Technical Paper Series

Congressional Budget Of¿ce

Washington, D.C.

Bequests, Inter Vivos Transfers, and

Wealth Distribution

Shinichi Nishiyama

Congressional Budget Of¿ce

December 2000

2000-8

Technical papers in this series are preliminary and are circulated to stimulate discussion and

critical comment. These papers are not subject to CBO¶s formal review and editing processes.

The analysis and conclusions expressed in them are those of the author and should not be interpreted as those of the Congressional Budget Of¿ce. Any reference to this paper in other

publications should be cleared with the author. Papers in this series can be obtained by sending

an email to techpapers@cbo.gov.
Bequests, Inter Vivos Transfers, and Wealth Distribution

Shinichi Nishiyama

Congressional Budget Of¿ce

Washington, DC 20515

e-mail: shinichi@cbo.gov

December 2000

Abstract

This paper extends the heterogeneous agent overlapping generations model with

bequests in Nishiyama (2000) by adding two-way intergenerational altruism and inter

vivos transfers. Calibrating the model to the U.S. economy, the paper measures time

preference and intergenerational altruism consistent with the economy¶s capital-output

ratio and the sizes of intergenerational transfers. In the model, households in the same

dynasty play a Nash game in each period to determine their optimal consumption, working hours, inter vivos transfers, and savings. The model suggests that when deciding the

level of bequests, a parent household considers the future utility of its child households,

on average, about 20 percent less than it considers its own future utility. But, the parent

household¶s motive for inter vivos transfers is much weaker than its motive for altruistic

bequests. The model replicates the wealth distribution of the United States fairly well

in terms of the Gini coef¿cient although the top 1 percent of the population holds proportionately less wealth than is observed in the data. The paper also analyzes the effect

of intergenerational transfers on wealth accumulation and distribution.

1 Introduction

Macroeconomic analyses usually rely on either an in¿nite horizon model or an overlapping

generations model. Those analyses implicitly assume a household is either perfectly altruistic

or completely sel¿sh. But, when economists evaluate ¿scal policies that involve income

redistribution between generations, the policy implication depends critically on the extent to

which the Ricardian Equivalence proposition holds. In other words, we need to know to what

extent households within a dynasty are altruistic toward each other and how those households

react when the government introduces a new policy.

WPrepared for the North American Winter Meeting of the Econometric Society, New Orleans, January 2001.

The title of my presentation is 3Measuring the Degree of Intergenerational Altruism by a Heterogeneous OLG

Model with Bequests and Inter Vivos Transfers. ́ For their valuable comments and suggestions, I would like

to thank José-Víctor Ríos-Rull, Andy Abel, Richard Rogerson, and Deborah Lucas. I am responsible for any

remaining errors. The analysis and conclusions expressed herein are solely mine and should not be interpreted

as those of the Congressional Budget Of¿ce.

1
Table 1: The Features of the Two Models

Nishiyama (2000) Present Paper

Intergenerational Altruism One-Sided Two-Way

Lifetime Uncertainty 6 or 7 Periods 6 or 7 Periods

Fertility Shock None 3 or q Children

Bequests Altruistic and Accidental Altruistic and Accidental

Inter Vivos Transfers None Two-Way (Altruistic)

In my previous paper (Nishiyama, 2000), I developed a heterogeneous agent overlapping

generations model with altruistic and accidental bequests, and I measured time preference

and parental altruism in the U.S. economy. That model considered both parental altruism

and lifetime uncertainty, and it captured the strategic behavior between a parent household

and its child households by assuming a Nash game. The result of the calibration showed that

a parent household, on average, cares about its adult child households roughly 30 percent

less than it cares about itself.1

However, several panel data sets (e.g., Gale and Scholz, 1994) show a signi¿cant number

of inter vivos transfers between a parent household and its child households, which the previous model did not consider. This simpli¿cation is justi¿ed if the capital market is perfect.

But, if a borrowing constraint exists, then it is also bene¿cial for an altruistic parent household to make transfers before it dies. Also, the possibility of adult child households making

gifts to their parent household affects both of their life-cycle saving schedules.

If we relax the model to allow households to make inter vivos transfers in each period,

how will the wealth accumulation and distribution in the United States change? This paper

extends the heterogeneous agent overlapping generations model by adding two-way intergenerational altruism, lifetime uncertainty, a fertility shock, and borrowing constraints, and

it measures time preference and intergenerational altruism through the calibration of the

model to the U.S. economy (see Table 1). The paper also analyzes, based on the obtained

parameters, the effect of bequests and gifts on wealth accumulation and inequality.

One of the features of this extended model is that it involves both an in¿nite horizon

economy (with borrowing constraints) and a pure life-cycle economy as two opposite cases.

It is likely that the economy has imperfectly altruistic households, and it can be shown as an

economy located between those two extremes.

Similarly to the previous paper, the main parameters 2 time preference and two-way intergenerational altruism 2 are obtained simultaneously. This is done through the calibration

of the model so that the steady-state equilibrium is consistent with the key statistics observed

in the United States: the capital-output ratio and the relative sizes of bequests and inter vivos

transfers. For the steady-state economy to be consistent, a parent household would have to

consider the future utility of its adult child households about 20 percent less than it considers

its own future utility.2 But, the parent household is actually less willing to make inter vivos

1

In that calibration, I included a half of inter vivos transfers to children or grandchildren in bequests from

parents to children as disguised bequests to avoid estate taxes. 2

The discount rate is lower in this paper than in the previous one (20 percent instead of 30 percent), partly

2
transfers to its adult child households, discounting the current utility of its adult children by

more than 50 percent.3

The model also replicates the wealth distribution of the United States fairly well. The

Gini coef¿cient of wealth distribution of the baseline economy turns out to be 0.701.4 But,

the share of wealth of the top 1 percent of households is 14.6 percent in the model, still lower

than the 29.6 percent in the data.5 The effects of bequests and inter vivos transfers on wealth

distribution are not very large. Under the parameter setting in this paper, those transfers, in

total, increase the inequality in a closed economy, but decrease it in a small open economy.

The rest of this paper is laid out as follows: Section 2 discusses previous literature about

bequests and inter vivos transfers, Section 3 describes the economy and the extended model,

and Section 4 shows the calibration of the model and the obtained main parameters. Section

5 uses policy experiments to examine the effects of altruistic and accidental bequests as well

as inter vivos transfers on wealth accumulation and inequality, and Section 6 concludes the

paper. The appendixes to the paper explain the algorithm of computing household decision

rules and the welfare measures.

2 Previous Literature

In this paper, I construct an altruistic model of bequests and inter vivos transfers based on the

strategic behavior between a parent household and its adult child households and measure

the degrees of altruism between those households. To my knowledge, except for Nishiyama

(2000), few analyses try to measure the degree of intergenerational altruism using a dynamic

general equilibrium model.

Lord and Rangazas (1991) used a partial equilibrium model to evaluate the effect of bequests on wealth accumulation by explicitly assuming their altruistic parameter to be unity,

i.e., parents care about their descendants as much as they care about themselves. Fuster, ÚImrohoroglu, and Ù ÚImrohoroglu (1999) introduced lifetime uncertainty and inter vivos transfers Ù

as well as bequests to their dynamic general equilibrium model, but they assumed perfect

risk sharing between parents and children. De Nardi (1999) introduced a 3warm glow ́ bequest motive to her dynamic general equilibrium model and chose the parameters to create

a reasonable size of bequests. But, since her model does not consider the state of recipients

(child households), it cannot measure the degree of parental altruism exactly.

The Share of Transfer Wealth. Many other papers have tried to measure the shares of

life-cycle wealth and transfer wealth. On the one hand, Kotlikoff and Summers (1981) constructed a measure of transfer wealth and showed that it accounts for at least 80 percent of

total wealth in the United States. On the other hand, Modigliani (1988) and others estimated

because I distinguish the elderly households with and without children in this paper. 3

The parent household may behave as if it is less altruistic when its adult child households are relatively

young, so that the parent will not discourage the children from working. I will discuss this result more in Section

6. 4

For example, according to the 1989 wealth data from the Panel Study of Income Dynamics (PSID), the

wealth Gini coef¿cient of married households is 0.71. 5

See Díaz-Giménez et al. (1997), Table 6.

3
that the share of bequeathed wealth is at most 20 percent.6 Gale and Scholz (1994) examined

inter vivos transfers, college expenditures, and trusts and life insurance purchases, and they

concluded that intentional transfers account for at least 20 percent of total wealth and more

than 50 percent if bequests are included.

Those estimates have two limitations, however: ¿rst, the transfer wealth measure is sensitive to the assumption of a steady-state interest rate and growth rate second, even if we

know the exact share of transfer wealth, we still cannot estimate to what extent national

wealth would be reduced if there were no intergenerational transfers. For example, if a 100

percent estate tax were introduced to eliminate all bequests, forward-looking child households would increase their life-cycle savings because they would not expect bequests from

their parents.7

Intergenerational Transfer Motives. Other papers have examined to what extent bequests

are intentional or accidental, and if they are intentional whether bequests and inter vivos

transfers are altruistic, sel¿sh, or strategic. Hurd (1987) compared the dissaving pattern of

old households with children with that of households without children and concluded that

bequests are mostly accidental. Wilhelm (1996) showed that parents tend to leave equal

bequests to each of their children even if the children¶s earnings differ signi¿cantly, and he

concluded that bequests are not altruistic.

In contrast, Menchik and David (1983) showed that elderly households do not dissave

and concluded that bequests are intentional. Bernheim (1991) used the same data set as Hurd,

the Longitudinal Retirement Household Survey, and concluded that bequests are intentional.

According to Bernheim¶s paper, households adjust their level of bequeathable wealth by

reducing their private annuity or increasing their life insurance when their public pension

increases.

Altruistic Hypotheses. This paper considers both intentional transfers (bequests and inter

vivos transfers) and accidental bequests due to lifetime uncertainty. Regarding the question

of whether the intentional transfers are altruistic, sel¿sh, or strategic, this paper assumes that

intentional transfers are motivated by altruism, although a part of bequests and gifts may be

sel¿sh or strategic.8

One of the main criticisms of the altruistic bequest model is that bequests are in many

cases divided equally by parents even if the earnings of their children differ signi¿cantly.

According to Wilhelm (1996), 76.6 percent of parents divided their estates almost equally

(within 	2 percent). This does not contradict the altruistic model, however, if we consider

the psychic cost of making unequal bequests. Stark (1998) introduced the notion of the

3relative deprivation ́ in children¶s utility function to show that the equal division of bequests

and the altruistic bequest model are not mutually exclusive.

6

See Modigliani (1988) for several estimates of others. The difference between those two conclusions is due

in part to different de¿nitions of transfer wealth and bequeathed wealth. 7

To what extent the savings of child households would change depends partly on how the government would

use the revenue from a 100 percent estate tax on accidental bequests. 8

Sel¿sh bequests are sometimes called 3joy-of-giving ́ bequests or 3bequests-as-consumption. ́ Also, 3warm

glow ́ bequests belong to that category. Strategic bequests are sometimes called 3gift-exchange ́ bequests. Inter

vivos transfers may be due to risk-sharing arrangements between households.

4
Another criticism of altruistic hypotheses is that inter vivos transfers from parents to children do not completely compensate for the income changes of parents and children. Altonji,

Hayashi, and Kotlikoff (1997) showed that parents increase transfers by only 13 cents when

their income increases by one dollar and that of their children decreases by one dollar. But, in

the presence of asymmetric information about children¶s working ability and efforts, it may

be optimal for parents to offer the partial insurance on children¶s income shocks to avoid

moral hazard. In fact, empirical analyses by Altonji et al. (1997) and Wilhelm (1996) show

that inter vivos transfers and bequests, respectively, are decreasing in the recipient¶s income

and imply that intergenerational transfers are at least partially motivated by altruism.

Implicit Insurance Contracts. Intergenerational transfers may be motivated by the income shock and lifetime uncertainty of households in the absence of perfect insurance and

annuity markets. Even if parents and children are not altruistic toward each other, it is bene-

¿cial for them to make a risk-sharing contract if they can avoid enforceability problems and

adverse selection. But, if parents and children are not altruistic at all, the sum of insurance

payments should be close to that of insurance bene¿ts. According to Gale and Scholz (1994),

inter vivos transfers from parents to children are about ten times larger than those from children to parents if we consider other transfers 2 such as bequests, trusts, and life insurance

2 that difference becomes much larger.

Those net transfers from parents to children are not explained solely by the risk-sharing

motive. Also, for the implicit annuity contract between parents and children, it is enough

to distinguish accidental bequests from other intentional transfers. This is because the price

of the annuity that parents have to pay is, on average, not very different from the amount of

accidental bequests.

Strategic Bequest Motive. Parents may want to keep their wealth in a bequeathable form,

even in the presence of perfect annuity markets, to attract their children¶s attention (e.g.,

Bernheim, Shleifer, and Summers, 1985). In that case, parents¶ bequests and other transfers

to their children are the payments for their children¶s services, such as telephone calls and

visits.9 But, as I mentioned before, Wilhelm (1996) showed that a majority of bequests

are divided equally. Also, Behrman and Rosenzweig (1998) showed that the relationship

between the amount of bequests and the number of visits across children is not signi¿cant

and rejected the framework in which parents use threats of disinheritance to elicit more visits

from their children.

3 Model

This section describes a four-period heterogeneous agent overlapping generations model with

bequests and inter vivos transfers. The model differs from the previous one (Nishiyama,

2000) in the following three ways. First, child households are also altruistic toward their

parent household second, both a parent household and its child households make inter vivos

9

Strategic bequests and altruistic bequests are not mutually exclusive because parents may value telephone

calls and visits by children only when the parents are altruistic.

5
transfers based on their degrees of altruism and an estate and gift tax and third, a household receives a fertility shock, i.e., some households have q child households and others

have none. Similarly to the previous model, this model considers both altruistic and accidental bequests. Also, households in the same dynasty behave strategically to determine their

consumption, working hours, gifts, and savings.

3.1 Two-Way Intergenerational Altruism

Before moving on to describe the model, I ¿rst want to show how to measure the degree

of intergenerational altruism, using a simple two-period overlapping generations economy

without uncertainty and population growth.

When there is no altruism, the lifetime utility of a generation j household, which lives

two periods, is shown as

x} @ x +f

}

, .  x+f

}

2,> (1)

where x+f

}

, denotes instantaneous utility from the consumption at age l> and  is a usual

time preference parameter of this household. Suppose that a child household (a generation

j . 4 household) is born to the household at the beginning of age 2 and that a household is

altruistic toward its descendants with a discount factor  per generation. Then the total utility

of a household of generation j is de¿ned as the in¿nite sum of the lifetime utility of each

generation, i.e.,

X }

 @ x} . + , x}n . + ,

2

x}n2 . + ,

 x}n . ===

@ x} . + , X }n

 >

where   3 and  ? 4= Using (1), we can arrange this total utility as the in¿nite sum

of instantaneous utility of the dynasty, which is a combination of a parent household and its

child household, i.e.,

X }

 @ x +f

}

, .  ["

'f

+ ,



q

x+f

}n

2 , .  x 

f

}nn



r =

The total utility of a generation j household of age 2 is also shown as

X }

2 @ ["

'f

+ ,



q

x+f

}n

2 , .  x 

f

}nn



r =

In this paper,  is the degree of parental altruism because it shows the relative importance of

the adult child¶s utility to its parent household.

Suppose that the degree of parental altruism differs when the parent is deceased  and

when the parent is alive f. The total utility of an age 2 parent household, X}

2 , becomes

X}

2 @

q

x+f

}

2, . f x



f

}n



r . X}n

2 = (2)

In other words, f represents the parent¶s inter vivos transfer motive and  represents its

bequest motive.

6
Suppose that the child household is also altruistic toward its parent household with a

discount factor = I call  the degree of the child¶s altruism toward its parent household. The

total utility of this age 4 child household is written as

X}n

 @

q

 x +f

}

2, . x



f

}n



r .  X}n

2 >

where the regularity condition is   3>   3> f  3>f  4> and ? 4=

Notice that the discount factor of the parent household on its child¶s utility from age 1 to

age 2,  @f, is different from the child¶s own discount factor . Still, equation (2) shows

that this is a time consistent problem.10

When  f @ 4 and f @ , we have X}

2 @  X}n , and this model becomes an in¿nite

horizon model (a dynasty model) as long as inter vivos transfers are allowed. But, when

 @  @ f @ 3> this model becomes an overlapping generations model (a pure life-cycle

model) in which households are completely sel¿sh.

In this paper, I do not use the model with 3two-sided ́ altruism, in which the total utility

of a household can be nested toward both its ascendant side and its descendant side, to avoid

the complexity from the 3hall of mirror ́ effects.11 In other words, a household does not care

about its ascendants that have already died.

This simpli¿cation is justi¿ed for the following reasons: First, in this setting, the model

involves both an in¿nite horizon model and a pure life-cycle model as two polar cases, and

this is enough to analyze the Ricardian Equivalence proposition. Second, in Section 4, I will

show that the parameter  is very small, and the difference between the results from this

model and a 3two-sided ́ altruism model is negligible.

3.2 Economy

The model is based on a standard growth economy that consists of a large number of households, a perfectly competitive ¿rm, and a government. Each household is assumed to act as

a single person.12

In each period, new households are born without any wealth. The life span of each

household is either three or four periods. One period in this model corresponds to 15 years

starting from the actual age of 30. A household dies either at the end of age 3 or at the end

of age 4. When a household reaches age 3, its q child households of age 1 are 3born ́ with

probability , and the former becomes a parent household (See Figure 1).

When a household is age 1 or 2, its working ability (labor productivity) at each age is

stochastically determined. It receives labor income (earnings) according to the market wage

rate, its working hours, and its working ability. A household of age 3 or 4 is assumed to

10The total utility of a generation } parent household is L}

2 ' S"

r'} Eq#

r3} 

ESr

2 n #f 



S

rn



,

and the total utility of its child household (generation } n ) is L}n

 ' 

4  ES

}

2 n 



S

}n



 n

q S"

r'}n Eq#

r3}3 

ESr

2 n #f 



S

rn



 When #f ' #, there is a conÀict of interests between the parent household and its child household about the child¶s consumption. Still, this model can be solved as a time

consistent problem. 11Here, I use the term 3two-way ́ altruism to distinguish it from 3two-sided ́ altruism. For 3two-sided ́ altruism, see Kimball (1987). 12In the calibration of the model, a household is assumed to be a married couple, but their decisions are made

jointly. Also, there is assumed to be no strategic interaction between siblings.

7
Figure 1: The Life Cycle of a Household

Age 1

(30-44)

Age 2

(45-59)

Age 3

(60-74)

Age 4

(75-89)

A household

is ‘born.’

It becomes

a ‘parent’

(its children 

are ‘born’) 

with prob. ξ.

It dies with

prob. 1-

It dies with

Its parent dies prob. 1.

with prob. 1- 

Its parent dies

with prob. 1.

be retired. Though it can work at home to produce a limited amount of consumption goods

and services, its working ability is assumed to be low and deterministic. A household can

hold only one kind of assets. It receives capital income according to its wealth level and the

market interest rate. The wealth of each household must be nonnegative.

A household pays federal income tax according to its total income. A household that

inherits any wealth from its parent also pays federal and state estate taxes.13 In addition, a

household of age 1 or 2 pays payroll tax for Social Security and Medicare. A household of

age 3 or 4 receives Social Security bene¿ts. The Social Security system is assumed to be the

de¿ned bene¿t type and, for simplicity, the size of the bene¿t is assumed to be the same for

all households.

At any time, there are two types of dynasties 2 the dynasties with both a parent household and its child households (Type I), and the dynasties without any overlapping generations

(Type II). Figure 2 shows two types of dynasties in this economy. Every parent household

is assumed to be equally altruistic and cares about its child households. In Type I dynasties,

both a parent household and its child households are altruistic, but to a different extent.

Beginning-of-period wealth of a parent household and its child households, and the working ability of the child households, are known to each other. A parent and its children choose,

simultaneously, their own optimal consumption, working hours, gifts (to each other), and

end-of-period wealth.

3.3 Households¶ Problem

For Type I dynasties, the state of each dynasty is shown by the ages of a parent household

and its child households i+6> 4,>+7> 5,j, the beginning-of-period wealth of the parent dR 5

D @ ^3> d4@ ` and that of its children d& 5 D> and the labor productivity (which determines

hourly wage) of the children h& 5 H @ ^H4?> H4@ ` = For Type II dynasties, the state of

each dynasty is shown simply by the age of a household i5> 6> 7j, the beginning-of-period

13In the baseline economy, the gift tax on inter vivos transfers is not considered and the tax rate is assumed to

be 0 percent. A couple (two parents) can avoid the federal gift tax on gifts of up to $20,000 per child per year.

8
Figure 2: Two Types of Dynasties

Generation 0:

Generation 1:

Generation 2: Type I (i = 1)

Type I (i = 2)

Type II (i = 2, no parents) 

Time

Generation 3: (not born)

Type II (i = 3 or 4, 

no children)

wealth d 5 D, and the working ability h 5 H. But, for the households of age 3 or 4, since

their working ability is assumed to be deterministic, their states are shown as +d>, instead

of +d> h,. In the calibration, the working ability of age l, h&c or h, is a member of ih

 > h2 > h j

for households of age 1 or 2, and it follows a Markov process.

For notational simplicity, let vU and vUU denote the states of a Type I dynasty and a Type

II dynasty, respectively, where

vU @ +dR> d&> h&,> vUU @ +d> h, or +d>,=

Then the value function of a Type I household of age l is denoted as yUc +vU ,, and that of a

Type II household of age l is denoted as yUUc +vUU ,.

3.3.1 Type I Households

An Age 3 Parent and Its Age 1 Children. Let fR, kR, jR, and d

R denote the parent household¶s consumption, working hours, inter vivos gifts to its child households, and the end-ofperiod wealth level, respectively. Similarly, let f&, k&, j&, and d

& denote each of its child

household¶s consumption, working hours, inter vivos gifts to the parent household, and the

end-of-period wealth level, respectively. Also, let & denote the parent household¶s conjecture about its child households¶ decision, q denote the number of child households,  denote

the survival rate at the end of age 3, u denote the rate of return on the capital, z denote the

wage rate per ef¿cient unit of labor,  denote the growth rate of the economy, 8 +=, be a federal income tax function, 7+=, be a payroll tax function for Social Security and Medicare,

.+=, be a federal and state estate and gift tax function, and wu77 denote Social Security

bene¿ts.

9
The value function of an age 3 parent household is shown as

yUc+vU > &, @ pd{ SRcRc}Rc@

R

q

x+fR> kR, . f q x+f&> k&,

. H 

 yUce+v

U , . +4  , qyUUc2+vUU ,m h&

 r

(3)

subject to

d

R @ 4

4 .  iz hRkR . +4 . u, dR . wu77  8 +u dR,  fR

+jR  q j&,j  3> (4)

where vU is the state of this dynasty,

vU @ +dR> 3> h&,>

& is the parent¶s conjecture of its child households¶ decision,

& @ +f&> k&> j&> d

&,>

and the law of motion of the state of this dynasty is

v

U @



d

R> d&> h&



>

v

UU @



d

& . dR@q  .



d

R@q

> h



= (5)

The parent household chooses its optimal consumption fR, working hours (housework

only) kR, inter vivos transfers jR, and end-of-period wealth level (normalized by the economic growth) d

R, taking the decision of its child households & as given. It discounts the

utility of each of q child households by f when it is alive. At the end of age 3, the parent

household dies with probability 4  . The value of this household at the beginning of the

next period is the weighted average of its own future value yUce (when this household is alive)

and its q children¶s future value q yUUc2 discounted by  (when this household is deceased).14

The term H ^ . m h&` denotes a conditional expectation given that the current working ability

of an age 1 child household is h&, i.e.,

H 

yUce+v

U ,m h&



@

]

.

yUce+v

U , c2+h& m h&, gh&>

where c2+h

& m h&, is a conditional probability of the working ability being h& in the next

period. The equation (4) is a budget constraint of this parent household, where jR  q j&

denotes the net gifts given to its child households. In the baseline economy, gift tax is not

considered. When the parent household dies, its end-of-period wealth d

R is split equally and

bequeathed to each of q child households.

14In this model, I assume that the utility the parent household receives from its child households is linear in

the number of child households.

10
Let R denote the child household¶s conjecture about its parent household¶s decision.

The value function of an age 1 child household is shown as

yUc+vU > R, @ pd{ S&c&c}&c@

&

q

x+f&> k&, .  q3x +fR> kR,

. H 

 yUc2+v

U , . +4  , yUUc2+vUU ,m h&

 r

(6)

subject to

d

& @ 4

4 . iz h&k& . +4 . u, d&  8 +z h&k& . u d&,  7+z h&k&,  f&

+j&  jR@q,j  3> (7)

where R is the child¶s conjecture of its parent household¶s decision,

R @ +fR> kR> jR> d

R,>

and the law of motion of the state is (5).

The child household chooses its optimal consumption f&, working hours k&, inter vivos

transfers j&, and end-of-period wealth level d

&> taking the decision of its parent household

R as given. It discounts the utility of its parent household by . The value of the child

household at the beginning of the next period is the weighted average of its own future value

when its parent is alive, yUc2, and its future value when its parent is deceased, yUUc2= The

equation (7) is a budget constraint of this child household, and j&  jR@q denotes the net

gifts given to its parent household.

Let gR and g& be the set of decisions of a parent household and each of its child households, respectively, i.e.,

gR @ +fR> kR> jR> d

R,> g& @ +f&> k&> j&> d&,=

Solving equations,

U +g&> vU , @ gR> U +gR> vU , @ g&>

where U +g&> vU , and U +gR> vU , are the best response functions of a parent and its children, respectively, Nash equilibrium decision rules are obtained as

gUc+vU , @ 

fUc+vU ,> kUc+vU ,> jUc+vU ,> d

Uc+vU ,



for vU 5 D2  H> where l @ 6 or 4=

An Age 4 Parent and Its Age 2 Children. An age 4 parent household is assumed to die at

the end of this period, and its child households become parent households with probability

 at the beginning of the next period= So, the value function of an age 4 parent household is

shown as

yUce+vU > &, @ pd{ SRcRc}Rc@

R

q

x+fR> kR, . f q x+f&> k&,

.qH 

 yUc+v

U , . +4  , yUUc



v

UU 

m h&

 r

(8)

11
subject to (4), where the law of motion of the state is

v

U @ +d& . dR@q  .+dR@q,> 3> h&,>

v

UU @



d

& . dR@q  .+dR@q,>



= (9)

The parent household considers its children¶s value at the beginning of the next period, which

is the weighted average of q yUc (when the child household becomes a parent) and q yUUc

(otherwise), discounted by .

15 Similarly, the value function of an age 2 child household is

shown as

yUc2+vU > R, @ pd{ S&c&c}&c@

&

q

x+f&> k&, .  q3x +fR> kR, (10)

. H 

 yUc+v

U , . +4  , yUUc



v

UU 

m h&

r

subject to (7), where the law of motion of the state is (9).

The decision rule of an age l household is obtained as

gUc+vU , @ 

fUc+vU ,> kUc+vU ,> jUc+vU ,> d

Uc+vU ,



for vU 5 D2  H> where l @ 7 or 5=

3.3.2 Type II Households

The value function of an age 2 household without its parent household is simply

yUUc2+vUU , @ pd{ Scc@

q

x+f> k, .  H  yUc+v

U , . +4  , yUUc



v

UU 

m h

 r

(11)

subject to

d @ 4

4 .  izhk . +4 . u, d  8 +zhk . u d,  7+zhk,  fj  3> (12)

where the law of motion of the state is

v

U@+d

> 3> h

&,> vUU@+d

> ,=

The value function of an age 3 household without its child households is

yUUc+vUU , @ pd{ Scc@

q

x+f> k, . yUUce



v

UU  r

(13)

subject to

d @ 4

4 .  izhk . +4 . u, d . wu77  8 +u d,  fj  3> (14)

where the law of motion of the state is

v

UU@+d

>,=

15To avoid introducing grandchildren into the model, I assume that a household does not know whether it will

be a parent household until the beginning of age 3.

12
Finally, the value function of an age 4 household without its child households is

yUUce+vUU , @ pd{ Scc@

x+f> k, (15)

subject to (14).

The household chooses its optimal consumption f, working hours k, and end-of-period

wealth level d

. The household¶s decision rules are obtained as

gUUc+vUU , @ 

fUUc+vUU ,> kUUc+vUU ,> d

UUc+vUU ,



for vUU 5 D  H and l 5 i5> 6> 7j =

3.4 The Measure of Households

Let {Uc+vU , denote the measure of Type I households of age l 5 i4> 5j, and let {UUc+vUU ,

denote the measure of Type II households of age l 5 i5> 6> 7j.

16 Also, let [Uc+vU , and

[UUc+vUU , be the corresponding cumulative measures. The population of age 1 child households is normalized to be unity, i.e.,

]

2f.

g[Uc+vU ,@4=

Let 4d@'+o be an indicator function that returns 1 if d @ | and 0 if d 9@ |= Then, the law

of motion of the measure of Type I households is

{

Uc2+vU , @ 

4 . 

]

2f.

4

@

R'@

UcEtU 

 4

@

&'@UcEtU 

 c2+h

& m h&, g[Uc+vU ,> (16)

and

{

Uc+vU , @ +4 . ,

]

2f.

4

@

R'@

Uc2EtU n@U ceEtU *?3.



@

U ceEtU *?

2c+h

& m h&, g[Uc2+vU ,

.

]

f.

4

@

R'@

UUc2EtUU 

 2c+h

& m h, g[UUc2+vUU ,



= (17)

The law of motion of the measure of Type II households is

{

UUc2+vUU , @ 4  

4 . 

]

2f.

4

@'@

U cEtU n@UcEtU *?3.



@

UcEtU *?

c2+h m h&, g[Uc+vU ,



> (18)

{

UUc+vUU , @ 4  

4 . 

]

2f.

4

@'@

Uc2EtU n@U ceEtU *?3.



@

U ceEtU *? g[Uc2+vU ,

.

]

f.

4

@'@

UUc2EtUU 

 g[UUc2+vUU ,



> (19)

16For Type I households, since the number of child households per parent household is ¿xed to ?, we don¶t

need the measure %Uc EtU  for  ' tc e 

13
and

{UUce



v

UU 

@ 4

4 . 

]



4

@'@

UUcEtUU 

 g[UUc+vUU ,=

The steady-state condition is

{

Uc+vU , @ {Uc+vU , for l 5 i4> 5j >

{

UUc+vUU , @ {UUc+vUU , for l 5 i5> 6> 7j > (20)

for all vU 5 D2  H and vUU 5 D  H=

3.5 The Firm¶s Problem

There is only one perfectly competitive ¿rm in this economy. In a closed economy, the stock

of ¿xed capital N is equal to the sum of total private wealth and the government net wealth

Z}. Total labor demand O is equal to total labor supply of households in ef¿ciency units.

N @ [

2

'

]

2f.

+dR@q . d&, g[Uc+vU , .[

e

'2

]

f.

d g[UUc+vUU , . Z}> (21)

O @ [

2

'

]

2f.

+hR kUcn2+vU ,@q . h& kUc+vU ,, g[Uc+vU ,

.

]

f.

h kUUc2+vUU , g[UUc2+vUU , .[

e

'

]



h kUUc+vUU , g[UUc+vUU ,= (22)

In a closed economy, the gross national product \ is determined by a production function,

\ @ I+N> DO,=

The pro¿t maximizing condition of the ¿rm is

u .  @ Ig+N> DO,> z +4 . 

7, @ Iu+N> DO,> (23)

where  is the depreciation rate of capital and 

7 is the marginal payroll (Social Security and

Medicare) tax rate.

In a small open economy, the gross national product \7 is de¿ned as

\7 @ u N . z O>

where N is the sum of total private wealth and the government net wealth, and u and z are

international factor prices.

14
3.6 The Government¶s Policy Rule

Government tax revenue consists of federal income tax W8 , payroll tax W7, and federal and

state estate taxes W.= These revenues are calculated as follows:

W8 @ [

2

'

]

2f.

i8 +u dR, @q . 8 +z h& kUc+vU , . u d&,j g[Uc+vU ,

.

]

f.

8 +zhkUUc2+vUU , . u d, g[UUc2+vUU ,

.[

e

'

]



8 +u d, g[UUc+vUU ,> (24)

W7 @ [

2

'

]

2f.

7 +z h& kUc+vU ,, g[Uc+vU ,

.

]

f.

7 +zhkUUc2+vUU ,, g[UUc2+vUU ,> (25)

W

. @ +4  ,

]

2f.

.



d

Uc +vU , @q

g[Uc+vU ,

.

]

2f.

.



d

Uce +vU , @q

g[Uc2+vU ,

.+4  ,

]



d

UUc +vUU , g[UUc+vUU ,= (26)

For simplicity, the wealth left by Type II households is assumed to be included in estate

tax. Total tax revenue is the sum of these three tax revenues and Social Security tax from

employers, i.e.,

W @ W8 . 5 W7 . W.=

The law of motion of the government wealth (debt if it is negative) is

Z

} @ 4

4 .  .  i+4 . u,Z} . W  F}  wu77 Qu(j > (27)

where F} is government consumption and Qu( is the population of households of age 3 or

4, i.e.,

Qu( @ 4

q



4 . 

4 . 



=

15
3.7 Recursive Competitive Equilibrium

The de¿nition of a steady-state recursive competitive equilibrium (which is also a Markov

perfect equilibrium) of this model is as follows:

De¿nition 1 Steady-State Recursive Competitive Equilibrium: Let vU and vUU be the state

of a Type I dynasty and that of a Type II dynasty, respectively, where

vU @ +dR> d&> h&,> vUU @ +d> h, or +d>,.

Given the time invariant government policy rules,

 @ i8 +=,> 5+=,> .+=,> wu77> F}>Z}j >

factor prices, u and z the value functions of households,

iyUc+vU ,je

' and iyUUc +vUU ,je'2 >

the decision rules of households,

q

fUc +vU ,> kUc +vU ,> jUc +vU ,> d

Uc +vU ,

re

'

and q

fUUc +vUU ,> kUUc +vUU ,> d

UUc +vUU ,

re

'2 >

and the measures of dynasties,

i{Uc +vU ,j2

' and i{UUc +vUU ,je'2

are in a steady-state recursive competitive equilibrium if, in every period,

1. a household solves the utility maximization problem, (3) ± (11), taking its counterpart¶s

(either its parent¶s or child¶s) decision as given,

2. the ¿rm solves the pro¿t maximization problem, and the capital and labor markets

clear, i.e., (21) ± (23) hold,

3. the government policy rules satisfy (24) ± (27),

4. the goods market clears, and

5. the measures of dynasties are constant, i.e., (20) holds.

4 Calibration

The four main parameters 2 the degree of time preference  and those of parental altruism

and child¶s altruism, , f, and  2 are determined simultaneously so that the steady-state

equilibrium of the model replicates the U.S. economy in terms of four key statistics: the

capital-output ratio and the relative sizes of bequests and two-way inter vivos transfers. The

functional forms and other parameters are chosen so as to be consistent with macroeconomic

and cross-section data in the United States.

16
4.1 The Choice of Functions and Parameter Values

The model uses the following Cobb-Douglas utility function with constant relative risk aversion (CRRA),

x+f> k, @



fk

 +k4@   k,3k3  4

4   >

and the Cobb-Douglas production function,

I+N|> D|O|, @ Nw

| +D|O|,

3w >

where D| @ h>|D and O| @ hD|O. Table 2 summarizes the choice of parameters. The model

also uses a progressive federal income tax function and a progressive estate tax function.17

The working ability in this model corresponds to the hourly wage of each household. Three

levels of ability h> h2> and h and their probabilities s and s2 are chosen so that the earnings

distribution is consistent with the U.S. data. The correlation of hourly wages of age 1 and

age 2 is assumed to be 0.80, and that of an age 2 parent and an age 1 child is assumed to be

0.40.18 For the fertility shock, I simply assumed that 90 percent of households have about

three children and that 10 percent of households have no children.19 For more details about

the choice of functions and parameter values, see Sections 3.1 to 3.3 in Nishiyama (2000).

Table 2: Parameters

Share Parameter for Consumption  0.765

Coef¿cient of Relative Risk Aversion  2.0

Capital Share of Output  0.32

Depreciation Rate of Capital Stock  0.046*

Long-Term Real Growth Rate  0.011*

Population Growth Rate  0.010*

Survival Rate at the End of Age 3  0.546

Fertility Rate at the Beginning of Age 3  0.9

* Annual Rate

4.2 Target Variables

The target value of the capital-output ratio is 2.81. The capital stock used here is measured

by μ¿xed reproducible tangible wealth¶ minus μdurable goods owned by consumers.¶ These

data are taken from the Survey of Current Business (1997). For the output data, the nominal

gross domestic product is used. So, the average capital-output ratio in 1990-96 is 2.81.

17Based on the formula by Gouveia and Strauss (1994).

18In this calibration, education spending paid by parent households is not included in inter vivos transfers,

since the model does not consider the households of age 29 or younger. The correlation of hourly wage of the

parent household and its child households, 0.4, is partly due to the parental investment in the children¶s schooling. 19This setting makes the population growth rate 1 percent per year.

17
For the relative sizes of bequests and inter vivos transfers, the model uses the Àow data

from Gale and Scholz (1994) based on the 1986 Survey of Consumer Finance (SCF) in

which each head of household was asked if he or she contributed $3,000 or more to other

households during 1983-85. Table 3 shows the annual Àows of intergenerational transfers

and their relative sizes as a percentage of net wealth.20

Table 3: The Annual Flows of Intergenerational Transfers 2 Gale and Scholz (1994)

Annual Flow

Transfer Category In Billions of Dollars As a Percentage

of Net WealthW

Support Given to:

Children or Grandchildren 37.74 0.32

Parents or Grandparents 3.44 0.03

Trusts 14.17 0.12

Life Insurance 7.84 0.07

Bequests 105.00 0.88

College Payments 35.29 0.29

*Aggregate net wealth in the 1986 Survey of Consumer Finance was $11,976 billion.

Table 4 shows the target values used in this calibration. First, both trusts and life insurance are included in bequests from parents to children. The relative size of bequests becomes

1.06 percent of total private wealth. Second, the gifts to grandchildren are included in the

gifts to children, and the gifts to grandparents are included in the gifts to parents. Then, the

relative size of the inter vivos transfers from parents to children becomes 0.32 percent and

that from children to parents becomes 0.03 percent.

Table 4: Target Variables and Values on Bequests

Transfer Category As a Percentage

of Net Wealth

AdjustedW

(17/20.8)

The Annual Flow of Bequests, Trusts,

Life Insurance, and a Part of Gifts 1.06 0.87

Inter Vivos Gifts to Children 0.32 0.26

Inter Vivos Gifts to Parents 0.03 0.02

*Adjusted to remove the effects of the supplemental high-income subsample of the SCF.

According to the estimate by Gale and Scholz, the stock of inter vivos transfers as a

percentage of net wealth declines from 20.8 percent to 17 percent if they do not include

the supplemental high-income subsample of the SCF. To avoid the inÀuence of very wealthy

20Excerpted from Table 4, p. 152, Gale and Scholz (1994), and rearranged.

18
households, the same rate of reduction is simply applied to the Àow data. The second column

of Table 4 shows the target variables used in this calibration.

4.3 Obtained Main Parameters

Table 5 shows the main parameters obtained through the calibration. For the altruistic parameters , f, and , the ¿rst column shows the parameters per recipient and the second

column shows the parameters per donor.

Table 5: The Obtained Main Parameters

Per Recipient Per Donor

Annual Time Preference  0.934  0.934

Parental Altruism (for Bequests)  0.529  q 0.792

Parental Altruism (for Gifts) f 0.323 fq 0.484

Child¶s Altruism  0.025 @q 0.017

Preference on the Next Generation f 0.069 fq 0.103

Note:  (the coef¿cient of relative risk aversion) = 2.0.

The annual time preference parameter  turned out to be 0.934. In other words, the

annual discount rate of a household¶s own future utility is 6.6 percent.

The degree of parental altruism on the child¶s well-being after the death of the parent

household  is 0.529. This number shows the relative importance of the future utility of each

of the child households to the future utility of the parent¶s own household and determines the

level of altruistic bequests. Since a parent household is assumed to have 1.5 child households

in this calibration, the degree of altruistic bequest motive is 0.792 in total. This means that a

parent household cares about its child households 21 percent less than it cares about itself.

The degree of parental altruism on the child¶s well-being while the parent household is

alive f is 0.323. This number determines the level of inter vivos transfers from a parent

household to each of its child households. Considering the number of child households per

parent household, the degree of inter vivos transfer motive is 0.484 in total.

The degree of child¶s altruism on the parent¶s well-being  is 0.025. This calibration

considers only tangible gifts and bequests. Since the annual Àow of inter vivos transfers

from child households to their parent household is only 0.02 percent of total private wealth,

the degree of child¶s altruism turned out to be a very small number. That number would be

higher if we also considered the transfer from child households to their parents in the form

of services.

In the main calibration, the coef¿cient of relative risk aversion  is assumed to be 2.0.

Table 6 shows the results under different assumptions of  from 1.0 to 4.0. If  were higher

(lower), both the parameter of time preference and the degree of altruism would be lower

(higher) to keep the capital-output ratio and the relative sizes of bequests and inter vivos

transfers at the same level.

19
Table 6: Obtained Parameters Under Different Assumptions of 

Coef¿cient of Relative

Risk Aversion 

1.0 2.0 4.0

Annual Time Preference  0.945 0.934 0.906

Parental Altruism (for Bequests)  0.707 0.529 0.341

Parental Altruism (for Gifts) f 0.522 0.323 0.127

Child¶s Altruism  0.117 0.025 0.001

4.4 The Comparison with Other Dynamic Models

Table 7 compares the obtained parameters with other conventional dynamic models. The

¿rst column shows the result of the model developed in this paper (the extended life-cycle

model) the second, third, and fourth columns show the results of an in¿nite horizon model

with liquidity constraints, an overlapping generations (OLG) model with accidental bequests,

and a pure life-cycle model, respectively.

Table 7: The Comparison with Other Dynamic Models 2 Obtained Parameters

Extended

Life-Cycle

Model

In¿nite Horizon Model

w/ Liquidity

Constraints

OLG

Model w/

Accidental

Bequests

Pure

Life-Cycle

Model

Annual Time Preference () 0.934 0.942 0.939 0.946

Parental Altruism +, 0.529 1.000 0.000 0.000

Parental Altruism +f, 0.323 1.000 0.000 0.000

Child¶s Altruism +, 0.025 1.000 0.000 0.000

Capital-Output Ratio 2.81 2.81 2.81 2.81

BequestsW 0.87% 2.75% 0.64% 0.00%

Parental GiftsW 0.26% 1.05% 0.00% 0.00%

Child¶s GiftsW 0.02% 4.91% 0.00% 0.00%

* Annual Flow as a Percentage of Net Wealth

The model in this paper becomes an in¿nite horizon model with liquidity constraints

when the parameters of altruism are changed to be unity. It becomes an OLG model with

accidental bequests when those parameters are changed to zero. Furthermore, it becomes

a pure life-cycle model when a perfect annuity market is introduced to the economy and

accidental bequests are eliminated.

The in¿nite horizon model predicts too high a percentage of bequests and inter vivos

transfers. The percentage of gifts from children to parents is especially high compared with

20
Table 8: The Comparison with Other Dynamic Models 2 Inequality

Extended

Life-Cycle

Model

In¿nite Horizon Model

w/ Liquidity

Constraints

OLG

Model w/

Accidental

Bequests

Pure

Life-Cycle

Model

U.S.

DataE

Gini Coef¿cients

Earnings 0.610 0.622 0.606 0.605

EarningsE2 0.381 0.400 0.375 0.373 E 0.51

Income 0.496 0.511 0.496 0.495

Wealth 0.701 0.782 0.687 0.680 Ee 0.78

Share of Wealth (%)

Top 1% 14.6 17.0 13.3 12.9 29.6

Top 5% 30.1 38.0 28.6 28.1 53.5

Top 10% 46.5 57.1 45.0 44.3 66.1

Top 20% 71.0 82.0 69.5 67.8 79.5

Top 40% 93.2 97.7 92.2 92.4 92.9

Top 60% 99.0 100.0 98.6 98.7 98.6

Top 80% 100.0 100.0 100.0 100.0 100.4

(1) Sources: Díaz-Giménez et al. (1997), Table 6 Quadrini et al. (1997), Tables 1 & 2.

(2) Households of age 1 and 2.

(3) For household heads ages 35-50.

(4) For married couples only, the number is 0.71 according to the PSID 1989 wealth data.

the U.S. economy.21 The overlapping generations model with accidental bequests replicates

the level of bequests fairly well. Of course, it cannot explain any inter vivos transfers.22

To show the performance of the extended life-cycle model, I also compare the distributions of earnings, income, and wealth produced by these four models in Table 8.

In the calibration, the distribution of the working ability of a household is similar to

that of the hourly wage of a married couple. The discrepancy between the earnings Gini

coef¿cient of the model and the U.S. data is partly because the U.S. data include the single

households. The wealth Gini coef¿cient generated by the extended model is 0.701, which

is lower than that of U.S. data (0.78) but almost the same as that of married couples in the

21However, the 4.91 percent predicted by the in¿nite horizon model may not be too high if we consider services

such as informal eldercare. 22The results under different assumptions of  are as follows:

In¿nite Horizon OLG Pure Life-Cycle



1.0 4.0 1.0 4.0 1.0 4.0

q 0.950 0.920 0.950 0.912 0.953 0.920

Bequests (%) 3.07 2.59 0.50 0.69 0.00 0.00

Parental Gifts (%) 1.11 1.02 0.00 0.00 0.00 0.00

Child¶s Gifts (%) 5.34 4.69 0.00 0.00 0.00 0.00

21
Panel Study of Income Dynamics (PSID) data (0.71).

If we look at the wealth distribution, the in¿nite horizon model seems to best explain the

inequality. The top 1 percent and 5 percent shares of wealth are, however, still much lower

than those of the U.S. data. But, the extended model explains the wealth shares of the top

40 percent, 60 percent, and 80 percent of households better than the in¿nite horizon model.

Also, it produces better wealth distribution compared with the two overlapping generations

models (the OLG model with accidental bequests and the pure life-cycle model).

The results also imply that both the bequest motive and the inter vivos transfer motive

tend to intensify inequality. For example, if we compare the in¿nite horizon model with the

pure life-cycle model, the wealth Gini coef¿cient in the former is higher by about 0.1 than

in the latter. But those intergenerational transfers and life-cycle savings cannot fully explain

the skewness of the wealth distribution in the United States if we look at the top 1 percent

share of the household wealth. We need to add some other forms of assets, such as chunky

assets as well as risky assets.

5 Policy Experiments

In this section, ¿rst, a 100 percent estate and gift tax is introduced to the economy to show

the effects of altruistic bequests and gifts on wealth accumulation and inequality. Next, a

perfect annuity market is introduced to show the role of accidental bequests due to lifetime

uncertainty. Finally, both a 100 percent estate and gift tax and a perfect annuity market are

introduced to show the total effect of intergenerational transfers.

5.1 A 100 Percent Estate and Gift Tax

How much would national wealth be reduced if altruistic bequests and gifts were eliminated?

How would the distribution of income and wealth change? In this experiment, the tax rate

was raised to 100 percent on both bequests and gifts. Government expenditure and wealth

level were assumed to be unchanged, and the increase in tax revenue (from a 100 percent tax

on accidental bequests) was transferred to all households in a lump-sum manner. The results

are shown in Table 9.

If there were no altruistic bequests and gifts, national wealth would be reduced by 11.1

percent in a closed economy and 16.4 percent in a small open economy.23 In a closed economy, the interest rate would rise by 1 percentage point.

Earnings inequality would be slightly lower because the estate tax would discourage

high-income households from working. Income and wealth inequalities would not change in

a closed economy but would be higher in a small open economy. This happens because of

the lump-sum transfers. Low-income households would save less as the lump-sum transfer

increases. The share of the wealth held by the top 1 percent of households would fall from

14.6 percent to 13.5 percent in a closed economy and to 14.2 percent in a small open economy. That result implies that altruistic bequests and transfers account only slightly for the

skewness of wealth distribution in the United States.

23In a small open economy, the interest rate and the wage rate are ¿xed at the levels of the baseline economy.

22
Table 9: When a 100 Percent Estate and Gift Tax Is Added to the Economy

Baseline

Economy

Closed

Economy

Small Open

Economy

% National Wealth 11.1 16.4

% Labor 0.2 0.5

% GNP 3.6 4.9

 Interest Rate 1.0 no change

% Wage Rate 3.7 no change

Gini Coef¿cients

Earnings 0.610 0.605 0.605

Earnings (Ages 1 & 2) 0.381 0.373 0.374

Income 0.496 0.498 0.510

Wealth 0.701 0.700 0.712

Share of Wealth (%)

Top 1% 14.6 13.5 14.2

Top 5% 30.1 29.4 30.5

Top 10% 46.5 46.1 47.8

Top 20% 71.0 71.0 72.8

Top 40% 93.2 93.5 94.2

Top 60% 99.0 98.9 99.0

Top 80% 100.0 100.0 100.0

5.2 A Perfect Annuity Market

If accidental bequests 2 instead of altruistic transfers 2 were eliminated from the economy,

how much would national wealth be reduced? And, how would the income and wealth inequalities change? In this experiment, a perfect annuity market is introduced to the economy

to eliminate precautionary savings for the uncertain life span. Table 10 shows the results.

National wealth would be reduced by a modest 0.2 percent in a closed economy and 1.0

percent in a small open economy. The change in the interest rate is negligible. The effect

of a perfect annuity market on earnings and income inequalities is very small. But, wealth

inequality would increase signi¿cantly. The Gini coef¿cient would rise about 0.02 points.

The share of wealth held by the top 1 percent of households would rise from 14.6 percent to

18.7 percent in a closed economy and to 18.8 percent in a small open economy.

The introduction of a perfect annuity market would reduce precautionary savings for

uncertain life span. But, at the same time, it would increase the marginal value of wealth at

the beginning of age 3 when the life-cycle saving is highest for most households. This result

implies that accidental bequests (and lifetime uncertainty) do not explain the skewness of

wealth distribution in the United States.

Even though the steady-state wealth level and GNP are lower than those of the baseline

economy, the introduction of a perfect annuity market would improve welfare. Table 10

also shows the welfare changes based on the compensating variation wealth measure and the

23
Table 10: When a Perfect Annuity Market Is Added to the Economy

Baseline

Economy

Closed

Economy

Small Open

Economy

% National Wealth 0.2 1.0

% Labor 0.4 0.4

% GNP 0.2 0.0

 Interest Rate 0.1 no change

% Wage Rate 0.2 no change

Welfare Changes (%)

Compensating VariationE 3.9 2.6

Equivalent VariationE2 4.0 2.7

Gini Coef¿cients

Earnings 0.610 0.608 0.608

Earnings (Ages 1 & 2) 0.381 0.379 0.379

Income 0.496 0.499 0.500

Wealth 0.701 0.720 0.721

Share of Wealth (%)

Top 1% 14.6 18.7 18.8

Top 5% 30.1 33.7 33.8

Top 10% 46.5 49.4 49.5

Top 20% 71.0 73.0 73.1

Top 40% 93.2 94.2 94.2

Top 60% 99.0 99.2 99.2

Top 80% 100.0 100.0 100.0

(1) (Total Wealth / Compensating Wealth ± 1)100

(2) (Equivalent Wealth / Baseline Total Wealth ± 1)100

equivalent variation measure. Households would be on average better off by 3.9±4.0 percent

in a closed economy and 2.6±2.7 percent in a small open economy. For the computation of

these welfare measures, see Appendix B.

The effect of a perfect annuity market would be much larger if households were not altruistic or, equivalently, if a 100 percent estate tax were introduced. In the economy without

any altruistic transfers, the introduction of a perfect annuity market would reduce national

wealth by 3.4 percent in a closed economy and by 4.6 percent in a small open economy. In

the presence of intergenerational transfers, both altruistic bequests and inter vivos transfers

would be a buffer for the risk of uncertain life span. If these altruistic transfers were prohibited for some reason, the risk of lifetime uncertainty would be larger, so the effect of perfect

annuity markets would be larger.

24
5.3 A Perfect Annuity Market with a 100 Percent Estate and Gift Tax

In total, how much would national wealth be reduced if there were no intergenerational

transfers? How much would they affect income and wealth inequalities? Table 11 shows the

effect of both a perfect annuity market and a 100 percent estate and gift tax.

Table 11: When a 100 Percent Estate and Gift Tax and a Perfect Annuity Market Are Added

to the Economy

Baseline

Economy

Closed

Economy

Small Open

Economy

% National Wealth 14.1 20.3

% Labor 0.3 0.7

% GNP 4.6 6.0

 Interest Rate 1.3 no change

% Wage Rate 4.8 no change

Gini Coef¿cients

Earnings 0.610 0.603 0.604

Earnings (Ages 1 & 2) 0.381 0.371 0.372

Income 0.496 0.491 0.507

Wealth 0.701 0.687 0.704

Share of Wealth (%)

Top 1% 14.6 13.1 13.8

Top 5% 30.1 28.8 30.4

Top 10% 46.5 45.3 47.5

Top 20% 71.0 68.9 71.4

Top 40% 93.2 92.7 93.6

Top 60% 99.0 98.7 98.9

Top 80% 100.0 100.0 100.0

National wealth would be reduced by 14.1 percent in a closed economy and by 20.3

percent in a small open economy. The interest rate would rise 1.3 percentage points in a

closed economy. Earnings inequality would be slightly lower, but income and wealth Gini

coef¿cients would fall in a closed economy and rise in a small open economy. The share

of wealth of the top 1 percent of households would fall from 14.6 percent to 13.1 percent

in a closed economy and to 13.8 percent in a small open economy. This result implies that

intergenerational transfers account to some extent for the skewness of wealth distribution in

the United States.

6 Concluding Remarks

Through the calibration of the model to the U.S. economy, I obtained the degrees of intergenerational altruism. The parent¶s motive for inter vivos transfers f turned out to be smaller

than its altruistic bequest motive . There are several possible explanations for this result.

25
First, a parent household may discount its child household¶s future utility less than the

child discounts its own future utility. In that case, even though the child household cares

about its current consumption relatively strongly, the parent household does not want to

make a gift to allow the child to consume as much as it wants. Second, in the real economy,

the working ability and the effort level of a child household may not be fully observable by

its parent. In that case, the parent household would transfer less in the form of inter vivos

transfers in order to avoid the moral hazard problem caused by intergenerational risk sharing.

Third, this result seems to support the altruistic model of transfers rather than 3joy-of-giving ́

type models because if the gift motive were sel¿sh, parents would make more gifts while they

were alive.

Based on the transfer data in the Survey of Consumer Finance, the child¶s inter vivos

transfer motive  turned out to be very small. But, there are other unmeasured gifts by

children that should be considered, such as informal caregiving. According to Arno, Levine,

and Memmott (1999), the national economic value of informal caregiving was estimated

at $196 billion in 1997. If we consider a part of that caregiving as the gifts from child

households to their parent households, the degree of child¶s altruism will be much higher

than the result obtained in the model.24

The introduction of inter vivos transfers and two-way altruism does not change the effect

of intergenerational transfers on wealth accumulation very much. The results are similar

to those in the previous paper (bequests only). National wealth would be reduced by 14

percent in a closed economy and by 20 percent in a small open economy if there were no

intergenerational transfers.

Regarding the wealth inequality, the effect of these transfers is not very strong. Based

on the parameters chosen and obtained in this paper, eliminating transfers would lower the

wealth Gini coef¿cient in a closed economy, but increase it slightly in a small open economy.

When we look at the share of wealth held by the top 1 percent of households, a 100 percent

estate and gift tax (to eliminate altruistic transfers) would reduce that share, but a perfect

annuity market (to eliminate accidental bequests) would rather increase inequality. In other

words, bequests and inter vivos transfers seem to contribute to the skewness of the wealth

distribution in the United States, but are not likely to be the main cause of it.

Appendixes

A The Computation of Equilibria

The equilibria of the model are obtained numerically. The state space of a dynasty is discretized. To ¿nd the optimal end-of-period wealth, the model uses the Euler equation method

and bilinear (for Type I households) or linear (for Type II households) interpolation of marginal value functions in the next period.

In this appendix, I only explain how to ¿nd the decision rule of households. For other

detatils of the computation, see the appendix to Nishiyama (2000).

24Annual Àow of $196 billion in 1997 corresponds to 0.86 percent of the (estimated) total private wealth. If

we consider 70 percent of this amount as inter vivos gifts from child households to their parent households, the

degree of child¶s altruism, 4, becomes 0.158.

26
A.1 The Decision Rule of Households

The algorithm to ¿nd the decision rule of Type I households is as follows. For simplicity, the explanation is abstracted from population growth, productivity growth, and lifetime

uncertainty.

1. Set the initial numbers of marginal values iyf

Uc+evU ,je

'2=

2. For each +avU > l, 5 De2  He  i4> 5> 6> 7j ¿nd the decision rule of all households,

gUc+evU , @ gR or g&, taking government policy rules  @ i8 +=,> 7+=,> .+=,> wu77>

F}> Z}j, factor prices +uf> zf,> and the marginal values as given.

(a) Set the initial values on the decision of the child household gf

& @ +ff&> kf&> jf&> df

& ,=

(b) Given the decision of the child household gf

&, ¿nd the optimal decision of its

parent household gf

R @ +ffR> kfR> jfR> df

R ,.

i. Set the initial value of the parent¶s end-of-period wealth df

R +gf

&, and the gift

to its child household jf

R+df

R > gf

&,=

ii. Find the level of consumption and working hours, ff

R+jfR> df

R >gf

&, and

kf

R+jfR> df

R >gf

&,, using the marginal rate of substitution of ff

R for kfR and aftertax marginal wage rate.

iii. Compare the marginal utility of consumption of its own and of its child

household. If x+ff

R> kfR,  f x+ff

&> kf&, with equality holds when jf

R A 3>

go to Step (iv). Otherwise, replace jf

R with jR that solves % @ duj plq

mf x



ff

& . %> kf&



x



ff

R  %> kfR



m subject to j

R @ jfR . %  3> and return

to Step (ii).

iv. Check the Euler equation of the parent household. If

C

CfR

x+ff

R> kfR, 

;

?

=

H Y

Y@

R

yUcn+ev

U , +if l @ 6,

H Y

Y@

R

yUc32+ev

U , +if l @ 7,

with equality holds when df

R A 3> go to step (c). Otherwise, replace dfR with

d

R where

d

R @

;

?

=

duj plq





H Y

Y@

R

yUcn+ev

U ,  Y

YSR x+ff

R> kfR,





 +if l @ 6,

duj plq





H Y

Y@

R

yUc32+ev

U ,  Y

YSR x+ff

R> kfR,





 +if l @ 7,

subject to d

R  3> and return to step (ii).

(c) Similarly, given the decision of the parent household gf

R obtained in step (b), ¿nd

the optimal decision of its child household g

& @ +f&> k&> j&> d

& ,.

(d) Compare the new decision of the child household, g

&, with the old one, gf&= If the

difference is suf¿ciently small, then go to step (e). Otherwise, replace gf

& with

g

& and return to step (b).

(e) Compute the marginal values +y

Uce+evU ,> yUc2+evU ,, or yUc+evU , using +gf

R>gf

&,=

3. Compare the new marginal values iy

Uc+evU ,je

'2 with iyf

Uc+evU ,je

'2. If the difference

is suf¿ciently small, then stop. Otherwise, replace iyf

Uc+evU ,je

'2 with iy

Uc+evU ,je

'2

and return to step 2.

27
B The Computation of Welfare Measures

Since the labor supply of households is endogenous in this paper, I constructed the following

welfare measures based on the wealth level of households and the government. The results

are shown in Section 5.2.

B.1 The Compensating Variation

Let yf

Ucn2 +dR> d&> h&,, yfUc +dR> d&> h&,, and yfUUc +d> h, be the value functions of a Type I

parent household, a Type I child household, and a Type II household, respectively, in the

baseline economy, where l 5 i4> 5j and m 5 i5> 6> 7j. Let [f

Uc +dR> d&> h&, and [fUUc +d> h,

be the cumulative measures of Type I households and Type II households, respectively, in the

baseline economy. Let y

Ucn2 +dR> d&> h&,, yUc +dR> d&> h&,, and yUUc +d> h, be the corresponding value functions in the alternative economy. Suppose that the functions d

Ucn2 +dR> d&> h&,,

d

Uc +dR> d&> h&,, and dUUc +d> h, solve

y

Ucn2 

d

Ucn2> dUc> h&



@ yf

Ucn2 +dR> d&> h&,>

y

Uc 

d

Ucn2> dUc> h&



@ yf

Uc +dR> d&> h&,>

y

UUc 

d

UUc> h

@ yf

UUc +d> h,

for all +dR> d&> h&, 5 D2  H and +d> h, 5 D  H. Then the compensating wealth N is

de¿ned as

N @ [

2

'

]

2f.

+d

Ucn2@q . dUc, g[fUc+vU , . [

e

'2

]

f.

d

UUc g[fUUc+vUU , . Zf} >

where Zf

} is the government wealth in the baseline economy. Let N be the total wealth (including the government wealth) in the alternative economy. The welfare change (in percent)

measured by the compensating variation is de¿ned as 

N@N  4



433. The alternative

economy is potentially Pareto preferred to the baseline economy if N A N=

B.2 The Equivalent Variation

Let [

Uc +dR> d&> h&, and [UUc +d> h, be the cumulative measures of Type I households and

Type II households, respectively, in the alternative economy. Suppose that the functions

d.

Ucn2 +dR> d&> h&,, d.Uc +dR> d&> h&,, and d.UUc +d> h, solve

yf

Ucn2 

d.

Ucn2> d.Uc> h&



@ y

Ucn2 +dR> d&> h&,>

yf

Uc 

d.

Ucn2> d.Uc> h&



@ y

Uc +dR> d&> h&,>

yf

UUc 

d.

UUc> h

@ y

UUc +d> h,

28
for all +dR> d&> h&, 5 D2  H and +d> h, 5 D  H. Then the equivalent wealth N. is de¿ned

as

N. @ [

2

'

]

2f.

+d.

Ucn2@q . d.Uc, g[Uc+vU , .[

e

'2

]

f.

d.

UUc g[UUc+vUU , . Z} >

where Z

} is the government wealth in the alternative economy. Let Nf be the total wealth

(including the government wealth) in the baseline economy. The welfare change (in percent)

measured by the equivalent variation is de¿ned as 

N.@Nf  4



 433. The alternative

economy is potentially Pareto preferred to the baseline economy if N. A Nf=

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30