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Learning from House Prices: Amplification and Business Fluctuations


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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 autonomous changes in sentiment without resorting to non-fundamental shocks or nominal rigidity.
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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
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
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:
Department of Economic Sciences, HEC Paris, 1 rue de la Liberation, 78350 Jouy en Josas. Email:
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.
1960 1969 1978 1987 1996 2005 2014
% 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.
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).
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
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
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.
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.
The representative household on island ichooses consumption, labor supply, and savings in
a risk-free nominal bond to maximize the utility function:
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, ∆|k,
constructed at time kand combined according to the Cobb-Douglas aggregator
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.
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
+1|k= (1 d)∆ |k
for τk(while, of course, ∆ |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)Hit1.
Housing consumption can now be written Hit = ∆1ψ
it1, which we use going forward.
The choices of the household are subject to the following budget constraint,
Bit Bit
+Cit +Pitit|tWitNit Bit1Πc
it 0 (3)
for t∈ {0,1,2...}with Bi1= 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|kit|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 =L1α
it Xα
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
combining local labor, Nh
it, with a traded commodity, Zit, with share parameter γ(0,1).
The housing firm maximizes profits,
it Pitit 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, ˜
ζt1+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.
from all sectors to maximize profits,
subject to the production function,
Yt=Ze˜µit Nc
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 = ˜µit1+ 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
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.
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
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,it1}.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{Bi1,Hi1,˜µi0}i(0,1) ,˜
ζi1o, 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 {˜
t=0, such that for each t0and
(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).
(ii) Cit,it are measurable with respect to {Pit,it1}
(iii) Nit, Bit are measurable with respect to {it};
(b) Housing producers optimize, i.e. {Nh
it, Zit , Lit,it }are solutions to max{Nh
it,Zit ,Lit,it }Πh
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
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. φC1
it =Eit|Pit ,it1]. 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.
Optimality of worker-shopper choices requires:
wit =λit (11)
λit =E[λit+1|it] + 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)
it +wit =pit +δit (14)
vit =pit +δit (15)
with production technology given by
δit =αγnh
it +α(1 γ)zit ˜
after imposing the fact that lit = 0.
Consumption producer optimality requires:
it = ˜µit η(wit wt) + nc
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
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 +bit1.(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|it] = ωµ˜µit+1 ωbbit and rt= 0 (21)
for any τ0and any i(0,1). In addition, ωµ>0and ωb>0, with limβ1ωb= 0.
Proof. Proved in Appendix B
10We linearize bond holdings in levels because Bit can take negative values.
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 β1 from below so that
λit is approximately exogenous. However, all the results in our propositions are stated for all
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 =˜
ζt1= 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-
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 β1 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 µiN(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
Following the related literature, we focus on linear equilibria. We therefore conjecture
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 (1γ)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)
1a(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
c(a) = a(1 γ)
1a(1 γ)2
σ2, (29)
where σ2
µand σ2σ2
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+1γ
1a(1 γ)ζ, (30)
whose precision with regard to µiis given by
τ(a)γ(1 a(1 γ))
(1 γ)σ2
We next compute the shopper’s optimal inference, taking the average weight of other
households as given. We seek an asuch that the covariance between the signal and forecast
error is zero, i.e. E[si(a)(µiasi(a))] = 0, which implies that information is used optimally.
The individual best-response weight is thus given by
a(a) = 1
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=aa), and there are as many
equilibria as intersections between a(a) and the 45line. 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 γ1/2and any β(0,1), there exists a unique REE equilibrium,
which is characterized by au(0, γ1). Moreover, limσ→∞ au= 0 and limσ0au=γ1with
∂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.
(a) γ= 0.75
(b) γ= 0.25
(c) γ= 0.75
(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).
Panel (c) of Figure 2 plots the variance of aggregate beliefs as a function of σ1. 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 σ0,
average beliefs exhibit no fluctuations. In this limit a=γ1and 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
aand =a+, respectively, both lying in the range ((1 γ)1, γ1). In the limit σ20:
i. the “high” equilibrium converges to a point with no aggregate volatility:
σ20a+= min 1
c(a+) = 0.
ii. the “low” and “middle” equilibria converge to the same point and exhibit non-trivial
aggregate volatility:
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 45line 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 β1 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,
since now γ1>(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 σ1goes 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
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
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
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. This implies
that the feedback from average expectations back into average expectations is bounded above
by (1 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. 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.
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
with ζn(N, σ2
ζn), ζs(N, σ2
ζs) and 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
µ, and 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
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.
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 ( 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=cp, (34)
1αγ pα(1 γ)
1αγ ζ. (35)
Moreover, because of the learning channel, we know that aggregate consumption shifts up-
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
(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)
1a(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.
With a few more lines of algebra, we have that
c=ZλiE[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 γ1/2 aggregate fluctuations still disappear in
the limit σ0. 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.
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.
(a) Comovements
(b) Amplification
(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.
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
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) 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)] = γ1τ(0)(1 + τ(0))1si(0), so that E[µi|si(0)]/∂ζ = (1 γ)γ1τ(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.
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:
τtlog M P Nt
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,
tlog(M P Nt/Wt) and τH
so that τt=τF
The first term is the “firm-side” wedge and describes the failure of marginal product to
(a) Comovements
(b) Amplification
(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
=ZE[λit|pit]di λt
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
t<0. This pattern for τHis exactly the qualitative pattern described by Karabarbounis
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 ain 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
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
-12 -6 0 6 12
(a) Income expectations at time tvs house
price experiences at time t+h.
-12 -6 0 6 12
(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
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.
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,
tZe˜µit Nc
Consumption productivity is a random walk with i.i.d. disturbance ζc
tN(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
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=λtλt+1 =ζc
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.
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 limβ1ωµ= 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
1a(1 γ)ζc
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
1a(1 γ)ζc
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
µ0, 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
ζc0aµ=γ1with limσ2
c= 0. For γ < 1/2instead three equilibria exist such that
ζc0ˆa∈ {a, a, a+}with lim
ca)∈ {σ2
c(a), σ2
c(a), σ2
Proof. Follows from the fact that the best response in (49) converges to the best response in
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,
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=λtp= (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)
1a(1 γ)ζ,
which is identical to (28). However, now we need two optimality restrictions to determine a
and b. These are
E[pi(µiE[µi|pi, ωij])] = 0 γσµa γ2σµ+(1 γ)2
(1 a(1 γ))2σζ!bγσµ= 0,
E[ωij(µiE[µi|pi, ωij])] = 0 σµaγσb(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
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, εN(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
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,t1+γtS1
i,t1pi,t (µi,t ai,t1pi,t) (52)
Si,t =Si,t1+γt+1 p2
i,t Si,t1,(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,0za), then limt→∞ ai,t
= ˆa; it is a globally learnable equilibrium if convergence happens
for any ai,0R.
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.
Proposition 6. For γ > 1/2the unique equilibrium auis globally learnable. For γ < 1/2
the “low” and the “high” equilibrium, aand a+, respectively, are always locally learnable,
whereas the middle equilibrium ais 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
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
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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
Cit,it ,Bit,N c
it,N h
βt log Cφ
it Θit vc(Nc
1 + χc
1 + χh!
subject to the constraints that (i) {Cit,it}can only depend on the information set {Pit ,it1};
(ii) {Bit, N c
it, N h
it}can only depend on Ωit; and (iii) the budget constraint,
Bit +Cit +Pitit Wc
it +Wh
it Nh
it +Bit1+ Πc
it + Πh
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
t)αcwith Nc
tZe˜µit Nc
Produtivity ˜
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
that Wc
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
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.
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 =βEti,t+1 Rt|it]
it = Λ1
it (Nc
it = Λ1
it Nh
it =Etit|Pit,it1],
(1 ψ)(1 φ)
(1 (1 d)βψ)11
it =EtitPit |Pit,it1]
where we use the fact that
= (1 ψ)(1 φit
((1 d)βψ)τt1
it = Θit
(1 ψ)(1 φ)
1(1 d)βψ 1
it .
The budget constraint
Bit +Cit +Pitit =Wc
it +Wh
it Nh
it +Bit1+ Πc
t+ Πh
holds with equality and the transversality condition
τ→∞ E"τ
Rt+κBit+τ|it#= 0,
must hold at the individual level.
The conditions for optimality in the consumption sector are:
it =e˜µit Wc
The conditions for optimality in the housing sector are:
ZitQt=α(1 γ)Pitit
it =γαPit it
VitLit = (1 α)Pitit
it =L1α
it Nh
Finally, market clearing requires
Lit = 1,ZCitdi =Yt,ZBitdi = 0,and ZZitdi =Zt.
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|it] + rt(A.1)
it =λit +wc
it (A.2)
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:
it = ˜µit η(wc
it wc
t) + nc
The conditions for the housing sector are:
zit +qt=pit +δit,(A.9)
it +wh
it =pit +δit (A.10)
vit =pit +δit (A.11)
δit = (1 α)li+ (αγ)nh
it +α(1 γ)zit ˜
Only the budget constraint must be approximated. Using expressions for profits, we have
+Cit +Pitit +Pit Hit1=Wc
it +Wh
it Nh
it +PitHit1+Bit1+
| {z }
+PitHit Wh
it Nh
it Qt(Zit Zt) + Vit
| {z }
This simplifies to
+ (Cit Yt)(Wc
it Wc
t) = Qt(Zit Zt) + Bit1.
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) + bit1,(A.13)
where capital letters denote steady states values.
Finally, market clearing conditions are:
0 = Zbitdi (A.14)
zt=Zzitdi (A.15)
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} ∪ it1. We continue to assume that shoppers only observe
{pit,it1}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,it]|it] = E[λit+j|it ] for all j1.
More generally, we denote Et[xit+j]E[xit+j|it] to capture worker expectations of any future
variable xit+j, letting the integer jspan future horizons j1.
Expectations of aggregates
Equation (A.7) combined with market clearing in the consumption market implies
tfor all t,
while combining and aggregating (A.2) and (A.4) implies that in expectation
t+j] = χcEt[nc
Comparing the two equations just above, we can conclude that
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[˜
Local budget constraint
Combining first order conditions (A.5) and (A.9) one finds that:
zit zt=cit ct.(B.2)
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) + bit1.(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
χc(nit nc
t) = χc˜µit ηχc(wc
it wc
Use the consumption demand condition in (A.4) to eliminate Lagrange multipliers, combine
the above two equations and take expectations to get
t+j] = χc
1 + ηχc
1 + ηχc
Taking expectations of (B.3), substituting in expression (B.4), and simplifying yields
Cη(1 + χc)
1 + ηχc
+QEt[cit+jct+j]C1 + χc
1 + ηχcEt[˜µit+j]
| {z }
+βEt[bit+j] = Et[bit+j1].
Using the definition of ∆˜cit+jabove, this reduces to
∆˜cit+j+βEt[bit+j] = Et[bit+j1].(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|it] = ˜µit+1
for for any τt. Hence, the ∆˜cit+jterm in (B.6) is constant across all horizons j1.
Calling this constant value ∆c, equation (B.6) can be solved forward to find
Et[bit+j1] = 1
Since this equation holds for all j1, bonds holdings must be expected to be constant going
forward, i.e.
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|it ]
Using Et[bit+j] = bit in equation (B.5) and Et[ct+1] = ˜
t+1, and solving for Et[cit+1 ] we get
Et[λit+1] = Et[cit+1 ] = ωµ˜µit+1 ωbbit ˜
1+ηχc+Q>0,and ωb=1β
As stated in the main text, notice that limβ1ωb= 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
tand wc
t+ (αc1)nc
tto get
(1 αc+χc)nc
Combining this with (A.8) gives a relation between the realized aggregate lambda and shop-
pers’ expectations
αcZE[λit|pit]di 1 + χc
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=λtE[λt+1|it] = 1αc+χc
αcZE[λit|pit]di 1 + χc
Given that (A.1) must also hold at the local level, we then have the following
λit =ωµ˜µit+1 ωbbit ζc
| {z }
=E[λit+1|it ]
=ωµ˜µit+1 ωbbit +1αc+χc
ct1 + χc
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
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,
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) (ζtqtE[λit|pit ])
=1α(1 γ)αγ
1 + χhE[λit|pit ] + α γ
1 + χhλit <