Exit strategies from quantitative easing: the role of the fiscal-monetary policy mix

Exit Strategies from Quantitative Easing:

The role of the ﬁscal-monetary policy mix

∗

Florencia S. Airaudo

†

January 15, 2023

Please download here the most recent version of this paper

Abstract

As a consequence of the policy responses to the COVID-19 crisis, central bank bal-

ance sheets, public debt, and liquidity increased in many developed economies. As the

economies recover and inﬂation far exceeds the target, central banks face a challenge

in how to manage their balance sheet. I study the macroeconomic eﬀects of reducing

the central bank balance sheet size, i.e., Quantitative Tightening (QT). I construct a

Regime-Switching New Keynesian DSGE model calibrated to the US economy. The

economy ﬂuctuates between a monetary-led regime, a ﬁscally-led regime, and the zero

lower bound on the monetary policy interest rate. The macroeconomic eﬀects of QT

crucially depend on the ﬁscal-monetary policy mix. In a monetary-led regime, QT

eﬀectively reduces inﬂation at the cost of increasing the government debt-to-GDP ra-

tio. In contrast, unwinding the central bank balance sheet in a ﬁscally-led regime has

little impact on inﬂation. The negative demand eﬀect driven by QT is not enough to

counteract the stimulative impact of negative real interest rates and ﬁscal stimulus.

Keywords: Monetary Policy; Fiscal and monetary policy mix; Quantitative Easing;

Quantitative Tightening

JEL Classiﬁcation: E31, E52, E58, E62, E63

∗

I am indebted to Hernán Seoane for his invaluable support and guidance and to Evi Pappa and Ulf

Söderström for insightful discussions. For helpful comments and suggestions, I am grateful to Carlo Galli,

Andrew Foerster, David Vestin, Roberto Billi, Anna Rogantini Picco, the Economics department of Univer-

sidad Carlos III, especially the participants of the Macroeconomics Reading group, and the Macro Work in

Progress. I thank the members of the Research Division at the Riksbank and the participants of the First

Ph.D. Workshop in Money and Finance at the Riksbank, the Doctoral Workshop on Quantitative Dynamic

Economics at the University of Konstanz, the Annual Meeting of the AAEP, Simposio de la Asociacion

Española de Economia 2022, members of the RedNIE, and participants of the ENTER Seminar at the Uni-

versity College London. I thank ﬁnancial support by PID2019-107161GB-C31/AEI/10.13039/501100011033

and Ministerio de Asuntos Económicos y Transformación Digital through grant Number ECO2016-76818-C3-

1-P, and to Álvaro Escribano Sáez for his ﬁnancial support through the Ministerio de Ciencia e Innovación

(Spain), grant RTI2018-101371-B-I00.

†

Department of Economics, Universidad Carlos III de Madrid, Calle Madrid 126, 28903 Getafe, Madrid,

Spain; E-mail address: [email protected].

“...I would just stress how uncertain the eﬀect is of shrinking the balance sheet..."

J. Powell, Federal Reserve Chairman, press conference May 2022.

1 Introduction

In the Great Recession and the COVID-19 crisis, short-term interest rates in the US ap-

proached the zero lower bound (ZLB), leaving the Federal Reserve without its conventional

monetary policy instrument. Due to this, the Federal Reserve began applying unconven-

tional monetary policies, such as Quantitative Easing (QE), to stimulate the economy by

lowering long-term interest rates.

QE consists of expanding the central bank balance sheet

by purchasing large amounts of assets and paying by issuing reserves.

At the same time,

the government embarked on a ﬁscal expansion that increased debt-to-GDP ratios to levels

not seen since World War II.

As the economy recovers and inﬂation reaches levels not seen in forty years, the Federal

Reserve faces the challenge of how to combine its two main policy tools (short-term interest

rate and balance sheet size) to stabilize the inﬂation rate. In this paper, I quantify the impact

of reducing the central bank balance sheet size on macroeconomic and ﬁnancial variables such

as inﬂation, public debt, and yield spreads. To this end, I construct a Regime-Switching

New Keynesian Dynamic Stochastic General Equilibrium (NK-DSGE) model where ﬁscal

and monetary authorities are subject to regime changes in their policy rules.

The main contribution of this paper is to show that the macroeconomic impact of QT

depends on how the public debt will be stabilized, i.e., through ﬁscal surpluses or inﬂation.

When the ﬁscal authority adjusts taxes to stabilize the debt, QT reduces inﬂation, even

under smaller increases in the short-term interest rate. However, shrinkage of the central

bank balance sheet has little impact on inﬂation if there is no appropriate ﬁscal framework

to stabilize government debt.

Other central banks in developed economies, like the Bank of England, the central bank of Sweden, and

the ECB, among others, implemented similar policies.

In practice, QE includes distinct policies. In this paper, QE consists of central bank purchases of

long-term public bonds by selling short-term assets (bank reserves). The main objective of this policy is

to stimulate output and inﬂation by lowering long-term yields. This policy was deﬁned as QE-type 2 by

Ricardo Reis in his talk “The original sin of QE."

The distinctive features of my model rely on the design of the ﬁnancial intermediaries

and the ﬁscal and monetary authorities. The model exhibits market segmentation in the

public debt market and a leverage constraint for ﬁnancial intermediaries, as in Elenev et al.

(2021). The ﬁscal authority issues long and short-term bonds, while the central bank issues

reserves. The central bank does conventional monetary policy, which consists in setting the

short-term interest rate, and unconventional monetary policy, which consists of central bank

purchases of long-term government bonds, paid by issuing reserves.

QE operates through two channels in this model. First, bank reserves and deposits

increase when the central bank expands its balance sheet. Second, the long-term yield falls

due to the central bank’s intervention in the long-term bond market. This yield decline

gives households incentives to rebalance their portfolio, selling long-term bonds in exchange

for deposits. QE transmits into the economy as a positive demand shock. The portfolio

revaluation eﬀect provides households with a wealth eﬀect, increasing consumption demand.

The fall in savings return generates a substitution eﬀect from savings to consumption. Both

channels operate by increasing aggregate demand. The extent to which QE increases output

and inﬂation depends on the ﬁscal-monetary policy mix.

Conventional monetary and ﬁscal policies consist of a Taylor rule for the short-term

interest rate (the return on short-term public bonds and reserves) and a ﬁscal rule for taxes.

These rules switch between three policy regimes, as in Bianchi and Melosi (2017) and Bianchi

and Melosi (2022): a monetary-led regime, a ﬁscal-led regime, and the ZLB regime. In the

ﬁrst regime, the central bank reacts strongly to inﬂation deviations from the target, and the

ﬁscal authority adjusts the primary ﬁscal surplus to stabilize the public debt.

The opposite

happens in the second regime, where the ﬁscal authority does not adjust the ﬁscal surplus

enough to stabilize the debt. As a result, the monetary authority allows the inﬂation rate to

deviate from the target to stabilize debt in real terms. Finally, the ZLB regime represents a

crisis regime, where the monetary authority is constrained by the eﬀective lower bound.

This behaviour was characterized as active monetary policy and passive ﬁscal policy in Leeper (1991)

and the following literature. The opposite happens in the ﬁscally-led regime, where monetary policy is

passive and ﬁscal policy active. The literature has also called the regimes monetary dominance and ﬁscal

dominance, respectively. These regimes represent, in a parsimonious way, the outcome of a game between

ﬁscal and monetary authorities.

The economy presents recurrent regime switches following a transition matrix. Diﬀerently

from Bianchi and Melosi (2017) and Bianchi and Melosi (2022), the transition to and from

the ZLB regime is endogenous and depends on macroeconomic conditions. This endogeneity

allows agents to form rational expectations regarding the occurrence of this regime based on

their knowledge of macroeconomic variables such as inﬂation, output, and nominal interest

rates. The endogenization of this transition probability constitutes one of the contributions

of this paper.

I calibrate the model for the US economy and show it can match volatility and cyclicality

features in the data. Thus, I use it as a laboratory to study central bank balance sheet

policies. I simulate a crisis that resembles the COVID-19 pandemic and the policy response.

I show that a QE program that increases central bank long term bonds purchases by 10p.p.

of GDP reduces the severity of the crisis. Output growth falls around 1p.p. less than without

the program and recovers much faster, allowing a shorter duration of the ZLB regime. QE

also has expansionary eﬀects in terms of inﬂation. Prices fall by 2.5p.p. less than without

the central bank’s intervention, at the cost of more signiﬁcant inﬂationary dynamics during

the recovery, where annualized inﬂation reaches a peak of 8.1. Without QE, inﬂation would

have still overshot the target, but with a lower peak of 6.6%.

I show that how the central bank balance sheet’s size is reduced matters for inﬂation,

output, and debt dynamics. In the crisis recovery, I study diﬀerent strategies regarding

the size of the central bank balance sheet: 1) maintaining the enlarged size after the crisis

(tapering), 2) decreasing the balance sheet size: a) by letting the bonds that mature, run oﬀ

the balance sheet (Quantitative Tightening (QT)), and b) at a faster speed by selling bonds

(Aggressive QT ).

When the central bank reduces its purchases of long-term public bonds, it triggers a fall

in their price and an increase in long-term yields. Increased returns result in a substitution

eﬀect from consumption to savings, reducing aggregate demand and output. At the same

time, public debt issuance increases for three reasons. First, the increase in interest rates

increases debt service. The fall in output triggers automatic stabilizers that increase the

ﬁscal deﬁcit. The fall in long-term bonds price reduces central bank remittances to the

treasury.

In the monetary-led regime, the increase in debt causes a rise in taxes due to the ﬁscal

rule. This exacerbates the fall in aggregate demand and disinﬂationary eﬀects of Quantitative

Tightening. In this regime, inﬂation falls even with lower increases in the policy rate. In the

ﬁscally-led regime, public debt increases actual and expected inﬂation, exacerbating the rise

in yield spreads and mitigating the disinﬂationary eﬀects of QT.

In a ﬁscally-led regime, thus, there are no clear advantages in reducing the balance sheet

size since it increases debt and spreads without helping to stabilize inﬂation. The negative

demand shock generated by QT is not enough to counteract the stimulative eﬀect of real

negative interest rates and ﬁscal deﬁcits.

Related Literature. This paper contributes to several strands of the literature. In

particular, it contributes to the literature that studies the real and inﬂationary eﬀects of

QE through diﬀerent channels.

Many authors studied the transmission mechanisms of QE

in general equilibrium models, either with ﬁnancial frictions, like in Gertler and Karadi

(2011), Sims and Wu (2021), Sims et al. (2020), Del Negro et al. (2017), among others; with

market segmentation or portfolio adjustment costs as the main mechanisms to break the

non-arbitrage condition between short-term and long-term bonds, as in Chen et al. (2012),

Harrison (2017), or information frictions as in Gaballo and Galli (2022). Furthermore, a

recent paper by Cui and Sterk (2021) studies the liquidity eﬀects of asset purchase programs

in a model with heterogeneous agents. The main contribution to this strand of the literature

is allowing the ﬁscal-monetary policy mix to play a role in shaping the macroeconomic eﬀects

of QE and showing alternative scenarios where ﬁscal and monetary policy interact.

In a related paper, Elenev et al. (2021) ask whether monetary policy can create ﬁscal

capacity. In their setting, ﬁscal capacity depends on the probability of shifting ﬁscal policy

from active to passive. In this sense, I share with them the objective of studying the eﬀects

of unconventional monetary policies while allowing ﬁscal policy to shift between regimes.

The main diﬀerence with this paper is that Elenev et al. (2021) assume and estimate a ﬁscal

limit beyond which the Treasury starts increasing taxes to ensure debt sustainability. At

the same time, the conventional monetary policy is active at all periods. These assumptions

prevent inﬂationary dynamics from arising in the model since all the agents in the economy

See Bhattarai and Neely (Forthcoming) and Kuttner (2018) for reviews of this literature.

are rational and know the government will never inﬂate away part of the debt. My main

contribution here is to study QT when ﬁscal and monetary authorities can follow a diﬀerent

policy conﬁguration.

By giving a central role to the interaction or coordination of ﬁscal and monetary policies,

this paper also relates to the literature that studies ﬁscal-monetary policy mix, as in Sims

(1994), Leeper (1991), Schmitt-Grohe et al. (2007), Leeper and Leith (2016), Reis (2017),

Bassetto and Sargent (2020), Barthélemy et al. (2021), among others, and to the literature

of the Fiscal Theory of the Price Level as in Sargent et al. (1981), Bassetto (2002), Cochrane

(2001), Sims (2016), Cochrane (2021), Brunnermeier et al. (2020). Allowing the policy mix

to alternate, this paper also relates to the literature on regime switches in policy rules, as in

Bianchi (2013), Bianchi and Melosi (2017), Bianchi and Melosi (2022), among others. The

contribution to this branch of the literature is the study of unconventional monetary policies

together with the conventional Taylor rule on short-term interest rates. By considering an

endogenous transition probability to get in and out of the ZLB, this paper relates to the

literature that studies monetary policy rules that depend on endogenous variables, as in

Barthélemy and Marx (2017), Davig and Leeper (2008).

This paper focuses on the macroeconomic eﬀects of reducing the central bank balance

sheet. In a broad sense, I share the question with, for example, Wen et al. (2014), Harrison

(2017), Sims et al. (2020), Bonciani and Oh (2021), Hall and Reis (2016), Foerster (2015)

and Benigno and Benigno (2022). However, this paper diﬀers from these in many aspects.

In Wen et al. (2014), the focus relies on the impact on ﬁrms, while there is no role for the

ﬁscal authority. Harrison (2017) studies the optimal QE policy in a DSGE-NK model with

portfolio adjustment costs, and Sims et al. (2020) studies optimal simple and implementable

QE rules through minimizing a quadratic loss-function in a DSGE-NK model with ﬁnancial

frictions. Bonciani and Oh (2021) extends the work of Sims et al. (2020) by showing that

the central bank’s loss function depends on its asset purchase volatility. Crucial diﬀerences

are that these articles focus on central bank purchases of corporate/ private sector bonds,

assume a limited role in ﬁscal policy, and do not allow for changes in the conduct of the ﬁscal

policy rule. Hall and Reis (2016) study how diﬀerent strategies for the exit from quantitative

easing in the face of interest-rate risk, as I do. However, the authors’ main objective is to

study the implications on central bank ﬁnancial stability, and they assume a passive ﬁscal

policy. Foerster (2015) examines the macroeconomic eﬀects of unwinding the central bank

balance sheet during and after a ﬁnancial crisis. He shows that private agents’ expectations

about the exit strategy from a QE program impact the initial eﬀectiveness of the policy in

a MS-DSGE model with a ﬁnancial sector. However, there is no role for parameter switches

in conventional policy rules, abstracting from the possibility of alternative ﬁscal-monetary

policy interactions, which is the main contribution of this paper.

Finally, in a recent related paper, Benigno and Benigno (2022) study monetary policy

normalization, deﬁned as the combination of lifting the policy rate and reducing the size of

central bank balance sheets. They study how these monetary policies interact and how they

depend on the behavior of the ﬁscal authority. However, they focus on optimal policy analysis

and does not account for policy uncertainty in the conﬁguration of ﬁscal and monetary policy

interactions, which constitutes a central contribution of this paper.

The remainder of the paper is as follows. In the following section, I provide motivating

evidence of the importance of the ﬁscal and monetary policies interaction during Quantitative

Easing programs, looking at data for the US during the COVID-19 crisis. Section 3 presents

the model, Section 4 discusses the calibration, functional forms, and solution method. In

Section 5, I present the quantitative results of the model, and in Section 6 explain the main

transmission mechanism of QE. Section 7 presents the paper’s main results, where I simulate

the crisis and present the diﬀerent exit strategies from QE. Finally, Section 8 concludes.

2 COVID-19 crisis and policy response in the US

This section provides data on the COVID-19 crisis and the policy response in the US. The

objective is to show the behavior of the main players in the bonds market, which motivates

some model decisions.

In March 2020, the COVID-19 pandemic hit economies worldwide, generating an unprece-

dented macroeconomic crisis. As a response, central banks in leading developed economies,

particularly the US, started to stimulate the economy through cuts in short-term interest

rates until they hit the zero lower bound. As a result, central banks began Quantitative

Easing programs. They consisted of purchasing assets, mainly government bonds of long

maturity, expanding their balance sheets at a breakneck speed. As shown in ﬁgure 1, the

assets in the Federal Reserve increased from $4 trillion in the third quarter of 2019 to over

$8 trillion one year later. Treasuries are the main component of this assets’ expansion at the

Federal Reserve, which increased from about $2 to $6 trillion in less than two years.

2007Q1 2009Q1 2011Q1 2013Q1 2015Q1 2017Q1 2019Q1 2021Q1

Trillions USD

Fed Assets Treasuries in Fed balance sheet

Figure 1: Federal Reserve Balance Sheet

Assets in the Federal Reserve balance sheet (orange line) and treasuries in the balance sheet

(dotted light-blue line). Variables in trillions of dollars. Source: FRED and US Financial

Accounts. Grey vertical lines show the start of a Quantitative Easing program.

The Quantitative Easing program took place with a substantial expansion in total debt

issuance from the Treasury. Figure 2 shows net purchases of treasuries from the start of the

COVID-19 crisis. They are net ﬂows of treasuries, i.e., net of revaluation eﬀects that take

into account the change in treasuries’ prices.

Diﬀerent colors represent diﬀerent agents in

the economy. When the bar is above zero, the agent had net purchases of treasuries during

the period, while if it is below zero, it represents net sales. The blue line represents the net

The conclusions do not change when we consider changes in treasuries. See Appendix, section 9.1.

debt issuance from the US Treasury, which is the sum of all the bars in a corresponding

period.

The ﬁgure shows the signiﬁcant increment in debt issuance during the whole period.

Furthermore, the monetary authority’s purchases of government bonds played a vital role

during this period, sustaining the demand for treasuries when private sectors like households,

mutual funds, and the rest of the world were selling their bond holdings.

2020Q1 2020Q2 2020Q3 2020Q4 2021Q1 2021Q2

-1000

-500

500

1000

1500

2000

2500

3000

Monetary Authority

Money Market Funds

Rest of the world

Household sector

Mutual funds

Others

Net debt issuance

Figure 2: Net purchases of treasuries

Source: US Financial Accounts. Data in billions of dollars. Flows, net of revaluation eﬀects.

Who sells treasuries to the central bank during a QE operation has implications regarding

the primary macroeconomic aggregates in the economy. When the Federal Reserve performs

quantitative easing policies, it purchases assets. The counterpart of this operation is the

issuance of bank reserves (or money in commercial banks’ hands) that increases the central

bank’s liabilities, expanding the central bank’s balance sheet. If commercial banks were the

original owners of the bonds (case 1), this operation would leave its balance sheet unchanged

since they increase one asset (reserves) while decreasing another (treasuries). However, sup-

pose the original owner of the bond was the non-bank private sector (case 2), like households

or mutual funds. In that case, the bank is a intermediary of this policy. The result would

be an increase in deposits or currency (money in private hands), together with an increase

in reserves and the central bank’s balance sheet. As a result, diﬀerent measures of money

increase after quantitative easing policies under diﬀerent cases; either bank money in the

form of reserves (case 1) or non-bank private money (case 2).

2019Q4

2020Q1

2020Q2

2020Q3

2020Q4

2021Q1

2021Q2

2021Q3

3.5

4.5

5.5

Debt purchases by FED

2019Q4

2020Q1

2020Q2

2020Q3

2020Q4

2021Q1

2021Q2

2021Q3

2.5

3.5

Reserves

2019Q4

2020Q1

2020Q2

2020Q3

2020Q4

2021Q1

2021Q2

2021Q3

Comercial banks deposits

2019Q4

2020Q1

2020Q2

2020Q3

2020Q4

2021Q1

2021Q2

2021Q3

1.5

2.5

3.5

Core Inflation (yoy)

Figure 3: Macroeconomic variables: public debt, liquidity, inﬂation

US Federal Debt, Reserves and Deposits, in trillions of dollars. Core inﬂation in %. Source

debt, deposits, and inﬂation data: FRED. Source reserves: US ﬁnancial accounts, released

December 2021.

In ﬁgure 3, we can see a great expansion in liquidity, both in the form of bank reserves

and deposits.

Other measures of monetary aggregates present signiﬁcant increases during

In the appendix, section 9.1, I show this last feature diﬀers from what happened after the Quantitative

Easing programs after the Great Recession.

this period too. Some of these variables, like public debt, are at values over GDP not seen

since the Second World War.

As economies recover and face inﬂationary dynamics not seen in the last 40 years, it is

unclear how to unwind the stimulus injected during the crisis. The combination of high inﬂa-

tion rates, large debt-to-GDP ratios, elevated liquidity, and expanded central bank balance

sheet presents an extra challenge for policymakers.

In the following section, I present a model able to generate the comovements that we ob-

served in aggregate macroeconomic variables during the COVID-19 crisis and the consequent

policy response and use it as a laboratory to study balance sheet policies.

3 Model

The model is a NK-DSGE model with ﬁve agents: Firms, households, Monetary Authority,

Fiscal Authority, and ﬁnancial intermediaries (FI). There are three assets in the economy:

Short-term public bonds B

with one-period maturity and price Q

; deposits D

which

provide liquidity services to households and with price Q

; and long-term public bonds: B

with geometrically decaying maturity δ, as in Hatchondo and Martinez (2009), that pay a

coupon κ every period, and their price is Q

As is standard in the literature that assesses the transmission mechanisms of quantitative

easing policies, there is segmentation in bond markets.

I follow Elenev et al. (2021) in as-

suming that households cannot invest in short-term public bonds, which are held exclusively

by ﬁnancial intermediaries and the central bank. In contrast, ﬁnancial intermediaries do

not invest in long-term public bonds. These assumptions align with the data presented in

section 2.

Short-term bonds in the model stand for treasury bills and central bank reserves indis-

tinctly since both assets share liquidity and return properties. The only diﬀerence between

them is which governmental institution issues them (i.e., central bank or treasury). The

See section 9.1 for a historical plot of this variable.

See, for instance, Chen et al. (2012), and Bhattarai and Neely (Forthcoming) for a comprehensive

analysis on diﬀerent mechanisms to break the non-arbitrage condition between diﬀerent assets in the economy,

and thus, the neutrality result from Wallace (1981).

assumption that ﬁnancial intermediaries exclusively hold them in the model makes this asset

close to “money in banks’ hands.”

Market segmentation interacts in the model with the following frictions. First, prices are

sticky, as in Rotemberg (1982) and the New-Keynesian literature. Second, households pay

a portfolio adjustment cost when purchasing long-term public bonds, and ﬁnancial interme-

diaries pay a convex cost when they raise new equity. Finally, ﬁnancial intermediaries are

subject to a leverage constraint that states that their debt (deposits) cannot be larger than

their assets (reserves and treasury bills). On top of this, the real value of deposits provides

liquidity services to households increasing their utility, as in the tradition of Money in the

Utility (MIU) function models. Including variables that stand for reserves and monetary

aggregates in the model, together with the frictions above, allows the model to generate the

main macroeconomic eﬀects of QE policies highlighted in the empirical literature and the

comovements in macroeconomic aggregates that we observed in section 2.

Two authorities constitute the government: the treasury and the central bank. Each

authority has its budget constraint and policy instrument(s). The presence of the two gov-

ernmental sectors allows the model to generate realistic policy interactions, which are at the

center of macroeconomic dynamics. I follow Bianchi and Melosi (2017), and Bianchi and

Melosi (2022) and assume that parameters in policy rules for conventional monetary policy,

the Taylor rule, and ﬁscal rule alternate between regimes.

Next, I describe each agent and its optimization problem.

3.1 Households

There is a continuum of measure one of homogeneous households that live inﬁnite periods

in the economy. A representative agent chooses consumption c

, labor n

, deposits D

, and

long-term public bonds B

L,H

to maximize its lifetime utility:

∞

t=0



, n



is a preference shock that follows an AR(1) process. Deposits are the most liquid asset

that a household can purchase. I assume agents derive utility from the liquidity services

this asset provides. The utility function is a monotone increasing function in consumption

and deposits, a monotone decreasing function in labor, and satisﬁes Inada conditions on all

variables.

Long-term bonds pay geometrically decaying coupons, as in Hatchondo and Martinez

(2009). A bond B

issued at time t pays the sequence of coupons: κ, κ(1 − δ), κ(1 − δ)

, ...,

where κ > 0 and δ ∈ (0, 1). This last parameter controls the debt maturity, where δ = 1

corresponds to a short-term bond (i.e., 1-period maturity), and δ = 0 represents a consol.

This maturity speciﬁcation reduces the number of state variables in the model. A bond

issued at time j − k is equivalent to (1 − δ)

bonds issued at period t, and hence the state

variable B

t−1

represents total long-term debt in equivalent newly issued long-term bonds.

Their price at period t is Q

Every period, the representative household pays lump-sum taxes τ

, receives nominal

dividend payments from ﬁnancial intermediaries Div

, ﬁrm’s proﬁts since households are

the owners Π

, and rebates

. When she invests in long-term bonds, she pays a portfolio

adjustment cost of Φ(.) on top of the asset’s price.

The optimization problem of a representative household is the following.

max

L,H

∞

t=0



, n



+ Q

+ B

L,H

+ Φ

L,H

= W

+ D

t−1

+ · · ·

L,H

t−1



κ + (1 − δ)Q



+ Π

+ Div

− τ

(1)

≥ 0

L,H

≥ 0

Notice that the deﬂated number of nominal bonds by the price level,

L,H

= b

L,H

is not the real value

of public debt. Instead, the real value of public bonds held by households is Q

L,H

. As discussed in Leeper

et al. (2021), introducing the variable b

L,H

allows to dispose of one variable, by working with b

L,H

instead

of P

and B

L,H

This assumption helps the model generate a positive term premium in long-term public bonds.

where P

is the price level, and W

is the nominal wage. Under this speciﬁcation, the expected

return on a long-term bond purchased at period t is R

t,t+1

= E

κ+(1−δ)Q

t+1

Notice that the non-negativity condition on deposits does not bind at the optimum even

if they pay a lower return, given the assumption that deposits increase utility. Finally, the

last constraint is the non-negativity conditions for public bonds since the household cannot

go short on them.

Associate the multiplier β

to the budget constraint. The system of equilibrium condi-

tions that characterize the households’ optimization problem solution is the following:

−U



, n





, n



= E

t,t+1



, n





, n



+ Φ

′

L,H

= E

t,t+1



κ + (1 − δ)Q

t+1



Where M

t,t+1

is the stochastic discount factor between period t and t + 1.

t,t+1

= β

t+1

= β

t+1



t+1

, n

t+1





, n



t+1

Together with the budget constraint (1), they characterize the solution to the household’s

optimization problem.

Deﬁne π

t−1

as the inﬂation rate of period t. Notice that

a spread will exist between the prices of deposits and long-term bonds even without the

presence of portfolio adjustment cost. Households are willing to invest in deposits, even

though they provide a lower return, because they derive utility from them. The presence

of portfolio adjustment costs in bonds generates a term spread and prevents the household

from fully exploiting the arbitrage opportunities in the assets’ markets. The literature has

shown this feature is vital for the transmission mechanism of QE policies.

The non-negativity condition in public bond purchases ensures the transversality condition is satisﬁed.

For future reference, it is convenient to deﬁne the return on a risk-free private asset in

this economy, which is in zero net supply, as given by:

= E

t,t+1

This expression is deﬁned as the natural interest rate in Benigno and Benigno (2022),

which diﬀers from the policy rate, and is key in the consumption-savings trade-oﬀ.

3.2 Firms

The productive sector comprises two levels: ﬁnal goods and intermediate goods producers.

3.2.1 Final good producer

A representative ﬁrm produces the domestic ﬁnal good y

from varieties y

, for i ∈ [0, 1].



ε−1

i,t



ε−1

Where ε is the elasticity of substitution between varieties. The optimization problem of the

representative ﬁrm is the following:

max

,{y

i,t

}

i∈[0,1]

−

i,t

s.t y



ε−1

i,t



ε−1

And the optimal demand function for variety i is given by the following expression:

i,t

= y



i,t



−ε

(2)

3.2.2 Intermediate goods ﬁrms

Intermediate goods ﬁrms are monopolistically competitive in the goods market. Each ﬁrm

produces a variety i according to a linear production function:

i,t

= z

i,t

(3)

Where z

is a mean reverting TFP shock, common to all varieties, with the law of motion:

ln (z

) = ρ

ln (z

t−1

) + σ

, ε

∼ N(0, 1)

When changing prices, ﬁrm i is subject to a quadratic adjustment cost in prices, as in

Rotemberg (1982):



i,t

i,t−1

− π

∗



Where ϕ

measures the degree of nominal price rigidity, and y

is aggregate output, given

by:

i,t

The nominal proﬁt of ﬁrm i at period t, transferred to households, is given by:

i,t

= P

i,t

− W

i,t

−



i,t

i,t−1

− π

∗



(4)

Firms maximize the discounted sum of proﬁts using the households’ discount factor, and

subject to technology 3, and demand 2.

Their optimization problem is the following:

max

i,t

∞

k=0

t,t+k

i,t+k

s.t. y

i,t

= z

i,t

= y



i,t



−ε

i,t

= P

i,t

− W

i,t

−



i,t

i,t−1

− π

∗



In equilibrium, all ﬁrms behave symmetrically, and we have P

i,t

= P

. Thus, aggregate

proﬁts are given by:

= P

− W

−

(π

− π

∗

)

Their optimization problem is characterized by the following conditions, where MC

the multiplier of equation 2, and it is the marginal cost.

1 − ε + ε

= ϕ

(π

− π

∗

) π

− ϕ



t,t+1

t+1

(π

t+1

− π

∗

)



(5)

= MC

(6)

The ﬁrst condition is the New-Keynesian Phillips curve.

3.3 Financial intermediaries

In this section, I follow a simpliﬁed version of the ﬁnancial intermediaries’ description in

Elenev et al. (2021).

A representative agent in this sector starts the period t with a net worth of W

. Every

period, it pays a fraction τ

of its net worth to households and raises new equity from them,

. The net payout to shareholders is then:

Div

= τ

− A

(7)

This agent can invest in short-term public bonds B

S,I

for the price Q

. B

S,I

is the

sum of treasury bills (treasuries with a maturity of up to a year, B

) and central bank

reserves (B

S,CB

). The reason for adding these assets into one variable is that they have the

same risk and return properties in the model, being perfect substitutes from the ﬁnancial

intermediaries’ point of view. Its liabilities are given by deposits from households D

. The

following expression then gives the balance sheet:

(1 − τ

+ A

− Φ

) + Q

= Q

S,I

(8)

Where Φ

(.) is a convex cost of issuing new equity, rebated lump-sum to Households. This

cost prevents ﬁnancial intermediaries from raising funds only through equity, going to a

corner solution with D

= 0.

The net wealth at period t is:

= B

S,I

t−1

− D

t−1

(9)

In line with Basel regulation, ﬁnancial intermediaries are subject to a leverage constraint.

It states that their debt (in this case, deposits) can be, at most, a fraction ζ of its assets.

≤ ζB

S,I

(10)

The optimization problem of a ﬁnancial intermediary consists of maximizing the dis-

counted sum of dividends subject to restrictions 8, 9, and 10. I assume the ﬁnancial inter-

mediary discounts its future ﬂows using the household’s stochastic discount factor.

I assume the ﬁnancial intermediary sector’s assets are composed only of treasury bills and central bank

reserves for simplicity. In a more complicated model, they could also purchase ﬁrms’ bonds or provide loans

to ﬁrms to ﬁnance capital purchases. This is the case in Elenev et al. (2021), and Benigno and Benigno

(2021). The qualitative conclusions of this paper do not change when considering a more complicated version

of the model, reducing to quantitative diﬀerences.

max

S,I

∞

k=0

t,t+k



t+k

− A

t+k



s.t. (1 − τ

+ A

− Φ

) + Q

= Q

S,I

= B

S,I

t−1

− D

t−1

≤ ζB

S,I

Deﬁne η

as the Balance sheet multiplier and µ

as the Leverage constraint multiplier.

Using the ﬁrst order condition to equity A

to substitute out the multiplier η

, we obtain

the system of equations that characterize the ﬁnancial intermediary’s optimization problem,

together with 7, 8, 9, 10:

= E

t,t+1

+ µ

(1 − Φ

′

)) (11)

= E

t,t+1

+ ζµ

(1 − Φ

′

)) (12)

Where

t,t+1

is the stochastic discount factor for ﬁnancial intermediaries, deﬁned as:

t,t+1

≡ M

t,t+1

(1 − Φ

′

))



1 − τ

1 − Φ

′

t+1

)



Notice that the spread between deposits and short-term bonds is a function of the leverage

constraint multiplier µ

. Since ζ < 1, the return on short-term bonds is higher than the

short-term bonds when the leverage constraint binds. If the leverage constraint does not

bind, then we get Q

= Q

, and from the Households’ problem, this could only be possible

in a situation where liquidity services are zero.

3.4 Monetary Authority

The Central Bank performs conventional and unconventional monetary policies. The con-

ventional monetary policy sets the short-term nominal interest rate R

, subject to a ZLB

See appendix, section 9.2 for further details.

restriction. This rate is the inverse of the short-term public debt price:

≡

(13)

The unconventional monetary policy consists of central bank balance sheet policies. In

particular, the central bank purchases long-term government debt B

L,CB

in exchange for

reserves (B

S,CB

), as in the data since the Great Recession. The rules for setting these two

instruments are provided in a following section.

The following expression gives the Central Bank’s budget constraint:

S,CB

t−1

L,CB

t−1



κ + (1 − δ)Q



= Q

S,CB

+ Q

L,CB

+ Λ

where Λ

are central bank remittances to the ﬁscal authority. I assume that proﬁts Λ

are transferred to the ﬁscal authority.

Finally, the central bank is subject to a revenue neutrality constraint, which is in line with

the data. It states that when the central bank increases its assets by purchasing long-term

bonds, it has to oﬀset the operation by decreasing its net position of short-term assets.

L,CB

+ Q

S,CB

= 0 (14)

When it performs QE, we have B

S,CB

< 0, representing the increase in reserves issuance.

3.5 Fiscal Authority

The treasury consumes g

and obtains resources from three diﬀerent sources. It collects tax

revenues from households in a lump-sun fashion, τ

, receives dividends from the central bank

, and issues debt, whose total real value is

. This debt comprises short-term bonds

For simplicity, in this paper, I abstract from asymmetries in the transfer of the central bank’s proﬁts to

the treasury, like the ones discussed in Hall and Reis (2016).

and long-term bonds B

. The total debt issuance is:

= Q

+ Q

(15)

The period budget constraint of the ﬁscal authority, in real terms, is:

− g

| {z }

≡s

+ Λ

t−1



κ + (1 − δ)Q



Where s

is the real primary ﬁscal surplus of period t. Replacing Λ

from the central

bank balance sheet and using 14, I obtain a consolidated budget constraint:

t−1

− B

S,CB

t−1

| {z }

t−1

− B

L,CB

t−1

| {z }



κ + (1 − δ)Q



(16)

The terms’ A’ and ‘B’ are the outstanding short and long-term public debt in the private’s

hands. Notice that through the purchases of long-term public bonds, the central bank relaxes

the budget constraint for the consolidated government. However, this is not necessarily

the case when considering the net position of short-term assets. For instance, if B

S,CB

0, implying reserves issuance, then the consolidated government debt of short maturity is

increasing with this policy. In this sense, quantitative easing policies can be interpreted as

a maturity swap, exposing the government to interest rate risk.

The following expression gives government consumption:

= θ(y

∗

− y

) + (1 − ρ

)¯g + ρ

t−1

+ σ

, ε

∼ N(0, 1)

0 < θ < 1 represents the government spending reaction to output deviations from its

steady state. This term generates a counter-cyclical behavior of government consumption,

introducing ﬁscal stimulus when the economy is in a recession. Government consumption

goods are thrown into the ocean. Finally, the maturity composition of newly issued govern-

ment debt is constant in book value terms, with a fraction ¯µ of debt being long-term.

Notice that the real amount of short-term bonds is Q

and the one of long-term bonds is Q

1 − ¯µ

¯µ

(17)

3.6 Market clearing conditions

Market clearing conditions are the following:

+ g

(π

− π

∗

)

= y

= D

− B

S,CB

= B

S,I

= B

L,H

+ B

L,CB

And households’ rebates by other agents in the model are equal to:

= Φ



L,H



+ Φ

)

3.7 Policy rules

To close the model, I assume that ﬁscal and monetary authorities follow policy rules to set

their instruments: τ

, R

, and b

L,CB

. First, I assume that central bank purchases of long-term

bonds follow an AR(1) process:

L,CB

= (1 − ρ

L,CB

∗

+ ρ

L,CB

t−1

+ σ

(18)

Where b

L,CB

∗

is the average amount of long-term bonds at the steady state. Increases in

the central bank balance sheet are random and unrelated to the economic conditions. I follow

this assumption for two reasons. First, this paper aims to study the eﬀects of QE policies

under diﬀerent interactions of conventional ﬁscal and monetary policies. Second, there is

no evidence or consensus of a clear rule for central bank purchases, a policy with a great

discretionary component. This assumption implies that QE constitutes a complementary

policy instrument of the central bank and not necessarily a substitute, in line with what we

observe in the data. Furthermore, it prevents the instrument from altering the determinacy

properties of the model.

I follow the literature on ﬁscal-monetary policy interactions and assume that ﬁscal and

monetary authorities follow rules to determine the conventional policy instruments. In par-

ticular, the nominal short-term interest rate follows the Taylor rule 20, reacting to output

and inﬂation deviations from its steady state values, together with an autoregressive pa-

rameter ρ

and a monetary shock ϵ

with standard deviation σ

. The monetary shock

stands for interest rate deviations from the reaction function. The intensity of interest rate

reaction to output and inﬂation deviations are characterized by policy parameters α

and

, respectively. Finally,

R is the mean of the nominal interest rate.

For ﬁscal policy, I assume that taxes τ

react to total real debt deviations from its steady

state value b

∗

and have an autoregressive coeﬃcient ρ

, as in 19. The elasticity of tax

deviations to debt deviations is characterized by the parameter γ.

− τ

∗

= ρ

(ξ

) (τ

t−1

− τ

∗

) + (1 − ρ

(ξ

)) γ (ξ

) (b

t−1

− b

∗

) (19)

R (ξ

)



t−1

R (ξ

)



(ξ

)



∗



(ξ

)



∗



(ξ

)

1−α

(ξ

)

(ξ

)ϵ

(20)

The parameters above depend on a discrete shock ξ

that follows a Markov process. I fol-

low Bianchi and Melosi (2017) and assume that this shock can take three values, representing

the three regimes through which the economy ﬂuctuates.

The ﬁrst regime, the monetary-led (M) regime, is characterized by a strong interest rate

reaction to inﬂation deviations (high α

) and a strong tax reaction to debt (high enough

γ).

This regime is associated to an active monetary policy and passive ﬁscal policy, in

Leeper (1991) terminology. Under this regime, the monetary authority adjusts the nominal

Assuming the ﬁscal rule reacts to real public debt in private hands, i.e., Q

L,H

+ Q

S,I

does not

change the conclusions in this paper. These results are available under request.

In Appendix, I present, for a given calibration, the combinations of values α

and γ that give rise

to determinacy at each regime. Notice, however, that the global stability of the system does not require

determinacy at each regime.

interest rate more than proportionally to changes in the inﬂation rate to stabilize inﬂation.

At the same time, the treasury passively adjusts taxes to stabilize the real debt.

The second regime is the ﬁscally-led regime (F) where the ﬁscal authority does not adjust

taxes enough to stabilize the real debt (low γ). Then, the central bank allows the inﬂation

rate to deviate from the target to stabilize debt in real terms. This regime can be deﬁned

as passive monetary policy and active ﬁscal policy in the sense of Leeper (1991).

The third regime is the Zero Lower Bound regime (ZLB). It is characterized by an un-

reactive nominal short-term interest rate that remains ﬁxed at its eﬀective lower bound and

by a ﬁscal policy that is unreactive to the real debt level. This regime is an extreme form

of the ﬁscally led regime since we have α

= γ = 0. As in Bianchi and Melosi (2017), this

regime represents a crisis regime, where the economy enters due to the realization of bad

shocks that drive the economy to a recession. However, diﬀerently from Bianchi and Melosi

(2017), the economy enters the ZLB regime endogenously, as I explain in the next section.

3.8 Transition probabilities

Assume the Markov-switching shock ξ

depends on the realization of two random variables

and ξ

. When ξ

= 1, the economy suﬀers a crisis and moves to the ZLB regime. When

= 0, the government can set ﬁscal and monetary policies without restriction. In this

case, the variable ξ

determines the regime in place stochastically. ξ

= M stands for the

monetary-led regime, ξ

= F for the ﬁscally-led regime, and it evolves according to the

transition matrix:

P =





1 − p





where p

= P



t+1

= j|ξ

= i



As seen from the transition matrix P, the probability of being in one regime or another

is constant and entirely exogenous. The realization of this shock represents, in a simpliﬁed

way, the outcome of a policy game between the ﬁscal and the monetary authority, where the

winner is the active authority.

Deﬁne q as the probability of entering to the ZLB regime, q = P (ξ

t+1

= 1|ξ

= 0) and r

as the probability of moving out of the crisis regime q = P (ξ

t+1

= 0|ξ

= 1).

The transition matrix for ξ

is then:

T =





(1 − q)P q[1; 1]

r[p

; (1 − p

)] (1 − r)





Where p

is the probability of exiting the ZLB regime towards a monetary-led regime. q

and r are endogenous processes. The probability of entering the ZLB regime is a decreasing

function of the nominal interest rate:

q = P (ξ

t+1

= 1|ξ

= 0) = f (R

)

Intuitively, it is a function that generates zero probability of switching when the gross

nominal interest rate is higher than one, and it increases as R

approaches the value of 1.

The probability of leaving the crisis regime, r, is a function of a shadow interest rate R

r = P (ξ

t+1

= 0|ξ

= 1) = g(R

)

The shadow interest rate represents the interest rate that would hold in the economy if

this were always in the monetary-led regime, without a lower bound restriction:



= M



t−1



= M



(

)



∗



(

)



∗



(

)

1−α

(

)

(

)

(21)

This assumption reﬂects that the probability of leaving the ZLB regime is not independent of

the economic conditions. For instance, when output and(or) inﬂation recovers, the Shadow interest

rate increases, increasing the likelihood that the central bank would start raising interest rates.

Endogenous transition probabilities to and out of the ZLB matter for agents’ expectations.

Contrary to an entirely exogenous transition matrix model, agents know the likelihood of tightening

monetary policy increases with output and inﬂation, even at the ZLB.

4 Functional forms, calibration and solution

4.1 Functional forms

In this section, I present the functional forms assumed in the numerical exercise. I assume the

following CRRA utility function for households, which depends positively on consumption and

deposits and negatively on labor.

U(c

, d

, n

) =

1−φ





1−σ

1 − σ

− ψ

Where d

are deposits deﬂated by the price level.

The portfolio adjustment cost for long-term bonds is quadratic:



L,H



L,H

where b

L,H

, and b

L,H

is the steady state value of public long-term bonds held by

households.

The convex cost of issuing equity is the following:

) =

Finally, the endogenous transition probabilities to and out of the ZLB regime are assumed to

follow a logistic distribution as in Benigno et al. (2020), and Bocola (2016). As in their models, the

economy’s transition between regimes is a logistic function of a subset of the model’s endogenous

variables. In this case, they are given by the following expressions:

q = P (ξ

t+1

= 1|ξ

= 0) =

exp {−γ

− 1)}

1 + exp {−γ

− 1)}

where γ

> 0 is a constant. And,

r = P(ξ

t+1

= 0|ξ

= 1) =

exp



−γ



− 1



1 + exp



−γ



− 1



with γ

< 0.

4.2 Calibration

I work with quarterly data for the US. In this section, I present the calibration. In table 1, I present

the calibration of structural parameters with their source or target. Some parameters have direct

counterparts in the data. For instance,

R, the average gross short-term nominal interest rate is set

equal to 1.011, as in quarterly data for the period 1980-2020.

The average of long-term public

bonds purchased by the central bank, b

L,CB

∗

is set to 0.014 to match the average 7% annual ratio

of Federal Reserve total treasuries to output ratio, in market value terms, for the period 1980-2020.

Finally, the parameter that characterizes the collateral ratio in the ﬁnancial intermediaries’ leverage

constraint, ζ, is set to 0.97 as in the Basel regulation.

Some parameters are taken from the literature. For instance, the risk aversion parameter σ is

set to 2, as is standard in the literature. The inverse of Frisch elasticity η, is set to 3 as in Leeper

et al. (2021), ¯µ, that is the proportion of long-term bonds in book values issued by the treasury,

is 0.67 as in Elenev et al. (2021) and θ, that is 0.27 as in Bianchi and Melosi (2017). The convex

cost of issuing equity for ﬁnancial intermediaries, χ is 22 as in Elenev et al. (2021). The coupon

payment of long-term bonds, κ, that includes the interest and the matured part fraction of bonds

is normalized to 1.

Other parameters are calibrated to match ﬁrst-order moments in the data. β takes the value

0.996 to match the average return of a real risk-free asset from Jordà et al. (2017). δ is set to 0.0357

to roughly match the average maturity of public bonds with a maturity longer than one year, seven

years. The parameter that characterizes the quadratic portfolio adjustment cost of long-term bonds,

is calibrated to 0.0039 to match the average spread between long and short-term bonds of 0.32%

in the period 1980Q1-2020Q1 at the steady state.

Notice that, in the data, this value corresponds to the average quarterly nominal interest rate, including

periods where the interest rate was at the eﬀective lower bound.

In the data, the balance sheet of ﬁnancial intermediaries includes a broader set of assets that can be

used as collateral, than the ones included in the model (T-bills and central bank reserves). This assumption,

although restrictive since it introduces a tight relationship between deposits and reserves, is maintained for

simplicity.

This spread in the model is the diﬀerence between the return on the long-term bond R

κ+(1−δ)Q

and the short-term interest rate .

Description Value Source or target

σ Risk aversion 2 Standard

η Inverse Frisch elasticity 3 Leeper et al. (2021)

κ Coupon Payment 1 Normalization

χ Equity cost 22 Elenev et al (2021)

θ Government spending 0.27 Bianchi and Melosi (2017)

ζ Leverage constraint FI 0.97 Basel regulation

ψ Preference parameter 1.339 Normalization labor

β Discount factor 0.996 Av. Inﬂation rate

∗

Average interest rate 1.011 Av. E. Fed Funds

¯µ Proportion of long-debt 0.67 Elenev et al. (2021)

δ Maturity parameter 0.0357 Maturity long bonds (7 years)

Portfolio adjustment cost 0.004 10-year yield

φ Preference parameter 0.0023 Debt/GDP

= 68%

Dividends distribution 0.84 Spread T-bill to deposits

Prices adjustment cost 150 Inﬂation volatility

ϵ Elasticity of subst. varieties 7 Markup 17%

L,CB

∗

Average CB Balance sheet 0.0140

LC,B

∗

= 7%

Table 1: Calibration: model parameters

The preference parameter φ is set to 0.0023 to generate a simulated mean of annualized debt

to GDP ratio as in the average data before the COVID-19 crisis, around 70% until 2020Q1. The

fraction of ﬁnancial intermediaries’ wealth paid to households as dividends is calibrated to 0.84 to

match the average spread between short-term interest rate and deposits of 0.31% from Drechsler

et al. (2017)

. The elasticity of substitution between varieties ϵ is set to 7, generating an average

markup of 17%, and the parameter ϕ

that characterizes Rotemberg adjustment costs is set to 150

to roughly match the average inﬂation standard deviation during the period 1980-2020. Finally, the

preference parameter ψ is calibrated to normalize labor to one at the steady state.

Table 2 presents the calibration for persistence and standard deviation of the exogenous processes

in the model. They were jointly calibrated to match some second-order moments in the data for

the period 1980-2020. The comparison between moments in the data with the simulated moments

is presented in the following section.

The model does not prevent the net return on deposits to be negative at the zero lower bound regime.

This could be motivated by the data, with the fact that during this period, although banks did not charge

fees on deposits, some of them increased their account maintenance charges to customers.

Parameter Description Value

Persistence QE 0.9

Persistence preference 0.89

Persistence TFP 0.92

Persistence gov. spending 0.96

Dispersion QE 0.0025

Dispersion preference 0.008

Dispersion TFP 0.0021

Dispersion gov. spending 0.0026

Table 2: Calibration: exogenous processes

Table 3 presents the regime-switching parameters correspondent to the ﬁscal and monetary

policy rules from section 3.7. The ﬁrst set of parameters corresponds to the Taylor rule 20, and

the correspondent parameters at each regime. The second set corresponds to the parameter values

for the Shadow interest rate 21. Parameters in the Shadow interest rate are equal to the Taylor

rule’s parameters at the monetary regime, and they are not regime-dependent. They were included

in the table for completeness. The third set of parameters in this table corresponds to the ﬁscal

rule 19. The parameters in this table, with the exception of mean interest rates in all regimes,

come from Bianchi and Melosi (2017). Since a parameter estimation goes beyond the scope of this

paper, I take the parameters correspondent to conventional ﬁscal and monetary policy rules from

this article that presents the same regimes and performs a Bayesian Estimation of the correspondent

parameters using data for the US until the Great Recession. The average interest rate out of the

ZLB is

R, explained in table 1. At the zero lower bound, I set the average interest rate to 1.0005,

as the average quarterly Eﬀective federal funds rate observed in the period: 2008Q4-2017Q1 and

2020Q1-2022Q1.

Description MD FD ZLB

Taylor rule 0.86 0.67 0.2

Taylor rule 1.6 0.64 0

Taylor rule 0.51 0.27 0

Taylor rule 0.25/100 0.25/100 0.25/1000

R Taylor rule R

∗

1.0005

R,s

Shadow R - - 0.86

π,s

Shadow R - - 1.6

Shadow R - - 0.9

M,s

Shadow R - - 0.0025

Shadow R R

∗

γ Fiscal rule 0.0712 0 0

Fiscal rule 0.96 0.69 0.69

Table 3: Calibration: regime-dependent policy parameters .

Table 4 presents parameters relative to transition probabilities between regimes. Exogenous

parameters from matrix P, p

and p

come from Bianchi and Melosi (2022). The probability of

exiting the ZLB towards a monetary-led regime, p

, is set to 0.7031, and it comes from Bianchi

and Melosi (2022). In this paper, the authors show that this probability signiﬁcantly decreased after

the COVID-19 crisis. Since this paper aims to study the exit strategies from the crisis and policies

applied during that period, I considered the latest value of this estimated probability. However, the

results do not signiﬁcantly diﬀer in a model with lower p

Parameters γ

and γ

give the steepness of the logistic function. They were calibrated to obtain

a similar ergodic probability of the ZLB regime as in Bianchi and Melosi (2022) and to minimize

the cases in which the economy is at this regime with a gross interest rate below one.

Parameter Value Source or target

0.9923 Bianchi and Melosi (2022)

0.7031 Bianchi and Melosi (2022)

500 Average prob. of ZLB regime

-200 Average prob. of ZLB regime

Table 4: Calibration: transition probabilities

Figure 4 shows the corresponding probabilities for values of the nominal interest rate (left) or

shadow interest rate (right) between 0.95 and 1.02. A positive value for γ

generates a higher

probability of entering the ZLB when the interest rate is below one and lower when it is above one.

Notice that for this calibration, the probability of entering this regime when R is 0.987 or lower is

one.

0.96 0.98 1 1.02

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

q=P(

t+1

=1|

=0)

0.96 0.98 1 1.02

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

r=P(

t+1

=0|

=1)

Figure 4: Endogenous transition probabilities to and out the zero lower bound.

For the probability of exiting the zero lower bound, the parameter γ

is negative, generating a

monotone increasing probability of leaving the crisis regime when the Shadow interest rate is higher.

4.3 Solution method

I solve the model in real terms through second-order perturbation methods for endogenous Markov

Switching DSGE models, following Benigno et al. (2020). In the solution, I assume the leverage

constraint for ﬁnancial intermediaries is binding in all regimes. The approximation point for the

solution method consists of a weighted average of steady states in diﬀerent regimes, as authors in

Benigno et al. (2020) explain. The weights are the ergodic means of corresponding regimes. In

appendix 9.4, I present the steady state equations and describe the approximation point.

5 Quantitative Results

5.1 Second order moments

In this section, I present some second-order moments for selected variables to evaluate how the model

performs in terms of generated volatility and cyclical properties. Data variables were demeaned to

make them comparable with their model counterpart, where there is no growth. Empirical moments

were calculated using quarterly data for the period 1980Q2-2020Q1. NIPA variables are real and per

capita. Debt, inﬂation, and interest rate are annualized, both in the model simulations and in the

data. Data moments were calculated from a simulation of one million periods, and they correspond

to the averages of the three regimes.

The following table compares the standard deviation (in %) and correlation with output growth.

Output and public debt are logarithmic diﬀerences.

∆GDP ∆ public debt Bal. Sheet Inﬂation Spread

Standard deviation (in %)

Data 0.8 1.7 3.7 2.8 1.6

Model 0.7 1.5 3.1 2.4 1.5

Correlation with ∆GDP

Data - -0.33 0.71 0.44 -0.11

Model - -0.27 0.22 0.25 -0.04

Selected second-order moments in data and model

Note: Growth rates for output and debt. Inﬂation is gross and annualized. The term spread is

the annualized diﬀerence between the 10-year treasury yield and the federal funds rate. Model

moments were obtained from a simulation with one million periods.

As can be seen from the table, the model generates the correct ranking in volatilities and

standard deviations similar to the ones in the data for the main endogenous variables. In terms

of cyclical properties, the model generates the correct sign in correlation with output for all the

variables in the table and close magnitudes to the empirical ones.

5.2 Conditional second order moments at diﬀerent regimes

Table 5 show means and volatility measures for debt to GDP ratio, inﬂation, nominal short-term

interest rate (R

) and nominal return on long-term bonds (R

), conditional on regimes. They reﬂect

features typically highlighted by the literature that studies ﬁscal-monetary policy mix.

The ﬁscal

regime is characterized by higher debt to GDP ratio, interest rates, and more volatile inﬂation and

interest rate. Real debt to GDP, on the contrary, is more volatile in the monetary-led regime. The

ZLB is characterized by a very stable short-term interest rate close to one and almost nil inﬂation

but quite volatile (standard deviation 2.7%).

M F ZLB

Mean Std(%) Mean Std(%) Mean Std(%)

Debt to GDP 72% 7.0 80% 3.8 74% 5.9

Inﬂation 1.02 1.7 1.02 3.8 1.01 2.7

Interest rate (R) 1.03 1.5 1.04 2.5 1.00 0.2

Long-run return (R

) 1.04 1.5 1.05 3.1 1.02 1.8

Table 5: Data moments conditional on regimes

Note: Data generated moments, from a sample of one million periods. The model is simulated

for a long sample where the regime is stochastic. Moments at each regime are obtained condi-

tioning the economy being on the corresponding regime at a given period. Debt to GDP is

inﬂation, and returns are annualized since the model is solved quarterly.

6 The transmission mechanism of Quantitative Easing

This section sheds light on the model’s transmission mechanism of an increase in central bank bond

purchases, b

L,CB

, given by an exogenous shock ϵ

, following 18. I show the log deviations (in

%) of a simulated path for endogenous variables when there is a one standard deviation shock in

the central bank purchases of long-term bonds to a counterfactual without the shock. This shock

implies increasing the real balance sheet to GDP ratio by 1.3p.p., i.e., rising from its steady state

of 7% to 8.3%, a conservative shock. I consider three diﬀerent scenarios, where the economy is at a

given regime and remains at the same regime for 16 quarters in both paths (with and without the

shock). Even though there is no regime change in the simulation exercise, agents in the economy

expect the economy to evolve according to the transition matrix 3.8.

See, for instance Bianchi et al. (2020), where the authors estimate a Markov-Switching VAR with three

regimes for the period 1960-2014.

6.1 Regime dependent Quantitative Easing shock

Figures 5 and 6 show the evolution of endogenous variables at monetary-led regime (continuous blue

line, named ‘M’), ﬁscally-led regime (dotted red line, named ‘F’) and ZLB regime (dashed yellow

line, named ‘ZLB’). They are presented as log deviations from a path without an increase in central

bank purchases, in percentage

When the central bank increases its purchases of long-term bonds (b

L,CB

), it increases the size of

its balance sheet, deﬁned as b

L,CB

. Due to the revenue neutrality constraint, it does it through the

issuance of reserves to ﬁnancial intermediaries. QE transmission mechanism operates through two

channels in this model. First, when the central bank introduces reserves in the intermediary sector,

it relaxes its leverage constraint 10, allowing them to increase the deposit supply to households.

Second, the central bank intervention in the long-term bond market drives its price up, and then

the long-term yield (E

t,t+1

) falls. This price increase gives households incentives to rebalance

their portfolio, selling their long-term bonds and exchanging them for deposits. These mechanisms

operate as a demand shock in this economy. The portfolio revaluation eﬀect provides households

with a wealth eﬀect, increasing consumption. The fall in savings returns generates a substitution

eﬀect from savings to consumption. Both channels operate, increasing aggregate demand in this

economy.

The increase in aggregate demand is inﬂationary. Since ﬁrms operate under monopolistic com-

petition, they can react in two ways to the rise in demand: either increase prices or produce more.

In the ﬁrst case, they have to pay a price adjustment cost due to the presence of nominal rigidities

in this model. In the second case, given a constant TFP, they need to hire more labor. This pressure

Notice that, as shown in the previous section, the ergodic mean of endogenous variables diﬀers among

regimes. In the absence of shocks, the resting point of these variables would be the ergodic mean. For this

reason, I show the log deviations from the correspondent mean, not from the steady state, broadly deﬁned

as an equilibrium without shocks.

Given their leverage constraint, ﬁnancial intermediaries increase deposits for a fraction ζ of the new

amount of reserves and raise equity from their shareholders, households, for the remainder fraction, net of

cost of issuing equity.

A complementary interpretation for the substitution between consumption and saving decision in the

model comes from looking at the "natural interest rate" as deﬁned in Benigno and Benigno (2022). As

the authors highlight in their paper, a key reason why QE policies have real eﬀects on the economy is that

the policy rate diﬀers from the natural interest rate. The later is the one that drives consumption/savings

decision, deﬁned as R

t,t+1

, i.e., the return on a risk-free private asset assumed to be in zero net

supply in this paper. When the central bank increases its purchases of long-term bonds, the natural interest

rate falls, even though it is not under the direct control of the central bank. This shifts households’ savings

towards consumption and explains the increase in aggregate demand. The most signiﬁcant fall in this variable

occurs at the ZLB, where we can see that QE is more expansionary.

on labor markets generates an increase in real wages and, thus, in ﬁrms’ marginal costs. We see a

combination of these eﬀects in equilibrium, so salaries, labor, and output also increase.

5 10 15

quarters

CB purchases (b

L,CB

)

5 10 15

quarters

CB balance sheet/y

5 10 15

quarters

Reserves (b

S,FI

)

5 10 15

quarters

Deposits (d)

5 10 15

quarters

-3

-2.5

-2

-1.5

HH long-debt (b

L,H

)

5 10 15

quarters

-60

-50

-40

-30

-20

Term Spread

5 10 15

quarters

Equity (A)

5 10 15

quarters

-0.1

0.1

0.2

Interest rate (R)

5 10 15

quarters

0.2

0.4

Output (y)

5 10 15

quarters

-0.2

0.2

0.4

0.6

0.8

Inflation ( )

M F ZLB

Figure 5: Impact of a Quantitative Easing shock conditional on a regime

Note: log deviations (in %) in a simulated path with a one standard deviation shock in the

central bank purchases of long-term bonds (ϵ

= 1) to the counterfactual path without

shock (ϵ

= 0). Path conditional on the monetary-led regime (continue blue line),

ﬁscally-led regime (dotted red line), and zero lower bound regime (dashed yellow line) for 16

periods. The balance sheet to output ratio, interest rates, and inﬂation are annualized.

The intensity of these expansionary eﬀects critically depends on the regime, being more expan-

sive at the ZLB and, in second place, in the ﬁscally-led regime. At the ZLB, the ﬁxed short-term

interest rate exacerbates the substitution eﬀect between savings and consumption. The short-term

nominal interest rate increases in the ﬁscal and monetary-led regimes, following the corresponding

Taylor rule. This return increase ameliorates the expansionary and inﬂationary eﬀects of an increase

in central bank purchases, especially in the monetary-led regime where the real short-term interest

rate becomes positive.

Figure 6 shows the ﬁscal variables. Government debt behavior depends on the evolution of the

three components of the government budget constraint: ﬁscal surplus, central bank remittances to

the treasury, and debt service. First, the expansionary eﬀect of central bank purchases triggers a

fall in government spending due to automatic stabilizers. Hence, primary ﬁscal surplus increases

on impact in all regimes. This eﬀect is more signiﬁcant in the ZLB, where output increases more.

Central bank proﬁts (Λ

) increase from period two in all scenarios, alleviating the pressure on

ﬁscal accounts. Finally, total debt service behavior diﬀers among regimes. Initially, it falls due to

larger ﬁscal surpluses and remittances from the central bank, allowing the treasury to issue less

debt (b

, b

). This eﬀect is larger in the ZLB and ﬁscally led regime, where higher inﬂation rates

give the government seigniorage revenues. However, the revaluation of long-term bonds is so strong

in the ﬁscally-led regime that it dominates the eﬀect on real debt (b

), which increases.

The ﬁnal result is an increase in ﬁscal space, interpreted as a fall in real debt (b

) at the ZLB and

the monetary regime. This eﬀect is milder in the second, where taxes fall following debt issuance.

In the ﬁscally-led regime, however, the revaluation eﬀect on long-term bonds is so strong that the

debt ratio to GDP increases.

5 10 15

quarters

-1

-0.5

0.5

Real debt (b)

5 10 15

quarters

-2

-1.5

-1

-0.5

Short-term debt (b

)

5 10 15

quarters

-2

-1.5

-1

-0.5

Long-term debt (b

)

5 10 15

quarters

-1

-0.5

Debt to GDP

5 10 15

quarters

-0.4

-0.3

-0.2

-0.1

Tax ( )

5 10 15

quarters

P. fiscal surplus

5 10 15

quarters

-1

-0.5

0.5

Debt service

5 10 15

quarters

100

200

300

CB Profits (

)

M F ZLB

Figure 6: Impact of a Quantitative Easing shock conditional on a regime: ﬁscal variables

Note: log deviations (in %) in a simulated path with a one standard deviation shock in the

central bank purchases of long-term bonds (ϵ

= 1) to the counterfactual path without

shock (ϵ

= 0). Path conditional on the monetary-led regime (continue blue line),

ﬁscally-led regime (dotted red line), and zero lower bound regime (dashed yellow line) for 16

periods. Debt to GDP is annualized.

This exercise shows that the expansionary eﬀects of QE policies are of considerate size under

a crisis regime. At the ZLB, 1.3p.p. temporal increase in the central bank balance sheet increases

output by almost 0.6 p.p. and inﬂation 0.9p.p. to the counterfactual scenario without intervention.

However, its eﬀects on output and inﬂation are milder in ﬁscal and monetary dominance regimes,

where output increases by 0.3 and 0.2 p.p. and inﬂation by 0.6 and 0.5p.p., being considerably less

persistent in the ﬁrst case. This analysis provides some insights into the power of QE to expand

the economy and how it diﬀers among regimes. Furthermore, it highlights the fact that the ﬁscal

implication of this policy entails substantial diﬀerences across policy scenarios.

7 Exit strategies from Quantitative easing programs

As stated by policymakers, ﬁnancial analysts, and academics, there is high uncertainty regarding

the impact of implementing Quantitative Easing programs and, even more, the unwinding of the

economic stimulus introduced through these measures. The main objective of this paper is to

contribute to understanding the macroeconomic eﬀects of diﬀerent strategies regarding the size of

the central bank balance sheet. In this section, I simulate the economy to generate a crisis that

resembles the one in the US from the ﬁrst quarter of 2020 when the COVID-19 pandemic became

widespread worldwide. First, I show the model can generate a crisis with similar characteristics

to the data: a substantial fall in output and short-term interest rate until it reaches the zero

lower bound, together with a rise in the debt-to-GDP ratio and a fall in the inﬂation rate. In

this context, I simulate a Quantitative Easing program that increases the ratio of the central bank

balance sheet over output by 10p.p. and compare the macroeconomic dynamics with and without

this program. Then, I study three diﬀerent strategies for managing the central bank balance sheet

size: 1) maintaining the enlarged size after the crisis (Tapering), 2) decreasing the balance sheet

size: a) by letting the bonds that mature, run oﬀ the balance sheet (Quantitative Tightening (QT)),

and b) at a faster speed by selling bonds (Aggressive QT ).

Tapering is deﬁned as a situation where the stock of long-term bonds at the central bank (b

L,CB

)

remains constant, implying b

L,CB

= b

L,CB

t−1

. With that objective, the central bank engages in new

purchases of bonds to make up for maturing bonds. The size of the central bank balance sheet over

output ratio is, however, endogenous since it is aﬀected by the price of long-term bonds (Q

) and

output y

, which are not under the control of the central bank.

Quantitative tightening is a strategy where the central bank does not repurchase the bonds that

mature every period, letting them run oﬀ the balance sheet. The stock of bonds at the central

bank will then decrease at rate δ, the average maturity of long-term bonds, the unique kind of

bonds the monetary authority purchases in the model. This implies the stock of bonds follows:

L,CB

= (1 − δ) b

L,CB

t−1

Finally, the strategy called Aggressive QT consists of the central bank actively selling public

bonds. It is any path of shocks that generates b

L,CB

< (1 − δ) b

L,CB

t−1

, implying that the stock of

bonds at the central bank decreases at a faster rate than the maturity rate.

7.1 The crisis development and the QE program

I simulate the model in 50,000 samples of 40 periods under two scenarios, “Baseline” and “Quantita-

tive Easing,” with the following characteristics. In the baseline, the economy is at the approximation

point at t=1, and in the monetary-led regime. During periods 2 to 4, the economy is hit by strong

negative preference and TFP shocks. Since both shocks are persistent, they remain below their

steady state value for the whole sample. From t=2 onward, the regime at place is stochastic, follow-

ing the transition matrix presented in section 3.8. This scenario makes QE, monetary policy, and

ﬁscal policy shock random. After the initial hit during periods 2 to 4, preference and TFP shocks

follow random paths.

The quantitative easing scenario shares the same characteristics as the baseline, except that the

path for QE shocks is not random. It is imposed to generate an increase in the annualized central

bank balance sheet to 20% of GDP. in the ﬁrst six periods. After period 6, the balance sheet remains

constant (tapering strategy).

Figures 7 and 8 show the mean of simulated variables for the ﬁrst 20 periods for the economy

without QE (continuous blue line) and with QE (dotted orange line). The negative preference and

technology shocks generate a substantial fall in output growth that reaches its bottom of -8.8p.p. in

period four. Besides, it does not fully recover for eight periods. The short-term interest rate starts

falling immediately until it reaches the ZLB. The primary ﬁscal surplus falls by almost 6p.p. in the

baseline scenario.

This exercise aims to analyze the eﬀects of the quantitative easing program. This baseline exercise

excludes the increase in ﬁscal transfers that took place during that period of turmoil. In the model, it can

be easily generated through a ﬁscal shock in g

, generating more signiﬁcant primary ﬁscal deﬁcits.

The main features of the crisis with quantitative easing are the following. First, there is a

slightly milder fall in output than without the program. Second, there is a faster recovery within

less than a year. QE also has expansionary eﬀects in terms of inﬂation. Prices fall less during the

crisis than in a scenario without the intervention of the central bank, however, this is at the cost of

more important inﬂationary dynamics in the recovery. Inﬂation reaches a peak of 8.4% in period

nine, two periods after the balance sheet reaches its maximum value of 20.8% of GDP. Without this

program, inﬂation increases less during the recovery and reaches a peak of 6.9% three periods later.

5 10 15 20

quarters

0.7

0.8

0.9

Pref shock

5 10 15 20

quarters

0.94

0.96

0.98

TFP shock

5 10 15 20

quarters

0.015

0.02

0.025

0.03

0.035

Treasuries in CB

5 10 15 20

quarters

CB Bal. sheet/GDP

5 10 15 20

quarters

-8

-6

-4

-2

Output growth

5 10 15 20

quarters

-5

Inflation ( )

5 10 15 20

quarters

Interest rate (R)

5 10 15 20

quarters

Long-run yield (R

)

5 10 15 20

quarters

-6

-4

-2

P. fiscal surplus/GDP

5 10 15 20

quarters

Debt/GDP

Crisis Crisis + QE

Figure 7: Simulated crisis

Note: Average from 50,000 samples. The continuous blue line is the scenario without the QE

program. The Orange dotted line is the scenario with a QE program that increases the

annualized central bank balance sheet to output by around 13p.p. Interest rate (R), long-run

return (R

), and inﬂation are annualized gross rates. Debt-to-GDP is annualized.

In both cases, real debt increases quickly when the crisis starts, increasing almost 20p.p. of GDP.

However, its real value decreases fast following the output and price recovery. The fall in real debt

is more signiﬁcant under the QE program for three reasons: ﬁrst, central bank purchases increase

central bank proﬁts transferred to the treasury; second, the central bank intervention decreases

the long-run yield, and thus interest payments on debt are lower; and third, the faster recovery

triggers a fall in government spending driven by automatic stabilizers so that the primary ﬁscal

deﬁcit is lower. These three factors allow the government to issue less debt during the crisis than

in a counterfactual without the QE program.

5 10 15 20

quarters

0.5

1.5

CB profits/GDP (

)

5 10 15 20

quarters

Interest payments/GDP

5 10 15 20

quarters

Debt/GDP

5 10 15 20

quarters

0.07

0.075

0.08

0.085

5 10 15 20

quarters

0.15

0.16

0.17

0.18

5 10 15 20

quarters

0.05

0.06

0.07

New debt issuance

5 10 15 20

quarters

0.22

0.225

0.23

0.235

Taxes ( )

5 10 15 20

quarters

-6

-4

-2

P. fiscal surplus/GDP

Crisis Crisis + QE

Figure 8: Simulated crisis

Note: Average from 50,000 samples. The continuous blue line is the scenario without the QE

program. The Orange dotted line is the scenario with a QE program that increases the

annualized central bank balance sheet to output by around 10p.p.

Finally, ﬁgure 9 shows, per period, the percentage of samples in which the economy is at the

ZLB regime. Notice it is zero in periods one (initial condition) and two. The reason is that the

endogenous probability of moving to the zero lower bound regime at t + 1 depends on the interest

rate at t. From period three onward, the economy is at the ZLB with a high probability. The

frequency of this regime is considerably lower under the scenario with the QE program, and it is

almost nil from period twelve.

2 4 6 8 10 12 14 16 18 20

Quarter

Crisis Crisis + QE

Figure 9: Frequency ZLB regime

Note: Percentage of simulated samples at the ZLB regime, per period, from 50,000 samples,

in the scenario without the QE (continuous blue line), and with QE program (orange dotted

line).

To summarize, unconventional monetary policy intervention reduces the severity of the crisis

and implies a faster recovery and a shorter duration of the ZLB regime. Additionally, the debt stock

is lower while inﬂation is higher.

7.2 Central bank balance sheet exit strategies

This section compares the path for endogenous variables under three scenarios after the initial crisis.

The continuous green line is the scenario with Quantitative Tightening (QT) from period 9 onward,

where the central bank allows mature treasuries to run oﬀ its balance sheet. Light-blue dashed line

is the scenario with sales of bonds from period 9 onward.

2728

The orange dotted line corresponds

to Tapering, identical to the one in ﬁgure 7. Figure 10 shows the evolution of the treasuries in the

central bank (b

L,CB

) and the central bank balance sheet (Q

L,CB

) as a GDP ratio.

10 20 30 40

quarters

0.7

0.75

0.8

0.85

0.9

0.95

Pref shock

10 20 30 40

quarters

0.94

0.95

0.96

0.97

0.98

0.99

TFP shock

10 20 30 40

quarters

0.015

0.02

0.025

0.03

0.035

Treasuries in CB

10 20 30 40

quarters

CB Balance sheet/y(%)

Tapering QT Aggressive QT

Figure 10: Crisis and exit strategies from QE.

Note: Average from 50,000 samples, plots since period t=8. The Orange dotted line is the

tapering scenario. The continuous green line is the scenario with Quantitative Tightening

(QT) from period 9 onward. Light-blue dashed line is the scenario with Aggressive QT or

sales of bonds from period 9 onward.

The ﬁgure shows mean values for each variable across samples. This implies an average behavior

where the regime is stochastic, i.e., which could leave the ZLB regime towards a ﬁscally-led regime

or a monetary-led regime following the ergodic probabilities of matrix presented in section 3.8.

For clarity of exposition, I show the variables from period t=8. The unwinding of the balance

sheet is assumed to start almost two years after the crisis when most of the simulated samples are

endogenously out of the ZLB (as shown in ﬁgure 4).

When the central bank unwinds its balance sheet, it decreases the total demand for long-term

bonds, pushing their price down, and generating a rise in the long-term yield. This price eﬀect causes

incentives in households to rebalance its portfolio away from deposits and toward long-term public

bonds. At the same time, since households were already investors in these assets, they perceive a

For this scenario, I assume a QE shock equal to -0.2 for every t from t=9.

The policies denominated “QT” and “Aggressive QT” in this paper have been classiﬁed as “Passive QT”

and “Active QT” by the Bank of England. See, for instance, the speech of Andrew Hauser at the ECB’s 2022

Conference on Money Markets, available at: www.bis.org/review/r221104k.pdf.

The main results are robust to start the unwinding of the central bank balance sheet some periods after

(period t = 12 or t = 15). These exercises are available under request.

negative wealth eﬀect from the revaluation of their portfolio, decreasing consumption and aggregate

demand in the economy, to a situation without the QT. This eﬀect complements the substitution

eﬀect that generates incentives to save more in the context of rising interest rates.

10 20 30 40

quarters

0.5

1.5

Output growth

10 20 30 40

quarters

0.72

0.74

0.76

0.78

Consumption

10 20 30 40

quarters

Inflation ( )

10 20 30 40

quarters

Interest rate (R)

10 20 30 40

quarters

Long-run return (R

)

10 20 30 40

quarters

Debt/GDP

10 20 30 40

quarters

-2

P.Fiscal Surplus/GDP

10 20 30 40

quarters

0.22

0.225

0.23

0.235

0.24

Taxes

10 20 30 40

quarters

0.5

1.5

CB profits/GDP

10 20 30 40

quarters

0.135

0.14

0.145

0.15

Interest payments

Tapering QT Aggressive QT

Figure 11: Crisis and exit strategies from QE.

Note: Average from 50,000 samples, since period t=8. The Orange dotted line is the tapering

scenario. The continuous green line is the scenario with Quantitative Tightening (QT) from

period 9 onward. Light-blue dashed line is the scenario with Aggressive QT or sales of bonds

from period 9 onward. Output growth, balance sheet, and Debt to GDP are in percentages.

Interest rate (R), long-run return (R

), inﬂation, and debt-to-GDP are annualized.

Firms respond to this fall in aggregate demand through a combination of a fall in production

and prices. In this context, unwinding the central bank balance sheet generates a fall in output

growth and faster inﬂation stabilization, allowing the central bank to stop increasing the short-term

nominal interest rate. The disinﬂationary forces of reducing the central bank balance sheet are more

substantial as the unwinding speed gets faster. Under an aggressive QT, after four years of applying

this strategy (period t=27) and in the absence of further shocks, the inﬂation rate is 3.3%, while

it remains at 4.1% under Tapering. Moreover, this is accomplished with a short-term interest rate

of 1p.p. lower than in the tapering scenario. In a longer horizon (40 periods from the beginning

of the simulation), we can see that the inﬂation rate remains at a higher level when the size of the

balance sheet remains constant after the expansion during the crisis. In other words, more liquidity

permanently circulating in the economy increases the long-term inﬂation rate.

On the ﬁscal side, we observe that the debt stock increases with the balance sheet unwinding.

There are three reasons. First, the negative eﬀects on output trigger government spending through

automatic stabilizers, decreasing the ﬁscal surplus even as taxes increase. Second, interest payments

increase due to higher interest rates. Third, central bank remittances to the treasury decrease in the

scenario of sales of bonds. Notice that when the short-term interest rate increases, the central bank

starts paying interest rates on reserves, decreasing its proﬁts. With the unwinding, bank reserves

which constitute the liabilities in its balance sheet, decline, reducing this cost. However, at the

same time, it generates a fall in the return it receives for its holdings of long-term bonds. Under

QT, the ﬁrst eﬀect dominates, and central bank proﬁts increase compared to the tapering scenario.

The opposite happens in an aggressive QT, where the unwinding is faster.

In conclusion, selling bonds from the central bank eﬀectively reduce the inﬂation rate to the

target, requiring smaller increases in the short-term interest rate. However, it has the downside

of a slower deleveraging of the economy. As I show in the next section, these conclusions are not

independent of the policy regime at the time of the unwinding.

7.3 Unwinding the balance sheet in a ﬁscally-led regime

In this section, I restrict the analysis to the samples that exit the ZLB toward a ﬁscally-led regime

and stay there for at least one year (4 quarters). I compare it with the selected samples where the

exit from the ZLB is toward the monetary regime. Since both ﬁscally and monetary-led regimes are

substantially persistent, the probability of staying there for a good proportion of the sample is high.

The exit of the crisis regime is stochastic and could happen at any period. I present the dynamics

corresponding to these subsamples in ﬁgures 12 and 13.

10 20 30 40

quarters

Interest rate (R)

10 20 30 40

quarters

Inflation ( )

10 20 30 40

quarters

Debt/GDP

10 20 30 40

quarters

-2

P. fiscal surplus/GDP

10 20 30 40

quarters

0.2

0.4

Output growth

10 20 30 40

quarters

Long-run return (R

)

Tapering QT Aggressive QT

Figure 12: Quantitative Tightening in the monetary-led regime.

Note: Simulation of 50,000 samples, since period t=8. The ﬁgure restricts the analysis to

samples that exit the zero lower bound toward a monetary-led regime and stay there for at

least one year (4 quarters). The Orange dotted line is the tapering scenario, the same as the

one in ﬁgure 7. The continuous green line is the scenario with Quantitative Tightening (QT)

from period 9 onward. Light-blue dashed line is the scenario with Aggressive QT or sales of

bonds from period 9 onward. Debt to GDP is in percentages. Interest rate (R), long-run

return (R

), inﬂation, and debt-to-GDP are annualized.

Exiting the crisis towards a ﬁscally-led regime brings a higher inﬂation rate that peaks at 11% in

period nine and an expansionary period where output growth is above trend. As a result, interest

rates are higher, even under lower reactions to inﬂation and output deviations. Higher inﬂation

rates help to quickly reduce the stock of real debt, which falls 20p.p. of GDP in 7 quarters.

The most remarkable diﬀerence between regimes is that QT in a ﬁscally-led regime has little

impact on inﬂation, while it has a sizable disinﬂationary impact in a monetary-led regime. In a

monetary-led regime, the inﬂation rate in scenarios with QT is persistently below the one in the

case with tapering. This diﬀerence tops at 1.5p.p. Notice that this is attained with lower increases

in the short-term interest rate. In the case of aggressive QT, the conventional monetary policy tool

changes the stance from contractionary to expansionary 10 quarters before the case of constant size

in the central bank balance sheet. Figure 14 shows the accumulated inﬂation since period t = 8

in both regimes under the alternative scenarios. In the monetary-led regime, the aggressive QT

reduces accumulated inﬂation by 34% at quarter 40.

10 20 30 40

quarters

Interest rate (R)

10 20 30 40

quarters

Inflation ( )

10 20 30 40

quarters

Debt/GDP

10 20 30 40

quarters

-2

-1

P. fiscal surplus/GDP

10 20 30 40

quarters

Output growth

10 20 30 40

quarters

Long-run return (R

)

Tapering QT Aggressive QT

Figure 13: Quantitative Tightening in the ﬁscally-led regime.

Note: Simulation of 50,000 samples, since period t=8. The ﬁgure restricts the analysis to

samples that exit the zero lower bound toward a ﬁscally-led regime and stay there for at least

one year (4 quarters). The Orange dotted line is the tapering scenario, the same as the one in

ﬁgure 7. The continuous green line is the scenario with Quantitative Tightening (QT) from

period 9 onward. Light-blue dashed line is the scenario with Aggressive QT or sales of bonds

from period 9 onward. Debt to GDP is in percentages. Interest rate (R), long-run return

), inﬂation, and debt-to-GDP are annualized.

The diﬀerence in inﬂationary dynamics after QT programs is the following. As shown in the

average scenario, real debt increases with QT. In the monetary-led regime, the increase in debt

causes a rise in taxes, given by the ﬁscal rule. This exacerbates the fall in aggregate demand and

the disinﬂationary eﬀects of QT. In the ﬁscally-led regime, the rise in public debt increases actual

and expected inﬂation, aggravating the rise in yield spreads and mitigating the disinﬂationary eﬀects

of QT. In this case, the negative demand eﬀect driven by the reduction in central bank purchases

is not enough to counteract the stimulative impact of negative interest rates and ﬁscal stimulus.

10 15 20 25 30 35 40

quarters

Price level

Monetary regime

10 15 20 25 30 35 40

quarters

Fiscal regime

Figure 14: Accumulated inﬂation under diﬀerent regimes and exit strategies from QE.

Note: Simulation of 50,000 samples, since period t=6. The ﬁgure restricts the analysis to

samples that exit the zero lower bound toward a monetary (left) or ﬁscally-led regime (right)

and stay there for at least one year (4 quarters). The continuous green line is the scenario

with Quantitative Tightening (QT) from period 9 onward. Light-blue dashed line is the

scenario with Aggressive QT or sales of bonds from period 9 onward. This ﬁgure shows the

CPI in the monetary vs. ﬁscal regime, with base P

= 1.

In conclusion, in a regime where the ﬁscal authority is not committed to stabilizing the govern-

ment debt, unwinding the central bank balance sheet does not generate clear economic advantages.

8 Concluding Remarks

I study diﬀerent exit strategies for reducing the central bank balance sheet in a model that generates

ﬁscal-monetary policy trade-oﬀs.

The impact of central bank balance sheet policies (QE or QT) depends on the ﬁscal-monetary

policy mix. First, QE reduces the need for new debt issuance. Second, as a counterpart of the central

bank’s balance sheet expansions, liquidity increases substantially, both in the hand of commercial

banks (reserves) and non-bank private agents (deposits). The extent to which the increase in

liquidity aﬀect output and inﬂation rate depends on the ﬁscal-monetary policy mix. QE is more

expansive at the ZLB and under the ﬁscally-led regime when the inﬂationary impact is also more

signiﬁcant.

How the central bank balance sheet’s size is reduced matters for inﬂation, output, and debt

dynamics. In an average simulation, when the monetary authority starts shrinking its holdings of

long-term government bonds, it decreases inﬂation at the cost of an increase in the ratio of debt to

GDP. These eﬀects increase when the exit strategy is more aggressive.

When the unwinding occurs in a ﬁscally-led regime, reducing the balance sheet’s size does not

help bring the inﬂation back to target. In this regime, the rise in public debt increases actual and

expected inﬂation, exacerbating the rise in yield spreads and mitigating the disinﬂationary eﬀects

of QT. This implies that, with unconventional policies, coordination between ﬁscal and monetary

authorities is necessary to stabilize the inﬂation rate.

9 Appendix

9.1 Extra plots for US case

Figure 15: Change in Treasuries’ holdings

Source: US Financial Accounts. Data in billions of dollars. Contain revaluation eﬀects.

Figure 16: Public debt to GDP in US: historical perspective

Source: FRED. Annual Gross Federal Debt as a Percent of GDP.

Figure 17: Macroeconomic variables: public debt, liquidity, inﬂation. Great recession and

COVID-19 crisis.

US Federal Debt, Reserves and Deposits, in trillions of dollars. Core inﬂation in %. Source

debt, deposits and inﬂation data: FRED. Source reserves: US ﬁnancial accounts, release

December 2021.

9.2 Model: Financial intermediary optimization problem

The optimization problem of a representative ﬁnancial intermediary can be written in sequential

form as follows:





= max

S,I

− A

+ E



t,t+1



t+1



s.t. (1 − τ

+ A

− Φ

) + Q

= Q

S,I

= B

S,I

t−1

− D

t−1

≤ ζB

S,I

Deﬁne η

as the Balance sheet multiplier and µ

as the Leverage constraint multiplier. The ﬁrst

order conditions are given by the following system:

: − 1 + η

− η

= 0

: η

− µ

− E

t,t+1



t+1



= 0

S,I

: − η

+ µ

ζ + E

t,t+1



t+1



= 0

And the envelope condition:



t+1



= τ

+ η

(1 − τ

)

Using the ﬁrst order condition with respect to A

to substitute out the multiplier η

, we obtain

the system of equations that characterize the ﬁnancial intermediary’s optimization problem, together

with 7, 8, 9, ??:

= E

t,t+1

+ µ

(1 − χa

) (22)

= E

t,t+1

+ ζµ

(1 − χa

) (23)

Where

t,t+1

is the stochastic discount factor for ﬁnancial intermediaries, deﬁned as:

t,t+1

≡ M

t,t+1



1 − χ



1 − τ

1 − χ

t+1

9.3 Equilibrium conditions

In this section I present the system of equilibrium conditions with functional forms and in real

terms.

Deﬁnitions: d

, b

j,i

, w

, div

Div

, π

, a

, mc

, w

t−1

, for j ∈ {L, S}, i ∈ {H, CB, F I}.

9.3.1 Households

+ Q

+ b

L,H

= w

− τ

t−1

L,H

t−1



κ + (1 − δ)Q



+ div

+ π

(1)

γn

1−φ





−σ

(1 − φ)c

−φ

)

= w

(2)

= E

t,t+1

φc

(1 − φ)d

(3)

+ ϕ

L,H

)

= E

t,t+1



κ + (1 − δ)Q

t+1



(4)

t,t+1

βλ

t+1

= βE

t+1

1−φ

t+1



t+1



−σ

−φ

t+1

)

1−φ





−σ

−φ

)

t+1

(5)

9.3.2 Firms

= z

(6)

= y

− w

−

(π

− 1)

(7)

1 − ε + εmc

= ϕ

(π

− π

∗

) π

− ϕ



t,t+1

t+1

(π

t+1

− π

∗

)



(8)

= mc

(9)

9.3.3 Financial intermediaries

div

= τ

− a

(10)

(1 − τ

+ a

−

+ Q

= Q

S,I

(11)

S,I

t−1

−

t−1

(12)

≤ ζb

S,I

(13)

= E

t,t+1

+ µ

(1 − χa

) (14)

= E

t,t+1

+ ζµ

(1 − χa

) (15)

t,t+1

= M

t,t+1

(1 − χa

)



1 − τ

1 − χa

t+1



(16)

9.3.4 Government

= Q

(17)

S,CB

+ Q

L,CB

= 0 (18)

= Q

+ Q

(19)

= (1 − ¯µ)/¯µb

(20)

= θ(y

∗

− y

) + (1 − ρ

)¯g + ρ

t−1

+ σ

, ε

∼ N(0, 1) (21)

9.3.5 Market clearing conditions

+ g

(π

− π

∗

)

= y

(22)

= d

(23)

= b

S,I

+ b

S,CB

(24)

= b

L,H

+ b

L,CB

(25)

9.3.6 Policy rules

− τ

∗

= ρ

(ξ

) (τ

t−1

− τ

∗

) + (1 − ρ

(ξ

)) γ (ξ

) (b

t−1

− b

∗

) (16)

R (ξ

)



t−1

R (ξ

)



(ξ

)



∗



(ξ

)



∗



(ξ

)

1−α

(ξ

)

(ξ

)ϵ

(27)

L,CB

= (1 − ρ

L,CB

∗

+ ρ

L,CB

t−1

+ σ

(28)

Law of motion for exogenous processes: z

, ν

Transition matrix for Markov-switching shock ξ

9.3.7 Deﬁnition of auxiliary variables used in ﬁgures and tables

Return on long-term bond:

t,t+1

= E

κ + (1 − δ)Q

t+1

Convenience yield:

= R

− E

t,t+1

Annual debt to GDP ratio:

Primary ﬁscal surplus:

= τ

− g

Total interest rate payments:

t−1

− 1)

t−1

+ (κ − δ)

t−1

Private interest rate payments:

t−1

− 1)

S,F I

t−1

+ (κ − δ)

L,H

t−1

Total debt service:

t−1

+ κ

t−1

9.4 Steady state and approximation point

At the steady state, shocks are equal to zero: ϵ

= 0, ϵ

= 0.

Then: z

= 1, ν

= 1, g

= ¯g, b

L,CB,ss

= b

L,CB

∗

There is one endogenous variable which steady state value diﬀers among regimes. For monetary

(ξ

= 0 and ξ

= M) or ﬁscal (ξ

= 0 and ξ

= F ) dominance regimes, we have: R

= R

∗

absence of shocks. For the Zero lower bound regime (ξ

= 1), R

= 1.0005. Deﬁne p

, p

as the ergodic probability of monetary dominance, ﬁscal dominance or crisis regime, respectively.

Where: p

+ p

= 0. The short-term nominal interest rate at the approximation point is

obtained as follows:



+ p



∗



1 − p

− p



1.0005

9.5 Additional results

9.5.1 Determinacy regions for ﬁscal and monetary policy rules

The following ﬁgure shows combinations for policy parameters γ and α

that give rise to a unique

equilibrium (determinacy, yellow areas), given the calibration for the rest of the parameters, in the

monetary-led regime. The blue areas represent parameter combinations that generate either no

equilibrium, or multiple equilibria.

Figure 18: Determinacy regions

Note: Yellow areas represent combinations of parameters α

and γ that give rise to determinacy.

Blue areas represent either multiple equilibria or no equilibrium.

9.5.2 Impulse response functions for additional shocks

This section presents ﬁgures showing the impact of the diﬀerent shocks in the model, under diﬀerent

regimes. They show log deviations (in %) in a simulated path with a one standard deviation shock,

to the counterfactual path without the shock.

Figure 19: Impact of a Quantitative Easing shock conditional on a regime

Note: log deviations (in %) in a simulated path with a one standard deviation shock in the

government spending (ϵ

= 1) to the counterfactual path without shock (ϵ

= 0). Path

conditional on the monetary-led regime (continue blue line), ﬁscally-led regime (dotted red

line), and zero lower bound regime (dashed yellow line) for 16 periods.

Figure 20: Impact of a Quantitative Easing shock conditional on a regime

Note: log deviations (in %) in a simulated path with a one standard deviation shock in the

government spending (ϵ

= 1) to the counterfactual path without shock (ϵ

= 0). Path

conditional on the monetary-led regime (continue blue line), ﬁscally-led regime (dotted red

line), and zero lower bound regime (dashed yellow line) for 16 periods.

Figure 21: Impact of a Quantitative Easing shock conditional on a regime

Note: log deviations (in %) in a simulated path with a one standard deviation shock in the

preferences (ϵ

= 1) to the counterfactual path without shock (ϵ

= 0). Path conditional on

the monetary-led regime (continue blue line), ﬁscally-led regime (dotted red line), and zero

lower bound regime (dashed yellow line) for 16 periods.

Figure 22: Impact of a Quantitative Easing shock conditional on a regime

Note: log deviations (in %) in a simulated path with a one standard deviation shock in the

preferences (ϵ

= 1) to the counterfactual path without shock (ϵ

= 0). Path conditional on

the monetary-led regime (continue blue line), ﬁscally-led regime (dotted red line), and zero

lower bound regime (dashed yellow line) for 16 periods.

Figure 23: Impact of a Quantitative Easing shock conditional on a regime

Note: log deviations (in %) in a simulated path with a one standard deviation monetary

shock (ϵ

= 1) to the counterfactual path without shock (ϵ

= 0). Path conditional on the

monetary-led regime (continue blue line), ﬁscally-led regime (dotted red line), and zero lower

bound regime (dashed yellow line) for 16 periods.

Figure 24: Impact of a Quantitative Easing shock conditional on a regime

Note: log deviations (in %) in a simulated path with a one standard deviation monetary

shock (ϵ

= 1) to the counterfactual path without shock (ϵ

= 0). Path conditional on the

monetary-led regime (continue blue line), ﬁscally-led regime (dotted red line), and zero lower

bound regime (dashed yellow line) for 16 periods.

Figure 25: Impact of a Quantitative Easing shock conditional on a regime

Note: log deviations (in %) in a simulated path with a one standard deviation TFP shock

(ϵ

= 1) to the counterfactual path without shock (ϵ

= 0). Path conditional on the

monetary-led regime (continue blue line), ﬁscally-led regime (dotted red line), and zero lower

bound regime (dashed yellow line) for 16 periods.

Figure 26: Impact of a Quantitative Easing shock conditional on a regime

Note: log deviations (in %) in a simulated path with a one standard deviation TFP shock

(ϵ

= 1) to the counterfactual path without shock (ϵ

= 0). Path conditional on the

monetary-led regime (continue blue line), ﬁscally-led regime (dotted red line), and zero lower

bound regime (dashed yellow line) for 16 periods.

9.6 Data sources

The following table presents data sources for plots in section 2 and calibration.

Variable Source Table

Deposits in depositary institutions FRED

Monetary aggregate M1 FRED

CPI FRED

US GDP Implicit Price Deﬂator, Index 2015=100 FRED

Federal reserve Assets US Financial Accounts L109

Federal Reserve Total treasuries US Financial Accounts L110

Federal Reserve Treasury bills US Financial Accounts L111

Federal Reserve Other treasuries US Financial Accounts L112

Federal Reserve Total liabilities US Financial Accounts L113

Federal Reserve Reserves US Financial Accounts L114

Checkable Accounts in Federal Reserve US Financial Accounts L115

Total public debt FRED

Eﬀective federal funds rate FRED

10-year Yield FRED

1-year Yield FRED

US GDP BEA NIPA

Private consumption BEA NIPA

Government spending BEA NIPA

US Population FRED

Table 6: Data sources

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