GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 324

Thankworthy Parental Altruismand Children’s Habit-Formation

Lin Zhang and Shinsuke Ikeda

November 2013

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanThankworthy Parental Altruism and Children’s Habit-Formation*
Lin Zhang† and Shinsuke Ikeda‡

Abstract
We propose a model of parental altruism in relation with child habit formation, where children
are unaware of their developing habits while young, and become cognizant of them only on
growing up. We show that an altruistic mother (i) maintains the amount of income transferred toher child lower than the child would desire and (ii) reduces further income transfer upon an
exogenous increase in the child’s performance of a particular habit. The child, when grown, may
end up grateful for the small income transfer if the mother is sufficiently rich and altruistic:
when evaluated by the realized habitual preferences, a small income transfer leads to greater
child welfare than the greater income transfer preferred by the child would have generated. Thisimplies that parents from richer families, ceteris paribus, tend to guard against their children’s
profligacy.

JEL Classification: D1; D9.
Keywords: Habit Formation, parental altruism, thankworthy, time preference.

* We thank Masao Ogaki and Yoshiyasu Ono for their comments. This work was supported from the

Ministry of Education, Culture, Sports, Science and Technology, Japan by Grants for Excellent Graduate
Schools; a Grants-in-Aid for Scientific Research (B No. 21330046) from the Japan Society for the
Promotion of Science; and the Joint Usage/ Research Center Project of ISER.
† Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,

Japan. Tel.: 81-6-6850-6111; Fax: 81-6-6850-5205.
Email: <kge017zl@mail2.econ.osaka-u.ac.jp>.
‡ The Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka

567-0047, Japan. Tel.: 81-6-6879-8568, Fax: 81-6-6879-8583.
E-mail: <ikeda@iser.osaka-u.ac.jp>.

1.

Introduction
Parents discipline their young children by keeping income transfer (e.g., allowances) to thechildren lower than what the children themselves desire. This causes conflict between theparents, who do so from an altruistic motive, and the children, who believe that they will bebetter off with more money. When the children grow up, some come to understand that theirparents had altruistic intentions and feel grateful; others remain resentful of their parent’s
thriftiness. Why do altruistic parents not transfer as much income to their young children as thelatter would like? Conversely, why do grown children not appreciate their parents' altruism?
Although it is important to understand how parent-child interactions affect theconsumption/saving behavior and related preference formation of the family members,
economics has not yet examined these issues.
To fill this knowledge gap, the present paper aims to address two questions:(i) Why do altruistic parents keep income transfers to young children lower than what thechildren desire?
(ii) Whether, and under which conditions does the parents' behavior of restricting income
transfers enhance the children's welfare and is, therefore, thankworthy?
To do so, we consider an altruistic parent's behavior toward her habit-forming child (Becker,
1974; Barro, 1974): the parent obtains utility from his or her own consumption, as well as thechild's utility. In line with Weinberg (2001), and Bhatt and Ogaki (2012), the altruistic parent—amother, for example—can determine the consumption level of her young child—a son, in thiscase—by controlling the amount of income transferred to him. A unique feature of our model is1that the child is habit-forming. By controlling income transfer to her young son, the mother inour model can influence his consumption habits, and hence, his future consumption behavior.
We posit two key assumptions. First, as is the case of actual children, the young child isassumed to be unaware that consumption is habit-forming, whereas his mother is assumed toknow this. The son thus tends to consume excessively. His altruistic mother has an incentive tokeep her income transfer low, so as to restrict the son’s excess consumption. The model ofmisperceived habit formation is reasonable because the psychological literature provides
empirical support for the importance of parental intervention in children's habits, in contexts asdiverse as food consumption (e.g., Baumrind, 1991), passive leisure such as TV viewing (Walshet al., 1998), and saving (Webley and Nyhus, 2006).1 Moreover, studies on the Easterlin
paradox have provided empirical evidence of people experiencing welfare loss because theyfailed to incorporate the effects of habit formation (e.g., Easterlin, 1974).
We also assume that, after reaching adulthood, the child becomes aware of the truemechanisms of habit formation. We are interested in how the grown-up child re-evaluates his
mother's strict upbringing retrospectively, by using his realized habitual preferences. In
particular, we regard the mother's income transfer to her young son as thankworthy from the(grown-up) son’s perspective if—when evaluated according to his true preferences—greater
welfare was attained under the transfer than would have been attained under the greater income
transfer that he sought in his youth.
For problem (i), we show analytically that an altruistic parent keeps income transfers to his or

1

However, there has been some controversy regarding the empirical validity of habit
formation models (see, e.g., Dynan, 2000).

2

her child lower than he wants to give. Thus, our model describes a conflict between altruistic
parents who raise their children strictly, and the children who complain about it. It is crucial toremember that children's habitual preference formation is known to their parents, but not to thechildren themselves.
Our solution to problem (ii) may be more important, and will reveal whether the parent's
income transfer to a child will induce gratitude in the child, to him as his parent is sufficiently
rich and altruistic or not. This is because reducing a marginal unit of income transfer to a childhas two countervailing effects. First, it directly reduces the child's felicity throughout the period.
Second, it weakens the child's consumption habit in the following period, and mitigates excess
consumption that comes from the child being unaware of his or her habit formation. The
parental behavior of keeping income transfer low enhances the child's welfare only when thesecond positive effect dominates the first negative effect. Note, however, that unless parental
income, as well as the degree of parental altruism are large enough—and hence, unless theincome transfer level is large enough—the marginal utility of income transfer is necessarily sohigh that the first negative effect dominates the second positive effect.
As a casual observation, some rich parents indulge their children by giving them too much,
while

some

poor

parents

strictly

discipline

theirchildrenbycontrolling

theirconsumption/saving behavior. Depending on our results regarding the thankworthiness ofparents’ altruistic interventions, the difference could be attributable to whether the parents
themselves experienced thankworthy transfers or not. Even when a mother is, at present, poor,
she could be strict about her children's profligacy if she comes from a rich family andappreciates her own parents’ strict rearing. In contrast, a rich mother may spoil her son if she is3

from a poor family, and does not appreciate her own strict childhood.
We also show that when the child's habit parameter exogenously shifts upward, the altruistic
parent necessarily reduces the unexpectedly low income transfer further, so as to mitigate thechild’s over-consumption. The result is comparable with parents' "tough love" behavior that
Ogaki and Bhatt (2012) derive by specifying an endogenous time-discounting function. Theyshow that an altruistic parent reduces income transfer in response to an exogenous rise in thechild's time preference. Since an increase in the habit parameter is shown to raise the child'stime preference induced by the consumption habit, our comparative static result may beconsidered a version of the tough love result. However, there are two key differences. First, inthe present model, time preference is endogenously generated by habit formation, rather than by
an exogenously given time preference formation. Second and more importantly, our tough loveis thankworthy to the child if his parent is sufficiently altruistic and rich. This contrasts with the
Ogaki and Bhatt model, wherein a mother's tough love necessarily reduces her son's welfare,because the mother evaluates her son's future utility by using a different time preference fromthat of the child.2
Our study is also related to that of Weinberg (2001). By using a static principal-agency modelof the child-parent interaction, he explains the positive relationship between parental income
and frequency of using pecuniary punishment, rather than corporal punishment. Unlike his

2Ogaki and Bhatt (2012) also examine an "endogenous altruism model," in which the parent evaluates
the child’s future utility by using the same time discounting function that the child uses. In this case, theincome transfer that the altruistic parent chooses necessarily equals what the child expects to receive.
Therefore, even when the parent reduces income transfer due to a shock, it is not “tough” but always
reasonable, even to the young child. This contrasts with our case, in which a reduction in the parent's
income transfer after a shock always seems harmful, and hence tough, to the child at that time, because heis then unaware of the habit effect. It is, therefore, only reasonable and thankworthy to the child after
reaching adulthood.

4

model with parent-child information asymmetry, however, our dynamic model supposes
cross-generation asymmetry in cognitive power to identify future habit effects. We emphasize
parental cognitive power that corrects children's habitual over-consumption.3
The results of this model jointly imply that it is important for the parents to control children’s
impulses and habitual consumption, because it is beneficial to children’s academic, economic,
and health outcomes in the long run. A series of empirical studies support this conclusion (see,for example, Shoda, Mischel and Peake, 1990; Heckman, 2006; Chabris et al., 2008; andMoffitt et al., 2011).
The remainder of this paper is organized as follows. Section 2 introduces the model. Section
3 describes the expected transfer by the child. Section 4 compares the parental income transfers
and the child’s welfare. Section 5 discusses the endogenous time preference and tough love.
Section 6 concludes the paper.

2.

The Model
Consider a family consisting of a parent and a child, both of whom live for three periods. The

parent (a mother) gives birth to a child (a son) in her second period, which therefore overlaps
with the child’s first period.

The Child’s Problem
In the child’s first period, he receives transfer 𝑇 from the parent. In the second period, the3

Despite the differences, our result—suggesting that income transfers of richer parents are likely to be
thankworthy to children—seems consistent with the positive relationship that Weinberg (2001) reports
between parental income and the frequency of using pecuniary punishment.

5child receives an exogenous income 𝑦2 . 4 Variables 𝑐𝑖 (𝑖 = 1,2,3) denote the child’s
consumption in period 𝑖. The child is assumed to face borrowing constraints in period 1:
𝑐1 ≤ 𝑇. We assume that his disposable income 𝑇 is small enough that the borrowing constraint
is binding:𝑐1 = 𝑇,(1)

His intertemporal budget in periods 2 and 3 is given by
𝑐3 = 𝑅(𝑦2 − 𝑐2 ).

(2)where 𝑅 is the gross interest rate.
The child forms consumption habits. His preferences are given by
𝑈 = 𝑢(𝑐1 ) + 𝛽𝑢(𝑐2 − 𝜃1 ℎ2 ) + 𝛽 2 𝑢(𝑐3 − 𝜃2 ℎ3 ),(3)where ℎ2 = 𝑐1 and ℎ3 = 𝑐2 ,which captures the habit effects of consumption in the previous period. Parameters 𝜃1 , 𝜃2 > 0
represent the strength of habit formation, and 𝛽 > 0 denotes the discount factor.
Equation (3) implies that while the existence of habit formation raises the marginal utility of
consumption, it also reduces the levels of period utilities. However, the child does not realize
until his third period that his consumption in his first and second periods was habit-forming:

Assumption 1. The child is unaware of his habit in his first and second periods. In his third
period, he becomes aware of the habit.

4

We assume away the first-period income 𝑦1 to the child. This, however, does not affect our main
results below.

6Under Assumption 1, the child's consumption behavior is assumed to proceed as follows. In
the first period, the child naively maximizes his lifetime utility𝑢(𝑐1 ) + 𝛽𝑢(𝑐2 ) + 𝛽 2 𝑢(𝑐3 ).

(4)

In the second period, the child notices that the utility is actually different from what was
expected in the previous period. However, instead of realizing the true mechanism of habit
formation, he incorrectly understands it to be caused by a permanent preference shock5 andmaximizes the utility𝑢(𝑐2 − 𝜃1 𝑐1∗ ) + 𝛽𝑢(𝑐3 − 𝜃1 𝑐1∗ ),(5)where 𝑐1∗ denotes the child’s actual consumption level in the first period. In his third (the last)period, the child becomes aware of his true preferences with habit formation as in (3).

The Parent’s Problem
Since the parent can affect the child’s consumption—and, thereby, welfare—though income
transfer 𝑇, we focus on her decision in her second period. In the second period, she receives
endowment income 𝑦𝑝 , and maximizes the sum of the second and third period utilities bychoosing consumption basket (𝑐2,𝑝 , 𝑐3,𝑝 ) and transfer 𝑇 to the child. Her budget constraint isgiven by𝑐3,𝑝 = 𝑅(𝑦𝑝 − 𝑐2,𝑝 − 𝑇).

(6)

Assumption 2. The parent knows that her child is unaware of his own habit formation until hislast period.

5

For the case where the misperception disappears, see Appendix A4.

7

The parent is altruistic toward her child. We specify her utility function as a convex sum of
the felicity from her own consumption (𝑢(𝑐2,𝑝 ) + 𝛽𝑢(𝑐3,𝑝 )) and her child’s true utility (𝑈):𝑉 = (1 − 𝛾)[𝑢(𝑐2,𝑝 ) + 𝛽𝑢(𝑐3,𝑝 )] + 𝛾𝑈, 6(7)where 𝛾 ∈ [0,1] denotes the degree of parental altruism.
She is aware that her child, who is ignorant of the effect of habit formation, will consume toomuch in his first and second periods. The altruistic parent therefore has an incentive to keep her
child’s consumption level lower by restricting the amount of income transfer.
In order to obtain closed-form solutions, we specify the period utility function as follows:

Assumption 3. 𝑢(𝑥) =

𝑥 1−𝛼
1−𝛼

(𝛼 > 0).

We guarantee that the arguments of the utility functions are positive by assuming thefollowing:
1

𝑅𝑦

2
Assumption 4. When θ2 < 1, (𝛽𝑅)− ⁄𝛼 < 1, 𝑦𝑝 < 𝜃 (1+𝑅)
;
1

1

1

1

[1−𝜃2 (𝛽𝑅)− ⁄𝛼 ]𝑅𝑦22

𝜃1 (𝜃2 +𝑅)[1−(𝛽𝑅)− ⁄𝛼 ]when θ2 ≥ 1, (𝛽𝑅)− ⁄𝛼 < 𝜃 , 𝑦𝑝 <1

.7

Intuitively, the above assumption ensures that the child’s consumption in the second and third
periods is sufficiently low compared to its value in the previous period. This is achieved by

6

Without loss of generality, we assume that the parent’s period utility function is the same as the child’s.

7

When the child gets the first-period income 𝑦1 , the corresponding assumption to Assumption 4 should
1

be as follows: when θ2 < 1 , (𝛽𝑅)− ⁄𝛼 < 1 , 𝑦𝑝 + 𝑦1 <𝑦𝑝 + 𝑦1 <1[1−𝜃2 (𝛽𝑅)− ⁄𝛼]𝑅𝑦21.

𝜃1 (𝜃2 +𝑅)[1−(𝛽𝑅)− ⁄𝛼 ]8𝑅𝑦2
𝜃1 (1+𝑅)1; when θ2 ≥ 1 , (𝛽𝑅)− ⁄𝛼 <

1
𝜃2

,

restricting parameters, such that (i) the sum of the parent’s income and the child’s first-period
income is sufficiently is lower than the child’s second-period income and (ii) the child’s
discount factor is sufficiently high.
To show the welfare implications of the altruistic transfer, we distinguish and compare two
types of the income transfers to the child: (i) the actual transfer that the parent determines by
maximizing her altruistic utility (i.e., (7)), and (ii) the income transfer that is expected by thechild. Because the parent knows of the child's habit formation, but the child does not, the twotransfers will differ.
The parent understands her child’s habit-formation, and considers it when she determines thetransfer to her child. In this context, the parent indirectly steers her child away from falling into
unintended excessive consumption due this habit effect. The child, however, does not know of
the parent's altruistic intervention.

3.

The Income Transfer Expected by the Child
We consider how much income transfer the child expects to receive from his parent. The

expected level of income transfer could be a reference point for us to evaluate the parent's
intervention through income transfer: it enables us to (i) show that, consistent with daily
observations, the parent's income transfer is unexpectedly small for her young child, and (ii)examine whether the unexpectedly small transfer is appreciable to the grown-up child.
Consider the child’s consumption choice prior to receiving an income transfer from the parent
in his first period. Because he does not foresee the transfer level perfectly, it is a sort of
expected consumption choice based on his expectations regarding parental income transfer. The

9obtained solutions for expected consumption, 𝑐𝑖𝑒 ( 𝑖 = 1,2,3), will not be realized—first
because the parent's income transfer differs from he expects, and second, because his
preferences would shift due to the habit effect of the transfer in the first period.
With an expected level of the parent's income transfer 𝑇 𝑒 being given, the 𝑐𝑖𝑒 are obtainedby maximizing the child’s lifetime utility (4) subject to budget constraints (1) and (2). The
optimality conditions are given by
𝑐1𝑒 = 𝑇 𝑒 ,(8)𝑢′ (𝑐2𝑒 ) = 𝛽𝑅𝑢′(𝑐3𝑒 ).

(9)Equation (8) determines 𝑐1𝑒 . Budget constraints (1) and (2) and first-order condition (9) jointly
determine 𝑐2𝑒 and 𝑐3𝑒 . Note that 𝑐2𝑒 and 𝑐3𝑒 are independent of the expected parent’s transfer
𝜕𝑐2𝑒
𝜕𝑐 𝑒
= 0, 3 = 0), because the child does not incorporate the habit effect from Assumption
𝜕𝑇
𝜕𝑇

𝑇𝑒 (
1.

Given that the expected consumption behavior 𝑐𝑖𝑒 is conditional on parental income transfer,the child, in turn, computes expected income transfer 𝑇 𝑒 by anticipating his parent's
utility-maximizing behavior. When anticipating the parent’s behavior, however, he does notincorporate his own habit from Assumption 1. From the anticipated utility-maximizing behaviorof the parent, the expected transfer 𝑇 𝑒 is determined jointly with expected parent’s𝑒𝑒consumption (𝑐2,𝑝, 𝑐3,𝑝) by the budget constraint𝑒𝑒𝑐3,𝑝= 𝑅(𝑦𝑝 − 𝑐2,𝑝− 𝑇𝑒)

and the first-order conditions𝑒𝑒(1 − 𝛾)[𝑢′ (𝑐2,𝑝)] = 0,) − 𝛽𝑅𝑢′(𝑐3,𝑝

10

(10)

𝑒
𝛽𝑅(1 − 𝛾)𝑢′ (𝑐3,𝑝) = 𝛾𝑢′ (𝑇 𝑒 ),

𝜕𝑐 𝑒

(11)

𝜕𝑐 𝑒

where we set 𝜕𝑇2 = 0 and 𝜕𝑇3 = 0 in (11) from Assumption 1.

4.

The Altruistic Intervention of the Parent

4.1. The Actual Transfer Decided by the Parent
The parent behaves differently from the child’s expectations because she knows that her
child’s preferences will shift due to the effect of habit formation, even though the child himself
does not recognize it in his early periods. Hence, when deciding an income transfer level for her
young son, the mother takes into account the habit effect that the transfer will have on his
consumption in his second and third periods. Formally, the parent chooses the transfer level toher child by maximizing her altruistic utility (7), which incorporates the child’s true utility
function (3). We denote the actual income transfer level that the parent decides optimally by 𝑇 ∗ .
In the child’s second period, he takes the habit as a permanent preference shock. Maximizing(5) leads to the first-order condition of the child in the second period:𝑢′ [𝑐2 − 𝜃1 𝑐1∗ ] = 𝛽𝑅𝑢′ [𝑐3 − 𝜃1 𝑐1∗ ],(12)

where the first-period consumption is given by the borrowing constraint𝑐1∗ = 𝑇 ∗ .
Under budget constraints (13) and𝑐3∗ = 𝑅(𝑦2 − 𝑐2∗ ),

11

(13)

the actual consumption levels 𝑐2∗ and 𝑐3∗ are determined by (12). Because the first-period
consumption 𝑐1∗ depends on the income transfer level from (13), the actual consumption levels𝑐2∗ and 𝑐3∗ are functions of the income transfer 𝑇 from (12).
With the function 𝑐2∗ and 𝑐3∗ being given, the parent chooses ( 𝑐2,𝑝 , 𝑐3,𝑝 , 𝑇 ∗ ) so as tomaximize (7). The first-order conditions are∗∗(1 − 𝛾)[𝑢′ (𝑐2,𝑝) − 𝛽𝑅𝑢′ (𝑐3,𝑝)] = 0,∗𝛽𝑅(1 − 𝛾)𝑢′ (𝑐3,𝑝) = 𝛾[𝑢′ (𝑇 ∗ ) + 𝛽𝑢′ (𝑐2∗ − 𝜃1 𝑐1∗ ) (

(14)

𝜕𝑐2∗
𝜕𝑐1∗
−
𝜃
)
1
𝜕𝑇 ∗
𝜕𝑇 ∗

𝜕𝑐3∗
𝜕𝑐 ∗
− 𝜃2 2∗ ).
∗𝜕𝑇𝜕𝑇+𝛽 2 𝑢′ (𝑐3∗ − 𝜃2 𝑐2∗ ) ((15)

First-order conditions (14) and (15) and budget constraint (6) jointly determine the parental
income transfer level. We denote it by 𝑇 ∗ (𝛼, 𝛾, 𝛽, 𝑅, 𝑦𝑝 , 𝜃1 , 𝜃2 , 𝑦2 ).
In the following subsections, we characterize the income transfer 𝑇 ∗ of the altruistic parent
by comparing it with the young child's expectation on income transfer 𝑇 𝑒 . By substituting
functions 𝑐𝑖∗ (𝑇) into (3), we obtain the child's indirect utility as a function of income transfer T,
𝑈(𝑇). Notice that there exists a maximum of 𝑈(𝑇). We denote the transfer level thatmaximizes the child’s utility by 𝑇̅. By definition, it satisfies 𝑈′(𝑇̅) = 0. Using Assumption 3,
we express 𝑇̅ as 𝑇̅(𝛼, 𝛽, 𝑅, 𝜃1 , 𝜃2 , 𝑦2 ).
Since the child’s utility is maximized at the point of 𝑇̅, too much income transfer (more than𝑇̅) harms him. Note that the parent necessarily chooses income transfer level 𝑇 ∗ such that it is12smaller than 𝑇̅: 𝑇 ∗ < 𝑇̅. This choice is made because if 𝑇 ∗ were higher than 𝑇̅, the parent
could enhance both her own welfare and the child's welfare by reducing the income transfer.
However, the expected transfer 𝑇 𝑒 can be either smaller or larger than 𝑇̅, depending on theparent’s income level 𝑦𝑝 . To compare the actual and expected transfers (𝑇 ∗ and 𝑇 𝑒 ) and thecorresponding welfare levels, it would, therefore, be helpful to separate two cases: (i) 𝑦𝑝 > 𝑇̅and (ii) 𝑦𝑝 ≤ 𝑇̅, which represent the rich-parent and poor-parent cases, respectively.

4.2. Comparison of Actual and Expected Transfers
The parent is aware of her child’s habit formation and incorporates it in her decision making,while the child does not. Thus, the actual transfer 𝑇 ∗ deviates from the expected one 𝑇 𝑒 .
Proposition 1 states the relationship between 𝑇 ∗ and 𝑇 𝑒 .
Proposition 1. For any degree of parental altruism 𝛾 ∈ (0,1), the parental income transfer is
unexpectedly small to the young child, that is, 𝑇 ∗ < 𝑇 𝑒 .

Proof. See Appendix A1.
Proposition 1 implies that, insofar as the parent is not perfectly selfish or perfectly altruistic,
she keeps the transfer level to her child lower than he expects. The parent knows that moretransfer leads the child to form deeper habits, and hence consume more excessively in the next
period. Hence, she intervenes in the child’s behavior by keeping the transfer low.
Note that when the parent is perfectly selfish (𝛾 = 0) or perfectly altruistic (𝛾 = 1), 𝑇 ∗ canequal 𝑇 𝑒 . In particular, when 𝛾 = 0, we have trivially 𝑇 ∗ = 𝑇 𝑒 = 0 (i.e., purely selfish
parents do not give, and are not expected to give, transfers to their children).

13

When 𝛾 = 1, we have 𝑇 ∗ = 𝑇̅ if the parent is poor, 𝑦𝑝 ≤ 𝑇̅. Since the poor parent cannot
afford to transfer 𝑇̅, she just gives her whole income to her child, as he expects. In the poor
parent case, therefore, we have 𝑇 ∗ = 𝑇 𝑒 = 𝑦𝑝 . When the parent is rich, 𝑦𝑝 > 𝑇̅; however, theparental transfer 𝑇 ∗ is smaller than what the child expects and again, 𝑇 ∗ < 𝑇 𝑒 . This isbecause 𝑇 ∗ maximizes the child’s utility, 𝑇 ∗ = 𝑇̅, whereas the naïve child expects the purely
altruistic parent to transfer her whole income to him, 𝑇 𝑒 = 𝑦𝑝 (> 𝑇̅ by constuction).

4.3. The Welfare of the ChildAs shown in the previous section, the parental income transfer is unexpectedly small in theeyes of the young child. Our interest here is to evaluate it from the viewpoint of the child's
welfare; we shall show that the unexpectedly small transfer could make the child better off.
We compare utility values of actual transfer 𝑇 ∗ (𝑈(𝑇 ∗ )) and expected transfer 𝑇 𝑒 (𝑈(𝑇 𝑒 ))by employing the true utility function (3). Because the child is incognizant of the true utility
function 𝑈(∙) until his third period, the utility function 𝑈(∙) can be regarded as a retrospective
welfare measure of income transfer T.8
Of particular interest is whether the parental income transfer generates higher welfare of thechild than the child’s expected transfer does, when evaluated by the retrospective measure. If it
does, the grown-up child is thankful to his parent for receiving the unexpectedly small transfer.
Hence, we include the following definition.
Definition 1. When 𝑈(𝑇 ∗ ) > 𝑈(𝑇 𝑒 ) , parental income transfer 𝑇 ∗ is referred to as
thankworthy to the child.

8

Pollack (1978) recommends "unconditional welfare ordering", which incorporates the effects ofendogenous preference formation—we used the same welfare criterion here.

14

We first consider the case of a poor parent, where 𝑦𝑝 ≤ 𝑇̅ (Figures 1 and 2). In this case, theparent cannot provide the child with a transfer exceeding 𝑇̅. As Figure 1 shows, both 𝑇 ∗ and𝑇 𝑒 are less than 𝑦𝑝 , and hence are less than 𝑇̅. Recall from Proposition 1 that 𝑇 ∗ < 𝑇 𝑒 for𝛾 ∈ (0,1). Since the child’s welfare increases with transfers in the region of T < 𝑇̅, we have𝑈(𝑇 ∗ ) < 𝑈(𝑇 𝑒 ) for 𝛾 ∈ (0,1) (i.e., the parental income transfer is not thankworthy to thechild).9<Figure 1> & <Figure 2>

When the parent is rich, 𝑦𝑝 > 𝑇̅, the child’s welfare first increases and then decreases with
the income transfer level. As Figure 3 shows, the rich parent can provide the child with moretransfer than 𝑇̅, but she maintains transfer level 𝑇 ∗ lower than this. However, the expectedtransfer 𝑇 𝑒 can be more than 𝑇̅. Too much transfer (more than 𝑇̅) causes the child’s welfare
𝑈 to decrease. Therefore, the retrospective utility of the expected transfer 𝑈(𝑇 𝑒 ) can be lower
than that of actual transfer 𝑈(𝑇 ∗ ).
Recall that when 𝛾 = 0, 𝑇 ∗ = 𝑇 𝑒 = 0, and that when 𝛾 = 1, 𝑇 ∗ = 𝑇̅, which is an interior
solution, and 𝑇 𝑒 = 𝑦𝑝 , which is a corner solution (Figure 4). As 𝛾 increases from 0 to 1, theactual transfer 𝑇 ∗ increases from 0 to 𝑇̅, and the expected transfer 𝑇 𝑒 increases from 0 to 𝑦𝑝 .
The parent has to balance between her own utility and the child’s utility. Under the condition of
the rich parent, the child’s welfare increases with the actual transfer 𝑇 ∗, and therefore also9

When the parent is purely selfish (𝛾 = 0) or purely altruistic (𝛾 = 1), we have 𝑈(𝑇 ∗ ) = 𝑈(𝑇 𝑒 )(Figure 2).

15increases with the degree of parental altruism 𝛾. However, the child’s retrospective utility levelof 𝑇 𝑒 decreases with 𝑇 𝑒 when it exceeds 𝑇̅ . Therefore, there exists a certain degree ofparental altruism, which equalizes 𝑈(𝑇 ∗ ) and 𝑈(𝑇 𝑒 ).
Lemma 1. When the parent is rich, there exists a degree of parental altruism 𝛾̃ ∈ (0,1) suchthat 𝑈(𝑇 ∗ ) = 𝑈(𝑇 𝑒 ).

Proof. See Appendix A2.
For all degrees of parental altruism higher than 𝛾̃, the actual transfer leads to higher welfareof the child, and hence, the actual transfer is thankworthy for the child.

<Figure 3> & <Figure 4>

Proposition 2. When the parent is rich ( 𝑦𝑝 > 𝑇̅ ), the parental income transfer 𝑇 ∗ isthankworthy for the child, that is, 𝑈(𝑇 ∗ ) > 𝑈(𝑇 𝑒 ), if the parent is so altruistic that 𝛾 > 𝛾̃.

Proof. See Appendix A2.
Reducing parental income transfer (i) directly lowers the child’s utility in each period and (ii)
weakens the child’s consumption habit. When the parent’s income and the degree of parental
altruism are large enough, and hence the income transfer is large enough, the second negative
effect dominates the first positive one. Therefore, the unexpectedly low transfer makes the childbetter off. In this context, in his third period, the child becomes aware of the habit and thanks
the parent for having given him less transfer than he expected.

16

5.

An Increase in the Habit ParameterAs shown in Proposition 1, the altruistic parent keeps income transfer to the child lower thanwhat the child expects to receive. When the child’s consumption habit and his consumption
propensity are exogenously enhanced, the parent would further reduce the income transfer so as
to mitigate the resulting increase in the child’s over-consumption.
We can show the validity of this conjecture by examining the effect of an exogenous increasein the child’s habit parameter in his second period 𝜃1 .
Proposition 3.

A higher degree 𝜃1 of the child’s habit in the second period implies a lower

level of parental income transfer 𝑇 ∗, that is,

𝑑𝑇 ∗
< 0.
𝑑𝜃1

Proof. See Appendix A3.
Proposition 3 shows that the parent penalizes the child for an increase in habitualconsumption by reducing income transfer. This strict behavior, however, comes from theparent's altruistic motive.
Note that the apparently strict yet altruistic behavior of the parent is similar to the parents'“tough love” behavior that Bhatt and Ogaki (2012) derived. By assuming an endogenous time
discounting function, they show that, in response to an exogenous rise in a child's future time
preference, his altruistic parent reduces income transfer so as to mitigate the increase in his
degree of impatience. Our result in Proposition 3 is comparable to this tough love result,because an increase in the habit parameter would raise the child's time preference.
Formally, following the literature on time preference (e.g., Obstfeld, 1990), we define the17child’s pure subjective rate of time preference in his second period as follows:𝜒2,3 (𝑐) = log 𝑀𝑅𝑆(𝑐2 , 𝑐3 ) |𝑐2 =𝑐3 =𝑐𝑢′ (𝑐 − 𝜃1 ℎ2 )𝜃2= 𝜌 + log [ ′−],𝑢 (𝑐 − 𝜃2 ℎ3 ) 1 + 𝜌(16)
where 𝜌 = 1⁄𝛽 − 1 is the subjective discount rate of the child. It is then is easy to see thatwhen degree of the habit formation in the child’s second period 𝜃1 exogenously increases, thechild’s time preference function 𝜒2,3 (𝑐) shifts upward. Because the parent reduces income
transfer 𝑇 ∗ (see Proposition 3) upon the increase in 𝜃1 , the parent's behavior above seems toduplicate the tough love behavior discussed by Bhatt and Ogaki (2012).
Our discussion is distinct in two aspects. First, in the present model, time preference isendogenously generated by habit formation rather than by an exogenously given time
preference formation. Second and more importantly, our tough love is thankworthy to the child
if his parent is altruistic and rich enough. This contrasts with the case explored in the Ogaki and
Bhatt model, wherein a mother's tough love necessarily reduces her son's welfare, because themother evaluates her son's future utility by using different time preference than that of the child.

6.

Concluding Remarks
By employing a parental altruism model where the child is ignorant of his own habit until hereaches adulthood, we show that the parental income transfer seems unexpectedly low to theyoung child. However, the unexpectedly low parental income transfer can generate higher
welfare for the child than the expected transfer, when evaluated retrospectively. In other words,
when the parent is rich and sufficiently altruistic, the parent’s income transfer is thankworthy to18the grown-up child. The predictions of this model are consistent with daily observations.
This research may have important implications for understanding how parent-childinteractions affect the consumption/saving behavior and related preference formation of family
members. For example, our result is consistent with the observation that richer parents, orparents from richer families, tend to be stricter than poorer parents (or those from poorer
families) when safeguarding against their children’s profligacy.

19

Appendix
A1. Proof of Proposition 1
When ∀𝛾 ∈ (0,1), we compare the actual transfer 𝑇 ∗ with the expected transfer 𝑇 𝑒 bydividing the first order condition (11) with the first order condition (15). By substituting
first-order conditions (9) and (14) as well as Assumption 1 we have𝑦𝑝 − 𝑇 ∗ −𝛼
𝑇 ∗ −𝛼()=()
𝑦𝑝 − 𝑇 𝑒
𝑇𝑒
1

−

1

1

[1 + 𝑅(𝛽𝑅)−𝛼 ]1−𝛼 (𝑇 𝑒 )−𝛼1−1𝛽𝜃1 (𝑅 + 1)(𝛽𝑅)−𝛼 [𝑅𝑦2 (𝛽𝑅)−𝛼 − 𝜃1 (𝑅 + 1)(𝛽𝑅)−𝛼 (𝑇 ∗ )]−𝛼
1

1

−𝛼

𝛽 2 𝜃2 (𝜃2 + 𝑅) [1 − (𝛽𝑅)−𝛼 ] [𝑅𝑦2 (1 − 𝜃2 (𝛽𝑅)−𝛼 ) − 𝜃1 (𝜃2 + 𝑅)(1 − (𝛽𝑅)−𝛼 )(𝑇 ∗ )]1

[1 + 𝑅(𝛽𝑅)−𝛼 ]1−𝛼 (𝑇 𝑒 )−𝛼

.
(A1)

𝑇∗

Equation (A1) implies that 𝑇 𝑒 < 1. Therefore, we have ∀𝛾 ∈ (0,1), 𝑇 ∗ < 𝑇 𝑒 .

A2. Proof of Lemma 1 and Proposition 2
The child’s utility 𝑈 is continuous in the degree of parental altruism 𝛾. When 𝑇 < 𝑇̅, 𝑈 is
increasing in 𝑇; and when 𝑇 > 𝑇̅, 𝑈 is decreasing in 𝑇. The income transfers 𝑇 ∗ and 𝑇 𝑒
are both continuous and increasing in 𝛾.
When 𝑦𝑝 > 𝑇̅, we obtain lim𝛾→1 𝑇 ∗ = 𝑇̅, which is an interior solution and that lim𝛾→1 𝑇 𝑒 =𝑦𝑝 , which is a corner solution. We also have lim𝛾→1 𝑈′(𝑇 ∗ ) = 0 and lim𝛾→1 𝑈′(𝑇 𝑒 ) < 0.
Hence, in the neighborhood of 𝛾 = 1, 𝑈(𝑇 ∗ ) > 𝑈(𝑇 𝑒 ).
When 𝛾 decreases from 1, both 𝑇 ∗ and 𝑇 𝑒 decrease, as does 𝑈(𝑇 ∗ ). However, 𝑈(𝑇 𝑒 )increases until 𝑇 𝑒 equals to 𝑇̅. Hence, there exists a certain 𝛾̃ such that 𝑈(𝑇 ∗ ) = 𝑈(𝑇 𝑒 ).

20

When 𝛾 > 𝛾̃, we have 𝑈(𝑇 ∗ ) > 𝑈(𝑇 𝑒 ).

A3. Proof of Proposition 3
First-conditions (10) and (11) and budget constraint (4) jointly determine the optimal transferof the parent 𝑇 ∗ (𝛼, 𝛾, 𝛽, 𝑅, 𝑦𝑝 , 𝜃1 , 𝜃2 , 𝑦1 , 𝑦2 ).
Substituting first-order condition (10) into (11) and taking the total derivative of the optimaltransfer by the parent 𝑇 ∗ with respect to the habit parameter 𝜃1 leads to−𝛼−1{(1 − 𝛾)𝛼𝛽𝑅1−𝛼 (𝑦𝑝 − 𝑇 ∗ )1−𝛼𝛾 [1 + 𝑅(𝛽𝑅) ]1++ 𝛼(𝑇 ∗ )−𝛼−1

−𝛼

1

1

𝛼𝛽(𝛽𝑅)−𝛼 𝜃1 (1 + 𝑅)[𝑅𝑦2 (𝛽𝑅)−𝛼 − 𝜃1 (1 + 𝑅)(𝛽𝑅)−𝛼 (𝑇 ∗ )]−𝛼−11[1 + 𝑅(𝛽𝑅)−𝛼 ]1−𝛼

1

1

1

𝛼𝛽 2 𝜃1 𝜃2 (𝜃2 + 𝑅)2 [1 − (𝛽𝑅)−𝛼 ]2 [𝑅𝑦2 (1 − 𝜃2 (𝛽𝑅)−𝛼 ) − 𝜃1 (𝜃2 + 𝑅)(1 − (𝛽𝑅)−𝛼 )(𝑇 ∗ )]−𝛼−1 𝑑𝑇 ∗
+
}
1
𝑑𝜃1
[1 + 𝑅(𝛽𝑅)−𝛼 ]1−𝛼
1

=−

1

1

𝛽(𝛽𝑅)−𝛼 (1 + 𝑅)[𝑅𝑦2 (𝛽𝑅)−𝛼 − 𝜃1 (1 + 𝑅)(𝛽𝑅)−𝛼 (𝑇 ∗ )]−𝛼−11[1 + 𝑅(𝛽𝑅)−𝛼 ]1−𝛼111

∙ {[𝑅𝑦2 (𝛽𝑅)−𝛼 − 𝜃1 (1 + 𝑅)(𝛽𝑅)−𝛼 (𝑇 ∗ )] + (1 + 𝑅)(𝛽𝑅)−𝛼 (𝑇 ∗ )}
1

−

1

1

𝛼𝛽 2 𝜃2 (𝜃2 + 𝑅)2 [1 − (𝛽𝑅)−𝛼 ]2 [𝑅𝑦2 (1 − 𝜃2 (𝛽𝑅)−𝛼 ) − 𝜃1 (𝜃2 + 𝑅)(1 − (𝛽𝑅)−𝛼 )(𝑇 ∗ )]−𝛼−11[1 + 𝑅(𝛽𝑅)−𝛼 ]1−𝛼∙ (𝜃2 + 𝑅)(𝑇 ∗ ),
(A2)
𝑑𝑇 ∗

where, from Assumption 4, the coefficient of 𝑑𝜃 is positive; and the right hand side is negative.
1𝑑𝑇 ∗Therefore, we have ∀𝛾(0,1), 𝑑𝜃 < 0.
1

21

A4. When Misperception Disappears in the Child’s Second Period
Here, we show that our main results are robust even when the child is assumed to becomeaware of his own habit formation in the second period, rather than in the third period.

In order to guarantee that the arguments of the utility functions are positive, we assume, thefollowing, instead of Assumption 1:
𝑅𝑦

2
Assumption A1. 𝑦𝑝 < 𝜃 (𝑅+𝜃
.
)
1

2

We denote optimal solutions in the present case by using an apostrophe, instead of an asteriskin the text.
In the second period, the misperception of the child disappears and the child maximizes theutility 𝑢(𝑐2 − 𝜃1 𝑐1′ ) + 𝛽𝑢(𝑐3 − 𝜃2 𝑐2 ). The first-order condition is given by𝑢′ [𝑐2 − 𝜃1 𝑐1′ ] = 𝛽(𝑅 + 𝜃2 )𝑢′ [𝑐3 − 𝜃2 𝑐2 ].

(A3)With 𝑐1′ being given by the parent's income transfer 𝑇 ′ , (A3) and budget constraint (3) jointly
determine the consumption levels 𝑐2′ and 𝑐3′ .
The parent’s first-order conditions of the parent are′′(1 − 𝛾)[𝑢′ (𝑐2,𝑝) − 𝛽𝑅𝑢′ (𝑐3,𝑝)] = 0,′𝛽𝑅(1 − 𝛾)𝑢′ (𝑐3,𝑝) = 𝛾[𝑢′ (𝑇 ′ ) + 𝛽𝑢′ (𝑐2′ − 𝜃1 𝑐1′ ) (

𝜕𝑐 ′

(A4)

𝜕𝑐2′
𝜕𝑐1′
−
𝜃
)
1
𝜕𝑇 ′
𝜕𝑇 ′

𝜕′

+𝛽 2 𝑢′ (𝑐3′ − 𝜃2 𝑐2′ ) (𝜕𝑇3′ − 𝜃2 𝜕𝑇 ′ ).
(A5)

22

First-order conditions (A4) and (A5) and budget constraint (6) jointly determine the parental
income transfer 𝑇 ′ . By noting that the expected transfer 𝑇 𝑒 is the same as in the text, we
obtain the following result:
Proposition A1. For any degree of parental altruism 𝛾 ∈ (0,1), the parental income transfer is
unexpectedly small to the young child, that is, 𝑇 ′ < 𝑇 𝑒 .

Proof.
By substituting (9) into (10), (A4) into (.5), and dividing (A5) with (10), we have𝑦𝑝 − 𝑇 ′ −𝛼
𝑇 ′ −𝛼()=()
𝑦𝑝 − 𝑇 𝑒𝑇𝑒−

𝛽 2 𝜃1 (𝑅 + 𝜃2 )[𝑅𝑦2 − 𝜃1 (𝑅 + 𝜃2 )(𝑇 ′ )]−𝛼1

−𝛼{1 + (𝑅 + 𝜃2 )[𝛽(𝑅 + 𝜃2 )]−𝛼 }(𝑇 𝑒 )−𝛼(A6)𝑇′

Equation (A6) implies that 𝑇 𝑒 < 1. Therefore, we have ∀𝛾 ∈ (0,1), 𝑇 ′ < 𝑇 𝑒 .
We can also show that there is a critical degree of parental altruism 𝛾̂ ∈ (0,1), such that𝑈(𝑇 ′ ) = 𝑈(𝑇 𝑒 ). As in Proposition 2, we have:
Proposition A2. When the parent is rich (𝑦𝑝 > 𝑇̅ ), the parental income transfer 𝑇 ′ isthankworthy for the child, that is, 𝑈(𝑇 ′ ) > 𝑈(𝑇 𝑒 ), if the parent is so altruistic that 𝛾 > 𝛾̂.

Proof.
The child’s utility 𝑈 is continuous in the degree of parental altruism 𝛾. When 𝑇 < 𝑇̅, 𝑈 is
increasing in 𝑇; and when 𝑇 > 𝑇̅, 𝑈 is decreasing in 𝑇. The income transfers 𝑇 ′ and 𝑇 𝑒 areboth continuous and increasing in 𝛾.

23

When 𝑦𝑝 > 𝑇̅, we obtain lim𝛾→1 𝑇 ′ = 𝑇̅, which is an interior solution and that lim𝛾→1 𝑇 𝑒 =𝑦𝑝 , which is a corner solution. We also have lim𝛾→1 𝑈′(𝑇 ′ ) = 0 and lim𝛾→1 𝑈′(𝑇 𝑒 ) < 0.
Hence, in the neighborhood of 𝛾 = 1, 𝑈(𝑇 ′ ) > 𝑈(𝑇 𝑒 ).
When 𝛾 decreases from 1, both 𝑇 ′ and 𝑇 𝑒 decrease, and 𝑈(𝑇 ′ ) also decreases. However,𝑈(𝑇 𝑒 ) increases until 𝑇 𝑒 equals to 𝑇̅ . Hence there exists a certain 𝛾̃ such that 𝑈(𝑇 ′ ) =𝑈(𝑇 𝑒 ). When 𝛾 > 𝛾̃, we have 𝑈(𝑇 ′ ) > 𝑈(𝑇 𝑒 ).

24

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Figure 1. The utility of the child as a function of 𝑇 when the parent is poor

27

Figure 2. The utility of the child as a function of 𝛾 when the parent is poor

28

Figure 3. The utility of the child as a function of 𝑇 when the parent is rich

29

Figure 4. The utility of the child as a function of 𝛾 when the parent is rich30