#  Idiosyncratic Risk, Government Debt and Inflation

Matthias Hänsel Department of Economics, Stockholm School of Economics. I
thank Lars Ljungqvist and David Domeij for guidance and support as well as
Tore Ellingsen, Richard Foltyn, Philipp Hochmuth and Zoltan Racz for helpful
comments. An earlier version of this paper was circulated under the title
“Fiscal Inflation According to HANK”. Email:
<EMAIL_ADDRESS>
(March 1, 2024)

###### Abstract

How does public debt matter for price stability? If it is useful for the
private sector to insure idiosyncratic risk, government debt expansions can
increase the natural rate of interest and create inflation. As I demonstrate
using a tractable model, this holds in the presence of an active Taylor rule
and does not require the absence of future fiscal consolidation. Further
analysis using a full-blown 2-asset HANK model reveals the quantitative
magnitude of the mechanism to crucially depend on the structure of the asset
market: under standard assumptions, the effect of public debt on the natural
rate is either overly strong or overly weak. Employing a parsimonious way to
overcome this issue, my framework suggests relevant effects of public debt on
inflation under active monetary policy: In particular, persistently elevated
public debt may make it harder to go the “last mile of disinflation” unless
central banks explicitly take its effect on the neutral rate into account.

Keywords: Monetary policy, Fiscal Policy, Inflation, HANK
JEL Classification: E31, E52, E63

##  1 Introduction

In the aftermath of the Covid-19 pandemic as well as the economic fallout
following Russia’s invasion of Ukraine, public debt levels have risen to
historic highs in many advanced countries. Should central banks be concerned
about this? In standard macroeconomic models, Ricardian Equivalence suggest
that they should only if said debt is “unfunded”, i.e. not backed by future
government revenue. In the words of ECB board member Schnabel (2022), “if
governments do not credibly signal their commitment to responsible fiscal
policies, the private sector may eventually expect that higher inflation is
needed to ensure the sustainability of public debt”. But if the debt is
expected to be eventually paid back by budget surpluses at some point in the
future, it doesn’t need to affect the conduct of their policies. Arguably,
potential support for this notion may be lent by the observation that
conventional monetary policy appears to have been broadly successful in
reigning in the recently high inflation, although concerns remain that “the
disinflation process during the last mile will be more uncertain, slower and
bumpier” (Schnabel, 2023).
However, things get more complex if government bonds have additional value for
the private sector, e.g. as a means of insurance against idiosyncratic risk.
In that case, the amount of public debt will generally affect the inflation-
neutral interest rate, as it imperfectly crowds out private demand and induces
households to require a higher real return if they are to hold more government
debt. Thus, if a central bank pursues an interest rate rule satisfying the
Taylor principle, this can create inflation even if a country’s fiscal
authority is committed to raise enough surpluses to eventually pay back the
debt (i.e. it is “funded”). All that is needed for this is the monetary
authority not (or imperfectly) adjusting its reaction function in response to
the government debt expansion.111It is worth emphasizing that this _does not_
mean the central bank not reacting at all. Indeed, I will always allow the
monetary authority to react according to an interest rate rule satisfying the
Taylor principle.
I first establish these results using a tractable New Keynesian model enriched
with idiosyncratic income risk. As obtaining analytical results for such
models is notoriously difficult in the presence of a positive net supply of
assets, this framework naturally relies on many simplistic assumptions, such
as households being ex-ante identical and subject to income risk only in a
single period. However, this simplicity also has the virtue of clarifying that
the mechanism does not rely on fiscal policy inducing any ex-ante
redistribution towards constrained households, a channel that received
substantial attention by the recent literature on Heterogeneous Agents New
Keynesian (HANK) models. Additional assumptions elucidate that it does neither
require the government to consume the resources it acquires through issuing
debt nor is it relying on distortionary taxation to consolidate its finances,
both of which could also affect inflation in Ricardian models. Furthermore, it
is consistent with the fiscal authority being committed to raise any amount of
surplus necessary to pay back its debt.
Naturally, the analytical insights beget the question to what extent they may
be quantitatively relevant. As their general equilibrium nature precludes
empirical identification, I approach this issue by relying on a calibrated
2-asset HANK model, in which households require liquid assets to insure
themselves against skill-, unemployment- as well as business risk but also
have access to illiquid capital assets yielding higher returns.
The calibrated framework matches well various micro-data moments emphasized by
the recent literature, such as a fairly realistic income- and wealth
distribution as well as empirically credible Marginal Propensities to Consume
(MPCs). However, under standard assumptions on the asset markets, it is not
able to generate a relation between government debt supply and real interest
rates of a magnitude in line with various empirical findings: If liquid and
illiquid assets are traded on segmented markets, as e.g. in Kaplan et al.
(2018) or Bayer et al. (forthcoming), then a higher government debt supply
yields much stronger increases in liquid bond rates. In contrast, if both
bonds and capital can be freely held either as liquid or illiquid asset, as
e.g. in Auclert et al. (2020, 2023), then more public debt is associated with
a much weaker rise in rates. Intuitively, if the private sector has access to
assets with superior returns to satisfy its longer-term savings needs,
substantially higher rates are necessary for it to be willing to hold a
correspondingly higher amount of liquid assets. But if the additional
government debt may just crowd out a fraction of the much larger (illiquid)
aggregate capital stock, its impact on equilibrium interest rates will be very
limited.
Nevertheless, I demonstrate that set-ups in which capital can imperfectly
serve for liquidity provision can resolve the tension. While important for my
results on inflation, this finding should be of independent interest for other
work analyzing fiscal policy using heterogeneous agents models: Getting
consumption behavior as well as the effects of fiscal expansions on government
financing costs right seems clearly desirable for such exercises.
Armed with the suitably calibrated model, I first analyze how fiscal policy
affects the time path of inflation in response to an inflationary supply
shock. This is interesting not only because such shocks were argued to be an
important factor behind the recent inflation experience, but also helpful to
pin down the mechanism at hand, as such shocks tend to generate similar real
responses for both HANK- and more conventional models (Kaplan and Violante,
2018). Nevertheless, I find that a public debt expansion in the aftermath of
an adverse supply shock can exert noticeable though moderate effects on
inflation for the HANK model but not in Representative (RA)- or Two Agent
(TA)- frameworks lacking idiosyncratic risk. In particular, if the debt level
remains persistently elevated, inflation can remain elevated as well and going
the “last mile of disinflation” take a long time. Comparisons with an
alternative fiscal rule as well as said RA- and TA- versions of the model
strongly suggest that this is indeed due to the same mechanism as in the
analytical model. I also demonstrate that these insights are similarly
relevant for the inflationary effects to expansionary fiscal shocks, another
purported driver of recent price level dynamics.
Still, since the 2-asset HANK model does _not_ feature fiscal dominance, such
an outcome is not set in stone but dependent on the conduct of monetary
policy: For example, more “hawkish” central bank reactions can still speed up
the disinflation process at the cost of less favorable real dynamics. However,
my framework suggests that a central bank explicitly taking into account the
“neutral rate” pressure generated by public debt can achieve faster
disinflation at lower costs in terms of aggregate consumption and
unemployment.

###  1.1 Related Literature

On the one hand, the paper connects to a long tradition in macroeconomics
studying monetary-fiscal policy interactions, going back to the seminal works
of Sargent and Wallace (1981) and Leeper (1991). Leeper and Leith (2016) and
Cochrane (2023) offer summaries of this literature, including its modern
incarnation as Fiscal Theory of the Price Level (FTPL). Notable recent
contributions include the work by Bianchi et al. (2023), who find fiscal
policy important to explain inflation persistence in the US, as well as Kaplan
et al. (2023), who study FTPL in a heterogeneous agent setting featuring
uninsurable idiosyncratic risk. As already indicated above, most these works
differ from mine in that they focus on the effects of unfunded government debt
(i.e. not backed by future surpluses).
On the other hand, my paper is part of the sprawling HANK literature: In
addition to the already mentioned papers, my work naturally relates to other
works studying fiscal policy: The work of Bayer et al. (2023a) is particularly
related in that it emphasizes the effects of expansionary fiscal policy on the
relative return of different assets and uses a two-asset model with many
similar features as mine. However, it mostly focuses on real outcomes and does
not analyze the role of the asset market structure. Similarly, other HANK
research on fiscal policy such as Hagedorn et al. (2019) or Seidl and Seyrich
(2023) mostly restrict attention to its real effects.
Moreover, a few other studies have also noticed the importance of the asset
market in quantitative HANK models: Dominguez Diaz (2021) finds model
responses to a shock to households’ income uncertainty to depend substantially
on whether financial intermediaries face constraints in the provision of
liquid assets. However, he does not consider a role for government debt.
Chiang and Żoch (2023) similarly study a HANK model with explicit financial
intermediation, which they calibrate using bank balance sheet data. Comparing
this structure with alternative settings, they also find the structure of the
asset market to be important for the aggregate effects of policy shocks.
However, they do not consider inflation as an outcome and find only limited
impact of the asset market structure for policies not targeted at the
financial sector, presumably due their different modelling choices.222For
instance, they assume the real interest rate on liquid assets is held fixed by
the central bank, while the effect of government debt on the former is an
important driver of my results.
In independent contemporary work, the recent working paper by Campos et al.
(2024) also analyzes how public debt effects the ”natural rate” and inflation
in a HANK economy. However, these authors focus on a one-asset HANK framework
and thus cannot provide for my insights on the role of the asset markets.
Moreover, in contrast to this work, they focus solely on the effects of a
permanent expansion of government debt and don’t analyze the channel’s
implications for business cycle dynamics.
Of course, my work also relates to studies analyzing fiscal policy in other
settings deviating from Ricardian Equivalence. In particular, related
inflationary effects of “funded” government debt were also noticed by Ascari
and Rankin (2013) and Aguiar et al. (2023) in the context of Overlapping
Generations (OLG)-models with nominal rigidities. Given their different
frameworks and focus, I view their work as complementary to mine.

The remainder of the paper is structured as follows: Section 2 presents the
tractable New Keynesian model enriched with income risk, derives the results
referred to above and discusses some empirical considerations regarding the
relationship between public debt and (liquid asset) interest rates. Section 3
then presents the 2-asset HANK model used for the quantitative analysis, the
calibration of which is detailed in Section 4. Section 5 studies the model
response to an adverse TFP shock, isolating the inflationary pressure from
government debt through comparisons with alternative model versions. The
resulting insights are then applied to fiscal policy shocks in Section 6
before Section 7 analyzes how alternative monetary policy rules can counteract
the inflationary effects of government debt. Ultimately, Section 8 discusses
various robustness checks before Section 9 concludes.

##  2 An Analytical Model

This section presents a simple New Keynesian model enriched with idiosyncratic
income risk, in which it is possible to analytically characterize the
mechanism mentioned above.

###  2.1 Model setup

Time is discrete and runs forever, starting from $t=0$. There is no aggregate
uncertainty, but households face idiosyncratic income risk as specified below.

####  2.1.1 Households

The model is inhabited by a unit mass of _ex-ante identical_ households (also
referred to as “agents” below), which gain utility from consumption and
leisure according to the utility function

$\displaystyle\mathbb{E}_{0}\sum^{\infty}_{t=0}\beta^{t}\left[\log(c_{it})+\gamma\log(1-N_{it})\right]~{}~{},$

where $N_{it}$ denotes time worked. The period felicity function corresponds
to the same analytically convenient balanced growth preferences as used by
Aguiar et al. (2023). Furthermore, it is assumed that in $t=0$, each
individual has the same labor productivity $z_{0}=1$, i.e. they supply one
efficiency unit of labor per unit of time worked.
Between periods $0$ and $1$, and at that time only, households face
_idiosyncratic_ income risk. In particular, they transition to a state of high
labor productivity $z^{h}>1$ with probability $\rho^{h}$ and to a state of low
labor productivity $z^{l}<1$ with probability $\rho^{l}=1-\rho^{h}$. These
labor productivities remain fixed for $t\geq 1$ onwards, so as of that time,
there will be a fraction $\rho^{h}$ of “high productivity” households and a
fraction $\rho^{l}$ of “low productivity” households.
For tractability, I restrict

$\displaystyle\rho^{h}z^{h}+(1-\rho^{h})z^{l}=1~{}~{},$ (1)

so that the economy’s average labor productivity is not affected by the time 0
risk.
In any period, a household with productivity $i\in\\{h,l\\}$ faces the budget
constraint

$\displaystyle
P_{t}w_{t}z^{i}N^{i}_{t}+(1+i_{t})B_{t-1}+P_{t}T_{t}=P_{t}c_{t}+P_{t}z_{t}\tau_{t}+B_{t}~{}~{},$

which can be stated in real terms as

$\displaystyle
w_{t}z_{t}N^{i}_{t}+\frac{1+i_{t}}{\pi_{t}}b_{t-1}+T_{t}=c_{t}+z_{t}\tau_{t}+b_{t}$
(2)

where $b_{t}:=B_{it}/P_{t}$ and $\pi_{t}:=P_{t}/P_{t-1}$. $P_{t}$ denotes the
current price level, $w_{t}$ the real wage and $T$ lump-sum transfers from the
government. $B_{t}$ denotes holdings of nominal bonds that each yield a gross
nominal return of $1+i_{t}$. Additionally, the government may levy a non-
distortionary tax proportional to individual labor productivity, which is
denoted by $\tau$. I additionally impose that in period $0$, each household
starts out without any bonds, i.e. $B_{-1}=0$: This is not only analytically
convenient, but also clarifies that, unlike for the FTPL, none of the results
derived here rely on “surprise” asset revaluations (cf. Niepelt, 2004).

####  2.1.2 Final good firms

The economy’s final good is produced by a representative firm, which combines
intermediate goods $y_{t}(j)$ according to the following CES production
function:

$\displaystyle
Y_{t}=\left[\int^{1}_{0}y_{t}(j)^{\frac{\epsilon-1}{\epsilon}}dj\right]^{\frac{\epsilon}{\epsilon-1}}$
(3)

Taking prices of intermediate goods $p_{t}(j)$ as given, the firm’s
optimization problem implies it will demand $y_{t}(j)$ according the familiar
demand structure

$\displaystyle
y_{t}(j)=\left[\frac{p_{t}(j)}{P_{t}}\right]^{-\epsilon}Y_{t}~{}~{},$ (4)

resulting in the price of its final good to be

$\displaystyle
P_{t}=\left[\int^{1}_{0}p_{t}(j)^{1-\epsilon}dj\right]^{\frac{1}{1-\epsilon}}~{}~{}.$

###  2.2 Intermediate good firms

Intermediate goods are produced by a unit mass of firms, each of which produce
a single variety $j$ as monopolists using labor purchased from households at
real wage $w_{t}$, which they take as given.
For simplicity, it is assumed that the intermediate goods firms are owned by
risk-neutral “capitalists” who cannot participate in the bond market and
discount the future at the same rate $\beta$ as the households.333Specifying a
different discount factor for the firms would not affect any of the results
below. Similar assumptions are common for so-called “tractable HANK” models in
the literature and aim to reduce the dependence of household behavior on firm
profits (e.g., Broer et al., 2020).
The intermediate goods firms are endowed with an identical initial price level
$p_{-1}(j)=P_{-1}$ and face a quadratic price adjustment cost à la Rotemberg
(1982), subject to which they maximize

$\displaystyle\sum^{\infty}_{t=0}\mathbb{E}_{0}\beta^{t}\left[(p_{t}(j)-w_{t})\left[\frac{p_{t}(j)}{P_{t}}\right]^{-\epsilon}Y_{t}-\frac{\phi}{2}\left(\frac{p_{t}(j)}{p_{t-1(j)}}-1\right)^{2}Y_{t}\right]~{}~{}.$

The first order conditions of the price-setting problem in period $t$ is

$\displaystyle(1-\epsilon)\left[\frac{p_{t}(j)}{P_{t}}\right]^{-\epsilon}Y_{t}+\epsilon\frac{w_{t}}{p_{t}(j)}\left[\frac{p_{t}(j)}{P_{t}}\right]^{-\epsilon-1}Y_{t}$
$\displaystyle-\phi\left(\frac{p_{t}(j)}{p_{t-1(j)}}-1\right)\frac{1}{p_{t-1}(j)}$
$\displaystyle+\beta\mathbb{E}_{t}\left(\frac{p_{t+1}(j)}{p_{t(j)}}-1\right)\frac{p_{t+1}(j)}{(p_{t}(j))^{2}}=0$

and by restricting focus to a symmetric equilibrium so that $p_{t}(j)=P_{t}$,
we obtain the following New Keynesian Phillips curve:

$\displaystyle\phi(\pi_{t}-1)\pi_{t}=(1-\epsilon)+\epsilon
w_{t}+\beta\mathbb{E}_{t}\frac{Y_{t+1}}{Y_{t}}\phi(\pi_{t+1}-1)\pi_{t+1}~{}~{}.$
(5)

####  2.2.1 Government

The government consists of two branches, a _monetary authority_ and a _fiscal
authority_. The _monetary authority_ determines the nominal interest rate
according to the standard Taylor rule

$\displaystyle i_{t+1}=r^{*}_{t}+\theta_{\pi}(\pi_{t}-1)~{}~{},$ (6)

where $r^{*}_{t}$ is taken as a parameter. This is consistent with typical
“textbook” formulations as e.g. in Galí (2015) but allows to account for the
neutral period being different in $t=0$. I restrict
$r^{*}_{t}=\frac{1}{\beta}-1~{}~{}\forall t>0$, which will below be shown to
be the neutral rate of interest once idiosyncratic risk has been resolved.
The _fiscal authority_ can provide lump-sum transfers to households, which are
financed by issuing nominal government bonds $B^{g}$ or levying taxes $\tau$.
In real terms, the budget constraint of the fiscal authority is thus

$\displaystyle
T_{t}+\frac{1+i_{t}}{\pi_{t}}b^{g}_{t-1}=b^{g}_{t}+\tau_{t}\int^{1}_{0}z_{t}(i)di~{}~{}.$
(7)

For the analysis, I focus on the following time path of fiscal policy: The
fiscal authority starts without initial debt, $b^{g}_{-1}=0$. In $t=0$, the
government pays out a lump-sum transfer to households that is entirely
financed by debt, i.e. $T_{0}=b^{g}_{0}$ and $\tau_{0}=0$. In $t=1$, the
government pays back all the debt, which requires taxes
$\tau_{1}=\frac{1+i_{1}}{\pi_{1}}b^{g}_{0}$. Afterwards, the fiscal authority
remains inactive, $b^{g}_{t}=0,~{}T_{t}=0$ as well as $\tau_{t+1}=0~{}\forall
t\geq 1$.444In the model, Ricardian equivalence will effectively hold once the
idiosyncratic risk has been resolved by $t=1$. Thus, the assumption of
inactive fiscal policy from period 1 onwards can be relaxed as long as the
uniform lump-sum transfers are not high enough to completely insure the
initial income risk.
Note that the initial transfer does not involve any ex-ante redistribution, as
households are homogeneous in period $0$. Additionally, it is obvious that in
this setting all government debt is backed by future surpluses, since the
fiscal authority will raise any amount of taxes necessary to pay back the debt
in period $1$.

###  2.3 Equilibrium Analysis

I begin with characterizing the equilibrium for the periods $t\geq 1$, during
which there is no more idiosyncratic risk and the government chooses
$b^{g}_{t}=0$:

######  Proposition 1.

For $t\geq 1$, the equilibrium is characterized by the following

  * •

aggregates:

$\displaystyle\pi_{t}=\pi_{ss}=1,~{}~{}w_{t}=w_{ss}=\frac{\epsilon-1}{\epsilon},$
$\displaystyle
i_{t+1}=i_{ss}=\frac{1-\beta}{\beta},~{}~{}Y_{t}=N_{t}=N_{ss}=\frac{1}{1+\gamma}$
(8)

  * •

Household policies:

$\displaystyle c_{t}^{i}$
$\displaystyle=\frac{1}{1+\gamma}\left(w_{ss}z^{i}+\frac{i_{ss}}{1+i_{ss}}\left((1+i_{1})b_{0}-z^{i}\tau_{1}\right)\right):=c^{i}_{ss}~{}~{}\forall
i\in\\{l,h\\}$ (9) $\displaystyle N_{t}^{i}$
$\displaystyle=\frac{1}{1+\gamma}\frac{1}{w_{ss}z^{i}}\left(w_{ss}z^{i}-\gamma\frac{i_{ss}}{1+i_{ss}}\left((1+i_{1})b_{0}-z^{i}\tau_{1}\right)\right):=N^{i}_{ss}~{}~{}\forall
i\in\\{l,h\\}$ (10)

###### Proof.

See Appendix A.1. ∎

Intuitively, since there is no aggregate uncertainty and no model variables
have any persistence, the model economy enters a steady state after the time
$0$ idiosyncratic risk is resolved.
Using the above results, we can now characterize the equilibrium in period
$0$:

######  Proposition 2.

In period $0$, we have

$\displaystyle
Y_{0}=N_{0}=\frac{1}{1+\gamma}~{},~{}c_{0}=\frac{w_{0}}{1+\gamma}~{}~{}\text{as
well as }~{}~{}w_{0}=\frac{\phi(\pi_{0}-1)\pi_{0}+\epsilon-1}{\epsilon}$ (11)

while the rate of inflation is implicitly characterized by

$\displaystyle\frac{\epsilon}{\epsilon-1+\phi(\pi_{0}-1)\pi_{0}}$
$\displaystyle=\beta\rho^{h}\frac{1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1)}{w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1)\right)b^{g}_{0}(1-z_{h})}~{}~{}$
$\displaystyle+\beta(1-\rho^{h})\frac{1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1)}{w_{ss}z^{l}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1)\right)b^{g}_{0}(1-z^{l})}~{}~{}.$
(12)

###### Proof.

See Appendix A.2. ∎

Labor supply is constant in period $0$, regardless of the realized wage rate:
This is a consequence of the chosen preferences, which imply that income- and
substitution effects of a wage change offset each other. So, as in Aguiar et
al. (2023), inflation and wage changes do not affect the level of output, but
only redistribute between households and the owners of intermediate goods
firms.
From (12), we immediately obtain the following result regarding the “natural”
rate of interest $r_{0}^{n}$ under which $\pi_{0}=1$:

######  Lemma 1.

In period $0$, the natural rate of interest $r_{0}^{n}$ is implicitly
characterized by

$\displaystyle\frac{\epsilon}{\epsilon-1}$
$\displaystyle=\beta\rho^{h}\frac{1+r_{0}^{n}}{w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{n}_{0}\right)b^{g}_{0}(1-z^{h})}~{}~{}$
$\displaystyle+\beta(1-\rho^{h})\frac{1+r^{n}_{0}}{w_{ss}z^{l}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{n}_{0}\right)b^{g}_{0}(1-z^{l})}~{}~{}.$
(13)

Equation (13) indicates that the natural rate of interest will in general
depend on the level of government debt $b^{g}_{0}$. While the functional forms
do unfortunately not provide for a closed-form solution, the implicit function
theorem allows us nevertheless to arrive at the following result:

######  Proposition 3.

Assume
$b_{g}^{0}\in\left[0,\frac{\epsilon-1}{\epsilon}\frac{\beta}{1-\beta}\right)$.
In that case,

$\displaystyle\frac{\partial r_{0}^{n}}{\partial b^{g}_{0}}>0~{}~{},$

i.e. the natural rate of interest is increasing in the level of government
debt issued.

###### Proof.

See Appendix A.3. ∎

The above result tells us that in the initial period featuring idiosyncratic
income risk, the natural rate of interest indeed increases in the level of
government debt, at least under some mild restrictions on the amount of the
latter.555If the amount of government debt issued becomes too high, the
proportional tax rule would eventually eliminate the difference between $h$
and $l$ worker consumption from period $1$ onwards. However, under typical
calibrations of New Keynesian models, that level would be very high.
Typically, $\epsilon$ would be at least $6$ and $\beta$ greater than $0.95$.
The flipside of this result is, of course, that if the government issues more
debt without the central bank adjusting the intercept of its Taylor rule,
inflation ensues. Formally:

######  Proposition 4.

Assume that $r^{*}_{0}$ is fixed at the natural rate $r^{n}_{0}(\bar{b})$, as
implicitly defined by (13), for some given level
$\bar{b}\in\left[0,\frac{\epsilon-1}{\epsilon}\frac{\beta}{1-\beta}\right)$ of
government debt to be issued in period 0. Then,

$\displaystyle\frac{\partial\pi_{t}}{\partial
b^{g}_{0}}\bigg{|}_{b^{g}_{0}=\bar{b}}>0~{}~{},$

i.e. inflation increases in the amount of government debt.

###### Proof.

See Appendix A.4. ∎

Summing up, from the simple model above we learned the following: If
households face idiosyncratic income risk, increases in government debt raise
the “natural” or “neutral” rate of interest, and that correspondingly, the
central bank would need to adjust its interest rule to the amount of
government debt if it wants to avoid inflation. As is clear from the model
structure, these results required neither the government consuming the
resources nor them being used for any _ex-ante_ redistribution. Of course, it
requires some ex-post redistribution, so that the debt issued can actually
serve insurance purposes: If the fiscal authority would repay its debt by
raising uniform lump-sum taxes instead, any type $i\in\\{h,l\\}$ household
would be taxed exactly equal to their savings in $t=1$ and the total amount of
debt be irrelevant for inflation.

###  2.4 Some simple intuition

At this point, it may be helpful to provide further intuition for why
increasing government debt influences inflation in the presence of a Taylor
rule. Assume that indeed, as in the model above, the “natural” gross interest
rate $R$ prevailing in an economy depends on the amount of government debt
$B^{g}$ in circulation. Assume furthermore that that economy’s central bank
aims to stabilize inflation around a target $\pi^{*}$ by setting the nominal
interest rate $i_{t}$ according to a Taylor rule of the form

$\displaystyle 1+i_{t}=\pi^{*}R^{*}+\theta_{\pi}(\pi_{t}-\pi^{*})~{}~{},$ (14)

which is a version of (6) allowing for a positive net inflation target.
$R^{*}$ denotes the natural (gross) rate consistent with some initial level of
government bonds $B^{g}_{0}$. Now, the amount of government debt in
circulation rises temporarily to $B_{1}^{g}>B_{0}^{g}$. Notice that we can re-
write (14) as

$\displaystyle
1+i_{t}=\tilde{\pi}_{1}R(B_{1}^{g})+\theta_{\pi}(\pi_{t}-\tilde{\pi}_{1})~{}~{}\text{with}~{}~{}\tilde{\pi}_{1}:=\pi^{*}\frac{\theta_{\pi}-R^{*}}{\theta_{\pi}-R(B_{1}^{g})}~{}~{}.$
(15)

If $R(B_{1}^{g})>R^{*}$, then $\tilde{\pi}_{1}>\pi^{*}$, so that if the
central bank sticks to rule (14), it will de facto operate as if targeting
higher inflation

###  2.5 Public Debt and interest rates: Empirical Considerations

(a) Levels

(b) Deviations from quadratic trends

Figure 1: Relationship between public debt and interest rates

Note: Data are from the MacroHistory Database (Jorda et al., 2017; Jorda et
al., 2019) for the years 1950-2020.

As demonstrated by the previous analysis, government debt can be inflationary
under active monetary policy if it induces demand pressure and raises the
neutral interest rate. While it will become clear below that the actual
realization of inflation also depends on central bank policy, a natural pre-
condition for there to be any effects is that increasing public debt actually
exerts upward pressure on liquid asset returns. While predicted by a range of
economic theories, one might object against the existence of such effects due
to empirical observations as in Figure 1(a): Using data from the MacroHistory
database (Jorda et al., 2017; Jorda et al., 2019) for the period 1950-2019, it
plots the real treasury bill rates of a range of developed countries against
their debt-to-GDP ratios. There is no obvious correlation between the both, a
simple linear regression of the former on the latter yields a slope
coefficient of $\approx 0$. But of course, this overlooks that in most
developed countries, rising public debt levels in more recent years were also
accompanied by other secular trends that exert negative pressure on interest
rates, such as demographic change or productivity growth slowdowns. In turn,
1(b) visualizes the relationship between both variables after purging them of
country-specific quadratic time trends, a common way to account for long-run
trends in empirical macroeconomics (cf. Ramey, 2016). We now observe a
positive correlation between both variables, with a 1% deviation of a
country’s debt-to-GDP ratio from trend now being associated with a 6.9 basis
points (bp) higher bill rate (relative to trend). As is perhaps needless to
say, the results of such a simple analysis should not be interpreted as a
causal relationship and could have a range of potential explanations. However,
they align well with an empirical literature attempting to measure the effects
of public debt on treasury rates empirically, exemplified e.g. by Engen and
Hubbard (2005) or Laubach (2009). According to a summary in Rachel and Summers
(2019), such estimates range between 3 and 6 bp. Thus, reasonable values
should be in a similar range, with the analysis above suggesting that slightly
higher values might be justifiable.
Another body of evidence relevant for the 2-asset model is work pointing to
increases in public debt raising spreads between government bonds and other
assets that are less liquid (and hence less well-suited for self-insurance
purposes): In particular, Krishnamurthy and Vissing-Jorgensen (2012) find that
increases in the debt-to-GDP yield lower spreads between treasury returns and
corporate bonds. Bayer et al. (2023a) also find rising spreads between
treasuries and various less liquid assets in response to identified fiscal
policy shocks that cause public debt expansions.

##  3 The Quantitative HANK model

While the analytically tractable model in the previous section allowed for a
sharp characterization of the inflationary effects of government policy, its
simplicity only allows for a qualitative assessment of the mechanism at hand.
But are they likely to be quantitatively relevant? While recent empirical work
has shown that fiscal expansions are indeed linked to subsequent inflation
(e.g., de Soyres et al., 2022; Jordà and Nechio, 2023), such reduced-form
evidence does not allow discriminating between different channels: Indeed,
instead of resulting from the usefulness of government debt for insurance
purposes, they could equally be due to the FTPL or a combination of the two.
In turn, I employ a quantitative model featuring only the former mechanism: In
particular, I use a framework similar to Consolo and Hänsel (2023), which
extends a 2-asset HANK model in the veins of Bayer et al. (forthcoming) with
Search-and-Matching (SaM) frictions on the labor market. This has several
attractive features: Firstly, the resulting time-varying unemployment rate
creates an additional realistic source of public debt expansions in response
to adverse business cycle shocks, as the government finances the temporarily
higher UI expenditures by issuing debt. Secondly, the previous literature has
argued time-variation in unemployment risk to be a potentially important
driver of real interest rates and inflation by the literature, e.g. Ravn and
Sterk (2017).666Papers employing quantitative HANK models with SaM frictions
include Gornemann et al. (2021), Graves (2021) or Kekre (2022). This
mechanism, not present in the analytical model, could potentially counteract
the inflationary effects of government debt and should thus be accounted for.
A two-asset structure is important to allow the model to both include capital
investment and generate empirically plausible MPCs. Additionally, it will turn
out that limited suitability of capital for providing liquid assets and the
resulting time-varying liquidity premia will be important for the quantitative
magnitude of the results. To allow for flexibility in this regard, I introduce
a financial intermediary referred to as the liquid asset fund below.
While it is always assumed that the fiscal authority is committed to
eventually consolidating its debt at the initial steady state level, I will
consider different policy rules for doing so in Section 5. Additionally, the
central bank is always following an “active” Taylor rule, with different
versions being considered in Section 7.

###  3.1 Period timing

In the model, time is discrete and runs forever, starting from $t=0$. Every
period, there is the following order of events, on which additional details
will be provided below:

  1. 1.

Aggregate shocks are revealed; job separations take place; Government policies
are announced

  2. 2.

The labor market opens: Labor agencies post vacancies; Unemployed search for
jobs; matches are formed

  3. 3.

The labor market closes; there is an even number of $M$ subperiods during
which production takes place and workers and labor agencies can negotiate over
wages

  4. 4.

Goods and asset markets open: Asset returns are paid out; consumption and
investment decisions are made

  5. 5.

Goods and asset markets close; shocks to idiosyncratic states $s_{it}$ and
$\Xi_{it}$ are revealed

###  3.2 Households

####  3.2.1 Idiosyncratic states

There is a unit mass of households, which I again also refer to as “agents”
interchangeably. These differ ex-post by several idiosyncratic states:

  * •

First of all, households vary in terms of their holdings of liquid and
illiquid assets $a_{it}$ and $k_{it}$. $k_{it}$ represents holdings of capital
and we require that $k_{it}\geq 0$ as well as $a_{it}\geq\underline{a}$, with
$\underline{a}$ representing an exogenous borrowing limit. Capital is illiquid
in that a household can change her stock $k_{it}$ only infrequently: In
particular, following Bayer et al. (forthcoming) and Auclert et al. (2023), I
assume that the opportunity to do so arises randomly in an i.i.d. fashion, in
that households only gets to participate in the market for illiquid assets
with probability $\lambda\in(0,1)$ every period.

  * •

Secondly, the agents can be workers ($\Xi_{it}=0$) or “entrepreneurs”
($\Xi_{it}=1$). The former participate in the frictional labor market, while
the latter don’t supply labor to the market but receive the profits generated
by the firms (to be described below), which, for simplicity, are assumed to be
shared equally among all households with $\Xi_{it}=1$. Transitions to and out
of the “entrepreneur” state are exogenous with probabilities $\zeta$ and
$\iota$.

  * •

Worker households ($\Xi_{it}=0$) additionally differ by their idiosyncratic
labor productivity or “skill”
$s_{it}\in\mathcal{S}=\\{s_{1},s_{2},...,s_{ns}\\}$, which evolves
stochastically according to a discrete Markov chain. I allow for transition
probabilities $\pi^{s}(s_{it+1}|s_{it},e_{it})$ to depend on employment status
$e_{it}$ (see next bullet point) in order to parsimoniously capture skill
accumulation (depreciation) while employed (unemployed). Workers who are
selected to become entrepreneurs lose their idiosyncratic $s_{it}$ state as
well as their job, while exiting entrepreneurs draw a new $s_{it}$ according
to exogenous probabilities $p_{s_{1}},p_{s_{2}},...$ and enter unemployment.

  * •

Finally, workers will either be employed $(e_{it}=1)$ or unemployed
($e_{it}=0$). I assume there to be no disutility from either work or job
search, so that all workers will be working or searching full time. Job
finding rates $p^{UE}_{t}$ and $p^{EU}_{t}(s_{it})$ will be _endogenously_
determined on the frictional labor market described in Section 3.4.2. Note
that the latter may depend on individual labor productivity.
Workers receive a wage $w_{t}(s_{it})$, while unemployed agents receive an
unemployment insurance (UI) benefit $b_{t}(s_{it})$. As outlined above, wages
$w_{t}$ will be the outcome of an AOB bargaining protocol to be described in
Appendix B.1, while $b_{t}(s_{it})$ is set by the government: its level is
assumed to depend on $s_{it}$ to introduce dependence on previous income
without adding additional state variables to the household problem.

Below, I will by denote by $m_{t}(\cdot)$ the mass of households that, at the
beginning of a period, are currently in the specified state, e.g.
$m_{t}(k,s,e)$ is the respective measure of households with capital holding
$k$, skill $s$ and employment status $e$. Additionally, I will use the
superscripts $e,u$ and $\Xi$ to refer to variables specific to the employed,
unemployed or entrepreneurs when it does not cause confusion. For example, I
will use $m^{e}_{t}$, $m^{u}_{t}$ and $m_{t}^{\Xi}$ to denote the masses of
agents that feature states $e_{it}=1$, $e_{it}=0$ or $\Xi_{it}=1$ at the
beginning of stage 4 of any period $t$ (compare Section 3.1 above).

####  3.2.2 The Household problem

Households value a consumption stream $\\{c_{t}\\}^{\infty}_{t=0}$ according
to standard time-separable CRRA preferences

$\displaystyle\mathbb{E}_{0}\sum^{\infty}_{t=0}\beta^{t}\frac{c_{it}^{1-\xi}-1}{1-\xi}~{}~{}.$
(16)

An agent who gets to adjust her illiquid capital stock will the face budget
constraint

$\displaystyle
c_{it}+q_{t}k_{it+1}+a_{it+1}=(1-\tau_{t}^{y})y_{it}(e_{it},s_{it},\Xi_{it})+(1+r_{t}^{a}(a_{it}))a_{it}+(q_{t}+r^{k}_{t})k_{it}$
(17)

while for non-adjusters, the constraint will be of the form

$\displaystyle
c_{it}+a_{it+1}=(1-\tau_{t}^{y})y_{it}(e_{it},s_{it},\Xi_{it})+(1+r_{t}^{a}(a_{it}))a_{it}+r^{k}_{t}k_{it}~{}~{}.$
(18)

Both budget constraint are already written in real terms. Furthermore,
$\tau^{y}$ represents a proportional income tax, $q_{t}$ the time $t$ price of
capital goods, $r_{t}^{k}$ the real net return of capital goods and
$r_{t}^{a}(a_{it})$ the real return on bonds $a_{it}$. The latter depends on
$a_{it}$ due to the presence of a borrowing penalty. In particular, we have

$\displaystyle
r_{t}^{a}(a_{it})=\begin{cases}r_{t}^{l}~{}~{}~{}~{}\text{if}~{}~{}a_{it}\geq
0\\\ r_{t}^{l}+\bar{R}~{}~{}~{}~{}\text{if}~{}~{}a_{it}<0\end{cases}$ (19)

where $r_{t}^{l}$ is the real return on liquid savings, which will be
effectively determined by the nominal central bank rate $R_{t}^{b}$ and
inflation $\pi_{t}=\frac{P_{t}}{P_{t-1}}$ as described below. $\bar{R}$ is a
real borrowing penalty.777My specification for the borrowing wedge implies
that every unit of debt held by a household incurs a real resource cost of
$\bar{R}$, e.g. due to costly monitoring. Finally, $y_{it}$ represents a
household’s labor-, transfer- or profit income which will be given by

$\displaystyle
y_{it}(e_{it},s_{it},\Xi_{it})=\begin{cases}w_{t}(s_{it})~{}~{}~{}~{}\text{if}~{}~{}e_{it}=1,~{}\Xi_{it}=0\\\
b_{t}(s_{it})~{}~{}~{}~{}\text{if}~{}~{}e_{it}=0,~{}\Xi_{it}=0\\\
\frac{\Pi_{t}}{m_{t}^{\Xi}}~{}~{}~{}~{}\text{if}~{}~{}\Xi_{it}=1\\\
\end{cases}~{}~{}.$ (20)

This formulation implies that unemployment benefits are subject to taxation,
as they indeed are in the US. Note that since the model does not feature an
intensive labor supply margin, the introduction of tax progressivity would not
have any related effects but be purely redistributive.
Letting $\Gamma_{t}$ denote a set containing the economy’s _aggregate_ state
at period $t$, we are now ready to state the Bellman equation corresponding to
the households’ dynamic utility maximization problem, which are

$\displaystyle V^{a}(a_{it},k_{it},e_{it},s_{it},\Xi_{it};\Gamma_{t})$
$\displaystyle=\max_{c_{it},k_{it+1},a_{it+1}}\left\\{\frac{c_{it}^{1-\xi}-1}{1-\xi}+\beta\mathbb{E}_{t}V(a_{it+1},k_{it+1},e_{it+1},s_{it+1},\Xi_{it+1};\Gamma_{t+1})\right\\}$
s.t. to (17), (20), $k_{it}\geq 0$ and $a_{it}\geq\underline{a}$ (21)

for an household able to adjust its capital stock and

$\displaystyle V^{na}(a_{it},k_{it},e_{it},s_{it},\Xi_{it};\Gamma_{t})$
$\displaystyle=\max_{c_{it},a_{it+1}}\left\\{\frac{c_{it}^{1-\xi}-1}{1-\xi}+\beta\mathbb{E}_{t}V(a_{it+1},k_{it},e_{it+1},s_{it+1},\Xi_{it+1};\Gamma_{t+1})\right\\}$
s.t. to (18), (20), $k_{it}\geq 0$ and $a_{it}\geq\underline{a}$ (22)

for an household that unable to do so. The ex-ante value function $V(\cdot)$
is given by

$\displaystyle V(a_{it+1},k_{it},e_{it+1},s_{it+1},\Xi_{it+1};\Gamma_{t+1})$
$\displaystyle=\lambda V^{a}(a_{it},k_{it},e_{it},s_{it},\Xi_{it};\Gamma_{t})$
$\displaystyle+(1-\lambda)V^{na}(a_{it},k_{it},e_{it},s_{it},\Xi_{it};\Gamma_{t})~{}~{}.$

###  3.3 Production

The model’s supply side is similar to standard “medium scale” DSGE models,
except the way the labor market is modelled: Production is vertically
integrated. There is again a final good, that can either be consumed or used
by capital goods producers to produce investment goods subject to adjustment
costs. This final good is assembled by a representative final goods producer,
that in turn requires differentiated inputs provided by a continuum of
retailers. The latter set prices in a monopolistic competitive fashion subject
to nominal rigidities and require intermediate goods to produce their output.
These are provided by a set of competitive intermediate goods producers that
require capital and labor services as inputs. However, the production of the
labor input requires hiring on a frictional labor market à la Diamond-
Mortensen-Pissarides, which is handled by labor agencies.
As Bayer et al. (forthcoming), I make the simplifying assumption that
entrepreneurs don’t make the dynamic decisions of the various firms directly
but instead outsource them to a group of risk-neutral managers with aggregate
measure $0$, which do not have access to asset markets and discount the future
at the same rate $\beta$ as the households.888Since I will linearize the model
with respect to aggregate shocks, only the steady-state value of the discount
factor in the firms’ dynamic problems will matter for the dynamic model
responses. Bayer et al. (2019) and Lee (2021) report that using different
specifications does not significantly affect results in their 2-asset HANK
models with many similar features.

####  3.3.1 Final goods production

The problem of the final goods producer is equivalent to the one described in
Section 2.1.2 and thus omitted. For notational convenience below, I define
$\mu:=\frac{\epsilon}{\epsilon-1}$.

####  3.3.2 Retailers

There is a unit mass of retailers, each of which produce a given variety of
the differentiated input as monopolist, taking into account demand schedule
(4). Their only input are intermediate goods, which they purchase at real
price $mc_{t}$ (also referred to as “marginal costs”) from the competitive
intermediate goods producers. However, they are subject to nominal rigidities
à la Calvo (1983) with price indexation, i.e. they can only re-set their price
if chosen with an exogenous probability $\lambda_{Y}$.
If not receiving the re-set opportunity, a retailer’s price is automatically
adjusted by the steady inflation rate $\pi_{SS}$.999This allows to normalize
$\pi_{SS}=1$. If receiving it, the retailer will choose a price to maximize
the corresponding expected net present value of real profits

$\displaystyle\mathbb{E}_{0}\sum^{\infty}_{t=0}\beta^{t}(1-\lambda_{Y})^{t}\left(\frac{p^{*}_{jt}\pi_{SS}^{t}}{P_{t}}-mc_{t}\right)\left(\frac{p^{*}_{jt}\pi_{SS}^{t}}{P_{t}}\right)^{\frac{-\mu}{\mu-1}}Y_{t}~{}~{}.$

Log-linearizing the first order conditions of the resulting price setting
problem gives rise to the standard log-linear Phillips curve

$\displaystyle\log\left(\frac{\pi_{t}}{\pi_{SS}}\right)=\kappa_{Y}\left(mc_{t}-\frac{1}{\mu}\right)+\beta\mathbb{E}_{t}\log\left(\frac{\pi_{t+1}}{\pi_{SS}}\right)$
(23)

with $\kappa_{Y}:=\frac{(1-\lambda_{Y})(1-\lambda_{Y}\beta)}{\lambda_{Y}}$.

####  3.3.3 Intermediate goods producers

The homogeneous intermediate good is produced by a continuum of firms that use
a constant-returns-to-scale technology represented by production function

$\displaystyle
F_{t}(u_{t}K_{t},H_{t})=Z_{t}F(u_{t}K_{t},H_{t})=Z_{t}(u_{t}K_{t})^{\alpha}H_{t}^{1-\alpha}~{}~{}.$
(24)

$K_{t}$ and $H_{t}$ denote the input of capital and labor services. $u_{t}$ is
the degree of capital utilization that determines capital depreciation
according to

$\displaystyle\delta(u_{t})=\delta_{0}+\delta_{1}(u_{t}-1)+\frac{\delta_{2}}{2}(u_{t}-1)^{2}~{}~{}$

and $Z_{t}$ is a shock to Total Factor Productivity (TFP). Taking the price
$h_{t}$ for labor services as well as the capital rental rate $r_{t}$ and its
output price $mc_{t}$ as given, an intermediate goods producer solves the
static profit maximization problem

$\displaystyle\max_{K_{t},H_{t},u_{t}}mc_{t}F_{t}(u_{t}K_{t},H_{t})-h_{t}H_{t}-(r_{t}+q_{t}\delta(u_{t}))K_{t}~{}~{},$

the solution of which can be characterized using the following first order
conditions:

$\displaystyle h_{t}$
$\displaystyle=(1-\alpha)mc_{t}Z_{t}(u_{t}K_{t})^{\alpha}H_{t}^{-\alpha}$ (25)
$\displaystyle r_{t}+q_{t}\delta(u_{t})$ $\displaystyle=\alpha
mc_{t}Z_{t}u_{t}(u_{t}K_{t})^{\alpha-1}H_{t}^{1-\alpha}$ (26) $\displaystyle
q_{t}(\delta_{1}+\delta_{2}(u_{t}-1))$ $\displaystyle=\alpha
mc_{t}Z_{t}(u_{t}K_{t})^{\alpha-1}H_{t}^{1-\alpha}~{}~{}.$ (27)

####  3.3.4 Capital goods producer

Capital goods producers use the final good as input and operate a technology
subject to adjustment costs: Using $I_{t}$ units of the final good, they can
produce

$\displaystyle\left[1-\frac{\phi}{2}\left(\frac{I_{t}}{I_{t-1}}-1\right)^{2}\right]I_{t}$

units of capital.
Taking the price of capital $q_{t}$ as given, the producers choose $I_{t}$ to
maximize the net present value of real profits

$\displaystyle\mathbb{E}_{0}\sum^{\infty}_{t=0}\beta_{t}\left(q_{t}\left[1-\frac{\phi}{2}\left(\frac{I_{t}}{I_{t-1}}-1\right)^{2}\right]I_{t}-I_{t}\right)~{}~{}$

and their optimal interior solution will fulfill first-order condition

$\displaystyle
q_{t}\left(\frac{\phi}{2}\left(\frac{I_{t}}{I_{t-1}}-1\right)^{2}+\phi\left(\frac{I_{t}}{I_{t-1}}-1\right)\frac{I_{t}}{I_{t-1}}\right)=\beta
q_{t+1}\phi\left(\frac{I_{t+1}}{I_{t}}-1\right)\left(\frac{I_{t+1}}{I_{t}}\right)^{2}.$
(28)

###  3.4 Labor market

####  3.4.1 Labor agencies

Labor services are produced by a continuum of homogeneous labor agencies, each
of which is matched with at most one worker of productivity $s$. Such a match
produces $s_{t}$ units of the labor service output, the price of which are
taken as given by an agency.101010Due to CRS and the market for labor services
being competitive, one could equivalently assume that intermediate goods firms
produce labor services “in-house” and handle hiring themselves. Job
separations are exogenous and take place either if (1) the match is subject to
an separation shock arriving with probability $\delta(s)$ or if (2) the worker
becomes an entrepreneur with probability $\zeta$. I allow for job separation
rates $\delta(s)$ to depend on skill, consistent with evidence that low-income
workers face higher job separation risk (see e.g. Birinci and See, 2021).
Given all the above, the recursive characterization of the value of a matched
agency is

$\displaystyle
J(s_{t};\Gamma_{t})=h_{t}s_{t}-w_{t}(s_{t})+(1-\zeta)(1-\delta(s_{t}))\beta\mathbb{E}_{t}J(s_{t+1};\Gamma_{t+1})~{}~{}.$
(29)

####  3.4.2 Job matching and vacancy creation

There is a single labor market, on which unmatched labor agencies can meet
unemployed workers by posting vacancies. The number of meetings is governed by
a Cobb-Douglas matching technology

$\displaystyle M_{t}(V_{t},U_{t})=A_{m}U_{t}^{\chi}V_{t}^{1-\chi}~{}~{}.$ (30)

$V_{t}$ represents the total number of vacancies posted and

$\displaystyle
U_{t}=m_{t}(e=0)+\sum_{s_{i}\in\mathcal{S}}\delta(s_{i})m_{t}(e=1,s=s_{i})$

the total mass of workers searching for a job. From (30), it follows that the
period-$t$ job-finding probability $p^{UE}_{t}$ and vacancy-filling
probability $p^{vf}_{t}$ are

$\displaystyle
p^{UE}_{t}=\frac{M_{t}(V_{t},U_{t})}{U_{t}}=A_{m}\theta_{t}^{1-\chi}~{}~{}~{}\text{and}~{}~{}~{}p^{vf}_{t}=\frac{M_{t}(V_{t},U_{t})}{V_{t}}=A_{m}\theta_{t}^{-\chi}~{}~{},$
(31)

respectively. $\theta:=V_{t}/U_{t}$ is the labor market tightness.
Hiring is costly in that posting a vacancy incurs a real resource cost of
$\kappa$. Labor market tightness is in turn pinned down by a free entry
condition of the form

$\displaystyle\kappa=p_{t}^{vf}\left(\sum_{s\in\mathcal{S}}\frac{U_{t}(s)}{U_{t}}J(s;\Gamma_{t})\right)$
(32)

with

$\displaystyle
U_{t}(s_{i})=m_{t}(e=0,s=s_{i})+\delta(s_{i})m_{t}(e=1,s=s_{i})$

denoting the mass of job-searchers of a given skill level
$s_{i}\in\mathcal{S}$. These terms are present because labor agencies take
into account which type of workers they are most likely to meet.

####  3.4.3 Wage determination

Wages are determined according to an _intra-period_ Alternative Offer
Bargaining (AOB) protocol in the veins of Christiano et al. (2016), imposing
the restriction of no intra-period bargaining break-downs used by Ljungqvist
and Sargent (2021):111111Ljungqvist and Sargent (2021) find this restriction
to hardly affect model dynamics in the rich representative agent New Keynesian
model studied by Christiano et al. (2016). As shown by Consolo and Hänsel
(2023), such a setting avoids the dependence of wages on the asset positions
of individual workers resulting from other bargaining protocols such as Nash
bargaining.
The wages resulting from the protocol are given by

$\displaystyle
w_{t}(s)=\frac{1}{2}\left(h_{t}s+\tilde{b}_{t}(s)\right)+\frac{M-2}{2M}\gamma(s)+(1-\zeta)(1-\delta(s_{t}))\beta\mathbb{E}_{t}J(s_{t+1},\Gamma_{t})~{}~{},$
(33)

with $\tilde{b}_{t}(s)$ denoting an outside income that a worker of skill $s$
receives while in disagreement with its employer. More details on the AOB
bargaining protocol as well as the derivation of (33) are provided in Appendix
B.1.

###  3.5 Liquid Asset Provision

While I assume a centralized market for claims to (illiquid) capital,
households obtain liquid assets from a set of competitive liquid asset funds
(LAFs). In contrast to households, these funds are able to trade claims to
capital every period and also have access to a technology to short-sell them.
Their objective is to maximize expected returns by investing the funds
$A_{t+1}^{l}$ received from the households in capital and government bonds. In
particular, the LAFs solve

$\displaystyle\max_{B^{l}_{t+1}}\left\\{\mathbb{E}_{t}\left[(r_{t+1}^{k}+q_{t+1})\frac{A_{t+1}^{l}-B_{t}^{l}}{q_{t}}+\frac{R^{B}_{t+1}}{\pi_{t+1}}B^{l}_{t+1}\right]-A_{t+1}^{l}\left(\varphi+\frac{\Psi}{2}\left(1-\frac{B_{t+1}^{l}}{A_{t+1}^{l}}\right)^{2}\right)\right\\}~{}~{}~{},$
(34)

where $A_{t+1}^{l}$ denotes the total amount of assets intermediated by the
LAF and $B_{t}^{l}$ the amount of government debt it chooses to acquire. A
fund faces costs for each unit of liquid asset it invests on behalf of the
households, which involve the linear component $\varphi$ and a part
$\frac{\Psi}{2}\left(1-\frac{B_{t}^{l}}{A_{t}^{l}}\right)^{2}$ that increases
in the relative amount of assets the fund does not invest in public debt
obligations. Its portfolio choice can be determined from the corresponding
F.O.C.

$\displaystyle\mathbb{E}_{t}\left(\frac{r^{k}_{t+1}-q_{t+1}}{q_{t}}\right)-\Psi\left(1-\frac{B_{t+1}^{l}}{A_{t+1}^{l}}\right)=\mathbb{E}_{t}\left(\frac{R^{B}_{t+1}}{\pi_{t+1}}\right)$
(35)

and the ex-post return to household’s liquid savings will be given by

$\displaystyle
r_{t}^{l}=\frac{(q_{t}+r_{t}^{k})(A_{t}^{l}-B_{t}^{l})+\frac{R^{B}_{t}}{\pi_{t}}B^{l}_{t}}{A_{t}^{l}}-\varphi-\frac{\Psi}{2}\left(1-\frac{B_{t}^{l}}{A_{t}^{l}}\right)^{2}-1~{}~{}.$
(36)

A few words on the above structure are in order: The aim of the perhaps
peculiar cost structure in (70) is not to provide a particularly realistic
model of financial intermediation, but rather to introduce a parsimonious way
to flexibly move between various assumption on liquid asset supply in the
literature. For this purpose, the parameter $\Psi$ has a simple interpretation
as determining how easily capital assets can be used for liquidity provision:
For $\Psi\rightarrow\infty$, the model nests the assumption of segmented
markets for liquid and illiquid assets as in Kaplan et al. (2018) or Bayer et
al. (forthcoming); for $\Psi\rightarrow 0$ it nests a completely integrated
market as in Auclert et al. (2023). While a micro-founded model of financial
intermediation as e.g. in Dominguez Diaz (2021) could also provide for
imperfect usefulness of capital for liquidity provision, it would be less
straightforward to map into the above limit cases and potentially harder to
interpret.

###  3.6 Government

The government again consists of two branches, a monetary authority and a
fiscal authority.

####  3.6.1 Monetary Authority

The monetary authority sets the nominal interest rate on government bonds,
which follow a Taylor rule of the form

$\displaystyle\frac{R_{t+1}^{B}}{R^{B}_{SS}}=\left(\frac{R_{t}^{B}}{R^{B}_{SS}}\right)^{\rho^{B}}\left[\left(\frac{\pi_{t}}{\pi_{SS}}\right)^{\theta_{\pi}}\exp(m_{t}^{u}-m^{u}_{SS})^{-\theta_{u}}\right]^{1-\rho^{B}}~{}~{}.$
(37)

The parameter $\rho^{B}$ introduces rate smoothing and if $\theta_{u}\neq 0$,
the rule reacts to unemployment in addition to inflation.

####  3.6.2 Fiscal Authority

The fiscal authority collects taxes, pays out unemployment insurance and
engages in government consumption $G_{t}$. Its budget constraint (in real
terms) is

$\displaystyle
B^{g}_{t+1}+\tau_{t}^{y}\left(\sum_{s\in\mathcal{S}}w_{t}(s)m^{e}_{t}(s)+\Pi_{t}\right)=G_{t}+\frac{R^{B}_{t}}{\pi_{t}}B^{g}_{t}+\sum_{s\in\mathcal{S}}(1-\tau^{y}_{t})b_{t}(s)m^{u}_{t}(s)~{}~{}.$
(38)

For simplicity, I assume the value of UI benefits to be equal to a fixed
fraction of real steady wages, i.e.

$\displaystyle b_{t}(s)=\Upsilon_{b}w_{ss}(s)~{}~{}.$ (39)

Furthermore, in the baseline model I assume government consumption $G_{t}$ to
remain constant at its Steady State level $G_{ss}$ and taxes to follow the
rule

$\displaystyle\left(\frac{\tau^{y}_{t}}{\tau^{y}_{s}s}\right)=\left(\frac{\tau^{y}_{t-1}}{\tau^{y}_{ss}}\right)^{\rho_{\tau}}\left(\frac{B^{g}_{t}}{B^{g}_{ss}}\right)^{(1-\rho_{\tau})\psi_{B}},$
(40)

with the fiscal authority to issue any amount of bonds $B^{g}_{t+1}$ necessary
to fulfill its budget constraint (38). Intuitively, policy (40) means that the
government will eventually raise taxes to pay back debt in surplus of its
long-run target, but may do so only slowly. In Appendix C.1, I will consider
the alternative scenario in which the fiscal authority consolidates its budget
by adjusting spending $G$ instead of taxes $\tau_{y}$. Furthermore, in the
baseline model, all government debt is short-term, i.e. consists of 1-period
bonds. Appendix C.2 will also discuss a version of the model in which the
fiscal authority issues long-term debt obligations. This affects some model
dynamics, but does not alter the main insights.121212The reason for selecting
the short-term debt version as the baseline is connected to the linearization
procedure used to solve the model: In the non-stochastic steady state, holders
of long-term nominal assets do not price in potential changes in inflation.
Thus, actually realized inflationary shocks result in overly large valuation
losses for households (and corresponding debt devaluation windfalls for the
government). While these hardly matters in representative agent models, the
size of such transfers can exerts real effects in models not providing for
Ricardian equivalence.

###  3.7 Market clearing conditions and equilibrium

The Definition of Equilibrium is standard, but tedious, given that the
quantitative model features multiple markets and also requires keeping track
of the evolution of measures $m_{t}(\cdot)$. In turn, I relegate these details
to Appendix B.2.

###  3.8 Numerical Approach

I approximate the dynamic equilibrium of the model using a version of the
method used by Bayer and Luetticke (2020), which conducts first-order
perturbation around the economy’s non-stochastic steady state, following a
dimension reduction step.
For obtaining that steady state, I use a multidimensional Endogenous Grid
Method similar to the algorithm described in Bayer et al. (2019) to solve the
households’ dynamic programming problem. The joint income- and asset
distribution is approximated as a histogram using the “lottery”-method
proposed by Young (2010).
However, the representations of the (marginal) value functions as well as the
joint distribution on a tensor grid are too large to be practically handled by
standard perturbation algorithms. In turn, the dimensionality of the
(marginal) value function is reduced by applying a Discrete Cosine Transform
(DCT) and perturbing only the coefficients most important for the shape of the
steady state marginal value function. Additionally, the joint distribution is
split into a copula and marginals and I only perturb the marginals as well as
the largest coefficients resulting from a similar DCT of the copula.
Further details on the numerical implementation are provided in Appendix B.4.

###  3.9 RANK/TANK versions

To gain further insights on how idiosyncratic income risk and high MPCs matter
for the effects of government debt on inflation, it is useful to compare the
results of the full-blown HANK economy with those of similar New Keynesian
models featuring only a representative agent (also called RANK models) or Two
Agents (so-called TANK models). In the former, Ricardian equivalence holds and
the household’s MPC is very low. The latter breaks Ricardian equivalence by
introducing a group of “constrained” agents unable to participate in asset
markets, which also results in high average MPCs. However, it still does not
feature any idiosyncratic income risk and will thus not provide for medium- to
long run effects of government debt on real interest rates.
I construct these alternative frameworks to resemble the baseline HANK economy
as close as possible. In particular, all frameworks will feature the same
parameters except for $\beta$ and $R^{B}_{SS}$: In RANK and TANK, the agents
with asset market access can overcome the illiquid asset adjustment friction
by trading state-contingent securities, so that a no-arbitrage condition
between government bonds and capital will have to hold. In turn, I always
choose $\beta$ and $R^{b}_{SS}$ to be consistent with the same steady state
return on capital as in the HANK economy.

##  4 Calibration of the quantitative model

A model period is interpreted to be a quarter. I aim for the model to be
consistent with the most relevant features of the US economy and first set a
range of parameters exogenously, relying on the previous literature: In
addition to standard preference- and technology parameters, this includes some
parameters exclusively affecting the dynamic model response to aggregate
shocks, for which I rely on previous papers estimating a HANK model.
Afterwards, the remaining parameter values are chosen to match various steady
state distribution- and labor market moments.

###  4.1 Externally calibrated

Parameter | Description | Value | Source
---|---|---|---
$\xi$ | risk aversion | 1.0 | Standard
$\iota$ | Exit prob. entrepreneurs | 1/16 | Bayer et al. (forthcoming)
$\alpha$ | Cobb-Douglas parameter | 0.32 | Standard
$\delta_{0}$ | Steady State depreciation | 0.015 | Standard
$\mu$ | SS Goods markup | 1.1 | Standard
$\kappa_{Y}$ | Slope of Phillips curve | 0.08 | Standard
$\phi$ | investment adjustment cost | 3.5 | Bayer et al. (forthcoming)
$\delta_{2}/\delta_{1}$ | utilization parameters | 1.0 | Bayer et al. (forthcoming)
$\chi$ | matching elasticity | 0.5 | Petrongolo and Pissarides (2001)
$M$ | no. bargaining periods | 60 | Christiano et al. (2016)
$(\rho_{\tau},\psi_{B})$ | Gov’t spending rule | (0.94, 0.35) | Bayer et al. (2023b)
$\tau^{y}$ | Proportional income taxes | $0.25$ | See text
$\Upsilon_{b}$ | SS UI replacement rate | 0.4 | Shimer (2005)
$(\rho_{R},\theta_{\pi},\theta_{u})$ | Taylor rule parameters | (0.75,1.5,0.0) | Standard
$\pi_{ss}$ | Inflation Target | 1.0 | Standard
Table 1: Externally set parameters

The household’s risk aversion parameter is set to $1.0$, a standard value in
the macroeconomic literature also used by Kaplan et al. (2018). Regarding
technology, I use the standard values of $\alpha=0.32$ for the Cobb-Douglas
parameter for capital and set a quarterly depreciation rate for capital of
$\delta=0.015$, implying approx. 6% annual depreciation. Similar, I set
$\mu_{t}$ to a conventional value of $1.1$, resulting in a steady state markup
of 10%. The number of subperiods during which bargaining can take place is
$M=60$, the same value as in Christiano et al. (2016): this reflects the
typical number of business days within a quarter. I furthermore set
$\chi=0.5$, a standard value for the matching elasticity going back to by
Petrongolo and Pissarides (2001).
Several other parameters governing the economy are calibrated following Bayer
et al. (forthcoming): First, I also set the probability of exiting the $\Xi=1$
state within a given period to be $6.25$%. The investment adjustment cost is
chosen to be $3.5$ and the ratio $\delta_{2}/\delta_{1}$ set to be $1$,
reflecting the results of their model estimation. For a given
$\delta_{2}/\delta_{1}$-ratio, I always set $\delta_{1}$ and $\delta_{2}$ to
achieve $u_{t}=1.0$ in steady state.
The government policies are parameterized as follows: I proceed as Shimer
(2005) by choosing an unemployment replacement rate of $0.4$. As is standard,
the central bank is assumed to target zero net inflation, i.e. $\pi_{ss}=1$,
and chooses $R^{b}_{SS}$ so that this is achieved in the economy’s non-
stochastic steady state. Regarding the interest rate rule, the baseline
calibration employs the standard value of $\theta_{\pi}=1.5$ and features some
nominal rate persistence with $\rho_{R}=0.75$. Since it is not crucial for
assessing the inflationary effects of public debt, I abstract from an
unemployment reaction in main text but consider a robustness exercise below.
Other parameterizations of the rule are also considered in Section 7. Next, I
assume all proportional income taxes to equal 25%: These values are close both
to the average US tax rates on labor incomes as well as profits and are
consistent with a reasonable long-run government consumption-to-GDP ratio of
$G/Y=0.158$. Finally, the policy rule for taxes is assumed to feature
substantial persistence with $\rho_{\tau}=0.94$ and $\psi_{B}=0.35$: Since
assessing the inflationary effects of government debt requires inducing
different time paths of it, this rule will naturally be contrasted with other
scenarios below.

###  4.2 Internal calibration

Parameter | Description | Value | Target
---|---|---|---
$\beta$ | Time discounting | $0.9838$ | $K/Y=11.44$
$\zeta$ | prob. entrepreneur state | 0.0003 | Wealth share top 10
$\lambda$ | prob. illiquid asset adjustment | $0.0901$ | $A^{l}/Y=1.04$
$\bar{R}$ | Borrowing penalty | 0.0449 | 16% borrower share
$B_{ss}$ | Government debt | 1.9269 | $B_{ss}=A^{l}_{ss}$
$G_{ss}$ | Government consumption | $0.2915$ |  Budget clearing (38)
$\varphi$ | Liquidity Cost | 0.0103 | $r^{l}_{ss}=0$
$\Psi$ | Liquidity Supply | 0.005 | See text
$\underline{a}$ | Borrowing limit | -0.873 | 100 % avg. quart. income
$A_{m}$ | Matching efficiency | 0.6522 |  Unemployment rate $5.5\%$
$\kappa$ | vacancy cost | 0.0473 | 7% of avg. hire wage
$s$ | Individual labor productivity |  See Appendix B.5 | See text
$\gamma(s)$ | Costs of delay |  See Appendix B.5 | See text
$\delta(s)$ | Separation rates |  See Appendix B.5 | See text
Table 2: Internally calibrated parameters

The remaining parameters are chosen so that the model matches various target
moments in the non-stochastic steady state. To clarify how they come about, I
present for each parameter the moment I use to identify it. While, if taking
other _parameters_ as given, any parameter will somewhat affect any of the
stationary equilibrium’s target moments, it is often the case that if one
assumes other target moments to have been realized, individual parameters can
be identified just by the respective moments. For the parameters for which
this is not true, it nevertheless turns out that achieving a good fit with the
target relies mostly on the stated parameter.
Several parameters are disciplined by moments related to the steady state
wealth distribution. I choose the household discount factor $\beta$ to match a
ratio of average steady state capital holdings to output of $11.44$ as in
Bayer et al. (2019), resulting in $\beta=0.9838$. $\lambda$ determines the
(il-)liquidity of capital and thus how much liquid bonds agents wish to
additionally hold for self-insurance purposes: I use it to target net liquid
asset holdings by households to equal 1.04 times quarterly GDP as in Kaplan et
al. (2018). This requires $\lambda=0.0901$. The borrowing penalty $\bar{R}$
determines the share of households with a negative liquid asset position: to
get a share of 16%, I set value of $0.0449$. The probability $\zeta$
determines the amount of “super rich” entrepreneur households and I use it to
target a Top 10% wealth share of 70%, approximately the value computed by
Krueger et al. (2016) using SCF data. This requires a value of approx.
$0.0003$. Additionally, I set the value of the borrowing constraint to the
equal average quarterly steady state labor- and transfer income, in line with
Kaplan et al. (2018).
In order to be able to flexibly consider different values for $\Psi$, I
require all liquid assets held by households in Steady State to be provided by
the government, i.e. $B_{ss}=A^{l}_{ss}$. While a Steady State Government Debt
to (annual) GDP ratio of $26\%$ may seem low for the US, it actually
corresponds well to the corresponding ratio for the federal debt held by the
public _minus_ the debt held by foreign and international investors. In
particular, for the period 1970-2019, the ratio of Federal Debt held by the
_domestic_ public to annual GDP was 28.6%, which is arguably the relevant
ratio for a closed economy model.131313These calculations are based on series
FYGFDPUN and FDHBFIN available on FRED. Next, I set $\phi=0.0103$ which will
induce the Steady State liquid return $r_{t}^{l}$ to equal $0$ as in Bayer et
al. (forthcoming). The calibration of $\Psi$ is detailed in 4.3. $G_{ss}$ is
eventually set as the value fulfilling (38) given the other targets.
I choose matching productivity $A_{m}=0.6522$ to achieve an average
unemployment rate of $5.5\%$. Furthermore, following Christiano et al. (2016),
I target steady state hiring costs $\kappa/p^{vf}_{SS}$ to be 7% of the
average wage of newly hired workers. This results in a vacancy posting cost of
0.0473. As them, I will also assume that a worker’s outside income during
bargaining is assumed to equal her unemployment benefit. Finally, it is
necessary to set the parameters connected to the individual labor productivity
levels $s$. To calibrate the values for $s$ and $\gamma(s)$, I build on the
literature estimating income processes: In particular, the recent paper by
Braxton et al. (2021) estimates a process in which the permanent component
$z_{i,t}$ of log individual income has an AR(1) form with labor market status-
specific parameters

$\displaystyle
z_{i,t+1}=\mu_{z}(e_{it})+\rho_{z}z_{i,t}+\sigma_{z}(z_{i,t})\varepsilon_{i,t}~{}~{},$

i.e. the drift and the innovation variance depend on whether an individual
works or not. Braxton et al. (2021) argue that such a set-up captures on-the-
job skill accumulation as well as human capital depreciation during
unemployment. In turn, I use their annual estimates $\rho_{z}=0.94$,
$(\mu_{z}(1),\mu_{z}(0))=(0.0038,-0.1472)$ as well as
$(\sigma_{z}(1),\sigma_{z}(0))=(0.2261,0.4171)$, transform them into quarterly
values and discretize the process onto a grid of 13 points following the
methodology outlined in their paper.141414For the transformation, I follow an
approach similar to Krueger et al. (2016): The persistence of the quarterly
process is set to $\hat{\rho}_{z}=\rho_{z}^{1/4}$ and we replicate the cross-
sectional variance of the AR(1) processes by setting
$\hat{\sigma}_{z}(e_{it})=\frac{1-\hat{\rho}_{z}^{2}}{1-\rho_{z}^{2}}\sigma_{z}(e_{it})$.
I adapt the drift so that the quarterly processes have the same mean as the
annual one, i.e.
$\hat{\mu}_{z}(e_{it})=\frac{1-\hat{\rho_{z}}}{1-\rho_{z}}\mu_{z}(e_{it})$.
This, however, provides us only with a discretized process for household’s
labor earnings, i.e. the wages $w_{t}(s_{it})$, while the calibration requires
the primitives determining them. Conveniently, the linearity of bargaining
outcome (33) provides an easy way of backing them out: To reduce the number of
parameters, I first restrict $\gamma(s)=\bar{\gamma}h_{ss}s$, i.e. that a
labor agency’s costs of delay are proportionally to the revenue generated by
the match in steady state. Then, together with a linear rule relating the
level of worker outside income
$\tilde{b}_{t}(s)=\tilde{\Upsilon}_{b}(s)w_{ss}(s)$ to the steady state wage
level, the steady state match revenue $h_{ss}s$ necessary to induce the wage
levels $w_{t}(s_{it})$ can be found by solving a linear system. I choose
$\bar{\gamma}$ by targeting a steady state vacancy filling rate of
$p^{vf}_{SS}=0.71$ (as in den Haan et al., 2000) and the $s$ levels themselves
are subsequently obtained by using that other target moments provide the
steady state level $h_{ss}$. The resulting values for $s$ and $\gamma(s)$ are
reported in Appendix B.5.
Finally, for the separation rates $\delta(s)$, I follow Birinci and See (2021)
by using the functional form $\delta(s)=\bar{\delta}\exp(\eta_{s}(s-1))$ and
choosing $\bar{\delta}$ to target an average quarterly EU flow rate of
$\bar{p}^{EU}_{SS}=3.5\%$ and an EU flow rate ratio of
$p^{EU}(s_{ns})/p^{EU}(s_{1})=0.2$. i.e. the most productive workers are 5
times less likely to loose their jobs than the least productive
ones.151515Birinci and See (2021) target $\bar{p}^{EU}_{SS}=1.2\%$ and
$p^{EU}(s_{ns})/p^{EU}(s_{1})=1/5.54\approx 0.18$ for a monthly calibration.

###  4.3 Calibrating Liquidity Supply

The “fiscal inflation” mechanism explained in Section 2 depended on higher
public debt stimulating demand or, vice versa, exerting upward pressure on the
real (liquid asset) interest rate. Thus, for obtaining reasonable quantitative
results, it will be important to ensure that this will be of a reasonable
magnitude. A natural candidate to discipline this aspect of the model is to
ensure consistency with the empirical evidence summarized in Section 2.5, i.e.
that in the medium- to long run a 1 percentage point (p.p.) increase of the
economy’s annual debt-to-gdp ratio is associated with an increase of the
annual real treasury rate of roughly 3-7 bp, allowing for some leeway due to
the uncertainty surrounding the various estimates.

Figure 2: Model long-run effects of gov’t debt on bond returns

Can the 2-asset HANK generate a relationship of this magnitude under the limit
assumptions from the literature, i.e. either $\Psi\rightarrow\infty$ or
$\Psi\rightarrow 0$? Computing new steady states for different Debt-to-GDP
ratios under the parameterization specified in Section 4 yields Figure
2.161616For this exercise, the following assumptions on government policy are
made: The central bank adjusts its nominal rate target so that $\pi_{ss}=1$ is
also achieved in the new steady state, while the fiscal authority adapts its
consumption $G$ so that (38) clears in the new steady state. The latter
implies that the resulting rate changes constitute an upper bound compared to
scenarios with partial tax adjustments: Under higher (lower) taxes, households
would choose to hold less (more) liquid assets at a given real interest rate.
The results are rather stark: Under segmented asset markets, a 1 p.p. higher
annual Debt-to-GDP ratio causes the annual steady state real treasury rate to
increase by more 20 bp, approx. 3 times _more_ than the upper end of the
empirical estimates. The opposite is true of the integrated ($\Psi\rightarrow
0$) asset markets, in which the response of the rate is much _smaller_ and
hardly noticeable, not even a third of the empirical estimates. These findings
can be rationalized as follows: In the model, households hold liquid assets
only to the extent necessary to insure their idiosyncratic risk, as illiquid
capital yields superior returns. Additionally, constrained households will
hardly adjust their savings in response to small rate changes. Thus, if the
private sector is to hold more liquid government bonds, substantial rate
increases are necessary. In contrast, if liquid assets can alternatively be
held in illiquid form at the same return as capital, the increase in the debt-
to-GDP ratio will only crowd out a bit of the much larger capital stock,
resulting in small changes of the equilibrium interest rate. We have thus to
conclude that the assumption of either perfectly segmented or flexible asset
markets, although attractive due to their simplicity, fail to generate a
reasonable long-run relationship between government debt supply and real
treasury rates.171717Naturally, this is also true for RANK and TANK model, in
which the steady state real interest rate is independent of the amount of
government debt.
However, we can still obtain a reasonable long-run real interest rate if we
allow for a restricted usefulness of capital for liquidity provision as
determined by $\Psi$. Computing new steady states for different values, we
obtain Figure 3, which displays by how much a 1 p.p. higher annual Debt-to-GDP
ratio increases the treasury rate compared to the original steady state.

Figure 3: Increase SS treasury rate after 1 p.p. annual $B/Y$ increase

For a $\Psi$ in the range between $0.002$ and $0.007$, we obtain a difference
roughly in line with the empirical results roughly in line with the empirical
results summarized in Section 2.5. I choose $\Psi=0.005$ as the baseline which
implies a long-run response of 6 bp, close to my own estimate. Naturally, I
will consider the impact of different values below.

###  4.4 Distributional Moments

In this section, I validate the internal calibration by analyzing various
model-generated moments that were not directly targeted.
Table 3 compares various untargeted moments of the model’s Steady State
income- and wealth distributions with their empirical counterparts as reported
by Krueger et al. (2016). The latter are based on the 2006 Panel Survey of
Income Dynamics (PSID) and the 2007 Survey of Consumer Finance (SCF),
respectively.181818Disposable Income is defined as the sum of after-tax
earnings plus unemployment benefits plus income generated by assets held. In
both model and data, Net Worth relates to both liquid and illiquid assets.
Overall, the model achieves a fairly good fit, in particular for Net Worth.
Using the “entrepreneur” status to target the Top 10 wealth share results in
the model featuring somewhat higher income inequality.
Since I am employing a two-asset model, it is not only relevant to assess how
closely the framework matches data moments related to the distribution of
overall net worth, but also the different asset classes held by the
households. I do so in Table 4: First, I am considering moments of the
illiquid- and liquid wealth distribution separately. In particular, I compare
them with statistics reported by Kaplan et al. (2018), who rely on the 2004
SCF. As in the data, the model generates a more unequal distribution of liquid
assets and ownership of both asset classes is concentrated in their respective
Top 10%, with the bottom 50% holding hardly any. Also, the model moments of
the illiquid asset distribution are close to the data, mildly under-predicting
the share of the Top 10%. However, for liquid assets, I generate a comparably
more equal asset distributions, with the share held by the Top 10% not as high
and the share of the Next 40% substantially larger than in the SCF data. But,
as noted by Kaplan et al. (2018), it is “notoriously challenging” to match the
extreme right tail of wealth distributions with income risk alone. From that
perspective, I view my model’s performance as satisfactory.

| Disposable Income | Net Worth
---|---|---
| Model | Data | Model | Data
Quint. 1 | 4.6 | 4.5 | 0.0 | -0.2
Quint. 1 | 8.6 | 9.9 | 1.6 | 1.2
Quint. 3 | 13.4 | 15.3 | 4.7 | 4.6
Quint. 4 | 20.8 | 22.8 | 12.0 | 11.9
Quint. 5 | 52.7 | 47.5 | 81.4 | 82.5
Gini | 0.48 | 0.42 | 0.79 | 0.78

  * •

Note: “Data” refers to moments computed by Krueger et al. (2016) using PSID
and SCF.

Table 3: Distributional moments comparison Moments | Model | Data (incl. source)
---|---|---
Illiquid asset shares |  | (from Kaplan et al., 2018)
Top 10% | 65.5 | 70
Next 40% | 31.5 | 27
Bottom 50% | 3.0 | 3
Liquid asset shares |  | (from Kaplan et al., 2018)
Top 10% | 75.5 | 86
Next 40% | 23.5 | 18
Bottom 50% | 1.0 | -4
Hand-to-Mouth (HtM) Status |  | (from Kaplan et al., 2014)
Share HtM | 30.5 | 31.2
Share Wealthy HtM | 21.0 | 19.2
Share Poor HtM | 9.5 | 12.1
Table 4: Portfolio moments comparison

Finally, I analyze how many households are Hand-to-Mouth (HtM) in the sense of
Kaplan et al. (2014), i.e. whether their liquid asset holdings are less than 2
weeks ($\approx$ 1/6 of a model period) of current household income above of
either $0$ or the borrowing constraint. I also classify them as “Wealthy HtM”
if they additionally hold illiquid assets and “Poor HtM” if they do not. The
model matches the empirical evidence on the size of either group of agents
well. As visualized in Figure 4, these low liquid-wealth agents tend to have
particularly high MPCs. In turn, my framework is able to generate an average
quarterly MPC of 15.7% and an average annualized MPC of 39%.191919I compute
individuals’ annualized MPCs $aMPC$ as $aMPC=1-(1-qMPC)^{4}$ following Carroll
et al. (2017). Note that these _annualized_ MPCs will not exactly equal
individuals’ _annual_ MPCs. The former is of a similar magnitude as the
corresponding value in Kaplan et al. (2018).

Figure 4: Model MPC distribution

###  4.5 Calibration for RANK/TANK

As already indicated above, I use the same parameters for the RANK/TANK
versions of the model wherever possible. Thus, I only re-calibrate $\beta$ and
$R^{B}_{SS}$: To be consistent with the same steady state return on capital as
in the HANK economy, I set $R^{b}_{SS}=1+r^{k}_{ss}$ as implied by the
$K/Y$-target and $\beta=1/R^{b}_{SS}$. Additionally, for the TANK economy, it
is necessary to specify the fraction of “spender” households that are unable
to participate in asset markets. Following Debortoli and Galí (2017), I assume
it to be 20%. Note that since all “spenders” have an instantaneous MPC of 1,
the quarterly MPC of the TANK model actually exceeds the corresponding value
for the baseline HANK model.

##  5 Inflation after Supply Shocks

We can now use the calibrated 2-asset HANK model to study whether the time
path of fiscal policy and government debt affects inflation dynamics in
response to different shocks. I start doing so in the context of an
inflationary supply shock, in particular a 1% decrease of TFP $Z$, which
reverts to its steady state value according to an AR(1)-process in logs with
persistence $0.9$. Studying such a simple scenario has both pragmatic and
expositional motivations: While Federal Reserve chairman Powell (2022)
concluded that “you couldn’t get this kind of inflation without a change on
the supply side”, matching recent inflation dynamics is typically found to
require a combination of different shocks as well as allowing for non-linear
model dynamics (cf. Amiti et al., 2023), which would complicate isolating the
effects of fiscal policy and require even more involved solution methods,
respectively. Furthermore, starting with a TFP shock is instructive as it
yields relatively similar aggregate responses also for models not featuring
idiosyncratic risk and/or high MPCs.

###  5.1 Response of the Baseline Model

Figure 5: Model IRFs to 1% TFP shock

The blue solid lines in Figure 5 display Impulse Response Functions (IRFs) of
the linearized baseline HANK economy in response to the TFP shock. Overall,
they are mostly in line with the literature: Upon realization of the shock,
output, consumption as well as investment drop since the economy’s potential
is reduced. The investment reaction follows an $U$-shape due to the adjustment
costs. However, the sluggish reaction of the central bank (which determines
the nominal bond rate) does initially not crowd out demand as much as the fall
in potential, which moderates the consumption drop on impact and actually
induces a short-lived decrease in unemployment. Eventually though, the latter
increases substantially, given that the AOB bargaining protocol induces only
limited real wage responses. Inflation jumps up on impact as lower TFP
increases marginal costs and initially falls relatively quickly once the
central bank starts increasing the nominal rate. At the same time, public debt
starts growing as the shock reduced tax revenues and requires more spending on
unemployment benefits: due to the assumed slow pace of fiscal consolidation,
the ratio of public debt to (annualized) GDP peaks at approximately 1
percentage point (p.p.) and remains above trend long after the shock.
Furthermore, after a while, we observe inflation to plateau above its steady
state level and hardly going down from there onwards, which induces the
central bank to keep the nominal rate elevated as well. This is accompanied by
a decrease in the liquidity premium, i.e. the real return on (liquid) bonds
increase relative to the one on (illiquid) capital.202020The (expected)
liquidity premium is defined as
$\mathbb{E}_{t}\left[\frac{q_{t+1}+r^{k}_{t+1}}{q_{t}}\middle/\frac{R^{b}_{t+1}}{\pi_{t+1}}\right]$.
Overall, the results of the quantitative model are in line with the insights
from the analytical model: The long-lived government debt expansion pushes up
the neutral nominal rate on liquid assets, so that a central bank following a
Taylor rule as in (37) will be confronted with similarly persistent inflation.
While the quantitative magnitude appears to be of a moderate size, it
effectively impedes going the “last mile” on disinflation after the shock.
Recall also that the analyzed shock scenario induced a raise in the annual
Public Debt-to-GDP ratio of only one percentage point. Of course, for a more
substantial debt expansion we would expect the absolute effects to be larger.

###  5.2 Response under alternative fiscal policy

To further support the notion that the public debt dynamics after the shock
are indeed the reason behind the persistently elevated inflation, I
additionally consider a model counterfactual in which the fiscal authority
holds _both_ its debt and government consumption constant over time, i.e.
$B_{t}=B_{ss}$ and $G_{t}=G_{ss}~{}\forall t$, and adjusts the tax rates to
balance its budget without delay: The resulting dynamics are displayed as red-
dashed lines in Figure 5. We see very similar dynamics for several real
variables, such as output, unemployment and wages, although the model features
a steeper consumption drop caused by the immediate tax increase upon impact of
the shock. The change in the Debt-to-GDP ratio is only due to the impact of
the TFP shock on its nominator. Now, in line with the intuition from the
simple model, we see that under the alternative fiscal policy, inflation goes
up less and is less persistent: it is approximately back at its steady state
level after 10 quarters. In turn, the central bank also does not end up
raising rates as much as in the baseline model. The faster decline of the real
rate on liquid assets also stabilizes investment, which is not crowded out by
additional government debt in this scenario.

###  5.3 Comparison with RANK/TANK

While the policy counterfactual above reaffirms the important role of public
debt in shaping the inflation response to the supply shock, one may still have
doubts whether the difference is actually driven by the mechanism outlined
above. For example, the counterfactual fiscal policy also implies a
substantially different time path for investment, which affects final goods
demand and could in turn influence inflation.

Figure 6: Inflation response in different model versions

As an additional sanity check, it is therefore useful to consider the RANK and
TANK versions of the model, which do not provide for effects of public debt on
the natural rate of interest due them not featuring idiosyncratic income risk.
The IRFs of the RANK model are additionally displayed as the green dot-dashed
line in 5: Except the initial drop in consumption, they are broadly similar to
the HANK model but feature inflation dynamics much closer to the HANK model
under tax financing. Figure 13 in Appendix D additionally provides the IRFs of
the TANK model and yields a similar picture, supporting the analytical result
that the presence of income risk and not just high MPCs are the key driver of
that “fiscal inflation”.
Finally, Figure 6 shows the inflation responses of all different model
versions under the different fiscal policy scenarios, which we see to affect
inflation dynamics only in the HANK economy.

###  5.4 The role of the asset market structure

As demonstrated in Section 4.3, the (long-run) elasticity of the model’s bond
return with respect to government debt was crucially determined by how useful
productive capital assets are for providing liquid assets suitable for self-
insurance. In the model, this is determined by the cost parameter $\Psi$,
which effectively stands in for various financial market frictions. Given that
the model was intentionally calibrated so that this parameter does not affect
its steady state, we are in a position to assess how the model’s aggregate
response changes depending on the strength of said friction. To do so, I
additionally compare the response of the baseline calibration to alternatives
with $\Psi=0.0025$ and $\Psi=0.01$. As can be seen in Figure 3 in Section 4.3,
the former induces a long-run response of the liquid bond rate of around 3
b.p., which corresponds to the lower end of the estimates from the literature.
The latter is an ad hoc higher value.

Figure 7: Model responses for different $\Psi$ values

Figure 7 presents model IRFs for these different cases: As we see, the precise
value of $\Psi$ does only have a marginal impact on the real responses to the
TFP shock but induces a noticeable higher (lower) medium-run inflation for an
higher (lower) value of $\Psi$. This suggests the structure of the asset
market to be a important factor for shaping inflation dynamics in HANK models.

Overall, the analysis in this section identified a noticeable effect of public
debt on inflation dynamics in the aftermath of a negative supply shock. As the
shock induced only a limited government debt expansion by itself, its overall
quantitative magnitude was limited but very persistent. The latter is
naturally a function of the exact fiscal policy rule in place and faster
consolidation would make the “fiscal inflation” more short-lived.

##  6 Inflation after fiscal shocks

In addition to various supply side shocks, previous or concurrent government
spending expansions have been argued to be additional drivers behind the
2022/2023 inflation bout (see e.g. de Soyres et al., 2022). As such policies
can be inflationary also according to many theories not providing for effects
of government debt on the natural rate of interest, it is relevant to evaluate
the potential of that particular mechanism to influence inflation dynamics in
the aftermath of such shocks. In this section, I do so in the context of a
persistent shock to government consumption $G$, a standard scenario in the
literature. However, similar insights apply to a “stimulus check” shock (see
Appendix C.3).

###  6.1 Response to a Government spending shock

I begin by studying the response to a government spending shock that
persistently increases $G$, which is assumed to increase by 0.5% of steady
state GDP on impact and then revert back according to

$\displaystyle\log(G_{t+1})=(1-\rho_{G})\log(G_{ss})+\rho_{G}\log(G_{t})$

with $\rho_{G}=0.94$: This value implies an annual persistence of around
$0.78$, similar to the value chosen by Auclert et al. (2023).

Figure 8: Model responses to the gov’t spending shock

Figure 8 displays the corresponding model responses, contrasting them with
those of the TANK model version: Recall that the latter also deviates from
Ricardian equivalence but does not allow for the additional feedback of public
debt on the natural rate. Overall, in the HANK model the spending shock has
approximately a unit multiplier on impact and substantially reduces
unemployment. However, consumption declines after the shock, due to a) the
central bank inducing a higher real interest rates due to the ensuing
inflation and b) the eventually increasing tax rates necessary to consolidate
the government budget. This is qualitatively consistent with the results of
other HANK models (Kaplan and Violante, 2018). Overall, the government’s
(annual) debt-to-GDP ratio increases by about 1.5% five years after the
initial shock and is decreased slowly afterwards. At the same time, as for the
TFP shock, inflation remains elevated long after the shock.
The results of the TANK model are qualitatively similar but several
differences are apparent: The spending shock stimulates the economy less and
consumption drops more on impact. Additionally, inflation does not only go up
less but also lacks persistence. Of course, the different real dynamics align
well with previous findings that households’ intertemporal MPCs matter
substantially for the real effects of fiscal expansions and that these can be
quite different between HANK and TANK models (Auclert et al., 2023). In
contrast, the latter suggests that the inflationary pressure created by higher
government debt may matter substantially for price dynamics after the fiscal
shock, too.

###  6.2 Varying asset market structure

To further back up this conclusion, we can compare the aggregate model
responses for different $\Psi$-values: Recall that this parameter determines
the magnitude of the effect of public debt supply on liquid bond returns. So,
if pressure on the neutral rate due to persistently higher public debt is an
important driver of inflation after the fiscal shock, we should expect the
response of the latter change substantially for different values of $\Psi$.

Figure 9: Model responses to spending shock for different $\Psi$

This is exactly what we see on Figure 9, with a lower (higher) $\Psi$
substantially decreasing (increasing) inflation after the shock. Of course,
the result is also again testament to the importance of the asset market
structure for inflation outcomes. But even for the low $\Psi=0.0025$,
inflation remains noticeably elevated after the shock.

In conclusion, the inflationary effects of government debt present in HANK
models are at least as relevant for the aggregate response to fiscal shocks as
they are for supply shocks. Indeed, they appear even more pronounced for the
latter. While this is partly mechanical due to the fiscal shock scenario
resulting in a stronger debt expansion, the different unemployment response
may also play a factor: in HANK models, the resulting higher idiosyncratic
risk depresses interest rates and inflation (see e.g. Bayer et al., 2019), so
that the effect should be stronger when a debt increase is accompanied by
lower unemployment risk.

##  7 Implications for monetary policy

The previous two Sections established that according to HANK, increasing
public debt in the aftermath of macroeconomic shocks generates additional
inflationary pressure that lasts as long as debt is high. Naturally, central
banks tasked with ensuring price stability would want to counteract such
pressure. But how?
A natural possibility would be for the central bank to act more “hawkish”,
i.e. react stronger to deviations of inflation from its target. This is also
suggested by equation (15) that provided us intuition for the inflationary
effects of public debt in Section 2.4: If the central bank operates a Taylor
rule with a very high reaction to inflation $\theta_{\pi}$, $\tilde{\pi}_{1}$
will be affected less by changes in the “natural rate”. As “hawkish” policy
for the model experiment below, I will consider the alternative value
$\theta_{\pi}=2.0$ in interest rate rule (37).
However, we now know that the underlying issue is the increased government
debt generating pressure on the neutral rate of interest. What if the monetary
authority were to take this into account directly? Of course, the neutral rate
is an unobservable object that can at best be estimated, but central banks
obviously know the current state of public finances and could thus react to
deviations of public debt from trend directly. In the model (and perhaps
reality), this can be operationalized by adding a public debt term to interest
rate rule (37):

$\displaystyle\frac{R_{t+1}^{B}}{R^{B}_{SS}}=\left(\frac{R_{t}^{B}}{R^{B}_{SS}}\right)^{\rho^{R}}\left[\left(\frac{\pi_{t}}{\pi_{SS}}\right)^{\theta_{\pi}}\left(\frac{B_{t}}{B_{SS}}\right)^{\theta_{B}}\right]^{1-\rho^{R}}~{}~{}.$
(41)

Below, I will refer to (41) as the “debt targeting” rule and use
$\theta_{B}=0.004$, an ad hoc value that serves illustration purposes.

Figure 10: Model responses to gov’t spending shocks under different monetary
rules

To assess the potential of the different interest rate rules to overcome the
inflationary pressure from higher government debt, I contrast the resulting
IRFs to the government spending shock in Figure 10. Clearly, either of the
alternative rules is able to reduce the inflationary impact of the shock.
However, we also see that a central bank that acts more “hawkish” but does not
take into account public debt will still face inflation staying persistently
above target, albeit at a lower level compared to the Baseline policy rule.
Additionally, the “hawkish” policy comes at the cost of a stronger reduction
in private consumption at impact and smaller responses of output and
unemployment. This contrasts with the “debt targeting” rule, which yields
slightly higher inflation at the time of the shock but achieves disinflation
faster without any long-lasting “last mile” delays. Moreover, compared to the
“hawkish” reaction, it gives rise to more favorable real dynamics, which are
initially barely distinguishable from the Baseline response. Similar results
arise for the case of the TFP shock, with an equivalent to Figure 10 being
provided in Appendix D (Figure 16): The “debt targeting” rule achieves faster
disinflation at lower real cost. So, according to HANK, fast disinflation
after shocks accompanied by substantial increases in public debt requires
monetary policy makers to sufficiently take into account the resulting upward
pressure on liquid asset interest rates.

##  8 Robustness

Given the complexity of the 2-asset HANK model used for the analysis above,
one may naturally wonder whether the respective results hinge on a peculiar
modelling assumption or calibration choice. To assuage related concerns, this
section briefly discusses varies robustness checks I have conducted, with most
details relegated to the Appendix.
Given the subject of the paper, one would hope that the results do not
crucially depend on the particular choices of how fiscal policy is modeled.
For example, the main text assumed that the fiscal authority consolidates it
budget by raising the level of taxes $\tau$, even though we do not routinely
observe government raising taxes in the aftermath of recessions. However, the
discussion in Appendix C.1 reveals that similar results obtain if one assumes
the government to never change tax rates but to adjust its spending $G$
instead.
Next, all government debt in the baseline model consisted of one-period bonds,
which is clearly not the case in reality. While it is difficult to include a
realistic maturity structure of public debt into a quantitative DSGE model,
Appendix C.1 analyzes the implications of alternatively assuming a geometric
maturity structure as used in the previous literature. While this induces
somewhat larger valuation losses for bondholders at impact of an inflationary
shock, it does not substantially affect model dynamics afterwards.
When thinking about the recent inflationary episode, one may also wonder to
what extent the fiscal shock analyzed in Section 6 is a relevant scenario,
given that many countries increased transfers instead of government
consumption around that period. But as discussed in C.3, the insights from
said exercise also apply to a “stimulus check” shock in which the fiscal
authority pays out debt-financed surprise transfers to the model households.
Aside from fiscal policy itself, one may also worry about other aspects of the
calibration: While many parameter choices are standard, this is not the case
for the capital adjustment- and utilization cost parameters $\phi_{k}$ and
$\delta_{2}/\delta_{1}$, the estimates for which vary considerably in the
literature. I thus repeated the analysis for the alternative calibration
$\varphi_{k}=7.5$ and $\delta_{2}/\delta_{1}=3.0$. However, as can be seen by
inspecting Figure 21 in Appendix D, the different capital adjustment costs do
not meaningfully impact inflation dynamics compared to the baseline model.
This is also true for the fiscal policy shock results presented in Figure 22.
Another concern regarding the calibration may be that the baseline interest
rate rule does not include a response to unemployment or the output gap. Would
a “dual mandate” central bank also targeting either face a similar debt-driven
inflation persistence? Figures 23 and 24 in Appendix D also display the
aggregate responses if I set $\theta_{u}=0.15$ as in Gornemann et al. (2021).
If anything, the unemployment response further increases inflation persistence
after either shock.

##  9 Concluding Remarks

This work has theoretically studied the interaction between public debt and
inflation, highlighting an additional channel for the former to affect the
latter: If government debt obligations are useful beyond merely being a
vehicle for intertemporal substitution, temporary increases in their supply
affect equilibrium interest rates, which creates inflationary pressure if not
explicitly taken into account by central banks.
For the purposes of this paper, the mentioned usefulness is micro-founded
through households facing idiosyncratic risk and requiring liquid debt
obligations for self-insurance due to the absence of a complete market for
state-contingent assets. Other settings, in which different actors such as
firms or banks value liquid government debt for self-insurance reasons, should
have similar implications. Studying the quantitative magnitude of the channel
in a state-of-the-art 2-asset HANK model required deviating from common
assumptions on asset market structure in such models, which would have implied
them to be either overly strong or weak. Eventually, I found the mechanism to
be quantitatively relevant for both supply- and fiscal policy shocks: Although
it implies its magnitude to be moderate compared to the overall inflation
generated by these business cycle shocks, it can result in persistently
elevated inflation in their aftermath so that going the “last mile” to
disinflation takes a long time. However, according to the model, a central
bank can avoid this outcome by explicitly reacting to deviations of public
debt from its long-run trend. So, even if the sustainability of government
budgets is taken as granted, monetary authorities should pay attention to
government debt dynamics and calibrate their policy stance accordingly.
Ultimately, the conducted analyses also suggest topics for further work. In
particular, it seems important to further study the implications of asset
market arrangements for monetary and fiscal policy, which seem to be
particularly pronounced if accounting for household heterogeneity and
incomplete markets. I hope to do so in future work.

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##  Appendix A Proofs for analytical model

###  A.1 Proof of Proposition 1

Let us first consider the First order condition for labor supply of an
household of type $i\in\\{h,l\\}$:

$\displaystyle\frac{w_{t}z^{i}}{c^{i}_{t}}=\gamma\frac{1}{1-N^{i}_{t}}$
$\displaystyle\Leftrightarrow(1-N^{i}_{t})w_{t}z^{i}=\gamma\left(w_{t}z^{i}N^{i}_{t}+\frac{(1+i_{t})}{\pi_{t}}b_{it-1}-b_{it}-z^{i}\tau\right)$
(42) $\displaystyle\Leftrightarrow
N^{i}_{t}=\frac{1}{1+\gamma}\frac{1}{w_{t}z^{i}}\left(w_{t}z^{i}+\gamma(b_{it}+z^{i}\tau-\frac{(1+i_{t})}{\pi_{t}}b_{it-1})\right)$
(43)

where the $c_{t}$ is substituted using the budget constraint in the second
step. Summing up the labor supplied by both groups yields aggregate labor
supply

$\displaystyle N_{t}$
$\displaystyle=\rho^{h}z^{h}N_{ht}+(1-\rho^{h})z^{l}N_{lt}$
$\displaystyle=\frac{1}{1+\gamma}\frac{\rho^{h}}{w_{t}}\left(w_{t}z^{h}+\gamma(b_{ht}+z^{h}\tau-\frac{(1+i_{t})}{\pi_{t}}b_{ht-1})\right)$
$\displaystyle~{}+\frac{1}{1+\gamma}\frac{1-\rho^{h}}{w_{t}}\left(w_{t}z^{l}+\gamma(b_{lt}+z^{l}\tau-\frac{(1+i_{t})}{\pi_{t}}b_{lt-1})\right)$
$\displaystyle=\frac{1}{1+\gamma}$ (44)

The second step follows uses (1), bond market clearing condition

$\displaystyle b^{g}_{t}=0=\rho^{h}b_{ht}+(1-\rho^{h})b_{lt}~{}~{}\forall
t\geq 1$

and in turn also $\tau_{t}=\frac{1+i_{t}}{\pi_{t}}b^{g}_{t-1}$ by the
government budget constraint (7) and the assumed policy path.
Next, it is easy to see that either group of household’s intertemporal Euler
equation will be of the form

$\displaystyle c_{it+1}=\beta\frac{(1+i_{t+1})}{\pi_{t+1}}c_{it}~{}~{}.$ (45)

Using again the budget constraints, we further obtain

$\displaystyle\rho^{h}c_{ht}+(1-\rho^{h})c_{lt}=$
$\displaystyle\rho^{h}\left(w_{t}z_{h}N_{ht}+\frac{(1+i_{t})}{\pi_{t}}b_{ht-1}-b_{ht}-z_{h}\tau_{t}\right)$
$\displaystyle+(1-\rho^{h})\left(w_{t}z_{h}N_{ht}+\frac{(1+i_{t})}{\pi_{t}}b_{ht-1}-b_{ht}-z_{h}\tau_{t}\right)$
$\displaystyle=\frac{w_{t}}{1+\gamma}$

after again using (1), (7), the bond market clearing condition and
additionally (44). Naturally, absent profit incomes and capital investments,
total household consumption must equal household labor income.
In turn, we can sum up (45) over types to obtain

$\displaystyle\rho^{h}c_{ht+1}+(1-\rho^{h})c_{lt+1}$
$\displaystyle=\beta\frac{(1+i_{t+1})}{\pi_{t+1}}(\rho^{h}c_{ht}+(1-\rho^{h})c_{lt})$
$\displaystyle\implies w_{t+1}$
$\displaystyle=\beta\frac{(1+i_{t+1})}{\pi_{t+1}}w_{t}$ (46)

Together with (6) and (5), (46) forms a system that characterizes the
equilibrium of the model for $t\geq 1$. Since
$r_{t}^{*}=\frac{1}{\beta}-1~{}~{}\forall t\geq 1$, it is easy to verify that
$\pi_{t}=1$, $w_{t}=\frac{\epsilon-1}{\epsilon}$ and $i_{t}=r_{t}^{*}$ indeed
solve that system. Local stability of this equilibrium is ensured by the
Taylor principle.
Also, from the Euler equation it follows that in such an equilibrium, the
consumption either household type $i\in\\{h,l\\}$ will be constant over time.
Hence, we can back out their savings choice from the budget constraint for
$t=1$ and $t=2$:

$\displaystyle c^{i}_{1}$ $\displaystyle=c^{i}_{2}$
$\displaystyle\Leftrightarrow
w_{1}z^{i}N^{i}_{ss}+(1+i_{1})b_{i0}-b_{i1}-z^{i}\tau_{1}$
$\displaystyle=w_{2}z^{i}N^{i}_{ss}+i_{ss}b_{i1}$
$\displaystyle\Leftrightarrow b_{i1}$
$\displaystyle=\frac{1}{1+i_{ss}}\left((1+i_{1})b_{i0}-z^{i}\tau_{1}\right)$

The second step uses that for $t\geq 1$, we need to have $b_{it+1}=b_{i+2}$
and since $c_{it}=c_{ss}$ as well as $w_{t}=w_{ss}\forall t\geq 1$, also
$N^{i}_{t}=N^{i}_{ss}$ due to the first-order condition for labor supply.
Using that result in (43), we obtain the labor supply as stated in the
Proposition, which we can in turn use in the labor supply optimality condition
as in (42) to back out the stated consumption schedule.

###  A.2 Proof of Proposition 2

Using that $1-N^{i}_{ss}=\gamma\frac{c^{i}_{ss}}{w_{ss}z^{i}}$ according to
(9) and (10), we can derive the continuation value of a household that enters
period $1$ with $b_{1}$ bonds and productivity draw $i\in\\{h,l\\}$ to be

$\displaystyle
V^{i}_{1}(b_{0})=\frac{1}{1-\beta}\left[(1+\gamma)\log\left(\frac{1}{1+\gamma}\left(w_{ss}z_{i}+\frac{i_{ss}}{1+i_{ss}}\left((1+i_{1})b_{0}-z^{i}\tau_{1}\right)\right)\right)+\gamma\log\frac{\gamma}{w_{ss}z_{i}}\right]$

This allows us to state the problem of a household in period $0$ as

$\displaystyle\max_{b_{0},N_{0}}\left\\{\log(w_{0}N_{0}+T_{0}-b_{0})+\gamma\log(1-N_{0})+\beta\left[\rho^{h}V_{1}^{h}(b_{0})+(1-\rho^{h})V_{1}^{l}(b_{0})\right]\right\\}$

which has the following first order conditions:

$\displaystyle N_{0}~{}:~{}~{}\frac{w_{0}}{w_{0}N_{0}+T_{0}-b_{0}}$
$\displaystyle=\frac{\gamma}{1-N_{0}}$ $\displaystyle
b_{0}~{}:~{}~{}\frac{1}{w_{0}N_{0}+T_{0}-b_{0}}$
$\displaystyle=\beta\rho^{h}\frac{dV_{1}^{h}(b_{0})}{db_{0}}+\beta(1-\rho^{h})\frac{dV_{1}^{l}(b_{0})}{db_{0}}$
$\displaystyle=\frac{\beta}{1-\beta}\rho^{h}\frac{\frac{i_{ss}}{1+i_{ss}}(1+i_{1})}{\frac{1}{1+\gamma}\left(w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left((1+i_{1})b_{0}-z^{h}\tau_{1}\right)\right)}$
$\displaystyle+\frac{\beta}{1-\beta}\rho^{h}\frac{\frac{i_{ss}}{1+i_{ss}}(1+i_{1})}{\frac{1}{1+\gamma}\left(w_{ss}z^{l}+\frac{i_{ss}}{1+i_{ss}}\left((1+i_{1})b_{0}-z^{l}\tau_{1}\right)\right)}$

Since households are identical in period $0$, they will make the same choices
regarding bond-holdings. Thus, asset market clearing requires
$b_{0}=b^{g}_{0}$ and in turn $b_{0}=T_{0}$ by the fiscal authority’s budget
constraint. Using that in the first order condition for labor supply, we
obtain $N_{0}=\frac{1}{1+\gamma}$. Since $c_{0}=w_{0}N_{0}+T_{0}-b_{0}$, it
further follows that $c_{0}=\frac{w_{0}}{1+\gamma}$. Substituting these
results into the first order condition for bond holdings, it simplifies to

$\displaystyle\frac{1}{w_{0}}$
$\displaystyle=\beta\rho^{h}\frac{(1+i_{1})}{\left(w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left((1+i_{1})b_{0}-z^{h}\tau_{1}\right)\right)}$
$\displaystyle+\beta(1-\rho^{h})\frac{(1+i_{1})}{\left(w_{ss}z^{l}+\frac{i_{ss}}{1+i_{ss}}\left((1+i_{1})b_{0}-z^{l}\tau_{1}\right)\right)}$
(47)

which furthermore uses that $\frac{i_{ss}}{1+i_{ss}}=1-\beta$ due to the
assumptions made above. Since we will have $\pi_{1}=1$ according to
Proposition (1), it follows immediately from the Phillips curve (5) that

$\displaystyle
w_{0}=\frac{\phi(\pi_{0}-1)\pi_{0}+\epsilon-1}{\epsilon}~{}~{}.$ (48)

Additionally, because of the assumed government policies, the tax rate in
$t=1$ needs to fulfil

$\displaystyle\rho^{h}z^{h}\tau_{1}+(1-\rho^{h})z^{l}\tau_{1}=\frac{1+i_{1}}{\pi_{1}}b^{g}_{0}\Leftrightarrow\tau_{1}=(1+i_{1})b^{g}_{0}$

where the second step uses (1) and $\pi_{1}=1$. By substituting the above as
well (48) and the Taylor rule (6) into (47), we obtain the equation that
characterizes $\pi_{0}$ as stated in the Proposition:

$\displaystyle\frac{\epsilon}{\epsilon-1+\phi(\pi_{t}-1)\pi_{t}}$
$\displaystyle=\beta\rho^{h}\frac{1+r^{*}_{0}+\theta_{\pi}(\pi_{t}-1)}{w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{*}_{0}+\theta_{\pi}(\pi_{t}-1)\right)b^{g}_{0}(1-z_{h})}~{}~{}$
$\displaystyle+\beta(1-\rho^{h})\frac{1+r^{*}_{0}+\theta_{\pi}(\pi_{t}-1)}{w_{ss}z^{l}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{*}_{0}+\theta_{\pi}(\pi_{t}-1)\right)b^{g}_{0}(1-z^{l})}~{}~{}.$

###  A.3 Proof of Proposition 3

By the Implicit Function Theorem, since $r_{0}^{n}(g^{0}_{b})$ is implicitly
defined by $F(g^{b}_{0},r_{0}^{n}(g^{0}_{b}))=0$ with

$\displaystyle F(g^{b}_{0},r_{0}^{n})$
$\displaystyle=\beta\rho^{h}\frac{1+r_{0}^{n}}{w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{n}_{0}\right)b^{g}_{0}(1-z^{h})}~{}~{}$
$\displaystyle+\beta(1-\rho^{h})\frac{1+r^{n}_{0}}{w_{ss}z^{l}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{n}_{0}\right)b^{g}_{0}(1-z^{l})}-\frac{\epsilon}{\epsilon-1}$

we have

$\displaystyle\frac{\partial r^{n}_{0}}{\partial
b^{0}_{g}}=-\left.\frac{\partial F}{\partial b^{0}_{g}}\middle/\frac{\partial
F}{\partial r^{n}_{0}}\right.~{}~{}.$ (49)

The derivatives in (49) are given by

$\displaystyle\frac{\partial F}{\partial b^{0}_{g}}$
$\displaystyle=\beta\rho^{h}\frac{\frac{i_{ss}}{1+i_{ss}}(1+r_{0}^{n})^{2}(z^{h}-1)}{\left(w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{n}_{0}\right)b^{g}_{0}(1-z^{h})\right)^{2}}$
$\displaystyle+\beta(1-\rho^{h})\frac{\frac{i_{ss}}{1+i_{ss}}(1+r_{0}^{n})^{2}(z^{l}-1)}{\left(w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{n}_{0}\right)b^{g}_{0}(1-z^{l})\right)^{2}}$
(50)

and

$\displaystyle\frac{\partial F}{\partial r^{n}_{0}}$
$\displaystyle=\beta\rho^{h}\frac{w_{ss}z^{h}}{\left(w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{n}_{0}\right)b^{g}_{0}(1-z^{h})\right)^{2}}$
$\displaystyle+\beta(1-\rho^{h})\frac{w_{ss}z^{l}}{\left(w_{ss}z^{l}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{n}_{0}\right)b^{g}_{0}(1-z^{l})\right)^{2}}~{}~{}.$
(51)

Since $w_{ss}z^{h}>0$ and $w_{ss}z^{l}>0$, it is clear that $\frac{\partial
F}{\partial r^{n}_{0}}>0$. However, things are less obvious for (50), as its
first term is positive and its second term is negative since $z^{l}<1$.
However, notice that if

$\displaystyle\underbrace{w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{n}_{0}\right)b^{g}_{0}(1-z^{h})}_{=(1+\gamma)c^{h}_{ss}}>\underbrace{w_{ss}z^{l}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{n}_{0}\right)b^{g}_{0}(1-z^{l})}_{=(1+\gamma)c^{l}_{ss}}$
(52)

then

$\displaystyle\frac{\partial F}{\partial b^{0}_{g}}$
$\displaystyle=\beta\rho^{h}\frac{\frac{i_{ss}}{1+i_{ss}}(1+r_{0}^{n})^{2}(z^{h}-1)}{\left((1+\gamma)c^{h}_{ss}\right)^{2}}+\beta(1-\rho^{h})\frac{\frac{i_{ss}}{1+i_{ss}}(1+r_{0}^{n})^{2}(z^{l}-1)}{\left((1+\gamma)c^{l}_{ss}\right)^{2}}$
$\displaystyle<\beta\rho^{h}\frac{\frac{i_{ss}}{1+i_{ss}}(1+r_{0}^{n})^{2}(z^{h}-1)}{\left((1+\gamma)c^{h}_{ss}\right)^{2}}+\beta(1-\rho^{h})\frac{\frac{i_{ss}}{1+i_{ss}}(1+r_{0}^{n})^{2}(z^{l}-1)}{\left((1+\gamma)c^{h}_{ss}\right)^{2}}=0~{}~{}$

(the final equality follows from (1)). This means that if (52) holds, then
$\frac{\partial F}{\partial b^{0}_{g}}<0$ and thus $\frac{\partial
r^{n}_{0}}{\partial b^{0}_{g}}>0$. For that to be the case, we require

$\displaystyle
w_{ss}-\frac{i_{ss}}{1+i_{ss}}(1+r^{n}_{0}(b^{g}_{0}))b^{0}_{g}>0~{}~{}.$ (53)

Under our restriction $0\leq
b^{g}_{0}<\frac{\epsilon-1}{\epsilon}\frac{\beta}{1-\beta}=\frac{w_{ss}}{i_{ss}}$,
this will be the case. Notice first that $r^{n}_{0}(b^{g}_{0})$ as implicitly
defined by (13) fulfills

$\displaystyle
r^{n}_{0}(0)=\frac{1}{\beta\left(\frac{\rho^{h}}{z^{h}}+\frac{1-\rho^{h}}{z^{l}}\right)}-1<\frac{1}{\beta}-1=r^{n}_{0}\left(\frac{w_{ss}}{i_{ss}}\right)=i_{ss}$
(54)

as
$\frac{\rho^{h}}{z^{h}}+\frac{1-\rho^{h}}{z^{l}}>\frac{1}{\rho^{h}z^{h}+(1-\rho^{h})z^{l}}=1$
due to Jensen’s Inequality. Now, guess that (53) does hold for any
$b^{g}_{0}\in[0,w_{ss}/i_{ss})$. In that case, our previous results imply that
$r^{n}_{0}$ is increasing at any of these values. But then, as there is no
discontinuity around $w_{ss}/i_{ss}$, (54) requires that we also must have
$r^{n}_{0}(b^{g}_{0})<i_{ss}$ at these values. In turn, (53) is indeed true
and we have $c^{h}_{ss}>c^{l}_{ss}$. So, it is also clear that $\frac{\partial
r^{n}_{0}}{\partial b^{0}_{g}}<0$, establishing the proposition that
$\frac{\partial r^{n}_{0}}{\partial b^{0}_{g}}>0$.

###  A.4 Proof of Proposition 4

We proceed similar as in Appendix (A.3). Given that (12) implicitly defines
$\pi_{0}$ as a function of $b^{g}_{0}$ for given parameters,

$\displaystyle\frac{\partial\pi_{0}}{\partial b^{0}_{g}}=-\left.\frac{\partial
F^{\pi}}{\partial b^{0}_{g}}\middle/\frac{\partial
F^{\pi}}{\partial\pi_{0}}\right.~{}~{}.$ (55)

with

$\displaystyle F^{\pi}(g^{b}_{0},\pi_{0})$
$\displaystyle=\beta\rho^{h}\frac{1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1)}{w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1)\right)b^{g}_{0}(1-z_{h})}~{}~{}$
$\displaystyle+\beta(1-\rho^{h})\frac{1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1)}{w_{ss}z^{l}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1)\right)b^{g}_{0}(1-z^{l})}-\frac{\epsilon}{\epsilon-1+\phi(\pi_{0}-1)\pi_{0}}$

Similarly to (50), we have

$\displaystyle\frac{\partial F^{\pi}}{\partial b^{0}_{g}}$
$\displaystyle=\beta\rho^{h}\frac{\frac{i_{ss}}{1+i_{ss}}(1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1))^{2}(z^{h}-1)}{\left(w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1)\right)b^{g}_{0}(1-z^{h})\right)^{2}}$
$\displaystyle+\beta(1-\rho^{h})\frac{\frac{i_{ss}}{1+i_{ss}}(1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1))^{2}(z^{l}-1)}{\left(w_{ss}z^{h}+\frac{i_{ss}}{1+i_{ss}}\left(1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1)\right)b^{g}_{0}(1-z^{l})\right)^{2}}~{}~{}.$
(56)

Additionally,

$\displaystyle\frac{\partial
F^{\pi}}{\partial\pi_{0}}=\beta\rho^{h}\frac{\theta_{\pi}w_{s}sz^{h}}{(1+\gamma)c^{h}_{ss}}+\beta(1-\rho^{h})\frac{\theta_{\pi}w_{s}sz^{l}}{(1+\gamma)c^{l}_{ss}}+\frac{\epsilon\phi(2\pi_{0}-1)}{(\epsilon-1+\phi(\pi_{0}-1)\pi_{0})^{2}}~{}~{}.$

Since we are considering the derivative around $r^{*}=r^{n}_{0}(b^{0}_{g})$ at
which $\pi_{0}=1$, $2\pi_{0}-1>0$ so all the terms are positive and
$\frac{\partial F^{\pi}}{\partial\pi_{0}}>0$.
Regarding (56), we can use the same argument as in Appendix (A.3). If
$c^{h}_{ss}>c^{l}_{ss}$, then

$\displaystyle\frac{\partial F^{\pi}}{\partial b^{0}_{g}}$
$\displaystyle=\beta\rho^{h}\frac{\frac{i_{ss}}{1+i_{ss}}(1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1))^{2}(z^{h}-1)}{\left((1+\gamma
c^{h}_{ss})\right)^{2}}+\beta(1-\rho^{h})\frac{\frac{i_{ss}}{1+i_{ss}}(1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1))^{2}(z^{l}-1)}{\left((1+\gamma
c^{l}_{ss})\right)^{2}}$
$\displaystyle<\beta\rho^{h}\frac{\frac{i_{ss}}{1+i_{ss}}(1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1))^{2}(z^{h}-1)}{\left((1+\gamma
c^{h}_{ss})\right)^{2}}+\beta(1-\rho^{h})\frac{\frac{i_{ss}}{1+i_{ss}}(1+r^{*}_{0}+\theta_{\pi}(\pi_{0}-1))^{2}(z^{l}-1)}{\left((1+\gamma
c^{h}_{ss})\right)^{2}}=0$

Again, we are considering the case with $r^{*}_{0}=r^{n}_{0}(b^{0}_{g})$ for
some $b^{0}_{g}\in[0,w_{ss}/i_{ss})$, so from the analysis in Appendix A.3, we
know that indeed $c^{h}_{ss}>c^{l}_{ss}$ in this case. Thus $\frac{\partial
F^{\pi}}{\partial b^{0}_{g}}<0$ and hence $\frac{\partial\pi_{0}}{\partial
b^{0}_{g}}>0$, which establishes the proposition.

##  Appendix B Details on quantitative HANK model

###  B.1 Wage determination

During the $M$ subperiods of a period’s production stage (compare Section
3.1), the worker and the labor agency take turns extending wage offers: I will
be denoting variables on a per-subperiod basis with a $\Delta$, e.g.
$h_{\Delta,t}:=\frac{h_{t}}{M}$ is the revenue a labor agency for producing
during one of the sub-periods etc..
I assume that in any given period, the labor agency gets to make the first
offer. If the worker rejects it, she can make a counter-offer in the next
period that the firm can reject or accept, and so on. Once a wage agreement
has been reached, the match starts producing labor services and the worker is
paid the agreed wage rate for the remainder of the period. However, before
that happens, an agency matched with a skill $s$-worker incurs a cost of delay
$\gamma_{\Delta}(s)$ per sub-period, while the worker will receive an outside
income $\tilde{b}_{\Delta,t}(s)$ per sub-period. Both these values will have
to depend on the respective’s workers productivity $s$ to avoid high (low)
productivity workers being able to bargain wages that are disproportionately
low (high) comparatively to their skill. If no wage is accepted before the
last period $M$, the worker gets to make a _take-it-or-leave-it_ offer during
the last sub-period ($M$ is even). If rejected by the firm, the match
irreparably dissolves and the worker enters the pool of the unemployed.
Now, to characterize the wage outcome, we first note that independently of
worker wealth, a worker (agency) in our model would always like the wage to be