Discussion Paper No. 1038

MONEY-FINANCEDFISCAL STIMULUS:
THE EFFECTS
OF IMPLEMENTATION LAG

Takayuki Tsuruga
Shota Wake

August 2018

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Money-Financed Fiscal Stimulus:
The Eﬀects of Implementation Lag∗
Takayuki Tsuruga† and Shota Wake‡
First draft: March 2016
This version: August 2018

Abstract
Previous studies argue that, based on the New Keynesian framework, a ﬁscal stimulus ﬁnanced by money creation has a strong positive eﬀect on output under a reasonable degree of
nominal price rigidities. This paper investigates the eﬀects of implementation lag in the moneyﬁnanced ﬁscal stimulus on output. We show that if a money-ﬁnanced government purchase has
a time lag between the decision and the implementation: (1) it may cause a recession rather
than a boom when the economy is in normal times; (2) it may deepen a recession when theeconomy is caught in a liquidity trap; (3) the longer is the implementation lag, the deeper is therecession; and (4) the depth of the recession depends on the interest semi-elasticity of money
demand. Our results imply that to strengthen the eﬃcacy of the money-ﬁnanced ﬁscal stimulus,
policymakers should shorten the implementation lag based on detailed knowledge of the money
demand function.
JEL Classiﬁcation: E32; E52; E62
Keywords: Anticipation eﬀect, Fiscal multiplier, Government spending, Seigniorage∗
We would like to thank Bill Dupor, Ryo Horii, Kenneth Kuttner, Taisuke Nakata, Makoto Saito, and Masahiko
Shibamoto and seminar and conference participants at Kobe University, Shiga University, the University of Tokyo,
and the Asian Meeting of the Econometric Society for helpful discussions and comments. This paper was previously
circulated as “Money-Financed Fiscal Stimulus: The Eﬀects of an Anticipated Shock.” Takayuki Tsuruga acknowledges the ﬁnancial support from Grants-in-Aid for Scientiﬁc Research (15H05729 and 15H05728), the Murata Science
Foundation, and the Zengin Foundation for Studies on Economics and Finance. The views expressed in this paper
are those of the authors and do not necessarily reﬂect the oﬃcial views of the Government of Japan.
†
Osaka University; Cabinet Oﬃce, the Government of Japan; Centre for Applied Macroeconomic Analysis‡
Cabinet Secretariat, the Government of Japan1

1

Introduction
Government spending to stimulate the economy is typically ﬁnanced through government debt

and eventually taxes. The global ﬁnancial crisis in the last decade led to high levels of the government debt-to-GDP ratio in many countries and these countries’ policy interest rates hit the zero
lower bound (ZLB). Against this backdrop, some argue that a ﬁscal stimulus ﬁnanced by money
creation can be considered a policy option to boost aggregate demand (Galı́ 2017, English, Erceg
and López-Salido 2017, among others).1 This policy option relies neither on lowering the nominalinterest rate nor on further issuance of government debt, because the ﬁscal stimulus is ﬁnanced
by seigniorage. Using the standard New Keynesian model with a reasonable degree of nominal
rigidities, Galı́ (2017) shows that an unexpected shock to money-ﬁnanced government purchases ortax cuts can have a large positive impact on output along with a crowding-in eﬀect on consumption. English, Erceg and López-Salido (2017) also argue that a money-ﬁnanced ﬁscal stimulus is a
powerful tool for boosting aggregate demand if such ﬁscal policy is credible. Overall, these studies
ﬁnd that the money-ﬁnanced ﬁscal stimulus can increase output and inﬂation as well as reduce thedebt-to-GDP ratio substantially.
This paper investigates the role of an implementation lag in the eﬀect of money-ﬁnanced government purchases. The introduction of an implementation lag of ﬁscal policy into the standard
New Keynesian model is a straightforward extension of the model. However, it captures an important aspect of ﬁscal policy and thus has been widely discussed in the literature. Ramey (2011)provides empirical evidence that government purchases tend to involve long lags between the decision to spend and the actual increase in spending. Cwik and Wieland (2011) assess the eﬀect ofdebt-ﬁnanced ﬁscal policy with an implementation lag, using a variety of empirical New Keynesian models. Yang (2005), Leeper, Walker and Yang (2010), Leeper, Richter and Walker (2010),
Mertens and Ravn (2011), and Fujiwara and Waki (2017), among others, analyze a distortionary
tax and ﬁnd that model responses to an announced tax change diﬀer substantially from those toan unanticipated tax change as well as their policy implications.
We present four ﬁndings, in all of which the implementation lag plays an important role for theeﬃcacy of the money-ﬁnanced ﬁscal stimulus. The ﬁrst ﬁnding is that, if the money-ﬁnanced ﬁscal
stimulus involves an implementation lag, the ﬁscal stimulus may cause a recession rather than aboom. This is in sharp contrast to a money-ﬁnanced ﬁscal stimulus without an implementation
1

For more practical discussions on the eﬃcacy of money-ﬁnanced ﬁscal stimulus, see also Turner (2015).

2

lag, in which the ﬁscal stimulus boosts aggregate demand strongly as discussed in the literature.
The mechanism behind our ﬁrst ﬁnding can be understood from an inﬂuential study by Ball
(1994), who ﬁnds that a credible disinﬂationary announcement in the presence of nominal rigidities
causes a boom rather than a recession. In his analysis, if a decline in the money growth rate is
anticipated, forward-looking ﬁrms reduce prices prior to the actual decline in the money growthrate. Ball (1994) assumes that households’ consumption is constrained by their real money balances(i.e., the cash-in-advance (CIA) constraint). In this case, the lower aggregate prices result in higher
real money balances and relax the CIA constraint. The resulting higher aggregate demand leads
to a boom prior to the decline in the money growth rate.
Ball’s (1994) argument is applicable to our context, but in an opposite direction. In our analysis,
if the money-ﬁnanced ﬁscal stimulus is anticipated in advance because of an implementation lag,forward-looking ﬁrms increase prices prior to the actual increase in money supply that ﬁnances theﬁscal stimulus. The higher aggregate prices result in lower real money balances and tighten the CIA
constraint. The resulting lower aggregate demand leads to a recession prior to the implementationof the money-ﬁnanced ﬁscal stimulus.
The second ﬁnding of this paper is that such a negative eﬀect on output remains even when theeconomy is caught in a liquidity trap. If the money-ﬁnanced ﬁscal stimulus is implemented withouta lag, the economy experiences a faster recovery from a recession caused by the liquidity trap thanthe case of no policy change. However, an implementation lag in the money-ﬁnanced ﬁscal stimulus
may result in a slower recovery from the recession than the case of no policy change.
The mechanism behind our second ﬁnding is slightly diﬀerent from that behind the ﬁrst ﬁnding.
Because the nominal interest rate is zero in the liquidity trap, there is no opportunity cost of holding
money. In this case, it is not necessarily optimal for the household to reduce money holding to
the level required for its consumption and thus the CIA constraint does not bind. Once the CIA
constraint does not bind, current consumption is inﬂuenced by the dynamic path of the real interest
rate through the Euler equation.
When the money-ﬁnanced ﬁscal stimulus is announced, households and ﬁrms know that the
future government spending will raise the real interest rate at the time of implementation. A
high real interest rate in the future crowds out current consumption through the Euler equation.
As long as the real interest rate increases suﬃciently, current consumption is crowded out by
the money-ﬁnanced ﬁscal stimulus with an implementation lag. By contrast, the money-ﬁnanced
ﬁscal stimulus without an implementation lag immediately generates higher inﬂation because of3the actual increases in the money supply and government purchases. Higher inﬂation under theliquidity trap leads to higher aggregate demand so that consumption is higher than in the case of
no policy change.
Turning to the third ﬁnding, we show that the negative eﬀect becomes stronger as the implementation lag of the money-ﬁnanced ﬁscal stimulus lengthens. The intuition is simple, especiallyin the case of the binding CIA constraint. If the implementation lag lengthens, the ﬁrms have more
opportunity to increase prices before the implementation. As a result, the aggregate price level
increases and the CIA constraint becomes tighter, reducing consumption. Therefore, the recession
is deeper, as the implementation lag is longer.
Finally, our analysis also suggests the importance of the interest semi-elasticity of money demand
for the eﬃcacy of the money-ﬁnanced ﬁscal stimulus with an implementation lag. If money demandis inelastic with respect to the nominal interest rate, the money market equilibrium condition
suggests that a reduction in real money balances resulting from higher prices needs to be absorbed
by a large increase in the nominal interest rate. Therefore, if the increase in the nominal interest
rate transmits to the real interest rate, it weakens aggregate demand. By contrast, if money demandis elastic with respect to the nominal interest rate, the reduced real money balances can be absorbed
by a small increase in the nominal interest rate. As a result, even when the money-ﬁnanced ﬁscal
stimulus is implemented with a lag, the real interest rate can be kept low, supporting aggregate
consumption. Therefore, the money-ﬁnanced ﬁscal stimulus under a high interest semi-elasticity ofmoney demand may have a strong positive eﬀect on output, even with an implementation lag.
We thus conclude that the eﬃcacy of a money-ﬁnanced ﬁscal stimulus depends on the lengthof the implementation lag and the interest semi-elasticity of money demand. An implication forﬁscal policy is that policymakers should consider the interest semi-elasticity of money demand inevaluating the money-ﬁnanced ﬁscal stimulus. If policymakers are faced with a low interest semielasticity of money demand, they should shorten the implementation lag of the ﬁscal stimulus. If
the implementation lag is diﬃcult to shorten, the money-ﬁnanced ﬁscal stimulus would be a usefulpolicy option only under a high interest semi-elasticity of money demand.

2

The model
The model we consider is a stylized New Keynesian model with an occasionally binding ZLB

constraint on the nominal interest rate. We also use a CIA constraint in modeling money demand.

4

Previous studies such as English, Erceg, and López-Salido (2017) and Galı́ (2017) examine a moneyﬁnanced ﬁscal stimulus that is implemented without a lag and compare the equilibrium dynamics
with a debt-ﬁnanced ﬁscal stimulus. By contrast, we focus on an implementation lag in a moneyﬁnanced ﬁscal stimulus and investigate the impact of the implementation lag on the eﬀect of theﬁscal stimulus. Following Galı́ (2017), we assume a deterministic environment other than Calvo
pricing. We also consider a model with an occasionally binding ZLB constraint under a deterministic
environment.

2.1

Households and ﬁrms∑∞ t
A representative household maximizes its lifetime utility given by
t=0 β U (Ct , Nt ) ,where][
U (Ct , Nt ) = Ct1−σ / (1 − σ) − Nt1+ϕ / (1 + ϕ) Zt . Here, Ct , Nt , and Zt represent consumption,
labor supply, and an exogenous preference shifter, respectively. The parameters σ > 0, ϕ > 0,
and 0 < β < 1 denote the degree of relative risk aversion, the inverse of the Frisch elasticity oflabor supply, and the discount factor, respectively. Throughout the model, the unit of time t is one
quarter. The household faces a budget constraint and a CIA constraint as follows:
Mt + Bt = (1 + It−1 )Bt−1 + (Mt−1 − Pt−1 Ct−1 ) + Wt Nt + Pt Dt − Pt Tt ,Mt ≥ Pt Ct ,

(1)
(2)

where Mt is nominal money holdings, Bt denotes households’ nominal holdings of one-period government bonds that pay the nominal interest rate It , Pt represents the aggregate price index,
and Wt is nominal wages. Furthermore, Dt denotes ﬁrms’ real proﬁts and Tt is lump-sum taxes
in real terms. The left-hand side of (1) is the nominal value of total wealth that the household
brings into period t. The right-hand side indicates that the household receives the nominal income
Wt Nt + Pt Dt and pays taxes Pt Tt . It also carries the nominal bond and cash holdings remaining
after consumption into period t. As usual, a no Ponzi game condition holds. Equation (2) is the
CIA constraint.
The ﬁrst-order conditions are standard:
UC,tUC,t+1= β(1 + It )
,
Pt
Pt+1
UN,t
Wt
−
,
=
UC,t
Pt
Mt − Pt Ct ≥ 0,
5

(3)
(4)
(5)

where UC,t = Ct−σ Zt and UN,t = −Ntϕ Zt . When It > 0, (5) holds with equality. This is because
holding money beyond the nominal transaction Pt Ct is not optimal under the strictly positive
opportunity cost of money holding. When It = 0, however, Mt is a perfect substitute for Bt intransferring the wealth into the next period and the CIA constraint is not necessarily binding. Thatis, the CIA constraint may not be binding if the ZLB on the nominal interest rate is binding (andvice versa).
The representative ﬁrm produces the ﬁnal good in a perfectly competitive market. It combines)ϵ/(ϵ−1)(∫1(ϵ−1)/ϵ djY(j)a continuum of intermediate goods, using the technology Yt =, wheret
0
Yt (j) denotes output produced by the intermediate good producers j ∈ [0, 1] and ϵ (> 1) is theelasticity of substitution among diﬀerentiated intermediate goods. The aggregate price index Pt(∫)1/(1−ϵ)1
is associated with intermediate good prices Pt (j) by Pt = 0 Pt (j)1−ϵ dj. The production
function for intermediate good producers is Yt (j) = Nt (j)1−α , where α ∈ (0, 1]. Each intermediateﬁrm is allowed to reset its price with probability 1 − θ. Let Pt∗ be the optimal nominal price set by
an intermediate good producer. The optimal nominal price solves the maximization problem:
max
∗
Pt

∞
∑
k=0

)
(
1
1−α
∗
θ Qt, t+k (1/Pt+k ) Pt Yt+k|t − Wt+k Yt+k|t ,ksubject to the demand function Yt+k|t = (Pt∗ /Pt+k )−ϵ Yt+k , where Yt+k|t denotes an intermediate
good’s output in period t + k for the producer that last reset its price in period t and Qt, t+k ≡βUC,t+k /UC,t is the discount factor of the ﬁrm. The ﬁrst-order condition for Pt∗ is∞∑(
θ Qt, t+k (1/Pt+k )Yt+k|t Pt∗ −
k

k=0

α
ϵ
1−α
Wt+k|t Yt+k|t
ϵ−1

)
= 0.

(6)

Along with the price index, log-linearizing this equation around the zero steady-state inﬂation rate
leads to the standard New Keynesian Phillips curve.

2.2

Government

Our experiment examines the impact of the money-ﬁnanced ﬁscal stimulus with an implementation lag of h periods. Following Galı́ (2017), let gt be government purchases, which exceed its
steady-state level, expressed as a fraction of the steady-state output (i.e., gt ≡ (Gt − G)/Y where
Gt is government purchases and G and Y are the steady-state value of government purchases and6

output, respectively). At t = 0, the government announces the following ﬁscal stimulus:

gt =




0

for t < h

 δ t−h for t ≥ h,(7)where h ≥ 0. Here δ ∈ [0, 1) measures the persistence of an exogenous increase in gt because gt+1 =δgt for t ≥ h. For any h ≥ 0, the magnitude of the initial increase in the government purchases isnormalized to one percent of the steady-state output, because gt is deﬁned as gt = (Gt − G) /Y .
Our experiment on the implementation lag allows for h > 0. The household and ﬁrms will takesome actions between 0 ≤ t < h before Gt actually increases. Obviously, when h = 0, the ﬁscal
stimulus is implemented without a lag.
To analyze the money-ﬁnanced ﬁscal stimulus, we follow the literature and consider the government as a single consolidated entity consisting of the ﬁscal authority and the central bank. Let
Mts be the money supply. The government’s consolidated budget constraint is
Pt Gt + (1 + It−1 ) Bt−1 = Pt Tt + Bt + ∆Mts ,(8)s . The left-hand side of (8) means that the government purchases thewhere ∆Mts = Mts − Mt−1

ﬁnal goods Gt and repays one-period government bonds with interest rate, (1 + It−1 ) Bt−1 . On
the right-hand side of (8), the government collects the lump-sum taxes and issues new government
bonds and noninterest-bearing money to ﬁnance its expenditure. Let us divide both sides of (8) by
Pt to obtain the consolidated budget constraint in real terms:
Gt + Rt−1 Bt−1 = Tt + Bt + ∆Mts /Pt ,(9)where Rt = (1 + It ) Pt /Pt+1 , and Bt = Bt /Pt . In the equation, ∆Mts /Pt is real seigniorage revenue.
We assume that Gt and Tt have their steady state values and seigniorage is zero in the steady state(∆Mts = 0). These assumptions imply that G + RB = T + B in the steady state, where a variablewithout a subscript indicates the steady-state value. Let τt and bt be the lump-sum taxes and theissuance of real government bonds exceeding its own steady-state level expressed as a fraction of
the steady-state output, respectively (i.e., τt = (Tt − T ) /Y , and bt = (Bt − B) /Y .). We take thediﬀerence between G + RB = T + B and (9) and express the resulting equation as a fraction of the7steady-state output. As a result, the linearly approximated consolidated budget constraint is
bt = Rbt−1 + Rb (it−1 − πt ) + gt − τt − V −1 ∆mst ,

(10)

)
(
s
. Furthermore,
where b = B/Y , it = ln [(1 + It ) /R], πt = ln (Pt /Pt−1 ), and ∆mst = ln Mts /Mt−1
V is the steady-state value of the income velocity of money (given by Vt = Pt Yt /Mts ). The ﬁrst andsecond terms of the right-hand side of (10) result from (Rt−1 Bt−1 − RB) /Y ≅ R (Bt−1 − B) /Y +(B/Y ) (Rt−1 − R) ≅ Rbt + Rb (it−1 − πt ). The last term is the linear approximation of the real
seigniorage revenue around ∆M s = 0 and π = 0. More speciﬁcally, real seigniorage expressed as afraction of the steady-state output is ∆Mts / (Pt Y ) and can be approximated ass∆Mts∆M s Pt−1 Mt−1
= st
≅ V −1 ∆mst .
Pt Y
Mt−1 Pt Pt−1 Y(11)

We now deﬁne the money-ﬁnanced ﬁscal stimulus as the monetization of government purchases
satisfying:
Gt − G =

∆Mts
.
Pt

That is, all government purchases exceeding its steady-state level is ﬁnanced by the seigniorage.2
By dividing both sides by the steady-state output, we have gt ≅ ∆mst /V from (11). Therefore, themoney supply under the money-ﬁnanced ﬁscal stimulus is approximately determined by∆mst = V gt .

(12)

The lump-sum tax in (10) is speciﬁed by the following reaction function:τt = ψb bt−1 .

(13)

Substituting (12) and (13) into (10), we have
bt = (R − ψb ) bt−1 + bR (it−1 − πt ) ,(14)

2
Galı́ (2017) considers a money-ﬁnanced ﬁscal stimulus that keeps the issuance of government bonds unchanged inreal terms. The money-ﬁnanced ﬁscal stimulus considered here involves money creation that ﬁnances the government
purchases in excess of its steady-state level. This is because Galı́’s (2017) speciﬁcation often violates the Blanchard–
Kahn condition, if it is combined with the CIA constraint. By contrast, ours robustly satisﬁes the Blanchard–Kahn
condition regardless of our assumption about the money demand function.

8

where we assume that R − ψb < 1 to ensure the stability of bt . The dynamics of the debt-to-GDP
ratio here are residually determined from the system of equations under (12) and (13). In thismodel, bt is driven by the interest payment on the government debt and its persistence depends on
how we specify ψb in (13).

2.3Market clearing

Equilibrium in the ﬁnal goods market requires Yt = Ct + Gt . The labor market clearing∫1
condition is Nt = 0 Nt (j) dj for all t. For the money market, we assume that the money supply
fully determines the household’s nominal holdings of money and Mts = Mt holds for all t. This
assumption is not inconsistent with an occasionally binding CIA constraint. If It > 0 in equilibrium
dynamics, the CIA constraint is binding and the household’s desired level of money equals themoney supply: Mt = Pt Ct = Mts . If It = 0, the CIA constraint is not binding and not all of themoney will be used to purchase goods: Mt > Pt Ct . Even if It = 0, however, the money can still be
used for transferring the household’s wealth into the next period. Therefore, Mts = Mt holds for
all t. Likewise, the government bond market also clears as usual if It ≥ 0. If It = 0, the nominalbond holdings by the household are fully determined by the money supply in the economy.

3

Results
To assess the impact of the implementation lag of the money-ﬁnanced ﬁscal stimulus on equi-

librium dynamics, we use the (log) linearized version of the model around the zero inﬂation steadystate.3 We perform two experiments, based on two initial states of the economy: (i) normal times
with It > 0 and (ii) the liquidity trap with It = 0. We conduct the ﬁrst experiment particularly
because it helps us better understand equilibrium dynamics in a liquidity trap. We then assess theeﬀects of the length of the implementation lag and the interest semi-elasticity of money demand.

3.1

Calibration

For the simulations, we select β = 0.995, σ = 1, ϕ = 5, α = 1/4, ϵ = 9, θ = 3/4, and ψb = 0.02.
These parameters are taken from Galı́ (2017) and all standard in the literature. Because inﬂationis zero in the steady state, β = 0.995 implies that the annualized real interest rate is two percent in
the steady state (i.e., R = 1.005). The steady-state share of government purchases to output is 20
3

The linearized equations are described in the Appendix A of this paper.

9percent (i.e., γ = G/Y = 0.2), which is consistent with the U.S. data.4 By deﬁnition, the income
velocity Vt = (Pt Yt ) /Mt = [(Pt Ct ) /Mt ] (Yt /Ct ). Assuming that the CIA constraint in the steadystate holds with equality, the steady-state income velocity is given by V = Y /C = 1/ (1 − γ) = 1.25.
In terms of implementation lag h, there seems no strong consensus about how long it takes for thegovernment to implement their policy in general or how far in advance agents become aware ofpossible changes to the money-ﬁnanced ﬁscal stimulus. For example, using the narrative-approachtax shock series of Romer and Romer (2010), Mertens and Ravn (2011) ﬁnd that the median
implementation lag is six quarters for a tax shock. Ramey (2011) argues that defense spending
shocks identiﬁed by vector autoregressions using the postwar U.S. data are forecastable four quarters
in advance. She also argues that even nondefense spending would be known at least a few months
in advance. Therefore, we consider a relatively wide range of values of h in the simulations. Taking
these previous studies into account, we set h = 6 as the benchmark and consider h = 1, 4, and 8
for robustness.

3.2
3.2.1

The eﬀect of the money-ﬁnanced ﬁscal stimulus on output
Experiment I: The economy in normal times

The ﬁrst experiment considers the money-ﬁnanced ﬁscal stimulus in normal times. More specifically, we assume that the economy for t < 0 is initially in the steady state with It > 0. Given thisstate of the economy, the money-ﬁnanced ﬁscal stimulus is announced at t = 0 and implemented
at t = h ≥ 0.
Figure 1 plots the output response to the money-ﬁnanced ﬁscal stimulus with and without animplementation lag (denoted by “h = 6” and “h = 0” in the ﬁgure). The blue line indicatesthe output response when the money-ﬁnanced ﬁscal stimulus is implemented immediately. In thiscase, the money-ﬁnanced ﬁscal stimulus can also be understood as an unanticipated ﬁscal stimulus
implemented at t = 0. In response to this unanticipated ﬁscal stimulus, output exhibits a substantial
expansion. In fact, when gt increases by one percent of the steady-state output, output in thesame period increases by 1.40 percent, meaning that the government spending multiplier is larger
than unity. The positive response is large because of expansionary ﬁscal policy and simultaneousmonetary expansion.
The red line in Figure 1 indicates that, if the same ﬁscal stimulus is implemented with a lag,4

For example, see Erceg and Lindé (2014).

10

the output responses diﬀer dramatically from the previous case. In this case, the money-ﬁnanced
ﬁscal stimulus is announced at t = 0 and the implementation at t = 6 is credibly anticipated.
In response to the announcement, output gradually decreases. That is, the money-ﬁnanced ﬁscal
stimulus causes a recession rather than a boom. The output decline is substantial at 0.77 percent
immediately before the implementation. Once the ﬁscal stimulus is implemented at t = 6, output
responses become positive for t ≥ h. However, when gt increases, output in the same period
increases by only 0.87 percent (compared with the steady state value), implying the ﬁscal stimulus
has a weaker eﬀect.
The reason for the recession is straightforward. Aggregate demand is depressed by the increases
in prices before the implementation of the money-ﬁnanced ﬁscal stimulus. To better understandthis, let us express the money and goods market equilibrium conditions in log-linearized form. The
money market equilibrium condition is ct = mst − pt and the goods market equilibrium condition
is yt = (1 − γ) ct + gt , where mst and pt represent the log-deviation from their (initial) values att = −1. Also, ct and yt are the log-deviation from the steady state. Combining these two equations
yieldsyt = (1 − γ)(mst − pt ) + gt .

(15)

By deﬁnition, ms−1 and p−1 are zero. Therefore, mst and gt are also zero before the money-ﬁnanced
ﬁscal stimulus is implemented (i.e., mst = gt = 0 for t < h). However, pt can change during
the period between announcement and implementation (i.e., 0 ≤ t < h). In response to theannouncement, forward-looking ﬁrms increase prices prior to the implementation. That is, higher
aggregate demand in the future increases prices pt . Therefore, we have
yt = −(1 − γ)pt < 0,(16)

for 0 ≤ t < h. This implies that the money-ﬁnanced ﬁscal stimulus for h > 0 causes a recession
rather than a boom.
The mechanism behind the output reduction is the same as that discovered by Ball (1994) in thecontext of the credible announcement of disinﬂationary policy. Ball (1994) shows that the credible
announcement of disinﬂationary monetary policy (i.e., an announcement of a decline in money
growth) causes a boom rather than a recession in a sticky price model. In our model with ﬁscal
policy, the announcement of the money-ﬁnanced ﬁscal stimulus is equivalent to an announcementof inﬂationary policy, causing a recession rather than a boom.
11

Figure 2 displays the responses of the other macroeconomic variables to the announcementof the ﬁscal stimulus with an implementation lag. As shown in the upper-right panel, inﬂation
responds immediately to the announcement because of the forward-looking property of inﬂation.
In the second row, consumption exhibits dynamic patterns similar to output, although the increasein consumption at the time of implementation is smaller than that in output. The nominal andreal interest rates are volatile, peaking in period 5. Furthermore, the nominal interest rate iscompletely insensitive before the implementation, based on our calibrated degree of relative risk
aversion of unity.5 This insensitive response of the nominal interest rate for t < h is preserved evenin the economy caught in a liquidity trap and thus the duration of ZLB spells is not substantiallyshortened by this ﬁscal stimulus.
3.2.2

Experiment II: The economy in a liquidity trap

The second experiment assesses the money-ﬁnanced ﬁscal stimulus at the time of the liquidity
trap. More speciﬁcally, the money-ﬁnanced ﬁscal stimulus is announced at t = 0 and implemented
at t = h ≥ 0 as before, but we assume that the government announces the money-ﬁnanced ﬁscal
stimulus when the economy is in the liquidity trap with It = 0
To generate the initial state of the liquidity trap, we follow the literature and introduce a
deterministic adverse demand shock.6 Here, the adverse demand shock appears in the log-linearized
Euler equation:
ct = ct+1 − σ −1 (it − πt+1 − ρt ) ,(17)

where ρt = − ln (Zt+1 /Zt ). We assume that the economy for t < −1 is in the steady state in which
the central bank implements the optimal discretionary monetary policy.7 In period t = −1, ρtdecreases so that the nominal interest rate hits the ZLB. In particular, ρt < log β = − log R for−1 ≤ t ≤ T and ρt = 0 for t > T . The decline in ρt is unexpected at the beginning of t = −1, but
once the decline is observed, the above path of ρt is fully known. The value of ρt is assumed tobe constant over t ∈ [−1, T ] and is calibrated to generate a three-percent decline in output at the5

The response of the nominal interest rate it can be seen from the log-linearized Euler equation in that period:∆ct+1 = σ −1 (it − πt+1 ). Before the implementation of the money-ﬁnanced ﬁscal stimulus, the CIA constraint implies
that ∆ct = −πt < 0 because ∆mst = V gt = 0 for t < h. As we calibrated σ equal to unity, ∆ct = it−1 − πt and∆ct = −πt lead to the fully insensitive nominal interest rate: it−1 = 0 for t < h. Once the ﬁscal stimulus is
implemented, ∆mst > 0. In this case, we have ∆ct = ∆mst − πt > −πt for t ≥ h resulting in it−1 = ∆mst > 0 fort ≥ h.
6
For example, see Jung, Teranishi and Watanabe (2005).
7
As discussed in Galı́ (2015, 2017), the allocation under the discretionary monetary policy with the ZLB constraint
is replicated by the Taylor rule, it = max (0, ρt + φπ πt ), where φπ > 1.

12

time of the decline in ρt . Given this decline in output, we examine how the government can help
the economy recover from the recession, using the money-ﬁnanced ﬁscal stimulus.
To measure the puriﬁed eﬀect of the money-ﬁnanced ﬁscal stimulus, we deﬁne the allocation
with no policy change as the reference level.8 In this allocation, the central bank continues toimplement the optimal discretionary monetary policy for t ≥ 0. Conveniently, this allocation
serves as the reference level in measuring the puriﬁed eﬀect of the money-ﬁnanced ﬁscal stimulus.
If the ZLB constraint is not binding, the optimal discretionary monetary policy can fully stabilize
both output and inﬂation because of the absence of cost-push shock. If the ZLB is binding, theoptimal discretionary monetary policy cannot fully stabilize the output and inﬂation because of
the central bank’s failure to lower the nominal interest rate. Note that the assumptions thatρt < log β = − log R for −1 ≤ t ≤ T and that ρt = 0 for t > T ensure that the ZLB constraint isbinding only for −1 ≤ t ≤ T under the optimal discretionary monetary policy. The black dashedline in Figure 3 is the output response under this allocation with no policy change. In the ﬁgure,
we set T = 5 so that the decreased ρt causes the liquidity trap during −1 ≤ t ≤ 5. Because of the
ZLB on the nominal interest rate, output is lower than the steady state for −1 ≤ t ≤ 5 and is kept
at the steady-state level for t > 5.
Figure 3 compares the output responses to the announcement of a money-ﬁnanced ﬁscal stimulus
between h = 0 and h = 6. The red line shows that the ﬁscal stimulus for h = 6 generates a slower
recovery from recession than the reference level of no policy change. In other words, it causes a“recession”compared with the reference level. Similar to Experiment I, this is in sharp contrast to
the case of h = 0. As shown by the blue line, the money-ﬁnanced ﬁscal stimulus without a lag
achieves a faster recovery from the recession than the reference level (i.e., a “boom” compared withthe reference level).
Note that, in Experiment II, the Euler equation matters for consumption rather than the CIA
constraint. This is because the CIA constraint (5) is not binding when the nominal interest rate
hits the ZLB. Given that the CIA constraint is not binding, (17) implies that the dynamic path of
the real interest rate is important for consumption under It = 0. In other words,
ct = −σ−1∞∑(rt+j − ρt+j ) ,

(18)

j=0

where rt = it − πt+1 .
8

In Galı́ (2017), the allocation with no policy change is denoted as the benchmark case of no ﬁscal response.

13

Figure 4 plots the impulse responses of the other macroeconomic variables under the liquidity
trap. Here, the adverse demand shock ρt (shown in the bottom row of the ﬁgure) is strong enough forthe nominal interest rate to hit the ZLB (the third row of the ﬁgure) so inﬂation drops substantially(the ﬁrst row). The shock to ρt is common irrespective of the presence of the ﬁscal stimulus. As aresult, the real interest rate is high during the ﬁrst few periods (the left panel of the third row of
the ﬁgure) in both the money-ﬁnanced ﬁscal stimulus and no policy change scenarios. This highreal interest rate reduces consumption in both cases.
To understand the mechanism behind the “recession” in Experiment II, we compare the realinterest rate between the cases of the money-ﬁnanced ﬁscal stimulus and no policy change. Comparisons reveal that the real interest rate under the money-ﬁnanced ﬁscal stimulus roughly follows
the reference level for t < 4. However, the real interest rate under the money-ﬁnanced ﬁscal stimulus exceeds the reference level substantially after t = 5. In particular, the real interest rate at t = 5
is 2.7 percentage points higher under the ﬁscal stimulus than under its reference level. The realinterest rate, which is higher than the reference level for t > 5, leads to weaker aggregate demandthan in the reference level.
Therefore, the intuition is that future government purchases gt crowds out current consumption
ct through the Euler equation. Likewise, yt is lower than the reference level for 0 < t < h because
only consumption is aggregate demand in these periods. Higher inﬂation expectations under themoney-ﬁnanced ﬁscal stimulus cause the nominal interest rate to exceed zero one period earlier thanthe case of no policy change. Consequently, the CIA holds with equality at t = 5 and consumption
in the subsequent periods is constrained by the CIA. At t = 5, the reason that consumption islower under the money-ﬁnanced ﬁscal stimulus than its reference level is the same as in Experiment
I. That is, because the aggregate prices are higher than the reference level, consumption becomes
lower than the reference level.
We also note that, unlike the case of the money-ﬁnanced ﬁscal stimulus in normal times, thedebt-to-GDP ratio is uniformly higher under the money-ﬁnanced ﬁscal stimulus with an implementation lag than under no policy change. Based on (14), a debt-to-GDP ratio higher than its
reference level results from a higher real interest than the reference level.

3.3

The role of the implementation lag

Because there is no broad consensus on the length of the implementation lag of the moneyﬁnanced ﬁscal stimulus, it is useful to consider the output responses for diﬀerent values of h.
14

Figure 5 plots the output responses to the announcement of the money-ﬁnanced ﬁscal stimulus
when the length of implementation lag varies: h = 0, 1, 4, 8. In the upper panel of the ﬁgure, we
assume that the policy is announced when the economy is initially in the steady state with It > 0.
In the lower panel, we assume that the policy is announced when the economy is in the liquidity
trap with It = 0. To facilitate comparison between the two cases, the output responses in the lower
panel are expressed as output responses relative to the reference level of no policy change.
In both panels of Figure 5, the recession is more severe for a larger h. In the upper panelof the ﬁgure, each line exhibits the largest output decline in the period immediately before theimplementation of the ﬁscal stimulus. The largest output decline is 0.41 percent for h = 1 and 0.78
percent for h = 8. In the lower panel, the largest output decline (relative to the reference level ofno policy change) is 0.40 percent for h = 1 and 0.84 percent for h = 8. Furthermore, for h = 8, we
observe another large output decline at the time of the announcement of the money-ﬁnanced ﬁscal
stimulus, making our results stronger.
When the economy is initially in normal times, (16) holds for 0 ≤ t < h. Therefore, therecession becomes more severe as forward-looking ﬁrms have more opportunities to adjust theirnominal prices upward prior to the implementation of the ﬁscal stimulus.
When the economy is in the liquidity trap, (18) matters for consumption for 0 ≤ t < h.
Therefore, the “recession” relative to the reference level becomes more severe as the extent towhich the real interest rate is higher than its reference level becomes greater. This relationshipholds as long as It = 0.
Figure 6 assumes that the economy is initially in the liquidity trap as in the lower panel of Figure
5 and numerically compares the extent to which the real interest rate exceeds its reference level.
The ﬁgure shows that, as h increases, the real interest rate is more likely to exceed the reference
level of no policy change in response to the announcement of the money-ﬁnanced ﬁscal stimulus.
Each panel presents the dynamics of the real interest rate for h = 0, 1, 4, 8 and its reference levelof no policy change. For h = 0 in the upper-left panel, the real interest rate is uniformly lower
than the reference level of no policy change. This implies that consumption exceeds its reference
level (see (18)). Next, for h = 1 in the upper-right panel, while the real interest rate is higher than
the reference level at t = 0, it is lower than the reference level afterward. Therefore, for h = 1,
while consumption falls below the reference level only at t = 0, it again exceeds the reference level
afterward. When h = 4, the real interest rate at t = 3 is 3.28 percent, or 2.90 percentage points
higher than the reference level. Because of the diﬀerences of the real interest rate from the reference15

level, the forward-looking household reduces its consumption at t = 0. As a result, output at t = 0
declines substantially in comparison with the cases of h = 0 and 1. For the same reason, the output
decline for h = 8 is even larger. In fact, the real interest rate at t = 7 is 5.33 percent, or 3.32
percentage points higher than the reference level.

3.4

The role of the interest semi-elasticity of money demand

The ﬁndings in the preceding subsections depend on how we specify the money demand function. More speciﬁcally, output declines during the periods between the announcement and theimplementation of the money-ﬁnanced ﬁscal stimulus result from the CIA constraint that leads to(15) and (16). When we replace the CIA by the money-in-the-utility (MIU) function, however,the policy implications may diﬀer because of the presence of the interest semi-elasticity of money
demand. In this subsection, we explore this possibility.9
To clarify the motivation of our analysis, let us consider the following log-linearized money
demand function (together with the money market equilibrium):mst − pt = ct − ηit ,(19)

where η > 0 denotes the interest semi-elasticity of money demand. Note that as suggested by
Eggertsson and Woodford (2003) and Jung, Teranishi and Watanabe (2005), the standard MIU
function without a satiation level of real money balances is not necessarily consistent with the ZLB
on the nominal interest rate. For this reason, we assume the MIU function with a satiation levelof real money balances and follow a convenient way of modeling money demand by English, Erceg
and Ló pez-Salido (2017). While we leave the derivation of (19) to the Appendix B, this money
demand function has nonzero η which depends on how much the steady-state consumption velocity
deviates from the consumption velocity evaluated at the satiation level of real money balances.
For simplicity, consider the case when the economy is in the steady state with It > 0 at thetime of the policy announcement. Using (19), (16) is now changed toyt = −(1 − γ)pt + (1 − γ)ηit ,(20)

for 0 ≤ t < h. This implies that the money-ﬁnanced ﬁscal stimulus does not necessarily cause
9

We also conduct many robustness analyses for the parameters in the model, including the size of the output
decline caused by the liquidity trap. As discussed in the not-for-publication appendix, our results in this paper arerobust to the parameter changes.

16

a recession. In fact, a boom takes place prior to the actual increase in government purchases, if
ηit > pt for 0 ≤ t < h. In other words, the policymakers could alleviate the recession, even if themoney-ﬁnanced ﬁscal stimulus has an implementation lag. A natural question is how large η should
be for them to successfully prevent the recession.
Figures 7 and 8 plot the output responses to the announcement of the money-ﬁnanced ﬁscal
stimulus under three values of η. In both ﬁgures, we newly introduce the output responses underη = 4 and 7 along with the benchmark result under η = 0. The left panel represents the caseof h = 6 while the right panel refers to the case of h = 0 for comparison. In each panel, theblack solid line corresponds to the case of η = 4 while the blue solid line indicates the case ofη = 7. The ﬁrst parameterization of the interest semi-elasticity (η = 4) is taken from Christiano,
Eichenbaum and Evans (2005) who estimate the semi-elasticity in the medium-scale New Keynesianmodel based on the U.S. data over the period 1965:Q3–1995:Q2.10 The second parameterization(η = 7) comes from Ireland (2009) who estimates η over the period 1980:Q1–2006:Q4.11 Thesesample periods are particularly important for our calibration, because the data during the ZLB
periods are explicitly excluded. In our model, when the ZLB constraint is binding, the household’s
real money holdings exceed the amount that it desires according to its money demand function.
That is, mst − pt > ct − ηit during the ZLB periods. If η is estimated under the assumption that(19) always holds with equality, the data including the ZLB periods may not be appropriate forcalibration.
Figure 7 assumes that the economy is in the steady state at the time of the announcement of
the money-ﬁnanced ﬁscal stimulus. The left panel of the ﬁgure reconﬁrms the predictions from(20). In the simulation, it > 0 for 0 ≤ t < h. Therefore, whether a recession takes place prior to
the implementation depends on η. As the black solid line of the left panel indicates (i.e., η = 4 &h = 6), the responses are small in magnitude. This is because the eﬀect of ηit cancels out the eﬀect
of pt . Therefore, the recession, if any, is not as deep under η = 4 as under η = 0. When η = 7 andh = 6 (the blue solid line in the left panel), the former tends to dominate the latter so that therecession does not take place. We also note that the sensitivity to η is observed only for h = 6: if
there is no implementation lag, the money-ﬁnanced ﬁscal stimulus is eﬀective in increasing output.
In Figure 8, the economy is assumed to be in the liquidity trap at the time of the announcement.
10
More precisely, they report that the interest semi-elasticity of money demand to the annualized interest rate is
0.96. Transforming this annualized interest semi-elasticity to the quarterly interest rate yields η = 3.84 ≅ 4.
11
He estimates the annualized interest semi-elasticity to range between 1.5 and 1.9. Translating this range in termsof the quarterly rate, it is between 6.0 and 7.6.

17

When η = 4 and h = 6, the output response follows a path similar to the reference level of no policychange prior to the implementation. In this case, output under η = 4 is slightly lower than thereference level of output. When η increases to 7, the slow recovery from the recession disappears.
In other words, the money-ﬁnanced ﬁscal stimulus is eﬀective even with an implementation lag.
Nevertheless, the money-ﬁnanced ﬁscal stimulus with an implementation lag achieves a much slower
recovery to the steady state than that without an implementation lag (see the right panel of thesame ﬁgure for the case of h = 0).
The responses of the other macroeconomic variables to the announcement under the liquidity
trap are shown in Figure 9 to highlight the diﬀerences between the cases of η = 0 and η > 0.
As shown in the third row of the ﬁgure, the nominal interest rate under η = 0 equals zero untilt = 5. Because the household is indiﬀerent between money and government bonds under It = 0, it
holds the money balances supplied by the central bank for the purpose of transferring its wealthinto the next period. Because consumption is low, inﬂation remains low and the real interest rate
remains high. The high real interest rate causes high interest payments of the government, so thedebt-to-GDP ratio continues to increase.
By contrast, the model dynamics are very diﬀerent under η = 7. The nominal interest rate
becomes positive at the time of the announcement at t = 0. The economy escapes from theliquidity trap and (19) holds with equality at and after t = 0. The nominal interest rate higher
than zero incentivizes the household not to hold money balances as a substitute for government
bonds. Real money holdings are reduced by the purchase of goods. As a result, consumption
increases. In fact, the second row of Figure 9 indicates that consumption under η = 7 increases
beyond the reference level. Likewise, inﬂation under η = 7 exceeds its reference level, as shown inthe ﬁrst row of the same ﬁgure. Such high inﬂation immediately lowers the real interest rate below
the reference level. The low real interest rate contributes to decreasing the interest payments of thegovernment, so the debt-to-GDP ratio is lower than the reference level and even starts to decline
around the time of the implementation.
The policy implications of our analysis for the money-ﬁnanced ﬁscal stimulus are threefold.
First, if the money-ﬁnanced ﬁscal stimulus has an implementation lag, policymakers should possess
detailed knowledge on the money demand function. Without this knowledge, they may fail toachieve a fast recovery from the recession in response to an adverse demand shock. Second, if
they are faced with a low interest semi-elasticity of money demand, they should use a moneyﬁnanced ﬁscal stimulus as immediately as possible. If policymakers do not give forward-looking
18households and ﬁrms many opportunities to make their decisions prior to the implementation of
the money-ﬁnanced ﬁscal stimulus, they can prevent the slower recovery of output. Finally, if they
face diﬃculties in shortening implementation lags, the money-ﬁnanced ﬁscal stimulus would bea reasonable policy option only when the interest semi-elasticity of money demand is suﬃciently
large.

4

Concluding remarks
Using the standard New Keynesian framework with nominal rigidities, we analyzed the eﬀectof a money-ﬁnanced ﬁscal stimulus with an implementation lag on output. The introduction of
implementation lags that are speciﬁc to ﬁscal policy may have nonnegligible impacts on the output
response to a shock in government purchases. While the money-ﬁnanced ﬁscal stimulus has astrongly positive eﬀect on output in the absence of an implementation lag, such a ﬁscal stimulus may
have a weak or even negative eﬀect on output if a reasonably long implementation lag exists.12 We
showed that a shorter implementation lag makes the money-ﬁnanced ﬁscal stimulus more eﬀective.
Our analysis also indicated that the eﬀect of the money-ﬁnanced ﬁscal stimulus varies, dependingon the interest semi-elasticity of money demand.13 An implication for policymakers is that detailed
knowledge of the money demand function is important for a better understanding of the eﬃcacy ofa money-ﬁnanced ﬁscal stimulus. In this regard, empirical studies on the money demand functionthat explicitly allows for the ZLB constraint are also important.14

12
Negative impacts of an implementation lag are related to the Fujiwara and Waki’s (2015) ﬁnding that thesocial welfare in the standard New Keynesian model reduces as the private sector is well informed about the future
fundamental shocks. See Fujiwara and Waki (2015) for additional details.
13
The importance of the money demand function for the eﬃcacy of ﬁscal policy was also pointed out by Holmes
and Smyth (1972) and Mankiw and Summers (1986), based on the old-fashioned Keynesian IS–LM model. However,they emphasize the roles of the scale variable (e.g., consumption vs. income) in the money demand function.
14
Some early examples of research include Bae, Kakkar and Ogaki (2006), Bae and de Jong (2007), and Nakashima
and Saito (2012), among others.

19References[1] Bae, Y. and R. M. de Jong (2007). “Money Demand Function Estimation by Nonlinear Cointegration,” Journal of Applied Econometrics, 22 (4), 767–793.
[2] Bae, Y., V. Kakkar and M. Ogaki (2006). “Money Demand in Japan and Nonlinear Cointegration,” Journal of Money, Credit, and Banking, 38 (6), 1659–1667.
[3] Ball, L. (1994). “Credible Disinﬂation with Staggered Price-Setting,” American Economic
Review, 84 (1), 282–289.
[4] Christiano, L. J., M. Eichenbaum and C. L. Evans (2005). “Nominal Rigidities and the Dynamic Eﬀects of a Shock to Monetary Policy,” Journal of Political Economy, 113 (1), 1–45.
[5] Cwik, T. and V. Wieland (2011). “Keynesian Government Spending Multipliers and Spillovers
in the Euro Area,” Economic Policy, 26 (67), 493-549.
[6] Eggertsson, G. B. and M. Woodford (2003). “The Zero Bound on Interest Rates and Optimal
Monetary Policy,” Brookings Papers on Economic Activity, 2003 (1), 139–233.
[7] English, W. B., C. J. Erceg and D. López-Salido (2017). “Money-Financed Fiscal Programs:
A Cautionary Tale,” Hutchins Center Working Paper No. 31.
[8] Erceg, C. and J. Lindé (2014). “Is There a Fiscal Free Lunch in a Liquidity Trap?” Journal of
the European Economic Association, 12 (1), 73–107.
[9] Fujiwara, I. and Y. Waki (2015). “Private News and Monetary Policy Forward Guidanceor (The Expected Virtue of Ignorance),” Federal Reserve Bank of Dallas Globalization andMonetary Policy Institute Working Paper No. 238.
[10] Fujiwara, I. and Y. Waki (2017). “Fiscal Forward Guidance: A Case for Selective Transparency,” RIETI-Discussion Paper Series 17-E-087.
[11] Galı́, J. (2015). Monetary Policy, Inﬂation, and the Business Cycle, Princeton University
Press, Princeton NJ.
[12] Galı́, J. (2017). “The Eﬀects of a Money-Financed Fiscal Stimulus,” University of Pompeu
Fabra Economic Working Paper Series, Working Paper No. 1441.

20

[13] Holmes, J. M. and D. J. Smyth (1972). “The Speciﬁcation of the Demand for Money and theTax Multiplier,” Journal of Political Economy, 80 (1), 179–185.
[14] Ireland, P. N. (2009). “On the Welfare Cost of Inﬂation and the Recent Behavior of Money
Demand,” American Economic Review, 99 (3), 1040–1052.
[15] Jung, T., Y. Teranishi and T. Watanabe (2005). “Optimal Monetary Policy at the ZeroInterest-Rate Bound,” Journal of Money, Credit and Banking, 37 (5), 813–835.
[16] Leeper, E. M., T. B. Walker and S.-C. S. Yang (2010). “Government Investment and Fiscal
Stimulus,” Journal of Monetary Economics, 57 (8), 1000–1012.
[17] Leeper, E. M. A. W. Richter and T. E. Walker (2010). “Quantitative Eﬀect of Fiscal Foresight,”
NBER Working Paper No. 16363.
[18] Mankiw N. G. and L. H. Summers (1986). “Money Demand and the Eﬀects of Fiscal Policies,”
Journal of Money, Credit and Banking, 14 (1), 27–54.
[19] Mertens, K. and M. O. Ravn (2011). “Understanding the Aggregate Eﬀects of Anticipated andUnanticipated Tax Policy Shocks,” Review of Economic Dynamics, 14 (1), 27–54.
[20] Nakashima, K. and M. Saito (2012). “On the Comparison of Alternative Speciﬁcations for
Money Demand: The Case of Extremely Low Interest Rate Regimes in Japan,”Journal of theJapanese and International Economies, 26 (3), 454–471.
[21] Ramey, V. A. (2011). “Identifying Government Spending Shocks: It’s All in the Timing,”
Quarterly Journal of Economics, 126 (1), 1–50.
[22] Romer, C. D. and D. H. Romer (2010). “The Macroeconomic Eﬀects of Tax Changes: Estimates
Based on a New Measure of Fiscal Shocks,” American Economic Review, 100 (3), 763–801.
[23] Turner A. (2015). “The Case for Monetary Finance – An Essentially Political Issue,” presented
at the 16th Jacques Polak Annual Research Conference, the International Monetary Fund,
November 5–6, 2015.
[24] Yang, S. S.-C. (2005). “Quantifying Tax Eﬀects under Policy Foresight,” Journal of Monetary
Economics, 52 (8), 1557–1568.

21

A

The system of linearized equations
The system of linearized equations is standard except that the CIA constraint is occasionallybinding. The equation for the money growth rate follows (12). The other equations in the system
are the same as Galı́ (2017).
Aggregate consumption of the households follows the Euler equation:
ct = ct+1 − σ −1 (it − πt+1 − ρt ) ,(A1)

as we discussed in the main text. Inﬂation evolves as
πt = βπt+1 − λµt ,(A2)where λ ≡ [(1 − θ)(1 − βθ)/θ] [(1 − α) / (1 − α + ϵα)]. In the above equation, µt is the average
price markup given byµt = −σct −

α+ϕ
yt .
1−α

(A3)

Note that the money demand function is given by the CIA constraint:lt ≥ ct ,(A4)

where lt = mt − pt is the real money holdings of the household. This equation holds with equality
if It > 0. We also have the law of motion for real money balances:lt = lt−1 + ∆mst − πt .

(A5)

If the CIA constraint is not binding (or It = 0), then (A4) is not used for computing the allocation.
Instead, we use
it = − ln R,(A6)

implying that the household passively holds money based on the money supply.
The equilibrium condition for the ﬁnal goods market isyt = (1 − γ) ct + gt ,

22

(A7)

where gt is exogenously given by (7). Because government purchases are ﬁnanced by seigniorage,
the money supply is automatically determined by∆mst = V gt .

(A8)

Due to the reaction function for the lump-sum tax rule τt = ψb bt−1 and the above money supply
process, the equation for the debt-to-GDP ratio reduces tobt = (R − ψb ) bt−1 + bRrt−1 ,(A9)where rt is the real interest rate given by the Fisher equation:
rt = it − πt+1 .

(A10)

The system of equations consists of the endogenous variables, yt , ct , it , rt , πt , lt , ∆mst , bt , and mut .
If It > 0 and the CIA constraint is binding, the model can be solved by (A1)–(A5) and (A7)–(A10)in which (A4) holds with equality. If It = 0 and the CIA constraint is not binding, (A4) must be
replaced by (A6). Whether or not the CIA constraint is binding depends on the complementaryslackness condition: (lt − ct ) (it − ln R) = 0, lt − ct > 0, and it − ln R > 0. For this complementaryslackness condition, see also Galı́ (2017).

B

The model with the MIU function
To derive money demand that is correlated with the nominal interest rate and is compatible

with the ZLB on the nominal interest rate, the literature has assumed an MIU function with asatiation level of liquidity (see Eggertsson and Woodford, 2003). This is because, if the marginalutility of real money balances is strictly positive, the standard optimality condition rules out thepossibility of It = 0 in equilibrium (given the ﬁnite marginal utility of consumption).
English, Erceg, and López-Salido (2017) propose a convenient way of modeling money demand.
Following their approach, we redeﬁne the representative instantaneous utility function and re[]place U (Ct , Nt ) by Ũ (Ct , Nt , Mt /Pt ) = Ct1−σ /(1 − σ) − Nt1+ϕ /(1 + ϕ) + h (Mt /Pt ) Zt . Using a23

quadratic function, we specify the function h(·) as follows:
(
h

Mt
Pt

)

[
(
)]
η0
Mt /Pt 2∗=−
max 0, k −
Zt ,
2
C̄t

(B1)

where k ∗ is the satiation level of real money balances and C̄t is the aggregate consumption thatis taken as given by the representative household and that equals the household’s consumption in
equilibrium. Although admittedly ad hoc, this function is convenient for analyzing the model with
the ZLB for at least two reasons. First, h (Mt /Pt ) reduces to a simple quadratic utility function
if (Mt /Pt ) /C̄t < k ∗ is satisﬁed for any t. In this case, the marginal utility of real balances isstrictly positive and thus the nominal interest rate is strictly positive. Second, h (Mt /Pt ) = 0 andh′ (Mt /Pt ) = 0 if the money supply is suﬃciently large so that (Mts /Pt ) /C̄t = (Mt /Pt ) /C̄t ≥ k ∗ .
In this case, the nominal interest can be zero and money becomes a perfect substitute as an asset
for transferring wealth into the next period.
Because we replace the CIA constraint by the money demand function based on the MIU
function, we change the budget constraint slightly. In particular, we replace Pt−1 Ct−1 in (1) by
Pt Ct :
Mt + Bt = (1 + It−1 )Bt−1 + Mt−1 − Pt Ct + Wt Nt + Pt Dt − Pt Tt .

(B2)

In contrast with (1), the nominal expenditure on the ﬁnal goods is ﬁnanced by the resource received
in the same period rather than by the money holdings carried over from the previous period.
Assuming that (Mt /Pt ) /C̄t < k ∗ , we replace (5) by
h′t
It
,
=
UC,t
1 + It

(B3)

)
](
[
where h′t ≡ h′ (Mt /Pt ) = η0 k ∗ − (Mt /Pt ) /C̄t Zt /C̄t . Furthermore, let kt ≡ (Mt /Pt ) /Ct be theinverse of the consumption velocity of money. Let k be the steady-state value of kt . We assume
that k satisﬁes k < k ∗ . Along with the money market clearing condition and the assumption ofσ = 1, (B3) can be log-linearized as (19), where η = (k ∗ /k − 1) (R − 1)−1 = (k ∗ /k − 1) [β/ (1 − β)].
Or equivalently, denoting the consumption velocity of money Vc = k −1 , we have
(
η=

Vc
−1
Vc∗

)

β
.
1−β

(B4)

In this speciﬁcation of the MIU function, the distance of the steady-state consumption velocity of24money from the consumption velocity at which real money balances are satiated aﬀects the interest
semi-elasticity of money demand.
The other ﬁrst-order conditions and resource constraints remain unchanged in solving the model.
As before, the ZLB needs to be considered with the money market equilibrium condition. In
particular, if It > 0, the household’s desired level of money holdings equals the money supply andmst − pt = ct − ηit holds. If It = 0 (or equivalently it = − ln R), however, the money holdings
may not be equal to their desired level and money is held as a perfect substitute for the oneperiod government bond. In this case, we compute the solution using it = − ln R instead of
mst − pt > ct − ηit .

25

Figure 1: Output responses to the announcement of the money-…nanced …scal stimulus: Normal
times
1.5

h=0
h=6
1

Output

0.5

0

-0.5

-1
0

5

10
Quarters

15

20

Notes: The money-…nanced …scal stimulus is announced in period 0 and implemented in period h. The initial
increase in government purchases is normalized to one percent of the steady-state output and government purchases
are …nanced by money creation speci…ed by (12). The blue line represents the case of h = 0 while the red line
corresponds to the case of h = 6. Each period is a quarter.

26

Figure 2: Responses to the announcement of the money-…nanced …scal stimulus for h = 6: Normal
times
OutputInf lation

2

1
1
0
0

-1
0

5

10

15

20

0

Consumption

1

5

10

15

20

Gov ernment purchases

1
0.5

0

0
-1
0

5

10

15

20

0

Real interest rate

6

6

2

4

10

15

20

Nominal interest rate

8

4

5

2

0
0

5

10

15

20

0

5

10

15

20

Quarters
Debt-to-GDP ratio

1
0
-1
-2
0

5

10

15

20

Quarters

Notes: In‡ation, the nominal interest rate, and the real interest rate are expressed at the annual rate. They are
drawn as the variation around their steady-state values of two percent (for the nominal interest rate and the realinterest rate) and zero percent (for in‡ation). While output and consumption are expressed as the log-deviation fromthe steady state, government purchases and the debt-to-GDP ratio are expressed as the di¤erence from the steadystate measured as a fraction of the steady-state output. See the notes to Figure 1 for additional details.

27

Figure 3: Output responses to the announcement of the money-…nanced …scal stimulus: Liquidity
trap
1.5
No policy change
h=0
h=6

1

0.5

0

Output

-0.5

-1

-1.5

-2

-2.5

-3
0

5

10
Quarters

15

20

Notes: The money-…nanced …scal stimulus is announced when the economy is in the liquidity trap with It = 0. The
dashed black line shows the reference level of output where the central bank continues to implement the optimal
discretionary monetary policy and makes no policy change. See the notes to Figure 1 for additional details.

28

Figure 4: Responses to the announcement of the money-…nanced …scal stimulus for h = 6: Liquidity
trap
OutputInf lation
2
0
-2
-4
-6

1
0
-1
-2
-3
0

5

10

15

20

0

Consumption

5

10

15

20

Gov ernment purchases

1

0
0.5
-2
0
-4
0

5

10

15

20

0

Real interest rate

5

10

15

20

Nominal interest rate

4

4

2

2

0

0
0

5

10

15

20

0

5

10

Debt-to-GDP ratio

15

20

t

0

10

-1
5

-2

No policy change
h=6

-3
0
0

5

10

15

20

0

Quarters

5

10

15

20

Quarters

Notes: The money-…nanced …scal stimulus is announced when the economy is in the liquidity trap with It = 0. The
dashed black line shows the reference level of output where the central bank continues to implement the optimal
discretionary monetary policy and makes no policy change. See the notes to Figures 1 and 2 for additional details.

29

Figure 5: E¤ects of implementation lags on output responses: Normal times and liquidity trap
1.5
h=0
h=1
h=4
h=8

Output

1
0.5
0
-0.5

Output relative to the reference level

-1
0

5

10
Quarters

15

20

0

5

10
Quarters

15

20

1.5
1
0.5
0
-0.5
-1

Notes: Both panels compare the responses of output for h = 0; 1; 4; 8. The solid blue line, the solid black line, thedashed purple line, and the solid red line correspond to the case of h = 0; 1; 4; 8, respectively. The upper panel
assumes that the money-…nanced …scal stimulus is announced when the economy is in the steady state with It > 0.
In the upper panel, each line shows the log-deviation of output from the steady state. The lower panel assumes thatthe money-…nanced …scal stimulus is announced when the economy is in the liquidity trap with It = 0. In the lower
panel, each line shows the log-deviation of output from its reference level of no policy change where the central bankcontinues to implement the optimal discretionary monetary policy. See the notes to Figure 1 for additional details.

30

Figure 6: Responses of the real interest rate under di¤erent values of h
6

6
h=0

Real interest rate

5
4

4

3

3

2

2

1

1

0

0

-1

-1
0

5

10

15

0

6

5

10

15

6
h=4

5
Real interest rate

h=1

5

h=8

5

4

4

3

3

2

2

1

1

0

0

-1

-1
0

5
Quarters

10

15

0

5
Quarters

10

15Notes: Responses of the real interest rate are shown for h = 0; 1; 4; 8. The money-…nanced …scal stimulus is announced
when the economy is in the liquidity trap with It = 0. The real interest rate is expressed at the annual rate. The
dashed black line shows the reference level where the central bank continues to implement the optimal discretionary
monetary policy and makes no policy change. See the notes to Figures 1 and 5 for additional details.

31

Figure 7: E¤ects of the interest semi-elasticity of money demand on output responses: Normal
times

2

2

= 0 & h=6
= 4 & h=6
= 7 & h=6

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

= 0 & h=0
= 4 & h=0
= 7 & h=0

-1
0

5

10
15
Quarters

20

0

5

10
15
Quarters

20

Notes: Output responses are based on the money demand function with the interest semi-elasticity of money demand( = 0; 4; 7). The solid red line, the solid black line, and the solid blue line correspond to the case of= 0; 4; 7,

respectively. The left panel shows the case of h = 6 while the right panel shows the case of h = 0. In each panel, themoney-…nanced …scal stimulus is announced when the economy is in the steady state with It > 0. See the notes to
Figure 1 for additional details.

32

Figure 8: E¤ects of the interest semi-elasticity of money demand on output responses: Liquidity
trap

1.5

1.5
No policy change= 0 & h=6
= 4 & h=6
= 7 & h=61
0.5

No policy change= 0 & h=0
= 4 & h=0
= 7 & h=0

1
0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

-2

-2.5

-2.5

-3

-3
0

5

10
15
Quarters

20

0

5

10
15
Quarters

20

Notes: Output responses are based on the money demand function with the interest semi-elasticity of money demand( = 0; 4; 7). In each panel, the money-…nanced …scal stimulus is announced when the economy is in the liquidity
trap with It = 0. See the notes to Figures 1, 3, 5, and 7 for additional details.

33

Figure 9: E¤ects of the interest semi-elasticity of money demand: Liquidity trap
OutputInf lation
2
0
-2
-4
-6

1
0
-1
-2
-3
0

5

10

15

20

0

Consumption

5

10

15

20

Gov ernment purchases

1

0
0.5
-2
0
-4
0

5

10

15

20

0

Real interest rate

5

10

15

20

Nominal interest rate

4

4

2

2

0

0
0

5

10

15

20

0

Debt-to-GDP ratio

5

10

15

20

t

0

10
-1
5

=0
=7
No policy change

-2
0
-3
0

5

10

15

20

0

Quarters

5

10

15

20

Quarters

Notes: Responses are based on the money demand function with the interest semi-elasticity of money demand ( = 0
and 7). The solid red line and the solid blue line correspond to the case of= 0 and= 7, respectively. In the …gure,

we assume that h = 6 and the money-…nanced …scal stimulus is announced when the economy is in the liquidity trapwith It = 0. See the notes to Figures 1, 2 and 4 for additional details.

34