Monetary Policy and Wealth Effects:
The Role of Risk and Heterogeneity∗
N ICOLAS C ARAMPD EJANIR H. S ILVAUC Davis

Purdue University

August 31, 2021

Abstract
We study the role of wealth effects, i.e. the revaluation of real and financial
assets, in the monetary policy transmission mechanism. We build an analytical
heterogeneous-agents model with two main ingredients: i) rare disasters andii) risky household debt. The model captures time-varying risk premia andprecautionary savings in a linearized setting that nests the textbook New Keynesian model. Quantitatively, the model matches the empirical response of
asset prices and the heterogeneous impact on borrowers and savers. We findthat wealth effects induced by time-varying risk and household debt accountfor the bulk of the output response to monetary policy. Defaultable private
debt amplifies the effects of monetary policy, but the effect decreases with its
duration. Controlling for the fiscal response to a monetary shock is crucial forthe quantitative results.

JEL Codes: E21, E52, E44
Keywords: Monetary Policy, Wealth Effects, Asset Prices, Heterogeneity∗ Caramp: UC Davis.

Email: ncaramp@ucdavis.edu. Silva: Krannert School of Management,
Purdue University. Email: dejanir@purdue.edu. We would like to thank Francesco Bianchi, Francisco Buera, Colin Cameron, James Cloyne, Bill Dupor, Gauti Eggertsson, Òscar Jorda, Rohan
Kekre (discussant), Narayana Kocherlakota, Steffan Lewellen (discussant), Dmitry Mukhin, Benjamin Moll, Sanjay Singh, Jón Steinsson, Gianluca Violante (discussant) and seminar participants at
EFA Annual Meeting, NBER Summer Institute, UC Berkeley, UC Davis, the Federal Reserve Bankof St. Louis, SED, and WFA Annual Meeting for their helpful comments and suggestions.

1Introduction

A long tradition in monetary economics emphasizes the role of wealth effects, i.e.
the revaluation of real and financial assets, in the economy’s response to changes
in monetary policy. Its importance can be traced back to both classical and Keynesian economists, such as Pigou, Patinkin, Metzler, and Tobin.1 Keynes himself
described the effects of interest rate changes as follows:
There are not many people who will alter their way of living because the rate of interest has fallen from 5 to 4 per cent, if their aggregate income is the same as before.
[...] Perhaps the most important influence, operating through changes in the rate ofinterest, on the readiness to spend out of a given income, depends on the effect of thesechanges on the appreciation or depreciation in the price of securities and other assets.
- John Maynard Keynes, The General Theory of Employment, Interest, and Money(emphasis added).

There is a large empirical literature documenting the impact of monetary policy on
asset prices. Bernanke and Kuttner (2005) and Kekre and Lenel (2020) study theeffects of monetary shocks on stock prices. Gertler and Karadi (2015) and Hanson
and Stein (2015) consider the effects on bonds. Moreover, Cieslak and VissingJorgensen (2020) shows that policymakers track the behavior of stock markets because of their impact on households’ consumption, while Chodorow-Reich et al.
(2021) establish the importance of this channel empirically.
Despite the evidence, relatively little work has been done to theoretically study
the role of asset price fluctuations on the monetary transmission mechanism. An
important reason for this is that incorporating these channels represents a challenge to standard monetary models. A robust finding of the empirical literatureis that changes in asset prices can be explained mainly by fluctuations in futureexcess returns, related to changes in the risk premia, rather than changes in therisk-free rate. However, the standard approach abstracts from risk premia andgenerates counterfactual asset-pricing dynamics. Moreover, models that featurericher asset-pricing dynamics require the use of complex global or high-order perturbation methods, which lack the insights of the role of the different channels oftransmission provided by analytically tractable models.
1 The revaluation of government liabilities was central to Pigou (1943) and Patinkin (1965), while

Metzler (1951) considered stocks and money. Tobin (1969) focused on how monetary policy interacted with the value of real assets.

1

In this paper, we propose a new framework that generates rich asset-pricing
dynamics and heterogeneous portfolios while preserving the simplicity of the textbook New Keynesian model, and study the role of wealth effects in the economy’s
response to monetary policy. The model has two main ingredients: i) rare disasters and ii) household debt. Rare disasters allow us to capture both a precautionarysavings motive and realistic risk premia. Barro (2009) and Gabaix (2012) argue thatthe risk of a rare disaster and, in particular, its time-varying component, can successfully explain major asset-pricing facts.2 Moreover, private debt is a significant
component of households’ portfolios, representing 75% of GDP, and, as recently
shown by Cloyne et al. (2020), borrowers account for the bulk of the response ofaggregate consumption to changes in interest rates. By incorporating private debt,
we are able to capture the role of revaluations in both gross and net asset positions. We also study the role of default risk and long maturities in household debt,
important features of most debt contracts, such as mortgages or student debt.
To capture the time-varying component of risk premia, we assume that theprobability of a disaster depends on the level of the nominal interest rate. Using
data on a panel of advanced economies since the nineteenth century, Schularick
et al. (2021) finds that contractionary monetary shocks significantly increases theprobability of a subsequent financial crisis. Based on this evidence, we provide
a new micro-foundation for the relationship between monetary policy, rare disasters, and asset prices. In an extension with financial intermediaries, we showthat banks’ exposure to interest rate risk makes them vulnerable to runs. A contractionary monetary shock then weakens banks’ balance sheets and increases theprobability of a financial crisis, an endogenous disaster. Our assumption on theprobability of disaster enables us to capture this mechanism in our baseline model.
We consider an economy populated by two types of households, borrowers
and savers. The model captures key features of heterogeneous-agents New Keynesian (HANK) models, such as precautionary savings and heterogeneous marginal
propensities to consume (MPCs), in a setting with positive private debt, a combination that has been elusive in the analytical HANK literature. Despite being
stylized, the model captures quantitatively central features of the monetary transmission mechanism, including important asset-pricing moments such as the term
2 Rare disasters have been widely used to explain a range of asset-pricing “puzzles”; see Tsaiand Wachter (2015) for a review.

2premium, the equity premium, and corporate spreads, as well as the differential
responses of borrowers and savers to monetary shocks observed in the data. Given
this success, we use the model to quantitatively assess the importance of the channels through which monetary policy affects households’ consumption. We findthat time-varying risk and private debt jointly account for more than 80% of the initial response of aggregate consumption to a monetary shock. These results reveal
the importance of accounting for risk and portfolio heterogeneity to understandthe economy’s response to monetary policy.
Our solution method allows us to obtain time-varying risk premia in a linearized setting and provide a complete analytical characterization of the channels involved. The method consists on perturbing the economy around a stationary equilibrium with positive aggregate risk instead of adopting the more common
approach of approximating around a non-stochastic steady state. By perturbing
around the stochastic stationary equilibrium, we are able to obtain time variationin precautionary motives and risk premia using a first-order approximation, while
the standard approach would require a third-order approximation (see e.g. Andreasen 2012). Moreover, by linearizing around an economy with zero monetaryrisk, we are able to solve for the stochastic stationary equilibrium in closed form,
avoiding the need to compute the risky steady state numerically, as in Coeurdacieret al. (2011). This hybrid approach allows us to capture the effect of aggregate risk
on asset prices in a linearized model.
The first result states that output satisfies an aggregate Euler equation, where its
sensitivity to interest rates depends on the disaster risk and the level of private
debt. With zero private liquidity and disaster probability, our economy features
a discounted Euler equation, where output is less sensitive to future interest ratechanges due to a precautionary motive, as in the incomplete-markets model of
McKay et al. (2017). The presence of private debt acts in the opposite direction,
as it pushes the economy towards compounding. We find that the second effectdominates in our calibration, so the aggregate Euler equation features compounding, even though, at the micro level, the savers’ Euler equation always features
discounting.
We turn next to the channels through which monetary policy affects the economy. We show that equilibrium output can be characterized as the sum of fourterms: an intertemporal-substitution effect (ISE); a time-varying risk effect; an inside3

wealth effect, i.e. the change in valuation of assets in zero net supply; and an outside
wealth effect, i.e. changes in the valuation of assets in positive net supply.3
The ISE corresponds to the output response that operates through changes inthe timing of output but not its overall (present value) level. While this channel
is quantitatively important in the textbook New Keynesian model, we find that it
has a marginal impact in the presence of heterogeneous agents and risk, consistent with the findings in Kaplan et al. (2018). Most of the output response can be
explained by time-varying risk and wealth effects.
The time-varying risk effect captures the change in the households’ precautionary motives due to changes in the probability of a disaster. When the probabilityof disaster is constant, the model is able to capture important unconditional assetpricing moments, such as the level of the equity premium and an upward-sloping
yield curve, but it fails to generate the observed response of risk premia to monetary shocks. This failure has significant real consequences, as aggregate risk hasthen only a minor impact on the response of output and inflation. With timevarying disaster risk, the model is able to simultaneously match how long-termbonds, corporate spreads, and equities respond to monetary shocks in the data,
and the impact on output increases almost threefold. This highlights the importance of matching the empirical response of asset prices to properly assess the roleof risk in determining how monetary policy affects the economy.
The inside wealth effect captures the aggregate implications of the differential
response of borrowers and savers to changes in interest payments.4 An increase innominal interest rates creates a positive wealth effect on savers, as they receive ahigher income from private lending, and a corresponding loss to borrowers. Given
the higher MPC for borrowers, this generates a negative aggregate response of
output on impact. We also show that allowing for time-varying default risk inhousehold debt substantially amplifies the effect of monetary policy, but this effectgets attenuated when debt has a high duration.
Finally, the outside wealth effect is the sum of the change in wealth for all
households. This includes the change in the value of stocks, government bonds,
3 The notion of inside/outside wealth is reminiscent of inside/outside money as used by Gurleyand Shaw (1960), and, more recently, inside/outside liquidity by Holmstrom and Tirole (2011).
4 Note that previous analytical HANK models focused on either heterogeneous MPC but noprivate debt, as in Bilbiie (2018), or positive private debt and no heterogeneity in MPCs, as in
Acharya and Dogra (2020).

4and human wealth, net of the impact of discount rates on the present discounted
value of consumption. An important result of our analysis is that the outside
wealth effect is tightly connected to the response of fiscal policy to monetary shocks.5
In particular, we show that the outside wealth effect is proportional to the revaluation of public debt and the fiscal backing, that is, the change in taxes and transfers,
in response to monetary shocks. Intuitively, in a closed economy, the governmentis the only trading counterpart to the household sector as a whole, so the outside
wealth effect can be inferred from the impact of monetary policy on government
finances. More importantly, this result implies that we can use standard estimation
techniques to identify the fiscal response to a monetary shock and discipline theability of the model to generate quantitatively meaningful wealth effects.
We find that when constrained to match the estimated fiscal response, the standard RANK model generates a substantially weaker output response to monetaryshocks than when fiscal backing is determined by a standard Taylor rule that restricts monetary shocks to follow an AR(1) process. Equivalently, the standard Taylor equilibrium requires a (passive) fiscal response that is counterfactually large.
Our results can be made consistent with a Taylor equilibrium by allowing a moregeneral specification of the monetary shock. We can then use both monetary andfiscal data to discipline the parameters of the interest rate rule. In this context, thepresence of heterogeneity and risk becomes particularly relevant, as these forces
can compensate for the missing fiscal response.
To quantify the importance of the channels present in the model, we decompose
the response of output by sequentially adding time-varying risk and private debt
to the standard RANK model. We find that time-varying risk accounts for morethan 50% of the output response, while private debt accounts for roughly 20%, andthe interaction between the two accounts for 10%.
Literature review.

Wealth effects have a long tradition in monetary economics.

Pigou (1943) relied on a wealth effect to argue that full employment could bereached even in a liquidity trap. Kalecki (1944) argued that these effects apply
only to government liabilities, as inside assets cancel out in the aggregate, while

5 See Caramp (2021) for the role of fiscal policy in the monetary transmission mechanism.

5

Tobin highlighted the role of private assets and high-MPC borrowers.6 Recently,
wealth effects have regained relevance. In an influential paper, Kaplan et al. (2018)
build a quantitative HANK model and find only a minor role for the standard
intertemporal-substitution channel, leading the way to a more important role for
wealth effects. Much of the literature has focused on the role of heterogeneous
marginal propensities to consume (MPCs) in settings with idiosyncratic income
risk. Instead, our focus is on aggregate risk and private debt.
Our work is closely related to two strands of literature. First, it relates to theanalytical HANK literature, such as Werning (2015), Debortoli and Galí (2017),
and Bilbiie (2018). While this literature focuses primarily on how the cyclicalityof income interacts with differences in MPCs, we focus instead on how heterogeneous asset positions interact with differences in MPCs. We see these two channels
as mostly complementary: even though Cloyne et al. (2020) does not find significant differences in income sensitivity across borrowers and savers, Patterson (2019)
finds a positive covariance between MPCs and the sensitivity of earnings to GDP
across different demographic groups, suggesting that the income-sensitivity channel is operative for a different cut of the data. We share with Eggertsson and Krugman (2012) and Benigno et al. (2020) the emphasis on private debt, but they abstractfrom a precautionary motive and focus instead on the implications of deleveraging. Iacoviello (2005) also considers a monetary economy with private debt but
focuses instead on the role of housing as collateral. Our work is also related to Auclert (2019), which studies the redistribution channel of monetary policy arisingfrom portfolio heterogeneity. Our paper emphasizes the redistribution channel inthe context of a general equilibrium setting with aggregate risk.
The paper is also closely related to work on how monetary policy affects the
economy through changes in asset prices, including models with sticky prices,such as Caballero and Simsek (2020), and models with financial frictions, suchas Brunnermeier and Sannikov (2016) and Drechsler et al. (2018). In recent contributions, Kekre and Lenel (2020) consider the role of the marginal propensity totake risk in determining the risk premium and shaping the response of the econ6 Tobin (1982) describes the role of inside assets: “The gross amount of these ’inside’ assets was

and is orders of magnitude larger than the net amount of the base. Aggregation would not matter
if we could be sure that the marginal propensities to spend from wealth were the same for creditorsand debtors. But if the spending propensity were systematically greater for debtors, even by asmall amount, the Pigou effect would be swamped by this Fisher effect.”6

omy to monetary policy, and Campbell et al. (2020) use a habit model to study
the role of monetary policy in determining bond and equity premia. Our model
highlights instead the role of heterogeneous MPCs, positive private liquidity, anddisaster risk in an analytical framework that preserves the tractability of standard
New Keynesian models.
Finally, a recent literature studies rare disasters and business cycles. Gabaix(2011) and Gourio (2012) consider a real business cycle model with rare disasters,
while Andreasen (2012) and Isoré and Szczerbowicz (2017) allow for sticky prices.
They focus on the effect of changes in disaster probability while we study monetary shocks in an analytical HANK model with rare disasters.

2

An Analytical Rare Disasters HANK ModelIn this section, we consider an analytical HANK model with two main ingredients:
the possibility of rare disasters and positive private liquidity. First, we describe thenon-linear model and later consider a log-linear approximation around a stochastic
stationary equilibrium.

2.1

The Model

Environment. Time is continuous and denoted by t ∈ R+ . The economy is populated by households, firms, and a government. There are two types of households,
borrowers and savers, who differ in their discount rates. A mass 0 ≤ µb < 1 ofhouseholds are borrowers and a mass µs = 1 − µb are savers. Households canborrow or lend at a riskless rate, but they are subject to a borrowing constraint.
Firms can produce final or intermediate goods. Final-goods producers operatecompetitively and combine intermediate goods using a CES aggregator with elasticity e > 1. Intermediate-goods producers use labor as their only input and face
Rotemberg pricing adjustment costs.7 Intermediate-goods producers are subject
to an aggregate productivity shock: with Poisson intensity λt ≥ 0, they receive
a shock that permanently reduces their productivity. This shock is meant to capture the possibility of rare disasters: low-probability, large drops in productivity
7 Rotemberg adjustment costs simplify the derivations, but they are not essential for our results.

7and output, as in the work of Barro (2006, 2009). We say that periods that predate the realization of the shock are in the no-disaster state, and periods that follow
the shock are in the disaster state. The disaster state is absorbing, and there are nofurther shocks after the disaster is realized. Assuming an absorbing disaster state
simplifies the presentation, but it can be easily relaxed, as shown in Appendix C.2.8
The government sets fiscal policy, comprising a sales tax on intermediate-goods
producers and transfers to borrowers and savers, and monetary policy, specified
by an interest rate rule subject to a sequence of monetary shocks. We assume thatthe government issues long-term nominal bonds that pay exponentially decaying
coupons, where the coupon in period t is given by e−ψL t . The rate of decay ψL isinversely related to the bond’s duration, where a perpetuity corresponds to ψL = 0
and the limit ψL → ∞ corresponds to the case of short-term bonds. We denote by
Q L,t the nominal price of the bond in the no-disaster state and by Q∗L,t the price ofthe bond in the disaster state, where the star superscript is used throughout the
paper to denote variables in the disaster state.
Households’ problem.

Households face a portfolio problem where they choose

how much to invest in short-term and long-term bonds. In this section, we assume
that borrowers issue only short-term risk-free bonds and the government issues
only long-term bonds. We study the case of defaultable long-term household debt
in Section 5. The nominal return on the long-term bond is given byQ∗L,t − Q L,t
Q̇ L,t1+− ψL dt +dNt ,dR L,t =Q L,t Q L,t
Q L,twhere Nt is a Poisson process with arrival rate λt .
L denote the total value of bonds (in real terms) held by a
Let Bj,t = BSj,t + Bj,ttype-j household, j ∈ {b, s}, that is, the sum of short-term (BSj,t ) and long-termL ) bonds. The problem of a household of type j is to choose consumption C ,
(Bj,t
j,t
L , given an initial real value of bonds B ,labor supply Nj,t , and long-term bonds Bj,tj,t8 Allowing for partial recovery after a disaster, as in Barro et al. (2013) and Gourio (2012), intro-

duces dynamics in the disaster state, but it does not change the main implications for the no-disasterstate, which is our focus.

8to solve the following problem:
Vj,t ( Bj,t ) =

max

L ]
[Cj,z ,Nj,z ,Bj,z
z≥t

Et 

ˆ t∗
t


e−ρ j (z−t) C1j,z−σ
1−σ

1+ φ

−

Nj,z

1+φ





∗
∗
 dz + e−ρ j (t∗ −t) Vj,t∗ ( B j,t∗ )  ,subject to the flow budget constraint
dBj,t =Wt
L
ej,t − Cj,t
N + Π j,t + T(it − πt ) Bj,t + r L,t Bj,t+Pt j,tL
dt + Bj,t

Q∗L,t − Q L,t
Q L,t

dNt ,

and the borrowing constraints
Bj,t ≥ − D P

and

L
Bj,t
≥ 0,

where ρb > ρs > 0, Wt is the nominal wage, Pt is the price level, Π j,t denotes
ej,t denotes government transfers, and r L,t ≡real profits from corporate holdings, T
Q̇ L,t
1
Q L,t + Q L,t − ψL − it is the excess return on long-term bonds conditional on no disasters. The random (stopping) time t∗ represents the period in which the aggregate∗ (·) and B∗ denote, respectively, the value function
shock hits the economy. Vj,t∗j,t∗and the real value of bonds in the disaster state. The non-negativity constraint on
L captures the assumption that only the government can issue long-term bonds.
Bj,t

We assume that Bs,0 > 0 and Bb,0 = − D P . For sufficiently large ρb , borrowers
are constrained in all periods. We also assume that Πb,t = 0, that is, firms are
entirely owned by savers.9
The labor supply is determined by the standard condition:Wt
φ σ= Nj,t Cj,t.
Pt
The Euler equation for short-term bonds, if Bj,t > − D P , is given by
Ċj,tλt= σ −1 ( i t − π t − ρ j ) +
Cj,t
σ

"

Cj,t
∗
Cj,t

!σ

#

−1 ,

(1)

∗ is the consumption of household j in the disaster state. The first term
where Cj,t
9 Alternatively, we could have assumed that households can trade shares of the firms.

In asteady state, borrowers would choose to sell their shares, and savers would entirely own the firms.

9captures the usual intertemporal-substitution force present in RANK models. The
second term captures the precautionary savings motive generated by the disaster risk,
and it is analogous to the precautionary motive that emerges in HANK models
with idiosyncratic risk.
L > 0, is given by
The Euler equation for long-term bonds, if Bj,tr L,t = λt|

!σ
Q L,t − Q∗L,t
Cs,t
.
∗
Cs,t
Q L,t
{z
}|
{z
}

price ofdisaster risk

(2)

quantity ofriskThis expression captures a risk premium on long-term bonds, which pins down
the level of long-term interest rates in equilibrium. The premium on long-termbonds is given by the product of the price of disaster risk, the compensation for aunit exposure to the risk factor, and the quantity of risk, the loss the asset suffers
conditional on switching to the disaster state.
Firms’ problem. Intermediate-goods producers are indexed by i ∈ [0, 1] and operate in monopolistically competitive markets. Final good producers are price takers and combine intermediate goods to produce the final good. Their demand for −ePvariety i is given by Yi,t = Pi,tt
Yt , and the equilibrium price level is given
1
´

1− e
1 1− ePt = 0 Pi,t.
diIntermediate-goods producers operate the linear technology Yi,t = At Ni,t . Productivity in the no-disaster state is given by At = A, and productivity in the disaster state is given by At = A∗ , where 0 < A∗ < A. Intermediate-goods producers
choose the rate-of-change of prices πi,t = Ṗi,t /Pi,t , given the initial price Pi,0 , to maximize the expected discounted value of real (after-tax) profits subject to Rotemberg
quadratic adjustment costs:"ˆt∗

Qi,t ( Pi,t ) = max Et[πi,z ]z≥t

t

ηz
ηt



#

Pi,z
Wz Yi,zϕ 2
ηt ∗ ∗(1 − τ )Yi,z −− πi,t dz +Q ∗ ( Pi,t∗ ) , (3)
Pz
Pz A
2
ηt i,tsubject to the demand Yi,t =



Pi,t
Pt

−e

Yt and Ṗi,t = πi,t Pi,t , where ηt denotes the∗ ( P ) denotes thestochastic discount factor (SDF) that is relevant to firms and Qi,tifirms’ value function in the disaster state. Note that the price Pi,t is a state variable10

in the firms’ problem and πi,t is a control variable. The parameter ϕ controls themagnitude of the pricing adjustment costs. We assume that these costs are rebatedto households, so they do not represent real resource costs. Moreover, as firms
are owned by savers, we assume that firms discount profits using the SDF ηt =−σe−ρs t Cs,t.

Combining the first-order condition and the envelope condition for problem(3), we obtain the non-linear New Keynesian Phillips curve:

π̇t =

η∗
it − πt + λt tηtπt − ϕ−1( e − 1)e Wt 1− (1 − τ ) Yt ,e − 1 Pt A

(4)assuming a symmetric initial condition Pi,0 = P0 , for all i ∈ [0, 1].
Time-varying risk.

To capture the effect of monetary policy on the market priceof risk, as documented e.g. by Gertler and Karadi (2015) and Hanson and Stein(2015), we assume that monetary shocks affect directly the probability of disasters:
λt = λ(it − rn ), for a given function λ(·). Importantly, we assume that λ(·) is anincreasing function, such that a contractionary monetary shock raises the probability of a disaster. This is consistent with recent evidence by Schularick et al. (2021)
on the effects of monetary policy on the probability of a financial crisis.
Motivated by this evidence, we provide a new micro-foundation for the relationship between monetary policy and asset prices, where monetary shocks endogenously affect the probability of a crisis and, ultimately, the market price ofrisk. In Appendix B, we extend our model to incorporate banks that borrow shortterm from savers to lend long-term to firms. Given this maturity mismatch, a contractionary monetary shock weakens the balance sheet of banks. As in Gertler et al.
(2020), a weaker balance sheet makes banks more vulnerable to bank runs, so monetary policy affects the probability of crises through its impact on banks’ balance
sheet positions. In particular, we show that the sensitivity of λt to monetary shocks
depends on features of the economy, such as bank’s leverage, the maturity of bankloans, and even the persistence of monetary shocks.10 Moreover, we show that the(linearized) model with banks and endogenous disasters is essentially identical tothe model without financial intermediaries but with a sensitivity that depends on
10 This implies that the sensitivity is not invariant to economic policy. This dependence does notaffect our positive analysis, but it is potentially relevant when considering normative questions.

11the financial sector’s characteristics in the stationary equilibrium. For this reason,
we describe the details of the micro-foundation in Appendix B, and here we simply
assume that λ(·) depends on monetary policy.
Government. The government’s flow budget constraint in the no-disaster stateis given by
ḊG,t = (it − πt + r L,t ) DG,t +∑ µ j Tej,t − τYt ,j∈{b,s}and the No-Ponzi condition limt→∞ E0 [ηt DG,t ] ≤ 0, DG,t denotes the real value ofgovernment debt and DG,0 = D G is given. We assume that government transfers
eb,t = Teb (Yt ), where transfers deto borrowers are determined by the policy rule T
eb (·) determines the cyclicality ofpend on aggregate output and the elasticity of T
government transfers to borrowers.
In the no-disaster state, monetary policy is determined by the policy ruleit = rn + φπ πt + ut ,(5)

where φπ > 1, ut is a monetary shock, and rn denotes the real rate when πt = ut =0 at all periods. We assume that in the disaster state there are no monetary shocks,
that is, it∗ = rn∗ + φπ πt∗ . By abstracting from the policy response after a disaster, we
isolate the impact of changes in monetary policy during “normal times.”
Market clearing. The market-clearing conditions for goods, labor, and bonds are
given by∑ µ j Cj,t = Yt ,j∈{b,s}2.2∑ µ j Nj,t = Nt ,∑ µ j Bj,tL = DG,t ,j∈{b,s}j∈{b,s}∑ µ j BSj,t = 0.

j∈{b,s}Equilibrium dynamics

Stationary equilibrium.

We define a stationary equilibrium as an equilibrium inwhich all variables are constant in each aggregate state. In particular, the economy
will be in a stationary equilibrium in the absence of monetary shocks, that is, ut = 0
for all t ≥ 0. Since variables are constant in each state, we drop time subscripts and∗ = C ∗ . For ease of exposition, we follow Bilbiie
write, for instance, Cj,t = Cj and Cj,tj

12

eb implements the(2018) and focus on a symmetric stationary equilibrium, where T
same consumption level for each household, and discuss the general case Cb 6= Cs
in the appendix.
The natural interest rate, the real rate in the stationary equilibrium, is given by
rn = ρs − λ

Cs
Cs∗

σ



−1 ,

where 0 < Cs∗ < Cs and, with a slight abuse of notation, λ = λ(0) > 0 is the disaster intensity when it = rn . The presence of a precautionary motive depresses thenatural interest rate relative to the one that would prevail in a non-stochastic economy, and the magnitude depends on the extent to which savers can self-insure.
In particular, holding everything else constant, a higher level of private debt D P
implies a weaker precautionary motive and a higher natural interest rate.
From Equation (2), we obtain the term spread, the difference between the yieldon the long-term bond and the short-term rate,i L − rn = λ

Cs
Cs∗

σ

Q L − Q∗L,
QL

where i L is the yield on the long-term bond in the stationary equilibrium.11 We
show in Appendix C.2.2 that the term spread i L − rn is strictly positive. Thus, ourmodel generates an upward-sloping yield curve, where the yield on the long-termbond exceeds the natural (short-term) rate, consistent with the data.12
Log-linear dynamics. Following the practice in the literature on monetary policy, we focus on a log-linear approximation of the equilibrium conditions. However, instead of linearizing around the non-stochastic steady state, we linearize theequilibrium conditions around the (stochastic) symmetric stationary equilibrium
described above. Formally, we perturb the allocation around the economy where
ut = 0 and λ > 0, while the standard approach would perturb around the economy where ut = λt = 0. This enables us to capture the effects of (time-varying)
11 Note that the yield on the bond is given by i

−1
L,t = Q L,t − ψL and, in a stationary equilibrium,

the expected excess return conditional on no disaster r L equals the term spread i L − rn .
12 The mechanism behind the upward-sloping yield curve is related to the lack of precautionarysavings in the disaster state. We would obtain similar results by introducing expropriation andinflation in a disaster, as in Barro (2006).

13

precautionary savings and risk premia in a linear setting.13
Let lower-case variables denote log-deviations from the stationary equilibrium,e.g., c j,t ≡ log Cj,t /Cj and n j,t ≡ log Nj,t /Nj . Borrowers’ consumption is given by
cb,t = (1 − α)(wt − pt + nb,t ) + Tb,t − (it − πt − rn )d P ,where 1 − α ≡ WPYN is the labor share in the stationary equilibrium, Tj,t ≡(6)
ej,t − TejTY ,and d P ≡ DYP . Using the fact that transfers satisfy Tb,t = Tb0 (Y )yt , and solving forthe real wage, we obtaincb,t = χy yt − χr d P (it − πt − rn ),where χy ≡(7)

Tb0 (Y )+(1−α)(1+φ)(1+φ−1 σ)and χr ≡ 1+(1−1α)φ−1 σ . The coefficient χy con1+(1−α)φ−1 σtrols the cyclicality of income inequality and has been extensively studied by theliterature on analytical HANK models. We focus throughout the paper on the case1
in which 0 < χy < µ−b , such that the consumption of both agents increases withyt .14 The second term is not present in the commonly studied case of zero private
liquidity, d P = 0, and it captures the impact of monetary policy on the consumption of constrained agents that is not directly mediated by aggregate output yt .
This term plays an important role in the analysis that follows.
Next, consider the savers’ problem. Recall that we assume that the disasterprobability depends on the interest rate, λt = λ(it − rn ). In our linearized setting,the only relevant parameter is the semi-elasticity of the disaster probability withrespect to monetary shocks, eλ ≡ λ0 (0)/λ(0). We focus on the case in which eλ ≥0, with eλ = 0 corresponding to a benchmark with constant probability.15 Then,the savers’ Euler equation is given byċs,t = σ−1

(i t − π t − r n ) + λ

Cs
Cs∗

σ
cs,t + χd eλ (it − rn ),(8)

13 This method differs from the procedure considered by Fernández-Villaverde and Levintal(2018) or Coeurdacier et al. (2011), as we linearize around a stochastic steady state of an economy
with no monetary shocks, instead of the stochastic steady state of the economy with both shocks.
14 The role of χ , including the case where χ > µ−1 , was originally considered by Bilbiie (2008).
yyb
15 See Appendix B for the micro-foundation of this relationship and how e depends on deeper
λstructural parameters.

14where χd ≡

h

Cs
Cs∗

σ

i

− 1 is a parameter capturing the strength of the precautionary motive in the stationary equilibrium. Importantly, time-varying disasterrisk introduces a new precautionary savings channel for savers, which ultimately
shapes the impact of changes in nominal rates on households’ consumption.
Combining condition (7) for borrowers’ consumption, equation (8) for savers’
Euler equation, and the market-clearing condition for goods, we obtain the evolution of aggregate output. Proposition 1 characterizes the dynamics of aggregate
output and inflation. All proofs are provided in Appendix A.
λσ

Proposition 1 (Aggregate dynamics). The dynamics of output and inflation is describedby the conditions:i. Aggregate Euler equation:ẏt = σ̃−1 (it − πt − rn ) + δyt + vt ,(1 − µ b ) σ −1 − µ b χ r d P r n,δ≡λ1− µ b χ y
1− µrn ) − i̇t ) + 1−µ χb y χd eλ (it − rn ).
bwhere σ̃−1 ≡



Cs
Cs∗

σ

(9)

µ b χr d Pb χr d P κ− µ1−µb χy and vt ≡ 1−µb χy ( ρ (it −ii. New Keynesian Phillips curve:π̇t = ρπt − κyt ,(10)

where ρ ≡ ρs + λ and κ ≡ ϕ−1 (e − 1)(1 − τ )(φ + σ)Y.
Condition (9) represents the aggregate Euler equation for this economy. The aggregate Euler equation has three terms. The first term, the product of the (aggregate) elasticity of intertemporal substitution (EIS) and the real interest rate, corresponds to the one present in RANK models. The dependence of the aggregate EIS
on the cyclicality of inequality is well-known in the literature, while the result thatprivate liquidity may reduce σ̃−1 is, to the best of our knowledge, new.16
The second term, δyt , captures how the impact of real interest rate changes
can be compounded or discounted in equilibrium. The sign of δ and, therefore,
whether the economy exhibits compounding or discounting, depends on two forces.
With disaster risk but in the absence of private debt, so that λ > 0 and d p = 0,
16 In our calibration, we obtain σ̃ −1 > 0, but most of our results do not rely on this.

15

we obtain δ > 0. This corresponds to the discounted Euler equation of McKay
et al. (2017), where aggregate disaster risk plays the role of idiosyncratic income
risk. In contrast, if λ = 0 but d P > 0, we get compounding, that is, δ < 0. As acontractionary monetary shock depresses the economy and reduces inflation in all
periods, it increases the real burden of debt for borrowers, amplifying the effect ofthe monetary shock. This amplification translates into a compounded response of
output to future interest rate changes. More generally, the aggregate Euler equation (9) can feature compounding or discounting. In particular, if λ is sufficiently
small, we can have that savers’ consumption satisfies a discounted Euler equation
while the aggregate Euler equation features compounding.
The third term in the aggregate Euler equation, vt , captures a direct effect ofmonetary policy on households that is not mediated by the intertemporal substitution motive or changes in aggregate demand. First, in an economy with household debt, monetary policy directly affects borrowers’ disposable income, the highMPC agents in this economy. Second, the time-varying component of the disasterrisk directly impacts the savers’ precautionary savings motive. Therefore, monetary policy has real effects even in the absence of intertemporal-substitution forces.
Finally, Proposition 1 derives the New Keynesian Phillips curve. The linearized
Phillips curve coincides with the one obtained from models with Calvo pricing. As
in a textbook New Keynesian model, inflation is given by the present discounted´∞
value of future output gaps, πt = κ t e−ρ(s−t) ys ds. One distinction relative to thestandard formulation is that future output gaps are not discounted by the naturalrate rn but by a higher rate ρ > rn . This is a consequence of the riskiness of thefirm’s value, so the appropriate discount rate incorporates an adjustment for risk.
Asset prices.

The response of asset prices to monetary policy depends cruciallyon the behavior of the price of disaster risk. In its log-linear form, the price ofdisaster risk is given by
pd,t ≡ σcs,t + eλ (it − rn ).

(11)

Note that this expression has two terms. The first term captures the change in theeffective size of the shock, represented by the drop in the savers’ marginal utilityof consumption if the disaster shock is realized. The second term represents thechange in the disaster probability after a monetary shock.

16

Given the price of risk, we can price any financial asset in this economy. For
example, the (linearized) price of the long-term bond in t = 0 is given byˆ ∞q L,0 = −

e
|0

ˆ ∞

−(ρ+ψL )t(it − rn )dt −e−(ρ+ψL )t r L pd,t dt .
{z
} |0
{z
}

path of nominal interest rates(12)

term premium

The yield on the long-term bond, expressed as deviations from the stationary equi1
librium, is given by − Q−
L q L,0 , which can be decomposed into two terms: the pathof nominal interest rates, as in the expectations hypothesis, and a term premium,capturing variations in the compensation for holding long-term bonds. The term
premium depends on the price of risk, pd,t , and the asset-specific loading r L . Because the term premium responds to monetary shocks, the expectation hypothesis
does not hold in this economy. This is important since the term premium accounts
for the bulk of the response of long-term rates to monetary policy in the data.
The pricing condition for stocks is analogous to the one for bonds:
Y
qS,0 =
Q
| S

ˆ ∞
0

e−ρt [(1 − τ )yt − (1 − α)(wt − pt + nt )] dt −{z}
dividends
ˆ ∞e−ρt [it − πt − rn + rS pd,t ] dt, (13){z}|0discount rate

where rS ≡ λ



Cs
Cs∗

σ

QS − Q∗S
is the (conditional) equity premium in the stationary
QS

equilibrium and QS is the value of a claim on firms’ profits. This expression showsthat the valuation of assets responds to changes in monetary policy through twochannels: a dividend channel, capturing changes in firms’ profits, and a discount ratechannel, capturing changes in real interest rates and risk premia. Note that the risk
premia depends on the price of risk, pd,t , like in the expression for the long-term
bond, but it has a loading rS rather than r L , capturing the different exposure to riskof the two assets.

17

2.3

The aggregate intertemporal budget constraint

By combining the intertemporal budget constraint for all households, we obtain
the economy’s aggregate intertemporal budget constraint:
E0

ˆ ∞
0


ˆ ∞
 
ηt 
ηt
et dt ,Ct dt = Q L,0 Dg,0 + E0
(1 − τ )Yt + Tη00 η0et ≡ ∑ j∈{b,s} µ j T
ej,t , and we used that ∑ j∈{b,s} µ jwhere Ct ≡ ∑ j∈{b,s} µ j Cj,t , ThWt
Pt Nj,t + Π j,t(1 − τ )Yt . This expression states that the present discounted value of aggregate consumption has to be equal to the initial value of assets in positive net supply (i.e. government bonds but not private debt) plus the present discounted value of aggregate
disposable income h(net of taxes
i and transfers). h´ i´ ∞ ηt∞ η
et dt . Note
Let QC,0 ≡ E0 0 η0 Ct dt and QY,0 ≡ E0 0 η0t (1 − τ )Yt + T
that QC,0 and QY,0 can be interpreted as the price of claims to the stream of consumption and disposable income, respectively. Thus, letting qC,0 and qY,0 denotethe log-linear approximations around the stationary equilibrium, we have
ˆ ∞
ˆ ∞
Y
−ρt
qC,0 =
e ct dt −e−ρt [it − πt − rn + rC pd,t ] dt,
QC 0
0
ˆ ∞
ˆ ∞
Y
−ρt
e
e−ρt [it − πt − rn + rY pd,t ] dt,
qY,0 =[(1 − τ )yt + Tt ] dt −QY 00
where rC = λ



Cs
Cs∗

σ

∗
QC − QC
and rY = λ
QC



Cs
Cs∗

σ

∗
QY − QY
QY , and we used that C = Y.

Note that these expressions have the same structure as the expression we derived
for stocks: a first term capturing the change in “dividends,” and a second termreflecting the change in the asset-specific discount rate, where the loading on theprice of disaster risk depends on each asset’s exposure to the disaster shock. Thus,
we get that the log-linear approximation of the aggregate intertemporal budget
constraint can be written as
ˆ ∞
e
0

−ρt

ˆ
QC ∞ −ρte
ct dt −
[it − πt − rn + rC pd,t ] dt = d g q L,0 +
Y 0
ˆ ∞
ˆ
QY ∞ −ρt−ρtee[it − πt − rn + rY pd,t ] dt.
[(1 − τ )yt + Tt ] dt −Y 00This expression reveals an important and often overlooked property of the effect
18

i

=

of changes in discount rates on households’ budget constraints. Changes in the
discount rate have two effects. First, they generate a revaluation of households’
assets. When the discount rate increases, the present discounted value of income
streams (e.g. dividends, coupons, wages) decreases, implying a negative wealtheffect. However, there is also an opposing effect. A higher discount rate reduces
the cost of the consumption bundle, generating a positive wealth effect. Thus, theoverall effect depends on which one of these two forces dominates. It turns outthat the answer depends on the government’s portfolio.
Lemma 1. The net effect of changes in the discount rate on the households’ aggregate
budget constraint is given by
QY
Y

ˆ ∞
e
0

−ρt

Q
[it − πt − rn + rY pd,t ] dt − CYˆ ∞e−ρt [it − πt − rn + rC pd,t ] dt =0ˆ ∞dGe−ρt [it − πt − rn + r L pd,t ] dt,
0where dG ≡ DYG is the public debt-to-GDP ratio. If government debt is short-term, i.e.
ψL → ∞, then r L = 0 and the households’ aggregate budget constraint is independent of
the price of disaster risk, pd,t .
Lemma 1 provides a novel insight on how discount rates affect the households’
aggregate wealth. Intuitively, the result reflects that, in a closed economy, the government is the only counterpart of the private sector as a whole. Thus, in theaggregate, the net effect of changes in the discount rate is given by the change inthe value of the assets in positive net supply, i.e. government bonds. Using thisresult, we can express the aggregate intertemporal budget constraint asˆ ∞0e−ρt [µb cb,t + (1 − µb )cs,t ] dt = Ω0 ,(14)

where Ω0 denotes the outside wealth effect, and it is given byΩ0 ≡ dG q L,0 +

ˆ ∞
0

h
i
e−ρt (1 − τ )yt + Tt + dG (it − πt − rn + r L pd,t ) dt.

(15)

Note that a simple rearrangement of (14) and (15) gives the government’s intertemporal budget constraint. By writing it this way, we make explicit the role of the out19

side wealth effect Ω0 , which captures the revaluation of assets in positive net supply:
stocks, human wealth, and government bonds. This is in contrast with the effect of
the revaluation of assets in zero net supply, such as private debt, which we refer toas the inside wealth effect. Interestingly, while the value of stocks and human wealthalso incorporate a risk premium and, therefore, are affected by changes in the priceof risk, only in the presence of long-term government bonds does the price of risk
directly affect the households’ wealth. This is once again the manifestation that
only the mismatch in exposure between the household sector as a whole and thegovernment (in terms of both maturity and risk) matters for the determination of
the change in the value of the economy’s wealth.

3

Monetary Policy and Wealth Effects

In this section, we study how households’ balance sheets determine the impact
of monetary policy on the dynamics of the economy. The main result presents a
decomposition that identifies the contribution of the different forces of the modelto the aggregate dynamics of the economy. In particular, we isolate the role ofintertemporal substitution, precautionary savings, and wealth effects in the transmission of monetary shocks. To derive this decomposition, we proceed in two
steps. First, we express the evolution of output and inflation in terms of equilibrium policy variables, that is, the path of nominal interest rates {it } and the corresponding fiscal backing { Tt }. Second, we derive an implementability result thatshows how to map the path of policy variables to the underlying monetary shockut in the interest rate rule (5).

3.1

The dynamic system

We can express output and inflation in terms of policy variables by solving thesystem of differential equations described in Proposition 1:
"

ẏt
π̇t

"

#

=

−σ̃−1
−κ
ρ
δ

20

#"

yt
πt

"

#

+

νt
0

#
,

(16)

where νt ≡ σ̃−1 (it − rn ) + vt depends only on the path of the nominal interest rate.
The eigenvalues of the system are given by
ω=

ρ+δ+

p

(ρ + δ)2 + 4(σ̃−1 κ − ρδ)
,
2

ω=

ρ+δ−

p

(ρ + δ)2 + 4(σ̃−1 κ − ρδ).
2

The following assumption, which we will assume holds for all subsequent analysis, guarantees that the eigenvalues are real-valued and that they have opposite
signs, that is, ω > 0 and ω < 0.
Assumption 1. The following condition holds: ρδ < σ̃−1 κ.
Assumption 1 implies that the system lacks exactly one boundary condition.
Next, we show that the missing boundary condition can be provided by an intertemporal budget constraint.
From equation (14), we have that the aggregate intertemporal budget constraint
is a necessary equilibrium condition. The next lemma establishes the sufficiency of
the aggregate intertemporal budget constraint for pinning down the equilibrium.
That is, it shows that if [yt , πt ]0∞ satisfies system (16) and the aggregate intertemporal budget constraint (in its log-linear form), then we can determine the value ofconsumption and labor supply for each household, wages, and prices such that all
equilibrium conditions are satisfied.
Lemma 2. Suppose that, given a path for the nominal interest rate [it ]0∞ , [yt , πt ]0∞ satisfy

system (16) and the aggregate intertemporal budget constraint (14). Then, [yt , πt ]0∞ canbe supported as part of a competitive equilibrium.
Therefore, the equilibrium dynamics can be characterized as the solution to the
dynamic system (16), subject to the boundary condition (14).

3.2Intertemporal substitution, risk and wealth effects

The next proposition characterizes the output response to a sequence of monetary
policy shocks for a given value of the outside wealth effect Ω0 . We provide afull characterization of Ω0 in Section 3.3. For ease of exposition, we focus on thecase of exponentially decaying nominal interest rates; that is, we assume it − rn =e−ψm t (i0 − rn ), where ψm determines the persistence of the path of interest rates.
21

Proposition 2 (Aggregate output in D-HANK). Suppose that it − rn = e−ψm t (i0 −rn ). The path of aggregate output is then given byµ χryt = σ−1 ŷt + χd eλ ŷt + b
ψ̃m d P ŷt + (ρ − ω )eωt Ω0 ,1 − µb| {z }
|
{z
} |
{z
}
| {z }
ISE

time-varying
risk

inside wealth effect(17)

GE multiplier×
outside wealth effectwhere ψ̃m ≡ ρ − rn + ψm , and ŷt is given by
1 − µb (ρ − ω ) eωt − (ρ + ψm ) e−ψm t( i0 − r n ),ŷt =1 − µb χy(ω + ψm ) (ω + ψm )and satisfies(18)´ ∞ −ρt∂ŷŷt dt = 0, ∂i00 < 0.
0 e

Proposition 2 shows that output can be decomposed into four terms: an intertemporalsubstitution effect (ISE), a time-varying risk channel, a revaluation of inside assets,
and a revaluation of outside assets. Note that the first three terms have a common structure. First, there is ŷt , which is uniquely determined by the path of thenominal interest rate. Figure 1 (left) shows its dynamics after a contractionary
monetary shock (we discuss the calibration in Section 4.1). After a contractionary
monetary shock, ŷt decreases on impact, but it eventually increases above its longrun level. Notably, the present value of ŷt is equal to zero. This implies that in
the absence of outside wealth effects, monetary policy affects only the timing of
output, but not its present value. Moreover, as can be seen from Figure 1 (left),ŷt responds more strongly to interest rate shocks in economies where inequality ismore counter-cyclical, i.e. when χy is higher.17 Second, there is the channel specificstrengths, given by σ−1 , χd eλ and 1−b µr ψ̃m d P . Figure 1 (middle) shows the dynamµ χb

ics of each of the first three channels of transmission in (17). In our calibration, thetime-varying channel has the largest impact on output, while the ISE has a significantly smaller effect. Finally, there is the outside wealth effect, plotted in Figure 1(right), which is mediated by a GE multiplier that captures a general equilibrium
amplification mechanism. Next, we consider each one of the channels separately.
The first term is the ISE, which captures the equilibrium implications of the intertemporal substitution channel. Similar in logic to the substitution effect in intro17 This amplification lies at the heart of the mechanism in analytical HANK models with zeroprivate liquidity and no aggregate risk.

22

Channel specific strength
0.1

0.0

0.0

0.1

0.1

0.2

0.2

0.3

0.0
0.1
0.2
0.3

0.3
y = 0.5
y = 1.0
y = 1.5

0.4
0.5

Outside Wealth Effect

%

0.1

%

%

The dynamics of ŷt

0

2

4

t

6

8

ISE
Inside WE
TVR

0.4
0.5

0

4

2

t

6

8

0.4
0.5

GE Outside WE
PE Outside WE
0

2

4

t

6

8

Figure 1: Aggregate output decomposition in HANK
ductory microeconomics, an increase in nominal interest rates reduces consumption today while it increases future consumption. In this sense, the intertemporalsubstitution channel of monetary policy operates simply by shifting demand overtime, and it is ineffective in the absence of an intertemporal-substitution motive;that is, we obtain σ−1 ŷt = 0 in all periods if σ−1 = 0.
The second term captures the role of time-varying risk. Given Ω0 , time-varying
risk amplifies the response of output in a way that is similar to that of the ISE,
and the magnitude of the amplification depends
cruciallyh 
i on the strength of the
Cs
Cs∗

σ

− 1 , and the degree of timevarying risk, as captured by eλ . Consider a contractionary monetary shock. Ifeλ > 0, the increase in the nominal interest rate increases the probability of a disaster shock, which increases savers’ precautionary motive. Thus, aggregate demanddecreases. As the nominal interest rate reverts to its long-run level, the probability of disaster decreases, and savers’ demand increases above its long-run target.
Thus, we have that the present discounted value of the time-varying risk term is´∞
zero, i.e. 0 e−ρt χd eλ ŷt dt = 0.
The third term corresponds to the inside wealth effect, and it is present only ineconomies with positive private debt and heterogeneous MPCs. The inside wealtheffect is analogous to the ISE and the time-varying risk in many respects, as it op´∞
erates by shifting demand over time, and it satisfies 0 e−ρt χr ψ̃m d P ŷt dt = 0. A
key distinction is that the strength of the inside wealth effect depends on the persistence of the monetary shock, and it is approximately equal to zero when themonetary shock is permanent, ψm = 0.18 An important implication of this result
precautionary motive, as captured by χd = λσ

18 The difference between ψ̃ and ψ is quantitatively small. The extra term ρ − r reflects movem
mnments in the precautionary motive due to consumption fluctuations in the no-disaster state. Be-23

is that the effectiveness of monetary policy depends on the persistence of monetary shocks. For instance, by promising to keep interest rates low for a very long
period of time, the monetary authority increases the persistence of the shock and,
therefore, reduces the importance of inside wealth effects and the overall output
response. To understand this result, note that an increase in interest rates has anegative impact on borrowers and a positive impact on savers. When the shock is
temporary, the impact of the change in interest rates is initially larger on borrowers, as savers respond less strongly to the change in wealth to smooth consumption. If the shock is permanent, however, there is no reason to smooth the shock.
In this case, the savers’ response coincides with the borrowers’ response, and theinside wealth effect is zero. Thus, it is the variability of interest rates rather than the
average level that matters for the inside wealth effect.
The last term in expression (17) plays a crucial role, as the outside wealth effectdetermines the average level of output. Holding everything else constant, the impact of a wealth effect Ω0 on consumption would be simply ρΩ0 , as households
attempt to smooth the impact of the change in wealth over time. However, theresponse of initial consumption is amplified in general equilibrium, as a positive
wealth effect generates inflation, which reduces real interest rates and shifts consumption to the present. Figure 1 (right) plots the outside wealth effect. Monetaryshocks usually have a small effect on households’ wealth. However, the GE multiplier can significantly amplify the effect of the change in outside wealth on theinitial level of output. In our calibration, the GE multiplier in period 0 is more than15, amplifying the impact of the wealth effect in general equilibrium.19
Inflation. The next proposition characterizes the response of inflation to monetary policy shocks in the context of our heterogeneous-agent economy.
Proposition 3 (Inflation in D-HANK). Suppose it − rn = e−ψm t (i0 − rn ). The path of
inflation is given by
πt = σ−1 π̂t + χd eλ π̂t +µ b χrψ̃m d P π̂t + κeωt Ω0 ,1 − µb(19)

cause the effect of the disaster is much larger than these fluctuations, the impact of these changes
in consumption on the precautionary motive is small and can be safely ignored.
19 Notably, the GE multiplier in period 0 is decreasing in the discounting parameter δ. Thisimplies that the precautionary savings motive dampens the effect of the outside wealth effect, while
positive private debt reduces the value of δ, which increases the effect of the outside wealth effect.

24κ (eωt −e−ψm t )1− µwhere π̂t = 1−µ χb y (ω +ψ )(ω +ψ ) (i0 − rn ) and π̂0 = 0.
mbm

Inflation can be analogously decomposed into four terms. The first three terms
capture the impact of the ISE, the time-varying risk, and the inside wealth effect, while the last term captures the impact of the outside wealth effect. Becauseπ̂0 = 0, the first three terms are initially zero. This implies that initial inflation
is determined entirely by the outside wealth effect, a consequence of the forwardlooking nature of the New Keynesian Phillips curve.

3.3

Outside Wealth Effects

We consider next the determination of the outside wealth effect Ω0 . The outside
wealth effect depends on the path of output, inflation and the price of risk, as wellas the initial price of government bonds:Ω0 = dG q L,0 +

ˆ ∞
0

i
h
e−ρt (1 − τ )yt + Tt + dG (it − πt − rn + r L pd,t ) dt.

(20)

But output, inflation, the price of risk and the price of the long-term bond in turndepend on the outside wealth effect,yt = χŷt + (ω − δ)eωt Ω0 ,pd,t = p̂d,t + χ pd ,Ω Ω0 ,

πt = χπ̂t + κeωt Ω0 ,(21)

q L,0 = q̂ L,0 + χq L ,Ω eωt Ω0 ,(22)

where χ, χ pd ,Ω and χq L ,Ω are constants defined in the appendix, and p̂d,t and q̂ L,0collect the terms that are a function only of [it ]0∞ . This simultaneity reflects the factthat spending decisions depend on the level of asset prices, as shown by (21), andthat asset prices react to the level of aggregate demand, as shown by equation (22).
By combining these expressions, we can express Ω0 in terms of policy variables,that is, the path of nominal interest rates, it , and the fiscal backing to the monetaryshock, Tt . In particular, we can express Ω0 as follows:Ω0 = (1 − eΩ )Ω0 + dG q̂ L,0 +|{z}|aggregate demand
effectˆ ∞0hie−ρt Tt + dG (it − χπ̂t − rn + r L p̂d,t ) dt,{z}direct effect25

where eΩ is a constant defined in the appendix. The first term captures the impactof aggregate demand on the valuation of stocks, bonds, and human wealth, while
the second term captures the impact of changes in monetary and fiscal variables
that are not mediated by aggregate demand. Assumption 2 guarantees that outside
wealth reacts less than one-to-one to aggregate demand.20
Assumption 2. The parameters of the model are such that eΩ ∈ (0, 1).
The next proposition shows that the outside wealth effect can be expressed as
the product of a multiplier and an autonomous term, that is, a term that does notdepend directly on Ω0 .
Proposition 4. Suppose Assumption 2 holds. The outside wealth effect is then given by

ˆ ∞
i 
h
1
−ρt
Ω0 =
dG q̂ L,0 +e
Tt + dG (it − χπ̂t − rn + r L p̂d,t ) dt .
eΩ
0

(23)

Proposition 4 introduces an important relationship between the model-implied
revaluation of assets in positive net supply, Ω0 , and the equilibrium path of policy
variables. For example, expression (23) shows that, in the absence of any fiscal
backing (Tt = 0) and government debt (dG = 0), the outside wealth effect is zero.21
Monetary policy still affects the value of stocks and human wealth, as can be seenin (13), but the reduction in the value of households’ assets is exactly offset by the
reduction in the value of households’ liabilities (in the form of consumption), as
discussed in Section 2.3. Under Assumption 2, the aggregate demand effect cannot
sustain a positive value of Ω0 in the absence of a direct effect of policy variables.
By incorporating fiscal data into the analysis, this relationship provides a way
to discipline the model’s economic forces. One can estimate the fiscal response toa monetary shock in the data and introduce the estimated values into expression(23) to obtain the model’s prediction for Ω0 . We follow this approach in Section 4.

20 Assumption 2 implies that either the primary surplus or the cost of servicing the debt respondto economic activity, as captured by Ω0 , so essentially monetary policy has fiscal consequences.
21 Adding capital does not qualitatively affect this result, as wealth effects from capital ownership are analogous to wealth effects from claims on profits. See Caramp (2021) for the details.

26

3.4

Implementability condition

The results in (17) and (19) express output and inflation in terms of the path of
nominal interest rates and the outside wealth effect Ω0 , while equation (23) givesΩ0 in terms of the underlying fiscal backing Tt . In combination, these results
demonstrate how the policy variables (it , Tt ) affect output and inflation. However,both the nominal interest rate and the associated fiscal backing are endogenous
variables and depend on the monetary policy rule (5). The next proposition shows
how we can determine the monetary rule that implements a particular equilibrium path of nominal interest rates and fiscal backing by appropriately choosing
the exogenous process for the monetary shock ut .
Proposition 5 (Implementability). Let yt be given by (17) and πt be given by (19), fora given path of nominal interest rates it − rn = e−ψm t (i0 − rn ), where ψm 6= −ω, and theassociated fiscal backing Tt . Let [it , yt , π t ]0∞ denote the (bounded) solution to the systemcomprising the Taylor rule (5), the aggregate Euler equation (9), and the New KeynesianPhillips curve (10), and suppose the monetary shock ut is given by
ut = ϑe−ψm t (i0 − rn ) + θeωt .

(24)Then, there exists parameters ϑ and θ such that it = it , yt = yt , and π t = πt .
Proposition 5 shows that the process for the monetary shock uniquely pins
down (it , Tt ), so one can equivalently express the solution either in terms of equilibrium policy variable or in terms of the underlying process for ut . The formulation in equation (24) generalizes the process for monetary shocks frequently usedin the literature, where the parameter θ is usually set to zero. While ϑ simply scales
the shock such that the initial nominal interest rate equals a given i0 , θ pins down
the outside wealth effect Ω0 and the underlying fiscal backing. An important feature of specification (24) is that the path of the nominal interest rate is the same forany value of θ, so the parameter θ affects only Ω0 .22
The extra degree of freedom given by the parameter θ will be important todiscipline the outside wealth effect empirically. If we impose θ = 0, we obtain
22 Note that the sign of the effect of a monetary shock on nominal interest rates depends on

ψm . If ψm < |ω |, a contractionary shock increases nominal rates, while it reduces nominal rates if
ψm > |ω |. Thus, the nominal interest rate do not react to a monetary shock when ψm = |ω |.

27

the standard process ut = e−ψm t u0 for some innovation u0 . This assumption determines a particular value of the fiscal backing that may be inconsistent with its
empirical counterpart, which implies that the outside wealth effect implied by themodel will also be counterfactual. By considering the generalized process (24),the model will be able to simultaneously match the persistence of the equilibrium
interest rate and the corresponding fiscal backing.

4

The Quantitative Importance of Wealth Effects

In this section, we study the quantitative importance of wealth effects in the transmission of monetary shocks. We calibrate the model to match key unconditionaland conditional moments, including asset-pricing dynamics and the fiscal response
to a monetary shock. We find that household heterogeneity and time-varying risk
are the predominant channels of transmission of monetary policy.

4.1

Calibration

The parameter values are chosen as follows. The discount rate of savers is chosen to match a natural interest rate of rn = 1%. We assume a Frisch elasticity ofone, φ = 1, and set the elasticity of substitution between intermediate goods to
e = 6, common values adopted in the literature. The fraction of borrowers is set toµb = 30%, and the parameter d P is chosen to match a household debt-to-disposable
income ratio of 1 (consistent with the U.S. Financial Accounts). The parameter dG
is chosen to match a public debt-to-GDP ratio of 66%, and we assume a durationof five years, consistent with the historical average for the United States. The tax
rate is set to τ = 0.27 and the parameter Tb0 (Y ) is chosen such that χy = 1, which
requires countercyclical transfers to balance the procyclical wage income. A valueof χy = 1 is consistent with the evidence in Cloyne et al. (2020) that the net incomeof mortgagors and non-mortgagors reacts similarly to monetary shocks. The pricing cost parameter ϕ is chosen such that κ coincides with its corresponding value
under Calvo pricing and an average period between price adjustments of threequarters. The half-life of the monetary shock is set to three and a half months toroughly match what we estimate in the data, and we set φπ = 1.5.

28

Tax Revenue (% GDP)

0.2

Government Expenditures

0.1
0.0
0.1

0.4

4

0.2

3

0.0

2
1

0.2

0.2

0

0.4
0

5

10

t

15

20

Value of government debt

5

1
0

5

10

t

15

20

0

5

10

t

15

20

Figure 2: Estimated fiscal response to a monetary policy shock
Note: IRFs computed from a VAR identified by a recursiveness assumption, as in Christiano et al. (1999). Variables included:
real GDP per capita, CPI inflation, real consumption per capita, real investment per capita, capacity utilization, hours
worked per capita, real wages, tax revenues over GDP, government expenditures per capita, federal funds rate, 5-year
constant maturity rate and the real value of government debt per capita. We estimate a four-lag VAR using quarterly data
for the period 1962:1-2007:3. The real value of government debt and the 5-year rate are ordered last, and the fed funds rateis ordered third to last. Gray areas are bootstrapped 95% confidence bands. See Appendix D for the details.

We calibrate the disaster risk parameters in two steps. For the stationary equilibrium, we choose a calibration mostly based on the parameters adopted by Barro(2009). We set λ (the steady-state disaster intensity) to match an annual disasterprobability of 1.7%, and A∗ to match a drop in output of 1 − YY∗ = 0.39.23 The
risk-aversion coefficient is set to σ = 4, a value within the range of reasonable values according to Mehra and Prescott (1985), but substantially larger than σ = 1, avalue often adopted in macroeconomic models. Our calibration implies an equity
premium in the stationary equilibrium of 6.1%, in line with the observed equity
premium of 6.5%. Moreover, by setting σ = 4 we obtain a micro EIS of σ−1 = 0.25,
in the ballpark of an EIS of 0.1 as recently estimated by Best et al. (2020). We discuss
the calibration of eλ , which determines the elasticity of asset prices to monetaryshocks, in the next subsection.
For the policy variables, we estimate a standard VAR augmented to incorporate fiscal variables and compute empirical IRFs applying the recursiveness assumption of Christiano et al. (1999). From the estimation, we obtain the path ofmonetary and fiscal variables: the path of the nominal interest rate, the change in
the initial value of government bonds, and the path of fiscal transfers. We provide the details of the estimation in Appendix D. Figure 2 shows the dynamics
of fiscal variables in the estimated VAR in response to a contractionary monetary
23 As discussed in Barro (2006), it is not appropriate to calibrate A∗ /A to the average magnitudeof a disaster, given that empirically the size of a disaster is stochastic. We instead calibrate A∗ /A tomatch E[(Cs /Cs∗ )σ ] using the empirical distribution of disasters reported in Barro (2009).

29

shock. Government revenues fall in response to the contractionary shock, while
government expenditures fall on impact and then turn positive, likely driven by
the automatic stabilizer mechanisms embedded in the government accounts. The
present value of interest payments increases by 69 bps and the initial value of government debt drops by 50 bps.24 In contrast, the present value of transfers Tt dropsby 12 bps.25 Moreover, we cannot, at the 95% confidence level, reject the possibility that the present discounted value of the primary surplus does not change in
response to monetary shocks and that the increase in interest payments is entirelycompensated by the initial reaction in the value of government bonds.

4.2

Asset-pricing implications of time-varying risk

Recall that the price of the long-term government bond is given byˆ ∞q L,0 = −0e−(ρ+ψL )t (it − rn + r L pd,t )dt,

where pd,t = σcs,t + eλ (it − rn ) is the price of the disaster risk. We use this expression and calibrate eλ to match the initial response of the 5-year yield on government bonds. Consistent with Gertler and Karadi (2015) and our own estimatesreported in Appendix D, we find that a 100 bps increase in the nominal interest
rate leads to an increase in the 5-year yield of roughly 20 bps. This procedure leads
to a calibration of eλ of 2.25, which implies an annual increase in the probabilityof disaster of roughly 95 bps after a 100 bps increase in the nominal interest rate.
Figure 3 shows the response of the yield on the long bond and the contributionsof the path of future interest rates and the term premium. We find that the bulk of
the reaction of the 5-year yield reflects movements in the term premium, a findingthat is consistent with the evidence.
The model is also able to capture the responses of asset prices that were not di24 The

present
discounted
valueofinterest
payments
iscalculatedast h gi−λ 4 d (î∑Tt=0 11+t L,t − π̂t ) , where T is the truncation period, î L,t is the IRF of the 5-yearρsrate estimated in the data, and π̂t is the IRF of inflation. We choose T = 60 quarters, when themain macroeconomic variables, including government debt, are back to their pre-shock values.
Other present value calculations follow a similar logic.
25 In the data, expenditures also include the response of government consumption and investment. When run separately, however, we cannot reject the possibility that the sum of these twocomponents is equal to zero in response to monetary shocks.
30

Government bond yield

0.20
0.15

0.0

0.06

0.05

0.5

%

0.10

0.00

Stocks0.5

Corporate spread
Risk-free rate
Risk premium

0.08

%

%

Corporate spread

Yield long bondRisk-free rate
Term premium

0.04

4

2

6

t

8

0.00

Stock priceRisk-free rate
Risk premium
Dividends

1.5

0.02

0

1.0

2.0
0

4

2

t

6

8

0

2

4

t

6

8

Figure 3: Asset-pricing response to monetary shocks with time-varying risk.
rectly targeted in the calibration. Consider first the response of the corporate spread,the difference between the yield on a corporate bond and the yield on a government bond (without risk of default) with the same promised cash flow. This corresponds to how the GZ spread is computed in the data by Gilchrist and Zakrajšek(2012). Let e−ψF t denote the coupon paid by the corporate bond. We assume thatthe monetary shock is too small to trigger a corporate default, but the corporatebond defaults if a disaster occurs, where lenders recover the amount 1 − ζ F in caseof default. We calibrated ψF and ζ F to match a duration of 6.5 years and a credit
spread of 200 bps in the stationary equilibrium, which is consistent with the estimates reported by Gilchrist and Zakrajšek (2012). Note that the calibration targets
the unconditional level of the credit spread. We evaluate the model on its ability togenerate an empirically plausible conditional response to monetary shocks.
The price of the corporate bond can be computed analogously to the computation of the long-term government bond:ˆ ∞q F,0 = −e0−(ρ+ψF )tˆ ∞(it − rn )dt −e0−(ρ+ψF )t  σQ F − Q∗F
Cs
pd,t dt,
λCs∗
QF

where Q F and Q∗F denote the price of the corporate bond in the stationary equilibrium in the no-disaster and disaster states, respectively. Given the price of thecorporate bond, we can compute the corporate spread. Figure 3 shows that thecorporate spread responds to monetary shocks by 8.9 bps. We introduce the excess
bond premium (EBP) in our VAR and find an increase in the EBP of 6.5 bps andan upper bound of the confidence interval of 10.9 bps, consistent with the model’s
prediction. Thus, even though this was not a targeted moment, time-varying risk
is able to produce quantitatively plausible movements in the corporate spread.
31Output in RANK and HANK

0.00

0.00

0.25

0.25

0.50

%

%

Decomposition in TVR-HANK

0.75

OutputISE
Inside WE
TVR
GE Multiplier x WE

0.75
1.00
0

2

4

t

6

0.50

RANK
TVR-RANKHANK
TVR-HANK

1.00

8

0

2

4

t

6

8

Figure 4: Output in RANK and HANK.
Note: In both plots, the path of the nominal interest rate is given by it − rn = e−ψm t (i0 − rn ), where i0 − rn equals 100 bps,
and the fiscal backing corresponds to the value estimated in Section 4.1.

Another moment that is not targeted by the calibration is the response of stocks
to monetary shocks. We find a substantial response of stocks to changes in interest
rates, which is explained mostly by movements in the risk premium. In contrast to
the empirical evidence, we find a positive response of dividends to a contractionary
monetary shock. This is the result of the well-known feature of sticky-prices models that profits are strongly countercyclical. This counterfactual prediction could
be easily solved by introducing some form of wage stickiness. Despite the positive
response of dividends, the model generates a decline in stocks of 2.15% in response
to a 100 bps increase in interest rates, which is smaller than the point estimate ofBernanke and Kuttner (2005) but is still within their confidence interval.26 Fixing
the degree of countercyclicality of profits would likely bring the response of stocks
closer to their point estimate.

4.3

Wealth effects in the monetary transmission mechanism

Figure 4 (left) presents the response of output and its components to a monetaryshock in the New Keynesian model with heterogeneous agents and time-varying
risk. We find that output reacts by −1.05% to a 100 bp increase in the nominalinterest rate, which is consistent with the empirical estimates of e.g. MirandaAgrippino and Ricco (2021). In terms of its components, time-varying risk (TVR)
and the outside wealth effect are the two main components determining the output dynamics, representing 39% and 47% of the output response, respectively. In
26 We follow standard practice in the asset-pricing literature and report the response of a levered

claim on firms’ profits, using a debt-to-equity ratio of 0.5, as in Barro (2006).

32

contrast, the ISE accounts for only 6.5% of the output response, indicating thatintertemporal substitution plays only a minor role in the monetary transmission
mechanism.
These findings stand in sharp contrast to the dynamics in the absence of heterogeneity and time-varying risk. Figure 4 (right) plots the response of output fordifferent combinations of heterogeneity (µb > 0 and µb = 0) and time-varying risk(eλ > 0 and eλ = 0). By shutting down the two channels, denoted by “RANK”
in the figure, the initial response of output would be −0.14%, a more than a sevenfold reduction in the impact of monetary policy. There are two reasons for thisresult. First, our calibration of σ = 4 implies an EIS that is one fourth of the standard calibration. This significantly reduces the quantitative importance of the ISE,
even if the intertemporal substitution channel represents a large fraction of theoutput response in the RANK model. Second, our estimate of the fiscal response
is substantially lower than the one implied by a standard Taylor equilibrium that
imposes an AR(1) process for the monetary shock. We discuss the role of fiscal
backing and the implications for the New Keynesian model in Section 4.5 below.
Figure 4 (right) also plots the response of output when there is household heterogeneity but not time-varying risk (“HANK” in the figure), and the response of
output when there is time-varying risk but not household heterogeneity (“TVRRANK” in the figure). We find that heterogeneity increases the response of output
by 22 bps while time-varying risk increases it by 54 bps. Notably, by combining both features, we get an increase in the response of output of 86 bps, which
is 10 bps larger than the sum of the individual effects. Thus, heterogeneity andtime-varying risk reinforce each other. In terms of the fraction of the response of
output that can be attributed to each channel, we find that 20.5% can be attributed
to household heterogeneity, 51.5% corresponds to time-varying risk, and 9.7% is
the amplification effect of heterogeneity together with time-varying risk (which isaround 50% larger than the contribution of the ISE), while the remainder represents the channels in the RANK model.
Finally, time-varying risk is essential for properly capturing the heterogeneous
response of borrowers and savers to monetary policy. Figure 5 shows that borrowers are disproportionately affected by monetary shocks. However, the magnitudeof the relative response of borrowers and savers is too large in the economy without time-varying risk. The drop in borrowers’ consumption is 7 times greater than33

Time-Varying Risk (eλ > 0)

Constant Risk (eλ = 0)

0.00

0.0

0.25

0.5

%

0.50
0.75

1.0
1.5

1.00
0

2

4

t

6

2.0

8

OutputBorrowers' consumption
Savers' consumption
0

2

4

t

6

8

Figure 5: Consumption of borrowers and savers with constant risk and timevarying risk.
Note: In both plots, the path of the nominal interest rate is given by it − rn = e−ψm t (i0 − rn ), where i0 − rn equals 100 bps,
and the fiscal backing corresponds to the value estimated in Section 4.1.

the decline in savers’ consumption with a constant disaster probability, while it is 3
times greater in the economy with time-varying risk. Cloyne et al. (2020) estimate
a relative peak response of mortgagors and homeowners of roughly 3.6. Therefore, allowing for time-varying risk is also important if we want to capture theheterogeneous impact of monetary policy.

4.4

The limitations of the constant disaster risk model

Consider the response of asset prices to a monetary shock in an economy that features constant disaster risk (i.e. λ > 0 but eλ = 0). Figure 6 (left) shows that theyield on the long bond increases by 6.5 bps, which implies a decline of the valueof the bond of 32 bps (given a 5-year duration), less than half of the response estimated by the VAR in Section 4.1. Moreover, movements in the long bond yield arealmost entirely explained by the path of nominal interest rates, while the term premium is indistinguishable from zero. This stands in sharp contrast to the evidence
reported in Gertler and Karadi (2015) and Hanson and Stein (2015). Similarly, it
can be shown that most of the response of stocks in the model is explained by
movements in interest rates instead of changes in risk premia, a finding that isinconsistent with the evidence documented in e.g. Bernanke and Kuttner (2005).
Figure 6 (right) shows how the presence of constant disaster risk affects the response of output to monetary shocks for the HANK and RANK economies. We
find that risk has only a minor impact on the response of output. Aggregate riskincreases the value of the discounting parameter δ, which reduces the GE multi34

Bond yield decomposition

Output dynamics
0.0

Yield long bondRisk-free rate
Term premium

0.06

0.1
0.04
0.2
0.02

D-HANK
HANK
D-RANK
RANK

0.3
0.00

0

2

4

t

6

8

0

2

4

t

6

8

Figure 6: Long-term bond yields and output for economies with and without risk.
Note: In both plots, the path of the nominal interest rate is given by it − rn = e−ψm t (i0 − rn ), where i0 − rn equals 100 bps, andthe fiscal backing corresponds to the value estimated in Section 4.1. D-HANK and D-RANK correspond to heterogeneousagent and representative agent economies with constant disaster risk (i.e. λ > 0 and eλ = 0). HANK and RANK correspondto economies with no disaster risk (i.e. λ = 0).

plier and dampens the initial impact of the monetary shock. Given that the term
premium barely moves, disaster risk plays only a small role in determining theoutside wealth effect. In contrast, the important role of heterogeneity can be seen
by comparing the response of the D-HANK and D-RANK economies.
Therefore, while introducing a constant disaster probability allows the model tocapture important unconditional asset-pricing moments, such as the (average) risk
premium or the upward-sloping yield curve, the model is unable to match key
conditional moments, in particular, the response of asset prices to monetary policy.
The limitations of the model with constant disaster probability in matching conditional asset-pricing moments were recognized early on in the literature, leading toan assessment of the implications of time-varying disaster risk, as in Gabaix (2012)
and Gourio (2012). This justifies our focus on time-varying disaster risk and how
it affects the asset-pricing response to monetary shocks and, ultimately, its impact
on real economic variables.

4.5

The role of fiscal backing and the EIS

We have found that time-varying risk and heterogeneity substantially amplify the
impact of monetary policy on the economy. To properly assess the importance of
these two channels, however, it was crucial to control for the implicit fiscal backing,as discussed in Section 3.3.
Figure 7 illustrates this point. In the three panels, we show the impact of a35

RANK (Fiscal Data)
0.00

0.00

0.25

0.25

0.25

0.50

0.50

0.50

0.75

D-HANK%%%RANK (Standard)
0.00

1.00

1.00

1.00
0

2

4

t

6

8

OutputISE
Inside WE
TVR
GE Multiplier x WE

0.75

0.75

0

4

2

t

6

8

0

2

4

t

6

8

Figure 7: Output in RANK vs D-HANK with time-varying risk.
Note: The first two panels show output in RANK (µb = λ = 0) with unit EIS (σ−1 = 1). In the left panel, fiscal backing isdetermined by a Taylor rule, while in the middle panel fiscal backing corresponds to the value estimated in the data. The
right panel corresponds to the D-HANK economy with time-varying risk and the estimated fiscal backing.

monetary shock that leads to an increase in nominal interest rates on impact of
100 bps. In the left panel, we consider a RANK economy (µb = λ = 0) withthe standard value for the EIS (σ−1 = 1) and fiscal backing implicitly determinedby a Taylor rule with a monetary shock that follows a standard AR(1) process,corresponding to the textbook New Keynesian model. In the middle panel, we
consider the same economy but the fiscal backing is set to the value estimatedin the data, corresponding to a Taylor equilibrium with a monetary shock that
follows the more general specification from equation (24). The right panel shows
our D-HANK model with time-varying risk and the calibrated value of the EIS,
σ−1 = 0.25.
The response of the textbook economy is only slightly smaller than that of our
D-HANK economy despite the lack of time-varying risk or heterogeneous agents.
An important reason for this is the difference in the value of the (implicit) fiscal
backing, which is almost ten times larger in the textbook economy compared withthe one we estimated in the data. When the fiscal backing is the same as in thedata, the response of output drops by almost half. The EIS also plays an importantrole. Even with fiscal backing directly from the data, the response of output isstill significant, only slightly less than that in our D-RANK with time-varying risk(see Figure 4). But this same response comes from very different channels. In the
RANK economy, the ISE accounts for roughly 40% of the output response, whilein our D-RANK the ISE accounts for less than 7% of that response.
These results suggest that the quantitative success of the RANK model is likely
the result of a counterfactually large fiscal backing in response to monetary shocks

36and a strong intertemporal-substitution channel, which compensate for missing
heterogeneous agents and risk channels. Once we discipline the fiscal backing withdata and calibrate the EIS to the estimates obtained from microdata, our model
suggests that heterogeneous agents and, in particular, time-varying risk are crucial
for generating quantitatively plausible output dynamics. However, it is important to note that our model made several simplifications to incorporate indebted
agents and time-varying aggregate risk without sacrificing the tractability of standard macro models. A natural extension would be to incorporate these channels
into a medium-sized DSGE model to better assess the quantitative properties of
the New Keynesian model.

5

The Effect of Risk and Maturity of Household Debt

We have assumed so far that households borrow using short-term riskless debt.
In practice, however, most household debt takes the form of long-term risky debt.
In this case, the effect of monetary policy on borrowers depends on how the term
spread and credit spread, the compensation for holding interest rate and default
risk, respond to changes in the short-term interest rate. In this section, we extend
the baseline model to allow for default risk and long-term maturities on household
debt and show how these two features affect the transmission of monetary policy
shocks to the real economy.

5.1

The model with long-term risky household debt

We describe next the model with long-term risky household debt. We highlightthe main differences with the model described in Section 2 and present a detailed
description in Appendix C. Households issue long-term debt that promises to pay
exponentially decaying coupons given by e−ψP t at period t ≥ 0, where ψP ≥ 0.
Importantly, households cannot commit to always repay their debts. In response
to a large shock, i.e. the occurrence of a disaster, households default and lenders
receive a fraction 1 − ζ P of the promised coupons, where 0 ≤ ζ P ≤ 1. We assume
that fluctuations in the no-disaster state are small enough such that they do not
trigger a default. Thus, households default only in the disaster state.

37

We denote the price of household debt in the no-disaster (disaster) state by Q P,t( Q∗P,t ), so the nominal return on household debt is given byQ∗ − Q P,tQ̇ P,t1
dR P,t =+− ψP dt + P,t
dNt ,Q P,t Q P,t
Q P,t

where i P,t ≡ Q1P,t − ψP is the yield on the bond. In a stationary equilibrium, thespread between the interest rate on household debt and the short-term interest
rate controlled by the central bank is given byrP = λCsCs∗σQ∗P − Q P.
QP

Note that the interest rate on household debt incorporates both a credit and aterm spread.27 We assume that households can borrow up to D P,t = Q P,t F, which
effectively puts a limit on the face value of household debt F.28 In a log-linear approximation of the economy around a zero-inflation stationary equilibrium, borrowers are constrained at all periods, and their consumption is given by
cb,t = (1 − α)(wt − pt + nb,t ) + Tb,t −ψP(i P,t − i P ) − πt d P .
i P + ψP

(25)

Equation (25) generalizes the expression for borrowers’ consumption given in Section 2. Monetary policy affects borrowers indirectly through its effect on the yieldon household debt i P,t . If we assume that debt is short-term, ψP → ∞, and riskless, ζ P = 0, we obtain i P,t = it and the expression above boils down to equation(6). At the other extreme, we have the case of a perpetuity, ψP = 0. In this case,households simply pay the coupon every period and there is no need to issue newdebt. Therefore, they are completely insulated from movements in nominal interest rates.

27 Let i ND denote the yield on a non-defaultable bond with coupons decaying at rate ψ . The termpP,tND − i and the credit spread to i
ND
ND
ND
spread corresponds to i P,ttP,t − i P,t , so r P = (i P − rn ) + (i P − i P ).
28 This formulation guarantees that, after an increase in nominal rates, the value of household

debt and the borrowing limit decline by the same amount. This specification of the borrowing
constraint, combined with the assumption of impatient borrowers, guarantees that borrowers areconstrained at all periods.

38

The price of household debt evolves according toq̇ P,t = (ρ + ψP )q P,t + it − rn + r P pd,t .
The price of the bond depends on the future path of short-term interest rates andthe risk premium. Quantitatively, the fluctuations in the risk premium are dominated by the time-varying risk component, while the term σcs,t gives a negligible
contribution, as shown in Figure 6. Motivated by this fact, we assume that r P σcs,tis negligible in a first-order approximation, such that we can write the price of thebond as follows29
q P,t = −1 + r P eλ
( i t − r n ),
ρ + ψP + ψm(26)

where ψm is the decaying rate of nominal interest rates.
Combining the behavior of borrowers’ consumption with savers’ Euler equation (8) and the Phillips curve (10), we can derive the response of aggregate output
to monetary shocks. In particular, we can extend the decomposition in Proposition
2 to the case of long-term risky debt.
Proposition 6 (Aggregate output with long-term risky debt). Suppose that it − rn =e−ψm t (i0 − rn ) and r P σ = O(i0 − rn ). The path of aggregate output is then given byµ χr d P ψP (1 + r P eλ )ψ̃m ŷt + (ρ − ω )eωt Ω0 , .
yt = σ−1 ŷt + χd eλ ŷt + b
1 − µb ρ + ψP + ψm
| {z }
|
{z
} |
{z
}
| {z }
ISE

time-varying
risk

inside wealth effect(27)

GE multiplier×
outside wealth effectProposition 6 shows that default risk and debt maturity have opposite effects
on the magnitude of the inside wealth effect. For instance, in the case of shortψ (1+r e )term debt, the term ρP+ψP +Pψmλ simplifies to 1 + r P eλ > 1, so the inside wealth effectis amplified relative to the case of riskless debt. The interest rate on household
debt now moves in response to changes in the short-term interest rate as well aschanges in the risk premium. In contrast, the inside wealth effect is dampened forlong-term bonds. In the limit case of a perpetuity, ψP = 0, the inside wealth goesto zero. Given that households do not issue new debt, they are not affected by the
29 Formally, we assume that the parameter r

P σ is of the same order as the (small) monetary shock,r P σ = O(i0 − rn ). Therefore, the term r P σcs,t is second-order in i0 − rn and it can be ignored in a
first-order approximation. The solution in the absence of this assumption is available upon request.

39

Output dynamics

Inside wealth effect
0.05

1.0

%

%

0.10
0.15

1.2

h = 0.00
h = 0.10
h = 0.25

0.20
0

5

10

Duration (quarters)

15

1.1

1.3

20

h = 0.00
h = 0.10
h = 0.25

0

5

10Duration (quarters)

15

20

Figure 8: Inside wealth effect and output as a function of duration and haircut ζ P .
change in interest rates, which eliminates the (inside) wealth effect.

5.2

Quantitative implications

We consider next the quantitative implications of default risk and maturity on
household debt. As shown in Proposition 6, these two features have opposing
effects on the response of output to monetary policy. To assess the quantitative impact of risk and maturity, we show in Figure 8 the inside wealth effect (left panel)and aggregate output (right panel) as a function of the duration of household debt
for different values of the haircut parameter ζ P . Greenwald et al. (2021) estimate
the duration of mortgage debt as 5.2 years, the duration of student debt as 4.50,
and the duration of consumer debt as 1.0 year. Therefore, we focus on values ofduration up to five years in Figure 8. We consider three different values for thehaircut parameter: riskless debt (ζ P = 0); risky debt with a spread in the stationary equilibrium of roughly 4.0% with a 5-year duration (ζ P = 0.10); risky debt
with a spread of 5.0% with a 5-year duration (ζ P = 0.25).
Default risk substantially amplifies the effect of monetary policy on output
when debt is short term. The inside wealth effect is almost three times larger inthe case of ζ P = 0.25 compared to ζ P = 0.0, which corresponds to an increase in
the initial response of output of almost 25%. However, this effect is strongly attenuated when household debt is long term. For even relatively small values ofduration, the inside wealth effect is smaller than in the case of short-term risklessdebt. For instance, in the case of a five-year duration, the response of output isroughly 10% smaller than the response in the case of short-term riskless debt.
It is important to note that even though the inside wealth effect is substan40

tially dampened with long-term debt, the presence of household debt still generates amplification through its impact on the compounding parameter δ, as shownin Proposition 1. The response of output when household debt is zero is roughly
35% smaller than in the economy with (positive) riskless debt, a much larger drop
relative to the one caused by introducing long-term bonds.

6

Conclusion

In this paper, we provide a novel unified framework to analyze the role of heterogeneity and risk in a tractable linearized New Keynesian model. The methods
introduced can be applied beyond the current model. For instance, they can beapplied to a full quantitative HANK model with idiosyncratic risk, extending theresults of Ahn et al. (2018) to allow for time-varying risk premia. Alternatively,
one could introduce a richer capital structure for firms and study the pass-throughof monetary policy to households and firms. These methods may enable us tobridge the gap between the extensive existing work on heterogeneous agents andmonetary policy and the emerging literature on the role of asset prices in the transmission of monetary shocks.

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Appendix: For Online PublicationA

Proofs

Proof of Proposition 1. Consider first the New Keynesian Phillips curveπ̇t =it − πt + λt

ηt∗
ηt



π t − ( e − 1 ) ϕ −1

e W wt − pte
− (1 − τ ) Yeyt .
e − 1 PA1
Linearizing the above expression, and using e−e 1 W
P A = (1 − τ ), we obtain


π̇t =


rn + λ

Cs
Cs∗

σ 

πt − ϕ−1 (e − 1)(1 − τ )(wt − pt ).

−1
Using the expression for wt − pt , we obtain π̇t= (ρs + λ) πt − κyt , where κ ≡ ϕ (e −1)(1 − τ )(φ + σ)Y and we used that rn + λ

Cs
Cs∗

σ

= ρs + λ.
Consider next the generalized Euler equation. From the market-clearing condition for1− µ χ
µ χdgoods and borrowers’ consumption, we obtain cs,t = 1−µb b y yt + 1b−rµbP (it − πt − rn ). Combining this condition
 withσ the Phillips Curve and savers’ Euler equation, and using theCsfact that rn = ρ − λ C∗ , we obtain ẏt = σ̃−1 (it − π − rn ) + δyt + vt , where the constantssσ̃−1 , δ, and vt are defined in the proposition.

Proof of Lemma 1. First, note that from the intertemporal budget constraint in the stationary
equilibrium, we have QYY − QYC = dG . Next, we have
QY
Q
rY
− rC C = λ
Y
Y



Cs
Cs∗

σ

( QY − QY∗ ) − ( QC − QC∗ )=λY

where Q L = i L +1ψL and Q∗L = i∗ +1ψL . Then, r L = λ
L



Cs
Cs∗

σ



Cs
Cs∗

σ

Q L − Q∗L
dG = r L dG ,QLrn∗ −rn σ∗
rn +ψL +λ CC∗ss

→ 0 as ψL → ∞.

Proof of Lemma 2. Suppose [yt , πt ]0∞ satisfies system (16) and the intertemporal budget con-straint (14) in the no-disaster state. We will show that [yt , πt ]0∞ can be supported as anequilibrium. Consider first the disaster state. The savers’ budget constraint implies Ts,t =−ρs bs bs,t∗ . All the remaining variables are equal to zero in the disaster state.
Consider now the no-disaster state. The real wage is given by wt − pt = (φ + σ )yt .
Borrowers’ consumption is given by cb,t = χy yt − χr d P (it − πt − r − n), while savers’ con1− µ χµ χdsumption are given by cs,t = 1−µb b y yt + 1b−rµbP (it − πt − rn ), and the labor supply is given1

by n j,t = φ−1 (wt − pt ) − φ−1 σc j,t .
By construction, the market-clearing condition for goods and labor are all satisfied. Because yt satisfies the aggregate Euler equation, the savers’ Euler equation is also satisfied.
Because πt satisfies the New Keynesian Phillips curve, the optimality condition for firms
is satisfied. Bond holdings by savers and government debt evolve according tobs ḃs,t = rn bs bs,t + (1 − α)(wt − pt + ns,t ) + Ts,t +(1 − τ )yt − (1 − α)(wt − pt + nt )+1 − µbL(i − πt − rn )bs + (r L,t − r L )bsL + r L bsL bs,t
− cs,t ,dG d˙G,t = dG (rn + r L )dG,t + Tt − τyt + (it − πt − rn + r L,t − r L )dG ,L

where bs,0 = bbss q L,0 and dG,0 = dG q L,0 . The value of cb,t is such that the flow budget
constraint for borrowers also holds.
Aggregating the budget constraint of borrowers and savers and using the market clearing condition for goods and labor, we obtainih(1 − µb )bs ḃs,t = rn (1 − µb )bs bs,t + Tt − τyt + (i − πt − rn ) (1 − µb )bs − µb d P +L)(1 − µb )bsL .
(r L,t − r L + r L bs,t
S + b L b L . We set bS = 0, so the market for short-term bonds clear at
Note that bs bs,t = bsS bs,t
s s,t
s,t

all periods. It remains to show that the market for long-term bonds also clears. Subtracting
the government’s flow budget constraint from the condition above, we obtain
L
L− dG d˙G,t = (rn + r L )((1 − µb )bsL bs,t
(1 − µb )bsL ḃs,t− dG dG,t ),L b L and (1 − µ ) b − µ d = (1 − µ ) b L = d . Integrating this expresusing bs bs,t = bs,t
G
b s
b s
b P
s,t
h
i

L −d d(r n +r L ) t (1 − µ ) b L b L − d dsion, we obtain (1 − µb )bsL bs,t
G G,t = e
G G,0 = 0, where theb s s,0equality uses the market clearing condition in period 0. Therefore, the market clearing condition for long-term bonds is satisfied in all periods. The only condition that remains to bechecked is the No-Ponzi condition for the government or, equivalently, the aggregate intertemporal budget constraint. Because condition (14) is satisfied, the No-Ponzi condition
for the government is also satisfied.
Proof of Propositions 2 and 3. We can write dynamic system (16) in matrix form as Żt =AZt + Bνt , where B = [1, 0]0 . "Applying the# spectral decomposition"#to matrix A," we ob#
ρ−ω
ρ−ω
ρ−ω
−
1
ω
0
κ
κ
κ
tain A = VΩV −1 where V =, V −1 = ω −κ ω.
ρ−ω , and Ω =
1
1
1 − κ
0 ω

2

Decoupling the system, we obtain żt = Ωzt + bνt , where zt = V −1 Zt and b = V −1 B.
Solving the equation with a positive eigenvalue forward and the one with a negative
eigenvalue backward, and rotating the system back to the original coordinates, we obtain


21

22



yt = V12 V y0 + V π0 eωt

− V11 V



11

ˆ ∞
e

−ω (z−t)tπt = V22 V 21 y0 + V 22 π0 eωt − V21 V 11ˆ ∞νz dz + V12 V

21

ˆ t

eω (t−z) νz dz

0e−ω (z−t) νz dz + V22 V 21ˆ tteω (t−z) νz dz,

0

where V i,j is the (i, j) entry of matrix V −1 . Integrating e−ρt yt and using the intertemporalbudget constraint,


21

22

Ω0 = V12 V y0 + V π0



1
1
−
V V 11ρ − ω ρ − ω 11

ˆ ∞

e

−ωt

0

−e

−ρt



1
V V 21
νt dt +
ρ − ω 12ˆ ∞e−ρt νt dt.

0

Rearranging the above expression, we obtain
ˆ ∞
ˆ ∞



ρ−ω
V11 V 11e−ρt νt dt.
V12 V 21 y0 + V 22 π0 = (ρ − ω )Ω0 +e−ωt − e−ρt νt dt − V12 V 21
ρ−ω00
ω −ρ ´ ∞Output is then given by yt = ỹt + (ρ − ω )eωt Ω0 , where ỹt = − ω −ω t e−ω (z−t) νz dz +´´∞ρ−ωω −δ t ω (t−z)νz dz − ω −ω eωt 0 e−ωz νz dz. Inflation is given by πt = π̃t + κeωt Ω0 , whereω −ω 0 e´∞´t´∞π̃t = ω −κ ω t e−ω (z−t) νz dz + ω −κ ω 0 eω (t−z) νz dz − ω −κ ω eωt 0 e−ωz νz dz.

If it − rn = e−ψm t (i0 − rn ), then i̇t = −ψm (it − rn ). This allows us to write ỹt = σ−1 ŷt +
µ χd

µ χd

χ p eλ ŷt + 1b−rµbp ψ̃m ŷt and π̃t = σ−1 π̂t + χ p eλ π̂t + 1b−rµbp ψ̃m π̂t , where ψ̃m ≡ ρ − rn + ψm , ŷt =hi
ρ−ω1− µ bψm +ρκ (eωt −e−ψm t )−ψm t +
ωt (i − r ), and π̂ = 1−µb
e
e
−
n
t
0
1− µ b χ y1−µb χy (ω +ψm )(ω +ψm ) (i0 −(ψm +ω )(ψm +ω )(ψm +ω )(ψm +ω )
´∞
∂ŷ
1− µ
rn ). Note that 0 e−ρt ŷt dt = 0, and ∂i00 = − 1−µb χb y ψm1+ω < 0. Moreover, π̂0 = 0.
Proof of Proposition 4. From Propositions 2 and 3, we have yt = χŷt + (ω − δ)eωt Ω0 , and
πt = χπ̂t + κeωt Ω0 , where χ ≡ σ−1 + χd eλ + 1−b µrb ψ̃m d P . Moreover, we can rewrite theµ χ

1− µ χ

µ χdprice of disaster risk as pd,t = p̂d,t + χ pd ,Ω eωt Ω0 , where p̂d,t ≡ σ 1−µb b y χŷt + σ 1b−rµbP (it −
1− µ χ
µ χdχπ̂t − rn ) + eλ (it − rn ) and χ pd ,Ω ≡ σ 1−µb b y (ρ − ω ) − 1b−rµbP κ . Finally, the price ofthe long-term bond in period 0 can be written as q L,0 = q̂ L,0 + χq L ,Ω Ω0 , where q̂ L,0 ≡´∞− 0 e−(ρ+ψL )t [it − rn + r L p̂d,t ] and χq L ,Ω ≡ − ρ+ψrLL −ω χ pd ,Ω . Introducing these expressions´∞
into (14), and using that 0 e−ρt yt dt = Ω0 , we obtainΩ0 = (1 − χΩ ) Ω0 + dG q̂ L,0 +

ˆ ∞
0

h
i
e−ρt Tt + dG (it − χπ̂t − rn + r L p̂d,t ) dt,

3ψ r dL L G
Gwhere χΩ ≡ τ + ρκd−ω − (ρ−ω )(ρ+ψ −ω ) χ pd ,Ω .
L

Proof of Proposition 5. We divide this proof in three steps. First, we derive the condition for
local uniqueness of the solution under the policy rule (5). Second, we derive the path of[yt , πt , it ]0∞ for a given path of monetary shocks. Third, we show how to implement a given
path of nominal interest rates it − rn = e−ψm t (i0 − rn ) and a given value of Ω0 , which maps´∞
to a given value of fiscal backing 0 e−ρt Tt dt.
hi

1− µ b
Equilibrium determinacy. First, note that we can write νt as νt = σ̃−1 1 + 1−
µb χy σ̃χd eλ φπ πt +µ b χr d P
1−µb χy φπ κyt + ν̃t , where ν̃t ≡h

1− µ b
−1 + 1− µ b χ e
1− µ b χ y σ
1− µ b χ y d λ

i

µ χd

ut − 1−b µrb χPy ((rn − ρ)ut + u̇t ).

The dynamic system for yt and πt can now be written as
"

ẏt

#

"

=

π̇t

−σ̃−1 (1 − φ̃π )
−κ
ρ
δ̃

#"

yt

#

"

1

+

0

πt

#
ν̃t ,

h
i
1− µ
µ χd
where δ̃ ≡ δ + 1−b µrb χPy φπ κ and φ̃π ≡ 1 + 1−µb χb y σ̃χd eλ φπ . The eigenvalues of the system√√ρ+δ̃− (ρ+δ̃)2 +4(σ̃−1 (1−φ̃π )κ −ρδ̃)
ρ+δ̃+ (ρ+δ̃)2 +4(σ̃−1 (1−φ̃π )κ −ρδ̃)are given by ω T =
and
ω
=
.
T
2
2
The system has a unique bounded solution if both eigenvalues have positive real parts. A
necessary condition for the eigenvalues to have positive real parts isµ χr d P
ρ+δ+ bφπ κ > 0 ⇐⇒ φπ > −(ρ + δ)
1 − µb χyµ b χr d P κ
1 − µb χy

! −1





= 1− ρ+λ

Cs
Cs∗

σ 

1 − µb χyµ b χr d P κ.

If the condition above is violated, then the real part of ω T is negative. Another necessary
condition for the eigenvalues to have positive real parts is"σ̃−1 (1 − φ̃π )κ < ρ δ +µ b χr d P
φπ κ
1 − µb χy#⇐⇒ φπ > 1 −ρλ 1−µb χyCs
1− µ bCs∗
χ d e λ + σ −1

χ d eλ + κσ.

If this condition is violated, then the eigenvalues are real-valued and ω T < 0. This establishes the necessity of the condition






ρλ 1−µ χχd eλ + κ 1−µb b y CC∗ss
φπ > max 1 −
−1

χ
e
+
σ
d λ


σ

,1− ρ + λ



Cs
Cs∗

σ 



1−µ χ b y

µ b χr d P κ .

If µb χy < 1, then φπ > 1 is sufficient to guarantee the local uniqueness of the solution.

4

Solution to the dynamic system. The dynamic system for [yt , πt ]∞t=0 is given by
"

ẏt

#

"

=

π̇t

−σ̃−1 (1 − φ̃π )
−κ
ρ
δ̃

#"

#

yt

"

+

πt

1
0

#
ν̃t .

In matrix form, the system is given by Z̃˙ t = Ã Z̃t + Bν̃t , where B = [1, 0]0"
. Applying the
#
ρ−ω T

ρ−ω T

κ
κ
spectral decomposition to matrix Ã, we obtain Ã = ṼΩ T Ṽ −1 where Ṽ =
,
1
1
#
"
"
#
ρ−ω T
−
1
ω
0
T
κ
, and Ω T =Ṽ −1 = ω T −κ ω
. Decoupling the system, we obtainρ−ωT
1 − κ T0 ωTz̃˙ t = Ω T z̃t + b̃ν̃t , where z̃t = Ṽ −1 Z̃t and b̃ = Ṽ −1 B. Solving the system forward, androtating the system back to the original coordinates, we obtainyt = −Ṽ11 Ṽ

11πt = −Ṽ21 Ṽ 11ˆ ∞e−ω T (z−t)

tˆ ∞ν̃z dz − Ṽ12 Ṽ

21

ˆ ∞

e−ω T (z−t) ν̃z dzte−ω T (z−t) ν̃z dz − Ṽ22 Ṽ 21ˆ ∞te−ω T (z−t) ν̃z dz.

t

We rewrite the above expression as follows:ˆ ∞ˆ ∞ωT − ρ
ωT − ρ
−ω T (z−t)
yt = −eν̃z dz +e−ω T (z−t) ν̃z dz
ωT − ωT t
ωT − ωT tˆ ∞κπt = −e−ω T (z−t) − e−ω T (z−t) ν̃z dz,
ωT − ωT t
where ν̃t ≡h

1− µ b
−1 + 1− µ b χ e
1− µ b χ y σ
1− µ b χ y d λ

i

µ χd

ut − 1−b µrb χPy ((rn − ρ)ut + u̇t ). Using that ut = e−ψm t u0 ,we obtain
ρ + ψm1 − µbyt = −(ω T + ψm )(ω T + ψm ) 1 − µb χyσ

κ
1 − µbπt = −(ω T + ψm )(ω T + ψm ) 1 − µb χy

σ

−1

−1

µ χr d P+ χ d eλ + bψ̃m
1 − µb!

µ χr d P+ χ d eλ + bψ̃m
1 − µb!

ut
ut ,where (ω T + ψm )(ω T + ψm ) = σ̃−1 κ (φ̃π − 1) + (δ̃ + ψm )(ρ + ψm ) > 0.


1− µ
µ χd
The wealth effect is given by Ω0 = − (ω +ψm )(1 ω +ψm ) 1−µb χb y σ−1 + χd eλ + 1b−rµbP ψ̃m u0 .
TT(δ+ψ )(ρ+ψ )−σ̃−1 κ

The nominal interest rate is given by it = rn + (δ̃+ψ )(ρm+ψ )+mσ̃−1 κ (φ̃ −1) ut . Note that ifmmπψm = −ω > 0, then it − rn , using the fact that ωω = ρδ − σ̃−1 κ and ω + ω = ρ + δ. Despite

the zero interest rate, the impact on output and inflation
is non-zero. In particular, the outside wealth effect is given by Ω0 = − (ω −ω )(1 ω −ω )TT

5

1− µ b
−1 + χ e ) + µ b χr d P ( ρ − r − ω )nd λ
1− µ b χ y ( σ1− µ b χ y



u0 .

Implementability condition. Suppose ut = ϑe−ψm t (i0 − rn ) + θeωt and denote by (it , yt , π t )the value of the nominal interest rate, output, and inflation under the Taylor rule. Given thelinearity of the system, the solution will be sum of the solutions for u1,t = ϑe−ψm t (i0 − rn )and u2,t = θeωt . After some algebra, we get that the nominal interest rate is given byit − rn = e−ψm t (i0 − rn ), using the fact that the nominal interest rate is zero under u2,t , and(δ̃+ψm )(ρ+ψm )+σ̃−1 κ (φ̃π −1)
ψm )(ω T +ψm )= (ω(ωT +.
+ψm )(ω +ψm )(δ+ψm )(ρ+ψm )−σ̃−1 κ ´∞ −ρtThe outside wealth effect, Ω0 = 0 e yt dt, is given bychoosing ϑ =Ω0 = −

1− µ b1− µ b χ y



µ χd

σ−1 + χd eλ + 1b−rµbP ψ̃m(ω T + ψm )(ω T + ψm )ϑ ( i0 − r n ) −To implement a Ω0 = Ω0 , we must choose θ =AR(1)1− µ bΩ0 ), where Ω0= − (ω T +ψm )(1 ω +ψm ) 1−µb χyT

1− µ b1− µ b χ y



µ χd

σ−1 + χd eλ + 1−b µrb χPy (ρ − rn − ω )(ω T − ω )(ω T − ω )1− µ b1− µ b χ yAR(1) (ω T +|ω |)(ω T +|ω |) (Ω−0µ b χr d P−
1
σ +χd eλ + 1−µ (ρ−rn +|ω |)bµ χσ−1 + χd eλ + 1−b µrb ψ̃m

ϑ ( i0 − r n ).

Given the process for ut and the values of ϑ and θ, output can be written as yt = σ−1 ŷt +µ χdχd eλ ŷt + 1b−rµbP ψ̃m ŷt + (ω − δ)eωt Ω0 , which coincides with (17), where we used ρ − ω =´∞´∞ω − δ. This result also implies that π t = πt , as π t = κ 0 e−ρt yt dt = κ 0 e−ρt yt dt = πt .

Fiscal transfers. Given that yt and πt are proportional to ut , we can write it − rn − πt =χr,u ut , pd,t = χ p,u ut , and q L,0 = χq L ,u u0 . Rearranging the aggregate intertemporal budget´ ∞ −ρtconstraint, we obtain the present discounted value of transfers:
0 e i µs Ts,t dt = ( τ −hϑ ( i0 −r n )0µb Tb (Y ))Ω0 − dG χq L ,u (ϑ (i0 − rn ) + θ ) − d g (χr,u + r L χ p,u ) ρ+ψm + ρ−θ ω .
Proof of Proposition 6. From the market clearing
condition for goods,
we obtain the con

sumption of savers: cs,t =1− µ b χ yµ b χr d P
1− µ b y t + 1− µ b

ψPi P +ψP (i P,t − i P ) − πtd P . Assuming exponen-tially decaying interest rates, we can write the expression above as
1 − µb χyµ χr d P
cs,t =
yt + b1 − µb1 − µbψP (1 + r P eλ )(it − rn ) − πt .
ρ + ψP + ψm

(A.1)

The Euler equation for savers can be written asċs,t = σ−1(it − πt − rn ) + λ

6

Cs
Cs∗

σ
cs,t + χd eλ (it − rn ).

(A.2)


θ

Combining equations (A.1) and (A.2), and using equation (C.8), we obtain
"
ẏt =

#
"  
#
Cs σ
µ b χr d Pµ b χr d P
1 − µ b −1−
σ −rn (it − πt − rn ) + λκ yt
1 − χy µb
1 − χy µbCs∗
1 − χy µb"#
1 − µbµ b χr d P
ψP (1 + r P eλ )+χ eλ +rn +(ρ − rn + ψm ) (it − rn ),1 − χy µb d
1 − χy µb
ρ + ψP + ψm

We can then write the aggregate Euler
as ẏt = σ̃−1 (it − πt −h rn ) + δyt + vt , where equation

1− µ

µ χd

σ̃−1 ≡ 1−χy µb b σ−1 − 1−b χry µPb rn , δ ≡ λ

Cs
Cs∗

σ

r P eλ )− µ1b−χrµdbP κ, and vt ≡ 1µ−b χχrydµPb rn + ψρP+(1ψ+P +
ψm

1− µ

(ρ − rn + ψm )] (it − rn ) + 1−χy µb b χd eλ (it − rn ). Therefore, output is given by
yt = σ−1 ŷt + χd eλ ŷt +µ b χr ψP (1 + r P eλ )ψ̃m d P ŷt + (ρ − ω )eωt Ω0 ,1 − µb ρ + ψP + ψm

where ψ̃m ≡ ψm + ρ − rn .

B

Monetary Policy and Endogenous Disasters

In this appendix, we consider an extension of the model in Section 2 where the probabilityof a disaster is endogenous and responds to changes in monetary policy. Building on thework of Gertler et al. (2020), we introduce financial intermediaries which are exposed toruns that lead to financial disentermediation and large economic losses. To avoid repetition, we describe only the main differences with respect to the baseline model.

B.1

Model

Firms’ capital structure. Firms’ pricing decision is analogous to the one described in
Section 2. Instead of being entirely equity financed, firms are now financed by a combination of equity, which is held by savers, and bank loans. Firms promise to pay an exponentially decaying coupon e−ψF t to banks at date t ≥ 0. Firms may default on their debt.
With Poisson intensity λ F , debt is restructured and banks suffer a haircut ζ F on corporateloans, where ζ F is drawn from a distribution GF (·) with support [ζ F , ζ F ]. Default events
are aggregate shocks, as they affect all firms at the same time.1

1 Alternatively, we could assume that a fraction ζ

event, which would deliver similar results.

7

F of firms default on their debts in a default

The return on bank loans is given by(1 − ζ F ) Q†F,t − Q F,t1
Q̇ F,tdR F,t =+− ψF dt +dN F,t ,
Q F,t
Q F,t
Q F,t


where Q F,t denotes the price of corporate debt prior to default and Q†F,t is the price ofcorporate debt (with unit face value) after a default event. Let DF,t denote the total valueDof corporate loans outstanding in nominal terms and FF,t ≡ QF,t
the face value of debt.
F,tCorporations maintain fix the face value, that is, FF,t = FF . This requires a debt repayment
1in the amount FF,t − ψF DF,t = i F,t DF,t every period, where i F,t = Q−
F,t − ψF . Firms also

issue new debt after a default event to keep the face value constant after the shock.

Banks’ problem. Banks issue deposits, which pay interest rate i D,t , and choose howmuch to lend to corporations. Banks only pay dividends at the terminal date, which arrives
with Poisson intensity λb . Every period a mass λb of new banks are created, which startwith funds B̃b,t given by savers, so the total mass of bankers is kept fixed.
The banker’s problem is given by
Vb,t ( Bb ) = max Et
BbC



ηT
B
ηt b,T

subject tohiF†dBb,t = (i D,t − πt ) Bb,t + r F,t Bb,t
dt + ( Bb,t
− Bb,t )dNt ,Q̇

F,twhere T is a stopping time with arrival rate λb , r F,t ≡ Q1F,t + QF,t− ψF − i D,t denotes the† denotes the bank’s net worth after the
spread with respect to the deposit rate, and Bb,tdefault event:

(
†
Bb,t
= max

F
Bb,t + Bb,t)
(1 − ζ F ) Q†F,t − Q F,t
,0 .
Q F,t

F > B ), so there is a chance that they will not
In equilibrium, banks operate leveraged (Bb,tb,thave enough resources to fully repay depositors in a default event. In this case, which werefer to as a bank failure, equity holders get zero, depositors suffer a loss, and the bank isultimately liquidated. We follow Gertler et al. (2020) and assume that the financial disintermediation caused by bank failures leads to real economic losses, which we capture by adrop in firms’ productivity from A to A∗ < A.

Bank runs. Bank failures can be caused by self-fulfilling reasons, as they create the possibility of bank runs: if depositors decide to keep funding the bank, Q†F,t is high enough for8banks to fully repay depositors; if depositors decide to run, then banks sell their loans at afire sale price Q†F,t = Q∗F,t , which could trigger losses to depositors, justifying the decision
to run. To focus on the implications of bank runs, we make the following assumptions: i)in the absence of a run, the default event is not enough to cause banks to fail; ii) for a smallenough haircut ζ F , banks do not fail even if depositors decide to run; iii) banks are immediately recapitalized after a default event that does not trigger a bank run. The first two
assumptions guarantee that banks fail only after large enough shocks which are followed
by a run. The third assumption allows us to abstract from the implications of small shocks
to bank net worth, focusing only on crises triggered by runs. We state the assumptions interms of parametric restrictions below.
Given the assumptions above, the price of a corporate loan does not change if there
is a default event but no runs, so Q†F,t = Q F,t in this case. In the case of a run, banks
sell the loans at a discount, that is, we assume that Q∗F,t = (1 − ζ d ) Q F,t , where 0 < ζ d <1 represents the fire-sale discount. Moreover, there exists a threshold ζ ∗F,t such that theprobability of a run is zero if ζ F ≤ ζ ∗F,t . If ζ F > ζ ∗F,t , then a run occurs depending on therealization of a sunspot variable. We follow Gertler et al. (2020) and assume a constantsunspot probability 0 < κ < 1. The probability of a run, conditional on a default event, isthen given by (1 − GF (ζ ∗F,t ))κ , where ζ ∗F,t satisfies the conditionFBb,t(1 − ζ ∗F,t ) Q∗F,t − Q F,t

Bb,t

Q F,t

= −1 ⇒ ζ ∗F,t =Bb,tF − ζdBb,t

1 − ζd,where ζ r must be sufficiently small such that ζ ∗F,t > 0. Moreover, ζ F <

Bb,t
F

Bb,t

, so there is nobank failure in the absence of a run. Note that the probability of a run is endogenous and
BF

b,t
.
increasing in bank’s leverage Bb,t

Banks’ optimality condition. The HJB equation is given by
i
∂Vb,t
∂V h
F
+ b,t (i D,t − πt ) Bb,t + r F,t Bb,t+ λb [ Bb,t − Vb,t ] +
∂t
∂Bb

ˆ ζF
ˆ ζF  ∗
 nr

ηt ∗
nr
λF
Vb,t − Vb,t dGF (ζ F ) + λ F κ
V − Vb,t dGF (ζ F ),ηt b,tζζ ∗F,t0 = max −(it − πt )Vb,t +BbF

Fnr = V ( B (1 + I ( ζ )) − ζ B F ) is the value function conditional on a no-run
where Vb,t
t F
F b,t
b,t
b,t

∗ is
default event, and It (ζ F ) represents the capital injection that happens at this period; Vb,tthe value function conditional on a bank run.
Let’s guess-and-verify that the value function is given by Vb,t = ξ b,t Bb,t , and the value9∗ = 0. The HJB equation can then be written asfunction conditional on a run is simply Vb,t"
0 = max ξ̇ b,t + ξ b,t i D,t − it + r F,t
BbF

F
Bb,t

#

Bb,t

+ λb [1 − ξ b,t ] + λc ξ b,t− λF ξ b κˆ ζF "
ζ ∗F,tˆ ζF "
ζF

−ζ F

F
Bb,t

Bb,t

1 + It (ζ F ) − ζ F#+ It (ζ F ) dGF (ζ F )

F
Bb,t

Bb,t

#
dGF (ζ F ).

We assume that banks internalize that they are fully recapitalized in the case of default thatF . From the first-orderdoes not trigger a run, that is, they internalize that Bb,t It (ζ F ) = ζ F Bb,tcondition with respect to BtF , we obtain r F,t = 0. Note that, because losses on loans are
passed to depositors at the margin in the case of a bank failure, the spread r F,t does notincorporate the effect of losses due to bank runs. The HJB can then be written asξ̇ b,t = λb [ξ b,t − 1] − ξ b,t i D,t − it + λ F κ(1 − GF (ζ ∗F,t )) .

Law of motion of banks’ net worth. Let Bb,t denote the aggregate net worth of theFbanking sector and Bb,t the aggregate holdings of bank loans. The law of motion of Bb,t isgiven by

dBb,t = (i D,t − πt ) Bb,t + λb B̃b,t − Bb,t dt − ι t Bb,t dNt ,where ι t is an indicator that is equal to one in the event of a bank run and zero otherwise.

Deposit rates and savers’ behavior. For simplicity, we assume that the deposit rate is
proportional to the short-term interest rate, i D,t = κ D it , where the parameter κ D captures
the pass-through of the policy rate to the deposit rate. Note that the spread between thedeposit rate and the short-term interest rate responds to changes in monetary policy, as the
spread is given by i D,t − it = (κ D − 1)it .
Deposits are provided by savers, so they are ultimately the owners of corporations, as
they own bank equity and deposits that finance lending to firms. Savers’ behavior willthen be the same as in the setup of Section 2, where we assumed that savers are the soleowners of firms.

10

B.2

Stationary equilibrium

We consider next a stationary equilibrium. The economy after a bank run coincides withthe disaster state described in Section 2, where the probability of a disaster is given byλ ≡ λ F κ (1 − GF (ζ ∗F )) ,where ζ ∗F = 1−1ζ d



Bb
− ζd
BbF



. The interest rate on deposits is i D = κ D rn and the price ofF

corporate debt is given by Q F = iD +1 ψF . The value of corporate loans is Bb = FF Q F . The
net worth of banks in the stationary equilibrium is given by Bb = λλbb−B̃ibD . We assume thatFB̃b = χb Bb , 0 < χb < 1, which allow us to solve for bank’s leverage:FBb
λ − iD
,
= b
λb χb
Bb
F

where we assume that λb > 1−iDχb , such that BBb > 1. To guarantee that ζ ∗F > 0, the fire-salebparameter must satisfy ζ d < λλBb−χibD . To guarantee there is no bank failure in the absence ofa run, the upper-bound of the support for ζ F must satisfy ζ F < λλBb−χibD .

B.3

Log-linear dynamics

In the presence of monetary shocks, banks’ leverage will be time-varying and so it willbe the probability of disaster. The probability of disaster λt = λ F χ(1 − GF (ζ ∗F,t )) and thethreshold ζ ∗F,t , up to first order, are given byλ̂t = −

gF (ζ ∗F )
ζ̂ ∗ ,
1 − GF (ζ ∗F ) F,t

where λ̂t ≡ λtλ−λ , ζ̂ ∗F,t = ζ ∗F,t − ζ ∗F , bb,t ≡two equations, we obtainλ̂t =ζ̂ ∗F,t =1 Bb
F(b − bb,t),
1 − ζ d B F b,tb

FF

Bb,t − Bb
F ≡ Bb,t − Bb . Combining the previous
and bb,t
F
Bb
BbgF (ζ ∗F )1 Bb F(b − bb,t ).
∗1 − GF (ζ F ) 1 − ζ d B F b,tb

F equals the real value of corporate debt D /P . Therefore, banks’In equilibrium, Bb,t
t
F,t
BF

leverage is given by Bb,t
=
b,t

n
DF,t
n , where D F,t is the nominal value of corporate debt and B b,t
Bb,t

is the nominal value of banks’ net worth. Given the assumption that corporations keep
F −bn
the face value of debt fixed, we have that bb,tb,t = q F,t − bb,t , where q F,t =11

Q F,t − Q F
is
QF

n =
the change in the nominal price of corporate debt and bb,t

N

N

Bb,t − BbNBb

is the change in thenominal value of banks’ net worth. From the pricing condition for loans, we have that−

κ D (it − rn )1
q F,t + q̇ F,t = κ D (it − rn ) ⇒ q F,t = −.
QF
i F + ψF + ψm

n is given by ḃn = (i − λ ) bn + κ (i − r ), and the initial
The law of motion of bb,t
n
D
D t
b b,t
b,t
F

n = Bb q . Integrating the law of motion of bn and assuming thatvalue bb,0 is given by bb,0
b,t
B F,0
b

nominal interest rates are exponentially decaying, it − rn = e−ψm t (i0 − rn ), we obtainFe−ψm t − e−(λb −iD )tn−(λb −i D )t Bb κ D (i0 − rn )+
κ D ( i0 − r n ).
bb,t = −eλb − i D − ψm
Bb i F + ψF + ψm

Leverage is then given byFnq F,t − bb,t=
where ϕt ≡

F

Bb
1
1
i F +ψF +ψm Bb + λb −i D −ψm

Bb
−1
Bb


!

κ D (it − rn )+ ϕt ,i F + ψF + ψme−(λb −iD )t − e−ψm t κ D (i0 − rn ).

The leverage of the banking sector is then given by two terms. First, a term that is
proportional to the nominal interest rate. Therefore, this term is initially positive after acontractionary monetary shock and decays at the rate ψm . The second term ϕt controls the
speed of convergence of leverage, which can be potentially different from the decay rate ofinterest rates. These properties imply that ϕt does not affect leverage on impact or in thelong run, but it affects how fast leverage converges back to the stationary equilibrium. For
instance, if ϕt > 0 for t > 0, then leverage decays more slowly than the monetary shockψm , and leverage decays faster if ϕt < 0.
Given the expression for leverage, we obtain the probability of a disasterϕλ̂t = eλ (it − rn ) + eλ ϕt ,ϕwhere eλ ≡ eλ

 F

Bb
−1
Bb

(B.1)

 −1


κD
i F +ψF +ψmandgF (ζ ∗F )1eλ ≡∗1 − GF ( ζ F ) 1 − ζ d

1−

12

Bb
F
Bb

!

κD
.
i F + ψF + ψm

(B.2)

A convenient special case. Expression (B.1) generalizes the expression for the disasterϕprobability given in Section 2, as it includes the additional term eλ ϕt . This term can bepositive or negative, for t > 0, depending on parameters, and it is equal to zero if
λb = (ψm + i D ) 1 −

Bb

!

− ψF

F
Bb

Bb

,

F

Bb

which after using BFb = λλbb−χibD , we obtainBb(1 − χb )(ψm + i D ) + i D − ψF χb +
λb =
2

√

∆λ

,

where ∆λ ≡ ((1 − χb )(ψm + i D ) + i D − ψF χ F )2 − 4(ψm + i D )i D . We assume that ψm is large
Fenough such that λb > 1−iDχb and BBb > 1.
b

In this case, the probability of a disaster simplifies to the expression given in Section 2:
λ̂t = eλ (it − rn ).
In this model with endogenous disasters, we are able to pin down the determinants ofeλ using expression (B.2). For instance, the probability of disaster is increasing in financial
intermediaries’ leverage in the stationary equilibrium, the fire-sale parameter ζ d , the passthrough of nominal interest rates to deposits κ D , and the maturity of corporate debt.

C

Derivations

C.1

The non-linear model

We consider the case where household debt is risky and long-term, as in Section 5. The
case of short-term riskless debt discussed in Section 2 corresponds to a particular special
case of this more general formulation.

Households’ problem. The household problem is given by
Vj,t ( Bj ) =max

L ,B P ][Cj,z ,Nj,z ,Bj,zj,z z≥t

Et 

ˆ t∗
t


e−ρ j (z−t) 

1− σ
Cj,z

1−σ

13

1+ φ

−

Nj,z

1+φ





∗
∗
 dz + e−ρ j (t∗ −t) Vj,t∗ ( B j,t∗ )  ,subject to the flow budget constraint
dBj,t =

Π j,t
Wj,t
L
P
ej,t − Cj,t dt+
Nj,t ++T(it − πt ) Bj,t + r L,t Bj,t
+ r P,t Bj,t
+
Pt
Pt


Q∗L,t − Q L,t
Q∗P,t − Q P,t
L
P
Bj,t
dNt ,
+ Bj,t
Q L,t
Q P,t


and the borrowing constraint and no-negativity constraint for long-term bonds
BSj,t ≥ 0,

Bj,t ≥ − D P,t ,

L
Bj,t
≥ 0,

given the initial condition Bj,t = Bj ≥ − D P,t , where DP,t = Q P,t F is the debt limit andQ̇rk,t ≡ Q1k,t + Qk,t− ψk − it is the excess return on the long-term bond of type k ∈ { L, P},k,tconditional on no disasters.
The first-order conditions with respect to consumption and labor are given by
−σ
Cj,t
=

∂Vj,t
,
∂B

φ

Nj,t =∂Vj,t Wj,t.
∂B Pt

L + B P ≤ B , and denoting the Lagrange
Writing the non-negativity condition on BSj,t as Bj,tj,tj,tmultiplier on the non-negativity constraint on short-term and long-term non-defaultablebonds by µSj,t and µ Lj,t , respectively, we obtain the first-order condition for the two types oflong-term bonds:∗∂Vj,t
Q L,t − Q∗L,t∂Vj,t− µ Lj,t + µSj,t ,r L,t = λt
∂B
∂B
Q L,t

∗
∂Vj,t
Q P,t − Q∗P,t∂Vj,t+ µSj,t .
r P,t = λt
∂B
∂B
Q P,t

The solution is also subject to the state-constraint boundary condition2∂Vj,t (− D P )
≥
∂B



Π j,t
Wj,t
ej,t
−(i P,t − πt ) D P +Nj,t +
+T
Pt
Pt

−σ
,

1
where i P,t = Q−
P,t − ψP .

Combining the first-order conditions for consumption and labor, we obtain the stanσtdard labor-supply condition W
Pt = Nj,t C j,t .
φAbstracting from the non-negativity constraints, the first-order conditions for long-

2 See Achdou et al. (2017) for a discussion of state-constraint boundary conditions in the context

of continuous-time savings problems with borrowing constraints.

14

term bonds can be written asr L,t = λt

!−σ

∗
Cj,t

Cj,t

Q L,t − Q∗L,t
,
Q L,t

!−σ

∗
Cj,t

r P,t = λt

Cj,t

Q P,t − Q∗P,t
.
Q P,t

(C.1)

The envelope condition with respect to Bj,t for an unconstrained household is given by

ρj

∂Vj,t
∂Vj,t
= (it − πt )+∂B
∂BEt [ d



∂Vj,t
∂B



dt

]
.

Combining this expression with the first-order condition for consumption, we obtain−σ
Et [dCj,t]dt−σ= −(it − πt − ρ j )Cj,t.

Expanding the expectation of the marginal utility, we obtain the Euler equation for savers
λt
Ċs,t= σ −1 ( i t − π t − ρ s ) +
Cs,t
σ

"

Cs,t
∗
Cs,t

!σ

#

−1 .

(C.2)

Firms’ problem. Final goods are produced according to the production function Yt =´e

1 e −10 Yi,tby Yi,t = e−e 1. The solution to final-good producers problem is a demand for variety i given´ 1−1 ePi,t −e1 1− e
.
Y
.
The
price
level
is
given
by
P
=
P
di
t
t
0 i,t
Pt

The intermediate-goods producers’ problem is given by"ˆt∗

Qi,t ( Pi ) = max Et[πi,s ]s≥t

subject to Yi,t =



t
Pi,t
Pt

−e

ηs
ηt



#

Pi,s
Ws Yi,sϕ 2
ηt ∗ ∗(1 − τ ) Yi,s −− πs ( j) ds +Q ∗ ( Pi,t∗ ) ,
Ps
Ps As
2
ηt i,tYt and Ṗi,t = πi,t Pi,t , given Pi,t = Pi . The first-order condition is

given by
∂Qi,t
Pi,t = ϕπi,t .
∂Pi
The change in πt conditional on no disaster is then given by 2
∂ Qi,t∂2 Qi,t∂Qi,t+πi,t Pi,t Pi,t +
πi,t Pi,t = ϕπ̇i,t .
2
∂t∂Pi
∂Pi
∂Pi

15

(C.3)

The envelope condition with respect to Pi,t is given by

0=

−e

∂2 Qi,t∂2 Qi,tYt++
πi,t Pi,t +
Pi,t
∂t∂Pi
∂Pi2
 ∗

∂Q j,t∂Qi,t∂Qi,tηt∗ ∂Qi,t
πi,t − (it − πt )+ λt−. (C.4)
∂Pi
∂Pi
ηt
∂Pi
∂Pi

P
Wt
(1 − e)(1 − τ ) i,t + e
Pt
Pt A



Pi,t
Pt

Multiplying the expression by Pi,t and using Equations (C.3) and (C.4), we obtain

0=

P
Wt
(1 − e)(1 − τ ) i,t + e
Pt
Pt A



Pi,t
Pt

−e
Yt + ϕπ̇t − (it − πt ) ϕπi,t + λtηt∗∗
ϕ(πi,t− πi,t ).
ηtRearranging the expression above, we obtain the non-linear New Keynesian Phillips curve

π̇t =

η∗
it − πt + λt
ηt
π t − ( e − 1) ϕ

−1




e Wt 1− (1 − τ ) Yt ,e − 1 Pt A

where we have assumed that Pi,t = Pt for all i ∈ [0, 1] and that πt∗ = 0.

C.2

Stationary equilibriumt
Introducing recurrent shocks. Suppose aggregate productivity follows dA
At = − ζd Nt ,given A0 = A and 0 < ζ < 1, where Nt is a Poisson process with arrival rate λt = λ(it −ρ; At ).3 The setting discussed in the paper corresponds to the case where λ(it − ρ; A) > 0
and λ(it − ρ; At ) = 0 for At < A.

C.2.1

Consumption and natural interest rate

Consumption determination in a stationary equilibrium. In a stationary equilibrium, all variables are independent of calendar time and depend only on aggregate productivity At , so they are constant between the realization of disasters. We can then write,
for instance, consumption and output as Cj,t = C j ( At ) and Yt = Y( At ).
Imposing Ċs,t = 0 in the Euler equation (C.2), we obtain the natural rate rn ( A)rn ( A ) = ρ s − λ ( A )Cs ( A )Cs ( A(1 − ζ ))

σ



−1 ,

where we abuse notation and write λ( A) instead λ(0; A) in a stationary equilibrium.
3 The process can be easily generalized to allow for trend growth, dA

t = gAt dt − ζ At d Nt . The
expressions in the text apply to this case as well after all variables are properly detrended.

16

The consumption of borrowers satisfies the conditionhCb ( A) = A(1 − τ )(1 − e−1)i 1+φ φ−σ

Cb φ ( A ) + Tb ( A ) − rn ( A ) D P ,where Tb ( A) represents the level of transfers as a function of productivity, and we usedt
the labor supply condition to solve for Nb and the fact that the real wage is given by W
Pt =At (1 − τ )(1 − e−1 ).4 The consumption of savers satisfies the conditionhi 1+ φ − σ
1 − (1 − τ )(1 − e−1 )µ T ( A ) rn ( A ) µ b
φφDP.
Cs ( A) = A(1 − τ )(1 − e−1 )Cs ( A ) +Y( A ) − b b+
1 − µb1 − µb1 − µbGiven that Y( A) = ∑ j∈{b,s} µ j C j ( A), the above conditions provide a pair of functional
equations that determine C j ( A).

Symmetric stationary equilibrium. In a symmetric equilibrium Cs ( A) = Cb ( A) =1+ φ  1
Y( A), hence Y( A) = A σ+φ (1 − τ )(1 − e−1 ) σ+φ . This equilibrium requires that Tb ( A) sat
h
σ
i


Y( A )
isfies Tb ( A) = 1 − (1 − τ )(1 − e−1 ) Y( A) + ρs − λ( A) Y( A(1−ζ )) − 1 D P . More-over, in a symmetric stationary
equilibrium, the real interest rate is given by rn ( A) =
ρ s − λ ( A ) (1 − ζ )φ−σ σ1++φ− 1 . These results hold for both the case of non-recurrent shocks,

where λ( A) > 0 and λ( A∗ ) = 0, and the case of recurrent shocks, where λ( A) is independent of productivity level A.

C.2.2

Risk premia

Equity premium. The value of the intermediate-goods firms is given by QS,t = Et

h´

∞ ηs
t ηt Πs ds ,twhere Πt = (1 − τ )Yt − W
Pt Nt . From this expression we get the expected return on the firm:∗ )−σ − C −σ Q − Q∗ (Cs,t
Et [dQS,t ]
Πt
S,t
s,t
S,t 

 dt.
dt +
=  (it − πt ) + λt
−
σ

QS,t
QS,t
Q
C
S,t


s,t
| {z } |{z}interest rate

(C.5)

risk premium

In a stationary equilibrium, profits are given by Πt = e−1 (1 − τ )Yt . Using that rn,t +4 For ease of exposition, we focus on the case of riskless household debt. The case of long-termrisky household debt can be handled analogously.

17

i


λt

Cs,t
∗
Cs,t

σ

= ρs + λt , we can write the dividend-yield as follows:ρs + λtΠt=  σ Q∗ ∗ ,
Cs,t
QS,t
S,t Πt1+λtΠ∗t Πt∗Cs,tSuppose λt > 0 in the no-disaster state and λt = 0 in the disaster state. Then, Π∗t =ρs Q∗S,t and Πt /QS,t can be written asΠtρs + λt
=
,
 σ
Cs,t
(1− ζ Y )
QS,t
1+λ
t

∗
Cs,t

ρs

∗

where 1 − ζ Y ≡ YY . The (unlevered) equity premium can then be written as
Πt
dt + λQS,t

Q∗S,t − QS,t
QS,t






(ρs + λt )(1 − ζ Y ) − r n = λ (1 − ζ Y ) − σ − 1 1 − σ  ,C
ρs + λt Cs,t∗s,tusing the fact that Π∗t /Πt = 1 − ζ Y .

Term spread. From the first-order condition (C.1), the excess return on the long-termbonds satisfies

1Q̇+ L,t − ψL − it = λt
Q L,t
Q L,t



Ct
Ct∗

σ

Q L,t − Q∗L,t
.
Q L,t

(C.6)

Let i L,t denote the yield on the long-term bond, then i L,t satisfies
ˆ ∞Q L,t =

e−(i L,s +ψL )(s−t) ds ⇒t

1Q̇
+ L,t = i L,t + ψL .
Q L,t
Q L,t

We consider next a stationary equilibrium with non-recurrent shocks. Combining theprevious two expressions, we obtain that the term
the difference between the long spread,and short interest rate, is given by i L − rn = λCsCs∗σ Q − Q∗L

Q L . The price of the long-termbond in the disaster state is given by Q∗L = i∗ +1ψL , where i∗L = rn∗ is the yield on the longLL

term bond in the disaster state. We can then express the term spread as
i L − rn = λ

Cs
Cs∗

σ

rn∗ − rn σ .
∗
rn + ψL + λ CC∗ss

Note that i L > rn and the difference is decreasing in ψL , so the yield increases with thebond duration.

18

Corporate bond premium. We can also price a corporate bond, which is a defautablebond. Let Q F,t denote the value of a bond that pays off coupons e−ψF t in nominal terms
in the absence of default. We assume that monetary shocks are too small to trigger default, so there is no default in the no-disaster state, and that the bond suffers a loss 1 − ζ Fconditional on a disaster.
A derivation analogous to the one for government bonds shows that the yield on the1
corporate bond, which is given by i F,t = Q−
F,t − ψF , can be expressed as follows in a sta-tionary equilibriumi F − rn = λCsCs∗

Q F − Q∗F.
QFσ

The value of the corporate bond in the disaster state is given by Q∗F = r1∗−+ζψFF . In the stationn

ary equilibrium, the value of the corporate bond in the no-disaster state is given by
1+λ
QF =



Cs
Cs∗

σ

Q∗F

ψF + ρ s + λ.

1
The yield on the corporate bond is given by i F,t = Q−
F,t − ψF . The corporate spread, thedifference between the yield on the corporate bond and a government bond with the same
coupons, is given by in the stationary equilibriumrF =ψF + ρ s + λψF + ρ s + λ
−
.
 σ
 σ
1 + λ CC∗sQ∗F
1 + λ CC∗s
Q∗F /(1 − ζ F )
s

s

C.3

Log-linear dynamics

C.3.1

Consumption of borrowers and savers

Labor supply and market clearing. The labor supply condition can be written as
wt − pt = φn j,t + σc j,t .
Log-linearizing the market-clearing conditions for consumption and labor, we obtainµcb cb,t + (1 − µcb )cs,t = yt ,where µcb ≡µnb nb,t + (1 − µnb )ns,t = nt ,

µb Nb
µb Cb
n
Y and µb ≡ N . Equating both agents’ labor-supply, we obtainns,t = nb,t + φ−1 σ(1 − µcb )−1 (cb,t − yt ),19

where we used the market-clearing condition for goods to eliminate cs,t . Plugging theabove expression into the market-clearing condition for labor, we obtainnb,t = (1 + φ−1 σ)yt − φ−1 σcb,t + φ−1 σµcb − µnb
(yt − cb,t ).
1 − µcb

The real wage is then given by
wt − p t = ( φ + σ ) y t + σµcb − µnb
(yt − cb,t ).
1 − µcb

Borrowers’ consumption. Since we assume that borrowers are sufficiently impatient,such that the borrowing constraint is always binding, their consumption is given by
Cb,t =

Wt
eb,t − (i P,t − πt ) D P,t .
N +T
Pt b,tLinearizing this expression, we obtain
W Nb
cb,t =(wt − pt + nb,t ) + Tb,t −PCbwhere Tb,t ≡ψP(i P,t − i P ) − πt d P ,i P + ψP

eb,t − TebTi P,t −i P
DPCb , and d P ≡ Cb , and we used that q P,t = − i P +ψP . Using the expression

for the real wage and labor supply, we obtainµcb − µnb
W Nb
−1
−1
−1
cb,t =(1 + φ )(φ + σ)yt − φ σcb,t + (1 + φ )σ(yt − cb,t ) +PCb
1 − µcbψP
Tb,t −(i P,t − i P ) − πt d P .
i P + ψP
Solving for cb,t , and using Tb,t = Tb0 (Y )yt , we obtaincb,t = χy yt − χrψP(i P,t − i P ) − πt d P ,i P + ψP

Tb0 (Y )+(1−α)(1+φ−1 )(φ+σ)
and χr ≡ 1+(1−1α)φ−1 σ . The expressions in the paper are
1+(1−α)φ−1 σobtained by imposing Cb = Cs = Y, so µcb = µnb = µb , and 1 − α ≡ WPYN .

where χy ≡Savers’ consumption. From the borrowers’ consumption and market clearing for goods,
we obtain
cs,t =

1 − µb χyµ χr d P
yt + b( i t − π t − r n ).
1 − µb1 − µb

20(C.7)

C.3.2

Asset pricing

Price of risk. The price of risk is given by Pd,t ≡ λt



Cs,t
∗
Cs,t

σ

. Approximating around thestationary equilibrium, we get
pd,t = σcs,t + eλ (it − rn ).

Stocks. Rearranging and log-linearizing expression (C.5) around the stationary equilibrium, we obtainq̇S,t = ρqS,t + (it − πt − rn ) + λ

Cs
Cs∗

σ

QS − Q∗S
Y
pd,t −((1 − τ )yt − (1 − α)(wt − pt + nt )) .
QSQSSolving this equation forward, we obtainY
qS,0 =QSˆ ∞0e−ρt [(1 − τ )yt − (1 − α)(wt − pt + nt )] dt−

 σ

ˆ ∞
QS − Q∗S
Cse−ρt it − πt − rn + λ
p
)
dt.
d,t
Cs∗
QS
0

Long-term bonds. Rearranging expression (C.6) and log-linearizing around the stationary equilibrium, we obtainq̇ L,t = (ρ + ψL )q L,t + it − rn + λσ

Q L − Q∗L
(σcs,t + eλ (it − rn )).
QL

σ


Q L − Q∗L
(σcs,t + eλ (it − rn )) dt.
QL

Cs
Cs∗

Solving the above equation forward, we obtainˆ ∞q L,0 = −e−(ρ+ψL )t
it − rn + λ

0

Cs
Cs∗

1
Using that Q−
L = i L + ψL , the yield on the long-term bond can then be written asˆ ∞i L,0 − i L = (i L + ψL )e−(ρ+ψL )t
it − rn + λ

0

Cs
Cs∗

σ


Q L − Q∗L
(σcs,t + eλ (it − rn )) dt.
QL

Household debt. The price of the household debt evolves according toq̇ P,t = (ρ + ψP )q P,t + it − rn + r P pd,t .

21

Under the assumption that r P σ = O(i0 − rn ), we can write the price of the bond asq̇ P,t = (ρ + ψP )q P,t + it − rn + r P eλ (it − rn ) ⇒ q P,t = −1 + r P eλ
( i t − r n ).
ρ + ψP + ψm

(C.8)

Corporate bonds. The linearized price of the corporate bond is given by an expression
analogous to the one for government bonds:
ˆ ∞q F,0 = −e−(ρ+ψF )tit − rn + λ

0

Cs
Cs∗

σ


Q F − Q∗F
(σcs,t + eλ (it − rn )) dt.
QF

1
The yield on the corporate bond is i F,0 − i F = − Q−
F q F,0 , which can be written asˆ ∞i F,0 − i F = (i F + ψF )e−(ρ+ψF )tit − rn + λ

0

Cs
Cs∗

σ


Q F − Q∗F
(σcs,t + eλ (it − rn )) dt,
QF

1
using the fact that Q−
F = i F + ψF .

The corporate spread is r F,0 = i F,0 − i F,0 , where i F,0 is the yield of a government bondwith the same coupons as the corporate bond:ˆ ∞r F,0 = r F0e−(ρ+ψF )t (it − rn )dt + (i F + ψF )(i F − i ) − (i F + ψF )(i F − i )ˆ ∞e−(ρ+ψF )t (σcs,t + eλ (it − rn ))dt,0where i F − i = λ



Cs
Cs∗

σ

Q F − Q∗F
is the difference between the corporate bond yield and the
QF

short-term nominal rate in the stationary equilibrium and i F − i is the corresponding object
for a bond without default risk.

D

Estimation of Fiscal Response to a Monetary ShockWe estimate the empirical IRFs using a VAR identified by a recursiveness assumption, as
in Christiano et al. (1999), extended to include fiscal variables. The variables included
are: real GDP per capita, CPI inflation, real consumption per capita, real investment percapita, capacity utilization, hours worked per capita, real wages, tax revenues over GDP,government expenditures per capita, the federal funds rate, the 5-year constant maturity
rate, and the real value of government debt per capita. We estimate a four-lag VAR using
quarterly data for the period 1962:1-2007:3. The identification assumption of the monetaryshock is as follows: the only variables that react contemporaneously to the monetary shock22

Federal Funds Rate

OutputInflation

0.25

0.9

0.2

0.00

0.6

0.25

0.3

0.0

0.50

0.0

0.2

0.75

0.3
0

5

10

t

15

20

0

5

Consumption

0.2

10

t

15

20

0

5

Investment

10

t

15

20

15

20

Hours
0.50

1

0.25

0.0

0

0.2

0.00
0.25

1

0.50

0.4
2
0.6

0.4

0

5

10

t

15

20

0.75
0

5

10

t

15

20

0

5

10

t

Figure D.1: Estimated IRFs.

are the federal funds rate, the 5-year rate and the value of government debt. All othervariables, including tax revenues and expenditures, react with a lag of one quarter.

Data sources. The data sources are: Nominal GDP: BEA Table 1.1.5 Line 1; Real GDP:
BEA Table 1.1.3 Line 1, Consumption Durable: BEA Table 1.1.3 Line 4; Consumption Non
Durable: BEA Table 1.1.3 Line 5; Consumption Services: BEA Table 1.1.3 Line 6; Private
Investment: BEA Table 1.1.3 Line 7; GDP Deflator: BEA Table 1.1.9 Line 1; Capacity Utilization: FRED CUMFNS; Hours Worked: FRED HOANBS; Nominal Hourly Compensation: FRED COMPNFB; Civilian Labor Force: FRED CNP16OV; Nominal Revenues:
BEA Table 3.1 Line 1; Nominal Expenditures: BEA Table 3.1 Line 21; Nominal Transfers:
BEA Table 3.1 Line 22; Nominal Gov’t Investment: BEA Table 3.1 Line 39; Nominal Consumption of Net Capital: BEA Table 3.1 Line 42; Effective Federal Funds Rate (FF): FREDFEDFUNDS; 5-Year Treasury Constant Maturity Rate:

FRED DGS5; Market Value ofGovernment Debt: Hall et al. (2018).
All the variables are obtained from standard sources, except for the real value of debt,
which we construct from the series provided by Hall et al. (2018). We transform the series
into quarterly frequency by keeping the market value of debt in the first month of thequarter. This choice is meant to avoid capturing changes in the market value of debt arisingfrom changes in the quantity of debt after a monetary shock instead of changes in prices.

VAR estimation. Figure D.1 shows the results. As is standard in the literature, wefind that a contractionary monetary shock increases the federal funds rate and reduces
output and inflation on impact. Moreover, the contractionary monetary shock reduces
consumption, investment, and hours worked.

23

Data

(1)

(2)

(3)

(4)

(5)

(1) - (2) - (3) + (4) - (5)Revenues

Interest Payments

Transfers &Expenditures

Debt in T

Initial Debt

Residual

-26

68.88

-12.092.91-49.7430.13[-72.89,20.89][30.01,107.75][-48.74,24.56][-12.79,18.62][-68.03,-31.46][-4.74,65]

Table D.1: The impact on fiscal variables of a monetary policy shock
Note: Calculations correspond to a a 100 bps unanticipated interest rate increase. Confidence interval at 95% level.

The Government’s Intertemporal Budget Constraint. The fiscal response in themodel corresponds to the present discounted value of transfers over an infinite horizon,
1− λtthat is, ∑∞t=0 β̃ Tt , where β̃ = 1+ρs . We next consider its empirical counterpart. First, we

calculate a truncated intertemporal budget constraint from period zero to T :

by b0
|{z}

debt
revaluation



T


−1
m
n 
= ∑ β̃t  τyt + τt − |β̃ by (it−1{z− πt − r }) −t =0| {z }
interest payments
tax revenue

T0,T + β̃T by bT
|
{z
}

(D.1)

other transfers/expenditures& final debt

The right-hand side of (D.1) is the present value of the impact of a monetary shock on
fiscal accounts. The first term represents the change in revenues that results from the realeffects of monetary shocks. The second term represents the change in interest paymentson government debt that results from change in nominal rates. The last two terms areadjustments in transfers and other government expenditures, and the final debt positionat period T , respectively. In particular, T0,T represents the present discounted value oftransfers from period 0 through T . Provided that T is large enough, such that (yt , τt , it )have essentially converged to the steady state, then the value of debt at the terminal date,bT , equals (minus) the present discounted value of transfers and other expenditures fromperiod T onward. Hence, the last two terms combined can be interpreted as the present
discounted value of fiscal transfers from zero to infinity. Finally, the left-hand side represents the revaluation effect of the initial stock of government debt.
Table D.1 shows the impact on the fiscal accounts of a monetary policy shock, both in
the data and in the estimated model. We first apply equation (D.1) to the data and check
whether the difference between the left-hand side and the right-hand side is different fromzero. The residual is calculated as
Residual = Revenues - Interest Payments - Transfers + Debt in T - Initial Debt

24

Federal Funds Rate

Excess Bond Premium

0.2

1.0
0.1
0.5
0.0

0.0
0.5

0

5

10

t

15

20

0.1

0

5

10

t

15

20

Figure D.2: IRFs for the federal funds rate and excess bond premium.
We truncate the calculations to quarter 60, that is, T = 60 (15 years) in equation (D.1).
The results reported in Table D.1 imply that we cannot reject the possibility that the residual is zero and, therefore, we cannot reject the possibility that the intertemporal budget
constraint of the government is satisfied in our estimation.
The adjustment of the fiscal accounts in the data corresponds to the patterns we observed in Figure 2. The response of initial debt is quantitatively important, and it accounts
for the bulk of the adjustment in the fiscal accounts.

EBP. To estimate the response of the corporate spread in the data, we add the EBP measure of Gilchrist and Zakrajšek (2012) into our VAR (ordered after the fed funds rate). Since
the EBP is only available starting in 1973, we reduce our sample period to 1973:1-2007:7.
The estimated IRFs are in line with those obtained for the longer sample. We find a significant increase of the EBP on impact, of 6.5 bps, in line with the estimates in the literature.

Appendix References
Achdou, Y., Han, J., Lasry, J.-M., Lions, P.-L., and Moll, B. (2017). Income and wealthdistribution in macroeconomics: A continuous-time approach.
Christiano, L. J., Eichenbaum, M., and Evans, C. L. (1999). Monetary policy shocks: What
have we learned and to what end? Handbook of Macroeconomics, 1:65–148.
Gertler, M., Kiyotaki, N., and Prestipino, A. (2020). Banking panics as endogenous disasters and the welfare gains from macroprudential policy. In AEA Papers and Proceedings,
volume 110, pages 463–69.
Gilchrist, S. and Zakrajšek, E. (2012). Credit spreads and business cycle fluctuations. American Economic Review, 102(4):1692–1720.
Hall, G., Payne, J., and Sargent, T. (2018). Us federal debt 1776-1960: Quantities and prices.

25