source: https://doi.org/10.7892/boris.115453 | downloaded: 9.8.2022
Macroeconomic Dynamics, 16, 2012, 230–239. Printed in the United States of America.
doi:10.1017/S1365100510000064
A CLOSED-FORM SOLUTION
TO A MODEL OF TWO-SIDED,
PARTIAL ALTRUISM
ANA FERNANDES
University of Bern
This paper presents a closed-form characterization of the allocation of resources in an
overlapping generations model of two-sided, partial altruism. Three assumptions are
made: (i) parents and children play Markov strategies, (ii) utility takes the CRRA form,
and (iii) the income of children is stochastic but proportional to the saving of parents. In
families where children are rich relative to their parents, saving rates—measured as a
function of the family’s total resources—are higher than when children are poor relative to
their parents. Income redistribution from the old to the young, therefore, leads to an
increase in aggregate saving.
Keywords: Closed Form, Value Function, Partial Altruism, Markov Strategies
1. INTRODUCTION
The ubiquitous presence of the family in many aspects of decision-making—from
human capital investments by the young [Becker (1993)] to the role of financial
transfers among kin in the distribution of wealth and persistence of inequality
[Becker and Tomes (1979), Loury (1981)]—has long been recognized in the
literature. Models of the family are especially important in analyzing topics that
involve resource redistribution across generations, such as the implementation of
social security programs; altruism has been a recurrent paradigm in this context
[see, e.g., Barro (1974), Becker (1974), Laitner (1979, 1988)].
Dynamic models of the altruistic family typically involve generalizations of the
overlapping-generations framework.
1
The utility of old agents at a given moment
in time no longer depends solely on their own consumption path; it is affected by
the well-being of their descendants. The models used range from assuming very
strong family ties—people value the utility of their descendants as much as their
own—to considering less extreme views of affection across kinship, whereby
the utility of a family member is somewhat discounted relative to one’s own.
2
In this literature, altruism is generally considered to be one-sided, with parents
This paper draws on chapter 7 of my thesis, Fernandes (1999). I thank Robert E. Lucas, Jr., Fernando Alvarez,
Gary Becker and Sherwin Rosen for guidance. Financial support from Banco de Portugal, Fundac¸
˜
ao Para a
Ci
ˆ
encia e a Tecnologia and Fundaci
´
on Ram
´
on Areces is gratefully acknowledged. Address correspondence to: Ana
Fernandes, Department of Economics, University of Bern, Schanzeneckstrasse 1, 3001 Bern, Switzerland; e-mail:
ana.fernandes@vwi.unibe.ch.
c
2012 Cambridge University Press 1365-1005/12
230
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CLOSED-FORM SOLUTION 231
caring about children and children caring about their children—but not about
their parents. One important reason for this one-sidedness has been the analytical
difficulty of obtaining tractable characterizations of the family’s choices otherwise.
The roots of the problem are sketched below.
3
Altruistic family members will donate income to less wealthy relatives to equal-
ize the marginal utility from consumption. This equalization will not be full in the
case of partial altruism, as the utility of relatives is less valued than one’s own.
In a static environment of two-sided altruism, this implies that transfers will flow
from parents to children when parents are wealthy relative to their children, from
children to parents otherwise, and that there will be an intermediate range where
no transfers take place, where parent and child have similar incomes and where
each would like to receive transfers from the other family member. Consider now
the utility the parent derives from consumption. Let us fix the child’s income and
examine what happens as the parent’s income increases. For low income values,
the parent receives a transfer from the child. An additional unit of income is
“taxed” by the child in the form of a transfer reduction. As parental income rises,
transfers decrease until they become exactly zero. At this point, additional parental
income reverts fully to parental consumption, as the child cannot impose a negative
transfer on the parent. The marginal utility from consumption therefore increases
locally in a discrete fashion (seen from the point of view of the parental utility from
consumption, the utility function has an upward kink at the point where transfers
from the child are exactly zero).
4
This causes concavity of the value function to
break down and prevents standard theorems—to demonstrate existence of a value
function summarizing parental future utility—from being used. Instead, people
have resorted either to full altruism (no kink exists, because transfers are always
positive and fully equalize consumption across family members) or to one-sided
altruism (there is a kink at the point where the parent’s income is high enough to
trigger transfers to a poorer child but the value function remains concave).
5
This paper presents a closed-form characterization of the allocation of resources
(consumption and saving) in an overlapping generations (OLG) model of two-
sided, partial altruism. Under the assumptions that (i) parents and children play
Markov strategies, (ii) utility takes the CRRA form, and (iii) the income of children
is stochastic but proportional to the saving of parents, it is possible to solve
explicitly for optimal consumption, transfers (from parents to children or from
children to parents), and saving, as well as for the value function summarizing the
utility parents derive from the future. As in the static setup, partial altruism causes
transfers to flow from wealthy relatives to relatively poorer family members, but
not to take place at all whenever incomes are sufficiently close. It is shown that
the saving decisions of young dynasty members are a function of the total income
of the family. In fact, when interfamily transfers flow from the old to the young
(bequests), the family saves a smaller share of its total resources compared to
the case where the young provide the older cohort with transfers (gifts). The
consumption of the old cohort is also a higher fraction of the family’s total income
when bequests are positive compared to the old’s consumption share when the
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232 ANA FERNANDES
young provide positive gifts. When intergenerational transfers are zero, family
members act independently and kinship is an irrelevant consideration for economic
decisions such as saving or consumption.
The paper is organized as follows. Section 2 describes the environment. Section
3 provides a parametric example. Section 4 concludes.
2. AN OLG MODEL WITH TWO-SIDED, PARTIAL ALTRUISM
2.1. Preferences
The environment is that of a standard OLG model with partial altruism [see
Laitner (1988)]. Each family member lives for two periods. When young, the
child coexists with the parent. In the second period, she raises her own child. In
the following period, that child will raise her grandchild. The extended family is
therefore infinitely lived.
Parent and child value each other’s utility but by less than they value their
own. Consider a parent–child pair. Let V
p
denote the parent’s forward-looking
utility associated with decisions concerning the present date and the entire future.
Similarly, let V
c
denote the child’s lifetime utility, and c
p
and c
c
stand for parent’s
and child’s consumption, respectively. For numbers β, λ
H
,λ
L
∈ (0, 1) and λ
H
>
λ
L
, consider the following system:
V
p
= λ
H
u(c
p
) + λ
L
V
c
, (1)
V
c
= λ
H
[u
(
c
c
)
+ βV
p
] + λ
L
V
p
. (2)
The function V
p
represents the utility accruing to the young agents in the second
period of their lives, because they will be old then and therefore assume the
role of parents. The assumption λ
H
>λ
L
reflects the partial nature of altruism:
individuals value their direct utility more than they value that of their relatives.
Solving the system (1) and (2), we get
V
p
=
λ
H
1 − λ
2
L
u(c
p
) +
λ
H
λ
L
1 − λ
2
L
[u
(
c
c
)
+ βV
p
], (3)
V
c
=
λ
H
λ
L
1 − λ
2
L
u(c
p
) +
λ
H
1 − λ
2
L
[u
(
c
c
)
+ βV
p
]. (4)
We may define
˜
λ
H
≡ λ
H
/(1 − λ
2
L
) and
˜
λ
L
≡ λ
H
λ
L
/(1 − λ
2
L
). There is no
guarantee that
˜
λ
H
and
˜
λ
L
are in the interval (0, 1) and, in particular, it is also
possible that the new discount factor in V
c
, βλ
H
/(1 − λ
2
L
), is not in this interval.
It may be convenient, therefore, to impose restrictions on λ
H
and λ
L
so that the
discount factor lies strictly in the unit interval. But the general idea that each agent
weighs more the utility from own consumption than the utility of other family
members goes through, in (3) and (4).
Also, parents discount the future more than their children, as the term multi-
plying β in V
p
, λ
H
λ
L
/(1 − λ
2
L
), is strictly smaller than the corresponding term in
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CLOSED-FORM SOLUTION 233
V
c
, λ
H
/(1 − λ
2
L
). The reason for this has to do with the fact that they value the
future indirectly, through their child’s future consumption, which they value less
than their own consumption.
In what follows, I will use the preference format of (3) and (4) to motivate the
functional forms
V
p
= λu(c
p
) +
(
1 − λ
)
[u
(
c
c
)
+ βV
p
], (5)
V
c
=
(
1 − λ
)
u(c
p
) + λ[u
(
c
c
)
+ βV
p
], (6)
where λ ∈ (0.5, 1).
2.2. A Dynasty
Equations (5) and (6) characterize the preferences of old and young agents, re-
spectively. Let us now turn to other aspects of the extended family. Nature acts at
the beginning of every period of this infinite-horizon game by drawing the income
of young agents, z
t
. Parental income is assumed nonstochastic, for simplicity. Old
agents in period t have income a
t
R, where a
t
represents assets accumulated from
period t − 1, which yields the exogenous return R.
6
Each agent moves in two adjacent periods. When he is born, he observes his
income and his parent’s resources and decides on a nonnegative transfer to the
parent, g
t
, and on the amount of savings to accumulate for the following period,
a
t+1
.Attimet, the parent (who was born in period t − 1) observes z
t
and
decides how to split the resources a
t
R into own consumption and a nonnegative
transfer to the child, b
t
. Both parent and child know the entire past history of
endowment realizations, transfers, and asset accumulation decisions made by
other dynasty members. Their choices of gifts, savings, and bequest—the agents’
strategies—map the entire set of past decisions and income realizations, as well as
the income realizations of the period in which they move, into the nonnegative real
numbers. In addition, strategies are additionally constrained to verify b
t
,g
t
≥ 0,
g
t
+ a
t+1
≤ z
t
+ b
t
, and b
t
≤ a
t
R + g
t
.
7
At time t, for fixed interest rate R, given resources a
t
,z
t
and decisions
b
t
,g
t
,a
t+1
, consumption is given by
c
pt
= a
t
R − b
t
+ g
t
, (7)
c
ct
= z
t
+ b
t
− g
t
− a
t+1.
(8)
2.3. Markov Perfect Equilibria
To characterize subgame-perfect equilibria, one could use the ideas of self-
generation and factorization, from Abreu et al. (1986, 1990), and determine the
set of lifetime discounted utilities associated with subgame-perfect strategies of
this infinite game. A more restrictive procedure, which also confines attention to
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234 ANA FERNANDES
Markov perfect strategies, would be to find a solution to a functional equation, as
follows.
Consider a space of functions with domain in R
2
, F(R
2
). Suppose that V ∈
F(R
2
) satisfies the functional equation
V(a,z)= max
b
λu
(
aR − b + g
)
+
(
1 − λ
)
u(z + b − g − a
) + β
V(a
,z
)µ(dz
)
(9)
subject to
g, a
∈ arg max
(
1 − λ
)
u
(
aR − b + g
)
+ λ
u(z + b − g − a
) + β
V(a
,z
)µ(dz
)
. (10)
Equations (9) and (10) characterize one Markov perfect equilibrium. The gain
from using this very restrictive approach will be shown in the next section: the para-
metric example solved below enables a complete characterization of the function
V(·) as well as of the optimal choices (b,g,a
) as functions of the state (a, z).
3. A PARAMETRIC EXAMPLE
In this section, I assume that the income of the child is stochastic but proportional
to her parent’s saving. Say the parent saves a, which earns the deterministic return
R. The child’s resources are the product of the stochastic shock z and the parent’s
saving a. Therefore, changes in a affect the income of both parent and child in
the same direction, something that would tend to reduce potential disincentives to
save; these could arise from the fact that, from altruism, children know they will
not appropriate the totality of their saved income if their own children turn out to
be poor, and, by saving more, the current generation also reduces the likelihood
that their children will transfers resources to them in the future. Together with
CRRA utility, the fact that changes in a do not affect the ratio of parent’s to
child’s resources suggests that decision rules will be proportional to a, therefore
eliminating the potential lack of global concavity of V(·) with respect to a.This
will indeed be the case, as shown below.
Is the assumption of stochastic proportionality of incomes plausible? Although
its analytical convenience is undeniable, it is not as restrictive as it may at first
appear. In fact, the overlapping-generations model as it now stands could be
calibrated to reproduce a certain degree of mean reversion around a by allowing the
distribution of the next period’s shocks, z
, to depend on today’s shock realization,
z, as in a Markov process.
8
As long as the distribution of future shocks does not
depend on current or past values of a, the analysis goes through.
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