IMES DISCUSSION PAPER SERIES

Monetary Policy According to HANK

Greg Kaplan, Benjamin Moll, and Giovanni L. Violante
Discussion Paper No. 2017-E-4

INSTITUTE FOR MONETARY AND ECONOMIC STUDIESBANK OF JAPAN
2-1-1 NIHONBASHI-HONGOKUCHOCHUO-KU, TOKYO 103-8660
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IMES Discussion Paper Series 2017-E-4
June 2017

Monetary Policy According to HANK
Greg Kaplan*, Benjamin Moll**, and Giovanni L. Violante***Abstract
We revisit the transmission mechanism of monetary policy for household
consumption in a Heterogeneous Agent New Keynesian (HANK) model. The
model yields empirically realistic distributions of wealth and marginal propensities
to consume because of two features: uninsurable income shocks and multiple
assets with diﬀerent degrees of liquidity and diﬀerent returns. In this environment,
the indirect eﬀects of an unexpected cut in interest rates, which operate through ageneral equilibrium increase in labor demand, far outweigh direct eﬀects such as
intertemporal substitution. This ﬁnding is in stark contrast to small- andmedium-scale Representative Agent New Keynesian (RANK) economies, where
the substitution channel drives virtually all of the transmission from interest rates
to consumption. Failure of Ricardian equivalence implies that, in HANK models,the ﬁscal reaction to the monetary expansion is a key determinant of the overall
size of the macroeconomic response.
Keywords: Monetary Policy; Heterogeneous
Consumption; Liquidity; Inequality

Agents;New

Keynesian;

JEL classification: D14, D31, E21, E52*University of Chicago and NBER (E-mail: gkaplan@uchicago.edu)**Princeton University and NBER (E-mail: moll@princeton.edu)***Princeton University, CEPR and NBER (E-mail: glv2@princeton.edu)

We thank Yves Achdou, Mark Aguiar, Fernando Alvarez, Adrien Au-clert, Jess Benhabib, Luca
Dedola, Emmanuel Farhi, Mark Gertler, Narayana Kocherlakota, Keith Kuester, David
Lagakos, Emi Nakamura, Larry Schmidt, Jon Steinsson, Mirko Wiederholt and seminar
participants at various institutions. Felipe Alves, Damien Capelle and Julia Fonseca provided
superb research assistance. The views expressed in this paper are those of the authors and do
not necessarily reflect the official views of the Bank of Japan.

1Introduction

A prerequisite for the successful conduct of monetary policy is a satisfactory understanding of the monetary transmission mechanism – the ensemble of economic forcesthat link the monetary policy instrument to the aggregate performance of the economy. This paper follows the tradition of treating the short-term nominal interest rate
as the primary policy instrument and is concerned with its transmission to the largest
component of GDP, household consumption.
Changes in interest rates affect household consumption through both direct andindirect effects. Direct effects are those that operate even in the absence of any change
in household disposable labor income. The most important direct effect is intertemporal
substitution: when real rates fall, households save less or borrow more, and therefore
increase their demand for consumption. In general equilibrium, additional indirect
effects on consumption arise from the expansion in labor demand, and thus in labor
income, that emanates from the direct impact of the original interest rate cut.
A full grasp of the monetary transmission mechanism requires an assessment of
the importance of direct and indirect effects. The relative magnitudes of each channel
are determined by how strongly household consumption responds to changes in realinterest rates, given income; and to changes in disposable income, given the real rate.
Our first result concerns Representative Agent New Keynesian (RANK) models. In
these commonly used benchmark economies, the aggregate consumption response to achange in interest rates is driven entirely by the Euler equation of the representative
household. Therefore, for any reasonable parameterization, monetary policy in RANK
models works almost exclusively through intertemporal substitution. As a consequence,
direct effects account for the full impact of interest rate changes on the macroeconomy,
and indirect effects are negligible.
The strong response of aggregate consumption to movements in real rates that
accounts for the large direct effects in RANK is questionable in light of empiricalevidence. Macroeconometric analysis of aggregate time-series data finds a small sensitivity of consumption to changes in the interest rate after controlling for income(Campbell and Mankiw, 1989; Yogo, 2004; Canzoneri et al., 2007). Crucially, this finding does not necessarily imply that the individual intertemporal elasticity of substitution is small, as other offsetting direct effects can be powerful. First, micro survey dataon household portfolios show that a sizable fraction of households (between 1/4 and1/3) hold close to zero liquid wealth and face high borrowing costs (Kaplan et al., 2014).
Since these households are at a kink in their budget set, they are insensitive to smallchanges in interest rates (consistent with evidence in Vissing-Jorgensen, 2002, that non
asset-holders do not react to interest rate cuts). Moreover, the possibility of being at a1

kink in the future, effectively shortens the time horizon and dampens the substitution
effect even for those households with positive holdings of liquid wealth. Second, simple
consumption theory implies that an interest rate cut has negative income effects on
the consumption of rich households. Third, these same survey data reveal vast inequality in wealth holdings and composition across households (Diaz-Gimenez et al.,2011). Some households may react to a short-term rate cut by rebalancing their asset
portfolio, rather than by saving less and consuming more.
The small indirect effects in RANK models follow from the property that the representative agent is, in essence, a permanent income consumer and so is not responsive totransitory income changes. This type of consumption behavior is at odds with a vast
macro and micro empirical literature (Jappelli and Pistaferri, 2010). The most convincing corroboration of this behavior is the quasi-experimental evidence that uncovers (i)an aggregate quarterly marginal propensity to consume (MPC) out of small transitory
government transfers of around 25 percent (Johnson et al., 2006; Parker et al., 2013)
and (ii) a vast heterogeneity in consumption responses across the population which islargely driven by the level of liquid wealth and by the composition of household balance
sheets (Misra and Surico, 2014; Cloyne and Surico, 2014; Broda and Parker, 2014).1
In light of this empirical evidence, we argue that the relative strength of the direct and indirect channels of monetary policy can be properly gauged only within aframework that offers a better representation of household consumption and household
finances than RANK. To this end, we develop a quantitative Heterogeneous Agent New
Keynesian (HANK) model that combines two leading workhorses of modern macroeconomics. On the supply side, we follow closely the standard New Keynesian model byassuming that prices are set by monopolistically competitive producers who face nominal rigidities and that monetary policy follows a Taylor rule. On the household side,
we build on the standard “Aiyagari-Huggett-İmrohoroğlu” incomplete market model,
with one important modification: as in Kaplan and Violante (2014), households cansave in two assets, a low-return liquid asset and a high-return illiquid asset that is
subject to a transaction cost. This extended model has the ability to be consistent
with both the joint distribution of earnings, liquid and illiquid wealth, and with thesizable aggregate MPC out of small windfalls.
Our main finding is that, in stark contrast to RANK economies, in our HANK modelthe direct effects of interest rate shocks are always small, while the indirect effects canbe substantial. Monetary policy is effective only to the extent that it generates a
general equilibrium response in household disposable income. In our framework, by
1

A recent body of work estimating the marginal propensity to consume out of changes in housing
net worth also documents consumption responses that are very heterogeneous and heavily dependent
on portfolio composition (e.g., Mian et al., 2013).

2

virtue of this indirect channel, overall consumption responses can be large, even though
the strength of the direct channel is modest.
The sharply different consumption behavior between RANK and HANK lies at
the heart of these results. Uninsurable risk, combined with the co-existence of liquid
and illiquid assets in financial portfolios leads to the presence of a sizable fraction ofpoor and wealthy hand-to-mouth households, as in the data. These households arehighly sensitive to labor income shocks but are not responsive to interest rate changes.
Moreover, the vast inequality in liquid wealth implies that, even for non hand-tomouth households, a cut in liquid rates leads to strong offsetting income effects on
consumption. Finally, with this multiple asset structure, to the extent that the spread
between asset returns widens after a monetary expansion, household portfolios adjust
away from liquid holdings and towards more lucrative assets, rather than towards higher
consumption expenditures. All these economic forces counteract the intertemporal
substitution effect and lower the direct channel of monetary policy in HANK.
A second important finding is that in HANK, because of a failure of Ricardianequivalence, the consequences of monetary policy are intertwined with the fiscal sideof the economy. Since the government is a major issuer of liquid obligations, a change
in the interest rate necessarily affects the intertemporal government budget constraint,
and generates some form of fiscal response that affects household disposable income.
Unlike in RANK models, the details of this response, both in terms of the timing
and the distributional burden across households, matter a great deal for the overallmacroeconomic impact of a monetary shock and for its split between direct and indirectchannels.2
We study the implications of these results for two key trade-offs policymakers facein the conduct of monetary policy. First, when attempting to stimulate the macroeconomy, the monetary authority faces a choice between large but transitory versus small
but persistent nominal rate cuts. In RANK models, transitory rate cuts and persistent
rate cuts are equally powerful, as long as the cumulative interest rate deviations are
the same. Instead, in HANK a more transitory, but larger, interest rate cut can be
more effective at expanding aggregate consumption because such a policy leads to amore immediate reduction in interest payments on government debt that can translate
into an additional fiscal stimulus. Second, we analyzed the inflation-activity trade-off
in RANK and HANK. Our experiments suggest that the slope of this relationship isquite similar in the two economies. Intuitively, it is the New-Keynesian side, common
2

The importance of government debt for the monetary transmission mechanism is also emphasizedby Sterk and Tenreyro (2015) in a model with flexible prices and heterogeneous households where
open market operations have distributional wealth effects, and by Eusepi and Preston (2017) in amodel in which Ricardian equivalence fails because of imperfect knowledge.

3

across models, that largely pins down that relationship. However, in HANK the slope
depends on the type of fiscal adjustment: more passive adjustment rules, where moreor less government debt absorbs the change in interest payments, are associated to amore favorable trade-off for the monetary authority.
We are not the first to integrate incomplete markets and nominal rigidities, andthere is a burgeoning literature on this topic.3 Relative to this literature, our paper adds
an empirically realistic model of the consumption side of the economy by exploiting
state-of-the art ideas for modeling household consumption and the joint distributionof income and wealth. As explained, the combination uninsurable earnings risk and a
two-asset structure is at the root of our finding that most of the monetary transmissionis due to indirect general equilibrium effects. In the paper, we show that the one-asset
model explored by the whole literature up to this point faces a daunting challenge whenused to study monetary policy. If calibrated to match total wealth in the economy,
it implies a very small MPC (similar to the one in RANK) and enormous income
effects on consumption because all wealth is liquid. If calibrated to match only liquid
wealth, it features a large aggregate MPC out of transitory income and reasonable
income effects. However, because such calibration misses over 95 pct of the wealth in
the economy, the model must completely abstract from some key sources of indirect
effects of monetary policy, such as those originating from firm investment and frommovements in the price of capital.
Additionally, the focus of our paper differs from that of earlier papers studying monetary policy in the presence of incomplete markets (Gornemann et al., 2014;
McKay et al., 2016) in that we inspect the transmission mechanism of monetary policy
and decompose it into direct and indirect general equilibrium effects. Our emphasison general equilibrium effects is shared by Werning (2015) who develops a useful theoretical benchmark where direct and indirect channels exactly offset so that the overall
effect of interest rate changes on consumption is unchanged relative to the RANK
benchmark. Werning’s assumptions do not hold in our economy, which explains why
the total effect of monetary policy, and not just its decomposition, is affected by thepresence of heterogeneity and incomplete markets. Conceptually, our decomposition issimilar to the one proposed by Auclert (2016).
Our paper is also related to the literature that studies New Keynesian models withlimited heterogeneity, building on the spender-saver model of Campbell and Mankiw(1989).4 The “spenders” in these models consume their entire income every period and3

See Guerrieri and Lorenzoni (2011),
Oh and Reis (2012),
Ravn and Sterk
McKay and Reis (2016), Gornemann et al. (2014), Auclert (2016), McKay et al.
Den Haan et al. (2015), Bayer et al. (2015), Luetticke (2015), and Werning (2015).
4
See, e.g., Iacoviello (2005), Gali et al. (2007), Bilbiie (2008) and Challe et al. (2015).

4(2012),(2016),

therefore share some similarities with our hand-to-mouth households in that they do notrespond to interest rate changes. However, these Two-Agent New-Keynesian (TANK)
models also feature “savers” who substitute intertemporally and are highly responsive
to interest rate changes. In contrast, in our model even high liquid-wealth households
do not increase consumption much in response to an interest rate cut mainly because
the risk of receiving negative income shocks and of binding liquidity constraints in the
future truncates their effective time horizon. We show that TANK models, when thefraction of spenders reflects the share of hand-to-mouth households in the data, stillfeature a monetary transmission mechanism with a large role for direct effects. Our
accent on indirect channels is shared by Caballero and Farhi (2014) which proposes analternative framework where the transmission of monetary policy works through its
general equilibrium impact on asset values.
Finally, we solve the model in continuous time building on Achdou et al. (2014).
In addition to imparting some notable computational advantages, continuous time
provides a natural and parsimonious approach to modeling a leptokurtic individual
earnings process, as recently documented by Guvenen et al. (2015): random (Poisson)
arrival of normally distributed jumps generates kurtosis in data observed at discrete
time intervals. This process, estimated by matching targets from Social Security Administration data, may prove useful in other contexts where an empirically realistic
representation of household earning dynamics is vital.
The rest of the paper proceeds as follows. Section 2 introduces the idea of decomposing the monetary transmission mechanism into direct and indirect effects, andapplies it to small- and medium-scale RANK models and spender-saver models. Section 3 lays out our HANK framework and Section 4 describes how we take it to thedata. Section 5 contains our quantitative analysis of monetary policy in HANK, andSection 6 examines some implications of our findings for some key trade-offs faced bypolicymakers in the conduct of monetary policy. Section 7 concludes.

2

Monetary Policy in Benchmark New-Keynesian Models

In this section, we introduce a formal decomposition of the overall consumption response to an interest rate change into direct and indirect effects.5 Since this decomposition is instrumental to our analysis of the transmission of monetary policy
in our larger quantitative model, we begin by applying it to a series of stylized models of monetary policy. We first demonstrate that, in representative agent economies,
5

This section benefitted greatly from detailed comments by Emmanuel Farhi and some of theresults directly reflect those comments.

5

conventional monetary policy works almost exclusively through direct intertemporal
substitution, and that indirect general equilibrium effects are unimportant. Next, we
illustrate how the monetary transmission mechanism is affected by the presence of
non-Ricardian hand-to-mouth households: (i) introducing hand-to-mouth households
increases the relative importance of indirect general equilibrium effects; (ii) the overall
effect of monetary policy now depends on the fiscal response that necessarily arises
because monetary policy affects the government budget constraint. Finally, we showthat these insights carry over to richer representative agent economies, such as typicalmedium-scale monetary DSGE models. Appendix A contains proofs of all the resultsin this section.

2.1

Representative Agent Model

Setup A representative household has CRRA utility from consumption Ct with parameter γ > 0, and discounts the future at rate ρ ≥ 0. A representative firm produces
output using only labor, according to the production function Y = N. Both the wage
and final goods price are perfectly rigid and normalized to one. The household commits
to supplying any amount of labor demanded at the prevailing wage so that its labor
income equals Yt in every instant. The household receives (pays) lump-sum governmenttransfers (taxes) {Tt }t≥0 and can borrow and save in a riskless government bond at rate
rt . Its initial bond holdings are B0 . In the absence of aggregate uncertainty, household
optimization implies that the time path of consumption satisfies the Euler equationĊt /Ct = γ1 (rt − ρ). The government sets the path of taxes in a way that satisfies its
intertemporal budget constraint.
Since prices are perfectly rigid, the real interest rate rt also equals the nominalinterest rate, so we assume that the monetary authority sets an exogenous time pathfor real rates {rt }t≥0 . We restrict attention to interest rate paths with the property
that rt → ρ as t → ∞ so that the economy converges to an interior steady state. Our
results place no additional restrictions on the path of interest rates. However, clean
and intuitive formulae can be obtained for the special casert = ρ + e−ηt (r0 − ρ),

t≥0

(1)

whereby the interest rate unexpectedly jumps at t = 0 and then mean reverts at
rate η > 0. In equilibrium, the goods market clears Ct ({rt , Yt , Tt }t≥0 ) = Yt , whereCt ({rt , Yt , Tt }t≥0 ) is the optimal consumption function for the household. We assume
that the economy returns to its steady state level in the long-run, Ct → C̄ = Ȳ as

6

t → ∞.6 7
Overall effect of monetary policy We can analyze the effects of a change in thepath of interest rates on consumption using only two conditions: the household Euler
equation, and our assumption that consumption
returnsback to its steady state level.
R
1 ∞It therefore follows that Ct = C̄ exp − γ t (rs − ρ)ds . When the path of interest
rates satisfies (1), this formula collapses to a simple expression for the elasticity of
initial consumption to the initial change in the interest rate1
d log C0
=− .
dr0
γη

(2)

The response of consumption is large if the elasticity of substitution 1/γ is high, andif the monetary expansion is persistent (η is low).
Note that if initial government debt is positive B0 > 0, then a drop in interest rates
necessarily triggers a fiscal response. This is because the time path of taxes must satisfy
the government budget constraint, and therefore depends on the path of interest rates:
Tt = Tt ({rs }s≥0). The government pays less interest on its debt and so will eventuallyrebate this income gain to households. However, Ricardian equivalence implies thatwhen the government chooses to do this does not affect the consumption response tomonetary policy. In present value terms, the government’s gain from lower interest
payments is exactly offset by the household’s loss from lower interest receipts.
Decomposition into direct and indirect effects We begin with the case of zero
government debt, Bt = 0 (and Tt = 0) for all t. We use a perturbation argument
around the steady state. Assume that initially rt = ρ for all t so that Yt = Ȳ forall t. Now consider a small change to the path of interest rates {drt }t≥0 , while holding the path of income {Yt }t≥0 constant. The effect of this change in interest rates
on consumption is the direct effect. In equilibrium, the consumption change induceschanges in labor income {dYt }t≥0 which lead to further changes in consumption. Thisis the indirect effect. Formally, these two effects are defined by totally differentiating
6

There are multiple equilibria in this economy. We select an equilibrium by anchoring the economy
in the long run and focusing only on paths for which Yt → Ȳ as t → ∞ for some fixed 0 < Ȳ < ∞.
For any value of steady state output Ȳ , the equilibrium is then unique. Since we are only concernedwith deviations of consumption and output from steady state, the level of Ȳ is not important for anyof our results.
7
Rather than assuming that wages and prices are perfectly rigid, our equilibrium could be viewed
as a “demand-side equilibrium” as in Werning (2015). In this interpretation, we characterize the setof time paths {rt , Yt }t≥0 that are consistent with optimization on the demand (household) side ofthe economy without specifying the supply (firm) side. Our results thus apply in richer environments
such as the textbook three-equation New Keynesian model.

7the initial consumption function C0 ({rt , Yt }t≥0 ):
dC0 =

Z ∞

|

0

Z ∞
∂C0
∂C0
drt dt +dYt dt .
∂rt
∂Yt
0
{z
}
|
{z
}

direct response to r

(3)indirect effects due to Y

The income innovations {dYt }t≥0 are equilibrium outcomes induced by the changes inR∞interest rates, which satisfy d log Yt = − γ1 t drs ds.8
The key objects in the decomposition (3) are the partial derivatives of the consumption function ∂C0 /∂rt and ∂C0 /∂Yt , i.e. the household’s responses to interest
rate and income changes. In this simple model, these two derivatives can be computed
analytically which leads to the main result of this section.9
Proposition 1 Consider small deviations drt of the interest rate from steady state.
R∞
The overall effect on initial consumption d log C0 = − γ1 0 drs ds can be decomposed as
1
d log C0 = −
γ
|

Z ∞
0

ρ
e drt dt −
γ
{z
} |
−ρt

direct response to r

Z ∞
0

−ρt

e

Z ∞
t

{z

drs dsdt .
}(4)indirect effects due to Y

The decomposition is additive, i.e. the two components sum to the overall effect.
This decomposition of the initial consumption response holds for any time path ofinterest rate changes {drt }t≥0 . The relative importance of the direct effect does notdepend on the intertemporal elasticity of substitution 1/γ.
When the interest rate path follows (1), the decomposition becomes:−

d log C0
1 h
=
dr0
γη

η
ρ+η
| {z }

direct response to r

+

ρ
ρ+η
| {z }

i.

(5)

indirect effects due to Y

The split between direct and indirect effect depends only on the discount rate ρ andthe rate of mean reversion η. A higher discount rate implies a smaller direct effect anda larger indirect general equilibrium effect. This reflects the fact that (i) in this model,
the marginal propensity to consume out of current income is equal to the discountrate; and (ii) the lower is η the larger is the impact of the interest rate change on
8Adjustments in income dYt can themselves be further decomposed into direct effects and indirect
general equilibrium effects. We nevertheless find this version of the decomposition especially useful.
In particular, it allows us to distinguish whether, following a change in interest rates, individual
households primarily respond through intertemporal substitution in and of itself or to changes intheir labor income.
9
See Theorem 3 in Auclert (2016) for a related decomposition.

8the permanent component of labor income. One important implication of equation(5) is that, for any reasonable parameterization, the indirect effect is very small, andmonetary policy works almost exclusively through the direct channel. For example in a
representative agent model, a quarterly steady state interest rate of 0.5% (2% annually,
as we assume in our quantitative analysis later in the paper) implies ρ = 0.5%. Suppose
the monetary policy shock mean reverts at rate η = 0.5, i.e. a quarterly autocorrelationof e−η = 0.61, then the direct effect accounts for η/(ρ + η) = 99% of the overall effect.10
Note that, even with a quarterly autocorrelation of 0.95 (η = 0.05), i.e. with an
implausibly persistent monetary shock, from an empirical standpoint, the contributionof the direct effect would still be above 90 pct.11
These results extend to the case where government debt is non-zero, B0 > 0. Whenthe government issues debt, in equilibrium a monetary expansion necessarily triggers
a fiscal response Tt = Tt ({rs }s≥0 ) in order to satisfy the government budget constraint.
This equilibrium feedback from fiscal policy affects household consumption which now
depends on taxes/transfers Ct ({rt , Yt , Tt }t≥0 ). In this case, the direct-indirect decomposition becomes:
dC0 =

Z ∞

|0


Z ∞
∂C0
∂C0
∂C0
drt dt +dYt +dTt dt .
∂rt
∂Yt
∂Tt
{z
} |0
{z
}

direct response to r

(6)indirect effects

Thus, in the special case (1) where interest rates mean-revert at rate η, we have:d log C0
1
−
=
dr0
γη



η
ρ+η
|



B0
1 − ργ+Ȳ{z}direct response to r

ρρ+η| {z }

indirect effects due to Y

+

η
B0
ργ
ρ + η Ȳ
|
{z
}


. (7)

indirect effects due to T

As already noted, due to Ricardian equivalence, the overall effect of monetary policy is
not impacted. Relative to (5), the presence of government debt reduces the direct effect.
This is because households now own some wealth and hence experience a negative(capital) income effect following an interest rate cut. Ricardian equivalence manifests
itself in the fact that the reduction in the direct effect is exactly offset by an additional
indirect effect due to changes in transfers. The split between these two components
depends on the debt-to-GDP ratio B0 /Ȳ . In principle, with large enough government
10This value implies the shock is fully reabsorbed after around 6 quarters. This speed of meanreversion is consistent with the dynamics of a shock to the federal fund rate commonly estimated by
VARs. See, e.g., Christiano et al. (2005) and Gertler and Karadi (2015).
11
As suggested by John Cochrane http://johnhcochrane.blogspot.com/2015/08/whitherinflation.html a better name for the standard New Keynesian model may therefore be the “stickyprice intertemporal substitution model.”9

debt, direct effects can be small even in RANK. However, for plausible debt levels, thedecomposition is hardly affected relative to (5). For instance, with a quarterly debtto-GDP ratio B0 /Ȳ = 1 (the number used in our calibration) and log-utility γ = 1,η
the direct effect accounts for ρ+η
1 − ργ BȲ0 = 98% of the overall effect.

2.2

Non-Ricardian Hand-to-Mouth Households

We now introduce “rule-of-thumb” households as in Campbell and Mankiw (1989,
1991) and Bilbiie (2008, 2017). The setup is identical, except that we assume that afraction Λ of households consume their entire current income, i.e. per-capita consumption of these “spenders” is given by Ctsp = Yt + Ttsp where Ttsp is a lump-sum transfer
to spenders. Spenders therefore have a marginal propensity to consume out of labor
income and transfers equal to one. The remaining fraction 1 − Λ of households optimize as before, yielding a consumption function for these “savers” Ctsa ({rt , Yt , Ttsa}t≥0 ).
Aggregate consumption is given by Ct = ΛCtsp + (1 − Λ)Ctsa . In equilibrium Ct = Yt .
The results from RANK extend in a straightforward fashion to this Two-Agent
New-Keynesian (TANK) economy. Consider first the case in which Bt = 0 for all t.
For brevity, we again only analyze the generalization of (5):1d log C0
−
=
dr0
γη





ρ
η
+ (1 − Λ)+Λ .
(1 − Λ)
ρ+η
ρ+η
{z
}
|
|
{z
}

direct response to r

(8)indirect effects due to Y

Note first that the total aggregate effect of monetary policy is exactly as in RANK.
The contribution of the direct effect and the indirect effect are each a weighted averageof the corresponding quantities for spenders and savers, with the weights equal toeach group’s population share. Since the direct effect for spenders is zero and theindirect effect is one, the overall share of the indirect effect approximately equals the
population fraction of spenders Λ. A reasonable estimate for the proportion of handto-mouth households in the U.S. is 0.3 (Kaplan et al., 2014), thus in TANK the shareof direct effects is roughly 0.7.
The result that the overall effect in TANK is the same as in RANK is due to thefact that the smaller direct effects due to the presence of hand-to-mouth households areexactly offset by larger indirect effects. To see this, note that aggregate consumption is
given by Ct = Yt = ΛYt + (1 − Λ)Ctsa where consumptionof savers is pinned
down from


R
1 ∞
sa
the time path of interest rates only Ct = C̄ exp − γ t (rs − ρ)ds . Equivalently,

1
Ct = M × (1 − Λ)Ctsa where M = 1−Λ> 1 is a multiplier. The presence of hand-tomouth households hence scales down direct effects by a factor 1 − Λ, but these then10

1
. This is the same logic thatget scaled up again by an exactly offsetting factor 1−Λlies behind a result of Werning (2015) who showed that, in a particular sticky priceeconomy with heterogeneous agents and incomplete markets, direct and indirect channels exactly offset so that the overall effect of interest rate changes on consumption isunchanged relative to the representative agent complete markets benchmark. In Werning’s economy, as well as in our toy model, labor is demand-determined and, therefore,
labor supply plays no role. Bilbiie (2008, 2017) studied the monetary transmission
mechanism in a TANK model with endogenous labor supply. His analysis implies that
this ‘as if’ result holds only in the knife-edge case of infinite elasticity. In general, theaggregate effect of a monetary policy shock in TANK can be smaller or larger than in
RANK.12Now consider the case where the government issues debt B0 > 0. As in Section 2.1,
a change in the path of interest rates affects the government budget constraint andinduces a fiscal response. Because Ricardian equivalence need not hold in the spendersaver economy, the effect of monetary policy depends crucially on the specifics of thisfiscal response. In particular, as long as the fiscal response entails increasing transfers to
the hand-to-mouth households, then this will increase the overall response of aggregate
consumption to monetary policy. This mechanism can be seen most clearly in the caseof the exponentially decaying interest rate path (1). If we assume that the government
keeps debt constant at its initial level, Bt = B0 for all t, and transfers a fraction ΛT of
the income gains from lower interest payments to spenders (and the residual fraction
to savers) so that ΛTtsp ({rs }s≥0 ) = −(rt − ρ)ΛT B0 , then initial consumption is13−

d log C0
ΛT B0
1
=
+ ,
dr0
1 − Λ Ȳγη

(9)Note the presence of the term ΛT (B0 /Y ): the overall effect of monetary policy differs from RANK only if there is both a debt-issuing government (B0 > 0) and NonRicardian households who receive a positive share of the transfers (ΛT > 0). It is only
under this scenario that the indirect component of the transmission mechanism could
be much larger in TANK, compared to RANK models (for the decomposition corresponding to (9) see equation (62) in the Appendix). As will become clear in Section 5,
12

The equivalence result between TANK and RANK derived in (8) also depends on the identity
Ct = Yt and hence on the fact that this model does not feature capital and investment. In the presenceof investment, the introduction of hand-to-mouth households has ambiguous effects on the elasticity
of aggregate consumption. In particular, we would have Ct + It = Yt = M × ((1 − Λ)Ctsa + It ) andΛ
hence Ct = Ctsa + 1−Λ
It and hence the elasticity may be larger or smaller depending on Λ/(1 − Λ) as
well as other factors determining the size of the investment response.
13
This is equivalent to assuming that the government maintains budget balance by adjusting lumpsum transfers, which is the baseline assumption we make in our full quantitative model.

11

Elasticity of C
P.E. elast. of C
Direct effects

RANK
B = 0 B > 0 S-W
(1)
(2)
(3)
−2.00 −2.00 −0.74−1.98 −1.96 −0.73
99%
98%
99%

B, K > 0
(4)
−2.07
−1.95
94%

TANK
B = 0 B > 0 B, K > 0
(5)
(6)
(7)
−2.00 −2.43−2.77−1.38 −1.39
−1.39
69%
57%
50%

Table 1: Elasticity of aggregate consumption and share of direct effects in several
versions of the RANK and TANK models.
Note: B = 0 denotes the simple models of Section 2 with wealth in zero net supply. B > 0 denotes
the extension of these models with government bonds in positive net supply. In RANK, we set
γ = 1, η = 0.5, ρ = 0.005, and B0 /Y = 1. In addition, in TANK we set Λ = ΛT = 0.3. ‘S-W’ is themedium-scale version of the RANK model described in Appendix A.4 based on Smets-Wouters.
‘B, K > 0’ denotes the richer version of the representative-agent and spender-saver New Keynesianmodel featuring a two-asset structure, as in HANK. See Appendix A.5 for a detailed description ofthis model and its calibration. In all cases lump-sum transfers adjust to balance the government
budget constraint in the economies with bonds in positive supply. ‘P.E. elast of C’ is the partial
equilibrium (or direct) elasticity computed as total elasticity times the share of direct effects.

in richer HANK models, the indirect effect is the larger component of monetary policy
even when the fiscal response following a monetary shock does not increase transfers
to hand-to-mouth households.

2.3

Richer RANK and TANK Models

Is our finding that conventional monetary policy works almost exclusively through direct intertemporal substitution special to these simple models? Compared to typicalmedium-scale DSGE models used in the literature, the RANK model in the present
section is extremely stylized. For instance, state-of-the-art medium-scale DSGE models
typically feature investment subject to adjustment costs, variable capital utilization,
habit formation, and prices and wages that are partially sticky as opposed to perfectly rigid. We therefore conducted a decomposition exactly analogous to that in (4)in one such state-of-the-art framework, the Smets and Wouters (2007) model (see theappendix for details). The result confirms our earlier findings: 99 percent of the consumption response to an expansionary monetary policy shock is accounted for by directintertemporal substitution effects. The reason is that none of the additional featuresof this richer model change the property that the consumption of the representative
agent is insensitive to the transitory income changes resulting from monetary shocks.14
14

With Smets and Wouters’ baseline parameterization, the total elasticity for consumption at impact
is −0.74, so substantially smaller than that of our stylized models. The key reason is that their modelfeatures habit formation in consumption which mutes the consumption response at impact. We
conducted a number of robustness checks, particularly with respect to the habit formation parameter12

We have also solved more complex versions of RANK and TANK models which,like the HANK model that follows, have both government debt and capital in positive
supply, and a New Keynesian production side with Rotemberg-style price adjustment
costs. These models, which are fully described in Appendix A.5, are designed to
be as close as possible to HANK, except for the nature of household heterogeneity.
Comparing column (4) of Table 1 with columns (1) and (2) (for RANK), and comparing
column (7) with columns (5) and (6) (for TANK) illustrates that the simple models of
Sections 2.1 and 2.2 approximate well these richer economies both in terms of the sizeof the total consumption response and decomposition into direct and indirect share.

3

HANK: A Framework for Monetary Policy Analysis

We now turn to our paper’s main contribution: the development and analysis of our
Heterogeneous Agent New Keynesian (HANK) model. Our main innovation is a richrepresentation of household consumption and saving behavior. Households face uninsurable idiosyncratic income risk which they can self-insure through two savings instruments with different degrees of liquidity. The model’s supply side is kept purposefully
simple, and we borrow a number of assumptions from the New Keynesian literature:
there is price stickiness and a monetary authority that operates a Taylor rule, and weanalyze the economy’s response to an innovation to this Taylor rule. For simplicity, we
consider a deterministic transition following a one-time zero-probability shock.

3.1

The Model

Households The economy is populated by a continuum of households indexed bytheir holdings of liquid assets b, illiquid assets a, and their idiosyncratic labor productivity z. Labor productivity follows an exogenous Markov process that we describe
in detail in Section 4.2.2. Time is continuous. At each instant in time t, the state of
the economy is the joint distribution µt (da, db, dz). Households die with an exogenous
Poisson intensity ζ, and upon death give birth to an offspring with zero wealth andlabor productivity equal to a random draw from its ergodic distribution.15 There are
perfect annuity markets so that the estates of the deceased are redistributed to otherindividuals in proportion to their asset holdings.16
which directly enters the representative agent’s Euler equation, and found that the share due to direct
effects never drops below 90 percent.
15
We allow for stochastic death to help in generating a sufficient number of households with zero
illiquid wealth relative to the data. This is not a technical assumption that is needed to guarantee
the existence of a stationary distribution, which exists even in the case ζ = 0.
16
The assumption of perfect annuity markets is implemented by making the appropriate adjustment
to the asset returns faced by surviving households. To ease notation, we fold this adjustment directly

13Households receive a utility flow u from consuming ct ≥ 0 and a disutility flow fromsupplying labor ℓt , where ℓt ∈ [0, 1] are hours worked as a fraction of the time endowment, normalized to one. The function u is strictly increasing and strictly concave in
consumption, and strictly decreasing and strictly convex in hours worked. Preferences
are time-separable and, conditional on surviving, the future is discounted at rate ρ ≥ 0:E0

Z ∞e−(ρ+ζ)t u(ct , ℓt )dt,

(10)

0

where the expectation is taken over realizations of idiosyncratic productivity shocks.
Because of the law of large numbers, and the absence of aggregate shocks, there is no
economy-wide uncertainty.
Households can borrow in liquid assets b up to an exogenous limit b at an interest
rate of rtb− = rtb +κ, where κ > 0 is an exogenous wedge between borrowing and lending
rates. With a slight abuse of notation, rtb (bt ) summarizes the full interest rate schedule.
Short positions in illiquid assets are not allowed.
Assets of type a are illiquid in the sense that households need to pay a cost for
depositing into or withdrawing from their illiquid account. We use dt to denote ahousehold’s deposit rate (with dt < 0 corresponding to withdrawals) and χ(dt , at ) todenote the flow cost of depositing at a rate dt for a household with illiquid holdings
at . As a consequence of this transaction cost, in equilibrium the illiquid asset pays ahigher return than the liquid asset, i.e. rta > rtb .
A household’s asset holdings evolve according toḃt = (1 − τt )wt zt ℓt + rtb (bt )bt + Tt − dt − χ(dt , at ) − ct

(11)ȧt = rta at + dt

(12)

bt ≥ −b,(13)

at ≥ 0.

Savings in liquid assets ḃt equal the household’s income stream (composed of labor
earnings taxed at rate τt , interest payments on liquid assets, and government transfers
Tt ) net of deposits into or withdrawals from the illiquid account dt , transaction costs
χ(dt , at ), and consumption expenditures ct . Net savings in illiquid assets ȧt equalinterest payments on illiquid assets plus net deposits from the liquid account dt . Notethat while we distinguish between liquid and illiquid assets, we follow much of theincomplete-markets literature and net out assets and liabilities within the two asset
classes. That is, ours is not a model of gross positions.
into the rates of return, which should therefore be interpreted as including the return from the annuity.

14

The functional form for the transaction cost χ(d, a) is given byχ(d, a) = χ0 |d| + χ1

d
a

χ2

a.

(14)

This transaction cost has two components that play distinct roles. The linear component generates an inaction region in households’ optimal deposit policies because
for some households the marginal gain from depositing or withdrawing the first dollar is smaller than the marginal cost of transacting χ0 > 0. The convex component(χ1 > 0, χ2 > 1) ensures that deposit rates are finite, |dt | < ∞ and hence household’s
holdings of assets never jump. Finally, scaling the convex term by illiquid assets adelivers the desirable property that marginal costs χd (d, a) are homogeneous of degree
zero in the deposit rate d/a so that the marginal cost of transacting depends on thefraction of illiquid assets transacted, rather than the raw size of the transaction.17
Households maximize (10) subject to (11)–(14). They take as given equilibrium
paths for the real wage {wt }t≥0 , the real return to liquid assets {rtb }t≥0 , the real return toilliquid assets {rta }t≥0 , and taxes and transfers {τt , Tt }t≥0 . As we explain below, {rtb}t≥0will be determined by monetary policy and a Fisher equation, and {wt }t≥0 and {rta }t≥0will be determined by market clearing conditions for capital and labor. In Appendix
B.1 we describe the household’s problem recursively with a Hamilton-Jacobi-Bellman
equation. In steady state, the recursive solution to this problem consists of decisionrules for consumption c(a, b, z; Γ), deposits d(a, b, z; Γ), and labor supply ℓ(a, b, z; Γ),with Γ ≡ (r b , r a , w, τ, T ).18 These decision rules imply optimal drifts for liquid andilliquid assets and, together with a stochastic process for z, they induce a stationary
joint distribution of illiquid assets, liquid assets, and labor income µ(da, db, dz; Γ). In
the appendix, we also describe the Kolmogorov forward equation that characterizes
this distribution. Outside of steady state, each of these objects is time-varying anddepends on the time path of prices and policies {Γt }t≥0 ≡ {rtb , rta , wt , τt , Tt }t≥0 .
Final-goods producers A competitive representative final-good producer aggregates a continuum of intermediate inputs indexed by j ∈ [0, 1]
Yt =

Z 1
0

ε−1
ε

yj,t dj

17

ε
 ε−1

Because the transaction cost at a = 0 is infinite, in computations we replace the term a withmax {a, a}, where the threshold a > 0 is a small value (always corresponding to less than $500 in all
calibrations) that guarantees costs remain finite even for households with a = 0.
18
In what follows, when this does not lead to confusion, we suppress the explicit dependence ofdecision rules on the vector of prices and policies Γ.

15where ε > 0 is the elasticity of substitution across goods. Cost minimization implies
that demand for intermediate good j isyj,t(pj,t ) =



pj,t
Pt

−ε

Yt ,

where Pt =

Z 1
0

p1−ε
j,t dj1 1−ε.

Intermediate goods producers Each intermediate good j is produced by a monopolistically competitive producer using effective units of capital kj,t and effective
units of labor nj,t according to the production function
α 1−αyj,t = kj,tnj,t .

(15)

Intermediate producers rent capital at rate rtk in a competitive capital market and hirelabor at wage wt in a competitive labor market. Cost minimization implies that themarginal cost is common across all producers and given by k α 
1−α
rt
wt
,
mt =
α
1−α

(16)

where factor prices equal their respective marginal revenue products.
Each intermediate producer chooses its price to maximize profits subject to priceadjustment costs as in Rotemberg (1982). These adjustment costs are quadratic in therate of price change ṗt /pt and expressed as a fraction of produced output Yt
 
 2
ṗt
θ ṗt
Θt
=
Yt ,
pt
2 pt

(17)

where θ > 0. Suppressing notational dependence on j, each intermediate producerchooses {pt }t≥0 to maximize
Z ∞R− 0t rsa ds

e

0

where
Π̃t (pt ) =


 
ṗt
Π̃t (pt ) − Θt
dt,
pt



pt
− mt
Pt



pt
Pt

−ε

Yt

(18)

are per-period profits. The choice of rta for the rate at which firms discount future
profits is justified by a no-arbitrage condition that we explain below.
Lemma 1, proved in Appendix B.2, characterizes the solution to the pricing problem
and derives the exact New Keynesian Phillips curve in our environment. The combina16

tion of a continuous-time formulation of the problem and quadratic price adjustment
costs allows us to derive a simple equation characterizing the evolution of inflation
without the need for log-linearization.
Lemma 1 The aggregate inflation rate πt = Ṗt /Pt is determined by the New KeynesianPhillips curve
Ẏt
rta −
Yt

!

πt =

ε
(mt − m∗ ) + π̇t ,
θ

m∗ =

ε−1
.
ε

(19)

The expression in (19) can be usefully written in present-value form as:
ε
πt =
θ

Z ∞

Rs

a

e− t rτ dτtYs(ms − m∗ ) ds.
Yt

(20)

Note that the marginal payoff to a firm from increasing its price at time s is Π′s (ps ) =εYs (ms − m∗ ). Firms raise prices when their markup Ms = 1/ms is below the flexibleεprice optimum M∗ = 1/m∗ = ε−1. Inflation in (20) is the rate of price changes that
equates the discounted sum of all future marginal payoffs from changing prices thisperiod to its marginal cost θπt Yt obtained from (17).
Composition of illiquid wealth Illiquid savings can be invested in two assets: (i)capital kt , and (ii) shares in the equity of the aggregate portfolio of intermediate firms,
which we denote by st . This equity represents a claim on the entire future stream ofmonopoly profits net of price adjustment costs, Πt ≡ Π̃t − θ2 πt2 Yt . Let qt denote theshare price. An individual’s illiquid assets can thus be expressed as at = kt + qt st . The
dynamics of capital and equity satisfyk̇t + qt ṡt = (rtk − δ)kt + Π̃t st + dt .

(21)

We assume that within the illiquid account, resources can be costlessly shifted between capital and shares. Hence a no-arbitrage condition must hold for the two assets,
implying that the return on equity equals the return on capital:
Πt + q̇t= rtk − δ ≡ rta .
qt

(22)

We can therefore reduce the dimensionality of the illiquid asset space and consider
only the combined illiquid asset a with rate of return r a given by any of the two

17

returns in (22) and with law of motion as in (12).19 Finally, note that (22) implies thatR ∞ Rτ aqt = t e− t rs ds Π̃τ dτ which justifies the use of rta as the rate at which future profits
are discounted by the intermediate firms and, thus, as the discount rate appearing in
the Phillips curve.
Monetary Authority The monetary authority sets the nominal interest rate on
liquid assets it according to a Taylor ruleit = r̄ b + φπt + ǫt

(23)

where φ > 1 and ǫt = 0 in steady state. Our main experiment studies the economy’s
adjustment after an unexpected temporary monetary shock ǫt .20
Given inflation and the nominal interest rate, the real return on the liquid asset isdetermined by the Fisher equation rtb = it − πt . The real liquid return rtb needs also to
be consistent with equilibrium in the bond market, which we describe in Section 3.2.
Government The government faces exogenous government expenditures Gt and administers a progressive tax and transfer scheme on household labor income wt zℓt thatconsists of a lump-sum transfer Tt and a proportional tax rate τt , with τt , Tt > 0. The
government is the sole issuer of liquid assets in the economy, which are real bondsof infinitesimal maturity Btg , with negative values denoting government debt. Its intertemporal budget constraint is
Ḃtg + Gt + Tt = τtZ

wt zℓt (a, b, z) dµt + rtb Btg(24)

Outside of steady state, the fiscal instrument that adjusts to balance the budget canbe either τt , Tt , or Gt . In our experiments, we consider various alternative possibilities
for the time path of these adjustments.
19
The no-arbitrage condition that allows us to reduce the illiquid portfolio to a single state variable,holds only in the absence of jumps in share price q, for example in steady state. In this case eachindividual’s illiquid asset portfolio composition between capital and equity is indeterminate, eventhough the aggregate composition is determined.
20
We assume that the monetary authority responds only to inflation. Generalizing the Taylor rule(91) to also respond to output gaps is straightforward and does not substantially affect our conclusions.
Since our focus is on understanding the transmission mechanism of conventional monetary policy innormal times, we do not consider cases in which the zero-lower bound on nominal interest rates
becomes binding.

18

3.2

Equilibrium

An equilibrium in this economy is defined as paths for individual household and firm
decisions {at , bt , ct , dt , ℓt , nt , kt }t≥0 , input prices {wt , rtk }t≥0 , returns on liquid and illiquid assets, {rtb , rta }t≥0 , the share price {qt }t≥0 , the inflation rate {πt }t≥0 , fiscal variables{τt , Tt , Gt , Bt }t≥0 , measures {µt }t≥0 , and aggregate quantities such that, at every t, (i)households and firms maximize their objective functions taking as given equilibrium
prices, taxes, and transfers, (ii) the sequence of distributions satisfies aggregate consistency conditions, (iii) the government budget constraint holds, and (iv) all markets
clear. There are five markets in our economy: the liquid asset (bond) market, markets
for capital and shares of the intermediate firms (that can be folded into a single illiquid
asset), the labor market, and the goods market.
The liquid asset market clears whenBth + Btg = 0,

(25)

R
where Btg is the stock of outstanding government debt and Bth = bdµt are total
household holdings of liquid bonds. The illiquid asset market clears when physicalcapital Kt plus the equity value of monopolistic producers qt (with the total numberRof shares normalized to one) equals households’ holdings of illiquid assets At = adµt ,Kt + qt = At ,

(26)

Z

(27)

The labor market clears whenNt =zℓt (a, b, z)dµt .

Finally, the goods market clearing condition is:
Yt = Ct + It + Gt + Θt + χt + κZ

max{−b, 0}dµt .

(28)Here, Yt is aggregate output, Ct is total consumption expenditures, It is gross additions
to the capital stock Kt , Gt is government spending, Θt are total price adjustment costs,
and the last two terms reflects transaction and borrowing costs (to be interpreted as
financial services).

19

3.3

Monetary Transmission in HANK

We are interested in analyzing the response of the economy to a one-time unexpected
expansionary monetary shock. We assume that the economy is initially in steadystate with monetary policy following the Taylor rule (91) with ǫt = 0. Then at time
t = 0 there is an innovation to the Taylor rule ǫ0 < 0 with some deterministic decay
back to zero. To examine the economy’s response to this shock, we generalize themethodology proposed in Section 2 to decompose the total effect of a monetary shock
into direct/partial equilibrium and indirect/general equilibrium effects. Our focus is on
the transmission mechanism of the shock on the dynamics of aggregate consumption at
impact, but it is clear that our decomposition can be extended to any other aggregate
variable at any horizon.
Let us begin by writing aggregate consumption Ct explicitly as a function of thesequence of equilibrium prices, taxes and transfers {Γt }t≥0 , with Γt = {rtb , rta , wt , τt , Tt },induced by the path of the monetary shock {ǫt }t≥0 from its initial innovation until its
full reversal back to zero:Z
Ct ({Γt }t≥0 ) = ct (a, b, z; {Γt }t≥0 )dµt .
(29)Here ct (a, b, z; {Γt }t≥0 ) is the household consumption policy function and µt (da, db, dz;{Γt }t≥0 ) is the joint distribution of liquid and illiquid assets and idiosyncratic income.21
Totally differentiating (29), we decompose the consumption response at t = 0 as
dC0 =

Z ∞
|

0


Z ∞
∂C0 b
∂C0
∂C0 a ∂C0
∂C0
dr dt +
dwt + a drt +dτt +dTt dt .
∂wt
∂rt
∂τt
∂Tt
∂rtb t
0
{z
} |
{z
}

direct effect(30)

indirect effects

The first term in the decomposition reflects direct effects of a change in the path ofthe liquid return, holding the wage, the illiquid return, and fiscal policy constant.22
Since the path of liquid rates enters the budget constraint (11), households respond
21

Strictly speaking, because households are forward-looking the consumption policy function at time
t is only a function of the sequence of prices from time t onwards {Γs }s≥t . Similarly, the distribution
is backward-looking and is only a function of the sequence of prices up to time t, {Γs }s<t . We chosethe somewhat less precise notation above for simplicity.
22
We define the direct effect of a monetary policy with respect to changes in rtb because this is
the relevant price from the point of view of households. Alternatively, we could define it “even more
directly” with respect to the monetary policy shock ǫt . With this alternative decomposition, the direct
effect in (30) would be split further into a direct due to ǫt and an indirect effect due to inflation πt .
This follows because rtb = r̄b + (φ − 1)πt + ǫt from the Taylor rule and the Fisher equation. Figure3(a) in our quantitative analysis shows that the drops in rtb and ǫt are almost equal so that the two
decompositions are quantitatively similar.

20directly to interest rate changes. This direct effect itself consists of both intertemporal
substitution and income effects. The latter arise because aggregate liquid assets areunequally distributed in the cross section and in positive net supply.
The remaining terms in the decomposition reflect the indirect effects of changes inwages, the illiquid return, and the government budget constraint that arise in generalequilibrium. There are three separate indirect channels at work in response to an
expansionary monetary policy shock. First, when the liquid return falls, intertemporal
substitution causes non hand-to-mouth households to increase consumption. In order
to meet this additional demand for goods, intermediate firms increase their demand forlabor, which pushes up the wage. Households respond to the increase in labor income
by further increasing their consumption expenditures.
Second, when the illiquid return changes in response to the change in the liquid
return, consumption may be further affected as households choose to rebalance theirasset portfolio with deposits into or withdrawals from the illiquid account.23
Third, there is a fiscal response to changes in the liquid rate through the government
budget constraint. A fall in r b reduces government’s interest payments on its debt
and results in higher tax revenues because of the additional labor income from theeconomic expansion. Both forces loosen the government budget constraint, and lead toan adjustment in one of the fiscal instruments. As will become clear from our numericalexperiments, both the total size of the macroeconomic effects of monetary policy andthe split between direct and indirect components depends on the type of fiscal response,
a consequence of the non-Ricardian nature of this class of HANK economies.
In practice, we need to compute each of these components numerically. For example,
the formal definition of the first term in (30), the direct effect of changes in the liquid
return {rtb }t≥0 , is
Z ∞
0

∂C0 b
dr dt =
∂rtb tZ ∞ Z0∂c0 (a, b, z; {rtb , r̄ a , w̄, τ̄ , T̄ }t≥0 ) rb
dµ0 drtb dt
∂rtb

(31)

b

where µr0 = µ0 (da, db, dz; {rtb , r̄ a , w̄, τ̄ , T̄ }t≥0 ). That is, this term is the aggregate
partial-equilibrium consumption response of a continuum of households that face atime-varying interest rate path {rtb }t≥0 but paths for illiquid asset return r̄ a , wage w̄,23

At impact, the share price q0 jumps (reflecting the change in expected future profits) inducing arevaluation of illiquid wealth. With a slight abuse of notation, the derivative of C0 with respect to r0a(29) embeds this effect (and all the results we report later in the paper take this initial instantaneous
jump into account). For t > 0, changes in qt are already embedded in rta through the no-arbitrage
condition (22). In order to quantify the effect of this price movement on the value of households
illiquid assets, we need to make an assumption about the portfolio composition between shares andcapital. We simply assume that every agent has the same portfolio composition as the aggregate.

21

and taxes and transfers (τ̄ , T̄ ) constant at their steady-state values. We calculate thisterm from the model by feeding these time paths into the households’ optimization
problem, computing c0 for each household, and aggregating across households using
the corresponding distribution.
The other terms in the decomposition are computed in a similar fashion. There
are similarities between our decomposition and one proposed by Auclert (2016). We
discuss these in more detail in Section 5.3.

4

Taking the Model to the Data

In this section, we explain how we map our framework to the data. We begin bypresenting a small extension of HANK that allows the model to generate more volatileand pro-cyclical investment in response to a monetary shock. Next, we describe ourcalibration strategy and illustrate our parameterization. Finally, we demonstrate thatthe model offers a realistic representation of microeconomic consumption behavior.

4.1

Distribution of Monopoly Profits

In models of monopolistic competition with price rigidities only, countercyclical markups are at the heart of economic fluctuations. Our framework is no exception. Because
prices are sticky but nominal marginal costs are not, expansionary monetary shocks
shrink mark-ups, causing firm profits to fall. In RANK models these fluctuations in
profits are typically borne lump-sum by the representative household. But in HANK
models, additional assumptions are needed about how profits are distributed acrosshouseholds, and in two-asset HANK models further assumptions are needed about how
profits are distributed between liquid and illiquid assets. These assumptions, which areneutral in RANK models, can have a large effect on the volatility and cyclicality ofinvestment.
For example, in our baseline HANK model of section 3.1 the fluctuations in profits
manifest as movements in the share price qt . Since equity is a component of illiquid
assets, rather than liquid assets, the fall in profits associated with an expansionary
monetary shock creates a downward pull on investment at a time when output isexpanding.24 This feature is in stark contrast with the data where, quantitatively,
investment is the most volatile and procyclical component of output.
24
Aggregating (21) across households and using the relevant market clearing conditions we haveK̇t = (rtk − δ)Kt + Πt + Dt or equivalently that aggregate investment is It = rtk Kt + Πt + Dt whereΠt is the profit flow and Dt are aggregate net deposits. When net deposits Dt do not move muchover the business cycle as is the case in practice, countercyclical fluctuations in profits Πt have a large
effect on investment It .

22

To avoid this counterfactual implication of sticky prices and restore an empiricallyrealistic co-movement between output, consumption and investment, we make a simple
modification to the baseline HANK model: we add one parameter, ω ∈ [0, 1], that
controls the fraction of profits reinvested directly into the illiquid account. Then, if we
aggregate the total illiquid income flow across all households, we obtain(rtk − δ)Kt + ωΠt = αmt Yt + ω(1 − mt )Yt .

(32)

The profit distribution scheme that fully sterilizes the impact of fluctuating mark-ups
corresponds to ω = α, the capital share of output. In this case, the total income
flow accruing into the illiquid account becomes αYt , which is independent of mt and isalways procyclical.
We assume that the residual share of profits (1 − ω)Πt is paid in liquid form toevery individual i as a lump-sum transfer in proportion to household productivity, i.e.
πitb = zz̄it (1 −ω)Πt , where z̄ is average productivity. We interpret this additional income
as the profit-sharing component of worker compensation from bonuses, commissions,and gains from exercising stock options, and in what follows the term labor income
should be interpreted as the sum of wage payments wt zit ℓit and bonuses πitb , wheneverω < 1. Appendix B.4 contains more details of this extension of the model.25

4.2

Calibration Strategy

We have four broad goals in choosing parameters for the model. First, we need todevelop a mapping between our aggregated two-asset (liquid-illiquid) structure anddata on the complex balance sheet of the U.S. household sector. Second, we seek acalibration of the exogenous stochastic process for labor earnings, which is the ultimate
source of inequality in the model. Third, in order to obtain quantitatively realistic
consumption behavior at the microeconomic level, our model must generate realistic
distributions of liquid and illiquid assets. Of particular importance is the skewnessof liquid wealth holdings: matching the fraction of households with low liquid wealthbears directly on the sensitivity of consumption to income changes, whereas matching
the top of the liquid wealth distribution is key to generate plausible redistributive
effects of interest rate changes. Finally, since the production side of the model isessentially a textbook New Keynesian model, we want to remain as close as possible
25

The importance of countercyclical profits has recently been highlighted by Broer et al. (2016) who
argue that the resulting income effects on labor supply greatly amplify the New Keynesian monetarytransmission mechanism in RANK. As we discuss in the next section, the parameter ω also allows us
to discipline the strength of such income effects which, under standard balanced-growth preferences,in our model are present whenever ω < 1.

23

Liquid B hRevolving consumer debt
Deposits
Corporate bonds
Government bonds
Total

Illiquid (A = K + q)
−0.03
0.23
0.04
0.02

Net housing
Net durables
Corporate equity
Private equity

0.26

Total

1.09
0.22
1.02
0.59
2.92

3.18

Table 2: Summary of taxonomy of assets
Notes: Categorization of assets into liquid versus illiquid. Values are expressed as a multiple of 2004
GDP($12,300B). See Appendix C for details of all calculations.

to the parameterization that is well accepted in that literature.
4.2.1

Categorization of Assets into Liquid and Illiquid

Mapping the model to data requires classifying assets held by US households as liquidversus illiquid. We label an asset as liquid or illiquid based on the extent to which buying or selling the asset involves transaction costs. We define net liquid assets B h as all
deposits in financial institutions (checking, saving, call, and money market accounts),
government bonds, and corporate bonds net of revolving consumer credit. We define
illiquid assets A as real estate wealth net of mortgage debt, consumer durables net of
non-revolving consumer credit, plus equity in the corporate and non-corporate businesssectors. We have chosen to include equity among illiquid assets, because nearly 3/4
of total equity is either indirectly held (in tax-deferred retirement accounts) or held in
the form of private businesses. Both of these assets are significantly less liquid than all
the other asset classes included in our definition of B h .26
We measure the aggregate size of each category of assets and liabilities using datafrom the Flow of Funds (FoF) and the Survey of Consumer Finances (SCF). We use
data from 2004, since this is the last SCF survey year before the Great Recession.
In Appendix C, we undertake a comprehensive comparison between these two data
sources for each component of the balance sheet. Based on this analysis, we choose
to use FoF measures for all assets and liabilities except for the three main categories
of liquid assets – deposits, government bonds and corporate bonds – for which we
use estimates from the SCF. Table 2 summarizes our preferred estimate, expressed as
26

In a former version of the paper (Kaplan et al., 2016a), we also separated illiquid assets between
productive (equity) and non-productive (housing and durables). In that version of the model, theproductive assets served as input into production and paid the rate of return ra , whereas the nonproductive ones yielded a utility flow to households. Our key quantitative findings are largely invariant
to adopting this richer classification, once the model is properly recalibrated.

24

fractions of annual 2004 GDP ($12,300B). The total quantity of net liquid assets B hamounts to $2,700B (26% of annual GDP). The total quantity of net illiquid assets A
amounts to $36,000B (2.92 times annual GDP).
4.2.2

Continuous Time Earnings Dynamics

When households have a choice between saving in assets with different degrees of liquidity, as in our model, the frequency of earnings shocks is a crucial input for determining
the relative holdings of the two assets. Households who face small, but frequent, shocks
have a strong incentive to hold low-return liquid assets to smooth consumption, whilehouseholds who face large infrequent shocks would prefer to hold high-return illiquid
assets that can be accessed at a cost in the unlikely event of a sizable windfall or asevere income loss.
In standard discrete-time error component models (e.g., the classic persistenttransitory model), the frequency of arrival of earnings shocks is dictated by the assumed time period. In continuous-time models, the frequency at which shocks arrive
is a property of the stochastic process, and must be estimated alongside the size andpersistence of shocks. Empirically, the challenge in estimating the frequency of earnings shocks is that almost all high quality panel earnings data are available only atan annual (or lower) frequency. It is thus challenging to learn about the dynamics ofearnings at any higher frequency. Our strategy to overcome this challenge is to infer
high frequency earnings dynamics from the high-order moments of annual earnings
changes. To understand why this identification strategy has promise, consider two
possible distributions of annual earnings changes, each with the same mean and variance, but with different degrees of kurtosis. The more leptokurtic distribution (i.e. thedistribution with more mass concentrated around the mean and in the tails) is likely
to have been generated by an earnings process that is dominated by large infrequent
shocks; the more platykurtic distribution (i.e. the distribution with more mass in theshoulders) by a process that is dominated by small frequent shocks.
Motivated by these observations, we model log-earnings as the sum of two independent components
log zit = z1,it + z2,it(33)

where each component zj,it evolves according to a “jump-drift” process. Jumps arrive′
at a Poisson rate λj . Conditional on a jump, a new log-earnings state zj,itis drawn
2′2from a normal distribution with mean zero and variance σj , zj,it ∼ N (0, σj ). Between

25

Moment
Variance: annual log earns
Variance: 1yr change
Variance: 5yr change
Kurtosis: 1yr change
Kurtosis: 5yr change
Frac 1yr change < 10%
Frac 1yr change < 20%
Frac 1yr change < 50%

Data
0.70
0.23
0.46
17.8
11.6
0.54
0.71
0.86

Model
0.70
0.23
0.46
16.5
12.1
0.56
0.67
0.85

Table 3: Earnings Process Estimation Fit
jumps, the process drifts toward zero at rate βj . Formally, the process for zj,it isdzj,it = −βj zj,it dt + dJj,it,(34)

where dJj,it captures jumps in the process.27
The process for each component is closely related to a discrete time AR(1) process.28
The key difference is that in our continuous time formulation, the arrival of eachinnovation is stochastic, and hence each process has an additional parameter, λj , which
captures the frequency of arrival.29
Estimation with male earnings data We estimate the earnings process in (33)(34) by Simulated Method of Moments using Social Security Administration (SSA)
data on male earnings from Guvenen et al. (2015).30 These authors report eight key
moments that we target in the estimation (see Table 3).31 Moments of the distribution(z)of the two compoEven more formally, the infinitesimal generators Aj f (z) := limt↓0 E[f (zt )]−ft
R∞nents j = 1, 2 are given by Aj f (z) = −βj zf ′ (z) + λj −∞ (f (x) − f (z))φj (x)dx where φj is the densityof a Normal distribution with mean zero and variance σj2 .
28
In particular, if the earnings innovations always arrived at regular intervals (say, annually), rather
than stochastically at rate λj , then each component would follow an AR(1) process. The drift parameter βj would correspond to (one minus) the discrete time auto-regressive parameter and the innovation
variance σj2 would describe the size of innovations. In this sense, the model is only a minimal departurefrom the familiar persistent-transitory process used to model discrete time earnings data.
29
Schmidt (2015) models earnings dynamics as a discrete-time compound Poisson process, using asimilar logic.
30
The main benefits of targeting moments from administrative earnings data such as the SSA arethat they are based on a very large sample and so are less prone to measurement error than survey
data, and that they are not top-coded. Both features are important: the sample size and absence ofmeasurement error allows a precise estimate of higher-order moments, and the absence of top-coding
allows for an accurate portrayal of the right-tail of the income distribution, which is important for
capturing the skewness in wealth holdings.
31
We restrict attention to a symmetric process since Guvenen et al. (2015) find only a small amountof negative skewness in 1-year and 5-year annual changes. It is possible to generate skewness in annualchanges by allowing the drift parameters βj to differ based on the sign of zj,it .
27

26

Parameter
Arrival rate
Mean reversion
St. Deviation of innovations

λj
βj
σj

Component
j=1
0.080
0.761
1.74

Component
j=2
0.007
0.009
1.53

Table 4: Earnings Process Parameter Estimates. Rates expressed as quarterly values.
of earnings changes at multiple durations are needed to separately identify the twocomponents. Since these data refer to annual earnings, we simulate earnings from themodel at a high frequency, aggregate to annual earnings and compare moments frommodel and data.
The fitted earnings process matches the eight targeted moments well. The estimatedparameter values, reported in Table 4, are consistent with the existence of a transitory
and a persistent component in earnings. The transitory component (j = 1) arrives on
average once every three years and has a half-life of around one quarter. The persistentcomponent (j = 2) arrives on average once every 38 years and has a half-life of around18 years. Both components are subject to relatively large, similar sized innovations.
In the context of an infinite horizon model, the estimated process thus has the naturalinterpretation of a large and persistent “career” or “health” shock that is perturbed byperiodic temporary shocks. Note that relative to a discrete-time model, our estimatedtransitory shock is both less frequent, and more temporary than an IID annual shock.
In our model, flow earnings are given by yit ≡ wt zit ℓit and are thus determined byboth the realization of productivity shocks zit and the choice of labor supply ℓit . In
Appendix D.1 we explain how we convert the estimated process for individual maleearnings to a discrete-state process for idiosyncratic productivity that is consistent
with our assumption of a household as the unit of observation.32 Relative to typicalearnings process calibrations based on survey data, and consistent with the crosssectional earnings distribution in SSA data, the resulting earnings process features alarge amount of right-tail inequality. The top 10, 1, and 0.1 percent shares of grosshousehold labor earnings in the steady state are 32%, 7% and 2% respectively. Thisskewed earnings distribution is an important factor in the model’s ability to generateskewed distributions of liquid and illiquid assets. However, unlike much of the existing
literature that has generated wealth concentration at the top of the distribution fromad-hoc skewed earnings distributions, here both inequality and dynamics of earnings
32

Our discrete approximation to the estimated productivity process uses 11 points for the persistentcomponent and 3 points for the transitory component. In Appendix D.1 we describe the discretization
process in detail and report further statistics from the discretized distribution, including plots of theLorenz curves for the ergodic distributions from the continuous and discretized processes.

27

Mean illiquid assets
Mean liquid assets
Frac. with b = 0 and a = 0
Frac. with b = 0 and a > 0
Frac. with b < 0

Data
2.92
0.26
0.10
0.20
0.15

Model
2.92
0.23
0.10
0.19
0.15

Moment
Top 0.1% share
Top 1% share
Top 10% share
Bottom 50% share
Bottom 25% share
Gini coefficientLiquid Wealth
Data Model
17%
2.3%
47%
18%
86%
75%
-4%
-3%
-5%
-3%
0.98
0.86

Illiquid Wealth
Data Model
12%
7%
33%
40%
70%
88%
3%
0.1%
0%
0%
0.81
0.82

Table 5: Left panel: Moments targeted in calibration and reproduced by the model.
Means are expressed as ratios to annual output. Right panel: Statistics for the top andbottom of the wealth distribution not targeted in the calibration. Source: SCF 2004.
are disciplined directly by high quality data.33
4.2.3

Adjustment Cost Function and Wealth Distribution

We set the steady-state real return on liquid assets r̄ b at 2% per annum and steady-state
inflation to zero. Given values for the capital share, demand elasticity and depreciation
rate (all set externally as described in Section 4.2.4) and for the unsecured borrowing
limit — our target for the illiquid assets of 2.9 times output yields a steady-state returnto illiquid assets r a of 5.7% per annum.
Given these returns, and the exogenous process for idiosyncratic labor income, thekey parameters that determine the incentives for households to accumulate liquid andilliquid assets are the borrowing limit b, the discount rate ρ, the intermediation wedge
κ, and the three parameters of the adjustment cost function χ0 , χ1 and χ2 .
Borrowing in the model should be interpreted as unsecured credit, so we set theborrowing limit b exogenously at 1 times quarterly average labor income.34 We thenchoose the remaining five parameters (ρ, κ, χ0 , χ1 , χ2 ) to match five moments of the
distribution of household wealth from the SCF 2004: (i)-(ii) the mean of the illiquidand liquid wealth distributions; (iii)-(iv) the fraction of poor and wealthy hand-tomouth households, since these are the most important moments of the liquid wealthdistribution for determining household consumption responses to income shocks; and(v) the fraction of households with negative net liquid assets, which serves to identify
33
The existing literature reverse-engineers a process for earnings risk in order to match data on
wealth inequality. This approach typically requires an implausibly extreme characterization of risk,
with a top income state around 500 times as large as the median, and a high probability of a dramaticfall in earnings once the top state is reached. See Benhabib and Bisin (2016) and De Nardi et al.
(2016) for a discussion of this issue. In our discretized process, instead, the highest productivity
realization is around 13 times as large as the median, and is realized by only 0.03% of the population.
34
In the steady state ergodic distribution only 0.02% of households are at the limit.

28

the borrowing wedge.35
The calibrated annual discount rate ρ is 5.1%, and the annual wedge κ is 6.0%
(implying an annual borrowing rate of 8.0%). The calibrated transaction cost function
is displayed in Figure D.3 in Appendix D. In the resulting ergodic distribution, roughly
80% of households are adjusting at any point in time. Conditional on making a deposit
or withdrawal, the mean absolute quarterly transaction as a fraction of the stock of
illiquid assets is 1.7%. The quarterly transaction cost for a transaction this size is 23%of the transaction. In steady-state the equilibrium aggregate transaction costs, which
one can interpret as financial services, amount to less than 4% of GDP.
The model replicates the five targeted moments well (left panel of Table 5).36 Figure
1 displays the distributions of liquid and illiquid wealth in the model. Despite only
targeting a handful of moments of each distribution, the model successfully matches
the distributions of liquid and illiquid wealth up to the very top percentiles, as is clear
from the right-panel of Table 5 which reports top wealth shares from the model anddata. Both Gini coefficients in the model are close to their data counterparts. The
reason for this success is a combination of the realistically skewed earnings distribution
and the heterogeneity in effective returns on wealth because of the two-asset structure:a fraction of households end up spending a long time in high earnings states, holdhigh-return illiquid assets, and accumulate a lot of wealth. These households populate
the upper tail of the wealth distribution.37
4.2.4

Remaining Model Parameters

Demographics We set the quarterly death rate ζ to 1/180 so that the average
lifespan of a household is 45 years.
35

We define hand-to-mouth households in the model as those with zero liquid wealth. The targets of 10% and 20% are chosen to replicate the fraction of households with net liquid wealth∈ [−$1, 000, $1, 000] with zero and positive illiquid assets, respectively. These targets are similar
to estimates in Kaplan et al. (2014). The target of 15% of households with negative liquid wealthreproduces the fraction of households with net liquid wealth < −$1, 000 in the data.
36
Besides matching the share of hand-to-mouth agents in the population, the model also does wellwith respect to their relative importance in terms of consumption share (20 pct vs. 25 pct in thePSID)) and wealth share (2.5 pct vs. 4.4 pct in the SCF).
37
To put this result in the context of the literature, see the survey by Benhabib and Bisin (2016) on
the sources of skewed wealth distributions in macroeconomic models. Note that our model is not able
to match the extreme right tail of the liquid wealth distribution and also does not feature a Pareto
tail for the distribution of total wealth as arguably observed in the data. It is notoriously challenging
to match the extreme right tail of wealth distributions with labor income risk alone and our model isno exception.

29

(a) Liquid wealth distribution(b) Illiquid wealth distribution

Figure 1: Distributions of liquid and illiquid wealth.
Preferences Households have instantaneous utility that is separable over consumption and hours worked:u (c, ℓ) =

c1−γ
ℓ1+ν
−ϕ
1−γ
1+ν

(35)

with γ ≥ 0 and ν ≥ 0. We set 1/γ, the intertemporal elasticity of substitution (IES),
to one and 1/ν, the Frisch elasticity of labor supply at the household level, to one.38
The weight on the labor supply component of utility, ϕ, is set so that average hours
worked are equal to 1/3 of the time endowment in steady-state.
Production The elasticity of substitution for final goods producers is set to ε = 10,
implying a steady state markup 1/(ε − 1) of 11%. Intermediate goods producers have aweight on capital of α = 0.33, which yields a capital share of 30%, a labor share of 60%,
and a profit share of 10%. We set the constant θ in the price adjustment cost function
to 100, so that the slope of the Phillips curve in (19) is ε/θ = 0.1.39 The fraction ω ofaggregate profits reinvested into the illiquid accounts is set to α, as explained in Section
4.1. This choice which, as explained, neutralizes the countercyclicality of mark ups,
happens to be also roughly in line with the data. In 2004, the sum of undistributed
corporate profits (the empirical counterpart of profits reinvested in the illiquid account38

This number is slightly higher than the 0.82 identified by Chetty et al. (2011) as the representative
estimate from existing studies of the micro elasticity at the individual level, accounting for intensive
and extensive margins of adjustment. At the household level though, the marginally attached worker isoften the wife and a Frisch labor supply elasticity of one is in line with the estimates of Blundell et al.
(2016) for married women.
39
See e.g. Schorfheide (2008) who survey many studies using the labor share as a proxy to measuremarginal costs, an approach suggested by Gali and Gertler (1999).

30

in the model) and dividend income (the counterpart of profits paid to households) was$946B. Of these, undistributed profits amounted to $384B, thus about 40 pct of thetotal.40
Government policy We set proportional labor income tax rate τ to 0.30 and the
lump-sum transfer T to be 6% of output (equivalent to around $7, 000 per year). In
steady state just over 9% of households receive a net transfer from the government.
In our model, the government is the only provider of liquid assets. Given our calibration of household liquid holdings, government debt amounts to 23.3% of annual GDP.
Government expenditures are then determined residually from the government budget
constraint (24).
Monetary Policy We set the Taylor rule coefficient φ to 1.25, which is in the middleof the range commonly used for New Keynesian models.
Table D.2 in the Appendix summarizes our parameter values. In Section 5.2, weverify the robustness of our results with a series of sensitivity analyses.

4.3

Micro Consumption Behavior

How successful is the calibrated model at generating empirically realistic distributions of household responses to changes in labor income? Some of the most convincing empirical evidence on marginal propensities to consume (MPCs) comes fromhousehold consumption responses to the tax rebates of 2001 and fiscal stimulus payments of 2008 (see e.g. Johnson et al., 2006; Parker et al., 2013; Misra and Surico,
2014; Broda and Parker, 2014). While the estimates are often imprecise because of thesmall sample size, this collective quasi-experimental evidence concludes that households spend approximately 15-25 percent of these payments (which average between$500 and $1, 000 depending on the episode) on nondurables in the quarter that they
are received.
Let MPCxτ (a, b, z) be the MPC over a period of length τ quarters out of one-time
inflow of x additional dollars of liquid wealth. This is the notion of an MPC thatis comparable to the empirical evidence cited above (as opposed to the slope of theconsumption function with respect to liquid wealth). In Appendix B.3 we state theformal definition and show how to compute it directly from households’ consumption
policy functions using the Feynman-Kac formula.
40

See NIPA Tables 2.1 and 5.1. Also over the period 1990-2016, for example, this fraction fluctuatesaround 40 pct.

31

0.5
One quarter
Two quarters
One year

0.45
0.4

0.3
0.25

0.35

0.2

0.3

0.15
0.1

0.25

0.05

0.2

0
400

0.15

300

0.1

20
10

200

0.05

0

100
0

0
0

200

(a)

R

400

600

800

-10

1000

(b) MPC$500(a, b, z)
1

MPCxτ (a, b, z)dµ by τ, x

Figure 2: MPC Heterogeneity

The average quarterly MPC out of a $500 transfer is 16% in the model, which iswithin the range of typical empirical estimates, and an order of magnitude larger thanits counterpart in one-asset incomplete-market models (Kaplan and Violante, 2014).
As seen in Figure 2(a) the fraction consumed decreases with the size of the transfer,and increases sharply as the horizon increases.
The average MPCs in Figure 2(a) mask important heterogeneity across the population. This heterogeneity can be seen in Figure 2(b), which plots the function
MPCxτ (a, b, z) for a x = $500 payment over one quarter as a function of liquid andilliquid assets, averaged across labor productivity z. The figure illustrates the strong
source of bi-modality in the distribution of consumption responses in the population.
In the model, the average response of 16% is composed of a group of households withpositive net liquid wealth and very low consumption responses, and another groupof hand-to-mouth households with no liquid wealth who display strong consumption
responses. Of these hand-to-mouth households, roughly two-third are have positive
illiquid wealth.
Several recent empirical papers have documented patterns of the distribution of
MPCs that are consistent with Figure 2. Broda and Parker (2014) find much stronger
consumption responses to the 2008 fiscal stimulus payments among households withlow easily accessible liquid funds. Misra and Surico (2014) use quantile regression
techniques to study the consumption responses in the tax rebate episodes of 2001 and2008 and document the presence of high-income households both in the low MPC andthe high MPC group, a fact consistent with the presence of wealthy HtM households.
Kaplan et al. (2014) use the method proposed by Blundell et al. (2016) to estimate the32consumption response to transitory income shocks on PSID data for three groups —nonhand-to-mouth (HtM), poor HtM and wealthy HtM— and uncover much higher MPCs
for both types of HtM households. Baker (2016) finds that households with similar net
asset positions behave differently with respect to income shocks if they hold varying
shares of liquid and illiquid wealth. Finally, Fagereng et al. (2016) examine MPCs outof lottery prizes using Norwegian administrative data. They find that MPCs vary withthe amount of households’ liquid assets, and that households with close to zero liquid
assets have high MPCs even if they are wealthy in terms of their illiquid asset positions.
This striking heterogeneity in MPCs underlines the importance of obtaining a realistic distribution of both wealth components. With such distributions in hand, we
now turn to the monetary transmission mechanism.

5

Monetary Transmission: Quantitative Results

Our main results concern the response of the economy to a one-time unexpected monetary shock. We consider an experiment in which at time t = 0, there is a quarterly innovation to the Taylor rule (91) of ǫ0 = −0.25% (i.e. −1% annually) that mean-reverts
at rate η, i.e. ǫt = e−ηt ǫ0 . We set η = 0.5, corresponding to a quarterly autocorrelationof e−η = 0.61, a value consistent with the VAR-based empirical evidence, as argued in
Section 2.1.
In our baseline specification, lump-sum transfers Tt adjust so as to keep the budgetbalanced, with government consumption and debt fixed at their steady-state level, as
we did in Section 2.2. In Section 5.2 we provide results under alternative assumptions,
including allowing government expenditure or government debt to adjust in the wakeof an unexpected shock.

5.1

Impulse Response to a Monetary Shock

Figure 3(a) displays the exogenous time path for the innovation ǫ and the impliedchanges in the liquid interest rate and rate of inflation. Figure 3(b) displays the corresponding impulse responses for aggregate quantities.
In response to an expansionary monetary policy shock, the real return on liquid
assets rtb falls, which stimulates consumption and investment, and leads to an increase inboth output and inflation. The magnitudes of these responses are, at least qualitatively,
consistent with empirical evidence from VARs: consumption increases by less than
output and by much less than investment, with an elasticity to the change in r b overthe first year after the shock equal to -2.9.41
41

See e.g. Figure 1 in Christiano et al. (2005). Our model cannot generate hump-shaped impulse

33

1.5

2.5

1

2

0.5

1.5

0

1

-0.5

0.5

-1

0

-1.5

0

5

10

15

20

(a) Monetary Shock, Interest Rate, Inflation

-0.5

0

5

10

15

20

(b) Aggregate Quantities

Figure 3: Impulse responses to a monetary policy shock (a surprise, mean-reverting
innovation to the Taylor Rule)
How does this magnitude compare to the corresponding response in the RANK
models analyzed in Section 2.1? Table 1 shows that, across RANK models, the totalelasticity is always around −2. Thus, in our baseline specification of HANK (withlump-sum transfers adjusting) the elasticity is almost 50 pct higher than in RANK.
Notably, this discrepancy implies that the ‘as if’ result of Werning (2015) does not holdin our framework.42
In the next section, we decompose this total effect of the monetary shock on aggregate consumption into direct and indirect components through the lens of our methodology developed in Section 3.3.

5.2

The Size of Direct and Indirect Effects

The equilibrium time paths for prices and government transfers induced by the monetary shock that we feed into the household problem to compute each element of thedirect and indirect effects are displayed in Figure 4(a), alongside the resulting decomposition in Figure 4(b). In the bottom panel of Table 6 we explicitly report thecontribution of each component to the overall consumption response over the first year
following the shock.43
responses since we abstract from the modeling ingredients in typical medium-scale DSGE models thatgenerate these dynamics, such as external habits and investment-rate adjustment costs.
42
Werning (2015) studies deviations from his benchmark incomplete markets economy and argues
that, in plausible cases, consumption becomes more sensitive than in RANK to current and futureinterest rate changes, as we find.
43
In principle, the contribution of the components need not add to 100%, since the exact decomposition holds only for infinitesimal changes in prices, as in Proposition 1 for the stylized model of
Section 2. In practice, though, they almost exactly do.

34

5

0.5

4

0.4

3

0.3

2

0.2

1

0.1

0

0

-1

0

5

10

15

-0.1

20

(a) Prices

0

5

10

15

20

(b) Consumption Decomposition

Figure 4: Direct and Indirect Effects of Monetary Policy in HANK
Notes: Returns are shown as annual percentage point deviations from steady state. Real wage andlump sum transfers are shown as log deviations from steady state.

The decomposition reveals the first novel quantitative insight into the monetarytransmission mechanism. The combined indirect effects are much larger than the direct
effect. In our HANK model, the indirect component account for 80 percent of theconsumption response while the direct component accounts for only 20 percent of the
response. This is in stark contrast to typical RANK models, as argued in Section 2.
This finding is very robust, as evident from the remaining columns of Table 6 thatreport analogous results from alternative model specifications. In the baseline model weallocate a fraction ω = α of profits to illiquid equity, in order to neutralize the effectof counter-cyclical profits on investment as explained in Section 4.1. Columns (2)and (3) show that this assumption is important for generating pro-cyclical investment
and a positive output response. When all profits are allocated to equity in the illiquid
account (ω = 1), the fall in profits following the monetary shock substantially dampens
the response of investment and thus reduces the total consumption response (line 4 in
the table) because with lower investment there is a smaller increase in labor demand,and hence in wages. However, the decomposition between direct and indirect effects isbarely affected. When profits are nearly all paid as dividends to households (ω = 0.1),
consumption responds very aggressively but the decomposition between direct andindirect effects is virtually unchanged.44
Columns (4) and (5) of Table 6 show that the two key parameters that determine
the strength of the New Keynesian elements in the model – the Taylor rule coefficient44

Within indirect effects, the role of the illiquid return component is now much more important.
We come back to the reasons in Section 5.3.2.

35

Baseline
(1)

ω=1
(2)

ω = 0.1(3)εθ = 0.2(4)

φ = 2.0(5)1
ν = 0.5

Change in r b (pp)-0.28%-0.34%-0.16%-0.21%-0.14%-0.25%

Elasticity of Y
Elasticity of I

-3.96
-9.43

-0.13
7.83

-24.9
-105

-4.11
-9.47

-3.94
-9.72

-4.30
-9.79

Elasticity of C
Partial Eq. Elast. of C

-2.93
-0.55

-2.06
-0.45

-6.50
-0.99

-2.96
-0.57

-3.00
-0.59

-2.87
-0.62

22%
56%
38%
-16%

15%
51%
19%
15%

19%
51%
31%
-2%

20%
51%
31%
-2%

22%
38%
45%
-4%

Component of change in C due to:
Direct effect: r b
19%
Indirect effect: w
51%
Indirect effect: T
32%
Indirect effect: r a & q
-2%

(6)

Table 6: Decomposition of the effect of monetary shock on aggregate consumption
Notes: Average responses over the first year. Column (1) is the baseline specification. In column (2)profits are all reinvested into the illiquid account. In column (3) 10% of profits are reinvested in theilliquid account and 90% of profits are paid as a labor subsidy into the liquid account. In column (4)we reduce the stickiness of prices by lowering the cost of price adjustment θ. In column (5) weincrease φ, which governs the responsiveness of the monetary policy rule to inflation. In column (6)we lower the Frisch elasticity of labor supply from 1 to 0.5.

φ and the degree of price stickiness θ – do not substantially affect either the overall sizeof the consumption elasticity or its decomposition between direct and indirect effects.
Rather, these elements primarily affect the inflation response for a given monetaryshock, and hence the extent of movements in the real interest rate.45
In column (6) of Table 6, we set the Frisch elasticity to 0.5, half its baseline value.
The decomposition between direct and indirect effects is unaffected by this change.
The effect of lowering the Frisch elasticity is to shift the composition of the indirect
effects away from the wage component towards the transfer component.
In a previous version of this paper (Kaplan et al., 2016a), we showed that replacing
the separable preference specification with GHH utility (Greenwood et al., 1988) yields
an elasticity of aggregate consumption C to r b that is almost twice as large as in
the baseline.46 Moreover, the indirect general equilibrium effects account for over 90
percent of the total effect. The key feature of GHH utility that drives these results
45

Table 6 reports results from a more aggressive monetary policy rule and lower price stickiness; aless aggressive policy rule or higher price stickiness have similarly sized opposite effects.
1+ν
46
The intra-period utility function is specified as log(c − ψ h1+ν ), with ν = 1 and ψ set to replicate
average hours equal to 1/3 of time endowment.

36

(a) Elasticity with respect to rb(b) Consumption Change: Indirect and direct

Figure 5: Consumption Responses by Liquid Wealth Positionis the strong complementarity between hours worked and consumption. As aggregate
demand, and hence the wage rate, increases, households increase their labor supply.
Because of the complementarity, this leads to an increase in desired consumption for
all households, even for non hand-to-mouth households with low marginal propensities
to consume.
In Tables E.3 and E.4 in Appendix E we report results from a comprehensive
robustness analysis of the key parameters that govern behavior in the “heterogeneous
agent block” of the model. We show that changes in the tightness of borrowing limits,
the cost of borrowing and the adjustment cost function can have large effects on thelevel of liquid wealth holdings and the fraction of poor hand-to-mouth and wealthhand-to-mouth households. However, in all cases the decompositions of the monetary
policy shocks into direct and indirect effects remain essentially unchanged from thefindings reported in Table 6.
These experiments (including those done under alternative fiscal adjustment that
we discuss below) reveal another, related, robust feature of HANK models. The partialequilibrium (or direct) elasticity of aggregate consumption —the consumption responseto changes in the liquid rate, keeping all other prices and taxes/transfers unchanged—
never deviates too much from 0.55, its baseline value, even across configurations where
the total elasticity differs greatly.47 This magnitude is considerably lower than in all
the versions of RANK, where the direct elasticity is always above 1.9 (see Table 1).
47

We here purposely exclude the results in column (3) of Table 6 for ω = 0.1 where the aggregate
partial equilibrium elasticity is 0.99 because these are from an extremely unrealistic parameterization
with an investment elasticity of 105 and an output elasticity of 25.

37

5.3

The Distribution of the Monetary Transmission

To better understand why the direct effects of an unexpected reduction in interest
rates are small, and the indirect effects are large, in HANK relative to RANK, it isinstructive to inspect the consumption response to the monetary policy shock acrossthe entire distribution of liquid wealth holdings.
Figure 5(a) shows the elasticity of average consumption of households with a given
liquid wealth level to the change in the interest rate at each point in the liquid wealthdistribution (black line, left axis), along with the corresponding consumption shares ofeach liquid wealth type (light blue histogram, right axis).48 The distribution of consumption responses features big spikes at the borrowing constraint b = b and at b = 0.
Our model features few households at the borrowing limit, but those with zero liquid
wealth account for 20 percent of total consumption and have an elasticity of around
6. Because many of these households have moderate income and own illiquid assets,
i.e. they are wealthy hand-to-mouth, their consumption share is much larger than inmodels where hand-to-mouth are income- and wealth-poor. All other households withpositive liquid assets, representing around 80 percent of total consumption expenditures, contribute an elasticity that is around 2.0. A back of the envelope calculation
yields 0.2 × 6 + 0.8 × 2 = 2.8, which is roughly the overall impact elasticity.
Figure 5(b) separates the total elasticity into the direct and indirect elasticities.
These two additive components measure the strength of the direct and general-equilibrium
channels of monetary policy. We now examine each of them separately.
5.3.1

Why are Direct Effects Small?

Figure 5(b) reveals that the direct effects are highest for households close to the borrowing constraint, then decline to zero for households with no liquid wealth. As liquid
wealth grows, the direct effects also increase until the direct elasticity peaks just below
a value of two. After a sufficiently high level of liquid wealth, direct effects start toslowly decline. Most of the population and consumption distribution is between 0 and$20,000 of liquid wealth, and in that range the direct elasticity is quite small. This
explains why the aggregate partial-equilibrium elasticity is only about 0.55 (see column(1) of Table 6).
48

Note that the figure reports the elasticity of consumption on impact of the monetary policy shock,in contrast to the numbers in Table 6 which report elasticities over the first year. This is because theimpact elasticities are considerably easier to compute. Integrating the elasticities in the figure weighted
by the consumption shares yields (the negative of) the overall impact elasticity to the monetary shock,which is R−2.81. The average consumption of households with a given liquid wealth
R ∞ level b is defined as
Ct (b) = ct (a, b, z)µt (da, b, dz) so that aggregate consumption satisfies Ct = b Ct (b)db. Therefore
the overall elasticity is a consumption weighted average of the elasticities at each level of liquid wealth.

38

To better understand these patterns, we further decompose the direct elasticity
into a substitution effect and an income effect.49 Our decomposition is conceptually
identical to the one in Auclert (2016), but we build on recent work by Olivi (2017)
who substantially generalizes Auclert’s approach to allow for persistent price changes
and a more general stochastic process for idiosyncratic risk. We here only briefly lay
out the relevant result. The derivations as well as some additional results can be foundin Kaplan et al. (2017). The time-zero consumption response d log c0 (a, b, z) to smallinterest rate deviations {drtb }t≥0 of an individual with asset portfolio (a, b) and labor
income z can be decomposed into substitution and income effects as:
1
d log c0 (a, b, z) = − E0
γ
|

Z ∞
0

R
− 0t ̺s dse



Mt drtb dt{z

substitution effect

+ E0
} |

Z ∞
0

R
− 0t ̺s ds

e

Mt
{z



∂b ct
ct





bt drtbdt

income effectRu′ (ct ) − 0t (ρ−rsb )ds
where ̺t := ∂b ct + (1 + χd (dt , at ))∂b dt − ∂a dt and Mt := u′ (c0 ) e

}

(36)

where the expectations are over sample paths of (at , bt , zt ) starting from (a0 , b0 , z0 ) =(a, b, z), and where ∂b ct is short-hand notation for ∂b ct (at , bt , zt ), i.e. the (instantaneous)
marginal propensity to consume (MPC) out of liquid wealth, and similarly for ∂b dt and∂a dt .
The first term in (36) is the substitution effect, which is negative and scales with
the intertemporal elasticity of substitution (IES) 1/γ. The second term is the income
effect, which depends on the time paths of both liquid wealth bt and the (instantaneous)
MPC ∂b ct . To better understand (36) it is useful to consider the corresponding formulain a one-asset model. In this case the formula remains unchanged, except that deposits
are zero (dt ≡ 0) so that the effective discount rate simplifies to ̺t = ∂b ct . Because∂b ct is always weakly higher than ρ, the MPC in RANK, the substitution effect is adampened version of its counterpart in RANK.50
The first term in (36) is therefore a natural dynamic generalization of Auclert’s
expression for the substitution effect in response to a transitory one-period interest
rate change which he writes as −IES × (1 − MPC). It also captures in a transparent
fashion the intuition of McKay et al. (2016) that a high likelihood of being constrainedin the future is equivalent to a shorter planning horizon.
49

Or more precisely the combination of different types of income effects, in particular a classic
income effect and a wealth/endowment effect.
R∞
50
Specifically, in RANK the substitution effect can be written as − γ1 0 e−ρt E0 (Mt )drtb dt. In the
i
Rt
Rt
R∞h
one asset model, it can be written as − γ1 0 E0 (e− 0 ∂b cs ds )E0 (Mt ) + cov(e− 0 ∂b cs ds , Mt ) drtb dt.
The substitution effect is higher in RANK because ∂b ct ≥ ρ and the covariance term is negative by
concavity of the consumption function.

39

(a) Breakdown of direct effect(b) Breakdown of indirect effectFigure 6: Consumption Responses by Liquid Wealth Position
In our two-asset model, the effective discount rate ̺t differs from that in a one-asset
model by the term (1 + χd (dt , at ))∂b dt − ∂a dt . In our computations this term is always
strictly positive and hence the effective discount rate is strictly larger than that in a
one-asset model, which further dampens the intertemporal substitution effect. Thisadditional effect is linked to portfolio rebalancing between liquid and illiquid asset.
In a one-asset model a fall in the liquid rate is an incentive for households to reducesavings and consume more. In contrast, in a two-asset model, households also have theoption to shift funds from their liquid to illiquid accounts. If the gap between illiquidand liquid rates becomes sufficiently large, then households respond to the fall in r b byrebalancing their portfolios rather than by increasing their consumption. Intuitively,
the fact that individuals may rebalance their portfolios in the future further shortens
their effective time horizon. The symmetric logic explains how the income effect, thesecond term in term in (36) is affected by portfolio reallocation.
To summarize, in the two-asset model an interest rate change induces income andsubstitution effects, just as in a one-asset model. But the size of both effects differ
from those in a one-asset model because the effect discount rate ̺ accounts for future
portfolio rebalancing in addition to future MPCs. Under plausible calibrations, theincome and substitution effects are larger in one-asset models than in the two-asset
model both because of the absence of portfolio rebalancing effect, and because MPCs
are lower. We return to this point in Section 5.5.
Figure 6(a) implements the decomposition in (36). The solid blue line plots the
direct effect, i.e. the same line as in Figure 5(b). The Figure then breaks down
this direct effect into the substitution effect (dashed red line) and the income effect(dash-dotted green line), as defined in (36). Households with zero holdings of liquid40

wealth do not substitute intertemporally because they are at a kink in their budget
constraint. For households with positive holdings of liquid wealth, the substitution
effect gradually increases. It levels off at 1.95 for households with high holdings ofliquid wealth, which is the size of the substitution effect for a household who is fully
insured against idiosyncratic income risk, as in RANK.
The income effect is a monotonically decreasing function of liquid wealth. It is
positive for borrowers since lower interest payments on their debt translate into higher
consumption and is negative for lenders with positive liquid wealth.51 For householdswith sufficiently high holdings of liquid wealth (outside the range plotted in the graph),the income effect becomes so strong that the direct elasticity becomes negative.
As noted above, the direct response of households with positive, but moderate,
amounts of liquid wealth ($1,000 to $20,000) is small and this is what accounts formost of the small aggregate direct elasticity. Why do these low MPC households not
respond more strongly to the reduction in interest rates? The decomposition in 6(a)suggests that the reason is twofold. First, portfolio rebalancing (solid pink line) has aconsiderably negative effect exactly for households with positive but moderate amountsof liquid wealth.52 Second, although these households currently have low MPCs, they
face the possibility of having higher MPCs in the future.
Relationship to Auclert (2016). As already noted, our decomposition is closely
related to the one proposed by Auclert (2016, Theorems 1 and 3). Auclert’s Theorem
1 is analogous to our decomposition and breaks down households’ micro consumption
response into income and substitution effects. He refers to the income effect of theinterest rate change as “unhedged interest rate exposure.” Auclert’s Theorem 3 thenaggregates the micro decomposition into an aggregate decomposition as we do in equation (30). The combination of indirect effects from wages and fiscal policy can beinterpreted as his “earnings heterogeneity channel” expanded to include income fromthe government besides labor income. At the same time there are also some importantdifferences between the two decompositions. First, Auclert emphasizes heterogeneity
in asset maturities, whereas our emphasis is on asset liquidity. Second, our model does
not feature his Fisher channel because all assets in our model are real. Third, buildingon Olivi (2017), our decomposition can handle persistent dynamics of the economy.
51

Di Maggio et al. (2014), Flodén et al. (2016) study borrowers with adjustable rate mortgages who
faced changes in monthly interest payments, and find evidence of a positive consumption response to adrop in monthly payments. In addition, Cloyne et al. (2015) offer supporting evidence that the directchannel is small relative to indirect effects occurring through changes in household labor income.
52
Figure 6(a) isolates the contribution of portfolio reallocation at different points of the liquid wealthdistribution. The line plots the difference between the income and substitution effects in the figure
and the analogous quantities computed as in a one-asset model (by setting the effective discount rateequal to ̺t = ∂b ct ).

41This is important because in models like ours and Auclert’s even purely transitory
one-time shocks typically lead to endogenous persistence through movements in thewealth distribution, a case to which Auclert’s decomposition does not apply. Finally,
his decomposition relies heavily on being able to collapse period budget constraints
into a single present-value constraint: as a result, his approach cannot handle binding borrowing limits, a wedge between borrowing and savings rates or illiquid assets– features that are at the heart of our model. For further details see Kaplan et al.
(2017).
5.3.2

Why are Indirect Effects Large?

Figure 5(b) reveals that the indirect effects are very large for households with zeroliquid wealth. The presence of hand-to-mouth households is thus a key determinant of
the transmission mechanism of monetary policy on the macroeconomy.
Figure 6(b), which offers a breakdown of the indirect effect among its three components, shows that these households respond sharply to the change in both labor
income and government transfers that occur in equilibrium in the wake of a monetaryshock. The rise in labor income is a consequence of an expansionary monetary shockthat increases demand for final goods. Transfers rise because the interest payments on
government debt fall and because the rise in aggregate income increases tax revenues.
This mechanism shares similarities with TANK models with government debt where,
like in HANK, the presence of non-Ricardian households means that the fiscal response
can play an important role in the indirect effects of monetary policy.
Finally, the combined indirect effect due to changes in r a and q is slightly negative,
but very small, everywhere in the distribution. Our model inherits the typical featureof the standard New Keynesian model that markups and profits falls in a monetary
expansion. Since the stock price q is the present discounted value of future profits, q
drops as well. A sizable literature examines the response of equity prices to monetary
policy shocks and finds positive, but only weakly significant, responses of stock prices toexpansionary monetary policy shocks.53 There are a number of potential strategies forgenerating procyclical profits, and hence stock prices, in New Keynesian models thatwould also apply in HANK. Chief among these is the introduction of sticky wages.54
While capital gains on equity are always countercyclical, our model is capable ofgenerating both procyclical and countercyclical returns through their dividend com53

See Rigobon and Sack (2004), Bernanke and Kuttner (2005) and Gürkaynak et al. (2005). In
the words of Rigobon and Sack (2004) the literature has been “somewhat inconclusive about thesignificance of the response of stock prices to monetary policy actions”.
54
Indeed, this is the route taken by Challe and Giannitsarou (2014) in the context of a RANK
model.

42

ponent depending on the assumption made on the fraction of profits ω reinvested in
illiquid accounts. As shown in columns (2) and (3) of Table 6, this parameter changes
the share of the indirect effect due to the illiquid return r a , but the direct effects always account for a small fraction of the total consumption response.55 More clearcut
empirical evidence on the response of various asset prices to monetary policy shocks
as well as the design of a HANK model that is consistent with this evidence should bea priority for future research.56

5.4

The Role of the Fiscal Response to a Monetary Shock

We now discuss some important implications of Ricardian non-neutrality in HANK.
In Table 7 we report the overall response and decomposition for alternative assumptions about how the government satisfies its intertemporal budget constraint after amonetary shock.
Column (1) contains the baseline case, in which government expenditures and debt
are held constant, and transfers adjust in every instant. When, instead, government expenditures adjust, the overall impact of monetary policy on aggregate output is stronger(column (2)). This is because when transfers adjust, only high MPC households increase consumption, and by less than one-for-one with the transfer; when government
expenditures adjust, the reduced interest payments on debt translate one-for-one into
an increase in aggregate demand, which contributes directly to an increase in output.
The elasticity of private consumption is similar to the baseline, but the bulk of theindirect effects are accounted for by higher labor income rather than a combination oflabor income and transfers.
In column (3), we let the tax rate adjust. Compared to the case where transfers rise
there are offsetting forces: on the one hand, a lower tax rate expands labor supply acrossthe board, whereas the higher transfers have a small negative impact on hours worked;
on the other hand, lowering taxes is less redistributive than more generous lump-sumtransfers and, thus, spurs a smaller demand for private consumption. Overall, results
are similar to the baseline.
The remaining alternative is to let government debt absorb the majority of the
fiscal imbalance in the short run. In the economies of columns (1) and (2), a sizable
55

Some readers may argue that the the indirect effects due to illiquid returns ra should be counted
as direct effects because all effects working through changes in asset returns are intimately linked due
to arbitrage considerations. Note that even in the case of column (3) in Table 6 where the indirect
effects due to ra are large, the combined effect due to ra and rb are still only 30% of the overall effect.
56
That correctly modeling asset price movements is potentially important is also consistent with aresult in Werning (2015) who shows that the consumption response to monetary policy depends onthe cyclicality of asset prices.

43

T adjusts(1)

G adjusts(2)

τ adjusts(3)

B g adjusts(4)

Change in r b (pp)-0.28%-0.23%-0.33%-0.34%

Elasticity of Y
Elasticity of I

-3.96
-9.43

-7.74
-14.44

-3.55
-8.80

-2.17
-5.07

Elasticity of C
Partial Eq. Elast. of C

-2.93
-0.55

-2.80
-0.60

-2.75
-0.56

-1.68
-0.71

21%
81%
-2%

20%
62%
18%
0%

42%
49%
9%
0%

Component of Change in C due to:
Direct effect: r b
19%
Indirect effect: w
51%
Indirect effect: T
32%
Indirect effect: τaIndirect effect: r & q-2%

Table 7: Importance of Fiscal Response to Monetary ShockNotes: Average responses over the first year. Column (1) is the baseline specification in whichtransfers T adjust to balance the government budget constraint. In column (2) government
expenditure G adjusts, and in column (3) the labor income tax τ adjusts. In column (4) government
debt adjusts, as described in main text.

fraction of the overall effect of monetary policy is due to additional government transfers
or expenditures from reduced debt payments. Without this additional stimulus toaggregate demand, labor income does not increase as much and indirect effects accountfor a smaller share of the total (60 pct compared to 80 pct in the other two scenarios,
but still an order of magnitude larger than in RANK). As a result, when government
debt absorbs the slack, the monetary shock has a much smaller impact on the economy,roughly half of the baseline value.57
To sum up, our second quantitative insight into the transmission mechanism ofmonetary policy is that the type of fiscal adjustment following the shock matters for theeffectiveness of monetary policy. This result represents another important deviationfrom RANK and from versions of the HANK model, such as the one developed by
Werning (2015), where the overall effectiveness of monetary policy does not depend on
liquidity constraints and incomplete markets.
57
In this experiment, we assume that lump-sum transfers jump by a very small amount on impact
and then decay back to their steady state level at a slow exogenous rate. Given the assumed rate ofdecay, the initial jump is chosen so that the government’s budget constraint holds in present value
terms. In Column (4) of Table 7 the transfer decays at a quarterly rate of 0.02. We experimented
with smaller and bigger decay rates and our main conclusions are unchanged.

44

5.5

The Role of Two Assets and Micro Heterogeneity

At this stage of our analysis, two questions naturally arise: (i) What do we gain fromthe two-asset version of HANK relative to the one-asset versions that have been studied
in the existing literature? and (ii) What do we gain from a realistic model of household
heterogeneity relative to the simpler spender-saver structure of TANK models?
5.5.1

Two-Asset vs. One-asset HANK Models

In this section we compare our model to a one-asset HANK model. We choose a version
of HANK as in McKay et al. (2016) in which all wealth is held as liquid government
bonds, and we let transfers adjust to balance the government budget constraint following the monetary shock.58
Recall that in our calibrated two-asset HANK model the wealth-to-output ratio wasover 3 (Table 2) and the average quarterly MPC out of $500 was 0.16 (Figure 2). In
one-asset HANK models, however, there is a well-known tension between matching thehigh observed aggregate wealth-to-output ratios and generating a large average MPC.
Figure 7(a), which plots aggregate wealth and the average quarterly MPC out of $500
in the one-asset HANK model for values of the discount rates ρ between 2.5% p.a.
to 7.5% p.a., illustrates this tension: the one-asset model can generate high average
wealth, or a high MPC, but not both simultaneously.59
Notwithstanding this failure of the one-asset model, Figure 7(b) plots the direct
and total elasticities of aggregate consumption following a monetary policy shock (theanalogues for the two-asset model are in Table 6). For low discount rates, in which
there is a large amount of liquid wealth in the economy, the direct elasticity becomes
negative because of the strong wealth effects that pull down the direct channel. For
high discount rates, the direct elasticity becomes larger but is always a small share of
the overall elasticity. This is because even though wealth effects are now modest due
to the smaller amount of wealth in the economy, there is a larger fraction of hand-tomouth households and so the intertemporal substitution channel is muted. The totalelasticity is hump-shaped with respect to ρ because of two offsetting forces: as thediscount rate increases, MPCs rise (panel (a)) resulting in larger indirect effects; at thesame time, the amount of liquid wealth (all of which is government debt) decreases,
58

The model can be thought of as the limit of our two-asset model as the capital share α goes tozero, implying that the marginal product of capital is zero. Because we assume that a fraction ω = α
of firm profits are reinvested directly into illiquid assets (see Section section 4.1), this also implies thatall profits are paid out in liquid form. As a result, the illiquid return rta goes to zero and all wealth isheld as liquid assets.
59
The rest of the calibration is identical to the one in our baseline model, except for α = ω = 0.

45

4

0.2

4

3

0.15

2

0.1

3

2

1
1

0.05

0

0

-1

0
2

3

4

5

6

2

7

(a) Average MPC and Wealth-to-GDP Ratio

3

4

5

6

7

(b) Total and Direct Effects

Figure 7: Key Features of One-Asset Model for Different Calibrations
resulting in a weaker fiscal response to the monetary shock, and hence smaller indirect
effects.
Perhaps surprisingly, one calibration of the one-asset model replicates many featuresof our two-asset model closely: with a discount rate of 7%, the average MPC is 0.16,
the direct elasticity is just above 0.5, and the overall elasticity is just below 3. The
one moment of the data that this calibration misses completely is the total amount ofwealth in the economy (0.25 versus 3). But if one interprets wealth as liquid wealth
only, this calibration performs well in that dimension (a value of 0.25 just as in Table2). This calibration can therefore be thought of as the “liquid-wealth-only calibration,”
advocated by Carroll et al. (2014).
This result then raises the question of what is to be gained from studying monetary
policy through the lens of a two-asset HANK model, rather than a one-asset HANK
model calibrated only to liquid wealth? The answer is that this latter model completely
abstracts from capital, and the responses of quantity and price of capital greatly matterfor the monetary transmission, through the indirect channel. To illustrate this point,columns (2) and (3) of Table 6 report results of our experiments in the two-asset model
for alternative values of the parameter ω that disciplines what fraction of firm profits
get reinvested in illiquid assets (our baseline results in column (1) are for ω = α =1/3). When ω = 0.1, strong procyclical movements in the illiquid rate of return r a ,stock price q and investment I result in the total consumption response more thandoubling relative to the baseline. Instead, when ω = 1, the illiquid price falls sharply
in the wake of a monetary expansion and the total consumption elasticity shrinks to
2/3 of the baseline. While these two alternative calibrations are counterfactual insome dimensions (in particular, with regard to the extreme investment responses),46

they illustrate qualitatively an important point: the effects of monetary policy depend
strongly on how investment and equity returns move in equilibrium. It is hard to see
how the liquid-wealth-only calibration of the one-asset model could ever accommodate
these effects.
5.5.2

Micro-Heterogeneity vs. Spender-Saver structure

In this section, we compare our model to the TANK models analyzed in Section 2.2.
The total consumption elasticity in TANK models with wealth in positive supply(columns (6) and (7) of Table 1) is somewhat smaller than that in HANK, and theshare of direct effects is roughly three times larger than in HANK. As a result, thedirect elasticity of aggregate consumption in TANK is always around 1.38, which is
two and a half times higher than in HANK.
There are three main reasons that account for this discrepancy between the simple
TANK model without wealth (column (5) in Table 1) and HANK. The first is a dampening of the substitution effect, even for low MPC households, because the prospectof having a high MPC in the future shortens their planning horizon. The second isincome effects. The third is portfolio rebalancing. The fact that this direct elasticity is
the same also in the version with wealth (where income effects are present, see column(6)) and in the two-asset version (where portfolio rebalancing is present, see column(7)) suggests that the last two forces are small in TANK. In particular, income effects
are negligible in TANK because the MPC of the savers is very small (equal to ρ, thediscount rate).
We conclude that the key reason for why direct effects are small in HANK is the one
associated with occasional periods of high MPC behavior, arising from the interaction
between the two-asset structure and the wedge between the interest rates on liquidborrowing and liquid savings. This result underscores the importance of modelingheterogeneity through uninsurable earnings shocks as opposed to via built-in differencesin preferences.

6

Monetary Policy Tradeoffs in HANK

We have thus far emphasized two main results. First, in our HANK model the indirect
effects of monetary policy on aggregate consumption far outweigh the direct effects thatare dominant in RANK models. Second, the overall response of aggregate consumption
to a cut in interest rates may be larger or smaller than in RANK models, depending
on a number of factors that are neutral in RANK, in particular the fiscal reaction to
the monetary expansion.
47

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2
0.4

0.6

0.8

1

0.4

0.6

0.8

1

(b) B g adjusts(a) T adjusts

Figure 8: Cumulative Elasticity of Aggregate Consumption by Persistence of the Shock
We now highlight two implications of these differences for some key tradeoffs policymakers face in the conduct of monetary policy. First, we study the choice between
sharper but more transitory versus smaller but more persistent interest rate cuts. Second, we analyze the most classical tradeoff of monetary policy: the one between inflation and real activity.

6.1

Tradeoff between Size and Persistence of Monetary Shocks

Our aim is to compare, within RANK and HANK models, a transitory drop in theinterest rate of a given size and persistence with a smaller but more persistent drop.
Recall, from Section 2, that the aggregate Euler equation in RANK implies:


Z
1 ∞
C0 = C̄ exp −(rs − ρ)ds .
γ 0

(37)

The integral, which we thereafter denote as R0 , is the cumulative deviation of the realinterest rate from the natural rate ρ. The elasticity of aggregate consumption at impact
with respect to R0 , −d log C0 /dR0 , is always equal to 1/γ and equal to one under ourcalibrated value for the IES. Crucially, this cumulative elasticity is independent of theparticular path of the real rate. More or less persistent paths with the same cumulative
deviation R0 have the same impact on aggregate consumption. Put differently, RANK
models feature a neutrality property with respect to the timing of monetary policy anddo not feature a size-persistence tradeoff.
This neutrality property of RANK models does not hold in HANK. Figure 8(a)plots the cumulative elasticity of aggregate consumption at impact with respect todifferent values for the persistence of the innovation e−η for our baseline fiscal policy

48

scenario in which transfers adjust. Intuitively, persistence is irrelevant for the non
hand-to-mouth households, as in RANK, but it does affect the response of hand-tomouth households. When shocks are persistent, a large portion of the interest rate cut,and the associated relaxation of the government budget constraint, occurs in the future.
Hence the hand-to-mouth households receive a smaller increase in transfers upon theimpact of the shock, and so their consumption response is weaker. As a result, thecumulative elasticity – which is invariant to persistence in RANK – declines sharply
with persistence in HANK. The failure of Ricardian equivalence implies that not only
the timing of fiscal policy matters but also that of monetary policy. For comparison,
we also plot the cumulative elasticity −d log C0 /dR0 for the simple TANK model of
Section 2.2. As in HANK the timing of monetary policy matters. This is again due tothe failure of Ricardian equivalence. However, the difference is much smaller and, in
contrast to HANK, the consumption response in TANK is always weakly larger thanthat in RANK.
Figure 8(b) repeats the exercise for the case where government debt adjusts. As
explained in the context of Table 7, in this case the consumption response in HANK
is considerably diminished because of the lack of transfers accruing to hand-to-mouthhouseholds. Even in the absence of transfers, highly persistent interest rate cuts are
considerably less potent than sharp but transitory ones. The reasons are related toour discussion of why the direct effects are so much smaller in HANK, compared to
RANK (Section 5.3.1). First, households recognize that their MPCS may be larger inthe future, which effectively reduces the horizon over which the substitution effects is
active. Second, portfolio reallocation is more potent in response to persistent changes
in the relative returns (because more durable changes in returns justify paying thetransaction cost).
Summarizing, in RANK models, transitory and persistent interest rate cuts areequally powerful as long as the time average of the interest rate deviations is the same.
Instead, in HANK a more transitory but sharper interest rate cut is more effective at
stimulating aggregate consumption.

6.2

Inflation-Activity Tradeoff

In New Keynesian models, any desired increase in aggregate output can be achievedby an appropriate choice of the size of the monetary innovation. A relevant question
is about the cost of such monetary stimulus in terms of inflation. That is, the proper
conduct of monetary policy requires knowledge of the tradeoff between inflation andreal activity.

49

2.5

2.5

4

2

2

3

1.5

1.5

1

1

0.5

0.5

0

0

0

-0.5

-0.5

-1

-1

-1

-1.5

-1.5

-2

-2

-2.5
-1.5

-2.5
-4

2
1

-2

-1

-0.5

0

0.5

1

(a) Inflation-Output Gap

1.5

-3

-3

-2

-1

0

1

2

3

(b) Inflation-Marginal Cost

4

-4
-1.5

-1

-0.5

0

0.5

1

1.5

(c) Marginal Cost-Output

Figure 9: Analysis of Inflation-Activity Tradeoff
Figure 9 graphically examines this tradeoff in RANK and HANK.60 Panel (a) plots
the inflation-output tradeoff, panel (b) the inflation-marginal cost relationship, andpanel (c) the marginal cost-output relationship. For each model, we feed monetary
policy shocks ε0 ranging from −2% to +2% annually into the Taylor rule.
Let’s begin by comparing the T-adjust case in HANK, our baseline, with RANK.
The main result, visible form panel (a), is that the inflation-activity tradeoff is similar
between RANK and HANK. Panels (b) and (c) illustrate that the way movements
in marginal costs induced by policy shocks translate into movements in inflation andoutput is basically identical across models. The reason is that the inflation-activity
relationship is largely determined by the New Keynesian side (summarized by thePhillips curve and Taylor rule), which is the same in RANK and HANK.
However, although the slopes of inflation-activity trade-off is the same across thetwo models, the length of the lines in panel (a) differs sharply. This is a reflection of thedifferent elasticities of economic activity to the monetary shock in the two models. As
explained in Section 5, the elasticity of C in HANK under the T-adjust case is higher
than in RANK. As a consequence, the same expansionary policy shock generates more
inflation and a larger output gap in HANK.
An examination of the inflation-activity tradeoff across different types of fiscal adjustments in HANK (T-adjusts vs. B-adjusts) reveals an additional finding. As opposed to RANK, where Ricardian neutrality implies that fiscal adjustment is irrelevant,in HANK it matters for the slope of this tradeoff. Panel (a) implies that a more passive
fiscal adjustment rule, where debt absorbs the change in interest payments following
the monetary shock, is associated to a more favorable tradeoff, i.e. a flatter line. The
reason, which can be seen in panel (b), is a the different marginal cost-inflation equilibrium relationship: in the B-adjust case, changes in marginal costs are spread out
60

For this set of results, we use the richer RANK model outlined in Section 2.3 and in Appendix
A.5.

50

over a longer period of time and, as a result, inflation reacts less at impact.
Finally, note that in our version of HANK the mapping from the output gap tomarginal costs is the same as in RANK (see panel (c)). An interesting avenue for
future research is to examine whether this is true more generally. In principle, thismapping may depend on the heterogeneity in the economy, which would lead to adifferent inflation-output tradeoff (panel (a)) between HANK and RANK.

7

Conclusion

In our Heterogeneous Agent New Keynesian (HANK) framework, monetary policy affects aggregate consumption primarily through indirect effects that arise from a generalequilibrium increase in labor demand. This finding is in stark contrast to Representative Agent New Keynesian (RANK) economies, where intertemporal substitution
drives virtually all of the transmission from interest rates to consumption. Moreover,in HANK, the way that fiscal policy responds to an interest rate change profoundly
affects the overall effectiveness of monetary policy – a result that is also at odds with
the Ricardian nature of standard RANK economies.
These differences between HANK and RANK matter for the conduct of monetary
policy. First, they imply different trade-offs in the choice between a large but transitory interest rate change versus a small but persistent change. Second, although the
inflation-output gap trade-off is not too different in RANK and HANK (because the
New Keynesian block is common across the two models), in HANK the slope of thisrelationship depends on the type of fiscal adjustment to movements in the nominal
rate.
Third, and perhaps most importantly, when direct effects are dominant as in a
RANK model, for the monetary authority to boost aggregate consumption it is sufficient to influence real rates: intertemporal substitution ensures that consumption will
respond. In contrast, when this direct transmission mechanism is small, as in HANK,
the monetary authority must rely on general equilibrium feedbacks that boost household income in order to influence aggregate consumption. Reliance on these indirectchannels implies that the overall effect of monetary policy may be more difficult tofine-tune by simply manipulating the nominal rate. The precise functioning of complex institutions, such as labor and financial markets, and the degree of coordination
with the fiscal authority play an essential role in mediating the way that the monetary
impulse affects aggregate consumption.
Our model’s ability to match the cross-section of household portfolios, wealth distribution, and microeconomic consumption behavior lies at the heart of this set of results.

51

Nonetheless, the household side of the model could be improved in a number of dimensions. The model lacks a distinction between net and gross positions, which would benecessary to assess the affects of household leverage on monetary transmission. The
model also lacks a distinction between real and nominal assets, which is a consequenceof all assets in our economy being infinitely short duration. This distinction would benecessary to study re-valuation (or Fischer) effects of monetary policy. Together these
two abstractions mean that our model cannot generate a commonly observed household
portfolio: illiquid housing assets together with long-term nominal mortgage debt witheither fixed or variable nominal coupon payments. Such a balance-sheet configuration
brings additional channels of monetary transmission; recent progress in this area has
been made, for example, by Garriga et al. (2015) and Wong (2016).
There are several other open areas for the next generation of HANK models to
address. First, in our version of HANK, the price of illiquid assets (which can beinterpreted as stock and house prices) co-moves slightly negatively with a monetaryshock. The empirical evidence on this co-movement is inconclusive, but if anything
points to a positive correlation. Getting this co-movement right is important for thesize of the portfolio rebalancing behavior that mutes the intertemporal substitution
channel in HANK.
Second, we have only studied deterministic transitional dynamics of the economy
following one-time monetary shocks. The computational method recently developedby Ahn et al. (2017) will allow future HANK models to directly incorporate aggregate
fluctuations into the economic environment.
Third, we have focused on the macroeconomic effects of conventional monetary policy, i.e. shocks to the Taylor rule, in economies that are far from the zero lower boundon nominal interest rates. When the lower bound is binding, the relevant monetary
instrument switches from short term rates to forward guidance and asset purchases.
Our experiments on monetary shocks with different levels of persistence suggest thatin HANK models, forward guidance may be less effective than conventional monetary
policy, providing a possible solution to the forward guidance puzzle (Giannoni et al.,2012). In Kaplan et al. (2016b) we fully articulate this point following the lead of
McKay et al. (2016) and Werning (2015). The presence of assets with different degrees
of liquidity also makes the framework a natural one to analyze the macroeconomic
effects of large-scale asset purchases (quantitative easing).
Finally, in RANK models there is a clear pecking order between monetary andfiscal policies: in economies that are away from the zero lower bound, monetary policy
can by itself restore the first-best equilibrium allocation (what Blanchard and Galı́,
2007, have termed the “divine coincidence”). An important question that remains
52unanswered is the design of optimal policy in HANK economies where the presenceof incomplete markets and distributional concerns, in addition to nominal rigidities,
breaks such “divine coincidence”.

53

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57

Appendix: For Online PublicationA

Proofs and Additional Details for Section 2

This Appendix spells out in more detail the simple RANK and TANK models in Section
2 and proves the results stated there.

A.1

Details for Section 2.1

A representative household has preferences over utility from consumption Ct discounted
at rate ρ ≥ 0
Z ∞
C 1−γ
e−ρt U(Ct )dt,
U(C) =,
γ > 0.
(38)1−γ
0There is a representative firm that produces output using only labor according to the
production function Y = N. Both the wage and final goods price are perfectly rigid
and normalized to one. The household commits to supplying any amount of labordemanded at the prevailing wage so that its labor income equals Yt in every instant.
The household receives (pays) lump-sum government transfers (taxes) {Tt }t≥0 and canborrow and save in a riskless government bond at rate rt . Its initial bond holdings are
B0 . The household’s budget constraint in present-value form is
Z ∞R− 0t rs dse

Ct dt =

0

Z ∞

Rt

e− 0 rs ds (Yt + Tt )dt + B0 .

(39)

0

The government sets the path of taxes/transfers in a way that satisfies its budget
constraint
Z ∞ Rte− 0 rs ds Tt dt + B0 = 0.
(40)
0As described in Section 2, the monetary authority sets an exogenous time path for real
rates {rt }t≥0 .
An equilibrium in this economy is a time path for income {Yt }t≥0 such that (i)the household maximizes (38) subject to (39) taking as given {rt , Yt , Tt }t≥0 , (ii) the
government budget constraint (40) holds, and (iii) the goods market clears
Ct ({rt , Yt , Tt }t≥0 ) = Yt ,(41)

where Ct ({rt , Yt , Tt }t≥0 ) is the optimal consumption function for the household.
The overall effect of a change in the path of interest rates on consumption is determined from only two conditions. First, household optimization implies that thetime path of consumption satisfies the Euler equation Ċt /Ct = γ1 (rt − ρ). Second, by
58

assumption, consumption returns back to its steady state level Ct → C̄ = Ȳ as t → ∞.
Therefore, we have1
Ct = C̄ exp −
γ

A.2

Z ∞

(rs − ρ)ds

t



⇔

1
d log Ct = −γZ ∞drs ds.

(42)t

Proof of Proposition 1

The proof covers both the case B0 = 0 as in Proposition 1 and the case B0 > 0 as
in (7). A key virtue of the simple model we consider is that it admits a closed-form
solution for the household’s optimal consumption function.
Lemma A.2 For any time paths {rt , Yt , Tt }t≥0 , initial consumption is given by
Z ∞ R
1− 0t rs ds(Yt + Tt )dt + B0 ,e
C0 ({rt , Yt , Tt }t≥0 ) =χ0Z ∞γ−1 R t1χ=e− γ 0 rs ds− γ ρt dt.

(43)
(44)

0

The derivatives of the consumption function evaluated at (rt , Yt , Tt ) = (ρ, Ȳ , T̄ ) are:61∂C0
1= − Ȳ e−ρt + ρB0 e−ρt
∂rt
γ

∂C0
∂C0
=
= ρe−ρt .
∂Yt
∂Tt

(45)

Proof of Lemma A.2 Integrating the Euler equation forward in time, we have1
log Ct − log C0 =γ

Z t

(rs − ρ)ds

⇒

0


 Z t
1
Ct = C0 exp(rs − ρ)ds
γ 0

Substituting into the budget constraint (39):
C0

Z ∞
0

R
R
− 0t rs ds+ γ1 0t (rs −ρ)ds

e

dt =

Z ∞

Rτ

e− 0 rs ds (Yτ + Tτ )dτ + B0 ,0or, equivalently, (43) with χ defined in (44).
Next, consider the derivatives ∂C0 /∂rt , ∂C0 /∂Yt and ∂C0 /∂Tt . Differentiating C0
Rt
in (43) with respect to Yt yields ∂C0 /∂Yt = χ1 e− 0 rs ds . Evaluating at the steady state,
we have∂C0= ρe−ρt .
(46)
∂Yt
61

In our continuous-time model the interest rate rt and income Yt are functions of time. Strictly
speaking, the consumption function C0 ({rt , Yt , Tt }t≥0 ) is therefore a functional (i.e. a “functionof a function”). The derivatives ∂C0 /∂rt , ∂C0 /∂Yt and ∂C0 /∂Tt are therefore so-called functional
derivatives rather than partial derivatives.

59

The derivative with respect to Tt is clearly identical.
Next consider ∂C0 /∂rt . Write (43) as
1
Y P DV + T P DV + B0 ,
χ
Z ∞ R
Z ∞ R
τ
P DV
− 0τ rs ds
P DV
Y
=
e
Yτ dτ,
T=e− 0 rs ds Tτ dτ.
C0 =

0

(47)

0

We have
∂C0
1
=
∂rt
χ





∂Y P DV∂T P DV
+
∂rt
∂rt


1 ∂χ
Y P DV + T P DV + B0 .
2
χ ∂rt

−

(48)

We calculate the different components in turn. From (47)∂
∂Y P DV
=
∂rt
∂rt

Z ∞

R
− 0τ rs dse0∂Yτ dτ =
∂rt

Z ∞

Rτ

e− 0 rs ds Yτ dτ

(49)

t

Rτ

where we used that e− 0 rs ds Yτ does not depend on rt for τ < t. Next, note that for
τ >t
Z τ
Rτ
Rτ
∂ − R τ rs ds
∂
r
ds
−
s
= −e 0e 0
rs ds = −e− 0 rs ds
∂rt
∂rt 0
R
τ
where the second equality uses ∂r∂ t 0 rs ds = 1 for t < τ . Substituting into (49), we
have
Z ∞ R
τ
∂Y P DV=−e− 0 rs ds Yτ dτ.
∂rtt
Similarly

and∂T P DV
=−
∂rt
∂χ
∂
=
∂rt
∂rt

Z ∞
t

− γ−1
γ

e

Rτ

1
0 rs ds− γ ρτ

Z ∞

Rτ

e− 0 rs ds Tτ dτ,

(50)

t

γ−1
dτ = −γZ ∞γ−1 R τ1
0 rs ds− γ ρτ

e− γ

dτ.

t

Plugging these into (48)
∂C0
1
=−
∂rt
χ

Z ∞
t

R
− 0τ rs dse

1 γ −1(Yτ +Tτ )dτ + 2
χ γ

Z ∞
t

γ−1 R τ1
0 rs ds− γ ρτe− γdτ Y P DV + T P DV + B0 .

Evaluating at the steady state and using χ̄ = 1/ρ, Y P DV = Ȳ /ρ, T P DV = T̄ /ρ andR ∞ −ρτe dτ = e−ρt /ρ:t∂C0
γ − 1 −ρt= −(Ȳ + T̄ )e−ρt +eȲ + T̄ + ρB0 .
∂rt
γ
60

(51)

The government budget constraint is T P DV +B0 = 0, so that in steady state T̄ = −ρB0
and hence (51) reduces to the expression in (45).
Conclusion of Proof Plugging (45) into (6), we have

Z ∞
Z ∞
Z ∞
1
−ρt
−ρt
dC0 = − Ȳ + ρB0e drt dt + ρ
e dYt dt + ρ
e−ρt dTt dt.
γ
0
0
0

(52)

It remains to characterize dYt and dTt and to plug in. First, from (42) in equilibriumZ ∞1
d log Yt = −γ

drs ds.

(53)t

Next, totally differentiate the government budget constraint
Z ∞
0

∂
∂rt

Z ∞

R
− 0τ rs ds

e

0

Tτ dτ



drt dt +Z ∞Rτ

e− 0 rs ds dTτ dτ = 0.

0

R∞
R∞
Using (50) and evaluating at the steady state − 1ρ 0 T̄ e−ρt drt dt+ 0 e−ρt dTτ dτ . Usingthat T̄ = −ρB0 ,
Z
Z
∞

∞

e−ρt dTτ dτ = −B00

e−ρt drt dt

(54)

0

Plugging (53) and (54) into (52), we have

Z ∞
Z
Z
Z
B0
1
ρ ∞ −ρt ∞B0 ∞ −ρt−ρtd log C0 = − + ρedrs dsdt − ρ
e drt dt −e drt dt.
γ
γ 0
Ȳ
Ȳ 0
t
0
{z
} |
{z
}
|
{z
} |
direct response to r

indirect effects due to Y

indirect effects due to T

(55)

Equation (4) in Proposition 1 is the special case with B0 = 0.
To see that this decomposition is additive, consider the second term in (55) andintegrate by parts:
ρ
γ

Z ∞
0

−ρt

e

Z ∞
t

Z
Z ∞
Z Z
∞
ρ ∞ −ρsρ ∞ ∞ −ρsdrs dsdt = −e ds
drs ds −e dsdrt dt
γ t
γ 0
t
t
0
Z
Z
∞
ρ 1 −ρt ∞ρ 1 ∞ −ρt=− e
drs ds −e drt dt
γρ
γρ 0
t
0
Z
Z
1 ∞
1 ∞ −ρt=
drs ds −e drt dt.
γ 0
γ 0Therefore it is easy to see that the first, second and third terms in (55) sum toR∞− γ1 0 drs ds.61Remark: The fact that second term in (4) scales with 1/γ —and therefore the result that with B0 = 0 the split between direct and indirect effects is independent of
1/γ— is an equilibrium outcome. In particular, without imposing equilibrium, thedecomposition with B0 = 0 (4) is1
d log C0 = −
γ
|

Z ∞

−ρt

0

e drt dt + ρ{z} |direct response to r

But in equilibrium d log Yt = − γ1

R∞
t

Z ∞
0

e−ρt d log Yt dt .
{z}

GE effects due to Y

drs ds which scales with 1/γ. Also see footnote 8.

R∞
Derivation of (5): In the special case (1), we have drt = e−ηt dr0 . Hence 0 e−ρt drt dt =
R∞
R∞
R∞
R∞
R ∞ −(ρ+η)t1
dr0 . Similarly 0 e−ρt t drs dsdt = 0 e−ρt t e−ηs dsdtdr0 =e
dtdr0 = ρ+η
0R
1 ∞ −(ρ+η)t1
e
dtdr0 = η1 ρ+η
dr0 . Plugging these into (4) yields (5).
η 0

A.3

Details for Section 2.2

In the environment described in Section 2.2, aggregate consumption is given by
Ct = ΛCtsp + (1 − Λ)Ctsa .

(56)

Savers face the present-value budget constraint
Z ∞R− 0t rs dse0Ctsa dt =

Z ∞
0

Rt

e− 0 rs ds (Yt + Ttsa )dt + B0sa ,The government budget constraint is
Z ∞
0

Rt

e− 0 rs ds (ΛTtsp + (1 − Λ)Ttsa )dt + B0 = 0,(57)

where Bt is government debt. The market clearing condition for government debt is
Bt = (1 − Λ)Btsa .

(58)

We additionally assume that the economy starts at a steady state in which Ctsp =Ctsa = C̄ = Ȳ (and hence T̄ sp = 0). As before, we also assume that the economy ends
up in the same steady state (and hence in particular Ttsp → T̄ sp = 0 as t → ∞).
We now show how to derive the results of Section 2.2. First, consider the overall
effect of interest rate changes on aggregate consumption. Asbefore, the consumptionR∞
response of savers is given by Ctsa = C̄ exp − γ1 t (rs − ρ)ds . From (56) and because
62spender consumption equals Ctsp = Yt + Ttsp , therefore
Ct = Λ(Yt + Ttsp ) + (1 − Λ)C̄ exp



Z
1 ∞
−
(rs − ρ)ds .
γ tUsing that in equilibrium Ct = Yt :


Z
Λ
1 ∞
sp
Ct =
T + C̄ exp −(rs − ρ)ds
1−Λ t
γ t

(59)

We first show how (9) is derived. The government budget constraint (57) can be
written in flow terms as Ḃt = rt Bt + ΛTtsp + (1 − Λ)Ttsa. Under the assumption thatthe government keeps debt constant at its initial level, Bt = B0 , we needΛ(Ttsp − T̄ sp ) + (1 − Λ)(Ttsa − T̄ sa ) + (rt − ρ)B0 = 0
Alternatively, denoting by ΛT the fraction of income gains that is rebated to spenders
and using the assumption that T̄ sp = 0:
ΛTtsp = −ΛT (rt − ρ)B0
Substituting into (59) and totally differentiating
ΛT B0
1d log Ct = −drt −1 − Λ Ȳγ

Z ∞drs ds(60)tEquation (9) is obtained by specializing to the interest rate time path (1). WhenB0 = 0, the total response of aggregate consumption and income in this simple TANK
model is therefore identical to that in the RANK version above.
Finally, we show how equation (8) is derived. Because the savers in our TANK
model solve the same problem as the representative agent in the RANK model above,
their consumption satisfies the analogue of (52):dC0sa =


Z ∞
Z ∞
Z ∞
1
sa
−ρt
−ρt
− Ȳ + ρB0e drt dt + ρ
e dYt dt + ρ
e−ρt dTtsadt
γ
0
0
0
B sa

From (60) and using Yt = Ct and (57), their income satisfies d log Yt = −ΛT Ȳ0 drt −R
1 ∞drs ds. Since spenders receive a fraction ΛT of the government’s income gains
γ tfrom expansionary monetary policy, savers receive the rest and hence (1 − Λ)Ttsa =−(1 − ΛT )(rt − ρ)B0 or from (57) Ttsa = −(1 − ΛT )(rt − ρ)B0sa and hence dTtsa =63

−(1 − ΛT )B0sa drt . Therefore
Z ∞

−ρt

e

0

Z ∞

dTtsa dt = −(1 − ΛT )B0sa

e−ρt drt dt

0

Substituting these expressions into the one for saver consumption:

Z ∞
 Z ∞

Z ∞
sa
1
B0sa
1
−ρt
T B0
−ρt
− +ρ
e drt dt − ρedrs ds + Λ
drt dt
γ
γ t
Ȳ
Ȳ
0
0
sa Z ∞
T B0− ρ(1 − Λ )e−ρt drt dt
Ȳ 0d log C0sa =Next, characterize spenders’ consumption response
Z ∞

dC0sp
dY0 + dT0sp
ΛT B0
1
=
=−
dr0 −
C0
Y0
1 − Λ Ȳ
γ
1
ΛT
B0
dr0 −
=−
Λ(1 − Λ) Ȳγ
dY +dT sp

drt dt −

0

Z ∞

ΛT B0
dr0
Λ Ȳ

drt dt0

From (56) d log C0 = (1 − Λ)d log C0sp + Λ 0 Y0 0 . Therefore, the analogue of Proposition 1 is
Z ∞

1−Λ
B0
e−ρt drt dt
d log C0 = −
+ρ
γ
Ȳ
0
Z ∞
Z ∞
Z ∞
ρ(1 − Λ)
ΛΛT B0
−ρt
T B0
−ρt
e
drs dsdt − ρΛ−e drt dt −dr0 (61)γ
1 − Λ Ȳ
Ȳ 0
0
t
Z
Z ∞
Λ ∞
B0
T B0
−ρt
drt dt − ΛT dr0− ρ(1 − Λ )e drt dt −
γ 0
Ȳ 0
Ȳ
The first line is the direct response to r, the second line are indirect effects due to Y ,and the third line are indirect effects due to T . An instructive special case is the one
without government debt, Bt = 0 for all t. In that case1−Λd log C0 = −
γ
|

Z ∞
0

−ρt

e
{z

ρ(1 − Λ)drt dt −γ} |direct response to r

Z ∞
0

−ρt

e

Z ∞

Λ
drs dsdt −γt{z

indirect effects due to YZ ∞t

drs ds .
}Equation (8) then follows from the fact that in the special case (1), drt = e−ηt dr0 .
For completeness, we also derive the split between direct and indirect effects forour analytic example drt = e−ηt dr0 in the case with both hand-to-mouth agents Λ > 0
and government debt B0 > 0. Collecting some of the indirect effects on the second and64third lines of (61) and specializing to drt = e−ηt dr0 , we haved log C0
1
=
−
dr0
γη



η
ρ+η
|




B0
ρ
η
ΛT B0
B0
1 − Λ − ργ+(1−Λ)
+
+Λ+
ργ
.
ρ+η
ρ + η Ȳ
1 − Λ ȲȲ{z}direct response to r

A.4

(62)

Details on Medium-Scale DSGE Model (Section 2.3)

The Smets-Wouters model is a typical medium-scale DSGE RANK model with a variety of shocks and frictions. The introduction of Smets and Wouters (2007) provides auseful overview and a detailed description of the model can be found in the paper’s online Appendix.62 We here only outline the ingredients of the model that are important
for the purpose of our decomposition exercise (reported in Table 1) as well as somedetails on the implementation of this exercise.
An important difference relative to the stylized model of Section 2.1 is that the
representative household’s utility function features external habit formation:
∞
X

1
E0
β
(Ct (j) − hCt−1 )1−σc exp
1
−
σ
c
t=0
t



σc − 1
Lt (j)1+σl
1 + σl(63)

where Ct (j) is consumption of one of a continuum of individual households and Ct
is aggregate consumption (in equilibrium the two are equal). The parameter h ∈[0, 1] disciplines the degree of external habit formation. As mentioned in the main
text, the model also features investment with investment adjustment costs and capitalutilization, as well as partially sticky prices and wages.
Our starting point for the decomposition are the impulse response functions (IRFs)
to an expansionary monetary policy shock in a log-linearized, estimated version of
the model. We set each of the model’s parameters to the mode of the corresponding
posterior distribution (see Table 1 in Smets and Wouters (2007) for the parametervalues). The IRFs are computed in Dynare using an updated version of the replication
file of the published paper.63 For our purposes, the relevant IRFs are the sequences{Ct , Rt , Yt , It , Gt , UCt , Lt }∞t=0 for consumption Ct , interest rates Rt , labor income Yt ,investment It , government spending Gt , capital utilization costs UCt = a(Zt )Kt−1 andlabour supply Lt . We further denote consumption at the initial steady state by C̄.
Given these IRFs, we decompose the overall consumption response to an expansionary monetary policy shock into direct and indirect effects as follows. Suppressing
62
63

Available at https://www.aeaweb.org/aer/data/june07/20041254_app.pdf
Available at http://www.dynare.org/phpBB3/viewtopic.php?f=1&t=3750.

65

j-indices for individual households, the budget constraint of households is
Ct +

Bt−1
Bt
+ Tt ≤
+ Mt
Rt Pt
Pt
Divt Πt
W h Lt Rth Kt−1 Zt+− a(Zt )Kt−1 ++− It
Mt = t
Pt
Pt
Pt
Pt

(64)
(65)

where the reader should refer to the online Appendix of Smets and Wouters (2007)
for an explanation of each term (the budget constraint is their equation (9)).64 In
present-value form
∞
∞
X
X
1
1
Ct =
(Mt − Tt )t−1
t−1
Πk=0 R̃kΠk=0 R̃kt=0
t=0
Rtdenotes the real interest rate. Households maximize (63) subject towhere R̃t = Πtthis budget constraint. For any price sequences, initial consumption C0 then satisfies:

!
∞
B−1 X
1
X+
(Mt + Tt )
+
t−1
P0
R̃
Π
k
k=0
t=0
 k !
t
∞
X
X
1
h
x
χ=
t−k
g
Πt−1
k=1 R̃kt=0
k=0
 k+1
∞
t−1
X
X
1
h
X=
xt−k
C̄
t−1
g
Πk=1 R̃k k=0
t=0


1/σc

σc − 1s s−1(Ls − L0 )exp
xs = β̄ Πk=0 R̃k
σc (1 + σl )

1
C0 =
χ

(66)

where β̄ = gβσc and g is the gross growth rate of the economy. The direct effectof consumption to interest rate changes is then computed from (66) by feeding in the∞equilibrium sequence of real interest rates {R̃t }∞t=0 while holding {Mt , Tt , Lt }t=0 at theirsteady state values. When computing this direct effect in practice, we simplify the righthand side of (66) further taking advantage of the fact that most terms are independentof the sequence of real interest rates {R̃t }∞t=0 . In particular, in equilibrium, profits andlabor union dividends are Πt = Pt Yt − Wt Lt − Rth Zt Kt−1 and Divt = (Wt − Wth )Lt andtherefore, substituting into (65)Mt = Yt − a(Zt )Kt−1 − It .
64

(67)

Note that Smets and Wouters’ budget constraint features some typos: it does not include dividends
from firm ownership Πt and there is a “minus” in front of Tt suggesting it is a transfer even though
it enters as a tax in the government budget constraint (equation (24) in their online Appendix).

66

Further, the government budget constraint in present-value form is∞X1

Πtk=0 R̃k
t=0

Tt =

∞
X

1

Πtk=0 R̃k
t=0

Gt .

(68)

Substituting (67) and (68) into (66), we have
C0 =


1
X + Y P DV − I P DV − GP DV − UC P DV
χ(69)

where Y P DV , I P DV , GP DV and UC P DV are the present values of {Yt , It , Gt , UCt }∞t=0∞
discounted at {R̃t }t=0 .
Note that although the series {Ct , R̃t , Yt , It , Gt , UCt , Lt }∞t=0 are generated using a
log-linearized approximation around the trend, our decomposition uses a non-linear
solution. In particular, both the overall and direct elasticities of consumption to interest
rate changes in Table 1 are computed using the exact non-linear Euler equation but
evaluated at the equilibrium prices from the linearized models – see the formula (69).
The fraction due to direct effects is the ratio of this direct elasticity to the overall
elasticity, with both numerator and denominator computed in this non-linear fashion.
In our baseline exercise in Table 1 this fraction equals 99 percent. For small shocks theoverall elasticity of consumption computed with the exact formula is very close to theelasticity computed using the linearized output from Dynare. For larger shocks, thetwo can differ somewhat. We have also recomputed the share of direct effects as theratio of the direct elasticity computed in a non-linear fashion and the overall elasticity
computed in a linear fashion. For the baseline exercise in Table 1, this yields a shareof direct effects of 91%.
As already stated in the main text, our main result is that – at the estimated parameter values of Smets and Wouters (2007) – the direct effect amounts for 99 percentof the total response of initial consumption to an expansionary monetary policy shock.
We have conducted a number of robustness checks with respect to various parametervalues, and in particular with respect to the habit formation parameter h. The results
are robust. In the case without habit formation h = 0, 95.1 percent of the overall
effect are due to direct intertemporal substitution effects. Finally, note that a difference between (63) and the specification of preferences in textbook versions of the New
Keynesian model is the non-separability between consumption and labor supply. We
have conducted an analogous decomposition exercise with a separable version of (63).
The decomposition is hardly affected.

67

A.5

Details on the Two-Asset RANK and TANK Models

A.5.1

Model

We begin by outlining the two-agent, spender-saver version of the model (TANK). The
representative agent is a special case with the fraction of spenders equal to zero. The
model is written and solved in discrete time.
Households . A fraction Λ of households are spenders indexed by “sp” and a fraction
1 − Λ are savers indexed by “sa”.
Savers . Savers derive utility from consuming csa
t and have disutility from supplyingblabor ℓsat . Savers are able to borrow and save in a liquid government bond at rate rt .
They also have access to an illiquid asset at with rate of return rta . Assets of typea are illiquid in the sense that households need to pay a cost for depositing into orwithdrawing from their illiquid account. Let dt denote the deposit decision and χ(dt )the cost of depositing dt . The saver’s problem in its sequential formulation is therefore
given by
max

sa
∞
{csa
t ,ℓt ,dt ,bt+1 ,at+1 }t=0

S.t.

∞
X

sa
β t u(csat , ℓt )

t=0

sa
sa
sa
b
csa
t + bt+1 + dt + χ(dt , at ) = (1 − τ )(wt ℓt + Γt ) + Tt + (1 + rt )bt(λ)

at+1 = (1 + rta )at + dt

(η)

where
ℓ1+ν
1+ν
χ2
χ(d) = χ1 |d| ,χ1 > 0, χ2 > 1.

u(c, ℓ) = log c − ϕ68The first-order conditions for the consumer’s problem can be written as
1=
ηt =



b
λt,t+1 1 + rt+1

(

h





a
λt,t+1 ηt+1 1 + rt+1

ηt = 1 + sign(dt ) × χ̃ dt

where

(70)
i

)

χ2 −1

(71)

,

χ̃ = χ1 χ2

ν sa
ϕ ℓsa
ct = (1 − τ )wt
t

(72)
(73)

λt+1
=β
λt,t+1 :=λt

 sa −1
ct+1
.
csa
t

(74)

Note that, by combining (70) and (71), one obtains that in steady state rb = ra .
Spenders . Spenders are hand-to-mouth, i.e. consume their labor income every period.
Their only margin of adjustment is labor supply ℓsp . The spender’s problem is
sp
max
u(csp
t , ℓt ) s.t.
sp sp

ct ,ℓt

sp
sp
sp
csp
t = (1 − τ )(wt ℓt + Γt ) + Tt

with first-order conditions
sp
sp
sp
csp
t = (1 − τ )(wt ℓt + Γt ) + Tt
ν sp
ϕ
ct .
ℓsp
wt =
t
1−τ

(75)
(76)

Firms . There is a continuum of intermediate-goods monopolistic firms, each producing a variety j using a constant returns to scale production function
yt (j) = kt (j)α nt (j)1−α .

(77)Each intermediate producer chooses its price pt (j) and inputs kt (j), nt (j) to maximize
pt (j)
yt (j) − wt nt (j) − rtk kt (j) − ΘPtpt (j)
pt−1 (j)(78)taking into account that the demand for its product depends on the price pt (j) charged.
The function Θ(·) is a quadratic adjustment cost for the price change and is expressed69

as a fraction of final good output Yt
Θt



pt
pt−1



θ
=
2



pt
−1
pt−1

2

Yt .

We divide the problem of the firm in two parts. First, the cost minimization problem
of producing y units of variety j delivers the following optimality conditions
wt = (1 − α)mt
rtk = αmtykt (j)ynt (j)(79)(80)

where marginal cost mt is the same across firms
 k α 
1−α
rt
wt
mt =
.
α
1−α
Since all firms face the same marginal cost, we drop the j subscript from now onwards. Taking cost minimization decisions as given, each intermediate producer chooses{pt }∞
t=0 to maximize discounted profitsmax{pt }∞t=0∞X(λt ηt )t=0s.t. yt =



pt
Pt

(

−ε



)
pt
pt
− mt yt − Θt
Pt
pt−1

Yt

(81)
(82)

where the discount factor used by the firm reflects that dividends will accrue to theilliquid account of savers. In a symmetric equilibrium, all firms will choose the same
price, which will be also the aggregate price Pt . That gives rise to the following Phillipstcurve relating aggregate inflation πt = PPt−1− 1 and marginal costs

1 − θπt 1 + πt + θ

λt+1 η+1
λt ηt

πt+1 1 + πt+1

 Yt+1= 1 − mt ε.
Yt

(83)

Note that, from equation (71), effectively firm discount at rate ra , which is also thediscounting that appears in the Phillips curve, exactly as in our HANK model.
Moreover, since in equilibrium all firms choose the same price, they all produce thesame quantity and hire the same amount of input on factor markets. Hence we can70

aggregate production function of each firm to getyt (j) = kt (j)α nt (j)1−α ⇒ Yt = Ktα Nt1−α .
Finally, profits are then given byθ 2Πt = Yt (1 − mt ) − π .
2(84)Illiquid Assets . As in HANK, illiquid assets at consist of both capital holdings (ktsa )and equity claims (st ) to a fraction ω of profits. Since there is no aggregate uncertainty,
no arbitrage dictates that the return to capital must be equal to the return on equity.
We denote this return by rta
ωΠt +
rta ≡

qt − qt−1
qt−1

= rtk − δ(85)which restricts how asset prices qt evolves over time:
qt =



1
ωΠ
+
q
.
t+1
t+1
a
1 + rt+1

In the event of an unexpected shock, however, the realized returns between capitaland shares do not need to be equalized at the moment of impact. What no-arbitrage
pricing requires, in such a circumstance, is for the stock price to jump so as to makethe return from holding shares the same as the return from holding capital from that
period onwards. Realized returns at impact, though, need not to be equalized, sinceasset positions are pre-determined. Hence, it is useful to write the law of motion of
illiquid assets by keeping track of portfolio compositionsa
at+1 ≡ kt+1
+ st+1 qt
= 1 + rta at + dt
= 1 + rtk − δ ktsa + st ωΠt + qt + dt .

(86)

By combining (85) and (86), it is easy to see that, in steady-state, savers withdraw
from the illiquid account an amount d = −r a a.
As in HANK, the remaining fraction (1 − ω) of profits are distributed to households(both spenders and savers) as a direct transfer Γt to agents liquid budget constraint(since there is no difference in productivity between the two groups, they are distributed

71

lump-sum):saΓspt = Γt = Γt .

Γt = (1 − ω)Πt ,(87)

We set ω = α so as to neutralize the role of countercyclical profits as explained in themain text.
Government . The government issues bonds denoted by B g , with the convention
that negative values denote government debt. Its budget constraint is therefore given
by



g
Bt+1
= 1 + rtb Btg + τ wt Nt + Γt − Tt − Gt(88)with government transfers Tt given by

Tt = ΛTtsp − (1 − Λ)Ttsa
ΛT Tt = ΛTtsp

(89)
(90)

Note that we allow for ΛT 6= Λ, i.e. spenders may receive bigger or smaller share oftransfers than their population share.
Monetary authority . Monetary policy follows a Taylor rule for the nominal interest
rate
it = r̄tb + φπt + ǫt ,ǫt+1 = ρεt + ut+1 ,
ut+1 ∼ N (0, σǫ2)
(91)
Given inflation and the nominal interest rate, the real return realized on liquid assets
hold by savers is given by
1 + it−1.
(92)
1 + rtb =1 + πtEquilibrium . To close the model, we state market clearing conditionsg
0 = (1 − Λ)bsa
t+1 + Bt+1

(93)

1 = (1 − Λ)st+1

(94)

sa
Kt+1 = (1 − Λ)kt+1(95)

It = Kt+1 − (1 − δ)Kt

(96)

Yt = Ct + It + Gt + 1 − Λ χt + Θt

(97)

72

whereA.5.2

ParameterizationCt = Λctsp + 1 − Λ csa
t
 sa
sp
Nt = Λℓt + 1 − Λ ℓt .

(98)
(99)

Some parameter values are set exactly as in the simple TANK models of Section 2. In
particular, we set Λ = ΛT = 0.3, risk aversion γ to 1, and the discount rate to 5%
annually. In this model, the liquid real rate also equals 5% in steady state.
Other parameters (Frisch elasticity, transaction cost, demand elasticity, price adjustment cost, share of profits paid as dividends, government policy parameters, andTaylor rule coefficient) are set as in HANK. Only two parameters are calibrated internally. The disutility of labor is set so that on average 1/3 of the time endowment isspent working, and the depreciation rate is set to match the same illiquid wealth to
GDP ratio as in HANK (13, quarterly). Table A.8 summarizes the parameterization.
A.5.3

Simulations and decompositions

We use Dynare to solve for the model’s steady state, its transitional dynamics and thedecompositions. The model is solved globally (i.e. without local linearization) fromthe equilibrium system of nonlinear equations. We always analyze monetary shocks ofthe same size as in HANK, i.e., 25 basis points, with a quarterly persistence of 0.5. It
is worth emphasizing that η = 0.5 is the correct choice also for discrete time, if we wish
to compare across models. To see this, consider that the cumulative deviation of theR∞R∞
interest rate path from t = 0 is 0 (rs − ρ) ds = 0 exp (−ηs) ds = 1/η. In discretePttime the cumulative deviation is ∞t=0 ρ = 1/(1 − ρ). Thus, a proper comparison witha continuous time model where η = 0.5 requires setting ρ = 0.5.
Figure A.10 reports the IRFs and decomposition in RANK and TANK for thebaseline fiscal policy scenario (T-adjust case). It is the counterpart of Figure 4 for
HANK. The elasticities and share of direct effects for the two-asset RANK and TANK
reported in Table 1 are obtained from these experiments.

73

Table A.8: Parameter values (period length is one quarter)
Preferences
Fraction of HtM
Discount factor
Risk aversion
Disutility of labor
Frisch elasticity

Λ
β
γ
ϕ
1/ν

0.30
1.0125−1
1.00

Deposit costs
Scaling
Curvature

χ1
χ2

0.956
1.402

Production
Demand elasticity
Capital share
Depreciation rate
Price adjustment cost
Share of profit paid as liquid dividends

ε
α
δ
θ
ω

10
0.33
0.017
100
0.33

τ
T
ΛT
g
|B |/(4Y )

0.25
0.06
0.30
0.23

rb
φ
u0
ρ

0.05
1.25
0.00250.50

Government policy
Proportional labor tax
Lump sum transfer (rel GDP)
Fraction transfer to HtM
Govt debt/annual GDP
Monetary Policy
Steady state real liquid return (pa)Taylor rule coefficient
Shock size at impact
Shock persistence

74

1.00

 0.3

3.5
                           

3

    !    !   !   

0.25

2.5
0.2

1



 

1.5

0.15

 

2

0.1

0.5
0.05
0
0
-0.5

-1

-0.05
0

5

10

15

20

0

5



10

15

20



 
0.35

4
                           

3.5

            

0.3

3
0.25
2.5
0.2



1.5

 

 

2

1

0.15

0.1
0.5
0.05
0
0
-0.5

-1

-0.05
0

5

10

15

20

0



5

10

15

20



Figure A.10: Impulse Response Functions and Decompositions in RANK and TANK

75

B
B.1

Additional Details on the HANK Model
HJB and Kolmogorov Forward Equations for Household’s
Problem

We here present the households’ HJB equation, and the Kolmogorov forward equation
for the evolution of the cross-sectional distribution µ. We focus on the stationary
versions of these equations under the assumption that the logarithm of income yit =log zit follows a “jump-drift process”
dyit = −βyit dt + dJit .
Jumps arrive at a Poisson arrival rate λ. Conditional on a jump, a new log-earnings
state yit′ is drawn from a normal distribution with mean zero and variance σ 2 , yit′ ∼N (0, σ 2 ). The stationary version of households’ HJB equation is then given by(ρ + ζ)V (a, b, y) = max u(c, ℓ) + Vb (a, b, y) (1 − τ )wey ℓ + r b (b)b + T − d − χ(d, a) − cc,ℓ,d+ Va (a, b, y)(r a a + d)

+ Vy (a, b, y)(−βy) + λ

(100)

Z ∞

(V (a, b, x) − V (a, b, y)) φ(x)dx−∞

where φ is the density of a normal distribution with variance σ 2 .
Similarly, the evolution of the joint distribution of liquid wealth, illiquid wealth andincome can be described by means of a Kolmogorov forward equation. To this end,
denote by g(a, b, y, t) the density function corresponding to the distribution µt (a, b, z),but in terms of log productivity y = log z. Furthermore, denote by sb (a, b, y) andsa (a, b, y) the optimal liquid and illiquid asset saving policy functions, i.e. the optimal
drifts in the HJB equation (100). Then the stationary density satisfies the Kolmogorov
forward equation
0 = − ∂a (sa (a, b, y)g(a, b, y)) − ∂b (sb (a, b, y)g(a, b, y))Z ∞− ∂y (−βyg(a, b, y)) − λg(a, b, y) + λφ(y)g(a, b, x)dx

(101)

−∞

− ζg(a, b, y) + ζδ(a − a0 )δ(b − b0 )δ(y − y0 ),where δ is the Dirac delta function and (a0 , b0 , y0) are starting assets and income.
Achdou et al. (2014) explain in detail how to solve (100) and (101), including how tohandle the state constraints, using a finite difference method.

76

B.2

Proof of Lemma 1 (Derivation of Phillips Curve)

The firm’s problem in recursive form isar (t)J (p, t) = maxπp− m(t)
P (t)



p
P (t)

−ε

θ
Y (t) − π 2 Y (t) + Jp (p, t) pπ + Jt (p, t)2

where J (p, t) is the real value of a firm with price p. The first order and envelope
conditions for the firm are
Jp (p, t) p = θπYp  p −ε−1 Y p −ε Y(r a − π)Jp (p, t) = −−m ε++ Jpp (p, t) pπ + Jtp (p, t).
P
P
P
P
P
In a symmetric equilibrium we will have p = P , and hence
Jp (p, t) =

θπY
p

(102)

(r a − π)Jp (p, t) = − (1 − m) εYY+ + Jpp (p, t) pπ + Jtp (p, t).
p
p

(103)

Differentiating (102) with respect to time gives
Jpp (p, t) ṗ + Jpt (p, t) =θY π̇ θẎ π θY ṗ
+
−
.
p
p
p p

Substituting into the envelope condition (103) and dividing by θY /p gives
ra −

Ẏ
Y

!

π=

1
(− (1 − m) ε + 1) + π̇.
θ

Rearranging, we obtain equation (19) in the main text.

B.3

Computation of Marginal Propensities to Consume

We begin by stating a notion of an MPC in our model that is directly comparable tothe empirical evidence:
Definition 1 The Marginal Propensity to Consume over a period τ for an individual77

with state vector (a, b, z) is given by
MPCτ (a, b, z) =∂Cτ (a, b, z)
,
Z ∂b

where

τ

Cτ (a, b, z) = E

(104)


c(at , bt , zt )dt|a0 = a, b0 = b, z0 = z .

0

(105)

Similarly, the fraction consumed out of x additional units of liquid wealth over a period
τ is given byCτ (a, b + x, z) − Cτ (a, b, z)
MPCxτ (a, b, z) =
.
(106)
x
The conditional expectation Cτ (a, b, z) in (105) and, therefore, the MPCs in Definition 1 can be conveniently computed using the Feynman-Kac formula. This formulaestablishes a link between conditional expectations of stochastic processes and solutions to partial differential equations. Applying the formula, we have Cτ (a, b, z) =Γ(a, b, y, 0), with y = log z, where Γ(a, b, y, t) satisfies the partial differential equation
0 = c(a, b, y) + Γb (a, b, y, t)sb(a, b, y) + Γa (a, b, y, t)sa (a, b, y)
Z ∞+ Γy (a, b, y)(−βy) + λ[Γ(a, b, x, t) − Γ(a, b, y, t)] φ(x)dx−∞

on [0, ∞) × [b, ∞) × [ymin, ymax ] × (0, τ ), with terminal condition Γ(a, b, y, τ ) = 0, andwhere c, sb and sa are the consumption and saving policy functions that solve (100).

B.4

Extension with firms’ profits allocated to both a and b

Under this extension, a fraction ω of aggregate profits is paid into the illiquid accounts
proportionately to the shares owned by each household and the remaining 1 − ω fraction is paid in liquid form to every individual i as a lump-sum rescaled by householdproductivity, i.e., πtb (zit ) = zz̄it (1 − ω)Πt where z̄ is average productivity. As explainedin the main text, we interpret πtb (zt ) as bonuses and commissions and wt zt ℓt + πtb (zt ))as total compensation. Labor income taxes are levied on total compensation.
Therefore, omitting the subscript i to ease notation, a household’s holdings of liquid
assets bt evolve according toḃt = (1 − τt ) wt zt ℓt + πtb (zt ) + rtb (bt )bt + Tt − dt − χ(dt , at ) − ct

(107)

The dynamics of illiquid assets are still given by (11).
To solve their optimization problem, households take also as given Πbt = (1−ω)Πt ,z̄
the rescaled fraction of aggregate profits that is paid out proportionally to individual78

productivity. It is useful to define Wt = (wt , Πbt ), the vector of aggregates that characterizes worker’s compensation. Then, in the vector Γt , wt should be replaced by Wt .
Similarly, in our decomposition, the term capturing the indirect effects from changes
in labor income induced by the monetary shock —the third term of equation (30)—becomes:
′

Z ∞
Z ∞
∂C0
∂C0 b
∂C0
dWt dt =dwt +dΠ dt.
(108)
∂Wt
∂wt
∂Πbt t00
Finally, the arbitrage condition between shares of the intermediate producers and capital and the government budget constraint becomeωΠt + q̇t
= rtk − δ.
qt

(109)

and the government budget constraint reads
Ḃtg + Gt + Tt = τtCZwt zℓt (a, b, z) + πtb (z) dµt + rtb Btg(110)

Details on 2004 SCF and FoF data

Our starting point is the balance sheet for U.S. households in 2004 (Flow of Funds
Tables B.100, and B100e for the value of market equity). An abridged version of thistable that aggregates minor categories into major groups of assets and liabilities isreproduced in Table C.1 (columns labelled FoF).
The columns labelled SCF in Table C.1 report the corresponding magnitudes, foreach asset class, when we aggregate across all households in the 2004 Survey of Consumer Finances (SCF). The comparison between these two data sources is, in many
respects, reassuring. For example, aggregate net worth is $43B in the FoF and $49B
in the SCF, and the FoF ranking (and order of magnitude) of each of these major categories is preserved by the SCF data.65 Nevertheless, well known discrepancies exist
across the two data sources.66
On the liabilities side, credit card debt in FoF data is roughly half as large as in
SCF data. The reason is that SCF measures outstanding consumer debt, whereas theFoF measures consumer credit, which includes current balances, whether or not they65

This is remarkable, since the underlying data sources are entirely different. The SCF is a household
survey. The macro-level estimates of U.S. household sector net worth in the FoF are obtained as aresidual with respect to all the other sectors of the economy, whose assets and liabilities are measured
based on administrative data derived from aggregate government reports, regulatory filings as well as
data obtained from private vendors and agencies such as the Bureau of Economic Analysis (BEA),
the Census Bureau, and the Internal Revenue Service (IRS).
66
For systematic comparisons, see Antoniewicz (2000) and Henriques and Hsu (2013).

79Assets

Liabilities

Real estateConsumer Durables

FoF
21,0004,100

SCF
27,7002,700

LiquidN
N

Deposits
Treasury Bonds
Corporate Bonds

5,800
700
900

2,800
200
500

Y
Y
Y

Corporate Equity
Equity in Noncorp. Bus.
Total

12,6007,300
52,40014,200
11,10059,200N
N

Mortgage Debt
Nonrev. Cons. Credit
Revolving Cons. Credit

FoF
7,600
1,400
800

SCF
8,500
1,200
400

Total

9,800

10,100

Table C.1: Balance sheet of US households for the year 2004.
Sources: Flow of Funds (FoF) and Survey of Consumer Finances (SCF). Values are in Billions of2004 US$. Y/N stands for Yes/No in the categorization of that asset class as liquid.

get paid in full. Thus, the SCF estimate seems more appropriate, given that a negativevalue of b in the model means the household is a net borrower.
On the asset side, real estate wealth in the SCF is 30 pct higher than in the FoF.
The SCF collects self-reported values that reflects respondents’ subjective valuations,whereas the FoF combines self-reported house values, from the American Housing
Survey (AHS) with national housing price index from CoreLogic and net investment
from the BEA. However, during the house-price boom, AHS owner-reported values
were deemed unreliable and a lot more weight was put on actual house price indexes,
an indication that SCF values of owner-occupied housing may be artificially inflated
by households’ optimistic expectations.
The valuation of private equity wealth is also much higher in the SCF, by a factor
exceeding 1.5. Once again, the FoF estimates appear more reliable, as it relies on
administrative intermediary sources such as SEC filings of private financial businesses(security brokers and dealers) and IRS data on business income reported on tax returns,
whereas, as with owner-occupied housing, the SCF asks non-corporate business owners
how much they believe their business would sell for today.67
Finally, deposits and bonds are more than twice as large in the FoF.68 Antoniewicz(2000) and Henriques and Hsu (2013) attribute this discrepancy to the fact that theFoF “household sector” also includes churches, charitable organizations and personal
trusts (that are more likely to hold wealth in safe instruments) and hedge-funds (that67

According to Henriques and Hsu (2013), another reason why the SCF data on private businessvalues is problematic is the combination of a very skewed distribution and the small sample size ofthe survey that make the aggregate value obtained in the SCF very volatile.
68
The SCF does not contain questions on household currency holdings, but SCF data summarized
above contain an imputation for cash. See Kaplan and Violante (2014) for details.

80

Liquid
N
N
Y

2

8

0

.5

1

Density

1.5

6
4

Density

2
0

-4

-2

0

2

4

-5

0

5

5 Year Log Earnings Changes

1 Year Log Earnings Changes

Figure D.1: Growth Rate Distribution of Estimated Earnings Process
may hold large amount of cash to timely exploit market-arbitrage opportunities).

D
D.1

Further Details on Calibration
Earnings Process

Figure D.1 displays histograms of one- and five-year log earnings changes generated
by our estimated earnings process (33)-(34), overlaid with Normal distributions withthe same means and variances. The leptokurtosis of annual income changes is clearly
evident from these figures. For a comparison with the analogous figures from SSA maleearnings data, we refer readers to Figure 1 in Guvenen et al. (2015).
In order to translate the estimated earnings processes (33)-(34) to a form that canbe used in the households’ consumption-saving problem (100), we take the following
steps.
First, we approximate the estimated continuous-time continuous-state processes
with continuous-time discrete-state processes. For each of the two components (j =1, 2) we construct a grid for zj . We use 11 grid points for the persistent component
and 3 grid points for the transitory component. We then construct the associatedcontinuous time transition matrix based on a finite difference approximation of theprocesses in (33)-(34), evaluated at the estimated parameters. We choose the grid
widths and spacing so that the annual moments produced by simulating the combined
discrete-state process are as close as possible to the annual earnings moments fromthe combined continuous-state process. These moments are reported in Table D.1.
The Lorenz curves for the ergodic distributions associated with the continuous anddiscretized process are shown by the black dashed line and the green dash-dot line in
Figure D.2, respectively. The two Lorenz curves are very close, as are the moments of81

Moment

Data

Variance: annual log earns
Variance: 1yr change
Variance: 5yr change
Kurtosis: 1yr change
Kurtosis: 5yr change
Frac 1yr change < 10%
Frac 1yr change < 20%
Frac 1yr change < 50%

0.70
0.23
0.46
17.8
11.6
0.54
0.71
0.86

Model
Estimated
0.70
0.23
0.46
16.5
12.1
0.56
0.67
0.85

Model
Discretized
0.74
0.21
0.49
15.5
13.2
0.63
0.71
0.83

Table D.1: Earnings Process Estimation Fit
1
Labor productivity: continuous (before adjustment)
Labor productivity: discretized (before adjustment)
Labor productivity: discretized (after adjustment)
Gross earnings: discretized

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

0.2

0.4

0.6

0.8

1

Figure D.2: Earnings Lorenz Curveearnings changes, which demonstrates that the discrete approximations are accurate.
Second, since earnings in the model are determined by both idiosyncratic productivity zit and endogenous labor supply decisions ℓit , we make an adjustment to the productivity grid so that the resulting cross-sectional distribution of earnings yit = wt zit ℓit
is as dispersed as in the data. This adjustment is necessary because with our preference
specification, optimal labor supply decisions ℓit are positively related to individual productivity zit . Hence earnings inequality in the model with labor supply is larger than
productivity inequality. To bring earnings inequality in line with the data we shrink
the log productivity grid by a factor 1 + ζ ν1 , where ν1 is the Frisch elasticity of labor
supply. We set the constant ζ equal to 0.85, which generates a standard deviation

82

Figure D.3: Calibrated Adjustment Cost Functionof log earnings in the model log yit equal to the standard deviation of log household
earnings in the data. To estimate the standard deviation of log household earnings
implied by the SSA data (which we cannot observe directly), we rescale the standard
deviation of log male earnings in the SSA data by the ratio of the standard deviationof log household earnings to the standard deviation of log male earnings in the Panel
Study of Income Dynamics from 2002 to 2006.
The red dash-dot line in Figure D.2 shows the Lorenz curve for the productivity
distribution once it has been re-scaled in this way. Note that it is less dispersed than
the raw productivity process (green dash-dot line). The blue solid line shows the Lorenz
curve for gross labor income, taking into account the optimal labor supply decisionsof households. Note that gross labor income is more dispersed than the adjusted
productivity process (because of the substitution effect), but is less dispersed than theraw productivity process (because of the distinction between household earnings andindividual male earnings).

D.2

Adjustment Cost Function

The calibrated transaction cost function is shown in Figure D.3. Consider first panel(a). The horizontal axis shows the quarterly transaction expressed as a fraction of ahousehold’s existing stock of illiquid assets, d/a. The vertical axis shows the cost ofwithdrawing or depositing this amount in a single quarter expressed as a fraction of thestock of illiquid assets, χ(d, a)/a. For values of a above the threshold a, this function
does not depend on the level of illiquid assets. From (14) for a > a, χ(d, a)/a =χ0 |d/a| + χ1 |d/a|χ2 . The light-blue histogram displays the stationary distribution ofadjustments d/a. Roughly 20% percent of households are inactive and neither deposit
nor withdraw. Of the remaining households, some deposit and some withdraw. The

83

histogram shows that, on average, households in the stationary distribution withdraw,
taking advantage of the fact that the income generated by the high return of illiquid
assets replenishes the illiquid account.
Panel (b) provides an alternative view of the adjustment cost function. The horizontal axis shows the quarterly transaction expressed as a fraction of illiquid assets,
d/a, as in panel (a). The vertical axis shows the cost of withdrawing or depositing expressed as a fraction of the amount being transacted, χ(d, a)/d, i.e. the “fee” for eachtransaction. For values of a above the threshold a, this function also does not dependon the level of illiquid assets. From (14) for a > a, χ(d, a)/d = |d/a| + χ1 |d/a|χ2 −1 .
The overlaid histogram is the same as in panel (a).
These two panels illustrate that, for the most common transaction sizes, the cost is
at most 25 percent of the value of the transaction, or at most 0.05 percent of the stockof illiquid wealth.

84

Description

Value

Target / Source

ζ
1/γ
1/ν
ϕ
ρ

Preferences
Death rate
Intertemporal elasticity of subst.
Frisch elasticity of labor supply
Disutility of labor
Discount rate (pa)

1/180
1
1
27
4.8%

Avg. lifespan 45 years

ε
θ
α
δ̄
δu

Production
Demand elasticity
Price adjustment cost
Capital share
Steady state depreciation rate (p.a.)
Elasticity of capital utilization

10
100
0.33
7%
1.2

Profit share of 10 %Slope of Phillips curve, ǫ/θ = 0.1

τT

Government
Proportional labor tax
Lump sum transfer (rel GDP)

0.25
0.06

φ
r̄ b

Monetary Policy
Taylor rule coefficient
Steady state real liquid return (pa)

1.25
2%

r borrb

Unsecured borrowing
Borrowing rate (pa)Borrowing limit

7.9%$16,500

Internally calibrated
1 × quarterly labor income

χ0
χ1
χ2
a

Adjustment cost function
Linear component
Convex component
Convex component
Min a in denominator

0.0438
0.956
1.402
$360

Internally calibrated
Internally calibrated
Internally calibrated

Avg. hours worked equal to 1/3
Internally calibrated

40% hh with net govt. transfer

Table D.2: List of Calibrated Parameter Values

85

E

Additional Monetary Policy Experiments

In this appendix we report results on the overall effectiveness of monetary policy andits decomposition between direct and indirect effects, when we vary the key parametersthat govern the “heterogenous agent block” of the model. These features include theborrowing limit, the cost of borrowing and the parameters of the adjustment costfunction. Unlike the robustness experiments conducted in the main text, changing
these parameters affects the level and distribution of wealth in the steady state. Henceto maintain comparability across experiments, in each case we re-calibrate the discountrate ρ so as to keep the mean of the illiquid asset distribution (and hence the equilibrium
K
ratio, wage rate w, interest rates (r, r a ), and output Y constant. The distribution ofY
illiquid wealth, as well as the mean and distribution of liquid wealth, necessarily differ
across the experiments, hence we report these features of the alternative economies
alongside the results of the monetary policy shock
Table E.3 reports robustness analyses on the borrowing environment —the tightnessof the borrowing limit and the wedge between the interest rates on borrowing andsaving in the liquid assets. Reasonable changes in these features of the model have asignificant effect on the level and distribution of liquid wealth holdings, but none of
the main findings about the size and decomposition of the monetary policy shock areaffected by these changes.
Table E.4 reports robustness analyses on the adjustment cost function. As with theborrowing environment, changes in the adjustment cost function affect the level anddistribution of liquid wealth holdings but none of the main findings about the size anddecomposition of the monetary policy shock are affected.

86Baseline

(1)

Loose
BorrowLimit
b = 4 × Y qu(2)

Low
BorrowCostsκ = 4% pa(3)HighBorrowCostsκ = 8% pa(4)Very High
BorrowCostsκ = 20% pa
(5)

0.23
0.10
0.19
0.15
0.16

0.21
0.10
0.17
0.17
0.16

0.22
0.07
0.19
0.25
0.14

0.24
0.12
0.18
0.09
0.16

0.25
0.09
0.15
0.04
0.15

Change in r b (pp)-0.28 %

-0.29 %

-0.28 %

-0.27 %

-0.27 %

Elasticity of Y
Elasticity of I

-3.96
-9.43

-3.75
-9.05

-3.72
-8.94

-4.17
-9.89

-4.17
- 9.94

Elasticity of C
Partial Eq. Elast. of C

-2.93
-0.55

-2.81
-0.64

-2.75
-0.62

-3.08
-0.52

-3.08
-0.52

23 %
50 %
29 %
-1 %

23 %
49 %
30 %
-2 %

17 %
53 %
33 %
-2 %

17 %
53 %
33 %
-2 %

Mean b (rel. to GDP)
Frac with b = 0, a = 0
Frac with b = 0, a > 0
Frac with b < 0
Quarterly $500 MPC

Component of change in C due to:
Direct effect: r b
19 %Indirect effect: w
51 %Indirect effect: T
32 %Indirect effect: r a q-2 %Table E.3: Robustness: borrowing environment
Notes: Average responses over the first year. Column (1) is the baseline specification.Column (2)loosens the borrowing limit from 1 times quarterly GDP to 4 times quarterly GDP. Column (3)lowers the wedge between the liquid borrowing and liquid savings rates from 6%pa to 4%pa. Column(4) raises the wedge between the liquid borrowing and liquid savings rates from 6%pa to 8%pa.
Column (5) raises the wedge between the liquid borrowing and liquid savings rates from 6%pa to
20%pa. All experiments re-calibrate the discount rate ρ so that mean illiquid assets relative to GDP
is held constant.

87

Baseline(1)No
LinearCost
χ0 = 0(2)

HighLinearCost
χ0 = 0.10(3)

Low
ConvexCostχ2 = 0.10(4)High
ConvexCostχ2 = 1
(5)

0.23
0.10
0.19
0.15
0.16

0.22
0.09
0.19
0.15
0.16

0.26
0.12
0.18
0.14
0.15

0.44
0.22
0.05
0.12
0.11

0.11
0.01
0.19
0.13
0.14

Change in r b (pp)-0.28 %

-0.28 %

-0.27 %

-0.27 %

-0.26 %

Elasticity of Y
Elasticity of I

-3.96
-9.43

-3.99
-9.68

-3.94
-9.20

-3.64
-7.36

-5.52
-18.64

Elasticity of C
Partial Eq. Elast. of C

-2.93
-0.55

-2.88
-0.54

-3.02
-0.55

-3.47
-0.56

-2.59
-0.43

19 %
52 %
31 %
-2 %

18 %
51 %
33 %
-2 %

16 %
43 %
42 %
0%

17 %
62 %
26 %
-4 %

Mean b (rel. to GDP)
Frac with b = 0, a = 0
Frac with b = 0, a > 0
Frac with b < 0
Quarterly $500 MPC

Component of change in C due to:
Direct effect: r b
19 %Indirect effect: w
51 %Indirect effect: T
32 %Indirect effect: r a q-1 %Table E.4: Robustness: adjustment cost function
Notes: Average responses over the first year. Column (1) is the baseline specification. Column (2)sets the linear component of the adjustment cost function to zero. Column (3) increases the linear
component of adjustment cost function from 4.4% to 10%. Column (4) reduces the exponent on theconvex component of the adjustment cost function from 0.4 to 0.1. Column (5) increases the
exponent on the convex component of the adjustment cost function from 0.4 to 1. All experiments
re-calibrate the discount rate ρ so that mean illiquid assets relative to GDP is held constant.

88

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