1. science
2. physics
3. thermodynamics

Price Dispersion, Private Uncertainty
and Endogenous Nominal Rigidities
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Gaetano Gaballo∗
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June 18, 2005
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Abstract
This paper demonstrates that, when agents learn from local prices, the introduction of vanishingly small dispersion in fundamentals can lead to price rigidity
as an equilibrium outcome. We identify general conditions for the result in the
context of a stylized real economy. Then, we show that such fragility naturally
obtains in an otherwise frictionless version of workhorse monetary models. In particular, we look at the case where producers set prices before demand is realized,
while observing the prices of their local inputs. The introduction of arbitrarily
small idiosyncratic disturbances, which blur the inference of producers, generates
two equilibrium outcomes: one that inherits, by continuity, the neutrality of money
typical of the unique perfect-information equilibrium, and another where monetary
aggregate shocks have sizable real effects and the impact of idiosyncratic shocks is
magnified. We also show that only the latter has strong stability properties, i.e. a
convergent process in higher-order beliefs or adaptive learning can select it.
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Keywords: price signals, expectational coordination, dispersed information.
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JEL Classification: D82, D83, E3.
∗
Banque de France, Monetary Policy Research Division [DGEI-DEMFI-POMONE), 31 rue Croix des
Petits Champs 41-1391, 75049 Paris Cedex 01, France. Email: gaetano.gaballo@banque-france.fr. This
paper partly incorporates results from ”Endogenous Signal and Multiplicity”. I would like to thank George
Evans, Roger Guesnerie, Christian Hellwig, David K. Levine, Ramon Marimon and Michael Woodford for
their support at different stages of this project, and participants to seminars at Banca d’Italia, Columbia
University, CREST, Einaudi Institute for Economics and Finance, European University Institute, New
York University, Paris School of Economics, University of Pennsylvania, Toulouse School of Economics
and various conferences for valuable comments on previous drafts. It goes without saying that any errors
are my own. The views expressed in this paper do not necessarily reflect the opinion of the Banque de
France or the Eurosystem.
1
”The mere fact that there is one price for any commodity - or rather that
local prices are connected in a manner determined by the cost of transport,
etc. - brings about the solution which (it is just conceptually possible) might
have been arrived at by one single mind possessing all the information which is
in fact dispersed among all the people involved in the process.” Hayek (1945)
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Introduction
In a series of seminal papers Lucas (1972, 1973, 1975) attributes the temporary real
effects of monetary shocks to the local nature of learning. This idea became one of
the most influential of the past century. On the other hand, an even older tradition in
Economics celebrates the competitive price system as a powerful source of information
(Hayek, 1945). The quotation above suggests that, as long as there is at least one good
that is priced in the wide economy or rather that local prices are closely related, then
information about a single aggregate shock should be readily aggregated and revealed.
Thus, one could be tempted to conclude that private uncertainty can have significant
macro-consequences, as envisaged by Lucas, only insofar the economy consists of severely
segmented markets.
This paper assesses Lucas’ insight in a Hayekian economy where agents learn from
competitive prices. It shows that even a vanishing degree of fundamental heterogeneity can disrupt the transmission of information, making prices endogenously sticky. In
particular, we show that this form of fragility can be naturally embedded in otherwise
frictionless monetary models.
The main result of this paper relates to the typical behavior of input demand, and so
potentially pertains to a large class of macro models. In section 2, we present a minimal
real economy to isolate the key mechanism. Final producers, located on different islands,
choose a quantity of local capital before knowing its productivity, while observing its price.
Local capital is produced by local intermediate producers, who transform a homogeneous
endowment that is traded across islands. Final and intermediate production are subject
to the same aggregate productivity shock, whereas only intermediate production is hit by
idiosyncratic productivity shocks. Hence, final producers are unsure to which extent the
prices they see reflect local rather than global productivity conditions. Thus, the price
of local capital provides final producers with an endogenous1 private signal of aggregate
productivity.
Without idiosyncratic shocks, only the perfect information equilibrium exists in which
local prices reveal the aggregate shock as they perfectly comove with it. Nevertheless,
with vanishingly small dispersion of productivity shocks, two additional equilibria, tagged
1
That is, the precision of the price signal depends on the stochastic properties of the average action.
This is different from the literature on information acquisition where the precision of the information is
endogenous to an individual choice as in Angeletos and La’O (2013) and Colombo et al. (2014).
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dispersed information limit equilibria, can arise where producers imperfectly infer aggregate productivity. The new equilibria, which at the limit characterize the same outcome,
feature stochastic properties in stark contrast to the perfect information benchmark: i)
input prices become endogenously sticky to the aggregate shock, and ii) despite vanishingly small price dispersion, equilibrium quantities exhibit sizeable cross-sectional variance across local capital markets.
The existence of dispersed information limit equilibria relies on the possibility that
pushing the uninformative idiosyncratic component of input prices to the limit of zero
variance, the common stochastic component endogenously shrinks to a sufficiently small
variance to generate confusion between the global and local nature of price realizations.
An equilibrium obtains when the reduction of sensitivity of local prices to the aggregate
shock is a consequence of producers being partial informed. To understand how this
happens, consider a positive productivity shock that increases the aggregate supply of
local capital. The typical average decrease of input prices due to a larger average supply is contrasted by the increase of producers’ demand boosted by information about
higher productivity. Whenever under perfect information the demand effect overcomes
the supply effect (i.e. demand and price positively comove!), then there also exits a finite
precision of information for which the two forces exactly offset each other. The latter case
entails dispersed information limit equilibria, where input prices are almost unreactive to
the aggregate shock and producers get only partially informed about them.
In addition, we prove that whenever dispersed-information limit equilibria exist, they
are outcomes of a convergent process under higher-order beliefs or adaptive learning,
whereas the perfect-information limit equilibrium is not. The reason is that, when at
perfect-information limit equilibrium producers demand more input when they see higher
prices - which is the condition for the existence of dispersed information equilibria small out-of-equilibrium perturbations lead to divergent paths. Eventually such nonconvexity vanishes away from the perfect information limit equilibrium, when prices loose
information content.
The conditions for the existence of dispersed information limit equilibria do not require any strong complementarity in actions, as usually assumed in the literature about
sunspots and multiple equilibria2 . On the contrary, they can be naturally reproduced in
otherwise unique-equilibrium models entailing neutrality of money under perfect information. In fact, when money is neutral, higher input prices signal higher nominal value of
production, which stimulates producers’ demand fulfilling the conditions for the existence
of dispersed information limit equilibria.
We show this in section 3, in the context of a fully microfounded economy replicating
the flexible price version of workhorse models for the quantitative assessment of monetary
effects, like Christiano et al. (2005). On each island, producers set a price before observing the demand for their differentiated good, on the basis of the prices of their inputs,
local labor and local capital. A representative household provides local labor, whereas
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Two popular examples are Benhabib and Farmer (1994), Cooper and John (1988)).
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intermediate producers transform homogeneous raw capital held by the household into
local capital. The economy is perturbed by an aggregate shock to the value of money
and two types of idiosyncratic shocks, one to the disutility of local labor and another
to the productivity of the intermediate technology. The presence of idiosyncratic preference shocks prevents the local wage, which is informationally equivalent to an exogenous
signal, from revealing the aggregate shock. The presence of idiosyncratic productivity
shocks blurs the local capital price, which is informationally equivalent to an endogenous
signal.
Without idiosyncratic productivity shocks the prices for local capital are homogeneous, the aggregate shock is revealed and the neutrality of money holds. Nevertheless,
at the limit of no idiosyncratic productivity shocks, a dispersed information limit outcome also exists where: i) aggregate monetary shocks have real effects comparable to the
medium-term impact of VAR estimates in Woodford (2003), Christiano et al. (2005) and
Smets and Wouters (2007) among others, and ii) final good prices are highly reactive
to idiosyncratic shocks, in line with empirical findings of Boivin et al. (2009). The sole
condition for the multiplicity result is that the precision of the exogenous information,
i.e. information inferred from the local wage, is below a certain threshold. Exactly like
in the stylized economy, also in this case, whenever dispersed information limit equilibria
exist, they are the only stable outcomes.
The existence of dispersed information limit equilibria sheds new light on the nature of
price rigidities. Previous work typically relied on some form of friction in the availability
or use of information about marginal costs to explain why producers do not readily
adjust their prices after a shock. Calvo pricing (Calvo, 1983), menu costs (Sheshinski
and Weiss, 1977) and more recently Inattentiveness (Reis, 2006) and Rational Inattention
(Mackowiak and Wiederholt, 2009) are a few popular examples of a vast literature. Here,
in contrast, producers perceive their marginal costs precisely and are not constrained in
their price setting; nevertheless, the signaling role of input prices lead to nominal rigidity
as an equilibrium outcome, with strong stability properties.
The effects of dispersed exogenous information over the real-business-cycle has been
extensively studied by Angeletos and La’O (2010). Amador and Weill (2010) look at the
effect of learning from prices on welfare in the context of a fully microfounded macromodel. They also prove the possibility of a multiplicity that, contrary to the one characterized in this paper, vanishes for sufficiently small dispersion of prices or with low
enough pay-off complementarities.3
The relationship between private uncertainty, prices and multiplicity has been discussed in the global games literature. Morris and Shin (1998) demonstrate that arbitrarily
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Most of the papers looking at multiple noisy REE either restrict the coefficient region to focus on a
unique equilibrium or characterize a multiplicity that vanishes with small enough dispersion of signals.
A non-exhaustive list includes: Angeletos et al. (2010), Angeletos and La’O (2008), Ganguli and Yang
(2009), Manzano and Vives (2011), Vives (2014) and Desgranges and Rochon (2013). Numerous examples
of multiplicity can be found in the CARA asset pricing literature, which often rely on risk aversion to
conditional variance of assets returns (see Walker and Whiteman (2007)).
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small private uncertainty over fundamentals leads to a unique equilibrium in an otherwise multiple-equilibrium model; Angeletos and Werning (2006), and Hellwig et al. (2006)
clarify that the result holds only in the absence of sufficiently informative public prices.
The current paper, in contrast, demonstrates that arbitrarily small price dispersion can
cause multiplicity in an otherwise unique-equilibrium model.
Benhabib et al. (2015) show that partly revealing equilibria - called sentiments - can
coexist with a fully revealing equilibrium because of the endogenous nature of information.
Nonetheless, their multiplicity collapses on the perfect-information outcome in the limit
of no dispersion. Moreover, Chahrour and Gaballo (2015) show that sentiment equilibria
can also be characterized as dispersed information limit equilibria pushing the variance
of a common - rather than an idiosyncratic - informational shock to the limit of zero.
In the game-theoretic language of Bergemann and Morris (2013), dispersed information equilibria are Bayes Nash equilibria, i.e. Bayes correlated equilibria decentralized
by a particular information structure. Bergemann et al. (2015) explore the stochastic
properties of the set of Bayes correlated equilibria by means of an exogenous information
structure. This paper contributes to that agenda showing how equilibria on the boundary
of the set of Bayes correlated equilibria (i.e. dispersed information limit equilibria) can
be decentralized by means of an endogenous information structure which is microfounded
within a system of competitive prices.
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Two-market two-shock model
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This section provides a simple illustration of the core results of the paper. We present
a real economy constituted of two markets where final producers are noisily informed
about aggregate productivity by the prices of their local inputs, which also react to local
disturbances. We will provide conditions for the existence and stability of dispersed
information limit equilibria.
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2.1
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A problem of decentralized cross-sectional allocation
Consider a one-period economy inhabited by atomistic agents who appreciate consumption according to the same linear utility function. Agents are of two types: intermediate and final producers. Both are evenly distributed across a continuum of islands
indexed in the unit interval. The representative intermediate producer in island i ∈ (0, 1)
can transform a quantity4 Z(i) of homogeneous endowment into a quantity
Kis ≡ eθ+ηi Z(i) ,
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(1)
From here onwards, subscripts in brackets denote the quantity of a homogeneous good acquired on
an island, whereas simple subscripts are used for the differentiated good produced on an island.
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of an island-specific local input.5 The transformation is subject to normally and independently distributed productivity shocks θ ∼ N (0, 1) and ηi ∼ N (0, σ 2 ), accounting
for variations in aggregate and island-specific conditions, respectively.6 The intermediate
producer on island i, chooses Z(i) to maximize profits:
Kis Ri − Z(i) Q,
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where Ri is the real price of the intermediate good type i and Q is the real price of the
endowment, which is traded across islands at a single price.7 The endowment is available
in a total quantity Z > 0 and is owned in equal shares by the intermediate producers. The
intermediate good is demanded by final producers on the same island. Final producers
type i can transform a quantity Ki into a quantity
Y(i) ≡ eµθ (Ki )1−α ,
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(2)
(3)
of homogeneous consumption good. Final production has decreasing returns to scale as
α ∈ (0, 1), and it is only affected by an aggregate productivity shock with an intensity
µ ≥ 0. The final producer on island i chooses Ki to maximize expected profits:
E[Y(i) |Ri ] − Ki Ri ,
(4)
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where E[Y(i) |Ri ] is the expectation of Y(i) conditional on the realization of Ri . Final
producers need to forecast production as they do not observe the aggregate shock when
they trade the local input.
Timing is as follows. First, the markets for the endowment and all local inputs open
and close simultaneously. At this stage, final producers observe the equilibrium price of
their local input, but not the aggregate shock. Then, the consumption good is actually
produced and the aggregate shock finally reveals to final producers. At the end of the
period, consumption occurs.
Notice that σ 2 introduces a metric of market segmentation. With σ 2 = 0 the economy
can be represented as an homogeneous one where the law of one price for capital holds.
However, we will show that, at the limit σ 2 → 0, a discontinuity in the set of equilibria
emerges.
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2.2
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Allocative vs. informative role of local prices
The distribution of local prices provides incentives for the cross-sectional allocation
of the endowment. In equilibrium, the endowment should be allocated such that it
5
In appendix C.1 we discuss the extension with decreasing returns to scale. As we will see, decreasing
returns restrict the conditions for the existence of dispersed information limit equilibria, but it is not a
general feature, that is, it emerges here due to the specific microfundations of this simple model. See
forward note 14.
6
N (m, var) conventionally denotes a Normal distribution with mean m and variance var.
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The price of the homogeneous consumption good, i.e the slope of the utility function, is taken as a
numeraire. Hence, Q and each Ri are prices in units of the consumption good.
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equalizes the marginal productivity of final producers across islands. Nevertheless, local
prices depend on final producers’ expectations, which are in turn formed conditionally on
local prices.
As usual in the literature on noisy rational expectations from Grossman (1976) and
Hellwig (1980) onward, I restrict my analysis to equilibria with a log-linear representation.
A formal definition of an equilibrium follows.
Definition 1. A rational expectation equilibrium is a collection of prices (Q, {Ri }1i=0 ),
quantities {Y(i) , Ki , Kis , Z(i) }1i=0 and island-specific expectations {E[Y(i) |Ri ]}1i=0 contingent
on the stochastic realizations (θ, {ηi }10 ), such that:
- (optimality) agents optimize their actions according to the prices they observe;
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- (market clearing) all markets clear, i.e. 0 Z(i) di = Z, Kis = Kid in each i ∈ (0, 1);
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- (log-linearity) prices and quantities are log-linearly distributed.
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Producers operating in island i maximize their profits when prices and quantities
satisfy the first-order conditions of their problems:
ri = q − θ − ηi ,
(5)
ri = µE[θ|ri ] − αki ,
(6)
where lower case denote log-deviations from the deterministic component.8 On the one
hand, the intermediate producer supplies whatever quantity of the local input is required,
provided its price covers its marginal cost, according to (5). On the other hand, the higher
the expected productivity and the lower the price, the higher the quantity of local input
demanded by final producers, according to (6). In particular, (6) defines a demand for
local capital ki (E[θ|ri ], ri ) which is increasing in the expected productivity E[θ|ri ] and
decreasing in the price ri . Crucially, expected productivity relies on the realization of
the local price.9 Therefore, ri has a twofold role: allocative as it clears the market,
and informative as it informs about the stochastic productivity of the local asset. The
importance of the latter is measured by µ, whereas the strength of the former is given
by α. Ceteris paribus, the higher the value of µ, the higher the elasticity of prices to
expectations, whereas, the higher the value of α, the steeper the demand schedule.
8
In particular, according to the log-linearity structure of the equilibrium, ri , in analogy with any
other variable, is defined as ri ≡ logRi − logR¯i where Ri = R¯i eri with R¯i and eri being, respectively, the
deterministic and the stochastic (log-normal) component of Ri . For more details see log-linear algebra
in appendix B.1.
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Note that ki is known by the final producer, but since it is a function of ri it does not convey additional
information. Specifically, suppose that ki provide some additional information, so that E[θ|ri , ki ] is a
linear combination of ri and ki . Then, given (6), ki is collinear with ri , so that focusing on E[θ|ri ] only
does not result in a loss of generality.
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2.3
Dispersed-information limit equilibria
Combining (5) and (6) we can recover q − θ =
into (5), gives
Z
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0
(µE[θ|ri ] − αki )di, that, plugged back
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E[θ|ri ]di − αθ − ηi ,
ri = µ
(7)
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where we also used the relation 0 ki di = θ (which holds because of (1) and the market
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clearing condition 0 z(i) di = 0). From the point of view of the final producer in island
i, the local price of its input constitutes a private signal of an underlying aggregate
endogenous state which is a linear combination of the average expectation of θ and its
actual realization. If agents were able to directly observe such an endogenous state,
the aggregate shock would be perfectly revealed. Nevertheless, the presence of private
uncertainty introduced by ηi generates confusion between local and aggregate fluctuations
in signal extraction.10
To find the set of equilibria one needs to solve the fixed point problem embedded in
(7) and write down the profile of final producers’ expectations as a function of shocks
only. There are many ways to tackle the problem, we will choose the one that provides
for a well defined out-of-equilibrium characterization.
Let us start from the observation that the optimal forecasting strategy when all shocks
are Gaussian is linear in the realization of the signal. Hence, agent i’s forecast is written
as E[θ|ri ] = bi ri , that is
Z 1
E[θ|ri ] = bi µ
E[θ|ri ]di − αθ − ηi ,
(8)
0
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where bi is a deterministic coefficient weighting the signal type i. Given that all agents
use the rule above, then by definition the aggregate expectation is
Z 1
bα
E[θ|ri ]di = −
θ,
(9)
1 − bµ
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provided by b 6= 1 where b ≡ 0 bi di denotes the average weight across agents.11 Therefore
the price signal can be rewritten as
α
ri = −
θ − ηi .
(10)
1 − µb
The private signal ri provides agents with information about θ of a relative precision
τ (b) =
α2
,
(1 − bµ)2 σ 2
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(11)
Notice that, in contrast to the popular setting introduced by Grossman (1976), the input price
responds to the actual fundamental realization also when all producers are uninformed. This happens
because the price embodies information from intermediate producers who actually observe the realization
at the time of trade. This avoids problems of implementability of the perfect information equilibrum
often discussed in the finance literature.
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Note that 0 bi ηi di = 0 given that 0 bi ηi di 6= 0 violates linearity in the realization of the signal.
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b
(a) µ = 1, α = 0.3
(b) µ = 0.5, α = 0.75
Figure 1: Fix point equation for: σ 2 → ∞ in solid light gray, σ 2 = 0.5 in solid grey, σ 2 = 0.02
in solid dark grey and σ 2 = 0 in dashed black.
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which depends non-linearly on the average weight. In particular, it decreases when the
absolute value of b is sufficiently large. This is a property of private endogenous signals
that does not arise in the case of a public endogenous signal, i.e. with a common ηi
instead.12 Relations (8)-(11) hold for whatever given profile of individual weights {bi }10 .
Expectations comply with Bayesian updating when each weight bi satisfies the requirement E[ri (θ − bi ri )] = 0, that is, the forecast error is orthogonal to the signal. This
entails the individual best weight function
bi (b) =
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α(1 − µb)
cov(θ, ri )
=− 2
,
var(ri )
α + (1 − µb)2 σ 2
(12)
which gives the individual weight as an optimal response to the average one. In game
theoretic-terms bi (b) pins down the unique strictly-dominant action in response to b, a
sufficient statistics of the profile of others’ actions. A REE is characterized by a symmetric
profile of weights bi (b) = b for each i ∈ (0, 1).
Panels (a) and (b) in figure 1 plot the optimal individual weight function (12) for
different values of σ 2 in two different cases: µ > α and µ < α, respectively. Equilibria
lie at the intersection with the bisector. From (12) observe that, independently of µ and
α, the curve (in solid light gray) approaches the x-axis when σ 2 → ∞, yielding a unique
equilibrium. As σ 2 decreases the curve approaches the perfect-information line (in dotted
black) (bµ − 1)/α, which is obtained exactly when σ 2 = 0. The perfect information line
12
The interested reader can easily go through the previous steps and check that in the case of a
public signal its relative precision is independent of the average weight. In fact, given an homogeneous
information set, agents would be able to predict the average forecast and disentangle the endogenous
component from the public signal.
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cuts the bisector once at b∗ = (µ − α)−1 ; it does that from below if and only if µ > α,
as in panel (a) in contrast to panel (b). When σ 2 is close to zero, but not zero, the
curve approximates the perfect information line around b∗ , although there now exists a
sufficiently large absolute value of b such that the optimal weight bi approaches zero.
This implies that if and only if µ > α two other intersections exist by continuity, as the
following proposition states.
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Proposition 1. In the perfect information case σ 2 = 0 a unique equilibrium exists, which
is characterized by b∗ = (µ−α)−1 . For arbitrarily small values of σ 2 , provided µ > α, three
equilibria exist: a perfect-information limit equilibrium, characterized by a b◦ arbitrarily
close to b∗ , and two dispersed-information limit equilibria, characterized by b+ > 0 and
b− < 0 respectively, such that |b± | > M where M is an arbitrarily large finite value.
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Proof. See appendix A.1.
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The perfect-information limit equilibrium inherits the properties of the equilibrium
under perfect information: the local prices ri = (µ − α)θ are perfectly correlated and
transmit information with infinite precision, that is E[θ|ri ] = θ. The picture changes
dramatically for the dispersed-information limit equilibria.
Proposition 2. The dispersed-information limit equilibria both feature: almost sticky
local prices ri → 0 which transmit information with a finite precision
τˆ ≡ lim
lim τ (b) =
2
σ →0 b→b±
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lim
lim
2
σ →0 b→b±
0
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1
Z
E[θ|ri ] −
lim lim
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(14)
and sizable cross-sectional variance of island-specific expectations
σ 2 →0 b→b±
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α
θ;
µ
1
E[θ|ri ]di =
Z 1
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(13)
under-reactive aggregate expectation
Z
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α
;
µ−α
0
0
2
α(µ − α)
E[θ|ri ]di di =
.
µ2
(15)
Proof. See appendix A.2.
To illustrate the forces that sustain dispersed information limit equilibria, it is useful
to discuss figure 2 which plots three different equilibria for the case of a positive θ and
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µ > α. On the x-axis we measure the average price r ≡ 0 ri di and on the y-axis the
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average quantity k ≡ 0 ki di. The downward sloping curve denotes an inverse aggregate
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demand schedule, recovered averaging (6), with slope α and intercept D ≡ µ 0 E[θ|ri ]di.
The equilibrium average price r is pinned down at the intersection of the demand schedule
with the supply which is fixed and equal to θ. Note, the higher the D the higher the
equilibrium r.
10
(a)
(b)
D = µθ
D è αθ
r = (µ-α)θ
r è 0
0
D=0
(c)
r = -αθ
k =θ
k =θ
k =θ
Figure 2: Configuration of average variables: a) no information equilibrium, b) perfect informaR1
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tion equilibrium, c) dispersed information limit equilibrium. D ≡ µ 0 E[θ|ri ]di and r ≡ 0 ri di
denote the intercept of the inverse aggregate demand schedule and the average local price,
respectively.
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The no information scenario is illustrated in panel (a), where E[θ|ri ] = 0 and 0 ri di =
−αθ. The local price does not transmit any information when the volatility of idiosyncratic shocks is so large volatility that any movement in local prices is interpreted as
being strictly local. As a consequence, because of decreasing returns to scale, the market
clearing of a higher average supply is induced by a lower average local price. In this case,
θ and r are negatively correlated as the allocative role prevails.
The perfect information scenario is illustrated in panel (b), where E[θ|ri ] = θ and
R1
r di = (µ − α)θ. The local price fully reveals θ when idiosyncratic shocks have zero
0 i
volatility, so that any movement in local prices is interpreted as being strictly global.
A higher local price entails a more costly local input, but also informs about higher
productivity. In particular, with µ > α, the aggregate productivity shock overturns the
loss of productivity due to greater production. As a consequence, final producers tend to
demand more local input, which triggers an increase in local prices driven by competition
for the endowment. In this case, θ and r are positively correlated as the informative role
prevails.
Therefore, provided µ > α, r is negatively correlated with θ in the case of no information and positively correlated in case of perfect information. A decrease in the precision of
information corresponds to a downward shift in the intercept D in panel (b) towards the
one in panel (a). Panel (c) plots one of such configurations, corresponding to a dispersed
information limit equilibria. For a specific finite value of precision, exactly τˆ given by (13),
the average expectation is described by (14), hence the average reaction of local prices to
the aggregate shock tends towards the steady state. At that point, the reaction of local
prices to the aggregate shock is infinitesimally small. Nevertheless, local prices may still
11
5
transmit information as island-specific shocks also have a vanishingly small variance. In
this case, the precision of the information transmitted by the price signal, namely τˆ, is
given by the ratio of two infinitesimal variances.
In the following we will clarify how such a configuration can emerge from an out-ofequilibrium initial condition.
6
2.4
1
2
3
4
Out-of-equilibrium convergence
11
This section demonstrates that, whenever dispersed-information limit equilibria exist,
they are also locally stable under higher-order belief dynamics and adaptive learning,
whereas perfect information limit equilibria are not. Hence, theoretical arguments indicate dispersed information limit equilibria, in contrast to the perfect information equilibrium, are the ones likely to be observed.
12
Convergence in higher-order beliefs
7
8
9
10
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
The following analysis is inspired by the work on Eductive Learning by Guesnerie
(2005, 1992) and has connections with the usual rationalizability argument used in the
Global Games literature (Carlsson and van Damme, 1993; Morris and Shin, 1998). Eductive Learning assesses whether or not rational expectation equilibria can be selected as
locally unique rationalizable outcome according to the original criterion formulated by
Bernheim (1984) and Pearce (1984). The difference here is that I deal with a well-defined
probabilistic structure. I consider beliefs on the average weight b characterizing the
equilibria, rather than beliefs directly specified in terms of price forecasts, as generally
assumed by Guesnerie in settings of perfect information.
In the economy, rationality and market clearing are common knowledge among agents,
meaning that nobody doubts that individual weights comply with (12). Nevertheless,
this is still not enough to determine which equilibrium, if any, is going to prevail. For
instance, suppose that possible values of b can be restricted to a neighborhood =(ˆb)
of an equilibrium characterized by a fixed point ˆb of (12). Common knowledge of =(ˆb)
implies, as first conjecture, that the rational individual weights, and therefore its average,
must actually belong to bi (=(ˆb)). But given that bi (=(ˆb)) is also common knowledge, then
agents should conclude, in a second-order conjecture, that rational individual weights and
therefore its average, must actually belong to b2i (=(ˆb)). Iterating the argument we have
that the νth.-order conjecture about b belongs to bνi (=(ˆb)). Agents can finally conclude
that ˆb will prevail if the equilibrium is a locally unique rationalizable outcome as defined
below.
Definition 2. A REE characterized by ˆb ∈ (b◦ , b+ , b− ) is a locally unique rationalizable
outcome if and only if
lim bνi (=(ˆb)) = ˆb,
ν→∞
34
i.e. bi entails a local contraction around the equilibrium.
12
5
4
3
2
bi
1
0
−1
−2
−3
−4
−5
−5
−4
−3
−2
−1
0
1
2
3
4
5
b
Figure 3: Higher-order belief convergence to dispersed information limit equilibria.
1
2
From an operational point of view local uniqueness requires that |b0i (ˆb)| < 1. The
following states the result.
6
Proposition 3. Whenever dispersed-information limit equilibria exist, they are locally
unique rationalizable outcomes whereas the perfect-information limit equilibrium is not.
The perfect-information limit equilibrium is a locally unique rationalizable outcome when
it is the unique limit equilibrium.
7
Proof. See appendix A.3.
3
4
5
8
9
To shed some light on the tatonnement process, let us rewrite the best individual
weight function
µb − 1 τ (b)
bi (b) =
,
(16)
α
1
+
τ
(b)
| {z } | {z }
scale
10
11
12
13
14
15
16
17
18
19
20
precision
as composed of a scale factor and a precision factor. On the one hand, for a given signal
precision, a lower signal variance requires a higher scale factor. On the other hand, for
a given signal variance, a higher signal precision requires a higher precision factor. The
average weight b affects both factors.
In a neighborhood of the perfect information equilibrium the precision factor is one, so
the dynamics of beliefs is governed by the scale factor. Notice that the scale factor entails
a first-order divergent effect led by high complementarity, i.e. limb→b◦ b0i (ˆb) = µ/α > 1. In
other words, if agents contemplate the possibility that the average weight is higher than
the equilibrium value b◦ , then their best individual weight must be further away from the
equilibrium. This happens because, despite the fact that the precision does not change in
the neighborhood, the scale factor reacts more than one to one to a change in b. Hence,
13
14
the perfect information limit equilibrium is inherently unstable under higher order belief
dynamics.13
Since the supply of capital is fixed at θ, this divergence process leads to lower and
lower reactions of local prices (consistently with a larger and larger scale factor) to the
aggregate shock, approaching the configuration (c) in figure 2. This evolution corresponds
to the divergent dynamics in higher-order beliefs displayed in figure 3.
Nevertheless, as conjectures about b further diverge from b◦ , the reaction of local prices
to θ will eventually shrink, to the point where they achieve the same order of magnitude
of island-specific noise. At that point, a second-order effect driven by the precision factor
is activated. The informativeness of the signal finally decreases, so that the divergent
effect of the scale factor is overturned by a lower and lower precision effect, up to the
point (τ (b) = τˆ for b → ∞) where a new equilibrium emerges. In particular, the sequence
of higher-order beliefs finally enters a contracting dynamics, which, as the proposition
proves, converges towards a dispersed-information limit outcome.
15
Convergence under adaptive learning
1
2
3
4
5
6
7
8
9
10
11
12
13
16
17
18
19
Consider now an infinite-horizon economy where each period is an i.i.d. realization
of the one-period economy described above. Suppose agents learn over time the right
calibration of their forecast rule. In particular, they individually set their weights in
accordance with
−1
bi,t = bi,t−1 + t−1 Si,t−1
ri,t (θt − bi,t−1 ri,t )
−1
2
Si,t = Si,t−1 + (t + 1)
ri,t
− Si,t−1 ,
20
21
22
23
24
25
26
27
28
29
which is a recursive expression for a standard OLS regression, where the matrix Si,t is
the estimate variance of the local price. Notice that, in contrast to the previous setting,
here agents ignore that learning is an ongoing collective process. In this sense this is a
bounded rationality approach. The following formally defines adaptive stability.
Definition 3. A REE characterized by ˆb is a locally learnable equilibrium if and only if
there exists a neighborhood z(ˆb) of ˆb such that, given an initial estimate bi,0 ∈ z(ˆb), it is
a.s.
limt→∞ bi,t = ˆb.
In other words, once estimates are close to the equilibrium value of a locally adaptively
stable equilibrium, then convergence towards the equilibrium will almost surely occur.
The result is stated by the proposition below.
13
An interesting observation is that the existence of a multiplicity of limit equilibria does not hinge
on the unboundedness of the domain of bi , but just on the slope of the best individual weight function
around the perfect-information limit equilibrium. To see this suppose that we arbitrarily restrict the
support of feasible individual weights bi in a neighborhood of b◦ , namely = (b◦ ) ≡ [b, b]. Whenever µ > α
it is easy to check by a simple inspection of figure 3 that two other equilibria beyond b◦ would arise as
corner solutions. I thank George-Marios Angeletos for driving my attention to this point.
14
4
Proposition 4. Whenever dispersed-information limit equilibria exist, they are locally
adaptively stable whereas the perfect-information limit equilibrium is not. The perfectinformation limit equilibrium is locally adaptively stable when it is the unique limit equilibrium.
5
Proof. See appendix A.4.
1
2
3
6
7
8
9
The asymptotic behavior of statistical learning algorithms can be studied by stochastic
approximation techniques as shown by Marcet and Sargent (1989a,b) and Evans and
Honkapohja (2001)). The proof uses well known results to show that the asymptotic
stability of the system around an equilibrium ˆb is governed by the differential equation
db
= bi (b) − b,
dt
10
11
12
13
14
15
16
17
(17)
which is stable if and only if b0i (ˆb) < 1. This condition is known as the E-stability principle
where bi (b) corresponds to the ”projected T-map” in the adaptive learning literature.
Therefore, adaptive stability implies eductive stability. Referring to figure 3, the slope
of the curves at the intersection of the bisector determines the stable or unstable nature
of the equilibrium. In our case it is straightforward to prove adaptive stability in that
the existence of dispersed information limit equilibria relies precisely on the condition
b0i (b◦ ) > 1, and necessarily b0i (b± ) < 1.
3
The Fragile Neutrality of Money
27
This section presents a fully microfounded monetary economy close to the flexible
price version of workhorse DSGE models, like Christiano et al. (2005). The main twist
is the introduction of a timing friction in the production of the final consumption good.
According to Lucas (1973), producers fix a selling price before they observe their actual
demand. However, in the spirit of Hayek (1945), producers see the price of their local
inputs and use them to infer aggregate conditions.
We will show that the conditions for the existence of dispersed-information limit equilibria naturally arise in this context. From a technical point of view, the new setting
extends the previous signal extraction analysis to cases where agents forecast an endogenous variable and exogenous information is also available.
28
3.1
29
Preferences
18
19
20
21
22
23
24
25
26
30
31
32
33
Model
Consider an economy composed of a continuum of islands indexed by i ∈ (0, 1). On
each island, a differentiated consumption good is produced by competitive producers
who employ island-specific inputs: local labor and local capital. Local labor is supplied
by a representative household, which also consumes a bundle of differentiated goods,
15
1
2
3
4
5
6
appreciates the services of money and sells raw capital to intermediate producers of local
capital.
At each period t, the household decides a level of composite consumption Ct , the
supply of labor Lsi,t of each type i ∈ (0, 1), and future money holdings Mt+1 , in order to
maximize a stochastic utility function
!
"∞
!#
Z 1
1−ψ
X
C
−
1
M
t
t
Et
δ t eθt
eξi,t Lsi,t di + log
−
,
(18)
1
−
ψ
P
t
0
t=0
subject to a budget constraint for each period
Z 1
Rt
Wi,t s
Mt
Mt+1
+
Li,t di +
= Ct +
,
Pt
Pt
Pt
Pt
0
(19)
where: δ ∈ (0, 1) is the discount factor; Wi,t is hourly nominal wage type i;
Z
Ct ≡
1
−1
Ci,t
−1
di
Z
and
Pt ≡
1
1
1−
1−
Pi,t
di
0
0
20
are a composite consumption good and its price, respectively; Ci,t is the consumption of
local good type i whose nominal per-unit price is Pi,t ; and > 1 is a CES parameters
measuring the degree of substituability of local goods. The lack of convexity in labor
disutility greatly simplifies the exposition, however, we will consider the convex case in
the main proofs and explore its impact in the numerical analysis in section 3.3. The
operator Et [·] denotes the mathematical expectation conditional on current realizations
of preference shocks θt and ξi,t , which are discussed below. The supply of money is fixed
at M . Finally R is the nominal price of one unit of an endowment in raw capital, which
expires and renews in each period in the hands of the household. In a richer version of
this model, raw capital can be modeled as the legacy of a capital accumulation choice
made by the household at the end of the previous period, as in Christiano et al. (2005).
Nevertheless, what matters to the main results of this paper is that the quantity of raw
capital is predetermined at the beginning of each period. Therefore, for the sake of
simplicity, we will interpret raw capital as being an endowment of a renewable resource.
21
Technology
7
8
9
10
11
12
13
14
15
16
17
18
19
22
23
24
25
Raw capital is acquired in an inter-island market in order to be transformed into
s
island-specific capital Ki,t
. The transformation is operated by competitive intermediate
producers maximizing profits
s
Ri,t Ki,t
− Rt Z(i),t ,
(20)
under the constraint of the following linear technology
s
Ki,t
≡ eηi,t Z(i),t ,
16
(21)
1
2
3
4
5
6
7
8
9
where eηi,t is the stochastic island-specific productivity factor discussed below. The local
s
is produced using Z(i),t units of the endowment which are acquired in a global
capital Ki,t
market at price Rt . The absence of decreasing returns to scale is adopted for the sake
of simplicity14 , however, the general case will be considered in the main proofs and its
effects discussed in the numerical analysis in section 3.3.
Local capital and local labor are used by the representative final producer on island
i to produce a differentiated consumption good type i. Competitive firms set prices to
maximize profits
Pi,t Yi,t − Wi,t Li,t − Ri,t Ki,t ,
(22)
under the constraint of a Cobb-Douglas technology with constant returns to scale
α 1−α
Li,t ,
Yi,t (Ki,t , Li,t ) ≡ Ki,t
(23)
14
with α ∈ (0, 1), where Ki , Li , and Yi denote respectively the demand for local capital, the
demand for local labor and the produced quantity of the consumption good, generated
on island i. Notice that production is island-specific, that is, each producer hires labor
and capital from his own island only. Input markets are segmented and there is one price
for each input on each island.
15
Shocks
10
11
12
13
16
17
In each period t, the economy is hit by i.i.d. aggregate and island-specific disturbances.
An aggregate source of randomness
θt ∼ N (0, 1),
18
19
20
21
22
23
24
25
26
equally concerns the utility of consumption and leisure across islands. Actually, θ has
the features of a shock to the value of money. An increase in θ decreases the marginal
utility of real money in terms of marginal utility of consumption and labor, without
altering the ratio between the latter. In particular, given a positive (resp. negative) θ,
an appropriate increase (resp. decrease) in the price level can leave allocation unaffected.
In this respect, the aggregate shock is isomorphic to a velocity shock. Nevertheless, this
formulation allows an exact log-transformation of FOCs without altering the economic
effect of the shock.15
The disutility of working hours type i varies according to
ξi,t ∼ N (0, σξ2 ),
27
28
(24)
(25)
which introduces differences in labor supply across islands. The productivity of the
intermediate sector type i is affected by stochastic productivity
ηi,t ∼ N (0, σ 2 ),
14
(26)
In contrast to the model of section 2, considering decreasing returns to scale in the intermediate sector
in this economy do not alter the conditions for the existence of dispersed information limit equilibria.
15
An exact log-transformation is also possible in the case of a velocity disturbance assuming permanent
shocks, as in Amador and Weill (2010).
17
2
where ηi is an island-specific realization distributed independently across islands. The
size of σ 2 measures cross-section heterogeneity in intermediate production.
3
Timing of actions and information acquisition
1
Each period t is divided into three stages.
4
5
6
7
8
9
10
11
i) In the first stage the shocks hit. The household observes the preference shocks and
each intermediate producer observes her own productivity shock.
ii) In the second stage, the markets for raw capital and local capitals open and clear
simultaneously. On each island, final producers observe the equilibrium price of
their local inputs, and fix their selling price before their demand realizes.16
iii) In the last stage, final producers observe the demand for their product at the fixed
price, and hire the quantity of labor needed to clear the market.
18
The absence of convexity in the disutility of labor allows a local equilibrium wage,
(observable in the second stage) to emerge irrespective of the quantity of working hours
(traded in the third stage). To allow quantity adjustments otherwise, one should assume
that at the second stage producers can observe the wage contingent on labor quantities, i.e.
the labor supply schedule posted by the household.17 This is informationally equivalent
to observing the equilibrium wage in the current specification. This case is fully worked
out in the main proofs of appendix B.1.
19
3.2
20
Definition of Equilibrium and first-order conditions
12
13
14
15
16
17
21
22
23
Equilibria
As in the previous section, I restrict the analysis to equilibria with a log-linear representation which, as we will see, obtains with no approximation in our model. A formal
definition of an equilibrium follows.
27
Definition 4. A log-linear rational expectation equilibrium is a distribution of prices
s
{{Ri,t , Wi,t , Pi,t }(0,1) , R}, quantities {Ci,t , Yi,t , Mi,t Li,t , Lsi,t , Ki,t , Ki,t
, Z(i),t }1i=0 and expectations {E[Yi,t |Ri,t , Wi,t ]}1i=0 , contingent on the stochastic realizations (θt , {ξi,t , ηi,t }(0,1) ),
such that:
28
- (optimality) agents optimize their actions according to the prices they observe;
24
25
26
16
If producers were able to observe the demand for their differentiated good when they set prices, then
the irrelevance of dispersed information obtains as explained by Hellwig and Venkateswaran (2014).
17
A more cumbersome solution is to introduce a third production factor whose market opens and clears
at the last stage, and let labor be traded in the second stage.
18
3
R
- (market clearing) the endowment market clears, Z(i),t di = 1; demand and supply
s
; the final markets clear, Yi,t = Ci,t ;
in local markets match, Li,t = Lsi,t and Ki,t = Ki,t
and the money market clears, Mi,t = M ;
4
- (log-linearity) prices and quantities are log-linearly distributed.
1
2
5
6
7
8
9
The first condition requires agents to optimize their actions according to the information conveyed by the equilibrium prices that they observe. The requirement of a log-linear
equilibrium allows the tractability of aggregate and island-specific relations.
The household chooses a level of aggregate and island-specific consumption, islandspecific working hours and future money holdings such that:
Λt = eθt Ct−ψ ,
−
Pi,t
Ci,t =
Ct ,
Pt
Λt
eθt eξi,t = Wi,t ,
P
t
Λt
1
Λt+1
+δ
= δEt
Pt
Pt+1
Mt+1
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
(27)
(28)
(29)
(30)
respectively, where Λt is the lagrangian multiplier associated with the budget constraint
of the representative household at time t. Intermediate producers supply any quantity of
local capital provided:
Ri,t = e−ηi,t Rt ,
(31)
that is, the price of the local capital equals the cost of the raw capital augmented by the
productivity disturbance. Final producers hire local capital and fix a price according to:
1−α
αWi,t
Ki,t =
E [Yi,t |Ri,t , Wi,t ]
(32)
(1 − α) Ri,t
1−α
α
Wi,t
Ri,t
Pi,t =
(33)
− 1 (1 − α)1−α αα
representing the minimal-expenditure-for-constant-production demand for local capital
and the optimal pricing function, respectively. The presence of E [Yi,t |Ri,t , Wi,t ] emphasizes that final producers fix Ki,t and Pi,t before knowing Yi,t while observing Ri,t
and Wi,t . In fact, local labor is determined in the third stage of the period so that
α 1−α
Ci,t = Yi,t = Ki,t
Li,t , i.e. final good markets clears.
Learning from prices
Here, we spell out the signal extraction problem faced by final producers. To start,
notice that producers’ uncertainty is resolved at the end of the period, once they observe
their own demand. Therefore, at the end of the period, all agents in the economy have
the same information about the next periods. Moreover, given that shocks are i.i.d.
19
1
2
3
4
and the supply of money is fixed, agents’ expectations at time t over the future course
of the economy is the unique stochastic steady state at each future date. Hence, as in
Amador and Weill (2010), the only intertemporal first-order condition - the one for money
- collapses to the one-period equilibrium relation
Λt
δ 1
=
Pt
1−δM
5
6
7
8
9
10
where I substituted the market clearing condition Mt = M . This means that final producers face a one-period signal extraction problem which renews each time. From here
onward, I will omit time indexes as the following relationships are all simultaneous.
After plugging (34) into (29), we conclude that observing deviations of the wage from
its steady state provides information on the preference shock on each island. That is, the
log-deviation of the equilibrium local wage from its deterministic component
wi = θ + ξi ,
11
12
13
1
16
17
19
20
21
22
23
24
(37)
is the endogenous common component of each island-specific demand, which is unobserved
by producers at the time of their price choice. Combining (32),(33) and (36), we can
express the local capital demand in log-deviations terms as:
ki = E[x|ri , wi ] − ( − 1)(1 − α)wi − (1 + ( − 1)α)ri .
18
(36)
where Pi is fixed (and so known) by final producers according to (33) and
X ≡ Y P,
15
(35)
constitutes a private exogenous signal about θ with precision σξ−2 . Concerning the equilibrium price for local capital, notice that at the equilibrium Yi,t = Ci,t , so that, using
(28) we can express the expectation as:
E [Yi |Ri , Wi ] = E [X |Ri , Wi ] Pi−
14
(34)
(38)
It is instructive to contrast the demand schedule (38) with (6). In both, the correlation
between ki and ri can be reversed when the average expectation is sufficiently reactive
R1
to ri . Unlikely in the model of section 2, here ki = z(i) − ηi , and 0 z(i) di = 0, so
the aggregate increment in the demand for local capital is fixed at zero. Nevertheless,
R
now wages contains information about the aggregate shock, in particular wi,t di = θ.
Therefore, (38) entails an expression for the average price of capital r as depending on
R1
the average expectation 0 E[x|ri , wi ]di and θ. Finally, we can use (31), to get
1
Z
E[x|ri , wi ]di − (φ − 1)θ − ηi ,
ri = φ
(39)
0
25
where
φ≡
> 1,
1 + ( − 1)α
20
(40)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
for whatever > 1 being φ ∈ (1, α−1 ), i.e. φ > 1 for all feasible calibrations. The price of
local capital (39) constitutes a private endogenous signal, similarly to (7), with one major
difference: final producers form an expectation about an endogenous aggregate state X
which is a function of the aggregate fundamental θ, and not about θ directly.
The price of the local capital exhibits opposite reactions to an aggregate shock in
the cases of perfect information (σ 2 = 0) or no information (σ 2 → ∞ and σξ2 → ∞).
In the latter case, all movements in ri are interpreted as movements in ηi , so we have
E[x|ri , wi ] = 0 and hence r = (1 − φ) θ; in the former case instead, all movements in ri
are interpreted as movements in θ given that E[x|ri , wi ] = x = p = θ which implies ri = θ.
The underlying economic intuition is naturally linked to the mechanics of neutrality of
money. Suppose a positive shock θ occurs. Under perfect information, the shock produces
a neutral inflationary effect: all prices go up at the same average rate θ, whereas quantities
are unchanged. In particular, the aggregate demand for capital increases because of final
producers’ expectations of a higher general price level. Conversely, when final producers
miss the global nature of a rise in local wages, the aggregate demand for capital falls
as each final producer anticipates a loss of competitiveness of their own product. As
a consequence, the average price of capital goes down to clear the aggregate supply of
capital, which is fixed by the given quantity of endowment.
An expression for the log-deviation of the aggregate state x can be obtained as follows.
First, recover from (33) an expression for the aggregate price as a function of the average
ri and wi , and hence, as a function of the average expectation and the aggregate shock.
Then, use the latter into (27) – where λ = p because of (34) – to obtain the actual
aggregate demand. Finally plug the actual aggregate demand and the aggregate price
jointly into (37). The final expression for x reads as
Z 1
x=β
E[x|ri , wi ]di − (β − 1)θ,
(41)
0
25
where
β≡
α( − ψ −1 )
< 1.
1 + ( − 1)α
(42)
30
Note that (41) has a structure similar to (39), except for the strength of the expectational
feedback, which, in contrast to (39), is smaller than one. This feature prevents a switch in
the sign of the correlation between x and θ, which always remains positive. In particular,
in the two scenarios of perfect information and no information we have: E[x|ri , wi ] = x =
p = θ, and E[x|ri , wi ] = 0 with x = (1 − β)θ, respectively.
31
Characterization and existence of an equilibrium
26
27
28
29
32
33
34
35
All first order conditions in the model have a multiplicative form, so they can be
log-linearized and solved without any approximation. In particular, the requirement of a
log-rational equilibrium implies that the aggregate state X, as with any other variable in
¯ x , where x ∼ N (0, var (x))
the model, is distributed log-normally according to X = Xe
21
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
¯ The stochastic
is the stochastic log-deviation of X from its deterministic component X.
1
var(x)
¯2
steady state of X is given by X
. To better enlighten the origins of the multiplicity,
let me state the following proposition which characterizes an equilibrium in terms of a
profile of final producers’ expectations about x.
Proposition 5. Given a profile of weights {eθ , eξ , eη } such that log-linear expectations
for final producers are described by
¯ E[x|ri ,wi ] ,
E[X|Ri , Wi ] = Ee
(43)
E[x|ri , wi ] = eθ θ + eξ ξi + eη ηi ,
(44)
¯ =X
¯ 21 var(x|ri ,wi ) with
where E
then there exists a unique log-linear conditional deviation and a unique steady state for
each variable in the model.
Proof. See appendix B.1.
In practice, an equilibrium is characterized by a distribution of firms’ expectations
about x, the stochastic aggregate component of the individual demand. Each individual
expectation type i is conditional on the observation of θ+ξi and ri , denoting the stochastic
log-components of, respectively, eθ+ξi and Ri inferred from the input markets. Both are
log-linear functions of the shocks, so producers’ expectations shall be a linear function of
the two signals. In particular, all agents use the rule
Z 1
E[x|ri , wi ] = bi φ
E[x|ri , wi ]di + (1 − φ) θ + ηi + ai (θ + ξi )
(45)
0
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
where ai and bi denote the weights put by agent i on the local wage and price for local
capital, respectively. Let us denote by a and b the average weights. As in section (2),
a log-linear equilibrium has to be symmetric. Hence, a profile of the optimal weights
given to these two pieces of information maps into a profile of weights {eθ , eξ , eη }. The
characterization of an equilibrium follows straightaway once the requirement of rational
expectations is imposed.
Definition 3 A log-linear rational expectation equilibrium is characterized by a profile
of weights {eθ , eξ , eη } such that (45) are rational expectations of (41) conditional to (35)
and (39), with
(1 − φ)b + a
eθ =
, eξ = a, eη = b
1 − bφ
being the corresponding weights in (44).
In other words, the number of equilibria of the model corresponds to the number
of solutions to the signal extraction problem. In the cases of no information (σ 2 → ∞
and σξ2 → ∞) and full information (σ 2 = 0) the economy has a unique equilibrium
characterized by (a, b) = (0, 0) and (a, b) = (0, 1), respectively. The following proposition
establishes the existence of multiple equilibria at the limit of a vanishing dispersion of
local prices for capital, σ 2 → 0.
22
1
2
3
4
5
6
Proposition 6. Consider the problem of agents forecasting (41) conditionally on the
information set {ri , wi }, given by (35) and (39). If the variance of preference shocks σξ2
satisfies
φ−1
σξ−2 < τˆ =
,
(46)
1−β
then in the limit of no productivity shocks, σ 2 → 0, there exist:
- a unique perfect-information limit equilibrium, characterized by a = 0 and b = 1;
where neutrality of money holds, i.e.
r = p = θ and y = 0,
7
8
9
10
for any realization of θ;
- and two dispersed-information limit equilibria, characterized by a = (1 − β)(φσξ2 )−1
and b = b± with limσ2 →0 b2± σ 2 = τˆ−1 (σξ − τˆ−2 ) σξ−2 , where shocks to the value of money
yield real effects, in particular,
r → 0, p = (1 − α)θ and y = αψ −1 θ,
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
for any realization of θ.
Otherwise, only the perfect-information limit equilibrium exists.
Proof. See appendix B.2.
The intuition for the new condition (46) is simple. The dispersed-information limit
equilibria originate in the limit of σ 2 → 0 when the precision of the private information
about the exogenous shock θ is τˆ. The possibility of achieving this value is prevented
when the precision of exogenous signals σξ−2 is higher than this threshold. Obviously,
the endogenous informational channel can eventually only increase the precision of information, but cannot in any case make agents lose information. In appendix C.2 we
prove that (46) is still the relevant condition when considering extended versions of the
additional signal (35) including correlated noise and/or the average expectation. In fact,
the existence of dispersed information limit equilibria only hinges on the possibility that
the variance of the common component of the endogenous signals shrinks to a sufficiently
small number, regardless the presence of common noise into it.
Condition (46) also enlightens the role of preference and technological parameters.
Notice that
( − 1)(1 − α)
τˆ =
,
1 − α + αψ −1
is defined as a simple combination of the CES parameters. In particular, the restriction for
the existence of dispersed information limit equilibria relaxes with higher substituability
across goods, i.e. higher , lower return to scale, i.e. lower α, and higher relative risk
aversion, i.e. higher ψ.
23
1
2
3
4
5
6
7
8
In the dispersed information limit equilibria a positive θ (equivalent to an increase
in the stock of money) have expansionary real effects as in the Lucas’ island model. A
general discussion of the impact of shocks to the value of money is provided in section
3.3.
To conclude the analysis of dispersed information equilibria we state the following
proposition which demonstrates the amplification of private uncertainty.
Proposition 7. In contrast to the perfect-information equilibrium, the dispersed-information
limit equilibria feature sizable cross-sectional variance of expectations:
Z 1
E[θ|ri , wi ] −
lim lim
σ 2 →0 b→b±
9
Z
0
0
1
2
(1 − β)(φ − 1)
E[θ|ri , wi ]di di =
,
φ2
(47)
Proof. See appendix B.3.
18
At the dispersed information equilibria, idiosyncratic productivity shocks are greatly
amplified, so that although vanishingly small they account for a fraction of the overall
cross-section volatility of expectations. Moreover, the idiosyncratic preference shocks
also play a role in pinning down firms’ expectations. Specifically, the contribution of
idiosyncratic preference shocks to overall cross-section volatility of expectations is given
by a2 σξ2 = (1 − β)2 (φ−2 σξ−2 ) which converges towards (47) when (46) is binding. The
contribution of idiosyncratic productivity shocks is instead given by (47) minus a2 σξ2 .
The relation between the cross-sectional variance of expectations and the cross-sectional
variance of prices and quantities in the model will be discussed in section 3.3.
19
Out-of-equilibrium selection
10
11
12
13
14
15
16
17
20
21
22
23
24
25
26
27
28
29
30
31
In this section we investigate the stability properties of the equilibria. The following
analysis extends that in section 2.4 to a case where agents forecast an endogenous state
and exogenous information is also available.
Let us start by assessing the iterative process of higher-order beliefs. Suppose that it
is common knowledge that the individual weights given to the signals {(ai , bi )}10 lie in a
neighborhood =(ˆ
a, ˆb) of the equilibrium characterized by (ˆ
a, ˆb). Common knowledge of
(ai,0 , bi,0 ) ∈ =(ˆ
a, ˆb) for each i implies that the average weights (a0 , b0 ) belong to =(ˆ
a, ˆb).
Call B : (a, b) → (ai , bi ) the mapping18 that gives, for a given couple of average weights
(a, b), a couple of individual optimal weights (ai , bi ). Since B(a, b) is common knowledge
then (a0 , b0 ) ∈ =(ˆ
a, ˆb) implies that a second-order belief is rationally justified for which
(ai,1 , bi,1 ) = B (a0 , b0 ) for each i, so that (ai,1 , bi,1 ) ∈ B(=(ˆ
a, ˆb)) and as a consequence
(a1 , b1 ) ∈ B(=(ˆ
a, ˆb)). Iterating the argument we have that (aν , bν ) ∈ B ν (=(ˆ
a, ˆb)).
Definition 4 A REE characterized by (ˆ
a, ˆb) is a locally unique rationalizable outcome
if and only if
lim B ν (=(ˆ
a, ˆb)) = (ˆ
a, ˆb).
ν→∞
18
Given by (62)-(63), see appendix B.4
24
From an operational point of view local uniqueness requires that
||J (B)(ˆa,ˆb) || < 1
1
2
3
where J (B)(ˆa,ˆb) is the Jacobian of the map B calculated at the equilibrium values (ˆ
a, ˆb).
The following proposition states a result which holds for all the cases investigated in this
section.
5
Proposition 8. Whenever dispersed-information limit equilibria exist, they are locally
unique rationalizable outcomes whereas the perfect-information limit equilibrium is never.
6
Proof. See appendix B.4.
4
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
The intuition for the result is similar to the one discussed for the unidimensional
case in section 2.4. The local instability of the perfect information outcome relies on
the high elasticity of price signals to expectation, which is maximal when close to the
perfect information scenario. This generates a circle of high complementarity between
input demand and price signal, that pushes the signal towards steady state, as aggregate
supply is predetermined. At the point where the variance of the common component of the
price signal is of the same order of magnitude as the one of the idiosyncratic component,
the price signal loses informativeness. Thus, the circle of high complementarity eventually
dampens and converges to a fixed point featuring a dispersed information equilibrium.
Let us turn attention now to the adaptive learning process. At the end of each period
producers see the realized demand, so they can eventually revise their beliefs in light
of the realized forecast errors. Following the approach described in section 2.4, suppose
that, at time t, agents set their weights χi,t = [ai,t bi,t ]0 in accordance with
−1
χi,t = χi,t−1 + t−1 Si,t−1
ωi,t xt − χ0i,t−1 ωi,t
0
Si,t = Si,t−1 + (t + 1)−1 ωi,t ωi,t
− Si,t−1 ,
where ωi,t = [wi,t ri,t ]0 , which is a recursive expression for a bivariate OLS regression, where
Si,t is the estimate variance-covariance matrix of price signals. The following formally
define adaptive stability.
Definition 5. A REE characterized by (ˆ
a, ˆb) is a locally learnable equilibrium if and only
if there exists a neighborhood z(ˆ
a, ˆb) of (ˆ
a, ˆb) such that, given initial estimates (ai,0 , bi,0 ) ∈
a.s.
z(ˆ
a, ˆb), it is limt→∞ (ai,t , bi,t ) = (ˆ
a, ˆb).
In other words, once estimates are close to the equilibrium value of a locally adaptively
stable equilibrium, then convergence towards the equilibrium will almost surely occur.
29
Proposition 9. Whenever dispersed-information limit equilibria exist, they are locally
adaptively stable whereas the perfect-information limit equilibrium is never.
30
Proof. See appendix B.5.
28
25
As before, the proof uses well known results to show that the asymptotic stability of
the system around an equilibrium (ˆ
a, ˆb) obtains if and only if
J (B)(ˆa,ˆb) < 1
3
which is a bi-dimensional transposition of the E-stability principle mentioned previously.
Also in this case, it is straightforward to prove that equilibria which are locally unique
rationalizable outcomes are also adaptively stable.
4
3.3
1
2
5
6
7
8
9
10
11
12
13
14
15
16
17
Extension and out-of-the-limit exploration
In this section we explore an extended version of the model beyond the limit of zero
cross-sectional variance of productivity shocks. The aim is threefold:
1) to demonstrate that the findings are robust to natural generalizations of the economic environment;
2) to show that the qualitative properties of equilibria demonstrated at the limit are
inherited, by continuity, by equilibria outside the limit;
3) to argue that that the economic implications of the model are consistent with the
widely accepted empirical finding that, although prices are flexible, monetary shocks
have temporary real effects.
For the sake of generality, we will focus on an extended version of the model which
includes: a more general specification of utility function
"∞
!
!#
Z 1
1−ψ
s 1+γ
X
L
C
M
t
i,t
t
Ut ≡ Et
δ t eθt
eξi,t
−
di + log
,
(48)
1
−
ψ
1+γ
Pt
0
t=0
instead of (18), where γ>0 is the inverse of the Firsh elasticity of labor; and a more
general specification of intermediate technology
1−ζ
s
Ki,t
≡ eηi,t Z(i),t
,
(49)
21
instead of (21), where ζ ∈ (0, 1) measures decreasing returns to scale in intermediate
production. The values γ = ζ = 0 feature the case studied above. The qualitative results
of our analysis are robust to these extensions. The proof of proposition 5 in appendix
B.1 already considers these more general specifications.
22
Equilibrium weights and cross-sectional variance of expectations
18
19
20
23
24
25
26
Figure 4 illustrates the profile of weights and the variance of the individual expectation, as functions of σ 2 for a fairly standard calibration = 8, α = 0.33, ψ = 2, γ = 0.75,
ζ = 0 and two different values of σξ2 , namely 0.4 (solid) and 1.4 (dotted). The three
weights {eθ , eξ , eη } can be interpreted as the response of an individual price expectation
26
32
to a unitary increase in the aggregate shock, the idiosyncratic preference shock and the
idiosyncratic productivity shock, respectively.
In all panels, we can observe the existence of three equilibria values for a sufficiently
low σ 2 . In the first two panels, we see two equilibria values converging towards the
same point on the y-axis at the limit σ 2 → 0: these correspond to dispersed information
equilibria. The perfect information limit equilibrium is represented by the other intersection with the y-axis, which lies far away from the dispersed information limit equilibria.
The third panel shows that eη goes respectively towards plus and minus infinity in the
dispersed-information limit equilibria (bear in mind that this is a reaction to a unitary
realization of a shock that in fact has infinitesimal variance).
In the last panel, we plot Vi the cross-sectional variance of conditional expectations,
which is zero in the perfect information limit equilibrium and it is given by (47) in the
dispersed information limit equilibria, irrespective of σξ2 . Notice the large multiplier effect
of the price feedback on private uncertainty: a vanishing idiosyncratic shock can generate
a large cross-sectional variance of beliefs.
Three remarks are in order here. First, in analogy with the insights of figure 1, only
one equilibrium out of three, survives in the whole range spanned by a strictly positive
value of σ 2 . This equilibrium entails dispersed-information at the limit σ 2 → 0. Hence,
the unique perfect-information equilibrium is not obtained, by continuity in σ 2 , from the
unique equilibrium emerging for a sufficiently high σ 2 .
Second, at the dispersed information limit equilibria, changes in σξ2 do not affect firms’
reaction to an aggregate shock as long as (46) holds. Nevertheless, a higher σξ2 decreases
the parameter region where a multiplicity exists for strictly positive σ 2 values.
Third, at the dispersed information limit equilibria, an individual expectation reacts
to a type-specific preference shock ξi by an amount that is inversely proportional to
its volatility. The second panel shows that, for an higher value of σξ2 , the equilibrium
reaction decreases. This effect occurs because, as long as σξ2 increases, the information
transmitted by the local wage becomes looser. This is an essential feature of learning
from prices. Notice that rational inattention (Mackowiak and Wiederholt, 2009) implies
the opposite comparative statics: the higher the variance of idiosyncratic shocks, the
higher the attention agents are willing to pay to them, and so the higher the sensitivity
of expectations to idiosyncratic conditions.
33
Impact of aggregate money shocks on aggregate variables
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
34
35
36
37
38
39
Let us turn our attention to the impact of aggregate shocks on aggregate variables.
Figure 5 illustrates the log-variation from the steady state of average price, production,
price for local capital and wage, induced by a positive unitary aggregate shock to the value
of money. In analogy to figure 4, the analysis is extended to values of σ 2 strictly above
zero. In particular, the picture sheds light on the role of a change in the elasticity of labor
supply. The plot is obtained for the same baseline calibration: = 8, ψ = 2,α = 0.33 and
27
eθ
1
eξ
0.5
0.9
eη
10
vari(E[x|r i,wi])
0.06
8
0.05
6
0.8
0.4
4
0.7
0.04
2
0.6
0.3
0
0.03
0.5
-2
0.2
0.4
0.3
-6
0.1
0.2
0.1
0.02
-4
0.01
-8
0
0.025
0
0.05
0
σ2
0.025
-10
0.05
0
σ2
0.025
0.05
0
0
σ2
0.025
0.05
σ2
Figure 4: Weights and cross-sectional variance of expectations for = 8, α = 0.33, ψ = 2,
γ = 0.75, ζ = 0 for: σξ2 = 0.4 (solid) and σξ2 = 1.4 (dashed).
p
1
y
0.8
r
1
0.8
0.4
w
1.5
1.4
0.6
0.6
1.3
0.4
0.4
0.2
1.2
0.2
0.2
0
1.1
0
0
0.025
0.05
σ2
0
0
0.025
0.05
-0.2
σ2
0
0.025
σ2
0.05
1
0
0.025
0.05
σ2
Figure 5: Impact on aggregate variables for = 8, α = 0.33, ψ = 2, σξ2 = 0.4, ζ = 0 for: γ = 0
(solid) and γ = 0.75 (dashed).
28
vari(pi)
vari(y i)
vari(ri)
vari(w i)
3
0.05
3
0.05
2
2
1
1
0
0
0.025
0.05
0
0
σ2
0.025
0
0.05
0
σ2
0.025
0.05
0
0
σ2
0.025
0.05
σ2
Figure 6: Cross sectional dispersion of local prices and produced quantities for = 8, α = 0.33,
ψ = 2, σξ2 = 0.4, γ = 0.75 and ζ = 0. The dotted line denotes the contribution of idiosyncratic
preference shocks to overall dispersion.
vari(pi)
vari(y i)
vari(ri)
vari(w i)
3
0.05
3
0.05
2
2
1
1
0
0
0.025
σ2
0.05
0
0
0.025
0.05
σ2
0
0
0.025
0.05
σ2
0
0
0.025
0.05
σ2
Figure 7: Cross sectional dispersion of local prices and produced quantities for = 8, α = 0.33,
ψ = 2, σξ2 = 0.4, γ = 0.75 for: ζ = 0 (solid) and ζ = 0.2 (dashed).
29
25
σξ2 = 0.4 and two different values of γ, namely 0 (solid) and 0.75 (dotted).19
At the perfect-information equilibrium all prices react one-to-one to a unitary positive
aggregate shock whereas average real quantities remain at steady state ( that is p = w =
r = θ and l = y = 0). At the dispersed-information equilibria instead, shocks to the value
of money feature real effects: aggregate price and production go up together, although the
price level rises less than under money neutrality. The temporary effect of money obtains
because, as local prices lose precision about aggregate conditions, firms are induced to
overestimate adverse local conditions; as a result, final producers set lower than optimal
prices, which boost actual demand. Notice that, at the dispersed information equilibrium,
the model predicts a rise in the real average wage (w−p) and a decrease in the real average
return on capital (r − p). This is because producers are rather pessimistic when they fix
their demand for capital, but then, once actual demand realizes, they hire more labor
than expected.
All these effects are consistent with the original insights of the Lucas island model
(Lucas, 1973) and the medium-term impact of monetary shocks empirically documented
in seminal works like Woodford (2003), Christiano et al. (2005) and Smets and Wouters
(2007), among others. Nevertheless, non-neutrality of money is obtained here only by
means of an arbitrarily small departure from the law of one price, in an economy where
final producers perfectly observe their marginal cost and optimally set their price.
Finally, concerning the role of the convexity in labor disutility, note that a higher γ
magnifies non-neutral effects on aggregate price, production and average wage, whereas
it has little effect on the price of capital. A higher γ also decreases the parameter region
where a multiplicity exists for strictly positive σ 2 -values.
Note that, in the case represented in figure 5, the qualitative features of the equilibria
at the limit of σ 2 → 0 also hold, by continuity, outside of the limit.
26
Impact of idiosyncratic shocks on island-specific variables
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
27
28
29
30
31
32
33
34
35
36
37
Perfect information entails homogeneous expectations about the aggregate component
of final demand. On the contrary, dispersed information limit equilibria feature sizable
dispersion of individual expectations, which magnifies the cross-sectional volatility of
prices and quantities in the economy. This increase in dispersion is generated by a tiny
cross-sectional variance of productivity shocks, so small that an external observer looking
for the fundamental source of firms’ disagreement could miss it. This results offers a
different interpretation of the price dispersion puzzles documented, for example, by Eden
(2013) and Kaplan and Menzio (2015), for which search and matching models (see Burdett
and Judd (1983)) generally offer an alternative explanation. We have showed, that, in
principle, it is possible to build a model where non-measurable dispersion in fundamentals
can generate sizable price dispersion in final markets without assuming any search friction.
19
Note that ζ does not affect the determination of the impact of the aggregate shock on aggregate
variables due to market clearing condition in the market for raw capital.
30
28
Figure 6 plots with a solid line the cross-sectional volatility of local price, production,
price of local capital and wage, as a function of σ 2 for the baseline calibration: = 6, α =
0.33, ψ = 2, σξ2 = 0.4, γ = 0.5 and ζ = 0. The dotted line denotes the fraction of overall
cross-sectional volatility which is only due to idiosyncratic preference shocks. At the
perfect information limit equilibrium, which corresponds to the y-intercept, idiosyncratic
preference shocks explain all the dispersion in the economy. At the dispersed information
limit equilibria instead, which correspond to the point of tangency of the curve with
the y-intercept, the relative contribution of preference shocks to the overall dispersion of
final prices falls, although their absolute contribution increases. Hence, on the one hand,
vanishingly small productivity shocks explain a sizable fraction of price dispersion; on the
other hand, the impact of preference shocks is also magnified.
This finding is in line with the evidence put forward by Boivin et al. (2009) that
whereas prices respond little to aggregate shocks they respond largely to idiosyncratic
shocks, so that, the latter explain most overall price volatility. The same picture holds
for output and wage cross-sectional variance, but not for the price of local capital whose
dispersion is exogenously fixed by σ 2 .
Figure 7 explores the role of decreasing returns to scale in intermediate production.
The solid line plots the same solid line in figure 6, whereas the dashed line plots the
curve for ζ = 0.2. The amount of cross-sectional variance generated by a vanishing σ 2 is
measured by the distance between the intercept on the y-axis and the point of tangency
which features dispersed-information. A higher ζ generally increases the cross-sectional
volatility of final prices, output, wages and the prices of capital, although decreases dispersion of final prices and output when close to the perfect information limit equilibrium.
In particular, note that the prices of local capital, which are now also a function of traded
quantities, reach a higher dispersion in the perfect-information equilibrium than in the
dispersed-information equilibria.
Again, in both the cases represented in figure 6 and 7, the qualitative feature of the
equilibria at the limit of σ 2 → 0 hold, by continuity, outside of the limit.
29
4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
30
31
32
33
34
35
36
37
38
39
Conclusion
This paper set out the conditions under which the introduction of an arbitrarily small
degree of dispersion in fundamentals can feature price rigidity as an equilibrium outcome.
When agents learn from local prices, no matter how small their dispersion, the tension
between the allocative and informative role of prices can dramatically alter their functioning. We showed that this mechanism is naturally reproduced by a typical input demand
function, and so potentially applicable to a large class of macro models. In particular,
we presented a standard monetary model with flexible prices with a unique equilibrium
under perfect information where money neutrality holds. We showed that, beyond the
perfect information limit equilibrium, the introduction of vanishingly small heterogeneity
in input markets produces price rigidity to aggregate monetary shocks and overreaction
31
1
2
3
4
5
6
7
8
9
10
11
12
to idiosyncratic shocks as an equilibrium outcome. This equilibrium, which I tagged as
a dispersed information limit equilibrium, has strong stability properties, in contrast to
the perfect-information equilibrium.
We left some important issues for future research. The model I presented is essentially
static in nature, as the realization of the shock is not informative about the future course
of the economy. When this is not the case, agents accumulate additional correlated
information over time. The extent to which non-neutralities can persist in a economy
where agents learn from current and past prices is a question that hopefully this paper
can help to address in a near future. Finally, despite the fact that the model can naturally
generate nominal rigidities and even price stickiness in the limit, it cannot reproduce the
frequency of price adjustments, which is the focus of most of the micro econometric
literature on price setting.
32
1
Appendix
2
A
3
A.1
Two-market two-shock model
Proof. of proposition 1
With σ 2 = 0, bi (b) is linear, so it is trivial to prove b∗ = (µ − α)−1 to be the unique solution
of the fix point equation bi (b) = b. No matter how small is σ 2 , the fix point equation bi (b) = b
is instead a cubic, so that it has at most three real roots. By continuity there must exist a
positive solution b◦ such that limσ2 →0 bi (b◦ ) = b◦ = b∗ . The condition µ > α is necessary and
sufficient to have
µ
lim b0i (b∗ ) = lim b0i (b◦ ) = > 1.
2
2
α
σ →0
σ →0
23
In such a case, since for any non-null σ 2 we have limb→+∞ bi (b) = 0, then by continuity at least
an other intersection of bi (b) with the bisector at a positive value b+ must exist too. Notice also
that bi (0) < 0. This, jointly with the fact that for σ 2 6= 0 it is limb→−∞ bi (b) = 0, implies that
by continuity at least an intersection of bi (b) with the bisector at a negative value b− exists too.
Therefore (b− b◦ , b+ ) are the three solutions we were looking for.
Suppose now that b− and b+ take finite values - that is |b± | ≤ M with M being an arbitrarily
large finite number - then limσ2 →0 σ 2 (1 − b± )2 = 0 and so necessarily b± = b◦ > 0. Nevertheless
b− < 0 so a first contradiction arises. Moreover if b+ is arbitrarily close to b◦ then by continuity
limσ2 →0 b0i (b+ ) > 1 which, for the same argument above, would imply the existence of a fourth
root; a second contradiction arises. Hence we conclude |b± | > M with M arbitrarily large.
Finally let me prove that a multiplicity arises for a σ 2 small enough if and only if µ > α.
For µ < α, two cases are possible, either 0 < limσ2 →0 b0i (b◦ ) < 1 or limσ2 →0 b0i (b◦ ) < 0. The case
0 < limσ2 →0 b0i (b◦ ) < 1 provides, for a σ 2 small enough, at least one intersection of bi (b) with
the bisector must exist by continuity. Nevertheless in this case such intersection is unique as
the existence of a second one would require max b0i (b) > 1 given that limb→±∞ bi (b) = 0± but
supb limσ2 →0 b0i (b) = limσ2 →0 b0i (b◦ ). With limσ2 →0 b0i (b◦ ) < 0 instead - for a σ 2 small enough
- the curve bi (b) either is strictly decreasing in the first quadrant (bi (b) > 0, b > 0) and never
lies in the fourth quadrant (bi (b) < 0, b < 0) or is strictly decreasing in the fourth quadrant
and never lies in the first quadrant. Hence limσ2 →0 bi (b) can only have one intersection with
the bisector.
24
A.2
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
25
Proof. of proposition 2
Using (12) we get
lim lim (1 − µb)2 σ 2 = −α2 − lim
σ 2 →0
26
27
b→b±
b→b±
α(1 − µb)
= α(µ − α),
b
(50)
which is positive provided µ > α. The cross-sectional variance of individual expectations is
given by b2 σ 2 . Notice that
lim lim (1 − µb)2 σ 2 = lim lim µ2 b2 σ 2 ,
σ 2 →0 b→b±
σ 2 →0 b→b±
33
(51)
and hence,
lim lim b2 σ 2 =
σ 2 →0 b→b±
α(µ − α)
µ2
Given (9) it is easy to show that
lim lim E[θ|ri ] = − lim lim
σ 2 →0
σ 2 →0
b→b±
b→b±
αb
α
θ = θ,
1 − µb
µ
so that the local price in the limit becomes
lim lim ri = µ lim lim E[θ|ri ] − αθ − lim ηi = 0.
1
A.3
σ 2 →0
σ 2 →0 b→b±
σ 2 →0 b→b±
Proof. of proposition 3
The derivative of (12) with respect to b is given by
b0i (b)
=−
αµ µ2 b2 σ 2 − 2µbσ 2 − α2 + σ 2
(µ2 b2 σ 2 − 2µbσ 2 + α2 + σ 2 )2
.
For the perfect information limit equilibrium, we have
lim lim b0i (b) =
σ 2 →0 b→b◦
µ
> 1.
α
For the dispersed information limit equilibria notice that
lim lim
σ 2 →0 b→b±
2
b0i (b)
=
−αµ α(µ − α) − α2
(α(µ − α) + α2 )2
after using (50) and (51) to compute the limit limσ2 →0 limb→b± µ2 b2 σ 2 . Finally we can prove
lim lim b0i (b) > −1,
σ 2 →0 b→b±
lim lim b0i (b), < 1
σ 2 →0 b→b±
3
given that, respectively,
2
αµ α2 + α (α − µ) − α2 − α (α − µ)
= −2α2 µ (µ − α) < 0,
2
αµ α2 + α (α − µ) + α2 − α (α − µ)
= 2α3 µ > 0.
4
A.4
Proof. of proposition 4
To check local learnability of the REE, suppose we are already close to the rest point of the
R
system. That is, consider the case limt→∞ bi,t di = ˆb where ˆb is one among the equilibrium
points {b− , b◦ , b+ } and so
lim Si,t = var(ri,t ) =
t→∞
34
α2
(1 − µˆb)2
+ σ2,
1
2
according to (10). From standard results in the stochastic approximation theory, we can write
the associated ODE governing the stability around the equilibria as
Z
h
i
dbt
−1
=
lim E Si,t−1
ri,t (θt − bi,t−1 ri,t ) di =
t→∞
dt
Z
= var(ri,t )−1 E [ri,t (θt − bi,t−1 ri,t )] di =
= var(ri,t )−1 (cov(θ, ri,t ) − bt var(ri,t )) =
= bi (bt ) − bt ,
3
where we used relation (12).
4
B
5
B.1
6
7
8
9
10
11
12
The Fragile Neutrality of Money
Proof. of proposition 5
First-order conditions and information in the extended model. The aim here
is to show that for each variable in the model there exists a unique steady state and a unique
log-linear deviation from a deterministic component implied by fixing a profile of coefficients
(eθ , eξ , eη ). We will consider the extended version of the economy with (48) and (49) instead of
(18) and (21). The baseline configuration obtain for γ = ξ = 0. Let us start providing the full
list of first-order conditions and discuss how final producers’ learning is affected in the extended
model.
The household optimizes her problem when
Yi = Pi− P Y,
θ
P =e Y
Wi =
13
14
15
16
17
18
19
20
21
22
23
−ψ
(foc1)
,
(foc2)
eθ+ξi Lγi ,
(foc3)
¯ )−1 being normalized to one without any loss of
where we already used (34) with δ((1 − δ)M
generality.
Because of the convexity in the disutility of labor, the wages cannot be determined independently from the quantity of labor hired. In this case, there is a conflict between observing
the equilibrium wage at stage two and leaving working hours determined by actual demand in
stage three. To circumvent this difficulty we assume that, at the time when final producers
make their decision (Ki , Pi ), they are informed about the labor supply schedule (foc3) posted
by workers, i.e. final producers know the equilibrium wage contingent to each possible quantity
traded. As a consequence, knowing the the labor supply schedule is informationally equivalent
to observing eθ+ξi , which is consistent with our baseline model.
The intermediate producers maximize their profits when
ζ
Ri = (1 − ζ)−1 e−ηi RZ(i)
.
24
25
26
27
(foc4)
Decreasing returns to scale on intermediate production makes Ri depends on Z(i) . At the same
1−ζ
time Ki = Kis = eηi Z(i)
is known, as it is final producers’ own action, and does not convey
additional information (see note 9 in analogy). Hence, the information that final producers get
from the local capital market is summarized by
35
ˆ i ≡ (1 − ζ)Ri K −ζ/(1−ζ) = e−ηi /(1−ζ) R,
R
i
which is identical to (31) up to a re-scaling of the variance of ηi . Therefore, final producˆ i , eθ+ξi ) that constitutes the information
ers optimize their expected profits conditionally to (R
transmitted by the local input markets. The first order condition of final producers relative to
the choices (Ki , Pi ) that they carry out in the second stage are
ˆ i , eθ+ξi ],
Ki = α1−α (1 − α)α−1 E[Wi1−α Riα−1 Yi |R
ˆ i , eθ+ξi ].
Pi = ( − 1)−1 α−α (1 − α)α−1 E[Riα W 1−α |R
i
(foc5)
(foc6)
1
In the third stage final producers hire working hours to satisfy actual demand.
2
Log-linear algebra: preliminaries. This section aims to clarify some issues linked to
3
4
5
6
7
8
the manipulation of log linear relations. The requirement that the equilibrium has a symmetric
log-linear representation implies that the individual production Yi , as any other variable in the
model, has the form
Yi (θ, ξi , ηi ) = Y¯i eyi (θ,ξi ,ηi ) ,
(54)
where yi ≡ log Yi − log Y¯i is defined as the distance of the logarithm of the particular realization
Yi from a constant deterministic component Y¯i . In particular, the log deviation yi is a linear
combination of the shocks
yi (θ, ξi , ηi ) = yi,θ θ + yi,ξ ξi + yi,η ηi ,
9
10
11
12
13
where we use roman characters to index the relative weight of a shock in the combination.
Notice that the stochastic steady state of individual production is given by its unconditional
mean, Y¯i evar(yi )/2 , with var (yi ) being the unconditional variance of yi . In analogy, let us denote
by ya,θ , ya,ξ and ya,η the log-deviations of the aggregate Y relative to a unitary increase in θ, ξ
and η, respectively. Of course, by definition we have
Y (θ) = Y¯ eya,θ θ ,
14
15
Yi
17
18
(56)
that is ya,ξ = ya,η = 0. At the same time, the aggregate production can be obtained by aggregating (54) across agents
Z
16
(55)
θ−1
θ
θ
θ−1
θ−1
di
= Y¯i eyi,θ θ+ 2θ var(yi,ξ ξi +yi,η ηi ) ,
(57)
so that, contrasting (54) and (57), we conclude yi,θ = ya,θ for any θ > 1. In other words, the
average log deviation of the individual production is equal to the log deviation of the aggregate
production. This is true for whatever CES aggregation.
Therefore, in the following we are going to drop the distinction between individual and
aggregate focusing on (yθ , yξ , yη ) with the understanding that yθ = yi,θ = ya,θ , yξ = yi,ξ and
yη = yi,η . Regarding the deterministic component of (56) and (57), notice that
θ−1
Y¯ = Y¯i e 2θ var(yi,ξ ξi +yi,η ηi ) ,
36
1
2
3
4
5
6
7
8
9
10
11
12
that is, both coincide only in the deterministic scenario, in which case they denote the deterministic steady state.
Uniqueness. Under the restriction of log linearity the logarithm of each variable is determined by four components, which lie in the four independent complementary subspaces spanned
by (1, θ, ξi , ηi ). We can therefore study the reaction of the system to each component separately.
First, we are going to consider the reaction to an unitary aggregate shock θ. Let us determine
the variables fixed at stage 2, for a given eθ , which represents the weight on θ in the stochastic
1
component of the individual expectation about X = P Y . To do that in the extended model
we need to introduce expectations of final producers over equilibrium wage and labor, which
in turn will depend on expected demand for the fixed price. This corresponds to the following
system of relations derived from (foc1), (foc3),(foc4),(foc5), (foc6), the market clearing condition
R
z(i) di = 0 and the final technology (23):20
yθe = (eθ − pθ )
wθe = γleθ + 1
kθ = 0 = (1 − α)(wθe − rθ ) + yθe
pθ = αrθ + (1 − α)wθe ,
(1 − α)leθ = yθe ,
13
14
15
16
where wθe , leθ and yθe denote the impact of an unitary increase in θ on final producers’ expectations
about Wi , Li and Yi , respectively. The expectation about Li is needed only with γ 6= 0 as the
producers needs to forecast the equilibrium wage to determine an optimal price. The system
yields a unique solution
yθe = ∆θ ((1 − α)(eθ − 1))
wθe = ∆θ (γeθ + (1 − α + α))
rθ = ∆θ ( (1 + γ) eθ − ( − 1)(1 − α))
pθ = ∆θ ( (α + γ) eθ + 1 − α),
leθ = ∆θ (eθ − 1)
17
18
19
with ∆θ ≡ (1 − α + (α + γ))−1 . Then, we can determine the equilibrium variables fixed at the
third stage. The relations (foc2),(foc3), the final technology (23) and the definition of X (37)
determine, for a given pθ , the actual impact on xθ , yθ , lθ and wθ in the last stage, so that
1 − ψyθ = pθ ,
wθ = γlθ + 1.
yθ = (1 − α)lθ ,
xθ = −1 yθ + pθ ,
In particular, note that
xθ = (( − ψ −1 ) (α + γ) eθ + 1 − α + ψ −1 (α + γ))∆θ
20
foc2 is needed only if one uses expectations about either aggregate price P or aggregate demand Y
rather than expectation on the combination of the two X, as we do here.
37
1
2
3
4
5
6
which is consistent with (41). Moreover, for eθ = 1 we have pθ = rθ = wθ = wθe = 1 and
yθ = yθe = lθ = lθe = 0, which entail the neutrality-of-money outcome.
Let turn attention now to the impact of idiosyncratic deviations, which only concern islandspecific variables. Let us start with the variables fixed at the second stage. The system relative
to ξ and η is determined by (foc1),(foc3),(foc4),(foc5),(foc6) and technologies (23) and (49),
yielding:
e
yξ,η
= (eξ,η − pξ,η ),
e
wξ,η
= γleξ,η + 1ξ
rξ,η = ζzξ,η − 1η ,
(1 − ζ)zξ,η = kξ,η − 1η
e
e
kξ,η = yξ,η
+ (1 − α)(wξ,η
− rξ,η )
e
pξ,η = αrξ,η + (1 − α)wξ,η
e
(1 − α)leξ,η = yξ,η
− αkξ,η .
7
8
where I index the direct impact of the two shocks ξi and ηi on an individual expectation about
X with 1ξ and 1η , respectively. There is a unique solution to the system given by
yξe = ∆( (αγ − (1 + γ) αζ + 1) eξ − (1 − α))
rξ = ∆(ζ (γ + 1) eξ − ζ (1 − α) ( − 1))
wξe = ∆(γeξ + 1 + ζ ( − 1) α)
pξ = ∆( (γ (1 − α) + αζ (1 + γ)) eξ + 1 − α)
leξ = ∆(eξ − α − (1 − α) − ζα ( − 1))
zeξ = ∆( (γ + 1) eξ − (1 − α) ( − 1))
kξ = ∆( (γ + 1) (1 − ζ) eξ − (1 − α) ( − 1) (1 − ζ)).
9
and
yηe = ∆(( + γα − ζ(1 + γ)α)eη + α (γ + 1))
rη = ∆(ζ (γ + 1) eη − 1 − γ (α + (1 − α)))
wηe = ∆(γeη + αγ( − 1))
pη = ∆((γ (1 − α) + αζ(1 + γ))eη − α (1 + γ))
leη = ∆(eη + α ( − 1))
zeη = ∆( (1 + γ) eη + α (1 + γ) ( − 1))
kη = ∆( (1 + γ) (1 − ζ) eη + 1 + ( − 1) α + γ).
10
11
with ∆ ≡ (1 + γ (α (1 − ) + ) + ζ ( − 1) (1 + γ) α)−1 . Finally, the relations (foc1),(foc3) and
(23) determine the actual cross section of yi , li and wi as
yξ,η = −pξ,η ,
wξ,η = γlξ,η + 1ξ ,
yξ,η = αkξ,η + (1 − α)lξ,η ,
38
1
2
3
4
5
6
7
8
9
10
11
12
13
for given pξ,η and kξ,η .
Finally, to pin down the deterministic component we can easily solve the model in the deterministic case. The height relations (foc2), (foc3), (foc4), (foc5), (foc6), (23), (49) and the
R
market clearing condition z(i) di = 0, form a linear system in height unknown, namely log Y¯ ,
¯ i , log W
¯ i , log R,
¯ log L
¯ i , log K
¯ i , log Z¯(i) (implicitly, expectations on deterministic
log P¯ , log R
components are trivially correct) which define a unique deterministic steady state. The unique
stochastic steady state of Y , as of any other variable in the model, is easily given by Y¯ evar(y)/2 .
Therefore, also the stochastic steady state is uniquely determined with island-specific and aggregate steady state variables differing for a second-order constant term.
B.2
Proof. of proposition 6
Given that all agents use the rule (45) then the average expectation is
Z 1
a + (1 − φ) b
E[x|wi , ri ]di =
θ.
1 − φb
0
Hence, an individual expectation can be rewritten as
a + (1 − φ) b
E[x|wi , ri ] = b φ
θ + (1 − φ) θ + ηi + a (θ + ξi ) ,
1 − φb
1 − φ + φa
E[x|wi , ri ] = b
θ + ηi + a (θ + ξi ) .
1 − φb
15
17
18
20
a + (1 − φ) b
θ,
1 − φb
− b 1−φ+φa
1 − β + β a+(1−φ)b
1−φb
1−φb
1 + σξ2
1−φb
1−φb
(1 − β) 1−φ+φa
+ β a+(1−φ)b
−
a
1−φ+φa
1−φ+φa
bi (a, b) =
2
1−φb
1 + 1−φ+φa
σ2
(61)
(62)
(63)
provided b 6= φ.
From (62) we can recover the expression for the equilibrium a = ai (a, b) which is
a=
19
(60)
as functions of weights and exogenous shocks only. Now we have a bidimensional individual
best weight function written as
ai (a, b) =
16
(59)
The actual law of motion of the aggregate component of island-specific demand (41) is given by
x = (1 − β) θ + β
14
(58)
(1 − β) (1 − b)
.
1 − β + (1 − bφ) σξ2
(64)
Notice that, with σξ2 6= 0, a takes finite values for any b. Plugging the expression above at the
numerator or (63) we obtain a fix point equation in b:
bi (b) =
(φ−β)σξ2 −(1−β)
b
(φ−1)σξ2 −(1−β)
1+
−
(1−β)σξ2
(φ−1)σξ2 −(1−β)
1−φb
1−φ+φa
39
2
σ2
(65)
1
2
3
4
5
which is a cubic continuous function. With σ 2 = 0, bi (b) is linear, so it is trivial to prove
(a∗ , b∗ ) = (0, 1) to be the unique solution of the fix point equation (65). No matter how small
is σ 2 , the fix point equation (65) is instead a cubic which can be satisfied at most by three real
roots. By continuity there must exist a positive solution b◦ such that limσ2 →0 bi (b◦ ) = b◦ = b∗ .
The condition
1−β
σξ2 >
with φ > 1
(66)
φ−1
is necessary and sufficient to have
lim b0i (b∗ ) =
σ 2 →0
(φ − β) σξ2 − (1 − β)
(φ − 1) σξ2 − (1 − β)
>1
for any β < 1. In such a case given that
lim b0i (b◦ ) = lim b0i (b∗ )
σ 2 →0
6
7
8
9
10
11
12
13
14
15
16
σ 2 →0
and that for σ 2 6= 0 we have limb→+∞ bi (b) = 0, then by continuity at least an other intersection
of bi (b) with the bisector at a positive value b+ must exist too. Moreover, (66) also implies
bi (0) < 0. This, jointly with the fact that for σ 2 6= 0 it is limb→−∞ bi (b) = 0, implies that by
continuity at least an intersection of bi (b) with the bisector at a negative value b− also exists.
Therefore (b− b◦ , b+ ) are the three solutions we were looking for.
Now suppose that b− and b+ take finite values - that is |b± | ≤ M with M being an arbitrarily
large finite number - then, since a is finite too, limσ2 →0 σ 2 ((1 − φb± ) /(1 − φ + φa))2 = 0 and
so necessarily b± = b◦ > 0. Nevertheless b− < 0 so a first contradiction arises. Furthermore, if
b+ is arbitrarily close to b◦ then by continuity limσ2 →0 b0i (b+ ) > 1 which, for the same argument
above, would imply the existence of a fourth root; a second contradiction arises. Hence we can
conclude |b± | > M with M arbitrarily large.
26
Finally let me prove that a multiplicity arises for a σ 2 small enough if and only if (66) holds.
Out of these conditions two cases are possible, either 0 < limσ2 →0 b0i (b◦ ) < 1 or limσ2 →0 b0i (b◦ ) <
0. The case 0 < limσ2 →0 b0i (b◦ ) < 1 yields, for a σ 2 small enough, at least one intersection of
bi (b) with the bisector must exist by continuity. Nevertheless in this case such intersection
is unique as the existence of a second one would require supb limσ2 →0 b0i (b) > 1 given that
limb→±∞ bi (b) = 0± but supb limσ2 →0 b0i (b) = limσ2 →0 b0i (b◦ ). With limσ2 →0 b0i (b◦ ) < 0 instead
- for a σ 2 small enough - the curve bi (b) either is strictly decreasing in the first quadrant
(bi (b) > 0, b > 0) and never lies in the fourth quadrant (bi (b) < 0, b < 0), or is strictly
decreasing in the fourth quadrant and never lies in the first quadrant. Hence limσ2 →0 bi (b) can
only have one intersection with the bisector.
27
B.3
17
18
19
20
21
22
23
24
25
Proof. of proposition 7
i) From (64) we get that
lim a =
b→±∞
28
1−β
.
φσξ2
Using (65) we have that
2
(φ − β) σξ2 − (1 − β)
(1 − β) σξ2
1 − φb
lim lim
σ2 =
−
1
=
.
(φ − 1) σξ2 − (1 − β)
(φ − 1) σξ2 − (1 − β)
σ 2 →0 b→b± 1 − φ + φa
40
(69)
Using the relation above we can write the total relative precision of private information about
θ is given by
σξ−2
lim lim τ = lim lim
σ 2 →0 b→b±
σ 2 →0 b→b±
+
1 − φb
1 − φ + φa
!
−2
σ
−2
=
= σξ−2 +
(φ − 1) σξ2 − (1 − β)
(1 −
β) σξ2
=
φ−1
.
1−β
which is positive provided φ > 1. ii) The cross sectional variance of individual expectations is
given by a2 σξ2 + b2 σ 2 . Note that
lim lim
σ 2 →0
b→b±
1 − φb
1 − φ + φa
=
2
σ 2 (1 − φ + φa)2 = lim lim (1 − φb)2 σ 2 =
σ 2 →0 b→b±
(1 − β) σξ2
(φ −
1) σξ2
− (1 − β)
1−β
1−φ+φ
φσξ2
!2
=
1−β
(φ − 1) σξ2 − (1 − β) .
2
σξ
We also have,
lim lim
σ 2 →0 b→b±
(1 − φb)2 σ 2
= lim lim b2 σ 2 .
φ2
σ 2 →0 b→b±
Using the relations above we get
lim lim a2 σξ2 + b2 σ 2 =
σ 2 →0 b→b±
=
(1 − β)2
1−β
2
−
(1
−
β)
+
(φ
−
1)
σ
=
ξ
σξ2 φ2
φ2 σξ2
= (1 − β)
1
which is positive provided φ > 1.
2
B.4
φ−1
φ2
Proof. of proposition 8
To check convergence in higher order beliefs we need to build up the Jacobian computed
around the equilibria using (62)-(63). It is
!
lim lim J =
σ 2 →0 b→b±
41
∂ai
∂a
∂bi
∂a
∂ai
∂b
∂bi
∂b
,
1
with
J11 =
φb
1−φb
σξ2
1
β 1−φb
−
1+
,
J12 =
(β − 1) (aφ + 1 − φ)
,
1 + σξ2 (1 − bφ)2
J21 =
a+(1−φ)b
(1−φb)2 σ 2
1−φb
1−φb
− (1−β)(1−bφ)
2φ
(1
−
β)
+
β
−
a
2
1−φ+φa
1−φ+φa
1−φ+φa
(aφ−φ+1)
(1−φ+φa)3
+
,
2
2
2
1−φb
2
1−φb
1 + 1−φ+φa σ
1+
σ2
1−φ+φa
J22 =
2
(1−φb)σ 2
1−φb
1−φb
2φ (1 − β) 1−φ+φa
+ β a+(1−φ)b
−
a
1−φ+φa
1−φ+φa
(1−φ+φa)2
.
+
2
2 2
1−φb
1−φb
σ2
1 + 1−φ+φa
2
1 + 1−φ+φa σ
β−φ+aφ
1−φ+φa
The latter terms become
2
J21 =
b(1−φb)
2
− (1−β)(1−bφ)
2φ (1−φ+φa)
3σ
(aφ−φ+1)2
+
2
2 ,
1−φb
1−φb
1 + 1−φ+φa
σ 2 1 + 1−φ+φa
σ2
J22 =
b(1−φb)
2
2φ (1−φ+φa)
2σ
+
2
2 .
1−φb
1−φb
2
1 + 1−φ+φa σ
1 + 1−φ+φa σ 2
β−φ+aφ
1−φ+φa
after substituting for (63). Notice that
lim lim J21 J12 = 0
σ 2 →0 b→b±
since a is always finite, J21 goes to infinity of order b± whereas J12 goes to an infinitesimal
2
of order b−2
± (as remember limσ 2 →0 b± σ is a finite value). Therefore the eigenvalues of the
Jacobian are
1
lim lim J11 =
∈ (0, 1)
2
b→b
1 + σξ2
σ →0
±
and
lim lim J22 = 1 − 2
σ 2 →0 b→b±
Γ
∈ (−1, 1)
1+Γ
where (see also (69))
Γ ≡ lim lim
σ 2 →0 b→b±
3
4
1 − φb
1 − φ + φa
2
σ2 =
(1 − β) σξ2
(φ − 1) σξ2 − (1 − β)
>0
if and only if (66) holds. Therefore, whenever dispersed information limit equilibria exists they
are stable outcome of a convergent process in higher order beliefs.
At the perfect-information limit equilibrium instead the Jacobian is given by
!
β−φ
1−β
−
2
2
)(1−φ)
(1+σξ )(1−φ) .
lim lim J = (1+σξ1−β
β−φ
σ 2 →0 b→1
− 1−φ
1−φ
42
The product of its eigenvalues (the determinant of J) is
∆ (J) = 1
1 − 2β + φ
,
1 + σξ2 (φ − 1)
and their sum (the trace of J) is
2 + σξ2 (φ − β)
Tr (J) = .
1 + σξ2 (φ − 1)
Provided Tr (J)2 > 4∆ (J), the largest real eigenvalue is greater than one whenever
q
1
1
Tr (J) +
Tr (J)2 − 4∆ (J) > 1
2
2
that is Tr (J) > 1 + ∆ (J), which requires σξ2 (1 − β) > 0 provided φ > 1, which is true for
whatever β < 1 (this also implies ∆ (J) > 0). Let us prove now Tr (J)2 > 4∆ (J), that is
2
2 + σξ2 (φ − β)2
> 4 (1 − 2β + φ) ,
1 + σξ2 (φ − 1)
2
that is equivalent to
(β − φ)2 σξ2 + 4 (β − φ)2 − 4 (φ − 1) (φ − 2β + 1) σξ2 +
+ 4 (β − φ)2 − 4 (φ − 1) (φ − 2β + 1) > 0
3
4
5
6
7
which holds whenever (φ − 1) (φ − 2β + 1) < 0 that is always true for β < 1 and φ > 1. Hence,
the perfect information limit equilibrium entails a divergence in higher order belief dynamics.
Finally, notice that J22 alone characterizes the univariate case putting a = 0 which obtains
at the limit σξ2 → ∞
B.5
Proof. of proposition 9
To check local learnability of the REE, suppose we are already close to the rest point of the
R
R
system. That is, consider the case limt→∞ ai,t di = a
ˆ and limt→∞ bi,t di = ˆb where (ˆ
a, ˆb)
characterizes an equilibrium, so that


1−φ+aφ
1 + σξ2
1−bφ


.
lim Si,t = S = 
2


t→∞
1−φ+aφ
1−φ+aφ
+ σ2
1−bφ
1−bφ
8
9
From standard results in the stochastic approximation theory, we can write the associated ODE
governing the stability around the equilibria as
Z
h
i
dχi,t
−1
0
0
=
lim E Si,t−1
ωi,t
xt − ωi,t
χi,t−1 di =
t→∞
dt
Z
0
−1
0
= S
E ωi,t
xt − ωi,t
χi,t−1 di =
= S−1 (cov(x, ωi,t ) − χi,t−1 =
= B (ai,t−1 , bi,t−1 ) − [ai,t−1 bi,t−1 ]0 ,
43
1
where we used relation (62)-(63). The proposition directly follows from the conducted in (B.4).
2
C
3
C.1
Other Robustness Checks
4
5
Two-market two-shock model: decreasing returns to scale in intermediate production
Consider the problem of intermediate producers in island i being:
1−ϑ
max{eθ+ηi Z(i)
Ri − Z(i) Q},
(71)
Z(i)
6
7
where ϑ ∈ (0, 1) measure returns to scale on intermediate production. The problem (71) generalizes (1)-(2) in the text which obtains for ϑ = 0. The first order condition of (3) is
−ϑ
(1 − ϑ)eθ+ηi Z(i)
Ri = Q,
8
which can be expressed in log-deviations from the steady state as
ri = q + ϑz(i) − θ − ηi .
9
The latter combined with (6) and market clearing in the endowment market
Z 1
E[θ|ri ]di − αθ + ϑz(i) − ηi .
ri = µ
R1
0
z(i) di = 0 gives
(72)
0
10
11
Given that the quantity traded in equilibrium ki = θ + ηi + (1 − ϑ)z(i) is known by the final
producer, then ri is observationally equivalent to
Z 1
ϑ
ki = µ
E[θ|˜
ri ]di − α
˜ θ − η˜i ,
r˜i = ri −
1−ϑ
0
where
(1 − ϑ)α + ϑ
1
, η˜i ≡
ηi ,
1−ϑ
1−ϑ
and E[θ|ri ] = E[θ|˜
ri ]. Therefore the analysis in the text can be exactly replicated. A multiplicity
arises in this case when µ > α
˜ , that is,
α
˜≡
12
13
µ−α>
ϑ
,
1−ϑ
(73)
17
which becomes more and more restrictive as ϑ approaches 1. At that point in fact, the term α
˜,
which measures the strength of the allocation effect, goes to infinity so that θ is fully revealed
irrespective of any finite value of µ. One other way to see it is to notice that for ϑ = 1 it is
ki = θ, meaning that each producer knows the fundamental.
18
C.2
14
15
16
19
The fragile neutrality of money: Extensions of the signal extraction
problem
Here we reconsider the signal extraction problem solved by proposition 6 assuming a more
general specification of the additional signal provided, in that context, by the local wage. Suppose agents forecast (41) while observing (39) and an additional signal:
Z 1
sˆi = θ + ϑe
E[x|ˆ
si , ri ]di + ϕˆ + ξˆi
0
44
1
2
where ϕˆ is normally distributed shock orthogonal to the fundamental θ, and ϑe is a constant.
Combining linearly21 sˆi and ri we can obtain:
φ
ϑe −1
ri − sˆi = θ + ϑηi + ϕ + ξi ,
si ≡ 1 − φ −
φ
ϑe
3
4
5
6
7
(74)
where ϕ and ξi are re-scaled realizations of ϕˆ and ξˆi , respectively, and ϑ is a constant.
An equilibrium is characterized by the two coefficients (ˆ
a, ˆb) that weight sˆi and ri , respectively, such that the individual expectation expressed as a linear combination of these two
signals is a rational expectation. Then, for each couple (ˆ
a, ˆb) there must exist another couple
(a, b) where: b weights ri , and a weights si , such that
Z 1
(75)
E[x|ri , si ]di + (1 − φ)θ + ηi + a (θ + ϑηi + ϕ + ξi )
E[x|ri , si ] = b φ
0
8
9
10
11
12
is still a rational expectation. Let us denote by σϕ2 the variance of the common stochastic
component ϕ which measures the commonality of the information structure. This specifications
nests several cases:
i) σϕ2 = 0, ϑ = 0 is when sˆi is the wage signal studied in proposition 6;
ii) σϕ2 6= 0, ϑ = 0 is when sˆi is an exogenous private signal correlated across agents;
13
iii) σϕ2 = 0, ϑ 6= 0 is when sˆi is an endogenous private signal like ri ;
14
iv) σϕ2 6= 0, ϑ 6= 0 is when sˆi is an endogenous private signal correlated across agents.
15
16
17
18
We will show here that, in all these cases, the conditions for the existence of dispersed limit
equilibria - characterized as b → b± where b2± σ 2 = b± with κ finite - are identical to the ones
uncovered in proposition 6.
Given that all agents use the linear rule above the average expectation is
Z 1
a
a + (1 − φ) b
E[x|ri , si ]di =
θ+
ϕ.
(76)
1 − φb
1 − φb
0
Hence, an individual expectation can be rewritten as
φa
1 − φ + φa
E[x|ri , si ] = b
θ+
ϕ + ηi + a(θ + ϑηi + ϕ + ξi ).
1 − φb
1 − φb
19
20
The actual law of motion of the aggregate component of island-specific demand (41) is given by
a
a + (1 − φ) b
θ+
ϕ ,
(77)
x = (1 − β) θ + β
1 − φb
1 − φb
as functions of weights and exogenous shocks only. The fix point equation is defined by
E[si (x − bri − asi )] = 0,
E[ri (x − bri − asi )] = 0,
21
Of course, the expectation operator in ri should be written conditional to sˆi instead of wi .
45
1
that gives,
1−β+β
a=
a+(1−φ)b
1−φb
+
a
2
1−φb σϕ
1 + ϑ2 σ 2 +
2
(1−β)(1−φb)
1−φ+φa
+β
a+(1−φ)b
1−φ+φa
+
φa2
σ2
(1−φ+φa)2 ϕ
b=
1+
3
1−φb
1−φ+φa
−a
2
1−φ+φa
1−φb
2
σϕ + σξ2
−b
σ2 +
1−φb
1−φ+φa
+
φa
2
1−φb σϕ
+ ϑσ 2
φa(1−φb) 2
σ
(1−φ+φa)2 ϕ
+
2
φa
1−φ+φa
(78)
+
1−φb
1−φ+φa
2
ϑσ 2
σϕ2
(79)
provided b 6=
It is easy to prove by substitution that for
→ 0, (a, b) = (0, 1) is a solution.
Let now prove that the condition for the existence of dispersed information limit equilibria are the same in this extended version. From (78) we can recover the expression for the
equilibrium which is
(1 − β) (1 − b) − b (1 − bφ) σ 2 ϑ
a=
(1 − β)(1 + σϕ2 ) + (1 − bφ) σξ2 + σ 2 ϑ2
φ−1 .
σ2
Let us look now for the solution at the limit σ 2 → 0 such that b → b± where limσ2 →0,b→b± b2 σ 2 →
κ with κ being finite (so that b± → ±∞). In such a case, the equilibrium value of a takes finite
values for any b, provided σξ2 6= 0, in particular:
lim
σ 2 →0,b→b±
4
5
7
8
1−β
φσξ2
which is identical to the dispersed information limit equilibrium value found in proposition 6.
Taking the same limit limσ2 →0,b→b± on both the right-hand side and left-hand side of (79)
we obtain:


(1−β)φ
φ
φ
aφ2
2
− 1−φ+φa
+ β − 1−φ+φa
− a − 1−φ+φa
− (1−φ+φa)
σ
2 ϕ


b± = 
 b±
2
2
φ
φa
1 + 1−φ+φa κ + 1−φ+φa σϕ2
that finally pins down the finite value of κ
κ=
6
a=
(φ − 1) σξ2 − (1 − β)
φ − 1 − aφ
=
,
φ2
φ2 σξ2
(80)
and a condition for its non negativity σξ2 > (1 − β)(φ − 1)−1 , which is (46). Note in particular,
that the condition is independent of ϑ or σϕ2 , i.e. wether or not the signal embeds the average
expectation or a common noise term, respectively.
46
1
2
3
4
5
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