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Working Papers Series
Wealth Inequality and Intergenerational
Links
By: Mariacristina De Nardi
Working Papers Series
Research Department
WP 99-13
Wealth Inequality and Intergenerational Links
Mariacristina De Nardi
First Draft: January 1998
This draft: February 2000
Abstract
Empirical studies have shown that, for many countries, the distribution of wealth is
much more concentrated than the one of labor earnings and that households with higher
levels of lifetime income have higher lifetime saving rates. Previous models have had diÆculty in generating these features. I construct a computable general equilibrium model with
overlapping generations in which parents and children are linked by bequests and earnings
persistence within families. I show that voluntary bequests are important to explain the
emergence of large estates that characterize the top of the wealth distribution, while accidental bequests are not. In addition, the introduction of a bequest motive generates lifetime
saving pro les more consistent with the data. Allowing for earnings persistence within families generates an even more concentrated wealth distribution. A cross-country comparison
between the U.S. and Sweden shows that intergenerational linkages are important to explain
the upper tail of the wealth distribution also in economies where redistribution programs
are more prominent and there is less inequality. Moreover Sweden, with its generous social
safety net, has a larger fraction of people with zero or negative wealth. The model is capable
of reproducing this feature as well.
University
of Chicago and Federal Reserve Bank of Chicago. Comments Welcome. I am grateful to Gary
S. Becker, Lars P. Hansen, Jose A. Scheinkman, Nancy L. Stokey, and especially Thomas J. Sargent for helpful
comments. My work bene ted from many conversations with Marco Bassetto, as well as from his constant
support. I also pro ted from discussions with Lisa Barrow, Martin Floden, Alex Monge, Guglielmo Weber, Chao
Wei and especially Marco Cagetti. I am thankful to Paul Klein and David Domeij for discussions about the
Swedish data. Neither the Federal Reserve Bank of Chicago nor the Federal Reserve System are responsible for
the views expressed in this paper. All errors are my own. Email address: nardi@bali.frbchi.org
1
1
Introduction
Empirical studies (e.g. Hurst, Luoh and Sta ord [20]; Wol [39]; Lillard and Willis [28]; DazGimenez, Quadrini and Ros{Rull [10]) have shown that labor earnings, income and wealth are
signi cantly concentrated, with distributions skewed to the right. However, wealth is the most
concentrated of the three variables with a Gini coeÆcient of .72, earnings rank second with a
Gini coeÆcient of .46, and income is the most disperse of the three with a Gini coeÆcient of
.44. While these empirical regularities are observed in many countries, previous models do not
provide a satisfactory explanation of how the observed earnings distribution leads to the observed
distribution of wealth (Quadrini and Ros{Rull [35]).
In the data, a signi cant fraction of the dispersion in earnings, income and wealth across
households is attributable to their di erent positions in the life cycle. Moreover, in the aggregate,
the intergenerational transmission of wealth is substantial. Kotliko and Summers [24] calculate
that the majority of the current value of the U.S. capital stock (at least 80%) can be attributed
to intergenerational transfers rather than to accumulation out of earnings, which is the emphasis
of the basic life-cycle model of capital accumulation. Gale and Scholz [12] use direct measures
of intergenerational ows and attribute 63% of the current value of the U.S. capital stock to
intergenerational transfers. Mulligan [29] shows that intergenerational links are essential to
explain the emergence of very large estates.
Other empirical work (e.g. Dynan, Skinner and Zeldes [11] and Lillard and Karoly [27])
highlights the fact that households with higher levels of lifetime income have higher saving rates.
Carroll [8] shows that it is diÆcult to explain the behavior of these consumers using either a
standard life-cycle model or a dynastic model.
The goal of this research is to study how wealth is accumulated in a life-cycle economy with
intergenerational links and how the characteristics of the accumulation process in uence the
distribution of wealth, given the distribution of labor earnings. I consider an incomplete markets
life-cycle model with earnings uncertainty, life-span risk, and links between parents and children:
the parents care about leaving bequests to their o spring, and the children partially inherit their
parents' productivity. In this setup households have several motives to save: to self-insure against
income and life-span risk, for retirement and possibly to leave a bequest to their children. The
characteristics of the accumulation process impact the life-cycle pattern of wealth accumulation,
the dispersion of wealth within cohorts and the overall wealth distribution.
I construct a computable general equilibrium model to study how these di erent saving
motives help in understanding the dispersion of wealth across households, both in the U.S. and
Swedish economies, taking as given the distributions of labor earnings. I also calibrate the model
to the Swedish economy to investigate whether intergenerational linkages are important even in
economies where redistribution programs are more prominent and there is less inequality.
My results show that voluntary bequests are important to explain the emergence of large estates that are usually accumulated in more than one generation and that characterize the upper
tail of the wealth distribution in the data, while accidental bequests do not generate more wealth
concentration. I also show that the introduction of a bequest motive generates lifetime saving
pro les more consistent with the data. Saving over the life cycle is the primary factor in understanding how wealth is accumulated at the lower tail of the distribution, while intergenerational
2
links signi cantly a ect the shape of the upper tail. Moreover, the introduction of a humancapital link in which the children partially inherit the productivity of their parents can generate
a yet more concentrated wealth distribution. In this case more productive parents accumulate
larger estates and leave larger bequests to their children who, in turn, are more successful than
average in the workplace. The cross-country comparison between the U.S. and Sweden shows
that intergenerational linkages are also important in economies where redistribution programs
are more prominent and there is less inequality. The calibration to the Swedish data also reveals
that the model reproduces well the fact that Sweden, with its more generous social insurance
net, has a larger fraction of people with zero or negative wealth.
Section 2 reviews some related literature, section 3 brie y discusses the main features of the
U.S. and Swedish data, and section 4 describes the model. Section 5 is a road map of the experiments that I run in order to understand the quantitative importance of each intergenerational
link. Sections 6 and 7 describe the calibration and the results of the various experiments for
the U.S. and the Swedish economies respectively. Section 8 discusses other factors that may be
important to explain the distribution of wealth and explains how the assumptions I make are
likely to a ect my results. Section 9 concludes and discusses some directions for future research.
2
The Literature
Previous attempts at studying how the distribution of wealth is determined fall broadly into
two categories. The rst group of papers studies overlapping-generations economies where all
savings arise over the life cycle.1 The second group of paper studies economies with in nitely
lived dynasties.
Huggett [19] and Gokhale et al. [13] are the only papers within the rst group to focus
primarily on the distribution of wealth.
Gokhale et al. [13] aim at evaluating how much wealth inequality arises from inheritance
inequality. To do so, they construct an overlapping-generations model and focus on intragenerational inequality of households whose head is age 66. Their model allows for random
death, random fertility, assortative mating, heterogeneous human capital, progressive income
taxation and social security. All of these elements are exogenous and calibrated to the data. The
families are assumed not to care about their o spring, hence all bequests are involuntary. To
solve the model, they impose that individuals are in nitely risk averse and that the rate of time
preference equals the interest rate. As a consequence, the families in the model have a constant
per capita consumption pro le, resulting in a large aggregate ow of bequests from people who
die before reaching the maximum lifespan. Moreover, families do not take into account expected
bequests when making consumption and saving decisions. Gokhale et al. nd that inheritances
in the presence of social security play an important role in generating intra-generational wealth
inequality in the cohort they consider. The intuition is that social security annuitizes completely
the savings of poor and middle-income people but is a very small fraction of the wealth of richer
people, who thus keep assets to insure against life-span risk. In this setup, were a market for
1 Cf.
_
_
Imrohoro
glu, Imrohoro
glu and Joines [21], Hubbard, Skinner and Zeldes [18].
3
annuities available, rich people would completely annuitize their wealth and no bequest would
be left.
In Huggett [19], the workers face uninsurable income shocks and uncertain life span. The
government taxes bequests at 100% and redistributes them equally to all agents alive. As in most
papers that address the distribution of wealth, the skewness is generated by the introduction of
a borrowing constraint. The paper succeeds in matching the U.S. Gini coeÆcient for wealth, but
the concentration is obtained by having more people with zero or negative wealth and a much
thinner upper tail than observed in the actual distribution.
The fact that people hit the borrowing constraint too often, leading to a large fraction of
people at zero or negative wealth, is a common problem of models with idiosyncratic income
shocks. In overlapping-generations models this problem is aggravated by the assumption that
young agents are born without wealth and hence need time to accumulate precautionary savings
to hedge against income shocks.
The proportion of people at zero wealth is less of a problem for the second group of papers,
which tries to explain the distribution of wealth in economies populated by in nitely lived dynasties. In this case the precautionary savings have already been accumulated in steady state,
hence the borrowing constraints bind less often. These models disregard the fact that the lower
tail of the distribution of wealth is mainly comprised of young and old households; they succeed
in lowering the proportion of households at zero or negative wealth by treating all agents as if
they were middle aged.
As for the second group of papers, Krusell and Smith [25] study an economy populated
by in nitely lived dynasties that face idiosyncratic income shocks. These dynasties also face
a stochastic process for their discount factor and thus have heterogeneous preferences. The
discount factor changes on average every generation and is meant to recover the fact that parents
and children in the same dynasty may have di erent preferences. Krusell and Smith nd that
allowing for di erent discount factors among agents helps in matching the cross-sectional wealth
distribution.
Casta~neda, Daz-Gimenez and Ros{Rull [9] consider a model of earnings and wealth inequality and use it to study the e ect of tax reforms. Their model economy is populated by dynastic
households that have some life-cycle avor; workers have a constant probability of retiring at
each period and once they are retired they face a constant probability of dying. They care about
their o spring. These newborns enter the model as workers and inherit the family's after-tax
capital; in equilibrium their utility is the same as that of old workers. The paper employs a
large number of free parameters to match some features of the U.S. data that are considered
particularly signi cant, which include measures of the wealth distribution. However, the simple
structure of the model does not allow proper accounting for the life-cycle pattern of savings and
the role of bequests in generating wealth inequality.
Quadrini [34] constructs an in nitely-lived agent model in which agents at each period decide
whether to be entrepreneurs or not. Three elements in the model are crucial. First, the existence
of capital market imperfections induces workers that have entrepreneurial ideas to accumulate
more wealth to reach minimal capital requirements. Second, in the presence of costly nancial
intermediation, the interest rate on borrowing is higher than the return from saving, therefore an
4
entrepreneur whose net worth is negative faces a higher marginal return from saving and reducing
his debt. Third, there is additional risk associated with being an entrepreneur, hence risk averse
individuals will save more. As in Casta~neda, Daz-Gimenez and Ros{Rull [9] the model uses a
large number of free parameters to match features of the earnings and wealth distribution.
In a recent paper Heer [17] adopts a life-cycle setup in which parents care about leaving bequests to their children. In his framework the bequest motive does not a ect much the distribution of wealth. His results di er from mine because his income process is much less representative
of the actual process faced by households and because he assumes that children can perfectly
observe their parent's characteristics and wealth.
In contrast with the papers that study economies with in nitely lived dynasties, I explicitly model the life-cycle structure, which contributes to a signi cant fraction of the dispersion
in earnings, income and wealth across households. Compared to Huggett's [19] paper I add
intergenerational transmission of wealth and ability. In contrast with Gokhale et al. [13], and
consistent with the data, my model generates higher saving rates for people with higher lifetime income and age-savings pro le consistent with the empirical observations. Compared to
Heer [17], my paper does a better job of modeling the earnings process and the bequest motive.
It also explores the relevance of intergenerational transmission of ability and applies the model
to two countries.
Rather than focussing on the wealth distribution, Carroll [8] concentrates on the fact that in
the data households with higher levels of lifetime income have higher lifetime saving rates (see
Dynan, Skinner and Zeldes [11] and Lillard and Karoly [27]). He shows that neither standard
life-cycle, nor dynastic models can recover the saving behavior of rich and poor families at the
same time. To solve this puzzle he suggests a \capitalist spirit" model, in which nitely lived
consumers have wealth in the utility function. This can be calibrated to make wealth a luxury
good, thus rendering nonhomothetic preferences. In my model, nonhomotheticity arises because
parents care about leaving bequests to their children (I calibrate this bequest motive taking into
account the children's utility of receiving the bequest). This setup allows me to test whether the
assumptions I make are consistent not only with the saving behavior of single individuals but
also with the wealth distribution as a whole.
3
On the Empirical Facts
3.1 The U.S. Economy
Capital Transfer
Percentage wealth in the top Percent with
output wealth Wealth
negative or
ratio
ratio
Gini 1% 5% 20% 40% 80% zero wealth
3.0
.63
.72
28
49
75
89
Table 1: U.S. wealth data.
5
99
5.8-15.0
In table 1 I present various statistics on wealth and wealth distribution in the U.S.
The measure of the capital-output ratio depends on the concept of capital one has in mind.
The most comprehensive notion of capital typically used includes residential structures, plant
and equipment, land and consumer durables. This implies a capital output ratio of about 3 for
the period 1959-1992 (Auerbach and Kotliko [3]). A narrower de nition of capital obviously
implies a lower capital-output ratio. Stokey and Rebelo [37] exclude land, consumer durables
and residential structures owned by the government and obtain a ratio below 2.
The notion of transfer wealth as a fraction of total wealth is the one computed by Kotliko
and Summers [24]. The authors distinguish two components of total wealth: transfers and
life-cycle savings. They compute the transfer wealth component for a person alive at a given
point in time as the current value of all non-government transfers received by that person. The
current value is computed using the realized after-tax rates of return on wealth holdings. The
life-cycle component is the residual one. They estimate that transfer wealth accounts for at
least 80% of total wealth. A more recent study by Gale and Scholz [12] uses direct measures of
intergenerational ows from the Survey of Consumer Finances and nds that intergenerational
transfers account for about 63% of the current value of wealth.
The data on the wealth distribution are from Wol [39]. His estimates are based on the 1983
Survey of Consumer Finances. Wealth is de ned as owner-occupied housing, other real estate,
cash, nancial securities, unincorporated business equity, insurance, and pension cash surrender
value, less mortgage and other debt. Wol makes adjustments to account for consumer durables,
household inventories and underreporting of nancial assets and equities. Hurst, Luoh and
Sta ord [20] provide data on the wealth distribution using di erent data sources for 1989. They
piece together the 1989 PSID household wealth up to the 98.6 percentile and then use the IRS
data for the 98.2 to the 99.6th percentile and the Forbes data for the balance of the 32 most
wealthy families. They adopt this strategy because the PSID oversamples poor people (Juster,
Smith and Sta ord [22] compare the PSID with the SCF data and nd that the PSID is accurate
only up to the richest 5% of the population). The wealth measure obtained by Hurst, Luoh and
Sta ord is very close to the one adopted by Wol , but they do not perform any adjustment.
They nd similar numbers for the wealth distribution.
Percentage earnings in the top
Percent with
Gini
negative or
coe . 1% 5% 10% 20% 40% 80% zero income
.46
6
19
30
48
72
98
7.7
Table 2: U.S. data on gross earnings.
Table 2 is computed using data from the Luxembourg Income Study (LIS) data set, which
collects income data sets from di erent countries (it is based on the CPS for the U.S.) and makes
them comparable. The table is computed using data for households whose head is 25 to 60 years
of age, and the de nition of gross earnings includes wages, salaries and self-employment income.
We can see from tables 1 and 2 that earnings display much lower concentration than wealth.
6
3.2 The Swedish Economy
Capital Transfer
Percentage wealth in the top Percent with
output wealth Wealth
negative or
ratio
ratio
Gini 1% 5% 20% 40% 80% zero wealth
2.0
> :51
.73
17 37 75
99
100
30
Table 3: Swedish wealth data.
Table 3 presents various statistics on wealth and wealth distribution as in the U.S. section.
The capital/gdp ratio is computed analogously that of U.S. (see Hansson [16]).
The wealth distribution, the Gini index and the number of people at zero or negative wealth
refers to 1984-85 data (Palsson [32]).
The inheritance-wealth ratio number is from Laitner and Ohlsson [26]. They compute the
current value of household inheritances in Sweden as a fraction of household wealth; the result
they obtain, conditional on the age of the household head varies from a low of .34 (for households
50-59) to a high of .85 (for households 70 and older). Their number for the economy as a whole
is .51. However, this number does not include inter-vivos transfers, therefore the actual present
value of intergenerational transfers to wealth is higher.
Table 4 is also computed from the LIS data set, in the same way described for Table 2.
Percentage earnings in the top Percent with
Gini
negative or
coe . 1% 5% 20% 40% 80%
zero income
Gross earnings
.40
4 15 42
68
98
7.6
Table 4: Swedish data on gross earnings.
As we can see from tables 1-4, the Swedish distribution of earnings is more equally distributed
than the U.S. distribution of earnings, but the Gini coeÆcient for wealth is the same in the two
countries (.72 in the U.S. and .73 in Sweden). However, the high Gini coeÆcient for wealth in
Sweden results from di erent reasons than in the U.S. In the U.S. the top 1-5% of people hold a
large fraction of total wealth, while the fraction of people with zero or negative wealth is relatively
small. On the contrary in Sweden the top 1-5% of people do not hold not as much of total wealth,
but a much larger fraction of people is at zero or negative wealth. This may be due to the fact
that social security and unemployment bene ts are more generous in Sweden than in the U.S.,
and these social insurance programs are a disincentive to save, especially for poorer people. In
fact, individuals for whom social security bene ts are high compared to their life-time income
will not save for retirement in presence of a redistributive social security system. Moreover if
security nets (such as unemployment insurance) are substantial, precautionary savings will be
lower.
7
4
The Model
The economy is populated by overlapping generations of people and an in nitely lived government. The agents may di er in their productivity level. The members of successive generations
are linked to one another by the altruism of the parents toward their children and the o spring's
inheritance of part of their parent's productivity. At age 20 each person enters the model and
starts consuming, working and paying labor and capital income taxes. At age 25 the consumer
procreates and he cares about leaving bequests to his children when he dies. After retirement
the agent no longer works but receives social security bene ts from the government and interest
from accumulated assets. The government taxes labor earnings, capital income, and estates and
pays pensions to the retirees.
4.1 Demographics
During each model period, which is ve years long, a continuum of people2 is born. I assume
that each person does not make saving or consumption decisions until he is 20, when he begins
working. Thus, I model the agent's behavior starting from age 20 and de ne age t = 1 as 20
years old, age t = 2 as 25 years old, and so on. After one model period, at t = 2, the agent's
children are born, and four periods later (when the agent is 45 years old) they are 20 and start
working. Since there are no inter-vivos transfers in this model economy, all individuals start o
their working life with no wealth. Total population grows at a constant, exogenous, rate (n), and
each agent has the same number of children.3 The agents retire at t = tr = 9 (i.e., when they are
65 years old) and die for sure by the end of age T = 14 (i.e., before turning 90 years old). From
t = tr 1 (i.e., 60 years of age) to T , each person faces a positive probability of dying given by
(1
t ); since death is assumed to be certain after age T , T = 0. The assumption that people
do not die before 60 years of age reduces computational time and does not in uence the results
because the number of people dying between the ages of 20 and 60 is small.
Since I consider only stationary environments, the variables are indexed only by age, t, and
the index for time is left implicit.
4.2 Preferences and Technology
Preferences are assumed to be time separable, with a constant discount factor. The utility from
consumption in each period is given by u(ct ) = c1t =(1 ).
Parents care about their sel sh children. The particular form of altruism I consider is called
\warm glow": the parents' derive utility from leaving a bequest (net of estate taxes) to their
o spring. The utility from leaving a net bequest bt , is (bt ). Considering a more sophisticated
form of altruism would increase the number of state variables (already 4 in this setup) and, in
some cases, would generate strategic parent/child interaction.
2 In
the theoretical sections, I will use the terms \agent", \person", \consumer" and \household" interchangeably. Each household is taken to be composed of one person and dependent children.
3 The number of children is thus n5 if n is the growth rate of the population over 5 years, or n25 if n is expressed
in yearly terms.
8
In this economy all agents face the same exogenous age-eÆciency pro le, t , during their
working years. This pro le is estimated from the data and recovers the fact that productive
ability changes over the life cycle. Workers also face stochastic shocks to their productivity level.
These shocks are represented by a Markov process fyt g de ned on (Y; B(Y )) and characterized
by a transition function Qy where Y <++ and B(Y ) is the Borel -algebra on Y . This Markov
process is the same for all households. The total productivity of a worker of age t is given by
the product of his stochastic productivity in that period and his deterministic eÆciency index
at the same age: yt t . The parent's productivity shock at age 40 is transmitted to children at
age 20 according to a transition function Qyh , de ned on (Y; B(Y )). What the children inherit
is only their rst draw; from age 20 on, their productivity yt evolves stochastically according
to Qy . An alternative possibility is to assume that agents face heterogeneous income processes
(both ex-ante and ex-post heterogeneity as opposed to only ex-post heterogeneity) or \education
levels" and that children partially inherit from their parents these di erent income processes.
While this is a sensible extension, it would introduce an additional state variable and is left for
future research.
After retirement, the agents do not work any more but live o pensions and accumulated
assets.
I assume that children cannot observe directly their parent's assets, but only their parent's
productivity when the parent is 40 and the o spring are 15, i.e., the period before they \leave
the house" and start working. Based on this information, they infer the size of the bequest they
are likely to receive.4 I will discuss the relevance and the qualitative e ects of relaxing all of
these assumptions in section 7.
The household can only invest in physical capital, at a rate of return r. The depreciation
rate is Æ , so the gross-of-depreciation rate of return on capital is r + Æ .5 I assume also that the
agents face borrowing constraints that do not allow them to hold negative assets at any time.
I assume that the U.S. are a closed economy with an aggregate production function F (Kt ; Lt ) =
AKt L1t , where Kt is aggregate capital and Lt is aggregate labor. I instead assume that Sweden
is a small open economy, so the interest rate net of taxes is taken as exogenous and equal to that
of the U.S. For each country, I normalize the units of labor and of the good so that, in steady
state, the average labor income of a worker per period and the wage are 1.
4.3 Government
The government is in nitely lived and taxes labor earnings, capital income and estates to nance
the exogenous public expenditure and to provide pensions to the retired agents.
Labor earnings are taken as exogenous and calibrated to the data, matching the after-tax
Gini coeÆcient. Since the U.S. tax system is progressive, this Gini coeÆcient is lower than the
one computed from pre-tax labor earnings. In the model, I introduce a constant tax rate l , in
order to balance the government budget, while all the progressive features of the tax system are
4 Once
again, this is done to keep the number of state variables at a minimum.
this linear technology, Æ is irrelevant for the agents in the economy if we hold r xed; it is only important
for measuring gross revenues pertaining to capital that are included in the de nition of GNP.
5 Given
9
already re ected in the calibrated earnings distribution.
Income from capital is taxed at a at rate a .
Estates larger than a given value exb are taxed at rate b on the amount in excess of exb .
The structure of the social security system is the following: the retired agents receive a lumpsum transfer from the government each period until they die. The amount of this transfer is
linked to the average earnings of a person in the economy.
4.4 The Household's Problem
I consider an environment in which, during each period, a t-year-old individual chooses consumption c and risk-free asset holdings for the next period, a0 . The state variables for an agent are
denoted as x = (t; a; y; yp), where t is his age, a are the assets he carries on from the previous
period, y is the current realization of his productivity process, and yp is the value of his parent's
productivity at age 40 until the worker inherits and zero thereafter. This latter variable takes
on two purposes. First, when it is positive, it is used to compute the probability distribution
on bequests that the household expects from his parent. Second, it distinguishes the agents
that have already inherited, for whom we set yp = 0, from those who have not, for whom yp is
strictly positive. The agents inherit bequests only once in a lifetime, at a random date which
depends on their parent's death. Since there is no market for annuities, part of the bequests the
child receives are \accidental bequests," linked to the fact that people's life span is uncertain
and therefore they accumulate precautionary savings to o set the life-span risk. The optimal
decision rules are functions for consumption, c(x), and next period's asset holding, a0 (x), that
solve the dynamic programming problem described below.
Let's consider the agent's recursive problem distinguishing four subperiods in his life to clarify
the problem he faces in each phase of his life.
(i) From age t = 1 to age t = 3, (from 20 to 30 years of age) the agent works and will survive
with certainty until next period. Moreover, he does not expect to receive a bequest soon
because his parent is younger than 60 and will survive at least one more period for sure.
Since the law of motion of yp is dictated by the death probability of the parent, for this
subperiod yp0 = yp.
n
o
V (t; a; y; yp) = max u(c) + Et V (t + 1; a0 ; y 0; yp)
c;a
0
subject to:
h
i
c 1 + r (1 a ) a + (1 l ) t y
h
a0 = 1 + r (1
i
a ) a c + (1 l ) t y
(1)
(2)
(3)
r is the interest rate on assets, and the evolution of y 0 is described by the transition function
Qy .
10
(ii) From t = 4 to t = 8, i.e. from 35 to 55 years of age the worker will survive for sure up to
next period. However, his parent is at least 60 years old and faces a positive probability
of dying any period; hence a bequest might be received at the beginning of next period.
Iyp>0 is the indicator function for yp > 0; it is 1 if yp > 0 and zero otherwise.
n
V (t; a; y; yp) = max u(c) + Et V (t + 1; a0 ; y 0 ; yp0)
c;a
o
(4)
0
subject to (2) and :
h
i
a0 = 1 + r (1
yp0 =
a ) a c + (1 l ) t y + b0 Iyp>0 Iyp =0
(5)
0
yp with probability
0
t+5
with probability (1
(6)
t+5 )
where Et is the conditional expectation based on the information available at time t.
The conditional distribution of b0 is given by b (x; :).6 b represents the bequest distribution
a person expects if his parent dies; in equilibrium it will have to be consistent with the
behavior of the parent. Since the evolution of the state variable yp is dictated by the death
process of the parent, yp0 jumps to zero with probability t+5 (5 periods is the di erence
in age between each parent and his children).7 I assume the following processes to be
independent: the survival/death of the decision maker; the survival/death of his parent;
the size of the bequest received from the parent, conditional on the parent dying; and the
future labor income, conditional on the current one.
(iii) tr 1, i.e., 60 years old: the period before retirement. The individual faces a positive
probability of dying and hence has a bequest motive to save. De ne after tax bequests as
b(a0 ) = a0 b max(0; a0 exb ).
n
V (t; a; y; yp) = max u(c) +
c;a
0
t
Et V (t + 1; a0 ) + (1
o
0
t ) (b(a ))
(7)
where
b 1
(b) = 1 1 +
2
(8)
subject to (2), (5) and (6).
I choose the functional form for (b) to make it roughly consistent with the child's utility
from receiving the bequest. To derive it, assume that the child consumes a constant amount
(say the average labor income, 1) in each period, and if he inherits, he consumes the bequest
6 The
7 If
probability distribution b depends on x only through t and yp, not through y .
yp = 0, eq.(7) implies yp0 = 0 for sure, which we want.
11
in 2 periods, in equal amounts. Under these assumptions, the additional utility he would
derive from receiving a bequest b would be:
(1 + b2 )1
(1 )
1
(1 )
(1 + b2 )1
+
(1 )
1
:::
(1 )
+ ::: +
2
1
(1
2
1
1
)
(1 + b2 )1
(1 )
:
Collecting terms, dropping the constant and de ning 1 appropriately we obtain (b). 1
can be thought of as a measure of how much the parent values the child's utility. I will
discuss how I chose 1 and 2 in sections 5 and 6.
(iv) From tr to T , i.e. from 65 to 85, after retirement. In the model economy, the agent will
not inherit after turning 65 years old because his parent is dead at that time. Moreover,
after retirement I assume that people no longer work and just live o pensions and interest.
This implies that we can drop two state variables from the retired people's value function,
y and yp, and that the only uncertainty the retired agents face is the time of their death.
n
V (t; a) = max u(c) +
c;a
0
t
V (t + 1; a0 ) + (1
o
0
t ) (b(a ))
(9)
subject to (8) and:
h
i
c 1 + r (1 a ) a + p
h
i
a0 = 1 + r (1 a ) a c + p
(10)
(11)
p is the pension payment from the government. The terminal period value function V (T +
1; a) is set to equal (b(a)).
4.5 Transition Function
From the policy rules, the bequest distribution and the exogenous Markov process for produc~ (x; ). M~ (x; ) is thus the probability distribution
tivity, we can derive a transition function M
0
of x (the state in the next period), conditional on x, for a person that behaves according to the
~ is de ned is (X;
~ X~ ), with:
policy rules c(x) and a(x).8 The measurable space over which M
X f1; ::::; T g <+ Y
(Y [ f0g);
X P f1; :::; T g B <+ B(Y ) B(Y [ f0g);
8 For
~ on c; a; b implicit.
simplicity of notation I keep the dependence of M
12
n o
X~ X [ D
n
X~ ~ : ~ = X [ d; X
o
2 X ; d 2 f;; fDgg
where P is the cardinal set of f1; :::; T g, and D indicates that a person is dead.
~ , it is enough to display it for the sets
To characterize M
L(t; a; y; yp
) (t0 ; a0 ; y 0; yp0) 2 X : t0 t ^ a0 a ^ y 0 y ^ yp0 yp
~ is de ned by M
~ x; L(t; a; y; yp
On such sets M
) =
8
>
>
>
>
<
>
>
>
>
:
h
if x 6= D
t It+1t Ia(x)a Iyp=0 + Iypyp
t+5 +
i
b (x; [0; a a(x)])(1 t+5 )Iyp>0 Qy (y; [0; y] \ Y )
if x = D
0
where I is an indicator function, which equals 1 if the subscript property is true and zero
otherwise.
~ , notice that t is the probability of surviving into the next period. CondiTo understand M
tional on survival, a person currently of age t will be of age t + 1 next period, hence the presence
of It+1t. If his parents are already dead, i.e., yp = 0, he cannot receive bequests anymore, and
his assets next period are a(x) for sure (as discussed above, this is always the relevant case for
people 65 and older). If, instead, his parents are still alive, i.e., yp > 0, they can survive into
the next period with probability t+5 ; in that case, tomorrow's assets for the worker will be a(x)
and yp0 = yp. Alternatively, the parents may die, with probability 1
t+5 ; under this scenario,
the person inherits next period, yp0 = 0, and the probability that next period's assets are no
more than a is the probability of receiving a bequest between 0 and a a(x). Qy describes the
evolution of income; note that the evolution of income, one's survival and the survival of the
parent are independent of each other. Finally, death is an absorbing state.
~ , I can de ne an operator RM~ that maps probability distributions on (X;
~ X~ ):
Based on M
(RM~ m
~ )( ~)
Z
M~ (x; ~)m
~ (dx); 8 ~ 2 X~ :
This operator describes the probability distribution of nding a person in state x0 tomorrow, given
the probability distribution of the state today. Such an operator has a unique xed point, which
is the probability distribution that attributes probability 1 to fDg: everybody dies eventually.
However, in the economy as a whole, we are not interested in keeping track of dead people,
so I will de ne a modi ed operator on measures on (X; X ). Furthermore, it is necessary to
take into account that new people enter the economy in each period. The transition function
corresponding to the modi ed operator RM is thus:
M (x; L(t; a; y; yp
)) =
M~ (x; L(t; a; y; yp
)) + n5 It=5 Qyh (y; [0; y] \ Y )Iyyp
n
13
M modi es M~ in two ways. First, it accounts for population growth; when population grows
at rate n, a group that is 1% (say) of the population becomes 1=n% in the subsequent period.
Second, it accounts for births, which explains the second term in the numerator. If a person is
40 years old (t = 5), his children (there are n5 of them), will enter the economy next period.
All of those children have age t = 1 and zero assets.9 Their stochastic productivity is inherited
from their parent's at 40, according to the transition function Qyh ; y (which is part of x) is their
parent's productivity at 40.
The operator RM is thus de ned as
(RM m)( )
Z
M (x; )m(dx)
8 2 X
RM maps measures on (X; X ) into measures on (X; X ), but it does not necessarily map
probability measures into probability measures. Unless the population is at a demographic
steady state, the total measure of people alive may grow at rate faster or slower than n, which
implies that (RM m)(x) 6= 1 even if m(x) = 1.
4.6 De nition of Stationary Equilibrium
A stationary equilibrium is given by:
8
>
>
>
>
<
>
>
>
>
:
an interest rate r,
allocations c(x); a0 (x),
government tax rates and transfers,( a ; l ; b ; exb ; p),
a family of probability distributions for bequests b (x; ),
and a constant distribution of people over the state variables x: m (x)
such that, given the interest rate and the government policy:
(i) the functions c and a0 solve the maximization problem described above, taking as given
the interest rate, the government tax rates and transfers, and the bequest distribution he
expects to receive from his parent, given as a function of his characteristics x;
(ii) given a per capita exogenous government expenditure g and the structure of the social
security system, the government policy is such that the government budget constraint
balances at every period:
g=
Z h
a r a + l t yIt<t
r
+ b (1
9 Since
t
p Itt
r
0
1 ) max(0; a (x)
t 1 and a 0, I do not need to include I1t and I0t.
14
i
exb ) dm (x);
(iii) m is an invariant distribution for the economy, i.e. it is a xed point of the operator RM
de ned in subsection 1.4.5:
RM m = m :
I normalize m so that m (X ) = 1, which implies that m ( ) is the fraction of people alive
that are in a state 2 X .
(iv) For the U.S., the share of income going to capital is , i.e.
(r + Æ ) K
= :
(r + Æ ) K + w L
R
Aggregate capital, K , is given by a dm (x). Due to the normalizations, at the steady
state, w = 1 and L is the fraction of working age people in the population.
Sweden is treated as a small open economy, so r is taken as exogenous.10
(v) the family of expected bequest distributions b (x; ) is consistent with the bequests that are
actually left by the parents. Let's now characterize this statement using formulas. De ne
rst the marginal distribution of age and income in the population, which is a probability
distribution on
(f1; :::::; T g Y; P (f1; :::; T g) B(Y )) :
mt;y ( t;y ) m (fx 2 X : (t; y ) 2 t;y g)
8 t;y 2 P (f1; :::; T g) B(Y )
De ne m (jt; y ) as the conditional distribution of x given t and y . For any given (t; y ),
m (jt; y ) is a probability distribution11 on (X; X ). For any set 2 X , m ( jt; y ) is
measurable with respect to P (f1; :::; T g) B(Y ) and is such that
Z
Xt;y
m ( jt; y )mt;y (dt; dy ) = m ( )
8 2 X 8 t;y 2 P (f1; :::; T g) B(Y )
The child observes his parent's income at 40. The conditional distribution of the characteristics of the parent at age 40, given an income level yp, is12 m (jt = 5; y = yp). I want
the characteristics of the parent at later ages, conditional on his income as of age 40 being
yp and conditional on not having died. Denote by l(jt; yp) these conditional distributions.
They can be obtained recursively as follows: l( j5; yp) = m ( j5; yp) and
R
~
x M (x; )l(dx t; yp ) :
l( t + 1; yp)
t
10 K represents the average capital held by Swedish citizens, which may di er from average capital present in
Sweden.
11 fm ( jt; y )g is uniquely de ned up to sets of m -measure zero.
t;y
12 I use the letter y to distinguish both from y and yp: y plays the role of income for the parent (state variable
p
p
y ) and the parent's income for the child (state variable yp).
j
15
j
The conditional distributions l(jt; yp) imply conditional distributions of assets la (jt; yp) on
(<+ ; B(<+ )) which are given by
la ( a jt; yP ) l(fx 2 X : a 2 a gjt; yp):
Since the probability of death is independent of income and assets, the distribution of assets
that are bequeathed by dying parents is the same as the distribution of assets of surviving
parents. We thus have
b ((t; a; y; yp); a) =
t 1
Y
l a 2 <+ : n5 (
s=1
s)
"
1
8 a 2 B(<+ ) 8a 2 <+
a exb
aIaex + exb +
I
(1 b ) a>ex
8y; yp 2 Y; t = 1; :::; T 5
#
b
b
2 a jt + 5; yp
!
(12)
In equation (12), I take into account the assumptions made before about the structure of
bequest taxation and the assumption that the bequest is distributed evenly among surviving
children.
I need now to de ne b when t = T 4, which is the last age a person can inherit. Since
there are no survivors at age T + 1, I cannot use the survivor's assets to compute the assets
that are bequeathed. Instead, let's use the policy function a(x) to de ne:
la ( a jT + 1; yp)
Z
X
Ia(x)2 l(dxjT; yp)
a
8 a 2 B(<+ ):
With this de nition, equation (12) can be extended to t = T
thus the formal requirement of consistency on b .
4 as well. Equation (12) is
4.7 The Algorithm
The following steps are used to solve the model:
(i) Solve the household's value functions.
Assume a functional form for (b) (the utility of leaving a bequest) and start from the last
period, T ; next period the agent will be dead for sure, hence he will derive utility only
from the bequests he will leave. Solve backward for the value function at T 1. Continue
analogously, taking as given the value function for next period until the rst period is
reached.
The diÆcult part of solving this model is linked to the curse of dimensionality; there are
four state variables. To manage this problem, keep track of the value function on a coarser
grid (90-150 points) for capital (the grid is not uniform and has more points concentrated
at low levels of capital). The maximization problem is solved for a household that starts
with an initial level of assets on this grid. However, future investment is allowed to lie on a
16
ner grid; this requires the household to evaluate the value function at points that do not
lie on the initial grid, which is accomplished by interpolation. The resulting investment
policy is thus de ned on a ner capital grid.
Keep track of the transition function and invariant distribution for this economy on the
coarser grid. To do so, take the asset level given by the investment policy function, nd
the two closest asset levels that include it on the coarser capital grid, and attribute to each
of these points a weight according to their relative distance from the original capital level
on the investment policy grid.
Choose the number of grid points for capital so that the results are neither sensitive to the
number of grid points nor to the linear interpolation procedure.
(ii) Taking as given the Markov processes for the productivity and productivity inheritance and
the agents' policy functions, compute the transition matrix and the associated invariant
distribution. Since the agents' policy functions are de ned on a ner grid, we need to
map them to the coarser grid used for the value function, transition matrix and invariant
distribution. To do so, take the agents' optimal decision, given his state variables, and nd
the adjacent values that include his optimal choice in the coarser grid. Then attribute to
these two points a weight given by the relative distance between each of the two points and
the agent's optimal choice.
(iii) Iterate on the tax rate on labor income until the government budget constraint is balanced.
(iv) Iterate on bequests until the equilibrium condition described by equation (12) is met.
5
The Experiments
To understand the quantitative importance of these intergenerational links, I construct several
simulations that I run both for the U.S. and the Swedish economies.
I start with an experiment in which the model is stripped of all intergenerational links: an
overlapping-generations model with lifespan and earnings uncertainty. The accidental bequests
left by the people who die prematurely are seized by the government and equally redistributed
to all people alive.13 The idea is to see how much wealth inequality can be generated by the
life-cycle structure when only lifespan and earnings uncertainty are activated.
The second experiment modi es the rst one: the unplanned bequests left are distributed to
the children of the deceased, rather than equally to everybody alive. This experiment is meant to
assess whether an unequal distribution of estates is quantitatively important when all bequests
are involuntary.
13 This exercise uses Huggett's setup but adapts it to the length of the periods and the productivity process
that I use throughout this paper in order to make the results comparable to the other simulations I run. I cannot
use the same time period and income process as Huggett, since the simulations with altruism require a higher
number of state variables and the model would require huge computing resources to solve.
17
Fixed
Parameter
t
t
n
g
a
r
p
Qy
Qyh
Calibrated
Parameter
b
exb
1
2
Value
*
*
1.5
1.2% yearly
19% of GDP
20%
6%
40% average income
+
+
Source(s)
Bell, Wade and Goss [6]
Hansen [15]
Attanasio et al. [2]
Econ. Report of the President [31]
Econ. Report of the President [31]
Kotliko et al. [23]
see text
Kotliko et al. [23]
Huggett [19], Lillard et al. [28]
Zimmerman [40]
Value
10%
40 years of average earnings
.95{.97
-9.5
8
Chosen to Match
see text
see text
capital-output ratio
intergenerational transfers share
\altruistic feedback", see text
Table 5: Parameters for the U.S. economy and their sources.
* refers to a vector
+ see description in the text
The third experiment introduces the bequest motive: parents care about their children and
leave them bequests. This allows us to see whether the fact that some of the bequests left are
voluntary matters.
The fourth exercise activates both the bequest motive and parent's productivity inheritance
in order to evaluate the importance of the family background.
6
Numerical Simulations for the U.S. Economy
Most of the parameters of the model are taken from other sources, while few of them are chosen
to match some aspects of the data. I summarize these choices in Table 5.
For people older than 60, t is the vector of conditional survival probabilities. The series I
use corresponds to the conditional survival probabilities of the cohort born in 1965. People 60
years old and younger survive for sure into the next period.
t is the age-eÆciency pro le vector.
I take the risk aversion parameter from Attanasio et al. [2] and Gourinchas and Parker [14],
who estimate it using consumption data. This value falls in the range (1-3) commonly used in
the literature.
18
The rate of population growth, n is set to equal the average population growth from 1950 to
1997, 1.2%
g is government expenditure excluding transfers (about 19% of GDP).
a is the capital income tax, 20%.
r is the interest rate on capital, net of depreciation and gross of taxes. In models without
aggregate uncertainty it is commonly chosen to be between the risk free rate and the rate of
return on risky assets. I assume an interest rate of 6% so that the capital share of output is
about .36.
Pensions (p) are such that the social security replacement rate is 40% and the implied government transfers to GDP ratio in the model is consistent with the one reported in the Economic
Report of the President [31].
The logarithm of the productivity process is assumed to be an AR(1). I choose its persistence
to be consistent with the one used by Huggett [19] adapted to a ve year period model and its
variance to match a Gini coeÆcient for earnings of workers of about .43 (Lillard and Willis [28]).
The implied autocorrelation parameter is .83 and its variance .41.
The logarithm of the productivity inheritance process (for yp) is also assumed to be an
AR(1). I take its persistence from Zimmerman's [40] estimates and its variance so that the
standard deviation of the logarithm of earnings is in the ballpark provided by Zimmerman [40].
The resulting autocorrelation parameter is .67 and its variance is .42.
I convert both the productivity and the productivity inheritance processes to a discrete
Markov chain according to Tauchen and Hussey [38]. I use three values for the income process.
The resulting income distribution is reported and compared with the data in appendix A.
The remaining parameters are chosen to match features of the U.S. economy as follows.
b is the tax rate on estates that exceed the exemption level exb . According to U.S. law, each
individual can make an unlimited number of tax-free gifts of $10,000 or less per year, per recipient;
therefore, a married couple can transfer $20,000 per year to each child, or other bene ciary. For
larger gifts and estates, there is a \uni ed credit", i.e., a credit received by the estate of each
decedent, against lifetime estate and gift taxes. For the period between 1987 and 1997, each
taxpayer received a tax credit that eliminated estate tax liabilities on estates valued less than
$600,000. The marginal tax rate applicable to estates and lifetime gifts above that threshold is
progressive, starting from 37% (Poterba [33]). However, the revenue from estate taxes is very
low (in the order of .2% of GDP in 1985-97) as there are many e ective ways to avoid such taxes
(see for example Aaron and Munnell [1]); moreover, only about 1.5% of decedents pay estate
taxes. Therefore, in the model I set exb to be 40 times the median income and b to be 10% to
match the observed ratio of estate tax revenues to GDP and the proportion of estates that pay
estate taxes. I discuss the sensitivity of the model to the choice of these two parameters when
describing the results.
I use the discount factor, , to match a capital to GDP ratio of 3. In the calibrations in which
the bequest motive is activated, I use 1 to get a reasonable share of the bequests to aggregate
capital and 2 to make the utility from leaving bequests, (b), roughly consistent with a \truly
altruistic model" in the sense that it is reasonably close to the utility of the child from receiving
the bequest. Figure 22 compares the function with the true value of receiving the bequest for
19
a 65 year old at the 10% (\poor"), 50% (\median") and 90% (\rich") quantiles of the wealth
distribution.14 We see that is a reasonable approximation of the value a truly altruistic parent
would derive from the bequest he leaves. However, for the parent of a poor child, the marginal
utility of leaving a bequest is higher in the fully altruistic model, and the bene t is more concave.
The opposite is true for the parent of a rich child.
Table 6 summarizes the results.15
Capital Transfer
Percentage wealth in the top Percent with
output wealth Wealth
negative or
ratio
ratio
Gini 1% 5% 20% 40% 80% zero wealth
U.S. data
3.0
.63
.72
28 49 75
89
99
5.8-15.0
No intergenerational links, equal bequests to all
3.0
N/A
.64
6 23 64
90
100
17
No intergenerational links, unequal bequests to children
3.0
.37
.65
6 24 65
90
100
17
One link: parent's bequest motive
3.0
.57
.71
12 34 72
93
100
19
Both links: parent's bequest motive and productivity inheritance
3.0
.61
.73
15 38 75
94
100
19
Table 6: Results for the U.S. calibration.
6.1 The Experiment with no Intergenerational Links and Equal Bequests to All
The results of this experiment show that an overlapping-generations model with no dynastic
links and equal distribution of bequests has serious diÆculties in generating enough skewness to
match the distribution of the U.S. wealth. For the parameter values in Table 5, I obtain a Gini
coeÆcient that is below what we observe in the data. Moreover, the concentration is mainly
achieved by having a lower tail which is too fat and an upper tail which is far too thin.
As discussed in the introduction, overlapping-generations models tend to have a large fraction
of people against the borrowing constraint. People are born without savings that could be used
to absorb negative income or productivity shocks. As a consequence, all the young consumers
that get a bad productivity shock hit the borrowing constraint. During their working age, the
households gradually accumulate assets both as life-cycle savings for their old age and as a form
14 The
functions are normalized by adding a constant to t on the graph.
all experiments I exclude 20 year old people from the computations on the wealth distribution. I do so
because in this paper I assume that people start o with zero wealth and I therefore do not propose a theory of
the distribution of wealth for them. To explain the data for 20 year old people, a theory of inter-vivos transfers
would, in my opinion, be required.
15 For
20
of precautionary savings. As a consequence, the fraction of people with zero wealth gradually
declines until retirement.
At the other end of the distribution, the absence of intergenerational links implies that it is
very hard to account for large estates, as a lifetime is too short a period for most households to
accumulate such large fortunes. With the current parameterization the top 1% of the population
holds just about 5.5% of total wealth. With a richer income process, the top quantiles are
somewhat higher, but it is still true that few people have suÆciently high income to accumulate
such large estates over a lifetime. With 7 productivity states (instead of 3) the top 1% of people
hold 8% of the aggregate wealth (which is still lower than obtained in the model with links and
only three income states). My main interest, however, is to show that, with the same income
process, experiments with intergenerational links fare much better than an environment in which
such links are not present.
I the model the richest 2% of people hold about 20 times the average annual labor income
in assets or about 7 times the highest annual labor income. In the U.S. data, the richest 2% of
people held about 35 years of average labor income in assets in 1994.16
The transfer-wealth ratio is not de ned for this experiment, as bequests are collected by the
government and redistributed lump sum to all the population, independently of family links.
The overlapping-generations model with no intergenerational links also fails to recover the
age-asset pro le observed in the data. All households in the model economy ( gure 1) run down
their assets during retirement until they are left with zero wealth at the time we assume they die
for sure. This implies a much larger dissaving than we observe in the data ( gure 45), especially
for richer households. This also suggests that a substantial fraction of wealth that is accumulated
in this economy is linked to the uncertainty over the life span. In such an economy a market
for annuities would thrive and would reduce savings signi cantly; by purchasing annuities the
households would avoid leaving large accidental bequests when they die in their early retirement
years.
6.2 The Experiment with No Intergenerational Links and Unequal
Bequests to Heirs
In this experiment the accidental bequests left by the deceased are inherited by their own children,
rather than being redistributed by the government to all people alive. As we can see from table
6, there is little change in the distribution of wealth. The intuition is simple: some people
in the economy inherit some wealth, some other people do not, but nobody cares about leaving
bequests. Even high income households do not plan to share their fortune with their children but
do so only if they die early in their retirement years. As a consequence, the transfer-wealth ratio
is low, no persistence in wealth across families is generated and no large fortunes are accumulated
in this economy.
Figure 5 depicts the probability of the parent dying for each age of the o spring, conditional
on the parent dying before the o spring. In this model economy, the individuals do not die
before age 60; therefore the probability of the parent dying before the o spring reaches age
16 Sources:
Economic Report of the President [31] and Hurst, Luoh and Sta ord [20].
21
35 is set to zero. Compare the saving behavior of the average agent who expects to receive
some bequest, with his behavior in the hypothetical case in which he does not expect to receive a
bequest ( gure 6). The fact that parent's assets are not observable in uences the saving behavior
of people that have not yet inherited.17 Since in this economy children do not become orphans
before age 40, and everybody attributes positive probability to receiving some bequest until their
parent die, the comparison starts for agents of age 40. For them, we can compare the behavior
of people whose parent died, and did not leave them any asset (these children do not expect an
inheritance anymore), with the behavior of people that still expect to receive something. The
top line refers to the age-saving pro le for the average agent in the case in which he does not
expect to inherit. We see that the expectation of inheriting, conditional on all other variables
being the same, decreases the saving rate of the agent. However these saving rates converge
over time because the parent has no bequest motive and by age 90 runs down all of his assets
( gure 4). Therefore the average size of the expected bequest also goes to zero by that time.
Since the probability of dying is assumed to be independent of other individual characteristics,
the distribution of bequests is simply the distribution of assets among the population of the
parent's generation at age 65, rescaled because of population growth (each person has more
than one child). Figure 6 shows the strictly positive range of the bequest distribution that an
agent faces at 40 years of age should the parent die at that moment, conditional on the observed
productivity of his parent at age 40. I do not plot the probability of receiving a bequest of zero
because this would make the graph very diÆcult to read. This probability is about 27, 8 and
0% for a 40-year old whose parent had the lowest, middle or highest productivity level at age 40,
respectively. Since age 65 is the peak of wealth accumulation, the children of those who die at
age 65 (the 40 year olds) are the ones that receive the largest bequest. At this age, the average
bequests are, respectively, 2.9, 6.7, and 15.3 years of average labor income, After the parents
retire they run down all of their assets by age 90, so the expected bequest declines, and the
people whose parents live up to the nal age of the model economy do not receive any bequest.
6.3 The Experiment with only one Intergenerational Link: Bequest
Motive
The bequest motive leads to a large increase in the concentration of wealth in the economy and
explains the emergence of large estates that are accumulated by more than one generation of
savers and are transmitted because of altruism. The Gini coeÆcient increases from .64 to .71
and the fraction of the total wealth by the people in the upper tail of the distribution increases
signi cantly. The top 1% of the population holds 12% of total wealth, and the top 5% holds
34% of total wealth. The households at the top 2% quantile hold about 25 years of average
labor income18 in assets (compared with 20 years in the model with no intergenerational links),
or about 8 years of the highest income level in the model.
Figure 13 compares the distribution of wealth by age conditional on having or not having
17 All
people have a positive probability of getting a bequest, not only those whose parent will actually leave
some.
18 The average labor income for an agent in this economy is one over a ve years period, therefore it is .2 yearly.
22
received a bequest.19 The model predicts that the upper tail of the wealth distribution will be
mainly made of households that have already received a bequest.
The age-assets pro les for various quantiles of the wealth distribution are displayed in gure 10. From this gure we notice a substantial di erence in the main motives that lead the
household to save.
The median household mainly saves for retirement; the peak in its wealth holdings occurs at
age 65 and is about 6.5 times the average annual labor income in the population. If he reaches
the age of 85, the median agent consumes all of his assets (before dying at 90) and does not leave
any bequest. The median consumer mostly leaves unintended bequests. This becomes clearer
when comparing the age-assets pro les for the bottom 10, 30 and 50% with the model with no
bequest motive ( gure 1): the pro les are very close.20
Those who are wealthy, either as a consequence of large inheritances or of a successful working
life, plan on sharing their luck with their o spring. As we see from gure 10, at the top of the
wealth distribution a lot of the accumulation is done especially in order to leave bequests, and
large bequests are left even when the parents die in advanced age. Comparing these top quantiles
with the ones in the model with no altruistic links, we see how the introduction of a bequest
motive produces an age-wealth pro le more consistent with the U.S. data ( gure 45), especially
in the second part of the agent's lifetime. In this setup, the absence of an annuity market is
a less severe restriction on the behavior of the richest households as most of their savings are
primarily accumulated to be left to the next generation.
As in the previous simulation, gure 12 shows that the expectation of receiving a bequest
reduces the saving rate of the average agent. Here, however, this e ect does not decrease over
time. In fact the richest parents do not run down all of their assets by age 90 because of the
bequest motive and, over time, the reduction in the size of the expected bequest is balanced by
the increased probability of the parent dying.
In gure 15 we can see the strictly positive range of the bequest distribution for a 40 year old
person, conditional on his observed parent's productivity level, should his parent die during that
period. The probabilities of receiving zero bequests are respectively 27, 11 and 0%, for individuals
with parents of low, middle and high productivity level. The average bequests expected are
respectively 4.7, 7.6 and 15.5 years of average labor income. Even in presence of a bequest
motive, the parents run down their assets after retirement, so the expected bequest declines.
The fraction of people whose parent lives up to the nal age of the model economy and who do
not receive a positive bequest are 93, 87 and 53%, respectively. The average bequest that they
expect at that point in life is about 1.4, 1.9 and 5 years of average labor income respectively.
19 The
bequest a person has received may be zero if his parent dies with no assets.
model without bequest motive generates somewhat steeper pro les until retirement. This happens
because to match the capital/output ratio, the model with no intergenerational links requires a higher discount
factor ( = :97 compared with = :96). The steeper consumption pro le early in their lives counterbalances
the steep decumulation of assets later in life, when the increased probability of dying implies very high e ective
discount factors. The bequest motive keeps the e ective discount factor high in old age. As a consequence, we
can allow for a lower implying a atter consumption pro le at younger ages.
20 The
23
6.4 The Experiment with both Intergenerational Links
Here I explore how the results of the previous section change when parent and children are linked
not only by the bequest the parent intends to leave to his children, but also through transmission
of productivity.
The introduction of an additional link increases the Gini coeÆcient further to .73. This
happens because success in the workforce is now correlated across generations: more productive
parents accumulate larger estates and leave their bequests to their children who are in turn more
successful than average in the workforce.
The introduction of a link in the productivity (or \human capital") of di erent generations
within a family tends to have two opposing e ects on the accumulation of assets at the tails of
the wealth distribution. First, consider the individuals that are at the lowest levels of wealth.
These people tend to be the least productive in the workforce. In particular, the people who
are least productive at 20 (who tend to be less productive than the others also at later ages)
are more likely to have less productive and hence poorer parents. As a result, people with low
productivity in their young age will, on average, receive smaller bequests, which will contribute
to lowering their pro le of asset accumulation. On the other hand, the anticipation of smaller
bequests will lead the same people to save more to compensate for the smaller transfer wealth.
The two e ects will be exactly reversed at the upper end of the distribution of wealth.
The direct e ect on transfer wealth dominates in our results. At the upper tail, the top 1%
of the population hold 15% of total wealth, up from 12% in the previous example, and the top
5% hold 38% of total wealth, up from 34%. The agents at the top 2% of the population hold 29
times the average yearly labor income in wealth and 9 times the maximum level of labor income
in this model, compared respectively with 25 and 8 times in the model with bequests only.
In gure 21 we can see the strictly positive range of the bequest distribution for a 40 year old
person, conditional on his observed parent's productivity level, should his parent die during that
period. The probabilities of receiving zero bequests are respectively 29, 11 and 0%, for individuals
with parents of low, middle and high productivity level. The average bequests expected are
respectively 4.8, 8.1 and 16.2 years of average labor income. Comparing these numbers with
the ones for the experiment with bequest motive only, we can see that introducing productivity
inheritance has small e ects on the average expected bequests.
The introduction of the \human capital" link in the form I consider here leads overall to
modest changes in the results of the previous section. The reason for this result might stem from
the weakness of the link introduced. In the current model, children inherit from their parent only
their initial productivity level (they do so with probability smaller than one, according to the
Markov process Qyh ), and then productivity evolves independently and stochastically over time
for all agents. This implies that children of poorer households tend to enter in the labor force
at the lowest levels of the income process, but expect an improvement later in life and hence
tend not to save. To better evaluate the productivity link, I also run an experiment in which the
children's initial productivity level is their parent's one at 40 (inheritance with probability one).
As a result, the share of wealth held by the top 1% of the population increases by a couple of
percentage points, and the Gini coeÆcient for wealth increases to .75. Ideally, one would like the
agents to be born with di erent income processes, to recover the fact that more-educated people
24
Fixed
Parameter
1
n
g
a
r
p
Qy
Qyh
Value
*
*
.95{.97
1.5
-9.5
.8% yearly
25% of GDP
30%
6.86%
50% average income
+
+
Source(s)
Statistical Yearbook of Sweden [36]
same as U.S.
same as U.S.
same as U.S.
same as U.S.
OECD Economic Surveys, Sweden [30]
OECD Economic Surveys, Sweden [30]
OECD Economic Surveys, Sweden [30]
see text
OECD Economic Surveys, Sweden [30]
see text
Zimmerman [40]
b
exb
2
Value
15%
10 years average earnings
3.6
Chosen to Match
see text
see text
\altruistic feedback", see text
t
t
Calibrated
Parameter
Table 7: Parameters for the Swedish economy and their sources.
* refers to a vector
+ see description in the text
have a permanent advantage over less-educated ones, and that the age-eÆciency pro le tends
to be atter for less-educated households, leaving them again to save more in earlier periods.
Unfortunately, these considerations would require the introduction of a further state variable in
our problem, making computations even more involved.
7
Numerical Simulations for the Swedish Economy
As in the calibration of the U.S. economy, most of the parameters for the Swedish economy are
taken from other sources, and a few are chosen to match some aspects of the data. I summarize
the parameter choices in Table 7.
The Statistical Yearbook of Sweden [36] provides the mortality probabilities for people at
di erent ages in 1991-1995. In the calibration of the U.S. economy I use the mortality probabilities of people born in 1965 (which are for the most part projected, since these people are still
young). Since life expectancy is increasing, the Swedish data underestimate the life expectancy
at the various ages with respect to the one faced by people born in 1965 in Sweden. To correct
for this problem I use the U.S. data to compute the relative increase in life expectancy for the
relevant period. I then correct the Swedish data assuming that the increase in life expectancy is
25
the same in the two countries. However, as a check, I also us the U.S. conditional probabilities
in the simulation of the Swedish economy. It turns out that this has a negligible impact on the
results even if the life expectancy of Swedish people is about three years longer than that of U.S.
people.
I take the age-eÆciency pro le t and the preference parameters , and 1 to be the same
as those for the U.S.
The rate of population growth, n is set to equal the average population growth from 1950 to
1997, .8%
The interest rate on capital, net of depreciation and gross of taxes, r, is taken to be 6.86%
so that the interest rate net of taxes in the U.S. and Sweden coincide.
Pensions (p) are such that the social security replacement rate is 50% and the implied government transfers to GDP ratio in the model is consistent with the one (net of taxes) reported
in the OECD Economic Surveys, Sweden [30].
The persistence of the income and productivity inheritance processes are taken to be the
same in Sweden and the U.S. Bjorklund and Jantti [7] estimate the degree of intergenerational
income mobility in Sweden and do not reject the hypothesis that it is the same as in the U.S. I
take the ratio of the variances of the two processes to be the same one adopted for the calibration
of the U.S. economy and vary their levels in order to match the Gini coeÆcient for the income
process, which in Sweden is somewhat lower than in the U.S.
b is the tax rate on estates that exceed the exemption level exb . I take the e ective tax rate
to be higher than the one for the U.S., 15%, and the exemption level to be lower, 10 years of
average labor earnings. In Sweden taxes are paid on inheritances, rather than on estates, and
the revenue from inheritance and gift taxes is approximately .1% of GDP. The statutory tax rate
for children's inheritance is higher than in the U.S. (for the rst bracket it is about 50%) and
the exemption level is much lower (in the order of $5,000), but there are legal ways, for example
bequeathing an apartment or a large rm, of obtaining a much larger exemption level. It is
therefore more diÆcult than in the U.S. to de ne the statutory exemption level. The combined
choice of b and exb matches the revenues from bequests and gift taxes.
As in the calibration of the U.S. economy, I choose 2 to make the utility from leaving bequests
(b) roughly consistent with a \truly altruistic" model, in the sense that it is reasonably close to
the utility of the child receiving the bequest. The value functions of the Swedish model economy
turn out to be more concave than the ones in the U.S. one; therefore the value of 2 that I adopt
in this simulation is di erent. As a sensitivity check, I compare the results of the model with
both intergenerational linkages when adopting the same 2 used in the U.S. simulation, instead.
As for the level of altruism as measured by 1 , it turns out that over the relevant range for assets,
the levels of the U.S. and Swedish value function of the bequest receiver are very close. In this
sense the assumed intensity of the bequest motive is the same in both economies.
The results of the various experiments calibrated to the Swedish economy are summarized in
Table 8 and discussed below.
26
Capital Transfer
Percentage wealth in the top Percent with
output wealth Wealth
negative or
ratio
ratio
Gini 1% 5% 20% 40% 80% zero wealth
Swedish data
2.0
> :51
.73
17 37 75
99
100
30
No intergenerational links, equal bequests to all
2.1
N/A
.64
5 23 64
89
100
24
No intergenerational links, unequal bequests to children
1.9
.38
.67
6 25 67
91
100
26
One link: bequest motive
2.0
.76
.71
8 29 73
95
100
30
Both links: bequest motive and productivity inheritance
2.0
.77
.73
9 31 75
95
100
30
Table 8: Results for the Swedish calibration.
7.1 The Experiments with no Intergenerational Links
Compared to the U.S. calibration, people in the Swedish model economy face less earnings
uncertainty (to match a lower Gini coeÆcient for earnings, we need to reduce the variance of
the income process) and a higher social security replacement rate. The rst element tends to
reduce precautionary saving and the second one to reduce life-cycle saving. The model predicts
a lower wealth-to-GDP ratio than in the U.S. The ratio predicted by the model is very close to
the capital-output ratio in Swedish data.
In the rst experiment, the only intergenerational transfers stem from accidental bequests;
since there is life-span uncertainty and there are no annuity markets, people accumulate assets
to self-insure against the risk of living for a long time. When they die earlier, they leave their
assets behind. These bequests are distributed equally to all people in the economy.
Looking at the distribution of wealth, we can see that even in an economy in which there is
less wealth and earnings inequality and the government redistributes more, the basic version of
the model generates an upper tail of the wealth distribution which is too thin; the top 1% of
people hold only 5% of total wealth, compared with 14% in the data. Unlike the results for the
U.S. model economy, the model for Sweden does not generate too many people at zero wealth.
In fact in the data about 30% of the population is in this situation while the model generates
24%.
When I assume that involuntary bequests are left to the children of the deceased (third row
in table 8), the distribution of wealth is almost the same as in the case in which bequests are
evenly distributed to all people alive. This is analogous to what I found for the U.S.
27
7.2 The Experiment with only one Intergenerational Link: Bequest
motive
As in the U.S simulations, the introduction of the bequest motive helps in generating a more
skewed wealth distribution by increasing the share of total wealth held by the rich. The forces
discussed for the U.S. model economy that generate this result are also at work for the Swedish
model economy.
The share of wealth held by the top 1% of people rises from 5% to 8%, and the share held
by the top 5% increases from 22% to 29%. Moreover, we can see from gures 35 and 36 that the
wealth quantiles of the people who do receive a bequest are signi cantly higher than those who
do not get a positive transfer of wealth from their parent.
Figure 37 shows the strictly positive range of the bequest distribution for a 40 year old
person, should his parent die today, conditional on his parent's productivity at 40 years of age.
Conditional on the parent's productivity level from the lowest to the highest, the probability of
receiving a zero bequest is 29, 14 and 0% and the average bequests are 2.9, 4.4, and 8.5 years
of average labor earnings. Compared to the U.S. simulation, therefore, the number of people
expecting to receive no intergenerational transfer is slightly higher, and the average bequest size,
for all parent's levels of ability, is lower.
7.3 The Experiment with both Intergenerational Links
The introduction of the second linkage, the intergenerational transmission of productivity, helps
further in matching the top 20% of the wealth distribution: the share held by the top 1, 5, 10
and 20% increase to 9, 31, 51 and 75%. However, as discussed previously, this linkage is not very
strong and hence does not change the results dramatically.
Figure 43 shows the strictly positive range of the bequest distribution for 40 year old agents,
should their parent die this period.
For people whose parent was at the lowest productivity level at age 40, the average bequest
is 2.6 years of average labor earnings; for these agents the probability of receiving no intergenerational transfer is 34%. For the individuals whose parent at 40 was at the middle productivity
level, the average bequest is 4.4 and their probability of receiving zero bequests is 15%. For those
whose parent was at the highest productivity level at age 40, these numbers are respectively 8.7
and 0%.
As mentioned in the calibration, I use di erent values for 2 in the U.S. and Swedish model
economies. As a sensitivity check I report the results for the calibration of the Swedish economy
using the 2 adopted in the U.S. simulations, all other parameters staying the same. In this
case, the capital-output ratio is 1.65, the transfer-wealth ratio .46, and the fraction of people
at zero wealth 31%. The top 1, 5, 10, 20, 40, 80% hold respectively 8, 28, 46, 71, 94 and 99%.
However, with this parameterization, the warm-glow utility of leaving a bequest is much atter
than the value function of the children receiving it, even for the richest children. Moreover, the
age-saving pro les show fast decumulation after retirement for people at all wealth levels, which
is in contrast with the empirical evidence.
28
8
Discussion of the Assumptions
In order to make the model manageable and solvable I have made several simplifying assumptions.
In this section I discuss the assumptions and their likely qualitative implications on the wealth
distribution.
It is widely recognized (e.g., Becker and Tomes [4, 5]) that the time and resources that
parents devote to children's education are very important in understanding the distribution
of earnings and wealth. In this setup the simpli cation that children partially inherit their
parents' productivity is meant to recover the fact that education and human capital are closely
related to the family background of each person. However, when explicitly modeling human
capital investment, the return is commonly assumed to be decreasing, and up to a given level
of investment (which may depend on the child's abilities), greater than the rate of return on
physical capital. This implies that parents will begin to invest in their children's human capital
and then invest in physical capital when the return from human capital reduces to the return
on physical capital. In this setup, in the presence of borrowing constraints, the poorest families
only invest in their children's human capital, and they may not even be able to invest up to the
optimal amount. The richest families not only invest in their children's human capital but also
leave them physical capital. Poorer families will tend to have poorer children, thus generating
persistence in the lower end of the wealth distribution. At the upper tail of the distribution,
rich children might want to save less because they expect large bequests. However, they will also
be richer (because of their dominant income process and the bigger transfers they receive from
their parents), hence they might want to save more. Depending on which e ect dominates, this
will increase or decrease persistence at the upper end of the wealth distribution. Most likely,
if the altruism toward one's children is very strong for the richer people, the desire of leaving
large estates to children will o set the reduction in saving because of the large bequests received.
Which e ect dominates thus depends on how wealth a ects savings at high levels of wealth.
As discussed previously, I made restrictive assumptions on the information available to the
children on their parent's wealth and income. These assumptions are made for computational
reasons but are also likely to a ect the results. In particular, I expect the model in the current
version to display fatter tails at both ends of the wealth distribution, compared with a model
in which the parent's assets and income are observable by the child. With perfect observability
children of poor parents will save more, since they are aware that no bequest will be left to them.
On the other hand, children of richer parents will save less. If wealth of poor parents is easier
to measure than wealth of rich parents, only the lower tail of the distribution of wealth would
become thinner.
Another important assumption is that there are no inter-vivos transfers. In the data these
transfers often have a compensatory nature: parents tend to give when the children need money
the most. This may happen when they go to college, start a new job, get married, buy a house
or get a sequence of bad shocks. This assumption is probably most relevant when children are
20 to 35 years of age and are starting o their own life. Allowing for inter-vivos transfers would
help reduce the number of people at zero wealth, especially among the young.
I take fertility to be exogenous and independent of the agent's wealth. This is likely to
be a reasonable assumption for the U.S., but not a good one for other countries, especially
29
developing countries. If poorer families on average are more proli c than richer families then
the concentration of the wealth distribution will be increased: poor families have to divide their
scarce resources among more children who, in turn, will be most likely poor.
Labor earnings are also assumed to be exogenous and people can only invest in a riskless asset.
The U.S. data show that there is a noticeable correlation between high wealth and income from
running a business. Introducing entrepreneurial choice in the model, for example in the form of
investment in a risky asset in presence of minumum investment size and borrowing constraints,
would generate more heterogeneity in the people's income processes and more precautionary
savings.
9
Directions for Future Research
There is considerable debate about abolishing estate taxation. In the U.S., Sweden and many
other countries, estate and gift taxes produce little revenue (in the order of 0.1% of GDP) and
possibly distort the savings decision of the few people that do most of the capital accumulation
in the economy: the rich. I plan to study the e ects of abolishing the estate taxes in the context
of this model. I am interested in looking both at the macroeconomic consequences (e.g., what
would happen to total capital) as well as the distributional e ects (e.g., would wealth dispersion
increase and by how much? Would the poor people become even poorer in absolute terms, or
could they also bene t if total capital in the economy increased?).
In addition, many rich people are entrepreneurs, and entrepreneurial activity is likely to be
an important factor in understanding the distribution of wealth. I plan to study entrepreneurial
income and the degree of wealth inequality across di erent countries to better assess the importance of entrepreneurial income in evaluating the wealth distribution.
I also plan to use my model to study the demand for annuities. In fact, the market for
annuities is remarkably thin. The literature provides several explanations, such as moral hazard,
altruism and social security provision. In my setup, social security annuitizes all or most of
the savings of poorer people and the bequest motive provides a reason for richer people not to
annuitize their wealth completely. It will be interesting to introduce annuities in my setup to
study the quantitative importance of these two elements.
30
A
Calibration of the income processes
As discussed in the calibration section, I convert both the productivity and the productivity
inheritance processes to discrete Markov chains according to Tauchen and Hussey [38]. I use
three values for the income process. Since I want the possible realizations for the initial inherited
productivity level to be the same as the possible realizations for productivity during the lifetime,
I choose the quadrature points to be the same for the two Markov processes.
Tables 9 and 10 report data on earnings distributions, respectively for the U.S. and Sweden,
and compare them with the earnings distributions used in the model. The tables are computed
using data for households whose head is 25 to 60 years of age. In row 1 the de nition of gross
earnings includes wages, salaries and self-employment income. Row 2 adds social insurance
transfers to gross earnings. Row 3 refers to the earnings distribution used in the simulations.
Since the setup does not explicitly model other social insurance transfers other than social
security, we should compare earnings used in the simulations to data that include social insurance
transfers. From this comparison we can see that the model understates both the earnings of the
upper tail, and the fraction of people at zero earnings.
Percentage earnings in the top
Percent with
Gini
negative or
coe . 5% 10% 20% 40% 60% 80% zero income
U.S. earnings data
.46 19 30
48
72
89
98
7.7
U.S. earnings + social insurance transfers
.44 19 30
47
71
88
97
5.3
U.S. simulated earnings
.43 13 25
48
72
87
96
0.0
Table 9: U.S. earnings.
Percentage earnings in the top
Percent with
Gini
negative or
coe . 5% 10% 20% 40% 60% 80% zero income
Swedish earnings
.40 15 25
42
68
86
98
7.6
Swedish earnings + social insurance transfers
.32 13 22
37
62
80
94
1.0
Swedish simulated earnings
.32 10 21
40
63
81
92
0.0
Table 10: Swedish earnings.
31
Figures
U.S. calibration. Experiment with no links and equal bequests to all
6
5
Wealth
4
3
2
1
0
20
30
40
50
60
70
80
90
Age
Figure 1: Wealth .1 .3 .5 .7 .9 quantiles, by
age
1
0.9
0.8
0.7
Wealth Gini
B
0.6
0.5
0.4
0.3
0.2
0.1
0
30
40
50
60
70
80
Age
Figure 2: Wealth Gini for di erent cohorts
32
U.S. calibration. Experiment with no links and unequal bequests to heirs
1
6
0.9
0.8
5
0.7
Wealth Gini
Wealth
4
3
0.6
0.5
0.4
2
0.3
0.2
1
0.1
0
20
30
40
50
60
70
80
0
90
Age
30
40
50
60
70
80
Age
Figure 3: Wealth .1 .3 .5 .7 .9 quantiles, by
age
Figure 4: Wealth Gini for di erent cohorts
0.22
0.4
0.2
0.35
0.18
0.3
0.16
0.25
0.14
0.2
0.15
0.12
0.1
0.1
0.05
0.08
0
0
10
20
30
40
Age of the child
50
0.06
20
60
Figure 5: Probability of the parent dying at
each age of the child
25
30
35
40
Age
45
50
55
60
Figure 6: Saving for people who expect or
not to inherit
33
6
6
5
5
4
4
Wealth
Wealth
U.S. calibration. Experiment with no links and unequal bequests to heirs
3
3
2
2
1
1
0
25
30
35
40
45
50
55
0
25
60
30
35
40
Age
45
50
55
60
Age
Figure 7: Wealth quantiles conditional on
not having inherited
Figure 8: Wealth quantiles conditional on
having inherited
0.06
0.05
Probability
0.04
0.03
0.02
0.01
0
0
1
2
3
4
Bequest Size
5
6
7
Figure 9: Strictly positive range of the expected bequest distribution at age 40, conditional on the productivity of the parent
34
U.S. calibration. Experiment with bequest motive
1
6
0.9
0.8
5
0.7
Wealth Gini
Wealth
4
3
2
0.6
0.5
0.4
0.3
0.2
1
0.1
0
20
30
40
50
60
70
80
0
90
30
40
50
Age
60
70
80
Age
Figure 10: Wealth .1 .3 .5 .7 .9 quantiles,
by age
Figure 11: Wealth Gini for di erent cohorts
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
20
25
30
35
40
Age
45
50
55
60
Figure 12: Saving for people who expect or
not to inherit
35
6
6
5
5
4
4
Wealth
Wealth
U.S. calibration. Experiment with bequest motive
3
3
2
2
1
1
0
25
30
35
40
45
50
55
0
25
60
30
35
40
Age
45
50
55
60
Age
Figure 13: Wealth quantiles conditional on
not having inherited
Figure 14: Wealth quantiles conditional on
having inherited
0.06
0.05
0.04
0.03
0.02
0.01
0
0
1
2
3
4
5
Figure 15: Strictly positive range of the expected bequest distribution at age 40, conditional on the productivity of the parent
36
U.S. calibration. Experiment with bequest motive and productivity inheritance
1
6
0.9
0.8
5
0.7
Wealth Gini
Wealth
4
3
2
0.6
0.5
0.4
0.3
0.2
1
0.1
0
20
30
40
50
60
70
80
0
90
30
40
50
Age
60
70
80
Age
Figure 16: Wealth .1 .3 .5 .7 .9 quantiles,
by age
Figure 17: Wealth Gini for di erent cohorts
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
20
25
30
35
40
Age
45
50
55
60
Figure 18: Saving for people who expect or
not to inherit
37
6
6
5
5
4
4
Wealth
Wealth
U.S. calibration. Experiment with bequest motive and productivity inheritance
3
3
2
2
1
1
0
25
30
35
40
45
50
55
0
25
60
30
35
40
Age
45
50
55
60
Age
Figure 19: Wealth quantiles conditional on
not having inherited
Figure 20: Wealth quantiles conditional on
having inherited
0.06
4
0.05
3
2
0.04
1
0.03
0
0.02
−1
−2
0.01
−3
0
0
1
2
3
4
0
5
Figure 21: Strictly positive range of the expected bequest distribution at age 40, conditional on the productivity of the parent
5
10
15
20
25
Bequest
Figure 22: Warm glow and true altruism compared: warm glow (solid), poor
(dashed), median (dash-dot), rich (dots)
38
Swedish calibration. Experiment with no links and equal bequests to all
6
5
Wealth
4
3
2
1
0
20
30
40
50
60
70
80
90
Age
Figure 23: Wealth .1 .3 .5 .7 .9 quantiles,
by age
1
0.9
0.8
Wealth Gini
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
30
40
50
60
70
80
Age
Figure 24: Wealth Gini for di erent cohorts
39
Swedish calibration. Experiment with no links and unequal bequests to heirs
1
6
0.9
0.8
5
0.7
Wealth Gini
Wealth
4
3
0.6
0.5
0.4
2
0.3
0.2
1
0.1
0
20
30
40
50
60
70
80
0
90
Age
30
40
50
60
70
80
Age
Figure 25: Wealth .1 .3 .5 .7 .9 quantiles,
by age
Figure 26: Wealth Gini for di erent cohorts
0.14
0.4
0.35
0.12
0.3
0.1
0.25
0.08
0.2
0.06
0.15
0.04
0.1
0.02
0.05
0
0
10
20
30
Age of the son
40
50
0
20
60
Figure 27: Probability of the parent dying
at each age of the child
25
30
35
40
Age
45
50
55
60
Figure 28: Saving for people who expect or
not to inherit
40
6
6
5
5
4
4
Wealth
Wealth
Swedish calibration. Experiment with no links and unequal bequests to heirs
3
3
2
2
1
1
0
25
30
35
40
45
50
55
0
25
60
30
35
40
Age
45
50
55
60
Age
Figure 29: Wealth quantiles conditional on
not having inherited
Figure 30: Wealth quantiles conditional on
having inherited
0.06
0.05
Probability
0.04
0.03
0.02
0.01
0
0
1
2
3
4
Bequest Size
5
6
7
Figure 31: Strictly positive range of the expected bequest distribution at age 40, conditional on the productivity of the parent
41
Swedish calibration. Experiment with bequest motive
1
6
0.9
0.8
5
0.7
Wealth Gini
Wealth
4
3
2
0.6
0.5
0.4
0.3
0.2
1
0.1
0
20
30
40
50
60
70
80
0
90
30
40
50
Age
60
70
80
Age
Figure 32: Wealth .1 .3 .5 .7 .9 quantiles,
by age
Figure 33: Wealth Gini for di erent cohorts
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
20
25
30
35
40
Age
45
50
55
60
Figure 34: Saving for people who expect or
not to inherit
42
Swedish calibration. Experiment with bequest motive
3
3
2.5
2.5
Wealth
Wealth
2
1.5
2
1.5
1
1
0.5
0.5
0
25
30
35
40
45
50
55
0
25
60
30
35
40
Age
Figure 35: Wealth quantiles conditional on
not having inherited
50
55
60
Figure 36: Wealth quantiles conditional on
having inherited
0.06
0.05
0.04
Probability
45
Age
0.03
0.02
0.01
0
0
1
2
3
Bequest Size
4
5
6
Figure 37: Strictly positive range of the expected bequest distribution at age 40, conditional on the productivity of the parent
43
Swedish calibration. Experiment with bequest motive and productivity inheritance
1
6
0.9
0.8
5
0.7
Wealth Gini
Wealth
4
3
2
0.6
0.5
0.4
0.3
0.2
1
0.1
0
20
30
40
50
60
70
80
0
90
30
40
50
Age
60
70
80
Age
Figure 38: Wealth .1 .3 .5 .7 .9 quantiles,
by age
Figure 39: Wealth Gini for di erent cohorts
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
20
25
30
35
40
Age
45
50
55
60
Figure 40: Saving for people who expect or
not to inherit
44
Swedish calibration. Experiment with bequest motive and productivity inheritance
3
3
2.5
2.5
Wealth
Wealth
2
1.5
2
1.5
1
1
0.5
0.5
0
25
30
35
40
45
50
55
0
25
60
30
35
Age
40
45
50
55
60
Age
Figure 41: Wealth quantiles conditional on
not having inherited
Figure 42: Wealth quantiles conditional on
having inherited
4
0.06
2
0.05
0
Probability
0.04
−2
0.03
0.02
−4
0.01
−6
0
0
1
2
3
Bequest Size
4
5
6
Figure 43: Strictly positive range of the expected bequest distribution at age 40, conditional on the productivity of the parent
−8
0
1
2
3
4
Bequest
5
6
7
Figure 44: Warm glow and true altruism compared: warm glow (solid), poor
(dashed), median (dash-dot), rich (dots)
45
200
Wealth
150
100
50
0
30
40
50
60
Age
70
80
90
Figure 45: U.S. data, wealth quantiles: .1, .3, .5, .7, .9
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@FedFRASER