Discussion Paper No. 890

WELFARE-ENHANCING
PARENTAL ALTRUISM
AND
CHILDREN’S HABIT FORMATION

Lin Zhang
Shinsuke Ikeda

November 2013
August 2016

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanWelfare-Enhancing Parental Altruism and Children’s Habit
Formation†
Lin Zhang* and Shinsuke Ikeda**

Abstract
We propose a model of parental altruism in relation to children’s habit formation, as children are
unaware of their developing habits while young and only become cognizant with age. We show thatan altruistic mother (1) maintains the amount of income transferred to her child lower than the childwould desire and (2) reduces further income transfer upon an exogenous increase in case of aparticular negative habit of the child. The child, when grown, may end up being grateful for thesmall income transfer if the mother is sufficiently altruistic and has a relatively high income: Whenevaluated by the realized habitual preferences, a small income transfer leads to greater welfare. Thisimplies that parents from richer families tend to appreciate the authoritative parenting practice.

Keywords: Habit Formation, Parental Altruism, Welfare-Enhancing, Time Preference.
JEL Classification: D10; D60; D91; J13.

Acknowledgements We thank Koichi Futagami, Tatsuro Iwaisako, Masao Ogaki, Yoshiyasu Ono, andTadashi Yagi for their comments. We are also grateful to the two anonymous referees for their valuablesuggestions. We acknowledge financial support from: Grants-in-Aid for Scientific Research (C No.
26380239) from the Japan Society for the Promotion of Science; the Global COE Program of Osaka
University; the Joint Usage/ Research Center Project of ISER from the Ministry of Education, Culture,
Sports, Science and Technology; and Research Subsidy of Kanazawa Seiryo University.
† Published in the International Review of Economics, September 2016, Vol. 63(3), pp. 281–303.

DOI: 10.1007/s12232-016-0255-2.
*

Faculty of Economics, Kanazawa Seiryo University, 10-1 Ushi, Gosho-machi, Kanazawa-shi, Ishikawa, 920-8620,
Japan. Tel.: 81-7-6253-3929; Fax: 81-7-6253-3995.
Email: <z-lin@seiryo-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1.

Introduction
Parents discipline their young children by keeping income transfer (e.g., allowances) lower than what

the children themselves desire. This causes conflicts between the parents, who act on altruistic motives,
and the children, who believe that they will benefit from more money. The scope of this paper is totheoretically address parents’ regulating behavior and the intergenerational conflict from the viewpoint ofchildren’s welfare.
Even though empirical research on parents’ restrictive attitude toward children is still sparse ineconomics, some psychological studies provide evidence of parents’ restrictive practice of money transfer
and feeding. A typical example is research on pocket money, a proxy of parents’ money transfer: A seriesof studies show that the difference in the amount of children’s pocket money is much smaller than theirfamily’s socioeconomic status (e.g., West et al. 2006; Barnet-Verzat and Wolff 2002; Furnham andThomas 1984). In other words, the amount of money transfer for children from high-income families issmaller than what they expect from their family’s socioeconomic status. Moreover, Furnham (1999)
reports that most of the children (over 80%) in his questionnaire claim that their parents will not givethem more money if they spent all of their allowance. It is also shown that parents do not give children
allowances or pocket money unconditionally, and they intentionally reduce the amount to cultivate betterself-regulation (e.g., Otto 2013; Furnham 2001; Miller and Yung 1990). As for feeding practices, parents’restrictions on unhealthy food consumption are often observed (e.g., van der Horst et al. 2007; Patrick et
al. 2005; Gable and Lutz 2000). Overall, all these restrictive actions correspond to an authoritative
parenting style (Baumrind 1967, 1971, and 1978), which represents a parenting behavior with high levelof control and maturity demands through active communication and induction. Doepke and Zilibotti(2014) show that “thrift,” as a typical feature of the authoritative parenting style is frequently observed
across countries. Meanwhile, several studies find that poverty has a considerable effect on children
because children are able to understand and anticipate their family economic status through social
comparison (e.g., Rosenberg and Pearlin 1978; Duncan et al. 1998). Furthermore, consistent with casual
observation, there is empirical evidence showing that the authoritative parenting is likely to cause
negative feelings from children (Fisher and Birch 2000). By conducting parent-child pair experiments,
Akabayashi et al. (2014) show that, unlike the standard intergenerational altruism model, conflict ofinterest exists between parents and children when they are making joint decision about payments to thechildren.
Given these empirical facts, the questions that this study is trying to answer are why do altruistic
parents not transfer as much income to their young children as the latter would like, and whether grown2children appreciate their parents’ altruism. Consequently, in order to understand how parent-childinteractions affect the consumption/saving behavior and related preference formation of the family
members, this paper aims to theoretically address the following problems:
1. Why do altruistic parents keep income transfers to young children lower than the children desire?
2. Whether, and under which conditions, does the parents’ behavior of restricting income transfers
enhance children’s welfare?
Therefore, we consider an altruistic parent’s behavior toward his/her habit-forming child by modifying
the framework of Becker (1974) and Barro (1974), in which the parent obtains utility from his or her own
consumption, as well as the child’s utility. In line with Weinberg (2001) and Bhatt and Ogaki (2012), thealtruistic parent (e.g., a mother) 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 is that thechild is habit forming. By controlling income transfers to her young son, the mother in our model can
influence his consumption habits, and hence, his future consumption behavior.
We posit two key assumptions when specifying the model in section 2. First, as is the case of actual
children, the young child is assumed to be unaware that consumption is habit forming, whereas his
mother is assumed to be aware. The assumption of the child’s misperception about habit formation isreasonable and reinforced by psychological literature, which provides empirical evidence for thenecessity of parental intervention in children’s habits, in contexts as diverse as food consumption (e.g.,
Baumrind 1991) and passive leisure such as TV viewing (Walsh et al. 1998).1 These studies show thatparents directly or indirectly regulate children’s habitual good consumption in case of unawarenessregarding habituation that may cause children to consume excessively. Moreover, the framework ofmisperceived habit formation is also motivated by an interpretation of the Easterlin paradox from the
aspect of habituation (Clark et al. 2008). 2 Clark et al. (2008) explain the failure of happiness increase(despite rise in GNP per capita) by incorporating the internal backward-looking reference point (i.e.,
habituation). However, people did not expect this failure of rising in happiness. It indicates that the
habituation is misperceived.
We also assume that, after reaching adulthood, the child becomes aware of the true mechanisms ofhabit formation. We are interested in how the grown 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 welfare enhancing from the (grown) son’s perspective if, when evaluated1

However, there has been some controversy regarding the empirical validity of habit formation models (e.g., Dynan
2000).
2 Easterlin (1974) found that happiness has remained constant over time despite rises in GNP per capital in Westerncountries over the last 50 years.

3according to his true preferences, greater welfare was attained under the transfer than would have been
under greater income transfer.
Relative to problems (1) and (2), Propositions 1 and 2 are elaborated in section 4. For problem (1), weshow analytically in Proposition 1 that an altruistic parent keeps income transfers to his or her child lower
than the child wants. Thus, our model describes a conflict between altruistic parents who raise theirchildren strictly, and the children who complain about it. It is crucial to note that children’s habitual
preference formation is known to their parents, but not to the children themselves.
Our solution to problem (2), summarized in Proposition 2, explains the welfare implications ofauthoritative parenting. It is shown that a parent’s income transfer to her child will induce child gratitude
if and only if the parent is sufficiently altruistic and rich compared with the child. This is because amarginal unit of income transfer reduction to a child has two countervailing effects. First, it directlyreduces the child’s felicity throughout the period. Second, it weakens the child’s consumption habit in thefollowing period and mitigates excess consumption that comes from the child being unaware of his or her
habit formation. The parental behavior of keeping income transfers low enhances the child’s welfare only
when the second positive effect dominates the first negative one. Consequently, if parental income, as
well as the degree of parental altruism, is not large, and hence, if the income transfer level is not large, themarginal utility of income transfer is sufficiently high so that the first negative effect dominates thesecond positive effect, and hence that the strict parenting is not appreciated by the child.
The result of Proposition 2 is related to theoretical and empirical literature showing that authoritative
parenting is more likely to be appreciated by high-income samples. Based on a static principal-agent
model of the child-parent interaction, Weinberg (2001) empirically supports the hypothesis that thefrequency of using pecuniary punishment, rather than corporal punishment, is positively associated withparental income. Unlike his model with parent-child information asymmetry, our dynamic model assumes
cross-generational asymmetry in cognitive power to identify future habit effects. We emphasize parental
cognitive power that corrects children’s habitual over-consumption.3 By parents directly choosing thetime preference parameter for children’s endogenous patience to affect their choices of occupation andeducational investment, Doepke and Zilibotti (2014) theoretically show that authoritative parenting
depends on economic status. Using data from World Value Survey, they empirically report that the rise of
authoritative parenting occurs with economic development (Doepke and Zilibotti 2014).
As for our main results, we provide two further discussion points in section 5. First, it is shown that3

Despite the differences, our results—suggesting that income transfers of richer parents are likely to be welfare
enhancing to children—seems consistent with the positive relationship that Weinberg (2001) reports between parental
income and children’s outcome. Both results relate child-rearing practices to parental income.

4when the child’s habit parameter exogenously shifts upward, the altruistic parent reduces theunexpectedly low income transfer further, so as to mitigate the child’s over-consumption. The result iscomparable with parental "tough love" behavior that Bhatt and Ogaki (2012) derive, by specifying an
endogenous time-discounting function. They show that an altruistic parent reduces income transfer inorder to affect the child’s rate of patience. Since an increase in the habit parameter is shown to raise thechild’s time preference induced by the consumption habit, our comparative static result could beconsidered a version of tough love. However, there are two key differences. First, in the present model,
time preference is endogenously generated by habit formation, rather than by an exogenously given time
preference formation. Second, our tough love is welfare-enhancing to the child if his parent is sufficiently
altruistic and relatively rich. This contrasts with the Bhatt and Ogaki model, wherein a mother’s toughlove reduces her son’s welfare, because the mother evaluates her son’s future utility by using a different
time preference than the child (Bhatt and Ogaki 2012).4
Second, we discuss how the main results change in the case of “good” habits, in which parents’ transfer
involves children’s good-habit formation that raises their future gratification (e.g., transfers in the form ofbooks and piano lessons). In this case, parents are shown to give unexpectedly large transfers to children,reflecting that the children are unaware of the future welfare-improving effect of the current consumptionthat would occur due to good-habit formation (Proposition 4). Thus, in contrast to the negative habit case,grown-up children appreciate the parental income transfer they received during childhood.
Our results show strong importance of parents’ control of children’s impulsive and habitualconsumption, because it is beneficial to children’s academic, economic, and health outcomes in the longrun. A series of empirical studies support this conclusion (see, e.g., Shoda et al. 1990; Heckman et al.
2006; Chabris et al. 2008; and Moffit 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 implications for two related issues (first, the effect of an exogenous increasein the habit parameter is analyzed, and second, we consider the case in which the parent’s income transfer
involves children’s good habits, rather than a negative habit); and section 6 concludes the paper.
4

Bhatt and Ogaki (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, the income transfer that thealtruistic 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, andhence tough, to the child at that time, because he is then unaware of the habit effect. It is, therefore, only reasonable
and welfare-enhancing to the child after reaching adulthood. Bhatt and Ogaki (2008) show that parental tough love
could be appreciated by the child in adulthood if welfare were retrospectively evaluated using time preference thatwas formed under the tough love. Note, however, that the attained welfare level under endogenous altruism isnecessarily higher than under tough love: children definitely prefer endogenous altruism to tough love.

5

2.

The Model
Consider a family consisting of a parent and a child, both of whom live for three periods. The parent (amother) 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, the childreceives an exogenous income 𝑦# .5 Variables 𝑐% (𝑖 = 1,2,3) denote the child’s consumption in period 𝑖.
The child is assumed to face borrowing constraints in period 1: 𝑐/ ≤ 𝑇. In order to focus on authoritative
parenting to reshape children’s intertemporal preferences, we follow Bhatt and Ogaki (2012) in assumingthat his disposable income 𝑇 is small enough that the borrowing constraint is binding:6
𝑐/ = 𝑇,(1)

His intertemporal budget in periods 2 and 3 is given by
𝑐2 = 𝑅(𝑦# − 𝑐# ).

(2)

where 𝑅 is the gross interest rate.
The child forms consumption habits. His preference is given by
𝑈 = 𝑢 𝑐/ + 𝛽𝑢 𝑐# − 𝜃/ ℎ# + 𝛽 # 𝑢(𝑐2 − 𝜃# ℎ2 ),
where

(3)

ℎ# = 𝑐/ and ℎ2 = 𝑐# ,which captures the habit effects of consumption in the previous period. Parameters 𝜃/ , 𝜃# > 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.

5

We assume away the first-period income 𝑦/ to the child. This, however, does not affect our main results below.
We can show that our main results below do not change if consumption in each period is a normal good even whentransfer T is large enough that the borrowing constraint is not binding: 𝑐/ < 𝑇. For the case where borrowing
constraint is not binding (𝑐/ < 𝑇), see Appendix A6. However, in that case, the parent’s control over income transfer
becomes less direct and less effective in authoritatively regulating the intertemporal consumption allocation of thehabit-forming child, because the transfer directly influences the child’s lifetime wealth and affects 𝑐/ , 𝑐# , and 𝑐2
equally.
For the similar reason, we assume that the child receive no transfer in his second period. An income transfer in thechild’s second period will also increase his lifetime wealth and affect consumption of each period equally. The effectof the parent’s restrictive action on the child’s habit will thus be diluted.
66

Assumption 1. The child is unaware of his habit in his first and second periods. In his third period, hebecomes aware of the habit.
Under Assumption 1, the child’s consumption behavior is assumed to proceed as follows.
In the first period, since the child is unaware of his habit, he just naively maximizes his lifetime utility
𝑢 𝑐/ + 𝛽𝑢 𝑐# + 𝛽 # 𝑢(𝑐2 ).

(4)

In this period, the child decides nothing, but receives the income transfer from his parent. However, hefinds that this actually received transfer deviates from his expectation, which will be discussed in the later
sections.
In the second period, the child notices that the utility is also actually different from what was expectedin the previous period. However, instead of recognizing the true mechanism of habit formation, he
incorrectly regards it as being caused by a permanent preference shock7 and hence maximizes the utility𝑢(𝑐# − 𝜃/ 𝑐/∗ ) + 𝛽𝑢(𝑐2 − 𝜃/ 𝑐/∗ ),(5)

where 𝑐/∗ denotes the child’s actual consumption level in the first period.8
In his third (the last) period, the child becomes aware of his true preference with habit formation as in(3). Therefore, he can re-evaluate his lifetime utility with the true preference (3).

The Parent’s Problem
Since the parent can affect the child’s consumption—and, thereby, welfare—through income transfer
𝑇, we focus on her decision in her second period. In the second period, she receives endowment income
𝑦@ , and maximizes the sum of the second and third period utilities by choosing consumption basket(𝑐#,@ , 𝑐2,@ ) and transfer 𝑇 to the child. Her budget constraint is given by
𝑐2,@ = 𝑅(𝑦@ − 𝑐#,@ − 𝑇).

(6)

Assumption 2. The parent knows that her child is unaware of his own habit formation until his lastperiod.
The parent is altruistic toward her child. We specify her utility function as a convex sum of the felicity

7

For the case where the misperception disappears in period 2, see Appendix A5.
In a more reasonable setting, by comparing equation (5) and actual transfer 𝑇 ∗ , the child could learn that his beliefof equation (5) is actually incorrect. Although we simplify our main analysis by putting the permanent preference
shock assumption (5), it can be shown that our main result holds valid even when the child gets aware of the true
preference in period 2. For the details, see Appendix A5.
8

7

from her own consumption (𝑢 𝑐#,@ + 𝛽𝑢 𝑐2,@ ) and her child’s true utility (𝑈): 9
𝑉 = 1 − 𝛾 𝑢 𝑐#,@ + 𝛽𝑢 𝑐2,@

+ 𝛾𝑈,

(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 too much inhis 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 =

G HIJ
/KL

(𝛼 > 0).

We guarantee that the arguments of the utility functions are positive by assuming the following:/

Assumption 4. When θ# < 1, (𝛽𝑅)K L < 1, 𝑦@ <when θ# ≥ 1, (𝛽𝑅)K

/

L <

/
RQ

OPQ
RH (/SO), 𝑦@ <;H[/KRQ (UO)I J ]OPQH
RH (RQ SO)[/K UO I J ].10Intuitively, the above assumption ensures that the child’s consumption in the second and third periods
is sufficiently low compared to the level in the previous period. This is achieved by restricting parameters,such that (i) the sum of the parent’s income and the child’s first-period income is sufficiently lower thanthe 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 theincome 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 the child. Because the parent knows thechild’s habit formation, while the child does not, the two transfers will differ.
The parent understands her child’s habit-formation, and considers it when she determines the transfer
to her child. In this context, the parent indirectly steers her child away from falling into unintended
excessive consumption due to this habit effect. The child, however, does not know the parent’s altruistic
intervention.

9

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

10

When the child gets the first-period income 𝑦/ , the corresponding assumption to Assumption 4 should be as

follows: when θ# < 1 , (𝛽𝑅)K[/KRQ (UO)I

H

J ]OP
Q
H
J]

RH (RQ SO)[/K UO I/

L <1 ,

𝑦@ + 𝑦/ <OPQ
RH (/SO).

8; when θ# ≥ 1 , (𝛽𝑅)K

/

L <

/
RQ

, 𝑦@ + 𝑦/ <

3.

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

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, theparent’s income transfer is unexpectedly small for her young child, and (ii) examine whether theunexpectedly small transfer is welfare-enhancing 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 obtained solutions for expectedconsumption, 𝑐%V (𝑖 = 1,2,3), will not be realized—first because the parent’s income transfer differs fromhe expects, and second, because his preferences would shift due to the habit effect of the transfer in thefirst period.
Let 𝑇 V denote the income transfer that the child expects to receive in his first period. With the
expected transfer 𝑇 V being given, the 𝑐%V are obtained by maximizing the child’s lifetime utility (4)subjecting to budget constraints (1) and (2). The optimality conditions are given by
𝑐/V = 𝑇 V ,(8)𝑢 W 𝑐#V = 𝛽𝑅𝑢′(𝑐2V ).

(9)Equation (8) determines 𝑐/V . Budget constraints (1) and (2) and first-order condition (9) jointly determine
𝑐#V and 𝑐2V . Note that 𝑐#V and 𝑐2V are independent of the expected parent’s transfer 𝑇 V (

YZQ[
Y\

= 0,

YZ][
Y\

=

0), because the child does not incorporate the habit effect from Assumption 1.
Given that the expected consumption behavior 𝑐%V is conditional on parental income transfer, the child,in turn, computes expected income transfer 𝑇 V by anticipating his parent’s utility-maximizing behavior.
When anticipating the parent’s behavior, however, he does not incorporate his own habit due toAssumption 1. From the anticipated utility-maximizing behavior of the parent, the expected transfer 𝑇 V
VV
is determined jointly with expected parent’s consumption (𝑐#,@, 𝑐2,@) by the budget constraintVV𝑐2,@= 𝑅(𝑦@ − 𝑐#,@− 𝑇V)

and the first-order conditionsVV
1 − 𝛾 [𝑢 W 𝑐#,@− 𝛽𝑅𝑢′(𝑐2,@)] = 0,(10)V
𝛽𝑅 1 − 𝛾 𝑢 W 𝑐2,@
= 𝛾𝑢 W 𝑇 V ,

(11)

9

where we set

4.

YZQ[
Y\

= 0 andYZ][
Y\= 0 in (11) from Assumption 1.

The Altruistic Intervention of the Parent

4.1. The Actual Transfer Decided by the Parent
The parent behaves differently from the child’s expectation because she knows that her child’s
preferences will shift due to the effect of habit formation, even though the child himself does notrecognize it in his early periods. Hence, when deciding an income transfer level for her young son, themother takes into account the habit effect that the transfer will have on his consumption in his second andthird periods. Formally, the parent chooses the transfer level to her 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, the child takes the habit as a permanent preference shock. Maximizing (5)leads to the first-order condition of the child in the second period:𝑢 W 𝑐# − 𝜃/ 𝑐/∗ = 𝛽𝑅𝑢 W 𝑐2 − 𝜃/ 𝑐/∗ ,(12)where the first-period consumption is given by the borrowing constraint𝑐/∗ = 𝑇 ∗ .

(13)

Under budget constraints (13) and𝑐2∗ = 𝑅(𝑦# − 𝑐#∗ ),the actual consumption levels 𝑐#∗ and 𝑐2∗ are determined by (12). Because the first-period consumption𝑐/∗ depends on the income transfer level from (13), the actual consumption levels 𝑐#∗ and 𝑐2∗ arefunctions of the income transfer 𝑇 from (12).
With the function 𝑐#∗ and 𝑐2∗ being given, the parent chooses (𝑐#,@ , 𝑐2,@ , 𝑇 ∗ ) so as to maximize (7). The
first-order conditions are∗∗
1 − 𝛾 𝑢 W 𝑐#,@− 𝛽𝑅𝑢 W 𝑐2,@

= 0,∗
𝛽𝑅 1 − 𝛾 𝑢 W 𝑐2,@= 𝛾[𝑢 W 𝑇 ∗ + 𝛽𝑢 W 𝑐#∗ − 𝜃/ 𝑐/∗+𝛽 # 𝑢 W 𝑐2∗ − 𝜃# 𝑐#∗

10

YZ]∗

Y\ ∗

− 𝜃#

YZQ∗

Y\ ∗

(14)

𝜕𝑐#∗
𝜕𝑐/∗
− 𝜃/ ∗
∗
𝜕𝑇
𝜕𝑇
.

(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, 𝑦@ , 𝜃/ , 𝜃# , 𝑦# .
In the following subsections, we characterize the income transfer 𝑇 ∗ of the altruistic parent bycomparing it with the young child’s expectation on income transfer 𝑇 V . By substituting functions 𝑐%∗ 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 that maximizes the child’s utility by 𝑇. By
definition, it satisfies 𝑈′ 𝑇 = 0. Using Assumption 3, we express 𝑇 as 𝑇(𝛼, 𝛽, 𝑅, 𝜃/ , 𝜃# , 𝑦# ).
Since the child’s utility is maximized at the point of 𝑇, too much income transfer (more than 𝑇) harmshim. Note that the parent necessarily chooses income transfer level 𝑇 ∗ no larger than 𝑇: 𝑇 ∗ ≤ 𝑇. This isbecause if 𝑇 ∗ were higher than 𝑇, it would not enhance the child’s welfare but would rather cost theparent’s consumption.
However, the expected transfer 𝑇 V can be either smaller or larger than 𝑇, depending on the parent’s
income level 𝑦@ . To compare the actual and expected transfers (𝑇 ∗ and 𝑇 V ) and the corresponding
welfare levels, it would, therefore, be helpful to separate two cases: (i) 𝑦@ > 𝑇 and (ii) 𝑦@ ≤ 𝑇, which
represent the rich-parent and poor-parent cases, respectively. 11

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 thechild does not. Thus, the actual transfer 𝑇 ∗ deviates from the expected one 𝑇 V . Proposition 1 states therelationship between 𝑇 ∗ and 𝑇 V .
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 V .
Proof. See Appendix A1.
Proposition 1 implies that, insofar as the parent is not purely selfish or purely altruistic, she keeps thetransfer level to her child lower than he expects. The parent knows that more transfer leads the child toform deeper habits, and hence consume more excessively in the next period. Hence, she intervenes in thechild’s behavior by keeping the transfer low.
Note that when the parent is purely selfish (𝛾 = 0) or purely altruistic (𝛾 = 1), 𝑇 ∗ can be equal to 𝑇 V .
In particular, when 𝛾 = 0, we have trivially 𝑇 ∗ = 𝑇 V = 0 (i.e., purely selfish parents do not give, andSince both the critical value 𝑦@ and 𝑇 are linear in 𝑦# , 𝑦@ > 𝑇 implies that 𝑦@ is relatively high comparedwith 𝑦# . Therefore, when referring to “the rich parent”, we stress that the parent’s income is high enough relative tothe child’s income, and ceteris paribus for “the poor parent”.
11

11

are not expected to give, transfers to their children).
When 𝛾 = 1, we have 𝑇 ∗ = 𝑦@ if the parent is poor, 𝑦@ ≤ 𝑇. Since the poor parent cannot afford totransfer 𝑇, she just gives her whole income to her child, as he expects. In the poor parent case, therefore,
we have 𝑇 ∗ = 𝑇 V = 𝑦@ . When the parent is rich, 𝑦@ > 𝑇; however, the parental transfer 𝑇 ∗ is smaller
than what the child expects and again, 𝑇 ∗ < 𝑇 V . This is because 𝑇 ∗ maximizes the child’s utility, 𝑇 ∗ =𝑇, whereas the naive child expects the purely altruistic parent to transfer her whole income to him, 𝑇 V =𝑦@ (> 𝑇 by construction).

4.3. The Welfare of the ChildAs shown in the previous section, the parental income transfer is unexpectedly small in the eyes of theyoung child. Our interest here is to evaluate it from the viewpoint of the child’s welfare; we shall showthat the unexpectedly small transfer could make the child better off.
We compare utility values of actual transfer 𝑇 ∗ (𝑈(𝑇 ∗ )) and expected transfer 𝑇 V (𝑈(𝑇 V )) 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 ofincome transfer T.12
Of particular interest is whether the parental income transfer generates higher welfare of the child thanthe 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 thefollowing definition.
Definition 1. When 𝑈(𝑇 ∗ ) > 𝑈(𝑇 V ), parental income transfer 𝑇 ∗ is referred to as welfare-enhancing to
the child.
We first consider the case of a parent with relatively low income, where 𝑦@ ≤ 𝑇 (Fig. 1 and Fig. 2). In
this case, the parent cannot provide the child with a transfer exceeding 𝑇. As an example of this case, Fig.
1 shows that both 𝑇 ∗ and 𝑇 V are less than 𝑦@ , and hence are less than 𝑇. Recall from Proposition 1 that𝑇 ∗ < 𝑇 V for 𝛾 ∈ 0,1 . Since the child’s welfare increases with transfers in the region of T < 𝑇, we have
𝑈 𝑇 ∗ < 𝑈 𝑇 V for 𝛾 ∈ 0,1 (i.e., the parental income transfer is not welfare-enhancing to the child).13

12

Pollak (1978) recommends "unconditional welfare ordering", which incorporates the effects of endogenous
preference formation. We used the same welfare criterion here.
13
When the parent is purely selfish (𝛾 = 0) or purely altruistic (𝛾 = 1), we have 𝑈 𝑇 ∗ = 𝑈 𝑇 V (See Fig. 2).

12

Fig. 1 The utility of the child as a function of 𝑇 when 𝑦@ ≤ 𝑇

Fig. 2 The utility of the child as a function of 𝛾 when 𝑦@ ≤ 𝑇

When the parent’s income is relatively high, 𝑦@ > 𝑇, the child’s welfare first increases and thendecreases with the income transfer level. As an example of this case, Fig. 3 shows that the rich parent can
provide the child with more transfer than 𝑇, but she maintains transfer level 𝑇 ∗ lower than this.
However, the expected transfer 𝑇 V can be more than 𝑇. Too much transfer (more than 𝑇) causes the13child’s welfare 𝑈 to decrease. Therefore, the retrospective utility of the expected transfer 𝑈(𝑇 V ) can belower than that of actual transfer 𝑈(𝑇 ∗ ).

Fig. 3 The utility of the child as a function of 𝑇 when 𝑦@ > 𝑇

Recall that when 𝛾 = 0, 𝑇 ∗ = 𝑇 V = 0, and that when 𝛾 = 1, 𝑇 ∗ = 𝑇, which is an interior solution,
and 𝑇 V = 𝑦@ , which is a corner solution (See Fig. 4). As 𝛾 increases from 0 to 1, the actual transfer 𝑇 ∗increases from 0 to 𝑇, and the expected transfer 𝑇 V increases from 0 to 𝑦@ . 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 hence also increases with the degree of parental altruism 𝛾.
However, the child’s retrospective utility level of 𝑇 V decreases with 𝑇 V when it exceeds 𝑇. Therefore,
there exists a certain degree of parental altruism, which equalizes 𝑈(𝑇 ∗ ) and 𝑈 𝑇 V .

14

Fig. 4 The utility of the child as a function of 𝛾 when 𝑦@ > 𝑇

Lemma 1. When the parent’s income is high enough relative to the child’s income, there exists a degreeof parental altruism 𝛾 ∈ (0,1) such that 𝑈 𝑇 ∗ = 𝑈(𝑇 V ).
Proof. See Appendix A2.
For all degrees of parental altruism higher than 𝛾, the actual transfer leads to higher welfare of thechild, and hence, the actual transfer is welfare-enhancing for the child.
Proposition 2. When the parent’s income is relatively high (𝑦@ > 𝑇), the parental income transfer 𝑇 ∗ iswelfare-enhancing for the child, that is, 𝑈(𝑇 ∗ ) > 𝑈(𝑇 V ), if the parent is so altruistic that 𝛾 > 𝛾.
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 positive effect dominates the firstnegative one. Therefore, the unexpectedly low transfer makes the child better 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.

15

5.

Discussions

5.1 An Increase in the Habit ParameterAs shown in Proposition 1, the altruistic parent keeps income transfer lower than the child expects toreceive. 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 increase in thechild’s habit parameter in his second period 𝜃/ .
Proposition 3.

A higher degree 𝜃/ of the child’s habit in the second period implies a lower level ofparental income transfer 𝑇 ∗ , that is,

`\ ∗
`RH

< 0.

Proof. See Appendix A3.
Proposition 3 shows that the parent penalizes the child for an increase in habitual consumption by
reducing income transfer. This strict behavior, however, comes from the parent’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 the child’s puresubjective rate of time preference in his second period as follows:𝜒#,2 (𝑐) = log 𝑀𝑅𝑆 𝑐# , 𝑐2 |ZQhZ]hZ= 𝜌 + log𝑢 W 𝑐 − 𝜃/ ℎ#𝜃#−,𝑢 W 𝑐 − 𝜃# ℎ2
1+𝜌
(16)

where 𝜌 = 1 𝛽 − 1 is the subjective discount rate of the child. It is then easy to see that when degree ofthe habit formation in the child’s second period 𝜃/ exogenously increases, the child’s time preferencefunction 𝜒#,2 𝑐 shifts upward. Because the parent reduces income transfer 𝑇 ∗ (see Proposition 3) upon
the increase in 𝜃/ , the parent’s behavior above seems to duplicate the tough love behavior discussed by
Bhatt and Ogaki (2012).

16

Our discussion is distinct in two aspects. First, in the present model, time preference is endogenously
generated by habit formation rather than by an exogenously given time preference formation. Second andmore importantly, our tough love is welfare-enhancing to the child if his parent is altruistic and richenough. This contrasts with the case explored in the Bhatt and Ogaki model, wherein a mother’s toughlove reduces her son’s welfare, because the mother evaluates her son’s future utility by using different
time preference than that of the child.

5.2 The Case of “Good” Habits
In the previous sections, we consider the case of “negative” habits by assuming habit parameters to bepositive (𝜃/ , 𝜃# > 0), where children’s current consumption harms their future utility. Alternatively, we
can think about parents’ income transfer that involves children’s “good” habits: Parents would give books
to children, or take children to piano lessons, which would increase the children’s future gratification. The
implications for the “good-habit” case could be obtained by assuming negative habit parameters,
𝜃/ , 𝜃# < 0.
Proposition 4. When the parent’s income transfer involves the child’s “good” habits, where 𝜃/ , 𝜃# < 0,
the following results are obtained.
(i) The parental income transfer is unexpectedly large to the young child, that is, 𝑇 ∗ > 𝑇 V .
(ii) The parental income transfer 𝑇 ∗ is welfare-enhancing for the child, that is, 𝑈(𝑇 ∗ ) > 𝑈(𝑇 V ).
Proof. See Appendix A4.
Proposition 4 shows that when habit parameters are negative and hence when the parental income
transfer contributes to the formation of the child’s good habit, parental income transfers are larger thanwhat the children expect for any degree of parental altruism. Because parents know that current
consumption enhances future utility, they tend to transfer more income to invest in children’s habit.14
When the habits of the child are “good”, increasing income transfer causes two positive effects onchildren’s utility. It (i) directly increases the child’s felicity throughout the period (ii) and enhances his“good” habit in the subsequent period. In this context, the parent tends to give more income transfer andthe grown-up child is always grateful toward the unexpectedly large income transfer.

6.

Concluding Remarks

14

More natural modeling would be that children consume two goods, a negative-habit good and a good-habit good.
In the setting, the results would be easily conjectured from our results insofar as the two types of goods are additively
separable: Parents will give unexpectedly small amount of the negative-habit goods and unexpectedly large amountof the good-habit goods.

17By employing a parental altruism model where the child is ignorant of his own habit until he reaches
adulthood, we show that the parental income transfer seems unexpectedly low to the young child.
However, the unexpectedly low parental income transfer can generate higher welfare for the child thanthe expected transfer, when evaluated retrospectively. In other words, when the parent’s income is highenough relative to that of the child and sufficiently altruistic, the parent’s income transfer iswelfare-enhancing to the grown child.
This research may have important implications for understanding how parent-child interactions affectthe consumption behavior and related preference formation of family members. In the framework of thismodel, the authoritative parenting practice is referring to the lower level of actual transfer 𝑇 ∗ than the
expected transfer 𝑇 V (𝑇 ∗ < 𝑇 V ). Although the result 𝑇 ∗ < 𝑇 V holds both for rich and for poor parents,
the lower actual transfer 𝑇 ∗ than 𝑇 V is welfare-enhancing to the child only when the parent issufficiently rich. Parents from rich families thus tend to appreciate this authoritative parenting practicethey were subject to when they were young, which might contribute to the transmission of these practices
in these families.
To give evidence to the theoretical predictions, we are conducting empirical analysis by using
questionnaire survey data (Zhang and Ikeda 2014). We find that there are significantly positive relations
between respondents’ strict upbringing attitude and their living standards when 15 years old in the casethat they are aware of habit. Besides, our theoretical prediction is also supported by Doepke and Zilibotti(2014): They show the rise of authoritative parenting in recent decades with the process of economic
development.
More work remains for the future research. By putting the simplifying assumption that the childmisperceives the effect of habit as a permanent preference shock, we assume away the possibility that thechild may learn about his true preference by observing how much transfer the parent gave. It is of interest
to investigate parent-child interaction in the presence of such learning process, possibly under someasymmetric information. For example, if children do not know how altruistic their parents are, they may
mistakenly attribute small income transfers to the parents’ selfishness, rather than to their tough love tocontrol children’s consumption habits. Due to this misperception, it would take longer time for thechildren to learn their true preferences from the parents’ behavior. The interesting topic is beyond thescope of the current work, however. We shall tackle this issue in the future.

18

Appendix

A1. Proof of Proposition 1
When ∀𝛾 ∈ 0,1 , we compare the actual transfer 𝑇 ∗ with the expected transfer 𝑇 V by dividing thefirst order condition (11) with the first order condition (15). By substituting first-order conditions (9) and(14) as well as Assumption 1 we have(𝑦@ − 𝑇 ∗ KL
𝑇 ∗ KL)=()
𝑦@ − 𝑇 V
𝑇V

H

−

−

H

H

𝛽𝜃/ (𝑅 + 1) 𝛽𝑅 KJ [𝑅𝑦# 𝛽𝑅 KJ − 𝜃/ 𝑅 + 1 𝛽𝑅 KJ (𝑇 ∗ )]KL

𝛽 # 𝜃/ 𝜃# + 𝑅

H[1 + 𝑅 𝛽𝑅 KJ ]/KL (𝑇 V )KLH

H

H

1 − 𝛽𝑅 KJ 𝑅𝑦# (1 − 𝜃# 𝛽𝑅 KJ ) − 𝜃/ 𝜃# + 𝑅 (1 − 𝛽𝑅 KJ ) 𝑇 ∗H[1 + 𝑅 𝛽𝑅 KJ ]/KL (𝑇 V )KL

KL

.
(A1)

The second and third items on the RHS are both negative, so we have (equation (A1) implies that\∗\[Pk K\ ∗ KL

Pk K\ [)\∗< ( [)KL . Hence\< 1. Therefore, we have ∀𝛾 ∈ 0,1 , 𝑇 ∗ < 𝑇 V .

A2. Proof of Lemma 1 and Proposition 2
The child’s utility 𝑈 is continuous in the degree of parental altruism 𝛾 . When 𝑇 < 𝑇 , 𝑈 is
increasing in 𝑇; and when 𝑇 > 𝑇, 𝑈 is decreasing in 𝑇. The income transfers 𝑇 ∗ and 𝑇 V are both
continuous and increasing in 𝛾.
When 𝑦@ > 𝑇, we obtain lim 𝑇 ∗ = 𝑇, which is an interior solution and that lim 𝑇 V = 𝑦@ , which is a
n→/

n→/

∗

V

corner solution. We also have lim 𝑈′(𝑇 ) = 0 and lim 𝑈′(𝑇 ) < 0. Hence, in the neighborhood of 𝛾 =
n→/

∗

n→/

V

1, 𝑈(𝑇 ) > 𝑈(𝑇 ).
When 𝛾 decreases from 1, both 𝑇 ∗ and 𝑇 V decrease, as does 𝑈(𝑇 ∗ ). However, 𝑈(𝑇 V ) increases
until 𝑇 V equals to 𝑇. Hence, there exists a certain 𝛾 such that 𝑈 𝑇 ∗ = 𝑈(𝑇 V ). When 𝛾 > 𝛾, we have𝑈(𝑇 ∗ ) > 𝑈(𝑇 V ).

A3. Proof of Proposition 3
First-conditions (10) and (11) and budget constraint (4) jointly determine the optimal transfer of theparent 𝑇 ∗ (𝛼, 𝛾, 𝛽, 𝑅, 𝑦@ , 𝜃/ , 𝜃# , 𝑦# ).

19

Substituting first-order condition (10) into (11) and taking the total derivative of the optimal transfer by
the parent 𝑇 ∗ with respect to the habit parameter 𝜃/ leads to{+ 𝛼(𝑇 ∗ )KLK/

H KL

𝛾 1 + 𝑅 𝛽𝑅 KJ

H

+

KLK/

1 − 𝛾 𝛼𝛽𝑅/KL 𝑦@ − 𝑇 ∗HH𝛼𝛽[ 𝛽𝑅 KJ 𝜃/ 1 + 𝑅 ]# [𝑅𝑦# 𝛽𝑅 KJ − 𝜃/ (1 + 𝑅) 𝛽𝑅 KJ (𝑇 ∗ )]KLK/H[1 + 𝑅 𝛽𝑅 KJ ]/KL
H

H

H

𝛼𝛽 # 𝜃/# (𝜃# + 𝑅)# [1 − 𝛽𝑅 KJ ]# [𝑅𝑦# 1 − 𝜃# 𝛽𝑅 KJ − 𝜃/ 𝜃# + 𝑅 (1 − 𝛽𝑅 KJ )(𝑇 ∗ )]KLK/ 𝑑𝑇 ∗
+
}
H
𝑑𝜃/
[1 + 𝑅 𝛽𝑅 KJ ]/KL
H

=−

H

H

𝛽 𝛽𝑅 KJ (1 + 𝑅)[𝑅𝑦# 𝛽𝑅 KJ − 𝜃/ (1 + 𝑅) 𝛽𝑅 KJ (𝑇 ∗ )]KLK/H[1 + 𝑅 𝛽𝑅 KJ ]/KL
H

H

∙ { 𝑅𝑦# 𝛽𝑅 KJ − 𝜃/ 1 + 𝑅 𝛽𝑅 KJ 𝑇 ∗H−H+ 𝛼 1 + 𝑅 𝛽𝑅 KJ 𝑇 ∗ }HH

𝛽 # (𝜃# + 𝑅)[1 − 𝛽𝑅 KJ ][𝑅𝑦# 1 − 𝜃# 𝛽𝑅 KJ − 𝜃/ 𝜃# + 𝑅 (1 − 𝛽𝑅 KJ )(𝑇 ∗ )]KLK/H[1 + 𝑅 𝛽𝑅 KJ ]/KL
H

H

∙ [𝑅𝑦# 1 − 𝜃# 𝛽𝑅 KJ − (1 − 𝛼)𝜃/ 𝜃# + 𝑅 (1 − 𝛽𝑅 KJ )(𝑇 ∗ )],(A2)
where, from Assumption 4, the coefficient of
Therefore, we have ∀𝛾(0,1),

`\ ∗
`RH

`\ ∗
`RH

is positive; and the right hand side is negative.

< 0.

A4. Proof of Proposition 4
From equation (A1), we have that with negative 𝜃/ , 𝜃# it implies that\∗\[< 1. Therefore, we have∀𝛾 ∈ 0,1 , 𝑇 ∗ > 𝑇 V .
When 𝜃/ , 𝜃# > 0, the child’s retrospective utility is always increasing in parental income transfer 𝑇.
Hence, we always have 𝑈(𝑇 ∗ ) > 𝑈(𝑇 V ).

A5. 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 become aware 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, the following,
instead of Assumption 4:
Assumption A1. 𝑦@ <OPQ
RH (OSRQ ).

20

We denote optimal solutions in the present case by using an apostrophe, instead of an asterisk in thetext.
In the second period, the misperception of the child disappears and the child maximizes the utility𝑢(𝑐# − 𝜃/ 𝑐/W ) + 𝛽𝑢(𝑐2 − 𝜃# 𝑐# ). The first-order condition is given by𝑢 W 𝑐# − 𝜃/ 𝑐/W = 𝛽(𝑅 + 𝜃# )𝑢 W 𝑐2 − 𝜃# 𝑐# .

(A3)

With 𝑐/W being given by the parent’s income transfer 𝑇 W , (A3) and budget constraint (2) jointly determine
the consumption levels 𝑐#W and 𝑐2W .
The parent’s first-order conditions of the parent areWW
1 − 𝛾 𝑢 W 𝑐#,@− 𝛽𝑅𝑢 W 𝑐2,@

YZ]s

Y\ s

− 𝜃#

(A4)

𝜕𝑐#W
𝜕𝑐/W
− 𝜃/ WW
𝜕𝑇
𝜕𝑇W
𝛽𝑅 1 − 𝛾 𝑢 W 𝑐2,@= 𝛾[𝑢 W 𝑇 W + 𝛽𝑢 W 𝑐#W − 𝜃/ 𝑐/W+𝛽 # 𝑢 W 𝑐2W − 𝜃# 𝑐#W

= 0,

YZQs

Y\ s

.
(A5)

First-order conditions (A4) and (A5) and budget constraint (6) jointly determine the parental income
transfer 𝑇 W . By noting that the expected transfer 𝑇 V 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 W < 𝑇 V .
Proof.
By substituting (9) into (11), (A4) into (.5), and dividing (A5) with (10), we have(−𝑦@ − 𝑇 W KL
𝑇 W KL)=()
𝑦@ − 𝑇 V
𝑇V

U Q RH (OSRQ )[OPQ KRH OSRQ (\ s )]IJ
H IJI/S(OSRQ ) U(OSRQ ) J.

(\ [ )IJ(A6)
Equation (A6) implies that\s\[

< 1. Therefore, we have ∀𝛾 ∈ 0,1 , 𝑇 W < 𝑇 V .

We can also show that there is a critical degree of parental altruism 𝛾 ∈ (0,1), such that 𝑈 𝑇 W =𝑈(𝑇 V ). As in Proposition 2, we have:
Proposition A2. When the parent’s income is high relative to the child’s income (𝑦@ > 𝑇), the parental

21

income transfer 𝑇 W is welfare-enhancing for the child, that is, 𝑈(𝑇 W ) > 𝑈(𝑇 V ), if the parent is so
altruistic that 𝛾 > 𝛾.
Proof.
The child’s utility 𝑈 is continuous in the degree of parental altruism 𝛾 . When 𝑇 < 𝑇 , 𝑈 is
increasing in 𝑇; and when 𝑇 > 𝑇, 𝑈 is decreasing in 𝑇. The income transfers 𝑇 W and 𝑇 V are both
continuous and increasing in 𝛾.
When 𝑦@ > 𝑇, we obtain lim 𝑇 W = 𝑇, which is an interior solution and that lim 𝑇 V = 𝑦@ , which is a
n→/

n→/

W

V

corner solution. We also have lim 𝑈′(𝑇 ) = 0 and lim 𝑈′(𝑇 ) < 0. Hence, in the neighborhood of 𝛾 =
n→/

W

n→/

V

1, 𝑈(𝑇 ) > 𝑈(𝑇 ).
When 𝛾 decreases from 1, both 𝑇 W and 𝑇 V decrease, and 𝑈(𝑇 W ) also decreases. However, 𝑈(𝑇 V )increases until 𝑇 V equals to 𝑇. Hence there exists a certain 𝛾 such that 𝑈 𝑇 W = 𝑈(𝑇 V ). When 𝛾 > 𝛾,
we have 𝑈(𝑇 W ) > 𝑈(𝑇 V ).

A6. When the Child’s Borrowing Constraint Is Not Binding (c/ < 𝑇)
When the child’s borrowing constraint is not binding, his life-time intertemporal budget is given by
𝑅 # 𝑐/ + 𝑅𝑐# + 𝑐2 = 𝑅 # T + R𝑦# ,(A7)

so parental transfer 𝑇 constitutes a part of the child’s wealth.
The transfer 𝑇 V expected by the child is determined by budget constraint (A7) and the following
first-order conditionsVV
1 − 𝛾 [𝑢 W 𝑐#,@− 𝛽𝑅𝑢′(𝑐2,@)] = 0,V
𝛽𝑅 1 − 𝛾 𝑢 W 𝑐2,@
=𝛾

YZH[

Y\ [

𝑢 W 𝑐/V .

(A8)
(A9)

By maximizing life-time utility (4) subjecting to budget constraint (A7), we could obtain the expectedconsumption 𝑐/V (𝑇 V ). Providing that 𝑐/V is a normal good, 𝑐/V is an increasing function of wealth andhence of 𝑇 V .
Under the permanent preference shock assumption∗∗
Under the permanent preference shock assumption, the parent’s consumption 𝑐#,@, 𝑐2,@and the actual

transfer 𝑇 ∗ are determined by budget constraint (A7) and the following first-order conditions∗∗
1 − 𝛾 [𝑢 W 𝑐#,@− 𝛽𝑅𝑢′(𝑐2,@)] = 0,22(A10)𝜕𝑐/∗ W ∗
{𝑢 𝑐/
𝜕𝑇 ∗

∗
𝛽𝑅 1 − 𝛾 𝑢 W 𝑐2,@
=𝛾
H

−

−

H

H

𝛽(𝜃/ + 𝜃/ 𝑅 + 𝑅 # ) 𝛽𝑅 KJ [𝑅𝑦# 𝛽𝑅 KJ − 𝜃/ 𝑅 + 1 𝛽𝑅 KJ (𝑐/∗ )]KLU Q RH RQ SO

H[1 + 𝑅 𝛽𝑅 KJ ]
/K UO

I

H
H
H
I
I
J SO Q (/KRQ UO J ) OPQ /KRQ UO J KRH RQ SO/SO UO

I

/K UO

I

H
J

ZH∗

H
J

IJ

}.

(A11)
By maximizing life-time utility (4) subjecting to budget constraint (A7), we could obtain the actual
consumption 𝑐/∗ (𝑇 ∗ ). Insofar as 𝑐/∗ is a normal good, 𝑐/∗ is an increasing function of wealth and henceof 𝑇 ∗ .
By comparing equations (A8) and (A9) with (A10) and (A11) respectively, we could find that all items
are the same except the RHS of equation (A9) and (A11). Because the second and third items of equation(A11) are both negative, when the parent decides the transfer, the marginal utility of the child in equation(A11) is lower than that in equation (A9). In other words, the actual transfer is smaller than the expected
one. Therefore, the result of Proposition 1 that 𝑇 ∗ < 𝑇 V is still valid.
When the misperception of habit disappears
In the case where the misperception of habit disappears in child’s second period (see Appendix A5), theWWparent’s consumption 𝑐#,@,𝑐2,@
and the actual transfer 𝑇 W are determined by budget constraint (A7) andthe following first-order conditionsWW
1 − 𝛾 [𝑢 W 𝑐#,@− 𝛽𝑅𝑢′(𝑐2,@)] = 0,W
𝛽𝑅 1 − 𝛾 𝑢 W 𝑐2,@
=𝛾

I

−

(A12)

𝜕𝑐/W W W
{𝑢 𝑐/
𝜕𝑇 W

H

U Q O Q SRH RQ SO S OSRQ O Q SORH SRH RQ (U(OSRQ )) J OPQ /KRQ UOIIHJ KRH RQ SO

/K UO

I

H
J

ZHs

H

IJ

}.

/S(OSRQ )(U(OSRQ )) J(A13)
By maximizing life-time utility (4) subjecting to budget constraint (A7), we could obtain the actual
consumption 𝑐/W (𝑇 W ). As a normal good, 𝑐/W is an increasing function of wealth and hence 𝑇 W .
Except the second item of the RHS in equation (A13), other items in equations (A12) and (A13) are thesame with equations (A8) and (A9). It implies that when the parent decides the transfer the marginal23

utility of child in equation (A13) is lower than that in equation (A9). Hence, the result that actual transfer
is smaller than the expected on is still valid.
Furthermore, similar to the proof in Appendix A2, we could find that when 𝑦@ > 𝑇, we obtainlim 𝑇 ∗ = 𝑇, which is an interior solution and that lim 𝑇 V = 𝑦@ , which is a corner solution. We also have

n→/

n→/

lim 𝑈′(𝑇 ∗ ) = 0 and lim 𝑈′(𝑇 V ) < 0. Hence, in the neighborhood of 𝛾 = 1, 𝑈(𝑇 ∗ ) > 𝑈(𝑇 V ).

n→/

n→/

When 𝛾 decreases from 1, both actual transfer and expected transfer decrease, as does 𝑈(𝑇 ∗ ).
However, 𝑈(𝑇 V ) increases until 𝑇 V equals to 𝑇. Hence, there exists a certain 𝛾 such that 𝑈 𝑇 ∗ =𝑈(𝑇 V ). When 𝛾 > 𝛾, 𝑈(𝑇 ∗ ) > 𝑈(𝑇 V ). Therefore, the conclusion of Proposition 2 still holds.

Funding:
Shinsuke Ikeda has received Grants-in-Aid for Scientific Research (C No. 26380239) from the Japan
Society for the Promotion of Science; the Global COE Program of Osaka University; the Joint Usage/
Research Center Project of ISER from the Ministry of Education, Culture, Sports, Science andTechnology. Lin Zhang has received Research Subsidy from Kanazawa Seiryo University.

24References
Akabayashi H., Kamesaka A., Nakamura R., Ogaki M., Tamura T. (2014) An Experimental Study ofIntergenerational Altruism with Parent-Child Pairs. Keio-IES Discussion Paper Series DP2014-005,
Institute for Economic Studies, Keio University.
Barnet-Verzat C., Wolff F. C. (2002) Motives for Pocket Money Allowance and Family Incentives.
Journal of Economic Psychology, Vol. 23, pp. 339-366.
Barro J. R. (1974) Are Government Bonds New Wealth? Journal of Political Economy, Vol. 82, pp.
1095-1117.
Baumrind D. (1967) Child Care Practices Anteceding Three Patterns of Preschool. Genetic Psychology
Monographs, Vol. 75, No. 1, pp. 43-88.
Baumrind D. (1971) Current Patterns of Parental Authority. Developmental Psychology, Vol. 4, pp. 1-103.
Baumrind D. (1978) Parental Disciplinary Patterns and Social Competence in Children. Youth & Society,Vol. 9, No. 3, pp. 239-276.
Baumrind D. (1991) The Influence of Parenting Style on Adolescent Competence and Substance Use.
Journal of Early Adolescence, Vol. 11, No. 1, pp. 56-95.
Becker G. (1974) A Theory of Social Interactions. Journal of Political Economy, Vol. 82, No. 6, pp.
1063-1093.
Bhatt V., Ogaki M. (2008) Tough Love and Intergenerational Altruism. Rochester Center for Economic
Research Working Paper, No. 544.
Bhatt V., Ogaki M. (2012) Tough Love and Intergenerational Altruism. International Economic Review,Vol. 53, No. 3, pp. 791-814.
Chabris C. F., Laibson D., Morris C. L., Schuldt J. P., Taubinsky D. (2008) Individual
Laboratory-Measured Discount Rates Predict Field Behavior. Journal of Risk and Uncertainty, Vol. 37,
pp. 237-269.
Clark A., Frijters P., Sields M. A. (2008) Relative Income, Happiness, and Utility: An Explanation for the
Easterlin Paradox and Other Puzzles. Journal of Economic Literature, Vol. 46, pp. 95-144.
Doepke M., Zilibotti F. (2014) Parenting with Style: Altruism and Paternalism in Intergenerational
Preference Transmission. IZA Discussion Paper, No. 7108, Institute for the Study of Labor.
Duncan G. J., Yeung W. J., Brooks-Gunn J., Smith J. R. (1998) How Much Does Childhood Poverty
Affect the Life Chances of Children? American Sociological Review, Vol. 63, No. 3, pp. 406-423.
Dynan K. E. (2000) Habit Formation in Consumer Preferences: Evidence from Panel Data. The American
Economic Review, Vol. 90, No. 3, pp. 391-406.

25

Easterlin R. A. (1974) Does Economic Growth Improve the Human Lot? Some Empirical Evidence. In
Nations and Households in Economic Growth: Essays in Honor of Moses Abramovitz, ed. Paul A.
David and Melvin W. Reder. New York: Academic Press, Inc..
Fisher J. O., Birch L. L. (2000) Parents’ Restrictive Feeding Practices Are Associated with Young Girls’
Negative Self-Evaluation of Eating. Journal of the American Dietetics Association, Vol. 100, pp.
1341-1346.
Furnham A. (1999) The Saving and Spending Habits of Young People. Journal of Economic Psychology,
Vol. 20, pp. 677-697.
Furnham A. (2001) Parental Attitudes to Pocket Money Allowances for Children. Journal of Economic
Psychology, Vol. 22, pp. 397-422.
Furnham A., Thomas P. (1984) Pocket Money: A Study of Economic Education. British Journal of
Developmental Psychology, Vol. 2, pp. 205-212.
Gable S., Lutz S. (2000) Household, Parent and Child Contributions to Childhood Obesity. Family
Relations, Vol. 49, pp. 293-300.
Heckman J. J., Stixrud J., Urzua S. (2006) The Effects of Cognitive and Noncognitive Abilities on Labor
Market Outcomes and Social Behavior. Journal of Labor Economics, Vol. 24(3), pp 411-482.
van der Horst K., Kremers S., Ferreira I., Singh A., Oenema A., Brug J. (2007) Perceived Parenting Styleand Practices and the Consumption of Sugar-Sweetened Beverages by Adolescents. Health Education
Research, Vol. 22, pp. 295-304.
Miller J., Yung S. (1990) The Role of Allowances in Adolescent Socialization. Youth and Society, Vol. 22,
pp. 137-159.
Moffit T. E., Arseneault L., Belsky D., Dickson N., Hancox R. J., Harrington H., Houts R., Poulton R.,
Roberts B. W., Ross S., Sears M. R., Thomson W. M., Caspi A. (2011) A Gradient of Childhood
Self-Control Predicts Health, Wealth, and Public Safety. Proceedings of the National Academy of
Sciences, Vol. 108, pp. 2693-2698.
Obstfeld M. (1990) Intertemporal Dependence, Impatience, and Dynamics. Journal of Monetary
Economics, Vol. 26, pp. 45-75.
Otto A. (2013) Saving in Childhood and Adolescence: Insights from Developmental Psychology.
Economics of Education Review, Vol. 33, pp. 8-18.
Patrick H., Nicklas T. A., Hughes S. O., Morales M. (2005) The Benefits of Authoritative Feeding Style:
Caregiver Feeding Styles and Children’s Food Consumption Patterns. Appetite, Vol. 44, pp. 243-249.
Pollak R. A. (1978) Endogenous Tastes in Demand and Welfare Analysis. The American Economic
Review, Vol. 68, No. 2, pp. 374-379.

26

Shoda Y., Mischel W., Peake P. K. (1990) Predicting Adolescent Cognitive and Self-Regulatory
Competencies

fromPreschool

Delay

of

Gratification:

Identifying

Diagnostic

Conditions.

Developmental Psychology, Vol. 26, pp. 978-986.
Rosenberg M., Pearlin L. I. (1978) Social Class and Self-Esteem among Children and Adults. American
Journal of Sociology, Vol. 84, pp. 54-58.
Walsh A. D., Laczniak P. N., Carlson L. (1998) Mothers’ Preferences for Regulating Children’s Television.
Journal of Advertising, Vol. 27, Issue 3, pp. 23-36.
Weinberg A. B. (2001) An Incentive Model of the Effect of Parental Income on Children. Journal of
Political Economy, Vol. 109, No. 2, pp. 266-280.
West P., Sweeting H., Young R., Robins M. (2006) A Material Paradox: Socioeconomic Status, Young
People’s Disposable Income and Consumer Culture. Journal of Youth Studies, Vol. 9, No. 4, pp.
437-462.
Zhang L., Ikeda S. (2014) Strict Parents Come From Wealthy Families: An Empirical Study.
Unpublished.

27