly informed about
While we stop short of quantifying these
heterogeneity of information has a very different
certainty about fundamentals
we hope that they
results,
is
the
mark on macroeconomics outcomes than the un-
— a point that we further elaborate on
Indeed, what drives our results
how
at least highlight
in
Angeletos and La'O (2009b).
not per se the level of uncertainty about the underlying technology
or other shocks, but rather the lack of
common knowledge about
them: our
effects are consistent
with an arbitrarily small level of uncertainty about the underlying fundamentals.
At the same time, the
lack of
common knowledge
does not alone explain the magnitude of our
Rather, this depends crucially on the strength of trade linkages
effects.
our economy. This idea
is
linkages in our economy,
among
formahzed by our game-theoretic representation.
namely the
the firms and workers
A
measure
elasticity of substitution across different goods,
one to the degree of strategic complementarity
in the
game
of the trade
maps one-to-
that represents our economy.
One can
then extrapolate from earlier more abstract work on games of strategic complementarity (Morris
and Shin, 2002, Angeletos and Pavan, 2007a) that the strength
may
of trade linkages in our
economy
We
play a crucial role in determining the equilibrium effects of heterogeneous information.
conclude that our findings hinge on the combination of heterogeneous information with strong trade
linkages
We
— but they do not hinge on the
finally seek to
level of uncertainty
about the underlying fundamentals.
understand the normative content of the aforementioned findings.
that a planner could improve welfare by aggregating the information that
or otherwise providing the agents with
more information. But
is
We know
dispersed in the economy,
this provides
no guidence on whether
the government should stabilize the fluctuations that originate in noise, or otherwise interfere with
the
way the economy responds
to available information.
To address
this issue,
one has to ask
whether a planner can improve upon the equilibrium allocations without changing the inform.ation
structure.
We
show that the answer
to this question
is
essentially negative.
case of our model where firms are competitive, there
raise welfare
without changing the information that
is
is
In particular, in the special
indeed no way in which the planner can
available to the economy.
general case where firms have monopoly power, the best the planner can do
monopoly
We
distortion,
much
alike
what he
is
conclude that, insofar the information
is
supposed to do when information
is
As
for the
more
merely to undo the
is
commonly
shared.
taken as exogenous, the key normative lessons of the
pertinent business-cycle theory survive the introduction of dispersed information, no matter
how
severely the positive lessons might be affected.
Layout. The remainder
of the introduction discusses the related literature. Section 2 introduces
the model. Section 3 characterizes the general equilibrium. Sections 4 and 5 explore the implications
for business cycles. Section 6 studies efficiency. Section 7 concludes.
Related literature. The macroeconomics
tory, a revived present,
in
and
literature
— hopefully— a promising
on informational
future.'^
Among
frictions has a
this literature,
most
long
his-
influential
our thinking have been Morris and Shin (2002), Woodford (2003a), and Angeletos and Pavan
(2007, 2009). Morris
and Shin (2002) were the
first
to highlight the potential implications of
asym-
metric information, and higher-order beliefs, for settings that feature strategic complementarity.
Woodford (2003a) exploited the
inertia of higher-order beliefs to generate inertia in the response of
prices to nominal shocks in a stylized
model
of price setting. Finally, Angeletos
and Pavan (2007a,
2009) provided a methodology for studying the positive and normative properties of a more general
games with
class of
strategic complementarity
The framework we
use in this paper
is
and dispersed information.
a variant of the ones we use in two companion papers
Angeletos and La'O (2009b) we elaborate on the broader insight that the
for different purposes. In
heterogeneity of information introduces a distinct t>-pe of uncertainty about economic activity, and
show how
this
can sustain sunspot-like fluctuations
in a
unique-equilibrium economy. In Angeletos
and La'O (2008), on the other hand, we extend the analysis
of the present
paper by allowing
information to get aggregated through certain price and quantity indicators and by mtroducing
nominal
we then explore the implications
frictions;
for
optimal
Part of the combined contribution of these papers
is
to
fiscal
and monetary
policy.
show how the equilibrium and
effi-
cient allocations of fully micro-founded business-cycle economies can be represented as the Perfect
Bayesian equilibrium of a game similar to those considered
in
the more abstract setting of Morris
and Shin (2002) and Angeletos and Pavan (2007a, 2009). This representation
cilitates a translation of
of business cycles.
some
of the
more abstract
At the same time, the
insights of this earlier
models.
Indeed,
it
is
of the welfare implications conjectured
Finally,
it
is
of
is
it
fa-
work within the context
understanding
complementarity featured
only these micro-foundations that explain either
complementarity turns out to be irrelevant when information
none
useful, as
specific micro-foundations are crucial for
both the positive and the normative implications of the particular form
in business-cycle
is
commonly
shared, or
why
why
it
this
has
by Morris and Shin (2002).
worth noting how our approach
differs
from those of Beaudry and Portier (2004,
2006), Christiano, Motto, and Rostagno (2006), Gilchrist and Leahy (2002), Jaimovich and Rebelo
(2008),
and Lorenzoni (2008). These papers consider certain types
^See, e.g.,
Adam
(2007),
Amador and
2007a, 2007b, 2009), Bacchetta and
Weill (2007, 2008),
Wincoop
of expectations-driven, or noise-
Amato and Shin
(2005), Coibion and
(2006), Angeletos
and Pavan (2004,
Gorodnichenko (2008), Collard and Delias (2005),
Hellwig (2002, 2005), Hellwig and Veldkamp (2008), Hellwig and Venkateswara (2008), Klenow and Willis (2007),
Lorenzoni (2008, 2009), Luo (2008), Mackowiak and Wiederholt (2008, 2009),
and Shin (2002, 2006), Moscarini (2004), Nimark (2008), Reis (2006, 2008),
Mankiw and
Rodma
Reis (2002, 2006), Morris
(2008),
Sims (2003, 2006), Van
Nieuwerburgh and Veldkamp (2006), Veldkamp (2006), Veldkamp and Woolfers (2007), and Woodford (2003a, 2008).
driven, fluctuations.
However, these fluctuations originate merely form uncertainty about future
technological opportunities, obtain within representative-agent models, and do not rest on the het-
erogeneity of information. Interestingly, Kydland and Prescott (1982) had also allowed for certain
expectational shocks; but they, too, did not consider heterogeneous information.
numerous papers that consider geographical and trading structures similar
are
model
(e.g.,
Similarly, there
to the one in our
Lucas and Prescott, 1974; Rios-Rull and Prescott, 1992; Alvarez and Shimer, 2008),
but once again rule out heterogeneous information about the aggregate economic fundamentals.
To
recap,
it is
this particular type of informational heterogeneity that
is
the distinctive feature
companion papers. This
of our approach in either the present paper or the aforementioned
also
distinguishes our approach from the Mirrless Uterature, which allows for heterogeneous information
about idiosyncratic shocks but also rules out heterogeneous information about aggregate shocks.
The model
2
There
a
is
a (unit-measure) continuum of households, or "families", each consisting of a consumer and
continuum
of workers.
There
is
a continuum of "islands", which define the boundaries of local
labor markets as well as the "geography "of information: information
but asymmetric across islands. Each island
in the
by
symmetric within an
=
[0. 1];
firms and commodities by {i,j) & I x J: and periods by
Each period has two
stages.
In stage
1,
island,
inhabited by a continuum of firms, which specialize
production of differentiated commodities. Households are indexed hy h £
e /
2
is
is
i
£
{0,
H=
[0, 1];
islands
1, 2, ...}.
each household sends a worker to each of the islands.
Local labor markets then open, workers decide how
much
labor to supply, firms decide
labor to demand, and local wages adjust so as to clear the local labor market.
At
how much
this point,
workers and firms in each island have perfect information regarding local productivity, but imperfect
information regarding the productivities in other islands. After employment and production choices
are sunk, workers return
home and
the economy transits to stage
that was previously dispersed becomes publicly known, and
are
now pre-determined by
made during
stage
1,
At
2.
commodity markets open. Quantities
the exogenous productivities and the endogenous employment choices
but prices adjust so as to clear product markets.
Households. The
this point, all information
utility of
household h
is
given by
UiC^t)^
1 S,,tVinh,,t)di,
(=0
ith
U{C) =
^
1
—
and
7
V{n)
=
-—
1
6
-l-
e
Here, 7
>
parametrizes the income elasticity of labor supply,''
elasticity of labor supply, n^j.t
1
of period
Sfi,t is
i,
is
who
the labor of the worker
e
>
parameterizes the Frisch
gets located on island
an island-specific shock to the disutility of labor, and Ch,t
the commodities that the household purchases and consumes during stage
This composite, which also defines the numeraire used
CES
by the following nested
is
durmg stage
a composite of
all
2.
wages and commodity
for
i
prices,
structure:
is
given
•
p-i
=
Ch.t
Ui
^klt
d^
where
and where
j
on island
by any
the quantity household h consumes in period
c/„j,( is
Here,
i.
is
rj^t
islands. Letting the within-island elasticity
of trade linkages
by the
is
<
differ
77
is
commodity produced by firm
will
a case of special interest that
(77
—
»
the elasticity of substitution across different
from the across-islands
we
and mark-up shocks, thus
will
be determined
much
of our analysis
complementarity (which
will
concentrate on for
00) while the strategic complementarity remains
RBC
00); this case nests a canonical, competitive
letting the within-island elasticity to be finite
p permits us to
elasticity
be determined by the former) from the strength
of strategic
the limit where monopoly power vanishes
non-trivial (p
while p
i,
monopoly power (which
and the associated degree
latter). In fact,
of the
a random variable that determines the period-i elasticity of demand faced
indi\'idual firm within a given island
distinguish the degree of
t
and random permits us
At the same time,
economy.
to introduce
monopoly power
facilitating a translation/extension of our results to the
New-Keynesian
framework.
Households own equal shares of
is
firms in the economy.
all
The budget
constraint of household h
thus given by the following;
Pij,tChzj.td{j,k)
/
+
<
Bh,t+i
JlxJ
where
pij^t is
is
B/,_( is
objective of each household
is
and informational constraints faced by
""Note that risk aversion
is
wunht,tdk
no
the
amount
of
i,
is
RtBk,t,
in
period
is
the period-f
t.
utility subject to the
members. Here, one should think
role in
i, tt,jj
the period-f nominal gross rate of
bonds held
and intertemporal substitution play no
capital. Therefore,
Rt
firm j on island
simply to maximize expected
its
+
Jl
the period-f wage on island
return on the riskless bond, and
insurable and there
TTtjjd{iJ)+
commodity produced by
the period-i price of the
profit of that firm, Wit
The
/
Jjx!
of the
our setting because
all
budget
worker-members
idiosyncratic risk
is
7 only controls the sensitivity of labor supply to income for given wage.
of each family as solving a
different information sets
off
during stage
a function of
1 its
(i)
team problem: they share the same objective (family
when making
their labor-supply choices.
market.
In stage
The
is
the
2,
consumer-member
how much
to
consume
in
to supply labor as
(ii)
the
wage that
collects all the
income
each of the commodities
to save (or borrow) in the riskless bond.
Asset markets. Asset markets operate
information
how
the information that will be available to them at that stage and
that the worker-member has collected and decides
and how much
Formally, the household sends
workers to different islands with bidding instructions on
will prevail in their local labor
but have
utility)
commonly
sole role of the
in stage 2,
when
along with commodity markets,
all
shared. This .guarantees that asset prices do not convey any information.
bond market
in
the model
then to price the risk-free rate. Moreover, because
is
our economy admits a representative consumer, allowing households to trade risky assets in stage 2
would not
affect
any of the
Firms. The output
where
A^^t is
results.
'
!
'
'
of firm j on island
the productivity in island
z,
i
nijj
'
,
during period
is
t
The
maximize
valuation of
expectation of U'{Ct)'Kij^f
its
Labor and product markets.
operate in stage
markets are as
2.
its
is
(0, 1)
parameterizes
given by
•
Labor markets operate
in
stage
1,
while product markets
Because labor cannot move across islands, the clearing conditions
follows:
-
.
for labor
.
/ n.j^tdj
=
n.hi.tdh Vi
/
Jh
JJ
On
"
/
expectation of the representative consumer's
is
namely,
given by
firm's realized profit
Finally, the objective of the f^rm
its profit,
is
the firm's employment, and 9 G
the degree of diminishing returns in production.
to
-_.,
'
the other hand, because commodities are traded beyond the geographical boundaries of islands,
the clearing conditions for the product markets are as follows:
:
/
,
chij,tdh
=
q^j^t
V(i,j)
_
Jh
'
Fundamentals and information. Each
island in our
shocks: shocks to the technology used by local firms
(TFP
economy
causing variation in their monopoly power (mark-up shocks).
components
to these shocks.
subject to three types of
shocks); shocks to the disutility of labor
faced by local workers (taste shocks); and shocks to the elasticity of
idiosyncratic
is
We
demand
faced by local firms,
allow for both aggregate and
The aggregate fundamentals
of the shocks
(.4^;, 5i(,7/,,()
standard practice
period
t.
we consider
In contrast,
signals
about ^^
choices.
meet
m
On
in stage
1,
to
assume that
assume that
when they have
'I'f
m
is
commodity and
"if
and others
is
the beginning of
imperfect and, most
employment and production
becomes common known
Alternatively,
make any
in stage 2,
when agents
special assumptions
about the
For example, we can impose a Gaussian structure as
we could allow some
be imperfectly informed, mimicking the idea
to
'it
in
financial markets.
available to each island.
Morris and Shin (2002).
commonly known
is
The
denote this distribution.
4'(
their decentralized
For our main theoretical results we do not need to
information that
Let
different islands observe only noisy private (local)
make
to
we assume that
the other hand,
the centralized
is
by the joint distribution
are identified
t
situations where information about
We thus
importantly, heterogeneous.
in period
the cross-section of islands.^
in
macroeconomics
in
economy
of the
in
islands to be perfectly informed
Mankiw and
Reis (2002) that only
a fraction of the agents update their information sets in any given point of time. To some extent,
we could even
(2003) and
interpret the noise in these signals as the product of rational inattention, as in
Woodford
More
(2003a).
generally,
we do not expect
Sims
the details of the origins of noise
to be crucial for our positive results.
We
thus start by allowing a rather arbitrary information structure, as in the more abstract work
of Angeletos
and Pavan (2009).
This variable encodes
all
First,
we
let ujt
denote the "type "of an island during period
the information available to an island about the local shocks as well as
about the cross-sectional distribution of shocks and information
denote the distribution of W(
state of the
(signals). Finally,
we
distributions over
S^^,
We
in
'it
let
t;
note that the aggregate state
S^ denote the
set of possible types for
then uses Qt to make independent draws of
make
their current-period
In special cases (as with Assumption
mean
1
cvt
now
Ut
includes not only the cross-
each island,
Sq
the set of probability
€
S.^,
Sq
one
In the beginning of period
using the measure V{Qt\^t-\)-'
for
later on), this distribution
f,
and
Nature
each island. In the beginning of period
employment and production
choices, agents in
any given
might be conveniently parameterized by the
values of the shocks; but in general the aggregate fundamentals are identified by the entire distribution.
^To avoid getting distracted by purely technical
However, none of our results hinges on this
'
let
and P(|-) a probability measure over Sq.^
then formalize the information structure as follows.
before they
economy. Next, we
of the shocks but also the cross-sectional distributions of the information
conditional on Qt-i, Nature draws a distribution Qt G
t,
in the
the cross-section of islands. This variable identifies the aggregate
economy during period
sectional distribution
t.
issues,
if
we
let
Su,
and Sn as
if
they were
finite sets.
restriction.
Note that we have imposed that the aggregate state
notation, nothing changes
our proofs treat
fit
follows a
Markov
process; apart from complicating the
the aforementioned probability measure depend on
all
past aggregate states.
island get to see only their
own
cot',
informs them perfectly about their local shocks,
in general, this
but only imperfectly about the underlying aggregate state Hj. In the end of the period, however,
Qt becomes commonly known (ensuring that
To
'^t
also
model
recap, the key informational friction in our
the underlying aggregate state 0(.
immaterial
Whether they
type of effects we analyze
for the
that the agents of an island learn their
we denote with
as functions of w^:
own
becomes commonly known).
is
that agents face uncertainty about
about their own
face uncertainty
in this
local
paper. Merely for convenience, then,
local shocks in stage
We
1.
shocks
is
we assume
can thus express the shocks
A{uJi) the local productivity shock, with S(ujt) the local taste
shock, and with 77(^4) the local mark-up shock.
Equilibrium
3
In this section
we characterize the equilibrium by providing a game-theoretic representation that
turns out to be instrumental for our subsequent analysis.
Definition
3.1
Because each family sends workers to every island and receives
each family's income
is
fully diversified
during stage
2.
profits
from every firm
This guarantees that our model admits a
To simplify the
representative consumer and that no trading takes place in the financial market.
we thus
exposition,
of the
any
symmetry
set
Bt
—
and abstract from the
of preferences, technologies
loss of generality to
impose symmetry
Finally, because of the absence of capital
summarizes
all
and information within each
in the choices of
and the Markov
aggregate outcomes, do depend on
An
1.
strategy q
S^ x Sq
:
-^ M.+
,
fit.
the other hand, the
We
-^
R+,
a
wage function
c
:
R^ —
»
M.+
,
is
without
t.
state,
^(-i
Recall then
only Ut. As a result,
depend on 0(_i and
all
prices in stage 2,
Wf,
and
thus define an equilibrium as follows.
w
an aggregate employment function
and a consumption strategy
1 is
commodity
equilibrium consists of an employment strategy
Definition
Q Sq
On
it
on the aggregate
restriction
the local levels of labor supply, labor demand, wage, and output can
:
island,
workers and firms within each island.
that the additional information that becomes available to an island in stage
all
Furthermore, because
financial market.
the payoff-relevant public information as of the beginning of period
but not the current aggregate state 0(.
the economy,
in
:
N
S^ x Sq -^
:
Sq
-^
n
IR+,
:
S^.
x
a production
an aggregate output function
R+. a price function p
such that the following are true:
10
Sq —* R+
:
S^,
x
Sq
^ R+,
The price function
(i)
normalized so that
is
=
P{Qt,^t-i)
for
=
fp{u,nunt-i?-''dnt{uj)
1
all (fit: ^(.-i)(ii)
The quantity c{p,p',Q)
whose price
is
p when the price of
output (mcom.e)
(Hi)
When
clears the
of that firm are.
levels
:
=
and
/
I
The quantities
firm in island
u>t
,
Qt
wage
is
n{uJt.D,t-i)
'i'S
the aggregate
p{u}t,V.t,flt-\): the
o-nd q{ijJt.,flt--\),
and employment indices
r
'
and
employment
with q{ujt.^t-i)
m
—
are, respectively,
N{nt,^t-\)
=
n(w, f](_i)(iQ,(w).
/
and g(w(,fi(_i) are optimal from the perspective of
taking into account that firms
the typical
other islands are behaving according to the
same
given by w{ujt- Q,t-i), that prices will be determined in stage 2 so as
and that aggregate income
The
local
wage w{u>t,^t-i)
Note that condition
(i)
m
I'S
will be given by Q{D,t^^t-i)-
such that the quantity n{uJt,^t-i)
for the
The
preferences.
for
our economy
rest of the conditions
among
different islands in stage
Let us expand on what we
return to labor.
is
the
mean by
CES
composite
then represent a hybrid of a
for the
in
stage
2,
once
incomplete-hiformation
1.
this.
When
employ and how much to produce during stage
will sell their
also the optimal labor
complete-information exchange economy that obtains
production choices are fixed, and a subgame-perfect equilibrium
plaj'ed
^s
an island of type iOf
simply means that the numeraire
when we introduced
Walrasian equihbrium
they
and
the past aggregate state is fi(-i, the price that
o.'^^d
respectively.
g(w,f2(_i)^dr2,(cj)
supply of the typical worker
game
is p'
product markets, that the representative consumer will behave according to consumption
to clear all
defined
is
the aggregate output
n(u)t,flt~i)
strategies, that the local
(v)
other commodities from the same island
the current aggregate state
Q(Qi,f](_i)
c,
all
demand for any commodity
Q.
is
A{ii>t)n{iOt,flt-i)^
strategy
the representative consumer's optimal
market for the product of the typical firm from islanduJt
and output
(iv)
is
1,
firms in an island decide
how much labor
to
they face uncertainty about the prices at which
product during stage 2 and hence they face uncertainty about the marginal
Similarly,
uncertainty about the real
when workers
mcome
their
in
an island decide how
household
will
have
much
in stage 2
labor to supply, they face
and hence face uncertainty
about the marginal value of the wealth that they can generate by working more. But then note that
firms and workers in each island can anticipate that the prices that clear the
and the realized
level of real
income
are, in
commodity markets
equilibrmm, determined by the level of employment
11
and production
in other islands.
economy by reducing
depend on
This suggests that we can solve
game, where the incentives of firms and workers
to a certain
it
for the general equilibrium of the
and workers
their expectations of the choices of firms
We
other islands.
in
an island
in
implement
this solution strategy in the following.
Remark. To simplify notation, we often use qu as a short-cut
a short-cut for
for Q{Q.t,Qt-i), Ejt as
and so on;
E[-|a;(, f](_i],
Qt as a short-cut
for q{uit,Q.t-i)^
we drop the
also,
indices h
and
j,
because we know that allocations are identical across households, or across firms within an island.
Characterization
3.2
Towards solving
demand
for the equilibrium, consider first
of the representative
price of other
where Pt
—
I
commodities
in
consumer
the same island
is
given by c{p,p',Q)
function faced by a firm during period
Consider now stage
is
a commodity from island
is pj(
is
t
=
U'{Ct) and that Ct
=
(1),
we conclude the
=
the above objective
firm's
problem
sufficient.
^^
^itn^f
is
is
by a particular
follows that the equilibrium
It
of
nominal income
for the representative
equihbrium, the objective of the firm
-
is
simply
Witritt)]
'
'
maximizes the following objective:
1
>
(1
—
^)6'
>
(2)
(which we assume to be always the case),
unique and that the corresponding first-order condition
is
is
both necessary and
simply given by equating the expected marginal cost and revenue of
this condition, note that
island; the
,,
U'{Qt)[{p'J~'^'Qt"'<l,r-' -^rtnu
^^'^S as
the
a strictly concave function of n,, which guarantees that the solution to the
This condition
To understand
Qt-
.
typical firm on island ut
E,t
where qu
=
,;,
El, [U' {Qt) {pitqu
Using
when
is p,t
',
is
in
Qt
whose price
i
The optimal
2.
p~^ {p'Y'~^ Q. Equivalently, the inverse demand
Given that the marginal value
1.
stage
in
given by the following:
by our choice of numeraire.^ In equilibrium. Ct
consumption strategy
household
for
how the economy behaves
demand
for the
cj,
=
I
-^
commodity
)
Ci
is
the
demand
for the
busket of commodities produced
of a particular firm in that islands
12
is
then
c,i
=
(
^
)
c',,.
labor, evaluated under local expectation of the equilibrium pricing kernel:
{^^^]Eu
Eu[U'{Q^t)]wu=
Next, note that, since
Pit
—
Pit-
Along with
all
'9A,K-').
firms within an island set the
^.
(1), this gives
same
price in equilibrium,
it
(3)
must be that
^.r—
This simply states that the equilibrium price of the typical commodity of an island relative to the
numeraire
is
equal to the
MRS
between that commodity and the numeraire. Fmally, note that the
optimal labor supply of the typical worker on island
MRS
between the numeraire and
i
is
given by equating the local wage with the
leisure:
^^^
""''-E^tlU'iQt)]
Conditions
cation.
and
(4)
(5) give
the equilibrium prices and wages as functions of the equilibrium allo-
Using these conditions into condition
we conclude that the equilibrium
(3),
allocation
is
pinned down by the following condition:
..,„,.
(-_i).„U,„
This condition has a simple interpretation:
island.
To
U' {Qt) [q^]
i;
as for the right-hand side,
(6)
.
simply the marginal disutility of an extra unit of
is
'^"~'
is
the reciprocal of the local monopolistic mark-up,
I.
''is
the marginal utility of an extra unit of the typical local commodity, and OAitrt^^
the corresponding marginal product of labor.
Note that condition
Qit
OAun^-^
equates the private cost and benefit of effort in each
see this, note that the left-hand side
labor in island
_
is
it
(I)
m
(6)
expresses the equilibrium levels of local employment n,( and local output
relation to the local shocks
production function,
q,t
=
and so
on),
local expectations of aggregate
Aun^f, to eliminate
precise notation of Definition
A{ijJt),
and the
1
we reach the
(i.e.,
replacing
n,f
q^t
in this condition,
output Qt-
and reverting
l-e + e + TB
13
to the
more
with q(iJt,^t-i), Qt with Q(f]t,ilt_i), Ait with
following result.
1\
Using the
9
1+e
be a composite of all the local shocks hitting
an island of type
u>
and define
the coefficient
'--1
p+
The
e
and aggregate output are
equilibrium, levels of local
the solution to the following fixed-point
problem:
log q {ut, nt-i)
=
(1
-
a) fi'^t)
Q[Qt,^t-i)
+a
=
log
{
E
Qi^t, f^t-i)^
q{uj,Qt-i)
V(tuf,n,_i)
(7)
V(Q,,a_i).
dnt{uj)
''
^-M
fit-l
iOt,
(8)
This result establishes that the general equilibrium of our economy reduces to a simple fixed-point
relation
between local and aggregate output. In so doing,
it
of our economy, similar to the one established in Angeletos
with capital. To see
in IR+.
Identify'
this,
consider a
game with a
game with an
a "player "in this
output of that island as the "action
"of the
offers
and La'O (2009b)
number
large
a game-theoretic representation
for
a variant
economy
of players, each choosing an action
economy and
island in our
interpret the level of
corresponding player. Next, identify the "types "of these
players with ut, which encodes the local shocks and local information sets in our economy. Finally,
let their
"best responses "be given by condition (7).
equilibrium of this
game
It is
then evident that the Perfect Bayesian
identifies the general equilibrium of our
Note then that the variable /(wf) conveniently summarizes
all
economy.
the local economic fundamentals,
To
while the coefficient a identifies the degree of strategic complementarity in our economy.
more
this
:•
clearly, consider a log-linear
log
,
g^f
=
approximation to conditions
const
logQf
-K (1
=
-
const
+
a)/j,
+
/
E,, [log Q(]
(8):
:':
-^
;
,
'
logqifd?',
where const capture second- and higher-order terms. ^
identifies the slope of
Q'
and
(7)
It
is
see
(9)
(10)
then e\'ident that the coefficient a
an island's best response to the activity of other islands
—which
is
the standard
definition of the degree of strategic complementarity.
Finally, note that Proposition
While much
ture
(e.g.
1
holds no matter the information structure. This
of the recent literature has focused
Mankiw and
on
Reis, 2002; Sims, 2003;
^In general, these second- and higher-order terms
is
important.
specific formalizations of the information struc-
Woodford, 2003a), our
may depend
on the underlying
result
.state
mdicates that the
and the above
is
only an
approximation. However, when the underlying shocks and signals are jointly log-normal with fixed second moments
(as
imposed by Assumption
1
in the
next section), these terms are invariant, the approximation error vanishes, and
conditions (9) and (10) are exact.
14
down
information structure typically matters only by pinning
We
activity.
would thus
the agents' forecasts of economic
pay more attention on the theoretical and
invite future researchers to
empirical properties of these forecasts as opposed to the details of the information structure.
and strategic complementarity
Ti-ade links
3.3
As evident from Proposition
the degree of complementarity, a,
1,
elasticity of substitution across the
commodities
is
a monotone function of the
what
In
of different islands, p.
follows,
the convention that variation in q represents variation in p for given other parameters.
a
interpret
as a
measure of the strength
motivated by the following observations.
household
lives
Proposition
—
First,
if
we
but also on
e,-),
and
also
These choices are
consider a variant of our model where each
one island and consumes only the products of that island, then
0;
in this sense,
it
is
islands that introduces strategic interdependence (q
p,
We
in
and works only
holds with a
1
of trade linkages in our economy.
we adopt
precisely the trade linkages across different
^
Second, while
0).
q-
depends, not only on
these other parameters affect the composite shock / and matter for
9.
equilibrium allocations whether islands (agents) are linked or not; in contrast, p affects only q. For
these reasons,
we henceforth
use the notions of strategic complementarity, elasticity of substitution
across islands, and strength of trade linkages, as
synonymous
to
one another. However, we also note
that strong complementarity in our model does not strictly require low
labor supply
(9
—
>
1),
The
is
small (7
^0),
the Frisch elasticity
then the degree of complementarity
is
is
high
high (q
—
>
(e
^
1)
no matter what p
0),
the wealth effect of
and production
is
nearly linear
is.
insight that trade introduces a form of strategic complementarity even in neoclassical,
perfectly-competively settings
is
extend well beyond the boundaries of the model we have
likely to
considered here or the variant in Angeletos and La'O (2009b).
under-appreciated
m
have thought us that
as games.
And
prior
it
work on business
rarely helps,
and
it
when information
is
To understand what we mean by the
symmetric aggregate shock
(i.e,,
We
believe that this insight has been
cycles for two reasons. First, the
commonly
last point,
we
highlight here
is
simply irrelevant
shared.
consider the response of the
economy
to a
a shock that keeps the level of heterogeneity invariant). Formally,
denote the cross-sectional average of the composite fundamental
that varies the average fundamental,
idiosyncratic
two welfare theorems
can often be misleading, to think of Walrasian settings
second, the type of strategic complementarity
for the business cycle
let ft
if
p:
/(,
/,t
and consider any shock
without varying the cross-sectional distribution of the
components of the fundamentals,
^,(
15
=
J^t
—
ft-
When
all
information
is
commonly
shared, aggregate output
also
is
commonh' known
^
logq.t
It is
{I
-
equilibrium. Condition (7) then reduces to
in
+
a){ft
^it)
+ alogQt.
(11)
then immediate that the entire cross-sectional distribution of logq-u moves one-to-one with
ft,
which establishes the following.
Proposition
2.
Then
invariant.
Suppose that information
is
commonly shared and
the equilibrium levels of aggregate output
log Qt
Recall that, by
its
definition, the
=
const
+
is
that the level of heterogeneity
is
given by
ft.
composite shock depends on
e
and 7 but not on
p.
It is
then
evident that the response of the economy to the underlying aggregate productivity, taste, or mark-
up shocks
of strategic
The
is
independent of
is
p.
In this sense, the business cycle
complementarity that
is
intuition behind this result
is
indeed independent of the degree
induced by trade.
is
further explained in Angeletos
that the strength of trade linkages matters only for
how much
and La'O (2009b). The key
agents care about forecasting
the level of economic activity relatively to forecasting the underlying economic fundamentals. But
when information
of
is
symmetric (commonly shared), any uncertainty the agents face about the
economic activity reduces
to the
one that they face about the underlying economic fundamentals,
which renders the degree of strategic complementarity
irrelevant. In contrast,
asymmetric (dispersed), agents can face additional uncertainty about the
The
bej'ond the one they face about the fundamentals.
precisely the impact on equilibrium
This
distinct
is
important.
It is
level
outcome
precisely the aforementioned property that
—
for
it
level of
economic
is
activity,
strength of trade linkages then dictates
of this residual uncertainty
from uncertainty about the fundamentals
when information
is
about economic
activity.
makes dispersed information
only the heterogeneity of information
that breaks the coincidence of forecasts of economic activity with the forecasts of the underlying
fundamentals when the equilibrium
is
unique.
We
further elaborate on this point in Angeletos
La'O (2009b), showing how dispersed information can open the door
fluctuations.
We
refer the reader to that
paper
for a
and
to a certain type of sunspot-like
more thorough discussion
of this important,
broader insight. In what follows, we concentrate on how this broader insight helps understand
why
the combination of dispersed information with the aforementioned type of complementarity can
have a significant impact on the positive properties of the
16
RBC
paradigm.
3.4
.
The
Relation to complementarity in New-Keynesian models
(e.g.,
Woodford, 2003b) looks
like
the following:
^
Pr,t
where
2,.(
pi^t is
{I
- oyt +
the target price of a firm (in logs), yt
with condition
(9) in
complementarity
is
+
z,,t,
shocks, and ^
is
striking.
is
pt
is
the aggregate price level,
a coefficient that
we compare
If
The only
interpreted
is
the above condition
noticeable diiTerence seems
New-Keynesian model, while
a price in the
(12)
.,.
in pricing decisions.
our model, the resemblance
to be that the rele\'ant choice variable
ipt
nominal GDP,
is
demand
captures idios\-ncratic productivity or
as the degree of strategic
in
New-Keynesian paradigm
familiar condition that characterizes optimal target prices in the
it is
a quantity
our model. However, there are some crucial differences behind this resemblance.
First, condition (12)
does not alone pin
down
other conditions regarding the determination of
(9) offers a
the equilibrmm. Rather,
3^^,
GDP
the nominal
it
must be combined with
In contrast, condition
level.
complete, self-contained, representation of the equilibrium in our model.
Second, the endogeneity of yt undermines the meaning of condition (12). For example, letting
denote real
GDP
and using yt
= pt+yt,
condition (12) can also be restated as p^j
but then the degree of complementarity appears to be
is
more informative when money
is
neutral, because yt
condition determines only relative prices. But even
degree of complementarity
at the very least
is
1,
in pricing decisions
not
is
In contrast, in our
when money
is
non-neutral, ^
simply because nominal
is
model the degree
only by preferences and technologies, and
is
+ (l— Oyf+^i,*!
In fact, this alternative representation
because monetary policy responds to variation
to policy parameters.
P(
then exogenous to nominal factors and this
incorporated, the complementarity in pricing decisions
down
^.
—
yt
in pt
different
GDP
and
is
yt-
fails to
far
identify the
from exogenous
Once
this
endogeneity
from ^ and becomes sensitive
of strategic complementarity
is
pinned
completely invariant to monetary policy.
Third, the comparative statics of the complementarity in our model (q) with respect to deeper
preference and technology parameters are different from those of
In particular, note that
decreases with
e
a
the labor and production side, as
what contributes
in
to strong
is
to
do with
is
important to bear
either the degree of
complementarity
crucial, along
in
mind that our notion
monopoly power
S,
of
in the
in
our model
is
with high substitutability
Hansen (1986) and King and Rebelo
discussants highlighted, the opposite comparative statics hold for
it
(^).
(the inverse of the Firsch elasticity of labor supply), and increases with 6 (the degree
low substitutability in the commodity side, so that trade
Nevertheless,
New-Keynesian counterpart
decreases with p (the elasticity of substitution across different goods),
of diminishing returns to labor). Hence,
in
its
(2000).
As one
of our
New-Keynesian paradigm.
complementarity may have
little
or the price elasticities of individual demands. In
17
our model, that latter are pinned down by
degree of strategic complementarity
is
(the within- island elasticity of substitution), while the
77
pined down by p (the across-island elasticity).
Last, but not least, the complementarity highlighted in the
vanish
in
is
New-Keyenesian framework would
New-Keyenesian complementarity
firms were setting real (indexed) prices. In this sense, the
if
a nominal phenomenon, whereas ours
a real phenomenon.
is
Dispersed information and the business cycle
4
we seek
In this section
to illustrate
positive properties of the
RBC
how
the introduction of dispersed information can impact the
paradigm. To facilitate this task, we impose a Gaussian specification
on the shocks and the information structure, similar to the one
Pavan
(2003a), Angeletos and
Assumption
1.
The shocks and
The aggregate shock
(i)
(2007),
ft
and many
and variance al
The
(ii)
follows a Gaussian
=
local
shock
=
I/k/,
ft is
i.i.d.
ft,
and
i.i.d.
sian sufficient statistic
where Qt
is
£,it:
all
^
ut is
and
—
fit
+
<^tt
r
.
1
Normally distributed with m.ean zero and variance a^,
x^ such
ft is
summarized
m
a Gaus-
that
=
ft
+ ^it,
,
.
Normally distributed with mean zero and variance
across islands.^^
i.i.d.
Normal innovation, with
a
across islands.
The public information about
statistic yt
where
noise,
process:
i^t,
-
^It
(iv)
+
random walk
The private information of an island about the aggregate shock
(ill)
and
ipft-i
or
given by
a purely idiosyncratic shock,
is
^-u
orthogonal to
ft
AR(1)
over time.
ft
where
others.
parameterizes the persistence of the composite shock and
ip
mean
Morris and Shin (2002), Woodford
the available information satisfy the following properties:
ft
where
in
;
cj^
=
I/k^,,
orthogonal to both
'
•
,
the aggregate shock ft
is
summarized
in a
-
„
Gaussian sufficient
such that
--.'•
St is noise,
yt^
ft
+
^t,
Normally distributed with mean zero and variance a^
=
l/^y, and orthogonal to
other variables.
Note that the
local
fundamental
/,t
is
itself
a private signal of
/(.
However, by the
sufficient statistic of all the local private information, the informational content of /,(
18'
is
fact that
we
define x,, as
already included in Xn.
a
This specification imposes a certain correlation
up shocks:
composite shock fu to follow a univariate process as above,
for the
three type of shocks are
We
simplicity.
underlying productivity, taste and mark-
in the
moved
bj'
a single underlying factor. However, this
it
is
must be that
all
the
only for expositional
can easily extend our results to a situation where each of the shocks follows an
independent Gaussian process, or consider a more general correlation structure among the shocks.
Closed-form solution
4.1
Under Assumption
1,
we can
identify W( with the vector {ft,xt.yt)-
normal distribution with mean
can guess and verify that there
and \ogQt
familiar
is
linear in (ft-i, ft,yt)-
method
of
undetermined
out any other equilibrium.
Proposition
We
log9j(
We
then a joint
which logg,(
in
is
(ft,yt)-
Next, we
linear in {ft-i,
fit, x%t, Vt)
then find the coefficients of these linear functions by the
we can use an independent argument to
coefficients. Finally,
rule
thereby reach the following result.
Under Assumption
3.
from n^ to the more convenient vector
always an equilibrium
is
is
and an invariant variance-autocovariance matrix, we can
ift,ft,yt)
also reduce the aggregate state variable
Because Qt
=
the equilibrium,
1,
const
+
+
(yJ_i/f_i
of local output
level,
<fffit
+ fxXit +
is
given by
(13)
'Pyyt,
where the coefficients {ip-i,tpf,ipx,tpy) are given by
fx
-
{
[ (1
^T
- a)Ki
+
Ky
+
^v
) Oi
Kf
=
}
^
{
S + ^/
1(1- Q)^x +
This result gives a closed-form solution of the equilibrium
level of
output
^
»
(14)
J
in
each island as
a log- linear function of the past aggregate fundamental ft-i, the current local fundamental
the local (private) signal
output
IS
Xit,
and the public
signal
i/(.
Note then that the equilibrium
necessarily an increasing function of the local fundamental
interpret this sign, note that higher /
a lower monopolistic distortion.
means a higher
But whether and how
f,t'-
VJ/
>
fit,
level of
necessarily.
To
productivity, a lower disutility of labor, or
local
output depends on ft-i,
x^t
and
yt is
determined by the degree of strategic complementarity a.
To understand
this,
note that local output depends on these variables only because these vari-
ables contain mformation about the current aggregate shocks and, in so doing, help agents forecast
the aggregate level of output. But
when a =
0,
the
demand- and supply
earlier perfectly offset each other, so that at the
local incentives
depend only the
local
side effects that
we discussed
end economic decisions are not interdependent:
fundamentals and not on expectations of aggregate
19
activity.
follows that the
It
hand,
if
a
^
0,
dependence of
local
>
0),
output to /t-i, xn and
output depends on ft-i,
help predict aggregate output.
(a
local
Xit
In particular,
and
yt
j/t
vanishes
when a =
On
0.
because, and only because, these variables
when economic
complements
decisions are strategic
the equilibrium level of output in each island responds positively to expectations of aggre-
gate output; in this case, the coefficients
decisions are strategic substitutes (q
>
and
(/?_i,y)j.,
0),
<py
are
all
positive.
When
instead economic
the equilibrium level of output in each island responds
negatively to expectations of aggregate output; in this case, the coefficients ^p-\,i^x: and
negative.
As mentioned
earlier,
more
rather,
it
yt,
is
or
0,
and hence
in
all
which economic
as the empirically
most
interpretation of noise and comparative statics
we proceed, we would
the signal
a >
are
generally.
Remark on
Before
the case in which
ipy
For this reason, our subsequent discussion will focus on this case; however, our
relevant scenario.
results apply
we view
good news about aggregate fundamentals,
activity responds positively to
4.2
the other
its
noise
like to
emphasize that one should not give a narrow interpretation to
This signal
£(.
is
not meant to capture only purely public information;
a convenient modeling device for introducing correlated errors in beliefs of aggregate
fundamentals. Indeed, the results we document below can easily be re-casted with a more general
information structure, one that allows agents to observe multiple private signals and introduce
imperfect cross-sectional correlation in the errors of these private signals; the origin of noise, then,
is
not only the public signal, but also the correlated errors in the private signals of the agents.
more general interpretation
invite the reader to keep this
it is
a
acronym
Similarly,
of
what
We
"noise "stands for in our model;
for all sources of correlated errors in expectations of the fundamentals.^'
we would
like to
warn the reader not to focus on the comparative
equilibrium with respect to the precisions of private and public information,
discussants suggests. These comparative statics
fail
to isolate the distinct
Kj-
and
statics of the
K.y,
as one of our
impact of the heterogeneity
of information, simply because they confound a change in the heterogeneity of information with a
change
in the overall precision of information.'-
signals with correlated errors,
''in fact, one could go further
do not move at
all
it
Furthermore,
if
we had allowed
would be unclear whether an increase
and interpret "noise
'as a certain
cis
robustly once information
dispersed. See Angeletos and
is
ours, such shocks cannot exist
in the precision of
a certain
type of sentiment shocks, namely shocks that
the agents' beliefs about the fundamentals and nevertheless
a unique-equilibrium model
for multiple private
move equilibrium outcomes. With
when information
is
commonly
shared; but emerge
La'O (2009b).
'^For example, an increase in Ki would increase the heterogeneity of information, but would also increase the overall
precision of information; and while the former effect would tend to amplifv' the volatility effects
here, the latter eifect
would work
in
the opposite direction.
20
we have documented
signal raises or reduces the heterogeneity of information.
on the comparative
statics
how
follows
we
focus
of information.-^^
and noise
to fundamentals
the dispersion of information and the strength of trade linkages affect aggregate
Towards
fluctuations.
what
in
simply because the degree of complementarity matters for aggregate
Macroeconomic responses
study
mind,
model only by regulating the impact of the heterogeneity
fluctuations in our
We now
this in
with respect to a. These comparative statics best isolate the distinct
role of dispersed information,
4.3
With
we aggregate condition
this goal,
(13)
and use the
fact that ft
=
4'ft-i
+
i^t
to obtain the following characterization of aggregate output.
Corollary
1.
Under Assumption
the equilibrium level of aggregate output
1,
=
\ogQi
const
+
i' ft-\
+
LpuVt
+
is
given by
(15)
^E^t,
where
=
V3^
<^/
and where
+
Ut
fundamentals,
+
V^a;
=
St
ft
=
Condition (15)
(/^y
—
yt
=
1
-
i'ft-i
—
aKf
T;
^p^
=
Lpy
aKy
=
is
4-'
£t-
by the
aggregate noise.
the equilibrium level of aggregate output as a log-linear function of the
Consider the impact
coefficient
ipi,.
effect of
This
same
essentially the
effect of
Because the
have that the impact
is
latter
an innovation
is
(e.g.,
underlying shocks, the
an innovation
fundamentals.
in
fundamentals decreases with the
—
is
measured
Reis, 2002): the less informed
we
level of noise.
monetary shocks
frictions (e.g., Lucas, 1972; Barro, 1976)
they respond to these shocks.
agents interact with one another
and the current
This effect
insight as the one that drives the real effects of
Mankiw and
less
in
i^t,
a decreasing function of the precisions k^ and Ky,
both the older macro models with informational
recent descendants
(16)
the persistence in the
past aggregate fundamentals, ft-\, the current innovation in the fundamentals,
noise,
,
f
^
\
the innovation in the fundamentals,
is
ft ts the
gi\'es
and
\
^;
in
and their
economic agents are about the
Clearly, this
is
true no matter whether
true even in a single-agent decision problem.
it is
''Angeletos and Pavan (2007a) propose that a good measure of the "commonahty" of information (an inverse
measure of the heterogeneity of information)
is
the cross-sectional correlation of the errors
in
the agents' forecasts of
the fundamentals: holding constant the variance of these forecast errors, an increase in the correlation implies that
agents can better forecast one another's actions, even though they cannot better forecast the fundamentals. Following
this alternative route
would deliver similar insights
as the ones
21
we document
here.
More
interestingly,
we
find that
ip^,
a decreasing function of a. That
is
is,
the more economic
agents care about aggregate economic activity, the weaker the response of the economy to innova-
At the same time, we
tions in the underlying fundamentals.
of a.
That
find that f^
is
an increasing function
the more economic agents care about aggregate economic activity, the stronger the
is,
equilibrium impact of noise. These properties originate from the interaction of strategic complementarity with dispersed information. Indeed,
if
the underlying shock was
common knowledge
here can be nested by taking the limit as the public signal becomes infinitely precise, Hy
would cease to depend on
then both
y?^
and
(p^
a reduces
^Pi,
and
raises
(p^-
business cycle once information
Corollary
2.
When
how
This highlights
is
information
But
a.
as long as information
strategic complementarity
is
(which
—
>
oo),
dispersed, a higher
becomes
crucial for the
dispersed.
is
dispersed,
and only
then, stronger
complementarity dampens the
impact of innovations in the fundamentals on eqmlibrimn output arid employment, while amplifying
the impact of noise.
The key
...__.
intuition behind this result
is
the same as the one in the more abstract work of Morris and
Shin (2002) and Angeletos and Pavan (2007a). Public information and past fundamentals (which
here determine the prior about the current fundamentals) help forecast the aggregate level of output
relativel}' better
in
than private information. The higher a
any given island depends on the
more anchored
why
information.
The anchoring
why
similar anchoring effect of the
common
more
less
it
depends on the
in
of each island
sensitive to public information,
effect of past
aggregate output responds less to any innovation
sensitivity to noisy public information explains
output and the
a induces the equilibrium output
to the past aggregate fundamentals,
less sensitive to private
the more the equilibrium level of output
local forecasts of aggregate
local current fundamentals. It follows that a higher
to be
is,
and
aggregate fundamentals explains
the fundamentals, while the heightened
aggregate output responds more to noise.
prior underlies the inertia effects in
A
Woodford (2003a),
Morris and Shin (2006), and Angeletos and Pavan (2007a), while the heightened sensitivity to
public information
we
favor a
is
the
same
as the
one
more general interpretation
in
Morris and Shin (2002). However, as mentioned before,
of the signal
j/(,
not as a public signal, but rather as a source
of correlated noise in forecasts of economic fundamentals.
As another way
to appreciate the aforementioned result, consider following variance-decomposition
exercise.
Let logQt be the projection of logQt on past fundamentals.
by logQt
=
logQt — logQt
aggregate output.
Its total
—
The
residual,
which
is
given
puVt +p}^et, can be interpreted as the "high-frequency component
variance
is
of the innovation in the fundamentals
Var(log Q;
—
)
plal+'pfa'^, where a^ (= 1/k/)
and a^ (= I/k^)
22
is
the variance of the noise,
is
''of
the variance
The
fraction of
the high-frequency variation in output that originates in noise
Var {log Qtli^t) _
_
—
^Mioise
Var(\ogQt)
Since a higher
Q'
raises
about the aggregate
is
and reduces
i/?£
level of
economic
i/Pj/,
it
is
thus given by the following ratio
2^2
fjcr]
E" E
^Wu + ^l^Y
more agents care
necessarily raises this fraction: the
more the high-frequency
activity, the
output that
volatility in
driven by noise.
We
can then further highlight the distinct nature of dispersed information by showing that, as
long as
Q
is
high enough, the contribution of noise to short-run fluctuations can be large even
level of noise
small.
is
Note that the
fundamentals
is
given by
Proposition
4.
When
k,
=
k.o
+
overall precision of an agent's posterior
+
Hx
information
is
K,y
We
w
a
is
not be possible when information
tion of noise on the business cycle
as this precision
becomes
infinite.
this possible
is
is
can be arbitrarily
sufficiently high, agents
is
~
1).
commonly
shared. In that case, the contribu-
when information
In contrast,
noise in the business cycle can be high even
What makes
about the underlying
tightly connected to the precision of information
is
when the
is
and vanishes
dispersed, the contribution of
precision of information
arbitrarily high.
is
the combination of heterogeneous information with a sufficiently strong
degree of strategic complementarity induced by trade linkages. Note then
how
this result also con-
would have
trasts with our earlier observation that this particular type of strategic complementarily
been irrelevant
Finally,
it is
for the business cycle
had information been commonly shared.
worth noting how the dispersion of information and trade hnkages
behavior of aggregate employment. The latter
=
logA^(
where
0( is
of
employment
proportional to that of output.
is
const
the aggregate productivity shock
immediate that the response
to
The same
+
(i.e.,
lilogQt -
at),
the cross-sectional average of log/1,
an aggregate shock
is
common
in either tastes or
>
0.
When
information
an innovation to aggregate productivity
"This fraction equals
may now
1
minus the R-square
is
is
simply
commonly
/3,
23
It is
then
interestingly,
turn from a positive
To
see this, let
shared, the sensitivity of output to
and that of employment
of the regression of log
().
monopoly power
More
true for the response to noise.
information to a negative sign under dispersed information.
P = -^ = ^_g^^,g
affect the cyclical
given by
the response of employment to an aggregate productivity shock
sign under
the
co) and, yet, the high-frequency variation in aggregate
output can be driven almost exclusively by noise (Rnmse
Clearly, this
if
can then show the following.
dispersed and
well informed about the fundamentals {k
is
14
Q, on the innovation
is
u,.
g(/?
—
1).
When,
instead, information
employment,
for
dispersed, the corresponding sensitivities are
is
witli
Suppose
as in (16).
ipi,
common
positively to a productivity shock under
RBC
framework. As noted
that,
when information
and may actually turn
is
it
earlier,
ipi,
/3
>
1,
for
output and ^{ip^(3
—
1)
which means that employment responds
information, as in any plausible calibration of the
necessarily lower than
is
</3^/3
1
and
dampen
dispersed, stronger trade linkages
is
decreasing in a.
It
follows
employment
the response of
negative.
Slow learning
5
The preceding has focused on
common knowledge
a setting where the underlying shocks become
within a period. Although this permitted a sharp theoretical analysis of the distinct implications
of dispersed information,
and of
its
interaction with trade linkages,
results to either empirical business cycles or calibrated
RBC
models.
it
map
our
seek to illustrate
how
makes
We now
hard to
it
incorporating slower learning can facilitate a better mapping between our analysis and the data.
Towards
licly
this goal,
to relax the assumption that the aggregate state,
fif,
becomes pub-
revealed at the end of each period, and instead allow for more interesting learning dynamics.
Accommodating
ized
we need
commodity
micro-founded way would require that there
this possibility in a fully
no central-
is
trading: with centralized trading, equilibrium prices are likely to reveal the state.
However, allowing
for decentralized trading
would complicate the analysis by introducing informa-
tional externalities and/or by letting the relevant state space explode as in
are currently exploring
some
possibilities along these lines.
However,
(1983).
We
purposes,
we
Townsend
for the current
opt for tractability and expositional simplicity.
In particular,
we assume that
period, nor do they learn about
Rather, they
type as
in
onl}'
it
firms
and workers do not observe the state
from observing past aggregate economic outcomes or past prices,
1,
and they use these signals to update each period
underlying state. Think of this as follows. Each firm has two managers: one
employment and production; and another who
realized profits to the firm's shareholders.
—but
Similarl}',
the consumers,
this information to the
solely
sells
on the exogenous
who
their beliefs
who
The two managers share the same
He
about the
decides the level of
the product, receives the revenue, and sends the
do not communicate with one another.
receives any signals on economic activity.
signals.
end of each
keep receiving exogenous signals about the current fundamentals, of the same
Assumption
firm valuation
at the
Moreover, the
objective
first
— maximize
manager never
only observes the exogenous local private and public
observe
all
the prices in the economy,
fail
to
communicate
workers in their respective famihes. The workers also base their decisions
signals.
24
Needless to say, this specification of the learning process
would
also be naive to take
to receive each period are
that these agents
knowledge too
may
it
too
literally:
meant
have.
not particularly elegant. However,
it
the exogenous signals that we allow firms and workers
more generally the multiple sources
to capture
of information
To the extent that the underlying shocks do not become common
more plausible formalizations
fast,
is
of the learning process, albeit highly desirable,
need not impact the qualitative properties we wish highlight here.^^
Under the aforementioned
the
same
specification, equilibrium behavior continues to be characterized
best-response-like condition as in the basehne model:
logq,j
=
(1
where we have normalized the constant
-
+
Q)fi,t
Q'E,,t
The
to zero.
[logQf]
(17)
only difference
is
in
the information that
underlies the expectation operator in this condition. Fnially, for concreteness,
on productivity shocks as the only shock to fundamentals:
The procedure we
is
by
summarized
in a vector
Woodford
(2003a).
Xt comprised
We
=
/SlogA,,;, with
dynamics
follow to soh'e for the equilibrium
similar to the one in
/,,(
we henceforth focus
is
based on
/3
=
^_qI^^,q
Kalman
filtering
and
guess and verify that the aggregate state can be
fundamental and aggregate output:
of the aggregate
s,
Xt
(18)
logQt
Firms and workers
in
any given island never observe the
state, but instead receive the following
vector of signals each period:
ft
+
(;it
It
+
et
before, yt should not be taken too literally
—
(19)
Zit
Vt
As emphasized
introducing
and
common
is
a 2 x 2 matrix, while
equilibrmm values of
Finally,
we guess
Xt follows a simple law of motion:
A';
M
a convenient modeling device for
noise in the agents' forecasts of the state of the economy.
verify that the state vector
where
it is
M,m^, and
- MXt-i +
m^ and
m,. are
m.iyVt
+ rn^Et
2x1
vectors.
(20)
We
then seek to characterize the
m.^.
^^The learning process we assume here
is
similar to the one in
Woodford
(2003a).
We
refer the reader to
Amador
and Weill (2008), Angeletos and La'O (2008), Angeletos and Pavan (2009), Hellwig (2002), and Lorenzoni (2008)
some
alternative formalizations of the learning process.
the positive results
we document
to the extent that they
make the
in this section.
None
of these alternative formalizations
would
for
crucially affect
However, some of them would have distinct normative implications,
learning endogenous to the actions of other agents. See Angeletos and La'O (2008)
and Angeletos and Pavan (2009) on
this issue.
25
In each period
t,
firms and workers start with
that they receive in the beginning of period
prior about
Xt and use the new signals
update their behefs about Xt-
to
t
some
Local output
is
then determined Condition (17) then givens local output as a function of the local belief about
Xt-
Aggregating across islands, we obtain the aggregate
of motion that aggregate output follows
the equilibrium
is
We
must match the one believed by the
a fixed point between the law of motion believed by agents
firms.
making following
decisions that firms and workers are
characterize the fixed point of this problem in the
and used to form
their signal extraction problem.
Appendix and use
its
solution to numerically
simulate the impulse responses of output and employment to positive innovations
in Vt
For our numerical simulations, we interpret a period as a quarter. Accordingly, we
for the
we
standard deviation of the productivity innovation and
set 6
=
.60
and
e
=
Less standard
implying that labor income
is
irrelevant,
is
is
our choice of 7.
period
is
=
0.02
Next,
persistence.
is
no capital,
the only source of wealth, the elasticity of intertemporal substitution
bay.
level of
specific value for
We
elasticity of labor supply.
^^
=
accordingly set 7
.2
to
Next, we set the standard deviations
These values are arbitrary, but they are not implausible: when the
a
economic activity
is
likely to
(equivalently, p). Rather,
be very limited. Finally, we do not pick
we study how
the variance decomposition of the
high-frequency components of output and employment varies as we vary q from
mind
=
interpreted as a quarter, the information about the current innovations to fundamentals
and/or the current
any
(Ty
its
Recall that in our setting there
ensure an empiricallj' plausible income effect on labor supply.
=
0.99 for
let a^,
£(.
These parameter values are broadly consistent with
2.
and 7 only controls the income
of the noises as a^
=
-i/'
and
which correspond to an income share of labor equal to 60% and a
.5,
Frisch elasticity of labor supply equal to
the literature.
Therefore
and the law of motion induced by the optimal output and
their forecasts of the aggregate state,
employment
In equilibrium, the law
level of output.
that a higher
a means
to
1
(keeping in
stronger trade linkagess or, equivalentlj', a lower p).
Impulse responses to productivity and noise shocks
5.1
Figure
1
plots the impulse responses of aggregate output
of productivity, for various degrees of a.
responses we report,
common
is
Size of the innovation here,
equal to one standard deviation.)
Clearly,
if
and
in all other
nnpulse
aggregate productivity were
knowledge, then output would follow the same AR(1) process as aggregate productivity
This
is
simply because there
information
is
dispersed but there
itself.
(The
and employment to a positive innovation
Woodford (2003a) uses a
similar
is
is
number
no capital
in
our model.
The same thing happens when
no strategic complementarity
for calibrating
26
in
output decisions (a
the New-Keynesian model
in
=
the absence of capital.
0).
Impulse Response
ol
Output
to Productivity
Shock
0.03
0,025
"
*
">
•.-,,«
0.02-
.^^^^
**
0.015-
- - a=0, common knowledge
'
u=.5
0.01
0.005
10
5
(
Impulse Response
M
15
periods
B
of
Employment
to Productivity
Shock
'
^^^^
(•
- " a=0, common knowledge
/
-
- - -a=.5
-0.04
20
10
periods
Figure
This
is
island
if
Impulse responses to a positive innovation
1:
a =
simply because when
knows perfectly
its
own
In contrast,
and output
islands.
when information
result,
is
knowledge.
dispersed but islands are interconnected (a
less to
the shock than what
it
^
0),
employment
in other
would have done had the shock been
knowledge, precisely because each island expects output
Note then that the key
for the
response of each island
is
tangle an aggregate shock from an idiosyncratic shock.
informed about the aggregate shock, as long as q
than under
to the aggregate shock as
even though each island remains perfectly informed about their local funda-
mentals, each island responds
common
common
economy responds
one island depends crucially on expectations of employment and output
in
As a
islands are effectively isolated from one another; but as each
productivity, the entire
the aggregate shock had been
in productivity.
common knowledge
if it
>
is
other islands to respond
less.
not per se whether the island can disen-
Even
if
a particular island was perfectly
the island will respond less to this shock
expects the other island to respond
the other island has imperfect information about the shock.
response of aggregate outcomes
in
less,
presumably because
Thus, the key for the inertia
m
the
the uncertainty islands face about one another's response, not
necessarily the uncertainty they themselves face about the aggregate shock.
As evident
in
Figure
complementaritj'. This
less
is
1,
the equilibrium inertia
because of two reasons.
is
higher the higher the degree of strategic
First, there
is
a direct
effect:
the higher
a
is,
the
the incentive of each island to respond to the underlying shock for any given expectation of the
response of other islands. But then there
is
also
an indirect, multiplier-like,
27
efTect: as all islands
are
Impulse Response of Output to Noise Shock
0.02
0.01
10
periods
Impulse Response of Employment
Figure
expected to respond
less to
At the same time, the
to the shock
is
2:
to
Noise Shock
Impulse responses to noise.
the underlying shock, each island finds
inertia vanishes in the long-run:
the same as with
common
it
optimal to respond even
the long-run response of the
is
First,
note that agents are always perfectly informed about their
no learning
in this
is
that's only part of the
own fundamentals,
so there
dimension. Second, recall that agents do not care per se about the aggregate
fundamentals, so the fact that they are learning more about them
the key
economy
knowledge. This seems intuitive: as time passes, agents
become better informed about the underlying aggregate shock. However,
stor3'.
is
per se inconsequential. Rather,
that agents in each island are revising their forecasts of the output of other islands.
then drives the result that inertia vanishes
in
the long-run
is
What
merely that forecasts of aggregate
output eventually converge their common-knowledge counterpart.'^^
Finally, a salient property of the
Q, the short-run
happens
for
dynamic response
of
'
employment
is
that, for sufficiently high
impact of a productivity shock on employment turns from positive to negative; this
parameters values
for
had information been symmetric.
^^It
less.
may be hard
which the model would have generate a strong positive response
We
find this striking.
to fully appreciate this point, because
icnowledge counterpart
is itself
how
The
fast
baseline
RBC
paradigm has long been
output forecasts converge to their common-
pinned down by the speed of learning about the underlying aggregate productivity
shock. However, with richer information structures, one can disentangle the speed of adjustment in output forecasts
from the speed of learning about the fundamentals.
Angeletos and La'O (2009a)
for a related
It
is
then only the former that matters
for
the result.
example within the context of a Calvo-like monetary model.
28
See
criticized for generating a near perfect correlation
data this correlation
negative
q
if
is
Some authors
near zero. In our setting, this correlation could be close to zero or even turn
is
paradigm
information
VAR
the structural
shocks lead to a reduction
RBC
Of
sufficiently high.
in
(e.g.,
may
between employment and output, whereas in the
in
course, correlations confound the effects of multiple shocks.
literature have thus sought to
show that
employment and have then argue that
Gah, 1999; Gah and Rabanal, 2004).
potentially help the
RBC
identified technolog}'
this as a clear rejection of the
we have shown that dispersed
Here,
paradigm accommodate
this fact
without any need to
invoke sticky prices.
Indeed, note that
as
to
fall.
If
agents had been imperfectly informed about the produc-
shock but information had been common, then they could
much
as they
employment
—
is
for
how could they respond
in
the
first
to the shock
place? Thus,
Turning to the
in
to increase their
effects of noise, in
by reducing employment
employment
well informed about the shock but the shock
ployment
fail
employment
would have done with perfect information, but they would not have reduced their
aware of the shock
agent
the dispersion of information, not the uncertainty about the productivity
employment
shock, that causes
tivity
it is
is
falls
not
m
if
they were not
our model precisely because each
common
knowledge.
Figure 2 we consider the impulse responses of output and em-
response to a positive innovation
in £(.
As emphasized
before, this should be interpreted
as a positive error in expectations of aggregate output, rather than as an error in expectations of
When a =
aggregate fundamentals.
0,
such forecast errors are irrelevant, simply because individ-
ual incentives do not depend on forecasts of aggregate activity.
But when a
a positive response in output and employment, thus becoming partly
—
0,
thej'
self-fulfilling.
generate
Furthermore,
the stronger the complementarity, the more pronounced the impact of these errors on aggregate
employment and output.
The
figure considers a positive noise shock,
economic
activity.
The impact
these shocks occur, output,
movement
type of
in
TFP. The
demand
shocks.
of a negative shift in expectations
employment and consumption move
resulting
We
not. This
5.2
is
is
in the
shift in
expectations about
symmetric.
same
Note that when
direction, without
any
recessions could thus be (mis)mterpreted as a certain
a moment. Finally, note that the impact of
employment can be quite
persistent, even
though the noise
itself
simply because the associated forecast errors are themselves persistent.
Variance decomposition and forecast errors
Comparing the responses
former
booms and
will return to this point in
these noise shocks on output and
is
which means a positive
is
of
employment with those
of
output to the two shocks, we see that the
smaller than the latter in the case of productivity shocks but quite larger in the case of
29
Fraction ot Outpul Variance
due
to
Noise
Fraction ot
Employment Variance Oue
to
Noise
10
Figure
This
noise.
is
12
Variance decomposition.
3;
simply because productivity siiocks have a double
effect
on output, both directly
and indirectly through employment, while the noise impacts output only through employment. But
then the response of employment to noise
there are diminishing returns to labor [6
<
is
bound
1),
to
be stronger than that of output as long as
and the more show the lower
6.
It
follows that noise
contributes to a higher relative volatility for employment, while productivity shocks contribute in
the opposite direction. In the standard
RBC
framework, employment
may
exhibit a higher volatility
than output to the extent that there are powerful intertemporal substitution
have ruled out since we have also ruled out
in this
RBC
capital).
dimension. Our results here indicate
framework
in this
Comparing Figures
how
RBC
However, the
effects
framework
(which here
is
known
we
to lack
noise could help improve the performance of the
dimension.
1
and
2,
it
is
evident that low- frequency
movements
in
employment and
output are dominated by the productivity shocks, while noise contributes relatively more to highfrequency movements.
To further
illustrate this property, in Figure 3
we
plot the variance de-
composition of output and employment at different time horizons. For sufficiently strong strategic
complementarity, productivity shocks explain only a small fraction of the high-frequency variation
in
output
of noise
—short-run fluctuations are driven mostly by
is
is
As
for
employment, the contribution
quite dramatic.
Finally, Figure 4 plots the
level of
noise.
dynamics of the average forecast
of aggregate
output and the true
aggregate output in response to a productivity or noise shock. The average forecast error
the distance between the two aforementioned variables.
forecast errors are smallest
when
A
salient feature of this figure
the degree of strategic complementarity
30
is
highest.
is
that
Impulse Responses
to Produclivily
Impulse Responses
Shock, tt=.5
0,025
0.02
0,02
016
0,015
0,01
to
Noise Shock, a= 5
- - - average
0,01
..
torecasi
aciual output
:
\
005
average lotecasi
•
s
actual output
0.005
5
(
15
10
10
5
(
Impulse Responses
15
periods
periods
to Productivity
Shock, a= 9
Impulse Responses
0.025
0,025
0.02
0,02
0015
0015
lo
Noise Shock, a=,9
- - - average
001
Z^'^..
005
,
This
is
of output
- - - average
crucial.
4:
We
lorecast
-
Q, the
10
10
periods
is
earlier
showed that a higher a leads to both more
to productivity shock,
a guarantees
and
to a bigger
is
inversely related to the
magnitude
the impact of noise become nearly
is
highest
inertia in the response
impact of
when a
is
noise.
In this sense,
highest.
However, one
in large forecast errors.
more driven by
To the
contrary,
forecasts of economic activity, so
that the forecast errors are smaller.
of the deviations of actual
Combined, these
-•
;-"
Forecast errors in response to producti\'ity and noise shocks.
implies that actual economic activity
magnitude
...
...
periods
and employment
that at the end a higher
-
^^
should not expect that these large deviations will show up
o
lorecast
,;. .
005
the deviation from the common-knowledge benchmark
a higher
.
\ok
01
.
Figure
•.
\.
It
follows that, as
we vary
outcomes from their common-knowledge counterparts
of the associated forecast errors. Indeed, both the inertia
self-fulfilling as
results illustrate the distinct
q
gets closer to
and
1.
and powerful mark that the dispersed information
can have on macroeconomic outcomes once combined with strong strategic complementarity. Not
only can the macroeconomic effects we have documented be quite significant, but they are also
consistent with small errors in the agents' forecasts of either the underlying economic fundamentals
or the level of economic activity.
VARs
5.3
Demand
Many
economists have found the idea that short-run fluctuations are driven primarily by technology
shocks, new-Keynesian models, and structural
shocks implausible either on a priori grounds or on the basis of certain structural VARs. Blanchard
and Quah (1989) were the
first
to
attempt to provide some evidence that short-run fluctuations are
31
"demand
driven by
"rather than "supply "shocks, albeit with the caveat that one cannot
know what
the shocks they identify really capture. Subsequent contributions by Gali (1999), Basu, Fernald and
Kimball (2006), Gali and Rabanal (2004), and others have tried to improve
way
dimension.
in that
One
or another, though, this basic view that business cycles are not driven by technology shocks
appears to underly the entire New-Keynesian literature.
Our
iindings here are consistent with this view.
In our environment, technology shocks
may
explain only a small fraction of the high-frequency volatility in macroeconomic outcomes. However,
the residual fluctuations have nothing to do with monetary shocks. Rather, they are the product of
the noise in the agents' information. Importantly, to the extent that information
dispersed and
is
trade linkages are important, this noise might be quite small and nevertheless explain a big fraction
macroeconomic outcomes.
of the high-frequency volatility in
Furthermore, the noise-driven fluctuations we have documented here, albeit being purely neoclassical in their nature, they could well
shocks in the following sense. This
is
be interpreted as some kind of "demand" or "monetarj'"
because they share
many
of the features often associated with
such shocks: they contribute to positive co-movement in employment, output and consumption;
they are orthogonal to the underlying productivity shocks; they are closely related to shifts in expectations of aggregate demand; and they explain a large portion of the high-frequency variation in
employment and output while vanishing
To
at low frequencies.-'^
we generate data from our model using
better appreciate this, suppose that
specification for the productivity shock
keynesian type
would then
— to
run a structural
correctlj' identify the
and
VAR
let
an applied macroeconomist
as in
— preferably of the new-
like;
is
One
allowed
and the underlymg noise shocks by the
and Quah, the productivity shocks would be interpreted
as "supply shocks "and the noise shocks as
and the
random- walk
underlying innovations to productivity by the shock that
In the language of Blanchard
relation to sticky prices
a
'
Blanchard and Quah (1989) or Gali (1999).
to have a long-run effect on output or labor productivity,
residual. ^^
'
"demand shocks", however, the
to the contrary, both type of shocks
latter
would have no
emerge from a purely
supply-side mechanism. In the language of Gali (1999), on the other hand, the productivity shocks
would be interpreted as "technology shocks". Furthermore,
of
employment
to these identified shocks
as already noted, the short-run response
would be negative
for
high enough a; but this would no
favor a sticky-price interpretation.
'*A
priori,
our predictions regarding real quantities seems consistent with any predictions about nominal prices.
Further exploring under what conditions our noise-driven fluctuations could be associated also with procyclical
nominal prices requires a monetary extension of the model.
"incidentally, note that the econometric issues studied in Blanchard, L'Huillier, and Lorenzoni (2009) do not apply
to the type of noise fluctuations obtained in our model.
32
As mentioned
Gilchrist
Beaudry and Portier (2004, 2006), Christiano
in the introduction,
et al.
(2008),
and Leahy (2002), Jaimovich and Rebelo (2008), and Lorenzoni (2008) have explored the
idea that noisy news about future productivity contribute to short-run fluctuations. Furthermore,
how
Lorenzoni (2008) interprets the resulting fluctuations as "demand shocks" and discusses
help match related facts.
However,
they
these papers focus on fluctuations that originate from un-
all
certainty about a certain type of fundamentals (namely future productivity), not on the distinct
type of uncertainty that emerges when information
is
paper. ""^ Second, as often the case with new-keynesian models, Lorenzoni's
found
real shocks
in his
model
(as in ours), expectations of future productivity
policy
away from the one that would
shocks "obtain
in
RBC
an
setting
would have been irrelevant
is
and are completely unrelated
shock
in
no capital
for
to
current
news about
that they cause an expansion in
replicate flexible-price allocations.
Finally, note that a positive productivity
In contrast, our
monetary
"demand
monetary pohcy.
our model induces a small impact on output
foUowed by a large persistence response
at high frequencies,
is
prices been flexible; the only reason then that
future productivity cause demand-like fluctuations
this
"demand shocks" con-
with monetary shocks. By this we mean the following. Since there
macroeconomic outcomes had nominal
m
heterogeneous and that we highlight
at lower frequencies,-^
Again these
properties are consistent with the estimated dynamics of "technology" shocks.
More
prices
generally,
dampen
is
it
many
interesting to note that in
empirical new-keynesian models sticky
the response of output to productivity shocks relative to the
RBC
framework and
help get a negative response for employment. According to some researchers, these properties seem
to be
more consistent with the data than
these models
is
their
RBC
is
that monetary policy
optimal thing to do.
sticky properties
Perhaps as
fails to
model
a success for
policy often reacts to
approach
may have
is
typically the
is
based on the idea that monetary
error in the level of aggregate economic activity (Bernanke
is
more plausible
of a producti\ity shocks
mean-reverting.
becomes
If
instead
we assume that
at
positive, while the rest of the results
33
is
a
random
In
an
calibration of the model,
^^In our numerical exercises, the impact of the productivity shock vanishes asymptotically, only because
Oi is (slowly)
and
that measurement
model, Lorenzoni considers a representative-agent model with symmetric information.
extension, he allows for dispersed information, but only to facilitate a
assumed that
RBC
intriguing implications for the identification of
Mihov, 1995: Christiano, Eichenbaum and Evans, 1999), In particular, the idea
his baseline
in the baseline
for policy.
of the standard identification strategies
measurement
from that
economy
same empirical properties without introducing
and without presuming any suboptimality
interestingly, our
differs
replicate flexible-price allocations, which
Here, instead, we obtain the
monetary shocks. One
^'^In
is
only a failure for monetary policy: the only reason that the response of the
to productivit}' shocks in the baseline new-keynesian
model
counterparts. However, what
we have
walk, then the long-run impact
remain unaffected.
error justifies the existence of
random shocks
true underlying state of the economy.
If
monetary pohcy, which are orthogonal to the
to
one then traces the impact
of these particular
shocks on
subsequent aggregate outcomes, one can escape the endogeneity problem and identify the impact
of
monetary shocks. However, these measurement
errors, or
more generally any
forecast errors that
the central bank makes about current and future economic activity, are likely to be correlated with
the corresponding forecast errors of the private sector. But then the so-identified monetary shocks
may
actually be proxying for the real effects of the forecast errors of the private sector, which
unfortunately are not observed by the econometrician.
5.4
Labor wedges and Solow residuals
Many
authors have argued that a good theory of the business cycle must explain the observed
variation in the labor
wedge and the Solow residual
(e.g..
We now
1999; Chari, Kehoe, and McGrattan, 2007; Shimer, 2009).
model
for these
two key characteristics of the business
Following the literature, we define the labor wedge
The
left
Hall, 1997;
Rotemberg and Woodford,
consider the implications of our
cycle.
r„,f
implicitly by
panel of Figure 5 plots the impulse response of the labor wedge to a positive productivity
and a positive noise shock.
two types
The
labor wedge follows very different dynamics in response to the
of shocks. In particular, a positive productivity shock induces a positive response in the
labor wedge, implying positive
comovement
of the labor
wedge with output.
On
the other hand, a
positive noise shock produces a negative response in the observed labor wedge, implying a negative
comovement with output.
Multiple authors have documented that variation
in
the labor wedge plays a large role in ac-
counting for business-cycle fluctuations during the post-war period. Importantly, the labor wedge
is
highly countercyclical, exhibiting sharp increases during recessions.
facts
and the multiple explanations that have been proposed
the labor.
These include taxes, shocks to the
for the
disutility of labor,
observed countercyclicality of
mark-up shocks, fluctuations
wage-setting power, and Shimer's preferred explanation, search frictions
we have found that
We
wards
this goal,
this input
Solow
noise offers another possible explanation for the
finally consider the potential implications of
we now introduce
responds to shocks, but
residual.
As
in
Shimer (2009) surveys the
the labor market. Here,
in
same
in
fact.
our results for observed Solow residuals. To-
a variable input in the production function; the optimal use of
is
unobserved by the econometrician and
is
King and Rebelo (2000), our preferred interpretation of
34
thus absorbed in the
this
input
is
capital
Impulse Response ol Labor
Wedge
to Productivity
Impulse Response
Stiock
ol
Solow Residual
to Productivity
Shock
0.02
----.«'-,_-,
f
"
•
0,01
- - a=0
I
10
5
0^
^^^
'
- - '
-0.005
Wedge
to
Impulse Response ol Solow Residual to Noise Shock
Noise Shock
:^
0,01
.005
-0.01
-0.015
15
10
periods
periods
Impulse Response of Labor
-
-a=5
- -
y*^
- -
/
-
\
- a=0
- - -
cz=.5
.
^^
if
,.=0
- - - n=.5
.
,.= ,9
,
-0.02
10
10
5
periods
periods
Figure
The only
utilization.
caveat
is
5:
Labor wedges and Solow
residuals.
that in our model capital exogenously fixed.
However, we could
introduce capital following the same approach as Angeletos and La'O (2009b), without affecting the
quahtative points we seek to
We
make
here.
denote the unobserved input by
and we specify the cost
of this input
net product of a firm
then
The optimal
(1
— 6)^ =
is
input
level of this
(5
(1
-I- (J)
Xif
q^t
—
Qit
we
x,(;
m
terms of
—
-i^
)
9
and
given by equating
is
=
i
its
-r^
]
A","'
.
Our
=
King and Rebelo, 2000), which implies
,
where
marginal product with
S,,5
>
0.
The
marginal cost:
its
9
=
(21)
analysis then remains intact, provided
we
We
.88.
set ^
of the
model continues
=
.6
and
^
=
.1
we
reinterpret
(a preferred value
also re-calibrate the underl3'ing aggregate
=
—
6'logA^t) implied
by the
to have a standard deviation of 0.02
and a
productivity shocks so that the observed Solow residual {SRt
common-knowledge version
(5\jj
-4,;,n„
the production function in the above way. Accordingly,
in
product as
y4j(\'j~ nf,;
obtain obtain the following reduced-form production function:
^^'^ ^'^"^s
A,t
final
=
Solving out for the optimal level of this input,
^Xut
q,t
where 9
the gross product of a firm be qu
let
logQt
persistence of 0.99.
The
right panel of Figure 5 plots the
noise shock.
Both shocks
has a persistent
effect.
raise the
dynamic response
measured Solow
of the
Solow residual to a productivity or a
residual, but only the innovation in productivity
Moreover, these responses of the Solow residual mirror those of output.
follows that the Solow residual
and output move tightly together, much
35
alike in a
standard
It
RBC
model, although employment has the more distinct behavior we mentioned
Finally,
it
from variation
worth noting that additional variation
is
in
surveys
is
5.5
is left
Note
that the observed heterogeneity in
in particular
highly countercyclical, suggesting that the dispersion of information
how such
countercyclical. Exploring
cycle
measured Solow residuals could obtain
the dispersion of information, simply because the dispersion of information affects
the cross-sectional allocations to resources.
foreccist
in
earlier.
may
also be
variation in the dispersion of information affects the business
work.
for future
Discussion
While the characterization
of equilibrium in Section 3 allowed for arbitrary information structures,
the more concrete positive results that
information structure (Assumption
emphasized
We
to
we documented
thereafter
presumed a
However, we do not expect any of the predictions we have
1).
be unduly sensitive to the details of the information structure.
build this expectation on the following observations. Proposition
economy
Gaussian
specific,
to a class of
games with hnear best responses,
(2002) and Angeletos and
Pavan (2007, 2009).
permits us to
1
map
those studied in Morris and Shin
like
In this class of games, one can
show under arbitrary
information structures that a stronger strategic complementarity makes equilibrium outcomes
sensitive to first-order beliefs (the forecasts of the fundamentals)
that higher-order beliefs are
more
sensitive to the initial
less
and more sensitive to higher-order
One can then proceed
beliefs (the forecasts of the forecasts of others).
our
common
to
prior,
show quite generally
to public signals,
and
to signals with strongly correlated errors, than lower-order beliefs, simply because these pieces of
information are relatively better predictors of the forecasts of others.
beliefs are less sensitive to innovations in the
noise than lower-order beliefs.
dampens the response
of noise
We
—which
of the
It
follows that higher-order
fundamentals and more sensitive to
common
sources of
Combined, these observations explain why stronger complementarity
economy
to innovations in fundamentals while amplifj'ing the impact
we documented
are the key properties that drive the results
in Sections 4
and
5.
conclude that these results are not unduly sensitive to the details of the underlying information
structure; rather, they obtain from robust properties of higher-order beliefs
and the very nature of
the general-equilibrium interactions in our economy.
Our
analysis has implications, not only for aggregate fluctuations, but also for the cross-sectional
dispersion of prices and quantities.
As evident from condition
(24), a higher
a
necessarily reduces
the sensitivitj' of local output to local fundamentals, while increasing the sensitivity to expectations
of aggregate output.
When
of aggregate output,
and hence heterogeneity
information
is
commonl}- shared,
in
all
agents share the same expectation
output (and thereby
36
in prices)
can originate only
from heterogeneity
in
fundamentals (productivities,
tastes, etc).
necessarily reduces cross-sectional dispersion in output
and
only source of heterogeneity. However, once information
prices,
simply because
dispersed, there
is
higher
tlren follows that a
It
it
dampens
a
the
an additional source
is
of heterogeneity: different firms have different expectations of aggregate economic activitj'. It then
follows that a higher
We
a dampens
the former source of heterogeneity while amplifying the latter.
conclude that, once information
dispersion
is
ambiguous
and quantities may
dispersed, the impact of complementarity on cross-sectional
—which also implies that evidence on the cross-sectional dispersion of prices
guidance
pro\'ide little
Similarly, evidence
is
on the
for
a quantitative assessment of our results."^
monopolistic mark-ups, or the elasticity of demands faced by
size of
individual firms, do not necessarily discipline the magnitude of our results. This
First, in our
it
is
is
for
two reasons.
model, the mark-up and the elasticity of individual demands identify only
p that matters for complementarity.
complementarity
in
our model
is
And
second,
evident from the definition of q, a high
cis
consistent with any value of p, provided that there
is
a sufficiently
small wealth effect on labor supply in the short run, a sufficiently high Frisch elasticity (as
1985), and nearly linear returns to labor in the short run (as in
Finally,
it is
New-Keynesian model
which firms cannot
in
demand
room
tell
King and Rebelo, 2000).
own
analysis.
That paper considers a
apart aggregate monetary shocks from idiosyncratic
monetary shocks even when these shocks are unobserved, thus lea^ing
These findings are interesting on
for real effects.
our motivation
for focusing
on
However,
real shocks.
their
it
own
right
— and indeed complement
was unfortunate
for
our discussant to
extrapolate from that paper to the likely quantitative importance of our results.
mechanism
for local
paper does not apply to our context:
of that
m
ones
if
number
^^One of our discussants made the opposite argument. But
necessarily reduces cross-sectional dispersion.
introduced in Assumption
1
his
First, the core
firms were to confuse aggregate shocks
our model, this confusion would only reinforce our results.
quantitative findings of that paper are based on a
a
Hansen,
shocks. For a particular calibration, this confusion induces firms to adjust
their prices a lot in response to
little
in
worth noting that, unlike what suggested by one of our discussants, the findings of
Hellwig and Venkateswaran (2009) bear httle relevance to our
productivity or
whereas
77,
'^'^
of heroic assumptions,
And
second, the
which might serve
argument was based on the premise that a higher
This happens to be true under the specific signal structure we
but, as just explained,
is
not true
in general.
We
refer the reader to
Angeletos and
La'O (2007) for a stark example where higher complementarity increases both the non-fundamental component of the
business cycle and the cross-sectional dispersion of quantities and prices.
^''To see this, recall
shock
Q
,j_^
in
from Proposition 3 and Corollary 3 that the response of equilibrium output to an idiosyncratic
fundamentals
+,t
+^
•
is
^^ long
given
as
b)'
a >
idiosyncratic shock only helps
v^/
0, ipf
dampen
=
1
is
smaller than
—
Q, while
its
yPi,,
response to an aggregate shock
given hy
ipi,
which means that mistaking an aggregate shock
the response of the economy to the aggregate shock.
37
is
=
for
\
—
an
certain purposes but are completely out of place in our
With
these observations
we
own
context.^''
are not trying to escape the need for a serious quantitative exercise,
nor are we ready to speculate on the outcome of such an exercise.
guidance
in this
and
(ii)
for
any future quantitative exploration of our
paper hinge only on
(i)
results.
We
are only trying to provide
The key
effects
some
we have documented
the sensitivity of individual output to forecasts of aggregate output
the sensitivity of these forecasts to the underlying shocks.
We
are thus skeptical that micro
evidence on prices or quantities can alone provide enough guidance on the quantitative importance
of our results.
We
instead propose that a quantitative assessment of our results should rely
more
heavily on sur^'ey evidence about the agents' forecasts of economic activity. Indeed, these forecasts
summarize
concisely
all
the informational effects in our model, and their joint stochastic behavior
with actual outcomes speaks to the heart of our
we
In this regard,
find the
approach taken
results.
Coibion and Gorodnichenko (2008) particularly
in
promising. This paper uses survey evidence to study
how the
agents' forecasts of certain macroe-
conomic outcomes respond to certain structural shocks (with the
structural
VARs).
latter being identified
In effect, the exercises conducted in that paper are empirical analogues of the
we conducted
theoretical exercise
in
Figure 4 for the case of productivity shocks.
paper focuses on how the stochastic properties of the forecasts alone could help
cific
However, that
apart spe-
tell
formalizations of the informational fictions and ignores the fact that the forecasts and actual
Here, instead, we propose that the emphasis should be shifted
outcomes are jomtly determined.
from the details of the underlying informational
forecasts
'^''in
frictions to the joint stochastic properties of these
and the actual macroeconomic outcomes.
particular, Hellwig
and Venkateswaran (2009) assume that workers are perfectly informed about the monetary
shocks, so that nominal wages adjust one-to-one with them.
elastic
demands,
firms cannot
tell
this
see no
good reason
hitting the economy.
their
for
firms face constant real marginal costs
will
and
iso-
move one-to-one with monetary shocks even
if
nominal wages have moved because of nominal or idiosyncratic reasons. Clearly, the
is
assuming a
questionable even within the context of that paper. As for our
priori that
own
context,
workers are perfectly informed about the aggregate real shocks
Furthermore, Hellwig and Venkateswaran (2009) assume that firms are free to adjust their
action at no cost and at a daily or weekly frequency.
When
paper), this assumption serves a useful pedagogical purpose:
But once that action
When
assumption can alone guarantee that prices
whether
empirical relevance of this assumption
we
by specific
is
interpreted as a real
employment
that action
it
is
interpreted as a nominal price (as in that
helps isolate information frictions from sticky prices.
or investment choice (as in our model), this
assumption
stops making sense even from pedagogical perspective: the "stickiness "of real employment and investment decisions
is
merely a matter of technology
in the
model.
38
6
Efficiency
The
positi^'e properties
we have documented
However, their normative content
are intriguing.
is
unclear. Is the potentially high contribution of noise to business-cycle fluctuations, or the potentially
high inertia in the response of the economy to innovations in productivity, a
More
generally,
the information that
But
information.
is
it
is
this
obvious that a planner could improve welfare
symptom
if
of inefficiency?
he could centralize
dispersed in society and then dictate allocations on the basis of
would endow the planner with a power that seems
powers that policy makers have
in reality.
far
all
this
all
remote from the
Furthermore, the resulting superiority of centralized
allocations over their decentralized equilibrium counterparts would not be particularly insightful,
since
it
would be driven mostly by the assumption that the planner has the superior power
overcome the information
frictions
(2007a. 2009) and Angeletos and
to
imposed on the market. Thus, following Angeletos and Pavan
La'O
both practical and conceptual grounds
(2008),
—
is
we contend that a more
interesting question
to understand whether a planner could improve
—on
upon
the equilibrium while being subject to the same informational frictions as the equilibrium.
This motivates us to consider a constrained
efficiency'
concept that permits the planner to choose
am- resource-feasible allocation that respects the geographical segmentation of information
economy
in the
— by which we simply mean that the planner cannot make the production and employment
choices of firms and workers in one island contingent on the private information of another island.
A
formal definition of this efficiency concept and a detailed analysis of
efficient allocations
can be
found, for a variant model, in Angeletos and La'O (2008). Here we focus on the essence.
Because of the concavity of preferences and technologies,
sumption across households,
Using these
facts,
as well as
:
5^,
x
[
5n
JSn
— M+
»
,
of local production
and employment
Q Sq
:
^>
strategies, q
R+
,
so as to
j^^s(u)n{iv,nt-i)'+'dnt{ij)\dp{nt\n,^i
Js^
I
in con-
are interested in as follows.
and an aggregate output function,
\u(Q(nt,nt-i))- f
symmetry
across firms and workers within any given island.
we can represent the planning problem we
Planner's problem. Choose a pair
R+ and n
symmetry
efficiency dictates
S^ x Sq
^
maximize
(221
J
subject to
q{u.',nt-i)
Q(n,.n,_i)
= A{oj)n{u,nt-if
q{ijj,rtt-i)
p
dQt(^)
where P(Q(jf](_i) denotes the probability distribution of
39
y[u.'Mt-i)
fit
V(Q,,n,_i)
conditional on ^t-i-
(23)
(24)
This problem has a simple interpretation.
the representative household; jS{Lo)n{u>,flt-iY
worker
in
U{Q{rit,^t-i)
is
the utility of consumption for
the marginal disutility of labor for the typical
is
a given island; and the corresponding integral
is
the overall disutility of labor for the
representative household. Furthermore, note that, once the planner picks the production strategy
q,
the employment strategy n
down by
(23),
is
pnmed down by
The reduced-form
(23)
objective in (22)
is
and the aggregate output function
Q
is
pinned
thus a functional that gives the level of welfare
implied by any arbitrary production strategy that the planner dictates to the economy.
Because
down by
this
problem
strictly concave,
is
it
has a unique solution and this solution
is
pinned
the following first-order condition:"^
SitiT-lt
=
git
OA^n^r)-
U'iQt)
Ej,
(25)
This condition simply states that the planner dictates the agents to equate the social cost of employ-
ment
in their island
employment.
with the
Essentially the
expectation of the social value of the marginal product of that
local
same condition characterizes
The only
sj'mmetric-information paradigm.
on the commonly-available information
information
As with
equilibrium,
we can
use
=
q^t
is
standard,
that there expectations are conditional
while here they are conditional on the locallj'-available
,,
..,;.
.,-
sets.
set,
difference
(first-best) efficiency in the
,/?..;;.:,,'
Aun^^t ^° eliminate
•
.:.'-'
nu
in the
above condition, thereby
reaching the following result.
Proposition
Let
5.
r(u;)
= log{0i^
g+^-l ( A{lj)\ f+T-i
A(w)\
SH
S{^^)
be a
the fixed point to the following:
\ogq{u,t,nt-i)
IS
=
(l
-
"^{iOtMt-i
(26)
A
number
of
.
a)f*{u;t)
+ a]oglE\Q(ntMt-i)'"' uJt,^t-i]^']
role for the efficient allocation as the
simplicity,
p~\
q{Lj,nt^})
remarks are worth making.
Because of the continuum, the
we bypass the almost
:
'
Q(a,a_i) =
"
S^ x Sq
composite of the local productivity and taste shocks. The efficient strategy q
First,
p
is
ft
(27)
/,*
played for the equilibrium:
determined only
qualification throughout the paper.
40
V(Q,,Q,_i).
note that the composite shock
composite shock
efficient allocation
dQt{^)
for
almost every
uj.
plays a similar
it
identifies the
For expositional
fundamentals that are relevant from the planner's point of view. This
above
result,
but also directly from the planner's problem: using
expression for welfare given
of the production strategy
m
qt
=
we can express
the planner's problem,
and the composite shock f*
is
evident, not only
Afuf to eliminate
much
what Proposition
alike
1
alone.
did for equilibrium:
coincides with the Bayes-Nash equilibrium of a
economy and
islands of the
game
in the
rit
welfare as a simple function
Second, note that Proposition 5 permits a game-theoretic interpretation of the
tion,
from the
in
efficient alloca-
economy
the efficient allocation of the
which the different players are the different
their best responses are given
by
(26).
Third, note that, apart from the different composite shock, the structure of the fixed point
that characterizes the efficient and the equilibrium allocation
with
f{u.'t),
f*{wt)
=
condition (26) coincides with
for every
f{ijJt)
Corollary
ujt if
and only
if
is
the same: once
equilibrium counterpart, condition
its
there
is
is
replace
efficient,
f*{iOt')
And because
(7).
no monopoly power, the following
In the absence of m.onopoly distortions, the equilibrium
3.
we
immediate.
is
no matter the
information structure.
This result estabhshes that neither the presence of noise nor the dispersion of information are
per se sources of inefficiency. This result might sound bizarre in light of our earlier results that the
economy can
feature extreme amplification elTects, with a tiny
aggregate fluctuations.
effects
is
However,
it
amount
of noise contributing to large
What
should be ex post obvious.
causes these large positive
the combination of dispersed information and strong complementarity.
But neither one
introduces a wedge between the equilibrium and the planner. Indeed, the geographical segmentation of information
is
similar to a technological constraint that impacts equilibrium
allocations in a completely symmetric way.
and technologies, not any type
of
market
As
for the
inefficiency,
complementarity,
We
can generalize this result
of noise,
for situations
it
origin
is
It
follows that,
efficient
preferences
guaranteeing that private motives
nating economic activity are perfectly aligned with social motives.
complementarity amplifies the impact
it's
and
in
coordi-
when stronger
does so without causing any inefficiency.^^
where firms have monopoly power, to the extent that
there are no aggregate shocks to monopol}' power, as follows.
Corollary
4.
Suppose that information
mark-up shocks
(f*
—
ft
is fixed).
is
Gaussian (Assumption
Then, the the business cycle
logQt — log Q* between the equilibrium and
If
But
we allow
this
is
for
mark-up shocks, then
earlier, this is
is
efficient in the sense the
the efficient level of output
is
gap
invariant.
clearh' the equilibrium business cycle ceases to be efficient.
true irrespectively of whether information
As mentioned
holds) and. there are no aggregate
1
is
dispersed or
commonly
the opposite of what happens in Morris and Shin (2002).
41
shared.
We conclude
that the dispersion of information per se
competitive
RBC
for
New-Keynesian model.
We
further discuss the implications of
optimal pohcy and the social value of information
this result for
We
or a monopohstic
not a source of inefficiency, whether one considers a
is
in
Angeletos and La'O (2008).
conclude this section with an important qualification. While our efficiency results allowed
an arbitrary information structure, they restricted the information structure to be exogenous
to the underlying allocations.
This ignores the possibility that information gets endogenously ag-
gregated through prices, macro indicators, and other channels of social learning
an important omission.
We
— which
who
clearly
address this issue, too, in Angeletos and La'O (2008), by allowing
information to get partly aggregated through certain price and quantity indicators.
that a planner
is
We
first
show
internalizes the endogeneity of the information contained in these indicators will
choose a different allocation than the equiUbrium. This typically means that the planner likes to
increase the sensitivity of allocations to private information, so as to increase the precision of the
information that gets revealed by the available macroeconomic indicators.
We
then explore policies
that could help in this direction.
Concluding remarks
7
The
pertinent macroeconomics literature has used informational frictions to motivate
agents
may happen,
or choose, to be partly
times the informational friction
main modeling
is
why economic
unaware about the shocks hitting the economy. Some-
exogenous, sometimes
role of informational frictions
it is
endogenized. Invariably, though, the
seems to remain a simple and basic one: to limit the
knowledge that agents have about the underlying shocks to economic fundamentals.
Our approach,
very distinct
instead, seeks to highlight that the heterogeneity of information
mark on macroeconomic outcomes than
highlighted this in this paper by showing
inertia
how
the uncertainty about fundamentals.
In Angeletos
when
We
the heterogeneity of information can induce significant
m the response of the economy to productivity shocks,
driven fluctuations, even
may have a
and can
also generate significant noise-
the agents are well informed about the underlying fundamentals.
and La'O (2009b), we further show that the heterogeneity
the door to a novel type of sentiment shocks
— namely shocks
of information
can open
that are independent of either the
underlying fundamentals or the agents' expectations of the fundamentals and nevertheless cause
variation in the agents' forecasts of economic activity and thereby in actual economic activity,
despite the uniqueness of equilibrium. This in turn permits a broader interpretation of
stood
for in
what noise
the present paper: noise could be interpreted more generally as any variation in the
forecasts of economic activity that
is
orthogonal by fundamentals.
42
In this paper,
we focused on the dispersion
of information
omy, ruhng out sticky prices and dismissing any lack of
monetary
however, does not
policy. This,
persed information and nominal
to
assume common knowledge
mean
frictions. It
we
monetary
monetary policy
real
shocks hitting the econ-
common knowledge about
we
policy.
is
innovations to
no interesting interaction between
see
only means that
of the current
interaction between our approach and
that
about the
find
it
a good modeling
Where we
dis-
benchmark
instead see an intriguing
when there
the following dimension:
is
dispersed information about the underlying real shocks hitting the economy and nominal prices are
rigid, the
may be
response of monetary policy to any information that becomes available about these shocks
crucial for
emphasized
at a
how
the
economy responds
more abstract
level
to these shocks in the first place. This point
by Angeletos and Pavan (2007b, 2009) and
is
was
first
further explored
by Angeletos and La"0 (2008) and Lorenzoni (2009) wdthin new-Keynesian variants of the economy
we have studied
We
in this
conclude with a
paper.
comment on
the alternative formalizations of informational frictions.
For
certain questions, one formalization might be preferable to another."' For other questions, however,
the specifics of any particular formalization
we have emphasized
on the
in this
may
prove unnecessary, or even distracting.
tion structure matters for economic
We
To highlight
outcomes only through
would thus
its
this,
we showed that the informa-
impact on the agents' forecasts of
invite other researchers not to
commit
to any partic-
ular formalization of the information structure (includmg ours), but rather to take a
approach to the modeling
results
paper appear to hinge only on the heterogeneity of information, not
specific details of the information structure.
aggregate economic activity.
The
of informational frictions.
After
all,
more
flexible
the data cannot possibly inform us
about the details of the information structure. What, instead, the data can do
the stochastic properties of the agents' forecasts of economic activity
is
to inform us about
— which, as mentioned,
is
the
only channel through which the dispersion of information matters of economic behavior. Thus, in
our view,
it is
^For example,
only this evidence that should help discipline the theory.
if
one wishes to understand which particular pieces of information agents are
attention to, Sims (2003) offers an elegant methodology.
43
likely to
pay more
Appendix
Proof of Proposition
sion in the
main
1.
The
characterization of
equilibrium follows directly from the discus-
tlie
and uniqueness can be obtained by showing that the equilibrium
text. Its existence
coincides with the solution to a concave planning problem. For the case that there
power
result
[t]
=
from our analysis
oo), this follows directly
can be obtained
for the case
and
in Section 6
in
no monopoly
is
Proposition
A
5.
similar
with monopoly power.
Proof of Proposition
2.
This follows from the discussion
Proof of Proposition
3.
Suppose
main
text.
on Uf and ^t-i, Qi^t,^t-i)
that, conditional
with variance independent of wt; that this
in the
log-normal,
is
true under the log-normal structure for the underlying
is
shocks and signals we will prove shortly. Using log-normality of
Q
in condition (7),
we
infer that
the equilibrium production strategy must satisfy condition (9) with
consi
= -
f
- -
Var[logQ(Q(,n(_i)|cj£,nt_i]
7J
=
and Var[logQ(r!f,nt_i)K,Q(_]]
We now
output
is
'^'^
fi,_i)|fi,_i].
=
ipo
+ f-ift-i
+ipfft
Suppose the equilibrmm production
+ ipx'Xt + ^yyt^
for
some
strateg)' takes a log-linear
coefficients {(f-i,ipf,Lpx,'fiy).
Aggregate
then given by
logQ{ntMt-i) = ^o +
where
..
guess and verify a log-linear equilibrium under the log-normal specification for the
shock and information structure.
form: logqt
Var [logQ(n,,
\\\._
=
ipo
+
^
'^-ift-i
It
(^^ j
+
i'-Pf
+ ^x}ft + ^yyt
follows that Q{Qt,^t-i)
(28)
is
indeed log-normal.
with
E[logQ(n,,fit-i)|wt,nf-i]
=
^'Q
Var[\ogQ{^t,^t-i)\^t,^t-i]
=
{^f
where E [/t|wi,nt-i]
=
^^j+^fqr^^V'/t-i
+
+
i^-Jt-i
+
{'^f
+ ^S~{
\Kf
+
'f,)^[ft\iOt,^t-i]+Vyyt
T"^-+
-r fix
l^y
(29)
(30)
)
/
~^xt + ^j^^^^y,- Substitutmg
these expressions
into (9) gives us
\ogq{ujt,^t-i)
=
const+
+a{ipf
{I
+
- a) f
Lpx)
[uj)
+
a{tpQ'
—+ +—
\ Kf
Kx
K.y
44
+
'fi-ift-i+ipyyt)
iift-i
+
—+
K.f
Kx
—
+
Ky
X,
+
—+
Kf
'
Kx
—
+
Hy
yt
For this to coincide with \ogq
and
(a;)
=
ipo
+ ip-ift-i+fff + fxX + ifyy
sufficient that the coefficients {(fo,
f-i,
(po
—
const
ipf
=
I
<Px
=
aiff
+
aifQ
+
Px
=
iPy
a{ipf
!</
+
+
ipx
and
Kx
+
.
Kf
V'
ifx)
^j
solution to this system for {ip-\,ipf,px^Py)
"t~
^x
^y
"r
,
the one given in the proposition;
is
lpq is
equation of this system along with the definition of const
first
p'q.
Proof of Proposition
and
0;
/?.
necessary
K.7,
+ a{ipf +
Q(Py
then uniquely determined from the
y), it is
Py) solve the following system:
,
The unique
every {f.x,
—a
= Q(p-i+
p-i
'Pf, ^Px,
for
—
1.
*
That
finally,
is,
The
4.
take
by a
result follows
00. It
is
First, take
triple limit.
a —
>
1;
easy to check that this triple hmit implies k
the precision of the agents posterior about the fundamentals (the
next, take
^
.do
and
mean squared
forecast error) converges to zero, while the fraction of the high-frequency variation in output that
is
due
to noise converges to
Kalman
filtering for
similar to that found in
State Vector and
of the
economy
dynamic extension.
The method we
t
of Motion.
are
Claim. The dynamics of
the
/(
We
is
guess and verify that the relevant aggregate state variables
and logQ( and thus define state vector
economy
Xt
are given by the following law of
— MXt-\ +
m.„i't
with
+
p
M
i'
A'( in (18)
accordingly.
motion
m^St
(31)
-1
-1
r-
1
,m„ s
A/21
The
use in solving this equilibrium
Woodford (2003b).
Law
at time
100%.
,m£
M,
=
m„2
(32)
nie2
coefficients {M2i,M22,my2,'rn.^2) cire given by
M21
=
^-(^'21
M22
=
^
(1
rnu2
-
1
-
^2
=
aK:.
Q'
+
A'22)
(33)
-
A'22)
(34)
-
(35)
/\2]
(1
-
A'21
A'22)
(36)
45
and
K=
K22
K2\
is
the matrix of
K = E [{Xt
We
kalman
-
gains, defined by
E,,(_i [Xt])
-
{zi,t
E,,t-i
In each period
Observation Equation.
and public
E
[(z,,f
-
E,,t_i
[z^,t]) (^,,t
and describe the procedure
verify this claim in the following
in (19), of private
[^^,t])']
firms and workers on island
i,
~'
E,,t_i
(37)
[z,,t])']
for finding the fixed point.
i
observe vector z^f as
and error terms, island
In terms of the aggregate state
signals.
-
i's
observation equation takes the form
^\'
'
1
Xt +
Zi,t
+
Qf
where
Cj is
defined as a column vector of length two where the j-th entry
•
are
-
0.
^
:
Forecasting and Inference. Island
E,;
where
E^^t-i [Xt]
is
island's
=
follows that Ei,t_i [X,]
i's t
—
ME,,f_i
1
i's t
i-1
.
—
1
forecast of
—
l-t.tj
kalman
island use the
filter
to
is
the
2x2
.;
zl is
E^,i-l
forecast of Xt.
given by
.
all
other entries
,••,-
,
,.
,
.
,
[A^t]
Combining
this
with the law of motion (31),
it
'•
[X,._i].
update their
=
E,:, f^i
of the current state, firms
forecasts.
[Xt]
K
+
matrix of Kalman gams, defined
Updating
[z,,t
-
is
E,,f_i
and workers on each
done via
[z,^t])
(39)
,
in (37). Substitution of island
i's i
—
1
forecast
'
,
E,,t [Xt]
=
and
•
oi z] into (39) gives us
Let Et [Xt]
1
'
E,,, [Xt]
K
is
'
To form minimum mean-squared-error estimates
where
(38)
£(
1
.''^
Jj Ej_< [Xt] di
^\I^K
be the time
t
ME,,(_i [Xt-i]
+
A'2,,t
average expectation of the current state. Aggregation over
(40) implies
^
E,
[A',
I
-
K
(40)
^ e'
MEt-i
46
[Xt-i]
+
K
/
z,,tdi
Finally, using the fact that aggregration over signals yields
/
Xt
2,,fdz
+
£(, it
follows
that the average expectation evolves according to
K
% \Xt
MXt^i +
+K
VlyVt
+
I
\
K
-
ME,_i [AVi]
"e'l"
K
+
nir
(41)
£(
1
e'l
_
where AI.viy,m^ are given by
(32).
Characterizing Aggregate Output. Local output
hke condition
in (17),
over this condition,
we
which
may
be rewritten as
find that aggregate
log
=
Qt
[{l-a)i>
+
+aM22Et_i
ai) (A'2i
=
-
(1
determined by the best-responsea)
ft
+
Qe2Ej,t [Xt].
Aggregating
output must satisfy
from
[A'(]
log(j,,t
is
= [l~a)ft+ae'2Et[Xt]
\ogQt
Substituting our expression for E(
each island
in
+
[logQ,_i]
(41) into (42), gives us
A'22)] It-\
+
(42)
[(1
-
+
+Q
q)
-
[oMsi
(A'21
+
ail'
{K21
A'22)] vt
=
Moreover, rearranging condition (42), we find that E( [logQj]
+
+
K22)] Et-i [ft-i]
QK22et
^ (logQt
—
(1
—
cv) ft).
Finally,
using this condition in the above equation gives us
logQt
=
[(1
-a)^' +
+ [aM2i -
aV'(Ar2i
atp (A'21
+A'22)-^/22(l-Q)]/t-l +-A/22logQ(-l
+
A'22)]
Et-i [ft-i]
For this to coincide with the law of motion conjectured
it is
necessary and sufficient
tliat
Af2i
=
(1
mv2
—
1
m^2
=
cvA'22
- Q')^ +
- Q +Q
a'!^'(A'2i
(A'21
+
in (31)
- a+a
and
(A'21
+
A"22)l Vt
+ aK22£t
(32) for every [ft-\..\ogQt^i.vt,et),
atl' (A'21
Therefore, given the Icalman gains matrix A',
+
A'22)
-
Af22 (1
-
a)
A'22)
+
solution to this system for (M21, A/22,
motion of Xt
[I
the coefficients [M2i,M22,rny2,m.e2) solve the following system:
= QM21 —
The unique
+
A'22)
'^1^2^
"^£2)
we can uniquely
.
47
is
the one given in the proposition.
identify the coefficients of the law of
Kalman
Filtering. Let us define the variance-covariance matrices of forecast errors as
V = E[{Xt-E,A^t]){Xt-E,,t\Xt]y]
These matrices
be the same for
will
islands
all
i,
since their observation errors are
have the same stochastic properties. Using these matrices, we
may
write
K
assumed to
two
as the product of
components:
- Ea-i
E, [{Xt
[Xt])
iz,,t
- Kt-i
[z^,t]y]
=£
6]
CTrTTT-E
ei
1
and
E^
-
[{Z,^t
1
"
lz^,t]) (z«,(
E^,t-l
E,,t_i
[z,,,])']
ei
+
ei
(43)
crl
'
'
e'l
+ C7t
m.;
+
1
m'.
ei
ei
+
1
K
Therefore,
is
given by
K
where a^
=
Finally,
and
V
E,
1
-
[(z,,(
E,,f_i
what remains
= E
ei
[zi^t]) {zi,t
-
to determine
a'^m.^
ei
1
)k?)"
(44)
'
E,,f_i
[2i,(])']
is
the matrix S.
is
given by (43).
The law
of
motion implies that matrices
S
satisfy
E =
MVM' + alm.ym'.^, + a'^'iriem'^,
In addition, the forecasting equation (40) imply these matrices
V=E
ei
ei
\
+
ON
cx^m^
must further
satisfy
-1
+
1
(^;
1
Combining the above two equations, we obtain the stationary
E
= MT.M' -
M
+almyTn'^
where
M,
ruy,
m^
E
ei
ei
a'^m^
1
Ricatti Equation for E:
{^'X'
rn
_-
+ a^m^m'^
are functions of the
(45)
kalman gains matrix K, and
K
is
itself
a function of
and m^. The variance-covariance matrix E, the kalman gains matrix K, and the law
matrices A/, m„,
by (33)-(36),
m^
(44),
M'
of
E
motion
are thus obtained by solving the large non-linear system of equations described
and
(45).
This system
is
too complicated to allow further analytical results;
thus solve for the fixed point numerically,
48
we
Proof of Proposition
is
unique and
is
5.
The
planner's problem
pinned down by
its
is
strictly convex,
first-order conditions.
guaranteeing that
The Lagrangian
of this
written as
its
solution
problem can be
•
,
,
.
,
=
A
Js„
+
The
A(a)
/
Q{^) and
first-order conditions with respect to
q{uj) are given bj' the following:
i7'(Q(Qt,nt-i))
+ A(fi()(^^)Q(Q(,Qt_i)-p
=
0(46)
=
(47)
P
-^5(w)e-'^°9(c..,n,_i)^'Sn
where
p-l
x{n,
q{uj.nt-i)
p
J^{9.t\uj.nt^i)
L
J^ {Qt\^,Qt-i) denotes the posterior
Restating condition (46) as Xi^t)
(^j =
into condition (47), gives condition (26),
about Qt
(or. equivalently,
about
ft
-U' {Q(9.tMt-i)) Qi^tMt-i)" and
and
yt)
given wt-
substituting this
which concludes the proof.
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