ISSN (Print) 0473-453X

Discussion Paper No. 1196

ISSN (Online) 2435-0982

SHOULD THE FISCAL AUTHORITY
AVOID IMPLEMENTATION LAG?
Masataka Eguchi
Hidekazu Niwa
Takayuki Tsuruga

November 2022

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

Should the fiscal authority avoid implementation lag?∗
Masataka Eguchi†, Hidekazu Niwa‡, and Takayuki Tsuruga§
November 11, 2022

Abstract
Implementation lags are one of policymakers’ concerns about fiscal policies, as these
may reduce their efficacy. Using a standard New Keynesian model with an effective
lower bound on the nominal interest rate, we compare the impacts of fiscal stimulus on
output across various lengths of implementation lag. We show that despite concerns
among policymakers, a fiscal authority can enhance the efficacy of government purchases on output with implementation lags when the economy is caught in a liquidity
trap.
JEL Classification: E32; E52; E62
Keywords: Fiscal multiplier, Effective lower bound, Government spending, Liquidity trap∗

The authors gratefully acknowledge the financial support of the Joint Usage/Research Centers at theInstitute of Social and Economic Research (ISER) and the Kyoto Institute of Economic Research (KIER),
and Grants-in-Aid for Scientific Research (20H05631, 20H05633, 21H00700, and 21H04397).
†
Faculty of Economics, Komazawa University‡
Osaka School of International Public Policy, Osaka University§
Institute of Social and Economic Research, Osaka University; Correspondence to: Institute of Socialand Economic Research, Osaka University 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. E-mail address:
tsuruga@iser.osaka-u.ac.jp1

1

Introduction

A broad consensus on discretionary fiscal policies is that they require long lags to implement.
Following seminal work by Friedman (1953), it has been argued that fiscal policies with
implementation lags may be ineffective or even destabilize the economy. While the argument
has become less pessimistic, implementation lags in discretionary fiscal policies remain aconcern in mitigating recession. For example, while Blanchard et al. (2009) argued for fiscal
stimulus during the 2008–09 financial crisis, they also recognized the long implementation
lags of fiscal policies as a risk. Their justification for fiscal stimulus with implementation
lags relies on the fact that a financial crisis in practice lasts for many quarters. Christiano,
Eichenbaum, and Rebelo (2011) also regard the long implementation lags as a risk. Theyfind that the efficacy of fiscal policies depends on the state of the economy in which they
are implemented. In particular, the effect of a fiscal stimulus on output is large when theeffective lower bound (ELB) on the nominal interest rate is binding but becomes smallerwhen the ELB is not binding. Their finding implies that the fiscal multiplier is small if thegovernment implements fiscal policy in some future period in which the ELB is no longer
binding. Therefore, their analysis supports the conventional view that the fiscal authority
should avoid implementation lag.
This note shows that a fiscal authority may not need to avoid implementation lags.
Using the standard New Keynesian model with an ELB, we examine the efficacy of fiscal
policy over various lengths of implementation lags. We find that implementation lags could
enhance the efficacy of fiscal stimulus when the ELB is binding. Thus, a certain length of
implementation lag may be desirable for the efficacy of fiscal stimulus. We also note, as
Christiano, Eichenbaum, and Rebelo (2011) argued, that policymaker concerns remain valid
when the ELB is no longer binding or when the economy has already recovered. That is,
implementation lags in fiscal stimulus deteriorate the efficacy of fiscal stimulus on output
when the nominal interest rate is positive. In such normal times, the fiscal authority should
avoid implementation lags. Consequently, the desirability of implementation lags critically2depends on whether the ELB is binding.

2

The model

The model we consider here is a standard closed-economy New Keynesian model with an
ELB. The model shares various features discussed in Galı́ (2015), consisting of the consumption Euler equation, the New Keynesian Phillips curve (NKPC), and the Taylor rule with the
ELB. Here we explicitly introduce fiscal policy with implementation lags into this standard
model. For simplicity, we consider the deterministic version of the model because we focus
only on the impulse response functions to the steady state. In what follows, we describe thekey linearized equations of the model and leave the details to the appendix.
The consumption Euler equation in the representative agent model is given by

ct = ct+1 − (rt − πt+1 − %t ) ,(1)

where ct denotes the log-deviation of consumption from the steady state, rt is the deviationof the net nominal interest rate from the steady state, and πt is inflation. Here we assume
zero inflation in the steady state and that the intertemporal elasticity of substitution is one.
Following Krugman (1998), Eggertsson and Woodford (2003), and Jung, Teranishi, and
Watanabe (2005), we introduce a deterministic preference “shock” %t into the consumption
Euler equation to generate a liquidity trap in which the nominal interest rate hits the ELB.
The NKPC takes the following form:
α+ψ
πt = κ ct +yt + βπt+1 ,
1−α

(2)

where yt is the log-deviation of output from the steady state, ψ > 0 denotes (the inverseof) the Frisch elasticity of labor supply, and β ∈ (0, 1) is the subjective discount factor ofhouseholds. We assume the firm’s production function with decreasing returns in labor,3where 1 − α is the returns to labor.1 Also, κ ≡ [(1 − θ)(1 − βθ)/θ)][(1 − α)/(1 − α + εα)]represents the slope of the NKPC, where θ is the probability that firms cannot reset theirprices in each period and ε is the elasticity of substitution across differentiated goods. In(2), ct + [(α + ψ)/(1 − α)]yt represents the log-deviation of the average real marginal cost.2
The monetary authority sets the nominal interest rate according to a simple Taylor rule
where rt responds only to inflation: rt = απ πt with απ > 1. Together with the ELB, thenominal interest rate is given by

rt = max (απ πt , ln β) .

(3)

Recall that rt is the deviation of the nominal interest rate. The steady-state nominal interest
rate equals ln(1/β). Thus, the nominal interest rate becomes zero if and only if rt decreasesto − ln(1/β) = ln β.
We are interested in the impact of a fiscal policy shock on the economy. Following Galı́,López-Salido, and Vallés (2007) and Galı́ (2020), denote gt as the deviation of government
purchases from the steady-state value expressed as a fraction of steady-state output (i.e.,
gt = (Gt − G)/Y where Gt , G, and Y are government purchases in period t, steady-state
government purchases, and steady-state output, respectively).
The goods market satisfies the following market-clearing condition:yt = (1 − γ)ct + gt ,(4)

where γ is the steady-state government purchases to output ratio G/Y . As discussed in thenext section, gt is exogenously determined.
1
Following Galı́ (2015), we assume that intermediate good producers produce differentiated goods using
the production function Yt (i) = Nt (i)1−α , where Yt (i) and Nt (i) are the output of firm i and labor demand
for firm i, respectively.
2Here, ct + ψ/(1 − α)yt is the log-deviation of real wages expressed by the marginal rate of substitution
between consumption and labor, and α/(1 − α)yt is the log-deviation of aggregate labor productivity.

4

3

Policy experiments of implementation lags

3.1

Simulations

Our simulations aim to compare the effect of government purchases on output across various
lengths of implementation lag. In simulations, the government makes an announcement
at t = 0 that government purchases are to increase by one percent of steady-state output
relative to steady-state government purchases. The increase in government purchases takes
place in period h ≥ 0 where h represents the length of the implementation lag. Government
purchases gt take the following values:

gt =




0

for t < h,δ t−h gh

for t ≥ h,(5)where δ ∈ [0, 1) measures the persistence of gt because gt+1 = δgt for t > h. Throughout
this paper, we consider fiscal stimulus, meaning that the government purchases in period h
are always positive: gh > 0.
We measure the overall effect of fiscal stimulus with implementation lag h on output by
P∞ t
β (Yt − YtR ).
ϕh = Pt=0∞t
t=h β (Gt − G)(6)

The numerator represents the cumulative changes in output from its reference level YtR , which
we define later. It measures changes in output from the period of announcement (t = 0).
The denominator is the cumulative changes in government purchases relative to the steadystate. This measures changes in government purchases from the period of implementation(t = h). Both the numerator and the denominator are expressed as the present value.3 As
a function of h, we assess the overall effect of fiscal stimulus on output.
Note that ϕh shown in (6) includes the output responses from the periods of announce3

We approximate (6) by

P60

P60

t
R
t=0 β (Yt − Yt )/[tt=h β (Gh − G)].

5

ment (i.e., from period 0 to h − 1) as well as from the periods after implementation (i.e.,from period h to ∞). Our measure looks like, but differs from, the net present value fiscal multiplier discussed in Uhlig (2010). The net present value fiscal multiplier is given byPP[ st=0 β t (Yt − Y )]/[ st=0 β t (Gt − G)], where s is the horizon for the fiscal multiplier and thereference level of output is YtR = Y . This net present value fiscal multiplier is designed tomeasure how the cumulative effect of an unanticipated shock to government purchases varies
with the increase in horizon. By contrast, ϕh fixes the horizon at infinity and evaluates
how the cumulative effect of an anticipated shock to government purchases varies as theimplementation lag increases.
As we proceed, it is useful to specify YtR in (6) for our two experiments. The firstexperiment investigates the impact of government purchases on the economy in a normal
time in which the ELB is not binding. In particular, we assume that %t = 0 for all t ≥ 0, andthus the economy is initially in the steady state where the nominal interest rate is greaterthan zero (i.e., rt > ln β). In this sense, the economy is in the normal time. Because theinitial state of the economy is in the steady state, this experiment specifies the reference
level of output as YtR = Y for all t.
The next experiment considers the economy under a liquidity trap in which the nominalinterest rate temporarily hits the ELB. In particular, we assume that a negative preference
shock occurs in period 0 unexpectedly and continues to be negative for the next three years.
After three years, the negative shock disappears. In equation %t ' −0.003 for t = 0, 1, 2, ..., 11
and %t = 0 for t ≥ 12. We parameterize the size of the preference shock to ensure thatthe weakened aggregate demand temporarily generates the liquidity trap.4 Because the
ELB prevents the nominal interest rate from falling below zero, the recession caused by
the preference shock is deeper in the case with the ELB than in that without the ELB. In
response to the declines in aggregate demand, the government announces the fiscal stimulus
in period 0 (the period when the negative preference shock hits the economy) and implements
4

The parameterization for Figure 4 with this size of %t leads to a decline in output by five percent and aliquidity trap that lasts for eight quarters.

6its policy in period h. We define the reference level of output YtR in the second experimentas the equilibrium output where the preference shock disturbs the economy without fiscal
stimulus. More specifically, YtR is the equilibrium output arising from fluctuations in %t while
keeping Gt at G for all t.
The remaining parameterization for the simulations is standard. We set β at 0.995, which
implies a steady-state annualized real interest rate of two percent under zero steady-state
inflation. The elasticity of substitution across differentiated goods ε is set to nine. The Frischelasticity of labor supply is 1/5 so that ψ = 5 and the returns to labor in the production
function 1 − α is 3/4. In addition, we set θ such that the average duration of price changes
is four quarters (i.e., θ = 3/4). Following Galı́, López-Salido, and Vallés (2007), we set γ at
0.2. The persistence of government spending δ is set to 0.5. Finally, απ = 1.01. Later, we
confirm the robustness of our results to the coefficient on inflation in the Taylor rule.

3.2

Results

3.2.1

Experiment I: The economy in normal times

Figure 1 plots ϕh against h to compare the overall effect of the fiscal stimulus on output
across various lengths of the implementation lag h. In this experiment, the initial state of
the economy is the steady state in which rt > ln β.5 Thus, the ELB is not binding when thegovernment announces its fiscal stimulus.
[Figure 1 about here.]
The figure shows that implementation lags are not desirable in terms of the overall effectof a fiscal stimulus on output because the curve for ϕh decreases with h. This result suggeststhat fiscal stimulus without implementation lags performs best in terms of the overall effecton output. One important message in the first experiment is that the fiscal authority should
avoid any implementation lags when the economy is in normal times. Quantitatively, the5The definitions of yt ' (Yt − Y )/Y and gt ' (Gt − G)/Y imply that ϕh =

7

P∞

t
t=0 β yt /P∞tt=h β gt .

decrease in efficacy is nonnegligible in that if a policymaker postpones the implementation of
government purchases by one year after the announcement, the efficacy of the fiscal stimulus
falls by almost 35 percent from ϕ0 = 0.81 to ϕ4 = 0.53.
The intuition is straightforward from (1). Government purchases with an h-period implementation lag raise the real interest rate in period h. An increase in the real interest
rate decreases consumption in the same period (ch < 0). Put differently, government purchases gh crowd out consumption ch as in the standard New Keynesian model. The decline in ch is directly transmitted to the decline in the previous period ch−1 because of theforward-looking property of the consumption Euler equation (1), ch−1 = ch − (rh−1 − πh ).
Furthermore, the reduction in consumption can be repeated backwards: the decline in ch−1
is directly transmitted to the decline in the previous period ch−2 . The reduction in ct isthen repeated if 0 ≤ t ≤ h. If the real interest rate during t ∈ [0, h) were constant, we
would have ch = ch−1 = ... = c0 < 0. As the implementation lag lengthens, consumption
has more opportunities to decline before the implementation of fiscal stimulus. Thus, a long
implementation lag weakens the overall effect of the fiscal stimulus on output.6

3.2.2

Experiment II: The economy in a liquidity trap

We next turn to the second experiment. Figure 2 plots ϕh against h to compare the overall
effect of the fiscal stimulus on output across various lengths of the implementation lag h. In
contrast to the first experiment, the second experiment assumes that the preference shock%t < 0 lasts for three years and then returns to zero.7[Figure 2 about here.]
6

In simulations, the real interest rate between period 0 and h − 1 is slightly lower than the steady-statevalue because a reduction in consumption causes inflation to decline, which in turn, lowers the nominalinterest rate according to the Taylor rule rt = απ πt . As a result, ch < ch−1 < ... < c0 holds under ourparameterization. Numerically, the effect of the future consumption dominates that of the real interest rate
on ct in (1). Therefore, equilibrium consumption declines before the implementation of the fiscal stimulus(i.e., ct < 0 for 0 ≤ t ≤ h) as discussed in the main text.
P∞P∞
7Note that ϕh in this experiment reduces to ϕh = t=0 β t (yt − ytR )/[ t=h β t gt ], where ytR denotes the
log-deviation of YtR from the steady state Y .

8In contrast to the result of the first experiment, the figure suggests that a certain lengthof implementation lag is desirable. In the second experiment, ϕh exhibits a hump shapethat peaks at h = 5. This shape suggests that a fiscal stimulus with no implementation lag
does not necessarily perform best in terms of the efficacy of the fiscal stimulus on output.
Furthermore, the increment in efficacy is substantial, such that if a policymaker postpones
the fiscal stimulus by five quarters after the announcement, the efficacy of the fiscal stimulus
will approximately triple from ϕ0 = 1.26 to ϕ5 = 3.90. An important message in the second
experiment is that the fiscal authority does not always need to avoid implementation lag
when the ELB is binding.
We emphasize that the main result from this experiment is robust to changes in theparameters. Among others, the coefficient on inflation in the Taylor rule may influence ourresults because απ directly affects aggregate demand through the nominal interest rate rt inthe consumption Euler equation (1). Figure 3 indicates how ϕh changes across four valuesof απ = {1.01, 1.25, 1.50, 1.75}. In all cases, ϕh is hump-shaped against h. In particular, ϕhincreases with h up to h ≤ 5 and is maximized at h = 5, regardless of the value of απ . Onceh exceeds five, the overall effect of the fiscal stimulus on output declines over h and becomes
more significant as απ becomes larger.
[Figure 3 about here.]
Our finding on the desirability of implementation lags sharply contrasts with the broadconsensus among policymakers. Since Friedman (1953), lags in implementing fiscal policies
have long been a concern for policymakers. Even in recent studies, this concern remains,
although it has become less irrelevant. For example, Blanchard et al. (2009) argue for fiscal
stimulus during a financial crisis, but also recognize that implementation lags remain a risk
for fiscal stimulus. Christiano, Eichenbaum, and Rebelo (2011) point out that while thefiscal multiplier is large under the liquidity trap, it is small when the nominal interest rate
becomes positive. Their result means that the fiscal stimulus are effective especially when9the ELB is binding. Our finding differs from these previous studies in pointing out thedesirability of implementation lags when the ELB is binding.

3.3

Model dynamics of the economy in a liquidity trap

In this section, we explain the mechanism behind the hump shape in Figure 3, based onmodel responses to shocks. Figure 4 presents the model dynamics of output, consumption,
the nominal interest rate, and the real interest rate in experiment II. These variables areshown in terms of the responses to an anticipated increase in government purchases. The
nominal and real interest rates in the figure are multiplied by four to express them at anannual rate. We employ απ = 1.5 to detect significant differences in the model responses fordifferent values of h. In each panel of Figure 4, the dashed line presents the responses of avariable arising from declines in %t while keeping Gt at G for all t. By definition, the dashedline corresponds to the model responses without government intervention (i.e., the reference
level of a variable such as ytR ). Here, negative preference shocks lasting for three years
weaken aggregate demand such that the nominal interest rate hits the ELB. In particular,
negative preference shocks cause output to decrease by five percent and the nominal interest
rate to decline by two percent. However, the nominal interest rate cannot decrease below
ln β because of the ELB. The ELB binds for two years during 0 ≤ t ≤ 7. Given a decline
in inflation from weakened aggregate demand, the real interest rate increases strongly when
the nominal interest rate hits the ELB.
[Figure 4 about here.]
We are now ready to discuss the effect of a fiscal stimulus with implementation lag. Figure
4 selects two values for the implementation lags: h = 4 (the solid line with circles) and h = 12
(the solid line with diamonds). For reference, the vertical lines in each panel represent theperiod of implementation of the fiscal stimulus. Note that an implementation lag of one year
means that the government implements the fiscal stimulus under a liquidity trap because10rt = ln β for 0 ≤ t ≤ 7 without government intervention. By contrast, an implementation
lag of three years implies that the government implements the fiscal stimulus when the ELB
is no longer binding. Recall that the overall effect of fiscal stimulus on output almost peaks
in h = 4, but is considerably smaller when h = 12 (See Figure 3). In particular, under ourcalibration of απ = 1.5, ϕ4 = 3.30 and ϕ12 = −3.28.8
Let us first discuss the effect of a fiscal stimulus on output when implemented under aliquidity trap (h = 4). The solid line with circles in Figure 4 details the model responses
to both an increase in gt and a decrease in %t . In the upper panel, the solid line withcircles (yt ) is located uniformly above the dashed line (ytR ), especially several quarters after
the announcement of a fiscal stimulus in period 0. That is, government purchases with an
implementation lag of one year raise output relative to the reference level. The positive effecton output is summarized by ϕ4 = 3.30 in Figure 4.
To obtain the intuition behind the positive effect on output, define the relative consumption Euler equation asc̃t = c̃t+1 − (r̃t − π̃t+1 ) ,(7)

where the variable with a tilde denotes the variable relative to the reference level. In particular, c̃t = ct − cRt is consumption relative to its reference level. Analogously, we can definer̃t by rt − rtR and π̃t by πt − πtR . Equation (7) compares two consumption Euler equations
by taking the difference from the reference level. The preference shock %t disappears in (7)because %t is common to the two consumption Euler equations and they thus cancel eachother out.
Government purchases with h = 4 increase inflation without affecting the nominal interest
rate because the economy is caught in a liquidity trap (see the solid line with circles for0 ≤ t ≤ 7 in the lower-left panel). Higher inflation caused by the implementation of the
fiscal stimulus then lowers the real interest rate relative to the reference level, as shown in
8

We chose απ = 1.5 for Figure 4 because a large value of απ enables us to compare the model responses
more easily than would be the case with a low value of απ (e.g., απ = 1.01).

11

the lower-right panel of Figure 4. This low real interest rate, in turn, stimulates consumption
in the same period (see the upper-right panel in period t = h). Put differently, government
purchases increase relative consumption because of the crowding-in effect of government
purchases. In terms of (7), gh leads to c̃h−1 > 0 through a lower real interest rate r̃h−1 − π̃h <0. This crowding-in effect is well known in the literature (Woodford (2011) and Christiano,
Eichenbaum, and Rebelo (2011)).
The forward-looking property of the relative consumption Euler equation (7) creates thedesirability of implementation lags. The increased consumption is directly transmitted tothe increase in the previous period because c̃h−2 = c̃h−1 − (r̃h−2 − π̃h−1 ). The increase in
consumption can be repeated backwards to c̃h−3 and c̃h−4 (= c̃0 ). If the real interest rate wereconstant, we would have c̃h−1 = c̃h−2 = ... = c̃0 > 0. Households can enjoy more consumption(and output) from the time of the announcement of the fiscal stimulus rather than the timeof its implementation. Thus, increases in consumption c̃t for 0 ≤ t < h strengthen the overall
effect of fiscal stimulus on output. As the implementation lag lengthens, consumption has
more opportunities to increase before the implementation of the fiscal stimulus.9
We next turn to the case of h = 12 where the fiscal stimulus is implemented after aliquidity trap (h = 12). In contrast to the case of h = 4, the effect of the fiscal stimulus on
output is negative. As shown by the solid line with diamonds in the upper panels of Figure
4, yt and ct are now located below ytR and cRt , respectively, in the period before the fiscal
stimulus is implemented at t = h = 12. Thus, government purchases with an implementation
lag of three years lower output relative to the reference level. The negative effect is reflectedin ϕ12 = −3.28 in Figure 4.
When the fiscal stimulus is implemented after a liquidity trap, the implementation lag is
no longer desirable. Not surprisingly, the intuition in experiment I is applicable to this result.
The increase in gh raises the real interest rate relative to the reference level, which creates9

Numerically, the real interest rate for 0 < t < h − 1 is lower than the reference level (see the lower-rightpanel of Figure 4) given the high inflation caused by c̃t > 0 for 1 < t < h. The falls in the real interest
rate over 0 < t < h − 1 further stimulate consumption relative to the reference level. Thus, allowing forfluctuation in the real interest rate does not influence our interpretation.

12the crowding-out effect, as the ELB is no longer binding in period t = h = 12. Because
the forward-looking property is preserved in the relative consumption Euler equation (7),relative consumption begins decreasing relative to the reference level from the time of theannouncement (t = 0). Once again, as the implementation lag becomes longer, consumption
has more opportunities to decline before the implementation of the fiscal stimulus.10 As aresult, ϕh tends to decline with the implementation lag for t ≥ h.
Before closing this section, two remarks are in order. First, using a New Keynesianmodel with a fixed nominal interest rate, Farhi and Werning (2016) analytically prove that
consumption in the initial period (i.e., c̃0 in our notation) is more strongly crowded in bygovernment purchases the farther the distance in time of implementation. In our model,such a strong crowding-in effect does not occur. As shown by the solid lines with diamonds
in Figure 4, relative consumption in the initial period c̃0 is crowded out by government
purchases. The reason is that the nominal interest rate is endogenous and no longer fixed
when the economy has almost recovered.
Second, Ngo (2021) shows that the fiscal multiplier is not monotonic in the persistenceof government purchases in the New Keynesian model with ELB. We can interpret his results from the viewpoint of implementation lag. In his model, if increases in government
purchases become negligible after a liquidity trap, an increase in the persistence of government purchases increases the fiscal multiplier. By contrast, if increases in government
purchases remain substantial even after a liquidity trap, an increase in the persistence ofgovernment purchases decreases the fiscal multiplier. In our model, the former corresponds
to the case in which government purchases are implemented during a liquidity trap, and thelatter corresponds to the case in which they are implemented after a liquidity trap.
10

In simulations, the ELB is binding for 0 ≤ t ≤ 8 and is not binding for 9 ≤ t ≤ 11. In the former period,
inflation relative to the reference level is low due to the weak aggregate demand and thus the real interest
rate is high relative to the reference level. The higher real interest rate during the binding ELB further
decreases c̃t and lowers the overall effect of fiscal stimulus on output.

13

4Conclusion

Since Friedman (1953), lags in fiscal policies have been a concern of policymakers because
they may reduce the efficacy of fiscal policies. Even recent studies support the conventionalview that the fiscal authority should avoid implementation lag.
We showed that implementation lags could enhance the efficacy of fiscal stimulus on
output when the ELB is binding. The efficacy exhibits a hump shape against the length of
implementation lags, suggesting that the fiscal authority can improve efficacy by delaying
the implementation of its fiscal stimulus. The desirability of implementation lags criticallydepends on whether the ELB is binding.

References
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“Fiscal policy for the crisis.” CEPR Discussion Paper No. DP7130.
Christiano, Lawrence J, Martin Eichenbaum, and Sérgio Rebelo. 2011. “When is the government spending multiplier large?” Journal of Political Economy 119 (1):78–121.
Eggertsson, Gauti B and Michael Woodford. 2003. “Zero bound on interest rates and optimalmonetary policy.” Brookings Papers on Economic Activity 2003 (1):139–233.
Farhi, Emmanuel and Iván Werning. 2016. “Fiscal multipliers: Liquidity traps and currency
unions.” In Handbook of Macroeconomics, vol. 2. Elsevier, 2417–2492.
Friedman, Milton. 1953. “The effects of a full-employment policy on economic stability: A
formal analysis.” In Essays in Positive Economics. University of Chicago Press Chicago,
117–132.
Galı́, Jordi. 2015. Monetary Policy, Inflation, and the Business Cycle: an Introduction to
the New Keynesian Framework and Its Applications. Princeton University Press.
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———. 2020. “The effects of a money-financed fiscal stimulus.” Journal of Monetary Economics 115:1–19.
Galı́, Jordi, J David López-Salido, and Javier Vallés. 2007. “Understanding the effects ofgovernment spending on consumption.” Journal of the European Economic Association
5 (1):227–270.
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the zero-interest-rate bound.” Journal of Money, Credit, and Banking 37 (5):813–835.
Krugman, Paul R. 1998. “It’s baaack: Japan’s slump and the return of the liquidity trap.”
Brookings Papers on Economic Activity 1998 (2):137–205.
Ngo, Phuong V. 2021. “Fiscal multipliers at the zero lower bound: The role of government
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American Economic Journal: Macroeconomics 3 (1):1–35.

A

The model description

In this appendix, we describe the details of the model in Section 2.
PtHouseholds A representative household maximizes its lifetime utility given by ∞t=0 β U (Ct , Nt ; Zt ) ,hiwhere U (Ct , Nt ; Zt ) = log Ct − Nt1+ψ / (1 + ψ) Zt . Here, Ct , Nt , and Zt represent consumption, labor supply, and an exogenous preference shifter, respectively. The parameterψ > 0 denotes the inverse of the Frisch elasticity of labor supply and 0 < β < 1 is the
subjective discount factor of the household. The household faces a budget constraint:

Bt + Pt Ct = (1 + Rt−1 )Bt−1 + Wt Nt + Pt Dt − Pt Tt ,15(8)

where Bt denotes the household’s nominal holdings of one-period government bonds paying anominal interest rate Rt , Pt is 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 household
maximizes its lifetime utility subject to (8), yielding the following first-order conditions:

1 = β(1 + Rt )Ct Zt+1 Pt,
Ct+1 Zt Pt+1

Ct Ntψ =

Wt
.
Pt

(9)

(10)

Log-linearizing (9) around the steady state gives (1). In deriving (1), we define the preference
shock %t as %t = − ln(Zt+1 /Zt ).
Firms The representative firm produces the final good in a perfectly competitive market. ItR
ε/(ε−1)
1
combines a continuum of intermediate goods, using the technology Yt = 0 Yt (i)(ε−1)/ε di,where Yt (i) denotes output produced by the intermediate good producers i ∈ [0, 1] and ε (> 1)
is the elasticity of substitution among differentiated goods. The aggregate price index Pt
R1/(1−ε)1
is associated with intermediate good prices Pt (i) by Pt = 0 Pt (i)1−ε di. The production function for intermediate good producers is Yt (i) = Nt (i)1−α , where α ∈ (0, 1].Each
intermediate good producer can reset its price with probability 1 − θ in any given period.
Let Pt∗ be the optimal nominal price set by an intermediate good producer. The optimal
nominal price solves the maximization problem:

max
∗
Pt

∞
X



1
1−α
∗
θ Qt, t+k (1/Pt+k ) Pt Yt+k|t − Wt+k Yt+k|t ,k

k=0

subject to the demand function Yt+k|t = (Pt∗ /Pt+k )−ε Yt+k , where Yt+k|t denotes an intermediate good producer’s output in period t + k given that the producer last reset its price in16period t and Qt, t+k ≡ β k Ct /Ct+k is the discount factor. The first-order condition for Pt∗ is∞Xkθ Qt, t+k (1/Pt+k )Yt+k|t



Pt∗ −

k=0

α
ε
1−α
Wt+k|t Yt+k|t
ε−1


= 0.

(11)

Along with the equation for the price index, the log-linearization of the above equation yields
the NKPC (2).

Market clearing Equilibrium in the final goods market requires Yt = Ct + Gt . Using
the definitions of gt = (Gt − G)/Y , we obtain (4). The labor market clearing condition is
R1
Nt = 0 Nt (i) di for all t.

17

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2
0

5

10

15

h

Figure 1: The overall effect of fiscal stimulus on output in normal times
Note: An increase in government purchases is announced in period 0 and implemented in period h. The
increase in government purchases is normalized to one percent of the steady-state output. At the time ofannouncement, the economy is in the steady state. The unit of time is a quarter.

18

4

3.5

3

2.5

2

1.5

1
0

5

10

15

h

Figure 2: The overall effect of fiscal stimulus on output in a liquidity trap
Note: An increase in government purchases is announced in period 0 and implemented in period h. The
increase in government purchases is normalized to one percent of the steady-state output. At the time ofannouncement, the economy is caught in a liquidity trap. The unit of time is a quarter.

19

4

3

2

1

0

-1

-2

-3

-4

-5

-6
0

5

10

15

h

Figure 3: The overall effect of fiscal stimulus on output in a liquidity trap under different απNote: The solid line replicates ϕh under απ = 1.01 in Figure 2 for comparisons. The dashed, dotted, anddash-dotted lines represent ϕh under απ = 1.25, 1.5, and 1.75, respectively. See the note accompanying
Figure 2 for the remaining details.

20

0.02

0.02

0

0

-0.02

-0.02

-0.04

-0.04

-0.06

-0.06
h=4
h=12
no policy change

-0.08

-0.08

-0.1

-0.1
0

5

10

15

0

5

0

5

10

15

10

15

0.02
0.1

0.01
0

0.05

-0.01
-0.02

0
-0.03
-0.04
0

5

10

15

quarters

quarters

Figure 4: Responses of variables to changes in %t and gt
Note: Each panel of the figure compares the model responses to %t and gt for an implementation lag of
between one and three years (h = 4 and h = 12), along with the reference level of the corresponding
variables. The upper-left and the upper-right panels refer to output and consumption, respectively. The
lower-left and the lower-right panels present the nominal interest rate and the real interest rate,
respectively. The dashed line in each panel represents the equilibrium responses of a variable arising fromfluctuations in %t while keeping Gt at G for all t (denoted by “no policy change”). The solid lines in eachpanel refer to the model response to decreases in the preference shock and increases in government
purchases. The solid line with circles is the model responses under h = 4, and the solid line with diamonds
is the model responses under h = 12. The horizontal axis is the quarters after the preference shocks. The
vertical lines in each panel represent the timing of the implementation of the fiscal stimulus. In the figure,απ = 1.5 and ϕh = 3.30 at h = 4 and ϕh = −3.28 at h = 12.

21