A Model of Anticipated Consumption Tax Changes(Please click here for the latest version)

Masashi Hino∗
November, 2020

Abstract
This paper studies household spending responses to anticipated changes in the consumption tax. To do so, I construct a life-cycle heterogeneous-agent general equilibrium model withdurables. The model features a wedge in durable transactions that reflects the actual consumption tax system: households pay the tax when buying the durables but do not receive thetax when selling them. There are three main findings. First, the baseline model reproduces
an empirically consistent dynamic pattern of tax elasticity of the taxable spendings. Second,
I find that life-cycle is a key component to match the level of tax elasticity of durable spending. Third, the baseline model generates smaller stockpiling of durables based on realistic
motive than a model without the wedge. I then use the model for two counter-factual experiments.The first counter-factual experiment finds that the effect of a consumption tax cut is not
symmetric to the tax hike. The second counter-factual experiment which compares a one-time
tax hike and a multiple-times tax hike shows the multiple-times tax hike scheme generates
smaller welfare cost than one-time tax hike.

JEL code: D15, E21, E62, H31.
Keywords: Durables, Anticipated Consumption Tax Change, Tax Wedge, Stockpiling, Lifecycle, Tax Elasticity.

∗ The Ohio State University. hino.4@buckeyemail.osu.edu. I am grateful to Kyle Dempsey, Aubhik Khan, Pok-sang
Lam, Julia Thomas and workshop participants at the Ohio State University for their helpful feedback and suggestions.
I also thank seminar participants at Kobe University, Kyoto University, Hitotsubashi University, DSGE workshop, andvirtual SWET. I have particularly benefited from discussions with Masao Fukui, Munechika Katayama, Taisuke Nakata,
Shinichi Nishiyama, Shin-ichi Nishiyama, Ryota Ogaki, Satoshi Tanaka, Kazuhiro Teramoto. I appreciate Michael Irwin
and Sayoudh Roy for their generous help.

11. Introduction
Preannounced consumption tax hikes cause salient intertemporal substitution of household
expenditures. Households largely increase the expenditures right before the consumption tax
hikes and household expenditures plummet afterwards. From a policy perspective, the use of
the preannounced future consumption tax hikes has been proposed as a household expenditures
stimulus policy tool, labeled unconventional fiscal policy.1 As such, understanding the effectand the mechanism of consumption tax changes on household expenditures is of the significant
importance, particularly in a low interest rate environment when conventional monetary policy
loses the efficacy.
The intertemporal substitution due to the consumption tax changes becomes even stronger
in the presence of consumer durables. Many empirical studies show that the effects of the consumption tax changes are prominent in long-lasting goods or durable goods (e.g., Cashin andUnayama 2016; D’Acunto et al. 2016, 2018; Baker et al. 2019, forthcoming; Nguyen et al. 2020).
Therefore, it is a natural extension of a macroeconomic model to incorporate the durable goods
into the analysis of the consumption tax changes.
Motivated by the empirical studies, this paper studies the effects of the anticipated future consumption tax changes on household expenditures using a life-cycle heterogeneous-agent generalequilibrium model with consumer durables. A remarkable feature of my baseline model is a tax
wedge which reflects the actual tax system. That is, the tax wedge is introduced to take into consideration that households pay the consumption tax when buying the durables but do not receive
the tax when selling them. Since this is the case internationally, the tax wedge is a qualitatively
necessary friction to analyze the consumption tax in a model of the durables.
Furthermore, for the quantitative analysis with the model to be credible, the model must be
disciplined by key empirical regularities. The most important empirical moment for this analysis
is the tax elasticity of household expenditures, since it measures how much the households respond to the consumption tax changes. I find that the tax wedge plays quantitatively central roles
in reproducing the empirically consistent household expenditure responses for the following two
reasons.
First, the tax wedge helps account for the estimated dynamic pattern of the taxable spendings,
reported in Baker et al. (forthcoming). As Figure 1 shows, Baker et al. (forthcoming) find that theelasticity of taxable spendings spike up right before the tax hike, plummet in the period of thetax hike and gradually recover to the new stationary states.2 I find that the baseline model with
1 The unconventional fiscal policy is, for example, proposed by Feldstein (2002) and Hall (2011), and extensively
studied by Correia et al. (2013), Baker et al. (forthcoming) and Baker et al. (2019).
2 This pattern can be confirmed even in the aggregate consumptions. See Figure 18 in the appendix 11. More2

Figure 1: Dynamic Tax Elasticity of Taxable Spendings

Note: Reported in Baker et al. (forthcoming). Dashed vertical line indicates the timing of
the consumption tax hike. It can be confirmed that household taxable spendings sharply
rise one period before the tax hike, fall in the period of the tax hike, and gradually recoverto the new steady state level afterwards.

the tax wedge is able to reproduce this dynamic pattern of taxable spendings and the tax wedge
is the only investment friction that accounts for this pattern. The intuition of the slow recovery
is simple. Since the households stock up the durables, they do not buy as much durables forsome periods afterwards. In this way, the baseline model generates swift adjustment before
the tax hike and gradual recovery after the tax hike. Surprisingly, a frictionless model (i.e., amodel without the tax wedge) does not generate the slow convergence after the tax hike. Models
with other investment adjustment costs, such as quadratic adjustment costs or fixed cost, also
fail to reproduce the dynamic pattern. Thus, the tax wedge is the only investment friction that
reproduces the dynamic pattern of the empirical tax elasticity. The model-implied tax elasticity
of taxable spendings from my baseline model is 1.87 one period before the tax hike, −1.44 in the
period of the tax hike, −1.23 in the following period, falling into the confidence interval of Baker
et al. (forthcoming) without targeting them in calibration. The baseline model also generates thetax elasticity of durable spendings of 10.6, which Baker et al. (2019) estimate 8.1 − 12.8.3
generally, other stimulative policies to the durables also witness this slow recovery in the household spendings. For
instance, Mian and Sufi (2012) confirms that CARS program induces the fewer automobile purchases for ten months
after the expiration of the program.
3 8.1 is from Column (3) in Table 2 and 12.8 comes from Table 3. As they discuss timing issue of purchase andregistration, I sum up to 9.05 and 3.70 in Column (1), and obtain the value of 12.8.

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Second, the tax wedge helps predict true stockpiling behavior. Without the tax wedge, themodel allows the households to receive the consumption tax when they sell the durables. This
unrealistic receipt of the consumption tax generates the following speculation opportunity overthe dynamics. Suppose that an automobile is worth $10, 000 and the tax hike from 5% to 10%
will take place tomorrow. The household buys the automobile and pays $10, 500 today. Whenselling the automobile with the tax rate of 10% next day, the household in the model without
the tax wedge receives $11, 000. Thus, $11, 000 − $10, 500 = $500 minus the depreciation is amonetary profit of speculation by taking advantage of the tax hike in the model without the
tax wedge. Knowing this speculation opportunity, the households excessively stockpile for the
speculative motive but not for their own use. In this way, the model without the tax wedge
generates strong stockpiling behavior based on the unrealistic mechanism. This misspecification
problem influences the existing quantitative analysis. For instance, Cashin and Unayama (2016)
estimate the intertemporal elasticity of substitution (IES) using the Japanese consumption tax
hike. They construct a model of durables with the consumption tax but did not introduce the tax
wedge. To reconcile with observed moderate stockpiling behavior using their model, they reacha conclusion that IES is low. But, their estimated value of IES must be biased downward due tothe misspecification problem.
Next, incorporating life-cycle into the model is important to match the magnitude of thetax elasticities as the life-cycle motives largely dampen the elasticity. Recently, Koby and Wolf
(2020) and Winberry (2020) show that (S,s) investment models generate infinitely high interest
rate elasticity of investment. This high interest rate elasticity also emerges in a model of durables
with (S,s) rules as McKay and Wieland (2019) study. As shown in Correia et al. (2008), thechange in the interest rate has a similar implication on the household expenditures as the change
in the consumption tax. Hence, resolution of this puzzle is a central problem this paper must
address.4 I find that incorporating the life-cycle structure makes the interest rate elasticity of
the aggregate durable expenditure in the baseline model fit empirically reported values closely,
without any additional frictions. This is because there is a mass of households who do notrespond to changes in the interest rate (and consumption tax) in a life-cycle model. For instance,
the young purchase the durable goods regardless of interest rate (and consumption tax) as they
have very high marginal utility of durables. In contrast, the old do not buy durables whatever thecurrent interest rate (and consumption tax) is, since they will die soon. In this way, the life-cycle
motive dampens the model-generated elasticity. As discussed above, adding other adjustment
costs (i.e., quadratic adjustment cost) to control the interest rate elasticity is not suitable here,
4 Intuitively, standard Euler equation in a model with the consumption tax, u ( c ) = β (1 + r 0 ) 1+τ c u0 ( c0 ) shows the1+(τ c )0

connection between r and 1 + τ.

4since the adjustment costs influence the dynamic pattern of the spendings, which makes themodel behavior inconsistent with the empirical findings.
Heterogeneity matters for welfare implications of the consumption tax hike over the transition
since the tax-induced inflation decreases the real value of preexisting wealth. Thus, wealthy
households incur a higher burden of the tax hike than poor households. This mechanism works
not only in the liquid asset but also in the durables. Thus, the presence of the durable widens
the inequality of the welfare loss from the tax hike. These progressive effects do not exist when
focusing on the stationary environment only. While the studies on the stationary environment
may have contributed to establishing a common sense that consumption tax is regressive, thispaper finds that the consumption tax hike is progressive on the transition dynamics.
Finally, I consider two counter-factual experiments. First, I consider an anticipated tax cut. Recently, Germany decreased the consumption tax rate temporarily during the on-going pandemiccrisis since 2020 July. Since the model is highly non-linear due to the combination of the presenceof the durables and the tax wedge, the prediction of the tax decrease is asymmetric to the tax
hike. This finding suggests two implications. If the model is linearized, then the solution would
not capture this asymmetricity. Furthermore, when an econometrician runs a regression withthe tax change experiment, the theory suggests that the empirical analysis should be separately
implemented between the tax hike case and the tax cut case.
Second, I compare two tax hike schemes: a multi-times stepwise tax hike scheme and a onetime tax hike scheme. The multiple-times stepwise tax hikes scheme causes stockpiling before
the first tax hike and the drop in the durable expenditure after the final tax hike. In reality,
the economic slump is persistent and thus policy makers hope to avoid the drop in the durable
expenditures in the midst of the recession. My result suggests that the policy makers can postpone
the largest drop in the household expenditures by keeping raising the tax rate until the economic
slump ends. Moreover, I find that the multi-times tax hikes scheme is better than the one-time tax
hike scheme in terms of social welfare since the multi-times tax hikes scheme causes moderatechange in price, which induces smoother adjustment.
The baseline model of this study is a two-asset life-cycle model with discrete choices. This
class of the models is known to be computationally expensive to solve. I extend the latest endogenous grid method (hereafter EGM), Nested EGM, which is developed by Druedahl (2020)and solve the entire analysis, including the transition dynamics of the general equilibrium withmy laptop. A novel feature of Nested EGM is by nesting the timing of the household’s problem,
we can divide the multi-dimensional optimization over the asset and durable into two steps ofunidimensional problems. The algorithm applies the off-the-shelf EGM to this uni-dimensionalpart and solves the dynamic optimization problem fast.

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1.1. Relation to Literature
To my best knowledge, this is the first paper to study the dynamic responses to the anticipatedconsumption tax changes with a general equilibrium model of consumer durables with the tax
wedge. This paper aims to contribute to the three strands of literature mainly: unconventional
fiscal policy, consumption tax reform, and the lumpy durable literature.
This paper contributes to the theoretical and quantitative understanding of the unconventional
fiscal policy. Correia et al. (2013) theoretically prove that the unconventional fiscal policy resolves
the limitation of conventional monetary policy in the zero lower bound using a representative
agent New Keynesian model and show that it also achieves the efficient allocation. My model isan extension of Correia et al. (2013) with regard to the durables, the tax wedge, life-cycle, andheterogeneity. Once the durables are introduced into a model, responses to the preannounced
consumption tax changes are significantly changed.
Baker et al. (forthcoming) also studies the unconventional fiscal policy both empirically andtheoretically. They develop a consumer inventory model with shopping costs to account for theobserved responses to the anticipated sales tax changes in the U.S. They focus on the U.S. sales
tax system, i.e., local sales tax changes while I study changes in the nation-wide consumption tax
rate. Therefore, I build an extended Bewley-Huggett-Aiyagari type model which is commonlyused in the macroeconomic literature. Also, my model is able to reproduce the reported slow
convergence to a new steady-state after the tax hike.
From the empirical side of the studies on the unconventional fiscaly policy, Baker et al. (2019)
empirically estimate the tax elasticity of automobile purchases in the U.S. D’Acunto et al. (2016,2018) also study the effects of anticipated consumption tax changes and show that the readinessof the durable purchase increased before the consumption tax hike in Germany and Poland. To
sum up, these empirical papers on consumption tax changes emphasized the roles of durable
goods. This paper builds a bridge between these empirical findings and macroeconomic theory
by introducing consumer durable goods into the theoretical model.
The second strand of the literature that I aim to contribute to is on consumption tax reforms.5
Among them, Altig et al. (2001) and Nishiyama and Smetters (2005) studies a transition over
the replacement of a realistic progressive income tax system with a flat consumption tax system,
using a life-cycle model. They study the intergenerational and heterogeneous welfare costs of
the introduction of the consumption tax. My paper revisits their analysis with a model of the
durables.
My baseline model is the most similar to Parodi (2018, 2019) from the structural modeling
5 Krusell et al. (1997), Ventura (1999), Erosa and Ventura (2002), Correia (2010), Kitao (2011), Nakajima and Takahashi(2017a), Laczó and Rossi (2019) and Motta and Rossi (2019).

6

perspective. Parodi (2018) constructs a rich life-cycle heterogeneous agent partial equilibrium
model with durable goods and the tax wedge to study the value-added tax system reform in Italy
that changes from the current consumption tax system with reduced tax rates to a uniform tax
rate. Parodi (2019) also uses a similar environment to study the optimal consumption tax rates for
the Italian economy in the presence of reduced tax rates in a stationary environment. My paper
differs from Parodi (2018, 2019) in that my focus is on the dynamic responses of the households
to the anticipated tax change. I will argue that effects from the tax-induced inflation which causes
heterogeneous welfare implication over the dynamics only, and therefore this paper complements
her works by adding this dynamic perspective.
The third related literature is on the lumpy durable adjustments. There are a considerable
amount of papers that contributed to this literature.6 Among them, Berger and Vavra (2015) isa recent and particularly influential paper in the literature. They show that the response of the
durable expenditure is more sluggish in recessions. Berger and Vavra (2015) also shows that
this responsiveness is robust to the assumption of partial equilibrium or general equilibrium.
My study applies the lumpiness to the responses to the anticipated consumption tax changes.
Also, Fernández-Villaverde and Krueger (2007, 2011) and Yang (2009) study both empirically andtheoretically the life-cycle profile of the durable expenditure. I find that life-cycle structure candampen the interest rate elasticity of aggregate durable expenditure in a model and help matchthe observed elasticities.
Finally, my paper relates to the conventional monetary policy because a permanent consumption tax hike can be viewed as lowering the real interest rate in a single period from the household
perspective. This equivalence between the fiscal and monetary policy is formally established inCorreia et al. (2008). Thus, my work potentially contribute to the understanding intertemporal
shifting due to the conventional monetary policy in a rich heterogeneous-agent model.
Many Heterogeneous Agent New Keynesian (HANK) models recently feature roles of durable,
housing, or illiquid asset. For instance, McKay and Wieland (2019) and Zorzi (2020) study transmission mechanisms of monetary policy with a model of lumpy durable adjustments.
Doepke and Schneider (2006), Sterk and Tenreyro (2018), Auclert (2019) and Doepke et al.
(2018) argue that inflation has a redistribution effect between the asset rich and the asset poor.
Nishiyama and Smetters (2005) analyze the heterogeneous welfare cost of the tax-induced inflation on the consumption tax context. I follow and revisit those insights with a model of the
durables.
6 For instance, Lam (1989, 1991), Grossman and Laroque (1990), Caballero (1990, 1993), Eberly (1994), Leahy andZeira (2005), Luengo-Prado (2006), Favilukis et al. (2017), Guerrieri and Lorenzoni (2017), Gavazza and Lanteri (2018).

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1.2. Layout
In what follows, I explain the baseline model in the next section. Section 3 overviews the computational solution. Section 4 discusses parametrization. Section 5 illustrates the basic result in astationary equilibrium. From section 6, I discuss the results of the consumption tax change. The
section 6 discusses the long-run effects of the consumption tax change. Section 7 is a main sectionof this paper that focuses on the dynamic responses to the anticipated consumption changes, assuming the partial equilibrium. Section 8 solves the general equilibrium dynamics and confirmsthat equilibrium assumption does not affect the quantitative results significantly.

2. Baseline Model
2.1. Households
Time is discrete. Let t = 0, 1, . . . be time and j be age. Households live for finite periods J.
They start working as soon as they are born and continue working until period J R − 1 < J. Afterthe retirement age J R , they receive social security ss. The households have no bequest motives
and are born with neither asset a nor durable stocks d. There exists bequest in the form of the
durable despite the deterministic lifetime and no bequest motive because some households own
the durables in the last period of the life. The sum of the durables that deceased households own
equals the total bequest, that is D J = B. This total bequest B is equally distributed to all the alivehouseholds at the end of every period. Let b denote the bequest each household receives. For
notational simplicity, I omit time and age subscript if not necessary.
The households supply labor inelastically. In beginning of every period, the working households draw an idiosyncratic labor productivity from a Markov process π (e0 |e) and receive laborR1earnings wκ j e where {κ j } jJ=−
1 is a deterministic age-dependent component of the labor earnings.

This κ j is introduced to account for the hump-shaped labor earnings. An initial productivity e0
at the age of 0 is assumed to be drawn from the ergodic distribution of the Markov process. For
notational simplicity, let y j (e) denote the labor earnings and social security. That is,y j (e) =wκ j e

if j < J Rss

if j ≥ J R .

Also, the households receive an interest income (1 + rt ) a−1 , and spend the sum of labor andinterest income and the bequest on either non-durables c ≥ 0 or durable purchases x d , or save in8liquid asset a. The durables depreciate at the rate of δd . When purchasing the non-durables andthe durables, the households face the same consumption tax rate τ c . The flat consumption tax
rate is assumed because the consumption tax rate is uniform across goods and across regions inJapan during sample periods I study.7 The households cannot borrow, i.e. a ≥ 0 is imposed.
Furthermore, the households are subject to a friction for the durable adjustments. The households face the tax wedge which reflects the realistic consumption tax system. The consumption
tax is charged only when the seller is business. That is, when the households sell their own
durable goods to the market, this transaction is not subject to the consumption tax. The tax
wedge takes the following form:

T (d, d−1 ) = (1 + τ c ) x d

if x d ≥ 0

qx dif x d < 0,where x d = d − (1 − δd )d−1 ,where q is an indicator whether the tax wedge in a model is turned on. I set q = 1 in a benchmark
case. I refer to frictionless or a symmetric tax case when q = (1 + τ c ) in which the tax wedge isturned off. In this manner, the households in the baseline model face different consumer prices
between buying and selling the durables.
To sum up, the households face the budget constraint below,(1 + τ c )c + a + T (d, d−1 ) = (1 + r ) a−1 + y j (e) + b.

(1)

and non-negativity constraints ( a, c, d) ≥ 0. The household state space is ( j, a−1 , d−1 , e). Letµ j ( a−1 , d−1 , e) be the distribution of the households at the age of j. Note that without the tax
wedge (i.e. setting q = 1 + τ c ), the households receive the consumption tax τ c x d when selling the
durables (x d < 0).8 I will discuss the importance of this misspecification later in detail in section
7.2.
The households’ dynamic optimization problem can be written as
v j ( a−1 , d−1 , e) = max u(c, d) + β ∑ v j+1 ( a, d, e0 )π (e0 |e)( a,c,d)≥0

(2)

e0

s.t. (1)7 As of 2019 October, Government of Japan introduced a reduced tax system.
8 That is, suppose q = 1 + τ c . Then, when x d < 0, the budget constraint implies that (1 + τ c ) c + a = (1 + r ) a−1 +y j −(1 + τ c ) x d .
|
{z
}

>0

9

where v J +1 ( a, d, e0 ) = 0 for all ( a, d, e0 ).
Alternatively, it seems to be more intuitive to rewrite the optimization problem explicitlywith discrete choices in the following way. The households choose either upward adjusting the
durables (i.e., buy the durables), inaction, or downward adjusting the durables (i.e., sell the
durables),upv j ( a−1 , d−1 , e) = max{v j ( a−1 , d−1 , e), vinact( a−1 , d−1 , e), vdown( a−1 , d−1 , e)}.
jj
Each choice-conditional problem is solved separately in the following manner.
When upward adjusting the durables, the households solveupv j ( a−1 , d−1 , e) = max u(c, d) + βE[v j+1 ( a, d, e0 )|e]
( a,c,d)≥0s.t. (1 + τ c )[c + x d ] + a= ( 1 + r ) a −1 + y j ( e ) + b
x d > 0.
Notice that x d > 0 is explicitly imposed.
When choosing the inaction, the households solve
vinact( a−1 , d−1 , e) = max u(c, (1 − δd )d−1 ) + βE[v j+1 ( a, (1 − δd )d−1 , e0 )|e]j
( a,c)≥0s.t. (1 + τ c )c + a= (1 + r ) a−1 + y j (e) + b.
The households who choose inaction simply carry the depreciated durable (1 − δ)d−1 to the next
period. Thus, this optimization is essentially one-dimensional as in standard incomplete market
models. This becomes key in computation. This inaction problems allows us to apply the EGM
technique for this choice-specific Bellman equation with special treatment for the kinky valuefunctions due to the discrete choices. The Nested EGM heavily takes advantage of this structure
and I will discuss the detail in appendix 10.1.

10Finally when downward adjusting the durables, the households solve
vdown( a−1 , d−1 , e) = max u(c, d) + βE[v j+1 ( a, d, e0 )|e]j
( a,c,d)≥0s.t. (1 + τ c )c + qx d + a= ( 1 + r ) a −1 + y j ( e ) + b
x d < 0.
Again, notice that x d < 0 is imposed.

2.2. Goverment
The government collects the consumption tax as a revenue and use it to finance total pension
R
SS = ∑ j≥ J R ssdµ j and government expenditure G.
d
G + SS = τ c (C + X+)

where C = ∑ j

Rd =cdµ j is an aggregate non-durable consumption and X+
∑j

(3)

R

x d >0x d dµ j is a sumof positive aggregate expenditures on the durables because consumption tax is paid only when
the durable expenditure is positive.
In my model, additional tax revenue from the tax hike is spent on the government expenditure
G. The government expenditure G is not used anywhere and wasteful. An alternative modeling of the use of the tax revenue is to transfer it to the households or mix of the government
expenditure and the transfer. I choose the government expenditure as a benchmark for threereasons. First, I want to isolate the effect of anticipated consumption tax hike from effects of thetransfer. It is well known that Hand-to-Mouth (including wealthy Hand-to-Mouth) households
strongly react to the lump-sum transfer in two asset models as Kaplan and Violante (2014) pointout. Therefore, the lump-sum transfer potentially boosts the stockpiling behavior in response to
anticipated consumption tax change in a non-trivial way. This makes interpretation of the household responses complicated and we cannot identify whether the heterogeneous response is dueto the tax hike or the transfer. Also, if the government transfers the revenue to the households,
the model would generate stronger stockpiling responses. In other words, my result can be seen
as lower bound in this respect. Second, this modeling is intended to reflect the fact that the additional tax revenue is mostly for the sustainability of the Japanese Government Debt and used
for the repayment of the debt. While the accumulated government debt is unmodeled, the government expenditure can be interpreted as expenditures on such a unmodeled factor.9 Third, this
9 Other unmodeled objects include frequent natural disasters (e.g.

11typhoons, earthquakes) and spending on themodeling is computationally easy. Given the OLG structure with two assets and discrete choices,burden of the computation, especially for the dynamics, is non-negligible.

2.3. Firm
The firm side is standard. I assume a representative competitive firm which owns CobbDouglas technology for the final output. This firm follows
r + δ = αK α−1 N 1−α(4)w = (1 − α ) K α N − α ,where K and N are aggregate capital and labor respectively. This output is used for the nondurable and the durable consumption, the capital investment and the government expenditure.

2.4. Dealers
It is worth noting that the households always pay the consumption tax when they buy the
durable goods. This implicitly means that households are not allowed to trade durables between
households in a model. In reality, the direct and private trades between households are a way to
avoid the consumption tax. If such an opportunity exists in a model, the households choose it
and avoid the consumption tax.
I assume that the competitive dealers manage all durable transactions. That is, the dealerspurchase the durables from both the firm and the households, and sell the durable goods to thehouseholds. When the households sell the durables to the dealer, the households receive q perunit of the durables. Similarly, when the households buy the durables from the dealers, they pay
1 per unit to the dealers and τ c per unit to the government.10 The used and new durables are
assumed to be perfect substitute. Therefore, the households do not distinguish the new and theused durables.

2.5. A Recursive Competitive Equilibrium
A recursive competitive equilibrium of this model consists of a pair of functions,{v j , gdj , g aj } j , {µ j }, C, D, K, N, w, r

on-going pandemic crisis.
10 Since I assume perfect pass-through of the consumption tax, the households incur full incidents of the consumption
tax.

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that solves the households and the firm optimization problems and clear markets for asset, laborand output. That is, the equilibrium satisfies(i) v j solves age j household problem (2) given prices (w, r ). gdj and g aj are policy functions forthe durables and the assets respectively.
(ii) The firm solves the static optimization and follows (4).
(iii) The government observes its budget constraint (3).
(iv) The distribution {µ j } followsµ j+1 (s, e0 ) =Z{( a−1 ,d−1 ,e)|

gdj ( a−1 ,d−1 ,e),g aj ( a−1 ,d−1 ,e)∈s}π (e0 |e)dµ j ( a−1 , d−1 , e).

(v) The labor market clears:
J R −1 Z

N= ∑eκ j dµ j .

j =1(vi) The asset market clears:
J

K=∑

Z

a−1 dµ j .

j =1(vii) The good market clears:
C + D + G + K 0 = K α N 1 − α + ( 1 − δ ) K + ( 1 − δ d ) D− 1 ,where D−1 = ∑ jJ=1R

d−1 dµ j is an aggregate durable stock.

3. Computation
Solving this class of the models is computationally demanding in the presence of the discrete
choices and a multidimensional optimization. The discrete choices cause non-concave regions
on the both unconditional and choice-specific value functions and therefore derivative based
optimization methods become less accurate or unstable. Researchers may need to worry the local
optima. Moreover, the multidimensionality in the optimization often suffers the curse of thedimensionality. When solving perfect foresight dynamics with a simulation period of T, we must13

solve J + T − 1 generations’ dynamic optimization problems. Hence, efficient solution method
for the dynamic programming is necessary.
Druedahl (2020) develops a novel algorithm to solve this class of models with multidimensional optimization problem with the discrete choices. I applied his Nested EGM to the modelwith the tax wedge and show that this algorithm gives a well-behaved solution to the richer model
than the benchmark model he used for exposition. While details of the algorithm are explainedin the appendix 10.1, this section briefly overviews ideas of Nested EGM.
The critical insight of the Nest EGM is nesting a model to reduce a dimension of choiceconditional optimization problems, introducing the additional timing structure within a period.
That is, the Nest EGM views both upward and downward adjusting problem as sequential. The
households are assumed to, first, decide how much durables they buy, and then decide how muchnon-durables. Importantly, this nesting timing assumption allows us to use the consumptionfunction derived by the inaction problem for the both adjusting problems to reduce a dimensionof the optimization problems. Thanks to this dimension reduction technique, the resulting optimization problems for both adjusting become one dimensional. These maximization problems
are solved with standard VFI.11
Since the dimension of the optimization for the inaction problem is essentially one in computation, we can use EGM with upper envelope algorithm, developed by Druedahl and Jørgensen(2017) and Iskhakov et al. (2017). We extend these techniques for the model with the tax wedge
or the partial irreversibility constraint.12
I use this algorithm for solving both the stationary equilibrium and the perfect foresightequilibrium path.

4. Parameterization
This section discusses the parametrization of the models. The model frequency is annual.
consumption tax rate τ c is set 0.05 and 0.08 in the baseline which were actual consumption tax
rates in Japan until 2014 April and since then until 2019 October.
I choose Japan as an application of my analysis for two reasons. First, there was no reduced
tax system in Japanese consumption tax system with few exceptions until 2019 October.13 This
11 Due to the discrete choice and a kinky value function, this optimization problem may have many local maxima.
I use Rowan’s Subplex algorithm (a local optimization method) provided by NLopt toolbox on very finely-spaced
subintervals and find a local maximum in the subintervals. And then I take the maximum from these local maxima inpractice.
12 Also, I solved a model with fixed-cost and quadratic adjustment cost with this solution method.
13 The exceptional transactions are medical expenses, tuition fees, funeral fees, and rent.

14

consumption tax system allows to keep the analysis simple and transparent. Second and more
importantly, the Japanese consumption tax hike from 5% to 8% was full pass-through to consumer
prices because the government prohibited the tax change related price changes.14 Therefore, I
focus on the effect of the consumption tax on the households.
The Cobb-Douglas utility function is commonly assumed in the durable literature, following
Ogaki and Reinhart (1998). However, the Cobb-Douglas utility specification in a simple lifecycle model with the consumer durables does not reproduce empirically observed hump-shapein durable expenditure as Fernández-Villaverde and Krueger (2011) study. Thus, I adopt nonhomothetic utility function below as with Parodi (2019),c θ ( d + ed )1− θu(c, d) =1−σ

 1− σwhere ed > 0 and ed is not close to 0. This Stone-Geary functional form implies that the durable
stocks are luxury, consistent to the finding in Pakoš (2011). Typically, since the households in alife-cycle model get rich in middle-age in terms of both asset and earnings, the middle-age households increase the durable purchase. In this manner, this utility functional form helps reproduce
the hump-shaped durable expenditure. I pin down θ and ed to simultaneously match the ratio
between initial level and peak of average durable expenditure, and aggregate non-durable anddurable share. Then, I set σ = 2.0 and β = 0.977 which are standard values in the literature.15
Next, there is no official data for the durable depreciation in Japan. Thus, I take a depreciation
rate of each durable goods (e.g. furniture, appliances, automobiles) from Fraumeni (1997), andcompute the weight average of the depreciation rates using the share of the expenditure from
National Survey of Family Income and Expenditure.
The tax wedge parameter q is set to be 1.0 in a benchmark case. But, to examine the role ofthe tax wedge, I compare the baseline model with the model without the tax wedge in which I set
q = (1 + τ c ). Also, q < 1.0 has been interpreted as a resale adjustment cost in the investment anddurable literature and has been extensively studied.16 Thus, I discuss the alternative value casein the appendix 13.2. The choice of q affects the wealth formation in the stationary equilibrium
because the durables play a role as an insurance if q is high. Hence, the higher the q is, the morehouseholds save in durables in the stationary equilibrium. But, I confirm that the stockpilingbehavior over the transition is almost irrelevant to choice of q.
As for the calibration of idiosyncratic shocks, I assume the simplest AR(1) labor productivity
14 See the following link for the datail. https://www.mof.go.jp/tax_policy/summary/consumption/250910tenka.

htm
15 High discount factor β is common in Japanese economy literature, for example, Hayashi and Prescott (2002) andYamada (2012).
16 For the literature of the lumpy durables with the partial irreversibility constraint, see Lam (1989) for instance.

15

Table 1: Parameters in the Baseline Modelparameters
β
σ
δ
δd
ed
θ
ss
q
ρ
σe

values
0.977
2.0
0.1
0.15
1.47
0.77
0.64
1.0
0.9
0.2072

description
discount factor
risk aversion
annual depreciation rate of the capitalannual depreciation rate of the durables
Stone-Geary preference parameter
utility share of the non-durables
social security
resale adjustment friction
persistence of the idiosyncratic shocks
std of the idiosyncratic shocks

process.
ln e0 = ρ ln e + e0 .
I follow a calibration strategy of Nakajima and Takahashi (2017b) for the parameters of the idiosyncratic shocks. There is no suitable panel data for the estimation of the earnings in Japan.
Thus, they set persistence ρ of the shock to be standard value of 0.9 a priori.17 Then, the agedependent components {κ j } j is set so that labor earnings in the model match observed humpshape in average labor earning reported in Yamada (2012). I make it annual by interpolatingwith the cubic spline. Lise et al. (2014) combines four major data in Japan to estimate volatilityof shocks: Basic Survey on Wage Structure (BSWS), the Family Income and Expenditure Survey(FIES), the National Survey of Family Income and Expenditure (NSFIE), and the Japanese Panel
Surveyof Consumers (JPSC). To be consistent with Lise et al. (2014), the earnings dispersion conditional on {κ j } is generated by idiosyncratic shocks and then I set σe = 0.2072. Finally I use
the Rouwenhorst (1995) method to discretize this AR(1) process with 13 grid points. The social
security is set at ss = 0.64 following the report of the OECD (2017), which includes voluntarycomponents.

5. Stationary Equilibrium
Figure 2 is a policy function of the durable expenditure x d = gdj ( a−1 , d−1 , e) − (1 − δd )d−1 .
Since the tax wedge generates a non-linearity in the budget constraint, the durable expenditure
17 For example, Floden and Lindé (2001) finds 0.914 for the U.S. Alonso-Ortiz and Rogerson (2010) report 0.94 for
the US.

16

Figure 2: Durable Expenditure Function x d = gdj ( a−1 , d−1 , e) − (1 − δd )d−1 .

Note: The policy function of the durable expenditure shown is that of age of 55 (j = 30)
with the lowest idiosyncratic shock realization. The durable expenditure function exhibitshigh non-linearity due to the tax wedge.

function exhibits high non-linearity. The durable expenditure function consists of three regions:
upward adjustment, inaction, and downward adjustment. Movement across different regions
means the extensive margin.
Next, I assume the population of each generation is 10, 000 in the model economy so that
total population is large enough (J × 10, 000 = 550, 000) and run the stochastic simulation. Figure
3 shows the average of non-durable consumptions, durable stocks, assets, and positive durable
expenditure profiles over the lifetime. The asset profile is much taller than other two profiles andthus scaled down while the durable expenditure profile is also scaled up. Due to the extensive
margin the individual profiles show some spikes over the life-cycle but the spikes in individuallevel are canceled out in aggregation. Therefore, the Figure 3 looks smooth despite the presenceof the discrete choice.
The Figure 3 shows that the average of the households increase the non-durable c and durable
consumptions d in youth until late 30s or early 40s. The growth of the durable consumptions
d is more persistent than non-durables c because of the Stone-Geary utility function. Since themiddle-age households become richer in both earnings and assets, the middle-age households
buy more luxurious durables than the young. Due to this effect, the baseline model replicates the
hump-shaped positive durable expenditure x d profile. After the age of 40, the average of bothconsumptions continue declining steadily toward the near end of life. At the end of the life, the17Figure 3: Average Life-cycle Profiles: the Non-Durables, the Durables Stock and Asset

Note: Average life-cycle is obtained from stochastic simulation. Each generation consistsof 10, 000 households. The durable expenditure profile exhibits hump-shape thanks to theStone-Geary utility function.

dying households substitute the durables with the non-durables and increase the non-durable
consumption but decrease durables expenditure. The average households start accumulating theliquid asset a around the age of 35. In their middle-age, the liquid assets a sharply rise and hit apeak before the retirement age. After the retirement the households decumulate the assets sincesocial security is less than the average of the labor earnings.
These life-cycle profiles turn out to be important for short-run responses to the anticipatedconsumption tax hike. I will show that while the young do not exhibit the strong stockpiling of
the durables in response to the anticipated hike, all the other generation hoard the durable well.
I will discuss this again later in the subsection 7.2.4.
Also, it is worth reporting the wealth distribution in my baseline model and from the observation. The wealth distribution I refer to is reported by Kitao and Yamada (2019). They use the
National Survey of Family Income and Expenditure (NSFIE). The NSFIE is a cross-sectional date
which has a very large sample size with 55, 000 to 60, 000 households in each survey year without
top-coding. The survey is collected every five year since 1959. This paper compares with data
in 2014 because it is the latest one in Kitao and Yamada (2019). In particular, I compare wealthshare from their report with the asset share in my model economy for three reasons. First, theirdefinition of the household wealth is the sum of financial assets and thus does not include either

18

Table 2: Wealth Share owned by each quintile

Data (2014)
Baseline

1st
0.3%
0.1%

2nd
3.7%
2.4%

3rd
9.8%
12.6%

4th
21.3%
27.3%

5th
64.9%
57.6%

top10%
45%
35.8%

top1%
10.2%
5.7%

Note: Data (2014) is the figure reported in Kitao and Yamada (2019). The 1st quintile, forexample, means the bottom of the distribution.

real assets or durables. Second, they also exclude debt associated with purchases of real estate.
In my model economy, the borrowing is strictly prohibited, and therefore the net assets coincidewith wealth. Third, they report the share of the wealth but not that of net wealth.
Table 2 is the wealth share both in the model and the NSFIE. Overall, the distribution generated from the baseline model matches the observed wealth distribution well. It is noteworthy thatJapanese wealth share is much less concentrated than that of the U.S. Since common difficulty is
to reproduce the fat right tail of the wealth distribution, our model also misses right tail of the
distribution.18 Also note that my baseline model does not incorporate adjustment frictions typically used to match the wealth distribution such as heterogeneous discount factor or idiosyncratic
shocks to the return. Overall, the baseline model matches the wealth share.

6. Long-Run Effects of the Consumption Tax ChangeThis section studies influences of the consumption tax hike on life-cycle consumption profiles
and distribution in the long-run. Specifically, I will compare two stationary equilibria under thetwo consumption tax rate, 5% and 8%. 5% consumption tax rate was the Japanese consumption
tax rate until 2014 March, and 8% was between 2014 April and 2019 September.

6.1. Intergenerational Effects
In figure 4, solid lines are the profiles with the consumption tax rate of 5% and dashed lines are
the profiles with τ c = 8%. The figure demonstrates both the non-durable and durable consumptions profile are shifted downward almost in parallel. These observations imply that burdens of
the consumption tax on the consumptions are borne by all the generations fairly equally in thelong-run. Also, as known in the literature, consumption tax change in the long-run do not affectasset accumulation in my baseline model as well. This is because the intertemporal relative price
18 For the survey of the wealth distribution literature, for example, see De Nardi and Fella (2017) and Benhabib andBisin (2018).

19

Figure 4: Life-cycle Consumptions Profile in response to Consumption Tax Hike.

Note: Solid lines are when τ c = 5% and dotted lines are when τ c = 8%. The tax hike
influences the composition of consumptions disproportionally since the durable is luxury.

remains constant in the stationary equilibrium.

6.2. Cross-sectional Effects
Next, I am interested in the cross-sectional effects in this model. In the literature, it has been
controversial whether or not fundamental tax reform which replaces current complex income tax
with a flat consumption tax induces exacerbate inequality.19 Regarding the tax reform, there are
two policy changes: (i) an increase in consumption tax rate and (ii) elimination of the income tax.
This section focuses on the former effect with a model of the durables.
Figure 5 shows distributional changes of consumptions against the consumption tax hike inthe long-run. A left panel shows the changes in the non-durable distribution and a right panel
shows that in the durable distribution. Both distributions see the larger impacts in the right thanleft, and thus result in being more right-skewed. This is because the poor and have stronger
incentive to maintain the level of consumptions than the rich. The resulting right-skewness has
a potential policy implication that the consumption tax affects the rich more than the poor in
the long-run. In fact, standard deviation of non-durables and durables decrease by 2.81% and2.71% respectively, implying that the increase in the consumption tax decreases the consumption
19 For instance, Ventura (1999) and Altig et al. (2001) argue the tax reform causes further inequality in earning andwealth and the poor lose with the flat tax, whereas Correia (2010) argue opposite.

20

Figure 5: Changes in the Stationary Distributions

Note: Left Non-Durable distribution, Right Durble distribution.

inequality. Regarding the change in asset distribution, there was no change and thus I do notdisplay it here.

7. Short-Run Impacts: Partial Equilibrium
The focus of this paper is on the study of stockpiling behavior in response to the anticipatedconsumption tax change. The analysis of the stockpiling in response to the anticipated consumption tax changes is implemented by solving the perfect foresight path. First, I test whether thebaseline model can replicate dynamic pattern of empirical tax elasticities.
I start the analysis with the assumption of the partial equilibrium for the perfect foresightpath in this section since the partial equilibrium dynamics is a simpler than general equilibrium
and thus is interpretable more clearly. The economy faces constant prices (r, w) over the sample
periods and this price path (r, w) are those at the initial stationary equilibrium. I show the resultsfrom the general equilibrium in the next section.

7.1. Tax Elasticities
7.1.1. Taxable Spendings
D + C) from the baseline model andFigure 6 compares the elasticity of taxable spendings (X+the empirical result reported in Baker et al. (forthcoming). I assume that tax hike is announced
before the t = 1 and implemented t = 4 in the model.20 Since their data frequency is monthly and
20 As will be clear in the next subsection, the announcement of the tax hike has a negative income effect. But we
do not see the effects in the Baker et al. (forthcoming) result because local governments had different lags between
announcement and implementation of the tax change in their sample. In this way, empirical result mutes the negative21

D + C)
Figure 6: Tax Elasticity of Taxable Spendings (X+Note: BJK refers to the point estimate in Figure 1 also from Baker et al. (forthcoming).
Numbers in the figure indicate the model-generated tax elasticity in every period. The
baseline model generates the empirically plausible dynamic pattern of the taxable spendings.

my model frequency is annual, I do back-of-envelope calculation for the frequency adjustment.21
Figure 6 shows that the baseline model replicates both dynamic pattern of tax elasticity and levelsof these values.
First, the baseline model replicates the dynamic pattern of the empirical elasticity. Key dynamic patters of the empirical tax elasticity are two fold. First, the empirical tax elasticity spikes
up only in period 3 but not in period 1 and 2. Second, the empirical elasticity plummets in period 4 and slowly recover to the new stationary equilibrium level. The baseline model exactly
replicates dynamic pattern.
Second, the baseline model is roughly consistent to the reported point estimate and all the values of the model-implied elasticity falls within empirical 95% confidence interval. It is notewothythat these moments in the model are not targeted in calibration. The model sees some discrepancies in the elasticity. A potential reason for this discrepancy is compositions in sample. Since
income effects. By assuming the announcement before t = 1, I also mute the negative income effect in the simulation
period shown.
21 First, I increase the tax rate by 12% in the annual model that corresponds to 1% increase in a monthly basis. I
compute the elasticity eX D +C |12% of 12% tax hike and then obtain eX D +C |1% = eX D +C |12% /12.
+

+

22

+

D ) from the baseline model.
Figure 7: Tax Elasticity of Durable Expenditure (X+Note: Baker et al. (2019) report that the tax elasticity of automobile purchases one period before the tax hike is estimated between 8.1 − 12.8. The baseline model predicts theelasticity of 10.6 one period before the tax hike.

Baker et al. (forthcoming) uses Nielsen which mainly collects data on groceries and sundries, theirsample set do not contain many of typical consumer durables such as autos, appliances, furnitureetc. Thus, the share of the durables in their sample must be smaller than aggregate level share,
which dampens the elasticity. Since my model is calibrated to the macroeconomic level data, it isnatural that my model predicts higher elasticity than the result in Baker et al. (forthcoming).
Overall, my model replicates the dynamic pattern of the empirical tax elasticity. This is thestrong evidence that the baseline model is correctly specified for the consumption tax change.
7.1.2. Durable Expenditure
Next, I compare the tax elasticity of the aggregate durable expenditure with the result from
Baker et al. (2019). They estimate the tax elasticity of the auto sales and obtain the value of
8.1 − 12.8 one month before the tax hike.22
Figure 7 shows the elasticity of the aggregate durable expenditure from the baseline model.
I did same back-of-envelope calculation as the previous subsection 7.1.1. I find the elasticity
one period before the tax hike from the baseline model is 10.6, which again is consistent to theempirical finding in Baker et al. (2019).
22 8.1 is from Column (3) in Table 2 and 12.8 comes from Table 3.

As they discuss timing issue of purchase andregistration, I sum up to 9.05 and 3.70 in Column (1), and obtain the value of 12.8.

23

Figure 8: Consumption Tax Hike 5% → 8%

Note: The economy start at stationary equilibrium at t = 1. In beginning of t = 2, the government announces that it will increase the tax in t = 6. Solid lines depict baseline resultand dashed lines show results of a model without the partial irreversibility constraint. The
values are normalized by the initial stationary equilibrium values.

7.2. Baseline Experiment: 5% → 8%
We then turn to the baseline experiment that mimics Japanese consumption tax in 2014 inwhich consumption tax rate was raised from 5% to 8%. At the same time, this section examines
how the tax wedge help account for the dynamic pattern of the empirical tax elasticity discussed
in the previous subsection 7.1.1. To investigate the roles of frictions, I compare the baseline modelwith a model without the tax wedge, where q = (1 + τ c ).
In the following experiments, the economy is assumed to be in the stationary equilibriumin the first period, the government announces in the beginning of the second period that theconsumption tax is increased in the sixth period, and the households fully understand this announcement.
Figure 8 shows the result of the experiment in which consumption tax is raised from 5%
to 8%. All the variables are normalized by the initial stationary equilibrium values (e.g. nondurable consumption in the figure in period t is Ct /C1 ). The solid lines depict dynamics of one
with the partial irreversibility (baseline) and dashed line show that in the model without the tax
wedge, that is when q = 1 + τ c . Regarding the dynamics of the durable expenditure, I reportRD ≡X+∑ j xd >0 x d dµ j .
24Figure 9: Other Investment Frictions

Note: Left: Comparison with the Quadratic adjustment cost model. Right: Comparison with Fixed Cost model. The functional form of the quadratic adjustment cost isd2d
QC (d, d−1 ) = QC
2 [ d − (1 − δ ) d−1 ] , and that of the fixed cost is F ( d, d−1 ) = 1x d 6=0 F (1 −δ d ) d −1 .

In the Figure 8, the non-durables stay almost constant in both models. Also, the both models
predict a large durable purchase right before the tax hike t = 5 and a large drop in the durable
expenditure in common in the period of the tax hike, implying that the stockpiling is due toshort-run intertemporal substitution. The two models, however, exhibit distinct predictions inspeed of the recovery after the tax hike and the magnitude of stockpiling in durable expenditure
D in fifth period.
X+

7.2.1. Slow Recovery
D after the tax hike, it can be confirmed the baseline
Looking at the durable expenditure X+

model exhibits gradual recovery. This is because of (S,s) nature resulted from the tax wedge,
which works in the way the partial irreversibility constraint functions. That is, once households
accumulate enough amounts of durable stocks before the tax hike, the they do not adjust the
durables for some periods afterwards. This keeps the households to wait inactively until thestockpiled durables depreciate enough. The fact that the partial irreversibility constraint increases
the persistence of aggregate expenditure is known in the literature (e.g. Lam (1989)). A new
finding here is that higher persistence comes from the slow convergence only after the tax change(or shock) but not before it. That is, persistence is asymmetric before and after the tax change.
As discussed, this asymmetric persistence is key observed pattern in empirical tax elasticity oftaxable spendings.
Readers may wonder if other investment frictions, for instance fixed-cost and quadratic adjustment cost, works in the same way. This concern is critical because if a model with otherfictions behaves in the same manner, the researcher cannot identify the model correctly. To re25

solve the concern, I construct a model that added to the baseline model (i) quadratic adjustmentd2
cost, QC (d, d−1 ) = QC
2 [ d − (1 − δ ) d−1 ] or (ii) fixed-cost,

F (d, d−1 ) = F d (1 − δ d ) d −1

if d 6= (1 − δd )d−1

0otherwise

which follows Berger and Vavra (2015). QC and F d are fixed parameters that governs degree ofadjustment cost respectively.23
Figure 9 shows the results from two models. The left panel compares the baseline model andthe model embedded with the quadratic adjustment costs. The quadratic adjustment cost model
D but gradual stockpilingexhibits slow recovery in the aggregate positive durable expenditure X+

in third and forth period which is inconsistent to Baker et al. (forthcoming) for aforementioned
reason.
The right panel of Figure 9 shows the results from the fixed-costs model. This model also
D dips in third and forth period
generates slow recovery but the aggregate durable expenditure X+

which is also counter-factual. In addition, the fixed-cost model exhibits overshooting in tenth,
fourteenth and eighteenth periods due to negative persistence that fixed-cost generates.
With these properties in the quadratic adjustment and fixed-cost, it seems that the tax wedge(or the partial irreversibility constraint) is the best investment frictions for the study of the consumption tax change.
7.2.2. Overprediction
The baseline model predicts lower stockpiling in the t = 5. Alternatively a model without the
tax wedge generates larger stockpiling. It is due to the unrealistic revenue from the tax system inthe model without the wedge. If ignoring the tax wedge, the budget constraint is simply(1 + τ c )[c + x d ] + a = (1 + r ) a−1 + y j (e) + b
regardless of sign of x d . Therefore, if the household sells durables (i.e. x d < 0), the household
receive τ c x d as a revenue on top of x d . This becomes more problematic on the dynamics. Toccdemonstrate the problem, consider a simple example. Suppose τtoday= 5% today and τtomorrow=

8% tomorrow. Importantly, the households in a model without the tax wedge can make anunrealistic revenue by doing the following speculation. That is, if the households buy the durablecctoday and sell it tomorrow, they will make a profit due to tax differential τtomorrow− τtoday= 3%.

This revenue from the tax differential partly offsets the cost of holding the durables i.e., the23 QC = 0.09 taken from Cashin (2017) and F d = 0.05 taken from Berger and Vavra (2015).

26

depreciation and the cost of large utility fluctuation due to the stockpiling. In this way, theunrealistic revenue induces the speculative stockpiling in the model without the tax wedge.
Notice that this problem is more critical as (i) the tax differential becomes large, (ii) the depreciation rate is low or (iii) the utility cost of fluctuation is low. For instance, if a researcher constructa monthly model, she would set δd very low. Or, if the researcher introduces very long-lasting
goods (e.g., housing) into a model, very low δd is also set.24 These discussions mean that choiceof frequency of the model and coverage of the durable goods may exacerbate the misspecification
problem critically. My model frequency is annual and the model does not include housing so far.
That is, the potential problem due to misspecification is more critical and I view my result as alower bound of the problem.
Cashin and Unayama (2016) and Cashin (2017) estimate intertemporal elasticity of substitution
1/σ using a representative household model with durables but without the tax wedge. If true
data generation process is my baseline model, their models overpredict the stockpiling behavior
due to the misspecification. This overprediction must lead to a bias in their estimation results.
For instance, Cashin and Unayama (2016) should have the lower IES than true value due to themisspecification.
7.2.3. Stockpiling with Different Magnitude of the Tax Hike
The previous subsection discussed the speculative stockpiling largely depends on the degreeof the tax hike. This subsection quantitatively study how much the stockpiling behavior can bebiased due to the misspecification problem with different scale of the tax hike.
Before demonstrate it, let be clear about how this section examines the speculation. As discussed in previous sections, the stockpiling in the benchmark model takes place (i) in the form ofD (5) /X D (1)durables only (ii) only one period before the tax hike. Thus, the Figure 10 reports X++
cfor different new tax rates τnew
= 6, 7, . . . , 15%. The Figure 10 reports results of both baselinemodel and the model without the tax wedge.
In the Figure 10, two models generate clearly distinct predictions. On the one hand, thebaseline model predicts relatively moderate and almost linearly increasing stockpiling in the newtax rate. On the other hand, the model without the tax wedge shows far large and non-linearly
D (5) /X D (1), 2 in the y-axis, for instance,increasing stockpiling. Note that since the scale is X++

means double but not 2 % increase. This large difference cannot be ignored for the analysis of tax
hike.
This analysis suggests an application of this work to future fiscal problem in Japan. Government of Japan is advised by IMF to increase consumption tax rate to 20% in the end (See24 The purchase of houses is subject to consumption tax in Japan.

27

c= 6, 7, . . . , 15%.
Figure 10: Stockpiling over τnextc = 5%. Then, the tax is increased to τ cNote: The tax in period 1 is τold
new = 6, 7, . . . , 15% in
D (5) /X D (1) occurs
t = 6. The lines depict how much stockpiling of durable X++

Georgieva (2019)). Also, macroeconomic research (e.g. Braun and Joines (2015) and Hansen andİmrohoroğlu (2016)) suggest the need of further consumption tax increase to around 30%. Thus,
it is not unrealistic that Government of Japan need to implement large scale consumption tax hikeat some point. To the credibly assess the large scale consumption tax hike, I argue the model withthe tax wedge is necessary.
7.2.4. Age-Dependency
There are, in theory, several age-dependent effects in this dynamics. First, as shown in Figure
3, different generations have different amount of assets. Cash-on-Hands are essential for thestockpiling and thus the life-cycle asset profile determines the degree of stockpiling for eachgeneration. Second, different generations have different degree of needs of the durables. For
instance, most of the young need the durables immediately because they do not own them andthus have the high marginal utility of the durables. The middle-age and old have weaker motivein this respect. Third, different generations pay the higher consumption tax rate for different
duration of periods. That is, the young live longer under the new tax rate than the old and thus
they pay the the high tax rate longer. This, in turn, implies the young incurs the heavier burdenof the tax hike.
Figure 11 shows four generations consumption spending responses. A top-left panel in Figure11 illustrates the response of those who are born in t = −5 (i.e. age of 11 + 25 = 36 in t = 5).
28

Figure 11: Age-Dependent Responses to the Consumption Tax Hike 5% → 8%

This generation of households show relatively weak response to the anticipated consumption tax
hike. There are two reasons for this. First, the young have high marginal utility of the durables
soon and purchased large amount of durables not only in t = 5 but also t = 1, 2, 3, 4. That
D ( t ) /X D (1)is large. Second, this generation of theis, the denominator of the variable shown X++

households do not own liquid assets a much, which prevent them from stockpiling large amountof the durables. As a result, this generation of households do not show the strong stockpilingin the panel. Next, a top-right panel is expenditure responses of those who are born t = −15(i.e. who are age of 21 + 25 = 46 at t = 5). This generations of households illustrate strong
stockpiling behavior for two reasons. First, since the average of this generation of households areasset-wealthy, they afford the durable more freely than the young. Second, this households already accumulated enough durable stocks and thus did not have strong needs for the durables inperiod t = 1 unlike the young generation. Therefore, this households stock up the large amountof durables for the short-term intertemporal substitution motive in t = 5. A bottom-left figure
shows another middle-age generation who are born in t = −30 (i.e. who are age of 36 + 25 = 61
at t = 5). The responses of this middle-age generation in t = 5 is stronger than those of who areborn in t = −15 (top right). This is because they are very rich in asset and thus buy the luxurious
durables more than those who are born in t = −15. Finally, a bottom-right figure demonstrates
how the old (who are age of 51 + 25 = 76 in t = 5) react to the tax hike. Because of the high29

Figure 12: Welfare Changes by Consumption Equivalent Variation

Note: Each line depicts welfare costs of the tax hike 5% → 8%. The histogram behindthe lines shows the asset distribution in the stationary equilibrium. The households withhigher productivity incur higher welfare costs of the tax hike. Also, the asset wealthy
households lose more.

durable depreciation rate, this old households still stockpile the durables to a certain degree.
Overall, Figure 11 confirms that stockpiling behavior is highly age-dependent and most of theaggregate stockpiling behavior is explained by the middle-age generations.
7.2.5. Welfare Analysis
Next, I analyze welfare changes over the transition with consumption equivalent variation(CEV, hereinafter). The welfare evaluation over the transition is of the first order interest in thispaper. The tax-induced inflation effect, for instance, only appears over the dynamics.
Figure 12 shows the welfare changes of two generations, age of 30 at t = 1 and age of 45
at t = 1, along asset dimension and idiosyncratic shock over the transition. Overall, all thehouseholds in the Figure 12 incur the welfare costs, simply because the policy change we consider
here is tax hike with no transfer. What this paper is interested in here is which households loose
more.
First, lines in both left and right panels of the Figure 12 show that asset poor households
lose less. This is because the tax-induced inflation effect. Asset wealthy households see large
decline in their assets value, and thus reduces the consumption more than the asset poor. In
other words, the consumption tax hike is a policy tool to tax preexisting wealth. Also, both ofthe panels confirms that highly productive households lose more as well. This is negative income
effect. That is, given the high persistence of labor productivity, the highly productive households
lose the future potent high wage. This results in the larger decline in the consumption of highearners. These analysis does not support regressivity of the consumption tax burden, which
many of popular discussions argue.
Second,I find that young households potentially lose more by comparing two panels in 12.
30

That is, the young households in the left panel see more than 2.7% decline in the CEV while themiddle-age households in the right panel see less than 2.7%. This is because the young live longer
in the remaining life and thus see a larger decline in the present value of labor earnings.
Third, these lines only shows the decline in the welfare given state spaces (a, d, e, j), but does
not reflect the distribution. Thus, I overlay the distribution on the both of panels. The distributions clearly show that there is no asset wealthy young households in the model thus only a few
young households loose −2.8% in CEV while many of middle-age households accumulate theasset and thus face −2.8% decline in CEV.
Overall, welfare analysis shows burden of the consumption tax is highly heterogeneous, andasset wealthy and highly productive young households incur the largest burden of the consumption tax hike.

7.3. Consumption Tax Decrease: Asymmetricity
This section examines a counter-factual experiment in which the tax is decreased. The resultof this analysis can be insightful for the recent policies implemented in Germany and U.K in 2020
in order to combat the on-going pandemic crisis. As the model features the high non-linearity,
this section confirms that the tax decrease is highly asymmetric to the tax hike.
Figure 13 shows the results of when consumption tax is decreased from 8% to 5%. For thecomparison, I also depict the result when tax is increased from 5% to 8%. The tax decreases
is highly asymmetric to the tax hike. As the households know the future tax decreases, they
D to postpone the purchase by letting the durable stock depreciate. Then,
gradually decreases X+

they refill the stock of durables once the tax is decreased in period t = 6.
This asymmetricity suggests one lesson for empirical strategy. Suppose an econometrician
runs a regression ∆ log(yt ) = ∑i β i ∆ log(1 + τtc+i ) + zt + et , where yt are taxable spendings or
durable purchases, zt here is a sum of controls and fixed effects, and β i is a coefficient of interest.25
If the econometrician separate sample into the tax hike and decrease cases, the resulting coeffient
β in the lead lag is hard to interpret. Thus, the theory suggests that the tax hike and decrease be
separately analyzed.
Another implication of asymmetricity is that linearized solution is not appropriate for the
study of consumption tax changes. If the model is linearized, then the solution does not capture
the asymmetricity.
25 For instance, see Baker et al. (2019), and Baker et al. (forthcoming) for the regression analysis in this literature.

31

Figure 13: The Tax Hike vs Decrease 8% → 5%.

The dashed lines illustrate the household expenditures when the tax is decreased. Thiscounter-factual experiment shows that the tax decrease has the asymmetric effects compared with the tax hike.

7.4. One-Time vs Two-Times
Many of the economic downturn are persistent. When considering the tax hike as the aggregate demand stimulus policy in the recessions as Feldstein (2002) and Hall (2011) proposed,
the natural concern is whether the policy simply induces the short-run intertemporal shifting of
the households expenditures. If the consumption tax hike induces the short-run intertemporaldemand shifting only, the economy may see the plummet of the households expenditure in the
midst of the recovery from the crises. Thus, a natural question is how to design the policy so thatit causes longer intertemporal shifting.26
To answer the question, this section compares two experiments: an experiment with one time
tax hike and the other with two tax hikes. The idea borrows from the original proponent of
the unconventional fiscal policy, Feldstein (2002). If the government keeps increasing the tax
successively, the households do not stop purchasing the durable right after the first tax hike.
Figure 14 simulates the both experiments. In the one time hike case, the government increases
the tax to 8% in t = 6. The effects of the tax hike is similar to previous section. When thegovernment increases two times, I assume it first increases to 6.5% in t = 6, and then 6.5% to 8%
in t = 7.
Figure 14 illustrates the effects of two tax hikes. The stockpiling behavior in the two times
26 Also, Government of Japan increased consumption tax rate with two steps: from 5% to 8% in 2014 and 8% to 10%
in 2019. This Japanese experience is one motivation for this experiment.

32

Figure 14: One Time Hike vs Two Times Hikes.

Note: In one time case: the government raises the tax rate 5% to 8% in t = 6. In two times
case: the government raises it 5% → 6.5% in t = 6 and 6.5% → 8% in t = 7.

hikes case is a bit weaker than one time. However, importantly, the drop in the durable expenditure after the tax hike gets (i) moderate and (ii) is postponed to the future. This result imply that
if the government keep increasing the tax hike until the end of the recession, it can postpone thetiming of the drop in the households expenditure.
Also, welfare analysis in Figure 15 indicates the two-times tax hike scheme is welfare improving for almost all of the households except young borrowing constrained households. The welfare
gains comes from the fact that the households change the durable stock less in the two-times tax
hike case. Thus, they do not incur the utility costs of violation of the consumption smoothing
due to the large stockpiling behavior.

8. Short-Run: General Equilibrium
My baseline tax hike experiment is to raise the tax by 3% and this experiment is arguably amacroeconomic-scale policy change. Therefore, it seems natural to consider how macroeconomic
variables including prices respond to the policy change. This section finally shows the resultsfrom the general equilibrium and compare them with the partial equilibrium results. It turns outthat the equilibrium assumption does not alter the results much. This section discuss why two
equilibrium assumptions predict the very similar consequence.
Figure 16 compares the dynamics of partial and general equilibrium. Overall, the dynamics

33

Figure 15: Welfare Costs of One Time Hike vs Two Times Hikes.

Note: In one time case: the government raises the tax rate 5% to 8% in t = 6. In two times
case: the government raises it 5% → 6.5% in t = 6 and 6.5% → 8% in t = 7.

of the two look very similar. The main reason why general equilibrium shows the similar resultsof the partial equilibrium is low interest rate elasticity of the aggregate durable expenditure.
Koby and Wolf (2020) and Winberry (2020) analytically shows that interest rate elasticity is thesufficient statistics for whether the results of the general equilibrium is close to those of the partial
equilibrium. The intuitive explanation follows that the general equilibrium is the environment
where prices move. Therefore, if the aggregate quantities respond to the price movement only alittle due to the low elasticity, the general equilibrium effects are tiny.
Recent literature argue that standard (S,s) models generate infinitely high interest rate elasticity. (e.g., Koby and Wolf (2020), McKay and Wieland (2019) and Winberry (2020)) In my benchmark model, the model-generated interest rate elasticity is as low as 4.1. On the contrary, manyof fixed-cost models generate the elasticity of 50 − 1, 075.27 Empirical literature on the durables
shows the interest elasticity is between 1.1 − 5.0 (Baker et al. (2019), Mian and Sufi (2012)). Thus,
the mechanism of lowering the model-implied interest rate elasticity is of interest in the lumpiness
literature. The baseline model generates the low elasticity for the following three reasons.
Successfully matching the interest rate elasticity is crucial not only for the equilibrium concept
but also for the tax change experiment because the consumption tax change has a profoundconnection with the change in the interest rate as the Euler equation below shows,28
u 0 ( c ) ≥ β (1 + r 0 )1 + τc 0 0u ( c ).
1 + (τ c )0

Thus, the tax change has a very similar implication to the interest rate in my baseline model as
27 See Koby and Wolf (2020) and Winberry (2020) for the lumpy investment, and McKay and Wieland (2019) for the
lumpy durables.
28 I omit the durables from the Euler equation here since the Stone-Geary is the additive separable when it is monotonically transformed.

34

Figure 16: Partial Equilibrium and General Equilibrium

Note: The dashed line shows the household expenditure in the general equilibrium where(r, w) clear the markets. The general equilibrium does not alter the quantitative results
significantly.

well. As a result, replicating the low elasticity in line with the empirical studies is key for themain focus of this study.
First, and most importantly, it turns out that life-cycle motives dampens the interest rateelasticity. This is first because there is a mass of households who adjusts durables regardless ofthe change in interest rate, which reduces the interest sensitivity of the durable expenditure in amodel. For instance, the young buy the durables because of the infinitely high marginal utility
while the old do not buy the durable because they will die soon. In this way, the model impliedinterest elasticity of the aggregate durable expenditure is dramatically lowered.
Second, two asset structure is key to reduce the elasticity, as explored in Berger and Vavra(2015). That is, if the agents have only one asset as in Khan and Thomas (2008), the good market
clearing condition Y = C + Ik implies that a change in Ik propagates a change in C. At thesame time, C is governed by the household smoothing motive in the equilibrium. Thus strong
households consumption smoothing motive smooths the investment Ik through the good market
clearing condition or large price adjustment. By contrast, if the households have two assets, alumpiness in one type of assets may not be smoothed by the consumption smoothing motive
because households now have an additional measure to smooth consumption. In this way, thetwo asset structure weakens the elasticity.
Third, since I do not include housing in my analysis, the depreciation of the durables ishigher than the value typically used in the literature. As House (2014) discusses, the higher
the depreciation is, the lower the interest sensitivity is. This is because, if the depreciation is35high, the households make durable spendings for a short-term use. This, in turn makes the
durable expenditure frequent and small-scale. As a result, the high depreciation rate reduces thesensitivity of the durable expenditure to the change in the interest rate.

9. Concluding Remarks
This paper is, to my best knowledge, a first paper that studies the stockpiling response to
the preannounced consumption tax changes using a life-cycle heterogeneous-household generalequilibrium model of durables and the tax wedge. The tax wedge is introduced to reflect theactual consumption tax system that households face only when buying the durables but not whenselling them. Using the novel model, this paper explores the effects of anticipated consumption
tax changes.
The model replicates the dynamic pattern of the tax elasticity of taxable spending very well,owing to presence of the tax wedge. I also show that either a frictionless model or models withother types of investment frictions (i.e. convex adjustment costs, and fixed-costs) do not reproduce
the dynamic pattern. Second, life-cycle structure is key to lower the elasticities. Combined withthe tax wedge and life-cycle structure, my baseline model accounts for the empirical tax elasticitiesof taxable spendings and durable expenditures over the dynamics well.
Also, the tax wedge is qualitatively necessary to generate stockpiling behavior with true stockpiling motive in response to a preannounced tax change. Without the tax wedge, the resulting
stockpiling is not only due to households own use but also for the speculation. Thus, the model
without the tax wedge largely overpredicts the stockpiling behavior. If researcher estimates structural parameters (e.g., elasticity of intertemporal substitution) using consumption tax hike andthe model without the tax wedge, the estimated parameters must be biased due to the speculative
stockpiling.
Welfare analysis finds that the tax hike over the transition is progressive. This is because theincrease in the consumption tax rate imposes a tax on the preexisting wealth. In my baselinemodel, the presence of the durables further widens the welfare costs inequality.
Finally, I conduct two couter-factual experiments. The first experiment is the consumption
tax decrease. I find that consumption tax decrease is asymmetric to the tax hike. In the tax
decrease case, many of the households stop buying the durables far before the tax hike and let the
durable depreciate until the period of tax decrease. The second experiment is the multiple-times
tax hike. The multiple-times tax hike induces smoother intertemporal shifting in the household
expenditure and thus welfare improving for most of the households except young borrowing

36constrained households. Overall, the paper explored effects of the anticipated consumption tax
with the novel model.

References
Alonso-Ortiz, J. and R. Rogerson (2010): “Taxes, transfers and employment in an incomplete
markets model,” Journal of Monetary Economics, 57, 949–958. [page 16.]
Altig, D., A. J. Auerbach, L. J. Koltikoff, K. A. Smetters, and J. Walliser (2001): “Simulating
fundamental tax reform in the United States,” American Economic Review, 91, 574–595. [pages 6
and 20.]

Auclert, A. (2019): “Monetary policy and the redistribution channel,” American Economic Review,
109, 2333–67. [page 7.]
Baker, S. R., S. Johnson, and L. Kueng (forthcoming): “Shopping for lower sales tax rates,” .
[pages 2, 3, 6, 21, 22, 23, 26, 31, and 48.]
Baker, S. R., L. Kueng, L. McGranahan, and B. T. Melzer (2019): “Do household finances
constrain unconventional fiscal policy?” Tax Policy and the Economy, 33, 1–32. [pages 2, 3, 6, 23, 31,
and 34.]

Benhabib, J. and A. Bisin (2018): “Skewed wealth distributions: Theory and empirics,” Journal of
Economic Literature, 56, 1261–91. [page 19.]
Berger, D. and J. Vavra (2015): “Consumption dynamics during recessions,” Econometrica, 83,
101–154. [pages 7, 26, 35, and 42.]
Braun, R. A. and D. H. Joines (2015): “The implications of a graying Japan for government
policy,” Journal of Economic Dynamics and Control, 57, 1–23. [page 28.]
Caballero, R. J. (1990): “Expenditure on durable goods: a case for slow adjustment,” The Quarterly Journal of Economics, 105, 727–743. [page 7.]——— (1993): “Durable goods: An explanation for their slow adjustment,” Journal of Political
Economy, 101, 351–384. [page 7.]
Cashin, D. and T. Unayama (2016): “Measuring intertemporal substitution in consumption:
Evidence from a VAT increase in Japan,” Review of Economics and Statistics, 98, 285–297. [pages 2,4, and 27.]

Cashin, D. B. (2017): “The household expenditure response to a consumption tax rate increase,”. [pages 26 and 27.]
Correia, I. (2010): “Consumption taxes and redistribution,” American Economic Review, 100, 1673–
94. [pages 6 and 20.]37

Correia, I., E. Farhi, J. P. Nicolini, and P. Teles (2013): “Unconventional fiscal policy at thezero bound,” American Economic Review, 103, 1172–1211. [pages 2 and 6.]
Correia, I., J. P. Nicolini, and P. Teles (2008): “Optimal fiscal and monetary policy: Equivalence
results,” Journal of Political Economy, 116, 141–170. [pages 4 and 7.]
D’Acunto, F., D. Hoang, and M. Weber (2016): “The effect of unconventional fiscal policy on
consumption expenditure,” Tech. rep., National Bureau of Economic Research. [pages 2 and 6.]——— (2018): “Unconventional fiscal policy,” in AEA Papers and Proceedings, vol. 108, 519–23.
[pages 2 and 6.]
De Nardi, M. and G. Fella (2017): “Saving and wealth inequality,” Review of Economic Dynamics,
26, 280–300. [page 19.]
Doepke, M. and M. Schneider (2006): “Inflation and the redistribution of nominal wealth,”
Journal of Political Economy, 114, 1069–1097. [page 7.]
Doepke, M., M. Schneider, and V. Selezneva (2018): “Distributional effects of monetary policy,”
Unpublished manuscript. [page 7.]
Druedahl, J. (2020): “A Guide On Solving Non-Convex Consumption-Saving Models,” . [pages 5,
14, 41, and 42.]

Druedahl, J. and T. H. Jørgensen (2017): “A general endogenous grid method for multidimensional models with non-convexities and constraints,” Journal of Economic Dynamics andControl, 74, 87–107. [page 14.]Eberly, J. C. (1994): “Adjustment of consumers’ durables stocks: Evidence from automobilepurchases,” Journal of Political Economy, 102, 403–436. [page 7.]
Erosa, A. and G. Ventura (2002): “On inflation as a regressive consumption tax,” Journal of
Monetary Economics, 49, 761–795. [page 6.]
Favilukis, J., S. C. Ludvigson, and S. Van Nieuwerburgh (2017): “The macroeconomic effects
of housing wealth, housing finance, and limited risk sharing in general equilibrium,” Journal of
Political Economy, 125, 140–223. [page 7.]
Feldstein, M. (2002): “The role for discretionary fiscal policy in a low interest rate environment,”
Tech. rep., National Bureau of Economic Research. [pages 2 and 32.]
Fernández-Villaverde, J. and D. Krueger (2007): “Consumption over the life cycle: Facts fromconsumer expenditure survey data,” The Review of Economics and Statistics, 89, 552–565. [page 7.]——— (2011): “Consumption and saving over the life cycle: How important are consumer
durables?” Macroeconomic dynamics, 15, 725–770. [pages 7 and 15.]
Floden, M. and J. Lindé (2001): “Idiosyncratic risk in the United States and Sweden: Is there arole for government insurance?” Review of Economic Dynamics, 4, 406–437. [page 16.]
Fraumeni, B. (1997): “The measurement of depreciation in the US national income and product38

accounts,” Survey of current business-United States Department of Commerce, 77, 7–23. [page 15.]
Gavazza, A. and A. Lanteri (2018): “Credit shocks and equilibrium dynamics in consumer
durable goods markets,” Economic Research Initiatives at Duke (ERID) Working Paper, 275. [page 7.]
Georgieva,

K.

ArticleIV(2019):“Japan:Mission,”

Staff

Concluding

Statement

of

the

2019

https://www.imf.org/en/News/Articles/2019/11/24/

mcs-japan-staff-concluding-statement-of-the-2019-article-iv-mission,accessed:

2020-8-18. [page 28.]
Grossman, S. and G. Laroque (1990): “Asset Pricing and Optimal Portfolio Choice in the Presence of Illiquid Durable Consumption Goods,” Econometrica, 58, 25–51. [page 7.]
Guerrieri, V. and G. Lorenzoni (2017): “Credit crises, precautionary savings, and the liquidity
trap,” The Quarterly Journal of Economics, 132, 1427–1467. [page 7.]
Hall, R. E. (2011): “The long slump,” American Economic Review, 101, 431–69. [pages 2 and 32.]
Hansen, G. D. and S. İmrohoroğlu (2016): “Fiscal reform and government debt in Japan: A
neoclassical perspective,” Review of Economic Dynamics, 21, 201–224. [page 28.]
Hayashi, F. and E. C. Prescott (2002): “The 1990s in Japan: A lost decade,” Review of Economic
Dynamics, 5, 206–235. [page 15.]
House, C. L. (2014): “Fixed costs and long-lived investments,” Journal of Monetary Economics, 68,
86–100. [page 35.]
Iskhakov, F., T. H. Jørgensen, J. Rust, and B. Schjerning (2017): “The endogenous grid methodfor discrete-continuous dynamic choice models with (or without) taste shocks,” Quantitative
Economics, 8, 317–365. [page 14.]
Kaplan, G. and G. L. Violante (2014): “A model of the consumption response to fiscal stimulus
payments,” Econometrica, 82, 1199–1239. [page 11.]
Khan, A. and J. K. Thomas (2008): “Idiosyncratic shocks and the role of nonconvexities in plant
and aggregate investment dynamics,” Econometrica, 76, 395–436. [page 35.]
Kitao, S. (2011): “Macroeconomic and redistributional effects of consumption taxes in the usa,”
The Japanese Economic Review, 62, 63–81. [page 6.]
Kitao, S. and T. Yamada (2019): “Dimensions of inequality in Japan: Distributions of earnings,
income and wealth between 1984 and 2014,” . [pages 18 and 19.]
Koby, Y. and C. K. Wolf (2020): “Aggregation in Heterogeneous-Firm Models: Theory andMeasurement,” . [pages 4 and 34.]
Krusell, P., V. Quadrini, and J.-V. Ríos-Rull (1997): “Politico-economic equilibrium and economic growth,” Journal of Economic Dynamics and Control, 21, 243–272. [page 6.]
Laczó, S. and R. Rossi (2019): “Time-consistent consumption taxation,” Journal of Monetary Economics. [page 6.]39Lam, P.-s. (1989): “Irreversibility and consumer durables expenditures,” Journal of Monetary Economics, 23, 135–150. [pages 7, 15, and 25.]
Lam, P.-S. (1991): “Permanent income, liquidity, and adjustments of automobile stocks: Evidence
from panel data,” The Quarterly Journal of Economics, 106, 203–230. [page 7.]
Leahy, J. V. and J. Zeira (2005): “The timing of purchases and aggregate fluctuations,” The Review
of Economic Studies, 72, 1127–1151. [page 7.]
Lise, J., N. Sudo, M. Suzuki, K. Yamada, and T. Yamada (2014): “Wage, income and consumption
inequality in Japan, 1981–2008: From boom to lost decades,” Review of Economic Dynamics, 17,
582–612. [page 16.]
Luengo-Prado, M. J. (2006): “Durables, nondurables, down payments and consumption excesses,” Journal of Monetary Economics, 53, 1509–1539. [page 7.]
McKay, A. and J. F. Wieland (2019): “Lumpy durable consumption demand and the limited
ammunition of monetary policy,” Tech. rep., National Bureau of Economic Research. [pages 4, 7,
34, and 42.]

Mian, A. and A. Sufi (2012): “The effects of fiscal stimulus: Evidence from the 2009 cash forclunkers program,” The Quarterly Journal of Economics, 127, 1107–1142. [pages 3 and 34.]
Motta, G. and R. Rossi (2019): “Optimal Fiscal Policy with Consumption Taxation,” Journal of
Money, Credit and Banking, 51, 139–161. [page 6.]
Nakajima, T. and S. Takahashi (2017a): “The Effectiveness of Consumption Taxes and Transfers
as Insurance against Idiosyncratic Risk,” Journal of Money, Credit and Banking. [page 6.]——— (2017b): “The optimum quantity of debt for Japan,” Journal of the Japanese and International
Economies, 46, 17–26. [page 16.]
Nguyen, A. D. M., L. Onnis, and R. Rossi (2020): “The Macroeconomic Effects of Income andConsumption Tax Changes,” American Economic Journal: Economic Policy. [page 2.]
Nishiyama, S. and K. Smetters (2005): “Consumption taxes and economic efficiency with idiosyncratic wage shocks,” Journal of Political Economy, 113, 1088–1115. [pages 6, 7, and 47.]——— (2014): “Analyzing fiscal policies in a heterogeneous-agent overlapping-generations economy,” in Handbook of Computational Economics, Elsevier, vol. 3, 117–160. [page 47.]
OECD (2017): “Old-age dependency ratio,” Pensions at a Glance 2017: OECD and G20 Indicators.
[page 16.]
Ogaki, M. and C. M. Reinhart (1998): “Measuring intertemporal substitution: The role ofdurable goods,” Journal of Political Economy, 106, 1078–1098. [page 15.]
Pakoš, M. (2011): “Estimating intertemporal and intratemporal substitutions when both income
and substitution effects are present: the role of durable goods,” Journal of Business & Economic
Statistics, 29, 439–454. [page 15.]40

Parodi, F. (2018): “Value-Added Tax, Household Consumption Dynamics, and Redistribution,” .
[pages 6 and 7.]——— (2019): “Taxation of Durables, Non-durables and Earnings with Heterogeneous Preferences,” . [pages 6, 7, and 15.]
Rouwenhorst, G. K. (1995): “Asset Pricing Implications of Equilibrium Business Cycle Models
Chapter 10, , T,” in Frontiers of Business Cycle Research, ed. by T. F. Cooley, Princeton University
Press. [page 16.]
Sterk, V. and S. Tenreyro (2018): “The transmission of monetary policy through redistributions
and durable purchases,” Journal of Monetary Economics, 99, 124–137. [page 7.]
Ventura, G. (1999): “Flat tax reform: A quantitative exploration,” Journal of Economic Dynamics
and Control, 23, 1425–1458. [pages 6 and 20.]
Winberry, T. (2020): “Lumpy investment, business cycles, and stimulus policy,” . [pages 4 and 34.]
Yamada, T. (2012): “Income risk, macroeconomic and demographic change, and economic inequality in Japan,” Journal of Economic Dynamics and Control, 36, 63–84. [pages 15 and 16.]
Yang, F. (2009): “Consumption over the life cycle: How different is housing?” Review of Economic
Dynamics, 12, 423–443. [page 7.]
Zorzi, N. (2020): “Investment Dynamics and Cyclical Redistribution,” . [page 7.]

10. Appendix A: Computation
10.1. Nested EGM Algorithm
This section explains algorithm of the Nested EGM used in the paper. The algorithm is originally developed by Druedahl (2020) and I extend it so as to incorporate the partial irreversibility
constraint into the model. The critical advantage of this algorithm is the nesting assumption which
reduces the dimensions of the maximization problem in choice-specific value functions. I startwith the exposition of the nesting assumption. For simplicity, I focus on the solution of stationary
equilibrium where prices (r, w) and the taxτ c are fixed. Extending the algorithm to the dynamics
is straightforward.
Also, the algorithm is general enough to have fixed-cost without any additional computational
efforts. Thus, I add the following form of fixed-cost here:

F (d, d−1 ) = F d (1 − δ d ) d −1

if d 6= (1 − δd )d−1

0otherwise41

where F d is a parameter. This is a common functional form in the lumpy durable literature (e.g.,
Berger and Vavra (2015), McKay and Wieland (2019)).
In order to apply the Nested EGM, we need to rewrite the model in the following way. The
notation here from now on follows the old version of Druedahl (2020).29 Let m be a cash-on-hand,
and which is defined as
m = ( 1 + r ) a −1 + y j ,where y j = (1 − 1 j< J R )wκ j e + 1 j< J R ss + b.
where a−1 is the asset stocks at the end of previous period and y j is the either labor earning or
the social security depending on the age plus received bequest.
In the beginning of every period, households own a set of (m, d,¯ e) and compute
mup = m + (1 + τ )(1 − δd )d¯ − F d (1 − δd )d¯
mdown = m + q(1 − δd )d¯ − F d (1 − δd )d¯where mup and mdown are choice-specific cash-on-hands for up and down respectively, and d¯ is a stockof the durable at the beginning of the period.
Given these cash-on-hands (m, mup , mdown ), the households compute choice-specific valuefunctions (vinact (·), vup (·), vdown (·))in the midst of the period. Particularly, I assume that theinaction problem is solved first. This is the assumption nesting the timing within a period so thatthe households first decide how much non-durable consume, and then decide later how muchdurable.
vinact(m, d,¯ e) = max u(c, (1 − δd )d¯) + w j ( a, (1 − δd )d,¯ e)ja ≥0s.t. (1 + τ c )c + a = m.
Given the choice of the non-durable consumption cinact (·), the households choose intensive
margin of the durable. When upward adjusting, the households use this consumption function
29 The title of his old version was “A Fast Nested Endogenous Grid Method for Solving General Consumption-Saving
Models.”42

and solve the problem below,upv j (m, d,¯ e) = max u(c, d) + w j ( a, d, e)ds.t. (1 + τ c )[c + d] + a = mupc = cinact (mup − (1 + τ )d, d, e)
d > (1 − δ)d.¯
Notice that the dimension of the maximization is only one, thanks to the nesting assumption.
This significantly reduces the computational burden and we are not bothered by the curse of thedimensionality to solve this optimization problem.30
When downward adjusting,vdown(m, d,¯ e) = max u(c, d) + w j ( a, d, e)jds.t. (1 + τ c )[c + d] + a = mdownc = cinat (mdown − qd, d, e)d < (1 − δ)d¯
Also notice that these Bellman equation does not contain v j+1 (·). We used instead post-decision
value function wt (·) and its domain is different from value function v j (·). The inclusion of postdecision value function allows pre-computation.
This post-decision value function is defined as
w j ( a, d, e) = βE[v j+1 (m0 , d¯0 , e0 )|e]m 0 = ( 1 + r ) a + y j +1 .
We are ready to explain the algorithm.
¯ m, mup , mdown . In practice, we use the same grid for(i) Grid spacing: Space the grids for a, d,
mup and mdown but they are conceptually different.31
30 In practice, w (·) can be a very kinky function due to the discrete choices. Thus, I use many multi-starts and ajlocal-maximization method for this problem.
31 It is not computationally costly to use a large number of grid points for ( a, m ) because the inaction problem issolved fast. But an increase in the number of grid points for d¯ is very costly.

43

¯ e ),(ii) Last period problem: Solve the last period problem.32 That is, for any (m, d,(1 + τ c ) c = m¯v J (m, d) = u


m
¯
,d .
1 + τc(iii) Loop for age: For any j = J − 1, . . . , 1, repeat the below.
(i) Inaction Problem:(i) Pre-computation:(i) Next Period Values: For any (e, a, d¯0 ), compute33

m 0 = ( 1 + r ) a + y j +1

(mup )0 = m0 + (1 + τ c )(1 − δd )d¯0 − F d (1 − δd )d¯0(mdown )0 = m0 + q(1 − δd )d¯0 − F d (1 − δd )d¯0(ii) Interpolations: Evaluate the value functions on (e, a, d). That is, compute34
0
inact 00 ¯00
¯0vinactj+1 ( e , a, d ) = v j+1 ( e , m , d )where d¯00 = (1 − δd )d¯0upup
v j+1 (e0 , a, d¯0 ) = v j+1 (e0 , (mup )0 , d¯0 )down 0
down 0 ¯0¯0vdown) ,d )
j+1 ( e, a, d ) = v j+1 ( e , ( mand then take a maximum of the discrete choiceupdown
v j+1 (e0 , a, d¯0 ) = max{vinact
j+1 (·), v j+1 (·), v j+1 (·)}.

and here store the unconditional next-period consumption function c0 (e, a, d¯0 )and durable function d0 (e, a, d¯0 ).
(iii) Post-Decision Value Function: For (e, a, d),w j (e, a, d¯0 ) = βE[v j+1 (e0 , a, d¯0 )|e]q j (e, a, d¯0 ) = β(1 + r )E j [α(c0 )α(1−ρ)−1 (d0 )(1−α)(1−ρ) ].
32 If solving an infinitely lived agent model, this computation can be also applicable if you set v0 = 0.

33 Notice that d¯0 is the durable stock at the beginning of next period, and d is the durable stock at the end of thisperiod. Thus, d = d¯0 . Also, a is asset stocks at the end of the period.
34 Regarding d¯00 , we need to count the depreciation here only for inaction because depreciation for up and down are
already taken into account in mup and mdown .

44

(ii) EGM: Apply EGM to the inaction problem. Here, notice d = d¯0 is used due to theinaction.
1 α(1−ρ)−1q(e, a, d¯0 )α(d¯0 )(1−α)(1−ρ)
m(e, a, d¯0 ) = (1 + τ c )c(e, a, d¯0 ) + ac(e, a, d¯0 ) = u−1



The m(e, a, d¯0 ) is the endogenous grid for the cash-on-hand.
(iii) Borrowing Constraint: This part is standard. Denoting m as an exogenous grid

point, if m(e, a, d¯0 ) ≥ m, then set(1 + τ c ) c = mvinact (e, m, d¯0 ) = u c, d¯ + w(e, 0, d¯0 )(iv) Upper Envelop Algorithm (and Interpolations): Let ia be the ia-th grid point ofgrid a. Similarly, ie, im, id are defined as well. We can create the following function,c∗ (ie, ia, id, im) = c(ie, ia, id) +m(im) − m(ie, ia, id)(c(ie, ia + 1, id) − c(ie, ia, id)),m(ie, ia + 1, id) − m(ie, ia + 1, id)

where m(im) is im-th exogenous grid point value. Subsequently define a∗ = m −(1 + τ c )c∗ (ie, ia, id, im), and thenw∗ (ie, ia, id, im) = w(ie, ia, id) +a∗ − a(ia)(w(ie, ia + 1, id) − w(ie, ia, id)).
a(ia + 1) − a(ia)Finally, take the max over a as follows
vinact(ie, im, id) = max u(c∗ (ie, ia, id, im), d(id)) + w∗ (ie, ia, id, im).
jia

The obtained choice-specific value function vinact(·) here is the globally maximizingjchoice-specific value function. Also, the consumption function for the inaction
problem cinact(·) is found here and will be used to solve up and down problems.
jNote that this procedure does not take so much time, because these maximization
are taken only over the candidates that satisfies necessary conditions for maximization (i.e. the Euler equation with inequality and the budget constraint), but
not over entire a.

45

(ii) Upward Adjusting Problem: Solve the upward adjusting problem with standard VFI.
upv j (e, mup , d¯) = max u(c, d) + w j (e, a, d)ds.t. (1 + τ c )[c + d] + a = mupc = cinact (e, mup − (1 + τ )d, d)d > (1 − δ)d¯
From practical perspective, kinks in the value function are caused by the discrete choice
and amplified through v j+1 over the iteration of age. Thus, w j is supposed to be nondifferentiable and have many local-maxima. To deal with this, we separate the grid dvery finely, compute local maximum in the subinterval, and take the maximum of thelocal maxima.
(iii) Downward Adjusting Problem: Similarly, solve the downward adjusting problem

with standard VFI.
vdown(e, mdown , d¯) = max u(c, d) + w j (e, a, d)jds.t. (1 + τ c )c + qd + a = mdownc = cinact (e, mdown − qd, d)d < (1 − δ)d¯
Almost all of the codes are written in C++ and they are called by MATLAB through MEX compilation.
When solving upward and downward adjusting problem with value function iteration, I used
NLopt Library for the optimization toolbox. I tried several optimization methods and find that
the Subplex algorithm is the best for solving this model in practice.

10.2. Perfect Foresight Path of The Life-cycle Model
Compared with solving the perfect foresight equilibrium path of an infinitely-lived agent
model, it is by far more computationally costly to solve that of the life-cycle model. This isbecause there exist J generations of households at any periods t ≥ 0.
This implies that all the generations Bellman equations must be solved to solve the partialRequilibrium dynamics. Let be Cj,t ≡ c j,t ( a−1 , d−1 , e)dµ j and as follows. Then the quantities for46

all the ages (Cj,t , D j,t , A j,t ) j are aggregated at every period, i.e.,J

Ct = ∑ Cj,t ,j =1

JDt = ∑ D j,t ,j =1

JAt = ∑ A j,t .
j =1

Notice that there are some generations who are born after the tax change and some who die
before the tax change. Thus, I must solve the T + J − 1 Bellman equations conditional on (w, r )where T is the time horizon for the simulation.
My solution method to solve the perfect foresight follows Nishiyama and Smetters (2005) andNishiyama and Smetters (2014), except that I use stochastic simulation for solving the distribution.
(i) Choose sufficiently long simulation periods T so that the economy reach the new steadystate at T. Guess a path of prices {rt0 , w0t }tT=1 .
(ii) For all the generations i = 1, . . . , J + T − 1, solve the Bellman equation from J to 1. Then,get {ci,t , di,t , ai,t } .
(iii) Aggregate the individual quantities for those who live at time tJ

Ct = ∑ Cj,t ,j =1

JDt = ∑ D j,t ,j =1

JAt = ∑ A j,t .
j =1(iv) Check if the markets clear for all t, that is

A t = Kt .
Note that since the labor supply is inelastic, the labor market is cleared by construction. By
Walras’s law, the good market clears if the asset market clearing condition is satisfied. If themarkets are not cleared, update prices {rt`+1 , wt`+1 }∞t=1 and iterate the procedure until themarkets clear.

11. Appendix B: Japanese Consumption Tax Hike
This paper applies the baseline mode to the Japanese economy. This section discusses thebasic effects of the consumption tax changes in Japan.
Figure 17 shows the cyclical components of HP filtered GDP and aggregate consumptions
since 1994 Q1. Aggregate consumptions have been smoother than GDP for most of the time in47

Figure 17: Cyclical Components of GDP and Aggregate Consumption in Japan

Note: HP filtered GDP and aggregate consumptions since 1994 Q1 until 2020 Q1. The
aggregate consumptions have been smooth except some periods, such as consumption tax
changes or large events (e.g. Great Recession and The Tohoku Earthquake). Data source:
the national account.

the sample periods, but they sometimes largely swing. For example, the consumptions plunged
in 2008 and 2011 because of the Great Recession and the Tohoku earthquake. Also, it can benoticed that large spikes in aggregate consumptions are accompanied with the consumption tax
changes in Japan. In 1997 April, the Government of Japan increased the consumption tax from
3% to 5% and aggregate consumption spikes up before the tax hike and fall afterwards. Similarly,
the Government of Japan increased the consumption tax from 5% to 8% in 2014 April and 8% to
10% in 2019 October. This aggregate level data shows the large impacts of the consumption tax
changes on the aggregate consumptions.
Next, Figure 18 shows the non-durables and durable spendings since 2013 October until 2014
December. The data source of this figure 18 is Family Income and Expenditure Survey (FIES)
which is collected in a monthly basis by the government to construct the price indices.35
Looking at the Figure 18, there are three noticeable features. First, the non-durable spendings
are relatively stable over the sample periods, which is consistent to the theory. Second, the durable
spendings spikes up right before the consumption tax hike. Third, the durable spendings after
the tax hike is persistently lower than that of the initial level in the sample periods. These threefindings are almost in line with the more cleanly identified pattern in Baker et al. (forthcoming)as discussed in the introduction with the Figure 1. Also, these findings are the facts which I
35 In short, FIES is comparable to CEX in the U.S.

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Figure 18: Non-Durable and Durable Spendings around 2014 April

Note: Taxable non-durable and durable spendings around 2014 April when the consumption tax is raised from 5% to 8%. Data source: FIES.

replicated using the baseline model.

12. Appendix C: Other Counter-Factuals
Since this paper develops the novel model for the policy analysis, it must be of certain interest
to conduct a thought-experiments. This section discusses some experiments.

12.1. Unanticipated Tax Changes
This paper focused on the anticipated tax changes because it is the anticipation of the future
tax hike that causes the stockpiling behavior of the durable goods. That is, if the consumption
tax is suddenly implemented without a preannouncement, the households simply decrease theconsumptions due to negative income effects. This subsection confirms this is the case.
Figure 19 shows the results of the experiment when the unanticipated tax hike is suddenly
implemented in period 2 without the preannouncement. The unanticipated tax hike simply decreases the consumption levels (c, d) in the period of the tax hike, and the expenditures stay
constant after the tax hike t = 3, 4, . . ..
A clear difference from the anticipated tax change is that the unanticipated tax hike generate
the larger negative income effects on the durable expenditure in periods 2. There are two reasonsfor this strong negative income effects. First, everyone face the reduction in the income and the49Figure 19: Anticipated Tax Hike vs Unanticipated Tax Hike

Note: When the tax hike is anticipated, the tax hike is preannounced in period 2 andimplemented in period 6. When the tax hike is surprise (i.e., unanticipated), the tax hike issuddenly implemented in period 2 without the preannouncement. The unanticipated case
exhibits the larger negative income effects.

asset when the tax hike is surprise. By contrast, when the tax hike is announced and implemented
in the future, there is some old generations who pass away by the tax hike. These generations do
not incur the negative income effects. Second, when the tax hike is announced, the stockpilingof the durable right before the tax hike is a way to mitigate the negative income effects. But, thismeasure is not available when the tax hike comes as the surprise.

12.2. Temporary Tax Cut: Germany in 2020
Next, I run an experiment designed to mimic the Germany’s VAT cut implemented during theongoing pandemic crisis. The German government announced in June 3rd that it would decrease
the VAT rate by from 19% to 16% for six months between July and December.36
Figure 20 shows the results of the temporary tax cut experiment in a partial equilibrium inwhich tax cut from 19% to 16% is announced in period 2 and implemented between period 3 andperiod 9.
The result resembles the combination of the tax cut and hike. In the period 2 when tax cut
is announced, the durable expenditure plummet since many households postpone the durable
36 The reduced tax rate applicable to the grocery is also reduced by 2% in Germany. Also, the U.K. cut hospitality
and tourism VAT by 15% for six months.

50

Figure 20: Temporary Tax Cut: Germany in 2020

Note: VAT is set 16% initially and economy is assumed in the stationary equilibrium inthe first period. The government announces in period 2 that it would cut the tax between
period 3 and period 9. Prices are fixed at stationary equilibrium levels.

purchase. Then, they refill the durable stock in the period 3. The aggregate durable expenditure
remains higher than the stationary equilibrium level because of the temporary VAT cut. In the
period 8, the households understand the temporary VAT cut ends in the next period and thus
stock up the durables. The aggregate durable expenditure falls in the period 9 and graduallyconverges to the original stationary equilibrium afterwards.

13. Appendix D: Sensitivity Check
13.1. Cobb-Douglas Utility Function
As discussed in the section 4, the Cobb-Douglas utility function is the most commonly used.
Thus, it might be useful to show what if the Cobb-Douglas function is adopted.
Figure 21 shows the life-cycle profiles with the Cobb-Douglas utility function. There are threenoteworthy differences from the Stone-Geary case. First, the durable expenditure profile with
the Cobb-Douglas utility function is almost decreasing in age. This is because the new bornhouseholds do not hold the durable stocks, have infinitely high marginal utility of durables, andthus purchase large amount of durables. Second, the consumption share between non-durables
and durable stocks is almost stable after around j ≥ 40. The constant share is directly coming
51

Figure 21: Life-cycle Profiles with Cobb-Douglas Utility

Note: There are three noticeable differences from the Stone-Geary utility. First, the durable
expenditure profile with the Cobb-Douglas utility becomes almost decreasing in age. Second, the share between non-durable c and durable stock d becomes constant once theborrowing constraint becomes slack for most of households (above j ≥ 40). Third, thedying households j = 80 hold plenty of durable stock on average.

52

Figure 22: Dynamics when q = 0

Note: The dashed line shows the results of the baseline experiment with a parameter valueq = 0. The baseline is when q = 1.

from the Cobb-Douglas nature u(c, d) =( c θ d 1− θ )1− σ
where θ governs the constant share as long as
1− σ

the borrowing constraint is not binding. Third, the very old households at age of 80 also own
the positive amount of durable stocks with the Cobb-Douglas utility since Cobb-Douglas utility
function satisfies Inada condition (limd→0 ud (c, d) = ∞).
Other change due to the Cobb-Douglas can be found over the dynamics. The Cobb-Douglas
utility function, in general, generates less elastic dynamic responses in the aggregate durable
expenditure since the Stone-Geary assumes the durable goods are luxury while the Cobb-Douglas
does not. For example, the interest rate elasticity of the aggregate durable expenditure with theStone-Geary is 4.1 while that of Cobb-Douglas is 0.6.

13.2. Choice of q
I set q = 1 in the benchmark so as to maintain the model as much simple as possible. But
this parameter choice of q may look ad-hoc. This section confirms that the choice of q have little
influence on the results.
Figure 22 compares the baseline case q = 1 and the other extreme calibration q = 0. This
comparison confirms that the choice of q has little influence on the dynamics. Thus, I conclude
that the choice of q is almost irrelevant for dynamic analysis.37

37 I find the choice of q affects the share between the durables d and the assets a. The higher the q is, the durable
play a role as a saving measure. Hence, the higher q leads to higher d and lower a.

53