Discussion Papers In Economics
And Business

Dynamic effects of anticipated and temporary tax
changes in a R&D-Based growth model
Kizuku Takao
Discussion Paper 14-10

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDynamic effects of anticipated and temporary tax
changes in a R&D-Based growth model
Kizuku Takao
Discussion Paper 14-10

February 2014

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDynamic eﬀects of anticipated and temporary tax
changes in a R&D-Based growth model∗
Kizuku Takao†

Abstract
Tax changes are often announced before the implementations and are not permanent but only temporary. R&D ﬁrms will optimally adjust their investment decision
to a tax schedule accordingly. This paper analyzes how anticipated and temporary
tax changes dynamically aﬀect the innovation activities. For the purpose, we consider adjustment costs for the investment process and allow ﬁrms to make a forward
looking investment decision in the framework of an R&D-based endogenous growthmodel. Calibrating the model with U.S. data, we obtain new insights on how todesign the corporate taxation policy. A dividend tax cut is not an eﬀective policy
instrument irrespective of how it is implemented. On the other hand, a capital gains
tax cut and a rise of the R&D tax credit rate are an eﬀective policy instrument irrespective of how they are implemented. However, the implementation lags of thesetax changes worsen the eﬀectiveness of them.
Keywords: Fiscal policy, R&D, Economic growth
JEL classiﬁcation: E62, O32, O41∗

I would like to express my sincere gratitude to Koichi Futagami, Been-Lon Chen, and Ching-Chong
Lai for the beneﬁcial discussions and suggestions. I would also like to thank Juin-Jen Chang, Takeo Hori,
Tatsuro Iwaisako, Yukihiro Nishimura, Katsunori Yamada and the participants at the 6th Conferenceof Macroeconomics for Young Professionals, the 2012 Spring Annual Meeting of the Japanese Economic
Association at the University of Hokkaido, and the IEAS seminar at the Academia Sinica. This paper was
written while visiting the Institute of Economics, Academia Sinica. The ﬁnancial support from Strategic
Young Researcher Overseas Visits Program for Accelerating Brain Circulation (Japan Society for the
Promotion of Science) is gratefully acknowledged. All remaining errors are mine.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail address: kge006tk@mail2.econ.osaka-u.ac.jp1

1

Introduction

Technological progress through commercial R&D activities is a major source of economic
growth. Firms make an R&D investment decision, considering the cost and the beneﬁt of
R&D. Those values are aﬀected by diﬀerent taxation policies. As Hall and Van Reenen(2000) surveys, ﬁscal incentives for R&D investments diﬀer across countries and change overtime. The purpose of this paper is to provide the clear policy implications of tax changes
by using an R&D-based endogenous growth model. In our model economy, technologicalprogress is driven from in-house R&D performed by long-lived value-maximizing ﬁrms. The
novel feature of our study is that we consider how ﬁrms make a forward-looking investment
decision of in-house R&D in reaction to future or temporary tax changes. In the real world,tax changes are usually announced before the implementations and are not permanent
but only temporary. In these situations, ﬁrms and households have an opportunity toadjust their intertemporal behavior to a tax schedule. For better understanding of taxation
policy, it is important to consider what diﬀerences of the policy eﬀects of tax changes arise
depending on how the tax changes are implemented.
Calibrating the model with U.S. data, we obtain the following main ﬁndings. First, weshow that an anticipated dividend tax cut stimulates in-house R&D and aggregate growthduring the announcement phase. After the implementation, the tax cut is detrimental to inhouse R&D and aggregate growth. The welfare eﬀect of the tax cut are negative irrespectiveof how it is implemented. However, the implementation lags diminish the welfare looses.
It allow ﬁrms to adjust the timing of dividend payments to the tax schedule. The future
dividend tax cut increases the future cost of in-house R&D. Hence ﬁrms have an incentive toincrease their current in-house R&D investments in order to raise the subsequent dividend
payments. Besides, households can adjust more smoothly to the tax schedule. Second,
we show that an anticipated capital gains tax cut and a rise of the R&D tax credit rate
have a negative impact on in-house R&D and aggregate growth during the announcement
phase. After the implementation, these tax changes stimulate in-house R&D and aggregate
growth. The welfare eﬀect of these tax changes is both positive irrespective of how they are
implemented. However, the implementation lags reduce the welfare gains. Future these tax

2changes reduce the future cost of in-house R&D. Hence ﬁrms have an incentive to reduce
their current in-house R&D investments and raises their current dividend payments. The
negative announcement eﬀect on growth exceeds the positive eﬀect derived from moresmooth path of the consumption and the leisure time of households.
To summarize our results, we suggest the following policy implications. A dividend tax
cut is not eﬀective policy instrument irrespective of how it is implemented. On the other
hand, a capital gains tax cut and a rise of the R&D tax credit rate are an eﬀective policy
instrument irrespective of how they are implemented. However, the implementation lagsof these tax changes worsen the eﬀectiveness of them.
Our analyses are based on that of Peretto (2007, 2011). Speciﬁcally, the model of
Peretto (2007, 2011) considers the economy where long-lived value-maximizing ﬁrms continuously improve the quality of their own product through in-house R&D, while at thesame time new ﬁrms also enter into the market. The advantage of the model of Peretto(2007, 2011) is to eliminate the well-known undesirable scale eﬀect property while keeping
the policy eﬀect property supported by recent growing empirical literatures.1 Increases
in the scale of the aggregate economy are perfectly fragmented by endogenous productproliferation. In-house R&D investments are only related to an average ﬁrm-level scale.
As a result, the channel through the undesirable scale eﬀect is removed. Only the channel
through the ﬁnancial market remains, which yields non-negligible eﬀect of ﬁscal policy on
the long-run growth.2
However, in the model of Peretto (2007, 2011), ﬁrm’s investment decision of in-house1

The ﬁrst generation R&D-based endogenous growth model (e.g. Romer(1990) and Grossman andHelpman (1991)) indicates that the equilibrium growth rate is increasing in the labor endowment. However,Jones (1995a) refutes this property by the time-series data of the post-war period. And then, the following
two prominent types of model are developed. The former type is referred to as the semi-endogenous growth
type (e.g. Jones (1995b) and Segerstrom (1998)). They resolves the undesirable scale eﬀect property byassuming the diminishing returns in R&D production technologies. This speciﬁcation yields that thesteady state growth rate is only pinned down to population growth rate. By contrast, the latter typeis referred as the fully-endogenous type (e.g. Peretto (1998), Howitt (1999) and Futagami and Ohkusa(2003)). They assumes that both of vertical innovation and horizontal innovation occur. This speciﬁcation
yields that the steady state growth rate is also dependent of the other parameters and policy variables.
Recent growing empirical literatures (e.g. Laincz and Peretto (2006), Ha and Howitt (2007), and Ang andMadsen (2011)) report that the latter type performs well rather than the former type.
2
There is no consensus about whether the long-run growth responds to taxation policy or not. However,recent empirical studies (e.g. Romer and Romer (2010) and Mountford and Uhilg (2009)) support thatthe macroeconomic eﬀects of tax changes are much higher than the conventional thoughts.

3

R&D turns out to be static problem. The reason is that the model assumes that the
production function of in-house R&D is linear. This assumption implies that current inhouse R&D investments are only dependent of the current market condition and tax rates.
As a result, if we consider anticipated and temporary tax changes in the setting, we fail tocapture how ﬁrm’s investment decision of in-house R&D dynamically reacts to anticipated
and temporary tax changes. To examine the macroeconomic impacts of such tax shocks, we
also incorporate the framework of adjustment costs of investment used in the literaturesof investment theory.3 More speciﬁcally, we assume that ﬁrms confront on the convex
adjustment costs associated with in-house R&D investments. This speciﬁcation is indeed
more realistic. Some empirical literature points out the existence of high adjustment costs
for R&D investments (e.g. Nadiri (1989), Himmelberg and Petersen (1994) and Brown
and Petersen (2011)). In the presence of the adjustment cost, ﬁrm’s investment decisionof in-house R&D alters to a forward-looking setting. Therefore, the dynamic system of
the economy is characterized also by the (tax-adjusted) shadow value of in-house R&D,
which determines the level of in-house R&D investments.4 The shadow value summarizes
all informations relevant to the investment decision of in-house R&D. The ﬂexibility ofthe shadow value is very useful to analyze how the investment decision of in-house R&D
dynamically reacts to anticipated and temporary tax changes.
Peretto (2007, 2011) have already examined eﬀects of various tax changes related tocorporate activities. In particular, Peretto (2007, 2011) mainly focus on eﬀects of a dividend
tax cut like that conducted in the Jobs Growth and Taxpayer Relief Reconciliation Act
of 2003 (JGTRRA) in the U.S..5 To be more detail, Peretto (2007) analyzes the revenueneutral tax changes in the environment where the rate of the dividend tax is endogenously
determined to ﬁnance the tax changes and balance the budget constraint of the government.
The analyses suggest that policy makers should conduct the other taxation policy (such as
3See, for example, Hayashi (1982), Abel (1982), and Abel and Blanchard (1983)
In Peretto (2007), the dynamic system of the economy is characterized by only one state variable(the number of ﬁrms per capita). In Peretto (2011), it is characterized by one state variable (the numberof ﬁrms per capita) and one jump variable (the consumption ratio). On the other hand, the dynamic
system of the model in our analysis is characterized by not only the number of ﬁrms per capita and theconsumption ratio but also one additional jump variable (the shadow value of the innovation).
5
To be precise, this tax reform substantially reduced the rate of the dividend tax with the moderate
cut of capital gains tax.
44

a capital gains tax cut and a rise of the R&D tax credits) rather than a dividend tax cut.
Peretto (2011) extends to the case where the government ﬁnances tax changes with debt.
Calibrating the model with U.S. data, Peretto (2011) quantitatively shows that a dividend
tax cut produces the slowdown of aggregate growth and considerable welfare losses.
The diﬀerences between our paper and Peretto (2007, 2011) are as follows. Peretto(2007, 2011) only focus on the macroeconomic impacts of unanticipated and permanent
tax changes. On the other hand, we consider eﬀects of anticipated and temporary tax
changes in the environment where ﬁrms dynamically determine the level of their R&D
investments. Besides, we examine the eﬀectiveness of the alternative policy instruments
rather than that of a divided tax cut in the environment where the government ﬁnances thetax changes with debt just like Peretto (2011). As Peretto (2011) claims, such a ﬁnancing
scheme is more appropriate to consider policy experiments of the taxation policy.
Our paper is also related to the following previous studies. Zeng and Zhang (2002) and
Peretto (2003) also study eﬀects of tax changes on the basis of non-scale R&D-based growthmodel. However, both of the paper analyze only unanticipated and permanent tax changes
and do not consider transitional dynamics and welfare implications. Summers (1981) and
Abel (1982) analyze how anticipated and temporary tax changes aﬀect a forward-looking
investment decision of ﬁrms by using the framework of the adjustment costs for investments.
However, their analysis are based on the partial equilibrium approach. As a result, theiranalysis can not consider eﬀects on aggregate growth and welfare. Strulik and Trimborn(2010) studies eﬀects of anticipated and temporary tax changes in the general equilibrium
setting. Their model is based on the neoclassical growth model with endogenous corporateﬁnance and then the steady state growth rate is exogenous in this setting.
The rest of the paper is organized as follows. Section 2 describes the model. Section 3
characterizes the dynamic system and the steady state equilibrium of the market economy.
Section 4 quantitatively analyzes the transitional adjustment of the aggregate economy
to tax changes, calibrating the model with U.S. data. Section 5 conducts the sensitivity
analysis of the numerical analysis. Finally, Section 6 discusses the policy implications andstates the conclusion remarks.

5

2

The model

In this section, we set up the model based on Peretto (2011). Except for the presenceof adjustment costs associated with in-house R&D investment for quality improving, theenvironment is the same as Peretto (2011). Time is continuous. The economy is closed
and consists of the ﬁnal goods sector, the intermediate goods sector, households, and thegovernment. Long-lived value-maximizing ﬁrms produce their speciﬁc intermediate goodand continuously improve the quality of their own product through in-house R&D, whileat the same time new ﬁrms also enter into the market. All ﬁscal variables are treatedparametrically since they change either only at discrete events or not at all. And so weomit the time index, t, on ﬁscal variables.

2.1

The ﬁnal goods sector

The ﬁnal goods, Yt , is produced by the following production function:
∫ Nt
Yt =

)1−θ
(
di, 0 < α, θ < 1,
Xitθ Zitα Z̄t1−α Lit(1)0where Nt , Xit , and Lit respectively represent the mass of intermediate goods, the input ofintermediate good i (produced by ﬁrm i), and the input of labor who use the intermediate
good i. The productivity of Lit depends not only on the quality of the intermediate good∫Ni, Zit , but also on the average level of quality across intermediate goods, Z̄t ≡ 0 t N1t Zjt dj.
Final goods are consumed by households and the government and used only as one factor
of all activities in the intermediate goods sector. Perfect competition prevails in the ﬁnal
goods sector. The price of ﬁnal goods is normalized to one. Therefore, we obtain thefollowing optimal conditions:
(
Xit =
(
Lit =

θ
Pit

1
) 1−θ

1−θ
Wt

) θ1

)
Zitα Z̄t1−α Lit ,

(2)

(
) 1−θ
Xit Zitα Z̄t1−α θ ,

(3)

(

6

where Pit and Wt respectively represent the price of the intermediate good i and the wage
rate of labor.

2.2

The intermediate goods sector

Monopolistic competition prevails in the intermediate goods sector. There is the continuumof goods indexed by type i ∈ [0, Nt ]. The ﬁrm i exclusively produces its diﬀerentiated goodwith its quality Zit . Monopoly of each ﬁrm is permanently protected by the perfect patent
protection. Producing one unit of intermediate goods requires one unit of ﬁnal goods. And
ﬁxed operating costs, ϕZ̄t (ϕ > 0), are required at each point in time.
Each ﬁrm improves its product quality through in-house R&D. The law of motion of
the ﬁrm-speciﬁc quality isŻit = Rit .

(4)

In contrast to Peretto (2007, 2011), we assume that given increases of the ﬁrm-speciﬁc
quality level, Rit ≥ 0, involve adjustment costs associated with the innovation, following
the speciﬁcation of Hayashi (1982). Speciﬁcally, we assume that the total amount of R&D
expenditure is given byΦ(Rit , Zit ) = Rit +h Rit2, h > 0,
2 Zit

(5)

where the case of h = 0 corresponds to the speciﬁcation of Peretto (2007, 2011).6
The gross cash ﬂow of ﬁrm i is Fit = Xit (Pit − 1) − ϕZ̄t , where the ﬁrst term represents
revenue minus variable production costs and the second term represents ﬁxed operating
costs. Let σ be the rate of the R&D tax credit (the fraction of R&D expenditure whichﬁrms are allowed to subtract from their corporate taxable amount).7 The total amount ofcorporate tax imposing on ﬁrm i is τΠ [Fit − σΦ(Rit , Zit )], where τΠ represents the rate of
the corporate tax. The gross cash ﬂow, Fit , distributes as follows:
Fit = τΠ [Fit − σΦ(Rit , Zit )] + Eit dit + Jit ,6This functional form is based on Turnovsky (2000).
Although σ is assumed to be zero for simpliﬁcation in Peretto (2011), we follow the speciﬁcation of
Peretto (2007) in order to see the eﬀects of the tax credit policy for R&D investment as well.
77

where Eit , dit , and Jit respectively represent the number of equities authorized by ﬁrm i,
the pre-tax dividends per share of ﬁrm i, and retained earnings of ﬁrm i. The ﬁrst term of
the RHS of the above identity represents the amount of corporate tax, the second terms
represents the amount of dividends delivered to equity shareholders, and the last term
represents retained earnings. Financial constraint of ﬁrm i is written by Jit + Ėit vit =Φ(Rit , Zit ), where Ėit and vit respectively represent the number of newly issued equity ofﬁrm i and the market value per share of ﬁrm i. Since here we do not consider ﬁnancing bybond issue, the above identity indicates that in-house R&D investment must be ﬁnanced
by retaining earnings, newly issued equities, or both.8 Along the lines of Peretto (2011),
we focus on the scenario where the marginal source of in-house R&D is only limited toretaining earnings. This scenario is called as “New view” in the literature of corporateﬁnance. In this scenario, Φ(Rit , Zit ) = Jit because of Ėit = 0.
Let Vit ≡ Eit vit and Dit ≡ Eit dit . Without loss of generality, Eit is normalized to one.
Dividends of ﬁrm i is given by
Dit = (1 − τΠ )Fit − (1 − στΠ )Φ(Rit , Zit ).

(6)

The return on equity is rewritten by
rt = (1 − τD )

V̇it
Dit
+ (1 − τV ) .
Vit
Vit

(7)

where τD is the rate of the dividend tax and τV is the rate of the capital gains tax.
Integrating (7) forward yields the value of ﬁrm i as follows:
(∫ s

∫ ∞
Vit =

exp
t

t

1
rv dv
−
1 − τV)(1 − τD1 − τV)[(1 − τΠ )Fis − (1 − στΠ )Φ(Ris , Zis )] ds.

We consider a symmetric equilibrium by assuming that any new ﬁrm starts with thesame level of technologies as incumbents so that the subscript i can be dropped. In theequilibrium, Zt = Z̄t holds. Each ﬁrm maximizes its value, subject to (2) and (4), given Z̄.
To solve the inter-temporal maximization problem, we deﬁne the following current-value
8

See Turnovsky (1990) for the detailed discussion.

8

Hamiltonian asH≡

1 − τD[(1 − τΠ )Ft − (1 − στΠ )Φ(Rt , Zi )] + qt [Rt ] ,1 − τV

where the co-state variable, qt , represents a shadow value of in-house R&D. We obtain thefollowing optimal conditions:1
Pt = ,
θ[q̃ ≡ Sqt = 1 + h

(8)

]

Rt
,
Zt

(9)

(1 − τV )h(1 − τV ) ∂Ft 1+
rt =Sη ∂Zt qt
2S

(

Rt
Zt

)2

q̇t
1
+ (1 − τV ) .
qt
qt

(10)

where for simplifying notation, we deﬁneS≡(1 − τV )1 − στΠand η ≡.
(1 − τD )(1 − στΠ )1 − τΠ

()∫s
1
The transversality condition is lims→∞ exp − 1−τ
rdv
Zs qs = 0. From (9), the ratevtVof quality growth is given by1Żt  (q̃t − 1) , if q̃t > 1,
ẑt ≡
= h
Zt 

0, if q̃ ≤ 1.

(11)

t

(8) represents the pricing rule with constant mark-up. (9) indicates that ﬁrms undertake
in-house R&D investment up to the point where a tax-adjusted shadow value of in-house
R&D (the RHS) is equal to the cost of the innovation (the LHS). Hereafter, q̃t is called as
modiﬁed q along the lines of Hayashi (1982). Where there is no adjustment costs (h = 0),
modiﬁed q always pins down to one.9 By contrast, in our setting, modiﬁed q is endogenously
determined and it has a transitional process in equilibrium. (10) represents the no-arbitrage
condition of return on in-house R&D. (11) shows that the rate of quality growth is thefunction of only modiﬁed q. Since modiﬁed q is derived from the intertemporal optimization
9

See Peretto (2007, 2011).

9

problem of ﬁrms, all informations relevant to in-house R&D decision are summarized bymodiﬁed q.
Development of new product requires βZt (β > 1). New entry ﬁrms ﬁnance by issuingequity. Free-entry condition yields
Vt = βZt ⇔ Ṅt > 0.

(12)

From (6) and (12), the return on equity, (7), is rewritten by]Ft
Φ(Rt , Zt )Żt
− (1 − στΠ )+ (1 − τV ) .
rt = (1 − τD ) (1 − τΠ )
βZt
βZt
Zt
[

2.3

(13)

Households

The economy has identical households who provide labor supply elastically and purchaseassets. The labor market and the asset market are supposed to be competitive. Each individual member of households is identically endowed with one unit of time. The population
exogenously grows over time at the constant rate, λ > 0. Without loss of generality, thenumber of population at time 0 is normalized to one. Hence, the number of population attime t is eλt . The household maximizes the following utility function:
∫ ∞
Ut =

[
]
e−(ρ−λ)(s−t) log Cs e−λs + ζ log (1 − ls ) ds,t

where Ct , lt , ζ > 0, and ρ (> λ) respectively represent the aggregate consumption, thefraction of the time allocated to work per capita, the measure of preference for leisure, andthe rate of the time preference. The budget constraint of the household is given by][Ṅt Vt = Nt (1 − τD )Dt − τV V̇t + (1 − τL )Wt lt eλt − (1 + τC )Ct − Tt ,where τL , τC , and Tt respectively represent the rate of the labor income tax, the rate of theconsumption tax, and the lump-sum tax. Solving the inter-temporal optimization problem

10

yields the following optimal conditions:Ċt= rt − ρ + λ,Ct
lt = 1 −(1 + τC )ζCt.
(1 − τL )Wt eλt

(14)

(15)

The transversality condition is lims→∞ e−(ρ−λ)(s−t) as µs = 0, where µt represents a shadow
value of holdings of assets.

2.4

The government

The government spending is given by Gt = gYt (0 < g < 1). The share of the government
spending to output is assumed to be an exogenously given rate. Along the lines of Peretto(2007, 2011), it is assumed that the government spending does not aﬀect the utility ofhouseholds and the eﬃciency of production activities in order to isolate the eﬀects ofdistortionary taxes from the eﬀects of the government expenditure. The budget constraintof the government is given by
Gt = τL Wt Nt Lt + τC Ct + τΠ Nt [Ft − σΦ(Zt , Rt )] + τD Nt Dt + τV Nt V̇t + Tt .
Since the Ricardian equivalence holds, we can see the same equilibrium dynamics as the
economy with public debt.

3

The market equilibrium

3.1

The equilibrium dynamics

In this section, we derive the dynamic system of the market equilibrium. The market
equilibrium condition of ﬁnal goods is given by
Yt = Gt + Ct + Nt [Xt + ϕZt + Φ(Zt , Rt )] + βZt Ṅt .

11

(16)

Deﬁne the number of ﬁrms per capita as nt ≡ Nt /eλt and the ratio of the aggregate
consumption to outputs as ct ≡ Ct /Yt . In Appendix 1, we show that the labor supply percapita is given byl(ct ) =1,
1 + ΓctΓ≡(1 + τC )ζ
> 0.
(1 − τL )(1 − θ)(17)

The reduced-form aggregate production function of ﬁnal goods is given by
Yt = Ωl(ct )eλt Zt ,2θΩ ≡ θ 1−θ(18)

In Appendix 2, we show the following simultaneous diﬀerential equation constitutes the
dynamical system of the economy (in the case where q̃t > 1):[][] Ωl(ct )(Sqt )2 − 1nt
2ṅt = 1 − θ − g − ct
− ϕ+
+ βλ
,
β
2h
β
[
]
Sqt − 1ċt = ct [1 + Γct ] rt − ρ −,h1αθ(1 − θ) Ωl(ct ) (Sqt − 1)2q̇t =r t qt −−,1 − τV
Sη
nt
2Sh

(19)
(20)
(21)

where from (13), the interest rate is given by[](1 − τV )Ωl(ct )
(Sqt )2 − 1
Sqt − 1rt =
θ(1 − θ)−ϕ−η
+ (1 − τV )
.
βSη
nt
2h
h

(22)

See Appendix 3 for the dynamic system in the case where q̃t ≤ 1.

3.2

The steady state equilibrium

Let yt ≡ Yt /lt eλt , which represents outputs per worker. From (18), the growth rate ofoutputs per worker is given by ŷt ≡ ẏt /yt = ẑt = (q̃t −1)/h. In what follows, we characterize
the steady state equilibrium, {n∗ , c∗ , q̃ ∗ , l∗ , r∗ , ŷ ∗ }. From (20), ċt = 0 and c∗ > 0 implies (ifq̃ ∗ > 1)q̃ ∗ − 1
r =ρ+
.
h
∗

12

(23)

From (21), q̇t = 0 and q̃ ∗ > 1 impliesr∗ = (1 − τV )αθ(1 − θ) Ωl(c∗ ) 1(q̃ ∗ − 1)2 1
+
(1
−
τ
)
.
V
η
n∗ q̃ ∗
2h
q̃ ∗

(24)

This equation represents the no-arbitrage condition of return on in-house R&D in thesteady state equilibrium. It turns out that a dividend tax cut does not directly aﬀect
incentives to in-house R&D. On the other hand, a corporate tax cut, a capital gains tax
cut, and a rise in the rate of the tax credit directly enhances incentives to in-house R&D.
From (23) and (24), eliminating r∗ yields (if q̃ ∗ > 1)ηΩl(c∗ )=∗nαθ(1 − θ){[]}q̃ ∗ − 1 ∗ (q̃ ∗ − 1)2
1
ρ+
q̃ −
.
1 − τVh2h(25)

Substituting (23) and (25) into (22), we ﬁnd that q̃ ∗ is derived from solving f (q̃) = 0 withrespect to q̃ where




1
(q̃ − 1)2q̃ − 1q̃ 2 − 1





(Sαβ
−
q̃)
+
ρ
+
+
α



1 − τV
h
2h
2h




q̃ − 1 αϕf (q̃) ≡−Sαβ+, if q̃ > 1,


h
η





αϕ
ρ



(Sαβ − q̃) +, if q̃ ≤ 1.
1 − τVη(26)

If Sαβ ≤ 1 − (1−τηρV )αϕ < 1, f (1) < 0 and f ′ (q̃) < 0. In the case, there is no steady state
equilibrium with a positive rate of quality growth. If 1 − (1−τηρV )αϕ < Sαβ, f (1) > 0. In thecase, f (q̃) is depicted as shown by Figure 1. Figure 1 shows that if 1 − (1−τηρV )αϕ < Sαβ, q̃ ∗is uniquely determined at the point where q̃ ∗ is higher than 1. In what follows, we focus
on the case where 1 − (1−τηρV )αϕ < Sαβ. In the case, there exists an unique steady state
equilibrium with a positive rate of quality growth. See Appendix 2 for the proof.
Since the Jacobian matrix derived from the linear approximation of (19)-(21) in theneighborhood of the steady state equilibrium is too complicated, we cannot analyticallyexamine the local stability of the dynamic system. However, our numerical simulations
conﬁrm that the unique steady state is locally saddle-point stable in the benchmark setting13

f (q̃)

f (1)

0

f (1)

q̃ ∗

1

q̃

0

1

q̃ ∗

q̃

Figure 1: The steady state equilibrium: the left (right) ﬁgure represents the case where f (q̃) isinverted U-shapes (monotonically decreasing in q̃) for q̃ > 1.

and the subsequent sensitivity analysis as we will see below.10
From (19) and (25), ṅt = 0 and n∗ > 0 implies[][] αθ(1 − θ)q̃ ∗2 − 12
c = 1−θ −g −ϕ++ βλ ,φ(q̃ ∗ )
2h
∗

where

[
]
q̃ ∗ − 1 ∗(q̃ ∗ − 1)2ηρ+q̃ + η
.
φ(q̃ ) ≡1 − τVh
2h∗

Rewriting (25) yieldsn∗ =αθ(1 − θ)Ωl(c∗ ).
φ(q̃ ∗ )The mechanism eliminating the scale eﬀect on the steady state growth rate of outputs
is consistent with the case where there is no adjustment cost (see Peretto (2007, 2011)). In
the steady state equilibrium, modiﬁed q is independent of the scale factor of the economy,l(c∗ ) (see (26)). An increase in production volumes allows in-house R&D expenditure to10

Since the dynamic system has one state variable (nt ) and two jump variables (ct and qt ), it must have
two positive characteristic roots and one negative characteristic root. Our numerical simulation reportsthat the value of three characteristic root corresponding to the dynamic system are −0.4129, 0.2240, and
0.1478 in the benchmark parameter setting.

14

be spread over more units of goods. Therefore, given the number of ﬁrms per capita,
more aggregate demand for intermediate goods has a direct positive eﬀect on incentives toin-house R&D. This eﬀect is called as the cost-spreading eﬀect. However, more aggregate
demand also attracts entry of a new ﬁrm as the value of a ﬁrm rises. As a result, the market
share per ﬁrms shrinks. This results in the lower scale of production of intermediate goods
at the level of an individual ﬁrm, which lowers incentives to in-house R&D. This eﬀect iscalled as the market share eﬀect. In the steady state equilibrium, the market share eﬀect
perfectly cancels out the cost-spreading eﬀect (see the discussion in Peretto (2007)).11

3.3

The steady state eﬀect of tax changes

How a permanent change of tax variables aﬀects the steady state growth rate is also
consistent with Peretto (2007, 2011). We can summarizes as follows:
The steady state growth rate of outputs is increasing in the dividend tax rate, the corporate
tax rate (if σ ≥ 1), and the R&D tax credit rate. On the other hand, it is decreasing in thecorporate tax rate (if σ = 0). The eﬀect of changes in the corporate tax rate (if σ ∈ (0, 1))
and the capital gains tax rate on the long-run growth is ambiguous.
See Appendix 4 for the proof. As previously discussed, a dividend tax cut does notdirectly aﬀect the return on in-house R&D. On the other hand, a dividend tax cut directly increases the return on equity. Given the aggregate market demand for intermediate
goods, the number of ﬁrm per capita increases. The production proliferation lowers incentives to in-house R&D through the market share eﬀect. Consequently, a dividend tax cut
unambiguously has a negative eﬀect on the long-run growth.12
11

Furthermore, we conﬁrm that comparative statics of parameters in the steady state equilibrium get
the similar results as Peretto (2007, 2011). Increases in α, β, and ϕ respectively enhance the steady state
growth rate of outputs. Increases in α allow each ﬁrm to internalize the positive return derived by its
own in-house R&D more intensely. Increases in β and ϕ make it harder for potential entrants to go into
the market, reallocating resources from product proliferation to quality improving. On the other hand,the eﬀect of time discount rate, ρ, on growth is ambiguous. Lower household’s incentive to hold equities
due to an increase in the time discount rate reduces the number of ﬁrms per capita, which has positiveeﬀect on incentives to in-house R&D, while the corresponding higher interest rate also has negative eﬀect
on incentives to in-house R&D. Increases in h reduce the steady state growth rate of output because it
directly increases the cost of in-house R&D.
12
If σ = 1, the corporate tax cut also has the same qualitative eﬀect as the dividend tax cut. Whenin-house R&D expenditures are fully deductible against corporate tax, there is no diﬀerence between the15

On the other hand, the higher rate of the R&D tax credit has an unambiguously positiveeﬀect on the long-run growth. It works just like the direct subsidy for in-house R&D.
The growth eﬀect of both a corporate tax cut (if σ ∈ (0, 1)) and a capital gains tax cut
is ambiguous. These tax cut directly increase both the return on in-house R&D and thereturn on equity. However, it is shown that if σ = 0, a corporate tax cut unambiguously
enhances the long-run growth. And if β and α is suﬃciently low, a capital gains tax cut
also enhances the long-run growth.

4

Numerical analysis

4.1

Data and Methodology

Since it is too complicated to analytically examine the transitional adjustment of the aggregate economy to tax changes, we carry out numerical simulation by using relaxation
algorithm method developed by Trimborn, Koch, and Steger (2008).13 We calibrate themodel with U.S. data. The benchmark value of ﬁscal variables and parameters is summarized in Table 1 and Table 2, which hereafter we call as the benchmark parametersetting.
As the benchmark, we choose the value of all tax variables along the lines of Peretto(2011).14 θ and ρ is respectively set to 0.30 and 0.04, which is conventional value in themacroeconomic literature. λ is set to 0.01, which is consistent with the average annual
population growth rate in the U.S. economy. The choice of the parameter associatedwith adjustment costs, h, is less clear. According to Schubert and Turnovsky (2011), theparameter of adjustment costs for physical capital investment is generally assumed within
dividend tax and the corporate tax.
13
Trimborn, Koch, and Steger (2008) details the relaxation algorithm. They also provide MATLAB programs for the relaxation algorithm, which are downloadable for free at http://www.wiwi.unisiegen.de/vwli/forschung/relaxation/matlab applications.html?lang=de. By using this method, Strulik
and Trimborn (2010) examines how anticipated and temporary tax reforms aﬀect on the aggregate economy in the framework of the neoclassical (exogenous) growth model with endogenous corporate ﬁnance.
14
R&D costs is in fact fully deductible against the corporate tax liability in the U.S. tax code. However,the setting of σ = 0 allows us to clearly see the fundamental distinction between the corporate tax andthe dividend tax. If R&D costs is assumed to be fully deductible (σ = 1.0), a corporate tax cut has thesame qualitative eﬀects on the economy as a dividend tax cut.

16

10-15 in the literature (e.g., Ortigueira and Santos (1997) and Auerbach and Kotlikoﬀ(1987)). Nadiri (1989) and Himmelberg and Petersen (1994) report that the extent towhich adjustment costs for R&D investment is the same or more than that of physicalcapital investment. And so we set h = 12.0 as the benchmark. The parameter associatedwith entry costs, β, is also less clear. Following Peretto (2011), we set β = 6.55 as thebenchmark.15 α and ϕ is respectively set to 0.141 and ϕ = 0.266 so that the steady state
consumption ratio and the steady state growth rate of outputs respectively is 0.69 and 0.02.
ζ is set to 1.459 so that the fraction of time devoted to labor supply is 0.33. Table 3 reports
the value of the steady state equilibrium, {n∗ , c∗ , q̃ ∗ , l∗ , r∗ , ŷ ∗ }, which is characterized underthe benchmark parameter setting.
Table 1: Tax variables (benchmark)
g

τD

τΠ

τV

σ

τC

τL

0.143

0.35

0.335

0.20

0

0.05

0.256

Table 2: Parameters (benchmark)
α

θ

h

ϕ

β

ζ

ρ

λ

0.141

0.30

12.0

0.266

6.55

1.459

0.04

0.01

Table 3: Steady state equilibrium values (benchmark)
n∗

c∗

q̃ ∗

l∗

r∗

ŷ ∗

0.0256

0.69

1.24

0.33

0.06

0.02

In what follows, we investigate what the transitional adjustment of key macro variables
and welfare consequences are induced by the following speciﬁc tax changes: 10 percentage
point reduction of the dividend tax rate, the corporate tax rate, and the capital gains tax
rate, and 20 percentage point rise of the tax credit rate. Besides, with respect to howeach tax change comes into eﬀect, we consider the following three diﬀerent implementation
scenarios: (1) an unanticipated and permanent change, (2) an anticipated and permanentchange, and (3) an unanticipated and temporary change. In the every scenario, the economy initially (at t = 0) stays in the steady state equilibrium before the tax change. In the15

See Peretto (2011) for the detailed explanation for this estimation.

17

implementation scenario of (1), each tax change suddenly hits at t = 5 and lasts forever
from then on. In the implementation scenario of (2), all economic agent expect at t = 0
that each tax change implements at t = 5 and last forever from then on. In the implementation scenario of (3), each tax change hits unexpectedly at t = 0 and reverts to the initiallevel after t = 10. The implementation period of this reversion is expected by all economic
agents at t = 0.
Figure 2-5 show the transitional path of key macro variables in response to each tax
change with the diﬀerent implementation scenarios as mentioned above in the benchmark
parameter setting. Speciﬁcally, each panel of these ﬁgures respectively represent the transitional path of the number of ﬁrms per capita (nt ), the consumption ratio (ct ), modiﬁed q(q̃t ), hours worked per capita (lt ), the market interest rate (rt ), the growth rate of outputs
per worker (ŷt ), the ratio of the after-tax dividends to the value of a ﬁrm, and the ratio ofthe distortionary tax revenue to GDP. The horizontal axis in each panel measures years. In
the vertical axis, rt and ŷt are measured by the actual value and all the other variables aremeasured by the percentage deviation from the level of the initial steady state equilibrium
before the tax change.
Table 4 reports the welfare consequences of the tax changes. The welfare evaluation ismeasured as consumption equivalent: how much constant relative increases in the annual
consumption per capita must be required so that the intertemporal utility of households
in the case when the economy remains in the initial steady state equilibrium before thetax changes becomes equal to that in the case when the economy moves to the new steadystate equilibrium from the initial due to the tax changes.16

4.2

The dividend tax cut

Figure 1-(a) shows the transitional path of the key macro variables in response to thepermanent dividend tax cut by 10 percentage points comparing the unanticipated and
16

More formally, the welfare evaluation is conducted as follows. U0O (cO , lO , ŷ O , nO ) is deﬁned as thelevel of the intertemporal utility of households in the case when the economy remains in the initial steadyNN
N
state equilibrium before a tax change. And U0N (cNt , lt , ŷt , nt ) is deﬁned as that in the case when theeconomy moves to the new steady state equilibrium from the initial due to a tax change. Here, wemeasure consumption equivalent by δ which is deﬁned as the value to satisfy U0O (cO (1 + δ), lO , ŷ O , nO ) =N
N
N
U0N (cNt , lt , ŷt , nt ). See Appendix 5 for how to calculate the value of intertemporal utility of households.

18

Table 4: Welfare gains of tax changes (benchmark)
Policy change

Unanticipated(Permanent)

Anticipated(Permanent)∆tD = −0.1∆tΠ = −0.1
∆tV = −0.1
∆σ = 0.2

−14.0
5.03
13.56
10.86

−11.17
5.67
10.78
9.58

Temporary
−8.19
0.64
7.86
4.87

Note: Welfare gains are measured in consumption equivalent and expressed in percentage points.

anticipated case under the benchmark parameter setting. If the permanent tax cut isunanticipated, the consumption ratio falls by around 10 percentage points when the tax
cut is implemented (that is, at t = 5). And then it gradually rises toward the new steadystate level. Hours worked reacts conversely. This is because the lump-sum tax increases(or the lump-sum transfer decreases) to ﬁnance the tax cut. Actually, the ratio of the
distortionary tax revenue to GDP decreases. The number of ﬁrms per capita starts to rise
at t = 5, while modiﬁed q instantaneously falls. After the implementation of the tax cut,they converge to the new steady state level. The growth rate of outputs per capita falls
to 0.0148 at t = 5 and then it converges to 0.0152. The unanticipated tax cut does notdirectly change ﬁrm’s investment decision, while it directly increases the value of a ﬁrm.
Therefore, it has a negative impact on in-house R&D through the market share eﬀect aspreviously discussed. The interest rate jumps up at t = 5 because the tax cut directly
raises the after-tax dividends. However, the interest rate gradually decreases during thetransition and it eventually falls below the initial steady state level. As Table 4 shows, thistax change yields welfare costs of around 14 percentage points of annual consumption percapita. The negative welfare consequence is derived from not just the lower consumption
and leisure time of households but also the slowdown of economic growth.
The impulse responses become quite diﬀerent in the case where the permanent tax cut
is expected in advance. Households and ﬁrms take into account the future tax cut. Hencethey change their decisions when the news arrives (that is, at t = 0). At t = 0, all variables
rather than the state variable (the number of ﬁrms per capita) instantaneously change. The

19

consumption ratio falls by around 2 percentage points. And it further decreases during theannouncement phase. After the tax cut is implemented (at t = 5), it gradually converges
to the new steady state level. Surprisingly, at t = 0, modiﬁed q jumps up. The growthrate of outputs per worker rises to 0.0250. During the announcement phase, the modiﬁed
q further increases and the growth rate of outputs per worker continues to rise. Afterthe implementation of the tax cut, the modiﬁed q drastically drops to a lower value than
the initial steady state level and the growth rate of outputs per worker falls from 0.02
to 0.015. And then the both values converge to the new steady state level. During theannouncement phase, the number of ﬁrms per capita gradually rises through the generalequilibrium eﬀect.
Why the implementation lag of the tax cut has a positive eﬀect on in-house R&D
investment during the announcement phase? A lower dividend tax rate directly raises thevalue of a ﬁrm given its dividend payments. The future dividend tax cut proportionately
raises the contribution of future gross cash ﬂows to the value of a ﬁrm and the future
cost of in-house R&D. By contrast, it does not change the current cost of in-house R&D.
As a result, ﬁrms adjust the timing of dividend payments by changing their investment
schedule of in-house R&D. That is, ﬁrms increase in-house R&D investment during theannouncement phase in order to raise the subsequent dividend payments.
Although we take into account the announcement eﬀect, the welfare eﬀect remains in
negative. The welfare costs are estimated to be the loss of around 11.17 percentages pointsof annual consumption per capita. However, the welfare costs are reduced compared tothe unanticipated case. This is because the growth rate of outputs per worker temporarilyrises during the announcement phase and the consumption and hours worked adjust moresmoothly.
Figure 2-(b) shows the transitional path in the case of the temporary tax cut. Whenthe tax cut is implemented (that is, at t = 0), all variables rather than the state variableinstantaneously change. After the tax cut is terminated (that is, at t = 10), all variables
reverts to the initial steady state level. Remarkably, during the implementation, modiﬁed
q declines more sharply compared to the case of the permanent tax. At t = 0, the growthrate of outputs per worker falls to 0.0125. And then it further decreases until the tax
20

cut is terminated. This is because ﬁrms have an incentive to alter the timing of dividend
payments. During the implementation, ﬁrms intend to raise their dividend payments.
Hence ﬁrms reduce their R&D investment. Consequently, the temporary dividend tax cut
also yields welfare costs. They are estimated to be around 8.2 percentage points of annual
consumption per capita.

4.3

The corporate tax cut

Figure 3-(a, b) shows impulse responses to the corporate tax cut by 10 percentage points
under the benchmark parameter setting. The transitional path of nt , ct , lt is respectively
similar to that in the case of the dividend tax cut. On the other hand, modiﬁed q adjusts
somewhat diﬀerently. If the tax cut is unanticipated, at t = 5, the modiﬁed q jumps up andthe growth rate of outputs per workers rises to 0.0219. And then they gradually increases
to the new steady state level. If the tax cut is anticipated, at t = 0, modiﬁed q jumps
up and the growth rate of outputs per worker rises to 0.0210. And then they gradually
increases to the new steady state level. If the tax cut is temporary, the growth eﬀect isalways positive during the transition.
Why the tax cut always has a positive eﬀect on growth? Recall that in-house R&D
investment is not deductible against the corporate tax under the benchmark parametersetting. The tax cut increases the contribution of the gross cash ﬂow to the value of a ﬁrm
while it does not change the cost of in-house R&D. Hence the tax cut directly enhances
an incentive to in-house R&D. The tax cut also increases the number of ﬁrms per capita.
However, the negative eﬀect through the product proliferation does not perfectly oﬀset thepositive direct eﬀect on growth. It is shown that welfare eﬀects are positive irrespectiveof how it is implemented. If in-house R&D investment is fully deductible, however, thecorporate tax has the same qualitative eﬀects as the dividend tax.

4.4

The capital gains tax cut

Figure 4-(a) shows impulse responses due to the unanticipated (or anticipated) and permanent capital gains tax cut by 10 percentage points under the benchmark case. Remarkably,
21

the tax cut increases both the number of ﬁrms per capita and the growth rate of economy
in the new steady state. If the permanent tax cut is unanticipated, modiﬁed q initiallyjumps up and then converges to the new steady state level. During the transition, thegrowth rate of per capita is always higher than the initial level. This tax cut yields welfare
gains of around 13.56 percentage points of annual consumption per capita.
If the permanent tax cut is anticipated, however, the growth rate slows down during theannouncement phase. Future capital gains tax cuts reduces the future cost of R&D. Henceﬁrms have an incentive to reduce their current in-house R&D investments and raise theircurrent dividend payments. The temporary slowdowns of growth during the announcement
phase has a negative eﬀect on welfare. Anticipated tax cuts make households behavior
more smoothly, yielding a positive eﬀect on welfare. However, the positive eﬀect cannot
oﬀset the negative eﬀect derived from the temporary slowdown of growth. As a result, theanticipated tax cut reduces welfare gains by 2.77 percentage points.
Figure 4-(b) shows impulse response due to the temporary capital gains tax cut by 10
percentages points under the benchmark case. Remarkably, the growth rate of economy
accelerates during the implementation. The temporal acceleration of growth is sharply
higher than the permanent eﬀect. The temporary tax cut reduces the cost of in-house
R&D during the implementation. Hence ﬁrms increase in-house R&D investment during
the implementation. Consequently, the temporary capital gains tax cut also yields welfare
gains. They are estimated to be around 7.86 percentage points of annual consumption percapita.

4.5

Increases in the rate of the R&D tax credit

Figure 5-(a) shows impulse responses due to the unanticipated (or anticipated) and permanent rise of the R&D tax credit rate by 20 percentage points under the benchmark case.
The tax change increases the steady state growth rate of economy but reduces the steadystate number of ﬁrm per capita. Besides, in the steady state, the tax changes is shown
to be self-ﬁnancing. The ratio of distortionary tax revenues to GDP is higher than theinitial level. Hence the steady state consumption ratio is higher than the initial level. If

22the permanent tax change is unanticipated, during the transition, the growth rate and theconsumption ratio are also higher than the initial revel. As a result, it yields welfare gains
which is estimated to be around 10.86 percentage points of annual consumption per capita.
On the other hand, if the permanent tax changes is anticipated, modiﬁed q decreases
and so the growth rate falls during the announcement phase. The reason is parallel tothat in the case of the capital gains tax cut. Future rises of the R&D tax credit rate
directly reduce the future cost of in-house R&D. As a result, the implementation lag of thetax change reduce the welfare gains by 1.28 percentage points of annual consumption percapita.
Figure 5-(b) shows impulse response due to the temporary rise of the tax credit rate by
20 percentages points under the benchmark case. The growth eﬀect is also parallel to thatin the case of the temporary capital gains tax cut. The temporal acceleration of growthis sharply higher than the permanent eﬀect. The temporary rise of the tax credit rate
reduces the cost of in-house R&D during the implementation. Firms have an incentive
to change the timing of dividend payments. Hence ﬁrms increases their in-house R&D
investment during the implementation. Consequently, the temporary capital gains tax cut
also yields welfare gains. They are estimated to be around 4.87 percentage points of annual
consumption per capita.

5

Sensitivity analysis

5.1

Parameter changes

We conduct the robustness check by changing some parameters. Firstly, we consider increasing or decreasing the value of less clear parameter, h and β. Secondly, we also consider
the case of σ = 1.0 as the real tax code in the U.S. sets to σ = 1.0. In any case, we reestimate α and ϕ so that the consumption ratio and growth rate of output in the steady state
before tax changes keep the same level under the benchmark setting. Only results of thecorporate tax cut in σ = 1.0 qualitatively changes. This is because a corporate tax cut has
the same eﬀects as a divided tax cut if in-house R&D investment is fully deductible. For

23

the other cases, our main ﬁndings in the benchmark analysis qualitatively hold. Besides,
we also consider the case in which labor supply is inelastic (that is, ζ = 0). In this case,our main ﬁndings in the benchmark analysis qualitatively hold. Table 5 reports the welfare
consequences of tax changes in those alternative parameter setting.

5.2

Social returns to product variety

In the model as described thus far, the number of ﬁrms (product variety) per capita does
not directly contribute to the production of ﬁnal goods. It only aﬀects degree of the market
competition among intermediate goods ﬁrms. This indirect channel distorts incentives toin-house R&D. In what follows, we relax this somewhat extreme feature. Along the linesof Peretto (2007, 2011), we consider the case where there exists social positive returns to
product variety of intermediate goods in the production of ﬁnal goods as follows:∫ Nt
Yt = nvt)1−θ
(di, v > 0,
Xitθ Zitα Z̄t1−α Lit

0where the contribution to product variety on output of ﬁnal goods is assumed to be external
to all agents. In this case, the reduce-form production function of ﬁnal goods is rewrittenbyYt = nκt Ωl(ct )eλt Zt , κ ≡

v
.
1−θ

The dynamic system of the economy is modiﬁed as follows:][[] Ωl(ct )nt(Sqt )2 − 1
2ṅt = 1 − θ − g − c+ βλ,−κ − ϕ +
2h
β
βnt
[
]
Sqt − 1ṅtċt = ct [1 + Γct ] rt − ρ −
,
−κ
h
nt
1
αθ(1 − θ) Ωl(ct ) (Sqt − 1)2q̇t =r t qt −−,1 − τV
Sη
2Sh
n1−κ
t
where
[
]
(1 − τV )Ωl(ct )
(Sqt )2 − 1
Sqt − 1rt =
θ(1 − θ) 1−κ − ϕ − η+ (1 − τV )
.
βSη
2h
h
nt

24

The growth rate of output per worker is given by q̃th−1 + κ ṅntt . In the steady state, thenumber of ﬁrms per capita is constant. The steady state growth rate of output is only
dependent of modiﬁed q as is the case η = 0. If κ > 0, the steady state number of ﬁrms
1

per capita is given by (n∗ ) 1−κ , where n∗ is the value of the steady state number of ﬁrms
per capita in the case κ = 0. Other steady state values coincide with those in the case
κ = 0. Therefore, permanent taxation eﬀects on variables in the steady state equilibrium
are consistent with the case κ = 0.
The transitional responses of macro variables rather than the growth rate of output
per worker are qualitatively consistent with the case κ = 0. The growth rate of output
per worker is also dependent of the growth rate of the number of ﬁrms per capita during
the transition. If the intensity of the growth rate of product variety dominates that of thegrowth rate of quality, the transitional path of the growth rate of output per worker is
modiﬁed compared to the case κ = 0. As an example, Figure 6 (a, b) shows that impulse
responses to the dividend tax cut by 10 percentages points in the case κ = 0.3. In thiscase, even if the tax cut is unanticipated and permanent (or temporary), the growth rateof output per worker initially jumps up. However, the other variables moves in the similar
way as the case κ = 0. That is, in this case, the positive growth rate of product variety
initially oﬀsets the growth rate of quality, resulting in the initial jump of the growth rateof output per worker. Our main ﬁndings about investment decision of in-house R&D ofﬁrms holds qualitatively.
However, in the case η > 0, welfare of households is also dependent of the number ofﬁrms per capita. Higher product variety directly increases welfare of households. So weneed to check the robustness about welfare consequences for tax changes. Table 6 reports
the welfare consequences of tax changes in the case η = 0.1, 0.3, 0.5, and 0.7. The tableshows that as the value of the spillover parameter increases the welfare losses from thedividend tax cut diminish. The welfare gains from the corporate tax cut and the capitalgains tax cut rise. And the welfare gains from rise of the R&D tax credit rate diminish.
However, the sign of those welfare eﬀects does not change. And, in any tax change, thewelfare eﬀect of the implementation lag qualitatively holds.

25

6

Policy implications and Conclusion remarks

In what follows, we summarize our results. The dividend tax cut is not an eﬀective policy
instrument irrespective of how it is implemented. The intuition of the negative eﬀect of
the tax cut on the long-run growth is same as Peretto (2007, 2011). The tax cut yields
product proliferation and reduce incentives to in-house R&D of ﬁrms through the market
share eﬀect. However, the welfare losses of the tax cut diminishes by the implementation
lags of the tax cut. The implementation lags allow ﬁrms to adjust the timing of theirdividend payments to the tax schedule by changing their investment decisions of in-house
R&D. The future dividend tax cut increases both the contribution of future gross cash ﬂow
to the value of ﬁrms and the future cost of in-house R&D. Hence ﬁrms increases current
in-house R&D investment during the announcement phase in order to raise the subsequent
dividend payments. Besides, the announcement also allows households to adjust the timingof consumption and leisure time more smoothly. If the tax cut is temporary, during theimplementation, ﬁrms reduce the current investment in order to raise the current dividend
payments. As a result, the temporary tax cut also yields a negative eﬀect on growth andwelfare.
On the other hand, the capital gains tax cut and rises of the R&D tax credit rate areeﬀective policy instrument. The intuition of the long-run eﬀect is also same as Peretto(2007, 2011).17 However, the eﬀectiveness of these policy is worsened by the implementation lags. The announcement of the future tax changes reduces the future cost of in-house
R&D. Hence ﬁrms increases the current dividend payments and reduce the current R&D
investment in order to increases the future R&D investment. If these tax changes aretemporary, during the transition, the current cost of in-house R&D are reduced. Hence,ﬁrms increases the current in-house R&D investment. As a result, the temporary thesetax changes yields positive eﬀect on growth and welfare.18
Therefore, when considering how to design the corporate taxation policy, the policy
17

Theoretically, the long-run eﬀect of the capital gains tax cut is ambiguous. However, in our calibration,
the long-run eﬀect is positive.
18
The corporate tax cut seems to be also eﬀective policy instruments. However, if in-house R&D
investment is fully deductible against the corporate tax liability, the qualitative eﬀect of the corporate tax
coincides with the dividend tax cut.

26

makers are careful about the dynamic responses of forward looking investment decisionsof ﬁrms which consider the timing of dividend payments.

A

Appendix

A.1

Appendix 1

The perfect distribution in the ﬁnal goods sector yields (letting Lit = Lt ):θ2 Yt = Nt Xt ,(A-1)

(1 − θ)Yt = Wt Nt Lt .

(A-2)

Using the deﬁnition of ct , (A-2), and the market equilibrium condition of labor, Nt Lt =eλt lt , (15) can be rewritten to (17). Substituting (2) and the market equilibrium conditionof labor into (1) yields (18).

A.2

Appendix 2

Dividing both sides of (16) by Yt and using the deﬁnition of nt and ct , (A-1), (18), and(12), we obtain
[
(
)]
ṅt
Φ(Rt , Zt )
nt
ϕ+
+β
+λ .
1 − θ − g − ct =Ωl(ct )
Zt
nt
2

(A-3)

Dividing (5) by Zt and using (11), we obtain
Φ(Rt , Zt )(Sqt )2 − 1
=
.
Zt
2h
Substituting (A-4) into (A-3) and using (17), we obtain (19).
From (A-1), the deﬁnition of nt , and (18), we obtain
Ft
=
Zt

(

1−θ
θ

)

27

Xt
− ϕ,
Zt

(A-4)

n

modified qchours worked

1

1.07

0.99

1.12

1.08

1.06

0.98
1.06

1.1

0.97

1.08

0.96

1.04

0.95

1.02

1.06

1.05
1.04
1.03

0.94
1

0.93

1.04
1.02

1.02

0.92

0.98

0.91

0.96

1.01

1

1
0

10

20

0

interest rate

10

20

0

growth rate of y1020

dividend ratio

0.03

0.068

0

10

20

tax revenue ratio
1

1.25

0.995

0.066
0.99

1.2
0.025

0.064

0.985

1.15

0.98

0.062

1.1

0.06

0.02

0.975
0.97

1.05

0.965

0.058

1
0.96

0.056

0.95

0.015
0

10

20

0

10

20

0.955
0

10

20

0

10

20

(a) Anticipated vs. unanticipated permanent reduction of the dividend tax rate by 10
percentage points in the benchmark setting. Solid (Dashed) lines plots the impulse response ofeach variable to the anticipated (unanticipated) tax cut. The circle marks on the left (right) verticalaxis indicates the steady state level before (after) the tax cut.

ncmodified q

1.04

1

1.02

0.98

1.08

1

0.96

1.06

0.98

0.94

0.96

0.92

hours worked
1.07

1.1

1.06
1.05
1.04
1.03
1.02

1.04
0.94
1.02

1.01
1

0.9

0.99

0.92
0.88

0.98

0.9

1
0

10

20

0

interest rate

10

20

0

10

20

dividend ratio

growth rate of y
0.02

0.066

0.016

0.06

0.014

tax revenue ratio
1

0.985

1.15

0.98
1.1

0.975

0.058

0.012

1.05

0.97

0.056

0.01

1

0.965

0.054

0.95

0.96

0.9

0.955

0.008

0.052
0

10

20

20

0.99

1.2

0.062

10

0.995

1.25

0.018

0.064

0

0

10

20

0

10

20

0

10

20

(b) Temporary reduction of the dividend tax cut by 10 percentage points in the benchmark
setting. The circle mark on the vertical axis indicates the initial level.

Figure 2

28

c

n

modified q

1

1.1

hours worked

1.02
1.05

0.99
1.08

0.98

1.015

1.04

0.97

1.06

1.03
0.96

1.04

1.01
1.02

0.95
1.005

0.94

1.02

1.01
0.93
1

1
0

10

20

interest rate
0.072

0

10

20

1
0

growth rate of y1020

dividend ratio

0

10

20

tax revenue ratio
1

0.022
0.995

1.2

0.02180.07

0.99

0.02160.068

1.15

0.02140.985

0.0212

0.066

0.021
0.0208

0.064

1.1

0.98

1.05

0.975

0.02060.062

0.97

0.02041
0.965

0.0202

0.06

0.95
0

10

20

0

10

20

0.96
0

10

20

0

10

20

(a) Anticipated vs. unanticipated permanent reduction of the corporate tax rate by 10
percentage points in the benchmark setting. Solid (Dashed) lines plots the impulse response ofeach variable to the anticipated (unanticipated) tax cut. The circle marks on the left (right) verticalaxis indicates the steady state level before (after) the tax cut.

n

c

1.07

1.02

1.06

1.01

modified q
1.012
1.04

1

1.05

1.01
1.03

0.99
1.04

0.98

1.03

0.97

1.008
1.02
1.006
1.01

0.96
1.02

hours worked
1.05

1.004
1

0.95

1.01

1.002

0.94

0.99

0.93

1
0

10

20

interest rate
0.07
0.068

1
0

10

20

0

10

20

dividend ratio

growth rate of y
0.02140.0212

1.2

0.021

1.15

0

10

20

tax revenue ratio
1
0.995
0.99

0.066

0.985

0.02081.1

0.0206

1.05

0.064

0.98

0.062

0.975
0.0204

1

0.97

0.95

0.965

0.06
0.0202
0.058
0

10

20

0

10

20

0

10

20

0

10

20

(b) Temporary reduction of the corporate tax cut by 10 percentage points in the benchmark
setting. The circle mark on the vertical axis indicates the initial level.

Figure 3

29

n

c

modified q

1
1.02

hours worked

1.04

0.998

1.01
1.02

0.996
1.015

1.008

0.994

1

0.992
1.01

1.006
0.98

0.99

1.004

0.988
1.005

0.96

0.986

1.002
0.94

0.984
1

1
0

10

20

0

interest rate

10

20

0

20

dividend ratio

growth rate of y

0.066

10

0

10

20

tax revenue ratio
1

0.024
1.02

0.064

0.998

0.022

0.062

1.01

0.02

1

0.996

0.018

0.99

0.994

0.06
0.058

0.98

0.016

0.992
0.97

0.056
0.014

0.99

0.96

0.054
0

10

20

0

10

20

0

10

20

0

10

20

(a) Anticipated vs. unanticipated permanent reduction of the capital gains tax rate by 10
percentage points in the benchmark setting. Solid (Dashed) lines plots the impulse response ofeach variable to the anticipated (unanticipated) tax cut. The circle marks on the left (right) verticalaxis indicates the steady state level before (after) the tax cut.

ncmodified q

1
1.001

0.999

1

1.004
1.0035

1.1

1.003

0.998

0.999

hours worked

1.12

1.08
1.0025

0.997

0.998

1.06

1.002

1.04

1.0015

0.996
0.997

1.001

0.995
1.02

0.996

1.0005

0.994
0.995

1
0

10

20

interest rate

0

10

20

1
0

growth rate of y1020

dividend ratio
1.01

0.072

0.032

0.07

0.028

10

20

tax revenue ratio
1

1.005

0.999

1

0.998

0.03

0.068

0

0.997

0.995

0.996

0.066

0.026

0.99

0.064

0.024

0.985

0.994

0.98

0.993

0.062

0.022

0.995

0.992

0.975
0

10

20

0

10

20

0

10

20

0

10

20

(b) Temporary reduction of the capital gains tax cut by 10 percentage points in the benchmark setting. The circle mark on the vertical axis indicates the initial level.

Figure 4

30

n

c

1

modified q

1.01

1.03

1.008

1.02

hours worked
1.001
1
0.999

0.995

0.99

0.985
0

10

1.006

1.01

1.004

1

0.997

1.002

0.99

0.996

1

0.98

0.998

0.97

20

interest rate

0.998

0.994
0

10

20

10

20

dividend ratio
1.015

0.021

1.005

0.02

1

0.019

0.995

0

10

20

tax revenue ratio
1.0025
1.002

1.01

0.022

0.062

0.993
0

growth rate of y
0.023

0.063

0.995

1.0015
1.001

0.061

1.0005

0.06
0.059

0.018

0.99

0.017

0.985

1
0.9995
0.999

0.058
0

10

20

0

10

20

0.9985
0

10

20

0

10

20

(a) Anticipated vs. unanticipated permanent rise of the tax credit rate by 20 percentage
points in the benchmark setting. Solid (Dashed) lines plots the impulse response of each variable
to the anticipated (unanticipated) change. The circle marks on the left (right) vertical axis indicatesthe steady state level before (after) rise of the tax credit.

nchours worked

modified q

1

1.07
1.004

0.998
1.06
1.01

0.996

1.002
1.05

0.994

1
1.005

0.992

1.04
0.998

0.99

1.03

0.996

1

0.988

1.02

0.994

1.01

0.992

0.986
0.995

0.984
0.982

1
0

10

20

interest rate

0

10

20

0

0.027

0.065

0.026

20

dividend ratio

growth rate of y

0.066

10

tax revenue ratio

1.0035

1.01

1.003

1.005

1.0025

0.024
0.063
0.023

0.061

20

1.004

1.015

0.025

0.062

10

1.025
1.02

0.064

0

1

1.002

0.995

1.0015

0.022

0.99

0.021

0.985

1.001
1.0005

0.98
0.02
0

10

20

1
0

10

20

0

10

20

0

10

20

(b) Temporary rise of the tax credit rate by 20 percentage points in the benchmark setting.
The circle mark on the vertical axis indicates the initial level.

Figure 5

31

n

c

1.2

modified q

1

1.03

1.06

0.99

1.15

hours worked
1.035

1.08

1.025
1.04

0.98

1.02
1.02

1.1

1.015

0.97
1

1.01
1.05

0.96

0.98

1.005

0.96

0.95

1

1
0

10

20

0

interest rate

10

20

0

growth rate of y1020

dividend ratio

0

10

20

tax revenue ratio

0.036
0.066

1

0.034

1.2

0.995

0.032
0.064

0.99

0.03

1.15

0.985

0.028

0.062

0.98

0.026

1.1

0.975

0.024

0.06

0.97

0.022
0.058

1.05

0.965

0.02

0.96

0.018
0.056

1

0.955

0.016
0

10

20

0

10

20

0

10

20

0

10

20

(a) Anticipated vs. unanticipated permanent reduction of the dividend tax rate by 10
percentage points in the case of κ = 0.3. Solid (Dashed) lines plots the impulse response of eachvariable to the anticipated (unanticipated) tax cut. The circle marks on the left (right) vertical axis
indicates the steady state level before (after) the tax cut.

nmodified q

c

1.14

hours worked

1

1.04

0.98

1.03

1.02

1.12

1.01

1.1

1

1.08

0.99

0.96

1.02

0.94
1.01

0.98

1.06

0.92
0.97

1

1.04
1.02

0.96

0.9

0.95

0.88

0.99

1
0

10

20

0

interest rate

10

20

0

growth rate of y1020

dividend ratio

0.022

0.064
0.062

0.02

1.2

0.018

1.15

0.056

tax revenue ratio
1

0.99

1.1

0.012

1.05

0.01

1

0.98
0.975
0.97

0.008

0.054

20

0.985

0.014
0.058

10

0.995

0.016

0.06

0

0.965
0.95

0.006
0.052

0.96
0.9

0.004
0

10

20

0

10

20

0.955
0

10

20

0

10

20

(b) Temporary reduction of the dividend tax cut by 10 percentage points in the case ofκ = 0.3. The circle mark on the vertical axis indicates the initial level.

Figure 632

= θ(1 − θ)Ωl(ct )
− ϕ.
nt

(A-5)

Using (A-4), (A-5), and (11), (13) is rewritten by[][](1 − τD )(1 − τΠ )Ωl(ct )
(1 − τD )(1 − στΠ ) (Sqt )2 − 1rt =θ(1 − θ)
−ϕ −
β
nt
β
2h
+ (1 − τV )Sqt − 1.
hThen, from the deﬁnition of S and η, rearranging the above equation yields (interest rate)
From logarithmic diﬀerentiation of ct with respect to time yields
c˙tĊt Ẏt=−= rt − ρ + λ −
ct
Ct Yt

{

˙ t)
l(c
Żt
+λ+
l(ct )
Zt}.

Using (17) and (11), the above equation is rewritten byc˙t
Γċt
Sqt − 1= rt − ρ +−.
ct
1 + Γcth(A-6)

Rearranging (A-6) yields (20).
From (2), (8), and the market equilibrium condition of labor, given Z̄t and Lt , we obtain∂FtΩl(ct )= αθ(1 − θ).
∂Ztnt
Using the deﬁnition of S and η, (11), and (A-7), (10) is rewritten byαθ(1 − θ) Ωl(ct ) (Sqt − 1)2
1+=
rt qt − q̇t .
Sη
nt
2Sh
1 − τV
Then, using (17), it yields (21).

33

(A-7)

A.3

Appendix 3

The dynamical system of the economy where qt ≤ 1 is constituted by[] Ωl(ct )nt− [ϕ + βλ] ,ṅt = 1 − θ2 − g − ctββċt = ct [1 + Γct ] [rt − ρ] ,q̇t =1αθ(1 − θ) Ωl(ct )rt qt −,
1 − τVSη
nt

where the interest rate is given by[](1 − τV )Ωl(ct )rt =−ϕ .
θ(1 − θ)
βSη
nt

A.4

Appendix 4

Diﬀerentiating (26) with respect to q̃ yields′

f (q̃) ≡






2
1
Sαβ + 1ρq̃
1 + α −

 +

−
1
−
,

1−τ
h
1−τ
h
1−τ
V






−

ρ
1 − τV,V

if q̃ > 1,V

if q̃ ≤ 1.

And second order diﬀerentiating (26) with respect to q̃ yields

f ′′ (q̃) ≡






1 + α −




0,


2
1 − τV



1
h

< 0,

if q̃ > 1,

if q̃ ≤ 1.

Here,lim f ′ (q̃) =q̃→1+01[τV (Sαβ − 1) − (1 − α) − hρ] .
h(1 − τV )If Sαβ ≤ 1 − (1−τηρV )αϕ (< 1), f (1) < 0 and limq̃→1+0 f ′ (q̃) < 0. Then, f ′ (q̃) < 0 forq̃ > 1 as f ′′ (q̃) < 0 for q̃ > 1. Therefore, in this case, f (q̃) has only one solution of q̃ which
value is less than one. That is, there is no steady state equilibrium with positive growth34rate of output. On the other hand, if 1 − (1−τηρV )αϕ < Sαβ, f (1) > 0. No matter whether
limq̃→1+0 f ′ (q̃) < 0 is positive or negative, f (q̃) has unique solution of q̃ which value ishigher than one as depicted in Figure 1.

A.5

Appendix 5

Diﬀerentiating the LHS of (26) with respect to τD , τΠ , tV , and σ respectively yields}{[]∂f (q̃ ∗ )q̃ ∗ − 1
1
αβ
1
=
ρ+
− 1 αβ
S > 0,∂τD
1 − τV1 − τVh(1 − τD ){[]}∂f (q̃ ∗ )αβ1q̃ ∗ − 1τΠαϕσρ+− 1 αβ
=
S+ 2
> 0,
∂σ
1 − τV1 − τVh(1 − στΠ )η (1 − τΠ ){[]}∂f (q̃ ∗ )αβ1q̃ ∗ − 1σαϕ(1 − σ)=ρ+− 1 αβS−T 0.,∂τΠ
1 − τV1 − τVh(1 − στΠ )(1 − στΠ )2= −αϕ < 0, if σ = 0,
{
[
]
}
αβ
1
1
q̃ ∗ − 1=ρ+− 1 αβ
S > 0, if σ = 1,
1 − τV1 − τVh(1 − τΠ )[]∂f (q̃ ∗ )ρq̃ ∗ − 1
q̃ ∗
∗
=−
q̃ +
Sαβ −
≡ Γ(q̃ ∗ ) T 0.
∂τV
(1 − τV )2h(1 − τV )1 − τVρ< 0,(1 − τV )2
1[−2q̃ ∗ − hρ + 1 + Sαβ(1 − τV )] T 0,
Γ′ (q̃ ∗ ) =2(1 − τV ) h2Γ′ (q̃ ∗ ) = −< 0.
(1 − τV )2 h
Γ(1) = −Because f (q̃ ∗ ) is decreasing function of q̃ ∗ in the neighborhood around the steady statesolution, this derivation implies that the steady state growth rate is increasing functionof τD , τΠ (if σ = 1) and is decreasing σ, τΠ (if σ = 0) and the eﬀect of tax change of τΠ
(if σ ∈ (0, 1)) and τV is ambiguous. However, if Sαβ < (1 + hρ)/(1 − τV ), Γ(q̃ ∗ ) < 0 forq̃ ∗ ≥ 1. Then, it is shown that if Sαβ < (1 + hρ)/(1 − τV ), the steady state growth rate isdecreasing function of τV .

35

A.6

Appendix 6

1
We deﬁne Ψt ≡ Ut − ρ−λlog Zt . From the deﬁnition of ct , (11), and (18), diﬀerentiating Ψtwith respect to time yields
Ψ̇t = (ρ − λ)Ψt − log Ω − log ct − log lt − ζ log (1 − lt ) −1 Sqt − 1.
ρ−λ hIn the steady state, Ψt is constant over time. Calculating the dynamic path of Ψt numerically by using the relaxation algorithm, we can obtain the initial value of Ψt , that1
is Ψ0 = U0 − ρ−λlog Z0 . Without loss of generality, Z0 is normalized to one. Hence, we

obtain U0 = Ψ0 .

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Model of Investment”, Journal of Monetary Economics, Vol. 9, No. 3, pp. 353-373.
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36

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37

[17] Peretto, P. F. (2003) “Fiscal Policy and Long-run Growth in R&D-based Models with
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Table 5: Welfare gains of tax changes (parameter changes)
Tax change

Unanticipated(Permanent)

Anticipated(Permanent)

Temporaryβ = 3.275 (with α = 0.277 and ϕ = 0.125)∆tD = −0.1∆tΠ = −0.1
∆tV = −0.1
∆σ = 0.2−14.94
5.09
15.33
11.55

−12.21
5.76
12.63
10.3

−8.52
0.58
8.3
5.05

β = 9.825 (with α = 0.0945 and ϕ = 0.408)∆tD = −0.1∆tΠ = −0.1
∆tV = −0.1
∆σ = 0.2

−13.7
5.03
13.04
10.66

−10.85
5.65
10.25
9.36

−8.08
0.66
7.72
4.81

h = 8.0 (with α = 0.133 and ϕ = 0.267)∆tD = −0.1∆tΠ = −0.1
∆tV = −0.1
∆σ = 0.2−15.44
5.87
15.1
12.37

−11.41
6.55
11.05
10.37

−9.98
0.89
9.7
6.05

h = 18.0 (with α = 0.153 and ϕ = 0.265)∆tD = −0.1∆tΠ = −0.1
∆tV = −0.1
∆σ = 0.2

−12.5
4.2
12.02
9.39

−10.55
4.77
10.15
8.59

−6.64
0.44
6.29
3.88

σ = 1.0 (with α = 0.0955 and ϕ = 0.260)∆tD = −0.1∆tΠ = −0.1
∆tV = −0.1
∆σ = 0.2−13.86
−13.6
12.95
17.06

−11.05−10.83
10.15
15.0

−8.08
−7.95
7.69
7.55

ζ=0
∆tD = −0.1∆tΠ = −0.1
∆tV = −0.1
∆σ = 0.2−14.23
4.81
13.5
10.87

−9.93
4.64
9.16
8.20

−7.12
0.53
6.72
4.18

Note: Welfare gains are measured in consumption equivalent and expressed in percentage points.
Other values of the tax variables and parameters are sames as the benchmark setting.

39

Table 6: Welfare gains of tax changes (positive social spillover of product variety)Tax change

Unanticipated(Permanent)

Anticipated(Permanent)

Temporaryκ = 0.1∆tD = −0.1∆tΠ = −0.1
∆tV = −0.1
∆σ = 0.2−13.08
5.88
13.73
10.72

−10.17
6.59
10.98
9.43

−7.95
0.83
7.86
4.82

κ = 0.3∆tD = −0.1∆tΠ = −0.1
∆tV = −0.1
∆σ = 0.2−10.75
8.07
14.17
10.35

−7.78
8.88
11.53
9.1

−7.12
1.42
7.85
4.66

κ = 0.5∆tD = −0.1∆tΠ = −0.1
∆tV = −0.1
∆σ = 0.2

−7.57
11.08
14.76
9.87

−4.65
11.93
12.25
8.68

−5.86
2.35
7.85
4.43

κ = 0.7∆tD = −0.1∆tΠ = −0.1
∆tV = −0.1
∆σ = 0.2

−3.07
15.34
15.54
9.23

−0.22
16.22
13.24
8.13

−4.11
3.61
7.86
4.13

Note: Welfare gains are measured in consumption equivalent and expressed in percentage points.
Other values of tax variables and parameters are same as the benchmark setting.

40