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Learning from House Prices:

Ampliﬁcation and Business Fluctuations

Ryan Chahrour∗

Boston College

Gaetano Gaballo†

HEC Paris and CEPR

November 2, 2020

Abstract

We formalize the idea that house price changes may drive rational waves of optimism

and pessimism in the economy. In our model, a house price increase caused by aggregate

disturbances may be misinterpreted as a sign of higher local permanent income, leading

households to demand more consumption and housing. Higher demand reinforces the

initial price increase in an ampliﬁcation loop that drives comovement in output, labor,

residential investment, land prices, and house prices even in response to aggregate supply

shocks. The qualitative implications of our otherwise frictionless model are consistent

with observed business cycles and it can explain the economic impact of apparently au-

tonomous changes in sentiment without resorting to non-fundamental shocks or nominal

rigidity.

Keywords: demand shocks, house prices, imperfect information, animal spirits.

JEL Classiﬁcation: D82, D83, E3.

∗Department of Economics, Boston College, Chestnut Hill, MA 02467, U.S.A. Email:

ryan.chahrour@bc.edu

†Department of Economic Sciences, HEC Paris, 1 rue de la Liberation, 78350 Jouy en Josas. Email:

gaballo@hec.fr

“Housing is the Business Cycle.”

Edward E. Leamer,

Jackson Hole Symposium, 2007

1 Introduction

House prices provide valuable information about ongoing changes in economic activity, both

at the aggregate and regional levels.1Over the last half century in the US, real house prices

and output have moved together at least half of the time (Figure 1). However, people have

very diﬀerent real-time information about these variables. Precise information about local

house prices is readily available and relevant to individual choices, while the earliest measures

of GDP are imprecise, released with delay, and may be less relevant to individual choices. For

these reasons, people observing higher house prices may rationally become more optimistic

about their own economic prospects. Through this learning channel, factors driving house

price movements may also drive waves of economic optimism or pessimism.

This paper proposes a new model of housing’s informational role in generating and am-

plifying demand-driven business ﬂuctuations. The essence of the model is a price-optimism

feedback channel: higher house prices beget economic optimism, which begets even higher

house prices, and so on. Since any aggregate shock can activate this loop, price-quantity

comovement can emerge in our model even in response to supply shocks. In this way, our

learning channel oﬀers a new source of ampliﬁcation for fundamental shocks and blurs the

traditional dichotomy between disturbances to supply and demand.

We embed our learning mechanism within a neoclassical model with housing. Households

are located on islands and consume a traded consumption good and local housing. Traded

consumption is produced using labor from all islands, while local housing is produced using

land, local labor, and a traded productive factor (commodity good) whose supply is ﬁxed.

Local house prices can move either because of an increase in the future product of local labor,

or because of a current aggregate disturbance to housing production.

Most ﬂuctuations in local house prices are driven by local labor productivity, so people

1Leamer (2007) and Leamer (2015) make the point forcefully for aggregates, while Campbell and Cocco

(2007) and Miller et al. (2011) provide evidence at the regional level.

1

1960 1969 1978 1987 1996 2005 2014

Year

-8

-6

-4

-2

0

2

4

6

8

% dev. from HP trend

Figure 1: Real gross domestic product and the Shiller national house price index.

observing high house prices become optimistic about their own labor income prospects. How-

ever, a fall in the productivity (or availability) of the commodity factor also drives an increase

in house prices across islands. In this case, the increase is misinterpreted by households as

good news about future wages, increasing demand for both consumption and housing on all is-

lands. Increasing aggregate demand further increases house prices, and consequently the price

of the commodity factor, reinforcing the initial price increase. In equilibrium, what started as

a small change in housing supply leads to an economy-wide increase in house prices, a boom

in aggregate demand for consumption and housing, and a spike in the commodity input price.

For this increase in aggregate demand to aﬀect real quantities, it must be associated

with an intratemporal “labor wedge.” We achieve this in our model by assuming that the

household is split between shoppers, who make consumption and housing decisions based only

on market prices, and worker-savers, who make labor supply and savings decisions based on full

information. As the shopper becomes overly optimistic about wages, he spends more, thinking

that the worker-saver is working less. However, since wages have not actually changed, the

worker-saver is induced to work more to avoid an ineﬃcient fall in household wealth. In this

way, workers’ optimal labor choices allow shoppers’ elevated demand to be met with higher

actual output and a boom in real quantities occurs.2

2Chahrour and Gaballo (2017) show that a similar wedge appears when some households are fully informed

and others base all of their decisions — savings, labor, and consumption — just on market prices.

2

Our model of learning from prices has several features that make it an appealing model

of the business cycle. First, our mechanism works in a ﬂexible price model with competitive

markets. This means that ﬂuctuations in housing demand, and their real eﬀects, are not

driven by competitive or nominal frictions, or by suboptimal monetary policy. Indeed, our real

economy can be interpreted as a monetary economy with a ﬁxed nominal price level. Hence,

our model aligns with recent experience in developed economies, where real ﬂuctuations have

coincided with small, largely acyclical ﬂuctuations in inﬂation (e.g. Angeletos et al.,2020).

Second, the logic of our model extends to other sorts of macroeconomic fundamentals and

to learning from any local price. Hence, the mechanism we propose can explain why business

cycle comovements emerge irrespective of the particular type of shock hitting the economy.

We show this generality by exploring an extension with an aggregate shock to consumption

— rather than housing — productivity, but also refer the reader to earlier drafts of this paper

that demonstrate how the mechanism works for shocks to the nominal money supply, and

when learning occurs from local consumption prices.3

Finally, the signal structure faced by households is fully microfounded without introducing

any extrinsic noise. Instead, we explicitly derive the house price signal as the outcome of

competitive markets and show how fundamental shocks play the role of aggregate noise in

people’s inference. Thus, the model explains how people’s beliefs become coordinated rather

than assuming coordination, as in the literature on sunspots (e.g. Cass and Shell,1983).

The fact that information comes from market prices, rather than from exogenously-

speciﬁed signals, is crucial for our mechanism. First, it means that higher house prices can

actually spur demand, for both consumption and housing. Indeed, learning from prices leads

housing demand to be upward sloping in our model, leading to realistic comovement in house

prices, housing investment, and non-house consumption.

Second, the feedback of the commodity price into local house prices allows the model

to deliver strong ampliﬁcation. For some calibrations, ampliﬁcation can be so strong that

aggregate prices and quantities exhibit sizable ﬂuctuations in the limit of arbitrarily small

aggregate shocks. To an econometrician, the ﬂuctuations emerging at the limit of no aggregate

3In Chahrour and Gaballo (2017) people learn from the price of local consumption. There, we show that

total factor productivity shocks can drive the business cycle and still be weakly correlated with business cycle

variables, as found in the data (Angeletos et al.,2018,2020).

3

shocks would appear to be driven by something akin to “animal spirits” (Shiller,2007), “noise”

(Gazzani,2019), or “sentiment” (Angeletos and La’O,2013;Benhabib et al.,2015).

After characterizing equilibrium in closed form, we examine the qualitative features of the

economy. We show that the model implies positive comovement between output, employment,

hours in the consumption and housing sectors, house prices, and land prices for any calibration

and any equilibrium so long as aggregate shocks are small enough. Hence, the model provides

a robust foundation for macroeconomic comovement across a wide range of variables.

We next enrich the model so that a portion of housing productivity is common knowl-

edge. This allows the model to exhibit typical “supply-like” comovement in response to the

common knowledge portion of the shock, while still experiencing “demand-like” ﬂuctuations

in response to the surprise component that blurs households’ inference. A calibrated version

of the extended model delivers qualitatively realistic (i.e. positive but imperfect) correlations

among many real variables. Moreover, even though it has a unique equilibrium, the model

both ampliﬁes housing market ﬂuctuations and generates strong ﬂuctuations in consump-

tion, which would disappear under full information. Indeed, ampliﬁcation is strong enough

that demand ﬂuctuations dominate unconditional comovements even when the majority of

productivity shocks are common knowledge.

We augment our discussion of real comovements with some non-structural evidence fa-

voring house prices as the source of people’s economic learning. For this, we use Michigan

Survey of Consumer Expectations data to show that people’s past house price experiences are

a far better predictor of their forecasts of their own income than are people’s reports about

aggregate economic news that they have heard. Moreover, house price experiences modestly

lead income expectations, a timing that is consistent with information ﬂowing from house

prices to income expectations. While this evidence is only suggestive, we think it indicates

that our model can help guide more structural interpretations of expectations survey data.

Literature review

This paper follows an extensive literature that oﬀers diﬀerent foundations for business cycles

caused by waves of economic optimism and pessimism. In this paper, changes in housing

supply drive initial consumer optimism through their eﬀects on house prices. The literature

4

has considered other origins for waves of consumer optimism, including: Lorenzoni (2009) with

news about future TFP; Ilut and Schneider (2014) with uncertainty shocks; and Angeletos

and Lian (2020) with discount factor shocks. Several others have also modeled belief-driven

ﬂuctuations that originate on the part of ﬁrms or producers, including Angeletos and La’O

(2009), Angeletos and La’O (2013) and Benhabib et al. (2015).

Unlike Lorenzoni (2009) and Ilut and Schneider (2014), the real eﬀects of consumer op-

timism in our model, as in Angeletos and Lian (2020), do not rely on nominal frictions or

suboptimal monetary policy. Angeletos and Lian (2020) show how a discount rate shock – a

shock to the intertemporal margin – can be ampliﬁed when consumers’ intertemporal substi-

tution operates under imperfect information and aggregate supply is upward sloping in the

real interest rate. By contrast, in our model, consumers’ uncertainty maps into distortions to

the intratemporal margin. As a result, people’s correlated mistakes about private conditions

can propagate, while aggregate intertemporal shocks aﬀect only real interest rates, just as

they would in a frictionless Real Business Cycle model (see section 5.1).

This paper shows that rational learning from prices can help explain business cycle and

housing comovements, but price-based learning has a long tradition in macroeconomics and

ﬁnance, starting with Lucas (1972) and Grossman and Stiglitz (1976,1980). Recent examples

in macroeconomics include Amador and Weill (2010), Benhima and Blengini (2020), Benhima

(2019), Gaballo (2016,2018), L’Huillier (2020), Nimark (2008) and Venkateswaran (2013).

Several ﬁnance papers show that price-based learning can deliver asset price ampliﬁcation or

multiple equilibria, including Burguet and Vives (2000), Barlevy and Veronesi (2000), Albagli

et al. (2014), Manzano and Vives (2011), and Vives (2014).

Among these papers, we are the ﬁrst to show extreme ampliﬁcation in limit cases of

noisy rational expectations equilibria. This result is connected to the sentiment equilibria of

Benhabib et al. (2015), a link that we explore in Section 5.3. Other papers have documented

ampliﬁcation when allowing for departures from rational expectations, including Eusepi and

Preston (2011) and Hassan and Mertens (2017), and Adam et al. (2011) in the housing context.

Our theory is consistent with a range of empirical evidence on housing and the business

cycle. Early housing macro models, like Davis and Heathcote (2005), struggle to explain

price-quantity comovement and authors have introduced housing demand shocks to match

5

these moments (e.g. Iacoviello and Neri,2010). Though our model is close to Davis and

Heathcote (2005), the learning in our model causes prices and quantities to positively comove.4

Our model also qualitatively accounts for the high volatility of the price of land (Davis and

Heathcote,2007) and for its strong comovement with labor markets (Liu et al.,2016).

Our paper also contributes to a long debate about the nature and size of housing wealth

eﬀects. Frictionless models typically imply that house prices should have no causal impact on

consumption (e.g. Buiter,2010) but many empirical studies suggest otherwise. For example,

Muellbauer and Murphy (1990) argue that the 1980’s spike in UK consumption was driven

by rising house prices, while King (1990), Pagano (1990), Attanasio and Weber (1994), and

Attanasio et al. (2009) argue consumption and house prices reﬂected people’s perceptions of

permanent income. In our model, these competing views coexist: high house prices drive

increased consumption not because consumers expect to sell their houses at the high price,

but because consumers interpret them as signaling higher permanent income.

Evidence from cross-sectional studies is also largely consistent with our theory. For ex-

ample, Campbell and Cocco (2007) ﬁnd that a 1% increase in an individual’s home value is

associated with a 1.22% increase in their non-durable consumption in the UK, while Miller

et al. (2011) ﬁnd a positive eﬀect of local house prices on metropolitan-level growth in the

US. The recent studies by Mian et al. (2013) and Mian and Suﬁ (2014) also present evidence

that falling house prices are associated with consumption reductions at the ZIP code level.

Other theoretical mechanisms for a direct consumption eﬀect of house prices have been

proposed in the literature, including borrowing constraints (Iacoviello,2005) and wealth het-

erogeneity with incomplete markets (Berger et al.,2017;Kaplan et al.,2017). The learning

channel we formalize oﬀers a complimentary explanation. One diﬀerence is that our channel

does not depend on actual new house sales or credit contracts, which might imply a longer

delay between house prices and their eﬀects on consumption.

4Recently, Nguyen (2018) and Fehrle (2019) have also proposed particular types of segmentation in capital

markets as solutions to these comovement challenges.

6

2 A housing model with learning from prices

In this section, we present a simple real business cycle model with housing. We aim as much

as possible to provide analytical results and make simplifying assumptions to this end. Most

of these assumptions can be relaxed; we discuss when and how as we proceed.

2.1 Preferences and technology

The economy consists of a continuum of islands, indexed by i∈(0,1). Each island is inhabited

by a continuum of price-taking households who consume local housing and a traded numeraire

consumption good. Households provide local labor which is used in the production of both

goods. On each island, a mass of competitive construction ﬁrms combine local labor and land

with a traded commodity good to construct new houses, while an aggregate consumption

sector combines all islands’ labor to produce the traded consumption good.

Households

The representative household on island ichooses consumption, labor supply, and savings in

a risk-free nominal bond to maximize the utility function:

Ui0≡

∞

X

t=0

βtnlog(Cφ

itH1−φ

it )−vNito.(1)

In the utility function above, Cit denotes household i’s consumption of the tradable consump-

tion good, Hit measures the total quantity of housing consumed, and Nit is the household’s

supply of labor. The household discount factor is β∈(0,1), the share of housing in the

consumption basket is φ∈(0,1), and vparameterizes the household’s disutility of labor.5

We assume that housing consumption is composed of a sequence of housing vintages, ∆iτ|k,

constructed at time kand combined according to the Cobb-Douglas aggregator

Hit ≡

t

Y

k=−∞

∆(1−ψ)ψt−k

it|k,(2)

where ψ∈(0,1). This formulation for housing utility adds a realistic dimension to the model,

since housing vintages can have very diﬀerent characteristics and are not perfect substitutes.

5We allow for convex disutility of labor in the Appendix.

7

More importantly for our purposes, however, this formulation in conjunction with log-utility

implies that every housing vintage has an additive-separable impact on intertemporal utility,

allowing us to analyze the dynamic model in closed form.

Each vintage of housing depreciates at a constant rate d∈(0,1), so that

∆iτ+1|k= (1 −d)∆iτ |k

for τ≥k(while, of course, ∆iτ |k= 0 for τ < k). The aggregate housing stock, deﬁned as

Hit =Pt

k=−∞ ∆it|k, then evolves according to a standard equation,

Hit = ∆it|t+ (1 −d)Hit−1.

Housing consumption can now be written Hit = ∆1−ψ

it|t(1−d)Hψ

it−1, which we use going forward.

The choices of the household are subject to the following budget constraint,

Bit ≡Bit

Rt

+Cit +Pit∆it|t−WitNit −Bit−1−Πc

t−Πh

it ≤0 (3)

for t∈ {0,1,2...}with Bi−1= 0. Household resources come from providing local labor at

wage Wit, from past bond holdings, from proﬁts Πh

it of locally-owned housing ﬁrms, and from

proﬁts Πc

tof the representative consumption ﬁrm, which is evenly held across islands. The

household uses its funds to purchase numeraire consumption, to acquire new housing at price

Pit, and to save in a zero-net-supply aggregate bond with a real risk-free return Rt. We denote

the price of the local housing vintages as Pit|kand deﬁne the price of the total housing stock

as PH

it =Pt

k=−∞ Pit|k∆it|k/Hit.

Notice, however, that only the price of the current vintage, Pit ≡Pit|t, appears in (3). This

happens because the local household is the only potential buyer and seller of past vintages,

meaning that trade in houses can never generate net resources for the island. For this reason,

housing wealth is not wealth in the sense of Buiter (2010). The literature has proposed several

strategies to break this irrelevance; our goal is to describe a potentially complementary channel

through which house prices can have a causal eﬀect on consumption.

Housing producers

House-producing ﬁrms construct new houses using a Cobb-Douglas technology,

∆it =L1−α

it Xα

it,(4)

8

that combines land (Lit) with new residential structures (Xit) to generate new residential

units ∆it ≡∆it|t. Residential structures have share α∈(0,1) and are produced, in turn, via

a Cobb-Douglas production function

Xit = (Nh

it)γ(e−˜

ζtZit)1−γ(5)

combining local labor, Nh

it, with a traded commodity, Zit, with share parameter γ∈(0,1).

The housing ﬁrm maximizes proﬁts,

Πh

it ≡Pit∆it −WitNh

it −Qt(Zit −Z)−VitLit

subject to (4) and (5). In the above, Vit is the local price of land, Wit is the price of local

labor, and Qtis the price of the commodity good. We assume that housing ﬁrms are endowed

each period with Zunits of the commodity good, which trades freely across islands and

depreciates fully at the end of the period. Land supply is exogenous: each period a ﬁxed

amount of residential land — normalized to one — becomes available to housing producers

on the island.6Without loss of generality, we assume that new land is endowed to local ﬁrms.

The only aggregate shock aﬀecting our baseline economy is a shock to productivity of the

commodity good, ˜

ζt.7This shock evolves according to a random walk, ˜

ζt=˜

ζt−1+ζt, with

i.i.d. innovation ζtdistributed according to N0, σ2

ζ. We focus our presentation on this shock

because it has no eﬀect on consumption under full information. Still, other shocks could play

a similar role: We consider an extension with an aggregate shock to consumption productivity

in Section 5 and show that ˜

ζtis isomorphic to a shock to the endowment of Zin the Appendix

B.2 (see Remark 3).

Consumption sector

The numeraire consumption good is traded freely across islands and is produced by a contin-

uum of identical competitive ﬁrms. The representative consumption producer combines labor

6These assumptions do not imply that land supply grows over time. Provided an appropriate transforma-

tion of the depreciation rate, this formulation is equivalent to a model in which structures are placed on a

ﬁxed stock of land and existing land becomes free as those structures depreciate. See Davis and Heathcote

(2005) for details.

7Notice that with our sign normalization in (5), a positive ˜

ζcorresponds to lower productivity.

9

from all sectors to maximize proﬁts,

Πc

t≡Yt−ZWitNc

itdi

subject to the production function,

Yt=Ze˜µit/η Nc

it

1−1

ηdi1

1−1

η.(6)

The quantity of local labor used is denoted by Nc

it, and labor types can be substituted with

elasticity η > 0. Island-speciﬁc labor productivity is a random walk, and evolves according

to ˜µit = ˜µit−1+ ˆµit, where ˆµit is i.i.d. and drawn from the normal distribution N(0,ˆσµ).

Market clearing

Clearing in the local land and labor markets requires

Lit = 1 and Nit =Nc

it +Nh

it.(7)

Per the discussion above, we omit market clearing conditions for all past housing vintages,

since their trade is irrelevant at the island level. Finally, clearing in the aggregate markets

for bonds, consumption, and the commodity good requires

Yt=ZCitdi, 0 = ZBit di, and Z=ZZitdi. (8)

2.2 Timing and equilibrium

The only friction that we introduce is uncertainty in households’ demand. To model this

in a parsimonious way, we use the family metaphor also adopted by Angeletos and La’O

(2009) and Amador and Weill (2010). The household is composed of two types of agents: a

shopper, who uses household resources to buy consumption and housing, and a worker-saver,

who chooses the number of hours to supply and the quantity of bonds to buy.

Both family member types act in the interest of the household, but they cannot pool

their information within a time period. Hence, choices of ∆it and Cit are conditioned on the

information set of shoppers, while Nit and Bit are conditioned on the full information set of

workers. Each period is composed of four stages:

1. The household splits into shoppers and worker-savers.

10

2. Shocks realize, namely future local productivity innovations, {ˆµi,t+1 }i∈(0,1), and the cur-

rent aggregate shock, ζt. The “best available” information set, Ωit ≡ {{ˆµi,τ }t+1

0,{ζτ}t

0},

is observed by ﬁrms and worker-savers on each island, but not shoppers.

3. Production and trade take place. Shoppers and workers make their choices based on the

information they have, which includes the competitive equilibrium prices in the markets

in which they trade. Firms make production choices in light of realized productivity

and input prices; and all markets clear.

4. Family members share information, revealing Ωit to the shoppers.

Because shoppers do not immediately observe Ωit, they make choices under uncertainty.

However, they do observe the local price of housing in their island, Pit, which they use to

make inference; shoppers’ information set is therefore {Pit,Ωit−1}.8We derive the information

about current conditions contained in Pit shortly.

The family metaphor is convenient but not essential. What is essential is that some

agents have access to information about realized shocks: Prices cannot reveal information

unless that information is already available, perhaps noisily, to some agents in the economy

(Hellwig,1980). We could have achieved the same eﬀect by assuming that only a fraction of

households on each island are informed in the spirit of Grossman and Stiglitz (1980). Nothing

crucial about our results would change if did this, though the algebra is more cumbersome.9

The formal deﬁnition of equilibrium is the following.

Deﬁnition 1 (Equilibrium).Given initial conditions n{Bi−1,Hi−1,˜µi0}i∈(0,1) ,˜

ζi−1o, a ratio-

nal expectations equilibrium is a set of prices, {{Pit, Vit , Wit}i∈(0,1) , Qt, Rt}∞

t=0, and quantities,

{{Bit, N c

it, N h

it, Nit , Cit, Hit ,∆it, Xit, Lit, Zit}i∈(0,1), Yt}∞

t=0, which are contingent on the realiza-

tion of the stochastic processes {{˜µit}i∈(0,1) }∞

t=0 and {˜

ζt}∞

t=0, such that for each t≥0and

i∈(0,1):

(a) Shoppers and worker-savers optimize, i.e. {Cit,∆it, Nit, Bit}are solutions

to max{Cit,∆it ,Nit,Bit }E[Uit]subject to

(i) Bit ≤0

8Shoppers also observe the price of old vintages for which trade does not occur in equilibrium. Nevertheless,

these prices convey no new information to shoppers as these prices are a function of shoppers’ local demand.

We show this formally in our discussion before Proposition 3.

9We took this approach in our working paper, Chahrour and Gaballo (2017). Earlier drafts also showed

that our mechanism could arise on the supply side of the economy, more like Lucas (1972).

11

(ii) Cit,∆it are measurable with respect to {Pit,Ωit−1}

(iii) Nit, Bit are measurable with respect to {Ωit};

(b) Housing producers optimize, i.e. {Nh

it, Zit , Lit,∆it }are solutions to max{Nh

it,Zit ,Lit,∆it }Πh

it

subject to (4) and (5);

(c) Consumption producers optimize, i.e. {Nc

it}i∈(0,1) are solutions to max{Nc

it}i∈(0,1) Πc

t

subject to (6);

(d) Markets clear, i.e. equations (7) - (8) hold.

The measurability constraints above imply that the consequences of a particular choice

must be evaluated by averaging across states which remain uncertain under the relevant mea-

sure. For example, let Λit be the Lagrange multiplier associated with constraint (i), which

the worker-saver correctly evaluates based on Ωit. Shopper optimality requires equating the

marginal utility of consumption with the expectation of this multiplier conditional to the

shopper’s information set, i.e. φC−1

it =E[Λit|Pit ,Ωit−1]. This condition equates the average

beneﬁts and costs of a marginal change in Cit, weighted by the probability of states indistin-

guishable to the shopper.

2.3 Linearized model

We now derive conditions describing an approximation to equilibrium in the economy, in which

we assume that deviations from the deterministic steady state of the economy are suﬃciently

small. Going forward, lower-case variables refer to log-deviations from this steady-state and

we refer to the shoppers’ information set as pit.

Shoppers demand consumption and housing goods according to the following:

cit =−E[λit|pit] (9)

δit =−E[λit|pit]−pit,(10)

where λit is the marginal value of household i’s resources — known by the worker but not the

shopper — and E[·|pit] denotes the shopper’s expectation conditional on the local house price

pit and observations of the past. Equations (9) and (10) show that, ceteris paribus, a higher

perceived value of resources lowers shoppers’ demand for both consumption and housing.

12

Optimality of worker-shopper choices requires:

wit =−λit (11)

λit =E[λit+1|Ωit] + rt.(12)

The worker provides any quantity of labor demanded, so long as the oﬀered wage equals

the household Lagrangian, and purchases bonds until the interest rate reﬂects the diﬀerence

between the current and the expected future marginal value of budget resources, which the

worker-saver forecasts based on Ωit, the full current information set.

We pause here to observe that the condition for intratemporal optimality (marginal rate

of substitution equals marginal product of labor) will hold only on average in our economy

because consumption and labor choices are conditioned on diﬀerent information (see Remark

1 in Appendix A.2). This means that our model can generate a time-varying labor wedge, an

observation we explore in Section 4.3.

Housing producer optimality conditions are standard:

zit +qt=pit +δit,(13)

nh

it +wit =pit +δit (14)

vit =pit +δit (15)

with production technology given by

δit =αγnh

it +α(1 −γ)zit −˜

ζt,(16)

after imposing the fact that lit = 0.

Consumption producer optimality requires:

nc

it = ˜µit −η(wit −wt) + nc

t(17)

yt=nc

t(18)

wt= 0,(19)

where wtdenotes the average log-wage in the economy. Condition (17) captures ﬁrms’ demand

for island-speciﬁc labor. Firms demand more of a type of labor whenever its productivity is

high or its wage is low compared to the average, or if they demand more labor overall. Notice

13

that the wage for the aggregate labor bundle is constant, since there are no shocks to aggregate

consumption productivity; we relax this assumption in the Appendix.

All relations above obtain as exact log transformations. Only the island resource constraint

needs to be log-linearized as follows:10

βbit +C(cit −ct) = C(wit −wt) + C(nc

it −nc

t)−Qzit +bit−1.(20)

In equation (20), Cand Qrepresent the deterministic steady state values used in the lineariza-

tion. We reiterate here that neither the local housing stock (hit) nor new house production

(δit and nh

it) appear in (20): since housing is non-tradable, housing adjustments can never be

used to raise island-level consumption. Market clearing conditions 0 = Rzitdi, 0 = Rbitdi,

and n=Rnc

itdi +Rnh

itdi complete the description of equilibrium in the linearized economy.

3 Learning from prices

This section presents the main theoretical results regarding the inference problem of shoppers.

We derive the value of household resources as a function of exogenous shocks, characterize

the shoppers’ price signal, and then show the implications for inference.

3.1 Marginal value of budget resources

The only friction in the economy is shoppers’ uncertainty regarding the marginal value of

household budget resources. Without this friction, the model is a standard real business

cycle economy. Lemma 1 expresses the value of resources, λit , as determined by the choices of

worker-savers. It depends on the income prospects of the household and end-of-period wealth.

Lemma 1. In equilibrium,

λit =E[λit+τ|Ωit] = −ωµ˜µit+1 −ωbbit and rt= 0 (21)

for any τ≥0and any i∈(0,1). In addition, ωµ>0and ωb>0, with limβ→1ωb= 0.

Proof. Proved in Appendix B

10We linearize bond holdings in levels because Bit can take negative values.

14

Intuitively, the intertemporal arbitrage carried out by worker-savers allows them to equal-

ize the marginal value of budget resources across time. One important implication of Lemma

1 is that the real interest rate does not react to housing productivity shocks. This is again a

consequence of the fact that housing wealth cannot be sold across islands.

By contrast, local labor and bonds can be traded across islands in exchange for consump-

tion. Therefore, islands with more productive labor or higher savings have better consumption

prospects and a lower marginal value of resources. Thus, Lagrangian multipliers depends on

future labor productivity, ˜µit+1, and on bond holdings at the end of the period, bit.

As βapproaches one, λit becomes independent of bond holdings. This happens because,

as βtends towards unity, bond wealth generates no interest earnings and is rolled over indeﬁ-

nitely. To simplify exposition, we present derivations in the case of β→1 from below so that

λit is approximately exogenous. However, all the results in our propositions are stated for all

β∈(0,1).

We conclude this section with a remark on the distinction between local and aggregate

productivity in the consumption sector. Our model resembles a standard real business cycle

model, in that an aggregate shock to future productivity in the consumption sector would

drive the future value of resources and the real interest rate in opposite directions, leaving λit

and current consumption unchanged. This is why papers looking for business cycle eﬀects of

productivity news require either real adjustment frictions (e.g. Jaimovich and Rebelo,2009)

or nominal frictions along with suboptimal monetary policy (e.g. Lorenzoni,2009). In our

environment, however, local news has an eﬀect on λit. The information friction we describe

below transforms the eﬀects of local news into ﬂuctuations in aggregate demand.

3.2 Local housing price

We now derive the signal that shoppers use to make their inferences about ˆµit+1. To economize

notation, we solve for equilibrium assuming that at time t, ˜µit =˜

ζt−1= 0, so that ˜µit+1 =

ˆµit+1 and ˜

ζt=ζt. Since past shocks are common knowledge, nothing in the description of

equilibrium changes when we relax this.

Rearranging ﬁrst order conditions from the housing sector, we recover the standard Cobb-

15

Douglass result that the price is a linear combination of input costs weighted by their elasticity:

pit = (1 −α)vit +αγwit +α(1 −γ) (ζt+qt).(22)

We wish to rewrite (22) in terms of the exogenous variables and expectations thereof. We

substitute (21) into the local wage in (11) and, recalling that β→1 implies ωb= 0, conclude

wit =ωµˆµit+1 ≡µi.(23)

Equation (23) says that workers who expect higher future local productivity demand higher

wages today, while (22) shows that higher wages increase house prices. Going forward, we use

the deﬁnition of µi∼N(0, σ2

µ) above and drop time subscripts for contemporaneous relations.

Importantly, the price of local land only reﬂects shoppers’ local housing demand, since

equations (10), (15) and (21) can be combined to get vi=E[µi|pi]. Hence, although shoppers’

do not observe vi, they can predict it exactly. By contrast, the price of the traded commodity

good varies with the aggregate appetite for housing across islands, since market clearing for

the commodity good and (13) together imply

q=Zvidi =ZE[µi|pi]di. (24)

Using (23) and (24), shoppers’ observation of the house price pit is informationally equivalent

to observing the signal:

si=γµi+ (1 −γ)ζ+ZE[µi|pi]di.(25)

The crucial feature of the signal in (25) is that it conﬂates house prices changes caused by

local conditions with those caused by aggregate shocks. Moreover, since the correlated portion

of the price signal contains an endogenous component, a common change in expectations feeds

back into local prices, thereby further shifting the inference of all consumers.

3.3 Equilibrium

We now solve the shopper’s inference problem. The main challenge is the self-referential

nature of the signal, as its precision depends on the equilibrium volatility of the commodity

price.

Following the related literature, we focus on linear equilibria. We therefore conjecture

16

that the optimal individual expectation is linear in siand takes the form

E[µi|pi] = asi=aγµi+ (1 −γ)ZE[µi|pi]di +ζ.(26)

In (26), ameasures the weight the shopper places on the price signal in forming his forecast.

Since the signal is ex ante identical for all shoppers, each uses a similar strategy. Integrating

across the population yields

ZE[µi|pi]di =a(1 −γ)ZE[µi|pi]di +ζ.(27)

Equation (27) is useful for summarizing how changes in aggregate expectations are ampliﬁed

by the endogenous signal structure: as the weight agrows from to zero towards (1−γ)−1, initial

changes in expectations experience increasingly strong ampliﬁcation. The case where a=

(1 −γ)−1is particularly extreme, as any initial perturbation (i.e. by a non-zero productivity

shock ζ) must lead to inﬁnitely large ﬂuctuations in RE[µi|pi]di.

When adoes not equal (1 −γ)−1, equation (27) can be solved for the average expectation,

ZE[µi|pi]di =a(1 −γ)

1−a(1 −γ)ζ, (28)

which is a nonlinear function of the weight a. The fact that the average expectation is normally

distributed conﬁrms the conjectured form of the optimal individual forecast.

Integrating consumption demand in (9) shows that aggregate consumption equals the

average forecast, i.e c=RE[µi|pi]di. Hence, as long as households put nonzero weight on

their signal si, aggregate consumption moves with housing productivity, and its variance is

σ2

c(a) = a(1 −γ)

1−a(1 −γ)2

σ2, (29)

where σ2

c≡var(RE[µi|pi]di)/σ2

µand σ2≡σ2

ζ/σ2

µare the variances of the average expecta-

tion and the aggregate shock after each is normalized by the variance of the idiosyncratic

fundamental. Substituting (28) into the price signal described in equation (25), we get an

expression for the local signal exclusively in terms of exogenous shocks:

si(a) = γµi+1−γ

1−a(1 −γ)ζ, (30)

whose precision with regard to µiis given by

τ(a)≡γ(1 −a(1 −γ))

(1 −γ)σ2

.

17

We next compute the shopper’s optimal inference, taking the average weight of other

households as given. We seek an a∗such that the covariance between the signal and forecast

error is zero, i.e. E[si(a)(µi−a∗si(a))] = 0, which implies that information is used optimally.

The individual best-response weight is thus given by

a∗(a) = 1

γτ(a)

1 + τ(a).(31)

The function a∗(a) captures the individual’s best reply to the proﬁle of others’ actions. An

equilibrium of the model is characterized by a ﬁxed point, ˆa=a∗(ˆa), and there are as many

equilibria as intersections between a∗(a) and the 45◦line. In the two top panels of Figure 2

we plot the best-response weight a∗(a) for two diﬀerent values of σ. The case γ > 1/2 appears

in panel (a) and the case γ < 1/2 in panel (b). We now provide existence conditions for these

equilibria and provide intuition for the diﬀerent cases.

Unique equilibrium

Our ﬁrst proposition concerns the case in which local house prices respond relatively strongly

to local conditions, i.e. the labor share in construction is greater than one half. In this case,

the model always has a unique equilibrium.

Proposition 1. For γ≥1/2and any β∈(0,1), there exists a unique REE equilibrium,

which is characterized by au∈(0, γ−1). Moreover, limσ→∞ au= 0 and limσ→0au=γ−1with

∂au/∂σ < 0.

Proof. Given in Appendix C.

The negative slope of the best response in the range a∈[0,(1 −γ)−1] is crucial for under-

standing the forces behind the equilibrium. Negative slope entails substitutability in people’s

use of information: a higher average response to the signal lowers the individual’s optimal

weight. This happens because a higher aampliﬁes the eﬀect of aggregate noise, making si

less informative about private conditions. This result contrasts with the complementarity

featured by other models, like Amador and Weill (2010), and explains why our model can

deliver a unique equilibrium for any variance of the the aggregate shock.

18

0

0

(a) γ= 0.75

0

0

(b) γ= 0.25

0

0.5

(c) γ= 0.75

0

0.5

(d) γ= 0.25

Figure 2: Top panels illustrate the best weight function a∗(a) in a case with unique equilibrium

(a) and with multiplicity (b) for two diﬀerent values of σ. Bottom panels show the evolution of

aggregate consumption volatility in a case with a unique equilibrium (c) and with multiplicity

(d) as the relative standard deviation of the aggregate shock, σ, ranges from roughly ten (left

extreme) to approximately zero (right extreme).

19

Panel (c) of Figure 2 plots the variance of aggregate beliefs as a function of σ−1. The

ﬁgure shows that the relationship is non-monotonic, as the equilibrium weight on ζgrows as

σshrinks. Nevertheless, the latter eﬀect eventually dominates so that, in the limit σ→0,

average beliefs exhibit no ﬂuctuations. In this limit a=γ−1and the local price signal is

perfectly informative about µi.

Multiple equilibria

When local house prices respond strongly to aggregate conditions, i.e. the local labor share

in construction is less than one half, the feedback loop between demand and the commodity

price can be so strong that multiple equilibria exist. Proposition 2 summarizes this result.

Proposition 2. For γ < 1/2there always exists a “low” REE equilibrium characterized by

a−∈(0,(1 −γ)−1); in addition, there exists a threshold ¯σ2(β)with ∂¯σ2(β)/∂β ≥0such that,

for any σ2∈(0,¯σ2(β)), a “middle” and a “high” REE equilibrium also exist characterized by

a◦and =a+, respectively, both lying in the range ((1 −γ)−1, γ−1). In the limit σ2→0:

i. the “high” equilibrium converges to a point with no aggregate volatility:

lim

σ2→0a+= min 1

γ,1

1−γlim

σ2→0σ2

c(a+) = 0.

ii. the “low” and “middle” equilibria converge to the same point and exhibit non-trivial

aggregate volatility:

lim

σ2→0a◦,−=1

1−γlim

σ2→0σ2

c(a◦,−) = γ(1 −2γ)

(1 −γ)2.(32)

Proof. Given in Appendix C.

The best weight function in this case is plotted in panel (b) of Figure 2. It shows that

the function yields three intersections with the 45◦line provided the variance of productivity

shocks σis suﬃciently low. We demonstrate in the proof that a lower βis isomorphic to

considering a larger σat any a, so β→1 turns out to be the case most favorable to multiplicity.

Importantly, the qualitative features of the unique equilibrium also hold for the “low”

equilibrium described above: (i) there is substitutability between individual and average

weights; and (ii) ampliﬁcation increases as the variance of aggregate shocks falls. However,

20

since now γ−1>(1 −γ)−1, the “low” equilibrium must be distinct from the full-information

equilibrium, implying the model has multiple equilibria in the limit.

Substitutability in information use is key in generating incomplete information in the limit

of zero noise. In models like Amador and Weill (2010), which feature complementarity in the

use of public information, decreasing exogenous aggregate noise always improves the precision

of the signal so that any limit equilibrium is fully revealing. By contrast, here and in Gaballo

(2018), strengthening feedbacks oﬀset the direct eﬀect of reducing exogenous noise, generating

substitutability in information use and potentially leading to noisy equilibria even in the limit.

Panel (d) of Figure 2 illustrates the volatility of these equilibria. Consumption volatility in

the “high” equilibrium case converges to zero as σ−1goes to inﬁnity. By contrast, consump-

tion volatility in the “middle” and “low” equilibria converges to a positive, ﬁnite number.

Surprisingly, the low and middle limit equilibria have the same stochastic properties as the

extrinsic sentiment equilibrium described by Benhabib et al. (2015). In our case, however,

ﬂuctuations are driven by inﬁnitesimally-small fundamental shocks, whose realizations coor-

dinate sizable ﬂuctuations in agents’ expectations. We elaborate on this connection in Section

5.3.

3.4 The economic forces behind equilibrium

We now summarize the economic forces behind the equilibria described in Propositions 1 and

2. The goal is to clarify the propagation of belief ﬂuctuations to real variables, and illuminate

the role of the traded good in generating ampliﬁcation and multiple equilibria.

To start, consider an island experiencing a positive idiosyncratic shock to tomorrow’s

productivity (µt+1 >0). The current worker-saver observes this, and demands a higher wage

to supply her labor: she is more productive tomorrow and, since the shopper is already out

shopping, today’s consumption is not directly aﬀected by her labor supply. High local wages

feed into high local house prices, however, which is observed by the shopper. So from the

perspective of the shopper, a high local house price could signal high future local productivity,

and thereby encourage higher consumption expenditures today.

Now, suppose the economy has experienced an aggregate negative shock to construction

productivity. That also leads to a higher house price, now on all islands. Since housing

21

is separable and non-tradable, the shopper does not wish to adjust their consumption in

response to this shock. But, since the shopper observing higher house prices cannot be sure

of the source, he attributes at least some of the change to improved local productivity. Because

of this mistake, the shopper increases his demand for both housing and consumption.

In order for quantities to rise, the shopper’s optimism must drive an actual increase in labor

supply, rather than a change in relative prices. Barro and King (1984) show such real eﬀects

cannot happen if all workers and shoppers have the same information. In our model, however,

the worker-saver understands that the shopper is making a mistake by spending too much and

that, other things equal, savings will fall. This potential fall in wealth increases the marginal

utility of budget resources and induces the worker to supply more labor. The worker’s choices

thereby support the increase in consumption demand with higher actual output. Thus, even

though household savings and labor choices are taken under full information, information

heterogeneity within the household drives a wedge in the standard intratemporal optimality

condition between consumption and leisure. We expand on the model’s implications for this

labor wedge in Section 4.3.

The mechanism described thus far does not require particular assumptions about housing

production, but the addition of the tradable input provides for strong ampliﬁcation. For,

whenever optimism drives up demand for housing across islands, demand for the traded

input also rises along with the input price, q. The higher input price pushes up all housing

prices, thereby driving an even larger change in expectations, and so on. The strength of this

aggregate feedback depends on the elasticity of housing production to inputs, as captured by

γ, and the reaction of shoppers to the house price, the weight a. In particular, a unit increase

in expectations triggers an increase in the housing price signal of 1 −γ, and hence an increase

in average expectations of a(1 −γ).

Our propositions show that, when production relies more on traded than on local factors,

ampliﬁcation be can so strong that multiple equilibria arise. To see the intuition for this

multiplicity, it is helpful to focus on the situation in which the ﬂuctuations of housing pro-

ductivity are vanishingly small. Consider ﬁrst the case in which we conjecture that qdoes

not move. Then the price signal is perfectly informative about local conditions and and no

correlated ﬂuctuations in house or commodity pries may emerge. Hence, an equilibrium in

22

which all shoppers are perfectly informed — and qnever moves — must exist.

Consider now the case in which we conjecture that qﬂuctuates. In this case, shoppers’

price signals will be “polluted” by changes in qthat act as aggregate noise in inference. This

noise correlates demand across islands, drivings further ﬂuctuations in q. When individual

expectations react exactly one-to-one to changes to in q, self-fulﬁlling ﬂuctuations become

possible and an equilibrium with imperfect information — and volatile q— also exists.

We have demonstrated that a one-to-one reaction can only occur with a large enough share

for the traded input, i.e. γ < 1/2. To see why this cutoﬀ is important, observe ﬁrst that the

largest weight rational shoppers could ever place on their price signal is 1/γ. This implies

that the feedback from average expectations back into average expectations is bounded above

by (1 −γ)/γ. When γ > 1/2, this bound implies that aggregate feedback is always strictly

less than one and ampliﬁcation can never translate arbitrarily small productivity shocks into

non-trivial ﬂuctuations.

By contrast, when the commodity share is large (γ < 1/2) even vanishing productivity

shocks engender substantial ﬂuctuations. To see why, note that with small ﬂuctuations, agents

would place a weight on their signal close to the upper bound of 1/γ. However, this makes

the aggregate feedback larger than one. Without a countervailing force, such a situation

would imply that any correlated ﬂuctuation in expectations would be indeﬁnitely magniﬁed

via the qfeedback. Yet, optimal signal extraction provides such a countervailing force: as

the eﬀects of aggregate disturbances transmitted by qgrow larger, shoppers’ optimal reaction

to the price signal must shrink. In this way, signal extraction picks the unique variance for

aggregate outcomes that is consistent with one-for-one feedback, i.e. a(1 −γ) = 1, and belief

ﬂuctuations are sustained.

4 Business cycle ﬂuctuations

In this section, we show that many features of the business cycle can be explained by our

model. Our analysis also demonstrates that the learning-from-prices mechanism can qualita-

tively change the comovement properties of fundamental shocks, implying that many strate-

gies for disentangling shocks may give misleading results if learning from prices is important.

23

4.1 Public news

Before proceeding to our analysis, we introduce an anticipated (common-knowledge) compo-

nent of aggregate housing productivity. The decomposition of productivity into a common-

knowledge and surprise component serves two purposes. First, it allows us to isolate the

eﬀects of the learning channel in our model, as the common-knowledge component of produc-

tivity transmits as a standard supply-side shock. Second, by combining the responses of the

economy to forecasted and surprise productivity shocks, the model can generate a rich and

realistic correlation structure among business cycle variables.

We assume that housing productivity is composed of two independent components

ζ=ζn+ζs;

with ζn∼(N, σ2

ζn), ζs∼(N, σ2

ζs) and σ2

ζn+σ2

ζs=σ2

ζ. The ﬁrst term (ζn) is “news”;

it corresponds to the common-knowledge component of productivity, and is known to all

agents before they make consumption choices. The second term (ζs) is the “surprise”; it is

unknown to shoppers and they seek to forecast it using their observation of prices.11 For

future reference, let σ2

n≡σ2

ζn/σ2

µ, and σ2

s≡σ2

ζs/σ2

µbe the normalized variances of the news

and surprise components of productivity respectively.

Only modest modiﬁcations are necessary to characterize equilibrium in this case. Shoppers

reﬁne the information contained in the price signal by “partialing-out” the known portion of

productivity. We can thus rewrite households’ expectations as

E[µi|pi] = a(si−(1 −γ)ζn),(33)

where si−(1 −γ)ζncaptures the information available to the shopper, after he has controlled

for the eﬀect of ζn. The equilibrium values ˆaand the conditions for their existence are the

same as in the baseline economy once σ2

stakes the place of σ2.

For a given total variance of productivity, σ2=σ2

s+σ2

n, we can now span the space between

two polar cases, from the case in which productivity occurs as a pure “surprise” to the case in

which the productivity shock is common-knowledge “news”. Thus, overall comovements in the

economy will represent a mix of demand and supply shocks. We note here that, because the

11Chahrour and Jurado (2018) show that this information structure is equivalent to assuming that agents

observe a noisy aggregate signal, s=ζ+ϑ.

24

Table 1: Business Cycle Comovements

GDP Cons Hours ResInv House Pr ResInv Pr Cons TFP

GDP 1.00 0.93 0.88 0.64 0.51 0.53 -0.17

Cons 1.00 0.80 0.65 0.47 0.47 -0.02

Hours 1.00 0.50 0.54 0.66 -0.35

ResInv 1.00 0.62 0.37 -0.11

House Pr 1.00 0.81 -0.37

ResInv Pr 1.00 -0.43

Cons TFP 1.00

Note: Data are real per-capita output, real per-capita consumption, p er-capita hours in the non-farm business sector, real per-capita residential

investment, Case-Schiller real house price index, real price of residential investment, and relative TFP in the construction sector from the World

KLEMS databse (http://www.worldklems.net/data.htm). All data are annual log-levels, HP-detrended using smoothing parameter λ= 10. Date

range: 1960 to 2018, except for construction TFP which ends in 2010. Details on data construction can be found in Appendix D.

two components of productivity transmit very diﬀerently in the economy, moments generated

by projecting variables onto total productivity ζcould give very misleading inference on

productivity’s eﬀects. Econometric identiﬁcation of the distinct components of productivity

represents a substantial empirical challenge, for which Chahrour and Jurado (2019) provide

some guidance in related contexts.

4.2 Demand-driven ﬂuctuations

Table 1 summarizes unconditional correlations between business cycle variables in US data.

Although these are simple raw statistics, the table summarizes several facts that have been

documented by more sophisticated empirical analysis. In particular, the table demonstrates

that business cycles are dominated by demand-like ﬂuctuations with real quantities, house

prices, and residential investment all substantially comoving. Meanwhile, construction pro-

ductivity is at most weakly negatively related to any of these variables.

In the model, the emergence of demand ﬂuctuations can be seen intuitively by analyzing

the aggregate demand and aggregate supply schedules. Using equations (9), (10), (14) and

(16), we can express aggregate demand and supply in the housing market as

δ=c−p, (34)

δ=αγ

1−αγ p−α(1 −γ)

1−αγ ζ. (35)

Moreover, because of the learning channel, we know that aggregate consumption shifts up-

25

wards in response to a correlated increase in price signals across island,

c=ZE[µi|pi]di =a(s−(1 −γ)ζn).

Note this expression implies cdoes not move with the news component of housing productivity,

as ζnis being removed from the price signal.

To derive the implications of shopper inference for housing demand, use p= (1 −α)v+αs

and v=cto express s= (p+ (1 −α)a(1 −γ)ζn)/((1 −α)a+α). Substituting the expression

for cinto (34) we get

δ=α(a−1)

(1 −α)a+αp+αa(1 −γ)

(1 −α)a+αζn.(36)

When aggregate conditions do not feed into shoppers’ beliefs (a= 0), equation (36)

entails a standard downward-sloping aggregate demand relation in the housing market, while

consumption and working hours are invariant to housing sector productivity. By contrast,

when learning from prices is suﬃciently important—i.e. whenever ais larger than one—

equation (36) shows that δand pmust comove in response to surprise shocks.

We can now solve for equilibrium consumption, residential investment, and the price of

new housing as functions of shocks and the equilibrium inference coeﬃcient:

c=a(1 −γ)

1−a(1 −γ)ζs(37)

p=α(1 −γ)ζ+ (1 −αγ)c(38)

δ=−α(1 −γ)ζ+αγc. (39)

Expressions (37) - (39) above are useful for disentangling the direct eﬀects of productivity

from the learning channel. Equation (37) shows that a correlated mistake due to a surprise in

aggregate productivity moves consumption. Equations (38) and (39) show how this change in

beliefs transmits into the housing market, moving prices for new housing and residential in-

vestment in the same direction. Under full information (a= 0) these spillovers across markets

would disappear. Meanwhile, the appearance of housing productivity ζin (38) and (39) is

independent of a, and captures the standard neoclassical channel through which productivity

changes drive prices and quantities in opposite directions.

26

With a few more lines of algebra, we have that

c=Zλi−E[λi|pi]di =Znh

idi =Znc

idi. (40)

Equation (40) implies that an increase in consumption corresponds to an increase in working

hours in both sectors. In times of optimism, shoppers’ spending increases but wages do not,

so production increases.

Since empirical house price measures include both new and existing homes, we also derive

the connection between the price of new housing, p, and the price of the total housing stock,

pH. In the Appendix, we show that the price of each vintage moves with shoppers’ expected

Lagrangian, pi|k=−E[λi|pi]. This happens because the supply of past vintages is ﬁxed and

prices must completely absorb any change in expectations. We therefore ﬁnd that pH=

κp + (1 −κ)E[−λi|pi] where κ∈(0,1) is the steady state fraction of new houses in the total

housing stock.

Collecting these results, it is straightforward to demonstrate the following:

Proposition 3. For σ2

ssuﬃciently small, surprise aggregate productivity shocks drive positive

comovement of consumption, employment (in both sectors), residential investment, prices for

new and existing housing, commodity prices, and the price of land.

Proof. Given in appendix C.

In sum, our model exhibits comovements of aggregate business cycle variables in response

to suﬃciently small productivity shocks, in any equilibrium and for any conﬁguration of

parameters. To an outside observer, the economy would appear to be buﬀeted by recurrent

shocks to aggregate demand.

Proposition 3 requires aggregate shocks to be “suﬃciently small”. Intuitively, this is

needed because price signals must be informative enough that shoppers put substantial weight

on them. Yet, Proposition 1 shows that for γ≥1/2 aggregate ﬂuctuations still disappear in

the limit σ→0. Taken together, these results raise the question: can the unique equilibrium

model deliver comovement and realistically large business cycle ﬂuctuations at the same time?

The answer is yes. As we show in the following section, even if the surprise component ac-

counts for a small fraction of realized productivity, demand driven ﬂuctuations may dominate

unconditional comovements.

27

4.3 Business cycles under unique equilibrium

In this section, we discuss the model’s business cycle properties when it has a unique equi-

librium. We organize the discussion around three pictures illustrating its implications for

business cycle comovements, ampliﬁcation, and correlations with productivity. Our goal is to

show that our model can qualitatively account for the empirical patterns reported in Table 1.

While we do not undertake a full quantitative evaluation of the model, we wish to demon-

strate the mechanism can be very powerful for reasonable parameterizations. To this end, we

calibrate a set of parameters to standard values and/or long run targets in the data. We set

the model period to one year. We set β= 0.96 consistent with an annual real interest rate of

roughly 4%. We set φ= 0.66, to be consistent with 2013-2014 CPI relative importance weight

placed on shelter. Estimates of η, the elasticity of local labor demand, range in the literature

from below one (Lichter et al.,2015) to above twenty (Christiano et al.,2005). We use η= 2

as a baseline, and note that the aggregate eﬀects of changing ηcan be oﬀset one-for-one by

changing the volatility of local productivity.

For the housing sector, we follow Davis and Heathcote (2005) in ﬁxing α= 0.89 to match

the evidence that land accounts about 11% of new home prices.12 We pick the residential

investment labor share parameter γ=.526 by computing the ratio of labor input costs

to materials and energy costs in the construction sector, using Bureau of Labor Statistics

data from 1997-2014. Finally, we select the volatility of local productivity shocks relative to

aggregate shocks std(ˆµi)/std(ζ) = 10, implying σ= 0.228.

Comovement in business cycle variables

Figure 3 plots the unconditional correlations and volatilities of several variables in the econ-

omy. On the horizontal axis of each panel we vary the ratio between the forecastable and

non-forecastable components of productivity, going from pure “surprise” on the left to pure

“news” on the right, while holding the total variance σconstant.

Panel (a) of the ﬁgure plots the correlation of consumption and house prices with resi-

dential investment. Towards the left of the panel, when productivity is mostly unanticipated,

12For existing homes, Davis and Heathcote (2007) ﬁnd that land prices accounts for a larger portion of

home prices.

28

-1

0

1

(a) Comovements

0

1

2

(b) Ampliﬁcation

-1

0

1

(c) Relation with productivity

Figure 3: Panels illustrate unconditional correlation and volatility of business cycle variables

as a function of the ratio between volatility of the forecastable and non forecastable component

for the baseline case of γ= 0.526.

our learning channel dominates: residential investment, house prices and consumption all

perfectly comove. Given the results derived above, this also implies comovement in hours in

both sectors, the average price of land, and the price of commodities.

By contrast, when productivity is largely common knowledge, prices and quantities in the

housing market exhibit the negative correlation associated with supply-driven ﬂuctuations,

while consumption does not move. Therefore, the more housing productivity is anticipated,

the more the economy behaves like a standard real business cycle model. In between these

two extremes, the model generates positive but imperfect correlations, consistent with the

data reported in Table 1.

Ampliﬁcation

What is the role of the endogeneity of the signal in generating ampliﬁcation? Panel (b) of

Figure 3 plots the standard deviation of consumption relative to that of aggregate productivity,

as a function of the share of productivity that is forecastable. The panel contrasts two

cases (i) the baseline model and (ii) the counter-factual case in which the price signal, ˜si=

γµi+ (1 −γ)ζs, excludes its dependence on q. This comparison is useful to evaluate the

role of qin amplifying the impact of surprise shocks. To highlight this aspect we also draw

the standard deviation of the surprise component of productivity, which by construction falls

from one to zero going from left to right.

The comparison is striking. With a completely exogenous price signal, the volatility of

29

consumption, while positive, would be strictly less than the volatility of the surprise com-

ponent of productivity. This is not the case for our baseline calibration, when the signal is

endogenous. The surprise component is ampliﬁed substantially, such that consumption re-

mains more volatile than aggregate productivity even when more than 90% of productivity

ﬂuctuations are anticipated (near the middle of the horizontal axis)!

The source of ampliﬁcation can also be seen in our analytical results via equation (37).

That equation shows there is a range of parameters where aggregate consumption responds

more than one-to-one to productivity shocks.13 This result depends on the endogenous pre-

cision of the signal and, in particular, on having the commodity price qenter in local house

prices. One can easily verify that, with a constant q, the reaction of expectations to produc-

tivity shocks cannot exceed unity, provided γ > 1/2.14

Relationship with construction TFP

In our model, the noise in people’s inference comes from a fundamental shock: housing produc-

tivity. One major advantage of our approach to microfounding information is that it provides

testable implications about how beliefs ﬂuctuations should relate to measurable economic fun-

damentals. In this section, we explore this potential by showing that the data are generally

consistent with the model’s implications for one direct (i.e. model-independent) measure of a

fundamental shock: construction TFP. Other shocks may play an important role in the cycle

and, as we show in Section 5.1, can induce the same comovements via the learning channel.

However, here we emphasize how learning from prices qualitatively changes the transmission

of supply shocks and oﬀers one possible interpretation of TFP’s contractionary eﬀects.15

To this end, the last column of Table 1 reports business cycle correlations with relative

productivity in the construction sector — the data analogue to ζ— using the USA KLEMS

productivity data of Jorgensen et al. (2012). Overall, the column shows that this measure of

housing-sector productivity is negatively, but weakly, correlated with business cycle variables.

13This occurs when ˆa∈(1/2(1 −γ),1/γ) with γ∈(1/2,2/3) then ∂c/∂ζs>1.

14To see, suppose that qis ﬁxed, so that the price signal corresponds to si(0) in (30) having a precision

τ(0). Then E[µi|si(0)] = γ−1τ(0)(1 + τ(0))−1si(0), so that ∂ E[µi|si(0)]/∂ζ = (1 −γ)γ−1τ(0)(1 + τ(0))−1<1.

15Gal´ı (1999) and Basu et al. (2006) ﬁnd that aggregate productivity is contractionary for hours, while Basu

et al. (2014) ﬁnd evidence that investment-speciﬁc productivity has contractionary eﬀects across many vari-

ables. Angeletos and La’O (2009) propose a diﬀerent dispersed information mechanism by which employment

can fall in response to positive productivity shocks.

30

Most notably, residential investment is somewhat negatively correlated with this measure of

productivity, a result that would be diﬃcult to reproduce in a full information environment.

Panel (c) of Figure 3 illustrates the correlations of residential investment, the price of

housing, and consumption as a function of the ratio between the volatilities of the news and

surprise components of productivity. These correlations depend on the fraction of anticipated

productivity and, as in the data, are generally not perfect. Correlations with total produc-

tivity are imperfect because the two components of productivity – surprise and news – are

transmitted very diﬀerently in the economy. In particular, so long as a suﬃcient portion of

productivity is unanticipated, all of these variables are negatively correlated with productiv-

ity. When instead productivity is mostly common knowledge, consumption and hours do not

move while residential investment and house prices move in opposite directions.

Implications for the labor wedge

How does our model address the (Barro and King,1984) challenge and generate realistic

business-cycle comovement without relying on contemporaneous changes to productivity?

The answer is that the model generates a counter-cyclical distortion of the intratemporal

margin or “labor wedge,” so that both hours and consumption can rise at the same time. We

draw out this implication below.

Frictionless real business cycle models usually include, as a condition of intratemporal

optimality, that the marginal product of labor should equal the household marginal rate of

substitution. The labor wedge measures deviations from this condition:

τt≡log M P Nt

MRSt,(41)

where M P Ntis the marginal product of labor and M RStis the marginal rate of substitution

between leisure and consumption. Several authors have argued that empirical analogues to

this quantity are counter-cyclical, i.e. that τtis high during recessions. Following Karabar-

bounis (2014) (and ignoring labor taxes) this wedge can be decomposed into two terms,

τF

t≡log(M P Nt/Wt) and τH

t≡log(Wt/MRSt),(42)

so that τt=τF

t+τH

t.

The ﬁrst term is the “ﬁrm-side” wedge and describes the failure of marginal product to

31

-1

0

1

(a) Comovements

0

1

2

(b) Ampliﬁcation

-1

0

1

(c) Relation with productivity

Figure 4: Panels illustrate correlation and the unconditional volatility of business cycle vari-

ables as a function of the ratio between volatility of the forecastable and non forecastable

component for the case of γ= 0.45.

equal the wage. The second is the “household-side” labor wedge, and corresponds with failure

of the marginal rate of substitution to equal the wage. Since labor markets in our model are

competitive with full information, we immediately know that τF

t= 0.

Using our functional forms and information assumptions, we have that

τH

t=wt−ct

=ZE[λit|pit]di −λt

=ZE[µi|pit]di.

In words, booms in our economy correspond to moments when people are optimistic about

their local conditions — RE[µi|pit]di is positive —, where ctgrows faster than wt, and where

τH

t<0. This pattern for τHis exactly the qualitative pattern described by Karabarbounis

(2014).

4.4 Multiple equilibria: supply shocks or animal spirits?

In this section, we explore the properties of one equilibrium when γ < 1/2 as an illustration of

the ampliﬁcation power of our mechanism. We focus on the “low” equilibrium, characterized

by a−in Proposition 2, since this equilibriums turns out to be learnable in the sense of the

adaptive learning literature (see Section 5.4.)

In Figure 4, we present correlations and ampliﬁcation plots for the case of the “low”

equilibrium, changing only γ= 0.45 with respect to our baseline calibration. Panel (b) shows

32

that, in contrast to our original calibration, consumption remains roughly twice as volatile

as realized productivity even as the variance of its surprise component goes to zero. This

happens because even inﬁnitesimal surprises drive large ﬂuctuations in beliefs. Note also

that the endogeneity of the price signal is crucial to this result: if inference were based on

the counter-factual signal ˜sithat excludes q, the model could deliver large ﬂuctuations in

consumption, but these would disappear as σsshrinks.

The housing demand and supply relations in (35) and (36) provide an alternative per-

spective on this powerful ampliﬁcation. As aapproaches 1/(1 −γ), the slope of the curves

coincide, implying the two curves overlie one another. In this case, the model exhibits extreme

ampliﬁcation of vanishingly-small shocks, as any point along the coincident upward-sloping

curves represents a market clearing allocation and equilibrium volatilities are pinned down by

the conditions for optimal inference.

Since belief ﬂuctuations do not disappear with σsin this parameterization, the model

has very diﬀerent implications for aggregate comovements. First, panel (a) shows that house

prices and consumption remain positively correlated with residential investment even when

nearly all of realized productivity is anticipated. Second, as shown in panel (c), when more of

productivity is anticipated, the correlation of consumption and house prices with productivity

becomes very small. This happens because, though consumption and house prices move

substantially with surprise productivity, surprise shocks themselves play a small role in total

productivity. Hence, it is with more public information that consumption and house prices

appear most disconnected from fundamentals!

In the limit of a small surprise component, house prices and residential investment are

moved by inﬁnitesimal productivity surprises. An econometrician looking at the data gen-

erated by our model would be unable to measure such small revisions in productivity and

would probably conclude that the housing market is moved by animal spirits in the vein of

Burnside et al. (2016); Shiller (2007) or sentiments as in Angeletos and La’O (2013) and

Benhabib et al. (2015). Our model shows how demand-driven waves can be the result of

extreme ampliﬁcation of small fundamental shocks sustained by the feedback loop of learning

from prices. With respect to earlier models of sentiments, the diﬀerent is sharp: the degree

of optimism or pessimism in the economy in our model is fully determined by (potentially

33

-12 -6 0 6 12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) Income expectations at time tvs house

price experiences at time t+h.

-12 -6 0 6 12

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Income expectations at time tvs economic

news heard t+h.

Figure 5: Auto-correlations of survey measure of own income expectations with respect to

own house price experience (panel a) and with respect to news heard about the economy

(panel b).

small) fundamental changes rather being totally erratic or “animal”.

4.5 Evidence from survey data

The essential feature of our model is that people’s expectations about their future prospects

depend on their own market experiences, particularly housing. We provide here one piece of

evidence from survey data that suggests this mechanism may be important in practice.

To this end, we use evidence from the Michigan Survey of Consumer Expectations. Survey

participants are asked each month about (i) their perceptions of local house price growth over

the last year (ii) whether they have heard good or bad news about overall economic conditions

and (iii) what they expect regarding their own real income growth over the coming year.

The survey then produces index numbers from the answers to these questions, essentially

subtracting those who experienced/heard/expect about negative outcomes from those who

have experienced/heard/expect positive ones.

Panel (a) of Figure 5 plots the autocorrelation structure of people’s current expectations

34

about future income, with respect to their recent experiences in the housing market. Negative

numbers on the horizontal axis reﬂect past responses to the housing experience question, while

positive numbers reﬂect future responses. Panel (a) shows that the two series are extremely

strongly correlated, with past housing experiences leading income expectations by roughly

half a year (as measured by the peak correlation.) This result suggests a strong connection

between peoples’ past experiences in the housing market and their expectations about their

own income, exactly as our model predicts.

By contrast, Panel (b) of the ﬁgure plots the correlation structure of people’s current ex-

pectations of their own income with respect to what they report having heard about aggregate

economic developments. The correlation in this picture is much smaller than in Panel (a), sug-

gesting that what people have heard about the aggregate economy (if they’ve heard anything)

plays a much smaller role in forming people’s expectations about their own prospects.

While these results are far from dispositive on the merits of our mechanism, we think they

provide some initial evidence that learning from prices is plausible in the context of housing.

5 Extensions

This section presents several extensions that demonstrate the mechanism is robust to various

modeling details. In Section 5.1, we explore the impact of contemporaneous and future

aggregate shocks to consumption production. In Section 5.2, we allow households to observe

additional private information about local conditions and show that our results do not rely

on excluding exogenous sources of information. In Section 5.3, we explore whether extrinsic

noise may drive ﬂuctuations jointly with aggregate productivity and conclude that this is

never the case. Finally, Section 5.4 studies the issue of stability under adaptive learning for

the various equilibria of the baseline model.

35

5.1 Aggregate shocks in consumption production

For this extension, we modify the production function of the consumption sector to allow for

aggregate shocks to labor productivity,

Yt=˜

ζc

tZe˜µit/η Nc

it

1−1

ηdi1

1−1

η.(43)

Consumption productivity is a random walk with i.i.d. disturbance ζc

t∼N(0, σζ). To simplify

our exposition, we focus on time tand assume that workers in island i, but not shoppers,

know {ζc

t+1, ζ c

t, µit+1}and abstract from the presence of other aggregate shocks. A few lines

of algebra shows that

λit =−ωµµit+1 −ωbbit −ζc

t(44)

We note immediately that a contemporaneous productivity shock in consumption is equiv-

alent to an increase in consumption spending (measured in consumption units). Given the

properties of log utility, an increase in consumption spending induces an increase in hous-

ing spending as well. In other words, a productivity shock to consumption production is

equivalent to an exogenous demand shock in the housing sector.

Including the future realization of aggregate productivity helps to clarify that the model

cannot generate demand shocks in the form of news about aggregate productivity as in Loren-

zoni (2009). To see this, notice that

rt=λt−λt+1 =−ζc

t+ζc

t+1.(45)

Thus, the real interest rate adjusts to equalize the return on savings in the two periods and

anticipation of higher productivity in the future has no eﬀect on the intertemporal margin,

i.e. on consumption choices today (see also Remark 2 in Appendix B.2). This is a feature that

our model shares with frictionless real economies, as Angeletos (2018) clariﬁes. By contrast,

news about future productivity creates a demand shock in Lorenzoni (2009) because of the

presence of nominal rigidities and monetary policy that is suboptimal. A corollary to this

result is that no current variable in the economy, other than the real interest rate, moves with

anticipated aggregate consumption shocks, so shoppers will not be able to learn about them

in advance.

36

Contemporaneous consumption productivity shocks, by contrast, decrease the marginal

value of households resources, pushing up the real wages demanded by workers. In the Ap-

pendix we show that the price signal in this case is:

sit =γ(µit+1 +ζc

t) + (1 −γ)ZE[µit+1 +ζc

t|sit]di, (46)

where again we present the case limβ→1ωµ= 0 with ˜µit+1 normalize by ωµ.

One again, correlated fundamentals generate confusion between the idiosyncratic and com-

mon components of the signal. As before, the individual expectation of a household is formed

according to the linear rule E[µit+1 +ζc

t|sit] = asi. Hence, the signal embeds the average

expectation, which causes the precision of the signal to depend on the average weight a.

Following our earlier analysis, the realization of the price signal can be rewritten as

si=γµit+1 +γ

1−a(1 −γ)ζc

t,(47)

where arepresents the average weight placed on the signal by other shoppers. The average

expectation is given by

ZE[µit+1 +ζc

t|sit]di =γa

1−a(1 −γ)ζc

t,(48)

which is slightly diﬀerent from (29). The shopper’s best response function is now given by

a∗(a) = 1

γ(1 −a(1 −γ))2+ (1 −a(1 −γ)) σ2

(1 −a(1 −γ))2+σ2.(49)

While the best-response function in equation (49) is slightly diﬀerent than in (31), the

characterization of the limit equilibria is identical.

Proposition 4. In the limit σ2

µ→0, the equilibria of the economy converge to the same

points as the baseline economy. For γ > 1/2: there exists a unique equilibrium ˆasuch that

limσ2

ζc→0aµ=γ−1with limσ2

ζc→0σ2

c= 0. For γ < 1/2instead three equilibria exist such that

lim

σ2

ζc→0ˆa∈ {a−, a◦, a+}with lim

σ2

ζc→0σ2

c(ˆa)∈ {σ2

c(a−), σ2

c(a◦), σ2

c(a+)}.

Proof. Follows from the fact that the best response in (49) converges to the best response in

(31).

The proposition has a straightforward intuition. In the limit of small productivity shocks,

it does not matter if perturbations emerge from the consumption or housing sector. Hence,

37

Proposition 3 applies in this case as well, and consumption productivity drives the same

broad-based comovement among aggregates.

The important diﬀerence with respect to our baseline model is that, in this case, our mech-

anism is amplifying an otherwise smaller demand driven ﬂuctuation in the housing market.

In other words, under perfect information a shock to consumption productivity would already

translate into a smaller, but still correlated, movement in business cycle variables. To see

this, rewrite aggregate consumption of residential investment and consumption in the case

of perfect information: c=ζc

tand δt=−λt−p= (1 −γ)ζc

t, which says that residential

investment, the price of new housing and consumption move together even under perfect in-

formation. Therefore, our baseline of aggregate shocks to housing productivity has the merit

of showing that our mechanism can both generate strong ampliﬁcation of fundamental shocks

and dramatically change the qualitative transmission of shocks in the economy.

5.2 Signal extraction with private signals

Here we show that the signal extraction problem, and corresponding equilibria, are not qual-

itatively aﬀected by the availability of a private signal about the local shock. Instead, the

addition of private information maps into our analysis of Section 3.3 as an increase in the

relative variance of aggregate shocks.

Let us assume that a household j∈(0,1) in island ihas a private signal

ωij =µi+ηij (50)

where ηij ∼N(0, ση) is identically and independently distributed across households and is-

lands. In this case, households form expectations according to

E[µi|pi, ωij] = aγµi+ (1 −γ)ZE[µi|pi, ωij ]di −ζ+b(µi+ηij ),

where bmeasures the weight given to the additional private signal. Averaging out the relation

above and solving for the aggregate expectation gives

ZE[µi|pi, ωij]di =−a(1 −γ)

1−a(1 −γ)ζ,

which is identical to (28). However, now we need two optimality restrictions to determine a

38

and b. These are

E[pi(µi−E[µi|pi, ωij])] = 0 ⇒γσµ−a γ2σµ+(1 −γ)2

(1 −a(1 −γ))2σζ!−bγσµ= 0,

E[ωij(µi−E[µi|pi, ωij])] = 0 ⇒σµ−aγσ−b(σµ+ση) = 0,

which identify the equilibrium aand bsuch that each piece of information is orthogonal with

the forecast error. Solving the system for a, we get a ﬁx point equation written as

a=γ

γ2+(1−γ)2

(1−a(1−γ))2

σµ+ση

ση

σζ

σµ

.(51)

For ση→ ∞, the right-hand side of the relation above matches (31). In particular, it follows

that a lower σηin (51) is equivalent to considering a larger σζin (31). The analysis of the

baseline model thus applies directly to this generalization, and small amounts of exogenous

private information do not qualitatively change any of our earlier results.

5.3 Relation with sentiments

Authors such as Benhabib et al. (2015) have found that extrinsic (non-fundamental) sentiment

shocks may emerge in environments with endogenous signals. A natural question, given the

results in Proposition 2, is whether any equilibria exist in which errors are driven by extrinsic

shocks in addition or instead of productivity. The next proposition states that, in fact,

extrinsic sentiments are always crowded-out by common shocks to productivity.

Proposition 5. Suppose that

ZE[µi|pi]di =φζζ+φεε,

where φεis the equilibrium eﬀect of an extrinsic sentiment shock, ε∼N(0, σ2

˜ε), not related to

fundamentals. Then, φε= 0 for any σ2>0.

Proof. Given in Appendix C.

Fundamental shocks always dominates extrinsic shocks because the former have two chan-

nels — one endogenous and one exogenous — through which they inﬂuences people’s informa-

tion. Intuitively, conjecture that the average action reﬂects a response to both fundamental

and extrinsic shocks. In equilibrium, agents respond to the average expectation, and therefore

39

proportionally to the conjectured endogenous coeﬃcients for each shock. But agents also re-

spond to the exogenous component of the fundamental that appears in the price signal. Thus,

any equilibrium must depend somewhat more-than-conjectured on the fundamental relative

to the extrinsic shock. This guess and update procedure cannot converge unless the weight

on the extrinsic shock is zero.

This logic highlights the fragility of the extrinsic version of sentiments, which are coor-

dinated by endogenous signal structures. For, any shock which tends to coordinate actions

for exogenous reasons will also beneﬁt from the self-reinforcing nature of learning, thereby

absorbing the role of belief shock for itself. Indeed, the same results emerge if local shocks µi

have any common component, as we consider in Section 5.1.

5.4 Stability analysis

Here, we examine the issue of out-of-equilibrium convergence, that is, whether or not an

equilibrium is a rest point of a process of revision of beliefs in a repeated version of the static

economy. We suppose that agents behave like econometricians. At time tthey set a weight ai,t

that is estimated from the sample distribution of observables collected from past repetitions

of the economy.

Agents learn about the optimal weight according to an optimal adaptive learning scheme:

ai,t =ai,t−1+γtS−1

i,t−1pi,t (µi,t −ai,t−1pi,t) (52)

Si,t =Si,t−1+γt+1 p2

i,t −Si,t−1,(53)

where γtis a decreasing gain with Pγt=∞and Pγ2

t= 0,and matrix Si,t is the estimated

variance of the signal. A rational expectations equilibrium ˆais a locally learnable equilibrium

if and only if there exists a neighborhood z(ˆa) of ˆasuch that, given an initial estimate

ai,0∈z(ˆa), then limt→∞ ai,t

a.s

= ˆa; it is a globally learnable equilibrium if convergence happens

for any ai,0∈R.

The asymptotic behavior of statistical learning algorithms can be analyzed by stochastic

approximation techniques (see Marcet and Sargent,1989a,b;Evans and Honkapohja,2001,

for details.) Below we show that the relevant condition for stability is a0

i(a)<1, which can

easily checked by inspection of Figure 2.

40

Proposition 6. For γ > 1/2the unique equilibrium auis globally learnable. For γ < 1/2

the “low” and the “high” equilibrium, a−and a+, respectively, are always locally learnable,

whereas the middle equilibrium a◦is never.

Proof. Given in Appendix C.

It turns out that the unique equilibrium is globally learnable: revisions will lead agents

to coordinate on the equilibrium regardless of initial beliefs. With multiplicity, the “high”

and “low” equilibrium are locally learnable, whereas the middle equilibrium is not. Instead,

the middle equilibrium works as a frontier between the basins of attraction of the “low” and

“high” equilibria.

6 Conclusion

Learning from prices has played an important role in our understanding of ﬁnancial markets

since at least Grossman and Stiglitz (1980). Yet, learning from prices appeared even earlier in

the macroeconomics literature, including in the seminal paper of Lucas (1972). Nevertheless,

that channel gradually disappeared from models of the business cycle, in large part because

people concluded that fundamental shocks would be eﬀectively revealed before incomplete

knowledge about them could inﬂuence relatively slow-moving macroeconomic aggregates.

In this paper we have shown that, even if aggregate shocks are nearly common knowl-

edge, learning from prices may still play a crucial role driving ﬂuctuations in beliefs. In

fact, the feedback mechanism we described may be strongest precisely when the aggregate

shock is almost, but not-quite-fully, revealed. Endogenous information structures can deliver

strong multipliers on small common disturbances, and thus oﬀer a foundation for coordinated,

expectations-driven economic ﬂuctuations that are entirely rational. Moreover, the key fea-

ture of our theory is also a feature of reality: agents observe and draw inference from prices

that are, themselves, inﬂuenced by aggregate conditions.

We have applied this idea to house prices, because these are among the most salient prices

in the economy. Even if the economy is driven only by productivity shocks, this mechanism

captures several salient features of business cycles and its close correlation with the housing

41

market while remaining consistent with the evidence that productivity and endogenous out-

comes are weakly correlated. Hence, our results suggest that the relationship between supply

and demand shocks is more subtle than typically assumed in the empirical literature and

future empirical work may wish to take in account the implications of price-based learning.

7 Acknowledgments

We are grateful to the Editor, Veronica Guerrieri, and three anonymous referees for their

insightful guidance in revising the paper. We also thank Elena Afanasyeva, Jess Benhabib,

Edouard Challe, Mehmet Ekmekci, Roger Guesnerie, Christian Hellwig, Peter Ireland, Pierre-

Olivier Wiell, Jianjun Miao, Patrick Pintus, Kristoﬀer Nimark, Richard Tresch, Robert Ul-

bricht, Rosen Valchev, Laura Veldkamp, Venky Venkateswaran, Xavier Vives, and seminar

participants at the Toulouse School of Economics, UCLA, Cornell University, Barcelona GSE

Summer Forum, Society for Economic Dynamics, and Boston Green Line Macro conferences

for valuable suggestions and comments. Laura Veronika G´ati and Serge Bechara provided

excellent research assistance.

8 Funding

The research leading to these results has received ﬁnancial support from the European Re-

search Council under the European Community’s Seventh Framework Program FP7/2007-

2013 grant agreement No.263790 and the Laboratoire d’excellence in Economics and Decision

Sciences (LabEx EDODEC).

9 Data Availability

The data underlying this article are available in Zenodo at https://doi.org/10.5281/zenodo.4110947.

42

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48

A Model: general case

We present our baseline model, extended to include convex costs of supplying labor, preference

shocks, and shocks to productivity in the consumption sector.

The representative household living on island isolves

max

Cit,∆it ,Bit,N c

it,N h

it

E0

∞

X

t=0

βt log Cφ

itH1−φ

it Θit −vc(Nc

it)1+χc

1 + χc

−vhNh

it1+χh

1 + χh!

subject to the constraints that (i) {Cit,∆it}can only depend on the information set {Pit ,Ωit−1};

(ii) {Bit, N c

it, N h

it}can only depend on Ωit; and (iii) the budget constraint,

1

Rt

Bit +Cit +Pit∆it ≤Wc

itNc

it +Wh

it Nh

it +Bit−1+ Πc

it + Πh

t,

must hold state-by-state.

In the above problem, the household may be hit by an island-speciﬁc “taste” shock Θit,

which we use to demonstrate that consumers’ forecast errors are equivalent to taste shocks

so deﬁned. The convex disutility of labor in each sector is parametrized by χcand χh. Since

workers are imperfectly mobile across sectors, we track sector-speciﬁc wages Wc

it and Wh

it and

drop the labor market clearing condition.

In the competitive consumption sector we allow for an aggregate productivity shock and

decreasing return to scale. The production function is

Yt=e˜

ζc

t(Nc

t)αcwith Nc

t≡Ze˜µit/η Nc

it

1−1

ηdi1

1−1

η.

Produtivity ˜

ζc

t=˜

ζc

t−1+ζc

twhere ζc

tis an iid innovation drawn from a normal distribution

N(0, σζc) and αc∈(0,1) measures returns to scale. We denote by Wc

tthe price of Nc

tsuch

that Wc

tNc

t=RWc

itNc

itdi.

We allow for exogenous variation in the supply Ztto clarify that housing productivity

shocks are isomorphic to changes in the supply the input. Market clearing for the traded

input is requires

Zt=ZZitdi.

Remaining portions of the model are the same as in the main text. The model in the main

text is nested here by setting χc=χh= 0, vc=vh,σζc= 0, αc= 1, and Zt=Z.

i

A.1 Complete list of equilibrium conditions

We list here all of the necessary equilibrium conditions at a given time t. The ﬁrst order

conditions for the household are:

Λit =βEt[Λi,t+1 Rt|Ωit]

Wc

it = Λ−1

it (Nc

it)χ

c

Wh

it = Λ−1

it Nh

itχh

ΘitφC−1

it =Et[Λit|Pit,Ωit−1],

Θit

(1 −ψ)(1 −φ)

(1 −(1 −d)βψ)−1∆−1

it =Et[ΛitPit |Pit,Ωit−1]

where we use the fact that

∂Ui0

∂∆it

= (1 −ψ)(1 −φ)Θit

∞

X

τ=t

((1 −d)βψ)τ−t∆−1

it = Θit

(1 −ψ)(1 −φ)

1−(1 −d)βψ ∆−1

it .

The budget constraint

1

Rt

Bit +Cit +Pit∆it =Wc

itNc

it +Wh

it Nh

it +Bit−1+ Πc

t+ Πh

it

holds with equality and the transversality condition

lim

τ→∞ E"τ

Y

κ=0

Rt+κBit+τ|Ωit#= 0,

must hold at the individual level.

The conditions for optimality in the consumption sector are:

Nc

it =e˜µit Wc

it

Wc

t−η

Nc

t

Nc

tWc

t=αcYt

Yt=eζc

t(Nc

t)αc.

The conditions for optimality in the housing sector are:

ZitQt=α(1 −γ)Pit∆it

Nh

itWh

it =γαPit ∆it

VitLit = (1 −α)Pit∆it

∆it =L1−α

it Nh

itφe−˜

ζtZit1−φα

.

Finally, market clearing requires

Lit = 1,ZCitdi =Yt,ZBitdi = 0,and ZZitdi =Zt.

ii

A.2 Log-linearized model

We now provide the log-linear relations that describe an approximate equilibrium. The ﬁrst

order conditions for the household are:

λit =E[λit+1|Ωit] + rt(A.1)

χcnc

it =λit +wc

it (A.2)

χhnh

it =λit +wh

it (A.3)

θit −cit =E[λit|pit] (A.4)

θit −δit =E[λit|pit] + pit,(A.5)

The conditions for the consumption sector are:

nc

it = ˜µit −η(wc

it −wc

t) + nc

t(A.6)

nc

t+wc

t=yt(A.7)

yt=˜

ζc

t+αcnc

t.(A.8)

The conditions for the housing sector are:

zit +qt=pit +δit,(A.9)

nh

it +wh

it =pit +δit (A.10)

vit =pit +δit (A.11)

δit = (1 −α)li+ (αγ)nh

it +α(1 −γ)zit −˜

ζt.(A.12)

Only the budget constraint must be approximated. Using expressions for proﬁts, we have

Bit

Rt

+Cit +Pit∆it +Pit Hit−1=Wc

itNc

it +Wh

it Nh

it +PitHit−1+Bit−1+

Yt−Wc

tNc

t

| {z }

Πc

t

+PitHit −Wh

it Nh

it −Qt(Zit −Zt) + Vit

| {z }

Πh

it

.

This simpliﬁes to

Bit

Rt

+ (Cit −Yt)−(Wc

itNc

it −Wc

tNc

t) = −Qt(Zit −Zt) + Bit−1.

We consider a linearization around a non-stochastic steady-state in which Bit = 0 for all i,

hence we linearize around Bit and log-linearize for other variables. In such a steady-state, the

terms in parenthesis above are zero, so that the linearization is

βbit +C(cit −ct) = C(wc

it −wc

t) + C(nc

it −nc

t)−Q(zit −zt) + bit−1,(A.13)

where capital letters denote steady states values.

Finally, market clearing conditions are:

0 = Zbitdi (A.14)

zt=Zzitdi (A.15)

ct=yt.(A.16)

iii

Remark 1. Inspection of the ﬁrst order conditions (A.1)-(A.16) shows that consumers’ fore-

cast errors are equivalent to a shock to the intratemporal margin, in particular, an individual

consumption-housing taste shock. In particular, note that any equilibrium in the incomplete

information economy without taste shocks can be implemented in a ﬁctitious full information

economy in which taste shocks are equal to the forecast errors of the corresponding incomplete

information economy, i.e. θit ≡E[λit|pit]−λit.

B Equilibrium

This section shows the analytical solution of the extended model. In light of Remark 1, we

ignore taste shocks going forward. We also generalize our information structure by introducing

news about future aggregate productivity, in line with the extend model in section 5.1. We

focus on time tand we assume that in the second stage the worker-saver iknows the current

housing productivity, current and future consumption productivity and local productivity,

i.e. Ωit ={˜µit+1, ζt, ζ c

t, ζc

t+1} ∪ Ωit−1. We continue to assume that shoppers only observe

{pit,Ωit−1}at time t.

B.1 Expectations of the saver-worker from t+ 1 onwards

Here we characterize the equilibrium of the economy from t+ 1 onwards, conditional on the

information set of the worker-savers at time t. Throughout, we make extensive use of the law

of iterated expectations, especially the result

E[E[λit+j|pit+1,Ωit]|Ωit] = E[λit+j|Ωit ] for all j≥1.

More generally, we denote Et[xit+j]≡E[xit+j|Ωit] to capture worker expectations of any future

variable xit+j, letting the integer jspan future horizons j≥1.

Expectations of aggregates

Equation (A.7) combined with market clearing in the consumption market implies

ct−wc

t=nc

tfor all t,

while combining and aggregating (A.2) and (A.4) implies that in expectation

Et[ct+j−wc

t+j] = −χcEt[nc

t+j].

Comparing the two equations just above, we can conclude that

Et[nc

t+j] = 0.

Combining Et[nc

t+j] = 0 with (A.7) and (A.8), we can then establish that

Et[ct+j] = Et[wc

t+j] = Et[˜

ζc

t+j].(B.1)

Local budget constraint

Combining ﬁrst order conditions (A.5) and (A.9) one ﬁnds that:

zit −zt=cit −ct.(B.2)

iv

Plugging (A.6) and (B.2) into the budget constraint we have:

(C+Q)(cit −ct) + βbit =C˜µit + (1 −η)C(wc

it −wc

t) + bit−1.(B.3)

Similarly, we can use equations (A.2) and (A.6) to relate island and aggregate labor,

χc(nit −nc

t) = (λit −λt)+(wc

it −wc

t)

χc(nit −nc

t) = χc˜µit −ηχc(wc

it −wc

t)

Use the consumption demand condition in (A.4) to eliminate Lagrange multipliers, combine

the above two equations and take expectations to get

Et[wc

it+j−wc

t+j] = χc

1 + ηχc

˜µit+j+1

1 + ηχc

(cit+j−ct+j).(B.4)

Taking expectations of (B.3), substituting in expression (B.4), and simplifying yields

Cη(1 + χc)

1 + ηχc

+QEt[cit+j−ct+j]−C1 + χc

1 + ηχcEt[˜µit+j]

| {z }

≡∆˜cit+j

+βEt[bit+j] = Et[bit+j−1].

(B.5)

Using the deﬁnition of ∆˜cit+jabove, this reduces to

∆˜cit+j+βEt[bit+j] = Et[bit+j−1].(B.6)

Use of the Euler equation and transversality

A ﬁrst observation involves the Euler equation (A.1). Subtracting (A.1) from its aggregated

version establishes that

Et[cit+j+1 −ct+j+1] = Et[cit+1 −ct+1 ] (B.7)

Moreover, since local productivity is a random walk, we also have that

E[˜µit+j|Ωit] = ˜µit+1

for for any τ≥t. Hence, the ∆˜cit+jterm in (B.6) is constant across all horizons j≥1.

Calling this constant value ∆c, equation (B.6) can be solved forward to ﬁnd

Et[bit+j−1] = 1

1−β∆c.

Since this equation holds for all j≥1, bonds holdings must be expected to be constant going

forward, i.e.

bit+j=bit.

This is the unique equilibrium path for bonds, since any other solution satisfying (B.6) implies

expected bond holdings grow unboundedly over time, violating transversality.

Derivation of E[λit+1|Ωit ]

Using Et[bit+j] = bit in equation (B.5) and Et[ct+1] = ˜

ζc

t+1, and solving for Et[cit+1 ] we get

Et[λit+1] = −Et[cit+1 ] = −ωµ˜µit+1 −ωbbit −˜

ζc

t+1,

v

with

ωµ=

C1+χc

1+ηχc

Cη(1+χc)

1+ηχc+Q>0,and ωb=1−β

Cη(1+χc)

1+ηχc+Q>0.

As stated in the main text, notice that limβ→1ωb= 0.

B.2 Equilibrium at time t

Derivation of λit (Lemma 1)

Our ﬁrst objective is ﬁnding the equilibrium mapping from fundamentals to bit. Let us ﬁnd

their common component λt. One can use the aggregate version of (A.2),(A.7) and (A.8) to

get χcnc

t=λt+wc

tand wc

t=˜

ζc

t+ (αc−1)nc

tto get

(1 −αc+χc)nc

t=λt+˜

ζc

t.

Combining this with (A.8) gives a relation between the realized aggregate lambda and shop-

pers’ expectations

λt=−1−αc+χc

αcZE[λit|pit]di −1 + χc

αc

ζc

t.

Note that this expression is valid also for future times. Using the law of iterated epectations,

we have that Et[λt+1 ] = −ζc

t+1 which is consistent with what we have found above. In this

case, the aggregate version of the Euler equation (A.1) implies,

rt=λt−E[λt+1|Ωit] = −1−αc+χc

αcZE[λit|pit]di −1 + χc

αc

ζc

t+ζc

t+1.

Given that (A.1) must also hold at the local level, we then have the following

λit =−ωµ˜µit+1 −ωbbit −ζc

t+1

| {z }

=E[λit+1|Ωit ]

+rt

=−ωµ˜µit+1 −ωbbit +1−αc+χc

αc

ct−1 + χc

αc

ζc

t,(B.8)

which reduces to expression (21) of Lemma 1 under our baseline assumptions that αc= 1,

χc= 0, and ζc

t= 0.

Remark 2. Equation (B.8) shows that the anticipation of future consumption productivity

does not aﬀect the marginal valuation of current consumption. This is a standard ﬁnding in

real business cycle models, since real interest rates neutralize the eﬀect of anticipated produc-

tivity changes. By contrast, current productivity does move the marginal valuation of current

consumption.

vi

Price of new housing

Here we derive the expression for the equilibrium price of new housing. By using (A.3), (A.5),

(A.9), and (A.10) we get

pit +δit =−E[λit|pit],

zit =−E[λit|pit]−qt,

nh

it =1

1 + χh

(λit −E[λit|pit]) .

The housing price is then

pit =−E[λit|pit]−αγnit −α(1 −γ) (−ζt+zit)

=−E[λit|pit]−αγ 1

1 + χh

(λit −E[λit|pit])−α(1 −γ) (−ζt−qt−E[λit|pit ])

=1−α(1 −γ)−αγ

1 + χhE[−λit|pit ] + α −γ

1 + χhλit <