Discussion Paper No.

595

OPTIMAL NONLINEAR INCOME AND
INHERITANCE TAXATIONIN AN INFINITE HORIZON MODELWITH QUASI-LINEAR PREFERENCE

Hisahiro Naito

September 2003

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

Optimal Nonlinear Income and Inheritance Taxation in an Infinite
Horizon model with quasi-linear preference

Hisahiro Naito∗
Institute of Social Economic Research
Osaka University
andDepartment of Economics
University of California Irvine
Current Version March 31, 2003

Abstract
This paper analyzes optimal nonlinear income and inheritance taxation by incorporating
two types of models that were developed independently in the public finance literature: an
infinite horizon representative agent model such as Judd (1995), Chamley (1986) and Lucas(1992), and asymmetric information model analyzed by Mirrlees (1971) and Stiglitz (1982).
In this paper, by using an infinite horizon model with heterogenous agents and quasi-linear
preference under an asymmetric information environment we characterize optimal income
and inheritance taxation. This paper shows that, contrary to the general perception that
inheritance taxation should be progressive to some extent, the expected tax liability of those
who have a higher level of assets is lower than the expected tax liability of those who have alower level of assets. Thus, the optimal inheritance tax is regressive.

Keywords: Capital income taxation, heterogenous agents, redistribution, inheritance taxation.
∗

Institute of Social Economic Research,Mihogaoka 6-1, Ibaraki City, Osaka, Japan, Postal-Code 567-0047
Tel:81-6-6879-8581; fax: 81-6-6878-2766.
E-mail address: naito@iser.osaka-u.ac.jp

1

Introduction

Inheritance taxation is one of the most controversial issues in modern economic policy. Conservatives often argue that inheritance taxes discourage saving, entrepreneurial activity and labor
supply and that they have large negative effects on the size of economic output and its growth.
Liberals argue that those taxes are necessary to put people on the same start line, to reduceincome inequality and to decrease the concentration of wealth. In addition, sometimes it is argued that such redistributive taxes work as social insurance if we consider being rich or poor
as idiosyncratic shocks (Varian 1980). Furthermore, a recent increase of inequality of wealthdistribution in the US accelerated the policy debate on the effect of inheritance taxes (Gale etal. 2000).
According to the Survey of Consumer Finance in 1995 (Wolf 2000), the top 1% of families(as ranked by marketable wealth) own 45% of total household financial wealth and the top 20%
own 92 % of the marketable financial wealth. The average financial wealth is $7,000,000 for the
top 1% while the average financial wealth of the population is $155,000. In addition, half of the
top 1% receive an inheritance whose average value is $800,000 while 20% of the total population
receive an inheritance whose value is $96,000. This implies that the rich are already richer atthe beginning than the average population. Thus, on wealth inequality and the transmission of
wealth, the argument of the liberals is not groundless. However, those top 1% rich are also very
productive people. According to the same data, among the top 1% rich, 69% are self-employed
while only 17% are so among the total population. 40% of the top 1% go to graduate school
while among the total population only 11% do so. Hence, it is also quite possible that the
top 1% are knowledgeable entrepreneurs and their wealth is the reward for their entrepreneurial
activity as the conservative argue. Given the fact that half of the top 1% receive inheritance, it is
not surprising that inheritance taxes affect incentive for entrepreneurial activity due to businessowner’s incentive to leave their business to their family members. In such a case, inheritancetaxes can have large negative effect on the economy. 1
Given those arguments, it is interesting to look at how the previous public finance literature
treated the issue of inheritance taxation. There are several papers that should be discussed insuch a context. In seminal papers, Chamley (1986) and Judd (1985) showed that in an infinite
horizon representative agent model, the constrained Pareto-efficient tax rate on capital income
should be zero at the steady state and that there should be no intertemporal distortion at thesteady state. In a static analysis with heterogenous agent, Atkinson and Stiglitz (1975) showed1

For other statistics, the top 1% of families (as ranked by marketable wealth) owned 36% of the total household
wealth and the top 20% percent of households owned 84% of it. The total wealth is the financial wealth plus thenet equity of owner occupied housing and automobiles.

1that if leisure is weakly separable with consumptions in different periods, then an intertemporal
distortion is not optimal by using a framework developed by Mirrlees (1971). Since inheritancetaxes can be interpreted as one form of intertemporal distortion, those results suggest that the
government should not use inheritance taxation.
However, the previous literature is limited in several ways. First, the representative agent
model does not seem to be appealing when we need to discuss redistribution of wealth from those
who have to those who do not have. Second, the assumption on the weak separability between
consumption and labor supply is useful only when labor is supplied in the first period. The resultof Atkinson and Stiglitz (1975,1980) does not hold when labor is supplied in multiple periods.
Third, the two period overlapping generation model is limited in the analysis of capital income
taxation. In the public finance literature, Summers (1981) showed that using the two period
overlapping generation model is misleading for the analysis of capital income taxation because
the two period overlapping generation model does not capture the income effect for the future
income that is caused by a change of discount rate. Fourth, the literature ignores the important
fact that the transfer of assets between generations is also accompanied by intangible assets suchas knowledge on management. If it is so, discouraging to continue business might imply a largesocial welfare loss.
In this paper, we examine the issue of inheritance taxation in an infinite horizon model withheterogenous agents, asymmetric information and quasi-linear preference (linear in labor supply).
We assume an infinite horizon model given that the substantial size of inheritance among the richexists in reality. It seems difficult to explain those observed inheritances among the rich by theaccidental bequest model. Also using a heterogenous model is needed to discuss redistribution of
the unequal wealth distribution. We use quasi-linear preference assumption to make the analysis
under the environment of asymmetric information tractable. 2
In this paper, we first characterize the optimal income and inheritance tax system and showthat in the environment of asymmetric information and quasi-linear preference the tax liabilityof those who have a higher level of capital is on average lower than those who have a lower levelof capital, contrary to the general perception that the inheritance should be progressive to someextent.
In the public finance literature, to our knowledge there are two papers that are close to
our analysis: Pirttila and Tuomala (2001) Boadway, Merchand and Pestieau (2000). Pirttila
and Tuomala analyzed capital income taxation under nonlinear income tax in an OLG model.
Boadway et al. analyzed capital income taxation with nonlinear income tax and accidental
2

In the mechanism design literature, the quasi-linear preference is assumed in many cases for tractability. See
Fudenberg and Tirole (1993).

2bequest. But neither of the papers analyzed the capital income taxation in a dynastic framework
where the amount of investment depends on the structure of income taxes in the future.
Methodologically, this paper borrows an approach used in a dynamic contract theory suchas Spear and Srivastava (1987) and Abreau, Pearce, and Stacchetti (1990), which showed thatby using the future’s expected discounted utility as state variables many interesting dynamic
contracts can be analyzed in a tractable way.

2

The model

2.1

Set upIn this economy, time is infinite. Before period 1, there are 2 types of agents denoted by index H
and index L. Type H has a higher level of capital K H and type L has a lower level of capital K L .
Types of capital is observable and verifiable to a social planner. Then, at the beginning of period
1, type H and type L agents turn out to be skilled workers or unskilled workers. This implies
that at the middle of each period, there are four types of workers, Hs, Hu, Ls, Lu. Throughout
this paper, we use an index i to indicate capital types(i = H, L) and indices j and m (m 6= j) toindicate labor types (j, m = s, u). We assume that the probability that an agent becomes skilled
or unskilled is i.i.d. and it does not depend on his past history or past labor supply. We also
assume that the skilled worker is θs /θu times productive than unskilled workers (θs > θu ). We
denote the population of type ij(i = H, L and j = s, u) at period t by nij
t . The type ij agent
supplies ltij units of labor and earns z ij t = θj wltij units of labor income and Fk K i units of capitalincome where w and Fk is the marginal product of labor and the marginal product of capital,
respectively. A production function F (·) has two arguments (capital and labor) and exhibits
constant returns to scale. Given type ij’s capital income and labor income, the social planner
determines his tax liability. We assume that the tax liability of each agent does not depend on
information of their parents due to the social planner’s concern about equity. Once after tax
income is determined, the agent decides how much to invest for future and how much to consume
for today. Each agent has a dynastic utility function:∞X
v1 = u(c1 ) − l1 + E1 [γ t {u(ct+1 ) − lt+1 ]}t=1where ct and lt are consumption and labor supply at period t and u0 (c) > 0 and u00 (c) < 0. αjis the probability to become type j (j = s, u) worker and αs + αu = 1. We assume that ct andlt take any value on the real number. This implies that there is no corner solution for lt and ct
at the optimum. We assume that one generation lives for only one period. Thus, we interpret γ
as a parameter exhibiting a degree of altruism. At the end of each period, each agent makes a3consumption and investment decision. Because each agent lives for one period, we assume thatthe agent cannot borrow money using children’s future income as collateral. At the end of periodt, given after tax income xit for agent i he decides how much to invest out of his after tax income.
At the beginning of the period t + 1, some investment will succeed and some investments will fail.
As a result, some agents will receive a high level of capital K H becoming type H and other agentswill receive K L becoming type L. We assume that the amount of investment is not observableto the social planner. Thus, there is a moral hazard problem regarding investment. We assume
that given type ij’s investment Itij , the probability of his child receiving K H at period t + 1 ispi (Itij ) and(A1)

pH (Itij ) ≥ pL (Itij )This implies that the probability of successfully leaving capital to his/her child for those whohave a high level of capital is higher than for those who have a low level of capital. This can be so
because of the effect of intangible assets and reputation of conducting business. The assumption(A1) means that whether the agent can leave his business successfully to his child does not dependon the agent’s genetic ability once it is conditioned by the agent’s investment. In other words,
other than physical investment, there is no factor that causes intergenerational transmission of
wealth inequality. After receiving capital at the beginning of period t + 1, the child again turns
out to be a skilled worker or an unskilled worker and the same game will be repeated.
The social planner’s objective is to maximize the social welfare function evaluated at t = 1 with
the intertemporal government budget constraint. Thus, given an initial population distributionL{nH
1 , n1 }, the social planner solves the following problem:

Primary programmax Ψ(v1H , v1L ){v1H , v1L }s.t.

HE(v1L , v1H ; nL
1 , n1 ) ≤ A

where Ψ(·) is a social welfare function for the social planner. It is concave and strictly increasingLwith respect to its arguments. E(v1H , v1L ; , nH
1 , n1 ) is the expenditure function and it is the

additionally necessary resource to achieve a lifetime utility v1i for each i agent when the populationof type i is ni1 . The above Primary Program says that the social planner will choose v1i for each i
to maximize his social welfare with the constraint that the additionally necessary cost to achievea life time utility v1i for each i is less than A. We normally set A to zero. For the analysis, define
H , v L ) as follows:
W ij (x, vt+1
t+1
H
L
H
L
W ij (x, vt+1
, vt+1) ≡ max u(x − I) + γpi (I)vt+1+ γ(1 − pi (I))vt+1{I}

4

H , v L ) is the sub-indirect utility when the type ij agent receives after tax income x andW ij (x, vt+1
t+1

the optimal investment is chosen. Given this individual optimization problem, since investment
H and v L , we can write it as I(x, v H , v L ).
I is a function of x, vt+1
t+1
t+1 t+1

Then, the expenditure function, E(·) is defined recursively as follows:
Sub-program
H
E(vtL , vtH ; nLt , nt ) =

X

s.t.

min

{cij ,,Itij ,vtj :j=1,2,...,J}L
H
H
Q(bt ) + γE(vt+1, vt+1; nL
t+1 , nt+1 )

ij
H
L
i
αj {W ij (xijt , vt+1 , vt+1 ) − lt } − vt = 0 for i = H, L

(PUi)

j=s,u

θm im
l for i = H, L ;j, m = s, u and m 6= j
θj t(ICij)
X X
X
X
X
X
i
bt +
αj nijnit αj ltij wθj ≥nit αj xij(RC)t K (1 + R) +
t

ij
H
L
ij im HL
W ij (xijt , vt+1 , vt+1 ) − lt ≥ W (xt , vt+1 , vt+1 ) −i=H,L; j=s,u

i=H,L j=s,u

X X

H
L
i
αj pi (I ij (xijt , vt+1 , vt+1 ))Nt

(TRNH)HLi
αj [1 − pi (I ij (xijt , vt+1 , vt+1 ))]nt(TRNL)

nH
t+1 =i=H,L j=s,u

i=H,L j=s,u

nL
t+1 =

X X
i=H,L j=s,uhlhl
I ij (x, vt+1
, vt+1) = arg max u(x − I) + γpi (I)vt+1+ γ(1 − pi (I))vt+1

t = 0, 1, 2, ...;
where Q(bt ) is a penalty function from lending. We assume that Q(bt ) = bt for bt < 0 andQ(bt ) = δG(bt ) for bt where δ > 0,G(bt ) ≥ Bt , G0 (bt ) > 1 and G00 (bt ) > 0.
The above two programming problems deserve several comments. First, we consider the
problem of the social planner in two steps. First the social planner chooses v1i for each i tomaximize his social welfare function with the total discounted resource constraint. Then, given
those chosen v1i for each i, the social planner will choose xijt for all i and t to minimize thediscounted resource cost to achieve v1i for each i. Second, Q(bt ) is a function that captures
the social planner’s accessibility to international capital market (openness of the economy). For
example, if this economy is closed, we can obtain the solution by setting δ at quite a large number.
If Q(bt ) is equal to bt for all bt , it means that the economy is open for the social planner andthe social planner can lend and borrow at the same price. Third, (PUi) is the promised utility
constraint. It says that for those who have K i level of capital, the expected utility from todayj, wemust be equal to vti for each i. Summarizing the effect of all policies in the future in vt+1

can design today’s tax policies when the agents’ behavior also depends on tax policies in the
future. (ICis) is the incentive compatibility constraint for those who have K i level of capital andwho are skilled workers. Because of the hidden types assumption, at each period t, the social
5

planner cannot know whether an agent is a skilled worker or an unskilled worker. Therefore, thetax system must be designed so that each type self-selects an allocation that the social plannerintended. The constraint says that the type ij worker has an incentive to announce that he istype j worker, to work ltij hours, to earn ztij ≡ wθj lij dollars and to receive xijt units of incomemrather than to announce that he is type m worker, to work wwj ltim hours, to earn ztim ≡ wθm ltim
units of income and to receive xim
units of income. (RC) is the resource constraint for thetsocial planner.

H
L
Definition of I(xijt , vt+1 , vt+1 ) requires that the investment is consistent withintertemporal maximization.

2.2

Analysis

Before analyzing the case with incentive problems, it would be useful to know the first best caseij
where the social planner can control xij
t and It perfectly. In this case, it is straightforward toshow that at the steady state (i) the consumption levels are equal for all types of agents (ii) theinvestment levels of all types of agents are equal (iii) the first order condition of the investment
implies that P 0 (I ij ) × Fk × (K H − K L ) = 1/γ where Fk is the marginal product of capital.
Thus, at the first best solution, the consumption is perfectly smoothed for all types of agents andinvestment is made so that the expected marginal product of investment is equal to the discountrate.
Now consider the second best situation where the social planner cannot observe individuallabor types and investment but can observe earned income, which was initially analyzed byMirrlees (1971) and Stiglitz (1982). In this case, the social planner will use a nonlinear income
tax system to distinguish skilled workers from unskilled workers. The social planner will give andrequire a higher consumption level and higher labor supply to those who announced that they
are skilled and will give a lower consumption level and lower labor supply to those who announce
that they are unskilled.
Let µt , λit and φij
t , be the Lagrangian multipliers of RC, PUi and ICij of the sub-program,ij
respectively. Then, the first-order conditions for xij
t , lt are

Bt : Q0 (bt ) + µt = 0.
ij
j ∂W

ij
im
∂I
i
ij ∂W
im ∂W
xij
:
λ
α
+
φ
−
φ
− µt ni αj + ij p0i (I ij )nit αj
t
t
ij
ij
ij
∂xt
∂xt
∂xt
∂xt

(

∂E
∂E
−
H
∂nt+1 ∂nL
t+1

)
=0

i
im θ
ltij : −αj λit − φij+ µt ni αj θj w = 0
t +φθmi = H, L ;j, m = s, u and j 6= miAs for the FOC of vt+1
and the population transition equation, it would be useful to write in6

matrix form:
 H 
λt+1
=
λL
t+1


ϕt
+
−ϕt
 H 
nt+1
=
nL
t+1
where ϕt =

!
P j

Lj
j p (I Hj )
α
p
(I
)
α
λH
H t
L t
j
j
t
P j
P j
Hj
Lj
λL
t
j α (1 − pH (It ))j α (1 − pL (It ))P(1)αj pH (ItHj )P jj
Hjj α (1 − pH (It ))αj pL (ItLj )P jj
Ljj α (1 − pL (It ))

P

X X

ij

φ

P

(pi (Itij ) − pi (Ibtij,m )) +

!

nH
t
nL
t


(2)

X X ∂I ij
p0i (Itij )nit αjH∂v
t+1
i=H,L j=s,u

j=s,u i=H,L
ij
ij ij hlhl
It = I (x , vt+1 , vt+1), and Ibtij,m = I ij (xim , vt+1
, vt+1
)

(

∂E
∂E
−
H
∂nt+1 ∂nL
t+1

)
,

(3)

Note that the matrix of the RHS of (2) is the Markov matrix. For the property of the Markov∂Ematrix, see Simon and Blume (1994). From the envelope theorem, ∂n∂E
can be calculated
H − ∂nL
t+1

t+1

as follows:
X
X
X
X
∂E
∂E
−
= µt+1 {[Fk K H + w
αj lHj −αj xHj ] − [Fk K L + w
αj lLj −αj xLj ]}HL∂nt+1 ∂nt+1
j=s,u
j=s,u
Hs
Ls
Hu
Lu
+ γ{αs (p(It+1) − pLs (It+1)) + αu (p(It+1) − p(It+1
))}{

∂E
∂E
−
}.
H
∂nt+2 ∂nL
t+2

ij
Hj
i
Let Fk K i +wltHj −xijis the total income of type ij agent and xij
t be Tt . Since Fk K +wlt
t isafter tax income, Ttij can be interpreted as the tax liability of the type ij agent at period t. Thus,PP
j Hj
j Lj is the difference of the expected tax liability of type H and type L
j=s,u α Tt −j=s,u α Tt
P
µ∗ P
∂E
∂E
j Hj
j Lj
agents. In addition, in the steady state, ∂n
H − ∂nL = ∆∗ {j=s,u α T∗ −j=s,u α T∗ } wherett∗ indicates variables at the steady state and ∆∗ = [1 − γ{αs (pH (I∗Hs ) − pL (I∗Ls )) + αu (pH (I∗Hu ) −∂E∂E
pL (I∗Lu ))}] > 0. Thus, the sign of ∂n
H − ∂nL will determine the sign of the difference of thett

expected tax liability between type H and type L agents at the steady state.
Now we are going to ask whether there should be an inheritance tax in this model. Optimalityof inheritance tax can be interpreted as the difference of the tax liability of type H and type L
agents for the same labor type. We characterize the structure of inheritance tax in the following
steps:
Claim 1 µt and λit are strictly negative.
Proof. Suppose that RC constraint is not binding. Then, by decreasing bt , the social planner
can decrease the cost. Thus, µt is strictly negative. Next we prove that λit is strictly negative.
Suppose that at the optimum, one of (PUi) is not binding. Then, by increasing ltis and liu the7social planner can decrease bt . Since (RC) is binding, this will decrease the total cost to the social
planner. This is a contradiction. Thus, (PUi) must be binding.



iu
is
iu
Claim 2 xis
t ≥ xt and zt ≥ zt

Proof. From the assumption on the utility function, a single crossing property is guaranteed inija dimension of xij
t and zt . The incentive compatibility constraint and a single crossing property
iu
isiu
imply that xis
t ≥ xt and zt ≥ zt for each i = H, L.



Claim 3 φis
t is strictly negative.
Suppose that neither ICHs nor ICLs are binding. Then increase ltis by dlis and decrease ltiu
by dliu = (αs /αu )dlis so that the expected utility is the same. On the other hand, the change ofthe total labor supply is θs αs dlis + θu αu ((αs /αu )dliu = αs (θs − θu )dlis . Thus, as long as θs > θu ,the social planner can decrease the total cost. This will continue until both ICHs and ICLs bind.
isiu
Claim 4 φiu
t is equal to zero as long as xt > xt .

Proof. Now we will show that if ICis is binding, then ICiu is automatically satisfied as longiu
as ztis ≥ ztiu . Note that from a single crossing property and the ICis and ICiu,xist ≥ xt implies

ztis ≥ ztiu . Now suppose that ICis is binding. Then,(xis , z is ) and (xiu , z iu ) must be on the sameindifference curve of type is. Since ztis ≥ ztiu ,(xis , z is ) is located to the right of (xiu , z iu ) or at thesame point of (xiu , z iu ). Because of a single crossing property and the shape of the indifference
curve, the indifference curve of type iu crosses any indifference curve of type is once from theabove. This implies that ICiu is also satisfied.



From this point, we consider a problem that ignores ICHu and ICLu but includes a constraintiu
xist ≥ xt .

iu
Note that from Claim 2 xis
t ≥ xt is a necessary condition from ICis and ICiu undera single crossing property. Thus, we look for the solution with smaller constraints. After finding
the solution with smaller constraints, we will check that the solution also satisfies ICHu and
ICLu.
L
L
H
Claim 5 λHt /nt = λt /nt .

Note that we can ignore ICHu and ICLu. Since ICHs and ICLs bind, from the first ordercondition of lis and liu ,i s s−λit − φist + µt n α θ w = 0 for i=H,Lθu−λit + φis s + µt ni αu θu w = 0 for i=H,Lθ8

H
Hs /nH = λL /nL +φLs /nL and λH /nH +(θ u φHs )/(θ s nH ) =Those four equations imply that λHt /nt +φ
t
t
t
t
t
t
t
L
u Ls
s L
H
H
L
L
λL
t /nt + (θ φ )/(θ nt ) . This implies that λt /nt = λt /nt .

Claim 6 ϕt = 0
H
L
L
Proof. Note from the FOC of l1ij , λH
1 /n1 = λ1 /n1 at period 1. On the other hand, onceni1 is determined, ni2 is determined by equation (TRNH) and (TRNL), and λi2 is determined byH
L
L
equation (1). Thus, when λHt /nt = λt /nt for t = 2, ϕt = 0. Extending this logic, it is obviousthat it must be true for all t ≥ 1.

Claim 7 ∂n∂E
− ∂n∂E≥ 0 for all t
H
L
t+1

t+1

Proof. Note that from Claim 4, φiu {p(Itiu ) − p(Ibtiu,s )} = 0 for i = H,L. From Claim 6 it
implies that
ϕt =

X

ij

φ

(pi (Itis ) − pi (Ibtis,u )) +i=H,L

X X ∂I ij
p0i (Itij )nit αjH∂v
t+1
i=H,L j=s,u

(

∂E
∂E
−
L
∂nH
∂n
t+1
t+1

)
=0

iu
is
bis,u ) = 0 if xis = xiu .
As for the first term, pi (Itis ) − pi (Ibtis,u ) > 0 if xis
t > xt and pi (It ) − pi (Ittt

Thus, we have ∂n∂E
− ∂n∂E
≥0
H
L
t+1



t+1

iu
Claim 8 xis
t > xtiu
Proof. Consider the problem with ICHs, ICLs and xis
t ≥ xt . Let τt be the Lagrangianiu
isiu
multiplier of the constraint of xis
t ≥ xt where τt ≤ 0. The first order conditions of x and xof this problem are
is
is
∂I
s ∂W
is ∂W
λH
+
φ
− µt ni αs + is p0i (I is )nit αs
t α
t
is
is
∂x
∂x
∂xt

(

is
∂W iu
∂I
is ∂W
λit αu−φ− µt ni αu + iu p0i (I iu )nit αu
t
iu
iu
∂x
∂x
∂xt

(

∂E
∂E
−
H
∂nt+1 ∂nL
t+1

)

∂E
∂E
−
H
∂nt+1 ∂nL
t+1)

+ τt = 0− τt = 0iu
It is easy to check that xis
t = xt is not compatible with the above first order conditions. Thus,iuxist > xt .

Note that from Claim 4, ICHu and ICLu are also satisfied. This means that it was appropriate∂E∂E
to ignore ICHu and ICLu. Also, from the argument of Claim 7, we have ∂n
H − ∂nL > 0. This∂E
∂E
implies that at the steady state, ∂n
H − ∂nL must be positive. On the other hand, at the steady
P
µ∗ P
∂E
∂E
j Hj
j Lj
state ∂nH − ∂nL = ∆∗ {j=s,u α T∗ −j=s,u α T∗ }. Since µ∗ is the Lagrangian multiplier ofthe resource constraint at the steady state, µ∗ is strictly negative. This means that the inside of
the bracket must be strictly negative. In other words, the excepted tax liability of those who have9

a higher level of assets is smaller than the tax liability of those who have lower level of assets.
Thus, we obtain the following proposition:
Proposition 1 The expected tax liability of those who have a high level of capital is lower thanthe expected tax liability of those who have a low level of capital.
The Proposition 1 can be understood as follows. When the investment is not observable tothe social planner, but the social planner wants to redistribute income, there is a moral hazard
problem and the incentive for investment is reduced. In order to keep the incentive for theinvestment, there must be some difference between the expected utilities of those who have ahigh level of capital and those who have a low level of capital. However, the difference between
those expected utilities can be generated by the difference between consumption levels or by thedifference between labor supply levels and the answer depends on the curvature of the utility
function with respect to consumption and labor supply. The idea of the Proposition 1 is thatwhen the marginal disutility of labor is constant, the difference between v H and v L should be
generated by the difference between labor supply levels. This implies that low level capital owners
needs to work more on average than high level capital owners. Other things being equal, thiseffect will generate a high level of labor income for low capital owners and a low level of labor
income for high capital owners. As a result, low level capital owners pays more taxes on averagethan high level capital owners.

3

Conclusion

The initial motivation of this paper was to examine whether the social planner should emphasize
redistributive and social insurance aspects of inheritance taxation or incentive problems to investment by inheritance taxation. To do so, I developed a model of infinite horizon heterogenous
agents with asymmetric information. By using this framework, this paper shows that in thecase of quasi-linear preference the social planner will emphasize the negative incentive effects ofinheritance taxation rather than redistributive effects. This paper suggests that the relative sizeof the curvature of utility functions with respect to consumption and labor, which are essentiallythe elasticity of saving and labor supply, are important for designing an efficient intertemporaltax system.

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4

Appendix

References[1] Gale, William, James Hines and Joel Slemrod, Rethinking Estate and Gift Taxation, Brooking
Institute Press, 2000
[2] Boadway, Robin; Maurice Marchand, Pierre Pestieau, “Redistribution with Unobservable
Bequests: A Case for Taxing Capital Income”, Scandinavian Journal of Economics, June
2000.
[3] Chamley, Christopher, “Optimal Taxation of Capital Income in General Equilibrium with
Infinite Lives”, Econometrica, 1986, Vol 54(3), pp[4] Judd Kenneth, L. ”Redistributive taxation in A Simple Perfect Foresight Model”, Journal of
Public Economics, Vol.28, pp59-83[5] Atkinson,Anthony and Joseph Stiglitz, 1980, Lecture on Public Economics, McGrawhill[6] Cremer,Helmuth, Pierre Pestieau, ”Non-linear Taxation of Bequests, Equal Sharing Rules
and the Trade-off between Intra- and Inter-family Inequalities” Journal-of-Public-Economics
79(1), January 2001, pp 35-53.
[7] Grossman, Sandy J., Oliver Hart, ”An Analysis of the Principal-Agent Problem,” Econometrica, 1982, 51, pp7-45.
[8] Feldstein, Martin, ” College Scholarship Rules and Private Saving, American Economic Review; 85(3), June 1995, pages 552-66.
[9] Fudernberg, Drew and Jean Tirole, Game Theory, MIT Press, 1993[10] Mirrlees, James, ” The Theory of Moral Hazard and Unobservable Behavior: Part I” Review
of Economic Studies, 1999, 66, pp3-21.
[11] Rogerson, Willian P. ”The First Order Approach to Principal-Agent Problems”, Econometrica, 1985, Vol 53(6), pp 1357-1367.
[12] Simon, Carl, Lawrence Blume, Mathematics for Economics, 1994, W.W. Norton.
[13] Spear, Stephen E., Sanjay Srivastava, ”On Repeated Moral Hazard with Discounting” Review
of Economics Studies, 1987, Vol54(4), pp599-617.

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[14] Summers, Lawrence-H ”Capital Taxation and Accumulation in a Life Cycle Growth Model”,
American-Economic-Review, 1981, September 71(4),pp533-544.
[15] Varian, Hal R., ”Redistributive Taxation as Social Insurance”, Journal of Public Economics,
1980, 14(1), pp 49-68.
[16] Wolf, Edward N. ”WHo Are the Rich? A Demographic Profile of High-Income and HighWealth Americans, Chapter 3 in Does Atlas Shrug? The Ecoomic Consequeces of Taxing theRich, edited by Joel B. Slemrod, Harvard University Press, 2000

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