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Optimal Monetary Policy with Uncertain Fundamentals
and Dispersed Information
Guido Lorenzoni∗
March 2008

Abstract
This paper studies monetary policy in an economy where output fluctuations are driven
by the private sector’s uncertainty about the economy’s fundamentals. I consider an economy where information on aggregate productivity is dispersed across agents and there are
two aggregate shocks: a standard productivity shock and a “noise shock” affecting public beliefs about aggregate productivity. Neither the central bank nor individual agents
can distinguish the two shocks when they hit the economy. The main results are: (1)
despite the lack of superior information, an appropriate monetary policy rule can change
the economy’s response to the two aggregate shocks; (2) monetary policy can achieve “full
aggregate stabilization,” that is, an equilibrium where aggregate activity is the same as
in the case of full information; (3) under optimal monetary policy, the economy achieves
a constrained efficient allocation; (4) optimal monetary policy is typically different from
full aggregate stabilization. Behind these results there are two crucial ingredients. First,
agents are forward looking. Second, as time passes, better information on past fundamentals becomes available. The central bank can then adopt a backward-looking policy rule,
based on more precise information about past fundamentals. By announcing its response
to future information, the central bank can influence the expected real interest rate faced
by agents with different beliefs and thus induce an optimal use of the information dispersed
in the economy.
Keywords: Optimal monetary policy, imperfect information, consumer sentiment.
JEL Codes: E52, E32, D83.

∗

MIT, Federal Reserve Bank of Chicago and NBER. Email: glorenzo@mit.edu. A previous version of this
paper circulated with the title “News Shocks and Optimal Monetary Policy.”I wish to thank for very useful
comments and suggestions Kjetil Storesletten, three anonymous referees, Marios Angeletos, Ricardo Caballero,
Marvin Goodfriend, Veronica Guerrieri, Ivan Werning and seminar participants at the SED Meetings (Budapest), UQAM (Montreal), the Kansas City Fed, MIT, UC San Diego, Chicago GSB, Northwestern, Cornell,
and U. of Texas at Austin.

1

Introduction

Suppose a central bank observes an unexpected expansion in economic activity. This could
be due to a shift in fundamentals, say an aggregate productivity shock, or to a shift in public
beliefs with no actual change in the economy’s fundamentals. If the central bank could tell
apart the two shocks the optimal response would be simple: accommodate the first shock
and offset the second. In reality, however, central banks can rarely tell apart these shocks
when they hit the economy. What can the central bank do in this case? What is the optimal
monetary policy response? In this paper, I address these questions in the context of a model
with dispersed information, which allows for a micro-founded treatment of fundamental shocks
and “sentiment shocks.”
The US experience in the second half of the 90s has fueled a rich debate on these issues.
The run up in asset prices has been taken by many as a sign of optimistic expectations about
widespread technological innovations. In this context, the advice given by different economists
has been strongly influenced by the assumptions made about the ability of the central bank
to identify the economy’s actual fundamentals. Some, e.g. Cecchetti et al. (2000) and Dupor
(2002), attribute to the central bank some form of superior information and advocate early
intervention to contain an expansion driven by incorrect beliefs. Others, e.g. Bernanke and
Gertler (2001), emphasize the uncertainty associated with the central bank’s decisions and
advocate sticking to a simple inflation-targeting rule. In this paper, I explore the idea that,
even if the central bank does not have superior information, a policy rule can be designed to
take into account, and partially offset, aggregate mistakes by the private sector regarding the
economy’s fundamentals.
I consider an economy with heterogeneous agents and monopolistic competition, where
aggregate productivity is subject to unobservable random shocks. Agents have access to a noisy
public signal of aggregate productivity, which summarizes public news about technological
advances, aggregate statistics, and information reflected in stock market prices and other
financial variables. The error term in this signal introduces aggregate “noise shocks,” that is,
shocks to public beliefs which are uncorrelated with actual productivity shocks. In addition to
the public signal, agents have access to private information regarding the realized productivity
in the sector where they work. Due to cross-sectional heterogeneity, this information is not
sufficient to identify the value of the aggregate shock. Therefore, agents combine public and
private sources of information to forecast the aggregate behavior of the economy. The central

1

bank has access only to public information.
In this environment, I obtain two sets of results. First, I show that the monetary authority,
using a policy rule which responds to past aggregate shocks, can affect the relative response
of the economy to productivity and noise shocks. Actually, there exists a policy rule which
achieves “full aggregate stabilization,” that is, an equilibrium where aggregate activity is the
same as in the case of full information. Second, I derive the optimal policy rule and show
that full aggregate stabilization is typically suboptimal. As long as the coefficient of relative
risk aversion is greater or equal than 1, it is optimal to let output respond less than one for
one to underlying changes in aggregate fundamentals and to let it respond positively to noise
shocks. At the optimal policy rule, the economy achieves a constrained efficient allocation,
where agents make optimal use of public and private sources of information.
The fact that monetary policy can tackle the two shocks separately is due to two crucial
ingredients. First, agents are forward looking. Second, productivity shocks are unobservable
when they are realized, but become public knowledge in later periods. At that point, the
central bank can respond to them. By choosing an appropriate policy rule the monetary
authority can then alter the way in which agents respond to private and public information.
In particular, the monetary authority can announce that it will increase the target for aggregate
nominal spending tomorrow, following an actual increase in aggregate productivity today, so
as to generate inflation. Under this policy, consumers observing an increase in productivity
in their own sector expect higher inflation than consumers who only observe a positive public
signal. Therefore, they expect a lower real interest rate and choose to consume more. This
makes consumption more responsive to private information and less to public information and
moderates the economy’s response to noise shocks. This result points to an idea which applies
more generally in models with dispersed information. If future policy is set contingent on
variables that are imperfectly observed today, this can change the agents’ reaction to different
sources of information, and thus affect the equilibrium allocation.
In the model presented, the power of policy rules to shape the economy’s response to
aggregate shocks is surprisingly strong. Namely, by adopting the appropriate rule the central
bank can support an equilibrium where aggregate output responds one for one to fundamentals
and does not respond at all to noise in public news. However, such a policy is typically
suboptimal, since it has undesirable consequences in terms of the cross-sectional allocation. In
particular, full stabilization generates an inefficient compression in the distribution of relative
prices.
2

To define the appropriate benchmark for constrained efficiency, I consider a social planner
who can dictate the way in which individual consumers respond to the information in their
hands, but cannot change their access to information.1 My constrained efficiency result shows
that, in a general equilibrium environment with isoelastic preferences and Gaussian shocks, a
simple linear monetary policy rule, together with a non-state-contingent production subsidy,
are enough to eliminate all distortions due to dispersed information and monopolistic competition. In particular, a policy rule which only depends on aggregate variables is enough to
induce agents to make an optimal use of public and private information.2
Finally, I use the model to ask whether better public information regarding the economy
fundamentals can have destabilizing effects on the economy and whether it can lead to social
welfare losses.

3

I show that increasing the precision of the public signal increases the response

of aggregate output to noise shocks and this can potentially increase output gap volatility.4
However, as agents receive more precise information on average productivity, they can set
relative prices that are more responsive to their idiosyncratic productivity shocks. Therefore,
a more precise public signal can improve welfare by allowing a more efficient allocation of
consumption and labor effort across sectors. What is the total welfare effect of increasing the
public signal’s precision? If monetary policy is kept constant, then a more precise public signal
may, for some set of parameters, reduce total welfare. This provides an interesting general
equilibrium counterpart to Morris and Shin’s (2002) “anti-transparency” result. However, if
monetary policy is chosen optimally, then a more precise signal is always welfare improving.
This follows from the fact that, as pointed out by Angeletos and Pavan (2007a), more precise
information is always desirable when the equilibrium is constrained efficient.
In this paper, equilibrium allocations and welfare are derived in closed form. This is possible
thanks to an assumption about the random selection of consumption baskets. In particular,
I maintain the convenience of a continuum of goods in each basket, while, at the same time,
I allow for baskets that differ from consumer to consumer. This technical solution may be
usefully adapted to other models of information diffusion with random matching, as it allows
1

See Hellwig (2005) and Angeletos and Pavan (2007a) for similar notions of constrained efficiency.
Angeletos and Pavan (2007b) derive a similar result in the context of quadratic games with optimal policy.
See Angeletos, Lorenzoni, and Pavan (2008) for an application of the same principle to a model of investment
and financial markets.
3
On the social value of public information in game-theoretic settings, see Morris and Shin (2002), Angeletos
and Pavan (2007a, 2007b), Hellwig and Veldkamp (2007), Amador and Weill (2007). This literature has sparked
a lively debate on the merits of transparency in monetary policy, see Amato, Morris, and Shin (2002), Svensson
(2005), Hellwig (2005), Morris and Shin (2005).
4
Here the output gap is measured with respect to the equilibrium under full information.
2

3

to construct models where each agent interacts with a large number of other agents, but does
not fully learn about aggregate behavior.
A number of recent papers, starting with Woodford (2002) and Sims (2003), have revived
the study of monetary models with imperfect common knowledge, in the tradition of Phelps
(1969) and Lucas (1972).5 In particular, this paper is closely related to Hellwig (2005) and
Adam (2006), who study monetary policy in economies where money supply is imperfectly
observed by the public. In both papers consumers’ decisions are essentially static, as a cashin-advance constraint is present and always binding. Therefore, the forward-looking element,
which is crucial in this paper, is absent in their models. In the earlier literature, King (1982)
was the first to recognize the power of policy rules in models with imperfect information. He
noticed that “prospective feedback actions” responding to “disturbances that are currently
imperfectly known by agents” can affect real outcomes.6 However, the mechanism in King
(1982) is based on the fact that different policy rules change the informational content of
prices. As I will show below, that channel is absent in this paper. Here, policy rules matter
because they affect agents’ incentives to respond to private and public signals.
The existing literature on optimal monetary policy with uncertain fundamentals has focused
on the case of common information in the private sector. This includes Aoki (2003), Svensson
and Woodford (2003, 2005), and Reis (2003). A distinctive feature of the environment in this
paper is that private agents have access to superior information about fundamentals in their
local market but not in the aggregate. The presence of dispersed information generates a novel
tension between aggregate efficiency and cross-sectional efficiency in the design of optimal
policy.
There is a growing literature on the effect of expectations and news on the business cycle.
In particular, Christiano, Motto, and Rostagno (2006) and Lorenzoni (2006) show that shocks
to expectations about productivity can generate realistic aggregate demand disturbances in
business cycle models with nominal rigidities.7 In Christiano, Motto, and Rostagno (2006) the
monetary authority has full information regarding aggregate shocks and can adjust the nominal
interest rate in such a way so as to essentially offset the effect of the news shock and replicate
the behavior of the corresponding flexible price economy. Moreover, this offsetting is optimal
5

See also Moscarini (2004), Milani (2005), Nimark (2005), Bacchetta and Van Wincoop (2005), Luo (2006),
Ma´kowiak and Wiederholt (2006). Mankiw and Reis (2002) and Reis (2006) explore the complementary idea
c
of lags in informational adjustment as a source of nominal rigidity.
6
King (1982), p. 248.
7
See Beaudry and Portier (2006) and Jaimovich and Rebelo (2006) for flexible price models of cycles driven
by news about future productivity.

4

in their model.8 This leads to the question: are expectations-driven cycles merely a symptom
of a suboptimal monetary regime, or is there some amount of expectations-driven volatility
that survives under optimal monetary policy? This paper addresses this question in a setup
with dispersed information, as in Lorenzoni (2006), and shows that optimal monetary policy
does not eliminate noise-driven cycles. One may think that this result comes immediately from
the assumption that the monetary authority has limited information. That is, it would seem
that the central bank cannot intervene to bring output towards its “natural” level, given that
this natural level is unknown. The analysis in this paper shows that the argument is subtler.
The monetary authority could eliminate the aggregate effect of news shocks by announcing
an appropriate monetary rule. However, this rule is not optimal due to its undesirable crosssectional consequences.
Finally, from a methodological point of view, this paper is related to a set of papers who
exploit isoelastic preferences and Gaussian shocks to derive closed-form expressions for social
welfare in heterogeneous agent economies, e.g., Benabou (2002) and Heathcote, Storesletten, and Violante (2008). The main novelty here is the presence of differentiated goods and
consumer-specific consumption baskets.
The model is introduced in Section 2. In Section 3, I characterize stationary, linear rational
expectations equilibria. In Section 4, I show how the choice of the monetary policy rule affects
the equilibrium allocation. In Section 5, I derive the welfare implications of different policies,
characterize optimal monetary policy and prove constrained efficiency. In Section 6, I study
the welfare effects of public information. Section 7 concludes. All the proofs not in the text
are in the appendix.

2

The Model

2.1

Setup

I consider a dynamic model of monopolistic competition ` la Dixit-Stiglitz with heterogeneous
a
productivity shocks and imperfect information regarding aggregate shocks. Prices are set at
the beginning of each period, but are, otherwise, flexible.
There is a continuum of infinitely lived households uniformly distributed on the unit interval
[0, 1]. Each household i is made of two agents: a consumer and a producer who is specialized
8

See Appendix B of Christiano, Motto, and Rostagno (2006).

5

in the production of good i. Preferences are represented by the utility function
∞

β t U (Cit , Nit ) ,

E
t=0

with
U (Cit , Nit ) =

1
1
1−γ
1+η
Cit −
Nit ,
1−γ
1+η

where Cit is a consumption index and Nit is the labor effort of producer i.9 The consumption
index is given by
σ−1
σ

Cit =

Jit

Cijt dj

σ
σ−1

,

where Cijt denotes consumption of good j by consumer i in period t, and Jit ⊂ [0, 1] is a
random consumption basket, which is described in detail below. The elasticity of substitution
between goods, σ, is greater than 1.
The production function for good i is
Yit = Ait Nit .
Productivity is household-specific and labor is immobile across households. The productivity
parameters Ait are the fundamental source of uncertainty in the model. Let ait denote the log of
individual productivity, ait = log (Ait ).10 Individual productivity has an aggregate component
at and an idiosyncratic component

it ,

ait = at +
with

1
0 it di

it ,

= 0. Aggregate productivity at follows the AR1 process
at = ρat−1 + θt ,

with ρ ∈ [0, 1].
At the beginning of period t, all households observe the value of aggregate productivity
in the previous period, at−1 . Next, the shocks
9

it

and θt are realized. Agents in household

If γ = 1, the per-period utility function is
U (Cit , Nit ) = log Cit −

10

1
N 1+η .
1 + η it

Throughout the paper, a lowercase variable will denote the natural logarithm of the corresponding uppercase
variable.

6

i cannot observe

it

and θt separately, they only observe the sum of the two, that is, the

individual productivity innovation
xit = θt +

it .

Moreover, all agents observe a noisy public signal of the aggregate innovation
st = θt + et .
The random variables

it ,

θt and et are independent, serially uncorrelated, and normally dis-

2
2
tributed with zero mean and variances σ 2 , σθ , and σe .11 I assume throughout the paper that σ 2
2
2
2
and σθ are strictly positive, and I study separately the cases σe = 0 and σe > 0, corresponding,

respectively, to full information and imperfect information on θt .
Summarizing, there are two aggregate shocks: the productivity shock θt and the “noise
shock” et . Both are unobservable during period t, but are fully revealed at the beginning
of t + 1, when at is observed. The second shock is the source of correlated mistakes in this
economy, as it induces households to temporarily overstate or understate the current value of
θt . The vector of past aggregate shocks is denoted by
ht ≡ (θt−1 , et−1 , θt−2 , et−2 , ..., θ0 , e0 ) .
Let me turn now to the random consumption baskets. Each period, nature selects a random
set of goods Jit ⊂ [0, 1] with correlated productivity shocks. In this way, even though each
consumer consumes a large number of goods (a continuum), the law of large numbers does
not apply, and consumption baskets differ across consumers. In the appendix, I give a full
description of the matching process between consumers and producers. Here, I summarize the
properties of the consumption baskets that arise from the process. Each consumer receives a
“sampling shock” vit (unobserved by the consumer) and the goods in Jit are selected so that
the distribution of the shocks

jt

for j ∈ Jit is normal with mean vit and variance σ 2|v . The

sampling shocks vit are normally distributed across consumers, with zero mean and variance
2
σv . They are independent of all other shocks and satisfy

of the matching process the variances

2
σv , σ 2|v

and

σ2

1
0 vit di

= 0. To ensure consistency

2
have to satisfy σv + σ 2|v = σ 2 . Therefore,

2
the variance σv is restricted to be in the interval 0, σ 2 . Let me introduce here the parameter
11
2
σθ

In the cases where γ = 1 and productivity is a random walk, ρ = 1, it is necessary to impose a bound on
to ensure that expected utility is finite, namely
2
σθ < 2

γ+η
(1 − γ) (1 + η)

7

2

(− log β) .

2
χ = σv /σ 2 which lies in [0, 1] and reflects the degree of heterogeneity in consumption baskets.

The limit case χ = 0 corresponds to the standard case where all consumers consume the same
representative sample of goods.

2.2

Trading, financial markets and monetary policy

The central bank acts as an account keeper for the agents in the economy. Each household
holds an account denominated in dollars, directly with the central bank. The account is debited
whenever the consumer makes a purchase and credited whenever the producer makes a sale.
The balance of household i at the beginning of the period is denoted by Bit . All households
begin with a zero balance at date 0. At the beginning of each period t, the bank sets the
(gross) nominal interest rate Rt , which will apply to end-of-period balances. Households are
allowed to hold negative balances at the end of the period and the same interest rate applies to
positive and negative balances. However, there is a lower bound on nominal balances, which
rules out Ponzi schemes.
To describe the trading environment, it is convenient to divide each period t in three
stages, (t, 0) , (t, I) , and (t, II). In stage (t, 0), everybody observes at−1 , the central bank
sets Rt , and the households trade one-period state-contingent claims on a centralized financial
market. These claims will be paid in (t + 1, 0). In stages (t, I) and (t, II), the market for
state-contingent securities is closed and the only assets traded are dollar balances in the central
bank’s payment system and non-state-contingent bonds payable in (t + 1, 0). By arbitrage, the
price of the bonds must be equal to 1/Rt at all stages. Since balances with the central bank and
non-state contingent bonds are perfect substitutes and are in zero net supply, I simply assume
that holdings of non-state-contingent bonds are always zero. In stage (t, I), all aggregate and
individual shocks are realized, producer i observes st and xit , sets the dollar price of good i, Pit ,
and stands ready to deliver any quantity of good i at that price. In stage (t, II), consumer i
observes the prices of the goods in his consumption basket, {Pjt }j∈Jit , chooses his consumption
vector, {Cijt }j∈Jit , and buys Cijt from each producer j ∈ Jit . In this stage, consumer i and
producer i are spatially separated, so the consumer does not observe the current production
of good i. Figure 1 summarizes the events taking place during period t.
In stages (t, I) and (t, II), households are exposed to idiosyncratic uncertainty and do not
have access to state-contingent claims. Therefore, they will generally end up with different
end-of-period balances. However, households can fully insure against these shocks ex ante, by
trading contingent claims in (t, 0). This implies that the nominal balances Bit will be constant
8

(t,0)

(t,I)

(t,II)

(t+1,0)

Everybody observes at-1

Household i observes

Household i observes price

State-contingent claims

Central bank sets Rt

st

vector

Agents trade state-

xit

contingent claims

et

t
t

Sets price Pit

it

Pjt

j J it

are settled

and

chooses consumption
vector

Cijt

j J it

Figure 1: Timeline
and equal to 0 in equilibrium.12 In this way, I can eliminate the wealth distribution from the
state variables of the problem, which greatly simplifies the analysis.13
Let Zit+1 (ωit ) denote the state-contingent claims purchased by household i in (t, 0), where
ωit ≡ ( it , vit , θt , et ). The price of these claims is denoted by Qt (ωit ). The household balances
at the beginning of period t + 1 are then given by
Bit+1 = Rt Bit −

R4

Qt (˜ it ) Zit+1 (˜ it ) d˜ t + (1 + τ ) Pit Yit −
ω
ω
ω

Jit

Pjt Cijt dj − Tt +Zit+1 (ωit ) ,

where τ is a proportional subsidy on sales and Tt is a lump-sum tax.
Let me define aggregate indexes for nominal prices and real activity. For analytical conve12
13

The balances Bit are computed after the claims from t − 1 have been settled.
The use of this type of assumption to simplify the study of monetary models goes back to Lucas (1990).

9

nience, I use simple geometric means,14
1

Pt ≡ exp

log Pit di ,

0
1

Ct ≡ exp

0

log Cit di .

The behavior of the monetary authority is described by a policy rule. In period (t, 0), the
central bank sets Rt based on the past realizations of the exogenous shocks θt and et , and
on the past realizations of Pt and Ct . The monetary policy rule is described by the map R,
where Rt = R (ht , Pt−1 , Ct−1 , ..., P0 , C0 ). Allowing the monetary policy to condition Rt on the
current public signal st would not alter any of the results. The only other policy instrument
available is the subsidy τ , which is financed by the lump-sum tax Tt . The government runs a
balanced budget so

1

Tt = τ

Pit Yit di.

0

As usual in the literature, the subsidy τ will be used to eliminate the distortions due to
monopolistic competition.

2.3

Equilibrium definition

Household behavior is captured by three functions, Z, P and C. The first gives the optimal
holdings of state-contingent claims as a function of the initial balances Bit and of the vector of past aggregate shocks ht , that is, Zit+1 (ωit ) = Z (ωit ; Bit , ht ). The second gives the
optimal price for household i, as a function of the same variables plus the current realization of individual productivity and of the public signal, Pit = P (Bit , ht , st , xit ). The third
gives optimal consumption as a function of the same variables plus the observed price vector,
Cit = C(Bit , ht , st , xit , {Pij }j∈Jit ). Before defining an equilibrium, I need to introduce two
other objects. Let D (.|ht ) denote the distribution of nominal balances Bit across households,
conditional on the history of past aggregate shocks ht . The price of a ωit -contingent claim in
period (t, 0), given the vector of past shocks ht , is given by Q (ωit ; ht ).
14

Alternative price and quantity indexes are
ˆ
Pt

1

≡
0

ˆ
Yt

≡

1
0

1−σ
Pit di

1
1−σ

,

Pit Yit di
.
ˆ
Pt

ˆ
ˆ
All results stated for Pt and Ct hold for Pt and Yt , modulo multiplicative constants.

10

A symmetric rational expectations equilibrium under the policy rule R is given by an array
{Z, P, C, D, Q} that satisfies three conditions: optimality, market clearing, and consistency.
Optimality requires that the individual rules Z, P and C are optimal for the individual household, taking as given: the exogenous law of motion for ht , the policy rule R, the prices Q,
and the fact that all other households follow Z,P, and C, and that their nominal balances are
distributed according to D. Market clearing requires that the goods markets and the market
for state-contingent claims clear for each ht . Consistency requires that the dynamics of the
distribution of nominal balances, described by D, are consistent with the individual decision
rules.

3

Linear equilibria

In this section, I characterize the equilibrium behavior of output and prices. Given the assumption that agents trade state-contingent claims in periods (t, 0), I can focus on equilibria
where beginning-of-period nominal balances are constant and equal to zero for all households.
That is, the distribution D (.|ht ) is degenerate for all ht . Moreover, thanks to the assumption
of separable, isoelastic preferences and Gaussian shocks, I can analyze linear rational expectations equilibria in closed form. In particular, I will characterize stationary linear equilibria
where the logs of individual prices and consumption levels take the following form
pit = φa at−1 + φs st + φx xit ,

(1)

cit = ψ0 + ψa at−1 + ψs st + ψx xit + ψx xit ,

(2)

where φ ≡ {φa , φs , φx } and ψ ≡ {ψ0 , ψa , ψs , ψx , ψx } are vectors of constant coefficients to be
determined and xit is the average productivity innovation for the goods in the basket of consumer i,
xit ≡

Jit

xjt dj = θt + vit .

I will explain in a moment why this variable enters (2). Summing (1) and (2) across agents, I
obtain the aggregate price and quantity indexes
pt = φa at−1 + φθ θt + φs et ,

(3)

ct = ψ0 + ψa at−1 + ψθ θt + ψs et ,

(4)

where φθ ≡ φs + φx and ψθ ≡ ψs + ψx + ψx .
In the rest of this section, I first characterize the optimal behavior of an individual household, assuming that the all other households follow (1) and (2). Then, I introduce a linear
11

monetary policy rule, and show that, under that rule, (1) and (2) form a rational expectations
equilibrium.

3.1

Optimal consumption and prices

As useful preliminary steps, let me derive the appropriate price index for household i and the
demand curve for good i. Individual optimization implies that, given Cit , the consumption of
good j by consumer i is
Cijt =

Pjt
P it

−σ

Cit ,

where P it is the price index
P it ≡

j∈Jit

1−σ
Pjt dj

1
1−σ

(5)

.

The assumptions made on consumption baskets imply that the productivity innovations xjt for
the goods in Jit are normally distributed with mean xit and variance σ 2|v . Using my conjecture
(1) for individual prices, I then obtain an exact expression for the log of the price index of
consumer i,15
pit = κp + pt + φx vit .

(6)

˜
Consider now the demand curve for good i. Let Jit denote the set of consumers buying good
−σ
i at time t. Aggregating their demand, gives Yit = Dit Pit , where Dit is the demand index

Dit ≡

σ

˜
j∈Jit

P jt Cjt dj.

Also for the demand index Dit , I can use my assumptions on consumption baskets and conjectures (1) and (2) to obtain an exact linear expression
dit = κd + ct + σpt + (ψx + σφx ) χ it .

(7)

Using these expressions, I can then derive the household’s first-order conditions for Pit and
Cit . The first takes the form
pit = κp + Ei,(t,I) [pit + γcit + ηnit ] − ait ,

(8)

where Ei,(t,I) [.] denotes the expectation of household i at date (t, I). The labor effort nit is
determined by the technological constraint nit = yit − ait , and the output yit by the demand
relation derived above, yit = dit − σpit . The expression on the right-hand side of (8) captures
15

This expression and expressions (7)-(9) below, are derived formally in the proof of Proposition 1, in the
appendix, where I also derive the constant terms κp , κd , κp , and κc

12

the expected nominal marginal cost for producer i. This depends positively on the price of
the consumption basket of consumer i, pit , and on the marginal rate of substitution between
consumption and leisure, γcit + ηnit , and it depends negatively on the productivity ait . To
compute the expectation in (8), notice that all the relevant information at (t, I) is summarized
by at−1 , st and xit , so Ei,(t,II) [.] can be replaced by E [.|at−1 , st , xit ].
The optimality condition for Cit takes the form
cit = κc + Ei,(t,II) [cit+1 ] − γ −1 rt − Ei,(t,II) pit+1 + pit ,

(9)

where Ei,(t,II) [.] denotes the expectation of household i at date (t, II). Apart from the fact
that expectations and price indexes are consumer-specific, this is a standard consumer’s Euler
equation: current consumption depends positively on future expected consumption and negatively on the expected real interest rate. To compute the expectations in (9), notice that
the consumer can observe the price vector {pjt }j∈Jit . However, if all producers follow (1),
these prices are normally distributed with mean φa at−1 + φx xit + φs st and variance φ2 σ 2|v .
x
Given that the consumer already knows at−1 and st , he can back out φx xit from the mean of
this distribution and this is a sufficient statistic for all the information on θt contained in the
observed prices. Therefore, Ei,(t,II) [.] can be replaced by E [.|at−1 , st , xit , φx xit ]. This result is
summarized in a lemma.
Lemma 1 If prices are given by (1), then the information of consumer i regarding the current
shock θt is summarized by the three independent signals st , xit and φx xit .
This confirms my initial conjecture that the individual consumption policy (2) is a linear
function of at−1 , st , xit , and xit .

3.2

Policy rule and equilibrium

To find an equilibrium, I substitute (1) and (2) in the optimality conditions (8) and (9), and
obtain a system of equations in φ and ψ.16 This system of equations does not determine φ and
ψ uniquely. In particular, for any choice of the parameter φa in R, there is a unique pair {φ, ψ}
which is compatible with individual optimality. To complete the equilibrium characterization
and pin down φa , I need to define a monetary policy rule.
Consider an interest rate rule which targets aggregate nominal spending. The nominal
interest rate is set to
rt = ξ0 + ξa at−1 + ξm (mt−1 − mt−1 ) ,
ˆ
16

See (32)-(39) in the appendix.

13

(10)

where mt ≡ pt + ct is an index of aggregate nominal spending and mt is the central bank’s
ˆ
target
mt = µ0 + µa at−1 + µθ θt + µe et .
ˆ

(11)

The parameters ξ = {ξ0 , ξa , ξm } and µ = {µa , µθ , µe } are chosen by the monetary authority.
The central bank’s behavior can be described as follows. At the beginning of period t, the
monetary authority observes at−1 and announces its current target mt for nominal spending.
ˆ
The target mt has a forecastable, backward-looking component µa at−1 , and a state-contingent
ˆ
part which is allowed to respond to the current shocks θt and et . During trading, each agent
i sets his price and consumption responding to the variables in his information set. At the
beginning of period t + 1, the central bank observes the realized level of nominal spending mt
and the realized shocks θt and et . If mt deviated from target in period t, in the next period
the nominal interest rate is adjusted according to (10).
Given this policy rule, I can complete the equilibrium characterization and prove the existence of stationary linear equilibria. In particular, the next proposition shows that the choice
of µa by the monetary authority pins down φa in the system of equations described above,
and thus the equilibrium coefficients φ and ψ. In the proposition, I exclude one possible
value for µa , denoted by µ0 , which corresponds to the pathological case where the equilibrium
a
construction would give φx = 0. This case is discussed in the appendix.
Proposition 1 For each µa ∈ R/ µ0
a

there is a pair {φ, ψ} and a vector {ξ0 , ξa , µ0 , µθ , µe }

such that the prices and consumption levels in (1)-(2) form a rational expectations equilibrium
under the policy rule (10)-(11), for any value of ξm ∈ R. If ξm > 1 the equilibrium is locally
determinate. The value of ψa is independent of the policy rule and equal to
ψa =

1+η
ρ.
γ+η

In equilibrium, monetary policy always achieves its nominal spending target, that is, mt = mt ,
ˆ
and the nominal interest rate is equal to
rt = ξ0 − (µa + (γ − 1) ψa ) (1 − ρ) at−1 .

4

The effects of monetary policy

Let me turn now to the effects of different monetary policy rules on the real equilibrium
allocation. By Proposition 1, the choice of the policy rule is summarized by the parameter
14

µa , so, from now on, I will simply refer to the “policy rule µa .” Proposition 1 shows that
in equilibrium mt = mt and the central bank always achieves its desired target for nominal
ˆ
output. In particular, by choosing µa the central bank determines the response of nominal
output to past realizations of aggregate productivity. This is done by inducing price setters
to adjust their nominal prices in equilibrium. Since at−1 is common knowledge, price setters
simply coordinate on setting prices proportionally to exp{µa at−1 }.17
The first question raised in the introduction can now be stated in formal terms. How does
the choice of µa affects the equilibrium response of aggregate activity to fundamental and noise
shocks, that is, the coefficients ψθ and ψs in (4)? More generally, how does the choice of µa
affects the vectors φ and ψ, which determine the cross-sectional allocation of goods and labor
effort across households? The rest of this section addresses these questions.

4.1

Full information

Let me begin with the case where households have full information on the aggregate shock θt .
2
This happens when st is a noiseless signal, σe = 0. In this case, households can perfectly

forecast current aggregate prices and consumption, pt and ct , by observing at−1 and st . Using
(6) and (7) to substitute for pit and dit in the optimal pricing condition (8), and taking the
expectation E[.|at−1 , st ] on both sides, gives
pt = (κp + κp + ηκd ) + pt + γct + η (ct − at ) − at .
This implies that aggregate consumption is
cf i = ψ0 +
t

1+η
at ,
γ+η

(12)

that is ψθ = (1 + η) / (γ + η). In the next proposition, I show that also ψ0 and the other
coefficients which determine the real equilibrium allocation, are uniquely determined and independent of µa . This is a baseline neutrality result: under full information the real equilibrium
allocation is independent of the monetary policy rule.18
Proposition 2 If households have full information on θt , the allocation of consumption goods
and labor effort in all stationary linear equilibria is independent of the monetary policy rule µa .
17
18

The response of real output to at−1 , instead, is independent of the policy rule, as shown in Proposition 1.
See McCallum (1979) for an early neutrality result in a model with pre-set prices.

15

4.2

Imperfect information: a special case

2
Let me turn now to the case of imperfect information on θt , which arises when σe is positive.

In this case, the choice of µa affects equilibrium prices and quantities. To understand how
monetary policy operates, it is useful to start from a special case.
Consider the case where the intertemporal elasticity of substitution γ is 1, the disutility of
effort is linear, η = 0, and productivity is a random walk, ρ = 1. In this case, the consumer’s
Euler equation can be rewritten as19
pit + cit = Ei,(t,II) pit+1 + cit+1 ,
that is, nominal spending is a random walk. Under the nominal output target (11), the
forecastable part of future nominal spending is equal to
Ei,(t,II) [pt+1 + ct+1 ] = µ0 + µa Ei,(t,II) [at ] .
Moreover, assume that χ is zero, so that all the consumers consume all the goods.20 Then, as I
will check below, the consumers can perfectly infer the value of θt from the observed values of
pt and st , so Ei,(t,II) [at ] = at . Putting together these results and using the fact that µ0 = ψ0 ,21
it follows that all consumers choose the same consumption
ct = ψ0 + µa at − pt .

(13)

To complete the equilibrium characterization, let me turn to price setting. Households still
have imperfect information when they set prices, since they only observe st and xit in (t, 0).
Given that η = 0 and substituting the optimal consumption derived above, the optimality
condition for prices (8) boils down to22
pit = µa E [at |st , xit ] − ait .
The expectation of at can be written as E [at |st , xit ] = at−1 + βs st + βx xit , where βs and βx are
positive inference coefficients which satisfy βs + βx < 1.23 Aggregating across producers and
19
To obtain this expression from (9), notice that, when ρ = 1, rt is constant and equal to ξ0 , by Proposition
1. Moreover, in the proof of the same proposition I show that ξ0 = γκc , so the terms κc and γ −1 rt in (9) cancel
out.
20
This shows that my basic positive results can be derived without introducing heterogeneous consumption
baskets. However, Proposition 6 below shows that heterogeneous consumption baskets are necessary to obtain
interesting welfare trade-offs.
21
See equation (42) in the appendix.
22
Equation (32) in the appendix shows that κp + κp + ψ0 = 0 when γ = 1 and η = 0, so there is no constant
term in this expression.
23
See (30) in the appendix.

16

rearranging gives
pt = (µa − 1) at−1 + µa βs st + (µa βx − 1) θt .

(14)

This shows that observing pt and st fully reveals θt , except in the knife-edge case where
µa = 1/βx . The following discussion disregards this case.
Combining (13) and (14), I get the equilibrium value of aggregate consumption
ct = ψ0 + at−1 + (1 + µa (1 − βs − βx )) θt − µa βs et ,

(15)

that is, in this economy ψθ = 1 + µa (1 − βs − βx ) and ψs = −µa βs . The choice of the policy
rule µa is no longer neutral. In particular, increasing µa increases the output response to
fundamental shocks and reduces its response to noise shocks. To interpret this result, it is
useful to look separately at consumers’ and price setters’ behavior. If the monetary authority
increases µa , (13) shows that, for a given price level pt , the response of consumer spending to
θt increases. A larger value of µa implies that, if a positive productivity shock materializes at
date t, the central bank will target a higher level of nominal spending in the following period.
This, given the consumers’ forward looking behavior, translates into higher nominal spending
in the current period. On the other hand, the consumers’ response to a noise shock et , for
given pt , is zero, irrespective of µa , given that consumers have perfect information on at and
place zero weight on the signal st .
Consider now the response of price setters. If the monetary authority chooses a larger
value for µa , price setters tend to set higher prices following a positive productivity shock θt
as they observe a positive st and, on average, a positive xit , and thus expect higher consumer
spending. However, due to imperfect information, they tend to underestimate the spending
increase. Therefore, their price increase is not enough to undo the direct effect on consumers’
demand, and, on net, real consumption goes up. Formally, this is captured by
∂ψθ
= 1 − βs − βx > 0.
∂µa
On the other hand, following a positive noise shocks, price setters mistakenly expect an increase
in demand, following their observation of a positive st , and tend to raise prices. Consumers’
demand, however, is unchanged. The net effect is a reduction in output, that is,
∂ψs
= −βs < 0.
∂µa
A further result which is easily established, is that monetary policy can achieve the full
information benchmark for aggregate activity, by picking the right µa . When γ = 1, (12) shows
17

that aggregate consumption under full information is cf i = ψ0 + at . Moreover, (15) shows that
t
the central bank can achieve the same aggregate consumption path by setting µa = 0. That
is, there is a value of µa which, at the same time, achieves ψθ = 1 and ψs = 0.24 This may
seem the outcome of the special assumptions made here and, in particular, of the fact that
consumers have full information. In fact, it is a result that holds more generally, as I will show
below.

4.3

Imperfect information: general results

The following two propositions extend the results derived above to the general case. First, I
extend the non-neutrality result and show that increasing µa increases the response of aggregate
consumption to fundamental shocks, ψθ , and reduces its response to noise shocks, ψs .
Proposition 3 If households have imperfect information on θ, the real equilibrium allocation
depends on µa . The equilibrium coefficients {φ, ψ} are linear functions of the policy parameter
µa , with

∂ψθ
> 0,
∂µa

∂ψs
< 0,
∂µa

∂φx
> 0.
∂µa

The simple example presented above helps to build the intuition for the general result.
Now, consumers no longer have perfect information on θt and form expectations based on the
imperfect signals st , xit , and xit . Consider two hypothetical scenarios. In case A, there is a
positive fundamental shock, both st and θt are positive, and the typical consumer receives both
a positive public signal and positive private signals xit , xit > 0. In case B, there is a positive
noise shock et , st is positive, θt is zero, and the typical consumer receives a positive public
signal and neutral private signals xit = xit = 0.
Suppose that µa increases. In both scenarios, consumers expect an increase in nominal
output at t + 1 and higher future prices. The increase in Ei,(t,II) pit+1 on the right-hand side
of (9) leads to an increase in consumer demand at time t, for given prices pit . Under both
scenarios, the producers forecast a demand increase and tend to raise current prices. However,
in case A the producers tend to underestimate the increase in Ei,(t,II) pit+1 which is driving
up demand, while in case B they tend to overestimate it. The reason for this is that, in case
A, the typical consumer is using both public and private information, while, in case B, he
is only using public information. In the first case, the producers can perfectly forecast the
24

This does not ensure that ψ0 will also be the same. However, the subsidy τ can be adjusted to obtain any
value for ψ0 .

18

demand increase associated to a positive st , but can only partially foresee the demand increase
due to the private signals. In the second case, they think that θt is positive and erroneously
forecast a demand increase driven by both public and private signals, while, in the aggregate,
only the public signal is operating. The underreaction of current prices in case A means that
Ei,(t,II) pit+1 − pit tends to increase. The overreaction of current prices in case B leads to the
opposite result. Therefore, consumers’ expected inflation goes up in case A and down in case
B, leading to an increase in real consumption in the first case and to a reduction in the second.
There are three crucial ingredients behind this result: dispersed information, forwardlooking agents, and a backward-looking policy based on the observed realization of past shocks.
The different information sets of consumers and price setters play a central role in the mechanism described above. The presence of forward-looking agents is clearly needed so that
announcements about future policy affect current behavior. The backward-looking policy rule
works because it is based on past shock realizations which were not observed by the agents
at the time they hit. To clarify this point, notice that the results above would disappear if
the central bank based its intervention at t + 1 on any variable that is common knowledge at
date t, for example on st . Suppose, for example, that the backward-looking component of the
nominal spending target (11) took the form µs st−1 instead of µa at−1 . Then, any adjustment
in the backward-looking parameter µs would lead to identical and fully-offsetting effects on
current prices and expected future prices, with no effects on the real allocation.25
The next proposition, extends the second result obtained in 4.2. There exists a policy rule
µa which achieves full aggregate stabilization, that is, an equilibrium where aggregate activity
perfectly tracks the full information benchmark derived in 4.1.
Proposition 4 There exists a monetary policy rule µf s which, together with the appropriate
a
subsidy τ f s , achieves full aggregate stabilization, that is, an equilibrium with ct = cf i .
t
To achieve the full information benchmark for ct , the central bank has to eliminate the
effect of noise shocks, setting ψs equal to zero, and ensure, at the same time, that the output
response to the fundamental shocks ψθ is equal to (1 + η) / (γ + η). Given that, by Proposition
3, there is a linear relation between µa and ψs and ∂ψs /∂µa = 0, it is always possible to find
a µa such that ψs is equal to zero.26 The surprising result is that the value of µa that sets ψs
25

On the other hand, it is not crucial that the central bank can observe θt perfectly in period t + 1. In fact,
it is possible to generalize the result above to the case where the central bank receives noisy information about
θt at t + 1, as long as this information is not in the agents’ information sets at time t.
26
In the proof of Proposition 4, I check that µf s is different from µ0 .
a
a

19

to zero does, at the same time, set ψθ equal to (1 + η) / (γ + η). This result is an immediate
corollary of the following lemma.
Lemma 2 In any linear equilibrium, ψθ and ψs satisfy
2
2
ψθ σθ + ψs σe =

1+η 2
σ .
γ+η θ

(16)

Proof. Starting from the optimal pricing condition (8), take the conditional expectation
E [.|at−1 , st ] on both sides and use the law of iterated expectations, to obtain
E [pit |at−1 , st ] = κp + E [pit + γcit + ηdit − ησpit − (1 + η) ait |at−1 , st ] .
I can then substitute for pit and dit using (6) and (7), and exploit the fact that all idiosyncratic
shocks have zero mean ex ante. Then, optimal pricing implies that ct satisfies
E ct − ψ0 −

1+η
at |at−1 , st = 0.
γ+η

(17)

Using (4) to substitute for ct and using E [at |at−1 , st ] = ρat−1 +E [θt |st ] and ψa = ρ (1 + η) / (γ + η),
this equation boils down to
E [ψθ θt + ψs et |st ] =

1+η
E [θt |st ] .
γ+η

2
2
2
2
2
2
Substituting for E [θt |st ] = (σθ /(σθ + σe ))st and E [et |st ] = (σe /(σθ + σe ))st , gives the linear

restriction (16).
The point of this lemma is that the output responses to the two shocks are tied together by
the fact that ex ante, conditional on at−1 and st , price setters must expect their prices to be in
line with their nominal marginal costs. This implies that aggregate consumption and output
are expected to be, on average, at their full information level, as shown by (17). In turns,
this implies that when ψθ increases ψs must decrease, otherwise the sensitivity of expected
output to st would be inconsistent with optimal pricing. This also implies that, if aggregate
consumption moves one for one with ((1 + η) / (γ + η)) θt , then the effect of the signal st (and
thus of the noise et ) must be zero.
To conclude this section, let me remark that the choice of µa also affect the sensitivity
of individual consumption and prices to idiosyncratic shocks. That is, the policy rule has
implications not only for aggregate responses, but also for the cross-sectional distribution of
consumption and relative prices. This observation will turn out to be crucial in evaluating the
welfare consequences of different monetary rules.
20

5

Optimal monetary policy

5.1

Welfare

I now turn to the welfare analysis and to the characterization of optimal monetary policy. In
a linear equilibrium, the consumption of good j by consumer i is given by
Cijt = exp {ψ0 + σκp + ψa at−1 + ψs st + ψx xit + ψx xit − σφx (xjt − xit )} ,
which follows substituting the equilibrium price and consumption decisions, (1) and (2), and
the price index (6), in equation (5). The equilibrium labor effort of household i is then given
by the market clearing condition
Nit =

˜
j∈Jit

Cjit dj

Ait

.

(18)

Using these expressions and exploiting the normality of the shocks, it is then possible to
compute the value of the expected utility of a representative household at the beginning of
period 0, as shown in the following lemma.
Lemma 3 Take any monetary policy µa ∈ R/µ0 and consider the associated linear equilibrium,
a
characterized by the coefficients {φ, ψ}. Assume that the subsidy τ is chosen optimally. Then
the expected utility of a representative household is given by
∞

β t U (Cit , Nit ) =

E
t=0

if γ = 1, and by

1+η
1
(1−γ) γ+η w(µa )
W0 e
,
1−γ

∞

β t U (Cit , Nit ) = w0 +

E
t=0

1
w(µa ),
1−β

if γ = 1. W0 and w0 are constant terms independent of µa , W0 is positive, and w(.) is a known
quadratic function, which depends on the model’s parameters.
The function w(µa ) can be used to evaluate the welfare effects of different policies in terms
of equivalent consumption changes. Suppose I want to compare two policies µa and µa by
finding the ∆ such that
∞

∞
t

β U (1 + ∆) Cit , Nit

E
t=0

β t U Cit , Nit

=E

,

t=0

where Cit , Nit and Cit , Nit are the equilibrium allocations arising under the two policies. The
value of ∆ represents the proportional increase in lifetime consumption which is equivalent, in
21

welfare terms, to a policy change from µa to µa . The following lemma shows that w (µa )−w (µa )
provides a first-order approximation for ∆.27
Lemma 4 Let ∆ (µa , µa ) be the welfare change associated to the policy change from µa to µa ,
measured in terms of equivalent proportional change in lifetime consumption. The function
∆ (., .) satisfies
d∆ (µa , µa + u)
du

5.2

= w (µa ) .
u=0

Constrained efficiency

To characterize optimal monetary policy, I will show that it achieves an appropriately defined
social optimum. I consider a planner who can choose the consumption and labor effort levels
Cijt and Nit facing only two constraints: the resource constraint (18) and the informational
constraint that Cijt be measurable with respect to at−1 , st , xit , xit and xjt . This requires that,
when selecting the consumption basket of consumer i at time t, the planner can only use the
information that would be available to the consumer in the market economy. Specifically, I
allow the planner to use the same information available to consumers in linear equilibria with
φx = 0.

28

An allocation that solves the planner problem is said to be “constrained efficient.”

The crucial assumption here is that the planner can determine how consumers respond to
various sources of information, but cannot intervene to change this information. This notion
of constrained efficiency is developed and analyzed in a broad class of quadratic games in
Angeletos and Pavan (2007a, 2007b). Here, it is possible to apply it in a general equilibrium
environment, even though agents extract information from prices and prices are endogenous,
because the matching environment is such that the information sets are essentially exogenous.
The following result shows that, with the right choice of µa and τ , the equilibrium found
in Proposition 1 is constrained efficient.
Proposition 5 There exist a monetary policy µ∗ and a subsidy τ ∗ such that the associated
a
stationary linear equilibrium is constrained efficient.
This proposition shows that a simple backward-looking policy rule, which is only contingent
on aggregate variables, is sufficient to induce agents to make the best possible use of the public
27

I am grateful to Kjetil Storesletten for suggesting this result.
In the proof of Proposition 5, I show that φx = 0 in the best linear equilibrium. When φx = 0, the consumer
can recover xit and xjt for all j ∈ Jit from {Pjt }j∈Jit . Therefore, I could allow each Cijt to be a function of
the entire distribution {xjt }j∈Jit . Making it just a function of xit and xjt simplifies the analysis and is without
loss of generality.
28

22

and private information available to them.
The resource constraint and the measurability constraint are satisfied by all the linear
equilibrium allocations. Therefore, an immediate corollary of Proposition 5 is that µ∗ is the
a
optimal monetary policy which maximizes w(µa ). However, the set of feasible allocations
for the planner is larger than the set of linear equilibrium allocations, since in the planner’s
problem Cijt is allowed to be any function, possibly non-linear, of at−1 , st , xit , xit and xjt , and
there are no restrictions on the responses of Cijt to these variables.29

5.3

Optimal accommodation of noise shocks

Having obtained a general characterization of optimal monetary policy, I can turn to more
specific questions: what is the economy’s response to the various shocks at the optimal monetary policy? In particular, is full aggregate stabilization optimal? That is, should monetary
policy completely eliminate the aggregate disturbances due to noise shocks, setting ψs = 0?
The next proposition shows that, typically, full aggregate stabilization is suboptimal.
Proposition 6 Suppose there is imperfect information and η > 0, χ ∈ (0, 1). If σγ > 1 full
aggregate stabilization is suboptimal and µ∗ < µf s . At the optimal policy, aggregate consumpa
a
tion is less responsive to fundamental shocks than under full information and noise shocks have
a positive effect on aggregate consumption,
∗
ψθ <

1+η
,
γ+η

∗
ψs > 0.

If σγ < 1 full aggregate stabilization is also suboptimal, but the opposite inequalities apply. Full
stabilization is optimal if one of the following conditions hold: η = 0, χ = 0, χ = 1, σγ = 1.
To interpret this result, I use the following expression for the welfare index w(µa ) defined
in Lemma 3,
1
w = − (γ + η) E[(ct − cf i )2 ] +
t
2
1
1
1
+ (1 − γ)
(cit − ct )2 di − (1 + η)
2
2
0
where nt is the employment index nt ≡

1
0 nit di.

1
0

(nit − nt )2 di + (ct − at − nt ) ,

(19)

This expression is derived in the appendix.

The first term in (19) captures the welfare effects of aggregate volatility. In particular, it
shows that social welfare is negatively related to the volatility of the “output gap” measure
29

In fact, it is possible to further generalize the result in Proposition 5, allowing the planner to use a general
time-varying rule for Cijt , conditional on all past shocks’ realizations.

23

ct − cf i , which captures the distance between ct and the full-information benchmark analyzed
t
in Section 4.1. A policy of full aggregate stabilization maximizes this expression, setting it
to zero. However, the remaining terms are also relevant to evaluate social welfare. Once
they are taken into account, full aggregate stabilization is no longer desirable. These terms
capture welfare effects associated to the cross-sectional allocation of consumption goods and
labor effort, conditional on the aggregate shocks θt and et . Let me analyze them in order.
The second and third term in (19) capture the effect of the idiosyncratic variances of cit
and nit . Since cit and nit are the logs of the original variables, these expressions capture both
level effects and volatility effects. In particular, focusing on the first one, when the distribution
of cit is more dispersed, Cit is, on average, higher, given that
E[Cit |at−1 , θt , et ] = exp ct +

1
2

1
0

(cit − ct )2 di ,

but Cit is also more volatile as
V ar [Cit |at−1 , θt , et ] = exp
This explains why the term

1
0 (cit

1
2

1
0

(cit − ct )2 di .

− ct )2 di is multiplied by (1 − γ). When the coefficient of

relative risk aversion γ is greater than 1, agents care more about the volatility effect than about
the level effect. In this case, an increase in the dispersion of cit reduces consumers’ expected
utility. The opposite happens when γ is smaller than 1. A similar argument applies to the
third term in (19), although there both the level and the volatility effects reduce expected
utility, given that the utility function is convex in labor effort.
The last term, ct − at − nt , reflects the effect of monetary policy on the economy’s average
productivity in consumption terms. Due to the heterogeneity in consumption baskets, a given
average level of labor effort, with given productivities, translates into different levels of the
average consumption index ct depending on the distribution of quantities across consumers
and producers. The following expression is also derived in the appendix.
1
σ (σ − 1)
˜
ct −at −nt = − V ar[cjt +σpjt |j ∈ Jit , at−1 , θt , et ]+
V ar [pjt |j ∈ Jit , at−1 , θt , et ] . (20)
2
2
To interpret the first term, notice that cjt + σpjt is the intercept in the log demand for good
i by consumer j. A producer who faces more volatile log demand has on average to put in
higher effort, to achieve the same average log output. To interpret the second term, notice that
consumers like price dispersion in their consumption basket, given that when prices are more
variable they can reallocate their expenditure from more expensive goods to cheaper ones.
24

Therefore, a given average effort by the producers translates into higher consumption indexes
when relative prices are more dispersed.

30

Summing up, when the dispersion in demand is

lower and the dispersion in prices is higher, a given average effort nt generates a higher average
consumption index ct .

5.4

A numerical example

To illustrate the various welfare effects just described, I will use a numerical example. The
parameters for the example are in Table 1. The coefficient of relative risk aversion γ is set to
1. The values for σ and η are chosen in the range of values used in the sticky-price literature.
2
2
The values for the variances σθ , σ 2 and σe are set at 1. The variance of the sampling shocks
2
2
σv must then be in [0, 1]. I pick the intermediate value σv = 1/2.

γ
σ
2
σθ
2
σe

1
7
1
1

η

2

σ2
2
σv

1
0.5

Table 1: Parameters for the numerical example.
Figure 2 shows the relation between µa and total welfare w. Figure 3 illustrates how µa
affects the various terms in (19). In particular, panel (a) plots the relation between µa and the
first term in (19), capturing the negative effect of aggregate volatility. Not surprisingly, the
maximum of this function is reached at the full-stabilization policy µf s , where it reaches zero.
a
With γ = 1, the second term in (19) is always zero, so I leave it aside. Panel (b) shows the
effects of monetary policy on the third term, the negative effect due to the dispersion in labor
supply. Panels (c) and (d) show the effect on the “productivity” term ct − at − nt , which is
further decomposed into two effects, using equation (20). In panel (c), I report the negative
effect of the demand dispersion faced by a given producer, in panel (d), the positive effect of
the price dispersion faced by a given consumer.
Figure 2 shows that the optimal monetary policy is to the left of the full-stabilization policy,
which is consistent with Proposition 6, given that γσ > 1. Figure 3 shows that the crucial
effect behind this result is the effect on price dispersion in panel (d). When moving from µf s to
a
µ∗ , there are welfare losses both in terms of aggregate volatility and in terms of labor supply
a
and demand dispersion, as shown in panels (a)-(c). But the welfare gain due to increased
30

Since prices are expressed in logs, an increase in the volatility of pjt has both a level and a volatility effect.
Given that σ > 1, the second always dominates.

25

0.45
0.44
0.43
0.42

w

0.41
0.4
0.39
0.38
0.37
0.36
0.35
0.7

0.8

0.9

*

µa

1

1.1

µ

a

1.2

fs

µa

1.3

1.4

1.5

Figure 2: Welfare effects of monetary policy.
price dispersion more than compensate for these losses. Let me provide some intuition for the
mechanism behind this picture.
In a neighborhood of µ∗ , increasing µa has the effect of reducing price dispersion by reducing
a
the value of |φx |, which determines the response of individual prices to individual productivity
shocks.

31

This reduction in price dispersion can be interpreted as follows. At the optimal

equilibrium, producers with higher productivity must set lower prices, to induce consumers to
buy more of their goods. This requires φ∗ < 0. By increasing µa , the central bank induces
x
household consumption to be more responsive to the private productivity signal xit .32 This
implies that a more productive household faces a lower marginal utility of consumption, and,
at the price-setting stage, has a weaker incentive to lower the price of its good. Through this
channel an increase in µa induces relative prices to be less responsive to differences in individual
productivities, leading to a more compressed price distribution, as shown in panel (d).
Under the parametric assumptions made, this mechanism also leads to a reduction in labor
supply dispersion and in demand dispersion, as shown in panels (b) and (c). In the economy considered, at the optimal policy, individual labor supply, nit , is increasing in individual
productivity, xit . When relative prices become less responsive to individual productivity, the
31

In the proof of Proposition 6, I show that φ∗ < 0 at µ∗ . This, together with ∂φx /∂µa > 0, from Proposition
x
a
3, implies that |φx | is decreasing in µa , in a neighborhood of µ∗ .
a
32
Expression (48) in the appendix shows that ∂ψx /∂µa > 0.

26

(a) aggregate volatility
0
−0.02
−0.04
0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.3

1.4

1.5

1.3

1.4

1.5

1.3

1.4

1.5

(b) labor supply dispersion
0
−0.2
−0.4
0.7

0.8

0.9

1

1.1

1.2

(c) demand dispersion
0
−0.2
−0.4
0.7

0.8

0.9

1

1.1

1.2

(d) price dispersion
1
0.8
0.6
0.7

0.8

0.9

µ*
a

1

1.1

µa

1.2

µfs
a

Figure 3: Welfare effects of monetary policy. Decomposition.

27

relation between xit and nit becomes flatter and this reduces the cross-sectional dispersion in
nit . Finally, an increase in µa leads to a compression in the distribution of demand indexes
cjt + σpjt faced by a given producer, because the price indexes pjt become less dispersed.
Summing up, if the central bank wants to reach full stabilization it has to induce households
to rely more heavily on their private productivity signals xit when making their consumption
decisions. By inducing them to concentrate on private signals the central bank can mute
the effect of public noise. However, in doing so, the central bank reduces the sensitivity of
individual prices to productivity, generating an inefficiently compressed distribution of relative
prices.
Notice that in standard new Keynesian models, relative price dispersion is typically harmful
for social welfare, because producers have identical productivities. Things are different here,
because there is heterogeneity in individual productivities. This does not mean that more price
dispersion is always better. An increase in price dispersion eventually leads to an excessive
increase in the dispersion of labor supply (as captured by panel (b) of Figure 3).
Using Lemma 4, it is possible to quantify the welfare costs associated to suboptimal policies.
Panel (a) in Figure 3 shows that, focusing purely on the aggregate output gap, the planner finds
that going from µ∗ to µf s leads to an approximate welfare gain of 1% in equivalent consumption.
a
a
However, when all cross-sectional terms are considered, Figure 3 shows that, in fact, this policy
change generates welfare losses of more than 2% in equivalent consumption. Although this is
just an example, these numbers show that disregarding the cross-sectional implications of
policy, in an environment with heterogeneity, can lead to serious welfare miscalculations.

5.5

The role of strategic complementarity in pricing

Proposition 6 identifies a set of special cases where full stabilization is optimal. An especially
interesting case is when η = 0, that is, when utility is linear in labor effort. In this case, there
is no strategic complementarity in price setting, under the nominal spending target (10)-(11).
Substituting the consumer’s Euler equation (9) on the right-hand side of the pricing condition
(8), and using the law of iterated expectations, after some manipulations, I obtain
pit =

µa −

ρ
γ

at−1 + µa +

γ−1
ρ Ei,(t,I) [θt ] − xit .
γ

(21)

This shows that in this case prices only depend on the agents’ first-order expectations regarding
the fundamental shock θt . The analysis in Section 4.2 shows that even in this simple case an
interesting form of non-neutrality is present, because of asymmetric information between price28

setters and consumers. However, in this case there is no significant interaction among pricesetters. That is, the strategic complementarity emphasized in Woodford (2002) and Hellwig
(2005) is completely muted.
When η = 0, the planner can reach the constrained efficient allocation by letting µa =
−((γ − 1)/γ)ρ.33 This policy implies that the marginal utility of expenditure, which is proportional to exp{−pit − γcit }, is perfectly equalized across households. At the same time, by
(21), relative prices are perfectly proportional to individual productivities. When η = 0 these
relative prices achieve an efficient cross-sectional allocation of labor effort. That is, in this
economy there is no tension between aggregate efficiency and cross-sectional efficiency. Actually, it is possible to prove that, under the optimal monetary policy, this economy achieves the
full-information first-best allocation.34
Once η = 0, producers must forecast their sales to set optimal prices and these sales depend
on the prices set by other producers. Now the pricing decisions of the producers are fully
interdependent. On the planner’s front, when η = 0, it is necessary to use individual estimates
of θt when setting efficient “shadow” prices. In this case, the optimal policy can no longer
achieve the unconstrained first-best. Therefore, the presence of strategic complementarity in
pricing is tightly connected to the presence of an interesting trade-off between aggregate and
cross-sectional efficiency.

6

The welfare effects of public information

So far, I have assumed that the source of public information, the signal st , is exogenous and
outside the control of the monetary authority. Suppose now that the central bank has some
control on the information received by the private sector. For example, it can decide whether
or not to systematically release some aggregate statistics, which would increase the precision
of public information. What are the welfare effects of this decision? To address this question
2
I look at the effects of changing the precision of the public signal, measured by πs ≡ 1/σe , on

total welfare. This exercise connects this paper to a growing literature on the welfare effects
of public information.35 I consider two possible versions of this exercise. First, I assume that
when πs changes the monetary policy rule µa is kept constant, while the subsidy τ is adjusted
33

This can be derived from equation (72) in the appendix.
To prove this, follow the same steps as in the proof of Proposition 5.1, but allow the consumption rule to
be contingent on θt . Then, it is possible to show that the optimal allocation is supported by the equilibrium
described above.
35
See the references in footnote 3.
34

29

to its new optimal level. Second, I assume that for each value of πs both µa and τ are chosen
optimally.
Suppose the economy’s parameters are those in Table 1 and suppose that µa is fixed at
its optimal value for πs = 1. Figure 4 shows the effect of changing πs on welfare. The solid
line represents total welfare, measured by w in (19).36 The dashed line represents the relation
between µa and the first component in (19), which captures the negative welfare effects of
aggregate volatility. Let me begin by discussing this second relation. When the signal st is
very imprecise agents disregard it and the coefficient ψs goes to zero. When the signal becomes
more precise, agents rely more on the public signal. So, although the volatility of et is falling,
the increase in ψs can lead to an increase in aggregate volatility. In the example considered, this
happens whenever log(πs ) is smaller than 1.8. In that region, more precise public information
has a destabilizing effect on the economy. Eventually, when the signal precision is sufficiently
large, the economy converges towards the full information equilibrium and output gap volatility
goes to zero. Therefore, there is a non-monotone relation between µa and aggregate volatility.
However, this only captures the first piece of the welfare function (19). The solid line in Figure
4 shows that, when all the other pieces are taken into account, welfare is increasing everywhere
in πs .
To understand the relationship between the two graphs in Figure 4, notice that, when
the public signal is very imprecise, agents have to use their own individual productivity to
estimate aggregate productivity. This makes them underestimate the idiosyncratic component
of productivity and leads to a compressed distribution of relative prices. An increase in the
signal precision helps producers set relative prices that reflect more closely the underlying
productivity differentials. The associated gain in allocative efficiency is always positive and
more than compensates for the welfare losses due to higher aggregate volatility, in the region
where log(πs ) ≤ 1.8.
The notion that more precise information about aggregate variables has important crosssectional implications is also highlighted in Hellwig (2005). In that paper, agents face uncertainty about monetary policy shocks and there are no idiosyncratic productivity shocks.
Therefore, the cross-sectional benefits of increased transparency are reflected in a reduction in
price dispersion. Here, instead, more precise public information tends to generate higher price
dispersion. However, the underlying principle is the same: in both cases a more precise public
36
To improve readability, the values of w in the plot are normalized subtracting the value of w at πs = 0, and
I use a log scale for πs .

30

0.1
total welfare
aggregate volatility

0.08

0.06

w

0.04

0.02

0

−0.02

−0.04
−10

−8

−6

−4

−2

0

2

4

6

8

10

log(πs)

Figure 4: Welfare effects of changing the public signal precision, η = 2.
signal leads to relative prices more in line with productivity differentials.
Let me now consider a more intriguing example, where social welfare can be decreasing
in πs . In Figure 5, I plot the relation between πs and w for an economy identical to the one
above, except that the inverse Frisch elasticity of labor supply is set to a much higher value,
η = 5. When η is larger, the costs of aggregate volatility are bigger, and, it is possible to have a
non-monotone relationship between πs and total welfare, as shown by the solid line in Figure 5.
For example, when log(π) increases from 0 to 1, social welfare falls by about one-half percent
in consumption equivalent terms. This result mirrors the result obtained by Morris and Shin
(2002) in a simple quadratic game. As stressed by Angeletos and Pavan (2007a), their result
depends crucially on the form of the agents’ objective function and on the nature of their
strategic interaction. In my model, the possibility of welfare-decreasing public information
depends on the balance between aggregate and cross-sectional effects. When η is large the
negative welfare effects of aggregate volatility become a dominant concern, and increases in
public signal precision can be undesirable.
This result disappears when I allow the central bank to adjust the monetary policy rule
to changes in the informational environment. In this case, more precise public information
is unambiguously good for social welfare. This is illustrated by the dotted line in Figure 5,
which shows the relation between πs and w, when µa is chosen optimally. By Proposition

31

0.15

total welfare, fixed policy
aggregate volatility, fixed policy
total welfare, optimal policy
aggregate volatility, optimal policy

0.1

w

0.05

0

−0.05

−10

−8

−6

−4

−2

0

2

4

6

8

10

log(πs)

Figure 5: Welfare effects of changing the public signal precision, η = 5.
5, the optimal µa induces agents to use information in a socially optimal way. Therefore, at
the optimal policy, better information always leads to higher social welfare. This provides a
general equilibrium counterpart to the results in Angeletos and Pavan (2007a), who apply the
same principle to quadratic coordination games. The following proposition summarizes.
Proposition 7 If µa is kept fixed, an increase in πs can lead to a welfare gain or to a welfare
loss, depending on the model’s parameters. If µa is chosen optimally for each πs , an increase
in πs never leads to a welfare loss.

7

Conclusions

In this paper, I have explored the role of monetary policy rules in an economy where information
about macroeconomic fundamentals is dispersed across the economy. The emphasis has been
on the ability of the policy rule to shape the economy’s response to different shocks. In
particular, the monetary authority is able to reduce the economy’s response to noise shocks
by manipulating agents’ expectations about the real interest rate. The principle behind this
result goes beyond the specific model used in this paper: by announcing that policy actions
will respond to future information, the monetary authority can affect differently agents with
different pieces of information. In this way, it can change the aggregate response to fundamental
and noise shocks even if it has no informational advantage over the private sector. A second
32

general lesson that comes from the model is that, in the presence of heterogeneity and dispersed
information, the policy maker will typically face a trade-off between aggregate efficiency and
cross-sectional efficiency. Inducing agents to be more responsive to perfectly observed local
information can lead to aggregate outcomes that are less sensitive to aggregate noise shocks,
but it can also lead to a worse cross-sectional allocation.
The optimal policy rule used in this paper can be implemented both under commitment
and under discretion. To offset an expansion driven by optimistic beliefs, the central bank
announces that it will make the realized real interest rate higher if good fundamentals do not
materialize. With flexible prices, this effect is achieved with a downward jump in the price level
between t and t+1. Since at is common knowledge at time t+1, this jump only affects nominal
variables, but has no consequences on the real allocation in that period. Therefore, the central
bank has no incentive to deviate ex post from its announced policy. In economies with sluggish
price adjustment, a similar effect could be obtained by a combination of a price level change
and an increase in nominal interest rates. In that case, however, commitment problems are
likely to arise, because both type of interventions have additional distortionary consequences
ex post. The study of models where lack of commitment interferes with the central bank’s
ability to deal with informational shocks is an interesting area for future research.
A strong simplifying assumption in the model is that the only financial assets traded in
subperiods (t, I) and (t, II) are non-state-contingent claims on dollars at (t+1, 0). Introducing
a richer set of traded financial assets would increase the number of price signals available to
both consumers and the monetary authority. In a simple environment with only two aggregate
shocks, this will easily lead to a fully revealing equilibrium. Therefore, to fruitfully extend the
analysis in that direction requires the introduction of a larger number of shocks, which make
financial prices noisy indicators of the economy’s fundamentals.
Finally, in the model presented, the information sets of consumers, producers, and of the
central bank, are independent of the monetary rule chosen. Morris and Shin (2005) have
recently argued that stabilization policies may have adverse effects, if they reduce the informational content of prices. Here this concern does not arise, as the information in the price
indexes observed by consumers is essentially independent of monetary policy. A natural extension of the model in this paper would be to introduce additional sources of noise in prices,
so as to make their informational content endogenous and sensitive to policy.

33

8

Appendix

8.1

Random consumption baskets

At the beginning of each period, household i is assigned two random variables, it and vit , independently
2
drawn from normal distributions with mean zero and variances, respectively, σ 2 and σv . These variables
are not observed by the household. The first random variable represents the idiosyncratic productivity
shock, the second is the sampling shock that will determine the sample of firms visited by consumer
i. Consumers and producers are then randomly matched so that the probability that a producer with
shock jt is matched to a consumer with shock vit is given by the bivariate normal density φ ( it , vjt ),
with covariance matrix
√
σ2
χσv σ
,
2
.
σv
where χ is a parameter in [0, 1]. Since the variable vit has no direct effect on payoffs, I can normalize
2
˜
its variance and set σv = χσ 2 . Let Jit denote the set of producers met by consumer i and Jit the set of
consumers met by producer i. Given the matching process above the following properties follow. The
˜
distribution { jt : j ∈ Jit } is a normal N (vit , σ 2|v ) with σ 2|v = (1 − χ) σ 2 . The distribution {vjt : j ∈ Jit }
2
2
2
is a normal N (χ it , σv| ), with σv| = χ(1 − χ)σ .

8.2

Proof of Proposition 1

The proof is split in steps. First, I derive price and demand indexes that apply in the linear equilibrium
conjectured. Second, I use them to setup the individual optimization problem and derive necessary
conditions for individual optimality. Third, I use these conditions to characterize a linear equilibrium.
Fourth, I show how choosing µa uniquely pins down the coefficients {φ, ψ} and derive the remaining
coefficients of the monetary policy rule that implements {φ, ψ}. The proof of local determinacy is in
the supplementary material.

8.2.1

Price and demand indexes

Lemma 5 If individual prices and quantities are given by (1) and (2) then the price index for consumer
i and the demand index for producer i are equal to (6) and (7) where κp and κd are constant terms
equal to
κp

=

κd

=

1−σ 2 2
φx σ |v ,
2
1 2 2 1
1−σ 2 2
2 2
ψx σ + (ψx + σφx ) σv| + σ
φx σ |v .
2
2
2

(22)
(23)

Proof. Recall that the shocks jt for j ∈ Jit have a normal distribution N (vit , σ 2|v ). Then, given
(1), the prices observed by consumer i are log-normally distributed, with mean pt + φx vit and variance
φ2 σ 2|v , therefore,
x
j∈Jit

1−σ
Pjt dj = e(1−σ)(pt +φx vit )+

(1−σ)2
2

φ2 σ 2|v
x

.

Taking both sides to the power 1/ (1 − σ) gives the desired expression for P it , from which (6) and
(22) follow immediately. Using this result and expression (2), the demand index for producer i can be
written as
σ
Dit =
Cjt P jt dj = ect +σpt eσκp
eψx jt +ψx vjt eσφx vjt dj.
˜
j∈Jit

˜
j∈Jit

˜
Recall that the distribution {vjt : j ∈ Jit } is a normal N (χ
It follows that
˜
j∈Jit

e ψx

jt +ψx vjt +σφx vjt

1

2

dj = e 2 ψx σ

34

2

2
it , σv|

+(ψx +σφx )χ

), and

jt

and vjt are independent.

2 2
1
it + 2 (ψx +σφx ) σv|

.

Substituting in the previous expression gives (7) and (23).

8.2.2

Individual optimization

Consider an individual household, who expects all other households to follow (1)-(2) and the central bank
to follow (10)-(11). In period (t, 0), before all current shocks are realized, the household’s expectations
about the current and future path of prices, quantities and interest rates depend only on at−1 and Rt .
Moreover, the only relevant individual state variable is given by the household nominal balances Bit .
Therefore, I can analyze the household’s problem using the Bellman equation
V (Bit , at−1 , Rt ) =

max

{Zit+1 (.)},{Bit+1 (.)},
{P (.,.)},{C(.,.,.)}

Et [U (Cit , Nit ) + βV (Bit+1 , at , Rt+1 )]

subject to the constraints
Bit+1 (ωit ) = Rt Bit −
−σ
Yit = Dit Pit ,

q (˜ it ) Zit (˜ it ) d˜ it + (1 + τ ) Pit Yit − P jt Cit − Tt + Zit (ωit ) ,
ω
ω
ω

Yit = Ait Nit ,

Pit = P (st , xit ) ,

Cit = C (st , xit , xit ) ,

and the law of motions for at and Rt+1 . Et [.] represents expectations formed at (t, 0) and, in the
equilibrium conjectured, it can be replaced by E [.|at−1 ]. This problem gives the following optimality
conditions for prices and consumption
−1

−γ
Ei,(t,I) (1 + τ ) P it Cit Yit −

σ
A−1 N η P −1 Yit
σ − 1 it it it

−1

−1

−γ
−γ
Ei,(t,II) P it Cit − βRt P it+1 Cit+1

=

0,

(24)

=

0,

(25)

where Ei,(t,I) [.] and Ei,(t,II) [.] denote the expectations of agent i at (t, I) and (t, II). Given the conjectured equilibrium and, given Lemma 1, they are equal to E [.|at−1 , st , xit ] and E [.|at−1 , st , xit , φx xit ].
By Lemma 5 all the random variables in the expressions above are log-normal, including the output
−σ
and labor supply of producer i which are equal to Yit = Dit Pit and Nit = A−1 Yit . Rearranging and
it
substituting in (24) and (25) gives (8) and (9) in the text, which I report here for completeness,
pit
pit + γcit

= κp + η Ei,(t,I) [dit ] − σpit − ait + Ei,(t,I) [pit + γcit ] − ait ,

(26)

= γκc − rt + Ei,(t,II) pit+1 + γcit+1 .

(27)

The constant terms κp and κc are equal to
κp
κc

= H (ψs , ψx , ψx , φx ) − log (1 + τ ) ,
= G (ψs , ψx , ψx , φx ) ,

(28)
(29)

and H and G are known quadratic functions of ψs , ψx , ψx and φx .

8.2.3

Equilibrium characterization

To check for individual optimality, I will substitute the conjectures made for individual behavior, (1)
and (2), in the optimality conditions (26) and (27) and obtain a set of restrictions on {φ, ψ}. Notice
that all the shocks are i.i.d. so the expected value of all future shocks is zero. Let me assume for now
that φx = 0, so that E [.|st , xit , xit ] can replace E [.|st , xit , φx xit ]. Let βs , βx and δs , δx , δx be coefficients
such that E [θt |st , xit ] = βs st + βx xit and E [θt |st , xit , xit ] = δs st + δx xit + δx xit . Defining the precision
−1
2 −1
2 −1
2 −1
parameters πθ ≡ σθ
, πs ≡ σe
, πx ≡ σ 2
, and πx ≡ σv
, the coefficients βs , βx and
δs , δx , δx are
πx
πs
, βx =
,
(30)
βs =
πθ + πs + πx
πθ + πs + πx
πs
πx
πx
δs =
, δx =
, δx =
.
(31)
πθ + πs + πx + πx
πθ + πs + πx + πx
πθ + πs + πx + πx

35

I use (6) and (7) to substitute for pit and dit in the optimality conditions (26) and (27), and then I use
(1)-(4) to substitute for pit , cit , pt and ct . Finally, I use it = xit −θit and vit = xit −θit , and I substitute
for E [θt |st , xit ] and E [θt |st , xit , xit ]. After these substitutions, (26) and (27) give two linear equations
in at−1 , st , xit , xit . Matching the coefficients term by term and rearranging gives me the following set
of equations.
(γ + η)ψ0

=

−(κp + κp + ηκd ),

(γ + η)ψa

=

(1 + η)ρ,

rt = γκc − (φa + γψa ) (1 − ρ) at−1 ,
(1 + ησ) φs
(1 + ησ) φx

=
=

η (ψs + σφs + (ψx + ψx + σφx ) βs − (ψx + σφx ) χβs ) + (φa + γψa ) βs ,
η ((ψx + ψx + σφx ) βx + (ψx + σφx ) χ (1 − βx ) − 1) + (φa + γψa ) βx − 1,
φs + γψs
γψx
φx + γψx

= (φa + γψa ) δs ,
= (φa + γψa ) δx ,
= (φa + γψa ) δx .

(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)

Notice that ψa is given immediately by (33) and is independent of all other parameters. Next, notice
that to ensure that (34) always holds in equilibrium, for any choice of ξm ∈ R, the following conditions
need to be satisfied
ξ0
ξa
µ0
µa
µθ
µe

8.2.4

=
=
=
=
=

γκc ,
− (φa + γψa ) (1 − ρ) ,
ψ0 ,
φ a + ψa ,
φs + φx + ψs + ψx + ψx ,

= φ s + ψs .

(40)
(41)
(42)
(43)
(44)
(45)

Constructing the linear equilibrium for given µa

Given µa , I immediately get φa = µa − ψa from (43). To find the values of the remaining parameters
in {φ, ψ} as a function of µa , I use (35)-(39) together with the following condition, which follows from
(43)
φa + γψa = µa + (γ − 1) ψa .
(46)
To simplify the notation, the right-hand side of this expression is denoted by
µ ≡ µa + (γ − 1) ψa .
˜
First, using (36), (38) and (39), I solve for φx , ψx and ψx ,
φx =

βx + ηγ −1 (δx βx + δx (βx + χ (1 − βx ))) µ − 1 − η
˜
,
−1 ) (β + χ (1 − β ))
1 + ησ − η (σ − γ
x
x
ψx
ψx

=
=

γ −1 δx µ,
˜
γ

−1

(δx µ − φx ) .
˜

(47)
(48)
(49)

Note that a solution for φx always exists since 1+ησ−η σ − γ −1 (βx + χ (1 − βx )) > 0. This inequality
follows from βx ∈ [0, 1] and χ ∈ [0, 1]. Next, combining (35) and (37), and using the expressions above
for φx , ψx and ψx , I find φs and ψs ,
φs

=

βs + ηγ −1 δs + ηγ −1 (δx + δx (1 − χ)) βs µ + η σ − γ −1 (1 − χ) φx βs
˜
,
−1
1 + ηγ

36

(50)

ψs = γ −1 (δs µ − φs ) .
˜

(51)

Substituting the values of ψs , ψx , ψx and φx thus obtained in (22), (23), (28), I obtain values for κp , κd
and κp . Substituting these values in (32), shows that ψ0 takes the form
ψ0 = J (ψs , ψx , ψx , φx ) +

log (1 + τ )
,
γ+η

(52)

where J is a known quadratic function of ψs , ψx , ψx and φx . To find the remaining parameters of the
monetary policy rule ξ0 , µ0 , µa , µs , µθ , use (40)-(42) and (44)-(45).
To show that the prices and quantities above form an equilibrium, I need to check that the market
for state-contingent claims clears and Bit is constant and equal to 0. Let f ( it , vit , θt , et ) denote the
joint density of the shocks it , vit , θt , and et . Recall that ωit ≡ ( it , vit , θt , et ) and let the prices of
state-contingent claims at (t, 0) be
−1
Q(ωit ) = Rt f (

it , vit , θt , et ) g (θt ) ,

(53)

where g (.) is a function to be determined. Suppose Bit = 0. Let the portfolio of state-contingent claims
be the same for each household and equal to
Zit+1 (ωit ) = Rt P it Cit − (1 + τ ) Yit Pit + Tt .
For each realization of the aggregate shocks θt and et , goods markets clearing and the government
budget balance condition imply that
∞

∞

1

Pit Yit − P it Cit di = 0.

Zit+1 ({ , v, θt , et }) f ( , v, θt , et ) d dv = Rt
−∞

−∞

0

This implies that the market for state-contingent claims clears for each aggregate state θt . It also
implies that the portfolio {Zit+1 (ωit )} has zero value at date (t, 0) given that
∞
R4

∞

∞

∞

−∞

−∞

−∞

−∞

Q (ωit ) Zit+1 (ωit ) dωit =

Zit+1 ({ , v, θ, e}) f ( , v, θ, e) g (θ) d dv dθ de = 0.

Substituting in the household budget constraint shows that Bit+1 = 0. Let me check that the portfolio
just described is optimal. The first-order conditions for Zit+1 (ωit ) and Bit+1 (ωit ) are, respectively,
λ (ωit ) =

Rt Q (ωit )

λ (ωit ) =

∂V (0, at , Rt+1 )
f (ωit ) ,
∂Bit+1

R3

λ (˜ it ) d˜ it ,
ω
ω

where λ (ωit ) is the Lagrange multiplier on the budget constraint. Combining them and substituting
for ∂V /∂Bit+1 , using the envelope condition
∂V (0, at , Rt+1 )
−1
−γ
= E P it+1 Cit+1 |at ,
∂Bit+1
I then obtain
∞

−1

−γ
E P it+1 Cit+1 |at f (ωit ) = Rt Q (ωit )

−∞

−1

−γ
˜
˜ ˜
E P it+1 Cit+1 |ρat−1 + θ f (θ)dθ.

Substituting (1), (2), and (53), and eliminating the constant factors on both sides, this becomes
e−(φa +γψa )(ρat−1 +θt ) = g (θt )

∞
−∞

37

˜
˜ ˜
e−(φa +γψa )(ρat−1 +θ) f (θ)dθ,

which is satisfied as long as the function g (.) is given by
g (θt ) ≡ exp − (φa + γψa ) θt −

1
2 2
(φa + γψa ) σθ .
2

Finally, to complete the equilibrium construction, I need to check that φx = 0. From (47), this
requires µa = µ0 where
a
µ0 ≡
a

1+η
1+η
− ρ (γ − 1)
.
βx + ηγ −1 (δx βx + δx (βx + χ (1 − βx )))
γ+η

Notice that when µa = µ0 a stationary linear equilibrium fails to exists. A stationary equilibrium with
a
φx = 0 can arise, but under a policy µa which is typically different from µ0 . If φx = 0 all the derivations
a
above go through, except that E [θt |st , xit , φx xit ] = βs st + βx xit . Therefore, it is possible to derive the
analogous of condition (47) and show that φx = 0 iff
1 + ηγ −1 βx βx µ − 1 − η
˜
= 0.
−1 ) (β + χ (1 − β ))
1 + ησ − η (σ − γ
x
x
This shows that an equilibrium with φx = 0 arises when µa = µa , where
ˆ
µa ≡
ˆ

1+η
1+η
− ρ (γ − 1)
.
−1 β )
βx (1 + ηγ
γ+η
x

However, µa is also consistent with an equilibrium with φx = 0. Summing up, if µa = µ0 there is no
ˆ
a
stationary linear equilibrium; if µa = µa there are two stationary linear equilibria, one with φx = 0 and
a
one with φx = 0; if µa ∈ R/ µ0 , µa , there is a unique stationary linear equilibrium.
a ˆ

8.3

Proof of Proposition 2

2
If σe = 0 then βs = δs = 1 and βx = δx = δx = 0. Substituting in (47)-(51) gives

φx

=

φs

=

1+η
,
1 + ησ − η (σ − γ −1 ) χ
η σ − γ −1 (1 − χ)
µ+
˜
φx ,
1 + ηγ −1

−

(54)

and
ψx = −γ −1 φx ,

ψx

=

0,

ψs

=

γ −1 (˜ − φs ) = −γ −1
µ

η σ − γ −1 (1 − χ)
φx ,
1 + ηγ −1

and ψ0 can be determined from (52). Notice φs is the only coefficient which depends on µa . However,
the real equilibrium allocation only depends on the consumption levels cit and on the relative prices
pit − pt , and, given (1) and (2), these are independent of φs .

8.4

Proof of Proposition 3

For the following derivations recall that under imperfect information all the coefficients βs , βx , δs , δx , δx
are in (0, 1) and χ ∈ [0, 1]. Differentiating (47) with respect to µa (recalling that µ = µa + (γ − 1) ψa
˜
and ψa is constant), gives
∂φx
βx + ηγ −1 (δx βx + δx (βx + χ (1 − βx )))
=
> 0,
∂µa
1 + ησ − η (σ − γ −1 ) (βx + χ (1 − βx ))

38

where the denominator is positive since βx + χ (1 − βx ) < 1 and η σ − γ −1 < ησ. Differentiating (50)
gives
ηβs σ − γ −1 (1 − χ) ∂φx
∂φs
ηγ −1 (δx + δx (1 − χ)) βs
βs + ηγ −1 δs
+
+
=
.
−1
−1
∂µa
1 + ηγ
1 + ηγ
1 + ηγ −1
∂µa
Some lengthy algebra, in the supplementary material, shows that the term in square brackets is positive,
so ∂φs /∂µa > 0. Next, differentiating (51) gives
∂ψs
∂φs
= γ −1 δs −
∂µa
∂µa

.

To prove that this quantity is negative notice that
∂φs
βs + ηγ −1 δs
>
> δs ,
∂µa
1 + ηγ −1
where the last inequality follows from βs > δs , which follows immediately from (30) and (31). To prove
that ∂ψθ /∂µa > 0 it is sufficient to use of the last result together with Lemma 2, which immediately
2
2
implies that ∂ψθ /∂µa = − σe /σθ ∂ψs /∂µa .

8.5

Proof of Proposition 4

The argument in the text shows that there is a µa that gives coefficients ψs = 0 and ψθ = (1+η)/(γ +η),
if one assumes that consumers form expectations based on at−1 , st , xit , and xit . It remains to check
that this value of µa is not equal to µ0 , so that φx = 0 and observed prices reveal xit . The algebra is
a
presented in the supplementary material.

8.6

Proof of Lemma 3

Let me consider the case γ = 1, the proof for the case γ = 1 follows similar steps and is presented in
1−γ
the supplementary material. First, I derive expressions for the conditional expectations E[Cit |at−1 ]
1+η
and E[Nit |at−1 ]. Substituting for cit in the first, using (2), I obtain
1−γ
E Cit |at−1

=

2
2 2
2 2
2 2
2 2
1
e(1−γ)(ψ0 +ψa at−1 )+ 2 (1−γ) (ψθ σθ +ψs σe +ψx σ +ψx σv ) .

Using (7) to substitute for dit , and the fact that ait = at +
equilibrium labor supply
Nit =

it

and pit − pt = φx

it ,

I derive the

−σ
Dit Pit
= eκd +ψ0 +ψa at−1 +ψθ θt +ψs et −at −(1+σφx −(ψx +σφx )χ) it .
Ait

(55)

From this expression, I obtain
2
2 2
2 2
2 2
1
1+η
E Nit |at−1 = e(1+η)(κd +ψ0 +(ψa −ρ)at−1 )+ 2 (1+η) ((ψθ −1) σθ +ψs σe +(1+σφx −(ψx +σφx )χ) σ ) .

Using the fact that ψa = ρ (1 + η) / (γ + η) to group the terms in at−1 , the instantaneous conditional
expected utility takes the form
E [U (Cit , Nit ) |at−1 ] =

1 (1+η)(k2 +ψ0 ) (1−γ)ψa at−1
1 (1−γ)(k1 +ψ0 )
e
−
e
e
.
1−γ
1+η

(56)

where
k1

=

k2

=

1
2 2
2 2
2
2 2
(1 − γ) ψθ σθ + ψs σe + ψx σ 2 + ψx σv ,
2
1
2 2
2
2 2
κd + (1 + η) (ψθ − 1) σθ + ψs σe + (1 + σφx − (ψx + σφx ) χ) σ 2 .
2

39

(57)
(58)

The equilibrium equation (52) shows that, for each value of µa , there is a one-to-one correspondence
between τ and ψ0 , and ψ0 is the only equilibrium coefficient affected by τ . Therefore, if τ is set optimally,
ψ0 must maximize the term in square brackets on the right-hand side of (56). Solving this problem
shows that the term in square brackets must be equal to exp{(((1 − γ) (1 + η))/(γ + η))(k1 − k2 )} and
ψ0 =

(1 − γ) k1 − (1 + η) k2
.
γ+η

(59)

Then, I can take the unconditional expectation of (56) and sum across periods to obtain
∞

∞

β t E [U (Cit , Nit )] =
t=0

(1−γ)(1+η)
(1−γ)(1+η)
γ+η
e γ+η (k1 −k2 )
β t E e γ+η ρat−1 .
(1 + η) (1 − γ)
t=0

Letting
w ≡ k1 − k2 ,
and
W0 ≡

γ+η
1+η

∞

βtE e

(60)

(1−γ)(1+η)
ρat−1
γ+η

,

t=0

I then obtain the expression in the text. Combining (57), (58), and (60), shows that w can be expressed
as follows
w

=

−

1 (1 − γ)(1 + η) 2 1
σθ − (γ + η)
2
γ+η
2

ψθ −

1+η
γ+η

2
2 2
2
σθ + ψs σe

+

1
1
2
2 2
2
+ (1 − γ) ψx σ 2 + ψx σv − (1 + η) (1 + σφx − (ψx + σφx ) χ) σ 2 ,
2
2
1
1
2 2
2
−
ψx σ 2 + (ψx + σφx ) σv| + σ (σ − 1) φ2 σ 2|v .
(61)
x
2
2
This shows that w is a quadratic functions of the equilibrium coefficients φ and ψ. Moreover, Proposition
3 shows that φ and ψ are linear functions of µa . Therefore, (61) implicitly defines w as a quadratic
function of µa . The discounted sum in W0 is always finite if ρ < 1, because each term is bounded by
2 2
exp{((1 − γ)ψa ρa−1 + (1/2)((1 − γ)2 /(1 − ρ2 ))ψa σθ } . If, instead, ρ = 1, to ensure that the sum is
2 2
finite it is necessary to assume that β exp (1/2)(1 − γ)2 ψa σθ < 1, which is equivalent to the inequality
in footnote 11.

8.7

Derivation of equations (19) and (20)

I will first show that (19) corresponds to (61) in the proof of Lemma 3. For ease of exposition, the
2
expression in the text omits the constant term −(1/2)((1 − γ)(1 + η)/(γ + η))σθ . The first two terms
in (19) can be derived from the two terms after the constant in (61), simply using the definitions of ct
1
2
and cit . Equation (55), can be used to derive 0 (nit − nt ) di, and check that the third term in (19)
equals the third term after the constant in (61). Finally, the last line of (61) corresponds to −κd , by
(23), while equation (55) implies that nt = κd + ct − at . This shows that the last terms in (19) and
(61) are equal. To derive (20) notice that, as just argued, the last line of (61) is equal to −κd . The
derivations in Lemma 5 can then be used to obtain the expression in the text.

8.8

Proof of Lemma 4

I concentrate on the case γ = 1, the case γ = 1 is proved along similar lines. Given two monetary policies
µa = µa and µa = µa + u, let Cit , Nit and Cit , Nit denote the associated equilibrium allocations, and
define the function
−1

∞

βtE e

f (δ, u) ≡
t=0

(1−γ)(1+η)
ρat−1
γ+η

∞

∞

β t E U eδ Cit , Nit
t=0

40

β t E [U (Cit , Nit )] .

−
t=0

Proceeding in as in the proof of Lemma 3, it is possible to show that
f (δ, u) =

(1+η)(1−γ)
1 (1−γ)(δ+k1 +ψ0 )
1 (1+η)(k2 +ψ0 )
γ+η
e
−
e
−
e γ+η w(µa +u) ,
1−γ
1+η
(1 + η) (1 − γ)

where k1 and k2 are defined in (57) and (58), for the coefficients {φ, ψ} associated to the policy µa , and
the function w (.) is defined by (61). Let the function δ (u) be defined implicitly by
f (δ (u) , u) = 0.
It is immediate that δ (0) = 0. Moreover,
∂f (δ, u)
∂δ
∂f (δ, u)
∂u

= e(1−γ)(k1 +ψ0 ) ,
δ=u=0

= e

(1+η)(1−γ)
w(µa )
γ+η

w (µa ) ,

δ=u=0

and (59) implies that
e(1−γ)(k1 +ψ0 ) = e

(1+η)(1−γ)
w(µa )
γ+η

.

It follows that
δ (u) = w (µa ) .
Since ∆ (µa , µa + u) ≡ exp {δ (u)} − 1, the result follows from differentiating this expression at u = 0.

8.9

Proof of Proposition 5

Let me begin by setting up and characterizing the planner’s problem. Then, I will show that there is a
monetary policy that reaches the constrained optimal allocation. Let a− be a given scalar representing
productivity in the previous period. Let θ be a normally distributed random variable with mean zero and
2
variance σθ and let s be a random variable given by s = θ + e, where e is also a normal random variable
2
with mean zero and variance σe . Let x, x and x be random variables given by x = θ + , x = θ + v, and
˜
2
x = x + ˜, where , v and ˜ are independent random variables with zero mean and variances σ 2 , σv , σ 2|v .
˜
˜
The planner’s problem is to choose functions C (s, x, x, x), C (s, x, x), and N (s, x, θ) that maximize
˜
E [U (C (s, x, x) , N (s, x, θ))]
subject to
C (s, x, x) =
x
eρa− +˜ N (s, x, θ) =
˜

E

˜
C (s, x, x, x)
˜

σ−1
σ

˜
E C (s, x, x, x) |s, x, θ
˜
˜

σ
σ−1

|s, x, x

for all s, x, x,

(62)

for all s, x, θ.
˜

(63)

Let Λ (s, x, θ) denote the Lagrange multiplier on constraint (63). Substituting (62) in the objective
˜
˜
function, one obtains the following first-order conditions with respect to C (s, x, x, x) and N (s, x, θ):
˜
˜
˜
C (s, x, x, x)
˜

1
−σ

1

(C (s, x, x)) σ

−γ
η

(N (s, x, θ))
˜

˜
= E [Λ (s, x, θ) |s, x, x, x] ,
˜

(64)

x
= eρa− +˜ Λ (s, x, θ) .
˜

(65)

The planner’s problem is concave, so (64) and (65) are both necessary and sufficient for an optimum.
To prove the proposition, I take the equilibrium allocation associated to a generic pair (µa , τ ), and I
derive conditions on µa and τ which ensure that it satisfies (64) and (65). An equilibrium allocation
immediately satisfies the constraints (62) and (63), the first by construction, the second by market

41

˜
clearing. Take a linear equilibrium allocation characterized by ϕ and ψ. Let C (., ., ., .) and N (., ., .)
take the form
˜
C (s, x, x, x) = exp {σκp + ψ0 + ψa a− + ψs s + ψx x + ψx x − σφx (˜ − x)} ,
˜
x

(66)

N (s, x, θ) = exp {κd + ψ0 + ψa a− + ψs s + (ψx + ψx ) θ − ρa− − x − (σφx − (ψx + σφx ) χ) (˜ − θ)}
˜
˜
x
I conjecture that the Lagrange multiplier Λ (s, x, θ) takes the log-linear form
˜
Λ (s, x, θ) = exp {λ0 + λs s + λx x + λθ θ} .
˜
˜

(67)

Let me first check the first-order condition for consumption, (64). Substituting (66) in (62) and using
the definition of κp , I get
C (s, x, x) = exp {ψ0 + ψa a− + ψs s + ψx x + ψx x} .
After some simplifications, the right-hand side of (64) then becomes
1

1

−
−γ
˜
˜
= exp {−κp + φx (˜ − x)} exp {−γ (ψ0 + ψa a− + ψs s + ψx x + ψx x)} .
x
C (s, x, x, x) σ C (s, x, x) σ

The left-hand side of (64), using (67), is equal to
1
E [Λ (s, x, θ) |s, x, x, x] = exp λ0 + λs s + λx x + λθ E [θ|s, x, x] + λ2 σθ ,
˜
˜
˜
ˆ2
2 θ
−1

where σθ is the residual variance of θ, equal to (πθ + πs + πx + πx )
ˆ2
holds for all s, x, x, x, the following conditions must hold,
˜
1
λ0 + λ2 σθ
ˆ2
2 θ
λs + λθ δ s
λx
λθ δx
λθ δx

. Therefore, to ensure that (64)

= −κp − γ (ψ0 + ψa a− ) ,
= −γψs ,
= φx ,
= −γψx ,
= −γψx − φx .

2
Set λθ = − (φa + γψa ) , λs = φs , λx = φx and λ0 = −κp − γ (ψ0 + ψa a− ) − (1/2)λ2 σθ . Then, the
θˆ
first and the third of these conditions hold immediately. The other three follow from the equilibrium
relations (37)-(39). Let me now check the first order condition for labor effort, (65). Substituting (67)
and matching the coefficients on both sides, gives

λ0 + ρa−
λs
λx + 1
λθ

=
=
=
=

η (κd + ψ0 + (ψa − ρ) a− ) ,
ηψs ,
−η (1 + σφx − (ψx + σφx ) χ) ,
η (ψx + ψx + σφx − (ψx + σφx ) χ) .

Substituting, the λ’s derived above, using ψa = ρ (1 + η) / (γ + η) and rearranging, gives
2

ˆ2
(γ + η) ψ0 + ηκd + κp + (1/2) (φa + γψa ) σθ = 0,
φs − ηψs = 0,
(1 + ησ) φx + 1 + η − η (ψx + σφx ) χ = 0,
φa + γψa + η (ψx + ψx + σφx − (ψx + σφx ) χ) =

0.

(68)
(69)
(70)
(71)

To complete the proof, I need to find µa and τ such that the corresponding equilibrium coefficients ϕ
and ψ satisfy (68)-(71). Setting µ = µ∗ where
˜ ˜
µ∗ ≡
˜

η σ − γ −1 (1 − χ) (1 + η)
,
(1 + ηγ −1 (δx + δx (1 − χ))) (1 + ησ − η (σ − γ −1 ) χ) + η 2 (σ − γ −1 ) (1 − χ) γ −1 δx χ

42

(72)

ensures that (69)-(71) are satisfied. To see why these three conditions can be jointly satisfied, notice
that the equilibrium conditions (35) and (36) can be rewritten as
φs − ηψs =
(1 + ησ) φx + 1 + η − η (ψx + σφx ) χ =

[η ((ψx + ψx + σφx ) − (ψx + σφx ) χ) + φa + γψa ] βs ,
[η ((ψx + ψx + σφx ) − (ψx + σφx ) χ) + φa + γψa ] βx ,

so that, in equilibrium, (71) implies the other two. Finally, the subsidy τ can be set so as to ensure
that (68) is satisfied. The value of φx at the optimal monetary policy is
φ∗ =
x

−1 − η + ηγ −1 δx χ˜∗
µ
.
−1 ) χ
1 + ησ − η (σ − γ

(73)

Substituting (72) in the expression ηγ −1 δx χ˜∗ , shows that this expression is strictly smaller than 1 + η,
µ
which implies that φ∗ < 0. This confirms that µ∗ = µ0 , so that, by Proposition 1, the associated
a
a
x
coefficients ϕ∗ and ψ ∗ form a linear equilibrium.

8.10

Proof of Proposition 6

Let me derive the value of ψs at the constrained efficient allocation. From condition (69) and the
equilibrium condition φs + γψs = µδs , I get
˜
∗
ψs =

1
δs µ ∗ .
˜
γ+η

∗
If χ = 0, the consumer extracts perfect information from xit = θt and δs = 0, which implies that ψs = 0.
∗
∗
If, instead χ > 0, ψs inherits the sign of µ . Inspecting (72) shows that if η > 0, χ < 1 and σγ = 1,
˜
µ∗ is not zero and has the sign of σγ − 1. In all other cases, µ∗ = 0. Therefore, if η > 0, χ ∈ (0, 1)
˜
˜
∗
∗
and σγ = 1, ψs is not zero and has the sign of σγ − 1. In all remaining cases ψs = 0. The inequalities
∗
∗
for ψθ follow from Lemma 2. To prove the inequalities for µa , notice that, by Proposition 3 there is a
decreasing relation between µa and ψs , and ψs = 0 at µa = µf s .
a

8.11

Proof of Proposition 7

The first part of the Proposition is proved by the two examples discussed in the text. Let me prove
the second part. By Proposition 5, social welfare under the optimal monetary policy is the value of
a single decision maker’s optimization problem (the planner’s). For a single decision maker facing a
2
normal signal st = θt + et , increasing the variance σe is equivalent to observing the original signal plus
an additional independent error, that is, observing st = st + ξt . Then, a decision maker who observes
st can always replicate the payoff of a decision maker with a less precise signal, by just adding random
noise to st and following the associated optimal policy. Therefore, the decision maker’s payoff cannot
2
increase when σe increases.

43

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47

9

Supplementary Material

[not for publication]

9.1

Proof of Proposition 1: local determinacy

Let variables with a tilde denote deviations from the equilibrium derived above. A first order approximation of the optimality conditions gives
˜
c
Ei,(t,I) (1 + ησ) pit − η dit + pit + γ˜it
˜

= 0,

Ei,(t,II) pit + γ˜it + rt − pit+1 − γ˜it+1
c
˜
c

= 0.

Taking expectations at time (T, 0) and integrating across agents (in that order) gives
ET [(1 + ησ) pt − η (˜t + σ pt ) + pt + γ˜t ] = 0,
˜
c
˜
˜
c
ET [˜t + γ˜t + rt − pt+1 − γ˜t+1 ] = 0,
p
c
˜
˜
c

(74)

for all t ≥ T . The first condition implies that
ET [˜t ] = 0 for all t ≥ T.
c

(75)

Moreover, notice that mt = mt − mt . Therefore, under the policy rule (10)
˜
ˆ
rt = ξm mt−1 for all t ≥ T.
˜
˜
Rewrite (74) as
ET [mt + (γ − 1) ct + rt − mt+1 − (γ − 1) ct+1 ] = 0.
˜
˜
˜
˜
˜
Using (75) and defining ht ≡ ET [mt ], this gives a difference equation for ht ,
˜
ht+1 − ht − ξht−1 = 0 for all t ≥ T
with initial condition hT −1 = mT −1 . The assumption ξm > 1 ensures that any hT −1 = 0 gives an
˜
explosive solution. This shows that any equilibrium in a neighborhood of the original equilibrium must
display mt = mt for all realizations of the aggregate shocks. Using this result one can show that the
ˆ
individual prices and consumption are the same as under the original equilibrium.

9.2

Algebra for the proof of Proposition 3

I need to prove that
γ −1 (δx + δx (1 − χ)) + σ − γ −1 (1 − χ)

∂φx
> 0,
∂µa

which is equivalent to proving
(δx + δx (1 − χ)) 1 + ησ − η σ − γ −1 (βx + χ (1 − βx )) +
(γσ − 1) (1 − χ) βx + ηγ −1 (δx βx + δx (βx + χ (1 − βx ))) > 0.
Let me show separately that
(δx + δx (1 − χ)) (1 + ησ − ησ (βx + χ (1 − βx ))) + (γσ − 1) (1 − χ) βx > 0,
and
(δx + δx (1 − χ)) (βx + χ (1 − βx )) + − (1 − χ) (δx βx + δx (βx + χ (1 − βx ))) > 0.

48

For the first, it is sufficient to observe that
δx + δx (1 − χ) − (1 − χ) βx > 0,
follows from δx + δx > βx . For the second, it is enough to see that
δx ((1 − χ) βx + χ) − (1 − χ) δx βx > 0.

9.3

Algebra for the proof of Proposition 4

To prove that µa = µ0 , I suppose the contrary and use the necessary conditions for an equilibrium to
a
obtain a contradiction. By definition, µa = µ0 implies φx = 0. Summing (38) and (39) and using
a
ψx + ψx =
shows that
φa + γψa =
Substituting in (39) then gives
ψx =

1+η
,
γ+η

(76)

1
γ (1 + η)
.
δx + δx γ + η

δx 1 + η
.
δx + δx γ + η

Substituting the last three expression in (36) then gives
0=η

1+η
δx 1 + η
βx γ (1 + η)
βx +
χ (1 − βx ) − 1 +
− 1.
γ+η
δx + δx γ + η
δx + δx γ + η

This leads to a contradiction because
η
δx
η
βx
γ
βx +
χ (1 − βx ) +
< 1,
γ+η
δx + δx γ + η
δx + δx γ + η
where the inequality follows because
βx +
and

δx
χ (1 − βx ) < 1
δx + δx
βx
< 1.
δx + δx

The last inequality follows from (30) and (31).

9.4

Proof of Lemma 3: case γ = 1

When γ = 1 steps analogous to the ones used in case γ = 1 lead to
E [U (Cit , Nit ) |at−1 ] = ψ0 + ψa at−1 −

1 (1+η)(k2 +ψ0 )
e
,
1+η

and the optimal choice of τ (and ψ0 ) gives the first-order condition
1 = e(1+η)(k2 +ψ0 ) ,
which implies that
ψ0 = k1 − k2 = w,

49

as γ = 1 implies that k1 = 0. The unconditional expected utility is then
∞

∞

1
1
1
β E [U (Cit , Nit )] = −
+
β t E [ρat−1 ] +
w,
1 + η 1 − β t=0
1−β
t=0
t

which gives the expression in the text, with
∞

1
1
w0 ≡ −
+
β t E [ρat−1 ] .
1 + η 1 − β t=0

50


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