QUANTITATIVE EASING∗

Wei Cui† and Vincent Sterk†‡

†University College London and CfM‡CEPR

June 14, 2019

Abstract

Is Quantitative Easing (QE) an effective substitute for conventional monetary pol-

icy? We study this question using a heterogeneous-agents model with nominal rigidities,

as well as liquid and partially liquid wealth. The direct effect of QE is determined

by the difference in marginal propensities to consume out of the two types of wealth,

which is large according to empirical studies. QE therefore emerges as a powerful policy

instrument to anchor expectations and stabilize output and inflation. Indeed, the esti-

mated model reveals that QE interventions greatly dampened the U.S. Great Recession.

However, we also find that QE may have strong side effects on inequality and welfare,

suggesting that conventional policy should remain the primary stabilization instrument

in the central bank’s toolkit.

JEL: E21, E30, E50, E58

Key words: Monetary Policy, Large-scale Asset Purchases, Estimated HANK model

∗Email: w.cui@ucl.ac.uk and v.sterk@ucl.ac.uk. We thank Klaus Adam, Marios Angeletos, RichardHarrison, Greg Kaplan, Benjamin Moll, our discussants Mei Dong, Carlos Thomas, and Gianluca Violante,as well as seminar and conference participants at the Deutsche Bundesbank, the Bank of England, Banco dePortugal, EIEF, University College London, University of Durham, University of Essex, University of Bristol,King’s College London, Copenhagen Business School, the 2018 HKUST-Jinan Conference on Macroeconomics,the 2018 Quantitative Macroeconomic Workshop at the Reserve Bank of Australia, AMCM conference inLillehammer, the 2019 EABCN conference in Warsaw, and the 2019 NYFED - HKMA joint conference forhelpful comments. We thank Jihyun Bae and Carlo Galli for excellent research assistance. An earlier versionof this paper is available as CEPR discussion paper 13322, and CfM discussion paper, CFM-DP2018-30,November 2018.

1 Introduction

It has been over ten years since the U.S. Federal Reserve (Fed) initiated a colossal expansion of

its balance sheet; the largest since the Great Depression. The 2008 financial crisis compelled

the Fed to start providing loans to the banking sector, which was suffering from a freeze of

interbank lending. However, as banks recovered from the crisis the Fed did not shrink its

balance sheet but instead expanded it further, buying up assets such as long-term government

debt in large quantities. This was done in a bid to stimulate aggregate demand, which slumped

during the Great Recession. Known as Quantitative Easing (QE), these interventions acted

as a placeholder for conventional monetary policy, which had become powerless as the policy

rate had hit the zero lower bound. Similar interventions took place in the UK and the Euro

Area, as well as in Japan during the early 2000s.

While conducting QE, central banks received little guidance from economic theory, as this

type of policy is completely ineffective in modern textbook models such as the standard New

Keynesian (NK) model.1 Nevertheless, central bankers have carried on with QE, presumably

believing that it is a useful instrument to manage aggregate demand. A decade into the

balance sheet expansion, it is still not well understood when to use QE, how aggressively to

use it, and when to roll it back. This leaves central banks in a precarious position in the face

of upcoming recessions, when the limits of conventional monetary policy might once again be

reached.

This paper presents a quantitative NK model to provide a better understanding of how

QE affects the macroeconomy, in a world with inequality. To this end, we extend the model

to allow for household heterogeneity and assets with different degrees of liquidity, following

a recent literature, cf. Kaplan, Moll and Violante (2017). Households make portfolio choices

between deposits, which offer low returns but are fully liquid, and mutual fund shares, which

generate higher returns but are subject to a withdrawal tax. In this setting, QE interven-

tions can have powerful effects on aggregate demand, but they may also create strong side

effects which exacerbate inequality. Because of the welfare costs associated with inequality,

conventional policy still emerges as the preferred stabilization instrument, provided that it is

available.

In the model, QE stimulates household spending by increasing the amount of liquidity

circulating in the economy. When conducting QE, the central bank buys long-term government

debt from mutual funds, which triggers the creation of additional deposits.2,3 Mutual funds,

1See for instance Woodford (2012).2Deposit creation is necessarily triggered because mutual funds –unlike banks– cannot directly hold reserves

at the central bank. In the US, only a small fraction of the QE assets were purchased from banks, see Section2 for more discussion and for empirical evidence.

3Aside from long-term government debt, the Fed also purchased mortgage securities. We do not explicitly

1

Figure 1: Reserves and deposits in the United States.

1990 1995 2000 2005 2010 20150

500

1000

1500

2000

2500

3000

Bill

ions

of D

olla

rsReserve Balances with Federal Reserve BanksTotal Checkable DepositsHouseholds and Nonprofit Organizations; Checkable Deposits and Currency

QE3QE2QE1

Source: Federal Reserve Board, Flow of Funds accounts. Grey areas denote rounds of QEpurchases by the Federal Reserve.

however, derive little value from liquidity and immediately trade the deposits for new long-

term government debt, which offers a higher yield. As a result, the liquidity flows out of the

mutual fund sector and ends up in the hands of households via various channels, who value

liquidity for self-insurance reasons.4 Because households have high Marginal Propensities to

Consume (MPCs) out of liquid wealth, the additional liquidity boosts aggregate spending.

Simultaneously, labor income increases, which enables households to finance both additional

consumption expenditures and additional liquid asset holdings.5

Before describing the model, we present a simple formula which captures the essence of the

QE transmission mechanism and which can be used for back-of-the-envelope calculations. The

key insight conveyed by this formula is that the direct effect of QE depends on the difference

between the MPCs out of deposits and less liquid sources of wealth. Empirical estimates in

the literature suggest that the gap between these two MPCs is large. An increase in household

liquidity may therefore boost aggregate demand substantially.

Figure 1 shows the evolution of reserves at the Fed, the aggregate amount of checkable

model such purchases, but they would create a similar transmission mechanism in the model. Fieldhouse,Mertens and Ravn (2018) provide empirical evidence that mortgage purchases by Government SponsoredEnterprises have expansionary macro effects.

4In the model, liquidity might flow to households via different channels, but we show that these channelsgenerate identical or similar effects. Our baseline model aligns with empirical evidence which suggests thatliquidity reached households primarily via government spending and/or transfers, as will be discussed inSection 2.1.

5This is also facilitated by a decline in taxes, as an increase in inflation reduces the real value of governmentdebt, which improves the government’s financial position (at the expense of the mutual funds).

2

deposits, and the amount of deposits/currency held by households and non-profits. As large-

scale asset purchases began, all three series increased sharply. This strongly suggests that QE

triggered the creation of additional deposits (through bank lending in practice), which in large

part ended up being held by households. In Section 2, we will discuss this empirical evidence

and the underlying channels in more detail. In Appendix A, we further draw a comparison

between the Flow of Funds data shown in Figure 1 and data from the Survey of Consumer

Finances, both of which show a similar pattern since 2009.

Sections 3 and 4 present the full quantitative model and its parametrization. We evaluate

the model’s implications for consumption at the micro level and show that it generates a large

gap in MPCs out of liquid and less liquid wealth, in line with empirical studies. A subset of

the parameters is estimated by Maximum Likelihood, using the data on household deposits

shown in Figure 1, as well as other macroeconomic time series.

A main finding of our study is that QE has strong stimulative macro effects. In particular,

QE had a very large and positive impact on U.S. output and inflation during the Great

Recession, preventing a much deeper downturn. This result follows from a counterfactual

simulation in which we shut down QE interventions. This exercise also reveals that the effects

of QE during its first round were stronger than during the second and the third round.

Section 5 proceeds to evaluate the relative efficacy of QE, compared to conventional policy,

as a stabilization instrument. As in much of the NK literature, we assume that monetary policy

follows a rule.6 In our case, this means that there is either a rule for interest rate policy or a

rule for QE, each depending on output and inflation.We compare the two policies along three

dimensions.

First, we consider their ability to anchor expectations, i.e. to rule out fluctuations driven

purely by changes in beliefs about the future. As is well known in the literature, conventional

policy does so only when the interest rate rule satisfies the “Taylor principle,” meaning that

the nominal interest rate responds strongly enough to changes in inflation (and output). We

find that, likewise, the QE rule is not always successful in anchoring expectations. Nonetheless,

it succeeds under a very wide range of realistically achievable values of the policy coefficients.

For example, expectations remain anchored under a special case of the QE rule in which the

level of real reserves is held completely constant, a policy which we refer to as Real Reserve

Targeting (RRT).

6We set aside the question of what are the optimal long-run levels of inflation, the nominal interest rates,the optimal long-run size and composition of the central bank balance sheet, and the optimal fiscal response tochanges in monetary policy. Instead, our objective is to evaluate the efficacy of interest rate policy and QE asinstruments for stabilization of the business cycle. Accordingly, we do not consider Ramsey optimal policy as,for instance, in Nuno and Thomas (2017). That said, we do consider the welfare effects of different stabilizationrules and in Section 5.3 we also consider the welfare implications of a one-time permanent expansion of thecentral bank’s balance sheet.

3

Second, we study the relative performance of QE in managing business cycles, i.e. in stabi-

lizing aggregate output and inflation. In order to draw a fair comparison between conventional

policy and QE, we give both policies the best possible chance in achieving the stabilization

objective. This is done by evaluating both policies under the optimal policy coefficients, given

the specified rule and objective function. We consider different types of shocks and different

weights on output versus inflation volatility, and then search numerically for the coefficients

which optimize the central bank’s objective.

We find that, under a wide range of configurations, QE is actually more effective in sta-

bilizing output and inflation than interest rate policy. This happens as QE tends to create a

positive co-movement between output and inflation, due to aggregate demand effects. This

co-movement in turn eases the trade-off between output and inflation volatility. We further

find that in many instances, RRT performs better than interest rate policy, even though RRT

is by itself a restricted and therefore suboptimal form of QE.

Finally, we consider the effects of QE on welfare. We find that QE rules tend to deliver

lower welfare than interest rate rules. Moreover, aggressive QE rules might be very detrimental

to welfare. This might seem surprising, given that QE is relatively effective in stabilizing

output and inflation. However, QE comes with strong side effects which adversely affect

social welfare. In particular, when the central bank creates movements in the amount of

reserves and hence the supply of deposits, it varies the extent to which households can insure

themselves against idiosyncratic income risk. The welfare costs of periods of low insurance

are relatively large; they can dominate the total welfare gains from periods of high insurance

and from reduced volatility of inflation and output. Thus, even though aggressive QE rules

can be powerful from a macro stabilization standpoint, such policies might not be advisable

from a welfare perspective. In fact, we find RRT to be approximately optimal from a welfare

perspective, when conventional policy is not available.

Given that liquidity is valuable to households, one may wonder if it might be optimal for a

government to permanently flood the market with liquidity, rather than adhering to a mean-

reverting rule. It turns out that this is not the case in the model. To show this, we consider

a permanent expansion of the central bank’s balance sheet via QE. Such an expansion can

move the economy to a new steady state with more liquidity and higher welfare. However,

the transition path to such a new steady state is costly, due to distributional side effects. We

therefore find that there is only limited scope for improving welfare through permanent QE.

The technical aspects of this paper may be of independent interest, as we make two contri-

butions to reduce the computational complexity of the model, which facilitates estimation and

optimal policy exercises. First we show that, in the computation, the time-varying distribu-

tion of partially liquid wealth may be dropped as a state if no ad-hoc lower bound beyond the

4

“natural limit” is imposed on this asset. Second, we show how to keep track of the distribu-

tion of liquid wealth in a parsimonious yet accurate way, exploiting the fact that households’

holdings of deposits are low in the data.

Related literature. The neutrality of central bank balance sheet policies in standard

complete-markets models was originally established by Wallace (1981), and reiterated more

recently by Woodford (2012). The underlying theoretical argument is a variation on the

Modigliani-Miller and Ricardian Equivalence theorems. Perhaps in part because of this strik-

ing neutrality result, much of the recent NK literature on unconventional monetary policy has

focused on Forward Guidance rather than on QE, see for instance Del Negro, Giannoni and

Patterson (2012) and McKay, Nakamura and Steinsson (2016).

That said, our model analysis of QE does have a number of precursors. Chen, Curdia and

Ferrero (2012) analyze QE in a medium-scale DSGE model with segmented asset markets.

They find that QE only has small effects. Large effects are found by Del Negro, Egertsson,

Ferrero and Kiyotaki (2017), who develop a quantitative model to evaluate the effects of

liquidity provisions during the financial crisis. In their model, liquidity interventions ease

financial constraints on the production side of the economy. A similar channel operates in

Gertler and Karadi (2012). Wen (2014) studies the QE exiting strategy and the impact on

firms. By contrast, we focus on the role of QE as a direct instrument to manage aggregate

demand, which has been used well beyond the financial crisis. Campbell (2014) considers

the implications of QE for the occurrence of liquidity traps, whereas Harrison (2017) studies

optimal QE policy in a representative-agent model with portfolio adjustment costs.

In our model, QE is a tool for liquidity management which comes with important distri-

butional effects. In particular, it affects the joint distribution of wealth, consumption, and

labor income. The existence of distributional effects of monetary policy has been extensively

documented in empirical work, for instance by Doepke and Schneider (2006) and Ampudia et

al. (2018). The theoretical literature on monetary policy and liquidity, cf. Lagos and Wright

(2005), typically does not focus on distributional effects, although such effects have been con-

sidered in recent work by Rocheteau, Weill and Wong (2018a); Rocheteau, Wright and Xiao

(2018b). We propose a computationally tractable yet standard incomplete-markets model to

evaluate liquidity policies and their impact on inequality. By including nominal rigidities, the

model allows for a quantitative comparison to conventional interest rate channels studied in

the New Keynesian literature. Also, it turns out that the transitional dynamics that arise

from nominal rigidities may render policies which fully satiate the economy with liquidity

undesirable from a social welfare perspective.

The importance of household liquidity for optimal monetary policy is emphasized by Bilbiie

and Ragot (2016). They show that liquidity frictions change the output-inflation trade-off,

5

as inflation affects the extent to which households can self-insure using nominal assets. Cui

(2016) studies the optimal monetary-fiscal policy mix in a model in which the liquidity of

different asset classes differs endogenously, but without QE. Finally, Caballero and Farhi

(2017) consider monetary policy –including QE– in a model with safe and risky assets and

heterogeneity in risk tolerance.

Unlike these studies, we use a quantitative model with incomplete markets in the Bewley-

Huggett-Aiyagari tradition, combined with sticky prices in the NK tradition. A number

of recent papers study the importance of household heterogeneity for the transmission of

conventional monetary policy in this type of models, see for instance Gornemann, Kuester

and Nakajima (2016), Kaplan et al. (2017), Auclert (2016), Luetticke (2015), Ravn and Sterk

(2016), Debortoli and Galı (2017), Challe (2017), and Hagedorn, Luo, Mitman and Manovskii

(2017). Our model fits into this category, but we instead estimate the effect of QE and its

welfare consequences. We exploit the model’s tractability to devise a fast solution method,

which might be of independent interest. Heterogeneity also plays a role in Sterk and Tenreyro

(2018), who study the distributional effects of open-market operations in a flexible-price model.

Finally, various authors have studied the empirical effects of large-scale asset purchases,

generally finding evidence for expansionary macro effects. For example, Weale and Wieladek

(2016) find that in the U.S., an asset purchase of one percent of annual GDP leads to an

increase in real GDP of 0.58 percent and an increase in inflation of 0.62 percent. A survey of

the broader literature on this topic can be found in Bhattarai and Neely (2016).

2 Transmission of liquidity: an empirical perspective

Before we present the full model, we provide a more detailed empirical account of the trans-

mission of Quantitative Easing to households in the United States. Drawing upon several

sources, we discuss the role of banks, mutual funds, households, and the fiscal authority. We

then present a simple formula to provide a first glance at the potential magnitude of the

macroeconomic effects of the increase in household liquidity.

2.1 The role of various actors

The role of banks. Banks played an important intermediary role in the transmission of

QE, for at least two important reasons. First, only banks can hold reserves at the Federal

Reserve and create deposits. Second, when purchasing assets the Fed trades with primary

dealers, which are typically banks.

However, these facts do not imply that banks were the owners of the assets that were

purchased under QE. Indeed, banks played only a modest role as sellers of assets to the Fed.

6

This is shown by for instance Carpenter, Demiralp, Ihrig and Klee (2015), who investigate in

detail which investors the Fed purchased from. They show that during the two decades prior

to QE, banks held only about 7% of the total amount of treasury debt and mortgage-backed

securities. They also provide more formal econometric evidence implying only a minor role for

banks. Indeed, the Fed could not even have purchased the QE assets exclusively from banks.

Flow of Funds data show that at the start of QE, commercial banks held about $1.2 trillion

worth of treasury debt and MBS. While this number may seem large, it is much less than

the total value of assets that were purchased under QE: Figure 1 shows that the increase in

reserves following QE was about $2.5 trillion.

The role of mutual funds and other non-bank entities. Using Flow of Funds data,

Carpenter et al. (2015) report that the major sellers of assets to the Fed during QE were

household-held mutual funds, pension funds, broker-dealers, and insurance companies. Im-

portantly, none of these entities has the ability to hold reserves at the Federal Reserve, which

implies that QE purchases from these entities must have led to deposit creation.

To understand this point, consider a mutual fund which sells $100 million dollars worth

of assets to the Fed. In turn, the Fed finances the purchase by issuing $100 million worth or

reserves. Immediately after the sale, the fund will hold $100 million of additional deposits at

a bank, which in turn balances this liability by holding $100 million of additional reserves at

the Fed. The bank thus serves at an intermediary which increases its deposit liabilities and

reserve holdings by the same dollar amount.

Of course, the mutual fund may subsequently choose to offload the additional cash/deposits

from its balance sheet, as cash offers a low return and the fund may have little use for extra

liquidity. Indeed, empirical evidence shows that this is what happened following QE in the

U.S.. Goldstein, Witmer and Yang (2015) use micro-level data on the behavior of mutual funds

following QE. They report that mutual funds did not increase their cash holdings following

the asset sales to the Fed.

The role of households and the fiscal authority. If mutual funds did not hold on to the

additional deposits, then where did the liquidity go? A direct way of answering this question

is to estimate the response of deposits held in various sectors to a QE announcement. We did

so using a local projection estimation, controlling for several macroeconomic variables. Figure

2 shows that QE announcements were followed by a surge in reserves and deposits. The latter

ended up being held mostly by households (middle panel). A smaller fraction was held by

non-households (right panel), a category which consists mainly of non-financial firms.7

7In the model, there is no motive for firms to hold liquidity. However, the corporate finance literature hasprovided empirical evidence that increased liquidity in firms may stimulate investment and hiring. This would

7

Figure 2: Reserves and deposits following a QE announcement.

0 5 10 15

quarter

-500

0

500

1000

1500

2000

2500

3000

$ bi

llion

reserves

0 5 10 15

quarter

-200

0

200

400

600

800

1000

1200deposits: households

0 5 10 15

quarter

-200

0

200

400

600

800

1000

1200deposits: non-households

Source: Federal Reserve Board, Flow of Funds accounts. Responses estimated using a localprojection controlling for unemployment, inflation, GDP growth and the Federal Funds Rate.Dash-dotted lines denote 90% confidence bands.

What did the mutual funds do with the additional liquidity and how did it end up in the

hands of households? One possibility is that, following QE, there was a direct net outflow of

cash from the mutual funds, which happens when the funds increase dividends to households,

when raises less new investment from households, or when households take wealth out of the

fund. The empirical importance of this channel seems rather limited, however, as outflows

from mutual funds who sold assets to the Fed were only moderate, see e.g., Goldstein et al.

(2015).8 They show that, instead, mutual funds mainly replaced the assets sold to the Fed

with newly issued government debt. This means that the liquidity left the mutual fund sector

via the government, who used it to lower taxes, to increase transfers, and/or to purchase

goods and services. Hence, the newly created liquidity flowed to the households (and to some

extent firms), who have a demand for liquidity, given the idiosyncratic risks that they face.

2.2 A first glance at the macroeconomic effects

Finally, we provide a simple formula to gauge the effects of QE on aggregate demand. To this

end, let us postulate an aggregate consumption demand function C(L, I,Γ), where L denotes

the (nominal) value of fully liquid assets held by households (e.g. deposits), and I denotes the

value of their illiquid, or partially liquid assets (e.g. assets owned via mutual funds). The third

argument, Γ, contains other relevant aspects of individual states and the overall economy,

be another complementary transmission channel of QE which we do not consider in this paper.8They also find little evidence that there was a rebalancing towards other asset classes such as firm equity

within mutual funds.

8

such as asset prices and wages, and is denoted by a scalar for simplicity. The (average)

marginal propensities to consume out of liquid and illiquid wealth are given by the respective

derivatives of the aggregate demand function, and will be denoted by MPCL ≡ CL(L, I,Γ)

and MPCI ≡ CI(L, I,Γ).

When the central bank conducts QE, it purchases I in exchange for L. The mechanics are

the following. The central bank purchases long-term government debt from mutual funds by

issuing reserves; the banks act as a pure middle man, sourcing the bonds from mutual funds

in exchange for deposits, and then selling bonds to the Fed in exchange for reserves. Since

these are all voluntary trades, QE does not directly change the total amount of wealth owned

by households, i.e. any increase in L is matched by a decrease in I of the same magnitude.

Denoting the value of assets purchased under QE by ∆QE, the consumption function becomes

C(L+∆QE, I−∆QE,Γ(∆QE)). By differentiating this function with respect to ∆QE, we obtain

the following formula for the marginal effect of QE on aggregate demand:

∂C

∂∆QE= MPCL −MPCI︸ ︷︷ ︸

direct effect

+ GE︸︷︷︸,indirect effect

where GE ≡ CΓ(L, I,Γ) ∂Γ∂∆QE denotes the general equilibrium effect. This formula splits the

effects of QE into “direct” and “indirect” GE effects, in the spirit of a decomposition proposed

by Kaplan et al. (2017) for conventional monetary policy.

The first term captures the direct effect. It is the difference between the MPCs out of

liquid and illiquid wealth. Intuitively, QE directly triggers a liquidity transformation: it lowers

households’ illiquid wealth holdings, while increasing their liquid wealth. The direct effect of

this transformation on consumption depends on the difference in the marginal propensities

to consume out of the two types of wealth. The second term captures the indirect general

equilibrium effects triggered by QE.

Simple as it looks, the formula conveys a number of important insights. First, if the two

types of wealth were equally liquid to households, as in many standard models, it would hold

that MPCL = MPCI , other things equal. In this case, QE would have no direct effect

on aggregate demand, echoing the neutrality result of Wallace (1981). Second, even in the

extreme case in which MPCI = 0, QE only has large effects to the extent that the marginal

propensity to consume out of liquid wealth, MPCL, is large. This point provides a way of

understanding why for instance Chen et al. (2012) find that QE has small effects on the real

economy, as it is well known that MPCs tend to be very small in representative-agent models.

On the other hand, models with incomplete markets and borrowing constraints are well known

to generate much higher MPCs out of liquid wealth. Moreover, when certain types of assets

are subject to liquidity frictions, the MPCs out of these types of wealth tend to be small, even

9

in incomplete-markets models.

Finally, the indirect GE effects depend crucially on the structure of the economy and

in particular on price stickiness. With flexible prices, an increase in aggregate consumption

demand is typically dampened by an increase in prices. With sticky prices, the increase in

aggregate demand might be further amplified.

Are strong direct effects of QE in line with the data, i.e. is the difference between MPCL

and MPCI large? A substantial body of empirical studies has found MPCs out of fully

liquid wealth to be very sizable. For example, Fagereng, Holm and Natvik (2018) estimate

an average MPC of 63 percent in the first year, based on high-quality administrative data

on Norwegian lottery participants.9 The literature on MPCs out of less liquid sources of

wealth is less extensive, but generally reports much smaller MPCs. Di Maggio, Kermani and

Majlesi (2018) use Swedish data to estimate MPCs out of changes in stock market wealth,

and estimate these to lie between 5 and 14 percent, much below typical estimates for the

MPCs out of fully liquid wealth. Moreover, they report that –among the same individuals–

MPCs out of fully liquid dividend payments are much higher. The empirical evidence is thus

consistent with sizable direct effects.

Based on the above formula, we can obtain a back-of-the-envelope estimate for the direct

effects of QE. This helps to get a sense of the quantitative importance of QE since the Great

Recession. Between 2007 and 2017, checkable deposits held by households and non-profit

organizations increased from about 1.5 to 6.3 percent of annual GDP.10 Figure 1 suggests that

this increase was largely driven by QE. Assuming MPCL = 0.63 following Fagereng et al.

(2018) and MPCI = 0.095, the mid point of estimates provided by Di Maggio et al. (2018),

this implies a direct effect of (6.3− 1.5) · (0.63− 0.095) = 2.57 percent of GDP.

Thus, the data suggest that the direct effects of QE on GDP are substantial. However,

the overall effect of QE depends also on the GE response to these direct effects. We will use

the model to evaluate the overall effects of QE.

3 QE in a HANK model

This section presents a fully-fledged general equilibrium model. We use the model to quantify

the macro effects of QE and contrast them to the effects of conventional policy. The model

is a so called Heterogeneous-Agents New Keynesian (HANK) model, cf. Kaplan et al. (2017).

9Parker, Souleles, Johnson and McClelland (2013) report an average quarterly MPC between 50 and 90percent for the U.S. during the Great Recession. For more discussion on the empirical evidence, see Kaplanet al. (2017).

10The amount of checkable deposits held by households and non-profits was $219 billions in 2007 and $1,219billions in 2017. Nominal GDP was $14,457 billions in 2007 and $19,485 billions in 2017.

10

Its key features are that nominal prices are sticky, that households face imperfectly insurable

income risk, and that households make a portfolio choice between fully liquid and partially

liquid assets (i.e., mutual-fund wealth), the latter being subject to a convex adjustment cost.

QE is a purchase by the central bank of long-term government debt held by mutual funds.

After describing the model, we provide two insights which render the model computation-

ally tractable, which facilitates estimation and optimal policy exercises. We argue that this

tractability preserves consistency with the micro data along key dimensions. In particular,

there is a rich joint distribution of the two types of assets, which evolves endogenously over

time.

3.1 The model

The model economy is populated by households, firms, banks, mutual funds, a treasury, and

a central bank.

Households. There is a continuum of infinitely-lived, ex-ante identical households, indexed

by i ∈ [0, 1]. Household i’s preferences are represented by:

E0

∞∑t=0

βtU (Ct(i), Nt(i)) , (1)

where Ct(i) is a basket of goods consumed in period t, Nt(i) denotes hours worked, supplied on

a competitive labor market, and β ∈ (0, 1) is the subjective discount factor. Moreover, Et is the

expectations operator conditional on information available in period t, and U (C,N) is a utility

function which is increasing and concave in consumption and decreasing in hours worked. The

consumption basket is given by Ct(i) ≡∫ 1

0

(Ct(i, j)

εt−1εt dj

) εtεt−1

, where Ct(i, j) denotes the

household’s consumption of good j and εt > 1 is the elasticity of substitution between goods,

which is exogenous. Following the NK literature, variations in εt can be thought of as “cost

push” shocks, since they affect mark-ups charged by firms. Household optimization implies

that the price of the consumption basket is given by Pt =∫ 1

0(Pt(j)

1−εtdj)1

1−εt , where Pt(j) is

the price of good j.

Households are subject to idiosyncratic unemployment risk. When unemployed the house-

hold cannot supply labor, i.e. Nt(i) = 0, so it has no labor income. When employed, the

household can freely choose the number of hours worked, earning a real wage rate wt per

hour. Unemployed households become employed with a probability pUE, whereas employed

households become unemployed with a probability pEU . These transitions are exogenous and

take place at the very end of each period. When unemployed, a household receives an un-

11

employment benefit given by Θt(i) = ΘU ≥ 0. This benefit is provided by a government

agency which runs a balanced budget by imposing an equal social insurance contribution on

the employed. That is, when employed the household receives a negative transfer given by

Θt(i) = ΘE = − uu−1

ΘU ≤ 0, where u = pEU/(pEU + pUE) is the unemployment rate.11

Households can hold deposits, denoted by Dt(i) in real terms, which pay a nominal interest

rate and are fully liquid, in the sense that there are no transaction costs involved. Deposits

provide households with a means of self insurance against the idiosyncratic income risks as-

sociated with unemployment, helping them to cushion the decline in consumption when they

lose their job. However, households must obey a borrowing constraint:

Dt(i) ≥ D, (2)

where −D is a borrowing limit.

Households can also own shares in mutual funds, which may generate higher returns but

are less liquid. The evolution of a household’s mutual-fund wealth, denoted At(i), is given by:

At(i) = (1 + rAt )At−1(i)−Xt(i) and At(i) ≥ A, (3)

where rAt is the real return on the funds, Xt(i) is a withdrawal by the household from the

fund (Xt(i) < 0 would be a deposit into the fund), and A is a lower bound. Withdrawals

come with a convex cost of adjustment, which is similar to Kaplan et al. (2017), although in

our case it takes the form of a tax.12,13 Due to the adjustment cost, wealth stored in mutual

funds is only partially liquid and is therefore only of limited use as a means of self insurance.

The adjustment cost is given by Ψt(i) = ωt(i)Ψ(Xt(i)), where Ψ(·) ≥ 0 is a convex function

with Ψ(0) = 0; ωt(i) = 1 for the employed and ωt(i) = ωU ∈ [0, 1) for the unemployed. The

idea behind the latter is that the unemployed might face a less steep tax schedule than the

employed.

In each period, household i chooses At(i), Ct(i), Dt(i), Nt(i), and Xt(i) to maximize (1)

subject to (2), (3), the adjustment cost function, the constraint that it can only choose Nt(i)

11McKay and Reis (2016) provide an in-depth analysis of the stabilization role of social insurance in a NKmodel with heterogeneous agents.

12Realistically, many of the costs triggered by mutual fund withdrawals are associated with taxation. Inparticular, withdrawals may trigger capital gains taxes or –in case of retirement accounts– early withdrawalpenalties. Such costs arguably do not reflect a loss of real resources. Also, back-end fees charged by mutualfunds upon withdrawal might best be thought of as a transfer, since the transaction itself requires few resources.The main purpose of back-end fees is to reduce the likelihood of large and sudden net outflows from the fund,which tend to complicate its investment strategy.

13Another difference is that Kaplan et al. (2017) let the adjustment cost depend not only on the size of thewithdrawal, but also on the stock of illiquid assets, which facilitates their calibration strategy.

12

when employed, and a budget constraint specified in real terms as:

Ct(i) +Dt(i) = wtNt(i) +Rt−1

Πt

Dt−1(i) + Θt(i)−Ψt(i) +Xt(i)− Tt, (4)

where Rt−1 is the gross nominal interest rate on deposits from period t − 1 to period t,

Πt = Pt/Pt−1 is the corresponding gross rate of inflation, and Tt is a lump-sum tax levied to

finance government expenditures other than benefits.

Firms. Each consumption good is produced by a different firm. The structure of household

preferences implies that firms are monopolistically competitive in the goods market. Firms

operate a linear technology using labor only, i.e. their output is given by Yt(j) = ZtNt(j).

Here, Zt denotes Total Factor Productivity (TFP).

Firms also face a quadratic cost of price adjustment following Rotemberg (1982), given

in real terms by Adjt(j) = φ2

(Pt(j)−Pt−1(j)

Pt−1(j)

)2

Yt, where φ ≥ 0 is a parameter which governs

the cost of price adjustment, and Yt =∫ 1

0Yt(j)dj denotes aggregate output. The dividends

paid by firm j are given, in real terms, by Divt(j) = Pt(j)PtYt(j) − wtNt(j) − Adjt(j) where in

equilibrium it holds that Pt(j) = Pt. Therefore, aggregate dividends satisfy

Divt = Yt − wtNt − Adjt, (5)

where Adjt =∫ 1

0Adjt(j)dj = φ (Πt − 1)2 Yt. Firms maximize the present value of profits,

which leads to the following relation, commonly known as the New Keynesian Phillips Curve:

1− εt + εtwtZt

= φ (Πt − 1) Πt − φEt[Λt,t+1

Yt+1

Yt(Πt+1 − 1) Πt+1

], (6)

where Λt,t+1 is the stochastic discount factor used by the firms.14 We assume that the distri-

bution of initial prices is the same across firms, so they behave symmetrically and we drop

the index j from now on.

Mutual funds. There is a representative mutual fund which owns shares in aggregate equity

of the firms (St) as well as long-term treasury debt (Bmt ). We model the latter following

Woodford (2001). A unit of long-term debt pays ρk dollars in any period t + k + 1 going

forward, where 0 ≤ ρ < 1. In the steady state, the duration of long-term government debt is

given by 11−βρ . Note that government debt is fully liquid to the mutual fund, as it does not

face any trading frictions. The equities of the representative mutual fund are the mutual fund

14Since we will linearize the model around a zero-inflation steady state, the precise specification of thestochastic discount factor is irrelevant for the results, as it drops out in the linearization.

13

shares owned by the households.

Let Xt ≡∫ 1

0Xt(i)di be the total amount withdrawn by households from the mutual fund.

The flow budget constraint of the mutual fund is given by:

Xt = (1 + ρqBt )Bmt−1

Πt

− qBt Bmt + qSt (St−1 − St) + St−1Divt, (7)

where Divt ≡∫ 1

0Divt(j)dj are aggregate dividends transferred from the firms to the fund, qBt

is the price of government debt issued in period t, and qSt is the price of a firm equity share.

The mutual fund allocates its budget over Bmt and St, in order to maximize expected returns.

This implies the following no-arbitrage relation:

EtqSt+1 +Divt+1

qSt= Et

(1 + ρqBt+1

)/Πt+1

qBt. (8)

The aggregate volume of firm equity shares is normalized to St = 1. The realized rate of

return of the mutual fund sector can then be expressed as:

rAt =(1 + ρqBt )Bm

t−1/Πt + qSt +DivtqBt−1B

mt−1 + qSt−1

− 1. (9)

Note that the mutual fund does not hold deposits on their balance sheets. In equilibrium,

the return on deposits is dominated by the return on long-term government debt. The reason

is that households value deposits for precautionary saving reasons, which drives down the real

interest rate on deposits.

Therefore, whenever the fund receives deposits in exchange for assets sold to the central

bank, it will immediately use these deposits to purchase other assets, in line with the empirical

micro evidence discussed in Section 2.1. As a result of this response, the liquidity flows to

other sectors in the economy, consistent with the Flow-of-Funds data. Moreover, this response

pushes up qBt , implying a decline in the long-term interest rate. To see this more clearly, one

can derive the following partial-equilibrium elasticity of qBt with respect to Bmt :

dqBt /qBt

dBMt /B

Mt

=1

ρ− 1< 0,

and note that Bmt declines as the fund sells long-term debt to the central bank, so that qBt

increases.15

15This partial-equilibrium elasticity is derived by taking a first-order approximation of Equation (7) arounda steady state with zero inflation, keeping all variables other than qBt and Bmt unchanged. In general equilib-rium, the mutual fund’s demand for long-term debt is also affected by changes in Πt and Divt.

14

Banks. There is a perfectly competitive banking sector. Banks can hold reserves at the

central bank, denoted by Mt in real terms, which pay a nominal interest rate Rt, controlled

by the central bank. In order to fund these assets, banks must create liabilities, i.e. deposits.

No-arbitrage implies that reserves and deposits carry the same nominal interest rate Rt. In

equilibrium banks therefore earn no profits. Consolidation of the balance sheet of the banking

sector implies that:16

1∫0

Dt(i)di = Mt. (10)

Treasury. Real government expenditures are exogenous and denoted by Gt. The treasury

targets a constant real level of long-term debt, denoted by Bt = B, during each period. The

budget constraint of the treasury is given by:

Gt = qBt B − (1 + ρqBt )B

Πt

+ T cbt + Tt + Ψt, (11)

where T cbt is a seigniorage transfer received from the central bank and Ψt ≡∫ 1

0Ψt(i)di is the

total revenue from taxation of mutual fund withdrawals. Note that the lump-sum component

of taxation (Tt) adjusts to balance the budget. In Appendix D, we consider alternative as-

sumptions on fiscal policy. Under these alternatives, the effects of QE on the macro economy

are similar or even larger than in our baseline model.

Central bank. The central bank targets the nominal interest rate on reserves (Rt) and the

real amount of reserves (Mt), depending on the policy regime. The budget constraint of the

central bank, in real terms, is given by:

T cbt +Rt−1

Πt

Mt−1 + qBt Bcbt = Mt + (1 + ρqBt )

Bcbt−1

Πt

, (12)

where Bcbt denotes the central bank’s holdings of long-term government debt.

We consider two versions of the model, each with a different conduct of monetary policy.

In the first version, the central bank conducts conventional interest rate policy. In this case,

the central bank targets the interest rate on reserves according to the following rule:

Rt = ΠtξRΠ Y

ξRYt , (13)

16It would also be straightforward to allow the banking sector to create additional deposits without holdingreserves. However, this would not impact directly on our key mechanism, which requires QE to trigger thecreation of additional deposits, as strongly suggested by Figure 1.

15

where hats denote variables relative to their steady-state values: Yt ≡ Yt/Y , Πt ≡ Πt/Π, and

Rt ≡ Rt/R (note: R, Π, and Y are the steady-state values of R, Π, and Y , respectively). In

the above policy rule, ξRΠ and ξRΠ are stabilization coefficients which determine the response

of monetary policy to fluctuations in output and inflation. Under conventional policy, the

central bank does not own any government debt (Bcbt = 0) and the real amount of reserves

(and hence aggregate deposits) is held at a constant level (Mt = M).17

In the second version of the model, the central bank conducts QE rather than interest

rate policy. When QE is used, the nominal interest rate is pegged at Rt = R, reflecting the

reality that QE is typically used when the nominal interest rate cannot be moved. We further

assume that if the central bank purchases government debt, it finances these purchases by

issuing reserves:

qBt Bcbt − (1 + ρqBt )

Bcbt−1

Πt

= Mt −Rt−1

Πt

Mt−1. (14)

In this case, QE targets the total amount of reserves according to the following rule:18

Mt = ΠξQEΠt Yt

ξQEY zQEt , (15)

where Mt = Mt/M is the amount of real reserves relative to the steady-state level and zQEt

is an exogenous shock to the QE rule, akin to conventional monetary policy shocks often

considered in the NK literature. We will study this shock to better understand the workings

of QE. In the above rule, ξQEΠ and ξQEY are policy coefficients which are, respectively, the

elasticities of real reserves with respect to inflation and output.

An interesting special case of the QE rule sets both stabilization coefficients to zero, i.e.

ξQEΠ = ξQEY = 0. In this case monetary policy directly targets a certain level of real reserves

given by Mt = MzQEt . We refer to this policy as Real Reserve Targeting (RRT). This policy

implies that, in the absence of QE shocks, the level of real reserves is constant and hence

the nominal amount of reserves moves one for one with the price level; but unlike under

conventional policy, Bcbt is not constrained to be zero.

3.2 Equilibrium

Given laws of motion for the exogenous states {εt, Zt, Gt, zQEt } and government policies {B,

Tt, Rt, Mt, Bcbt , T cbt }, the competitive equilibrium is defined as a joint law of motion for house-

holds’ choices {Nt(i), Ct(i), Dt(i), At(i), Xt(i),Ψt(i)}i∈[0,1], mutual fund choices {Bmt , St = 1},

17We abstract from the zero lower bound on the net nominal interest rate (Rt−1). However, we will assumethat the interest rate is pegged at zero in the model version with QE. Regarding QE policy, we similarly donot impose a lower bound on Bcbt , i.e. the central bank itself could in principle issue long-term debt.

18This rule can be reformulated as a rule for nominal reserves, being a function of the current and laggedprice level, and nominal output.

16

aggregate quantities {Yt, Nt, Divt} and prices {Πt, wt, qBt , q

St , r

At }, such that ∀t,

(i) Each household i ∈ [0, 1] maximizes (1) subject to the constraints (2), (3), and (4),

with Nt(i) = 0 when he/she is unemployed, and subject to the adjustment cost;

(ii) Firms in total produce Yt = ZtNt, pay out dividends according to (5), and set nominal

prices such that the New Keynesian Phillips Curve (6) holds;

(iii) The mutual fund’s budget constraint(7), and pricing conditions (8) and (9) hold, and

the mutual fund’s assets equal its liabilities:∫ 1

0

At(i)di = qBt Bmt + qSt St.

(iv) The banks create deposits such that (10) holds;

(v) The treasury’s and central bank’s budget constraints, (11) and (12), hold;

(vi) The markets for deposits/reserves clear, i.e., equation (10) holds. Households’ expec-

tations about the distribution of assets are consistent with the actual distribution. Also, the

markets for long-term government debt and labor clear, i.e.,19

B = Bcbt +Bm

t , Nt =

∫ 1

0

Nt(i)di.

3.3 Tractability

The model is in principle computationally complex, as the economic state contains a time-

varying joint distribution of liquid and partially liquid asset holdings. However, two insights

allow us to reduce the complexity considerably, each of which may be of independent interest.

To explain these, we anticipate a few elements of the calibration strategy, which will be

discussed extensively in the next section.

The distribution of partially liquid wealth. The first insight pertains to the distribution

of wealth in the mutual fund. The idea is simply to impose no ad hoc lower bound on A(i)

beyond the “natural” limit, i.e. the present value of the minimum stream of non-asset income,

which in our case is given by A = − ΘU

R/Π−1. When parametrizing the model, we will target a

steady-state inflation rate of Π = 1, as is often the case in New Keynesian models. We also

target R = 1, as we think of QE as being implemented during periods in which the zero lower

bound on the net nominal interest rate is binding. Together, these two targets imply a zero

19The goods market clearing is satisfied because of Walras law. To see this, we sum over all budgetconstraints from individual households, the mutual fund, the treasury, and the central bank; then, we usethe balance sheet of the banks and the market clearing conditions for government debt and labor to reach

Yt =∫ 1

0Ct(i)di+Gt + φ (Πt − 1)

2Yt.

17

net real interest rate in the steady state. The inequality in (3) then becomes:

A(i) ≥ −∞.

Alternatively, the absence of a lower limit on on A(i) may be directly motivated as a form

of partial insurance between households which has been agreed ex ante. Indeed, any more

stringent limit would further reduce households’ ability to self-insure against idiosyncratic

risks.

It can now be shown that households’ decisions do not depend on A(i) since, for lack

of a constraint, the shadow value of A(i) reduces to zero. Still, the household is prevented

from making unlimited withdrawals, but only because of the adjustment cost, which depends

on X(i) but not on the level of A(i). Hence, A(i) is irrelevant for consumption and the

distribution of A(i) can be dropped as an economic state even though it moves endogenously

over time, see Appendix B for a more detailed discussion.

Figure 3: Saving rates.

0 2 4 6 8 10

wealth (multiple of annual labor income)

0

0.1

0.2

0.3

0.4

0.5saving rates (employed)

net saving rategross saving rate

Notes: the gross saving rate is computed as rAA(i)−X(i)y+rAA(i)

whereas the net saving rate is computed as −X(i)y .

Here, y is set to average labor income and wealth is measured as A(i).

The disconnect between the withdrawal decision and the level of wealth has direct cross-

sectional implications which can be compared to the data. In particular, it implies that gross

saving rates (i.e. including capital gains) are increasing in wealth, as households save capital

gains rather than spending them. This is illustrated in Figure 3, which previews results from

the calibrated model. On the other hand, net saving rates (i.e., excluding capital gains) are

not systematically related to wealth. In the data, these two patterns have recently been

documented by Fagereng, Holm, Moll and Natvik (2019). They argue that this evidence is at

odds with standard models, but consistent with models in which households “save by holding”,

18

i.e. they do not spend out of capital gains, as is exactly the case in our model.

While the average level of At(i) is uniquely determined as∫ 1

0At(i)di = qBt B

mt + qSt St,

the steady state of the model is consistent with any initial distribution of At(i) (including

the one observed in the data). This is shown in Appendix B. At the same time, the model

does have unique predictions on the evolution of the distribution of At(i) over time, given

initial conditions and aggregate shocks. It can be shown that the distribution of At(i) is

non-stationary, with inequality rising over time. This property follows from the “saving by

holding” behavior in the model, and is also in line with the fact that US wealth inequality

has been steadily increasing for half a century by now, see for instance Piketty (2014).

The distribution of liquid wealth. Having dealt with the distribution of At(i), we now

show how to keep track of the distribution of liquid wealth, Dt(i), in a parsimonious way. To

this end, we set the length of a model period to one quarter and consider a calibration in which

the amount of liquidity in the steady state is not too large. In that case, those who become

unemployed exhaust their deposits within the first quarter of unemployment, and thus hit the

liquidity constraint.

When calibrating the model to average deposit holdings in the US economy this in fact

turns out to be an outcome that is naturally obtained. Indeed, the deposit holdings of most US

households do not exceed a few weeks of wage income, even among higher-income households.20

Also, we will argue that this calibration strategy generates MPCs that are close to the data.

The fact that households hit the liquidity constraint upon job loss has some important

consequences.21 It implies that all employed households with the same employment duration

behave identically, as do the newly unemployed with the same employment duration before

job loss, and those who have been unemployed for more than one quarter.

We exploit this outcome to solve the model as easily as a typical medium-scale DSGE model

with a representative agent. In particular, we group agents who were employed in quarter

t − 1 into cohorts, indexed by the length of the employment spell in the previous quarter,

denoted by k ≥ 0. The cohort with k = 0 was previously unemployed, is currently employed,

and enters the period with zero deposits. Hence they make identical decisions. Therefore,

agents in cohort k = 1 all start with the same level of deposits, Dt−1(i). Hence, conditional on

their employment status, all agents within cohort k = 1 make the same decisions. Extending

20In the data, a small fraction of households holds a large amount of deposits, for instance because theyhave set aside deposits in anticipation of a large upcoming durable purchase. Such a cash holding motive isoutside the scope of our model. Moreover, Campbell and Hercowitz (2018) argue that even those householdsmay have high MPCs out of liquid wealth, which is key for the mechanism in our model.

21This property is also exploited for tractability by Krusell, Mukoyama and Smith (2011) and Ravn andSterk (2017), who assume zero aggregate liquidity, as well as by Challe and Ragot (2016) who assume thatthe employed are on a locally linear segment of the utility function. In our model, there is positive aggregateliquidity and a globally concave utility function.

19

Figure 4: Decision rules (steady state).

0 0.02 0.04 0.06 0.08 0.1 0.120

0.02

0.04

0.06

0.08

0.1

0.12Deposits

0 0.02 0.04 0.06 0.08 0.1 0.120

0.05

0.1

0.15

0.2

Consumption

employed, following employment spell of k quartersnewly unemployed, following employment spell of k quartersunemployed for more than 1 quarter

k=2k=3

k>75

k>75

k>75

k>75

k=0

k=1

k=2

k=0 k=1 k=2

k=1

k=1 k=2 k=3

Notes: markers denote mass points of the liquid wealth distribution observed in the steady-state equilibrium.The black line is the 45-degree line.

this logic, within any cohort k ≥ 1 a fraction pEU of the agents has become unemployed

in the current quarter. They all behave identically and move out of the cohort in the next

quarter. The remaining fraction of the cohort 1 − pEU remains employed. Again, they all

behave identically and move on to become cohort k + 1 in the next quarter. Finally, turning

to the households who were unemployed in quarter t− 1, we note that all behave identically

as they have depleted their deposits.

Figure 4 illustrates the steady-state choices of deposits and consumption of the different

cohorts. Note that for larger values of the employment spell k, cohorts converge to a certain

level of deposits and consumption. We use a total of K cohorts, and group all cohorts with

k ≥ K into one bin. We thus need to keep track of K state variables characterizing the

wealth distribution. In our quantitative exercises, we set K = 75. The precise value of the

cutoff K is quantitatively irrelevant as long as it is not too small. To appreciate this point,

note that from Figure 4 it can be seen that the behavior of cohorts beyond k = 15 is almost

indistinguishable.

To solve the model, we apply a first-order perturbation method for dynamic analysis, using

the popular Dynare software package. More details are provided in Appendix B.22

22We keep track of K = 75 state variables characterizing the wealth distribution. However, we obtained

20

4 The power of QE: quantitative results

We calibrate the model to the U.S. economy and set the length of a period to one quarter.

We further normalize zQE = Z = 1 and will discuss the calibration of G and ε below. Table

1 presents the parameter values. More details are provided in Appendix B.

4.1 Calibration

We assume the following utility function:

U (C,N) =C1−σ − 1

1− σ− κ0

1 + κ1

N1+κ1 ,

where σ > 0 is the coefficient of relative risk aversion, which we set equal to one. Moreover,

κ1 > 0 is the inverse Frisch elasticity of labor supply, which is also set to one. Finally, κ0 > 0

is a parameter scaling the disutility of labor, which we calibrate such that employed workers

on average supply N = 1/3 units of labor in the steady state.

We further calibrate the steady-state elasticity of substitution between goods as ε = 9,

which implies a steady-state markup of 12.5 percent, and β = 0.99, which corresponds to an

annual subjective discount rate of four percent. We target an unemployment rate of u = 0.045

and an unemployment inflow rate of pEU = 0.044, corresponding to a monthly inflow rate of

about 1.5 percent, as measured in the Current Population Survey. The implied unemployment

outflow rate is pUE = 0.934. The unemployment benefit is targeted to be 25 percent of average

wage income in the steady state, which implies that ΘU = 0.25 ε−1εN = 0.074.23 The price

adjustment cost parameter is set to φ = 47.1, which corresponds to an average price duration

of three quarters in the Calvo equivalent of the model.

To facilitate comparison of the two policies, we calibrate the model such that the steady

states of the model version with QE and the version with conventional policy coincide. Specif-

ically, we assume that in both cases the central bank targets zero inflation in the steady

state, i.e. Π = 1. The implied nominal steady-state interest rate is R = 1.24 We further

almost identical results with as few as K = 20 state variables. This is a much lower number than requiredby similar, perturbation-based solution methods. For example, the popular method of Reiter (2009) typicallyrequires hundreds of state variables to obtain good accuracy. LeGrand and Ragot (2017) solve models bytruncating idiosyncratic histories. In our application, even with a truncation cutoff lowered to K = 20, thiswould still imply 220 state variables, i.e., more than a million.

23Statutory benefits are typically around 40 percent of labor income. However, the actual amount receivedby households is much lower due to limited eligibility and take-up. Chodorow-Reich and Karabarbounis(2016) argue that taking into account all these factors reduces the benefit to around 6 percent of income. Ourcalibration strikes a balance between their number and the statutory rate.

24Note that in the version with conventional policy, we abstract from the Zero Lower Bound on the nominalinterest rate. In our comparison exercises, we thus ask whether QE is more or less effective than a hypotheticalconventional policy that would not be subject to the ZLB. Alternatively, we could have calibrated the model

21

set ρ = 0.947, which implies a duration of government debt of four years. The borrowing

limit,−D, is set to zero.25

The steady-state values of government expenditures, deposits, and government debt, i.e.,

G, D = M, and B are chosen to hit the following targets. We target a ratio of government

expenditures to output of 23 percent, in line with national accounts data, and a deposit-to-

annual-output ratio of 7.5 percent, in line with data from the Flow Of Funds (FoF) accounts

for households and non-profits. The real return on long-term government debt is targeted to

be four percent annually. While not explicitly targeted, our model implies a ratio of the value

of government debt to annual output, i.e. qB

4Yof 49.6 percent.

It turns out that we do not have to take a stand on the specific functional form of the ad-

justment cost. In Appendix B, we show that all unemployed and employed households choose

to withdraw constant amounts, denoted by XU and XE < XU , respectively. In that appendix,

we also show that it is also possible to obtain a richer distribution of withdrawals, depending

on consumption, while still preserving tractability. This can be achieved for example by for-

mulating the adjustment cost as a utility cost rather than a tax. This, however, would require

one to fully specify the adjustment cost function, which we avoid here. Specifically, we exploit

that the withdrawal net of adjustment costs, which is the relevant object in the households’

budget constraints, can be summarized as:

Xt(i)−Ψt(i) =

XE ≡ XE −Ψ(XE)

if employed;

XU ≡ XU − ωUΨ(XU)

if unemployed.

Note that the aggregate withdrawal in the mutual fund’s budget constraint is now given by

X = uXU+(1−u)XE+Ψ, where Ψ = uωUΨ(XU)+(1−u)Ψ

(XE)

is the aggregate adjustment

cost, which also enters into the government’s budget constraint.

It follows that we can directly parameterize XE, XU , and Ψ, instead of the individuals’

adjustment cost function. We do so by targeting moments in the data. Note that these

parameters have distributional consequences and affect the degree of self-insurance. We target

a real interest rate on liquid assets of zero percent, as mentioned above. Moreover, we target

data on liquid wealth from the Survey of Consumer Finances (SCF). In particular, we target

the median amount of transaction accounts (deposits in the model) held by a household with

median income, as a fraction of (pre-tax) median income. This ratio is about 26 percent

in the SCF, averaged over the years 1989-2016. Finally, we set the parameters such that

version with conventional policy to be away from the ZLB, but this would make a clear comparison moredifficult since the steady states of the two model versions would be different.

25We have solved a model with a positive borrowing limit. We obtained very similar results to our baseline,since we target the same steady-state real interest rate. Details of this version are available upon requests.

22

Table 1: Parameter values and steady-state targets.

Parameter Description Value Notes

β subjective discount factor 0.99 subjective annual discount rate: 4%

σ coefficient of relative risk aversion 1 convention

κ0 labor disutility parameter 11.4296 average labor supply employed: 1/3

κ1 inverse Frisch elasticity 1 convention

pEU unemployment inflow rate 0.044 monthly rate: 1.5% (CPS)

pUE unemployment outflow rate 0.934 steady-state unemployment rate: 4.5%

ΘU unemployment benefit 0.0741 benefit 25% of avg. real wage

XE net mutual fund withdrawal: employed 0.0218 real interest rate: 0 %

XU net mutual fund withdrawal: unemployed 0.0852 median holdings liquid wealth (SCF), see text

Ψ total adjustment cost 0.0027 10% withdrawal cost: Ψ/X = 10%

B government debt parameter 0.0399 long-term interest rate: 4%

−D borrowing limit 0 see text

ε elasticity of substitution varieties 9 markup: 12.5%

φ price adjustment cost parameter 47.1 average price duration: 3 quarters

G real government expenditures 0.0732 expenditures-to-annual-output: 23%

ρ decay government debt 0.9470 duration of government debt: 4 years

D = M steady-state deposits (=reserves) 0.1009 deposits-to-annual-output (FoF): 7.5%

Π long-run inflation target 1 net inflation rate: 0%

Ψ/X = 10%, i.e. the average adjustment cost is ten percent of the size of the withdrawal,

which is the typical tax penalty when withdrawing funds from a U.S. retirement account. The

implied net withdrawal of the employed corresponds to about 7.5 percent of their average

wage income, whereas the unemployed withdraw about four times as much.

As a final remark, we assume that each of the stochastic driving forces z ∈ {ε, Z,G, zQE}follows an independent process of the form ln zt = (1−λz) ln z+λz ln zt−1+νzt . Here, λz ∈ [0, 1)

is a persistence parameter and νzt is an i.i.d. innovation, drawn from a Normal distribution

with mean zero and a standard deviation given by σz ≥ 0. We allow λz and σz to potentially

differ across the four types of shocks, and we will discuss their values below.

4.2 Model implications for micro-level consumption

We now explore the implications of the calibration for micro-level consumption behavior, and

in particular for the gap in Marginal Propensities to Consume (MPCs) out of liquid and illiquid

(partially liquid) wealth. This is important since the simple formula presented in Section 2

makes clear that this gap is a key determinant of the power of QE. We evaluate MPCs at

different horizons, the importance of which has been emphasized by Auclert et al. (2018).

Figure 5 shows the average MPC gap, cumulated over time in both the model and the data,

23

Figure 5: Gap in Marginal Propensities to Consume out of liquid and illiquid wealth.

MPC liquid - MPC illiquid

0 2 4 6 8 10 12 14 16 18

horizon (quarter)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cros

s-se

ctio

nal a

vera

ge

data

model

Notes: the figure shows the cumulative average MPCs across households. In the model, MPCs are computedas the response to a one-time surprise increase in additional wealth in deposits / mutual funds, keeping allprices constant. The range shown for “data” is computed as the dynamic MPC function for liquid assetsprovided by Fagereng et al (2018), minus a range of estimates for the MPC out of stock mutual funds in Table3 of Di Maggio et al (2018). The latter is only provided at a one year horizon.

where the latter is computed based on estimates by Fagereng et al. (2018) and Di Maggio et al.

(2018). The gap in the model is large and increases over time. Because of labor supply effects,

it does not converge to one. Relative to the data, the model initially somewhat undershoots,

whereas for medium-run horizons there is some overshooting. At longer horizons, the MPC

gap in the model and the data is very similar. Given that this MPC gap and its dynamic

shape have not been targeted in the calibration procedure, we conclude that overall the model

does a reasonable job in accounting for this key piece of empirical evidence.

One might also wonder about the ability of households to smooth consumption in the

face of unemployment shocks. The left panel of Figure 6 plots the model-implied drop in

consumption upon job loss, as a function of the household’s position in the distribution of

liquid wealth (deposits). The line is downward-sloping, as households with more liquid wealth

are better able to cushion the consumption effect of becoming unemployed. The average

consumption drop is 22 percent, which is very close to the empirical estimate of Chodorow-

Reich and Karabarbounis (2016), who report a 21 percent drop based on data from the

Consumer Expenditure Survey.

The right panel of Figure 6 shows the composition of the “consumption cushion” upon

job loss. The cushion is defined as the difference between the drop in labor income and the

drop in consumption upon job loss. Between 30 and 40 percent of the consumption cushion is

financed by unemployment benefits, depending on the amount of liquid assets owned by the

24

Figure 6: Consumption behavior upon job loss.

0 20 40 60 80 100position in liquid wealth distribution (%)

0

10

20

30

40

50

60

70

80

90

100

%

Consumption drop upon job loss

0 20 40 60 80 100position in liquid wealth distribution (%)

0

10

20

30

40

50

60

70

80

90

100

cont

ribut

ion

(%)

Consumption cushion upon job loss

model

Karabarbounis and Chodorow-Reich (2016)

withdrawal deposits

withdrawal mutual fund

unemployment benefits

Notes: the black line in the left panel plots 100 · [1− Ct(i)/Ct−1(i)] for households who lost their jobs inthe current quarter t. The right panel shows the contributions of the components of the “consumptioncushion,” for households who lost their job in the current quarter t. The consumption cushion is definedas cusht(i) ≡ wt−1Nt−1(i) − [Ct−1(i)− Ct(i)]. The contribution of unemployment benefit is computed as(ΘU − ΘE)/cusht(i), the contribution of liquidation of mutual funds as µ/cusht(i), and the contribution ofdeposit withdrawal is computed as [Dt(i)−Dt−1(i)] /cusht(i). Both panels show outcomes in the deterministicsteady state.

households. These benefits directly help households alleviate the fall in consumption. Around

30 percent of the consumption cushion is due to additional withdrawals from mutual funds

after job loss. The remainder of the cushion is due to the withdrawal of deposits.

4.3 Equilibrium responses to a QE shock

Before applying the model to the Great Recession, we conduct a simple experiment which helps

to understand how QE affects the macroeconomy. To this end, we consider an exogenous shock

to QE, i.e. a positive innovation to zQEt . For transparency, we consider a version with Real

Reserve Targeting (RRT, i.e. ξQEΠ = ξQEY = 0), so that there is no feedback from output

and inflation to real reserves. The shock is scaled such that the purchase of long-term debt

(and hence the increase in real reserves) is equivalent to one percent of annual steady-state

output. We further assume a persistence coefficient of λzQE = 0.9, which implies that the QE

expansion has a half life of about 1.7 years.

The black solid lines in Figure 7 plot the responses to the QE expansion in the base-

line model. Immediately after the central bank starts purchasing government debt, output

increases by 1.09 percent on impact and by 0.61 percent on average during the first year fol-

25

Figure 7: Responses to an expansionary QE shock.

0 5 10 15 20

quarter

0

0.5

1

1.5%

of s

.s. a

nnua

l out

put

real reserves

0 5 10 15 20

quarter

-1

0

1

2

3

%-p

oint

s

inflation

0 5 10 15 20

quarter

-0.5

0

0.5

1

1.5

%

output

baselineflexible pricesquick exit

0 5 10 15 20

quarter

-1

0

1

2

3

%

real wage

Notes: the shock is scaled such that real reserves increase by an amount equivalent to one percent of annualoutput on impact. The policy rule assumes ξQEΠ = ξQEY = 0 (Real Reserve Targeting). The baseline andflexible price responses assume a persistence coefficient of λzQE = 0.9, whereas the “quick exit” responseassumes a persistence coefficient of λzQE = 0.6.

lowing the intervention.26 This is a strong effect. For comparison, in a version of the model

with conventional policy, setting ξRΠ = 1.5 and ξRY = 0 and otherwise identical parameter

values, output increases by about 0.1 percent on impact, after an innovation to the interest

rule set to be equivalent to a 100 basis points fall in the annualized policy rate.

Inflation also responds strongly to QE. One year after the intervention, the price level has

increased by 1.16 percent. Real wages also increase substantially, reflecting the increase in

labor demand which ensues from the increase in goods demand. As QE is rolled back, this

increase dies out to almost zero after two years.

Given that households do not directly receive any of the additional liquidity from the

central bank or the mutual funds, how are they able to finance both an increase in consumption

and an increase in liquid asset holdings? Figure 8 decomposes the change in the income and

the expenditure side of the aggregated household budget constraint, in the initial quarter

following the shock. As anticipated, both consumption expenditures and deposit holdings

26If we use the simple formula for the same QE expansion, the direct effect on output amounts to 0.63-0.095= 0.54 percent.

26

increase (together by 1.3% of annual steady-state output).27 Most of the change on the

income side is due to the labor income component (about 62%), which increases by enough

to finance a large chunk of the additional deposits. A significant part (about 39%) is due to

a decline in taxes, which occurs as the government benefits from an increase of the price at

which it sells new debt and from a downward revaluation of its stock of existing debt due to

higher inflation. Finally, a downward revaluation of the households’ deposits reduces their

spending capacity, although this effect is small.

Figure 8: Decomposition of aggregate household budget constraint.

expenditure side

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

% o

f ann

ual s

tead

y-st

ate

outp

ut

DepositsConsumption

income side

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8Labor incomeTaxesRevaluation deposits

Notes: Changes in the component of the aggregated household budget constraint, in response to a QE shockin the baseline model. The decomposition shown is for the initial quarter.

Next, we consider a version of the model with flexible prices (i.e. setting φ = 0), illustrated

by the blue dashed lines in Figure 7. In this case, the effect on output is much smaller, whereas

there is a large spike in inflation on impact (2.5 percentage points). Intuitively, the increase

in prices strongly dampens the increase in goods demand following the QE intervention. That

is, indirect effects mostly offset the direct effects. Real wages remain constant under flexible

prices. The fact that the QE shock still creates a small increase in output under flexible prices

is associated with labor supply effects and re-distributions of nominal wealth.

Finally, the green solid line in Figure 7 shows the effects in the baseline model when the

QE expansion is less persistent, setting λzQE = 0.6, so that the exit is quicker. In that case,

the initial expansion in output and inflation is much smaller. Intuitively, the contractionary

effects associated with the quick unwinding of QE are immediately anticipated following the

intervention, which dampens its effectiveness on impact. Thus, the overall power of a QE in-

27Recall that in the experiment, all asset purchases happen in the initial quarter. Therefore, the initialincrease in deposits is relatively large compared to the consumption response. After the initial period, however,the exit sets in and the deposit response takes the opposite sign, while the consumption response is still positive.

27

tervention depends crucially not only on the degree of price stickiness, but also on expectations

regarding its persistence.

Alternative assumptions on mutual fund management and fiscal policy. In the

model, households optimally choose to keep their mutual fund withdrawals constant over

time, and hence Xt does not respond to QE. Given this and the fact that mutual funds

themselves have no demand for liquidity, the funds use the deposits obtained following QE

to purchase new long-term debt, effectively replacing the debt sold to the central bank. This

in turn drives up the price of debt, lowering borrowing costs for the government, allowing for

lower taxes, and freeing up budget for the households to hold more deposits, in additional to

the extra wage income obtained in equilibrium.

While the constant withdrawal appears in line with the empirical evidence on mutual fund

outflows, see Section 2, it is by no means crucial for the result. In Appendix D, we consider

a version of the model in which Xt is modeled as a payout. The payout is decided by the

representative mutual fund and can respond flexibly to QE. It turns out that the response of

macro variables is the same as in the baseline. Intuitively, it is still the case that the liquidity

ends up in the hands of households, although this time households finance the liquidity in

part with the extra payout from the fund. To understand this result in a more formal way, it

is useful to consolidate the budget constraints of mutual funds (7), the treasury (11), and the

central bank (12). This leads to:

Xt −Ψt − Tt +Gt +Rt−1

Πt

Mt−1 = Divt +Mt.

That is, QE has an impact on the net flow Xt − Ψt − Tt from the consolidated entity to the

households. Note that Xt − Ψt − Tt enters directly into the households’ budget constraint.

What matters for the household is thus the impact on the net flow Xt − Ψt − Tt from QE,

rather than variation in Xt, Ψt, and Tt individually. Note also that the amount of government

debt, Bt, drops out of the consolidated budget constraint.

We also consider a model version in which Xt, Ψt, and Tt all remain constant over time,

and government expenditures Gt adjust in order to satisfy the consolidated budget constraint

given above. In this case, QE raises aggregate demand directly via government expenditures

and we again find that QE has strong positive effects on output and inflation.28

28We finally consider a version in which all of Xt, Ψt, Tt, and Gt are kept at their steady-state levels, seethe Appendix. In this version, QE generates similar responses of macro variables compared to the case whrewe allow Gt to vary. One can see from the consolidated budget constraint that QE now works purely throughthe adjustments of inflation and dividends. Notice that in this version QE is essentially replacing long-termdebt with short-term debt of the consolidated entity.

28

Forward guidance and helicopter drops. In Appendix E, we consider two other uncon-

ventional policy options. The first one is Forward Guidance, i.e. statements about future

interest rate policy. We show that when the central bank also uses QE, there is no “Forward

Guidance Puzzle”, i.e. the policy only has small macroeconomic effects. We also consider

Helicopter Drops, i.e. outright transfers to the households (via the fiscal authority), financed

by the issuance of reserves. We find that the effects of Helicopter Drops on aggregate output

are smaller but slightly more persistent than the effects of QE.

4.4 The macro effects of QE since the Great Recession

We now quantify the macro effects of QE on the U.S. economy since the Great Recession, when

the nominal interest rate was (almost) at the zero lower bound, starting from 2008Q3. To this

end, we structurally estimate the model, using data on the deviation of real output from its

potential, the government-spending-to-output ratio, the deposits-to-output ratio, and year-

over-year CPI inflation. To measure potential output, we use estimates from the Congressional

Budget Office. The data, as differences relative to the 2008Q3 levels, are shown in Figure 9.

We estimate the version of the model with QE, and four shocks: cost push shocks, TFP

shocks, government expenditure shocks, and QE shocks. We assume that all four shocks follow

first-order autoregressive processes, as before, and we estimate the associated parameters.

One might think of QE shocks as discretionary policy interventions. At the same time we also

allow for systematic responses via the QE rule, and we estimate the stabilization coefficients on

output and inflation. The remaining parameters are calibrated as described above. The model

is estimated by Maximum Likelihood, using the Kalman filter combined with a perturbation

method.

Table 2: Estimated parameter values.

Parameter Description value std. error t-statistic

λε persistence cost push shock 0.979 0.036 27.203

λA persistence TFP shock 0.854 0.217 3.939

λG persistence G shock 0.996 0.006 189.626

λzQE persistence QE shock 0.714 0.071 10.107

σε st.dev. cost push innovation 0.033 0.037 0.890

σA st.dev. TFP innovation 0.009 0.001 6.433

σG st.dev. G innovation 0.007 0.001 7.647

σzQE st.dev. QE innovation 0.170 0.038 4.508

ξQEY QE coef. output 0.240 0.090 2.662

ξQEΠ QE coef. inflation -0.315 0.091 3.459

Notes: parameters have been estimated using Maximum Likelihood. See the main text and the appendix fora description of the data series and the sample period.

29

Table 2 displays the estimated parameter values. The implied magnitude of QE shocks is

substantial, with one standard deviation being 17% of the steady-state amount of reserves.

At the same time, we also find a systematic component to the QE rule. In particular, we

estimate the coefficient on inflation ξQEΠ to be significantly negative. When the annualized

inflation falls by 1 percentage points, the central bank buys assets and creates reserves worthy

of 0.315 percent of annual GDP (roughly 0.51 trillions of US dollars in 2012). The point

estimate for ξQEY is positive, though not as significant as ξQEΠ . When output falls by 1%, the

central bank creates reserves worthy of 0.24 percent of annual GDP (roughly 0.39 trillions of

US dollars in 2012). Thus, QE since the Great Recession appears best described as a mix of

systematic and discretionary interventions.

With the estimated model at hand, we quantify the effects of active QE on the macro

economy. We do so by simulating a counterfactual in which we both set ξQEY = ξQEΠ = 0

and shut down the QE shocks. In this case, real reserves and deposits remain fixed at their

steady-state levels throughout the sample period.

Figure 9: The impact of QE in the U.S. since the Great Recession.

Notes: data series and a counterfactual simulation without QE. For a description of the data series, see themain text and also Appendix A. In the counterfactual, we set ξQEY = ξQEΠ = 0 and shut down the QE shocks.Grey areas denote rounds of QE purchases by the Federal Reserve. Time series have been normalized around2008Q3.

Figure 9 shows the results of this counterfactual. The difference between the two lines in

30

the upper left panel captures the Fed’s asset purchases, which resulted in large-scale deposit

creation. The lower right panel shows that QE had a large positive impact on aggregate

output. Without active QE interventions, the recession would have been much deeper. For

example, the fall in output relative to potential would have been about 7.5%, compared to

about 1.7% in 2008Q4. Note that this effect is much larger than the direct effect computed

in Section 2.2. Thus, in the initial years the direct effects of QE were amplified by general

equilibrium effects.

However, after the recession the QE effects gradually fade out, and later on even switch

sign, even though the policy itself had not been rolled back. Over longer horizons, general

equilibrium effects turn from an amplifying into a dampening factor, as prices have had more

time to adjust.29 Indeed, the effects during the second and third round of QE are considerably

smaller than during the first round, as the later rounds had been anticipated by the private

sector. The counterfactual analysis implies that inflation could have been higher than the

data since the end of 2012. At the end of 2015, inflation would have been 3.5 percentage

points higher if the central bank never purchased any assets.

5 The efficacy of QE versus conventional policy

In this section, we evaluate the efficacy of QE, drawing a comparison to conventional policy.

We compare the two policies along three dimensions. First, we consider their ability to anchor

expectations and thereby rule out expectations-driven fluctuations. Second, we evaluate their

success in stabilizing output and inflation, traditionally a key objective of central banks.

Finally, we consider their ability to mitigate the welfare costs of business cycles. The latter

is affected not only by output and inflation volatility, but also by considerations regarding

self-insurance and inequality.

5.1 Anchoring expectations

A widely appreciated objective of monetary policy is to anchor expectations. When expecta-

tions become disanchored, high inflation or deflation may arise purely due to changes in beliefs

about the future. The “Taylor principle”, arguably the most celebrated policy recommenda-

tion of the NK model, concerns precisely this issue. The principle states that the central bank

should let the nominal interest rate respond sufficiently aggressively to movements in inflation

and/or output. When policy satisfies the Taylor principle, expectations remain anchored and

29Figure 9 also shows a large positive effect of QE on inflation, although this effect was relatively short-livedand switched sign during 2013. The latter result reflects the overshooting of inflation also visible in Figure 7.In these responses, the effect on inflation dies out much faster than the effect on output.

31

Figure 10: Determinacy under an interest rate rule and a QE rule.

QE rule

-5 0 5-5

-4

-3

-2

-1

0

1

2

3

4

5Interest rate rule

ind

etermin

acy

0 1 2 3 4 5R

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

R Y

determinacy determinacy

indeterminacy

instability

Notes: determinacy outcomes are obtained by analyzing the eigenvalues of the linearized system of modelequations, after a first-order perturbation around the deterministic steady state. “Determinacy” refers to anoutcome in which the number of eigenvalues inside the unit circle coincides with the number of state variablesin the system, whereas “Indeterminacy” (“Instability”) refers to an outcome in which there are more (fewer)eigenvalues inside the unit circle than there are state variables.

belief-driven fluctuations are ruled out. Clarida et al. (2000) argue that a switch from passive

to aggressive interest rate policy since Fed president Paul Volcker contributed to a dramatic

decline in inflation and output volatility.

In December 2008, the Federal Funds rate was reduced to (almost) zero and stayed there

until 2016. Potentially, this opened up the door to a disanchoring of expectations, as the ability

of interest rate policy to respond to output and inflation had been curtailed. However, this does

not happen if subsequent unconventional policy is able to successfully replace conventional

policy and thus re-anchor expectations.

We now use the model to investigate the ability of QE and conventional policy to avoid

belief-driven fluctuations. In more technical terms, we investigate whether the equilibrium is

locally determinate around the steady state under each of the two policy rules. To this end,

we consider a range of values for the stabilization coefficients of the interest rate rule and the

QE rule. For ease of interpretation, we introduce re-scaled versions of the QE stabilization

coefficients: ξQEY ≡ M4YξQEY and ξQEΠ ≡ M

16YξQEΠ . Here, ξQEY is the response of reserves –in units

of annual steady-state output– to a one percent increase in output. Moreover, ξQEΠ is the

response of reserves –again in units of annual steady-state output– to a one percentage point

increase in annualized inflation.

Figure 10 illustrates the outcomes regarding local determinacy under the various policy

configurations. The left panel shows outcomes under the QE rule, for a range of values of

32

ξQEY and ξQEΠ . The figure shows that local determinacy does not arise under all combinations

of the QE coefficients. In particular, it may fail to hold when the coefficient on inflation is

sufficiently negative. The threshold for determinacy lies at around ξQEΠ = −0.5.30 By contrast,

the output coefficient ξQEY has relatively little impact on equilibrium determinacy.

Still, the equilibrium is locally determinate under a wide range of realistically achievable

QE parameters. For example, determinacy is obtained under a Real Reserve Targeting (RRT)

policy which sets ξQEΠ = ξQEY = 0. Intuitively, by targeting the amount of real reserves, the

central bank creates a real anchor to the expectations of households and firms. Interestingly,

the determinacy frontier under QE is upward sloping, i.e. responsiveness to output is not

a substitute for responsiveness to inflation. This property is related to the (positive) co-

movement of output and inflation in response to shocks to be illustrated later.

The right panel of Figure 10 shows outcomes under conventional policy, i.e. an interest

rate rule. A substantial region of the parameter space implies local indeterminacy. Only if

ξRΠ and ξRY are sufficiently high do we obtain determinacy. This reflects the Taylor Principle,

which states that in a basic NK model ξRΠ > 1 is typically a necessary and sufficient condition

for determinacy (given ξRY = 0). Figure 10 shows that this result applies approximately also

to the incomplete-markets model considered here.

5.2 Managing aggregate fluctuations

Having established that QE can be an effective instrument to anchor expectations, we now

study its power in mitigating fluctuations in aggregate output and inflation, relative to con-

ventional policy. To this end, we set up a direct horse race between the two policy rules. Let

us introduce the following loss function:

L(ω) = ωV ar(Yt) + (1− ω)V ar(Πt).

The loss function is a weighted average of the unconditional variance of output and inflation,

where the parameter ω ∈ [0, 1] controls the relative weight given to output volatility.

Our purpose is to compare the ability of QE and conventional policy rules to minimize the

loss L. To compare the two types of policy on a fair basis, we evaluate the two rules at the

optimal values of the policy rule coefficients, given L(ω). In this way, the two policy rules are

each given the best possible chance in achieving the objective. To implement this strategy,

30Recall that this value of the policy coefficient means that in response to a 1 percentage point declinein annual inflation, the central bank buys government debt worth 0.5 percent of annual GDP. To betterunderstand why determinacy is not obtained under sufficiently negative values of ξQEΠ , note from Figure 7

that expansionary QE initially increases inflation, but reduces it over longer horizons. When ξQEΠ < 0, apositive feedback between QE and inflation arises over longer horizons, which undermines a unique and stableequilibrium path.

33

we first search over the values of, respectively, {ξQEΠ , ξQEY } and {ξRΠ , ξRY } which minimize L(ω),

and do so for a range of value of ω between zero and one, each time computing the minimized

objective. We also compute L(ω) under Real Reserve Targeting (RRT). In that case, we

simply fix ξQEΠ = ξQEY = 0 rather than searching for optimal parameters.

We consider, individually, three types of aggregate shocks: cost push shocks, TFP shocks,

and government expenditure shocks, as defined previously. For comparability, we assume

in all three cases a persistence parameter of λzQE = 0.9, and we set the volatility of the

shock innovations, σz, such that under an interest rate rule with ξRΠ = 1.5 and ξRY = 0, the

unconditional standard deviation of output is one percent.

Figure 11: Loss function under different shocks and policy configurations.

0 0.2 0.4 0.6 0.8 1

weight on Y

0

0.5

1

1.5

2

2.5

3

3.5

Loss

cost push shock

0 0.2 0.4 0.6 0.8 1

weight on Y

0

0.5

1

1.5

2

2.5

3 TFP shock

0 0.2 0.4 0.6 0.8 1

weight on Y

0

0.5

1

1.5

2

2.5

3

3.5

4G shock

QE ruleReal Reserve TargetingInterest rate rule

Notes: value of 100 · L(ω) as a function of the output weight ω, under the optimal interest rate rule, theoptimal QE rule, and the RRT rule. The optimal rules were found by searching for the values of the policycoefficients ({ξQEΠ , ξQEY } in case of QE, {ξRΠ , ξRY } in case of conventional policy) which minimize L(ω), givenω. To this end, we constructed a large grid for each of the two sets of policy coefficients, and solved the modelat each of the grid points. Next, we constructed a grid for ω and searched for the policy coefficients whichminimize L(ω) for a given value of ω. We did so individually for each of the three types of aggregate shocks,under our baseline calibration with sticky prices.

Figure 11 plots the objective under the optimal interest rate rule, the optimal QE rule,

and under RRT, for the three types of shocks and the full range of output volatility weights.

A striking outcome revealed by the figure is that, under a wide range of configurations, the

QE policy rule is substantially more successful in stabilizing business cycles than the interest

rate rule. This is particularly the case for intermediate values of ω, i.e. when the objective

is to stabilize both output and inflation. When the objective is mainly to stabilize inflation

(i.e. ω is close to zero) or output (i.e. ω is close to one), the performance of the two rules is

similar. We thus find that QE is not only an effective substitute for conventional policy to

34

anchor expectations, but also to simultaneously stabilize aggregate output and inflation.

RRT is by construction less successful than the optimal QE rule, since RRT is nested in

the QE rule. But interestingly, in a wide range of cases RRT actually performs better than

the optimal interest rate rule, even though under RRT the policy rule coefficients have been

fixed rather than optimized. Therefore, despite being a simple policy, RRT rivals conventional

interest rate policy, both in terms of anchoring expectations and in terms of stabilizing output

and inflation.

Figure 12: Responses to a cost push shock.

0 5 10 15 20-0.5

0

0.5

1

1.5

%-p

oint

s

inflation

Real Reserve Targetingoptimal QE rule (Y weight = 0.75)optimal R rule (Y weight = 0.75)

0 5 10 15 20-1

-0.5

0

0.5

1

%

output

0 5 10 15 200

0.5

1

1.5

%-p

oint

s

nominal interest rate

0 5 10 15 20-1

0

1

2

3

4

% o

f s.s

. out

put

real reserves

Notes: responses to a cost push shock (i.e. a shock to εt) under the optimal QE rule, the optimal interest raterule, both given an output weight of ω = 0.75, and the RRT rule. See the main text and the note of Figure11 for details.

To understand why the QE rule and RRT are relatively successful in stabilizing both

output and inflation, it is useful to consider the co-movement between the two variables.

When the two variables co-move imperfectly, it is generally difficult to stabilize both variables

with conventional monetary policy, given that a change in the interest rate tends to move both

variables in the same direction. This trade-off is illustrated in Figure 12, which shows the

responses to a cost push shock. Under the (optimal) interest rate rule, output and inflation

move in opposite directions following the shock. If the interest rate rule were more aggressive

on inflation, the volatility of inflation would be dampened but the volatility of output would

35

be increased, and vice versa.

Under QE and RRT, however, output and inflation co-move much more positively, which

eases the policy trade-off. To understand what gives rise to the positive co-movement, note

that the nominal interest rate is pegged under QE. Therefore, the expected real interest rate

is given by EtR

Πt+1, where R is fixed. Since a persistent cost push shock triggers a persistent

increase in inflation, the real interest rate must fall. A decline in the real interest rate in turn

stimulates aggregate consumption demand, pushing up output. Hence, output and inflation

move in the same direction. By contrast, under conventional policy the real interest rate is

given by EtRt

Πt+1, where Rt can move. Under the Taylor principle, the central bank increases the

nominal interest rate more than one-for-one with inflation, which tends to create an increase

rather than a decrease in the real interest rate. This in turn lowers consumption demand and

pushes down output, exacerbating the negative co-movement.

5.3 Inequality and welfare effects

So far, we have found that QE can be very effective in achieving two traditionally important

central bank objectives: anchoring expectations and stabilizing the aggregate business cycle.

We now consider the broader welfare implications. In representative-agent NK models, welfare

is typically well approximated by a weighted combination of only the volatility of output and

inflation. In that case, the sort of analysis we conducted in the previous subsection could also

be used to evaluate the broader welfare effects of monetary policy rules.

In heterogeneous-agents economies like the one considered here, there is no direct map-

ping from aggregate output and inflation volatility to welfare. With incomplete markets and

idiosyncratic risk, welfare also depends on factors concerning consumption insurance and in-

equality, as emphasized by for instance Bhandari et al. (2017). Therefore, policies which are

successful in stabilizing output and inflation might be undesirable from a broader welfare per-

spective. Moreover, optimal policy might sacrifice stability of output and inflation in order to

avoid undesirable side effects on inequality.

To investigate these issues, we introduce the following utilitarian welfare objective, taken

from a timeless perspective:

W = E∫ 1

0

U (C(i), N(i)) di,

where E is the unconditional expectation operator. Given this objective, we repeat the exercise

of the previous subsection. That is, we again search for the policy parameters which optimize

the objective (this time W rather than L), and then evaluate welfare under the optimal policy

coefficients. As before we consider cost push shocks, TFP shocks, and government expenditure

shocks, and we also evaluate the objective under RRT.

36

Figure 13: Welfare impact of business cycles (% of s.s. consumption).

QE rule

-0.012679

-0.013989

-1 -0.5 0 0.5 1 1.5 2-1

-0.5

0

0.5

1

1.5

2

-0.010074

-0.010211

-1 -0.5 0 0.5 1 1.5 2-1

-0.5

0

0.5

1

1.5

2

-0.010186

-0.010725

-1 -0.5 0 0.5 1 1.5 2-1

-0.5

0

0.5

1

1.5

2

Interest rate rule

-0.025-0.02

-0.02-0.015

-0.015

-0.0

1

-0.0

1

-0.0

05

-0.00032116

1 1.5 2 2.5 3 3.5 40

1

2

3

4

-0.011-0.011

-0.0

11

-0.01

-0.01-0

.01

-0.009

-0.009

-0.0

09-0.0081006

1 1.5 2 2.5 3 3.5 40

1

2

3

4

-0.0

18-0

.016

-0.0

14

-0.012

-0.012

-0.010514

1 1.5 2 2.5 3 3.5 40

1

2

3

4

<-0.3<-0.3

<-0.3

<-0.3

Notes: Welfare impact of business cycles as a function of the policy coefficients, expressed as percentageequivalents of average steady-state consumption. Yellow (dark blue) areas denote the highest (lowest) levelsof welfare. Red round markers indicated the optimal policy, whereas blue square markers indicate RRT. Weconstructed a grid for the policy coefficients, solved the model at each grid point, and computed welfare (W).We then considered a steady-state version of the model with an additional lump-sum tax, τ c. We then solvedfor the level of τ c which renders welfare in this steady-state model exactly equal to the model with shocks,on each of the grid points. The figure plots −τ c as a percentage of average steady-state consumption. Thetop row shows results for the model with cost push shocks, the middle row for a model with TFP shocks, andthe bottom row for a model with government expenditure shocks. Shock volatility parameters were calibratedsuch that under an interest rate rule with ξRΠ = 1.5 and ξRY = 0, the volatility of aggregate output is onepercent.

37

Figure 13 is a contour plot of welfare outcomes for different values of the policy coefficients

under QE and the interest rate rule. Red markers indicate the optimal policy coefficients,

and the numbers next to the markers denote the associated welfare outcome. The latter is

measured as a welfare cost of business cycles, expressed in percentages of average consumption

in the steady state. Blue markers indicate the outcomes of Real Reserve Targeting (RRT).

Three striking results follow from Figure 13. First, welfare under the optimal QE policy and

RRT is generally lower than welfare under the optimal interest rate rule. The only exception

is the government expenditure shock, under which the optimal QE policy performs marginally

better. Under the cost push shock, however, the optimal QE policy performs substantially

worse than the optimal interest rate rule. Second, under all three shocks the optimal QE

rule is still quite successful in mitigating the welfare costs of business cycles, generating a

loss equivalent to only 0.01 percent of consumption.31 However, away from the optimal policy

coefficients welfare can drop sharply. The dark blue areas in the left panels in Figure 13 denote

configurations for which the welfare cost exceeds 0.3 percent of steady-state consumption; in

some cases the cost is much more than that. Under the interest rule, by contrast, welfare is

much less sensitive to the precise policy coefficients. Third, the optimal QE policy is similar

to RRT, both in terms of the coefficients and in terms of the associated welfare outcome.

To understand these results, we consider more in detail the side effects that QE can have

on insurance and inequality. Note that when ξQEΠ and ξQEY are not zero, the aggregate amount

of reserves and hence the supply of deposits varies over the business cycle, as the central bank

adjusts the amount of QE in response to changes in inflation and output. The variation in

reserves in turn creates time variation in the supply of deposits, and hence in the extent to

which households can insure themselves against idiosyncratic income risks. In addition, QE

induces time variation in wages which creates redistributions.32 These adverse side effects on

welfare may prompt the central bank to keep real reserves more or less constant even if this

means that fluctuations in aggregate output and inflation are larger than they could be under

more active QE policy. By contrast, conventional policy does not directly affect the amount

of insurance, and hence conventional policy comes with less severe side effects on welfare.

These points are illustrated in Figure 14, which plots welfare in the baseline model as well

as in a version with flexible prices, both under different policy configurations. For simplicity,

we vary only the stabilization coefficients on inflation i.e. we set ξQEY = ξRY = 0. The right

column of Figure 14 show that under conventional policy, the removal of price stickiness creates

a much flatter welfare function. Intuitively, under flexible prices monetary policy is unable

31Recall that the shock volatility parameter σz was calibrated such that under an interest rate rule withξRΠ = 1.5 and ξRY = 0, the volatility of aggregate output is one percent.

32In particular, note that wages increase following expansionary QE. Therefore, the employed gain at theexpense of the unemployed, which reduces welfare.

38

Figure 14: Welfare impact of business cycles (% of s.s. consumption) under sticky and flexibleprices.

-0.2 -0.1 0 0.1 0.2-0.06

-0.04

-0.02

0

Cos

t Pus

h sh

ock

QE rule

-0.2 -0.1 0 0.1 0.2-0.04

-0.03

-0.02

-0.01

TF

P s

hock

-0.2 -0.1 0 0.1 0.2-0.0108

-0.01075

-0.0107

-0.01065

G s

hock

sticky pricesflexible prices

1 1.2 1.4 1.6 1.8 2-0.03

-0.02

-0.01

0Interest rate rule

1 1.2 1.4 1.6 1.8 2-0.014

-0.012

-0.01

-0.008

1 1.2 1.4 1.6 1.8 2R

-0.01074

-0.01072

-0.0107

-0.01068

Notes: welfare impact of business cycles, expressed in percentage consumption equivalents, in the baselinemodel and under flexible prices (φ = 0), setting ξQEY = ξRY = 0. See the note of Figure 9 for a description ofthe procedure.

to affect output and inflation via the Phillips Curve. As the traditional lever of monetary

policy has been removed, the precise aggressiveness of the policy becomes close to irrelevant

to welfare.

By contrast, removing price stickiness under the QE rule creates more curvature in the

welfare function. That is, when prices are flexible the values of the stabilization coefficients

matter even more for welfare. This indicates that under the QE rule, much of the welfare

effects of the stabilization policy operate not via the traditional monetary policy channel of

the NK model, but rather via direct side effects on welfare and insurance. Therefore, the

optimal QE rule is geared towards avoiding the side effects which come with time-variation in

deposit supply, rather than towards stabilizing aggregate output and inflation.33

Remark: The redistributional side effects of monetary policy depend on the precise struc-

ture of the economic environment as well on the behavior of fiscal policy. Nonetheless, the

mechanisms discussed above suggest a tendency for the side effects of QE to be relatively

33Under flexible prices, fluctuations in inflation are typically larger than under sticky prices. Given a QErule with a certain non-zero coefficient on inflation, this generates larger fluctuations in the amount of reserves,and hence deposits. Therefore, welfare becomes more sensitive to the QE policy coefficients.

39

strong, compared to those of conventional policy.34

Permanent QE. Finally, we explore the possibility of “permanent QE,” i.e. a purchase

of government debt which is never reversed. Potentially, such a policy could improve welfare

as it increases the amount of liquidity in the hands of households, enabling them to better

self-insure against idiosyncratic income risk. One may therefore wonder if it is optimal for the

central bank to conduct large permanent QE operations, flooding the market with liquidity.

We explore this possibility in the model, by simulating the effects of a (semi-) permanent

increase (or decrease) in real reserves implemented via QE. We do so for a range of magnitudes

of the QE intervention and compute the impact on welfare. Figure 15 shows the results of this

exercise. The red dashed line shows the level of welfare in the new steady state as a function

of the level of reserves in the new steady state, excluding the transition path. As expected,

welfare is increasing in the amount of steady-state liquidity (reserves).

Figure 15: Welfare impact of permanent QE.

0.085 0.09 0.095 0.1 0.105 0.11 0.115

real reserves

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08Welfare relative to initial steady state

including tansitionexcluding transition

Notes: The figure plots the effect of permanent QE on the present value of welfare, starting from the initialsteady state (red round marker), including and excluding the transition path. The effect of permanent QE on

the economy was computed by simulating a one-time permanent increase in zQEt .

The blue solid line again shows the effect of permanent QE on welfare, but this time

including welfare along the transition path towards the new steady state. Interestingly, it is

no longer the case that welfare is monotonically increasing in the size of the QE intervention.

This happens as the transition to the new steady state is costly from a welfare perspective, due

to the side effects created by QE. For instance, the announcement of permanent QE creates

a strong and persistent increase in aggregate demand, and hence the wage rate which in turn

34For example, an interest rate cut stimulates the economy but also generates an income loss for wealthysavers holding interest-bearing assets. For QE policy, however, wealthy savers benefit more by being able tosave even more.

40

reduces firms’ markups.35 The latter two effects create a redistribution from the unemployed to

the employed, which is relatively strong and persistent compared to a transitory QE expansion

we had before and which is unfavorable from a social welfare perspective.

We find that welfare is maximized at a level of reserves that is around 13 percent higher

than in the original steady-state level (red marker), which corresponds to a 3.8 percent in-

crease in reserves as a fraction of annual output. Thus, the scope for improving welfare by

permanently flooding the market with liquidity is rather limited, once the transition path to

the new steady state is taken into account.

6 Concluding remarks

We found that QE can be an effective tool to stabilize the economy and that it has greatly

dampened the Great Recession. However, it is generally not desirable to replace a conventional

interest rate policy with QE, as the latter tends to create strong side effects on inequality and

hence welfare.

In future work, it would be interesting to extend the model. For example, it would be

straightforward to extend the model with physical capital or to endogenize unemployment.

Also, it could be worthwhile to explore different specifications of the adjustment cost, as

suggested in Appendix B. These extension can add realism to the model without compromising

its tractability.

We conclude with a note on conventional monetary policy for future research. We have

assumed that the central bank directly controls the short-term interest rate, following the

NK literature. In practice, most central banks implement interest rate policy through open

market operations. That is, they lower short-term interest rates by purchasing T-bills. To the

extent that deposits are more liquid than T-bills, the stimulating effects of QE emphasized in

this paper may apply to conventional policy as well.

35Recall that the strength of impact of QE on aggregate outcomes depends on the persistence of the balancesheet expansion.

41

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AppendixSupplemental material for “Quantitative Easing” by Wei Cui and Vincent Sterk

A. Data

A1. Data on household liquidity

Flow of Funds versus SCF. The data shown in Figure 1 in the main text are taken from theFlow of Funds (FoF). An important advantage of the Flow of Funds data is that they are constructedfrom administrative sources, that they aggregate up to the macro level and that their frequency isrelatively high (quarterly). However, a possible concern regarding these data is that the sector“Households and Non-Profit Organizations” is constructed as a residual. For this reason, we drawa comparison to data from the Survey of Consumer Finances (SCF). A disadvantage of the SCF isthe relatively low frequency (once every 3 years) and the limited number of survey participants. Onthe other hand, the data do give a direct insight into households’ finances.

Table 3: Checkable deposits and currency on the Household Balance sheet.year ratio SCF to FoF year ratio SCF to FoF

1989 0.70 2007 8.42

1992 0.49 2010 2.68

1995 0.54 2013 1.62

1998 0.93 2016 1.60

2001 1.51 1989-2016 2.06

2004 2.10

Source: Batty et al. (2015), Table 1.

To compare household liquidity in the FoF to the SCF we draw on Batty et al. (2015), whodocument the ratio of households’ checkable deposits and currency in the FoF relative to the SCF,see Table 3. The table shows a highly unusual spike in 2007. From 2010 onward however, when mostof the increase in household deposits in the FoF took place, the ratio is relatively stable. Based onthe SCF data, we constructed an adjusted time series for household deposits. To do so, we multipliedthe FoF series by the ratio shown in Table 3, linearly interpolated between SCF releases. Figure 16shows the increase in the original flow of funds series, as well in the SCF adjusted series. In bothseries there is a similarly large increase in household deposits. We conclude that the increase inhousehold deposits since QE as observed in the FoF is consistent with the SCF.

Nonprofits. Another potential concern with the SCF data is that it combines households andnon-profit organizations. Figure 17 shows data from Holmquist (2019), who splits up the total seriesinto a household component and a nonprofit component. As can be seen from the figure, the relativecontribution of non-profits has been small and fairly constant over time.

A2. Data used in the calibration and estimation

For households’ deposits data, we use “Households and Nonprofit Organizations; Checkable Depositsand Currency” from the U.S. Flow-of-Funds accounts. For consumption data, we use “Personal

1

Figure 16: Household deposits: FoF versus SCF.

2010 2011 2012 2013 2014 2015 2016

year

-200

-100

0

100

200

300

400

500

600

700

800

Bill

ions

of D

olla

rs

Flow of FundsSCF adjusted

Source: Flow of Funds accounts, Survey of Consumer Finances and Batty et al. (2015). See textfor details.

Figure 17: Checkable deposits and currency in the Flow of Funds (annual).

2010 2011 2012 2013 2014 2015 2016

year

0

200

400

600

800

1000

1200

1400

Bill

ions

of D

olla

rs

totalhouseholdsnonprofits

Source: Holmquist (2019).

2

Consumption Expenditures” from U.S. BEA (Bureau of Economic Analysis) minus “Personal Con-sumption Expenditures: Durable Goods.” For government expenditures data, we use “GovernmentConsumption Expenditures and Gross Investment” from BEA. Output is defined as the sum of con-sumption and government expenditures. For inflation, we use the growth rate of “Consumer PriceIndex for All Urban Consumers”. The above four series are obtained from 1985Q1 to 2018Q2, andwe use the sample averages of each series to calibrate the model in the steady state.

For the estimation exercise, we only use the sub-sample period 2008Q3-2015Q4 because thenominal interest rate (i.e., the Fed Funds rate) is at (almost) zero during this period. We usethe deposits-to-output ratio, the government-expenditures-to-output ratio, and inflation. For theoutput deviation, we do not use detrended output because the sample is too short. Instead, wefirst obtain the real potential GDP estimated by the U.S. Congressional Budget Office (CBO); theoutput deviation is then the difference between observed real GDP in natural log terms and the CBOestimated potential real GDP in natural log terms. To simplify the estimation exercise, we normalizeall variables in 2008Q3 to zero.

For the Survey of Consumer Finance (SCF), we use 2016 SCF Chartbook.36 Specifically, we use“Median value of before-tax family income for families with holdings” Table on Page 7 and “Medianvalue of transaction accounts for families with holdings” Table on Page 151.

B. Tractability and computation

B1. Mutual fund withdrawals

We now discuss in more detail the household’s choices regarding partially liquid wealth stored inmutual funds. The first-order optimality condition for the household’s decision for the withdrawalXt(i) can be written as:

UC,t(i) = ωt(i)Ψ′(Xt(i))UC,t(i) + λt(i)

where UC,t(i) is the marginal utility of consumption and λt(i) ≥ 0 is the shadow value of mutualfund wealth, i.e. the Lagrange multiplier on the evolution of At(i) in (3). As discussed in the maintext, there is no lower bound on partially liquid wealth since the natural limit in the model impliesthat At(i) ≥ −∞. This in turn implies that the shadow value of illiquid wealth is zero, i.e. λt(i) = 0.Therefore, households’ decisions become independent of the distribution of At(i). Similarly, themutual fund’s choices do not depend on this distribution. We can therefore drop At(i) as a state inthe computation.

Specifically in our case, the first-order condition for Xt(i) reduces to:

1 = ωt(i)Ψ′(Xt(i)).

We can now solve for the withdrawal Xt(i) directly from the above equation. It follows that theemployed all withdraw a constant amount Xt(i) = XE , whereas the unemployed all withdraw Xt(i) =XU ≤ XE . In the calibration, we treat XE and XU directly as parameters. Thereby we avoid havingto make assumptions on the precise functional form of adjustment cost function. The first-ordercondition for At(i) can be expressed as

UC,t(i)λt(i) = βEt[UC,t+1(1 + rAt+1)λt+1(i)

].

Given λt(i) = 1−Ψ′t(Xt(i)) = 0 at any time t, the left-and the right-hand side collapse to zero. Theexpected return EtrAt+1 is then determined purely via the mutual funds.

36https://www.federalreserve.gov/econres/files/BulletinCharts.pdf.

3

Note that here, the withdrawal decision is not only disconnected from the level of partially liquidwealth but also from variables such as income and consumption. As explained in the calibrationsection of the main text, this has the advantage that we can directly treat the net withdrawalsXEand XU as parameters. On the other hand, this is not strictly necessary for tractability. Forexample, one could specify the adjustment cost as a utility cost rather than as a tax. The first-ordercondition would become (again using λt(i) = 0):

UC,t(i) = ωt(i)Ψ′(Xt(i)).

In this case, there would be a richer distribution of withdrawals which is connected to the distributionof consumption, which varies over time. But still, withdrawals do not depend on At(i), and hencetractability is preserved. Note that in this case, one does have to take a stand on the adjustmentcost function, however. We therefore opted for the approach described above, given that we do nothave household-level data on the distribution of mutual fund withdrawals, and also given that microevidence at the mutual fund level suggests that withdrawals did not respond much to QE.

B2. The distribution of partially liquid wealth

We now discuss in more detail the properties of the distribution of At(i). As noted in the main text,the average level of wealth is pinned down uniquely in the model as At ≡

∫ 10 At(i)di = qBt B

mt +Divt.

Given that the right-hand side variables are uniquely pinned down in the steady state, so is At.Next, we note that the distribution of At(i) is not uniquely pinned down in the steady state of

the model. To see why, first note that the mutual fund’s decisions do not depend on the distributionof At(i). Second, recall that households withdraw XE from the fund when employed and XU whenunemployed. Therefore, the decisions of households also do not depend on At(i), as noted above.It follows that the steady state of the model is consistent with any distribution of liquid wealth, aslong as its mean equals A = qBBm +Div.

At the same time, given an initial distribution for At(i) in the initial period t = 0, the evolutionof the distribution for any subsequent period t = 1, 2, ... is pinned down uniquely. Since the returnon mutual funds rAt responds to economic shocks, so will the distribution of At(i). But even withoutaggregate shocks, the distribution will evolve, as it is non-stationary. To see why, consider a steadystate with rA > 0. The cross-sectional variance of illiquid wealth evolves as:

V ar(At(i)) ≥ (1 + rA)2V ar(At−1(i)) + V ar(Xt(i)) > V ar(At−1(i)),

where the first inequality follows from the fact that the covariance between At−1(i) and Xt(i) maybe negative due to the persistence of employment and unemployment spells. It now follows thatthe wealth distribution is non-stationary and that its cross-sectional variance is ever increasing. Asnoted in the main text, this non-stationarity is due to the “saving by holding” property of the model,which is in line with cross-sectional evidence on saving rates and the fact that wealth inequality hasbeen steadily increasing for decades.

B3. The distribution of liquid wealth and model solution

We now turn to the distribution of liquid wealth (deposits). In the presence of aggregate shocks, thisdistribution fluctuates over time, which is relevant to the state of the economy. When solving forequilibrium dynamics, we therefore need to keep track of this distribution. In the calibrated model, itturns out that the liquid wealth distribution consists of only mass points. This happens as householdswho become unemployed spend all their liquid wealth in the initial quarter of unemployment, hitting

4

the no-borrowing constraint within the first quarter of unemployment. Thus, all the unemployedchoose Dt(i) = 0.

It follows that any household which transitions from unemployment to employment holds exactlyzero deposits. As a result, all employed households with the same employment duration behave iden-tically (see also the discussion in Section 3.3). Moreover, all households which have been unemployedfor more than one quarter consume simply their current net income, whereas the newly unemployedhouseholds consume their current income plus their liquid wealth (which in turn depends on theirprevious employment duration).

Let us introduce some notation indicating various “cohorts” of employed and unemployed house-holds. Let a superscript E denote the employed, EU the newly unemployed, and UU those whohave been unemployed for at least one quarter. Further, let k denote the employment duration ofa household up until the current period (i.e. excluding the current period). For example CEt (k)with k = 0 denotes the consumption level of a currently employed household who was unemployedin the previous quarter and CEUt (k) with k = 3 denotes a newly unemployed household, who hadcompleted an employment spell of 3 quarters upon job loss.

We can now characterize the household’s choices with the following system of equations. Foremployed households we have the following equations:

CEt (k) +DEt (k) = wtN

Et (k) + ΘE

t + XEt − Tt, k = 0, (16)

CEt (k) +DEt (k) = wtN

Et (k) +

Rt−1

ΠtDEt−1(k − 1) + ΘE

t + XEt − Tt, ∀k ≥ 1, (17)[

CEt (k)]−σ = βEt

[Rt

Πt+1

[(1− pEU

)(CEt+1(k + 1))−σ + pEU

[CEUt+1(k + 1)

]−σ]], ∀k ≥ 0, (18)

wt[CEt (k)

]−σ = κ0

[NEt (k)

]κ1 , ∀k ≥ 0. (19)

For the newly unemployed households (EU) and the remaining unemployed households (UU) wehave:

CEUt (k) +DEUt (k) =

Rt−1

ΠtDEt−1(k − 1) + ΘU

t + XUt − Tt, ∀k ≥ 1, (20)

DEUt (k) = 0, ∀k ≥ 1, (21)

CUUt +DUUt = ΘU

t + XUt − Tt, (22)

DUUt = 0. (23)

The above system contains three blocks of equations. Equations (16), (17), (20), and (22) are budgetconstraints. Moreover, (18), (21), and (23) characterize the optimal choices for deposits (using thefact that the unemployed are at the no-borrowing constraint, whereas the employed are on theEuler equation for deposits), and (19) is the first-order optimality condition for labor supply of theemployed households.

In practice, we truncate the above system at a certain employment duration, i.e. we let k =0, 1, 2, 3..,K, which renders the state-space finite dimensional. As can be seen from Figure 4, underour calibration, households converge fairly quickly to a maximum amount of assets. In our applica-tion, we set K = 75 and verify that results are insensitive to the truncation threshold.37 We close

37Setting the threshold as low as K = 20 delivers very similar results.

5

the system by setting for the final cohort of employed households:

CEt (K) +DEt (K) = wtN

Et (K) +

Rt−1

ΠtDEt−1(K) + ΘE

t + XEt − Tt,[

CEt (K)]−σ

= βEt[Rt

Πt+1

[(1− pEU

) [CEt+1(K)

]−σ+ pEU

[CEUt+1(K)

]−σ]],

wt[CEt (K)

]−σ = κ0

[NE(K)

]κ1 .

These equations impose that beyond an employment duration of k = 75 quarters (i.e., more than 18years), all households behave identically. This is not a very restrictive cutoff, since households alreadybehave practically identically beyond an employment duration of 10 to 20 quarters, see Figure 4. Wesolve the above system jointly with the remaining model equations, given by:

1− εt + εtwtZt

= φ (Πt − 1) Πt − φEt[βYt+1

Yt(Πt+1 − 1) Πt+1

],

Xt = Divt + (1 + ρqBt )Bmt−1

Πt− qBt Bm

t ,

Xt = uXUt + (1− u)XE

t + Ψt,

XUt = XU ,

XEt = XE ,

Ψt = Ψ,

Gt = qBt B − (1 + ρqBt )B

Πt+ T cbt + Tt + Ψ,

T cbt +Rt−1

ΠtMt−1 + qBt B

cbt = Mt + (1 + ρqBt )

Bcbt−1

Πt,

Bcbt +Bm

t = B,

K∑k=0

ψE(k)DEt (k) = Mt,

K∑k=0

ψE(k)NEt (k) = Nt,

Yt −Gt + φYt (Πt − 1)2 =

K∑k=0

ψE(k)CEt (k) +

K∑k=0

ψEU (k)DEUt (k) + ψUUCUUt ,

Yt = ZtNt,

where XU , XE , and Ψ are positive constants, and in addition to the policy equations for either QEor conventional policy and the exogenous evolution of (εt, Zt,, Gt, z

QEt ), as stated in the main text.

In the above equations, ψE(k), ψEU (k), and ψUU are population share parameters, which satisfyψUU = u(1−pUE)−pEU (1−u), ψE(k) = pUE(1−pEU )ku for k < K, ψE(K) = 1−

∑K−1k=0 ψE(k)−u,

ψEU (k) = pUE(1 − pEU )kpEUu for k < K, and ψEU (K) = pEUψE(K). For equilibrium dynamics,we use a first-order perturbation method to solve for the joint system.

Finally, we discuss how to verify easily that in equilibrium the unemployed hit the no-borrowing

6

constraint in deposits. This is the case if:

[CEUt (K)

]−σ > βEt[Rt

Πt+1

[pUE(CEt+1(0))−σ +

(1− pUE

)(CUUt+1)−σ

]].

This equation implies that the newly unemployed with the longest previous employment spell do notwant to save, i.e., they are at the constraint. If this condition holds, then the same is true for allthe other unemployed households, since these are less wealthy, which implies that CUUt ≤ CEUt (K)and CEUt (k) ≤ CEUt (K). See also Figure 4 for an illustration of this point. We verify that the aboveequation holds in the steady state.38

C. Steady state and calibration

This appendix shows how to solve for the steady-state economy in a systematic way, which is usefulfor the calibration exercise. The calibration strategy is shown after we discuss how to solve for thesteady state efficiently. A more “black-box” alternative is to solve the entire system of steady-stateequations all at once using a numerical solution routine. The procedure below, however, makes iteasier to hit certain calibration targets.

C1. Solving for the steady-state equilibrium

We first show how to solve for the steady-state equilibrium, given q, w, Π, XU , Ψ, G, B, M , ΘU ,and ΘE . We solve for the resulting tax policy T and interest rate policy R, together with the netoutflow XE . Suppose we have an initial guess of (T,R, XE).

For the unemployed agents without any savings, the budget constraint implies:

CUU = ΘU +XU − T.

There are K cohorts of employed agents. The labor supply decision of the kth cohort satisfies

w

CE(k)= κ0N

E(k),

with σ = 1, which means that the labor income is wNE(k) = w2

CE(k)κ0. To this end, we first solve the

consumption and saving choice.For the Kth cohort, the Euler equation for deposits is

1

CE(K)= βR

[pEU

1

CUU +DE(K)R+ (1− pEU )

1

CE(K)

],

and the budget constraint is

CE(K) =w2

CE(K)κ0+DE(K) (R− 1) + ΘE ,

where ΘE ≡ ΘE + XE − T . The above two equations pin down CE(K) and DE(K).39

Let us now guess CE(1). For the i = 0th cohort, the Euler equation and the budget constraint

38Under a local perturbation, the constraint then also holds outside the steady state.39There is even an analytical solution as CE(K) can be solved from a quadratic equation. To see this,

7

are

CE(0) = β−1R−1

[pEU

1

CUU +DE(0)R+ (1− pEU )

1

CE(1)

]−1

CE(0) =w2

CE(0)κ0−DE(0) + ΘE .

Since we know CE(1), the two equations solve the two unknowns CE(0) and DE(0).For any k = 1, 2, ...,K − 1 cohort we then obtain from the budget constraint

CE(k) =w2

CE(k)κ0+RDE(k − 1)−DE(k) + ΘE ,

or DE(k) = w2

CE(i)κ0+RDE(k − 1) + ΘE − CE(k). From the Euler equation, we obtain

1

CE(k)= βR

[pEU

1

CUU +DE(k)R+ (1− pEU )

1

CE(k + 1)

],

or CE(k+1) =[

1βR(1−pEU )CE(k)

− pEU

(1−pEU )[CUU+DE(k)R]

]−1. Therefore, given CE(1), we can calculate

DE(1), CE(2), DE(2), ..., CE(K − 1) and DE(K − 1). We look for CE(1) such that DE(K − 1) =DE(K)− ε where ε is an arbitrary small and positive number (i.e., the amount of savings convergeto the fixed point). This is effectively a shooting algorithm, which generates optimal consumptionand saving choices. Aggregate labor supply is then given by

N =

K∑k=0

ψE(k)NE(k) =

K∑k=0

ψE(k)w

κ0CE(k). (24)

Now, we turn to the government side. Notice that the debt held by the central bank is

Bcb =M(1−R)

qB − (1 + ρq),

rearrange the Euler equation

1− βR(1− pEU

)CE(K)

=βRpEU

CUU +RDE(K)→ DE(K) =

βpEU

1− βR(1− pEU )CE(K)− CUU

R,

which can be used to express the budget constraint as a quadratic equation of CE(K):[1− (R− 1)βpEU

1− βR(1− pEU )

] [CE(K)

]2+

(R− 1

RCUU − ΘE

)CE(K)− w2

κ0= 0.

Since[1− (R−1)βpEU

1−βR(1−pEU )

]> 0 and −w

2

κ0< 0, the only positive root is

CE(K) =ΘE − R−1

R CUU +

√(ΘE − R−1

R CUU)2

+ 4[1− (R−1)βpEU

1−βR(1−pEU )

]w2

κ0

2[1− (R−1)βpEU

1−βR(1−pEU )

] .

8

and the total debt held by the mutual fund is thus

Bm = B −Bcb.

Finally, after obtaining all equilibrium objects with a given (T,R, XE), we check the followingthree equations. We know that from the budget constraint of the mutual funds:

uXE + (1− u)XU + Ψ = N − wN + (1 + ρq − q)Bm. (25)

The market clearing for reserves is given by:

M =K∑k=0

ψE(k)DE(k). (26)

The goods market clearing is given by:

K∑k=0

ψE(k)CE(k) +K∑k=0

ψEU (k)DEU (k) + ψUUCUU +G = AN. (27)

These three equations above solve the three unknowns (T,R, XE). That is, if these three equationsdo not hold, we change our initial guess of (T,R, XE) and iterate the computation.

C2. Calibration

The above strategy for calculating the steady-state equilibrium objects will be used in the followingcalibration exercise.

The average labor supply is targeted (1/3 in our calibration), so N is known. Recall that the wagerate is w = (ε− 1)/ε, and we thus know the average labor income in the model. The unemploymentbenefit is calibrated to be a fraction (0.25 in our calibration) of average labor income, so ΘU isknown. Because of the budget-neutral unemployment insurance, ΘE is also known. Without lossof generality, we normalize the steady-state TFP to Z = 1, so that Y = N . We target the returnof long-term government debt which generates the steady-state q. We target the reserves-to-outputratio M/4Y (or M/Y ), the government-expenditure-to-output ratio G/Y , the real interest rate r,and the median deposits-to-income ratio. With these targets, we directly obtain G and M . Thefollowing discussion shows how we calibrate κ0 and XU (note: the level of government debt B willbe a free parameter).

First, we have an initial guess of (κ0, XU , Ψ). The lump-sum tax T , or equivalently the levelof government debt B, is set such that the model hits the median deposits-to-income ratio. To seethis, notice that the consolidated fiscal and monetary budget constraint implies that

G+ (R− 1)M + (1 + ρqB − qB)Bm = T + Ψ. (28)

For any given T , we obtain Bm = [T − (R− 1)M −G] /(1 + ρqB − qB); using (25) gives XE .Second, we now have (T , R, XE), and we can follow the strategy specified before to calculate

steady-state equilibrium objects. The lump-sum tax T is adjusted so that the model generates themedian deposits-to-income ratio as in the data. In addition, since the level of total debt satisfiesB = Bm + Bcb, we can also view that B is calibrated to hit the median deposits-to-income ratio,where B is given by

9

B = Bm +Bcb = Bm +M(1−R)

qB − (1 + ρqB).

Finally, to hit the steady-state labor supply N and the real interest rate r = R/Π = R, theguessed values of κ0, XU , and Ψ are adjusted such that the labor market clearing condition (24) andthe reserve market clearing condition (26) are satisfied, as well as that the ratio of the adjustmentcost Ψ to the withdrawals X is the same as the targeted percentage adjustment cost (i.e., 10% inthe calibration).40 Notice that for a different (κ0, XU , Ψ) one needs to re-calculate B.

D. Alternative assumptions on mutual funds and fiscal policy

In this part, we consider different assumptions on liquidation of mutual fund wealth, together withvarious assumptions on the behavior of fiscal policy.

D1. Modeling mutual fund flow management

Instead of having constant withdrawals from the mutual funds, we now model the mutual fund’sflow management. Specifically, the manager of the representative mutual fund now decides activelyon the payout policy Xt, while the adjustment cost for the households are fixed. For comparability,

the gap between the withdrawals XtU − XE

t = µ is kept the same as in the baseline, but now thefund’s payout policy will affect the levels of XU

t and XEt . With this alternative assumption, part of

the liquidity created by QE can be directly transferred to households via mutual funds.The mutual fund manager maximizes

∑∞t=0 β

tXt subject to a sequence of ((7)) by choosing Xt

and Bmt in each period. The first-order condition for government bond holdings and deposits imply

that:

qBt = βEt1 + ρqBt+1

Πt+1. (29)

Equation (29) implies that the manager will adjust the net outflow Xt freely, and QE policy islikely to induce her/him to increase the outflow from the fund directly, which pushes up aggregateconsumption demand from households.

D2. Robustness exercises

Following the discussion above, we verify robustness of the results by comparing the following fourversions of the model:

1. The baseline with a fixed mutual-fund payout Xt;

2. A version with a flexible mutual-fund payout Xt (as discussed above);

3. A version with both a fixed mutual-fund payout and a fixed tax (i.e., T is kept the sameas in the steady state). In this version, government expenditures (G) adjust to balance thegovernment budget;

4. The baseline, but with a rule for real government debt given by B/B = (Y/Y )ς , where B andY are the steady-state levels of respectively government debt and output. Taxes adjust tobalance the government budget. Government expenditures are kept fixed. We set ς = −0.5.

10

Figure 18: Responses to an expansionary QE shock: alternative assumptions on mutual fundsand fiscal policy.

0 5 10 15 200

0.5

1

1.5

quarter

real reserves

% o

f s.s

. ann

ual o

utpu

t

0 5 10 15 20−0.5

0

0.5

1

quarter

inflation

%−

poin

ts

0 5 10 15 20−0.5

0

0.5

1

1.5

2

2.5

quarter

%

output

1. Basline2. Flexible X3. Constant X and T4. B rule

0 5 10 15 20−1

0

1

2

3

4

quarter

real wage

%

0 5 10 15 20−30

−20

−10

0

10

quarter

payout from mutual fund (X)

%

0 5 10 15 20−15

−10

−5

0

5

10

quarter

Tax (T)

%

Notes: see text and notes on Figure 7.

Figure 18 shows the responses to a QE shock. As a general observation, note that a positive QEshock stimulates aggregate output and inflation in all cases.

In the baseline, X remains constant whereas T falls more on impact, followed by an increase.Intuitively, in the initial period the mutual funds try to replace debt sold to the Fed with new debt.This reduces borrowing costs for the government, allowing for lower taxes. In subsequent periods,QE is gradually reversed. This means that the demand for government debt by the mutual fundsfalls below steady state, which increases borrowing costs for the government. As a result, taxes mustrise.

In version 2, X increases somewhat upon impact whereas T falls. Importantly, the response ofX−T−Ψ is precisely the same as in the baseline, generating precisely the same reactions of inflation,output, and real wage. In fact, we have even experimented with a setup in which the mutual fundshave adjustment costs in choosing payouts to households, and we still obtain exactly the same result,

40Equation (27) is not used here as we have used the consolidated fiscal and monetary budget constraintin calibration. The households’ budget constraints and the consolidated government budget constraint implythe goods market clearing condition (27).

11

precisely because the tax T adjusts to achieve the same aggregate impacts.In version 3, we keep both the net outflow X and the tax T constant. There is a short-lived

rise in G, followed by a contraction. The stimulative effect on output and inflation is greater thanin the baseline on impact, but is shorter-lived. Finally, we consider version 4, which is the baselinebut with a rule for government debt. Responses are similar to the baseline. The effect on output issomewhat more muted early on, but also more persistent.

Overall, we conclude that the specific time variation in the mutual fund dividend is not at allcritical to the transmission of QE to the macro economy, and that stimulative effects are obtainedunder a wide range of assumptions on the behavior of the mutual fund and fiscal policy. Underlyingthis result is the fact that, in all cases, the households end up being the ultimate holders of the extraliquidity as a result of QE.

E. Other unconventional policies

We consider two popular proposals of unconventional policies: Helicopter Drops and Forward Guid-ance.

E1. Helicopter Drops (HD)

In this part, we compare QE to an alternative policy measures, known as a helicopter drop. Helicoptermoney is a theoretical and unconventional monetary policy tool that central banks use to stimulateeconomies. The term helicopter money is attributed to Milton Friedman, while former FederalReserve Chairman Ben Bernanke later popularized the notion. Helicopter money involves the centralbank supplying large amounts of money to the public, as if the money was being scattered from ahelicopter.

A money-financed tax cut is essentially equivalent to Friedman’s “helicopter drop” of money.Helicopter drop in our model is then an expansionary fiscal policy through a lower tax T , that isfinanced by an increase in reserves and thus deposits. We therefore impose the following helicopterpolicy restriction

Bcbt = 0.

Since the government targets a constant level of debt, the only way to finance the tax cut is throughmoney/reserve finance in the initial period, besides the revaluation of government debt by variationsin inflation, due to general equilibrium effects.

It turns out that the economic dynamics exhibit indeterminacy if we let the central bank targeta certain interest rate rule. Therefore, we let the central bank actively set interest rates, i.e., ξRΠand ξRY are not constrained at zero. One can view this type of policy as a conventional monetarypolicy without the constraint on the amount of reserves but with the constraint of the amount ofgovernment debt held by the central bank.

Figure 19 compares HD policy with QE. Both types have the same path for real reserves, butHD policy has a much smaller initial impact on the aggregate while generating only slightly morepersistent effect. Interestingly, inflation never drops below zero, which means that the price level ispermanently higher in the long run, and the nominal quantity of reserves/deposits is also permanentlyhigher, reflecting the quantity theory of money by Friedman. To understand the smaller impact ofHD, note that one can view reserves as another form of debt owed by the consolidated governmentof the treasury and the central bank. The tax cut is initially funded by more reserves, which impliesthat a tax increase in the future is needed to pay back the debt. QE, on the other hand, substitutesreserves for long-term government debt, and does not directly imply a significant or any tax increase

12

Figure 19: Responses to a Helicopter Drop.

0 5 10 15 20

quarter

0

0.2

0.4

0.6

0.8

1

1.2

% o

f s.s

. out

put

real reserves

0 5 10 15 20

quarter

-0.2

0

0.2

0.4

0.6

0.8

%-p

oint

s

inflation

0 5 10 15 20

quarter

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

%

output

QE

HD with R = 1.5 and RY

= 0

HD with R = 3 and RY

= 0

HD with R = 1.5 and RY

= 1.5

0 5 10 15 20

quarter

-0.5

0

0.5

1

1.5

2

2.5

%

real wage

Notes: responses to a Helicopter Drop shock compared to quantitative easing.

in the future (although there are tax implications via equilibrium effects on the government budgetconstraint).

Although more work on HD policy could certainly be done, our preliminary conclusion is thatdeterminacy is far from guaranteed under this policy, and also that its aggregate impact might berelatively small, compared to QE.

E2. Forward Guidance

Among unconventional policies, another alternative to QE is Forward Guidance: an announcementabout monetary policy in the future. In the standard NK model without QE, Forward Guidance isan extremely effective policy once the zero lower bound on the nominal interest rate binds. In fact,macroeconomic responses to Forward Guidance turn out to be so enormous that they might call intoquestion the basic tenets of the NK model, see Del Negro et al. (2012).

To address this puzzle, McKay et al. (2016) revisit the effects of Forward Guidance in anincomplete-markets NK model (without QE). They show that the output response to a five-year-ahead announcement is dampened substantially, relative to a representative-agent version of themodel. Nonetheless, the effects of Forward Guidance remain large in comparison to empirical evi-dence, as presented for instance in Del Negro et al. (2012). Hagedorn, Luo, Mitman and Manovskii(2017) consider an incomplete-markets NK model with a target for nominal expenditure growth andshow that the effects of Forward Guidance are much smaller.

We explore the effects of Forward Guidance in our model, with a QE policy on the part of the

13

central bank. For transparency, we assume that a QE rule with Real Reserve Targeting is in place,i.e. we set ξQEΠ = ξQEY = 0. We then consider a pre-announced decline in the nominal interest rateof 50 basis points (corresponding to about 2 percentage points on an annualized basis) which lastsfor one quarter. During all other periods, the net nominal interest rate remains fixed at zero. Weconsider a Forward Guidance announcement two years ahead, and another one five years ahead.

Figure 20: Responses to a Forward Guidance shock.

0 10 20 30

quarter

-0.6

-0.4

-0.2

0

%-p

oint

s

nominal interest rate

2 year ahead5 year ahead

0 10 20 30

quarter

-0.4

-0.3

-0.2

-0.1

0

0.1

%-p

oint

s

real interest rate

0 10 20 30

quarter

-0.04

-0.02

0

0.02

0.04

0.06

%

output

0 10 20 30

quarter

0

0.1

0.2

0.3

0.4%

price long-term debt

Notes: responses to a forward guidance shock, reducing the quarterly nominal interest rate by 50 basispoints for one quarter, and announced 2 or 5 years ahead. Responses were computed in the model versionwith QE, setting ξQEΠ = ξQEY = 0, and letting the nominal interest rate Rt vary with the forward guidanceshock, starting from Rt = R.

Figure 20 shows the effects of the two Forward Guidance shocks. The figure shows that once thenominal interest rate is actually reduced, there is a strong decline in the real interest rate. Duringthe quarters leading up to the implementation there is a small expansion in output, followed by aminor contraction after the implementation. Importantly, the output increase in the initial periodof the announcement is almost negligible.41 The impact response of the real interest rate (and henceinflation) is also extremely small. Moreover, the initial responses are declining in the announcementhorizon. Finally, the lower right panel of Figure 20 shows that the Forward Guidance shock doeshave a substantial initial effect on the price of long-run treasury debt.

We thus conclude that once we account for incomplete markets and QE policy, the effects offorward guidance on output and inflation are no longer puzzlingly large. Rather, they are close tonegligible. An implication of this finding is that, in comparison, QE stands out as the more effectivestabilization policy, at least when the nominal interest rate is immutable in the short run.

41The output response is less than 0.0002 percent. Putting this number in perspective, McKay et al. (2016)report an initial output increase of 0.25 (0.1) percent under complete (incomplete) markets, in response to aforward guidance shock to the real interest rate of 50 basis points, 20 quarters ahead.

14