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

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.