Household Consumption and Dispersed Information∗
Jonathan J. Adams†
University of Florida

Eugenio Rojas‡
University of Florida

October 14, 2022

WORKING PAPER
Link to Most Current Version
Abstract
We study the effects of aggregate income shocks in a small open economy heterogeneous agent model. By introducing a standard information friction, we are able toexplain two patterns of small economies experiencing large income changes: (1) excess
volatility in consumption and (2) household consumption elasticities that have low
correlation with income. With a standard dispersed information structure, households
cannot distinguish aggregate income shocks from idiosyncratic ones. Therefore theirconsumption responds excessively to aggregate income changes, which they forecast
as likely to be more persistent than they would if they had full information. We
demonstrate that this effect occurs at all points in the income distribution, lowering
the correlation of the consumption elasticity with income. Finally, we corroborate ourcentral mechanism using survey data on household expectations of their future income.

JEL-Codes: D84, E21, E32
Keywords: Heterogeneous Agents, Incomplete Information, Heterogeneous Beliefs, Business Cycles, Consumption Volatility∗
We thank Robert Ainsworth, German Bet, Yu-Ting Chiang, Elias Dinopoulos, Joel Flynn, Rafael Guntin,
Alexandre Kohlhas, Felipe Saﬀie, David Sappington, and seminar participants at the University of Florida
and SED 2022 for helpful comments and suggestions. All errors are our own.
†
Department of Economics, University of Florida, 333 Matherly Hall, PO Box 117140, Gainesville, FL
32611. Email: adamsjonathan@ufl.edu.
‡
Department of Economics, University of Florida, 331 Matherly Hall, PO Box 117140, Gainesville, FL
32611. Email: erojasbarros@ufl.edu.

11Introduction

Why is aggregate consumption so volatile? This is a classic question in macroeconomics.
Traditional models with full information and rational expectations (FIRE) predict that consumption will be relatively inelastic to aggregate income changes, as households smooth consumption (Kydland and Prescott, 1982). However, consumption volatility is much higher
than expected, even more so in developing open economies (Aguiar and Gopinath, 2007).
Possible explanations abound, including financial frictions: when agents face collateral or
borrowing constraints, they become more elastic to transitory income shocks (Carroll, 2001).
However, recent evidence suggests that standard household financial frictions cannot be themain mechanism. The financial friction mechanism gives a clear prediction: the consumption elasticity with respect to an aggregate income shock should decrease with householdincome. Yet, evidence for this pattern is weak; Guntin et al. (2020) (hereafter GOP) document that the consumption elasticity to large macroeconomic shocks remains high acrossthe entire income distribution, using household-level consumption data from five small open
economies.1 They conclude that consumers of all income levels appear to respond as ifchanges to aggregate income are permanent shocks.
We propose an explanation that is consistent with the finding by GOP: households facean information friction. Specifically, households are unable to accurately distinguish idiosyncratic income shocks from aggregate income shocks. Crucially, the idiosyncratic componentof household income is more persistent than aggregate income. Therefore, if household income rises due to an aggregate shock, households will expect their income improvement tobe more persistent than if they had full information. Households respond by increasing consumption by more than if they correctly predicted that the income improvement would beshort-lived.
We study this mechanism in a tractable heterogeneous agent model with dispersed information. Households in a small open economy receive stochastic income, and solve a standard
consumption-savings problem to mitigate their income risk and smooth consumption overtime. Income is determined by two stochastic components: an idiosyncratic income process, and a less persistent aggregate income process. Households have rational expectations,
1

Like Guntin et al. (2020), we are careful to distinguish the distribution of consumption elasticities toaggregate income shocks from the distribution of marginal propensities to consume (MPCs), which areelasticities to unexpected idiosyncratic windfalls. Empirical evidence is clear that the MPCs to transitory
income shocks are decreasing with income and especially household liquidity. For example, Johnson et al.
(2006) document this pattern in the response to 2001 tax rebates, Parker et al. (2013) do the same for2008 stimulus checks, as do Chetty et al. (2020) for COVID-19 relief. These relationships with income
are consistent with standard theories, including our own. Kaplan and Violante (2014) augment a standard
theory to explain why liquidity matters more than income or wealth.

2but incomplete information. They do not directly observe aggregate income shocks, and socannot accurately distinguish between the two components, which distorts their consumption and savings decisions. We find that the volatility of aggregate consumption growth is
one third larger when agents face the information friction. The additional risk induces astronger precautionary savings motive, so average savings is 43% higher than the full information baseline. And while consumption is more volatile, the elevated wealth also increases
the average consumption level.
In the cross-section, we also recover results that are in line with GOP’s finding of relatively homogeneous consumption responses to income shocks. Specifically, we document thatthe consumption elasticity to aggregate income is larger and more homogeneous across theincome distribution when agents face information frictions, where the slope of the consumption elasticity across log income levels is nearly zero. This is the first of several results that
require studying incomplete information and heterogeneous agents in a unified framework.
Moreover, we find the the information friction and financial friction interact in a variety ofrich ways: the frictions jointly attenuate inequality dynamics, reduce the sensitivity of thewealth distribution to the borrowing constraint, reverse the relationship between idiosyncratic risk and the aggregate consumption elasticity, and generate endogenous correlations
between aggregate forecasts and wealth.
Is our information friction realistic? To answer this question, we employ survey data on
household expectations, and document evidence corroborating our central mechanism. The
main implication of the information friction is that the response of households forecasts of
their own income should respond with the same elasticity to aggregate and idiosyncraticincome shocks. If instead households have full information, then their forecasts will be lesselastic to aggregate than to idiosyncratic shocks. This is a testable prediction. We employ
data on household forecasts from the NY Fed’s Survey of Consumer Expectations, anddecompose household income into aggregate and idiosyncratic components at the state level.
Our results are clear: household forecasts are at least as elastic to aggregate shocks as toidiosyncratic shocks. Thus we confirm that GOP’s characterization of consumption behavior
applies to household expectations as well: consumers’ forecasts respond to aggregate incomechanges as if they are more persistent than they really are.
Our paper contributes to several strands of the literature. First, we join a small but
promising new literature that synthesizes incomplete information theories with heterogeneous
agent models. Broer et al. (2021) and Broer et al. (2022) depart from the standard FIRE
structure by introducing a rational inattention decision that is endogenously heterogeneous.2
2

Other papers depart from FIRE in heterogeneous agent models by relaxing rational expectations rather
than of full information. For example, Auclert et al. (2020) and Carroll et al. (2020) assume agents have3Angeletos and Huo (2021) show that myopic effects of information frictions are exacerbatedby the MPC distribution typical of HANK models. Gallegos (2022) extends Bilbiie (2020)
to study a linear HANK model with dispersed information.3
Second, we contribute to the empirical literature that examines the effects of aggregate
shocks on forecasts of idiosyncratic variables. Many papers study forecasts of aggregate
quantities, about which survey evidence is bountiful. But less works exists for households’
or firms’ forecasts of factors that are specific to them. Among this group, Andrade et al.
(2022) study how firms’ forecasts of their own prices and production respond to aggregate
and industry-level shocks, finding support for the standard dispersed information structure.
Adams-Prassl et al. (2022) study workers’ perceptions of the returns to searching for a job,
and they document a channel through which incomplete information affects worker behavior:
workers who are more optimistic about the macroeconomy believe they are more likely toreceive a job offer.
Third, we contribute to a class of heterogeneous agent models attempting to understandlarge consumption responses to income. While we follow GOP and study the consumptionincome elasticity to aggregate shocks, many more papers focus on explaining large MPCs.
We consider our explanation of the consumption-income elasticity to be complementary tothis literature, which Kaplan and Violante (2022) survey. While our simple model cannot
explain the cross-sectional evidence on consumption out of idiosyncratic windfalls, our large
consumption elasticities to aggregate shocks are relevant for many of the same aggregate
applications.4 One advantage of our approach is that we attain large consumption-income
elasticities, while maintaining a realistic wealth distribution that does not suffer from the“missing middle” problem that Kaplan and Violante identify as plaguing most single asset
models that otherwise achieve large MPCs.
The remainder of the paper is organized as follows. In Section 2 we describe our model,
including the structure and intuition for the information friction. Section 3 describes our
main results and equilibrium behavior in the model. In Section 4 we explore the interactions
between the information and asset market frictions. Section 5 documents the empiricalevidence corroborating our information friction. Section 6 concludes.
sticky expectations, in the style of Mankiw and Reis (2002) and Carroll (2003).
3
Angeletos and Lian (2016) survey the broader literature of incomplete information in macroeconomics.
See Heathcote et al. (2009), Quadrini and Ríos-Rull (2015), Krueger et al. (2016) for broad surveys of thehousehold heterogeneity models, and Kaplan and Violante (2018) for a more recent survey that includes
HANK features.
4This includes monetary policy (Kaplan et al., 2018), fiscal policy (Auclert et al., 2018), and aggregate
shocks in general (Bilbiie, 2020) among others.

4

2

Model

In this section we describe our baseline model. Heterogeneous agents trade risk-free assets ina small open economy in order to self-insure against income risk and smooth consumption.
The agents face a standard friction: dispersed information in the style of Lucas (1972) thatprevents them from observing the aggregate state of the economy.

2.1

Households

There is a unit measure of identical and infinitely lived households. Households are indexedby i and time is indexed by t.
The household’s preferences over current and future consumption are represented by theutility function][∞
1−γ∑ Ci,t+s
−
1
(1)
Ei,t
βs
1
−
γ
s=0
where Ci,t is the household’s consumption in period t, β is its discount factor, and γ is thecoeﬀicient of relative risk aversion. The expectation operator Ei,t is conditional on household
i’s information set Ωi,t .
The household receives stochastic income Yi,t , which in logs is the sum of a mean zero
idiosyncratic component Yi,tI and a common aggregate component YtG :ln Yi,t = ln Yi,tI + ln YtG(2)

The idiosyncratic and aggregate components each follow an AR(1) process:I
ln Yi,tI = ρI ln Yi,t−1+ uIi,tG
ln YtG = ρG ln Yt−1
+ uG
t

(3)

2
with uIi,t ∼ N (0, σI2 ), uGt ∼ N (0, σG ), ρI ∈ (0, 1), and ρG ∈ (0, 1). Crucially, we assume ρI >
ρG so that the idiosyncratic component is more persistent than the aggregate component.
Household log income ln Yi,t is the sum of independent AR(1) processes, so ln Yi,t is an
ARMA(2,1). Appendix A derives the parameters of this composite time series.
The household may hold a risk-free asset At which pays exogenous interest rate r. The
household’s budget constraint is

Yi,t + (1 + r)Ai,t = Ci,t + Ai,t+1
with Ai,t+1 ≥ 0 for t ≥ 0. This implies that households cannot borrow.

5

(4)

2.2

The Information Friction

Households do not observe the incomes or choices of any other households, nor of the aggregate economy. They observe their income Yi,t but cannot independently observe theidiosyncratic and aggregate components Yi,tI and YtG . Thus if their income rises, they are
unsure to what extent the increase was specific to them or economy-wide. Formally, thehousehold’s information set evolves byΩi,t = {Ωi,t−1 , Yi,t , Ai,t }(5)

The information friction makes households over-estimate the persistence of an aggregate
income shock. The autocorrelation of the aggregate income component ρG is less than thatof idiosyncratic income component ρI . The sum of the two components, which is observed byhouseholds, has an autocorrelation between ρG and ρI : individual income is more persistentthan aggregate income. So when there is an aggregate income shock and households cannot
tell that the shock is aggregate, they expect their income to change more persistently thanthey would if they had full information.

(a) Different Idiosyncratic Persistences(b) Different Idiosyncratic Variances

Figure 1: Income Forecasts After Aggregate Shocks
Figure 1 demonstrates this over-estimation of an aggregate shock’s persistence. This figure plots households’ forecasts of their income multiple periods into the future after receiving
an aggregate income shock.5 The aggregate autocorrelation is ρG = 0.8; when households
have full information, they correctly forecast their future income which decays relatively
rapidly (the solid red curves).
5

Appendix A.3 derives expressions for these forecasts.

6

The information friction matters most when the aggregate and idiosyncratic autocorrelations are most dissimilar. Panel (a) plots forecasts under incomplete information fordifferent idiosyncratic autocorrelations ρI . When ρI = ρG , the information friction has noeffect and households’ forecasts are equivalent to the full information forecasts. When ρI islarger, individual income is more persistent, so households’ forecasts of future income decay
more slowly, and their expectations diverge from the full information case.
The information friction also has larger effects when the idiosyncratic shock has a larger
variance. The autocorrelation of individual income is somewhere between those of the aggregate and idiosyncratic components, and when the idiosyncratic component has a larger
variance, its larger autocorrelation makes individual income more persistent. Panel (b) plots
this effect for different idiosyncratic shock variances σI2 , while the other parameters are held2
at ρG = 0.80, ρI = 0.95, and the aggregate shock variance is σG= 1. When σI2 is larger,
the idiosyncratic process has higher weight in determining income which becomes more persistent, and forecasted income decays more slowly. When σI2 is small, household income ismostly driven by the aggregate process so households forecast accurately after an aggregate
shock; as σI2 goes to zero, the effect of the information friction disappears.

2.3

Equilibrium Definition

I
Given infinite sequences of exogenous variables {Yi,t , YtG , Yi,tI , uGt , ui,t } for all i ∈ I, a competitive equilibrium in this economy consists of infinite sequences of allocations {Ci,t , Ai,t }for all i ∈ I; and information sets Ωi,t for all i ∈ I such that:

1. Households maximize utility (1), subject to the budget constraint (4) and the noborrowing constraint.
2. Income is determined by (2) and (3).
3. Information sets evolve according to (5).

3Quantitative Analysis

In this section we document the model behavior. The information friction raises consumption volatility and nearly eliminates the correlation between household income and theirconsumption elasticity to aggregate shocks.

7

3.1

Calibration

We calibrate the model to match features of U.S. states, which we treat as small open
economies. The time frequency is annual, and we set the world annual real interest to 2%.
The discount factor β is set to be 0.946 in order to match the U.S. ratio of net worth tolabor income in the National Income and Product Accounts (NIPA) and the Federal Reserve’s
Flow of Funds Tables. The risk aversion parameter γ is equal to 1 so that agents have a
log utility function. To parameterize the stochastic process for idiosyncratic income, we set
the values of ρI and σI to match the income dynamics estimated by Guvenen et al. (2021),
which implies ρI = 0.97 and σI = 0.19. For the aggregate income shock process, we use theU.S. NIPA accounts to estimate a state-level aggregate income process (Appendix C). We
find that the autocorrelation is approximately 0.87 and the standard deviation is roughly
0.03. These statistics define the values for ρG and σG . Table 1 summarizes our baseline
calibration, although we explore some alternative parameter values in Section 4.1.
Table 1: Calibration
Parameter
Interpretation
Value
Referenceβ
Discount factor
0.946 U.S. net-worth-to-earnings ratio of 8rReal interest rate
0.02
Standard valueγ
Risk aversion
1
Standard value
ρI
Persistence of idiosyncratic income shock
0.97
Guvenen et al. (2021)
σIStandard deviation of idiosyncratic income shock 0.19
Guvenen et al. (2021)
ρG
Persistence of aggregate income shock
0.87
NIPA
σG
Standard deviation of aggregate income shock
0.03
NIPA

The discretization of the stochastic processes for idiosyncratic and aggregate income
is different for the full information and the incomplete cases. As shown in Appendix A,
the income process in the incomplete information case follows an ARMA(2,1) process. We
express this process as a VAR(1) and then use Tauchen (1986)’s approach. We solve themodel using a variation of Coleman (1990)’s time iteration method. See Appendix B for
more details about the solution method or grids. We consider an asset grid with a zero lower
bound for assets, so that in the baseline calibration agents are not able to borrow.

3.2

Main Results

3.2.1

Long-run Moments

In order to assess the role of incomplete information we compare the two models in terms of
their long-run moments, and in the way agents respond to idiosyncratic and aggregate income
shocks. Using the policy functions we simulate an economy composed of 2,000 individuals
8for 10,000 periods. We simulate this economy with both incomplete and full information,
employing the same sequences of shocks in each simulation. Table 2 presents a summary of
the long-run moments of aggregate and cross-sectional moments of the two models.
Table 2: Long-run Moments
Aggregate Dynamics
Consumption: Standard Deviation (log change)
Consumption: Autocorrelation
Assets: Standard Deviation (log change)
Assets: Autocorrelation
Cross-Sectional Statistics
Income: Mean
Income: Coeﬀicient of Variation
Consumption: Mean
Consumption: Coeﬀicient of Variation
Assets: Mean
Assets: Coeﬀicient of Variation

Full Information

Incomplete Information

0.00790.979
0.00700.998

0.01040.965
0.0040
0.997

1.38
0.95
1.54
0.81
7.73
1.46

1.38
0.95
1.61
0.80
11.09
1.12

Notes: Long-run moments are calculated from a simulation of 2,000 households and 10,000 periods.
We use the same sequences of aggregate and idiosyncratic shocks for both models.

The simulated statistics of Table 2 reveal how the information friction distorts consumption decisions. Aggregate consumption growth is 32% more volatile under incomplete
information, because households undersave in response to aggregate income shocks, which
they cannot distinguish from more persistent idiosyncratic shocks. Consumption is also lessautocorrelated, reflecting that households are less effective at smoothing consumption. Because the incomplete information households forecast income less accurately, they have astronger precautionary savings motive. Facing the same interest rate, they hold more assets
than they would under full information. The additional financial income allows them toafford higher average consumption as well.
In the cross-section, the information friction distorts consumption and assets in different
ways. The friction decreases wealth inequality because the increased precautionary savings
motive is strongest at lower asset levels: poor households have stronger incentives to save andmove away from the constraint, but rich households still act as if they are nearly unaffectedby the constraint. This effect is clear in Figure 2 panel (a), which presents the ergodic
distributions of aggregate assets for both models. The information friction distorts the
distribution most for low asset levels: the full information model has much more mass near
the borrowing constraint, but a similar right tail.

9(a) Assets(b) Consumption

Figure 2: Ergodic Distributions - Full and Incomplete Information Models
Notes: The ergodic distributions are each calculated from a simulation of 2,000 households and 10,000
periods. We use the same sequences of aggregate and idiosyncratic shocks for both models. Both distributions
extend outside the axis range, but the right tails are omitted for readability.

In contrast, the friction has little effect on consumption inequality (Figure 2 panel (b).)
All else equal, the lower wealth inequality would reduces consumption. But this force isoffset because households are less effective at consumption smoothing. Most of their income
is driven by idiosyncratic shocks, to which households oversave in the short run, before
appropriately increasing their consumption response once they realize that their incomechange was persistent. This delayed consumption response amplifies consumption dispersion
because households with large shocks have additional oversavings to draw down as excess
consumption. On net, the coeﬀicient of variation for consumption is almost as large with theinformation friction as it is without, even though wealth is much more equally distributed.
3.2.2

Consumption Volatility

Table 2 reports that the information friction increases aggregate consumption volatility. To
understand why, this section compares how the two economies respond to income shocks.
Figure 3 present the impulse response functions of aggregate consumption and assets toan aggregate income shock. The income shock is a one standard deviation increase in uGt .
In order to calculate the impulse response functions we follow a procedure similar to the oneused in Gilchrist et al. (2014). We simulate an economy where the aggregate shock is set toits long-run average for 400 periods. Then we shock the economy at period 401 and compare
it to a counterfactual economy where the aggregate shock remains at its long-run average.
10

The difference in responses is what we report as the impulse response function.

(a) Consumption(b) Assets

Figure 3: Impulse Responses to an Aggregate Income ShockNotes: Impulse response functions are calculated by subjecting the economy to a one standard deviation
aggregate income shock, and comparing with a counterfactual economy receiving no shock. The impulse
response functions are reported as the difference in consumption or assets, normalized by the size of shock.

The responses of consumption and assets differ substantially across models. The responseof aggregate consumption, on impact, is nearly 50% larger in magnitude when agents face
information frictions. For assets we see the opposite behavior: agents save more in the full
information setting. The full information case is the standard response that we would expect
to see when agents react to transitory income shocks: upon receiving the income shock,consumption should increase modestly while most of the additional income should be saved.
In the presence of financial frictions this behavior is partially mitigated since agents near the
borrowing constraint consume a large fraction of the additional income. Under incomplete
information, agents cannot initially distinguish if the income shock they are experiencing isan aggregate shock or the more persistent idiosyncratic shock, so their consumption responds
much more, and their savings responds less. In short, under incomplete information agentstend to undersave in response to aggregate shocks.
We also compute the responses to idiosyncratic income shocks. We follow a similar
procedure to the one employed to generate impulse responses to aggregate income shocks.
The idiosyncratic income shock consists of a one standard deviation increase to uIi,t . Figure
4 presents the impulse response functions to the idiosyncratic income shock.

11

10

0.9
0.8

8

0.7
6
0.6
4
0.5
0.4

2

Incomplete Info
Full Info

0.3

Incomplete Info
Full Info

0
5

10

15

20

Periods

5

10

15

20

Periods(a) Consumption(b) Assets

Figure 4: Impulse Responses to an Idiosyncratic Income ShockNotes: Impulse response functions are calculated by subjecting a households to a one standard deviation
idiosyncratic income shock, and comparing with a counterfactual household receiving no shock. The impulse
response functions are reported as the difference in consumption or assets, normalized by the size of shock.

Figure 4 shows that the relative responses to idiosyncratic income shocks have reversed
across models: under incomplete information agents tend to save more of the income shock
than under the full information scenario. The full information response is consistent withwhat we would expect to observe under the permanent income hypothesis, whereby “permanent” (in this case very persistent) increases in income translate nearly one-for-one intohigher consumption. Since agents cannot tell whether their shock is aggregate or idiosyncratic on impact, they tend to oversave in response to idiosyncratic shocks.
The economy features heterogeneous oversaving and undersaving across households. To
demonstrate, we calculate CIEi,t , the consumption-income elasticity (CIE) of household iin period t:log(Ci,t ) − log(Ci,t−1 )CIEi,t =(6)log(Yi,t ) − log(Yi,t−1 )We study the CIE rather than the well-known MPC in order to directly compare with GOP’s
evidence. Figure 5 presents the ergodic distribution of CIEs under incomplete and full information. The incomplete information economy features smaller CIEs on average because
agents in this economy are worse forecasters and thus have a stronger precautionary savings
motive. Idiosyncratic shocks drive most income changes, and agents with full information can
immediately observe that these shocks have persistent effects, so they change consumption
more elastically than the incomplete information agents. Some CIEs are negative because
agents may see their income increase, but by less than they expect, so they reduce consump12

tion in response. This is less common under full information, where the AR(1) structure
makes such events less likely. The full information distribution also has larger mass at one,because full information households are much more likely to be borrowing constrained, as
shown in Figure 2.

Figure 5: Ergodic Distributions of Consumption-Income Elasticities
Notes: The ergodic distributions of household-level CIEs are each calculated from a simulation of 2,000
households and 10,000 periods. We use the same sequences of aggregate and idiosyncratic shocks for bothmodels. CIEs in the plotted distributions are only included for households experiencing income changes of
one standard deviation or more in absolute value.

Why is aggregate consumption so much more volatile when information is incomplete if
the CIEs are lower than under full information? Crucially, the CIEs are different in response
to idiosyncratic versus aggregate income changes. When we distinguish the generic CIE fromthe consumption elasticity to aggregate income shocks, we find that the incomplete information households are more elastic to aggregate income changes. We explore this distinctionin the next section.
3.2.3

Elevation and Homogenization of Consumption Elasticities to Aggregate
ShocksGuntin et al. (2020) find that the elasticity of consumption to aggregate shocks is bothlarge and homogeneous across the income distribution. In effect, households respond totransitory aggregate shocks as if they perceive them to be permanent. This is exactly how
the information friction in our model affects households. Thus, we find that introducing the13

information friction to a heterogeneous agent model elevates and homogenizes consumption
elasticities to aggregate income.
To characterize this effect, we calculate the consumption-income elasticities to aggregate
income. Like GOP, we focus specifically on large aggregate shocks, which in our model affects
every agent across the income distribution proportionately.6 The consumption elasticity toaggregate income YtG for individual i at time t isG=
CIEi,tlog(Ci,t ) − log(Ci,t−1 )Glog(YtG ) − log(Yt−1)(7)

We calculate the elasticities for each agent in our simulation for each information structure.
To ascertain the cross-sectional relationships with income and wealth, Figure 6 presents theGwithin-decile averages of CIEi,t, across models. In both cases deciles are calculated fromthe ergodic distribution of the incomplete information model, so that levels are comparable
across information structures.

(a) Elasticities by Wealth(b) Elasticities by Income

Figure 6: Consumption-Income Elasticities to Aggregate Income
Notes: The solid and dashed curves are fit from quadratic regressions. The distributions of CIE G are eachcalculated from a simulation of 2,000 households and 10,000 periods. We use the same sequences of aggregate
and idiosyncratic shocks for both models. The elasticity is calculated at the household level and averaged
within household groups corresponding to the asset or income deciles of the incomplete information model’s
ergodic distribution. Households are grouped based on their position in period t − 1 for a shock that occurs
in period t. The plotted elasticities only include periods with aggregate shocks exceeding two standard
deviations in absolute value.

The elasticity of consumption to aggregate income is elevated in the incomplete infor6

We consider aggregate income changes larger than two standard deviations, although our conclusions
are not dependent on this particular threshold. In addition to following GOP, setting a threshold for incomechanges prevents us from occasionally calculating excessively large CIE’s when the denominator happens tobe small.

14

mation model, where the average CIE G is 0.35, versus 0.27 for full information. Figure 6
panel (a) plots the average elasticities within asset deciles, where the information friction(blue circles, with solid blue quadratic fit) substantially elevates the elasticities relative tofull information (red crosses, with dashed red quadratic fit) across most of the asset distribution. The elasticities are only similar at very low levels of wealth, where agents havehigh elasticities because they are likely to be constrained. How can this relationship be sodifferent from the general CIEs (Figure 5) which were much higher under full information?
The full information households are extremely elastic to idiosyncratic shocks which drive themajority of income changes, but less elastic to aggregate income changes which are muchless persistent. However incomplete information households have similar elasticities to bothtypes of shocks, because they cannot distinguish between them.
The information friction’s homogenization effect is clear in the relationship between the
consumption elasticity and household income (Figure 6 panel (b)). Homogenization occurs inpanel (a) as well, but we focus on the relationship with income in order to mirror the findings
by GOP: the consumption elasticity is homogeneously large across the income distribution,in contrast to the negative relationship implied by full information. Why does this occur?
Under full information, the response is heterogeneous due to the financial friction; low wealthindividuals are more elastic to transitory income shocks because they are near the borrowing
constraint. However, the response becomes more homogeneous as income becomes more
persistent; in the extreme case when all income shocks are permanent, all agents have thesame unit elasticity. Thus under incomplete information where households perceive aggregate
shocks as more persistent than they really are, they react more homogeneously.
Our results differ from GOP’s findings in two ways. First, we show that the information
friction elevates average CIE G , but not to levels as large as GOP’s estimates (0.7 − 1.2).
This is because agents in the model mistake aggregate income for idiosyncratic income, whichin the conservative baseline calibration only has autocorrelation ρI = 0.97. If idiosyncraticincome were more persistent, then the average CIE G would be even larger. Second, in somecountries GOP find that the relationship between the consumption elasticity and householdincome is distorted so much as to be upward-sloping in income. This is possible in ourmodel for some alternative calibrations. In particular, when idiosyncratic income has ahigher autocorrelation ρI , households mistakenly perceive aggregate shocks to be nearly
permanent, which elevates elasticities enough to be increasing with income. We study thiscase in Section 4.1.2.

15

4

Interactions Between the Frictions

The main purpose of introducing dispersed information into a heterogeneous agents framework was to understand how the information friction affect distributions, particularly theelevation and homogenization documented by GOP, but also the general patterns of consumption and wealth inequality that we discuss in Section 3.2.
However, the information friction and the financial friction interact in rich ways. For
example, in this section we show that the information friction attenuates the effects of thefinancial friction on the wealth distribution, and on the dynamics of inequality. But we also
show that the information friction reverses the effects of idiosyncratic risk on the aggregate
consumption elasticity, and introduces new cross-sectional heterogeneity, skewness, and correlations for household forecasts. These substantial interactions are further motivation to
study incomplete information and heterogeneous agents in a unified framework.

4.1

Parameter Sensitivity

4.1.1

The Borrowing Constraint

To ascertain the impact of the financial friction on our economy, we solve the model forseveral values of the borrowing constraint, ranging from the no-borrowing baseline to thenatural borrowing limit.7
Figure 7 plots asset distributions under incomplete and full information for each of these
two extreme values. In both cases, reducing the lower bound on assets weakens the distortion
that precautionary savings has on the asset distribution: the distribution shifts left of zero,with large masses of agents borrowing to smooth income shocks.
However, this effect is asymmetric across information structures. Under incomplete information, households have a strong precautionary savings motive because they are poor forecasters; few of them choose to be constrained even when unable to borrow, and so relaxing
the constraint has little effect on the distribution of assets. But full information households
have a weaker precautionary savings motive, often choosing to go to the constraint. Whenthe constraint is relaxed, many more households borrow than under incomplete information.
Therefore, we conclude that the information friction interacts to attenuate the financial
friction. By raising the precautionary savings motive, the information friction makes theborrowing constraint less distortionary.
The natural borrowing limit is −Ā = ymin /r, where ymin denotes the lowest possible value in the income
grid, which in logs is −4.5 standard deviations. We construct the income grids to imply the same naturalborrowing limit for both models.
7

16

(a) Incomplete Information(b) Full Information

Figure 7: Ergodic Distribution of Assets with the Natural Borrowing Limit
Notes: The ergodic distributions are each calculated from a simulation of 2,000 households and 10,000
periods, experiencing the same sequences of aggregate and idiosyncratic shocks. The light gray distributions
are the corresponding ergodic distributions when no borrowing is allowed (Figure 2).

4.1.2

Idiosyncratic Risk

How does the economy change when we adjust the dynamics of idiosyncratic income risk?
To address this question, we consider alternative values of ρI , the autocorrelation on theidiosyncratic component of income Y I . This parameter has no effect on how full information
agents forecast aggregate income, but under incomplete information, increasing ρI makes
forecasting aggregate income more diﬀicult by making the combined income process more
persistent (as demonstrated in Figure 1.) Figure 8 plots several summary statistics for arange of values of ρI .
When idiosyncratic income is more persistent, households increasingly mistake aggregate shocks for permanent income shocks. This strengthens the main mechanisms of ourmodel. Increasing ρI raises aggregate consumption volatility (Figure 1 panel (a)) and raises
consumption elasticities to aggregate income (panel (b)). Increasing ρI also increases thehomogenization effect documented by GOP: the slope of the CIE G -income relationship rises(panel (c)). If idiosyncratic income is suﬀiciently persistent, the slope can become positive.
GOP document a positive relationship for several countries, but this never occurs in our full
information model.
The autocorrelation ρI has a nonmonotonic effect on the precautionary savings motive.
Panel (d) makes this clear, plotting the ratio of average wealth to average income. At mostlevels, increasing ρI increases household income risk. However, when idiosyncratic income
170.0180.36

Incomplete Info
Full Info

0.016

0.34

0.014

0.32

0.012

0.3

0.01

0.28

0.008

0.26

0.006
0.95

Incomplete Info
Full Info

0.24
0.96

0.97

0.98

0.99

0.95

0.96

0.97

0.98

0.99

(b) Aggregate CIE G(a) Consumption Volatility
0.05

9

8

0

7
-0.05
6
-0.1

5

Incomplete Info
Full Info
-0.15
0.95

0.96

0.97

0.98

4
0.95

0.99

(c) Slope of CIE G -Income Curve

Incomplete Info
Full Info
0.96

0.97

0.98

0.99

(d) Wealth-Income Ratio

Figure 8: Sensitivity to Idiosyncratic Persistence
Notes: We consider the same sequence of shocks for both models, for every possible value of ρI . The baseline
ρI = 0.97 is marked with a dotted line in each panel. Each statistic is calculated from a simulation of 2,000
households and 10,000 periods, experiencing the same sequences of aggregate and idiosyncratic shocks.

becomes extremely persistent, the precautionary savings motive declines. For intuition,
consider the limit: when income shocks are completely permanent, there is no precautionarysavings motive at all, because consumption follows income one-for-one.
These effects are not common across information structures. The full information CIE G
is decreasing in ρI , while it rises under incomplete information. Why? Under full information, the CIE G moves inversely to the wealth-income ratio, because households consume
more when they hold greater wealth, so their consumption is less elastic to aggregate income
shocks. The information friction breaks this relationship: when information is incomplete,
ρI monotonically increases the CIE G . Under both information structures, a larger ρI makes
consumption more elastic to idiosyncratic income shocks. But with the friction, households
cannot distinguish aggregate from idiosyncratic shocks, so their consumption choice must be
more elastic to both types of shocks.

18

4.2

Inequality Dynamics

One valuable feature of heterogeneous agent models is the ability to study the dynamics ofinequality. Introducing the information friction changes these dynamics in nontrivial ways.
To demonstrate these effects, Figure 9 plots the average response of inequality measures
to a one standard deviation aggregate income shock uG
t . We calculate inequality as thestandard deviation of logs, and for this exercise alone we consider “assets” as cash-on-hand(i.e. financial assets plus current income) so that borrowing constrained households do nothave undefined log assets.

(a) Std. Deviation of Log Assets(b) Std. Deviation of Log Consumption

Figure 9: Inequality Response to Aggregate Shocks
Notes: Impulse response functions are calculated by subjecting the economy to a one standard deviation
aggregate income shock, and comparing with a counterfactual economy receiving no shock. The impulse response functions are reported as the percentage point difference in the standard deviation of log consumption
or log assets, relative to the counterfactual economy, and normalized by the size of shock.

Under full information, the standard model predicts that a positive aggregate income
shock should reduce consumption and asset inequality (Figure 9, dashed lines). This reduction occurs because all incomes increase proportionately, and income is distributed moreequally than assets. To understand this effect, it is useful to view a household’s “total wealth”
as the sum of financial wealth (i.e. the assets in the model) and human capital (i.e. thepresent value of future income Yi,t ) because, absent any financial friction, total wealth would
entirely determine consumption. The aggregate shock reduces the share of households’ total
wealth that is held as financial assets and increases the share held as human capital. As
usual, financial assets are distributed more unequally than income, so shifting towards human capital reduces consumption inequality (panel (a)). Similarly, the shift towards humancapital causes savings to be distributed more equally, reducing asset inequality (panel (b)).
19

Under incomplete information, agents have a stronger precautionary savings motive, sothey hold more financial assets. Therefore when the aggregate shock increases incomes, it
has a smaller effect on the shares of total wealth held as financial assets and human capital.
The shock induces a smaller shift towards the more equally distributed human capital thanunder full information, attenuating the reductions in consumption and asset inequality.

4.3

Forecast Heterogeneity

One implication of heterogeneity among agents is that there is heterogeneity of forecasts.
This is true of any model with a persistent income process. But the information friction
introduces an additional complication: there is heterogeneity of forecasts about aggregate
variables.
There is clear empirical evidence that households have heterogeneous forecasts about
the macroeconomy.8 This heterogeneity requires information frictions because FIRE agents
all form the same expectations. But there is an additional interaction between the friction
and the agent heterogeneity: in linear dispersed information models, the average agent
typically holds the average expectation, so the heterogeneity of expectations is irrelevant
for macroeconomic dynamics. A consequence of the heterogeneous agent framework is that
forecasts about aggregates are nonlinearly related to the endogenous distributions of wealthand consumption.
What is the mechanism? Income is persistent, so higher income individuals expect higher
income in the future. Because they cannot disentangle idiosyncratic from aggregate incomes,
agents that have higher forecasts of their own income also have higher forecasts of aggregate
income. This relationship is strictly mechanical, following from the assumed income process.9 But income is endogenously correlated with wealth and consumption, so forecasts ofaggregates are endogenously correlated as well.
Figure 10 plots the joint distributions of household forecasts of aggregate log incomes andother quantities in the incomplete information model. The joint distribution with income isplotted in panel (a): this relationship is mechanical, entirely implied by the assumed income
process and information friction. When households receive higher income, they tend to save,so wealth is positively correlated with income and thus the forecast in panel (b). However,households with higher income do not save it all; they also consume, which thus is positively
correlated with forecasts in panel (c).
8

A large literature documents how heterogeneous forecasts about macroeconomic variables are correlatedwith household decisions, including Vissing-Jorgensen (2003), Egan et al. (2021a), Egan et al. (2021b),
Coibion et al. (2021), and Coibion et al. (2022).
9
Appendix A.4 derives this relationship.

20

(a) Correlation with Income(b) Correlation with Assets(c) Correlation with Consumption

Figure 10: Distributions of Expectations
Notes: The heatmaps display the ergodic joint distributions in the incomplete information model of: (1.)
household forecasts of aggregate income and (2.) either income, assets, or consumption. Red regions indicate the highest density, while blue regions indicate the lowest. The dashed white lines mark the average
household, weighted by the x-axis quantity.

The joint distributions have three common patterns. First, the forecasts feature substantial heterogeneity. Second, optimism about the aggregate economy is positively correlatedwith income, wealth, and consumption. Third, the joint distributions are all skewed withrespect to the x-axis. This inequality is typical in heterogeneous agent models. But it has a
crucial interaction with the information friction: the skewness biases any weighted-averageof forecasts. This can be seen in Figure 10, where the dashed white lines plot the average
forecast, weighted by the corresponding x-axis variable. In all cases, this weighted averageis greater than the unweighted average, which is necessarily zero.
Our model is simple, but we expect these patterns hold in more general settings so longas income is suﬀiciently correlated with wealth and consumption. In other models, theconsequences of these patterns depend on what matters for the macroeconomy: for example,
if it is the forecast associated with the average asset (rather than the average household)that matters for the macroeconomy, then this unequal joint distribution can further distortaggregate dynamics.

5

Corroborating Evidence for the Mechanism

In this section we test whether household expectations of their future income respond differently to idiosyncratic and aggregate shocks.

21

5.1

Data

To document the response of household income forecasts to idiosyncratic and aggregate
shocks, we employ data from the New York Fed Survey of Consumer Expectations (SCE).
The SCE is a monthly survey of aggregate and household-level economic conditions andforecasts. It consists of a nationally representative rotating panel of approximately 1300
American households, which remain in the sample for up to 12 months. The survey has been
administered since 2013. For our purposes, we require data on both expected and realized
household earnings. Unfortunately, this pair is reported only in the auxiliary labor market
module of the survey, which is administered to participants every 4 months.
We primarily use two questions from the labor market module. First, to measure household expectations of future earnings, we use the household head’s forecast of their 4-monthahead earnings. This measure is the answer to:
What do you believe your annual earnings will be in 4 months?
dollars per year
which we interpret as the household’s forecast of instantaneous annualized earnings fourmonths into the future. The advantage of this measure relative to the earnings forecasts in
the general SCE survey, is that it is unconditional. The general SCE asks respondents toforecast their earnings over the following year, but do so conditional on holding a job. We
prefer to use an unconditional earnings process, which both fits the model and corresponds
to the income process that we estimate in the aggregate.
Second, to measure realized income, we use the household head’s current annualized
reported earnings. This measure is the answer to:
How much do you make before taxes and other deductions at your [main/current] job,on an annual basis? Please include any bonuses, overtime pay, tips or commissions.
dollars per year
measured as gross wages or salaries, which respondents are more likely to report accurately.
This question is not asked to individuals who are unemployed or out of the labor force, towhom we assign zero earnings. The earnings expectation question was added only in March
2015, and the most recently released data are for March 2021, giving us 19 time periods.
The data set contains 10,080 unique households, appearing on average in 2.2 editions of themodule. But not all respondents answer all questions; we are left with 12,332 observations
with suﬀicient data.
The general SCE survey contains additional household-level descriptors. Crucially, weobserve the state where respondents reside, so that we can connect households to state-level22

shocks. We use additional descriptors as controls: we observe the industry in which thehead of household either works or was employed most recently, we observe their age andeducation, and we observe demographic characteristics including ethnicity and gender.

5.2

Regressions

Idio
We divide a household’s log real labor earnings yi,s,t into an idiosyncratic component yi,s,tAggrand an aggregate component ys,t
:
Aggr
Idio
yi,s,t = yi,s,t+ ys,twhere i indexes households, s indexes their state, and t indexes the 4 month time period.
AggrThe aggregate component ys,tis the mean earnings in state s and period t. We aggregate
at the state level for several reasons. First, using states rather than the entire US economy
provides considerably more observations, which is essential given the short history of the
SCE. Second, state-level income is more volatile than aggregate income, which gives ouranalysis additional power. Finally, we treat states as small open economies, which matches
the structure of our model and the motivating evidence from Guntin et al. (2020).
Our main regression estimates how household forecasts depend on income:
Aggr
Idio
fi,s,t = β Idio yi,s,t+ β Aggr ys,t
+ Xi,s,t + εi,s,t(8)

where i indexes households, s indexes their state, and t indexes the 4 month time period.
Aggr
Idio
fi,s,t is the household-level forecast of their 4-month-ahead earnings, yi,s,t
and ys,t
are therealized aggregate and idiosyncratic earnings components, and Xi,s,t is a vector of householdlevel controls.
If households have full information, then their forecasts would be given by[FIRE]:

AggrIdio
fi,s,t = Et [yi,s,t+1] + Et [ys,t+1]

Idio
In the model, the earnings components are both AR(1). And if the income components yi,s,tAggrand ys,t
are each AR(1) with autocorrelation ρIdio and ρAggr respectively, then the forecast(8) under full information would satisfy[FIRE, AR(1)]:β Idio = ρIdio

β Aggr = ρAggr(9)

However, if households are unable to distinguish between aggregate and idiosyncratic earn-23

ings components, then the forecast (8) would satisfy[Incomplete Info., AR(1)]:β Idio = β Aggr(10)

We test these information structures when estimating regression (8). First, we test
whether we can reject the incomplete information assumption, i.e. whether β Idio = β Aggr .
This is a useful direct test, but failing to reject the null hypothesis is not a confirmation of
information frictions. Therefore we also test whether the full information assumption fails.
In order to test the full information relationship between forecast coeﬀicients (9), we
require values for the autocorrelations ρIdio and ρAggr . For the idiosyncratic autocorrelation,
we turn to estimates in the literature. Guvenen et al. (2021) use administrative panel data
on US workers from the Social Security Administration to estimate earnings dynamics. We
adopt ρIdio = 0.97, which is their most conservative (smallest) estimate for models with aGaussian AR(1) process. This result is consistent with Heathcote et al. (2010) who use datafrom the PSID and also estimate 0.97 as the autocorrelation of the persistent componentof individual earnings. For the aggregate autocorrelation, we estimate an AR(1) process
directly using state-level earnings series from the National Accounts, and find ρAggr = 0.87.
Appendix C details this estimation.
Wary of possible attenuation biases, we do not necessarily want to reject full information
if the forecast coeﬀicients β Idio and β Aggr are summarily lower than their corresponding
autocorrelations. Instead, we test whether β Idio is relatively larger than β Aggr , independentof their absolute scale. Specifically, define the scalar ψ as
( Aggr ) 13
ρ
ψ≡
ρIdio
The 13 exponent transforms the annual autocorrelations to match the thrice-yearly frequencyof the data used in our regression. Appropriately transformed, the full information relationship (9) implies that ψβ Idio = β Aggr . Furthermore, if ψβ Idio > β Aggr , then full information
fails, but in the opposite direction as implied by our model. Therefore our second statistical
test is whether ψβ Idio ≥ β Aggr .

5.3

Results

Table 3 presents the results of forecast regression (8). Column (1) is the basic regression withno additional controls. In column (2), we control for state-level effects on expectations. We
cannot include state-time controls, because aggregate earnings vary at the state level. Indus-

24

try fixed effects (3) control for the industry and employment status of the household head.
Human capital fixed effects (4) control for education and age. Demographic fixed effects(5) control for gender and ethnicity. Columns (6)-(8) include all fixed effects. Column (7)utilizes the panel dimension to control for lagged earnings; this is our preferred specification
because our theory assumes that households cannot distinguish between the idiosyncratic
and aggregate components of lagged earnings. Still, we also run the specification in column(8) which includes lagged idiosyncratic and aggregate earnings separately. Finally, column(9) uses household fixed effects.
Table 3: Effects of Log Earnings on Household Earnings Forecasts
(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

Idio. Log Earnings

0.677(0.0195)

0.676
0.665
(0.0195) (0.0206)

0.653(0.0207)

0.666(0.0200)

0.633(0.0223)

0.563(0.0412)0.563(0.0412)0.311(0.0106)

Aggr. Log Earnings

0.800(0.0388)

0.750
0.783
(0.0518) (0.0395)

0.764(0.0402)

0.785(0.0394)

0.702(0.0546)

0.634(0.0533)

0.605(0.0493)

0.292(0.0330)

Lag Log Earnings

0.205(0.0302)

Lag Idio. Log Earnings

0.203(0.0303)

Lag Aggr. Log Earnings

0.264(0.0434)

β Idio = β Aggr p-valueψβ Idio ≥ β Aggr p-value
Observations
R2
State F.E.
Industry F.E.
Human Capital F.E.
Demographic F.E.
Individual F.E.

0.002
0.000
12332
0.554

0.144
0.053
12332
0.556
X

0.003
0.000
12332
0.558

0.005
0.001
12332
0.563

0.003
0.000
12323
0.558

X
X
X

0.187
0.080
12323
0.572
X
X
X
X

0.074
0.021
6406
0.649
X
X
X
X

0.209
0.062
6406
0.650
X
X
X
X

0.543
0.794
12332
0.551

X

Notes: Standard errors in parentheses, clustered at the state-month level. In all cases, the dependent variable is the householdlevel log forecast of its 4-month-ahead annualized earnings.

In all cases except column (9), the coeﬀicient on aggregate log earnings exceeds that of
idiosyncratic log earnings. Household forecasts of future income are more sensitive to changes
in aggregate earnings, even though their idiosyncratic earnings are much more persistent!
The test results formalize this conclusion. When we test ψβ Idio ≥ β Aggr , we reject theinequality at the 5% level in our preferred specification (7). This means that the relative
response of household forecasts to aggregate earnings exceeds what it would be under full
information, or if full information failed but in the opposite direction than implied by our25model. And even when our statistical test fails to reject, our estimates remain a better fit
to the incomplete information model than the full information model. Indeed, when we test
the assumption of our model (β Idio = β Aggr ), we either fail to reject it, or if we do, it isbecause households are even more responsive to aggregate income than our friction implies.
Specification (9) demonstrates why we choose not to use panel-level fixed effects: theyattenuate the estimated coeﬀicients. This reflects a classic problem; households are in thepanel for at most 3 periods, so the panel component is very short. In a traditional dynamic
panel regression, short panels bias the coeﬀicients (Nickell, 1981). Our panel regression
does not have an explicit dynamic component, but it is not safe from Nickell bias concerns,
because the household expectations are forecasts of the panel component. Still, even in thisspecification, the estimated coeﬀicients are close enough that we fail to reject our information
structure.

6

Conclusion

In this paper we introduced dispersed information to an otherwise standard open economy
heterogeneous agents model. We demonstrated that the information friction increased consumption volatility and reduced heterogeneity in household’s response to aggregate income
shocks. Then we documented evidence for our central mechanism in US forecast data.
Our central findings are robust, but there is further work to be done. How would the friction interact with capital accumulation in the model? Or a richer asset market or householdrisks and decisions? What about closed economies? Regardless, it is clear that households
respond to aggregate shocks as if they are more persistent than they actually are, so this typeof information friction will be valuable in any setting where consumption volatility matters.

26

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29

A

The Sum of Independent Income Processes

In this appendix, we characterize the time series properties of log income in the model. Index
households by i and time by t. Household log income yi,t is given byI+ ytGyi,t = yi,tI
where yi,t
is idiosyncratic and mean zero in the population for all t, while ytG is aggregate
and common to all households.

A.1

The ARMA(2,1) Representation

The idiosyncratic and aggregate components are AR(1):I
Iyi,t= ρI yi,t−1+ uIi,tGGyi,t= ρG yi,t−1
+ uG
i,t
2
with uIi,t ∼ N (0, σI2 ), uGt ∼ N (0, σG ), ρI ∈ (0, 1), and ρG ∈ (0, 1).
It is helpful to use lag operator notation to define these time series, which become:y I = LρI y I + uI
y G = LρG y G + uG
y = Lϱ(L)y + w
where ϱ is a lag operator polynomial to be found, and w is a white noise process to be found.
It is well known that the sum of AR(1) processes is ARMA(2,1). The autoregressive
coeﬀicients parameters are ϱ0 = ρI + ρG and ϱ1 = −ρI ρG :y − (ρI + ρG )Ly + ρI ρG L2 y = yI + yG − (ρI + ρG )L(yI + yG ) + ρI ρG L2 (yI + yG )= ρI LyI + uI + ρG LyG + uG − (ρI + ρG )L(yI + yG ) + ρI ρG L2 (yI + yG )= uI + uG − ρG LyI − ρI LyG + ρI ρG L2 (yI + yG )= uI + uG − ρG L(yI − ρI LyI ) − ρI L(yG − ρG LyG )= uI + uG − ρG LuI − ρI LuG ≡ z

30

The object z is M A(1). What is the structure of the MA term?

cov(z, Lj z) =
var(uI )(1 + ρ2G ) + var(uG )(1 + ρ2I )−var(uI )ρG − var(uG )ρI



0

j=0
j=1
j>1

thus we can writez = w + θLw
To finish characterizing the ARMA process, we need to know the variance of w, and thevalue of θ. These quantities are related by two equations:var(z) = var(w) + θ2 var(w)cov(z, Lz) = θvar(w)which imply
0 = var(w)2 − var(z)var(w) + cov(z, Lz)2
Apply the quadratic formula and take the larger root:var(z)
var(w) =+2√(

var(z)
2)2− cov(z, Lz)2

Then calculate θ by
θ = cov(z, Lz)/var(w)Finally, w can be expressed analytically in terms of the underlying shocks:
w = (1 + θL)−1 z = z − θLz + θ2 L2 z − ...
w = (1 + θL)−1 (I − ρG L)uI + (1 + θL)−1 (I − ρI L)uG

A.2

The VAR(1) Representation

Stack the variables as such:




yt


y t ≡  yt−1 
wt

31

(11)




ϱ0 ϱ1 θThen y t is a VAR(1) with coeﬀicient matrix B =  1 0 0  and innovation Cwt for
0 0 0
 
1
 
C =  0 :
1
y t = By t−1 + Cwt

A.3

Income Forecasts at Various Horizons

This section derives the term structure of expectations under the information friction, which
are is used in Section 2.2.
Income follows the ARMA(2,1) process
yi,t = ϱ0 yi,t−1 + ϱ1 yi,t−2 + wi,t + θwi,t−1
so the one-period-ahead forecast is given by
E[yi,t+1 |Ωi,t ] = ϱ0 yi,t + ϱ1 yi,t−1 + θwi,tbecause E[wi,t+1 |Ωi,t ] = 0.
The two-period-ahead forecast is given by
E[yi,t+2 |Ωi,t ] = ϱ0 E[yi,t+1 |Ωi,t ] + ϱ1 yi,tbeyond this horizon, the h-period-ahead forecast can be found recursively by
E[yi,t+h |Ωi,t ] = ϱ0 E[yi,t+h−1 |Ωi,t ] + ϱ1 E[yi,t+h−2 |Ωi,t ]for h ≥ 2.

A.4

Forecasts of Aggregate Income

In the model, agents have no need to forecast the aggregate economy; they only need toforecast their own income. However, it is possible to construct the forecasts that agentswould make, given the information friction that they face.
Agent i’s forecast of aggregate income conditional on their information set Ωi,t isG
E[yt+1|Ωi,t ] = ρG E[ytG |Ωi,t ]

32

= ρG

∞
∑
ρjG E[uGt−j |Ωi,t ]j=0

How do agents form expectations over past shocks? Linear backcasting is easily expressedin terms of current and past orthogonal forecast errors wi,t . Let W G denote the lag operator
polynomial that gives the aggregate component of wi,t from current and past aggregateILshocks. Per equation (11), this polynomial is given by W G = 1−ρ
. Let WjG denote the j1+θL
coeﬀicient of this polynomial. Then the backcast is given by:
E[uGt−j |Ωi,t ] =j∑cov(uGt−j , wi,t−k )k=0var(wi,t−k )wi,t−k

j
j
G
G
2
∑
∑
cov(uGσG
t−j , Wj−k ut−j )Gwi,t−kwi,t−k =
Wj−k
=
var(w
)
var(w
)
i,t−k
i,t
k=0
k=0

Plugging in this backcasting formula gives the expression for the aggregate income forecast:G
E[yt+1|Ωi,t ] =

j
∞ ∑
2
∑
ρG σ Gρj W G wi,t−kvar(wi,t ) j=0 k=0 G j−k

B Solution Method
This section provides a description of the algorithm we use to numerically solve for theequilibrium of the full and incomplete information cases.
We start by discretizing the income processes using the approach of Tauchen (1986). For
the full information case we discretize the aggregate and idiosyncratic processes separately(we consider 11 points for each income process). For the incomplete information case wewrite the income process as a VAR(1) (see A.2 for more details) and then discretize it. Notethat the VAR(1) case contains three variables. We allow for 11 points for each variable, so
the number of exogenous states is 11 × 11 × 11 = 1131.
The asset grid is discrete and consists of 200 points.10 We skew the allocation of points
in the asset grid in order to have a better coverage over lower asset levels.
After discretizing the income processes, we proceed to our time iteration method, which
is similar to the one described in Coleman (1990). We start with a conjecture for the asset
holdings policy function, A′ , defined over the state space (Y, A), where Y represents thevector of exogenous states of each model.11 The steps of the solution algorithm are the10

Our results do not change substantially by increasing the number of asset grid points, as our solution
method relies on first order conditions.
11
Note that this guess corresponds to a matrix with dimensions NY × NA , where NY and NA correspond

33

following:
1. Start iteration j with a guess for A′j (Y, A) ≥ −Ā, where −Ā denotes the borrowing
limit. Using this guess construct:
Cj (Y, A) = Y + A(1 + r) − A′j (Y, A)(12)

Using (12), compute the discounted expected marginal utility][β(1 + r∗ )EY ′ |Y uj (Y ′ , A′j (Y, A)) ,(13)

where uj (Y, A) = Cj (Y, A)−γ .
2. Assume the borrowing constraint binds. Note that when the constraint binds we havethat consumption is Cj+1 (Y, A) = Y +A(1+r)+ Ā. We check whether this assumption
holds by calculating the residual of the Euler equation:[]
R(Y, A) = uj+1 (Y, A) − βR∗ EY ′ |Y uj (Y ′ , A′j (Y, A)) .

(14)

If R(Y, A) > 0, we keep the values for Cj+1 (Y, A). Otherwise, the constraint does notbind for that point of the state space and we discard Cj+1 (Y, A). We then solve forthe value of Cj+1 (Y, A) that satisfies[]
Cj+1 (Y, A)−γ = βR∗ EY ′ |Y uj (Y ′ , A′j (Y, A)) .

(15)

3. Use the resource constraint to obtain the updated conjecture for asset holdings A′j+1 (Y, A) =Y + A(1 + r) − Cj+1 (Y, A).
4. Check for convergence. If ||A′j+1 (Y, A) − A′j (Y, A)|| < ϵ, then the problem is solved.
Otherwise, discard A′j and use A′j+1 as the new guess for the problem (go back to step
1).

C Aggregate Income Persistence
In this section we estimate the autocorrelation of aggregate log labor earnings.
Aggr
We measure aggregate log real labor earnings ys,tat the state level from the National
Accounts, for state s and year t. We define labor earnings as the sum of wages and salaries,
to the number of elements in the grid of income states and assets, respectively.

34

supplements to wages and salaries, and proprietors’ income. We deflate by the PCE deflator,
to match the procedure used by Guvenen et al. (2021) for idiosyncratic earnings. We estimate
the following panel regression:
Aggr
Aggr
ys,t
= ρAggr ys,t−1+ ds + Xs,t + εi,s,twhere ds denote fixed state-level controls, while Xs,t are further controls that vary with time.
Aggr
Aggr
.
with ys,t−2
We estimate this regression both by OLS and by 2SLS, instrumenting for ys,t−1
The IV regressions allows for consistent estimation of ρAggr even when the error term εi,s,t
is MA(1). We take this approach in order to be comparable to Guvenen et al. (2021), whoallow individual earnings to contain an i.i.d. transitory term, which is equivalent to letting
earnings be ARMA(1,1).
Table 4 reports our estimates of ρAggr . Columns (1) report results for the regression withstate-specific as the only controls, which allows for the longest sample. We let our trends
be state-specific, given the well-known heterogeneity of growth rates across states (Barroet al., 1991). This is our simplest specification and yields the largest estimate, but even inthis case it is significantly lower the the idiosyncratic persistence. In specifications (2) and(3) we control for state-level demographics, as changes in worker composition affect average
earnings in a predictable way. We include data from the Current Population Survey on age,gender, race, and education. Controlling for these factors, we estimate an autocorrelationof 0.89 or 0.87 depending on whether our controls are additive or interacted with a timetrend, respectively. The specifications (4)-(5) use the 2SLS approach in order to allow
for transitory i.i.d. shocks to earnings. Allowing for these transitory shocks reduces theestimated magnitudes even further: 0.84 and 0.81 respectively. Lastly, we run our estimation
for the 1994-2013 subsample, in order to most directly compare our aggregate estimate with
the idiosyncratic autocorrelation estimated from this sample period by Guvenen et al. (2021),
which is the value for ρIdio that we use in our calibration. Columns (6) and (7) present theseresults, with autocorrelations of 0.80 or 0.56; during this time period, income was generally
less persistent than in the broader sample.

35

Table 4: Aggregate Earnings Autocorrelations
Lag Real Earnings
Observations
R2
State Trends
Demo. F.E.
Demo. Trends
Transitory Shocks
Sample Period

(1)
0.924
(0.006)
4599
0.991
X

(2)
0.887
(0.008)
2686
0.995
X
X

(3)
0.871
(0.009)
2686
0.995
X
X

1929-2020 1962-2020 1962-2020

(4)
0.837
(0.009)
2686
0.995
X
X

(5)
0.814
(0.009)
2686
0.995
X

X
X
X
1962-2020 1962-2020

(6)
0.798
(0.017)
1020
0.992
X
X

(7)
0.560
(0.029)
1020
0.992
X

X
X
X
1994-2013 1994-2013

Notes: Heteroskedasticity-consistent standard errors in parentheses. In all cases the dependent variable isstate-level average annual labor earnings.

Our preferred specification is column (5), given that we expect the effects of demographic
factors on income to change over time, due to female entry into the labor force, changing
attitudes towards race, and rising capital-skill complementarity (Krusell et al., 2000). However, we choose specification (3)’s 0.87 as our baseline calibration for ρAggr ; smaller values
will strengthen the effects of the information friction in our model, and we aim to be conservative in our approach. It might be reasonable to choose a lower value, given the lower
estimates for the 1994-2013 time period that informs our idiosyncratic process. But, we areconcerned that this shorter time period might be susceptible to Nickell bias attenuating theestimates, so we are wary of selecting the low estimates below 0.80.

36