Discussion Paper Series
Graduate School of Economics and School of Economics
Meisei University
Discussion Paper Series, No.41
December, 2018
Tax Evasion and Optimal Corporate Income Tax Rates
in a Growing Economy

Takeo Hori
（Tokyo Institute of Technology）
Noritaka Maebayashi（The University of Kitakyushu）
Keiichi Morimoto(Meisei University)

Hodokubo 2-1-1, Hino, Tokyo 191-8506
School of Economics, Meisei University
Phone: +81-(0)42-591-9479

Fax: +81-(042)-599-3024

URL: https://keizai.meisei-u.ac.jp/econ
E-mail: keizai@econ.meisei-u.ac.jp

Tax Evasion and Optimal Corporate Income Tax Rates
in a Growing Economy∗
Takeo Hori†

Noritaka Maebayashi‡

Keiichi Morimoto§

December 22, 2018

Abstract
We explore how tax evasion by firms aﬀects the growth- and welfare-maximizing rates
of corporate income tax (CIT) in an endogenous growth model with productive public service. We show that the negative eﬀect of CIT on growth is mitigated in the presence of tax
evasion. This increases the benefit of raising the CIT rate for public service provision. Thus,
in contrast to Barro (1990), the optimal tax rate is higher than the output elasticity of public
service. Through numerical exercises, we demonstrate that the role of tax evasion by firms
is quantitatively significant.

Keywords: corporate income tax, tax evasion, growth, welfare∗

We are grateful to Shingo Ishiguro, So Kubota, Kazutoshi Miyazawa, Ryoji Ohdoi, Makoto Saito, Daichi Shirai,
Koki Sugawara, Yuta Takahashi, Akira Yakita, and Taiyo Yoshimi for their helpful comments. We also thank theparticipants of the 2018 Public Economic Theory Conference, Journées LAGV 2018, 2018 Australasian Meetingof Econometric Society, MUETEI workshop at Meisei University, Nagoya Macroeconomic Workshop, Economic
Theory and Policy Workshop, the Macro Money Workshop at Hitotsubashi University, and the Economics Seminarat Tokyo International University. We are responsible for any remaining errors.
†
Department of Industrial Engineering and Economics, School of Engineering, Tokyo Institute of Technology,
2-12-1, Ookayama, Meguro-ku, Tokyo, 152-8552, Japan. E-mail address: hori.t.ag@m.titech.ac.jp‡
Faculty of Economics and Business Administration, The University of Kitakyushu, 4-2-1, Kitagata, Kokura
Minami-ku, Kitakyushu, Fukuoka 802-8577, Japan. E-mail address: non818mn@kitakyu-u.ac.jp§
Department of Economics, Meisei University, 2-1-1, Hodokubo, Hino, Tokyo 191-8506, Japan. E-mail address:
keiichi.morimoto@meisei-u.ac.jp

1 Introduction
Corporate income tax (hereafter, CIT) has detrimental eﬀects on investment by firms. Modern
prevailing opinions advocate that CIT should be cut to promote economic growth. In reality,
governments in developed countries have lowered the CIT rate in recent years: The average rateof CIT in OECD countries decreased from 32.5% to 24.2% between 2000 and 2017. 1 However,
because CIT is an important tax basis for public finance, cutting the CIT rate reduces public
investment in infrastructure (productive public services) and may lower the rate of economic
growth (e.g., Barro 1990).2 Therefore, setting CIT rates involves a trade-oﬀ between private
investment and the provision of productive public services for economic growth, and it is an
important policy issue.
In addition, tax evasion by firms is a serious problem of CIT. There is large-scale tax evasionin real economies. For example, in the US, the Internal Revenue Service (2016) reports about 44
billion dollars as the estimated tax gap of CIT on annual average from 2008 to 2010.3 Nonetheless, the actual audit coverage of taxed corporations is very limited, primarily because of fiscal
tightness; only 1.0% of taxed returns were examined in the US in 2016 and only 3.1% of all taxed
corporations were audited in Japan in 2015.4
One of the most general ways of tax evasion is to underdeclare income (e.g., Allingham andSandmo 1972). To underdeclare corporate income, firms may pretend to have lower productivity
and overstate their costs. In reality, there is institutional cause for such tax evasion behavior at
the microeconomic level. The CIT systems in many countries have reduction and exemption
measures; deficits of corporations can be carried forward and CITs of corporations with small
income are reduced or exempted. Such systems give corporations an incentive to underdeclaretheir profit intentionally by overstating cost. These are neither unusual nor insignificant at themacroeconomic level. For example, the National Tax Agency (2016) reports that more than twothirds of the ordinary corporations in Japan were loss corporations between 2011 and 2015 on

1

According to the policy stance of President Trump, the US government reduced the CIT rate to 21%. In Japan,
the government plans to cut the CIT rate to about 20% gradually under some conditions.
2
In fact, CIT occupies a considerable fraction of the annual revenue of the public sector. For example, the Internal
Revenue Service (2017) and the National Tax Agency (2017) report that the shares of CIT to total tax revenues wereabout 10% in the US in 2016 and 19% in Japan in 2015, respectively.
3Here, the “tax gap” is the “gross tax gap” in the Internal Revenue Service’s survey. This is defined as the amountof true tax liability that is not paid voluntarily and timely. For details, see Internal Revenue Service (2016).
4
The sources of these facts are Internal Revenue Service (2017) and National Tax Agency (2016).

1

average. The reported losses amounted to 11.3 billion yen in 2015 and occupied a quarter of thereported total income of the corporate sector.
With these problems of CIT as motivation, this study investigates an optimal CIT policy ina growing economy with tax evasion by firms. The evaded tax payments are utilized for private
investment, while a part of the tax revenue is lost. This aﬀects the trade-oﬀ between private
investment and the provision of productive public services. We are particularly interested in how
CIT evasion by firms changes the optimal rate of CIT.
To tackle this problem, we construct a variety-expansion model of growth in which private
firms invest to earn monopolistic profits and their investments sustain economic growth. 5 A
single final good is produced by using intermediate goods. Each intermediate good is produced
by a monopolistically competitive firm. Each intermediate good firm invests to enter into businessand pursues monopolistic profits. CIT discourages private investment because the monopolistic
profits are subject to CIT. On the other hand, productive public services have a positive eﬀect on
the monopolistic profits of intermediate good firms. Thus, CIT has a positive eﬀect on private
investment through productive public services financed by CIT.
In our model, facing CIT, firms have an incentive to evade tax payment by underdeclaring
their profits in the absence of a perfect tax enforcement system. Because each firm can avoid apart of tax payment, tax evasion weakens the discouraging eﬀect of CIT on private investment.
Simultaneously, tax evasion reduces the provision of productive public services. Thus, tax evasion by the corporate sector aﬀects private investment, provision of productive public services,and economic growth.
In a tractable model, we obtain a qualitative result that both the growth- and welfare-maximizing
CIT rates are higher than the output-elasticity of public service. This is in contrast to the familiar
Barro’s (1990) rule, which indicates that the tax rate should be set at the output elasticity of public
services. The mechanism behind our result is as follows. When the government raises the CIT
rate, the eﬀective CIT rate and the tax revenue rise.6 This increases the provision of productive
public services and promotes economic growth. Simultaneously, raising the CIT rate discourages private investment and has a detrimental eﬀect on growth. However, this negative eﬀect on
growth is weakened by tax evasion. This is because, in response to a tax hike, firms attempt to5

We follow Barro and Sala-ı́-Martin’s (2004) variety-expansion model.
Here, we mean that the eﬀective tax rate is the ratio of the actually collected CIT revenue to the true profits offirms.
62

secure profits by increasing tax evasion. It alleviates the reduction in private investment and thetax base of CIT. CIT evasion generates the benefits of raising the CIT rate for the provision of
productive public services.
Next, we extend the model to obtain quantitative results. Our quantitative analyses show thattax evasion significantly raises the optimal CIT rate. In particular, the optimal tax rate is 40%
for the benchmark case, which is much higher than the standard values of the output elasticity of
public service, e.g., 10%. This optimal CIT rate in our model, at 40%, is close to that in Aghion
et al. (2016) mentioned below. Besides, we decompose the diﬀerence between the optimal tax
rate and the output elasticity of public service into several parts including a part associated withthe eﬀect of CIT evasion. We find that the eﬀect of CIT evasion occupies more than half of thetotal diﬀerence between the optimal tax rate and the output elasticity of public service for a wide
range of reasonable parameter values. That is, even quantitatively, the eﬀect of CIT evasion is
the main source of the high optimal tax rate.

Related Literature
Largely, our study is part of the literature starting with Barro (1990), which explores optimal
taxation in an endogenous growth model with productive public service (capital) (e.g., Futagami
et al, 1993; Glomm and Ravikumar, 1994; Turnovsky 1997).7 These studies emphasize that thegrowth-maximizing income tax rate equals the output elasticity of public capital and it coincides
with the welfare-maximizing one in the balanced growth path. This is the so-called Barro rule.
More recent studies cast some doubts on the Barro rule. Some studies (e.g., Futagami et al.
1993; Ghosh and Roy 2004; Agénor 2008) show that the welfare-maximizing tax rate is lower
than the growth-maximizing one (output elasticity of public capital), while other studies (e.g.,
Kalaitzidakis and Kalyvitis 2004; Chang and Chang 2015) show the opposite result.8 While these
existing studies consider neither CIT nor tax evasion, our study investigates optimal taxation,
7

Some empirical studies suggest the importance of productive public expenditure for economic growth. Abiad
et al. (2016) show that increases in public investment in infrastructure raise output in both the short and long run.
8
Futagami et al.’s (1993) finding works if the policy eﬀects during the transition path caused by the accumulationof public capital is evaluated. Ghosh and Roy (2004) consider the composite output externality between the stockof public capital and the flow of public services. Kalaitzidakis and Kalyvitis (2004) consider the trade-oﬀ of public
expenditure between new investment and the maintenance of existing public capital. Agénor (2008) considers publichealth capital in addition to general infrastructure. Chang and Chang (2015) consider an endogenous growth modelwith market imperfection both in goods and labor markets and show that welfare-maximizing productive government
spending can be larger than growth-maximizing spending if it is financed by capital income taxes.

3theoretically and quantitatively, incorporating CIT evasion.9
Among the existing studies of tax evasion and growth, we should refer to Chen (2003) and
Kafkalas et al. (2014). They investigate tax evasion by household-firms in Barro’s (1990) typemodel, in which government incurs inspection expenditure to audit taxpayers. These authors
focus on the trade-oﬀ between the government’s inspection expenditure and public investment.
Chen (2003) shows that the growth-maximizing announced tax rate is higher than the output
elasticity of public service, under the assumptions that inspection expenditure is proportional
to output and household-firms must incur tax evasion costs. On the other hand, Kafkalas et al.
(2014) assume that the government’s inspection expenditure is proportional to tax revenues. Theyshow that, even with tax evasion, the growth-(and utility-) maximizing eﬀective tax rate equals
the output elasticity of public capital, in line with Barro’s rule.
In contrast to these studies, we do not focus on the role of inspection expenditure. Therefore,
we adopt the same specification of inspection expenditure as Kafkalas et al. (2014), which does
not aﬀect Barro’s rule. Thus, we can concentrate on the role of tax evasion behavior of firms withmarket power.
Although the findings on optimal tax rates from Chen (2003) and Kafkalas et al. (2014) are
interesting, there are some reservations. First, they consider tax evasion by household-firms,
where the firm is the same as a household. Since there is no diﬀerence between CIT and ahousehold’s income tax in their models, they do not consider the role of CIT evasion. Second,
they assume a competitive goods market, and therefore, ignore tax evasion associated with profit
maximization in an imperfectly competitive product market.10 Thus, although tax evasion aﬀects
the aggregate economy through the government budget in their models, these studies do notconsider the role of tax evasion in the production side directly.
Our study is also comparable to Aghion et al. (2016). They examine the relationship amongcorporate taxation, growth, and welfare, focusing on corruption. Aghion et al. (2016) construct9Agénor and Neanidis (2017) show empirically that public capital aﬀects growth through productivity and innovation (R&D) capacity.
10
Some theoretical studies investigate the role of public capital on growth under an imperfectly competitive product market. Pereto (2007) explores this issue in an R&D-based growth model where the market structure (entry andexit of firms) is endogenously determined. Pereto (2007) shows that productive public capital is neutral to growth,and therefore, no optimal tax exists. On the other hand, Chang and Chang (2015) advocate that infrastructure aﬀects
growth and optimal tax rates exist in an endogenous growth model, which includes monopolistic competition in the
goods market and unionization in the labor market. The recent empirical study of Agénor and Neanidis (2017) may
support the latter result.

4

an endogenous growth model in which public capital raises the expected returns to entrepreneurial
eﬀorts on R&D but tax revenue decreases due to the corruption of oﬃcers after tax collection.
They suggest that the relationship between CIT rate and growth is an inverted-U shape and thewelfare-maximizing CIT rate is 42% in the calibrated model. However, Aghion et al. (2016) do
not consider tax evasion by firms but focus on the corruption between government and households. In this study, we incorporate corporate tax evasion in an endogenous growth model andsuggest that the optimal CIT rate is as high as Aghion et al.’s (2016) estimate.

2 ModelTime is discrete and denoted by t = 0, 1, · · ·. The economy is inhabited by the following
four types of agents: producers of final output, producers of intermediate goods, a representative
household, and government. An infinitely lived representative household has perfect foresightand is endowed with L unit of labor. Labor moves freely across diﬀerent production sectors. The
number of intermediate good in period t is Nt . We assume N0 = 1 without loss of generality.

2.1 Producers of final goodA single final output is produced by perfectly competitive producers using the following technology:∫ Nt
Yt = AL1−αY,t(Gt xi,t )α di,

A > 0, 0 < α < 1,(1)0

where Yt is output, LY,t is labor input in the final goods sector, and xi,t is the input of intermediate
good i. Following Barro (1990), public services, Gt , increases the productivity of output. We
take the final output as the numeraire. Although α encompasses both (i) the output elasticityof public services and (ii) the price elasticity of intermediate goods 1/(1 − α), we examine theformer in this simple model. In the extended model (Section 5), we separate (i) and (ii). We
denote the price of intermediate good i and wage rate as pi,t and wt , respectively. The profit
maximization yields
wt = (1 − α)L−α
Y,t

∫ Nt

(Gt xi,t )α di = (1 − α)0
α α−1pi,t = αAL1−α
Y,t Gt xi,t .

Yt
,
LY,t

(2)
(3)

5

Solving (2) and (3) with respect to xi,t yields the demand function for the product of firm i:(1 − α)Ytx(pi,t ) ≡
wt

(

αAGαt
pi,t

1
) 1−α

.

2.2 Producers of intermediate goods
2.2.1 Entry into the intermediate goods market
Each intermediate good is produced by a monopolistically competitive firm. To operate in periodt, each intermediate good firm must invest η unit of the final good in period t − 1. Firms financethe cost of this investment by borrowing from households. Because firms must incur investment
costs in each period, the planning horizon of each firm is one period, as in Young (1998). When
incurring investment costs, each firm draws its productivity b > 0 from distribution F (b). We
assume that (i) b is independent and identically distributed (iid) over time as well as across firmsand (ii) b is private information. These two assumptions are useful in describing an environment
where the true tax base of each firm cannot be observable without auditing firms, as we explainin the next subsection. Besides, heterogeneity among firms’ productivity enables us to describe
the realistic firm size distribution for the quantitative analysis in Section 5.
Let us denote the expected after-tax operating profit of a firm with productivity b in period t byeee
in detail. The objective of intermediate
depends on b. The next subsection discusses πi,t. πi,tπi,tgood firm i that invests in period t − 1 is given by1
Πi,t−1 =
Rt−1

∫
e
πi,t
dF (b) − η,(4)where Rt−1 is the gross interest rate between periods t − 1 and t. Since b is iid across firms, allfirms face the same Πt−1 in period t − 1. Free entry into the intermediate goods market implies∫eπi,tdF (b) = Rt−1 η.

(5)

2.2.2 Maximization of operating profits
The price of each intermediate good pi,t is public information. A firm with productivity b needs
1/b unit of labor to produce one unit of intermediate good. The true operating profit of firm i is6

given by(
πi,t =

wt
pi,t −bi)x(pi,t ).

(6)

We use the word “true” to distinguish the true operating profit from the operating profit that firm
i declares, π̃i,t , to the government. Since b is private information, the government cannot directly
observe the true operating profit, πi,t , and thus, cannot know whether π̃i,t equals πi,t without
auditing firms.
Let us denote the announced CIT rate by τ ∈ (0, 1]. The after-tax profit of each firm is givenby πi,t − τ π̃i,t because CIT is imposed on declared profit π̃i,t . We denote the probability of auditby q̄ ∈ [0, 1]. We impose an assumption on q̄ later. Suppose that a firm is audited. If this firm
underdeclares its operating profit (π̃i,t < πi,t ), it has to pay a penalty of (1 + s)τ (πi,t − π̃i,t ),where s(≥ 0) is an additional tax rate. If this firm overdeclares its operating profit (π̃i,t > πi,t ),the overpayment, τ (π̃i,t − πi,t ), is refunded. The expected after-tax operating profit of firm i isgiven byeπi,t= πi,t − τ π̃i,t − q̄(1 + s)τ · max{0, πi,t − π̃i,t } + q̄τ · max{0, π̃i,t − πi,t }.

(7)

e
e
decreases with π̃i,t if π̃i,t > πi,t .
. Given πi,t , πi,tFirm i chooses π̃i,t and pi,t to maximize πi,tThus, no firms overdeclare their operating profits. The inequality π̃i,t ≤ πi,t must be satisfied in
the following discussion.
Our specification of tax evasion behavior is essentially similar to Allingham and Sandmo(1972). They consider tax evasion by a representative household, which evades income tax bychoosing declared income when its actual income is not known directly by the government. In
our setting, each firm chooses its declared profit by adjusting the price level.
Using (7), we have the following lemma.
Lemma 1: Suppose that a firm setting the price at p̃i,t declares its operating profits truthfully,
π̃i,t = πi,t . Then, the declared profit of this firm satisfies
π̃i,t = (1 − α)p̃i,t x(p̃i,t ).
Proof: See Appendix A.

7(8)Since the contraposition of a true claim is also true, we can conclude that if a firm does notdeclare (8), it does not declare its true profit. Thus, we assume that any firms whose declaredprofits do not satisfy (8) are audited with the probability of one. In addition, we assume that if a
firm declares (8), the firm is audited with a probability that is lower than one. Note that (8) is anecessary condition for true declaration of profits. Thus, a firm declaring (8) does not necessarily
declare its profit truthfully.
Assumption: Consider a firm that sets the price at p̃i,t .
(i) If the firm’s declared profit does not satisfy (8), the firm is audited with a probability of one,q̄ = 1.
(ii) If the firm’s declared profit satisfies (8), the firm is audited with a probability that is lower
than one, q̄ = q ∈ [0, 1).
e
Since π̃i,t ≤ πi,t , we now rewrite (7) as πi,t
= (1 − τ̃ )πi,t , where τ̃ is the eﬀective CIT rateand is defined as

}
{
π̃i,t
+ q̄(1 + s) τ.
τ̃ ≡ [1 − q̄(1 + s)]
πi,t

(9)

The following proposition shows each firm’s decisions on profit declaration and pricing.
Proposition 1. (i) Suppose that 1 − q(1 + s) ≤ 0. Then, all firms declare their true profit
π̃i,t = πi,t . Firms with productivity b set the price at p̃i,t = wt /(αbi ). The declared profit satisfies(8). The true and the expected after-tax operating profit are, respectively, given bye
πi,t = π̃i,t = (1 − α)p̃i,t x(p̃i,t ) and πi,t
= (1 − τ̃ )(1 − α)p̃i,t x(p̃i,t ).

(10)

The eﬀective CIT rate coincides with the announced CIT rate, τ̃ = τ .
(ii) Suppose that 1 − q(1 + s) > 0. Firms with productivity bi set the price at
p̃i,t =

1 − q(1 + s)τ
wt
Γ(τ ), where Γ(τ ) ≡> 1.
αbi
1 − q(1 + s)τ − (1 − α)[1 − q(1 + s)]τ(11)

The declared profit satisfies (8). The true and the expected after-tax operating profit are, respec-8

tively, given by()
πi,t = 1 − αΓ(τ )−1 p̃i,t x(p̃i,t ) > π̃i,t ,
(
)
e
πi,t
= (1 − τ̃ ) 1 − αΓ(τ )−1 p̃i,t x(p̃i,t ).

(12)
(13)

The inequality πi,t > π̃i,t indicates that all firms underdeclare their operating profits. The eﬀec](, satisfies that τ̃ < τ .
tive CIT rate, τ̃ ∈ 0, 1+αq(1+s)
1+αProof: See Appendix B.
Proposition 1 states that the firms control their declared profits according to whether tax
evasion is beneficial. If the expected additional tax rate is higher than the rate under honest
declaration (1 ≤ q(1 + s)), firms have no incentive to underdeclare profits. Thus, they declaretrue profits (π̃i,t = πi,t ).
In contrast, if the expected additional tax rate is lower than the rate under honest declaration(1 > q(1 + s)), firms choose to evade CIT by underdeclaring profits (π̃i,t < πi,t ). This is capturedby the term Γ(τ ) in (12), because the only diﬀerence between the true profit π in (10) and declaredprofit π̃ in (12) is the presence of Γ(τ ). 11
Note the following. When q(1 + s) > 1, because firms declare true profit and tax evasion
does not occur, the CIT has nothing to do with firms’ control of their profits. However, whenq(1 + s) < 1, the CIT rate aﬀects tax evasion substantially through the term, Γ(τ ). By Γ′ (τ ) > 0,
an increase in the CIT rate causes a larger diﬀerence between the true profit πi,t and declaredprofit π̃i,t . Thus, we find that it encourages tax evasion. Appendix C shows that d(τ /τ̃ )/dτ > 0
if and only if Γ′ (τ ) > 0. 12
Remarks
The properties dτ̃ /dτ > 0 and d(τ /τ̃ )/dτ > 0 are in line with Roubini and Sala-ı́-Martin (1995)
and Kafkalas et al. (2014).13 However, the tax evasion considered in our model departs from
11

They underdeclare profits by pretending to be lower productivity firms, which leads to a higher price setting(Γ(τ ) > 1) as represented by (11).
12Equation (9) indicates that the degree of tax evasion in response to a tax hike, d(πi,t /π̃i,t )/dτ , is reduced tod(τ /τ̃ )/dτ .
13
We obtain d(τ /τ̃ )/dq < 0 (see Appendix C) as in Roubini and Sala-ı́-Martin (1995) and Kafkalas et al. (2014).
Both studies assume that the diﬀerence between the announced and the eﬀective tax rate increases with the announced
tax rate τ and decreases with the detection probability q.

9these previous studies in the following respects.
First, tax evasion in our model is derived endogenously from the microfoundation of firms’
tax evasion behavior, in contrast to the ad-hoc expression by Roubini and Sala-ı́-Martin (1995).
Second, Chen (2003) and Kafkalas et al. (2014) do not consider tax evasion associated withprofit maximization in an imperfectly competitive product market. Thus, there is no relationshipbetween the tax rates and the firms’ choice of profits and declarations of profits in their models.
In contrast, in our model, the CIT rate aﬀects them substantially, as represented by Γ(τ ) > 1 andΓ′ (τ ) > 0.

2.3 Household
The population size is constant at one. The utility function of a representative household is
)t
∞ (
∑
1
U0 =
u(Ct ),1+ρ
t=0u(Ct ) =Ct1−σ, σ > 0.
1−σ(14)u(Ct ) = ln Ct , when σ = 1. Here, Ct , ρ(> 0) and 1/σ denote consumption in period t, thesubjective discount rate, and the intertemporal elasticity of substitution, respectively. The representative household supplies L unit of labor inelastically. The household’s budget constraint
is given by Wt = Rt−1 Wt−1 + wt L − Ct , where Wt−1 is assets at the end of period t − 1. The
household’s utility maximization yields
Ct+1
=
Ct

(

Rt
1+ρ

)1/σ
,

(15)

and the transversality condition (TVC) is
Ct−σ Wt−1= 0.
t→∞ (1 + ρ)t
lim

(16)

2.4 Government
We assume that the government keeps a balanced budget in each period. From (7) and (9), the∫N ∫aggregate tax revenue of the government is 0 t {τ π̃i,t + q(1 + s)τ [πi,t − π̃i,t ]}dF (b)di =∫∫N ∫τ̃ 0 t πi,t dF (b)di = τ̃ Nt πi,t dF (b). Here, we use the fact that b is iid across firms. This
revenue is allocated to productive government spending, Gt , and inspection expenditure to detect10tax evasion, Mt . Thus, the budget constraint of the government is given by∫τ̃ Nt

πi,t dF (b) = Gt + Mt .

(17)

We assume that spending a constant fraction, Q(q), of government revenue on Mt leads to a
successful detection of tax evasion by a firm with probability q, where Q′ (q) > 0, Q(q) ∈[0, 1) for q ∈ [0, 1], and Q(0) = 0. Therefore, the inspection expenditure is given by Mt =∫Q(q)τ̃ Nt πi,t dF (b). This specification of Mt is in line with Kafkalas et al. (2014). Thus, (17)reduces to∫
Gt = [1 − Q(q)]τ̃ Nt

πi,t dF (b).

(18)

As mentioned in Introduction, such a specification allows us to isolate the role of tax evasion byfirms with market power for pursuing Barro’s rule. This is because Kafkalas et al.’s (2014) result
indicates that Barro’s rule holds even with the tax evasion of perfectly competitive firms underinspection expenditure of the form in (18).

3 Equilibrium
The labor market clears as∫
L = LY,t + Ntxi,tdF (b).
b(19)

The entry cost of intermediate good market η is financed by borrowing from households. Since
Nt+1 firms invest in period t, the asset market equilibrium condition is given by Wt = ηNt+1 .
The final good market clears as Yt = Ct + ηNt+1 + Gt + Mt .
We next characterize the dynamic system and the steady state of the economy. Appendix D
derives the following dynamic system with respect to zt ≡ Ct /Nt .
η 1− σ [(1 − τ̃ )(1 − αΘ−1 )αΩ(τ̃ )/(1 + ρ)]1/σ zt
zt+1 =,[1 − τ̃ (1 − αΘ−1 )α]Ω(τ̃ ) − zt
1

where
Θ=



1

if 1 − q(1 + s) ≤ 0Γ(τ )

if 1 − q(1 + s) > 0,

11

(20)

and
[
(
) ] α
1
2α
Yt
= Ω(τ̃ ) ≡A 1−α (1 − Q(q))τ̃ 1 − αΘ−1 α 1−α (1 − α)Θα 1−α
Nt
1
[
] 1−α
[∫
]
α
L
×
b 1−α dF (b) .
(1 − α)Θ + α2(21)

From (20), we arrive at the following proposition.
Proposition 2. A unique steady state exists. In the steady state, zt takes the following constantvalue:
[
(
) ]
[
(
)
]1/σ
ẑ = 1 − τ̃ 1 − αΘ−1 α Ω(τ̃ ) − η (1 − τ̃ ) 1 − αΘ−1 α(1 + ρ)−1 η −1 Ω(τ̃ )∈ (0, z̄),(22)
where z̄ ≡ [1 − τ̃ (1 − αΘ−1 ) α] Ω(τ̃ ). In the steady state, Ct , Nt , and Yt grow at the same
constant rate:
[
(
)
]1/σ
ĝ = (1 − τ̃ ) 1 − αΘ−1 α(1 + ρ)−1 η −1 Ω(τ̃ ).

(23)The economy jumps to the steady state initially.
Proof: See Appendix E.

4 Optimal CIT Rates
As mentioned in Section 2, the CIT rate substantially aﬀects the tax evasion behavior of firms in
the imperfectly competitive market. Therefore, we analyze the growth- and welfare-maximizing
CIT rates under such CIT evasion. 14

4.1 Growth-maximizing CIT Rate
We obtain the following proposition.
14

Note that the parameters q and s are really endogenously selected by the governments in that the government
controls tax enforcement and punishment against tax evasion. However, to concentrate on the choice of optimal
CIT and keep the analyses simple, we ignore these policy dimensions and assume that they are exogenous parameters. This assumption is in line with Johansen (2010) who assume that parameters related to tax enforcement andpunishment against profit shifting by multinational firms are exogenous.

12

Proposition 3. Let us denote the growth-maximizing announced CIT rate and growth-maximizing
eﬀective CIT rate as τ GM and τ̃ GM , respectively.
1. When 1 − q(1 + s) ≤ 0 and each firm declares its true operating profit, τ̃ GM = τ GM = αholds.
2. When 1 − q(1 + s) > 0 and each firm underdeclares its operating profit, τ GM > τ̃ GM > αholds.
Proof: See Appendix F.
The first part of Proposition 3 is in line with Barro’s (1990) rule, that is, the tax rate thatmaximizes long-run growth equals the output elasticity of public services, α. This result is also
consistent with that of Kafkalas et al. (2014), indicating that even with government spending fordetection, the growth-maximizing tax rate coincides with the output elasticity of public services.
Importantly, the mechanism behind this result is the same as that of Barro (1990). In Barro’s
model, the growth-maximizing rule is attributed to the following trade-oﬀ between income tax
and growth. On the one hand, an increase in income tax decreases the net interest rate ((1−taxrate)× interest rate) and has a negative eﬀect on growth. On the other hand, an increase in income
tax boosts productive government spending and raises the interest rate. This has a positive eﬀect
on growth.
The interest rate in our model is determined through the free entry condition of intermediate
goods firms as
Rt−1 = η

−1

∫

e
πi,t
dF (b) = η −1 (1 − τ̃ )(1 − αΘ−1 )αYt /Nt .

(24)

Here, Yt /Nt , given in (21), is positively aﬀected by15∫
Gt = [1 − Q(q)]τ̃ Nt

πi,t dF (b) = [1 − Q(q)]τ̃ (1 − αΘ−1 )αYt .

(25)This induces essentially the same trade-oﬀ as Barro (1990). Thus, in the absence of tax evasion(Θ = 1), we obtain the same result as Barro (1990).

15

See Appendix D for the derivation of (24) and (25).

13The second part of Proposition 3 states that with underdeclaration of profit, the growthmaximizing announced CIT rate is higher than the growth-maximizing eﬀective CIT rate, τ GM >τ̃ GM . Moreover, both are higher than the output elasticity of public services, α. That is, tax evasion by firms increases the growth-maximizing tax rates, τ GM and τ̃ GM . τ GM > τ̃ GM stems
simply from the evasion of CIT by firms. Then, we consider the intuition behind the result ofτ̃ GM > α here.
As we have seen after Proposition 1, in response to a tax hike, each intermediate good firmeincreases tax evasion. This secures the expected operating after-tax profit, πi,t
, and private in-vestment. Because the true operating profit, πi,t , is also secured, the tax base for public service∫
provision (= Nt πi,t dF (b)) is maintained. 16 These are caused by the eﬀect of CIT evasion
associated with profit maximization in an imperfectly competitive product market, which arecaptured by the term 1 − αΘ−1 in (24) and (25), where Θ = Γ(τ ) and Γ′ (τ ) > 0. Hereafter, wecall this simply the eﬀect of CIT evasion. The eﬀect of CIT evasion mitigates the negative eﬀectof CIT on growth and increases the benefit of raising the CIT rate for the provision of productive
public services. 17
Our result (τ̃ GM > α) is diﬀerent from Kafkalas et al. (2014), who advocate that Barro’s
rule holds (τ̃ GM = α) even in the economy with tax evasion. While tax evasion does not aﬀect
firms’ decision-making in Kafkalas et al (2014), our model includes the eﬀect of CIT evasion as
mentioned above, which causes τ̃ GM > α. 18

16

Indeed, the disparity between the announced and eﬀective CIT rates is enlarged by the strong tax evasion due
to the tax hike. At a glance, this is likely to damage
public service provision. However, at the same time, such an∫increase in tax evasion holds the tax base, Nt πi,t dF (b) = (1 − αΘ−1 )αYt .
17
Except for the eﬀect of CIT evasion, there exist general equilibrium eﬀects in our model. An increase in p̃i,treduces the demands of intermediate good in the final good sector, x(p̃i,t ). This exerts the following opposite eﬀects
eon Rt−1 . On the one hand, a fall in x(p̃i,t ) reduces πi,tand lowers Rt−1 . On the other hand, a fall in x(p̃i,t ) causesa labor shift from the intermediate to final good sector, which increases final output and raises Rt−1 . However,
we find that these general equilibrium eﬀects are not the most important ones, as follows. By taking the logarithmof the growth rate (23) and diﬀerentiating it with respect to the eﬀective tax rate τ̃ , we find that the sum of the
opposing general equilibrium eﬀects is negative; the negative eﬀect of a fall in x(p̃i,t ) dominates the positive eﬀectof a rise in the final output by shifting labor to the final good sector. Therefore, the primary force of raising the
growth-maximizing eﬀective tax rate is the direct eﬀect, which we mentioned in the text.
18
In Chen’s (2003) model, although the growth-maximizing announced tax rate is higher than the output elasticityof public service, it is ambiguous whether so is the growth-maximizing eﬀective tax rate. This is because thehousehold-firms need to pay some tangible cost to evade tax and it is included in the definition of the eﬀective tax
rate of that model. Therefore, Chen’s (2003) result is not necessarily suitable for a rigorous comparison with thoseof Kafkalas et al. (2014) and this paper.

144.2 Welfare-maximizing CIT Rate
Next, we analyze the welfare-maximizing CIT rate. Using the balanced growth rate, ĝ, the equilibrium path of consumption is given by Ct = ĝ t C0 = ĝ t ẑ.19 Substituting it into the lifetime
utility function of the representative household, (14), we obtainU0 =ẑ 1−σ,(1 − σ) [1 − (1 + ρ)−1 ĝ 1−σ ](26)

where 1 > (1 + ρ)−1 ĝ 1−σ holds by the TVC. The social welfare is determined by the initial levelof consumption and the long-run growth rate, both of which depend on the announced CIT rate,τ . Let us denote the welfare-maximizing announced CIT rate by τ W M .
Proposition 4.
1. When each firm declares its true operating profit, 1 − q(1 + s) ≤ 0, τ W M > τ GM = αholds. Therefore, the welfare-maximizing announced CIT rate, τ W M , is higher than thegrowth-maximizing announced CIT rate, τ GM .
2. Suppose that 1 − q(1 + s) > 0 and q = 0. Then, each firm understates its operating profit
and a marginal increase in the announced CIT rate at the growth-maximizing CIT rate
improves social welfare.
Proof: See Appendix G.
Proposition 4 shows that the welfare-maximizing CIT is higher than the growth-maximizing
one. 20 On the one hand, a marginal increase in CIT from τ GM does not aﬀect the growthrate, because the first order eﬀect vanishes at τ = τ GM . On the other hand, it increases current
consumption, because labor income, which is exempt from taxation, is raised by increases in productive public services (see Appendix G). Therefore, the welfare-maximizing CIT rate is higher
than the growth-maximizing one.

19

The economy jumps onto the balanced growth path in the initial period, as explained in section 3. Rememberthat we assumed N0 = 1.
20
In the case of underdeclaration of profits by firms (1 − q(1 + s) > 0), we cannot reach such a statement whenq ̸= 0. However, in Section 5, we conjecture that the statement may be true for the other values of q because we
could not find any counterexample.

15

5 Quantitative Analysis
Propositions 3 and 4 summarize that the following three eﬀects make τ W M larger than the output
elasticity of public services with tax evasion by firms (1 − q(1 + s) > 0).
The first eﬀect is represented by τ W M > τ GM ; the welfare-maximizing announced tax rate
is higher than the growth-maximizing one. This is because the tax base is CIT, and wage income
is exempt from taxation as we have seen in Proposition 4. We call this the tax base eﬀect. The
second eﬀect is represented by τ GM > τ̃ GM . This stems from the degree of tax evasion by firms,(τ > τ̃ ). We call this the diﬀerence in tax rates. The third eﬀect is represented by τ̃ GM > α.
This is attributable to the eﬀect of CIT evasion, as mentioned in subsection 4.1
The objective of this section is to investigate how high the welfare maximizing CIT rate τ W M
is and which of the three eﬀects contributes most to it. To solve these quantitative problems, we
extend the above base model in this section.

5.1 Extended Model
We start with a small revision of the model, because the problematic restriction on the parameterlies in the previous form of production technology. The output-elasticity of public services α
must be set at the price elasticity of intermediate good, 1/(1 − α). To resolve this, we change
production technology (1) into
∫ Nt
Yt = AL1−αY,t(a(Gt , Nt )xi,t )α di,

(27)

0

where
a(Gt , Nt ) = Gϵt Nt1−ϵ ,0 < ϵ < 1.

(28)

The composite externality (28) represents a combination of the role of knowledge spillover, as in
Benassy (1998), together with productive public services, as in Barro (1990).
We adopt this form of the composite externality for the following two reasons. First, as will
become evident below and as stated by Chatterjee and Turnovsky (2012), it helps provide a plausible calibration of the aggregate economy, something that is generically problematic in the conventional one-sector endogenous growth model.21 Under (27) and (28), the output-elasticity of
21

Chatterjee and Turnovsky (2012) consider the composite externality from physical capital, as in Romer (1986)

16

public services is β ≡ αϵ(< α), which diﬀerentiates α from the output-elasticity of public services. Second, as the following Remark shows, although we take an additional externality (thespillover of knowledge) into account, the basic property of the benchmark model is maintained.
Remark. The qualitative results do not change in the extended model. By replacing α with β,
the same results as Proposition 3 and Proposition 4 hold.
Proof: See Appendix H.

5.2 Calibration
To conduct numerical exercises, we set the baseline parameter value as in Table 1. Appendix I
provides details of our calibration.
Insert Table 1 here.
The distribution of firms’ productivity is determined to make the curvature of the distributionfunction of firm sizes in the model equal that of the Pareto distribution estimated with US data by
Axtell (2001). This requires ψ = 1.059. We choose α = 0.8620 so that the markup rate of firmsµ (= Γ(τ )/α − 1) takes 20%, which is a standard value of markup rate of firms (e.g., Rotemberg
and Woodford (1999)).
We set the parameter to measure the knowledge spillover, ϵ, to 0.1160, so that the output
elasticity of public services, β, equals 0.1. Although the estimates of the elasticity vary amongsome empirical studies, 0.1 is one of the reasonable values of β. 22
In this subsection, we provide the baseline value of announced CIT rate, τ = 0.2706, to
determine the baseline balanced growth rate because the balanced growth rate depends on τ inthis model economy. This value of τ (= 0.2706) is the average CIT rate in OECD countries fromand productive public spending, as in Barro (1990) and Futagami, Morita and Shibata (1993) and conduct numericalanalyses, including growth and welfare eﬀects. Our application of (28) is in line with Chatterjee and Turnovsky(2012) because in our model, the stock of Nt replaces the role of physical capital and is both the engine of growth
and the source of spillover and positive social returns to variety, as discussed in Romer (1986) and applied in R&Dbased growth models as in Aghion and Howit (1998), Benassy (1998), Reretto (2007), and others.
22
From an empirical point of view, the output elasticity of infrastructure (or productive public services) has been
estimated and examined using data from many countries. Recent empirical studies (e.g., Röller and Waverman,
2001; Shioji, 2001; Esfahani and Ramı́rez, 2003; Kamps, 2006; Bom and Ligthart, 2014) indicate that the output
elasticity of infrastructure (or productive public services) lies in the range of 0.1–0.2, on average. More recent studies
by Bom and Ligthart, 2014 and Caldeón et al. (2015) indicate that the output elasticity of infrastructure is around
0.1.

17

2000 to 2017. 23 We set penalty tax rate, s, to 0.5, according to Fullerton and Karayannis (1994),
who take this value as a normal rate in the US.
We set the benchmark value of audit rate, q, to 0.096. To obtain this value, we utilize thestatistics provided by the Internal Revenue Service. Each IRS Data Book between 2000 and 2017provides the actual ratio of the examined corporations to all corporations under a classification by
firm size. 24 We choose the class of the smallest size of the large corporations. 25 This is becausesuch a class occupies a significantly large part (about 60%) of the large corporations, which arecorporations above a certain business size. Besides, the ratios of the audited corporations vary
greatly across the classes and so does the number of corporations in the classes. This meansthat taking the average audit rate among the classes is unreasonable. Therefore, we set the audit
rate in such a manner. Later, we confirm the robustness of our results for a range of q, including
q = 0.089, the value adopted by Fullerton and Karayannis (1994).
We specify the functional form of Q(·) by Q(q) = kq, where k is a constant. We set k = 0.167
to make Q(q) equal the ratio of inspection cost to the CIT revenue in the US on average from2000 to 2017.26 However, not only the value but also the functional form does not change ourresults because the growth rate of our model is independent of them: see (23).
We set σ = 1.5 and ρ = 0.0204 according to Jones et al. (1993). These are the standard values
used in quantifying growth models. We set L = 1 for normalization. Finally, we choose the scaleparameter, A, and the cost of developing one intermediate good, η such that the balanced growthrate equals 2%.27
The above benchmark parameter set realizes the case of tax evasion, q(1 + s) − 1 < 0.

23
The value of τ does not strongly aﬀect the levels of growth- and welfare-maximizing CIT rates. Therefore, thechoice of the baseline value of τ makes little diﬀerence in the quantitative results of our numerical exercises.
24
We can collect the data from the archive of the prior year IRS Data Books in the Internal Revenue Service’s
website. These are downloadable at https://www.irs.gov/statistics/soi-tax-stats-prior-year-irs-data-books.
25
According to the definition by the Internal Revenue Service, large corporations are those with assets greaterthan 10 million dollars. The upper bound of the asset size of the smallest class is 50 million dollars.
26
The source is IRS Data Book between 2000 and 2017.
27
We do not eliminate the scale eﬀect explicitly in this calibration because we obtain the same results in Section
5.3 if we do. Following the method in pp. 302 of Barro and Sala-ı́-Martin (2004), we can eliminate the scale eﬀect
1
2α
by imposing the relation η = Bα 1−β L 1−β , where B > 0. Given an arbitrary positive number B, we set η satisfying
the above equation. Given such η, we set the scale parameter A to fix the growth rate to 2%. This modification leads
to the same quantities in Table 3. We can also adjust the value of η by varying B.

18

5.3 Results
Main Results
Insert Table 2 here.
Table 2 provides the welfare-maximizing announced CIT rate, τ W M , the growth-maximizing
announced rate, τ GM , and the growth-maximizing eﬀective CIT rate, τ̃ GM for the benchmark
case. We find that τ W M = 0.4025, τ GM = 0.3921, and τ̃ GM = 0.3133. The value of the optimal
CIT rate, 0.4025, is close to the estimated value of 42% by Aghion et al. (2016). As Propositions
3 and 4 indicate, τ W M , τ GM , and τ̃ GM are all larger than β(= 0.1). Importantly, the diﬀerence
between τ W M and β is 0.3025, which is quite large.
The fourth, fifth, and sixth rows of Table 2 provide a decomposition of the total eﬀect, τ W M −β. The eﬀect of CIT evasion, τ̃ GM −β, amounts to 0.2133 and is largest of the three eﬀects. Thus,
the eﬀect of CIT evasion is the primary source of the high optimal CIT rate quantitatively.
Robustness
The eﬀect of CIT evasion depends on the markup rate µ (= Γ(τ )/α − 1) because it is related totheir market power. Therefore, we calculate the optimal CIT rates for the various markup rates:
see Table 3. 28
Insert Table 3 here.
Around the benchmark value (e.g., the case of markup rate = 0.1, ..., 0.5), we find that both
τ GM and τ W M are much higher than β. In particular, the eﬀect of CIT evasion, τ̃ GM − β, issignificantly large for the alternative markup rates. This indicates that the eﬀect of CIT evasionis the main source of the high optimal CIT rates.
Insert Table 4 here.
Unsurprisingly, the level of the optimal CIT rate substantially changes according to the output
elasticity of public services. However, we find that the impact of tax evasion remains relatively
28

We choose the values of α so that the markup rates take the values listed in Table 3. See also Appendix I for
details of the calibration.

19

strong for the alternative values of β, the output elasticity of public services. 29 In Table 4, we
provide the ratio of the welfare-maximizing CIT rate to the output elasticity of public services,τ W M /β. Remember that the criterion in the comparison with the Barro rule is β. Thus, forexample, although τ W M ≈ 0.1 in the case of β = 0.025, we can interpret this rate as a relatively
large value. Because the ratios in Table 4 show that the optimal CIT rates are much higher thanβ, we can confirm the eﬀect of tax evasion on the optimal CIT rate.
Next, we calculate the share of the eﬀect of CIT evasion in the total eﬀect, ττ̃W M−β, for the−βGMalternative values of β. In the benchmark case, the share is 70%. For any case, the eﬀect of CIT
evasion occupies more than half of the total eﬀect. This ensures the robustness of the relative
importance of the eﬀect of CIT evasion.
Finally, we conduct a robustness check of the main result with respect to audit rate q.
Insert Figure 1 here.
Figure 1 illustrates the result. Indeed, the level of the optimal CIT rate changes according to q.
However, for the various alternative values of q, we can confirm that the eﬀect of CIT evasion isthe dominant factor of the high optimal CIT rate relative to the output elasticity of public services.
30

6 Conclusion
This study investigates the optimal CIT in an endogenous growth model with productive public
services, incorporating tax evasion by monopolistically competitive firms of intermediate goods.
We show that the growth- and welfare-maximizing CIT rates are higher than the output elasticity of productive public services. Thus, in view of tax evasion by firms, CIT should be higher
than the output elasticity of public services. This is mainly because the eﬀect of CIT evasion
mitigates the negative eﬀect of CIT on growth and increases the benefit of raising the CIT rate
for the provision of productive public services.

29

As we mentioned in the calibration section, the estimations of β lie in the neighborhood of 0.10 in existing
empirical studies. Although the quantitative performance of the model heavily depends on β, the range consideredhere is suﬃciently wide to keep the model quantitatively plausible.
30
In particular, consider q = 0.089. This is the value chosen in Fullerton and Karayannis (1994). Table 5 showsthat the results do not change for the benchmark parameter (other than q).

20

Under the plausible parameter values, our numerical exercises show that the eﬀect of CIT
evasion is significantly large and the optimal level of the CIT rate is much higher than the output
elasticity of public services.

21

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25

Appendix
A Proof of Lemma 1
Suppose that a firm declares its profit truthfully (π̃i,t = πi,t ). Then, (7) can be rewritten aseπi,t= (1 − τ )πi,t . This is maximized at pi,t = wt /(αb). From (6), we know that the firm’s true

profit is given by πi,t = (1 − α)pi,t x(pi,t ). Thus, if a firm setting the price at p̃i,t declares its
operating profit truthfully, (8) holds for the firm.

B Proof of Proposition 1eProof of (i): Since π̃i,t ≤ πi,t , (7) shows that πi,t= {1 − q(1 + s)τ }πi,t − {1 − q(1 + s)}τ π̃i,t .
eThus, if 1 − q(1 + s) < 0, πi,tincreases with π̃i,t . Therefore, firms declare the largest π̃i,t , whichis equal to πi,t because of π̃i,t ≤ πi,t .
e= (1 − τ )πi,t . The maximization ofSince all firms declare their true profit, we have πi,te
= (1 − τ )πi,t yields p̃i,t = wt /(αbi ). We can obtain the true profit by substituting p̃i,t into
πi,te(6). The expected after-tax profit follows from πi,t
= (1 − τ )πi,t . Since π̃i,t = πi,t holds in (9),we have τ̃ = τ .
e= (1 − τ )πi,t . In this case, the expected afterNext, if 1 − q(1 + s) = 0, we obtain πi,ttax profit is indiﬀerent between whether a firm declares its profit truthfully. The maximizationeof πi,t= (1 − τ )πi,t yields the same results as in the case of truth-telling firms, π̃i,t = πi,t ,p̃i,t = wt /(αbi ), and τ̃ = τ .
Proof of (ii): We first prove the following lemma.
Lemma 2: Suppose 1 − q(1 + s) > 0. If a firm sets the price at p̃i,t , then the declared profit of
the firm satisfies (8).
Proof of Lemma 2: To prove this lemma, we consider (a) firms that declare their operating profits
truthfully and (b) firms that declare their operating profits dishonestly.
(a) Consider firms that declare their operating profits truthfully. Lemma 1 shows that thedeclared profit of these firms satisfies (8).
(b) We next consider the following two types of dishonest firms: those that do not declare (8)and those that declare (8).
˜i,t , which is not equal to (8). From
We begin with a firm of the first type that declares π̃26

Assumption (i), this firm is audited with probability one, q̄ = 1. Thus, the expected after-tax
profit of this firm is given by
(
)
e
˜i,t
˜i,t − 1 × (1 + s)τ πi,t − π̃
˜i,t .
π̃= πi,t − τ π̃(B.1)

We next consider a dishonest firm that declares (8). From Assumption (ii), this firm is auditedwith a probability lower than one, q̄ = q ∈ [0, 1). The expected after-tax profit of this type ofdishonest firms is given byeπ̃i,t= πi,t − τ π̃i,t − q(1 + s)τ (πi,t − π̃i,t ).

(B.2)

e
e
˜i,t
˜i,t , which is not equal
Suppose π̃i,t< π̃holds. This implies that a dishonest firm declares π̃ee
˜i,tto (8). Using (B.1) and (B.2), we rearrange π̃i,t< π̃
as follows:()
˜i,t − (1 + s)τ πi,t − π̃
˜i,t ,
πi,t − τ π̃i,t − q(1 + s)τ (πi,t − π̃i,t ) < πi,t − τ π̃
˜i,t ,⇔ (1 + s)(1 − q)πi,t − [1 − q(1 + s)]π̃i,t < sπ̃⇒ (1 + s)(1 − q)πi,t − [1 − q(1 + s)]π̃i,t < sπi,t ,⇔ [1 − q(1 + s)]πi,t < [1 − q(1 + s)]π̃i,t .
˜i,t ≤ πi,t . If 1 − q(1 + s) > 0, the inequality in the last line indicates
The third line uses the fact π̃
˜i,t ≤ πi,t . Thus, dishonest firms declare (8). Lemma 2 is proved.
π < π̃i,t , which contradicts π̃
eFrom Lemma 2, we have that πi,t
= (1 − τ̃ )πi,t = [1 − q(1 + s)τ ]πi,t − [1 − q(1 + s)]τ π̃i,t ,ewhere π̃i,t is given by (8). Firms maximize this πi,tby choosing pi,t . The first-order condition is

given by
e
α
∂πi,t
1
− 1
= LY,t (αA) 1−α Gt1−α p̃i,t1−α∂ p̃i,t{()}
1 − q(1 + s)τ
wt
×−αbi ++ [1 − q(1 + s)]τ α = 0(1 − α)bi
p̃i,t

(B.3)

Since 1 − q(1 + s) > 0 holds, we have that 0 < (1 − α)[1 − q(1 + s)]τ < [1 − q(1 + s)]τ <1 − q(1 + s)τ , which ensures the second-order condition. Solving this condition yields (11).
Since 1 − q(1 + s) > 0 and (1 − α)[1 − q(1 + s)]τ < 1 − q(1 + s)τ , we have Γ(τ ) > 1, and27

ehence, p̃i,t > wt /(αbt ). If we substitute (11) into (6), we obtain (12). From πi,t
= (1 − τ̃ )πi,tand (12), we obtain (13).
Here, let us defineγ(τ ) ≡ Γ(τ )−1 = 1 −(1 − α)[1 − q(1 + s)]τ(< 1).
1 − q(1 + s)τ

(B.4)

Substituting (8) and (12) into (9), we haveτ̃1−α
= [1 − q(1 + s)]+ q(1 + s)
τ
1 − αγ(τ )

(B.5)

1−α
First, we can easily show that τ̃ < τ because of [1 − q(1 + s)] 1−αγ(τ+ q(1 + s) − 1 = [1 −
)
[
]
1−α
q(1 + s)] 1−αγ(τ− 1 < 0, where γ(τ ) < 1 and 1 − q(1 + s) > 0.
)
1 + αq(1 + s)
, andSecond, from the definition of γ(τ ) and (B.5), lim τ̃ = 0 and τ̃ |τ =1 =
τ →0
1+α
(
]
therefore, we have τ̃ ∈ 0, 1+αq(1+s).
1+αC Relationship between τ̃ and τ
From (B.4), we obtain(1 − α)[1 − q(1 + s)]< 0,[1 − q(1 + s)τ ]2(C.1)(1 − α)(1 − τ )τ (1 + s)dγ(τ )=> 0,dq[1 − q(1 + s)τ ]2(C.2)

γ ′ (τ ) = −or both Γ′ (τ ) > 0 and dΓ(τ )/dq < 0. From (B.5) and (C.1), we obtain d(τ /τ̃ )/dτ > 0. From
(B.5), (C.2), and γ(τ ) < 1, we obtain d(τ /τ̃ )/d(q(1 + s)) < 0.
Finally, we prove dτ̃ /dτ > 0. From (B.5),[1 − q(1 + s)](1 − α)dτ̃=
[1 − αγ(τ ) + ατ γ ′ (τ )] + q(1 + s)dτ
[1 − αγ(τ )]2(C.3)

Thus, we have dτ̃> 0, if and only ifdτ
1 − αγ(τ ) + ατ γ ′ (τ ) > −[1 − αγ(τ )]2 q(1 + s)
[1 − q(1 + s)](1 − α)(C.4)

From γ ′ (τ ) < 0 and γ ′′ (τ ) < 0, dτd [1 − αγ(τ ) + ατ γ ′ (τ )] = −αγ ′ (τ ) + ατ γ ′′ (τ ) < 0 holds. This28

indicates that the LHS of (C.4) is decreasing in τ . Furthermore, the RHS of (C.4) is decreasingdτ̃in τ because of γ(τ ) < 1 and γ ′ (τ ) < 0. Thus, dτ
> 0 for any τ ∈ (0, 1) if the minimum

value of the LHS of (C.4), 1 − αγ(1) + ατ γ ′ (1), is larger than the maximum value of the RHS,2

q(1+s)1−α− [1−αγ(0)]. Using γ(0) = 1, γ(1) = α and γ ′ (1) = − 1−q(1+s), we obtain[1−q(1+s)](1−α){}[1 − αγ(0)]2 q(1 + s)
(1 − α)[1 − αq(1 + s)]
1 − αγ(1) + ατ γ (1) − −=> 0,[1 − q(1 + s)](1 − α)
1 − q(1 + s)
′

(C.5)

dτ̃
and therefore, dτ
> 0 for any τ ∈ (0, 1).

D Derivation of equilibrium conditions
Θ
The price level of firm i and j whose productivity is bi and bj is p̃i,t = αb
wt and p̃j,t = αbΘj wt ,i( )α−1pi,txi,twhere Θ = 1(Γ(τ̃ )) for 1−q(1+s) ≤ (>)0. Combining these with (3) yields pj,t = xj,t=1( ) α−1bjbΘ. Thus, we have xi,t = bji
xj,t . This together with (1), (2), and (3) rewrites p̃i,t = αb
wt
bi
i

into
xi,t =

α2 LY,t
.
1
∫ α
α−1
1−α
(1 − α)Θbi
b dF (b)Nt(D.1)

From (19) and (D.1), we obtain labor employed in the final good sector as follows:
LY,t =(1 − α)Θ
L.
(1 − α)Θ + α2

(D.2)

Substituting (D.2) into (D.1) leads to
[

1
]
α2
bi1−α
L
x(p̃i,t ) =.
∫ α
2(1 − α)Θ + α
b 1−α dF (b) Nt

From (1) and (3), we obtain∫ Nt
0pi,t xi,t di = αYt . Combining

∫ Nt
0

(D.3)

pi,t xi,t di = αYt withe= (1 − τ̃ ) (1 − αΘ−1 ) p̃i,t x(p̃i,t ), we obtain the total expected after-tax operating profit,
πi,t∫ edF (b) = (1 − τ̃ )(1 − αΘ−1 )αYt . By combining the total expected after-tax operating
Nt πi,tprofit with (5), we obtain the gross interest rate
Rt−1 =(1 − τ̃ )(1 − αΘ−1 )αYt
.
ηNt

29

(D.4)

Combining πi,t = (1 − αΘ−1 )p̃i,t x(p̃i,t ) with

∫ Nt

pi,t xi,t di = αYt , the budget constraint

0of the government (17) is rewritten into Gt + Mt = τ̃ (1 − αΘ−1 )αYt . Dividing both sidesof the final good market clearing condition, Yt = Ct + ηNt+1 + Gt + Mt , by Nt and using
Gt + Mt = τ̃ (1 − αΘ−1 )αYt yield
Nt+1
1
=
Nt
η

{

) ] Yt[(Ct
1 − τ̃ 1 − αΘ−1 α
−
Nt Nt

}
,

(D.5)

and substituting (D.4) into (15), we obtain
[
]1/σ
Ct+1
(1 − τ̃ ) (1 − αΘ−1 ) α Yt+1=.
Ct
(1 + ρ)η
Nt+1

(D.6)

Substituting πi,t = (1 − αΘ−1 ) p̃i,t x(p̃i,t ) into (18) reduces to Gt = [1 − Q(q)]τ̃ (1 − αΘ−1 )αYt .
Combining it with (1), (D.2), and (D.3), we obtain NYtt = Ω(τ̃ ). Substituting NYtt = Ω(τ̃ ) into(D.5) and (D.6) and dividing (D.6) by (D.5) leads to (20).

E Proof of Proposition 2
Let us define the right-hand side (RHS) of (20) as ϑ(zt ) ≡ η

11− σ[(1−τ̃ )(1−αΘ−1 )αΩ(τ̃ )/(1+ρ)]1/σ zt.
[1−τ̃ (1−αΘ−1 )α]Ω(τ̃ )−zt

Here, note that the denominator of ϑ(zt ): [1 − τ̃ (1 − αΘ−1 )α]Ω(τ̃ ) − zt must be positive, thatis, zt < z̄ ≡ [1 − τ̃ (1 − αΘ−1 )α]Ω(τ̃ ), otherwise Nt eventually equals to zero from (D.5):
Nt+1= η1 {[1 − τ̃ (1 − αΘ−1 ) α] Ω(τ̃ ) − zt }. When Nt = 0, both output and consumption equalNt

to zero, which violates the first order condition of the representative household.
The properties of ϑ(zt ) for zt ∈ [0, z̄) are as follows:ϑ(0) = 0,lim ϑ(zt ) = +∞,zt →z̄

η 1− σ [(1 − τ̃ )(1 − αΘ−1 )αΩ(τ̃ )/(1 + ρ)]1/σ [1 − τ̃ (1 − αΘ−1 )α]Ω(τ̃ )ϑ (zt ) => 0,{[1 − τ̃ (1 − αΘ−1 )α]Ω(τ̃ ) − zt }21′

η [(1 − τ̃ ) (1 − αΘ−1 ) α(1 + ρ)−1 η −1 Ω(τ̃ )][1 − τ̃ (1 − αΘ−1 ) α] Ω(τ̃ )1/σϑ′ (0) =ϑ′′ (zt ) =

2η

1− σ1

, lim ϑ′ (zt ) = +∞zt →z̄

[(1 − τ̃ )(1 − αΘ−1 )αΩ(τ̃ )/(1 + ρ)]1/σ [1 − τ̃ (1 − αΘ−1 )α]Ω(τ̃ )> 0. (E.1){[1 − τ̃ (1 − αΘ−1 )α]Ω(τ̃ ) − zt }3(E.1) indicates that ϑ(zt ) is monotonically increasing and convex function of zt and takes zero
when zt = 0.
30

On the other hand, the left-hand side (LHS) of (20) represents 45◦ line. Thus, we find that aunique steady state ẑ ∈ (0, z̄), which is unstable exists if and only if ϑ′ (0) < 1 holds. From (D.5),(D.6) and Yt /Nt = Ω(τ ), Ct , Nt , and Yt grow at the same constant rate, Ct+1 /Ct = Nt+1 /Nt =Yt+1 /Yt = ĝ in the steady state.
The rest of this appendix shows that the TVC (16) ensures ϑ′ (0) < 1. (23) and the asset
market clearing condition, Wt = ηNt+1 together with the assumption N0 = 1 transform the−σ t(1−σ)ĝ
TVC (16) into limt→∞ ẑ (1+ρ)
t= 0. To satisfy the TVC, 1 > (1 + ρ)−1 ĝ 1−σ must holds.
−1)(1−αΘ )α
1 > (1 + ρ)−1 ĝ 1−σ and (23) together with (1−τ̃< 1 lead to ϑ′ (0) < 1.
1−τ̃ (1−αΘ−1 )α

F Proof of Proposition 3
F.1 Proof of 1
In this case, since each firm does not evade CIT, Θ = 1 and τ̃ = τ . Then, by (21) and (23), we
obtain τ̃ GM = τ GM = α immediately.
F.2 Proof of 2
In the beginning, note that the decision of optimal announced CIT rate is equivalent to that of
the optimal eﬀective CIT rate. This is because τ̃ is the function of τ from (B.5), and τ̃ is strictlyincreasing in τ , dτ̃ /dτ > 0, for any τ ∈ (0, 1], as shown in Appendix C.
Because the growth rate converges to 0 as τ goes to 0 by the construction of the model, thegrowth rate is maximized at 1 or some interior point in (0, 1).
First, we prove that the growth-maximizing eﬀective CIT is higher than α when it is an interior
point in (0, 1). From the definition of Ω(τ̃ ) and (D.6), growth maximization with respect to τ̃ is
1
] 1−α
[
α
1
γ(τ )
1−α
1−α
equivalent to max f (τ̃ ) = ln(1 − τ̃ )τ̃[1 − αγ(τ )] γ(τ ) 1−α+α2 γ(τ ), subject to (B.5):
τ̃
τ̃
1−α
=
[1
−
q(1
+
s)] 1−αγ(τ+ q(1 + s). The first derivative of f (τ̃ ) is
τ
)

]
]
[
α dτ ′
1
α 1
1
1
α
+
.
f (τ̃ ) = −+γ (τ )−−
1 − α dτ̃γ(τ ) 1 − αγ(τ ) 1 − α + α2 γ(τ )| 1 − τ̃ {z 1 − α τ̃}
{z
}
|
[

′

≡Ψ1 (τ̃ )≡Ψ2 (τ )

(F.1)
1
α 1
It is obvious that Ψ1 (τ̃ ) = − 1−τ̃
+ 1−α
= (1−τ̃α−τ̃≥ 0 for τ̃ ≤ α. Next, we show that theτ)(1−α)τ̃31

sign of Ψ2 (τ ) is negative for τ̃ ≤ α.
signΨ2 (τ ) = [1 − αγ(τ )][1 − α + α2 γ(τ )] − γ(τ )[1 − α + α2 γ(τ )] − αγ(τ )[1 − αγ(τ )]= −α3 γ(τ )2 − (1 − α)[(1 + 2α)γ(τ ) − 1]

(F.2)

1
Here, signΨ2 (τ ) < 0 for γ(τ ) > 1+2α
. Furthermore, (B.4) and (B.5) indicate that γ(τ ) is
τ̃
τ̃
1−τ̃
increasing in q(1+s) and when q(1+s) = 0, γ(τ ) = 1−(1−α)τ , τ = 1−ατ̃and γ( 1−ατ̃) = 1−ατ̃
α−τ̃ +α(1−τ̃ )
1−τ̃
1
τ̃
1−τ̃
1
hold. From 1−ατ̃− 1+2α= (1−ατ̃> 0, we obtain γ( 1−ατ̃) = 1−ατ̃> 1+2αfor τ̃ ≤ α.
)(1+2α)Thus, signΨ2 (τ ) < 0 for τ̃ ≤ α. Combining Ψ1 (τ̃ ) ≥ 0 and Ψ2 (τ̃ ) < 0 for τ̃ ≤ α with γ ′ (τ ) < 0> 0, we obtain f ′ (τ̃ ) > 0 for τ̃ ≤ α. From the discussion so far, we find that((B.5)) and dτ̃dττ̃ GM > α holds.
Next, we consider the case of the corner solution of growth-maximization: τ GM = 1. Asadditionally, we ensure that τ̃ GM > α because τ̃ |τ =1 = 1+αq(1+s).31suming q(1 + s) > α(1+α)−1α
1+αG Proof and intuition of Proposition 4(i) Proof of 1
The maximization condition of social welfare is ∂U= 0. By (26), this is equivalent to∂τ[

1 − (1 + ρ)−1 ĝ 1−σ] ∂ ẑ∂ĝ+ (1 + ρ)−1 ẑĝ −σ
= 0.
∂τ
∂τ

(G.1)

1−α(1−α)τ
σ
From (22) and (23), with Θ = 1, we obtain ẑ = η−1 (1+ρ)− ηĝ. Diﬀerentiating−1 α(1−α)(1−τ ) ĝ[]
η(1+ρ) 1−α(1−α) σ
∂ĝ
∂ĝ
∂ ẑ
it with respect to τ yields ∂τ= α(1−α)ĝ + 1−α(1−α)τ
σĝ σ−1 ∂τ− η ∂τ
. Substituting it(1−τ )21−τ1−α(1−α)τ
σ
into (G.1) and rearranging it using ẑ = η−1 (1+ρ)−1 α(1−α)(1−τ ) ĝ − ηĝ, we have][
1 − (1 + ρ)−1 ĝ 1−σ (1+ρ)[1−α(1−α)]ĝ σ∂ĝ
α(1−α)(1−τ )2
=−
,
∂τ
K

(G.2)

The parameter restriction q(1 + s) > α(1+α)−1
holds in a plausible parameter region because the valid range ofαα, the output elasticity of public service is around 0.1, according to empirical studies. See the calibration in Section
5.2. The corner point τ = 1 can be optimal because the eﬀective CIT rate is lower than 1 when firms evade CIT. If
q(1 + s) is small and α is large, the eﬀective CIT rate becomes low, even though the growth-maximization CIT rate
is high. Thus, the corner solution τ = 1 is optimal in such a case.
31

32

]
1−α(1−α)where K = [1 − (1 + ρ)−1 ĝ 1−σ ] (1+ρ)[1−α(1−α)τ
σĝ σ−1 + α(1−α)(1−τ> 0. Therefore, by (G.2),α(1−α)(1−τ ))∂ĝ
we find that ∂τ|τ =τ W M < 0. This implies τ GM < τ W M because ĝ is a single-peaked function ofτ (see (23) and the definition of Ω(τ̃ ) with Θ = 1).
(ii) Proof of 2
∂ ẑ
∂ ẑ
Since sign{ dU| GM } = sign{ ∂τ|τ =τ GM }, we show ∂τ|τ =τ GM > 0 for q = 0. From (23) anddτ τ =τ
1−α[1−αγ(τ )]τ̃σ(22), with Θ = Γ(τ ) = γ(τ )−1 , we obtain ẑ = η−1 (1+ρ)−1 (1−τ̃ )[1−αγ(τ )]α ĝ − ηĝ, Diﬀerentiating

it with respect to τ , we obtaindτ̃
(1 − αγ(τ )) [1 − α(1 − αγ(τ ))] dτ+ α(1 − τ̃ )γ ′ (τ )∂ ẑ= ĝ σ∂τ τ =τ GM
η −1 (1 + ρ)−1 α [(1 − αγ(τ ))(1 − τ̃ )]2(G.3)

Here, let us define the numerator of (G.3) as J. Through simple algebra, we have γ(τ ) = 1 −[ 1−α ]
(1 − α)τ and τ̃ = 1−αγ(ττ . Hence, utilizing these, we have
)
[

]
[
]
1 − α(1 − αγ) (1 − α)3[] 1 − α + α(1 + α)τ .
J=(1 − αγ(τ )) 1 − α(1 − α)(1 − ατ )(G.4)

Equation (G.4) ensures that J > 0 for any τ ∈ (0, 1]. This completes the proof.
(iii) Intuition of Proposition 4
For an intuitive interpretation, we focus on the marginal eﬀect of raising the tax rate on thegrowth-maximizing rate, τ GM . By (26), the lifetime utility, U0 , depends on the long-run growthrate, ĝ, and the initial consumption, ẑ. At τ = τ GM , the marginal eﬀect of raising τ on ĝ
disappears. Then, the CIT rate aﬀects welfare only through the eﬀect on initial consumption:{}{ ∂ ẑ
}GM
GM . Note that by (22) and (D.4) in Appendix D, the initial
sign dU
|
=
sign
|
τ
=τ
τ
=τ
dτ
∂τ
consumption is decomposed into
)
[
(
]1/σ
) ]
[
(
.
ẑ = 1 − τ̃ 1 − αΘ−1 α Ω(τ̃ ) −η (1 − τ̃ ) 1 − αΘ−1 α(1 + ρ)−1 η −1 Ω(τ̃ )
|
|
{z
}
{z
}
=R/(1+ρ)disposable income
Because the second term depends on the interest rate, this is a Slutsky decomposition of the initialconsumption. Since the interest rate is also maximized at τ = τ GM , the marginal eﬀect on thesecond term disappears here. Consequently, the marginal eﬀect of raising τ on ẑ at τ = τ GM
33

equals that on the disposable income of household.
The disposable income of a household increases by raising τ from τ GM marginally. Thus,τ GM < τ W M . Raising τ increases before tax income Ω(τ̃ ). While Ω(τ̃ ) includes both laborand asset income, CIT is imposed only on the source of asset income, that is, the firms’ profits.
Thus, the disposable income of a household increases as a whole. It suggests that the break inthe coincidence of growth- and welfare-maximization tax rates is because the tax base is CIT.32

H Proof of Remark
The counterpart of Proposition 4 is proved the same way as Proposition 4 because our calculations
do not depend on the expression of the growth rate, ĝ, which is the unique diﬀerence between
the baseline and extended model. We can prove the counterpart of Proposition 3 in the Remark
by modifying Appendix F as follows.
Under the production technology of the final goods, (27) and (28), the wage rate, (2) and the∫ Ntαprice of intermediate goods, (3) are rewritten into wt = (1 − α)L−αY,t 0 (a(Gt , Nt )xi,t ) di =α α−1t(1 − α) LYY,tand pi,t = αAL1−αY,t a(Gt , Nt ) xi,t , respectively. Other equations remain unchanged

with the following exceptions. The definition of Ω(τ̃ ) changes intoΩ(τ̃ ) ≡A

1
1−β

[

(

) ] β
1−α
2α
(1 − Q(q))τ̃ 1 − αΘ−1 α 1−β [(1 − α)Θ] 1−β α 1−β

[∫
×

b

α
1−α

[

] 1−α
1−β
dF (b)

L(1 − α)Θ + α2

1
] 1−β

(H.1)

β

1

1−α

By (23) and (H.1), we have τ̃ GM = argmax (1−τ̃ )τ̃ 1−β (1−αΘ−1 ) 1−β Θ 1−β(

Θ−11−α+α2 Θ−1

1
) 1−β

.

(F.1) changes as follows:
[

]
1
β 1
α dτ ′f (τ̃ ) = −++γ (τ )Ψ2 (τ ).
1 − τ̃
1 − β τ̃1 − β dτ̃
|
{z
}
′

(H.2)

≡Ψ̃1 (τ̃ )≥ 0 for τ̃ ≤ β. As in Appendix F, sign Ψ2 (τ ) < 0 for τ̃ ≤ β
Immediately, Ψ̃1 = (1−τ̃β−τ̃)(1−α)τ̃
because of β < α. This indicates that τ̃ GM ≥ β in the case of the interior solution. Besides, we
32

When the tax system is household total income tax, as in Barro (1990) and Kafkalas et al. (2014), this eﬀect
vanishes. In fact, we find that the first term of ẑ is replaced by (1 − τ̃ )Ω(τ̃ ), and hence, welfare-maximization isequivalent to growth-maximization; see (23).

34

discuss the case of the corner solution (τ GM = 1) in the same way as Appendix F. 33

I Details of calibration
We seek to obtain quantitative implications for tax evasion in OECD countries.
• The distribution of productivity is set in such a way that the distribution of firm size is set
to the Pareto distribution, which is estimated by Axtell (2001).
By (D.1) and (D.2), letting N0 = 1, we have
]
[
α
L
xi
α2
1−α
=
b
,
∫
α
i
bi
(1 − α)Γ(τ ) + α2
b 1−α dF (b)
|
{z
}
≡B

where Γ(τ ) =1−q(1+s)τ
[] . This is the size of intermediate good firms. To

1−q(1+s)τ −(1−α) 1−q(1+s) ταmake its distribution a Pareto distribution, we set the distribution of b 1−α to Pareto distribution with scale parameter ϕ > 0 and shape parameter ψ > 1. Then, letting SCALE = Bϕ,∫ αbecause b 1−α dF (b) = ψ−1
ϕ in B, the distribution of firm size is the Pareto distribution
ψwith scale parameter
SCALE =

α2
ψ−1
L,
2
(1 − α)Γ(τ ) + α ψ(I.1)

and shape parameter ψ. Since the shape parameter is the most important factor of firm
distribution, we set ψ = 1.059 according to the estimate in Axtell (2001), which uses US
data.
• Next, we control the markup rate of intermediate good firms. Letting the markup rate beµ, by (11),Γ(τ )= 1 + µ.
α

(I.2)

As the benchmark value of µ, we adopt 0.2, a usual value of the macroeconomic model

An additional parameter restriction is q(1 + s) > β(1+α)−1. This is a quantitatively reasonable assumption, asαwe explain at the end of Appendix F.
33

35

with imperfect competition. 34• To determine the value of the remaining parameters, we set the benchmark value of τ, q
and s exogenously, as shown in Table 1. For their source, see Table 1 and the text.
• There are four undetermined parameters for (I.1) and (I.2), α, L, SCALE, and ϕ. Here, we
put L = ϕ = 1 and determine α and SCALE by (I.1) and (I.2). Note that productivity bαfollows the Pareto distribution with scale parameter ϕ 1−α and shape parameter 1−αψ, sinceαα

we assume b 1−α follows the Pareto distribution with scale parameter ϕ and shape parameterψ. Because the scale of productivity may be arbitrarily fixed whenever the distribution offirm size is properly controlled, we set ϕ = 1. We simply normalize L = 1. Since thenumber of the intermediate good firms is a continuum, the minimum of firm size among
them does not have to correspond to the minimal number of employees in actual data (andonly the shape of the distribution, the curvature of the density function, matters). Thus,
we do not care about the magnitude of parameter SCALE, which is determined by (I.1)
for given α and the other parameters. Parameter α is pinned down by the condition of themarkup rate, (I.2).
• We explain the determination of α. Through long but straightforward algebra, (I.2) can berearranged as the quadratic equation with respect to α as follows: Φ2 α2 + Φ1 α + Φ0 = 0,[][]
where Φ2 = (1 + µ) 1 − q(1 + s) τ , Φ1 = (1 + µ)(1 − τ ) and Φ0 = − 1 − q(1 + s)τ .
By Φ2 > 0 and Φ1 > 0, a necessary and suﬃcient condition for a unique solution in (0, 1)
is Φ0 < 0 and Φ2 + Φ1 + Φ0 > 0, which holds for any parameter set. The solution is given√−Φ1 + Φ21 −4Φ2 Φ0by α =.
2Φ2
1• Finally, by (23), we choose the value of A 1−αε /η to fix the long-run growth rate to 2%.
This completes the parameter specification for conducting the numerical exercises.

34

For example, Rotemberg and Woodford (1999) adopt this value. However, because markup rate is a key parameter of our analysis, we conduct numerical exercises for alternative values. See Table 3.

36

Table 1: Baseline Parameter Value

ψ
α
ϵ
τ
q
s
k
σ
ρ
L
1

A 1−αϵ /η

benchmark
1.059
.8620
.1160
.2706
.096
.5
.1667
1.5
.0204
1
34.2321sourceAxtell (2001)
markup rate = .2
∂ ln Y /∂ ln G = .1
average CIT rate of OECD countries
average audit rate of the majority class of the US large corporations
Fullerton and Karayannis (1994)
Q(q) = kq, enforcement budget and CIT revenue of the IRS
Jones et al. (1993)
Jones et al. (1993)
normalization
growth rate = .02

Table 2: The welfare- and growth-maximizing tax rates and the contributions to τ W M − β for thebenchmark
τWM

.4025

τ GM

.3921

τ̃ GM

.3133

β

.1000

the tax base eﬀect: τ W M − τ GM.0104

the diﬀerence in tax rates: τ GM − τ̃ GM.0787

the eﬀect of CIT evasion: τ̃ GM − β.2133

total eﬀect: τ W M − β

.3025

37

Table 3: The optimal CIT rates for alternative markup rates (benchmark: markup rate = .20)
markup rate

.10

.20

.30

.40

.50

.75

1.00

τWM

.5258

.4025

.3324

.2868

.2549

.2058

.1783

τ GM

.5077

.3921

.3248

.2806

.2493

.2007

.1733

τ̃ GM

.3761

.3133

.2718

.2422

.2201

.1835

.1617

τ W M − τ GM

.0181

.0104

.0075

.0062

.0056

.0050

.0050

τ GM − τ̃ GM

.1315

.0787

.0529

.0383

.0291

.0171

.0116

τ̃ GM − β

.2761

.2133

.1718

.1422

.1201

.0835

.0617

τWM − β

.4258

.3025

.2324

.1868

.1549

.1058

.0783

Table 4: We provide τ W M , the ratio of τ W M to β, and the shares of the eﬀect of CIT evasion in
the total eﬀect for alternative output elasticities of public services (benchmark: β = .10).
β

.025

.05

.10

.15

.20

τWM

.0986

.1980

.4025

.6188

.8488

τ W M /β

3.94

3.96

4.03

4.13

4.24

τ̃ GM −β
τ W M −β

.8833

.8225

.7025

.6000

.5084

38

Table 5: The optimal tax rates in the case of q = .089 for the benchmark

τWM

.4162

τ GM

.4056

τ̃ GM

.3203

β

.1000

the tax base eﬀect: τ W M − τ GM

.0105

the diﬀerence in tax rates: τ GM − τ̃ GM.0853

the eﬀect of CIT evasion: τ̃ GM − β.2203

total eﬀect: τ W M − β.3162

Figure 1: The growth- and welfare-maximizing tax rates (τ W M , τ GM , τ̃ GM , and contributions toτ W M > β for alternative values of q)
The growthï and welfareïmaximizing tax rates

0.7

τWM
τGM
~
τ GM

0.5

0.6

The tax base effect

0.3

0.4

The difference intax rates

0.2

The effect of CIT evasion
benchmark

β
0.00

0.05

0.10

q

39

0.15