ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E05−1

Capital Income Tax Evasion and Welfare Levels in
an Overlapping Generations Modelby
Hideya Kato
Mitsuyoshi Yanagihara

January 2005

ECONOMIC RESEARCH CENTERSCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Capital Income Tax Evasion and Welfare Levels in an
Overlapping Generations Model
Hideya Katoa and Mitsuyoshi Yanagiharab∗a,bGraduate School of Economics, Nagoya University, Nagoya, Aichi, 464-8601, Japan

January 18, 2005

Abstract
We construct an overlapping-generations model where individuals evade capitalincome tax, imposed on their savings and carry out the long-run analysis, as well as theshort-run analysis. We consider that the detection probability depends on the amountof undeclared savings, the interest rate and the detection probability parameter.
It is shown that a rise in the capital income tax rate (the detection probability
parameter or the penalty rate) lowers (raises) capital stock both in the short- andthe long-run. The change in undeclared savings depends on this indirect eﬀect, in
addition to the direct eﬀects of such parameters. It is also found that, as opposed to
the results of static models, a rise in the tax rate (the detection probability parameteror the penalty rate) can increase welfare in the short-run (the long-run).

Keywords: Tax evasion; Capital income tax; Capital accumulation; Overlappinggenerations model
JEL classification: D91; E22; H26∗

Corresponding author. Tel.: +81-52-789-5952; fax: +81-52-789-4924; e-mail: yanagi@soec.nagoya-u.ac.jp.

1

1Introduction

This paper constructs an overlapping-generations model in which individuals evade a capital income tax imposed on their savings, to examine how the changes in the capital income
tax rate, the detection probability of tax evasion and the penalty rate aﬀect the capitalstock levels, undeclared savings and the welfare levels of individuals.
As noted, extensive studies treating tax evasion have been carried out from a theoreticalstandpoint, examining the eﬀects of the actions of governments and tax authorities on the
tax evasion behavior of individuals 1 . The first theoretical study of note, which focuses on
the tax evasion behavior of individuals, is that of Allingham and Sandmo (1972), where
the probability that the concealment of income is detected is given and the amount of thepenalty imposed is determined by the level of concealed income 2 .
Contrary to this assumption, Yitzhaki (1987) assumes that the probability with which
the concealment of income is detected is increasing in the amount concealed. The results
obtained in this study are consistent with the facts found by Clotfelter (1983) and Poterba(1987) that a rise in the tax rate will encourage tax evasion behavior. Yitzhaki (1974),
on the other hand, presumes that the amount of the penalty depends not on the level ofconcealed income, but rather on the amount of tax evaded.
In contrast to the above studies which are made within a static and partial equilibrium
framework, Caballe and Panades (1997) and Lin and Yang (2001) examine the eﬀects of tax
evasion behavior on economic growth from a dynamic macroeconomics viewpoint, withina general equilibrium framework 3 . Caballe and Panades (1997), based on a Diamond1

A notable survey on tax evasion theory is that of Cowell (1990) and Andreoni, Brian and Feinstein(1998).
2

Much of the studies such as Yitzhaki (1974) and Yaniv (1994) have followed Allingham and Sandmo(1972).
3

Chen (2003) incorporates the tax evasion behavior into Barro (1990) to examine the optimal income2(1965) -type, overlapping-generations model, deals with the tax evasion of a lump-sum
tax imposed on the individuals of the young generation, of which penalty is levied during
the individual’s old period. Under the circumstance that all taxes and penalties areallocated to the public goods which generate a positive externality on the productivity ofthe private sector, it concludes that a rise in the penalty rate or the detection probabilityof the concealment of income has a negative eﬀect on “eﬀective savings” (the average levelof savings in the economy). Lin and Yang (2001) introduce both a Barro (1990) -typeutility function, on which public goods have an externality and an AK-type production
function into a portfolio selection model. Contrary to the results obtained by the portfolio
selection model in a static framework, a higher income tax rate may encourage tax evasion.
It is also revealed that, although a rise in the tax rate leads to an initial drop in economic
growth, it will raise economic growth in the long-run.
As shown above, research on the tax evasion behavior of individuals has recently been
extended in the direction of taking a dynamic viewpoint. However, past studies have notdealt with tax evasion in terms of savings, i.e., capital income tax evasion, and not inferto the eﬀects on welfare of individuals 4 . As it is well-known, capital income taxation will
lower the capital stock level, and therefore, the utility level in the absence of capital income
tax evasion; in the presence of capital income tax evasion, it is not clear, though it lowers
the capital stock levels, whether the rise in the capital income tax rate necessarily lowers
the utility levels of individuals. This is because the change in the capital accumulation,
which brings about the changes in the interest rate and wages, will make individuals’ tax
evasion behavior change in the long-run. As Poterba (1987), which analyzes the eﬀects ofmarginal tax rates on the reporting of capital gains, emphasizes the importance of a longtaxation
4

As stated in the appendix of Clotfelter’s (1983) study, while the declaration rate for wage income is100%, the declaration rate for income from interest and dividend income are 98.3% and 97.5%, respectively.

3

run eﬀect, as well as a short-run eﬀect, it is required that the capital income tax evasion
should be analyzed in a dynamic framework. Therefore, although Caballe and Panades(1997) and Lin and Yang (2001) have all focused on economic growth, the utility levels of
the individuals have to be adequately covered in the context of tax evasion behavior.
This paper introduces capital income tax evasion behavior by individuals into theoverlapping-generations model by Diamond (1965), and conducts a dynamic analysis thattakes capital accumulation into account. In this paper, the detection probability will be
endogenized in a Yitzhaki (1987)-fashion. Considering this detection probability and thepenalty by the tax authority, the individuals are capable of evading the capital income
tax by declaring a false amount of savings, i.e. concealing a part of their savings.
The main results of this paper are as follows: first, on one hand, a rise in the capitalincome tax rate lowers the level of capital stock in both the short- and the long-run. On
the other hand, a rise in the detection probability parameter or the penalty rate increases
the levels of capital stock. The latter finding is in contrast to that of Caballe and Panades(1997).
Next, the amount of undeclared savings may increase even when the capital income tax
rate falls, or the detection probability parameter or the penalty rate rises. This is because
the capital accumulation process is incorporated. That is, the eﬀect on the amount of
undeclared savings is determined by the eﬀect on the capital stock, above mentioned.
Lastly, we show that while a rise in the capital income tax rate can increase the welfare,or the utility levels, of individuals in the short-run, a rise in the detection probability
parameter or the penalty rate can raise utility in the long-run.
This paper is organized as follows: In Section 2, we will present a basic model. Section
3 examines both the short- and the long-run eﬀects caused by increases in the capitalincome tax rate, the detection probability and the penalty rate on the levels of capital4

stock and undeclared savings. In Section 4, we examine the short- and the long-run eﬀects
on welfare levels. In Section 5, we give numerical examples. Lastly, Section 6 contains theconclusion of this paper.

2

A Basic Model

The model developed in this paper extends the overlapping-generations model proposed
by Diamond (1965) to include circumstances where a capital income tax imposed on
individuals can be evaded. The economy begins from an initial period (t = 1) and lasts
forever. Individuals are identical and live for two periods: the young and the old periods.
In every period, there exists the young generation and the old generation. For the sake ofsimplicity, we assume that there is no population growth and that the population size ofeach generation is normalized to one.

2.1

Maximization of Individuals’ Expected Utility

In this subsection, focusing on the individuals who live in the t-th period as part of theyoung generation (henceforth called “the t-th generation”), we formulate the optimization
behavior of those individuals with an incentive to evade tax.
The individuals supply one unit of labor inelastically to obtain wages, wt , during their
young period. The wages are allocated between consumption during the young period, c1t ,and savings, st 5 . Therefore, the budget constraint in the young period can be expressed
as follows:

c1t = wt − st .
5

(1)

Subscripts generally indicate the period. However, in regards to the individuals’ consumption, theformer subscript represents the generation (1 and 2 correspond to the young and the old generation,
respectively) while the latter subscript represents the period.

5The savings of the individuals are allocated to consumption in the old period. Savings
bear interest at a rate of rt+1 > 0, on which a capital income tax is imposed at a fixed
rate, 0 ≤ τ ≤ 1. The net interest rate, which individuals receive, becomes (1 − τ )rt+1 .
Since the amount of capital income tax is calculated based on the amount of savings
as declared by the individuals, it is possible for them to evade the tax by declaring a false
amount of savings, i.e., concealing a part of their savings, xt+1 . The tax authority can
detect this tax evasion behavior by a probability of 0 < pt < 1. Following Allingham andSandmo (1972), when tax evasion is detected, a penalty of the amount θrt+1 xt+1 , will be
imposed, where θ(> τ ) is a fixed penalty rate, or a heavy additional tax rate 6 . Therefore,
the budget constraint in the old period becomes:
c2t+1 = (1 − pt+1 )[(1 + rt+1 )st − rt+1 τ (st − xt+1 )]+pt+1 [(1 + rt+1 )st − rt+1 τ (st − xt+1 ) − θrt+1 xt+1 ].

(2)

In order for an incentive to evade tax to exist, the expected return when the tax can beevaded should be larger than that in the case where the tax cannot be evaded, that is,
τ > pt+1 θ. In addition, when there exists an inner solution,st > xt+1(3)

must hold in each period. We assume that the detection probability of tax evasion isincreasing in the amount of the individuals’ undeclared savings, that is,

pt+1 = δ(rt+1 xt+1 )α ,6

Behavior of the government and the tax authority will be covered in detail in Section 2.3.

6(4)

where δ > 0 is a fixed detection probability parameter which the tax authority can control
as a policy variable and α ≥ 1 7 . This relation is assumed to be known to all individuals.
Individuals’ utility is assumed to be a time-separable log-linear function consisting of
the amount of consumption in the young period and the expected amount of consumption
in the old period, u(c1t , c2t+1 ) = log(c1t ) + log(c2t+1 ) 8 . The individual’s problem is tomaximize this utility subject to (1), (2) and (4). Then, the amount of undeclared savingsand the savings can be obtained as:

xt+1 =

st =

Φ
rt+1

,

wt
ατ Φ
−
,
2
2(1 + α)[1 + (1 − τ )rt+1 ]

(5)

(6)

where Φ ≡ {τ /[δθ(1 + α)]}1/α > 0.
The features of undeclared savings and savings can be stated as follows. First, undeclared savings are decreasing in the interest rate and neutral in regards to the wages.
The reason for the former feature is that a rise in the interest rate increases the detection
probability of tax evasion. The latter feature is attributed to the fact that the utility
function is time-separable.
Second, the savings function is increasing in both the interest rate and the wages. The
former feature and be explained as follows. In the absence of tax evasion, it is well-known
that, in a two-period setting in which the utility function is a log-linear, the amount of the7This assumption regarding the detection probability is, qualitatively, the same as the one in Yitzhaki(1987), which is p0 (x) > 0 and p00 (x) ≥ 0.
8This formation is the same as the one in Chen (2003).

7

savings will be independent of the interest rate. However, in the presence of tax evasion,
this dose not hold: the change in the interest rate will have an eﬀect on the intertemporal
allocation. More concretely, when the interest rate rises, the return from savings increases
and therefore, the amount of savings increases. The latter feature is the same one which
can be obtained from the log-linearity of the utility function, as the same with the case in
the absence of tax evasion.
Summarizing the above results, we arrive at the following Lemma.

Lemma 1 (1) If individuals’ utility function is log-linear and time separable and (2) the
detection probability parameter is increasing in the amount of undeclared savings:
1. Undeclared savings are decreasing in the interest rate and independent of wages.
2. Savings are increasing in the interest rate and increasing in wages.

Then, we will state the eﬀects of the variables given by the government and the tax
authority, τ , δ and θ, on the amounts of savings and undeclared savings. First, when thegovernment raises the capital income tax rate, undeclared savings will increase because
the incentive to evade tax rises. On the other hand, the rise in the capital income tax rate
decreases the amount of savings because it decreases the return from savings 9 .
Second, when the tax authority raises the detection probability parameter or the
penalty rate, individuals will increase their savings and decrease their undeclared savings. The return from tax evasion decreases as the detection probability parameter or the
penalty rate increases. A decrease in undeclared savings, contrary to the first case, leads
to the increase in the amount of savings.
9

Strictly speaking, there is another negative eﬀect of the rise in capital income tax rate on the amountof savings. As stated, the rise in capital income tax rate increases declared savings, which implies theincrease in the return from tax evasion. Therefore, the savings will be less needed.

8Summarizing the above results, we arrive at the following Lemma.

Lemma 2 (1) If individuals’ utility function is log-linear and time separable and (2) the
detection probability parameter is increasing in the amount of undeclared savings:
1. A rise in the capital income tax rate increases undeclared savings and decreases
savings.
2. A rise in the detection probability parameter or the penalty rate decreases undeclaredsavings and increases savings.

As we will see in the following sections, these results may reverse in the general equilibrium
framework where capital accumulation is considered. That is, a rise in the capital income
tax rate may decrease undeclared savings and a rise in the detection probability parameteror the penalty rate may increase them 10 .

2.2

Maximization of Firms’ Profit

Firms produce consumption goods from capital and labor, using a Cobb-Douglas type,
constant returns to scale production technology. Denoting the capital share and the technology parameter as 0 < γ < 1 and A > 0 respectively, produced goods per capita can be
expressed as a function of capital stock per capita shown as follows: yt = Aktγ . Assumingthat the price of the goods is normalized to one, that there is no capital depreciationand that production is carried out under a perfect competitive market, the first orderconditions for profit maximization can be written as:

rt = Aγktγ−1 ,10

See Proposition 1 and 2.

9(7)

wt = A(1 − γ)ktγ .

(8)

No tax is imposed on firms: firms’ tax evasion behavior is beyond the scope of this paper.

2.3

Government and Tax Authority Behavior

We have already referred to the taxation behavior of the government, and the tax evasion
detection and penalty levy behavior of the tax authority in Section 2.1. In this subsection,
we further elaborate on their behavior.
The government imposes a capital income tax at a constant rate on individuals’ interest income from savings. The tax authority attempts to detect individuals’ tax evasion
behavior and imposes a penalty on oﬀenders. We will not explicitly deal with the expenditure side of the government and the tax authority: all government’s income, capitalincome tax and penalty revenue, will be allocated to government consumption and it isassumed that the government keeps a balanced budget on the whole. In addition, anychanges to the detection probability parameter is assumed not to incur any costs 11 .

2.4

Equilibrium

The capital market equilibrium in each period is attained when individuals’ savings (supplyof capital) is equal to firms’ demand for capital, that is, kt+1 = st holds. Therefore, from
(6), (7) and (8), the capital market equilibrium condition can be expressed as:

kt+1 =111τ αΦ
.
A(1 − γ)ktγ −γ−122(1 + α)[1 + (1 − τ )Aγkt+1](9)Although many studies assume that a government’s income is transferred to the individuals in someway, we will consider the behavior of the government and the tax authority as, so to speak, “given”, inorder to focus on the individuals’ tax evasion behavior.

10This condition is clearly dependent upon kt and kt+1 . Then, substituting (7) into (5), thedeclared savings become

xt+1 =

Φ
γ−1
Aγkt+1.

(10)

Similarly, in the steady state, the capital market equilibrium can be rewritten by
substituting kt = kt+1 = k into (9):1τ αΦk = A(1 − γ)kγ −,2
2(1 + α)[1 + (1 − τ )Aγkγ−1 ](11)and the declared savings can be rewritten as

x=

3

Φ
.
Aγkγ−1(12)The Eﬀects of Changes to the Policy Parameters on Capital Stock

In this section, first, we will examine the short- and the long-run eﬀects on the capitalstock level and undeclared savings when the government raises the capital income tax rate
for period t + 1. Next, we will examine the eﬀects caused when the tax authority raises
the detection probability parameter or the penalty rate. It must be noted that we assume
that once the policy parameters have been changed, they will be kept constant.

11

3.1

The Eﬀects of the Capital Income Tax

As seen in Appendix, when the government raises the capital income tax rate, the shortand the long-run eﬀects on the capital stock are as follows:γ−1dkiΦ 1 + α + (1 + α − τ )Aγki
< 0,
=−
dτ
∆i 2(1 + α)[1 + (1 − τ )Aγkiγ−1 ]2(13)

where ∆i > 0, and i = t + 1 in the short-run and i = s in the long-run (the steady state)
12 . From (13), a rise in the capital income tax rate decreases the capital stock in both theshort- and the long-run.
A rise in the tax rate will have two eﬀects on savings incentive. One, a negative savings
incentive eﬀect: a rise in the tax rate will discourage the incentive to save, and decrease
the capital stock. In the short-run, being as this is the only eﬀect present, the capital stock
will always decrease. Two, a long-run savings incentive eﬀect: in addition to the negativeeﬀect, the above mentioned decrease in the capital stock (or the supply of capital) willraise the interest rate, which will in turn lower wages in the long-run. It should be notedthat a rise in the interest rate will attract savings, and hence, work in the direction ofraising the capital stock. Therefore, the long-run savings incentive eﬀect works in either
direction. However, since the first negative eﬀect dominates the second eﬀect, the capitalstock will decrease in both the short-and the long-run.
On the other hand, the eﬀects on undeclared savings are as follows:dxi(1 − γ)Φ dki
Φ
+
=
.
γ−1
dτ
Aγkiγ dτατ Aγki

12

Henceforth, we will use these expressions, unless otherwise noted.

12

(14)

While the first term of (14) is positive, as indicated in (13), the second term becomes
negative. Therefore, neither the signs in the short- nor the long-run is determined on
the whole. These terms can be interpreted as having two separate eﬀects on undeclaredsavings, the same as the aforementioned eﬀect on the capital stock. One, the first term
is a (direct) positive undeclared savings incentive eﬀect, as shown in Lemma 2: a rise in
the tax rate stimulates the incentive to evade tax. Two, the second term is a (indirect)
negative undeclared savings incentive eﬀect. As we have seen in (13), a rise in the capitalincome tax rate lowers the capital stock. This brings about an increase in the interest
rate, and subsequently, a rise in the detection probability. Consequently, as mentioned in
Lemma 1, undeclared savings will decrease.
In general, because the direct eﬀect will dominate the indirect eﬀect, a rise in thecapital income tax rate has the eﬀect of increasing undeclared savings. Oppositely, if thelatter dominates the former, the overall eﬀect will prove to be negative. That is, thedirection of the eﬀect is determined by the magnitude of the above eﬀects on the capitalstock.
The results obtained here can be summarized in the following Proposition.
Proposition 1 When the capital income tax rate rises, in both the short- and the longrun,
1. The capital stock decreases.
2. Undeclared savings decrease if the negative indirect eﬀect dominates the positive direct eﬀect.

13

3.2

The Eﬀects of Detection Probability Parameter and the Penalty
Rate

In this subsection, we will examine the eﬀects of a rise in the detection probability parameter or the penalty rate by the tax authority on the capital stock and undeclaredsavings.
The eﬀects of rises in the detection probability parameter and the penalty rate on thecapital stock are

dki
τΦ
1
> 0,
=
γ−1
dδ
∆i 2(1 + α)δ[1 + (1 − τ )Aγkt+1
]

(15)

dki
τΦ
1
> 0,
=
γ−1
dθ
∆i 2(1 + α)θ[1 + (1 − τ )Aγkt+1](16)respectively. In both cases, these signs are positive (in both the short- and the long-run).
These results can be divided into two eﬀects. The first eﬀect is a positive direct eﬀect
on savings (and capital stock): an increase in these parameters will induce an increase in
the return from savings as shown in Lemma 2. As this is the only eﬀect present in the
short-run, the capital stock will necessarily increase. The second eﬀect is an indirect eﬀect
brought on by a change in the interest rate, which we have seen in the previous section.
Since the former eﬀect dominates the latter eﬀect, the capital stock will necessarily increasein both the short- and the long-run.
Based on the above results, the eﬀects of an increase in the detection probability
parameter or the penalty rate on undeclared savings can be written as follows:14dxi(1 − γ)Φ dki
Φ
+
=−
,
γ−1
dδ
Aγkiγ dδαδAγki(17)dxi(1 − γ)Φ dkiΦ
=−.
γ−1 + Aγk γdθdθαθAγkii(18)

While the first terms are always negative, as seen in (15) and (16), the second terms arealways positive. As such, these signs are not decisive. The first negative eﬀect is to lower
the incentive for tax evasion directly and the second positive eﬀect is to stimulate the
incentive, as a result of a fall in the interest rate caused by an increase in the capital stock13.
In general, as it is the case with a rise in the capital income tax rate, since the direct

eﬀect generally dominates the indirect eﬀect, a rise in the detection probability parameteror the penalty rate tends to decrease undeclared savings. In contrast, if the eﬀect on thecapital stock is suﬃciently large enough that the indirect eﬀect dominates the direct eﬀect,
a rise in the policy parameters increases undeclared savings.
These results can be summarized as follows.

Proposition 2 When the detection probability parameter or the penalty rate rises, in boththe short- and the long-run:
1. The capital stock increases.
2. Undeclared savings increase if the positive indirect eﬀect dominates the negative direct
eﬀect.
13

It should be noted that the detection probability is dependent upon the interest rate

15

4

The Eﬀects of Changes to the Policy Parameters on Utility Levels

To see the eﬀects of a rise in the capital income tax rate, the detection probability parameter or the penalty rate on the utility levels of individuals, we first describe the utility
levels of individuals in the short-run can be written as:v(kt , kt+1 ; τ, θ, δ) = log[A(1 − γ)ktγ − kt+1 ]∙¸ατγ+ log kt+1 + (1 − τ )Aγkt+1 +
Φ .
1+α

(19)

In the same way, the utility levels of individuals in the long-run can be written as:v(k; τ, θ, δ) = log[A(1 − γ)k γ − k]∙¸ατγ+ log k + (1 − τ )Aγk +
Φ .
1+α

4.1

(20)

The Eﬀects of the Capital Income Tax Rate on Utility Levels

First, diﬀerentiating (19) with respect to τ , we can obtain the eﬀect of the capital income
tax rate on utility levels in the short-run,

γ−1 ½

Aγkt+1dv(kt , kt+1 ; τ, θ, δ)
=
dτ
c2t+1

−(kt+1 − xt+1 ) − (1 − τ )(1 − γ)

dkt+1
dτ

¾

,

(21)

where all values of the variables are the ones obtained in the former section. From (3), (10)16and kt+1 = st , the sign of the first term is negative, and from (13), the sign of the second
term is positive 14 . The first negative eﬀect is that the rise in the capital income tax rate
directly decreases utility levels, because it reduces the return from declared savings. The
second positive eﬀect is that the rise in the capital income tax rate indirectly increases
utility levels, because the decrease in capital stock raises the interest rate, and in turn,raises return from savings. In conclusion, the eﬀect of the capital income tax rate on
utility levels in the short-run depends on the direct and the indirect eﬀects, and therefore,
the sign of (21) is indeterminate.
Next, diﬀerentiating (20) with respect to τ , the eﬀect of the capital income tax rate
on utility levels in the long-run can be obtained by:

dv(k; τ, θ, δ)
Aγkγ−1
=
dτ
c2s

½
¾
γ−1 ∂k)]−(k − x) + (1 − γ)[1 − (1 − τ )(1 − Aγk
< 0.
∂τ

(22)

It is clear that the sign is always negative, that is, the rise of the capital income tax
rate unambiguously decreases utility in the long-run. This result can be also interpreted
into two eﬀects, the negative direct eﬀect and indirect eﬀects. The direct eﬀect, which isbrought by the decrease in the return from declared savings, is represented by the firstterm; the indirect eﬀect, which is caused by the decrease in the return from savings, isrepresented by the second term. Unlike the case of the short-run, the decrease in capitalstock lowers utility levels in the long-run, because the economy is always dynamicallyeﬃcient.
The results obtained here can be summarized in the following Proposition.

14

γ−1
It should be noted that because xt+1 = Φ/(Aγkt+1) and kt+1 = st , the first term in the curly brackets

represents the “declared savings”, (st − xt+1 ).

17

Proposition 3 When the capital income tax rate rises,
1. Utility levels always fall in the long-run.
2. Utility levels rise in the short-run if the positive indirect eﬀect dominates the negative
direct eﬀect.

4.2

The Eﬀect of the Detection Probability Parameter and the Penalty
Rate on Utility Levels

First, diﬀerentiating (19) with respect to δ and θ, we can obtain the eﬀects of such parameters as the detection probability parameter and the penalty rate on utility levels inthe short-run:

dv(ki , kj ; τ, θ, δ)
θE (1 − τ )(1 − γ) ∂kj
−
=−
< 0,
dδ
c2j
c2t+1
∂δ

(23)

dv(ki , kj ; τ, θ, δ)δΦ1−α (1 − τ )(1 − γ) ∂kj
−
=−
< 0,
dθ
c2j
c2j
∂θ

(24)

where the signs of (23) and (24) are always negative.
The eﬀects can be interpreted into two negative eﬀects: the direct and the indirect
eﬀects. The first eﬀect is brought by the decrease in the return from declared savings; thesecond eﬀect is caused by the decrease in the return from savings, because the increase incapital stock decreases the interest rate.
Next, the long-run eﬀects are calculated as:

18

dv(k; τ, θ, δ)
θE (1 − γ)Aγkγ−1 [1 − (1 − τ )(1 − Aγkγ−1 )] ∂k
+
=−
,
dδ
c2s
c2s
∂δ

(25)

dv(k; τ, θ, δ)
δE (1 − γ)Aγkγ−1 [1 − (1 − τ )(1 − Aγkγ−1 )] ∂k
+
=−
.
dθ
c2s
c2s
∂θ

(26)

The signs of the first term in (25) and (26) is negative and the sign of the second term is
positive. Hence, the signs of (25) and (26) become ambiguous, that is, the rise in theseparameters are independent upon the configuration of these direction and indirect eﬀects.
The direct eﬀect is brought by the decrease in the return from undeclared savings. On
the other hand, the indirect eﬀect is from the increase in capital stock.
The results obtained here can be stated in the following Proposition.

Proposition 4 When the detection probability parameter or the penalty rate rises,
1. Utility levels always fall in the short-run.
2. Utility levels rise in the long-run if the positive indirect eﬀect dominates the negative
direct eﬀect.

In Table 1 and 2, we summarize the results obtained in Lemma 2 under the subjective
equilibrium and in Proposition 1 to 4 under the general equilibrium.

19

5

Numerical Examples

In this section, we will present numerical examples for a comparative static analysis, thatis, the eﬀects of γ, A, τ , θ and δ on k, x and v, so as to clarify that the results mentioned
above might occur. Here, we use two types of capital share. The one is that in a “narrow
sense,” in which capital is defined only as physical capital, is about 0.3. The other one
is that in a “broad sense,” in which capital is defined not only as physical but humancapital. For example, it is 0.75 in Barro and Sala-i-Martin (1995) and is from 0.35 to 0.6
in Mankiw, Romer and Weil (1992). Therefore, we use the capital share in its “narrow
sense” in Table 3 and 4 (γ = 0.3), and the one in its “broad sense” in Table 5 (γ = 0.6),
respectively.
For each table, the second column contains the given (initial) values of τ , θ and δ,and the third to fifth columns contain the (initial) equilibrium values of the capital stock,k, undeclared savings, x, and utility level, v respectively. The sixth column indicates thecondition of dynamic stability, the value of dkt+1 /dkt (the upper figure), and the detection
probability, p (the lower figure). If 0 < ∆ < 1 holds, the stability condition is satisfied;
this condition holds for all cases given in the Table. In addition, the detection probability
ranges from 0 to 1 for all cases, which is consistent with the presumption. The resultsof the comparative static analysis are shown in the eighth to thirteenth columns: where
the parameters in the seventh column are changed, the short-run eﬀects of k, x and v
are written in the former three columns, and the long-run eﬀects are in the latter three,respectively.
Table 3 treats an “orthodox” case, as we have indicated in Proposition 1 to 4, where,
in general, a rise in τ increases x and decreases v, and a rise in θ or δ decreases both xand v.
On one hand, though the direction of change in the capital stock is the same in both
20

the short- and the long-run, the degree of change in the long-run is larger than that in
the short-run: by raising τ , the capital stock decreases more and by raising θ and δ, thecapital stock will decreases less in the long-run. This is expected because monotonous
convergence to the steady state is assumed. On the other hand, in regards to undeclaredsavings, the long-run eﬀects are smaller than the short-run eﬀects.
In regards to the eﬀects on utility levels, a rise in τ lowers them in the long-run morethan in the short-run; in contrast, the reverse is true for a rise in θ and δ. These findings
correspond to the changes in the capital stock discussed above: as noted in Proposition
1, whether in the short-run or the long-run, the capital stock level will decrease. To
elaborate, in the case of a rise in τ , the level of capital stock in the long-run becomes lower
than that in the short-run, while in the case of a rise in θ and δ, the level of capital stockin the long-run becomes higher than that in the short-run; therefore, in the long-run, theresources in the economy will decrease and increase, respectively 15 .
Table 4 and 5 show two “paradoxical” cases with regards to the eﬀects on utility levels,
as we have mentioned in Proposition 3 and 4: utility levels become higher in the short-run
when τ rises (Table 4), and they become higher in the long-run when θ and δ rise (Table5). In the former case, because δ is low enough that individuals can evade tax without
diﬃculty, it can be thought that a rise in the capital income tax rate increases undeclaredsavings; accordingly, (the expected) consumption in the old period also increases. In thelatter case, it can be recognized that the indirect eﬀect caused by capital accumulation
dominates the direct eﬀect 16 . In this case, it must be noted that, the larger the capitalshare of the production function, the more highly accumulated capital will be in the long-

15

This result can be applied to all of the following cases.

16

While the indirect eﬀect on the short-run utility level is brought about by a change in the interest rate,

the eﬀect on the long-run utility level and the change in the interest rate are brought about by a change
in wage levels. Also refer to the argument in Section 2.1.

21

run.

6

Conclusion

We have analyzed the capital income tax evasion behavior of individuals from a dynamic
viewpoint as opposed to a static one, in order to illustrate the long-run eﬀects of changesof the policy parameters from the standpoint of capital accumulation. We have also
incorporated tax evasion behavior into a general equilibrium framework. This makes it
possible to investigate the eﬀects of changes of the policy parameters on the economy asa whole, which could not have been examined in a partial equilibrium framework.
The main results obtained in this paper are as follows. First, while a rise in the capitalincome tax rate decreases the levels of capital stock in both the short- and the long-run, arise in the detection probability parameter or the penalty rate increases the levels of capitalstock in both the short- and the long-run. This diﬀerence results from how the incentive
to save is aﬀected. In the former case a rise in the capital income tax rate discourages the
incentive to save. Meanwhile, in the latter case, a rise in the policy parameters decreases
the expected return from savings, thereby stimulating the incentive to save.
Second, in regards to the eﬀects on undeclared savings, the magnitude of the eﬀects on
the capital stock works as a determinant factor, in all cases where the capital income tax
rate, the detection probability parameter or the penalty rate rises. More specifically, while
a rise in the capital income tax rate has the direct eﬀect of increasing undeclared savings,
since a fall in the (total amount of) savings will bring about an increase in the interest rate,
the indirect eﬀect will work in the direction of decreasing undeclared savings. Conversely,
when the detection probability parameter or the penalty rate rises, the opposite eﬀects
will occur.
Third, for the eﬀects of changes to the policy parameters on welfare, or utility levels,22

a rise in the capital income tax rate may have a positive eﬀect on welfare in the short-run
when tax evasion behavior is taken into consideration. Similarly, a rise in the detection
probability parameter or the penalty rate may bring about a positive eﬀect on welfare in
the long-run. These results imply that policy tools that discourage tax evasion behavior(decrease the amount of undeclared savings) are, under certain conditions, preferable toindividuals’ welfare.
Although this dynamic general equilibrium model has succeeded in presenting some
economic implications, our simplification of government and tax authority behavior may
have been excessive. One possibility for modification is to adopt an approach that takes
the expenditure side into consideration. In our analysis, all of the income collected by
the government and the tax authority is assumed to be disposed of, or consumed. Somecorrection will be required when considering cases where the income is to be transferred
to individuals in a lump-sum fashion, or allocated to the public goods supply. Anotherpossibility for modification is to endogenize the behavior of the government and tax authority. Also, it may be more plausible to assume that the government is capable ofcarrying out optimal taxation. Lastly, strategic settings, or game theoretical frameworks
could be introduced into individual and government (or tax authority) behavior.

A

Appendix

This appendix derives the results of the comparative static analysis in Section 3. First，
totally diﬀerentiating (9) we find that:23∆t+1 dkt+1 =∆t dkt −++γ−1
1 + α + (1 + α − τ )Aγkt+1

γ−1 2
2(1 + α)[1 + (1 − τ )Aγkt+1
]

τΦ
γ−1
2(1 + α)δ[1 + (1 − τ )Aγkt+1
]

τΦ
γ−1
2(1 + α)θ[1 + (1 − τ )Aγkt+1]

where, ∆t ≡ Aγ(1 − γ)ktγ−1 /2 > 0, ∆t+1 ≡ 1 +

dδ
dθ,

γ−1
τ (1−τ )αAγ(1−γ)kt+1
γ−12(1+α)[1+(1−τ )Aγkt+1]

We evaluate (27) in the steady state:∆s dk = −
+
+

Φdτ

γ−1
1 + α + (1 + α − τ )Aγkt+1

γ−1 2
2(1 + α)[1 + (1 − τ )Aγkt+1
]

τΦ
γ−1
2(1 + α)δ[1 + (1 − τ )Aγkt+1
]

τΦ
γ−1
2(1 + α)θ[1 + (1 − τ )Aγkt+1
]

(27)

h

τ
δθ(1+α)

i1/α

> 0.

Φdτ
dδ
dθ,

(28)

where ∆s is the value of ∆ ≡ ∆t+1 − ∆t evaluated in the steady state, and k is the capitalstock in the steady state. Furthermore, the condition for monotonic convergence to thesteady state is:

0<

dkt+1
∆t
=
<1
dkt
∆t+1
⇐⇒ ∆ > 0.

(29)

Under these conditions, the results of the comparative static analysis in the steady state
can be obtained from (28). The short-run eﬀects in the comparative static analysis canalso be obtained from (27) in the same manner: by substituting ∆s with ∆t+1 .

24

Next, totally diﬀerentiating (10), we arrive at the following:

dxt+1

½

¾
(1 − γ)Φ dkt+1
=
+
dτ
γ
γ−1
dτ
Aγkt+1ατ Aγkt+1¾½(1 − γ)Φ dkt+1
Φ
+
dδ
+ −
γ
γ−1
dδ
Aγkt+1αδAγkt+1¾½(1 − γ)Φ dkt+1
Φ
+
dθ.
+ −
γ
γ−1
dθ
Aγkt+1αθAγkt+1Φ(30)

Evaluating (30) in the steady states gives:½¾
Φ(1 − γ)Φ dk
dx =
+
dτ
ατ Aγk γ−1
Aγkγ dτ½¾Φ(1 − γ)Φ dk+ −+dδαδAγkγ−1
Aγkγ dδ¾½(1 − γ)Φ dk
Φ
+
dθ.
+ −
γ−1
αθAγk
Aγkγ dθ(31)

25

Acknowledgements

We would like to thank Kojun Hamada for his contributions at the Japanese Economic
Association Autumn Meeting, Kunihiko Kaneko, Yoshinao Sahashi, Terukazu Suruga,
Tatsuhiro Shichijo, Shingo Takagi, Yasutomo Murasawa, Akira Momota, and Shigeru
Watanabe for their contributions at the Seminar of Economic Theory and Econometrics
at Osaka Prefectural University, and Tatsuo Hatta, Ryo Horii, Nobuhiro Hosoe, Akihiko
Kaneko, Shinji Miyake, Yasusada Murata, Takumi Naito, Yoshitomo Ogawa, and Ryosuke
Okamoto for their valuable suggestions and helpful comments at the Seminar of Spatial
Public Economics at Tokyo University. We would also like to acknowledge the contributions of Hideo Hashimoto, Masaki Hashimoto and Nobuhito Takeuchi. The authors
assume responsibility for any remaining errors.
This research is supported in part by a grant from the Nitto Foundation.

References[1] Andreoni, J., Brian, E. and J. Feinstein, 1988, Tax Compliance, Journal of Economic
Literature 36, 818-860.
[2] Allingham, M. G. and A. Sandmo, 1972, Income tax evasion: A theoretical analysis,
Journal of Public Economics 1, 323-338.
[3] Barro, R. J., 1990, Government spending in a simple model of endogenous growth,
Journal of Political Economy 98, S103-S125.
[4] Barro, R. J. and X. Sala-i-Martin, 1995, Economic growth, (Cambridge, Massachusetts: MIT Press).

26

[5] Caballe, J. and J. Panades, 1997, Tax evasion and economic growth, Public Finance
52, 318-340.
[6] Chen, B. L., 2003, Tax Evasion in a Model of Endogenous Growth, Review of Economic Dynamics 6, 381-403.
[7] Clotfelter, C. T.,1983. Tax evasion and tax rates: An analysis of individual returns,
Review of Economics and Statistics 65, 363-373.
[8] Cowell, F. A., 1990, Cheating the government: The economics of tax evasion, (Cambridge, Massachusetts: MIT Press).
[9] Diamond, P. A., 1965, National debt in neoclassical growth model, American Economic Review 55, 1126-1150.
[10] Lin, W. Z. and C. C. Yang, 2001, A dynamic portfolio choice model of tax evasion:Comparative statics of tax rates and its implication for economic growth, Journal of
Economic Dynamics and Control 25, 1827-1840.
[11] Mankiw, N. G., Romer, D. and D. N. Weil, 1992, A contribution to the empirics of
economic growth, Quarterly Journal of Economics 107, 407-437.
[12] Poterba, J., 1987, Tax evasion and capital gains taxation, American Economic Review
77, 234-239.
[13] Yaniv, G., 1994, Tax evasion and the income tax rate: A theoretical reexamination,
Public Finance 49, 107-112.
[14] Yitzhaki, S., 1974, A note on income tax evasion: A theoretical analysis, Journal of
Public Economics 3, 201-202.
[15] Yitzhaki, S., 1987, The excess burden of tax evasion, Public Finance Quarterly 15,
123-137.
27

Table 1: Comparative Statics
Lemma 2

Proposition 1

Proposition 2

τ

δ, θ

τ

δ,θ

k (s)

−

+

−

+

x

+

−

∗1

∗2

∗1: This sign becomes negative (positive), if the negative indirect eﬀect dominates (isdominated by) the positive direct eﬀect.
∗2: This sign becomes positive (negative), if the positive indirect eﬀect dominates (isdominated by) the negative direct eﬀect.

Table 2: The Eﬀects on the Utility Levelsv

Proposition 3

Proposition 4

τ

δ, θ

Short

Long

Short

Long

∗3

−

−

∗4

∗3: This sign becomes positive (negative), if the positive indirect eﬀect dominates (isdominated by) the negative direct eﬀect.
∗4: This sign becomes positive (negative), if the positive indirect eﬀect dominates (isdominated by) the negative direct eﬀect.

28

Table 3: Comparative Statics (Orthodox Case)：γ = 0.3, A = 6.0
case

initial

No.

value

equilibrium
k

x

∆
v

chg.

/p

θ = 0.3

2.84

1.28

2.66

δ = 0.3

x

v

k

x

v

τ

-0.34

6.30

-0.24

-0.49

6.25

-0.33

0.70

θ

0.11

-4.24

-0.09

0.16

-4.22

-0.06

/0.33

δ

0.11

-4.24

-0.09

0.16

-4.22

-0.06

τ

-0.26

4.76

-0.31

-0.37

4.73

-0.37

0.70

θ

0.06

-2.40

-0.05

0.09

-2.39

-0.03

/0.25

δ

0.08

-3.19

-0.06

0.12

-3.19

-0.04

τ = 0.2
1-2

θ = 0.4

2.85

0.96

2.66

δ = 0.3

long-runkτ = 0.2
1-1

short-runTable 4: Comparative Statics (τ ↑ =⇒ Short-run v ↑)：γ = 0.3, A = 8.0
case

initial

No.

value

equilibrium
k

x

∆

chg.

v

θ = 0.3

4.21

3.80

3.50

δ = 0.1

long-run

k

x

v

k

x

v

τ

-1.02

18.36

0.02

-1.47

18.07

-0.16

0.70

θ

0.32

-12.46

-0.17

0.47

-12.37

-0.11

/0.33

δ

0.97

-37.39

-0.51

1.40

-37.12-0.34

τ = 0.2
2

short-runTable 5: Comparative Statics (θ,δ ↑ =⇒ Long-run v ↑)：γ = 0.6, A = 6.0
case

initial

No.

value

equilibrium
k

x

∆

chg.

v

θ = 0.3
δ = 0.3

1.54

0.37

2.13

long-run

k

x

v

k

x

v

τ

-0.18

1.82

-0.63

-0.45

1.79

-0.92

0.40

θ

0.05

-1.22

-0.08

0.14

-1.21

0.01

/0.33

δ

0.05

-1.22

-0.08

0.14

-1.21

0.01

τ = 0.2
3

short-run29