Technical Paper Series

Congressional Budget Of¿ce

Washington, D.C.

Measuring Time Preference and Parental Altruism

Shinichi Nishiyama

Congressional Budget Of¿ce

October 2000

2000-7

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.
Measuring Time Preference and Parental Altruism

Shinichi Nishiyama

Congressional Budget Of¿ce

Washington, D.C. 20515

e-mail: shinichi@cbo.gov

October 2000

Abstract

This paper extends a heterogeneous agent overlapping generations model by implementing bequests 2 both altruistic and accidental 2 and measures the degrees of

time preference and parental altruism through the calibration of the model to the U.S.

economy. In this model, a parent household and its adult child households behave consistently and strategically to determine their optimal consumption, working hours, and

savings. Based on the obtained parameters, the paper also shows the individual effects

of altruistic and accidental bequests on wealth accumulation by examining the impacts

of a 100 percent estate tax and a perfect annuity market in the model. To match the

economy¶s capital-output ratio and the relative size of bequests observed in the United

States, the parent household would have to discount its children¶s utility by about 30

percent relative to its own utility, according to the model. The model suggests that total bequests contribute about 14 percent to wealth accumulation in a closed economy

and 21 percent in a small open economy. Also, the effect of a perfect annuity market

depends on the degree of parental altruism in the economy.

1 Introduction

Most existing macroeconomic models assume either that people care about their descendants as much as they care about themselves (in¿nite horizon models) or that they don¶t care

about their descendants at all (life-cycle models). But when economists evaluate government

policies that involve income redistribution between generations, the effects of these policies

critically depend on the degree of the altruism between parents and children. If people are

mostly altruistic, in¿nite horizon models are appropriate. But if they are mostly sel¿sh,

overlapping generations models are better.

Previous analyses that evaluate the degree of bequest motives using panel data, however,

show several different conclusions, and it is still inconclusive to what extent parents are

WI would like to thank Andy Abel, Bob Dennis, Doug Hamilton, Richard Rogerson, and seminar participants

at the University of Pennsylvania and the Congressional Budget Of¿ce for valuable comments and suggestions.

I am especially indebted to Víctor Ríos-Rull for his guidance from the early stage of this paper. The views

expressed herein do not necessarily reÀect those of the Congressional Budget Of¿ce.

1
altruistic toward their children. For example, Hurd (1987) and Wilhelm (1996) show that

parental altruism is insigni¿cant and that bequests are likely to be accidental. In contrast,

Menchik and David (1983) and Bernheim (1991) show that the bequest motives of parents

are strong, although these are not necessarily caused by altruism. In fact, Hurd and Bernheim

actually used the same data set, the Longitudinal Retirement Household Survey, but reached

opposing conclusions using different approaches.

Previous studies also differ greatly with regard to estimating the importance of transfer

wealth versus life-cycle wealth. On the one hand, Modigliani (1988) and others estimate that

the share of bequeathed wealth among total wealth is at most 20 percent.1 On the other hand,

Kotlikoff and Summers (1981) show that transfer wealth (which includes inter vivos transfers) accounts for more than 80 percent of total wealth. The difference between those two

conclusions is signi¿cant, although it is due in large part to different de¿nitions of transfer

wealth.

This paper constructs a measure of time preference (how much people discount their own

future utility) and a measure of parental altruism (how much they discount their children¶s

utility), and it answers the same questions using an extended dynamic general equilibrium

model.

More speci¿cally, the paper uses a heterogeneous agent overlapping generations model,

extended to include altruistic and accidental bequests.2 In this model, a parent household and

its adult child households play a Cournot-Nash game and decide their optimal consumption,

working hours, and savings in each period. The model is calibrated to the U.S. economy. In

that calibration, the two main parameters, time preference and parental altruism, are determined simultaneously so that the steady-state equilibrium of the model replicates the U.S.

economy in terms of the following two key statistics 2 the capital-output ratio and the relative size of bequests. Finally, based on the parameters obtained, the paper calculates the

individual contribution of altruistic and accidental bequests to wealth accumulation.

For simplicity, the model abstracts from inter vivos transfers as well as education spending by parents. So, in this model, intergenerational transfers occur only at the time of death in

the form of bequests.3 But, in the calibration, both bequests and part of inter vivos transfers

in the data are regarded as bequests in the model.4 Altruism in this model is one-sided, i.e.,

parents care about their children but children don¶t care about their parents. In the baseline

economy, there are assumed to be no annuity markets other than the Social Security pensions

later, the model is extended to include a private annuity market to analyze the contribution

of precautionary savings.

1

See Modigliani (1988) for several estimates of others.

2

Intentional bequests are not necessarily motivated by parental altruism but also by risk sharing and by gift

exchange. According to the empirical studies, however, those last two motives are not very large. For example,

Wilhelm (1996) showed that parents tend to leave an equal amount of bequests to each of their children even

if the children¶s earnings are signi¿cantly different. 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. 3

This assumption is partly justi¿ed because uncertainty about earnings and lifetime implies incentives for

altruistic parents to defer transfers to their children. But, in the presence of borrowing constraints, there are

incentives for parents to make inter vivos transfers. 4

In the calibration, education spending paid by parents and part of inter vivos transfers are not regarded as

bequests in this model. Instead, I assume the parent-child working ability correlation, partly due to schooling,

using a Markov transition matrix.

2
The ¿ndings in this paper are as follows. With a coef¿cient of relative risk aversion of

2.0, the annual time preference parameter turns out to be 0.934, and the degree of parental

altruism is 0.51 for each child household and 0.69 in total.5 The last-mentioned number

indicates that a parent household cares about its child households 31 percent less than it cares

about itself. If a 100 percent estate tax was introduced to eliminate parental bequest motives,

national wealth would be reduced by 10 percent in a closed economy and by 15 percent in

a small open economy. In addition, if a perfect annuity market was introduced to eliminate

precautionary savings and accidental bequests, national wealth would be reduced, in total,

by 14.3 percent and 20.5 percent in a closed and a small open economy, respectively. The

effect of a perfect annuity market depends critically on whether parents are altruistic. With

parental altruism, the introduction of the perfect annuity market would increase national

savings slightly.

The remainder of the paper is laid out as follows. The extended overlapping generations

model is described in section 2. The model is calibrated to the U.S. economy, and two main

parameters are obtained in section 3. The contribution of bequests to wealth accumulation

is analyzed in section 4, and section 5 concludes the paper. The algorithm to calculate the

steady-state equilibria and an optimal annuity holding are explained in the appendix to this

paper.

2 Model

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

altruistic and accidental bequests. The model considers both parental altruism and lifetime

uncertainty, and households in this economy play a Cournot-Nash game to make an optimal

decision about their consumption, working hours, and savings.

2.1 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. In the calibration of the model, a household is assumed to be a married

couple, but all of their decisions are made jointly. Also, there is assumed to be no strategic

interaction between siblings.

The Life Cycle of Households. The life of a household is shown in Figure 1. 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. So, age 1 corresponds to 30-44 years old, age 2 corresponds to 45-59

years old, age 3 corresponds to 60-74 years old, and age 4 corresponds to 75-89 years old.6 A

household dies either at the end of age 3 or at the end of age 4. The mortality rate is known,

5

In the calibration, I assume the population growth rate as 1 percent per year and 35 percent per generation.

So, each parent household has 1.35 child households. 6

Because of this setting, households in this model are assumed to retire at the beginning of age 60, contrary

to the fact that most people retire at either age 62 or age 65 in the United States.

3
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’).

It dies with

prob. 1- .

It dies with

Its parent dies prob. 1.

with prob. 1-  .

Its parent dies

with prob. 1.

but for each household its own life span is uncertain. When a household reaches age 3, its

child households of age 1 are 3born, ́ and the former becomes a parent household.

Labor Income and Capital Income. 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 be retired. Though a household can work at home to

produce a limited amount of consumption goods and services, its working ability is assumed

to be low and deterministic. There are only one kind of assets (which are supposed to be a

mixture of bonds, stock, and real estate) a household can hold. It receives capital income

according to its wealth level and the market interest rate. There is assumed to be a borrowing

constraint, and the wealth of each household must be nonnegative.

Taxes and Social Security Bene¿ts. A household pays federal income tax according to

its total income (the sum of labor income and capital income). A household that inherits

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

of age 1 or 2 pays payroll tax for Social Security and Medicare based on its labor income.

A household of age 3 or 4 receives Social Security bene¿ts. The Social Security system is

assumed to be one of de¿ned bene¿t type. Every household of age 3 or 4 is eligible for Social

Security bene¿ts and, for simplicity, the size of the bene¿t is assumed to be the same for all

households.

Dynasty and Altruism. Since each household lives either three or four periods, at any

period of time there are two types of dynasties 2 the dynasties with both a parent household and its child households (Type I), and the dynasties with age 2 households only (Type

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

cares about its child households and is assumed to be equally altruistic. (Since a parent also

4
Figure 2: Two Types of Dynasties

Generation 0:

Generation 1:

Generation 2:

Type I (Age 3, 1)

Type I (Age 4, 2)

Type II (Age 2 Only)

Time

Generation 3:

Type I (Age 3, 1)

Type I (Age 4, 2)

knows its children are altruistic toward its grandchildren, it actually cares about all its descendants indirectly.) Thus, a parent household chooses end-of-period wealth, which will be

bequeathed to its child households if it dies. The wealth choice is made so as to maximize

the weighted sum of its own utility and its children¶s utility.

Strategy of a Parent and a Child. Beginning-of-period wealth of a parent household and

its child households, the working ability of the child households, and the mortality of the

parent household (at the end of age 3) are known to each other. A parent and its children

choose, simultaneously, their own optimal consumption, working hours, and end-of-period

wealth. Since a parent household is altruistic toward its child households, and the children

know their parents are altruistic, the decisions of a parent and its children are dependent on

each other. For example, if a parent knew its children¶s wealth was going to be higher in the

next period, it would reduce the amount of bequest since the marginal value of the bequest

would be smaller. Also, if a child knew the bequest of its parent was going to be higher, it

would reduce its own savings and consume more.7

2.2 The Households¶ Problem

2.2.1 The Preference of a Household

Consider a household that lives either three or four periods and call it a Generation 0 household. If this household is sel¿sh, the household¶s problem is shown as

pd{

iSf

 cf j

e

'

xf @ H

%

[



'

3x



ff

 > kf



. x



ff

e> kfe

&

7

In this model, a parent household and its child households behave strategically. But, this model doesn¶t

consider so-called strategic bequests (or gift exchanges) in which the child offers service to the parent in exchange

for future bequests.

5
subject to some budget constraint, where xf is the lifetime utility of Generation 0 x +=> =,

is an instantaneous utility function ff

 and kf are consumption and working hours, respectively, at age l  is a time preference factor and  is the survival rate at the end of age 3.

This household chooses its optimal consumption and working hours to maximize its lifetime

utility xf.

Suppose that Generation 0 considers not only the utility from its own consumption and

leisure (say, k4@  kf

 ) but also the utility of Generation 1, and that Generation 1 considers

the utility of Generation 2 as well, and so on. Then the total utility of Generation 0 (including

the utility from its descendants) Xf is shown as

Xf @ pd{

iSf

 cf j

e

'

H

k

xf . *X

l

@ pd{

iSf

 cf j

e

'

H

5

7xf . *

;

?

=

pd{

iS

 c j

e

'

x . *

;

?

=

pd{

iS2

 c2 j

e

'

x2 . ===

<

@

>

<

@

>

6

8 >

where * denotes the discount factor by the parent household on the utility of its child households. This paper assumes that * @ 2q? 4, where q is the number of child households

per parent household and  is the degree of parental altruism. Since the consumption and

leisure of the child household occur two periods later, the utility is discounted by 2. The

parent cares about its children proportionally to the number q of its child households. The

degree of parental altruism  shows how much the parent household cares about each of its

adult child households relative to how much it cares about itself in the same period.

This paper decomposes * and measures the degree of time preference  and the degree

of parental altruism  (per child) or  q (in total) through the calibration using the aggregate

statistics of national wealth and intergenerational transfers.

2.2.2 The State of a Dynasty

Since the utility maximization problem of a household is nested as shown above, in the

following sections the preference of households is described by value functions.

For Type I dynasties, the state of each dynasty is shown by the ages of parent and 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@ ` = In the calibration, h&c is a member of ih

&c> h2&c> h&cj

for age l @1 or 2, and it follows a Markov process. For Type II dynasties, the state of each

dynasty is simply shown by the age i5j, the beginning-of-period wealth, and the working

ability of the age 2 households +d> h,.

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&, and vUU @ +d> h,=

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

function of a Type II household of age 2 is denoted as yUUc2 +vUU ,.

6
2.2.3 Type I Households

An Age 3 Parent and Its Age 1 Children. The value function of an age 3 parent household

is shown as

yUc+vU > , @ pd{ SRcRc@

R

ix+fR> kR, . qx+f&> k&,

. H 

 yUce+v

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

 (1)

subject to

d

R @ 4

4 . iz hRkR . +4 . u, dR . wu77  8 +u dR,  fRj> (2)

where vU is the state of this dynasty,

vU @ +dR> 3> h&,>

 is the parent¶s conjecture of its age 1 child households¶ decision,

+vU ,@+f&> k&> 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



= (3)

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

only) kR, 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 . 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). 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 (2) is a budget constraint of this parent household, where  is the

growth rate of the economy, z is the wage per ef¿ciency unit of labor, u is the rate of return

on capital, wu77 denotes Social Security bene¿ts, and 8 +=> =,is a federal income tax function.

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

R is split equally and bequeathed

to each of q child households. In the law of motion (3), . +=, denotes an estate tax (both

federal and local) function.

Similarly, the value function of an age 1 child household is shown as

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

&



x+f&> k&, .  H  yUc2+v

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

 (4)

7
subject to

d

& @ 4

4 . iz h&k& . +4 . u,d&  8 +z h&k&>ud&,  7+z h&k&,  f&j> (5)

where  is the child¶s conjecture of its age 3 parent household¶s decision,

+vU ,@+fR> kR> d

R,>

and the law of motion of the state is (3).

The child household chooses its optimal consumption f&, working hours k&, and end-ofperiod wealth level d

&> taking the decision of its parent household  as given. 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 (5) is a budget constraint of this child household, and 7+=, is a Social

Security and Medicare tax (payroll tax) function.

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 each of its child households becomes a parent at the beginning of

the next period= So, the value function of an age 4 parent household is shown as

yUce+vU > 2, @ pd{ SRcRc@

R



x+fR> kR, . qx+f&> k&, . qH yUc+v

U ,m h&

 (6)

subject to (2), where the law of motion of the state is

v

U @ +d& . dR@q  .+dR@q,> 3> h&,= (7)

The parent household considers its children¶s value at the beginning of the next period q yUc

discounted by = Similarly, the value function of an age 2 child household is shown as

yUc2+vU > e, @ pd{ S&c&c@

&



x+f&> k&, .  H yUc+v

U ,m h&

 (8)

subject to (5), where the law of motion of the state is (7).

The Strategy of a Parent and Its Children. 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> d

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

The best response functions of an age 3 parent and an age 1 child are derived from the

value functions (1) and (4) as follows:

U+g&> vU , @ duj pd{ SRcRc@

R

ix+fR> kR, . qx+f&> k&,

. H 

 yUce+v

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



subject to (2), and

U+gR> vU , @ duj pd{ S&c&c@

&



x+f&> k&, .  H  yUc2+v

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



8
subject to (5), where the law of motion of the state is (3).

Similarly, the best response functions of an age 4 parent and an age 2 child are derived

from the value functions (6) and (8) as follows:

Ue+g&> vU , @ duj pd{ SRcRc@

R



x+fR> kR, . qx+f&> k&, . qH yUc+v

U ,m h&



subject to (2), and

U2+gR> vU , @ duj pd{ S&c&c@

&



x+f&> k&, .  H yUc+v

U ,m h&



subject to (5), where the law of motion of the state is (7).

Solving these best response functions, Nash equilibrium decision rules are obtained as

gUc+vU , @ 

fUc+vU ,> kUc+vU ,> d

Uc+vU ,



for l 5 i4> 5> 6> 7j. The consistency condition is +vU , @ gUc+vU , for all vU 5 D2  H and

l 5 i4> 5> 6> 7j =

2.2.4 Type II Households

The value function of an age 2 household is simply

yUUc2+vUU , @ pd{ Scc@



x+f> k, .  H yUc+v

U ,m h

 (9)

subject to

d @ 4

4 .  izhk . +4 . u, d  8 +z h k> u d,  7+zhk,  fj > (10)

where the law of motion of the state is

v

U@+d

> 3> h

&,=

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

gUUc2+vUU , @ 

fUUc2+vUU ,> kUUc2+vUU ,> d

UUc2+vUU ,



=

2.3 The Measure of Households

Let {Uc+vU , denote the measure of Type I households of age l 5 i4> 5> 6> 7j, and let {UUc2+vUU ,

denote that of Type II households of age 2= Also, let [Uc+vU , and [UUc2+vUU , be the corresponding cumulative measures.

9
2.3.1 Population of Households

The population of age 1 child households is normalized to be unity, i.e.,

]

2f.

g[Uc+vU ,@4=

Then, the population of Type I child households of age 2 is

]

2f.

g[Uc2+vU , @ 

4 . 

>

where  is a population growth rate and  is the survival rate of the parent household. The

population of Type II households of age 2 is

]

f.

g[UUc2+vUU , @ 4  

4 . 

=

Since every Type I parent household has q @ +4 . ,

2 child households, the population of

Type I parent households of age 3 is

]

2f.

g[Uc+vU , @ 4

q>

and the population of Type I parent households of age 4 is

]

2f.

g[Uce+vU , @ 

q +4 . ,

=

Since we don¶t have to use {Uc+vU , and {Uce+vU , because

{Uc+vU , @ 4

q

{Uc+vU , and {Uce+vU , @ 4

q

{Uc2+vU ,

for all vU 5 D2  H> from now on only i{Uc +vU ,> {Uc2 +vU ,> {UUc2 +vUU ,j is used as the

measure of dynasties.

2.3.2 The Law of Motion of the Measures

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 dynasties is

{

Uc2+vU , @ 

4 . 

]

2f.

4^@

R'@

RcEtU ` 4

@

&'@&cEtU 

 c2+h

& m h&, g[Uc+vU ,> (11)

and

{

Uc+vU , @ +4 . ,

]

2f.

4

@

R'@

&c2EtU n@

RceEtU *?3.+@RceEtU *?,



2c+h

& m h&, g[Uc2+vU ,

.

]

f.

4^@

R'@

2EtUU ` 2c+h

& m h, g[UUc2+vUU ,



> (12)

10
and the law of motion of the measure of Type II households is

{

UUc2+vUU , @ 4  

4 . 

]

2f.

4

@'@

&cEtU n@

RcEtU *?3.+@RcEtU *?,



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



= (13)

The steady-state condition is

{

Uc+vU , @ {Uc+vU , and {UUc2+vUU , @ {UUc2+vUU , (14)

for all vU 5 D2  H> vUU 5 D  H> and l 5 i4> 5j =

2.4 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 , . ]

f.

d g[UUc2+vUU , . Z}> (15)

O @ [

2

'

]

2f.

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

.

]

f.

h kUUc2+vUU , g[UUc2+vUU ,= (16)

In a closed economy, the total output \ 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,> (17)

where  is the depreciation rate of capital and 

7 is the marginal Social Security tax rate.

2.5 The Government¶s Policy Rule

Government tax revenue consists of federal income tax W8 , Social Security and Medicare

taxes W7, and federal and state estate taxes in the next period W

.= These revenues are calculated as follows:

W8 @ [

2

'

]

2f.

8 +z hR kUcn2+vU ,>udR, @q g[Uc+vU ,

.[

2

'

]

2f.

8 +z h& kUc+vU ,>ud&, g[Uc+vU ,

.

]

f.

8 +zhkUUc2+vUU ,>ud, g[UUc2+vUU ,> (18)

11
W7 @ [

2

'

]

2f.

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

.

]

f.

7 +zhkUUc2+vUU ,, g[UUc2+vUU ,> (19)

W

. @ +4  ,

]

2f.

.



d

Uc +vU , @q

g[Uc+vU ,

.

]

2f.

.



d

Uce +vU , @q

g[Uc2+vU ,= (20)

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 > (21)

where Qu( is the population of households of age 3 or 4, i.e.,

Qu( @ 4

q



4 . 

4 . 



=

2.6 Recursive Competitive Equilibrium

This paper considers the steady state of the economy only, because the main purpose of the

model is to measure the degrees of time preference and parental altruism and to estimate the

contribution of bequests to wealth accumulation. The transition analyses will be presented

in other papers. The de¿nition of a steady-state recursive competitive equilibrium (which is

also a Markov Perfect Equilibrium) for 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,.

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 yUUc2 +vUU , >

the decision rules of households,

q

fUc +vU ,> kUc +vU ,> d

Uc +vU ,

re

'

and q

fUUc2 +vUU ,> kUUc2 +vUU ,> d

UUc2 +vUU ,

r

>

the measures of dynasties,

i{Uc +vU ,j2

' and {UUc2 +vUU ,>

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

12
1. each household solves the utility-maximization problem, (1) ± (9), 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., (15) ± (17) hold,

3. the government policy rules satisfy (18) ± (21),

4. the goods market clears, and

5. the measures of dynasties are constant, i.e., (14) holds.

3 Calibration

The two main parameters, the degree of time preference  and that of parental altruism

, are determined simultaneously so that the steady state of the model replicates the U.S.

economy in terms of two key statistics 2 the capital-output ratio and the relative size of

bequests including a part of 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.

As is explained below, a Cobb-Douglas-CRRA utility function and a Cobb-Douglas production function are used for the calibration. Table 1 summarizes the choice of these parameters.

Table 1: 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

* annual rate

The following subsections describe the choice of functional forms and parameter values,

the choice of two target variables and values, and, ¿nally, the result of the calibration 2 the

degrees of time preference and parental altruism.

3.1 Households

There is assumed to be a large number of households, and the population of age 1 households

is normalized to be unity.

13
Table 2: The Working Hours of a Household (Head and Wife, including Housework)

Ages 1 and 2

(30-59 years old)

Mean 4,810

Standard Deviation 1,353

Skewness 0.121

Kurtosis 3.835

90th percentile 6,430

Source: PSID 1993 Family Data.

Utility Function. According to Cooley and Prescott (1995), the elasticity of substitution of

consumption for leisure is not far from unity. So, the model uses the following Cobb-Douglas

utility function with constant relative risk aversion (CRRA),

x+f> k, @



fk

 +k4@   k,3k3  4

4   =

Here  is the coef¿cient of relative risk aversion, and it is set to be 2.0 in the main calibration.

Later, I also show the results when the coef¿cient is toggled from 1.0 to 4.0. The maximum

working hours k4@ 

 depend on the age of the household and are explained below.

Working Hours. Table 2 shows the working hours of a household in the Panel Study of

Income Dynamics (PSID) 1993 Family Data. The annual working hours are the sum of the

working hours of a husband (Head) and a wife (Wife), including housework.

Total Work Hours @ Head¶s Market Work Hours . Head¶s Housework Hours

.Wife¶s Market Work Hours . Wife¶s Housework Hours.

Suppose the 90th percentile (6,430 hours) is regarded as the maximum working hours k4@ 



(l @ 1 or 2). Then, the share of average working hours (4,810 hours) is 0.75, and the share

of leisure is 0.25.8 So, the share parameter for consumption in the Cobb-Douglas utility

function should not be very far from 0.75. In this calibration, the parameter  is chosen to be

0.765 so that average working hours of age 1 and age 2 become approximately 4,810 hours

in the steady state.

Market Work and Housework. According to the PSID data, for married couples between

ages 30 and 59, about 68 percent of total working hours are declared as market work and

the remaining hours are declared as housework. This ratio is used to compute the taxable

earnings of each household in the model. Also, one hour of housework is assumed to produce

8

The maximum working hours in the PSID data are actually 11,400 hours in 1992. But, I use the 90th

percentile instead because the average of the maximum in 15 years must be signi¿cantly smaller.

14
the same value of consumption goods or services as the after-tax hourly market wage of this

household.

For simplicity, there is no retirement decision in this model, and all of the age 3 and age

4 households are retired. The maximum working hours of age 1 and age 2 are assumed to

be 6,430 hours, in which 68 percent are assumed to be market working hours. Subtracting

this amount, the maximum working hours of age 3 and age 4, k4@ 

 (l @ 3 or 4), are set to be

2,058 hours.

Working Ability. The working ability in this model corresponds to the hourly wage of

each household. According to the PSID data, the average hourly wage of a married couple

has the distribution shown in Table 3.9 Data are calculated by using the family weight in

PSID and the following formula,

Hourly Wage @ Head¶s Labor Income . Wife¶s Labor Income

Head¶s Market Work Hours . Wife¶s Market Work Hours=

Table 3: The Average Hourly Wage of a Married Couple (in U.S. dollars)

Age 1 Age 2

(30-44 years old) (45-59 years old)

Mean 18.181 18.663

Standard Deviation 19.475 17.403

Skewness 6.708 4.583

Kurtosis 66.332 35.280

Gini Index 0.380 0.385

Source: PSID 1993 Family Data.

Three discrete levels of working abilities in Table 4 are chosen based on the statistics of

the hourly wage. For each age, three levels of ability h> h2> and h and their probabilities s

and s2 are computed so that these ¿ve numbers attain the ¿ve statistics in Table 3.

According to Bureau of Labor Statistics data, average hourly earnings of production

workers have increased by 16.2 percent since 1992. Multiplying by 1.162, the numbers

h> h2> and h are converted into those corresponding to 1997 U.S. dollars.

Markov Transition Matrix. Since one period in this model corresponds to 15 years, to

compute a transition matrix of working ability from the data, we need to have 30 years of

longitudinal hourly wage data. But the PSID has at most 27 years of longitudinal series

(1968-1994). So, the transition matrix from age 1 to age 2 and the matrix from an age

2 parent to an age 1 child were constructed based on the statistics of hourly wages. The

steady-state distribution of each age is consistent with the ¿ve statistics in Table 3. Also, the

correlation of hourly wages of age 1 and age 2 is assumed to be 0.80, and that of an age 2

9

If we consider households that didn¶t work at all in the market in 1992, the distribution of actual working

ability will be slightly different from that of hourly wages.

15
Table 4: Working Abilities of a Household (in 1992 U.S. dollars per hour)

Age 1 Age 2

(30-44 years old) (45-59 years old)

h 6.94 8.06* 7.51 8.73*

h2 27.87 32.38* 29.45 34.22*

h 208.16 241.88* 170.19 197.76*

s 0.535 0.542

s2 0.457 0.450

4  s  s2 0.008 0.008

Note: R and R2 are the shares of e and e2 households,

respectively.  3 R 3 R2 is the share of e households.

* in 1997 U.S. dollars.

parent and an age 1 child is assumed to be 0.40.10 The Markov transition matrices used in

this calibration are

2 @

3

EC

3=;:< 3=454 3

3=48: 3=;73 3=336

3 3=4:4 3=;5<

4

FD > 2 @

3

EC

3=955 3=6:; 3

3=773 3=883 3=343

3 3=895 3=76;

4

FD

where 2+l> m, @ +h2 @ h



2 m h @ h

, and 2+l> m, @ +h @ h



 m h2 @ h

2,=

Population Growth and Mortality. The population growth rate is assumed to be annually

1.0 percent and 16.1 percent per period (15 years). Since new households are born to the

dynasty every 30 years, the number of child households per parent household q is 1.348

(@ 4=34f)= In the United States, the life expectancy of a 60-year-old male is 80.68 and that

of a 60-year-old female is 85.71.11 Taking the simple average of male and female, the life

expectancy becomes 83.20 years. The survival rate at the end of age 3 (75 years old)  is set

at 0.546 so that the life expectancy in this model is also 83.20.

3.2 The Firm

There is only one perfectly competitive ¿rm in this economy.

Production Function. The model uses the Cobb-Douglas production function,

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

| +D|O|,

3w >

where

D| @ h>|D> O| @ hD|O>

10See Solon (1992) for the father-son correlations in hourly wages.

11Source: http://www.retireweb.com/, which is based on the Group Annuity Mortality Table (GAM83).

16
 is the growth rate of labor productivity, and  is the population growth rate. The capital

share of output  is chosen by

 @ 4  Compensation of Employees . +4  ,  Proprietors¶ Income

National Income . Consumption of Fixed Capital =

The average of  in 1996-1998 is 0.32. The annual growth rate of total factor productivity

is assumed to be 0.74 percent. This number is the average in 1980-1995 computed from

the data. When the labor share of output, 4  > is equal to 0.68, the growth rate of labor

productivity is 1.1 percent. The growth rate of the economy  is also assumed to be 1.1

percent. The annual population growth rate  between 1970-1995 is around 1.0 percent.

The labor productivity D is chosen to be 0.974 so that the wage per unit of ef¿cient labor is

normalized to be unity.

Fixed Capital and Private Wealth. To compute the GNP and factor prices in the model,

the ¿xed capital N is computed by the following formula:

Fixed Capital  Fixed Reproducible Tangible Wealth

Durable Goods Owned by Consumers=

These data are taken from the Survey of Current Business (1997). In 1990-1996, ¿xed capital

accounted for 89.7 percent of ¿xed reproducible tangible wealth, and durable goods owned

by consumers accounted for the remainder.

To connect the total private wealth with the ¿xed capital, it is assumed that all of the

private capital is owned by households and that part of the government-owned ¿xed capital

is effectively owned by households in the form of government bonds.

Fixed Capital @ Private Wealth . Government-Owned Fixed Capital

 Government Bonds Owned by Private Sector

 Durable Goods Owned by Consumers=

Based on the data in 1990-1996, I use an approximate relationship,

Fixed Capital @ 3=<89  Private Wealth>

in the model.

The Depreciation Rate of Fixed Capital. The depreciation rate of ¿xed capital  is chosen

by

 @ Total Gross Investment

Fixed Capital  Productivity Growth  Population Growth.

In 1998, gross private domestic investment accounted, on average, for 16.1 percent of GDP,

and gross government investment (federal and state) accounted for 2.8 percent of GDP. Based

on these numbers and the capital-output ratio of 2.81, the gross investment - ¿xed capital

ratio is 6.7 percent. Subtracting the productivity and population growth rates, the annual

depreciation rate is assumed to be 4.6 percent.

17
3.3 Taxes and Transfers

Federal Income Taxes. For federal income tax, the model uses the following tax function

estimated by Gouveia and Strauss (1994),

Federal Income Tax @ !f



|  +|3 . !2,

3*



>

where | is the total income of a household after the standard deduction and exemption for

four people. In 1997, the sum of this deduction and exemption was $17,900. The parameters

! @ 3=:9; and !2 @ 3=364 are taken from Gouveia and Strauss¶s estimation of 1989 federal

income tax function. The parameter !f corresponds to the limit of the marginal tax rate. To

replicate the statutory income tax, the parameter must be around 0.4 (40 percent). But,

because of itemized deductions, the effective tax rate of high-income households is regarded

as much lower.12 Since in 1997 the ratio of total private income tax to nominal GDP was

0.091, !f is assumed to be 0.25 so that the income tax - GDP ratio is around 9 percent in the

steady-state equilibrium.

Payroll Tax. The Social Security tax rate is 6.2 percent and the Medicare tax rate is 1.45

percent. These are assumed to be taxed on labor income. In 1997, employee compensation

above $68,400 was not taxable for Social Security. But, for simplicity, this ceiling is not

considered in this analysis, i.e., it is assumed to be a 7.65 percent Àat payroll tax in total. The

same amount of taxes is also paid by employers. So the ¿rm¶s pro¿t-maximization problem

becomes

z  (4 . Marginal Social Security Tax Rate) @ DIu+N> DO,,

where the marginal Social Security (and Medicare) tax rate is 0.0765.

Estate Taxes. For federal estate tax, the model uses the same function as federal income

tax, i.e.,

Federal Estate Tax per Capita @ #f



e  +e3 . #2,

3*



=

In this function, e is the number of bequests per capita minus the exemption of $600,000.

The parameters #f @ 3=88> # @ 3=598, and #2 @ 3=37< are chosen so that the function

replicates the federal estate tax schedule. It is simply assumed that each household consists

of a married couple (two persons) and that a husband and a wife receive bequests from his

or her own parents separately. In total, a household can receive at most $1.2 million without

paying federal tax. The federal estate tax that each household pays is twice the number in

the above equation.

In addition, each household pays state estate taxes. Since the estate tax rate differs from

one state to another, in this calibration it is simply assumed to be a 6 percent Àat tax.13

12See Gouveia and Strauss (1994) for effective federal tax rates.

13For example, the estate tax in Pennsylvania is a 6 percent Àat tax and interspousal transfers are not taxable.

The estate tax in New York State is a progressive tax, ranging from 2 percent on amounts below $50,000 to 21

percent (marginal) on amounts above $10.1 million. The average tax rate when $1 million is bequeathed is 5.4

percent.

18
Social Security Bene¿ts. Social Security bene¿ts depend on average indexed monthly

earnings and the corresponding replacement rate. To calculate the bene¿t, we need to have

an additional state variable, the working history of the parent, for each dynasty. But, in this

calibration, for simplicity, the annual bene¿ts per married couple are assumed to be the same

for all eligible households. Table 5 shows the example of bene¿ts for four types of married couples.14 If we take an average of these four couples, the annual bene¿t per couple

is $13,161. If this number is adjusted using the consumer price index, the estimated bene¿t

per couple for 1997 is $18,600. Thus, the annual bene¿t is assumed to be $18,600 in the

steady-state equilibrium.

Table 5: Monthly Social Security Bene¿ts for Retiring Couples in 1987 (in U.S. dollars)

Couple A Couple B Couple C Couple D

Earnings

Husband 24,000 12,000 16,000 24,000

Wife 0 12,000 8,000 8,000

Monthly Bene¿ts

Husband 797 499 606 797

Wife 398 499 393 398

Total Bene¿ts 1,195 998 999 1,195

Source: Table 4-4 in Schulz (1995)

3.4 Target Variables

Capital-Output Ratio. 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 for 1990-1996 is 2.81.

The Relative Size of Bequests: Gale and Scholz (1994). For the relative size of bequests,

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 1983-1985. Gale and Scholz computed the Àow of

intergenerational transfers to estimate the stock of transfer wealth. Table 6 shows the annual

Àows of intergenerational transfers and their relative sizes as a percentage of net wealth.15

Table 7 shows the target values used in this calibration. First, I include both trusts and life

insurance in bequests from parents to children. Then, the relative size of bequests becomes

1.06 percent of total private wealth. Second, since no inter vivos transfers are assumed in

14Excerpted from Table 4-4 in Schulz (1995). Original source: Technical Committee on Earnings Sharing,

Earnings Sharing in Social Security: A Model for Reform (Washington, D.C.: Center for Women Policy Studies,

1988), Table A-1.3.1.2.3. 15Excerpted from Table 4, pp.152, Gale and Scholz (1994), and rearranged.

19
Table 6: 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 SCF in 1986 was $11,976 billion.

this model, I include a half of gifts to children or grandchildren in bequests from parents to

children as disguised bequests. The relative size of the total transfer becomes 1.22 percent.

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

percentage of net wealth goes down from 20.8 percent to 17 percent if they don¶t include

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

households, I simply apply this rate of reduction to the Àow data. The second column of the

table shows the target variables used in this calibration.

Table 7: 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 1.22 1.00

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

The Transfer Wealth Ratio: Kotlikoff and Summers (1981). Alternatively, we can use

the transfer wealth measure de¿ned by Kotlikoff and Summers (1981) after making some adjustments. In their de¿nition, the capital return on transfer wealth is also regarded as transfer

wealth. In this model, the interest rate at the steady-state equilibrium is approximately 6.8

percent, and the after-tax interest rate is 5.44 percent if the tax rate on interest is around 20

percent. Based on the table in their paper, at this interest rate the share of transfer wealth is

around 90 percent.

One of the dif¿culties in applying their measure to this calibration is that their transfer

wealth includes all transfers from parents to children above age 18. To distinguish inter

vivos transfers when children are below age 30 (such as college tuition and other expenses)

20
from others, the same ratio used for the SCF data is applied. Then, 73 percent of the total

Àow is transferred as bequests and inter vivos transfers when children are above age 30. If

the timing of these transfers is considered, 54 percent of transfer wealth can be regarded as

bequeathed assets.16 Then, the share of transfer wealth (bequests and disguised bequests)

that corresponds to this model becomes 49 percent. The target value of the transfer wealth in

this model is set to be 49 percent of total private wealth.

3.5 Obtained Main Parameters

Table 8 shows the key parameters obtained through the calibration when the target variable is

the annual Àow of bequests (= 1.0 percent). Since each parent household in a Type I dynasty

is assumed to have q @ 4=68 child households, for the altruistic parameter  the ¿rst column

shows the parameter per recipient and the second column shows the parameter per donor.

Table 8: When the Annual Flow of Bequests is 1.0 Percent of Total Private Wealth

Per Recipient Per Donor

Annual Time Preference  0.934

Parental Altruism  0.509  q 0.686

Preference on the Next Generation f 0.066 f q 0.089

Note:  (the coef¿cient of relative risk aversion) ' 2f.

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

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

The degree of parental altruism  is 0.509. This number shows the relative importance of

the current (future) utility of each child household to the current (future) utility of the parent¶s

own. Since a parent household is assumed to have 1.35 child households in this calibration,

the degree of parental altruism toward its child households is 0.686 in total. This means that

a parent household cares about its child households 31 percent less than it cares about itself.

Table 9 shows the obtained parameters when the target variable is the transfer wealth ratio

(= 49 percent). The degree of time preference is almost equal to the ¿gure in the previous

calibration, but the degree of parental altruism is a little higher.

Table 9: When the Transfer Wealth Is 49 Percent of Total Private Wealth

Per Recipient Per Donor

Annual Time Preference  0.934

Parental Altruism  0.580  q 0.782

Preference on the Next Generation f 0.074 f q 0.099

Note:  (the coef¿cient of relative risk aversion) ' 2f

16Here, I assumed the difference of timing is 25 years, and the gap between the after-tax interest rate and the

productivity and population growth rate is 3.34 percent.

21
In the main calibration, the coef¿cient of relative risk aversion  is assumed to be 2.0.

Table 10 shows the results under different assumptions of  from 1.0 to 4.0.

Table 10: Obtained Parameters Under Different Assumptions of 

Coef¿cient of Relative

Risk Aversion 

1.0 2.0 3.0 4.0

Annual Time Preference  0.945 0.934 0.921 0.906

Parental Altruism  0.697 0.509 0.393 0.328

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.

4 Policy Experiments

Policy experiments in this paper focus on the effect of altruistic and accidental bequests on

wealth accumulation. If there were no parental altruism nor lifetime uncertainty, there would

be no bequests in the economy.17 The wealth of parent households would be reduced. But, at

the same time, the savings of child households might be increased because those households

could not expect any bequests from their parents any more. The purpose of the following

policy experiments is to evaluate the overall effect of bequests on national wealth.

First, a 100 percent estate tax is introduced to the baseline economy. If the estate tax

rate was raised to 100 percent, all of the altruistic bequests would be eliminated. Thus, from

the donor¶s point of view, the remaining bequests would be accidental because of lifetime

uncertainty. From the recipient¶s point of view, accidental bequests would also disappear

because of the 100 percent tax. It is assumed that the increase in tax revenue from the estate

tax on these accidental bequests is transferred to all households in a lump-sum manner. Thus,

the change in national wealth can be regarded as the contribution of altruistic bequests.

Next, a perfect annuity market is introduced to the baseline economy to evaluate the

contribution of accidental bequests to wealth accumulation. The perfect annuity market has

two effects on private savings 2 (1) to increase the marginal value of savings, and (2) to

decrease the precautionary savings caused by lifetime uncertainty. It will be shown that,

in the presence of parental altruism, the net effect of the perfect annuity market on wealth

accumulation is small and may be opposite to the effect in the absence of altruism.

Finally, both the 100 percent estate tax and the perfect annuity market are introduced at

the same time, eliminating all of the bequests from the economy. In the absence of altruistic

bequests, the effect of the perfect annuity market is signi¿cantly large. In total, national

wealth would be reduced by 14.3 percent in a closed economy and 20.5 percent in a small

open economy.

17This is not necessarily true if bequests are motivated by risk sharing or by gift exchange.

22
4.1 A 100 Percent Estate Tax

If a 100 percent estate tax was introduced, there would be no incentive for parent households

to leave any bequests, but accidental bequests due to lifetime uncertainty would remain in the

absence of perfect annuity markets. Here, the tax increase from the estate tax on accidental

bequests is assumed to be transferred to all households in a lump-sum manner so that the

government wealth level as well as expenditure would not change. The result of this policy

experiment is shown in Table 11.

Table 11: Changes from the Baseline Economy When a 100 Percent Estate Tax Is Added

Closed

Economy

Small Open

Economy

( Capital Stock 10.0 15.0

( Labor 0.0 0.3

( GNP 3.3 4.6

( Bequests 33.6 42.1

( Interest RateE 0.8 no change

( Wage Rate 3.3 no change

 Annual Lump-Sum TransfersE2 1.010 0.807

(1) Change in Percentage Points

(2) $1,000 per household

The capital stock, which corresponds to national wealth, would be reduced by 10.0 percent in a closed economy and 15.0 percent in a small open economy. Thus, altruistic bequests

account for about 10 percent to 15 percent of national wealth, and life-cycle savings and accidental bequests (precautionary savings) account for the remainder. The reduction in wealth is

smaller in a closed economy because of the general equilibrium effect. In a closed economy,

the rate of return on capital would rise by 0.8 percentage points and encourage household

savings. The wage rate would fall by 3.3 percent in a closed economy.

The gross national product would be reduced by 3.3 percent in a closed economy and

by 4.6 percent in a small open economy. Bequests would be reduced by about 33.6 percent

and 42.1 percent, respectively, in these two economies.18 Because of the increase in the tax

revenue on accidental bequests, the annual lump-sum transfer would be increased by $1,010

in a closed economy and $807 in a small open economy.

4.2 A Perfect Annuity Market

In the previous policy experiment, even if a 100 percent estate tax was introduced, about 58

percent to 66 percent of the original level of bequests would remain as accidental bequests.

What would happen if a perfect annuity market was introduced to the economy to elimi18We cannot conclude that altruistic bequests account for 34 percent to 42 percent of total bequests and accidental bequests account for the remainder because these two types of bequests are not additively separable.

23
nate all accidental bequests? The extension of the model for the perfect annuity market is

described in the appendix to this paper.

First, only the effect of a perfect annuity market is evaluated. In this case, precautionary

savings for lifetime uncertainty would be reduced in the presence of annuity markets, but the

marginal value of savings would be increased. The total effect is shown in Table 12.

Table 12: Changes from the Baseline Economy When a Perfect Annuity Market Is Added

Closed

Economy

Small Open

Economy

( Capital Stock 0.5 0.5

( Labor 0.5 0.5

( GNP 0.5 0.5

( Bequests 43.6 43.6

( Interest RateE 0.0 no change

( Wage Rate 0.0 no change

 Annual Lump-Sum TransfersE2 0.205 0.205

(1) Change in Percentage Points

(2) $1,000 per household

The effect of a perfect annuity market on wealth accumulation is relatively small. The

capital stock (national wealth) would be increased by 0.5 percent in both a closed economy

and a small open economy. Because the marginal value of savings would increase, households are encouraged to work longer, and the gross national product would be increased by

0.5 percent both in a closed economy and in a small open economy. When accidental bequests were eliminated by the perfect annuity market, the level of bequests was reduced by

roughly 44 percent in both economies.

4.3 A Perfect Annuity Market with a 100 Percent Estate Tax

What is the total contribution of bequests, both altruistic and accidental, to the wealth accumulation in the United States? Table 13 shows the result.

If there were no bequests at all, national wealth would be reduced by 14.3 percent in

a closed economy and by 20.5 percent in a small open economy. The remainder can be

considered as life-cycle savings. Though labor supply would be increased slightly, the gross

national product would be reduced by 4.7 percent and 6.2 percent, respectively, in closed and

small open economies.

Two things should be noted about this result. First, in the calibration, the bequest in this

model includes altruistic and accidental bequests and half of inter vivos ¿nancial transfers

only, and it doesn¶t include another half of inter vivos transfers as well as educational spending paid by parents. This is the main reason why the reduction of national wealth is not

very large compared with the result of Kotlikoff and Summers (1981). Second, in the general equilibrium analysis, forward-looking households would increase their life-cycle savings

24
Table 13: Changes from the Baseline Economy When a Perfect Annuity Market and a 100

Percent Estate Tax Are Added

Closed

Economy

Small Open

Economy

( Capital Stock 14.3 20.5

( Labor 0.1 0.5

( GNP 4.7 6.2

( Bequests 100.0 100.0

( Interest RateE 1.3 no change

( Wage Rate 4.8 no change

 Annual Lump-Sum TransfersE2 0.534 0.555

(1) Change in Percentage Points

(2) $1,000 per household

when they could not expect to receive future bequests from their parents. This reaction of

the recipient households keeps the national wealth at a relatively high level.

In the previous subsection, the introduction of a perfect annuity market actually increased

national savings. But, is the effect of a perfect annuity market the same even if altruistic

bequests don¶t exist? Table 14 shows how the economy with only a 100 percent estate tax

would change in a perfect annuity market. In an economy without any altruistic bequests, or

equivalently, in the absence of parental altruism, the introduction of a perfect annuity market

would reduce national wealth by 4.8 percent in a closed economy and by 6.5 percent in a

small open economy. The gross national product would be reduced by 1.5 percent and 1.7

percent, respectively.

Table 14: Changes from the Economy with a 100 Percent Estate Tax When a Perfect Annuity

Market Is Added

Closed

Economy

Small Open

Economy

( Capital Stock 4.8 6.5

( Labor 0.1 0.2

( GNP 1.5 1.7

( Bequests 100.0 100.0

( Interest RateE 0.4 no change

( Wage Rate 1.6 no change

 Annual Lump-Sum TransfersE2 1.544 1.362

(1) Change in Percentage Points

(2) $1,000 per household

25
5 Concluding Remarks

This paper constructed a measure of time preference and one of parental altruism by extending a heterogeneous agent overlapping generations model. Then, the model was calibrated

to the U.S. economy, and the degree of time preference and that of altruism were obtained.

Those parameters depend on the choice of target variable of the relative size of bequests and

the variable¶s value. When the statistics from the Survey of Consumer Finance summarized

by Gale and Scholz (1994) were used, the degree of parental altruism turned out to be 0.51

(per child household) and 0.69 (in total). This means, on average, a parent household cares

about its adult child households about 31 percent less than it cares about itself. When the

transfer wealth measure de¿ned by Kotlikoff and Summers (1981) was used as the target

variable after proper adjustments, the degree of altruism was 0.58 and 0.78, respectively.

Though the obtained numbers are slightly different, these results show that the altruistic

model of bequests presented in this paper appears to be in harmony with the U.S. economy.

One of the main features of the model is that it captures the behavior of a parent household (a donor of bequests) and that of its child household (a receiver of bequests) consistently

in a dynamic context. This is the similar feature of the model by Abel (1985). He showed

the closed form solution for the decision of a household in the economy with lifetime uncertainty and imperfect annuity markets. The present paper also considered parental altruism

and showed the decision of households numerically. Regarding the mechanism to determine the optimal level of consumption, savings, and working hours, the model assumed the

Cournot-Nash game between a parent household and its child household, and it calculated

Markov Perfect Equilibria.

For future research projects, the following extensions of the model are planned. The

¿rst is to introduce inter vivos transfers because, in the presence of borrowing constraints,

the effect of inter vivos transfers is different from that of bequests even if the present values

of those transfers are the same. The second is to measure the degree of a child¶s altruism

toward its parents. If two-way intergenerational transfers are properly allowed, the model

will actually contain both an in¿nite horizon model and a pure life-cycle model as special

cases. Then, the Ricardian equivalence proposition can be re-evaluated in the presence of

intergenerational altruism and transfers. Other related topics include the education spending

by parents, the retirement decision of elderly households, and the working decision of wives.

6 Appendix

6.1 The Computation of Equilibria

The equilibria of the model were obtained numerically in the following way. This section

describes, ¿rst, how the state space of dynasties is discretized for the computation next,

the algorithm to ¿nd steady-state equilibria of this model and, ¿nally, the algorithm to ¿nd

the decision rules of households. For a variety of procedures used in the computation of

equilibria of heterogeneous agent economies, see Ríos-Rull (1995, 1997).

26
6.1.1 The Discretization of the State Space

In this model, the state of a dynasty is shown as vU @ +dR> d&> h&, 5 D2  H for Type I

dynasties, or vUU @ +d> h, 5 DH for Type II dynasties, where dR and d& are the beginningof-period wealth of a parent and a child, respectively, and h& is the working ability of a child,

and D @ ^3> d4@ ` and H @ ^h4?> h4@ `= To compute an equilibrium, the state space of a

dynasty is discretized as evU 5 De2  He and evUU 5 De  H> e where De @ id> d2> ===> d@j and

He @ ih> h2> ===> he j=

For all these discrete points, compute

1. the optimal decision of households, igUc+evU ,je

' and gUUc2+evUU ,, where gUc+evU , or

gUUc2+evUU , 5 +3> f4@ `  ^3> k 4@ `  D>

2. the marginal values, y

Uc+evU ,@+ Y

Y@R yUc+evU ,> YY@& yUc+evU ,, and y

UUc2+evUU , @ Y

Y@ yUUc2+evUU ,,

given the government policy rule and factor prices.

Note that d

Uc+evU , and dUUc2+evUU , belong to D @ ^3> d4@ ` instead of De @ id> d2>

===> d@j= 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.19

In this paper, Q@ is set at 84 and Qe is set at 6= The total number of discrete states for

Type I dynasties is 7,803 in each age for Type II dynasties, the total number is 153.

Table 15: The Choice of Discrete Wealth Levels for the Computation

The Wealth of a Household The Interval of The Number of

Above or Equal Below Discrete Points Discrete Points

$0  $30,000 $5,000 6

$30,000  $100,000 $10,000 7

$100,000  $200,000 $20,000 5

$200,000  $500,000 $50,000 6

$500,000  $1,000,000 $100,000 5

$1,000,000  $2,000,000 $200,000 5

$2,000,000  $5,000,000 $500,000 6

$5,000,000  $10,000,000 $1,000,000 5

$10,000,000  $20,000,000 $2,000,000 5

$20,000,000 1

Total 51

Table 15 shows the discrete wealth levels (1997 U.S. $1,000) used for the calibration.

Note that the wealth space is not divided equally. We need to have a smaller number of

discrete points as the wealth level increases because the curvature of the marginal value

decreases as the level rises.

19Because of this bilinear interpolation of marginal value functions, any equilibrium in this model is shown as

a mixed strategy equilibrium.

27
The discretized working ability ih> h2> hj for age 1 households and age 2 households

is shown in Table 4 in section 3=

6.1.2 Steady-State Equilibria

The algorithm to compute a steady-state equilibrium is as follows. Let  denote the government policy rules  @ i8 +=,> 7+=,> .+=,> wu77> F}>Z

}j=

1. Set the initial values of factor prices +uf> zf, and the government policy variables

+wuf

77> Ff

} > Zf

} , if these are determined endogenously.

2. Find the decision rule of households given factor prices and the government policy

variables, igf

Uc+evU ,je

' and gf

2+evUU ,> for all evU 5 De2  He and evUU 5 De  H= e

3. Find the steady-state measure of dynasties, i{f

Uc+evU ,j2

' and {f

UUc2+evUU ,, using the

decision rule obtained in step 2 and the Markov transition matrix on the working ability

of households.

4. Compute the aggregate capital stock and labor supply +N> O, and government aggregate variables and ¿nd factor prices +u> z, and the government policy variables

+wu

77> F

} > Z

} ,.

5. Compare +u> z> wu

77> F

} >Z

} , with +uf> zf> wuf

77> Ff

} >Zf

} ,= If the difference is

suf¿ciently small then stop. Otherwise, replace +uf> zf> wuf

77> Ff

} > Zf

} , with +u> z>

wu

77> F

} >Z

} , and return to step 2.

6.1.3 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  Ha  i4> 5> 6> 7j ¿nd the decision rules of all households,

gUc+evU , @ gR or g&, taking government policy rules  @ i8 +=,> 5+=,> .+=,> 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&> df

& ,=

(b) Given the decision of the child household gf

&, ¿nd the optimal decision of its

parent household gf

R @ +ffR> kfR> df

R ,.

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

R +gf

&,=

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

R+df

R >gf

&, and kf

R+df

R > gf

&,,

using the marginal rate of substitution of ff

R for kfR and after-tax marginal

wage rate.

28
iii. 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 +ff

R> kfR,





 +if l @ 6,

duj plq





H Y

Y@

R

yUc32+ev

U ,  Y

YSR +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&> 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.

6.2 Optimal Annuity Holdings of a Household

In this model, death is uncertain at the end of age 3= In the presence of perfect annuity markets, age 3 households choose the optimal level of end-of-period annuity holdings, t e

Rc+vU ,>

where t is the price of annuity and t @ = Clearly, it will not exceed the end-of-period wealth

level, i.e., 3  t e

Rc+vU ,  dRc+vU ,. If an age 3 household has t e

R of its wealth in the form

of annuity at the end of this period and if it is alive in the next period, its wealth at the beginning of age 4 is +d

R  t eR, . eR @ dR . +4  , eR. If it dies at the end of age 3, the wealth

inherited by its child is simply +d

R   eR, @ q.

The best response functions of an age 3 parent and an age 1 child are written as follows:

U+f&> k&> d

&> vU , @ duj pd{ SRcRc@

RcKR

q

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

. H 

 yUce



v

U



. +4  , qyUUc2



v

UU 

m h&

 r

>

U+fR> kR> d

R> eR> vU , @ duj pd{ S&c&c@

&

q

x+f&> k&,

. H 

 yUc2



v

U



. +4  , yUUc2



v

UU 

m h&

r

>

where the law of motion of the state is

v

U @



d

R . +4  , eR> d&> h&



>

29
v

UU @



d

& .



d

R   eR



@q  .

d

R   eR



@q> h



=

Solving these two functions for +fR> kR> d

R> eR, and +f&> k&> d

&,, we get a Nash equilibrium end-of-period wealth combination and the optimal annuity holding,

gUc+vU , @ 

fUc +vU ,> kUc +vU ,> d

Uc +vU ,> eUc +vU ,



>

gUc+vU , @ 

fUc +vU ,> kUc +vU ,> d

Uc +vU ,



>

for each state v 5 D2  H.

References

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[2] Jere R. Behrman and Mark R. Rosenzweig, 3In-Law Resources, Parental Resources and

Distribution within Marriage, ́ University of Pennsylvania, Population Aging Research

Center Working Paper, September 1998, no. 98-18.

[3] Douglas B. Bernheim, 3How Strong Are Bequest Motives? Evidence Based on Estimates of the Demand for Life Insurance and Annuities, ́ Journal of Political Economy,

October 1991, vol. 99, no. 5, pp. 899-927.

[4] Thomas F. Cooley and Edward C. Prescott, 3Economic Growth and Business Cycles, ́

in Frontiers of Business Cycle Research, edited by Thomas F. Cooley, 1995, pp. 1-38.

[5] William G. Gale and John Karl Scholz, 3Intergenerational Transfers and the Accumulation of Wealth, ́ Journal of Economic Perspectives, Fall 1994, vol. 8, no. 4, pp. 145-160.

[6] Miguel Gouveia and Robert P. Strauss, 3Effective Federal Individual Income Tax Functions: An Exploratory Empirical Analysis, ́ National Tax Journal, June 1994, vol. 47,

no. 2, pp. 317-339.

[7] Michael D. Hurd, 3Savings of the Elderly and Desired Bequests, ́ American Economic

Review, June 1987, vol. 77, no. 3, pp. 298-312.

[8] Laurence J. Kotlikoff, 3Intergenerational Transfers and Savings, ́ Journal of Economic

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[9] Laurence J. Kotlikoff and Lawrence H. Summers, 3The Role of Intergenerational Transfers in Aggregate Capital Accumulation, ́ Journal of Political Economy, August 1981,

vol. 89, no. 4, pp. 706-732.

[10] Paul L. Menchik and Martin David, 3Income Distribution, Lifetime Savings, and Bequests, ́ American Economic Review, September 1983, vol. 73, no. 4, pp. 672-690.

30
[11] Franco Modigliani, 3The Role of Intergenerational Transfers and Life-Cycle Saving in

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no. 2, pp. 1-40.

[12] José-Víctor Ríos-Rull, 3Models with Heterogeneous Agents, ́ in Frontiers of Business

Cycle Research, edited by Thomas F. Cooley, 1995, pp. 98-125.

[13] José-Víctor Ríos-Rull, 3Computation of Equilibria in Heterogeneous Agent Economies, ́ Federal Reserve Bank of Minneapolis, Research Department Staff Report,

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[14] James H. Schulz, 3The Economics of Aging (6th edition), ́ 1995.

[15] Gary Solon, 3Intergenerational Income Mobility in the United States, ́ American Economic Review, June 1992, vol. 82, no. 3, pp. 393-408.

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pp. 37-38.

[17] Mark O. Wilhelm, 3Bequests Behavior and the Effect of Heirs¶ Earnings: Testing the

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no. 4, pp. 874-892.

31